# Robust Regulation and Tracking Control of a Class of Uncertain 2DOF Underactuated Mechanical Systems

Authors: David I. Rosas Almeida; Carlos Gamez; Raul Rascón
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A strategy to design and implement a robust controller for a class of underactuated mechanical systems, with two degrees of freedom, which solves the problems of regulation and trajectory tracking, is proposed. This control strategy considers the partial measurement of the state vector and the presence of parametric uncertainties in the plant; these conditions are common in the implementation of a control system. The strategy is based on the use of robust finite time convergence observers to estimate the unmeasured state variables, unknown disturbances, and other signals needed for the control system implementation. The performance of the control strategy is illustrated numerically and experimentally.


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## Body

## 1. Introduction

Antecedents and Motivation. Control of underactuated mechanical systems, systems with fewer number of control inputs than their degrees of freedom, has received much attention in the last decades. This is because of the theoretical challenges as well as practical applicability; robots, aerospace vehicles, underwater vehicles, and surface vessels are some examples of underactuated mechanical systems. Some important papers which address this control problem for different situations are [1–9]. While many interesting techniques and results have been presented for this class of systems, the control of them still remains an open problem. Important issues are as follows: how control models can be formulated for such systems and how closed-loop control problems can be solved and implemented. These issues are addressed in this paper for a particular class of uncertain underactuated mechanical systems. These problems have been addressed by many authors and important solutions have been proposed, some of which are as follows.In [10] a sliding mode control method for a class of second-order underactuated mechanical systems is proposed; the controller has the double-layer structure. Firstly, the system states are divided into several different subsystems. For each of these subsystems, a first-layer sliding plane is constructed; from that, a second-layer sliding plane is constructed. By analyzing the features of the model of the plant, they derive the sliding control law. Here, the proposed controller only solves the regulation problem; furthermore, the implementation of the controller requires the measurement of full state vector; this condition is not satisfied in practice. For a similar class of systems, in [11], the Olfati transformation is applied first to represent the system into a special cascade form. Since, in general cases, some of the terms in the new space might become too complex to drive, they are regarded as uncertainties. A backstepping-like adaptive control based on function approximation technique is designed so that the system in the new space can be stabilized with uniformly ultimately bounded performance. This paper assumes knowledge of all system parameters and the measurement of all state variables and the perturbation terms that appear in the approach are well known, but experimental results are not presented.Other works deal with particular systems, for example, [1, 3, 12, 13], but many of them only deal with the regulation problem and present performance results through numerical simulations. Reference [14] addresses the observer-based multivariable control of a class of nonlinear, underactuated Lagrangian systems with application to trajectory tracking and sway control of a 3D overhead gantry crane subject to Coulomb friction. A second-order sliding mode observer is used for the estimation of velocities. Based on these estimates, the sliding function of a second-order sliding mode controller for trajectory tracking and antiswing control is proposed. This is a very important paper because it considers a very common situation in practice, the lack of measurement of the velocities, but it only show numerical results.An important work is presented in [15] where, for a class of second-order underactuated mechanical systems, a robust finite time control strategy is proposed. The robust finite time controller drives the tracking error to be zero at the fixed final time. By utilizing a Lyapunov stability theorem, the controller can achieve finite time tracking of desired reference signals for underactuated systems, which are subject to both external disturbances and system uncertainties. However, the complete measurement of the state vector is assumed and only stabilization problem is solved. Moreover, illustration controller performance is through numerical simulations.Main Contribution. We propose a strategy to design and implement a robust controller for a class of underactuated mechanical systems, with two degrees of freedom, which solves the problems of regulation and trajectory tracking. This control strategy considers the partial measurement of the state vector and the presence of parametric uncertainties in the plant; these conditions are common in the implementation of a control system.The strategy is based on the use of robust finite time convergence observers to estimate the unmeasured state variables, unknown disturbances, and other signals needed for the control system implementation. The performance of the control strategy is illustrated numerically and experimentally.Paper Structure. This paper is organized as follows: Section 2 provides the control problem, the model of the plant, and the control objective. In Section 3, we propose the solution to the problem; to implement such solution is necessary to know the velocities, the exact value of the disturbances, and the availability of auxiliary signals and their derivatives, which are unknown. One way to implement this control signal is presented in Section 4, where with the help of robust observers with finite time convergence we estimate all the terms needed for implementation. Section 5 shows the performance of the controller through numerical simulations and experimental results. Finally, in Section 6, we present some general conclusions.


## 2. Problem Statement

Consider a 2DOF underactuated mechanical system whose dynamics are given by(1)q¨1=f1q1,q˙1,q2,q˙2+g1q1,q2u+γ1·,q¨2=f2q2,q˙2+g2vq1+γ2·,y1=q1,y2=q2,where q1, q2, q˙1, and q˙2 are the generalized positions and velocities, respectively. f1·, f2·, and g1· are smooth functions; g1·≠0 for all q1 and q2. g2≠0 is a constant, vq1 is an invertible function for all q1 in the domain of operation of the system, and u is the control input. γ1· and γ2· are smooth terms due to parameter variations; based on Lagrangian model (1), these terms may depend on q1, q2, q˙1, q˙2, q¨1, and q¨2, but if these variables are bounded, γ1· and γ2· also are bounded [16]. Additionally, we considered that there is no measurement of the velocities q˙1 and q˙2.Some well known mechanisms that belong to this class of underactuated systems are the mass-spring-damper, magnetic suspension, and the ball and beam systems.A state space representation of system (1) is(2)x˙1=x2,x˙2=f1x1,x2,x3,x4+g1x1,x2u+γ1·,x˙3=x4,x˙4=f2x3,x4+g2vx1+γ2·,y1=x1,y2=x3.The control problem, for system (2), is to design a control input u such that the underactuated position x3 tracks a reference signal yrt in asymptotic form; in other words(3)limt→∞⁡y2-yrt=0,where yrt is a Ck function, for a sufficiently large k.To solve the control problem we define the error variablese1=x3-yrt and e2=x4-y˙rt, whose dynamics are given by(4)e˙1=e2,(5)e˙2=f2e1+yr,e2+y˙r+g2vx1+γ2·-y¨rt,(6)x˙1=x2,(7)x˙2=f1x1,x2,e1+yr,e2+y˙r+g1x1,x2u+γ1·,(8)ye=e1,(9)y1=x1.Now we can say that the control problem is to design a control input u such that the origin of the error variables of subsystems (4) and  (5) will be an asymptotic stable equilibrium point, while the variables x1 and x2 stay bounded.


