Balanced realizations near stable invariant manifolds - Semantic Scholar
Automatica 42 (2006) 653 – 659 www.elsevier.com/locate/automatica
Brief paper
Balanced realizations near stable invariant manifolds夡 W. Steven Gray a,∗ , Erik I. Verriest b a Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, Virginia 23529-0246, USA b School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250, USA
Received 15 May 2004; received in revised form 22 June 2005; accepted 6 December 2005
Abstract It is shown that the notion of balancing a state space realization about a stable, isolated equilibrium point can be generalized to systems possessing stable invariant regular submanifolds of arbitrary dimension, for example, a stable limit cycle. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Nonlinear systems; Balanced realizations; Invariant manifolds
1. Introduction The original notion of balanced realizations given by Moore (1981) applies to linear state space systems which are stable, minimal and time-invariant. Over the years the concept has been extended to accommodate unstable linear systems (Meyer, 1990; Ober & McFarlane, 1989; Verriest & Kailath, 1983), nonminimal linear systems (Safonov, Chiang, & Limebeer, 1990; Verriest & Kailath, 1983), periodic linear systems (Varga, 1997; Verriest & Helmke, 1998), general linear time-varying systems (Verriest & Kailath, 1983), and affine nonlinear systems (Scherpen, 1993, 1994; Verriest & Gray, 2000, 2001a,b). In the case of stable systems, either linear or nonlinear, a specific requirement for a well defined balanced realization in the sense of Scherpen is the existence of an asymptotically stable equilibrium point, which without loss of generality is normally taken to be the origin of the state space. In this context, the general idea behind balancing is the introduction of a (local) coordinate frame so that the input energy required to drive the state vector from the origin to any specific state is equal to the output energy generated by the system’s natural response when this same state value is used 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Franco Blanchini under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +1 757 683 4671; fax: +1 757 683 3220. E-mail addresses: [email protected] (W.S. Gray), [email protected] (E.I. Verriest).
0005-1098/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.12.007
as an initial condition. In this paper it is shown that this notion of balancing can be further generalized to systems which have stable or attracting invariant regular submanifolds, e.g., stable limit cycles, instead of isolated stable equilibria. The technique intrinsically produces a time-varying extension of the approach of Scherpen (1993, 1994). It also complements the flow balancing approach (Verriest & Gray, 2000, 2001a,b) in the sense that the method of balancing introduced here applies to energies generated over infinite time intervals, while flow balancing treats energies generated over infinitesimal time intervals. The paper is organized as follows. In Section 2, the definitions of the input and output energy functions for an invariant set are introduced. Then a pair of partial differential equations is described which these energy functions will satisfy when the equations are known to have a smooth solution with certain boundary conditions. In Section 3, the generalized notion of balancing using these energy function is presented. The final section demonstrates the technique using a simple system possessing a stable limit cycle. The mathematical notation used throughout is fairly standard. Given two sets A and B, A − B denotes the complement of B relative to A. The vector norm on Rn is taken √ T to be x = x x. L2 (a, b) represents the set of Lebesgue measurable functions, possibly vector-valued, with finite L2 b n 2 norm xL2 = a x(t) dt. If L : R → R is a differentiable function, then its partial derivative jL/jx will be the row vector of partial derivatives jL/jxi , where i = 1, . . . , n.
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W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
Furthermore, L generally denotes the mapping, while L(x) is a real number.
2. Energy functions Let M be an n dimensional
state space manifold, and let
(1)
be a system written in terms of local coordinates on M with t ∈ R, u(t) ∈ Rm and y(t) ∈ Rp . It is assumed throughout that f, g and h are C ∞ on M × R. For any t1 , t2 ∈ R with t2 > t1 let x(t2 ) = (t2 , t1 , x1 , u) denote the solution of (1) with initial condition x(t1 ) = x1 and input u : [t1 , t2 ] → Rm . An analogous statement holds when t2 < t1 . In this setting consider the following definitions. Definition 1 (Guckenheimer and Holmes, 1983). A set S ⊂ M is an invariant set under f if for any x0 ∈ S and t0 ∈ R the solution (t, t0 , x0 , 0) belongs to S for all t ∈ R. A closed invariant set S is called an attracting set or stable set if there exists a neighborhood W of S for which x0 ∈ W implies that (t, t0 , x0 , 0) ∈ W for all t > t0 , and (t, t0 , x0 , 0) → S as t → ∞. W is called the region of attraction. Definition 2. The null set for h is N(h) := {x ∈ M : h(x, t)=0, ∀t ∈ R}. It is assumed in all that follows that system (1) has a nonempty closed set, S, invariant under f. From the point of view of balancing, if the system is initialized on S then no input energy is required to keep the resulting trajectory contained in S. In a dual manner, if the system is initialized on S then no output energy should be observed for the resulting trajectory. This can only be assured if S is contained in N(h). The classic example of this situation is where (1) has an isolated equilibrium at x = 0 and h(0, t) = 0, ∀t ∈ R. A nontrivial example from linear system theory is the case where the system has an invariant subspace contained in the unobservable subspace of the system. In an application, this condition could result from designing a new output function from some given physical out˜ put y(t) ˜ = h(x(t), t). For example, one might let h(x(t), t) := ˜h(x(t), t) − h(P ˜ (x(t)), t), where P : W → S is a projection into S. Next consider the following pair of energy functions which is central to our generalized notion of a balanced realization. Definition 3. The controllability and observability functions for the system (f, g, h) with invariant set S are defined, respectively, as min
u∈L2 (−∞,t0 ),x∈S ˜ x(−∞)=x,x(t ˜ 0 )=x0
1 Lo (x0 , t0 ) = 2
∞
y(t)2 dt
t0
with x(t0 ) = x0 and u(t) = 0 for all t t0 . C∞
x(t) ˙ = f (x(t), t) + g(x(t), t)u(t), y(t) = h(x(t), t),
Lc (x0 , t0 ) =
and
1 2
t0
−∞
u(t)2 dt
Clearly, Lc (x0 , t0 ) has the interpretation of being the minimum amount of input energy required to drive the system from S at t = −∞ to x(t0 ) = x0 , while Lo (x0 , t0 ) is equivalent to the output energy generated by the natural response of the system initialized at x(t0 ) = x0 . These functions need not be well defined for all x0 ∈ M. For example, Lo will not be finite if the state does not approach the null set of h in the limit, or Lc may not be finite if the state x0 is not reachable from S at time t0 . But when these functions are known to exist, at least locally, and S is attracting then they necessarily satisfy a pair of partial differential equations described in the following theorem. These equations reduce to the usual Lyapunov equations describing the observability and controllability Gramians in the linear time-invariant case with S = {0}. Theorem 4. Suppose the system (f, g, h) has an attracting set S ⊂ N(h) with a region of attraction W and (x0 , t0 ) ∈ W × R. If the equation jLo jLo 1 f+ + hT h = 0 jx jt 2
(2)
has a smooth solution L¯ o on W × R with L¯ o (x, t) = 0 for all x ∈ S and t t0 then Lo (x0 , t0 ) = L¯ o (x0 , t0 ). Furthermore, if the equation jLc jLc 1 jLc T jLTc f+ + gg =0 jx jt 2 jx jx
(3)
has a smooth solution L¯ c on W ×R with L¯ c (x, t)=0 for all x∈S and t ∈ R, and S is an attracting set of −(f + gg T (jL¯ Tc /jx)) on W × R then Lc (x0 , t0 ) = L¯ c (x0 , t0 ). Proof. The claim regarding Lo is trivial when x0 ∈ S since S is invariant under f and S ⊂ N(h). So suppose instead that x0 ∈ W − S. Observe Lo (x0 , t0 ) 1 ∞ h((t, t0 , x0 , 0), t)2 dt = 2 t0 ∞ ¯ jLo =− f ((t, t0 , x0 , 0), t) jx t0 jL¯ o + ((t, t0 , x0 , 0), t) dt jt ∞ d ¯ Lo ((t, t0 , x0 , 0), t) dt =− dt to = − lim L¯ o ((t, t0 , x0 , 0), t) + L¯ o ((t0 , t0 , x0 , 0), t0 ) t→∞
= L¯ o (x0 , t0 ).
