非線性系統之倒階適應性類神經控制器設計

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(1)國立台灣師範大學工業教育學系碩士論文. 指導教授: 洪欽銘博士王偉彥博士. 非線性系統之倒階適應性類神經控制器設計 Adaptive Backstepping Neural Network Controller Design for Nonlinear Systems. 研究生: 李正皓撰中華民國九十八年七月.

(2) Acknowledgement 首先，非常由衷感謝指導教授洪欽銘老師、王偉彥老師與呂藝光老師，在我碩士班期間學業上辛苦的指導與生活上的教誨與幫助，也讓我們學習處理承辦國際研討會事項，使得在碩士班兩年的求學過程中，學習到很多寶貴的經驗與知識，對於老師們的辛苦栽培，學生將謹記於心。. 亦誠摯的感謝口試委員：李祖添教授、王文俊教授、以及呂藝光教授，的蒞臨指導，提供本論文寶貴的意見以及不同的思考方向與問題，使得論文內容能更臻完善。. 在碩士班的求學過程中，感謝兩年來一起努力的同學們與學長的教導，建宏、建佑、建豪、俊堯、名峰及宏建學長、銘滄學長，一起做研究、讀書、出遊，在生活與學業上互相砥礪，很高興能在碩士班這兩年裡跟你們一起成長茁壯。. 最後，感謝所有直接或是間接幫助過我的朋友們，有你(妳)們的協助，才能讓我完成這本碩士論文，順利取得碩士學位。. i.

(3) Adaptive Backstepping Neural Network Controller Design for Nonlinear Systems. Student: Zheng-Hao Lee. Advisors: Dr. Chin-Ming Hong Dr. Wei-Yen Wang. Department of Industrial Education National Taiwan Normal University. ABSTRACT. Three control methods for nonlinear systems are proposed in this thesis. The first controller design is about a B-spline adaptive backstepping controller for affine nonlinear systems. The controller is comprised of a B-spline neural network identifier and a robust controller. The B-spline neural network identifier is the main controller and the robust controller is developed to achieve L2 tracking performance to a desired attenuation level. B-spline neural networks have the advantage over other neural networks of local output adjustment, allowing them to more easily online estimate the system dynamics by tuning their interior parameters, including control points and knot points. To online adjust these parameters, a mean-value estimation technique is proposed to avoid the higher-order derivative problem. This problem generated by both the Taylor linearization expansion and the requirement of finding the derivatives of B-spline basis functions with respect to their parameters. The second controller design is about a B-spline adaptive backstepping controller for nonaffine nonlinear systems. The control scheme combines the backstepping design technique with mean-estimation B-spline neural networks. The mean-estimation B-spline neural networks use a mean estimation ii.

(4) technique to develop the update laws for the design of online adaptive controllers. The third controller design is about a B-spline adaptive backstepping controller for nonaffine nonlinear systems with first order filters. The backstepping design technique suffers from on explosion of complexity as order of system increases. In order to overcome this problem, the third controller design uses first order filter at each step of the backstepping design.. Key word: Adaptive Control, Neural Network, Nonlinear Systems. iii.

(5) 非線性系統之倒階適應性類神經控制器設計. 學生：李正皓. 指導教授：洪欽銘博士王偉彥博士台灣師範大學工業教育學系碩士班. 摘要本篇論文提出三種非線性系統的控制方法。首先，在第一章先提出一個 B-spline 適應性倒階典型非線性系統的控制器。這個控制器結合 B-spline 類神經近似器與強建控制器。主要控制器為 B-spline 類神經近似器而強建控制器符合 L2 的追蹤效能。B-spline 類神經再局部調整的能力比其他類神經來的優異許多，所以非常適合透過內部參數(控制點或是結點)的訓練來即時估測未知的動態系統，為了及時調整這些參數，本篇論文提出均值定理來取代泰勒級數展開的方式避免 B-spline 基底高次項微分的問題。在第二章我們提出一個 B-spline 適應性倒階非典型非線性系統的控制器。這個控制系統包含 B-spline 均值估測類神經網路倒階控制系統設計，而此系統利用均值的觀念來設計即時的更新律。最後，本篇論文提出結合 B-spline 適應性倒階與一階濾波器得設計概念來控制非典型非線性系統。在 n 階到階設系統設計的過程中都會發生運算量激增的問題，所以為了克服這問題在本章再傳統到階設計時的每一階步驟都引入一階濾波器的觀念來解決這類問題。. 關鍵字；適應性控制，類神經網路，非線性系統. iv.

(6) Table of Content Acknowledgement. i. Abstract in English. ii. Abstract in Chinese. iv. List of Tables……………………………………………………………… v List of Figures………………………………………………………………vi. 1. Introduction. 2. B-spline Adaptive Backstepping Controllers for Affine Nonlinear Systems 2.1 Problem Formulation……………………………………………………… 4 2.2 Design of Backstepping Controllers for Known systems……………….. 5 2.3 Design of Backstepping Controllers for Unknown Systems……………… 8 2.4 Simulation Results………………………………………………………...16. 3. B-spline Adaptive Backstepping Controllers for Nonaffine Nonlinear Systems 3.1 Problem Formulation……………………………………………………. 31 3.2 Design of Backstepping Controllers for Known Systems……………….. 32 3.3 Design of B-spline Adaptive Backstepping Controllers for Unknown Systems…………………………………………………………………. 34 3.4 Simulation Results……………………………………………………….. 39. 4. B-spline Adaptive Backstepping Controllers for Nonaffine Nonlinear Systems with First Order Filter 4.1 Problem Formulation……………………………………………………. 46 4.2 Design of B-spline Adaptive Backstepping Controllers with First Order Filter……………………………………………………………………… 47 v.

(7) 4.3 Stability Analysis………………………………………………………… 54 4.4 Simulation Results………………………………………………………. 59. 5. Conclusion……………………………………………………………….. 62. Reference……………………………………………………………………. 64. vi.

(8) List of Tables Table 2.1. Three cases of ρ = 0.1 ρ = 0.2 and ρ = 0.3 ……………………… 17. Table 2.2. Three cases of ρ = 0.2 ρ = 0.3 and ρ = 0.4 ……………………. 24. Table 3.1. Three cases of ρ = 0.1 , ρ = 0.3 and ρ = 0.5 ………………... 40. Table 3.2. Three cases of ρ = 0.15 , ρ = 0.175 and ρ = 0.2 ……………….. 43. vii.

(9) List of Figures Fig. 2.1. The B-spline neural network structure………………………….9. Fig. 2.2. Mean value theorem………………………………………… 10. Fig. 2.3. Output y follow reference signal ym with ideal backstepping controller………………………………………………………. 18. Fig. 2.4. Error ym − y with ideal backstepping controller…………………18. Fig. 2.5. Control input u with ideal backstepping controlle……………... 19. Fig. 2.6. Output y follow reference signal ym with B-spline adaptive backstepping controller ( ρ = 0.1 )………………………………. 19. Fig. 2.7. Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.1 )…………………………………………………………20. Fig. 2.8. Control input u with B-spline adaptive backstepping controller ( ρ = 0.1 )…………………………………………………………20. Fig. 2.9. Output y follow reference signal ym with B-spline adaptive backstepping controller ( ρ = 0.2 )……………………………… 21. Fig. 2.10. Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.2 )……………………………………………………….. 21. Fig. 2.11. Control input u with B-spline adaptive backstepping controller ( ρ = 0.2 )………………………………………………………. 22. Fig. 2.12. Output y follow reference signal ym with B-spline adaptive backstepping controller ( ρ = 0.3 )……………………………… 22. Fig. 2.13. Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.3 )……………………………………………………….. 23. Fig. 2.14. Control input u with B-spline adaptive backstepping controller ( ρ = 0.3 )……………………………………………………….. 23. Fig. 2.15. Output y follow reference signal ym with ideal backstepping….. 26. Fig. 2.16. Error ym − y with ideal backstepping controller…………………26. Fig. 2.17. Control input u with ideal backstepping controller…………… 26. Fig. 2.18. Output y follow reference signal ym with B-spline adaptive viii.

(10) backstepping controller ( ρ = 0.2 )………………………………. 28 Fig. 2.19. Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.2 )………………………………………………………… 29. Fig. 2.20. Control input u with B-spline adaptive backstepping controller ( ρ = 0.2 )………………………………………………………… 29. Fig. 2.21. Output y follow reference signal ym with B-spline adaptive backstepping controller ( ρ = 0.3 )……………………………… 30. Fig. 2.22. Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.3 )……………………………………………………….. 30. Fig. 2.23. Control input u with B-spline adaptive backstepping controller ( ρ = 0.3 )………………………………………………………... 31. Fig. 2.24. Output y follow reference signal ym with B-spline adaptive backstepping controller ( ρ = 0.4 )………………………………. 31. Fig. 2.25. Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.4 )……………………………………………………….. 32. Fig. 2.26. Control input u with B-spline adaptive backstepping controller ( ρ = 0.4 )………………………………………………………... 32 u a = −αsat ( z n ) ……………………. 44. Fig. 3.1. The simulation results when. Fig. 3.2. The simulation results when. ua =. − zn 2ρ 2. , where ρ = 0.1 ………… 44. Fig. 3.3. The simulation results when. ua =. − zn 2ρ 2. , where ρ = 0.3 ………… 45. Fig. 3.4. The simulation results when. ua =. − zn 2ρ 2. , where ρ = 0.5 ………… 45. Fig. 3.5. The simulation results when. u a = −αsat ( z n ) …………………….. Fig. 3.6. The simulation results when. ua =. − zn 2ρ 2. , where ρ = 0.15 ……….. 47. Fig. 3.7. The simulation results when. ua =. − zn 2ρ 2. , where ρ = 0.175 ………. 48. Fig. 3.8. The simulation results when. ua =. − zn 2ρ 2. , where ρ = 0.2 ………... 48. Fig. 4.1. Output. Fig. 4.2. Control output. y. 47. y m ………………………. 65. u …………………………………………….... 65. tracking reference signal. ix.

