# Adaptive Control for a Class of Nonlinear System with Redistributed Models

Authors: Haisen Ke; Jiang Li
Journal: Journal of Control Science and Engineering
 (2012)

---

## Abstract

Multiple model adaptive control has been investigated extensively during the last ten years in which the “switching” or “switching and tuning” have emerged as the mainly approaches. It is the “switching” that can improve the transient performance to some extent and also make it difficult to analyze the stability of the system with multiple models adaptive controller. Towards this goal, this paper develops a novel multiple models adaptive controller for a class of nonlinear system in parameter-strict-feedback form which not only improves the transient performance significantly, but also guarantees the stability of all the states of the closed-loop system. A simulation example is proposed to illustrate the effectiveness of the developed multiple models adaptive controller.


---

## Body

## 1. Introductions

The multiple model adaptive control was introduced to cope with the large parametric uncertainty [1] which always results in large and oscillatory responses or even instable when using the classical adaptive control methods. The multiple models adaptive control [1–7] employing both fixed model and adaptive model have been used to identify the characteristics of the plants, and numerous methods are currently available for controlling such plant satisfactorily. However, the methods mainly focus on the linear time invariant plant [1, 2, 4–6]. The multiple models adaptive controller for nonlinear system is firstly considered in [8], which uses a direct parameter update law to guarantee the stability of the closed-loop system. Then, Ciliz and Cezayirli [9] proposes a different nonlinear multiple models adaptive control which require the condition of persistence of excitation, so that the unknown parameter can be evaluated at the very beginning. Recently, an indirect multiple models adaptive control was developed in [7] which also demonstrated the global asymptotic stability of the closed-loop switching system.As illustrated in the literature that the “switching” (to the closest model) based on the index of performance results in fast response, and tuning (from the closet model) improves the identification and control errors on a slower time scale, which have the assumption that there are abundant models available. Otherwise, the results may be improved less if the number of the identification models is not adequate to achieve the satisfactory response.In this paper, a novel multiple models adaptive control was considered for the nonlinear system in parameter-strict-feedback form, which retains the advantages of the multiple models adaptive controller, meanwhile facilitate the procedure to analyze and synthesize the controller of the closed-loop system. The approach developed here in which the multiple models adaptive controller are used to play a significantly larger role in the decision making role, results in substantial improvement in performance. Besides, we also reduce the number of the identification models by redistributing the candidate models even as the system is in operation.


## 2. Problem Formulation

Consider the multiple models adaptive control of the following nonlinear parameter-strict-feedback (PSF) system:(1)ẋi=xi+1+φiT(x̅i)θ,1≤i≤n-1,ẋn=β(x)u+φnT(x)θ,y=x1,
where x̅i=[x1,…,xi]T∈Ri and x∈Rn are the state, u∈R is the control input, θ∈Rp is an unknown parameter vector belonging to a known compact set S. The functions φi(x̅i) and β(x) are known smooth functions with β(x)≠0, forallx∈Rn. The focus of this paper is to improve the transient performance in the presence of large parametric uncertainties.One easly way to improve the transient performance may be choosing sufficiently large high-frequency parameters in the conventional backstepping adaptive control design. Unfortunately, the control efforts can also be very large simultaneously [7]. Alternately, in cope with these difficulties, “switching” or “switching and tuning” have emerged as the leading methods during the last decade.


## 3. Multiple Models Adaptive Controller Design

In order to ensure the stability and transient performance of the system with larger parametric uncertainty, and consequently the boundedness of the statex(t), the well-established results from the classical adaptive control cannot be used directly. Our multiple models adaptive controller contains N parallel operating identification models on which the control law and the adaptive law are based. For improving the transient performance, it is necessary to distribute the initial estimate values of the unknown parameter {θ̂j(0)}j=1N uniformly in the compact set S to which the unknown parameter θ belongs. Therefore, at least one θ̂j(0) is close to θ, consequently there must exists one or more identification models in its neighborhood. Since adaptive control can perform well when parametric errors are small, it is naturally that the controller developed on the jth identification model can stabilize the system with satisfactorily transient performance.


