# Limit Cycle Analysis in a Class of Hybrid Systems

Authors: Antonio  Favela-Contreras; Francisco Beltrán-Carbajal; Alejandro Piñón; Angelo Raimondi
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

Hybrid systems are those that inherently combine discrete and continuous dynamics. This paper considers the hybrid system model 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.


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## Body

## 1. 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 [2–6]. 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 continuous-linear 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.


## 2. 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.


### 2.1. Hybrid Automata Model

A hybrid automaton is a formal model for a dynamic system with discrete and continuous components (Figure1). 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 Rr and, in each location, its continuous dynamic is modeled by flow conditions such as differential equations.Figure 1
Representation of hybrid automaton.Definition 1 (hybrid automata).
Based on [8, 11], we consider a hybrid automaton H=(X,S,flow,E,F,∑‍,init) consists of the following: (i)
X: variables, a finite ordered set of real-valued variables. The size r of X is called the automaton dimension.(ii)
S: locations: a finite set of n locations S={s1,s2,…,sn}.(iii)
flow: flow conditions: a labeling function flow that assigns a flow condition flow(si) to each location si∈S, where the continuous dynamic is usually represented by differential equations.(iv)
E: transitions, a finite set E of discrete jumps (edges) called transitions and represented by arcs. Each transition Ti=(si,sj) identifies a source location and a target location.(v)
F: jump conditions, a labeling function F(Ti) that assigns a jump condition to each transition Ti∈E. 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 eachevent in ∑‍ (including empty event) to every transition Ti∈E. The events can be used to define and model the parallel composition of hybrid automata.(vii)
init: initial conditions, a labeling function init(si) that assigns an initial condition to each location si∈S. 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 x˙≥B). 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 x˙=Ax.Definition 2 (linear hybrid automaton).
A hybrid automatonH is a linear hybrid automaton HL if (1)
all variables ofH are linear,(2)
for all locationssi∈S, the initial condition init(si) and the flow condition flow(si) are all convex,(3)
for all transitionTi∈E, the jump condition F(Ti) is convex.For example, in the hybrid automatonH of Figure 2, we obtain a linear hybrid automaton HL of dimension 2 setting r=2, ∀i∈{1,2}:(1)inits1⟹x0=x10x20Tflowsi⟹x˙t=Bi,FTi⟹Cixt≥Ki considering the constant values x(0),Bi,Ki∈R2 and Ci∈R2×R2.Figure 2
Two-location linear hybrid automaton.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 continuous-linear 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, continuous-linear 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 oner-dimensional first-order differential equation, as defined by [2, 7](2)x˙=ft,x,u,where x is the continuous state vector taking values in some subset of the Euclidean space and f(t,x,u) is a controlled vector field. We consider the state vector x(t)∈X⊆Rr and the control input vector u(t)∈U⊆Rm. The size r of x is called the automaton dimension. The size m of u is the control input size. Accordingly we define(3)x=x1x2⋯xrT;u=u1u2⋯umT;ft,x,u=f1t,x,uf2t,x,u⋯frt,x,uT.In hybrid systems, continuous-linear flows can be modeled considering in (2) f(x,u,t)=Ai(t)x(t)+Bi(t)u(t) to be the controlled vector field (in all si locations), where Ai(t)∈Rr×r, Bi(t)∈Rr×m, and the automaton dimension r is also called the dimension of Ai(t).Definition 3 (continuous-linear hybrid automata).
A hybrid automatonH is a linear hybrid automaton HCL if (1)
for all locationssi∈S, the initial condition init(si) is convex,(2)
the flow conditionflow(si) is continuous-linear,(3)
for all transitionTi∈E, the jump condition F(Ti) is convex.Thus, a continuous-linear hybrid automatonHCL is characterized by a set S of locations si, by oriented arcs Ti∈E called transitions, by convex relations in the initial and jump conditions, and by a continuous-linear flow equations usually expressed as x˙(t)=Ai(t)x(t)+Bi(t)u(t). From Figure 2, we will obtain a continuous-linear hybrid automaton HCL replacing the flow condition x˙(t)=Bi,∀i∈{1,2}, by(4)x˙t=Aitxt+Bitut.Definition 4 (time invariant hybrid automata).
A hybrid automatonH is a time invariant hybrid automaton HI if (1)
for all locationssi∈S, the initial condition init(si) and the flow condition flow(si) are all time invariant and(2)
for all transitionsTi∈E, the jump condition F(Ti) is time invariant.Remark 5.
Further after, we will call continuous-linear time invariant hybrid automatonHCLI to all hybrid automaton that satisfies Definitions 3 and 4. In this case, the vector field in (2) becomes f(x,u).Definition 6 (uncoupled hybrid automata).
A hybrid automatonH is uncoupled hybrid automaton HD if (1)
for all locationssi∈S, the flow condition flow(si) is uncoupled; that is, the variable x is flow independent and the vector field in (2) is given by(5)ft,x,u=f1t,x1,uf2t,x2,u⋯frt,r,uTand for all transition Ti∈E, the jump condition F(Ti) is function of only one state variable.Remark 7.
Hence, we will call uncoupled continuous-linear time invariant hybrid automatonHCLID all hybrid automata that satisfy Definitions 3, 4, and 6. In this case, for all locations si∈SAi 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 Ai for any si∈S and/or at least two different state variables in any jump condition predicate.


