# Patent

## Title
Method and apparatus for tuning a PID controller

## Abstract

In tuning a PID controller for a process in a feedback control system, a method is provided for bringing the system into symmetric self-excited oscillations for measuring the frequency and the amplitude of the oscillations, and tuning the controller in dependence on the measurements obtained. A control algorithm referred to as the modified relay feedback test is introduced into a system in series with a process to generate self-excited oscillations. Tuning includes the steps of selecting a desired gain margin or phase margin, generating oscillations with the algorithm parameter corresponding to a selected gain or phase margin, measurements of the amplitude and the frequency of these oscillations, and computing PID controller tuning parameters. Data and formulas are given for the computation of the specific parameter of the modified relay feedback test and for tuning the parameters of the PID controller. An apparatus for performing the method is disclosed.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS










U.S. Patent Documents:


















4,214,300
July, 1980
Barlow et al.
364/105.


4,549,123
October, 1985
Hagglund et al.
364/157.


4,602,326
July, 1986
Kraus
364/158.


4,754,391
June, 1988
Suzuki
364/157.


4,836,319
June, 1989
Haseda et al.
180/142.


Re33267
July, 1990
Kraus
364/158.


5,124,626
June, 1992
Thoen
364/161.


5,222,022
June, 1993
Adams et al.
364/431.


5,223,778
June, 1993
Svarovsky et al.
364/157.


5,229,699
July, 1993
Chu et al.
364/162.


5,283,729
February, 1994
Lloyd
364/157.


5,311,421
May, 1994
Nomura et al.
364/157.


5,325,288
June, 1994
Satou
364/162.


5,396,415
March, 1995
Konar et al.
364/162.


5,412,557
May, 1995
Lauw
363/37.


5,453,925
September, 1995
Wojsznis et al.
364/157.


5,535,117
July, 1996
Hiroi
364/158.


5,568,377
October, 1996
Seem et al.
364/157.


5,587,899
December, 1996
Ho et al.
364/157.


5,587,896
December, 1996
Hansen et al.
700/28


5,742,503
April, 1998
Yu
700/42


7,035,695
April, 2006
Boiko
700/28


7,249,882
July, 2007
Lin
374/101









Other References


    Cheng et al., 1999, Identification and Control of Nonlinear Systems Using Multiple Models: Relay Feedback Approach. Proc. American Control Conference 1999, 3327-3331.
    Chiang et al., 1992, Derivation of Transfer Function from Relay Feedback Systems. Ind. Eng. Chem. Res. 31, 855-860.
    I. Boiko, Discontinuous Control Systems: Frequency-Domain Analysis and Design, Boston, Birkhauser, 212 p., 2009.


STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT
Not Applicable
Reference to Sequence Listing, a Table, or a Computer Program Listing Compact Disk Appendix
Not Applicable
BACKGROUND OF THE INVENTION
Despite the apparent success of advanced process control, the proportional-integral-derivative (PID) control still remains the main type of control in the process industries. PID controllers and their variations such as proportional-integral (PI), proportional-derivative (PD), and proportional (P) controllers are implemented as either stand-alone controllers or configurable software modules within the distributed control systems (DCS). The DCS software is constantly evolving providing a number of new features, among which the controller autotuning functionality is one of most useful.
Tuning of PID controllers based on relay feedback tests proposed by Astrom and Hagglund in 1981 (respective U.S. Pat. No. 4,549,123 issued in 1985) received a lot of attention after that (W. L. Luyben, “Derivation of Transfer Functions for Highly Nonlinear Distillation Columns”, Ind. Eng. Chem. Res. 26, 1987, pp. 2490-2495; Tore Hagglund, Karl J. Astrom, “Industrial Adaptive Controllers Based on Frequency Response Techniques”, Automatica 27, 1991, pp. 599-609). It identifies the important dynamic information, ultimate gain and ultimate frequency, in a straightforward manner. The success of this type of tuners lies on the fact that they are simple and reliable. These features of the relay feedback autotuning have lead to a number of commercial tuners (Tore Hagglund, Karl J. Astrom, “Industrial Adaptive Controllers Based on Frequency Response Techniques”, Automatica 27, 1991, pp. 599-609) and industrial applications (H. S. Papastathopoulou, W. L. Luyben, “Tuning Controllers on Distillation Columns with the Distillate-Bottoms Structure”, Ind. Eng. Chem. Res. 29, 1990, pp. 1859-1868).
Despite the apparent success of relay based tuning, it can lead to reduced or vice versa excessively large gain and phase margins because of the choice of the test point (frequency of the oscillations) corresponding to the phase lag of the process equal to −180° (phase cross-over frequency ωπ) while inclusion of the controller in the loop introduces additional phase shift, which was not accounted for at the selection of the test point and designing the tuning rules.
BRIEF SUMMARY OF THE INVENTION
The present invention completely eliminates this source of inaccuracy via the selection of a different test point, development of the test for excitation of oscillation in the selected test point, and designing the tuning rules that account for the phase shift due to the controller introduction. Therefore, the proposed test and tuning rules ensure that either the specified gain margin of the system or the specified phase margin of the system will be provided. The present invention defines a method and an apparatus for bringing the system (comprising the process, and the developed control algorithm) into symmetric oscillations (further referred to as modified relay feedback test) for determining (measuring) quantities essential for the tuning of the controller. The method is based on the measurement of the frequency and the amplitude of the oscillations in the system under the modified relay feedback test and calculation of the controller tuning parameters in dependence on the measurements obtained. The invention includes all variations and combinations (P, PI, PID, etc.) of the control type of the PID controller.



BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING
FIG. 1. Nyquist plots for process (1) and first-order-plus-dead-time (FOPDT) approximation (3).
FIG. 2. Nyquist plots for open-loop system with PI controller and process.
FIG. 3. Modified relay feedback test.
FIG. 4. Finding periodic solution.
FIG. 5. Modified RFT and tuning with specification on phase margin.
FIG. 6. Nyquist plots of open-loop systems for Example 2.
FIG. 7. Embodiment of the controller and the process.



DETAILED DESCRIPTION OF THE INVENTION
Referring to the drawings, a description will be given of an embodiment of a controller tuning method according to the present invention.
The practice of the use of tuning algorithms shows that simple non-parametric methods (the ones that are not based on any model of the process) such as Ziegler-Nichols's closed-loop tuning method (J. G. Ziegler, and N. B. Nichols, “Optimum settings for automatic controllers”, Trans. Amer. Soc. Mech. Eng., Vol. 64, 1942, pp. 759-768) and Astrom-Hagglund's relay feedback test (RFT) (K. J. Astrom and T. Hagglund, “Automatic tuning of simple regulators with specifications on phase and amplitude margins,” Automatica, 20, 1984, p. 645-651) provide a satisfactory and consistent performance even in the conditions characterized by the presence of measurement noise and varying disturbances, when deterioration of more sophisticated methods becomes significant. However, the use of only ultimate gain and frequency cannot ensure sufficient accuracy of tuning There is one more factor that contributes to the issue of relatively low precision. This is the fact that the test point in the closed-loop test is selected to be the one in which the phase characteristic of the process is equal to −180° (phase cross-over frequency ωπ). This approach does not account for the change of frequency ωπ due to the controller introduction, which is the factor that contributes to the deterioration of tuning accuracy.
The present invention aims to introduce a tuning method in which the above-noted problem of the cross-over frequency shift is compensated for by the respective design of the modified relay feedback test (modified RFT). Respective tuning rules matching to the modified relay feedback test are also proposed that allows for the better accuracy of tuning
It has been a popular notion that the most important point on the frequency response of the system is the point where the phase characteristic of the process is equal to −180° (phase cross-over frequency ωπ). However, in publication (I. Boiko, “Modified relay feedback test and its use for non-parametric loop tuning” Proc. 2008 American Control Conference, Seattle, Jun. 11-13, 2008, pp. 4755-4760) it was shown that this point remains the most important one only in the system with the proportional controller, when introduction of the controller does not change the value of ωπ. This circumstance is often neglected, and the principle is applied to all types of PID control. We consider the following example that shows how the introduction of the controller affects the results of tuning Example 1. We assume that the process is given by the following transfer function (which was used in a number of works as a test process):




  
    
      
        
          
            
              W
              p
            
            ⁡
            
              (
              s
              )
            
          
          =
          
            
              e
              
                
                  -
                  2
                
                ⁢
                s
              
            
            ⁢
            
              1
              
                
                  (
                  
                    
                      2
                      ⁢
                      s
                    
                    +
                    1
                  
                  )
                
                5
              
            
          
        
        ,
      
    
    
      
        (
        1
        )
      
    
  




We find the first order plus dead time (FOPDT) approximating model Ŵp(s) to the process (1) based on matching the values of the two transfer functions at frequency ωπ:




  
    
      
        
          
            
              
                W
                ^
              
              p
            
            ⁡
            
              (
              s
              )
            
          
          =
          
            
              
                K
                p
              
              ⁢
              
                e
                
                  -
                  τs
                
              
            
            
              
                
                  T
                  p
                
                ⁢
                s
              
              +
              1
            
          
        
        ,
      
    
    
      
        (
        2
        )
      
    
  




where Kp is the process static gain, Tv is the time constant, and τ is the dead time. Let us apply method (J. G. Ziegler, and N. B. Nichols, “Optimum settings for automatic controllers”, Trans. Amer. Soc. Mech. Eng., Vol. 64, 1942, pp. 759-768) to the tuning of process (1) and note that both (1) and (2) should produce the same ultimate gain and ultimate frequency in the Ziegler-Nichols closed-loop test or the same values of the amplitude and the ultimate frequency in the RFT (K. J. Astrom and T. Hagglund, “Automatic tuning of simple regulators with specifications on phase and amplitude margins,” Automatica, 20, 1984, p. 645-651). This problem has infinite number of solutions, as there are three unknown parameters of (2) and only two measurements obtained from the test. Assume that the value of the process static gain is known: Kp=1, and determine Tp and τ. These parameters can be found from the following equation:

Ŵp(jωπ)=Wp(jωπ),

where ωπ is the phase cross-over frequency for both transfer functions. Therefore, arg Wp(jωπ)=−π. The value of ωπ is 0.283, which gives Wp(jωπ)=(−0.498,j0), and the first-order-plus-dead-time (FOPDT) approximation is, therefore, as follows (found via solution of the set of two algebraic equations):





  
    
      
        
          
            
              W
              ^
            
            p
          
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            
              e
              
                
                  -
                  7.393
                
                ⁢
                s
              
            
            
              
                6.153
                ⁢
                s
              
              +
              1
            
          
          .
        
