# Patent

## Title
Variable load sense spring setting for axial piston open circuit pump

## Abstract

A variable load sense spring setting for an axial piston open circuit having three subsystems. A backstepping control applied to the first and second subsystems to determine an operator control input and a load sense spool effective open area control input that stabilize the first and second subsystems. The calculating a feasible load sense area from the third subsystem and the control inputs.

## Background
CROSS REFERENCE TO RELATED APPLICATION
This application claims the benefit of U.S. Provisional Application No. 61/947,038 filed Mar. 3, 2014.


BACKGROUND OF THE INVENTION
This invention is directed to a hydraulic load sense system, and more particularly to a hydraulic load sense system for nonlinear applications.
Load sense systems are used widely throughout the power hydraulic industries. One of the main reasons of its popularity is the simplicity of control. Using Load Sense and Pressure Control appropriately, a load sense system is typically passive in nature which can be seen easily analytically. The difference between analytically found solutions and practical implementations lies in the choice of appropriate parameters such as clearance, spool area profile etc. Some of these parameters prescribe overall stability of the pump and in most cases these are figured out by several trial and error methods.
A complete hydraulic load sense system such as that found in an excavator is a naturally complex model. Components such as the engine, hydraulic pump, valves, uncertain load dynamics as well as uncertain operator behavior can instigate instability in various ways in a complex load-sense system. Many have confronted this situation in different ways. Most have ignored some practical aspects such as effect of load-sense control spool area etc. For a typical load sense system if the engine speed is assumed to be constant, the operator input to the valve is the only external input or effect into the system that drives the stability of the whole system. The variation in the load could be considered as external disturbance. Even though it is possible to formulate a control algorithm with one input for the whole system, the analytical solution would be very difficult to obtain because of the complexity of the involved nonlinear dynamics. Linearized models are good enough for predicted load situations, however the controls derived based on a linearized model may not be effective as nonlinear control under uncertain load conditions. Therefore, there exists a need in the art for a manner in which to address these deficiencies.
Therefore, an objective of the present invention is to prove a hydraulic load sense control that is based upon a nonlinearized model.
Another objective of the present invention is to provide a hydraulic load control actuated by operator command and load-sense spool effective opening area.
These and other objectives will be apparent to one of ordinary skill in the art based upon the following written description, drawings, and claims.
BRIEF SUMMARY OF THE INVENTION
A hydraulic load sense system for nonlinear applications having a first subsystem related to operator command, a second load system related to a load sense spool effective opening area, and third subsystem related to a load sense spool displacement. A first control input to stabilize the first subsystem and a second control input to stabilize the second subsystem are determined by a computer by applying backstepping controls to the first and second subsystems.
Using the first and second controls and the third subsystem a feasible load sense area is calculated by the computer. The computer then uses the feasible load sense area to control a hydraulic pump.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a side sectional view of a load sense pump;
FIG. 2 is a schematic view of a load sense system;
FIG. 3 is a graph showing a Load Sense spool variable area slot;
FIG. 4 is a graph comparing pump source and load pressure;
FIG. 5 is a graph of a phase plot of pump protection;
FIG. 6 is a graph of system pressure under load oscillation;
FIG. 7 is a graph showing calculated Load Sense spool area;
FIG. 8 is a graph showing operator command/valve orifice area;
FIG. 9 is a graph showing load flow from source;
FIG. 10 is a graph showing variable margin pressure; and
FIG. 11 is a graph showing cylinder position.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Referring to FIG. 1, shown is a sectional view of a load sense pump 10. Included in the pump 10 is a servo piston 12 that is slidably received within a servo piston guide 14. The guide 14 engages a swashplate 16 that is operatively connected to a rotor shaft 18. Opposite the servo piston 12, and also engaging the swashplate 16 is a bias piston 20 having a bias spring 22.
FIG. 2 shows a schematic view of a hydraulic load sense system. A motor 24 is connected to pump 10. Pump 10 delivers and pressurizes fluid from a tank 26 to a control valve 28 at system pressure. As an example, valve 28 is a variable area orifice controlled by operator input. A compensation circuit 30 is provided that includes a pressure limiting compensation valve 32 and a load sense compensation valve 34. The system also includes a torque control valve 36 such as a servo piston that adjusts the displacement of the swashplate 16. Connected to the system is a personal computer 38. Alternatively a load sense control can be used.
The nonlinear model of the hydraulic system is described with the following differential equations. The load sense spool dynamic is given by




  
    
      
        
          
            x
             ̈
          
          ls
        
        =
        
          
            1
            
              m
              ls
            
          
          ⁢
          
            {
            
              
                
                  c
                  ls
                
                ⁢
                
                  
                    x
                     ̈
                  
                  ls
                
              
              -
              
                
                  K
                  ls
                
                ⁢
                
                  x
                  ls
                
              
              +
              
                
                  A
                  ls
                
                ⁡
                
                  (
                  
                    
                      P
                      s
                    
                    -
                    
                      P
                      ls
                    
                    -
                    
                      P
                      
                        k
                        ls
                      
                    
                  
                  )
                
              
            
            }
          
        
      
    
    
      
        (
        1
        )
      
    
  




where Pkls=load sense spring force per unit area which also known as margin pressure. The Pressure Control spool dynamics is given by





  
    
      
        
          
            x
             ̈
          
          pc
        
        =
        
          
            1
            
              m
              pc
            
          
          ⁢
          
            {
            
              
                
                  -
                  
                    C
                    pc
                  
                
                ⁢
                
                  
                    x
                    .
                  
                  pc
                
              
              -
              
                
                  K
                  pc
                
                ⁢
                
                  x
                  pc
                
              
              +
              
                
                  A
                  pc
                
                ⁡
                
                  (
                  
                    
                      P
                      s
                    
                    -
                    
                      P
                      
                        k
                        pc
                      
                    
                  
                  )
                
              
            
            }
          
        
      
    
    
      
        (
        2
        )
      
    
  




where Pkpc is the Pressure Control spring setting as force per unit area and we will design it later in this paper. The swash plate kit dynamics is given by Equation (3).

Iaä=−Ca{dot over (a)}−Kaa+Tmswash+lbsPbsAbs+lbsKbsxbs−lsvPsvAsv  (3)

Tmswash is the swash moments generated by pump piston movements. We used tabulated test data to interpret this dynamics into simulations. Servo flow in and out of servo is defined as [8]

Qstsv=CdAstsv√{square root over (2/p(Ps−Psv))}

Qsvtt=CdAsvtt√{square root over (2/p(Psv−P1))}  (4)

FIG. 3 shows a graph of the variable area slot. For positive xls the servo flow is defined as flow from source to servo. On the other hand, for negative xls the flow is defined as servo to tank. The total horizontal length of the rectangular area is defined as LlsA and the vertical length is defined as h.





