Robust Output Feedback Control for Fractional Order Nonlinear Systems with Time-varying Delays

Transcription

1 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 3, NO. 4, OCTOBER Robust Output Feedback Control for Fractional Order Nonlinear Systems with Time-varying Delays Changchun Hua, Tong Zhang, Yafeng Li, and Xinping Guan Abstract Robust controller design problem is investigated for a class of fractional order nonlinear systems with time varying delays. Firstly, a reduced-order observer is designed. Then, an output feedback controller is designed. Both the designed observer and controller are independent of time delays. By choosing appropriate Lyapunov functions, we prove the designed controller can render the fractional order system asymptotically stable. A simulation example is given to verify the effectiveness of the proposed approach. Index Terms Fractional order systems, time-varying delays, Laypunov function, backstepping. I. INTRODUCTION Fractional calculus is an ancient concept, which can be dated back to the end of 7th century, the time when the classical integer order calculus was established. It is a generalization of the ordinary differentiation and integration to arbitrary order []. Although it has a long history, it has not attracted much attention until recently in the control field. It is found that many systems with memory feature or complex material can be more concisely and actually described by fractional order derivatives, such as the diffusion process, the heat transfer process and the effect of the frequency in induction machines. It also has been proved that fractional order controllers, like fractional order PID controllers and fractional order model reference adaptive controllers, can capture much better effect and robustness []. For some basic theory of fractional order calculus and fractional order systems, one can refer to [ 6] and the references therein. Stability analysis is one of the most fundamental and essential issues for the control system. In [7], Matignon firstly studied the stability of fractional-order linear differential systems with the Caputo definition. Since then, many further achievements have been obtained [8 ]. In [8 9], the authors presented the sufficient and necessary conditions for the asymptotical stability of fractional order interval systems with fractional order α satisfying 0 < α < and < α <, Manuscript received September 8, 05, accepted March 3, 06. This work was partially supported by National Natural Science Foundation of China 6903, 673, 63303, , , Hebei Province Applied Basis Research Project D, and Top Talents Project of Hebei Province and Yanshan University Project 3LGA00. Citation: Changchun Hua, Tong Zhang, Yafeng Li, Xinping Guan. Robust output feedback control for fractional order nonlinear systems with time-varying delays. IEEE/CAA Journal of Automatica Sinica, 06, 34: Changchun Hua, Tong Zhang, and Yafeng Li are with the Institute of Electrical Engineering, Yanshan University, Qinhuangdao , China cch@ysu.edu.cn; @63.com; y.f.li@foxmail.com. Xinping Guan is with the Department of Automation, Shanghai Jiao Tong University, Shanghai 00030, China xpguan@ysu.edu.cn. respectively. Reference [0] designed both state and output feedback controllers for fractional order linear systems in triangular form by introducing appropriate transformations of coordinates. Based on Gronwall-Bellman lemma and sector bounded condition, the stability and stabilization of fractional order linear systems subject to input saturation were studied in []. For the nonlinear fractional order systems, the stability analysis is much more difficult than that of the linear systems. We can find some sufficient conditions in [ 4]. In [5], the authors introduced Mittag-Leffler stability by using Lyapunov direct method. Time delay is an inherent phenomenon in the interconnected systems or processes, which makes the stability analysis and controller design challenging. Robust output control problem for a class of nonlinear time-delay systems was studied in [6]. The necessary