adaptive output feedback control olf nonlinear systems...

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 2, FEBRUARY 1996 177

Adaptive Output Feedback Control olf Nonlinear Systems Represented by Input-Output Models

Hassan K. Khalil, Fellow, IEEE

Abstract- We consider a single-input-single-output nonlinear system which can be represented globally by an input-output model. The system is input-output linearizable by feedback and is required to satisfy a minimum phase condition. The nonlinearities are not required to satisfy any global growth condition. The model depends linearly on unknown parameters which belong to a known compact convex set. We design a semiglobal adaptive output feedback controller which ensures that the output of the system tracks any given reference signal which is bounded and has bounded derivatives up to the nth order, where n is the order of the system. The reference signal and its derivatives are assumed to belong to a known compact set. It is also assumed to be sufficiently rich to satisfy a persistence of excitation condition. The design process is simple. First we assume that the output and its derivatives are available for feedback and design the adaptive controller as a state feedback controller in appropriate coordinates. Then we saturate the controller outside a domain of interest and use a high-gain observer to estimate the derivatives of the output. We prove, via asymptotic analysis, that when the speed of the high-gain observer is sufficiently high, the adaptive output feedback controller recovers the performance achieved under the state feedback one.

I. INTRODUCTION HE motivation for this work has come from recent de- T velopments in the use of high-gain observers in nonlinear

control design. The last 15 years or so have witnessed a great deal of progress in the design of feedback control for nonlinear systems. Most of the results, however, assume full state feedback. Efforts to extend some of these results to output feedback have naturally included the idea of designing an observer to estimate the state of the system from its output. Due to nonlinearity, one should not expect that the observer design can be carried out independent of the state feedback design. In other words, one should not expect a “separation principle” as in linear control theory. In fact, even in linear control design, when model uncertainties are taken into consideration, the design of the observer cannot be separated from the design of state feedback control. It turns out that for a class of feedback linearizable systems, a certain degree of separation can be achieved by designing high-gain observers which have certain disturbance rejection properties that would allow asymptotic recovery of the performance achieved under state feedback, where “asymptotic” here refers

Manuscript received January 5, 1994; revised October 15, 1994 and May 2, 1995. Recommended by Associate Editor, L. Praly. This work was supported in part by the National Science Foundation under Grants ECS-9121501 and ECS-9402187.

The author is with the Department of Electrical Engineering, Michigan State University, East Lansing, MI 48824-1226 USA.

Publisher Item Identifier S 0018-9286(96)00978-6.

to the behavior of the system as the poles of the observer approach infinity [I], [2]. High-gain observers, however, exhibit a peaking phenomenon in their transient behavior. As the observer poles approach infinity, its exponential modes will decay to zero arbitrarily fast, but the amplitudes of these modes will approach infinity thus producing an impulsive-like behavior. This peaking phenomenon is studied in [2] and a new idea to overcome peaking is introduced. The idea is to design the state feedback control to be globally bounded and then introduce the high-gain observer. During the short transient period when the state estimates exhibit peaking, the controller saturates, thus preventing peaking from being transmitted to the plant. Over the same period, the estimation error decays to small values, while the state of the plant remains close to its initial value. The validity of the idea is justified in [2] via asymptotic analysis from singular perturbation theory. Since the introduction of this idea, it has been used in a number of studies on the design of output feedback control for nonlinear systems [3]-[9]. A common theme in many of these studies is semiglobal output feedback stabilization where the controller is designed to include any given compact set in the region of attraction. The combined use of high-gain observer and globally bounded control makes such a design possible because the output feedback control asymptotically recovers the region of attraction achieved under state feedback. While most of these studies deal with certain classes of feedback linearizable systems, [9] adds a new dimension to the use of high-gain observers and globally bounded control. It considers a class of obseirvable systems from [lo] where the state of the system can be expressed as a function of the input, the output, and their derivatives up to a certain order. Extending the dynamics of the system by augmenting a series of integrators at the input side makes the derivatives of the input available for feedback. Then by using a high- gain observer to estimate the derivatives of the output, a globally bounded state feedback controller can be implemented using output feedback. This approach is used in [9] to show that global stabilizability (by state feedback) and observability imply semiglobal stabilizability by output feedback. The result of [9] suggests that we now have the tools to design output feedback control for a wider class of control problems, where an acceptable control can be designed as a function of the input, the output, and their derivatives. This has motivated the adaptive control problem studied in this paper.

We consider a class of single-input-single-output nonlinear systems with unknown constant parameters. The class of systems includes as a special (case the nonlinear systems treated

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178 EEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 2, FEBRUARY 1996

in [ l l ] and [12] for output feedback adaptive control and the linear systems treated in the traditional adaptive control literature, e.g., [13]-[15]. We extend the dynamics of the system by adding a series of integrators at the input side and represent the augmented system by a state-space model, where the states are the input, the output, and a number of their derivatives. We assume that the derivatives of the output are available for feedback and proceed to design an adaptive state feedback controller that achieves tracking of a given reference signal. The state feedback control is as simple as the one designed for linear systems, e.g., [14, Section 3.41, except that we assume that the unknown parameters belong to a known compact convex set and include a parameter projection feature in the parameter adaptation rule to ensure that the parameter estimates do not leave this set. This state feedback control works globally. For a given set of initial conditions, we estimate the region of interest of the states and saturate the control and the right-hand side of the adaptation rule outside this set. Then we use a high-gain observer to estimate the derivatives of the output and implement the controller using only measurement of the output. We use an asymptotic analysis to prove that the output feedback controller recovers the performance achieved under the state feedback controller. The parameter projection feature plays a crucial role in our analysis when we show boundedness of the trajectories of the closed-loop system in the presence of estimation error. A persistence of excitation condition ensures convergence of the tracking error and the parameter estimation error under both the state and output feedback. The result we obtain using output feedback is semiglobal in the sense that we can include any given compact set in the region of attraction. When applied to the nonlinear systems of [ 111 and [ 121 and the linear systems of [ 131-[ 151, our method produces a new adaptive controller. We apply our method to two examples from the literature for which other adaptive controllers are available and compare its performance with the other methods.

11. PROBLEM STATEMENT

We consider a single-input-single-output nonlinear system represented globally by the nth-order differential equation

P / P \

i=l / i=l )

where U is the control input, y is the measured output, ~ ( ~ 1 denotes the ith derivative of y, and m < n. The functions f; are known smooth nonlinearities which could depend on v,

The constant parameters go to g p are known, while the constant parameters 81 to Qp are unknown, but the vector 19 = [&, . . . , BPIT belongs to R, a known compact convex subset of RP. Let 6 be a convex subset of RP which contains R in its interior. We make the following assumption on the system.

