Multirate versions of sampled-data stabilization of nonlinear systems

Automatica 40 (2004) 1035 – 1041

www.elsevier.com/locate/automatica

Brief paper

Multirate versions of sampled-data stabilization of nonlinear systems
Ilia G. Polushina;∗;1 , Horacio J. Marquezb a Department

b Department

of Electrical Engineering, Lakehead University, Thunder Bay, Ont., Canada P7B 5J2 of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4

Received 27 November 2002; received in revised form 29 November 2003; accepted 16 January 2004

Abstract The problem of state feedback sampled-data stabilization of nonlinear systems is considered under the “low measurement rate” constraint and in the presence of (not necessarily small) time delay in the measurement channel. A multirate control scheme is proposed that utilizes a numerical integration scheme to approximately predict the current state from the delayed measurements. For both the controller emulation approach and the approach based on approximate discrete-time model of the system, we show that under standard assumptions the closed-loop multirate sampled data system is asymptotically stable in the semiglobal practical sense. An illustrative example of sampled-data control of vertical take o9 and landing aircraft is discussed. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Multirate sampled-data systems; Nonlinear control; Time delays; Limited data rates

1. Introduction Owing to the fact that modern control systems usually employ digital technology for controller implementation, the study of sampled-data control systems (Araki, 1993; Chen & Francis, 1995) has become an important part of control science. Signi@cant progress has been made in this area in recent years. See NesiAc and Teel (2001a, b), NesiAc and Laila (2002), NesiAc and Angeli (2002), NesiAc, Teel, and Sontag (1999b), Grune and NesiAc (2003). In particular, in NesiAc and Teel (2001b) two di9erent approaches to sampled-data
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Jessy W. Grizzle under the direction of Editor Hassan Khalil. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). ∗ Corresponding author. Tel.: +1-807-346-7921; fax: +1-807-3438928. E-mail addresses: [email protected] (I.G. Polushin), [email protected] (H.J. Marquez). 1 On leave from CCS Lab., Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia. The paper was written when the @rst author was with the Department of Electrical and Computer Engineering, University of Alberta.

0005-1098/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.01.017

controller design for nonlinear systems were described. The @rst one, so-called controller emulation, involves digital implementation of a continuous-time stabilizing control law at a suJciently high sampling rate. The second approach consists of discretizing the plant model and then proceed to design a discrete-time control law. Both of these approaches are essentially single-rate, i.e. sampling rates of the input and the measurement channels are assumed to be equal. In practical applications, however, hardware restrictions on input and measurement sampling rates can be essentially different. For example, the discrete–analog (D/A) converters are generally faster than the analog–discrete (A/D) converters, so the input sampling rate can be made higher than the measurement one. Moreover, in some cases, such as visual servo control systems (Hitchinson, Hager, & Corke, 1996; Fujimoto & Hori, 2001), the sensor channel contains complex image processing blocks that cause slowing down of the measurement sampling rate together with delays in the measurement process. Note that, even in the absence of the measurement delay, none of the mentioned approaches directly leads to controllers that can successfully operate under restrictions on measurement sampling rate. In this paper a di9erent scenario is considered, namely, the use of a multirate control scheme. We address the

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I.G. Polushin, H.J. Marquez / Automatica 40 (2004) 1035 – 1041

problem of sampled-data stabilization of nonlinear systems under the “low measurement rate” constraint and in the presence of delay in the measurement channel. The idea behind our approach is the following: we propose a multirate controller that contains a “fast” numerical integration scheme that allows us to reconstruct approximately state trajectories between samples. This scheme is updated periodically using the “low-rate delayed” measurements of the actual state. The control law, in turn, depends on the state of the model rather than the actual state of the plant. We address the design of multirate controllers based on the controller emulation approach as well as on the approach that utilizes the properties of approximate discrete-time models. In both cases we show that under the standard assumptions taken from NesiAc, Teel and KokotoviAc (1999a), and Teel, NesiAc, and KokotoviAc (1998), for any measurement delay and for an arbitrary slow output measurement sampling rate, the proposed multirate scheme makes the system asymptotically stable in the semiglobal practical sense. The following standard notations and de@nitions will be used throughout the paper. Denote R+ := [0; +∞). A continuous function : R+ → R+ is said to belong to class K( ∈ K) if (0) = 0 and it is strictly increasing. Also, a continuous function : R+ × R+ → R+ is said to belong to class KL( ∈ KL), if for each @xed t ¿ 0, (·; t) belongs to class K and for each @xed s ¿ 0, (s; t) decreases to zero as t → +∞.

unstable closed-loop sampled data system (see example in Section 5). On the other hand, let the D/A converter be described for each t ∈ [jTi ; (j + 1)Ti ) as a zero-order hold of the form u(t) = v(j), where Ti ¿ 0 is the input sampling period. We endeavour to solve the following problem: for any given minimal measurement sampling period Tm∗ ¿ 0 and any given minimal measurement delay ∗ ¿ 0 @nd a digital controller which makes (possibly after choosing Ti ¿ 0 appropriately small) the closed-loop system asymptotically stable with the prescribed (@nite) restriction and the prescribed (nonzero) o9set.

