Approximate reduction of dynamic systems

Transcription

1 Systems & Control Letters Approximate reuction of ynamic systems Paulo Tabuaa a,, Aaron D. Ames b, Agung Julius c, George J. Pappas c a Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA 90095, Unite States b Control an Dynamical Systems Department, California Institute of Technology, Pasaena, CA 925, Unite States c Department of Electrical an Systems Engineering, University of Pennsylvania, Philaelphia, PA 904, Unite States Receive 25 July 2007; receive in revise form 2 December 2007; accepte 9 December 2007 Available online February 2008 Abstract The reuction of ynamic systems has a rich history, with many important applications relate to stability, control an verification. Reuction of nonlinear systems is typically performe in an exact manner as is the case with mechanical systems with symmetry which, unfortunately, limits the type of systems to which it can be applie. The goal of this paper is to consier a more general form of reuction, terme approximate reuction, in orer to exten the class of systems that can be reuce. Using notions relate to incremental stability, we give conitions on when a ynamic system can be projecte to a lower imensional space while proviing har bouns on the inuce errors, i.e. when it is behaviourally similar to a ynamic system on a lower imensional space. These concepts are illustrate on a series of examples. c 2007 Elsevier B.V. All rights reserve. Keywors: Reuction; Approximate reuction; Dynamical systems; Incremental input-to-state stability. Introuction Moelling is an essential part of many engineering isciplines an often a key ingreient for successful esigns. Although it is wiely recognize that moels are only approximate escriptions of reality, their value lies precisely on the ability to escribe, within certain bouns, the moelle phenomena. In this paper we consier moeling of closeloop nonlinear control systems, i.e. ifferential equations, with the purpose of simplifying the analysis of these systems. The goal of this paper is to reuce the imensionality of the ifferential equations being analyse while proviing har bouns on the introuce errors. One promising application of these techniques is to the verification of hybri systems, which is currently constraine by the complexity of high imensional ifferential equations. Reucing ifferential equations an in particular mechanical systems is a subject with a long an rich history. This research was partially supporte by the National Science Founation, EHS awar Corresponing author. Tel.: ; fax: aresses: tabuaa@ee.ucla.eu P. Tabuaa, ames@cs.caltech.eu A.D. Ames, agung@seas.upenn.eu A. Julius, pappasg@ee.upenn.eu G.J. Pappas. The first form of reuction was iscovere by Routh in the 860 s; over the years, geometrical reuction has become an acaemic fiel in itself. One begins with a ifferential equation with certain symmetries, i.e. a ifferential equation invariant uner the action of a Lie group on the phase space. Using these symmetries, one can reuce the imensionality of the phase space by iviing out by the symmetry group an efine a corresponing ifferential equation on this reuce phase space. The main result of geometrical reuction is that one can unerstan the behaviour of the full-orer system in terms of the behavior of the reuce system an vice versa [,6,5]. While this form of exact reuction is very elegant, the class of systems for which this proceure can be applie is actually quite small. This inicates the nee for a form of reuction that is applicable to a wier class of systems an, while not being exact, is close enough. In systems theory, reuce orer moelling has also been extensively stuie uner the name of moel reuction [4,3]. The typical problem aresse in this literature consists in approximating a system Σ by a system Σ 2 while minimizing the L 2 norm: /$ - see front matter c 2007 Elsevier B.V. All rights reserve. oi:0.06/j.sysconle

