# Two-Valued Control for a Second-Order Plant with Additive External Disturbance

Authors: Alejandro Rincon; Felipe Londoño; Fabiola Angulo
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

In this work a two-valued state feedback control for a plant of second order with known constant coefficients and an additive bounded disturbance is designed. In this controller the control signal can take only two possible values. The controller design is based on Lyapunov-like function method, achieving the convergence of the tracking error to a user-defined residual set. A boundedness condition for the user-defined reference signal is defined, which is necessary to allow out-put tracking. The developed scheme avoids large commutation rate of the control input. The controller design and stability analysis have important contributions with respect to closely related controllers based on the direct Lyapunov method, namely, (i) conditions to guarantee the expected convergence of the tracking error are established. These conditions are imposed on the reference signal and the extreme values of the control input. The stability analysis is developed by means of the Lyapunov-like function method and the Barbalat's Lemma and includes (ii) the bounded nature of the Lyapunov function, (iii) the monotonic convergence of the Lyapunov function to a residual set, and (iv) the asymptotic convergence of the tracking error to a residual set of user-defined size.


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## Body

## 1. Introduction

The design of two-valued feedback controllers has attracted important research [1–8]. It is intended for two-valued actuators, which lead to a control input that takes only two possible preassigned finite values. Thus, the control input is discontinuous with respect to time [1, 9]. The two-valued actuators have the following advantages with respect to proportional ones [1, 10, 11]: they are simple, relatively cheaper, lead to fast output response, and overcomes the issue of actuator static gain. The electrical kiln [5], hydrogel valves [12], the compressor [6, 13], the satellite antenna [2, page 25] are some systems with two-value actuators.A basic method for two-valued control is the relay feedback with fixed hysteresis band. The aim of the hysteresis is to avoid high commutation rate of the control input, what is known as input chattering [10, 13] and may lead to high power consumption and wear of mechanical components (cf. [5, 6]). The width of the hysteresis band determines the commutation rate of the control signal and the size of the residual set to which the output error converges. The width has to be chosen to obtain a trade-off between commutation rate and size of the residual set, because one fact improves at the expense of the other [6]. An improper designed relay feedback may lead to overshoot, large amplitude oscillation of the output, or large settling time (cf. [10, 13]). These can be overcomed by two-valued control based on the direct Lyapunov method, as shown in [1]. In that paper the authors consider a single input single output (SISO) linear plant in controllable form, with time varying coefficients and additive disturbances. They ensure the convergence of the tracking error to a residual set whose size is user defined. They show that large commutation rate is avoided if the initial values of the tracking error and its time derivative are such that the initial value of the Lyapunov function is inside a target region of user defined size.In the case of input output stable plants with fixed preset values of the control input extremes, a two-valued input implies the following: (i) the output remains inside some bounded region, regardless the controller, (ii) a user defined trajectory with periodic behavior and large frequency or large magnitude cannot be reached by the system output. The suitability of the frequency and magnitude of the user defined trajectory depends on the extreme values of the control input, and the plant model coefficients, as it will show in this work. Nevertheless, as far as we know, there is not a condition that indicates such suitability, in the literature on finite-valued control based on the direct Lyapunov method. In this work such condition is established. The controller design and stability analysis have important contributions with respect to closely related controllers based on the direct Lyapunov method, namely, (i) conditions to guarantee the expected convergence of the tracking error are established. These conditions are imposed on the reference signal and the extreme values of the control input. The stability analysis is developed by means of the Lyapunov-like function method and the Barbalat’s Lemma and includes (ii) the bounded nature of the Lyapunov function, (iii) the monotonic convergence of the Lyapunov function to a residual set, and (iv) the asymptotic convergence of the tracking error to a residual set of user defined size.The work is only valid for a second-order plant with constant coefficients and an unknown external disturbance with known upper bounds. Upper bounds can be known from previous modelling tasks. A controller for the plant is developed, the controller is based on the direct Lyapunov method, ensuring the convergence of the tracking error to a residual set whose size are user defined. The work is organized as follows. In Section2 the plant model and the goal of the control design is detailed. In Section 3 the condition that indicates if the desired output is suitable to achieve tracking is established. In addition, a preliminary rough control law, which achieves the convergence of the tracking error to a residual set whose size is user defined is formulated. Nevertheless, it may lead to large commutation rate of the control input. This will remedy in subsequent sections. In Section 4 the control law to avoid large commutation rate of the control input is formulated. In Sections 5 and 6 the boundedness of the closed loop signals and the convergence of the tracking error are proven. In Section 7 numerical simulation is presented. Finally, in Sections 8 and 9 discussion and conclusions are presented.