Assumption 1: (go + BTg) # 0 V % E fi. The objective of this paper is to design an adaptive out-

put feedback controller which guarantees boundedness of all variables of the closed-loop system and tracking of a given reference signal yT(t). To state this objective precisely, we assume that yT( t ) is bounded, has bounded derivatives up to the nth order, and y?)(t) is piecewise continuous. Let

Y( t ) = [y(t), y(l)(t), ' * ' , y(n-l)(t)]T

YT(t) = [YT( t ) , Y$l'(t), . . . , y?-l)(t)lT

yR(t) = [ Y T ( ~ ) , yi"(t), ' ' ' , y?"-"(t), y?'(t)lT

and Y and YR be any given compact subsets of R" and Rn+', respectively. Our objective is to design the adaptive output feedback controller such that for all y(0) E Y , for all y ~ ( t ) E YR, and for all B E R, all variables of the closed-loop system are bounded for all t 2 0, and

lim / y ( t ) - yT(t)/ = 0.

The compact sets Y, YR, and R are arbitrary, but we assume that they are known. Borrowing the semiglobal terminology from recent nonlinear feedback control literature, e.g., [ 161, we may say that our goal is to design a semiglobal adaptive controller. The controller will use the knowledge of the sets

Throughout the paper we represent an extended version of (1) by a state-space model. We augment a series of m integrators at the input side of the system. We denote the states of these integrators by z1 = U , 2 2 = u(l), up to zm = ~ ( ~ - l ) , and set v = u ( ~ ) as the control input of the augmented system. By taking z1 = y, 2 2 = y(l), up to z, = y("-l), we can represent the augmented system by the state-space model

t+m

Y , YR, and R.

X % = z,+1, 1 5 i 5 n - 1 2, iz = z,+1, 1 5 i 5 m - 1 .im = V

Y = 5 1

= f o ( x , .) + Q T f ( x , 2 ) + (go + BTg)v

where T

2 = [XI, . . . , z,] z = [z1, . . ' ~ z,]T

f = [ f l , ' " > & I T , 9 = [91, " ' > & I T . ,

The initial states of the integrators are chosen such that z (0) E 20, a compact subset of R". Assumption 1 implies that (2) is input-output linearizable by full state feedback for every B E R. It also guarantees that for every 0 E R, there is a globally defined normal form for (2). In particular, the change of variables

transforms the last m state equations of (2) into

j z = ~ i + l , 1 5 i 5 m - l

which, together with the first n state equations of (2), define the global normal form. Setting x = 0 in (4) results in the


KHALIL ADAPTIVE OUTPUT FEEDBACK CONTROL 179

zero dynamics of (2), as defined in [17, 4.31. This leads to the following assumption which includes a minimum-phase condition.

Assumption 2: For every 0 E R, the system (4) has the property that for any z (0) E 20 and any bounded ~ ( t ) , the state [ ( t ) is bounded.

The input-output model ( l ) , with Assumptions 1 and 2 satisfied, includes as a special case the input-output model used in [ 111 and the class of state-space models used in [ 121 which is characterized by geometric conditions. In [ l l ] , the nonlinear system is described by the nth-order differential equation

m

4 D ) Y = B(D)b(Y)UI + Di[Pio(Y) +pT(Y)el] (5) i=O

where D = (d /d t ) , A(D) = D" + U,-~D"-~ + - . . + a 0 is a polynomial with unknown coefficients, B ( D ) = b,Dm +

+ bo (m < n) is a Hurwitz polynomial with unknown coefficients and b, # 0, 81 is an e-dimensional vector of unknown parameters, and a(y) and pij(y) are smooth known nonlinearities with a(y) # 0 for all y E R. It is straightforward to see that defining a vector 0 to include O1 as well as the unknown coefficients of the polynomials A(D) and B(D) , and taking ii = a(y)u as the control input, (5) can be represented in the form of (1). The constant go + OTg will be simply b,; hence Assumption 1 is satisfied. Equation (4) will take the form

1 j , = [i+1, l l i l m - 1

1 cm = - & CL;' bici+l - ~ [ p 0 ( 2 1 , * * 3 zm+l)

where po ( e ) and p ( .) are smooth functions of their arguments. Assumption 2 follows from the assumption that B(D) is Hurwitz. The class of state-space models treated in [12] has an input-output model of the form

y(") = B(D)[a(y)u] + &(y,. . . , y("-1) ) P

i=l

where 81 to 0, are the unknown constant parameters, B(D) = bmDm + . . . + bo (m < n) is a Hurwitz polynomial with coefficients bi(0) = pot + B j p j i , and a(y) and $~i(.) are smooth known nonlinearities with a(y) # 0 for all y E R. It is assumed that b,(B) # 0. Redefining the control input as ii = a(y)u, it can be seen easily that (7) is a special case of (l) , and Assumption 1 is satisfied since go + OTg = bm(8). Similar to the case of [ l l ] , (4) will be linear in [ with a Hunvitz matrix, leading to the fulfillment of Assumption 2. We note also that the class of linear systems treated in the traditional adaptive control literature, e.g., [13]-[15], is of the form (1) and satisfies Assumptions 1 and 2.

The input-output model (1) has two restrictions which are

(go + gTO) to be constant, arid second, the linear dependence on the unknown parameters 8. The first restriction is made for convenience. In fact, the result of this paper can be extended to the case when go and g are functions of y, y(l), ., y("-'), U, u(l), . . a , u ( ~ - ' ) , provided 1g0+gT81 is globally bounded from below. This, however, will complicate (4) which is now simple due to an explicit transformation of (2) into the normal form. The linear dependence on the parameters is a more serious restriction. Redefinition of physical parameters may be needed to arrive at ( l ) , as the following example shows.

Exumple I : A single link manipulator with flexible joints and negligible damping can be represented by [18]

where 41 and 42 are angular positions, and U is a torque input. The physical parameters g , I , J , k, L, and M are all positive. Taking y = 91 as the output, it can be verified that y satisfies the fourth-order differential equation

k I J I J sing + -U.

g k L M --

Taking

the equation takes the form of (1). Knowing the physical bounds on the parameters g , I , J , k, L, and M , one can define a compact convex set (1 such that 0 E R. If R is defined such that 64 is bounded from below on a convex set fi which contains Cl in its interior, then Assumption 1 will be satisfied. Assumption 2 is satisfied by default, since in this example m = 0.

111. STATE FEE~DBACK CONTROL Assuming the full state (2, z ) is available for feedback, we

proceed to design an adaptive state feedback controller. Taking

worthy of mention: first, the restriction of the coefficient


180 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 2, FEBRUARY 1996

where ( A , b ) and (A2, b2) are controllable canonical pairs of the form

. . . . . .