## 3. Control Strategy

In this section we present a strategy to solve the control problem considering that every disturbance terms and velocities are known; the next section will show its implementation.System ((4)–(9)) is formed by two subsystems; the unactuated part is (10)e˙1=e2,e˙2=f2e1+yr,e2+y˙r+g2vx1+γ2·-y¨rt,ye=e1,and the actuated part is (11)x˙1=x2,x˙2=f1x1,x2,e1+yr,e2+y˙r+g1x1,x2u+γ1·,y1=x1.The function v(x1) in (10) can be seen as a control input for this subsystem; therefore we rename v(x1) as ue or x1=v-1ue in (10):(12)e˙1=e2,e˙2=f2e1+yr,e2+y˙r+γ2·-y¨rt+g2ue,ye=e1.An ideal control ue that stabilizes the origin of system (12) is(13)ue=1g2-f2e1+yr,e2+y˙r-γ2·+y¨rt-k1e1-k2e2;substituting it in (12) results in(14)e˙1=e2,e˙2=-k1e1-k2e2,ye=e1.If k1 and k2 are positive constants the origin of system (14) is an exponentially stable equilibrium point.Nowv-1ue must be the reference signal for the position x1 in (11). Define xre=v-1ue and new error variables ɛ1=x1-xre and ɛ2=x2-x˙re whose dynamics are given by(15)ɛ˙1=ɛ2,ɛ˙2=f1x1,x2,e1+yr,e2+y˙r+g1x1,x2u+γ1·-x¨re.A control u that stabilizes the origin of system (15) is(16)u=1g1x1,x2-f1x1,x2,e1+yr,e2+y˙r-γ1·+x¨re-α1ɛ1-α2ɛ2-α3sign⁡ɛ1;substituting it in (15) we have(17)ɛ˙1=ɛ2,ɛ˙2=-α1ɛ1-α2ɛ2-α3sign⁡ɛ1.If the constants α1, α2, and α3 are positive the origin of system (17) is an exponentially stable equilibrium point.It is important to note that, because this section considered that we have the measurement of all terms, it is not necessary to incorporate the term discontinuous in the control (16); however this term is very useful when the implementation is done because it will give robustness to closed-loop system.


### 3.1. Stability Analysis

To prove the stability of the closed-loop system consider the error system(18)e˙1=e2,e˙2=f2e1+yr,e2+y˙r+g2vx1+γ2·-y¨rt,ɛ˙1=ɛ2,ɛ˙2=-α1ɛ1-α2ɛ2-α3sign⁡ɛ1.The last two equations of (18) form a subsystem uncoupled of the e1 and e2, regardless of the value of ue. Then, if α1, α2, and α3 are positive constants, the origin of this subsystem is exponentially stable [17].Becausex1 converges exponentially to ue, x1 can be expressed in the form(19)x1=v-11g2-f2e1+yr,e2+y˙r-γ2·+y¨rt-k1e1-k2e2+Δ·,where Δ· is the difference between x1 and xre and satisfies(20)Δ·≤ρ0e-σ0t,for some positive constants ρ0 and σ0. Substituting (19) in (18) we have(21)e˙1=e2,e˙2=f2e1+yr,e2+y˙r+γ2·-y¨rt+g2vv-11g2-f2e1+yr,e2+y˙r-γ2·+y¨rt-k1e1-k2e2+Δ·,ɛ˙1=ɛ2,ɛ˙2=-α1ɛ1-α2ɛ2-α3sign⁡ɛ1.This system can be rewritten in the form(22)e˙1=e2,e˙2=-k1e1-k2e2+μ·,ɛ˙1=ɛ2,ɛ˙2=-α1ɛ1-α2ɛ2-α3sign⁡ɛ1,where μ· is a term produced by Δ· that satisfies(23)μ·≤ρ1e-σ1t,for some positive constants ρ1 and σ1. System (22) is a piecewise linear system with a vanishing disturbance; therefore, there exist a set of constants k1, k2, α1, α2, and α3 such that the origin will be an asymptotically stable equilibrium point in a sufficient large region Ω⊂R4 [17, 18].


## 3.1. Stability Analysis

To prove the stability of the closed-loop system consider the error system(18)e˙1=e2,e˙2=f2e1+yr,e2+y˙r+g2vx1+γ2·-y¨rt,ɛ˙1=ɛ2,ɛ˙2=-α1ɛ1-α2ɛ2-α3sign⁡ɛ1.The last two equations of (18) form a subsystem uncoupled of the e1 and e2, regardless of the value of ue. Then, if α1, α2, and α3 are positive constants, the origin of this subsystem is exponentially stable [17].Becausex1 converges exponentially to ue, x1 can be expressed in the form(19)x1=v-11g2-f2e1+yr,e2+y˙r-γ2·+y¨rt-k1e1-k2e2+Δ·,where Δ· is the difference between x1 and xre and satisfies(20)Δ·≤ρ0e-σ0t,for some positive constants ρ0 and σ0. Substituting (19) in (18) we have(21)e˙1=e2,e˙2=f2e1+yr,e2+y˙r+γ2·-y¨rt+g2vv-11g2-f2e1+yr,e2+y˙r-γ2·+y¨rt-k1e1-k2e2+Δ·,ɛ˙1=ɛ2,ɛ˙2=-α1ɛ1-α2ɛ2-α3sign⁡ɛ1.This system can be rewritten in the form(22)e˙1=e2,e˙2=-k1e1-k2e2+μ·,ɛ˙1=ɛ2,ɛ˙2=-α1ɛ1-α2ɛ2-α3sign⁡ɛ1,where μ· is a term produced by Δ· that satisfies(23)μ·≤ρ1e-σ1t,for some positive constants ρ1 and σ1. System (22) is a piecewise linear system with a vanishing disturbance; therefore, there exist a set of constants k1, k2, α1, α2, and α3 such that the origin will be an asymptotically stable equilibrium point in a sufficient large region Ω⊂R4 [17, 18].


