A performance comparison between two design techniques for non linear output feedback control

A performance comparison between two design techniques for non-linear output feedback control

For a system possessing a non-linear output feedback normal form, an observer backstepping design is compared to a high gain observer design with respect to non-singular performance cost functional. If the initial error between the initial condition of the state and the initial condition of the observer is large, the high gain observer design is shown to have better performance than the observer backstepping design. An output feedback system with parametric uncertainty is then considered. It is shown that if ana prioriestimate for the bound of the uncertain parameter is conservative, then an adaptive observer backstepping design has better performance than the adaptive high gain observer design.

1. Introduction

In recent years several alternative constructive control techniques have been proposed for controlling non-linear systems using output feedback. In this paper we will be concerned with two major classes of control designs. The first class of controllers are based on high gain observers with saturated controls (see, e.g. Esfandiari and Khalil 1992, Khalil and Esfandiari 1993, Khalil 1996, Atassi and Khalil 1999). We refer to this class of control designs as Khalil designs. The second class of controllers are based on backstepping techniques (Kristic et al. 1995), and we refer to this class of controllers asKKKdesigns.

TheKhalildesigns are applicable to affine systems of full relative degree, whilst the KKK designs are appli-cable to an alternative class of systems, namely those which possess an output feedback normal form. By con-sidering systems which are both full relative degree and have an output feedback normal form, we can compare the behaviour of the controllers on common systems. (Note also that such systems are characterized in a coor-dinate free manner (Kristicet al. 1995).) This motivates the study of comparison of different designs, as initiated in Khalil (1999).

The results in Khalil (1999) are purely numerical, and give rise to many interesting questions, such as:

. When are the Khalil designs more sensitive to disturbances than the KKK designs, and vice versa?

. When do theKKKdesigns require greater control effort than theKhalildesigns, and vice versa? . When do theKhalildesigns have superior output

transients to theKKKdesigns, and vice versa?

In particular, by introducing suitable measures of performance and sensitivity we would like to be able to characterize situations in which one design is prefer-able to another. Such characterizations have obvious consequences for design choices, and also should give insight into the dynamics and trade-offs inherent in these controllers.

In this paper we will consider the latter two points, by considering a non-singular cost functional penalizing both the output transient and the control effort. It should be observed that whilst there are many results concerning the transient performance of the output (see, e.g. Kristicet al.1995), there is little work in the litera-ture on non-singular costs for non-optimal designs. See, however, French et al. (2000), French (2002) and Beleznay and French (2003) for related results and techniques.

In particular, for an output feedback system Swith inputuand outputy, and a controllerXmappingy!u, we consider the following cost which penalizes both the control and the output signal

PðS,XÞ ¼ ð

L2_ðT Þ

y2dtþ sup t2R

þ

juðtÞj

¼ kyk2_L2_ðT

Þþ kukL

1_ðR þÞ

where the time setT is defined by

T¼ t0jyðtÞj>

n o

andis a small positive number. Such a cost penalizes the input and output response of the system whilst

yðtÞ2 ½= ,, hence for a closed loop whose goal is to regulate y to zero, keeping y,u bounded, this cost is finite and is a reasonable penalty on the transient behav-iour. Note that whilst a directL2penalty on the output could be considered for the designs given, the relaxation of the output penalty is physically meaningful, and considerably simplifies the technical treatment.

In}2 we show that aKhalildesign out-performs with a KKK design when the information on initial state is poor and leads to a large initial observer error. In}3 we

International Journal of ControlISSN 0020–7179 print/ISSN 1366–5820 online#2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals

DOI: 10.1080/00207170310001657289

Received 15 April 2003. Revised 4 December 2003. * Author for correspondence. e-mail: [email protected]

establish a result in the reverse direction. We consider an output feedback system with an unknown parameter, and then show that an adaptive KKK design out-performs an adaptive Khalil design as the information on the size of the parameter becomes conservative.

2. Performance of output feedback system

In this section we study the performances of aKKK

design and aKhalildesign for a systemSðx0Þwhich can be expressed in the output feedback form

Sðx0Þ: xx_¼Axþ’ðyÞ þBu, xð0Þ ¼x0 ð1aÞ

y¼Cx ð1bÞ

where

x¼

x1

x2

.. .

x_n

0 B B B @

1 C C C

A, x0¼

x01

x02

.. .

x0n 0 B B B @

1 C C C

A, ’ðyÞ ¼

’1ðyÞ ’2ðyÞ

.. .

’nðyÞ 0 B B B @

1 C C C A

A¼

0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0

B B B B B @

1 C C C C C A

, B¼ 0

.. .

0 1 0 B B B B B @

1 C C C C C A ,

C¼ð1, 0,. . ., 0Þ

andu is the control input,y is the measured output,x0 is the initial condition of the state, and the functions’i are sufficiently smooth and Lipschitz continuous, more-over, we assume that’_ið0Þ ¼0, i¼1,. . .,nthrough this paper.

2.1. Initialization of the observer

Let us first consider a generic observer based controller Xðxx^0Þ, where xx^0 is the initial condition for the observer. The performance of the closed loop ðSðx0Þ,Xðxx^0ÞÞ is dependent on both the initial state x0 and the initial condition for the observerxx^0. Whilst the initial statex0 is the property of a system, the control designer has the freedom to choose the initial condition

^

x

x0 for the observer.

It is intuitive that good performance results from initializing the observer statexx^0to be close to the actual initial state x0. Of course, in practice, the initial state is often unknown, so it can be hard to initialize in this manner. Nevertheless standard practice is to try to minimize

kxx~0k ¼ kx0xx^0k

according to the best information available. To establish a rigorous justification for this intuitive idea (or more precisely: to characterize the situations when it is valid)

remains an open research problem; in this paper we simply illustrate the validity of this approach on a single example, as discussed next.