. . . . . . 81 F81

Choose a matrix K such that A, = A - bK is Hurwitz, and rewrite (8) as

6 =A,e + b{Ke + f o ( e + Y,, 2 ) + Q T f ( e + Y,,z) + (go + BTg)w - y p } . (10)

Let P = PT > 0 be the solution of the Lyapunov equation

PA,+AT,P=-Q, Q = Q T > O

and consider the Lyapunov function candidate

v = eTPe + + JTr-le" where r = rT > 0, e" = 8 - 8, and 8 is an estimate of 6' to be determined by the parameter adaptation rule. The derivative of V along the trajectories of the system is given by

V = - eTQe + 2eTPb[fo + QTf + (go + BTg)v

+ Ke - yp)] + ijTr-lO. Taking

-Ke+y?) - f o ( e + Y T , z ) -8* f ( e+Y , , z ) ' U =

go + 0Tg

(1 1) dlf - +(e , 2, YR, 0)

and setting

4 = 2eTpb[f(e + Y,, 2) + g+(e, z , Y R , @I = +(e, z , Y E , 8) (12)

we can rewrite the expression for V as

V = -eTQe + BTr-'[i - r+]. The parameter adaptation rule should be chosen to ensure that

e"Tr-$ - r411 o (13)

and 0(t) E R for all t 2 0. This can be achieved by standard adaptation rules with parameter projection; c.f. [ 191. The parameter projection of [19] results in an adaptation rule with discontinuous right-hand side. To alleviate this difficulty we use the smoothed projection idea of [20]. We will give two choices of the adaptation rule for the cases when R is a closed ball and a convex hypercube, respectively. Suppose R = {OT8 5 k } and let RA = { Q T 8 5 k + 6}, where 6 > 0 is chosen such that RA c 6. Define the projection Proj(8,d) by

b 4 , otherwise

where

On the other hand, suppose R is the convex hypercube

R = { B I a, 5 8, 5 b,}, 1 5 i 5 p } .

Let

526 = (6' I U ; -6 5 8, 5 b, + S } , 15 i L p } where S > 0 is chosen such that R,s c 6, and choose r to be a positive diagonal matrix. In this case the projection Proj(B,+) is taken as

'yZz4,, if a, I 4, I b, or if 8, > b, and 4, 5 0 or if 8, < a, and 4, 2 0

if 0, > b, and 4, > 0 I yi;&, if 0, < U+ and 4, < 0

(15) yi,&,

[proj(8,#)1i =

where

The choice of S such that 0 6 c fi ensures that (go + BTg) # 0 V 8 E 526. The adaptation rule is taken as

0 = Proj(0,+). (16)

It can be verified that this rule satisfies (13)

8(0) E R =+ 8(t) E RA, v t 2 0

and Proj(8,+) is locally Lipschitz in (e, 4 ) . The closed-loop system is defined by (8), (9), (1 l), and (16). Since the right- hand side function is locally Lipschitz in the state variables, for any given initial state, the system has a unique solution defined on some interval [O,To). Let [O,T) be the maximal interval of existence of this solution. Since (13) ensures that V 5 0, its components e and e" are bounded on [O,T). Since y, is bounded, IC is bounded on [0, T ) . From Assumption 2, we conclude that z is bounded on [0, T ) . This implies that T = ca. The boundedness of 0 follows also from the fact that 8 E RA. With all signals bounded, we conclude from V 5 -eTQe and the invariance theorem [21, Theorem 4.81 that

e ( t ) + 0 as t --+ ca. (17)

In preparation for the output feedback controller, we will saturate our control w and the vector 4 outside the domain of interest. We assume that all the initial conditions are bounded. In particular, O(0) E R, e(0) E Eo, and z ( 0 ) E 20, where Eo and 20 are compact sets. The sets Eo and ZO can be chosen large enough to cover any given bounded initial conditions, but once they are chosen we cannot allow initial conditions outside them. We have already seen that 0(t) E 0 6 for all t 2 0. Let

e1 = maxeTPe, and e2 = max (e - 6')TlT1(8-6') eEEo BEO2,8En;2s


KHALIL: ADAPTIVE OUTPUT FEEDBACK CONTROL 181

and c3 > c1 + c2. Then e(t) E E ef {eTPe 5 c3) for all t 2 0. With the bounds on YR(t) and 8 known, we can, in view of Assumption 2 and (3), find a compact set 2 such that

z(0) E 20 and e( t ) E E V t 2 0 + z ( t ) E 2 V t 2 0.

The control input w = $(e , z ,YR,d) defined by (11) and the vector q5(e, z , YR, e) defined by (12) are continuous functions of e,YR, z , and 0. Hence, they are bounded on compact sets of these variables. Let

where the maximization is taken over all e E El %f {eTPe 5 c4}, z E 2, YR E YR, 8 E fl6, where c4 > c3. Define the saturated functions $" and @ by

for 1 5 i <_ p , where sat(.) is the saturation function defined by

-1 for y < -1

1 for y > 1 y for IyI 5 1 .

For all e(0) E EO, z(0) E 20, and 8(0) E 0, we have 17111 6: S and Iq5il I Si for all t 2 0. Hence the saturation functions will not be effective, and the state feedback adaptive controller with $ and q5 replaced by 4" and 6 will result in the same performance.

IV. PARAMETER CONVERGENCE UNDER STATE FEEDBACK

In this section we show that under an additi:nal persistence of excitation condition, the parameter estimate e( t ) approaches the true unknown parameter vector 8. Let us start by stating the additional assumptions.

Assumption 3: For any 8 E R, let c(t) be the solution of (4) when x ( t ) = Yr(t) and z (0 ) = 0. For any bounded ~ ( t ) such that x( t ) + Yr(t) as t + 00 and any x(0) E 20, the state ( ( t ) + c(t) as t + 00.

Using (3) to define Z ( t ) as

15 i 5 m (20) Y P m + j - l ) %(t) = li(t) +

go + BTg it follows that

It can be verified that %(t) = G ( t ) is the solution of the mth-order differential equation that results from (1) when y ( t ) f yr(t) and d i ) (0) = 0, 0 5 i 5 m - 1. It can be also seen that Z i ( t ) = In the special case when (4) takes the form

with a Hunvitz matrix A0 and a continuous function Fc,, as in the class of systems treated in [11] and [12], Assumption 3 is always satisfied.

