簡化退火演算法基於模糊類神經網路控制器於非線性系統之控制

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(1)Reduced SA Fuzzy-neural Controller for Nonlinear Systems Student: Jian-Hao Liao. Adviser: Yih-Guang Leu Wei-Yen Wang. Department of Industrial Education National Taiwan Normal University. ABSTRACT In this thesis, a reduced simulated annealing algorithm used to tune the parameters of fuzzy neural networks is proposed for function approximation and adaptive control of nonlinear systems. For the design of adaptive controller, the reduced simulated annealing algorithm does not require the procedure of off-line learning and the complicated mathematical form. Compared with traditional adaptive controllers, computation loading can be effectively alleviated. In adaptive control procedure for nonlinear systems, the weights of the fuzzy neural controller are online adjusted by the reduced simulated annealing algorithm in order to generate the appropriate control input. For the purpose of on-line evaluating the stability of the closed-loop systems, an energy cost function derived from Lyapunov function is involved in the reduced simulated annealing algorithm. In addition, the system states may go into the unsafe region if the reduced simulated annealing algorithm can not instantaneously generate the appropriate weights. In order to guarantee the stability of the closed-loop nonlinear system, a supervisory controller is incorporated into the fuzzy neural controller.. I.

(2) Finally, some computer simulation examples and a servo motor experiment are provided to demonstrate the feasibility and effectiveness of the proposed method.. Keywords: simulated annealing algorithm, fuzzy neural networks, adaptive control, nonlinear systems.. II.

(3) 簡化退火演算法基於模糊類神經網路控制器於非線性系統之控制學生:廖建豪. 指導老師:呂藝光博士王偉彥博士國立台灣師範大學工業教育碩士班摘要. 本文提出一個利用簡化的模擬退火演算法來調整模糊類神經網路的參數，並將其應用於函數近似與非線性系統之適應控制器設計。此簡化的模擬退火演算法應用於適應控制器設計，不需要事先離線學習的程序和複雜的數學運算。相較於傳統非線性系統的適應控制器，可有效減少適應控制器所需複雜的數學運算。在非線性系統之適應控制過程中，模糊類神經控制器的權重値是經由模擬退火演算法來即時調整，以產生適當的控制輸入。為了即時評估閉迴路系統穩定的趨勢，本文從 Lyapunov 函數的推導過程中，提出一個能量成本函數於簡化的模擬退火最佳演算法中，藉著獲得較佳的閉迴路系統的穩定度。此外，由於簡化模擬退火法，可能在即時控制過程中使系統狀態進入不安全的區域。因此，加入監督控制器以限制閉迴路系統的狀態進入不安全的區域。本文藉由電腦模擬結果驗證所提出方法的可行性與效能。最後，將此模糊類神經控制器應用在直流伺服馬達追蹤控制實驗。. 關鍵詞:模擬退火演算法、模糊類神經、適應控制、非線性控制. III.

(4) 誌. 謝. 感謝指導教授呂藝光博士的諄諄教誨，教授的豐富學識讓學生在課業上受益良多，在專業領域的學習中更上層樓，此外，呂教授嚴謹的治學態度、謙沖為懷的素養，更是學生人格養成的最佳典範。在生活教育中，老師不時關心學生們的狀況，鼓勵學生運動以維持健康，支持學生在系辦工讀學習與人互動，更給予學生足夠的自由來規劃校外的瑣事，讓學生們在這個富有專業卻又不失人文氣度的環境中快樂的學習。同時也要感洪欽銘所長和王偉彥博士於學業的指導與生活的關心；再者，要感謝口試委員台灣科技大學蘇順豐博士、國立海洋大學吳政郎博士於口試時提供諸多寶貴的建議，使得本論文更為完整與充實。最後，還要感謝台灣師範大學能夠提供學生一個良好學習環境，讓學生能夠在充滿人文藝術與現代科技環境下學習成長，學生以最誠摯的心感謝曾經指導與教誨過學生的老師與曾經為這所學校貢獻心力的所有人。. 此外，還要感謝學長士恆、銘滄、宏見，同學建宏、建佑、俊堯、正皓、名峰、伯楷，學弟皓勇、嘉良、皓程、弨廣、小建宏、暉翔等人在碩士期間的相互幫助，更要感謝工教所助教耘盈、應電所助教琇文與琼姿，在學業與生活上提供許多幫忙與協助，濃情厚誼永誌不忘。. 最後，還要深深感謝雙親廖明雄先生與姚婉君女士的養育及栽培之恩，讓學生基礎的人格教育能健全發展，並養成積極進取負責任態度，也要感謝女友宜朔的陪伴及體諒，讓學生能夠專心於課業無後顧之憂。在此，願將成長之喜悅與他們共享，並貢獻自己所學於社會、國家。. IV.

(5) Contents ABSTRACT........................................................................................................I 摘. 要..........................................................................................................III. 誌. 謝..........................................................................................................IV. List of Figures ............................................................................................... VII List of Tables.................................................................................................... X Chapter 1 Introductions ..................................................................................1 1.1. Background......................................................................................1. 1.2. Motivation and Major Works...........................................................3. 1.3. Thesis Overview ..............................................................................4. Chapter 2 Evolutionary Learning of Fuzzy-Neural Networks Using a Reduced Simulated Annealing Optimization Algorithm............7 2.1. Fuzzy-Neural Networks...................................................................7. 2.2. Reduced Simulated Annealing Optimization (RSAO) Algorithm. for Off-line Learning ..................................................................................9 2.3. Simulation...................................................................................... 11. 2.4. Conclusions....................................................................................22. Chapter 3 Indirect RSA On-Line Tuning of Fuzzy-Neural Networks for Uncertain Nonlinear Systems......................................................23 3.1. Problem Formulation .....................................................................23 3.1.1 The Design of Certainty Equivalent Controller .......................23 3.1.2 Supervisory Control .................................................................26. 3.2. Description of Reduce Simulated Annealing Algorithm for On-line. Controllers ................................................................................................27 3.3 Simulation Examples of the RIAFC ...................................................29 3.4 Conclusions.........................................................................................35 Chapter 4 Backstepping Adaptive Control of Uncertain Nonlinear Systems Using RSA On-Line Tuning of Fuzzy-Neural Networks ..........................................................................................................37 4.1. Problem Formulation .....................................................................37. V.

(6) 4.1.1 The Design of Backstepping Controller...................................37 4.1.2 On-line Learning of Fuzzy-Neural Backstepping Control Using RSAOA .............................................................................................40 4.2 Simulation Examples of the RBAFC..................................................43 4.3 Conclusions.........................................................................................49 Chapter 5 Design of Fuzzy-neural Controller Using Reduce Simulated Annealing Algorithms for MIMO Nonlinear Systems....................50 5.1. 5.2. Problem Formulation and Fuzzy-Neural Networks ......................50 5.1.1. Problem Formulation and Backstepping Control Design...50. 5.1.2. Description of MIMO Fuzzy-neural Networks ..................53. Development of Simulated Annealing Adaptive Fuzzy-neural. Control Scheme.........................................................................................54 5.3. Simulation Examples.....................................................................57. 5.4. Conclusions....................................................................................62. Chapter 6 Design of Fuzzy-neural Controllers for DC Servo motors Using Reduced Simulated Annealing Algorithms................................64 6.1. Problem Description of DC Servo Motors ....................................64. 6.2. Simulation Results .........................................................................66. 6.3. The Hardware Structure and Experimental Results ......................70. 6.4. Conclusions....................................................................................77. Chapter 7 Summaries and Suggestions for Future Research ....................78 7.1 Summaries ..........................................................................................78 7.2 Suggestions for Future Research ........................................................79 References .......................................................................................................80. VI.

(7) List of Figures Fig.2-1. Configuration of fuzzy neural network…………………………….....8 Fig.2-2. Desired approximating surface……..……………………………….12 Fig.2-3. Output of the fuzzy-neural network trained by the proposed RSAOA after 40 iterations……………………………………….….13 Fig.2-4. Error curve of the fuzzy-neural network trained by the RSAOA with respect to it iterations……………………………….………….13 Fig.2-5. The series-parallel identification model………………………...…...16 Fig.2-6. Training curve of the nonlinear function g (u ) using training data from Table 2-2…………………………………………………16 Fig.2-7. Output of the nonlinear function g (u ) and the gradient descent identification method for 40 iterations of learning………....18 Fig.2-8. Error curves of the approximated fˆ [u (k )] using two method for 40 iterations of learning……………………………………....…18 Fig.2-9. The gradient descent identification method for 300 iterations of learning……………………………………………..………..……20 Fig.2-10. Error curve of the approximated fˆ [u (k )] using gradient descent method for 300 iterations of learning………….………..….20 Fig.2-11. Output of the nonlinear system (dotted line) and the identification model (solid line) using the proposed RSAOA……..21 Fig.2-12. Identification error of the approximated model of Fig.2-11…….....22 Fig.3-1. The block diagram of RIAFC……………………….……………... 29 Fig.3-2. The system output y (t ) and bounded reference yd (t ) ……..……......31 Fig.3-3. The tracking error e ………………………………...…………..….32 Fig.3-4. The control input u(t)…………………………………………….…32 Fig.3-5. The system output y (t ) and bounded reference ym (t ) …...…..….…...34 Fig.3-6. The tracking error e ………………………………………………..35 Fig.3-7. The control input u(t)………………………………………….....…35 Fig.4-1 The block diagram of RBAFC……………...……….…………..…..43. VII.

