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Robust stabilization of T–S fuzzy discrete systems with actuator saturation via PDC and non-PDC law
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Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law Ling Zhao, Li Li
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S0925-2312(15)00783-3 http://dx.doi.org/10.1016/j.neucom.2015.05.085 NEUCOM15617
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Received date: 3 February 2015 Revised date: 14 April 2015 Accepted date: 21 May 2015 Cite this article as: Ling Zhao, Li Li, Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.05.085 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law Ling Zhaoa,b , Li Lic,∗ a. Hebei Province Key Laboratory of Heveay Machinery Fluid Power Transmission and Control, Yanshan University, Qinhuangdao 066004, China. b. Key Laboratory of Advanced Forging & Stamping Technology and Science (Yanshan University) Ministry of Education of China, College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China. c. College of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China.
Abstract In this paper, the problem of robust stabilization of T-S fuzzy discrete systems with actuator saturation is studied. A set invariance condition is established as null controllable region. The robust stabilization results of the closed-loop T-S fuzzy system have a domain of attraction which is arbitrarily close to the null controllable region. The problem of estimating the domain of attraction of the T-S fuzzy closed-loop systems is formulated and solved as an optimization problem. By using parameter-dependent Lyapunov function, both parallel-distributed compensation (PDC) control law and nonparallel-distributed compensation control law are designed. And there is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T-S fuzzy system with actuator saturation. A numerical example is provided to illustrate the effectiveness of the proposed design techniques. Keywords: Actuator saturation, T-S fuzzy system, robust control, stability analysis, linear matrix inequality. 1. Introduction Fuzzy technique has been widely and successfully used in nonlinear system modeling and control for more than two decades. The T-S fuzzy model ∗
Corresponding author. Email: [email protected]
Preprint submitted to Elsevier
June 2, 2015
described by a family of fuzzy IF-THEN rules was firstly introduced in Takagi and Sugeno (1985). In other words, it formulates the complex nonlinear systems into a framework that interpolates some affine local models by a set of fuzzy membership functions. Based on this framework, a systematic analysis and design procedure for complex nonlinear systems can be possibly developed in view of the powerful control theories and techniques in linear systems. In recent years, based on the T-S fuzzy model, an intensive study on the stability issue of nonlinear systems has been made and many important results have been reported, such as in Hua et al., (2005, 2009); Yang et al., (2011, 2012, 2014); Yang, Shi, et al., (2014). Based on the interval type-2 T-S fuzzy model, some new fuzzy control results have also been reported, such as in Li et al., (2014, 2015); Li, Wu, et al., (2014, 2015), and references therein. A combined backstepping and small-gain approach to robust fuzzy adaptive output feedback control has been investigated in Tong et al., (2009). An adaptive control problem for strict-feedback nonlinear systems has been given based on a fuzzy observer method in Tong and Li, (2009). A largescale nonlinear systems with dynamical uncertainties has been considered by using fuzzy-adaptive decentralized output-feedback control approach in Tong et al., (2010). Reliable fuzzy control for nonlinear active suspension systems has been researched in Li et al., (2012, 2013). An efficient design technique, which is named parallel-distributed compensation (PDC) technique, was proposed to design fuzzy controllers and relax stability conditions in Wang et al., (1996). However, sufficient conditions derived using the fuzzy Lyapunov function and PDC law are usually in the forms of bilinear matrix inequalities which is not easily solved. To deal with the problem, a non-PDC controller have been adopted instead of the PDC controller in Lam et al., (2007). It has been shown in Guerra et al., (2004) that the non-PDC control laws derived by the parameter-dependent Lyapunov function-based approach could obtain less conservative stability conditions. A nonlinear controller has been designed by fuzzy non-PDC approach and non-quadratic Lyapunov function in Lee et al., (2010). An approach for discrete-time T-S fuzzy systems with time-varying state delay has been proposed via non-PDC scheme in Wu et al., (2011). By non-PDC control law derived by parameter-dependent Lyapunov function, constrained infinite-horizon model predictive control for fuzzy discrete systems has been investigated in Xia et al., (2010). Although many researchers have studied the stability issue of fuzzy systems with PDC or non-PDC control law for many years, the problem of non-PDC control law and parameter-dependent Lyapunov function in T-S fuzzy systems with 2
actuator saturation remains as an open research area. Actuator saturation are the key of control which is applicable to virtually all areas of engineering and science. However, majority of actuators are not strictly accord with linearity, most of them subject to saturation in real physical systems. During the past several decades, control systems with actuator saturation have received much attention, see for examples Hu et al., (2001, 2002); Zhou et al., (2011); Zhou, Zheng, et al., (2011); Zhou et al., (2012); Yang, Shi, et al., (2014); Yang, et al., (2014, 2015), and the references therein. The analysis and synthesis of T-S fuzzy systems with actuator saturation nonlinearities have received increasing attention recently (see, e.g., Ozgoli et al., (2009); Kim et al., (2009), and the references therein). A problem of fuzzy-scheduling control for nonlinear systems subject to actuator saturation has been considered in Cao et al., (2003). In Zhao et al., (2012), an overhead crane model is described by a class of T-S fuzzy systems with input delay and actuator saturation. Piecewise fuzzy anti-windup dynamic output feedback control of nonlinear processes with amplitude and rate actuator saturations has been investigated in Zhang et al., (2009). A nonlinear H∞ guidance law has been designed against maneuvering targets for fuzzy systems with actuator saturation Chen et al., (2002). A robust anti-windup controller has been designed for time-delay fuzzy systems with actuator saturation in Ting et al., (2011). Based on delta operator approach, the problems on robust fuzzy-scheduling control for nonlinear systems subject to actuator saturation have been considered in Yang et al., (2014). However, to the best of our knowledge, there have been few papers on nonPDC control law and parameter-dependent Lyapunov function for T-S fuzzy systems with actuator saturations, which motivates us to make an effort in this paper. In this paper, we consider the robust stabilization for a kind of uncertain T-S fuzzy discrete systems with actuator saturation. A wider class of parameter uncertainties named linear fractional parameter uncertainties than norm-bounded parameter uncertainties is used for the uncertain fuzzy system. The definition of the domain of attraction is introduced to analyze the stability of the closed-loop T-S fuzzy discrete system. Based on parameterdependent Lyapunov function approach, both PDC controller and non-PDC controller are designed via linear matrix inequalities. There is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T-S fuzzy system since the fuzzy non-PDC law controller uses more information about the simulation system than the fuzzy PDC con3
troller. Finally, a numerical example is provided to show the usefulness of the proposed techniques. The rest of the paper is organized as follows. Section 2 formulates the problem statement of uncertain T-S fuzzy discrete systems with actuator saturation. Section 3 covers the design of stabilizing PDC and non-PDC controllers for the class of uncertain T-S fuzzy discrete systems. Section 4 presents a numerical example to show the effectiveness of the developed results. The paper is concluded by Section 5. Notation: In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. Rn denotes the n-dimensional Euclidean space; the notation X > Y (X ≥ Y ) means that the matrix X −Y is positive definite (X − Y is semi-positive definite, respectively); I is the identity matrix of appropriate dimension; For any matrix A, AT denotes the transpose of matrix A, A−1 denotes the inverse of matrix A; · denotes the norm for vectors or the spectral norms of matrices. The shorthand diag{M1 M2 · · · Mr } denotes a block diagonal matrix with diagonal blocks being the matrices M1 , M2 · · · , Mr . The symmetric terms in a symmetric matrix are denoted by ∗. 2. Problem Statement In this paper, the fuzzy model is described by fuzzy IF-THEN rules, which represent local linear input-output relations of a nonlinear system. The ith rules of the fuzzy discrete-time models with uncertainty parameters and actuator saturation are of the following form. Plant Rule i: IF z1 (k) is Mi1 , z2 (k) is Mi2 , · · ·, zq (k) is Miq , THEN x(k + 1) = Ai (k)x(k) + Bi (k)sat(u(k)), i = 1, 2, · · · , r
