# Discrete-Time Sliding-Mode Control of Uncertain Systems with Time-Varying Delays via Descriptor Approach

Authors: Maode Yan; Aryan Saadat Mehr; Yang Shi
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

This paper considers the problem of robust discrete-time sliding-mode control (DT-SMC)
design for a class of uncertain linear systems with time-varying delays. By applying a descriptor
model transformation and Moon's inequality for bounding cross terms, a delay-dependent
sufficient condition for the existence of stable sliding surface is given in terms of linear matrix
inequalities (LMIs). Based on this existence condition, the synthesized sliding mode controller
can guarantee the sliding-mode reaching condition of the specified discrete-time sliding surface
for all admissible uncertainties and time-varying delays. An illustrative example verifies the
effectiveness of the proposed method.


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## Body

## 1. Introduction

Time delay
often appears in many engineering systems, such as networked control systems,
telecommunication systems, nuclear reactors, chemical engineering systems, and
so on. Since time delay is often an important source of poor
performance and instability, considerable attention has been dedicated to
stability analysis and control synthesis for time-delay systems. Many research
results have been reported and well documented in [1, 2], and the references therein.
A linear matrix inequalities (LMIs-) based approach to delay-dependent
stability for continuous-time uncertain systems with time-varying delays is
proposed by using the Leibniz-Newton formula in [3]. In [4], a new technique is studied
by incorporating both the time-varying-delayed state and the
delay-upper-bounded state to make full use of all available information in the
design. It is worth noting that the wide use of digital computers in practical
control systems requires the stability analysis and controller synthesis for
discrete-time time-delay systems. In [5],
delay-dependent sufficient conditions for the existence of the guaranteed cost
controller in terms of LMIs are derived for uncertain discrete-time systems
with constant delays. A necessary and sufficient condition is established for
stabilizing a networked control system by modeling both the
sensor-to-controller and controller-to-actuator delays as Markov chains in
[6]. In [7], Gao
et al. propose an output-feedback
stabilization method for discrete-time systems with time-varying delays. This
method is dependent on the lower and upper delay bounds, and is further
extended to discrete-time systems with norm-bounded mismatched uncertainties.
Further, a less conservative result on stability of discrete-time systems with
time-varying delays, which also incorporates both the time-varying-delayed
state and the delay-upper-bounded state, is obtained by defining new Lyapunov
functions in [8].It is well known that the sliding-mode control (SMC)
is an effective method to achieve robustness and invariance to matched
uncertainties and disturbances on the sliding surface [9]. In recent years, increasing
attention has been paid to the design of SMC for uncertain systems with time
delay. One approach based on lumped sliding surface has been applied to design
an SMC law for state-delay systems in [10]. In [11], the SMC method is proposed for systems with
mismatched parametric uncertainties and time delay, and a delay-independent
sufficient condition for the existence of linear sliding surface is derived in
terms of LMIs; the synthesized sliding-mode controller can guarantee the reaching
condition of the specified sliding surface. In [12], an observer-based sliding
mode control is studied for continuous-time state-delayed systems with
unmeasurable states and nonlinear uncertainties, and a sufficient condition of
asymptotic stability is derived for the overall closed-loop systems.Recently, a new descriptor model transformation is
introduced for stability analysis [13], controller design [5, 14], and filtering design
[15] of time-delay
systems. This approach can significantly reduce the overdesign entailed in the
existing methods because of the model equivalence and less conservative bound
on cross terms [14].
In [16], a
Lyapunov-Krasovskii techniques-based delay-dependent descriptor approach to
stability and control of linear systems with time-varying delays is proposed,
and is further combined with the sliding mode control result to deal with the
mismatched uncertainties and unknown nonlinear functions. However, to the best
of authors' knowledge, the descriptor approach has not been fully investigated
for research on delay-dependent DT-SMC design of linear systems with mismatched
uncertainties and time-varying delays. This motivates our research work in this
paper.In this paper, we first propose a delay-dependent
sufficient condition for the existence of stable sliding surface via the
descriptor approach. Then, based on this existence condition, the synthesized
discrete-time sliding-mode controller is designed to guarantee the sliding mode
reaching condition of the specified discrete-time sliding surface. The rest of
the paper is organized as follows. Section2 describes the problem formulation
and some necessary preliminary results. In Section 3, sufficient conditions for
the existence of stable sliding surface are presented in terms of LMIs and a
robust DT-SMC law is presented. This DT-SMC law can guarantee the sliding-mode
reaching condition of the specified discrete-time sliding surface for all
admissible uncertainties and time-varying delays. An illustrative example is
used to demonstrate the validity and effectiveness of the proposed method in
Section 4. Finally, Section 5 offers some concluding remarks.Throughout this paper, superscript  “T”  stands for matrix transposition, ℝn denotes the n-dimensional Euclidean space, ℝn×m is the set of all real matrices of dimension n×m,P>0  means that P is real symmetric and positive-definite
matrix, I and 0 represent identity matrix and zero matrix with
appropriate dimensions. In symmetric block matrices or long matrix expressions,
we use an asterisk (∗) to represent a term that is induced by
symmetry. diag{⋅} stands for a block-diagonal matrix. Matrices,
if their dimensions are not explicitly stated, are assumed to be compatible for
algebraic operations.