2 P. Tabuaa et al. / Systems & Control Letters y t y 2 t t where y is the output of Σ an y 2 is the output of Σ 2. This kin of reuction is not aequate when one is intereste in applications to formal verification of hybri systems. A typical safety verification problem consists in etermining if any trajectory of Σ starting in a given set of initial conitions S enters a given set of unsafe states B. If one solves this verification problem with the reuce orer moel Σ 2, then one cannot conclue, base on an upper boun on, if trajectories of Σ o enter the B. This motivates us to stuy reuction problems in which trajectories of Σ an its reuce moel Σ 2 are instea relate by the L norm: sup y t y 2 t. 2 t [0, [ More recent work consiere exact reuction of control systems [7,5] base on the notion of bisimulation which was later generalize to approximate bisimulation [7,3,8]. We evelop our results in the framework of incremental stability an our main result is in the spirit of existing stability results for cascae systems that proliferate the Inputto-State Stability ISS literature. See, for example, [2] an the references therein. A preliminary version of our results appeare in the conference paper [4]. 2. Preliminaries A continuous function γ : R + 0 R+ 0, is sai to belong to class K if it is strictly increasing, γ 0 = 0 an γ r as r. A continuous function β : R + 0 R+ 0 R+ 0 is sai to belong to class KL if, for each fixe s, the map βr, s belongs to class K with respect to r an, for each fixe r, the map βr, s is ecreasing with respect to s an βr, s 0 as s. A function ϕ : R n R m is sai to be smooth if it is infinitely ifferentiable. We enote by T ϕ the tangent map to ϕ an by T x ϕ the tangent map to ϕ at x R n. The map ϕ is sai to be a submersion at x R n if T x ϕ is surjective an is sai to be a submersion if it is a submersion at every x R n. When ϕ is a submersion we will also use the notation kert ϕ to enote the istribution: kert ϕ = {X : R n R n T ϕ X = 0}. The Lie bracket of vector fiels X an Y is enote by [X, Y ] an [ker T ϕ, Y ] enotes the istribution efine by all the vector fiels Z such that Z = [X, Y ] for some X ker T ϕ. Given a point x R n, x enotes the usual Eucliean norm while f enotes ess sup t [0,τ] f t for any given function f : [0, τ] R n, τ R Dynamic an control systems In this paper we shall restrict our attention to ynamic an control systems efine on Eucliean spaces. Definition. A vector fiel is a pair R n, X where X is a smooth map X : R n R n. A smooth curve x : I R n, efine on an open subset I of R incluing the origin, is sai to be a trajectory of R n, X if the following conition hols: xt = X xt t I. t When we want to emphasize the initial conition x0 = x we shall enote a trajectory as x, x. A vector fiel is sai to be forwar complete when for every x R n the trajectory x, x is efine on an interval of the form ] a, + [ for some a < 0. All the vector fiels in this paper are assume to be forwar complete. This assumption is always satisfie for problems of formal verification in which the vector fiel escribes the result of applying a stabilizing controller to the open-loop ynamics. Definition 2. A control system is a triple R n, R m, F where F is a smooth map F : R n R m R n. A smooth curve x : I R n, efine on an open subset I of R incluing the origin, is sai to be a trajectory of R n, R m, F if there exists a smooth input curve u : I R m such that the following conition hols: xt = Fxt, ut t I. t Similarly to vector fiels, we enote by x u, x the trajectory x of a control system associate with the input curve u an satisfying x0 = x. We have efine trajectories base on smooth input curves mainly for simplicity since the presente results hol uner weaker regularity assumptions. 3. Exact reuction For some ynamic systems escribe by a vector fiel X on R n it is possible to replace X by a vector fiel Y escribing the ynamics of the system on a lower imensional space, R m, while retaining much of the information about X. When this is the case we say that X can be reuce to Y. This iea of exact reuction is capture by the notion of ϕ-relate vector fiels. Definition 3. Let ϕ : R n R m be a smooth map. The vector fiel R n, X is sai to be ϕ-relate to the vector fiel R m, Y if for every x R n we have: T x ϕ X x = Y ϕx. 3 The following proposition, prove in [], characterizes ϕ-relate vector fiels in terms of their trajectories. Proposition. The vector fiel R n, X is ϕ-relate to the vector fiel R m, Y for some smooth map ϕ : R n R m iff for every x R n an for every t R + 0 we have: ϕ xt, x = yt, ϕx, 4 where x an y are the trajectories of X an Y, respectively.

