A direct adaptive nonlinear tracking control frame- work for multivariable nonlinear uncertain systems with actuator amplitude and rate saturation constraints is.
Proceedings of the American Control Conference Arlington, VA June 25-27, 2001
Adaptive Tracking for Nonlinear Systems with Control Constraints Alexander Leonessat ,Wassim M. Haddadj, and Tomohisa Hayakawat +Department of Ocean Engineering, Florida Atlantic University - Sea Tech, Dania Beach, FL 33004-3023 %khool of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150
I
rect adaptive control framework developed in 91 to guarantee partial asymptotic stability of the closed- oop trackA direct adaptive nonlinear tracking control frame- ing s stem. that is, asymptotic stability with respect to work for multivariable nonlinear uncertain systems with the c%sed-ioop system states associated with the tracking actuator amplitude and rate saturation constraints is error dynamics, in the face of actuator amplitude and rate developed. To guarantee asymptotic stability of the saturation constraints. Specifically, a reference dynamical closed-loop tracking error dynamics in the face of am- system is constructed to address tracking and regulation plitude and rate saturation constraints, the adaptive by deriving adaptive update laws that guarantee that the control signal to a given reference system is modified to error system dynamics are asymptotically stable and the effectively robustify the error dynamics to the saturation adaptive controller gains axe Lyapunov stable. In the case constraints. An illustrative numerical exam le is provided where the actuator amplitude and rate are limited, the adaptive control signal to the reference system is modified to demonstrate the efficacy of the proposecfapproach. to effectively robust the error dynamics to the saturation constraints and t us guaranteeing asymptotic stability of the error states. 1. Introduction In light of the increasingly complex and highl uncertain 2. Adaptive Tracking for Nonlinear Uncertain Systems nature of d namical systems requiring controg, it is not surprising t t a t reliable system models for many high perIn this section we consider the problem of characterizformance e n r g applications are unavailable. In the ing adaptive feedback tracking control laws for nonlinear face of such 'gh levels of system uncertainty, robust controllers may unnecessarily sacrifice s stem performance uncertain systems. Specifically, we consider the controlled whereas adaptive controllers are clear& appropriate since nonlinear f i e system (i given by they can tolerate far greater system uncertainty levels to S ( t ) = f ( 4 t ) ) + G(Z(t))U(t), @) = 2 0 , t L 0, (1) improve system performance. However, an implicit assumption inherent in most adaptive control frameworks is that the adaptive control law is implemented without where z(t) E R", t 2 0, is the state vector, U t E Rm, any regard to actuator am litude and rate saturation con- t 2 0, is the control input, f : R" + Rn, and : R" + straints. Of course, any ePectromechanicalcontrol m t u a IFxm.The control input U(.) in (1) is restricted to the tion device is subject to amplitude and/or rate constraints class of admissible controls such that (1) has a unique leading to saturation nonlinearities enforcing limitations solution forward in time. Fhthermore, we assume that on control amplitudes and control rates. As a conse- the reference trajectories to be tracked are generated by quence, actuator nonlinearities arise frequently in practice the reference system G, given by , and can severely de ade