2. Statement of the problem

Given h ¿ 0, denote fh (x; u) := x +hf(x; u). Thus, x(k + 1) = fh (x(k); u(k)) is the discrete-time Euler approximation of system (1) with integration period h ¿ 0. Here h ¿ 0 will always be taken from a set H de@ned as follows: Assuming without loss of generality that the measurement sampling period Tm ¿ Tm∗ and the delay  ¿ ∗ are chosen such that p1 Tm = p2  for some integers p1 ¿ 0, p2 ¿ 0, we de@ne H as follows:
 Tm H= ; kh = 1; 2; : : : : (2) kh p 2

Consider a nonlinear system of the form x˙ = f(x; u);

(1)

where x ∈ Rn is the state, u ∈ Rm is the input, and f(·; ·) is locally Lipschitz in both arguments. In sampled-data control systems, a continuous-time plant is connected with a digital controller via the A/D and D/A converters (Chen & Francis, 1995). In this paper, we will consider the important practical case where hardware restrictions are imposed on the “measurement-A/D conversion” process. More precisely, we address the problem of multi-rate sampled data state feedback stabilization of system (1) under “low measurement rate” constraint and in the presence of delay in the measurement channel, described as follows: We assume that the minimal output sampling period Tm∗ ¿ 0 as well as the minimal measurement delay ∗ ¿ 0 are given, so that the “measurement-A/D conversion” process is described for each i ∈ {0; 1; : : :} by the equation y(i) := x(iTm − ), where the measurement sampling period Tm and the delay  can be chosen by the designer but must satisfy the constraints Tm ¿ Tm∗ ,  ¿ ∗ . Informally speaking, we address the case where the constants Tm∗ , ∗ are large enough, i.e. the measurement sampling rate is low, and the measurement delay is large. In this case, the single rate design methods presented in NesiAc and Teel (2001b) may lead to

3. Multirate design based on continuous-time model In this section, we consider a multi-rate control scheme that based on the knowledge of the solution of the continuous-time stabilization problem for the system (1). More precisely, we address the problem formulated in the previous section under the following two assumptions. Assumption 1. There exists a state feedback locally Lipschitz control law : Rn → Rm such that the equilibrium x = 0 of the closed-loop continuous-time system x˙ = f(x; (x)) is globally asymptotically stable. Assumption 2. The value of input sampling period Ti ¿ 0 can be assigned arbitrarily.

Note that, if h ∈ H, then Tm = kh p2 h, and  = kh p1 h, where kh ∈ {1; 2; : : :}. Now, let us denote the sequence of “low-rate” delayed measurements as follows: y(j) = x(jTm − ) = x(hkh (jp2 − p1 )):

(3)

We propose a digital controller with the following structure: xc (i + 1) = Fh (i);

(4)

v(i) = (xc (i));

(5)

where Fh (i) = fh (x(i); ˜ (xc (i))) for i = jkh p2 − 1, j = 0; 1; : : :, Fh (i) = fh (xc (i); (xc (i))) otherwise, and x(i) ˜ = fh (: : : (fh (y(j); (xc (i + 1 − kh p1 ))); : : : ; (xc (i − 1))) is the Euler approximate estimate of the state x(ih) based on measurements (3). Finally, using assumption 2, we set Ti = h, so that u(t) = v(i)

for t ∈ [iTi ; (i + 1)Ti ):

(6)

I.G. Polushin, H.J. Marquez / Automatica 40 (2004) 1035 – 1041

Remark 3. The idea behind controller (4)–(5) is to use the Euler discretization to approximate the system’s current state based on the delayed low-rate measurements. Note that, due to the delay , each measurement y(j) becomes available to the controller at the (jp2 kh )th step. The following theorem is the main result of this section. For each t ∈ [ih; (i + 1)h), let us denote
 x(t) := max max |x(t)|; max |xc (j)| : s∈[t−; t]

j∈{i−kh p1 ; i}

Theorem 4. Consider the system (1)–(6). Under Assumptions 1 and 2, there exists  ∈ KL such that the following holds. Given Tm ¿ Tm∗ ,  ¿ ∗ ,  ¿ 0,  ¿ 0, there exists hmax ¿ 0 such that if h ∈ H ∩ (0; hmax ], then all trajectories with initial conditions x(0) 6  satisfy |x(t)| 6 (|x(0)|; t) +  |xc (i)| 6 (|x(0)|; ih) + 

for all t ¿ 0; for all i ∈ {0; 1; : : :}:

(7) (8)

Proof. We start with the following claim. Claim 5. The closed-loop sampled data system (1), (3)– (6) has the following property: given  ¿ 0,  ¿ 0, there exists h∗1 ¿ 0 such that if h ∈ H ∩ (0; h∗1 ], x(0) 6 , and maxs∈[0; t] |x(s)| 6  for some t ¿ 0, then |x(ih)−xc (i)| 6  for all i ∈ {0; 1; : : :} such that ih 6 t. This claim is a consequence of well-known properties of the Euler numerical approximation scheme (see, for example, Gear (1971)). The proof is omitted due to space reasons. Now, by Assumption 1, the equilibrium x = 0 of the closed loop continuous-time system is globally asymptotically stable, therefore (see, for example (Isidori (1999, Theorem 10.4.3)) there exist a smooth function V : Rn → R+ , a gain 0 ∈ K∞ , and a smooth invertible matrix function G(x) ∈ Rm×m such that 1 (|x|) 6 V (x) 6 2 (|x|), and V (x) ¿ 0 (|w|) implies (@V=@x)f(x; (x)+G(x)w) 6−3 (V (x)) for some i ∈ K∞ , i = 1; 2; 3. Claim 6. The closed-loop sampled data system (1),(3)–(6) has the following property: for any D2 ¿ 0, $ ¿ 0 there exists h∗2 ¿ 0 such that if h ∈ H ∩ (0; h∗2 ], and x(0) 6 D2 , then for any t ¿ 0 V (x(t)) ¿ $ ⇒ V˙ (x(t)) 6 − 3 (V (x(t))):

(9)

Proof. Let D2 ¿ 0, $ ¿ 0 be given. Without loss of generality assume that $ ¡ 2 (D2 ). Denote 1 := 21−1 ◦ 2 (D2 ). ˜ 1− Let ˜ ∈ K be a function such that |(x1 ) − (x2 )| 6 (|x x2 |) if |x1 |, |x2 | 6 1 . Such a function ˜ always exists due to continuity of (·) and compactness of the set −1 {x: |x| 6 1 }. Also, denote G0 := maxx∈B( (x)|, S 1 ) |G

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and let ˜ be a positive constant such that ˜ ¡ 1−1 ◦ 2 (), and $ ¿ 0 (G0 (2 ˜ )). ˜ Further, let h∗21 ¿ 0 be a constant given by Claim 5 corresponding to  = D2 , and ˜  = . ˜ Denote h∗21 = =max |x1 |;|x2 |61 |f(x1 ; (x2 ))|, and put h∗2 = min{h∗21 ; h∗22 }. Suppose h ∈ H ∩ (0; h∗2 ]. By assumption, there exists t0 ¿ 0 such that V (x(t)) 6 2 (D2 ) holds at least for all t ∈ [0; t0 ]. Therefore |x(t)| 6 1−1 ◦ 2 () = 1 =2 holds for all ∈ [0; t0 ]. Applying Claim 5, we get |x(ih) − xc (i)| 6 , ˜ therefore |xc (i)| 6 1 holds for all i ∈ {0; 1; : : :} such that ih 6 t0 . Due to our choice of h∗2 , we have |x(t) − x(ih)| 6 ˜ for all t ∈ [0; t0 ] ∩ [ih; (i + 1)h). Now, denote Tu(t) = (x(t)) − (xc (i)). Substituting G −1 (x)Tu for w, one gets for each t ∈ [0; t0 ] ∩ [ih; (i + 1)h) that V (x(t)) ¿ 0 (|G −1 (x(t))Tu(t)|) implies @V f(x(t); (xc (i))) 6 − 3 (V (x(t))): @x

(10)

