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1、Stability of hybrid system limit cycles: application to the compass gait biped RobotIan A. Hiskens Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana IL 61801 USAAbstract Limit cycles are common in hybrid systems. However the non-smooth dynamics of su
2、ch systems makes stability analysis difficult. This paper uses recent extensions of trajectory sensitivity analysis to obtain the characteristic multipliers of non-smooth limit cycles. The stability of a limit cycle is determined by its characteristic multipliers. The concepts are illustrated using
3、a compass gait biped robot example. 1 Introduction Hybrid system are characterized by interactions between continuous (smooth) dynamics and discrete events. Such systems are common across a diverse range of application areas. Examples include power systems l, robotics 2, 3, manufacturing 4 and air-t
4、raffic control 5. In fact, any system where saturation limits are routinely encountered can be thought of as a hybrid system. The limits introduce discrete events which (often) have a significant influence on overall behaviour. Many hybrid systems exhibit periodic behaviour. Discrete events, such as
5、 saturation limits, can act to trap the evolving system state within a constrained region of state space. Therefore even when the underlying continuous dynamics are unstable, discrete events may induce a stable limit set. Limit cycles (periodic behaviour) are often created in this way. Other systems
6、, such as robot motion, are naturally periodic. Limit cycles can be stable (attracting), unstable (repelling) or non-stable (saddle). The stability of periodic behaviour is determined by characteristic (or Floquet) multipliers. A periodic solution corresponds to a fixed point of a Poincare map. Stab
7、ility of the periodic solution is equivalent to stability of the fixed point. The characteristic multipliers are the eigenvalues of the Poincare map linearized about the fixed point. Section 4 reviews the connection between this linearized map and trajectory sensitivities. Poincare maps have been us
8、ed to analyse the stability of limit cycles in various forms of hybrid systems. However calculation of the underlying trajectory sensitivities has relied upon particular system structures, see for example 7, 8, or numerical differencing, for example 6. This paper uses a recent generalization of traj
9、ectory sensitivity analysis 9 to efficiently detemine the stability of limit cycles in hybrid systems.A hybrid system model is given in Section 2. Section 3 develops the associated variational equations. This is followed in Section 4 by a review of stability analysis of limit cycles. Conclusions and
10、 extensions are presented in Section 5.2 ModelDeterministic hybrid systems can be represented by a model that is adapted from a differential-algebraic (DAE) structure. Events are incorporated via impulsive action and switching of algebraic equations, giving the Impulsive Switched (DAIS) modelwhere a
11、re dynamic states and are algebraic states; is the Dirac delta; is the unit-step function; ；; some elements of each will usually be identically zero, but no elements of the composite g should be identically zero; the are defined with the same form as g in (2), resulting in a recursive structure for
12、g; are selected elements of y that trigger algebraic switching and state reset (impulsive) events respectively; and may share common elements.The impulse and unit-step terms of the DAIS model can be expressed in alternative forms: Each impulse term of the summation in (1) can be expressed in the sta
13、te reset formwhere the notation denotes the value of x just after the reset event, whilst and refer to the values of x and y just prior to the event. The contribution of each in (2) can be expressed aswith (2) becomingThis form is often more intuitive than (2). It can be convenient to establish the
14、partitionswhere are the continuous dynamic states, for example generator angles, velocities and fluxes; z are discrete dynamic states, such as transformer tap positions and protection relay logic states; are parameters such as generator reactances, controller gains and switching times.The partitioni
15、ng of the differential equations f ensures that away from events, evolves according to, whilst z and remain constant. Similarly, the partitioning of the reset equations ensures that and remain constant at reset events, but the dynamic states z are reset to new values given by .The model can capture
16、complex behaviour, from hysteresis and non-windup limits through to rule-based systems l. A more extensive presentation of this model is given in 9. Away from events, system dynamics evolve smoothly according to the familiar differential-algebraic modelwhere g is composed of together with appropriat
17、e choices of or , depending on the signs of the corresponding elements of yd. At switching events (2),some component equations of g change. To satisfy the new g = 0 equation, algebraic variables y may undergo a step change. Reset events (3) force a discrete change in elements of x. Algebraic variabl
18、es may also step at a reset event to ensure g = 0 is satisfied with the altered values of x.The flows of and y are defined respectively aswhere x(t) and y(t) satisfy (l),(2), along with initial conditions,3 Ikajectory Sensitivities Sensitivity of the flows and to initial conditions are obtained by l
19、inearizing (8),(9) about the nominal trajectory,The time-varying partial derivative matrices given in (12),(13) are known as trajectory sensitiuities, and can be expressed in the alternative formsThe form , provides clearer insights into the development of the variational equations describing the ev
20、olution of the sensitivities. The alternative form, highlights the connection between the sensitivities and the associated flows. It is shown in Section 4 that these sensitivities underlie the linearization of the Poincare map, and so play a major role in determining the stability of periodic soluti
