ISA Transactions 53 (2014) 298–304
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Research Article
Control design for one-sided Lipschitz nonlinear differential inclusions Xiushan Cai a,n, Hong Gao a, Leipo Liu b, Wei Zhang c a b c
College of Mathematics, Physics, and Information Engineering, Zhejiang Normal University, Jinhua 321004, China College of Electric and Information Engineering, Henan University of Science and Technology, Luoyang 471003, China Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science, Shanghai 201620, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 May 2012 Received in revised form 7 October 2013 Accepted 3 December 2013 Available online 1 January 2014 This paper was recommended for publication by Prof. A.B. Rad
This paper considers stabilization and signal tracking control for one-sided Lipschitz nonlinear differential inclusions (NDIs). Sufficient conditions for exponential stabilization for the closed-loop system are given based on linear matrix inequality theory. Further, the design method is extended to signal tracking control for one-sided Lipschitz NDIs. A control law is designed such that the state of the closed-loop system asymptotically tracks the reference signal. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design technique. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Nonlinear differential inclusions One-sided Lipschitz Exponential stabilization Linear matrix inequalities
1. Introduction Differential inclusions (DIs) have been widely studied in recent years [1]. The study mainly focuses on the linear differential inclusions (LDIs) [2–5], because the LDI is simpler than the DI but can include numerous nonlinear systems [5]. Hu and Lin defined the convex hull Lyapunov function (CHLF) and proved that the CHLF holds less conservative than the traditional Lyapunov function in the study of LDIs in [6]. Recently, many authors have studied the stabilization problem of LDIs via the CHLF. A nonlinear control design method for LDIs via the CHLF was obtained in [7]. The saturated stabilization problem for LDIs was studied based on the CHLF in [8]. A nonlinear feedback law was established for time-delay LDIs by using the CHLF to make the state asymptotically stable in [9]. The feedback law was constructed by the CHLF to stabilize LDIs with stochastic disturbance in [10]. Non-conservative matrix inequality conditions for stability and stabilizability of LDIs were presented in [11]. A sliding mode control design method for polytopic DI systems was investigated in [12]. For a class of feedback linearizable differential inclusions, globally asymptotical stabilization of the closed-loop system was dealt with in [13]. Robust stabilization of LDIs with affine uncertainty was considered in [14]. The results of [14] are considered with the Lipschitz nonlinearities. The Lipschitz constant is usually region based and often dramatically increases as the operating region is enlarged. To overcome this
n
Corresponding author. E-mail address: [email protected] (X. Cai).
drawback, in the literature of mathematics the Lipschitz continuity has been generalized to one-sided Lipschitz continuity which has important applications in numerical analysis and others areas, such as the stability analysis of ordinary differential equations [15]. For many problems, the one-sided Lipschitz constant is significantly smaller than the classical Lipschitz constant [15]. This makes it much more suited for estimating the influence of nonlinear part of the corresponding system. For the one-sided Lipschitz systems, nonlinear observer design is provided based on the linear matrix inequality (LMI) in [16]. Further, they presented full-order and reduced-order observers design for onesided Lipschitz nonlinear systems using Riccati equations in [17]. Based on interval observers, stabilization method was investigated for a class of quasi-one-sided nonlinear uncertain systems in [18]. Tracking is always a hot topic in the field of control theory, many papers have been published to present lots of designing methods of the tracking, for example [19–22]. For LDIs, by using the Hamilton– Caylay Theorem, an operator is constructed such that tracking problem is converted into a standard stabilization problem in [23], and asymptotic tracking control for a class of reference signals in [24]. To the best of authors knowledge, tracking control for one-sided Lipschitz NDIs has not been appeared. Motivated by the above statements, this paper considers control design for one-sided Lipschitz NDIs. Sufficient conditions for exponential stabilization for the closed-loop system are given based on LMIs. Moreover, we extend the design method to signal tracking control for this class of NDIs. A control law is designed such that the state of the closed-loop system asymptotically tracks the reference signal. The rest of the paper is organized as follows: Section 2 gives the description of the system and several necessary lemmas. Control law
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.12.005
X. Cai et al. / ISA Transactions 53 (2014) 298–304
design for one-sided Lipschitz NDIs is presented in Section 3. Numerical examples are given in Section 4. The paper is concluded in Section 5.
Then the CHLF is defined as !1
Consider the following NDIs: _ A cofAi xðtÞ þ Bi uðtÞ þ ϕðxÞ; i ¼ 1; 2; …Ng xðtÞ
ð1Þ n
where co denotes the convex hull of a set and xðtÞ A R , uðtÞ A Rm are the state and the input respectively. Ai A Rnn ; Bi A Rnm are given real matrices. ϕðxÞ represents a nonlinear function that is continuous with respect to x and ϕð0Þ ¼ 0. We first review the concept of Lipschitz property as well as the one-sided Lipschitz property for the nonlinear function ϕðxÞ. Further details about these concepts can be found in [15]. Definition 1 (Abbaszadeh and Marquez [15]). The nonlinear function ϕðxÞ is said to be locally Lipschitz in a region D including the origin with respect to x, if there exists a constant l 4 0 satisfying J ϕðx1 Þ ϕðx2 Þ J r l J x1 x2 J ;
8 x1 ; x2 A D:
ð2Þ
The smallest constant l 40 satisfying (2) is called the Lipchitz constant. If the condition (2) is valid in Rn , then the function ϕðxÞ is said to be globally Lipschitz. Definition 2 (Abbaszadeh and Marquez [15]). The nonlinear function ϕðxÞ is said to be one-sided Lipschitz if there exists a constant ρ A R such that 8 x1 ; x2 A D: 〈ϕðx1 Þ ϕðx2 Þ; x1 x2 〉 r ρ J x1 x2 J 2 where
ð3Þ
ρ is called the one-sided Lipschitz constant.
The one-sided Lipschitz condition generalizes the classical Lipschitz theory to a more general family of nonlinear systems [15]. It is worth mentioning that the one-sided Lipschitz constant can be zero or even negative while the Lipschitz constant must be positive. It is easy to see that any Lipschitz function is also onesided Lipschitz. However, the converse is not true. As it has been pointed out in [15,16], usually the one-sided Lipschitz constant can be found to be much smaller than the Lipschitz constant. Next concept is quadratic inner-bounded for the function ϕðxÞ, which will be used in the control design. Definition 3 (Abbaszadeh and Marquez [15]). The nonlinear function ϕðxÞ is called quadratic inner-bounded in the origin D if 8 x1 ; x2 A D, there exist β ; γ A R such that
ΔϕT Δϕ r β J x1 x2 J 2 þ γ 〈x1 x2 ; Δϕ〉
ð4Þ
with Δϕ ¼ ϕðx1 Þ ϕðx2 Þ. From the definition, the Lipschitz function is quadratically inner-bounded with γ ¼ 0 and β 4 0. So the Lipschitz continuity implies quadratic inner-bounded. But the converse is not true [15,25]. It should be pointed out that γ in (4) can be any real number. In fact, if γ is restricted to be positive, then from Definition 3, it is easy to see that ϕ must be Lipschitz which is only a special case of our proposed class of systems. For a positive-definite (semidefinite) matrix P, it is denoted as P 4 0 ðP Z 0Þ. When we say positive-definite (semidefinite), it is implied that the matrix is symmetric. Let P A Rnn , P 40. Then denote the 1-level set of the function xTPx as ɛðPÞ ¼ fx A Rn : xT Px r 1g. If VðxÞ ¼ xT Px, 1-level set of VðÞ, denoted LV is defined as LV ¼ fx A Rn : VðxÞ r 1g ¼ ɛðPÞ. The CHLF is constructed from a family of positive definite matrices. Let Q j A Rnn ; Q j ¼ Q Tj 4 0; j ¼ 1; 2; …J, and S J ¼ fs : j s ¼ ðs1 ; s2 ; …; sJ Þ : s1 þ s2 þ ⋯ þ sJ ¼ 1;
sj Z 0g:
J
V c ðxÞ ¼ min xT sAS
2. Preliminaries
299
∑ sj Q j
J
ð5Þ
x:
j¼1
It is obvious that Vc(x) is a positive definite function. From the definition of Vc(x), we have 8 9 !1 < = J J T V c ðxÞ ¼ min α : α Z x ∑ sj Q j x; sj A S : : ; j¼1 By the Schur complement, Vc(x) and the optimal value of s can be computed by solving an LMI constraint: V c ðxÞ ¼ min α s1 ;…sj
2
α
6 4x
s:t:
xT
3
7 N ∑ sj Q j 5 Z 0;
j¼1 N
∑ sj ¼ 1;
j¼1
sj Z 0
which is an optimization problem and can be easily solved with the techniques presented in [5]. Define a function sn ðxÞ as follows: !1 sn ðxÞ ¼ arg min xT sAS
J
J
∑ sj Q j
ð6Þ
x:
j¼1
We see that for a given x the optimal value sn ðxÞ is such that V c ðxÞ ¼ xT ð∑Jj ¼ 1 snj Q j Þ 1 x. Generally, sn ðxÞ is uniquely determined by x and is a continuous function of x except for some degenerated cases. For example, this may happen if some Qj can be expressed as the convex combination of other matrices in the set. For a compact convex set S, a point x on the boundary of S (denoted as ∂S) is called an extreme point if it cannot be represented as the convex combination of any other points in S. A compact convex set is completely determined by its extreme points. In what follows, we characterize the set of extreme points of LV c . From the definition of Vc, we know that LV c is the convex hull of ɛðQ j 1 Þ; j ¼ 1; 2; …J, an extreme point must be on the boundaries of both LV c and ɛðQ j 1 Þ for some jA f1; 2; …Jg. Denote Ek ¼ ∂LV c \ ∂ɛðQ k 1 Þ ¼ fx : V c ðxÞ ¼ xT Q k 1 x ¼ 1g. The exact description of Ek is given as follows. Lemma 1 (Hu and Lin [6]). For each k A f1; 2; …; Jg, it holds that Ek ¼ fx A ∂LV c : xT Q k 1 ðQ j Q k ÞQ k 1 x r 0; j ¼ 1; 2; …Jg: Lemma 2 (Hu and Lin [6]). Let x A Rn . For simplicity and without loss generality, assume that snk ðxÞ 40 for k ¼ 1; 2; …J 0 and snk ðxÞ ¼ 0 for k ¼ J 0 þ 1; …J. Denote J0
Q ðsn Þ ¼ ∑ snk Q k ; k¼1
xk ¼ Q k Q ðsn Þ 1 x;
k ¼ 1; 2; …J 0 :
Then V c ðxk Þ ¼ V c ðxÞ ¼ xTk Q k 1 xk and xk A ðV c ðxÞÞ1=2 Ek , for k ¼ 1; 2; …J 0 . J Moreover, x ¼ ∑k0 ¼ 1 snk xk , and for k ¼ 1; 2; …J 0 , ∇V c ðxÞT ¼ ∇V c ðxk ÞT ¼ 2Q k 1 xk ¼ 2Q ðsn Þ 1 x;
ð7Þ
where ∇V c ðxÞ denotes the gradient of Vc at x. Notations: Rn denotes the n-dimensional real Euclidean space. Rmn represents the set of all m n real matrices. 〈; 〉 is the inner product in Rn. J J denotes the Euclidean norm in Rn. In symmetric block matrices, we use an asterisk to represent a term that is induced by symmetry. I is an identity matrix with appropriate dimension. P 1 is the inverse matrix of a matrix P.