Note that |Tu(t)| 6 (|x(t) ˜ − xc (i)|) 6 (2 ˜ ). ˜ Combining all the above, due to choice of ˜ we see that (9) is valid at least for all t ∈ [0; t0 ]. Now, let us show that V (x(t)) 6 2 (D2 ) holds for all t ¿ 0. Indeed, assume the converse, then there exists t0 ¿ 0 such that V (x(t0 ))=2 (), and V (x(t)) ¿ 2 () for t ∈ (t0 ; t0 + $) for some $ ¿ 0. On the other hand, since (9) is valid for t = t0 , we get V˙ (x(t0 )) 6 − 3 ◦ 2 () ¡ 0. This contradiction shows that V (x(t)) 6 2 (D2 ) for all t ¿ 0, and therefore (9) is also valid for all t ¿ 0. This completes the proof of Claim 6. Let  ¿ 0,  ¿ 0 be given. Using Claim 5, @nd h∗1 ¿ 0 corresponding to D1 = 1−1 ◦ 2 () and  = 1=2. Then, using Claim 6, @nd h∗2 ¿ 0 corresponding to D2 =  and $ = 1 (1=2). Put h∗ = min{h∗1 ; h∗2 }. It is a well-known fact (see, for example, Khalil, 1996) that (9) implies the existence of 0 ∈ KL such that V (x(t)) 6 max{0 (V (x(0)); t); $}; (without loss of generality it is assumed that 3 (·) ∈ K∞ in (9) is locally Lipschitz). Therefore, |x(t)| 6 1−1 (max{0 (2 (|x(0)|); t); $}) 6 (|x(0)|; t) + =2;

(11)

where (·; t) := 1−1 ◦ 0 (2 (| · |); t) ∈ KL, which proves (7). On the other hand, due to the choice of h∗ , and using Claim 5, we get |x(ih) − xc (i)| 6 =2 for all i ∈ {0; 1; : : :}. The combination of the last inequality with (11) proves (8). The proof of Theorem 4 is complete. Remark 7. The proposed controller (4), (5), includes the Euler-type approximation model of system (1). We address the Euler-type numerical approximation because of it’s simplicity, but only the approximation properties of the Euler-type scheme (given by Claim 5) are actually utilized in the proof of Theorem 4. Any other approximation satisfying the conditions of Claim 5 can be used in the control algorithm (4), (5).

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I.G. Polushin, H.J. Marquez / Automatica 40 (2004) 1035 – 1041

Remark 8. It is easy to see that (7), (8) imply that x(t) 6 1 ( x(0) ; t) +  for some 1 ∈ KL, which corresponds to standard de@nition of (practical) asymptotic stability. The presented formulation of Theorem 4, however, gives slightly more precise estimates for the trajectories of the closed-loop sampled-data system. See also NesiAc et al. (1999b). 4. Multirate design based on discrete-time model In this section, we address an alternative approach to the design of multi-rate sampled-data systems, which utilizes the knowledge of properties of the set of approximate discrete-time models (see NesiAc & Teel, 2001a, NesiAc & Angeli, 2002, NesiAc & Laila, 2002, Grune & NesiAc, 2003). Let x(i + 1) = F(x(i); u(i))

(12)

be the exact discrete-time model of the continuous-time system (1) with sampling period T ¿ 0. Throughout this section, we assume that the input sampling period Ti is @xed and equal to the sampling period of the model (12), i.e. Ti = T . We will call Ti the “sampling period”. Following the general statement of our problem (see Section 2), we assume that the measurement sampling Tm and the measurement delay  satisfy the constraints Tm ¿ Tm∗ ,  ¿ ∗ , where Tm∗ ¿ 0 and ∗ ¿ 0 are given. Without loss of generality we assume that both Tm and  are multiples of T , i.e.  = l1 T , Tm = l2 T for some integers l1 ¿ 0, l2 ¿ 0. The sequence of measurements y(·) will be numbered as follows: y(j) = x(jl2 − l1 );

j = 0; 1; : : : :

(13)

Since the measurement signal y(·) comes with delay l1 steps, y(j) becomes available to the controller at the jl2 th step. The exact discrete-time model (12) of the nonlinear system (1) is usually unknown. Following NesiAc et al. (1999a), we assume that a family of approximate discrete-time models x(i + 1) = Fh (x(i); u(i)) corresponding to the sampling period T ¿ 0 is available, parameterized by the modelling parameter h ¿ 0. Usually, h ¿ 0 is the integration period of the numerical integration used to generate the family of approximate models. In the following, we assume that the family of approximate models Fh (·; ·) satis@es the following assumptions (which are similar to those used in NesiAc et al. (1999a)). Assumption 9. (Equi-global asymptotic stabilizability of approximate models with equi-Lipschitz Lyapunov functions). There exist 1 ; 2 ∈ K∞ , 3 ∈ K, and h∗ ¿ 0 such that for each h ∈ (0; h∗ ] there exist a function Vh : Rn → R+ and a control law h (·) with the following properties: (i) 1 (|x|) 6 Vh (x) 6 2 (|x|) for each h ∈ (0; h∗ ]. (ii) for each D ¿ 0 there exist M ¿ 0 and h∗1 ∈ (0; h∗ ] such that |Vh (x1 ) − Vh (x2 )| 6 M |x1 − x2 | for each S and each h ∈ (0; h∗1 ]. x1 ; x2 ∈ B(D)

