Limit Cycle Analysis in a Class of Hybrid Systems

Hybrid systems are those that inherently combine discrete and continuous dynamics.This paper considers the hybrid systemmodel to be an extension of the discrete automata associating a continuous evolution with each discrete state. This model is called the hybrid automaton. In this work, we achieve a mathematical formulation of the steady state and we show a way to obtain the initial conditions region to reach a specific limit cycle for a class of uncoupled and coupled continuous-linear hybrid systems. The continuous-linear term is used in the sense of the system theory and, in this sense, continuous-linear hybrid automata will be defined.Thus, some properties and theorems that govern the hybrid automata dynamic behavior to evaluate a limit cycle existence have been established; this content is explained under a theoretical framework.


Introduction
Many system structure changes lead to discontinuities in their dynamics.These changes may be caused by discrete events generated by discrete actuators, sensors, or inherent process discontinuities.In general, systems where continuous and discrete components interact are called hybrid systems.This interaction reflects the compositional properties underlying the hybrid system model [1].Several mathematical models have been proposed for hybrid systems with different modeling capabilities and different purposes [2][3][4][5][6].In this paper, a continuous dynamic is represented by a differential equation set and the discrete dynamic by a finite state machine.This model is called the hybrid automaton and was introduced by [7] that is probably the most powerful model.In this work some properties are determined for hybrid automata.Using reachability properties we have provided an analytical formulation of the steady state behavior that gives the necessary and sufficient conditions to reach a limit cycle in a continuous-linear hybrid system.Herein, the continuouslinear term is used in the sense of the system theory and in this sense continuous-linear hybrid automata will be defined.The framework used here is introduced in [7].This paper is organized as follows.Section 1 is an introduction to the topic; Section 2 presents the hybrid automaton model, defines different subclass of hybrid automata, and presents some definitions and properties for discrete transition enabling in the hybrid dynamic evolution.Section 3 briefly presents a reachability analysis of uncoupled case and provides the necessary conditions to obtain a specific limit cycle.In Section 4 a coupled case will be conducted and presents a detailed analysis of a limit cycle existence for the coupled case presented.Finally, Section 5 presents a brief summary and some issues for further research.

Theory about Hybrid Automata
In the discrete event systems analysis of hybrid systems, two important approaches are established: the automata approach [8] and the Petri net approach (PN).In this work the automata approach is considered.Hybrid automata take their origin from the finite automaton model [9].This basic model shows only commutation logic, that is, the discrete event sequences.With this model we can only express, for example, the event sequence generated until a marked state is reached, starting from an initial state.This model is simple but insufficient when time has to be taken into account.Therefore, to enrich the previous model, the timed (finite) automaton was proposed by [10].Thus, a timed automaton is a finite automaton with a finite set of real-valued clocks.
The clocks can be reset to 0 (independently of each other) with the transitions of the automaton and keep track of the time elapsed since the last reset.It provides a simple way to annotate state-transition graphs with timing constraints.On the other hand, the timed automata model is not enough if we wish to model not only the time evolution of the discrete transitions, but also the process variable continuous evolution.Therefore, if we wish to model the continuous evolution of a process model, a hybrid automata model is suggested.