## 4. Controller Implementation

The control given in (13) and (16) cannot be implemented directly because e2, ɛ2, γ1·, γ2·, and x¨re are not available. In this section we present a method to implement this control input. It is based on discontinuous observers with finite time convergence.


### 4.1. Estimation ofe2 and γ2·

To estimatee2 and γ2· we propose a finite time observer for the underactuated part:(24)e˙1=e2,e˙2=f2e1+yr,e2+y˙r+g2vx1+γ2·-y¨rt,ye=e1.The following observer is based on Levant’s exact deriver [19]:(25)e^˙1=z1+c1,1e1-e^11/2sign⁡e1-e^1,z˙1=c2,1sign⁡e1-e^1,y^1=e^1,e^˙2=g2vx1-y¨rt+z2+c1,2z1-e^21/2sign⁡z1-e^2,z˙2=c2,2sign⁡z1-e^2,y^2=e^2.To show the stability of the observer define the errors ϵ1=e1-e^1 and ϵ2=e2-e^2, whose dynamics are given by(26)ϵ˙1=e2-z1-c1,1ϵ11/2sign⁡ϵ1,z˙1=c2,1sign⁡ϵ1,ϵ˙2=f2e1+yr,e2+y˙r+γ2·-z2-c1,2z1-e^21/2sign⁡z1-e^2,z˙2=c2,2sign⁡z1-e^2.Making a change of variables in the first two equations (27)v1=ϵ1,v2=e2-z1,we obtain the subsystem(28)v˙1=v2-c1,1v11/2sign⁡v1,v˙2=e˙2-c2,1sign⁡v1.If e˙2≤δ1, where δ1 is a known constant, there exist constants c1,1 and c2,1 such that the trajectories converge in finite time to v1=0,v2=0, [20]; therefore(29)z1=e2in finite time.The criteria to choosing the constantsc1,1 and c2,1 are(30)c1,1>2c2,1+δ1c2,1+δ11+p1-p,c2,1>δ1,where 0<p<1.For the last two equations in (26),(31)ϵ˙2=f2e1+yr,e2+y˙r+γ2·-z2-c1,2ϵ2-v21/2sign⁡ϵ2-v2,z˙2=c2,2sign⁡ϵ2-v2.Making a change of variables (32)v3=ϵ2-v2,v4=f2e1+yr,e2+y˙r+γ2·-z2-v˙2,the dynamics of these variables are given by (33)v˙3=v4-c1,2v31/2sign⁡v3,v˙4=f˙2e1+yr,e2+y˙r+γ˙2·-v¨2-c2,2sign⁡v3.If f˙2x3,x4+γ˙2·-v¨2≤δ2 there exist constants c1,2 and c2,2 such that the trajectories converge in finite time to v3=0,v4=0; therefore (34)0=ϵ2-v2,0=f2e1+yr,e2+y˙r+γ2·-z2-v˙2.Considering that for (28) v2=0 in finite time, after this time, we estimate the velocity error e2 and disturbance terms f2e1+yr,e2+y˙r+γ2·:(35)e^2=e2,z2=f2e1+yr,e2+y˙r+γ2·.As we can see γ2· is estimated through z2.


### 4.2. Estimation ofɛ2, γ1·, and x¨re

Now we design an observer for system (15): (36)ɛ˙1=ɛ2,ɛ˙2=f1x1,x2,e1+yr,e2+y˙r+g1x1,x3u+γ1·-x¨re,yɛ=ɛ1.The observer is(37)ɛ^˙1=w1+a1,1ɛ1-ɛ^11/2sign⁡ɛ1-ɛ^1,w˙1=a2,1sign⁡ɛ1-ɛ^1,y^1=ɛ^1,ɛ^˙2=g1x1,x3u+w2+a1,2w1-ɛ^21/2sign⁡w1-ɛ^2,w˙2=a2,2sign⁡w1-ɛ^2,y^2=ɛ^2.To show the stability of the observer define the errors ϵ3=ɛ1-ɛ^1 and ϵ4=ɛ2-ɛ^2, whose dynamics are given by(38)ϵ˙3=ɛ2-w1-a1,1ϵ31/2sign⁡ϵ3,(39)w˙1=a2,1sign⁡ϵ3,(40)ϵ˙4=f1x1,x2,e1+yr,e2+y˙r+γ1·-u¨e-w2-a1,2w1-ɛ^21/2sign⁡w1-ɛ^2,(41)w˙2=a2,2sign⁡w1-ɛ^2.Making a change of variables for the first two equations ((38)-(39)), (42)v5=ϵ3,v6=ɛ2-w1,we obtain the subsystem(43)v˙5=v6-a1,1v51/2sign⁡v5,v˙6=ɛ˙2-a2,1sign⁡v5.According to (30), if ɛ˙2≤δ3 there exist constants a1,1 and a2,1 such that the trajectories converge in finite time to v5=0,v6=0 [20]; therefore(44)w1=ɛ2,in finite time.For the last two equations in ((40)-(41)),(45)ϵ˙4=f1x1,x2,e1+yr,e2+y˙r+γ1·-x¨re-w2-a1,2ϵ4-v61/2sign⁡ϵ4-v6,w˙2=a2,2sign⁡ϵ4-v6.Making a change of variables (46)v7=ϵ4-v6,v8=f1x1,x2,e1+yr,e2+y˙r+γ1·-x¨re-v˙6-w2,the dynamics of these variables are given by(47)v˙7=v8-a1,2v71/2sign⁡v7,v˙8=f˙1x1,x2,x3,x4+γ˙1·-x⃛re-v¨6-a2,2sign⁡v7.If f˙1x1,x2,e1+yr,e2+y˙r+γ˙1·-x⃛re-v¨6≤δ4 there exist constants a1,2 and a2,2 such that the trajectories converge in finite time to v7=0,v8=0 [20]; therefore (48)0=ϵ4-v6,0=f1x1,x2,e1+yr,e2+y˙r+γ1·-x¨re-v˙6-w2.Considering that for (43) v6=0 in finite time, after a finite time, we have(49)ɛ^2=ɛ2,w2=f1x1,x2,e1+yr,e2+y˙r+γ1·-x¨re.Now, the control inputs (13) and (16) may be implemented in the following form:(50)xre≈1g2-z2+y¨rt-k1e1-k2e^2,u≈1g1x1,x3-w2-α1ɛ1-α2ɛ^2-α3sign⁡ɛ1.The implementation of the controller must be in several stages. First we have to apply a signalu, in open loop, such that the behavior of the system will be bounded; in this way the observers can estimate the state and disturbances in finite time. After this time, the control signals (50) can be implemented and then close the control loop.