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Using Lemma 2 and by (15), it can be deduced that
3. Main results In this section, we present the main contribution of this paper.
V_ c ðxÞ r max
i ¼ 1;2;…;N
nn
Theorem 1. Given positive definite matrices Q k A R ; k ¼ 1; 2; …; J. Suppose that the function ϕðxÞ in system (1) satisfies the conditions (3) and (4) with constants ρ; β and γ. If there exist matrices F k A Rmn ; k ¼ 1; 2; …; J and positive scalars ɛ1 4 0; ɛ2 4 0; λ 4 0; lijk 4 0 such that the following LMIs are feasible: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 Γ I ɛ21 Q k þ γ2ɛ2 Q k Q k j ɛ 1 ρ þ ɛ2 β j J 6 7 4n 5 o 0; ɛ2 I 0 n
I
n
i ¼ 1; 2; …N; k ¼ 1; 2; …; J
ð8Þ
with Γ ¼ Ai Q k þ Bi F k þ Q k ATi þ F Tk BTi ∑Jj ¼ 1 lijk ðQ j Q k Þ þ λQ k , then for any initial state in D \ D, system (1) under the control law: n
n 1
uðtÞ ¼ Fðs ÞQ ðs Þ
¼
Q ðsn Þ ¼ ∑ snk Q k
ð16Þ where xk ðtÞ A ðV c ðxðtÞÞÞ1=2 Ek ; k ¼ 1; 2; …; J 0 and xðtÞ ¼ Since 0 o snk o 1; k ¼ 1; 2; …J 0 , then we have
and s ðxÞ is the function defined in (6). Moreover, if s ðxðtÞÞ is continuous then the function defined in (9) is continuous too. Proof. Let Vc(x) be composed of positive definite matrices Q k ; k ¼ 1; 2; …; J and given by (5). xðtÞ A D \ D is an arbitrary point. By Lemma 2, x(t) is a convex combination of a set of x0sk ðtÞ, each of 1=2
which belongs to a certain V c ðxðtÞÞ Ek . For simplicity, assume that snk ðxÞ 4 0 for k ¼ 1; 2; …J 0 and snk ðxÞ ¼ 0 for k 4 J 0 . Then xðtÞ ¼ J ∑k0 ¼ 1 snk xk ðtÞ. Recalling Lemma 2, Q ðsn Þ 1 x ¼ Q k 1 xk ; k ¼ 1; 2; …; J 0 ;
T
n 1
we have ∇V c ðxÞ ¼ 2Q ðs Þ
x and
furthermore,
J0
Fðsn ÞQ ðsn Þ 1 x ¼ ∑ snk F k Q k 1 xk :
i ¼ 1;2;…;N
ð11Þ
k¼1
max f∇V c ðxðtÞÞT ðAi xðtÞ þ Bi uðtÞ þ ϕðxÞÞg
i ¼ 1;2;…;N
max f2xT ðtÞQ ðsn Þ 1 ðAi xðtÞ þBi Fðsn ÞQ ðsn Þ 1 xðtÞ þ ϕðxÞÞg i ¼ 1;2;…;N ! " # xðtÞ Λ Q ðsn Þ 1 T T ¼ max ðx ðtÞ ϕ ðxÞÞ ð12Þ ϕðxÞ i ¼ 1;2;…;N Q ðsn Þ 1 0
r
n 1
8 9 J0 J0 > > > > > 2 ∑ snk xTk ðtÞQ k 1 ðAi þ Bi F k Q k 1 Þxk ðtÞ þ ðɛ1 ρ þ ɛ 2 βÞJ 0 ∑ snk xTk ðtÞxk ðtÞ > > > < k¼1 = k¼1 J0 J0 > ɛ 1 γ ɛ2 I > 1 n T n T > > : þ 2 ∑ sk xk ðtÞðQ k 2 I þ 2 ÞϕðxÞ ɛ 2 ∑ sk ϕ ðxÞϕðxÞ k¼1
Λ ¼ Q ðsn Þ 1 ðAi þ Bi Fðsn ÞQ ðsn Þ 1 Þ þ ðAi þ Bi Fðsn ÞQ ðsn Þ 1 ÞT
Q ðIs Þ . From (3), we get ρxT ðtÞxðtÞ xT ðtÞϕðxÞ Z 0, for any x A D. Therefore, for any positive scalar ɛ1, ! " # xðtÞ ρI 12I T T ɛ 1 ðx ðtÞ ϕ ðxÞÞ Z 0: ð13Þ ϕðxÞ n 0 Similarly, from (4) we have " # ! xðtÞ βI 2γ I T Z0 ɛ 2 ðxT ðtÞ ϕ ðxÞÞ ϕðxÞ n I
k¼1
for any x A D; and for any positive scalar ɛ2. Then, adding the terms on the left-hand sides of (13) and (14) to the time derivative of Vc(x) yields " # ! xðtÞ M Q ðsn Þ 1 ɛ21 I þ γ ɛ22 I T V_ c ðxÞ r max ðxT ðtÞ ϕ ðxÞÞ ϕðxÞ i ¼ 1;2;…;N n ɛ2 I ð15Þ with M ¼ Q ðsn Þ 1 ðAi þ Bi Fðsn ÞQ ðsn Þ 1 Þ þ ðAi þ Bi Fðsn ÞQ ðsn Þ 1 ÞT n 1 Q ðs Þ þ ɛ1 ρI þ ɛ 2 β I.
:
Multiplying (8) from left and from right with 2 1 3 0 0 Qk 6 7 4 0 I 0 5; 0
0
I
we have 2 1 Q k Γ Q k 1 6 4 n
Q k 1 ɛ21 I þ γ2ɛ2 I ɛ2 I
I
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 j ɛ1 ρ þ ɛ2 β j J 7 5o0 0
ð18Þ
I
n
with Γ ¼ Ai Q k þ Bi F k þQ k ATi þ F Tk BTi ∑Jj ¼ 1 lijk ðQ j Q k Þ þ λQ k , for i ¼ 1; 2; …; N; k ¼ 1; 2; …; J. From (18), we deduce that " # Q k 1 Γ Q k 1 þ ðɛ 1 ρ þ ɛ2 βÞJ 0 I Q k 1 ɛ21 I þ γ2ɛ2 I o0 ð19Þ n ɛ2 I for i ¼ 1; 2; …; N; k ¼ 1; 2; …; J. By (17) and (19), and Lemma 1, then it can be deduced that J0
J0
k¼1
k¼1
V_ c ðxÞ r ∑ snk xTk ðtÞQ k 1 ðQ j Q k ÞQ k 1 xk ðtÞ λ ∑ snk xTk ðtÞQ k 1 xk ðtÞ r λV c ðxðtÞÞ
ð20Þ
for any x A D \ D: Thus system (1) under the control law (9) is exponentially stable at the origin. Since Yðsn Þ and Q ðsn Þ are continuous in sn, and Q ðsn Þ 4 0, the continuity of u ¼ Fðsn ÞxðtÞ follows from that of sn ðxÞ. This completes the proof. □ We extend the stabilization design to signal tracking control. Assume that the reference signal xr(t) is generated from x_ r ðtÞ ¼ Ar xr ðtÞ;
ð14Þ
> > > > ;
ð17Þ
n
Thus the derivative of Vc(x) along trajectories of the closed-loop system (1) and (9) satisfies the following inequality:
with
J ∑k0 ¼ 1 snk xk ðtÞ.