      
    
    
      
        (
        3
        )
      
    
  




The Nyquist plots of the process 1 given by formula (1) and its approximation 2 given by formula (3) are depicted in FIG. 1. The point of intersection of the two plots (denoted as Ωhd 0) is also the point of intersection with the real axis. Also Ω0=ωπ for both process dynamics (1) and (3), and therefore Ŵp(jΩ0)=Wp(jω0). If the designed controller is of proportional type then the gain margins for processes (1) and (3) are the same. However, if the controller is of PI type then the stability margins for (1) and (2) are different. We illustrate that below. We design the PI controller given by the following transfer function:




  
    
      
        
          
            
              W
              c
            
            ⁡
            
              (
              s
              )
            
          
          =
          
            
              K
              c
            
            ⁡
            
              (
              
                1
                +
                
                  1
                  
                    
                      T
                      c
                    
                    ⁢
                    s
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        4
        )
      
    
  




using the Ziegler-Nichols tuning rules (J. G. Ziegler, and N. B. Nichols, “Optimum settings for automatic controllers”, Trans. Amer. Soc. Mech. Eng., Vol. 64, 1942, pp. 759-768). This results in the following transfer function of the controller:





  
    
      
        
          
            
              W
              c
            
            ⁡
            
              (
              s
              )
            
          
          =
          
            0.803
            ⁢
            
              (
              
                1
                +
                
                  1
                  
                    17.76
                    ⁢
                    s
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        5
        )
      
    
  




The Nyquist plot of the open-loop systems containing the process (1) and the controller (5) denoted as 1 and the Nyquist plot of the open-loop systems containing the process approximation (3) denoted as 2 are depicted in FIG. 2. It follows from the frequency-domain theory of linear systems and the used tuning rules that the mapping of point Ω0 in FIG. 1 into point Ω0 in FIG. 2 is done via clockwise rotation of vector {right arrow over (W)}p(jΩ0) by the angle ψ=arctan(1/(0.8·2π))=11.25° and multiplication of its length by such value, so that its length becomes equal to 0.408. However, for the open-loop system containing the PI controller, the points of intersection of the Nyquist plots of the system and of the real axis are different for the system with process (1) and with process approximation (3). They are shown as points Ω1 and Ω2 in FIG. 2. The mapping of these points to the Nyquist plots of the process and its approximation is shown in FIG. 1. Therefore, the stability margins of the systems containing a PI controller are not the same any more. It is revealed as different points of intersection of the plots and of the real axis in FIG. 2. In fact, the position of vector {right arrow over (W)}ol(jΩ0)={right arrow over (W)}c(jΩ0){right arrow over (W)}p(jΩ0) is fixed, but this vector does not reflect the stability of the system. As one can see in FIG. 2, the gain margin of the system containing the FOPDT approximation of the process is higher than the one of the system with the original process. The considered example illustrates the drawback of methods of tuning based on the measurements of process response in the critical point Ω0. This drawback is because of the shift of the critical point due to the introduction of the controller.
The present invention aims to eliminate the noted drawback by generating oscillations in the critical point that is not necessarily located on the real axis but in the point Ω0, which would correspond to certain desired phase lag of the process, so that the introduction of the controller is accounted for in the test itself.
It is proposed that the modified RFT should be designed as the following control:




  
    
      
        
          u
          ⁡
          
            (
            t
            )
          
        
        =
        
          {
          
            
              
                
                  
                    h
                    ⁢
                    
                      
                    
                    ⁢
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ≥
                  
                    
                      Δ
                      1
                    
                    ⁢
                    
                      
                    
                    ⁢
                    or
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            σ
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          >
                          
                            
                              -
                              
                                Δ
                                2
                              
                            
                            ⁢
                            
                              
                            
                            ⁢
                            and
                            ⁢
                            
                              
                            
                            ⁢
                            
                              u
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                
                                )
                              
                            
                          
                        
                        =
                        h
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    
                      -
                      h
                    
                    ⁢
                    
                      
                    
                    ⁢
                    if
                    ⁢
                    
                      
                    
                    ⁢
                    
                      σ
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                  
                  ≤
                  
                    
                      Δ
                      2
                    
                    ⁢
                    
                      
                    
                    ⁢
                    or
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            σ
                            ⁡
                            
                              (
                              t
                              )
                            
                          
                          <
                          
                            
                              Δ
                              1
                            
                            ⁢
                            
                              
                            
                            ⁢
                            and
                            ⁢
                            
                              
                            
                            ⁢
                            
                              u
                              ⁡
                              
                                (
                                
                                  t
                                  -
                                
                                )
                              
                            
                          
                        
                        =
                        
                          -
                          h
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




where Δ1=βσmax, Δ2=−βσmin, σmax and σmin are last “singular” points of the error signal 2 (FIG. 3) corresponding to the last maximum and minimum values of σ(t) after crossing the zero level, β is a positive constant.