### 3.1. Multiple Identification Models

We will run in parallelN identification models with the same structure which take the different initial parameter estimate values {θ̂j(0)}j=1N uniformly distributed in the compact set S to which the unknown parameter belongs. We first introduce the filters as follows:
(2)ξ̇0=(A0-λΞ(x)ΞT(x)P)(ξ0-x)+f(x,u),ξ0∈Rn,(3)ξ̇=(A0-λΞ(x)ΞT(x)P)ξ+Ξ(x),ξ∈Rn×p,
where(4)f(x,u)=[x2…xnβ(x)u]T,Ξ(x)=[φ1(x1)…φn(x)]T.λ>0, and A0 is a Hurwitz matrix such that the Lyapunov equation: PA0+A0TP=-I has a positive definite solution P.Define(5)ẽ=x-ξ0-ξθ,(6)ej=x-ξ0-ξθ̂j,j=1,…,N,(7)θ̃j=θ-θ̂j,j=1,…,N.It can be derived from (1)–(7) that
(8)ẽ̇=(A0-λΞ(x)ΞT(x)P)ẽ,(9)ej=ξθ̃j+ẽ,j=1,…,N.Sinceẽ converges to zero exponentially, (9) are called identification error equations.


### 3.2. Controller Design

The controller design involvesN models at total and is developed as [10], which can guarantee the asymptotic tracking when there is not identification error and avoid the finite time escape phenomenon when there exists bounded identification error. Now, the first identification model’s adaptive controller is given by
(10)u1=[α1,n(x,θ̂1,yr,…,yrn)]β(x),
where yr is the reference signal to be tracked and α1,n can be recursively designed by
(11)zi=xi-α1,i-1(x1,…,xi-1,θ̂1,yr,…yri-1),(12)α1,i=-zi-1-c1,izi-w1,iTθ̂1+yri-s1,izi+∑k=1i-1(∂α1,i-1∂xkxk+1+∂α1,i-1∂yrk-1yrk),(13)w1,i(x1,…xi,θ̂1,yr,…,yri-1)=φi-∑k=1i-1∂α1,i-1∂xkφk,(14)s1,i=k1,i|w1,i|2+g1,i|∂α1,i-1∂θ̂1|2,
We choose
(15)V1=12∑i=1nzi2.The time derivative ofV1, computed with (10)–(14), is given by
(16)V̇1=-∑i=1nc1,izi2+∑i=1n(w1,iTθ̃1-∂α1,i-1∂θ̂1θ̂̇1)zi-∑i=1n(k1,i|w1,i|2+g1,i|∂α1,i-1∂θ̂1|2)zi2≤-∑i=1nc1,izi2-∑i=1nk1,i|w1,izi-12k1,iθ̃1|2-∑i=1ng1,i|∂α1,i-1∂θ̂1zi-12g1,iθ̂̇1|2+∑i=1n14k1,i|θ̃1|2+∑i=1n14g1,i|θ̂̇1|2≤-∑i=1nc1,izi2+∑i=1n14k1,i|θ̃1|2+∑i=1n14g1,i|θ̂̇1|2,
with c1,i, k1,i, g1,i being designed parameters. Equation (16) implies the boundedness of the states of zi, 1≤i≤n, and which in turn indicates the boundedness of the states of xi, 1≤i≤n and control u1 on the conditions of θ̃ and θ̂̇ are bounded which will be proved later. The rest of N-1 controllers can be designed and analyzed similarly which can also guarantee the boundedness of the states of zi, 1≤i≤n, and which in turn indicates the boundedness of the states of xi, 1≤i≤n and control uj, j=1,…,N.