## 4.1. Estimation ofe2 and γ2·

To estimatee2 and γ2· we propose a finite time observer for the underactuated part:(24)e˙1=e2,e˙2=f2e1+yr,e2+y˙r+g2vx1+γ2·-y¨rt,ye=e1.The following observer is based on Levant’s exact deriver [19]:(25)e^˙1=z1+c1,1e1-e^11/2sign⁡e1-e^1,z˙1=c2,1sign⁡e1-e^1,y^1=e^1,e^˙2=g2vx1-y¨rt+z2+c1,2z1-e^21/2sign⁡z1-e^2,z˙2=c2,2sign⁡z1-e^2,y^2=e^2.To show the stability of the observer define the errors ϵ1=e1-e^1 and ϵ2=e2-e^2, whose dynamics are given by(26)ϵ˙1=e2-z1-c1,1ϵ11/2sign⁡ϵ1,z˙1=c2,1sign⁡ϵ1,ϵ˙2=f2e1+yr,e2+y˙r+γ2·-z2-c1,2z1-e^21/2sign⁡z1-e^2,z˙2=c2,2sign⁡z1-e^2.Making a change of variables in the first two equations (27)v1=ϵ1,v2=e2-z1,we obtain the subsystem(28)v˙1=v2-c1,1v11/2sign⁡v1,v˙2=e˙2-c2,1sign⁡v1.If e˙2≤δ1, where δ1 is a known constant, there exist constants c1,1 and c2,1 such that the trajectories converge in finite time to v1=0,v2=0, [20]; therefore(29)z1=e2in finite time.The criteria to choosing the constantsc1,1 and c2,1 are(30)c1,1>2c2,1+δ1c2,1+δ11+p1-p,c2,1>δ1,where 0<p<1.For the last two equations in (26),(31)ϵ˙2=f2e1+yr,e2+y˙r+γ2·-z2-c1,2ϵ2-v21/2sign⁡ϵ2-v2,z˙2=c2,2sign⁡ϵ2-v2.Making a change of variables (32)v3=ϵ2-v2,v4=f2e1+yr,e2+y˙r+γ2·-z2-v˙2,the dynamics of these variables are given by (33)v˙3=v4-c1,2v31/2sign⁡v3,v˙4=f˙2e1+yr,e2+y˙r+γ˙2·-v¨2-c2,2sign⁡v3.If f˙2x3,x4+γ˙2·-v¨2≤δ2 there exist constants c1,2 and c2,2 such that the trajectories converge in finite time to v3=0,v4=0; therefore (34)0=ϵ2-v2,0=f2e1+yr,e2+y˙r+γ2·-z2-v˙2.Considering that for (28) v2=0 in finite time, after this time, we estimate the velocity error e2 and disturbance terms f2e1+yr,e2+y˙r+γ2·:(35)e^2=e2,z2=f2e1+yr,e2+y˙r+γ2·.As we can see γ2· is estimated through z2.


## 4.2. Estimation ofɛ2, γ1·, and x¨re

Now we design an observer for system (15): (36)ɛ˙1=ɛ2,ɛ˙2=f1x1,x2,e1+yr,e2+y˙r+g1x1,x3u+γ1·-x¨re,yɛ=ɛ1.The observer is(37)ɛ^˙1=w1+a1,1ɛ1-ɛ^11/2sign⁡ɛ1-ɛ^1,w˙1=a2,1sign⁡ɛ1-ɛ^1,y^1=ɛ^1,ɛ^˙2=g1x1,x3u+w2+a1,2w1-ɛ^21/2sign⁡w1-ɛ^2,w˙2=a2,2sign⁡w1-ɛ^2,y^2=ɛ^2.To show the stability of the observer define the errors ϵ3=ɛ1-ɛ^1 and ϵ4=ɛ2-ɛ^2, whose dynamics are given by(38)ϵ˙3=ɛ2-w1-a1,1ϵ31/2sign⁡ϵ3,(39)w˙1=a2,1sign⁡ϵ3,(40)ϵ˙4=f1x1,x2,e1+yr,e2+y˙r+γ1·-u¨e-w2-a1,2w1-ɛ^21/2sign⁡w1-ɛ^2,(41)w˙2=a2,2sign⁡w1-ɛ^2.Making a change of variables for the first two equations ((38)-(39)), (42)v5=ϵ3,v6=ɛ2-w1,we obtain the subsystem(43)v˙5=v6-a1,1v51/2sign⁡v5,v˙6=ɛ˙2-a2,1sign⁡v5.According to (30), if ɛ˙2≤δ3 there exist constants a1,1 and a2,1 such that the trajectories converge in finite time to v5=0,v6=0 [20]; therefore(44)w1=ɛ2,in finite time.For the last two equations in ((40)-(41)),(45)ϵ˙4=f1x1,x2,e1+yr,e2+y˙r+γ1·-x¨re-w2-a1,2ϵ4-v61/2sign⁡ϵ4-v6,w˙2=a2,2sign⁡ϵ4-v6.Making a change of variables (46)v7=ϵ4-v6,v8=f1x1,x2,e1+yr,e2+y˙r+γ1·-x¨re-v˙6-w2,the dynamics of these variables are given by(47)v˙7=v8-a1,2v71/2sign⁡v7,v˙8=f˙1x1,x2,x3,x4+γ˙1·-x⃛re-v¨6-a2,2sign⁡v7.If f˙1x1,x2,e1+yr,e2+y˙r+γ˙1·-x⃛re-v¨6≤δ4 there exist constants a1,2 and a2,2 such that the trajectories converge in finite time to v7=0,v8=0 [20]; therefore (48)0=ϵ4-v6,0=f1x1,x2,e1+yr,e2+y˙r+γ1·-x¨re-v˙6-w2.Considering that for (43) v6=0 in finite time, after a finite time, we have(49)ɛ^2=ɛ2,w2=f1x1,x2,e1+yr,e2+y˙r+γ1·-x¨re.Now, the control inputs (13) and (16) may be implemented in the following form:(50)xre≈1g2-z2+y¨rt-k1e1-k2e^2,u≈1g1x1,x3-w2-α1ɛ1-α2ɛ^2-α3sign⁡ɛ1.The implementation of the controller must be in several stages. First we have to apply a signalu, in open loop, such that the behavior of the system will be bounded; in this way the observers can estimate the state and disturbances in finite time. After this time, the control signals (50) can be implemented and then close the control loop.