Assume that the reference signal r(t) 1 in the system containing the process 6, the control algorithm (modified RFT) 5, and the summer 7 (FIG. 3), is zero and show that in the steady mode, the motions of the control 3 provided by the algorithm 5 given by (6) and of the output 4 are periodic. Apply the describing function (DF) method (Derek P. Atherton, “Nonlinear Control Engineering”, Van Nostrand Reinhod New York, 1982) to the analysis of motions in FIG. 3. Assume that the steady mode periodic, and show that this is a valid assumption by finding parameters of this periodic motion (Note: a rigorous proof is presented in the article I. Boiko, L. Fridman, I. M. Castellanos, “Analysis of second-order sliding mode algorithms in the frequency domain,” IEEE Trans. Automat. Contr., Vol. 49, No. 6, pp. 946-950, 2004). If the motions in the system are periodic then σmax and σmin represent the amplitude of the oscillations: α0=σmax=−σmin, and the equivalent hysteresis value of the relay is Δ=Δ1=Δ2=βσmax=−βσmin. The DF of the hysteretic relay is given as the following function of amplitude α:




  
    
      
        
          
            N
            ⁡
            
              (
              a
              )
            
          
          =
          
            
              
                
                  4
                  ⁢
                  h
                
                
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  a
                
              
              ⁢
              
                
                  1
                  -
                  
                    
                      (
                      
                        Δ
                        a
                      
                      )
                    
                    2
                  
                
              
            
            -
            
              j
              ⁢
              
                
                  4
                  ⁢
                  h
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    a
                    2
                  
                
              
            
          
        
        ,
        
          a
          >
          Δ
        
      
    
    
      
        (
        7
        )
      
    
  




However, the system FIG. 3 with control (6) is not a conventional relay system. This system has the hysteresis value that is unknown a-priori and depends on the amplitude value: Δ=βα. Therefore, (7) can be rewritten as follows:




  
    
      
        
          
            N
            ⁡
            
              (
              a
              )
            
          
          =
          
            
              
                4
                ⁢
                h
              
              
                π
                ⁢
                
                  
                
                ⁢
                a
              
            
            ⁢
            
              (
              
                
                  
                    1
                    -
                    
                      β
                      2
                    
                  
                
                -
                jβ
              
              )
            
          
        
        ,
      
    
    
      
        (
        8
        )
      
    
  




The modified RFT will generate oscillations in the system under control (6). Parameters of the oscillations can be found from the harmonic balance equation:




  
    
      
        
          
            
              W
              p
            
            ⁡
            
              (
              
                jΩ
                0
              
              )
            
          
          =
          
            -
            
              1
              
                N
                ⁡
                
                  (
                  
                    a
                    0
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        9
        )
      
    
  




where α0 and Ω0 are the amplitude and the frequency of the periodic motions. The negative reciprocal of the DF is given as follows:





  
    
      
        
          -
          
            1
            
              N
              ⁡
              
                (
                a
                )
              
            
          
        
        =
        
          
            -
            
              
                π
                ⁢
                
                  
                
                ⁢
                a
              
              
                4
                ⁢
                h
              
            
          
          ⁢
          
            (
            
              
                
                  1
                  -
                  
                    β
                    2
                  
                
              
              +
              jβ
            
            )
          
        
      
    
    
      
        (
        10
        )
      
    
  




Finding a periodic solution in system FIG. 3 with control (6) has a simple graphic interpretation (FIG. 4) as finding the point of intersection of the Nyquist plot of the process 1 and of the negative reciprocal of the DF 2, which is a straight line that begins in the origin and makes a counterclockwise angle ψ=arcsin)β with the negative part of the real axis. In the problem of analysis, frequency Ω0 and amplitude α0 are unknown variables and are found from the complex equation (9). In the problems of tuning, Ω0 and α0 are measured from the modified RFT, and on the basis of the measurements obtained the tuning parameters are calculated from Ω0 and α0.
The modified RFT allows for the exact design of the gain margin. Since the amplitude of the oscillations α0 is measured from the test, the process gain at frequency Ω0 can be obtained as follows:




  
    
      
        
          
            
            
              
                W
                p
              
              ⁡
              
                (
                
                  jΩ
                  0
                
                )
              
            
            
          
          =
          
            
              π
              ⁢
              
                
              
              ⁢
              
                a
                0
              
            
            
              4
              ⁢
              h
            
          
        
        ,
      
    
    
      
        (
        11
        )
      
    
  




which after introduction of the controller will become the process gain at the critical frequency.

The PID controller transfer function is




  
    
      
        
          
            W
            c
          
          ⁡
          
            (
            s
            )
          
        
        =
        
          
            
              K
              c
            
            (
            
              1
              +
              
                1
                
                  
                    T
                    ic
                  
                  ⁢
                  s
                
              
              +
              
                
                  T
                  dc
                
                ⁢
                s
              
            
            )
          
          .
        
