Fuzzy adaptive H∞ control for a class of nonlinear systems

ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 3 (2007) No. 2, pp. 149-154

Fuzzy adaptive H∞ control for a class of nonlinear systems∗ Zhenbin Du1† , Zifang Qu2 1

School Of Computer Science and Technology, Yantai University, Yantai 264005, P. R. China 2 College of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai 264005, P. R. China (Received August 31 2006, Accepted November 15 2006)

Abstract. Combining both kinds of fuzzy logic forms including fuzzy T-S model and adaptive fuzzy logic systems, this paper presents an adaptive control scheme for a class of nonlinear systems. Firstly, the fuzzy T-S model is used to approximate the nonlinear systems, and the fuzzy control law of the fuzzy model is derived by the linear matrix inequality. Secondly, the adaptive fuzzy logic systems are constructed, and the modeling errors are eliminated by a compensator based on the adaptive fuzzy logic systems with three adjustable parameters: weights, centers and widths. It is proved that the closed loop system satisfies the anticipant performance. The simulation results demonstrate that the control scheme is effective. Keywords: fuzzy T-S model, adaptive fuzzy logic systems, nonlinear systems

1

Introduction

The H∞ control problem of nonlinear systems based on fuzzy T-S model has attracted considerable attention, mainly because the fuzzy T-S model provides an effective approach to linearize nonlinear systems .And this approach has been successful applied in [2, 3] . The stability problem of fuzzy control system is discussed in [9]. And the robust problem of fuzzy control system is researched in [5].However, the modeling error is neglected in these studies. Therefore, the designed controller does not always guarantee the stability of the original system. Besides, in nonlinear modeling, the paper[7] considers that the modeling error has the upper bound, the papers [1, 6] consider that the modeling error satisfies the matching condition, However, the upper bound and the matching condition are difficult to find in practice. The adaptive fuzzy logic systems have the universal approximation property and could uniformly approximate nonlinear continuous functions to an arbitrary accuracy. The adaptive fuzzy logic systems could sufficiently make use of the linguistic information and the expert information. The adaptive fuzzy logic systems are used to model nonlinear systems by a set of fuzzy “if-then” rules. When a proper control is given to the model, an anticipant output is produced from the nonlinear systems. At present, the adaptive fuzzy logic systems have been successfully used in nonlinear control[4, 8] . Combining fuzzy T-S model and adaptive fuzzy logic systems, this paper presents a new H∞ control scheme for a class of nonlinear systems. Firstly, the fuzzy T-S model is used to approximate the nonlinear systems, and the fuzzy controller is designed by the linear matrix inequalities to guarantee the stability of the fuzzy system. Secondly, the adaptive fuzzy logic systems are constructed, and the modeling error is eliminated by a compensator based on the adaptive fuzzy logic systems with three adjustable parameters: weights, centers and widths. It is proved that the closed-loop system satisfies the anticipant H∞ performance. The simulation results demonstrate that the control scheme is effective. ∗ †

This work is supported by the Natural Science key Foundation of China (60234010). Corresponding author. Tel.: +86-013808900192. E-mail address: [email protected]. Published by World Academic Press, World Academic Union

Z. Du & Z. Qu: Fuzzy adaptive H∞ control

150

2

Problem formulation Consider the following nonlinear system  x˙1 = x2    x˙2 = x3 ···    x˙n = f (x1 , x2 , · · · , xn , u) + d0

(1)

where x1 , x2 , · · · , xn ∈ Rn are the measurable state vectors, u ∈ Rn is the control input vector, f are 2 T the nonlinear vector function, d0 denotes the external disturbance. Denote x = [xT1 , xT2 , · · · , xTn ] ∈ Rn , T 0 d = [0, 0, · · · , 0, d T ] . The system equation (1) can be approximated by a fuzzy T-S model composed of rules. The L rule of fuzzy model is in the following form. If z1 (t) is F1i and, · · · , and zs (t) is Fsi ,Then x˙t = Ai x(t) + Bi u(t) + d,

i = 1, 2, · · · , L

(2)

where z1 (t), z2 (t), · · · , zs (t) are the premise variables, Fji (for j = 1, 2, · · · , s) are the fuzzy sets, L is the number of if-then rules, Ai , Bi are some constant matrices with compatible dimensions. 2 T Bi = [0, 0, · · · , 0, bTi ] ∈ Rn ×n , bi ∈ Rn×n . The final output of the fuzzy system is inferred as follows x˙t =