## 2.1. Hybrid Automata Model

A hybrid automaton is a formal model for a dynamic system with discrete and continuous components (Figure1). 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 Rr and, in each location, its continuous dynamic is modeled by flow conditions such as differential equations.Figure 1
Representation of hybrid automaton.Definition 1 (hybrid automata).
Based on [8, 11], we consider a hybrid automaton H=(X,S,flow,E,F,∑‍,init) consists of the following: (i)
X: variables, a finite ordered set of real-valued variables. The size r of X is called the automaton dimension.(ii)
S: locations: a finite set of n locations S={s1,s2,…,sn}.(iii)
flow: flow conditions: a labeling function flow that assigns a flow condition flow(si) to each location si∈S, where the continuous dynamic is usually represented by differential equations.(iv)
E: transitions, a finite set E of discrete jumps (edges) called transitions and represented by arcs. Each transition Ti=(si,sj) identifies a source location and a target location.(v)
F: jump conditions, a labeling function F(Ti) that assigns a jump condition to each transition Ti∈E. 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 eachevent in ∑‍ (including empty event) to every transition Ti∈E. The events can be used to define and model the parallel composition of hybrid automata.(vii)
init: initial conditions, a labeling function init(si) that assigns an initial condition to each location si∈S. 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 x˙≥B). 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 x˙=Ax.Definition 2 (linear hybrid automaton).
A hybrid automatonH is a linear hybrid automaton HL if (1)
all variables ofH are linear,(2)
for all locationssi∈S, the initial condition init(si) and the flow condition flow(si) are all convex,(3)
for all transitionTi∈E, the jump condition F(Ti) is convex.For example, in the hybrid automatonH of Figure 2, we obtain a linear hybrid automaton HL of dimension 2 setting r=2, ∀i∈{1,2}:(1)inits1⟹x0=x10x20Tflowsi⟹x˙t=Bi,FTi⟹Cixt≥Ki considering the constant values x(0),Bi,Ki∈R2 and Ci∈R2×R2.Figure 2
Two-location linear hybrid automaton.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 continuous-linear 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, continuous-linear 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 oner-dimensional first-order differential equation, as defined by [2, 7](2)x˙=ft,x,u,where x is the continuous state vector taking values in some subset of the Euclidean space and f(t,x,u) is a controlled vector field. We consider the state vector x(t)∈X⊆Rr and the control input vector u(t)∈U⊆Rm. The size r of x is called the automaton dimension. The size m of u is the control input size. Accordingly we define(3)x=x1x2⋯xrT;u=u1u2⋯umT;ft,x,u=f1t,x,uf2t,x,u⋯frt,x,uT.In hybrid systems, continuous-linear flows can be modeled considering in (2) f(x,u,t)=Ai(t)x(t)+Bi(t)u(t) to be the controlled vector field (in all si locations), where Ai(t)∈Rr×r, Bi(t)∈Rr×m, and the automaton dimension r is also called the dimension of Ai(t).Definition 3 (continuous-linear hybrid automata).
A hybrid automatonH is a linear hybrid automaton HCL if (1)
for all locationssi∈S, the initial condition init(si) is convex,(2)
the flow conditionflow(si) is continuous-linear,(3)
for all transitionTi∈E, the jump condition F(Ti) is convex.Thus, a continuous-linear hybrid automatonHCL is characterized by a set S of locations si, by oriented arcs Ti∈E called transitions, by convex relations in the initial and jump conditions, and by a continuous-linear flow equations usually expressed as x˙(t)=Ai(t)x(t)+Bi(t)u(t). From Figure 2, we will obtain a continuous-linear hybrid automaton HCL replacing the flow condition x˙(t)=Bi,∀i∈{1,2}, by(4)x˙t=Aitxt+Bitut.Definition 4 (time invariant hybrid automata).
A hybrid automatonH is a time invariant hybrid automaton HI if (1)
for all locationssi∈S, the initial condition init(si) and the flow condition flow(si) are all time invariant and(2)
for all transitionsTi∈E, the jump condition F(Ti) is time invariant.Remark 5.
Further after, we will call continuous-linear time invariant hybrid automatonHCLI to all hybrid automaton that satisfies Definitions 3 and 4. In this case, the vector field in (2) becomes f(x,u).Definition 6 (uncoupled hybrid automata).
A hybrid automatonH is uncoupled hybrid automaton HD if (1)
for all locationssi∈S, the flow condition flow(si) is uncoupled; that is, the variable x is flow independent and the vector field in (2) is given by(5)ft,x,u=f1t,x1,uf2t,x2,u⋯frt,r,uTand for all transition Ti∈E, the jump condition F(Ti) is function of only one state variable.Remark 7.
Hence, we will call uncoupled continuous-linear time invariant hybrid automatonHCLID all hybrid automata that satisfy Definitions 3, 4, and 6. In this case, for all locations si∈SAi 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 Ai for any si∈S and/or at least two different state variables in any jump condition predicate.