Hybrid Automata Model.
A hybrid automaton is a formal model for a dynamic system with discrete and continuous components (Figure 1).The vertices of a graph, called locations or control modes, model the state of the discrete system, and the arcs, called transitions or control switches, model the discrete dynamics.Thus, the hybrid automaton model (as an extension of the finite automaton) is a bipartite graph, that is, locations (discrete states usually represented by circles) and oriented arcs (transitions represented by arrows).It is compulsory for each oriented arc to have a location at its end.Therefore, locations are connected by arcs.The number of locations is finite and nonzero.The number of arcs is also finite and nonzero.The state of the continuous system is modeled by points in   and, in each location, its continuous dynamic is modeled by flow conditions such as differential equations.
(iii) flow: flow conditions: a labeling function flow that assigns a flow condition flow(  ) to each location   ∈ , where the continuous dynamic is usually represented by differential equations.
(iv) : transitions, a finite set  of discrete jumps (edges) called transitions and represented by arcs.Each transition   = (  ,   ) identifies a source location and a target location.
(v) : jump conditions, a labeling function (  ) that assigns a jump condition to each transition   ∈ .
The jump conditions relate the values of the variables and their tangents before a (discrete) transition with those after a transition.They also provide the conditions to fire a transition in general as boolean combinations of inequalities.
(vi) ∑: events, a finite set of events which made up ∑ and a labeling function which is assigned to each event in ∑ (including empty event) to every transition   ∈ .The events can be used to define and model the parallel composition of hybrid automata.
(vii) init: initial conditions, a labeling function init(  ) that assigns an initial condition to each location   ∈ .In the automata graph representations, initial conditions appear as labels on incoming arrows without a source location.
From the field of computer science, there are algorithmic techniques for checking certain properties of linear hybrid automata [7,12].These automata have linearity restrictions on transitions (linear inequalities between sources and targets of jumps) and continuous flows of variables (differential inequalities ẋ ≥ ).The definition of linearity used there is more restrictive than in system theory.For instance, linear hybrid automata cannot directly model continuous flows of the form ẋ = .

Definition 2 (linear hybrid automaton). A hybrid automaton
is a linear hybrid automaton   if (1) all variables of  are linear, (2) for all locations   ∈ , the initial condition init(  ) and the flow condition flow(  ) are all convex, (3) for all transition   ∈ , the jump condition (  ) is convex.
It is important to emphasize in the present work that the continuous-linear term is used in the sense of the system theory.Therefore, this paper considers the continuouslinear hybrid automaton to be a general model to represent continuous-linear dynamics in hybrid systems.On the other hand, as in the case of linear hybrid automata, continuouslinear hybrid automata have the same linearity restrictions on discrete transitions.Thus in this sense, continuous-linear hybrid automata is defined.As the majority of systems in control theory are modeled by dynamics using coupled differential equations, we have expressed mathematically the continuous coupled flow equations by means of a state space representation.Thus, in this sense, we will consider hybrid systems with continuous flows to be modeled as one dimensional first-order differential equation, as defined by [2,7] ẋ =  (, , ) , where  is the continuous state vector taking values in some subset of the Euclidean space and (, , ) is a controlled vector field.We consider the state vector () ∈  ⊆   and the control input vector () ∈  ⊆   .The size  of  is called the automaton dimension.The size  of  is the control input size.Accordingly we define (, , ) = [ 1 (, , )  2 (, , ) ⋅ ⋅ ⋅   (, , )]  . ( In hybrid systems, continuous-linear flows can be modeled considering in (2) (, , ) =   ()() +   ()() to be the controlled vector field (in all   locations), where   () ∈  × ,   () ∈  × , and the automaton dimension  is also called the dimension of   ().
Definition 3 (continuous-linear hybrid automata).A hybrid automaton  is a linear hybrid automaton   if (1) for all locations   ∈ , the initial condition init(  ) is convex, (2) the flow condition flow(  ) is continuous-linear, (3) for all transition   ∈ , the jump condition (  ) is convex.
Thus, a continuous-linear hybrid automaton   is characterized by a set  of locations   , by oriented arcs   ∈  called transitions, by convex relations in the initial and jump conditions, and by a continuous-linear flow equations usually expressed as ẋ () =   ()() +   ()().From Figure 2, we will obtain a continuous-linear hybrid automaton   replacing the flow condition ẋ () =   , ∀ ∈ {1, 2}, by ẋ () =   ()  () +   ()  () . ( Definition 4 (time invariant hybrid automata).A hybrid automaton  is a time invariant hybrid automaton   if (1) for all locations   ∈ , the initial condition init(  ) and the flow condition flow(  ) are all time invariant and (2) for all transitions   ∈ , the jump condition (  ) is time invariant.
Remark 5. Further after, we will call continuous-linear time invariant hybrid automaton   to all hybrid automaton that satisfies Definitions 3 and 4. In this case, the vector field in (2) becomes (, ).
Definition 6 (uncoupled hybrid automata).A hybrid automaton  is uncoupled hybrid automaton   if (1) for all locations   ∈ , the flow condition flow(  ) is uncoupled; that is, the variable  is flow independent and the vector field in ( 2) is given by and for all transition   ∈ , the jump condition (  ) is function of only one state variable.
Remark 7. Hence, we will call uncoupled continuous-linear time invariant hybrid automaton   all hybrid automata that satisfy Definitions 3, 4, and 6.In this case, for all locations   ∈   matrix in the vector field is diagonal.
In consequence, an automaton can be considered a coupled continuous-linear hybrid automaton if it is represented by at least one nondiagonal matrix   for any   ∈  and/or at least two different state variables in any jump condition predicate.