closed-loop system erformance, and in some cases g i v e the system to instagility. These Sr(t) = Arzr(t) + Brr(t), Zr(0) = Zro, t 2 0, (2) dects are even more pronounced for adaptive controllers which continue to adapt when the feedback loop has been severed due to the presence of actuator saturation causing where zr(t) E R", t 2 0, is the reference state vector, unstable controller modes to drift,which in turn leads to r ( t ) E Rm,t 2 0, is the reference input, and A, E RnXn severe windup effects. and Br E Rnxm are such that the pair (Ar,B,)is stabilizThe research literature on ada tive control with actu- able. For the statement of the following result define the ator saturation effects is rather k k t e d . Notable excep- tracking error e(t) s(t) - zr(t). tions include [1-6]. However, the results reported in [1-6] Theorem 2.1. Consider the nonlinear system B given are confined to h e a r plants with amplitude saturation. Many practical applications involve nonlinear dynamical by (1) and the reference system Br given by (2). Assume systems with simultaneous control amplitude and rate sat- there exists gain matrices k~E PXm and k; E l P x s , uration. The presence of control rate saturation may furand functions 6 : Rn + RmXm and F : R" + Rs such ther exacerbate the problem of control am litude saturation. For example, in advanced tactical &hter aircraft that with high maneuverability requirements, pilot induced os0 = G ( z ) G ( s ) R~ Br, x E Rnl (3) cillations [7,8]can cause actuator amplitude and rate saturation in the control surfaces, leading to catastrophic fail0 = f ( ~ + ) BrkZF(2) A+, z E R" , (4) ures. Abstract
%
0
In this paper we develop a direct adaptive control framework for adaptive trackin of multivariable nonlinear uncertain systems with am&tude and rate saturation constraints. In particular, we extend the Lyapunov-based di-
hold. Furthermore, let K E Rmx" be given by
This research was supported in part by the National Science Foundation under Grant ECS-9496249 and the Air Force Office of Scientific Research under Grant F49620-001-0095.
where the n x n positive definite matrix P satisfies
0-7803-6495-3/01/$10.00 0 2001 AACC
K = -&-lB:P,
1292
O=A:P+PAr-PBrh-'B,TP+R1,
(5)
(6)
and R1 E Etnxn and R2 E ItmXmare arbitrary positivedefinite matrices. Then the feedback control law
+
+
u(t) = G ( z ( t ) ) k l ( r ( t ) & ~ ( z ( t ) ) Ke(t)), (7)
yarantees that the zero solution e(t) G 0 of the error ynamics given by
G ( t ) = (f(s(t)) + G(%(t))u(t)) - (Arzr(t)+ Brr(t)), e(0) = zo - zr,, eo, (8)
Proof. With u(t), t 2 0, given by (11) it follows from (3), (4), (12), and (13), that the error dynamics e(t), t 2 0, are gwen by
+
&(t)= (A, B,K)e(t) +Br(I?cl - ~ ; ' ( t ) ) G - l ( z ( t ) ) u ( t ) +B,(Kz(t) - &)F(z(t)), e(0) = eo, t >_ 0. (14) To show Lyapunov stability of the closed-loop system (11)-( 14), consider the Lyapunov function candidate
v(e,K I ,K2)
is globally asymptotically stable.
= eTPe + tr (k;' - K ; ' ) ~ Q ; ' ( ~ ; ' -K ; ~ ) +tr ( ~ -2 I ? ~ ) ~ Q ; ' (-K2 ~ 21, (15)
Proof. Using the feedback control law given by (7), (8) becomes
&(t)= (A,
+ G(z(t))G(z(t))&lK)e(t)
+ ( G ( z ( t ) ) G ( z ( t ) ) r k 2 F ( z ( t+>f(z(t)) ) -Arz(t)) + ( G ( z ( t ) ) G ( z ( t ) )R Br)r(t), ~ e(0) = e o , t 2 0. (9) Now, using (3) and (4), it follows from (9) that k ( t ) = (A,
+ B,K)e(t),
e(0) = eo, t 2 0.
(10)
Finally, since (A,, B,)is stabilizable and R1 > 0, it follows from standard linear-quadratic regulator theory,that A, B,K, with K given by (5), is Hurwitz. Hence, the zero solution e(t) z 0 to (8) is globally asymptotically stable.
