Noninteracting Control with Stability for Hamiltonian Systems

Copyright ~ IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998

NONINTERACTING CONTROL WITH STABILITY FOR HAMILTONIAN SYSTEMS A. Astolfi * L. Menini **

* Dept. Electrical and Electronic Engineering, Imperial College,

Exhibition Road, London SW7 2BT, England. E-mail: a.astolfiClic.ac . uk .* Dip. Injormatica, Sistemi e Produzione, Univ. Roma Tor Vergata , via di Tor Vergata , 00133 Roma, Italy. E-mail: meniniCldisp.uniroma2.it

Abstract: The problem of noninteraction with stability via dynamic state-feedback is addressed and solved for a class of nonlinear Hamiltonian systems. A simple to check necessary and sufficient condition to solve the problem is proposed. Some properties of the P* and ~mix dynamics are investigated, in particular, it is shown that such dynamics are generally not Hamiltonian. Copyright~ 1998 IFAC Keywords: Hamiltonian systems, noninteraction, nonlinear control, geometric control

in the case of linear systems, the problem can be solved by means of dynamic state-feedback if the two problems of stabilization and of noninteraction are separately solvable.

1. INTRODUCTION AND MOTIVATIONS

The problem of noninteraction with stability for nonlinear systems has been widely studied, see (Wagner 1989, Isidori 1994) as well as the excellent monograph (Battilotti 1994) . Necessary and sufficient conditions for the existence of either static or dynamic state-feedback control laws which solve the problem have been proposed, and procedures for the design of such control laws have been given. Despite the elegant characterization of the problem, based on geometric control theory, the applicability of the theory to physical systems has not received (to the best of the authors knowledge) enough interest. A notable exception is the paper (Huijberts and van der Schaft 1990) . Therein, the problem of noninteraction with stability is addressed for nonlinear Hamiltonian systems by means of a particular class of state-feedback control laws.

The main result of this paper consists in a simple to check condition for the solvability of the problem of noninteraction with stability, which is necessary and sufficient for the class of systems considered. In what follows, all the proposed results are local, i.e., valid in a neighborhood of the equilibrium. On the way to the main result, contained in Section 4, it is also shown that, contrarily to what holds true for the zero dynamics, the P* and ~mix dynamics of general Hamiltonian systems are not necessarily Hamiltonian. Such facts are shown, in Section 3, by means of simple low-order examples. Finally, in Section 5, the problem of non interaction with (simple) stability is solved by means of dynamic state-feedback for a physical system having an unstable zero dynamics. As a result of the considerations carried out in this paper, the class of Hamiltonian systems for which the problem of noninteracting control with stability can be solved is enlarged with respect to the results available in the related literature .

The work reported in this paper stems from the consideration that, by using more general statefeedback control laws, either static or dynamic, the problem of noninteracting control with stability can be solved for larger classes of systems. It is well known (see (Wonham and Morse 1970)) that, 397

2. THE SYSTEM

a(x) iJ(x)

The problem of non interacting control with stability will be tackled for a class of 2 inputs-2 outputs Hamiltonian systems, described by oH(q , p, u)

qi =

Yi

o.

'

(5)

b( ) .= [LJhd X )] x . LJh 2 (x)'

and v [VI v2]T is the vector of the new inputs . If the vectors ~i are given by

i = 1, . . . , n , i = 1, . .. , n ,

Oqi i = 1, 2,

(1)

hi (x)]

~i = [ LJhi(x)

= Ho(q , p) -

ql Ul - q2 U2 ,

= [qiqi ] '

i = 1,2 ,

=

and the vectors ij and p are given by ij [q3 . . . qnf , p = [P3 . . . Pn], the closed-loop system can be written as (Huijberts and van der Schaft 1990)

where the Hamiltonian function H(q, p, u) is

H(q, p, u)

:=

=

oHr~, p , u)

= qi,

(A(X))-1 b(x) , (A(x))-1 ,

:= -

(2)

and is assumed to be sufficiently smooth. ~i

The first 2 components of the vector q, are directly actuated by means of the external inputs Ul , U2 , which constitute the input vector u , hence, they coincide with the "natural outputs" Yl, Y2 (see (Nijmeijer and van der Schaft 1990)) . A subclass of Hamiltonian systems of special importance is that of simple Hamiltonian systems, in which the function Ho(q, p) has the form :

Ho(q, p) =

21 pT G(q) p + V(q) .

