About the Lyapunov exponent of sampled-data systems with non-uniform sampling

About the Lyapunov exponent of sampled-data systems with non-uniform sampling L. Hetel ∗ A. Kruszewski ∗ J.P. Richard ∗∗ ∗

LAGIS, UMR CNRS 8146, Ecole Centrale de Lille, BP48, 59651 Villeneuve d’Ascq Cedex, France ∗∗ INRIA, Projet ALIEN, France

Abstract: In this paper we propose a method for evaluating the Lyapunov exponent of sampleddata systems with sampling jitter. We consider the case of systems in which the sampling interval is unknown, time-varying and bounded in a given interval. In order to take into account the inter-sampling behaviour of the system, the problem is addressed from the continuous time point of view. The approach exploits the fact that the command is a piecewise constant signal and leads to less conservative stability conditions. Using geometrical arguments, a lower bound of the Lyapunov exponent can be expressed as a generalized eigenvalue problem. Numerical examples are given to illustrate the approach.Copyright IFAC 2009

©

1. INTRODUCTION This paper is dedicated to the analysis of sampled data systems (˚ Astr¨om and Wittenmark (1997)) with non-uniform sampling periods. This problem represents an abstraction of more complex sample-and-hold phenomena that can be encountered in networked control / embedded systems (Zhang et al. (2001)). In the literature, several authors addressed the control under time-varying sampling period. This control problem is not easy even for linear time invariant (LTI) systems. In fact, it has been shown that, under variations of the sampling interval, the trajectory of a system may be unstable although for each value of the sampling period the equivalent discrete-time model is Schur stable (Nilsson (1998)). The stability problem for systems with sampling jitter has been addressed both from the discrete- and continuoustime point of view. In fact, most of the existing stability analysis approaches address the problem using discretetime tools. The first methods that are related to this problem can be found in the work of (Wittenmark et al. (1995)) and (Walsh et al. (1999)). In (Sala (2005), Skaf and Boyd (2009)), the authors used the gridding of the set of possible sampling intervals to derive LMI-based stability conditions and performance criteria. This simplified modelling approach has been applied by (Felicioni and Junco (2008)), in order to obtain sufficient algebraic conditions for existence of quadratic Lyapunov functions. In this case, the Lyapunov functions can be analytically constructed using Lie algebra arguments. However, the gridding based-models are not so accurate as, in practice, one encounters an infinite number of possible sampling periods. More accurate discrete-time models have been obtained by combining the gridding approach with a discrete time equivalent of the bounded real lemma (Balluchi et al. (2005), Fujioka (2008)) or by using a convex embedding ? This work was developed in the framework of GRAISyHM - projet Commande en Reseaux

modeling approach (Hetel et al. (2007, 2009)). The main drawback of the discrete-time analysis is the fact that it ignores the inter-sample behaviour of the system, in the sense that it does not take into account the existence of side-effects of sampled-data control, such as ripples. Moreover, these methods are numerically inaccurate when the sampling interval goes to zero (see Fujioka (2008), Oishi and Fujioka (2009)). This stability analysis of systems with sampling jitter has also been addressed in continuous time using a time-delay system modelling of the sample-and-hold function. The main advantage of such methods is the fact that they take into account the inter-sampling behaviour of the system. A notable approach is the descriptor system modeling applied by (Fridman et al. (2004)) for which stability conditions can be obtained using the Lyapunov-Krasovskii functional method. It has been shown by (Mirkin (2007)) that this approach is related to the one of (Kao and Lincoln (2004)) that is based on a norm bounded uncertainty modelling of the sample-and-hold operator. However, such an approach may induce conservatism, due to the symmetry of ellipsoidal norms used for norm bounded uncertainties. A different method, using impulsive delay differential equations, has been proposed in (Naghshtabrizi and Hespanha (2006), Naghshtabrizi et al. (2008)). In general, for the existing continuous-time approaches, the analysis does not take into account the particular variation of the sampling induced delay. This is why, faced to numerical benchmarks, they seem to be more conservative than the discrete-time approaches (see Fujioka (2008), Hetel et al. (2009)). From the literature survey, we can conclude that both continuous-time and discrete-time approaches have drawbacks and advantages. It is desirable to provide one method that is able to treat the problem in continuoustime (for inter-sampling performance issues) and to take into account the particular variation of the sampling in-