and sufficient stability conditions for linear fractional-order differential equations and linear time-delayed fractional differential equations have already been obtained in [7 9]. Reference [0] investigated the stability of α- dimensional linear fractional-order differential systems with order < α <. References [ ] contain the stability analysis of fractional order nonlinear time delay system based on Lyapunov direct method and by using properties of Mittag- Leffler function and Laplace transform. In recent years, the backstepping technique has attracted much attention as a powerful method for controlling the strict feedback nonlinear systems. There are a few works using backstepping technique to handle fractional order systems. Using Lyapunov indirect method, the authors of [3] presented a new method to design an adaptive backstepping controller for triangular fractional order nonlinear systems. In [4], a new adaptive fractional-order backstepping method is proposed for a class of commensurate fractional order nonlinear systems with uncertain constant parameters. However, for fractional order nonlinear systems with time-varying delays, there is none related work. Motivated by the mentioned situation, we devote to solve the stabilization problem of fractional order nonlinear systems with time-varying delays. The contributions of this paper are as follows: A reducedorder observer is designed to estimate the state of the system; Based on the backstepping method, we design a robust output feedback controller for a class of fractional order nonlinear time-varying delay systems; 3 With a novel class of fractional Lyapunov functions, we prove the stability of fractional order nonlinear systems. The remainder of this paper is organized as follows: Section II presents some basic concepts about fractional order calculus and the stability of fractional order nonlinear systems. In Sec-

2 478 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 3, NO. 4, OCTOBER 06 tion III, as the main part of this note, an adaptive controller is designed by using the backstepping method for fractional order nonlinear time-varying delayed systems. An example is presented to show the effectiveness of the proposed controller in Section IV. Finally, Section V gives the conclusion of this paper. Notations. Throughout this paper, R denotes the set of real numbers, R n for n-dimensional Euclidean vector space and R n n for the space of n n real matrices. X T and X represent the transpose and the inverse of matrix X, respectively. I denotes the unit matrix with proper dimensions. For any matrix A R n n, λ i A stands for the i-th eigenvalue of A. For simplicity, C 0 D α t is mentioned as D α. II. PRELIMINARIES In this section, we provide some basic knowledge of fractional calculus and fractional order systems details can be found in [ ]. There are several definitions of fractional order derivatives, among which the Riemann-Liouville and Caputo definitions are well known and most commonly used. In this paper, we choose the Caputo definition for the fractional order derivatives. The Caputo derivative and fractional integral are defined as follows. Definition Caputo fractional derivative. The Caputo fractional derivative of order α 0 for a function f : [0, ] R is defined as 0D α t ft = Γn α t 0 f n s ds, t > 0, t s α+ n n is the first integer that is larger than α and Γ is the well known Gamma function which is defined as follows: Γt = 0 x t e x dx. Definition Fractional integral. The fractional integral of order α 0 for a function f : [0, ] R is defined as 0I α t ft = Γα t 0 fs ds, t > 0. t s α Definition 3 Class-K function. A continuous function γ : [0, t [0, is said to belong to class-k function if it is strictly increasing and γ0 = 0. Here are some lemmas we could use in this paper. Lemma General Leibnitz s rule. If f and g are differentiable and continuous functions, the product f g is also differentiable and its α-th α 0 derivative is given by D α f g = D α f g + k= Γα + D α k f D k g Γk + Γα k +. 