L X

µi Ai x(t) +

i=1

PL

L X

µi Bi u(t) + d

(3)

i=1

where µi = νi (z(t))big/ i=1 νi (z(t)), νi (z(t)) = j = 1s Fji (zj (t)), Fji (zj (t)) is the grade of membership P of zj (t) in Fji . It is easy to find that µi ≥ 0 for i = 1, 2 · · · , L and L i=1 µi = 1 for all t . Therefore, the modeling error of the nonlinear system equation (1) can be expressed as     x2 0  0   x3  L L X     X     B∆(x) =  · · ·  = · · · − µi Ai x(t) − µi Bi u(t) (4)  0   xn  i=1 i=1 ∆f f Q

where B = [0 0 · · · 0 In×n ]T , 4(x) = [0 0 · · · 0 4f T ]T . Therefore, the nonlinear system equation (1) could be rearranged as x˙t =

L X

µi Ai x(t) +

i=1

3

L X

µi Bi u(t) + B 4 (x) + d

(5)

i=1

The design of the controller

If we do not consider the effect of the 4(x) . Denote 4(x) in equation (5). Design a controller to guarantee to stabilize the corresponding closed loop system of the linear part of the nonlinear system equation (5), and make the H∞ performance satisfied. The fuzzy state feedback controller is in the following form: if z1 (t) is F1i and, · · · , and zs (t) is Fsi , then u(t) = Ki x(t), where Ki are matrixes with proper dimensions. The overall state feedback controller is given by WJMS email for contribution: [email protected]

i = 1, 2, · · · , L

(6)

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World Journal of Modelling and Simulation, Vol. 3 (2007) No. 2, pp. 149-154

u(t) =

L X

µi Ki x(t)

(7)

i=1

Because of the existence of the fuzzy modeling error, it is difficult to make the system equation (1) satisfy the anticipant H∞ performance by only using the fuzzy state feedback controller. In this situation, we construct the fuzzy logic systems to eliminate the fuzzy modeling error. Therefore, the nonlinear system is stabilized and the desired H∞ performance is achieved. Combining the fuzzy state feedback control with the fuzzy-logic-system-based control, we design the control law u(t) = ul (t) − uf (t)

(8)

where ul (t) denotes the fuzzy control law in equation (6), uf (t) is the compensator based on the adaptive fuzzy logic systems. Choose the adaptive compensator uf (t) = E −1 u ˆ(x|Θ, α, δ) T −1

E T (I + EE )

(9)

u ˆ(x|Θ, α, δ)

(10)

If E is nonsingular, equation (9) is chosen, or else, equation (10) ˆ(x|Θ, α, δ) is constructed PL is chosen. Where u n×n by the adaptive fuzzy logic systems, Ei = bi ∈ R , E = i=1 µi bi . ¯ = −B. Substituting equation (8) into equation(5), we get a closed-loop system Denote B x˙t =

L X L X

µi µj A¯ij x(t) + d + (ˆ u(x|Θ, α, δ) − 4(x))

(11)