(1)
where r is the number of IF-THEN rules, z1 (k), z2 (k), · · ·, zq (k) are the premise variables, Mil (l = 1, 2, · · · , q) are the fuzzy sets, Ai (k) and Bi (k) are real-valued time-varying matrices of appropriate dimensions, x(k) ∈ Rn is the state variable, u(k) ∈ Rm is the control input. The function sat: Rm → Rm is the standard saturation function. It is defined as sat(u) = [sat(u1 ), sat(u2 ), · · · , sat(um )]T
4
where sat(ui ) = sgn(ui ) min{1, ui } We assume that the time-varying uncertainties enter the system matrices in the following manner Ai (k) = Ai + ΔAi (k), Bi (k) = Bi + ΔBi (k) where Ai and Bi are some constant matrices of compatible dimensions, ΔAi (k) and ΔBi (k) are real-valued matrix functions of compatible dimensions representing time-varying parameter uncertainties. The linear fractional parametric uncertainties ΔAi (k) and ΔBi (k) are time-varying matrices with appropriate dimensions, which are defined as follows: ˆ i (k) N1i N2i ΔAi (k) ΔBi (k) = Mi Θ (2) −1 ˆ i (k) = Θi (k) [I − Gi Θi (k)] Θ (3) where Mi , N1i , N2i and Gi are known constant real matrices with appropriate dimensions, and Θi (k) is unknown time-varying matrix satisfying ΘTi (k)Θi (k) ≤ I, ∀k ≥ 0. It is assumed that the matrix [I − Gi Θi (k)]−1 is invertible for any Θi (k) and I − GTi Gi > 0. Through the use of fuzzy ‘blending’, the resulting fuzzy system model is inferred as the weighted average of the local models of the form: x(k + 1) = Ap (k)x(k) + Bp (k)sat(u(k)) where Ap (k) =
r
pi (k)Ai (k) =
i=1
Bp (k) =
r i=1
r
pi (k)Ai +
i=1
pi (k)Bi (k) =
r
r
ˆ i (k)N1i pi (k)Mi Θ
i=1
pi (k)Bi +
i=1
r
ˆ i (k)N2i pi (k)Mi Θ
i=1
with p
ωi (zi (k)) , ωi (zi (k)) = pi (k) := pi (zi (k)) = r Mij (zi (k)) j=1 ωj (zi (k)) j=1 5
(4)
where Mij (zi (k)) is the grade of membership of zj (k) in Mij . Note that the time-varying parameter vector p(k) belongs to a convex polytope P, where r P := p(k) ∈ Rr : pj (k) = 1, 0 ≤ pj (k) ≤ 1 j=1
Therefore, when pi (k) = 1 and pj (k) = 0 for i, j ∈ [1, r], j = i, the fuzzy system (4) reduces to its ith linear time-invariant “local” model, i.e., (Ap (k), Bp (k)) = (Ai (k), Bi (k)). It is clear that as p(k) varies inside the polytope P, the system matrices of (4) vary inside a corresponding polytope Ω whose vertices consist of r local system matrices [Ap (k), Bp (k)] ∈ Ω = co {(Ai (k), Bi (k)) , i ∈ [1, r]} where “co” denotes the convex hull. For fuzzy system (4), we consider the overall non-PDC state feedback fuzzy control law which is represented by u(k) = Kp L−1 p x(k)
(5)
where Kp =
r
pi (k)Ki , Lp =
i=1
r
pi (k)Li
i=1
which is called fuzzy scheduling controller. With the fuzzy control law (5), the closed-loop fuzzy system is written as follows
x(k + 1) = Ap (k)x(k) + Bp (k)sat Kp L−1 (6) p x(k) For the matrices Kp ∈ Rm×n and L ∈ Rn×n , denote the qth row of Kp L−1 p as −1 Kpq Lp and define n −1 L(Kp L−1 p ) := x(k) ∈ R : |Kpq Lp | ≤ 1, q = 1, 2, · · · , m Let P ∈ Rn×n be a positive-definite matrix. Denoting Ω(P ) := x(k) ∈ Rn : xT (k)P x(k) ≤ 1
6
(7)
(8)
and letting V (x(tk )) = xT (k)P x(k), the ellipsoid Ω(P ) is said to be contractive invariant if ΔV (x(k)) < 0 for all x(k) ∈ Ω(P )\{0}. Note that the contractively invariant set will be treated as an estimate of the domain of attraction for system (6). Let V be the set of m × m diagonal matrices whose diagonal elements are either 1 or 0. Suppose that each element of V is labeled as El , l = 1, 2, · · · , 2m . Denote El− = I − El . Note that El− is also an element of V if El ∈ V. Remark 1. The model (2) and (3) describe a wider class of parameter uncertainties than norm-bounded parameter uncertainties. It is easy to see that the linear fractional parameter uncertainties can be reduced to norm-bounded parameter uncertainties when Gi = 0. The following lemma, which captures certain properties of dynamical system with actuator saturation, will be used in this paper. Lemma 1. Hu et al., (2001) Let F ∈ Rm×n and H ∈ Rm×n be given. If x(k) ∈ L(F ), then sat(F x(k)) can be represented as: m
sat(F x(k)) =
2
ηl (k) El F + El− H x(k)
l=1 m where η2lm(k) for l = 1, 2, · · · , 2 are some scalars which satisfy 0 ≤ ηl (k) ≤ 1 and l=1 ηl (k) = 1.
The following is a well-known lemma which will be used to prove our main results in the next Section. Lemma 2. Xie et al., (1996) For some given matrices Υ, D and E of appropriate dimension and with Υ symmetric, then ˆ T (k)D T ≤ 0, ˆ Υ + D Θ(k)E + ET Θ ˆ where Θ(k) is given in (3), if and only if there exists a scalar ε > 0 such that Υ+
−1
ε E
T
εD
I −G T −G I
7
−1
ε−1 E εD T
≤ 0.
3. Main Results In this Section, a set of sufficient conditions on the robust stability for uncertain fuzzy discrete systems with actuator saturation are provided by using PDC control law and non-PDC control law, respectively. 3.1. Non-parallel Distributed Compensation By using Lemma 1, system (6) is represented as m
x(k + 1) =
2
− ηs (k) Ap (k) + Bp (k)Es Kp L−1 p + Bp (k)Es Hp x(k)
(9)
s=1
The aim of this Section is to design fuzzy scheduling controller (5) such that the origin of the closed-loop fuzzy discrete system (9) is asymptotically stable in a region as large as possible. Theorem 1. For the given uncertain fuzzy discrete system (9), if there exists a set of positive scalars αi > 0, as well as matrices Li , Lj , Kj , Wj , Sij > 0 with appropriate dimensions such that the following conditions ⎤ ⎡ 0 −LTi − Li + Sij Ξ(1, 2) Ξ(1, 3) ⎢ 0 αi Mi ⎥ ∗ −Sμν ⎥<0 Ξ=⎢ (10) ⎣ ∗ ∗ −αi I αi Gi ⎦ ∗ ∗ ∗ −αi I with Ξ(1, 2) = LTj ATi + KjT Es BiT + WjT Es− BiT Ξ(1, 3) = LTj N1iT + KjT Es N2iT + WjT Es− N2iT hold for all i, j, μ, ν = 1, 2, · · · , r, s = 1, 2, · · · , 2m and Ω(Sij−1 ) ⊂ L(H), in which the set ri=1 rj=1 Ω(Sij−1 ) is contained in the domain of attraction in square sense of system (9), the closed-loop fuzzy system (9) is asymptotically stable at the origin with Ω(Sij−1 ) contained in the domain of attraction. Proof. The weighting-dependent fuzzy non-PDC Lyapunov function is chosen as follows: −1 x(k) V (x(k)) = xT (k)Spp
8
(11)
where Spp =
r r
pi (k)pj (k)Sij
i=1 j=1
Let Spp(k+1) =
r r
pi (k + 1)pj (k + 1)Sij
i=1 j=1
Taking the difference manipulations along the trajectory of system (9), we have −1 −1 ΔV (x(k)) = xT (k + 1)Spp(k+1) x(k + 1) − xT (k)Spp x(k) −1 −1 x(k) A1 (k) − Spp = xT (k) AT1 (k)Spp(k+1)
(12)
where m
A1 (k) =
2
− ηs (k) Ap (k) + Bp (k)Es Kp L−1 p + Bp (k)Es H
s=1
It follows from (12) that −1 −1 A1 (k) − Spp x(k) < 0 ΔV (x(k)) = xT (k) AT1 (k)Spp(k+1)
(13)
−1 −1 A1 (k) − Spp . Pre-multiplying and post-multiplying Let Ξ1 = AT1 (k)Spp(k+1) T Ξ1 < 0 by Lp and Lp , respectively, we have that −1 −1 A1 (k)Lp − LTp Spp Lp < 0 Ξ2 = LTp AT1 (k)Spp(k+1)
Using Schur’s complement, Ξ2 < 0 is changed to
m
Ξ3 =
2 s=1
ηs (k)
−1 −LTp Spp Lp [Ap (k)Lp + Bp (k)Es Kp + Bp (k)Es− HLp ] ∗ −Spp(k+1)
<0
9
T
Moreover, there exists the fact that −1 −1 −1 )Spp (Lp − Spp ) ≥ 0 ⇒ LTp + Lp − Spp ≤ LTp Spp Lp (LTp − Spp
(14)
By inequalities Ξ3 < 0 and (14), there exists
m
2 s=1
ηs (k)
−LTp − Lp + Spp [Ap (k)Lp + Bp (k)Es Kp + Bp (k)Es− HLp ] ∗ −Spp(k+1)
T
<0 which can be rewritten as m
Ξ4 =
r r 2 r r
pi (k)pj (k)pμ (k + 1)pν (k + 1)ηs (k)
i=1 μ=1 ν=1 j=1 s=1
−LTi − Li + Sij AT2 (k) ∗ −Sμν
×
<0
(15)
with A2 (k) = Ai (k)Lj + Bi (k)Es Kj + Bi (k)Es− HLj Letting HLj = Wj , inequality (15) is equal to
T −LTj − Lj + Sij [Ai (k)Lj + Bi (k)Es Kj + Bi (k)Es− Wj ] Ξ5 = < 0 (16) ∗ −Sμν From (2) and (16), it can be got that ˆ i (k)ζi + ζ T Θ ˆ T (k)ξ T < 0 Ξ5 = Ξ6 + ξ i Θ i i i where
Ξ6 =
ζi =
−LTi − Li + Sij Ξ6 (3, 3) ∗ −Sμν
, ξi =
[N1i Li + N2i Es Ki + N2i Es− Wj ] 0
10
T
T
(17)
0 Mi
with Ξ6 (3, 3) = Ai Li + Bi Es Kj + Bi Es− Wj According to the Lemma 2, there exists a scalar εi > 0 such that inequality (17) is equal to the following inequality Ξ7 = Ξ6 +
εi ζiT
ε−1 i ξi
−I Gi GTi −I
−1
εi ζi −1 T εi ξi
Using Schur’s complement, Ξ7 < 0 is changed to ⎡ 0 −LTi − Li + Sij Ξ8 (1, 2) Ξ8 (1, 3) −1 ⎢ ∗ −S 0 ε μν i Mi Ξ8 = ⎢ ⎣ ∗ ∗ −I Gi ∗ ∗ ∗ −I
< 0.
