Stabilization of LPV Systems

8/8/2019 Stabilization of LPV Systems

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Stabilization of LPV systems: state feedback, state estimation and duality

Franco Blanchini

DIMI, Universita di Udine

Via delle Scienze 208,

33100 Udine, Italy

tel: (39) 0432 558466

fax: (39) 0432 558499

[email protected]

Stefano Miani

DIEGM, Universita di Udine

Via delle Scienze 208,

33100 Udine, Italy

tel: (39) 0432 558262

fax: (39) 0432 558251

[email protected]

Abstract In this paper, the problem of the stabilizationof Linear Parameter Varying systems by means of GainScheduling Control, is considered This technique consists ondesigning a controller which is able to update its parametersonline according to the variations of the plant parameters.We first consider the state feedback case and we show adesign procedure based on the construction of a Lyapunov

function for discretetime LPV systems in which the parametervariations are affine and occur in the state matrix only. Thisprocedure produces a nonlinear static controller. We show that,differently from the robust stabilization case, we can alwaysderive a linear controller, that is nonlinear controllers cannotoutperform linear ones for the gain scheduling problem. Thenwe show that this procedure has a dual version which leads tothe construction of a linear gain scheduling observer. The twoprocedures may be combined to derive an observerbased gainscheduling compensator.

Keywords LPV systems, scheduling, Lyapunov design.

I. INTRODUCTION

The gain scheduling approach for the control of Linear

Parameter Varying (LPV) systems is a problem encoun-tered in several applications in industrial world. when the

plant parameter variations can be measured on line and

the compensator may take advantage from this knowledge

and improve its performances. However only quite recently

techniques which have been heuristically applied became a

subject of mathematical investigation. In [33] [34] a rigorous

stability analysis of some general gainscheduled schemes is

provided. A pole placementlike technique for gain schedul-

ing synthesis is proposed in [30]. In [16] it is considered the

problem of designing a nonlinear controller whose lineariza-

tions in several operating points match the linear controllers

designed for these points. An LMI technique for the gain

scheduled control design is proposed in [31]. A analysisapproach is proposed in [15]. A more recent technique based

on a settheoretic approach is presented in [35] for discrete

time LPV systems with bounded variations. The reader is

referred to [32] for a tutorial exposition.

Although the knowledge of the plant parameters is an

advantage for the compensator, an interesting exception has

been investigated in the literature which concerns the state

feedback case. Indeed for the class of controlaffine nonlin-

ear continuoustime systems in which the term associated

with the control is certain [22] or the so called convex

processes [8], the knowledge of the parameter is not an

advantage for the compensator as far as it concerns its

stabilization capability. As a particular case, gain scheduling

design for state feedback LPV systems can be handled

without restriction as a robust design, with the remarkable

advantage that no parameter measurement are needed..

In the discretetime case this property does not hold. As

we will see, trivial examples exist of LPV discretetime

systems that can be stabilized via gainscheduling controller

but not by means of state feedback controllers which ignore

the parameter. This facts motivated us to derive a procedure

to design a gain scheduling control in which the exploitation

of the parameter measurement is allowed. In particular we

consider a procedure for the robust stabilization case [6],

which is in principle unsuitable for the gain scheduling

design, and we show that it still comes into help if applied to

a proper subsystem evidenced by a suitable transformation.

Furthermore we show that in the gain scheduling case,

nonlinear compensator cannot outperform linear ones. This

is in contrast with the robust stabilization case in which

nonlinear controller can outperform linear ones [9] (see also

[21] for further implications). Thus limiting the attention to

linear compensator is not a restriction for gain scheduling

statefeedback stabilization.

Then we consider the dual problem of designing a

gain scheduling state observer of the type considered in

[26][20][36]. We show that necessary and sufficient condi-

tions for the existence of a linear observer are the dual of

those for the existence of a gain scheduling state feedback

(linear or not) compensator. We show how this kind of dualitydoes not hold for the robust state estimation problem, i.e.

when the parameters are not available to the observer.

Finally we show that the two procedures for state feedback

stabilization and state detection can be combined together to

solve the problem of stabilization by means of an observer

based compensator.

Due to lack of space the present version of the paper does

not report any of the proofs.

