LMI-based control of discrete-time systems with

Proceedings of the American control Conference Arlington, VA June 25-27,2001

LMI-Based Control of Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities Haizhou Pan and Vikram Kapila Department of Mechanical, Aerospace, and Manufacturing Engineering, Polytechnic University, Brooklyn, NY 11201 Abstract

In this paper, we address the control design problem for discrete-time systems subject to actuator amplitude and rate saturation. In particular, we develop linear matrix inequality formulations for the full-state feedback and dvnamic, outDut feedback control desinns for discretetim;! systems wiih simultaneous actuator amditude and rate saturation. Furthermore, we provide a direct methodology to determine the stability-multipliers that are essential for reducing the conservatism of the weighted circle criterion-based saturation control design. Finally, we give an illustrative numerical example to demonstrate the efficacy of the proposed control design framework. 1.

Introduction

A common assumption in many control designs is that the system actuators can generate the necessary level of control effort for closed-loop stabilization and performance. However, most actuators have physical constraints that limit the control amplitude and rate simultaneously. In fact, an Air Force investigation [l]identified control surface rate saturation as the cause of the crash of the YF-22A prototype fighter aircraft. Therefore, the need for control schemes which ensure stability and performance despite the presence of amplitude and rate-limited control efforts is fairly evident. The control problem for linear systems with actuator amplitude saturation has been a topic of considerable interest over the past several years. For continuoustime systems, an extensive literature is devoted to it (see, e.g., 12-51 and the numerous references therein). In addition, for continuous-time systems, the control design problem with simultaneous actuator amplitude and rate saturation has recently received significant attention [6-111. Since most physical processes evolve naturally in continuous time, it is not surprising that the bulk of the actuator saturation control theory has been developed for continuous-time systems. Nevertheless, it is the overwhelming trend to implement controllers digitally. The references that address the actuator amplitude saturation control issue for discrete-time systems include [12-151. In particular, for discretetime systems, Riccati equation-based global and semi-global stabilization techniques for actuator amplitude saturation control have been developed in 113,141. In addition, the application of an anti-windup actuator saturation control framework to discrete-time systems is given in [12]. In a recent paper [15], a Riccati equation-based global and local static, output feedback control design framework for discrete-time systems with time-varying, sectorbounded, input nonlinearities was developed. Unfortunately, however, in contrast to the continuous-time sysResearch supported in part by the NASA/New York Space Grant Consortium under Grant 32310-5891 and the Mechanical Engineering Department, Polytechnic University.

0-7803-64953/01/$10.000 2001 AACC

tems, the problem of stabilizing discrete-time systems in the presence of control signal amplitude and rate saturation has received scant attention. In this aper, we develop a Iinear matrix inequality (LMI) [16rformulation for full-state feedback and dynamic, output feedback control of discrete-time systems with actuator amplitude and rate nonlinearities. The LMI formulation for the aforementioned problem is motivated by a desire to provide a simple and numerically tractable approach to actuator amplitude and rate saturation control. Specifically, since the LMI-based feasibility and optimization problems are convex (for which commercial software i s available, e.g., [17]), the proposed actuator amplitude and rate saturation control scheme can be effortlessly im lemented. In addition, we provide a -direct approach to Rndina the stabilitv multidiers that are paramaunt to reducing the inherent consehatism of the weighted circle criterion-based saturation control design. All theorem proofs are omitted due t o page limitation. Nomenclature lR,R'XS,RP- real numbers, T x s real matrices, EXrx1 I,, 0, - r x T identity, zero matrices Pp - T x T positive-definite matrix na,nc,ii - n a = n + m ; l < n c I n , ; i i = n a - t - n , .MI,M z , - diag[Mlu1MI,], diag[n/lzu,MzVl, M, H - diag[Mu, M,], diag[H,, H,] 2.

fill-State Feedback Control

In this section, we introduce the stabilization problem for discrete-time, linear, dynamic systems with actuators containing a set Q, of time-varying, conic sector nonlinearities. The goal of the problem is t o determine an optimal, full-state feedback controller that stabilizes a given discrete-time, linear, dynamic system with actuator amplitude and rate nonlinearities & ( q ( k ) , k ) E Q, (where q refers to control amplitude, U ,or control rate, v) and minimizes a quadratic performance criterion involving weighted state and control variables. The structure of CP is specified later in this section. These objectives are addressed by developing a nonlinear matrix inequality (NMI) that guarantees global asymptotic stability of the closed-loop system for all & ( q ( k ) , k ) E CP and provides a guaranteed bound on the quadratic performance criterion. Full-State Feedback Stabilization Problem. Given the nth-order, stabilizable, discrete-time plant with actuator amplitude nonlinearities & ( u ( k ) ,k ) E @, kEN,

