Automatica 44 (2008) 3157–3161

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Automatica

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Brief paper

Input-to-state stabilization for nonlinear dual-rate sampled-data systems viaapproximate discrete-time modelI

Xi Liu a, Horacio J. Marquez b,∗, Yanping Lin a,ca Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1b Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4c Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Hom, Hong Kong, China

a r t i c l e i n f o

Article history:Received 26 June 2006Received in revised form5 May 2008Accepted 12 May 2008Available online 6 November 2008

Keywords:Input-to-state stabilityNonlinearDual-rateDiscrete-timeSampled-data

a b s t r a c t

The problem of state feedback stabilization of nonlinear sampled-data systems is considered under the‘‘low measurement rate’’ constraint. A dual-rate control scheme is proposed that utilizes a numericalintegration scheme to approximately predict the current state. Given an approximate discrete-timemodel of a sampled nonlinear plant and given a family of controllers that stabilizes the plant model ininput-to-state sense, we show that under some standard assumptions the closed loop dual-rate sampleddata system is input-to-state stable in the semiglobal practical sense.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The prevalence of digital controllers and the fact that mostsystemsof interest in control systems are oftennonlinear,motivatethe area of nonlinear sampled-data control systems. Significantprogress has been made in recent years (Chen & Francis, 1991; Li,Shah, & Chen, 2002; Nesic, Teel, & Kokotovic, 1999; Nesic & Teel,2001; Nesic & Laila, 2002; Nesic & Teel, 2004, 2006; Polushin &Marquez, 2004). There are two main approaches for the designof digital controllers (see Nesic and Teel (2001)): continuous-time design (CTD) and discrete-time design (DTD). The first oneinvolves digital implementation of a continuous-time stabilizingcontrol law. The second approach consists of discretizing theplant model and then designing a discrete-time controller. Bothof these approaches are essentially single-rate, i.e. sampling ratesof the input and the measurement channels are assumed to beequal. For single-rate sampled-data systems, Nesic and Laila (2002)investigated input-to-state (ISS) property proposed by Sontag

I This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Dragan Nešićunder the direction of Editor Hassan K. Khalil. This work was supported by theNatural Sciences and Engineering Research Council of Canada (NSERC).∗ Corresponding author. Tel. +1 780 492 3333; fax: +1 780 492 8506.E-mail addresses: xliu@math.ualberta.ca (X. Liu), marquez@ece.ualberta.ca

(H.J. Marquez).

0005-1098/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2008.05.016

(1989, 1998). They showed that if a digital controller input-to-state stabilizes the approximate discrete-time plant model,then it would also input-to-state stabilize the exact discrete-timemodel. However, this result requires fast sampling which meansthat it may not be implementable in cases when the requiredsampling period is too small to be realized with the availablehardware. In practical applications, hardware restrictions on inputand measurement sampling rate can be essentially different. Forexample, the D/A converters are generally faster than the A/Dconverters, so the measurement sampling rate is often madeslower than that of the input. In such cases, it makes sense toconfigure the control system so that several sample rates co-existto achieve better performance.In this paper, we address the problem of sampled-data

stabilization of nonlinear systems under the ‘‘low measurementrate’’ constraint. In this case, the single rate method (Nesic & Laila,2002) may lead to unstable closed loop performance (see examplein Section 3). We address the design of dual-rate controllerscontaining a fast-rate model to estimate the intersample statesbased on the DTD method. We show that under some standardassumptions the closed loop dual-rate sampled data systemis input-to-state stable in the semiglobal practical sense. Weemphasize that the result is prescriptive since it can be usedas a guide when designing controllers based on an approximatediscrete-time plant model.The paper is organized as follows. After a brief description of

problem statement and relevant definitions and notations, the

3158 X. Liu et al. / Automatica 44 (2008) 3157–3161

main result is presented and illustrated via an example. Finally, thepaper is closed with conclusions in Section 4.

2. Statement of the problem

The following notationswill be used in the sequel. DenoteZ+ asthe sets of nonnegative integer numbers. A continuous functionα :R≥0 → R≥0 is said to belong to classK if α(0) = 0 and it is strictlyincreasing. Also, a continuous function β : R≥0 × R≥0 → R≥0 issaid to belong to classKL if for each fixed t ≥ 0, β(·, t) belongstoK and for each fixed s ≥ 0, β(s, t) decreases to zero as t →∞.The Euclidean norm of a vector is denoted as | · |. For a functionw :R≥0 → Rn, we denote w[i] := {w(t) : t ∈ [iT , (i + 1)T ], i ∈ Z+}with the norm ‖w[i]‖∞ = esssup τ∈[iT ,(i+1)T ]|w(τ)| andw(i) is thevalue ofw(·) at t = iT , i ∈ Z+.