## 2. Problem Statement

The plant, the reference model, and the state goal of the control design are detailed in this section. Consider the following second-order plant (systems analyzed with this model can be found in [14–16])
(2.1)ÿ=-a1ẏ-aoy+bu+d,(2.2)u∈{umn,umx},umx>umn,
where y(t)∈ℝ is the system output, u(t)∈ℝ is the input, ao, a1, b are plant coefficients, being b the control gain, and d is an uncertain term that may result from modelling error or an external disturbance. Let us consider the following assumptions.(Ai) The coefficientsao, a1, b are constant, known, and positive. (Aii) The signals y, ẏ are available for measurement. (Aiii) The values of umn, umx are constant, user defined, satisfy umx>umn, and are not restricted to positive values. (Aiv) The uncertainty d is time varying and satisfies either(2.3)(i)|d|≤μo,μo>0,whereμoisaknownpositiveconstant,or(ii)d=μo,μo=0.
Now, the control goal can be established. Let
(2.4)e(t)=y(t)-yd(t),(2.5)ÿd=-am1ẏd-amoyd+amor,(2.6)Ωe={e∈R:|e|≤Cbe},
where am1, amo are positive constant of the user choice, the command signal r is bounded and user defined, yd is the desired output or reference signal, Cbe is a positive constant, user defined. It is important that the initial conditions yd(to), ẏd(to) be chosen such that the initial value of the Lyapunov function be inside the target set, to avoid large commutation rate (see [1]). This requirement is included in the control scheme. The aim of the reference model (2.5) is to provide an adequate nature of yd and ẏd, such that the control input can induce tracking while large input commutation rate is avoided.The objective of the control design is to formulate a control law for the control inputu, provided the plant model (2.1), subject to assumptions (Ai) to (Aiv), such that (Gi) the tracking error e(t) converges asymptotically to the residual set Ωe, (Gii) large commutation rate of the control input is avoided. Other goal of the control design is to develop a condition that indicates if a given desired output is suitable to achieve tracking.