  
    Maximum
    ⁢
    
      
    
    ⁢
    opening
    ⁢
    
      
    
    ⁢
    area
  
  =
  
    
      
        L
        lsA
      
      2
    
    ⁢
    h
  




and for a given Load Sense spool displacement the area is defined as xlsh for flow from source to servo and





  (
  
    
      
        L
        lsA
      
      2
    
    -
    
      x
      ls
    
  
  )




h as flow from servo to tank. This permits one to calculate a feasible Load Sense spool area through Lyapunov stability analysis such that it would help to calculate the size of choke and gain orifices with required. This is the motive of using Load Sense spool area as an external input. Now the servo pressure is defined by the following differential equation as





  
    
      
        
          
        
        ⁢
        
          
            
              
                P
                .
              
              sv
            
            =
            
              
                β
                
                  V
                  sv
                
              
              ⁢
              
                (
                
                  
                    
                      c
                      d
                    
                    ⁢
                    
                      A
                      stsv
                    
                    ⁢
                    
                      
                        
                          2
                          ρ
                        
                        ⁢
                        
                          (
                          
                            
                              P
                              s
                            
                            -
                            
                              P
                              sv
                            
                          
                          )
                        
                      
                    
                  
                  -
                  
                    
                      c
                      d
                    
                    ⁢
                    
                      A
                      svtt
                    
                    ⁢
                    
                      
                        
                          2
                          ρ
                        
                        ⁢
                        
                          (
                          
                            
                              P
                              sv
                            
                            -
                            
                              P
                              1
                            
                          
                          )
                        
                      
                    
                  
                
                )
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          or
        
      
    
    
      
        (
        5
        )
      
    
  
  
    
      
        
          
            P
            .
          
          sv
        
        =
        
          
            β
            
              V
              sv
            
          
          [
          
            
              
                c
                d
              
              ⁢
              
                x
                ls
              
              ⁢
              h
              ⁢
              
                
                  
                    2
                    ρ
                  
                  ⁢
                  
                    (
                    
                      
                        P
                        s
                      
                      -
                      
                        P
                        sv
                      
                    
                    )
                  
                
              
            
            -
            
              
                c
                d
              
              ⁢
              
                h
                ⁡
                
                  (
                  
                    
                      
                        L
                        lsA
                      
                      2
                    
                    -
                    
                      x
                      ls
                    
                  
                  )
                
              
              ⁢
              
                
                  
                    2
                    ρ
                  
                  ⁢
                  
                    (
                    
                      
                        P
                        sv
                      
                      -
                      
                        P
                        1
                      
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        6
        )
      
    
  




Note that both the servo and bias piston slide across the swash plate for nonzero swash angle a and therefore the effective length from the swash plate kit center to servo and bias piston contact points are described as

xsv=−lsv tan a 

xbs=lbs tan a  (7)

Source flow Qs, flow from source to load Qstld and flow from actuator to tank Qldtt is defined respectively as





  
    
      
        
          
            Q
            s
          
          =
          
            
              
                
                  
                    n
                    p
                  
                  ⁢
                  
                    A
                    p
                  
                  ⁢
                  
                    l
                    p
                  
                  ⁢
                  tan
                  ⁢
                  
                    
                  
                  ⁢
                  a
                
                π
              
              ⁢
              ω
            
            -
            
              Q
              stsv
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Q
            stld
          
          =
          
            
              C
              d
            
            ⁢
            
              A
              stld
            
            ⁢
            
              
                
                  2
                  ρ
                
                ⁢
                
                  (
                  
                    
                      P
                      s
                    
                    -
                    
                      P
                      ld
                    
                  
                  )
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Q
            ldtt
          
          =
          
            
              C
              d
            
            ⁢
            
              A
              ldtt
            
            ⁢
            
              
                
                  2
                  ρ
                
                ⁢
                
                  (
                  
                    
                      P
                      ld
                    
                    -
                    
                      P
                      t
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




Subsequently load pressure is defined as





  
    
      
        
          
            P
            .
          
          ld
        
        =
        
          
            β
            
              V
              ld
            
          
          ⁢
          
            (
            
              
                Q
                stld
              
              -
              
                
                  A
                  ld
                
                ⁢
                
                  
                    x
                    .
                  
                  ld
                
              
            
            )
          
        
      
    
    
      
        (
        9
        )
      
    
  




The hydraulic actuator load dynamics is defined as

{umlaut over (x)}ld=−Cld{dot over (x)}ld−Kldxld+AldPld−mldg−0.7AldPt  (10)

Since the cylinder ratio s 7:10 and the rod end of the actuator is directly connected to tank, it maintains the tank pressure of Pt.





  
    
      
        
          
            P
            .
          
          s
        
        =
        
          
            β
            
              V
              
                P
                s
              
            
          
          ⁢
          
            (
            
              
                Q
                s
              
              -
              
                Q
                stld
              
              -
              
                Q
                leak
              
            
            )
          
        
      
    
    
      
        (
        11
        )
      
    
  




Now defining the states as

x1=xpc, x2={dot over (x)}pc, x3=Ps 

x4=a, x5={dot over (a)}, x6=Psv 

x7=xls, X8={dot over (x)}ls,x9=Pld 

x10=xld, x11={dot over (x)}ld  (12)

Rearranging the Equations from (1) to (11) and implying the state variables one can rewrite the following





  
    
      
        
          
        
        ⁢
        
          
            
              x
              .
            
            1
          
          =
          
            x
            2
          
        
      
    
    
      
        (
        13
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              x
              .
            
            2
          
          =
          
            
              
                1
                
                  m
                  pc
                
              
              ⁢
              
                {
                
                  
                    
                      -
                      
                        C
                        pc
                      
                    
                    ⁢
                    
                      x
                      2
                    
                  
                  -
                  
                    
                      K
                      pc
                    
                    ⁢
                    
                      x
                      1
                    
                  
                  -
                  
                    
                      A
                      pc
                    
                    ⁢
                    
                      P
                      
                        k
                        pc
                      
                    
                  
                
                }
              
            
            +
            
              
                
                  A
                  pc
                
                
                  m
                  pc
                
              
              ⁢
              
                x
                3
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  
  
    
      
        
          
            x
            .
          
          3
        
        =
        
          
            
              β
              
                V
                
                  p
                  s
                
              
            
            ⁢
            
              (
              
                
                  
                    -
                    
                      C
                      d
                    
                  
                  ⁢
                  
                    A
                    stld
                  
                  ⁢
                  
                    
                      
                        2
                        ρ
                      
                      ⁢
                      
                        (
                        
                          
                            x
                            3
                          
                          -
                          
                            x
                            9
                          
                        
                        )
                      
                    
                  
                
                -
                
                  Q
                  leak
                
              
              )
            
          
          +
          
            
              β
              
                V
                
                  p
                  s
                
              
            
            ⁢
            
              
                
                  n
                  p
                
                ⁢
                
                  A
                  p
                
                ⁢
                
                  l
                  p
                
              
              π
            
            ⁢
            ω
            ⁢
            
              
            
            ⁢
            
              tan
              ⁡
              
                (
                
                  x
                  4
                
                )
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              x
              .
            
            4
          
          =
          
            x
            5
          
        
      
    
    
      
        (
        16
        )
      
    
  
  
    
      
        
          
            x
            .
          
          5
        
        =
        
          
            
              1
              
                l
                a
              
            
            ⁢
            
              {
              
                
                  
                    -
                    
                      C
                      a
                    
                  
                  ⁢
                  
                    x
                    5
                  
                
                -
                
                  
                    K
                    a
                  
                  ⁢
                  
                    x
                    4
                  
                
                +
                
                  T
                  mswash
                
                +
                
                  
                    l
                    bs
                  
                  ⁢
                  
                    A
                    bs
                  
                  ⁢
                  
                    x
                    3
                  
                
                +
                
                  
                    l
                    bs
                    2
                  
                  ⁢
                  
                    K
                    bs
                  
                  ⁢
                  
                    tan
                    ⁡
                    
                      (
                      
                        x
                        4
                      
                      )
                    
                  
                
              
              }
            
          
          -
          
            
              
                
                  l
                  sv
                
                ⁢
                
                  A
                  sv
                
              
              
                l
                a
              
            
            ⁢
            
              x
              6
            
          
        
      
    
    
      
        (
        17
        )
      
    
  
  
    
      
        
          
            x
            .
          