(iii) for any 1 ¿ 0 there exists h∗2 ∈ (0; h∗ ] such that sup{|h (x)|: h ∈ (0; h∗2 ]; |x| 6 1 } ¡ ∞. (iv) Vh (Fh (x; h (x)) − Vh (x) 6 − 3 (|x|) holds for each x ∈ Rn and each h ∈ (0; h∗ ]. Assumption 10. (Uniform local Lipschitz property of the approximate models). For any 1 ; 2 ¿ 0 there exist L ¿ 0 and h∗ ¿ 0 such that |Fh (x; u) − Fh (z; u)| 6 L|x − z| for any S 1 ), any u ∈ B( S 2 ), and any h ∈ (0; h∗ ]. x; z ∈ B( Assumption 11. (Consistency of the approximation scheme). For any 1 ; 2 ¿ 0 there exists a K-class function S 1) +(·) such that |F(x; u) − Fh (x; u)| 6 +(h) for any x ∈ B( S and any u ∈ B(2 ). To stabilize system (12) subject to the measurements (13), we will use a multi-rate control strategy similar to one presented in Section 3. The proposed multi-rate digital controller is described as follows: xc (i + 1) = Fh (i);

(14)

u(i) = h (xc (i));

(15)

where, again, Fh (i) = Fh (x(i); ˜ h (xc (i))) for i = l2 j − 1, j = 0; 1; : : :, Fh (i) = Fh (xc (i); h (xc (i))) otherwise, and x(i) ˜ := Fh (: : : (Fh (y(j); h (xc (i + 1 − l1 ))); : : : ; h (xc (i − 1))) is an approximate estimate of the state x(i). Denote
 x(i) := max max |x(k)|; max |xc (j)| : k∈{i−l1 ; i}

j∈{i−l1 ; i−1}

The following theorem is the main result of this section: Theorem 12. Consider a system (12) with sampling period T ¿ 0. Suppose a family of approximate models is available that satis=es Assumptions 9–11. Then there exists  ∈ KL such that the following holds. Given  ¿ 0,  ¿ 0,  = l1 T , where l1 ∈ {0; 1; : : :}, and Tm = l2 T , where l2 ∈ {1; 2; : : :}, there exists hmax ¿ 0 such that if h ∈ (0; hmax ], then all trajectories of the closed loop system (12)–(15) with initial conditions x(0) 6  satisfy max{|x(i)|; |xc (i)|} 6 (|x(0)|; iT ) +  for all i ∈ {0; 1; 2; : : :}. Proof. We begin with the following claim. Claim 13. Given D1 ¿ 0, 1 ¿ 0, there exists h1 ¿ 0 such that for all h ∈ (0; h1 ] the following holds. If x(0) 6 D1 , and maxi∈{0; 1; :::; k} |x(i)| 6 D1 for some k ∈ {0; 1; : : :}, then |x(k) − xc (k)| 6 1 , and |x(k + 1) − Fh (xc (k); h (xc (k)))| 6 1 . Proof. Let D1 ¿ 0, 1 ¿ 0 be given, and assume that 1 ¡ D1 . By Assumption 9(iii) there exist h11 ¿ 0, D2 ¿ 0 such that |h (x)| 6 D2 for all h ∈ (0; h11 ], x ∈ BS n (2D). Also, let h12 ¿ 0, L ¿ 0 be as in Assumption 10 corresponding to 1 = 2D1 , 2 = D2 , and let +(·) ∈ K∞