(8) Fig.4-2. The system output y (t ) and bounded reference yd (t ) ……………….45 Fig.4-3. The tracking error e …………………………………………….....45 Fig.4-4. The control input u(t)………………………………………….…....46 Fig.4-5. The system output y (t ) and bounded reference ym (t ) ……...…..…47 Fig.4-6. The tracking error e ……………………………………………......48 Fig.4-7. The control input u(t)…………………………………………..…...48 Fig.5-1. The system output y11 (t ) and bounded reference y1d (t ) (case 1)…...59 Fig.5-2. The system output y21 (t ) and bounded reference y2 d (t ) (case 1)........59 Fig.5-3. The control inputs u1 and u2 (case 1)………...………………….60 Fig.5-4. The tracking errors z11 (t ) and z21 (t ) (case 1)………………….......60 Fig.5-5. The system output y11 (t ) and bounded reference y1d (t ) (case 2)........61 Fig.5-6. The system output y21 (t ) and bounded reference y2 d (t ) (case 2)........61 Fig.5-7. The control inputs u1 and u2 (case 2)………………………...….62 Fig.5-8. The tracking errors z11 (t ) and z21 (t ) (case 2)………………..…..62 Fig.6-1. Circuits of the servomotor……………………...…………………...64 Fig.6-2. The system output y (t ) and bounded reference yd (t ) (case1)………..68 Fig.6-3. The tracking error e (case1)………………………………………..68 Fig.6-4. The control input u(t) (case1)………………………………………69 Fig.6-5. The system output y (t ) and bounded reference yd (t ) (case2)....…69 Fig.6-6. The tracking error e (case2)……………………………..………....70 Fig.6-7. The control input u(t)(case2)……………………...………………..70 Fig.6-8. The MT22R2-24 DC servomotor…………………………………..71 Fig.6-9. Relationship of the control modules output and motors output..…...71 Fig.6-10. The block diagram of hardware implementation……………….…73 Fig.6-11. Control system block diagram………………………………..…...73 Fig.6-12. The system output x1 and bounded reference r (case1).…….….74 Fig.6-13. The tracking error e (case1)……….……………………………....75 Fig.6-14. The control input u(t) (case1)…………………………………..….75. VIII.

(9) Fig.6-15 The system output x1 and bounded reference r (case2)……….....76 Fig.6-16. The tracking error e (case2)……………………...………………..76 Fig.6-17. The control input u(t) (case2)…………...…………………………77. IX.

(10) List of Tables Table 2-1. Error with respect to it iterations in Example1………………...…..14 Table 2-2. Training data and approximated data obtained by RSAOA for 40 iterations of learning……………………..…………………….17 Table 2-3. Error with respect to it iterations in Example2……………………19 Table 6-1. The specifications of MT22R2-24…………………………..…….72. X.

(11) Chapter 1 Introductions 1.1 Background Recently, by using universal approximators such as neural networks [1], fuzzy logic systems [2], etc., nonlinear function modeling via them has been widely developed for many practical applications [3-7]. Moreover, combining fuzzy logic with neural networks [5-7] has been developed to improve the effectiveness of function approximation and control system modeling. In fuzzy logic systems, the construction of membership functions is important in practical applications of fuzzy set theory [3][8][9]. In [3], the fuzzy B-spline membership functions (BMF), which possesses the local control property, have been applied to successfully construct the fuzzy-neural control. The design of fuzzy logic systems and/or neural networks for adaptive controllers [11-16] has been widely developed because of the universal approximation feature [17-18], and the stability analysis of the adaptive fuzzy logic and/or neural network controllers for nonlinear systems is generally provided by Lyapunov stability theory. To absolutely guarantee the stability of the closed-loop systems, a supervisory controller for fuzzy systems has been proposed in [14]. In general, these universal approximators as mentioned above are trained via gradient-based methods, which may only find a local minimum solution during the learning process. Unfortunately, such techniques also suffer from difficulties, such as the choice of starting guess and convergence. Moreover, since the cost function generally has multiple local minimal, the attainment of global optimum by these nonlinear optimization techniques is difficult [10]. To search for global optimal solutions, many heuristic learning algorithms have. 1.

(12) been proposed, such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Simulated Annealing (SA). The simulated annealing algorithm [19-48] have been incorporated into the design of fuzzy logic systems and/or neural networks systematically because they possess the simple implement ability and the capability of escaping from local optima. Simulated annealing (SA), which was proposed by N. Metropolis [19], is a kind of heuristic random searching method. Kirkpartick transformed simulated annealing model into a mathematical process [20]. The SA algorithm has some advantages of searching solution easily, escaping from the local optimum solution and approaching the global optimum solution [23]. Therefore, this heuristic algorithm has been applied in many fields, such as tuning methods for proportional-integral-derivative (PID) controllers [28-29] [33] [38], very large scale integration (VLSI) [21-22], and artificial neural networks [23-24]. However, the SA algorithm also has some drawbacks of solving the optimization problem. First, it is almost impossible to derive an analytic solution and thus the optimal cost function value is unknown in advance for tuning PID controllers. Second problem is that the robustness constraints for all proportional (P) control systems are very intractable. For solving with the aforementioned difficulties, many researchers have proposed efficient approaches to directly and simultaneously optimize the four fuzzy neural networks by using a newly developed orthogonal simulated annealing [28][30-31]. Third, in standard SA algorithm, a large share of the computation time is spent in randomly generating and evaluating solutions [39] [41-43]. It means that the standard SA algorithms convergence speed is low. In the field of unit commitment (UC) in power systems, many researchers provide absolutely stochastic SA methods to overcome the problem of the fuzzy UC. 2.

(13) [39] [44]. The same torment exists in the optimal design of electromagnetic devices. Therefore, adaptive simulated annealing has been proposed to accelerate the convergence speed and reinforce the efficiency of this algorithm [25-27]. In the past decade, there have been significant research efforts on control schemes for nonlinear systems via indirect adaptive control [58-61]. The fundamental concept of indirect adaptive control is to transform a nonlinear system into a linear one. Therefore, linear control techniques can be used to acquire the desired performance. Some preliminary results have been shown in [58]. As compared with indirect adaptive control methods [49], backstepping technique [50-51] 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. Its main design procedure is that an appropriate state and virtual control are selected for each smaller subsystem, then the state equation is rewritten in terms of them, and finally Lyapunov functions are chosen for these subsystems so that the true controller integrating the individual controls of these subsystems guarantees 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 [52-57] and indirect adaptive control [32] methods have been proposed to control nonlinear systems with unknown system dynamics by combining the intelligent control methods with backstepping or indirect design.. 1.2 Motivation and Major Works Since the complicated mathematical form for fuzzy logic systems and/or. 3.

(14) neural networks [62-65], such as the update laws and the Lyapunov condition for the system stability, must be solved, it is difficult to implement the control algorithms into real controllers. Moreover, the design of controllers incorporated into SA algorithms generally requires the procedure of off-line learning before they on-line control a plant. Also, the standard SA algorithm requires lots of computing. That results in difficulty in on-line control. Thus, in this thesis, to avoid solving complicated mathematical equations, a reduced simulated annealing optimum algorithm (RSAOA) fuzzy neural control without the procedure of off-line learning is developed for uncertain nonlinear systems, and the stability of the closed-loop system is guaranteed. More specifically, we propose an RSAOA-based fuzzy-neural controller for uncertain nonlinear systems. The weighting factors of the fuzzy neural controller are tuned on-line via the RSAOA approach, instead of solving complicated mathematical equations [66]. For the purpose of on-line tuning these parameters and evaluating the stability of the closed-loop system, a special cost function is included in the RSAOA approach. According to the special cost function, the RSAOA algorithm can online tune the parameters of the fuzzy-neural networks to control the uncertain nonlinear systems. In addition, in order to guarantee the stability of the closed-loop nonlinear system, the concept of supervisory control is incorporated into the RSAOA-based fuzzy-neural controller.. 1.3 Thesis Overview This thesis is organized as follows: Chapter 1: We describe the background, motivation, major works and framework of this thesis.. 4.

