# Invariant Sets of Hybrid Autonomous Systems with Disturbance

Authors: Li Jian-Qiang; Zhu Zexuan; Ji Zhen; Pei Hai-Long
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

The concept and model of hybrid systems are introduced. Invariant sets introduced by LaSalle are proposed, and the concept is extended to invariant sets in hybrid systems which include disturbance. It is shown that the existence of invariant sets by arbitrary transition in hybrid systems is determined by the existence of common Lyapunov function in the systems. Based on the Lyapunov function, an efficient transition method is proposed to ensure the existence of invariant sets. An algorithm is concluded to compute the transition mode, and the invariant set can also be computed as a convex problem. The efficiency and correctness of the transition algorithm are demonstrated by an example of hybrid systems.


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## Body

## 1. Introduction

Hybrid systems and invariant sets are novel topics in recent years. Hybrid systems are systems which include discrete and continuous dynamics. In many applications, hybrid systems have multiple operating modes; each is described by a different dynamic equation [1–3]. Invariant set of hybrid system plays an important role in the many situations when dynamics system is constrained in some way [4, 5]. For hybrid systems, the transition is very important for constructing an invariant set for the systems. For some transitions, the states of the systems may get out of each subsystem’s invariant set. In this paper, an efficient transition method which makes the states of the hybrid systems stay in the invariant sets is proposed. When the system is given an initial set, this method constructs transition regions for the hybrid system to ensure the invariant set existence. Then, the invariant set can be computed as ellipsoid set or polyhedral set [6–9].Some studies have been done in the area of hybrid systems in recent years. Controller design and invariant set of hybrid systems have been studied [10, 11]. The lecture notes [12] by Lygeros have talked about the basic theory of hybrid system including automata, existence, analysis and synthesis, model checking, and reachability. The set invariance in control provides a survey of the literature on invariant sets and their applications [13], and the stability of mode transitions has been studied in [14]. Research about invariant set of special hybrid systems has been done in [6, 7]. Polyhedral approximation computation of invariant set has been studied in [15].This paper proposes an efficient approach to compute the transition regions. By the efficient transition based on the Lyapunov function theory, the system will stay in the invariant sets. This method is attractive as the invariant set is useful to design the switching strategies and the controller. The rest of the paper is organized as follows. Section2 introduces the basic concept and model of the hybrid systems. The invariant sets, as well as the existence and computation of invariant sets are given in  Section 3. Section 4 presents the invariant sets in the hybrid system and proposes an efficient transition method for the invariant set. In Section 5, an example demonstrating the efficacy of the proposed method is introduced. Section 6 summarizes the results presented in this paper and discusses future research.


## 2. Hybrid Systems

Roughly speaking, hybrid systems are dynamic systems that involve the interaction of different types of dynamics. Consider a hybrid system(2.1)ẋ=fi(x(t)),i=1,2,…,m.
It can be modelled by a hybrid automaton [16].Definition 2.1 (Hybrid Automaton).
A Hybrid automatonH is the collection H=(Q,X,U,f,Init,D,E,G,R), where Q is a discrete state space, X:X⊂Rn is a continuous state space, U:U⊂Rλ is a set of inputs, f:Q×X→Rn is a vector field, Init is a set of initial states, D:Q→P(X) is a domain (the domain is sometimes called the invariant set in hybrid system), E⊂Q×Q is a set of edges, G is a guard condition, and R:E×X→P(X) is a reset map.Hybrid automaton defines possible evolution for their states. Starting from an initial value(x0,q0)∈Init, the continuous states x flow according to the vector field f(q0,·) in the discrete state q0 and remain in D(q0). If the states reach a guard G(q0,qi), the discrete state will change to qi. At the same time the states will reset to R(q0,qi,x) if the system has the reset condition. The discrete state qi is called mode in hybrid system. So in the hybrid automaton, the system starts from initial mode q0 to any mode qi. The process can be repeated.By the hybrid automaton, a temperature controller can be described as follows.In modeq0, ud and u are the inputs, the temperature x flows according to f(q0,ud,u), and state x stays in domain D(q0). If the temperature reaches G(q0,q1)={x>18}, the mode will change to q1. It is the same process in mode q1 as shown in Figure 1. By the hybrid controller, the temperature will stay in a domain which is called invariant set.Figure 1
Temperature controller automaton.


