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Input-to-state stabilization for nonlinear dual-rate sampled-data systems via approximate discrete-time modelI Xi Liu a , Horacio J. Marquez b,∗ , Yanping Lin a,c a

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada T6G 2G1

b

Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G 2V4

c

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Hom, Hong Kong, China

article

info

Article history: Received 26 June 2006 Received in revised form 5 May 2008 Accepted 12 May 2008 Available online 6 November 2008 Keywords: Input-to-state stability Nonlinear Dual-rate Discrete-time Sampled-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 numerical integration scheme to approximately predict the current state. Given an approximate discrete-time model of a sampled nonlinear plant and given a family of controllers that stabilizes the plant model in input-to-state sense, we show that under some standard assumptions the closed loop dual-rate sampled data 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 most systems of interest in control systems are often nonlinear, motivate the area of nonlinear sampled-data control systems. Significant progress 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 design of digital controllers (see Nesic and Teel (2001)): continuoustime design (CTD) and discrete-time design (DTD). The first one involves digital implementation of a continuous-time stabilizing control law. The second approach consists of discretizing the plant model and then designing a discrete-time controller. Both of these approaches are essentially single-rate, i.e. sampling rates of the input and the measurement channels are assumed to be equal. 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 was recommended for publication in revised form by Associate Editor Dragan Nešić under the direction of Editor Hassan K. Khalil. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). ∗ Corresponding author. Tel. +1 780 492 3333; fax: +1 780 492 8506. E-mail addresses: [email protected] (X. Liu), [email protected] (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 inputto-state stabilizes the approximate discrete-time plant model, then it would also input-to-state stabilize the exact discrete-time model. However, this result requires fast sampling which means that it may not be implementable in cases when the required sampling period is too small to be realized with the available hardware. In practical applications, hardware restrictions on input and measurement sampling rate can be essentially different. For example, the D/A converters are generally faster than the A/D converters, so the measurement sampling rate is often made slower than that of the input. In such cases, it makes sense to configure the control system so that several sample rates co-exist to achieve better performance. In this paper, we address the problem of sampled-data stabilization of nonlinear systems under the ‘‘low measurement rate’’ constraint. In this case, the single rate method (Nesic & Laila, 2002) may lead to unstable closed loop performance (see example in Section 3). We address the design of dual-rate controllers containing a fast-rate model to estimate the intersample states based on the DTD method. We show that under some standard assumptions the closed loop dual-rate sampled data system is input-to-state stable in the semiglobal practical sense. We emphasize that the result is prescriptive since it can be used as a guide when designing controllers based on an approximate discrete-time plant model. The paper is organized as follows. After a brief description of problem statement and relevant definitions and notations, the

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main result is presented and illustrated via an example. Finally, the paper is closed with conclusions in Section 4. 2. Statement of the problem +

The following notations will be used in the sequel. Denote Z as the sets of nonnegative integer numbers. A continuous function α : R≥0 → R≥0 is said to belong to class K if α(0) = 0 and it is strictly increasing. Also, a continuous function β : R≥0 × R≥0 → R≥0 is said to belong to class KL if for each fixed t ≥ 0, β(·, t ) belongs to K 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 function w : R≥0 → Rn , we denote w[i] := {w(t ) : t ∈ [iT , (i + 1)T ], i ∈ Z+ } with the norm kw[i]k∞ = esssup τ ∈[iT ,(i+1)T ] |w(τ )| and w(i) is the value of w(·) at t = iT , i ∈ Z+ . Definition 1. The system x(i + 1) = F˜T (x(i), w[i]) is said to be input-to-state stable if there exist β ∈ KL and γ ∈ K such that for any positive real numbers (∆1 , ∆2 ) there exists T ∗ > 0 such that for all |x(0)| ≤ ∆1 , kwk∞ ≤ ∆2 and T ∈ (0, T ∗ ], the solution of the system satisfy |x(i)| ≤ β(|x(0)|, iT ) + γ (kwk∞ ), ∀i ∈ Z+ . Consider the nonlinear continuous-time plant: x˙ (t ) = f (x(t ), u(t ), w(t ))

(1)

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

m

p

x(i + 1) = FTe (x(i), u(i), w[i])