duced delay (in order to provide less conservative stability conditions, such as in the case of discrete-time approaches) . Here we intend to propose such a method. In order to explicitly analyse the inter-sampling behaviour, the proposed method is based on the evaluation of the Lyapunov exponent (also called decay-rate, see Boyd et al. (1994)). It represents a measure that can be used both for stability and performance analysis. For linear time invariant system it can be directly deduced from the state-matrix eigenvalues. However, in the case of sampled-data systems with non-uniform sampling its computation is not a trivial task. The contribution of the paper is to show how to take into account the piecewise constant nature of the control signal in order to determine an efficient estimation of the Lyapunov exponent. This paper is structured as follows: In section II we mathematically formalize the problem under study. Section III presents a method for estimating the Lyapunov exponent in the case of sampling variation. Next, we introduce a LMI method for the computation of the Lyapunov exponent. In Section IV, we present numerical examples illustrating our method. Finally, conclusions are given in Section V. Notations: For a matrix M we denote by kM k the induced matrix norm. By M  0 or M ≺ 0 we mean that the symmetric matrix M is positive or negative definite respectively. We denote the transpose of a matrix M by M T . By I (or 0) we denote the identity (or the null) matrix with the appropriate dimension. By λmax (M ) we denote the maximum eigenvalue of a square matrix. For a given set S, co(S) denotes the convex hull of S. By diag(v1 , v2 , . . . , vn ) we denote a diagonal matrix with v1 , v2 , . . . , vn on the main diagonal and zeros elsewhere. 2. PROBLEM DESCRIPTION In this section we provide a mathematical description of the problem under study. Consider n, m, two positive integers and the matrices A ∈ Rn×n , B ∈ Rn×m . We are interested in the class of systems described by the equation: x(t) ˙ = Ax(t) + Bu(t), ∀t ∈ R+ (1) n x(0) = x0 ∈ R . (2) Here x(·) : R+ → Rn represents the system state and u(·) : R+ → Rm represents the control. In order to take the sampling effect into account, we consider that the command is a piecewise constant state feedback, i.e. u(t) = Kx(tk ), ∀t ∈ [tk , tk+1 ) (3) where  tk : tk ∈ R+ , ∀k ∈ N (4) represents the set of sampling instants such that lim tk = ∞, (5) k→∞

0 = t0 < t1 < . . . < tk < . . . , (6) that is, no accumulation is allowed. We denote by hk := tk+1 − tk the sampling interval and we consider that it is bounded parameter, i.e. 0 ≤ hmin ≤ tk+1 − tk ≤ hmax < ∞. (7) In this paper we assume that the gain K ∈ Rm×n in equation (3) represents a known constant matrix, computed such as the A + BK is Hurwitz.

The Lyapunov exponent (Boyd et al. (1994)) is defined to be the largest α such that lim eαt kx(t)k = 0. (8) t−→∞

It represents a basic measure of system performance. It shows how fast the norm of the state vector converges to zero, since kx(t)k ≤ e−αt ckx(0)k, ∀t > 0 (9) for some positive constant c ∈ R (for more details, see Boyd et al. (1994), chapter 5). It can also represent a parameter for analysis the system stability : a system is stable if and only if its Lyapunov exponent is strictly positive. The goal of the paper is provide a method for computing the Lyapunov exponent. This problem is formalized as follows : Problem : Consider the system (1) with the control law (3) and all the possible sequences of sampling under the assumptions (6) and (7). Obtain the maximum computable α that satisfies (9). 3. LOWER ESTIMATE OF THE LYAPUNOV EXPONENT In this section, we present a method for computing a lower estimate of the Lyapunov exponent for the system (1) with the control law (3). First we present theoretical results for the computation of the Lyapunov exponent using a quadratic Lyapunov function; next we show how this problem can be approached using classical numerical optimization tools. 3.1 Time-delay model of the system Following the work of (Fridman et al. (2004)), system (1) with the control law (3) can be represented as a time-delay system with piecewise-continuous time varying delay: dx(t) = Ax(t) + BKx(t − τ (t)) (10) dt with τ (t) := t − tk , ∀t ∈ [tk , tk+1 ) . (11) The delay is bounded in a interval specified by the maximum sampling period, i.e. τ (t) ∈ [0, hmax ]. It represents a piecewise continuous and piecewise derivable function. Notice that the derivative of the delay satisfies the relation τ˙ (t) = 1, ∀t ∈ [tk , tk+1 ) , and that τ (tk ) = 0, i.e the delay signal has a sawtooth form. 3.2 Theoretical evaluation As follows we show how to use quadratic Lyapunov functions in order to estimate a lower bound of the largest Lyapunov exponent of the system (10) with the particular definition of the delay given in (11). Proposition 1. Consider the following matrix operator Λ(.) : R → Rn×n , Z τ Λ(τ ) := I + esA ds (A + BK) . (12) 0

and let M[0,hmax ] ⊂ Rn×n denote the convex hull of Λ(τ ) for all τ ∈ [0, hmax ], i.e. M[a,b] := co {Λ(τ ), ∀τ ∈ [a, b]} , (13)

with a, b ∈ R. Moreover, consider the following optimization problem: max α subject to P  0 and