## 3. A Preliminary Rough Control Law

In this section, a preliminary control law for the control inputu, provided by the plant (2.1), subject to assumptions (Ai) to (Aiv) is developed. The main contribution is that the tracking error asymptotically converges to Ωe, being Ωe defined in (2.6). The Lyapunov-like function method, which is commonly used to design robust controllers for plants with continuous control inputs is used. A preliminary control law is developed. This control does not prevent large commutation rate but will be improved in subsequent sections.Subtractingÿd from both sides of  (2.1)(3.1)ÿ-ÿd=-a1ẏ-aoy+bu-ÿd+d,ÿ-ÿd=-a1(ẏ-ẏd)-ao(y-yd)-a1ẏd-aoyd+bu-ÿd+d,
since the first- and second-time derivatives of the tracking error are ė=ẏ-ẏd, ë=ÿ-ÿd, it is obtained:(3.2)ë=-a1ė-aoe+bu-ÿd-a1ẏd-aoyd+d=-a1ė-aoe+bv,(3.3)v=u-ÿd+a1ẏd+aoydb+db.
The term v is introduced with two objectives. The first one is for notational simplicity, and the second one is to simplify the design of a control law that overcomes the effect of yd and its time derivatives. Equation (3.2) can be rewritten as(3.4)ẋ1=x2,ẋ2=-a1x2-aox1+bv,(3.5)x1=e,x2=ė.
Consider the following Lyapunov function:
(3.6)V(x(t))=12aoS2+12x12,(3.7)x(t)=[x1(t)x2(t)]⊤,(3.8)S(x(t))=a1x1+x2.
The time derivative of V along trajectory (3.4) is(3.9)V̇=1aoSṠ+x1ẋ1=1aoS(a1x2+ẋ2)+x1x2=-a1x12+baoSv=-c1a1x12-(1-c1)a1x12+baoSv,
where v is defined in (3.3), whereas c1 is constant, user defined, and satisfies c1∈[0,1]. Constant c1 is introduced in (3.9) with the objective to change the commutation rate. A complete expression of the improved controller is given in (4.9) and it shows explicitly the usefulness of this constant. The above equation suggests that if the control law for u is properly defined, then
(3.10)-(1-c1)a1x12+(bao)Sv≤0⟹V̇≤-c1a1x12.
The condition V̇≤-c1a1x12 implies that the tracking error converges asymptotically to a small value. To find a rough control law that achieves this, the term Sv can be written as(3.11)Sv=-S(-u+ÿd+a1ẏd+aoydb)+dbS=-|S|(-usgn⁡(S)+ÿd+a1ẏd+aoydbsgn⁡(S))+dbS.
From assumption (Aiv) it follows that (d/b)S≤(μo/b)|S|. Substituting this into (3.11), it is obtained:(3.12)Sv≤-|S|(-usgn⁡(S)+ÿd+a1ẏd+aoydbsgn⁡(S))+μob|S|,Sv≤-|S|(-usgn⁡(S)+ÿd+a1ẏd+aoydbsgn⁡(S)-μob).
If the term in parenthesis is positive, then Sv≤0. It can be achieved with the following rule:
(3.13)u={u̅-δsgn⁡(S)ifS≠0doesnotchangeifS=0,
where(3.14)u̅=(12)(umn+umx),δ=(12)(umx-umn),
substituting (3.13) into (3.12), it is obtained(3.15)Sv≤-|S|[δ+sgn⁡(S)(ÿd+a1ẏd+aoydb-u̅)-μob],
the following property is needed.Proposition 3.1.
If(3.16)umn+μob≤ÿd+a1ẏd+aoydb≤umx-μob∀t≥to,
then
(3.17)δ+sgn⁡(S)(ÿd+a1ẏd+aoydb-u̅)-μob≥0∀t≥to,
the proof is presented in Appendix A.Remark 3.2.
A given desired output yd with excessive magnitude or excessive frequency would not fulfill (3.16). Nevertheless, since it is assumed that the coefficients ao,a1,b,umn,umx are constant, and the plant model (2.1) is usually predefined, condition (3.16) can be fulfilled by modifying the values of amo,am1, and r of the reference model (2.5). This can be carried out by means of simulation.Remark 3.3.
Condition (3.16) implies that the value of umn is low enough and the value of umx is high enough such that the control input can drive the output y towards the desired output yd.Substituting (3.17) in (3.15), it follows Sv≤0 for S≠0, and from (3.9) it follows that V̇≤-a1x12≤-c1a1x12. If S=0, thus Sv=0 and from (3.9) it follows that V̇≤-c1a1x12. Therefore, V̇≤-c1a1x12 if rule (3.13) is used. These results are summarized in the following Theorem.Theorem 3.4.
Consider the plant model (2.1) subject to assumptions (Ai) to (Aiv), the Lyapunov function  V(x(t)) defined in (3.6) and the signal  S defined in (3.8). If condition (3.16) is fulfilled, and rule (3.13) is used, then
(3.18)V̇≤-c1a1x12.Therefore, the control law (3.13) implies the convergence of the tracking error x1 to a small value.Remark 3.5.
The control rule (3.13) operates as follows. For t=to,if S(x(to))=0 the control input may take any on the values umn,umx.If S(x(to))≠0,the control input is defined by u=u̅-δsgn(S(x(to))) for t=to. Thecontrol input u retains its initial value until the signal sgn(S(x(t))) changes its value with respect to sgn(S(to)).Then, u changes according to the rule u=u̅-δsgn(S).The input u retains such value until the value of sgn(S) changes again, so that u changes according to u=u̅-δsgn(S).This is repeated successively. If S=0 in some instant time, the input u does not change its value.Remark 3.6.
According to [2, 7], the use of discontinuous control law may lead to (i) loss of trajectory unicity, (ii) sliding motion of trajectories along the discontinuity surface, what may imply chattering (see [2, pages 282-283]), and (iii) input chattering, which is an undesired large commutation rate component in the control input (see [2, page 292]). Large commutation rate may lead to high power consumption and wear of mechanical components (cf. [5, 6]). A rigorous design of a direct Lyapunov method should include the following tasks (cf. [7]): (i) ensure that trajectory unicity is preserved, (ii) develop the Filippov’s construction for the case that sliding motion occurs, in order to avoid chattering. In the case under study, there may be sliding motion of the states x1, x2 along the surface S=0. In [1] sliding motion for a closely related control scheme is illustrated. Thus, the control scheme (3.13) may lead to undesired large commutation rate in the input u when S takes on small values, so that goal (Gii) is not fulfilled. If large commutation rate is not a problem, sliding mode control could be used as an alternative approach [17–19].
In next section, the control law given by (3.13) is improved.