(11) Fig. 4.3. Error function. Fig. 4.4. Output and. y. z1 ………………………………………………. tracking reference signal. ym. with know function. f 2 ………………………………………………………….. Fig. 4.5. Control output. u. with know function. Fig. 4.6. Error function. z1. with unknown function. x. f1. and f1. 65 f1. 66. f 2 …………... 66. f 2 ……….. 66. and.

(12) Chapter 1 Introduction Neural networks have been widely used to model nonlinearities. A neural network is a universal approximator which, with sufficient size and complexity, can approximate any nonlinear function with arbitrary precision. Based on its capabilities, neural networks have also been widely adopted for nonlinear system identification and control [1]-[8]. The B-spline neural network (BSNN) is a kind of neural networks. B-spline function produces a piecewise polynomial and a BSNN operates with a local weight-updating scheme, with the advantage of fast convergence speed and low computational complexity. Because of these features, BSNNs are suitable for online tuning or modeling. BSNNs consist of B-spline basis functions and control points. BSNNs are suitable for online adaptive modeling and control [9]-[13]. In [14], a novel knot-optimizing B-spline network was proposed to approximate general nonlinear system behavior. In [15], a high performance neural network controller for an uninterruptible power supply was proposed based on a BSNN, and the controller is easy to design and simple to implement.. In recent adaptive and robust control literature, numerous approaches have been proposed for the design of nonlinear control systems. Since 1990s, backstepping has become one of the most popular design methods for a large class of nonlinear systems [16]-[24]. Compared with feedback linearization methods, the backstepping technique [16]-[18] has the advantage of avoiding the cancellation of useful nonlinearities in the design process. Thus, the backstepping technique has been widely used for nonlinear control systems. When designing such a system, first an appropriate state and virtual controller are selected for each smaller subsystem, then the state equation is rewritten in terms of the subsystems, and finally Lyapunov functions are chosen for these subsystems. This ensures that the true controller integrating the individual 1.

(13) controllers of these subsystems can guarantee the stability of the overall system. Recently, owing to the development of intelligent control methods, such as fuzzy logic control, neural network control, etc., many intelligent backstepping methods [16]-[22] have been proposed to control nonlinear systems with unknown system dynamics by combining the intelligent control methods with an adaptive backstepping design [25].. A backstepping based neural network controller for unknown nonlinear systems first came out in [26]. In [27]-[37], backstepping based neural networks were used in many kinds of controller design, e.g. time delay systems or MIMO systems. In [21], a robust and adaptive backstepping controller using an RBF neural network was proposed. An adaptive neural controller of uncertain MIMO nonlinear systems was proposed in [20]. In [11], a wavelet adaptive backstepping control system was proposed with simple second order nonlinear systems. The wavelet adaptive backstepping control system comprised a neural backstepping controller and a robust controller. Taylor expansion linearization was used to transform the nonlinear function into a partially linear form in [17], [27]-[28]. In thesis 2, an ideal backstepping controller and a B-spline adaptive backstepping controller for a complex affine nonlinear nth-order system is proposed. We propose a B-spline adaptive backstepping controller which is comprised of an estimate B-spline neural network identifier, a robust controller, and an adaptive law to tune the knot vectors of the B-spline basis functions. Finally, a mean-value estimation technique is proposed to avoid a higher-order derivative problem. In thesis 3, a B-spline adaptive backstepping scheme for nonaffine nonlinear systems is proposed. The control scheme incorporates the backstepping design technique with the mean-estimation B-spline neural networks which are utilized to estimate the system dynamics and use a mean estimation technique to develop the update laws for the design of online 2.

(14) adaptive controllers. Finally, B-spline adaptive backstepping scheme for nonaffine nonlinear systems with first order filter is proposed in thesis 4. In order to overcome the explosion of complexity problem, we use a first order filter at each step of the backstepping design. In thesis 5, we draw some conclusions.. 3.

(15) Chapter 2 B-spline Adaptive Backstepping Controllers for Affine Nonlinear Systems. An ideal backstepping controller and a B-spline adaptive backstepping controller for a complex affine nonlinear nth-order system are proposed in this chapter. First we use a B-spline adaptive backstepping controller which is comprised of an B-spline neural network identifier and a robust controller. Second we use an adaptive law to tune the knot vectors of the B-spline basis functions. Finally a mean-value estimation technique is proposed to avoid a higher-order derivative problem generated by both the Taylor linearization expansion and the requirement of finding the derivatives of B-spline basis functions with respect to their parameters.. This chapter is organized as follows. The system problem is formulated in section 2.1. An ideal backstepping controller and virtual controller is described in section 2.2. A B-spline adaptive backstepping controller is described in section 2.3. In section 2.4, two simulation examples are given. Finally, we draw some conclusions in section 2.5.. 2.1 Problem Formulation A. System Description The model of many affine nonlinear systems can be expressed in or transformed into a state-space form. 4.

(16) x1 = x2 x2 = x3. (2.1). xn = f (x) + g (x)u. where x = [ x1 , x2. xn ]T ∈ R n , u ∈ R , and f (x) and g (x) are unknown smooth. continuous functions. The control objective is to design a B-spline adaptive backstepping controller for system (2.1) such that all the signals in the closed-loop are uniformly stable and the state x1 can track a bound reference signal y m arbitrarily closely. In designing an adaptive B-spline neural controller of system (2.1), we make the following assumptions. Assumption 2.1: There exist positive constants, g u and g l , such that g l ≤ g (x) ≤ g u .. Assumption 2.2: There exists a constant, g L , such that g (x) ≤ g L .. 2.2 Design of Backstepping Controllers for Known Systems Theorem. 2.1:. Consider. the. nth-order. nonlinear. system. (2.1).. Let z1 = x1 − y m and zi = xi − α i −1 for 2 ≤ i ≤ n , where α1 = ym − c1 z1 , and k. k −1. i =1. j =1. α k = yd( k ) − ∑ ci zi( k −i ) − ∑ z (jk −1− j ) for 2 ≤ k ≤ n − 1 . Suppose that the control. law is given as uideal =. ⎤ 1 ⎡n ⎤ ⎡ n−1 ( ym( n ) − ⎢∑ ci zi( n−i ) ⎥ − ⎢∑ z (jn−1− j ) ⎥ − f (x)) , where g ( x) ⎣ i =1 ⎦ ⎣ j =1 ⎦. ci > 0 . Then, the state trajectory x1 can asymptotically track the bounded. signal ym .. ◢. proof : Step 1) Define the tracking error as z1 = x1 − ym 5. (2.2).

(17) Differentiating of z1 can be expressed as z1 = x1 − ym. (2.3). Define the virtual controller as. α1 = ym − c1 z1. (2.4). where c1 > 0 is a design parameter. From (2.2) and (2.3), if α1 = x1 , then lim z1 → 0 , that is, the state trajectory x1 can asymptotically track t →∞. the bounded signal y m . Define an error state as z 2 = x1 − α 1 = x 2 − α 1 . Then, our next goal is to force the error state z2 to decay to zero. By using (2.3) and the fact that x1 = z2 + α1 , equation (2.2) can be rewritten as z1 = z2 − c1 z1. (2.5). Step 2) Differentiating of z2 can be expressed as z 2 = x 2 − α 1 = x3 − ( −c1 z1 + y m ). (2.6). Similarly, define the virtual controller as. α 2 = y m − c1 z1 − c2 z 2 − z1. (2.7). where c2 > 0 is a design parameter. Moreover, define the error state as z 3 = x3 − α 2 . Then, by using (2.6) and the fact that x2 = z3 + α 2 , equation (2.7) can be rewritten as z 2 = z 3 − c 2 z 2 − z1. (2.8). Step 3) Let k be a positive integer. Define the error state as zk = xk − α k −1 . Then, the derivative of zk , where 3 ≤ k ≤ n − 1 , can be expressed as zk = xk − α k −1. (2.9). Define the virtual controller as ⎡ k −1. ⎤. ⎦ ⎣ j =1. ⎦. α k = y m( k ) − ⎡⎢∑ ci z i( k − i ) ⎤⎥ − ⎢∑ z kj −1− j ⎥ k. ⎣ i =1. (2.10). where ci > 0 is a design parameter. Moreover, define the error state as zk +1 = xk +1 − α k . Then, by using (2.10) and the fact that xk = zk +1 + α k , 6.

