Local Observer Design for Nonlinear Systems - Science Direct

MATHEMATICAL COMPUTER MODELLING

PERGAMON

Mathematical

and Computer

Modelling

35 (2002) 25-36 www.elsevier.com/locate/mcm

Local Observer Design for Nonlinear Systems V. SUNDARAPANDIAN Department of Mathematics, Indian Institute of Technology Kanpur-208016, Uttar Pradesh, India vsundaraQiitk.ac.in (Received April 2001; accepted June 2001) Abstract-This paper is a geometric study of the local observer design for nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable nonlinear systems. As an application of our local observer design, we consider a class of nonlinear systems with an input generator (ecosystem) and show that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs. @ 2001 Elsevier Science Ltd. All rights reserved.

Keywords-Nonlinear

observers, Exponential

observers, Stability theory.

1. INTRODUCTION During the past two decades, there has been an impressive variety of new techniques

developed

stabilization and output regulation of nonlinear systems [l-5]. Most of this literature is based on state feedback, and hence, the state of the system is assumed to be available through measurement. In the case of a linear system, the “separation principle” assures that for feedback

an estimate of the state may be used in lieu of the state, provided the estimation error decays exponentially. As far as feedback stabilization is concerned, a similar “separation principle” holds for nonlinear systems locally around a state equilibrium. Explicitly, it has been proved [6,7] that if a state feedback system is exponentially, or even critically asymptotically stable and if an exponentially good state estimator is known, then the composite feedback-state estimator scheme is locally asymptotically stable at the origin. Furthermore, elementary counter-examples to such a ‘separation principle” when the error is only critically asymptotically stable are known [7]. It is for this reason that the existence and construction of local exponential observers is of crucial importance in nonlinear observer design. The problem of designing observers for nonlinear systems was introduced by Thau [8]. Over the past three decades, considerable attention has been paid in the literature to the construction of observers for nonlinear systems. In [9], Xia and Gao obtained a necessary condition for the existence of an exponential observer for nonlinear systems. Explicitly, in [9], Xia and Gao showed that an exponential observer exists for a nonlinear system only if the linearization of the nonlinear system is detectable. On the other hand, sufficient conditions for nonlinear observers have 0895-7177/01/s - see front matter @ 2001 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(01)00145-5

Typeset

by A&-m

V. SUNDARAPANDIAN

26

been obtained from an impressive variety of points of view. In [lo], a Lyapunov-like method was presented for exponential observer design. In [ll-151, suitable coordinate transformations were found under which the nonlinear system is transformed into a canonical form where the observer design is carried out. In [16,17], sufficient Lyapunov-like conditions were derived for local asymptotic observer design. In [18], a harmonic-analysis method was presented for asymptotic observer design. In [19], geometric techniques were used to characterize local exponential observers for unforced nonlinear systems. Finally, in [20], the existence of a partial observer without stability requirements was shown to be locally equivalent to the existence of a conditioned invariant distribution. In this paper, we consider the local observer design only for the nonlinear systems which are Lyapunov stable. This is because of the conceptual problem, viz. what does the existence of a local exponential observer mean in terms of the nonlinear dynamics of the system to be observed? For example, it must mean that the trajectories do not have finite escape time, but what does local existence mean for unbounded trajectories ? For this reason, we have focused our efforts in treating the local existence problem on those nonlinear systems which are Lyapunov stable. This focus leads to a clearly posed problem for which necessary and sufficient conditions can be derived. This paper is organized as follows. In Section 2, we give the basic definitions for local asymptotic and exponential observers for nonlinear systems. In Section 3, we characterize the local exponential observers for unforced systems which are Lyapunov stable. In Section 4, we establish that the definition of local exponential observers can be weakened for neutrally stable nonlinear systems. Finally, in Section 5, we extend our results to a class of nonlinear systems with inputs, and establish that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies, and is implied by the existence of local exponential observers in the absence of inputs.

2. DEFINITIONS In this paper, we consider a nonlinear plant of the form k =

F(x, u),

Y=

h(x),

(1)

where x E IP is the state, u E R”

the input, and y E IwPthe output of plant (1). The state 2 belongs to an open neighborhood X of the origin of EP, and the input mapping u takes values in an open neighborhood U of the origin of IP. We assume that F : X x U --+ IP is a C1 vector field and also that F(O,O) = 0. We also assume that the output mapping h : X -+ iW is a C1

map, and also that h(0) = 0. Let Y A h(X). a class of functions U.