Limit Cycle Analysis of Uncoupled Hybrid Systems
In this section, our goal is to establish some definitions, properties, and theorems concerning limit cycle existence and uniqueness of the hybrid automata   .To reach a limit cycle in a dynamic behavior of an automaton, we must consider that at least one closed loop trajectory exists in the automaton model.

Basic Concepts.
In general, the analysis of continuous evolution of a hybrid system is not the main problem.The difficulty stems from the discrete behavior interaction.It is necessary to model the commutation instants as well as considering the continuous state variables evolution.This leads us to an important notion in hybrid system modeling that is called the transition enabling.Consider the automaton partial structure where   location has a single input and a single output transition, as in Figure 3.
In each   location, differential equation models flow(  ) for a continuous dynamic evolution for a fixed horizon defined in a closed interval   = [0,   ].
Definition 8 (residence time).For all hybrid automata , a residence time   is defined as the amount of time spent in location   before a transition   fires.Property 1 (transition enabling).For all hybrid automata being in location   ,   transition is enabled at  = 0 input instant, if and only if a finite time   ∈  ≥0 solution exists.This   solution is a function of the initial condition (initial point), the jump condition (final point), and the differential equations (the trajectory) defined for location   .
A first particular interesting analysis case exists when the dynamic behavior of the continuous-linear time invariant hybrid system is uncoupled; therefore the hybrid automaton model considers diagonal   matrices and () a constant control signal, ∀  ∈ .
Hypothesis 1.Let us consider a hybrid automaton   the dynamic representation of uncoupled continuous-linear time invariant hybrid systems.In this representation we suppose First, for each   location (Figure 4), let the jump condition be a simple predicate ,  ∈   , and let   be a point in some region   ⊆ .Now, let   be   () value at   transition instant.
The state vector evolution (); ∀ ∈   is given by For  = , we obtain, where ∀ ∈ [1, ], Herein,   and   are, respectively, th and th constant components of   and   vectors.Hence solving (8), we can calculate   residence time.Therefore, if   finite real positive solution of (8) exists, then   transition is enabled.It must be noted that, in the general case (7), that is, for any   matrix, it is difficult to find an explicit solution and in the majority of cases, only   numerical solution is possible.Property 2. The residence time   in all locations   ∈  is determined by where This last result could be obtained by solving (8) Theorem 9. Given a hybrid automaton   with   =   input (initial) condition and   =   jump condition predicate of   (output) transition, the necessary and sufficient condition to enable   is given by the following logical equations: The proof, from the above relations, is a simple one and it is based on the fact that for enabling a transition   it is enough that ∃  |   ∈  ≥0 .It must be noted in (9) that the equality   =   gives the lower bound of residence time (  = 0) in location   .On the other hand, a value   =   where   ̸ =   , will produce the upper limit of residence time (  = inf).In this last case,   transition will not be enabled; therefore   will be a sink location.For reachability analysis, we consider a region in the state space   as initial condition and a region   as jump condition of   transition.  and   are two points in   and   regions, respectively.The final region   is reachable from the initial region   ,   →   , if for each point   ∈   in the final region, there is at least one point   ∈   that satisfy the logical equations (10).
Figure 5:   hybrid automaton with a closed loop structure.
Hypothesis 2. Based on the Hypothesis 1, let us consider  closed loop trajectory obtained from automaton locations (Figure 5).
Theorem 10.For each  closed loop trajectory in   hybrid automaton, the necessary condition to reach a limit cycle is given by the logical expression: In this case where Considering all subindices (+) = ; ∀ ≥ 0, due the closed loop configuration.Theorem 11.For  closed loop structure   hybrid automaton (Figure 5), the initial condition point to reach a limit cycle must belong to the initial region defined by x(0) ẋ = −0.1x+ 5 Figure 6: Thermostat hybrid automaton model.
As we can see in Example 12, the limit cycle reachability is function of the automaton jump conditions as well as its initial condition region.Therefore, (11) and ( 14) enable us to evaluate in a rapid, easy, and direct way if () hybrid dynamics will reach a limit cycle.
Example 12.   hybrid automaton of Figure 6 models a thermostat that controls the temperature, (), of a heating system.The task is to evaluate its reachability of a limit cycle for () and find the region of initial conditions that make it possible.
Using (11) for   < 0,  ∈ {1, 2},  = 2,  = 0 (simplifying index notations because there is only one continuous variable   () = ()) and again considering index zero equal to , we get the necessary condition to reach the limit cycle (Theorem 10): x * 50 K 2  with  1 = 50,  2 = 0; we obtain (0 <  2 ≤  1 < 50) = 1.Therefore, it is possible to reach a limit cycle if the previous inequality holds true.Then, the region that defines the initial condition is given by

