Stability and Stabilization of Linear Sampled-data ...

Stability and Stabilization of Linear Sampled-data Systems with Multi-rate Samplers and Time Driven Zero Order Holds ? Miad Moarref, Luis Rodrigues Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, H3G 2W1, Canada

Abstract In multi-rate sampled-data systems, a continuous-time plant is controlled by a discrete-time controller which is located in the feedback loop between sensors with different sampling rates and actuators with different refresh rates. The main contribution of this paper is to propose sufficient Krasovskii-based stability and stabilization criteria for linear sampled-data systems, with multi-rate samplers and time driven zero order holds. For stability analysis, it is assumed that an exponentially stabilizing controller is already designed in continuoustime and is implemented as a discrete-time controller. For each sensor (or actuator), the problem of finding an upper bound on the lowest sampling frequency (or refresh rate) that guarantees exponential stability is cast as an optimization problem in terms of linear matrix inequalities (LMIs). Furthermore, sufficient conditions for controller synthesis are formulated as LMIs. It is shown through examples that choosing the right sensors (or actuators) with adequate sampling frequencies (or refresh rates) has a considerable impact on stability of the closed-loop system. Key words: Linear systems; Multi-rate sampled-data systems, Krasovskii functionals, Linear matrix inequalities.

Introduction

Plant

? This paper was not presented at any IFAC meeting. Corresponding author L. Rodrigues. Tel. +1 514 8482424x3135; Fax +1 514 8482802. Email addresses: m [email protected] (Miad Moarref), [email protected] (Luis Rodrigues).

Preprint submitted to Automatica

x(t )  Ax(t )  Bu (t )

...

Sensor

Sm

xm (t )

S2 Sensor

...

x1 (t ) x2 (t )

Sensing Block

u

S1

Z2 Zero Order Hold

un (t )

Sensor

Z1

Actuating Block

In multi-rate sampled-data systems, a continuous-time plant is controlled by a discrete-time controller which is located in the feedback loop between sensors with different sampling rates and actuators with different refresh rates (see Fig. 1). Control signals are computed as soon as new data becomes available from any of the sensors. If the actuators have high refresh rates, they almost instantly apply the updated control signals. Such actuators can be modeled as event driven zero order holds where the events are the controller update instants. Electrostatic and piezoelectric actuators that work at frequencies around 1 (MHz) are examples of these actuators. However, according to Zupan et al. (2002), actuators such as solenoids, electric cylinders, shape memory alloys, and electroactive polymers have low refresh rates of 10 (Hz) or less. The delay in applying the control signals using these actuators is not negligible and affects stability and performance of the closed-loop system. In this paper, we focus on actuators with low refresh rates and model them as time driven zero order holds.

Zero Order Hold

u1 (t ) u2 (t )

Zero Order Hold Z nu

1

Linear Feedback Controller Continuous Signal Digital Signal

Fig. 1. The schematic diagram of a linear multi-rate sampled-data system.

Stability analysis and controller synthesis of multi-rate sampled-data systems are practically relevant problems that have attracted researchers for several decades (see Chen and Qiu (1994); Longhi (1994); Lall and Dullerud (2001); Nagamune et al. (2005) and the references therein). The main drawback of these works is their restriction to sensors with known, uniform, and commensurate sampling intervals. However, these restrictions do not hold in practice. For instance, in the servo control of brushless DC motors via Hall-effect sensors, the sampling intervals depend on the motor speed and, therefore, are not predetermined or uniform; see Yen et al. (2002). Furthermore, all sensors are

11 June 2014

diagonal matrix with diagonal entries d1 , . . . , dm is denoted by diag(d1 , . . . , dm ). A matrix with rows r1 , . . . , rm is denoted by column(r1 , . . . , rm ). The notation 1 represents the vector column(1, . . . , 1). Where there is no confusion, a vector x(t) is written as x. The notation |.| denotes the norm of a vector. The Kronecker product is represented by ⊗.

prone to uncertain non-uniform samplings due to non-ideal communication links with delays and packet losses. Single-rate sampled-data systems with non-uniform sampling intervals have been studied extensively in the literature. Input-delay approach and impulsive systems modeling, combined with (discontinuous) Lyapunov-Krasovskii functionals, are among the main approaches to address sampleddata systems (see Fridman et al. (2004); Naghshtabrizi et al. (2008); Fridman (2010); Liu and Fridman (2012); Seuret (2012); Briat and Seuret (2012) and the references therein). To the best of the authors’ knowledge, the multi-rate sampled-data problem with non-uniform sampling intervals (for the sensors) and update intervals (for the actuators) has not received many research contributions. Dual-rate sampled-data systems with non-uniform sampling and update intervals are studied in Suplin et al. (2007). In that setup, however, the states are sampled by one sensor and the control signal is applied via a time driven zero order hold that works at a different rate than the sensor. In this paper, we address a more general problem where the states are sampled by dedicated sensors at different rates and the inputs are asynchronously applied to the plant through multiple time driven zero order holds.