(18)
⎤ ⎥ ⎥<0 ⎦
(19)
with Ξ8 (1, 2) = LTj ATi + KjT Es BiT + WjT Es− BiT Ξ8 (1, 3) = εi LTj N1iT + εi KjT Es N2iT + εi WjT Es− N2iT Pre-multiplying and post-multiplying (19) by the diagonal matrix −1 I, ε I , respectively, and letting αi = ε−2 diag I, I, ε−1 i i i , we have that (19) is rewritten as (10). This completes the proof. With all the ellipsoids satisfying the set invariance condition of Theorem 1, the “largest” one will be chosen to obtain the least conservative estimate of the domain of attraction. In the following, we will measure the largeness of the ellipsoids with respect to a shape reference set. Let XR ⊂ Rn be a prescribed bounded convex set containing origin, which can be represented as the polyhedron XR = Co{x10 , x20 , · · · , xυ0 } which is a priori given initial states in Rn . For a set S ⊂ Rn which contains origin, define βR (S) := sup {β > 0 : βXR ⊂ S} With the aforeIt is easy to get that if βR (S) ≥ 1, there exists s XRs ⊂ S. −1 mentioned reference sets, we can choose a set i=1 j=1 Ω(Sij ) from all that satisfy the condition such that the quantity βR si=1 sj=1 Ω(Sij−1 ) is maxi11
mized. To see if the initial states x0 ∈ Rn is in the domain of attraction, we formulate the following maximization problem: max
Sij >0,L,Kj ,Wj ,αi >0
⎧ ⎨ (i)
β
(20)
−1 βXR ⊂ si=1 sj=1 Ω(Sij ) (ii) Inequality (10) s.t. ⎩ (iii) |h x(k)| ≤ 1, ∀ x(k) ∈ r r Ω(S −1 ) iq i=1 j=1 ij
where hiq denotes the qth row of Hi . In the maximization problem (20), we have that condition (iii) is equivalent to Ω(Sij−1 ) ⊂ L(Hi ) as in Theorem 1. To facilitate the synthesis procedure, we need to formulate the problem (20) into a convex optimization problem. The condition (i) in (20) is equivalent to βxp0
∈
r r
Ω(Sij−1 ), p = 1, 2, · · · , υ.
i=1 j=1
Therefore, we have β 2 (xp0 )T Sij−1 xp0 ≤ 1, p = 1, 2, · · · , υ, i, j = 1, 2, · · · , r. which is equivalent to
−β −2 (xp0 )T ≤ 0, p = 1, 2, · · · , υ, i, j = 1, 2, · · · , r. xp0 −Sij One sufficient condition satisfying (iii) in (20) is xT (k)hTq hq x(k) ≤ xT (k)Sij−1 x(k), ∀ x(k) = 0 which is equivalent to hTq hq − Sij−1 ≤ 0, i, j = 1, 2, · · · , r; q = 1, 2, · · · , m Using Schur’s complements, we have
−Sij−1 hTq ≤ 0, i, j = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1
12
Pre- and post-multiply it with diag{LTi , 1} and diag{Li , 1}, respectively, then
−LTi Sij−1 Li LTi hTq ≤ 0, i, j = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1 By using inequality (14), it is obtained that
−LTi − Li + Sij LTi hTq ≤ 0, i, j = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1 Denote the qth row of Wi as wiq , then wiq = hq Li . By letting μ = β −2 , the optimization problem (20) can be transformed as the following linear matrix inequality problem: min
Sij >0,L,Kj ,Wj ,αi >0
⎧ ⎪ (i) ⎪ ⎪ ⎨
μ
(21)
−μ (xp0 )T ≤0 xp0 −Sij (ii) Inequality (10) s.t. ⎪ T −L +S T ⎪ −L i ij wiq ⎪ i ⎩ (iii) ≤0 ∗ −1
for all i, j, μ, ν = 1, 2, · · · , r, p = 1, 2, · · · , υ, q = 1, 2, · · · , m and s = 1, 2, · · · , 2m . If the obtained minimal value μmin < 1, which is equivalent to βmax > 1, then the concerned initial state x0 is in the domain of attraction. Moreover, we can see that the smaller the μmin is, the larger the βmax is, and therefore the obtained estimation of the domain of attraction is less conservative. 3.2. Parallel Distributed Compensation Based on PDC scheme, we consider the following fuzzy control law for the uncertain T-S fuzzy discrete systems (6). Control Rule i: IF z1 (k) is Mi1 , z2 (k) is Mi2 , · · ·, zq (k) is Miq , THEN u(k) = Fi x(k), i = 1, 2, · · · , r
13
(22)
The overall state feedback fuzzy control law is represented by u(k) =
r
pi (k)Fi x(k)
(23)
i=1
which is called fuzzy scheduling controller. The aim of this subjection is to design r local linear state feedback law (22) or the time-varying linear state feedback law (23) such that the origin of the closed-loop fuzzy system with actuator saturation is asymptotically stable in a region as large as possible. With control law (23) and letting Fp =
r
pi (k)Fi
i=1
the closed-loop fuzzy discrete system is written as follows x(k + 1) = Ap (k)x(k) + Bp (k)sat (Fp x(k))
(24)
By using Lemma 1, system (24) can be represented as m
x(k + 1) =
2
ηs (k) Ap (k) + Bp (k)Es Fp + Bp (k)Es− Hp x(k)
(25)
s=1
where Hp =
r
i=1
pi (k)Hi . Note that x(k) ∈ ri=1 L (Hi ) implies that r pi (k)Hi x(k) ∈ L i=1
since ri=1 pi (k) = 1 and 0 ≤ pi (k) ≤ 1. A set of sufficient conditions on the robust stability for system (25) are provided in the following theorem. Theorem 2. For the uncertain fuzzy discrete system (25), if there exists a set of positive scalars αi > 0, as well as matrices Pi > 0, X, Yj , Zj with
14
appropriate dimensions such that the following conditions ⎤ ⎡ 0 −X T − X + Pi Γ(1, 2) Γ(1, 3) ⎢ 0 αi Mi ⎥ ∗ −Pl ⎥<0 Γ=⎢ ⎣ ∗ ∗ −αi I αi Gi ⎦ ∗ ∗ ∗ −αi I
(26)
with Γ(1, 2) = X T ATi + YjT Es BiT + ZjT Es− BiT Γ(1, 3) = X T N1iT + YjT Es N2iT + ZjT Es− N2iT hold forall i, j, l = 1, 2, · · · , r, s = 1, 2, · · · , 2m and Ω(Pi−1 ) ⊂ L(Hi ), in which the set ri=1 Ω(Pi−1 ) is contained in the domain of attraction in square sense of system (25), the closed-loop fuzzy system (25) is asymptotically stable at the origin with Ω(Pi−1 ) contained in the domain of attraction. Proof. The fuzzy PDC Lyapunov function is chosen as follows: V (x(k)) = xT (k)Pp−1 x(k)
(27)
where Pp =
r
pi (k)Pi
i=1
Let Pp(k+1) =
r
pi (k + 1)Pi
i=1
Taking the difference manipulations along the trajectory of system (25), we have −1 x(k + 1) − xT (k)Pp−1 x(k) ΔV (x(k)) = xT (k + 1)Pp(k+1) −1 A1 (k) − Pp−1 x(k) = xT (k) A1T (k)Pp(k+1)
15
(28)
where m
A1 (k) =
2
ηs (k) Ap (k) + Bp (k)Es Fp + Bp (k)Es− Hp
s=1
It follows from (28) that −1 ΔV (x(k)) = x (k) A1T (k)Pp(k+1) A1 (k) − Pp−1 x(k) < 0 T
which means that −1 A1 (k) − Pp−1 < 0 A1T (k)Pp(k+1)
(29)
Using Schur’s complement, (29) is changed to
m
Γ1 =
2 s=1
ηs (k)
−Pp−1 [Ap (k) + Bp (k)Es Fp + Bp (k)Es− Hp ] ∗ −Pp(k+1)
T
<0
which is rewritten as
m
Γ1 =
r r r 2
pi (k)pj (k)pl (k + 1)ηs (k)
i=1 j=1 l=1 s=1
−Pi−1 A2T (k) ∗ −Pl
< 0 (30)
with A2 (k) = Ai (k) + Bi (k)Es Fj + Bi (k)Es− Hj Inequality (30) is equal to
T −Pi−1 [Ai (k) + Bi (k)Es Fj + Bi (k)Es− Hj ] <0 Γ2 = ∗ −Pl
(31)
From (2) and (31), it can be got that ˆ i (k)ζi + ζ T Θ ˆ T (k)ξ T < 0 Γ2 = Γ3 + ξ i Θ i i i
16
(32)
where
Γ3 =
ζi =
−Pi−1 ΓT3 (1, 2) ∗ −Pl
, ξi =
0 Mi
[N1i + N2i Es Fj + N2i Es− Hj ] 0
T
T
with Γ3 (1, 2) = Ai + Bi Es Fj + Bi Es− Hj By Lemma 2, there exists a scalar εi > 0 such that inequality (32) equal to the following inequality
−I Gi −1 εi ζi −1 T Γ4 = Γ3 + ε i ζ i ε i ξ i < 0. (33) T GTi −I ε−1 i ξi Using Schur’s complement, Γ4 < 0 is changed to ⎡ −Pi−1 Γ5 (1, 2) Γ5 (1, 3) 0 −1 ⎢ ∗ 0 εi Mi −Pl Γ5 = ⎢ ⎣ ∗ ∗ −I Gi ∗ ∗ ∗ −I
⎤ ⎥ ⎥<0 ⎦
(34)
with Γ5 (1, 2) = ATi + FjT Es BiT + HjT Es− BiT Γ5 (1, 3) = εi N1iT + εi FjT Es N2iT + εi HjT Es− N2iT For any matrix X, pre-multiplying and post-multiplying (34) by the diagonal matrices −1 diag X T , I, ε−1 I, ε I i i and
−1 diag X, I, ε−1 i I, εi I ,
respectively, and using the fact (X T − Pi−1 )Pi (X − Pi−1 ) ≥ 0 ⇒ X T + X − Pi ≤ X T Pi−1X
17
(35)
letting Fj X = Yj , Hj X = Zj and αi = ε−2 i , we have that (34) can be rewritten as (26). This completes the proof. If the initial states x0 ∈ Rn is in the domain of attraction, we formulate the following maximization problem: max
Pi >0,Pj >0,Yij ,Zjl ,αi >0
⎧ ⎨ (i)
β
(36)
βXR ⊂ ri=1 Ω(Pi−1 ) (ii) Inequality (26) s.t. ⎩ (iii) |h x(k)| ≤ 1, ∀x(k) ∈ r Ω(P −1 ) iq i=1 i
where hiq denotes the qth row of Hi . In the maximization problem (36), we have that condition (iii) is equivalent to Ω(Pi−1 ) ⊂ L(Hi ) in Theorem 2. To facilitate the synthesis procedure, we need to formulate the problem (36) into a convex optimization problem. The condition (i) in (36) is equivalent to βxp0 ∈
r
Ω(Pi−1 ), p = 1, 2, · · · , υ.
i=1
Therefore, we have β 2 (xp0 )T Pi−1xp0 ≤ 1, p = 1, 2, · · · , υ, i = 1, 2, · · · , r. which is equivalent to
−β −2 (xp0 )T ≤ 0, p = 1, 2, · · · , υ, i = 1, 2, · · · , r. xp0 −Pi One sufficient condition to satisfy (iii) in (36) is xT (k)hTiq hiq x(k) ≤ xT (k)Pi−1x(k), ∀ x(k) = 0 which is equivalent to hTiq hiq − Pi−1 ≤ 0, i = 1, 2, · · · , r; q = 1, 2, · · · , m
18
Using Schur’s complement, we have
−Pi−1 hTiq ≤ 0, i = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1
(37)
Pre-multiplying and post-multiplying (37) by the diagonal matrix diag{X T , I} and diag{X, I}, respectively, and using inequality (35), it is obtained that
−X T − X + Pi X T hTiq ≤ 0, i = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1 Denote the qth row of Zi as ziq , then ziq = hiq X. By letting μ = β −2 , the optimization problem (36) can be transformed as the following linear matrix inequality problem: min
Pi >0,L,Yj ,Zj ,αi >0
s.t.
