Adaptive Predictive Observer Design for Class of Uncertain Nonlinear Systems with Bounded Disturbance

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1 International Journal of Control Science and Engineering 2018, 8(2): DOI: /j.control Adaptive Predictive Observer Design for Class of Saeed Kashefi *, Majid Hajatipor Faculty of Electrical and Computer Engineering, Kashan University, Kashan, Iran Abstract In this paper a adaptive predictive observer design methodology for a class of uncertain nonlinear systems in the presence of unknown nonlinearity and bounded disturbances is proposed. At first, the tracking error is approximated in the receding horizon by its Taylor series expansion. Then an analytic solution to the predictive algorithm is developed and a predictive observer is proposed. The gain vector of observer between any gain vector that satisfies stability conditions, optimally chosen by predictive algorithm. The universal approximation property of neural networks is used in the predictive technique to develop adaptive predictive observer that ensure asymptotic convergence of the state estimation error to zero. Simulation results show the effectiveness of the proposed method in the state observation. Keywords Nonlinear System, Adaptive Observer, Predictive Control, Uncertain Dynamic System 1. Introduction The first observer was proposed by Luenberger [1-3]. Taue in [4] presented Luenberger-like state observers for a class of nonlinear systems, where the nonlinearities are assumed to be known and Lipschitz continuous. Designing of adaptive observers for nonlinear systems is one of the active areas of research, the significance of which cannot be underestimated in problems of system identification, failure detection, or output feedback control [5]. The design of adaptive observers for uncertain nonlinear systems has attracted much attention of researchers. In [6], Bastin proposed the first nonlinear adaptive observer based on certain coordinate trasfformation. Similar works have been presented in [7-9]; in which some change of coordinates can linearize the nonlinear systems. A simple but restricted observer is proposed based on the satisfaction of strict positive real (SPR) conditions in [7]. An universal adaptive observer for a class of single-output nonlinear systems presented which this nonlinear systems are linear with respect to an unknown constant parameter [8]. Marino and Tomei (1992, 1995) guarantee that the state estimation errors converge to zero [11]. However their assumption of linearity with respect to any unknown system parameter sand their conditions on transforming the original system into special canonical form. In [11], A Robust adaptive observers is * Corresponding author: s.kashefi@grad.kashanu.ac.ir (Saeed Kashefi) Published online at Copyright 2018 The Author(s). Published by Scientific & Academic Publishing This work is licensed under the Creative Commons Attribution International License (CC BY). presented for nonlinear systems with bounded disturbances that state and estimation error maintain bounded under persistency of excitation conditions. First adaptive observers for a class of nonlinear systems that satisfy a Lipschitz condition can be found in [12] and [13], in these works it is not necessary the concerned nonlinear system to be linearizable. Inspired by [12], adaptive observer design is extended for Lipschitz systems in [14-16]. In this paper, it is considered the problem of designing an adaptive predictive observer for a class of unknown nonlinear systems with unknown time-varying bounded disturbances. We have proposed an solution to predictive algorithm for the state tracking problem of class of nonlinear systems analytically. In this algorithm, the tracking error in the receding horizon is approximated by its Taylor series expansion. Stability of the proposed predictive observer is proved. Furthermore, a predictive algorithm for nonlinear observer system using the above state approximation technique is developed. The gain of observer