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IMA Journal of Mathematical Control and Information
, Volume Advance Article – Jan 27, 2017

22 pages

/lp/ou_press/robust-mathcal-q-mathcal-s-mathcal-r-gamma-dissipative-sliding-mode-hUmzh8zpB9

- Publisher
- Oxford University Press
- Copyright
- © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 0265-0754
- eISSN
- 1471-6887
- D.O.I.
- 10.1093/imamci/dnw074
- Publisher site
- See Article on Publisher Site

In this study, a robust sliding mode-based learning control strategy for a class of non-linear discrete-time descriptor systems with time-varying delay and external disturbance is developed. A new sliding function is proposed and a sufficient condition is derived to guarantee the sliding mode dynamics to be robustly admissible and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative. Moreover, a sliding-mode control law is proposed such that the reaching motion satisfies the discrete-time sliding mode reaching condition for all admissible uncertainties and time-varying delay. The appealing attributes of this approach include: (i) the closed-loop system exhibits a strong robustness against uncertain dynamics and (ii) the control scheme enjoys the chattering-free characteristic. Two representative examples are given to illustrate the theoretical developments. 1. Introduction Dynamical models of a wide class of physical processes meant to represent the interactions among process components in mathematical framework. Among these frameworks is the descriptor model, which provides a natural representation of the static and dynamic parts of several physical plants. In studying descriptor systems, it turns out that system regularity and causality (for discrete systems) or the absence of impulses (for continuous systems) need to be guaranteed (Dai, 1989). On another research front, considerable attention has been devoted to the analysis and synthesis of dynamical systems with time delays in which delays often occur in the transmission of information or material between different components of the physical process (Xia et al., 2009; Mahmoud, 2010). Several research results have been reported in the literature (see, for example, Chen et al., 2008; Kchaou et al., 2011; Gassara et al., 2014 and references therein). We emphasize that there are many practical systems can be modelled by descriptor systems with time-delay. These include transportation systems, communication systems, power systems, nuclear reactors and chemical processes are some examples. In view of the generality of descriptor models and time-delay phenomenon, some fundamental results based on the theory of state-space systems have been extended to this class of systems, such as the stability and the stabilization (see, for instance, Duan, 2010; Kchaou et al., 2013a,b). Among basic notions of state-space systems generalized to descriptor systems, dissipativity is one of the most important properties of dynamical systems and plays crucial roles in various problems of analysis and control design of linear and non-linear systems (Mahmoud, 2011; Mahmoud & Khan, 2014; Mahmoud & Saif, 2014). Since its introduction, it has been attracting a great deal of research interests and many results have been reported so far (Su et al., 2011; Wu et al., 2012; Cui et al., 2013). Sliding mode control (SMC) is one of robust control technique which has been widely applied to various practical engineering systems. It is a very effective approach to achieve robustness and invariance to matched uncertainties and external disturbances (Wu et al., 2008; Ding et al., 2011; Chang, 2012). Nowadays, with the advancement of digital computers most continuous systems are treated in their discretized forms. However, due to a finite sampling rate, some features available for the continuous-time SMC could be inappropriate for the discrete-time system. Hence, considerable amount of research interests has been received for the design of discrete-time SMC (Ma et al., 2009; Xi & Hesketh, 2010; Hu et al., 2012a; Liu et al., 2013). The original approach proposed by Gao et al. in (1995) has become very popular. In this paper, a reaching law approach-based discrete SMC was designed and the motion of such system confines to quasi sliding mode band. Therefore, by reason of the presence of the sign function in control input, the problem of chattering is the major drawback in this discrete-time (SMC) approach. Recently, several SMC strategies are proposed for time-delay systems with different delays and disturbance structures. In Yan & Shi (2008), Xia et al. (2010), a discrete-time