3 According to Lemma, the α-th order time derivative of V x = x T x can be extended as D α V x = D α x T x + x T D α x + γ, x is a column vector and γ = Γα+D α k x T D k x k= Γk+Γα k+. Lemma Fractional comparison principle. Let D α xt D α yt and x0 = y0, α 0,. Then xt yt. In particular, if xt = c, c is a constant, D α c = 0 and y0 = c, we will have yt c. Theorem [5]. Let x = 0 be an equilibrium point for D α xt = ft, x and D R n be a domain containing x = 0. Let V t, x : [0, D R be a continuously differentiable function such that for t 0, x D, 0 < α <, W x V t, x W x, D α V t, x W 3 x, 4 W x, W x, and W 3 x are class-k functions on D. Then x = 0 is asymptotically stable. Lemma 4 []. Let xt R n be a differentiable and continuous function. Then, for t t 0 and α 0, Dα x T txt x T td α xt. 5 Lemma 5 Schur complement lemma. The linear matrix inequality LMI [ ] A B M = B T < 0, C A = A T, C = C T and C is invertible, is equivalent to C < 0, A BC B T < 0. III. MAIN RESULTS Consider the following fractional order nonlinear system with 0 < α < and time-varying delays: D α x = x + F x + H yt, yt d t, D α x i = x i+ + F i x i + H i yt, yt d i t, D α 6 x n = u + F n x n + H n yt, yt d n t, y = x, xt = [x t, x t,..., x n t] T R n is the state and x i θ = φ i θ, θ [ d i 0, 0, i =,..., n, x0 = 0, ut R and yt R are the control input and the output of the system, respectively; x i t = [x t, x t,..., x i t] T ; F i and H i are smooth nonlinear functions and F i 0,..., 0 = H i 0, 0 = 0; d i t is the time-varying delay and there exists positive scalar η i such that d i t η i <. We impose the following assumptions on system 6. Assumption. Nonlinear functions H i ξ, ξ i =,,..., n satisfy the following inequality: H i ξ, ξ H i ξ ξ + H i ξ ξ, 7 H i and H i are known functions. Let H i ξ = H i ξ ξ, H i ξ = H i ξ ξ, then we can have the following inequality: H i ξ, ξ H i ξ ξ + H i ξ ξ. 8 Assumption is very common in nonlinear time delay systems, by which the term yt d i t can be separated from the delay-function, so that we can handle the delay problems. Assumption. For nonlinear functions F i, there exist some positive scalars l i such that the following inequalities hold for i =,, 3,..., n: F i ζ i F i ζi l i ζi ζi, 9 ζ i = [ζ, ζ,..., ζ i ] T, ζi = [ζ, ˆζ,..., ˆζ i ] T and l i is a known positive parameter. We can use the following expression for F i x i

3 HUA et al.: ROBUST OUTPUT FEEDBACK CONTROL FOR FRACTIONAL ORDER NONLINEAR SYSTEMS WITH TIME-VARYING DELAYS 479 F i x i = n i x + F i x i, i =,..., n, 0 n i is a constant which could be zero. In this paper, we focus on solving the following problem. For system 6 satisfying Assumptions and, design a reduced-order observer based memoryless output feedback controller to render the closed-loop system stable. Considering system 6 with unmeasured state variables, we propose the following reduced-order observer: D α λ i t = λ i+ t + k i+ x t + F i x i t k i λ t + k yt + F yt, D α λ n t = ut + F n x n t k n λ t + k yt + F yt, ˆx i t = λ i + k i yt, i =,..., n, x i t = [x t, ˆx t,..., ˆx i t] T and parameters k i i =,..., n are to be specified later. Similar to 0, we can change F i x i t into F i x i t = n i x + F i x i t. Remark. In this paper, we introduce the reduced-order observer instead of the full-order one. In this way, some of the states can be derived from the real output, making the results more precise and simplifying the structure and computation complexity. To our best knowledge, it is the first time that the reduced-order observer is introduced to fractional order nonlinear systems. The estimation errors are defined as From and, we can get e i t = x i t ˆx i t. D α et = Aet + F t + Ht, 3 et = [e t,..., e n t] T, k 0 0 k A = , k n F t = [F x t F x t,..., F n x n t F n x n t] T, 4 Ht = [H yt, yt d k H yt, yt d,..., H n yt, yt d n k n H yt, yt d ] T. 5 Next, we extend the backstepping technique to the fractional order nonlinear system with time-varying delays described by 6. The virtual controllers α i i =,..., n are developed at each step. Finally, at step n, the actual controller u is designed. First, we introduce the following transformation of states. z t = yt, z i t = λ i t α i, i =,..., n. 6 Then we choose the Lyapunov function as V = V e + V z + V d, 7 V e = I α e T P e, 8 and P is a real symmetric positive matrix. V z = I α M i = z i, V d = + n i= M i, 9 t ki + t t d it t d t η H ytytdt η i H i ytytdt. 