(2)

be the exact discrete-time model of (1) with the sampling period T > 0. We assume that the input sampling period is equal to the sampling period of system (2), that is Ti = T . Suppose that due to physical constraints, we cannot sample the measurement as fast as we wish. Without loss of generality, let the measurement sampling Tm be a multiple of T , i.e. Tm = lT for some integer l ≥ 1. For this setting we consider a dual-rate control scheme and for such a scheme to work, we need a model with fast sampling rate for the nonlinear plant. We emphasize that the exact discrete-time model FTe in (2) is unknown in most cases. Hence, let FTa,h (x(i), u(i), w[i]) be a family of approximate discrete-time plant model of (2). We assume that the approximate model with zero disturbance x(i + 1) = FTa,h (x(i), u(i), 0)

(3)

corresponding to the sampling period T is available, parameterized by the modeling parameter h > 0. The parameter h, which represents the integration period of the numerical integration used to generate the family of the approximate models, may be different from the sampling period T . Then the idea is the following: to compensate for the lack of information about the states which are fed to the fast-rate controller, we introduce a periodic switch which connects to the actual state x at times klT and connects to the estimate of the state at t = klT + jT , j ∈ {1, 2, . . . , l − 1}, which is reconstructed by the zero disturbance model with periodically updated initialization at sampling instant i = klT by the actual state. Thus the output of the switch is a fast rate signal given by

x(i + 1), i + 1 = kl, k ∈ Z+ a xc (i + 1) = FT ,h (xc (i), u(i), 0) with initialization x (kl) = x(kl), otherwise. c

(4)

where z ∈ Rnz and GT ,h , UT ,h are zero at zero. To summarize, the dual-rate control scheme uses a fast-rate approximate model, a fast-rate controller and a periodic switch. T T T To shorten our notation, we denote e h ix := (x , z ) , wf := w[i] and F˜Ta,h (e x, wf ) := following definitions.

FTa,h (x, UT ,h (x, z ), wf ) GT ,h (x, z )

. We now introduce the

Definition 2. The system x˜ (i + 1) = F˜Ta,h (˜x(i), w[i]) is equiLipschitz Lyapunov-ISS if there exist functions α1 , α2 , α3 ∈ K∞ , γ˜ ∈ K and for any positive real numbers (∆1 , ∆2 ) there exist T ∗ > 0 such that for each fixed T ∈ (0, T ∗ ] there exists h∗ ∈ (0, T ] such that for all x˜ ∈ Rnx˜ , kwk∞ ≤ ∆2 and h ∈ (0, h∗ ) there exists a function VT ,h : Rnx˜ → R≥0 with the following properties:

α1 (|˜x|) ≤ VT ,h (˜x) ≤ α2 (|˜x|) VT ,h (F˜Ta,h (˜x, wf )) − VT ,h (˜x) T

(7)

≤ −α3 (|˜x|) + γ˜ (kwk∞ )

(8)

¯ (∆1 ), there exists M > 0 such that |VT ,h (˜x1 ) − and, for all x˜ 1 , x˜ 2 ∈ B VT ,h (˜x2 )| ≤ M |˜x1 − x˜ 2 |. Definition 3. FTa,h (x, u, wf ) is said to be one-step consistent with FTe (x, u, wf ) if for any positive real numbers (∆1 , ∆2 , ∆3 ) there exist a K -class function ρ(·) and T ∗ > 0 such that for each fixed T ∈ (0, T ∗ ], there exists h∗ ∈ (0, T ] such that |FTe (x, u, wf ) − ¯ (∆1 ), u ∈ B¯ (∆2 ), kwk∞ ≤ ∆3 FTa,h (x, u, wf )| ≤ T ρ(h) for all x ∈ B and h ∈ (0, h∗ ). Definition 4. The control law (GT ,h , UT ,h ) is said to be uniformly locally Lipschitz if for any ∆1 > 0 there exist L1 , L2 > 0 and T ∗ > 0 such that for each fixed T ∈ (0, T ∗ ] there exists h∗ ∈ (0, T ] such ¯ (∆1 ) and h ∈ (0, h∗ ], We have |GT ,h (ξ1 ) − that for all ξ1 , ξ2 ∈ B GT ,h (ξ2 )| ≤ L1 |ξ1 − ξ2 |, |UT ,h (ξ1 ) − UT ,h (ξ2 )| ≤ L2 |ξ1 − ξ2 |, where ξ := (xTc , z T )T . 3. Main result In this section, we state and prove our main result. we consider a dual-rate control scheme that is based on a fast numerical integration approximation to predict the interstates between samples. Our result specifies conditions which guarantee that the dual-rate controller input-to-state stabilizes the closedloop sampled-data system in the semiglobal practical sense. More precisely, we address the stabilization problem under the following assumptions. Assumption. (1) F˜Ta,h (˜x, wf ) is equi-Lipschitz Lyapunov-ISS. (2) FTa,h (x, u, wf ) is one-step consistent with the exact discretetime model FTe (x, u, wf ). (3) The controller (5) and (6) is uniformly locally Lipschitz. Remark 1. By Assumption 3 and the property that UT ,h is zero at zero, we have that given positive numbers (∆1 , ∆2 ) there exist ¯ (∆1 ) and T ∗ > 0, h∗ > 0 such that for all ξ := (xTc , z T )T ∈ B h ∈ (0, h∗ ), |UT ,h (ξ )| ≤ ∆2 holds. That is, the output of the controller is locally uniformly bounded (see Khalil (1996)).