W e suppose that the system inputs u(.) belong to

DEFINITION 1. (See [9,101.) A C1 dynamical system described by

2 = G(z,Y, u),

(z

E

W)

is called a local asymptotic (respectively, exponential) observer system (l),(2) satisfies the following two requirements.

(2) for plant (1) if the composite

I. If x(O) = z(O), then x(t) = z(t),Vt 2 0 and VU(.) E U. II. There exists a neighborhood V of the origin of Rn such that for all z(0) - z(0) E V, the estimation error z(t) -x(t) decays asymptotically (respectively, exponentially) to zero. 1 REMARK 1. There are some important cases of interest included in Definition 1. First, the case U = (0) corresponds to the problem of finding local observers for unforced dynamical systems,

27

Local ObserverDesign

which is treated in Section 3. Other important cases of interest are constant inputs, and periodic inputs with any given period, both of which are treated in Section 5.

I

The estimation error e is defined by eez-z. Then e satisfies the differential equation t! = G(x + e, h(x), u) - F(x, u). We consider the composite system

jr = F(x,

u),

(3)

& = G(x + e, h(x), u) - F(x, u).

Next, we state a simple lemma which provides a geometric characterization of Condition (I) in Definition 1. LEMMA 1. (See [21].)

The following statements are equivalent.

(a) Condition I in Definition 1 holds for the composite system (l),(2). (b) G(z, h(z),u) = F(z, u), Vx E X, and VU(.) E U. (c) The submanifold defined via e = 0 is invariant under the Aow of the composite tem (3).

sysI

3. LOCAL EXPONENTIAL OBSERVERS FOR UNFORCED SYSTEMS In this section, we study the problem of finding local exponential observers for unforced systems. So, we suppose that U = (0). When the input u is set equal to zero, plant (1) reduces to 5.T= F(x,O)

ef(x),

(4

y = h(x). Also, the candidate observer (2) reduces to

2 = W,

Y, 0)

Adz, y).

,

(5)

Accordingly, the composite system (3) takes the simple form

25= f(x),

t = g(x

(6)

+ e, h(x)) - f(x).

We denote

A

e g(O),

E

e z(O,O),

and

K i $(O,O).

(7)

Next, we state a fundamental theorem that completely characterizes the existence of local exponential observers of form (5) for Lyapunov stable nonlinear plants of form (4). This theorem can be established using center manifold theory as in [21]. However, we state and prove a more general theorem in Section 5 using Lyapunov stability theory.

V. SUNDARAPANDIAN

28

THEOREM 1. Suppose that the plant dynamics in (4) is Lyapunov stable at the equilibrium x = 0. Then system (5) is a local exponential observer for plant (4) if, and only if, (a) the submanifold defined via e = 0 is invariant under the Aow of the composite system (6); (b) the dynamics 6 = g(e, 0) is locally exponentially

stable.

I

REMARK 2. Theorem 1 provides the necessary and sufficient conditions for a candidate observer of form (5) to be a local exponential observer for Lyapunov stable plants of form (4). Implicitly contained in the statement of a local exponential

of Theorem 1 is a necessary condition on plant (4) for the existence

observer, namely that the system linearization

pair (C, A) is detectable,

i.e., that there exists a matrix K such that A - KC is Hurwitz. This necessary condition was obtained by Xia and Gao [9]. The proof for this necessary condition is basically based on the observations

that Condition

(b) in Theorem 1 implies that the matrix E = gz(O,O) is Hurwitz

and also, Condition (a) in Theorem 1 implies that the equation E + KC = A holds. These two observations combined together yields the necessary condition that E = A - KC is Hurwitz, i.e., that the linearization

of plant (4) is detectable.

I

REMARK 3. Inspired by Theorem 1, one may try to give the following characterization asymptotic

for local

observers of form (5) for Lyapunov stable plants of form (4).

(a) The submanifold defined via e = 0 is invariant under the flow of the composite system (6). (b) The dynamics B = g(e, 0) is locally asymptotically

stable.