## 2. Problem Formulation

Consider the
following discrete-time uncertain system with time-varying delays in the
regular form:(1)x1(k+1)=(A11+ΔA11)x1(k)+(Ad11+ΔAd11)x1(k−d(k))+(A12+ΔA12)x2(k),x2(k+1)=∑i=12[A2ixi(k)+Ad2ixi(k−d(k))]+Bu(k)+f(k,x(k)),x(k)=φ(k),k=−dM,−dM+1,…,0,where x1∈ℝn−m, x2∈ℝm, x(k)=(x1,x2)T is the system state vector, u(k)∈ℝm is the control input, f(k,x(k))∈ℝm is an unknown nonlinear function representing
the unmodeled dynamics and external disturbances, d(k) is the unknown time-varying delays, A11,Ad11,A12,A21,A22,Ad21,Ad22,
and B represents real constant system matrices with
appropriate dimensions, ΔA11,ΔAd11,
and ΔA12 are time-varying matrix functions representing
parameter uncertainties, φ(k)∈ℝn represents initial values of x(k).
It is assumed that the system uncertainties are norm-bounded with the following
form:(2)[ΔA11ΔA12]=E1F1(k)[H1H3],ΔAd11=E2F2(k)H2,where E1, E2,
and Hi(i=1,2,3) are known constant matrices with compatible
dimensions, and the properly dimensioned matrix Fi(k) is an unknown and time-varying matrix of
uncertain parameters, but norm bounded as FiT(k)Fi(k)≤I,i=1,2.
Associated with system (2), we make the following assumptions.Assumption 1.
The time-varying delaysd(k) is assumed to satisfy dm≤d(k)≤dM,
where dm and dM are constant positive scalars representing the
lower and upper bounds on the time delay, respectively.Assumption 2.
The pair(A11,A12) in the nominal system of (2) is controllable.Assumption 3.
Nonlinear functionf(k,x(k)) is unknown but bounded in the sense of the
Euclidean norm.The design procedure of DT-SMC consists of two steps:
design of the stable sliding surface and the reaching motion control law. For
discrete-time uncertain systems with time-varying delays, the linear sliding
surface is chosen as(3)S(k)=C¯x(k)=[CI]x(k)=Cx1(k)+x2(k)=0with C¯∈ℝm×n and C∈ℝm×(n−m) is a real matrix to be designed. Once sliding
surface satisfies the discrete-time sliding mode reaching condition,
substituting x2(k)=−Cx1(k) into the first subsystem of (2), we obtain
the following sliding motion:(4)x1(k+1)=(A11−A12C+ΔA11−ΔA12C)x1(k)+(Ad11+ΔAd11)x1(k−d(k)),x1(k)=φ1(k),k=−dM,−dM+1,…,0.Here, φ1(k)∈ℝn−m stands for initial values of x1(k).
In order to derive our main results, the following two lemmas are necessary.Lemma 1 (See [17], Moon's Inequality).
Assume thata∈ℝna, b∈ℝnb,
and N∈ℝna×nb.
Then for any matrices X∈ℝna×na, Y∈ℝna×nb,
and Z∈ℝnb×nb,
the following holds:(5)−2aTNb≤[ab]T[XY−NYT−NTZ][ab],where(6)[XYYZ]≥0.Lemma 2 (See [18]).
For any matricesD∈ℝn×p, E∈ℝp×n, F∈ℝp×p with FT(k)F(k)≤I and scalars ε>0,(7)DFE+ETFTDT≤ε−1DDT+εETE.The main objective of this paper is to design the
sliding surfaceS(k) and a DT-SMC law u(k) such that(1)
the sliding surface is asymptotically stable;(2)
the DT-SMC law can guarantee discrete-time sliding
mode reaching condition.


