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A new approach to enlarging sampling intervals for sampled-data systems
Proceedings Proceedings of of the the 12th 12th IFAC IFAC Workshop Workshop on on Time Time Delay Delay Systems Systems June 28-30, Ann Arbor, USA Proceedings of the the 12th IFACMI, Workshop on Time Time Delay Delay Systems Systems June 28-30, 2015. 2015. Ann Arbor, MI, USA Proceedings of 12th IFAC Workshop on Available online at www.sciencedirect.com June June 28-30, 28-30, 2015. 2015. Ann Ann Arbor, Arbor, MI, MI, USA USA
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A new approach to enlarging sampling A new approach to enlarging sampling A new approach to enlarging sampling A intervals new approach to enlarging sampling for sampled-data systems intervals for sampled-data systems intervals for sampled-data systems intervals for sampled-data systems
∗ ∗∗ ∗ ∗∗∗ ∗ L. Burlion ∗∗∗ Ahmed-Ali F. Ahmed-Ali ∗∗ E. E. Fridman Fridman ∗∗ F. Giri Giri∗∗∗∗ L. Burlion ∗∗∗ ∗∗ ∗ ∗ ∗∗ ∗ Ahmed-Ali E. Fridman F. Giri L. Burlion F. Ahmed-Ali E. Fridman F. Giri∗∗∗∗ L. Burlion ∗∗∗ F. Lamnabhi-Lagarrigue Lamnabhi-Lagarrigue ∗∗∗∗ ∗∗∗∗ F. Lamnabhi-Lagarrigue F. Lamnabhi-Lagarrigue ∗ ∗ Laboratoire GREYC, UMR CNRS 6072, Universit´ Laboratoire GREYC, UMR CNRS 6072, Universit´ee de de Caen Caen ∗ ∗ Laboratoire GREYC, UMR CNRS 6072, Universit´ e de Caen Basse-Normandie , ENSICAEN, Caen, France Laboratoire GREYC, UMR CNRS 6072, Universit´ e de Caen Basse-Normandie , ENSICAEN, Caen, France ∗∗ Basse-Normandie , ENSICAEN, Caen, France ∗∗ School Basse-Normandie of Electrical Engineering, Tel-Aviv University, Tel-Aviv , ENSICAEN, Caen, France School of Electrical Engineering, Tel-Aviv University, Tel-Aviv ∗∗ ∗∗ School of Electrical Engineering, Tel-Aviv 69978, Israel Israel School of Electrical Engineering, Tel-Aviv University, University, Tel-Aviv Tel-Aviv 69978, ∗∗∗ 69978, Israel ∗∗∗ ONERA The French Aerospace Lab 31055 Toulouse, France 69978, Israel ONERA The French Aerospace Lab 31055 Toulouse, France ∗∗∗ ∗∗∗∗ ∗∗∗ ONERA The French Aerospace Lab 31055 Toulouse, France ∗∗∗∗ CNRS-INS2I, Laboratoire des Signaux et Systemes (L2S), ONERA The French Aerospace Lab 31055 Toulouse, France CNRS-INS2I, Laboratoire des Signaux et Systemes (L2S), ∗∗∗∗ ∗∗∗∗ CNRS-INS2I, Laboratoire des Signaux et Systemes (L2S), European Embedded Control Institute, EECI Supelec, 3 rue CNRS-INS2I, Laboratoire des Signaux et Systemes (L2S), European Embedded Control Institute, EECI Supelec, 3 rue Joliot Joliot European Institute, Supelec, Curie,Control 91190 Gif-sur-Yvette, Gif-sur-Yvette, France European Embedded Embedded Control Institute, EECI EECIFrance Supelec, 3 3 rue rue Joliot Joliot Curie, 91190 Curie, Curie, 91190 91190 Gif-sur-Yvette, Gif-sur-Yvette, France France Abstract: Abstract: This This paper paper provides provides exponential exponential stability stability results results for for aa family family of of nonlinear nonlinear ODE ODE Abstract: This paper provides exponential stability results for a family of ODE systems which involves sampled-data states and a time-varying gain. Sufficient conditions Abstract: Thisinvolves paper provides exponential results for again. family of nonlinear nonlinear ODE systems which sampled-data states stability and a time-varying Sufficient conditions systems which involves sampled-data states and a time-varying gain. Sufficient conditions ensuring global exponential stability are established in terms of Linear Matrix Inequalities systems which involves sampled-data states and a time-varying gain. Sufficient conditions ensuring global exponential stability are established in terms of Linear Matrix Inequalities ensuring global are in of Matrix Inequalities (LMIs) the Lyapunov-Krasvoskii functionals. established stability results ensuring globalon exponential stability are established established in terms termsThe of Linear Linear Matrix Inequalities (LMIs) derived derived onexponential the basis basis of ofstability Lyapunov-Krasvoskii functionals. The established stability results (LMIs) derived on the basis of Lyapunov-Krasvoskii functionals. The established stability results prove to be useful in designing exponentially convergent observers based on the sampled-data (LMIs) on the basis of Lyapunov-Krasvoskii functionals. The based established results prove toderived be useful in designing exponentially convergent observers on thestability sampled-data prove to be useful in designing exponentially convergent observers based on the sampled-data measurements. It is shown throughout simple examples from the literature that the introduction prove to be useful designing exponentially convergent observers based that on the measurements. It isin shown throughout simple examples from the literature thesampled-data introduction measurements. It is shown shown throughout simple examples from the literature that the the introduction of gains is beneficial for enlargement of sampling intervals while preserving measurements. is throughout from literature that introduction of time-varying time-varying It gains is quite quite beneficial simple for the the examples enlargement of the sampling intervals while preserving of gains is the stability system. of time-varying gains is quite quite beneficial beneficial for for the the enlargement enlargement of of sampling sampling intervals intervals while while preserving preserving thetime-varying stability of of the the system. the stability of the system. the stability of the system. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Sampled-data Sampled-data systems; systems; Sampled-data Sampled-data observers; observers; Time-varying Time-varying gain. gain. Keywords: Keywords: Sampled-data Sampled-data systems; systems; Sampled-data Sampled-data observers; observers; Time-varying Time-varying gain. gain. 1. INTRODUCTION sampling interval, compared 1. INTRODUCTION sampling interval, compared with with the the constant constant gain gain case case 1. sampling interval, compared with the constant gain case (corresponding to η = 0 ). It is worth noting that the 1. INTRODUCTION INTRODUCTION sampling interval, compared with the constant (corresponding to η = 0 ). It is worth noting gain that case the (corresponding to ηη stability = 00 ). It is worth noting that the present theoretical result can also be used in (corresponding to = ). It is worth noting that the present theoretical stability result can also be used in Designing sampled-data observers and controllers has been Designing sampled-data observers and controllers has been sampled-data present theoretical stability result can also be used in control design improving exponential stabilpresent theoretical stability result can also be used in sampled-data control design improving exponential stabilDesigning sampled-data observers and controllers has been aDesigning in see Fridman [2010], ssi´ observers controllers hasNeˇ been a hot hot topic topicsampled-data in recent recent years, years, see e.g e.gand Fridman [2010], Neˇ i´cc ity sampled-data control design improving exponential stabilproperties and enlarging sampling intervals. The paper sampled-data control design sampling improvingintervals. exponential stability properties and enlarging The paper aand hot topic in recent years, see e.g Fridman [2010], Neˇ s i´ c Teel [2004] H.Ye et