While (a) and (b) are indeed necessary conditions for local asymptotic sufficient. This can be easily seen by considering the plant

observers, they are not

i = 0, y =

x2

and the candidate observer i = -Z3 + yz. Next, as an application of Theorem 1, we state and prove a simple construction

I procedure for

local exponential observers for Lyapunov stable plants of form (4). First, we state the necessary condition due to Xia and Gao. THEOREM 2. (See 191.) If plant (4) has a local exponential observer, then the system linearization pair (C, A) must be detectable.

I

Next, we show that the above necessary condition is, in fact, sufficient if the plant dynamics in (4) is Lyapunov stable at x = 0. THEOREM 3. Suppose that the plant dynamics in (4) is Lyapunov stable at x = 0, and suppose also that for some n x p matrix K, A - KC is Hurwitz. Then the dynamical system defined by 2 = f(z) + K[y - h(z)]

(8)

is a local exponential observer for plant (4). Th us, if the system linearization pair (C, A) is detectable, there always exists a local exponential observer of form (8) for plant (4). PROOF. It is easy to check that and the candidate observer (8) satisfies Conditions in Theorem 1.

(a) and (b)

We combine the two previous theorems to obtain a necessary and sufficient condition Lyapunov stable system to have a local exponential observer as follows.

I for a

THEOREM 4. If the plant dynamics in (4) is Lyapunov stable, then a necessary and sufficient condition for plant (4) to have a local exponential observer is that the system linearization pair (C, A) is detectable. I Next, we illustrate our construction

procedure with an example.

Local Observer Design

EXAMPLE 1. Consider

the nonlinear

system

described

29

by

(9)

In polar coordinates, x1 = rcosl9, the plant

dynamics

assumes

x2 = rsint3,

a simple form

04

r2 sin

7: =

r

’

(10)

s = -1. Therefore,

T = l/n,

19= t (n E Z+) describes

a circular

orbit

with center

radius l/n. Any orbit passing though a point in the xixz-plane that diverges from the unit circle with increasin, u time since F > 0 when spiral outward in the counter-clockwise direction since 6 = 1. It is circular orbits with radii 1/(2n + 1) (n E 2,) are attractive, while the 1/(2n)(n

E 2,)

at the origin

is outside

the unit

T > 1. Also, the orbits also easy to see that the circular

orbits

with radii

are repulsive.

System (10) has a Lyapunov stable equilibrium at the origin since every neighborhood origin has a solution curve encircling the origin. This is a classical example of a Lyapunov system. Linearizing the plant (in its Cartesian coordinates), we obtain the system matrices, C=[l

O]

0

A=

and

-1

1

o

[ Clearly, A - KC

Hence,

and circle

(C, A) is an observable pair. In particular, is Hurwitz with eigenvalues -1 and -1. by Theorem

3, the dynamical

‘I 21

I

defined

-22 + 21 (.$ + .zi) sin

ZZ

.52

is a local exponential

system

zl + z2 (,zf + 2:)

observer

regulation

.

K

the matrix

= col(2,O)

is such that

by

sin

&?G[I >

+

;

[Y--l-g],

X

)I

(9).

I

4. EXISTENCE OF LOCAL OBSERVERS FOR NEUTRALLY As in the output

1

X

( &?z

(

for plant

of the stable

of nonlinear

systems

EXPONENTIAL STABLE SYSTEMS

[4], Condition

I in Definition

1 which states

that the zero error manifold is itself invariant can be considerably weakened in some cases. We note that all that is needed for asymptotic observation is the existence of an attractive, invariant submanifold contained in the zero error submanifold. DEFINITION 2. Lyapunov points

We say that the plant dynamics

2 = f(x)

stable at x = 0 and, for some neighborhood

of all trajectories

which are initialized

is neutrally

stable at x = 0, if it is

Q of x = 0, the set R of all positive

limit

in Q is such that 0 n Q is dense in Q.