3 540 P. Tabuaa et al. / Systems & Control Letters For ϕ-relate vector fiels, we can replace the stuy of trajectories x, x with the stuy of trajectories y, ϕx living on the lower imensional space R m. To illustrate the usefulness of this result in the context of formal verification, assume that one is intereste in showing that part of the state will never enter a set of unesirable states B R m. If the part of the state we are intereste in is given by y = ϕx with y R m, m < n an if X is ϕ-relate to Y then we can analyse the evolution of y by working with the reuce moel Y instea of working with the full-orer moel X. If a vector fiel an a submersion ϕ are given we can use the following result, prove in [0], to etermine the existence of ϕ-relate vector fiels. Proposition 2. Let R n, X be a vector fiel an let ϕ : R n R m be a smooth submersion. There exists a vector fiel R m, Y that is ϕ-relate to R n, X iff: [kert ϕ, X] kert ϕ. 5 When X is a linear vector fiel X x = Ax an ϕ is a linear map ϕx = Lx, conition 5 amits a simpler an intuitive escription. Recalling that [v, Ax] = Av for any v R n, 5 becomes Ak kerl for every k kerl or equivalently, AkerL kerl. A linear vector fiel Ax woul then be L-relate to another vector fiel iff kerl is an A-invariant subspace of R n. This is easily seen to be a quite restrictive conition. In orer to enlarge the class of vector fiels that can be reuce we introuce, in the next section, an approximate notion of reuction. 4. Approximate reuction The generalization of Definition 3 propose in this section requires a ecomposition of R n of the form R n = R m R k. Associate with this ecomposition are the canonical projections π m : R n R m an π k : R n R k taking R n x = y, z R m R k to π m x = y an π k x = z, respectively. Intuitively, R n correspons to the state space of the full moel an we will be intereste in the evolution of only the part of the state escribe by y = π m x, for which we will be seeking a reuce moel. Definition 4. The vector fiel R n, X is sai to be approximately π m -relate to the vector fiel R m, Y if there exist a class K function γ an a constant c R + 0 such that the following estimate hols for every x R n an for every t R + 0 : π m xt, x yt, π m x γ π k x + c, 6 where x an y are the trajectories of X an Y, respectively. Note that when X an Y are π m -relate we have: π m xt, x yt, π m x = 0, which implies 6. Definition 4 can thus be seen as a generalization of exact reuction capture by Definition 3. Similar ieas have been use in the context of approximate notions of equivalence for control systems [8]. Although the boun on the gap between the projection of the original trajectory x an the trajectory y of the approximate reuce system, given by 6, is a function of x, typical verification problems assume that initial conitions belong to a compact set S. The following result is therefore useful in those situations: Proposition 3. If R n, X is approximately π m -relate to R m, Y, then for any compact set S R n there exists a δ R + such that for all x S an all t R + 0 the following estimate hols: π m xt, x yt, π m x δ. 7 Proof. Let δ = max x C γ π k x + c. The scalar δ is well efine since γ π k + c is a continuous map an C is compact. From a practical point of view, approximate reuction is only a useful concept if it amits characterizations that are simple to check. In orer to erive such characterizations we nee to review several notions of incremental stability. 4.. Incremental stability In this subsection we review two notions of incremental stability which will be funamental in proving the main contribution of this paper. We follow [6] an [2]. Definition 5. A control system R n, R m, F is sai to be incrementally uniformly boune-input-boune-state stable IUBIBSS if there exist two class K functions γ an γ 2 an a constant R + 0 such that for each x, x 2 R n an for each pair of smooth input curves u, u 2 : R + 0 Rm the following estimate hols for all t R + 0 : x u t, x x u2 t, x 2 γ x x 2 + γ 2 u u A system is IUBIBSS when two ifferent trajectories x u an x u2, starting at ifferent but close initial conitions an associate with close but ifferent input curves, will remain close for all time. In the linear case IUBIBSS turns out to be equivalent to stability but it is a istinct concept in the nonlinear case [6]. In general it is ifficult to establish IUBIBSS irectly. A sufficient conition is given by the existence of an IUBIBSS Lyapunov function. Note, however, that IUBIBSS only implies the existence of a IUBIBSS Lyapunov function with very weak regularity conitions [6]. Definition 6. A C function V : R n R n R + 0 is sai to be an IUBIBSS Lyapunov function for control system R n, R m, F if there exist a ξ R + an class K functions α, α, an µ such that for every x, x 2 R n an u, u 2 R m the following hols: x x 2 ξ α x x 2 V x, x 2 α x x 2 ; 2 x x 2 µ u u 2 + ξ x V Fx, u + V x 2 Fx 2, u 2 0.