(18) equation (9) can be rewritten as zk = zk +1 − ck zk − zk −1. (2.11). Step 4) Differentiating of zn can be expressed as ⎤ ⎡ n −1 ⎤ ⎡ n−2 zn = xn − α n −1 = f (x) + g (x)u − ( ym( n ) − ⎢ ∑ ci zi( n−i ) ⎥ − ⎢ ∑ z nj −1− j ⎥ ) ⎣ i =1 ⎦ ⎣ j =1 ⎦. (2.12). Define the control law as ⎤ 1 ⎡ n ⎤ ⎡ n−1 ( ym( n ) − ⎢ ∑ ci zi( n−i ) ⎥ − ⎢ ∑ z (jn−1− j ) ⎥ − f (x)) g ( x) ⎣ i =1 ⎦ ⎣ j =1 ⎦. uideal =. (2.13). where cn > 0 is a design parameter. Then, from (2.13), equation (2.12) can be rewritten as zn = −cn zn − zn −1. (2.14). Step 5) Consider the Lyapunov function 1 n V = ∑ zi2 2 i =1. (2.15). By differentiating (2.15) and using (2.5), (2.8), (2.11) and (2.14), we have n. V = ∑ zi zi i =1. n −1. = z1 ( z2 − c1 z1 ) + ∑ zi ( zi +1 − ci zi − zi −1 ) + zn (−cn zn − zn−1 ) i=2. (2.16). n. = − ∑ ci zi2 i =1. ≤ −c1 z12. From (2.15) and (2.16), we can conclude that zi is bounded. Moreover, from (2.5), z1 is also bounded. Integrating (2.16), we get. ∫. ∞. 0. z1 (τ )dτ = −(V (∞) − V (0)) c1. (2.17). Because of the fact that the right side of (2.17) is bounded, we have z1 ∈ L2 . According to Barbalat’s Lemma [25], lim z1 = 0 , that is, the t →∞. state trajectory x1 can asymptotically track the bounded signal ym .. 7.

(19) 2.3 Design of Backstepping Controllers for Unknown Systems. Because the system functions f (x) and g (x) are unknown, the ideal backstepping controller (2.13) cannot be precisely obtained. To solve this problem, a B-spline neural network is utilized to approximate the ideal backstepping controller.. A. B-spline neural network [12] For p+1 control points {c0 , c1 ,..., c p } , the B-spline function f (λ ) can be defined as follows: p. f (λ ) = ∑ ci N i ,k (λ ). (2.18). i =0. where N i , k (λ ) =. (λ − λi ) N i ,k −1 (λ ). λi + k −1 − λi. +. (λi + k − λ ) N i +1,k −1 (λ ). λi + k − λi +1. , λi ≤ λ < λi + k. (2.19). ⎧ 1 if λi ≤ λ < λi +1 N i ,1 (λ ) = ⎨ ⎩ 0 otherwise. where. N i , k (λ ). denotes the ith B-spline basis function of order k . Define the. knot vector T as a sequence of real number. {λ0 , λ1 ,...} , i.e.,. T = {λ0 , λ1 , λ2 ...} .. Equation (2.19) is a recursive definition specifying how to construct the kth-order function from two basis functions of order k − 1 . On the basis of the above description, define the B-spline neural networks for the system dynamics f (q) as f (q) = θT ξ(q, T). (2.20). where ci = θi is the update weight, N i ,k = ξi (q, T) is the basis function, q is the input vector, T = [λ1 , λ2 ,..., λl ] is the update knot vector, where λi < λi +1. 8.

(20) The structure of the B-spline neural network approximator is shown in Fig. 2.1.. ξ 0 (T) θf. ξ1 (T). 0. θf. f. 1. q. θf. ξ 2 ( T). ∑ 2. θ fp. ξ p (T). Fig.2.1. The B-spline neural network structure. B. B-spline adaptive backstepping controller design The network structure of the B-spline neural network is shown in Fig. 2.1. The ideal backstepping controller can be expressed uideal =. n ⎤ 1 ⎡n ⎤ ⎡ n−1 ( ym( n ) − ⎢∑ ci zi( n−i ) ⎥ − ⎢∑ z (j n−1− j ) ⎥ − f (x)) = ∑ Pi N i ,k (q, T) = θT ξ(q, T) g ( x) i =0 ⎣ i =1 ⎦ ⎣ j =1 ⎦. (2.21) where θ ∈ R n is the ideal adaptive parameter vector, ξ(q, T) ∈ R is the vector of basis functions. Define the input vector q = {x, z1 ,…, zn , z1 ,. , zn−1 , ym( n ) } ∈ R m and. T is a knot vector. There exists an ideal B-spline neural network identifier u ∗ such that uideal = u * + σ = θ∗T ξ(q, T∗ ) + σ. (2.22). where σ denotes the approximation error T∗ is the optimal knot vector and θ∗ is the optimal adaptive parameter vector. In fact, the optimal parameter vectors are difficult to obtain. The B-spline neural network identifier is defined as 9.

(21) ˆ) uˆ = θˆ T ξ(q, T. (2.23). ˆ are the estimates of u ∗ and T∗ . In the following, some where uˆ and T. tuning laws will be derived to online tune the parameters of the B-spline neural network identifier to achieve this goal. Then the mean value theorem will be used to transform the nonlinear function into a linear form. ς ( p, h) ς ( p, hˆ). ς ( p , h* ). ς ( p, hˆ + δ ). h∗. hˆ Fig.2.2 Mean value theorem. hˆ + δ. h. In Fig.2.2 ς ( p, h) is a continuous function, and δ is a positive constant. According to the mean value theorem, there exists a point hˆ between h∗ and hˆ + δ , such that. ς ( p, h∗ ) − ς ( p, hˆ + δ ) ∂ς | ˆ = ∂h h= h h* − (hˆ + δ ) ∂ς |h =hˆ (h* − ( hˆ + δ )) ς ( p, h∗ ) − ς ( p, hˆ + δ ) = ∂h. ∂ς ∂ς |h =hˆ (h* − hˆ) − δ | ˆ ∂h ∂h h= h ∂ς (h* − hˆ) + ς ( p, hˆ + δ ) − ς ( p, hˆ) − δ | ˆ ∂h h= h. ς ( p, h∗ ) − ς ( p, hˆ + δ ) + ς ( p, hˆ) − ς ( p, hˆ) = ∂ς | ˆ ς ( p, h ) − ς ( p, hˆ) = ∂h h =h ∗. 10. (2.24).

(22) (2.24) can be expressed as ∂ς ς ( p, h∗ ) − ς ( p, hˆ) = (2.25) |h =hˆ (h* − hˆ) + d ∂h ∂ς where d = ς ( p, hˆ + δ ) − ς ( p, hˆ) − δ | ˆ is the minimum approximation error, ∂h h= h if δ → 0 then d → 0 . From (2.25), we have ⎡ ∂ξ1 ⎢ ⎡ξ 1 ⎤ ⎢ ∂ T ⎢ ~ ⎥ ⎢ ∂ξ 2 ~ ⎢ξ ⎥ ξ = ⎢ 2 ⎥ = ⎢ ∂T ⎢ ⎢ ⎥ ⎢ ~ ⎢⎣ ξ m ⎥⎦ ⎢ ∂ξ m ⎢ ∂T ⎣ ~. ⎤ ⎥ ⎥ ⎥ ~ ⎥ | T = Tˆ T + d ⎥ ⎥ ⎥ ⎥ ⎦. (2.26). ~ ˆ and ~ ˆ ). ξ = ξ(q, T∗ ) − ξ(q, T where Ti = T∗ − T. Let ⎡ ∂ξ1 ⎢ ∂t ⎢ 1 ⎢ ∂ξ1 … ∂ξ m ⎤ ⎡ ∂ξ1 ∂ξ 2 , ,.... | ˆ = ⎢ ∂t Φ=⎢ ∂T ⎥⎦ T=T ⎢ 2 ⎣ ∂T ∂T ⎢ ∂ξ ⎢ 1 … ⎢⎣ ∂t n. ∂ξ m ⎤ ∂t1 ⎥ ⎥ ⎥ ⎥ |T=Tˆ ⎥ ∂ξ n ⎥⎥ ∂t n ⎥⎦ n×m. (2.27). where m is the number of basis functions, n is the number size of knot vectors. Then (2.26) can be expressed as ~ ~ ξ = ΦT T + d Define the estimation error of u as. 11. (2.28).

(23) u~ = uideal − uˆ. = u ∗ − uˆ + σ ˆ ) + θˆ T ξ (q, T∗ ) − θˆ T ξ(q, T∗ ) + σ = θ∗T ξ(q, T∗ ) − θˆ T ξ(q, T T ˆ )) + σ = (θ∗ − θˆ T )ξ (q, T∗ ) + θˆ T (ξ (q, T∗ ) − ξ(q, T ~ ~ = θ T ξ(q, T∗ ) + θˆ T ξ + σ ~ ~ ~ ~ = θ T ξ(q, T∗ ) + θˆ T (ΦT T + d) + θ T ξ(q, Tˆ ) − θ T ξ(q, Tˆ ) + σ ~ ~ ~ ˆ )) + θˆ T (ΦT T = θ T (ξ(q, T∗ ) − ξ(q, T + d) + θ T ξ(q, Tˆ ) + σ ~ ~ ~ ~ ˆ) +σ = θ T (ΦT T + d) + θˆ T (ΦT T + d) + θ T ξ(q, T ~ ~ ~ ~ ~ = θ T ΦT T + θ T ξ(q, Tˆ ) + TT Φθˆ + ( θ T + θˆ T )d + σ ~ ~ ~ ˆ)+~ = θ T ΦT (T∗ − T θ T ξ(q, Tˆ ) + TT Φθˆ + ( θ T + θˆ T )d + σ ~ ~ ~ ~ ~ = θ T ΦT T∗ − θ T ΦT Tˆ + θ T ξ(q, Tˆ ) + TT Φθˆ + ( θ T + θˆ T )d + σ ~ ~ ~ ~ = − θ T ΦT Tˆ + TT Φθˆ + ( θ T + θˆ T )d + θ T (ΦT T∗ + ξ(q, Tˆ )) + σ ~ ~ ~ ~ = − θ T ΦT Tˆ + TT Φθˆ + ( θ T + θˆ T )d + θ T (ΦT T∗ + ξ (q, T∗ ) − ~ ξ (q, T)) + σ ~ ~ ~ T ˆ ~T ˆ T ˆ +T = − θ T ΦT T Φθ + ( θ + θ )d + θ T (ΦT T∗ + ξ (q, T∗ ) − ~ ΦT T − d ) + σ ~ ~ = − θ T ΦT Tˆ + TT Φθˆ + ε. (2.29). where θ = θ∗ − θˆ , the uncertainty term ~ ~ ~ ε = ( θ T + θˆ T )d + θ T (ΦT (T∗ − T) + ξ(q, T∗ ) − d) + σ. Design a controller which is comprised of an estimated B-spline neural network identifier uˆ and a robust controller ua . The B-spline adaptive backstepping controller u is developed as follows: u = uˆ + ua. (2.30). Substituting (2.30) into (2.1), we obtain xn = f (x) + g (x)(uˆ + ua ) = f (x) + g (x)uˆ + g (x)uideal − g (x)uideal + g (x)ua. ⎤ ⎡n ⎤ ⎡ n−1 = f (x) − g (x)u~ + ( ym( n ) − ⎢∑ ci zi( n−i ) ⎥ − ⎢∑ z (j n−1− j ) ⎥ − f ( x)) + g (x)ua ⎣ i=1 ⎦ ⎣ j =1 ⎦ ⎤ ⎡n ⎤ ⎡ n−1 = ym( n ) − g (x)u~ − ⎢∑ ci zi( n−i ) ⎥ − ⎢∑ z (jn−1− j ) ⎥ + g (x)ua ⎣ i=1 ⎦ ⎣ j =1 ⎦ 12. (2.31).