i=1 j=1

where A¯ij = Ai + Bi Kj (i, j = 1, · · · , L). If the fuzzy logic systems u ˆ(x|Θ, α, δ) could eliminate 4(x) , then the closed-loop system equation (11) is stable. It has been proven that the adaptive fuzzy logic systems have the universal approximation property. We construct the adaptive fuzzy logic systems to approximate the vector function 4(x) with n dimensions. The ˆ fuzzy logic systems have the form 4(x|Θ, α, δ) = Ψ (x, α, δ)Θ, where the fuzzy basis function Ψ (x, α, δ) = diag[ξ1 (x, α1 , δ1 ), · · · , ξn (x, αn , δn )], Θ = [θ1T , θ2T , · · · , θnT , ], α = diag[α1 , · · · , αn ], δ = diag[δ1 , · · · , δn ], θi (for i = 1, 2, · · · , n) are the column vectors, αi , δi (for i = 1, 2, · · · , n) are the row vectors. The weights Θ , the centers α and the widths are the adjustable parameters. We have the following theorem: ˜ = Θ− Theorem 1. Define the estimation errors of the parameter vector and the parameter matrix Θ ∗ ∗ ∗ ∗ ∗ ∗ ˜ Θ = (θi − θi )n×1 , α ˜ = α − α = diag(α1 − α1 , · · · , αn − αn ), δ = diag(δ1 − δ1 , · · · , δn − δn∗ ), ξ˜i = ξi (x, αi , δi ) − ξi∗ (x, αi , δi ) (for i = 1, 2, · · · , n), then the approximation error for the vector function ∆(x) = [∆1 (x), · · · , ∆n (x)]T can be expressed as ˜ δ (x))Θ + w1 ˆ ˜ + (˜ ∆(x|Θ, α, δ) − ∆(x) = (Ψ (x) − αΨα (x) − δΨδ (x))Θ αΨα (x) + δΨ where Ψ (x) = Ψ (x, α, δ), Ψα (x) = diag[ξ1α1 (x, α1 , δ1 ), · · · , ξnαn (x, αn , δn )], Ψδ (x) = diag[ξ1δ1 (x, α1 , δ1 ), · · · , ξnδn (x, αn , δn )], ξiαi (x, αi , δi ) and ξiδi (x, αi , δi ) denote partial derivatives of ξi with αi and δi ,respectively. w1 is the residual term. ˜i ) ˜ Proof. Denote (∆ n×1 = ∆(x|Θ1 , α, δ) − ∆(x) ˜i = ∆ ˜i (x|θi , αi , δi ) − ∆i (x) = ξi (x, αi , δi )θ˜i + ξ˜i θi − ξ˜i θ∗ + ε¯i ∆ i

(12)

WJMS email for subscription: [email protected]

Z. Du & Z. Qu: Fuzzy adaptive H∞ control

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˜i (x|θ∗ , α∗ , δ ∗ ) − ∆i (x). where ε¯i = ∆ i i i Using the Taylor expansion of ξi (x, αi , δi ) in (αi∗ , δi∗ ) ,we derive ξi (x, αi , δi ) = ξi (x, αi∗ , δi∗ ) + (αi − αi∗ )ξiα + (δi − δi∗ )ξiδ + o(x, αi − αi∗ , δi − δi∗ ) = ξi (x, α∗ , δ ∗ ) + α˜i ξiα + δ˜i ξiδ + o(x, α˜i , δ˜i ) i

(13)

i

where o(x, α˜i , δ˜i ) denotes the high order argument. Substituting equation (13) into equation (12) yields ∆˜i = (ξi (x, αi , δi ) − αi ξiαi − δi ξiδi )θ˜i + (α˜i ξiαi + δ˜i ξiδi )θi + (αi∗ ξiαi + δi∗ ξiδi )θ˜i + o(x, α˜i , δ˜i )θi∗ + ε¯i Denote w1 = (w1 i)n × 1,w1 i = (αi∗ ξiαi + δi∗ ξiδi )θ˜i + o(x, α˜i , δ˜i )θi∗ + ε¯i . We have ˜ δ (x))Θ + w1 ˆ ˜ + (˜ ∆(x|Θ, α, δ) = (Ψ (x) − αΨα (x) − δΨδ (x))Θ αΨα (x) + δΨ The proof is completed.  Denote w ¯1 = 0 0 x˙t =

L X L X

···

0

w1T

T

,w ¯ =d−w ¯1 . By use of Theorem 1, (11) is rearranged as

˜ δ (x))Θ] ¯ (x) − αΨα (x) − δΨδ (x))Θ ˜ + (˜ µi µj A¯ij x(t) + w ¯ + B[(Ψ αΨα (x) + δΨ

(14)

i=1 j=1

If the following parameter updating laws are chosen as ¯ T P x) Θ˙ = −η1 (Ψ (x) − αΨα (x) − δΨδ (x))T (B ¯ T P x ∗ (Ψα (x)Θ)T α˙ = −η2 B