4 P. Tabuaa et al. / Systems & Control Letters A stronger notion than IUBIBSS is incremental input-tostate stability. Definition 7. A control system R n, R m, F is sai to be incrementally input-to-state stable IISS if there exist a class KL function β an a class K function γ such that for each x, x 2 R n an for each pair of smooth curves u, u 2 : R + 0 R m the following estimate hols for all t R + 0 : x u t, x x u2 t, x 2 β x x 2, t + γ u u 2. 9 Since β is a ecreasing function of t we immeiately see that 9 implies 8 with γ r = βr, 0 an γ 2 r = γ r, r R + 0. In aition to require trajectories to remain close if initial conitions an input curves are close, IISS requires the istance between trajectories to converge to zero over time. In the linear case IISS is equivalent to asymptotic stability but it is a istinct concept in the nonlinear case [2]. The notion of IISS is also implie by the existence of an IISS Lyapunov function. See [2] for a converse result when the inputs take values in a compact set. Definition 8. A C function V : R n R n R + 0 is sai to be an IISS Lyapunov function for the control system R n, R m, F if there exist class K functions α, α, α, an µ such that for every x, x 2 R n an u, u 2 R m the following hols: α x x 2 V x, x 2 α x x 2 ; 2 x x 2 µ u u 2 x V Fx, u + V x 2 Fx 2, u 2 α x x Fiberwise stability In aition to incremental stability we will also nee a notion of partial practical stability. This notion will be use to ensure that the ynamics neglecte in the approximate reuction process is well behave. Definition 9. A vector fiel R n, X is sai to be fibrewise practically stable with respect to π k if there exist a class K function γ an a constant c R + 0 such that the following estimate hols for all x R n an t R + 0 : π k xt, x γ π k x + c. Fibrewise practical stability can be checke with the help of the following result: Lemma. A vector fiel R n, X is fibrewise practically stable with respect to π k if there exist two K functions, α an α, a constant R + 0, an a function V : Rn R such that for every x R n satisfying π k x we have: α π k x V x α π k x, X x 0. 2 V x 4.3. Existence of approximate reuctions In this subsection we prove the main result proviing sufficient conitions for the existence of approximate reuctions. Theorem. Let R n, X be a fibrewise practically stable vector fiel with respect to π k an let F = T π m X : R m R k R m, viewe as a control system with state space R m, be IUBIBSS. Then, the vector fiel R m, Y efine by: Y y = T y,0 π m X y, 0 = Fy, 0 for every y R m is approximately π m -relate to R n, X. Proof. By assumption, control system R m, R k, F = T π m X is IUBIBSS. If we enote by y a trajectory of F we have: y v t, y y v2 t, y 2 γ y y 2 + γ 2 v v 2 +. In particular, we can take: y = y 2 = π m x, v = π k x, x, v 2 = 0, to get: π m xt, x yt, π m x = y πk xt,xt, π m x y 0 t, π m x = y v t, π m x y 0 t, π m x γ 2 v + = γ 2 π k x, x +. But it follows from fiberwise practical stability of X with respect to π k that: π k x, x γ π k x + c. We thus have: π m xt, x yt, π m x γ 2 γ π k x + c + γ 2 λ γ π k x + γ 2 λ 2 c +, for some constants λ, λ 2 R + 0. This conclues the proof since γ 2 λ γ is a class K function an γ 2 λ 2 c+ R + 0. Theorem shows that sufficient conitions for approximate reuction can be given in terms of ISS-like Lyapunov functions an how reuce system can be