(24) Using Theorem 2. 1, we obtain ⎡ n−1. ⎤ ⎡ n−2. ⎤. ⎣ i =1. ⎦ ⎣ j =1. ⎦. α n−1 = ym( n−1) − ⎢ ∑ ci zi( n−i ) ⎥ − ⎢ ∑ z nj −1− j ⎥. (2.32). Substituting (2.32) into (2.31), we have xn − α n −1 = − g (x)u − cn zn − zn −1 + g (x)ua zn = − g (x)u − cn zn − zn −1 + g (x)ua. (2.33). Theorem 2 shows the properties of the proposed adaptive B-spline backstepping controller.. Theorem 2: If the (2.1) is an unknown function, the control input is designed as (2.30), where the estimated BSNN identifier uˆ is (2.23). The virtual controller. α k is (2.10). The adaptive laws of the B-spline neural network identifier are ~ ˆ θ = −θˆ = −γ 1 znΦT T ~ ˆ = γ z Φθˆ T = −T 2 n. (2.34). where γ 1 , γ 2 are the positive learning rates. The robust controller is ua = −. zn 2ρ 2. (2.35). where ρ is a prescribed attenuation constant. Then all the signals in the closed-loop are uniformly stable and the state x1 can track a bound reference signal y m arbitrarily closely... ▲. proof: Define a Lyapunov function as ~ ~ ~ ~ 1 n 2 θ T θ TT T V= ∑ zi + 2γ + 2γ 2 g (x) i=1 1 2. (2.36). Differentiating (2.36) and using (2.5), (2.8), (2.11) and (2.33)-(2.35), we obtain. 13.

(25) ~ ~ ~ ~ 1 n g (x) n 2 θ T θ TT T zi + V = + ∑ zi zi − 2 g 2 ( x ) ∑ γ1 γ2 g (x) i=1 i =1 n−1 1 ( z1 ( z2 − c1 z1 ) + ∑ zi ( zi+1 − ci zi − zi−1 ) + zn (− g (x)u~ − cn zn − zn−1 + g ( x)ua )) g ( x) i =2 ~ ~ ~ ~ θ T θ TT T g ( x) n − 2 ∑ zi2 + + γ1 γ2 2 g (x) i=1 ~T ~ ~ T ~ n θ θ T T g ( x) n 2 1 2 ~ (−∑ ci zi − g (x) znu + g (x) znua ) + = + − ∑z γ1 γ 2 2 g 2 (x) i=1 i g (x) i=1 ~T ~ ~ T ~ θ θ TT g ( x) n −1 n 2 ~ (∑ ci zi ) − znu + = + + znua − 2 ∑ zi2 γ1 γ2 2 g (x) i=1 g (x) i=1 ~T ~ ~ T ~ θ θ TT g ( x) n 2 −1 n ~T T ˆ ~ T ˆ 2 (∑ ci zi ) − zn (− θ Φ T + T Φθ + ε ) + = + + z n u a − 2 ∑ zi γ1 γ2 g (x) i=1 2 g (x) i=1 ~ ~ g ( x) n −1 n ~T θ ~T T T ˆ 2 (∑ ci zi ) + θ ( + znΦ T) + T ( − znΦT θˆ ) − znε + znua − 2 ∑ zi2 = γ1 γ2 2 g ( x) i=1 g (x) i=1. =. g ( x) n 2 −1 n ( ∑ zi ) + z n u a − z nε (∑ ci zi2 ) − 2 g (x) 2 i=1 g ( x) i=1 n gL −1 n (∑ ci zi2 ) + ( z 2 ) + znua − znε ≤ 2 ∑ i 2 g (x) i=1 g ( x) i=1 =. (2.37) Consider −. n n 1 gL 1 n 2 gL 2 (∑ ci zi2 ) + ( z ) zi ( = − ci ) ∑ i g ( x) ∑ g (x) i=1 2 g (x) 2 i=1 2 g ( x) i =1. Choose c* = ci , i = 1 n , such that c* ≥. (2.38). gL , then (2.38) can be express 2 g l ( x). 1 n 2 gL − c* ) ≤ 0 zi ( ∑ 2 g l ( x) g u (x) i=1 Then (2.37) becomes. 14. (2.39).

(26) V≤ ≤. 1 n 2 gL − c * ) + z n u a − z nε zi ( ∑ 2 g l (x) g u (x) i=1 z 1 n 2 gL − c * + z n ( − n 2 ) − z nε zi ( ∑ 2 g l (x) 2ρ g u (x) i=1. ρ 2ε 2 ρ 2ε 2 zn2 1 n 2 gL * ≤ ∑ zi ( 2 g (x) − c ) − 2 ρ 2 − znε + 2 − 2 g u (x) i=1 l. (2.40). ρ 2ε 2 ρ 2ε 2 zn2 1 n 2 gL * ε ≤ − − + + z c z ( ) ( )+ ∑ i 2 g (x) n 2 ρ g u (x) i=1 2 2 2 l ≤. ρ 2ε 2 1 n 2 gL 1 zn * 2 ρε − − + + ( ) ( ) z c ∑ i 2 g (x) 2 ρ 2 g u (x) i=1 l. ρ 2ε 2 1 n 2 gL * ≤ ∑ zi ( 2 g ( x ) − c ) + 2 g u (x) i=1 l Integrating (2.40) from t = 0 to t = T , we obtain V (T ) − V (0) ≤. T n 1 g ρ2 ( L − c* )∫0 ∑ zi2 (τ )dτ + gu (x) 2 gl (x) 2 i =1. T. ∫. 0. ε 2 (τ )dτ. (2.41). Since V2 (T ) ≥ 0 , (2.41) implies T n ρ2 1 gL (c * − ) ∫0 ∑ zi2 (τ )dτ ≤ V2 (0) + g u ( x) 2 g l ( x) 2 i =1. T. ∫. 0. ε 2 (τ )dτ. (2.42) T. Since V2 (0) is finite, if the approximation error ε ∈ L2 , that is ∫0 ε 2 (τ )dτ < ∞ . n. Using Barbalat’s lemma [19], lim | ∑ zi |= 0 , closed-loop signals are uniformly t →∞. i =1. stable and the state x1 can track the bound reference signal y m as close as possible.. The steps of B-spline adaptive backstepping controller design is summarized in the following. Step 1) Select the positive design parameters ci > 0 , i=1,2,…,n. Step 2) Choose an appropriate value for ρ in (2.35), and appropriate values for γ 1 and γ 2 in (2.34), respectively. 15.

(27) Step 3) Determine the order of the B-spline function and the number of knots for T . Then, compute the basis vector ξ in (2.23). Step 4) Obtain the update laws (2.34), and the B-spline adaptive backstepping controller (2.30), including the B-spline neural network identifier in (2.23) and the robust controller in (2.35). 2.4 Simulation Results. In this section, two examples are proposed. Example 1 is second-order unknown nonlinear system. Example 2 is third-order unknown nonlinear system. Theorem 1 is used to design the ideal backstepping controller and theorem 2 is used to design the B-spline adaptive backstepping controller. 2.1) Example: Consider the second-order unknown nonlinear system x1 = x2 x2 = −0.1x2 − x13 + 12 cos( x1 ) + (2 + cos( x1 ))u (t ). (2.43). By using theorem 1, the ideal backstepping controller can be expressed follow equation uideal =. 1 ( ym − c1 z1 − f ( x1 , x2 ) − c2 z2 − z1 ) . Because g ( x1 , x2 ). g ( x1 , x2 ) and f ( x1 , x 2 ) are unknown functions, the adaptive B-spline neural ˆ ) . By using network identifier controller can be designed as uˆ = θˆ Tξˆ ( q, T ~ ~ ˆ = γ z Φθˆ . ˆ and T = −T theorem 2, the adaptive laws are θ = −θˆ = −γ 1 z2Φ TT 2 2. Robust controller is u a = −. z2 . Reference signal is ym = sin(t ) . Learning 2ρ 2. rates are γ 1 = 5 , γ 2 = 0.5 . Initial condition point is (0.5,0.5). Damping constants are c1 = 2.5 , c2 = 2.5 . B-spline order is k = 3 . The knot vectors. 16.

