- Email: [email protected]
New Examples in Noninteracting Control for Hamiltonian Systems
Copyright@ IFAC Lagrangian and Hamiltonian Methods for Nonlinear Control, Princeton, New Jersey, USA, 2000
NEW EXAMPLES IN NONINTERACTING CONTROL FOR HAMILTONIAN SYSTEMS Alessandro Astolfi * Laura Menini **
* Dep. of Electrical and Electronic Engineering, Imperial College
Exhibition Road, London SW7 2BT, England E-mail: a.astolfi
Abstract: In this paper, the problem of noninteracting control with stability is studied for generalized Hamiltonian systems with reference to two relevant case studies: the control of the angular velocities of an underactuated rigid body and the control of a third order food chain. For the rigid body, under suitable conditions, noninteraction and asymptotic stability are jointly achlevable by means of a static state-feedback control law; whereas, for the food chain, noninteraction and asymptotic stability of a "critical" equilibrium are jointly achievable by means of dynamic state-feedback control laws. Copyright @2000 [FAC
1. INTRODUCTION The problem of asymptotic stabilization of Hamiltonian and generalized Hamiltonian systems has been widely studied in the control literature, see e.g. (van der Schaft, 1986; Ortega et al., 1998; Woolsey and Leonard, 1999; Bloch et al., 1999; Bloch, 1999). This is because, due to their special structure, it is possible to obtain general results, valid for a large set of physically motivated systems.
Fig. 1. Inclusion properties relative to the dynamics concerned with the problem of noninteraction with stability.
On the other hand, the problem of noninteracting control for nonlinear systems has received considerable attention (Isidori and Grizzle, 1988; Wagner, 1989; Zhan et al., 1991; Battilotti, 1994; Isidori, 1995). By means of geometric control theory, necessary and sufficient conditions for the existence of either static or dynamic state-feedback control laws, which allow to obtain stable noninteractive closed-loop systems, have been proposed, and systematic design procedures have been given. The essential results of the theory are related with the "so called" P* and .6. mix dynamics, contained in the zero dynamics .6. * as
depicted in the illustrative diagram in Fig. 1, see (Isidori, 1995, Chapter 7) for details. As a matter of fact, under suitable regularity and stabilizability assumptions, the problem of noninteracting control with stability is solvable by means of a regular static state-feedback if and only if the P* dynamics are asymptotically stable, whereas it is solvable by means of dynamic state-feedback if and only if the .6. mix dynamics are asymptotically stable. The problem of noninteracting control with stability for Hamiltonian systems characterized by
145
= (13 -It) W3 WI + U2, 13W3 = (II - h) WI W2 + b3 •1 UI + b3,2 U2,
a non-degenerate Poisson bracket has been dealt with in (Huijberts and van der Schaft, 1990; Astolfi and Menini, 1999; Astolfi and Menini, 2000), with particular attention to the so-called "simple" Hamiltonian systems. Therein, easy to check conditions for the solvability of the problem by means of either static or dynamic state-feedback control laws have been given. To the best of the authors' knowledge, the problem of noninteraction with stability for generalized Hamiltonian systems has not been considered yet, i.e., no general results exploiting the special structure of such systems are available in the literature. In this paper, two significant examples are studied in detail: an underactuated rigid body (Euler's equations) and a three dimensional food-chain. In both cases it is assumed that the state of the system is available for feedback.
h W2
Ba: (x) + b(x)u
YI=WI+Bw2+CW3,
(2d)
Y2=FwI +w2+Hw3'
(2e)
H(x)
= ~ (11 wi + 12 w~ + 13 W~) , 13 W3 11 12
o J(x)
h
=
1
---
W3
o
h
1 12 W2 It 13
It WI 12 h
12 W2 It 13 It WI I2 h
o
=
and R(x) O. Notice that J(x) has rank 2 for every x =I O.
Example 2. (Food chain). Consider the normalized third order preys-predators system
In this paper we consider systems described by:
- R(x»
(2c)
Setting x = [Wl W2 w3f, such a system can be written as a generalized Hamiltonian system, with
2. GENERALIZED HAMILTONIAN SYSTEMS
x = (J(x) y = hex),
(2b)
+ UI, X2 = -(XI, X2) - X2 + (X2, X3), X3 = -(X2, X3) - X3 + U2,
Xl
(la) (lb)
=
(Xl,
X2) -
Xl
= Xl, Y2 = X3,
where x E IRn, U E IRm, y E IRq, J(X) = -JT(x), and R(x) = RT(x) ~ 0, which are also called port-controlled Hamiltonian systems, see (Maschke and van der Schaft, 1992; van der Schaft, 1996; Ortega et al., 1999). Many physical systems can be described by equations (1), with the matrix J(x) describing the internal structure (the interconnection) of the system and the matrix R(x) describing the natural damping.
(3a) (3b) (3c) (3d)
Yl
(3e)
with Lotka-Volterra predation mechanism, i. e., with (Xi, Xj) = Xi Xj' Such a system is positive, i.e., Xi(O) ~ 0, i = 1, 2, 3 and Ui(T) ~ 0, i = 1, 2, for all T E [0, t), imply Xi(t) ~ 0, for all t > O. Moreover, it can be written in the form (1) with H(x) Xl X2 X3'X~;) di]ag (Xl, X2, X3),
= +[ 0+
Observe that the dimension of the state-space of system (1) is in general not even. This has strong influence on the structural properties of the system, as many interesting properties can be proved only if the Poisson structure is non-degenerate, i.e., if the matrix J(x) has full rank. Since the rank of any skew-symmetric matrix is even, any Poisson structure defined on an odd dimensional manifold is degenerate. Generalized Hamiltonian systems with degenerate Poisson structure are also called Poisson systems.
and J(x) =
-XlX2
o
=0
0 X2X3 -X2X3 0
.
