Sliding mode control for polytopic differential inclusion systems

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Mathematics and Computers in Simulation 79 (2009) 3018–3025

Sliding mode control for polytopic differential inclusion systems Leipo Liu a,∗ , Zhengzhi Han a , Xiushan Cai b b

a School of Electrical and Information Engineering, Shanghai Jiao Tong University, Shanghai 200240, China College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China

Received 16 June 2008; received in revised form 17 January 2009; accepted 29 January 2009 Available online 10 February 2009

Abstract The stabilization problem of polytopic differential inclusion (PDI) systems is investigated by using sliding mode control. Sliding surface is designed and sufficient conditions for asymptotic stability of sliding mode dynamics are derived. A novel feedback law is established to make the state of system reach the sliding surface in a finite time. Finally, an example is given to illustrate the validity of the proposed design. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Polytopic differential inclusion systems; Sliding mode control; Asymptotic stability

1. Introduction At the past 50 years, the theory of linear systems have been widely and deeply investigated. Since Rudolph E. Kalman proposed the state space method of linear systems, a lot of textbooks have been published, especially, [4] and [21] are two classical works. Recently, [1] gives a more present description of linear system theory. The main target of the paper is to extend some results for linear systems to the systems described by differential inclusions (DI). Recently, the study of DI systems has been paid much attention by many authors. The DI systems are considered a generalization of the system of differential equations. Many practical systems should be described by DI systems [3,5–20]. In [3], the asymptotic stability of linear differential inclusion (LDI) systems is investigated, especially, some specific families of LDIs, such as polytopic LDIs, norm-bound LDIs, diagonal norm-bound LDIs, are studied extensively. In [18], the control Lyapunov function approach is employed to solve the stabilizing problem for single-input PDI systems. In [9], a nonlinear control design method for LDIs is presented by using quadratic Lyapunov functions of their convex hull. In [14], a frequency-domain approach is proposed to analyze the globally asymptotic stability of DI systems with discrete and distributed time-delays. In [5], the problem of tracking control of nonlinear uncertain dynamical systems described by DIs is studied. In [10], a necessary and sufficient condition for the stability of polytopic LDIs is derived by bilinear matrix equations. As we know, sliding mode control is a very effective approach for the control of nonlinear systems. It has many attractive features such as fast response, good transient response and insensitivity to variations [6,11,19]. This paper will apply sliding mode control to the stabilization of a multi-input PDI system which is more general such that the systems dealt with in [12,13,16,20] can be regarded as especial forms of the system. A novel feedback law is ∗

Corresponding author. Tel.: +86 21 34202028; fax: +86 21 62932083. E-mail address: [email protected] (L. Liu).

0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.01.022

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given to make the closed-loop system stable asymptotically and to reduce the chattering phenomenon. Finally, we give a simulation result to show the advantages of this method. 2. Problem formulation Consider the following PDI system: ˙ x¯ 1 (t) = A11 x¯ 1 (t) + A12 x¯ 2 (t), (1)

x¯˙ 2 (t) ∈ co{fi (x) + gi (x)B(u(t) + w(t)), i = 1, 2, . . . , N}. T

where x¯ 1 = [x1 , x2 , . . . , xm ]T , x¯ 2 = [xm+1 , . . . , xn ]T and x = [¯x1T , x¯ 2T ] ∈ Rn is the system state, co{·} denotes the convex hull of a set, fi (x) ∈ Rn−m , gi (x) ∈ R, are smooth vector-valued function and function, respectively. u(t) ∈ Rn−m is the control input, B ∈ R(n−m)×(n−m) is nonsingular. A11 , A12 are matrices with compatible dimensions and (A11 , A12 ) is assumed to be completely controllable. w(t) is a bounded disturbance, i.e., w(t) ≤ γ, with a positive constant γ. This paper assumes that gi (x) > 0, for all i = 1, 2, . . . , N. By the conclusion established in convex analysis theory [15], this PDI system (1) is equivalent to the following uncertain system: ⎧˙ x¯ 1 (t) = A11 x¯ 1 (t) + A12 x¯ 2 (t), ⎪ ⎪ ⎨ N (2)  ⎪ ˙ ⎪ x ¯ (t) = αi [fi (x) + gi (x)B(u(t) + w(t))]. ⎩ 2 i=1

where αi , i = 1, 2, . . . , N, are uncertain parameters with the properties that αi ≥ 0 and

N

i=1 αi

= 1.

3. Main results The control law of the uncertain system (2) is designed by two phases. Firstly, an sliding surface is chosen, and secondly, a feedback law is designed such that the state of system (2) converges to the sliding surface in a finite time. The design is stated in the following two steps. Step 1. (A11 , A12 ) is controllable, hence we can find a matrix C1 ∈ R(n−m)×m such that A11 − A12 C1 is a Hurwitz matrix. Let C = [C1 I], the sliding surface is defined by s(t) = Cx(t).

(3)

Let s(t) = 0. Then from (3), we have x¯ 2 (t) = −C1 x¯ 1 (t).

(4)

Substituting Eq. (4) into the first equation of (2), the sliding mode dynamics is x¯˙ 1 (t) = (A11 − A12 C1 )¯x1 (t).