2

Problem Statement

Consider a stabilizable linear system x(t) ˙ = Ax(t) + Bu(t),

(1)

where x ∈ Rnx denotes the state vector, A ∈ Rnx ×nx , B ∈ Rnx ×nu , and u ∈ Rnu is the control input. Let u(t) = Kx(t), with K ∈ Rnu ×nx , be a continuous-time stabilizing control input. In practice, the controller is located in the feedback loop between a sensing block and an actuating block (see Fig. 1). The sensing block comprises m sensors S i , i ∈ {1, . . . , m}, where m ≤ nx . Each sensor S i is dedicated to sampling one component of the state vector denoted by xi . Each component xi can possibly be a vector (e.g. a camera provides the position of an object in a two dimensional space). The state vector is then written as x = column(x1 , . . . , xm ). The actuating block comprises nu actuators. Each actuator is modeled as a zero order hold Z j , j ∈ {1, . . . , nu }. The sensors and the zero order holds are time driven and asynchronous. Furthermore, the sampling frequency of the sensors and the refresh rate of the actuators are uncertain and non-uniform. For a study of linear multi-rate sampled-data systems where the zero order holds are event driven see Moarref and Rodrigues (2013).

The main contribution of this paper is to present sufficient Krasovskii-based stability and stabilization criteria for linear sampled-data systems with multi-rate samplers and time driven zero order holds. For stability analysis, it is assumed that an exponentially stabilizing controller is already designed in continuous-time and is implemented as a discretetime controller. If the sampling and update intervals are large, the continuous-time stability results are no longer applicable. Our objective is to find lower bounds on the maximum allowable sampling periods (MASPs) and maximum allowable update periods (MAUPs) that recover the exponential stability result. For each sensor (or actuator), the problem of finding an upper bound on the lowest sampling frequency (or refresh rate) that guarantees exponential stability is cast as an optimization problem in terms of linear matrix inequalities (LMIs). Furthermore, we formulate sufficient conditions for controller synthesis as LMIs. Note that LMIs can be solved in polynomial time using available optimization software such as SeDuMi (Strum (1999)) and YALMIP (L¨ofberg (2004)). It is shown through examples that choosing the right sensors (or actuators) with adequate sampling frequencies (or refresh rates) has a considerable impact on stability of the closed-loop system.

Assumption 1 The sensor S i , i ∈ {1, . . . , m}, samples the ith component of the state vector xi at sampling instants sik , where 0 < s < sik+1 − sik < τsi , ∀ k ∈ N. Assumption 2 The zero order hold Z j , j ∈ {1, . . . , nu }, is j updated at instants zkj , where 0 < z < zk+1 − zkj < τzj , ∀ k ∈ N. The positive constant s (respectively z ) models the fact that a sensor (respectively an actuator) cannot measure a particular phenomenon (respectively be updated) twice at the same instant. The scalars τsi , i ∈ {1, . . . , m}, and τzj , j ∈ {1, . . . , nu }, denote the longest interval between two consecutive samplings by the sensor S i and the longest interval between two consecutive updates of the actuator Z j , respectively. For each sensor S i , i ∈ {1, . . . , m}, the time elapsed since the sensor’s last sampling instant is denoted by a sawtooth function ρis (t) (see the top plot in Fig. 2) defined as ρis (t) = t − sik , ∀ t ∈ [sik , sik+1 ). (2) Similarly, the time elapsed since the last update of each zero order hold Z j , j ∈ {1, . . . , nu }, is denoted by a sawtooth function ρjz (t) (see the middle plot in Fig. 2) defined as

The rest of the paper is organized as follows. Section 2 is dedicated to problem statement. Stability analysis and controller synthesis results are presented in Section 3 and Section 4, respectively. Numerical examples are provided in Section 5, followed by the concluding remarks in Section 6. Notation. The zero matrix and the identity matrix are represented by 0 and I, respectively. The notation Z1 > Z2 (or Z1 < Z2 ), where Z1 and Z2 are symmetric matrices, denotes that Z1 − Z2 is positive (or negative) definite. A block

j ρjz (t) = t − zkj , ∀ t ∈ [zkj , zk+1 ).