3. NONINTERACTING CONTROL WITH STABILITY FOR THE RIGID BODY Consider the system in Example 1 and let Al = (12 - 13)/11 , A 2 = (13 - Id/I2, A 3 = (It I 2)/h Q: = b3,I/h (3 = b3.2/I3, and K = [AI A 2 A 3 Q: (3 B C F Hl T . Assume Ai =I 0, i = 1, 2, 3.
Example 1. (Rigid body). Consider a rigid body with principal moments of inertia I i > 0, i = 1, 2, 3, and let WI, W2 and W3 denote the angular velocities with respect to the principal axes. Assume that two actuators (e.g. gas jet actuators) provide control torques Ul and U2, which are the control inputs of the system. Assume also that the two outputs to be decoupled are linear combinations of the wi's. In particular, let the system be described by:
The problem of stabilization of the zero equilibrium of the rigid body angular velocity has been widely studied and solved, see e.g., (Brockett, 1983; Outbib and Sallet, 1992; Astolfi, 1999) and the references therein. On the other hand, the problem of noninteracting control is trivially solvable. Goal of this section is to investigate under which conditions both problems can be jointly solved. To be more general, we also investigate the possibility of regulating the two outputs to desired values Yi" and Y2*, obtaining, at the same time,
(2a)
146
both noninteraction and stability for the overall system.
If none of conditions (i)-(iv) holds, then the Llmix dynamics (coinciding with the P* dynamics) are described by the equation:
To this end, we make the simplifying assumption 1 F B ¥ 1 and we assume F B + BH a - H /3 - 1 Ca + C /3 F ¥ 0 so that the system has vector relative degree (1, 1). After a preliminary feedback, called standard noninteracting feedback, system (2) can be rewritten as
= Vi , il2 = V2, ill
W3 = Qo(K, yt, yz*)
== Llmix
= O.
c) System (5) has 2 coinciding equilibria, i.e., Ll = 0 and Q2 ¥ O. d) System (5) has 2 distinct equilibria, i.e., Ll and Q2 ¥ O.
= 0 if and
Remark 1. If Yt = 0 and Y2* c) and e) can occur.
3.1 A numerical example. If condition (i) of Lemma 1 holds and condition (iv) does not 2, then, in view of Corollary 1, the problem of noninteracting control with stability can be solved by means of static statefeedback. From equations (4), we have that Pt = span {92, [/,92]} and P; = span {9d. Then, the change of coordinates z = -91,3 Yl + W3, transforms system (4) into the following form:
The condition in Corollary 1 can be easily tested in view of the following lemma.
= 0 if and only if either
(i) Ai A 3 > 0, /3 = 0,
H
= -F/a,
or (ii) A 2 A 3
/3 £921
> 0, a
H
= -1//3.
= 0 if and only if either (iii) A 2 A 3 > 0, a = 0, /3 = ±.jA3 /A 2 , C = -B//3,
Y2 = V2,
(6b)
= Iz(z, Y2) + bV2,
(6c)
where b is real constant. The stabilization of subsystem (6a) is trivial, whereas, in order to deal with subsystem (6b)-(6c), a further change of coordinates z = z - bY2 is convenient. As a matter of fact, the origin Y2 = 0, Z = 0 of subsystem
Al A 3 > 0, /3 = 0, a = ±.jA 3 /A l , C = -1/a.
Note that (i) and (iii) cannot occurr simultaneously because Al A3 > 0 => A2 A3 < O.
2 I
(6a)
i
or (iv)
= VI,
ill
= 0,
= ±.jA 3 /A 2 ,
= 0 then only cases
Theorem 2. If L Y1 I ¥ 0 and L Y2 I ¥ 0, then the problem of noninteracting control with asymptotic stability is globally solvable if and only if case b) occurs, with Ql < 0, and it is locally solvable if case d) occurs. The problem of noninteracting control with simple stability is locally solvable in case e).
Corollary 1. If L y1 1 = 0 and L y2 1 ¥ 0 (or I = 0 and L 91 I ¥ 0), then the problem of noninteracting control with asymptotic stability is locally solvable for system (4) by means of static state-feedback.
= ±.jA 3 /A 1 ,
>0
e) System (5) has an infinite number of equilibria, i.e., Q2 = Ql = Qo = o.
£92
a
<0
b) System (5) has 1 equilibrium, i.e., Q2 = 0 and Ll > O.
Proof. The necessity is based on straightforward but tedious computations, which show that, for all possible values of the vector K, if L y , I ¥ 0, i = 1, 2, then dim(Llmix) = dim(P*) = 1. To show that LyJ = 0 implies P* = 0 (the case of L y2 1 = 0 can be treated similarly), notice that, in such a case, P; = span{9d, and Pt = span{92, [0 0 IjT}.
Lemma 1. L 91 I
(5)
a) System (5) has no equilibria, i.e., either Ll or Ql = Q2 = 0 and Qo ¥ O.
(4c)
where Vi and V2 are the new control inputs, and 91,3 and 92,3 are two real numbers, depending on the vector K. Letting I (Yl, Y2, W3) = [0 0 !J(Yl, Y2, W3)]T, 91 = [1 0 91,3jT and 92 = [0 1 92,3]T, we can state the following result.
Theorem 1. For system (4), P* only if either L Y1 I = 0 or L Y2 I
+
Therefore, we now study the properties of equation (5). Let Ll = Qi - 4Q2 Qo (for simplicity we drop the arguments), then, one of the following conditions occurs.
(4b)
+ 9l,3 Vl + 92,3V2,
yt, yz*) W3
Q2(K, yt, Y2*)W~,
(4a)
W3 = !J(Yl, Y2, W3)
+ Ql (K,
This assumption can be easily removed.
147
1.6r---~--~--~--~-~
..
o. 06
0.'