## 5. Control System Performance

This section shows the performance of the control system through numerical simulations and experimental results; the control systems are a ball and beam system and a spring-mass-damper mechanism.


### 5.1. Control of a Ball and Beam System

Consider the ball and beam system shown in Figure1; its model is given by (51)J+mx2α¨+2mxα˙x˙-mgxcos⁡α+δ1α˙=u,75x¨-xα˙2-gsin⁡α+δ2x˙=0,where x is the position of the ball, α is the angle of the frame, J=0.032 kg·m2 is the moment of inertia of the beam, m=0.06 kg is the mass of the ball, δ1 and δ2 are viscous friction coefficients, and g=9.8 m/seg2 is the gravitational force. Defining the state variables as x1=α, x2=α˙, x3=x, and x4=x˙ and substituting the values of the constants we have the model(52)α¨=mxα˙x˙J+mx2+mgxcos⁡αJ+mx2-δ1J+mx2α˙+uJ+mx2,x¨=57xα˙2-57δ2x˙+57gsin⁡α.A state variable representation is as follows:(53)x˙1=x2,x˙2=mx3x2x4J+mx32+mgx3cos⁡x1J+mx32-δ1J+mx2x2+uJ+mx32,x˙3=x4,x˙4=57x3x22-57δ2x4+57gsin⁡x1,y1=x1,y2=x3.In this example, without loss of generality, the model is free from uncertainties and external disturbances. It is important to note that the variable x1 is the argument of the sine function, so the control will have a bounded amplitude.Figure 1
Ball and beam mechanical system.The control objective is that the ball positionx3 tracks the reference signal yrt. Define the error variables e1=x3-yrt and e2=x4-y˙rt and the auxiliary control ue=sin⁡x1 to obtain the following system:(54)x˙1=x2,x˙2=mx3x2x4J+mx32+mgx3cos⁡x1J+mx32-δ1J+mx2x2+uJ+mx32,e˙1=e2,e˙2=57x3x22-57δ2x4-y¨rt+57gue,y1=x1,ye=e1,where ue only may take values in the -1,1 interval.The ideal controllerue to stabilize the origin of system (54) is(55)ue=75g-57x3x22+57δ2x4+y¨rt-k1e1-k2e2,where k1=30 and k2=10. Thus, the reference signal for x1 is(56)xre=arcsin⁡75g-57x3x22+57δ2x4+y¨rt-k1e1-k2e2.Define new error variables ɛ1=x1-xre and ɛ2=x2-x˙re whose dynamics are given by(57)ɛ˙1=ɛ2,ɛ˙2=mx3x2x4J+mx32+mgx3cos⁡x1J+mx32-δ1J+mx2x2-x¨re+uJ+mx32.Then, an ideal control to stabilize the origin of system (57) is (58)u=J+mx32-mx3x2x4J+mx32-mgx3cos⁡x1J+mx32+δ1J+mx2x2+x¨re+uo,uo=-σ1ɛ1-σ2ɛ2-σ3sign⁡ɛ1,where σ1=40, σ2=10, and σ3=0.7. Figure 2 shows the results when the reference yr takes different constant values; this is the case of regulation. Steady state error is practically zero and the control signal takes values suitable for a possible implementation.Figure 2
Simulation results with the ball and beam system; performance for the regulation problem.The same situation occurs when the reference is a time-varying signal. Figure3 shows that the output signal x3 of the nonactuated link converges to a time-varying signal with an almost zero steady-state error and a control signal with adequate performance for experimental implementation.Figure 3
Simulation results with the ball and beam system; performance for the tracking problem.