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

(5)

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 ∗ ] there exists h∗ ∈ (0, T ] such that for all |˜x(0)| ≤ ∆x˜ , kwk∞ ≤ ∆w and all h ∈ (0, h∗ ], the exact closed loop discrete-time model (2) and (4)–(6) satisfies |˜x(i)| ≤ β(|˜x(0)|, iT ) + γ (kwk∞ ) + δ .

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

(6)

We begin with the following claims.

The controller depends on the switch output xc (i) and is implemented using a zero-order hold. We consider a dynamic feedback controller:

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

Claim 1. Consider the exact closed loop discrete-time model (2) and (4)–(6). Given any strictly positive real numbers (D1 , D3 , ε), there exists T1 > 0 such that for any fixed T ∈ (0, T1 ] there exists h1 ∈ (0, T ] such that for all h ∈ (0, h1 ], |˜x(0)| ≤ D1 and kwk∞ ≤ D3 , the following holds: if maxi∈{0,1,...,k} |˜x(i)| ≤ D1 for some k ∈ {0, 1, . . .} then the exact discrete-time state of the plant satisfies: |x(k) − xc (k)| ≤ T ε + T λkwk∞ , for some λ > 0. Proof. Let (D1 , D3 , ε) be given. Define ∆1 = D1 + ε + 1. By Remark 1, for given D2 > 0 there exist T11 > 0 and h11 > 0 such ¯ (∆1 ). Let L > 0 be the that |UT ,h (xc , z )| ≤ D2 for all (xTc , z T )T ∈ B Lipschitz constant of function f . Also, let λ > 0 be a number such that eL(l−1)T − 1 ≤ λT for any T ∈ (0, T11 ]. Let T12 > 0 and h12 > 0 be as in Assumption 2 corresponding to ∆1 = D1 + ε + 1, ∆2 = D2 and ∆3 = D3 , and let ρ(·) ∈ K∞ be a function from Assumption 2. 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} and h1 = min{h11 , h12 , h13 }. Suppose T ∈ (0, T1 ], h ∈ (0, h1 ] and maxi∈{0,1,...,k} |˜x(i)| ≤ D1 for some k ∈ {0, 1, . . .}. First we claim that |(xTc (k), z T (k))T | ≤ ∆1 for some k ∈ {0, 1, . . .} follows by induction. Consider k in the following three cases. If k = jl for some j ∈ {0, 1, . . .}, then it is obvious that |x(k) − xc (k)| = 0. If k = jl + 1, then using Assumption 2 as well as triangle inequalities we have |x(k) − xc (k)| = |FTe (x(jl), UT ,h (x(jl), z (jl)), w[jl]) − FTa,h (x(jl), UT ,h (x(jl), z (jl)), 0)| ≤ |FTe (x(jl), UT ,h (x(jl), z (jl)), w[jl])