## 4. The Final Control Law

In previous section it was formulated a control law that achieves adequate convergence of the tracking error but leads to large commutation rate. In this section, the convergence of the quadratic functionV to a small residual set and adequate initial values of the Lyapunov function V are considered and the large commutation rate is overcome.In [1, 20] authors show that the convergence of Lyapunov function to a target manifold leads to the convergence of tracking error to a residual set of user defined size. From (3.6) it follows that if V converges to some small residual set of adequate size, then the tracking error e converges to the residual set Ωe defined in (2.6).Proposition 4.1.
Consider the functionV defined in (3.6), the tracking error  e defined in (2.4) and the set  Ωe defined in (2.6). Let
(4.1)Ωv={V(x(t))∈R:V(x(t))≤Cbv},(4.2)Cbv=(12)Cbe2max⁡{1,a12ao}.
The convergence of V and e are related as follows: If V converges to Ωv, then e converges to Ωe, where Cbe is a positive constant defined by the user. The proof is presented in Appendix B.Therefore, it is necessary to formulate a control law foru that ensures:(4.3)V̇≤-c1a1x12forV≥Cbv.
so as to achieve the expected convergence of the tracking error. Indeed, if the above condition is ensured, then: (i) V converges asymptotically to Ωv, where Ωv={V(x)∈ℝ:V(x)≤Cbv}, (ii) if in addition V reaches Ωv for some instant, V remains inside thereafter, (iii) the tracking error e converges asymptotically to Ωe. Control scheme (3.13) can achieve condition (4.3) as mentioned in Theorem 3.4. does not involve the case V<Cbv, then the control input u can take arbitrary values for V<Cbv without disrupting the convergence of e to Ωe. As in [1], it is advisable to stop the commutation of u when V<Cbv, in order to avoid large commutation rate:
(4.4)u={u̅-δsgn⁡(S)ifV≥Cbv,S≠0,stopscommutationotherwise.This control rule implies that large commutation rate is prevented forV≤Cbv, but not for V>Cbv. Indeed, there may be sliding motion of the trajectories along S=0 if V>Cbv, as occurs in [1]. A possible remedy is to impose a boundary layer around S=0 for V>Cbv. From (4.3) it follows that commutation can be stopped when the condition V̇≤-c1a1x12 is satisfied for V≥Cbv under arbitrary values of the control input u. The restrictions over x1 and x2 ensuring the above condition will be determined at the following. Expression (3.9) can be rewritten as(4.5)V̇≤-c1a1x12-(1-c1)a1x12+(bao)|S||v|.
The requirement (3.16) implies that |v| is bounded by a constant, as the following proposition shows.Proposition 4.2.
The signalV defined in  (3.3) satisfies
(4.6)|v|≤umx-umn.
The proof is presented in Appendix C.Substituting (4.6) into (4.5) yields(4.7)V̇≤-c1a1x12-(1-c1)a1x12+(bao)|S||umx-umn|.
Therefore,(4.8)V̇≤-c1a1x12forV≥Cbv,if-(1-c1)a1x12+(bao)|S||umx-umn|≤0,
where -(1-c1)a1x12+(b/ao)|S||umx-umn|≤0 defines a boundary layer around S=0 in the x1-x2 state space, for V>Cbv. The control law can take advantage of the above expression. Equation (4.8) indicates that it is possible to turn off commutation of the control input u while obtaining V̇≤-c1a1x12 for V≥Cbv if x1, x2 satisfy the condition therein. The control law for u is then formulated as follows: (i) u follows the rule u=u̅-δsgn(S) for the case that V≥Cbv and condition in (4.8) is not fulfilled, (ii) u follows the rule u=u̅-δsgn(S) for the case that V=Cbv and S≠0, and (iii) u stops commutation (stops cm) otherwise (ow). Equivalently:
(4.9)u={u̅-δsgn⁡(S)ifV≥Cbvand-(1-c1)a1x12+(bao)|S|(umx-umn)>0,u̅-δsgn⁡(S)ifV=CbvandS≠0,stopscmotherwise,
and for t=to: (4.10)(i)ifS(x(to))=0,ucantakeanyofthevaluesumnorumx,(ii)ifV(x(to))<Cbv,ucantakeanyon⁡thevaluesumn,umx,(iii)ifS(x(to))≠0andV(x(to))=Cbv,utakeson⁡thevalueu=u̅-δsgn⁡(S(x(to))),(iv)ifV(x(to))>Cbvand-(1-c1)a1x12+(bao)|S|(umx-umn)>0,utakeson⁡thevalueu=u̅-δsgn⁡(S(x(to)))
the signals necessary for the computation of u are: u̅ and δ (3.14), Cbv (4.2), S (3.8), e (3.5), yd provided by (2.1), c1 is a user defined positive constant.Remark 4.3.
The control law (4.9) operates as follows. For t=to,u follows (4.10). The control input u retains its initial value until some of the conditions in (4.9) is fulfilled. At that instant time, the control input u follows the rule u=u̅-δsgn(S),inducing the decrease of V.The input u retains such value until some of the conditions in (4.9) are fulfilled again. This procedure is repeated in the same way. Condition (3.16) should be fulfilled.Remark 4.4.
Equation (4.9) indicates that the control signal u commutes as less as possible. When the commutation stops, the control signal keeps the value acquired during previous commutation mode. The commutation rate of the control input u does not reach excessive values, because if S becomes zero, then u stops commutation.Remark 4.5.
The constantCbe cannot be zero, as we explain at the following. From (4.9) it follows that the control input u commutes when V=Cbv and S≠0.A value Cbe=0 would imply Cbv=0, as it follows from the definition (4.2). Therefore, the control input u would commute when V=0 and S≠0.Such condition is not possible, according to definition (3.6).If Cbe is overly small, then Cbv is also small, as it follows from the definition (4.2). Therefore, the time that V takes to reach V=Cbv is small, implying a larger commutation rate, according to (4.9).The discussion and simulation examples shown in [1] indicate that if condition V(x(to))≤Cbv is fulfilled, then V(x(t))≤Cbv for all t≥to and large commutation rate is avoided. Indeed, from definition (3.6) it follows that the condition V(x(to))≤Cbv is fulfilled if e(to), ė(to) have adequate magnitude, or equivalently, if the distance between yd(to) and y(to), and distance between ẏd(to) and ẏ(to) are adequate. Thus, the following control strategy is chosen:
(4.11)u={u̅-δsgn⁡(S)ifV=Cbv,≠0,stopscmotherwise.yd(to), ẏd(to) are chosen such that V(x(to))∈Ωv,Ωv={V(x(t))∈ℝ:V(x(t))≤Cbv}. For t=to, the signal control is computed as:
(4.12)(i)ifV(x(to))<Cbv,ucantakeanyon⁡thevaluesumn,umx,(ii)ifS(x(to))≠0,V(x(to))=Cbv,utakeson⁡thevalueu=u̅-δsgn⁡(S(x(to))),(iii)ifS(x(to))=0,V(x(to))=Cbv,ucantakeanyon⁡thevaluesumn,umx,Remark 4.6.
The control law (4.11) operates as follows. For t=to,u follows (4.12). The control input u retains its initial value until V=Cbv and S≠0.At that instant time, the control input u follows the rule u=u̅-δsgn(S),inducing the decrease of V.The input u retains such value until V=Cbv and S≠0 is fulfilled again. This procedure is repeated in the same way. If V=Cbv and S=0,then u does not change. Condition (3.16) should be fulfilled. Notice that the constant c1 is not necessary to formulate the control law (4.11).Remark 4.7.
Equation (4.11) indicates that the control signal u commutes as less as possible. When the commutation stops, the control signal keeps the value acquired during previous commutation mode. The commutation rate of the control input u does not reach excessive values, because if S becomes zero, then u stops commutation.