Proceedings of the 42nd IEEE

Conference on Decision and Control

Maui, Hawaii USA, December 2003 TuPPI-7

0-7803-7924-1/03/$17.00 2003 IEEE 1492


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I I . PROBLEM STATEMENT AND BASIC RESULTS

In this paper we consider LPV systems of the form

x(k+ 1) = A(w(k))x(k) +Bu(k)y(k) = Cx(k)

(1)

where x(k) IRn is the state, u(k) IRq is the control

input, y(k) IRp is the control output and w(k) is a timevarying parameter. We assume that the state matrix A(w(k))is constrained to belong to the matrix polytope:

A(w(k)) =m

h=1

A(h)wh(k) (2)

where, for every k, w(k) W= {w : wh 0, mh=1 wh = 1},

and A(h), h = 1,2, . . . ,m are assigned constant n n matrices.We consider the following assumption.

Assumption 1: The matrices B and Chave full column and

row rank respectively.

The basic problem considered in this paper is the stabilization

of system (1) by means of a state observer and an estimated-

state feedback compensator which are scheduled on the

parameter w(k). Definition 2.1: The system (1) is gain scheduling state

feedback (GSSF) stabilizable if there exists a (possibly dy-

namic) continuous statefeedback compensator whose equa-

tions are function of the timevarying parameter w(k)

z(k+ 1) = F(z(k),x(k),w(k))u(k) = G(z(k),x(k),w(k))

(3)

such that the resulting closedloop system is globally uni-

formly asymptotically stable.

The next definition is essentially the dual

Definition 2.2: The system (1) is gain scheduling de-tectable (GSDE) if there exists a (possibly dynamic) system

whose equations are function of the timevarying parameter

w(k)z(k+ 1) = F(z(k),y(k),w(k))

x(k) = G(z(k),y(k),w(k))(4)

and such that for all x(0), z(0), w() the condition e(k).

=x(k) x(k) 0 as k is assured.

With obvious meaning, we will say that (1) is robustly sta-

bilizable (RS) and robustly detectable (RD) if the equations

(3) and (4) do not depend on the parameter w. Furthermore,

we will distinguish the case in which (3) and (4) are linear

with respect to x(k) and z(k) (respectively y(k) and z(k)).

A. Robust stabilization and state detection

One of the main points of the paper is to show that there is

no apparent duality between the problem of robust detection

and robust state feedback stabilization while, in turn, a kind

of duality relationship exists between the gain scheduling

stabilization and detection.

Let us consider the following system

x(k+ 1) = A(w(k))x(k) +Bu(k)y(k) = x(k)

(5)

and let us define the dual as follows

x(k+ 1) = AT(w(k))x(k) + u(k)y(k) = BTx(k)

(6)

As it is well known, if A is a constant matrix, we have

a duality relation which says that (5) is reachable iff (6) is

observable. In our case such a relation does not exist as longas the parameter w is unknown as it can be shown by the

next counterexamples. Consider the systemx1(k+ 1)x2(k+ 1)

=

2 0

1 + w(k) 0

x1(k)x2(k)

y(k) = x2(k)

with |w(k)| w. Such a system is such that there is noobserver which can estimate asymptotically x1(k) ignoringw(k). Indeed, the second state equation x1(k) enters in theterm

.= (1 + w(k))x1(k). Even if (k) = x2(k+ 1) can be

determined with one step delay, it is impossible to estimate

x1(k) from (k), if w > 0, up to the fact it belongs to the

uncertainty interval /(1 + w) x1 /(1 w), which sizegrows arbitrarily if x1(0) = 0. Conversely, the dual of thissystem

x1(k+ 1)x2(k+ 1)

=

2 1 + w(k)0 0

x1(k)x2(k)

+

0

1

u(k)

with the control u(k) = 4x1(k) 2x2(k) is stable providedthat w is sufficiently small. Thus, roughly, robust stabilizabil-

ity does not imply the robust detectability of the dual.

It can be easily shown that robust detectability does not

imply the robust stabilizability of the dual, as well. To this

aim it is sufficient to consider the following simple example

with A(w(k)) = w(k), B = 1 and C= 1 (say the system andis dual coincide)

x(k+ 1) = w(k)x(k) + u(k) (7)

y(k) = x(k) (8)

with |w(k)| 2 which can not be robustly stabilized by anystate feedback, but is obviously detectable.