V_ c ðxÞ r max
ð10Þ n
> > > > ;
k¼1
k¼1
n
V_ ðxÞ r
J0 J0 ɛ1 γ ɛ2 I 1 n T n T > > : þ 2 ∑ sk xk ðtÞðQ k 2 I þ 2 ÞϕðxÞ ɛ 2 ∑ sk ϕ ðxÞϕðxÞ
i ¼ 1;2;…;N > >
J
k¼1
max
;
9 8 J0 J0 J0 > > > > > 2 ∑ sn xT ðtÞQ k 1 ðAi þ Bi F k Q k 1 Þxk ðtÞ þ ðɛ 1 ρ þɛ 2 βÞ ∑ snk xTk ðtÞ ∑ snk xk ðtÞ > > > = < k¼1 k k k¼1 k¼1 k¼1
is exponentially stable at the origin where Fðsn Þ ¼ ∑ snk F k ;
: þ 2xT ðtÞðQ ðsn Þ 1 ɛ21 I þ γ ɛ22 IÞϕðxÞ ɛ 2 ϕT ðxÞϕðxÞ
ð9Þ
xðtÞ
J
8 9 < 2xT ðtÞQ ðsn Þ 1 ðAi þ Bi Fðsn ÞQ ðsn Þ 1 ÞxðtÞ þ xT ðtÞðɛ 1 ρ þ ɛ 2 βÞxðtÞ =
xr ð0Þ ¼ xr0 n
ð21Þ nn
where xr ðtÞ A R , and Ar A R . Denote eðtÞ ¼ xðtÞ xr ðtÞ, the tracking error dynamics is given by _ A cofAi eðtÞ þ Bi uðtÞ þ ðAi Ar Þxr ðtÞ þ ϕðxÞ; i ¼ 1; 2; …Ng: eðtÞ
ð22Þ
The objective is to design a controller such that the state of system (1) asymptotically tracks the desired reference signal xr(t). Theorem 2. Given positive definite matrices Q k A Rnn ; k ¼ 1; 2; …; J. Suppose that the function ϕðxÞ in system (1) satisfies the conditions (3) and (4) with constants ρ; β and γ . If there exist matrices F k A Rmn ; k ¼ 1; 2; …; J and positive scalars ɛ 1 4 0; ɛ 2 4 0; λ 4 0; lijk 4 0
X. Cai et al. / ISA Transactions 53 (2014) 298–304
such that the following LMIs are feasible: 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Q k j ɛ1 ρ þ ɛ2 β j J 7 7 0 7r0 7 0 5 I
Γ ðAi Ar Þ þ ðɛ 1 ρ þ ɛ 2 βÞQ k I ɛ21 Q k þ γ2ɛ2 Q k ðɛ 1 ρ þ ɛ 2 βÞI ɛ21 I þ γ ɛ22 I
6 6n 6 6 4n
n
ɛ2 I
n
n
n
ð23Þ with Γ ¼ Ai Q k þ Bi F k þ Q k ATi þ F Tk BTi ∑Jj ¼ 1 lijk ðQ j Q k Þ þ λQ k , i ¼ 1; 2; …N; j; k ¼ 1; 2; …; J. Then for any initial state x(t) in D \ D, the state of system (1) under the control law: uðtÞ ¼ Fðsn ÞQ ðsn Þ 1 eðtÞ
with M 1 ¼ Q ðsn Þ 1 ðAi þ Bi Fðsn ÞQ ðsn Þ 1 Þ þ ðAi þ Bi Fðsn ÞQ ðsn Þ 1 ÞT Q ðsn Þ 1 þɛ 1 ρI þ ɛ 2 β I, M 2 ¼ Q ðsn Þ 1 ðAi Ar Þ þ ɛ 1 ρI þ ɛ2 βI. Using the arguments as the proof in Theorem 1, we have V_ c ðeÞ r max i ¼ 1;2;…;N 9 8 J0 J0 > > > > 1 1 n T n T > > > 2 ∑ sk ek ðtÞQ k ððAi þ Bi F k Q k Þek ðtÞþ ðɛ 1 ρ þ ɛ 2 βÞJ 0 ∑ sk ek ðtÞek ðtÞ > > > > > k ¼ 1 k ¼ 1 > > > > > > > > J0 > > > > > > þ 2 ∑ sn eT ðtÞ Q 1 ɛ 1 I þ γ ɛ 2 I ϕðxÞ > > > > k k k > > 2 2 > > k¼1 > > =
> > > > > > > > > > > > > > > > > ;
2
2
T 6 max ðeT ðtÞ xTr ðtÞ ϕ ðxÞÞ4 n
i ¼ 1;2;…;N
ɛ 1 γ ɛ2 I þ ∑ snk ð2xTr ðtÞ I þ 2 2 k¼1
with Γ ¼ i ¼ 1; 2; …; N; j; k ¼ 1; 2; …; J. From (31), we deduce that
eðtÞ
Λ Q ðsn Þ 1 ðAi Ar Þ Q ðsn Þ
J0
Ai Q k þ Bi F k þ Q k ATi
i ¼ 1;2;…;N
T
k¼1
ð30Þ
By Lemma 2, the derivative of Vc(e) along trajectory of the closedloop system (22) and (24) satisfies the following inequality: V_ c ðeÞ r max f∇V c ðeðtÞÞT ðAi eðtÞ þBi uðtÞ þ ðAi Ar Þxr ðtÞ þ ϕðxÞÞg
J0
þ 2 ∑ snk eTk ðtÞðQ k 1 ðAi Ar Þ þ ɛ 1 ρI þ ɛ 2 βIÞxr ðtÞ
> > > > > > > > > > > > > > > > > > :
ð24Þ
Proof. Let Vc(e) be composed of positive definite matrices Q k ; k ¼ 1; 2; …; J and given by (5). For any xðtÞ A D \ D, and xr(t) is the reference signal, denote eðtÞ ¼ xðtÞ xr ðtÞ. By Lemma 1, e(t) is a convex combination of a set of e0sk ðtÞ, each of which belongs to a certain V c ðeðtÞÞ1=2 Ek . For simplicity, assume that snk ðeÞ 4 0 for J k ¼ 1; 2; …J 0 and snk ðeÞ ¼ 0 for k 4 J 0 . Then eðtÞ ¼ ∑k0 ¼ 1 snk ek ðtÞ. RecalT n 1 ling Lemma 2, we have ∇V c ðeÞ ¼ 2Q ðs Þ e and Q ðsn Þ 1 e ¼ Q k 1 ek ; k ¼ 1; 2; …; J 0 ; furthermore,
301
30
1 eðtÞ 7B x ðtÞ C 5@ r A
ϕðxÞ
ð29Þ
x_ 1 ¼ α1 x1 m1 x1 þ rðt; x3 Þ; x_ 2 ¼ α1 x1 α2 x2 m2 x2 ; x_ 3 ¼ α2 x2 m3 x3 bðtÞuðtÞ;
ð34Þ
þ
where x1 ; x2 ; x3 A R stand for larvae, juveniles and adults respectively. The positive coefficients αi ; mi represent the growth and mortality rates respectively. Assume that the births in class x1 are generated only by the adults class x3 with a reproduction law of Beverton–Holt type [26] rðt; x3 Þ ¼ aðtÞx3 =ðc þ x3 Þ; aðtÞ 40; c 4 0. b(t) represents an harvesting effort on the adult population. We assume that the uncertain parameters satisfy the inequalities: a r aðtÞ r a þ ;
þ
b r bðtÞ r b :
302
X. Cai et al. / ISA Transactions 53 (2014) 298–304
15
x_ A cofAx þ Bu þ f ðx; γ ðtÞÞg
10
ð35Þ
where 2 3 0 0 ðα1 þ m1 Þ 6 α1 ð α 2 þ m2 Þ 0 7 A¼4 5; α2 m3 0 2 3 2 3 0 rðt; x3 Þ 6 7 607 B ¼ 4 5; f ðx; γ ðtÞÞ ¼ 4 0 5 b 0
5 0
ɛ2 ¼ 0:12;
λ ¼ 1;
F 2 ¼ ½0:0328; 0:0676; 2:2692;
l121 ¼ 100;
x2(t)
þ
by solving the LMI (8), we get
l221 ¼ 102;
8
10
12
14
16
18
12
14
16
18
12
14
16
18
15 10
0
0
2
4
6
8
10
time (sec) 20 15 10 5 0
0
2
4
6
8
10
time (sec) Fig. 1. State trajectory x(t) of the closed-loop system (34) and (36) (the initial state is ½15 25; 20T ).
l112 ¼ 150; 15
Vc(x) can be computed as follows:
s:t:
6
5
l212 ¼ 150:
10
u
V c ðxÞ ¼ min α 0 r s1 r 1 "
4
20
Assume that α1 ¼ 0:1; m1 ¼ 0:2; α2 ¼ 0:15; m2 ¼ 0:4; m3 ¼ 0:5; þ a ¼ 0:5; a þ ¼ 0:7; b ¼ 1; b ¼ 3; c ¼ 0:9. Let 2 3 0:2336 0:0056 0:0009 6 0:0056 0:2097 0:0060 7 Q1 ¼ 4 5; 0:0009 0:0060 0:2796 2 3 4:4419 0:0680 0:0052 6 7 Q 2 ¼ 4 0:0680 4:1760 0:0330 5; 0:0052 0:0330 4:3546
ɛ 1 ¼ 0:1;
2
25
and b is b or b . It is easy to know that the nonlinear function f ðx; γ ðtÞÞ is globally one-sided Lipschitz with the one-sided Lipschitz constant ρ ¼ a þ =2c, and it is globally Lipschitz with constant l ¼ a þ =c, and is globally quadratically inner-bounded with β ¼ a þ =c, γ ¼ 0.
F 1 ¼ ½0:0276; 0:2004; 1:2593;
0
time (sec)
x3(t)
x1(t)
From [18], it can be deduced that system (34) belongs to the following differential inclusion:
#
α
xT
x
s1 Q 1 þ ð1 s1 ÞQ 2
5
Z 0;
0
and let FðsÞ ¼ s1 F 1 þ ð1 s1 ÞF 2 ;
0
2
4
6
8
10
12
14
16
18
time (sec)
Q ðsÞ ¼ s1 Q 1 þ ð1 s1 ÞQ 2 :
Fig. 2. Control law u(t) of the closed-loop system (34) and (36).
