Design Method for Robust Supervisor Controller

Design Method for Robust Supervisor Controller

Copyright @ IFAC Adaptation and Learning in Control and Signal Processing. Cemobbio-Como. Italy. 2001 DESIGN METHOD FOR ROBUST SUPERVISOR CONTROLLER ...

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Copyright @ IFAC Adaptation and Learning in Control and Signal Processing. Cemobbio-Como. Italy. 2001

DESIGN METHOD FOR ROBUST SUPERVISOR CONTROLLER Toshihiko Nakajima *Hiromitsu Ohmori ** Akira Sano **

* School of Electric Engineering, Keio University •• Department of System Design Engineering, Keio University

Abstract: We suggest a method to design supervisor which guarantees robust stability for linear time-invariant systems in the presence of the uncertainty via output feedback. The system is a composite of continuous-time uncertain plant and switched controller. The controller is defined by a collection of given controllers whose state is shared. Furthermore the proposed supervisor controller determines the mode of the controller to guarantee robust stability of the total closed-loop system. A sufficient condition of robust stability is given in terms of an algebraic Riccati equation and bilinear matrix inequalities(BMI). Copyright © 2001lFAC Keywords: supervisory control, switching, robustness, bilinaer, hybrid

1. INTRODUCTION

Recently the effectiveness of supervisory controller is pointed out, which is capable of switching into feedback of a controller from a family of candidate controllers. One advantage of supervisory control is that it can achieve closed-loop performance objectives that cannot be achieved using classical linear or nonlinear controllers (J .A.De Dona and G.C.Goodwin, 1999), (N.H.Mcclamroch and I.Kolmanovsky, 2000). Moreover it is effective for real-time changes in plant dynamics (A.S.Morse, 1996), (A.S.Morse, 1992), (Edoardo Mosca, 1999), where the nominal plant is a member of a known class of admissible set. In some situations the choice of linear or nonlinear controllers available to the designer is limited and the design task is to use the available set of controllers. In this paper, we consider such a situation and output feedback robust stabilization problem for a class of uncertain linear time-invariant systems. The controller is selected from a set of given linear controllers whose state is shared.

tions of this paper are that solving BMI for designing the switching-law is used, and the controller is allowed to have dynamics. Then, a sufficient condition for robust stability is given in terms of the existence of solutions of an algebraic Riccati equation for an observer and BMI. In order to verify the effectiveness of the proposed switching system, numerical simulations are carried out.

2. PROBLEM STATEMENT We consider the linear time-invariant uncertain system P defined on the infinite time interval

[0,00),

= AXp(t) + B1e(t) + B 2 u(t), z(t) = C1xp(t), yet) = C 2 xp(t) + vet), xp(t)

P: {

(1)

where x p (·) E Rnp is the state of the plant, e(·) E RP, v( .) E Ri are uncertainty inputs, u(·) E Rh is the control input, z(·) E Rm is the performance output, and y(.) E ~ is the measured output, A E Rnpxnp,B 1 E Rn px P ,B2 E Rnpx\C1 E

We are motivated mainly by (A.V.Savkin and R.J.Evans, 1999). Compared with it, the contribu-

291

Controllers

/

p

y

Switching:

Observer

i

logic

:

'- ____________________ -'____________________ J

Supervisor

Fig. 1. Supervisory control system Rm x np ,

and C 2

E

~ x np

t

are given constant

t

j(lIe(t)1I + IIv(t)1I2)dt 2

matrices.

o

Assumption 1. P admits a realization (1) satisfying, (i) (C 2 , A) are detectable, (ii) (A , B 2 ) are stabilizable.

::; d +

j IIz(t)1I 2dt , 0

t E [0, 00). (3)

In this paper 11 · 11 denotes the standard Euclidean norm, L2[0 , t) denotes the Hilbert space of square integrable vector valued functions defined on [0, t) where t E (0,00] .

Suppose we have a collection of given linear output feedback controllers {Kd whose state is shared. The design task is to decide the switchingtime and the controller to switch. Then the controller is ,

We now introduce a corresponding notion of robust stability for the uncertain system Eqs. (1)- (3) . Here, we define the composite state

(4)

= ~~(t) Xc (t) + B~(t)y(t), u(t) = C(7(t )x c(t) , x c(t)

K(7 : {

0"(-) E M

Definition 1. The system Eqs. (1)- (3) is said to be stable, if there exists a constant c > 0 and a function V(x) : Rnp+ne ---> R such that V(x(O)) 2: 0, V(O) = 0, such that the following inequality holds.