0.'
(a)
(b)
Fig. 2. The rigid body example. (a): Wl(t) (solid), W2(t) (dashed) and W3(t) (dotted); (b) Yl(t) (solid) and Y2(t) (dashed).
~
Z
4. NONINTERACTING CONTROL WITH STABILITY FOR THE FOOD-CHAIN
(7a)
Y2 = V2,
= a2 z + al z Y2, -2
-
(7b)
The goal of this section is to regulate, if possible, the two outputs of system (3) to desired constant values Yt > 0 and Y 2- > 0, obtaining, at the same time, both noninteraction and stability for the overall system. The stabilization problem has been solved in (Ortega et al., 1999) by means of both state and output feedback control laws, by using the concepts of energy-shaping and damping injection. Notice that, after the static statefeedback
being at =I 0, can be easily stabilized by means of a static feedback control law from Y2 and z, whose design can be performed by means of standard backstepping techniques (Krstic et al., 1995). The controller proposed here, in order to obtain global asymptotic stability of the origin and noninteraction, is of the form:
=- k l Yl + Tl , V2 = ('1'1 + 2'1'2 z) (83 Z3 + ~: zY2) k 2 (Y2 - hI Z + '1'2 Z2)) - al Z2 + T2, VI
= Xl - Xl X2 + VI, U2 = X3 + X2 X3 + V2, Ul
system (3) can be written as
where '1't, '1'2 and 83 are suitable constants, and Tl and T2 new inputs with respect to which noninteraction is guaranteed. The results of a simulation with the proposed controller, with Al = 1, A 2 = -1, A 3 = 1.5 a = A 3 /A t , 13 = 0, B = 1, C = 0.5, F = 0.5, H = -F/a, at ~ -1.23, a2 ~ -1.61, '1't ~ -1.32, '1'2 ~ 612, 83 = -1.5, k l = 0.2, k2 = 0.2, Wl (0) = 1, W2(0) = 0, W3(0) = 1, are reported in Fig. 2. In order to emphasize the property of noninteraction, Tl (t) has been set to zero for t 1: [50, 100], and equal to a sinusoidal signal in such an interval, whereas the input T2(t) has been set to zero for t 1: [150, 200], and equal to a sinusoidal signal in such an interval.
Yl = VI,
(8a)
= V2, X2 = -X2 (1 + Yl -
(8b)
Y2
J
Y2) .
(8c)
From equation (8c), it is obvious that, if 1 + Yt > Y2-, then the problem is easily solvable (with limt-t+oo X2 (t) = 0), whereas, if 1 +Yt < Y 2-, then the problem is not solvable at all. Hence, only the special case 1 + Yt = Y2- needs to be investigated. By means of easy computations, it can be seen that Amix = 0, whereas, at any point of the state space where X2 =I 0, P- = span{%x2}' Hence, the problem of noninteraction with asymptotic stability around an equilibrium Xl = yt, X2 = Xi > 0, X3 = Y2- = 1 + Yt, is not solvable by means of static state-feedback, but it is solvable by means of dynamic state-feedback. The controller proposed here is based on a dynamic extension as described in (Isidori, 1995, Chapter 7), but has lower dimension than the controller which would result from applying the design technique reported there. Let J.Ll and J.L2 be the two state variables of the controller, and let:
As for the possibility of regulating the two outputs Yl and Y2 to constant reference values Yt and Y 2- , it is clear from the above equations that there are no restrictions on the admissible values for Yt· Furthermore, from equations (7), it is clear that, in general, for every Y2- =I 0, only local asymptotic stability of Y2 = Y 2-, Z = Z-, being Z- a real constant depending on K, can be achieved.
{J.l
148
=
-J.Ll
(1
+ Yl) + Wl,
(9a)
3
!\ !\
1\
3S I
------------------~
"
r
I
•
:
~
... - - - - - - - -
\! \i Il'
1\
\/
I1
.>
(a)
(b)
Fig. 3. The food chain example. (a): Xl(t) (solid), X2(t) (dashed) and X3(t) (dotted); (b) Ul(t) (solid) and U2(t) (dashed) (9b)
V2 = - (J.L2
= Vl, [J,l = -J.Ll (1 + yd + Wl, Yl = Vl,
(lOb)
+ W2,
(IOd)
Wl _ W2) . J.Ll J.L2
(lOe)
[J,2 = -J.L2 (1 - Y2)
~3 = -6
(1-
(lOc)
+ 1) J.Ll,
(lla)
W2
=-
(Y2* - 1) J.L2,
(llb)
+ T2,
Remark 2. It can be shown that, if X:; < 1 and the initial conditions Xl (0) and X3 (0) are in a suitable neighborhood of the desired values Yt and Y 2*, then Ul (t) ~ 0 and U2(t) ~ 0, for all t ~ O.
(lOa)
Remark 3. It is stressed that the special case 1 + Yt = Y 2* is the only one in which it is possible to obtain a steady state response in which also
From equation (lOe), it is clear that the choice Wl = (yt
J.L2,00)
with k2 > 0, guarantees asymptotic stability of the equilibrium Y2 = Y 2*, J.L2 = J.L2,oo, with domain of attraction J.L2 > 0, for the subsystem (lOc)(lOd)-(llb).
where the two fictitious inputs Wl and W2 will be specified later as functions of the state of the extended system. After the change of coordinates 6 = J.Ll J.L2/X2, which is valid as long as X2 =j:. 0, the extended system (8)-(9), has the following form: Yl
+ k 2) (Y2 - Y2* + J.L2 -
Xi> o. 4.1 Simulation results
The results of a simulation with the proposed controller, with Yt = 2, Y 2* = 3, X:; = 0.2, J.Ll,oo = J.L2,00 = 0.3 Xl(O) = X3(0) = 1, X2(0) = 0.5, kl = k 2 = 2, are reported in Fig. 3. In order to emphasize the property of noninteraction, the input Tl (t) has been set to zero for t i [15, 30], and equal to a sinusoidal function inside such an interval, whereas the input T2(t) has been set to zero for t i [45, 60], and equal to a sinusoidal function inside such an interval.