Then the state of the closed-loop system under the control law: uðtÞ ¼ FðsÞQ ðsÞ 1 xðtÞ
ð36Þ
is exponentially stable at the origin. For simulation the following values of parameters have been chosen:
A2 ¼
1
1
2
1
;
B2 ¼
1 10
:
aðtÞ ¼ 0:5 þ cos 2 ðtÞ=16; bðtÞ ¼ 1:8 þ 0:5 sin t: Figs. 1 and 2 show the time response of the state trajectories x(t) under state feedback control law (36) and the control law u(t) respectively. Remark 1. The merit of algorithm in this paper is that the convergent speed is very quick. The disadvantage of the algorithm is that some scalars need to be given, in order to solve nonlinearities in the matrix inequalities (8), and make them LMIs and can be solved by Matlab. Example 2. Consider a second-order NDI system: _ A cofA1 xðtÞ þB1 uðtÞ þ ϕðxÞ; A2 xðtÞ þ B2 uðtÞ þ ϕðxÞg xðtÞ where 1 A1 ¼ 1
ð37Þ
As it has shown in [15], the above equation can be used to describe the motion of a moving object. From (15), we know that the system is globally one-sided Lipschitz with the one-sided Lipschitz constant ρ ¼ 0. Also, the system is locally Lipschitz on any set D ¼ fx A R2 : J x J r rg, the Lipschitz constant l is 3r 2 . Assume that the reference signal xr1 ¼ cos t; xr2 ¼ sin t, they can be generated from " # # " x_ r1 1 0 1 xr1 ¼ ; xr ð0Þ ¼ : ð38Þ x_ r2 xr2 0 1 0 Let
1 ; 1
2 B1 ¼ ; 1
"
ϕðxÞ ¼
x1 ðx21 þ x22 Þ x2 ðx21 þ x22 Þ
#
ϑ ¼ min
rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi! γ 4 γ2 ; ; βþ 4 4
γ o 0; β þ
γ2 4
4 0:
X. Cai et al. / ISA Transactions 53 (2014) 298–304
Then one can verify the quadratically inner-bounded property of ϕðxÞ in D ¼ fx A R2 : J x J r ϑg [15]. As the system is globally onesided Lipschitz, i.e., D ¼ R2 , D \ D ¼ D. Note that the region D can be made arbitrarily large by choosing appropriate values for γ and β. If β ¼ 200, γ ¼ 141, derives ϑ ¼ 5:9372. Let 0:1130 0:0160 0:1008 0:0035 Q1 ¼ ; Q2 ¼ ; 0:0160 0:1290 0:0035 0:1031 by solving the LMI (23), we get F 1 ¼ ½ 1:2391; 0:6376; F 2 ¼ ½ 2:3221; 3:1070; ɛ 1 ¼ 0:1; ɛ 2 ¼ 0:12; λ ¼ 1; l121 ¼ 50; l112 ¼ 60; l221 ¼ 100; l212 ¼ 150: For each x0 A fx : J x J r 5:9372g, Vc(e) can be computed as follows: V c ðeÞ ¼ min α 0 r s1 r 1 " s:t:
eT
e
s1 Q 1 þ ð1 s1 ÞQ 2
Z 0;
FðsÞ ¼ s1 F 1 þ ð1 s1 ÞF 2 ;
Q ðsÞ ¼ s1 Q 1 þ ð1 s1 ÞQ 2 :
Then the state of the closed-loop system under the control law: uðtÞ ¼ FðsÞQ ðsÞ 1 eðtÞ
2
x1/xr1
1 0
1
2
3
4
5
3
4
5
time (sec) 1
x2/xr2
0.5 0 −0.5 −1 0
1
2 time (sec)
Fig. 3. State trajectory x(t) of the closed-loop system and the desired reference signal xr(t) (the initial state xð0Þ ¼ ½2; 1T and xr ð0Þ ¼ ½1; 0T .)
2
u
0 −2 −4 −6
0
1
In this paper, we present a control design method for one-sided Lipschitz NDIs. Based on LMIs, sufficient conditions for exponential stabilization for the closed-loop system are given. Further, we extend the design method to signal tracking control for this class of NDIs. A control law is designed such that the state of the closed-loop system asymptotically tracks the reference signal. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design technique.
The authors thank the anonymous reviewers for their many helpful suggestions. The authors are grateful for the support of the National Natural Science Foundation of China (Grant numbers 61074011 and 61374077).
ð39Þ
asymptotically tracks the desired reference signal xr(t). Figs. 3 and 4 show that the state x(t) of the closed-loop system asymptotically tracks the desired reference signal xr(t) under state feedback control law (39) and the control law u(t) respectively. The switching between A1 eðtÞ þ ðA1 Ar Þxr ðtÞ þ B1 uðtÞ þ ϕðxÞ and
−1.5
5. Conclusions
Acknowledgment
with e ¼ x0 xr and let
0
A2 eðtÞ þ ðA2 Ar Þxr ðtÞ þ B2 uðtÞ þ ϕðxÞ is chosen such that V c ðeðtÞÞ is minimized at each time instant. For any initial state x0 A fx : J x J r 5:9372g, it has a similar simulative result.
#
α
−1
303
2
3
4
time (sec) Fig. 4. Control law u(t) of the closed-loop system.
5
References [1] Smirnov GV. Introduction to the theory of differential inclusions. Graduate studies in mathematics. Philadelphia: SIAM; 2002. [2] Bichegkuev M. On a weakened Cauchy problem for a linear differential inclusion. Math Notes 2006;79:483–7. [3] Goebel R, Teel A, Hu T, Lin Z. Conjugate convex Lyapunov functions for dual linear differential equations. IEEE Trans Autom Control 2006;51:661–6. [4] Liu LP, Han ZZ, Cai XS, Huang J. Robust stabilization of linear differential inclusion system with time delay. Math Comput Simul 2010;80:951–8. [5] Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear matrix inequalities in systems and control theory, SIAM studies in applied mathematics. Philadelphia: SIAM; 1994. [6] Hu T, Lin Z. Composite quadratic Lyapunov functions for constrained control systems. IEEE Trans Autom Control 2003;48:440–50. [7] Hu T. Nonlinear control design for linear differential inclusions via convex hull of quadratics. Automatica 2007;43:685–92. [8] Cai XS, Liu LP, Zhang W. Saturated control design for linear differential inclusions subject to disturbance. Nonlinear Dyn 2009;58:487–96. [9] Cai XS, Huang J, Liu LP. Stability analysis of linear time-delay differential inclusion systems subject to input saturation. IET Control Theory Appl 2010;4:2592–602. [10] Huang J, Han ZZ, Cai XS, Liu L. Control of time-delayed linear differential inclusions with stochastic disturbance. J Frankl Inst 2010;347:1895–906. [11] Hu T, Blanchini F. Non-conservative matrix inequality conditions for stability/ stabilizaility of linear differential inclusions. Automatica 2010;46:190–6. [12] Liu LP, Han ZZ, Cai XS. Sliding mode control for polytopic differential inclusion systems. Math Comput Simul 2010;79:3018–25. [13] Cai XS, Liu LP, Huang J, Zhang W. Globally asymptotical stabilization for a class of feedback linearisable differential inclusion systems. IET Control Theory Appl 2011;5:1586–96. [14] Huang J, Han ZZ, Cai XS, Liu L. Robust stabilization of linear differential inclusions with affine uncertainty. Circuits Syst Signal Process 2011;30: 1369–82. [15] Abbaszadeh M, Marquez HJ. Nonlinear observer design for one-sided Lipschitz systems. In: Proceedings of the American control conference, Baltimore, USA, 2010. p. 5284–9. [16] Zhang W, Su H, Wang H, Han ZZ. Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations. Commun Nonlinear Sci Numer Simul 2012;17:4968–77. [17] Zhang W, Su H, Liang Y, Han ZZ. Nonlinear observer design for one-sided Lipschitz systems: an linear matrix inequality approach. IET Control Theory Appl 2012;6:1297–303. [18] Cai XS, Lv GY, Zhang W. Stabilisation for a class of non-linear uncertain systems based on interval observers. IET Control Theory Appl 2012;6:2057–62. [19] Chen M, Wu QX, Cui RX. Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISA Trans 2013;52:198–206. [20] Wang JJ. Stabilization and tracking control of XCZ inverted pendulum with sliding-mode control. ISA Trans 2012;51:763–70. [21] Travieso-Torres JC, Duarte-Mermoud MA, Estrada JL. Tracking control of cascade systems based on passivity: the non-adaptive and adaptive cases. ISA Trans 2006;45:435–45. [22] Purwar S, Kar I, Jha A. Adaptive output feedback tracking control of robot manipulators using position measurements only. Expert Syst Appl 2008;34: 2789–98.
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[23] Cai XS, Huang J, Xie QY. Output tracking and disturbance rejection of linear differential inclusion systems. Int J Syst Sci 2012;43:2072–8. [24] Cai XS, Liu Y, Zhang W. Asymptotic tracking control for a class of reference signals for linear differential inclusions. Int J Syst Sci, http://dx.doi.org/10. 1080/00207721.2012.748104, this issue.
[25] Zemouche A, Boutayeb M. Observer design for Lipschitz nonlinear systems: the discrete-time case. IEEE Trans Circuits Syst II: Express Briefs 2006;53:777–81. [26] Moisan M, Bernard O, Gouzé JL. Near optimal interval observers bundle for uncertain bioreactors. Automatica 2009;45:291–5.