Assumption 4: The reference signal gyr ( t ) has the additional properties that y?+l)(t) is bounded' for all t 2 0 and, for all 0 E Q, the signal

is persistently exciting2

as The closed-loop system under state feedback can be written

6 = Ame - beTw(t)

8 = 2f'(t)w(t)bTPe (24)

(23)

(25) i = A22 + bqq

where

w(t) = f ( e ( t ) + Yr(t), z ( t ) ) -k g$(e(t>, x(t),yR(t)i i ( t ) ) (26)

and f ( t ) is a bounded, locally integrable function of t , determined by the adaptation rule (16). Since we have already established that all variables are bounded, we know that t: and 8 are bounded. Moreover, using the assumption that y?+l)(t) is bounded, it can be shown that e is bounded. It follows from (17) and Barbalat's Lemma [21, Lemma 4.41 that

t:(t) 3 0 as t -+ 00. (27)

Using (17), (21), and (27) in (lo), we obtain

as t --t 00. Therefore

w(t) -+ wT( t ) as t + 00 (28)

which implies that w( t ) is persistently exciting.

varying system We rewrite (23) and (24) as the homogeneous linear time-

Am -bwT( t )] [i]. (29) 2f'(t)w(t)bTP 0

[I] = [ The derivative of V = eTPe+ +&'e" along the trajectories of (29) satisfies V 5 0. By an argument similar to the one used in the traditional adaptive control theory (see for example [21, Sections 4.4 and 5.6]), we can use the persistence of excitation of w ( t ) to show that

'This requirement can be weakened by requiring that yp) ( t ) belong to the class of signals defined in [22] which have bounded continuous derivatives almost everywhere on [0, m).

2See 121. Section 5.61 for the definition of oersistence of excitation.


182 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 2, FEBRUARY 1996

for some k > 0 and y > 0; hence

8(t) --+ 0 as t + 00.

Our conclusions for the closed-loop system under state feed-

remains in 0 6 for all t > 0, whenever 8(0) E R, even when e is replaced by Z. By taking

back are summarized in the following theorem. and J = [cl, . . . , & I T , we represent the closed-loop system in the standard singularly perturbed form Theorem I : Suppose that Assumptions 1 and 2 are satisfied

and consider the closed-loop system (23)-(25). Suppose that 8(0) E R, e (0 ) E Eo, and z (0 ) E 20. Then, all the state variables of the closed-loop system are bounded and the tracking error e ( t ) converges to zero. If Assumptions 3 and 4 are satisfied as well, then the tracking error e ( t ) and the parameter estimation error e"@) satisfy the exponentially decaying bound (30).

V. OUTPUT FEEDBACK CONTROL To implement the state feedback adaptive controller using

output feedback, we need to estimate e. Note that z is already available since its components are obtained by integrating the control signal U. With the goal of recovering the performance achieved under state feedback, we use the high-gain observer

8, = &+I + (a . / t i ) (e l - h), 1 < i < n - 1

e, = (an/Cn)(e l - e l )

where E is a small positive parameter to be specified. The positive constants cy, are chosen such that the roots of

sn + a1sn- l+ . . ' + an- ls + a, = 0 (32)

have negative real parts. To eliminate the peaking phenomenon associated with this high-gain observer, we use the saturated function 111" and @, as defined by (18) and (19), respectively. To eliminate peaking in the implementation of the observer, let

(33)

Then

6 = A,e + b { K e + f o ( e + Y T , z ) + Q T f ( e + Y,, z ) + (go + 6 " g ) V ( e - D(c)E, z , YR, e) - Y6")) (41)

(42)

(43) €4 = ( A - H C ) J + tb{fo(e + Y T , z ) + O T f ( e + Y T , z )

(44)

where H = [a1,. . . , cy,]', and D ( E ) is a diagonal matrix with en-' as the ith diagonal element. The characteristic equation of (A-HC) is (32); hence it is Hurwitz. The singularly perturbed system (41)-(44) has an exponentially stable boundary-layer model, and its reduced model is the closed-loop system under the adaptive state feedback controller. The t-dependent scaling (40) causes an impulsive-like behavior in E as E -+ 0, but since enters the slow equations (41)-(43) through bounded functions 4' ( e - D( e)<, z,, Y E , 8) and 4' ( e - D( E)(, z , YR, e), the slow variables ( e , z , 6') do not exhibit a similar impulsive like behavior.

The first part of our analysis is to establish that there is a short transient period during which the fast variables decay to O ( E ) values, while, due to the difference in speed, the slow variables remain within a bounded subset of the region of attraction. This part of our analysis has a lot of similarity with our previous work [2], [4], [7]; therefore we do it briefly. Let R, = {e E E } x { z E 2) x {e E 0 6 ) . Since V(e (O) , &O)) 5 c1+ c2 < c3 and 111' in (41), (42) is bounded uniformly in E , there exists a finite time T2, independent of E ,

such that for all t E [O,Tz], z ( t ) E 2 and V(e ( t ) , e " ( t ) ) 5 c3. Taking W = {'PE, where P = PT > 0 is the solution of the Lyapunov equation

z = A ~ Z + bz?lfs(e - D ( ~ ) E , z , Y R , 8) 8 = Proj (8 ,4"(e - D ( ~ ) J , z , Y R , 8))

+ (go + 6''g)V(e - D( t ) l , 2, YR, 8) - Y p }

E q n = an(e1 - 41)

The system (34) is a standard singularly perturbed model and will not exhibit peaking if the input el and the initial conditions are bounded functions of t. In summary, the output feedback adaptive controller is given by

'U = $"(2,z, YR, 8) 8 = Proj(8, $'(E, z , JJ,, 8))

(35)

(36) 2% = %+1, l < i < m - 1 (37)

zm = U (38) U = z1 (39)

andusing the boundedness of [fo(e+Y,,z)+OTf(e+YT,z)+ (go + QTg)$" - g?'] over R,, it can be shown that for any TI E (0, $T2] there exists EO > 0 such that for all E E (0, E O ]

and all t E [ T I , ~ T ~ ) , W ( [ ( t ) ) 5 ~'0, where T3 2 T2 is the first time ( e , z , 8 ) exits from the set R,; T3 may be 00. The time TI is a function of E and T I ( € ) t 0 as t -+ 0.

In the second part of the analysis, we study (41)-(43) over the time interval [TI, T3). Using the fact that E is O ( E ) , it can be shown that there exists a positive constant IC such that for all E E (0, E O ] , the derivative of V = e T P e + ;e"'r-'e" along the trajectories of (41)-(44) satisfies

V 5 -eTQe + I C E 5 -coeTPe + kt = - c o v + ; coe"Tr-18 + Ek

where Proj(Q,q$) is defined by (14) or (15), depending on the set R, 2 is given by (33) and (34), and qS and #' are defined by (18) and (19), respectively. It can be verified that the parameter

where CO = AmZn(Q)/Amaz(P). For all e E Rs, we have igTr-le" 5 c2. Hence

projection feature of the adaptation rule guarantees that 6 ( t ) V 5 -cov + cocz + Ek.