− (x(jl), UT ,h (x(jl), z (jl)), 0)|+ T ρ(h) ≤ T ρ(h)+(e − 1)kwk∞ . Otherwise, we obtain by induction that |x(k) − xc (k)| ≤ T ρ(h) + eLT |x(k − 1) − xc (k − 1)| + (eLT − 1)kwk∞ ≤ T ρ(h)(e(k−jl)LT − 1)/(eLT − 1) + (e(k−jl)LT − 1)kwk∞ holds for all k ∈ {jl + 2, . . . , (j + 1)l − 1}. By the choice of T and h, we have |x(k) − xc (k)| ≤ T ε + T λkwk∞ . This completes the proof of Claim 1. FTe

3159

|˜x(0)| ≤ α2−1 ◦ α1 (D), w ∈ W and all i ∈ Z+ , we have |˜x(i)| ≤ D and the solution of the exact closed loop discrete-time model exists and satisfies

|˜x(i)| ≤ β(|˜x(0)|, iT ) + α1−1 (d).

(9)

Proof. The definitions of d and D imply |˜x(0)| ≤ max{α1−1 ◦ VT ,h (˜x(0)), α1−1 (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} follows by induction and |˜x(i)| ≤ D holds as well. Then (9) follows, using an 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). There exists γˆ ∈ K∞ such that the following holds. For any strictly positive real numbers (Cx˜ , Cw , ν) with Cx˜ ≥ α1−1 (γˆ (Cw ) + ν), there exists T4 > 0 such that for each T ∈ (0, T4 ] there exists h4 ∈ (0, T ] such that for all h ∈ (0, h4 ], |˜x(0)| ≤ α2−1 ◦ α1 (Cx˜ ), kwk∞ ≤ Cw and all i ∈ Z+ we have max{VT ,h (˜x(i + 1)), VT ,h (˜x(i))} ≥ γˆ (kwk∞ ) + ν T

⇒ VT ,h (˜x(i + 1)) − VT ,h (˜x(i)) ≤ − α3 (|˜x(i)|). 4

LT

Claim 2. Consider the exact closed loop model (2) and (4)–(6). For any strictly positive real numbers (D01 , D03 ) there exists T2 > 0 such that for any fixed T ∈ (0, T2 ] there exists h2 ∈ (0, T ], ∆ > 0 such that for all h ∈ (0, h2 ], |˜x(0)| ≤ D01 and kwk∞ ≤ D03 , the following holds: if maxi∈{0,1,...,k} |˜x(i)| ≤ D01 for some k ∈ {0, 1, . . .} then we have the exact state of closed loop system x˜ (k + 1) and the ¯ (∆). approximation F˜Ta,h (˜x(k), w[k]) ∈ B Proof. Let (D01 , D03 ) be given. Take ε1 > 0. From Claim 1, let (D01 , D03 , ε1 ) generate T21 > 0, h21 > 0 and let λ > 0 be from in Claim 1. From Assumption 1, let T22 > 0, h22 > 0 be generated by ∆1 = D01 + ε1 + 1, ∆2 = D03 and let T23 , h23 , L1 , L2 > 0 be as in Assumption 3. Let T24 > 0, h24 > 0 and ε2 > 0 be such that T24 (ρ(h24 ) + L1 (ε1 + λkwk∞ ) + L2 (eLT24 − 1)(ε1 + λkwk∞ )) ≤ ε2 . Define ∆ = α1−1 (α2 (D01 ) + γ˜ (D03 )) + ε2 . Let T2 = min{T21 , T22 , T23 , T24 } and h2 = min{h21 , h22 , h23 , h24 }. Suppose T ∈ (0, T2 ], h ∈ (0, h2 ] and maxi∈{0,1,...,k} |˜x(i)| ≤ D01 for some