(28) are t0 = −5, t1 = −4, t2 = −3,. , t12 = 6 .. The. simulation. results. of. ideal. backstepping controller are show in Fig. 2.3-Fig. 2.5. The simulation results of adaptive B-spline neural network controller for ρ = 0.1 are shown in Fig. 2.6-Fig. 2.8, ρ = 0.2 are shown in Fig. 2.9-Fig. 2.11, ρ = 0.3 are shown in Fig. 2.12-Fig. 2.14. The mean square errors of ρ = 0.1 ,. ρ = 0.2 and. ρ = 0.3 are shown in Table 2.1. By simulation results, the better tracking performance can be achieved as ρ is chosen smaller. The mean square error (MSE) is shown as follow: MSE =. 1 m (mi − mˆ i ) 2 ∑ m i =1. (2.44). where i is the index of the m points over which the MES is computed, mi is the actual output of the system, and mˆ i is the estimated output of the system.. Table 2.1 Three cases of ρ = 0.1 ρ = 0.2 and ρ = 0.3 Attenuation constant ( ρ ). Mean square error. ρ = 0.1. 0.0007412. ρ = 0.2. 0.0016021. ρ = 0.3. 0.004711. 17.

(29) 1.5 y ym 1. 0.5. 0. -0.5. -1. -1.5 0. 5. 10 time (sec). 15. 20. Fig. 2.3 Output y follow reference signal y m with ideal backstepping controller 0.6 0.5. 0.4. error. 0.3. 0.2 0.1. 0. -0.1 0. 5. 10 time (sec). 15. 20. Fig. 2.4. Error ym − y with ideal backstepping controller. 18.

(30) -2 -2.5 -3 -3.5. u. -4 -4.5 -5 -5.5 -6 -6.5 -7 0. 5. 10 time (sec). 15. 20. Fig. 2.5 Control input u with ideal backstepping controller. 1.5 y ym 1. 0.5. 0. -0.5. -1 0. 2. 4. 6 time. 8. 10. (sec). Fig. 2.6 Output y follow reference signal y m with B-spline adaptive backstepping controller ( ρ = 0.1 ). 19.

(31) 0.5 0.45 0.4 0.35. error. 0.3 0.25 0.2 0.15 0.1 0.05 0 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.7 Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.1 ). 100 50. 0. u. -50. -100 -150. -200. -250. 0. 2. 4. 6 time. 8. 10. (sec). Fig. 2.8 Control input u with B-spline adaptive backstepping controller ( ρ = 0.1 ). 20.

(32) 1.5 y ym 1. 0.5. 0. -0.5. -1 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.9 Output y follow reference signal y m with B-spline adaptive backstepping controller ( ρ = 0.2 ). 0.5 0.45 0.4 0.35. error. 0.3 0.25 0.2 0.15 0.1 0.05 0. 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.10. Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.2 ). 21.

(33) 10 0. -10. u. -20. -30 -40. -50. -60 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.11 Control input u with B-spline adaptive backstepping controller ( ρ = 0.2 ). 1.5 y ym 1. 0.5. 0. -0.5. -1 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.12 Output y follow reference signal y m with B-spline adaptive backstepping controller ( ρ = 0.3 ) 22.

(34) 0.5 0.45 0.4 0.35. error. 0.3 0.25 0.2 0.15 0.1 0.05 0 0. 2. 4. 6 time. 8. 10. (sec). Fig. 2.13 Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.3 ). 5. 0. u. -5. -10. -15. -20. -25 0. 2. 4. 6 time. 8. 10. (sec). Fig. 2.14. Control input u with B-spline adaptive backstepping controller ( ρ = 0.3 ). 23.

(35) 2.2) Example: Consider the third-order unknown nonlinear system x1 = x2 x2 = x3. (2.45). x3 = ( x12 + x2 )( x22 − x3 ) + ( 2 + cos( x1 + x2 + x3 ))u. By using theorem 2.1, the ideal backstepping controller can be expressed follow. equation. uideal =. 1 (−c1 z1 − c 2 z2 − c3 z3 − z1 − z2 + ym − f (x)) g ( x). ,. where x = [ x1 , x2 , x3 ]T . Because of g (x) and f (x) are unknown functions, the ˆ). adaptive B-spline neural network controller can be designed as uˆ = θˆ Tξˆ ( q, T. By using theorem 2.2, the adaptive laws are. ~ ˆ , θ = −θˆ = −γ 1 z3ΦTT. ~ ˆ = γ z Φθˆ . The robust control is u = − z3 . The reference signal is T = −T 2 3 a 2ρ 2 ym = sin(t ) . The learning rates are γ 1 = 10 , γ 2 = 0.5 . Initial condition point is. (0.5,0.5). The damping constants are c1 = 5 , c2 = 5 , c3 = 5 . B-spline order is k = 3 , The knot vectors are t0 = −5, t1 = −4, t2 = −3,. , t12 = 6 . The simulation. results of ideal backstepping controller are shown in Fig. 2.15-Fig. 2.17. The simulation results of adaptive B-spline neural network controller for ρ = 0.2 are shown in Fig. 2.18~Fig. 2.20, ρ = 0.3 are shown in Fig. 2.21-Fig. 2.23,. ρ = 0.4 are shown in Fig. 2.24-Fig. 2.26. The mean square error of ρ = 0.2 , ρ = 0.3 and ρ = 0.4 are shown in Table 2.2. Table 2.2 Three cases of ρ = 0.2 ρ = 0.3 and ρ = 0.4 Attenuation constant ( ρ ). Mean square error. ρ = 0.2. 0.0026463. ρ = 0.3. 0.002701. ρ = 0.4. 0.0028352. 24.

(36) 0.6 0.5 0.4. error. 0.3 0.2 0.1 0 -0.1 0. 5. 10 time. 15. 20. (sec). Fig. 2.16 Error ym − y with ideal backstepping controller. 1 0.8 0.6 0.4. u. 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0. 5. 10 time (sec). 15. 20. Fig. 2.17 Control input u with ideal backstepping controller. 25.

(37) 1.5 y ym 1. 0.5. 0. -0.5. -1. -1.5 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.18 Output y follow reference signal y m with B-spline adaptive backstepping controller ( ρ = 0.2 ). 0.6 0.5. 0.4. error. 0.3. 0.2 0.1. 0. -0.1 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.19 Error y m − y with B-spline adaptive backstepping controller ( ρ = 0.2 ) 26.

(38) 40 20 0. u. -20 -40 -60 -80 -100 -120 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.20 Control input u with B-spline adaptive backstepping controller ( ρ = 0.2 ). 1.5 y ym 1. 0.5. 0. -0.5. -1. -1.5 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.21 Output y follow reference signal y m with B-spline adaptive backstepping controller ( ρ = 0.3 ). 27.

(39) 0.6 0.5. 0.4. error. 0.3. 0.2 0.1. 0. -0.1 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.22 Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.3 ). 30 20 10. u. 0 -10 -20 -30 -40 -50 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.23 Control input u with B-spline adaptive backstepping controller ( ρ = 0.3 ). 28.

(40) 1.5 y ym 1. 0.5. 0. -0.5. -1. -1.5 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.24 Output y follow reference signal y m with B-spline adaptive backstepping controller ( ρ = 0.4 ). 0.6 0.5. 0.4. error. 0.3. 0.2 0.1. 0. -0.1 0. 2. 4. 6 time (sec). 8. 10. Fig. 2.25 Error ym − y with B-spline adaptive backstepping controller ( ρ = 0.4 ) 29.

(41) 15 10 5 0. u. -5 -10 -15 -20 -25 -30 0. 2. 4. 6 time. 8. 10. (sec). Fig. 2.26 Control input u with B-spline adaptive backstepping controller ( ρ = 0.4 ). 30.

(42) Chapter 3 B-spline Adaptive Backstepping Controllers for Nonaffine Nonlinear Systems. In this chapter, B-spline adaptive backstepping scheme for nonaffine nonlinear systems is proposed. The control scheme incorporates the backstepping design technique with the mean-estimation B-spline neural networks which are utilized to estimate the system dynamics. The mean-estimation B-spline neural networks use a mean estimation technique to develop the update laws for the design of online adaptive controllers. In doing so, differentiating B-spline basis functions is not required, and therefore the computation burden can be mitigated. In addition, two kinds of robust controllers are used to compensate unmodeling dynamics. Finally, computer simulation results are shown to demonstrate the applicability of the proposed scheme, and comparison between the two robust controllers is given by the simulation results.. This chapter is organized as follows. The system problem is formulated in section 3.1. The ideal backstepping controller and virtual controller is described in section 3.2. In section 3.3, a B-spline adaptive backstepping controller is described. Computer simulation results are shown in section 3.4. Finally, some conclusions are given in section 3.5.. 31.

(43) 3.1 Problem Formulation Consider the nonaffine nonlinear systems expressed in or transformed into a state-space form xn = f (x, u ). where x = [ x1 , x2 , xn ]T = [ x, x, x ( n−1) ]T ∈ R n is the state vector, and. (3.1) u∈R. is. the control input. The system function f ( x, u ) is unknown continuous function, and it is assumed that 0 < ∂f ∂u < ∞ . The control objective is to design a controller for system (3.1) such that all the signals in the closed-loop system are stable and the state x1 can track a bound reference signal y m arbitrarily closely.. 3.2 Design of Backstepping Controllers for Known Systems Consider the nth-order nonlinear system (3.1) where the system function f ( x, u ) is known. By using backstepping technique, an ideal backstepping. controller can be expressed follow: Step 1) Define a tracking error as z1 = x1 − ym. (3.2). Then, differentiating z1 can be expressed as z1 = x1 − ym. (3.3). Define a virtual controller as α1 = ym − c1 z1. (3.4). where c1 > 0 is a design parameter. From (3.3) and (3.4), if α1 = x1 , then we have lim z1 → 0 , that is, the state trajectory x1 can t →∞. asymptotically track the reference signal ym . Thus, define an error state as z 2 = x1 − α1 = x2 − α1 . Then, our next goal is to force the error state z2 to decay to zero. By using (3.3) and the fact that 32.