Note in passing that the last example shows that robust

state feedback stabilization and the gain scheduling state

feedback stabilization are different problems for discrete

time systems. Indeed the system can be gainscheduling

stabilized (e.g. by u(k) = w(k)x(k)), but not robustly stabi-lized. This fact was pointed out in [8] and will motivate the

results of the next section in which we will present a pro-cedure for the gainscheduling statefeedback stabilization

by means of the procedures already available for the robust

control synthesis [6].

B. Some preliminary results

In this section we will remind some basic results concern-

ing the stability of linear LPV systems. We denote by C-set a

convex and compact set containing the origin as an interior

point. Given x IRn the (dual) one and infinity norms are

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defined as x1 =ni=1 |xi| and x = maxi |xi|, respectively,

and the corresponding induced norm for matrices are H1 =

supx=0Hx1x1

= supjni=1 |Hi j| and H = supx=0

Hxx

=

supinj=1 |Hi j|.

We define (symmetric) polyhedral function the Minkowski

functional of a 0-symmetric convex and compact set contain-

ing the origin as an interior point (in brief symmetric C-set).Such a function can be represented as

(x) = Fx, (9)

where F IRsn is a full column rank matrix, or in its dualform

(x) = inf{p1, s.t. x = X p} (10)

where now the matrix X IRnl is full row rank. There is aduality relation between (9) and (10) in the sense that the two

definitions give the same function if and only if the convex

hull of all the column vectors of X and its opposite X isthe unit ball of Fx.

Such duality holds also for the stability of an autonomoussystems, as per the following (duality) lemma.

Lemma 2.1: The system

x(k+ 1) = A(w(k))x(k), (11)

where A(w) is as in (2), is robustly stable for all admissiblew(k) if and only if any of the following holds:

i) there exists a full row rank matrix X IRnl and mmatrices P(h) IRll such that P(h)1 < 1 and

A(h)X = X P(h).

ii) there exists a full column rank matrix F IRsn and m

matrices H(h

) IRss

such that H(h

) < 1 and

FA(h) = H(h)F.The coefficient above turns out to be an index of the

speed of convergence. The proofs of the above lemma follows

immediately by the fact that if the system is stable, then it

admits a polyhedral Lyapunov function [1][2][25][23]. The

details can be found in [7].

This also means that (11) is robustly if and only if the

dual system is robustly stable.

III. SOLUTION OF THE GAIN SCHEDULING STATE

FEEDBACK STABILIZATION PROBLEM

In this section we consider the problem of determininga state feedback control of the gain scheduling type for the

system (1). To this aim we introduce the following important

notion of speed of convergence.

Definition 3.1: We say that the system (1) is Gain

Scheduling stabilizable with speed of convergence if thereexists a control such that

xcl(k) Ckxcl(0),

where Xcl is the closedloop system state.

Note that the definition above implies exponential conver-

gence of the state to 0. We will see that any stabilizable

system of the considered class can be indeed exponentially

stabilized.

A. Problem solvability

To solve our problem we consider the class of the poly-hedral functions as candidate Lyapunov functions. The next

theorem motivates this choice by showing that the existence

of such function is necessary and sufficient for the Gain

Scheduling stabilizability of system (1).

Theorem 3.1: The following statements are equivalent.

i The system (1) is gain scheduling stabilizable.

ii There exists a control (x,w) such that the poly-hedral function (10) is a Lyapunov function for the

system (1).

iii There exist a full row rank matrix X IRnl , mmatrices P(h) IRll and U(h) IRql such that

A(h)X+BU(h) = X P(h), with P(h)1 < < 1,(12)

The previous theorem states that eq. (12) is crucial, since

the existence of a solution in terms of X, P(h) and U(h) is

necessary and sufficient for the GS stabilization problem

to be solvable. The next theorem states that as long as

the parameter w is known to the controller, we can always

implement a linear controller for the system, namely GS

stabilizability implies GS stabilizability by means of a linear

controller. This result extends that in [8] to the discretetime

case. To introduce the theorem we augment the equation (12)

as follows

A(h) 0

0 0

XZ

+

B 00 I

U(h)V(h)

=

XZ

P(h),

(13)

with P(h)1 < 1 and where, Z is an arbitrary matrix such

that

X

Z

is square invertible and V(h)

.=ZP(h). We are now

able to state the following theorem which states that Gain

Scheduling stabilizability is equivalent to GainScheduling

stabilizability via linear control.