## 3. Main Results

In this section, the results of slidng surface design
and robust DT-SMC law will be presented for a class of uncertain linear systems
with time-varying delays. Let us first consider the problem of sliding surface
design. In order to reduce the overdesign entailed in the stability analysis
methods for time-delay systems, we employ the descriptor approach to derive the
delay-dependent sufficient conditions for the existence of the sliding surface
in terms of LMIs. The first result on the asymptotic stability of designing
sliding surface is presented in the following theorem.Theorem 1.
If, for certain prescribed positive numberκ,
there exist positive scalars δ1>0,δ2>0,(n−m)×(n−m),  matrices L1>0,L2,L3,U1>0,U2>0,W¯1,W¯2,W¯3,
and m×(n−m) matrix K such that the following LMIs
hold:(8)[Θ1Θ200L1ϑL2TL2T∗Θ3ς000L3TL3T∗∗−U1U1H2T0000∗∗∗−δ2I0000∗∗∗∗χ000∗∗∗∗∗−δ1I00∗∗∗∗∗∗−L10∗∗∗∗∗∗∗−dM−1U2]<0,(9)[W¯1W¯20∗W¯3κAdU1∗∗U1]≥0,where ϑ denotes L1H1T−KTH3T,ς   denotes (1−κ)AdU1, and  χ  denotes −(dM−dm+1)−1U1,(10)Θ1=L2+L2T+dMW¯1,Θ2=L3+L1(AT+κAdT−I)−KTA12T−L2T+dMW¯2,Θ3=−L3−L3T+dMW¯3+∑i=12δiEiEiT.Then, the system (4) with
Assumption 1 is asymptotically stable, and the sliding surface of (3) is given
by(11)S(k)=KL1−1x1(k)+x2(k)=0.Proof.
Letη(k)=x1(k+1)−x1(k),
the system (4) can be rewritten in an equivalent descriptor
form:(12)x1(k+1)=x1(k)+η(k),0=(A¯11(k)+A¯d11(k)−I)x1(k)−η(k)−A¯d11(k)∑i=k−d(k)k−1η(i),where(13)A¯11(k)=A11−A12C+ΔA11−ΔA12C,A¯d11(k)=Ad11+ΔAd11.Choose the Lyapunov-Krasovskii
functional candidate as(14)V(k)=V1(k)+V2(k)+V3(k)+V4(k).Here,(15)V1(k)=x1T(k)P1x1(k),V2(k)=∑i=k−d(k)k−1x1T(i)Qx1(i),V3(k)=∑j=−dM+2−dm+1∑i=k+j−1k−1x1T(i)Qx1(i),V4(k)=∑j=dM−1∑i=k+jk−1ηT(i)Gη(i),where P1,Q,G∈ℝ(n−m)×(n−m) are symmetric positive definite matrices to be
determined. The Lyapunov-Krasovskii functional candidate V(k) is positive definite for all x1(k)≠0.
Now, in order to evaluate the forward difference ΔV(k)=V(k+1)−V(k),
in what follows we calculate ΔV1,ΔV2,ΔV3,
and ΔV4,
respectively. For ΔV1(k),
by using the descriptor system form (12), we obtain(16)ΔV1(k)=ηT(k)P1η(k)+2x1T(k)P1η(k)=ηT(k)P1η(k)+2x¯1T(k)PT[η(k)0]=ηT(k)P1η(k)+2x¯1(k)PT[η(k)(A¯11(k)+A¯d11(k)−I)x1(k)−η(k)]−2x¯1T(k)PT∑i=k−d(k)k−1Kn−mA¯d11(k)η(i),where x¯1T(k)=[x1T(k)ηT(k)],Kn−m=[0I]T, and P=[P10P2P3].
If identifying N:=PTKn−mA¯d11(k), a:=x¯1(k),
and b:=η(i) in (16), then applying Lemma
1 gives rise to
the following inequality:(17)ΔV1(k)≤ηT(k)P1η(k)+2x¯1T(k)PT×[η(k)(A¯11(k)+A¯d11(k)−I)x1(k)−η(k)]+∑i=k−d(k)k−1[x¯1(k)η(i)]T[WM−PTKn−mA¯d11(k)∗G][x¯1(k)η(i)]≤ηT(k)P1η(k)+2x¯1T(k)PT[η(k)(A¯11(k)+A¯d11(k)−I)x1(k)−η(k)]+dMx¯1T(k)Wx¯1(k)+∑i=k−dMk−1ηT(i)Gη(i)+2x¯1T(k)[M−PTKn−mA¯d11(k)][x1(k)−x1(k−d(k))],where W, M∈ℝ(n−m)×(n−m) are constant matrices with appropriate