## 3. Invariant Set

An (positive) invariant set of a dynamic system is a subset of the state space such that once the state enters this set, it will remain in it for all the future time [6].That is,(3.1)x(0)∈M⟶x(t)∈M,t>0,
where x(t) is the subset of the dynamic system at time t and X is a subset of the state space. Consider the continuous dynamic system(3.2)ẋ(t)=f(x(t)),
where f is a continuous Lipschitz function with x. A sufficient and necessary condition for X in the invariant set is that the differential equation is directed into the set at every point of the boundary.Definition 3.1 (Bouligand, 1932).
Given a closed setκ⊂ℝn, the tangent cone to κ in x0 is the set
(3.3)κx0={z∈Rn:lim⁡inf⁡h→0dist⁡(x0+hz,κ)h=0}.The definition is due to Bouligand. As shown in Figure2, if x0∈int{κ}, the interior of κ, then κx0=ℝn; if x0∉int{κ}, then κx0=ϕ, and x0∈∂κ. The tangent cone is a cone which contains all the vectors directing into the set κ. According to Nagumo theorem, the closed set is a positively invariant set for the system (3.2) if and only if for all x∈κ,(3.4)f(x)∈κx.Figure 2
Tangent cone.Every point on the boundary is directed into the set. This can be expressed as follows:(3.5)nκ(x)Tf(x)≤0,∀x∈κ,
where nk(x) denotes a normal to ∂κ at x. The invariant set is described by an inequality(3.6)κ={x∈Rn∣V(x)≤b}.V(x) which defines the invariant set is a function of x. Considering the existence of a quadratic Lyapunov function for a system and the fact that the  level sets of Lyapunov functions are invariant sets, a lemma can be concluded as follows.Lemma 3.2.
A systemẋ=Ax, x∈ℝn, A∈ℝn, exists as invariant set if all eigenvalues of A lies in the open left half plane.Given a set of initial states, if there exists invariant set in the system from Lemma3.2, the invariant set can be computed as invariant ellipsoid or invariant polyhedron.Computed as an invariant polyhedron, the invariant set can be represented in the following form [11]:(3.7)δ={x:Fx≤1̅},
where F is a r×n matrix and 1̅∈Rr denotes vector of the form(3.8)1̅=[11⋯1]T.There are mainly two computation methods [17] which include eigenstructure analysis/assignment method [13] and the iterative method as follows.Algorithm 3.3.
One has the following.Initialization:k=0 and δ0=δRepeat:δk+1={x∈δ:∃u(x)∈U:Ax+Bu(x)+Ew∈δk,forallw∈W}    (3.9)δ∞=̇⋂i=0∞δiUntilδi+1=δi, then δ∞=δi.At each step, the algorithm computes the set of states for which all solutions of the system stay in theδk+1. The invariant set of the system is the intersection of δi until δi+1=δi.As an invariant ellipsoid, the invariant set can be represented as the following ellipsoid:(3.10)δ={x∈Rn∣(x-xa)TQ-1(x-xa)≤1},
where QT=Q∈ℝn and Q≻0. Given an initial set, the maximal invariant set in it can be computed as a convex optimization problem. xa is the center of the ellipsoid. If a set of initial states δ0 is given, it can be described as a polytope by linear inequalities first:(3.11)δ0={x∈Rn∣vkTx≤1,k=1,…,q}.Then this invariant set can be computed as a convex problem (CP), and it is more convenient to compute by LMI toolbox:(3.12)Minimizelog⁡det⁡Q-1SubjecttoQAT+AQ≤0,P≻0,vkTQ-1vk≤1.


## 4. Invariant Sets in Hybrid System

The computation for the existence of invariant sets in linear systems presented in the previous section can be used for computation for the existence of invariant set for the subsystems in the hybrid systems. In hybrid systems, the existence of invariant set can be determined by the common Lyapunov function first.


### 4.1. Common Lyapunov Function

In some cases, it is possible to prove the existence of invariant set. Based on the stability theorem [18, 19], the computation is based on the sufficient conditions.Theorem 4.1.
A hybrid systemqi, fi(x)=Aix exists as invariant sets by arbitrarily transition if a common Lyapunov function exists in each subsystem.By the common Lyapunov function, if there existsAiTPAi≤0, P≻0, the states of the systems will stay in an invariant set for a given initial set.Proof.
(1) Take an arbitrary mode qi, fi, and define v(t)∶=V(x(t))(4.1)v̇=∂v∂x(x)ẋ=∂v∂xfi≤0.
(2) There is
(4.2)v(t)∶=V(x(t))≤v(0)∶=V(x(0)).
So, if the hybrid system is given an initial set which can be found asV(x(0))≤b and V(x(t))≤b, the invariant set of the hybrid system exists.
In some cases, it is needed to verify that no common Lyapunov function in Theorem4.1 exists. If the systems satisfy the following condition, as matrices Ri, Ri≻0 can be found to satisfy
(4.3)∑qi∈QAiTRi+RiAi>0,
then there is not a solution P=PT existing in Theorem 4.1.