Definition 1. The system x(i + 1) = FT (x(i), w[i]) is said to beinput-to-state stable if there exist β ∈ KL and γ ∈ K such thatfor any positive real numbers (∆1,∆2) there exists T ∗ > 0 suchthat for all |x(0)| ≤ ∆1, ‖w‖∞ ≤ ∆2 and T ∈ (0, T ∗], the solutionof the system satisfy |x(i)| ≤ β(|x(0)|, iT )+ γ (‖w‖∞),∀i ∈ Z+.

Consider the nonlinear continuous-time plant:

x(t) = f (x(t), u(t), w(t)) (1)

where x ∈ Rnx , u ∈ Rm and w ∈ Rp are respectively the state,control input and exogenous disturbance and f is locally Lipschitz.Let

x(i+ 1) = F eT (x(i), u(i), w[i]) (2)

be the exact discrete-time model of (1) with the sampling periodT > 0. We assume that the input sampling period is equal to thesampling period of system (2), that is Ti = T . Suppose that due tophysical constraints, we cannot sample themeasurement as fast aswewish.Without loss of generality, let themeasurement samplingTm be a multiple of T , i.e. Tm = lT for some integer l ≥ 1. Forthis setting we consider a dual-rate control scheme and for such ascheme to work, we need a model with fast sampling rate for thenonlinear plant. We emphasize that the exact discrete-timemodelF eT in (2) is unknown in most cases. Hence, let F

aT ,h(x(i), u(i), w[i])

be a family of approximate discrete-time plant model of (2). Weassume that the approximate model with zero disturbance

x(i+ 1) = F aT ,h(x(i), u(i), 0) (3)

corresponding to the sampling period T is available, parameterizedby the modeling parameter h > 0. The parameter h, whichrepresents the integration period of the numerical integration usedto generate the family of the approximatemodels, may be differentfrom the sampling period T . Then the idea is the following: tocompensate for the lack of information about the states whichare fed to the fast-rate controller, we introduce a periodic switchwhich connects to the actual state x at times klT and connects to theestimate of the state at t = klT + jT , j ∈ {1, 2, . . . , l − 1}, whichis reconstructed by the zero disturbance model with periodicallyupdated initialization at sampling instant i = klT by the actualstate. Thus the output of the switch is a fast rate signal given by

xc(i+ 1) =

x(i+ 1), i+ 1 = kl, k ∈ Z+F aT ,h(xc(i), u(i), 0) with initializationxc(kl) = x(kl), otherwise.

(4)

The controller depends on the switch output xc(i) and isimplemented using a zero-order hold. We consider a dynamicfeedback controller:

z(i+ 1) = GT ,h(xc(i), z(i)) (5)

u(i) = UT ,h(xc(i), z(i)) (6)

where z ∈ Rnz and GT ,h,UT ,h are zero at zero. To summarize, thedual-rate control scheme uses a fast-rate approximate model, afast-rate controller and a periodic switch.To shorten our notation, we denote x := (xT, zT)T, wf := w[i]

and F aT ,h(x, wf ) :=[FaT ,h(x,UT ,h(x, z), wf )

GT ,h(x, z)

]. We now introduce the

following definitions.

Definition 2. The system x(i + 1) = F aT ,h(x(i), w[i]) is equi-Lipschitz Lyapunov-ISS if there exist functions α1, α2, α3 ∈

K∞, γ ∈ K and for any positive real numbers (∆1,∆2) there existT ∗ > 0 such that for each fixed T ∈ (0, T ∗] there exists h∗ ∈ (0, T ]such that for all x ∈ Rnx , ‖w‖∞ ≤ ∆2 and h ∈ (0, h∗) there existsa function VT ,h : Rnx → R≥0 with the following properties:

α1(|x|) ≤ VT ,h(x) ≤ α2(|x|) (7)

VT ,h(F aT ,h(x, wf ))− VT ,h(x)

T≤ −α3(|x|)+ γ (‖w‖∞) (8)

and, for all x1, x2 ∈ B(∆1), there existsM > 0 such that |VT ,h(x1)−VT ,h(x2)| ≤ M|x1 − x2|.