(2)

= {1,2,· ··, K},

where x c(-) ERne is the state of the controller, A~ ERne x ne, Bf ERne x t ,Cf E Rh X ne, i = 1, 2, . .. , K are given constant matrices, 0"( t) is the switch index which is active, and M is the set of symbols. Note that it is allowed for controllers to own dynamics against the paper (A .V.Savkin and R.J .Evans, 1999) .

00

j ( IIxp(t)II2 + 1I{(t) 112 + Ilv(t)1I2)dt o

where xp(O) is an expected value of initial condition xp(O).

We will consider uncertainty inputs e( ·), v( ·) such that the following Integral Quadratic Constraint holds,

3. Hoo OBSERVER

Assumption 2. (Integral Quadratic Constraint) For any locally square integrable control input u(·) for which any solution to Eq.(l) with [u( .), e(-) , v( ·)] exists on [0 , 00) and there exists a constant d 2: 0 such that,

For 10 > 0, we involve the following algebraic Riccati equation

292

initilllize a

The Hoo observer is described as

:i:p(t) = [A

+ P[C[C l + d - CrC 2 ]]x p (t) + B2U(t) . (7)

+ PCr y(t)

The following lemma plays an important rule to design supervisor.

no Fig. 2. Switching-law

Lemma 1. (A.V.Savkin and R.J.Evans, 1999) Let xp(O) be a given vector, t: > 0 and V(x(O)) be given constants. Suppose that a matrix P > 0 is a solution for the algebraic Riccati equation (ARE) (6) . Then if the condition

So our object will be achieved by designing switching-law such that condition (14) holds for the system (13) .

IN

j(lIC1X p(t)1I 2 + t:llxp(t)1I2 -IIC2xp(t) - y(t)1I2)dt o S; V(x(O)) ,

(8)

Now we consider the following BMI such that X = XT,Zi = Z;'I-l-i > O,Ti > 0 and 'Yi > 0, for Vi EM,

holds, the following condition holds for all solutions to system (1), with any inputs ~( . ) and v( ·) belong to L 2 [0,tN]. tN

j(lI z (t)1I 2 + t:llxp(t)1I2

4. SWITCffiNG-LAW AND SUPERVISORY CONTROL

_1I~(t)12 -lIv(t)1I2)dt

rA~ Xi + Xi~j. + c'i Cl + Zi

l

Bi Xi

o S; (xp(O) - xp(O)f p-l(Xp(O) - xp(O))

+ V(x(O)). (9)

If condition (9) holds, by combining it with the Integral Quadratic Constraint (3), we obtain condition of Definition 1, so the total closed system will be stable.

rA + P[C[C l

+d

where

u(t)

if . X(~)T~a(t - )X(~) > 0, m(t) else If x(t) Za(t-)x(t) = O.

= {U(C)

- CrC2] B2CfJ

(19)

• lOAf '

m(t) = {m E M

(10)

Figure 2 shows the computation diagram of the switching-law Eq. (19) .When x(t)TZa(t-)x(t) is 0, the switch index u(t) is set equal to the value m(t) . The next switch index m(t) is defined by Eq. (20), which is the switch index that maximizes the value x(tf ZiX(t), where Zi is calculated by BMI (15)-(17). Figure 3 shows an example of the switching surface and the state trajectory. The gray region implies the region {x : x T ZIX > O} U {x : xTZ2X > O}. The border xTZIX = 0 is the surface where switching (1 ~ 2) occurs, and x T Z2X = 0 is the surface where switching (2 ~ 1) occurs. Because the two switching surfaces are defferent from each other, this switching-law is robust about some swiching-delay.

(13)

Then, we can rewrite the inequality (8) as follows IN

j(llC l x(t)1I 2 -IIG 2 x(t) - y(t)1I2)dt

o S; V(x(O)).

(20)

iEM

Using these definitions enable us to obtain the differential equation

= Aa(t)x(t) + Ba(t)y(t).

I x(tf Zmx(t)

= max x(tf ZiX(t)} .