### 3.3. Construction of Equivalent Control

In this section, the crucial point is that the transient performance can be improved significantly, and at the same time the switching between the identification models can be avoided. Besides, the information provided by all the identification models is to be utilized efficiently. For the complement of the goals mentioned, instead of using the estimate values of the model with the minimum of performance criterion to reinitial an adaptive controller, a convex combination of all theN models is used to generate the control of the plant as
(17)u=∑j=1Nγjuj,
and the adaptive update law as
(18)θ̂̇j=ΓξTej1+v|ξ|2,Γ=ΓT>0,v>0,j=1,…,N,
where γj are nonnegative values satisfying ∑j=1Nγj=1, and γj can be calculated from
(19)γj=(1/Jj)∑j=1N(1/Jj),
where Jj is the performance indices of the form:
(20)Jj(t)=αej2(t)+β∫t0tej2(τ)dτ,a≥0,β>0,
with t0 can be reset when the identification models is redistributed.


### 3.4. Redistribution of the Identification Models

In this section, the goal is that the transient performance can be improved significantly as far smaller numbers of the identification models as possible. As is illustrated in the literature, the classical adaptive control can cope with the control of linear time invariant system with unknown parameters and achieve satisfactory closed-loop objective only if the plant parametric uncertainty is small. So if the number of the identification models that can be used is abundantly large, the “switching” or “switching and tuning” scheme may act on satisfactorily. Otherwise, the multiple models adaptive control cannot work as expected when the numbers of identification models available is relatively smaller compared with the size of the uncertainty region. Inspired by the “switching” techniques [11–13], we consider the method in which the location of the identification models can be redistributed. From (8) and (9), it can be concluded that the ẽ=0 can be achieved by choosing the initial values of ξ0 and ξ as long as the initial state x0 is known or there exists T > 0 such that
(21)ej=ξθ̃j,j=1,…,N,t>T.
It is obviously that the errors ej and θ̃j, j=1,…,N are linearly related. This implies that the index of the performance Jj(t) is a quadratic function of the unknown parameter vector θ̃j. Since ξTξ is not negative definite, it follows that the performance indices of all the models are merely points on a time-varying quadratic surface, whose minimum corresponds to the plant indicating the mostly closet identification model Mj (corresponds to the parameter θ̃j). So we can redistribute the other (N-1) models Mk(k≠j) as
(22)θ̅k=JkJk+Jjθj+JjJk+Jjθk.
By introducing the minimum of interval time Tmin into our switching scheme to ensure a finite number of switching.


## 3.1. Multiple Identification Models

We will run in parallelN identification models with the same structure which take the different initial parameter estimate values {θ̂j(0)}j=1N uniformly distributed in the compact set S to which the unknown parameter belongs. We first introduce the filters as follows:
(2)ξ̇0=(A0-λΞ(x)ΞT(x)P)(ξ0-x)+f(x,u),ξ0∈Rn,(3)ξ̇=(A0-λΞ(x)ΞT(x)P)ξ+Ξ(x),ξ∈Rn×p,
where(4)f(x,u)=[x2…xnβ(x)u]T,Ξ(x)=[φ1(x1)…φn(x)]T.λ>0, and A0 is a Hurwitz matrix such that the Lyapunov equation: PA0+A0TP=-I has a positive definite solution P.Define(5)ẽ=x-ξ0-ξθ,(6)ej=x-ξ0-ξθ̂j,j=1,…,N,(7)θ̃j=θ-θ̂j,j=1,…,N.It can be derived from (1)–(7) that
(8)ẽ̇=(A0-λΞ(x)ΞT(x)P)ẽ,(9)ej=ξθ̃j+ẽ,j=1,…,N.Sinceẽ converges to zero exponentially, (9) are called identification error equations.