Definition 3. F aT ,h(x, u, wf ) is said to be one-step consistent withF eT (x, u, wf ) if for any positive real numbers (∆1,∆2,∆3) thereexist aK-class function ρ(·) and T ∗ > 0 such that for each fixedT ∈ (0, T ∗], there exists h∗ ∈ (0, T ] such that |F eT (x, u, wf ) −F aT ,h(x, u, wf )| ≤ Tρ(h) for all x ∈ B(∆1), u ∈ B(∆2), ‖w‖∞ ≤ ∆3and h ∈ (0, h∗).

Definition 4. The control law (GT ,h,UT ,h) is said to be uniformlylocally Lipschitz if for any∆1 > 0 there exist L1, L2 > 0 and T ∗ > 0such that for each fixed T ∈ (0, T ∗] there exists h∗ ∈ (0, T ] suchthat for all ξ1, ξ2 ∈ B(∆1) and h ∈ (0, h∗], We have |GT ,h(ξ1) −GT ,h(ξ2)| ≤ L1|ξ1 − ξ2|, |UT ,h(ξ1)− UT ,h(ξ2)| ≤ L2|ξ1 − ξ2|, whereξ := (xTc , z

T)T.

3. Main result

In this section, we state and prove our main result. weconsider a dual-rate control scheme that is based on a fastnumerical integration approximation to predict the interstatesbetween samples. Our result specifies conditions which guaranteethat the dual-rate controller input-to-state stabilizes the closed-loop sampled-data system in the semiglobal practical sense.More precisely, we address the stabilization problem under thefollowing assumptions.

Assumption. (1) F aT ,h(x, wf ) is equi-Lipschitz Lyapunov-ISS.(2) F aT ,h(x, u, wf ) is one-step consistent with the exact discrete-time model F eT (x, u, wf ). (3) The controller (5) and (6) is uniformlylocally Lipschitz.

Remark 1. By Assumption 3 and the property that UT ,h is zero atzero, we have that given positive numbers (∆1,∆2) there existT ∗ > 0, h∗ > 0 such that for all ξ := (xTc , z

T)T ∈ B(∆1) andh ∈ (0, h∗), |UT ,h(ξ)| ≤ ∆2 holds. That is, the output of thecontroller is locally uniformly bounded (see Khalil (1996)).

Theorem 1. Under Assumption 1–3, there exist β ∈ KL and γ ∈K∞ such that the following holds. Given any positive real numbers(∆x,∆w, δ), there exists T ∗ > 0 such that for each T ∈ (0, T ∗] thereexists h∗ ∈ (0, T ] such that for all |x(0)| ≤ ∆x, ‖w‖∞ ≤ ∆w and allh ∈ (0, h∗], the exact closed loop discrete-time model (2) and (4)–(6)satisfies |x(i)| ≤ β(|x(0)|, iT )+ γ (‖w‖∞)+ δ.

We begin with the following claims.

X. Liu et al. / Automatica 44 (2008) 3157–3161 3159

Claim 1. Consider the exact closed loop discrete-time model (2) and(4)–(6). Given any strictly positive real numbers (D1,D3, ε), thereexists T1 > 0 such that for any fixed T ∈ (0, T1] there existsh1 ∈ (0, T ] such that for all h ∈ (0, h1], |x(0)| ≤ D1 and‖w‖∞ ≤ D3, the following holds: if maxi∈{0,1,...,k} |x(i)| ≤ D1 forsome k ∈ {0, 1, . . .} then the exact discrete-time state of the plantsatisfies: |x(k)− xc(k)| ≤ Tε + Tλ‖w‖∞, for some λ > 0.

