Dissipativity and Design of Unknown Input Observers for Nonlinear Systems

Copyright © IFAC Nonlinear Control Systems, Stuttgart, Gennany, 2004

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IFAC PUBLICATIONS www.elsevier.comllocatelifac

DISSIPATIVITY AND DESIGN OF UNKNOWN INPUT OBSERVERS FOR NONLINEAR SYSTEMS Edmundo Rocha-C6zatl,2 J aime A. Moreno 3

Instituto de Ingenieria, Universidad Nacional Autonoma de Mexico (UNA M), Ap. Postal 70-472, 04510 Mexico, D.F., Mexico. [email protected]

Abstract: Unknown input observers (DIO) are able to estimate perfectly the state of a system, despite of completely unknown input perturbations. Previous works of the authors have shown that for linear systems the existence of an UIO is equivalent to the possibility of rendering the plant dissipative by output injection. In this paper it will be shown that, under some weak regularity conditions, an UIO can be constructed for nonlinear systems if an error system, build as the difference of the plant and the system with zero unknown input, can be rendered dissipative by output injection. Copyright @2004 IFAC Keywords: Feedback Passivity, Unknown Input Observers, Strong* Detectability, Passivity, Output Injection Passivity, Passive Design, Dissipative Design.

1. INTRODUCTION

Dissipativity plays a fundamental role in control theory. This is related, on the one side, to its strong relationship with the physical concepts of energy and mass balance, making it physically appealing. On the other side, dissipative systems have important interconnection properties as the passivity theorem (Sepulchre et al., 1997; Van der Schaft, 2000), what has lead to the development of several strategies to design robust control for linear and nonlinear systems (Byrnes et al., 1991; Jiang and Hill, 1998; Sepulchre et al., 1997).

Since Unknown Input Observers (UIO) are able to estimate the state of a system robustly against completely unknown input perturbations, they are of great importance for fault detection and isolation, in the design of robust observers and in the control of decentralized systems. In the linear case, the existence conditions were completely established by (Hautus, 1983) and many different design methodologies are known (Chu, 2000; Hou and Miiller, 1994; Saberi et al., 2000). In the nonlinear case, these conditions are not very well established, but there are some recent works aiming at characterizing existence conditions (Moreno, 2000b; Moreno, 2oo0a).

Since dissipativity (passivity) stays at the roots of many robustness concepts, it is then not surprising that the existence of UIOs is strongly related to it. For the linear time invariant case it has been shown in (Moreno, 2001) that the existence of an UIO for a system is equivalent to the possibility of rendering it strictly dissipative, from the unknown inputs to the outputs and with respect to some class of supply rates, by using whether state feedback or output injection. This result is not only of theoretical interest, but establishes a bridge

1 This work has been done with the financial aid of DGAPA-UNAM under project PAPIIT INlO6002-2 and CONACyT under project 34934A. 2 Partial support through scholarships awarded by DGEPUNAM and CONACyT. 3 On leave at the ISR, University of Stuttgart, Germany

471

observer (UfO) for ~ if it is able to estimate asymptotically the state of ~ despite of the lack of information on w, Le. lim (x (t) - i: (t)) = 0 . t-oo

between both areas leading to their enrichment. In the case of square systems, i.e. those having the same number of unknown inputs as outputs, dissipativity specializes to passivity. It is then natural to wonder if in the nonlinear case a similar relationship is valid. For square systems (Rocha-C6zatl and Moreno, 2002) show that, under some additional regularity assumJ>tions, the existence of VIa implies the possibility of rendering the system passive by feedback or output injection, but that the converse is not true in general. This work shows that passivity of the plant is a very weak concept to allow for the existence of an VIa.

2.1 Error dynamics

The error dynamics of the plant ~e:

consists of two copies of the plant, one of which with zero unknown input, and its output is the difference of the individual ones. By the diffeomorphic change of coordinates

T: (x,() ........ (x,l:) ~e

(2) F(l:;x,u) = f (x,u) - f (x -l:,u) . H(l:,x)=h(x)-h(x-l:) . Note that F (0; x, u) = 0, and H (0, x) = 0 for all x and u. Moreover, F(-l:;x -l:,u) = -F(l:;x,u) and H (-I:,X - 10) = -H (l:,x).