I.G. Polushin, H.J. Marquez / Automatica 40 (2004) 1035 – 1041

be a function from Assumption 11 corresponding to 1 = 2D1 and 2 = D2 . Consider a sequence of functions +i (·) ∈ K, de@ned recursively by the formulas +1 (·) := +(·), +i+1 (·) := +(·) + L+i (·), i ∈ {1; 2; : : : ; l1 + l2 − 1}. Let h13 ¿ 0 be such that +i (h13 ) 6 1 for all i ∈ {1; 2; : : : ; l1 + l2 }. Put h1 = min{h11 ; h12 ; h13 }. Suppose h ∈ (0; h1 ], x(0) 6 D1 , and maxi∈{0; 1; :::; k} |x(i)| 6 D1 for some k ∈ {0; 1; : : :}. Let j ∈ {0; 1; : : :} be such that k ∈ {jl2 ; : : : ; (j + 1)l2 − 1}. We have |x(jl2 − l1 + 1) − Fh (y(j); h (xc (jl2 − l1 )))| 6 +(h) := +1 (h). Using Assumptions 10 and 11, as well as triangle inequalities, we obtain by induction that |x(i) − xc (i)| 6 +i−( jl2 −l1 ) (h) and |x(i + 1) − Fh (xc (i); h (xc (k)))| 6 +i+1−( jl2 −l1 ) (h) hold for all i ∈ {jl2 − l1 + 1; k}. Note that, by the choice of h, we have +i (h) 6 1 for all i ∈ {1; 2; : : : ; l1 + l2 }. This completes the proof of Claim 13. Claim 14. For any D2 ¿ 0 there exists h2 ¿ 0 such that if x(0) 6 D2 , h ∈ (0; h2 ], then for all i ∈ {0; 1; : : :} Vh (x(i)) 6 2 (D2 ):

(16)

Proof. Suppose D2 ¿ 0 is given. From Assumption 9(ii), let M ¿ 0 and h21 ¿ 0 be generated by D = 21−1 ◦ 2 (D2 ). Denote (·) := 3 ◦2−1 (·), and take any 1 ∈ (0; D2 ) which also satis@es the inequality −1 (2M1 ) + M1 ¡ 2 (D2 ). From Claim 13, let D1 =1−1 ◦2 (D2 ) and 1 ¿ 0 generate h22 ¿ 0. Put h2 = min{h21 ; h22 }. Finally, denote 2 = M1 . Suppose x(0) 6 D2 . Assume there exists i ∈ {0; 1; : : :} such that (16) is valid for i ∈ {0; 1; : : : i }, but Vh (x(i + 1)) ¿ 2 (D2 ). Then by Claim 13 we get |x(i ) − xc (i )| 6 1 , |x(i + 1) − Fh (xc (i ); h (xc (i )))| 6 1 :

(17) )| 6 21−1

By the choice of 1 we see that |xc (i therefore by Assumption 9(ii), it follows that

◦ 2 (D2 ),

Vh (xc (i )) 6 Vh (x(i )) + M1 6 2 (D2 ) + 2 :

(18)

By Assumption 9, we have Vh (Fh (xc (i ); h (xc (i )))) − Vh (xc (i )) 6 − (Vh (xc (i))), and it is easy to check that we always have Vh (Fh (xc (i ); h (xc (i )))) 6 2 (D2 )−2 . By Assumption 9(ii), the last implies that |Fh (xc (i ); h (xc (i ))) − z| ¿ 2 =M = 1 for any z ∈ Rn such that Vh (z) ¿ 2 (D2 ). In particular, we get |Fh (xc (i ); h (xc (i ))) − x(i + 1)| ¿ 1 , which contradicts (17). This contradiction proves Claim 14. Claim 15. There exists 0 ∈ KL such that the following holds. For any D3 ¿ 0, d ¿ 0 there exists h3 ¿ 0 such that if h ∈ (0; h3 ], then all trajectories of the closed loop system (12)–(15) with initial conditions x(0) 6 D2 satisfy for all i ∈ {0; 1; : : : ; } the inequality Vh (x(i)) 6 max{0 (Vh (|x(0)|); iT ); d}:

(19)

Proof. Let D3 ¿ 0, d ¿ 0 be given. From Claim 14, let h31 ¿ 0 be generated by D2 = D3 . From Assumption 9(ii), let h32 ¿ 0 and M ¿ 0 be generated by