with 1 + Yt = Y2*' implies that the variable 6 remains constant for all t ~ O. Hence, assuming that the two state variables J.Ll and J.L2 are regulated to suitable desired values J.Ll,oo and J.L2,oo, respectively, one has (t) J.L2,00 X2 (0) · 1lm X2 = J.Ll,oo t ....++oo J.Ll (0) J.L2(0) Therefore, not only it is possible to regulate Yl and Y2, but also the non-actuated state value X2 to a desired positive value X:;, by choosing J.ll (0) = J.Ll,ooVX2(0)/X:;, J.L2(0) = J.L2,ooVX2(0)/X:;. Now, in order to obtain asymptotic stability of the equilibrium Yl = yt, J.Ll = J.Ll,oo, with domain of attraction J.Ll > 0, for the subsystem (lOa)(lOb)-(lla), one can use standard backstepping techniques (Krstic et al., 1995), yielding the controllaw
4.2 Extensions
r;,. - J.ll + J.ll,oo) + Tl,
It is easy to see that the property 6 mix = 0 still holds for more general predation mechanisms than the above mentioned
where k l > 0 and Tl is the new input, with respect to which noninteraction has to be guaranteed. Analogously, the choice
X2 = -X2 lh (Xl) - X2 + X2 83(X3), where the 8i (·), i = 1, 2 are arbitrary functions. This fact suggest that it might be possible to
Vl = - (J.ll
+ kd (Yl
-
149
extend the results presented above also to this case.
Bloch, A. M., N. E. Leonard and J. E. Marsden (1999). Potential shaping and the method of controlled Lagrangians. In: Proceedings of the 38-th Conference on Decision and Control. Phoenix, AZ. Brockett, R. W. (1983). Asymptotic stability and feedback stabilization. Differential geometry control theory pp. 181-191. 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. (1995). Nonlinear control systems. Springer Verlag. Third edition. Isidori, A. and J. W. Grizzle (1988). Fixed modes and nonlinear noninteractive control with stability. IEEE Trans. Aut. Contr. AC-33, 907914. Krstic, M., 1. Kanellakopoulos and P. Kokotovic (1995). Nonlinear and Adaptive Control Design. John WHey & sons. New York. Maschke, B. M. and A. J. van der Schaft (1992). Port controlled Hamiltonian systems: modeling origins and system theoretic properties. In: Proc. 2nd IFAC Symp. on Nonlinear Control Systems design (NOLCOS'92). Bordeaux. pp. 282-288. Ortega, R., A. Astolfi, G. Bastin and H. Ro driguez Cortes (1999). Output feedback control of food-chain systems. In: New trends in nonlinear observer design (H. Nijmeijer and T. Fossen, Eds.). Springer-Verlag. Ortega, R., A. Loria, P. J. Nicklasson and H. SiraRamirez (1998). Passivity-Based Control of Euler-Lagrange Systems. Springer Verlag. Berlin. Outbib, R. and G. Sallet (1992). Stabilizability of the angular velocity of a rigid body revisited. Syst. f3 Contr. Lett. 18, 93-98. van der Schaft, A. J. (1986). Stabilization of Hamiltonian systems. Nonl. An. Th. Meth. Appl. 10, 1021 - 1035. van der Schaft, A. J. (1996). L 2 -Gain and Passivity Techniques in Nonlinear Control. Springer-Verlag. Berlin. Wagner, K. G. (1989). On nonlinear noninteraction with stability. In: Proceedings of the 28-th Conference on Decision and Control. Tampa, FL. Woolsey, C. A. and N. E. Leonard (1999). Global asymptotic stabilization of an underwater vehicle using internal rotors. In: Proceedings of the 38-th Conference on Decision and Control. Phoenix, AZ. Zhan, W., A. Isidori and T. J. Tarn (1991). A canonical dynamic extension algorithm for noninteraction with stability for affine nonlinear systems. Syst. f3 Contr. Lett. 17, 177184.
Another interesting extension is suggested by the fact that Amix = 0 also for higher order food chains, provided that a special input and output structure is selected. For example, limiting the attention to systems characterized by the LotkaVolterra predation mechanism
=
- Xl
+ Ul,
= -r/J(Xn-l, x n ) - X n + U2, Yl = X r , r E {1, 2, ... , n - 2}
Xn
after the change of coordinates ... , Zr
_ -
(r-l)
=
= YI, Z2 . = YI, Zr+2 = Y2, ... ,
Zl
Yl ,Zr+l Y2, (n-r-2) Y2 , Zn = Xr+l, and
. .
a smtable statIC state-feedback, the system becomes: Zn-l
=
Zr Zr+l
YI Y2
= Vl, =
Zr+2,
= Zl, = Zr+l·
It is easy to see that also in this case whereas po = span{8j8zn }.
Amix
= 0,
5. REFERENCES
Astolfi, A. (1999). Output feedback stabilization of the angular velocity of a rigid body. Syst. f3 Contr. Lett. Astolfi, A. and L. Menini (1999). Further results on decoupling with stability for Hamiltonian systems. In: Stability and Stabilization (D. Aeyels, A. van der Schaft and F. Lamnabhi-Lagarrigue, Eds.). Springer Verlag. Astolfi, A. and L. Menini (2000). Noninteracting control with stability for Hamiltonian systems. IEEE Trans. Automatic Control. To appear. Battilotti, S. (1994). Noninteracting control with stability for nonlinear systems. Springer Verlag. Bloch, A. M. (1999). Asymptotic stability in energy preserving systems. In: Proceedings of the 38-th Conference on Decision and Control. Phoenix, AZ.
