Stability and ℒ2 Gain Analysis for Discrete-Time LTI Systems with Controller Failures

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Copyright © IFAC Large Scale Systems: Theory and Applications, Osaka, Japan, 2004

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STABILITY AND £2 GAIN ANALYSIS FOR DISCRETE-TIME LTI SYSTEMS WITH CONTROLLER FAILURES Guisheng Zhai· Xinkai Chen·· Hai Lin •••

• Os aka Prefecture University, Os aka, Japan •• Shibaura Institute of Techno[0.9Y, Saitama, Japan ••• University of Notre Dame, Indiana, USA

Abstract: In this paper, we analyze stability and £2 gain properties for discretetime linear time-invariant (LTI) systems controlled by a pre-designed dynamical output feedback controller which fails from time to time due to physical or purposeful reason. Our aim is to find conditions concerning controller failure time, under which the system 's stability and £2 gain properties are preserved to a desired level. For stability, by using a piecewise Lyapunov function, we show that if the unavailability rate of the controller is smaller than a specified constant and the average time interval between controller failures (ATBCF) is large enough, then global exponential stability of the system is guaranteed. For £2 gain, also by using a piecewise Lyapunov function, we show that if the unavailability rate of the controller is smaller than a specified constant, then the system with an ATBCF achieves a reasonable weighted £2 gain level, and the weighted £2 gain approaches normal £2 gain when the ATBCF is sufficiently large. Copyright © 2004 IFA C Keywords: Linear time-invariant (LTI) system, dynamical output feedback, exponential stability, (weighted) £2 gain, unavailability rate, average time interval between controller failures (ATBCF), piecewise Lyapunov function

1. INTRODUCTION

that economical issue or system life consideration is concerned, we desire to suspend the controller purposefully from time to time.

In this paper, we consider some quantitative properties for discrete-time linear time-invariant (LTI) control systems with controller failures . The motivation of studying such problem stems from the fact that controller failures always exist in any real control systems due to various environmental factors. For example, in a feedback control system which is composed of a system and a feedback controller, controller failures occur when the signals are not transmitted perfectly on routes among the system, the sensors and the actuators , or when the controller itself is not available sometimes due to some known or unknown reason. Another important motivation concerning controller failures is that we can think about "failure" in a positive way: "suspension", i.e., in the situation

For feedback control systems, the problem of possessing integrity was considered in (Shimemura & Fujita, 1987), where it was proposed to design a state feedback controller so that the closed-loop system remains stable even when some part of the controller fails. In (Hassibi & Boyd, 1999) , similar control systems were dealt with using the name of asynchronous dynamical systems (ADS) , and two real systems, the control over asynchronous network and the parallelized algorithm, were discussed. In that context, a Lyapunov-based approach was proposed to construct the controller so that the system has desired properties. (Zhang, Branicky & Philli ps, 200 1) stated similar control

545

"controller failure" in this paper to mean complete breakdown of the controller (u( k) = 0) on certain time interval, neither as the one in (Shimemura & Fujita, 1987) that part of the controller fails, nor as the one in (Hassibi & Boyd, 1999) that the controller decays slowly at a rate .

problems in the framework of networked control systems (NCS), where various information (reference input, plant output, control input, etc.) is exchanged through a network among control system components (sensors, controller, actuators, etc.), and thus packet dropouts occuring inevitably due to unreliable transmission paths lead to controller failures.

To proceed, we first introduce some notation. For any given k > 1, we denote by Tu(k) the total time interval of controller failures during [0, k), and call the ratio Tut) the "unavailability rate" of the controller in the system. We denote by N k the number of times of controller failures during [0, k). If for some constant Tf > 0, the inequality Nk ~ ;/ holds for any k > 1, then T f is called as a lower bound of the "average time between controller failures" (ATBCF) . The idea is that the average time interval between subsequent controller failures is not less than Tf according to the equivalent inequality ;. ~ T f .