2

(3)

 si ( zkj ) *



2 s

zkj* t

ski

 s2 2 0 s1

s22

s32

s42

 zj (t ) Time

 szij (t ) Time

 1z

Fig. 3. The function ρij sz (t)

 1z 1 1

0

z

z

0

1 1

1 2

1 2

1 3

1 4

z

z

1 3

1 4

Considering Fig. 2, note that at each instant zkj ∗ , j ∈ {1, . . . , nu }, the function ρjz (t) vanishes and the sawtooth i j i function ρij sz (t) jumps to ρs (zk∗ ). Therefore, unlike ρs (t) j ij and ρz (t), the function ρsz (t) does not necessarily decrease to zero at its discontinuities. Based on (4) and (5), equation (8) yields

Time

 sz21

 sz21

j i ij 0 ≤ ρij sz < τz + τs = τsz .

z

z

z

z

Time

(9)

ij Here, τsz denotes the maximum allowable transfer interval from the sensor S i to the zero order hold Z j . The control signal at each input channel j ∈ {1, . . . , nu } is now computed as uj (t) = Kj xj (t), where Kj represents the j th row of K, i.e. K = column(K1 , . . . , Knu ), and

Fig. 2. The sawtooth functions ρ2s (t), ρ1z (t), and ρ21 sz (t) in a multi-rate sampled-data structure.

Therefore, equation (2) and Assumption 1 yield 0≤

ρis


0 and λ > 0 such that for any initial condition x0 ∈ W the solution is (globally) defined and satisfies |x(t)| ≤ δe−λt ||x0 ||W , ∀ t ≥ 0.

V (zk , xzk ) ≤ V (zk− , xz− ),

where V (zk− , xz− ) = limt%zk V (t, xt ). Inequality (14) k guarantees that V is non-increasing at the instants zk . The following theorem provides sufficient conditions for exponential stability of linear multi-rate sampled-data systems.

Consider the Lyapunov-Krasovskii functional V (t, xt ) =

4 X

Vl , t ∈ [zk , zk+1 ),

Theorem 1 Consider the closed-loop linear multi-rate sampled-data system defined in (1) and (11) under Assumptions 1 and 2. The system is exponentially stable with a decay rate greater than α/2 if there exist symmetric pos0 itive definite matrices P , R, Rij , Rij , i ∈ {1, . . . , m}, j ∈ {1, . . . , nu }, a symmetric matrix X1 , and matrices X2 , 0 00 N , N , N , and N , with appropriate dimensions, satisfying

(13)

l=1

where V1 =xT (t)P x(t), Z V2 =(zk+1 − t)

t

h i eα(s−t) x˙ T (s) x eT (s) xT (zk )



h iT × R x˙ T (s) x eT (s) xT (zk ) ds,

  

t−ρz

V3 =

nu X m Z X j=1 i=1

V4 =

nu X m Z X j=1 i=1

ij (τsz − t + s)eα(s−t) x˙ Ti (s)Rij x˙ i (s) ds,

t

t−ρij sz

Ψ + γM1 + (1 ⊗ F )T

!



?   < 0, ∀ γ ∈ {0, τz },  U1 − D1 (15) " # 0 T Ψ + τz M2 + (1 ⊗ F ) τ (R + R )(1 ⊗ F ) ? < 0, U2 − D2 (16)

t

t−ρij sz

(14)

k

ij 0 (τsz − t + s)eα(s−t) x˙ Ti (s)Rij x˙ i (s) ds,

0 where x e(t) is defined in (12), P , R, Rij , Rij , i ∈ {1, . . . , m}, j ∈ {1, . . . , nu }, are positive definite matrices, and α/2 is the desired bound on the decay rate. Also, for t ∈ [zk , zk+1 ], consider the functional

0

×τ (R + R )(1 ⊗ F )

h where τz is defined in (7), F = A

BK

i 0 , and

1j mj τ j = diag(τsz I, . . . , τsz I), τ = diag(τ 1 , . . . , τ nu ),

1j mj h i h iT E = diag(eατsz I, . . . , eατsz I), E = diag(E 1 , . . . , E nu ), j V5 (t, xt ) = (zk+1 −t) xT (t) xT (zk ) X xT (t) xT (zk ) , Rj = diag(R1j , . . . , Rmj ), R = diag(R1 , . . . , Rnu ), 0 0 0 0 0 0 " # Rj = diag(R1j , . . . , Rmj ), R = diag(R1 , . . . , Rnu ), X1 − X2 h i h iT and X1T = X1 . Nowhere X = Ψ = FT P 0 0 + P 0 0 F −X2T X2 + X2T − X1 h iT h i tice that the sum of the entries of X is equal to zero and +α I 0 0 P I 0 0 therefore V5 vanishes at the bounds of the interval [zk , zk+1 ].