constructe. Before illustrating Theorem with several examples in the next section we present an important corollary. Corollary. Let R n, X an R m, Y be vector fiels satisfying the assumptions of Theorem. Then, for any compact set S R n there exists a δ > 0 such that for any x S an y π m S the following estimate hols: π m xt, x yt, y δ. Proof. Using the same proof as for Theorem, except picking y = π m x an y 2 = y, it follows that: π m xt, x yt, y γ π m x y + γ 2 λ γ π k x + γ 2 λ 2 c +. The boun δ is now given by: δ = max x,y S π m S γ π m x y + γ 2 λ γ π k x + γ 2 λ 2 c +

5 542 P. Tabuaa et al. / Systems & Control Letters Step 2: Fiberwise practical stability We now consier a compact set C invariant uner the ynamics an restrict our analysis to initial conitions in this set. Such a set can be constructe, for example, by taking {x R 2 V x c} for some positive constant c. Note that stability of 0 implies fibrewise stability on C since π m C is compact. Fig.. Ball in a rotating hoop. which is well efine since S π m S is compact. 5. Examples In this section, we consier examples that illustrate the usefulness of approximate reuction. 5.. Ball in the hoop As a first example we consier the ball in a rotating hoop with friction, as escribe in Chapter 2 of [9] an isplaye in Fig.. For this example there are the following parameters: m mass of the ball, R raius of the hoop, g acceleration ue to gravity, µ friction constant for the ball, an ξ angular velocity for the ball. The equations of motion are given by: ω = µ m ω + ξ 2 sin θ cos θ g R sin θ θ = ω 0 where θ is the angular position of the ball an ω is its angular velocity. If π ω : R 2 R is the projection π ω ω, θ = ω, then accoring to Proposition 2 there exists no vector fiel Y on R which is π ω -relate to X as efine by 0. However, we will show that Y ω = T ω,0 π ω X ω, 0 is approximate π ω -relate to X. In applying Theorem we follow three steps: We show that X is forwar complete; 2 We show that X is fiberwise practically stable; 3 We show that F = T π m X : R m R k R m is IUBIBSS; 4 We construct the reuce moel Y Step : Forwar completeness We use: V = 2 m R2 ω 2 + mgr cos θ 2 m R2 ξ 2 sin 2 θ as a Lyapunov function to show that 0 is stable. Note that V ω, θ = 0 for ω, θ = 0, 0 an V ω, θ > 0 for ω, θ 0, 0 provie that Rξ 2 < g, which we assume. Computing the time erivative of V we obtain: V = µr 2 ω 2 0, thus showing stability of 0 an forwar completeness Step 3: IUBIBSS We will show that: T ω,θ π ω X ω, θ = µ m ω + ξ 2 sin θ cos θ g R sin θ is IUBIBSS on C with θ seen as an input by proving the stronger property of IISS. Consier the function: U = 2 ω ω 2 2. Its time erivative is given by: [ U = ω ω 2 µ m ω ω 2 + ξ 2 sin θ cos θ g R sin θ ξ 2 sin θ 2 cos θ 2 + g ] R sin θ 2 µ m ω ω ω ω 2 ξ 2 sin θ cos θ g R sin θ ξ 2 sin θ 2 cos θ 2 + g R sin θ 2 µ m ω ω ω ω 2 L θ θ 2 = µ 2m ω ω µ 2m ω ω ω ω 2 L θ θ 2, where the secon inequality follows from the fact that ξ 2 sin θ cos θ g R sin θ is a smooth function efine on the compact set π θ C an is thus globally Lipschitz on π θ C since its erivative is continuous an thus boune on any compact convex set containing π θ C with Lipschitz constant L. We now note that the conition: ω ω 2 > 2mL µ θ θ 2 makes the secon term in negative from which we conclue the following implication: ω ω 2 > 2mL µ θ θ 2 U µ 2m ω ω 2 2 showing that U is an IISS Lyapunov function for 0 an thus concluing IUBIBSS Step 4: Construction of the reuce moel Accoring to Theorem the approximate reuction of 0 is given by: ω = µ m ω. Projecte trajectories of the full-orer system as compare with trajectories of the reuce system can be seen in Fig. 2; here