Consider the two-dimensional system S0ðx0Þ: xx_1¼x2

_

x

x2¼’ðyÞ þu, xð0Þ ¼ ðx01,x02ÞT

y¼x1

where’ðyÞis a Lipschitz continuous function. We con-sider a KKK controller (see Kristic et al. 1995; Ch. 7, p. 291) defined as

X0Oðxx^0Þ: u¼2ðy,xx^1,xx^2Þ

_^

x x^

x

x1¼xx^2þk1ðyxx^1Þ

_^

x x^

x

x2¼k2ðyxx^1Þ þ’ðyÞ þu, xx^ð0Þ ¼ ðxx^01,xx^02ÞT where

1ðyÞ ¼y

1ðyÞ ¼ c11d11 2ðy,xx^1,xx^2Þ ¼xx^21ðy,xx^1Þ 2ðy,xx^1,xx^2Þ ¼ c221d2

@ ₁ @y

2 2

k2ðyxx^1Þ ’ðyÞ þ @1

@y xx^2

andk1,k2,ci,di, 1j2 are positive constants. Since we can measure x1, we can always take

^

x

x01 ¼x01. However, x02 may be unknown, and so it is meaningful to compare the behaviour of the closed loops with the alternative choices of

^

x

x02¼x02, xx^02¼0

We can then show the following proposition, whose proof is given in the Appendix.

Proposition 1: Consider the system S0_ðx

0Þ and the

controller X0ðxx^0Þ, then there exist ci,di,kiði¼1, 2Þ

such that

lim x02!1

PðS0ðx0Þ,X0Oððx01, 0ÞTÞÞ

PðS0ðx0Þ,X0Oððx01,x02ÞTÞÞÞ ¼ þ 1 This proposition shows that as x02 becomes large, the difference of performance between PðS0ðx0Þ,X0O ððx₀₁, 0ÞTÞÞ and PðS0ðx₀Þ,X0_Oððx₀₁,x₀₂ÞTÞÞ can be larger than any positive constant. Therefore, it is advantageous to initialize the second state of the observer close to actual state rather than to initialize it at zero.

are interested in studying the situation in which our estimate ofx0is not accurate and kxx~0k is large, in par-ticular how does poor information onx0(which causes ‘bad’ choices of xx^0), affect the performance of the controllers?

2.2. KKK design

We first consider aKKK design (Kristicet al.1995) which achieves global regulation of the output. Although theKKKdesign has a global region of attrac-tion (in ðx0,xx^0Þ), we will prove that the performance of the controller can degrade arbitrarily as the initial error kxx~0k becomes large for any fixed initial state conditionx0.

TheKKKdesign (Kristicet al.1995) for systemSðx0Þ is as follows.

First, an observer is defined by

_^

x x^

x

x¼Axx^þkðyyy^Þ þ’ðyÞ þBu, xx^ð0Þ ¼xx^0 ð2aÞ

^

y

y¼Cxx^ ð2bÞ

where

k¼ ðk1,k2,. . .,knÞT, ki>0, 1in is chosen such thatAkCis Hurwitz.

Then define 1ðyÞ ¼y

1ðyÞ ¼ c11d11’1ðyÞ iðy,xx^1,. . .,xx^iÞ ¼xx^ii1ðy,xx^1,. . .,xx^i1Þ iðy,xx^1,. . .,xx^iÞ ¼ ciii1di

@i1 @y

2 i

kiðyxx^1Þ ’iðyÞ

þ@i1

@y ðxx^2þ’1ðyÞÞ

þX

i1

j¼1 @i1

@xx^_j ðxx^jþ1þkjðyxx^1Þ þ’jðyÞÞ

i¼2, 3,. . .,n

whereci, di, 1in are positive constants. The con-troller is then defined as

XOðxx^0Þ: u¼nðy,xx^1,. . .,xx^nÞ

_^

x x^

x

x¼Axx^þkðyyy^Þ þ’ðyÞ þBu, xx^ð0Þ ¼xx^0

^

y y¼Cxx^

The following result summarizes the standard properties of this closed loop.

Proposition 2: Consider the closed loop systemðSðx0Þ, XOðxx^0ÞÞ. For any initial data x02Rn and xx^0 2Rn, the

following hold:

(1) The signals x,xx^,u and y and bounded;

(2) The output is regulated to zero

lim

t!1yðtÞ ¼0 (3) The performance is finite

PðSðx0Þ,XOðxx^0ÞÞ<1

Proof: The proof of 1, 2 can be found in Kristicet al. (1995). Let mðTÞ denote the Lebesgue measure of the set T. Note that mðTÞ<1 since yðtÞ !0 as t! 1 hence

kyk_L2_ðT

ÞmðTÞ

1=2_kyk L1_ðR

þÞ <1

by 1. The boundedness of the performance follows

directly. œ

We now establish the critical performance property for the KKKdesign, which states that the performance gets arbitrarily large as the initial observer error increases.

Theorem 1: For any choice of the controller gains ki, 1in,and for any fixed initial state x0of the system Sðx0Þ, the performance of the controller XOðxx^0Þ has the

property

lim sup kxx~0k!1

PðSðx0Þ,XOðxx^0ÞÞ ¼ 1 ð3Þ

Proof: For the convenience of notation, the following definitions are introduced

ið0Þ ¼iðy,xx^1,. . .,xx^iÞjt¼0 ið0Þ ¼iðy,xx^1,. . .,xx^iÞjt¼0

j¼1, 2,. . .,n

To prove this theorem, it suffices to show lim sup

kxx~0k!1

kuk_L1_ðR

þÞ ¼ 1

Sinceu(t) is continuous, to establish the above equation, we only need to show

lim sup kxx~0k!1

uð0Þ ¼lim sup kxx~0k!1

nð0Þ ¼ 1 ð4Þ

Let CRn1 _{be a compact set, define}

Cr¼ xx^02Rnj ðxx^01,. . .,xx^0,n1Þ 2C;xx^0n¼r

Consider the initial data of the observer xx^0 2Cr. Becausex0is fixed, if we can prove that

lim r!1_xx^sup

02Cr

_nð0Þ ¼ 1 ð5Þ

We now establish (5). Since all’_iand its derivatives are continuous functions it follows that _i and _i are continuous functions of their variables. Note that