4

h i   T  + (zk+1 − t) x˙ T x eT xT (zk ) I 0 −I I 0 −I     h iT T  − 0  0    N − N 0 0 0 0 × R x˙ T x (18) eT xT (zk ) − αV2 , 0 0 0 0 0 0 h iT T h i where h(t) ∈ R(2+nu )nx is an arbitrary time varying vector − 1⊗I −I 0 N −N 1⊗I −I 0 (see Moarref (2013) for more details). Similarly, we can h iT 0T h i write 0 − 0 −I 1⊗I N −N 0 −I 1⊗I nu X m   h i  h iT 00T X 00 ij T N −N 1⊗ I 0 −I − 1⊗ I 0 −I V˙ 3 = τsz x˙ i Rij x˙ i j=1 i=1 " #T " # Z t  I 0 0 I 0 0 − X , − eα(s−t) x˙ Ti (s)Rij x˙ i (s) ds − αV3 0 0 I 0 0 I t−ρij sz nu X m   T   X ij −1 ij T T ατsz A BK 0 A BK 0 ≤ τsz x˙ i Rij x˙ i + ρij Rij hij sz hij e     j=1 i=1    R M1 =  I 0 I 0 0 0  T hij − xi − xi (t − ρij sz ) 0 0 I 0 0 I   T ij " #T " # (19) − hij xi − xi (t − ρsz ) − αV3 , I 0 0 I 0 0 nu X m  +α X X ij T 0 0 0 I 0 0 I V˙ 4 = τsz x˙ i Rij x˙ i #T " # # " " #T " j=1 i=1 Z t−ρz F I 0 0 I 0 0 F , X + X + 0 − eα(s−t) x˙ Ti (s)Rij x˙ i (s) ds 0 0 0 I 0 0 I 0 t−ρij sz Z t  M2 = − diag(0, I, I)N T − N diag(0, I, I), α(s−t) T 0 − e x ˙ (s)R x ˙ (s) ds − αV4
i ij i T 0T U1 = column Eτ N , Eτ N , t−ρz nu X m  0
X ij D1 = diag Eτ R, Eτ R , 0−1 0 ij T 0 0T ατsz τsz x˙ i Rij x˙ i + (ρij Rij hij ≤ sz − ρz )hij e
T 0T 00T T j=1 i=1 U2 = column Eτ N , Eτ N , τz N , τz N ,  T 0 0 0
hij − xi (zk ) − xi (t − ρij D2 = diag Eτ R, Eτ R , τz e−ατz R, τz e−ατz R . sz )   0T ij − hij xi (zk ) − xi (t − ρsz ) T

ατz 0−1 00 + ρz h00T Rij hij − [xi − xi (zk )] h00ij ij e  − h00T (20) ij [xi − xi (zk )] − αV4 ,

P5

PROOF. First, we define W (t, xt ) = l=1 Vl and compute its time derivative in the interval t ∈ (zk , zk+1 ). The time derivative of V1 is V˙ 1 = x˙ P x + x P x. ˙ T

T

(17)

Applying the Leibniz integral rule to V2 and using Jensen’s inequality yields V˙ 2 = −

Z

t

t−ρz

h i eα(s−t) x˙ T (s) x eT (s) xT (zk )

h iT × R x˙ T (s) x eT (s) xT (zk ) ds h i + (zk+1 − t) x˙ T x eT xT (zk ) h iT × R x˙ T x eT xT (zk ) − αV2 h i ≤ρz hT eατz R−1 h − xT − xT (zk ) ρz x eT ρz xT (zk ) h h iT − hT xT − xT (zk ) ρz x eT ρz xT (zk )

5

where hij (t), h0ij (t), and h00ij (t), i ∈ {1, . . . , m}, j ∈ {1, . . . , nu }, are arbitrary time-varying vectors of the appropriate dimensions. Based on (8) and (9) we can ij ij write 0 ≤ ρij sz − ρz ≤ ρsz < τsz . We use this inequality to make (19) and (20) independent of the coefficient ρij sz , i ∈ {1, . . . , m}, j ∈ {1, . . . , nu }. Leaving ρij sz in these inequalities decreases conservatism of the sufficient conditions in Theorem 1, but increases the number of LMIs to 2nu m and causes scalability issues. Inequalities (19) and (20) are hence rewritten in a more compact form as V˙ 3 ≤

nu  X

−1

T

T

x˙ T τ j Rj x˙ + hj τ j E j Rj hj − [x − xj ] hj

j=1

 T − hj [x − xj ] − αV3 T

= (1 ⊗ x) ˙ T τ R(1 ⊗ x) ˙ + h τ ER T

− h [1 ⊗ x − x e] − αV3 ,

−1

T

h − [1 ⊗ x − x e] h (21)

V˙ 4 ≤

nu  X

0T

0

0−1 0

T

if zk+1 −zk is equal to 0 or τz . Similarly, for ρz = zk+1 −zk , LMI (15) with γ = 0 and LMI (16) imply (25) is valid if zk+1 − zk is equal to 0 or τz . Since (24) is affine in ρz and zk+1 − zk , LMIs (15) and (16) are sufficient conditions for (25) to hold for all possible values of ρz and zk+1 − zk . Solving (25) in the interval (zk , zk+1 ), k ∈ N, − yields W (zk+1 , xz− ) ≤ e−α(zk+1 −zk ) W (zk , xzk ). The