Encouraged by the above works, the authors considered in (Zhai, Takai & Yasuda, 2000a) a controller failure time analysis problem for exponential stability of LTI continuous-time systems. By using a piecewise Lyapunov function, we showed that if the unavailability rate of the controller is smaller than a specified constant and the average time interval between controller failures (ATBCF) is large enough, then global exponential stability of the system is guaranteed. In (Zhai, Takai & Yasuda, 2001a), the result of (Zhai et al., 2000a) was extended to LTI discrete-time systems. Furthermore, the authors extended the consideration to L2 gain analysis for LTI continuous-time systems with controller failures in (Zhai, Chen, Takai & Yasuda, 2001b). However, these references dealt with only state feedback. Although it is quite easy to extend the results to the case of static output feedback, the extension to dynamical output feedback is not the case.

For stability analysis, we will prove in Section 2 that if the unavailability rate of the controller is smaller than a specified constant and the ATBCF is large enough, then global exponential stability of the system is guaranteed. For L2 gain analysis, we show in Section 3 that if the unavailability rate of the controller is smaller than a specified constant, then the system with an ATBCF achieves a reasonable weighted L2 gain level, and the weighted L2 gain approaches normal L2 gain if the ATBCF is sufficiently large.

Motivated by the above observations, we extend our consideration to dynamical output feedback case. The system is described by equations of the form x(k

+ 1) =

Ax(k)

+ Blw(k) + B2U(k) + Du(k)

z(k) = C1x(k) {

2. STABILITY ANALYSIS (1)

y(k) = C2x(k) ,

where x(k) E 1Rn is the state, u(k) E )Rm is the control input, w(k) E )Rr is the disturbance input, y(k) E 1RP is the measurement output, z(k) E lR'1 is the controlled output, and A, B}, B 2 , Cl, C2 , D are constant matrices of appropriate dimension. Throughout this paper, we assume (i) A is not stable; (ii) the triple (A, B 2 , C2 ) is stabilizable and detectable; (iii) a dynamical output feedback controller xc(k {

+ 1) = u(k)

Acxc(k)

+ Bcy(k)

= Ccxc(k) + Dcy(k)

In this section, we set w (k) == 0 in the system (1) to analyze stability for the system with controller failures. More precisely, we assume that the con. troller (2) has been designed so that the closedloop system x(k

+ 1) =

Asx(k) ,

is exponentially stable, where i=[x T xn T is the state of the closed-loop system. We first give a definition concerning exponential stability of an autonomous system quantitatively.

(2)

Definition 1. The system x(k + 1) = f(x(k)) with f(O) = 0 is said to be globally exponentially stable with decay rate 0 < ,\ < 1 if IIx(k)1I ~ c,\kllx(O)II holds for 2Iny x(O), any k ~ 1 and a constant c > O. I

has been designed so that the closed-loop system composed of (1) and (2) has desired property (exponential stability with certain decay rate or certain L2 gain level) , where xe(k) E 1Rnc is the controller's state, ne is the controller's order, and A e , Be, C e, De are constant matrices. However, due to physical or purposeful reason, the designed controller sometimes fails with a (not constant necessarily) time interval until we recover it. In this setting, we derive the condition of controller failure time, under which the system's exponential stability or its L2 gain property is preserved to a desired level. As in (Zhai et al., 2000a; Zhai et al., 2001a; Zhai et al., 2001b), we use the word

We suppose that the designed controller (2) sometimes fails and we need a (not definitely constant) time interval to recover it. Obviously, when the controller fails, the closed-loop system assumes the form of x(k

+ 1) = A"x(k) ,

A"

= [ B:C2

1c]' .

(4)

which is obtained by substituting u(k) = 0 in (1) . Hence, the performance of the entire system

546

is dominated by the following piecewise difference equation: x(k

+ 1) =

Asx(k) whe n the controller works { Aux(k) whe n the controller fails .

Noting that V(k j ) :S p,V(kj) holds for any j > 0 according to (11), we obtain from (12) that for k 2j :S k < k 2j+1,

(5)

Since A. is stable and Au is unstable, we can always find two positive scalars As > 1 and Au > 1 such that As As remains stable and \: 1 Au becomes stable. As can be seen later, large As and small Au are desirable in real problems. Then , there exist p. > 0 and Pu > 0 such that

(13)

and then by induction that V(k) ~ 1'2Nk A.;-2(k - Tu (k» A~T.. (k)V(O) .