_ið0Þ ¼xx^0ii1ð0Þ

_ið0Þ ¼ c_iið0Þ _i1ð0Þ di @i1

@y

t¼0

2

_ið0Þ

k_iðx₀₁xx^₀₁Þ ’_iðx₀₁Þ

þX

i1

j¼1 @_i1

@xx^j

t¼0

ðxx^_0,jþ1þkjðx01xx^01Þ þ’jðx01ÞÞ

So, for 1in1,ið0Þ,ið0Þare independent ofxx^0n, i.e. bounded independently ofr. Therefore there exists

M> 0 dependent onCandx01 but not onr, for which sup

^

x x02Cr

jið0Þj M, sup

^

x x02Cr

jið0Þj M, 1in1

Now we computenð0Þ. First, we have nð0Þ ¼xx^0nn1ð0Þ ¼rn1ð0Þ and so

_nð0Þ ¼ c_inð0Þ _n1ð0Þ dn @n1

@y

t¼0

2

_nð0Þ

knðx01xx^01Þ ’nðx01Þ þ Xn1

j¼1

@_n1 @xx^j

t¼0

ðxx^0,jþ1þkjðx01xx^01Þ þ’jðx01ÞÞ

¼ c_nd_n @n1

@y

t¼0 2!

rþ @n1 @n1

t¼0

r

þFðx01,xx^01,. . .,xx^0,n1Þ where

Fðx01,xx^01,. . .,xx^0,n1Þ

¼ cnþdn @_n1

@y

t¼0

2

@n1 @xx^n1

t¼0 !

n1ð0Þ

þn1ð0Þ knðx01xx^01Þ ’nðx01Þ

þX

n2

j¼1 @n1

@xx^j

t¼0

ðxx^0,jþ1þkjðx01xx^01Þ þ’jðx01ÞÞ

is independent of xx^0n, namely r. Let us consider the second term of the expression fornð0Þ.

@i @xx^_i ¼ ci

@i @xx^_idi

@i1 @y

2 @i @xx^_iþ

@i1 @xx^_i1

¼ cidi @i1

@y

2 þ@i1

@xx^i1 :

Therefore, by recursive substitution we obtain @n1

@xx^_n1

¼X

n1

j¼2

cjdj @j1

@y

2! þ@1

@xx^1

¼X

n1

j¼2

cjdj @j1

@y

2!

since1 is independent ofxx^1. Hence,

nð0Þ ¼r Xn

j¼2

cjdj @_j1

@y

t¼0 2!

þFðx01,xx^01,. . .,xx^0,n1Þ:

Because cj and dj are all positive numbers, and F is independent of r, this establishes (5) as required. œ 2.3. Khalil design

It is well known that by a suitable coordinate trans-formation the systemSðx₀Þcan also be written as inte-grator chain with a matched nonlinearity. Concretely, we define a coordinate transformation

T:Rn_!Rn_{, z¼TðxÞ} by

T: z1 ¼x1, z2¼x2þ 1ðx1Þ,. . .,

z_n ¼x_nþ _n1ðx1,x2,. . .,xn1Þ where

iðx1,. . .,xiÞ ¼’iðx1Þ þ Xi1

j¼1 @ _i1

@xj

xjþ1þ’jðx1Þ

,

1in

Then in thezcoordinates,Sðx0Þis of the form

Sðz0Þ: zz_¼AzþBð ðzÞ þuÞ, zð0Þ ¼z0 ð6aÞ

y¼Cz ð6bÞ

where

z0¼Tðx0Þ

ðzÞ ¼ n T1ðzÞ

nðxÞ ¼ nðx1,. . .,xnÞ 9 > > > = > > > ;

ð7Þ

Since the systemSðx₀Þhas a uniform relative degree

n, the mappingTis a global diffeomorphism inRn_{; see} Isidori (1989). Hence the inverse mappingT1 exists.

Remark 1: Since the output y is unchanged by the

Hence the Khalil designs considered in Esfandiari and Khalil (1992), Khalil and Esfandiari (1993) and Atassi and Khalil (1999) can be applied to the system Sðz0Þ. Typical results establish semiglobal regulation of the output. The Khalil designs utilize a high gain observer and a nonlinear separation principle (Atassi and Khalil 1999) which allow the observer and a glob-ally bounded state feedback controller to be designed separately, and then combined using certainty equiva-lence, to ensure semiglobal results and closeness of the output feedback controller’s trajectory to the underlying state feedback controller’s trajectory. For the system Sðx0Þ, if ’i and its higher derivatives are globally bounded, it is straightforward to design a globally bounded state feedback controller achieving bounded performance. Hence through the high gain observer we can design an output feedback controller, which, for fixed initial condition of the state z0 ¼Tðx0Þ and any initial condition of the observer zz^0 also has bounded performance. Furthermore, if the initial error

kzz~₀k ¼ kz₀zz^₀k

becomes large, this design still achieves a bounded performance independent of the initial condition of the observer.

To design an output feedback controller, we first give a state feedback controller forSðx0Þ. The controller

u¼ ðzÞ þv ð8Þ

feedback linearizes the systemSðx0Þ, yielding

_

zz¼AzþBv, zð0Þ ¼z0 ð9aÞ

y¼Cz ð9bÞ

We first design a bounded state feedback controller for the linear system (9). From Sussmann et al.(1994) we have the following lemma.

Lemma 1: The system (9) is null controllable with

bounded control(ANCBC)if and only if:

(1) A has no eigenvalues with positive real part; (2) The pair (A,B) is stabilizable in the ordinary

sense.