k ∈ {0, 1, . . .}. From Assumption 1, we have F˜Ta,h (˜x(k), w[k]) ≤ α1−1 ◦ VT ,h (F˜Ta,h (˜x(k), w[k])) ≤ α1−1 (VT ,h (˜x(k)) + γ˜ (D03 )) ≤ ∆. Using Assumption 2–3 as well as triangle inequalities, we have |˜x(k + 1)− F˜Ta,h (˜x(k), w[k])| ≤ T (ρ(h))+ L1 |x(k)− xc (k)|+ L2 (eLT − 1)|x(k) − xc (k)|. Applying Claim 1 and from the choice of T24 , h24 and ∆, we have |˜x(k + 1)| ≤ |F˜Ta,h (˜x(k), w[k])| + ε2 = ∆. The proof of Claim 2 is complete. Claim 3. Let W = {w ∈ L∞ |kwk∞ ≤ Cw , ∀Cw > 0} and let α1 , α2 , α3 ∈ K∞ . Let strictly positive real numbers (d, D) be such that 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 exists a function VT ,h : Rnx˜ → R≥ 0 such that for all x˜ ∈ Rnx˜ , we have α1 (|˜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 following holds: VT ,h (˜x(i + 1)) − VT ,h (˜x(i)) ≤ − T4 α3 (|˜x(i)|). Then, for all

(10)

Proof. Let positive real numbers (Cx˜ , Cw , ν) be given. Define ε2 = 1 −1 ν α ( 2 ), ε3 = α2−1 ( 12 α1 (ε2 )), and ∆ = α1−1 (α2 (Cx˜ ) + γ˜ (Cw )) + ε2 . 2 2 Take any ε1 > 0 which satisfies the inequality: ML1 ε1 ≤ 14 α3 (ε3 ). From Claim 1, let (Cx˜ , Cw , ε1 ) generate T41 , h41 and let λ > 0 be as in Claim 1. Let T42 , h42 come from Claim 2 corresponding to (Cx˜ , Cw ) and also the following holds: T42 (ρ(h42 ) + L1 (ε1 + λkwk∞ ) + L2 (eLT42 − 1)(ε1 + λkwk∞ )) ≤ ε2 . Let positive real numbers T43 , h43 , T44 , h44 and T45 be such that: T43 ( 41 α3 (Cx˜ ) + γ˜ (kwk∞ ) + M (ρ(h43 ) + L1 (ε1 + λkwk∞ ) + L2 (eLT43 − 1)(ε1 + λkwk∞ ))) ≤ ν2 , M ρ(h44 ) + ML2 (eLT44 − 1)ε1 ≤

α3 (ε3 ) and T45 γ˜ (Cw ) ≤ 21 α1 (ε2 ). Let γˆ (s) = α2 ◦ α3 (4(γ˜ (s) + M λL1 s + ML2 λ(eL − 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)| ≤ α2−1 ◦ α1 (Cx˜ ) and kwk∞ ≤ Cw . First of all, we claim that, for any i ∈ Z+ , |˜x(i)| ≤ Cx˜ . −1

1 4

From now we suppose this to be true. Applying Claim 2, we have |F˜Ta,h (˜x(i), w[i])| ≤ ∆ and |˜x(i + 1)| ≤ ∆. Suppose VT ,h (˜x(i + 1)) ≥ γˆ (kwk∞ ) + ν2 . Using Assumption 1 and triangle inequalities, we have VT ,h (˜x(i + 1)) − VT ,h (˜x(i)) ≤ −T α3 (|˜x(i)|)+ T γ˜ (kwk∞ )+ M |˜x(i + 1)− F˜Ta,h (˜x(i), w[i])|. Applying Assumption 2–3 and triangle inequalities and from the choice of T41 and h41 , we see that VT ,h (˜x(i + 1))− VT ,h (˜x(i)) ≤ −T α3 (|˜x(i)|)+ T γ˜ (kwk∞ ) + TM (ρ(h) + L1 (ε1 + λkwk∞ ) + L2 (eLT − 1)(ε1 + λkwk∞ )). Denote µ1 := ML1 ε1 , µ2 := M (ρ(h) + L2 (eLT − 1)ε1 ) and κ(s) := γ˜ (s) + M λ(L1 + L2 (eLT − 1))s. Then we have VT ,h (˜x(i + 1)) − VT ,h (˜x(i)) T

T

≤ − α3 (|˜x(i)|) − α3 (α2−1 (VT ,h (˜x(i)))) + T κ(kwk∞ ) 4 | 4 {z } (∗)

T

T

− α3 (|˜x(i)|) + T µ1 − α3 (|˜x(i)|) + T µ2 . | 4 {z }| 4 {z } (∗∗)

(∗∗∗)