21、ons. Away from events, where system dynamics evolve smoothly, trajectory sensitivities and are obtained by differentiating (6),(7) with respect to .This giveswhere , and likewise for the other Jacobian matrices. Note that are evaluated along the trajectory, and hence are time varying matrices. It is
22、 shown in 19, 101 that the numerical solution of this(potentially high order) DAE system can be obtained as a by-product of numerically integrating the original DAE system (6),(7). The extra computational cost is minimal. Initial conditions for are obtained from (10) aswhere I is the identity matrix
23、. Initial conditions for follow directly from (17),Equations (16),(17) describe the evolution of the sensitivities and between events. However at an event, the sensitivities are generally discontinuous. It is necessary to calculate jump conditions describing the step change in and . For clarity, con
24、sider a single switching/reset event, so the model (1),(2) reduces (effectively) to the formLet （） be the point where the trajectory encounters the triggering hypersurface s(x,y) = 0, i.e., the point where an event is initiated. This point is called the junction point and r is the junction time. It
25、is assumed the encounter is transversal. Just prior to event triggering, at time , we haveSimilarly, are defined for time , just after the event has occurred. It is shown in 9 that the jump conditions for the sensitivities are given byThe assumption that the trajectory and triggering hypersurface me
26、et transversally ensures a non-zero denominator for The sensitivities . immediately after the event are given byFollowing the event, i.e., for , calculation of the sensitivities proceeds according to (16),(17) until the next event is encountered. The jump conditions provide the initial conditions fo
27、r the post-event calculations.4 Limit Cycle AnalysisStability of limit cycles can be determined using Poincare maps 11, 12. This section provides a brief review of these concepts, and establishes the connection with trajectory sensitivities. A Poincark map effectively samples the flow of a periodic
28、system once every period. The concept is illustrated in Figure 1. If the limit cycle is stable, oscillations approach the limit cycle over time. The samples provided by the corresponding Poincare map approach a fixed point. A non-stable limit cycle results in divergent oscillations. For such a case
29、the samples of the Poincare map diverge. To define a Poincare map, consider the limit cycle shown in Figure 1. Let be a hyperplane transversal to at . The trajectory emanating from will again encounter at after T seconds, where T is the minimum period of the limit cycle. Due to the continuity of the
30、 flow with respect to initial conditions, trajectories starting on in a neighbourhood of . will, in approximately T seconds, intersect in the vicinity of . Hence and define a mappingwhere is the time taken for the trajectory to return to. Complete details can he found in 11,12. Stability of the Pain
31、care map (22) is determined by linearizing P at the fixed point , i.e.,From the definition of P(z) given by (22), it follows that DP() is closely related to the trajectory sensitivities . In fact, it is shown in 11 thatwhere is a vector normal to .The matrix is exactly the trajectory sensitivity mat
32、rix after one period of the limit cycle, i.e., starting from and returning to . This matrix is called the Monodromy matrix .It is shown in 11 that for an autonomous system, one eigenvalue of is always 1, and the corresponding eigenvector lies along The remaining eigenvalues of coincide with the eige
33、nvalues of DP(), and are known as the characteristic multipliers mi of the periodic solution. The characteristic multipliers are independent of the choice of cross-section . Therefore, for hybrid systems, it is often convenient to choose as a triggering hypersurface corresponding to a switching or r
34、eset event that occurs along the periodic solution.Because the characteristic multipliers mi are the eigenvalues of the linear map DP(x*), they determine the stability of the Poincarb map P(), and hence the stability of the periodic solution. Three cases are of importance: 1. All lie within the unit
35、 circle, i.e., ,.The map is stable, so the periodic solution is stable. 2. All lie outside the unit circle. The periodic solution is unstable.3. Some lie outside the unit circle. The periodic solution is non-stable. Interestingly, there exists a particular cross-section , such thatwhere.This cross-s
36、ection is the hyperplane spanned by the n - 1 eigenvectors of that are not aligned with . Therefore the vector that is normal to is the left eigenvector of corresponding to the eigenvalue 1. The hyperplane is invariant under , i.e., maps vectors back into .5 Conclusions Hybrid systems frequently exh
37、ibit periodic behaviour. However the non-smooth nature of such systems complicates stability analysis. Those complications have been addressed in this paper through application of a generalization of trajectory sensitivity analysis. Deterministic hybrid systems can be represented by a set of differe
38、ntial-algebraic equations, modified to incorporate impulse (state reset) action and constraint switching. The associated variational equations establish jump conditions that describe the evolution of sensitivities through events. These equations provide insights into expansion/contraction effects at
39、 events. This is a focus of future research. Standard Poincar6 map results extend naturally to hybrid systems. The Monodromy matrix is obtained by evaluating trajectory sensitivities over one period of the (possibly non-smooth) cyclical behaviour. One eigenvalue of this matrix is always unity. The r
40、emaining eigenvalues are the characteristic multipliers of the periodic solution. Stability is ensured if all multipliers lieReferences l LA. Hiskens and M.A. Pai, “Hybrid systems view of power system modelling,” in Proceedings of the IEEE International Symposium on Circuits and Systems, Geneva, Swi
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