(15) Chapter 2: We describe the evolutionary learning of fuzzy-neural networks using a reduced simulated annealing optimization algorithm in this chapter. Offline learning for the fuzzy-neural network is considered by using the RSAOA. The simulation results show that it provides a suited way of learning for the fuzzy-neural network. Chapter 3: A simulated annealing indirect adaptive fuzzy-neural control scheme is proposed for a class of single-input single-output (SISO) nonlinear systems. The control scheme incorporates a reduced simulated annealing algorithm and fuzzy-neural networks into indirect adaptive control design. In addition, in order to guarantee that the system states are confined to the safe region, a safe control term is incorporated into the control scheme. Chapter 4: A simulated annealing adaptive fuzzy-neural control scheme is proposed for a class of single-input single-output (SISO) nonlinear systems. The control scheme incorporates a reduced simulated annealing algorithm and fuzzy-neural networks into backstepping design. In addition, in order to guarantee that the system states are confined to the safe region, a safe control term is incorporated into the control scheme. Chapter 5: A simulated annealing adaptive fuzzy-neural control scheme is proposed for a class of multiple-input multiple-output (MIMO) nonlinear systems. The control scheme incorporates a reduced simulated annealing algorithm and fuzzy-neural networks into backstepping design. In addition, in order to guarantee that the. 5.

(16) system states are confined to the safe region, a safe control term is incorporated into the control scheme. Chapter 6: Based on the reduced simulated annealing algorithm, A DC servomotor experiment is performed in order to verify the effectiveness of the proposed method in real-time control. Chapter 7: Conclusion in the final chapter in this thesis.. 6.

(17) Chapter 2 Evolutionary Learning of Fuzzy-Neural Networks Using a Reduced Simulated Annealing Optimization Algorithm A novel method of adjusting the weights of fuzzy-neural networks using a reduced SA optimization algorithm (RSAOA) is proposed in this chapter. This method can be used to search for the optimal parameters. The RSAOA is applied in function approximation. The simulation results show that the RSAOA is a good learning algorithm for fuzzy neural networks.. 2.1 Fuzzy-Neural Networks A fuzzy-neural network is generally a fuzzy inference system constructed from structure of neural networks. A learning algorithm is used to adjust the weights of the fuzzy inference system [2], [3]. Fig. 2-1 shows the configuration of a fuzzy-neural network [15]. The system has a total of four layers. Nodes at layer I are input nodes that represent input linguistic variables. Nodes at layer II, represent the values of the membership function of total linguistic variables. At layer III, nodes are the values of the fuzzy basis vector ξ . Each node at layer III is a fuzzy rule. Layer III and layer IV are fully connected by the weights, θ T = w p = [ w1p w2p L whp ]T , i.e., the adjustable parameters. Layer IV is the output layer y (x) .. 7.

(18) Fig. 2-1. Configuration of fuzzy neural network. The fuzzy inference engine uses fuzzy IF-THEN rules to perform a mapping from training input data xq , q = 1, 2,L , n, xT = [ x1 x2 L xn ] ∈ ℜ n to output data y p , y ∈ ℜ .The ith fuzzy rule has the following form: R ( i ) : if x1 is A1i and L and xn is Ani. (2-1). then y is B i. where i is a rule number, and Aqi and B i are fuzzy sets. By using product inference, center-averaging and singleton fuzzification, the output of the fuzzy-neural network can be expressed as: y (x | wp ) =. h. n. i =1. q =1. ∑ wip (∏ μ A ( xq )) n. h. ∑ (∏ μ i =1. i q. q =1. Aqi. (2-2). ( xq )). = θ ξ ( x) T. where μ A ( xq ) is the membership function value of the fuzzy variable xq , h i q. is the total number of IF-THEN rules, wqi is the point at which μ B ( wip ) = 1 , i. θ T = wp = [ w1p w2p L whp ]T is a weighting vector, and ξ T = [ξ 1 ξ 2 L ξ h ] is the. 8.

(19) fuzzy basis vector. ξ i is defined as n. ξ ( x) = i. (∏ μ A ( xq )). q =1 n h. i q. ∑ (∏ μ i =1. q =1. (2-3) Aqi. ( xq )). By adjusting the weights wip of the fuzzy-neural network, the learning algorithm attempts to minimize the error function as follows: e p ( wp ) = ( y p − y*p ) 2. (2-4). for a single output system, or E ( w) = Y − Y *. 2. (2-5). for a multiple output system, where Y = [Y1 Y2 L Ym ] is an m-dimensional vector of the actual outputs of the fuzzy-neural network, Y* = [Y1* Y2* L Ym* ] is an m-dimensional vector of the desired outputs, and w = [ w1T w2T L wmT ]T is a weighting vector of the fuzzy-neural network for m outputs.. 2.2 Reduced Simulated Annealing Optimization (RSAO) Algorithm for Off-line Learning In the standard SA algorithm [67], it initializes a solution space first and many solution sets exist in this solution space. Through some arrangements and selections, the SA algorithm would find the best solution set. This best solution set is the optimum solution set in SA algorithm. For fuzzy-neural networks, the SA algorithm, however, needs the process of off-line learning. This results is difficulty in real-time control application. Therefore, in this thesis, we propose a Reduced Simulated Annealing (RSA) algorithm to alleviate the computation load. The RSA algorithm is characterized by two features: (1) one perturbation in each temperature, and (2) perform compact perturbation mechanism and special cooling schedule on the state by the cost. 9.

(20) function. The cost function is defined as Φ=−. 1 1+ E. (2-6). where E is an estimation error function defined in (2-5). In Greed Search, in each temperature, the current solution w old perturbs to become a new solution w new . Let ΔΦ = Φ (w new ) − Φ (w old ) . If ΔΦ < 0 , the new solution w new is. accepted. Otherwise, we abandon the new solution w new and preserve the current solution w old . So this method just finds the local optimum solution but not global optimum solution. The SA algorithm, however, imports the conception of Boltzmann probabilistic distribution. So, it can escape from local optimum and approach the global optimum. The Boltzmann probabilistic distribution is defined as p = e. − ΔΦ Kb T. , where K b denotes Boltzmann constant. and T is temperature. If p > r , where r is given at random in the interval [0, 1], the new solution w new is accepted. Otherwise, the current solution w old is preserved. As a result of the probabilistic condition, the SA algorithm. can escape from local optimum and approach the global optimum.. The reduced SAO algorithm can be discussed as follow: Initial configuration space: Let w = [w1 w2 L wm ]T be a solution in the. solution space. Cost function: The cost function is given as (2-6) Perturbation mechanism: The goal of perturbation is to produce a new. solution according to the present solution. In this thesis, we design one perturbation in each temperature and the perturbation mechanism is given as follow:. 10.

(21) w new. ⎧w old + αγ β (w max − w old ), =⎨ β ⎩ w old − αγ (w old − w min ),. if ε > 0.5 if ε ≤ 0.5. (2-7). where the factors α and β are constants, and γ and ε are random in the interval [0,1]. Acceptance condition: The acceptance probability function for the new. solution is defined as ⎧1, ⎪ P (w p _ new ) ⎨ Φ ( w p _ old )−Φ ( w p _ new ) KbT ⎪e > r, ⎩. if if. Φ (w p _ new ) ≤ Φ (w p _ old ) Φ (w p _ new )>Φ (w p _ old ). (2-8). where r is random in the interval [0,1]. Cooling schedule: Cooling schedule is a crystallizing process. If. temperature decreases faster, the time of crystallization is shorter but the crystal would have defects easily. If temperature decreases slower, the time of crystallization is longer but the crystal would be more perfect. Here, the cooling schedule is given as follow: Ts +1 = e. (. − s2. σ. ). T0. (2-9). where s is the number of iterations of SA and σ is a constant.. 2.3 Simulation In this section, using the proposed RSAOA, two examples are illustrated to show the effects of training of the fuzzy-neural network for function approximation. Each input of the fuzzy-neural network has seven membership functions in the two examples.. Example 2-1: Here we describe a process of offline learning [32]. Two input. variables and one output variable are used to approximate a desired surface. 11.

(22) shown in Fig. 2-2. Forty-nine training data pairs are given. The adjustable parameters are in the intervals D = [ wmin wmax ] = [−10 10] and α =0.1, β =2,. σ =10, T0 = 1000 . In the fuzzy-neural network, the number of the weightings is 49. Fig. 2-3 shows the simulation results of RSAOA after 100 iterations of learning. The result confirms the effectiveness of RSAOA, and the fuzzy-neural network can approximate the desired surface with less iterations. The error curve of RSAOA for 100 iterations of learning is shown in Fig. 2-4. The error with respect to its iterations is given in Table 2-1.. Fig. 2-2 Desired approximating surface.. 12.

(23) Fig. 2-3 Output of the fuzzy-neural network trained by the proposed RSAOA after 100 iterations.. Fig. 2-4 Error curve of the fuzzy-neural network trained by the RSAOA with respect to iterations.. 13.