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D = 21−1 ◦ 2 (D3 ). Take any 1 ∈ (0; D3 ) satisfying 2M1 + 3 (1 ) 6 min{0 (d=2); d=2}. Further, from Claim 13, let h33 ¿ 0 be generated by D1 = 1−1 ◦ 2 (D3 ) and 1 . Put h3 = min{h31 ; h32 ; h33 }. By Claim 14, we have |x(i)| 6 1−1 ◦ 2 (D3 ) for all i ∈ {0; 1; : : : ; }. Applying Claim 13, we see that |x(i) − xc (i)| 6 1 and |x(i + 1) − Fh (xc (i); h (xc (i)))| 6 1 hold for all i ∈ {0; 1; : : : ; }. In particular, it follows that both the inequalities |xc (i)| 6 21−1 ◦ 2 (D3 ) and |Fh (xc (i); h (xc (i)))| 6 21−1 ◦ 2 (D3 ) hold for all i ∈ {0; 1; : : : ; }. Therefore, by Assumption 9(ii), for all i ∈ {0; 1; : : : ; } we get |Vh (x(i)) − Vh (xc (i))| 6 M1 and |Vh (x(i + 1)) − Vh (Fh (xc (i); h (xc (i))))| 6 M1 . Taking into account all the above, and using simple triangle inequality, it is easy to obtain that Vh (x(i + 1)) − Vh (x(i)) 6 2M1 − 3 (|xc (i)|): Since 3 ( 12 |x(·)|) 6 3 (|xc (·)|) + 3 (|x(·) − xc (·)|), we see that Vh (x(i+1))−Vh (x(i)) 6 2M1 +3 (1 )−20 (Vh (x(i))), where 0 (·) := 12 3 ( 12 2−1 (·)) ∈ K, and by the choice of 1 we have Vh (x(i + 1)) − Vh (x(i)) 6 min{0 (d=2); d=2} − 20 (Vh (x(i))). Using the last inequality, it is easy to check that max{Vh (x(i + 1)); Vh (x(i))} ¿ d implies Vh (x(i + 1)) − Vh (x(i)) 6 − 0 (Vh (x(i))). The last implies that there exists 0 ∈ KL independent on d and such that (19) holds (NesiAc et al., 1999a). This concludes the proof of the claim. Now, the proof of Theorem 12 can be @nalized as follows. Let  ¿ 0,  ¿ 0 be given. Let 0 ∈ KL be a function given by Claim 15, and let h1 ¿ 0 be generated by D3 =  and d = 1 (=2). From Claim 13, let h2 ¿ 0 be generated by D2 = 1−1 (0 (2 (; 0))), and 1 = =2. Take hmax = min{h1 ; h2 }. Applying Claim 15, and using Assumption 9(i), we see that |x(i)| 6 (|x(0)|; iT ) + =2, where (x; t) = 1−1 (0 (2 (|x|; t))) ∈ KL. On the other hand, by Claim 13, we have |x(i) − xc (i)| 6 =2. The proof of Theorem 12 is complete.

5. Example: VTOL aircraft An interesting area of application for multirate control is visual based control of autonomous vehicles. Indeed, the presence of a visual processing block in the measurement channel usually results in a larger measurement sampling period together with measurement delay. In this case, the stability and performance of the sampled-data system may be drastically improved using multirate control. To illustrate, consider the problem of hovering control of the vertical take o9 and landing (VTOL) aircraft. The following simpli@ed model taken from Hauser, Sastry, and Meyer (1992) describes the motion of the VTOL aircraft in the vertical

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(x; y)-plane xV = −u1 sin- + u2 cos-; yV = u1 cos- + u2 sin- − 1;

(20)

-V = u2 : Here, x; y are the position coordinates of the aircraft in the vertical–lateral plane, - is the roll angle, and the control inputs u1 , u2 are the thrust and the rolling moment, respectively. Also,  is a small positive coeJcient which represents the coupling between the rolling moment and the lateral acceleration of the aircraft. As shown in Sepulchre, Jankoric, and Kokotovic (1997), the origin (x; y; -) = (0; 0; 0) of system (20) can be stabilized by the following continuous time control law:  u1 = v12 (x; x) ˙ + (v2 (y; y) ˙ + 1)2 ;    −v1 (x; x) ˙ ˙ − k -; (21) u2 = −k 2 - − tan−1 v2 (y; y) ˙ +1 where v1 (x; x) ˙ = −k11 x − k12 x, ˙ v2 (y; y) ˙ = −k21 y − k22 y, ˙ and k11 ; k12 ; k21 ; k22 ¿ 0. Setting k = 10, k11 = k12 = k21 = k22 = 1 and  = 0:01, we studied a single-rate sampled data implementation of this controller using controller emulation. Assuming initial conditions are x(0) = 5, y(0) = 0, -(0) = 0 our simulation shows that, even without measurement delay, this single-rate sampled-data controller stabilizes the origin of the system only when the sampling period T ¡ 0:2 s. On the other hand, we studied a multirate sampled-data system with low measurement sampling rate Tm = 1 s and in presence of measurement delay  = 1 s. Setting the input sampling period Ti = 0:1 s, and the integration step of the controller h = 0:01 s, our simulations show that the proposed multirate controller successfully stabilizes the system. Moreover, the form of transient response curve is almost identical to the one obtained using continuous time controller. 6. Conclusions In this paper we have addressed the problem of sampled-data stabilization of a nonlinear system under “low-measurement rate” constraints and in presence of (not necessarily small) delay in the measurement channel. Our approach to the solution of this problem employs a multirate control law. The main feature of the scheme is that an approximate discrete-time model of the system is included into the controller, and the control action depends on the state of this model. The state of the model is, in turn, corrected from time to time using the “low rate” delayed measurements of the actual state of the plant. It is worth noting that, for linear systems and in absence of measurement delay, some close ideas were used in Fujimoto

and Hori (2001, 2002) to design high-performance multirate servo systems. Since the open-loop estimator is used to obtain approximations of state trajectories between samples, the robustness properties of the proposed scheme need further investigations. Also, the corresponding development of the output feedback multirate control schemes appears to be much more involved and challenging problem. These topics will be subjects of future research.