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Research Article
Control design for one-sided Lipschitz nonlinear differential inclusions Xiushan Cai a,n, Hong Gao a, Leipo Liu b, Wei Zhang c a b c
College of Mathematics, Physics, and Information Engineering, Zhejiang Normal University, Jinhua 321004, China College of Electric and Information Engineering, Henan University of Science and Technology, Luoyang 471003, China Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science, Shanghai 201620, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 May 2012 Received in revised form 7 October 2013 Accepted 3 December 2013 Available online 1 January 2014 This paper was recommended for publication by Prof. A.B. Rad
This paper considers stabilization and signal tracking control for one-sided Lipschitz nonlinear differential inclusions (NDIs). Sufficient conditions for exponential stabilization for the closed-loop system are given based on linear matrix inequality theory. Further, the design method is extended to signal tracking control for one-sided Lipschitz NDIs. A control law is designed such that the state of the closed-loop system asymptotically tracks the reference signal. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design technique. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Nonlinear differential inclusions One-sided Lipschitz Exponential stabilization Linear matrix inequalities
1. Introduction Differential inclusions (DIs) have been widely studied in recent years [1]. The study mainly focuses on the linear differential inclusions (LDIs) [2–5], because the LDI is simpler than the DI but can include numerous nonlinear systems [5]. Hu and Lin defined the convex hull Lyapunov function (CHLF) and proved that the CHLF holds less conservative than the traditional Lyapunov function in the study of LDIs in [6]. Recently, many authors have studied the stabilization problem of LDIs via the CHLF. A nonlinear control design method for LDIs via the CHLF was obtained in [7]. The saturated stabilization problem for LDIs was studied based on the CHLF in [8]. A nonlinear feedback law was established for time-delay LDIs by using the CHLF to make the state asymptotically stable in [9]. The feedback law was constructed by the CHLF to stabilize LDIs with stochastic disturbance in [10]. Non-conservative matrix inequality conditions for stability and stabilizability of LDIs were presented in [11]. A sliding mode control design method for polytopic DI systems was investigated in [12]. For a class of feedback linearizable differential inclusions, globally asymptotical stabilization of the closed-loop system was dealt with in [13]. Robust stabilization of LDIs with affine uncertainty was considered in [14]. The results of [14] are considered with the Lipschitz nonlinearities. The Lipschitz constant is usually region based and often dramatically increases as the operating region is enlarged. To overcome this
n
Corresponding author. E-mail address: [email protected] (X. Cai).
drawback, in the literature of mathematics the Lipschitz continuity has been generalized to one-sided Lipschitz continuity which has important applications in numerical analysis and others areas, such as the stability analysis of ordinary differential equations [15]. For many problems, the one-sided Lipschitz constant is significantly smaller than the classical Lipschitz constant [15]. This makes it much more suited for estimating the influence of nonlinear part of the corresponding system. For the one-sided Lipschitz systems, nonlinear observer design is provided based on the linear matrix inequality (LMI) in [16]. Further, they presented full-order and reduced-order observers design for onesided Lipschitz nonlinear systems using Riccati equations in [17]. Based on interval observers, stabilization method was investigated for a class of quasi-one-sided nonlinear uncertain systems in [18]. Tracking is always a hot topic in the field of control theory, many papers have been published to present lots of designing methods of the tracking, for example [19–22]. For LDIs, by using the Hamilton– Caylay Theorem, an operator is constructed such that tracking problem is converted into a standard stabilization problem in [23], and asymptotic tracking control for a class of reference signals in [24]. To the best of authors knowledge, tracking control for one-sided Lipschitz NDIs has not been appeared. Motivated by the above statements, this paper considers control design for one-sided Lipschitz NDIs. Sufficient conditions for exponential stabilization for the closed-loop system are given based on LMIs. Moreover, we extend the design method to signal tracking control for this class of NDIs. A control law is designed such that the state of the closed-loop system asymptotically tracks the reference signal. The rest of the paper is organized as follows: Section 2 gives the description of the system and several necessary lemmas. Control law
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.12.005
X. Cai et al. / ISA Transactions 53 (2014) 298–304
design for one-sided Lipschitz NDIs is presented in Section 3. Numerical examples are given in Section 4. The paper is concluded in Section 5.
Then the CHLF is defined as !1
Consider the following NDIs: _ A cofAi xðtÞ þ Bi uðtÞ þ ϕðxÞ; i ¼ 1; 2; …Ng xðtÞ
ð1Þ n
where co denotes the convex hull of a set and xðtÞ A R , uðtÞ A Rm are the state and the input respectively. Ai A Rnn ; Bi A Rnm are given real matrices. ϕðxÞ represents a nonlinear function that is continuous with respect to x and ϕð0Þ ¼ 0. We first review the concept of Lipschitz property as well as the one-sided Lipschitz property for the nonlinear function ϕðxÞ. Further details about these concepts can be found in [15]. Definition 1 (Abbaszadeh and Marquez [15]). The nonlinear function ϕðxÞ is said to be locally Lipschitz in a region D including the origin with respect to x, if there exists a constant l 4 0 satisfying J ϕðx1 Þ ϕðx2 Þ J r l J x1 x2 J ;
8 x1 ; x2 A D:
ð2Þ
The smallest constant l 40 satisfying (2) is called the Lipchitz constant. If the condition (2) is valid in Rn , then the function ϕðxÞ is said to be globally Lipschitz. Definition 2 (Abbaszadeh and Marquez [15]). The nonlinear function ϕðxÞ is said to be one-sided Lipschitz if there exists a constant ρ A R such that 8 x1 ; x2 A D: 〈ϕðx1 Þ ϕðx2 Þ; x1 x2 〉 r ρ J x1 x2 J 2 where
ð3Þ
ρ is called the one-sided Lipschitz constant.
The one-sided Lipschitz condition generalizes the classical Lipschitz theory to a more general family of nonlinear systems [15]. It is worth mentioning that the one-sided Lipschitz constant can be zero or even negative while the Lipschitz constant must be positive. It is easy to see that any Lipschitz function is also onesided Lipschitz. However, the converse is not true. As it has been pointed out in [15,16], usually the one-sided Lipschitz constant can be found to be much smaller than the Lipschitz constant. Next concept is quadratic inner-bounded for the function ϕðxÞ, which will be used in the control design. Definition 3 (Abbaszadeh and Marquez [15]). The nonlinear function ϕðxÞ is called quadratic inner-bounded in the origin D if 8 x1 ; x2 A D, there exist β ; γ A R such that
ΔϕT Δϕ r β J x1 x2 J 2 þ γ 〈x1 x2 ; Δϕ〉
ð4Þ
with Δϕ ¼ ϕðx1 Þ ϕðx2 Þ. From the definition, the Lipschitz function is quadratically inner-bounded with γ ¼ 0 and β 4 0. So the Lipschitz continuity implies quadratic inner-bounded. But the converse is not true [15,25]. It should be pointed out that γ in (4) can be any real number. In fact, if γ is restricted to be positive, then from Definition 3, it is easy to see that ϕ must be Lipschitz which is only a special case of our proposed class of systems. For a positive-definite (semidefinite) matrix P, it is denoted as P 4 0 ðP Z 0Þ. When we say positive-definite (semidefinite), it is implied that the matrix is symmetric. Let P A Rnn , P 40. Then denote the 1-level set of the function xTPx as ɛðPÞ ¼ fx A Rn : xT Px r 1g. If VðxÞ ¼ xT Px, 1-level set of VðÞ, denoted LV is defined as LV ¼ fx A Rn : VðxÞ r 1g ¼ ɛðPÞ. The CHLF is constructed from a family of positive definite matrices. Let Q j A Rnn ; Q j ¼ Q Tj 4 0; j ¼ 1; 2; …J, and S J ¼ fs : j s ¼ ðs1 ; s2 ; …; sJ Þ : s1 þ s2 þ ⋯ þ sJ ¼ 1;
sj Z 0g:
J
V c ðxÞ ¼ min xT sAS
2. Preliminaries
299
∑ sj Q j
J
ð5Þ
x:
j¼1
It is obvious that Vc(x) is a positive definite function. From the definition of Vc(x), we have 8 9 !1 < = J J T V c ðxÞ ¼ min α : α Z x ∑ sj Q j x; sj A S : : ; j¼1 By the Schur complement, Vc(x) and the optimal value of s can be computed by solving an LMI constraint: V c ðxÞ ¼ min α s1 ;…sj
2
α
6 4x
s:t:
xT
3
7 N ∑ sj Q j 5 Z 0;
j¼1 N
∑ sj ¼ 1;
j¼1
sj Z 0
which is an optimization problem and can be easily solved with the techniques presented in [5]. Define a function sn ðxÞ as follows: !1 sn ðxÞ ¼ arg min xT sAS
J
J
∑ sj Q j
ð6Þ
x:
j¼1
We see that for a given x the optimal value sn ðxÞ is such that V c ðxÞ ¼ xT ð∑Jj ¼ 1 snj Q j Þ 1 x. Generally, sn ðxÞ is uniquely determined by x and is a continuous function of x except for some degenerated cases. For example, this may happen if some Qj can be expressed as the convex combination of other matrices in the set. For a compact convex set S, a point x on the boundary of S (denoted as ∂S) is called an extreme point if it cannot be represented as the convex combination of any other points in S. A compact convex set is completely determined by its extreme points. In what follows, we characterize the set of extreme points of LV c . From the definition of Vc, we know that LV c is the convex hull of ɛðQ j 1 Þ; j ¼ 1; 2; …J, an extreme point must be on the boundaries of both LV c and ɛðQ j 1 Þ for some jA f1; 2; …Jg. Denote Ek ¼ ∂LV c \ ∂ɛðQ k 1 Þ ¼ fx : V c ðxÞ ¼ xT Q k 1 x ¼ 1g. The exact description of Ek is given as follows. Lemma 1 (Hu and Lin [6]). For each k A f1; 2; …; Jg, it holds that Ek ¼ fx A ∂LV c : xT Q k 1 ðQ j Q k ÞQ k 1 x r 0; j ¼ 1; 2; …Jg: Lemma 2 (Hu and Lin [6]). Let x A Rn . For simplicity and without loss generality, assume that snk ðxÞ 40 for k ¼ 1; 2; …J 0 and snk ðxÞ ¼ 0 for k ¼ J 0 þ 1; …J. Denote J0
Q ðsn Þ ¼ ∑ snk Q k ; k¼1
xk ¼ Q k Q ðsn Þ 1 x;
k ¼ 1; 2; …J 0 :
Then V c ðxk Þ ¼ V c ðxÞ ¼ xTk Q k 1 xk and xk A ðV c ðxÞÞ1=2 Ek , for k ¼ 1; 2; …J 0 . J Moreover, x ¼ ∑k0 ¼ 1 snk xk , and for k ¼ 1; 2; …J 0 , ∇V c ðxÞT ¼ ∇V c ðxk ÞT ¼ 2Q k 1 xk ¼ 2Q ðsn Þ 1 x;
ð7Þ
where ∇V c ðxÞ denotes the gradient of Vc at x. Notations: Rn denotes the n-dimensional real Euclidean space. Rmn represents the set of all m n real matrices. 〈; 〉 is the inner product in Rn. J J denotes the Euclidean norm in Rn. In symmetric block matrices, we use an asterisk to represent a term that is induced by symmetry. I is an identity matrix with appropriate dimension. P 1 is the inverse matrix of a matrix P.