          6
        
        =
        
          
            β
            
              V
              sv
            
          
          ⁢
          
            {
            
              
                
                  c
                  d
                
                ⁢
                
                  x
                  7
                
                ⁢
                h
                ⁢
                
                  
                    
                      2
                      ρ
                    
                    ⁢
                    
                      (
                      
                        
                          x
                          3
                        
                        -
                        
                          x
                          6
                        
                      
                      )
                    
                  
                
              
              -
              
                
                  
                    c
                    d
                  
                  ⁡
                  
                    (
                    
                      
                        
                          L
                          lsA
                        
                        2
                      
                      -
                      
                        x
                        7
                      
                    
                    )
                  
                
                ⁢
                h
                ⁢
                
                  
                    
                      2
                      ρ
                    
                    ⁢
                    
                      (
                      
                        
                          x
                          6
                        
                        -
                        
                          P
                          t
                        
                      
                      )
                    
                  
                
              
            
            }
          
        
      
    
    
      
        (
        18
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              x
              .
            
            7
          
          =
          
            x
            8
          
        
      
    
    
      
        (
        19
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              x
              .
            
            8
          
          =
          
            
              1
              
                m
                ls
              
            
            ⁢
            
              {
              
                
                  
                    -
                    
                      C
                      ls
                    
                  
                  ⁢
                  
                    x
                    8
                  
                
                -
                
                  
                    K
                    ls
                  
                  ⁢
                  
                    x
                    7
                  
                
                +
                
                  
                    A
                    ls
                  
                  ⁡
                  
                    (
                    
                      
                        x
                        3
                      
                      -
                      
                        x
                        9
                      
                      -
                      
                        P
                        
                          k
                          ls
                        
                      
                    
                    )
                  
                
              
              }
            
          
        
      
    
    
      
        (
        20
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              x
              .
            
            9
          
          =
          
            
              β
              
                V
                ld
              
            
            ⁢
            
              {
              
                
                  C
                  d
                
                ⁢
                
                  A
                  stld
                
                ⁢
                
                  
                    
                      
                        2
                        ρ
                      
                      ⁢
                      
                        (
                        
                          
                            x
                            3
                          
                          -
                          
                            x
                            9
                          
                        
                        )
                      
                    
                    -
                    
                      
                        A
                        ld
                      
                      ⁢
                      
                        x
                        11
                      
                    
                  
                
              
              }
            
          
        
      
    
    
      
        (
        21
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              x
              .
            
            10
          
          =
          
            x
            11
          
        
      
    
    
      
        (
        22
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              x
              .
            
            11
          
          =
          
            
              
                -
                
                  C
                  ld
                
              
              ⁢
              
                x
                11
              
            
            -
            
              
                K
                ld
              
              ⁢
              
                x
                10
              
            
            -
            
              
                A
                ld
              
              ⁢
              
                x
                9
              
            
            +
            
              0.7
              ⁢
              
                A
                ld
              
              ⁢
              
                P
                t
              
            
            +
            
              
                m
                ld
              
              ⁢
              g
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




Also considering the virtual state variable as δ39=x3−x9 and thus





  
    
      
        
          
            δ
            .
          
          39
        
        =
        
          
            
              β
              
                V
                
                  p
                  s
                
              
            
            ⁢
            
              (
              
                
                  
                    -
                    
                      C
                      d
                    
                  
                  ⁢
                  
                    A
                    stld
                  
                  ⁢
                  
                    
                      
                        2
                        ρ
                      
                      ⁢
                      
                        δ
                        39
                      
                    
                  
                
                -
                
                  Q
                  leak
                
              
              )
            
          
          +
          
            
              β
              
                V
                
                  p
                  s
                
              
            
            ⁢
            
              
                
                  n
                  p
                
                ⁢
                
                  A
                  p
                
                ⁢
                
                  l
                  p
                
              
              π
            
            ⁢
            ω
            ⁢
            
              
            
            ⁢
            
              tan
              ⁡
              
                (
                
                  x
                  4
                
                )
              
            
          
          -
          
            
              β
              
                V
                ld
              
            
            ⁢
            
              {
              
                
                  C
                  d
                
                ⁢
                
                  A
                  stld
                
                ⁢
                
                  
                    
                      
                        2
                        ρ
                      
                      ⁢
                      
                        δ
                        39
                      
                    
                    -
                    
                      
                        A
                        ld
                      
                      ⁢
                      
                        x
                        11
                      
                    
                  
                
              
              }
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




and assuming Vps≈Vld





  
    
      
        
          
            δ
            .
          
          39
        
        =
        
          
            
              β
              
                V
                
                  p
                  s
                
              
            
            ⁢
            
              
                
                  n
                  p
                
                ⁢
                
                  A
                  p
                
                ⁢
                
                  l
                  p
                
              
              π
            
            ⁢
            ω
            ⁢
            
              
            
            ⁢
            
              tan
              ⁡
              
                (
                
                  x
                  4
                
                )
              
            
          
          +
          
            
              β
              
                V
                
                  p
                  s
                
              
            
            ⁢
            
              A
              ld
            
            ⁢
            
              x
              11
            
          
          -
          
            
              β
              
                V
                
                  p
                  s
                
              
            
            ⁢
            
              Q
              leak
            
          
          -
          
            
              
                2
                ⁢
                
                  
                
                ⁢
                β
              
              
                V
                
                  p
                  s
                
              
            
            ⁢
            
              C
              d
            
            ⁢
            
              A
              stld
            
            ⁢
            
              
                
                  2
                  ρ
                
                ⁢
                
                  δ
                  39
                
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




This is based on the assumption that the volumes Vps, Vld are the same without any loss of generality.

From these equations subsystems are defined. As an example, a first subsystem includes equations (21), (22), and (23), a second subsystem includes equations (15), (16), and (17), and a third subsystem includes equations (13), (14), (19) and (20). The first and the second systems are actuated by operator commands and load-sense spool effective opening. The third subsystem includes Pressure Control and Load Sense spools and is a passive system.
To determine an operator input that would stabilize the first subsystem a backstepping control is applied.
In control theory backstepping is used for designing stabilizing controls for a special recursive class of nonlinear systems. Starting with Equation (22) one figures out an appropriate x11 so that Equation (22) will be stabilized. Now, Equation (22) can be written as

{dot over (x)}10=θ11(t)  (26)

θ11(t) is known as the desired control law for subsystem (13) and the change of variable is z11=x11−θ11(t). Then consider a Lyapunov function candidate V1 as

V1=1⁄2z112≧0  (27)

Differentiating (27) and as δ39=(x3−x9) one has





  
    
      
        
          
        
        ⁢
        
          
            
              
                V
                .
              
              1
            
            =
            
              
                z
                11
              
              ⁢
              
                
                  z
                  .
                
                11
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                V
                .
              
              1
            
            =
            
              
                z
                11
              
              ⁡
              
                (
                
                  
                    
                      x
                      .
                    
                    11
                  
                  -
                  
                    
                      
                        θ
                        .
                      
                      11
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                )
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              
                V
                .
              
              1
            
            =
            
              
                z
                11
              
              ⁢
              
                {
                
                  
                    
                      -
                      
                        C
                        ld
                      
                    
                    ⁢
                    
                      x
                      11
                    
                  
                  -
                  
                    
                      K
                      ld
                    
                    ⁢
                    
                      x
                      10
                    
                  
                  +
                  
                    
                      A
                      ld
                    
                    ⁡
                    
                      (
                      
                        
                          x
                          3
                        
                        -
                        
                          x
                          9
                        
                      
                      )
                    
                  
                  +
                  
                    0.7
                    ⁢
                    
                      
                    
                    ⁢
                    
                      A
                      ld
                    
                    ⁢
                    
                      P
                      t
                    
                  
                  -
                  
                    
                      A
                      ld
                    
                    ⁢
                    
                      x
                      3
                    
                  
                  +
                  
                    
                      m
                      ld
                    
                    ⁢
                    g
                  
                  -
                  
                    
                      
                        θ
                        .
                      
                      11
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                }
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              
                V
                .
              