SMC approach is proposed for uncertain linear systems with unknown time-varying delay. A sufficient condition for the existence of stable sliding surfaces depending is established and a discrete-time (SMC) which guarantee the sliding mode reaching condition of the specified discrete-time sliding surface is synthesized. In Hu et al. (2012b), the $$H_\infty$$ sliding mode observer design problem for a class of non-linear discrete time-delay systems is investigated. A new non-linear sliding mode observer is synthesized to estimate the unmeasured states and based on the delay-fractioning approach, a sufficient condition has been developed to guarantee the error dynamics to be asymptotically stable and the estimation error satisfies the specified $$H_\infty$$ performance requirement. However, to the best of authors’ knowledge, research on discrete-time SMC for discrete-time descriptor systems with uncertainties and time-varying state delay has not been fully investigated, which motivates this research work. In this paper, we aim to investigate the $$({\mathcal{Q}},\mathcal{S},\mathcal{R})$$-dissipativity SMC problem for a class of non-linear discrete-time descriptor systems with uncertainties and time-varying delay. The main contributions can be summarized as follows: (i) for the addressed non-linear discrete descriptor system, a new form of discrete switching function is designed such that the sliding motion is robustly admissible and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative; (ii) a reaching motion control law is designed such that the discrete sliding mode is reachable in finite time, and the states of closed-loop system evolve around a residual set of the origin. The rest of the paper is organized as follows. Section 2 presents the description of the system and some preliminaries. The main results, which consist of design a new switching function and a robust reaching motion control law, are developed in Section 3. Section 4, shows the effectiveness of the proposed strategy by two numerical examples. Some conclusion remarks are given in Section 5. Notation The notations in this paper are quite standard except where otherwise stated. The superscript âŁ∼TâŁTM stands for matrix transposition; $$X\in\mathbb{R}^{n}$$ denotes the $$n$$-dimensional Euclidean space, while $$X\in\mathbb{R}^{n\times m}$$ refers to the set of all $$n\times m$$ real matrices; $$X>0$$ (respectively, $$X\geq 0$$) means that the matrix $$X$$ is real symmetric positive definite (respectively, positive semi-definite); $$ L_2[0,\ \infty)$$ is the space of square summable vectors; I and 0 represent the identity matrix and a zero matrix with appropriate dimension, respectively; $$\text{sym}(X)$$ stands for $$X+X^T$$; $$\|.\|$$ denotes the Euclidean norm of a vector and its induced norm of a matrix. In symmetric block matrices or long matrix expressions, we use a star $$*$$ to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. System description and preliminaries Consider a class of descriptor discrete-time systems with state delay described by: {Ex(k+1) =(A+ΔA)x(k)+(Ad+ΔAd)x(k−d(k))+B(u(k)+f(x(k)))+Bww(k)z(k) =Cx(k)+Cdx(k−d(k))+Dww(k)x(k) =ϕ0(k), k∈[−dM,0], (2.1) where $$x(k) \in \mathbb{R}^n$$ is the state vector, $$u(k) \in \mathbb{R}^m$$ is the control input vector, $$w(k) \in \mathbb{R}^w$$ is the exogenous disturbance that belongs to $$L_2[0,\infty)$$, $$z(k)\in \mathbb{R}^q$$ is the controlled output vector, $$f(x(k))$$ is an unknown but bounded non-linear real-valued function, which represents any model uncertainties in the system including external disturbances and $${\phi_0}(k)$$ is a compatible initial condition. The time delay $$ d (k) $$ is a positive integer, is assumed to be time-varying in the whole dynamic process, and satisfies dm≤d(k)≤dM, (2.2) where $$d_m$$ and $$d_M$$ are constant positive integers representing the bounds of the delay. Matrix $$E \in \mathbb{R}^{n\times n}$$ may be singular, and we assume that $${\rm rank}(E)=r< n$$. $$A$$, $$A_d$$, $$B$$, $$B_w$$, $$C$$, $$C_d$$ and $$D_w$$ are known real constant matrices with appropriate dimensions. $${\it\Delta} A$$ and $${\it\Delta} A_d$$ are unknown matrices representing the parametric uncertainties. Without loss of generality, we introduce the following assumption for technical convenience: 1. Unmatched uncertainties $${\it\Delta} A$$ and $${\it\Delta} A_d$$ satisfy [ΔAΔAd] =MF(k)[NNd], (2.3) where $$M$$, $$N$$ and $$N_d$$ are known real constant matrices with appropriate dimensions. $$F(k)$$ is an unknown matrix function satisfying $$F^T(k)F(k)\leq I$$. 2. Exogenous signal, $$w(k)$$, is bounded. 