2.2 Partial stability

Results from partial stability theory are going to be useful in the developments, and some will be recalled here for completeness (Chellaboina and Haddad, 2002). Consider the nonlinear system

Xl = h (Xl,X2) , X2 = 12 (Xl,X2) ,

,

x (0)

= XQ

,

Xl (0) = XlO , x2 (0) = X20 ,

(3)

where Xl E V c;;;: ]Rn" V is an open set with 0 E V, X2 E ]Rn1 , h : V x ]Rn1 ........ ]Rn, is such that for every x2 E ]Rn1 , h(0,X2) = 0, and h(·,X2) is locally Lipschitz in Xl, 12 : V x ]Rn, ........ ]Rn, is such that for every Xl E V, 12 (Xl, .) is locally Lipschitz in X2. Recall the partial stability definitions:

Consider the nonlinear MIMa system

y=h(x)

X = f (x, u) + 9 (x) w , x (0) = Xo , i.:F(I:;x,u)+g(x)w, 10(0)=100, { e-H(l:,x) ,

where

2. PRELIMINARIES

.

= (x,x - ()

can be represented in a more useful form

Ee :

The paper is organized as follows: In Section 2 some preliminary results and definitions are introduced. In particular, the error system is defined and some necessary partial stability results are reviewed. Section 3 presents the main result, Le. the existence of an VIa follows from the possibility of rendering dissipative the error system. The necessary assumptions and some consequences are discussed. An example in Section 4 illustrates the results.

. { x = f (x, u) + 9 (x) w

= f (x, u) + 9 (x) w , x (0) = Xo , ( = f ((, u) , (0) = (0 , { e = h (x) - h (() , ~

In this paper, an error system, build as the difference of the plant and a copy of it with zero unknown input, will be considered. It will be shown that if the error system can be rendered strictly dissipative for some class of supply rates (or passive for square systems) by using output injection, and under one additional regularity assumption, then an VIa exists. The dissipativity of the error system can be also interpreted as an incremental dissipativity of the plant. In the linear case both concepts are equivalent, and the known results are recovered, and the condition is necessary too. For some material related to the concept of output injection dissipativity see (Rocha and Moreno, 2001).

~

~,

(1)

where x E ]Rn is the state vector, u E ]RP is the known input, w E ]Rq is the unknown input and y E ]RID is the output of the system. Let f (x, u), g(x) and h(x) be smooth functions of x E ]Rn. Note that fx (x) = af (x) lax.

Definition 1. The nonlinear system (3) is Lyapunov stable with respect to Xl uniformly in X20 if, for every c > 0, there exits b (c) > 0 such that IlxlOll < b implies that Ilxl (t)11 < c for all t ~ 0 and for all X20 E ]Rn,. The nonlinear system (3) is (globally) asymptotically stable with respect to Xl uniformly in X20 if it is Lyapunov stable with respect to Xl uniformly in X20 and there exists b > 0 such that IIxlOll < b (for every XlO E ]Rn,) implies that limt_oo Xl (t) = 0 for all X20 E ]Rn1.

The usual theory of observers deals with the problem of designing an state estimator of system (1) using information from inputs u (t) and w (t) and output y (t) of the system. However, when w (t) is not measurable a system n with input y and u, and output i: is called an unknown input

472

Remark 4. A slight generalization can be easily obtained by allowing output transformations in the plant, Le. diffeomorphic variable changes of the output. In principle, it is possible to generalize the dissipative condition by changing the term Se in (7) by a more general one p(e), with p(O) = O. However, because of the transformation in the next section it follows that p (e) has to be linear.

A Lyapunov characterization is given by

Theorem 2. (Chellaboina and Haddad, 2002, Theorem 1) Consider the nonlinear system (3). Then the following statements hold: (i) If there is a continuously differentiable function V : D x IRn, - 4 IR and class K functions {3 ('), , (-) satisfying for (Xl, X2) E D x IRn ,

° (.),

° (lI x III) :S V(XI,X2):S {3(lIxtll)

,

(4) System Ee is said to be partially dissipative because the storage function can depend only on f, and the behavior of the X subsystem is not very relevant. The output injection is called partial since it affects only the € subsystem and not the whole state of the system. Note that if the system Ee is POIPSD then E~i is (partially) strictly passive with respect to the new output Se. Partial strictly dissipativity corresponds to standard dissipativity (Byrnes et al., 1991), but with only positive semidefinite storage function.

and

V (Xl, X2) :S -, (lIxlll) , (5) then the nonlinear dynamical system (3) is asymptotically stable with respect to Xl uniformly in X20· (ii) If D = IRn , and there exists a continuously differentiable function V : IR n, x IR n, - 4 IR, a class K function, (.), and class K oo functions 0('), {3 ('), satisfying (4) and (5), then the nonlinear dynamical system (3) is globally asymptotically stable with respect to Xl uniformly in X20.