al. [1998], J. Hespanha and Xu a hotTeel topic in recent e.g Fridman [2010], Neˇ si´c ity and [2004] H.Ye years, et al. see [1998], J. Hespanha and Xu properties and enlarging sampling paper is organized follows: in II, the stability ity propertiesas and enlarging sampling intervals. The paper organized as follows: in Section Section II,intervals. the main mainThe stability and Teel H.Ye et [1998], Hespanha Xu [2007] and reference therein. regard, aa long and Teel [2004] H.Ye list et al. al. [1998],In J.this Hespanha and Xu is [2007] and[2004] reference list therein. InJ. this regard,and long is organized as follows: in Section II, the main stability result, concerning nonlinear ODE systems, is stated. This is organized as follows: in Section II, the main stability result, concerning nonlinear ODE systems, is stated. This [2007] and reference In regard, aa long standing is to enlarge sampling [2007] andissue reference list therein. In this this regard,intervals long result standing issue is how howlist to therein. enlarge the the sampling intervals result, concerning nonlinear ODE systems, is stated. This is then applied to observer design in Section III. A result, concerning nonlinear ODE systems, is stated. This result is then applied to observer design in Section III. A standing issue is how to enlarge the sampling intervals Heemels al. [2007],Donkers and Heemels standing is how to enlarge the sampling Heemels et etissue al. [2012],Tabuada [2012],Tabuada [2007],Donkers and intervals Heemels conclusion result is then applied to observer design in Section III. and reference list end the paper. The technical result is then applied to observer design in Section III. A A conclusion and reference list end the paper. The technical Heemels et al. [2012],Tabuada [2007],Donkers and Heemels [2012]. In a recent paper Cacace et al. [2014], it has been Heemels [2012],Tabuada [2007],Donkers Heemels [2012]. Inetaal. recent paper Cacace et al. [2014],and it has been conclusion and reference list end proof of our main result is conclusion and reference listappended. end the the paper. paper. The The technical technical proof of our main result is appended. [2012]. In a recent paper Cacace et al. [2014], it has been shown that the introduction of time-varying gains in a [2012]. a recent paper Cacace al. [2014], itgains has been shown In that the introduction of et time-varying in a proof proof of of our our main main result result is is appended. appended. shown introduction of gains in specific class of improves their conshown that the introduction of time-varying time-varying gains conin aa Notations and preliminaries specific that class the of observers observers improves their exponential exponential Notations and preliminaries specific class of observers improves their exponential convergence properties in of delay. In specific of observers improves their exponential convergenceclass properties in presence presence of measurement measurement delay. In Notations Notations and and preliminaries preliminaries vergence properties presence of In the work, some general of vergence properties inconsider presence of measurement measurement delay. In Throughout the paper the the present present work, we wein consider some general classes classesdelay. of nonnonThroughout the paper the superscript superscript T T stands stands for for matrix matrix the work, consider some general classes of n linear sampled-data analyze their exponential the paper the superscript T stands for matrix the present present work, we wesystems considerand some general classes of nonnon- Throughout n denotes transposition, R the n-dimensional Euclidean linear sampled-data systems and analyze their exponential Throughout the paper the superscript T stands for matrix transposition, R denotes the n-dimensional Euclidean n linear sampled-data systems and analyze their exponential n×mn-dimensional Euclidean stability. The common feature of these classes is that the n transposition, R denotes the linear sampled-data systems and analyze their exponential n×m space with vector norm |.|, R is the set of all n×m real stability. The common feature of these classes is that the space transposition, R denotes the n-dimensional Euclidean with vector norm |.|, R is the set of all n×m real n×m stability. The common feature of these classes is that the n×m systems are allowed to involve aa time-varying n×m space with vector norm |.|, R is the set of all n×m real stability. The include common feature of these classes is that the n×m matrices, and the notation P > 0, for P ∈ R means systems they they include are allowed to involve time-varying space with vector norm |.|, R is the set of all n×m real matrices, and the notation P > 0, for P ∈ R means −η(t−t ) k allowed to involve a time-varying n×m systems they include are (with ηηto> 0 as a tuning paramgain form ee−η(t−t k) n×m matrices, and the notation P > 0, for P ∈ R means systems they include are allowed involve a time-varying that P is symmetric and positive definite. Symmetric (with > 0 as a tuning paramgain of of the the form matrices, and the notation P > 0, for P ∈ R means −η(t−t ) that P is symmetric and positive definite. Symmetric k gain the ee−η(t−t 0 paramk ). (with eter), ttk (k = .. .. )) is sampling instant. Note P positive definite. Symmetric > 0 as as a a tuning tuning paramgain ofwhere the form form elements symmetric matrices of symmetric matrix eter),of where = 0, 0, 1, 1, . (with is ηηa a> sampling instant. Note that k (k that P is isin symmetric and positive definite. Symmetric elements in symmetric symmetric and matrices of the the symmetric matrix eter), where t (k = 0, 1, . . . ) is a sampling instant. Note that our class of systems is a generalization of that dealt k elements in symmetric matrices of the symmetric matrix eter), where t (k = 0, 1, . . . ) is a sampling instant. Note will be denoted by ∗. The notation (t ) to that our classkof systems is a generalization of that dealt elements k )k≥0 refers in symmetric matrices of the symmetric matrix will be denoted by ∗. The notation (t refers to k k≥0 that our class of systems is a generalization of that dealt with in Cacace et al. [2014]. In the present paper, sufficient be denoted by ∗. The notation (t ) refers to that ourCacace class of is In a generalization of that dealt will a strictly increasing sequence such that t = 0 and k k≥0 with in et systems al. [2014]. the present paper, sufficient 0 = will be denoted by ∗. The notation (t ) refers to a strictly increasing sequence such that t 0 and k k≥0 0 with in et In present sufficient conditions for exponential are a strictly increasing sequence such that t = 0 and with in Cacace Cacace et al. al. [2014]. [2014]. In the the stability present paper, paper, sufficient lim t = ∞. The sampling periods are bounded with 0 conditions for global global exponential stability are established established k→∞ k a strictly increasing sequence such that t = 0 and lim t = ∞. The sampling periods are bounded with 0 k→∞ k conditions exponential are for our of systems in Linear Inequali=tk∞.< sampling periods conditions for global exponential stability are established established 00 < ttk+1ttkk− h some 00 < < all for our class classfor of global systems in terms terms of ofstability Linear Matrix Matrix Inequali- lim lim The sampling periods areh bounded with