It is easy to confirm that the above inequality is also true for k2i+1 :S k < k 2j + 2 . Since V(k) ~ otllx(k)1I2 and V(O) :S 02I1x(0)11 2, we obtain from the above inequality that

Note that the above inequalities are LMls (Boyd, El Ghaoui, Fem n .'Ie Balakrishnan, 1994) with respect to Ps, Pu , and thus are easily solved using any of the existing softwares, such as the LMI Control Toolbox.

Il x(k) 1I

~

~I'NkA; (k-Tu( k»A;'u(k)llx(O)II .

V~

(15)

From now on, we consider the convergence property of x(k) in (15) for two different cases .

Using the solutions Ps and Pu of (6), we define the following piecewise Lyapunov function candidate

First, when p,

(7) IIx(k)1I

for the system. Here, Pu(k) is a two-valued piecewise constant matrix function as Ps when t.he controller works Pu k = { ( ) P.. when the controller fails ,

(14)

~

= 1,

we obtain from (15) that

~A;(k-Tu(k»A;'.. (k)llx(O)II.

V~

(16)

If there exists a positive scalar A satisfying A < 1 such that

(8)

T,,{k) < InA.+lnA k - In As + In Au '

and Vu(k)(x(k» is : ~fined correspondingly. Then, the following properties of (7) are obtained:

(17)

which is a condition on the unavailability rate of the controller, then we obtain easily from (17) that

= xT Psx , Vu(x) = xT Pux are continuous and their differences along solutions of the corresponding system satisfy

(i) V.(x)

V.(x(k

+ 1))

~ A;2V.(x(k))

Vu(x(k

+ 1))

~ A~V.. (x(k));

(ii) There exist constant scalars 02

and thus (9) ~

(19)

01 > 0

This implies that the entire system is globally exponentially stable with decay rate A.

such that alllxf~{V.(x),Vu(x)}~a21IxIl2 ,

(Hi) There exists a constant scalar p, that V,(x) ~ I'Vu (x) ,

v..(x) ~ I'V.(x) ,

IIx ;

~

Secondly, when p, > 1, in addition to (17), if there exists a scalar A· E (A, 1) such that

(10)

1 such

k NL< ~ - Tf ' IIx .

(20)

(11)

holds for all k

The first property is a straightforward consequence of (6) while the second and third properties hold, for example, with 01 = min {Am(Ps), Am(Pu)} , 02 = max{AM(Ps), AM(Pu )} , and p, = £:!., respectively. Here, AM(' ) (Am(-» et) denotes the largest (smallest) eigenvalue of a symmetric matrix.

> 1, then we obtain easily I'NkAk ~ (A")k ,

(21)

and thus (22)

Now, without loss of generality, we assume that the designed controller works during [k 2j , k2i+d, and the controller fails during rk2j +l, k2i+2), j = 0, 1" " , where ir{) - O. Then, for any k > 1, we obtain from (9) that

This means that the entire system is globally exponentially stable with decay rate A· . We observe that the condition (20) is the requirement on the ATBCF . More precisely, if the ATBCF in the system (1) is larger than or equal to Tf given in (20), then (21) and (22) hold as the same and thus the system's exponential stability is guaranteed.

547

with Cl being a fixed constant satisfying 0 ~ Cl < 1. In that case, if the closed-loop system composed of (1) and U = ClG(Z)Y is unstable, the discussions up to now are the same by making some notation modification. If this is not the case, then the entire system can be considered as a switched system composed of two stable subsystems, and thus it is globally exponential stable if the ATI3CF is large enough; see detailed discussions in (Zhai et al., 2001a; Zhai, Hu, Yasuda & Michel, 2000b) . •

The above discussions indicate that (7) with (6) constitutes a piecewise Lyapul10v function for the system (1) with controller failures satis(ying (17) and (20). We state this result in the following theorem.

Theorem 1.