Assumption 1. The error system plant is POIPSD.

3. SUFFICIENT CONDITIONS FOR EXISTENCE OF DIO

It will be useful to transform E~ into a global normal form. Although for LTI systems this is a consequence of Assumption 1, and for nonlinear systems local results are valid under weak conditions (Byrnes et al., 1991; Isidori, 1995), the global results are only valid under stronger requirements.

3.1 Output Injection Dissipativity of the error

dynamics The main ingredient in the relationship between existence of UIOs and dissipativity is the possibility of rendering the error system Ee (partially) dissipative by means of a (partial) output injection.

Assumption 2. For system E~i (6) the output variables Se = SH (f, x) can be completed by a set of smooth functions ~ (f, x) = 4J (x) - 4J (x - f) such that 4Jx (x)g(x) = O.

Definition 3. Ee (2) is said to be Partial Output Injection Partial Strictly Dissipative (POIPSD) if there exist a continuously differentiable function L (e; y, X - f, u), with L (0; y, X - f, u) = 0 for all y, X, f, u, a continuous differentiable storage function V (f,X), positive definite in f, uniformly in x, i.e. it is satisfied for all (f,X) E IR n x IR n

(1Ifl!) :S V (f, X) :S

02

and

is a global diffeomorphism. It follows easily that if P is a diffeomorphism then

r (x) £ [ S:(~)] is also a diffeomorphism. Using

for some class K oo functions 01 (.), 02 ('), and a full file rank matrix S E IRqxm, such that along the trajectories of the system X = f (x, u) + 9 (x) w , E=F(f;X,u)+L(e;y,x-f,U)+g(x)w, { e=H(f,x) , (6) (with initial conditions X (0) = Xo, f (0) = fO ) it is satisfied for all (f, x) E Rn X IR n E~:

for a class K function

,

(~) ~ (~) ~ (s~J'~)))

p

(!If!!)

V :S -03 (1Ifl!) + w T Se

(2) of the

3.2 Existence of global normal form

Sufficient conditions for the existence of UIO will be stated in this section, that, when specialized to the LTI case, are necessary and sufficient.

01

Ee

as new coordinates 17 = r (x) and z = r (x) r (x - f) system E~ is transformed into the global normal form

r,l = 11 (17,U) + r(17)W, r,2 = 12 (17, u) , i 1 = 4>1 (z; 17, u) + Al (e; y, 17 - z, u) + r (17) w i 2 = 4>2 (z;17,u) + A2 (e;y,17 - z,u) ,

= zl = Se, e2 = J1. (17) - J1. (17 - z) = Te .

el

(7)

(8)

03 (.).

473

with initial conditions 11 (0) where

= 110, z (0) = zo, and

It turns out that this condition is necessary for the existence of an UIa for

]1 (11, u) = Sh x (x) I (x, u) ]2 (11, u) = 1 (Z; 11, u) = A (11, u) - ]1 (11 <1>2 (Z; 11, u) = ]2 (11, u) - ]2 (11 -

Theorem 5. Suppose that there exists an UIa for Then for system (10) £ = 0 is an asymptotically stable equilibrium for every (x, u, w) trajectory of the plant ~. ~.

z, u) z, u) A1 (eiY,11- z,u) = Sh x (x - £) L (eiY,x - £,u) A 2 (e;Y,11- z,u) =
PROOF. Suppose there is an initial state

3.3 Some consequences of the assumptions

The property expressed in Theorem 5 is a kind of detectability and leads in the linear case to the strong detectability introduced in (Hautus, 1983). More precisely, consider the LTI system

3.3.1. Existence of an observer for vanishing unknown input. From Assumption 1, if the unknown input is zero, i.e. w = 0, then (7) implies, together with Theorem 2, that £ -+ 0 when t -+ 00 uniformly with respect to Xo. This means that the system

f ((, u) -

Y = h (()

H.

L (y -

y; Y, (, u)

,

i; =

Ax+ Bw

y=Cx

((0) = (0 ,

(9) constitutes an asymptotic convergent observer for the plant ~ when there is no unknown input. Note that (6) represents the composite dynamics of plant and estimation error when system (9) is used as an observer.

In (Hautus, 1983) this system is called strongly detectable if for every input and initial state such that Y (t) = 0 =* lim x (t) = O. This is t-oo equivalent to the algebraic condition rank [ sI ;; A

-oB] = n + q, V sECt,

ct

where is the closed right half plane of the complex plane, that is a necessary condition for existence of UIa.