              1
            
            =
            
              
                z
                11
              
              ⁢
              
                {
                
                  
                    
                      -
                      
                        C
                        ld
                      
                    
                    ⁢
                    
                      x
                      11
                    
                  
                  -
                  
                    
                      K
                      ld
                    
                    ⁢
                    
                      x
                      10
                    
                  
                  +
                  
                    
                      A
                      ld
                    
                    ⁢
                    
                      δ
                      39
                    
                  
                  +
                  
                    0.7
                    ⁢
                    
                      
                    
                    ⁢
                    
                      A
                      ld
                    
                    ⁢
                    
                      P
                      t
                    
                  
                  -
                  
                    
                      A
                      ld
                    
                    ⁢
                    
                      x
                      3
                    
                  
                  +
                  
                    
                      m
                      ld
                    
                    ⁢
                    g
                  
                  -
                  
                    
                      
                        θ
                        .
                      
                      11
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                }
              
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




Choosing a desired control law δ39d as

δ39d=Ald−1(Cldx11+Kldx10−0.7AldPt+Aldx3−mldg−γ11z11+{dot over (θ)}11(t))   (29)

If δ39=δ39d then

{dot over (V)}1=−γ11z112≦0,γ11>0  (30)

Note that z11 is the error variable for cylinder velocity x11 with the desired cylinder velocity x11d; and therefore negative definiteness of (30) will guarantee the tracking. Also note that this process develops a tracking problem around cylinder velocity so that for any operator command the load velocity could be decided based on guided trajectory. This behavior will also solve the problem of cavitation for runaway loads.

However at this point one cannot guarantee δ39=δ39d. Therefore another error variable is considered as z39=δ39−δ39d and δ39=z39+δ39d is substituted in the Equation (28); to

{dot over (V)}1=z11(−Cldx11−Kldx10+Ald{z39⇄δ39d}+0.7AldPt−Aldx3+mldg−{dot over (θ)}11(t))=γ11z112+Aldz11z39  (31)

Now, for differentiating error variable z39; one gets





  
    
      
        
          
            z
            .
          
          39
        
        =
        
          
            
              
                δ
                .
              
              39
            
            -
            
              
                δ
                .
              
              39
              d
            
          
          =
          
            
              
                β
                
                  V
                  
                    p
                    s
                  
                
              
              ⁢
              
                
                  
                    n
                    p
                  
                  ⁢
                  
                    A
                    p
                  
                  ⁢
                  
                    l
                    p
                  
                
                π
              
              ⁢
              ω
              ⁢
              
                
              
              ⁢
              
                tan
                ⁡
                
                  (
                  
                    x
                    4
                  
                  )
                
              
            
            +
            
              
                β
                
                  V
                  
                    p
                    s
                  
                
              
              ⁢
              
                A
                ld
              
              ⁢
              
                x
                11
              
            
            -
            
              
                β
                
                  V
                  
                    p
                    s
                  
                
              
              ⁢
              
                Q
                leak
              
            
            -
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  β
                
                
                  V
                  
                    p
                    s
                  
                
              
              ⁢
              
                C
                d
              
              ⁢
              
                A
                stld
              
              ⁢
              
                
                  
                    2
                    ρ
                  
                  ⁢
                  
                    δ
                    39
                  
                
              
            
            -
            
              
                δ
                .
              
              39
              d
            
          
        
      
    
    
      
        (
        32
        )
      
    
  




At this stage a second Lyapunov function is considered as V2 

V2=V1=1⁄2z392  (33)

and differentiating (33) one has





  
    
      
        
          
            V
            .
          
          2
        
        =
        
          
            
              
                V
                .
              
              1
            
            +
            
              
                z
                39
              
              ⁢
              
                
                  z
                  .
                
                39
              
            
          
          =
          
            
              
                -
                
                  γ
                  11
                
              
              ⁢
              
                z
                11
                2
              
            
            +
            
              
                A
                ld
              
              ⁢
              
                z
                11
              
              ⁢
              
                z
                39
              
            
            +
            
              
                z
                39
              
              ⁢
              
                {
                
                  
                    
                      β
                      
                        V
                        
                          p
                          s
                        
                      
                    
                    ⁢
                    
                      
                        
                          n
                          p
                        
                        ⁢
                        
                          A
                          p
                        
                        ⁢
                        
                          l
                          p
                        
                      
                      π
                    
                    ⁢
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    
                      tan
                      ⁡
                      
                        (
                        
                          x
                          4
                        
                        )
                      
                    
                  
                  +
                  
                    
                      β
                      
                        V
                        
                          p
                          s
                        
                      
                    
                    ⁢
                    
                      A
                      ld
                    
                    ⁢
                    
                      x
                      11
                    
                  
                  -
                  
                    
                      β
                      
                        V
                        
                          p
                          s
                        
                      
                    
                    ⁢
                    
                      Q
                      leak
                    
                  
                  -
                  
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        β
                      
                      
                        V
                        
                          p
                          s
                        
                      
                    
                    ⁢
                    
                      C
                      d
                    
                    ⁢
                    
                      A
                      stld
                    
                    ⁢
                    
                      
                        
                          2
                          ρ
                        
                        ⁢
                        
                          δ
                          39
                        
                      
                    
                  
                  -
                  
                    
                      δ
                      .
                    
                    39
                    d
                  
                
                }
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




Note that the proposed input as operator command to the valve has appeared in (34). For simplicity a simple variable area orifice as the valve is considered. Therefore operator command is assumed to directly proportional to the variable area of the orifice Astld. Since the term Astld appeared in (34), an appropriate input is Astld such that {dot over (V)}2≦0. One of the possible choices for Astld could be





  
    
      
        
          A
          stld
        
        =
        
          
            
              (
              
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    β
                  
                  
                    V
                    
                      p
                      s
                    
                  
                
                ⁢
                
                  C
                  d
                
                ⁢
                
                  
                    
                      2
                      ρ
                    
                    ⁢
                    
                      δ
                      39
                    
                  
                
              
              )
            
            
              -
              1
            
          
          ⁢
          
            {
            
              
                
                  A
                  ld
                
                ⁡
                
                  (
                  
                    
                      z
                      11
                    
                    +
                    
                      z
                      39
                    
                  
                  )
                
              
              -
              
                
                  β
                  
                    V
                    
                      p
                      s
                    
                  
                
                ⁢
                
                  Q
                  leak
                
              
              -
              
                
                  δ
                  .
                
                39
                d
              
              +
              
                
                  β
                  
                    V
                    
                      p
                      s
                    
                  
                
                ⁢
                
                  
                    
                      n
                      p
                    
                    ⁢
                    
                      A
                      p
                    
                    ⁢
                    
                      l
                      p
                    
                  
                  π
                
                ⁢
                ω
                ⁢
                
                  
                
                ⁢
                
                  tan
                  ⁡
                  
                    (
                    
                      x
                      4
                    
                    )
                  
                
              
              +
              
                
                  β
                  
                    V
                    
                      p
                      s
                    
                  
                
                ⁢
                
                  A
                  ld
                
                ⁢
                
                  x
                  11
                
              
            
            }
          
        
      
    
    
      
        (
        35
        )
      
    
  




so that

{dot over (V)}2=−γ11z112−Aldγ39z392≦0  (36)

Equation (36) guarantees the negative semi-definiteness of {dot over (V)}2, if (35) is feasible and implementable. Note that Astld in (35) is a function of the desired cylinder acceleration. Also, in the first subsystem, system pressure, servo pressure and load dynamics are involved and a desired valve area is derived in (35) to stabilize the first subsystem.

