Noninteracting Control for a Class of Nonlinear Differential-Algebraic Systems

2b-076

Copyright © I Y% IFAC 13th Triennial World Congress. San Francisco. USA

NONINTERACTING CONTROL FOR A CLASS OF NONLINEAR DIFFERENTIAL-ALGEBRAIC SYSTEMS Lingshan Chen and Xiaoping Liu

P.O.Box 1951 Dept. of Automatic Control} Northeastern University Shenyang J Liaoning, 110006, P.R. China

Abstract: This paper concerns nonlinear singular systems defined by a cla.ss of nonlinear differential-algebraic equations. Assumptions are given first for which initial value problem and the procedure of nonlinear coordinates transformation are well posed. The problem of noninterating control with internal stability via static state feedback is stated. Noninteracting control feedback law is then presented. An algorithmic procedure is developed to obtain the manifold on which there exists a unique differentiable solution to the dosed-loop system. The conditions for local internal stability of a single equilibrium solution are indicated, Nonintercating control problem of the DAE systems is solved based on the framework. Keywords: singular systems, differential-algebaraic equations, nonintera.ctig control, equilibrium, stability.

I.INTRODUCTION Singular systems arise naturally in a variety of control problems such as robotics (Mills and Goldenberg, 1989L contact or constrained problems in mechanics (Bloch and McClamroch, 1989; You and Chen , 1993), some chemical processes (Pant elides , et al., 1988). Such systems are often called differential-algebraic equations systems (DAEs) because they are often described in differential and algehaic equations (Campbell, 1990). From the view of state space model, the form of DAEs is nonstandard. Petzold {1982} pointed out that the approach to solve systems of DAE8 is different than the methodology for solving systems of ordinary differential

equations. The problem of solvability of DAEs systems has been investigated by Rheinboldt (1984, 1991) and Liu (1995). Lin and Ahmed (1991) studied controllability problems for singular systems. Some control problems for nonlinear singular systems have been investigated by Christodoulou and ]sik (1990), Liu(I993). In the control community) an important area of research recently has been focused on a special class of process models, see (McCla.mroch, 19nOa., Kumar and daoutidis, 1994). The control of such systems has been addressed recently. The design of feedback stabilizers for this problem has been concerned by McClamroch (1990b). The problem of tracking in control system has been

2126

addressed by Krishnan and McClamroch (1993). The feedback control of a more genera.l class of DAEs systems has been indicated by Kumar and Daoutidis (1994). The framework employed by the authors of above papers involves converting the DAEs to a nonlinear differential problem by solving for the algebraic variables in terms of the control inputs and states. Nonlinear design methodologies are then applied. In this paper, we will address the problem of nOllinteracting control by means of static state feedback for affine nonlinear DAEs systems and investigate the internal stability of the closed-loop system. Since there is the requirement of consistent initial data for DAEs systems (Panteiides, 1988)' the control loop cannot be arbitrarily closed at any operating point. This require ment impose the need to derive a certain manifold on which noninteracting control is effective. Based on the above ideal An algorithm is developed by which the problem of unique differentiable solutions to the corresponding closed-loop system under the consistent initial condition is solvable. Noninteracting control problem is then interpreted and leads to the stability conditions for the closed-loop system.

2.PROBLEM FORMULATION AND ASSUMPTIONS

i =

Assumptions: There are finite positive integers 1", ., p and Xo E M, such tha.t:

riJ

A1) Lg,L,I,(x) = 0, for all k = O,···,T. 1,· .. ,p, j 11 ' .. , m1 x E M n U, U neighborhood of Xo.