+

~ ( k1) = Az(lc) - B & ( u ( ~ k) ), , ~ ( 0= ) SO, k E NI(1) where u ( k ) E troller

4140

Wm, determine a full-state feedback con-

k- 1

= 40)+

@v(v(5),SI,

(2)

s=o

v(k) = K,+)

+ K,u(k),

(3)

where v(k) E I"and +,(v(k),k) E a, k 2 0, that satisfies the following design criteria z) the zero solution z ( k ) 0 of the closed-loop system (1)-(3) is globally asymptotically stable for all q5q(q(k),k)E Q , IC E N , and ii) the following quadratic performance functional is minimized

In order to reduce conservatism within the synthesis framework presented below, we introduce a constant, diagonal, positive-definite scaling matrix Hq E RmXmthat preserves the structure of the nonlinearities [5]. The following result provides the foundation for our full-state feedback controller synthesis framework. For the s t a t e ment of this result, we define the notation l& 2HM-l. Theorem 2.1. Let 2 m x 2 m diagonal matrices M I and Mz be given such that Mz - M1 is positivedefinite. In addition, let an m x n, matrix K and a scalar E , 0 < E < 1, be given. Suppose there exists a 2m x 2m diagonal, positive-definite matrix H and an n, x na positivedefinite matrix P satisfying

4

I

ATPA - P + C P+ STE ( H e - BTPA)T

+

+

where z ( k ) fElz(k) Ezuu(k) Ez,,v(k), z f Rp. Note that the feedback interconnection of (1)-(3) with actuator amplitude nonlinearities 4%( U , k) and actuator rate nonlinearities +,,(U,k ) represents a software rate limiter that ensures that no rate commands are sent to the actuators that are beyond their specified limits. To characterize the class Q of time-varying, sectorbounded, memoryless actuator amplitude and rate nonlinearities the following definitions are needed. Let M l q , Mzq E Rmxm be given diagonal matrices such that hllS = diag(Ml,,, . . . ,MI,,), Mzq = diag(Mzql,. . . ,Mzq,), and Mq f Mz, - Mlq is positive definite with diagonal entries Mq,,,i = 1 , . . . ,m. Next, we define the set of allowable nonlinearities dq(.,.) by

Rm x N + R m : MI&: 5 @q.(q,k)q*I MZq,q:, q , E R , i = l , ..., m , q E R m , k E N } . (5)

G?${+q:

Now, we provide a closed-loop NMI that guarantees lobal asymptotic stability of the closed-loop system b ( 3 ) for all actuator amplitude and rate nonlinearities dq(.,.) E Cp. First, we decompose the nonlinearity @,(.;) into linear and nonlinear parts so that d,(q(k),k) = @ q s ( d k ) , k )+ Mlqq(k). In this case, the closed-loop system (1)-(3) has a statespace representation

+

% ( k 1) = A % ( k )- B&(ii(k), k),i(0) = 20,k E N , (6) G ( k ) = Z'iqk), where

(7)

HC - BTPA

-Ro

+ BTPB

Then the function V(Z) = Z T P i is a Lyapunov function that guarantees that the zero solution Z(k) 0 of the closed-loop system (1)-(3) is globally asymptotically stable for all actuator nonlinearities dq(.,.) E Q. Furthermore, the performance functional (4)satisfies the bound J ( & , K ) < V(Z0). Note that J ( Z 0 , K ) < 3;fPZo = tr PZoZ;f, which has the same form as the Hz cost appearing in the standard LQR theory. Hence, we replace ?of: by D D T , where D E P a x d , and proceed by determining the controller gains that minimize t r PDDT = tr DTPD. Next, in the spirit of [15],J(P, K ) 5 tr DTPD can be interpreted as an auxiliary cost. Theorem 2.1 provides an efficient computational approach for closed-loop stability analysis when the controller K , scalar e, 0 < E < 1, and the sector-bounds MI, Mz for input nonlinearities r ~ 5 ~ ( .). , E @ are given. Specifically, since (9) is an LMI in the variables H and P , one can efficiently determine the feasibility of (9) to e s t a b lish the asymptotic stability of 1 -(3). In this paper, however, we focus on extending eorem 2.1 to design stabilizing feedback controllers for systems with actuator amplitude and rate saturation nonlinearities. Before proceeding, observe that 9) is an NMI since it contains product terms involving and P , H . Note that multiplier theory-based robust control design problems frequently result in NMIs when simultaneous determination of the controller and multiplier matrices is attempted [18,19]. In order to circumvent the technical difficulties arising from the numerical solution of such NMIs, in prior literature, many researchers have focused on an iterative solution of the closed-loop stability analysis and stabilizing controller synthesis subproblems 18,191. Specifically, [18,19] have shown that the multip ier theory-based robust control design can be accomplished by i) solving an LMI problem for closedloo stability analysis which provides the stability multip!er for a given controller and ii) solving an LMI problem for controller synthesis with a given stabilit multiplier. Note that although sub-problems z) and iijrconsidered separately are convex, the problem of simultaneous multiplier and controller determination is not. In addition, no claim can be made regarding the convergence of this iterative scheme. However, the aforementioned procedure offers attractive computational advantage by exploiting the convexity of the two LMI sub-problems and has been widely used with success in ractice. Unfortunately, however, does not the class of NMIs that arise in the stan ard multiplier theory-based