= A ~i + B Vi ;\. i

= 1, 2,

(6a)

:. ail - <"I c ,"i) - <.c, 1, 6 ) , (6b) q= op (-q, P, <.,2 + R 1 (-q, p,

p= - ~~ (ij , p, 6 , 6) + R2 (ij , p, ~1' 6) ,(6c) Yi

= [1 Ol ~i '

i = 1, 2,

(6d)

where the matrices A , Bare

(3)

The matrix G(q) E IR n xn is assumed to be positive definite for every q: this condition is satisfied by many Hamiltonian systems of practical interest , e.g. , by mechanical systems , in which the first term in (3) is the kinetic energy. It is also assumed an

il (ij , p, 6 , 6) is the function Ho(q , p), written in the new coordinates, and RI (ij , p, 6 , 6) , R 2 (ij, p, ~I' 6) are suitable functions such that R; (ij, p, 0, 0) = 0, i = 1, 2, for all ij, p in a neighborhood of the origin (Huijberts and van der Schaft 1990).

Let x := [qT pTf , let f(x) be the Hamiltonian vector field corresponding to the Hamiltonian function H(q , p, 0) , and let hi(x) := qi , i = 1, 2, so that equations (1) can be rewritten as

For simple Hamiltonian systems , the results in (Nijmeijer and van der Schaft 1990 , Huijberts and van der Schaft 1990) imply that, if A· denotes the largest locally controlled invariant distribution contained in Ker(dhd n Ker(dh2) (see (Isidori 1994)), then

oV

t hat aq(O)

= 0, so

that q = 0, p

= 0, is

equilibrium for system (1).

A· = span {0/ oijj,

x= f(x) + 91 Ul + 92 U2 , Yi = hi(X) ,

i = 1, 2,

= 1, . . . , n -

2} .

ail (q, p) qy oH (q, p) p= oq

ij=

In the case of simple Hamiltonian systems, it is well known (see (Nijmeijer and van der Schaft 1990)) that PI = P2 1, and that A(x) G 1l (q) , where Gll(q) is the 2 x 2 leading submatrix of G (q) . Therefore, the matrix A( x) is nonsingular everywhere. Consider the feedback control law

u = a(x) + f3(x) v,

i

Hence the coordinates q, p, can be used to describe the zero dynamics of the system (1), (3) :

where 9i equals the (n + i)-th column of the 2 ndimensional identity matrix, i = 1,2. Let the characteristic numbers PI and P2 at the point q = 0, p = 0, and the decoupling matrix A(x) be defined as in (Nijmeijer and van der Schaft 1990) .

=

0/ OPi,

(7)

where the restricted Hamiltonian (Nijmeijer and van der Schaft 1990) il (ij, p) := il (q , p, 0, 0) .

=

As observed in (Huijberts and van der Schaft 1990), the zero dynamics of general Hamiltonian systems of the form (1) are Hamiltonian ; for simple Hamiltonian systems this implies that, if n > 2, any "decentralized" feedback control law , described by the equations

(4)

where a(x) and iJ(x) are defined as: 398

Vi

= tilqi + t i24i ,

tit , ti2

E IR,

i

= 1, 2, (8)

non interacting feedback (10) , composed with a linear "decentralized" state-feedback , similar to (8) . With an abuse of notation , such a statefeedback will be called in the following "decentralized", or , in case of Pi = 1, "PD-like decentralized". Such a class of state-feedback has been used in (Huijberts and van der Schaft 1990) to solve the problem of noninteracting control with simple stability. Stability of the zero dynamics is not necessary to solve the noninteracting control problem with stability. In fact , as shown in (Isidori 1994, Battilotti 1994) , the problem is solvable, by means of general static state-feedback , if and only if the system itself is stabilizable in the first approximation and the P* dynamics are asymptotically stable in the first approximation. If this last condition is not satisfied , the problem might still be solvable by means of dynamic state-feedback . Under some regularity assumptions , in (Battilotti 1994, Isidori 1994) it is shown that a necessary and sufficient condition for the existence of a dynamic state-feedback control law which solves the problem is that the system itself is stabilizable in the first approximation and the ~mi x dynamics are asymptotically stable in the first approximation .

cannot achieve asymptotic stability for the closedloop system . Nevertheless , if (0 , 0) is a stable equilibrium point for (7) , (simple) stability can be achieved by a proper choice of the gains tij in (8) (see (Huijberts and van der Schaft 1990)). In this paper , it is shown that, allowing more general state-feedback control laws, static or dynamic , the problem of noninteracting control with stability (simple or asymptotic, depending on the properties of the given system) can be solved for a wider class of Hamiltonian systems. 3. SOME GEOMETRIC PROPERTIES Given a general non linear system of the form ;f = ](x) + g(x)u, Y = h(x),