T. T. T. T.
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x(t) ˙ = A0 x(t) + A1 e−η(t−tk ) x(tk ) + φ(x(t)),
t ∈ [tk , tk+1 ) (1) where x(t) ∈ Rn ; the scalar η ≥ 0; A0 , A1 are constant matrices with appropriate dimensions. As in Am and Fridman [2014], the function φ is supposed to be class C 1 with uniformly bounded Jacobian φx , satisfying φ(0) = 0 and φTx (x)φx (x) ≤ M ∀x (2) for some positive constant n×n-matrix M . Using Jensen’s inequality it is readily checked that (2) implies the following inequality: 1 1 T φx (sx)ds φx (sx)ds ≤ M 0
0
Remark 1: Equation (1) may represent a networked control system described by x(t) ˙ = A0 x(t) + φ(x(t)) + Bu(t), with the communication network placed between the sensor and the controller (but no network is placed between the controller and the actuator). Assuming that the discrete-time state measurements x(tk ) are transmitted through the communication network from the sensor to controller, consider the state-feedback,
u(t) = e−η(t−tk ) Kx(tk ), t ∈ [tk , tk+1 ), where K is a gain and η > 0 is a scalar. It turns out that the resulting closed-loop system fits equation (1) with A1 = BK. As in Cacace et al. [2014], introduce the following change of coordinates z(t) = eηt x(t) with η > 0. Then one gets z(t) ˙ = ηz(t) + A0 z(t) + A1 z(tk ) 1 + φx (sx(t))ds eηt x(t), t ∈ [tk , tk+1 ) 0
which rewrites as follows for all t ∈ [tk , tk+1 ) 1 φx (sx(t))ds z(t) z(t) ˙ = (ηIn + A0 ) z(t) + A1 z(tk ) + 0
(3) Following Fridman [2010], consider the following LyapunovKrasovskii functional for (3): (4) V (t) = V¯ (t, z(t), z˙t ) + VX (t, zt ) with V¯ (t, z(t), z˙t ) = z T (t)P z(t) + (tk+1 − t) and VX (t, zt ) =
t
z˙ T (s)U z(s)ds ˙
P +h
441
X + XT 2
hX1 − hX
∗
hX1T
−hX1 −
> 0 (5)
X + XT +h 2
Using the definition of z(t), we can see that the exponential stability of system (1) is guaranteed if:
V˙ (t) + 2αV (t) ≤ 0
t ∈ [tk , tk+1 )
(6)
for some scalar α ∈ (−η, 0] (note that the scalar α is allowed to be negative). Indeed, if (6) is satisfied one has, V˙ (t) ≤ −2αV (t) =⇒ |z(t)| ≤
V|t=0 e−αt λmin (P )
Then, using the fact that z(t) = eηt x(t), one gets: |x(t)| ≤
V|t=0
λmin (P )
e−(η+α)t
From the above inequality, one sees that the exponential convergence is guaranteed if η+α > 0. Since the parameter η is positive and free, it is sufficient to let α ∈ (−η, 0] for ensuring an exponential convergence with a decay rate η + α. In the following proposition, it is shown that the property (6), and resulting exponential stability with a decay rate η + α > 0, is actually ensured under well established sufficient conditions, expressed in terms of LMIs. Proposition 1. Consider the system (1) with possibly varying sampling-intervals subject to tk+1 − tk ≤ h with some scalar h > 0. Given η > 0 and α ∈ (−η, 0], let there exist n × n matrices P > 0, U > 0, X, X1 , P2 , P3 , T, Y1 , Y2 and a scalar λ > 0 that satisfy the LMI (5) and the following LMIs:
Φ11 − Xα Φ12 + Xτ (t) Φ13 + X1α P2T ∆ ∗ Φ22 + hU Φ23 − X1τ (t) P3T Ψ(0) = ∗ ∗ Φ33 − X2α 0 ∗ ∗ ∗ −λIn
tk
t ∈ [tk , tk+1 ), P > 0, U > 0,
τ (t)=0
<(7) 0
and
X + XT −X + X 1 2 (tk+1 − t)ξ T T ξ, X + X ∗ −X1 − X1T + 2 where ξ(t) = col{z(t), z(tk )}, X and X1 are n×n matrices.
The positiveness of (4) is ensured if the following LMI holds Fridman [2010]:
441
Φ11 − Xα| Φ12 Φ13 + X1α ∗ Φ22 Φ23 ∆ Ψ(h) = ∗ ∗ Φ33 − X2α ∗ ∗ ∗ ∗ ∗ ∗
where
hY1T P2T hY2T pT 3 hT T 0 −hU 0 ∗ −λIn
< 0, (8)
τ (t)=h
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Φ11 = (A0 + ηIn )T P2 + P2T (A0 + ηIn )
x(t) ˙ =
+ 2αP − Y1 − Y1T + λM
with K = −[3.75, 11.5]. It is well-known that the above system with η = 0 remains stable under constant sampling with h < 1.72 and becomes unstable for constant sampling periods with h > 1.73. Applying the result of Proposition 1, one obtains for η = 0.8 and α = −0.8 a maximal variable sampling interval value of h = 2.43 that preserves the stability.
Φ13 = Y1T + P2T A1 − T Φ22 = −P3 − P3T Φ23 = Y2T + P3T A1 Φ33 = T + T T X + XT 2
X + XT 2 = (h − τ (t))(X − X1 )
The above two examples illustrate the fact that, when using time varying gain, the sampling interval bound h that preserves the stability can be much larger compared to the constant gain case (η = 0).
Xα = (1 − 2α(h − τ (t))) X1τ (t)
X1α = (1 − 2α(h − τ (t)))(X − X1 ) X2α = (1 − 2α(h − τ (t)))
X + X T − 2X1 − 2X1T 2
.
3. APPLICATION TO SAMPLED-DATA OBSERVER DESIGN
Then the system (1) is exponentially stable with a decay rate η + α. Moreover, if the above LMIs hold with α = −η, then the system is exponentially stable with a small enough decay rate. See Appendix A for the proof. Remark 2: Proposition 1 clearly shows that the sampling interval bound h that preserves the exponential stability is made dependent on the design parameters η and α. Additional highlights will be provided in Example 1 (given hereafter) which will show that maximum h preserving the stability may be significantly enlarged by tuning η. On the other hand, Proposition 1 also confirms the somewhat obvious fact that the exponential decay rate η + α < η is also depending on η and α. Remark 3: In the linear case, x(t) ˙ = A0 x(t) + A1 e−η(t−tk ) x(tk ), t ∈ [tk , tk+1 ) the conditions of Proposition 1 apply with λ = 0. Then, the last columns and rows in Ψ(0) and Ψ(h) must be deleted. Example 1 To illustrate the result of Proposition 1, consider the following system x(t) ˙ = −x(t)+sin(x(t))−Ke−η(t−tk ) x(tk ), t ∈ [tk , tk+1 ) with K = 1. Here we have A0 = −1, A1 = −K = −1, and M = 1. Then, application of Proposition 1 with α = −η leads to the maximum values of h that preserves the exponential stability of the system given in Table 1. Thus, (2.1) with η = 1 achieves exponential stability under more than twice larger sampling interval than under the constant gain (with η = 0). η h
0 1.38
0.5 1.84
0.7 2.36
0 1 0 −η(t−tk ) x(t) + e Kx(tk ) 0 −0.1 0.1
t ∈ [tk , tk+1 )
Φ12 = P − P2T + (A0 + ηIn )T P3 − Y2
Xτ (t) = (h − τ (t))
1 3.17
Consider the class of nonlinear systems x(t) ˙ = Ax(t) + f (x(t)) y(t) = Cx(t)
(9)
where x ∈ Rn and A, C are matrices with appropriate dimensions. It is supposed that the function f satisfies assumption (2) and y is accessible to measurements only at instants tk . Then the following observer is proposed for all t ∈ [tk , tk+1 ) ˆ(tk ) − y(tk )) (10) x ˆ˙ (t) = Aˆ x(t) + f (ˆ x(t)) − Ke−η(t−tk ) (C x with η > 0 and the matrix gain has the form T , K = K 1 ... K n
where K i are matrices with appropriate dimensions. Our goal is to determine a matrix K and an upper bound of the maximum allowable sampling period h such that, the observation error is globally exponentially vanishing. To this end, consider the observation error x ˜(t) = x ˆ(t) − x(t). It is readily checked that this error satisfies the following equation for all t ∈ [tk , tk+1 ) x ˜˙ (t) = A˜ x(t) + f (ˆ x(t)) − f (x(t)) − Ke−η(t−tk ) (C x ˆ(tk ) − y(tk ))
Consider the change of coordinates z1 (t) = eηt x ˜(t) that leads to the equation z˙1 (t) = (ηIn + A)z1 (t) − KCz1 (tk ) + eηt (f (ˆ x(t)) − f (x(t))) (11) Using the fact that, 1 f (ˆ x(t)) − f (x(t)) = fx (x(t) + s(ˆ x(t) − x(t)))ds x ˜(t), 0
equation (11) can be rewritten as follows:
z˙1 (t) = (ηIn + A)z1 (t) + A1 z1 (tk ) 1 + fx (x(t) + s(ˆ x(t) − x(t)))ds z1 (t) (12) 0
, t ∈ [tk , tk+1 ).