≺ −2αM T P M, (14) for all M ∈ M[0,hmax ] . Then the largest Lyapunov exponent for the system (10) is at least α. Proof. Consider the following candidate Lyapunov function : V (x) = xT P x, where P = P T  0. (15) If the derivative of the Lyapunov function satisfies the relation dV (x) < −2αV (x) (16) dt for all system’s trajectories except x 6= 0, then V (x(t)) < e−2αt V (0) and s λmax (P ) −αt kx(t)k < e kx(0)k, λmin (P ) i.e. the Lyapunov exponent is at least α. The derivative of the Lyapunov function (15) along the solution of system (1) with the control law (3) is given by : dx(t) dV (x) dxT (t) = P x(t) + xT (t)P dt dt dt
= xT (t − τ (t))K T B T + xT (t)AT P x(t)

(17)

Notice that for all t ∈ [tk , tk+1 ) the solutions of system (1) with the control law (3) satisfy the relation: x(t) = e(t−tk )A x(tk ) +

Z

(t−tk )

esA dsBKx(tk )

(18)

0

= I+

Z

(t−tk )

Z

(t−tk )

0

= I+

esA dsA +

Z

(t−tk )

esA dsBK

0

0

!


K T B T + ΛT (τ )AT P Λ(τ ) +ΛT (τ )P (AΛ(τ ) + BK) x(t − τ ) 

< −2αxT (t − τ )ΛT (τ )P Λ(τ )x(t − τ )

(21)

holds for all x(t−τ ) 6= 0 and τ ∈ [0, hmax ] then the relation (20) is verified. This is the same as


K T B T + M T AT P M + M T P (AM + BK)

+xT (t)P (Ax(t) + BKx(t − τ (t))) .

xT (t − τ )

x(tk )

!

esA ds (A + BK) x(tk )

Using the definition of the delay given in equation (11) this is the same as ! Z τ (t) x(t) = I + esA ds (A + BK) x(t − τ (t)) (19) 0

From (17), (18) and (12) we obtain that
dV (x) = xT (t − τ ) K T B T + ΛT (τ ) AT P Λ (τ ) x(t − τ ) dt +xT (t − τ )ΛT (τ ) P (AΛ (τ ) + BK) x(t − τ ), (20) ∀τ ∈ [0, hmax] . In order to compute the Lyapunov exponent of the closedloop system under all possible sequences of sampling, notice that the derivative of the Lyapunov function should satisfy the relation (16) for all x 6= 0. Using (16), (18) and (12), notice that if


K T B T + ΛT (τ )AT P Λ(τ )

+ΛT (τ )P (AΛ(τ ) + BK) ≺ −2αΛT (τ )P Λ(τ ), (22) for all τ ∈ [0, hmax ] . Then one can show that condition (22) holds if and only if the inequality
K T B T + M T AT P M + M T P (AM + BK) ≺ −2αM T P M,

(23)

holds for all M ∈ M[0,hmax ] which ends the proof. 2 3.3 Numerical evaluation In this subsection, we show how to solve numerically the optimization problem presented in the previous subsection (the evaluation of the Lyapunov exponent). The difficulty of solving the problem is the fact that it leads to solving an infinite number of linear matrix inequalities, one for each point M in the set M[0,hmax ] defined in equation (13). To derive a finite number of LMI conditions, one has to deal with the non-linear representation of the exponential uncertainties (Hetel et al. (2007)) Z τ Ω(τ ) = eAs ds (24) 0

that appear in the definition of the set M[0,hmax ] = co {Λ(τ ), ∀τ ∈ [0, hmax ]} ,

with Λ(τ ) := I + Ω(τ ) (A + BK) defined in (12). Analytical methods exist in the literature for dealing with such uncertainties (Hetel et al. (2006, 2007), Cloosterman et al. (2007), Gielen et al. (2008), Donkers et al. (2009)). The basic idea is to embed this set in a polytopic set of matrices S, i.e. to find a set of N matrices Mi such that M[0,hmax ] ⊂ S = co {M1 , M2 , . . . , MN } . (25) In order to provide some insight about how such a convex embedding can be computed, consider the method proposed by Cloosterman et al. (2007), based on the Jordan normal form of the state matrix A. We chose present this method here because, in comparison with the other approaches, it can be explained in quite an easy manner. For the sake of simplicity we present the case when the matrix A has n real distinct eigenvalues λ1 , . . . , λn . In this case there exist an invertible matrix T such that A = T −1 diag (λ1 , . . . , λn ) T. The method leads to re-expressing the exponential uncertainty (24) under the form Z τ  Z τ −1 λ1 s λn s Ω(τ ) = T diag e ds, . . . , e ds T. 0