      
    
    
      
        
      
    
  




The tuning rules for a PID controller are then defined as follows:





  
    
      
        
          
            K
            c
          
          =
          
            
              c
              1
            
            ⁢
            
              
                4
                ⁢
                h
              
              
                π
                ⁢
                
                  
                
                ⁢
                
                  a
                  0
                
              
            
          
        
        ,
        
          
            T
            ic
          
          =
          
            
              c
              2
            
            ⁢
            
              
                2
                ⁢
                π
              
              
                Ω
                0
              
            
          
        
        ,
        
          
            T
            dc
          
          =
          
            
              c
              3
            
            ⁢
            
              
                2
                ⁢
                π
              
              
                Ω
                0
              
            
          
        
        ,
      
    
    
      
        (
        12
        )
      
    
  




where c1, c2, and c3 are constant parameters that define the tuning rule. It can be noted that if the tuning rules are assigned as (12) then the closed-loop system characteristics become invariant to the time constants of the process, so that if all time constants of the process were increased by the factor α then the critical frequency would decrease by the same factor α, and the product of every time constant by the critical frequency would remain unchanged. If the tuning rules (12) are used then the frequency response of the PID controller at Ω0 becomes





  
    
      
        
          
            
              W
              c
            
            ⁡
            
              (
              
                jΩ
                0
              
              )
            
          
          =
          
            
              c
              1
            
            ⁢
            
              
                4
                ⁢
                h
              
              
                π
                ⁢
                
                  
                
                ⁢
                
                  a
                  0
                
              
            
            ⁢
            
              (
              
                1
                -
                
                  j
                  ⁢
                  
                    1
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        c
                        2
                      
                    
                  
                
                +
                
                  j2π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    c
                    3
                  
                
              
              )
            
          
        
        ,
      
    
    
      
        (
        13
        )
      
    
  




so that if the tuning rules are established through the choice of parameters c1, c2, and c3, and the test provides oscillations of the frequency Ωu which is equal to the phase cross-over frequency ωπ of the open-loop system (including the controller), then the controller phase lag at the frequency ωπ=Ωu is constant and depends only on the values of c2, and c3:





  
    
      
        
          
            
              
                
                  φ
                  c
                
                ⁡
                
                  (
                  
                    ω
                    π
                  
                  )
                
              
              =
              
                arctan
                (
                
                  
                    
                      c
                      3
                    
                    ⁢
                    
                      T
                      u
                    
                    ⁢
                    
                      Ω
                      u
                    
                  
                  -
                  
                    1
                    
                      
                        c
                        2
                      
                      ⁢
                      
                        T
                        u
                      
                      ⁢
                      
                        Ω
                        u
                      
                    
                  
                
                )
              
            
          
        
        
          
            
              =
              
                arctan
                (
                
                  
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      c
                      3
                    
                  
                  -
                  
                    1
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        c
                        2
                      
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




Tuning Rules for Specification on Gain Margin.

Let the specified gain margin be γm>1 (in absolute values). Then taking absolute values of both sides of (13) and considering (11) we obtain the following equation:




  
    
      
        
          
            
              γ
              m
            
            ⁢
            
              c
              1
            
            ⁢
            
              
                1
                +
                
                  
                    (
                    
                      
                        2
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          c
                          3
                        
                      
                      -
                      
                        1
                        
                          2
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          
                            c
                            2
                          
                        
                      
                    
                    )
                  
                  2
                
              
            
          
          =
          1
        
        ,
      
    
    
      
        (
        15
        )
      
    
  




which is a constraint complementary to the tuning rules (11). To provide the specified gain margin, the modified RFT must be carried out with parameter





  
    
      
        
          
            
              β
              =
              
                
                  -
                  sin
                
                ⁢
                
                  
                
                ⁢
                
                  φ
                  c
                
                ⁢
                
                  Ω
                  0
                
              
            
          
        
        
          
            
              =
              
                
                  -
                  sin
                
                ⁢
                
                  
                
                ⁢
                arc
                ⁢
                
                  
                
                ⁢
                
                  tg
                  (
                  
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        c
                        3
                      
                    
                    -
                    
                      1
                      
                        2
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




In the example considered above, if we keep parameter c2 the same as in Ziegler-Nichols tuning rules (J. G. Ziegler, and N. B. Nichols, “Optimum settings for automatic controllers”, Trans. Amer. Soc. Mech. Eng., Vol. 64, 1942, pp. 759-768): c2=0.8, then to obtain, for example, gain margin γm=2 the tuning parameter c1 for the modified RFT should be selected as c1=0.49, and parameter)β for the test should be selected in a with (16) as)β=0.195. For any process, the system will have gain margin γm=2 (6 dB). Therefore, the modified RFT with parameter)β calculated as (16) and tuning rules (12) satisfying the constraint (15) can ensure the desired gain margin. However, (15) is an equation containing three unknown variables, which gives one a freedom to vary parameters c1, c2, and c3. Some optimal sets of tuning rule parameters c1, c2, and c3 (for gain margin γm=2) that were obtained via simulations, along with the values of φc (Ωo) and parameter)β for the test, are given in Table 1. One should note the difference between the values of the critical frequency of the conventional RFT and the frequency of oscillations in the modified RFT (except for the proportional controller). Therefore, even if the coefficients c2, c3 of Table 1 have the same values as corresponding coefficients of Ziegler-Nichols tuning rules (J. G. Ziegler, and N. B. Nichols, “Optimum settings for automatic controllers”, Trans. Amer. Soc. Mech. Eng., Vol. 64, 1942, pp. 759-768), they will actually produce different values of controller parameters Tic and Tdc. In fact, due to the negative value of φc (Ω0) for the PI controller (and consequently, lower frequency of oscillations of the modified RFT), one would get higher value of Tic computed through the modified RFT and data of Table 1. And vice versa, due to the positive value of φc (Ω0) for the PID controller, one would get lower values of Tic and Tdc, computed through the modified RFT and data of Table 1.