dimensions satisfying(18)[WM∗G]≥0.For ΔV2,
we have(19)ΔV2=∑i=k−d(k+1)+1kx1T(i)Qx1(i)−∑i=k−d(k)k−1x1T(i)Qx1(i)=x1T(k)Qx1(k)−x1T(k−d(k))Qx1(k−d(k))+∑i=k−d(k+1)+1k−1x1T(i)Qx1(i)−∑i=k−d(k)+1k−1x1T(i)Qx1(i)≤x1T(k)Qx1(k)−x1T(k−d(k))Qx1(k−d(k))+∑i=k−dM+1k−dmx1T(i)Qx1(i).For ΔV3,
we get(20)ΔV3=∑j=−dM+2−dm+1[x1T(k)Qx1(k)−x1T(k+j−1)Qx1(k+j−1)]=(dM−dm)x1T(k)Qx1(k)−∑j=k−dM+1k−dmx1T(j)Qx1(j).For ΔV4,
we obtain(21)ΔV4=∑j=−dM−1[ηT(k)Gη(k)−ηT(k+j)Gη(k+j)]=dMηT(k)Gη(k)−∑j=k−dMk−1ηT(j)Gη(j).Based on the above results in
(17),  (19)–(21), and using
Lemma 2, we
can derive(22)ΔV(k)=ΔV1(k)+ΔV2(k)+ΔV3(k)+ΔV4(k)≤x¯1T(k)Ψx¯1(k)+2x¯1T(k)×(PTKn−mAd11−M)x1(k−d(k))+x1T(k−d(k))(δ2−1H2TH2−Q)x1(k−d(k))=[x1T(k)ηT(k)x1T(k−d(k))]×[ΨPT[0Ad11]−M∗δ2−1H2TH2−Q][x1T(k)ηT(k)x1T(k−d(k))],where(23)Ψ=PT[0IA11−A12C−I−I]+[0A11T−CTA12T−II−I]P+dMW+[M0]+[MT0]+[ω00P1+dMG]+∑i=12δiPT[0Ei][0EiT]Pwhere ω denotes δ1−1(H1T−CTH3T)(H1−H3C)+(dM−dm+1)Q  and δ1, δ2 are any positive scalars. Let ξ=[x1T(k)ηT(k)x1T(k−d(k))] and(24)Φ=[ΨPT[0Ad11]−M∗δ2−1H2TH2−Q].Then(25)ΔV(k)≤ξT(k)Φξ(k).In order to obtain a convenient
LMI, we restrict the choice of M to be(26)M=κPT[0Ad11(k)].Define(27)P−1=L:=[L10L2L3],K=CL1,Q−1=U1,G−1=U2,W¯=(P−1)TW(P−1)=[W¯1W¯2W¯2TW¯3].Multiply (18) by diag[(P−1)T,U2] and diag[(P−1),U2] on the left- and right-hand sides,
respectively. Alternatively, multiply (24) by diag[(P−1)T,U1] and diag[(P−1),U1] on the left- and right-hand sides, respectively.
By using the Schur complement [19], we obtain that (8) is equivalent to Φ<0,
and (18) is equivalent to (9), which yields(28)ΔV(k)<0,∀ξ(k)≠0.It follows from Lyapunov
stability theory [20]
that the system (4) is asymptotically stable. Moreover, based on the above
definition of the sliding surface in (3), it can be obtained with C=KL1−1.Once the sliding surface is appropriately designed
according to Theorem1, the next result on the design of DT-SMC is summarized
in the following theorem.Theorem 2.
Consider the system (2) with Assumptions
1–3. If the
linear sliding surface is designed as in (11), then the following DT-SMC
law(29)u(k)=−B−1{∑i=12[(A2i+CA1i)xi(k)]−(1−qτ)S(k)+ϵτ⋅sgn(S(k))+(α0+β0+γ0)+(ρα+ρβ+ργ)⋅sgn(S(k))}can guarantee the discrete-time
sliding mode reaching condition. Here,(30)S(k)=[S1(k)S2(k)⋯Sm(k)],sgn(S(k))=[sgn(S1)sgn(S2)⋯sgn(Sm)],α0=αU+αL2,β0=βU+βL2,γ0=γU+γL2,ρα=αU−αL2,ρβ=βU−βL2,ργ=γU−γL2,C=KL1−1, αU and αL are the upper bound and lower bound of α˜(k)=CΔA11x1(k)+CΔA12x2(k), βU and βL are the upper bound and lower bound of β˜(k)=CΔAd11x1(k−d(k))+CAd11x1(k−d(k))+∑i=12[Ad2ixi(k−d(k))], γU and γL are the upper bound and lower bound of γ˜(k)=f(k,xk), q>0, ε>0, 1−qτ>0,
and τ>0 is the sampling period.Proof.
The design of DT-SMC law guarantees that the reaching
condition must be satisfied when there exist uncertainties and time-varying