As δ39d, has been designed, one now can replace Pkpc with δ39d. Thus the system will have a variable load sense or margin pressure setting which could be realized by using electronic control of Load Sense spool spring setting. A variable margin also helps to reduce the loss across the valve.
To determine a second input control as effective load-sense spool area, a backstepping technique is applied to the second subsystem. The second subsystem involves swashplate 16 and servo 36 dynamics.
Although these subsystems are treated separately for design control, they are interconnected separately and coupled through states. For the second subsystem a backstepping control approach is used to find out the appropriate height h(xls). Starting with (16)

{dot over (x)}4=x5=θ5(t)  (37)

where θ5 is the new virtual control law for the second subsystem (16). Therefore the error variable is defined as z5=x5−θ5(t) and similarly a Lyapunov function candidate is defined as

V3=1⁄2z52  (38)

Differentiating (38) one has





  
    
      
        
          
            V
            .
          
          3
        
        =
        
          
            
              z
              5
            
            ⁢
            
              
                z
                .
              
              5
            
          
          =
          
            
              
                z
                5
              
              ⁡
              
                (
                
                  
                    
                      x
                      .
                    
                    5
                  
                  -
                  
                    
                      
                        θ
                        .
                      
                      5
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                )
              
            
            =
            
              
                z
                5
              
              ⁡
              
                [
                
                  
                    
                      1
                      
                        I
                        a
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            -
                            
                              C
                              a
                            
                          
                          ⁢
                          
                            x
                            5
                          
                        
                        -
                        
                          
                            K
                            a
                          
                          ⁢
                          
                            x
                            4
                          
                        
                        +
                        
                          T
                          mswash
                        
                        +
                        
                          
                            l
                            bs
                          
                          ⁢
                          
                            A
                            bs
                          
                          ⁢
                          
                            x
                            3
                          
                        
                        +
                        
                          
                            l
                            bs
                            2
                          
                          ⁢
                          
                            K
                            bs
                          
                          ⁢
                          
                            tan
                            ⁡
                            
                              (
                              
                                x
                                4
                              
                              )
                            
                          
                        
                      
                      }
                    
                  
                  -
                  
                    
                      
                        
                          l
                          sv
                        
                        ⁢
                        
                          A
                          sv
                        
                      
                      
                        I
                        a
                      
                    
                    ⁢
                    
                      x
                      6
                    
                  
                  -
                  
                    
                      
                        θ
                        .
                      
                      5
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        39
        )
      
    
  




Even though the swash moments are nonlinear in nature, this example used is test branch data for accuracy. Similarly, pump leakage flow Qleak is also computed as tubular data. In (37) and (39), θ5 and {dot over (θ)}5 is defined as desired angular velocity and acceleration for the swash plate respectively. Therefore one can choose any desired and guided swash trajectory for the control law derived in (51).
If one defines θ6 as




  
    
      
        
          θ
          6
        
        =
        
          
            
              (
              
                
                  
                    1
                    sv
                  
                  ⁢
                  
                    A
                    sv
                  
                
                
                  I
                  a
                
              
              )
            
            
              -
              1
            
          
          ⁢
          
            {
            
              
                z
                5
              
              -
              
                
                  
                    θ
                    .
                  
                  5
                
                ⁡
                
                  (
                  t
                  )
                
              
              +
              
                
                  1
                  
                    I
                    a
                  
                
                ⁢
                
                  (
                  
                    
                      
                        -
                        
                          C
                          a
                        
                      
                      ⁢
                      
                        x
                        5
                      
                    
                    -
                    
                      
                        K
                        a
                      
                      ⁢
                      
                        x
                        4
                      
                    
                    +
                    
                      T
                      mswash
                    
                    +
                    
                      
                        I
                        bs
                      
                      ⁢
                      
                        A
                        bs
                      
                      ⁢
                      
                        x
                        3
                      
                    
                    +
                    
                      
                        1
                        bs
                        2
                      
                      ⁢
                      
                        K
                        bs
                      
                      ⁢
                      
                        tan
                        ⁡
                        
                          (
                          
                            x
                            4
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
            }
          
        
      
    
    
      
        (
        40
        )
      
    
  




and if x6=θ6 then one can easily find out that

{dot over (V)}3=−γ6z52≦0,γ6>0  (41)

This does not guarantee x6=θ6 at this point and therefore another error variable is used z6=x6−θ6(t) and substituting x6=z6+θ6 in (39) one gets




  
    
      
        
          
            V
            .
          
          3
        
        =
        
          
            z
            5
          
          ⁡
          
            [
            
              
                
                  1
                  
                    I
                    a
                  
                
                ⁢
                
                  {
                  
                    
                      
                        -
                        
                          C
                          a
                        
                      
                      ⁢
                      
                        x
                        5
                      
                    
                    -
                    
                      
                        K
                        a
                      
                      ⁢
                      
                        x
                        4
                      
                    
                    +
                    
                      T
                      mswash
                    
                    +
                    
                      
                        l
                        bs
                      
                      ⁢
                      
                        A
                        bs
                      
                      ⁢
                      
                        x
                        3
                      
                    
                    +
                    
                      
                        l
                        bs
                        2
                      
                      ⁢
                      
                        K
                        bs
                      
                      ⁢
                      
                        tan
                        ⁡
                        
                          (
                          
                            x
                            4
                          
                          )
                        
                      
                    
                  
                  }
                
              
              -
              
                
                  
                    
                      l
                      sv
                    
                    ⁢
                    
                      A
                      sv
                    
                  
                  
                    I
                    a
                  
                
                ⁢
                
                  (
                  
                    
                      z
                      6
                    
                    +
                    
                      θ
                      6
                    
                  
                  )
                
              
              -
              
                
                  
                    θ
                    .
                  
                  5
                
                ⁡
                
                  (
                  t
                  )
                
              
            
            ]
          
        
      
    
    
      
        (
        42
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              V
              .
            
            3
          
          =
          
            
              
                -
                
                  γ
                  6
                
              
              ⁢
              
                z
                5
                2
              
            
            -
            
              
                
                  
                    l
                    sv
                  
                  ⁢
                  
                    A
                    sv
                  
                
                
                  I
                  a
                
              
              ⁢
              
                z
                5
              
              ⁢
              
                z
                6
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




Equation (43) is not negative semi-definite as the sign of the term z5z6 is unknown. One then proceeds with the next step of backstepping until a proposed input variable shows up in the Lyapunov proof. To continue differentiating the variable z6




  
    
      
        
          
            z
            .
          
          6
        
        =
        
          
            
              
                x
                .
              
              6
            
            -
            
              
                
                  θ
                  .
                
                6
              
              ⁡
              
                (
                t
                )
              
            
          
          =
          
            
              
                β
                
                  V
                  sv
                
              
              ⁢
              
                {
                
                  
                    
                      c
                      d
                    
                    ⁢
                    
                      x
                      7
                    
                    ⁢
                    h
                    ⁢
                    
                      
                        
                          2
                          ρ
                        
                        ⁢
                        
                          (
                          
                            
                              x
                              3
                            
                            -
                            
                              x
                              6
                            
                          
                          )
                        
                      
                    
                  
                  -
                  
                    
                      
                        c
                        d
                      
                      ⁡
                      
                        (
                        
                          
                            
                              L
                              lsA
                            
                            2
                          
                          -
                          
                            x
                            7
                          
                        
                        )
                      
                    
                    ⁢
                    h
                    ⁢
                    
                      
                        
                          2
                          ρ
                        
                        ⁢
                        
                          (
                          
                            
                              x
                              6
                            
                            -
                            
                              P
                              t
                            
                          
                          )
                        
                      
                    
                  
                
                }
              
            
            -
            
              
                
                  θ
                  .
                
                6
              
              ⁡
              
                (
                t
                )
              
            
          
        
      
    
    
      
        (
        44
        )
      
    
  
  
    
      
        
          
            z
            .
          