Note, however, that from (7) it cannot be concluded that (9) is an UIa for the plant, since for its convergence it is required that the dissipation term w T Se -+ 0 as t -+ 00, what is in general impossible if the perturbation w does not vanish, as it is clear from the normal form (8).

(10) represents the zero dynamics of (6), and it has to be weakly stable in the particular sense given by Theorem 5. This corresponds to a partial minimum phaseness of the error system t e (2), i.e. e (t) = 0 =* lim £ (t) = o. Note that the t-oo

satisfaction of this property is independent of the output injection term L O.

3.3.2. Minimum Phaseness: A necessary condition lor existence of UfO. Although the foregoing assumptions are going to be shown to be sufficient for the existence of an UIa for the plant, Assumption 1 implies a necessary condition for this existence, that will be now clarified. Consider that in (6) it eventually happens that the error output e (t) = 0, Vt 2: 0 (this is for example the case if £0 = 0 and w = 0). In that case L (0; Y, (, u) = 0 and (6) reduces to the Differential-Algebraic Equation (DAE)

= I (x, u) + 9 (x) w, x (0) = Xo , f = F(£;x,u) + g(x)w , £(0) = £0, O=H(£,x) .

£0

and a trajectory of the plant (x,u,w) such that the corresponding (£, x, u, w) satisfy (10) and for which £ (t) -+t 0 as t -+ 00. Then this implies that (x - £, u, 0) is also a trajectory of the plant that is indistinguishable from (x, u, w) and this contradicts the existence of an UIa.

using x = r- 1 (11), and x - £ = r- 1 (11- z), and [ST, TT] a regular matrix. Note that r (11) is not assumed to be regular, i.e. the relative degree does not need to be well defined.

...... { ( =

~.

It follows that in order to be able to satisfy Assumption 1 it is necessary (as in the linear case) that the number of outputs is bigger or equal to the number of unknown inputs, i.e. m 2: q.

3.3.3. Controlled Minimum Phaseness The same arguments of the previous paragraph show that Assumption 1 implies that if the error output e1 (t) = Se (t) = 0, Vt 2: 0, then it follows from (7) and (6) that for the system

i;

i;

(10)

f

= I (x, u) + 9 (x) w , = F (£; x, u) + L (er; Y, X

-

£, u)

+ 9 (x) w ,

O=SH(£,x) ,

It follows from (7) that for system (10) the set

(ll)

0 is an asymptotically stable equilibrium uniformly in (x, u, w), where (x, u, w) correspond to trajectories of the plant (see also Theorem 2).

where er corresponds to the restriction of the variable e to values such that Se = 0, the set £ = 0 is an asymptotically stable equilibrium

£

=

474

uniformly in (x, u, w), where (x, u, w) correspond to trajectories of the plant (see also Theorem 2). This property can be interpreted again as a partial minimal phaseness of the error system, with two main differences: In (11) i) only a reduced number of outputs is considered, Le. e\ = Se, and ii) the dynamics of the error system can be changed by an output injection depending on the outputs not set to zero, er' Consequently, (10) and (11) are in general not equivalent, except when S is a square matrix. The assignment of the zero dynamics will be an important issue in the observer design.

dered dissipative by output injection. In the linear case, however, these two conditions coincide.

Remark 8. In the linear case Assumption 2 follows from Assumption 1, and this latter condition is sufficient for the existence of an VIO. Moreover, for LTIs the condition is also necessary. Remark 9. A similar idea of passivating the error equation has been used in (Shim et al., 2003) for the design of observers for systems without unknown inputs. This method can be interpreted in the context of Theorem 6: by defining a dummy unknown input and if the error equation can be rendered strictly passive, then system 0 (9) is an observer for the plant (without unknown inputs) in accordance with Section 3.3.1.

It is clear that the solution set of system (10) is contained in the solution set of system (11), and it follows that the later condition is stronger than the former one. In the transformed coordinates (8) (when Assumption 2 is satisfied), the previous condition implies that for

Z2

=

(0, Z2; 1],u)

+ A2 (O,e2;y,1]I,1]2

- Z2,U) , (12) with initial condition Z2 (0) = Z20, the point Z2 = 0 is an asymptotically stable equilibrium uniformly in (1], u). Moreover, the (transformed) storage function V (ZI' Z2, 1]), from the Definition 3, restricted to Z\ = 0, is a Lyapunov function for (12) that satisfies the conditions of Theorem 2. 4>2

4. EXAMPLE To illustrate the results, the model of an induction machine will be used (Marino and Tomei, 1995) 1 = - 2aXTJI I X2 - ]TL X = Ax + wNx + Bu

w.

