Fixed-Structure Controller Design for Systems with Actuator Amplitude ...

Proceedings of the 37th IEEE Conference on Decision 8, Control Tampa, Florida USA December 1998

WM13 14:OO

Fixed-Structure Controller Design for Systems with Actuator Amplitude and Rate Nonlinearities Vikram Kapilal and Wassim M. Haddad2 'Department of Mechanical, Aerospace, and Manufacturing Engineering, Polytechnic University, Brooklyn, NY 11201 2School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150 Abstract

tems often involve simultaneous control amplitude and control rate saturation resulting in loss of closed-loop system performance and, in some cases, instability. In addition, the presence of control rate saturation may further exacerbate the problem of control amplitude saturation. For example, in advanced tactical fighter aircrafts with high maneuverability requirements, actuator amplitude and rate saturation in the control surfaces can cause pilot induced oscillations [24,25] leading to catastrophic failures. Specifically, according to an Air Force investigation reported in [26], control surface rate saturation led to the crash of the YF-22A prototype fighter aircraft. Similar investigations have attributed the crash of the Swedish Gripen prototype fighter aircraft to large pilot inputs in pitch/roll which led to rate saturation in the elevons [27]. In the current literature, the control rate saturation problem is typically treated by considering a first-order positionfeedback internal model approximation for the actuator dynamics [11,28 301. However, if the actuator rate dynamics do not conform to a first-order behavior, the rate saturation controllers proposed in [ll,29,301 do not guarantee closed-loop stability. In this paper, we develop a framework to design feedback controllers for systems with control amplitude and rate saturation constraints. The proposed framework addresses actuator rate saturation without resorting to the approximate first-order position-feedback internal model for rate saturation nonlinearities proposed in [11,28-301. Specifically, the proposed design methodology employs a rate limiter as part of the controller architecture. A related but different approach to the present paper using a software rate limiter for rate saturation is given in [25]. Unlike the framework of [25] which is predicated on designing control amplitude and then computing control rates, the proposed amplitude and rate saturation control framework directly accounts for control rate constraints. Furthermore, it is shown that the combined control amplitude and rate saturation problem can be collapsed t o a control amplitude saturation problem for an augmented system model. The problem of control amplitude saturation for the augmented system capturing amplitude and rate saturation for the original system is then addressed using the absolute stabilization framework for systems with

In this paper we develop fixed-order (i.e., full- and reduced-order) dynamic output feedback compensators for systems with actuator amplitude and rate saturation constraints. The proposed design methodology employs a rate limiter as part of the controller architecture. The problem of simultaneous control amplitude and rate saturation is embedded within an optimization problem by constructing a Riccati equation whose solution guarantees closed-loop global asymptotic stability in the face of sector bounded actuator amplitude and rate nonlinearities. Application of the proposed framework is demonstrated via a numerical example.

1. Introduction The design of linear feedback controllers for linear systems has significantly matured over the past several decades. However, actual controller implementation can often lead to nonlinear system behavior. In particular, uncertain large amplitude exogenous disturbances can drive system actuators into saturation forcing the system to operate in a nonlinear region for which it was not designed. Hence, t o ensure system stability and performance actuator saturation needs to be accounted for within the control system design process. There exists an extensive literature devoted to the control saturation problem (see for example, [l--231 and the references therein). Specifically, [l-3,5,16] concentrate on extensions of classical optimal control theory to account for control amplitude constraints while anti-windup compensation techniques are developed in [4,6,8-11,221 to desensitize the effects of actuator saturation. References [7,12,14,15,17,18]provide various Riccati and Lyapunov equation-based local and semiglobal stabilization methods for systems with actuator constraints. Alternatively, the authors in [19-21,231 develop saturation controllers based on absolute stability theory. Many practical applications of feedback control sysThis research was supported in part by the NASA/New York Space Grant Consortium under subcontract 32310-5891, the National Science Foundation under Grant ECS-9496249, and the Air Force Office of Scientific Research under Grant F49620-96-1-0125.