On the boundary V = c3 we have V < 0 whenever c3 > c2 + ek/co. Since c3 was chosen strictly greater than c1 + c2, we cqnclude that for sufficiently small E , the set {V 5 cg} n (6' E n6) is a positively invariant set. Inside this set e E E. As long as e E E , z will remain in 2. Thus the trajectory ( e , z , 6) is trapped inside R, = { e E E } x { z E 2) x {e E as). Therefore, T3 = 00, and we conclude that all the state variables of (41)-(44) are bounded for all t 2 0 and is '('1 for 2 ' 1 . Moreover, since e(t) to E = r: C3}r the functions '@ and 4' are saturated outside El = {e'pe 5 c4) where c4 > c3, we conclude that for sufficiently small e the saturation components of q5' and qY will not be effective for t 2 TI. We note that the inequality V 5 -eTQe + kc and the boundedness of V

are bounded and the mean square tracking error is of order O ( E ) . If Assumptions 3 and 41 are satisfied as well, then (45) is satisfied, showing that

(46) z ( t ) - Z ( t )

e(t) - e ( t )

Moreover, as E + 0, the trajectories (e@), z ( t ) , e(t)) approach the ones that would have been achieved had the controller been implemented using e l , d l , . . . , instead of their estimates, and this is uniform in for all t l , for any fixed tl > o.

imply that 4 P I

which shows that the mean-square tracking error is of order

In the third, and last, part of the analysis we use the persistence of excitation condition to show convergence results. Suppose that Assumptions 3 and 4 are satisfied. In the previous section we saw that under state feedback control e ( t ) and 6( t ) satisfy the exponentially decaying bound (30). By singular perturbation analysis of the closed-loop system under output feedback, it can be shown that, for sufficiently small E , e ( t ) , 6( t ) , and [ ( t ) satisfy the exponentially decaying bound

' ( E ) .

for some k1 > 0, y1 > 0, and any tl > ' 1 . The singular perturbation analysis leading to (45) represents a mild variation of the standard analysis that can be found in the singular perturbation literature, but it does not follow from that literature because we do not require z ( t ) - Z ( t ) to decay exponentially. Another technical complication is that the right- hand side function in the current case is only locally Lipschitz and not continuously differentiable, as it is usually assumed in the singular perturbation literature. For completeness, we give the singular perturbation argument needed to arrive at (45) in Appendix A. Inequality (45) shows in particular that e ( t ) --t 0 as t + m, uniformly in E , which in view of Assumption 3 implies that z ( t ) + %(t) as tA+ 00, uniformly in E . The limiting behavior of (e( t ) ,z( t ) ,O(t)) as t + m, together with the fact that (' is ' ( E ) for all t > 7'1, show that ( e ( t ) , z ( t ) , e(t)) under output feedback approach their corresponding trajectories under state feedback as E + 0, uniformly in t for all t 2 tl for any fixed tl > 0.

VI. EXAMPLES

We illustrate the adaptive controller of the previous section via two examples: a second-order linear example from [13, pp. 147-1511 and a third-order nonlinear example from [23]. We chose both examples from the adaptive control literature to have a reference against which we can judge the performance of the proposed adaptive controller.

Exumple2: Consider a linear system represented by the transfer function

G(s) = A

where a and c > 0 are unknown constants, and the reference model

s(s + a )

We want to design an adaptive controller such that the output of the system y asymptotically tracks the output of the reference model y, for any "well-behaved" command signal r(t). A traditional model reference ad,aptive controller (MRAC) design is given in [13, Example 4.81. The system can be represented by the second-order differential equation

y = --ay + c u

which takes the form of (1) with 6' = [a c]'. We assume that -1.9 5 a 5 1.9 and 1.1 5 c 5 2.9 so that 6' belongs to a compact convex hypercube R, and we take S = 0.1. Assumption 1 is satisfied for all 6' E Q6. Assumption 2 is satisfied by default since: there are no zero dynamics. Moreover, we do not add integrators at the system's input; hence w = U. The reference signal y, is generated from the command signal ~ ( t ) using the second-order differential equation

We can now summarize our conclusions in the following yr = -25wyr - w2y, + W 2 T ( t ) . theorem.

Theorem 2: Suppose that Assumptions 1 and 2 are satisfied and consider the closed-loop system (41)-(44) formed of the plant (1) and the output feedback adaptive control (35)-(39). Suppose that e(0) E R, e(0) E EO, z(0) E 20, and q(0)

0 < E < E * , all the state variables of the closed-loop system

We assume that ~ ( t ) and T ( t ) are bounded, so that yT, y,, yT, and d3' are bounded. Persistence of the excitation condition (Assumption 4) requires

is bounded. Then there exists E* > 0 such that for all wr(t) =



to be persistently exciting for all -1.9 5 a 5 1.9 and 1.1 5 c 5 2.9. It is not hard to see that this will be the case if and only if

is persistently exciting. The latter condition holds if ~ ( t ) is a stationary signal that is sufficiently rich of order two [15, Theorem 2.7.21. To design the state feedback controller of Section 3 we take e1 = y - yr and e2 = y - yr. We choose K = [w2 2(w] so that

is Hunvitz and its eigenvalues are the poles of the reference model Gm(s). The matrix

satisfies the Lyapunov equation PA, + AZP = -Q for some Q > 0. The functions $ and 4, defined by (11) and (12), are given by

where ii and 6 are the parameter estimates. Since R is a convex hypercube, we use (15) with 711 = yzz = y > 0. Taking U = $ completes the description of the adaptive state feedback controller. To proceed toward the output feedback controller, we need to saturate ?l,, $1, and $2 outside the region of interest. Suppose that Eo = { e I / e l / 5 1 and le21 5 l} includes the sets of initial conditions of interest to us. Suppose also that 5 = 0.7 and w = 1. It can be verified that c1 = maxeEEo { e T P e } 5 7.4~ and c2 = maxocn,Jcn{ 118 - B112/2y} 5 1O/y. Choosing Q = 0.5, y = 10, and c4 = 4.8 defines the region of interest for e as El = { e T P e 5 cq}. By calculating upper bounds on I$/, 1411, 1421 for e E El and 8 E %, IyT(t)l I 1.5, and Iyr(t)l 5 1.5, we choose the saturation levels of 4, 41, and $2 as S = 10.1, SI = 10, and SZ = 32.2, respectively. The state estimates are taken as 61 = 41 and & = Q Z / E , where 41 and qz satisfy the singularly perturbed equation

€41 = q 2 + e1 - 41

4 2 = el - 41

and t > 0 is small. By replacing el and e2 in the expressions of $ and 4 be their estimates 61 and 62, the description of the adaptive output feedback controller becomes complete except for the choice of E . Theorem 2 guarantees certain properties if E is chosen small enough. Typically, we do not choose E based on the analysis of Theorem 2 because such a choice would be conservative. Instead, we rely on simulations to choose e. We start with a certain value of E and then decrease it in steps until acceptable performance is achieved. Theorem 2 guarantees that

-2 0 5 10 15 20 25 30 35 40 45 50

(a)

1, h I

Time

(b)

Fig. 1. Simulation results for Example 2 with a = 1, c = 2 (known), E = 0.01, and zero initial conditions. (a) Output y and reference signal yr. @) Control U .