300
X. Cai et al. / ISA Transactions 53 (2014) 298–304
Using Lemma 2 and by (15), it can be deduced that
3. Main results In this section, we present the main contribution of this paper.
V_ c ðxÞ r max
i ¼ 1;2;…;N
nn
Theorem 1. Given positive definite matrices Q k A R ; k ¼ 1; 2; …; J. Suppose that the function ϕðxÞ in system (1) satisfies the conditions (3) and (4) with constants ρ; β and γ. If there exist matrices F k A Rmn ; k ¼ 1; 2; …; J and positive scalars ɛ1 4 0; ɛ2 4 0; λ 4 0; lijk 4 0 such that the following LMIs are feasible: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 Γ I ɛ21 Q k þ γ2ɛ2 Q k Q k j ɛ 1 ρ þ ɛ2 β j J 6 7 4n 5 o 0; ɛ2 I 0 n
I
n
i ¼ 1; 2; …N; k ¼ 1; 2; …; J
ð8Þ
with Γ ¼ Ai Q k þ Bi F k þ Q k ATi þ F Tk BTi ∑Jj ¼ 1 lijk ðQ j Q k Þ þ λQ k , then for any initial state in D \ D, system (1) under the control law: n
n 1
uðtÞ ¼ Fðs ÞQ ðs Þ
¼
Q ðsn Þ ¼ ∑ snk Q k
ð16Þ where xk ðtÞ A ðV c ðxðtÞÞÞ1=2 Ek ; k ¼ 1; 2; …; J 0 and xðtÞ ¼ Since 0 o snk o 1; k ¼ 1; 2; …J 0 , then we have
and s ðxÞ is the function defined in (6). Moreover, if s ðxðtÞÞ is continuous then the function defined in (9) is continuous too. Proof. Let Vc(x) be composed of positive definite matrices Q k ; k ¼ 1; 2; …; J and given by (5). xðtÞ A D \ D is an arbitrary point. By Lemma 2, x(t) is a convex combination of a set of x0sk ðtÞ, each of 1=2
which belongs to a certain V c ðxðtÞÞ Ek . For simplicity, assume that snk ðxÞ 4 0 for k ¼ 1; 2; …J 0 and snk ðxÞ ¼ 0 for k 4 J 0 . Then xðtÞ ¼ J ∑k0 ¼ 1 snk xk ðtÞ. Recalling Lemma 2, Q ðsn Þ 1 x ¼ Q k 1 xk ; k ¼ 1; 2; …; J 0 ;
T
n 1
we have ∇V c ðxÞ ¼ 2Q ðs Þ
x and
furthermore,
J0
Fðsn ÞQ ðsn Þ 1 x ¼ ∑ snk F k Q k 1 xk :
i ¼ 1;2;…;N
ð11Þ
k¼1
max f∇V c ðxðtÞÞT ðAi xðtÞ þ Bi uðtÞ þ ϕðxÞÞg
i ¼ 1;2;…;N
max f2xT ðtÞQ ðsn Þ 1 ðAi xðtÞ þBi Fðsn ÞQ ðsn Þ 1 xðtÞ þ ϕðxÞÞg i ¼ 1;2;…;N ! " # xðtÞ Λ Q ðsn Þ 1 T T ¼ max ðx ðtÞ ϕ ðxÞÞ ð12Þ ϕðxÞ i ¼ 1;2;…;N Q ðsn Þ 1 0
r
n 1
8 9 J0 J0 > > > > > 2 ∑ snk xTk ðtÞQ k 1 ðAi þ Bi F k Q k 1 Þxk ðtÞ þ ðɛ1 ρ þ ɛ 2 βÞJ 0 ∑ snk xTk ðtÞxk ðtÞ > > > < k¼1 = k¼1 J0 J0 > ɛ 1 γ ɛ2 I > 1 n T n T > > : þ 2 ∑ sk xk ðtÞðQ k 2 I þ 2 ÞϕðxÞ ɛ 2 ∑ sk ϕ ðxÞϕðxÞ k¼1
Λ ¼ Q ðsn Þ 1 ðAi þ Bi Fðsn ÞQ ðsn Þ 1 Þ þ ðAi þ Bi Fðsn ÞQ ðsn Þ 1 ÞT
Q ðIs Þ . From (3), we get ρxT ðtÞxðtÞ xT ðtÞϕðxÞ Z 0, for any x A D. Therefore, for any positive scalar ɛ1, ! " # xðtÞ ρI 12I T T ɛ 1 ðx ðtÞ ϕ ðxÞÞ Z 0: ð13Þ ϕðxÞ n 0 Similarly, from (4) we have " # ! xðtÞ βI 2γ I T Z0 ɛ 2 ðxT ðtÞ ϕ ðxÞÞ ϕðxÞ n I
k¼1
for any x A D; and for any positive scalar ɛ2. Then, adding the terms on the left-hand sides of (13) and (14) to the time derivative of Vc(x) yields " # ! xðtÞ M Q ðsn Þ 1 ɛ21 I þ γ ɛ22 I T V_ c ðxÞ r max ðxT ðtÞ ϕ ðxÞÞ ϕðxÞ i ¼ 1;2;…;N n ɛ2 I ð15Þ with M ¼ Q ðsn Þ 1 ðAi þ Bi Fðsn ÞQ ðsn Þ 1 Þ þ ðAi þ Bi Fðsn ÞQ ðsn Þ 1 ÞT n 1 Q ðs Þ þ ɛ1 ρI þ ɛ 2 β I.
:
Multiplying (8) from left and from right with 2 1 3 0 0 Qk 6 7 4 0 I 0 5; 0
0
I
we have 2 1 Q k Γ Q k 1 6 4 n
Q k 1 ɛ21 I þ γ2ɛ2 I ɛ2 I
I
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 j ɛ1 ρ þ ɛ2 β j J 7 5o0 0
ð18Þ
I
n
with Γ ¼ Ai Q k þ Bi F k þQ k ATi þ F Tk BTi ∑Jj ¼ 1 lijk ðQ j Q k Þ þ λQ k , for i ¼ 1; 2; …; N; k ¼ 1; 2; …; J. From (18), we deduce that " # Q k 1 Γ Q k 1 þ ðɛ 1 ρ þ ɛ2 βÞJ 0 I Q k 1 ɛ21 I þ γ2ɛ2 I o0 ð19Þ n ɛ2 I for i ¼ 1; 2; …; N; k ¼ 1; 2; …; J. By (17) and (19), and Lemma 1, then it can be deduced that J0
J0
k¼1
k¼1
V_ c ðxÞ r ∑ snk xTk ðtÞQ k 1 ðQ j Q k ÞQ k 1 xk ðtÞ λ ∑ snk xTk ðtÞQ k 1 xk ðtÞ r λV c ðxðtÞÞ
ð20Þ
for any x A D \ D: Thus system (1) under the control law (9) is exponentially stable at the origin. Since Yðsn Þ and Q ðsn Þ are continuous in sn, and Q ðsn Þ 4 0, the continuity of u ¼ Fðsn ÞxðtÞ follows from that of sn ðxÞ. This completes the proof. □ We extend the stabilization design to signal tracking control. Assume that the reference signal xr(t) is generated from x_ r ðtÞ ¼ Ar xr ðtÞ;
ð14Þ
> > > > ;
ð17Þ
n
Thus the derivative of Vc(x) along trajectories of the closed-loop system (1) and (9) satisfies the following inequality:
with
J ∑k0 ¼ 1 snk xk ðtÞ.
