# Asymptotic Tracking Control for a Class of Nonlinear Systems with Unknown Failures of Hysteretic Actuators

Authors: Chuanjing Hou; Lisheng Hu; Yingwei Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

An adaptive failure compensation controller for a class of nonlinear systems preceded by hysteretic actuators is proposed in this paper. Three types of high-gain functions are constructed to counteract the effects of the hysteresis, bounded modeling errors, and bounded disturbances. It is shown that the proposed controller not only ensures bounded signals and asymptotic tracking but also avoids possible chattering, despite the presence of unknown hysteretic actuator failures. Simulation results verify the desired failure compensation performance.


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## Body

## 1. Introduction

The hysteresis phenomenon occurs in all the smart material-based actuators and sensors. With such nonlinearity in control systems, it may lead to undesirable inaccuracies or oscillations. On the other hand, actuator failures seem inevitable in practical systems. Thus, failure compensation of hysteretic actuators is an important and challenging problem. Adaptive failure compensation has received great attention in recent years [1–12]. However, available results based on adaptive approaches to address hysteretic actuator failures are very limited [13–16].To address such a challenging problem, it is important to find an appropriate model for the hysteresis. As mentioned in [17], several models were proposed such as Duhen model, Preisach model, Prandtl-Ishlinskii hysteresis operator, Bouc-Wen differential model [17, 18]. The Bouc-Wen differential model is one of the most widely accepted models of the hysteresis. Actually, it can be shown that the hysteresis model presented in [13–16] is a special case of the Bouc-Wen hysteresis model.In [13], an adaptive failure compensation scheme for a class of nonlinear systems was studied, where control gains are constants. To avoid possible chattering, the sign (·) functions were involved in the backstepping controller. When the control gains are nonlinear functions of system states, the effects of the actuator hysteresis can no longer be assumed bounded as in [13]. How to handle such effects is a challenging issue especially when there are possible actuator failures. In [14, 15], the idea is to separate such effects into two parts by applying Young’s inequality. It is noted that the control methods in [14, 15] are very complicated and the tracking error cannot asymptotically converge to zero but to the so-called predefined bound. Furthermore, the existence of the estimators increases the order of the closed-loop system.In this paper, we develop a backstepping [19, 20] adaptive compensation controller for a class of nonlinear systems preceded by hysteretic actuators described by Bouc-Wen model. Three types of high-gain functions are incorporated into the controller to counteract the effects of the hysteresis, bounded modeling errors, and bounded disturbances, respectively. In our design, the sign (·) function and a priori knowledge on the bounds of control gains are not required. Besides showing the stability of the closed-loop system, the tracking error is also ensured to achieve zero asymptotically.The rest of the paper is outlined as follows. In Section2, the control problem is formulated. In Section 3, a robust adaptive compensation scheme with high-gain functions is proposed. In Section 4, the stability analysis is presented. Simulation results are presented to show the proposed scheme is effective in Section 5. Finally, this paper is concluded in Section 6.