0-7803-4394-8198 $10.000 1998 IEEE

909

tectable plant with actuator amplitude and rate nonlinearities # ( u ( t ) , t )E @ and $(w(t),t) E @, t 0, respectively,

input nonlinearities developed in [21]. Unlike the results of [ll,29,301, the proposed framework is not limited to actuator amplitude and rate saturation non1inearities. Specifically, the proposed design framework provides fixed-structure dynamic controllers for systems with general sector-bounded actuator amplitude and rate nonlinearities, and hence is directly applicable to systems with amplitude and rate saturating actuators. Furthermore, unlike the semi-global stabilizatioc. technique of [12,14,15] based on unverifiable a priori assumptions that the system initial conditions and system states lie in a predefined compact set so that the magnitude of the control signal lies in a bounded region, the proposed framework provides guaranteed domains of attraction. In particular, our results are based on the assumption that the initial condition of the closed-loop system with amplitude and rate saturating actua,tors lies in a specified subset of the domain of attraction for the closed-loop system and hence the control signal can amplitude and rate saturate during closed-loop operation without destroying the asymptotic stability of the closed-loop system. Finally, to account for closed-loop system performance degradation, we also consider the minimization of a quadratic performance criterion over the allowable class of actuator amplitude and rate nonlinearities. All proofs are omitted due t o space constraint.

>

k ( t ) = Az(t) - B 4 ( ~ ( t ) , t )~, ( 0=) 20, t 2 0 , (1) Y(t) = W t ) , (2) where ~ ( tE )R" and y ( t ) E R', determine an nLh-order linear time-invariant dynamic compensator

& ( t ) = ACzc(t)+ &y(t), w ( t ) = CCG(t), U(t) = 4 0 )

(4)

4(w(s),s w s ,

(5)

ii) the quadratic performance functional

1

ca

,n

J(Ac,Bc,Cc)

SUP

m Z ( t > &

9(.>.)E 0 0

+

(6)

+

where z(t):Elz(t) EzUu(t) EzUv(t), z ( t ) E Rp, ETE221= ETEz, = ,OX, and E,T,E2u = Om,,, is minimized. Note that the input rate rnodel given by (5) represents a software rate limiter that ensures that no rate commands are sent t o the actuators that are beyond their specified limits. To characterize the class @ of time-varying sector bounded memoryless actuator amplitude and rate nonlinearities the following definitions are needed. Let Mlq, Mzq E W m x m be given diagonal matrices such that MIq = diag(M1, ..., Mlq,), Mzq = diag(Mz,,, . . . ,Adz,,,,), and Mq A'Mzq - MIq is positive definite with diagonal entries M q , % , i= 1,.. . , m. Next, define the set of allowable nonlinearities 4(-,.) by

- real numbers, r x s real matrices, R r x l - r x r symmetric, nonnegative-definite,

positive-definite matrices n, = n m; 1 5 nc 5 na; 6 = n, n, ETEi, E2T,E2u, E , T , E 2 u > 0 U (control amplitude) or w (control rate) m x m diagonal matrices m x m diagonal matrices diag[Ml,, MI,], diag[Mzu,MzVl diag[Mu,M u ] diag[Hu, , &I

+

I'

(3)

where w ( t ) E R", that satisfies the following design criteria: 2) the closed-loop system (1) (5) is asymptotically stable for all $ ( q , t ) E @, t 2 0; and

Nomenclature B,R'X",R' S',N',P'

+

~ ( 0 =) GO,

+

@ f{4 : W"

xR+ +R"

2

: Mlqt9, I 4, (9,t ) q ,

>

5 MZq,9?2,

9% E R, i = 1,.. . ,m, g E Bm, &.a. t 0, and 4,(q,.) is Lebesgue measurable for all g E B", i = 1,.. . ,m}. (7)

Fixed-Structure Dynamic Compensation for Systems with Actuator Amplitude and Rate Nonlinearities In this section we introduce the fixed-structure dynamic compensation problem for systems with actuator amplitude and rate nonlinearities. The goal of the problem is to determine an optimal output feedback controller that stabilizes a given linear dynamic system witE. actuator amplitude and rate nonlinearities $ ( g ( t ) , t ) E @ (where q refers to control amplitude, U, or control rate, w ) and minimizes a quadratic performance criterion involving weighted state and control variables. The structure of @ is specified later in the section. Dynamic Output Feedback Stabilization Problem. Given the nth-order stabilizable and de2.