3. Matched non-linearity $$f(x(k))$$ is unknown but bounded in the sense of the Euclidean norm. First of all, we recall some definitions for the following unforced linear discrete-time descriptor system with time-delay: {Ex(k+1) =Ax(k)+Adx(k−d(k))x(k) =ϕ0(k), k∈[−dM,0]. (2.4) Definition 2.1 (Dai, 1989; Xu & Lam, 2006) 1. Pair $$(E,A)$$ is said to be regular if $$\det(zE-A)$$ is not identically zero. 2. Pair $$(E,A)$$ is said to be causal, if it is regular and $${\rm deg}({{\rm det}(zE -A)}) = {\rm rank}(E)$$. 3. For given positive scalars $$d_m$$ and $$d_M$$, discrete singular time-delay system (2.4) is said to be regular and causal for any time delay $$d(k)$$ satisfying $$d_m \leq d(k) \leq d_M$$, if pair $$(E,A)$$ is regular and causal. 4. System (2.4) is said to be admissible if it is regular, causal and stable. 5. Discrete-time descriptor system (2.4) is said to be stable if, for any scalar $$\varepsilon>0$$, there exists a scalar $$\delta(\varepsilon)>0$$ such that, for any compatible initial condition $$\phi(k)$$, satisfying $$\sup_{-d_M\leq k\leq 0}||\phi(k)||\leq \delta(\varepsilon)$$, solution $$x(k)$$ to system (2.4) satisfies $$||x(k)||\leq \varepsilon$$ for any $$k\geq 0$$; moreover $$\displaystyle \lim_{k\rightarrow \infty}x(k)=0$$. Definition 2.2 (Wu et al., 2012) Given some scalar $$\gamma >0$$, matrices $$\mathcal{Q}=\mathcal{Q}^T$$, $$\mathcal{R}=\mathcal{R}^T$$ and $$\mathcal{S}$$, system (2.4) is called $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative if for any integer $$T$$, the following inequality: ∑s=0T(zT(s)Qz(s)+2zT(s)Sw(s)+wT(s)Rw(s))≥γ∑s=0TwT(s)w(s),∀T≥0 (2.5) holds under zero initial condition. Remark 2.1 From Definition 2.2, the notion of $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-dissipativity includes $$H_{\infty}$$ performance and strict passivity as special cases by choosing different values for $$\mathcal{Q} $$, $$\mathcal{R} $$ and $$\mathcal{S}$$. If $$\mathcal{Q}=-I$$, $$\mathcal{R}=\gamma ^2 I$$ and $$\mathcal{S}=0$$, inequality (2.5) reduces to an $$H_{\infty}$$ performance requirement. If $$\mathcal{Q}=0$$, $$\mathcal{R}=0$$ and $$\mathcal{S}=I$$, inequality (2.5) correspond to a strict passivity or strictly positive realness. If $$\mathcal{Q}=-\theta I$$, $$\mathcal{R}=\theta \gamma ^2I$$ and $$\mathcal{S}=(1 -\theta) I$$, $$\theta\in [0,1]$$ be a given scalar weight representing a trade off between $$H_{\infty}$$ and positive real performance, then $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-dissipativity reduces to the mixed $$H_{\infty}$$ and positive real performance. Without loss of generality, it is assumed that $$\mathcal{Q}<0$$. Before proceeding, we introduce the following lemmas that will be used in deriving our main results. Lemma 2.1 (Petersen, 1987) Given matrices $$M$$, $$N$$ and $$P$$ of appropriate dimensions, with $$P$$ symmetrical, then P+MF(k)N+NTFT(k)MT<0 for any $$F(k) $$ satisfying $$F^T(k)F(k)\leq I$$, if and only if there exists a scalar $$\varepsilon>0$$ such that P+εMMT+ε−1NTN<0. (2.6) Lemma 2.2 (Zhu et al., 2009) For any matrix $$M >0$$, integers $$p$$ and $$q$$ satisfying $$q>p$$, and vector function $$x:\mathbb{N}[p,q]\rightarrow\mathbb{R}^n$$ such that the sums concerned are well-defined, then: −(q−p+1)∑s=pqxT(s)Mx(s)≤−(∑s=pqx(s))TM(∑s=pqx(s)). Lemma 2.3 (Park et al., 2011) Let $$f_1$$, $$f_2$$, ..., $$f_N$$: $$\mathbb{R}^m\mapsto\mathbb{R}$$ have positive values in an open subset $$\mathbf{D}$$ of $$\mathbb{R}^m$$. Then, the reciprocally convex combination of $$f_i$$ over $$\mathbf{D}$$ satisfies min{αi|αi>0,∑iαi=1}∑i1αifi(t)=∑ifi(t)+maxgi,j(t)∑i≠jgi,j(t)subject to {gi,j:Rm↦R;gj,i(t)=gi,j(t),[fi(t)gi,j(t)gi,j(t)fi(t)]≥0}. (2.7) 3. Main results This work aims to solve the addressed robust SMC problem for system (2.1) in presence of parameter uncertainties, time-varying delay and non-linear input. Firstly, a new sliding surface is designed and a sufficient condition is developed that ensures for the sliding mode dynamics to be robustly admissible and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative. Secondly, a sliding mode controller is designed to drive the state onto the sliding mode region. 