Limit Cycle Analysis for a Coupled Case
The fact of having the automaton jump condition predicates related to two different state variables gives a certain coupling between them and consequently a more difficult analytical solution.Therefore for this case,   variables in the jump condition predicates will not be solvable in the solvability sense defined in [7] and as a result a more difficult reachability analysis.Our goal in this section is then to provide some theorems and properties to reach a limit cycle for this coupled case.
Hypothesis 3. Let Figure 9 be   location of two dimensions for   hybrid automaton.In this coupled case for  1 = 0,  2 ̸ = 0, where Property 3. The discrete dynamic relation between  2 and  1 at each   transition instant is given by for  1 = 0,  2 ̸ = 0,   Remark 13.The existence and uniqueness of a limit cycle are concluded from the convergence analysis of   values at commutation instants ∀ ∈  and  ∈ {1, 2} according to the fixed point theorem.
Theorem 14 (about the existence of a limit cycle).For  locations closed loop  ∈   with jump conditions predicate   () =   |  ∈ {1, 2, . . ., }, the necessary condition to obtain a limit cycle is given by This result is a natural extension of (11) for the case of  locations and  commutation variables.Therefore, the existence of a limit defined, by (11), must be true for each commutation variable that gives the logical product of (20).Property 5.   convergence is assured by ∀ ∈ {1, 2, . . ., } and ∀ ∈ {1, 2, . . ., } if in the hybrid system behavior, the state vector trajectory enters and remains inside the state space region defined by (20).In this case, the limit cycle will be uniquely defined for a system region of initial conditions.For each particular case, this region of initial conditions could be found by means of a system phase portrait (isoclines) analysis considering the transitions enabling conditions imposed by (20).A geometric signification of (20) and Property 5 will be conducted in Section 5. Property 6.If Theorem 14 has been satisfied, we can find an initial condition region to reach a limit cycle replacing Mathematical Problems in Engineering x(0) Remark 15.Thus, we must note that a sufficient condition to reach a limit cycle is given by Theorem 14 plus the initial condition region found from Property 6.
Property 7.For the case where the diagonal matrix   is stable ∀  ∈  and jump condition predicates   () =   |  ∈ {1, 2, . . ., }, the () values are bounded.For the limit cycle case, the bounds are defined by ( 20) and for another case by the sink location equilibrium point.
The previous analysis procedure can be applied to automata for -location case.In Section 5, we present an analysis of the two-location case.A geometric interpretation of results is then conducted.
One has () = [ 1 ()   , ( 1 ) ⇒  1 () =  11 , and ( 2 ) ⇒  2 () =  22 . 11 and  22 are real constants of known magnitude.So without loss of generality, because the analysis procedure is the same, we will consider stable continuous equations (  < 0 ∀  |  ∈ {1, 2} and ∀ ∈ {1, 2}).According to Property 5, to reach a limit cycle from a given initial condition region, the state vector dynamic must always enter and remain in a state space region (transition enabling conditions) defined by where and it gives 4 bounded regions of the state space, where a cyclic dynamic will take place, as in Figure 11.
In Figure 11, the trajectory attraction points ẋ * =  1 and  * =  2 (equilibrium points) for locations  1 and  2 , respectively, are shown.Thus, a unique limit cycle is assured if the system dynamic starts from an initial state space region that makes that () enters the shadow zone and remains inside.Hence, it can be seen in this case that the shadow zone respects the invariant principle and consequently all initial point from this zone will produce a limit cycle.According to Property 6, a sufficient initial condition in the state space region to reach a unique limit cycle is obtained replacing by [ 1 (0)  2 (0)]  in (21).Because  22 is a known constant, we obtain The necessary and