Table 1. Example 1: max. values of h vs. η preserving the stability Example 2 Consider the following much studied system (see e.g. Zhang et al. [2001], Fridman [2010]): 442
(13)
with A1 = −KC. Clearly, equation (12) falls in the class of systems described by equation (1). Therefore, the result of Proposition 1 can directly be applied to get sufficient conditions for the observer (10) to be exponentially convergent. This is illustrated in the next example.
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Example 3 Consider the following nonlinear system x˙ 1 (t) = x2 (t) x˙ 2 (t) = −2x1 (t) − x2 (t) + 0.2sin(x2 (t)) y(t) = Cx(t) = x1 (t) Clearly, this system is of the form (9) with A = 0 1 T , C = (1, 0), and f (x(t)) = ( 0 0.2sin(x2 (t)) ) . −2 −1 It is readily checked that the function f satisfies inequality (2) with M = diag{0, 0.04}. Then the observer (10) writes as follows: ˆ˙ 1 (t) = x ˆ2 (t) − K 1 e−η(t−tk ) (ˆ x1 (tk ) − y(tk )) x x1 (t) − x ˆ2 (t) + 0.2sin(ˆ x2 (t)) x ˆ˙ 2 (t) = −2ˆ 2 −η(t−tk ) (ˆ x1 (tk ) − y(tk )) −K e −K 1 0 . Applying PropoHere one has A1 = −KC = −K 2 0 T sition 1 with K = [1.65, −0.85] and α = −η, one obtains the results given in Table 2. η h
0 0.99
1 1.64
2 2.78
443
1.5
1
0.5
0
−0.5
2.4 2.86
Table 2. Example 3: max. values of h vs. η preserving the stability
−1 0
10
20
30
40
50
60
70
80
The above Table shows that the sampling interval bound h can be significantly increased thanks to the time-varying gain feature of the observer. Finally, we illustrate this Table by performing some numerical simulations : figure (1) (res. figure (2)) illustrates the case where η = 0 and h = 0.99 (resp. η = 2.4 and h = 2.86). These figures have been appended to appendix B. We have also derived from numerical simulations the real maximum bounds of h for both η = 0 and η = 2.4 which are respectively h = 1.1 and at least h = 100. This fact confirms that the improvements of the time-varying gain are important. 4. CONCLUSION In this paper, a novel stability result, namely Proposition 1, is established for a large class of sampled-data nonlinear systems which involve an exponentially decaying gain e−η(t−tk ) allowing to enlarge the sampling interval ; the stability conditions are expressed in terms of LMIs. We have also shown that our result can be applied to a wide class of nonlinear sampled-data observers. It has to be noticed that the results presented here can be easily extended to networks with Round Robin protocols. In future work, this idea will be extended to some classes of infinite dimensional systems. REFERENCES N. Bar Am and E. Fridman. Network-based distributed h∞ -filtering of parabolic systems. Automatica, 50, 2014. F. Cacace, A. Germani, and C. Manes. Nonlinear systems with multiple time-varying measurement delays. SIAM J. Control Optim., 52(3):1862–1885, 2014. M.C.F. Donkers and W.P.M.H. Heemels. Output- based event-triggered control with guaranteed l-gain and improved and decentralised event-triggering. IEEE Trans. on Automatic Control, 57(6):1362–1376, 2012. 443
Fig. 1. x2 (solid line), x ˆ2 (dashed line) for η = 0 and h = 0.99 E. Fridman. New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems. 43(4):309–319, 2001. E. Fridman. A refined input delay approach to sampleddata control. Automatica, 46:421–427, 2010. W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada. An introduction to event-triggered and self-triggered control. IEEE Conference on Decision and Control, 46: 3270–3285, 2012. H.Ye, A.N Michel, and L.Hou. Stability analysis of systems with impulse effects. IEEE Transaction on Automatic and Control, 43:1719–1723, 1998. P. Naghshtabrizi J. Hespanha and Y. Xu. A survey of recent results in networked control systems. IEEE Special Issue on Technology of Networked Control Systems, 95 (1):138–162, 2007. D. Neˇsi´c and A.R. Teel. A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time. IEEE Transactions on Automatic Control, 49:1103–1122, 2004. P. Tabuada. Event-triggered real-time scheduling of stabilizing control tasks. IEEE Trans. on Automatic Control, 52(9):1680–1685, 2007. W. Zhang, M. Branicky, and S. Phillips. Stability of networked control systems. Automatica, 21:84–99, 2001.
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V˙ (t) + 2αV (t) +
1
T
λz (t) M −
0
φTx (sx)ds
1
φx (sx)ds z(t) 0
≤ 2z T (t)P z(t) ˙ + (tk+1 − t)z˙ T (t)U z(t) ˙ t z˙ T (s)U z(s)ds ˙ − tk
X + XT −X + X 1 2 − ξT ξ X + XT T ∗ −X1 − X1 + 2 + (tk+1 − t)[z˙ T (t)(X + X T ))z(t)
1.4 1.2 1
+ 2z˙ T (t)(−X + X1 )z(tk )] + 2αz T (t)P z(t)
0.8
+ 2α(tk+1 − t)ξ T X + XT 2 ×
0.6 0.4 0.2 0
∗
−0.2 −0.4 −0.6 0
2
4
6
8
10
12
14
16
18
−X + X1
−X1 −
+ λz T (t)M z(t) 1 T T φx (sx)ds − λz (t)
20
X + XT + 2
X1T
0
1
ξ
φx (sx)ds z(t) 0
(A.1)
Using the well known Jensen’s inequality, one has, t z˙ T (s)U z(s)ds ˙ ≥ (t − tk )v1T U v1 tk
t T 1 with v1 = t−t z˙ (s). According to the description tk k approach Fridman [2001], the left-hand sides of the equations, Fig. 2. x2 (solid line), x ˆ2 (dashed line) for η = 2.4 and h = 2.86
0 = 2 z T (t)Y1T + z˙ T (t)Y2T + z T (tk )T T × − z(t) + z(tk ) +(t − tk )v1 0 = 2 z T (t)P2T + z˙ T (t)P3T × (A0 + ηIn )z(t) + A1 z(tk ) 1 + φx (sx)ds z(t) − z˙ 0
are added to V˙ (t) + 2αV (t) where Y1 , Y2 , T, P2 , P3 are free matrixes. Let us define the augmented vector, 1 µ = col{z(t), z(t), ˙ z(tk ), v1 , φx (sx)ds z(t)}. 0
Combining (A.2) and (A.1), one gets: V˙ (t) + 2αV (t) +λz T (t) M −
Appendix A. PROOF OF PROPOSITION 1
Let α ∈ (−η, 0] and differentiate V (t). Then, after some simple computations one obtains: 444
1 0
φTx (sx)ds
1 0
φx (sx)ds z(t)
≤ µT Ψµ < 0 provided that the following matrix Ψ is negative definite T T
Φ11 − Xα Φ12 + Xt−tk Φ13 + X1α ∗ Φ22 + (tk+1 − t)U Φ23 − X1(t−tk ) ∗ ∗ Φ33 − X2α ∗ ∗ ∗ ∗ ∗ ∗
(t − tk )Y1 P2 (t − tk )Y2T pT 3 (t − tk )T T 0 −(t − tk )U 0 ∗ −λIn
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Writing the last matrix inequality, for τ (t) → 0 and τ (t) → h leads to LMIs (7) and (8), respectively. Now, using arguments of Lemma 2 in Fridman [2010], it is not difficult to see that, if the LMIs (5), (7) and (8) are feasible ¯ ∈ (0, h]. for some h > 0 then, they also are feasible for all h This completes the proof of Proposition 1.