⎧ ⎪ (i) ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
μ
(38)
−μ (xp0 )T ≤0 xp0 −Pi (ii) Inequality (26) T −X T − X + Pi ziq ≤0 (iii) ∗ −1
for all i, j, l = 1, 2, · · · , r, p = 1, 2, · · · , υ, q = 1, 2, · · · , m and s = 1, 2, · · · , 2m . In this case, as a by-product, we can get the controller gains Fi = Yi X −1 . Remark 2. From the Theorems, it can be seen that By using parameterdependent Lyapunov function, both PDC control law and non-PDC control law have been designed. And there is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T-S fuzzy system with actuator saturation. Two weighting-dependent fuzzy Lyapunov functions used in the paper have greater degree of freedom. Furthermore, the real control input is the saturated control input in control systems. 4. Numerical Example In the following, we will consider the problem of balancing and swing-up of an inverted pendulum on a cart to demonstrate the effectiveness of the proposed methods in this paper. 19
Example: The inverted pendulum model considered in this example is shown in Fig. 1, in which I is inertia of the pendulum rod, φ is the angle between the pendulum rod and the vertical, x is the horizontal displacement of the cart, u is the control input.
Fig. 1. Inverted pendulum model.
Following Cao et al., (2003) and Yang et al., (2014), the equation of motion for the pendulum can be approximated by the following two-rule fuzzy continuous-time model: Rule 1: IF x1 is about 0, THEN x(t) ˙ = A1s x(t) + B1s sat(u(t))
Rule 2: IF x1 is about ± π3 |x1 | < π3 , THEN x(t) ˙ = A2s x(t) + B2s sat(u(t)) with
! A1s = ! A2s =
0 g
4l −aml 3
0
g 2 π ( 4l −amlβ ) 3
1 0
! " 1 0 , B1s = " ! , B2s =
0
"
−μa
4l −aml 3
0
"
−μaβ
4l −amlβ 2 3
where x(t) = [xT1 (t) xT2 (t)]T , x1 (t) denotes the angle (in radians) of the pendulum from the vertical, x2 (t) is the angular velocity, g = 9.8m/s2 is 20
the gravity constant, M = 8.0kg is the mass of the cart, m = 2.0kg is the mass of the pendulum, 2l = 1.0m is the length of the pendulum, u(t) is the force applied to the cart (in Kilo-Newtons), a = 1/(m + M), μ = 1000 and β = cos(51◦ ). Taking T = 0.2, the continuous-time T-S fuzzy system in the simulation is changed to the following uncertain discrete-time T-S fuzzy system Rule 1: IF x1 (k) is about 0, THEN ˆ ˆ 1 (k)N21 sat(u(k)) x(k + 1) = A1 + M1 Θ1 (k)N11 x(k) + B1 + M1 Θ
Rule 2: IF x1 (k) is about ± π3 |x1 | < π3 , THEN ˆ 2 (k)N12 )x(k) + (B1 + M2 Θ ˆ 2 (k)N22 )sat(u(k)) x(k + 1) = (A2 + M2 Θ The membership functions for Rule 1 and Rule 2 are p1 (k) = cos(x1 (k)), p2 (k) = 1 − p1 (k) The parameters of the uncertain discrete-time T-S fuzzy system with actuator saturation is given as
1.3663 0.2239 −3.7376 A1 = , B1 = 3.8716 1.3663 −39.5064
1.1014 0.2067 −0.6653 A2 = , B2 = 1.0312 1.1014 −6.7635 T G1 = G2 = 0.2, M1 = M2 = 0.1 0.1 N11 = N12 = 0.1 0.1 , N21 = N22 = 0.1 To design the fuzzy non-PDC law, we solve the optimization problem (21) to get μ1,min = 0.6760 and
24.3946 −33.9855 19.0955 −44.2745 S11 = , S12 = −33.9855 108.5738 −44.2745 153.2287
26.7577 −20.7635 33.6875 −20.8143 , S22 = S21 = −20.7635 68.3062 −20.8143 51.7275 The domain of attraction obtained by the fuzzy non-PDC control law is 21
represented by the intersection of the four ellipsoids in Fig. 2.
10
x2
5
0
−5
−10
−10
−5
0 x
5
10
1
Fig. 2. Estimates of domain of attraction for fuzzy non-PDC control law
The following fuzzy non-PDC controller parameters are obtained as K1 = 3.2378 −0.3743 , K2 = 3.5816 0.4021
28.5927 −33.3302 28.1503 −17.1570 L1 = , L2 = −23.1270 85.6184 −27.7949 70.5024 By using the above controller parameters, the initial condition u(0) = 0.32, the control input of system (9) is given in Fig. 3. 0.05 u(k) 0
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
0
1
2
3
4
5
6
7
8
9
10
Fig. 3. Input of inverted-pendulum model for fuzzy non-PDC controller.
22
The state vector of the closed-loop fuzzy system (9) with initial conditions x(0) = [−3 7.9]T is given in Fig. 4. 8 x1(tk) x2(tk) 6
4
2
0
−2
−4
0
1
2
3
4
5
6
7
8
9
10
Fig. 4. States of inverted-pendulum model for fuzzy non-PDC controller.
It is shown from Fig. 4 that the closed-loop fuzzy system is convergent. For the fuzzy PDC law, we solve the optimization problem (38) to get μ2,min = 0.7726 and
21.2074 −43.6701 38.5650 −30.1296 P1 = , P2 = −43.6701 146.7205 −30.1296 68.9698
10
x2
5
0
−5
−10
−10
−5
0 x
5
10
1
Fig. 5. Estimates of domain of attraction for fuzzy PDC control law
23
The domain of attraction obtained by fuzzy PDC control law is represented by the intersection of the two ellipsoids in Fig. 5. The following controller parameters is obtained F1 = 0.1738 0.0511 , F2 = 0.1738 0.0511 By using the above controller parameters, the initial condition u(0) = 0, the control input of system (25) is given in Fig. 6. 0.25 u(k) 0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
0
1
2
3
4
5
6
7
8
9
10
Fig. 6. Input of inverted-pendulum model for fuzzy PDC controller.
The state vector of the closed-loop fuzzy system (9) with initial conditions x(0) = [−3 7.9]T is given in Fig. 4. 10 x1(k) x (k) 2
8
6
4
2
0
−2
−4
0
1
2
3
4
5
6
7
8
9
10
Fig. 7. States of inverted-pendulum model for fuzzy PDC controller.
24
It is shown from Fig. 7 that the closed-loop fuzzy system is also not divergent. Remark 3. For the reason of that there exists μ1,min = 0.6760 < 0.7726 = μ2,min in this simulation, there is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T-S fuzzy system with actuator saturation. The simulation result also showed that the fuzzy nonPDC controller has characteristics of small overshoot and longer convergence time than the fuzzy PDC controller by comparing Figs. 3-4 and Figs. 67. This is reasonable since the fuzzy non-PDC law controller uses more information about the simulation system than the fuzzy PDC controller. Remark 4. It has been shown in this paper that control of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law has been considered. The similar conclusion in delta domain is also gotten in Yang et al., (2014). However, we obtain some less conservative results for the reason of that the restriction on sampling period is removed in this paper. Furthermore, the two weighting-dependent fuzzy Lyapunov functions used in this paper have greater degree of freedom than the ones in reference Yang et al., (2014). That is, the convergence speeds are faster by PDC or non-PDC law in this paper than Yang et al., (2014). Some comparison results on convergence time are given in Table 1. Table 1: Comparisons on convergence time of this paper with reference Yang et al., (2014)
PDC non-PDC Reference Yang et al., (2014) 7.5∼8.5 6.3∼6.8 This paper 3.0∼4.0 2.1∼2.5
5. Conclusion The paper has considered the design of robust controller for a family of fuzzy discrete systems with saturated nonlinear feedback. A set invariance condition in the T-S fuzzy representation has been established. Based on the set invariance condition, the problem of estimating the domain of attraction of the T-S fuzzy discrete system has been considered. By using a class of parameter-dependent Lyapunov function, a type of non-PDC control law has been designed using linear matrix inequality approach. Moreover, PDC control law has also been considered to obtain another type of controllers. It 25
can be seen that there is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T-S fuzzy system since the fuzzy non-PDC law controller uses more information about the system than the fuzzy PDC controller. Numerical simulation results have been given to illustrate the effectiveness of the developed techniques. 6. Acknowledgment The authors would like to thank the anonymous reviewers for their detailed comments which helped to improve the quality of the paper. This work was supported by the National Key Basic Research Program (973), China (2014CB046405). The work of Ling Zhao was supported partially by the Hebei Provincial Natural Science Fund under Grand E2014203122. The work of Li Li was supported by the National Natural Science Foundation of China under Grant 61403330, the Postdoctoral Science Foundation of China under Grant 2014M551052, the Hebei Provincial Natural Science Fund under Grand F2015203163, and the Natural Science Foundation of Hebei Education Department under Grant QN2014068, respectively. References T. Takagi and M. Sugeno, “Fuzzy identification of systems and its application to modeling and control,” IEEE Transactions on Fuzzy Systems, 15, pp. 116-132, 1985. C. Hua, X. Guan and P. Shi, “Adaptive fuzzy control for uncertain interconnected time-delay systems,” Fuzzy Sets and Systems, 153(3), pp. 447-458, 2005. C. Hua, Q. Wang and X. Guan, “Adaptive fuzzy output feedback controller design for nonlinear time delay systems with unknown control direction,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 39(2), pp. 363-374, 2009. H. Yang, Y. Xia, B. Liu and G.-P. Liu, “Fault detection for T-S fuzzy discrete systems in finite frequency domain,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 41(4), pp. 911-920, 2011.