2,i = IS

some

A2) Lh,L'I.(x) = 0, for all k = O,···.T. - 2,i = 1,"',p,j= 1,"',p, andxEMnU. A3) rank A(xo) = p, where A(x) is p * p matrix who,e element on the i~th row and j-th column is

where

al;(x) Lfl,(x) = --f(x) 8x

and

Assumptions A1)-A3} are essential that the constraint rela.tive degree vector of (I, g,l) is (rI"'" rp) and is not greater than the control relative degree vector of (J,h,l) componentwise, where g(x) = [g.(x) ... g.,,(x)[,

h(x) = [hdx) ... hp(x)[. We now make the additional assumption that there exists a unique differentiable solution to the system (1) and (2) for any consistent initial condition data in a neighborhood of (xO I vG) and any integrable inputs.

2.1 Preiiminar£es and Assumptions

2.2 Problem of Nonintemcting Control with Stability

We consider control systems described by a class of nonlinear differential-algebraic equations as follows:

Defin£tionl: Given the differential-algebraic equations of the form (1)-(3) and all equilibrium solution of equtions (1) and (2) (xO, uO) corresponding to u = 0, i.e. f(x") = 0, q(XO) = 0, and g(XO) has full rank, find a regular static state feedback control law:

:i: = f(x)

+

m

p

;=1

i=l

L g,(x)vd L h;(x)u,

Y. =q,(x) =0, i~l,·,m z. = l;(x), ,= 1, ., P

(1) (2) (3)

where x E M I with M an open subset of Rn I tJ = [Vl ... Vm]t E Rfn,u, (Ul"'U,.]! E RV ,y = [Yl ... Ym]t E Rm"z = [Zl ... Zp!t E RP. Here /(x),gi(xLi = 1, ·",m,hdx),i = l,"',P, are smooth functions on M. The above description of DA Es systems is affine in both the control inputs u. and algebraic variables VI and the algebraic equations are implicit in the algebraic variables. There exist a large mumber of physical systems modelled as the description. For instance, constraint robot systems (McClamroch and Wang, 1988), mobile robots (d' Andrea-Novel, 1991), constrained mechanical systems (Bloch, et al., 1992). For the convenience of analysis, the following fundamental assumptions are introduced for the class of nonlinear constrained systems described by equations (1)and{3).

u =

",(x)

+ .B(,)" + 1(X)V,

(4)

defined in a neighborhood eof (xO, v°), with a(xO) 0, 1(xO) = 0 such that:

i) The equilibrium (xo) vU) of the followimg system: :i: = f(x)

+ h(x)",(x) + (g(x) + h(xh(x))v

qdx) =0,

i=I,· ',",'

(5)

is asymptotically stable.

ii) In the closed-loop system f(x) + h(x)",(x) + (g(x) l-h(x).B(x)a qi(X) =0, t=l,···,m z,=l;(x), ,=l, .. ·,p

:i; =

+ h(xh(x))v (6)

each output channel Zi is affected only by the corresponding input channel t1i and not by ili, if J' 0:1 i.

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iii) Under the feedback law, system (5) and (6) in i) and ii) has a unique differentiable solution for any consistent

following form:

initial condition data in a neighborhood of (xo! vOl.

elO) ~ F, (e) + C , (e)v + H , (e)"

If there exists such a feedback function of the form (4) which is smooth and requirements i), iiL in) are satisfied, then we say noninteracting contol problem is solvable and the closed-loop system is internally asymptotically stable for the DAEs system in question.

t;O)

F,(E) + C,(E)v + H,(E)"

=

,,(1) _

,(1)

t,.1

1,.2

-

Remark: ni) is necessary for Definition 1 to be wellposed. In the following section we will present an algorithm in which the algebraic equation is differentiated, consistent initial value condition therefor should

(8) ,,(I') _ ,(I') 1,.1 -"'2

be guaranteed.

For the rest part of this paper, we will be dealing with the control systems under the assumptions presented in the subsection 2.1. Our objective is to derive the static state feedba.ck law under which noninteracting control is achieved, internal stability at the equilibrium solution is also obtained under certain conditions.