TABLE 1







Tuning rules for gain margin γm = 2














Controller
c1
c2
c3
φc (Ω0)
β


















P
0.50
0
0
0
0



PI
0.49
0.80
0
−11.2°
0.195



PID
0.46
0.50
0.12
23.5°
−0.399










Tuning Rules for Specification on Phase Margin.

The relationship that would allow one to tune PID controllers with specification on phase margin for the open-loop system can also be derived. Using the same format of the tuning rules (12), and considering that if the parameter β of the modified RFT is calculated from the sum of φc (Ω0) and the phase margin φm as:




  
    
      
        
          
            
              β
              =
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      φ
                      m
                    
                    -
                    
                      
                        φ
                        c
                      
                      ⁡
                      
                        (
                        
                          Ω
                          0
                        
                        )
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                =
                
                  sin
                  (
                  
                    
                      φ
                      m
                    
                    +
                    
                      arc
                      ⁢
                      
                        
                      
                      ⁢
                      
                        tg
                        (
                        
                          
                            1
                            
                              2
                              ⁢
                              π
                              ⁢
                              
                                
                              
                              ⁢
                              
                                c
                                2
                              
                            
                          
                          -
                          
                            2
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              3
                            
                          
                        
                        )
                      
                    
                  
                  )
                
              
              ,
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




we formulate the constraint for the tuning rules ensuring φm as follows:





  
    
      
        
          
            c
            1
          
          ⁢
          
            
              1
              +
              
                
                  (
                  
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        c
                        3
                      
                    
                    -
                    
                      1
                      
                        2
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          c
                          2
                        
                      
                    
                  
                  )
                
                2
              
            
          
        
        =
        1.
      
    
    
      
        (
        18
        )
      
    
  




The graphical interpretation of modified RFT and tuning with specification on phase margin are presented in FIG. 5, where the Nyquist plot of the process 5 intersects with the negative reciprocal of
the DF of the modified RFT 2 in the point that defines the circle of radius




  R
  =
  
    
      
        π
        ⁢
        
          
        
        ⁢
        
          a
          0
        
      
      
        4
        ⁢
        h
      
    
    ⁢
    3.
  











TABLE 2







Tuning rules for phase margin φm = 45°














Controller
c1
c2
c3
φc (Ω0)
β


















P
1.0
0
0
0°
0.707



PI
0.98
0.80
0
−11.2°
0.831



PID
0.92
0.50
0.12
23.5°
0.367










Therefore, if tuning rules (12) are subject to constraint (18) then at frequency Ω0 of the modified RFT: (a) the absolute value of the open-loop frequency response, in accordance with (11), (13), is




  
    
      
        
          
            
              
                
                
                  
                    W
                    ol
                  
                  ⁡
                  
                    (
                    
                      jΩ
                      0
                    
                    )
                  
                
                
              
              =
              
                
                  
                  
                    
                      W
                      c
                    
                    ⁡
                    
                      (
                      
                        jΩ
                        0
                      
                      )
                    
                  
                  
                
                ⁢
                
                  
                  
                    
                      W
                      p
                    
                    ⁡
                    
                      (
                      
                        jΩ
                        0
                      
                      )
                    
                  
                  
                
              
            
          
        
        
          
            
              =
              
                
                  
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      a
                      
                        
                      
                    
                  
                  
                    4
                    ⁢
                    h
                  
                
                ⁢
                
                  c
                  1
                
                ⁢
                
                  
                    4
                    ⁢
                    h
                  
                  
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      a
                      
                        
                      
                    
                  
                
                ⁢
                
                  
                    1
                    +
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            
                              c
                              3
                            
                          
                          -
                          
                            1
                            
                              2
                              ⁢
                              π
                              ⁢
                              
                                
                              
                              ⁢
                              
                                c
                                2
                              
                            
                          
                        
                        )
                      
                      2
                    
                  
                
              
            
          
        
        
          
            
              
                =
                
                  
                    
                      c
                      1
                    
                    ⁢
                    
                      
                        1
                        +
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                π
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  c
                                  3
                                
                              
                              -
                              
                                1
                                
                                  2
                                  ⁢
                                  π
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    c
                                    2
                                  
                                
                              
                            
                            )
                          
                          2
                        
                      
                    
                  
                  =
                  1
                
              
              ,
            
          
        
      
    
    
      
        
      
    
  




that constitutes the magnitude cross-over frequency, and (b) the phase of the open-loop frequency response is

arg Wol(jΩ0)=arg Wc(jΩ0)+arg Wp(jΩ0)=−180°+(φm−φc(Ω0))+φc(Ω0)=−180°+φm,

which shows that the specification on the phase margin is satisfied. Assuming that the controller at frequency Ω0 of the modified RFT should provide the same phase response as at critical frequency of the conventional RFT, we can obtain the following values of parameters c1, c2, c3 (see Table 2 for φm=45°. Like in the tuning with specification on gain margin, one should note the difference between the values of the critical frequency of the conventional RFT and the frequency of oscillations in the modified RFT, which will result in different values of the controller parameters. The proposed method of tuning is illustrated by the following example.