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
Next let L¯ c be a smooth solution of Eq. (3) with L¯ c (x, t) = 0 for any (x, t) ∈ S × R. Again the result is trivial when x0 ∈ S, so assume that x0 ∈ W − S. Then it follows via a completion of the squares argument (e.g., see Lee & Markus, 1967) that for any x˜ ∈ S and any u with (t0 , −∞, x, ˜ u) = x0 (while staying in the neighborhood W) ˜ u), t) L¯ c (x0 , t0 ) − lim L¯ c ((t, −∞, x, =
−∞
d ¯ ˜ u), t) dt Lc ((t, −∞, x, dt x(t)
¯ ¯ jLc jL c f (x(t), t) + g (x(t), t)u(t) jx jx −∞
=
t→−∞
t0
t0
jL¯ c (x(t), t) dt + jt
¯ t0 jLc 1 jL¯ c T jL¯ Tc = − (x(t), t) + gg (x(t), t) jt 2 jx jx −∞ ¯ jL¯ c jLc g (x(t), t)u(t) + (x(t), t) dt + jx jt 2 t0 ¯T 1 1 2 T jLc u(t) − u(t) − g (x(t), t) = dt. 2 jx −∞ 2 Since L¯ c (x, ˜ t) = 0 when x˜ ∈ S, and Lc is smooth, then 1 L¯ c (x0 , t0 ) 2
t0
−∞
u(t)2 dt
with equality if and only if the input u(t) = u∗ (t) := (g T (jL¯ Tc /jx))(x(t), t). This lower bound on the input energy is achievable since u∗ is an admissible input in the definition of Lc . That is, the closed loop system f + gu∗ is assumed here to have the property that x(t) → S as t → −∞ if x(t0 ) = x0 ∈ W . Hence, L¯ c (x0 , t0 ) = Lc (x0 , t0 ). From the definitions it is immediate that Lc and Lo are always nonnegative. Conditions are now developed under which these functions are positive on (W −S)×R. The following definitions are either classic or obvious variations of classic definitions with the exception of asymptotic reachability. This latter notion was introduced in Scherpen and Gray (2000). Definition 5. Consider system (1) and any set S ⊂ M. • The system is reachable from S if for any x¯ ∈ M there exists a t¯ t0 , an input u, and x0 ∈ S such that x¯ = (t¯, t0 , x0 , u). • The system is asymptotically reachable from S on a neighborhood W ⊃ S if for all x ∈ W there exists an input u ∈ L2 (t0 , ∞) and x0 ∈ S such that (t, t0 , x0 , u) ∈ W for all t t0 , and limt→∞ (t, t0 , x0 , u) = x, or in other words, for all x ∈ W and any > 0, there exists a t¯ < ∞ such that (t, t0 , x0 , u) ∈ W for all t0 t t¯ and (t¯, t0 , x0 , u) − x < .
655
• The system is S-state observable if S ⊂ N(h) and any trajectory where u(t) ≡ 0 and y(t) ≡ 0 implies that x(t) ∈ S for all t t0 , i.e., for all x0 ∈ M, the condition h((t, t0 , x0 , 0), t) = 0 for all t t0 implies that (t, t0 , x0 , 0) ∈ S for all t t0 . • The system is locally S-state observable if S ⊂ N(h) and there exists an open set U ⊃ S such that if x0 ∈ U then the condition h((t, t0 , x0 , 0), t) = 0 for all t t0 implies that (t, t0 , x0 , 0) ∈ S for all t t0 . With these definitions in place, the following theorems address the positivity issue. These results are generalizations of those which appear in Scherpen (1993, 1994). Theorem 6. Suppose S is an attracting set of f with region of attraction W. If a smooth Lc exists on W ×R then Lc (x0 , t0 ) > 0 for all (x0 , t0 ) ∈ (W −S)×R if the system (1) is asymptotically reachable from S on W. Proof. The method of proof which appeared in Scherpen and Gray (2000) can be directly extended to show that asymptotic reachability from S on W and the existence of a smooth Lc on W implies that S is an attracting set of −(f + gg T (jL¯ Tc /jx)) on W × R. In which case, Theorem 4 applies and the value of Lc (x0 , t0 ) for any (x0 , t0 ) ∈ W × R can be computed as 1 t0 Lc (x0 , t0 ) = u∗ (t)2 dt 2 −∞ 2 ¯T 1 t0 g T jLc (x(t), t) dt. = 2 jx −∞
Now specifically, choose any (x0 , t0 ) ∈ (W − S) × R and assume that Lc (x0 , t0 ) = 0. The latter implies that u∗ (t) = 0 on −∞ t t0 . In which case this is a situation where the input / S. Since S is an invariant set is zero, x(−∞) ∈ S and x(t0 ) ∈ under f, this is a contradiction. Theorem 7. Suppose S is an attracting set of f with a region of attraction W. If S ⊂ N(h) and a smooth Lo exists on W × R then Lo (x0 , t0 ) > 0 for all (x0 , t0 ) ∈ (W − S) × R if system (1) is locally S-state observable on W. Proof. Choose any (x0 , t0 ) ∈ (W −S)×R. Since S is attracting on W then (t, t0 , x0 , 0) is well defined for all t t0 . Now if Lo (x0 , t0 ) = 0 then h((t, t0 , x0 , 0), t) = 0 for all t t0 . But since the system (1) is S-state observable on W this implies that x0 ∈ S, a contradiction. 3. Locally balanced realizations In this section, the notion of locally balancing a realization (f, g, h) near an invariant set S is developed under the following set of assumptions: (A1) S is a regular C ∞ submanifold of M with dimension r. (A2) S is invariant under f. (A3) S is an attracting set with a region of attraction W.
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
(A4) The system is asymptotically reachable from S on W. (A5) The system is locally S-state observable on W. (A6) Lc and Lo are well defined and smooth functions on W × R. (A7) rank(j2 Lc /jx 2 ) = rank(j2 Lo /jx 2 ) = n − r everywhere on S × R. Given that Lc and Lo are well defined, as per (A6), one can fix the time argument, t, of these energy functions and introduce state space coordinate transformations to modify their original form. As in the linear time-varying case, where the Gramians are dependent on t, the transformed energy functions are (pseudo-)quadratic forms written in terms of matrices parameterized by t. Theorem 8. Let t ∈ R be fixed. Choose any s ∈ S. Then there exists a smooth local coordinate transformation x =t (x¯t ) such that s = t (0) and on some neighborhood U ⊂ W of s the function Lc in the new coordinates x¯t = −1 t (x) has the form Lc (t (x¯t ), t) =
1 2
x¯tT diag(In−r , 0)x¯t ,
x¯t ∈ −1 t (U ).