,,(I')

_

<"r,,-l -

El;)

=

Q,( E) Zi

,I,,) I"r~

Wp(E) ~

+ C,,(E)v + Dp(E)"

0, i ~ 1, ... , m

,Ii) ="'1 ) t. = 1, ... ,p

Where

3. NON-INTERACTING CONTROL WITH STABILITY Consider a control system of the form (1}-(31, under the a.bove assumptions a.nd proposition indicated by !sidori (1989), there exists a mapping X .~ I; ~ 1>(x), x E M n U, which is diffeomorphism on M n U, define the mapping as follows:

fdx) Based on A3)' we choose the following static sta.te feedback control law:

f,(x) ~(())

Id x )

~(l)

~

1;=

1>(x)

~

Lj'- ll ,(X)

(7) Where

1;(1')

1,,( x) C(E) ~ [C, (E)'" Cp(e)l'

where 8 = n - E~J=l ri_ Using the diffeomorphism to make a. non linear coordinates change, the differentialalgebaic equations (1)-(3) are transformed into the

D(O

=

[DdE)

Dp(E)]'.

Under the above feedback law) the closed-loop system

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QoW = Q(el , rank dQo(E") = So· Differentiate the

of (8) becomes

equation (12) once to obtain the following algebraic equation:

dOl = F\ W + Gdc)v + H,( c)a ~;()I =

F,te) + Gd~)v + f{, Wa Soppo.e the matrix [LgQo(E") LhQO(CO}] has constant rank say ro, t:> E M8. If Yo = rn, stop, otherwise proceed to next itera.tion.

,,(11 _
-

'2

Step 1 Define an (su - ro).

" 0

matrix Ro(e) such that

(10) setting n(€) = Ro(c)L / Qo( li ), then for some neighborhood U. of EO , the set

Q,(E)

= 0,

_ .Iil

Zj -

\;. 1

•

If the smooth mapping col(Qo(e), o(c)) has constant rank say So + SI, near se.

eo,

i = 1"" m

._

~ -

1, ·

,p

Where

Fi(E)

Differentiate the above equ;Ltion once to o btain the following algebraic equation:

= F,(E) - HdE)D-I(E)W(E)

G,(E) = aM) - H,(E)D - '(E)C(E) H,W = H;(E)D - I(EL • = 1" " ,6. The stucture of the system (lO) shows that the noninteracting required has been achieved. The condensed form of the closed-loop system (10) is:

~

= fW + g(c)v + ii(€)a

y, ~Q;(E) = 0, i = 1, ',m

(11)

(12)

Consider the equilibrium solution (en, vOL our objective is t.o find a submanifold Z· cM , with M an open s ubset of Rn such that if E(O) = €" E Z' , the solution satisfied c(t) E Z' for each t for which the solution t o (11)-(12) is denned uniquely. So the requirement in iii) of Def.1 will be sat.isfied. One would therefore intuitively expect to employ algorithm similar to the one (Pantelides, 1988) which is bailed on Hirschorn's {1979} inversion algorithm , to caculaie Z " . Through the algorithm ,the DAEs (11)-(12) C<>ll be derived into the differential equations realization which is useful in determining the stability of closed-loop system.

Algorithim: Initially, Mo is defined as the set of points where the mapping Q is zero, U(J is neighborhood of then the set

eo,

Mo' = Mo n Uo

= {( E Uo ,Qo(C) = 01,

Suppose the matrix [L 9 QI (c") Lr,QdeO)] has constant rank rI' If rI = rn, stop , otherwise, proceed to next step. Step k The iteration is started with mappings Qk- l (0 and k_,(e), where Qk - l is such that the rank of dQk _. is exactly equal to the number 50+·· ·+sk- l of its rows) and for some neighborhood r..~k of eO,

M k· = MknUk = (E E Uk ,Qk- ,(E)

=0

and .-,(e)

= 01

(1 3 1 The .mooth mapping COI(Qk _ ,(E) k-l(e)) has constant rank (so·t I "k) nea.r eO, set

Differentiate the above equation once to obtain the following algebraic equa.tioD

L,Q.(c)

+ L. Q.( El,· + LhQk(C)il =

Ifrank [L"Q.(~U) LhQ.(EU)] = r. proceed to next iteration .