Theorem 3.2: If system (1) is GS stabilizable, then it is

GS stabilizable via linear control. A stabilizing control is

given by

u(k) = K(w)x(k) +H(w)z(k)z(k+ 1) = G(w)x(k) + F(w)z(k), (14)

whereK(w) H(w)G(w) F(w)

=

m

h=1

K(h) H(h)

G(h) F(h)

wh (15)

with K(h) H(h)

G(h) F(h)

=

U(h)

V(h)

X

Z

1(16)

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B. Computation of the Lyapunov function

The results above are nonconstructive as long as we

cannot provide algorithms to compute the matrices X(h), P(h)

and U(h) in equation (12). As observed in [7] such type of

equations are not convenient to solve the problem, since they

are bilinear as long as X and P(h) are both unknown. Note

also that the same problem holds in the robust stabilizationcase, where we have to cope with the same equation with the

difference that [7] the matrices U(h) are all equal according

to the following proposition.

Proposition 1: [6]. The system

x(k+ 1) = A(w(k))x(k) +B(w(k))u(k)y(k) = x(k)

(17)

where A(w(k)) is as in (2) and

B(w(k)) =m

h=1

B(h)wh(k) (18)

and B(h)

IRnq, h = 1,2, . . . ,m are assigned constant matri-

ces, is robustly stabilizable if and only if there exist a full

row rank matrix X IRnl , m matrices P(h), and a singlematrix U IRql such that

A(h)X+B(h)U = XP(h), with P(h)1 < 1, (19)In [6] an iterative procedure is proposed to solve the prob-

lem of the determination of X(h), P(h) and U. Clearly this

procedure may be applied in a conservative way to our

problem since if a system is robustly stabilizable then it

is also GS stabilizable. However, for discretetime systems

this is a conservative way of proceeding, since, as we have

seen, the knowledge of w can be an advantage for the

compensator. The next result will allow us to exploit the

existing procedures for the solution of the robust stabilization

problem for GS stabilizability.

In view of Assumption 1 we can consider the system in

the following form:

x1(k+ 1) = A11(w) x1(k) + A12(w) x2(k) (20)

x2(k+ 1) = A21(w) x1(k) + A22(w) x2(k) + u(k) (21)

It is immediately seen that this form is achieved by

applying to all the generating matrices A(h) the linear trans-

formation T =

B B

, where B is any matrix such that T is

invertible.

By means of this form, we can recast the GS stabiliza-

tion problem in a robust stabilization problem for the pair( A11(w(k)), A12(w(k))) according to the following theorem.

Theorem 3.3: System (1) is GS stabilizable if and only if

the system

x1(k+ 1) = A11(w) x1(k) + A12(w) x2(k), (22)

(where x2 has now to be thought to as input), is robustly

stabilizable.

Thus the above result allows us to use existing algorithms

(properly modified) to compute a solution of (12).

IV. SOLUTION OF THE GAIN SCHEDULING STATE

ESTIMATION PROBLEM

In the previous section we have considered the problem

of designing a statefeedback gain scheduling compensator.

Among the results we have seen that if a system is gain

scheduling stabilizable, then is gain scheduling stabilizable

via linear state feedback control. Now we cope with the dual

problem of designing a state observer for the system

x(k+ 1) = A(w(k))x(k) + v(k)y(k) = Cx(k)

For this system we consider a linear observer of the form:

z(k+ 1) = P(w(k))z(k) L(w(k))y(k) + T1(w(k))v(k)x(k) = Q(w(k))z(k) +R(w(k))y(k)

(23)

with z IRs. Definition 4.1: The system (23) is an asymptotic observer

if and only if

i for all w(k), v(k) and x(0) and z(0) we have x(k)x(k) 0 as k ;

ii If x(0) = 0, z(0) = 0 then x(k) =x(k) for all w(k) W and v(k).

It is known that, for a given constant w, the system (23)

represents the most general form of a linear observer [26].

Furthermore, for a given constant w, for (23) to be an

observer the following necessary and sufficient conditions

must hold

P(w)T1(w) T1(w)A(w) = L(w)CQ(w)T1(w) +R(w)C = I

(24)

and

P(w) is a stable matrix (i.e. its eigenvalues are in theopen unit disk);

T1(w) has full column rank

The main problem is now to see what happens when w(k)is timevarying as in our case. To this aim introduce the

assumption that the family of matrices P(w) does not containcommon invariant subspaces included in the kernel of Q(w).