## 2. Problem Formulation

Consider a class of nonlinear systems in the following form [15]:(1)x˙i=xi+1+aTφi(x-i)(i=1,2,…,ρ-1)⋮x˙ρ=φ0(x,ξ)+aTφρ(x-ρ)+∑j=1mbjβj(x)uj+d(t)+η(x,t)ξ˙=ψ(x,ξ)+aTϕ(x,ξ),y=x1,where ρ is the relative degree of the system; uj∈R, j=1,2,…,m are the inputs whose actuators may fail during system operation; x-i=[x1,x2,…,xi]T(i=1,…,n-1), x=[x1,x2,…,xn]T are the state vectors; a=[a1,…,aq]Tare unknown constant parameters; bj∈R(j=1,2,…,m) are unknown constant parameters with known signs; ψ0∈R, φj∈Rρ(j=1,…,ρ), βj(x)≠0∈R(j=1,…,m), ψ∈Rn-ρ, and ϕ∈R(n-ρ)×ρ are known smooth functions; d(t)≤D- denotes bounded disturbance; and η(x,t) is an unknown nonlinear function representing system modeling errors. There exists a known function δ(x,t) such that η(x,t)≤δ(x,t).The hysteresis nonlinearity can be described by Bouc-Wen model [17, 18]. Consider(2)vi=μikiuci+1-μikiζi=μi1uci+μi2ζiζ˙i=u˙ci-χi1u˙ciζiLi-1-χi2u˙ciζiLi,where 0<μi<1(i=1,…,m) are weighting parameters, kii=1,…,m are stiffness coefficients, μi1=μiki, μi2=1-μikii=1,…,m are constants; uci is the input of the ith (i=1,…,m) actuator, χi1, χi2i=1,…,m describe the shape and amplitude of the ith hysteresis, respectively, Li governs the smoothness of the transition from initial slope to the slope of the asymptote, and χi1≥χi2, Li≥1(i=1,2…,m). By Lemma  1 in [17], ζi(i=1,2…,m) are bounded.The actuator failure can be modeled as [8, 9, 13–15](3)ui=ρiμi1vi+di+uki,∀t≥tiF,ρiuki=0,i=1,…,m,where ρi∈0,1, uki, and tiF are all unknown constants and di=ρiμi2ζi(i=1,2…,m) are bounded. For different values of ρi, three types of failures are included:(1)
ρi=1, where the actuator works normally; namely, ui=uci, which is regarded as a failure-free actuator;(2)
0<ρi<1; it implies ui=ρiuci; the ith actuator is called partial loss of effectiveness (PLOE);(3)
ρi=0; it indicates ui=uki; the ith actuator is called total loss of effectiveness (TLOE).Remark 1.
The values ofρi can change only from ρi=1 to some values with 0≤ρi<1. This means that possible changes from normal to any one of the failure cases are unidirectional. The uniqueness of tiF indicates that a failure occurs only once on the ith actuator.
Substituting (2), (3) into (1), we have(4)x˙i=xi+1+aTφix-ii=1,2,…,ρ-1.⋮x˙ρ=φ0(x,ξ)+aTφρ(x-ρ)+∑j=1mbjβj(x)(ρjμj1ucj+ukj+dj)hhhh+d(t)+η(x,t)ξ˙=ψ(x,ξ)+aTϕ(x,ξ)y=x1.
To derive a suitable adaptive control scheme, the following assumptions are made.Assumption 2.
When TOLE type of actuator failures up tom-1, the remaining actuators can still achieve a desired control objective.Remark 3.
Note that all actuators are allowed to have partial loss of effectiveness simultaneously.Assumption 4.
The zero dynamics ofξ˙=ψ(x,ξ)+aTϕ(x,ξ) is input to state stable with respect to x as its input.
LetT0=0. Suppose that there are pk(0≤pk≤m) actuators failing at time instants tk, k=1,2,…,q, and t0<t1<⋯tq<∞. In other words, all actuators work normally in time interval t0,t1 and no new failure will occur after time tq. Let the set QjT denote the actuators of total failure in interval tj,tj+1 and use the set Q-jT to represent other normal actuators. It can be concluded that QjT∪Q-jT={1,2,…,m}.Our objective is to design a control lawuci(t) for the nonlinear systems with p unknown actuator failures, when p changes at time instants tk, k=1,2,…,q, such that the output y(t) asymptotically tracks a given reference signal yr(t) with up to ρth order derivatives bounded and that all closed-loop signals are bounded.