Example 2
Consider the process transfer function (1) that was used in Example 1. (a) Apply the modified RFT with amplitude h=1, parameter)β=0.195 and c1, c2 values from Table 1 for tuning a PI controller with specification on gain margin γm=2. (b) After that use the modified RFT with amplitude h=1, parameter)β=0.659 and c1, c2 values from Table 2 for tuning a PI controller with specification on phase margin φm=45°. The controller tuning that is done according to the presented method produces the following results. (a) The modified RFT gives Ω0=0.188 and α0=0.393; for tuning with specification on gain margin the controller parameters calculated per (12) are Kc=0.978, Tic=26.74; (b) The modified RFT gives Ω0=0.133 and α0=0.544; for tuning with specification on gain margin the controller parameters calculated per (12) are Kc=1.416, Tic=37.85. The frequency response (Nyquist plots) of the open-loop systems for the system tuned with specification on gain margin 1 and for the system tuned with specification on gain margin 2, is presented in FIG. 6. One can see that, indeed, the gain margin is two for option “a”, and the phase margin is 45° for option “b”.
It was assumed above that the reference signal 1 in FIG. 3 was zero. Therefore, it can be further noted that the values of the control and the process variable should be understood as increments from the corresponding values in a steady operating point. To realize that, the system must be brought to a steady state first (by means of not optimally tuned but stable PID controller). After that the modified RFT should be carried out with increments of the control +h and −h from the value in the steady state. The amplitude of the oscillations must be measured with respect to the value of the process variable in the steady state.
Eventually, the designed self-tuning PID (or another type) controller is supposed to be realized as a processor based (micro-computer or controller) device and all above formulas, the nonlinear element, the tuning rules are realized as computer programs with the use of applicable programming languages. The preferred embodiment of the controller is depicted in FIG. 7. The controller 1 has two A/D converters 2 and 3 on its input for the process output and set point signals respectively (alternatively it may have only one A/D converter for the process output signal, and the set point may be realized within the controller software), a processor (CPU) 4, a read-only memory (ROM) 5 for program storage, a random access memory (RAM) 6 for buffering the data, an address/data/control bus 7 for data transfer to/from the processor, and an D/A converter 8 that converts digital control signal generated by the controller into analog format. The analog control signal is applied to the process 9 (to a control valve, etc.). All elements of the controller interact with each other in a known manner. Some elements of the controller listed above (for example A/D and D/A converters) may be missing as well as the controller may also contain elements other than listed above-depending on specific requirements and features of the control system.

## Claims

1. A method of tuning a proportional-integral-derivative controller, further referred to as a PID controller, having a transfer function




  
    
      
        
          
            
              W
              c
            
            ⁡
            
              (
              s
              )
            
          
          =
          
            
              K
              c
            
            (
            
              1
              +
              
                1
                
                  
                    T
                    ic
                  
                  ⁢
                  s
                
              
              +
              
                
                  T
                  dc
                
                ⁢
                s
              
            
            )
          
        
        ,
      
    
    
      
        
      
    
  



where s is a Laplace variable and Kc, Tic, and Tdc are tuning parameters, for a process in a feedback control system comprising the steps of:
(a) selection of a desired gain margin γm of the system for tuning for a gain margin specification or desired phase margin φm, of the system for tuning for a phase margin specification;
(b) determining the values of the parameters c1, c2, and c3 that define optimal tuning rules of the PID controller from the desired gain margin γm of the system for tuning for a gain margin specification or the desired phase margin φm of the system for tuning for a phase margin specification selected at the previous step;
(c) bringing the control system into symmetric self-excited oscillations of a frequency at which a phase response of the process is equal to the desired value equal to ψ−180°, where −90°<ψ<90°;
(d) measurement of a frequency of the oscillations Ω0, and an amplitude of the oscillations α0;
(e) calculation of the tuning parameters of the PID controller in dependence on the measurements obtained at the previous step.