### 4.2. Problem Formulation

In a hybrid system, there may not exist an invariant set by arbitrary transition if common Lyapunov function does not exist, although an invariant set exists in each subsystem according to Lemma3.2. It can be shown as the example below.Example 4.2.
One has the following:(4.4)q1:f1(x)=A1x,q2:f2(x)=A2x,A1=(-0.11-10-0.1),A2=(-0.110-1-0.1).This is a simple transition system. In modeq1, it is easy to compute that all eigenvalues of A1 lie in the open left half plane max{Re(λ(A1))}=-0.1<0, max{Re(λ(A2))}=-0.1<0. So an invariant set exists in subsystem 1 and subsystem 2 if given an initial set as Figure 3.Trajectory of systemẋ=f1(x),ẋ=f2(x).
(a)(b)However, after arbitrary transitions between the two modes, the states of the system may not shrink to an invariant set. As Figure4 shows, the states get out of the initial set after transitions. There is no invariant set existing.Figure 4
No invariant set exists.


### 4.3. Efficient Transition

As the example in Section4.2 shows, the invariant set does not exist by arbitrary transitions although it exists in each subsystem. If there is not an invariant set in the subsystems, how to design the transition strategy to construct the invariant set for the hybrid systems is very important such as the temperature controller in Section 1. There is not an invariant set existing in each mode of the temperature controller, but the temperature stays in a domain after transitions between the two modes. Based on the stability theorem by Li [20], the transition method constructing for an invariant set can also be established as follows.Theorem 4.3.
A hybrid system(4.5)ẋ(t)=Aix(t)+Ei(t)x(t),‖Ei(t)‖≤ηi,i∈Q
if there exist a positive matrix P(P=PT) and positive number β satisfying the matrix inequalities below:
(4.6)(∑i=1mδiAi)TP+P(∑i=1mδiAi)+P2+βI≤0,(∑i=1mδi)=1,(4.7)η≤β2.
Then, there exist invariant sets in this hybrid system by efficient transition.In Theorem4.3, (∑i=1mδi)=1, Ei(t)x(t) is the disturbance of each systemi, and ||Ei(t)||≤ηi, ηiis a constant, η=max(ηi). The domain of each system can be computed as follows:(4.8)Di={x(t)∣xT(t)(AiTP+PAi+P2+βI)x(t)≤0}.Proof.
Construct the Lyapunov function as follows:(4.9)V(x(t))=xT(t)Px(t)+γ∫0txT(τ)x(τ)d(τ),γ>0.
so
(4.10)V̇(x(t))=xT(t)(ATP+PA)x(t)+2xT(t)PE(t)x(t)+γxT(t)x(t)≤xT(t)(ATP+PA)x(t)+xT(t)P2x(t)+xT(t)ET(t)E(t)x(t)+γxT(t)x(t)=xT(t)(ATP+PA+P2+2γI)x(t)+xT(t)(η2-γ)x(t).
From (4.3),
(4.11)δ1(A1TP+PA1)+δ2(A2TP+PA2)+⋯+δm(AmTP+PAm)+P2+βI≤0.
Considering the condition δ1+δ2+⋯+δm=1, the inequality can also be written as
(4.12)δ1(A1TP+PA1+P2+βI)+δ2(A2TP+PA2+P2+βI)+⋯+δm(AmTP+PAm+P2+βI)≤0.
Suppose that
(4.13)Di={x(t)∣xT(t)(AiTP+PAi+P2+βI)x(t)≤0}.
Taking β=2γ, when x(t)∈Di,
(4.14)V̇(x(t))≤xT(t)(AiTP+PAi+P2+βI)x(t)+xT(t)(η2-β2)x(t).
Then in conditions (4.6) and (4.7), V̇(x(t))≤0, the states stay in the level set of Lynapunov function V(x). So, an invariant set exists by the efficient transition. From (4.7), we know that η determines the bound of E(t), so η is bigger and the robustness of the system is better.From common Lyapunov function and the concept of Theorem4.3, the computation for the efficient transition can be concluded as follows.(1)
Compute whether there exists common Lyapunov function by Theorem4.1 and inequality (4.3). If it exists, invariant sets exist in the hybrid systems by arbitrary transitions in a given set. Otherwise go to step (3.1).(2)
Computeβmax and matrix P in (4.6), and inequality (4.6) is equivalent to the linear matrix inequalities (LMI) as below:
(4.15)P>0,β>0,((∑i=1mδiAi)TP+P(∑i=1mδiAi)+P2+βIPP-I)<0.βmax and P can be computed by LMI toolbox in Matlab.(3)
Compute whetherη≤βmax/2; if the inequality is true, the invariant set exists in the hybrid system (4.5).(4)
Compute the transition domain by inequality (4.8), and the computation of invariant set in hybrid systems can be extended to  the computation methods in Section 3.