(ll)

x(t)

+ T2Z2 + .. . + TkZK > 0,(17)

Then, we propose the following switching-law,

Next, for each i EM, we define matrices

A.=

T1Zl

xiBil < 0 ,(15) -I J Xi - 'YiZi > 0,(16)

(14)

293

T

-T

-

+ Xu(t)Au(t) -T - T + Xu(t )Bu(t)Bu(t)Xo(t) + Cl Cdx(t) ::; O. x (t) (Au(t)Xu(t)

(23) Here, we define matrix Hi as follows

H,x=O

-T

- H, = A, Xi

-

+ X,A i -T - T + XiBiBi Xi + Cl Cl.

(24)

Moreover, we consider the following Riccati differential equation

o Fig. 3. State trajectory

.

- Xi (t)

5. STABILITY ANALYSIS OF TOTAL CLOSED LOOP SYSTEM

-T

-

= Ai X,(t) + Xi(t)Ai -

-T

+ Xi(t)BiBi Xi(t)

- T -

+ Cl Cl + Hi

= 0, (25)

In the sequel, the following theorem is obtained. it follows Theorem 1. Consider the system Eqs . (1)-(3) supervisory controlled by Eqs. (7), (19). Suppose that there exist I: > 0, X = XT, Zi = ZT,J-Li > 0, T, > 0 and ri > 0 such that the following conditions hold: (i) There exits a solution P > 0 to the algebraic such that the matrix Riccati equation Eq. A + p[CiC l + d - C 2 C 2 ] is stable. (ii) Vi EM, BMI Eqs. (15)-(17) hold. Then, the uncertain system Eqs. (1)-(3) can be stabilized by the supervisor Eqs. (7), (19) in the sense of Definition 1.

J ti+1

+

(x(t)Xj(t)x(t). + x(t)Xj(t)x(t)

t,

(tp

+ x(t)Xj(t)x(t))dt

= O.

(26)

t Adding It :+'(lIC\x(t)1I 2-IIC2x(t)-y(t)11 2) dt to Eq.(26), and combining it with Eqs.(23) and (25) , we get the following

Now, we give a lemma that is used in this section.

ti+l

Lemma 2. (S-procedure). Let e, Sb S2,"', Srn E Rnxn be symmetric matrices. Then, c"T ec" < 0, foral! c" =f 0 such that c"T Sic" ::; 0 (i = 1,2,···, m), if there exist scalars Ti > 0 such that e L:':l TiSi < O.

J(

IIC lx(t)1I 2 - IIC2x(t) - y(t)112)dt

ti ti+l

-J

(11[C 2 + n:(t.)xu(t;)]x(t) - y(t)1I2)dt.

t,

The derived switching sequence would be of the form

S

= (a(to), to), (a(td, td, "

(27) Applying inequality Eq. (27) for i we obtain it

" (a(t n ), t n ),· ··, (21)

which means the controller switches from a(t i to a(t,) at time t i .

= 0, 1, 2, ... , N ,

tN sup [ j(lIC l x(t)1I 2 -IIC2x(t) - y(t)112)dt] to

l )

::; x(tof Xu(to)x(to) - x(tlf Xu(to)x(t l ) Proof 1. (Theorem1). Suppose that the condi-

+ x(tlf Xu(t,Jx(td - x(t2f X u(t,)X(t2) + ... + X(tN-lf Xu(tN_,)X(tN - l)

tions (i),(ii) hold. Then, we get -T

Ai Xi

-

--T

+ XiAi + XiBiBi Xi -T-

.

+ClCl+Zi
- X(tN)T Xu(tN_,)X(tN).

(22)

(28)

From the definition of Xi in Eq.(18), it follows

The switching-law (19) implies x(t)T Zu(t)x(t) 2 0, and applying Lemma2 (S-procedure) for (15), it follows

294

= x(ti+lf(/-Lu(t;}ZU(ti) - /-Lu(ti+dZu(t.+d)X(ti+l)

= -X(ti+I)T /-Lu(tHdZu(tHdX(ti+d (29)

~O.

Combining Eq. (29) with Eq. (28), it follows tN

sup [ j(IICIX(t)1I2 -IIC2x(t) - y(t)1I2)dtj to

~ x(tof Xu(to)x(to) - X(tN)T Xu(tN _dx(tN), time

(30)

Fig. 4. time vs states (Case.l)

Eq.(16) implies x(t)T Xu(t)x(t) ~ O. Then, we get tN

sup [ j(IICIX(t)1I2 -IIC2X(t) - y(t)1I2)dtj to N

(31)

c..