Proof. Let (D1,D3, ε) be given. Define ∆1 = D1 + ε + 1. ByRemark 1, for given D2 > 0 there exist T11 > 0 and h11 > 0 suchthat |UT ,h(xc, z)| ≤ D2 for all (xTc , z

T)T ∈ B(∆1). Let L > 0 be theLipschitz constant of function f . Also, let λ > 0 be a number suchthat eL(l−1)T − 1 ≤ λT for any T ∈ (0, T11]. Let T12 > 0 and h12 > 0be as in Assumption 2 corresponding to∆1 = D1+ε+1,∆2 = D2and ∆3 = D3, and let ρ(·) ∈ K∞ be a function from Assumption2. Let T13 > 0, h13 > 0 be such that ρ(h13)(eL(l−1)T13 − 1)/(eLT13 −1) ≤ ε. Finally we define T1 = min{T11, T12, T13, 1/λD3, 1} andh1 = min{h11, h12, h13}. Suppose T ∈ (0, T1], h ∈ (0, h1] andmaxi∈{0,1,...,k} |x(i)| ≤ D1 for some k ∈ {0, 1, . . .}. First we claimthat |(xTc (k), z

T(k))T| ≤ ∆1 for some k ∈ {0, 1, . . .} follows byinduction. Consider k in the following three cases. If k = jl forsome j ∈ {0, 1, . . .}, then it is obvious that |x(k) − xc(k)| = 0. Ifk = jl+ 1, then using Assumption 2 as well as triangle inequalitieswe have |x(k) − xc(k)| = |F eT (x(jl),UT ,h(x(jl), z(jl)), w[jl]) −F aT ,h(x(jl),UT ,h(x(jl), z(jl)), 0)| ≤ |F

eT (x(jl),UT ,h(x(jl), z(jl)), w[jl])

−F eT (x(jl),UT ,h(x(jl), z(jl)), 0)|+Tρ(h) ≤ Tρ(h)+(eLT−1)‖w‖∞.

Otherwise, we obtain by induction that |x(k) − xc(k)| ≤ Tρ(h) +eLT |x(k − 1) − xc(k − 1)| + (eLT − 1)‖w‖∞ ≤ Tρ(h)(e(k−jl)LT −1)/(eLT −1)+(e(k−jl)LT −1)‖w‖∞ holds for all k ∈ {jl+2, . . . , (j+1)l − 1}. By the choice of T and h, we have |x(k) − xc(k)| ≤Tε + Tλ‖w‖∞. This completes the proof of Claim 1. �

Claim 2. Consider the exact closed loop model (2) and (4)–(6). Forany strictly positive real numbers (D′1,D

′

3) there exists T2 > 0 suchthat for any fixed T ∈ (0, T2] there exists h2 ∈ (0, T ],∆ > 0such that for all h ∈ (0, h2], |x(0)| ≤ D′1 and ‖w‖∞ ≤ D

′

3, thefollowing holds: if maxi∈{0,1,...,k} |x(i)| ≤ D′1 for some k ∈ {0, 1, . . .}then we have the exact state of closed loop system x(k + 1) and theapproximation F aT ,h(x(k), w[k]) ∈ B(∆).

Proof. Let (D′1,D′

3) be given. Take ε1 > 0. From Claim 1, let(D′1,D

′

3, ε1) generate T21 > 0, h21 > 0 and let λ > 0 be from inClaim 1. From Assumption 1, let T22 > 0, h22 > 0 be generatedby ∆1 = D′1 + ε1 + 1,∆2 = D′3 and let T23, h23, L1, L2 > 0be as in Assumption 3. Let T24 > 0, h24 > 0 and ε2 > 0 besuch that T24(ρ(h24) + L1(ε1 + λ‖w‖∞) + L2(eLT24 − 1)(ε1 +λ‖w‖∞)) ≤ ε2. Define ∆ = α−11 (α2(D

′

1) + γ (D′

3)) + ε2. Let T2 =min{T21, T22, T23, T24} and h2 = min{h21, h22, h23, h24}. SupposeT ∈ (0, T2], h ∈ (0, h2] and maxi∈{0,1,...,k} |x(i)| ≤ D′1 for somek ∈ {0, 1, . . .}. From Assumption 1, we have F aT ,h(x(k), w[k]) ≤α−11 ◦ VT ,h(F

aT ,h(x(k), w[k])) ≤ α−11 (VT ,h(x(k)) + γ (D

′

3)) ≤ ∆.Using Assumption 2–3 as well as triangle inequalities, we have|x(k+1)− F aT ,h(x(k), w[k])| ≤ T (ρ(h))+L1|x(k)−xc(k)|+L2(e

LT−

1)|x(k) − xc(k)|. Applying Claim 1 and from the choice of T24, h24and∆, we have |x(k+1)| ≤ |F aT ,h(x(k), w[k])|+ε2 = ∆. The proofof Claim 2 is complete. �

Claim 3. Let W = {w ∈ L∞|‖w‖∞ ≤ Cw,∀Cw > 0} and letα1, α2, α3 ∈ K∞. Let strictly positive real numbers (d,D) be suchthat α1(D) ≥ d and let T3 > 0 be such that for each fixed T ∈ (0, T3]there exists h3 ∈ (0, T ] such that for any h ∈ (0, h3] there existsa function VT ,h : Rnx → R≥0 such that for all x ∈ Rnx , we haveα1(|x|) ≤ VT ,h(x) ≤ α2(|x|) and for all x ∈ Rnx with |x| ≤ D,all w ∈ W and max{VT ,h(x(i + 1)), VT ,h(x(i))} ≥ d, the followingholds: VT ,h(x(i + 1)) − VT ,h(x(i)) ≤ − T4α3(|x(i)|). Then, for all

|x(0)| ≤ α−12 ◦ α1(D), w ∈ W and all i ∈ Z+, we have |x(i)| ≤ Dand the solution of the exact closed loop discrete-timemodel exists andsatisfies

|x(i)| ≤ β(|x(0)|, iT )+ α−11 (d). (9)

Proof. The definitions of d and D imply |x(0)| ≤ max{α−11 ◦VT ,h(x(0)), α−11 (d)} ≤ D. So either VT ,h(x(1)) ≥ d which,from the condition of Claim 3, implies VT ,h(x(1)) ≤ VT ,h(x(0))or else VT ,h(x(1)) ≤ d. Then, in either case, VT ,h(x(1)) ≤max{VT ,h(x(0)), d}. Hence VT ,h(x(i)) ≤ max{VT ,h(x(0)), d} followsby induction and |x(i)| ≤ D holds as well. Then (9) follows, usingan argument similar to the proof of Theorem 2 in Nesic et al.(1999). �

Claim 4. Consider the exact closed loop model (2) and (4)–(6). Thereexists γ ∈ K∞ such that the following holds. For any strictly positivereal numbers (Cx, Cw, ν) with Cx ≥ α−11 (γ (Cw) + ν), there existsT4 > 0 such that for each T ∈ (0, T4] there exists h4 ∈ (0, T ] suchthat for all h ∈ (0, h4], |x(0)| ≤ α−12 ◦ α1(Cx), ‖w‖∞ ≤ Cw and alli ∈ Z+ we have

max{VT ,h(x(i+ 1)), VT ,h(x(i))} ≥ γ (‖w‖∞)+ ν

⇒ VT ,h(x(i+ 1))− VT ,h(x(i)) ≤ −T4α3(|x(i)|). (10)

Proof. Let positive real numbers (Cx, Cw, ν) be given. Define ε2 =12α−12 (

ν2 ), ε3 = α

−12 (

12α1(ε2)), and∆ = α

−11 (α2(Cx)+ γ (Cw))+ε2.

Take any ε1 > 0 which satisfies the inequality: ML1ε1 ≤ 14α3(ε3).

From Claim 1, let (Cx, Cw, ε1) generate T41, h41 and let λ > 0 be asin Claim1. Let T42, h42 come fromClaim2 corresponding to (Cx, Cw)and also the following holds: T42(ρ(h42) + L1(ε1 + λ‖w‖∞) +L2(eLT42 − 1)(ε1 + λ‖w‖∞)) ≤ ε2. Let positive real numbersT43, h43, T44, h44 and T45 be such that: T43( 14α3(Cx) + γ (‖w‖∞) +M(ρ(h43)+ L1(ε1+ λ‖w‖∞)+ L2(eLT43 − 1)(ε1+ λ‖w‖∞))) ≤ ν

2 ,Mρ(h44)+ML2(eLT44 − 1)ε1 ≤ 1

4α3(ε3) and T45γ (Cw) ≤12α1(ε2).