### 4.1. Implementation Issues

In experimental implementation could be difficult to detect the exact moment whenV=Cbv, so that it could be difficult to use the control scheme (4.11). Then, with the aim to apply the control strategy to one system, it is possibility to use the control scheme given by (4.9) instead of (4.11), because (4.9) includes the case V>Cbv. Other possibility is to use a threshold δV in (4.11) as follows:
(4.13)u={u̅-δsgn⁡(S)ifV∈[Cbv-δV,Cbv],S≠0,stopscmotherwise,
where δV is a positive constant that satisfies δV<Cbv, and yd(to) and ẏd(to) are chosen such that V(x(to))∈Ωv,Ωv={V(x(t))∈ℝ:V(x(t))≤Cbv}.


## 4.1. Implementation Issues

In experimental implementation could be difficult to detect the exact moment whenV=Cbv, so that it could be difficult to use the control scheme (4.11). Then, with the aim to apply the control strategy to one system, it is possibility to use the control scheme given by (4.9) instead of (4.11), because (4.9) includes the case V>Cbv. Other possibility is to use a threshold δV in (4.11) as follows:
(4.13)u={u̅-δsgn⁡(S)ifV∈[Cbv-δV,Cbv],S≠0,stopscmotherwise,
where δV is a positive constant that satisfies δV<Cbv, and yd(to) and ẏd(to) are chosen such that V(x(to))∈Ωv,Ωv={V(x(t))∈ℝ:V(x(t))≤Cbv}.


## 5. Boundedness Analysis

In this section we analyze the boundedness properties of the closed loop signals. As in [21], the notation (·)∈L∞ is used. This notation indicates that (·) is bounded.Theorem 5.1 (boundedness of the closed loop signals).
Consider the plant model  (2.1), subject to assumptions (Ai) to (Aiv), the tracking error  e, the desired output  yd and the function  S provided by  (2.4), (2.5), (3.8), respectively. If condition  (3.16) is fulfilled and the controller  (4.9) is applied, then the signals  x1,  x2,  S remain bounded.Proof.
From (4.8) it follows that
(5.1)V̇≤-c1a1x12ifV>Cbv,-(1-c1)a1x12+(bao)|S||umx-umn|≤0.
From (4.9) it follows that
(5.2)u=u̅-δsgn⁡(S),ifV>Cbv,-(1-c1)a1x12+(bao)|S||umx-umn|>0,orV=Cbv,S≠0.
Therefore, it follows from Theorem 3.4 that
(5.3)V̇≤-c1a1x12,ifV>Cbv,-(1-c1)a1x12+(bao)|S||umx-umn|>0orV=Cbv,S≠0.,
and controller (4.9) is applied. From (4.7) it follows that
(5.4)V̇≤-c1a1x12,ifV>Cbv,-(1-c1)a1x12+(bao)|S||umx-umn|≤0,orV=Cbv,S=0.
From (5.3), (5.4) it follows that
(5.5)V̇≤-c1a1x12ifV≥Cbv.
Since the above expression is not valid for V<Cbv, it does not lead to a straightforward proof of the boundedness and convergence of V. In order to show that the Lyapunov function V is bounded, a Lyapunov-like function fa=fa(V) that satisfies
(5.6)(i)fa≥0∀t≥to,(ii)V≤ca+cbfacc(V)∀t≥to,(iii)ḟa≤0∀t≥to,
will be used, being ca, cb, cc positive constants. If fa satisfies the above three conditions, then fa∈L∞ and consequently V∈L∞. One example of such function is
(5.7)fa(V)={(12)(V-Cbv)2ifV≥Cbv,0otherwise.
The reader is referenced to [2, page 309], [22, 23] for closely related functions. Its time derivative is:
(5.8)ḟa=∂fa∂VV̇,∂fa∂V={V-CbvifV≥Cbv,0otherwise,(5.9)⟹ḟa={(V-Cbv)ifV≥Cbv,0otherwise,
since V-Cbv is positive or zero for V≥Cbv, it can be multiplied by (5.12) without changing the order of the inequality:
(5.10)(V-Cbv)V̇≤-a1x12(V-Cbv)ifV≥Cbv,
substituting it into (5.9), it is obtained:
(5.11){ḟa≤-a1x12(V-Cbv)≤0ifV≥Cbv,ḟa=0otherwise.
Then, fa satisfies properties (5.6), as discussed at the following. From the definition (5.7) it follows that fa≥0, V≤Cbv+2fa, so that properties (5.6)(i) and (5.6)(ii) are satisfied. From (5.11) it follows that property (5.6)(iii) is true, so that fa∈L∞. Since (5.6)(ii) is true, then V∈L∞. From (3.6) it follows that x1∈L∞, S∈L∞, and from (3.8) it follows that x2∈L∞. This completes the proof.If the control law (4.11) is applied to the plant (2.1), and the initial value of the function V is located inside the target region, then V remains inside the target region thereafter, as is proven at the following.Theorem 5.2 (boundedness of the Lyapunov function forV≤V(x(to))).
Consider the plant model  (2.1), subject to assumptions (Ai) to (Aiv), the tracking error  e, the desired output  yd and the function  S provided by  (2.4), (2.5), and  (3.8), respectively. If condition  (3.16) is fulfilled, the controller  (4.11) is applied and  V(x(to))≤Cbv, then  V(x(t))≤Cbvforallt≥to,  x1∈L∞,  x2∈L∞.Proof.
The nature ofV̇ for V=Cbv must be examined considering each of the cases S=0 and S≠0 separately. From Theorem 3.4 it follows that if the control law (3.13) is used, S≠0 and V=Cbv, then V̇≤-a1x12. To show that this expression is also valid for the case when S=0 and V=Cbv, (3.9) is used. From (3.9) it follows that V̇≤-a1x12  if S=0, regardless the value of V. Consequently, if the control law (4.11) is used, S=0 and V=Cbv, then V̇≤-a1x12. So far, it has been shown that if the control law (4.11) is used and V=Cbv, then
(5.12)V̇≤-c1a1x12,
regardless the value of S. This implies that if V(x(t1))=Cbv for any t1≥to, then V(x(t1+δt))≤V(x(t1)) for a small value of δt. Consequently, V(x(t2))≤V(x(t1))forallt2≥t1. Moreover, if V(x(to))≤Cbv, then V(x(t))≤Cbvforallt≥to. From (3.6) it follows that x1∈L∞, S∈L∞, and from (3.8) it follows that x2∈L∞. This completes the proof.