150
NEW EXAMPLES IN NONINTERACTING CONTROL FOR HAMILTONIAN SYSTEMS Alessandro Astolfi * Laura Menini **
* Dep. of Electrical and Electronic Engineering, Imperial College
Exhibition Road, London SW7 2BT, England E-mail: a.astolfi
Abstract: In this paper, the problem of noninteracting control with stability is studied for generalized Hamiltonian systems with reference to two relevant case studies: the control of the angular velocities of an underactuated rigid body and the control of a third order food chain. For the rigid body, under suitable conditions, noninteraction and asymptotic stability are jointly achlevable by means of a static state-feedback control law; whereas, for the food chain, noninteraction and asymptotic stability of a "critical" equilibrium are jointly achievable by means of dynamic state-feedback control laws. Copyright @2000 [FAC
1. INTRODUCTION The problem of asymptotic stabilization of Hamiltonian and generalized Hamiltonian systems has been widely studied in the control literature, see e.g. (van der Schaft, 1986; Ortega et al., 1998; Woolsey and Leonard, 1999; Bloch et al., 1999; Bloch, 1999). This is because, due to their special structure, it is possible to obtain general results, valid for a large set of physically motivated systems.
Fig. 1. Inclusion properties relative to the dynamics concerned with the problem of noninteraction with stability.
On the other hand, the problem of noninteracting control for nonlinear systems has received considerable attention (Isidori and Grizzle, 1988; Wagner, 1989; Zhan et al., 1991; Battilotti, 1994; Isidori, 1995). By means of geometric control theory, necessary and sufficient conditions for the existence of either static or dynamic state-feedback control laws, which allow to obtain stable noninteractive closed-loop systems, have been proposed, and systematic design procedures have been given. The essential results of the theory are related with the "so called" P* and .6. mix dynamics, contained in the zero dynamics .6. * as
depicted in the illustrative diagram in Fig. 1, see (Isidori, 1995, Chapter 7) for details. As a matter of fact, under suitable regularity and stabilizability assumptions, the problem of noninteracting control with stability is solvable by means of a regular static state-feedback if and only if the P* dynamics are asymptotically stable, whereas it is solvable by means of dynamic state-feedback if and only if the .6. mix dynamics are asymptotically stable. The problem of noninteracting control with stability for Hamiltonian systems characterized by
145
= (13 -It) W3 WI + U2, 13W3 = (II - h) WI W2 + b3 •1 UI + b3,2 U2,
a non-degenerate Poisson bracket has been dealt with in (Huijberts and van der Schaft, 1990; Astolfi and Menini, 1999; Astolfi and Menini, 2000), with particular attention to the so-called "simple" Hamiltonian systems. Therein, easy to check conditions for the solvability of the problem by means of either static or dynamic state-feedback control laws have been given. To the best of the authors' knowledge, the problem of noninteraction with stability for generalized Hamiltonian systems has not been considered yet, i.e., no general results exploiting the special structure of such systems are available in the literature. In this paper, two significant examples are studied in detail: an underactuated rigid body (Euler's equations) and a three dimensional food-chain. In both cases it is assumed that the state of the system is available for feedback.
h W2
Ba: (x) + b(x)u
YI=WI+Bw2+CW3,
(2d)
Y2=FwI +w2+Hw3'
(2e)
H(x)
= ~ (11 wi + 12 w~ + 13 W~) , 13 W3 11 12
o J(x)
h
=
1
---
W3
o
h
1 12 W2 It 13
It WI 12 h
12 W2 It 13 It WI I2 h
o
=
and R(x) O. Notice that J(x) has rank 2 for every x =I O.
Example 2. (Food chain). Consider the normalized third order preys-predators system
In this paper we consider systems described by:
- R(x»
(2c)
Setting x = [Wl W2 w3f, such a system can be written as a generalized Hamiltonian system, with
2. GENERALIZED HAMILTONIAN SYSTEMS
x = (J(x) y = hex),
(2b)
+ UI, X2 = -(XI, X2) - X2 + (X2, X3), X3 = -(X2, X3) - X3 + U2,
Xl
(la) (lb)
=
(Xl,
X2) -
Xl
= Xl, Y2 = X3,
where x E IRn, U E IRm, y E IRq, J(X) = -JT(x), and R(x) = RT(x) ~ 0, which are also called port-controlled Hamiltonian systems, see (Maschke and van der Schaft, 1992; van der Schaft, 1996; Ortega et al., 1999). Many physical systems can be described by equations (1), with the matrix J(x) describing the internal structure (the interconnection) of the system and the matrix R(x) describing the natural damping.
(3a) (3b) (3c) (3d)
Yl
(3e)
with Lotka-Volterra predation mechanism, i. e., with (Xi, Xj) = Xi Xj' Such a system is positive, i.e., Xi(O) ~ 0, i = 1, 2, 3 and Ui(T) ~ 0, i = 1, 2, for all T E [0, t), imply Xi(t) ~ 0, for all t > O. Moreover, it can be written in the form (1) with H(x) Xl X2 X3'X~;) di]ag (Xl, X2, X3),
= +[ 0+
Observe that the dimension of the state-space of system (1) is in general not even. This has strong influence on the structural properties of the system, as many interesting properties can be proved only if the Poisson structure is non-degenerate, i.e., if the matrix J(x) has full rank. Since the rank of any skew-symmetric matrix is even, any Poisson structure defined on an odd dimensional manifold is degenerate. Generalized Hamiltonian systems with degenerate Poisson structure are also called Poisson systems.
and J(x) =
-XlX2
o
=0
0 X2X3 -X2X3 0
.