Let γ (s) = α2 ◦ α−13 (4(γ (s) + MλL1s + ML2λ(e

L− 1)s)). Take

T4 = min{T41, T42, T43, T44, T45} and h4 = min{h41, h42, h43, h44}.Consider any T ∈ (0, T4], h ∈ (0, h4], |x(0)| ≤ α−12 ◦ α1(Cx) and‖w‖∞ ≤ Cw . First of all, we claim that, for any i ∈ Z+, |x(i)| ≤ Cx.From now we suppose this to be true. Applying Claim 2, we have|F aT ,h(x(i), w[i])| ≤ ∆ and |x(i+ 1)| ≤ ∆.Suppose VT ,h(x(i + 1)) ≥ γ (‖w‖∞) +

ν2 . Using Assumption

1 and triangle inequalities, we have VT ,h(x(i + 1)) − VT ,h(x(i)) ≤−Tα3(|x(i)|)+T γ (‖w‖∞)+M|x(i+1)−F aT ,h(x(i), w[i])|. ApplyingAssumption 2–3 and triangle inequalities and from the choice ofT41 and h41, we see that VT ,h(x(i+1))−VT ,h(x(i)) ≤ −Tα3(|x(i)|)+T γ (‖w‖∞) + TM(ρ(h) + L1(ε1 + λ‖w‖∞) + L2(eLT − 1)(ε1 +λ‖w‖∞)). Denote µ1 := ML1ε1, µ2 := M(ρ(h) + L2(eLT − 1)ε1)and κ(s) := γ (s)+Mλ(L1 + L2(eLT − 1))s. Then we have

VT ,h(x(i+ 1))− VT ,h(x(i))

≤ −T4α3(|x(i)|)−

T4α3(α

−12 (VT ,h(x(i))))+ Tκ(‖w‖∞)︸ ︷︷ ︸

(∗)

−T4α3(|x(i)|)+ Tµ1︸ ︷︷ ︸

(∗∗)

−T4α3(|x(i)|)+ Tµ2︸ ︷︷ ︸

(∗∗∗)

.

Wededuce that VT ,h(x(i+1)) ≥ γ (‖w‖∞)+ ν2 implies γ (‖w‖∞)+

ν2 ≤ VT ,h(F aT ,h(x(i), w[i])) − VT ,h(x(i)) + |VT ,h(x(i + 1)) −VT ,h(F aT ,h(x(i), w[i]))| + VT ,h(x(i)) ≤ T γ (‖w‖∞) + MT (ρ(h) +L1(ε1 + λ‖w‖∞)+ L2(eLT − 1)(ε1 + λ‖w‖∞))+ VT ,h(x(i)). From

3160 X. Liu et al. / Automatica 44 (2008) 3157–3161

the choice of T43 and h43, we have γ (‖w‖∞) + ν2 ≤

ν2 + Vh(x(i)).

Hence we have

VT ,h(x(i+ 1)) ≥ γ (‖w‖∞)+ν

2⇒ VT ,h(x(i)) ≥ γ (‖w‖∞).

From the definition of γ (·), we have Term (∗) ≤ 0 holds. Bysupposition VT ,h(x(i + 1)) ≥ γ (‖w‖∞) + ν

2 , we have x(i + 1) ≥α−12 (

ν2 ) = 2ε2. Then from the choice of T42 and h42, we obtain

|F aT ,h(x(i), w[i])| ≥ |x(i + 1)| − |x(i + 1) − FaT ,h(x(i), w[i])| ≥

2ε2 − ε2 = ε2. Using our choice of T45, it follows that

α2(|x(i)|) ≥ VT ,h(F aT ,h(x(i), w[i]))− T γ (Cw)

≥ α1(|F aT ,h(x(i), w[i])|)− T γ (Cw)

≥ α1(ε2)−12α1(ε2) =

12α1(ε2),

which implies |x(i)| ≥ α−12 (12α1(ε2)) = ε3 ≥ α

−13 (4µ1) and then

Term (∗∗) ≤ 0 holds. Moreover, from the choice of T44 and h44, wehave |x(i)| ≥ ε3 ⇒ − T4α3(|x(i)|)+Tµ2 ≤ 0. Hence, By suppositionVT ,h(x(i+1)) ≥ γ (‖w‖∞)+ ν

2 , we haveVT ,h(x(i+1))−VT ,h(x(i)) ≤−T4α3(|x(i)|).Suppose VT ,h(x(i + 1)) ≤ γ (‖w‖∞) +

ν2 and VT ,h(x(i)) ≥

γ (‖w‖∞) + ν. From our choice of T43, it follows that: VT ,h(x(i +1))−VT ,h(x(i)) ≤ γ (‖w‖∞)+ ν

2−VT ,h(x(i))+ν2−

ν2 ≤ γ (‖w‖∞)+

ν − VT ,h(x(i))− ν2 ≤ −

ν2 ≤ −

T4α3(|x(i)|).