+
where P > 0 satisfies (6). Note that V(0,k ~ Rz) , = 0 and, since P , &I, and Q2 are positive definite, V ( e ,K1, K2) > 0 for a~ ( e , K l , K z )# (olk1,k2). NOW, letting e(t), t >_ 0 , denote the solution to (14), using (6), (11)-(13), and using the fact that $(KC1(t))= -K;l(t)kl(t)KC1(t),it follows that the Lyapunov derivative along the closed-loop system trajectories is given by
Wt)lKl(t),K2(t)) = eT(t)Pe(t) iT(t)Pe(t) +2tr (kc'- K,-'(t))TQ,'~(-K:l(t))
0
Theorem 2.1 provides sufficient conditions for characterizing trackin controllers for the nonlinear dynamical system Q: In t8e next result we show how to construct adaptive gains K l ( t ) E EtmXm, t 2 0, and K2 t ) E Itmxs, t 2 0, for achieving tracking control in the ace of system uncertainty. For this result y e do no! require explicit knowledge of the gain matrices K1 and K2; d that is required is the existence of k1 and Kz such that (3) and (4) hold. Furthermore, we shall require that det 3 1 # 0.
i
+
+2tr ( ~ 2 ( t )K 2 ) T ~ ; 1 k 2 ( t )
+
+
+ B,K)TPe(t) +2eT(t)~~,(R:l - ~ ~ ' ( t ) ) G - ' ( ~ ( t ) ) u ( t ) + 2 e * ( t ) ~ ~ , ( ~z (Rt 2) ) ~ ( z ( t ) ) -2tr (I?;' - K;' (t>)TB,TPe(t).uT(t)~:-T(z(t)) -2tr ( ~ 2 ( t )1 t 2 ) ~ B , T p e ( t ) ~ ~ ( z ( t ) ) = -eT(t)(R1 KTR2K)e(t) +2tr G-' (z(t))21(t)eT(t)pBr(k~l - KC' ( t ) ) +2tr F(z(t))eT(t)PBr(K2( t )- k2)
= eT(t)P(A, B,K)e(t) eT(t)(A,
+
Theorem 2.2. Consider the nonlinear system G given by (1) and the reference system Qr gven by (2). Assume - 2 t r G - l ( ~ ( t ) ) u ( t ) e ~ ( t ) ~ ~, (~R;;'' ( t ) ) there exists gain matrices KIE R y x m and K2 E Rmxs, -2tr~(z(t))e~(t)~~,(~z(t) - k2) with d e t k 1 # 0, and functions G : Iwn + and F : Iw" + F, with detG(z) # 0, 2 E R", such that (3) = -eT(t)(R1 ~ ~ ~ 2 ~ ) e ( t ) and (4) hold. Furthermore, let K E RmXnbe given by (5), I 0, (16) where P > 0 satisfies 6). In addition, let & I , Q2 E BmXm be positive definite. r! hen the adaptive feedback control which proves that there*exi?ts a neighborhood V c Etn x law ItmXmx Rmxq of (O,Kl,K2)such that if (e(O>,Kl(O), W = G ( z ( t ) ) ~ l ( t ) ( r+( t~) 2 ( t ) ~ ( z +( tKe(t)), )) (11) K2(0)) E D, then the solution ( e ( t ) , K l ( t ) , K z ( t )E) (0, k1,&2) of the closed-loop system given by (11)-( 14) is where K l ( t ) E EtmXm, t 2 0, and K 4 t ) E Rmxs, t 2 0, Lyapunov stable. Furthermore, since RI KTR2 K > 0, it follows from Theorem 4.4 of [lo] that e(t) -+ 0 as t + 00 with update laws for all (e(O),Kr(O), Kz(0))E V. U k l ( t ) = -K~(t)Q1B,TPe(t)21T(t)G-T(Z(t))K~(t)l (12) Remark 2.1. Note that the conditions in Theorem 2.2 &(t) = -Q2BTPe(t)FT(z(t)), (13) imply that e(t) -+ 0 as t -+ 00 and hence it follows from (12) and (13) that (e(t),Kl(t),Kz(t))-+ M ! i guarantees that there exists a neighborhood V c Rn x { ( e , . ~ 1 , ~E2 v ) c IP x ~~~m x 1 ~ x :8e = 0, I;rl = R m x m x !Emx8 of ( O , k ~ , k z such ) that if (e(O),Kl(O), Kz(0)) E V , then the solution (e(t),K1(t),Kz(t)) 0, K2 =O} as t + 00. ( O l I ? 1 , k 2 ) of the closed-loop system given by (11)-(13) Remark 2.2. Note the Lyapunov function candidate is Lyapunov stable and e(t) -+ 0 as t -+ 00. (15) is not radially unbounded with respect to K1, and
+
+
=
1293
@ZIT,
hence Theorem 2.2 provides local stability guarantees. However, if G(z), x E Rn,is known, then K l ( t ) ,t 0, can be taken to be the constant ain matrix k1 so that (12 is su erfluous. In this case, t%eadaptive feedback contro law 5 1 ) with update law (13) guarantees that the solution ( e ( t ) K2(t)) , E (0, k2) of the closed-loop system given by ( l l ) , (13), and (14) is Lyapunov stable and e ( t ) + 0 as t + 00 for all eo E Rn. For further details see [9].