3. NONINTERACTING CONTROL WITH STABILITY FOR THE RIGID BODY Consider the system in Example 1 and let Al = (12 - 13)/11 , A 2 = (13 - Id/I2, A 3 = (It I 2)/h Q: = b3,I/h (3 = b3.2/I3, and K = [AI A 2 A 3 Q: (3 B C F Hl T . Assume Ai =I 0, i = 1, 2, 3.
Example 1. (Rigid body). Consider a rigid body with principal moments of inertia I i > 0, i = 1, 2, 3, and let WI, W2 and W3 denote the angular velocities with respect to the principal axes. Assume that two actuators (e.g. gas jet actuators) provide control torques Ul and U2, which are the control inputs of the system. Assume also that the two outputs to be decoupled are linear combinations of the wi's. In particular, let the system be described by:
The problem of stabilization of the zero equilibrium of the rigid body angular velocity has been widely studied and solved, see e.g., (Brockett, 1983; Outbib and Sallet, 1992; Astolfi, 1999) and the references therein. On the other hand, the problem of noninteracting control is trivially solvable. Goal of this section is to investigate under which conditions both problems can be jointly solved. To be more general, we also investigate the possibility of regulating the two outputs to desired values Yi" and Y2*, obtaining, at the same time,
(2a)
146
both noninteraction and stability for the overall system.
If none of conditions (i)-(iv) holds, then the Llmix dynamics (coinciding with the P* dynamics) are described by the equation:
To this end, we make the simplifying assumption 1 F B ¥ 1 and we assume F B + BH a - H /3 - 1 Ca + C /3 F ¥ 0 so that the system has vector relative degree (1, 1). After a preliminary feedback, called standard noninteracting feedback, system (2) can be rewritten as
= Vi , il2 = V2, ill
W3 = Qo(K, yt, yz*)
== Llmix
= O.
c) System (5) has 2 coinciding equilibria, i.e., Ll = 0 and Q2 ¥ O. d) System (5) has 2 distinct equilibria, i.e., Ll and Q2 ¥ O.
= 0 if and
Remark 1. If Yt = 0 and Y2* c) and e) can occur.
3.1 A numerical example. If condition (i) of Lemma 1 holds and condition (iv) does not 2, then, in view of Corollary 1, the problem of noninteracting control with stability can be solved by means of static statefeedback. From equations (4), we have that Pt = span {92, [/,92]} and P; = span {9d. Then, the change of coordinates z = -91,3 Yl + W3, transforms system (4) into the following form:
The condition in Corollary 1 can be easily tested in view of the following lemma.
= 0 if and only if either
(i) Ai A 3 > 0, /3 = 0,
H
= -F/a,
or (ii) A 2 A 3
/3 £921
> 0, a
H
= -1//3.
= 0 if and only if either (iii) A 2 A 3 > 0, a = 0, /3 = ±.jA3 /A 2 , C = -B//3,
Y2 = V2,
(6b)
= Iz(z, Y2) + bV2,
(6c)
where b is real constant. The stabilization of subsystem (6a) is trivial, whereas, in order to deal with subsystem (6b)-(6c), a further change of coordinates z = z - bY2 is convenient. As a matter of fact, the origin Y2 = 0, Z = 0 of subsystem
Al A 3 > 0, /3 = 0, a = ±.jA 3 /A l , C = -1/a.
Note that (i) and (iii) cannot occurr simultaneously because Al A3 > 0 => A2 A3 < O.
2 I
(6a)
i
or (iv)
= VI,
ill
= 0,
= ±.jA 3 /A 2 ,
= 0 then only cases
Theorem 2. If L Y1 I ¥ 0 and L Y2 I ¥ 0, then the problem of noninteracting control with asymptotic stability is globally solvable if and only if case b) occurs, with Ql < 0, and it is locally solvable if case d) occurs. The problem of noninteracting control with simple stability is locally solvable in case e).
Corollary 1. If L y1 1 = 0 and L y2 1 ¥ 0 (or I = 0 and L 91 I ¥ 0), then the problem of noninteracting control with asymptotic stability is locally solvable for system (4) by means of static state-feedback.
= ±.jA 3 /A 1 ,
>0
e) System (5) has an infinite number of equilibria, i.e., Q2 = Ql = Qo = o.
£92
a
<0
b) System (5) has 1 equilibrium, i.e., Q2 = 0 and Ll > O.
Proof. The necessity is based on straightforward but tedious computations, which show that, for all possible values of the vector K, if L y , I ¥ 0, i = 1, 2, then dim(Llmix) = dim(P*) = 1. To show that LyJ = 0 implies P* = 0 (the case of L y2 1 = 0 can be treated similarly), notice that, in such a case, P; = span{9d, and Pt = span{92, [0 0 IjT}.
Lemma 1. L 91 I
(5)
a) System (5) has no equilibria, i.e., either Ll or Ql = Q2 = 0 and Qo ¥ O.
(4c)
where Vi and V2 are the new control inputs, and 91,3 and 92,3 are two real numbers, depending on the vector K. Letting I (Yl, Y2, W3) = [0 0 !J(Yl, Y2, W3)]T, 91 = [1 0 91,3jT and 92 = [0 1 92,3]T, we can state the following result.
Theorem 1. For system (4), P* only if either L Y1 I = 0 or L Y2 I
+
Therefore, we now study the properties of equation (5). Let Ll = Qi - 4Q2 Qo (for simplicity we drop the arguments), then, one of the following conditions occurs.
(4b)
+ 9l,3 Vl + 92,3V2,
yt, yz*) W3
Q2(K, yt, Y2*)W~,
(4a)
W3 = !J(Yl, Y2, W3)
+ Ql (K,
This assumption can be easily removed.
147
1.6r---~--~--~--~-~
..
o. 06
0.'
0.'