The next result says that if the unforced plant dynamics is neutrally for local exponential observers can be weakened for this plant.

stable,

then the definition

I

30

V. SUNDARAPANDIAN

THEOREM 5. Suppose that the plant dynamics in (4) is neutrally stable at x = 0. Then any system of form (5) satisfying Condition II also satisfies Condition I in Definition 1 for local exponential observers, and hence, is a local exponential observer for plant (4). PROOF. In view of Lemma 1, we need only to prove that the submanifold defined via e = 0 is invariant under the flow of the composite system (6). Using (7), the composite system (6) can be locally expressed as = where 4(x)

E+KC-A A

(11)

E

and $(x, e) are C1 functions vanishing at 5 = 0 and (x, e) = (O,O), respectively,

together with all their first-order derivatives. By hypothesis, matrix A has all eigenvalues with zero real part. By Condition II in Definition 1 for local exponential observers, it follows that the dynamics 6 = g(e, 0) is locally exponentially stable. Hence, the linearization matrix E = gz(O, 0) is Hurwitz. Since the matrices A and E have disjoint spectrum, we know that the Sylvester equation TA=ET+(E+KC-A) has a unique solution T. Under a linear change of variables d = e - TX, system (11) takes the form

[b]=[t 11[;I+[qg)]

(12)

3

where 4(x) and $( e, e-) are C1 functions vanishing at x = 0 and (x,E) together with their first partial derivatives.

= (O,O), respectively,

Hence, system (12) has a center manifold at (x, a) = (O,O), the graph of a C’ mapping (defined on a neighborhood N) f? = ii(x), with ii satisfying ii(O) = 0 In the original coordinates

and

D%(O) = 0.

(x, e), this center manifold is the graph of the mapping r(x)

= %(x) + TX,

(13)

with n satisfying n(0) = 0

and

On(O) - T = 0.

(14)

We define F = V n N n Q, where V is the neighborhood stated in Condition II in Definition 1 for local exponential observers and Q is the open neighborhood stated in Definition 2. It is clear that the positive limit set off is dense in F. We denote the positive limit set off in F by P. Let p E P be arbitrary. Then there exists a point 2’ E F and a sequence t, E R such that t, -+ +oc~ and x(t,; x0) -+ p as n -+ foe. Hence, ~ (z (%~“))

--+ T(P),

asn++oo,

i.e., e (G z”) -+ r(p),

asn-++co.

Since x0 E V, we know that e(t,;xO) + 0, as n + 00. Hence, w(p) = 0, for each p E P. Since P is dense in F, it follows that ?r = 0 in F. Hence, the submanifold defined via e = 0 is locally invariant under the flow of system (11). I

Local Observer Design

31

5. LOCAL EXPONENTIAL OBSERVERS FOR A CLASS OF NONLINEAR SYSTEMS WITH INPUTS In this section, we suppose that the class Lf consists of inputs u(.) of the form u =

r(w),

(15)

where w satisfies the autonomous system (exosystem) LJ = s(w). The state w of exosystem (16) lies in an open neighborhood

(16) W of the origin in KY. One can

view equations (16) and (15) as an input generator. Thus, we consider a class of nonlinear systems with inputs of the form j: = F(x, r(w)),

ti = s(w), Y

(17)

= h(x),

where x is the state and y the output of the plant. We suppose that F, r, s, and h are all C’ vector fields and also that F(O,O) = 0, r(0) = 0, s(O) = 0, and h(0) = 0. In this paper, we assume that the exosystem dynamics (16) is Lyapunov stable. Thus, the class 6f inputs that we consider in this paper include the important cases of constant inputs, and periodic inputs with any given period. The candidate observer (2) defined in Definition 1 takes the form i = G(z, y, r(w)). The error vector e A z - x now satisfies the dynamics d = G(x + e, h(x), r(w)) - F(x, r(w)).

(19)

The composite system (3) takes the form 2 = F(x, r(w)), l.2= s(w),

(20)

6 = G(x + e, h(x), r(w)) - F(x, r(w)). We denote the linearization matrices of plant (17) by A A g(o,o),

B A g(o,

o),

R g z(o),

and

When w is set equal to zero, (i.e., when no input is present), plant (17) reduces to i = F(x,O), y = h(x).

(21)

Next, we prove a fundamental theorem that completely characterizes the existence of local exponential observers of form (18) for Lyapunov stable plants of form (17).

32

V. SUNDARAPANDIAN

THEOREM 6. Suppose that the plant Then system (18) is a local exponential

dynamics observer

(a) the submanifold

defined via e = 0 is invariant

(b) the dynamics

d = G(e, 0,O) is locally

PROOF. The necessity local exponential theory

in (17) is Lyapunov stable at (x,w) for plant (17) if and only if

part

observers).

under the flow of the composite

exponentially

follows immediately The sufficiency

(19) can be written

system

(20);

stable.

from Conditions

I and II in Definition

part can be established

as in [al]. Here, we give a new proof using Lyapunov

the error dynamics

= (0,O).

using

stability

the center

theory.