Now since all the eigenvalues ofAare zero, namely, without positive real parts, and the pair (A,B) is stabi-lizable, the system (9) is null controllable with bounded control, and, furthermore, there exists bounded state feedback controllers for the system (9). An explicit example (Sussmann et al. 1994) of such a bounded state feedback controller is given by

v¼ X

n

i¼1

isatðhiðzÞÞ ð10Þ

where 0< 1₄, eachh_i:Rn_!R_{, 1in, is a linear} function, and satðÞis the saturation function defined by

satðwÞ ¼

1, w<1

w, 1w1

1, w>1 8

< :

This controller achieves global asymptotic stability for the resulting closed-loop system (Sussmannet al. 1994).

Consequently, the state feedback controller Xs: u¼ ðzÞ

Xn

i¼1

isatðhiðzÞÞ ð11Þ

globally asymptotically stabilizes the origin of system Sðz0Þ.

Now we design a output feedback controller for Sðx0Þ. Following Esfandiari and Khalil (1992) and Atassi and Khalil (1999), we define the high gain observer as

_^

zz^

zz¼Azz^þHðyzz^1Þ; zz^ð0Þ ¼zz^0 ð12Þ where

H¼HðÞ ¼ 1

, 2 2,. . .,

n n

T

ð13Þ andis a positive constant to be specified. The positive constants i, 1in, are chosen such that the roots of the equation

snþ1sn1þ þn1sþn¼0 are in the open left-half plane.

To apply the non-linear separation principle, the state feedback controller is required to be globally bounded. Generally, this property can be achieved by saturating the controller outside some set. But in our case we are interested in the initial condition of the observer becoming large. Instead, we introduce further assumptions on’_ito ensure that is globally bounded. Lemma 2: For system Sðx0Þ, suppose ’i2CniðRÞ, ’ð_ikÞ 2L1_{ðRÞ, 1in; 1kn, then defined by} (7)lies in L1_ðRn_Þ.

Proof: Since ’i2CniðRÞ, ’ðikÞ2L1ðRÞ, from (6) we have that nðxÞ is continuous and in L1ðRnÞ. Note that the mapping T is a global diffeomorphism, we know that ðzÞalso is continuous and inL1_ðRn_{Þ. œ} Suppose that the conditions of Lemma 2 are satis-fied, then the state feedback controller (11) is globally bounded, so an output feedback controller for system Sðx₀Þcan be taken as

XHðÞðzz^0Þ: u¼ ðzz^Þ Xn

i¼1

isatðhiðzz^ÞÞ ð14aÞ

_^

zz^

For the system Sðz₀Þ and the output feedback con-troller X_HðÞðzz^0Þ, relevant properties of the closed loop are summarized below.

Proposition 3: For system Sðz0Þ, suppose that z0 ¼

Tðx0Þ, x0 is fixed, and the assumption of Lemma 2 is

satisfied. Then for any zz~0¼z0zz^0 there exists such

that for all :0< < _{the output feedback controller} XHðÞðzz^0Þguarantees:

(1) The signals z, zz^, u and y are bounded; (2) The output is regulated to zero

lim

t!1yðtÞ ¼0 (3) The following limit

lim

!0zðt,Þ ¼zzðtÞ

holds uniformly in t for all t0,where zðt,Þis the solution of the closed systemðSðz₀Þ,X_HðÞðzz^₀ÞÞ;and

zzðtÞ is the solution of the state feedback control closed systemðSðz0Þ,XsÞ.

(4) The performance is finite PðSðz0Þ,XHðÞðzz^0ÞÞ<1 Proof: First, the function

pðzÞ ¼ ðzÞ X n

i¼1

isatðhiðzÞÞ

is locally Lipschitz continuous since ðzÞ is continuous and Pn_i¼1 isatðhiðzÞÞ is bounded. Second, pðzÞ is bounded from Lemma 2. Third, the origin is an asymp-totically stable equilibrium of the closed-loop of the state feedback control. Hence Assumption 2 in Atassi and Khalil (1999) is satisfied. Assumptions 1 and 3 for the system in Atassi and Khalil (1999) are also satisfied. Take any compact set C2Rn _{and CC}^₂Rn _such that z02C and zz^02CC^, then 1, 2, 3 follow directly from Theorem 1 and Theorem 2 in Atassi and Khalil (1999). As to 4, the finiteness of kyk_L2_ðT

Þ is obtained

from 2. Note that is continuous and zz^ is bounded by 1. Hence, kukL1_ðR

þÞ is also finite. So, PðSðx0Þ,

XHðÞðzz^0ÞÞis finite. œ

Now it is straightforward to uniformly bound the performance of systemSðx0Þfor theKhalildesign.

Theorem 2: Let x0 be fixed and consider the system

Sðx0Þ. Let z0¼Tðx0Þ. Let ’i2CniðRÞ, ’iðkÞ2L1ðRÞ, 1in; 1kn. Then there is a positive constant M, such that for any zz~0 there exists >0 for which

the controller XHðÞðzz^0Þ achieves a uniformly bounded

performance

PðSðz0Þ,XHðÞðzz^0ÞÞ<M ð15Þ

Proof: First note that

PðSðz0Þ,XHðÞðzz^0ÞÞ ¼ ð

T

jyj2dtþ kuk_L1ðR_þÞ

¼ ð

T

jz1ðt,Þj2dtþ kukL1ðR_þÞ

ð16Þ From Lemma 2, we know that ðzz^Þis bounded. So, the control inputuhas a bound which is independent of

^

zz0. By Proposition 3, if is small enough, thenz1ðt,Þ tends uniformly inttozz1ðtÞ, which is independent ofzz^0 and uniformly bounded. Hence, zz₁ðtÞhas a bound that is independent of zz^₀. Similarly the measure of the time set T is also independent of zz^0 and finite. Hence the integral in (16) is finite and the bound is independent ofzz^0. Therefore, we can find a constantMsuch that (15)

holds. œ

2.4. Comparison

Theorem 1 shows that for fixed initial statex0, when the initial errorkxx~0kbecomes large, the performance of theKKKdesign is not uniformly bounded even if’iand its higher derivatives are globally bounded. On the other hand, Theorem 2 shows for theKhalil design, if’i and its higher derivatives are globally bounded, then for any initial error zz~0, through the high gain factor, we can design a globally bounded controller, achieving a uni-formly bounded performance.