## 4.1. Common Lyapunov Function

In some cases, it is possible to prove the existence of invariant set. Based on the stability theorem [18, 19], the computation is based on the sufficient conditions.Theorem 4.1.
A hybrid systemqi, fi(x)=Aix exists as invariant sets by arbitrarily transition if a common Lyapunov function exists in each subsystem.By the common Lyapunov function, if there existsAiTPAi≤0, P≻0, the states of the systems will stay in an invariant set for a given initial set.Proof.
(1) Take an arbitrary mode qi, fi, and define v(t)∶=V(x(t))(4.1)v̇=∂v∂x(x)ẋ=∂v∂xfi≤0.
(2) There is
(4.2)v(t)∶=V(x(t))≤v(0)∶=V(x(0)).
So, if the hybrid system is given an initial set which can be found asV(x(0))≤b and V(x(t))≤b, the invariant set of the hybrid system exists.
In some cases, it is needed to verify that no common Lyapunov function in Theorem4.1 exists. If the systems satisfy the following condition, as matrices Ri, Ri≻0 can be found to satisfy
(4.3)∑qi∈QAiTRi+RiAi>0,
then there is not a solution P=PT existing in Theorem 4.1.


## 4.2. Problem Formulation

In a hybrid system, there may not exist an invariant set by arbitrary transition if common Lyapunov function does not exist, although an invariant set exists in each subsystem according to Lemma3.2. It can be shown as the example below.Example 4.2.
One has the following:(4.4)q1:f1(x)=A1x,q2:f2(x)=A2x,A1=(-0.11-10-0.1),A2=(-0.110-1-0.1).This is a simple transition system. In modeq1, it is easy to compute that all eigenvalues of A1 lie in the open left half plane max{Re(λ(A1))}=-0.1<0, max{Re(λ(A2))}=-0.1<0. So an invariant set exists in subsystem 1 and subsystem 2 if given an initial set as Figure 3.Trajectory of systemẋ=f1(x),ẋ=f2(x).
(a)(b)However, after arbitrary transitions between the two modes, the states of the system may not shrink to an invariant set. As Figure4 shows, the states get out of the initial set after transitions. There is no invariant set existing.Figure 4
No invariant set exists.