445
445
ScienceDirect IFAC-PapersOnLine 48-12 (2015) 440–445
A new approach to enlarging sampling A new approach to enlarging sampling A new approach to enlarging sampling A intervals new approach to enlarging sampling for sampled-data systems intervals for sampled-data systems intervals for sampled-data systems intervals for sampled-data systems
∗ ∗∗ ∗ ∗∗∗ ∗ L. Burlion ∗∗∗ Ahmed-Ali F. Ahmed-Ali ∗∗ E. E. Fridman Fridman ∗∗ F. Giri Giri∗∗∗∗ L. Burlion ∗∗∗ ∗∗ ∗ ∗ ∗∗ ∗ Ahmed-Ali E. Fridman F. Giri L. Burlion F. Ahmed-Ali E. Fridman F. Giri∗∗∗∗ L. Burlion ∗∗∗ F. Lamnabhi-Lagarrigue Lamnabhi-Lagarrigue ∗∗∗∗ ∗∗∗∗ F. Lamnabhi-Lagarrigue F. Lamnabhi-Lagarrigue ∗ ∗ Laboratoire GREYC, UMR CNRS 6072, Universit´ Laboratoire GREYC, UMR CNRS 6072, Universit´ee de de Caen Caen ∗ ∗ Laboratoire GREYC, UMR CNRS 6072, Universit´ e de Caen Basse-Normandie , ENSICAEN, Caen, France Laboratoire GREYC, UMR CNRS 6072, Universit´ e de Caen Basse-Normandie , ENSICAEN, Caen, France ∗∗ Basse-Normandie , ENSICAEN, Caen, France ∗∗ School Basse-Normandie of Electrical Engineering, Tel-Aviv University, Tel-Aviv , ENSICAEN, Caen, France School of Electrical Engineering, Tel-Aviv University, Tel-Aviv ∗∗ ∗∗ School of Electrical Engineering, Tel-Aviv 69978, Israel Israel School of Electrical Engineering, Tel-Aviv University, University, Tel-Aviv Tel-Aviv 69978, ∗∗∗ 69978, Israel ∗∗∗ ONERA The French Aerospace Lab 31055 Toulouse, France 69978, Israel ONERA The French Aerospace Lab 31055 Toulouse, France ∗∗∗ ∗∗∗∗ ∗∗∗ ONERA The French Aerospace Lab 31055 Toulouse, France ∗∗∗∗ CNRS-INS2I, Laboratoire des Signaux et Systemes (L2S), ONERA The French Aerospace Lab 31055 Toulouse, France CNRS-INS2I, Laboratoire des Signaux et Systemes (L2S), ∗∗∗∗ ∗∗∗∗ CNRS-INS2I, Laboratoire des Signaux et Systemes (L2S), European Embedded Control Institute, EECI Supelec, 3 rue CNRS-INS2I, Laboratoire des Signaux et Systemes (L2S), European Embedded Control Institute, EECI Supelec, 3 rue Joliot Joliot European Institute, Supelec, Curie,Control 91190 Gif-sur-Yvette, Gif-sur-Yvette, France European Embedded Embedded Control Institute, EECI EECIFrance Supelec, 3 3 rue rue Joliot Joliot Curie, 91190 Curie, Curie, 91190 91190 Gif-sur-Yvette, Gif-sur-Yvette, France France Abstract: Abstract: This This paper paper provides provides exponential exponential stability stability results results for for aa family family of of nonlinear nonlinear ODE ODE Abstract: This paper provides exponential stability results for a family of ODE systems which involves sampled-data states and a time-varying gain. Sufficient conditions Abstract: Thisinvolves paper provides exponential results for again. family of nonlinear nonlinear ODE systems which sampled-data states stability and a time-varying Sufficient conditions systems which involves sampled-data states and a time-varying gain. Sufficient conditions ensuring global exponential stability are established in terms of Linear Matrix Inequalities systems which involves sampled-data states and a time-varying gain. Sufficient conditions ensuring global exponential stability are established in terms of Linear Matrix Inequalities ensuring global are in of Matrix Inequalities (LMIs) the Lyapunov-Krasvoskii functionals. established stability results ensuring globalon exponential stability are established established in terms termsThe of Linear Linear Matrix Inequalities (LMIs) derived derived onexponential the basis basis of ofstability Lyapunov-Krasvoskii functionals. The established stability results (LMIs) derived on the basis of Lyapunov-Krasvoskii functionals. The established stability results prove to be useful in designing exponentially convergent observers based on the sampled-data (LMIs) on the basis of Lyapunov-Krasvoskii functionals. The based established results prove toderived be useful in designing exponentially convergent observers on thestability sampled-data prove to be useful in designing exponentially convergent observers based on the sampled-data measurements. It is shown throughout simple examples from the literature that the introduction prove to be useful designing exponentially convergent observers based that on the measurements. It isin shown throughout simple examples from the literature thesampled-data introduction measurements. It is shown shown throughout simple examples from the literature that the the introduction of gains is beneficial for enlargement of sampling intervals while preserving measurements. is throughout from literature that introduction of time-varying time-varying It gains is quite quite beneficial simple for the the examples enlargement of the sampling intervals while preserving of gains is the stability system. of time-varying gains is quite quite beneficial beneficial for for the the enlargement enlargement of of sampling sampling intervals intervals while while preserving preserving thetime-varying stability of of the the system. the stability of the system. the stability of the system. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Sampled-data Sampled-data systems; systems; Sampled-data Sampled-data observers; observers; Time-varying Time-varying gain. gain. Keywords: Keywords: Sampled-data Sampled-data systems; systems; Sampled-data Sampled-data observers; observers; Time-varying Time-varying gain. gain. 1. INTRODUCTION sampling interval, compared 1. INTRODUCTION sampling interval, compared with with the the constant constant gain gain case case 1. sampling interval, compared with the constant gain case (corresponding to η = 0 ). It is worth noting that the 1. INTRODUCTION INTRODUCTION sampling interval, compared with the constant (corresponding to η = 0 ). It is worth noting gain that case the (corresponding to ηη stability = 00 ). It is worth noting that the present theoretical result can also be used in (corresponding to = ). It is worth noting that the present theoretical stability result can also be used in Designing sampled-data observers and controllers has been Designing sampled-data observers and controllers has been sampled-data present theoretical stability result can also be used in control design improving exponential stabilpresent theoretical stability result can also be used in sampled-data control design improving exponential stabilDesigning sampled-data observers and controllers has been aDesigning in see Fridman [2010], ssi´ observers controllers hasNeˇ been a hot hot topic topicsampled-data in recent recent years, years, see e.g e.gand Fridman [2010], Neˇ i´cc ity sampled-data control design improving exponential stabilproperties and enlarging sampling intervals. The paper sampled-data control design sampling improvingintervals. exponential stability properties and enlarging The paper aand hot topic in recent years, see e.g Fridman [2010], Neˇ s i´ c Teel [2004] H.Ye et al. [1998], J. Hespanha and Xu a hotTeel topic in recent e.g Fridman [2010], Neˇ si´c ity and [2004] H.Ye years, et al. see [1998], J. Hespanha and Xu properties and enlarging sampling paper is organized follows: in II, the stability ity propertiesas and enlarging sampling intervals. The