(a)
(b)
Fig. 2. The rigid body example. (a): Wl(t) (solid), W2(t) (dashed) and W3(t) (dotted); (b) Yl(t) (solid) and Y2(t) (dashed).
~
Z
4. NONINTERACTING CONTROL WITH STABILITY FOR THE FOOD-CHAIN
(7a)
Y2 = V2,
= a2 z + al z Y2, -2
-
(7b)
The goal of this section is to regulate, if possible, the two outputs of system (3) to desired constant values Yt > 0 and Y 2- > 0, obtaining, at the same time, both noninteraction and stability for the overall system. The stabilization problem has been solved in (Ortega et al., 1999) by means of both state and output feedback control laws, by using the concepts of energy-shaping and damping injection. Notice that, after the static statefeedback
being at =I 0, can be easily stabilized by means of a static feedback control law from Y2 and z, whose design can be performed by means of standard backstepping techniques (Krstic et al., 1995). The controller proposed here, in order to obtain global asymptotic stability of the origin and noninteraction, is of the form:
=- k l Yl + Tl , V2 = ('1'1 + 2'1'2 z) (83 Z3 + ~: zY2) k 2 (Y2 - hI Z + '1'2 Z2)) - al Z2 + T2, VI
= Xl - Xl X2 + VI, U2 = X3 + X2 X3 + V2, Ul
system (3) can be written as
where '1't, '1'2 and 83 are suitable constants, and Tl and T2 new inputs with respect to which noninteraction is guaranteed. The results of a simulation with the proposed controller, with Al = 1, A 2 = -1, A 3 = 1.5 a = A 3 /A t , 13 = 0, B = 1, C = 0.5, F = 0.5, H = -F/a, at ~ -1.23, a2 ~ -1.61, '1't ~ -1.32, '1'2 ~ 612, 83 = -1.5, k l = 0.2, k2 = 0.2, Wl (0) = 1, W2(0) = 0, W3(0) = 1, are reported in Fig. 2. In order to emphasize the property of noninteraction, Tl (t) has been set to zero for t 1: [50, 100], and equal to a sinusoidal signal in such an interval, whereas the input T2(t) has been set to zero for t 1: [150, 200], and equal to a sinusoidal signal in such an interval.
Yl = VI,
(8a)
= V2, X2 = -X2 (1 + Yl -
(8b)
Y2
J
Y2) .
(8c)
From equation (8c), it is obvious that, if 1 + Yt > Y2-, then the problem is easily solvable (with limt-t+oo X2 (t) = 0), whereas, if 1 +Yt < Y 2-, then the problem is not solvable at all. Hence, only the special case 1 + Yt = Y2- needs to be investigated. By means of easy computations, it can be seen that Amix = 0, whereas, at any point of the state space where X2 =I 0, P- = span{%x2}' Hence, the problem of noninteraction with asymptotic stability around an equilibrium Xl = yt, X2 = Xi > 0, X3 = Y2- = 1 + Yt, is not solvable by means of static state-feedback, but it is solvable by means of dynamic state-feedback. The controller proposed here is based on a dynamic extension as described in (Isidori, 1995, Chapter 7), but has lower dimension than the controller which would result from applying the design technique reported there. Let J.Ll and J.L2 be the two state variables of the controller, and let:
As for the possibility of regulating the two outputs Yl and Y2 to constant reference values Yt and Y 2- , it is clear from the above equations that there are no restrictions on the admissible values for Yt· Furthermore, from equations (7), it is clear that, in general, for every Y2- =I 0, only local asymptotic stability of Y2 = Y 2-, Z = Z-, being Z- a real constant depending on K, can be achieved.
{J.l
148
=
-J.Ll
(1
+ Yl) + Wl,
(9a)
3
!\ !\
1\
3S I
------------------~
"
r
I
•
:
~
... - - - - - - - -
\! \i Il'
1\
\/
I1
.>
(a)
(b)
Fig. 3. The food chain example. (a): Xl(t) (solid), X2(t) (dashed) and X3(t) (dotted); (b) Ul(t) (solid) and U2(t) (dashed) (9b)
V2 = - (J.L2
= Vl, [J,l = -J.Ll (1 + yd + Wl, Yl = Vl,
(lOb)
+ W2,
(IOd)
Wl _ W2) . J.Ll J.L2
(lOe)
[J,2 = -J.L2 (1 - Y2)
~3 = -6
(1-
(lOc)
+ 1) J.Ll,
(lla)
W2
=-
(Y2* - 1) J.L2,
(llb)
+ T2,
Remark 2. It can be shown that, if X:; < 1 and the initial conditions Xl (0) and X3 (0) are in a suitable neighborhood of the desired values Yt and Y 2*, then Ul (t) ~ 0 and U2(t) ~ 0, for all t ~ O.
(lOa)
Remark 3. It is stressed that the special case 1 + Yt = Y 2* is the only one in which it is possible to obtain a steady state response in which also
From equation (lOe), it is clear that the choice Wl = (yt
J.L2,00)
with k2 > 0, guarantees asymptotic stability of the equilibrium Y2 = Y 2*, J.L2 = J.L2,oo, with domain of attraction J.L2 > 0, for the subsystem (lOc)(lOd)-(llb).
where the two fictitious inputs Wl and W2 will be specified later as functions of the state of the extended system. After the change of coordinates 6 = J.Ll J.L2/X2, which is valid as long as X2 =j:. 0, the extended system (8)-(9), has the following form: Yl
+ k 2) (Y2 - Y2* + J.L2 -
Xi> o. 4.1 Simulation results
The results of a simulation with the proposed controller, with Yt = 2, Y 2* = 3, X:; = 0.2, J.Ll,oo = J.L2,00 = 0.3 Xl(O) = X3(0) = 1, X2(0) = 0.5, kl = k 2 = 2, are reported in Fig. 3. In order to emphasize the property of noninteraction, the input Tl (t) has been set to zero for t i [15, 30], and equal to a sinusoidal function inside such an interval, whereas the input T2(t) has been set to zero for t i [45, 60], and equal to a sinusoidal function inside such an interval.