          6
        
        =
        
          
            
              β
              
                V
                sv
              
            
            ⁢
            
              {
              
                
                  
                    c
                    d
                  
                  ⁢
                  
                    x
                    7
                  
                  ⁢
                  h
                  ⁢
                  
                    
                      
                        2
                        ρ
                      
                      ⁢
                      
                        (
                        
                          
                            x
                            3
                          
                          -
                          
                            x
                            6
                          
                        
                        )
                      
                    
                  
                
                -
                
                  
                    
                      c
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            L
                            lsA
                          
                          2
                        
                        -
                        
                          x
                          7
                        
                      
                      )
                    
                  
                  ⁢
                  h
                  ⁢
                  
                    
                      
                        2
                        ρ
                      
                      ⁢
                      
                        (
                        
                          
                            x
                            6
                          
                          -
                          
                            P
                            t
                          
                        
                        )
                      
                    
                  
                
              
              }
            
          
          -
          
            
              
                (
                
                  
                    
                      l
                      sv
                    
                    ⁢
                    
                      A
                      sv
                    
                  
                  
                    I
                    a
                  
                
                )
              
              
                -
                1
              
            
            ⁢
            
              d
              
                d
                ⁢
                
                  
                
                ⁢
                t
              
            
            ⁢
            
              {
              
                
                  z
                  5
                
                -
                
                  
                    
                      θ
                      .
                    
                    5
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                +
                
                  
                    1
                    
                      I
                      a
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          -
                          
                            C
                            z
                          
                        
                        ⁢
                        
                          x
                          5
                        
                      
                      -
                      
                        
                          K
                          a
                        
                        ⁢
                        
                          x
                          4
                        
                      
                      +
                      
                        T
                        mswash
                      
                      +
                      
                        
                          l
                          bs
                        
                        ⁢
                        
                          A
                          bs
                        
                        ⁢
                        
                          x
                          3
                        
                      
                      +
                      
                        
                          l
                          bs
                          2
                        
                        ⁢
                        
                          K
                          bs
                        
                        ⁢
                        
                          tan
                          ⁡
                          
                            (
                            
                              x
                              4
                            
                            )
                          
                        
                      
                    
                    )
                  
                
              
              }
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




Rewriting (45) one has





  
    
      
        
          
            z
            .
          
          6
        
        =
        
          
            
              β
              
                V
                sv
              
            
            ⁢
            
              {
              
                
                  
                    C
                    d
                  
                  ⁢
                  
                    x
                    7
                  
                  ⁢
                  h
                  ⁢
                  
                    
                      
                        2
                        ρ
                      
                      ⁢
                      
                        (
                        
                          
                            x
                            3
                          
                          -
                          
                            x
                            6
                          
                        
                        )
                      
                    
                  
                
                -
                
                  
                    
                      C
                      d
                    
                    ⁡
                    
                      (
                      
                        
                          
                            L
                            lsA
                          
                          2
                        
                        -
                        
                          x
                          7
                        
                      
                      )
                    
                  
                  ⁢
                  h
                  ⁢
                  
                    
                      
                        2
                        ρ
                      
                      ⁢
                      
                        (
                        
                          
                            x
                            6
                          
                          -
                          
                            P
                            t
                          
                        
                        )
                      
                    
                  
                
              
              }
            
          
          -
          
            
              
                (
                
                  
                    
                      l
                      sv
                    
                    ⁢
                    
                      A
                      sv
                    
                  
                  
                    I
                    a
                  
                
                )
              
              
                -
                1
              
            
            ⁢
            
              {
              
                
                  z
                  5
                
                -
                
                  
                    
                      θ
                      .
                    
                    5
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                +
                
                  
                    1
                    
                      I
                      a
                    
                  
                  ⁢
                  
                    (
                    
                      
                        
                          -
                          
                            C
                            a
                          
                        
                        ⁢
                        
                          
                            x
                            .
                          
                          5
                        
                      
                      -
                      
                        
                          K
                          a
                        
                        ⁢
                        
                          
                            x
                            .
                          
                          4
                        
                      
                      +
                      
                        
                          T
                          .
                        
                        mswash
                      
                      +
                      
                        
                          l
                          bs
                        
                        ⁢
                        
                          A
                          bs
                        
                        ⁢
                        
                          
                            x
                            .
                          
                          3
                        
                      
                      +
                      
                        
                          l
                          bs
                          2
                        
                        ⁢
                        
                          K
                          bs
                        
                        ⁢
                        
                          
                            sec
                            2
                          
                          ⁡
                          
                            (
                            
                              x
                              4
                            
                            )
                          
                        
                      
                    
                    )
                  
                
              
              }
            
          
        
      
    
    
      
        (
        46
        )
      
    
  




Note that Load Sense spool displacement xls or x7 is one of the dominant factors in (46). But one can always choose to discretize the servo flow based on x7 as





  
    
      
        
          
            
              
                
                  
                    Q
                    stsv
                  
                  =
                  
                    
                      C
                      d
                    
                    ⁢
                    
                      x
                      7
                    
                    ⁢
                    h
                    ⁢
                    
                      
                        
                          2
                          ρ
                        
                        ⁢
                        
                          (
                          
                            
                              x
                              3
                            
                            -
                            
                              x
                              6
                            
                          
                          )
                        
                      
                    
                  
                
                }
              
              ⁢
              
                ∀
                
                  
                    x
                    7
                  
                  >
                  0
                
              
            
            ⁢
            
              

            
            ⁢
            
              
                Q
                stsv
              
              =
              
                
                  
                    C
                    d
                  
                  ⁡
                  
                    (
                    
                      
                        
                          L
                          lsA
                        
                        2
                      
                      -
                      
                        x
                        7
                      
                    
                    )
                  
                
                ⁢
                h
                ⁢
                
                  
                    
                      2
                      ρ
                    
                    ⁢
                    
                      (
                      
                        
                          x
                          6
                        
                        -
                        
                          P
                          t
                        
                      
                      )
                    
                  
                
              
            
          
          }
        
        ⁢
        
          ∀
          
            
              x
              7
            
            ≤
            0
          
        
      
    
    
      
        (
        47
        )
      
    
  




Considering another Lyapunov function as

V4=V3+1⁄2z62  (48)

and differentiating (48) with respect to time one has





  
    
      
        
          
        
        ⁢
        
          
            
              V
              .
            
            4
          
          =
          
            
              
                
                  V
                  .
                
                3
              
              +
              
                
                  z
                  6
                
                ⁢
                
                  
                    z
                    .
                  
                  6
                
              
            
            =
            
              
                
                  -
                  
                    γ
                    6
                  
                
                ⁢
                
                  z
                  5
                  2
                
              
              -
              
                
                  
                    
                      l
                      sv
                    
                    ⁢
                    
                      A
                      sv
                    
                  
                  
                    I
                    a
                  
                
                ⁢
                
                  z
                  5
                
                ⁢
                
                  z
                  6
                
              
              +
              
                
                  z
                  6
                
                ⁢
                
                  
                    z
                    .
                  
                  6
                
                ⁢
                
                  
                
                ⁢
                or
              
            
          
        
      
    
    
      
        (
        49
        )
      
    
  
  
    
      
        
          
            V
            .
          