paper organized as follows: in Section Section II,intervals. the main mainThe stability and Teel H.Ye et [1998], Hespanha Xu [2007] and reference therein. regard, aa long and Teel [2004] H.Ye list et al. al. [1998],In J.this Hespanha and Xu is [2007] and[2004] reference list therein. InJ. this regard,and long is organized as follows: in Section II, the main stability result, concerning nonlinear ODE systems, is stated. This is organized as follows: in Section II, the main stability result, concerning nonlinear ODE systems, is stated. This [2007] and reference In regard, aa long standing is to enlarge sampling [2007] andissue reference list therein. In this this regard,intervals long result standing issue is how howlist to therein. enlarge the the sampling intervals result, concerning nonlinear ODE systems, is stated. This is then applied to observer design in Section III. A result, concerning nonlinear ODE systems, is stated. This result is then applied to observer design in Section III. A standing issue is how to enlarge the sampling intervals Heemels al. [2007],Donkers and Heemels standing is how to enlarge the sampling Heemels et etissue al. [2012],Tabuada [2012],Tabuada [2007],Donkers and intervals Heemels conclusion result is then applied to observer design in Section III. and reference list end the paper. The technical result is then applied to observer design in Section III. A A conclusion and reference list end the paper. The technical Heemels et al. [2012],Tabuada [2007],Donkers and Heemels [2012]. In a recent paper Cacace et al. [2014], it has been Heemels [2012],Tabuada [2007],Donkers Heemels [2012]. Inetaal. recent paper Cacace et al. [2014],and it has been conclusion and reference list end proof of our main result is conclusion and reference listappended. end the the paper. paper. The The technical technical proof of our main result is appended. [2012]. In a recent paper Cacace et al. [2014], it has been shown that the introduction of time-varying gains in a [2012]. a recent paper Cacace al. [2014], itgains has been shown In that the introduction of et time-varying in a proof proof of of our our main main result result is is appended. appended. shown introduction of gains in specific class of improves their conshown that the introduction of time-varying time-varying gains conin aa Notations and preliminaries specific that class the of observers observers improves their exponential exponential Notations and preliminaries specific class of observers improves their exponential convergence properties in of delay. In specific of observers improves their exponential convergenceclass properties in presence presence of measurement measurement delay. In Notations Notations and and preliminaries preliminaries vergence properties presence of In the work, some general of vergence properties inconsider presence of measurement measurement delay. In Throughout the paper the the present present work, we wein consider some general classes classesdelay. of nonnonThroughout the paper the superscript superscript T T stands stands for for matrix matrix the work, consider some general classes of n linear sampled-data analyze their exponential the paper the superscript T stands for matrix the present present work, we wesystems considerand some general classes of nonnon- Throughout n denotes transposition, R the n-dimensional Euclidean linear sampled-data systems and analyze their exponential Throughout the paper the superscript T stands for matrix transposition, R denotes the n-dimensional Euclidean n linear sampled-data systems and analyze their exponential n×mn-dimensional Euclidean stability. The common feature of these classes is that the n transposition, R denotes the linear sampled-data systems and analyze their exponential n×m space with vector norm |.|, R is the set of all n×m real stability. The common feature of these classes is that the space transposition, R denotes the n-dimensional Euclidean with vector norm |.|, R is the set of all n×m real n×m stability. The common feature of these classes is that the n×m systems are allowed to involve aa time-varying n×m space with vector norm |.|, R is the set of all n×m real stability. The include common feature of these classes is that the n×m matrices, and the notation P > 0, for P ∈ R means systems they they include are allowed to involve time-varying space with vector norm |.|, R is the set of all n×m real matrices, and the notation P > 0, for P ∈ R means −η(t−t ) k allowed to involve a time-varying n×m systems they include are (with ηηto> 0 as a tuning paramgain form ee−η(t−t k) n×m matrices, and the notation P > 0, for P ∈ R means systems they include are allowed involve a time-varying that P is symmetric and positive definite. Symmetric (with > 0 as a tuning paramgain of of the the form matrices, and the notation P > 0, for P ∈ R means −η(t−t ) that P is symmetric and positive definite. Symmetric k gain the ee−η(t−t 0 paramk ). (with eter), ttk (k = .. .. )) is sampling instant. Note P positive definite. Symmetric > 0 as as a a tuning tuning paramgain ofwhere the form form elements symmetric matrices of symmetric matrix eter),of where = 0, 0, 1, 1, . (with is ηηa a> sampling instant. Note that k (k that P is isin symmetric and positive definite. Symmetric elements in symmetric symmetric and matrices of the the symmetric matrix eter), where t (k = 0, 1, . . . ) is a sampling instant. Note that our class of systems is a generalization of that dealt k elements in symmetric matrices of the symmetric matrix eter), where t (k = 0, 1, . . . ) is a sampling instant. Note will be denoted by ∗. The notation (t ) to that our classkof systems is a generalization of that dealt elements k )k≥0 refers in symmetric matrices of the symmetric matrix will be denoted by ∗. The notation (t refers to k k≥0 that our class of systems is a generalization of that dealt with in Cacace et al. [2014]. In the present paper, sufficient be denoted by ∗. The notation (t ) refers to that ourCacace class of is In a generalization of that dealt will a strictly increasing sequence such that t = 0 and k k≥0 with in et systems al. [2014]. the present paper, sufficient 0 = will be denoted by ∗. The notation (t ) refers to a strictly increasing sequence such that t 0 and k k≥0 0 with in et In present sufficient conditions for exponential are a strictly increasing sequence such that t = 0 and with in Cacace Cacace et al. al. [2014]. [2014]. In the the stability present paper, paper, sufficient lim t = ∞. The sampling periods are bounded with 0 conditions for global global exponential stability are established established k→∞ k a strictly increasing sequence such that t = 0 and lim t = ∞. The sampling periods are bounded with 0 k→∞ k conditions exponential are for our of systems in Linear Inequali=tk∞.< sampling periods conditions for global exponential stability are established established 00 < ttk+1ttkk− h some 00 < < all for our class classfor of global systems in terms terms of ofstability Linear Matrix Matrix Inequali- lim lim The sampling periods areh bounded with