## 6. Convergence of the Tracking Error

In this section it is proven that if the controller (4.11) is applied to the plant model (2.1), the tracking error e(t) converges to a residual set Ωe={e∈ℝ:|e|≤Cbe}.Equation (5.11) will be arranged into a single expression. Using(6.1)fb={V-CbvifV≥Cbv,0otherwise.
Equation (5.11) can be rewritten as
(6.2){ḟa≤-a1x12fb≤0ifV≥Cbv,ḟa=0=-a1x12fb,otherwise⟹ḟa≤-a1x12fb.
A difficulty is that dfb/dt is not continuous, as ∂fb/∂V is discontinuous at V=Cbv. Consequently, the Barbalat’s Lemma can not be applied on fb. One remedy is to express (6.2) in terms of a new function whose first derivative with respect to V is continuous. One instance of that function is
(6.3)fc={(V-Cbv)2ifV≥Cbv,0otherwise.
The reader is referenced to [2, page 309], [22, 24, 25] for closely related functions. To express (6.2) in terms of fc, the following property is needed.Proposition 6.1.
The function  (6.3) satisfies
(6.4)fb≥fc,
the proof is presented in Appendix D.Substituting (6.4) into (6.2), it is obtained(6.5)ḟa≤-a1x12fb≤-a1x12fc≤0.
Arranging and integrating, as in [26, 27], it is obtained(6.6)a1∫totx12fcdτ≤fa(Vo)-fa(V),Vo=V(x(to)),fa(V)+a1∫totx12fcdτ≤fa(Vo),
since fc≥0, then x12fc∈L1. As in [21], (·)∈L1 is used to indicate that ∫tot|(·)|dτ is bounded.Proposition 6.2.
The termx12fc satisfies:  x12fc∈L∞,  d(x12fc)/dt∈L∞. The proof is presented in  Appendix E.In view of the above proposition and applying the Barbalat’s Lemma (cf. [21, page 76], [26–28]), it is obtained
(6.7)lim⁡t→∞x12fc=0⟹lim⁡t→∞x12=0orlim⁡t→∞fc=0.
If fc is the one that converges towards zero, it follows from (6.3) that V converges to Ωv, where Ωv={V(x)∈ℝ:V(x)≤Cbv}. From Proposition 4.1 it follows that e converges to Ωe, where Ωe={e∈ℝ:|e|≤Cbe}. The results can be summarized as follows.Theorem 6.3 (convergence of the tracking error).
Consider the plant model  (2.1), subject to assumptions (Ai) to (Aiv), the tracking error  e, the desired output  yd and the function  S provided by  (2.4), (2.5), and (3.8), respectively. If condition  (3.16) is fulfilled and the controller  (4.11) is applied, then  e converges to  Ωe, where  Ωe={e∈ℝ:|e|≤Cbe}, being  Cbe a positive user defined constant.