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Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law Ling Zhao, Li Li
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)00783-3 http://dx.doi.org/10.1016/j.neucom.2015.05.085 NEUCOM15617
To appear in:
Neurocomputing
Received date: 3 February 2015 Revised date: 14 April 2015 Accepted date: 21 May 2015 Cite this article as: Ling Zhao, Li Li, Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.05.085 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Robust stabilization of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law Ling Zhaoa,b , Li Lic,∗ a. Hebei Province Key Laboratory of Heveay Machinery Fluid Power Transmission and Control, Yanshan University, Qinhuangdao 066004, China. b. Key Laboratory of Advanced Forging & Stamping Technology and Science (Yanshan University) Ministry of Education of China, College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China. c. College of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China.
Abstract In this paper, the problem of robust stabilization of T-S fuzzy discrete systems with actuator saturation is studied. A set invariance condition is established as null controllable region. The robust stabilization results of the closed-loop T-S fuzzy system have a domain of attraction which is arbitrarily close to the null controllable region. The problem of estimating the domain of attraction of the T-S fuzzy closed-loop systems is formulated and solved as an optimization problem. By using parameter-dependent Lyapunov function, both parallel-distributed compensation (PDC) control law and nonparallel-distributed compensation control law are designed. And there is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T-S fuzzy system with actuator saturation. A numerical example is provided to illustrate the effectiveness of the proposed design techniques. Keywords: Actuator saturation, T-S fuzzy system, robust control, stability analysis, linear matrix inequality. 1. Introduction Fuzzy technique has been widely and successfully used in nonlinear system modeling and control for more than two decades. The T-S fuzzy model ∗
Corresponding author. Email: [email protected]
Preprint submitted to Elsevier
June 2, 2015
described by a family of fuzzy IF-THEN rules was firstly introduced in Takagi and Sugeno (1985). In other words, it formulates the complex nonlinear systems into a framework that interpolates some affine local models by a set of fuzzy membership functions. Based on this framework, a systematic analysis and design procedure for complex nonlinear systems can be possibly developed in view of the powerful control theories and techniques in linear systems. In recent years, based on the T-S fuzzy model, an intensive study on the stability issue of nonlinear systems has been made and many important results have been reported, such as in Hua et al., (2005, 2009); Yang et al., (2011, 2012, 2014); Yang, Shi, et al., (2014). Based on the interval type-2 T-S fuzzy model, some new fuzzy control results have also been reported, such as in Li et al., (2014, 2015); Li, Wu, et al., (2014, 2015), and references therein. A combined backstepping and small-gain approach to robust fuzzy adaptive output feedback control has been investigated in Tong et al., (2009). An adaptive control problem for strict-feedback nonlinear systems has been given based on a fuzzy observer method in Tong and Li, (2009). A largescale nonlinear systems with dynamical uncertainties has been considered by using fuzzy-adaptive decentralized output-feedback control approach in Tong et al., (2010). Reliable fuzzy control for nonlinear active suspension systems has been researched in Li et al., (2012, 2013). An efficient design technique, which is named parallel-distributed compensation (PDC) technique, was proposed to design fuzzy controllers and relax stability conditions in Wang et al., (1996). However, sufficient conditions derived using the fuzzy Lyapunov function and PDC law are usually in the forms of bilinear matrix inequalities which is not easily solved. To deal with the problem, a non-PDC controller have been adopted instead of the PDC controller in Lam et al., (2007). It has been shown in Guerra et al., (2004) that the non-PDC control laws derived by the parameter-dependent Lyapunov function-based approach could obtain less conservative stability conditions. A nonlinear controller has been designed by fuzzy non-PDC approach and non-quadratic Lyapunov function in Lee et al., (2010). An approach for discrete-time T-S fuzzy systems with time-varying state delay has been proposed via non-PDC scheme in Wu et al., (2011). By non-PDC control law derived by parameter-dependent Lyapunov function, constrained infinite-horizon model predictive control for fuzzy discrete systems has been investigated in Xia et al., (2010). Although many researchers have studied the stability issue of fuzzy systems with PDC or non-PDC control law for many years, the problem of non-PDC control law and parameter-dependent Lyapunov function in T-S fuzzy systems with 2
actuator saturation remains as an open research area. Actuator saturation are the key of control which is applicable to virtually all areas of engineering and science. However, majority of actuators are not strictly accord with linearity, most of them subject to saturation in real physical systems. During the past several decades, control systems with actuator saturation have received much attention, see for examples Hu et al., (2001, 2002); Zhou et al., (2011); Zhou, Zheng, et al., (2011); Zhou et al., (2012); Yang, Shi, et al., (2014); Yang, et al., (2014, 2015), and the references therein. The analysis and synthesis of T-S fuzzy systems with actuator saturation nonlinearities have received increasing attention recently (see, e.g., Ozgoli et al., (2009); Kim et al., (2009), and the references therein). A problem of fuzzy-scheduling control for nonlinear systems subject to actuator saturation has been considered in Cao et al., (2003). In Zhao et al., (2012), an overhead crane model is described by a class of T-S fuzzy systems with input delay and actuator saturation. Piecewise fuzzy anti-windup dynamic output feedback control of nonlinear processes with amplitude and rate actuator saturations has been investigated in Zhang et al., (2009). A nonlinear H∞ guidance law has been designed against maneuvering targets for fuzzy systems with actuator saturation Chen et al., (2002). A robust anti-windup controller has been designed for time-delay fuzzy systems with actuator saturation in Ting et al., (2011). Based on delta operator approach, the problems on robust fuzzy-scheduling control for nonlinear systems subject to actuator saturation have been considered in Yang et al., (2014). However, to the best of our knowledge, there have been few papers on nonPDC control law and parameter-dependent Lyapunov function for T-S fuzzy systems with actuator saturations, which motivates us to make an effort in this paper. In this paper, we consider the robust stabilization for a kind of uncertain T-S fuzzy discrete systems with actuator saturation. A wider class of parameter uncertainties named linear fractional parameter uncertainties than norm-bounded parameter uncertainties is used for the uncertain fuzzy system. The definition of the domain of attraction is introduced to analyze the stability of the closed-loop T-S fuzzy discrete system. Based on parameterdependent Lyapunov function approach, both PDC controller and non-PDC controller are designed via linear matrix inequalities. There is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T-S fuzzy system since the fuzzy non-PDC law controller uses more information about the simulation system than the fuzzy PDC con3
troller. Finally, a numerical example is provided to show the usefulness of the proposed techniques. The rest of the paper is organized as follows. Section 2 formulates the problem statement of uncertain T-S fuzzy discrete systems with actuator saturation. Section 3 covers the design of stabilizing PDC and non-PDC controllers for the class of uncertain T-S fuzzy discrete systems. Section 4 presents a numerical example to show the effectiveness of the developed results. The paper is concluded by Section 5. Notation: In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. Rn denotes the n-dimensional Euclidean space; the notation X > Y (X ≥ Y ) means that the matrix X −Y is positive definite (X − Y is semi-positive definite, respectively); I is the identity matrix of appropriate dimension; For any matrix A, AT denotes the transpose of matrix A, A−1 denotes the inverse of matrix A; · denotes the norm for vectors or the spectral norms of matrices. The shorthand diag{M1 M2 · · · Mr } denotes a block diagonal matrix with diagonal blocks being the matrices M1 , M2 · · · , Mr . The symmetric terms in a symmetric matrix are denoted by ∗. 2. Problem Statement In this paper, the fuzzy model is described by fuzzy IF-THEN rules, which represent local linear input-output relations of a nonlinear system. The ith rules of the fuzzy discrete-time models with uncertainty parameters and actuator saturation are of the following form. Plant Rule i: IF z1 (k) is Mi1 , z2 (k) is Mi2 , · · ·, zq (k) is Miq , THEN x(k + 1) = Ai (k)x(k) + Bi (k)sat(u(k)), i = 1, 2, · · · , r