1 (for manifold

By Condition

8 = Ee + cy(z, w, e)e, where a(~, w, e)e consists By Also, i! = (z,w,

(a),

in the form

of the nonlinear

terms

(22)

in the error dynamics

and a(0, 0,O) = 0.

hypothesis, (z, w) = (0,O) is a Lyapunov stable equilibrium of the plant dynamics in (17). by Condition (b), e = 0 is a locally exponentially stable equilibrium of the dynamics G(e,O,O). Hence, from a total stability result [22, p. 446, Corollary], it follows that e) = (O,O,O) is a Lyapunov stable equilibrium of the composite system (20) (by its tri-

angular structure). Hence, for any E > 0, there exists a 6 > 0 such that (23) By Condition

(b), it follows immediately

that

the linearization

E of the dynamics

matrix

i! = G(e, 0,O) is Hurwitz. Lyapunov

Hence, there

exists a real, symmetric,

positive

definite

matrix

P solving

the matrix

equation ETP $ PE = -I.

To show that candidate

the error

Lyapunov

dynamics

(19) is locally

(24)

exponentially

stable

at e = 0, we take the

function V(e) = (e, Pe) = eTPe.

Clearly,

V is a quadratic

positive

definite

function.

Differentiating

(25) V along

the trajectories

of (19), we get V(e) = (k, Pe) + (e, Pk)

= (Ee + a(~, w, e)e, Pe) + (e, PEe + Pa(z, w, e)e) = (e, (ETP + PE) e) + (Pe, a.(~, w, e)e) + (e, Pa(s, w, e)e) I (e, e) C-1 +

Wll~),

(by (23) and (24)).

Thus, choosing E sufficiently small, p can be made a negative definite function in a small open neighborhood of e = 0. Hence, it follows from Lyapunov stability theory that e = 0 is a locally exponentially stable equilibrium for (19). This completes the proof. I EXAMPLE 2. Suppose that the unforced plant dynamics in (21) is locally asymptotically and also that the class U consists of inputs u of the form u = T(W), where iI = SW,

O

s=

[ ti = 0.

u

-” 0 1

’

stable,

Local ObserverDesign

33

Clearly, U consists of periodic inputs with any desired period. Thus, the plant dynamics in (17) is Lyapunov stable at (5,~)

= (0,O) (by its triangular structure).

Hence, the hypothesis of

Theorem 6 holds for this example.

I

As an application of Theorem 6, we establish the following result which states that when the plant dynamics in (17) is Lyapunov stable at (z, w) = (0,0), the existence of a local exponential observer for plant (17) in the presence of inputs implies and is implied by the existence of a local exponential observer for plant (17) in the absence of inputs. THEOREM 7. Suppose that the plant dynamics in (17) is Lyapunov stable at (5, w) = (0,O). If the system f = G(z, Y, T(W)) is a local exponential observer for plant (17), then the system defined by 2 = G(z, Y, 0) 2 9(2, Y) is a local exponential observer for the unforced plant (21). Conversely, if the system 22= 9(&Y)

(26)

is a local exponential observer for the unforced plant (21), then the system defined by f = G(z, Y, +J))

i g(z, Y) + F(z, +))

- J’(z, 0)

(27)

is a local exponential observer for plant (17). PROOF. The first part of this theorem is straightforward. So, we prove only the converse. For this, suppose that (26) is a local exponential observer for the unforced plant (21). We will establish that the system defined by (27) is a local exponential observer for plant (17). Since the plant dynamics in (17) is Lyapunov stable at (z, w) = (0,0), it suffices to show that the candidate observer (27) satisfies Conditions (a) and (b) in Theorem 6. Using the definition of G(.z, y, T(W)) in (27), we find that G(x, h(z), T(W)) = 9(x, h(z)) + F(z, T(W)) - F(z, 0).

(28)

Since (26) is a local exponential observer for the unforced plant (21), we know from Lemma 1 that VXEX.

9(x, h(z)) = F(z, O), Hence, equation (28) takes the form G(x, h(z), r(w)) = F(x, T(W))>

VXEX

and

QwEW.