## 7. Simulation Example

The aim of the following example is to show that the controller (4.11) achieves the benefits mentioned in Theorem 6.3. To that end, assumptions (Ai) to (Aiv) of Section 2 and condition (3.16) have to fulfill. Considerations of Section 4 are taken into account. Consider the plant:
(7.1)ÿ=-ẏ-y+u+d,y(to)=0.97,ẏ(to)=0,u∈{-1.2,1.2},d=0.1(1+0.1sin⁡(2π3t)),|d|≤0.11⟹μo=0.11,
which is an example of the plant (2.1) with a1=1, ao=1, b=1, umn=-1.2, umx=1.2, μo=0.11. Therefore, assumptions (Ai) to (Aiv) stated in Section 2 are fulfilled. Since ao>(1/4)a12, the linear part of the plant is underdamped. The aim is that y converges towards the value r=cos(0.5t) with an accuracy of 0.05. Thus, set Cbe=0.05 and yd is defined by means of the second order system:(7.2)yd=λr2(p+λr)2r,
which is a special case of (2.1). V(x(to))≤Cbv is fulfilled choosing yd(to)=1 and ẏd(to)=0. By means of simulation it is possible to check that yd, ẏd, ÿd, and λr=12 satisfy condition (3.16), as shown in Figure 1. The chosen values of λr, yd(to), ẏd(to) imply that yd≈rforallt≥to. Since a1=1, the signal S defined in (3.8) is then S=a1e+ė=e+ė. The closed loop behavior of y, e, u is shown in Figure 2, whereas the state plane is shown in Figure 3. The state plane confirms that V(x(to))≤Cbv, so that V(x(t))≤Cbvforallt≥to and u only commutes when V=Cbv. The figure of the control input u and the state plane indicate that large commutation rate is avoided. This confirms the importance of choosing yd(to), ẏd(to) such that V(x(to))≤Cbv.Figure 1
Example 1, top horizontal line:umx-μo/b, bottom horizontal line: umn+μo/b, thin line: (1/b)(ÿd+a1ẏd+aoyd).Example 1, (a) outputy (continuous line) and desired output yd (dashed line); (b) tracking error e(t); (c) control input u.
(a)(b)(c)Figure 3
Example 1, the solid line representsV=Cbv.


## 8. Discussion

The discussion presented here is only valid for the plant model (2.1), which is a model second order with known constant coefficients and an additive disturbance. The time derivative of the function V defined in (3.6) results in a negative semidefinite term, so that it is necessary to use Lyapunov-like function method to prove the convergence of the tracking error. Hereafter, some conclusions based on both stability proofs and numerical simulations are stated. In the classical relay feedback method [10, 13] the commutation of the control input is function of the tracking error, but it does not take into account the time derivative of the tracking error.The developed condition (3.16) indicates wether a given desired output yd is suitable to achieve the convergence of the tracking error e to a residual set of user defined size, that is, Ωe, for the plant (2.1). For any instance of the plant (2.1), one may modify the coefficients of the reference model (2.5) to satisfy condition (3.16), by means of trial and error. This procedure does not involve the output y, nor the input u, and is previous to the implementation of the controller.It is important to set the initial values of the desired output and its time derivative such thatV(x(to))≤Cbv. This implies that the Lyapunov function V is always located inside the target region Ωv={V∈ℝ:V≤Cbv}, so that it avoids convergence towards this region, and consequently large commutation rate of the control input are avoided.The contribution of the scheme with respect to classical relay feedback control based on hysteresis is to ensure the convergence of the tracking errore(t) to a residual set whose size is user defined. The main contribution with respect to closely related control based on the direct Lyapunov method is that the condition that the desired output yd has to fulfill in order to achieve tracking is defined.


## 9. Conclusions

The controller achieves the convergence of the tracking error to a residual set of user defined size if the desired output satisfies the formulated condition. This condition uses the upper bound of the additive disturbance, as in a basic nonadaptive robust controller. It allows us to develop a rigorous proof of the tracking error convergence to a residual set that is user defined, by means of the Lyapunov-like function. If the initial values of the desired output and its time derivative are properly defined, the Lyapunov function is located inside a target region at initial time and thereafter. In this case, the control input only commutes when the Lyapunov function reaches the boundary.



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*Source: 527413-2012-03-27.xml*