0

x˙ T τ j Rj x˙ + hj τ j E j Rj hj − [x(zk ) − xj ] hj

j=1 0T

00T

0−1 00

− hj [x(zk ) − xj ] + ρz hj eατz Rj hj  00T T 00 − [x − x(zk )] hj − hj [x − x(zk )] − αV4 0T

0

= (1 ⊗ x) ˙ T τ R (1 ⊗ x) ˙ + h τ ER T

0

0T

− [1 ⊗ x(zk ) − x e] h − h 00T ατz

+ ρz h 00T

−h

e

R

0−1 00

0−1 0

h

k+1

functional W strictly decreases in intervals (zk , zk+1 ) that have a nonzero length (note that, according to Lemma 1, for any time interval with length z , there exists at least one interval (zk , zk+1 ), with a length greater than or equal to z /(nu + 1)). Since V5 is continuous and V5 (zk+1 , xzk+1 ) = V5 (zk , xzk ) = 0, we can write − eα(zk+1 −zk ) V (zk+1 , xz− ) − V (zk , xzk ) < 0. There-

[1 ⊗ x(zk ) − x e]

00

h − (1 ⊗ [x − x(zk )])T h

(1 ⊗ [x − x(zk )]) − αV4 ,

(22)

0

0

where τ j , τ , E j , E, Rj , R, Rj , and R are defined in Theorem 1, xj and x e are defined in (10) and (12), 0 hj = column(h1j , . . . , hmj ), hj = column(h01j , . . . , h0mj ),

k+1

fore, inequality (14) yields eα(zk+1 −zk ) V (zk+1 , xzk+1 ) − V (zk , xzk ) < 0. Hence, the conditions of Theorem 4 in Seuret (2012) are satisfied and the closed-loop sampled data system is exponentially stable with a decay rate greater than α/2. Note that the Zeno phenomenon does not occur since, by Assumption 2, for any time interval with a length z , there exists a finite number of (at most nu ) instants zk , k ∈ N. 2

00

and hj = column(h001j , . . . , h00mj ) are vectors in Rnx , and 0

0

0

h = column(h1 , . . . , hnu ), h = column(h1 , . . . , hnu ), 00 00 00 and h = column(h1 , . . . , hnu ) are vectors in Rnu nx . The time derivative of V5 is computed as i h iT xT (zk ) X xT xT (zk ) h i h iT + (zk+1 − t) x˙ T 0 X xT xT (zk ) h i h iT + (zk+1 − t) xT xT (zk ) X x˙ T 0 .

h V˙ 5 = − xT

Remark 1 The conditions of Theorem 1 can be further relaxed by using Wirtinger’s inequality as an alternative to Jensen’s inequality; see Seuret and Gouaisbaut (2013). (23)

Based on Theorem 1, the problem of finding a lower bound on the MASP τsi or the MAUP τzj such that exponential stability is preserved, can be formulated as a convex optimization problem in terms of LMIs.

We now define an augmented vector
ζ(t) ∈ R(2+nu )nx as ζ(t) = column x(t), x e(t), x(zk ) , t ∈ [zk , zk+1 ), where x e is defined in (12). Recalling (1) and (11), the ˙ = h closed-loop i vector field can be written as x(t) A BK 0 ζ(t). Replacing x˙ in (17), (18), and (21)-(23), 0

T

Problem 1 maximize τsi (or τzj ) 0 subject to Rij > 0, Rij > 0, i ∈ {1, ..., m}, j ∈ {1, ..., nu },

0T

setting h(t) = N T ζ(t), h(t) = N ζ(t), h (t) = N ζ(t), 00 00T 0 00 h (t) = N ζ(t), where N , N , N , and N are matrices of the appropriate dimensions, and using Lemma 2 yields

P > 0, R > 0, X1 = X1T , (15) and (16).

˙ + αW = P5 (V˙ l + αVl ) W l=1 ≤ ζ T Ψ + (zk+1 − zk − ρz )M1 + ρz M2 + ρz N eατz R−1 N T 0

+ (1 ⊗ F )T τ (R + R )(1 ⊗ F ) + N τ ER 0

+ N τ ER

0−1

N

0T

00 ατz

+ ρz N e

R

0−1

N

−1

00T

N

T

4

! ζ,

Controller Synthesis

(24) When the controller gain K is unknown, the LMIs in Theorem 1 turn into bilinear matrix inequalities that cannot be solved efficiently. The following theorem addresses this issue by providing sufficient conditions for the controller synthesis problem in terms of LMIs.

where Ψ, M1 , M2 , and F are defined in Theorem 1. From (6), for t ∈ (zk , zk+1 ), the function ρz varies between 0 and zk+1 − zk , which in turn varies between 0 and τz (see Assumption 2). Using Schur complement, for ρz = 0, LMI (15) with γ ∈ {0, τz } implies ˙ (t, xt ) + αW (t, xt ) < 0, t 6= zk , W

We denote the computed lower bound on the MASP τsi (or i the MAUP τzj ) that preserves exponential stability by τs,max j (or τz,max ). The number of optimization variables in Problem 1 is O(4.5n2u n2x ).