0.05 I

-0.15l 0 2 4 6 8 10 12 14 16 18 20

(a)

Estimate of a

0.5 l':L!?EZl 0 2 4 6 8 10 12 14 16 18 20

Time

(b) Fig. 2. Simulation results for Example 2 with a = 1, c = 2 (known), E = 0.01, and zero initial conditions. (a) Tracking error e = y - yr. (b) Parameter estimate 2

this tuning procedure will succeed. If an upper bound E* can be conveniently calculated from the analysis, we will use it as a starting point, but we will increase E in steps until we reach the maximum value of E beyond which performance will not be acceptable. Of course, in choosing E we are interested in the largest value that will do the job because the bandwidth of the observer and some of its coefficients are proportional to 1 / ~ .

Figs. 1 and 2 show simulation results for a = 1, c = 2, E = 0.01, and zero initial conditions, when c is known; that is, the adaptation for c is turned off.

The command signal ~ ( t ) is a square wavefom3 Simulation 3?( t ) is bounded almost everywhere.



2

-2

0.5 e=y-y-r

2

1 -

0-

-1

-2.

results for the same case using a traditional MRAC are given in [13]. A comparison of our results with those of [13] shows that the performance achieved by our controller is as good as the one achieved by the MRAC, if not better. Of course we do not imply that our controller is superior to the MRAC, because in either case the performance achieved is dependent on several parameters chosen in the course of the design. It might very well be the case that one can go back and redesign the MRAC to produce performance better than the one reported here. The point we want to make, however, is that at least in this example, it has been possible to design our controller to produce performance comparable to what is achieved by other designs. It should be mentioned also that the MRAC controller of [13] uses a ninth-order filter in addition to the parameter adaptation, while our controller uses a second-order observer in addition to the parameter adaptation.

The simulation of Figs. 1 and 2 might not be a good repre-

Estimate of a -1

-2

-

P

(a)

0.05 0.1 0.15 0.2 0.25 0.3 ;;E-/ 0

-20 0 0.05 0.1 0.15 0.2 0.25 0.3

Time

sentation of our controller because with zero initial conditions there is no peaking in the high-gain observer. TO give a better picture of the performance of the controller, we show in Figs. 3-5 the simulation results for a = -1, c = 2, E = 0.01,

(b) Fig. 5. Simulation results for Example 2 with a = -1, c 2 (unknown), E = 0.01, and zero initial conditions except y(0) = 1 and &(0) = 1.1. (a) Control U . (b) State e2 and its estimate 62.

and all initial conditions equal to zero except y(0) = 1 and &(o> = 1.1. w e also assume that c is unknown, so the adaptation for c is used.

The nonzero initial condition of y induces peaking, and the choice a = -1 makes the open-loop system unstable. We think the figures speak for themselves, but we would like to draw attention to Fig. 5 which shows the control input U and the state estimate 22 over a short period of time. Fig. 5(b) illustrates the peaking in 22 that is induced by y(0) = 1. During this peaking the control U saturates at the predetermined saturation level S = 10.1, thus preventing this peaking from being transmitted to the plant. Once this short period passes, the estimation error becomes small and the saturation mechanism becomes idle.

Example 3: Consider the nonlinear system

61 = (2 + 6'6; i 2 = U + 5 3

53 = - 4 3 + Y

y = <l

treated in [23], where 6' is an unknown constant parameter. The system can be represented by the third-order differential equation

= (U + y - i ) $- 28(y$ + y2 + yji) + i which takes the form of (1) with n = 3 and m = 1. We add an integrator at the input of the system, take z = U as the state of



the integrator, and set v = U as the new control input. It cm be seen easily that Assumptions 1-3 are satisfied. Assumption 4 requires yS4'(t) to be bounded and

to be persistently exciting. This will be satisfied, for example, with y,(t) = asinwt, but not with y,(t) constant. We define el = y - yr, e2 = ?j - y r , and e3 = y - y,. The matrix K is chosen as K = [2 4 31 to assign the eigenvalues of A, = ( A - bK) at -1 and -1 ij. The matrix P is obtained by solving the Lyapunov equation PA, + AZP = -I. The functions $ and q5 are given by

$ = -Ke - U - y + y - 28(yy + y2 + ye) + yi3)

$ = 4(eTpb)(yy + y2 + yy).

We assume that B belongs to R = { B 1 0 5 B 5 2) and take 6 = 0.1. We use the projection (15) with y = 10. Noting that maxHEbl,~Ebls{&(8 - B ) ' } = 0.2205, we choose e4 = 2.4 which allows for a choice of c1 = 2 and the set of permissible initial conditions of e will be Eo = {eTPe 5 el}. The region of interest for e is El = {eTPe 5 2.4). Using the zero dvnamics eauation

Tracking error

0.5

-0.5' I

(a)

2 4 6 8 10 12 14

I 0 2 4 6 8 10 12 14

Time

(b)

Fig. 6. Simdation results for Example 3 with 6 = 1, yr = 0.1 s int , E = 0.01, and zero initial conditions except y(0). (a) Trackmg error e = y - y,. for y(0) = 0.45 and y(0) = 1. (b) Control U for y(0) = 1.

5. = -c - [y + 2B(yY + y2 + yy)]

where = z - y, taking z (0) = 0, and assuming that the magnitude of y, and its first two derivatives are bounded by 0.1, it can be shown that for all e E El, IzI is bounded by 33. Then, by calculating upper bounds on ?(I and Q, for all e E El , IzI 5 33 and 8 E &, the saturation levels S = 75 and SI = 25 are chosen for $ and q5, respectively. The state estimates are given by 81 = 41, = q2/E, and &, = q3/e2, where 41, q2, and q3 satisfy the equation

€41 = q2 + 2(e1 - 41) €42 = q3 + 4(el - q1) €43 = 3(el - 41).

The output feedback controller is obtained by replacing e, in $ and q5 by their estimates E % , and taking v = $' and U = v.

Figs. 6-8 show simulation results for 6' = 1, y, = O.lsint, t = 0.01, and zero initial conditions for all variables except y(0). Simulation results under the same conditions are given in [23]. A comparison of the tracking error of Fig. 6 with the results of [23] shows that we have obtained comparable results despite the fact the controller of [23] uses full state feedback, while ours uses only output feedback from y. Fig. 7 illustrates again the effective saturation mechanism during peaking of the state estimates. It shows w, the control input of the extended system, on a short period as well as on a long one. The signal w goes through four cycles of saturation before peaking diminishes .