6 P. Tabuaa et al. / Systems & Control Letters Fig. 2. A trajectory of the full orer system re vs. a trajectory for the reuce system blue for R = 5, 0, 20, 40 from left to right an top to bottom, respectively. For interpretation of the references to colour in this figure legen, the reaer is referre to the web version of this article. µ = m = an ξ = 0.. Note that as R, the reuce system converges to the full-orer system or the full-orer system effectively becomes ecouple Penulum on a cart We now consier a penulum attache to a cart which is mounte to a spring see Fig. 3. For this example, there are the following parameters: M mass of the cart, m mass of the penulum, R length of the ro, k spring stiffness, g acceleration ue to gravity, friction constant for the cart, an b friction constant for the penulum. The equations of motion are given by: ẋ = v θ = ω v = M + m sin 2 θ m Rω 2 sin θ + mg sin θ cos θ kx v + br cos θ ω = RM + m sin 2 m Rω 2 sin θ cos θ θ m + Mg sin θ + kx cos θ + v cos θ + M b m R ω 2 where x is the position of the cart, v its velocity, θ is the angular position of the penulum an ω its angular velocity. If π x,v : R 4 R 2 is the projection: π x,v x, θ, v, ω = x, v an X is the vector fiel as efine in 2, the goal is to reuce X to R 2 by eliminating the θ an ω variables. Fig. 3. A graphical representation of the penulum on a cart mounte on a spring Steps an 2: Forwar completeness an fibrewise practical stability Forwar completeness an fibrewise practical stability of 2 are prove like in the previous example by noting that X is Hamiltonian for = b = 0 an using the Hamiltonian as a Lyapunov function V Step 3: IUBIBSS Control system F = T π x,v X x, θ, v, w, in which θ an ω are regare as inputs, is given by: F x, v, θ, ω = T π x,v X x, θ, v, ω = M + m sin 2 m Rω 2 sin θ kx θ + mg sin θ cos θ v + b R cos θ. 3

7 544 P. Tabuaa et al. / Systems & Control Letters Fig. 4. A projecte trajectory of the full-orer system re an a trajectory for the reuce system blue for = 0.00, 0.0, 0., from left to right an top to bottom, respectively. For interpretation of the references to colour in this figure legen, the reaer is referre to the web version of this article. To show that F is IUBIBSS we first rewrite 3 in the form: F x, v, θ, ω = m Rω 2 sin θ kx v m R ω cos θ M + m an consier the following IISS caniate Lyapunov function: U = 2m + M x x v v 2 2. Its time erivative is given by: U = m + M v v m R m + M ω 2 sin θ ω cos θ ω2 2 sin θ 2 + ω 2 cos θ 2 v v 2. Using an argument similar to the one use for the previous example, we conclue that: v v 2 2m RL θ, ω, ω θ 2, ω 2, ω 2, with L the Lipschitz constant of the function ω 2 sin θ ω cos θ, implies: U 2m + M v v 2 2, thus showing that X is IISS an in particular also IUBIBSS Step 4: Construction of the reuce moel The reuce moel Y x, v is given by: ẋ = Y x, v = T v x,0,v,0 π x,v X x, 0, v, 0 v =. v + kx M In orer to illustrate some of the interesting implications of approximate reuction, we compare the reuce system, Y, an the full-orer system, X, in the case when R = m = k = b = an M = 2. It follows that the equations of motion for the reuce system are given by the linear system: ẋ 0 x = v v 2, so we can completely characterize the ynamics of the reuce system: every solution spirals into the origin. This is in stark contrast to the ynamics of X see 2 which are very complex. The fact that X an Y are approximately relate, an more specifically Theorem, allows us to unerstan the ynamics of X through the simple ynamics of Y. To