A, = [g, where 0, E Rmxnis a known matrix, let k2 E Itmxs,where s = n q, be given by
It is important to note that if (1) is in normal form with asymptotically stable internal dynamics [ll],then we can always construct functions 6 : R* + Rmxmand F : I P -+ Rs,with det G ( x ) # 0, x E R*, and a stabilizable pair A,, B r ) such that (3) and (4) hold without requiring know edge of the system dynamics. To see this assume that the nonlinear uncertain system 0 is generated by
In this case, it follows that, with 6 ( x ) = k; = @-lBrs, G(~)G(x)k= l B, and
>
\
\
c
12'2
+
=BrT1[Q,
-@cy
-Qd],
(19)
and let
and (21)
m
qi(r')(t) = fui(q(t))
+
Gs(i,j)(q(~))%(4,
j=l
q(0) = 40,
t 2 0, i = l , * * - , m ,(17)
where qi(r') denotes the r:" derivative of qs, ri denotes the relative degree with respect to the output 4i, fuui(q) = (rl-1) ( r -I)), 1 * * 1 qm, * * i qmm fui(q1, * * , and Gs(j,j)(q)= G,(i,j)(ql,..-, q1(r1-1) l-.-lqm,..-,q$m-l)). Here we assume that the square matrix function G,(q) composed of the entries Gs(i,j)(q), i,j = 1,. ,m, is such that rmis the detG,(q) # 0, q E It9, where r^ = r1 (vector) relative degree of (17). Furthermore, since (17) is in a form where it does not possess internal dynamics, it follows that i: = n. The case where 17) possesses inputtestate stable internal dynamics can e handled as shown in [9].
+ + e . .
6
Next, define xi 4 [ p i , . . . ,qiri-')] Zm+1
z2+,]',
4 [qp-l)l...lqmm (r
-l)IT,
T
,i = l , - . - , m ,
and z 4 [x:,
.-.,
so that (17) can be described as (1) with
= Arx, (22) where A, is in multivariable controllable canonical form. Hence, choosing A, and Br such that (A,, B,)is stabilizable and choosing RI > 0 and F2 > 0, it follows that there exists a positive-definite matnx P satisfying the Riccati equation (6).