between any gain that satisfied stability conditions, is optimally chosen. In continuation by using NN approximation properties of continuous unknown functions an adaptive predictive observer present for class of uncertain nonlinear system with bounded disturbances. Using adaptive predictive algorithm, the state estimation is achived asymptotically, in the presence of unknown nonlinearity. The rest of the paper is organized as follows; In Section 2 we give mathematical preliminaries and nonlinear predictive observer. Stability analysis is presented in Section 3. We discuss the nero-adaptive predictive observer design in Section 4, and simulation results are given in

2 32 Saeed Kashefi et al.: Adaptive Predictive Observer Design for Class of Section 5. Finally, conclusion remarks will be shown in the last section. 2. Nonlinear Predictive Observer An SISO nonlinear system that can be described by x (t) = Ax(t) + B(f(x, u) + d(t)) y(t) = Cx(t) (1) x R n, u R, y R, are state, input and output respectively. d(t) is bounded disturbance. Assumption 1: pair (A,C) is observerable In this section without loss of generality, it assumed that d(t) = 0 for all t. the system (1) can expressed as x 1(t) = x 2 (t) x 2(t) = x 3 (t) (2) x n(t) = x 1 [n] (t) = f(x, u) y(t) = Cx(t) approximating the state x i (t) in the receding horizon by its Taylor series expansion x 1 (t + τ) x 1 (t) + τx 1(t) + + τr r x [r] 1 (t), r = n τ (3) x 1 (t + τ) = τ 1 X 1 (t) X 1 (t) x 1 (t), x 1 [1] (t),, x 1 [n] (t) T = [x 1 (t), x 2 (t),, x n(t)] T The proposed observer is then given by x (t) = Ax (t) + B(f(x, u) + u o ) y (t) = Cx (t) x 1 (t) = x 2 (t) x 2 (t) = x 3 (t) (4) x n (t) = x 1 [n] (t) = f(x, u) + u o y (t) = Cx (t) u o is yield in continuation. similar to (3), observer states has x 1 (t + τ) x 1 (t) + τx 1 (t) + + τr r x [r] 1 (t), r = n τ x 1 (t + τ) = τ 1 X 1 (t) (5) X 1 (t) x 1 (t), x 1 [1] (t),, x 1 [n] (t) T = x 1 (t), x 2 (t),, x n(t) T after approximating the state x 1 (t) of nonlinear system (1) and the state x 1 (t) of proposed observer by series expansion the tracking error then can expressed by e(t + τ) = x 1 (t + τ) x 1 (t + τ) e(t + τ) τ 1 [X 1 X 1 ] (6) τ 1= [1, τ,, τn ] The quadratic cost function in this paper, as follows: J = 1 e(t + 2 τ)2 dτ (7) and are the lower and upper prediction time respectively. The cost function (7) can be approximated by J 1 2 X 1 (t) X 1 (t) T τ τ 1 ( X 1 (t) X 1 (t))dτ J 1 X 2 1 (t) X 1 (t) T τ τ 1 dτ ( X 1 (t) X 1 (t))(8) J 1 2 X 1 (t) X 1 (t) T Λ ( X 1 (t) X 1 (t)) Λ = τ τ 1 dτ = τ 11 T T 1 ()! Λ = T 2 T1 2 T 2 1 ()! (2) The necessary condition for optimization is given by J = 0; (10) Theorem 1: Consider a nonlinear systems (1) and (2) satisfying Assumption 1 and with the quadratic cost function (7). After Approximating tracking error (7) by its Taylor series expansion up to n, the predictive algorithm u o in (4) is obtained by u o = (f(x, u) f(x, u) + KE n 1 ) (11) where x 1 (t) x 1 (t) E n 1 =, (12) x [n 1] [n 1] 1 x 1 where K R n is calculated by K = [k 1,, k n ] = τ 1 T 22 = 2 T 2 1 (13) (2) = T 1 T,, 2 T1 () (n 1)!() Proof: The necessary condition (10) can be written as J = X T 1 (t) X 1 (t) Λ X 1 (t) X 1 (t) = 0 X 1 X 1 = E n 1 (14) H H = f(x, u) f(x, u) + u o (15) (9)

3 International Journal of Control Science and Engineering 2018, 8(2): which implies that H T Λ ( X 1 (t) X 1 (t)) = 0 u 0 H T τ 11 u τ 0 12 T τ E n 1 22 H = 0 H = τ 1 22 τ 2 E n 1 = KE n 1 (16) K = τ 1 T 22 Invoking (15) into (16) lead to Finally u o yield f(x, u) f(x, u) + u o = KE n 1 (17) u o = (f(x, u) f(x, u) + KE n 1 ) (18) Subtracting (4) and (2) leads to error dynamic e 1(t) = e 2 (t) e 2(t) = e 3 (t) e n(t) = f(x, u) f(x, u) + u o (19) E = AE + B(f(x, u) f(x, u) + u o ) Invoking (18) into error dynamic (19) obtains E = A c E 0 1 A c =, K = [k 1,, k n ] (20) k 1 k n according to (2) and (4) output error can expressed as e y = CE (21) Then (18) can be written as u o = B f(x, u) f(x, u) + Le y (22) A c = A LC (23) 3. Stability Analysis of Predictive Observer Error dynamic (21) can be written as where e 1 [n] + k n e 1 [n 1] + k n 1 e 1 [n 2] + + k 1 e 1 = 0 (24) K = [k 1,, k n ] = τ 1 T 22 (25) and in (9) can be written as Or = 2 2 ( + 1) = (),, T1 (n 1)!