We deduce that VT ,h (˜x(i + 1)) ≥ γˆ (kwk∞ )+ ν2 implies γˆ (kwk∞ )+ ν

≤ VT ,h (F˜Ta,h (˜x(i), w[i])) − VT ,h (˜x(i)) + |VT ,h (˜x(i + 1)) − VT ,h (F˜Ta,h (˜x(i), w[i]))| + VT ,h (˜x(i)) ≤ T γ˜ (kwk∞ ) + MT (ρ(h) + L1 (ε1 + λkwk∞ ) + L2 (eLT − 1)(ε1 + λkwk∞ )) + VT ,h (˜x(i)). From 2

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the choice of T43 and h43 , we have γ˜ (kwk∞ ) + ν2 ≤ ν2 + Vh (˜x(i)). Hence we have VT ,h (˜x(i + 1)) ≥ γˆ (kwk∞ ) +

ν 2

⇒ VT ,h (˜x(i)) ≥ γˆ (kwk∞ ). From the definition of γˆ (·), we have Term (∗) ≤ 0 holds. By supposition VT ,h (˜x(i + 1)) ≥ γˆ (kwk∞ ) + ν2 , we have x˜ (i + 1) ≥

α2−1 ( ν2 ) = 2ε2 . Then from the choice of T42 and h42 , we obtain |F˜Ta,h (˜x(i), w[i])| ≥ |˜x(i + 1)| − |˜x(i + 1) − F˜Ta,h (˜x(i), w[i])| ≥ 2ε2 − ε2 = ε2 . Using our choice of T45 , it follows that α2 (|˜x(i)|) ≥ VT ,h (F˜Ta,h (˜x(i), w[i])) − T γ˜ (Cw ) ≥ α1 (|F˜Ta,h (˜x(i), w[i])|) − T γ˜ (Cw ) 1

≥ α1 (ε2 ) − α1 (ε2 ) = 2

1 2

Fig. 1. The performance of the closed-loop system without disturbance under two control schemes.

α1 (ε2 ),

which implies |˜x(i)| ≥ α2 ( 2 α1 (ε2 )) = ε3 ≥ α3−1 (4µ1 ) and then Term (∗∗) ≤ 0 holds. Moreover, from the choice of T44 and h44 , we have |˜x(i)| ≥ ε3 ⇒ − T4 α3 (|˜x(i)|)+T µ2 ≤ 0. Hence, By supposition VT ,h (˜x(i + 1)) ≥ γˆ (kwk∞ )+ ν2 , we have VT ,h (˜x(i + 1))− VT ,h (˜x(i)) ≤ −1 1

− T4 α3 (|˜x(i)|).

ν

Suppose VT ,h (˜x(i + 1)) ≤ γˆ (kwk∞ ) + 2 and VT ,h (˜x(i)) ≥ γˆ (kwk∞ ) + ν . From our choice of T43 , it follows that: VT ,h (˜x(i + 1))− VT ,h (˜x(i)) ≤ γˆ (kwk∞ )+ ν2 − VT ,h (˜x(i))+ ν2 − ν2 ≤ γˆ (kwk∞ )+ ν − VT ,h (˜x(i)) − ν2 ≤ − ν2 ≤ − T4 α3 (|˜x(i)|). It remains to establish our initial claim: |˜x(i)| ≤ Cx˜ for any i ∈ Z+ . This claim follows by induction. Indeed, it clearly holds for i = 0, since |˜x(0)| ≤ α2−1 ◦ 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 = γˆ (kwk∞ ) + ν.), we have |˜x(1)| ≤ Cx˜ . That is, 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 finalized as follows. Let (∆x˜ , ∆w , δ) be given and let all conditions in Theorem 1 hold. Let γˆ ∈ K∞ come from Claim 4. We define (Cx˜ , Cw , ν) as: Cw := ∆w , ν > 0 is such that sups∈[0,∆w ] [α1−1 (γˆ (s) + ν) − α1−1 (γˆ (s))] ≤ δ, Cx˜ :=