0

Computing the different minimumR and the maximum τ bounds on the scalar terms δi (τ ) = 0 eλi s ds, i = 1, . . . , n

τ ∈[0,hmax ]

τ ∈[0,hmax ]

0

0

3.5

3

2.5

δ2 (τ )

for τ ∈ [0, hmax ] and replacing for all the possible combinations of extreme values leads to a polytopic representation, a hypercube with N = 2n vertex: Ω(τ ) ∈ co {Ωi , i = 1, . . . , N = 2n } , with Ωi = T −1 diag (ω1 , ω2 , . . . , ωn ) T (26) and   Z τ Z τ ωj ∈ min eλi s ds, max eλi s ds ,

2

1.5

∀j = 1, . . . , n.

Example 1. Consider that the state matrix A is a second order diagonal matrix   λ1 0 A= 0 λ2

with λ1 = 2, λ2 = −1.5 and that the unknown parameter τ is bounded, i.e τ ∈ [0, 1]. In this case the exponential uncertainty Ω(τ ) is also a diagonal matrix of the form   δ1 (τ ) 0 0 δ2 (τ )  R τ λ s e 1 ds R 0 = 0 τ λ2 s ds 0 0 e
 1 λ1 τ  e −1 0 λ 1
= 1 λ2 τ 0 −1 λ2 e

Ω(τ ) =

(28) (29)

In the case of the Jordan normal form, the polytopic set (25) can be obtained using equations (26), (12) with Mi = I + Ωi (A + BK) . Using such a polytopic set, with a finite number of N vertices Mi , a finite number of LMI conditions can be obtained and the computation of the Lyapunov exponent can be expressed as the following generalized eigenvalue minimization problem (Boyd et al. (1994)) in P and α max α subject to P  0, T

K B +

MiT AT ≺




P Mi +

(30) MiT P

−2αMiT P Mi ,

0.5

0

(AMi + BK)

0

0.1

0.2

0.3

δ1 (τ )

0.4

0.5

0.6

0.7

Fig. 1.

Illustration of a polytopic embedding for the exponential uncertainty Ω(τ ) from Example 1. The exponential uncertainty is represented with the bold blue curve and the vertex of the polytopic descriptions are marked by stars.

(27)

An illustration of the polytopic description for this example (based on the Jordan development) is given in Figure 1. For the given numerical values, the unknown scalar parameters δ1 (τ ), δ2 (τ ) are bounded in the intervals [0, 0.51] and [0, 3.2], respectively. The convex description (26) can be obtained using the 4 vertex of the rectangle in the figure, i.e. Ω1 = diag(0, 0), Ω2 = diag(0, 0.51), Ω3 = diag(3.2, 0) and Ω4 = diag(3.2, 0.51).

T

1

to the authors knowledge, there is no explicit method in computational geometry for generating a convex polytopic embedding the matrix exponential with arbitrary small error. For this reasons we propose here an ad-hoc method that leads to a more precise polytopic approximation. However, this is done at the price of complexity. For any of the existing methods (Hetel et al. (2006), Cloosterman et al. (2007), Gielen et al. (2008)), a tighter over-approximation of the exponential uncertainty can be obtained by combining the convex embedding method with the gridding method proposed by (Sala (2005), Fujioka (2008)). The idea is to define a grid 0 = h1 < h2 < . . . < hP = hmax over the interval [0, hmax] and to use the existing analytical methods for constructing locally a convex embedding Sj , Sj = co {Mj1 , Mj2 , . . . , MjN ∗ } ⊃ M[hj ,hj+1 ] , for each sub-interval [hj , hj+1 ] with j = 1, . . . , P. The global convex embedding S ⊃ M[0,hmax ] is obtained with S = co {Sj , j = 1, . . . , P } . In this case, some of the local vertex can be removed using numerical methods for convex hull and vertex computation (as described in Avis and Fukuda (1992)). An illustration of this method for the case presented in Example 1 is given in Figure 2 for the case of the Jordan normal form and in Figure 3 for the case of the Taylor development method (Hetel et al. (2006)). The method is applied by replacing the Mi matrices in (31) with the vertex of the polytope S.