If the unavailability rate of the controller in the system (1) is small in the sense of satisfying (17) for some positive scalar>' < 1, then for any positive scalar>.· satisfying>. < >'. < 1, there exists a finite constant Tt such that the system (1) is globally exponentially stable with decay rate>.· , for any ATBCF larger than or equal to Tt . I

3. L2 GAIN ANALYSIS

In this section, we assume that the dynamical output feedback controller (2) has been designed so that the closed-loop system

The next two remarks give more precise discussion about the conditions (17) and (20).

Remark 1. The condition (17) implies that if we expect the entire system has the potentiality of decay rate close to >.;1 (i.e., >. -+ >.;1), where >.; 1 is known to be kind of decay rate of the closed-loop system (3), we should restrict the total controller failure time small enough (Le ., Tu(k) -+ 0) . This is reasonable when we consider the control system with the designed controller breaks down very occasionally and we can recover it very soon. In this case, we definitely expect that the system stability does not degenerate greatly.

i(k {

= A,i(k)

z(k)

=

+ Eh w(k)

(23)

C.i(k) ,

is stable and the £2 norm of the transfer function from w to z in (23) is smaller than a prespec-

of,

ified constant ,,(, where 8 1 = [Br Cs = [Cl + DDcC2 DCc]. Since our interest here is to analyze £2 gain property of the system, we assume i(O) = 0 in (23) . Also, we suppose that the designed controller (2) sometimes fails and we need a (not constant necessarily) time interval to recover it. When the controller fails, the closed-loop system assumes the form of

Concerning the other two stability indices >'s and >'u, we observe that according to the unavailability rate condition (17), comparatively long controller failure time Tu(k) is tolerable for large >'s and small >'u. This is reasonable since the closed-loop system has large decay rate (thus good stability property) when the controller works with large >'s, and the open-loop system does not diverge greatly when the controller fails with small >'u . Therefore, if we concentrate on stability property of the system, we should design the original output feedback controller so that a large >'s can be obtained. •

i(k {

+ 1) =

A"i(k)

+ alw(k)

z(k) = C"i(k)

(24)

where C u = [Cl 0]. Then, the behavior of the entire system is dominated by the piecewise LTI system: the system (23) when the controller works and the system (24) when the controller fails. Since As is stable and the £2 gain of the transfer function from w to z in (23) is smaller than ,,(, according to the well known I30unded Real Lemma for discrete-time LTI systems (Iwasaki, Skelton & Grigoriadis, 1998), there exists Ps > 0 such that

Remark 2. While the unavailability rate condition (17) of the controller is easy to imagine, the ATBCF condition (20) is not so straightforward. The key point is that if the open-loop system (when the controller fails) has poor stability property and the controller failures occur very frequently, then the entire system will not perform well even when the total controller failure time interval is not long. If we expect that the entire system has decay rate close to >., we should require Tt to be large enough and thus Nk to be small enough, which means that the controller does not fail very frequently. Therefore, the condition (20) is a balanced requirement of decay rate and the number of controller failure times. • Remark 3. Although we concentrated on the case of complete controller breakdown (u = 0) in this paper, it is an easy matter to extend the discussion to the case where due to various reason the output feedback controller (2) (write as u = G(z)y shortly) decays in the sense of u -+ ClU

+ 1)

(25)

which is equivalent to. T

A .• P.,A. - P. [

1 T + :;C s Cs

arp.A.

Thus, there always exists a scalar (s

>

1 such that

< o. (27) Now we consider the case when the controller fails . In this case, we can always find a scalar (u > 1

548

such that (;; I Au is Schur stable and the C2 gain

(Cl Au, El,

of the transfer function

(;;ICu ) is

L k- I

smaller than "(. Thus, there exists Pu > 0 such that p,,(;;l Au) P"Fh

[

- Po,

(,~l A,,)T P"

0

-T BI P"

- P" 0

--.,1

0

(,:IC,,)

0

«~'~o,)T 1< 0

(~(k - I - m)r(m)

(34)

according to (32). Using the fact V(kd ~ J1.V(k jinduction that

0 . (28)

) ,

we obtain by

- -.,1 V(k)

or equivalently,

:<::: J1.2N k A .; 2(k-Tu (k» A~l'u (k) V(O)

-L

k-I

T A"P"BI

]

< O.