Theorem 2 confirms that the performance of the closed-loop system under output feedback approaches the performance achieved under state feedback as E approaches zero, where state feedback here means measurement of y, c, and y. Fig. 8 illustrates this point by showing trajectories of y and U for both state and output feedback.

- - I

-1001 0 0.05 0.1 0 15 0.2 0.25 03

(a)

-20' I

(b)

Fig. 7. Simulation results for Example 3 with 6 = 1, yr = 0.1 sint, E = 0.01, and zero initial conditions except y(0) = 1: Control input of the extended system.

2 4 6 8 10 12 14 Time

VII. CONCLUSIONS

This paper has extended the theory of adaptive output feedback control of nonlinear systems. We have relaxed some of the restrictions imposed in [ 111 and [ 121. In particular, we have allowed the right-hand side of (1 ) to depend nonlinearly on U instead of the linear dependence used in [l l] ; see (5) and (7). We have also relaxed the requirement that the zero dynamics be linear and exponentially stable. Similar to [ 111 and [12], we do not impose global growth conditions on the nonlinearities. While the results of [ l l ] and [12] are global, however, our results are only semiglobal. We do not see this as a serious drawback since practical controllers will have


KHALIL: ADAPTIVE OUTPUT FEEDBACK CONTROL 187

21 ,-. I I

I 2 4 6 8 10 12 14

-0.5’

(a)

i I - r-’

I 2 4 6 6 10 12 14

Time

(b)

Fig. 8. Simulation results for Example 3 with 6’ = 1, yr = O.lsint, E = 0.01, and zero initial conditions except y(0) = 1: Comparison of state (solid) and output (dashed) feedback. (a) Output y. (b) Control U.

to be globally bounded anyway. A second difference is that our design uses a priori information (bounds on parameters, on initial conditions, and on the reference signals) which is not required in [ l l ] and [12]. A third, and more important, difference is that our result requires persistence of excitation not only for parameter convergence, but even for tracking error convergence. This is an unusual requirement in adaptive control results, including those of [ 113 and [ 121, where tracking error convergence is shown without persistence of excitation. We have shown that in our case the mean-square tracking error will be of order O(E) without persistence of excitation. Since E is a design parameter, we can, in essence, make the mean- square error arbitrarily small. This, however, is weaker than showing error convergence. This drawback of our result may be partially balanced by the fact that we provide a verifiable persistence of excitation condition for parameter convergence; a subject that was not pursued in [ll] or [12].

The new adaptive controller proposed here is simpler than the traditional output feedback adaptive controllers, since we do not use filtering or error augmentation ideas. Our controller is simply a state feedback design with a linear observer. Our simulations show that the performance of the new controller is comparable to the performance of traditional controllers. What could be a potential problem with our controller is the fact that we have to estimate the saturation levels used to saturate the control and the right-hand side of the adaptation rule outside the domain of interest. Those estimates could be conservative which would force us to use a very small value of E in the observer. This would be a potential problem because in practical implementation E will be limited by various factors, like sampling rates, finite word length in computers, and unmodeled high-frequency dynamics. These issues as well as the robustness of our adaptive controller to bounded disturbances and unmodeled dynamics will be the subject of future research.

After the first draft of this paper was written, we became aware of the work in [3] and [24]. Paper [24] presents a completely different adaptive control approach to the class of systems represented by (1). Paper [3] considers a single- input-single-output nonlinear system represented by

xi = 2,+1, i 5 T - 1 x, =z2fl+$2(21,...,2T)+~T(51,...,5r)B, i 2 T

i n = P(51,. . 7 &)U + $ n ( m , . . . , $ 7 ) + C ( Z 1 , . . . , 5 T ) O

y = 2 1

where 1 5 T L: n, 0 E RP is a vector of unknown parameters and IP(-)l > 0. It can be verified that the system has an input-output model of the form

y(”) = p(y,. . . ,y(r-l))U + &/, . . . ,p)) + $T(y, * . . , #-1))0.

Except for the fact that ,O is allowed to depend on y(’) to this input-output model is a special case of (1) with

no uncertainty in the input coefficient and no zero dynamics. Paper [3] derives an output feedback adaptive controller using a high-gain observer, combined with saturation, to estimate the states 22 to zr. This is thie common idea between [3] and our paper. Otherwise, the design approaches are completely different. Paper [3] uses a modified version of the observer- based identification scheme of [25]. The design is not based on a “separation principle” like ours. The result of [3] is semiglobal and requires a persistence of excitation condition for tracking error and parameter error convergence. Unlike our result, however, the persistence of excitation condition of [3] is dependent on the operation of the adaptive system.

APPENDIX A Consider the system

(47)

where F and G are piecewise continuous in t and locally Lipschitz in ( z , z ) , uniformlly in t for all t 2 to. Suppose that for every ( z 0 , x o ) E 620 (a compact set) the solution ( ~ ( t ) , ~ ( t ) ) starting at ( t o , q], zo) satisfies

(z( t ) ,z( t ) ) E R (a compact set) for all t 2 t o . Ilz(t)II I Ice--y(t-to)llxOII, v t 2 to .

Then there is a function V(t , 2, z ) such that

x i = = F(t1x~9 G(t, x,.)

II4I 1 V ( t , z, 2) I ~1141 (48) I V ( t , Z , Z ) -- V ( t , Z , Z ) ) I5 LlllZ - 311 + L2112 - 511 (49)

(50)

for all (z ,x ) E Ro, (Z,,T) E: Ro and t 2 to, where L1, Lz , and X < y are positive constants. The notation q4q denotes the derivative of V along thie trajectories of (47). The proof of this claim is similar to the proof of the standard converse Lyapunov theorem for exponential stability; see for example [26, Theorem 19.21. The function V ( t , x,.) is taken as

$47)(t, 2, .) rS -XV(t, 5 , .)

V ( t , 2, .) = SUP Il$l(t + T, t , 5 , 411eX‘ 720


188 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO 2, FEBRUARY 1996

where 41 (t + 7 , t , z, z ) is the 2-component of the solution of (47) that starts at (t , 2 , z ) , and X < y is a positive constant.

Consider now the singularly perturbed system

} (51) x 2 ~ l j = AY + ~ H 3 ( t , 2 , z , y, E )

= F ( t , 2, 2) + H l ( t , 2, z , y, €) = G(t , 2, 2) + Hz( t , 2, z , y, E )

where A is Hurwitz. Suppose the following additional conditions are satisfied:

For every (xo,zo, yo) E f20 x ro (a compact set), there is

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[I21 R. Marino and P. Tomei, “Global adaptive output-feedback control of nonlinear systems, part I: Linear parameterization,” IEEE Trans. Automat. Contr., vol. 38, no. 1, pp. 17-32, Jan. 1993.