## 3. Adaptive Compensation Control Schemes

The backstepping technique [19, 20] is applied to derive an adaptive actuator failure compensation controller. The following change of coordinates is required:(5)z1=x1-yrzi=xi-yri-1-αi-1i=2,…,ρ,where z1 is the tracking error and αρ is the ρth stabilizing function. To illustrate the backstepping procedures, only the last step of the design is elaborated in details.Step i. Consider i=1,…,ρ-1; the ith stabilizing function αi, the ith regressor ωi, and the ith tuning function τi are chosen as(6)α1=-c1z1-φ1(x1)Ta^αi=-zi-1-cizi-ωiTa^+∑k=1i-1∂αi-1∂xkxk+1+∂αi-1∂yr(k-1)yr(k)+∂αi-1∂a^Γτi+∑k=2i-1∂αk-1∂a^Γωizk,(i=2,…,ρ-1)ωi=φi-∑k=1i-1∂αi-1∂xkφki=1,…,ρ-1τi=τi-1+ωizii=1,…,ρ-1,where ci(i=1,…,ρ-1) are positive design parameters and a^∈Rq is estimator of the unknown vector a.Step
ρ
. From (6), the derivative of zρ is(7)z˙ρ=φ0x,ξ+φρ(x-ρ)Ta+∑i=1mbiβixρiμi1uci+uki+di+ηx,t+d(t)-∑k=1ρ-1∂αρ-1∂xkxk+1+φkTa-∂αρ-1∂a^a^.-∑k=1ρ-1∂αρ-1∂yrk-1yrk-yrρ.If the effects of hysteresis are treated as disturbances, it should be noted that the disturbances in (7) can be classified into three types: (1) d(t) is bounded by an unknown constant D; (2) η(x,t) is bounded by a known function; (3) ∑j=1mbjβjxdi cannot be bounded by any known function but βj(x)(j=1,…,m) are known and bjdj(j=1,2…,m) are bounded. According to their different characteristics, three high-gain functions will be proposed to counteract the effects of the disturbances. At the final step, the stabilizing function αρ is given by(8)αρ=-φ0-ωρTa^-zρ-1-cρzρ+yrρ+∑k=1ρ-1∂αρ-1∂xkxk+1+∂αρ-1∂yrk-1yrk+∂αρ-1∂a^Γτρ-eftzρ-eftδx2zρ-eft∑k=1mβi2xzρ,where (9)ωρ=φρ-∑k=1ρ-1∂αρ-1∂xkφkτρ=τρ-1+ωρzρ.f(t) is bounded and limt→∞∫0te-f(τ)dτ exists, and cρ is a positive design parameter.Let(10)ω=αρ,β1,…,βmT.The control law and parameter update laws are obtained as follows(11)uci=signbi1βixk^Tω,(i=1,…,m)(12)a^.=Γτρ(13)k^.=-Γkωzρ,where k^=k^1,k^2TT, k^2=k^21,k^22T,…,k^2mTT, k^ and a^ are the estimates of k and a, respectively, and Γ,Γk are positive definite matrices chosen by users. If k is a desired constant vector which can be chosen to satisfy(14)∑i=1,i∉Q-jTmbiρikTw=αρ-∑j∈QjTbjβjukj,this gives(15)k1=1∑i=1,i∉Q-jTmbiρi,k2,j=-bjukj∑i=1,i∉Q-jTmbiρi.Substituting (11)–(13) and (8) into (7), we have(16)z˙ρ=-zρ-1-cρzρ+ωρTa~+∑i∈Q-jTbiρiμi1k~Tω-eftzρ-dt-eftδx2zρ-ηx,t-eft∑k=1mβk2xzρ-∑k=1mβkxbkdk.


## 4. Stability Analysis

To prepare for the stability analysis, we rewrite the error system as(17)z˙=Azz,tz+Wθz,tTa~+Dteρ,where the system matrices Az(z,t), Wθ(z,t), D(t), and eρ are given by(18)Azz,t=-c110⋯⋯01-c21+σ23σ24⋯σ2ρ⋮-1-σ23⋱⋱⋱⋮⋮-σ24⋱⋱⋱σρ-2ρ⋮⋮⋱⋱⋱1+σρ-1,ρ0-σ2ρ⋯-σρ-2,ρ-1-σρ-1,ρ-cρWεz,t=ω1ω2,…,ωρT∈Rρ.Wθz,t=Wεz,tωT-ρ^y˙r+α-1e1e1T∈Rρ×ρDt=∑i∈Q-jTbiρiμi1k~Tω-eftzρ-dt-eftδx2zρ-ηx,teρ=0,…,0,1T∈Rρσjk=-∂αj-1∂a^Γωk.The closed-loop system has the following desired properties.Theorem 5.
With the ISS of the zero dynamics of system (1) and m hysteretic actuators modeled in (2) with possible unknown failures by (3), the controller (11) with the adaptive laws (12) and (13) ensures the boundedness of the closed-loop signals and the asymptotic output tracking: limt→∞(y-yr)→0.Proof.
For each time intervaltk,tk+1, k=0,1,2,…,q, we have a Lyapunov function V in the following form:(19)V=12∑i=1ρzi2+12a~TΓa~+∑i∈Q-jTρiμi1bi2k~jTΓkk~j.Taking the derivative of (19) yields(20)V˙=∑i=1ρziz˙i-a~TΓa^.-∑i∈Q-jTρiμi1bik~jTΓkk^.j.Substituting (11)–(13) and (8) into (7), we have(21)V˙=-∑i=1ρcizi2-eftzρ2-dtzρ-eftδx2zρ2-ηx,tzρ-eft∑k=1mβk2xzρ2-∑k=1mβkxbkdkzρ.From d(t)≤D-, η(x,t)≤δx,t, and bounded dii=1,2,…,m, we get(22)V˙≤-∑i=1ρcizi2-eftzρ2-D-zρ-eftδx2zρ2-δxzρ-ef(t)∑k=1mβk2xzρ2-∑k=1mβkxbkdkzρ≤-∑i=1ρcizi2+D-24e-f(t)+14e-f(t)+14∑k=1mbk2dk2e-f(t)=-∑i=1ρcizi2+∑k=1mbk2dk2+D-2+14e-f(t)=-∑i=1ρcizi2+D0e-ft,where(23)D0=∑k=1mbk2dk2+D-2+14.
From (22), we conclude that V(t)∈L∞, t∈t0,t1, so that zi(i=1,…,ρ), z˙i(i=1,…,ρ), and a~,k~ are bounded for t∈t0,t1. It follows from (8) and (10) that αρ and ω are bounded. Therefore, all closed-loop signals are bounded for t0,t1. In order to prove the asymptotic tracking, considering the last time interval tq,∞. From (22), we can obtain(24)ci∫0tzi2τdτ≤V0-Vt+D0∫0te-fτdτ.
Because of the boundedness ofD0∫0te-fτdτ, we have zi∈L2, (i=1,…,ρ). By Barbalat’s lemma, limt→∞(zi(t))→0(i=1,…,ρ) can be obtained. This completes the proof.