## 3. 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 automataHCLID. 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.


### 3.1. 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 wheresj location has a single input and a single output transition, as in Figure 3.Figure 3
SISO transition location of a hybrid automaton.In eachsj location, differential equation models flow(sj) for a continuous dynamic evolution for a fixed horizon defined in a closed interval Ij=[0,δj].Definition 8 (residence time).
For all hybrid automataH, a residence time δj is defined as the amount of time spent in location sj before a transition Tj fires.Property 1 (transition enabling).
For all hybrid automata being in locationsj, Tj transition is enabled at t=0 input instant, if and only if a finite time δj∈R≥0 solution exists. This δj solution is a function of the initial condition (initial point), the jump condition (final point), and the differential equations (the trajectory) defined for location sj. 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 Aj matrices and u(t) a constant control signal, ∀sj∈S.Hypothesis 1.
Let us consider a hybrid automatonHCLID the dynamic representation of uncoupled continuous-linear time invariant hybrid systems. In this representation we suppose ∀sj∈S a diagonal matrix Aj=diag⁡aj1⋯ajr and a real constant vector Bju=bj1⋯bjrT. First, for each sj location (Figure 4), let the jump condition be a simple predicate xq=xq(δj)=Kjq such as xq(t)=^x(t)[q] is a definite function R≥0→R,∀q∈[1,r], t∈Ij, and let Kjq be a point in some region Rjq⊆R. Now, let Kjq be xq(t) value at Tj transition instant.Figure 4
SISO transition location of aHCLI  hybrid automaton.The state vector evolutionx(t);∀t∈Ij is given by(6)xt=eAjtx0+∫0teAjt-φBjuφdφ.For t=δj, we obtain, where ∀q∈[1,r],(7)Kj=eAjδjx0+∫0δjeAjδj-φBjdφ,(8)Kjq=eajqδjKiq+∫0δjeajqδj-φbjqdφ,∀ajq≠0Kjq=Kiq+bjq∫0δjdφ,ajq=0.Herein,Kip and Kjq are, respectively, pth and qth constant components of Ki and Kj vectors. Hence solving (8), we can calculate δj residence time. Therefore, if δj finite real positive solution of (8) exists, then Tj transition is enabled. It must be noted that, in the general case (7), that is, for any Aj matrix, it is difficult to find an explicit solution and in the majority of cases, only δj numerical solution is possible.Property 2.
The residence timeδj in all locations sj∈S is determined by(9)δj=1ajqln⁡Kjq-PjqKiq-Pjq,ifajq≠0orδj=Kjq-Kiqbjq,ifajq=0,where Pjq=-bjq/ajq.This last result could be obtained by solving (8) at δj residence time. For ajq∈R≠0 it must be noted that the corresponding equilibrium point x∗q∈R to xq is given by relation x∗q=Pjq=-bjq/ajq. In general ∀sj∈S and for this uncoupled case, the equilibrium point of x(t) in the Euclidean space Rr is given by x∗=Pj1Pj2⋯PjrT.Theorem 9.
