FUZZY CONTROL OF NONLINEAR SYSTEMS WITH INPUT SATURATION USING MULTIPLE MODEL STRUCTURE. Min Zhang and Shousong Hu

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1 ICIC Express Letters ICIC International c 2008 ISSN X Volume 2, Number 2, June 2008 pp FUZZY CONTROL OF NONLINEAR SYSTEMS WITH INPUT SATURATION USING MULTIPLE MODEL STRUCTURE Min Zhang and Shousong Hu College of Automatic Engineering Nanjing University of Aeronautics and Astronautics Nanjing , P. R. China anne.zm@163.com Received October 2007; accepted February 2008 Abstract. For a class of nonlinear systems with input saturation, a kind of adaptive fuzzy control law based on multiple-model structure is presented in this paper. First, a basic fuzzy controller is designed with adaptive weight parameters determined by multiplemodel switching performance indexes. Then a dynamic structure adaptive neural network is introduced for ensuring the system stable, while the control hedging scheme is also adopted to prevent the system from being influenced by the actuator saturation and maintain working normally. Finally, the simulation results show the control method presented is effective by demonstrating the full envelope tracking control for a puddle-jumper. Keywords: Input saturation, Fuzzy control, Multiple models, Adaptive neural network 1. Introduction. Since most actual engineering systems operate in multiple environments which may change abruptly from one to another, it is hard to maintain the control performance. One of powerful schemes to control such system is the multiple-model control strategy as [1-5]. For instance, in [1], the linear system is controlled by two adaptive models and finite fixed models, and in [2], the nonlinear discrete system is studied in the way by using several adaptive linear models to ensure the stability and one adaptive nonlinear model to improve the control performance. Both [1] and [2] adopt the multiple-model method with some drawbacks like the complex adaptive algorithm, the discontinuous control process that can not ensure the system stable, and the dissatisfied tracking precision. While in [3], also adopting the multiple-model structure, the controlled system works well under just one fuzzy controller which uses the switching performance indexes to adaptively update its weight parameters, so this method is referenced by the paper to design the basic fuzzy control law for avoiding the high frequency noise produced by switching controllers, and improving the real-time control performance. For dealing with modeling error, inherent nonlinearity and uncertain external disturbance, modified from the fully tuned adaptive RBF neural network and the dynamic structure neural network, a dynamic structure adaptive RBF neural network (DRBF NN) is presented in this paper, which has the outstanding specialty that the number of hidden units can increase on line from original few to appropriate scalar with the approximation errorgrowinguntilitstayswithinthetolerantrangestablyexceptofon-lineadjusting network s parameters such as weight, center and width [6]. So the relationship between the convergence rate and the network structure complexity is balanced to make the NN operate more coordinately with the better approximation ability. In addition, the input saturation often occurs in actual engineering systems resulting in poor control performance. So for it, the presented basic controller is further modified by augmenting a component as control hedging to prevent the saturation signal from influencing the normal control process and maintain the system works stably with satisfactory performance. 131

2 132 M. ZHANG AND S. HU The paper is organized as follows: After we formulate the control problem in Section 2, how to design the fuzzy controller with control hedging based on multiple-model structure is described in Section 3. The approach of designing the DRBF NN compensator is introduced in Section 4, and experimental tracking control results of a puddle-jumper are presented to demonstrate the control scheme in Section 5. Finally, the paper is concluded in Section Problem Formulation. Let the nonlinear system dynamics [7] be given by: ẋ(t) =f(x(t),u(t)) y(t) =Cx(t) (1) where x(t) R n are states, u(t) R m and y(t) R p are control and measurement variables, and f( ) is continuous nonlinear function vector. Its linear model can be obtained at the equilibrium point, and the tracking error is defined as e = y y d = Cx y d,where y d is the given output, so the augmented linear system dynamics I i with input saturation caused by actuator saturating is constructed as follows: I i : x(t) =Āi x(t)+ B i δ(t)+ w(t) (2) h ³ R i T where x = t e(τ) dτ T x T (t) 0 R n+p, δ =[δ 1,, δ m ] T 0 C R m, Āi = 0 A i 0 R (n+p) (n+p), Bi = R (n+p) m, w(t) = y T Bi d w T (t) T,andw(t) L2 [0, ) is the nonlinear function with high orders caused by linearization, and A i, B i, C are known real matrixes with appropriate dimensions. The actuator model is defined as follows: ½ ui, u i δ 0i δ i = g(u i )= u δ i 0i, u u i i > δ, i =1,,m (3) 0i where δ 0i is the maximal input limit corresponding to the ith actuator. 3. Design of Fuzzy Controller with Control Hedging Based on Multiple-model Structure. This section surveys the scheme which uses only one fuzzy controller with control hedging based on multiple-model structure to control the system continuously, mainly describing the way of how the weight of the controller updates adaptively according to the switching performance indexes of all the models that are simultaneously calculated at every interval, and the design of the control hedging. First, the switching performance index is defined as follows: J i (t) =α ke i (t)k 2 + β Z t 0 exp( κ(t τ))ke i (τ)k 2 dτ (4) where α 0, β > 0, and memory factor κ > 0. Then the weight of the fuzzy controller is designed as: exp( J i (t) 2 /σ) α i (t) = P N exp( J (5) i(t) 2 /σ) Obviously, (5) satisfies 0 < α i < 1and P N α i = 1. Here, the weight defined shows the matching extent of each linear model approximating to the plant. When α i 1and α j 0(j 6= i), the plant is optimally approximated by the linear model I i,andcanbe described as the form of I i. So the controlled fuzzy system is obtained as the following expression based on the fuzzy logic theory [9] [12] : x(t) = α i (t)āi x(t)+ B(δ(t)+ ( x(t), δ,t)) (6)