Hence we obtain the following comparative result.

Corollary 1: Consider the system Sðx0Þ, where

’i2CniðRÞ,’ ðkÞ i 2L

1

ðR_{Þ, 1in. Then there exist} > 0andxx^0such that for anyzz^0 we have

PðSðz0Þ,XHðÞðzz^0ÞÞ<PðSðx0Þ,XOðxx^0ÞÞ

Proof: The result follows directly from Theorems 1

and 2. œ

3. Performance of parametric output feedback system

In this section we study the performances of the

KKKdesign and theKhalildesign for system

Sð,x0Þ: xx_¼AxþBð ðyÞ þuÞ, xð0Þ ¼x0 ð17aÞ

y¼Cx ð17bÞ

where

x¼ ðx1,x2,. . .,xnÞT

x₀¼ ðx₀₁,x₀₂,. . .,x_0nÞT

for which both KKK and Khalil designs can be given to achieve regulation of the output and bounded performance.

To design a Khalil-type output feedback controller with a high gain observer, we need first to design a globally bounded state feedback controller. Generally, this is achieved by saturation of the state feedback con-troller. But we also require that the saturated controller stabilizes the system. For this purpose, we need to deter-mine suitable saturation levels. However, the required saturation levels are typically dependent on , the unknown constant. Therefore, we have to first quantify

a priori estimates for the magnitude of . Since is assumed to be unknown our knowledge of it is typically poor. Hence we have to estimate conservatively. But when oura prioriupper bound forjjis conservative, we will show that the performance of the Khalil design becomes poor.

However, for a KKK design, the performance is independent of the any a priori upper bound for jj. Therefore, the performance keeps uniformly bounded as thea prioriupper bound forjjbecomes conservative. Hence, for this system we will establish a result with the contrary performance relationship to that in}2.

3.1. KKK design

TheKKKdesign for the parametric output feedback system Sð,x0Þ as follows (Chapter 7 in Kristic et al. 1995).

Choose a vector K such that A0¼AKC is Hurwitz, and define the filters

_

0 ¼A0_0þKy

_

1 ¼A01þB ðyÞ

_

vv0 ¼A0v0þenu

The controller is defined by: XAð#0,xx^0Þ: u¼n

_

#

#1¼G!1ðy,ð2Þ,vvð2ÞÞ1

_

#

#2¼G !2ðy,ð2Þ,vvð2Þ,##ð2ÞÞ þ1e2

2

_

#

#_i¼G!_iðy,ðiÞ,vvðiÞ,##ði1ÞÞ_i, i¼3,. . .,n

^

x

xð0Þ ¼xx^₀¼ ðxx^₀₁,xx^₀₂,. . .,xx^_0nÞT #ð0Þ ¼#0¼ ð#01,#02,. . .,#0nÞT

where ei denotes the ith coordinate vector in R2_{, and} i, !i, i, i¼1,. . .,n are defined by the recursive

expressions 1¼y

i¼v0,ii1ðy,##iÞ 1¼ #T1!1

2¼ c22#2, 21d2 @1

@y

2 2þ

@1 @y 0, 2

#T2!2þk2v0, 1þ @1 @0

ðA00þKyÞ

þ@1 @1

ðA01þB ðyÞÞ þ @1 @v0

A0v0þ @1 @#1

G!11

i¼ ciidi @i1

@y

2 iþ

@i1

@y 0, 2#

T

i!iþkiv0, 1

þ@i1 @0

ðA00þKyÞ þ @i1

@1

ðA01þB ðyÞÞ

þ@i1 @v0

A0v0þ @i1

@#1

G!11þ @i1

@#2

Gð!2þ1e2Þ2

þ X

i1

j¼3 @i1

@#j

G!jj, i¼3,. . .,n

!T1 ¼ ðc11þd11þ0, 2,v0, 2Þ !Ti ¼

@i1

@y ð1, 2,v0, 2Þ, i¼2,. . .,n1

!Tn ¼ @n1

@y ð’þ1, 2,v0, 2Þ

ðiÞ¼ ð0, 1,. . .,0,i,. . .,1, 1,. . .,1,iÞ, i¼1,. . .,n

vvðiÞ¼ ðv0, 1,. . .,v0,iÞ, i¼1,. . .,n

#

#ðiÞ¼ ð#T1,. . .,#TiÞ, i¼1,. . .,n

We summarize the relevant well-known properties of this controller in the following proposition.

Proposition 4: For the system Sð,x0Þ the controller XAð#0,xx^0Þ guarantees the global boundedness of all

sig-nals and the regulation of output

lim

t!1yðtÞ ¼0

Moreover,the controller achieves bounded performance PðSð,x0Þ,XAð#0,xx^0ÞÞ<1

3.2. Khalil design

the controller out off some sets based on our a priori

knowledge of the worst case bounds for the closed loop signals; next replace the unmeasurable state variables by the estimated states from a high gain observer. This defines an output feedback control.

3.2.1. Controller design. Firstly we design a state feed-back controller based on Lyapunov theory, and obtain

a prioriworst case estimates for the bounds of the closed loop signals.

We chose a vector

K¼ ðk1,k2,. . .,knÞ

such that matrixAþBKis Hurwitz, and let matrixPbe the positive definite symmetric solution of the Lyapunov equation

ðAþBKÞTPþPðAþBKÞ ¼ I

By considering the Lyapunov function

Vðx,Þ ¼^ xTPxþ1₂ðÞ^2 we define a state feedback controller

Xsð^0,x0Þ: u¼ðx,Þ ¼^ Kx ð^ yÞ ð18aÞ

_^

^

¼ ðx,Þ ¼^ 2xTPB ðyÞ, ð^0Þ ¼^0 ð18bÞ noting that along the solution of the closed loop, we have

_

V

V¼ xTx0

This suffices to show global stability and regulation of the output to zero.