Given a hybrid automatonHCLID with xq=Kiq input (initial) condition and xq=Kjq jump condition predicate of Tj (output) transition, the necessary and sufficient condition to enable Tj is given by the following logical equations:(10)Pjq<Kjq≤Kiq∨Pjq>Kjq≥Kiq=1;ifajq<0,Pjq<Kiq≤Kjq∨Pjq>Kiq≥Kjq=1;ifajq>0,Kjq≤Kiqbjq<0∨Kjq≥Kiqbjq>0=1;ifajq=0.The proof, from the above relations, is a simple one and it is based on the fact that for enabling a transitionTj it is enough that ∃δj∣δj∈R≥0. It must be noted in (9) that the equality Kjq=Kiq gives the lower bound of residence time (δj=0) in location sj. On the other hand, a value Kjq=Pjq where Kjq≠Kiq, will produce the upper limit of residence time (δj=inf). In this last case, Tj transition will not be enabled; therefore sj will be a sink location. For reachability analysis, we consider a region in the state space Ri as initial condition and a region Rj as jump condition of Tj transition. Kiq and Kjq are two points in Ri and Rj regions, respectively. The final region Rj is reachable from the initial region Ri, Ri→Rj, if for each point Kjq∈Rj in the final region, there is at least one point Kiq∈Ri that satisfy the logical equations (10).Hypothesis 2.
Based on the Hypothesis1, let us consider β closed loop trajectory obtained from automaton locations (Figure 5).Figure 5
H
C
L
I
D hybrid automaton with a closed loop structure.Theorem 10.
For eachβ closed loop trajectory in HCLID hybrid automaton, the necessary condition to reach a limit cycle is given by the logical expression:(11)⋃j=1nCjq¯∧Cj-1q∧Cj+1q∗=1.In this case(12)Cj+1q∗=1,ifn=2,⋂i=1n-2Cj+iq∨Cj+iq¯,ifn>2,where(13)Cjq=ajq<0∧Pjq<Kjq≤Kj-1q∨ajq>0∧Pjq<Kj-1q≤Kjq∨ajq=0∧bjq<0∧Kjq≤Kj-1q,Cjq¯=ajq<0∧Pjq>Kjq≥Kj-1q∨ajq>0∧Pjq>Kj-1q≥Kjq∨ajq=0∧bjq>0∧Kjq≥Kj-1q.
Considering all subindices(n+i)=i;∀i≥0, due the closed loop configuration.Theorem 11.
Forβ closed loop structure HCLID hybrid automaton (Figure 5), the initial condition point to reach a limit cycle must belong to the initial region defined by(14)Rinit=Cinit∨Cinit¯,(15)Cinit=a1q<0∧P1q<K1q≤x0q∨a1q>0∧P1q<x0q≤K1q∨a1q=0∧b1q<0∧K1q≤x0q,Cinit¯=a1q<0∧P1q>K1q≥x0q∨a1q>0∧P1q>x0q≥K1q∨a1q=0∧b1q>0∧K1q≥x0q.As we can see in Example12, 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 x(t) hybrid dynamics will reach a limit cycle.Example 12.
H
C
L
I
D hybrid automaton of Figure 6 models a thermostat that controls the temperature, x(t), of a heating system. The task is to evaluate its reachability of a limit cycle for x(t) and find the region of initial conditions that make it possible.Figure 6
Thermostat hybrid automaton model.Using (11) for ajq<0,j∈{1,2},n=2,q=0 (simplifying index notations because there is only one continuous variable xq(t)=x(t)) and again considering index zero equal to n, we get the necessary condition to reach the limit cycle (Theorem 10):(16)⋃j=12Cj¯∧Cj-1=1,C1¯∧C2∨C2¯∧C1=1,P1>K1≥K2∧P2<K2≤K1∨P1<K1≤K2∧P2>K2≥K1=1,P2<K2≤K1<P1∨P2>K2≥K1>P1=1with P1=50,P2=0; we obtain (0<K2≤K1<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 (17)Rinit=Cinit∨Cinit¯=P1<K1≤x0∨P1>K1≥x0=50>K1≥x0.Thus, if we considerx(0)=10,K1=22, and K2=18, all the previous relations are satisfied, and, therefore, the temperature response x(t) will be cyclic (Figure 7).Figure 7
Cyclic behavior of temperature.
(a)
Phase
(b)
TimeOn the other hand, if we considerx(0)=-15,K1=-10, and K2=-5,x(0) is in the initial condition region, but Theorem 10 is not satisfied. Therefore, in steady state, the temperature x(t) will not reach a limit cycle (Figure 8).Figure 8
Temperaturex(t) with noncyclic behavior.
(a)
Phase
(b)
Time