### 5.2. Control of a Mass-Spring-Damper System

Consider the 2DOF underactuated mass-spring-damper mechanical system shown in Figure4, with the model(59)m1x¨=-k1x-δ1x˙+z-xk2+z˙-x˙δ2+kmu+γ1·,m2z¨=-z-xk2-z˙-x˙δ2+γ2·,where x, x˙, and x¨ are the position, velocity, and acceleration of the first mass, z, z˙, and z¨ are the position, velocity, and acceleration of the second mass, u is the control input, and γ1· and γ2· are disturbances that include terms produced by parameter uncertainties.Figure 4
Underactuated mass-spring-damper mechanical systems.The nominal parameters arek1=k2=189.65 N/m, δ1=10.54 kg/sec, δ2=1.19 kg/sec, m1=0.77 kg, m2=0.60 kg, and km=2.85 N/V; these are the nominal parameter values for the mass-spring-damper system manufactured by Educational Control Products Inc. A state representation of system (59) is(60)x˙1=x2,x˙2=-491.74x1-15.23x2+10245.87x3+1.55x4+3.69u+γ1·,x˙3=x4,x˙4=312.72x1+1.97x2-312.72x3-1.97x4+γ2·,y1=x1,y2=x3.The control objective is(61)limt→∞⁡x3-yrt=0,with a bounded behavior in x1, x2, and x4.Define the errore1=x3-yrt whose dynamics are given by (62)e˙1=e2,e˙2=-312.72e1-1.97e2-312.72yrt-1.97y˙rt-y¨rt+1.97x2+γ2·+312.72x1,ye=e1,x˙1=x2,x˙2=-491.74x1-15.23x2+10245.87x3+1.55x4+3.69τ+γ1·,y1=x1.In this case the signal ue=xre and is given by(63)xre=3.19×10-3-c1e1-c2e2-1.97x2-γ2·+Θ,where c1=c2=10 and Θ=312.72yrt+1.97y˙rt+y¨rt.To design the controlu define the errors (64)ɛ1=x1-xre,ɛ2=x2-x˙re,with dynamics given by(65)ɛ˙1=ɛ2,ɛ˙2=-491.74x1-15.23x2+10245.87x3+1.55x4+3.69u+γ1·-x¨re,and the control input for this subsystem is (66)u=0.270491.74x1+15.23x2-10245.87x3-1.55x4-γ1·+x¨re+uo,uo=-α1ɛ1-α2ɛ2-α3sign⁡ɛ1.The observer used to estimate the unknown signalse2 and γ2· is(67)e^˙1=z1+c1,1e1-e^11/2sign⁡e1-e^1,z˙1=c2,1sign⁡e1-e^1,y^1=e^1,e^˙2=-312.72e1-1.97e2-312.72yrt-1.97y˙rt-y¨rt+312.72x1+z2+c1,2z1-e^21/2sign⁡z1-e^2,z˙2=c2,2sign⁡z1-e^2,y^2=e^2,where z1≈e2 and z2≈γ2·.The observer to estimateɛ2, γ1·, and x¨re is(68)ɛ^˙1=w1+a1,1ɛ1-ɛ^11/2sign⁡ɛ1-ɛ^1,w˙1=a2,1sign⁡ɛ1-ɛ^1,y^1=ɛ^1,ɛ^˙2=-491.74x1+10245.87x3+3.69u+w2+a1,2w1-ɛ^21/2sign⁡w1-ɛ^2,w˙2=a2,2sign⁡w1-ɛ^2,y^2=ɛ^2,where w1≈ɛ2 and w2≈-491.74x1-15.23x2+10245.87x3+1.55x4+γ1·-x¨re.The control inputs (13) and (16) may be implemented in the following form:(69)xre≈1g2-z2+y¨rt-k1e1-k2e^2,u≈1g1x1,x3-w2-α1ɛ1-α2ɛ^2-α3sign⁡ɛ1,where k1=10, k2=1, α1=20, α2=20, and α3=0.4.


#### 5.2.1. Numerical Results

Figure5 shows the results when the reference yr, red line, takes different constant values; this is the case of regulation. As can be seen, the output signal x3 (black line) reaches asymptotically the reference after a short transient. The steady state error x3-yrt is practically zero and the control signal u takes values suitable for a possible implementation.Figure 5
Simulation results with the mass-spring-damper system; performance for the regulation problem.The same situation occurs when the reference is a time-varying signal. Figure6 shows that the output signal x3 (black line) of the unactuated link converges to a time-varying signal yrt (red line) with an almost zero steady-state error, x3-yrt, and a control signal u with good characteristics for experimental implementation.Figure 6
Simulation results with the mass-spring-damper system; performance for the tracking problem.For this last case, we analyze the behavior of the state observers. Figure7 shows the behavior of the errors e^2-e2 and γ^2·-γ2·; these errors converge to zero in few seconds.Figure 7
Estimation errors ofe2 and γ2⋅.For the observer that estimatesɛ2 and the term that includes γ1· and x¨re, it is not possible to compare the actual values with the estimate values. Therefore we check the behavior of the errors ɛ1-ɛ^1 and ɛ2-ɛ^2; as these errors go to zero, as we can see in Figure 8, the estimation of ɛ2 and the term that includes γ1· and x¨re is correct.Figure 8
Behavior of the internal errors of the observer that estimatesɛ2, γ1⋅, and x¨re.


#### 5.2.2. Experimental Results

The proposed controller is applied to a spring mass damping system manufactured by Educational Control Products Inc., shown in Figure9. In this experiment it is assumed that the plant has the same parameters as those considered in the numerical simulation of the previous section and did not conduct a rigorous procedure to estimate the parameters of the real plant, creating a significant challenge to the controller. This situation was resolved by tuning the parameters of each observer, the internal control signal ue, and the total control u; where k1=20, k2=2, α1=20, α2=5, and α3=0.4.Figure 9
Mass damper spring system used in the experiments.Experimental results are shown with a reference signalyr with constant values at different times; that is, the control objective is regulation. The results are shown in Figure 10, where we can see that the transient takes about one second; the amplitude of the steady-state error, x3-yr, has a maximum of 8×10-6 meters and the control signal u takes values which are in the permissible range of the control system, ±3 volts.Figure 10
Experimental results and the regulation case.In the second experiment we apply a time varying signal; a sine function, that is, the control target, is tracking. The results are shown in Figure11, where we can see that the transient takes about 5.8 seconds with an initial error of about 0.01 meters, the steady-state error amplitude, x3-yrt, has a maximum of ±3×10-4 meters, and the control signal takes values which are in the permissible range control system, as in the previous case, between ±3 volts.Figure 11
Experimental results and tracking case.