To design an output feedback controller through a high gain observer, the functions and should be globally bounded (Atassi and Khalil 1999). So, we sat-urateand outside some suitably defined sets which ensure that the modified controller still stabilizes the system. For this purpose, we utilize a priori estimates ofxand^.

Firstly, fromVV_ 0, we get 1

2ðÞ^ 2

VðtÞ Vð0Þ ¼xT0Px0þ12ð^0Þ2 Hence

jj ^ mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðPÞ2

mþ ðmþ j^0jÞ2 q

¼:Y0 ð19Þ where m and m are the a priori estimates of upper bound for the magnitude of the unknown parameter and the magnitude of the initial state jx0j, and ðPÞ is the largest eigenvalue ofP.

Similarly,

kxk ¼ ðxTxÞ1=2 1 ðPÞx

T

Px

1=2

1

ðPÞVð0Þ 1=2

1

ðPÞ ðPÞ 2 mþ

1

2ðmþ j^0jÞ 2

1=2

¼:X0 ð20Þ

whereðPÞis the smallest eigenvalue of Pand

jyj ¼ jx1j kxk X0 ð21Þ Finally, from (18a)

jj nkX0þY0C0¼:U0 ð22Þ where

k¼ max

1jnfjkjjg C0¼ sup

jx1jX0

fj ðx1Þjg

On the other hand, suppose that p is the biggest element in the last row ofP, then by (18b) we obtain

j j npkxkC0npX0C0 ¼:V0 ð23Þ Now we saturate and as

sðx,Þ ¼^ U0sat

ðx,Þ^

U0 !

sðx,Þ ¼^ V0sat

ðx,Þ^

V0 !

to obtain a globally bounded state feedback controller Xb_sðm,m,^0,x0Þ: u¼sðx,Þ^ ð24aÞ

_^

^

¼ _sðx,Þ^, ð^0Þ ¼^0 ð24bÞ Consequently aKhalilcontroller can be obtained as XHðÞðm,m,^0,xx^0Þ: u¼sðxx^,Þ^ ð25aÞ

_^

^

¼ sðxx^,Þ^, ð^0Þ ¼^0 ð25bÞ

_^

x x^

x

x¼Axx^þHðyxx^1Þ, xx^ð0Þ ¼xx^0 ð25cÞ The properties of this controller are summarized in the following Proposition.

Proposition 5: For the systemSð,x0Þ,ifjj m,then

when is small enough, the controller XHðÞðm,m,

^

0,xx^0Þ guarantees the global boundedness of all signals

and the regulation of output

lim

Moreover,the controller achieves bounded performance PðSð,x0Þ,XHðÞðm,m,^0,xx^0ÞÞ<1

Proof: The system Sð,x0Þ is global normal form,

hence satisfies Assumptions 1 and 2 in Khalil (1996). The state feedback control input sðx,Þ^ and update law sðx,Þ^ are defined the same as those in Khalil (1996). So, by Theorem 2 of Khalil (1996), we obtain the boundedness of all signals and the regulation of output.

The proof of the boundedness of the performance follows from the boundedness of the closed-loop

signals. œ

3.2.2. Performance. First we are going to establish the following lemma.

Lemma 3: Let

e0j¼x0jxx^0j, 1jn

and suppose that at least one of e0j, 1jn1, is

not equal to zero. Then for the closed loop ðSð,x0Þ, X_HðÞ ðm,m,^0,xx^0ÞÞ,we have

lim !0kukL

1_ðRþ_Þ¼U₀ ð26Þ

Proof: From the definition of the controller XHðÞðm, m,^0,xx^0Þ, it suffices to prove that

lim !0 _tsup₂Rþ

kxx^ðtÞk !

¼ 1 ð27Þ

Now let

ej¼xjxx^j, 1jn

j¼ 1

njej, 1jn

Then the closed loop ðSð,x0Þ,XHðÞðm,m,^0,xx^0ÞÞ is given by

_

x

x¼AxþBð ð^ yÞ þsðxx^,ÞÞ^ , xð0Þ ¼x0 ð28aÞ

_^

^

¼ sðxx^,Þ^, ð^0Þ ¼^0 ð28bÞ _¼DþBð ð^ yÞ þsðxx^,ÞÞ^ , ð0Þ ¼0 ð28cÞ where L¼ ð1,2,. . .,nÞT, the matrix D¼ ALC is Hurwitz and

0¼

e01 n1,. . .,

e0,n1 ,e0n

T

To prove (27) it is enough to show

lim !0 _tsup₂Rþ

keðtÞk !

¼ 1 ð29Þ

Since_n¼e_n, it is sufficient to show

lim !0 _tsup₂Rþ

jnðtÞj !

¼ 1 ð30Þ

On the other hand, let t¼, then (28c) can be writ-ten into

d

d ¼DþBð ð^ yÞ þ s_ð^

x

x,ÞÞ^ , ð0Þ ¼0 ð31Þ When is small enough, the output y¼x1 converges uniformly in t to the solution of state feedback closed system, and hence is uniformly bounded, therefore ðyÞ is uniformly bounded. So, the term Bð ð^ yÞþ sðxx^,ÞÞ^ in (31) is bounded uniformly in . Therefore, when!0, the solution of (31) is convergent uniformly in to the solution of

d

d¼D, ð0Þ ¼0 ð32Þ Hence, we only need to show

lim !0 _tsup₂Rþ

jnðtÞj !