3. Sufficient Conditions for Stabilization In this section we provide a Riccati equation that guarantees asymptotic stability of the closed-loop system (1) (5) for all actuator amplitude and rate nonlinearities 4 ( . , . ) E @. First, however, we decompose the nonlinearity 4(.,.) into a linear and nonlinear part so that 4 ( q ( t ) , t )= 4s(q(t),t) + Mlqq(t). (8) In this case the closed-loop system (1) (5) has a statespace representation

i(t) =

A q t ) - B&(G(t)),

G ( t ) = &(t),

910

2(0) = io, t 2 0, (9)

(10)

Fixed-Structure Dynamic Controllers for Systems with Actuator Amplitude and Rate Nonlinearities In this section we present our main result characterizing fixed-order controllers for systems with actuator amplitude and rate nonlinearities. For design flexibility the compensator order n, may be less than the augmented plant order na. For convenience in stating this result define the notation

4.

1.

Note that the transform-

ed actuator amplitude and rate nonlinearities &(., belong to the set asgiven by

e)

a s f { ~ , : R " x R + ~ R ":~0 5 $ , , ( q , t ) q i I Mqiqp,qi E R , i = 1,. . . ,m, q E R", 8.8.t 2 0, and $si (q, .) is Lebesgue measurable for all q E R", i = 1,.. . ,m}. (11) In order to reduce conservatism within the synthesis framework presented below we introduce constant diagonal positive definite scaling matrices Hq E R""" that preserve the structure of the nonlinearity. Specifically, note that since each nonlinearity &(., .), i = 1 , . . . , m, satisfies (7) and Hq E Pm is diagonal, it follows that if q5s E asthen @Hq(M;'$, - q ) I 0. The following result provides the foundation for our fixed-structure dynamic output feedback compensation framework. For the statement of this result define RO f 2HM-1 and R diag[R1, R2u, CTR2uCc].

e

for arbitrary P, Q , P E Rnax n a , diagonal positive defi, and scalar E > 0. Furthernite matrix H E R2m~x2m more, define ab C such that the input nonlinearity is time-invariant, that is, $ ( q , t ) = $(q) and $(q) is contained in for a finite range of its argument q, that is,

Theorem 3.1. Let (Ac,B,, C,) be given, let H , M be 2m x 2m positive-definite diagonal matrices, and PI and a scalar E > 0 satisfysuppose there exists P E ' ing

where gi < 0 and iji Finally, define

Then, the function V ( 2 )= 2 T p 2 is a Lyapunov function that guarantees that the closed-loop system (1)-(5) is globally asymptotically stable for all actuator amplitude and rate nonlinearities 4(., -) E a. Furthermore, the performance functional (6) satisfies the bound

> 0, i

= 1,.

. . , m , are

given.

i E {m + 1,.. . ,2m},

K+ f Note that J ( 5 0 ,A,, B,, C,) < Z T P Z O = tr P202:, which has the same form as the H2 cost appearing in standard LQG theory. Next, we replace 502: by 5 diag[Vl,BCV2BT],where VI E RnaXna and E R r Xare r arbitrary design weights such that VI 2 0 and V2 > 0, and proceed by determining controller g@:s (A,, B,,C,) that minimize J ( p ,A,, B,, Cc) f tr P V , where P E Rfixii,P > 0, satisfies (12).