## 5.1. Control of a Ball and Beam System

Consider the ball and beam system shown in Figure1; its model is given by (51)J+mx2α¨+2mxα˙x˙-mgxcos⁡α+δ1α˙=u,75x¨-xα˙2-gsin⁡α+δ2x˙=0,where x is the position of the ball, α is the angle of the frame, J=0.032 kg·m2 is the moment of inertia of the beam, m=0.06 kg is the mass of the ball, δ1 and δ2 are viscous friction coefficients, and g=9.8 m/seg2 is the gravitational force. Defining the state variables as x1=α, x2=α˙, x3=x, and x4=x˙ and substituting the values of the constants we have the model(52)α¨=mxα˙x˙J+mx2+mgxcos⁡αJ+mx2-δ1J+mx2α˙+uJ+mx2,x¨=57xα˙2-57δ2x˙+57gsin⁡α.A state variable representation is as follows:(53)x˙1=x2,x˙2=mx3x2x4J+mx32+mgx3cos⁡x1J+mx32-δ1J+mx2x2+uJ+mx32,x˙3=x4,x˙4=57x3x22-57δ2x4+57gsin⁡x1,y1=x1,y2=x3.In this example, without loss of generality, the model is free from uncertainties and external disturbances. It is important to note that the variable x1 is the argument of the sine function, so the control will have a bounded amplitude.Figure 1
Ball and beam mechanical system.The control objective is that the ball positionx3 tracks the reference signal yrt. Define the error variables e1=x3-yrt and e2=x4-y˙rt and the auxiliary control ue=sin⁡x1 to obtain the following system:(54)x˙1=x2,x˙2=mx3x2x4J+mx32+mgx3cos⁡x1J+mx32-δ1J+mx2x2+uJ+mx32,e˙1=e2,e˙2=57x3x22-57δ2x4-y¨rt+57gue,y1=x1,ye=e1,where ue only may take values in the -1,1 interval.The ideal controllerue to stabilize the origin of system (54) is(55)ue=75g-57x3x22+57δ2x4+y¨rt-k1e1-k2e2,where k1=30 and k2=10. Thus, the reference signal for x1 is(56)xre=arcsin⁡75g-57x3x22+57δ2x4+y¨rt-k1e1-k2e2.Define new error variables ɛ1=x1-xre and ɛ2=x2-x˙re whose dynamics are given by(57)ɛ˙1=ɛ2,ɛ˙2=mx3x2x4J+mx32+mgx3cos⁡x1J+mx32-δ1J+mx2x2-x¨re+uJ+mx32.Then, an ideal control to stabilize the origin of system (57) is (58)u=J+mx32-mx3x2x4J+mx32-mgx3cos⁡x1J+mx32+δ1J+mx2x2+x¨re+uo,uo=-σ1ɛ1-σ2ɛ2-σ3sign⁡ɛ1,where σ1=40, σ2=10, and σ3=0.7. Figure 2 shows the results when the reference yr takes different constant values; this is the case of regulation. Steady state error is practically zero and the control signal takes values suitable for a possible implementation.Figure 2
Simulation results with the ball and beam system; performance for the regulation problem.The same situation occurs when the reference is a time-varying signal. Figure3 shows that the output signal x3 of the nonactuated link converges to a time-varying signal with an almost zero steady-state error and a control signal with adequate performance for experimental implementation.Figure 3
Simulation results with the ball and beam system; performance for the tracking problem.


## 5.2. Control of a Mass-Spring-Damper System

Consider the 2DOF underactuated mass-spring-damper mechanical system shown in Figure4, with the model(59)m1x¨=-k1x-δ1x˙+z-xk2+z˙-x˙δ2+kmu+γ1·,m2z¨=-z-xk2-z˙-x˙δ2+γ2·,where x, x˙, and x¨ are the position, velocity, and acceleration of the first mass, z, z˙, and z¨ are the position, velocity, and acceleration of the second mass, u is the control input, and γ1· and γ2· are disturbances that include terms produced by parameter uncertainties.Figure 4
Underactuated mass-spring-damper mechanical systems.The nominal parameters arek1=k2=189.65 N/m, δ1=10.54 kg/sec, δ2=1.19 kg/sec, m1=0.77 kg, m2=0.60 kg, and km=2.85 N/V; these are the nominal parameter values for the mass-spring-damper system manufactured by Educational Control Products Inc. A state representation of system (59) is(60)x˙1=x2,x˙2=-491.74x1-15.23x2+10245.87x3+1.55x4+3.69u+γ1·,x˙3=x4,x˙4=312.72x1+1.97x2-312.72x3-1.97x4+γ2·,y1=x1,y2=x3.The control objective is(61)limt→∞⁡x3-yrt=0,with a bounded behavior in x1, x2, and x4.Define the errore1=x3-yrt whose dynamics are given by (62)e˙1=e2,e˙2=-312.72e1-1.97e2-312.72yrt-1.97y˙rt-y¨rt+1.97x2+γ2·+312.72x1,ye=e1,x˙1=x2,x˙2=-491.74x1-15.23x2+10245.87x3+1.55x4+3.69τ+γ1·,y1=x1.In this case the signal ue=xre and is given by(63)xre=3.19×10-3-c1e1-c2e2-1.97x2-γ2·+Θ,where c1=c2=10 and Θ=312.72yrt+1.97y˙rt+y¨rt.To design the controlu define the errors (64)ɛ1=x1-xre,ɛ2=x2-x˙re,with dynamics given by(65)ɛ˙1=ɛ2,ɛ˙2=-491.74x1-15.23x2+10245.87x3+1.55x4+3.69u+γ1·-x¨re,and the control input for this subsystem is (66)u=0.270491.74x1+15.23x2-10245.87x3-1.55x4-γ1·+x¨re+uo,uo=-α1ɛ1-α2ɛ2-α3sign⁡ɛ1.The observer used to estimate the unknown signalse2 and γ2· is(67)e^˙1=z1+c1,1e1-e^11/2sign⁡e1-e^1,z˙1=c2,1sign⁡e1-e^1,y^1=e^1,e^˙2=-312.72e1-1.97e2-312.72yrt-1.97y˙rt-y¨rt+312.72x1+z2+c1,2z1-e^21/2sign⁡z1-e^2,z˙2=c2,2sign⁡z1-e^2,y^2=e^2,where z1≈e2 and z2≈γ2·.The observer to estimateɛ2, γ1·, and x¨re is(68)ɛ^˙1=w1+a1,1ɛ1-ɛ^11/2sign⁡ɛ1-ɛ^1,w˙1=a2,1sign⁡ɛ1-ɛ^1,y^1=ɛ^1,ɛ^˙2=-491.74x1+10245.87x3+3.69u+w2+a1,2w1-ɛ^21/2sign⁡w1-ɛ^2,w˙2=a2,2sign⁡w1-ɛ^2,y^2=ɛ^2,where w1≈ɛ2 and w2≈-491.74x1-15.23x2+10245.87x3+1.55x4+γ1·-x¨re.The control inputs (13) and (16) may be implemented in the following form:(69)xre≈1g2-z2+y¨rt-k1e1-k2e^2,u≈1g1x1,x3-w2-α1ɛ1-α2ɛ^2-α3sign⁡ɛ1,where k1=10, k2=1, α1=20, α2=20, and α3=0.4.