!rL

B

\

In addition, the performance variable z(k) is given by z ( k ) = &(IC), where 981 + EzUK,2 1 f [El Ez,], and K f [ K , K,] . Note that the transformed actuator amplitude and rate nonlinearities q5q, (., .) belong to the set Q, given by : R" x N If&m : 0 5 4qs aq;. (17)

Now, suppose there exists a 2m x 2m diagonal, positivedefinite matrix H and an n, x n, positive-definite matrix P satisfying (9) with given m x n, matrix K , scalar e, 0 < E < 1, Mlq = 0, and Mzq = 1. Then, with q replaced by U and U , (5) captures control amplitude saturation and control rate saturation, respectively. In this case, since &(.) E Q, Theorem 3.1 can be used to guarantee global asymptotic stability of the closed-loop system (1)(3) for all &(.) satisfying (17). Alternatively, suppose there exists a 2m x 2m diagonal, positive-definite matrix

4142

H and an n, x n , positive-definite matrix P satisfying (9) with given mxn, matrix K , scalar E , 0 < E < 1, Mlq > 0, and Mzq = I > MI,> 0. Then, with q replaced by U and v , (13) captures control amplitude saturation and control rate saturation, respectively. In particular, with Mlq > 0, take q, = -e = i = 1,. . . , m , in (13). In this case, since &(.) E a b , Theorem 3.1 and (14) can be used to guarantee local asym totic stability of the closed-loop system (1)-(3) for alf &(.) satisfying (17) with a guaranteed domain of attraction.

e,

4.

The following result provides the foundation for our dynamic, output feedback compensation framework. For t_he statement of this result, we define the notation R 5 ETB. T h e o r e m 4.1. Let 2m x 2m diagonal matrices M1 and Mz be given such that MZ - M I is positive-definite. In addition, let (A,, B,, C,) and a scalar E , 0 < E < 1, be iven. Suppose there exist a 2m x 2 m diagonal, positiveSefinite matrix H and an fi x fi positive-definite matrix P satisfying

Dynamic O u t p u t Feedback Control

In this section, we introduce the problem of dynamic, output feedback control of discrete-time, linear systems with actuator amplitude and rate nonlinearities. Dynamic O u t p u t Feedback Stabilization Problem. Given the nth-order, stabilizable and detectable, discretetime plant with actuator amplitude nonlinearities &(u(k),k ) E a, k E NI '

(21)

as in section 2, it follows that < 2:&. Next, J(i5.0,A,, B,, C,) < tr 5rP30 = tr I%&', which has the same form as the H2 cost appearing in the standard LQG theory. Hence, ~ DDT, where D 2 [DF DTB,T]T, we replace Z O Z by Di E W n a x d , Dz E E t i x d , and DzD: > 0, and proceed by determining the controller gains _that minimize the auxiliary cost J ( p ,A,, B,, C,) f tr PDDT.

(22)

5.

~ ( tk1) = A z ( k ) - B & ( u ( k ) , k ) , ~ ( O = )z o j k E N,(18)

Y(k)

= CS(k),

(19)

where u ( k ) E Rm, y(k E E!,determine an nLh-order, linear, time-invariant, ynamc compensator

d

xc(k + 1) = Acsc(k)+ BCy(k), v ( k ) = C,Z,(k),.-

Then the function V ( 5 )= 5'l% is a Lyapunov function that guarantees that the zero solution Z(k) = 0 of the closed-loop system (18)-(22) is globally asymptotically stable for all actuator amplitude and rate nonlinearities @q(., ,) E @. Furthermore, the performance functional (23) satisfies the bound J ( E 0 , A,, B,, C,) < V(50).