          4
        
        =
        
          
            
              -
              
                γ
                6
              
            
            ⁢
            
              z
              5
              2
            
          
          -
          
            
              
                
                  l
                  sv
                
                ⁢
                
                  A
                  sv
                
              
              
                I
                a
              
            
            ⁢
            
              z
              5
            
            ⁢
            
              z
              6
            
          
          +
          
            
              z
              6
            
            ⁡
            
              [
              
                
                  
                    β
                    
                      V
                      sv
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          C
                          d
                        
                        ⁢
                        
                          x
                          7
                        
                        ⁢
                        
                          
                            
                              2
                              ρ
                            
                            ⁢
                            
                              (
                              
                                
                                  x
                                  3
                                
                                -
                                
                                  x
                                  6
                                
                              
                              )
                            
                          
                        
                      
                      -
                      
                        
                          
                            C
                            d
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  L
                                  lsA
                                
                                2
                              
                              -
                              
                                x
                                7
                              
                            
                            )
                          
                        
                        ⁢
                        
                          
                            
                              2
                              ρ
                            
                            ⁢
                            
                              (
                              
                                
                                  x
                                  6
                                
                                -
                                
                                  P
                                  t
                                
                              
                              )
                            
                          
                        
                      
                    
                    }
                  
                  ⁢
                  h
                
                -
                
                  
                    
                      (
                      
                        
                          
                            l
                            sv
                          
                          ⁢
                          
                            A
                            sv
                          
                        
                        
                          I
                          a
                        
                      
                      )
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          z
                          .
                        
                        5
                      
                      -
                      
                        
                          
                            θ
                             ̈
                          
                          5
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      +
                      
                        
                          1
                          
                            I
                            a
                          
                        
                        ⁢
                        
                          (
                          
                            
                              
                                -
                                
                                  C
                                  a
                                
                              
                              ⁢
                              
                                
                                  x
                                  .
                                
                                5
                              
                            
                            -
                            
                              
                                K
                                a
                              
                              ⁢
                              
                                
                                  x
                                  .
                                
                                4
                              
                            
                            +
                            
                              
                                T
                                .
                              
                              mswash
                            
                            +
                            
                              
                                l
                                bs
                              
                              ⁢
                              
                                A
                                bs
                              
                              ⁢
                              
                                
                                  x
                                  .
                                
                                3
                              
                            
                            +
                            
                              
                                l
                                bs
                                2
                              
                              ⁢
                              
                                K
                                bs
                              
                              ⁢
                              
                                
                                  sec
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    x
                                    4
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        50
        )
      
    
  




The second proposed input h shows up in (50). Therefore one has a chance to define h such that {dot over (V)}4≦0 and one way to achieve this is as follows





  
    
      
        h
        =
        
          
            
              [
              
                
                  β
                  
                    V
                    sv
                  
                
                ⁢
                
                  {
                  
                    
                      
                        C
                        d
                      
                      ⁢
                      
                        x
                        7
                      
                      ⁢
                      
                        
                          
                            2
                            ρ
                          
                          ⁢
                          
                            (
                            
                              
                                x
                                3
                              
                              -
                              
                                x
                                6
                              
                            
                            )
                          
                        
                      
                    
                    -
                    
                      
                        
                          C
                          d
                        
                        ⁡
                        
                          (
                          
                            
                              
                                L
                                lsA
                              
                              2
                            
                            -
                            
                              x
                              7
                            
                          
                          )
                        
                      
                      ⁢
                      
                        
                          
                            2
                            ρ
                          
                          ⁢
                          
                            (
                            
                              
                                x
                                6
                              
                              -
                              
                                P
                                t
                              
                            
                            )
                          
                        
                      
                    
                  
                  }
                
              
              ]
            
            
              -
              1
            
          
          ⁢
          
             
            
              [
              
                
                  
                    
                      
                        l
                        sv
                      
                      ⁢
                      
                        A
                        sv
                      
                    
                    
                      I
                      a
                    
                  
                  ⁢
                  
                    z
                    5
                  
                
                -
                
                  
                    γ
                    7
                  
                  ⁢
                  
                    z
                    6
                  
                
                +
                
                  
                    
                      (
                      
                        
                          
                            l
                            sv
                          
                          ⁢
                          
                            A
                            sv
                          
                        
                        
                          I
                          a
                        
                      
                      )
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          z
                          .
                        
                        5
                      
                      -
                      
                        
                          
                            θ
                             ̈
                          
                          5
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      +
                      
                        
                          1
                          
                            I
                            a
                          
                        
                        ⁢
                        
                          (
                          
                            
                              
                                -
                                
                                  C
                                  a
                                
                              
                              ⁢
                              
                                
                                  x
                                  .
                                
                                5
                              
                            
                            -
                            
                              
                                K
                                a
                              
                              ⁢
                              
                                
                                  x
                                  .
                                
                                4
                              
                            
                            +
                            
                              
                                T
                                .
                              
                              mswash
                            
                            +
                            
                              
                                l
                                bs
                              
                              ⁢
                              
                                A
                                bs
                              
                              ⁢
                              
                                
                                  x
                                  .
                                
                                3
                              
                            
                            +
                            
                              
                                l
                                bs
                                2
                              
                              ⁢
                              
                                K
                                bs
                              
                              ⁢
                              
                                
                                  sec
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    x
                                    4
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                
              
            
          
        
      
    
    
      
        (
        51
        )
      
    
  




Also using (51) in (50) one has

{dot over (V)}4=−γ6z52−γ7z62  (52)

Therefore if h defined in (51) can be realized practically then the second subsystem is also stable when swash plate angular velocity and acceleration is guided.

The third subsystem includes Pressure Control and spool dynamics. While passive, the Load Sense spool displacement determines the servo flow which controls the pump's swashplate subsystem.
To begin one considers the Load Sense spool dynamics from (19)-(20) and one can rewrite it as




  
    
      
        {
        
          
            
              
                
                  
                    x
                    .
                  
                  7
                
                =
                
                  x
                  8
                
              
            
          
          
            
              
                
                  
                    x
                    .
                  
                  8
                
                =
                
                  
                    1
                    
                      m
                      ls
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          -
                          
                            C
                            ls
                          
                        
                        ⁢
                        
                          x
                          8
                        
                      
                      -
                      
                        
                          K
                          ls
                        
                        ⁢
                        
                          x
                          7
                        
                      
                      +
                      
                        u
                        1
                      
                    
                    }
                  
                
              
            
          
        
      
    
    
      
        (
        53
        )
      
    
  




where u1=Als(x3−x9−Pkls) From [13], the following theorems are applicable for passive systems.

Theorem 1: The feedback connection of two passive systems is passive.
Theorem 2: Consider a feedback connection of two dynamical systems. When input=0, the origin of the closed-loop system is asymptotically stable if each feedback component is either strictly passive, or output strictly passive and zero-state observable. Furthermore, if the storage function for each component is radially unbounded, the origin is globally asymptotically stable.
Theorem 3: consider a feedback connection of a strictly passive dynamical system with a passive memory-less function. When input=0, the origin of the closed-loop system is uniformly asymptotically stable. If the storage function for the dynamical system is radially unbounded, the origin will be globally uniformly asymptotically stable.
In view of the Theorems 1-3, one considers a Lyapunov function as

Vx=1⁄2(Klsx72+x82)  (54)

From this it is easy to prove that while Vx is radially unbounded, the origin globally asymptotically stable if u1=0.

Next consider Pressure Control spool dynamics which is also very similar to Load Sense spool dynamics. Rewriting (2) one has




  
    
      
        {
        
          
            
              
                
                  
                    x
                    .
                  
                  1
                
                =
                
                  x
                  2
                
              
            
          
          
            
              
                
                  
                    x
                    .
                  
                  2
                
                =
                
                  
                    
                      1
                      
                        m
                        pc
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            -
                            
                              C
                              pc
                            
                          
                          ⁢
                          
                            x
                            2
                          
                        
                        -
                        
                          
                            K
                            pc
                          
                          ⁢
                          
                            x
                            1
                          
                        
                      
                      }
                    
                  
                  +
                  
                    u
                    2
                  
                
              
            
          
        
      
    
    
      
        (
        55
        )
      
    
  




with





  
    u
    2
  
  =
  
    
      (
      
        
          
            
              A
              pc
            
            
              m
              pc
            
          
          ⁢
          
            x
            3
          
        
        -
        
          
            
              A
              pc
            
            
              m
              pc
            
          
          ⁢
          
            P
            
              k
              pc
            
          
        
      
      )
    
    .
  