(24) Table 2-1 Error with respect to iterations.. Iterations. Error. 1. 0.27105. 2. 0.087261. 3. 0.045344. 4. 0.030357. 5. 0.019737. 10. 0.0075411. 15. 0.0032227. 20. 0.0016179. 30. 0.00038908. 40. 0.00027872. 50. 0.00023119. 100. 0.00021729. Example 2-2: We perform another procedure which is offline learning and. online test in this example [32]. For a nonlinear system, the unknown nonlinear item is approximated by the fuzzy-neural network via RSAOA. First, some of training data from the unknown function are collected for an offline initial learning process of the fuzzy-neural network. After offline learning, the trained fuzzy-neural network replaces the unknown nonlinear function for online test. Consider a nonlinear system in [2] as y ( k + 1) = 0.3 y ( k ) + 0.6 y (k − 1) + g[u (k )]. (2-10). We assume that the unknown nonlinear function has the form: g (u ) = sin(2π u ) + 0.6sin(4π u ) + 0.2sin(6π u ). 14. (2-11).

(25) For offline learning, 21 training data are regularly collected from u = -1 to 1 are provided. The offline learning configuration of the 21 training data points is shown in Fig. 2-5(a). In order to show the approximated effect for the unknown nonlinear function g (u ) , a series-parallel model shown in Fig. 2-5(b) is defined as yˆ(k + 1) = 0.3 y (k ) + 0.6 y (k − 1) + fˆ [u (k )]. (2-12). where fˆ [u (k )] is the approximated function for g[u (k )] by the fuzzy-neural network.. The. adjustable. parameters. are. in. the. interval. D = [ wmin wmax ] = [−10 10] , and α =0.1, β =2, σ =10, T0 = 1000 . In the. fuzzy-neural network, the number of weightings is 7. Fig. 2-6 shows the exact curve of g (u ) driven by 21 training data in Table 2-2.. u. e. fˆ (u ). (a) Offline learning by 21 training data. 15.

(26) y (k + 1) g z −1. η0. u (k ). e z. −1. η1 fˆ ( x). yˆ(k + 1) z. η0. −1. z −1. η1. (b) Online test for real u (k ) = sin(2π k / 250). Fig. 2-5 The series-parallel identification model 0.8 0.6 0.4. g(u). 0.2 0 -0.2 -0.4 -0.6 -0.8 -1. -0.8. -0.6. -0.4. -0.2. 0 u. 0.2. 0.4. 0.6. 0.8. 1. Fig. 2-6 Training curve of the nonlinear function g (u ) using training data from Table 2-2. 16.

(27) Table 2-2 Training data and approximated data obtained by RSAOA for 100 iterations of learning g (u ). u. -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0. -2.4493e-016 -0.52812 -0.63799 -0.47812 -0.3943 -0.4 -0.3943 -0.47812 -0.63799 -0.52812 0 0.52812 0.63799 0.47812 0.3943 0.4 0.3943 0.47812 0.63799 0.52812 2.4493e-016. fˆ ( x). 0.00066069 -0.51926 -0.63768 -0.47811 -0.39432 -0.40006 -0.39456 -0.4784 -0.63808 -0.52836 -0.00041581 0.52841 0.63998 0.478 0.39446 0.39949 0.39485 0.47614 0.63806 0.52809 0.024094. Besides, the back-propagation gradient descent method is also built for learning the weightings of fuzzy-neural network [3]. The initial values of the weightings are randomly generated from the interval -10 to 10, which is the same as that of RSAOA method, and learning rate of gradient descent method is 0.005. After 100 iterations of learning, as demonstrated in Fig. 2-7, the approximated function fˆ (u ) learned by RSAOA method is much closer to the exact function g (u ) than the gradient descent method. The error curves with respect to iterations are shown in Fig. 2-8. It is obvious from Fig. 2-8 that RSAOA method converges faster than the gradient descent method. Note that. 17.

(28) all of the error curves shown in Fig. 2-8 are calculated by (2-6), and error with respect to iterations is shown in Table 2-3.. Fig. 2-7 Two method approach to nonlinear function g (u ) for 100 iterations of learning. Fig. 2-8 Error curves of the approximated fˆ [u (k )] using two method for 100. 18.

(29) iterations of learning. Table 2-3 Error with respect to iterations. Iterations. error Gradient descent. RSAOA. 1. 2.0158. 0.60798. 2. 1.9212. 0.44276. 3. 1.8316. 0.39962. 4. 1.7465. 0.3459. 5. 1.6659. 0.2589. 10. 1.5894. 0.13168. 15. 1.5167. 0.11088. 20. 1.4476. 0.066867. 30. 1.382. 0.034946. 40. 1.3197. 0.015955. 50. 1.2604. 0.003084. 100. 1.2041. 0.0019518. From Fig. 2-7, Fig. 2-8 and Table2-3 show that the effect of gradient descent method is bad for 100 iterations. Fig. 2-9 shows that the effect of gradient descent method for 300 iterations and the error curves with its iterations are shown in Fig. 2-10.. 19.

(30) Fig. 2-9 The gradient descent identification method for 300 iterations of learning. Fig. 2-10 Error curve of the approximated fˆ [u (k )] using gradient descent method for 300 iterations of learning. 20.

(31) For online test, we assume that the series-parallel model shown in Fig. 2-5(b) is driven by u (k ) = sin(2π k / 250) and α =1, β =1, σ =10, T0 = 1000 . The response of the nonlinear system in which g[u (k )] is. approximated by the fuzzy-neural network, and learned offline for 10 iterations of learning by the RSAOA is shown in Fig. 2-11. The error curve is shown in Fig. 2-12. As demonstrated in Fig. 2-11 and Fig. 2-12, the fuzzy-neural network using the proposed RSAOA successfully approximates the unknown nonlinear function g[u (k )] .. Fig. 2-11. Output of the nonlinear system (dotted line) and the identification model (solid line) using the proposed RSAOA. 21.

(32) Fig. 2-12. Identification error of the approximated model of Fig. 2-11. 2.4 Conclusions This chapter proposed a RSA, which can be successfully applied in the fuzzy-neural network to search for the optimal parameters, in spite of a vast number of adjustable parameters of the fuzzy neural networks. Two examples for RSA verify that it has outstanding efficacy in learning and approximation.. 22.

(33) Chapter 3 Indirect RSA On-Line Tuning of Fuzzy-Neural Networks for Uncertain Nonlinear Systems In this chapter, an RSA indirect adaptive fuzzy-neural controller (RIAFC) for uncertain nonlinear systems is proposed by using a reduced simulated annealing algorithm (RSA). The weighting factors of the adaptive fuzzy-neural controller are tuned on-line via the RSA approach. For the purpose of on-line tuning these parameters and evaluating the stability of the closed-loop system, a cost function is included in the RSA approach. In addition, in order to guarantee that the system states are confined to the safe region, a supervisory controller is incorporated into the RIAFC. To illustrate the feasibility and applicability of the proposed method, two examples of nonlinear systems controlled by the RIAFC are demonstrated.. 3.1 Problem Formulation In this section, we describe the control problem for a class of nonlinear systems, and then design the controller.. 3.1.1 The Design of Certainty Equivalent Controller Here we consider another nth-order nonlinear system of the form x&1 = x2 x& 2 = x3 L x& n = f ( x1 ,K, xn ) + g ( x1 ,K, xn )u. (3-1). y = x1. where u ∈ R and y ∈ R are the input and output of the system, respectively, 23.

(34) and x = [ x1 , x2 ,K , xn ]T is the state vector of the system, which is assumed to be available for measurement. We assume that f and g are uncertain continuous functions, and g is, without loss of generality, a strictly positive function. In [49], these systems are in normal form and have a relative degree n. Our objective is to design an indirect adaptive fuzzy-neural controller so that the system output y follows a given bounded reference signal ym . First, let e = y − ym , δ = [e, e&,K , e( n−1) ]T and k = [kn ,K , k1 ]T ∈ R n such that all roots. of a polynomial h( s ) = s n + k1s n −1 + L + kn are in the open left half-plane. If the functions f and g are known, then the optimal control law is 1 [ ym( n ) − k T δ − f ( x)] u= g ( x). (3-2). From (3-1) and (3-2), we have e( n ) + k1e( n −1) + L + kn e = 0. (3-3). However, since f and g are uncertain, the optimal control law (3-18) cannot be obtained. To solve this problem, we use the fuzzy-neural systems as approximators to approximate the uncertain continuous functions, f and g. First, we replace f and g in (3-2) by fuzzy-neural networks , i.e., fˆ ( x | w f ) and gˆ ( x | wg ), respectively. Based on certainty equivalent controller [64], the resulting control law is uc =. 1 [ ym( n ) − k T δ − fˆ ( x | w f )] gˆ ( x | wg ). (3-4). Substituting (3-4) in (3-1) and after some manipulations, we obtain the error equation e( n ) = −k T δ + [ f ( x) − fˆ ( x | w f )] + [ g ( x) − gˆ ( x | wg )]uc. (3-5). or equivalently. δ& = Λ cδ + bc [( f ( x) − fˆ ( x | w f )) + ( g ( x) − gˆ ( x | wg ))uc ]. 24. (3-6).