V_ c ðxÞ r max
ð10Þ n
> > > > ;
k¼1
k¼1
n
V_ ðxÞ r
J0 J0 ɛ1 γ ɛ2 I 1 n T n T > > : þ 2 ∑ sk xk ðtÞðQ k 2 I þ 2 ÞϕðxÞ ɛ 2 ∑ sk ϕ ðxÞϕðxÞ
i ¼ 1;2;…;N > >
J
k¼1
max
;
9 8 J0 J0 J0 > > > > > 2 ∑ sn xT ðtÞQ k 1 ðAi þ Bi F k Q k 1 Þxk ðtÞ þ ðɛ 1 ρ þɛ 2 βÞ ∑ snk xTk ðtÞ ∑ snk xk ðtÞ > > > = < k¼1 k k k¼1 k¼1 k¼1
is exponentially stable at the origin where Fðsn Þ ¼ ∑ snk F k ;
: þ 2xT ðtÞðQ ðsn Þ 1 ɛ21 I þ γ ɛ22 IÞϕðxÞ ɛ 2 ϕT ðxÞϕðxÞ
ð9Þ
xðtÞ
J
8 9 < 2xT ðtÞQ ðsn Þ 1 ðAi þ Bi Fðsn ÞQ ðsn Þ 1 ÞxðtÞ þ xT ðtÞðɛ 1 ρ þ ɛ 2 βÞxðtÞ =
xr ð0Þ ¼ xr0 n
ð21Þ nn
where xr ðtÞ A R , and Ar A R . Denote eðtÞ ¼ xðtÞ xr ðtÞ, the tracking error dynamics is given by _ A cofAi eðtÞ þ Bi uðtÞ þ ðAi Ar Þxr ðtÞ þ ϕðxÞ; i ¼ 1; 2; …Ng: eðtÞ
ð22Þ
The objective is to design a controller such that the state of system (1) asymptotically tracks the desired reference signal xr(t). Theorem 2. Given positive definite matrices Q k A Rnn ; k ¼ 1; 2; …; J. Suppose that the function ϕðxÞ in system (1) satisfies the conditions (3) and (4) with constants ρ; β and γ . If there exist matrices F k A Rmn ; k ¼ 1; 2; …; J and positive scalars ɛ 1 4 0; ɛ 2 4 0; λ 4 0; lijk 4 0
X. Cai et al. / ISA Transactions 53 (2014) 298–304
such that the following LMIs are feasible: 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Q k j ɛ1 ρ þ ɛ2 β j J 7 7 0 7r0 7 0 5 I
Γ ðAi Ar Þ þ ðɛ 1 ρ þ ɛ 2 βÞQ k I ɛ21 Q k þ γ2ɛ2 Q k ðɛ 1 ρ þ ɛ 2 βÞI ɛ21 I þ γ ɛ22 I
6 6n 6 6 4n
n
ɛ2 I
n
n
n
ð23Þ with Γ ¼ Ai Q k þ Bi F k þ Q k ATi þ F Tk BTi ∑Jj ¼ 1 lijk ðQ j Q k Þ þ λQ k , i ¼ 1; 2; …N; j; k ¼ 1; 2; …; J. Then for any initial state x(t) in D \ D, the state of system (1) under the control law: uðtÞ ¼ Fðsn ÞQ ðsn Þ 1 eðtÞ
with M 1 ¼ Q ðsn Þ 1 ðAi þ Bi Fðsn ÞQ ðsn Þ 1 Þ þ ðAi þ Bi Fðsn ÞQ ðsn Þ 1 ÞT Q ðsn Þ 1 þɛ 1 ρI þ ɛ 2 β I, M 2 ¼ Q ðsn Þ 1 ðAi Ar Þ þ ɛ 1 ρI þ ɛ2 βI. Using the arguments as the proof in Theorem 1, we have V_ c ðeÞ r max i ¼ 1;2;…;N 9 8 J0 J0 > > > > 1 1 n T n T > > > 2 ∑ sk ek ðtÞQ k ððAi þ Bi F k Q k Þek ðtÞþ ðɛ 1 ρ þ ɛ 2 βÞJ 0 ∑ sk ek ðtÞek ðtÞ > > > > > k ¼ 1 k ¼ 1 > > > > > > > > J0 > > > > > > þ 2 ∑ sn eT ðtÞ Q 1 ɛ 1 I þ γ ɛ 2 I ϕðxÞ > > > > k k k > > 2 2 > > k¼1 > > =
> > > > > > > > > > > > > > > > > ;
2
2
T 6 max ðeT ðtÞ xTr ðtÞ ϕ ðxÞÞ4 n
i ¼ 1;2;…;N
ɛ 1 γ ɛ2 I þ ∑ snk ð2xTr ðtÞ I þ 2 2 k¼1
with Γ ¼ i ¼ 1; 2; …; N; j; k ¼ 1; 2; …; J. From (31), we deduce that
eðtÞ
Λ Q ðsn Þ 1 ðAi Ar Þ Q ðsn Þ
J0
Ai Q k þ Bi F k þ Q k ATi
i ¼ 1;2;…;N
T
k¼1
ð30Þ
By Lemma 2, the derivative of Vc(e) along trajectory of the closedloop system (22) and (24) satisfies the following inequality: V_ c ðeÞ r max f∇V c ðeðtÞÞT ðAi eðtÞ þBi uðtÞ þ ðAi Ar Þxr ðtÞ þ ϕðxÞÞg
J0
þ 2 ∑ snk eTk ðtÞðQ k 1 ðAi Ar Þ þ ɛ 1 ρI þ ɛ 2 βIÞxr ðtÞ
> > > > > > > > > > > > > > > > > > :
ð24Þ
Proof. Let Vc(e) be composed of positive definite matrices Q k ; k ¼ 1; 2; …; J and given by (5). For any xðtÞ A D \ D, and xr(t) is the reference signal, denote eðtÞ ¼ xðtÞ xr ðtÞ. By Lemma 1, e(t) is a convex combination of a set of e0sk ðtÞ, each of which belongs to a certain V c ðeðtÞÞ1=2 Ek . For simplicity, assume that snk ðeÞ 4 0 for J k ¼ 1; 2; …J 0 and snk ðeÞ ¼ 0 for k 4 J 0 . Then eðtÞ ¼ ∑k0 ¼ 1 snk ek ðtÞ. RecalT n 1 ling Lemma 2, we have ∇V c ðeÞ ¼ 2Q ðs Þ e and Q ðsn Þ 1 e ¼ Q k 1 ek ; k ¼ 1; 2; …; J 0 ; furthermore,
301
30
1 eðtÞ 7B x ðtÞ C 5@ r A
ϕðxÞ
ð29Þ
x_ 1 ¼ α1 x1 m1 x1 þ rðt; x3 Þ; x_ 2 ¼ α1 x1 α2 x2 m2 x2 ; x_ 3 ¼ α2 x2 m3 x3 bðtÞuðtÞ;
ð34Þ
þ
where x1 ; x2 ; x3 A R stand for larvae, juveniles and adults respectively. The positive coefficients αi ; mi represent the growth and mortality rates respectively. Assume that the births in class x1 are generated only by the adults class x3 with a reproduction law of Beverton–Holt type [26] rðt; x3 Þ ¼ aðtÞx3 =ðc þ x3 Þ; aðtÞ 40; c 4 0. b(t) represents an harvesting effort on the adult population. We assume that the uncertain parameters satisfy the inequalities: a r aðtÞ r a þ ;
þ
b r bðtÞ r b :
302
X. Cai et al. / ISA Transactions 53 (2014) 298–304
15
x_ A cofAx þ Bu þ f ðx; γ ðtÞÞg
10
ð35Þ
where 2 3 0 0 ðα1 þ m1 Þ 6 α1 ð α 2 þ m2 Þ 0 7 A¼4 5; α2 m3 0 2 3 2 3 0 rðt; x3 Þ 6 7 607 B ¼ 4 5; f ðx; γ ðtÞÞ ¼ 4 0 5 b 0
5 0
ɛ2 ¼ 0:12;
λ ¼ 1;
F 2 ¼ ½0:0328; 0:0676; 2:2692;
l121 ¼ 100;
x2(t)
þ
by solving the LMI (8), we get
l221 ¼ 102;
8
10
12
14
16
18
12
14
16
18
12
14
16
18
15 10
0
0
2
4
6
8
10
time (sec) 20 15 10 5 0
0
2
4
6
8
10
time (sec) Fig. 1. State trajectory x(t) of the closed-loop system (34) and (36) (the initial state is ½15 25; 20T ).
l112 ¼ 150; 15
Vc(x) can be computed as follows:
s:t:
6
5
l212 ¼ 150:
10
u
V c ðxÞ ¼ min α 0 r s1 r 1 "
4
20
Assume that α1 ¼ 0:1; m1 ¼ 0:2; α2 ¼ 0:15; m2 ¼ 0:4; m3 ¼ 0:5; þ a ¼ 0:5; a þ ¼ 0:7; b ¼ 1; b ¼ 3; c ¼ 0:9. Let 2 3 0:2336 0:0056 0:0009 6 0:0056 0:2097 0:0060 7 Q1 ¼ 4 5; 0:0009 0:0060 0:2796 2 3 4:4419 0:0680 0:0052 6 7 Q 2 ¼ 4 0:0680 4:1760 0:0330 5; 0:0052 0:0330 4:3546
ɛ 1 ¼ 0:1;
2
25
and b is b or b . It is easy to know that the nonlinear function f ðx; γ ðtÞÞ is globally one-sided Lipschitz with the one-sided Lipschitz constant ρ ¼ a þ =2c, and it is globally Lipschitz with constant l ¼ a þ =c, and is globally quadratically inner-bounded with β ¼ a þ =c, γ ¼ 0.
F 1 ¼ ½0:0276; 0:2004; 1:2593;
0
time (sec)
x3(t)
x1(t)
From [18], it can be deduced that system (34) belongs to the following differential inclusion:
#
α
xT
x
s1 Q 1 þ ð1 s1 ÞQ 2
5
Z 0;
0
and let FðsÞ ¼ s1 F 1 þ ð1 s1 ÞF 2 ;
0
2
4
6
8
10
12
14
16
18
time (sec)
Q ðsÞ ¼ s1 Q 1 þ ð1 s1 ÞQ 2 :
Fig. 2. Control law u(t) of the closed-loop system (34) and (36).