(29)

BTp"BI - -.,1

J1.2(Nk _ l - N~)(;2(k - l - m - (Tu(k-I)-T" ("'»)

m=O

(35)

Note that the above inequalities are LMls (Boyd et al., 1994) with respect to Ps and Pu , and thus can be easily solved.

When J1. = 1, we obtain from (35) with x(O) = 0 and V(k) 2: 0 that

Using the solutions p. and Pu , we define the same piecewise Lyapunov function candidate (7) for the system (I), and consider the difference of the piecewise Lyapunov function candidate along the trajectories of the system (23) or (24) . When the controller works, Y.(k

+ 1) -

Y, (k) = iT(k

+ l)P,i(k + 1) -

L

k-I

X(~(Tu(k-I)-Tu("'»r(m) :<::: O.

:<:::

=

BT P. :\..

~c; C. + «;2 -

[i(k)] w(k)

T

-.!.r(k)

+ «;2 -

-.,

[ _

BT P.BI l)Ps

0

w(k)

0] -.,1

i(k) ] [ w(k)

1(;2(k - I - m-(Tu (k - I)-Tu (m») (~(T" (k-I) _Ol;, (m»

(37)

holds with c

:<:::

(;2Y.(k) - .!.r(k).

-.,

= V~, a

under the assumption that

l

where r(k) = zT(k)z(k) _"(2wT (k)w(k) and (27) was used to obtain the inequality. Therefore, in the case where the designed controller works, we obtain

+ 1)

I

(30)

l)Vs(k).

6-

V.(k

(36)

Note that the summation term before f(m) in the above inequality is the transition matrix from time instant m to k - 1. Then , according to the stability analysis result in the previous section, the inequality

iT(k)P.i(k)

= [i(k) ] T [A; P.A. - Ps A; P. BI] [i(k)]

w(k)

(;2(k - I - ",-(Tu (k-I) - Tu ("'»)

there exists a positive scalar ( T..(m) m

< 1 such

<

In«s) + In() - In«.) + In(,,)

that (38)

for any m > 1, which is the condition on the unavailability rate of the controller. Combining (36) and (37), we obtain

(31)

In a similar manner . when the controller fails, we obtain

k-I " " " . - 2(k - I - m)

L

(32)

~.,

z T( m ) z ( m )

m=O

k- I

:<:::

Now, without loss of generality, we assume that the designed controller works during [k2j , k2J+d, and the controller fails during [k2j+1, k2j+2) , j = 0,1, ··· . where ko = O. Then, for any k 2: 1 in the interval [k 2j , k2j +l), we obtain easily from (31) that

L

c2 -.,2

L

(2(k-I-m)w 1' (m)w(m).

(39)

m=D

\Ve sum both sides of the above inequality from k = 1 to k = +00 to obtain (by rearranging the double-summation area)

k-I

__1

...,

) (,-2(k-l- m )r( m.

(33)

which means that the C2 gain level JI~5<~2 C"( is achieved under the unavailability rate condition

m=k2j

(38) .

549

where the output feedback controller (2) (write as u = G(z)y shortly) decays in the sense of u -+ au with a being a fixed constant satisfying 0 ~ a < l. In that case, if the closed-loop system composed of (1) and u = aG(z)y is unstable , the discussions up to now are the same by making some notation change. If this is not the case, then the entire system can be viewed as a switched system composed of two stable subsystems, and thus a weighted £2 gain level is achieved under an ATBCF scheme (without consideration of unavailability rate of the controller) and the achieved weighted £2 gain level approaches normal £2 gain level if the ATBCF is large enough; refer to (Zhai, Hu, Yasuda & Michel, 2001c; Zhai, Hu, Yasuda & l\'fichel , 2002) I for detailed discussions.

Next, when Jl > 1, we rewrite (35) as k-l '"

L/l-

- 2N

_-2(k-l-m-(T.. (k-l)-Tu (m»)

~(, .,

X(~(T.. (k-l)-Tu(m))f(m) ~ O.