Theorem 2 Consider the closed-loop linear multi-rate sampled-data system defined in (1) and (11) under Assumptions 1 and 2. There exists an exponentially stabilizing

(25)

6

−1

h i h i T 0T U2 = column N − 1 ⊗ Q 0 0 , N − 1 ⊗ 0 0 Q , i h i  h 00T N − 1 ⊗ Q 0 − Q , τ 1 ⊗ AQ BY 0 , T 0T 00T
τ EN , τ EN , τz N T , τz N ,
1 e e−ατz τz Q , D2 = diag Q, Q, Q, τ Q, QEτ , QEτ , e−ατz τz Q, 2

linear state feedback gain K = Y Q , if there exist symmetric positive definite matrices Q, Qij , i ∈ {1, . . . , m}, 0 00 j ∈ {1, . . . , nu }, matrices Y , N , N , N , and N , with appropriate dimensions, and scalars 1 and 2 , satisfying "

"

#

Φ + γM1

?

U1

− D1

Φ + τz M2

?

U2

− D2

< 0, ∀ γ ∈ {0, τz },

(26)

and τ and E are defined in Theorem 1.

# < 0,

(27) PROOF. Here, we prove that inequalities (26) and (27) are sufficient conditions for LMIs (15) and (16). Suppose there exist matrices Q > 0, Qij > 0, i ∈ {1, . . . , m}, 0 00 j ∈ {1, . . . , nu }, matrices Y , N , N , N , and N , with appropriate dimensions, and scalars 1 and 2 , satisfying the stabilization criteria in (26) and (27). Let

where τz is defined in (7), and Qj = diag(Q1j , . . . , Qmj ),

(28)

Q = diag(Q1 , . . . , Qnu ), e = diag(Q, Q, Q), Q

(29) (30)

0 −1 e −1 , Rij = Rij P = Q−1 , R = Q = Q−1 , ij , X1 = 1 Q

Y = column(Y1 , . . . , Ynu ), Y = diag(Y1 , . . . , Ynu ),

−1

(31) iT h i h iT Φ = AQ BY 0 I 0 0 + I 0 0 h i h iT h i AQ BY 0 + α I 0 0 Q I 0 0  T   I 0 −I I 0 −I     T  − 0  0  0 0  N − N 0 0  0 0 0 0 0 0 h iT h T 0T 0 + 0 I 0 (N + N ) + (N + N ) 0 I   1 Q 0 − 2 Q   , − 0 0 0   −2 Q 0 (22 − 1 )Q T   1 AQ 1 BY 1 AQ 0     M1 =  0 0  0  + 0 −2 AQ − 2 BY 0 −2 AQ   1 Q 0 − 2 Q   , + α 0 0   0 −2 Q 0 (22 − 1 )Q h

e −1 N Q e −1 , N = Q e −1 N Q , X2 = 2 Q−1 , N = Q 0 e −1 N 0 Q−1 , N 00 = Q e −1 N 00 Q−1 , K = Y Q−1 , N =Q (32)

×

e and Y where i ∈ {1, . . . , m}, j ∈ {1, . . . , nu }, and Q, Q, are defined in (29)-(31). Multiplying (26) from left and right e −1 , I) and using Schur complement yields by diag(Q  0

i

1 BY 0 − 2 BY

Ψ + γM1 + (1 ⊗ F )T

  ×τ (R + R0 )(1 ⊗ F ) + Υ   T  Eτ N  0T Eτ N



! ?

?

− Eτ R

?

0

− Eτ R

0

    < 0,  

(33) where Ψ, M , and F are defined in Theorem 1, with the 1  0 change of variables (32), and   h iT −1  h i 0  Υ= 1⊗ I 0 0 Q 1⊗ I 0 0 0  h iT −1  h i + 1⊗ 0 0 I Q 1⊗ 0 0 I h i  h iT −1  Q 1⊗ I 0 −I + 1⊗ I 0 −I T

T

M2 = − diag(0, I, I)N − N diag(0, I, I), h i h i T 0T U1 = column N − 1 ⊗ Q 0 0 , N − 1 ⊗ 0 0 Q , h i  h i 00T N − 1 ⊗ Q 0 − Q , τ 1 ⊗ AQ BY 0 ,   AQ BY 0  
 , τ EN T , τ EN 0T , τz  0 Q 0   0 0 Q
1 e QEτ , QEτ , D1 = diag Q, Q, Q, τ Q, τz Q, 2

7

0

+ N QN + N QN

0T

00

+ N QN

00T

.