(Here Ii ∈ Ri×i denotes an identity matrix.) Furthermore, in the new coordinates, one can write Lo in the form Lo (t (x¯t ), t) =
1 2
x¯tT M(x¯t , t)x¯t ,
where M(0, t) =
j2 Lo (s, t) jx 2
with M(x¯t , t) an n × n symmetric matrix such that its entries are smooth functions of x¯t and t. Proof. Since S is an r dimensional C ∞ submanifold of M, it follows that for any s ∈ S there exists a local coordinate neighborhood U and coordinate transformation x˜ = (x) such that (s) = 0, (U ) = Cn (0) (an n dimensional cube centered at 0 and with edges of length 2, > 0) and (U ∩ S) = {x˜ ∈ Cn (0) : x˜1 = · · · = x˜n−r = 0}. (See Boothby, 1975, p. 75 and Fig. 1.) In this coordinate frame it is easily verified that Lc (−1 (0), t) = 0 and that 0 is a critical point of Lc (−1 (·), t) (since (A2)–(A4) and Theorem 6 imply that 0 is a local minimum of Lc (−1 (·), t)). In light of (A7), the Splitting Lemma (a generalized Morse Lemma, e.g., see Poston & Stewart, 1978) applies and provides for the existence of a coordinate transformation x˜ = t (x¯t ) such that t (0, . . . , 0, x¯t,n−r+1 , . . . , x¯t,n ) = (0, . . . , 0, x˜n−r+1 , . . . , x˜n ) and Lc (−1 ◦ t (x¯t ), t) t
2 2 = 21 (x¯t,1 + · · · + x¯t,n−r ) + p(x¯t,n−r+1 , . . . , x¯t,n ),
W S
ψ(U)=C2ε (0) x~2
U ψ
s
x~1
0
ψ(U S) U
656
Fig. 1. The local coordinate transformation in the case where n = 2 and r = 1.
where p : Rr → R is smooth. However, in light of the fact that Lc (t (¯s ), t) = 0 for all s¯ = (0, . . . , 0, x¯t,n−r+1 , . . . , x¯t,n ) ∈ −1 −1 t (U ∩ S) then p ≡ 0 on t (U ), and the first part of the theorem is proven. The second part of the theorem is proven in a manner perfectly analogous to the proofs of Scherpen (1993, 1994) except that the smoothness of M(x¯t , t) in the argument t follows from (A6). In order to produce an input-normal form where M(x¯t , t) is a diagonal matrix, the following technical lemma is required. Lemma 9. If on the set −1 t (U ), as defined in Theorem 8, the number of distinct nonzero eigenvalues of M(x¯t , t) is constant then the eigenvalues i (x¯t , t), i = 1, . . . , n, are smooth functions of x¯t on −1 t (U ), as well as the associated orthonormal eigenvectors. Proof. With t ∈ R a fixed parameter, the proof is exactly as it appears in Kato (1976). Theorem 10. Let t ∈ R be fixed. Choose any s ∈ S. Assume that the condition of Lemma 9 is fulfilled. Then there exists a smooth local coordinate transformation x = t (zt ) such that s = t (0) and on some neighborhood V ⊂ W of s the function Lc in the new coordinates zt = −1 t (x) has the form L˘ c (zt , t) := Lc (t (zt ), t) =
1 T 2 zt diag(In−r , 0)zt ,
and the function Lo is of the form L˘ o (zt , t) := Lo (t (zt ), t) = 21 ztT diag(1 (zt , t), . . . , n−r (zt , t), 0, . . . , 0)zt , where 1 (zt , t) · · · n−r (zt , t) > 0 are smooth functions of zt , referred to as the singular value functions. Proof. The proof is perfectly analogous to that which appears in Scherpen (1993, 1994), where the Lemma 9 simply insures the smooth diagonalizability of M(x¯t , t).
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
657
The form of the controllability and observability functions is not yet entirely balanced. For that, one additional coordinate transformation is required. Consider the smooth transformation
and the corresponding open covering of W:
z¯ t,i = t,i (zt,i ) := i ((0, . . . , 0, zt,i , 0, . . . , 0), t)1/4 zt,i ,
Ub = {(x1 , x2 ) ∈ R2 : x2 < 0},
for i = 1, . . . , n − r and t,i (zt,i ) := zt,i otherwise. Then z¯ t = t (zt ) := (t,1 (zt,1 ), . . . , t,n (zt,n )) on Vt,b = t,b (V ) := t ◦ −1 t (V ). In the new coordinates it follows then that
Uc = {(x1 , x2 ) ∈ R2 : x1 < 0},
2 i (¯zt,i , t)−1 , Lˆ c ((0, . . . , 0, z¯ t,i , 0, . . . , 0), t) = 21 z¯ t,i 2 Lˆ o ((0, . . . , 0, z¯ t,i , 0, . . . , 0), t) = 21 z¯ t,i i (¯zt,i , t),
zt,i ), 0, . . . , 0), t)1/2 for where i (¯zt,i , t) := i ((0, . . . , 0, −1 i (¯ i = 1, . . . , n − r and zero otherwise. In these coordinates the system is said to be in balanced form.
x˙1 = x2 , x˙2 = −(x12 + x22 − 1)x2 − x1 + u, y = 2(x12 + x22 − 1)x2 .
(4)
The autonomous system has an unstable equilibrium at x = 0 and a stable limit cycle, S, consisting of the unit circle centered at x = 0 with region of attraction W = R2 − {0}. Therefore, the assumptions (A1)–(A3) are satisfied with n = 2 and r = 1. The null set of h contains S, and it can be easily verified that (A5) holds. Next observe that everywhere on W, the functions Lc (x, t) = Lo (x, t) = 21 (x12 + x22 − 1)2 satisfy Eqs. (2) and (3), including the required boundary conditions on S × R. In addition, a simple geometric analysis show that S is also an attracting set for the vector field
T −x2 0 T jLc − f + gg + . (x, t) = −(x12 + x22 − 1)x2 x1 jx tangential near S
Thus, from Theorem 4, Lo and Lc are indeed the energy functions for system (4), and consequently, assumptions (A4) and (A6) are verified. (Since Lc and Lo do not depend explicitly on time in this example, the t argument will be suppressed throughout.) Finally, observe that 2 2 j Lc
j Lo
rank = rank
jx 2 S jx 2 2 S x1 x1 x2 = rank = 1, x1 x2 x22 so (A7) is valid. Taken in total, these assumptions guarantee that about any point s ∈ S, there must exist a local balancing transformation. Consider the set of points: sb = (0, −1),
sc = (−1, 0),
In light of Theorem 10, set Lc (x) = Lo (x) = 21 (x12 + x22 − 1)2 = 1 2 ˘
˘
2 (z1 ) = Lc (z ) = Lo (z ), where ∈ A := {a, b, c, d} (note that Lemma 9 applies by default). Therefore, the following coordinate transformations are valid local balancing transformations on their respective coordinate patches:
a
Consider the second-order system
sa = (1, 0),
Ud = {(x1 , x2 ) ∈ R2 : x2 > 0}.
4. Example: system with a stable limit cycle
attractive near S
Ua = {(x1 , x2 ) ∈ R2 : x1 > 0},
sd = (0, 1),
z1a = x12 + x22 − 1, ⇐⇒ z2a = x2 ,
x1 =
1 + z1a − (z2a )2 ,
x2 = z2a ,
x1 = z2b , ⇐⇒ b b z2 = x 1 , x2 = − 1 + z1b − (z2b )2 , c z1 = x12 + x22 − 1, x1 = − 1 + z1c − (z2c )2 , ⇐⇒ c c z2 = x2 , x2 = z2c , d x1 = z2d , z1 = x12 + x22 − 1, ⇐⇒ d d z2 = x1 , x2 = 1 + z1d − (z2d )2 . z1b = x12 + x22 − 1,
The family U = {U , } ∈A constitutes a C ∞ structure on W. The corresponding balanced realizations are ⎤ ⎡ ⎧
a −2z1a (z2a )2 ⎪ 2z2 ⎪ ⎨ z˙ a = ⎣ ⎦+ u, 2 a a a a −z1 z2 − 1 + z1 − (z2 ) 1 a (Ua ) ⎪ ⎪ ⎩ y = 2z1a z2a , ⎧ −2z1b (1 + z1b − (z2b )2 ) ⎪ b ⎪ ⎪ z˙ = ⎪ ⎪ ⎪ − 1 + z1b − (z2b )2 ⎪ ⎪ ⎨ −2 1 + z1b − (z2b )2 b (Ub ) u, + ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ y = −2z1b 1 + z1b − (z2b )2 ,
⎧ −2z1c (z2c )2 2z2c ⎪ ⎨ z˙ c = + u, 1 c (Uc ) −z1c z2c + 1 + z1c − (z2c )2 ⎪ ⎩ y = 2z1c z2c , ⎧ −2z1d (1 + z1d − (z2d )2 ) ⎪ d ⎪ ⎪ ⎪ z˙ = ⎪ ⎪ 1 + z1d − (z2d )2 ⎪ ⎪ ⎨ 2 1 + z1d − (z2d )2 d (Ud ) u, + ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ y = 2z1d 1 + z1d − (z2d )2 .