We say that a subspace S of IRn is P(w)invariant, ifx Simplies P(w)x S for all w W. Furthermore we call thekernel of Q(w) the set of all vectors r such that Q(w)r= 0for all w W.

Assumption 2: There are no P(w)invariant subspaces in-

cluded in the kernel of Q(w).The assumption below essentially means that the system

cannot be reduced by the transformation [S S], where S isa basis of S, the invariant subspace and S is a complement

matrix, to the form

P(w) =

P11(w) P12(w)

0 P22(w)

, Q(w) =

0 Q2(w)

Note that for known P and Q this is just an observability as-

sumption. The restriction above is obviously nonrestrictive,

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since, if the decomposition above can be achieved, then we

can eliminate the nonobservable part of the system.

Consider now a new variable r(k) such that

z(k) = r(k) + T1(w(k))x(k) (25)

With simple computations if we replace (25) in (23) in viewof (24) we get the following equation

r(k+ 1) = P(w(k))r(k) + [T1(w(k+ 1))T1(w(k))]x(k+ 1)

x(k) x(k) = Q(w(k))r(k)(26)

We have now the following result that concerns the structure

of (23).

Lemma 4.1: The system (23) is an observer only if the

matrix T1(w) = T1 is a constant.

Given that T1 must be constant, without restriction we can

consider observer of the form

z(k+ 1) = P(w(k))z(k) L(w(k))y(k) + T1v(k)x(k) = Q(w(k))z(k) +R(w(k))y(k) (27)

whose associated error equation, derived from (26), is

r(k+ 1) = P(w(k))r(k)e(k)

.= x(k) x(k) = Q(w(k))r(k)

(28)

The previous equation together with Assumption 2 leads us

immediately to the following basic result.

Lemma 4.2: The system (27) under Assumption 2 is an

observer only if the system

r(k+ 1) = P(w(k))r(k)

is stable and there exists a full row rank matrix T1

P(w)T1 T1A(w) = L(w)CQ(w)T1 +R(w)C = I

(29)

The proof of the above Lemma is immediate and thus it is

omitted. We are in the position now to prove the main result

of this section.

Theorem 4.1: The following statements are equivalent.

i The system (27) is an observer.

ii There exists full column rank matrix F, m matrices

H(h) and m matrices Y(h) such that

FA(h) +Y(h)C= H(h)F, with H(h) < 1, (30)

iii The dual system

x(k+ 1) = AT(w(k))x(k) +CTu(k)

is gain scheduling stabilizable.

The previous result is constructive, since, by duality, we

can always apply the procedure to design a gainscheduling

state feedback stabilizer to the dual system (AT(w),CT) todesign the observer.

V. A SEPARATION PRINCIPLE FOR DESIGN

We briefly describe a way to stabilize the system by means

of a linear observer and gain scheduling estimated state

feedback. The next theorem is a simple consequence of the

results of the previous section. It shows that one can always

synthesize a stabilizing compensator by separately designing

an observer and a state feedback control if the system is

statefeedback stabilizable and detectable.

Theorem 5.1: Assume that (A(w),B) is gain schedulingstabilizable and (C,A(w)) is gain scheduling detectable. Thenthe dynamic controller (we dropped the time-dependence in

w(k) for clarity)

zc(k+ 1) = G(w) x(k) + F(w)zc(k)zo(k+ 1) = P(w)zo(k) L(w)y(k) + T1Bu(k)

x(k) = Q(w)zo(k) +R(w)y(k)u(k) = K(w) x(k) +H(w)zc(k),

(31)

where the matrices in the above equations are as in theorem

3.2 and lemma 4.2, asymptotically stabilizes the plant.

V I . CONCLUDING DISCUSSIONS

In this paper we focused our attention on the stabilization

of Linear Parameter Varying (LPV) discrete-time systems.

We showed by very simple examples that the robust sta-

bilization and the gain scheduling stabilization problems,

differently from the continuous-time case, are rather different

problems. We also showed the existence of a duality relation

between the gain scheduling control and the gain scheduling

observation problem for LPV. Finally a separation principle

to derive output feedback stabilizing controllers was derived.

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