## 3.2. Controller Design

The controller design involvesN models at total and is developed as [10], which can guarantee the asymptotic tracking when there is not identification error and avoid the finite time escape phenomenon when there exists bounded identification error. Now, the first identification model’s adaptive controller is given by
(10)u1=[α1,n(x,θ̂1,yr,…,yrn)]β(x),
where yr is the reference signal to be tracked and α1,n can be recursively designed by
(11)zi=xi-α1,i-1(x1,…,xi-1,θ̂1,yr,…yri-1),(12)α1,i=-zi-1-c1,izi-w1,iTθ̂1+yri-s1,izi+∑k=1i-1(∂α1,i-1∂xkxk+1+∂α1,i-1∂yrk-1yrk),(13)w1,i(x1,…xi,θ̂1,yr,…,yri-1)=φi-∑k=1i-1∂α1,i-1∂xkφk,(14)s1,i=k1,i|w1,i|2+g1,i|∂α1,i-1∂θ̂1|2,
We choose
(15)V1=12∑i=1nzi2.The time derivative ofV1, computed with (10)–(14), is given by
(16)V̇1=-∑i=1nc1,izi2+∑i=1n(w1,iTθ̃1-∂α1,i-1∂θ̂1θ̂̇1)zi-∑i=1n(k1,i|w1,i|2+g1,i|∂α1,i-1∂θ̂1|2)zi2≤-∑i=1nc1,izi2-∑i=1nk1,i|w1,izi-12k1,iθ̃1|2-∑i=1ng1,i|∂α1,i-1∂θ̂1zi-12g1,iθ̂̇1|2+∑i=1n14k1,i|θ̃1|2+∑i=1n14g1,i|θ̂̇1|2≤-∑i=1nc1,izi2+∑i=1n14k1,i|θ̃1|2+∑i=1n14g1,i|θ̂̇1|2,
with c1,i, k1,i, g1,i being designed parameters. Equation (16) implies the boundedness of the states of zi, 1≤i≤n, and which in turn indicates the boundedness of the states of xi, 1≤i≤n and control u1 on the conditions of θ̃ and θ̂̇ are bounded which will be proved later. The rest of N-1 controllers can be designed and analyzed similarly which can also guarantee the boundedness of the states of zi, 1≤i≤n, and which in turn indicates the boundedness of the states of xi, 1≤i≤n and control uj, j=1,…,N.


## 3.3. Construction of Equivalent Control

In this section, the crucial point is that the transient performance can be improved significantly, and at the same time the switching between the identification models can be avoided. Besides, the information provided by all the identification models is to be utilized efficiently. For the complement of the goals mentioned, instead of using the estimate values of the model with the minimum of performance criterion to reinitial an adaptive controller, a convex combination of all theN models is used to generate the control of the plant as
(17)u=∑j=1Nγjuj,
and the adaptive update law as
(18)θ̂̇j=ΓξTej1+v|ξ|2,Γ=ΓT>0,v>0,j=1,…,N,
where γj are nonnegative values satisfying ∑j=1Nγj=1, and γj can be calculated from
(19)γj=(1/Jj)∑j=1N(1/Jj),
where Jj is the performance indices of the form:
(20)Jj(t)=αej2(t)+β∫t0tej2(τ)dτ,a≥0,β>0,
with t0 can be reset when the identification models is redistributed.


## 3.4. Redistribution of the Identification Models

In this section, the goal is that the transient performance can be improved significantly as far smaller numbers of the identification models as possible. As is illustrated in the literature, the classical adaptive control can cope with the control of linear time invariant system with unknown parameters and achieve satisfactory closed-loop objective only if the plant parametric uncertainty is small. So if the number of the identification models that can be used is abundantly large, the “switching” or “switching and tuning” scheme may act on satisfactorily. Otherwise, the multiple models adaptive control cannot work as expected when the numbers of identification models available is relatively smaller compared with the size of the uncertainty region. Inspired by the “switching” techniques [11–13], we consider the method in which the location of the identification models can be redistributed. From (8) and (9), it can be concluded that the ẽ=0 can be achieved by choosing the initial values of ξ0 and ξ as long as the initial state x0 is known or there exists T > 0 such that
(21)ej=ξθ̃j,j=1,…,N,t>T.
It is obviously that the errors ej and θ̃j, j=1,…,N are linearly related. This implies that the index of the performance Jj(t) is a quadratic function of the unknown parameter vector θ̃j. Since ξTξ is not negative definite, it follows that the performance indices of all the models are merely points on a time-varying quadratic surface, whose minimum corresponds to the plant indicating the mostly closet identification model Mj (corresponds to the parameter θ̃j). So we can redistribute the other (N-1) models Mk(k≠j) as
(22)θ̅k=JkJk+Jjθj+JjJk+Jjθk.
By introducing the minimum of interval time Tmin into our switching scheme to ensure a finite number of switching.