Thus, applying Lemma 1 again, we see that system (27) satisfies Condition (a) in Theorem 6. Also, since (26) is a local exponential observer for the unforced plant (21), we know from Theorem 1 that the dynamics 6 = g(e, 0) is locally exponentially stable. But G(e, 0,O) = g(e, 0) + F(e,O) - F(e, 0) = g(e, 0). Hence, it is immediate that the dynamics i: = G(e, 0,O) is also locally exponentially stable. This shows that system (27) also satisfies Condition (b) in Theorem 6. This completes the proof. I Under the standing assumption that the plant dynamics in (17) is Lyapunov stable at (xc,W) = (0,0), the next theorem shows that a necessary and sufficient condition for plant (17) to have a local exponential observer is that the system linearization pair (C, A) of the unforced plant (21) is detectable. This theorem is a simple consequence of Theorems 6 and 7.

V. SUNDARAPANDIAN

34

THEOREM 8. Suppose that the plant dynamics in (17) is Lyapunov stable at (z,w) = (0,O). If plant (17) has a local exponential observer, then the system linearization pair (C, A) of the unforced plant (21) is detectable. Conversely, if the system linearization pair (C, A) of the unforced plant (21) is detectable, and if K is a matrix such that A - KC is Hurwitz, then the dynamical system defined by

4 = G(z, y, r(w)) fi F(z,r(w)) + K[y - h(z)] is a local exponential

(29)

observ& for plant (17).

PROOF. If plant (17) h as a local exponential observer, then it follows that the unforced plant (21) also has a local .exponential observer. By Theorem 2, the system linearization pair (C, A) of the unforced plant (21) is detectable. Conversely, if (C, A) is detectable,

by Theorem 3, a local exponential observer for the unforced

plant (21) is given by i- = g(z, y) 2 F(z, 0) + K[y - k(z)]. Hence, by Theorem 7, a local exponential observer for plant (17) is given by 4 = G(z, y, r(w))

b g(z,

y) + F(z, T-(W)) - F(z, 0).

Note that G(z, Y, T(W)) = F(z, 0) + I((Y - h(z)) + F(z, T(W)) - F(z, O), i.e., G(z, Y,T(w)) = F(z, T(W)) + K[Y - h(z)]. This completes the proof. Next, we illustrate our construction EXAMPLE

procedure with an example.

3. Consider the nonlinear system defined by ii =

-2X2

i,

Xl

=

+X2X3

-

ci3 = x1x2 Ljl

=

lb2 = ti =

-

Xf + Ufl/”

x;

+ WlW2U,

x1x3

-

-

+ Y2 + w;,

2;

-

W,3,

vw2,

(30)

-vw1, 0,

y1 = x3 + x:x;,

y2 = x1 - x2 + XT. The unforced plant is easily obtained as kl

=

-2X2

+X2X3

iJ2 = x1 -

x123

53 = 51x2

-

Yl

-

-

X!,

x;,

x;,

(31)

2 3 = X3 fXlX2,

y2 = 21 - x2 + XT. It can be easily checked that V(x) = xf + 2~; + x$ is a Lyapunov function for the plant dynamics in the unforced system (31).Hence, by Lyapunov stability theorem, x = 0 is a locally asymptotically stable equilibrium of the plant dynamics in (31). Also, the exosystem is easily

Local Observer Design

seen to be Lyapunov

Corollary],

stable

in (30) (by its triangular Linearizing

the

at w 4 (WI, ~2, V) = 0. Hence, from a total stability (z, w) = (0,O) is a Lyapunov

it follows that

plant

(31), we obtain

K=

Hence,

the matrix

by Theorem

A - KC is Hurwitz 8, a local exponential i =

i.e.,

[I [ Zl

Z2

23

=

stable

equilibrium

result

[22, p. 446,

of the plant

dynamics

structure).

unforced

the system

It, is easy to see that the pair (C, A) is observable.

we find that

35

matrices

1

In particular,

0 [0

2 -2

2

0

choosing

,

with the eigenvalues observer

for plant

-2,

-2,

and -2.

(30) is given by

F(z,T(w)) + K[y - h(z)],

-222 + Z2Z3 - Zf + W;2Y2 Z1 - ZlZ3 - 2; + WlW2V z~Z~-z~+u2+W;

Wi

0 I[ I[ +

0

2

2 -2 0

y1 - z3 - z;z2”

yz-zlfq-zf

1 .

I

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