3 ICIC EXPRESS LETTERS, VOL.2, NO.2, where B 0 =, ( x(t), δ, t)= f( x, δ, t)+ g(δ, t)+ w(t), g(δ, t)= I m (n+p) m P N α i(z(t))( B i B)δ, and f denotes the modeling error of the fuzzy system (6). The reference model I F of the fuzzy system (6) is get with the uncertain item ( x, u, t) = 0: I F : X N ˆx(t) = α i (z(t))āiˆx(t)+ Bu fh (7) The fuzzy controller u f canbedesignedas: u f = α i (z(t))k iˆx (8) Thesignalofcontrolhedgingisdefined as: u h = u n δ (9) u fh = u f u h (10) where u n is the output of the neural network which will be discussed later. Theorem 3.1. For the reference system (7), if there exists a symmetrical positive definition matrix P which satisfies the following equation: Āi + BK T i P + P Āi + BK i = Q, (11) where Q is a given positive definition matrix, the system will ultimately closed-loop bounded stable. Proof: Choose the Lyapunov candidate as V = 1 2 ˆxT (t)p ˆx(t), which has the derivation satisfying the following inequality: V 1 2 α i (z(t))ˆx T (t)qˆx(t)+kˆx(t)kkp Bkku h k Let λ = λ min (Q), and if kˆx(t)k kp Bk(ku n k+kδk),itholdsthat V < 0. In view of the λ above condition, the system can be stable if the input signal u n and the saturation signal δ are both limited. Due to the definition of (3), it has kδk δ 0,andthesignalu n is also limited based on the property of RBF neural network which is chosen in this paper. Consequently, the system (7) can be bounded stable. The proof is completed. 4. Design of DRBF NN Compensator. The error system dynamics is defined as: ẽ(t) = x(t) ˆx(t) = α i (z(t))āiẽ Bu f + Bu n + B ( x, u, t) (12) where u n = u nn + u ns, u nn is the output of the DRBF NN, and u ns is its compensation for the approximation error. Substituting (8) into (12), the following expression is achieved: ẽ(t) = α i (z(t))(āi + BK i )ẽ + B(u nn + u ns + ( x, u, t) K i x) (13) Based on the neural network s approximation property, we can let u nn = Ŵ T Ĝ(X, ˆξ, ˆη), u ns = sgn(b T P ẽ)ˆϕ(t), ( x, u, t) K i x = W T G (X, ξ, η )+ε(x), so the expression