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
zd2
ψd(sd)
za2
t=-1
t=-2
t=-3 -1
-0.5
0 zd1
0.5
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
1
t=-1
ψa(sa)
zd2
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
t=0 t=-6
-1
-0.5
0 za1
0.5
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
t=6
ψd(sd) t=5
t=4 -1
1
za2
658
-0.5
0 zd1
0.5
1
t=-2 1
t=-1
-0.5
0 za1
0.5
1
x2
t=-4
sa t=-6
sb
-1
sd
t=4
sb
-1
-2
-2 -2.5
-2.5 -2 -1.5 -1 -0.5 0
0.5
1
1.5
2
2.5
-2 -1.5 -1 -0.5 0
x1
zb2 t=-4 t=-5 0
zc1
0.5
1
t=-6
ψb(sb) t=-5
t=-4 -1
-0.5
0
0.5
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
t=5 t=4 ψc(sc) t=3 t=2 -1
1
-0.5
z1b
Fig. √2. The state trajectory x(t) = x (t, −∞, S, u∗ ) corresponding to Lc (( 2, 0)) and its images z (t) in the balanced coordinate patches (U ),
∈ A.
0 zc1
0.5
1
1.5
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
2
2.5
t=1
ψb(sb)
t=2
t=3 -1
1
-0.5
0 zb1
0.5
1
√ Fig. √ 3. The state trajectory x(t) = x (t, 0, ( 2, 0), 0) corresponding to
Lo (( 2, 0)) and its images z (t) in the balanced coordinate patches (U ),
∈ A.
1.2
Clearly, the singular value function 1 (z ) = 1 on (U ) for √ each ∈ A. Take as an example the test point x = ( 2, 0), ˘ a −1 where L˘ ac (−1 a (x)) = Lo (a (x)) = 0.5. The center plot in Fig. 2 shows the state trajectory from S to the test point (at time t = 0) in the x-coordinate frame. It was determined numerically by computing the input u∗ in the proof of Theorem 4. Surrounding this plot are the four balanced coordinate patches showing the corresponding trajectories, z (t), entering and exiting the patch (U ), the region strictly to the right of the parabola 1 + z1 − (z2 )2 = 0. In this case (U ∩ S) is the open line segment connecting (0, 1) and (0, −1). Fig. 3 shows the natural response of the system when the same test point is used as an initial condition at t =0. Using these numerical results, the energy functions L˘ c and L˘ o were evaluated at the test point by plotting t the cumulative energy functions Eu (t) := 21 −∞ u∗ ()2 d t and Ey (t) := 21 0 yn ()2 d (yn is the natural output
0.5 x1
zb2
t=-3
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
zc2
t=-2
t=1
t=2
t=-5 -1.5
t=0
sa
t=3
-0.5
-1.5
ψc(sc)
t=6
sc
0
t=0 x2
sc
-0.5
t=5
0.5
t=-3
0
zc2
t=0
t=1 -1
1
sd
0.5
-0.5
ψa(sa)
1.5
1.5
-1
t=6
2
2
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
1 0.8 u*
0.6 0.4
Eu
0.2 0 -0.2 -6
-5
-4
-3 -2 time
-1
0
0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2
Ey
yn
0
1
2
3 4 time
5
6
Fig. 4. The cumulative energy functions Eu and Ey for the respective signals u∗ and yn .
response) as a function of t as shown in Fig. 4. As predicted, ˘ a −1 L˘ ac (−1 a (x)) = Eu (0) = 0.5 and Lo (a (x)) = Ey (∞) = 0.5. Finally, it should be noted that if one linearizes the original
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
system along S, the resulting time-varying system will fail to be either controllable or observable in the conventional sense. Thus, a straightforward application of linear balancing methods as in Verriest and Kailath (1983) is not possible. References Boothby, W. M. (1975). An introduction to differentiable manifolds and Riemannian geometry. Orlando, FL: Academic Press. Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer. Kato, T. (1976). Perturbation theory for linear operators (2nd ed.), Heidelberg: Springer. Lee, E. B., & Markus, L. (1967). Foundations of optimal control theory. New York: Wiley. Meyer, D. G. (1990). Fractional balanced reduction—model reduction via a fractional representation. IEEE Transactions on Automatic Control, 35(12), 1341–1345. Moore, B. C. (1981). Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control, 26(1), 17–32. Ober, R., & McFarlane, D. (1989). Balanced canonical forms for minimal systems: A normalized coprime factor approach. Linear Algebra and Its Applications, 122–124, 23–64. Poston, T., & Stewart, I. N. (1978). Catastrophe theory and its applications. London: Pitman Publishing. Safonov, M. G., Chiang, R. Y., & Limebeer, D. J. N. (1990). Optimal Hankel model reduction for nonminimal systems. IEEE Transactions on Automatic Control, 35(4), 496–502. Scherpen, J. M. A. (1993). Balancing for nonlinear systems. Systems & Control Letters, 21(2), 143–153. Scherpen, J. M. A. (1994). Balancing for nonlinear systems. Doctoral dissertation, University of Twente, The Netherlands. Scherpen, J. M. A., & Gray, W. S. (2000). Minimality and local state decompositions of a nonlinear state space realization using energy functions. IEEE Transactions on Automatic Control, 45(11), 2079–2086. Varga, A. (1997). Solution of positive periodic discrete Lyapunov equations with applications to the balancing of periodic systems. Proceedings of the 1997 European control conference, Brussels, Belgium.
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Verriest, E. I., & Gray, W. S. (2000). Flow balancing nonlinear systems. Proceedings of the 2000 international symposium on mathematical theory of networks and systems, Perpignan, France. Verriest, E. I., & Gray, W. S. (2001a). Discrete-time nonlinear balancing. The fifth IFAC nonlinear control systems design symposium (pp. 515–520), Saint Petersburg, Russia. Verriest, E. I., & Gray, W. S. (2001b). Nonlinear balanced realizations. Proceedings of the 40th IEEE conference on decision and control (pp. 3250–3251), Orlando, FL. Verriest, E. I., & Helmke, U. (1998). Periodic balanced realizations. The IFAC conference on structure and control (pp. 519–524), Nantes, France. Verriest, E. I., & Kailath, T. (1983). On generalized balanced realizations. IEEE Transactions on Automatic Control, 28(8), 833–844. W. Steven Gray received the B.S. degree in electrical engineering from Purdue University in 1983. He then received the M.S. degree in Electrical Engineering in 1985, the M.S. degree in Applied Mathematics in 1988, and the Ph.D. in Electrical Engineering in 1989, all from the Georgia Institute of Technology. Currently, he is on the Electrical and Computer Engineering Faculty of Old Dominion University in Norfolk, Virginia. His research interests are in modeling and control theory for nonlinear systems. Erik I. Verriest has been with the Electrical and Computer Engineering Department at the Georgia Institute of Technology since 1980. He received the degree of ‘Burgerlijk Electrotechnisch Ingenieur’ from the State University of Ghent, Ghent, Belgium in 1973, and the M.Sc. and Ph.D. degrees from Stanford University in 1975 and 1980, respectively. He has served on several IPC’s and is a member of the IFAC Committee on Linear Systems. He is co-editor (with L. Dugard) of Stability and Control of Time-delay Systems, Springer, 1998. Dr. Verriest’s current interests are in mathematical system theory, with a focus on delay systems, nonlinear system balancing, optimal control and control with information constraints.