sufficient initial condition region could be found using the automaton trajectory equations, that is, from its phase portrait analysis.In this particular case, the following relations give the initial condition region: where relations ⊳ = (<, >) and are defined according to the (known) automaton relations  11 ⊳ 1  11 and  22 ⊳ 2  22 .This result will be obtained by the automaton phase portrait analysis using trajectories equations (18), still under the restrictions imposed by (21).Now, a limit cycle existence can be determined the  12 and  21 convergence analysis (Remark 13).Its dynamic behaviors are governed by (Property 3) Considering  12 =  1 ( 21 ) and  21 =  2 ( 12 ) from (25), Figure 12 shows  1 ( 21 ) and the curves generated under restrictions imposed by (21).Herein, we can observe the 4 possible cases to reach a limit cycle.Additionally, in this particular case we will note that the automaton dynamic behavior will reach one of two possible fixed points in  12 −  21 space.The limit cycle existence and uniqueness demonstration can be done starting from  12 and  21 fixed point convergence analysis.This is the formal method that proves the cyclic steady state convergence.
Therefore, from (25), by means of  12 and  21 and a fixed point analysis, we find two convergence possibilities.possibilities: first, the possibility of having only one fixed point (Figure 12) and second the possibility of having two.In case of one stable fixed point the residence time in each location is 0 <   < inf, ∀  ∈ , and for an unstable point, a boundary case, where  12 =  22 and  21 =  11 , the residence times are zero; that is,   = 0, ∀  ∈ . . ( By mean of preceding terms analysis (restricted of limit cycle condition (20)), we find always |  | < 1, ∀ ∈ {1, 2, 3, 4}.Thus, we can guarantee the fixed point existence and uniqueness assuring the constant value || < 1, but this fixed point produces a zero period limit cycle (Figures 12(a), 12(b), 12(c), and 12(d)).Therefore, in general, the desirable solution to avoid the zero period limit cycle is one which gives two fixed points (Figures 12(a     (b) For the same jump conditions, find the necessary and sufficient initial region in  1 that will give a limit cycle system behavior.
(c) Design a 10-second limit cycle period by considering steady state residence times of  1 = 6 sec.and  2 = 4 sec.Find  11 and  22 values as well as the initial condition region that guarantees it.
(d) For  22 = 8, determine the state space region of  11 values to allow obtaining a nonzero period limit cycle.
For this example, the trajectory attraction (equilibrium) points corresponding to each location are given by Thus according to Property 5, to guarantee the limit cycle existence and uniqueness, the hybrid state vector evolution must enter and remain in the state space region defined by Because the necessary and sufficient conditions are satisfied, we can assure the limit cycle existence and uniqueness.As in part (b), the necessary and sufficient initial condition region to guarantee the limit cycle existence (24) are given by System simulations (using MATLABÝ) for different initial conditions are shown in Figure 13.
In Figure 13(a) the system dynamic starts from a point in the resulting initial condition region and the system reaches a limit cycle.Otherwise, for the boundary case, Figure 13(b), where  1 (0) = 14 and  2 (0) = 2 + 6[4 − 0.2 1 (0)] 5 = 16.9, the automaton dynamic behavior will reach its second (unstable) fixed point.In this case, the residence time for the found locations will be equal to zero and  12 =  22 and  21 =  11 .
(36) Figure 14 shows system simulations using the above results.Therein, how the period specifications are obtained should be observed.
Finally, using the results obtained from recursive equations  12 and  21 fixed point analysis, to assure the existence of two fixed points, it is necessary that  > 1 (sufficient condition).Therefore and also, to assure the limit cycle existence and uniqueness, initial state variables values must be an element of the following initial condition region:

Conclusion
The main theoretical results in this paper are concerned to establish limit cycle properties for uncoupled and a class of coupled hybrid systems.In this way, we have provided an analytical formulation of the steady state behavior that gives the necessary and sufficient conditions to reach a limit cycle.
We have established that when the hybrid automaton has a cyclic structure and when the steady state exists, there will be a transient behavior followed by a steady state.This transient behavior depends on the automaton initial condition region as well as its jump conditions.The limit cycle in the steady state behavior is obtained from an initial region and is independent of the initial point within the region.In this paper we also have shown a way to design a specific limit cycle for a coupled hybrid system case.The sufficient and necessary conditions have been formulated.One goal of current research is to extend the analysis to a more general x 2 (t) hybrid dynamic evolution x 1 (t) hybrid dynamic evolution x 2 (t) x 1 -x 2 space (a) x 2 (t) hybrid dynamic evolution class of coupled linear hybrid automata with more general jump conditions.This will extend the model applicability to a wide process class of dynamic hybrid systems.

Figure 3 :
Figure 3: SISO transition location of a hybrid automaton.

Figure 4 :
Figure 4: SISO transition location of a   hybrid automaton.

Figure 9 :
Figure 9: An automaton location for a coupled case.
), 12(b), 12(c), and 12(d)).So we must assure the relation || > 1; that is  > 1 (because in the four regions defined in Figure11,  is positive).Then, choosing adequately  11 and  22 values to achieve  > 1, we can assure the discrete values convergence toward the stable fixed point for a defined initial condition region.Example 16.Consider the hybrid automaton from Figure10.Therein, a continuous dynamic behavior is modeled by
at   residence time.For   ∈  ̸ =0 it must be noted that the corresponding equilibrium point  *  ∈  to   is given by relation  *  =   = −  /  .In general ∀  ∈  and for this uncoupled case, the equilibrium point of () in the Euclidean space   is given by Hypothesis 4. Let  be a closed loop obtained by  single output locations   ∈  |  ⊆  of   hybrid automaton.In the jump condition predicates, the enabled   transitions are function of   (), for  ∈ {1,2}.It is important to note that in each location   there are two unknown discrete variables  1 and  2 at input and output commutation instants, respectively.In general, for each location   we have one equation obtained from (19), with two unknowns ( 1 and  2 ) that will determine (by mean of its convergence analysis) the existence of a limit cycle.
This is an analytical result obtained by finding the commutation point in the state space trajectory equation (18), at   transition firing instant.Now, for limit cycle analysis we must consider that there exists a closed loop  obtained from   hybrid automaton with jump condition predicates function of  1 or  2 state variables.Property 4. For all closed loops  ∈   , dynamic relations among state variables in discrete commutation instants are governed by a set of equations obtained from (19), ∀  ∈  and  ∈ {1, 2, . . ., }.Thus, in general, in  closed loop structure we will have the same number of equations (and unknowns) as locations in the loop.