T. T. T. T.
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x(t) ˙ = A0 x(t) + A1 e−η(t−tk ) x(tk ) + φ(x(t)),
t ∈ [tk , tk+1 ) (1) where x(t) ∈ Rn ; the scalar η ≥ 0; A0 , A1 are constant matrices with appropriate dimensions. As in Am and Fridman [2014], the function φ is supposed to be class C 1 with uniformly bounded Jacobian φx , satisfying φ(0) = 0 and φTx (x)φx (x) ≤ M ∀x (2) for some positive constant n×n-matrix M . Using Jensen’s inequality it is readily checked that (2) implies the following inequality: 1 1 T φx (sx)ds φx (sx)ds ≤ M 0
0
Remark 1: Equation (1) may represent a networked control system described by x(t) ˙ = A0 x(t) + φ(x(t)) + Bu(t), with the communication network placed between the sensor and the controller (but no network is placed between the controller and the actuator). Assuming that the discrete-time state measurements x(tk ) are transmitted through the communication network from the sensor to controller, consider the state-feedback,
u(t) = e−η(t−tk ) Kx(tk ), t ∈ [tk , tk+1 ), where K is a gain and η > 0 is a scalar. It turns out that the resulting closed-loop system fits equation (1) with A1 = BK. As in Cacace et al. [2014], introduce the following change of coordinates z(t) = eηt x(t) with η > 0. Then one gets z(t) ˙ = ηz(t) + A0 z(t) + A1 z(tk ) 1 + φx (sx(t))ds eηt x(t), t ∈ [tk , tk+1 ) 0
which rewrites as follows for all t ∈ [tk , tk+1 ) 1 φx (sx(t))ds z(t) z(t) ˙ = (ηIn + A0 ) z(t) + A1 z(tk ) + 0
(3) Following Fridman [2010], consider the following LyapunovKrasovskii functional for (3): (4) V (t) = V¯ (t, z(t), z˙t ) + VX (t, zt ) with V¯ (t, z(t), z˙t ) = z T (t)P z(t) + (tk+1 − t) and VX (t, zt ) =
t
z˙ T (s)U z(s)ds ˙
P +h
441
X + XT 2
hX1 − hX
∗
hX1T
−hX1 −
> 0 (5)
X + XT +h 2
Using the definition of z(t), we can see that the exponential stability of system (1) is guaranteed if:
V˙ (t) + 2αV (t) ≤ 0
t ∈ [tk , tk+1 )
(6)
for some scalar α ∈ (−η, 0] (note that the scalar α is allowed to be negative). Indeed, if (6) is satisfied one has, V˙ (t) ≤ −2αV (t) =⇒ |z(t)| ≤
V|t=0 e−αt λmin (P )
Then, using the fact that z(t) = eηt x(t), one gets: |x(t)| ≤
V|t=0
λmin (P )
e−(η+α)t
From the above inequality, one sees that the exponential convergence is guaranteed if η+α > 0. Since the parameter η is positive and free, it is sufficient to let α ∈ (−η, 0] for ensuring an exponential convergence with a decay rate η + α. In the following proposition, it is shown that the property (6), and resulting exponential stability with a decay rate η + α > 0, is actually ensured under well established sufficient conditions, expressed in terms of LMIs. Proposition 1. Consider the system (1) with possibly varying sampling-intervals subject to tk+1 − tk ≤ h with some scalar h > 0. Given η > 0 and α ∈ (−η, 0], let there exist n × n matrices P > 0, U > 0, X, X1 , P2 , P3 , T, Y1 , Y2 and a scalar λ > 0 that satisfy the LMI (5) and the following LMIs:
Φ11 − Xα Φ12 + Xτ (t) Φ13 + X1α P2T ∆ ∗ Φ22 + hU Φ23 − X1τ (t) P3T Ψ(0) = ∗ ∗ Φ33 − X2α 0 ∗ ∗ ∗ −λIn
tk
t ∈ [tk , tk+1 ), P > 0, U > 0,
τ (t)=0
<(7) 0
and
X + XT −X + X 1 2 (tk+1 − t)ξ T T ξ, X + X ∗ −X1 − X1T + 2 where ξ(t) = col{z(t), z(tk )}, X and X1 are n×n matrices.
The positiveness of (4) is ensured if the following LMI holds Fridman [2010]:
441
Φ11 − Xα| Φ12 Φ13 + X1α ∗ Φ22 Φ23 ∆ Ψ(h) = ∗ ∗ Φ33 − X2α ∗ ∗ ∗ ∗ ∗ ∗
where
hY1T P2T hY2T pT 3 hT T 0 −hU 0 ∗ −λIn
< 0, (8)
τ (t)=h
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Φ11 = (A0 + ηIn )T P2 + P2T (A0 + ηIn )
x(t) ˙ =
+ 2αP − Y1 − Y1T + λM
with K = −[3.75, 11.5]. It is well-known that the above system with η = 0 remains stable under constant sampling with h < 1.72 and becomes unstable for constant sampling periods with h > 1.73. Applying the result of Proposition 1, one obtains for η = 0.8 and α = −0.8 a maximal variable sampling interval value of h = 2.43 that preserves the stability.
Φ13 = Y1T + P2T A1 − T Φ22 = −P3 − P3T Φ23 = Y2T + P3T A1 Φ33 = T + T T X + XT 2
X + XT 2 = (h − τ (t))(X − X1 )
The above two examples illustrate the fact that, when using time varying gain, the sampling interval bound h that preserves the stability can be much larger compared to the constant gain case (η = 0).
Xα = (1 − 2α(h − τ (t))) X1τ (t)
X1α = (1 − 2α(h − τ (t)))(X − X1 ) X2α = (1 − 2α(h − τ (t)))
X + X T − 2X1 − 2X1T 2
.
3. APPLICATION TO SAMPLED-DATA OBSERVER DESIGN
Then the system (1) is exponentially stable with a decay rate η + α. Moreover, if the above LMIs hold with α = −η, then the system is exponentially stable with a small enough decay rate. See Appendix A for the proof. Remark 2: Proposition 1 clearly shows that the sampling interval bound h that preserves the exponential stability is made dependent on the design parameters η and α. Additional highlights will be provided in Example 1 (given hereafter) which will show that maximum h preserving the stability may be significantly enlarged by tuning η. On the other hand, Proposition 1 also confirms the somewhat obvious fact that the exponential decay rate η + α < η is also depending on η and α. Remark 3: In the linear case, x(t) ˙ = A0 x(t) + A1 e−η(t−tk ) x(tk ), t ∈ [tk , tk+1 ) the conditions of Proposition 1 apply with λ = 0. Then, the last columns and rows in Ψ(0) and Ψ(h) must be deleted. Example 1 To illustrate the result of Proposition 1, consider the following system x(t) ˙ = −x(t)+sin(x(t))−Ke−η(t−tk ) x(tk ), t ∈ [tk , tk+1 ) with K = 1. Here we have A0 = −1, A1 = −K = −1, and M = 1. Then, application of Proposition 1 with α = −η leads to the maximum values of h that preserves the exponential stability of the system given in Table 1. Thus, (2.1) with η = 1 achieves exponential stability under more than twice larger sampling interval than under the constant gain (with η = 0). η h
0 1.38
0.5 1.84
0.7 2.36
0 1 0 −η(t−tk ) x(t) + e Kx(tk ) 0 −0.1 0.1
t ∈ [tk , tk+1 )
Φ12 = P − P2T + (A0 + ηIn )T P3 − Y2
Xτ (t) = (h − τ (t))
1 3.17
Consider the class of nonlinear systems x(t) ˙ = Ax(t) + f (x(t)) y(t) = Cx(t)
(9)
where x ∈ Rn and A, C are matrices with appropriate dimensions. It is supposed that the function f satisfies assumption (2) and y is accessible to measurements only at instants tk . Then the following observer is proposed for all t ∈ [tk , tk+1 ) ˆ(tk ) − y(tk )) (10) x ˆ˙ (t) = Aˆ x(t) + f (ˆ x(t)) − Ke−η(t−tk ) (C x with η > 0 and the matrix gain has the form T , K = K 1 ... K n
where K i are matrices with appropriate dimensions. Our goal is to determine a matrix K and an upper bound of the maximum allowable sampling period h such that, the observation error is globally exponentially vanishing. To this end, consider the observation error x ˜(t) = x ˆ(t) − x(t). It is readily checked that this error satisfies the following equation for all t ∈ [tk , tk+1 ) x ˜˙ (t) = A˜ x(t) + f (ˆ x(t)) − f (x(t)) − Ke−η(t−tk ) (C x ˆ(tk ) − y(tk ))
Consider the change of coordinates z1 (t) = eηt x ˜(t) that leads to the equation z˙1 (t) = (ηIn + A)z1 (t) − KCz1 (tk ) + eηt (f (ˆ x(t)) − f (x(t))) (11) Using the fact that, 1 f (ˆ x(t)) − f (x(t)) = fx (x(t) + s(ˆ x(t) − x(t)))ds x ˜(t), 0
equation (11) can be rewritten as follows:
z˙1 (t) = (ηIn + A)z1 (t) + A1 z1 (tk ) 1 + fx (x(t) + s(ˆ x(t) − x(t)))ds z1 (t) (12) 0
, t ∈ [tk , tk+1 ).