Adaptive Trackig with Actuator Amplitude and Rate Saturation Constraints In this section we extend the adaptive control framework presented in Section 2 to account for actuator amplitude and rate saturation constraints. Recall that T h e 0, orem 2.1 guarantees that the tracking error e ( t ) , t converges to zero; that is, the state vector s(t), t 2 0, converges to the reference state vector sr(t),t 2 0. Furthermore, it is important to note that xr(t), t >_ 0, does not directly appear in the control signal u(t), t 2 0, given (111, which depends on the re erence input r ( t ) , t >_ 0. owever, since for a ftred set o initial conditions there exists a one-to-one mapping between the reference input r(t), t 2 0, and the reference state z,(t), t 0, it follows that the control signal (11) guarantees convergence of the state z(t), t 0, to the reference state s,(t), t 0, corresponding to the specified reference input r ( t ) , t 0. Of course, the reference input ~ ( t )t , 0, should be chosen so as to guarantee asymptotic convergence to a desired state vector Z d ( t ) , t 2 0. However, the choice of such a reference input r(t), t >_ 0, is not unique since the reference state vector zr(t),t 0, can converge to the desired state vector z d ( t ) , t 2 0, without matching its transient behavior. Next, we rovide a framework wherein we construct a family of regrence inputs r ( t ) , t 2 0, with associated reference state vectors z r ( t ) , t 2 0, that arantee that a given reference state vector within this gmily converges to a desired state vector Zd(t), t 2 0, in the face of actuator amplitude and rate saturation constraints. We begin by differentiating both sides of the control law ( l l ) , and using the assumption that Kl (t),t 0, is nonsingular, to obtain the time rate of change of the control input given bY k(t) = H(s(t),i(t)) 4 & ( Z ( t ) ) K l (t)[i(t) k 2 ( t ) F ( Z ( t ) ) 3.
>
2
f
>
>
>
where
>
&E is a known matrix of zeros and ones c a p turing the multivariable controllable canonical form representation [12], and fu :Rn + Rm and G, : Rn + are unknown functions. Here, we assume that fu(x) and G,(z) are unknown and are parameterized as A(%)= @cs + q,~fd(s), where fd : Rn + RqlG , ( x ) = where G, : Et"' + Rmxmand satisfies detG,(s) # 0, z E R", and 0~E Rmxn,0 d E Rmxq,and @ E Etmxm, with det @ # 0, are matrices of uncertain constant parameters. Next to ap ly Theorem 2.2 to the uncertain system (1) with f(x) and G(z) given by (18), let B r = T [O~n-m~xm, Br?] , where B,, E Rmxmis invertible, let
@e,(%),
1294
>
+
> >
+
+K2(t)F1(z(t))k(t)K ( k ( t )- kr(t))]
.KT'(t)G'-l(z(t))u(t), t 2 0,
(23)
UtmXm and F : Utn -+ Rs,with det G(z) # 0,z E Rn, such that (3) and (4) hold. Furthermore, let K E Rmxn be given by (5), where P > 0 satisfies (6). In addition, for a given desired reference input T d ( t ) , t 2 0, let the reference input r ( t ) , t 2 0, be given by
where s ( t ) 4 (z(t>,zr(t),Kl(t),K2(t),r(tt) 2), 0, F'(z) i ( t ) = H-' ( s ( t ) ,C(u(t),U*@))U* ( t ) ) , denotes the Fkech6t derivative of F ( z ) , and k ( t ) , &(t), r(0) = rd(o), t 1 0, (28) 4t), k 2 ( t ) ,t 2 0, are given by (1)l (21, ( W , (1211 and (13), respectively. Using (23), the time rate of change where of the reference input i ( t ) ,t 2 0, is given by U*@) = H ( s ( t ) ,i * ( t ) ) , (29) i(t) = iT1(S(t),U(t)) i * ( t ) = fd(t) A(r(t)- T d ( t ) ) , t 20, (30)
m),
+
9 K;l(t)G-l(z(t)) [U@) - (G(z(t))k&)
4
and where ~ ( t=) (.(t),zr(t),Ki(t),K2(t),r(t)), t 2 0, and A E Rmxmis Hurwitz. Then the adaptive feedback control law (ll),with update laws (12) and (13) and reference input r ( t ) , t 0, satisfying (28)-(30) guarantees that:
>
2 (t)F(z(t)) - K2 ( t ) F ' ( W ) i
(t)
- K ( i ( t ) - k r ( t ) ) , t 2 0.