(9)

with x E IRn , U , y E IRm , ](0) = 0, h(O) = 0, satisfying suitable regularity assumptions (see (Battilotti 1994, Isidori 1994)) , several approaches can be adopted in order to solve the problem of noninteraction with stability. The main results of the general theory will be now summarized with reference to the case in which the stability requirement is that of asymptotic stability in the first approximation . Moreover, we assume that system (9) is stabilizable, the characteristic numbers Pi , i 1, 2, .. . , rn , can be defined, and the m x rn decoupling matrix A(x) is nonsingular at x = O. Consider any regular static state-feedback control law

Hamiltonian systems are not asymptotically stable at any equilibrium , although they can be stable. A sufficient condition for stability of an Hamiltonian system is that the Hamiltonian function has an isolated local minimum at the equilibrium (Huijberts and van der Schaft 1990) . Hence it is of interest to know whether the dynamics of certain subsystems are Hamiltonian . Despite the fact that the zero dynamics of Hamiltonian systems are Hamiltonian, it will be now shown that, for Hamiltonian systems of the form (1), (2) , neither the p. dynamix nor the ~mi x dynamics are Hamiltonian, in general. This will be done by means of simple counterexamples, all derived from the following Example.

=

u = a(x) + ".8(x)'U,

(10)

such that the closed-loop system is noninteractive (such a control law exists by virtue of the assumptions on the matrix A(x), see (Isidori 1994)) and rewrite the closed-loop system (9), (10) as ~ = }(x)

+ g(x)'U,

Y = h(x) ,

Example 1. Consider the nonlinear system :

(ll)

t.e. define }(x) := 7(x) + g(x)a(x) and 9(x) := g(x)".8(x) . Let the distributions Pi, i = 1, .. . , rn, P* and ~mix be defined as in (Isidori 1994). In the following , the dynamics of system (11) restricted to S· (the integral submanifold of p. containing x = 0) will be called p. dynamics, the restriction to L· (the integral submanifold of ~mix containing x = 0) will be called ~mix dynamics.

qi=Pi ,

i=I , 2,

Pi =

i = 1, 2,

43

Vi ,

( 12)

= b P3 + alql + a2q2 + a qlq2 ,

= -cq3- dlql-d 2q2- 0 qlq2, with outputs Yi = qi , i = 1, 2, obtained by means P3

of the static state-feedback control law:

= alP3 + a q2P3 + d I q3 + 0 q2q3 + Vl , U2 = a2P3 + aqIP3 + d2q3 + 0 qIq3 + V2, Ul

To decide which class of control laws has to be used to solve the problem of noninteracting control with stability, and to decide whether the problem is indeed solvable, the following considerations are to be made. If the zero dynamics of the system are stable in the first approximation, the problem can be solved by means of the standard

applied to the Hamiltonian system (1) with 1 H (q, p, u) = 2

(2 2 2) PI + P2 + b P3 + 1

?

+ a2q2 + et qIq2) P3 + 2cq3 + (dlql + d2q2 + 0 qlq2) q3 - qlUl - Q2 U2, (aIql

399

where aj, b, c, dj, 0' , 0, are real parameters. Note that the zero dynamics of such a system can be simply written as q3 = bP3 , P3 = -cq3 '

convergence 0', being 0' a positive real number , is required , respectively. Moreover , let
(13)

• Now, by properly choosing the parameters in Example 1, the following two facts can be proven.

[er rf,

of the form (1) need not be Hamiltonian .

=

=

1, c -1, Consider Example (1) . Let b dl 1, a2 1, d2 0, 0' 0 1. The distributions Pi and Pi can be easily computed for system (12), obtaining

=

=

=

p.

=

= =

= P; n P; = span {es -

=

where the vector z E IR 2n - 4 is given by z := F , L , M are real matrices of suitable dimensions, and the vector valued function t9(6 , 6, z) is zero, together with its first order derivatives, at (0 , 0, 0) .

Fact 1. The p. dynamics of Hamiltonian systems

al

=

A ~j + B Vi , i 1, 2, Z = Fz+L6+M6+1?(6 , 6 , z) , (15) Yi = [1 Ol~i ' i= 1, 2, ~j

In order to consider only systems which require dynamic state-feed~.ack control laws in order to be rendered stable and in.oninteractive, the following two assumptions are made.

e6} ,

where ej denotes the i-th column of the 6 dimensional identity matrix.