References Araki, M. (1993). Recent developments in digital control theory. In 12th IFAC world congress, Sydney, Australia, Vol. 9, July 1993 (p. 251–260). Chen, T., & Francis, B. (1995). Optimal sampled-data control systems. London: Springer. Fujimoto, H., & Hori, Y. (2001). Visual servoing based on intersample disturbance rejection by multirate sampling control. In Proceedings of the 40th IEEE conference on decision and control, Orlando, FL, (pp. 334–339). Fujimoto, H., & Hori, Y. (2002). High-performance servo systems based on multirate sampling control. Control Engineering Practice, 10, 773–781. Gear, C. W. (1971). Numerical initial value problems in ordinary di@erential equations. Englewood Cli9s, NJ: Prentice-Hall. Grune, L., & NesiAc, D. (2003). Optimization based stabilization of sampled-data nonlinear systems via their approximate discrete-time models. SIAM Journal on Control and Optimisation, 42, 98–122. Hauser, J. H., Sastry, S., & Meyer, G. (1992). Nonlinear control design for slightly nonminimum phase systems: Application to v/stol aircraft. Automatica, 28(4), 665–679. Hitchinson, S., Hager, G. D., & Corke, P. I. (1996). A tutorial on visual servo control. IEEE Transactions on Robotics and Automation, 12(5), 651–670. Isidori, A. (1999). Nonlinear control systems II. London: Springer. Khalil, H. K. (1996). Nonlinear systems (2nd ed.). Upper Saddle River, NJ: Prentice-Hall. NesiAc, D., & Angeli, D. (2002). Integral versions of ISS for sampled-data nonlinear systems via their approximate discrete-time models. IEEE Transactions on Automatic Control, 47, 2033–2038. NesiAc, D., & Laila, D. S. (2002). A note on input-to-state stabilization for nonlinear sampled-date systems. IEEE Transactions on Automatic Control, 47, 1153–1158. NesiAc, D., & Teel, A. R. (2001a). Backstepping on the Euler approximate model for stabilization of sampled-data nonlinear systems. In Proceedings of the 40th IEEE conference on decision and control Orlando, FL, (pp. 1737–1742). NesiAc, D., & Teel, A. R. (2001b). Sampled-data control of nonlinear systems: an overview of recent results. In R. S. O. Moheimani (Ed.), Perspectives on robust control (pp. 221–239). New York: Springer. NesiAc, D., Teel, A. R., & KokotoviAc, P. V. (1999a). SuJcient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations. Systems and Control Letters, 38, 259–270. NesiAc, D., Teel, A. R., & Sontag, E. D. (1999b). Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems. Systems and Control Letters, 38, 48–60. Sepulchre, R., Jankovic, M., & Kokotovic, P. (1997). Constructive nonlinear control. Series on communications and control engineering (CCES). London: Springer.

I.G. Polushin, H.J. Marquez / Automatica 40 (2004) 1035 – 1041 Teel, A. R., NesiAc, D., & KokotoviAc, P. V. (1998). A note on input-to-state stability of sampled-data nonlinear systems. In Proceedings of the 37th IEEE conference on decision and control, Tampa, FL, USA, (pp. 2473–2478). Ilia G. Polushin was born in 1968 in Kirov, Russia. He obtained his Ph.D. (Candidate of Science) degree in Automatic Control from St. Petersburg Electrotechnical University (LETI) in 1996. He was aJliated with St. Petersburg Electrotechnical University and the University of Alberta. Currently he is a research fellow in the Department of Electrical Engineering, Lakehead University, and on leave from Institute for problems in Mechanical Engineering of Russian Academy of Science.

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Horacio J. Marquez received the M.Sc.E and Ph.D. degrees in electrical engineering from the University of New Brunswick, Fredericton, Canada, in 1990 and 1993. 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, where he is currently a Professor and Associate Chair of Graduate Studies. His interests are in control theory and its applications. He is the author of “Nonlinear Control Systems: Analysis and Design” (Wiley, 2003). He received the 2003/2004 University of Alberta McCalla Professorship.