Brief paper
Balanced realizations near stable invariant manifolds夡 W. Steven Gray a,∗ , Erik I. Verriest b a Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, Virginia 23529-0246, USA b School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250, USA
Received 15 May 2004; received in revised form 22 June 2005; accepted 6 December 2005
Abstract It is shown that the notion of balancing a state space realization about a stable, isolated equilibrium point can be generalized to systems possessing stable invariant regular submanifolds of arbitrary dimension, for example, a stable limit cycle. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Nonlinear systems; Balanced realizations; Invariant manifolds
1. Introduction The original notion of balanced realizations given by Moore (1981) applies to linear state space systems which are stable, minimal and time-invariant. Over the years the concept has been extended to accommodate unstable linear systems (Meyer, 1990; Ober & McFarlane, 1989; Verriest & Kailath, 1983), nonminimal linear systems (Safonov, Chiang, & Limebeer, 1990; Verriest & Kailath, 1983), periodic linear systems (Varga, 1997; Verriest & Helmke, 1998), general linear time-varying systems (Verriest & Kailath, 1983), and affine nonlinear systems (Scherpen, 1993, 1994; Verriest & Gray, 2000, 2001a,b). In the case of stable systems, either linear or nonlinear, a specific requirement for a well defined balanced realization in the sense of Scherpen is the existence of an asymptotically stable equilibrium point, which without loss of generality is normally taken to be the origin of the state space. In this context, the general idea behind balancing is the introduction of a (local) coordinate frame so that the input energy required to drive the state vector from the origin to any specific state is equal to the output energy generated by the system’s natural response when this same state value is used 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Franco Blanchini under the direction of Editor Roberto Tempo. ∗ Corresponding author. Tel.: +1 757 683 4671; fax: +1 757 683 3220. E-mail addresses: [email protected] (W.S. Gray), [email protected] (E.I. Verriest).
0005-1098/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.12.007
as an initial condition. In this paper it is shown that this notion of balancing can be further generalized to systems which have stable or attracting invariant regular submanifolds, e.g., stable limit cycles, instead of isolated stable equilibria. The technique intrinsically produces a time-varying extension of the approach of Scherpen (1993, 1994). It also complements the flow balancing approach (Verriest & Gray, 2000, 2001a,b) in the sense that the method of balancing introduced here applies to energies generated over infinite time intervals, while flow balancing treats energies generated over infinitesimal time intervals. The paper is organized as follows. In Section 2, the definitions of the input and output energy functions for an invariant set are introduced. Then a pair of partial differential equations is described which these energy functions will satisfy when the equations are known to have a smooth solution with certain boundary conditions. In Section 3, the generalized notion of balancing using these energy function is presented. The final section demonstrates the technique using a simple system possessing a stable limit cycle. The mathematical notation used throughout is fairly standard. Given two sets A and B, A − B denotes the complement of B relative to A. The vector norm on Rn is taken √ T to be x = x x. L2 (a, b) represents the set of Lebesgue measurable functions, possibly vector-valued, with finite L2 b n 2 norm xL2 = a x(t) dt. If L : R → R is a differentiable function, then its partial derivative jL/jx will be the row vector of partial derivatives jL/jxi , where i = 1, . . . , n.
654
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
Furthermore, L generally denotes the mapping, while L(x) is a real number.
2. Energy functions Let M be an n dimensional
state space manifold, and let
(1)
be a system written in terms of local coordinates on M with t ∈ R, u(t) ∈ Rm and y(t) ∈ Rp . It is assumed throughout that f, g and h are C ∞ on M × R. For any t1 , t2 ∈ R with t2 > t1 let x(t2 ) = (t2 , t1 , x1 , u) denote the solution of (1) with initial condition x(t1 ) = x1 and input u : [t1 , t2 ] → Rm . An analogous statement holds when t2 < t1 . In this setting consider the following definitions. Definition 1 (Guckenheimer and Holmes, 1983). A set S ⊂ M is an invariant set under f if for any x0 ∈ S and t0 ∈ R the solution (t, t0 , x0 , 0) belongs to S for all t ∈ R. A closed invariant set S is called an attracting set or stable set if there exists a neighborhood W of S for which x0 ∈ W implies that (t, t0 , x0 , 0) ∈ W for all t > t0 , and (t, t0 , x0 , 0) → S as t → ∞. W is called the region of attraction. Definition 2. The null set for h is N(h) := {x ∈ M : h(x, t)=0, ∀t ∈ R}. It is assumed in all that follows that system (1) has a nonempty closed set, S, invariant under f. From the point of view of balancing, if the system is initialized on S then no input energy is required to keep the resulting trajectory contained in S. In a dual manner, if the system is initialized on S then no output energy should be observed for the resulting trajectory. This can only be assured if S is contained in N(h). The classic example of this situation is where (1) has an isolated equilibrium at x = 0 and h(0, t) = 0, ∀t ∈ R. A nontrivial example from linear system theory is the case where the system has an invariant subspace contained in the unobservable subspace of the system. In an application, this condition could result from designing a new output function from some given physical out˜ put y(t) ˜ = h(x(t), t). For example, one might let h(x(t), t) := ˜h(x(t), t) − h(P ˜ (x(t)), t), where P : W → S is a projection into S. Next consider the following pair of energy functions which is central to our generalized notion of a balanced realization. Definition 3. The controllability and observability functions for the system (f, g, h) with invariant set S are defined, respectively, as min
u∈L2 (−∞,t0 ),x∈S ˜ x(−∞)=x,x(t ˜ 0 )=x0
1 Lo (x0 , t0 ) = 2
∞
y(t)2 dt
t0
with x(t0 ) = x0 and u(t) = 0 for all t t0 . C∞
x(t) ˙ = f (x(t), t) + g(x(t), t)u(t), y(t) = h(x(t), t),
Lc (x0 , t0 ) =
and
1 2
t0
−∞
u(t)2 dt
Clearly, Lc (x0 , t0 ) has the interpretation of being the minimum amount of input energy required to drive the system from S at t = −∞ to x(t0 ) = x0 , while Lo (x0 , t0 ) is equivalent to the output energy generated by the natural response of the system initialized at x(t0 ) = x0 . These functions need not be well defined for all x0 ∈ M. For example, Lo will not be finite if the state does not approach the null set of h in the limit, or Lc may not be finite if the state x0 is not reachable from S at time t0 . But when these functions are known to exist, at least locally, and S is attracting then they necessarily satisfy a pair of partial differential equations described in the following theorem. These equations reduce to the usual Lyapunov equations describing the observability and controllability Gramians in the linear time-invariant case with S = {0}. Theorem 4. Suppose the system (f, g, h) has an attracting set S ⊂ N(h) with a region of attraction W and (x0 , t0 ) ∈ W × R. If the equation jLo jLo 1 f+ + hT h = 0 jx jt 2
(2)
has a smooth solution L¯ o on W × R with L¯ o (x, t) = 0 for all x ∈ S and t t0 then Lo (x0 , t0 ) = L¯ o (x0 , t0 ). Furthermore, if the equation jLc jLc 1 jLc T jLTc f+ + gg =0 jx jt 2 jx jx
(3)
has a smooth solution L¯ c on W ×R with L¯ c (x, t)=0 for all x∈S and t ∈ R, and S is an attracting set of −(f + gg T (jL¯ Tc /jx)) on W × R then Lc (x0 , t0 ) = L¯ c (x0 , t0 ). Proof. The claim regarding Lo is trivial when x0 ∈ S since S is invariant under f and S ⊂ N(h). So suppose instead that x0 ∈ W − S. Observe Lo (x0 , t0 ) 1 ∞ h((t, t0 , x0 , 0), t)2 dt = 2 t0 ∞ ¯ jLo =− f ((t, t0 , x0 , 0), t) jx t0 jL¯ o + ((t, t0 , x0 , 0), t) dt jt ∞ d ¯ Lo ((t, t0 , x0 , 0), t) dt =− dt to = − lim L¯ o ((t, t0 , x0 , 0), t) + L¯ o ((t0 , t0 , x0 , 0), t0 ) t→∞
= L¯ o (x0 , t0 ).