## 4. Stability Analysis

Theorem 1.
Suppose the multiple models adaptive controller (17) and adaptive law (18) presented in this paper is applied to system (1). Then, for all initial conditions, all closed-loop states are bounded on [0,∞), and asymptotic tracking can be achieved, that is, Limt→∞z(t)=0 or y(t)=yr(t) as t→∞.Proof.
Since allN models are identical structure and only with different initial estimate parameters, it follows that each controller acts on the system is only different from each other at the weight (each of the controllers can be designed with the same structure and designed parameters).
When we choose the whole candidate Lyapunov function as(23)V=12∑i=1nzi2.
It is obvious that (23) can be divided into
(24)V=12∑i=1nzi2=∑j=1Nγj12∑i=1nzi2=∑j=1NγjVj.
As illustrated by (16), each of the controllers can guarantee the boundedness of the states of zi, 1≤i≤n at its portion, which accompanied with the control (17), and weighting coefficient (19) can establish the boundedness of the states of zi, 1≤i≤n.
Next, we prove the controller (17) and adaptive law (18) can also guarantee the asymptotic tracking of the closed-loop system states. It can be computed from (3) that(25)ddt(ξPξT)=-ξξT-2λξPΞTΞPξT+ξPΞT+ΞPξT=-ξξT-2λ(ΞPξT-12λI)T(ΞPξT-12λI)+12λ,
which shows ξ is bounded regardless of the state x. Let Vj=(θ̃jTΓ-1θ̃j+ẽTẽ)/2, it can be derived that
(26)V̇j=-ejT(ej-ẽ)1+v|ξ|2+ẽT(A0-λΞ(x)ΞT(x)P)ẽ≤-34ejTej1+v|ξ|2,
without loss of generality, we can design the parameter satisfies A0-λΞ(x)ΞT(x)P>I, Iis a unit matrix. Therefore, ẽ, θ̃j, j=1,…,N are all bounded, which companied with the boundedness of ξ, further yields from ej=ξθ̃j+ẽ that ej is bounded. It can be also concluded from (26) that ej is squarely integrable on [0,∞). Furthermore, we can also conclude from (18) that θ̂̇j is bounded, which can accomplish the assumption that it is bounded. We can now give the asymptotically tracking control analysis.
The time derivate of identification error is given by(27)ėj=(A0-λΞΞTP)ej+Ξθ̃j-ξθ̂̇j.
Due to the boundedness of all the closed-loop system states ėj, ė̇j, j=1,…,N are also bounded, so by Barbalat’s lemma, we must have Limt→∞ej(t)=0 and since Limt→∞∫t1tėj(τ)dτ=limt→∞ej(t)-ej(t1)<∞, we further have Limt→∞ėj(t)=0. Then, it can be concluded from (18) that Limt→∞θ̂̇j=0 which accompanied with (27) implies Limt→∞Ξθ̃j=0 and in turn leads to
(28)Limt→∞Nj(z,θ̂,yr)Ξθ̃j=0,
where
(29)Nj(z,θ̂j,yr)=[100-∂αj,1∂x110⋮⋱0-∂αj,n-1∂x1-∂αj,n-1∂x21].
By direct calculating, the differentiation of zwith parametric controller uj can be described, in z coordination, by
(30)ż=-∑i=1ncj,izi+Nj(z,θ̂,yr)Ξθ̃j-∂αj,i-1∂θ̂jθ̂̇j-∑i=1n(kj,i|wj,i|2+gj,i|∂αj,i-1∂θ̂j|2)zi.
From (28), accompanied with Limt→∞θ̂̇j=0 and the designed parameters are all positive, it can be easily concluded that Limt→∞z(t)=0, and thus Limt→∞z1(t)=Limt→∞(y(t)-yr(t))=0. The proof is completed.