## 5. Simulations

We consider a second order nonlinear [15] system with two inputs described as(25)x˙1=x2+φ1Tx1θx˙2=φ2Txθ+∑i=12biβixui+ηx,t,where φ1x1=1+x12, φ2x=1+sinx1, β1x=1.9+0.1sinx1, β2(x)=ex1, θ=2, b1=b2=1, b1, b2 are unknown constants, u1, u2 are the outputs of two hysteretic actuators, x1,x2 are the states, and η(x)=0.1x12(1+sint)(1+sinx2) is bounded by δ(x)=0.4x12. The reference signal is set as yr(t)=sin(3t). The backlash-like hysteresis is described by (2) with parameters μ11=μ21=μ12=μ22=1, χ11=χ21=1, χ12=χ22=0, and L1=L2=1. The high-gain function is chosen as f(t)=e2.5arctan(t). For simulation, we consider three actuator failure cases.Case 1.
There are no actuator failures.
By Theorem5, we can obtain the actual control law and the update laws. The initial conditions are set as follows:(26)a^=1.5,Γ=0.0005,k^=0.4,0,0T,Γk=0.0001*I3x0=0.3,0T,c1=60,c2=60,where I3 is the 3rd order identity matrix.The simulation results including outputy(t), reference output yr(t), and tracking error e(t) are shown in Figure 1; the actuators outputs u1,u2 are shown in Figure 2. The system responses are as expected. At the beginning, there is a transient response in tracking errors. But, as time goes on, the tracking errors become smaller and ultimately vanish. The proposed controller guarantees that asymptotic tracking is achieved.Figure 1
Outputy, reference yr, and error e(t).Figure 2
u
1 and u2.Case 2.
Actuatoru1 is stuck at u1=45 from t=3 s, thus undergoing a TLOE type of failure. By Theorem 5, we can obtain the actual control law and the update laws. The initial conditions are set as follows.
The other parameters are the same as those in Case1.The simulation results including outputy(t), reference output yr(t), and tracking error et are shown in Figure 3; the actuators outputs u1,u2 are shown in Figure 4. The system responses are as expected. When one of the actuators fails, there is a transient response in tracking errors. But, as time goes on, the tracking errors become smaller and ultimately vanish. The proposed controller guarantees that asymptotic tracking is achieved.Figure 3
Outputy, reference yr, and error e(t).Figure 4
u
1 and u2.Case 3.
Actuatoru2 is stuck at u2=40 from t=3 s and actuator u1 loses 60% from t=6 s. Thus, u1 undergoes a PLOE type of failure while u2 is a TLOE type of failure.
The other parameters are the same as those in Case1.The simulation results including outputy(t), reference output yr(t), and tracking error et are shown in Figure 5; the actuators outputs u1,u2 are shown in Figure 6. The system responses are as expected. When the actuators fail, there is a transient response in tracking errors. But, as time goes on, the tracking errors become smaller and ultimately vanish. The proposed controller guarantees that asymptotic tracking is achieved.Figure 5
Outputy, reference yr, and error e(t).Figure 6
u
1 and u2.


## 6. Conclusions

This paper presents an adaptive failure compensation controller for a class of uncertain nonlinear systems dominated by the hysteresis actuator nonlinearity. We propose three types of high-gain functions to deal with the unknown bounded disturbances, unknown modeling errors, and unknown actuator failures. It has been shown that the tracking errors can converge to zero asymptotically while all the closed-loop signals remain bounded. Furthermore, the proposed scheme can avoid possible chattering. Simulation results illustrate the effectiveness of our proposed scheme.



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