(35) where ⎡ 0 ⎢ 0 ⎢ Λc = ⎢ L ⎢ ⎢ 0 ⎢⎣ − k n. 1 0 L 0 − k n −1. 0 ⎤ ⎡0⎤ 1 0 L 0 0 ⎥ ⎢L⎥ ⎥ L L L L L ⎥ , bc = ⎢ ⎥ ⎢0⎥ ⎥ 0 0 L 0 1 ⎥ ⎢ ⎥ ⎣1⎦ L L L L − k 1 ⎥⎦ 0. 0. L. 0. (3-7). Since Λ c is a stable matrix (| sI − Λ c |= s ( n ) + k1 s ( n −1) + L + k n ), we know that there exists a unique positive definite symmetric n × n matrix P, which satisfies the Lyapunov equation: ΛTc P + PΛ c = −Q. (3-8). where Q is a positive matrix. Let Vδ = 12 δ T Pδ . Using (3-4) and (3-8), we have 1 1 V&δ = δ&T Pδ + δ T Pδ& 2 2 1 = − δ T Qδ + δ T Pbc [ f ( x) − fˆ ( x | w f ) + ( g ( x) − gˆ ( x | wg ))uc ] 2. (3-9). In order to confine the state xi to a bounded value, Vδ must be bounded, which means we require that V&δ ≤ 0 , when Vδ is greater than a large constant V > 0 . Next, we assume the following.. Assumption 3.1: We can determine functions f U ( x), g U ( x) and g L ( x). such that | f ( x) |≤ f U ( x) and. g L ( x) ≤ g ( x) ≤ g U ( x). for x ∈ U c , where. f U ( x) < ∞, g U ( x) < ∞ , and g L ( x) > 0 for x ∈ U c .. Based on Assumption 3.1, equation (3-9) can be modified as 1 V&δ ≤ − δ T Qδ + | δ T Pbc | f ( x) − δ T Pbc fˆ ( x | w f ) 2 + | δ T Pbc | g ( x)uc − δ T Pbc gˆ ( x | wg )uc 1 ≤ − δ T Qδ + | δ T Pbc | f U ( x) − δ T Pbc fˆ ( x | w f ) 2 + | δ T Pbc | g U ( x)uc − δ T Pbc gˆ ( x | wg )uc. (3-10). The solution must move in the direction of smaller values of Vδ if V&δ < 0 . 25.

(36) Therefore, according to (3-10), we define a cost function for the RSA as Φ = −δ T Pb fˆ ( x | w ) − δ T Pb ( gˆ ( x | w )u (3-11) c. f. c. g. c. in order to on-line tune weightings, a solution with the smallest cost function denotes the optimal solution. So, a better crystal can be obtained according to (3-11).. 3.1.2 Supervisory Control By incorporating a control term us into uc , the control law becomes u = uc + u s. (3-12). where us is a supervisory control [63]. The supervisory control us is turned on when the error function Vδ is greater than a positive constant V . If Vδ ≤ V , then the supervisory control us is turned off. That is, if the system. tends to be unstable ( Vδ > V ) , then us forces Vδ ≤ V . In this way, us acts as a supervisor. Substituting (3-12) into (3-1), the error equation becomes. δ& = Λ cδ + bc [( f ( x) − fˆ ( x | w f ) + ( g ( x) − gˆ ( x | wg )uc − gˆ ( x | wg )us ]. (3-13). Using (3-13) and (3-14), we have 1 V&δ = − δ T Qδ + δ T Pbc [( f ( x) − fˆ ( x | w f )) + g ( x)uc − gˆ ( x | wg )uc − gˆ ( x | wg )us ] 2 1 ≤ − δ T Qδ + | δ T Pbc | [(| f ( x) | + | fˆ ( x | w f ) |)+ | g ( x)uc | + | ( gˆ ( x | wg )uc |] 2 T +δ Pbc g ( x)us. (3-14) Based on Assumption 3.1 and (3-14), we choose the supervisory control us as us = − I1* sgn(δ T Pbc ). 1 [| fˆ ( x | w f ) | + f U ( x)+ | gˆ ( x | wg )uc | + | g U ( x)uc |] g L ( x). 26.

(37) (3-15) where I 1* =1 if Vδ > V , V is a constant specified by the designer, I 1* =0 if Vδ ≤ V , and sgn( y ) = 1( −1) if y ≥ 0(< 0) . Substituting (3-15) into (3-14) and. considering the case Vδ > V , we have 1 g V&δ ≤ − δ T Qδ + | δ T Pbc | [| fˆ | + | f | + | gˆu c | + | gu c | − (| fˆ | + f U + | gˆuc | + | g U u c |)] 2 gL 1 ≤ − δ T Qδ ≤ 0 (3-16) 2. Equation (3-16) ensures the bounded stability of the RIAFC for the nonlinear system in (3-1). Remark 3.1 : The concept of the supervisory control is added into our. design mainly because the system states may go into the unsafe region if the RSA operations can not simultaneously generate the appropriate weightings. To safely control the uncertain nonlinear systems, the supervisory controller must be turned on when the system states go into the unsafe region.. 3.2 Description of Reduce Simulated Annealing Algorithm for On-line Controllers For the purpose of speeding up the computation of the simulated annealing operation, the mechanism of the reduced simulated annealing algorithm has two simplified parts: (1) one perturbation in every temperature, and (2) perform compact perturbation mechanism and special cooling schedule on the solution by the cost function. The details are discussed in the following. First, the adjustable parameters of the p-th output of the fuzzy-neural networks are as follow. w p = ⎡⎣ w Tp ⎤⎦. (3-17). where w p denotes a set of weighting factors in the interval D p = [− d p , d p ] , d p > 0 . Each element represents the adjustable parameter of the fuzzy-neural 27.

(38) networks. Note that the solution is accepted by the cost function, that is, the optimum solution denotes the best solution of cost function. Next, to instantaneously evaluate the stability of the closed-loop system, define the p-th cost function as. Φ = −δ T Pbc fˆp ( x | w f ) − δ T Pbc gˆ p ( x | wg )uc. (3-18). where fˆp (x | w f ) is the estimation of the unknown dynamic f p (x | w f ) and gˆ p (x | w g ) is the estimation of the unknown dynamic g p (x | w g ) A state. with smallest cost function denotes the optimal solution. The detail explanation of the cost function is given later. Then, according to the cost function, perturbation mechanism operators are performed. The operation procedure of the perturbation mechanism is as follows. w p _ new. β ⎪⎧w p _ old + αγ (w p max − w p _ old ), =⎨ β ⎪⎩ w p _ old − αγ (w p _ old − w p min ),. if ε > 0.5 if ε ≤ 0.5. (3-19). where the factors α and β are the constants and γ and ε are random in the interval [0,1]. The acceptance probability function for the new solution vector is defined as if Φ ( w p _ new ) ≤ Φ (w p _ old ) ⎧1, ⎪ Φ ( w )−Φ ( w ) P (w p _ new ) ⎨ p _ old p _ new K bT ⎪e > r , if Φ ( w p _ new )>Φ (w p _ old ) ⎩. where r. (3-20). denotes a probability in the interval [0,1], K b represents. Boltzmann constant and T is temperature. And then, we design cooling schedule as follow:. Ts +1 = e. (. − s2. σ. ). where s is iterations of SA and σ is a constant.. 28. T0. (3-21).

(39) The block diagram of RIAFC is given as follow: if V ≤ V 0 +. Supervisory C ontroller u s. y. Sim ulated annealing adaptive backstepping fuzzy-neural controller u c. e. −. Plant. +. if V > V. yd. u. +. y. Fuzzy-neural N etw orks O utput fˆ gˆ W. Initial w (0) determ ined in the interval D = [− d d ]. R educed SA W eighting Factors W. Fig. 3-1 The block diagram of RIAFC. The design algorithm of RIAFC is as follows:. Step 1: Construct fuzzy-neural networks for. fˆ ( x | w f ) and gˆ ( x | wg ),. including fuzzy sets for x(t), the weighting vectors w f and wg .. Step 2: Adjust the weighting vectors by using the RSA approach with the cost function (3-11).. Step 3:. Compute fˆ ( x | w f ) and gˆ ( x | wg ) . Then, obtain the control law (3-4).. 3.3 Simulation Examples of the RIAFC This section presents the simulation results of the proposed on-line RIAFC for a class of uncertain nonlinear systems to illustrate the stability of the closed-loop system is guaranteed, and all signal involved are bounded.. 29.