max{α1−1 (γˆ (∆w ) + ν), α1−1 ◦ α2 (∆x˜ )}. From the choice of (Cx˜ , Cw , ν), we have Cx˜ ≥ α1−1 (γˆ (Cw ) + ν) and |˜x(0)| ≤ α2−1 ◦ α1 (Cx˜ ). Using Claim 4, let (Cx˜ , Cw , ν) generate T ∗ > 0, h∗ > 0 such that (10) holds. Let D = Cx˜ and d = γˆ (kwk∞ ) + ν , then we have α1 (D) ≥ d. With the definition of (D, d), we have all conditions of Claim 3 are satisfied. Therefore for all h ∈ (0, h∗ ), |˜x(0)| ≤ ∆x˜ and kwk∞ ≤ ∆w , we have |˜x(i)| ≤ β(|˜x(0)|, iT ) + α1−1 (d) ≤ β(|˜x(0)|, iT ) + α1−1 (γˆ (kwk∞ ) + ν) ≤ β(|˜x(0)|, iT ) + γ (kwk∞ ) + δ , where γ (s) := α1−1 ◦ γˆ (s). This completes the proof of Theorem 1. Remark 2. Following the proof of Theorem 1, it is easy to see that if we relax Assumption 1 slightly to the assumption of Practical Lyapunov-ISS, that is, VT ,h (F˜Ta,h (˜x, wf )) − VT ,h (˜x) ≤ −T α3 (|˜x|) + T γ˜ (kwk∞ ) + T δ1 holds, then Theorem 1 still holds. Example. Consider the continuous-time plant x˙ (t ) = x3 (t ) + u(t )+w(t ). Let x(i + 1) = FTe (x(i), u(i), w[i]) be the exact discretetime model of the continuous-time plant with the sampling period T . Let fh (x, u, w) represent one step of the numerical integration routine on the sampling interval [iT , (i + 1)T ) defined R iT +h by fh (x, u, w) = x + h(x3 + u) + iT w(s)ds := fh1 (x, u, w). Then

we can generate its numerically integrated approximate model FTa,h (·, ·, ·) by fh (k, x, u, w) := x + h(x3 + u) +

iT +(k+1)h

Z

w(s)ds

iT +kh

fhk+1 (x, u, w) := fh (k + 1, fhk , u, w) FTa,h (x, u, wf ) := fhN (x, u, w),

k = 1, 2, . . .

where h represents the integration period, T is the sampling period and N = Th . Moreover, the approximate model with disturbance free, which reconstructs the missing plant states between samples, is generated by FTa,h (x, u, 0). Consider a digital controller: u(i) =

−x(i) − x3 (i). We first check the consistency of the approximation scheme. By Lemma Π .2 in Nesic and Laila (2002), fh is one-step

consistent with Fhe where Fhe is the exact discrete-time model with the sampling period h. Also, the multi-step consistency is guaranteed by the one-step consistency plus the uniform Lipschitz condition on fh (see Remark 13 in Nesic and Teel (2004)). Then following closely the conclusions of Corollary 4 and Remark 14 in Nesic and Teel (2004), we have that FTa,h (x, u, wf ) is one-step consistent with FTe (x, u, wf ). Take VT ,h (x) = |x|, it follows that the approximate discrete-time model x(i + 1) = FTa,h (x(i), u(i), w[i]) with u(i) = −x(i) − x3 (i) is practical Lyapunov-ISS with α3 (|x|) := |x| and γ˜ (kwk∞ ) := kwk∞ . Moreover it is easy to see that Assumption 3 is also satisfied. We conclude from Theorem 1 that the exact closed loop system is semiglobally practically inputto-state stable. Assume x(0) = 3.4. Our simulation shows that the single-rate method stabilizes the system without disturbance only when T < 0.205 s. On the other hand, consider the dualrate method with low measurement rate Tm = 1.5 s. Setting Ti = 0.15 s and h = 0.0075 s (N = 20), our simulation shows that the dual-rate controller stabilizes the system successfully (Fig. 1). Comparing with the response obtained under single rate scheme, we see that advantages of dual-rate control system, which can render a stable closed loop using much lower sampling rate. In Fig. 2 we take a sinusoidal disturbance of amplitude 0.8 and frequency 2 rad/s and our simulation shows that the closed-loop system is practical ISS. This example shows that the dual-rate inferential system is indeed more robust than the corresponding fast single-rate system. 4. Conclusions In this paper, we concentrate on the problem of sampleddata input-to-state stabilization of nonlinear systems under low measurement constraint. Our approach to the solution of this problem employs a dual-rate scheme based on discrete-time