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
Next let L¯ c be a smooth solution of Eq. (3) with L¯ c (x, t) = 0 for any (x, t) ∈ S × R. Again the result is trivial when x0 ∈ S, so assume that x0 ∈ W − S. Then it follows via a completion of the squares argument (e.g., see Lee & Markus, 1967) that for any x˜ ∈ S and any u with (t0 , −∞, x, ˜ u) = x0 (while staying in the neighborhood W) ˜ u), t) L¯ c (x0 , t0 ) − lim L¯ c ((t, −∞, x, =
−∞
d ¯ ˜ u), t) dt Lc ((t, −∞, x, dt x(t)
¯ ¯ jLc jL c f (x(t), t) + g (x(t), t)u(t) jx jx −∞
=
t→−∞
t0
t0
jL¯ c (x(t), t) dt + jt
¯ t0 jLc 1 jL¯ c T jL¯ Tc = − (x(t), t) + gg (x(t), t) jt 2 jx jx −∞ ¯ jL¯ c jLc g (x(t), t)u(t) + (x(t), t) dt + jx jt 2 t0 ¯T 1 1 2 T jLc u(t) − u(t) − g (x(t), t) = dt. 2 jx −∞ 2 Since L¯ c (x, ˜ t) = 0 when x˜ ∈ S, and Lc is smooth, then 1 L¯ c (x0 , t0 ) 2
t0
−∞
u(t)2 dt
with equality if and only if the input u(t) = u∗ (t) := (g T (jL¯ Tc /jx))(x(t), t). This lower bound on the input energy is achievable since u∗ is an admissible input in the definition of Lc . That is, the closed loop system f + gu∗ is assumed here to have the property that x(t) → S as t → −∞ if x(t0 ) = x0 ∈ W . Hence, L¯ c (x0 , t0 ) = Lc (x0 , t0 ). From the definitions it is immediate that Lc and Lo are always nonnegative. Conditions are now developed under which these functions are positive on (W −S)×R. The following definitions are either classic or obvious variations of classic definitions with the exception of asymptotic reachability. This latter notion was introduced in Scherpen and Gray (2000). Definition 5. Consider system (1) and any set S ⊂ M. • The system is reachable from S if for any x¯ ∈ M there exists a t¯ t0 , an input u, and x0 ∈ S such that x¯ = (t¯, t0 , x0 , u). • The system is asymptotically reachable from S on a neighborhood W ⊃ S if for all x ∈ W there exists an input u ∈ L2 (t0 , ∞) and x0 ∈ S such that (t, t0 , x0 , u) ∈ W for all t t0 , and limt→∞ (t, t0 , x0 , u) = x, or in other words, for all x ∈ W and any > 0, there exists a t¯ < ∞ such that (t, t0 , x0 , u) ∈ W for all t0 t t¯ and (t¯, t0 , x0 , u) − x < .
655
• The system is S-state observable if S ⊂ N(h) and any trajectory where u(t) ≡ 0 and y(t) ≡ 0 implies that x(t) ∈ S for all t t0 , i.e., for all x0 ∈ M, the condition h((t, t0 , x0 , 0), t) = 0 for all t t0 implies that (t, t0 , x0 , 0) ∈ S for all t t0 . • The system is locally S-state observable if S ⊂ N(h) and there exists an open set U ⊃ S such that if x0 ∈ U then the condition h((t, t0 , x0 , 0), t) = 0 for all t t0 implies that (t, t0 , x0 , 0) ∈ S for all t t0 . With these definitions in place, the following theorems address the positivity issue. These results are generalizations of those which appear in Scherpen (1993, 1994). Theorem 6. Suppose S is an attracting set of f with region of attraction W. If a smooth Lc exists on W ×R then Lc (x0 , t0 ) > 0 for all (x0 , t0 ) ∈ (W −S)×R if the system (1) is asymptotically reachable from S on W. Proof. The method of proof which appeared in Scherpen and Gray (2000) can be directly extended to show that asymptotic reachability from S on W and the existence of a smooth Lc on W implies that S is an attracting set of −(f + gg T (jL¯ Tc /jx)) on W × R. In which case, Theorem 4 applies and the value of Lc (x0 , t0 ) for any (x0 , t0 ) ∈ W × R can be computed as 1 t0 Lc (x0 , t0 ) = u∗ (t)2 dt 2 −∞ 2 ¯T 1 t0 g T jLc (x(t), t) dt. = 2 jx −∞
Now specifically, choose any (x0 , t0 ) ∈ (W − S) × R and assume that Lc (x0 , t0 ) = 0. The latter implies that u∗ (t) = 0 on −∞ t t0 . In which case this is a situation where the input / S. Since S is an invariant set is zero, x(−∞) ∈ S and x(t0 ) ∈ under f, this is a contradiction. Theorem 7. Suppose S is an attracting set of f with a region of attraction W. If S ⊂ N(h) and a smooth Lo exists on W × R then Lo (x0 , t0 ) > 0 for all (x0 , t0 ) ∈ (W − S) × R if system (1) is locally S-state observable on W. Proof. Choose any (x0 , t0 ) ∈ (W −S)×R. Since S is attracting on W then (t, t0 , x0 , 0) is well defined for all t t0 . Now if Lo (x0 , t0 ) = 0 then h((t, t0 , x0 , 0), t) = 0 for all t t0 . But since the system (1) is S-state observable on W this implies that x0 ∈ S, a contradiction. 3. Locally balanced realizations In this section, the notion of locally balancing a realization (f, g, h) near an invariant set S is developed under the following set of assumptions: (A1) S is a regular C ∞ submanifold of M with dimension r. (A2) S is invariant under f. (A3) S is an attracting set with a region of attraction W.
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
(A4) The system is asymptotically reachable from S on W. (A5) The system is locally S-state observable on W. (A6) Lc and Lo are well defined and smooth functions on W × R. (A7) rank(j2 Lc /jx 2 ) = rank(j2 Lo /jx 2 ) = n − r everywhere on S × R. Given that Lc and Lo are well defined, as per (A6), one can fix the time argument, t, of these energy functions and introduce state space coordinate transformations to modify their original form. As in the linear time-varying case, where the Gramians are dependent on t, the transformed energy functions are (pseudo-)quadratic forms written in terms of matrices parameterized by t. Theorem 8. Let t ∈ R be fixed. Choose any s ∈ S. Then there exists a smooth local coordinate transformation x =t (x¯t ) such that s = t (0) and on some neighborhood U ⊂ W of s the function Lc in the new coordinates x¯t = −1 t (x) has the form Lc (t (x¯t ), t) =
1 2
x¯tT diag(In−r , 0)x¯t ,
x¯t ∈ −1 t (U ).