3.1. Sliding surface design The design procedure of discrete-time SMC consists on two steps. The first one consists to design a sliding surface such that, in the sliding mode, the system response acts like the desired dynamics performance. In the second step a SMC law is synthesized to guarantee that the sliding mode is reached and the system states maintain in the sliding mode thereafter. Now, consider the following sliding surface: S(k)=GEx(k)−∑s=0k−1G(A+BK−E)x(s), (3.1) where $$\mathbb{G}\in \mathbb{R}^{m \times n}$$ is a matrix such that $$\mathbb{G}B$$ is invertible. $$K$$ is a gain matrix to be designed later. The sliding mode satisfies S(k+1)=S(k)=0. (3.2) From (2.1) and (3.1), we get ΔS(k)=G(ΔAx(k)+(Ad+ΔAd)x(k−d(k))+Bww(k))+GB(u(k)+f(x(k))−Kx(k))=0. (3.3) Then the equivalent control law is obtained as follows: u(k)= −(GB)−1G(ΔAx(k)+(Ad+ΔAd)x(k−d(k))+Bww(k))−f(x(k))+Kx(k). (3.4) Substituting (3.4) into the system (2.1) and denoting $$\bar {\mathbb{G}}=I-B{(\mathbb{G}B)}^{-1}\mathbb{G}$$, the sliding mode dynamics and the output equation can be formulated as {Ex(k+1) =(A+ΔA)x(k)+(Ad+ΔAd)x(k−d(k))+Bww(k)z(k) =Cx(k)+Cdx(k−d(k))+Dww(k), (3.5) where A =A+BK,Ad =G¯Ad,Bw =G¯Bw,ΔA =MF(k)N,ΔAd =MF(vk)Nd,M =G¯M. (3.6) 3.2. Sliding mode dynamics dissipativity analysis In the sense of Definitions 2.1 and 2.2, we aim in this section to provide a sufficient condition under which sliding mode dynamics (3.5) is robustly admissible and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative. Theorem 3.1 Consider system (2.1) with sliding surface (3.1). For given positive integers $$d_{m}$$ and $$d_{M}$$, a positive scalar $$\gamma$$ and matrices $$\mathcal{Q}$$, $$\mathcal{R}$$ and $$\mathcal{S}$$, the sliding mode dynamics (3.5) is robustly admissible and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative, if there exist a positive scalar $$\varepsilon$$ and matrices $$P>0$$, $$Q>0$$, $$Q_{1}>0$$, $$Q_{2}>0$$, $$Z_{1}>0$$, $$Z_{2}>0$$, $$Z_{3}>0$$, $$S$$, $$R$$, $$G_{i}$$, $$i=1,\ldots,3$$, such that the following inequalities hold: [ΦΓ1εΓ2T∗−εI0∗∗−εI] <<0, (3.7) [Z3RRTZ3] >0, (3.8) where Φ =[Φ11ETZ2EΦ13Φ14Φ15G1TBw−CTSCT−Q∗Φ22Φ23ETRTE000∗∗Φ33Φ34Φ35G2TBw−CdTSCdT−Q∗∗∗Φ44000∗∗∗∗Φ55G3TBw0∗∗∗∗∗Φ66DwT−Q∗∗∗∗∗∗−I]Γ1 =[MTG10MTG20MTG300]TΓ2 =[N0Nd0000] (3.9) Φ11=Q1+Q2+(τ+1)Q+sym(G1T(A−E))−ETZ1E−ETZ2EΦ22= −Q1−ETZ3E−ETZ2EΦ13=G1TAd+(A−E)TG2Φ23= −ETRTE+ETZ3EΦ33= −Q+sym(G2TAd)−2ETZ3E+sym(ETRE)Φ14=ETZ1EΦ34= −ETRE+ETZ3EΦ44= −Q2−ETZ1E−ETZ3EΦ15=ETP+SΩT−G1T+(A−E)TG3Φ35= −G2T+AdTG3Φ55=dM2Z1+dM2Z2+τ2Z3−sym(G3)Φ66= −(R−γI)−sym(DwTS) (3.10)$${\it\Omega}\in \mathbb{R}^{{n}\times{n-r}}$$ is any matrix with full column rank satisfying $$E^T{\it\Omega}=0$$ and $$\tau=d_{M}-d_{m}.$$ Proof. Assume that $${\it\Delta}\mathbb A=0$$ and $${\it\Delta}{\mathbb A_d}=0$$. Under conditions (3.7)–(3.8), we first prove that the nominal case of (3.5) is admissible and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative. Since $$\text{rank}(E)=r< n$$, there always exist two non-singular matrices $$\bar M$$ and $$\bar N\in \mathbb{R}^{n \times n}$$ such that E¯ =M¯EN¯=[Ir000]. (3.11) Then, $${\it\Omega}$$ can be characterized as ${\it\Omega}=\bar M^T \begin{bmatrix} 0\\ \bar{\it\Phi} \end{bmatrix} $, where $$\bar{\it\Phi}\in \mathbb{R}^{(n-r)\times(n-r)}$$ is any non-singular matrix. We also define A¯ =M¯AN¯=[A¯11A¯12A¯21A¯22],S¯ =N¯TS=[S¯11S¯21],A¯d =M¯AdN¯=[A¯d11A¯d12A¯d21A¯d22]. (3.12) It follows from (3.7) that [Ψ11Ψ12∗Ψ22]<0, (3.13) where Ψ11=sym(G1T(A−E))−ETZ1E−ETZ2E,Ψ12=ETP+SΩT−G1T+(A−E)TG3,Ψ22= −sym(G3). Pre- and post-multiplying (3.13) by $$\left[{I,\mathbb A^T}\right]$$ and its transpose, respectively, we obtain sym(ET(P−G3−G1)A−G1E−ETZ1E+SΩTA)<0. (3.14) Pre- and post-multiplying (3.14) by $$\bar N^T$$ and $$\bar N$$, respectively, and then using expression (3.11) and (3.12), yields sym(S¯21Φ¯TA¯22)<0 (3.15) and thus $$\overline A_{22}$$ is non-singular. Otherwise, suppose that matrix $$\overline A_{22}$$ is singular, then there must exist a non-zero vector $$\psi\in \mathbb{R}^{{n-r}}$$ which ensures $$\overline A_{22}\psi=0$$. As a consequence, we conclude that $$\psi^T\text{sym}{(\overline S_{21}{\it\Phi}^T\overline A_{22})}\psi=0$$ which contradicts (3.15), so $$\overline A_{22}$$ is non-singular. Then, pair $$(E,\mathbb A)$$ is regular and causal. Next, we show that system (3.5) is stable. To this end, we select a Lyapunov–Krasovskii functional candidate as V(k)=V1(k)+V2(k)+V3(k)+V4(k),V1(k)=xT(k)ETPEx(k),V2(k)=∑s=k−dmk−1xT(s)Q1x(s)+∑s=k−dMk−1xT(s)Q2x(s),V3(k)=∑θ=−dM−dm∑s=k+θk−1xT(s)Q1x(s),V4(k)=dM∑θ=−dM−1∑s=k+θk−1ηT(s)ETZ1Eη(s)+dm∑0=−dm−1∑s=k+θk−1ηT(s)ETZ2Eη(s),+∑θ=−dM−1∑s=k+θk−1ηT(s)ETZ3Eη(s) (3.16) where $$\eta\left(k\right)={x}\left({k+1}\right)-x\left(k\right)$$. By calculating the difference $${\it\Delta}{V}{(k)}= V{(k+1)}-V(k)$$ along the trajectory of sliding mode dynamics (3.5), we get ΔV1(k)=ηT(k)ETPEη(k)+2xT(k)ETPEη(k) (3.17) ΔV2(k)=xT(k)(Q1+Q2)x(k)−xT(k−dm)Q1x(k−dm)−xT(k−dM)Q2x(k−dM) (3.18) ΔV3(k) =(τ+1)xT(k)Qx(k)−∑s=k−dMk−dmxT(s)Qx(s) ≤(τ+1)xT(k)Qx(k)−xT(k−d(k))Qx(k−d(k)) (3.19) ΔV4(k)=ηT(k)ET(dM2Z1+dm2Z2+τ2Z3)Eη(k)−dM∑s=k−dMk−1ηT(s)ETZ1Eη(s) −dm∑s=k−dmk−1ηT(s)ETZ2Eη(s)−τ∑s=k−dMk−dm−1ηT(s)ETZ3Eη(s). (3.20) According