(44) x1 = z2 + α1 , Eq. (3.3) can be rewritten as z1 = z2 − c1 z1. (3.5). Step 2) Differentiating z2 can be expressed as z 2 = x2 − α1 = x3 − (−c1 z1 + ym ). (3.6). Likewise, define a virtual controller as α 2 = ym − c1 z1 − c2 z2 − z1. (3.7). where c2 > 0 is a design parameter. Moreover, define an error state z3 = x3 − α 2 . Then, by using (3.6) and the fact that x2 = z3 + α 2 ,. equation (3.6) can be rewritten as z2 = z3 − c2 z2 − z1. (3.8). Step 3) Let k be a positive integer. Define an error state as zk = xk − α k −1 . Then, differentiating zk is zk = xk − α k −1. (3.9). where 3 ≤ k ≤ n − 1 . Define a virtual controller as k. k −1. i =1. j =1. α k = ym( k ) − ∑ ci zi( k −i ) − ∑ z (jk −1− j ) , where ci > 0 is a design parameter. Moreover, define an error state as zk +1 = xk +1 − α k . Then, by using the fact that xk = zk +1 + α k , equation (3.9) can be rewritten as zk = zk +1 − ck zk − zk −1. (3.10). Step4) Differentiating zn can be expressed as zn = xn − α n−1. (3.11). Define ϕ = f (x, u ) + cn z n − α n−1 + zn−1 . By the assumption that 0 < ∂f ∂u < ∞ and implicit function theorem, there exists a ideal. backstepping controller uideal such that ϕ (x, uideal ) = 0 . By adding and subtracting the term cn zn − α n−1 + zn−1 in (3.1) and using the fact that. ϕ (x, uideal ) = 0 , we obtain xn = −cn zn − zn−1 + α n −1 . As a result, Eq. (3.11) can be rewritten as zn = −cn zn − zn −1 33. (3.12).

(45) Step5) Consider the Lyapunov function as follows V=. 1 2. n. ∑z. 2 i. (3.13). i =1. By differentiating (3.13) and using (3.5), (3.8), (3.10) and (3.12), we have n. n. V = ∑ zi zi = −∑ ci zi2 ≤ −c1 z12 i =1. i =1. From (3.13) and (3.14), we can conclude that Moreover, from (3.5),. ∫. ∞. 0. (3.14). z1. zi. is bounded.. is also bounded. Integrating (3.14), we get. z12 (τ )dτ = −(V (∞) − V (0)) c1. (3.15). Because of the fact that the right side of (3.15) is bounded, we have z1 ∈ L2 . According to Barbalat’s Lemma, we have lim z1 = 0 , that is, t →∞. the state trajectory x1 can asymptotically track the reference signal ym .. 3.3 Design of B-spline Adaptive Backstepping Controllers for Unknown Systems Because the system function is unknown, the ideal backstepping controller can not be precisely obtained. To solve this problem, the B-spline adaptive backstepping controller is proposed for the unknown nonlinear system (3.1), instead of the ideal backstepping controller. The network structure and features of the b-spline neural network are shown in section 2.3. According to the universal approximation theorem, let uideal = u * + σ = θ∗T ξ(q, T∗ ) + σ , where σ denotes an approximation error, T∗ is. an optimal knot vector, θ∗ is an optimal parameter vector. According to the definition of the B-spline neural networks, define the B-spline neural control input as 34.

(46) ˆ) uˆ = θˆ Tξ(q, T. (3.16). ˆ are the estimation of u ∗ , and T ∗ , respectively. In the following, where uˆ, and T ˆ will be developed for the B-spline neural control the update laws of uˆ, and T. input (2.16) by using the mean estimation technique, and such B-spline neural networks with the mean estimation technique is called the mean estimation B-spline neural networks.. ˆ - δ between T* and By using the mean value theorem, there exist a point T ˆ , such that T. ˆ ) = ∂ξ | ˆ (T* - T ˆ) ξ (q,T* ) − ξ (q,T T=T-δ ∂T. (3.17). ∂ξ | ˆ . Then, by adding Define ξ = ξ (q, T* ) − ξ (q, Tˆ ) , T = T* − Tˆ , and Φ = ∂T T=T-δ ˆ , we obtain and subtracting the term Φ*T + ΦT ˆ + ΦT + d ξ = ΦT. (3.18). ˆ , d = (Φ − Φ* )T . Therefore, according to (3.18), let where Φ = Φ* − Φ ˆ ) in u~ = uideal − uˆ . By substituting uideal = θ∗T ξ(q,T∗ ) + σ and uˆ = θˆ Tξ(q, T u~ = uideal − uˆ , and adding and subtracting the term θˆ T ξ (q,T* ) , we obtain. u = θT ξ (q,T* ) + θˆ T ξ + σ. (3.19). ~ ˆ ) , Eq. where θ = θ* − θˆ . Then, by adding and subtracting the term θT ξ (q,T. (3.19) can be rewritten as ˆ ) +σ u = θT ξ + θˆ T ξ + θT ξ (q,T. (3.20). ˆ + ΦT + d in (3.20) and after some manipulations, we By substituting ξ = ΦT. obtain m. n. ˆ)−Φ ˆ TT ˆ ) + TT Φθ ˆ ˆ − ∑∑ Φ θˆ Tˆ + ε u = θ (ξ (q,T ij j i T. (3.21). i =1 j =1. ~ ~ ~ ~ where ε = θ T ΦT T + θˆ T ΦT T* + θ*d + σ , and Φij = Φij* − Φˆ ij , which are the. elements in the ith row and jth column of the matrix. Φ.. Next, utilizing (3.21) to derive the update law for the mean estimation B-spline neural networks and developing the B-spline adaptive backstepping 35.

(47) controller for the nonaffine systems (3.1) are given as follows. By using mean value theorem, the function ϕ = f (x, u ) + cn zn − α n−1 + zn−1 can be rewritten as ϕ (u ) = ϕ (uideal ) + g (x)(u − uideal ) , where uideal < u < u and g (x) = ∂ϕ ∂u |u =u . Because of the fact that ϕ (uideal ) = 0 , we have. ϕ (u ) = g (x)(u − uideal ). (3.22). For developing the B-spline adaptive backstepping controller for the nonaffine systems (3.1), we assume that there exist positive constants of g u and g l such that g l ≤ g (x) ≤ g u .. From (3.22), Eq. (1) can be rewritten as xn = g (x)(u − uideal ) − cn zn + α n−1 − zn −1. (3.23). Define the B-spline adaptive backstepping controller as u = uˆ + u a. (3.24). where ua is a robust controller. By substituting (3.11) into (3.23) zn = − g (x)u + g (x)ua − cn zn − zn−1. (3.25). On the basis of the above discussion, the following theorem can be obtained. Theorem3.1 : Consider the nonlinear nonaffine systems (3.1). Suppose that the. update laws of the mean estimation B-spline neural networks are ˆ)−Φ ˆ TT ˆ) θ = −θˆ = γ 1 zn (ξ (q,T ˆ = γ z Φθ ˆˆ T = −T 2 n. (3.26). ˆ ij = −γ 3 znθˆ jTˆi Φ ij = −Φ. where γ 1 , γ 2 , γ 3 are the positive learning rates. The B-spline adaptive backstepping controller with the update laws (3.26) is designed as (3.24), where the B-spline neural control input uˆ is given as (3.16). Then, according to the design of the robust controller, the following properties are guaranteed. (a). Suppose that ua =. − zn 2ρ 2. (3.27). where ρ is a prescribed attenuation constant. Then, all the signals in 36.

(48) the closed-loop are bounded, and the following inequality is satisfied. 1 gl. ∫. T. 0. 1 where υ = 2g l. e 2 (t )dt ≤ υ +. ρ2 2. ∫. T. 0. τ 2 (t )dt. (3.28). 2 θT (0)θ(0) TT (0)T(0) m n Φ ij (0) z (0) + + + ∑∑ ∑ 2γ 1 2γ 2 i =1 i =1 j =1 2γ 3 n. 2 i. (b). Suppose that u a = −αsign( z n ). (329). where α is a constant described later. Then, all the signals in the closed-loop are bounded, and lim z1 ( t ) = 0 . t →∞. ▲ Proof of (a): Define a Lyapunov function as. 1 V= 2g l. 2 θT θ TT T m n Φ ij z + + + ∑∑ ∑ 2γ 1 2γ 2 i =1 j =1 2γ 3 i =1 n. 2 i. (3.30). By differentiating (3.30) with respect to time and using (3.5) (3.8) (3.10) and (3.23), and after some manipulations, we obtain zn θT θ TT T m n Φ ij Φ ij −1 n 2 + + ∑∑ V = (∑ ci zi ) + (− g (x)u + g (x)ua ) + gl i =1 gl γ1 γ2 γ3 i =1 j =1. (3.31). By adding and subtracting the term u in (3.31), we have −1 n θT θ TT T m n Φ ij Φ ij zn 2 + + ∑∑ + ( g (x)ua + u ( g l − g (x))) V = (∑ ci zi ) − znu + γ1 γ2 γ3 gl i =1 gl i =1 j =1. (3.32) Substituting (3.26) into the above equation, we obtain V=. where ζ = (1 −. −1 n g ( x) (∑ ci zi2 ) + zn ( ua + ζ ) gl i =1 gl. (3.33). g ( x) )u − ε . By substituting (3.37) into (3.33), using the fact gl. that g (x) gl ≥ 1 , and completing the squares, we obtain 37.