(Here Ii ∈ Ri×i denotes an identity matrix.) Furthermore, in the new coordinates, one can write Lo in the form Lo (t (x¯t ), t) =
1 2
x¯tT M(x¯t , t)x¯t ,
where M(0, t) =
j2 Lo (s, t) jx 2
with M(x¯t , t) an n × n symmetric matrix such that its entries are smooth functions of x¯t and t. Proof. Since S is an r dimensional C ∞ submanifold of M, it follows that for any s ∈ S there exists a local coordinate neighborhood U and coordinate transformation x˜ = (x) such that (s) = 0, (U ) = Cn (0) (an n dimensional cube centered at 0 and with edges of length 2, > 0) and (U ∩ S) = {x˜ ∈ Cn (0) : x˜1 = · · · = x˜n−r = 0}. (See Boothby, 1975, p. 75 and Fig. 1.) In this coordinate frame it is easily verified that Lc (−1 (0), t) = 0 and that 0 is a critical point of Lc (−1 (·), t) (since (A2)–(A4) and Theorem 6 imply that 0 is a local minimum of Lc (−1 (·), t)). In light of (A7), the Splitting Lemma (a generalized Morse Lemma, e.g., see Poston & Stewart, 1978) applies and provides for the existence of a coordinate transformation x˜ = t (x¯t ) such that t (0, . . . , 0, x¯t,n−r+1 , . . . , x¯t,n ) = (0, . . . , 0, x˜n−r+1 , . . . , x˜n ) and Lc (−1 ◦ t (x¯t ), t) t
2 2 = 21 (x¯t,1 + · · · + x¯t,n−r ) + p(x¯t,n−r+1 , . . . , x¯t,n ),
W S
ψ(U)=C2ε (0) x~2
U ψ
s
x~1
0
ψ(U S) U
656
Fig. 1. The local coordinate transformation in the case where n = 2 and r = 1.
where p : Rr → R is smooth. However, in light of the fact that Lc (t (¯s ), t) = 0 for all s¯ = (0, . . . , 0, x¯t,n−r+1 , . . . , x¯t,n ) ∈ −1 −1 t (U ∩ S) then p ≡ 0 on t (U ), and the first part of the theorem is proven. The second part of the theorem is proven in a manner perfectly analogous to the proofs of Scherpen (1993, 1994) except that the smoothness of M(x¯t , t) in the argument t follows from (A6). In order to produce an input-normal form where M(x¯t , t) is a diagonal matrix, the following technical lemma is required. Lemma 9. If on the set −1 t (U ), as defined in Theorem 8, the number of distinct nonzero eigenvalues of M(x¯t , t) is constant then the eigenvalues i (x¯t , t), i = 1, . . . , n, are smooth functions of x¯t on −1 t (U ), as well as the associated orthonormal eigenvectors. Proof. With t ∈ R a fixed parameter, the proof is exactly as it appears in Kato (1976). Theorem 10. Let t ∈ R be fixed. Choose any s ∈ S. Assume that the condition of Lemma 9 is fulfilled. Then there exists a smooth local coordinate transformation x = t (zt ) such that s = t (0) and on some neighborhood V ⊂ W of s the function Lc in the new coordinates zt = −1 t (x) has the form L˘ c (zt , t) := Lc (t (zt ), t) =
1 T 2 zt diag(In−r , 0)zt ,
and the function Lo is of the form L˘ o (zt , t) := Lo (t (zt ), t) = 21 ztT diag(1 (zt , t), . . . , n−r (zt , t), 0, . . . , 0)zt , where 1 (zt , t) · · · n−r (zt , t) > 0 are smooth functions of zt , referred to as the singular value functions. Proof. The proof is perfectly analogous to that which appears in Scherpen (1993, 1994), where the Lemma 9 simply insures the smooth diagonalizability of M(x¯t , t).
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
657
The form of the controllability and observability functions is not yet entirely balanced. For that, one additional coordinate transformation is required. Consider the smooth transformation
and the corresponding open covering of W:
z¯ t,i = t,i (zt,i ) := i ((0, . . . , 0, zt,i , 0, . . . , 0), t)1/4 zt,i ,
Ub = {(x1 , x2 ) ∈ R2 : x2 < 0},
for i = 1, . . . , n − r and t,i (zt,i ) := zt,i otherwise. Then z¯ t = t (zt ) := (t,1 (zt,1 ), . . . , t,n (zt,n )) on Vt,b = t,b (V ) := t ◦ −1 t (V ). In the new coordinates it follows then that
Uc = {(x1 , x2 ) ∈ R2 : x1 < 0},
2 i (¯zt,i , t)−1 , Lˆ c ((0, . . . , 0, z¯ t,i , 0, . . . , 0), t) = 21 z¯ t,i 2 Lˆ o ((0, . . . , 0, z¯ t,i , 0, . . . , 0), t) = 21 z¯ t,i i (¯zt,i , t),
zt,i ), 0, . . . , 0), t)1/2 for where i (¯zt,i , t) := i ((0, . . . , 0, −1 i (¯ i = 1, . . . , n − r and zero otherwise. In these coordinates the system is said to be in balanced form.
x˙1 = x2 , x˙2 = −(x12 + x22 − 1)x2 − x1 + u, y = 2(x12 + x22 − 1)x2 .
(4)
The autonomous system has an unstable equilibrium at x = 0 and a stable limit cycle, S, consisting of the unit circle centered at x = 0 with region of attraction W = R2 − {0}. Therefore, the assumptions (A1)–(A3) are satisfied with n = 2 and r = 1. The null set of h contains S, and it can be easily verified that (A5) holds. Next observe that everywhere on W, the functions Lc (x, t) = Lo (x, t) = 21 (x12 + x22 − 1)2 satisfy Eqs. (2) and (3), including the required boundary conditions on S × R. In addition, a simple geometric analysis show that S is also an attracting set for the vector field
T −x2 0 T jLc − f + gg + . (x, t) = −(x12 + x22 − 1)x2 x1 jx tangential near S
Thus, from Theorem 4, Lo and Lc are indeed the energy functions for system (4), and consequently, assumptions (A4) and (A6) are verified. (Since Lc and Lo do not depend explicitly on time in this example, the t argument will be suppressed throughout.) Finally, observe that 2 2 j Lc
j Lo
rank = rank
jx 2 S jx 2 2 S x1 x1 x2 = rank = 1, x1 x2 x22 so (A7) is valid. Taken in total, these assumptions guarantee that about any point s ∈ S, there must exist a local balancing transformation. Consider the set of points: sb = (0, −1),
sc = (−1, 0),
In light of Theorem 10, set Lc (x) = Lo (x) = 21 (x12 + x22 − 1)2 = 1 2 ˘
˘
2 (z1 ) = Lc (z ) = Lo (z ), where ∈ A := {a, b, c, d} (note that Lemma 9 applies by default). Therefore, the following coordinate transformations are valid local balancing transformations on their respective coordinate patches:
a
Consider the second-order system
sa = (1, 0),
Ud = {(x1 , x2 ) ∈ R2 : x2 > 0}.
4. Example: system with a stable limit cycle
attractive near S
Ua = {(x1 , x2 ) ∈ R2 : x1 > 0},
sd = (0, 1),
z1a = x12 + x22 − 1, ⇐⇒ z2a = x2 ,
x1 =
1 + z1a − (z2a )2 ,
x2 = z2a ,
x1 = z2b , ⇐⇒ b b z2 = x 1 , x2 = − 1 + z1b − (z2b )2 , c z1 = x12 + x22 − 1, x1 = − 1 + z1c − (z2c )2 , ⇐⇒ c c z2 = x2 , x2 = z2c , d x1 = z2d , z1 = x12 + x22 − 1, ⇐⇒ d d z2 = x1 , x2 = 1 + z1d − (z2d )2 . z1b = x12 + x22 − 1,
The family U = {U , } ∈A constitutes a C ∞ structure on W. The corresponding balanced realizations are ⎤ ⎡ ⎧
a −2z1a (z2a )2 ⎪ 2z2 ⎪ ⎨ z˙ a = ⎣ ⎦+ u, 2 a a a a −z1 z2 − 1 + z1 − (z2 ) 1 a (Ua ) ⎪ ⎪ ⎩ y = 2z1a z2a , ⎧ −2z1b (1 + z1b − (z2b )2 ) ⎪ b ⎪ ⎪ z˙ = ⎪ ⎪ ⎪ − 1 + z1b − (z2b )2 ⎪ ⎪ ⎨ −2 1 + z1b − (z2b )2 b (Ub ) u, + ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ y = −2z1b 1 + z1b − (z2b )2 ,
⎧ −2z1c (z2c )2 2z2c ⎪ ⎨ z˙ c = + u, 1 c (Uc ) −z1c z2c + 1 + z1c − (z2c )2 ⎪ ⎩ y = 2z1c z2c , ⎧ −2z1d (1 + z1d − (z2d )2 ) ⎪ d ⎪ ⎪ ⎪ z˙ = ⎪ ⎪ 1 + z1d − (z2d )2 ⎪ ⎪ ⎨ 2 1 + z1d − (z2d )2 d (Ud ) u, + ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ y = 2z1d 1 + z1d − (z2d )2 .