## 3.1. 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 wheresj location has a single input and a single output transition, as in Figure 3.Figure 3
SISO transition location of a hybrid automaton.In eachsj location, differential equation models flow(sj) for a continuous dynamic evolution for a fixed horizon defined in a closed interval Ij=[0,δj].Definition 8 (residence time).
For all hybrid automataH, a residence time δj is defined as the amount of time spent in location sj before a transition Tj fires.Property 1 (transition enabling).
For all hybrid automata being in locationsj, Tj transition is enabled at t=0 input instant, if and only if a finite time δj∈R≥0 solution exists. This δj solution is a function of the initial condition (initial point), the jump condition (final point), and the differential equations (the trajectory) defined for location sj. 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 Aj matrices and u(t) a constant control signal, ∀sj∈S.Hypothesis 1.
Let us consider a hybrid automatonHCLID the dynamic representation of uncoupled continuous-linear time invariant hybrid systems. In this representation we suppose ∀sj∈S a diagonal matrix Aj=diag⁡aj1⋯ajr and a real constant vector Bju=bj1⋯bjrT. First, for each sj location (Figure 4), let the jump condition be a simple predicate xq=xq(δj)=Kjq such as xq(t)=^x(t)[q] is a definite function R≥0→R,∀q∈[1,r], t∈Ij, and let Kjq be a point in some region Rjq⊆R. Now, let Kjq be xq(t) value at Tj transition instant.Figure 4
SISO transition location of aHCLI  hybrid automaton.The state vector evolutionx(t);∀t∈Ij is given by(6)xt=eAjtx0+∫0teAjt-φBjuφdφ.For t=δj, we obtain, where ∀q∈[1,r],(7)Kj=eAjδjx0+∫0δjeAjδj-φBjdφ,(8)Kjq=eajqδjKiq+∫0δjeajqδj-φbjqdφ,∀ajq≠0Kjq=Kiq+bjq∫0δjdφ,ajq=0.Herein,Kip and Kjq are, respectively, pth and qth constant components of Ki and Kj vectors. Hence solving (8), we can calculate δj residence time. Therefore, if δj finite real positive solution of (8) exists, then Tj transition is enabled. It must be noted that, in the general case (7), that is, for any Aj matrix, it is difficult to find an explicit solution and in the majority of cases, only δj numerical solution is possible.Property 2.
The residence timeδj in all locations sj∈S is determined by(9)δj=1ajqln⁡Kjq-PjqKiq-Pjq,ifajq≠0orδj=Kjq-Kiqbjq,ifajq=0,where Pjq=-bjq/ajq.This last result could be obtained by solving (8) at δj residence time. For ajq∈R≠0 it must be noted that the corresponding equilibrium point x∗q∈R to xq is given by relation x∗q=Pjq=-bjq/ajq. In general ∀sj∈S and for this uncoupled case, the equilibrium point of x(t) in the Euclidean space Rr is given by x∗=Pj1Pj2⋯PjrT.Theorem 9.