Now the main result of the work can be stated.

[ U a Ub] T the stator voltages and torque. Parameters are given by

Theorem 6. Suppose that for system ~ Assumption 1 and Assumption 2 are satisfied. Then there exists an VIO. Moreover, a reduced order VIO is given by

A

Y2

= I-t (171, (2)

e2 = Y2 -

1

,

H [-aIlH a\2 ] , N a21H -a22H

[~~]

,JI

= [ ~1 ~]

=

TL

is the load

[°

2 nlJI ] , O2 -npJI

,0 2 =

[~ ~]

,

H is the identity matrix of dimension 2, and all, a12, a21, a22, b, a and n\ are positive constants depending on the physical parameters.

ih

171 = Sy , X = r- I (Sy, (2)

with initial condition (2 (0)

=

B =

~2 = i2 (1]1, (2, u) - A 2 (0, e2; Y, 171, (2, u) , Ora:

f,

[xi

xf] T and x\ = [i a ib X2 [.,pa .,pb] T are the stator currents and the rotor fluxes, respectively. w is the rotor speed, u =

where x = 3.4 Main result: Existence of UfO

Suppose that

as an unknown perturbation, i.e. and that wand XI are measured, i.e. YI = W, Y2 = Xl. Taking S = [1,0,0] and P as the identity map, it easily follows that the Assumption 2 is satisfied, and the error system, with output injection £, is given in global normal form (8) as

(13)

w

= (20.

PROOF. Assumption 2 allows the error equation to be written in the normal form (8). Note that the estimation error given by the observer (13) is given by (12), but Assumption 1 implies that the point Z2 = 0 is an asymptotically stable equilibrium uniformly in (1], u) for this system. Therefore limt--+oo ((2 - 172) = O. Since

TL

= [- j-TL],

W = -2axiJIx2

+w

x = Ax + wN X + Bu

T

~ = -2axfJIx2 + 2a (XI - xd JI (X2 - X2) i = Ai + ylNx - (YI - w) N (x - i) + £2

it follows easily that limt--+oo (x (t) - x (t)) = O.

el

=W

e2

= Ci

+ £1 + w

= Xl

(14)

Remark 7. The Assumption 1 is in general stronger than the requirement that the plant can be ren-

475

where C = [J[,02J, and w = w - w, X X i. To show that Assumption 1 is satisfied take V (w, i) = ~w2 + ~iT i as a candidate storage

function for the error system. Selecting the output injection L as L 1 = -"fW

Lz

+ 2axfJrxz

= - (Ko + ylKd (yz - Cx) - wNx

where

Ko

(k - an) )H] k 0 K _ [ Oz ] = [ (alZ + aZ1 H' > , 1 - _ nl.u11

,

then the time derivative of V along the trajectories of (14) is

V.

aXfJrR] [w]+_ = - [--T][ W X T"f T Q _ ww aR Jr Xl X

.

M

with R = [Oz, H], and Q = diag {kH, azzH}. Matrix M is positive definite iff "f > 0 and a

Z

T

T

Q - -R Jr XlxlJrR> O. "f

stays in a compact set, then selecting "f big enough M can be made positive definite, and the dissipation inequality (7) will be satisfied. If

Xl

The reduced order UIO (13) is therefore Ora:

i: = AX+Y1Nx+Bu+(Ko + ylKl ) (yz

that is robust against

TL,

- Cx) ,

J, and a.

5. CONCLUSIONS The possibility of rendering the error system partially strictly dissipative by means of partial output injection, under suitable regularity conditions, is shown to be a sufficient condition for the possibility of constructing an UIO for a nonlinear system. This generalizes to the nonlinear case the known equivalence of existence of UIO and the possibility of rendering a LTI system dissipative by output injection. This allows to interpret the possibility of constructing UIOs in "physical" terms, i.e. in terms of mass or energy balance properties of systems. This seems to be appropriate in particular for electro-mechanical systems. Moreover, for certain classes of systems it should be possible to interpret these conditions in terms of LMI, and allowing for a computationally effective way to design DIOs. Finally, the necessity of these conditions for the existence of UIOs has to be clarified, although some results have been already obtained by the authors (Rocha-C6zatl and Moreno, 2002).

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