VS

DA

v

where

Qi(Zi),

v,- 4* i ( i i i ) ,

f

min {min(K+,y-)}, -z=l,.. . ,2m

4

{ji. E

Ci

i E (1,. . . ,2m},

,

R'

: V(Z)

< vS, gi 5 CiZ I Zi,,

i = m + 1 , . . .,2m},

(15)

C,

a f A + B M l C , and

is the ith row of

R'"', P > 0, satisfies (12) for a given compensator (AC,Bc, CC).

P

911

E

Theorem 4.1. Let n, 5 n,, E > 0, and let H , M be 2m x 2rn positive-definite diagonal matrices. Furthermore, suppose there exist n, x n, nonnegative-definite matrices P, (3,P , and Q satisfying

+

+ +

+ +

+ +

+m(4m(t))lT where +7(47(t)), t 2 0, i E is characterized by h ( q t ( t ) ) = qz(t)i

47(q2(t))

+

0 = ATP PAP RI C,THRilHCa PBa .RilB,TP- P,TRY,P, + T T P , T R ; ~ P , T ~(16) , 0 = AQQ QA; Vi - QCQ T ~ Q C Q T T , (17) 0 = A:@ PAP PB,Ri'B,TP P,TRg,Pa

+

P,TRT:P,TL, = A& + QA; + QCQ -

o

*

(18)

7 1 ~ ~ ~ T T ,(19) A

rank Q = rank P = rank QP = n,,

QP

=

r

5

GTih', GTr,

rGT = Inc, n

71 =

n;r E IEg%

In - 7 ,

(20) x "c I

(21)

and let A,, 13,, and C, be given by - BTP)

(23)

[ P-GP+ P

(25)

1

-PGT GPG~

satisfies (X!) and (A,,B,,C,) is an extrema1 of J ( F , A,, B,, C,).Furthermore, the closed-loop system (1) (5) is globally asymptotically stable for all actuator amplitude and rate nonlinearities d(.,.) E a. In addition, if +(-) cf @b then the closed-loop system (1) (5) is locally asymptotically stable, and V A defined by (15) is a subset of the domain of attjaction of the closed-loop system. Finally, the cost J ( P , A,, B,, C,)is given by

J ( F ,A,, Bc, Cc)=

(28)

Remark 4.3. It is important to note that the estimate of the domain of attraction D A given by (15) for the closed-loop system is predicated on both cl_osed and open Lyapunov surfaces. Specifically, since C 7 B # 0, i E { 1 , . . . ,m}, domain of attraction computations based on open Lyapunov surfaces lead to extreme algebraic complexity [31]. Thus, for simplicity, for i E ( 1 , . . . , m } , we use closed Lyapunov surfaces for computing the domains of attraction in the presence of control amplitude nonlinearities. Alternatively, since C,B= 0, i E { m 1 , . . . , 2 m } , we use open Lyapunov surfaces for computing the domain of attraction in the presence of control rate nonlinea'rities. Hence, a combination of closed and open Lyapunov surfaces are used to compute the estimate of the domain of attraction V A which yields a considerable improvement over domains of attraction predicated on only closed Lyapunov surfaces. See [23,31]for a detailed discussion on the distinction between open versus closed Lyapunov surfaces for estimating domains of attraction.

(24)

Then

F=

>

1

Remark 4.2. In the full-order case, set n, = n, so that G = = T = In, and 71 = 0. In this case the last term in each of (16) (19) can be deleted and G and r in (23) (25) can be taken to be the identity. Furthermore, Q plays no role so (19) is superfluous.

(22)

A, = r [ A , - QaVFICYT- B,&l(HC, -Bp R;: Pa] GT, B, = r Q ( I V , l , c, = -R;;P,G*.