Given a hybrid automatonHCLID with xq=Kiq input (initial) condition and xq=Kjq jump condition predicate of Tj (output) transition, the necessary and sufficient condition to enable Tj is given by the following logical equations:(10)Pjq<Kjq≤Kiq∨Pjq>Kjq≥Kiq=1;ifajq<0,Pjq<Kiq≤Kjq∨Pjq>Kiq≥Kjq=1;ifajq>0,Kjq≤Kiqbjq<0∨Kjq≥Kiqbjq>0=1;ifajq=0.The proof, from the above relations, is a simple one and it is based on the fact that for enabling a transitionTj it is enough that ∃δj∣δj∈R≥0. It must be noted in (9) that the equality Kjq=Kiq gives the lower bound of residence time (δj=0) in location sj. On the other hand, a value Kjq=Pjq where Kjq≠Kiq, will produce the upper limit of residence time (δj=inf). In this last case, Tj transition will not be enabled; therefore sj will be a sink location. For reachability analysis, we consider a region in the state space Ri as initial condition and a region Rj as jump condition of Tj transition. Kiq and Kjq are two points in Ri and Rj regions, respectively. The final region Rj is reachable from the initial region Ri, Ri→Rj, if for each point Kjq∈Rj in the final region, there is at least one point Kiq∈Ri that satisfy the logical equations (10).Hypothesis 2.
Based on the Hypothesis1, let us consider β closed loop trajectory obtained from automaton locations (Figure 5).Figure 5
H
C
L
I
D hybrid automaton with a closed loop structure.Theorem 10.
For eachβ closed loop trajectory in HCLID hybrid automaton, the necessary condition to reach a limit cycle is given by the logical expression:(11)⋃j=1nCjq¯∧Cj-1q∧Cj+1q∗=1.In this case(12)Cj+1q∗=1,ifn=2,⋂i=1n-2Cj+iq∨Cj+iq¯,ifn>2,where(13)Cjq=ajq<0∧Pjq<Kjq≤Kj-1q∨ajq>0∧Pjq<Kj-1q≤Kjq∨ajq=0∧bjq<0∧Kjq≤Kj-1q,Cjq¯=ajq<0∧Pjq>Kjq≥Kj-1q∨ajq>0∧Pjq>Kj-1q≥Kjq∨ajq=0∧bjq>0∧Kjq≥Kj-1q.
Considering all subindices(n+i)=i;∀i≥0, due the closed loop configuration.Theorem 11.
Forβ closed loop structure HCLID hybrid automaton (Figure 5), the initial condition point to reach a limit cycle must belong to the initial region defined by(14)Rinit=Cinit∨Cinit¯,(15)Cinit=a1q<0∧P1q<K1q≤x0q∨a1q>0∧P1q<x0q≤K1q∨a1q=0∧b1q<0∧K1q≤x0q,Cinit¯=a1q<0∧P1q>K1q≥x0q∨a1q>0∧P1q>x0q≥K1q∨a1q=0∧b1q>0∧K1q≥x0q.As we can see in Example12, 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 x(t) hybrid dynamics will reach a limit cycle.Example 12.
H
C
L
I
D hybrid automaton of Figure 6 models a thermostat that controls the temperature, x(t), of a heating system. The task is to evaluate its reachability of a limit cycle for x(t) and find the region of initial conditions that make it possible.Figure 6
Thermostat hybrid automaton model.Using (11) for ajq<0,j∈{1,2},n=2,q=0 (simplifying index notations because there is only one continuous variable xq(t)=x(t)) and again considering index zero equal to n, we get the necessary condition to reach the limit cycle (Theorem 10):(16)⋃j=12Cj¯∧Cj-1=1,C1¯∧C2∨C2¯∧C1=1,P1>K1≥K2∧P2<K2≤K1∨P1<K1≤K2∧P2>K2≥K1=1,P2<K2≤K1<P1∨P2>K2≥K1>P1=1with P1=50,P2=0; we obtain (0<K2≤K1<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 (17)Rinit=Cinit∨Cinit¯=P1<K1≤x0∨P1>K1≥x0=50>K1≥x0.Thus, if we considerx(0)=10,K1=22, and K2=18, all the previous relations are satisfied, and, therefore, the temperature response x(t) will be cyclic (Figure 7).Figure 7
Cyclic behavior of temperature.
(a)
Phase
(b)
TimeOn the other hand, if we considerx(0)=-15,K1=-10, and K2=-5,x(0) is in the initial condition region, but Theorem 10 is not satisfied. Therefore, in steady state, the temperature x(t) will not reach a limit cycle (Figure 8).Figure 8
Temperaturex(t) with noncyclic behavior.
(a)
Phase
(b)
Time