delays in systems. We consider the reaching condition for the DT-SMC proposed
in [21] as
follows:(31)ΔS(k)=S(k+1)−S(k)≤−ϵτ⋅sgn(S(k))−qτS(k),ifS(k)>0,ΔS(k)=S(k+1)−S(k)≥−ϵτ⋅sgn(S(k))−qτS(k),ifS(k)<0.From the designed sliding surface(32)S(k)=C¯x(k)=[CI]x(k),we have(33)ΔS(k)=(CA11+CΔA11)x1(k)+(CAd11+CΔAd11)x1(k−d(k))+(CA12+CΔA12)x2(k)+∑i=12[A2ixi(k)+Ad2ixi(k−d(k))]+Bu(k)+f(k,xk).Since it is assumed that ΔA11, ΔAd11, ΔA12,
and f(k,xk) are bounded, using the discrete-time version
of improved Razumikhin theorem in [22], for any solution x(k−d(k)) of (2), there exists a constant θ such that the following inequality is
satisfied:(34)∥x(k−d(k))∥<θ∥x(k)∥,∀k,dm≤d(k)≤dM.We can also obtain that α˜(k)=CΔA11x1(k)+CΔA12x2(k), β˜(k)=C(Ad11+ΔAd11)x1(k−d(k))+∑i=12[Ad2ixi(k−d(k))],
and γ˜(k)=f(k,xk) will be bounded with upper and lower bounds.
Let the bounds be(35)αL≤α˜(k)=CΔA11x1(k)+CΔA12x2(k)≤αU,βL≤β˜(k)=C(Ad11+ΔAd11)x1(k−d(k))+∑i=12[Ad2ixi(k−d(k))]≤βU,γL≤γ˜=f(k,xk)≤γU.Here, the
inequality(36)αL=[αL,1αL,2⋯αL,m]T≤α˜(k)=[α˜1α˜2⋯α˜m]T≤αU=[αU,1αU,2⋯αU,m]Timplies that αL,i≤α˜i(k)≤αU,i,i=1,2,…,m;
similar notations are used for β˜(k) and γ˜(k).
Then, substituting (29) into (33), we
have(37)ΔS(k)=α˜(k)+β˜(k)+γ˜(k)−qτS(k)−ϵτ⋅sgn(S(k))−(α0+β0+γ0)−(ρα+ρβ+ργ)⋅sgn(S(k)).According to previous
discussion, the following relations hold:(38)α˜(k)≤α0+ρα⋅sgn(S(k))ifS(k)>0,α˜(k)≥α0+ρα⋅sgn(S(k))ifS(k)<0,β˜(k)≤β0+ρβ⋅sgn(S(k))ifS(k)>0,β˜(k)≥β0+ρβ⋅sgn(S(k))ifS(k)<0,γ˜(k)≤γ0+ργ⋅sgn(S(k))ifS(k)>0,γ˜(k)≥γ0+ργ⋅sgn(S(k))ifS(k)<0.Thus, the sign change of ΔS(k) in (37) is opposite to that of S(k),
irrespective of the value of the uncertainties α˜(k), β˜(k),
and γ˜(k).
Moreover, the closed-loop control system (2) satisfies the reaching condition
(31).Remark 1.
The descriptor system approach [13, 14, 16] can greatly reduce the overdesign entailed in the
existing methods because the descriptor model leads to a system that is
equivalent to the original one (from the point of view of stability) and
requires bounding of fewer cross-terms. Therefore, in this work, we apply the
descriptor system approach to the SMC design for discrete-time systems with
uncertainties and time-varying delays.Remark 2.
SMC approach is an effective method to achieve the
robustness and invariant property to matched uncertainties and disturbance on
the sliding surface [9, 21]. In this paper, the results of
Theorem 1 guarantee
that the sliding motion dynamics will be robustly asymptotically stable.
Furthermore, the descriptor-system-approach-based DT-SMC in Theorem 2 provides
a solution to satisfy the sliding mode reaching condition of the designed
discrete-time sliding surface for discrete-time systems with uncertainties and
time-varying delays.