(49) −1 n 1 zn ρ 2ζ 2 2 2 V ≤ (∑ ci zi ) − ( − ρζ ) + 2 ρ 2 g l i =1. (3.34). ρ 2ζ 2 −1 n 2 ≤ (∑ ci zi ) + 2 gl i =1 By integrating both sides of (3.34) from t = 0 to t = T and after some manipulations, we obtain 1 gl. T. n. 0. i =1. ∫ ∑c z. 2 i i. (t )dt ≤V (0) − V (T ) +. ρ2 2. ∫. T. 0. ζ 2 (t )dt. (3.35). Because of the fact that V (T ) ≥ 0 , we have 1 gl. where υ = V (0) =. 1 2g l. ∫. T. 0. e (t )dt ≤ υ + 2. ρ2 2. ∫. T. 0. τ 2 (t )dt. (3.36). 2 θT (0)θ(0) TT (0)T(0) m n Φ ij (0) . z (0) + + + ∑∑ ∑ 2 2 2 γ γ γ i =1 i =1 j =1 1 2 3 n. 2 i. Proof of (b): It is assumed that ζ ≤ ω , where ω is a bound constant.. Because of the assumption that g l ≤ g (x) ≤ g u , and the fact that u is bounded by using projection method to avoid the parameter drift phenomenon, the assumption that ζ ≤ ω is reasonable. Let α ≥ ω . By substituting (3.29) into (3.33), we obtain V≤. −1 n −1 2 (∑ ci zi2 ) ≤ z1 (t ) ≤ 0 g l i =1 gl. (3.37). According to (3.34) and Barbalat’s lemma, we can easily prove that all the closed-loop signals are bounded and lim z1 ( t ) = 0 , i.e., the system output x1 t →∞. can track the reference signal ym . This completes the proof. ▲. Note that to avoid the chattering problem, the sign(⋅) in (3.29) can be replaced by saturation function sat (⋅) defined as 38.

(50) ⎧1 , z n > β ⎪ ⎪z sat ( z n ) = ⎨ n , z n ≤ β ⎪β ⎪⎩− 1 , z n < β. (3.38). where β > 0 . The design procedure of the B-spline adaptive backstepping controller is summarized in the following.. Step 5) Select the positive design parameters ci > 0 , i=1,2,…,n. Step 6) Choose a constant α in (3.29) or an appropriate value for ρ in (3.27), and appropriate values for γ 1 and γ 2 in (3.26), respectively. Step 7) Determine the order of the B-spline function and the number of knots for T . Then, compute the basis vector ξ in (3.16). Step 8) Obtain the update laws (3.26), and the control law, including the B-spline neural control input in (3.16) and the robust controller in (3.24) or (3.25).. 3.4 Simulation Results. Example3.1: Consider the second-order unknown nonlinear system. x1 = x2 x2 = −0.1x2 − x13 + 12cos( x1 ) + (2 + cos( x1 ))u (t ). (3.39). ˆ)−Φ ˆ TT ˆ ) ,T ˆ = −γ z Φθ ˆ ˆ, Φ ˆ = γ z θˆ Tˆ . The adaptive laws are θˆ = −γ 1 z2 (ξ (q,T 2 2 ij 3 2 i j. The reference signal is given as ym = sin(t ) . Let γ 1 = 5 , γ 2 = 5 , c1 = 5 , c2 = 5 , and c3 = 5 . The initial states are set as x(0) = [0.5,0.5] . The order of the B-spline. neural network approximator is selected as k = 3 , and the number of its knot points is twelve. The robust controller is selected as u a = −αsat ( z n ) , where 39.

(51) β = 3 and α = 5 , and in this case, the simulation results are shown in Fig. 3.1.. The mean square errors (MSE) of u a = −αsat ( z n ) is 0.0029445. In addition, the robust control is selected as ua =. − zn , and in this case, simulation results 2ρ 2. are shown in Figs. 3.2-3.4, and the MSE of the tracking output error for. ρ = {0.1,0.3,0.5} are shown in Table 3.1. Figs. 3.2-3.4 show the results of ρ = 0.1 , ρ = 0.3 , and ρ = 0.5 , respectively. From simulation results, as ρ is. chosen smaller, the better tracking performance can be achieved at the expense of the larger control input u.. Table 3.1 Three cases of ρ = 0.1 , ρ = 0.3 and ρ = 0.5 Attenuation constant ( ρ ) Mean square error ρ = 0.1. 0.0026799. ρ = 0.3. 0.0033743. ρ = 0.5. 0.0051655. 40.

(52) 2. 20. y. 1 0 u. 0. ym. -20. -1 -2. 0. 5 time(sec). -40. 10. 0.6. 10. 0. 5 time(sec). 10. 1 z2. z1. 5 time(sec). 2. 0.4 0.2. 0. 0 -0.2. 0. 0. 5 time(sec). -1. 10. Fig. 3.1. The simulation results when u a = −αsat ( z n ) .. 2. 50. y1. 0 -50. u 0. -100. ym -1. 0. 5 time(sec). 10. -150. 0.8. 10. 0. 5 time(sec). 10. 1 z2. z1. 5 time(sec). 2. 0.6 0.4. 0. 0.2 0. 0. 0. 5 time(sec). 10. -1. Fig. 3.2. The simulation results when ua =. − zn , where ρ = 0.1 2ρ 2. 41.

(53) 5 0. 0. -5. u. 2. y1 ym -1 -2. -10. 0. 5 time(sec). -15. 10. 0.6. 10. 0. 5 time(sec). 10. z2. 1. 0.2. 0. 0. 0. 5 time(sec). 10. -1. Fig. 3.3. The simulation results when ua =. 2. − zn , where ρ = 0.3 2ρ 2. 5. y u. 1 0. 0. ym -1. 0. 5 time(sec). -5. 10. 0.6. 2. 0.4. 1 z2. z1. z1. 5 time(sec). 2. 0.4. -0.2. 0. 0.2 0 -0.2. 0. 5 time(sec). 10. 0. 5 time(sec). 10. 0 -1. 0. 5 time(sec). -2. 10. Fig. 3.4. The simulation results when ua =. − zn , where ρ = 0.5 2ρ 2. 42.

(54) Example3.2: Consider the third-order unknown nonlinear system x1 = x2 x2 = x3 x3 = x1 x2 x3 + 2u +. (3.40) 1 sin(0.5 x1 x2u ) x1 x2. ˆ TT ˆ ),T ˆ = −γ z Φθ ˆˆ, Φ ˆ = γ z θˆ Tˆ . The The adaptive law θˆ = −γ 1 z3 (ξ (q,Tˆ ) − Φ 2 3 ij 3 3 i j. reference signal is given as ym = sin(0.5t ) + cos(t ) . Let γ 1 = 2 ,. γ 2 = 2 , c1 = 1.5 , c2 = 1 , and c3 = 1 . The initial states are set as x(0) = [0.5,0.5] . The order of the B-spline neural network approximator is selected as k = 3 , and the number of its knot points is twelve. The robust control is selected as u a = −αsat ( z n ) , where β = 3 and α = 20 , and in this case, the simulation. results are show in Fig. 3.5. The MSE of u a = −αsat ( z n ) is 0.0093624. In addition, the robust control is selected as ua =. − zn , and in this case, 2ρ 2. simulation results are shown in Figs. 3.6-3.8, and the MSE of the tracking output error for ρ = {0.15,0.175,0.2} are shown in Table 3.. 2. Figs.. 3.6-3.8 show the results of ρ = 0.15 , ρ = 0.175 and ρ = 0.2 , respectively. From simulation results, as ρ is chosen smaller, the better tracking performance can be achieved at the expense of the larger control input u.. Table 3.2 Three cases of ρ = 0.15 , ρ = 0.175 and ρ = 0.2 Attenuation constant ( ρ ). Mean square error. ρ = 0.15. 0.0092555. ρ = 0.175. 0.0096391. ρ = 0.2. 0.0178290. 43.

(55) 5. ym. 0. y -5. 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 50 u. 0 -50. z1. 0.5 0 -0.5. Fig. 3.5. The simulation results u a = −αsat ( z n ) .. when. 5. ym. 0 -5. y 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. u. 20 0 -20. z1. 0.5 0 -0.5. Fig. 3.6. The simulation results when ua =. − zn , where ρ = 0.15 2ρ 2. 44.

(56) 2. ym. 0 -2. y 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 10 u. 0 -10. z1. 0.5 0 -0.5. Fig. 3.7. The simulation results when ua =. 2. ym. 0 -2. − zn , where ρ = 0.175 . 2ρ 2. y 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. 0. 2. 4. 6. 8. 10 12 time(sec). 14. 16. 18. 20. u. 10 0 -10. z1. 0.5 0 -0.5. Fig. 3.8. The simulation results when ua =. 45. − zn , where ρ = 0.2 2ρ 2.