## 4. 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,xq 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 Figure9 be sj location of two dimensions for HCLI hybrid automaton. In this coupled case Aj=diag⁡aj1aj2, vectors Bj=bj1bj2T, Kj1, and Kj2 are real constants, and the jump condition predicates are related to one of two different state variables xq(t)=Kjq∣q∈{1,2}.Figure 9
An automaton location for a coupled case.The state vector at transition instants is defined asx(δj)=Kj1Kj2T. The discrete values Kj1 and Kj2 are obtained at each Tj transition instant. The trajectory equation x2=f(x1),∀t∈[0,δj] in the state space (for each sj location) is given by(18)x2=x20+bj2bj1x1-x10foraj1=aj2=0,x2=Pj2+x20-Pj2eaj2/bj1x1-x10,foraj1=0,aj2≠0,x2=Pj2+x20-Pj2x1-Pj1x10-Pj1aj2/aj1,foraj1≠0,aj2≠0,where x1(0)x2(0)T=Ki1Ki2T is sj initial condition.Property 3.
The discrete dynamic relation betweenKj2 and Kj1 at each Tj transition instant is given by(19)Kj2=Ki2+bj2bj1Kj1-Ki1,foraj1=aj2=0,Kj2=Pj2+Ki2-Pj2eaj2/bj1Kj1-Ki1,foraj1=0,aj2≠0,Kj2=Pj2+Ki2-Pj2Kj1-Pj1Ki1-Pj1aj2/aj1,foraj1≠0,aj2≠0.This is an analytical result obtained by finding the commutation point in the state space trajectory equation (18), at Tj transition firing instant. Now, for limit cycle analysis we must consider that there exists a closed loop β obtained from HCLI hybrid automaton with jump condition predicates function of x1 or x2 state variables.Hypothesis 4.
Letβ be a closed loop obtained by n single output locations sj∈S¯∣S¯⊆S of HCLI hybrid automaton. In the jump condition predicates, the enabled Tj transitions are function of xq(t), for q∈{1,2}. It is important to note that in each location sj there are two unknown discrete variables Ki1 and Kj2 at input and output commutation instants, respectively. In general, for each location sj we have one equation obtained from (19), with two unknowns (Ki1 and Kj2) that will determine (by mean of its convergence analysis) the existence of a limit cycle.Property 4.
For all closed loopsβ∈HCLI, dynamic relations among state variables in discrete commutation instants are governed by a set of equations obtained from (19), ∀sj∈S¯ and j∈{1,2,…,n}. Thus, in general, in β closed loop structure we will have the same number of equations (and unknowns) as locations in the loop.Remark 13.
The existence and uniqueness of a limit cycle are concluded from the convergence analysis ofKjq values at commutation instants ∀s∈S¯ and q∈{1,2} according to the fixed point theorem.Theorem 14 (about the existence of a limit cycle).
Forn locations closed loop β∈HCLI with jump conditions predicate xq(t)=Kjq∣q∈{1,2,…,r}, the necessary condition to obtain a limit cycle is given by(20)⋂q=1r⋃j=1nCjq¯∧Cj-1q∧Cj+1q∗=1.This result is a natural extension of (11) for the case of n locations and r 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.
K
j
q convergence is assured by ∀j∈{1,2,…,n} and ∀q∈{1,2,…,r} 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 Theorem14 has been satisfied, we can find an initial condition region to reach a limit cycle replacing Kn1Kn2⋯KnrT by x1(0)x2(0)⋯xr(0)T in (20).Remark 15.
Thus, we must note that a sufficient condition to reach a limit cycle is given by Theorem14 plus the initial condition region found from Property 6.Property 7.
For the case where the diagonal matrixAj is stable ∀sj∈S and jump condition predicates xq(t)=Kjq∣q∈{1,2,…,r}, the x(t) 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 forn-location case. In Section 5, we present an analysis of the two-location case. A geometric interpretation of results is then conducted.