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
zd2
ψd(sd)
za2
t=-1
t=-2
t=-3 -1
-0.5
0 zd1
0.5
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
1
t=-1
ψa(sa)
zd2
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
t=0 t=-6
-1
-0.5
0 za1
0.5
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
t=6
ψd(sd) t=5
t=4 -1
1
za2
658
-0.5
0 zd1
0.5
1
t=-2 1
t=-1
-0.5
0 za1
0.5
1
x2
t=-4
sa t=-6
sb
-1
sd
t=4
sb
-1
-2
-2 -2.5
-2.5 -2 -1.5 -1 -0.5 0
0.5
1
1.5
2
2.5
-2 -1.5 -1 -0.5 0
x1
zb2 t=-4 t=-5 0
zc1
0.5
1
t=-6
ψb(sb) t=-5
t=-4 -1
-0.5
0
0.5
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
t=5 t=4 ψc(sc) t=3 t=2 -1
1
-0.5
z1b
Fig. √2. The state trajectory x(t) = x (t, −∞, S, u∗ ) corresponding to Lc (( 2, 0)) and its images z (t) in the balanced coordinate patches (U ),
∈ A.
0 zc1
0.5
1
1.5
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
2
2.5
t=1
ψb(sb)
t=2
t=3 -1
1
-0.5
0 zb1
0.5
1
√ Fig. √ 3. The state trajectory x(t) = x (t, 0, ( 2, 0), 0) corresponding to
Lo (( 2, 0)) and its images z (t) in the balanced coordinate patches (U ),
∈ A.
1.2
Clearly, the singular value function 1 (z ) = 1 on (U ) for √ each ∈ A. Take as an example the test point x = ( 2, 0), ˘ a −1 where L˘ ac (−1 a (x)) = Lo (a (x)) = 0.5. The center plot in Fig. 2 shows the state trajectory from S to the test point (at time t = 0) in the x-coordinate frame. It was determined numerically by computing the input u∗ in the proof of Theorem 4. Surrounding this plot are the four balanced coordinate patches showing the corresponding trajectories, z (t), entering and exiting the patch (U ), the region strictly to the right of the parabola 1 + z1 − (z2 )2 = 0. In this case (U ∩ S) is the open line segment connecting (0, 1) and (0, −1). Fig. 3 shows the natural response of the system when the same test point is used as an initial condition at t =0. Using these numerical results, the energy functions L˘ c and L˘ o were evaluated at the test point by plotting t the cumulative energy functions Eu (t) := 21 −∞ u∗ ()2 d t and Ey (t) := 21 0 yn ()2 d (yn is the natural output
0.5 x1
zb2
t=-3
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
zc2
t=-2
t=1
t=2
t=-5 -1.5
t=0
sa
t=3
-0.5
-1.5
ψc(sc)
t=6
sc
0
t=0 x2
sc
-0.5
t=5
0.5
t=-3
0
zc2
t=0
t=1 -1
1
sd
0.5
-0.5
ψa(sa)
1.5
1.5
-1
t=6
2
2
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
1 0.8 u*
0.6 0.4
Eu
0.2 0 -0.2 -6
-5
-4
-3 -2 time
-1
0
0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2
Ey
yn
0
1
2
3 4 time
5
6
Fig. 4. The cumulative energy functions Eu and Ey for the respective signals u∗ and yn .
response) as a function of t as shown in Fig. 4. As predicted, ˘ a −1 L˘ ac (−1 a (x)) = Eu (0) = 0.5 and Lo (a (x)) = Ey (∞) = 0.5. Finally, it should be noted that if one linearizes the original
W.S. Gray, E.I. Verriest / Automatica 42 (2006) 653 – 659
system along S, the resulting time-varying system will fail to be either controllable or observable in the conventional sense. Thus, a straightforward application of linear balancing methods as in Verriest and Kailath (1983) is not possible. References Boothby, W. M. (1975). An introduction to differentiable manifolds and Riemannian geometry. Orlando, FL: Academic Press. Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer. Kato, T. (1976). Perturbation theory for linear operators (2nd ed.), Heidelberg: Springer. Lee, E. B., & Markus, L. (1967). Foundations of optimal control theory. New York: Wiley. Meyer, D. G. (1990). Fractional balanced reduction—model reduction via a fractional representation. IEEE Transactions on Automatic Control, 35(12), 1341–1345. Moore, B. C. (1981). Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control, 26(1), 17–32. Ober, R., & McFarlane, D. (1989). Balanced canonical forms for minimal systems: A normalized coprime factor approach. Linear Algebra and Its Applications, 122–124, 23–64. Poston, T., & Stewart, I. N. (1978). Catastrophe theory and its applications. London: Pitman Publishing. Safonov, M. G., Chiang, R. Y., & Limebeer, D. J. N. (1990). Optimal Hankel model reduction for nonminimal systems. IEEE Transactions on Automatic Control, 35(4), 496–502. Scherpen, J. M. A. (1993). Balancing for nonlinear systems. Systems & Control Letters, 21(2), 143–153. Scherpen, J. M. A. (1994). Balancing for nonlinear systems. Doctoral dissertation, University of Twente, The Netherlands. Scherpen, J. M. A., & Gray, W. S. (2000). Minimality and local state decompositions of a nonlinear state space realization using energy functions. IEEE Transactions on Automatic Control, 45(11), 2079–2086. Varga, A. (1997). Solution of positive periodic discrete Lyapunov equations with applications to the balancing of periodic systems. Proceedings of the 1997 European control conference, Brussels, Belgium.
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Verriest, E. I., & Gray, W. S. (2000). Flow balancing nonlinear systems. Proceedings of the 2000 international symposium on mathematical theory of networks and systems, Perpignan, France. Verriest, E. I., & Gray, W. S. (2001a). Discrete-time nonlinear balancing. The fifth IFAC nonlinear control systems design symposium (pp. 515–520), Saint Petersburg, Russia. Verriest, E. I., & Gray, W. S. (2001b). Nonlinear balanced realizations. Proceedings of the 40th IEEE conference on decision and control (pp. 3250–3251), Orlando, FL. Verriest, E. I., & Helmke, U. (1998). Periodic balanced realizations. The IFAC conference on structure and control (pp. 519–524), Nantes, France. Verriest, E. I., & Kailath, T. (1983). On generalized balanced realizations. IEEE Transactions on Automatic Control, 28(8), 833–844. W. Steven Gray received the B.S. degree in electrical engineering from Purdue University in 1983. He then received the M.S. degree in Electrical Engineering in 1985, the M.S. degree in Applied Mathematics in 1988, and the Ph.D. in Electrical Engineering in 1989, all from the Georgia Institute of Technology. Currently, he is on the Electrical and Computer Engineering Faculty of Old Dominion University in Norfolk, Virginia. His research interests are in modeling and control theory for nonlinear systems. Erik I. Verriest has been with the Electrical and Computer Engineering Department at the Georgia Institute of Technology since 1980. He received the degree of ‘Burgerlijk Electrotechnisch Ingenieur’ from the State University of Ghent, Ghent, Belgium in 1973, and the M.Sc. and Ph.D. degrees from Stanford University in 1975 and 1980, respectively. He has served on several IPC’s and is a member of the IFAC Committee on Linear Systems. He is co-editor (with L. Dugard) of Stability and Control of Time-delay Systems, Springer, 1998. Dr. Verriest’s current interests are in mathematical system theory, with a focus on delay systems, nonlinear system balancing, optimal control and control with information constraints.