(57) Chapter 4 B-spline Adaptive Backstepping Controllers for Nonaffine Nonlinear Systems with First Order Filter. In this chapter, B-spline adaptive backstepping scheme for nonaffine nonlinear systems with first order filters are proposed. The control scheme incorporates the backstepping design technique with the mean-estimation B-spline neural networks which are utilized to estimate the system dynamics. The backstepping design scheme has the explosion of complexity problem as the order n of system increases. This explosion of complexity is caused virtual controller include basis function and every step of backstepping technique need differentiate the basis of virtual controller. In order to overcome this problem, this chapter uses first order filter at each step of the backstepping design.. This chapter is organized as follows. The system problem is formulated in section 4.1. In section 4.2, a B-spline adaptive backstepping controller with first order filters is described. The stability analysis is shown in section 4.3. Computer simulation results are shown in section 4.4.. 4.1 Problem Formulation Consider the nonaffine nonlinear systems expressed in or transformed into a state-space form x1 = f1 ( x1 ) + x 2 x 2 = f 2 ( x1 , x 2 ) + x3. (4.1) x n = f n ( x, u ) 46.

(58) where x = [ x 1 , x 2. xn ]∈ R. n. is the state vector, and u ∈ R is the. control input. The system functions f1 , f 2 ,. , f n are unknown continuous. functions, and it is assumed that 0 < ∂f n / ∂u < ∞ . The control objective is to design a controller for system (4.1) such that all the signals in the closed-loop system are stable and the state x1 can track a bound reference signal y m arbitrarily closely.. 4.2 Design of B-spline Adaptive Backstepping Controllers with First Order Filter. In this section, we use basckstepping technique to design a B-spline neural network controller for system (4.1). Step 1: At this step, we consider the first equation of system (4.1) x1 = f1 ( x1 ) + x2. (4.2). Define the tracking error as z1 = x1 − ym. (4.3). Then, differentiating z1 can be expressed as z1 = f1 ( x1 ) − ym + x2. (4.4). Define the virtual controller as X 2 = − f1 ( x1 ) + ym − c1 z1. (4.5). where c1 > 0 is a design parameter. From (4.4) and (4.5), if X 2 = x2 , then we have lim z1 → 0 , that is, the state trajectory x1 can asymptotically track t →∞. the reference signal y m .. According to the universal approximation theorem, let f1 ( x) = f1* + σ 1 = θ1*T ξ1 (q, T1* ) + σ 1 . An estimate function is defined as. ˆ are the estimations of f * , θ* , and T* . ˆ ) , where fˆ , θˆ , and T fˆ1 = θˆ 1ξ1 (q, T 1 1 1 1 1 1 1 47.

(59) Let fˆ1 pass through a first-order filter and introduces a new state variable Fˆ1 . Fˆ1 = fˆ1 / h1s + 1. (4.6). where s is a Laplace variable, h1 is a Laplace constant. Define the filter error between fˆ1 and Fˆ1 is e1 = Fˆ1 − fˆ1 . Then, differentiating e1 can be expressed as fˆ − Fˆ1 ˆ e e1 = Fˆ1 − fˆ1 = 1 − f1 = − 1 − fˆ h1 h1. (4.7). The estimated virtual controller is Xˆ 2 = − Fˆ1 + ym − c1 z1. (4.8). First, the illustration of the mean estimation technique is shown in Fig. 2.1. ˆ − δ between By using the mean value theorem, there exist a point T 1 1 ˆ , such that T1* and T 1. ˆ ) = ∂ξ1 | ˆ (T* − T ˆ) ξ1 (q, T1* ) − ξ1 (q, T 1 1 1 ∂T1 T =T −δ1 1. (4.9). 1. ~ ~ ˆ , and Φ = ∂ξ1 | ˆ . Then, Define ξ1 = ξ1 (q, T1* ) − ξ1 (q, Tˆ 1 ) , T1 = T1* − T 1 1 ∂T1 T =T −δ 1. 1. 1. ~ ˆ ~ by adding and subtracting the term Φ*1 T1 + Φ 1T1 , we obtain ~ ˆ T~ ~T~ ξ1 = Φ1 T1 + Φ1 T1 + d1 ~ ~ ˆ , d = (Φ − Φ* )T where Φ1 = Φ1* − Φ 1 1 1 1 1. (4.10). ~ Using (4.10) and the estimate error is defined as f1 = f1 − fˆ1 , and adding. and subtracting the term θˆ 1Tξ(q, T1* ) , we obtain ~ ~T f1 = θ1 ξ(q 1 , T1* ) + θˆ 1Tξ 1 + σ 1 (4.11) ~ ~ ˆ ) , Eq. where θ1 = θ1* − θˆ 1 . Then, by adding and subtracting the term θ1T ξ(q, T 1 (4.11) can be rewritten as ~ ~T ~ ˆ T ~ ~T ~ ˆ )+σ f1 = θ1 ξ1 + θ1 ξ1 + θ1 ξ1 (q, T 1 1. (4.12). By substituting (4.10) in (4.12) and after some manipulations, we obtain m n ~ ~T ~T ˆ ˆ ~ T ˆ ˆ ˆ f1 = θ1 (ξ1 (q, T1 ) − Φ1 T1 ) + T1 Φ1θ1 − ∑∑ Φ1,klθˆ1,lTˆ1,k + ε 1 (4.13) k =1 l =1 ~ ~ ~ ~ ~ where θ1 = θ1* − θˆ 1 , ε 1 = θ1T Φ1T T1 + θˆ 1T Φ1T T1* + θ1*d1 + σ 1 . 48.

(60) Finally, we define z 2 = x2 − Xˆ 2. (3.14). Subtracting (4.14) into (4.4), we obtain z1 = f1 − ym + z 2 + Xˆ 2 = f1 − Fˆ1 − c1 z1 + z 2 ~ = f1 − c1 z1 − e1 + z 2. (4.15). Step 2: At this step, we consider the second equation of system (4.1) x2 = f 2 ( x1 , x2 ) + x3. (4.16). Then, differentiating z2 can be expressed as z 2 = x2 − Xˆ 2 = f 2 ( x1 , x2 ) + x3 − Xˆ 2. (4.17). Define the virtual controller as X 3 = − f 2 ( x1 , x2 ) + Xˆ 2 − c2 z 2 − z1. (4.18). Because the system function and Xˆ 2 are unknown, the virtual controller can not be precisely obtained. To solve this problem, we assume f 2' = f 2 ( x1 , x2 ) − Xˆ 2 . According to the universal approximation theorem, let f 2' = f 2* + σ 2 = θ*2T ξ 2 (q, T2* ) + σ 2 . An estimate function is defined as ˆ ) , where fˆ , θˆ , and T ˆ are the estimation of f * , θ* , and fˆ2 = θˆ 2ξ 2 (q, T 2 2 2 2 2 2. T2* , Let fˆ2 pass through a first-order filter and introduces a new state. variable Fˆ2 . Fˆ2 = fˆ2 / h2 s + 1. (4.19). where s is a Laplace variable, h2 is a Laplace constant. Define the filter error between fˆ2 and Fˆ2 is e2 = Fˆ2 − fˆ2 . Differentiation e2 can be expressed as fˆ − Fˆ2 ˆ e − f 2 = − 2 − fˆ2 e2 = Fˆ2 − fˆ2 = 2 h2 h2. (4.20). Then the estimated virtual controller is Xˆ 3 = − Fˆ2 − c2 z 2 − z1 49. (4.21).

(61) ˆ − δ between By using the mean value theorem, there exist a point T 2 2 ˆ , such that T2* and T 2. ˆ ) = ∂ξ 2 | ˆ ˆ ) (T2* − T (4.22) ξ 2 (q, T2* ) − ξ 2 (q, T 2 2 T =T −δ 2 ∂T2 ~ ∂ξ 2 ~ * ˆ ˆ ), T | ˆ . Then, Define ξ2 = ξ 2 (q, T2* ) − ξ 2 (q, T 2 2 = T2 − T2 , and Φ 2 = ∂T2 T =T −δ 2. 2. 2. 2. 2. ~ ~ ˆ T by adding and subtracting the term Φ*2 T2 + Φ 2 2 , we obtain ~ ~ ~T~ ˆ TT ξ2 = Φ 2 2 + Φ 2 T2 + d 2 ~ ~ ˆ , d = (Φ − Φ* )T where Φ 2 = Φ*2 − Φ 2 2 2 2 2. ~ Using similar math skill of step 1, and f 2 = f 2' − fˆ2 we obtain m n ~ ~T ~T ˆ ˆ ~ T ˆ ˆ ˆ f 2 = θ2 (ξ 2 (q, T2 ) − Φ 2 T2 ) + T2 Φ 2θ 2 − ∑∑ Φ2,klθˆ2 ,lTˆ2,k + ε 2 ~ ~ T ~ T ~ ˆ Tk =~1 Tl =1 * * ˆ where θ2 = θ 2 − θ 2 , ε 2 = θ2 Φ 2 T2 + θ 2 Φ 2 T2 + θ*2d 2 + σ 2 .. (4.23). (4.24). Finally, we define z3 = x3 − Xˆ 3. (4.25). Subtracting (4.25) into (4.17), we obtain z2 = f 2 − X 2 + x3 = f 2' + x3 = f 2' + z3 + Xˆ 3 ~ = f 2 − c2 z2 − e2 + z3 − z1. (4.26). Step i: At this step, we consider the ith equation of system (4.1) xi = f i ( x1 , x2 , … , xi ) + xi +1. (4.27). zi = xi − Xˆ i. (4.28). Define. Then, differentiating z i can be expressed as zi = xi − Xˆ i = f i ( x1 , x2 ,. , xi ) + xi +1 − Xˆ i. (4.29). Define the virtual controller as X i +1 = − f i ( x1 , x2 ,. , xi ) + Xˆ i − ci zi − zi −1. (4.30). Because the system function and Xˆ i are unknown, the virtual controller 50.