## 5. Results Interpretation

Consider two-locationsj∣j∈{1,2} of a continuous-linear hybrid automaton, as in Figure 10.Figure 10
H
C
L, two-location coupling case.One hasx(t)=x1(t)x2(t)T, A1=a11a12, A2=a21a22, B1=b11b12T, B2=b21b22T, F(T1)⇒x1(t)=K11, and F(T2)⇒x2(t)=K22. K11 and K22 are real constants of known magnitude. So without loss of generality, because the analysis procedure is the same, we will consider stable continuous equations (ajq<0∀sj∣j∈{1,2} and ∀q∈{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(21)⋂q=12C1q¯∨C2q∧C2q¯∨C1q=1,where (22)C1q¯=P21<K21<K11<P11,C2q=P21>K21>K11>P11,C2q¯=P22<K22<K12<P12,C1q=P22>K22>K12>P12, and it gives 4 bounded regions of the state space, where a cyclic dynamic will take place, as in Figure 11.Figure 11
State space regions where cyclic dynamics must take place.
(a)
(b)
(c)
(d)In Figure11, the trajectory attraction points x˙∗=P1 and x∗=P2 (equilibrium points) for locations s1 and s2, respectively, are shown. Thus, a unique limit cycle is assured if the system dynamic starts from an initial state space region that makes that x(t) 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 K21K22T by x1(0)x2(0)T in (21). Because K22 is a known constant, we obtain(23)P21<x10<K11<P11∨P21>x10>K11>P11∧x20=K22=1.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:(24)x10⊳1K11,x20⊳2P12+K22-P12x10-P11K11-P11a12/a11,where relations ⊳=(<,>) and are defined according to the (known) automaton relations K11⊳1P11 and K22⊳2P22. 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 K12 and K21 convergence analysis (Remark 13). Its dynamic behaviors are governed by (Property 3)(25)K12=P12+K22-P12K11-P11K21-P11a12/a11,K21=P21+K11-P21K22-P22K12-P22a21/a22.ConsideringK12=g1(K21) and K21=g2(K12) from (25), Figure 12 shows g1(K21) 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 K12-K21 space. The limit cycle existence and uniqueness demonstration can be done starting from K12 and K21 fixed point convergence analysis. This is the formal method that proves the cyclic steady state convergence.Figure 12
g
1
(
K
21
) and g2-1(K21) curves: (a), (b), (c), and (d) with two fixed points’ convergence; (a′), (b′), (c′), and (d′) with only one fixed point.Therefore, from (25), by means of K12 and K21 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<δj<inf,∀sj∈S, and for an unstable point, a boundary case, where K12=K22 and K21=K11, the residence times are zero; that is, δj=0,∀sj∈S.Thus, according to the fixed point theory, the fixed point uniqueness is assured if(26)dK12i+1dK12i<1,dK21i+1dK21i<1, where(27)dK12i+1dK12i=cf1K12f2K12,dK21i+1dK21i=cf3K21f4K21 with (28)c=K11-P21K11-P11K22-P12K22-P22a12a11a21a22,f1K12=P21-P11K11-P11+K11-P21K11-P11K22-P22K12-P22a21/a22-1+a12/a11,f2K12=K22-P22K12-P221+a21/a22,f3K21=P12-P22K22-P22+K22-P12K22-P22K11-P11K21-P11a12/a11-1+a21/a22,f4K21=K11-P11K21-P111+a12/a11.By mean of preceding terms analysis (restricted of limit cycle condition (20)), we find always fi<1,∀i∈{1,2,3,4}. Thus, we can guarantee the fixed point existence and uniqueness assuring the constant value c<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), 12(b), 12(c), and 12(d)). So we must assure the relation c>1; that is c>1 (because in the four regions defined in Figure 11, c is positive). Then, choosing adequately K11 and K22 values to achieve c>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 A1=diag⁡-0.1-0.5, A2=diag⁡-0.2-0.1, B1=21T, B2=11T, and x(0)=100T considering the following:(a)
If jump conditions areF(T1)⇒x1(t)=K11=15 and F(T2)⇒x2(t)=K22=8, determine the limit cycle existence and uniqueness.(b)
For the same jump conditions, find the necessary and sufficient initial region ins1 that will give a limit cycle system behavior.(c)
Design a 10-second limit cycle period by considering steady state residence times ofδ1=6 sec. and δ2=4 sec. Find K11 and K22 values as well as the initial condition region that guarantees it.(d)
ForK22=8, determine the state space region of K11 values to allow obtaining a nonzero period limit cycle.For this example, the trajectory attraction (equilibrium) points corresponding to each location are given by(29)P1=P11P12T=202T,P2=P21P22T=510T.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(30)5<K21<15<20∧2<K12<8<10.So consideringx(0)=100T as an initial point, we will verify if this point is in the initial condition region. Therefore, using (24) and considering K11<P11 and K22<P22,(31)x10<K11⟹10<15,x20<P12+K22-P22x10-P11K11-P11a12/a11⟹0<194.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(32)x10<15,x20<2+64-0.2x105.System simulations (usingMATLAB®) for different initial conditions are shown in Figure 13.Figure 13
Limit cycle convergence inx1-x2 space with x1(0)x2(0)T (a) 1410T, and for the boundary case (b) 1416.9T.
(a)
(b)In Figure13(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 x1(0)=14 and x2(0)=2+64-0.2x1(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 K12=K22 and K21=K11.For (c), the residence time in each location is deduced by(33)δ1=1a1qln⁡K1q-P1qK2q-P1q,δ2=1a2qln⁡K2q-P2qK1q-P2qfor q∈{1,2}. Thus the discrete evolution of K1q and K2q may be obtained as(34)K1q=11-ea1qδ1+a2qδ2P1q+P2q-P1qea1qδ1-P2qea1qδ1+a2qδ2,K2q=11-ea2qδ2+a1qδ1P2q+P1q-P2qea2qδ2-P1qea2qδ2+a1qδ1.In this way, to obtain a 10-second period limit cycle by consideringδ1=6 sec. and δ2=4 seconds, it is necessary that(35)K11=11-ea11δ1+a21δ2P11+P21-P11ea11δ1-P21ea11δ1+a21δ2=13.983,K22=11-ea22δ2+a12δ1P22+P12-P22ea22δ2-P12ea22δ2+a12δ1=4.7285.The initial condition region is given by(36)x10<13.983,x20<2+2.733.32-0.16x105.Figure14 shows system simulations using the above results. Therein, how the period specifications are obtained should be observed.Figure 14
Dynamic simulation of a limit cycle design,x1-x2 space, and time response for x1(0)x2(0)T=88T.Finally, using the results obtained from recursive equationsK12 and K21 fixed point analysis, to assure the existence of two fixed points, it is necessary that c>1 (sufficient condition). Therefore(37)K11-P21K11-P11K22-P12K22-P22a12a11a21a22>1⟹K11>5.48and also, to assure the limit cycle existence and uniqueness, initial state variables values must be an element of the following initial condition region:(38)x10<15,x20<2+6x10-20K11-205.Figure15 shows the simulation results considering x1(0)x2(0)T=56T and K11=5.6 that satisfy the previous inequality and K11=5.2 that does not.Figure 15
Automaton hybrid dynamics designed to have a limit cycle period: (a) nonzero(K11=5.6), (b) zero (K11=5.2).
(a)
(b)


## 6. 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 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.



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*Source: 2981952-2016-03-10.xml*