(1)
where r is the number of IF-THEN rules, z1 (k), z2 (k), · · ·, zq (k) are the premise variables, Mil (l = 1, 2, · · · , q) are the fuzzy sets, Ai (k) and Bi (k) are real-valued time-varying matrices of appropriate dimensions, x(k) ∈ Rn is the state variable, u(k) ∈ Rm is the control input. The function sat: Rm → Rm is the standard saturation function. It is defined as sat(u) = [sat(u1 ), sat(u2 ), · · · , sat(um )]T
4
where sat(ui ) = sgn(ui ) min{1, ui } We assume that the time-varying uncertainties enter the system matrices in the following manner Ai (k) = Ai + ΔAi (k), Bi (k) = Bi + ΔBi (k) where Ai and Bi are some constant matrices of compatible dimensions, ΔAi (k) and ΔBi (k) are real-valued matrix functions of compatible dimensions representing time-varying parameter uncertainties. The linear fractional parametric uncertainties ΔAi (k) and ΔBi (k) are time-varying matrices with appropriate dimensions, which are defined as follows: ˆ i (k) N1i N2i ΔAi (k) ΔBi (k) = Mi Θ (2) −1 ˆ i (k) = Θi (k) [I − Gi Θi (k)] Θ (3) where Mi , N1i , N2i and Gi are known constant real matrices with appropriate dimensions, and Θi (k) is unknown time-varying matrix satisfying ΘTi (k)Θi (k) ≤ I, ∀k ≥ 0. It is assumed that the matrix [I − Gi Θi (k)]−1 is invertible for any Θi (k) and I − GTi Gi > 0. Through the use of fuzzy ‘blending’, the resulting fuzzy system model is inferred as the weighted average of the local models of the form: x(k + 1) = Ap (k)x(k) + Bp (k)sat(u(k)) where Ap (k) =
r
pi (k)Ai (k) =
i=1
Bp (k) =
r i=1
r
pi (k)Ai +
i=1
pi (k)Bi (k) =
r
r
ˆ i (k)N1i pi (k)Mi Θ
i=1
pi (k)Bi +
i=1
r
ˆ i (k)N2i pi (k)Mi Θ
i=1
with p
ωi (zi (k)) , ωi (zi (k)) = pi (k) := pi (zi (k)) = r Mij (zi (k)) j=1 ωj (zi (k)) j=1 5
(4)
where Mij (zi (k)) is the grade of membership of zj (k) in Mij . Note that the time-varying parameter vector p(k) belongs to a convex polytope P, where r P := p(k) ∈ Rr : pj (k) = 1, 0 ≤ pj (k) ≤ 1 j=1
Therefore, when pi (k) = 1 and pj (k) = 0 for i, j ∈ [1, r], j = i, the fuzzy system (4) reduces to its ith linear time-invariant “local” model, i.e., (Ap (k), Bp (k)) = (Ai (k), Bi (k)). It is clear that as p(k) varies inside the polytope P, the system matrices of (4) vary inside a corresponding polytope Ω whose vertices consist of r local system matrices [Ap (k), Bp (k)] ∈ Ω = co {(Ai (k), Bi (k)) , i ∈ [1, r]} where “co” denotes the convex hull. For fuzzy system (4), we consider the overall non-PDC state feedback fuzzy control law which is represented by u(k) = Kp L−1 p x(k)
(5)
where Kp =
r
pi (k)Ki , Lp =
i=1
r
pi (k)Li
i=1
which is called fuzzy scheduling controller. With the fuzzy control law (5), the closed-loop fuzzy system is written as follows
x(k + 1) = Ap (k)x(k) + Bp (k)sat Kp L−1 (6) p x(k) For the matrices Kp ∈ Rm×n and L ∈ Rn×n , denote the qth row of Kp L−1 p as −1 Kpq Lp and define n −1 L(Kp L−1 p ) := x(k) ∈ R : |Kpq Lp | ≤ 1, q = 1, 2, · · · , m Let P ∈ Rn×n be a positive-definite matrix. Denoting Ω(P ) := x(k) ∈ Rn : xT (k)P x(k) ≤ 1
6
(7)
(8)
and letting V (x(tk )) = xT (k)P x(k), the ellipsoid Ω(P ) is said to be contractive invariant if ΔV (x(k)) < 0 for all x(k) ∈ Ω(P )\{0}. Note that the contractively invariant set will be treated as an estimate of the domain of attraction for system (6). Let V be the set of m × m diagonal matrices whose diagonal elements are either 1 or 0. Suppose that each element of V is labeled as El , l = 1, 2, · · · , 2m . Denote El− = I − El . Note that El− is also an element of V if El ∈ V. Remark 1. The model (2) and (3) describe a wider class of parameter uncertainties than norm-bounded parameter uncertainties. It is easy to see that the linear fractional parameter uncertainties can be reduced to norm-bounded parameter uncertainties when Gi = 0. The following lemma, which captures certain properties of dynamical system with actuator saturation, will be used in this paper. Lemma 1. Hu et al., (2001) Let F ∈ Rm×n and H ∈ Rm×n be given. If x(k) ∈ L(F ), then sat(F x(k)) can be represented as: m
sat(F x(k)) =
2
ηl (k) El F + El− H x(k)
l=1 m where η2lm(k) for l = 1, 2, · · · , 2 are some scalars which satisfy 0 ≤ ηl (k) ≤ 1 and l=1 ηl (k) = 1.
The following is a well-known lemma which will be used to prove our main results in the next Section. Lemma 2. Xie et al., (1996) For some given matrices Υ, D and E of appropriate dimension and with Υ symmetric, then ˆ T (k)D T ≤ 0, ˆ Υ + D Θ(k)E + ET Θ ˆ where Θ(k) is given in (3), if and only if there exists a scalar ε > 0 such that Υ+
−1
ε E
T
εD
I −G T −G I
7
−1
ε−1 E εD T
≤ 0.
3. Main Results In this Section, a set of sufficient conditions on the robust stability for uncertain fuzzy discrete systems with actuator saturation are provided by using PDC control law and non-PDC control law, respectively. 3.1. Non-parallel Distributed Compensation By using Lemma 1, system (6) is represented as m
x(k + 1) =
2
− ηs (k) Ap (k) + Bp (k)Es Kp L−1 p + Bp (k)Es Hp x(k)
(9)
s=1
The aim of this Section is to design fuzzy scheduling controller (5) such that the origin of the closed-loop fuzzy discrete system (9) is asymptotically stable in a region as large as possible. Theorem 1. For the given uncertain fuzzy discrete system (9), if there exists a set of positive scalars αi > 0, as well as matrices Li , Lj , Kj , Wj , Sij > 0 with appropriate dimensions such that the following conditions ⎤ ⎡ 0 −LTi − Li + Sij Ξ(1, 2) Ξ(1, 3) ⎢ 0 αi Mi ⎥ ∗ −Sμν ⎥<0 Ξ=⎢ (10) ⎣ ∗ ∗ −αi I αi Gi ⎦ ∗ ∗ ∗ −αi I with Ξ(1, 2) = LTj ATi + KjT Es BiT + WjT Es− BiT Ξ(1, 3) = LTj N1iT + KjT Es N2iT + WjT Es− N2iT hold for all i, j, μ, ν = 1, 2, · · · , r, s = 1, 2, · · · , 2m and Ω(Sij−1 ) ⊂ L(H), in which the set ri=1 rj=1 Ω(Sij−1 ) is contained in the domain of attraction in square sense of system (9), the closed-loop fuzzy system (9) is asymptotically stable at the origin with Ω(Sij−1 ) contained in the domain of attraction. Proof. The weighting-dependent fuzzy non-PDC Lyapunov function is chosen as follows: −1 x(k) V (x(k)) = xT (k)Spp
8
(11)
where Spp =
r r
pi (k)pj (k)Sij
i=1 j=1
Let Spp(k+1) =
r r
pi (k + 1)pj (k + 1)Sij
i=1 j=1
Taking the difference manipulations along the trajectory of system (9), we have −1 −1 ΔV (x(k)) = xT (k + 1)Spp(k+1) x(k + 1) − xT (k)Spp x(k) −1 −1 x(k) A1 (k) − Spp = xT (k) AT1 (k)Spp(k+1)
(12)
where m
A1 (k) =
2
− ηs (k) Ap (k) + Bp (k)Es Kp L−1 p + Bp (k)Es H
s=1
It follows from (12) that −1 −1 A1 (k) − Spp x(k) < 0 ΔV (x(k)) = xT (k) AT1 (k)Spp(k+1)
(13)
−1 −1 A1 (k) − Spp . Pre-multiplying and post-multiplying Let Ξ1 = AT1 (k)Spp(k+1) T Ξ1 < 0 by Lp and Lp , respectively, we have that −1 −1 A1 (k)Lp − LTp Spp Lp < 0 Ξ2 = LTp AT1 (k)Spp(k+1)
Using Schur’s complement, Ξ2 < 0 is changed to
m
Ξ3 =
2 s=1
ηs (k)
−1 −LTp Spp Lp [Ap (k)Lp + Bp (k)Es Kp + Bp (k)Es− HLp ] ∗ −Spp(k+1)
<0
9
T
Moreover, there exists the fact that −1 −1 −1 )Spp (Lp − Spp ) ≥ 0 ⇒ LTp + Lp − Spp ≤ LTp Spp Lp (LTp − Spp
(14)
By inequalities Ξ3 < 0 and (14), there exists
m
2 s=1
ηs (k)
−LTp − Lp + Spp [Ap (k)Lp + Bp (k)Es Kp + Bp (k)Es− HLp ] ∗ −Spp(k+1)
T
<0 which can be rewritten as m
Ξ4 =
r r 2 r r
pi (k)pj (k)pμ (k + 1)pν (k + 1)ηs (k)
i=1 μ=1 ν=1 j=1 s=1
−LTi − Li + Sij AT2 (k) ∗ −Sμν
×
<0
(15)
with A2 (k) = Ai (k)Lj + Bi (k)Es Kj + Bi (k)Es− HLj Letting HLj = Wj , inequality (15) is equal to
T −LTj − Lj + Sij [Ai (k)Lj + Bi (k)Es Kj + Bi (k)Es− Wj ] Ξ5 = < 0 (16) ∗ −Sμν From (2) and (16), it can be got that ˆ i (k)ζi + ζ T Θ ˆ T (k)ξ T < 0 Ξ5 = Ξ6 + ξ i Θ i i i where
Ξ6 =
ζi =
−LTi − Li + Sij Ξ6 (3, 3) ∗ −Sμν
, ξi =
[N1i Li + N2i Es Ki + N2i Es− Wj ] 0
10
T
T
(17)
0 Mi
with Ξ6 (3, 3) = Ai Li + Bi Es Kj + Bi Es− Wj According to the Lemma 2, there exists a scalar εi > 0 such that inequality (17) is equal to the following inequality Ξ7 = Ξ6 +
εi ζiT
ε−1 i ξi
−I Gi GTi −I
−1
εi ζi −1 T εi ξi
Using Schur’s complement, Ξ7 < 0 is changed to ⎡ 0 −LTi − Li + Sij Ξ8 (1, 2) Ξ8 (1, 3) −1 ⎢ ∗ −S 0 ε μν i Mi Ξ8 = ⎢ ⎣ ∗ ∗ −I Gi ∗ ∗ ∗ −I
< 0.