(15)

T

¯ T P x ∗ (Ψδ (x)Θ) δ˙ = −η3 B

where η1 , η2 andη3 are positive constants. We have the following theorem: Theorem 2. For the nonlinear system equation (1), the control law is chosen as equation (8) composed of the fuzzy state feedback control law equation (6) and the compensator equation (9), (10) based on the adaptive fuzzy logic systems, and the parameter updating laws are chosen as equation (15), then the closed-loop system equation (11) satisfies the H∞ performance Z T 1 ˜T ˜ xT (t)Qx(t)dt ≤ xT (0)P x(0) + Θ (0)Θ(0) η 1 0 Z T 1 1 T T 2 ˜ ˜ (w ¯ T w)dt ¯ + tr(˜ α (0)˜ α(0)) + tr(δ (0)δ(0)) + ρ (16) η2 η3 0 where ρ > 0, P, Q are some symmetric and positive definite matrices. P and Q satisfy the following matrix inequalities 1 T A¯ij P + P A¯ij + 2 P P + Q < 0 ρ Proof. Choose the Lyapunov function V + 12 xT (t)P x(t) + V˙

(i, j = 1, · · · , L)

1 ˜T ˜ 2η1 Θ Θ

+

1 αT α ˜) 2η2 tr(˜

(17) +

1 ˜T ˜ 2η3 tr(δ δ)

=

1 T 1 ˜ T ˜˙ 1 1 ˜˙ [x˙ (t)P x(t) + xT (t)P x(t)] ˙ + Θ Θ + tr(˜ αT α ˜˙ ) + tr(δ˜T δ) 2 η1 η2 η3

=

L L L X L X 1 XX [( µi µj A¯ij x(t)) P x(t) + xT (t)P ( µi µj A¯ij x(t)) + w ¯ T P x(t) + xT (t)P w] ¯ 2

T

i=1 j=1

i=1 j=1

1 ¯ (x) − αΨα (x) − δΨδ (x))Θ ˜+ 1Θ ˜ T Θ] ˜ + [xT P B ¯α +[xT P B(Ψ ˜ Ψα (x)Θ + tr(˜ αT α ˜ )] η1 η2 ˜ δ (x)Θ + 1 tr(δ˜T δ)] ˜˙ ¯ δΨ +[xT P B η3 WJMS email for contribution: [email protected]

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World Journal of Modelling and Simulation, Vol. 3 (2007) No. 2, pp. 149-154

˙ ˜˙ = Θ, ˙ α Because of Θ ˜˙ = α, ˙ δ˜ = δ˙ using (15), we yield L

V˙

=

L

1 XX 1 1 T ¯ P x(t) + xT (t)P w ¯ µi µj [xT (t)A¯Tij P x(t)] + w 2 2 2 i=1 j=1

1 1 1 1 − ρ2 w ¯T w ¯ − 2 xT (t)P P x(t) + ρ2 w ¯T w ¯ + 2 xT (t)P P x(t) 2 2ρ 2 2ρ L L 1 XX = { µi µj [xT (t)A¯Tij P x(t) + xT (t)pA¯ij x(t)] 2 i=1 j=1

T 1 1 1 −( P x(t) − ρw) ¯ ( P x(t) − ρw) ¯ + ρ2 w ¯2w ¯ + 2 xT (t)P P x(t) ρ ρ 2ρ L L 1 1 XX 1 ¯T w ¯ ≤ µi µj xT (t)(A¯Tij P + P A¯ij + 2 P P )x(t) + ρ2 w 2 2ρ 2 i=1 j=1

From equation (12), we derive V˙ ≤ − 12 xT (t)Qx(t) + 12 ρ2 w ¯T w ¯. Integrating the above equation from t = 0 to T yields equation (16). The proof is completed. By the Schur complements, the inequalities equation (17) can be transformed into the linear matrix inequalities. Denote W = P −1 , Yj = Kj W . The inequalities equation (17) are equivalent to the linear matrix inequalities   −1 Ai W + W ATi + Bi Yj + (Bi Yj )T + (ρ2 ) I W