## 4. Numerical Example

In this section, an illustrative example is given to
verify the design method proposed in this paper. Consider the discrete-time
system (2) with the following description:(39)x1(k)=[x11(k)x12(k)],A11=[0.700.050.8],Ad11=[−0.200−0.1],A12=[0.30],A21=[−0.50],A22=2.0,Ad21=[00.1],Ad22=0.05,B=1,ΔA11=[0.02sin(0.01kπ)0.02sin(0.01kπ)0.01cos(0.01kπ)0.04sin(0.01kπ)],ΔA12=[0.050.04],ΔAd11=[0.01sin(0.01kπ)0.04sin(0.01kπ)0.01cos(0.01kπ)0.05sin(0.01kπ)],f(k,x(k))=0.4sin(x11(k)).Initial states of the system are x1(k)=[1.0−0.5]T,x2(k)=0.6,
for k∈[−dM0].
Using Theorem 1 and choosing κ=0.5,
we obtain C=[0.93050.5168].
Then, the linear sliding surface is S(k)=[C1]x(k).
According to Theorem 2, the robust DT-SMC law is designed as(40)u(k)=−B−1{∑i=12[(A2i+CA1i)xi(k)]−(1−qτ)S(k)+(ϵτ+ρα+ρβ+ργ)⋅sgn(S(k))},where ρα=0.0238|x11(k)|+0.0393|x12(k)|+0.0672|x2(k)|, ρβ=0.5014|x11(k)|+0.2785|x12(k)|+0.125|x2(k)|, ργ=0.4|sinx11(k)|, τ=0.1, q=5,
and ϵ=0.1.
The time-varying delay 1≤d(k)≤3 is shown in Figure 1. System state
trajectories are illustrated in Figure 2. The resulting sliding surface is in
Figure 3. Figure 4 depicts the control input signal.Figure 1
Time-varying delaysd(k).Figure 2
System statesx11,x12,andx2.Figure 3
Sliding surfaceS(k).Figure 4
Control inputu(k).It is observed from Figure2 that the state
trajectories of the system all converge to the origin quickly. The system can
be stabilized quickly by the proposed method and the reaching motion satisfies
the sliding reaching condition in spite of the time-varying delays and
uncertainties. Simulation results illustrate that the proposed approach in this
paper is feasible and effective for discrete-time uncertain linear systems with
time-varying delays.


## 5. Conclusion

In this paper, the problem of robust DT-SMC for
uncertain systems with time-varying delays has been studied. By using the
descriptor model transformation and a recent result on bounding cross products
of vectors, a delay-dependent sufficient condition for the existence of stable
sliding surfaces is constructed for all admissible uncertainties. Based on this
existence condition, the corresponding reaching motion controller is developed
such that the reaching motion satisfies the discrete-time sliding mode reaching
condition for uncertain systems with time-varying delays. An example shows the
validity and effectiveness of the proposed DT-SMC design method.



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*Source: 489124-2008-07-22.xml*