It remains to establish our initial claim: |x(i)| ≤ Cx for anyi ∈ Z+. This claim follows by induction. Indeed, it clearly holds fori = 0, since |x(0)| ≤ α−12 ◦ α1(Cx) ≤ Cx by the definition of x(0).Then (10) holds for i = 0 from the deduction above. By Claim 3(Take D = Cx and d = γ (‖w‖∞) + ν.), we have |x(1)| ≤ Cx. Thatis, this claim holds for i = 1 as well. Then |x(i)| ≤ Cx, i ∈ Z+follows by induction. The proof of Claim 4 is complete. �

Proof of Theorem 1. Now the proof of Theorem 1 can be finalizedas follows. Let (∆x,∆w, δ) be given and let all conditionsin Theorem 1 hold. Let γ ∈ K∞ come from Claim 4.We define (Cx, Cw, ν) as: Cw := ∆w, ν > 0 is suchthat sups∈[0,∆w ][α

−11 (γ (s) + ν) − α−11 (γ (s))] ≤ δ, Cx :=

max{α−11 (γ (∆w) + ν), α−11 ◦ α2(∆x)}. From the choice of(Cx, Cw, ν), we have Cx ≥ α−11 (γ (Cw) + ν) and |x(0)| ≤ α−12 ◦

α1(Cx). Using Claim 4, let (Cx, Cw, ν) generate T ∗ > 0, h∗ > 0 suchthat (10) holds. Let D = Cx and d = γ (‖w‖∞) + ν, then we haveα1(D) ≥ d. With the definition of (D, d), we have all conditions ofClaim 3 are satisfied. Therefore for all h ∈ (0, h∗), |x(0)| ≤ ∆xand ‖w‖∞ ≤ ∆w , we have |x(i)| ≤ β(|x(0)|, iT ) + α−11 (d) ≤β(|x(0)|, iT )+ α−11 (γ (‖w‖∞)+ ν) ≤ β(|x(0)|, iT )+ γ (‖w‖∞)+δ, where γ (s) := α−11 ◦ γ (s). This completes the proof ofTheorem 1. �

Remark 2. Following the proof of Theorem 1, it is easy to see thatif we relax Assumption 1 slightly to the assumption of PracticalLyapunov-ISS, that is, VT ,h(F aT ,h(x, wf )) − VT ,h(x) ≤ −Tα3(|x|) +T γ (‖w‖∞)+ Tδ1 holds, then Theorem 1 still holds.

Example. Consider the continuous-time plant x(t) = x3(t) +u(t)+w(t). Let x(i+1) = F eT (x(i), u(i), w[i]) be the exact discrete-time model of the continuous-time plant with the samplingperiod T . Let fh(x, u, w) represent one step of the numericalintegration routine on the sampling interval [iT , (i+ 1)T ) definedby fh(x, u, w) = x+ h(x3+ u)+

∫ iT+hiT w(s)ds := f 1h (x, u, w). Then

Fig. 1. The performance of the closed-loop systemwithout disturbance under twocontrol schemes.

we can generate its numerically integrated approximate modelF aT ,h(·, ·, ·) by

fh(k, x, u, w) := x+ h(x3 + u)+∫ iT+(k+1)h

iT+khw(s)ds

f k+1h (x, u, w) := fh(k+ 1, f kh , u, w)

F aT ,h(x, u, wf ) := fNh (x, u, w), k = 1, 2, . . .

where h represents the integration period, T is the sampling periodand N = T

h . Moreover, the approximate model with disturbancefree, which reconstructs themissing plant states between samples,is generated by F aT ,h(x, u, 0). Consider a digital controller: u(i) =−x(i)− x3(i). We first check the consistency of the approximationscheme. By Lemma Π .2 in Nesic and Laila (2002), fh is one-stepconsistent with F eh where F