¼ 1 ð33Þ

Note that

D¼

d1 1 0 0

d2 0 1 0

dn1 0 0 1

d_n 0 0 0

0 B B B @

1 C C C A

and by induction we can show

Dj¼

_1j 1 0 0

1 0

1

_n 0 0 0

0 B B B B B B @

1 C C C C C C A

, 1jn

ð34Þ where the s are elements which do not need to be specified. Let

s¼ min

1jn1fjje0j6¼0g and consider the time

t¼

where

¼ðnsÞ=s

and

¼1 2 min

ns sþ1, 1

Note the solution of (32) is given by ðÞ ¼eD0 i.e. equivalently by

ðtÞ ¼eDðt=Þ0 ð35Þ Hence

ðtÞ ¼eD0 Now

Noting thate0j ¼0 for 1js1, yields

nðtÞ ¼ n s

s!þo

sþð1=2Þ

e0s nsþo

sþð1=2Þ

_e

0,sþ1 ns1

þ þosþð1=2Þ e0,n1

þ 1þo sþð1=2Þ

e_0n

¼ ne0s

s! þe0nþoðÞ ð36Þ

where

¼nsþ 2s >0

But by assumption,n >0, ande0s6¼0. Therefore, (36) implies (33). This completes the proof. œ From this lemma, we can obtain the following theorem.

Theorem 3: Let ,x0 be fixed, and supposejx0j m.

Consider the system Sð,x0Þ and the controller XHðÞðm,m,^0,xx^0Þ. Suppose that at least one of the

initial errors e0j, 1jn1,is not equal to zero. Then for the closed loop systemðSð,x0Þ,XHðÞðm,m,^0,xx^0ÞÞ,

we have

lim !0 mlim!1

P Sð,x0Þ,XHðÞðm,m,^0,xx^0Þ

¼ 1

Proof: For the closed loop system ðSð,x0Þ,

XHðÞðm,m,^0,xx^0ÞÞ, the saturation levels U0 and V0 for the output feedback controller are dependent onm, thea priori estimate of upper bound for the unknown parameter. Whenm is large, from (19)–(22),U0and

V0are large. By Lemma 3, as the high gain factor is small, kuk_L1_ðRþ_Þ is also large, that is the performance

becomes large. œ

3.3. Comparison

For the system Sð,x0Þ, as the a priori estimate of upper boundm for the uncertain parameterbecomes conservative, Proposition 4 shows that theKKKdesign guarantees uniform bounded performance of the con-troller, whereas Theorem 3 shows that the performance of the Khalil design becomes large. Here we have the following comparative result.

Corollary 2: Let ,x0 be fixed, and suppose jx0j m.

For the system Sð,x0Þ,if the bound m for the unknown

parameteris conservative enough,and the gain factor

is small enough,then

PðSð,x0Þ,XAð#0,xx^0ÞÞ<PðSð,x0Þ,XHðÞðm,m,^0,xx^0ÞÞ Proof: The result follows directly from Proposition 4

and Theorem 3. œ

4. Conclusion

Through the comparison of performances for KKK

and Khalil designs, we have established the following results:

. For output feedback system, the performance of

KKKdesign is sensitive to the initial datum of the observer. The performance of the KKK design is not uniformly bounded in the initial error between the initial datum of the state and the initial datum of the observer. When the initial error becomes large, the performance becomes large. Whereas, for the Khalil design, for any initial error, by choosing small high gain factor, we can design a globally bounded controller, achieving a uni-formly bounded performance. Therefore, if the initial error is large or in the case that we have poor information for the initial datum of the state, the Khalil design has better performance than theKKKdesign.

. For parametric output feedback system, the per-formance of the KKK design is independent of the a priori estimate bound of the uncertain parameter. When the a priori estimate becomes conservative the performance remains uniformly bounded. Whilst, for theKhalildesign, the perfor-mance is dependent on the saturation levels for the eD_¼Iþ

Dþ

2 2!D

2

þ þ s

s!D

s

þo sþð1=2Þ

¼

_1s

nðs=s!Þ þo sþð1=2Þ

o sþð1=2Þ

o sþ1=2

1þo sþð1=2Þ 0

@

controller and the adaptive law, that is dependent on the a priori estimate bound of the uncertain parameter, and the performance becomes large as the a priori estimate becomes conservative. Hence, if we have poor information for the unknown parameter and the a priori estimate bound is conservative, theKKKdesign has better performance than theKhalildesign.

The primary contribution of this paper is to provide rigorous statements and proofs of the intuitively reason-able trade-offs in performance between the differing classes of designs. The results have been expressed in qualitative terms only. The purpose of the paper is to illustrate the asymptotic differences between the designs. It should also be noted that the results are asymptotic in nature, that is they require some parameter (either an initial condition or an uncertainty level) to be large in order to make the required comparison. Of course, in practice these parameters cannot be arbitrarily large without causing the control to run into physical limits. A more quantitative approach is challenging, as achiev-ing tight bounds on non-sachiev-ingular performance is diffi-cult. This is an interesting avenue for future research.

Acknowledgement

The authors are grateful to the reviewers for their helpful comments and constructive suggestions. Appendix. The proof of proposition 1

Consider the closed loop ðS0ðx0Þ,X0Oðx0ÞÞ. First, observe that the observation errorxx~¼xxx^ satisfies

_~

x x~

x

x1¼ k1xx~1þxx~2 ð37aÞ

_~

x x~

x

x2¼ k2xx~1, xx~ð0Þ ¼xx~0 ð37bÞ hence satisfies equation

€~

x x~

x

x1þk1xxxx_~~1þk2xx~1¼0 ð38aÞ

~

x

x1ð0Þ ¼x01xx^01 ð38bÞ

_~

x x~

x

x1ð0Þ ¼x02xx^02k1ðx01xx^01Þ ð38cÞ where

~

x

x0 ¼x0xx^0¼ ðx01xx^01,x02xx^02ÞT

Second, note that the control signal u can be expressed as

X0_Oðxx^0Þ: u¼2ðy,xx^1,xx^2Þ ¼k2xx^1h2xx^2hy’ðyÞ ð39Þ where

h¼ c2þd2ðc1þd1Þ2

ðc1þd1Þ þk2þ1

h2¼c2þd2ðc1þd1Þ2þc1þd1

So, the closed loop system can be written as

_

x

x₁¼x₂ ð40aÞ

_

x

x2¼ hx1þk2xx^1h2xx^2 ð40bÞ

_^

x x^

x

x1¼k1x1k1xx^1þxx^2 ð40cÞ

_^

x x^

x

x2¼ h1x1h2xx^2 ð40dÞ where

h1 ¼hk2¼ c2þd2ðc1þd1Þ2

ðc1þd1Þ þ1 Consider the first situation xx^0¼x0, namely,xx~0¼0. The solution of (37) is xx~¼0, so xx^ðtÞ xðtÞ, and the closed system (40) reduces to

