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Journal of Control Science and Engineering
Volume 2012 (2012), Article ID 409139, 6 pages
http://dx.doi.org/10.1155/2012/409139
Research Article

Adaptive Control for a Class of Nonlinear System with Redistributed Models

College of Mechanical and Electrical Engineering, China Jiliang University, Zhejiang, Hangzhou 310018, China

Received 3 November 2011; Accepted 11 April 2012

Academic Editor: Chengyu Cao

Copyright © 2012 Haisen Ke and Jiang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [17] 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, 46]. 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: ̇𝑥𝑖=𝑥𝑖+1+𝝋𝑇𝑖𝐱𝑖𝜽,1𝑖𝑛1,̇𝑥𝑛=𝛽(𝐱)𝑢+𝝋𝑇𝑛(𝐱)𝜽,𝑦=𝑥1,(1) where 𝐱𝑖=[𝑥1,,𝑥𝑖]𝑇𝑅𝑖 and 𝐱𝑅𝑛 are the state, 𝑢𝑅 is the control input, 𝜽𝑅𝑝 is an unknown parameter vector belonging to a known compact set 𝑆. The functions 𝝋𝑖(𝐱𝑖) and 𝛽(𝐱) are known smooth functions with 𝛽(𝐱)0, forall𝐱𝑅𝑛. 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 state 𝐱(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 {𝜽𝑗(0)}𝑁𝑗=1 uniformly in the compact set S to which the unknown parameter 𝜽 belongs. Therefore, at least one 𝜽𝑗(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 parallel N identification models with the same structure which take the different initial parameter estimate values {𝜽𝑗(0)}𝑁𝑗=1 uniformly distributed in the compact set S to which the unknown parameter belongs. We first introduce the filters as follows: ̇𝝃0=𝐀0𝜆Ξ(𝐱)Ξ𝑇𝝃(𝐱)𝐏0𝐱+𝑓(𝐱,𝑢),𝝃0𝑅𝑛̇𝐀,(2)𝝃=0𝜆Ξ(𝐱)Ξ𝑇(𝐱)𝐏𝝃+Ξ(𝐱),𝝃𝑅𝑛×𝑝,(3) where𝑥𝑓(𝐱,𝑢)=2𝑥𝑛𝛽(𝐱)𝑢𝑇,𝜑Ξ(𝐱)=1𝑥1𝜑𝑛(𝐱)𝑇.(4)

𝜆>0, and 𝐀0 is a Hurwitz matrix such that the Lyapunov equation: 𝐏𝐀0+𝐀𝑇0𝐏=𝐈 has a positive definite solution P.

Define ̃𝐞=𝐱𝝃0𝐞𝝃𝜽,(5)𝑗=𝐱𝝃0𝜽𝝃𝑗𝜽,𝑗=1,,𝑁,(6)𝑗𝜽=𝜽𝑗,𝑗=1,,𝑁.(7)

It can be derived from (1)–(7) that ̇̃𝐀𝐞=0𝜆Ξ(𝐱)Ξ𝑇̃𝐞(𝐱)𝐏𝐞,(8)𝑗𝜽=𝜉𝑗+̃𝐞,𝑗=1,,𝑁.(9)

Since ̃𝐞 converges to zero exponentially, (9) are called identification error equations.

3.2. Controller Design

The controller design involves N 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 𝑢1=𝛼1,𝑛𝜽𝐱,1,𝑦𝑟,,𝑦𝑛𝑟𝛽(𝐱),(10) where 𝑦𝑟 is the reference signal to be tracked and 𝛼1,𝑛 can be recursively designed by 𝑧𝑖=𝑥𝑖𝛼1,𝑖1𝑥1,,𝑥𝑖1,𝜽1,𝑦𝑟,𝑦𝑟𝑖1𝛼,(11)1,𝑖=𝑧𝑖1𝑐1,𝑖𝑧𝑖𝑤𝑇1,𝑖𝜽1+𝑦𝑖𝑟𝑠1,𝑖𝑧𝑖+𝑖1𝑘=1𝜕𝛼1,𝑖1𝜕𝐱𝑘𝐱𝑘+1+𝜕𝛼1,𝑖1𝜕𝑦𝑟𝑘1𝑦𝑘𝑟,𝑤(12)1,𝑖𝑥1,𝑥𝑖,𝜽1,𝑦𝑟,,𝑦𝑟𝑖1=𝝋𝑖𝑖1𝑘=1𝜕𝛼1,𝑖1𝜕𝑥𝑘𝝋𝑘𝑠,(13)1,𝑖=𝑘1,𝑖||𝑤1,𝑖||2+𝑔1,𝑖||||𝜕𝛼1,𝑖1𝜕𝜽1||||2,(14) We choose 𝑉1=12𝑛𝑖=1z2𝑖.(15)