### 5.2.1. Numerical Results

Figure5 shows the results when the reference yr, red line, takes different constant values; this is the case of regulation. As can be seen, the output signal x3 (black line) reaches asymptotically the reference after a short transient. The steady state error x3-yrt is practically zero and the control signal u takes values suitable for a possible implementation.Figure 5
Simulation results with the mass-spring-damper system; performance for the regulation problem.The same situation occurs when the reference is a time-varying signal. Figure6 shows that the output signal x3 (black line) of the unactuated link converges to a time-varying signal yrt (red line) with an almost zero steady-state error, x3-yrt, and a control signal u with good characteristics for experimental implementation.Figure 6
Simulation results with the mass-spring-damper system; performance for the tracking problem.For this last case, we analyze the behavior of the state observers. Figure7 shows the behavior of the errors e^2-e2 and γ^2·-γ2·; these errors converge to zero in few seconds.Figure 7
Estimation errors ofe2 and γ2⋅.For the observer that estimatesɛ2 and the term that includes γ1· and x¨re, it is not possible to compare the actual values with the estimate values. Therefore we check the behavior of the errors ɛ1-ɛ^1 and ɛ2-ɛ^2; as these errors go to zero, as we can see in Figure 8, the estimation of ɛ2 and the term that includes γ1· and x¨re is correct.Figure 8
Behavior of the internal errors of the observer that estimatesɛ2, γ1⋅, and x¨re.


### 5.2.2. Experimental Results

The proposed controller is applied to a spring mass damping system manufactured by Educational Control Products Inc., shown in Figure9. In this experiment it is assumed that the plant has the same parameters as those considered in the numerical simulation of the previous section and did not conduct a rigorous procedure to estimate the parameters of the real plant, creating a significant challenge to the controller. This situation was resolved by tuning the parameters of each observer, the internal control signal ue, and the total control u; where k1=20, k2=2, α1=20, α2=5, and α3=0.4.Figure 9
Mass damper spring system used in the experiments.Experimental results are shown with a reference signalyr with constant values at different times; that is, the control objective is regulation. The results are shown in Figure 10, where we can see that the transient takes about one second; the amplitude of the steady-state error, x3-yr, has a maximum of 8×10-6 meters and the control signal u takes values which are in the permissible range of the control system, ±3 volts.Figure 10
Experimental results and the regulation case.In the second experiment we apply a time varying signal; a sine function, that is, the control target, is tracking. The results are shown in Figure11, where we can see that the transient takes about 5.8 seconds with an initial error of about 0.01 meters, the steady-state error amplitude, x3-yrt, has a maximum of ±3×10-4 meters, and the control signal takes values which are in the permissible range control system, as in the previous case, between ±3 volts.Figure 11
Experimental results and tracking case.


## 5.2.1. Numerical Results

Figure5 shows the results when the reference yr, red line, takes different constant values; this is the case of regulation. As can be seen, the output signal x3 (black line) reaches asymptotically the reference after a short transient. The steady state error x3-yrt is practically zero and the control signal u takes values suitable for a possible implementation.Figure 5
Simulation results with the mass-spring-damper system; performance for the regulation problem.The same situation occurs when the reference is a time-varying signal. Figure6 shows that the output signal x3 (black line) of the unactuated link converges to a time-varying signal yrt (red line) with an almost zero steady-state error, x3-yrt, and a control signal u with good characteristics for experimental implementation.Figure 6
Simulation results with the mass-spring-damper system; performance for the tracking problem.For this last case, we analyze the behavior of the state observers. Figure7 shows the behavior of the errors e^2-e2 and γ^2·-γ2·; these errors converge to zero in few seconds.Figure 7
Estimation errors ofe2 and γ2⋅.For the observer that estimatesɛ2 and the term that includes γ1· and x¨re, it is not possible to compare the actual values with the estimate values. Therefore we check the behavior of the errors ɛ1-ɛ^1 and ɛ2-ɛ^2; as these errors go to zero, as we can see in Figure 8, the estimation of ɛ2 and the term that includes γ1· and x¨re is correct.Figure 8
Behavior of the internal errors of the observer that estimatesɛ2, γ1⋅, and x¨re.


## 5.2.2. Experimental Results

The proposed controller is applied to a spring mass damping system manufactured by Educational Control Products Inc., shown in Figure9. In this experiment it is assumed that the plant has the same parameters as those considered in the numerical simulation of the previous section and did not conduct a rigorous procedure to estimate the parameters of the real plant, creating a significant challenge to the controller. This situation was resolved by tuning the parameters of each observer, the internal control signal ue, and the total control u; where k1=20, k2=2, α1=20, α2=5, and α3=0.4.Figure 9
Mass damper spring system used in the experiments.Experimental results are shown with a reference signalyr with constant values at different times; that is, the control objective is regulation. The results are shown in Figure 10, where we can see that the transient takes about one second; the amplitude of the steady-state error, x3-yr, has a maximum of 8×10-6 meters and the control signal u takes values which are in the permissible range of the control system, ±3 volts.Figure 10
Experimental results and the regulation case.In the second experiment we apply a time varying signal; a sine function, that is, the control target, is tracking. The results are shown in Figure11, where we can see that the transient takes about 5.8 seconds with an initial error of about 0.01 meters, the steady-state error amplitude, x3-yrt, has a maximum of ±3×10-4 meters, and the control signal takes values which are in the permissible range control system, as in the previous case, between ±3 volts.Figure 11
Experimental results and tracking case.


## 6. Conclusions

The control strategy proposed formally guarantees the control objective, either regulation or trajectory tracking, and at the same time establishes a strategy for its implementation considering partial measurement of the state variables and parametric uncertainties. Although stability is not global, the subspace that can ensure stability can be made as large as needed in practice. Some of its limitations are the number of parameters to adjust, both in the observers and in the controller, and the need to use a real-time platform to implement the controller to ensure a sample time less than or equal to one millisecond, and thus the actual sliding mode, produced by discontinuous terms, enough approaches the ideal sliding mode.



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*Source: 429476-2015-10-11.xml*