() 2 = 1 T ( 2 ( + 1) ) T n 2 (26) Let = 1 T1 () T1,, 1 diag(1, T n 1 T () 2,, 2 ) (27) (n 1)! M n = 1,, 1, M () () μ = ( 1 Then one has 2 T1 (2) ) (28) K = τ 1 22 τ T 12 = T n 2 M 1 μ M n diag(1,,, n 1 ) (29) (n 1)! Let m = [m 1,, m n ] denote the vector M 1 n M m. Since K is determined by τ 1 22 τ 2, it follow from (29) that n 1+i k i = m i i = 1,, n, (i 1)! m = [m 1,, m n ] = M μ 1 M n (30) Substituting k i from (30) into the error dynamic (24) yields e 1 [n] + m n 1 ne 1 [n 1] + m n 1 2 n(n + 1)e 1 [n 2] + + m 1 n e 1 = 0 (31) It can concluded that the stability of error dynamic is determined by polynominal s [n] + m n 1 ns [n 1] + m n 1 2 n(n + 1)s [n 2] + + m 1 n s = 0 i.e., (32) n s [n] + m n n 1 ns [n 1] + m n 1 n 2 n(n + 1)s [n 2] + + m 1 s = 0 By using the transform p = s the polynomial (32) becomes p [n] + m n np [n 1] + m n 1 n(n + 1)p [n 2] + + m 1 p = 0 (33) The error dynamic (24) is stable if and only if all roots of the polynomial in (33) has negative real parts. Since the transform p = s does not change the sign of roots. Therefore the stability of error dynamic only is determined by the coefficient m i, i = 1,, n, i.e., The stability of observer (4) depend on the prediction time and. Furdurmore, if is selected as zero, it follow from (21) the coefficient in polynomial (26) does not depend on the prediction time. This shows that in this case, the stability depend on n. 4. Nero- Adaptive Predictive Observer In this section a stable linear in-parameter neural network (LPNN) nero- adaptive predictive observer is present for nonlinear system whose dynamics are described by (1). Suppose in nonlinear system (1), f(x, u) is unknown and d(t) is bounded disturbance. Then u o = B(f(x, u) f(x, u) d(t)) + Le y (34)

4 34 Saeed Kashefi et al.: Adaptive Predictive Observer Design for Class of LPNN have been used for approximation of unknown function in (34). The LPNN is obtained by fixing the weights of the first layer as V T Δ = I and unknown function in (34) can be expressed as a multilayer neural network with sufficiently large number of hidden layer neurons. This unknown function also can be estimated as follows: f = w T σ x, V T Δ w, σ R nq 1, V T Δ R n q n f (35) σ(. ) is the transfer function of the hidden neurons. σ(x, V T Δ ) = exp ( V T Δ x m 0 /σ 0 )) x R n f 1 (36) x, w T are the input and the weights vectors of neural network respectively, and σ 0, m 0 are constant value. Then unknown function in (34) can expressed as f(x, u) = w T σ(x ) f(x, u) = w T σ(x ) + ε (37) x = [x, u], ε is the bounded neural network approximation error, Invoking (37) into (34) obtains u o = B(w T σ(x ) w T σ(x ) + ε d(t)) + Le y (38) adding w T σ(x ) σ(x ) to (38) leads to u o = B(w T σ(x ) + ϵ) + Le y (39) w T = w T w T and ϵ = w T σ(x ) σ(x ) + ε d(t) is bounded. invoking (27) in (14) error dynamic obtains. E = A c E + B(w T σ(x ) + ϵ) e y = CE (40) For the neural network that is used in this paper, there is no need to data for training and the NN weights are tuned on-line. The essential idea of neural network observers stability is to define the learning rule (the weight updating mechanism) such that the stability of the observer is guaranteed. Furthermore, by using standard Lyapunov technique and based on the error backpropagation algorithm, the weights updating mechanism is performed. In [17] an equation is proposed for training the neural network to maintain the stability of observer. Outcome of this equation provides the value of network weight for system at each moment. Therefore, learning rule of this neural network is achieved from the following equation. w = γσ(x )e y k f e y w w = w w (41) where γ is any constant design parameter that can influence the speed of convergence. k f also is a design parameter. y(t) = Cx(t), f(x) is unknown function; d(t) is bounded disturbance, A = , B = 0, C = [1 