The time derivative of 𝑉1, computed with (10)–(14), is given by ̇𝑉1=𝑛𝑖=1𝑐1,𝑖𝑧2𝑖+𝑛𝑖=1𝑤𝑇1,𝑖𝜽1𝜕𝛼1,𝑖1𝜕𝜽1̇𝜽1𝑧𝑖𝑛𝑖=1𝑘1,𝑖||𝑤1,𝑖||2+𝑔1,𝑖||||𝜕𝛼1,𝑖1𝜕𝜽1||||2𝑧2𝑖𝑛𝑖=1𝑐1,𝑖𝑧2𝑖𝑛𝑖=1𝑘1,𝑖||||𝑤1,𝑖𝑧𝑖12𝑘1,𝑖𝜽1||||2𝑛𝑖=1𝑔1,𝑖||||𝜕𝛼1,𝑖1𝜕𝜽1𝑧𝑖12𝑔1,𝑖̇𝜽1||||2+𝑛𝑖=114𝑘1,𝑖||𝜽1||2+𝑛𝑖=114𝑔1,𝑖|||̇𝜽1|||2𝑛𝑖=1𝑐1,𝑖𝑧2𝑖+𝑛𝑖=114𝑘1,𝑖||𝜽1||2+𝑛𝑖=114𝑔1,𝑖|||̇𝜽1|||2,(16) with 𝑐1,𝑖, 𝑘1,𝑖, 𝑔1,𝑖 being designed parameters. Equation (16) implies the boundedness of the states of 𝑧𝑖, 1𝑖𝑛, and which in turn indicates the boundedness of the states of 𝑥𝑖, 1𝑖𝑛 and control 𝑢1 on the conditions of 𝜽 and ̇𝜽 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, 1𝑖𝑛, and which in turn indicates the boundedness of the states of 𝑥𝑖, 1𝑖𝑛 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 the N models is used to generate the control of the plant as 𝑢=𝑁𝑗=1𝛾j𝑢𝑗,(17) and the adaptive update law as ̇𝜽𝑗𝝃=𝚪𝑇𝐞𝑗||𝝃||1+𝑣2,𝚪=𝚪𝑇>0,𝑣>0,𝑗=1,,𝑁,(18) where 𝛾j are nonnegative values satisfying 𝑁𝑗=1𝛾j=1, and 𝛾j can be calculated from 𝛾j=1/𝐽𝑗𝑁𝑗=11/𝐽𝑗,(19) where 𝐽𝑗 is the performance indices of the form: 𝐽𝑗(𝑡)=𝛼𝑒2𝑗(𝑡)+𝛽𝑡𝑡0𝑒2𝑗(𝜏)𝑑𝜏,𝑎0,𝛽>0,(20) 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 [1113], 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 ̃𝐞=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 𝐞𝑗𝜽=𝝃𝑗,𝑗=1,,𝑁,𝑡>𝑇.(21) It is obviously that the errors 𝐞𝑗 and 𝜽𝑗, 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 𝜽𝑗. 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 𝜽𝑗). So we can redistribute the other (𝑁1) models 𝑀𝑘(𝑘𝑗) as 𝜽𝑘=𝐽𝑘𝐽𝑘+𝐽𝑗𝜽𝑗+𝐽𝑗𝐽𝑘+𝐽𝑗𝜽𝑘.(22) 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, Lim𝑡𝑧(𝑡)=0 or 𝑦(𝑡)=𝑦𝑟(𝑡) as 𝑡.