(40) Example 3-1: Consider the three-order nonlinear system described as x&1 = x2 x&2 = x3 x&3 =. (3-22). 1− e + sin x1 − x2 + 0.1x3 + u = f ( x1 , x2 , x3 ) + u 1 + e− x − x1. 1. Thus, the RIAFC is suitable to control the system. The adjustable parameters, w f of fˆ ( x1 , x2 , x3 ) are in the intervals D1 =[-2,2] α =0.01, β =20, σ =10. respectively. The reference signal is given as yd (t ) = sin(t ) in the following simulations. The initial states are set as x(0) = [0.3,1,0.5] . The membership functions for xi , i=1,2 are given as. μ A ( xi ) = 1/(1 + exp[5 x + 3)] 1 i. μ A ( xi ) = exp[−( x + 2) 2 ] 2 i. μ A ( xi ) = exp[−( x + 1) 2 ] 3 i. .. μ A ( xi ) = exp[− x 2 ] 4 i. μ A ( xi ) = exp[−( x − 1) 2 ] 5 i. μ A ( xi ) = exp[−( x − 2)2 ] 6 i. μ A ( xi ) = 1/(1 + exp[5 x − 3)] 7 i. To apply the RIAFC to the system, the bounds f U should be obtained: 1 − e− x f ( x1 , x2 , x3 ) = + sin x1 − x2 + 0.1x3 ≤ 2 + x2 + 0.1 x3 ≡ f U ( x1 , x2 , x3 ) −x 1+ e 1. 1. (3-23) The design parameters are set as k1 = 40 , k2 = 40 and k3 = 40 ,. Q=diag(10,10,10) and V = 1 . Then, solve (3-8) and obtain ⎡ 20.384 10.384 0.125⎤ P = ⎢⎢10.384 20.644 0.384 ⎥⎥ ⎢⎣ 0.125 0.384 0.134 ⎥⎦. The simulation results are shown in Figs. 3-2, 3-3, 3-4 the RIAFC can control. 30.

(41) the uncertain nonlinear systems to follow the desired trajectories very well. In Fig. 3-3, the tracking error reaches a bounded error ( Vδ ≤ V = 1 ). Therefore, the tracking performance is very good as shown in Fig. 3-2, in which yd is the reference trajectory and x1 is the system output. As shown in Fig. 3-4, the chattering effect of the control input ( uc + us ) almost disappears after 2 seconds, respectively. In 2 seconds, the RSA searches the neighborhood for the optimal parameters of the RIAFC.. Fig. 3-2. The system output y(t) and bounded reference yd (t ). 31.

(42) Fig. 3-3. The tracking error e. Fig. 3-4. The control input u(t). Example 3-2: Consider the dynamic equations of the inverted pendulum. system as [18]. 32.

(43) x&1 = x2 cos x1 mlx22 cos x1 sin x1 g sin x1 − mc + m mc + m x&2 = + u = f ( x1 , x2 ) + g ( x1 , x2 )u 2 4 m cos x1 4 m cos 2 x1 ) ) l( − l( − 3 mc + m 3 mc + m. (3-24) where g=9.8 meter/sec 2 is the acceleration due to gravity, mc is the mass of the cart, l is the half-length of the pole, m is the mass of the pole and u is the control input. In this example, we assume mc =1 kg, m=0.1 kg and l=0.5 meter. Thus, the RIAFC is suitable to control the system. The adjustable parameters, w f of fˆ ( x1 , x2 ) are in the intervals D1 =[-2,2] α =0.5, β =20, σ =10. respectively. The adjustable parameters of gˆ ( x1 , x2 ) are in the intervals. D2 =[1,2], respectively. The reference signal is given as ym (t) = 0.1* sin(t ) in the following simulations. The initial states are set as x(0) = [. π 60. ,0] . The. membership functions are the same as Example1. To apply the RIAFC to the system, the bounds f U , g U , and g L should be obtained: mlx22 cos x1 sin x1 0.025 2 g sin x1 − 9.8 + x2 mc + m 1.1 ≤ | f ( x1 , x2 ) |= 2 0.05 4 m cos 2 x1 − l( − ) 3 1.1 3 mc + m. (3-25). = 15.78 + 0.0366 x22 ≡ f U ( x1 , x2 ) 1 mc + m 1 | g ( x1 , x2 ) |= ≤ = 1.46 ≡ g U ( x1 , x2 ) 2 2 0.01 4 m cos x1 ) ) 1.1( − l( − 3 1.1 3 mc + m cos x1. (3-26). If we require that | x1 |≤ π / 6 , then. g ( x1 , x2 ) ≥. cos π 6 = 1.12 ≡ g L ( x1 , x2 ) 2π 1.1( 23 + 0.05 6) 1.1 cos. 33. (3-27).

(44) The design parameters are set as k1 = 1 and k2 =2, Q=diag(10,10) and ⎡15 5⎤ V = 0.01 . Then, solve (3-8) and obtain P = ⎢ ⎥ . As shown in Figs. 3-5, 5 5 ⎣ ⎦. 3-6, 3-7 the RIAFC can control the inverted pendulum to follow the desired trajectories very well. In Fig. 3-6, the position error reaches a bounded error ( Vδ ≤ V = 0.01 ). Therefore, the tracking performance is very good as shown in Fig. 3-5, where ym is the reference trajectory and x1 is the system output.. Fig. 3-5. The system output y(t) and bounded reference ym (t ). 34.

(45) Fig. 3-6. The tracking error e. Fig. 3-7. The control input u(t). 3.4 Conclusions In this chapter, an RSA indirect adaptive fuzzy-neural controller (RIAFC) has been proposed. The free parameters of the adaptive fuzzy-neural controller. 35.

(46) can be successfully tuned on-line via the RSA approach with a special evaluation mechanism, instead of solving complicated mathematical equations. The RIAFC with the supervisory controller guarantees the bounded stability of the closed-loop system. The simulation results show that the RSA-based adaptive fuzzy-neural controller performs on-line tracking successfully.. 36.

(47) Chapter 4 Backstepping Adaptive Control of Uncertain Nonlinear Systems Using RSA On-Line Tuning of Fuzzy-Neural Networks In this chapter, an RSA backstepping adaptive fuzzy-neural controller (RBAFC) for uncertain nonlinear systems is proposed by using a reduced simulated annealing algorithm (RSA). The weighting factors of the adaptive fuzzy-neural controller are tuned on-line via the RSA approach. For the purpose of on-line tuning these parameters and evaluating the stability of the closed-loop system, a cost function is included in the RSA approach. In addition, in order to guarantee that the system states are confined to the safe region, a supervisory controller is incorporated into the RBAFC. To illustrate the feasibility and applicability of the proposed method, two examples of nonlinear systems controlled by the RBAFC are demonstrated.. 4.1 Problem Formulation In this section, we describe the control problem for a class of nonlinear systems, and then design the backstepping controller.. 4.1.1 The Design of Backstepping Controller Here we consider another nth-order nonlinear system of the form x&1 = x2 x&2 = x3 M. (4-1). x&n = f ( x1 , x2 ,L , xn ) + g ( x1 , x2 ,L , xn )u. Where f and g are unknown smooth continuous functions, u ∈ R is the. 37.

(48) system input, and x = [ x1 , x2 L xn ]T ∈ R n is the state vector. The control objective is to design the backstepping controller for system (4-1) such that all the signals in the closed-loop are uniformly stable and the state x1 can track a bound reference signal ym arbitrarily closely. Next, the detail design procedure of the backstepping controller is described as follows. Step 1) Define the tracking error as z1 = x1 − ym. (4-2). Then, the derivative of z1 can be expressed as z&1 = x&1 − y& m. (4-3). Define the virtual control as. α1 = y& m − c1 z1. (4-4). where c1 > 0 is a design parameter. From (4-2) and (4-3), if α1 = x&1 , then lim z1 → 0 , that is, the state trajectory x1 can asymptotically track t →∞. the bounded signal ym . Define an error state as z2 = x&1 − α1 = x2 − α1 . Then, our next goal is to force the error state z2 to decay to zero. By using (4-4) and the fact that x&1 = z2 + α1 , equation (4-3) can be rewritten as z&1 = z2 − c1 z1. (4-5) Step 2) The derivative of z2 can be expressed as z&2 = x&2 − α&1 = x3 − (−c1 z&1 + && ym ). (4-6). Similarly, define the virtual control as. α 2 = &&ym − c1 z&1 − c2 z2 − z1 (4-7) where c2 > 0 is a design parameter. Moreover, define the error state as z3 = x3 − α 2 . Then, by using (4-6) and the fact that x&2 = z3 + α 2 ,. 38.