with 1 + Yt = Y2*' implies that the variable 6 remains constant for all t ~ O. Hence, assuming that the two state variables J.Ll and J.L2 are regulated to suitable desired values J.Ll,oo and J.L2,oo, respectively, one has (t) J.L2,00 X2 (0) · 1lm X2 = J.Ll,oo t ....++oo J.Ll (0) J.L2(0) Therefore, not only it is possible to regulate Yl and Y2, but also the non-actuated state value X2 to a desired positive value X:;, by choosing J.ll (0) = J.Ll,ooVX2(0)/X:;, J.L2(0) = J.L2,ooVX2(0)/X:;. Now, in order to obtain asymptotic stability of the equilibrium Yl = yt, J.Ll = J.Ll,oo, with domain of attraction J.Ll > 0, for the subsystem (lOa)(lOb)-(lla), one can use standard backstepping techniques (Krstic et al., 1995), yielding the controllaw
4.2 Extensions
r;,. - J.ll + J.ll,oo) + Tl,
It is easy to see that the property 6 mix = 0 still holds for more general predation mechanisms than the above mentioned
where k l > 0 and Tl is the new input, with respect to which noninteraction has to be guaranteed. Analogously, the choice
X2 = -X2 lh (Xl) - X2 + X2 83(X3), where the 8i (·), i = 1, 2 are arbitrary functions. This fact suggest that it might be possible to
Vl = - (J.ll
+ kd (Yl
-
149
extend the results presented above also to this case.
Bloch, A. M., N. E. Leonard and J. E. Marsden (1999). Potential shaping and the method of controlled Lagrangians. In: Proceedings of the 38-th Conference on Decision and Control. Phoenix, AZ. Brockett, R. W. (1983). Asymptotic stability and feedback stabilization. Differential geometry control theory pp. 181-191. 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. (1995). Nonlinear control systems. Springer Verlag. Third edition. Isidori, A. and J. W. Grizzle (1988). Fixed modes and nonlinear noninteractive control with stability. IEEE Trans. Aut. Contr. AC-33, 907914. Krstic, M., 1. Kanellakopoulos and P. Kokotovic (1995). Nonlinear and Adaptive Control Design. John WHey & sons. New York. Maschke, B. M. and A. J. van der Schaft (1992). Port controlled Hamiltonian systems: modeling origins and system theoretic properties. In: Proc. 2nd IFAC Symp. on Nonlinear Control Systems design (NOLCOS'92). Bordeaux. pp. 282-288. Ortega, R., A. Astolfi, G. Bastin and H. Ro driguez Cortes (1999). Output feedback control of food-chain systems. In: New trends in nonlinear observer design (H. Nijmeijer and T. Fossen, Eds.). Springer-Verlag. Ortega, R., A. Loria, P. J. Nicklasson and H. SiraRamirez (1998). Passivity-Based Control of Euler-Lagrange Systems. Springer Verlag. Berlin. Outbib, R. and G. Sallet (1992). Stabilizability of the angular velocity of a rigid body revisited. Syst. f3 Contr. Lett. 18, 93-98. van der Schaft, A. J. (1986). Stabilization of Hamiltonian systems. Nonl. An. Th. Meth. Appl. 10, 1021 - 1035. van der Schaft, A. J. (1996). L 2 -Gain and Passivity Techniques in Nonlinear Control. Springer-Verlag. Berlin. Wagner, K. G. (1989). On nonlinear noninteraction with stability. In: Proceedings of the 28-th Conference on Decision and Control. Tampa, FL. Woolsey, C. A. and N. E. Leonard (1999). Global asymptotic stabilization of an underwater vehicle using internal rotors. In: Proceedings of the 38-th Conference on Decision and Control. Phoenix, AZ. Zhan, W., A. Isidori and T. J. Tarn (1991). A canonical dynamic extension algorithm for noninteraction with stability for affine nonlinear systems. Syst. f3 Contr. Lett. 17, 177184.
Another interesting extension is suggested by the fact that Amix = 0 also for higher order food chains, provided that a special input and output structure is selected. For example, limiting the attention to systems characterized by the LotkaVolterra predation mechanism
=
- Xl
+ Ul,
= -r/J(Xn-l, x n ) - X n + U2, Yl = X r , r E {1, 2, ... , n - 2}
Xn
after the change of coordinates ... , Zr
_ -
(r-l)
=
= YI, Z2 . = YI, Zr+2 = Y2, ... ,
Zl
Yl ,Zr+l Y2, (n-r-2) Y2 , Zn = Xr+l, and
. .
a smtable statIC state-feedback, the system becomes: Zn-l
=
Zr Zr+l
YI Y2
= Vl, =
Zr+2,
= Zl, = Zr+l·
It is easy to see that also in this case whereas po = span{8j8zn }.
Amix
= 0,
5. REFERENCES
Astolfi, A. (1999). Output feedback stabilization of the angular velocity of a rigid body. Syst. f3 Contr. Lett. Astolfi, A. and L. Menini (1999). Further results on decoupling with stability for Hamiltonian systems. In: Stability and Stabilization (D. Aeyels, A. van der Schaft and F. Lamnabhi-Lagarrigue, Eds.). Springer Verlag. Astolfi, A. and L. Menini (2000). Noninteracting control with stability for Hamiltonian systems. IEEE Trans. Automatic Control. To appear. Battilotti, S. (1994). Noninteracting control with stability for nonlinear systems. Springer Verlag. Bloch, A. M. (1999). Asymptotic stability in energy preserving systems. In: Proceedings of the 38-th Conference on Decision and Control. Phoenix, AZ.
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