(18)
⎤ ⎥ ⎥<0 ⎦
(19)
with Ξ8 (1, 2) = LTj ATi + KjT Es BiT + WjT Es− BiT Ξ8 (1, 3) = εi LTj N1iT + εi KjT Es N2iT + εi WjT Es− N2iT Pre-multiplying and post-multiplying (19) by the diagonal matrix −1 I, ε I , respectively, and letting αi = ε−2 diag I, I, ε−1 i i i , we have that (19) is rewritten as (10). This completes the proof. With all the ellipsoids satisfying the set invariance condition of Theorem 1, the “largest” one will be chosen to obtain the least conservative estimate of the domain of attraction. In the following, we will measure the largeness of the ellipsoids with respect to a shape reference set. Let XR ⊂ Rn be a prescribed bounded convex set containing origin, which can be represented as the polyhedron XR = Co{x10 , x20 , · · · , xυ0 } which is a priori given initial states in Rn . For a set S ⊂ Rn which contains origin, define βR (S) := sup {β > 0 : βXR ⊂ S} With the aforeIt is easy to get that if βR (S) ≥ 1, there exists s XRs ⊂ S. −1 mentioned reference sets, we can choose a set i=1 j=1 Ω(Sij ) from all that satisfy the condition such that the quantity βR si=1 sj=1 Ω(Sij−1 ) is maxi11
mized. To see if the initial states x0 ∈ Rn is in the domain of attraction, we formulate the following maximization problem: max
Sij >0,L,Kj ,Wj ,αi >0
⎧ ⎨ (i)
β
(20)
−1 βXR ⊂ si=1 sj=1 Ω(Sij ) (ii) Inequality (10) s.t. ⎩ (iii) |h x(k)| ≤ 1, ∀ x(k) ∈ r r Ω(S −1 ) iq i=1 j=1 ij
where hiq denotes the qth row of Hi . In the maximization problem (20), we have that condition (iii) is equivalent to Ω(Sij−1 ) ⊂ L(Hi ) as in Theorem 1. To facilitate the synthesis procedure, we need to formulate the problem (20) into a convex optimization problem. The condition (i) in (20) is equivalent to βxp0
∈
r r
Ω(Sij−1 ), p = 1, 2, · · · , υ.
i=1 j=1
Therefore, we have β 2 (xp0 )T Sij−1 xp0 ≤ 1, p = 1, 2, · · · , υ, i, j = 1, 2, · · · , r. which is equivalent to
−β −2 (xp0 )T ≤ 0, p = 1, 2, · · · , υ, i, j = 1, 2, · · · , r. xp0 −Sij One sufficient condition satisfying (iii) in (20) is xT (k)hTq hq x(k) ≤ xT (k)Sij−1 x(k), ∀ x(k) = 0 which is equivalent to hTq hq − Sij−1 ≤ 0, i, j = 1, 2, · · · , r; q = 1, 2, · · · , m Using Schur’s complements, we have
−Sij−1 hTq ≤ 0, i, j = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1
12
Pre- and post-multiply it with diag{LTi , 1} and diag{Li , 1}, respectively, then
−LTi Sij−1 Li LTi hTq ≤ 0, i, j = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1 By using inequality (14), it is obtained that
−LTi − Li + Sij LTi hTq ≤ 0, i, j = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1 Denote the qth row of Wi as wiq , then wiq = hq Li . By letting μ = β −2 , the optimization problem (20) can be transformed as the following linear matrix inequality problem: min
Sij >0,L,Kj ,Wj ,αi >0
⎧ ⎪ (i) ⎪ ⎪ ⎨
μ
(21)
−μ (xp0 )T ≤0 xp0 −Sij (ii) Inequality (10) s.t. ⎪ T −L +S T ⎪ −L i ij wiq ⎪ i ⎩ (iii) ≤0 ∗ −1
for all i, j, μ, ν = 1, 2, · · · , r, p = 1, 2, · · · , υ, q = 1, 2, · · · , m and s = 1, 2, · · · , 2m . If the obtained minimal value μmin < 1, which is equivalent to βmax > 1, then the concerned initial state x0 is in the domain of attraction. Moreover, we can see that the smaller the μmin is, the larger the βmax is, and therefore the obtained estimation of the domain of attraction is less conservative. 3.2. Parallel Distributed Compensation Based on PDC scheme, we consider the following fuzzy control law for the uncertain T-S fuzzy discrete systems (6). Control Rule i: IF z1 (k) is Mi1 , z2 (k) is Mi2 , · · ·, zq (k) is Miq , THEN u(k) = Fi x(k), i = 1, 2, · · · , r
13
(22)
The overall state feedback fuzzy control law is represented by u(k) =
r
pi (k)Fi x(k)
(23)
i=1
which is called fuzzy scheduling controller. The aim of this subjection is to design r local linear state feedback law (22) or the time-varying linear state feedback law (23) such that the origin of the closed-loop fuzzy system with actuator saturation is asymptotically stable in a region as large as possible. With control law (23) and letting Fp =
r
pi (k)Fi
i=1
the closed-loop fuzzy discrete system is written as follows x(k + 1) = Ap (k)x(k) + Bp (k)sat (Fp x(k))
(24)
By using Lemma 1, system (24) can be represented as m
x(k + 1) =
2
ηs (k) Ap (k) + Bp (k)Es Fp + Bp (k)Es− Hp x(k)
(25)
s=1
where Hp =
r
i=1
pi (k)Hi . Note that x(k) ∈ ri=1 L (Hi ) implies that r pi (k)Hi x(k) ∈ L i=1
since ri=1 pi (k) = 1 and 0 ≤ pi (k) ≤ 1. A set of sufficient conditions on the robust stability for system (25) are provided in the following theorem. Theorem 2. For the uncertain fuzzy discrete system (25), if there exists a set of positive scalars αi > 0, as well as matrices Pi > 0, X, Yj , Zj with
14
appropriate dimensions such that the following conditions ⎤ ⎡ 0 −X T − X + Pi Γ(1, 2) Γ(1, 3) ⎢ 0 αi Mi ⎥ ∗ −Pl ⎥<0 Γ=⎢ ⎣ ∗ ∗ −αi I αi Gi ⎦ ∗ ∗ ∗ −αi I
(26)
with Γ(1, 2) = X T ATi + YjT Es BiT + ZjT Es− BiT Γ(1, 3) = X T N1iT + YjT Es N2iT + ZjT Es− N2iT hold forall i, j, l = 1, 2, · · · , r, s = 1, 2, · · · , 2m and Ω(Pi−1 ) ⊂ L(Hi ), in which the set ri=1 Ω(Pi−1 ) is contained in the domain of attraction in square sense of system (25), the closed-loop fuzzy system (25) is asymptotically stable at the origin with Ω(Pi−1 ) contained in the domain of attraction. Proof. The fuzzy PDC Lyapunov function is chosen as follows: V (x(k)) = xT (k)Pp−1 x(k)
(27)
where Pp =
r
pi (k)Pi
i=1
Let Pp(k+1) =
r
pi (k + 1)Pi
i=1
Taking the difference manipulations along the trajectory of system (25), we have −1 x(k + 1) − xT (k)Pp−1 x(k) ΔV (x(k)) = xT (k + 1)Pp(k+1) −1 A1 (k) − Pp−1 x(k) = xT (k) A1T (k)Pp(k+1)
15
(28)
where m
A1 (k) =
2
ηs (k) Ap (k) + Bp (k)Es Fp + Bp (k)Es− Hp
s=1
It follows from (28) that −1 ΔV (x(k)) = x (k) A1T (k)Pp(k+1) A1 (k) − Pp−1 x(k) < 0 T
which means that −1 A1 (k) − Pp−1 < 0 A1T (k)Pp(k+1)
(29)
Using Schur’s complement, (29) is changed to
m
Γ1 =
2 s=1
ηs (k)
−Pp−1 [Ap (k) + Bp (k)Es Fp + Bp (k)Es− Hp ] ∗ −Pp(k+1)
T
<0
which is rewritten as
m
Γ1 =
r r r 2
pi (k)pj (k)pl (k + 1)ηs (k)
i=1 j=1 l=1 s=1
−Pi−1 A2T (k) ∗ −Pl
< 0 (30)
with A2 (k) = Ai (k) + Bi (k)Es Fj + Bi (k)Es− Hj Inequality (30) is equal to
T −Pi−1 [Ai (k) + Bi (k)Es Fj + Bi (k)Es− Hj ] <0 Γ2 = ∗ −Pl
(31)
From (2) and (31), it can be got that ˆ i (k)ζi + ζ T Θ ˆ T (k)ξ T < 0 Γ2 = Γ3 + ξ i Θ i i i
16
(32)
where
Γ3 =
ζi =
−Pi−1 ΓT3 (1, 2) ∗ −Pl
, ξi =
0 Mi
[N1i + N2i Es Fj + N2i Es− Hj ] 0
T
T
with Γ3 (1, 2) = Ai + Bi Es Fj + Bi Es− Hj By Lemma 2, there exists a scalar εi > 0 such that inequality (32) equal to the following inequality
−I Gi −1 εi ζi −1 T Γ4 = Γ3 + ε i ζ i ε i ξ i < 0. (33) T GTi −I ε−1 i ξi Using Schur’s complement, Γ4 < 0 is changed to ⎡ −Pi−1 Γ5 (1, 2) Γ5 (1, 3) 0 −1 ⎢ ∗ 0 εi Mi −Pl Γ5 = ⎢ ⎣ ∗ ∗ −I Gi ∗ ∗ ∗ −I
⎤ ⎥ ⎥<0 ⎦
(34)
with Γ5 (1, 2) = ATi + FjT Es BiT + HjT Es− BiT Γ5 (1, 3) = εi N1iT + εi FjT Es N2iT + εi HjT Es− N2iT For any matrix X, pre-multiplying and post-multiplying (34) by the diagonal matrices −1 diag X T , I, ε−1 I, ε I i i and
−1 diag X, I, ε−1 i I, εi I ,
respectively, and using the fact (X T − Pi−1 )Pi (X − Pi−1 ) ≥ 0 ⇒ X T + X − Pi ≤ X T Pi−1X
17
(35)
letting Fj X = Yj , Hj X = Zj and αi = ε−2 i , we have that (34) can be rewritten as (26). This completes the proof. If the initial states x0 ∈ Rn is in the domain of attraction, we formulate the following maximization problem: max
Pi >0,Pj >0,Yij ,Zjl ,αi >0
⎧ ⎨ (i)
β
(36)
βXR ⊂ ri=1 Ω(Pi−1 ) (ii) Inequality (26) s.t. ⎩ (iii) |h x(k)| ≤ 1, ∀x(k) ∈ r Ω(P −1 ) iq i=1 i
where hiq denotes the qth row of Hi . In the maximization problem (36), we have that condition (iii) is equivalent to Ω(Pi−1 ) ⊂ L(Hi ) in Theorem 2. To facilitate the synthesis procedure, we need to formulate the problem (36) into a convex optimization problem. The condition (i) in (36) is equivalent to βxp0 ∈
r
Ω(Pi−1 ), p = 1, 2, · · · , υ.