(49) equation (4-7) can be rewritten as z&2 = z3 − c2 z2 − z1. (4-8). 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 z&k = x&k − α& k −1. (4-9). Define the virtual control as ⎡. k. ⎤ ⎡ k −1. ⎤. ⎦ ⎣ j =1. ⎦. α k = ym( k ) − ⎢ ∑ ci zi( k −i ) ⎥ − ⎢ ∑ z kj −1− j ⎥ ⎣ i =1. (4-10). where ci > 0 is a design parameter. Moreover, define the error state as. zk +1 = xk +1 − α k . Then, by using (4-10) and the fact that. x&k = zk +1 + α k , equation (4-9) can be rewritten as z&k = zk +1 − ck zk − zk −1. (4-11). Step 4) The derivative of zn can be expressed as. z&n = x&n−1 − α& n −1 = f (x) + g (x)u − ( y. (n) m. ⎡ n−1 ( n−i ) ⎤ ⎡ n−2 n−1− j ⎤ − ⎢ ∑ ci zi ⎥ − ⎢ ∑ z j ⎥ ) (4-12) ⎣ i =1 ⎦ ⎣ j =1 ⎦. Define the control law as u =. ⎤ 1 ⎡ n ⎤ ⎡ n −1 ( ym( n ) − ⎢ ∑ ci zi( n−i ) ⎥ − ⎢ ∑ z (jn−1− j ) ⎥ − f (x)) g ( x) ⎣ i =1 ⎦ ⎣ j =1 ⎦. (4-13). where cn > 0 is a design parameter. Then, from (4-13), equation (4-12) can be rewritten as z&n = −cn zn − zn −1. (4-14). Step 5) Consider the Lyapunov function. V=. 1 n 2 ∑ zi 2 i =1. (4-15). By differentiating (4-15) and using (4-5), (4-8), (4-11) and (4-14), we have. 39.

(50) n. V& = ∑ zi z&i i =1. n −1. = z1 ( z2 − c1 z1 ) + ∑ zi ( zi +1 − ci zi − zi −1 ) + zn (−cn zn − zn−1 ) i =2. (4-16). n. = − ∑ ci zi2 i =1. ≤ −c1 z12. We can conclude that the state trajectory x1 can asymptotically track the bounded signal ym .. 4.1.2 On-line Learning of Fuzzy-Neural Backstepping Control Using RSAOA In practical applications, since f and g are uncertain, the optimal control law (4-13) cannot be obtained. To solve this problem, we use the fuzzy-neural systems as approximators to approximate the uncertain continuous functions f and g. First, we replace f and g in (3-13) by fuzzy-neural networks, i.e., fˆ ( x | w) and gˆ ( x | w) . The resulting control law uc =. ⎤ 1 ⎡ n ⎤ ⎡ n −1 ( ym( n ) − ⎢ ∑ ci zi( n−i ) ⎥ − ⎢ ∑ z (jn−1− j ) ⎥ − fˆ (x)) gˆ (x) ⎣ i =1 ⎦ ⎣ j =1 ⎦. (4-17). Substituting (4-17) in (4-1) and after some manipulations, we obtain the error equation z&n = −cn zn − zn −1 + f ( x ) − fˆ ( x | w f ) + ⎡⎣ g ( x) − gˆ ( x | wg ) ⎤⎦ uc. Let Vδ =. (4-18). 1 n 2 ∑ zi . Using (4-18) we have 2 i =1. n. V&δ = ∑ zi z&i i =1. n −1. = z1 ( z2 − c1 z1 ) + ∑ zi ( zi +1 − ci zi − zi −1 ) i =2. + zn ⎡⎣( f ( x) − fˆ ( x | w f ) + ( g ( x) − gˆ ( x | wg ) ) uc − cn zn − zn −1 ) ⎤⎦. 40. (4-19).

(51) In order to confine the state xi to a bounded value, Vδ must be bounded, which means we require that V&δ ≤ 0 , when Vδ is greater than a large constant V > 0 . Next, we assume the following.. Assumption 4.1: We can determine functions f U ( x) , g U ( x) and g L ( x) such that. | f ( x) |≤ f U ( x) and g L ( x) ≤ g ( x) ≤ g U ( x ) ,. for. x ∈U c ,. where. f U ( x) < ∞ , g U ( x) < ∞ , x ∈ U c and g L ( x) > 0 , x ∈ U c . Based on Assumption 4.1, equation (4-20) can be modified as n. V&δ ≤ −∑ ci zi2 + zn f ( x) − zn fˆ ( x | w f ) + zn g ( x)uc − zn gˆ ( x | wg )uc i =1. (4-20). n. ≤ −∑ c z + zn f ( x) − zn fˆ ( x | w f ) + zn g ( x)uc − zn gˆ ( x | wg )uc i =1. 2 i i. U. U. The states must move in the direction of smaller values of Vδ if V&δ < 0 . Therefore, according to (4-20), we define a cost function for the RSA as. Φ = − zn fˆ ( x | w f ) − zn gˆ ( x | wg )uc. (4-21). in order to on-line tune weightings. Note that a state with the smallest cost function denotes the optimal solution. By incorporating a control term us into uc , the control law becomes. u = uc + u s (4-22) where us is supervisory control. The supervisory control us is turned on when the error function Vδ is greater than a positive constant V .. If. Vδ ≤ V , then the supervisory control us is turned off. That is, if the system. tends to be unstable ( Vδ > V ) , then us forces Vδ ≤ V . Substituting (4-22) into (4-1), the error equation becomes z&n = f ( x) − fˆ ( x | w) − cn zn − zn−1 + g ( x)uc − gˆ ( x | w)uc − gˆ ( x | w)us. Using (4-23) and (4-19), we have. 41. (4-23).

(52) n. V&δ = − ∑ ci zi2 + z n [ f ( x ) − fˆ ( x | w f ) + g ( x )uc − gˆ ( x | wg )uc − gˆ ( x | wg )u s ] i =1 n. ≤ − ∑ ci zi2 + z n [ f ( x ) + fˆ ( x | w f ) + g ( x ) u c + gˆ ( x | wg ) u c ] + z n gˆ ( x | wg )u s i =1. (4-24) Based on Assumption 4.1 and (4-24), we choose the supervisory control us as us = − I1* sgn( zn ). where. I1* =1. 1 U [ f ( x) + fˆ ( x | w) + g U uc + gˆ ( x | w)uc ] gL. (4-25). if Vδ > V , V is a constant specified by the designer, I1* =0 if. Vδ ≤ V , and sgn( y ) = 1(−1) if y ≥ 0(< 0) . Substituting (4-25) into (4-24) and. considering the case Vδ > V , we have n. V&δ ≤ −∑ ci zi2 i =1. + zn [ f ( x) + fˆ ( x | w) + gˆ ( x | w) uc + g ( x)uc −. (. ). g f U ( x) + fˆ ( x | w) + gˆ ( x | w) uc + g U ( x)uc ] gL. (4-26). n. ≤ −∑ ci zi2 ≤ 0 i =1. Equation (4-26) ensures the asymptotical stability of the RBAFC for the nonlinear system in (4-1).. 42.

(53) The block diagram of RBAFC is given as here (for example in two-order nonlinear system) if V ≤ V 0 +. supervisory control u s. z1. −. d. d. −. y& d +. α1. dt. &y&d. x1. Simulated annealing adaptive backstepping fuzzy-neural controller u c. c1. +. Plant. +. if V > V. yd. u. d Fuzzy-neural Networks Output fˆ gˆ. − z2. dt +. dt. x2. W. Initial w (0) determined in the interval D = [− d d ]. Reduced SA Weighting Factors W. Fig. 4-1 The block diagram of RBAFC The design algorithm of RBAFC is as follows:. Step 1: Construct the fuzzy-neural networks for fˆ ( x | w) and gˆ ( x | w) , including fuzzy sets for x(t), and the weighting vectors w f , wg .. Step 2: Adjust the weighting vectors by using the RSA approach with the cost function (4-21).. Step 3: Compute. fˆ ( x | w) and. gˆ ( x | w). according to output of the. fuzzy-neural network. Then, obtain the control law (4-17).. 4.2 Simulation Examples of the RBAFC This section presents the simulation results of the proposed on-line RBAFC for a class of uncertain nonlinear systems to illustrate the stability of the. 43.

(54) closed-loop system is guaranteed, and all signal involved are bounded. Example 4-1: Consider the three-order nonlinear system described as x&1 = x2 x&2 = x3 x&3 =. (4-27). 1 − e− x + sin x1 − x2 + 0.1x3 + u = f ( x1 , x2 , x3 ) + u 1 + e− x 1. 1. Our objective is to control the system state x1 to track the reference trajectory yd . Clearly, (4-27) is in the form of (4-1). Thus, the RBAFC is suitable to control the system. The adjustable parameters, w f of fˆ ( x1 , x2 , x3 ) are in the intervals D1 =[-2,2] α =0.01, β =20, σ =10, T0 = 1000 . The reference signal is given as yd (t ) = sin(t ) in the following simulations. The initial states are set as x(0) = [0.5,1,0.5] . The membership functions for xi ,. i=1,2 are given as. μ A ( xi ) = 1/(1 + exp[5 x + 3)] 1 i. μ A ( xi ) = exp[−( x + 2) 2 ] 2 i. μ A ( xi ) = exp[−( x + 1) 2 ] 3 i. μ A ( xi ) = exp[− x 2 ] 4 i. μ A ( xi ) = exp[−( x − 1) 2 ] 5 i. μ A ( xi ) = exp[−( x − 2)2 ] 6 i. μ A ( xi ) = 1/(1 + exp[5 x − 3)] 7 i. To apply the RBAFC to the system, the bounds f U should be obtained: 1 − e− x + sin x1 − x2 + 0.1x3 ≤ 2 + x2 + 0.1 x3 ≡ f U ( x1 , x2 , x3 ) f ( x1 , x2 , x3 ) = −x 1+ e 1. 1. (4-28) The design parameters are set as c1 = c2 = 5 , c3 = 10 and V = 0.05 . The simulation results are shown in Figs. 4-2, 4-3, 4-4. As shown in Figs. 4-2, 4-3 the RBAFC can control the uncertain nonlinear systems to follow the desired. 44.