_

x

x1¼x2 ð41aÞ

_

x

x2¼ h1x1h2x2 ð41bÞ

Thus we have

€

x

x1þh2xx_1þh1x1¼0 ð42aÞ

x1ð0Þ ¼x01, xx_1ð0Þ ¼x02 ð42bÞ Write the solution of the above equation asx01ðtÞ, and observe thatx0₁ðtÞcan be expressed as

x01ðtÞ ¼x01q1ðtÞ þx02q2ðtÞ

where q1,q2 are functions which are independent of

x01,x02. Moreover, we can choosey ci,di,i¼1, 2 such thatq₂ðtÞ>0 fort> 0, and furtherzx0₁ðtÞ>0 fort> 0 ifx₀₂ >0.

Now consider the second situation xx^01¼x01 and

^

x

x02 ¼0, namely,xx~01¼0 andxx~02 ¼x02. So, the problem (38) becomes

€~

x x~

x

x1þk1xxxx_~~1þk2xx~1¼0 ð43aÞ

~

x

x₁ð0Þ ¼0, xxxx_~~₁ð0Þ ¼x₀₂ ð43bÞ The solution of the above problem can be written as

~

x

x₁¼x₀₂f₁ðtÞ ð44Þ

yFor example, we can choose c_i,d_i,i¼1, 2 such that

h2₂>4h₁, and let ₁¼ 1

2 h2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2

24h1

q

, ₂¼ 1

2 h2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2

24h1

q

thenq₁ðtÞandq₂ðtÞcan be written as

q₁ðtÞ ¼ 1

₁₂ 1e

2t 2e1t

,

q2ðtÞ ¼ 1 ₁₂ e

1t_e2t

>0, t>0

zHere,x01ðtÞcan also be written as

x0₁ðtÞ ¼ 1

₁₂ ðx021x01Þe

1t _x

022x01

ð Þe2t

It can verify that ifx₀₂>0 thenxx_₁0ðtÞ>0 fort>0. Note that

where f₁ðtÞ is a continuous function which is indepen-dent ofx02. At the same time, xx~2 can be written as

~

x

x2¼x02 f2ðtÞ ð45Þ

also where f₂ðtÞ is a continuous function which is independent ofx02.

Now substitutexx^¼xxx~ into the closed loop (40), and rewrite the first two equations as

_

x

x1¼x2 ð46aÞ

_

x

x2¼ h1x1h2x2k2xx~1þh2xx~2 ð46bÞ or

€

x

x1þh2xx_1þh1x1¼x02fðtÞ ð47Þ where

fðtÞ ¼ k2f1ðtÞ þh2f2ðtÞ

is also independent ofx02. Again we can choose} k1,k2 such thatfðtÞ>0.

Solve the following problem

€

x

x1þh2xx_1þh1x1¼x02fðtÞ ð48aÞ

x1ð0Þ ¼x01, xx_1ð0Þ ¼x02 ð48bÞ We can express the solution of (48) as

x1ðtÞ ¼x01ðtÞ þ ðt

0

ðtÞx02 fðÞd

where ðtÞ is the solution of (42a) which satisfies ð0Þ ¼0 and_ð0Þ ¼1, namelyðtÞ ¼q₂ðtÞ. Write

gðtÞ ¼ ðt

0

q2ðtÞfðÞd then

x1ðtÞ ¼x01ðtÞ þx02 gðtÞ

wheregðtÞ>0. Writing

T¼ t0jx1ðtÞj>

n o

T0¼nt0jx0₁ðtÞj> o

thenT0 T since x₁ðtÞ>x0₁ðtÞ. Hence kx1k2L2_ðT

Þ kx

0 1k2L2_ðT0

Þ

ð

T0

ðx1ðtÞÞ2 ðx01ðtÞÞ2

dt

¼x202aþx02b ð49Þ

where

a¼ ð

T0

gðtÞ2þ2q2ðtÞgðtÞ

dt

is a positive constant sincegðtÞ,q2ðtÞ>0 and

b¼ ð

T0

2x01q1ðtÞdt is a constant which is independent ofx₀₂.

Write the control input of controller X0_Oðx₀₁,x₀₂Þas

u0, and the control input of controller X0_Oðx01, 0Þ as u1. Then by a tedious calculation, we can obtain

ku0k x02a0þb0 ð50aÞ ku1k x02a1þb1 ð50bÞ since’is Lipschitz continuous, wherea_i,b_i,i¼1, 2, are positive constants which are independent of x02. Therefore, from (49) and (50), we obtain

lim x02!1

1

x2 02

PðS0ðx0Þ,X0Oðx01, 0ÞÞ PðS0ðx0Þ,X0Oðx01,x02ÞÞ

¼ lim x02!1

1

x2₀₂ðkx1k

2 L2_ðT

Þ kx

0 1k2L2_ðT0

ÞÞ

þ 1

x2 02

ðku1k ku0kx202Þ a>0 So, finally, we obtain that

lim x02!1

PðS0ðx₀Þ,X0_Oðx₀₁, 0ÞÞ

PðS0ðx0Þ,X0Oðx01,x02Þ

¼ þ 1

This completes the proof.

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}We choosek22>4k2, and let 1¼

1 2ðk1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2

14k2

q

Þ, 2¼

1 2ðk1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2

14k2

q

Þ

then we get

f₁ðtÞ ¼ 1

12

e1t_e2t

, f₂ðtÞ ¼ 1

12

₂e1t_þ

1e2t

fðtÞ ¼ 1

₁₂ ðh22k2Þe

1t_{þ ðh}

21þk2Þe2t

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