to Lemma 2.2, we get −dM∑s=k−dMk−1ηT(s)ETZ1Eη(s)≤ −γ1(k)ETZ1Eγ1(k)−dm∑s=k−dmk−1ηT(s)ETZ2Eη(s)≤ −γ2(k)ETZ2Eγ2(k) (3.21) and −τ∑s=k−dMk−dm−1ηT(s)ETZ3Eη(s)= −τ∑s=k−dMk−d(k)−1ηT(s)ETZ3Eη(s)−τ∑s=k−d(k)k−dm−1ηT(s)ETZ3Eη(s) ≤−1α1ψ 1T(k)ETZ3Eψ1(k)−1α2ψ 2T(k)ETZ3Eψ 2T(k), (3.22) where γ1(k) =x(k)−x(k−dM),γ2(k) =x(k)−x(k−dm),ψ1(k) =x(k−d(k))−x(k−dM),ψ2(k) =x(k−dm)−x(k−d(k)),α1 =dM−d(k)dM−dm,α2 =d(k)−dmdM−dm. Performing a congruence transformation to (3.8) by $${\rm diag}\left({E\psi_1(k),E \psi_2(k)}\right)$$, yields, [ψ 1T(k)ETZ3Eψ1(k)ψ 1T(k)ETREψ2(k)ψ 2T(k)ETRTEψ1(k)ψ 2T(k)ETZ3Eψ2(k)]≥0. (3.23) According to Lemma 2.3, we conclude −τ∑s=k−dMk−dm−1ηT(s)ETZ3Eη(s) ≤−[ψ1(k)ψ2(k)]T[ETZ3EETREETRTEETZ3E][ψ1(k)ψ2(k)]. (3.24) Note that when $$d(k)={d_{m}}$$ or $$d(k)=d_{M}$$, we have $$\psi_1(k)=0$$ or $$\psi_2(k)=0$$, respectively. Thus, (3.24) still holds. Let $$\xi \left(k \right)={{\left[{{x}^{T}}\left(k \right){{x}^{T}}\left(k-{{d}_{m}} \right){{x}^{T}}\text{(}k-d(k)\text{)}\,x\left(k-{{d}_{M}} \right){{\eta }^{T}}\left(k \right){{E}^{T}} \right]}^{T}}$.$ From (3.5), it is easy to see that the following equation holds for any matrices $$G_1$$, $$G_2$$ and $$G_3$$ with appropriate dimensions 2[xT(k)G1T+xT(k−d(k))G2T+ηT(k)ETG3T][(A−E)x(k)+Adx(k−d(k))−Eη(k)]=0. (3.25) On the other hand, it is clear that 2xT(k)SΩTEη(k)=0. (3.26) From (3.17)–(3.26), we have ΔV(k)≤ξT(k)Φ^ξ(k), (3.27) where Φ^ =[Φ11ETZ2EΦ13Φ14Φ15∗Φ22Φ23ETRTE0∗∗Φ33Φ34Φ35∗∗∗Φ440∗∗∗∗Φ55]. (3.28) According to Lyapunov stability theory, then there exists a scalar $$\alpha>0$$ such that ΔV(k)≤−α‖x(k)‖2. (3.29) Therefore, we have ∑i=0k‖x(i)‖2≤1αV(0)<∞ (3.30) that is, the series $$\sum\limits_{i=0}^{k} \|x(i)\|^2$$ converges, which implies that $$\displaystyle \lim_{k\rightarrow \infty} x(k)=0$$. Thus, according to Definition 2, system (3.5) is stable. To prove the dissipativity of system (3.5), we introduce the following performance index: Jzw =∑k=0T(zT(k)Qz(k)+2zT(k)Sw(k)+wT(k)(R−γI)w(k)). (3.31) Define ${\zeta}(k)= \begin{bmatrix} {\xi^{T}_1}(k)& w^T(k) \end{bmatrix}^T$. The following null equation holds 2ζT(k)G[(A−E)x(k)+Adx(k−d(k))+Bww(k)−Eη(k)]=0, (3.32) where $$\mathbf{G}=\left[{{G_1\,0 \, G_2\,0 \, G_3\,0}} \right]^T$$. Then, according to (3.7), we have ΔV(k)−(zT(k)Qz(k)+2zT(k)Sw(k)+wT(k)(R−γI)w(k))=ζT(k)Φζ(k)<0. (3.33) Summing (3.33) over the range $$\left[{{0,T}} \right]$$ with initial condition $$ V(0) = 0$$, yields ∑k=0TΔV(k)−Jzw≤V(T+1)−Jzw<0. (3.34) Since $$ V(T+1)\geq 0$$, we conclude ∑k=0T(zT(k)Qz(k)+2zT(k)Sw(k)+wT(k)(R−γI)w(k))>0. (3.35) Thus, systems (3.5) is $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative in the sense of Definition 2.2. In the light of the same reasoning, we can conclude that the sliding mode dynamics (3.5) is robustly admissible and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative for all admissible uncertainties. In fact by considering (3.6), we can establish that Φ+sym(Γ1F(k)Γ2)<0. (3.36) Then, according to Lemma 2.1, inequality (3.7) holds applying the Schur complement. This completes the proof. □ 3.3. Sliding mode dynamics dissipativity synthesis Based on the previous results, we focus on developing a method to synthesize gain $$K$$ such that sliding mode dynamics (3.5) is robustly admissible and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative. Theorem 3.2 For given positive integers $${d_{m}}$$ and $${d_{M}}$$, a positive scalar $$\gamma$$, matrices $$\mathcal{Q}$$, $$\mathcal{R}$$ and $$\mathcal{S}$$, and tuning parameters $${\lambda_{i}}$$, $$(i =1,2,3)$$, system (3.41) is robustly admissible and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipative, if there exist a positive scalar $$\varepsilon$$ and appropriate dimensions matrices $$P>0$$, $$Q>0$$, $${Q_{1}}>0$$, $${Q_{2}}>0$$, $${Z_{1}}>0$$, $${Z_{2}}>0$$, $${Z_{3}}>0$$, $$S$$, $$R$$, $${G_{i1}}$$, $${G_{i2}}$$, $$F\in \mathbb{R}^{m \times n} $$ and $$Y\in \mathbb{R}^{m \times n}$$ such that (3.8) and the following LMI hold: [Φ¯Γ¯1εΓ¯2∗−εI0∗∗−εI] <0, (3.37) where Φ¯ =[Φ¯11E¯TZ2E¯Φ¯13Φ14Φ¯15Bw1−C¯TSC¯T−Q∗Φ¯22Φ¯23E¯TRTE¯000∗∗Φ¯33Φ¯34Φ¯35Bw2−C¯dTSC¯dT−Q∗∗∗Φ¯44000∗∗∗∗Φ¯55Bw30∗∗∗∗∗Φ¯66DwT−Q∗∗∗∗∗∗−I]Γ¯1 =[M1T0M2T0M3T00]TΓ¯2 =[N¯0N¯d0000]T (3.38) Φ¯11=Q1+Q2+(τ+1)Q+sym(A1)−E¯TZ1E¯−E¯TZ2E¯Φ¯22=−Q1−E¯TZ3E¯−E¯TZ2E¯Φ¯13=Ad1+A2TΦ¯23=−E¯TRTE+E¯TZ3E¯Φ¯33=−Q+sym(Ad2)−2E¯TZ3E¯+sym(E¯TRE¯)Φ¯14=E¯+E¯TZ1E¯Φ¯34=−E¯TRTE¯+E¯TZ3E¯Φ¯44=−Q2−E¯TZ1E¯−E¯TZ3E¯Φ¯15=E¯TP+SΩT−G1T+A3TΦ¯35=−G2T+Ad3Φ¯55=dM2Z1+dm2Z2+τ2Z3−sym(G3)Φ¯66=−(R−γI)+sym(SDw) (3.39) Ai=[Gi1(A−E)+λiIYGi1B−λiIFGi2(A−E)+λiYGi2B−λiF],Adi=[Gi1Ad0Gi2Ad0],Bwi=[Gi1BwGi2Bw],Mi=[Gi1MGi2M], (i=1,2,3). (3.40) Furthermore, $$K= {F^{ - 1}}Y$$. Proof. If we view input $$u(k)$$ as a state component, system (3.5) can be rewritten as {E¯x¯(k+1) =(A¯+ΔA¯)x¯(k)+(A¯d+ΔA¯d)x¯(k−d(k))+B¯ww(k)z(k) =C¯x¯(k)+C¯dx¯(k−d(k))+Dww(k), (3.41) where ${\bar{x}}(k) =\begin{bmatrix} x^T(k)&u^T(k) \end{bmatrix}^T$, $${\it\Delta}\bar{\mathbb{A}}=\bar{\mathbb{M}}F(k) \bar N$$, $${\it\Delta}\bar{\mathbb{A}}_d=\bar{\mathbb{M}}F(k) \bar N_d$$ and E¯ =[E000],A¯ =[ABK−I],A¯d =[Ad000],B¯w =[Bw0],C¯ =[C0],C¯d =[Cd0],M¯ =[MT0]T,N¯ =[N0],N¯d =[Nd0]. (3.42) Applying Theorem 3.1 to system (3.41) with the following particular structures of matrices $$G_i$$, $$(i = 1, 2, 3)$$ G1T =[G11λ1IFG12λ1F],G2T =[G21λ2IFG22λ2F],G3T =[G31λ3IFG32λ3F],I =[Im0(n−m)×m], (3.43) condition (3.37) holds by setting $$Y=FK$$. However, we must also emphasize, under the conditions of Theorem 3.2, a feasible solution satisfies the condition $$\bar{{\it\Phi}}_{55}<0$$. This implies that $$ G_3$$ is non-singular and thus $$F$$ is also non-singular. This completes the proof. □ 3.4. SMC law synthesis Once the sliding surface is appropriately designed according to Theorem 3.2, the robust discrete-time SMC control signal $$ u(k) $$ can be derived to satisfy the sliding mode reaching condition. This control law can force the state trajectory to move monotonically toward the switching surface and then oscillate around the sliding surface in zigzag type motion. In fact, when the state trajectory crosses the sliding surface for the first time, it will cross the surface again in each sampling period. The state trajectory is required to enter and stay within a neighbourhood of the pre-specified switching surface called sliding-mode band. Since it is assumed that $${\it\Delta} A$$, $${\it\Delta} A_d$$, $$ w(k)$$ and $${f(x(k))}$$ are bounded and using the discrete-time version of improved Razumikhin Theorem (for any $$d(k)$$ satisfying (2.2), there exists a constant $$\theta>1$$ such the inequality $$\lVert x(k-d(k))\rVert\leq \theta\lVert x(k)\rVert$$ holds), the following functions $$\phi(k)=\mathbb{G} {\it\Delta} A x(k) $$, $$\varphi(k)=\theta\mathbb{G}(A_d+ {\it\Delta} A_d) x(k) $$, $$\chi(k)=\mathbb{G}B_w w(k) $$ and $$\psi(k)=\mathbb{G}B{f(x(k))}$$ are also bounded with upper and lower bounds. Note that the $$\phi(k)\in \mathbb{R}^m$$, $$\varphi(k)\in \mathbb{R}^m$$, $$\chi(k)\in \mathbb{R}^m$$ and $$\psi(k)\in \mathbb{R}^m$$. We assume that elements of $$\phi(k)$$, $$\varphi(k)$$, $$\chi(k)$$ and $$\psi(k)$$ are defined as: ϕ(k) =[ϕ1(k),ϕ2(k),…,ϕm(k)]T,φ(k) =[φ1(k),φ2(k),…,φm(k)]T,χ(k) =[χ1(k),χ2(k),…,χm(k)]T,ψ(k) =[ψ1(k),ψ2(k),…,ψm(k)]T. (3.44) We assume also that there exist $$ \phi_{L,i}$$, $$ \varphi_{L,i}$$, $$ \chi_{L,i}$$, $$\psi_{L,i}$$, $$\phi_{U,i}$$, $$\varphi_{U,i}$$,$$ \chi_{U,i}$$ and $$\psi_{U,i}$$ such that ϕL,i≤ϕi(k)≤ϕU,i,φL,i≤φi(k)≤φU,i,χL,i≤χi(k)≤χU,i,ψU,i≤ψi(k)≤ψL,i, (3.45) where $$\phi_i(k)$$, $$\varphi_i(k)$$, $$\chi_i(k)$$ and $$\psi_i(k)$$ are the $$i$$th element in $$ \phi (k)$$, $$\varphi (k)$$, $$ \chi (k)$$ and $$ \psi (k)$$, respectively. Thus the mean and the spread of $$ \phi (k)$$, $$\varphi (k)$$, $$ \chi (k)$$ and $$ \psi (k)$$ can be computed as ϕ¯i =ϕU,i+ϕL,i2φ¯i =φU,i+φL,i2,ψ¯i =ψU,i+ψL,i2χ¯i =χU,i+χL,i2,ϕ^i =ϕU,i−ϕL,i2φ^i =φU,i−φL,i2,ψ^i =ψU,i−ψL,i2χ^i =χU,i−χL,i2. (3.46) We propose the following theorem to summarize the design technique of the robust discrete-time SMC. Theorem 3.3 For system (2.1), assume that the condition in Theorem 3.2 has a feasible solution. Then with the designed sliding surface given in (3.1), the discrete-time sliding mode reaching condition can be satisfied with the following control law u(k) =Kx(k)−(GB)−1{(ϕ¯+φ¯+χ¯+ψ¯)+(ϕ^+φ^+χ^+ψ^)sgn(S(k))}, (3.47) where ϕ¯ =[ϕ¯1,ϕ¯2,…,ϕ¯m]T,φ¯ =[φ¯1,φ¯2,…,φ¯m]T,ψ¯ =[ψ¯1,ψ¯2,…,ψ¯m]T,χ¯ =[χ¯1,χ¯2,…,χ¯m]T,ϕ^ =diag(ϕ^1,ϕ^2,…,ϕ^m),φ^ =diag(φ^1,φ^2,…,φ^m),ψ^ =diag(ψ^1,ψ^2,…,ψ^m),χ^ =diag(χ^1,χ^2,…,χ^m). (3.48) Proof. In (3.3), the incremental change of $$S(k)$$ can be written as ΔS(k)=ϕ(k)+φ(k)+χ(k)+ψ(k)+GB(u(k)−Kx(k)). (3.49) As in Niu et al. (2010), we propose the following reaching law for the uncertain descriptor system described by (2.1). ΔS(k)=Γ(k)−Γ¯−Γ^sgn(S(k)), (3.50) where $${\it\Gamma}(k)=\left({\phi(k)+\varphi(k)+\chi(k)+\psi(k)}\right)$$, $$\bar {\it\Gamma}=\left({\bar \phi+\bar \varphi+\bar \chi+\bar \psi}\right)$$ and $$\hat{\it\Gamma}=\left({\hat\phi +\hat\varphi+\hat\chi +\hat\psi }\right)$$. Solving (3.49) for $$ u(k) $$, using (3.50), the discrete-time SM control law (3.47) is easily designed. Further, it