i=1
Therefore, we have β 2 (xp0 )T Pi−1xp0 ≤ 1, p = 1, 2, · · · , υ, i = 1, 2, · · · , r. which is equivalent to
−β −2 (xp0 )T ≤ 0, p = 1, 2, · · · , υ, i = 1, 2, · · · , r. xp0 −Pi One sufficient condition to satisfy (iii) in (36) is xT (k)hTiq hiq x(k) ≤ xT (k)Pi−1x(k), ∀ x(k) = 0 which is equivalent to hTiq hiq − Pi−1 ≤ 0, i = 1, 2, · · · , r; q = 1, 2, · · · , m
18
Using Schur’s complement, we have
−Pi−1 hTiq ≤ 0, i = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1
(37)
Pre-multiplying and post-multiplying (37) by the diagonal matrix diag{X T , I} and diag{X, I}, respectively, and using inequality (35), it is obtained that
−X T − X + Pi X T hTiq ≤ 0, i = 1, 2, · · · , r; q = 1, 2, · · · , m ∗ −1 Denote the qth row of Zi as ziq , then ziq = hiq X. By letting μ = β −2 , the optimization problem (36) can be transformed as the following linear matrix inequality problem: min
Pi >0,L,Yj ,Zj ,αi >0
s.t.
⎧ ⎪ (i) ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
μ
(38)
−μ (xp0 )T ≤0 xp0 −Pi (ii) Inequality (26) T −X T − X + Pi ziq ≤0 (iii) ∗ −1
for all i, j, l = 1, 2, · · · , r, p = 1, 2, · · · , υ, q = 1, 2, · · · , m and s = 1, 2, · · · , 2m . In this case, as a by-product, we can get the controller gains Fi = Yi X −1 . Remark 2. From the Theorems, it can be seen that By using parameterdependent Lyapunov function, both PDC control law and non-PDC control law have been designed. And there is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T-S fuzzy system with actuator saturation. Two weighting-dependent fuzzy Lyapunov functions used in the paper have greater degree of freedom. Furthermore, the real control input is the saturated control input in control systems. 4. Numerical Example In the following, we will consider the problem of balancing and swing-up of an inverted pendulum on a cart to demonstrate the effectiveness of the proposed methods in this paper. 19
Example: The inverted pendulum model considered in this example is shown in Fig. 1, in which I is inertia of the pendulum rod, φ is the angle between the pendulum rod and the vertical, x is the horizontal displacement of the cart, u is the control input.
Fig. 1. Inverted pendulum model.
Following Cao et al., (2003) and Yang et al., (2014), the equation of motion for the pendulum can be approximated by the following two-rule fuzzy continuous-time model: Rule 1: IF x1 is about 0, THEN x(t) ˙ = A1s x(t) + B1s sat(u(t))
Rule 2: IF x1 is about ± π3 |x1 | < π3 , THEN x(t) ˙ = A2s x(t) + B2s sat(u(t)) with
! A1s = ! A2s =
0 g
4l −aml 3
0
g 2 π ( 4l −amlβ ) 3
1 0
! " 1 0 , B1s = " ! , B2s =
0
"
−μa
4l −aml 3
0
"
−μaβ
4l −amlβ 2 3
where x(t) = [xT1 (t) xT2 (t)]T , x1 (t) denotes the angle (in radians) of the pendulum from the vertical, x2 (t) is the angular velocity, g = 9.8m/s2 is 20
the gravity constant, M = 8.0kg is the mass of the cart, m = 2.0kg is the mass of the pendulum, 2l = 1.0m is the length of the pendulum, u(t) is the force applied to the cart (in Kilo-Newtons), a = 1/(m + M), μ = 1000 and β = cos(51◦ ). Taking T = 0.2, the continuous-time T-S fuzzy system in the simulation is changed to the following uncertain discrete-time T-S fuzzy system Rule 1: IF x1 (k) is about 0, THEN ˆ ˆ 1 (k)N21 sat(u(k)) x(k + 1) = A1 + M1 Θ1 (k)N11 x(k) + B1 + M1 Θ
Rule 2: IF x1 (k) is about ± π3 |x1 | < π3 , THEN ˆ 2 (k)N12 )x(k) + (B1 + M2 Θ ˆ 2 (k)N22 )sat(u(k)) x(k + 1) = (A2 + M2 Θ The membership functions for Rule 1 and Rule 2 are p1 (k) = cos(x1 (k)), p2 (k) = 1 − p1 (k) The parameters of the uncertain discrete-time T-S fuzzy system with actuator saturation is given as
1.3663 0.2239 −3.7376 A1 = , B1 = 3.8716 1.3663 −39.5064
1.1014 0.2067 −0.6653 A2 = , B2 = 1.0312 1.1014 −6.7635 T G1 = G2 = 0.2, M1 = M2 = 0.1 0.1 N11 = N12 = 0.1 0.1 , N21 = N22 = 0.1 To design the fuzzy non-PDC law, we solve the optimization problem (21) to get μ1,min = 0.6760 and
24.3946 −33.9855 19.0955 −44.2745 S11 = , S12 = −33.9855 108.5738 −44.2745 153.2287
26.7577 −20.7635 33.6875 −20.8143 , S22 = S21 = −20.7635 68.3062 −20.8143 51.7275 The domain of attraction obtained by the fuzzy non-PDC control law is 21
represented by the intersection of the four ellipsoids in Fig. 2.
10
x2
5
0
−5
−10
−10
−5
0 x
5
10
1
Fig. 2. Estimates of domain of attraction for fuzzy non-PDC control law
The following fuzzy non-PDC controller parameters are obtained as K1 = 3.2378 −0.3743 , K2 = 3.5816 0.4021
28.5927 −33.3302 28.1503 −17.1570 L1 = , L2 = −23.1270 85.6184 −27.7949 70.5024 By using the above controller parameters, the initial condition u(0) = 0.32, the control input of system (9) is given in Fig. 3. 0.05 u(k) 0
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
0
1
2
3
4
5
6
7
8
9
10
Fig. 3. Input of inverted-pendulum model for fuzzy non-PDC controller.
22
The state vector of the closed-loop fuzzy system (9) with initial conditions x(0) = [−3 7.9]T is given in Fig. 4. 8 x1(tk) x2(tk) 6
4
2
0
−2
−4
0
1
2
3
4
5
6
7
8
9
10
Fig. 4. States of inverted-pendulum model for fuzzy non-PDC controller.
It is shown from Fig. 4 that the closed-loop fuzzy system is convergent. For the fuzzy PDC law, we solve the optimization problem (38) to get μ2,min = 0.7726 and
21.2074 −43.6701 38.5650 −30.1296 P1 = , P2 = −43.6701 146.7205 −30.1296 68.9698
10
x2
5
0
−5
−10
−10
−5
0 x
5
10
1
Fig. 5. Estimates of domain of attraction for fuzzy PDC control law
23
The domain of attraction obtained by fuzzy PDC control law is represented by the intersection of the two ellipsoids in Fig. 5. The following controller parameters is obtained F1 = 0.1738 0.0511 , F2 = 0.1738 0.0511 By using the above controller parameters, the initial condition u(0) = 0, the control input of system (25) is given in Fig. 6. 0.25 u(k) 0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
0
1
2
3
4
5
6
7
8
9
10
Fig. 6. Input of inverted-pendulum model for fuzzy PDC controller.
The state vector of the closed-loop fuzzy system (9) with initial conditions x(0) = [−3 7.9]T is given in Fig. 4. 10 x1(k) x (k) 2
8
6
4
2
0
−2
−4
0
1
2
3
4
5
6
7
8
9
10
Fig. 7. States of inverted-pendulum model for fuzzy PDC controller.
24
It is shown from Fig. 7 that the closed-loop fuzzy system is also not divergent. Remark 3. For the reason of that there exists μ1,min = 0.6760 < 0.7726 = μ2,min in this simulation, there is less conservative by using non-PDC law than using PDC law to control the uncertain discrete-time T-S fuzzy system with actuator saturation. The simulation result also showed that the fuzzy nonPDC controller has characteristics of small overshoot and longer convergence time than the fuzzy PDC controller by comparing Figs. 3-4 and Figs. 67. This is reasonable since the fuzzy non-PDC law controller uses more information about the simulation system than the fuzzy PDC controller. Remark 4. It has been shown in this paper that control of T-S fuzzy discrete systems with actuator saturation via PDC and non-PDC law has been considered. The similar conclusion in delta domain is also gotten in Yang et al., (2014). However, we obtain some less conservative results for the reason of that the restriction on sampling period is removed in this paper. Furthermore, the two weighting-dependent fuzzy Lyapunov functions used in this paper have greater degree of freedom than the ones in reference Yang et al., (2014). That is, the convergence speeds are faster by PDC or non-PDC law in this paper than Yang et al., (2014). Some comparison results on convergence time are given in Table 1. Table 1: Comparisons on convergence time of this paper with reference Yang et al., (2014)
PDC non-PDC Reference Yang et al., (2014) 7.5∼8.5 6.3∼6.8 This paper 3.0∼4.0 2.1∼2.5
5. Conclusion The paper has considered the design of robust controller for a family of fuzzy discrete systems with saturated nonlinear feedback. A set invariance condition in the T-S fuzzy representation has been established. Based on the set invariance condition, the problem of estimating the domain of attraction of the T-S fuzzy discrete system has been considered. By using a class of parameter-dependent Lyapunov function, a type of non-PDC control law has been designed using linear matrix inequality approach. Moreover, PDC control law has also been considered to obtain another type of controllers. It 25
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