0] 1 f(x) = 0.5 tanh(x 1 ) + 0.1x (0.6 +sin( 1.8x 1 x 3 2 )(1 0.4exp ( 0.5t)) d(t) = sint Choosing x(0) = [0.55,0.6] T, x (0) = [0,0] T, We obtained the simulation given in Figures. Convergence of the state observation error to zero for proposed observer is demonstrated in figure 1 and 2. It can be seen that proposed adaptive predictive observer make the state estimations approach the actual states exactly, despite the presence of unknown nonlinear parameters and disturbance. Moreover, the transient responses of the proposed predictive observer is acceptable. The parameters in weight updating (41) and gain vector of observer are tuned as follows: γ = 500, k f = 0.5, L = [50; ] 5. Simulation Results Consider a process that is described by the closed loop system [5] x (t) = Ax(t) + B(f(x) + d(t)) Figure 1. State trajectory for proposed adaptive predictive observer

5 International Journal of Control Science and Engineering 2018, 8(2): REFERENCES [1] D.G. Luenberger, Observing the state of a linear system, IEEE Transaction on Military Electronics, 8 (1964), [2] D.G. Luenberger, Observers for multivariable systems, IEEE Transaction Automatic Control, 1 (1966), [3] D.G. Luenberger, An Introduction to Observers, IEEE Transaction Automatic Control, 16 (1971), [4] F.E. Thau, Observing the state of non-linear dynamic systems. International Journal of Control, 17(1973), [5] V. Stepanyan, N. Hovakimyan, Robust adaptive observer design for uncertain systems with bounded disturbances, IEEE Transaction on Neural Network. 18, (2007). [6] G. Bastin, M. R. Gevers, Stable adaptive observers for nonlinear time-varying systems, IEEE Transaction Automatic Control 33(1988) [7] R. Marino, Adaptive Observers for Single Output Nonlinear Systems, IEEE Transaction on Automatic Control, 35(1990) [8] R. Marino, P. Tomei, Global Adaptive Observers for Nonlinear Systems via Filtered Transformations, IEEE Transaction on Automatic Control 37(1992) [9] R. Marino, P. Tomei, Nonlinear Control Design, Geometric, Adaptive and Robust, New York Prentice Hall [10] R. Marino, P. Tomei, Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems. IEEE Transaction on Automatic Control 40(1995) Figure 2. State trajectory x 2 for proposed adaptive predictive observer 6. Conclusions In this study, for a class of nonlinear uncertain systems with unknown functions and time varying bounded disturbances a adaptive predictive observer design methodology is presented. This approach ensure asymptotic convergence of the state estimation error to zero. The gain of observer between any gain that satisfies stability conditions, optimally chosen by predictive algorithm. This paper provide a universal and reusable observer for the class of uncertain nonlinear system and for nonlinear system in presence of time-varying unknown parameters. The benefits of the approach are illustrated in simulation studies. Developing adaptive predictive observer for general uncertain nonlinear system with bounded disturbance can be a subject for further research. [11] R. Marino, P. Tomei, G. Santosuosso, Robust adaptive observers for nonlinear systems with bounded disturbances. Transaction on Automatic Control 46(2001) [12] Y. M. Cho, R. Rajamani, A systematic approach to adaptive observer synthesis for nonlinear systems, Transaction on Automatic Control 42(1997) [13] R. Rajamani, K. Hedrick, Adaptive Observer for Active Automotive Suspensions: Theory and experiment, Transaction on Automatic Control 3(1995) [14] D. Ya-Li, M. Sheng-Wei, Adaptive observer for a class of nonlinear Systems, Acta Automatica Sinica 33(2007) [15] Z. Lili, L. Xiuliang, L. Ping, Adaptive observer design for a class of MIMO nonlinear Systems, In Intelligent Control and Automation [16] M. Ekramian, F. Sheikholeslam, S. Hosseinnia, M.J. Yazdanpanah M.J, Adaptive state observer for lipschitz nonlinear systems, Sys Ctrl Letters. 62(2013) [17] Y.H. Kim., F.L. Lewis, C.T. Abdallahs, A dynamic recurrent neural network based adaptive observer for a class of nonlinear systems, Automatica 33(1997)