is observed that Γ(k)−Γ¯−Γ^sgn(S(k))<0,S(k)>0,Γ(k)−Γ¯−Γ^sgn(S(k))>0,S(k)<0. (3.51) From (3.49), we have S(k)ΔS(k)=S(k)(Γ(k)−Γ¯)−Γ^|S(k)|<0 (3.52) which yields S(k+1)<S(k)if S(k)>0,S(k+1)>S(k)if S(k)<0. (3.53) Furthermore, if $$ S(k) >2\hat{\it\Gamma} $$, we have S(k+1) =S(k)+(Γ(k)−Γ¯)−Γ^>2Γ^+(Γ(k)−Γ¯)−Γ^>0 (3.54) which implies, according to (3.54), that $$0<S(k+1)<S(k) $$. Similarly, if $$ S(k) <-2\hat{\it\Gamma} $$, we can show that $$S(k+1)<S(k) <0$$. Besides, when $$ 0 \leq S(k) \leq 2\hat{\it\Gamma} $$, one has −2Γ^≤S(k+1)=S(k)+(Γ(k)−Γ¯)−Γ^≤2Γ^ (3.55) and, when $$ -2\hat{\it\Gamma} \leq S(k) \leq 0 $$, one obtains −2Γ^≤S(k+1)=S(k)+(Γ(k)−Γ¯)+Γ^≤2Γ^ (3.56) thus, it follows from (3.55) and (3.56) that $$ |S(k + 1)|\leq 2 \hat{\it\Gamma} $$ as $$ |S(k)|\leq 2 \hat{\it\Gamma} $$. As a conclusion, the reaching conditions provided in Niu et al. (2010) are satisfied by the reaching law (3.50). □ Remark 3.1 Due to the discontinuity of the signum function $$ {\rm sgn}() $$, chattering may occur in the control input. The boundary layer can be adopted to alleviate the chattering phenomenon. It consists to replace the signum function in (3.47) with the saturation function as sat(S(k),Δ)={sgn(S(k))if |S(k)|>ΔS(k)Δif |S(k)|≤Δ, (3.57) where the boundary layer thickness $${\it\Delta}$$ ensures that $$ S(k) $$ is bounded by $$\pm {\it\Delta}$$. The discrete time sliding law becomes u(k) =Kx(k)−(GB)−1{(ϕ¯+φ¯+χ¯+ψ¯)+(ϕ^+φ^+χ^+ψ^)sat(S(k),Δ)}. (3.58) Remark 3.2 Based on the following sliding surface S(k)=GEx(k)−G(A+BK)x(k−1) (3.59) a discrete-time SMC law is designed in Liu et al. (2013): u(k) =K(x(k)−x(k−1))−(GB)−1{GAx(k−1)−GEx(k)+αS(k)+βe−μksgn(S(k)) +(ϕ¯+φ¯+χ¯+ψ¯)+(ϕ^+φ^+χ^+ψ^)sgn(S(k))}, (3.60) where term $$e^{-\mu k}$$ is added as a convergent factor to reduce the chattering. 4. Numerical examples In the sequel, we demonstrate the applicability of the suggested discrete-time SMC strategy by means of two simulation examples. Example 4.1 Consider the uncertain time-delay descriptor system (2.1) with parameters as E =[100000011],A =[1.51100.5−1.600.31.5],Ad =[0.20.20000.100.1−0.1],B =[100],Bw =[0.10.51],C =[0.10.10],Cd =[0.10.10],Dw =0.1,M =[100],N =[0.100.1],Nd =[0.1−0.10.1]. Choose $G=\begin{bmatrix} 1& 1 &0.1 \end{bmatrix}$, $$\mathcal{Q}=-1$$, $$\mathcal{S}=0$$, $$\mathcal{R}=1$$, $$\gamma=0$$, $$d_m=1$$ and $$d_M=3$$. With $$\lambda_1=5$$, $$\lambda_2=0$$ and $$\lambda_3=9$$, Theorem 3.2 produces a feasible solution to the corresponding LMIs with the following parameters: F =0.082849,Y =[−0.077347−0.0955010.12099]. The associate controller gain is K=[−0.93359−1.15271.4604]. (4.1) It is assumed that the non-linear function input is $$f(x(k))=2x_3^2(k)+0.25x_2(k)\cos(0.25x_2^2(k))$$. It is further assumed that the disturbance acting on the system is equal to $$\displaystyle{w(k)}=0.5e^{-3k}\sin(k)$$. From Theorem 3.3, the robust discrete-time SMC can be designed, where $$\bar \phi,\ \bar \varphi,\ \bar \chi,\ \bar \psi,\ \hat\phi ,\ \hat\varphi,\ \hat\chi$$ and $$\hat\psi $$ are calculated at each step of the simulation from (3.46) using the following bounded functions: ϕL=−ϕU=−‖GM‖‖Nx(k)‖,φL=−φU,i=−1.05(‖GAd‖+‖GM‖‖Nd‖)‖x(k)‖,χL=−χU=−0.1,ψL=−ψU=−‖GBf(x(k))‖. (4.2) As in in Liu et al. (2013), robust discrete time SMC (3.60) is designed by selecting $$\alpha=0.02$$, $$\beta=0.1$$ and $$\mu=500$$. The sampling period and initial states in this example are chosen as follows: $$T_e=0.1$$ s and $$\phi_0(k)=[0.5\ -0.3\ -1.5]^T$$, $$k=-3,\ldots,0$$. Figure 1 depicts the numerical simulation where control laws (3.58) and (3.60) are adopted. Figure 1a, b show the evolutions of state variables upon applying both control laws, while Fig. 1c shows values of the control signals. Fig. 1. View largeDownload slide Simulation results for Example 4.1. (a) State trajectories with controller (3.58). (b) State trajectories with controller (3.60). (c) Input trajectories. Fig. 1. View largeDownload slide Simulation results for Example 4.1. (a) State trajectories with controller (3.58). (b) State trajectories with controller (3.60). (c) Input trajectories. It is observed from these figures that control law (3.58), proposed in this paper, guarantees the convergence of the system state to zero and requires a less control effort in the sliding mode than control law (3.60). Based on Fig. 1c, we can calculate $$(\Vert{u}\Vert_\infty)_{(3.58)}=1.0686<(\Vert{u}\Vert_\infty)_{(3.60)}= 2.6593$$ and $$(\Vert{u}\Vert_2)_{(3.58)}=1.5694<(\Vert{u}\Vert_2)_{(3.60)}= 5.3240$$. Note from the preceding results that the proposed approach is not only robust, in spite of the time-varying delay and