=

0

m stop, otherwise,

If the above algebra.ic converges to the step k"! we have following proposition by Isidori(1989):

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Proposition: Assume the following for system (11), (12)

e)

e

eO,

a) dQ( has constant rank for all around and for some choice of Ro(e)'···, Rk.-de), the differentials of the mappings COI(QkW ~dE)), i.e. the matrices col (dQdO d~k(E)) have constant rank for all e around EO , OSkSk' - l.

b) The matrices [LgQ.(e) L ,.Q.( e)] have constant rank for all EE Mk around eO, 0 S k S k' - 1. c) The matrix L'JQk' (e" ) has rank rn, then '0 = m,'I = 80 - '-0, and for all k > 1, Sk+l = Sk - rk + f'k + l- As a consequence

QoW

=

Q(e)

and

Moreover,

any

other

choice

of

matrices

Rk--dE) is such that the conditions a) b)

Ro(e)"

static state feedback control la.w (9), we can investigate the system obtained by setting the feedback inputs t£ ,i 1,· ' ,p in equation (15) equal to zero,namely:

=

Notice € = ~(x),suppo,e E" is a regular p oint of the algorithm, then the system (5) is asymtotically s t a ble at the equillbrium (xO, tP) if and only if the system (16) is asymptotically stable at EO. Based on the above analysis, we have following result:

Th eo rem: Consider differential-algebraic equations systom (1)- (3) with x E Z' ,the noninteracting control probJem is solvable if the s tatic state feedback law (9) is taken. In addition, suppose x O is the regu lar point of the algorithm for th e dosed-Joop s ystem , the dosedloop system is asymptotically stable at the equi)jbrium point (.:z:0, VO) if and only if the differentia] equations realization (16) is asymptotically stable at th e point x,o.

c) are still satisfied. De/in1°tion2: A point EO is a regular point of the above zero dynamics algorithm if t.h_e conditions a ) b) c} of the

4.CONCLUSION

proposition a re satisfied .

Sop pose ~o is a regular point, then th e a.lgorithm converges to the k" step, at this step, M~ ~ = Mp n Uk • = {E E Uk -, Q.- - 1 = 0 and ~k- - l = aj, for some neighborhood Uk " of eO, let

Z" = Mko , then Z· if> the submanifold on which the system (U) (12) has a unique differentiable solution for any consistent initial value around (eO J vO). So the requirement oi) in definit ion 1 has been met if we define ~ E

Nonintera.cting cont rol problem with internal stabilty via static state feedback have been solved for a class of nonlinear systems described by a set of differentialalgebraic equations. Based on the non interacting equations, we employ an algorithm to derive a. certain ma.nifold on which the consistent initial conditions for a unique differentiable solution t.o the dosed-loop system is satisfied. Furthermore, we show the conditions und er wh ic h the local internal stability of a single equilibrium solu tion is achieved.

Z·.

ACKNOWLEDGEMENTS

At the step k"', algebraic equation takes the form of

Since rankLfJQko (e) = variables v as:

rn, we ca.n derive algebraic

The specification of the submanifold Z'" and reconstruction of the algebraic variables v yields the differential equations realization of DAEs (11) (1 2) as foUows: ~ where

=

EE

f(El - [ L~Q.- ( E WILfQk ·(E) +(Ii(E) - ' L.Qk- (~ W l L"Q._ (mu

This research is supported by NSF of C hin a, by NSF of Li aoning, China, by the Sta.te Education Commission of China, by the Ministry of Metallurgic Industry of China, by State Key Laboratory of Industrial Control Technology, Hangzhou, China, and by t.he Open Laboratory of Complex Systems and Control l the C hinese Academy of Sciences .

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(15)

Z*.

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