[13] K. J. Astrom and B. Wittenmark, Adaptive Control. Reading, MA: Addison-Wesley, 1989.

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1151 S. Sastry and M. Bodson, Adaptive Control. Englewood Cliffs, NJ: Prentice-Hall. 1989.

a unique solution ( ~ ( t ) , ~ ( t ) , y(t)) E 1;2 x r (a compact

* \\Hi\\ 5 k i \ \ y \ \ , \\Hz\l 5 k, \ \y \ \ , and \)H3\\ 5 k3) ) z l ) + Taking W ( t , x , z , y ) = V ( t , z , z ) + Q d m , where

1161 H. J. Sussmann and P. V. Kokotovic, “The peakmg phenomenon and the global stabilization of nonlinear systems,” IEEE Trans Automat. Contr , vol. 36, no. 4, pp. 424440, Apr. 1991.

[17] A. Isidori, Nonlinear Control Systems, 2nd ed. New York Springer- Verlag, 1989.

[I81 M. W. Spang and M. Vidyasagar, Robot Dynamics and Control. New York Wiley, 1989.

[I91 G. C. Goodwin and D. Q. Mayne, “A parameter estimation perspective of continuous time model reference adaptive control,” Automatica, vol.

[20] J.-B. Pomet and L. Praly, “Adaptive nonlinear regulation Estimation from the Lyapunov equation,” IEEE Trans. Automat. Contr., vol. 37,

set) for all t 2 t o .

bl ly l l , for all ( z , z , y ) E 0.

PA + ATP = -I and a > 0, it can be shown that a 23, pp. 57-70, 1987.

2 E d m J‘yJJ w(51) 5 -AV + (Llkl + L2kZ)\\y\\ -

a! pp. 729-740, June 1992. l lP l l (~311~11 + ~ 4 l l Y l l > 1211 H. K. Khalil, Nonlinear Systems. New York. Macmillan, 1992.

1221 J. S-C. Yuan and W. M. Wonham, “Probing signals for model reference identification,” ZEEE Trans. Automat. Contr , vol. AC-22, no. 4, pp. 530-538, Aug. 1977.

[23] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizahle systems,” IEEE Trans. Automat. Contr., vol. 36, no. 11, pp. 1241-1253, Nov. 1991.

[24] H. B. Penfold, I. M. Y. Mareels, and R J Evans, “Adaptively controlling nonlinear systems, using trajectory approximations,” Znt. J. Adaptive Contr. Signal Processing, vol. 6, pp. 395411, 1992.

[25] M. Krstic, P. V. Kokotovic, and I. Kanellakopoulos, “Adaptive nonlinear output-feedback control with an observer-based identifier,” in Proc. Amer. Contr. Con$, San Francisco, CA, June 1993, pp. 2821-2825.

[261 T. Yoshawa, Stabil@’ Theory bY LzaPunOv’s &condMethod Tokyo: The Mathematical Society of Japan, 1966

Jxzm + and there are positive constants a and E* such that for all 0 <E< E * , T/ii(51) 5 - $Aw. Thus there exist positive constants k l and y1 such that

Z( to ) , tr t 2 t o . 1 1 [;;:;I 1 1 k 1 e - q [,(to)] I1 ACKNOWLEDGMENT

The author is grateful to M. jankovic and 1. K ~ ~ ~ ~ & ~ ~ ~ ~ ~ ~ ~ for pointing out errors in preliminary drafts of this paper. He also acknowledges L. Praly for many helpful suggestions, including suggesting the use,of (27) in the proof.

I

Hassan K. Khalil (S’77-M’78-SM’SS-F’89) received the B.S and M.S degrees from Cairo Uni- versity, Cairo, Egypt, and the Ph.D. degree from the University of Illinois, Urbana-Champaign, in 1973, 1975, and 1978, respectively, all in electrical engineering.

Since 1978, he has been with Michigan State University, East Lansing, where he is currently Professor of Electrical Engineering He has been a Consultant for General Motors and Delco Products He has published several papers on singular per-

turbation methods, decentralized control, robustness, nonlinear control, and adaptive control. He is author of the book Nonlinear Systems (Englewood Cliffs, NJ: F’rentice-Hall, 1996), coauthor of the book Szngular Perturbation M&“.Y In control: Analysis and DeWF (New York: Academic, 1986), and coeditor of the book Singular Perturbation in Systems and Control (New York:

Dr. Khalil was the recipient of the 1983 Michigan State University Teacher Scholar Award, the 1989 George s. Axelby Outstanding Paper Award of the LEEE TRANSACTIONS ON AUTOMATIC CONTROL, the 1994 Michigan State University Withrow Distinguished Scholar Award, and the 1995 Michigan State University Distinguished Faculty Award He served as Associate Editor of IEEE TRANSACTIONS ON AUTOMATIC CONTROL, 1984-1985; Registration Chairman of the IEEE-CDC Conference, 1984; CSS Board of Governors, 1985; Program C o m t t e e member, IEEE-CDC Conference, 1986; Finance C h m a n of the 1987 American Control Conference (ACC); Program Chair- man of the 1988 ACC; Program Committee member, 1989 ACC; and General Chalr of the 1994 ACC. He is now serving as Associate Editor of Automatzca.

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[2] F Esfandiari and H. K. Khalil, “Output feedback stabilizaton of fully linearizable systems,” Int. J. Contr., vol. 56, pp. 1007-1037, 1992.

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[5] H. K, Khalil and F, Esfandiari, ‘cSemglobal stabillzaton of a ,-lass of nonlinear systems using output feedback,” IEEE Trans. Automat. Contr., vol. 38, no. 9, pp. 1412-1415, 1993.

[6] Z. Lin and A. Saberi, “Robust semi global stabilization of minimum- phase Input-output lineaizable systems via partial state and output feedback,” IEEE Trans. Automat. Contr., vol. 40, no. 6, pp. 1029-1041, 1995.

[7] N. A Mahmoud and H K Khalil, “Asymptohc regulation of minimum phase nonlinear systems using output feedback,” in Proc. Amer. Contr. Con$, San Francisco, CA, June 1993, pp. 1490-1494.

[SI A. Tee1 and L. Praly, “Tools for semiglobal stabilization by partial state and output feedback,” SIAM J. Contr. Optzm., 1995.

[9] -, “Global stabilizability and observability imply semi-global stabilizability by output feedback,” Syst. Contr. Lett., vol. 22, pp. 313-325, 1994.

l986).


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