## 5. Simulation

Consider the following second-order nonlinear system:(31)ẋ1=x2+θ1x1+θ2x12,ẋ2=u,y(t)=x1(t),
where θ1∈[1,5] and θ2∈[1,40] are unknown parameters. The output y(t)=x1(t) is to asymptotically track the reference signal yr(t)=sin2t.In simulation, the parametric controller is developed as (10)–(14) and (17)–(19) with v=0, Γ=5, cj,1=cj,2=4, kj,1=kj,2=gj,2=0.1, j=1,…,N, α=β=1, Tmin is 5 units of time. Since in (31), the unknown parameter appears only in the first equation, the filter can be constructed as [1] to reduce filter dynamic order:(32)ξ̇0=-c(ξ0-x)+x2,ξ0∈R1,ξ̇=-cξ+[x1,x12],ξ∈R1×2,wherec=10.The unknown parameter is[θ1,θ2]=[4.4,38.5]; the number of the multiple identification models is N=4; for convenience to comparison with [7], the initial plant state is [x1(0),x2(0)]=[0.5,-10]; the same initial filter states are ξ0=0.5, ξ=[00], and the initial estimate parameters for model 1, model 2, model 3, and model 4 are θ̂1(0)=[1,1]T,  θ̂2(0)=[1,5]T, θ̂3(0)=[5,1]T, θ̂4(0)=[5,40]T, respectively. Figures 1–4 depict the simulation results.Figure 1
Outputy(t). dash-dotted line for the classical adaptive control, dashed line for the multiple model case (N=200) as in [7], and solid line for the multiple identification model developed in this paper.Figure 2
Controlu(t) on the time interval [0, 0.06]. dash-dotted line for the classical adaptive control, dashed line for the multiple model case (N=200) as in [7], and solid line for the multiple identification model developed in this paper.Figure 3
Controlu(t) on the time interval [0.6, 20]. dash-dotted line for the classical adaptive control, dashed line for the multiple model case (N=200) as in [7], and solid line for the multiple identification model in this paper.Figure 4
The redistribution trajectory of the identification models.These simulation results clearly showed that the multiple models adaptive controller presented in this paper guarantees the boundedness of all the states in the closed-loop system and achieves the asymptotic tracking of the output.Figure1 is the output y(t), which demonstrates that the multiple models adaptive controller developed in this paper has the similar property as shown in [7] and is significantly better than using the classical adaptive control. Figures 2 and 3 are the control inputs which show that the multiple model adaptive control can reduce the maximum control input dramatically. Besides, it seems to conclude that the multiple model adaptive control proposed in this paper has the similar property and so the trajectory is nearly to overlap from Figures 1–3 because the method in [7] uses more identification models than ours. Figure 4 is the trajectory of the redistribution of the identification models which can find the most suitable identification model and enhance the transient performance.Next, we can compare the approach presented in this paper with the method developed in [7] with the multiple identification models (N=200) is set to (N=4), which is the same identification models used in our approach. Figures 5–7 depict the simulation results.Figure 5
Outputy(t). dashed line for the multiple model case (N=4) as in [7], solid line for the multiple identification model developed in this paper.Figure 6
Controlu(t) on the time interval [0, 0.06]. dashed line for the multiple model case (N=4) in [7], and solid line for the multiple model developed in this paper.Figure 7
Controlu(t) on the time interval [0.06, 10]. dashed line for the multiple model case (N=4) in [7], and solid line for the multiple model developed in this paper.Figure5 is the output y(t) with the multiple models adaptive controller, which shows the approach developed in this paper is superior to the method presented in [7]. Figures 6 and 7 are the control inputs which show that the multiple model adaptive control developed in this paper has better properties than the method presented in [7] which has switching and larger control input.


## 6. Conclusions

In this paper, a novel multiple models adaptive controller was developed for a class of nonlinear systems. The multiple models technique was used to describe the most appropriate model at different environments. If the number of the identification models that can be used is abundantly large, the “switching” or “switching and tuning” scheme may act on satisfactorily. Otherwise, the multiple models adaptive control cannot work as expected when the number of identification models available is relatively small compared with the size of the uncertainty region. So we consider the method in which the location of the identification models can be redistributed. Unlike previous results, we do not require a switching scheme to guarantee the most appropriate model to be switched into the controller design which can simplify the analysis of the stability of the closed-loop system.



---

*Source: 409139-2012-06-06.xml*