(24)
The above expression relates the time rate of change of the reference input to the time rate of change of the control input. Next, we assume that the control signal is amplitude and rate limited SO that [ui(t)l5 umax and @(t)l5 Umm, t.2 0, i = 1, ...,m, where um, > 0 and > 0 are ven. For the statement of our main result the following refinitions are needed. For i E { 1 , . m} define e ,
U(Ui(t),&i(t))
4
if lui(t)l = uma and Ui(t)Ui(t> > 0, 1 otherwise,
E)
There exists a neighborhood 2) c Utn x EtinaXmx Itmxs of 0 such that u*(t), t > t*,
ii4 IUi(t)l iu)
t 2 0,
does not violate the amplitude and rate saturation constraints, then lit,, r(t) = l"d(t).
Proof. i) is a direct consequence of Theorem 2.2 with r(t),t 2 0, satisfying (28)-(30). To prove ii) and iii) note that it follows from (23), (24), and (28) that
U(t) = H(s(t),i(t)) = H(s(t),H-l(s(t), C(u(t),U*(t))U*(t))) = C(U(t),ti*(t))U*(t), t >_ 0, (31)
where ui,Ui E n$ i E { l , - * - , m } ,denote the ith component of u E Rm and U E Rm, respectively. Note that for i E { l , . . - , m ) and t = ti'> 0, the function a*(.,.) is such that the following properties hold:
4) If lui(tl)l = um, tii(t&T*(Ui(tl), id&))
and ui(tl)Ui(tl)
> 0, then
>
= 0.
ii) If Izii(t1)l > 1m, a d lui(ti)l < U m U or if Itii(ti))> t i m s and IUi(ti)l = am, and Ui(ti)Ci(ti) 5 0, then Ui(tl)c*(ui( t l ),Ui( t i ) ) = Um,Sgn(Ui ( t i ),) where S g n U i k IUil/Ui.
iii) If no constraint is violated, 'O*(Ui(tl), U&)) = Ui(t1).
which implies zii(t) = a*(ui(t), U T ( t ) ) U t ( t ) , i = 1 , . -.,m. Hence, if the control input ui(t) with a rate of change Uf(t), i = l,-..,m,t 0, does not violate the amplitude and rate saturation constraints then it follows from (26) that a*(ui(t),Ut(t))= 1 and hence tii(t) = ti;@), i = l , . . - , m . Alternatively, if the pair (ui(t),U!(t)), i = 1,. m, t 2 0, violates one or more of the input amplitude and/or rate constraints then (25), (26), and (31) imply
then
Ui(t1)
Finally, we define the component decoupled diagonal nonlinearity C(u,&) by C(u(t),id(t))?! diag[a*(ul(t),til(t)),a*(uz(t),&(t)), . 1 a*(um(t),Uina(t))l, t 2 0. (27)
-
Theorem 3.1. Consider the controlled nonlinear sys-
tem G given by (1) and the reference system Gr given by (2). Assume there exist gain matrices I?1 E R*xm and K2 E Utmxs, with detK1 # 0, and functions G : R" +
e ,
i) U i ( t ) = 0 for all t 2 0 if lUi(t)l = ui(t)Uif(t)> 0; and
U,
and
ii) tii(t) = tim=Sgn(tit(t)) for dl t 1O if Iidr(t)l > tim, and I~i(t)l< Umax or if IUt(t)l > timaxand Iui(t)l = uma
and ~ i ( t ) U i ( t5) 0;
which guarantee that Iui(t)l 5 U,,,, and ltii(t)l 5 &,a for all t 2 0 and i = l , . * . , m . Finally, to show iv) let t = t* > 0 such that C(u(t),U*(t))= I,,,, t > t*. In this case, i ( t ) = H-'(z(t),U*(t))= i * ( t ) ,t > t*. Hence, it fOllOWS from (30) that f ( t ) - f d ( t ) = A(r(t) - rd(t)),
1295
t > t*, which, since by assumption A E Rmxmis Hurwitz, = Td(t)0
bt+ca T(t)
Note that it follows from Theorem 3.1 that if the desired reference input ?‘d(t), t 0, is such that the actuator amplitude and/or rate saturation constraints are not violated, then r(t) = rd(t), t 0, and hence z(t), t 0, converges to 2d(t) t 0. Alternatively, if there exists t = t* > 0 such tfiat the desired reference input drives one or more of the control inputs to the saturation boundary, then ~ ( t#) Td(t)r t > t*. In this case however, (30) guarantees that limtAmT(t) = rd(t) so long as the time interval over which the control input remains saturated is finite. If this is not the case, then our approach cannot guarantee convergence of input reference r(t), t 2 0, to the desired reference rd(t , t 3 0. Of course,-if there exists a solution to the tr a C Lg problem wherein the input amplitude and rate saturation constraints are not violated when the tracking error is within certain bounds, then our approach is guaranteed to always work.