eh is the exact discrete-time model

with the sampling period h. Also, the multi-step consistency isguaranteed by the one-step consistency plus the uniform Lipschitzcondition on fh (see Remark 13 in Nesic and Teel (2004)). Thenfollowing closely the conclusions of Corollary 4 and Remark 14in Nesic and Teel (2004), we have that F aT ,h(x, u, wf ) is one-stepconsistent with F eT (x, u, wf ). Take VT ,h(x) = |x|, it follows that theapproximate discrete-time model x(i + 1) = F aT ,h(x(i), u(i), w[i])with u(i) = −x(i)− x3(i) is practical Lyapunov-ISS with α3(|x|) :=|x| and γ (‖w‖∞) := ‖w‖∞. Moreover it is easy to see thatAssumption 3 is also satisfied. We conclude from Theorem 1 thatthe exact closed loop system is semiglobally practically input-to-state stable. Assume x(0) = 3.4. Our simulation shows thatthe single-rate method stabilizes the system without disturbanceonly when T < 0.205 s. On the other hand, consider the dual-rate method with low measurement rate Tm = 1.5 s. SettingTi = 0.15 s and h = 0.0075 s (N = 20), our simulation showsthat the dual-rate controller stabilizes the system successfully(Fig. 1). Comparing with the response obtained under single ratescheme, we see that advantages of dual-rate control system, whichcan render a stable closed loop using much lower sampling rate.In Fig. 2 we take a sinusoidal disturbance of amplitude 0.8 andfrequency 2 rad/s and our simulation shows that the closed-loopsystem is practical ISS. This example shows that the dual-rateinferential system is indeed more robust than the correspondingfast single-rate system.

4. Conclusions

In this paper, we concentrate on the problem of sampled-data input-to-state stabilization of nonlinear systems under lowmeasurement constraint. Our approach to the solution of thisproblem employs a dual-rate scheme based on discrete-time

X. Liu et al. / Automatica 44 (2008) 3157–3161 3161

Fig. 2. The closed-loop performance for dual-rate control systemwith disturbance.

controller design, since the fast sampling results may not beimplemented due to hardware limitations. The main idea is tointroduce a controller that includes an approximate discrete-timemodel of the plant. The control action depends on the state ofthis model which is corrected from time to time using the lowrate measurements of the actual state of the plant. We showthat if one designs a discrete-time controller for an approximatediscrete-time plant model so that the closed-loop system is input-to-state stable, then the input-to-state stability property will bepreserved for the exact discrete-time plant model based on a dual-rate control scheme in a semi-globally practical sense.

Acknowledgements

The authors are grateful to the reviewers and editor forsuggesting the strengthened version of the approximate numericalintegration model used here.

References

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Xi Liu received the B.Sc. degree in applied mathematicsfrom Liaoning University, People’s Republic of China, andthe M.Sc. degree in control theory from NortheasternUniversity, People’s Republic of China, in 2001 and 2004,respectively. Currently, she is a Ph.D. candidate in theDepartment of Mathematical and Statistical Sciences atthe University of Alberta. Her current research interestsare dynamical systems, sampled-data nonlinear controlanalysis and design, and network based control systems.

Horacio J. Marquez received the B.Sc. degree from theInstituto Tecnologico de Buenos Aires (Argentina), and theM.Sc.E and Ph.D. degrees in electrical engineering fromthe University of New Brunswick, Fredericton, Canada, in1987, 1990 and 1993, respectively.From 1993 to 1996 he held visiting appointments at

the Royal Roads Military College, and the University ofVictoria, Victoria, British Columbia. Since 1996 he hasbeen with the Department of Electrical and ComputerEngineering, University of Alberta, Edmonton, Canada,where he is currently a Professor and Department Chair.

Dr. Marquez is currently an Area Editor for the International Journal of Robustand Nonlinear Control and Associate Editor of the J. of the Franklin Institute. He isthe Author of ‘‘Nonlinear Control Systems: Analysis and Design’’ (Wiley, 2003). Heis the recipient of the 2003–2004 McCalla Research Professorship awarded by theUniversity of Alberta. His current research interests include nonlinear dynamicalsystems and control, nonlinear observer design, robust control, and applications.

Yanping Lin received the B.Sc. degree in applied mathe-matics from Northeastern University, People’s Republic ofChina, in 1982, and the M.Sc. and Ph.D. degrees in appliedmathematics from Washington State University, USA, in1985 and 1988, respectively. After post-doctorial fellow-ships from McGill University and University of Waterloofrom 1988 to 1990, he has been with the Department ofMathematical and Statistical Sciences, University of Al-berta since 1991, and he is currently a Professor and a Pro-fessor at the Hong Kong Polytechnic University. He is therecipient of the 2004–2005 McCalla Research Professor-

ship awarded by the University of Alberta. His current research areas are mathe-matical modeling, scientific computing, numerical analysis and control theory.