Table 1. Example 1: max. values of h vs. η preserving the stability Example 2 Consider the following much studied system (see e.g. Zhang et al. [2001], Fridman [2010]): 442
(13)
with A1 = −KC. Clearly, equation (12) falls in the class of systems described by equation (1). Therefore, the result of Proposition 1 can directly be applied to get sufficient conditions for the observer (10) to be exponentially convergent. This is illustrated in the next example.
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Example 3 Consider the following nonlinear system x˙ 1 (t) = x2 (t) x˙ 2 (t) = −2x1 (t) − x2 (t) + 0.2sin(x2 (t)) y(t) = Cx(t) = x1 (t) Clearly, this system is of the form (9) with A = 0 1 T , C = (1, 0), and f (x(t)) = ( 0 0.2sin(x2 (t)) ) . −2 −1 It is readily checked that the function f satisfies inequality (2) with M = diag{0, 0.04}. Then the observer (10) writes as follows: ˆ˙ 1 (t) = x ˆ2 (t) − K 1 e−η(t−tk ) (ˆ x1 (tk ) − y(tk )) x x1 (t) − x ˆ2 (t) + 0.2sin(ˆ x2 (t)) x ˆ˙ 2 (t) = −2ˆ 2 −η(t−tk ) (ˆ x1 (tk ) − y(tk )) −K e −K 1 0 . Applying PropoHere one has A1 = −KC = −K 2 0 T sition 1 with K = [1.65, −0.85] and α = −η, one obtains the results given in Table 2. η h
0 0.99
1 1.64
2 2.78
443
1.5
1
0.5
0
−0.5
2.4 2.86
Table 2. Example 3: max. values of h vs. η preserving the stability
−1 0
10
20
30
40
50
60
70
80
The above Table shows that the sampling interval bound h can be significantly increased thanks to the time-varying gain feature of the observer. Finally, we illustrate this Table by performing some numerical simulations : figure (1) (res. figure (2)) illustrates the case where η = 0 and h = 0.99 (resp. η = 2.4 and h = 2.86). These figures have been appended to appendix B. We have also derived from numerical simulations the real maximum bounds of h for both η = 0 and η = 2.4 which are respectively h = 1.1 and at least h = 100. This fact confirms that the improvements of the time-varying gain are important. 4. CONCLUSION In this paper, a novel stability result, namely Proposition 1, is established for a large class of sampled-data nonlinear systems which involve an exponentially decaying gain e−η(t−tk ) allowing to enlarge the sampling interval ; the stability conditions are expressed in terms of LMIs. We have also shown that our result can be applied to a wide class of nonlinear sampled-data observers. It has to be noticed that the results presented here can be easily extended to networks with Round Robin protocols. In future work, this idea will be extended to some classes of infinite dimensional systems. REFERENCES N. Bar Am and E. Fridman. Network-based distributed h∞ -filtering of parabolic systems. Automatica, 50, 2014. F. Cacace, A. Germani, and C. Manes. Nonlinear systems with multiple time-varying measurement delays. SIAM J. Control Optim., 52(3):1862–1885, 2014. M.C.F. Donkers and W.P.M.H. Heemels. Output- based event-triggered control with guaranteed l-gain and improved and decentralised event-triggering. IEEE Trans. on Automatic Control, 57(6):1362–1376, 2012. 443
Fig. 1. x2 (solid line), x ˆ2 (dashed line) for η = 0 and h = 0.99 E. Fridman. New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems. 43(4):309–319, 2001. E. Fridman. A refined input delay approach to sampleddata control. Automatica, 46:421–427, 2010. W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada. An introduction to event-triggered and self-triggered control. IEEE Conference on Decision and Control, 46: 3270–3285, 2012. H.Ye, A.N Michel, and L.Hou. Stability analysis of systems with impulse effects. IEEE Transaction on Automatic and Control, 43:1719–1723, 1998. P. Naghshtabrizi J. Hespanha and Y. Xu. A survey of recent results in networked control systems. IEEE Special Issue on Technology of Networked Control Systems, 95 (1):138–162, 2007. D. Neˇsi´c and A.R. Teel. A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time. IEEE Transactions on Automatic Control, 49:1103–1122, 2004. P. Tabuada. Event-triggered real-time scheduling of stabilizing control tasks. IEEE Trans. on Automatic Control, 52(9):1680–1685, 2007. W. Zhang, M. Branicky, and S. Phillips. Stability of networked control systems. Automatica, 21:84–99, 2001.
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V˙ (t) + 2αV (t) +
1
T
λz (t) M −
0
φTx (sx)ds
1
φx (sx)ds z(t) 0
≤ 2z T (t)P z(t) ˙ + (tk+1 − t)z˙ T (t)U z(t) ˙ t z˙ T (s)U z(s)ds ˙ − tk
X + XT −X + X 1 2 − ξT ξ X + XT T ∗ −X1 − X1 + 2 + (tk+1 − t)[z˙ T (t)(X + X T ))z(t)
1.4 1.2 1
+ 2z˙ T (t)(−X + X1 )z(tk )] + 2αz T (t)P z(t)
0.8
+ 2α(tk+1 − t)ξ T X + XT 2 ×
0.6 0.4 0.2 0
∗
−0.2 −0.4 −0.6 0
2
4
6
8
10
12
14
16
18
−X + X1
−X1 −
+ λz T (t)M z(t) 1 T T φx (sx)ds − λz (t)
20
X + XT + 2
X1T
0
1
ξ
φx (sx)ds z(t) 0
(A.1)
Using the well known Jensen’s inequality, one has, t z˙ T (s)U z(s)ds ˙ ≥ (t − tk )v1T U v1 tk
t T 1 with v1 = t−t z˙ (s). According to the description tk k approach Fridman [2001], the left-hand sides of the equations, Fig. 2. x2 (solid line), x ˆ2 (dashed line) for η = 2.4 and h = 2.86
0 = 2 z T (t)Y1T + z˙ T (t)Y2T + z T (tk )T T × − z(t) + z(tk ) +(t − tk )v1 0 = 2 z T (t)P2T + z˙ T (t)P3T × (A0 + ηIn )z(t) + A1 z(tk ) 1 + φx (sx)ds z(t) − z˙ 0
are added to V˙ (t) + 2αV (t) where Y1 , Y2 , T, P2 , P3 are free matrixes. Let us define the augmented vector, 1 µ = col{z(t), z(t), ˙ z(tk ), v1 , φx (sx)ds z(t)}. 0
Combining (A.2) and (A.1), one gets: V˙ (t) + 2αV (t) +λz T (t) M −
Appendix A. PROOF OF PROPOSITION 1
Let α ∈ (−η, 0] and differentiate V (t). Then, after some simple computations one obtains: 444
1 0
φTx (sx)ds
1 0
φx (sx)ds z(t)
≤ µT Ψµ < 0 provided that the following matrix Ψ is negative definite T T
Φ11 − Xα Φ12 + Xt−tk Φ13 + X1α ∗ Φ22 + (tk+1 − t)U Φ23 − X1(t−tk ) ∗ ∗ Φ33 − X2α ∗ ∗ ∗ ∗ ∗ ∗
(t − tk )Y1 P2 (t − tk )Y2T pT 3 (t − tk )T T 0 −(t − tk )U 0 ∗ −λIn
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T. Ahmed-Ali et al. / IFAC-PapersOnLine 48-12 (2015) 440–445
Writing the last matrix inequality, for τ (t) → 0 and τ (t) → h leads to LMIs (7) and (8), respectively. Now, using arguments of Lemma 2 in Fridman [2010], it is not difficult to see that, if the LMIs (5), (7) and (8) are feasible ¯ ∈ (0, h]. for some h > 0 then, they also are feasible for all h This completes the proof of Proposition 1.
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445