(a) the two pairs:

= = = =

Hence S· is given by {ql P1 q2 P2 0, P3 = -q3} and the p. dynamics can be written as

are controllable, (b) O'(F) n
(14)

Notice that assumption (a) implies that P* == == span{o;oz} , hence assumption (b) implies that the problem of noninteraction with stability is not solvable by means of static state-feedback.

which is clearly not Hamiltonian . Since (14) is asymptotically stable, and , moreover , the given system is stabilizable , in this case the problem of noninteracting control with asymptotic stability in the first approximation can be solved by means of static state-feedback.

Fact 2. The

~.

V:' vt

Now , let denote the two F-invariant subspaces of IR 2n - 4 such that

~mix

dynamics of Hamiltonian systems of the form (1) need not be Hamiltonian .

IR 2n -

Consider again the system in Example 1, with the values of the parameters chosen as above . The following vector field

[91 , adj92] = e5 -

1

4 - VF -g

ffi v.F <:V b ,

0' (

Flv:) c
0' (

Flvt ) c
Let a linear coordinate transformation be defined on p. such that, if the coordinates are given by

z

e6 ,

z = T z,

certainly belongs to ~mix' As ~mix C p., it is evident that, in this example, ~mix == p., hence the ~mix dynamics are not Hamiltonian.

then one has

4. MAIN RESULT The problem of noninteracting control with stability will be dealt with for the class of simple Hamiltonian systems given by (1) , (2), (3), using dynamic state-feedback control laws.

with

Let the symbol
O'(Fg) =

0' (

O'(Fb) =

0'

Flv: ) ,

(Flvt ) .

Let z = : [zJ zl'] T be the partition of z corresponding to the block partition of P. Finally, let the vector J (~1 ' 6, z) , defined by

J(6 , 6 , z):= T1?(6, 6 , T-lz) , 1 The symbol 0'(-) denotes the spectrum of the matrix at argument .

400

be partitioned according to the partition of

z:

simple stability are jointly achievable, if the use of dynamic state-feedback control is allowed. Example 2. Consider the system represented in Fig. 4, which is composed of four heavy dimensionless carts, denoted by Cl, C 2 , C 3 and C 4 , subject to the gravitational field , of magnitude g, and constrained to move on a vertical plane. The four carts, Cl and C 2 having mass rn , C 3 and C4 having mass M, interact through mechanical couplings involving other massless objects, as shown in Fig. 4. Any kind of friction is neglected . The two carts C3 and C4 are constrained to slide along a guide-rail, whose shape can be described as the union of the two curves, f; , i = a , b, given below, each of them parameterized through its curvilinear abscissa S E [-2, 2] :

t9(6, 6, z) = [19 r(66, z) lJf(C,l, 6, z)f · The following assumption (c) considerably simplifies the problem. (c) The vector lJb(C,l , 6, z) is a function of the variables C,l , 6 only:

19 b (6, 6 , z) =: lP(6, 6), for all (C,l , 6 , z) in a neighborhood of the origin. The following result provides a condition to solve the noninteracting control problem with stability by means of dynamic state-feedback, which is, in general , much easier to check than the necessary and sufficient conditions based on the explicit computation of the distribution .6. m ix . In order to apply the results recalled in Section 3 , the following technical assumption is introduced:

.' {Xi(S) =

(d) The origin (C,l , 6 , z) = (0 , 0 , 0) is a regular point of the distribution .6. m ix of system (15).

Yi ( s) =

Ci (

=

Proposition 1. Under assumptions (a) , (b) , (c) and (d), a dynamic state-feedback control law which solves the noninteracting control problem with either

(A) (simple) stability, or (B) asymptotic stability in the first approximation, or (C) asymptotic stability with a prescribed convergence rate ,

=

Xi = (l2i(XI, X2,

~ Fb [lPqlq, (6,6) + lPq,q, (C,l, 6)] +

Sa, Sb)

+ ui)/rn ,

°

i = 1, 2,

Sa =P3/ M ,

lPq,q,(6,6)-

Sb =P4/M,

[ql lPq,q, q, (6,6) + q2lPq,q2 q, (6, 6)] = 0.