(41)

In this case, if in addition to (38) there exists a positive scalar (* E ((, 1) such that T' = f

In(/l-) In((')-ln(()

(42)

holds for all m > 1, then we know /l-N~ (m ~ (C)m

=

p.-N~ Cm ~ (C)-m

(43)

holds for any m > 1 . Using this inequality in (41), we obtain

REFERENCES Boyd, S. , L. El Ghaoui, E. Feron and V. Balakrishnan (1994) . Linear Matrix Inequalities in System and Control Theory , SIAM, Philadelphia. Hassibi, A. and S.P. Boyd (1999). Control of asynchronous dynamical systems with rate constraints on events. In Proceedings of the 38th IEEE Conference on Decision and Contro~ Phoenix, USA, pp. 1345-1351. Iwasaki, T ., R.E. Skelton and KM. Grigoriadis (1998). A Unified Algebmic Approach to Linear Control Design, Taylor & Francis, London. Lin, H., G. Zhai and P.J. Antsaklis (2003) . Robust stability and disturbance attenuation analysis of a class of networked control systems. In Proceedings of the 42nd IEEE Conference on Decision and Contro~ Hawaii, USA, pp. 1182-1187. Shimemura, E . and M. Fujita (1987). Control systems possessing integrity. Journal of the Society of Instrument and Control Engineers 26, 40Q--405. Zhai, G ., S. Takai and K Yasuda (2000a). Controller failure time analysis for linear time-invariant ~'Ystems. Transactions of the Society of Instrument and Control Engineers 36, 1050--1052. Zhai, G ., 8. Hu, K Yasuda and A.N. Michel (2000b). Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach. In Proceedings of the American Control Conference, Chicago, USA, pp. 200--204. Zhai, G., S. Takai and K. Yasuda (2001a). Stability analysis of linear time-invariant systems with controller failures. In Proceedings of the European Control Conference, Porto, Portugal , pp. 138-142. Zhai, G ., X. Chen, S. Takai and K Yasuda (200lb). Controller failure time analysis for ?-loo control syst.ems. In Proceedings of the 40th IEEE Conference on Decision and Contro~ Orlando, USA, pp. 1029--1030. Zhai, G., B. Hu, K Yasuda and A.N. Michel (2001c). Disturbance attenuafion properties of time-controlled switched systems. Journal of the Fronklin Institute 338, 765-779. Zhai, G., B. Hu, K Yasuda and A.N. Michcl (2002). Stability and £.2 gain analysis of discrete-time switched systems. Transactions of the Institute of Systems, Control and Information Engineers 15, 117- 125. Zhai, G. , H. Lin and P.J. Antsaklis (2003). Cont.roller failure time analysis for symmetric ?-loo control systems. In Proceedings of the 42nd IEEE Conference on Decision and Contro~ Hawaii, USA, pp. 2459-2464. Zhang, W., M.S. Branicky and S.M. Phillips (2001) . Stability of networked control systems. IEEE Control Systems Magazine 20, 84-99.

k-l

~

c2

-l

L

C

2 (k-l - m)w T (m)w(m) .

(44)

Tn=O

Summing both sides of the above inequality from k = 1 to k = 00 yields 1 1-

(C )-2m L"( 00

--2 (,.. m=O

2

2

zT(m)z(m)

00

< ~2 ' " wT(m)w(m) , - 1- ( L

(45)

?Tl=O

which means a weighted £2 gain level is achieved .

.j \-5(~2 Cl

We observe that the condition (42) is the requirement on the ATBCF . More precisely, if the ATBCF in the system (1) is larger than or equal to given in (42), then (45) holds as the same and thus the system achieves the same weighted £2 gain level. We summarize the above discussions in the following theorem.

Ti

Theorem 2. If the unavailability rate of the controller in the system (1) is small in the sense of satisfying (38) for some positive scalar ( < 1, then there exists a finite constant such that the system (1) achieves a weighted £2 gain level

Ti

.j\-5{2 Cl 2

in the sense of (45), for any ATBCF larger than or equal to I

Ti .

Remark 4. The inequality (45) describes a weighted £2 gain level due to the existence of (

~)

- 2m.

When (* is close enough to (, which

means the ATBCF is sufficiently large according to (42), obviously the inequality (45) approaches the normal £2 gain . I

Remark 5. Same as in the previous section, we can easily extend the discussion here to the case

550