(5)

A11 − A12 C1 is a Hurwitz matrix, hence x¯ 1 (t) → 0 as t → ∞. It follows x¯ 2 (t) → 0 by (4). Thus we conclude that the sliding mode dynamics (5) is asymptotically stable. Step 2. We present a novel feedback law in this step to drive the state to the sliding surface s(t) = 0 in a finite time. The novel feedback law is considered as

B−1 s(t) u(t) = − C1 A11 x¯ 1 (t) + C1 A12 x¯ 2 (t) + k(x) , a(x) s(t)

(6)

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where



b(x) k(x) = − 1 C1 A11 x¯ 1 (t) + C1 A12 x¯ 2 (t) + d(x) + b(x)Bγ + ηe−λt s(t)1−α , a(x) a(x) = min{g1 (x), . . . , gN (x)} > 0, b(x) = max{g1 (x), . . . , gN (x)}, d(x) = max{f1 (x), . . . , fN (x)},

and η > 0, λ > 0, 0 < α ≤ 1. In the following, we show that the trajectory of the closed-loop system combined by (2) and (6) converges to the sliding surface s(t) = 0 within a finite time T and remains in it thereafter. The reach time T satisfies that T ≤ −(1/λ)ln[1 − (λ/(αη))s(0)α ], where 0 ≤ (λ/(αη))s(0)α < 1. Consider a Lyapunov function candidate as follows: V (t) =

1 T s (t)s(t), 2

(7)

Calculating the time derivative of V (t) along the trajectory of (2), we have

 N N   T ˙ V (t) = s (t) C1 A11 x¯ 1 (t) + C1 A12 x¯ 2 (t) + αi fi (x) + αi gi (x)B(u(t) + w(t)) ⎡⎛ ⎢⎜ ⎢⎜ ⎢⎜ = s (t) ⎢⎜1 − ⎢⎜ ⎣⎝ T

N 

i=1

⎞

αi gi (x) ⎟ ⎟ ⎟ ⎟ (C1 A11 x¯ 1 (t) + C1 A12 x¯ 2 (t)) a(x) ⎟ ⎠

i=1

N 

+

N 

αi (fi (x) + gi (x)Bw(t)) −

i=1

Define: k(x) =



i=1

⎤ αi gi (x)

i=1

a(x)

⎥ s(t) ⎥ ⎥ k(x) ⎥. s(t) ⎥ ⎦

b(x) − 1 C1 A11 x¯ 1 (t) + C1 A12 x¯ 2 (t) + d(x) + b(x)Bγ + ηe−λt s(t)1−α a(x)

(8)

(9)

Substituting (9) into (8) yields to V˙ (t) ≤ −ηe−λt s(t)2−α .

(10)

Next, we will give the reach time. In terms of the matrix theory [2], we have ds(t) 1 ds(t)2 V˙ (t) = sT (t)˙s(t) = = s(t) , 2 dt dt

(11)

From (10) and (11), we have ds(t) ≤ −ηe−λt s(t)1−α . dt Let s(T ) = 0, integrating (12) from 0 to T, we get

1 λ T ≤ − ln 1 − s(0)α , λ αη

(12)

(13)

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Fig. 1. State x(t) (α1 = α2 = 0.5).

Fig. 2. State x(t) (α1 = 0, α2 = 1).

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Fig. 3. Control u1 (t) (α1 = 0, α2 = 1).

Fig. 4. Control u1 (t) (α1 = 0, α2 = 1) in [13].

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Fig. 5. Control u2 (t) (α1 = 0, α2 = 1).

Fig. 6. Control u2 (t) (α1 = 0, α2 = 1) in [13].

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where the parameters α, η and λ are selected such that the following inequality is satisfied: 0≤

λ s(0)α < 1. αη

According to sliding mode theory, we conclude that the trajectory of the closed-loop system combined by (2) and (6) converges to the sliding surface s(t) = 0 in a finite time T and remains in it thereafter. Thus, the proof is completed. Remark. Comparing with [13], we add a term of ηe−λt s(t)1−α to the feedback law (12) in order to reduce the chattering. In the next section, we give an example to illustrate the effect of the feedback law. 4. Numerical example In this section, a simulation of a nonlinear system is provided to show the effectiveness of the method proposed in this paper. System considered is ˙ x¯ 1 (t) = A11 x¯ 1 (t) + A12 x¯ 2 (t), (14) x¯˙ 2 (t) ∈ co{fi (x) + gi (x)B(u(t) + w(t)), i = 1, 2}.







 0 0 1 0 0 0 where A11 = , A12 = , f2 (x) = , g1 (x) = 1 + x22 , g2 (x) = 1, B = , f1 (x) = 2 0 0 1 0 x2 x1



 1 0 sin t , w(t) = and α1 + α2 = 1, α1 , α2 ≥ 0. That is 0 1 0 ⎧ x˙ 1 = x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 2 = x3 (15) ⎪ ⎪ x˙ 3 = [α1 (1 + x22 ) + α2 ](u1 + sin t) ⎪ ⎪ ⎪ ⎪ ⎩ x˙ 4 = α1 x12 + α2 x2 + [α1 (1 + x22 ) + α2 ]u2

 2 3 such that the eigenvalues of the matrix A11 − A12 C1 are ξ1 = −1, ξ = −2. The sliding We choose C1 = 0 0 surface is designed as s1 (t) = 2x1 + 3x2 + x3 and s2 (t) = x4 (t). The parameters in the controller (6) are chosen as η = 2, λ = 2, α = 0.4. The initial value x(0) = [2 − 1 − 21]T . Figs. 1 and 2 show the time response of the state x(t) for the different parameters α1 , α2 [13] gives a typical design of traditional slide mode control. To present the advantages of the method proposed in this paper, we would like to give a comparison. Figs. 3 and 4 give the input u1 (t) for these two designs, and Figs. 5 and 6 give the input u2 (t). It is obvious that the feedback law presented in this paper reduces the chattering effectively. 5. Conclusion This paper presents a novel slide mode control for PDI systems. The advantage of the method is to reduce the chattering effectively. From the design, we can see that the novel controller also makes the state reach the sliding surface in a finite time. Finally, an example illustrates the validity of the proposed method. Acknowledgement The authors are grateful for the support of the National Natural Science Foundation of China under grants 60774011 and 60674024.

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