(34)

Since Q > 0, one can conclude that Υ is positive semidefinite. Therefore, comparing (33) and (15), it can be seen that inequality (33) implies LMI (15). Hence, LMI (26) is a sufficient condition for LMI (15). Similarly, multiplying e −1 , I) and using LMI (27) from left and right by diag(Q Schur complement yields LMI (16) with the change of variables (32). The proof is complete since for any set of matrix variables satisfying inequalities (26) and (27), there exists a set of matrix variables (32) that satisfy the stability criteria in Theorem 1. 2

Table 1 2 The MASP τs,max that guarantees exponential stability in a multirate scenario for Example 1 with α = 0.001.

Remark 2 The stabilization criteria in Theorem 2 are sufficient conditions for the stability criteria in Theorem 1 and therefore are more conservative. However, they can be used to design linear controllers by solving a convex optimization program that can be solved efficiently using available software packages. h

Known upper bounds

i

Proposition 1 Let the pair of system matrices Ω = A B be unknown Pp but satisfy the polytopic Pp uncertainty condition Ω ∈ { β Ω , 0 ≤ β ≤ 1, l l l l=1 l=1 βl = 1}, where Ωl = h i

Al Bl , l ∈ {1, ..., p}, denote the vertices of a convex polytope. Assume that the LMIs in Theorem 2 hold for each Ωl , l ∈ {1, ..., p}, with the same variables Q, Qij , i ∈ {1, . . . , m}, j ∈ {1, . . . , nu }, Y , 1 , and 2 . Then the closedloop linear multi-rate sampled-data system described in (1) and (11) under Assumptions 1 and 2 can be exponentially −1 stabilized with the linear state feedback gain K = Y Q , where Q and Y are defined in (29) and (31), respectively.

Based on Theorem 2, the problem of designing a state feedback controller that gives a larger lower bound on the MASP τsi (or the MAUP τzj ), such that exponential stability is guaranteed, can be formulated as a convex optimization problem in terms of LMIs.

maximize τsi (or τzj ) subject to Qij > 0, i ∈ {1, . . . , m}, j ∈ {1, . . . , nu }, Q > 0, (26) and (27).

5

Example 1 Consider the closed-loop system defined in (1) and (11) with the following matrix parameters

A=

#

0

1

0

− 0.1

" , B=

0

#

0.1

h , K = − 3.75

= 2 (s) and

τz1

= 0.1 (s)

0.42 (s)

τs1 = 2 (s) and τz1 = 0.4 (s)

0.02 (s)

τs1 = 3 (s) and τz1 = 0.0001 (s)

0.16 (s)

τs1

0.01 (s)

= 3 (s) and

τz1

= 0.1 (s)

Example 2 Consider the closed-loop " system#defined in" (1) # 0 1 0 and (11) with parameters A = , B = , −1 − 2 1 h i and K = −1 1 , which are taken from Fridman (2010). In a multi-rate scenario, suppose that the first and the second elements of the state vector are to be sampled by different sensors at unknown non-uniform sampling intervals smaller than τs1 = 4 (s) and τs2 = 0.1 (s), respectively. Furthermore, assume that the actuator is updated in nonuniform intervals smaller than τz1 = 1/3 (s). In this case, the LMIs of Theorem 1 are infeasible and simulation results show that the system is unstable for the mentioned values of MASPs and MAUP. Using Theorem 2, with α = 0.001 0 and h 1 = 2 = 1,iwe find a new controller gain K = − 0.0088 0.0406 that guarantees exponential stability of the closed-loop multi-rate sampled-data system.

.

Numerical Examples

"

0.56 (s)

τs1

As a special case, when the MAUP τz1 approaches zero, the control signal is applied to the system as soon as new data arrives from any of the sensors. Equivalently, the zero order holds can be assumed to be event-driven. This case was studied in Moarref and Rodrigues (2013). As expected, 2 when τz1 = 0.0001 (s) are the values computed for τs,max very close (absolute error = 0.01) to the values computed in Moarref and Rodrigues (2013).