Proof. Since all N 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 as1𝑉=2𝑛𝑖=1z2𝑖.(23) It is obvious that (23) can be divided into 1𝑉=2𝑛𝑖=1z2𝑖=𝑁𝑗=1𝛾j12𝑛𝑖=1z2𝑖=𝑁𝑗=1𝛾j𝑉𝑗.(24) As illustrated by (16), each of the controllers can guarantee the boundedness of the states of 𝑧𝑖, 1𝑖𝑛 at its portion, which accompanied with the control (17), and weighting coefficient (19) can establish the boundedness of the states of 𝑧𝑖, 1𝑖𝑛.
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) that𝑑𝑑𝑡𝝃𝐏𝝃𝑇=𝝃𝝃𝑇2𝜆𝝃𝐏Ξ𝑇Ξ𝐏𝝃𝑇+𝝃𝐏Ξ𝑇+Ξ𝐏𝝃𝑇=𝝃𝝃𝑇2𝜆Ξ𝐏𝝃𝑇1𝐈2𝜆𝑇Ξ𝐏𝝃𝑇1𝐈+12𝜆,2𝜆(25) which shows 𝝃 is bounded regardless of the state 𝐱. Let V𝑗𝜽=(𝑇𝑗Γ1𝜽𝑗+̃𝐞𝑇̃𝐞)/2, it can be derived that ̇V𝑗𝐞=𝑇𝑗𝐞𝑗̃𝐞||𝝃||1+𝑣2+̃𝐞𝑇𝐀0𝜆Ξ(𝐱)Ξ𝑇̃𝐞3(𝐱)𝐏4𝐞𝑇𝑗𝐞𝑗||𝝃||1+𝑣2,(26) without loss of generality, we can design the parameter satisfies 𝐀0𝜆Ξ(𝐱)Ξ𝑇(𝐱)𝐏>𝐼, 𝐼is a unit matrix. Therefore, ̃𝐞, 𝜽𝑗, 𝑗=1,,𝑁 are all bounded, which companied with the boundedness of 𝝃, further yields from 𝐞𝑗𝜽=𝝃𝑗+̃𝐞 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 ̇𝜽𝑗 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 bẏ𝐞𝑗=𝐀0𝜆ΞΞ𝑇𝐏𝐞𝑗𝜽+Ξ𝑗̇𝜽𝝃𝑗.(27) Due to the boundedness of all the closed-loop system states ̇𝐞𝑗, ̇̇𝐞𝑗, 𝑗=1,,𝑁 are also bounded, so by Barbalat’s lemma, we must have Lim𝑡𝐞𝑗(𝑡)=0 and since Lim𝑡𝑡𝑡1̇𝐞𝑗(𝜏)𝑑𝜏=lim𝑡𝐞𝑗(𝑡)𝐞𝑗(𝑡1)<, we further have Lim𝑡̇𝐞𝑗(𝑡)=0. Then, it can be concluded from (18) that Lim𝑡̇𝜽𝑗=0 which accompanied with (27) implies Lim𝑡Ξ𝜽𝑗=0 and in turn leads to Lim𝑡𝑁𝑗𝑧,𝜽,𝑦𝑟Ξ𝜽𝑗=0,(28) where 𝑁𝑗𝜽𝑧,𝑗,𝑦𝑟=100𝜕𝛼𝑗,1𝜕𝑥1100𝜕𝛼𝑗,𝑛1𝜕𝑥1𝜕𝛼𝑗,𝑛1𝜕𝑥21.(29) By direct calculating, the differentiation of z with parametric controller 𝑢𝑗 can be described, in z coordination, by ̇z=𝑛𝑖=1𝑐𝑗,𝑖𝑧𝑖+𝑁𝑗𝑧,𝜽,𝑦𝑟Ξ𝜽𝑗𝜕𝛼𝑗,𝑖1𝜕𝜽𝑗̇𝜽𝑗𝑛𝑖=1𝑘𝑗,𝑖||𝑤𝑗,𝑖||2+𝑔𝑗,𝑖|||||𝜕𝛼𝑗,𝑖1𝜕𝜽𝑗|||||2𝑧𝑖.(30) From (28), accompanied with Lim𝑡̇𝜽𝑗=0 and the designed parameters are all positive, it can be easily concluded that Lim𝑡𝑧(𝑡)=0, and thus Lim𝑡𝐳1(𝑡)=Lim𝑡(𝑦(𝑡)𝑦𝑟(𝑡))=0. The proof is completed.

5. Simulation

Consider the following second-order nonlinear system: ̇𝑥1=𝑥2+𝜃1𝑥1+𝜃2𝑥21,̇𝑥2𝑦=𝑢,(𝑡)=𝑥1(𝑡),(31) 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:̇𝝃0𝝃=𝑐0𝐱+𝑥2,𝝃0𝑅1,̇𝑥𝝃=𝑐𝝃+1,𝑥21,𝝃𝑅1×2,(32)

where 𝑐=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 𝜽1(0)=[1,1]𝑇,  𝜽2(0)=[1,5]𝑇, 𝜽3(0)=[5,1]𝑇, 𝜽4(0)=[5,40]𝑇, respectively. Figures 14 depict the simulation results.

409139.fig.001
Figure 1: Output 𝑦(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.
409139.fig.002
Figure 2: Control 𝑢(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.
409139.fig.003
Figure 3: Control 𝑢(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.
409139.fig.004
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.

Figure 1 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 13 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 57 depict the simulation results.

409139.fig.005
Figure 5: Output 𝑦(𝑡). dashed line for the multiple model case (𝑁=4) as in [7], solid line for the multiple identification model developed in this paper.
409139.fig.006
Figure 6: Control 𝑢(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.
409139.fig.007
Figure 7: Control 𝑢(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.

Figure 5 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.

Acknowledgments

The authors would like to thank the reviewers and the editors for their helpful and insightful comments for further improvement of the quality of this work. The work is supported by the National Natural Science Foundation of China (nos. 60674023 and 60905034) and the Zhejiang Provincial Natural Science Foundation of China (no. Y12F030119).

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