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

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Nesic, D., & Laila, D. S. (2002). A note on input-to-state stabilization for nonlinear sampled-data systems. IEEE Transactions on Automatic Control, 47, 1153–1158. Nesic, D., & Teel, A. R. (2004). A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models. IEEE Transactions on Automatic Control, 49, 1103–1122. Nesic, D., & Teel, A. R. (2006). Stabilization of sampled-data nonlinear systems via backstepping on their Euler approximate model. Automatica, 42, 1801–1808. Polushin, I. G., & Marquez, H. J. (2004). Multirate versions of sampled-data stabilization of nonlinear systems. Automatica, 40, 1035–1041. Sontag, E. D. (1989). Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control, 34, 435–443. Sontag, E. D. (1998). Comments on integral variants of ISS. System and Control Letter, 34, 93–100.

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

controller design, since the fast sampling results may not be implemented due to hardware limitations. The main idea is to introduce a controller that includes an approximate discrete-time model of the plant. The control action depends on the state of this model which is corrected from time to time using the low rate measurements of the actual state of the plant. We show that if one designs a discrete-time controller for an approximate discrete-time plant model so that the closed-loop system is inputto-state stable, then the input-to-state stability property will be preserved for the exact discrete-time plant model based on a dualrate control scheme in a semi-globally practical sense. Acknowledgements The authors are grateful to the reviewers and editor for suggesting the strengthened version of the approximate numerical integration model used here. References Chen, T., & Francis, B. A. (1991). Input–output stability of sampled-data systems. IEEE Transactions on Automatic Control, 36, 50–58. Khalil, H. K. (1996). Nonlinear systems. New Jersey: Prentice-Hall. Li, D. G., Shah, S. L., & Chen, T. (2002). Analysis of dual-rate inferential control systems. Automatica, 38, 1053–1059. Nesic, D., Teel, A. R., & Kokotovic, P. V. (1999). Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations. System and Control Letter, 38, 259–270. Nesic, D., & Teel, A. R. (2001). Sampled-data control of nonlinear systems: An overview of recent results. In Perspectives on robust control (pp. 221–239). New York: Springer.

Xi Liu received the B.Sc. degree in applied mathematics from Liaoning University, People’s Republic of China, and the M.Sc. degree in control theory from Northeastern University, People’s Republic of China, in 2001 and 2004, respectively. Currently, she is a Ph.D. candidate in the Department of Mathematical and Statistical Sciences at the University of Alberta. Her current research interests are dynamical systems, sampled-data nonlinear control analysis and design, and network based control systems.

Horacio J. Marquez received the B.Sc. degree from the Instituto Tecnologico de Buenos Aires (Argentina), and the M.Sc.E and Ph.D. degrees in electrical engineering from the University of New Brunswick, Fredericton, Canada, in 1987, 1990 and 1993, respectively. From 1993 to 1996 he held visiting appointments at the Royal Roads Military College, and the University of Victoria, Victoria, British Columbia. Since 1996 he has been with the Department of Electrical and Computer Engineering, 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 Robust and Nonlinear Control and Associate Editor of the J. of the Franklin Institute. He is the Author of ‘‘Nonlinear Control Systems: Analysis and Design’’ (Wiley, 2003). He is the recipient of the 2003–2004 McCalla Research Professorship awarded by the University of Alberta. His current research interests include nonlinear dynamical systems and control, nonlinear observer design, robust control, and applications. Yanping Lin received the B.Sc. degree in applied mathematics from Northeastern University, People’s Republic of China, in 1982, and the M.Sc. and Ph.D. degrees in applied mathematics from Washington State University, USA, in 1985 and 1988, respectively. After post-doctorial fellowships from McGill University and University of Waterloo from 1988 to 1990, he has been with the Department of Mathematical and Statistical Sciences, University of Alberta since 1991, and he is currently a Professor and a Professor at the Hong Kong Polytechnic University. He is the recipient of the 2004–2005 McCalla Research Professorship awarded by the University of Alberta. His current research areas are mathematical modeling, scientific computing, numerical analysis and control theory.

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