8 P. Tabuaa et al. / Systems & Control Letters be more specific, because the istance between the projecte trajectories of X an the trajectories of Y is boune, we know that the projecte trajectories of X are essentially be spirals. Moreover, the friction constant will irectly affect the rate of convergence of these spirals. Examples of this can be seen in Fig. 4 where is varie to affect the convergence of the reuce system, an hence the full orer system. References [] R. Abraham, J. Marsen, T. Ratiu, Manifols, Tensor Analysis an Applications, in: Applie Mathematical Sciences, Springer-Verlag, 988. [2] D. Angeli, A Lyapunov approach to incremental stability properties, IEEE Transactions on Automatic Control [3] A.C. Antoulas, D.C. Sorensen, S. Gugercin, A survey of moel reuction methos for large-scale systems, Contemporary Mathematics [4] C.L. Beck, J. Doyle, K. Glover, Moel reuction of multiimensional an uncertain systems, IEEE Transactions on Automatic Control [5] A.M. Bloch, P.S. Krishnaprasa, J.E. Marsen, R. Murray, Nonholonomic mechanical systems with symmetry, Archive for Rational Mechanichs an Analysis [6] A. Bacciotti, L. Mazzi, A necessary an sufficient conition for bouneinput boune-state stability of nonlinear systems, SIAM Journal on Control an Optimization [7] A. Girar, G.J. Pappas, Approximate bisimulations for nonlinear ynamical systems, in: Proceeings of the 44th IEEE Conference on Decision an Control, Seville, Spain, [8] A. Girar, G.J. Pappas, Approximate bisimulation relations for constraine linear systems, Automatica [9] Jerrol E. Marsen, Tuor S. Ratiu, Introuction to Mechanics an Symmetry, 2n e., in: Texts in Applie Mathematics, vol. 7, Springer- Verlag, 999. [0] Giuseppe Marmo, Alberto Simoni, Bruno Vitale, Eugene J. Saletan, Dynamical Systems, John Wiley & Sons, 985. [] J.E. Marsen, A. Weinstein, Reuction of symplectic manifols with symmetry, Reports on Mathematical Physics [2] E.D. Sontag, Input to state stability: Basic concepts an results, in: P. Nistri, G. Stefani Es., Nonlinear an Optimal Control Theory, 2006, pp Electronically available at: sontag/. [3] P. Tabuaa, Approximate simulation relations an finite abstractions of quantize control systems, in: A. Bempora, A. Bicchi, G. Buttazzo Es., Hybri Systems: Computation an Control 2006, in: Lecture Notes in Computer Science, vol. 446, Springer-Verlag, Pisa, Italy, 2007, pp [4] P. Tabuaa, A. Ames, A. Julius, G.J. Pappas, Approximate reuction of ynamical systems, in: Proceeings of the 45th IEEE Conference on Decision an Control, San Diego, CA, December [5] P. Tabuaa, G.J. Pappas, Bisimilar control affine systems, Systems an Control Letters [6] A. van er Schaft, Symmetries an conservation laws for hamiltonian systems with inputs an outputs: A generalization of noether s theorem, Systems an Control Letters [7] A. van er Schaft, Equivalence of ynamical systems by bisimulation, IEEE Transactions on Automatic Control