I allt

In this case, with q replaced by U and w , (28) captures control amplitude saturation and control rate saturation, respectively. Furthermore, in this case, Theorem 4.1 can be used to guarantee asymptotic stability of the closed-loop system (1) (5) for all $(.) satisfying (14) with a guaranteed domain of attraction. In particular, if MI, > 0 and Mz, = I 2 MI, > 0 and there exists a positive-definite matrix P satisfying (12), then take q2 = - q -a = % i = l , ..., m , i n ( 1 4 ) . Miqt '

+

-T:

sgn(q*(t)),

=

Iqz(t) I

{L...lflLTL),

+

tr[(P+i))Vi +PQaVFIQT]- (27)

Theorem 4.1 provides constructive sufficient conditions that yield dynamic controllers for systems with actuator amplitude and rate nonlinearities. When solving (16)- (19) numerically, the matrices MI,,, Mz,, ,Ml,, Mz,, H,, , and H, appearing in the design equations can be adjusted to examine trade-offs between performance and allowable sector bounded actuator nonlinearities +(.,.) E or f#(.) E @ b . Furthermore, as in [23], to further reduce conservatism, one can view the scaling matrices H , and H,, as free parameters and optimize the performance bound J with respect to

Illustrative Numerical Example In this section we provide a numerical example to demonstrate the proposed stabilization approach for systems with actuator amplitude and rate saturation constraints. The designed controller was tested using the actuator amplitude and rate saturation model given in Figure 1, where ul,(t), t 2 0, denotes an amplitude and rate saturated control signal. For simplicity we consider the design of fixed-structure full-order controllers. The design equations (16) (19) were solved using a homotopy continuation algorithm. For details of a similar algorithm see [23]. The example considered illustrates the application of Theorem 4.1 for the design of linear dynamic controllers 5.

H. Remark 4.1. A key application of Theorem 4.1 is the case in which +(q) represents a vector of timeinvariant a c x a t o r amplitude and rate saturation nonlinearities. Specifically, let + ( q ( t ) ) = [ + I ( q l ( t ) ) ,. . . ,

912

with integrators for tracking a step input. The example is adopted from [30,32] which consider step command tracking dynamic controllers for a bank-to-turn missile with saturating actuators. Specifically, the state-space realization for a two-input two-output yaw/roll dynamics of a bank-to-turn missile is given by

0 ~ ~ with 1 , the step input command T = [4.2,-4.2IT. Note that Z:l%o = 3.0598 x lo5 so that 20 $! V A . Figure 2 shows the response of the controlled system output, in the presence of control amplitude and rate saturation constraints, with the controller designed using Theorem 4.1 and the controller given by [30]. From Figure 2 note that although the yaw rate response ~ ( t ) , t 0, of Theorem 4.1 design exhibits a larger overshoot as compared to the design given by [30], it provides slightly better settling time and steady state error. Specifically, the steady state value of ~ ( t t) , 0, for Theorem 4.1 design is -4.1996 while the controller of [30] gives a steady state value of -4.145. Furthermore, the side slip response ,tJ(t),t 2 0, for Theorem 4.1 design exhibits faster rise time and settling time. In addition, the steady state value of P ( t ) , t 2 0, for the design given by Theorem 4.1 and the design given by [30] is found t o be 4.1606. In both cases, the slight offset in the steady state behavior is attributed to the fact that 20 $! V A .

>

>

where

,tJ : side slip (deg) T : yaw rate (deg/sec)] p : roll rate (deg/sec)

,

yp =

[@],

U,: rudder (deg) ua: aileron (deg)

6.

-2176

Conclusion

A fixed-architecture controller synthesis framework for systems with actuator amplitude and rate saturation constraints was developed. The simultaneous actuator amplitude and rate saturation constraint problem was modeled as an equivalent actuator amplitude constraint problem for an augmented dynamic systems. A closed-loop Riccati equation was formulated to develop a controller synthesis framework for the above class of input nonlinear system. Finally, a design example involving actuator amplitude and rate saturation nonlinearities was presented to demonstrate the effectiveness of the proposed approach.