2. The method as recited in claim 1, wherein said parameters c1, c2, and c3 that define the tuning rules are selected from the following table for tuning for the gain margin specification γm=2 for a proportional, proportional-integral, and PID controllers, respectively,
















Controller
c1
c2
c3





















Proportional
0.50
0
0



Proportional-integral
0.49
0.80
0



PID
0.46
0.50
0.12

3. The method as recited in claim 1, wherein said parameters c1, c2, and c3 that define the tuning rules are selected from the following table for tuning for the phase margin specification Φm=45° for a proportional, proportional-integral, and PID controller, respectively,
















Controller
c1
c2
c3





















Proportional
1.0
0
0



Proportional-integral
0.98
0.80
0



PID
0.92
0.50
0.12

4. The method as recited in claim 1, wherein said parameters c2, and c3 that define the tuning rules are selected from the table for a proportional-integral and PID controller, respectively,















Controller
c2
c3




















Proportional-
0.80
0



integral





PID
0.50
0.12









plus or minus 20% and parameter c1 is calculated for tuning for a gain margin specification as follows:





  
    c
    1
  
  =
  
    
      1
      
        
          γ
          m
        
        ⁢
        
          
            1
            +
            
              
                (
                
                  
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      c
                      3
                    
                  
                  -
                  
                    1
                    
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        c
                        2
                      
                    
                  
                
                )
              
              2
            
          
        
      
    
    .

5. The method as recited in claim 1, wherein said parameters c2, and c3 that define the tuning rules are selected from the table for a proportional-integral and PID controller, respectively,















Controller
c2
c3




















Proportional-
0.80
0



integral





PID
0.50
0.12









plus or minus 20%, and parameter c1 is calculated for tuning for a phase margin specification as follows:





  
    c
    1
  
  =
  
    
      1
      
        
          1
          +
          
            
              (
              
                
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    c
                    3
                  
                
                -
                
                  1
                  
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      c
                      2
                    
                  
                
              
              )
            
            2
          
        
      
    
    .

6. The method as recited in claim 1, wherein said bringing the control system into symmetric self-excited oscillations is done via the introduction into the system the following control algorithm further referred to as modified relay feedback test with parameter β, in series with the process, so that the error signal σ is the difference between the set point and the process control variable, the error signal is an input to the modified relay feedback test, and the output u of the modified relay feedback test is an input to the process:




  
    u
    ⁡
    
      (
      t
      )
    
  
  =
  
    {
    
      
        
          h
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              
                σ
                ⁡
                
                  (
                  t
                  )
                
              
            
            ≥
            
              
                Δ
                1
              
              ⁢
              
                
              
              ⁢
              or
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      σ
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    >
                    
                      
                        -
                        
                          Δ
                          2
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      and
                      ⁢
                      
                        
                      
                      ⁢
                      
                        u
                        ⁡
                        
                          (
                          
                            t
                            -
                          
                          )
                        
                      
                    
                  
                  =
                  h
                
                )
              
            
          
        
      
      
        
          
            -
            h
          
        
        
          
            
              if
              ⁢
              
                
              
              ⁢
              
                σ
                ⁡
                
                  (
                  t
                  )
                
              
            
            ≤
            
              
                Δ
                2
              
              ⁢
              
                
              
              ⁢
              or
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      σ
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    <
                    
                      
                        Δ
                        1
                      
                      ⁢
                      
                        
                      
                      ⁢
                      and
                      ⁢
                      
                        
                      
                      ⁢
                      
                        u
                        ⁡
                        
                          (
                          
                            t
                            -
                          
                          )
                        
                      
                    
                  
                  =
                  
                    -
                    h
                  
                
                )
              
            
          
        
      
    
  



where t is current time, Δ1=βσmax, Δ2=−βσmin, σmax and σmin are last singular points of the error signal σ(t) corresponding to the last maximum and minimum values of σ(t), respectively, after crossing the zero level, h is the control amplitude, u(t−) is the output of the modified relay feedback test at time immediately preceding current time, and parameter βof the modified relay feedback test is computed as β=sinψ, ψ is a parameter.

7. The method as recited in claim 6, wherein said parameter ψ is calculated as follows:




  ψ
  =
  
    
      -
      arc
    
    ⁢
    
      
    
    ⁢
    
      tg
      (
      
        
          2
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            c
            3
          
        
        -
        
          1
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              c
              2
            
          
        
      
      )
    
  



for tuning for the gain margin specification.

8. The method as recited in claim 6, wherein said parameter ψ is calculated as follows:




  ψ
  =
  
    
      φ
      m
    
    +
    
      arc
      ⁢
      
        
      
      ⁢
      
        tg
        (
        
          
            1
            
              2
              ⁢
              π
              ⁢
              
                
              
              ⁢
              
                c
                2
              
            
          
          -
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              c
              3
            
          
        
        )
      
    
  



for tuning for the phase margin specification.

9. The method as recited in claim 6, wherein said tuning parameters are calculated as follows:




  
    
      K
      c
    
    =
    
      
        c
        1
      
      ⁢
      
        
          4
          ⁢
          h
        
        
          π
          ⁢
          
            
          
          ⁢
          
            
              a
              0
            
            
              
            
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      T
      ic
    
    =
    
      
        c
        2
      
      ⁢
      
        
          2
          ⁢
          π
        
        
          Ω
          0
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      T
      dc
    
    =
    
      
        c
        3
      
      ⁢
      
        
          
            2
            ⁢
            π
          
          
            Ω
            0
          
        
        .

10. The method as recited in claim 6, wherein said output of the modified relay feedback test and the process control variable are increments from the corresponding values in a steady operating point to which in a preferred embodiment the control system is brought by means of not optimally tuned but stable PID controller.