(20)

Note

J(Z0,

that,

A,, B,, C,)

k-1

44 = 4 0 ) +

4 u ( 4 s ) , 3),

Dynamic O u t p u t Feedback Controller Synthesis

s=o

Next, we rovide an NMI that guarantees 4obal asymptotic statility of the closed-loop system (187-(22) for all # q ( . , .) E @. Using a similar procedure as in Section 2, the closed-loop system is given by

In this section, we present our main theorem characterizing dynamic, output feedback controllers for discretetime systems with actuator amplitude and rate nonlineaxities. In order to state this result, as in section 3, we assume that a scalar E , 0 < E < 1, and a 2m x 2m diagonal, positivedefinite matrix H are given. In addition, we assume that M I and Mz are given 2m x 2 m dia o nal matrices such that Mz - M I is positive-definite. the remainder of this section, we assume that n, = n,. It follows from 211 that a procedure similar to Theorem 4.1 given be ow can be used to design reduced-order controllers. For convenience in stating the main result of this section, recall the definitions of W,X , Y,2 and define the notation

Z(k + 1) = A Z ( k ) - B&(fi(k),k),f ( 0 ) = 50,k E N,(24)

A,

that satisfies the following design criteria i) the zero solution of the closed-loop system (18)-(22) is globally asymptotically stable for all $,(q(k), k) E @, k E N,and ii) the following quadratic performance functional is minimized

80;

m

u ( k ) = &(k),

I

B

f

O nxm

(25)

where

so that

A

-BII.flu

Onxnc

e = [ c, we,1 .

(27)

Next, without loss of-generality, consider the following partitioning of P and P-' In addition, the performance variable z ( k ) is given by z ( k ) = E Z ( k ) , where E 2 [ EzUCc

1.

4143

'

=

[

R

A

0

]

= t

[

s l$T

e]

@ .

7

(28)

/

where

k ,S E P"-.In addition, we define

=

Using P P - l = I R , it now follows that PlI1 = I I z . With a slight modification of [21], we define a change of controller variables as follows

A K fh;rAcNT + h;rBcCYT> + RZM1,CcNT

BK t M B , ,

Then P and ( A c ,Bc,Cc) satisfy (26) and the zero solution 5(k) 0 of the feedback interconnection of linear system with input amplitude and rate nonlinearities &(.,.) E CP given by (18)-(22) is globally asymptotically stable for all input amplitude and rate nonlinearities &(,,-) E @. In addition, if &(.) E CPb then the zero solution Z ( k ) z 0 of the closed-loop system (18)(22) is locally asymptotically stable and V A defined by (32) is a subset of the domain of attraction of the closedE @, the auxiliary loop system. Finally, for all &(.,.) cost J ( p ,A,, B,, C,) satisfies J ( P , A,, B,, C,) < t r Q , where Q E Pd is such that the LMI variable 8,3 E PnU and BK satisfying (33) additionally satisfy

+ RAaS,

CKfCcNT.

(30)

By defining the variables A ~ I I ~ A ~f IIIyPB, , C E I I , , E f Enl, D f nrd,P J I ~ P I as I ~in, [21], we obtain the identities

e

4

C = [ C a s + W C K Ca] , E

=

[EIS+ E ~ , C K El] ,

[ B y ] >o.

Remark 5.1. It is important to note that the estimate of the domain of attraction V A given by (32) for the closed-loop system (18)-(22) is predicated on open Lyapunov surfaces. See [5] for further details. Illustrative Numerical Example

6.

Before proceeding, note that the variables A, B , C, E , D , and P are affine in (RISIA K ,B K ,C K ) . Finally, we define

Consider a valve-control system [22Jin which the valve is constructed with a spring on the flapper" so that if power is removed the valve closes. The control input is a torque applied to the flapper. The dynamics of the valve with control amplitude saturation is

Y(t) =

v,+

f Qz(Z*), y- 5 P z ( i i z ) , Vs f min {min(y+,y-)}, z = 1 , ,2m

V A

5

(2 E R%: V(Z)
0, satisfies (26) for a given compensator

( 4Bc, , CC). Theorem 5.1. Let 2m x 2m diagonal matrices M I and Mz be given such that M2 - M1 is positive-definite. Furthermore, let a 2m x 2m diagonal, positive-definite matrix H a9d: scalar e, 0 < E < 1, be given. Suppose there exist R, S E Pn-and ( A K ,B K ,C K ) satisfying

I

- ( l l ~ ) P AT

CTH

06xzm 0pmx6 -Ro

05 06

A HC

-0.5P

0%

0%

B

-0.5p

E

OpxR

Opxzm

opxii

In addition, let

P

BT

Ofixp

1

O z m x p