0<

Combining this inequality with the Integral Quadratic Constraint (3), we obtain the condition of (5) . This completes the proof of this theorem. .(I,

·1

.(I,

...

.(11

X

0

02

04

••

..

p1

Fig. 5.

6. NUMERICAL SIMULATIONS

Xpl

vs

Xp2

(Case.l)

Consider the uncertain system (1)-(3) with the coefficients

A = [

-1~25 ~] ,B

I

=

[n

,B2 =

[~] ,

Cl = [0.10] ,C2 = [1 -2]. This plant will be controlled by (2) with the coefficients, X

A~ =

[

p2

-;1 ~7]' B~ = [ ~1] ,C~ = [3 -6)'

lime

Fig. 6. time vs states (Case.2)

A~ = [ -~i5 -~.5]' B~ = [ ~1] ,C~ = [ -1 2] .

Z -

Note that this plant can't be stabilized even if we use whichever in these two controllers. We apply Theoreml to design a stabilizing supervisor. We nwnerically solve the Riccati equation (6) by setting t = 0.01, then we get

p

=

1-

Z2

=

[2.1550.537] 0 0.537 0.857 >

From the P > 0, it follows the Hoc observer (7) .

-0.521 4.116 1.468 -3.919

-0.328 1.468 0.805 -1.305

0.874] -3.919 -1.305 4.360

0.501 -2.270 -1.429 7.536

-0.462 -1.429 -1.193 2.460

-0.1931 7.536 2.460 ' -8.896

0.174 -0.601 0.448 [ 0.253

-0.601 2.827 1.697 -8.804

0.448 1.697 1.415 -2.858

0.253 1 -8.804 -2.858 10.484

1O- 3 ,/-L2

= 2.881

/-LI

=5.152

Tl

= 4.40, T2 =

'"'(1

= 5.22 X

We solve BMI (15),(16),(17), and we have 0.892 X = -0.521 -0.328 [ 0.874

-0.043 0.501 -0.462 [ -0.193

X

J

J

X

10- 3 ,

3.91,

10- 2 , '"'(2

= 11.1l X

10- 2 •

Here we take initial condition xp(O) = [-0.4 0.6jT. We present the simulation results as follows,

295

x

'""-

I

I r • xpl •

Fig. 7.

Xpl

vs

Xp2

(Case.2)

• Case.l. e(t) = sin(1Ot)z(t), v(t) = 0.1 cos(t)z(t) (Fig.4 and Fig.5) . • Case.2. e(t) = sin(lOt)z(t) + 3e- o.Olt , v(t) = 0.1 cos(t)z(t) + 3e- o.Olt (Fig.6 and Fig.7).

7. CONCLUSIONS We proposed a design method of supervisor that. robustly stabilizes a class of uncertain system. A sufficient condition for stabilizability is described. The result is given in terms of the existence of suitable solutions to an algebraic Riccati equation of the H 00 filter and BMI. Compared with existing result, the main contribution of this paper is that we proposed solving BMI, which get it simply to design switching-law.

8. REFERENCES A.S.Morse (1992) . Applications of hysteresis switching in parameter adaptive control. IEEE Trans.Automat.Contr vol.37, 00.9, 1343-1354. A.S.Morse (1996) . Supervisory control of families of linear set-point controllers -part 1:exact IEEE Trans. Automat. Contr matching. vo1.41, 00.10, 1413-1431. A.V.Savkin, E.Skafidas and R .J.Evans (1999). Robust output feed back stabilizability via controller switching. Automatica 35, 69-74. Edoardo Mosca, Tommaso Agnoloni (1999). Adaptive gain scheduling control:inference of feedback-loop behavior. Proceeding of 14th IFAC pp. 313-318. J.A.De Dona, S.O.Reza Moheimani and G.C.Goodwin (1999). Allowing for over-saturation in robust switching control of a class of uncertain systems. Proceedings 38th CDC pp. 3053-3058. N.H.Mcclamroch and I.Kolmanovsky (2000). Performance benefits of hybrid control design for linear and nonlinear systems. Proceedings IEEEvol.88, 00.7, 1083-1096.

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