uncertainties, but also able to alleviate chatter without creating an extra control burden. Example 4.2 Consider a discrete-time delayed descriptor system with the following parameters: E =[1000],A =[011.20],Ad =[0.10.40.10],B =[11.5],Bw =[0.10],C =[0.1−0.1],Cd =[0.10.1],Dw =0.1,M =[11.5],N =[00.5],Nd =[0.5−0.5]. Because the $$(2,2)$$th entry of $$A$$ is $$0$$, it follows that the matrix pair $$(E,A)$$ must not be causal, and hence the unforced part of the considered system is not admissible for all delay $$d(k)$$. Set $$\mathcal{Q}=-0.1$$, $$\mathcal{S}=0.1$$, $$\mathcal{R}=0.5$$ and $$\gamma=0.1$$, $$d_m=2$$ and $$d_M=5$$. According to Theorem 3.2, choosing $$G=[-0.5\ 1]$$, $$\lambda_1=0.04$$, $$\lambda_2=0.05$$$$\lambda_3=0.2$$ and solving LMI (3.37), the desired variables are solved as follows: P =[21.3990004.3550004.355],F =0.76122,Y =[−3.54743.0248]. The associate controller gain is K=[−4.66023.9736]. (4.3) For simulation purpose, the non-linear input and the exogenous disturbance are supposed to be, respectively, $$f(k,x(k))=0.25\sin(2x_1(k))+0.5x_1^3(k)$$ and $$\displaystyle{w(k)}=\frac{0.5 k}{5k^2+1}$$. Based on (3.58), the robust discrete time SMC is then designed as follows: u(k) =[−4.6602 3.9736]x(k)−{(ϕ¯+φ¯+χ¯+ψ¯)+(ϕ^+φ^+χ^+ψ^)sat(S(k),0.07)}, (4.4) where $$\bar \phi,\ \bar \varphi,\ \bar \chi,\ \bar \psi,\ \hat\phi ,\ \hat\varphi,\ \hat\chi$$ and $$\hat\psi $$ can be calculated at each step of the simulation from (3.46) using the following bounded functions: ϕL=−ϕU=−‖GM‖‖Nx(k)‖,φL=−φU,i=−1.1(‖GAd‖+‖GM‖‖Nd‖)‖x(k)‖,χL=−χU=−0.3,ψL=−ψU=−‖GBf(x(k))‖. (4.5) Let $$T_e=0.05$$ s and $$F(k)=0.5\sin(k)$$. The simulation results are shown in Fig. 2. For initial condition $$\phi_0(k)=[-1.5\ 0.5]^T$$, $$k=-5,\ldots,0$$, Fig. 2a–c display the time responses of the state, sliding function, control input, respectively. Moreover, Fig. 2d shows a repeating of sequence $$[5,3,4,4,2,2,4]$$ representing time-varying delay $$d(k)$$. It is concluded that the proposed method is effective where all state trajectories of the system converge asymptotically to the origin, the sliding variable is driven to converge to zero and the control signal is chattering free. Fig. 2. View largeDownload slide Simulation results for example. (a) The trajectory of state $${x(k)}$$. (b) The trajectory of sliding variable $${s(k)}$$. (c) Control signal $${u(k)}$$. (d) Time-varying delay $$d(k)$$. Fig. 2. View largeDownload slide Simulation results for example. (a) The trajectory of state $${x(k)}$$. (b) The trajectory of sliding variable $${s(k)}$$. (c) Control signal $${u(k)}$$. (d) Time-varying delay $$d(k)$$. To illustrate the effect of the parameter uncertainties, we assume that $$f(k,x(k))=0$$ and at $$k\geq 20$$ the model parameters abruptly change with uncertainty function $$F(k)=2.2\sin(0.1k)$$. Figure 3a, b plot the state evolutions when control law (4.4) is applied to the system with and without sliding mode term, respectively. It is clear that with only linear controller, the stability and the performance of system is degraded. We can conclude that the proposed discrete-time (SMC) law yields a good performance and stabilizes the non-linear system with time varying delay and unknown parameters uncertainties. Fig. 3. View largeDownload slide System responses with/without sliding mode term. (a) State trajectories with sliding mode term. (b) State trajectories without sliding mode term. Fig. 3. View largeDownload slide System responses with/without sliding mode term. (a) State trajectories with sliding mode term. (b) State trajectories without sliding mode term. To emphasize the influence of the choice of the sampling period, Fig. 4 presents the state trajectories for sampling period $$T_e=0.15$$ s. We remark that for a worse choice of the sampling period, the chattering phenomenon is clear. Fig. 4. View largeDownload slide System state signals for sampling period $$T_e=0.15$$ s. Fig. 4. View largeDownload slide System state signals for sampling period $$T_e=0.15$$ s. 5. Conclusion In this paper, the problem of SMC for uncertain discrete-time descriptor systems with time-varying delays has been investigated. Based on a new discrete-time sliding function, a delay-dependent sufficient condition, has been derived to guarantee the admissibility and $$(\mathcal{Q},\mathcal{S},\mathcal{R})$$-$$\gamma$$-dissipativity for the sliding mode dynamics. A sliding-mode controller is designed such that the reaching condition is satisfied and the chattering can be reduced. The ensuing results have been verified by two numerical examples. 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