(55) trajectories very well. In Fig. 4-3, the tracking error reaches a bounded error ( Vδ ≤ V = 0.05 ). Therefore, the tracking performance is very good as shown in Fig. 4-2, in which yd is the reference trajectory and x1 is the system output. As shown in Fig. 4-4, the chattering effect of the control input ( uc + us ) almost disappears after 1 seconds, respectively. In 1 second, the RSA searches the neighborhood for the optimal parameters of the RBAFC.. Fig. 4-2. The system output y(t) and bounded reference yd (t ). 45.

(56) Fig. 4-3. The tracking error e. Fig. 4-4. The control input u(t). Example 4-2: Consider the two-order nonlinear system described as [18] x&1 = x2 cos x1 mlx22 cos x1 sin x1 g sin x1 − (4-29) mc + m mc + m x&2 = + u = f ( x1 , x2 ) + g ( x1 , x2 )u 2 2 4 m cos x1 4 m cos x1 l− l− mc + m mc + m 3 3. Our objective is to control the system state x1 to track the reference trajectory ym . Clearly, (4-29) is in the form of (4-1). Thus, the RBAFC is suitable to control the system. The adjustable parameters, w f of fˆ ( x1 , x2 ) are in the intervals D1 =[-2,2] and wg of gˆ ( x1 , x2 ) are in the intervals D2 =[1,1.5] α =0.01, β =20, T0 = 1000 . The reference signal is given as ym (t) = 0.1 *sin(t ) in the following simulations. The initial states are set as. x(0) = [. −π ,0] . The membership functions are the same as Example1. 60. To apply the RBAFC to the system, the bounds f U , g U and g L should. 46.

(57) be obtained: mlx22 cos x1 sin x1 0.025 2 x2 9.8 + mc + m 1.1 ≤ 2 0.05 4 m cos 2 x1 − ) l( − 3 1.1 3 mc + m. g sin x1 − | f ( x1 , x2 ) |=. (4-30). = 15.78 + 0.0366 x22 ≡ f U ( x1 , x2 ) 1 mc + m 1 ≤ = 1.46 ≡ g U ( x1 , x2 ) | g ( x1 , x2 ) |= 2 2 0.01 4 m cos x1 ) l( − ) 1.1( − 3 1.1 3 mc + m cos x1. (4-31). If we require that | x1 |≤ π / 6 , then g ( x1 , x2 ) ≥. cos π 6 = 1.12 ≡ g L ( x1 , x2 ) 2π 1.1( 23 + 0.05 cos ) 6 1.1. The design parameters are set as c1 = c2 = 15 ,. (4-32) and V = 0.02 . The. simulation results are shown in Figs. 4-5, 4-6, 4-7. As shown in Figs. 4-5, 4-6, the RBAFC can control the uncertain nonlinear systems to follow the desired trajectories very well. In Fig. 4-6, the tracking error reaches a bounded error ( Vδ ≤ V = 0.02 ). Therefore, the tracking performance is very good as shown in Fig. 4-5, in which ym is the reference trajectory and x1 is the system output.. 47.

(58) Fig. 4-5. The system output y(t) and bounded reference ym (t ). Fig. 4-6. The tracking error e. 48.

(59) Fig. 4-7. The control input u(t). 4.3 Conclusions In this chapter, an RSA backstepping adaptive fuzzy-neural controller (RBAFC) has been proposed. The free parameters of the adaptive fuzzy-neural controller can be successfully tuned on-line via the RSA approach with a special evaluation mechanism, instead of solving complicated mathematical equations. The RBAFC with the supervisory controller guarantees the bounded stability of the closed-loop system. The simulation results show that the RSA-based backstepping adaptive fuzzy-neural controller performs on-line tracking successfully.. 49.

(60) Chapter 5 Design of Fuzzy-neural Controller Using Reduce Simulated Annealing Algorithms for MIMO Nonlinear Systems In this chapter, a simulated annealing adaptive fuzzy-neural control scheme is proposed for a class of multiple-input multiple-output (MIMO) nonlinear systems. The control scheme incorporates a compact simulated annealing algorithm and fuzzy-neural networks into backstepping design. The reduce simulated annealing (RSA) algorithm is used to adjust the parameters of the fuzzy-neural networks in order to instantaneously generate the appropriate control strategy. The reduce simulated annealing algorithm has a simplified procedure with an energy cost function which is used to evaluate the real-time stability of the closed-loop systems. The state represents an adjustable parameter of the fuzzy-neural networks. To illustrate the feasibility and applicability of the proposed method, an example of the double inverted pendulums controlled by the proposed method is provided.. 5.1 Problem Formulation and Fuzzy-Neural Networks In this section, we describe the control problem for a class of MIMO nonlinear systems, and then design the backstepping controller. In addition, the structure of the fuzzy-neural networks is briefly reviewed in this section.. 5.1.1 Problem Formulation and Backstepping Control Design First, consider the MIMO nonlinear systems as. 50.

(61) x& p1 = x p 2 x& p 2 = x p 3. , p = 1, 2,K, m. M. (5-1). x& pn p = f p (x1 , x 2 ,..., x h ) + bpu p. where f p is the unknown system dynamics of the p-th subsystem, u p is the input of the p-th subsystem, bp. is a positive unknown constant,. x = [x1 , x 2 , ...x m ]T is the state vector, and x p = [ x p1 , x p 2 ,...x pn p ]T is the state. vector of the p-th subsystem. Our control objective is to develop the backstepping controller so that the state trajectory x p1 can asymptotically track a bounded command y pd . Next, the detail design procedure of the backstepping controller under the assumption of the known system dynamics f p is described as follows. Step 1) Define a tracking error as z p1 = x p1 − y pd. (5-2). Then, differentiating z p1 can be expressed as z& p1 = x& p1 − y& pd. (5-3). Define a virtual control as. α p1 = y& pd − c p1 z p1. (5-4). where c p1 > 0 is a design parameter. From (5-3) and (5-4), if α p1 = x& p1 , then lim z p1 → 0 , that is, the state trajectory x p1 can asymptotically t →∞. track the bounded command y pd . Thus, define an error state as z p 2 = x& p1 − α p1 = x p 2 − α p1 . Then, our next goal is to force the error state z p 2 to decay to zero. By using (5-4) and the fact that x& p1 = z p 2 + α p1 ,. equation (5-3) can be rewritten as z& p1 = z p 2 − c p1 z p1. Step 2) Differentiating. z p2. (5-5) can be expressed as. 51.

(62) z& p 2 = x& p 2 − α& p1 = x p 3 − (−c p1 z& p1 + && y pd ). (5-6). Similarly, define a virtual control as. α p 2 = &&y pd − c p1 z& p1 − c p 2 z p 2 − z p1. (5-7). where c p 2 > 0 is a design parameter. Moreover, define an error state as z p 3 = x p 3 − α p 2 . Then, by using (5-7) and the fact that x& p 2 = z p 3 + α p 2 ,. equation (5-6) can be rewritten as z& p 2 = z p 3 − c p 2 z p 2 − z p1. (5-8). Step 3) Let k be a positive integer. Define an error state as z pk = x pk − α p ( k −1) . Then, differentiating z pk , where 3 ≤ k ≤ n p − 1 , can be expressed as z& pk = x& pk − α& p ( k −1). (5-9). Define a virtual control as k. k −1. α pk = y (pdk ) − ∑ c pi z (pik −i ) − ∑ z (pjk −1− j ) i =1. (5-10). j =1. where c pi > 0 is a design parameter. Moreover, define an error state as z p ( k +1) = x p ( k +1) − α pk . Then, by using (5-10) and the fact that x& pk = z p ( k +1) + α pk , equation (5-9) can be rewritten as z& pk = z p ( k +1) − c pk z pk − z p ( k −1). (5-11). Step 4) Differentiating z pn p can be expressed as z& pn p = x& pn p − α& p ( n p −1) = f p + bp u p − α& p ( n p −1). (5-12). Define a control law as up = −. 1 ( f p − α& p ( n p −1) ) − c pn p z pn p − z p ( n p −1) bp. (5-13). = − f p − c pn p z pn p − z p ( n p −1). where f p =. 1 ( f p − α& p ( n p −1) ) and c pn p > 0 is a design parameter. Then, bp. from (5-13), equation (5-12) can be rewritten as z& pn p = bp (−c pn p z pn p − z p ( n p −1) ). (5-14). 52.