4 134 M. ZHANG AND S. HU (13) is altered as: ẽ(t) = μ i (z(t)))āi BK i )ẽ(t)+ B W T Ĝ + BŴ T (Ĝ0 ξ ξ + Ĝ0 η η) + BE Bsgn(B T P ẽ)ˆϕ(t) (14) where X = x T e T t T R n+p+1,andx A d ( a large enough compact set [7] ), ε(x) is the bounded approximation error, G is chosen as commonly used Gaussian functions. W = W Ŵ, ξ = ξ ˆξ, η = η ˆη, E = W T (Ĝ0 ξ ξ + Ĝ0 η η) +W T o(x, ξ, η) +ε(x), and o(x, ξ, η) is the sum of high-order arguments in Taylor s series expansion. Based on the property of RBF NN, it is easy to prove that is bounded [8],andbeingsetaskEk ϕ here. To avoid the case that the change of NN s topological structure influences the real-time property of the system, the threshold logic unit (TLU) is designed which works parallel with the updating of the network s parameters. The input of TLU is the tracking error, and the sampling frequency is the same as the parameters updating. The output of TLU determines whether it is necessary to add a new hidden unit according to the given growth criterion. Define the TLU as the following form that includes two main components. One is called the operation rule as the form: ρ = α exp(e tra E 1 )+(1 α)exp(e rms E 2 ), and the other is the growth criterion of hidden units described by logical comparing: ½ ρ > 1, L = L +1 (15) ρ 1, L = L where L q is the number of hidden units, e tra = k x(n) ˆx(n)k is the approximation error, Pn e rms = i=n (M 1) k x(i) ˆx(i)k2 /M denotes the error over a sliding window(m)which can ensure the change of the number of hidden units more smooth, E 1 and E 2 are the given bound values, and 0 < α < 1istheinfluence factor. The parameters associated with the new hidden unit are given initially: ξ L+1 = x(n), η L+1 = λe tra,whereλis the optional regulation factor. Theorem 4.1. For the system (12), if there exists the fuzzy controller (8) with the DRBF NN compensator u n, and the following inequality is satisfied: (Āi + BK i ) T P + P (Āi + BK i ) < 0 (16) the system will be closed-loop asymptotically stable. The growth criterion of the NN s hidden units is defined as (15), and the parameters such as weight, center and width of the NN update as the following expressions: Ŵ = σ 1 Ĝẽ T PB, ˆξ = σ 2 (ẽ T PBŴ T Ĝ 0 ξ) T ˆη = σ 3 (ẽ T PBŴ T Ĝ 0 η) T, ˆϕi = σ 4 kẽ T PBk (17) where σ 1, σ 2, σ 3, σ 4 are given positive constants. Proof: Chose the following Lyapunov function: V = 2ẽT 1 (t)p ẽ(t)+ 1 tr( 2σ W T 1 1 W )+ ξt ξ + η T η + 1 ϕ T ϕ 1 2σ 2 2σ 3 2σ 4 where ϕ = ϕ ˆϕ is the estimation error of the compensator u ns. The derivation of the above expression satisfies the following inequality: V 1 α i (z(t))ẽ T [(Āi + 2 BK i ) T P + P (Āi + BK i )]ẽ. Based on Theorem 4.1, the inequality (16) can be satisfied, so there exists V 0, that is to say, the system (12) is closed-loop asymptotically stable. The proof is completed.

5 ICIC EXPRESS LETTERS, VOL.2, NO.2, Simulation Result. In this section, the proposed fuzzy controller with control hedging and the DRBF NN compensator is used on fully envelope tracking control for a kind of puddle-jumper. Define the state variables as x =[V, α, β, p,q,r,ψ, θ, H] T,the control variables as u =[δ e, δ a, δ r ] T, the output variable as y = θ, ande = θ θ d. So h R i T t theaugmentedstatevariablesisalteredas x = e(τ) dτ, V,α, β, p,q,r,ψ, θ, H 0. We choose three flight working point to build three corresponding fixed linear models for the actual plant. The parameters used in this paper are given as L =3,λ =1, σ 1 = σ 2 = σ 3 =1,σ 4 =0.8, α =1,β =1,κ =6,δ =0.01. The simulation of tracking the pitch angle command θ d is carried through at two flight working points to show the efficiency of the control scheme presented, which are 1) V =45m/s, H =1219.2m; 2) V =45m/s, H = m. The responding curves of pitch angle θ, pitch rate q, roll rate p and control input δ e are shown in Figure 1 and Figure 2. Among those figures, Figure 1(a) and Figure 2(a) show the tracking results with control hedging for actuator saturation, and Figure 1(b) and Figure 2(b) show the results without control hedging. From the simulation results, we can see that the system under the controller with control hedging has better dynamical performance, and tracks more quickly and precisely. Figure 1. The response of the longitudinal states (V = 45m/s,H = m, (a) with control hedging, (b) without control hedging) 6. Conclusions. In this paper, an adaptive fuzzy control scheme with control hedging and the DRBF NN compensator is presented for a class of complex nonlinear systems with input saturation based on multiple-model structure which is constructed by several fixed linear models and one fuzzy reference model. The controlled system can continuously work under the control of just one controller, but not switch among different controllers. The DRBF NN introduced is to eliminate the uncertainties and nonlinearities to ensure the system is stable, while the control hedging for input saturation can make the system work normally when the actuators are saturating. From the simulation results, it is clear that the control scheme proposed is reasonable and effective. Acknowledgment. Supported by the National Natural Science Foundation of China ( ), and Aviation Science Foundation (05E52031).

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