-1093

To design side slip and yaw rate step input tracking controllers we introduce the integrator states ZIas in [30]. Furthermore, as in [30] for design purposes, consider a pseudo-equivalent feedback interconnection with I 2 subsystem interchanged with the amplitude and rate saturation nonlinearities. This yields the augmented dynamic system

References

[l] M. Athans and P. L. Falb, Optimal Control, A n Introduction to the Theory and Its Applications. New York: McGraw-Hill, 1966. [2] A. E. Bryson and Y. C. Ho, Applied Optimal Control: Optimization, Estimation, and Control. New York: Hemisphere Publishing, 1975. [3] J. F. Frankena and R. Sivan, “A non-linear optimal control law for linear systems,” Znt. Journal of Control, vol. 30, pp. 457-480, 1979. [4] N. J. Krikelis, “State feedback integral control with 5ntelligent” integrators,” Int. Journal of Control, vol. 32, pp. 465-473, 1980. [5] E. P. Ryan, “Optimal feedback control of saturating systems,” Int. Journal of Control, vol. 35, pp. 521-534, 1982. [6] N. J. Krikelis and S. K . Barkas, “Design of tracking systems subject to actuator saturation and integrator wind-up,” Int. Journal of Control, vol. 39, pp. 667-682, 1984. [7] P. 0. Gutman and P. Hagander, “A new design of constrained controllers for linear systems,” ZEEE Tkans. Automat. Control, vol. 30, pp. 22-33, 1985. [8] P. Kapasouris, M. Athans, and G. Stein, “Design of

where e(t) = yp(t) - T and T is the vector of step reference inputs. Now we interpret equations (29), (30) as equations (l),(2) and proceed with the tracking controller synthesis. However, first note that the amplitude saturation nonlinearity + ( u ( t ) ) , t 2 0, in (29) is given by (28) with q = U, i = 2, and a,, = auz = 10 and the rate saturation nonlinearity 4 ( u ( t ) ) ,t 0, is given by (28) with q = U , i = 2, and a,, = avz = 4. Next, choosing RI = diag[I3,5000,50000], R2, = I 2 , R2, = 1 2 , VI = diag[R1,I2], V2 = I 2 , MlU = 0.99912, Mi, = 0.99312, M2u = M2, = 1.0I2, H , = 1.2 x 10612, and H, = 2 x lo5& a full-order (n, = n, = 7) dynamic compensator was designed using Theorem 4.1 -with guaranteed domain of attraction V A = { 2 : Z T P Z < 4.1211 x lo2, IC‘i21 5 4, i = 3,4}. To illustrate the closed-loop behavior of the system with the controller designed using Theorem 4.1 let [ x ~ ( 0 ) , e T ( O )= ] [ 0 1 x 3 , - ~ T ] ,4 0 ) = 02x1, and xc, =