>
> >
Illustrative Numerical Example In this section we present a numerical example to demonstrate the utility of the proposed direct adaptive control framework for adaptive stabilization and tracking in the face of actuator amplitude and rate saturation constraints. Consider the nonlinear dynamical system representing a controlled rigid spacecraft given by
+ I
4.
k ( t ) = -IC’XIbZ(t)
R = 10013, and R2 = 0.113, so that K and P satisfying (5) are gwen by
-6
1,
[l]F. Ohkawa and Y. Yonezawa, “A discrete model reference adaptive control system for a lant with input amplitude constraints,” Int. J. &ntr., vol. 36, pp. 747-753,1982. [2] D.Y. Abramovitch, R. L. Kosut, and G. F. Franklin, “Adaptive control with saturating inputs,” in PTOC. IEEE Conf. Dec. Contr., pp. 848-852, Athens, Greece, 1986. [3] A. N. Payne, “Adaptive onestep-ahead control subject to an input-amplitude constraint,” Int. J. Contr., vol. 43, pp. 1257-1269,1986. [4] C. Zhang and R. J. Evans, “Amplitude constranied adaptive control,” Int. J. Contr., vol. 46, pp. 53-64, 1987. [5] S. P. Kkason and A. M. Annaswamy, “Ada tive control in the presence of input constraints,” &El3 Dam. Autom. Contr., vol. 39, pp. 2325-2330,1994. [6] A. M. Annaswamy and S. P. Khason, “Discretetime adaptive control in the presence of input constraints,” Automaticu, vol. 31, pp. 1421-1431,1995.
[7]K. McKay, “Summary of an AGARD workshop on pdot induced oscillation,” in PTOC.AIAA Guid. Nav. Contr., pp. 1151-1161,1994. 1296
[8]R. A. Hess and S.A. Snell, “Flight control design with rate saturating actuators,” AIAA J. Guid. Contr. Dyn., vol. 20, pp. 90-96, 1997. [9] W. M. Haddad and T. Hayakawa, “Direct adaptive control for nonlinear uncertain systems with exogenous disturbances,” in Proc. Amer. Contr. Conf., pp. 4425-4429, Chicago, E,June 2000.
[lo] H. K.Khalil, Nonlinear Systems. Upper Saddle River, NJ: Prentice-Hall, 1996. [ll] A. Isidori, Nonlinear Control Systems. New York, NY: Springer-Verlag, 1995. [12] C.T.Chen, Linear S stem Theory and Design. New
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York, N Y : Holt, Rinetart, and Winston, 1984.
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-. ---__
-
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Figme 4.3: Control rate versus time
- Saturated
0.4 -
e
- saturated
unsaturated-
0-
5
10
IS
20
25
Time
30
Figure 4.1: Angular velocities versus time 1 8 0
5
I
r I‘ ’\ I
I
- saturated - -. Unsaturated
Figure 4.2: Control signal versus time
1297
io