P3 = h (So(XI ' P4 = k (Xl

In cases (B) and (C), conditions (i), (ii) and (iii) are also sufficient for the existence of a solution, whereas, in case (A), a set of sufficient conditions is given by (i), (ii) , (iii) and

0,

Zg,

0)

X2) -

+ X2 -

with the outputs Yl

2 Sb)

=

M 9 Sa/ 2 , + M 9 Sb/ 2 ,

sa) -

Xl -

Xe,

Y2

= X2 + Xe·

After a first state-feedback of the form (4), if C,i := [Xi xif , with Xi := Xi - X e , i = 1, 2, the origin X of the state vector X [E,[ Sa Sb P3 P4jT is an equilibrium point, hence the closed-loop system can be written in the form (15), with Z = [Sa Sb P3 P4jT, as usual. Having checked that the given system satisfies assumption (a) , in order to study the stability properties, in the first approximation, of the zero dynamics , the matrix F has been computed. If k < gM /4, the matrix F has a real eigenvalue >. with positive real part .

=°

(iv) the equilibrium of the dynamical system

= Fg Zg + lJ g (O,

,

%.

lPq,q,(6,6) =0 , lPq, q2 (6 , 6) = lPq, q, (6 , 6) ,

Zg

+ S~)

=

°

(iii)

-1

A suitable vector of configuration coordinates for the described system is given by q [Xl X2 Sa sbjT . An equilibrium point is qe := [xe -Xe ojT , q = 0, where Xe := La - 2 + In the following , the motion of the system around such an equilibrium point is considered. The system is described by the equations

exists only if the following conditions hold in a neighborhood of 6 = 0, 6 = (it is recalled that C,l = [ql (h]T, 6 = [q2 (hjT) :

(i)

(~g + arcsin G)) , .,

where Ca 1, Cb -1. Carts Cl and C 2 are subject to two external forces having direction parallel to the x axis and intensity UI and U2 , respectively. UI and U2 are the only control inputs of the system. The springs with elastic constant k have length La, when undeformed, whereas the spring with elastic constant h has length equal to zero , when undeformed; all such springs are linear and lie on the same guide-rails along which the carts are constrained to slide.

We are now ready to state the main result of this section, whose proof is omitted , for brevity.

(ii)

Ci

r. .

(16)

is stable.

5. EXAMPLE The physical example presented in this section is an unstable system , for which noninteraction and

401

=

a

p

Figure 1. The mechanical system considered in Example 2 3r---~------~-------'

\. ,

O . 2.----~--~--...........--~--_,

,

.:. ,."" "". ., , . ,, ., , . ,." .,

0 . 15

.

"

H

, ," .

,,

,,

, , ,,

'.5

. . - I",, ,

, , , , ,

0 .5

o

-- - - - "---:-,,;.-------'---1

-0. 5 O!:----:S----:':,0=------=,'="5---;;2~O---..J (a)

o

5

15

20

(b)

Figure 2. Simulation results for Example 2:(a) time behavior of the output variables Yl (continuous) and Y2 (dashed); (b) time behavior of the state variables Sa (dashed) and Sb (continuous) . hence stability of the closed-loop system cannot be achieved by any static state-feedback control law which guarantees noninteraction. Since the eigenvalues of Fare {). , -). , JW , -Jw}, with wEIR, w > 0, and J being the imaginary unit , it makes sense to check for the existence of a dynamic statefeedback control law guaranteeing noninteraction and simple stability. Since hypotheses (i) , (ii), (iii) and (iv) are satisfied, the problem of noninteraction with simple stability can be solved by means of dynamic state-feedback.

6. REFERENCES

Battilotti, S. (1994) . Noninteracting control with stability for nonlinear systems. Springer Verlag. Huijberts, H. J . C . and A. J . van der Schaft (1990) . Input-output decoupling with stability for hamiltonian systems. Math . Control Signal Systems pp. 125-138. Isidori , A. (1994) . Nonlinear control systems. Springer Verlag. Third edition. Nijmeijer, H. and A. J. van der Schaft (1990) . Nonlinear Dynamical Control Systems. Springer Verlag. Wagner, K. G. (1989) . On nonlinear noninteraction with stability. In: Proceedings of the 28-th Conf. on Decision and Control. Tampa, FL . Wonham, W . M. and A. S. Morse (1970) . Decoupling and pole assignment in linear multivariable systems: a geometric approach . SIAM 1. Control 8(1), 1-18.

A compensator which solves the problem has been designed on the basis of the methodologies in (Isidori 1994) . The results of a significant simulation of the closed-loop noninteractive and stable control system are reported in Fig . 2. The values of the physical parameters are 9 9.81 , m = 1, M 2, k 1, h 1/2, Lo 5 - 7r/2, L = 3.5. In Fig. 2(a) one can see that it is possible to control separately the two outputs (two"shifted" piecewise constant inputs have been applied in this simulation), whereas in Fig. 2(b) it is possible to appreciate the stable time behavior of the positions Sa and Sb of two carts C3 and C4 . •

=

=

=

=

=

402