>

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feedback control systems for stable plants with saturating xtuators,” in Proc. IEEE Conf. o n Dec. and Control, (Austin, TX), pp. 46S479, 1988. [9] K. J. astrom and L. Rundqwist, “Integrator windup and how to avoid it,” in Proc. of American Control Conf., (Pittsburgh, PA), pp. 1693-1698, 1989. [lo] P. J. Campo, M. Morari, and C. N. Nett, “Multivariable anti-windup and bumpless transfer: A general theory,” in Proc. of American Control Conf., (Pittus burgh, ]?A), pp. 1706-1711, 1989. [ll] P. Kapilsouris and M. Athans, “Control systems with rate ancl magnitude saturation for neutrally stable open loop sy~:tems,”in Proc. IEEE Conf. on Dec. and Control, (Honolulu, HI), pp. 3404-3409, 1990. [12] Z. Lin and A. Saberi, “Semi-global exponential stabilizaticm of linear systems subject to “input saturation”,” Systems and Control Letters, vol. 21, pp. 225239, 1903. [13] D. S. Bernstein and W. M. Haddad, “Nonlinear controllers for positive real systems with arbitrary input nonlineitrities,” IEEE Trans. Automat. Control, vol. 39, pp. 151:3-1517, 1994. [14] Z. Lin, A. Saberi, and A. R. Teel, “Simultaneous L, stabilization and internal stabilization of linear systems subject to input saturation: State feedback case,” in Proc. IEEE Conf. on Dec. and Control, (Orlando, FL), pp. 380r3-3813, 1994. (151 A. R. Teel, “Semi-global stabilizability of linear null controllable systems with input nonlinearities,” in Proc. of American Control Conf., (Baltimore, MD), pp. 947.-951, 1994. [16] D. S. Bernstein, “Optimal nonlinear, but continuous, feedback control of systems with saturating actuators,” Int. Journal of Control, vol. 62, pp. 1209-1216, 1995. [17] Z. Lin and A. Saberi, “A semi-global low-and-high gain design technique for linear systems with input saturation: S3:abilization and disturbance rejection,” Int. J. Robust and Nonlinear Contr., vol. 5, pp. 381-398,1995. [18] A. Saberi, Z. Lin, and A. R. Teel, “Control of linear systems; with saturating actuators,” in Proc. of American Control Conf., (Seattle, WA), pp. 285-289, 1995. [19] F. Tyan and D. S. Bernstein, “Anti-windup compensator synthesis for systems with saturation actuators,” Int. J . Robust and Nonlinear Contr., vol. 5, pp. 521537, 19’35. [20] F. Tyan and D. S. Bernstein, “Dynamic output feedback compensation for systems with input saturation,” in Proc:. of American Control Conf., (Seattle, WA), pp. 3916-3920, 1995. (211 W. M. Haddad and V. Kapila, “Antiwindup controllers for systems with input nonlinearities,” A I A A J. Guid., Contr., and Dyn., vol. 19, pp. 1387-1390, 1996. [22] K. J. h r o m and B. Wittenmark, Computer Controlled Systems: Theory and Design. Englewood Cliffs, NJ: Prentice-Hall Inc., 1997. [23] W. M. 13addad and V. Kapila, “Fixed-architecture controller synthesis for systems with input-output timevarying nonlinearities,” Int. J. Robust and Nonlinear Contr., vol. 7, pp. 675-710, 1997. [24] K. McKay, ‘‘Summaryof an AGARD workshop on pilot induced oscillation,” in AIAA, (Paper 94-3668), 1994. (251 R. A. Hess and S. A. Snell, “Flight control system design with rate saturating actuators,” A I A A J. Guid., Contr., and Dyn., vol. 20, pp. 90-96, 1997.

[26] M. A. Dornheim, “Report pinpoints factors leading to YF-22 crash,” Aviation Week and Space Technology, pp. 53-54, 1992. [27] J. M. Lenorovitz, “Gripen control problems resolved through in-flight, ground simulations,” Aviation Week and Space Technology, pp. 74-75, 1990. [28] C. Zhang and R. J. Evans, “Rate constrained adaptive control,” Int. Journal of Control, vol. 48, pp. 21792187, 1988. [29] Z. Lin, “Semi-global stabilization of linear systems with position and rate limited actuators,” Systems and Control Letters, vol. 30, pp. 1-11, 1997. [30] F. Tyan and D. S. Bernstein, “Dynamic output feedback compensation for systems with independent amplitude and rate saturations,’’ Int. Journal of Control, vol. 67, pp. 89-116, 1997. [31] W. M. Haddad, V. Kapila, and V. S. Chellaboina, “Guaranteed domains of attraction for multivariable Lure systems via open Lyapunov surfaces,” Int. J . Robust and Nonlinear Contr., vol. 7, pp. 935-949, 1997. [32] A. A. Rodriguez and J. R. Cloutier, “Control of a bankto-turn (BTT) missile with saturating actuators,” in Proc. of American Control Conf., (Baltimore, MD), pp. 166G1664, 1994.

Figure 1: Amplitude and Rate Saturating Actuator

Model

aj o y

I

,

,

,

,

0

2

4

6

8

, 10

, 12

I

,

I

14

16

18

0

Time (sec)

Figure 2: Comparison of [30] and Theorem 4.1 Designs

914