Then the state of the closed-loop system under the control law: uðtÞ ¼ FðsÞQ ðsÞ 1 xðtÞ
ð36Þ
is exponentially stable at the origin. For simulation the following values of parameters have been chosen:
A2 ¼
1
1
2
1
;
B2 ¼
1 10
:
aðtÞ ¼ 0:5 þ cos 2 ðtÞ=16; bðtÞ ¼ 1:8 þ 0:5 sin t: Figs. 1 and 2 show the time response of the state trajectories x(t) under state feedback control law (36) and the control law u(t) respectively. Remark 1. The merit of algorithm in this paper is that the convergent speed is very quick. The disadvantage of the algorithm is that some scalars need to be given, in order to solve nonlinearities in the matrix inequalities (8), and make them LMIs and can be solved by Matlab. Example 2. Consider a second-order NDI system: _ A cofA1 xðtÞ þB1 uðtÞ þ ϕðxÞ; A2 xðtÞ þ B2 uðtÞ þ ϕðxÞg xðtÞ where 1 A1 ¼ 1
ð37Þ
As it has shown in [15], the above equation can be used to describe the motion of a moving object. From (15), we know that the system is globally one-sided Lipschitz with the one-sided Lipschitz constant ρ ¼ 0. Also, the system is locally Lipschitz on any set D ¼ fx A R2 : J x J r rg, the Lipschitz constant l is 3r 2 . Assume that the reference signal xr1 ¼ cos t; xr2 ¼ sin t, they can be generated from " # # " x_ r1 1 0 1 xr1 ¼ ; xr ð0Þ ¼ : ð38Þ x_ r2 xr2 0 1 0 Let
1 ; 1
2 B1 ¼ ; 1
"
ϕðxÞ ¼
x1 ðx21 þ x22 Þ x2 ðx21 þ x22 Þ
#
ϑ ¼ min
rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi! γ 4 γ2 ; ; βþ 4 4
γ o 0; β þ
γ2 4
4 0:
X. Cai et al. / ISA Transactions 53 (2014) 298–304
Then one can verify the quadratically inner-bounded property of ϕðxÞ in D ¼ fx A R2 : J x J r ϑg [15]. As the system is globally onesided Lipschitz, i.e., D ¼ R2 , D \ D ¼ D. Note that the region D can be made arbitrarily large by choosing appropriate values for γ and β. If β ¼ 200, γ ¼ 141, derives ϑ ¼ 5:9372. Let 0:1130 0:0160 0:1008 0:0035 Q1 ¼ ; Q2 ¼ ; 0:0160 0:1290 0:0035 0:1031 by solving the LMI (23), we get F 1 ¼ ½ 1:2391; 0:6376; F 2 ¼ ½ 2:3221; 3:1070; ɛ 1 ¼ 0:1; ɛ 2 ¼ 0:12; λ ¼ 1; l121 ¼ 50; l112 ¼ 60; l221 ¼ 100; l212 ¼ 150: For each x0 A fx : J x J r 5:9372g, Vc(e) can be computed as follows: V c ðeÞ ¼ min α 0 r s1 r 1 " s:t:
eT
e
s1 Q 1 þ ð1 s1 ÞQ 2
Z 0;
FðsÞ ¼ s1 F 1 þ ð1 s1 ÞF 2 ;
Q ðsÞ ¼ s1 Q 1 þ ð1 s1 ÞQ 2 :
Then the state of the closed-loop system under the control law: uðtÞ ¼ FðsÞQ ðsÞ 1 eðtÞ
2
x1/xr1
1 0
1
2
3
4
5
3
4
5
time (sec) 1
x2/xr2
0.5 0 −0.5 −1 0
1
2 time (sec)
Fig. 3. State trajectory x(t) of the closed-loop system and the desired reference signal xr(t) (the initial state xð0Þ ¼ ½2; 1T and xr ð0Þ ¼ ½1; 0T .)
2
u
0 −2 −4 −6
0
1
In this paper, we present a control design method for one-sided Lipschitz NDIs. Based on LMIs, sufficient conditions for exponential stabilization for the closed-loop system are given. Further, we extend the design method to signal tracking control for this class of NDIs. A control law is designed such that the state of the closed-loop system asymptotically tracks the reference signal. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design technique.
The authors thank the anonymous reviewers for their many helpful suggestions. The authors are grateful for the support of the National Natural Science Foundation of China (Grant numbers 61074011 and 61374077).
ð39Þ
asymptotically tracks the desired reference signal xr(t). Figs. 3 and 4 show that the state x(t) of the closed-loop system asymptotically tracks the desired reference signal xr(t) under state feedback control law (39) and the control law u(t) respectively. The switching between A1 eðtÞ þ ðA1 Ar Þxr ðtÞ þ B1 uðtÞ þ ϕðxÞ and
−1.5
5. Conclusions
Acknowledgment
with e ¼ x0 xr and let
0
A2 eðtÞ þ ðA2 Ar Þxr ðtÞ þ B2 uðtÞ þ ϕðxÞ is chosen such that V c ðeðtÞÞ is minimized at each time instant. For any initial state x0 A fx : J x J r 5:9372g, it has a similar simulative result.
#
α
−1
303
2
3
4
time (sec) Fig. 4. Control law u(t) of the closed-loop system.
5
References [1] Smirnov GV. Introduction to the theory of differential inclusions. Graduate studies in mathematics. Philadelphia: SIAM; 2002. [2] Bichegkuev M. On a weakened Cauchy problem for a linear differential inclusion. Math Notes 2006;79:483–7. [3] Goebel R, Teel A, Hu T, Lin Z. Conjugate convex Lyapunov functions for dual linear differential equations. IEEE Trans Autom Control 2006;51:661–6. [4] Liu LP, Han ZZ, Cai XS, Huang J. Robust stabilization of linear differential inclusion system with time delay. Math Comput Simul 2010;80:951–8. [5] Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear matrix inequalities in systems and control theory, SIAM studies in applied mathematics. Philadelphia: SIAM; 1994. [6] Hu T, Lin Z. Composite quadratic Lyapunov functions for constrained control systems. IEEE Trans Autom Control 2003;48:440–50. [7] Hu T. Nonlinear control design for linear differential inclusions via convex hull of quadratics. Automatica 2007;43:685–92. [8] Cai XS, Liu LP, Zhang W. Saturated control design for linear differential inclusions subject to disturbance. Nonlinear Dyn 2009;58:487–96. [9] Cai XS, Huang J, Liu LP. Stability analysis of linear time-delay differential inclusion systems subject to input saturation. IET Control Theory Appl 2010;4:2592–602. [10] Huang J, Han ZZ, Cai XS, Liu L. Control of time-delayed linear differential inclusions with stochastic disturbance. J Frankl Inst 2010;347:1895–906. [11] Hu T, Blanchini F. Non-conservative matrix inequality conditions for stability/ stabilizaility of linear differential inclusions. Automatica 2010;46:190–6. [12] Liu LP, Han ZZ, Cai XS. Sliding mode control for polytopic differential inclusion systems. Math Comput Simul 2010;79:3018–25. [13] Cai XS, Liu LP, Huang J, Zhang W. Globally asymptotical stabilization for a class of feedback linearisable differential inclusion systems. IET Control Theory Appl 2011;5:1586–96. [14] Huang J, Han ZZ, Cai XS, Liu L. Robust stabilization of linear differential inclusions with affine uncertainty. Circuits Syst Signal Process 2011;30: 1369–82. [15] Abbaszadeh M, Marquez HJ. Nonlinear observer design for one-sided Lipschitz systems. In: Proceedings of the American control conference, Baltimore, USA, 2010. p. 5284–9. [16] Zhang W, Su H, Wang H, Han ZZ. Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations. Commun Nonlinear Sci Numer Simul 2012;17:4968–77. [17] Zhang W, Su H, Liang Y, Han ZZ. Nonlinear observer design for one-sided Lipschitz systems: an linear matrix inequality approach. IET Control Theory Appl 2012;6:1297–303. [18] Cai XS, Lv GY, Zhang W. Stabilisation for a class of non-linear uncertain systems based on interval observers. IET Control Theory Appl 2012;6:2057–62. [19] Chen M, Wu QX, Cui RX. Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISA Trans 2013;52:198–206. [20] Wang JJ. Stabilization and tracking control of XCZ inverted pendulum with sliding-mode control. ISA Trans 2012;51:763–70. [21] Travieso-Torres JC, Duarte-Mermoud MA, Estrada JL. Tracking control of cascade systems based on passivity: the non-adaptive and adaptive cases. ISA Trans 2006;45:435–45. [22] Purwar S, Kar I, Jha A. Adaptive output feedback tracking control of robot manipulators using position measurements only. Expert Syst Appl 2008;34: 2789–98.
304
X. Cai et al. / ISA Transactions 53 (2014) 298–304
[23] Cai XS, Huang J, Xie QY. Output tracking and disturbance rejection of linear differential inclusion systems. Int J Syst Sci 2012;43:2072–8. [24] Cai XS, Liu Y, Zhang W. Asymptotic tracking control for a class of reference signals for linear differential inclusions. Int J Syst Sci, http://dx.doi.org/10. 1080/00207721.2012.748104, this issue.
[25] Zemouche A, Boutayeb M. Observer design for Lipschitz nonlinear systems: the discrete-time case. IEEE Trans Circuits Syst II: Express Briefs 2006;53:777–81. [26] Moisan M, Bernard O, Gouzé JL. Near optimal interval observers bundle for uncertain bioreactors. Automatica 2009;45:291–5.