Defining a similar Lyapunov function as

Vy=1⁄2(Kpcx12+x22)  (56)

one can prove that the third subsystem is as a whole passive. For bounded u1 and u2 one can also easily show that x1, x2, x7 and x8 are all bounded. To prove the boundedness of u1 and u2 as well as for the whole system let us consider a Lyapunov function V5 as

V5=V4+V2  (57)

Now differentiating V5 from (57) we have

{dot over (V)}5≦−γ7z52−γ7z62−γ11z112−Aldγ39z392  (58)

if Equations (36) and (51) hold. Therefore the whole load sense system is bounded and Lyapunov stable.

A simulation was carried out to verify the proposed results. The simulation was written as a Matlab script file to solve the stiff differential equations given by Equation (1) to (25). Therefore without any loss of generality, a simple example was considered which is less difficult and complex to implement as a script file.
In the simulation example, the stability of the pump pressure and other vital pump parameters under high frequency load oscillations is shown, and there is an assumption that the load is moving away from the bottom of the cylinder, assumed asxld=0.
FIG. 4 and FIG. 5 show the pump and load pressure characteristics as well as phase plane plot of the source pressure respectively. It can be seen from the phase plane plot that the system pressure is Lyapunov stable under oscillating load conditions given in FIG. 6.
FIG. 7 and FIG. 8 show the calculated Load Sense spool area and operator command as valve orifice area respectively. It can be noted that, to maintain practicality of the obtained result, saturations were used for the inputs and thereby observed whether Lyapunov stability could be achieved under given restrictions. FIG. 9 and FIG. 10 demonstrate the flow and variable margin pressure of the system.
FIG. 11 describes the cylinder position under oscillating load conditions.
Accordingly, stability analysis for a load sense pump is considered. Backstepping methodology has implemented on a nonlinear pump model to observe pump stability under oscillating load conditions while computing valve orifice area and Load Sense spool area as external input. While the former one is a real external input, but the later isn't. Desired is to find out a spool area profile to adjust gain and choke orifice size under oscillating load situation.

## Claims

1. A method of stabilizing a hydraulic load sense system, comprising the steps of:
defining a first subsystem related to operator command, a second subsystem related to a load sense spool effective opening area, and a third subsystem related to load sense spool displacement;
applying with a computer a first backstepping control to determine a first control input to stabilize the first subsystem;
applying with a computer a second backstepping control to determine a second control input to stabilize the second subsystem;
calculating a feasible load sense area based upon the first and second control inputs and the third subsystem; and
controlling a hydraulic pump based upon the feasible load sense area.

2. The method of claim 1 wherein the first backstepping control includes using a first and a second error variable.

3. The method of claim 1 wherein the first subsystem includes cylinder acceleration, system pressure, servo pressure, and load dynamics.

4. The method of claim 1 further comprising the step of determining a margin pressure setting.

5. The method of claim 1 wherein the first subsystem includes states {dot over (x)}9, {dot over (x)}10, and {dot over (x)}11; wherein {dot over (x)}9, is defined as




  
    
      
        x
        .
      
      9
    
    =
    
      
        β
        
          V
          ld
        
      
      ⁢
      
        {
        
          
            C
            d
          
          ⁢
          
            A
            stld
          
          ⁢
          
            
              
                
                  2
                  ρ
                
                ⁢
                
                  (
                  
                    
                      x
                      3
                    
                    -
                    
                      x
                      9
                    
                  
                  )
                
              
              -
              
                
                  A
                  ld
                
                ⁢
                
                  x
                  11
                
              
            
          
        
        }
      
    
  
  ,




{dot over (x)}10 is defined as {dot over (x)}10=x11, and {dot over (x)}11 is defined as {dot over (x)}11=−Cldx11−Kldx10−Aldx9+0.7AldPt+mldg.

6. The method of claim 1 wherein the second subsystem includes states {dot over (x)}3, {dot over (x)}4, and {dot over (x)}5; wherein {dot over (x)}3 is defined as




  
    
      
        x
        .
      
      3
    
    =
    
      
        
          β
          
            V
            
              p
              s
            
          
        
        ⁢
        
          (
          
            
              
                -
                
                  C
                  d
                
              
              ⁢
              
                A
                stld
              
              ⁢
              
                
                  
                    2
                    ρ
                  
                  ⁢
                  
                    (
                    
                      
                        x
                        3
                      
                      -
                      
                        x
                        9
                      
                    
                    )
                  
                
              
            
            -
            
              Q
              leak
            
          
          )
        
      
      +
      
        
          β
          
            V
            
              p
              s
            
          
        
        ⁢
        
          
            
              n
              p
            
            ⁢
            
              A
              p
            
            ⁢
            
              l
              p
            
          
          π
        
        ⁢
        ω
        ⁢
        
          
        
        ⁢
        
          tan
          ⁡
          
            (
            
              x
              4
            
            )
          
        
      
    
  
  ,




{dot over (x)}4 is defined as {dot over (x)}4=x5, and {dot over (x)}5 is defined as




  
    
      x
      .
    
    5
  
  =
  
    
      
        1
        
          I
          a
        
      
      ⁢
      
        {
        
          
            
              -
              
                C
                a
              
            
            ⁢
            
              x
              5
            
          
          -
          
            
              K
              a
            
            ⁢
            
              x
              4
            
          
          +
          
            T
            mswash
          
          +
          
            
              l
              bs
            
            ⁢
            
              A
              bs
            
            ⁢
            
              x
              3
            
          
          +
          
            
              l
              bs
              2
            
            ⁢
            
              K
              bs
            
            ⁢
            
              tan
              ⁡
              
                (
                
                  x
                  4
                
                )
              
            
          
        
        }
      
    
    -
    
      
        
          
            l
            sv
          
          ⁢
          
            A
            sv
          
        
        
          I
          a
        
      
      ⁢
      
        
          x
          6
        
        .

7. The method of claim 1 wherein the third subsystem includes states {dot over (x)}1, {dot over (x)}2, {dot over (x)}7, and {dot over (x)}38; wherein {dot over (x)}1 is defined as {dot over (x)}1=x2, {dot over (x)}2 is defined as




  
    
      
        x
        .
      
      2
    
    =
    
      
        
          1
          
            m
            pc
          
        
        ⁢
        
          {
          
            
              
                -
                
                  C
                  pc
                
              
              ⁢
              
                x
                2
              
            
            -
            
              
                K
                pc
              
              ⁢
              
                x
                1
              
            
            -
            
              
                A
                pc
              
              ⁢
              
                P
                
                  k
                  pc
                
              
            
          
          }
        
      
      +
      
        
          
            A
            pc
          
          
            m
            pc
          
        
        ⁢
        
          x
          3
        
      
    
  
  ,




{dot over (x)}7 is defined as {dot over (x)}7=x8, and {dot over (x)}8 is defined as




  
    
      x
      .
    
    8
  
  =
  
    
      1
      
        m
        ls
      
    
    ⁢
    
      
        {
        
          
            
              -
              
                C
                ls
              
            
            ⁢
            
              x
              8
            
          
          -
          
            
              K
              ls
            
            ⁢
            
              x
              7
            
          
          +
          
            
              A
              ls
            
            ⁡
            
              (
              
                
                  x
                  3
                
                -
                
                  x
                  9
                
                -
                
                  P
                  
                    k
                    ls
                  
                
              
              )
            
          
        
        }
      
      .

8. The method of claim 1 wherein the third subsystem is passive.