Problem 2

−1

τs1 = 2 (s) and τz1 = 0.0001 (s)

(i.e. x1 and x2 ) is sampled by a dedicated sensor (S 1 and S 2 , respectively) at different unknown non-uniform sampling intervals. Furthermore, the control signal is applied via an actuator Z 1 whose update time is not synchronized with any of the two sensors. Assume that the sampling intervals of the sensor S 1 and the refresh rate of the actuator Z 1 have known upper bounds, i.e. τs1 and τz1 are fixed. Solving Problem 1, the computed lower bound on the MASP for 2 sensor S 2 (τs,max ) that guarantees exponential stability is presented in Table 1. It can be seen that, in this example, the sampling intervals of sensor S 1 can be longer than the limit for the single-rate case if sensor S 2 performs samplings at a faster rate. In other words, we can decrease the controller’s dependency on the data from the first sensor by increasing the sampling rate of the second sensor. According to Table 1, 2 decreases to as the MAUP τz1 increases the MASP τs,max compensate for the late updates of the actuator.

PROOF. The LMIs in Theorem 2 are affine in A, B, N , 0 00 N , N , and N . The proof follows from the properties of convex hulls. 2

The controller gain is then computed as K = Y Q

2 τs,max

i 11.5 .

It is known (see e.g. Naghshtabrizi et al. (2008)) that the MASP in a single-rate scenario for this example is 1.7 (s) (in a single-rate scenario all the elements of the state vector are sampled at the same sampling instants and the eventdriven actuator is updated instantly). Now consider a multirate scenario where each of the two states of the system

8

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Conclusion

Naghshtabrizi, P., Hespanha, J. P., Teel, A. R., 2008. Exponential stability of impulsive systems with application to uncertain sampled-data systems. Systems & Control Letters 57 (5), 378–385. Seuret, A., 2012. A novel stability analysis of linear systems under asynchronous samplings. Automatica 48 (1), 177– 182. Seuret, A., Gouaisbaut, F., 2013. Wirtinger-based integral inequality: application to time-delay systems. Automatica 49 (9), 2860–2866. Strum, J. F., 1999. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 11-12, 625–653. Suplin, V., Fridman, E., Shaked, U., 2007. Sampled-data control and filtering: nonuniform uncertain sampling. Automatica 43 (6), 1072–1083. Yen, J.-Y., Chen, Y.-l., Tomizuka, M., 2002. Variable sampling rate controller design for brushless DC motor. In: Proc. 41st IEEE Conference on Decision and Control. pp. 462–467. Zupan, M., Ashby, M. F., Fleck, N. A., 2002. Actuator classification and selection–the development of a database. Advanced Engineering Materials 4 (12), 933–939.

Linear sampled-data systems with multi-rate samplers and time driven zero order holds were studied. Sufficient Krasovskii-based stability and stabilization criteria were proposed as a set of LMIs. For each sensor (or actuator), the problem of finding an upper bound on the lowest sampling frequency (or refresh rate) that guarantees exponential stability was cast as an optimization problem in terms of LMIs. It was shown that choosing the right sensor (or actuator) with adequate sampling frequency (or refresh rate) has a considerable impact on stability of the closed-loop system. We consider the case of dynamic sampled-data feedback is an interesting area for future work. 7

Acknowledgements

The authors would like to acknowledge NSERC for funding this research. References Briat, C., Seuret, A., 2012. A looped-functional approach for robust stability analysis of linear impulsive systems. Systems & Control Letters 61 (10), 980–988. Chen, T., Qiu, L., 1994. H∞ design of general multirate sampled-data control systems. Automatica 30 (7), 1139– 1152. Fridman, E., 2010. A refined input delay approach to sampled-data control. Automatica 46 (2), 421–427. Fridman, E., Seuret, A., Richard, J.-P., 2004. Robust sampled-data stabilization of linear systems: an input delay approach. Automatica 40 (8), 1441–1446. Lall, S., Dullerud, G., 2001. An LMI Solution to the Robust Synthesis Problem for Multi-Rate Sampled-Data Systems. Automatica 37 (12), 1909–1922. Liu, K., Fridman, E., 2012. Wirtingers inequality and Lyapunov-based sampled-data stabilization. Automatica 48 (1), 102–108. L¨ofberg, J., 2004. YALMIP: a Toolbox for modeling and optimization in MATLAB. In: Proc. IEEE International Symposium on Computer Aided Control Systems Design. Taipei, Taiwan, pp. 284–289. Longhi, S., 1994. Structural properties of multirate sampleddata systems. IEEE Transactions on Automatic Control 39 (3), 692–696. Moarref, M., 2013. Sampled-data networked control systems: a Lyapunov-Krasovskii approach. Ph.D. thesis, Concordia University, Department of Electrical Engineering. Moarref, M., Rodrigues, L., 2013. Exponential stability and stabilization of linear multi-rate sampled-data systems. In: Proc. American Control Conference. pp. 158–163. Nagamune, R., Huang, X., Horowitz, R., 2005. Multirate track-following control with robust stability for a dualstage multi-sensing servo system in HDDs. In: Proc. 44th IEEE Conference on Decision and Control. pp. 3886– 3891.

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