0 Next, we will give the derivative of V e, V z, V d, and V in turn. From Lemma 4, the derivative of V e is: V e = D α et T P et D α et T P et + e T tp D α et = Aet + F t + Ht T P et + e T tp Aet + F t + Ht = e T ta T P + P Aet + F T tp et + e T tp F t + H T tp et + e T tp Ht According to Assumption, we can get F T t F t = F i x i t F i x i t l i xi x i ρ xn x n, ρ = n l i. V e e T ta T P + P Aet + ρe T tet + H T t Ht + e T tp P et e T ta T P + P A + ρi + P P et + H T t Ht e T ta T P + P A + ρi + P P et + Hi + ki H 3 and P and k i i =,..., n satisfy A T P + P A + ρi + P P < n ɛ I. 4 To solve inequality 4, we decompose A = Ā + kb with [ ] k 0 In n Ā =, k =. 0 0., B = [, 0,..., 0] n. k n n

4 480 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 3, NO. 4, OCTOBER 06 According to Lemma 5, inequality 4 is equivalent to the following LMI [ P Ā + W B + B T W T + ĀT P + ρ + n ] ɛ I P P 0.5I < 0, 5 W = P k. Further, we can use LMI toolbox in Matlab to obtain P and k. The derivative of V z is: V z = D α M i. 6 i= Next, by the backstepping method, the virtual controllers α i i =,,..., n and controller u are designed respectively. Step. D α M = Dα z z D α z = z z + α + k x + e + F yt + H yt, yt d t 4 + ɛ z + e + H yt, yt d t ɛ + z α + k x + F x t + z z, 7 ɛ is a positive constant. Choose α = c ɛ + k + n z F yt ki + H yt + H yt η H i yt + H i yt η i = K z ᾱ, 8 c i i =,,..., n are positive constants, Step. K = c ɛ + k + n ᾱ = F yt + ki + H yt + η H yt + + η i H i yt. D α M = Dα z z D α z H i yt = z z 3 + α + k 3 x + F x t k λ + k x + F x D α α z z 3 + ɛ e + z α + k 3 x + F x t k λ + k x + F x + D α ᾱ + K z + α + k x + F yt + K ɛ z. 9 Choose α = z c + K ɛ z k 3 x n z F x t K z + α + k x + F yt + k λ + k x t + F x t D α ᾱ = K z ᾱ, 30 K = + K k n + k + n + k 3 ᾱ = c + K ɛ z + F x t + K k z + α + F yt + D α ᾱ. Step i. D α M i = Dα z i z i D α z i = z i z i+ + α i + k i+ x + F i x i t k i λ + k x t + F x t D α α i = z i z i+ + ɛ e + z i α i + k i+ x + F i x i t k i λ + k x t + F x t + D α ᾱ i + K i z + α + k x + F yt + K i ɛ zi. 3 Choose α i = z i c i + K i ɛ z i k i+ x n i z F i x i t + k i λ + k x t + F x t D α ᾱ i K i z + α + k x + F yt = K i z ᾱ i, 3 K i = K i k i n + k + n i + k i+ ᾱ i = z i + c i + K i ɛ z i + F i x i t + K i k i z + α + F yt + D α ᾱ i. Step n. D α M n = Dα z n z n D α z n = z n u + F n x n t k n λ + k x t + F x t D α α n = e + K n ɛ zn + z n u + F n x n t ɛ + D α ᾱ n k n λ + k x t + F x t + K n z + α + k x + F yt. 33 Then u = z n c n + K n ɛ z n F n x n t + k n λ + k x t + F x t K n z + α + k x + F x t D α ᾱ n. 34

5 HUA et al.: ROBUST OUTPUT FEEDBACK CONTROL FOR FRACTIONAL ORDER NONLINEAR SYSTEMS WITH TIME-VARYING DELAYS 48 The derivative of V d is: V d = ki + d t H ytytdt dt t d t η d t + H i ytytdt dt t d it η i ki + H yt yt η H yt d tyt d t + H i ytyt η i H i yt d i tyt d i t. 35 Then, we have Then, K =, ᾱ = 4.5x 3 + x 5, the controller can be designed as ut = 0.6x t λ t +.8α x t D α ᾱ. The simulation results are shown in Figs. and, from which we can see that the constructed controller renders the closed-loop system stable. V = V e + V z + V d So e T t A T P + P A + ρ + n I + P P et ɛ c i zi. 36 i= V W et, z n, 37 Fig.. The output response of the closed-loop system. z i = [z, z,..., z n ] and W and W are class-k functions. According to the definition of the Caputo fractional derivative, Lemma and 37 D α V = I α V I α W et, z n W et, z n Finally, we present the main result of this paper as follows: Theorem. For a system described by 6 satisfying Assumptions and, controller 34 can render the closedloop system asymptotically stable. IV. SIMULATION In this section, an example is given to show the effectiveness of the proposed controller. Consider the following system: { D α x t = x t 0.8x t + 0.5x t d t sin t, D α x t = u 0.8x t + 0.5x 3 t d t sin t, d t = sin t, d t = cos t. We can see that the aforementioned system satisfies the above assumptions with P = I, η = η = 0.5, l = 0.8, ρ = 0.64, n = 0.8, F x = 0, n = 0, F x = 0.8x, H t = H t = 0, H t d t = 0.5x 3 t d t and H t d t = 0.5 x 5 t d t. Choosing ɛ =, k =, c = 0.05 and c = 0.8, we can obtain the reduced-order observer and the function β x t: D α λ t = u.8λ t 4x t, α x t = x t 4.5x 3 t x 5 t. Fig.. The trajectories of x and ˆx. V. CONCLUSION In this paper, we study the controller design problem for fractional order nonlinear time-varying delay systems, using the well known backstepping method. Also, we extend the Lyapunov method to fractional order systems. Both the designed observer and controller are independent of time delays. Through the simulation presented in Section IV, the effectiveness of the proposed controller has been verified. As put in [5], fractional order systems have a memory feature, which could make difficulties in the process of controller design. In the future, we will further consider the memory feature and its influence. Based on the result of this paper, we will study the stability and stabilization problems for fractional nonlinear delayed systems with fractional order < α < and the stability of fractional order nonlinear systems with fractional order α >.

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Los Angeles, CA: IEEE, [5] Syta A, Litak G, Lenci S, Scheffler M. Chaotic vibrations of the duffing system with fractional damping. Chaos: An Interdisciplinary Journal of Nonlinear Science, 04, 4: 0307 Changchun Hua received the Ph. D. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 005. He was a research fellow in National University of Singapore from 006 to 007. From 007 to 009, he worked in Carleton University, Canada, funded by Province of Ontario Ministry of Research and Innovation Program. From 009 to 0, he worked in University of Duisburg- Essen, Germany, funded by Alexander von Humboldt Foundation. Now he is a full professor at Yanshan University, China. He is the author or coauthor of more than 0 papers in mathematical, technical journals, and conferences. He has been involved in more than 0 projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, and other important foundations. His research interests include nonlinear control systems, control systems design over network, teleoperation systems, and intelligent control. Corresponding author of this paper. Tong Zhang received the B. S. degree in electrical engineering from Yanshan University, Qinhuangdao, China, in 04. She is currently working toward the M. S. degree in electrical engineering. Her research interests include fractional order system control. Yafeng Li received the B. S. degree in electrical engineering from Hebei University of Science and Technology, Shijiazhuang, China, in 03. He is currently working toward the Ph. D. degree in Yanshan University. His research interests include nonlinear system control and multi-agent system control. Xinping Guan received the B. S. degree in mathematics from Harbin Normal University, Harbin, China, and the M. S. degree in applied mathematics and the Ph. D. degree in electrical engineering, both from Harbin Institute of Technology, in 986, 99, and 999, respectively. He is with the Department of Automation, Shanghai Jiao Tong University. He is the coauthor of more than 00 papers in mathematical, technical journals, and conferences. As aan co-investigator, he has finished more than 0 projects supported by National Natural Science Foundation of China NSFC, the National Education Committee Foundation of China, and other important foundations. He is Cheung Kong Scholars Programme Special appointment professor. His current research interests include networked control systems, robust control and intelligent control for complex systems and their applications. Dr. Guan is serving as a reviewer of Mathematic Review of America, a member of the Council of Chinese Artificial Intelligence Committee, and chairman of Automation Society of Hebei Province, China.