(31)

4. NUMERICAL EXAMPLES

∀i = 1, . . . , N 3.4 Numerical Improvements For a given α, the conditions (31) provide sufficient conditions for the existence of a solution to conditions (14). The quality of the solution depends on the quality of the polytopic approximation (25) of the exponential uncertainty (24). Clearly, any polytope can be refined using using a more precise description. Still, for the moment,

As follows we presents several examples illustrating our method. Example 2. We consider the case where the continuoustime system (1),(3) is described by the matrices     1 2 1 A= , B= . 2 2 .6

and

K = − (1 6) .

3

1.2

2.5

1

α (hmax )

1.4

δ2 (τ )

3.5

2

1.5

0.8

0.6

1

0.4

0.5

0.2

0

0

0.1

0.2

0.3

δ1 (τ )

0.4

0.5

0.6

Fig. 2.

Illustration of a polytopic embedding for the exponential uncertainty Ω(τ ) from Example 1 using the Jordan normal form with 5 sub-interval. The global polytope is represented in gray, its vertex are marked by stars. The local polytopes are represented using thin lines.

3.5

3

δ2 (τ )

2.5

2

1.5

1

0.5

0

0

0.1

0.2

0.3

δ1 (τ )

0.4

0.5

0.6

0

0.7

0.7

Fig. 3.

Illustration of a polytopic embedding for the exponential uncertainty Ω(τ ) from Example 1 using the Taylor expansion method with 5 sub-interval. The global polytope is represented in gray, its vertex are marked by stars. The local polytopes are represented using thin lines.

We study the influence of the variation of the sampling period, given by hmax on the estimation of the Lyapunov exponent when the minimum sampling period is hmin = 0. The results, illustrated in Figure 4, allow to certify theoretically the well known fact that jitter affects the control performances. Example 3. As follows we use the method proposed in this paper for stability analysis. Consider a continuous-time system described by the following matrices:     1 15 1 A= and B = . −15 1 1

The state matrix A has complex eigenvalues λ = 1 ± 15i. The gain K = − (5.33 9.33) is obtained by pole placement, in such way that the ideal closed-loop system is stabilized and oscillations are reduced : the matrix A + BK has the

0

0.02

0.04

0.06

0.08

0.1

hmax

0.12

0.14

0.16

0.18

Fig. 4.

The estimation of the Lyapunov exponent as a function of the maximum sampling period with hmin = 0.

eigenvalues at −1 ± i. We intend to use our method in order to characterize the maximum allowable sampling interval hmax for which the stability is ensured despite sampling interval variations. This comes to finding the maximum value of hmax for which the Lyapunov exponent is positive. For this example, the method of Mirkin (2007) and Naghshtabrizi et al. (2008) show that the system is stable for the intervals [0, 0.014] and [0, 0.033], respectively. Using the method proposed here, the stability can be ensured for a time-varying sampling period in the interval [0, 0.08] which shows a significant conservatism reduction. An example of system evolution is given in Figure 5. Notice however that using discrete-time methods such as (Hetel et al. (2009)), we can show that the system is stable for more important values of sampling period, provided that the minimum sampling period is larger than zero. For this example the method in (Hetel et al. (2009)) is able to show that the system is stable for any time-varying sampling period in the interval [0.91, 0.95] which means that improvents are still possible in continuous-time 1 . For both of the examples we used a polytopic embedding based on Taylor method (10th order developement), with 100 local polytopes obtained from an equidistant gridding. The optimisation problems were solved using SEDUMI and YAMLIP (Lofberg (2004)), a PC with Intel Centrino 2 and 4G RAM, in less than 1 minute. 5. CONCLUSION This paper proposed a method for computing the Lyapunov exponent of sampled-data systems with sampling jitter. The problem was addressed from the continuous time point of view. The basic idea was to take into account the evolution of the delay τ by using an integration operator Λ(τ ) that can be treated by means of the exponential uncertainty approach. Numerical examples are given to present the approach and illustrate the improvement in comparison with other classical approaches. In the future, this preliminary research will be used as a basis for a 1

However, the discrete-time approach does not guarantee the intersampling behaviour and, in this case, ripples may occur.

x1 (t)

1.4 1.2 1

x2 (t)

2

u(t)

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0.7

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0

0.1

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0 −20 0.1

hk

0.1

0 −2

0.05 0 0.1

V (t)

0

0.05 0

t

Fig. 5.

Simulation with hmin = 0 and hmax = 0.08 for the Example 3. The sampling instances are marked by stars.

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