## 4.3. Efficient Transition

As the example in Section4.2 shows, the invariant set does not exist by arbitrary transitions although it exists in each subsystem. If there is not an invariant set in the subsystems, how to design the transition strategy to construct the invariant set for the hybrid systems is very important such as the temperature controller in Section 1. There is not an invariant set existing in each mode of the temperature controller, but the temperature stays in a domain after transitions between the two modes. Based on the stability theorem by Li [20], the transition method constructing for an invariant set can also be established as follows.Theorem 4.3.
A hybrid system(4.5)ẋ(t)=Aix(t)+Ei(t)x(t),‖Ei(t)‖≤ηi,i∈Q
if there exist a positive matrix P(P=PT) and positive number β satisfying the matrix inequalities below:
(4.6)(∑i=1mδiAi)TP+P(∑i=1mδiAi)+P2+βI≤0,(∑i=1mδi)=1,(4.7)η≤β2.
Then, there exist invariant sets in this hybrid system by efficient transition.In Theorem4.3, (∑i=1mδi)=1, Ei(t)x(t) is the disturbance of each systemi, and ||Ei(t)||≤ηi, ηiis a constant, η=max(ηi). The domain of each system can be computed as follows:(4.8)Di={x(t)∣xT(t)(AiTP+PAi+P2+βI)x(t)≤0}.Proof.
Construct the Lyapunov function as follows:(4.9)V(x(t))=xT(t)Px(t)+γ∫0txT(τ)x(τ)d(τ),γ>0.
so
(4.10)V̇(x(t))=xT(t)(ATP+PA)x(t)+2xT(t)PE(t)x(t)+γxT(t)x(t)≤xT(t)(ATP+PA)x(t)+xT(t)P2x(t)+xT(t)ET(t)E(t)x(t)+γxT(t)x(t)=xT(t)(ATP+PA+P2+2γI)x(t)+xT(t)(η2-γ)x(t).
From (4.3),
(4.11)δ1(A1TP+PA1)+δ2(A2TP+PA2)+⋯+δm(AmTP+PAm)+P2+βI≤0.
Considering the condition δ1+δ2+⋯+δm=1, the inequality can also be written as
(4.12)δ1(A1TP+PA1+P2+βI)+δ2(A2TP+PA2+P2+βI)+⋯+δm(AmTP+PAm+P2+βI)≤0.
Suppose that
(4.13)Di={x(t)∣xT(t)(AiTP+PAi+P2+βI)x(t)≤0}.
Taking β=2γ, when x(t)∈Di,
(4.14)V̇(x(t))≤xT(t)(AiTP+PAi+P2+βI)x(t)+xT(t)(η2-β2)x(t).
Then in conditions (4.6) and (4.7), V̇(x(t))≤0, the states stay in the level set of Lynapunov function V(x). So, an invariant set exists by the efficient transition. From (4.7), we know that η determines the bound of E(t), so η is bigger and the robustness of the system is better.From common Lyapunov function and the concept of Theorem4.3, the computation for the efficient transition can be concluded as follows.(1)
Compute whether there exists common Lyapunov function by Theorem4.1 and inequality (4.3). If it exists, invariant sets exist in the hybrid systems by arbitrary transitions in a given set. Otherwise go to step (3.1).(2)
Computeβmax and matrix P in (4.6), and inequality (4.6) is equivalent to the linear matrix inequalities (LMI) as below:
(4.15)P>0,β>0,((∑i=1mδiAi)TP+P(∑i=1mδiAi)+P2+βIPP-I)<0.βmax and P can be computed by LMI toolbox in Matlab.(3)
Compute whetherη≤βmax/2; if the inequality is true, the invariant set exists in the hybrid system (4.5).(4)
Compute the transition domain by inequality (4.8), and the computation of invariant set in hybrid systems can be extended to  the computation methods in Section 3.


## 5. Example

In this sector, the hybrid system which includes two continuous dynamics inℝ2 is chosen as the trajectories and sets can be easily visualized.Consider a class of hybrid system as follows.Example 5.1.
One has the following:(5.1)ẋ(t)=Aix(t)+Ei(t)x(t),(5.2)A1=(-30-22),E1=(0.6sint00-0.3cos⁡t),A2=(160-6),E2=(-0.1sint000.5sint).
For the system, we  know that
(5.3)η=max⁡{η1,η2}=0.6.
The transition domain can be computed by the computation algorithm in Section 4.3.(1) Common Lyapunov function. As in the modes q1 and q2,
(5.4)max⁡{Re(λ(A1))}=2,max⁡{Re(λ(A2))}=1.
There exists no Lyapunov function in the subsystems and no common Lyapunov function in the hybrid system, so an invariant set exists neither in each subsystem nor in the hybrid system by arbitrary transitions.(2) Compute βmax and matrix P. Fist, δi can be selected as δ1=0.3, δ2=0.7. Then, compute the linear matrix inequalities (4.7) by LMI toolbox in Matlab. The variables can be computed as below:
(5.5)βmax⁡=0.993,P=[0.61740.79780.79782.8776].(3) Compute the maximum range of the disturbance, βmax/2=0.7069, η≤βmax/2, and the disturbance is inside the bound. So, there exists an invariant set in the hybrid system.(4) Compute the transition domain of the hybrid system by the inequality(4.6)(5.6)D1={x(t)∣xT(t)(A1TP+PA1+P2+βI)x(t)<0}={x(t)∣xT(-4.8786-3.7647-3.764721.4268)x<0}={(x1,x2)T∣(-4.8786x12-7.5294x1x2+21.4268x22<0)},D2={x(t)∣xT(t)(A2TP+PA2+P2+αI)x(t)<0}={x(t)∣xT(3.25182.50372.5037-15.0412)x<0}={(x1,x2)T∣(3.2518x12-5.0074x1x2-15.0412x22<0)}.By the transition method, there exist invariant sets in the system (5.1). The simulation of the system by efficient transition is shown as Figure 5.Figure 5
Invariant set exists by efficient transitions.In Figure5, it is shown that there exists an invariant set for a given initial set (rectangle:5×5) by the efficient transition which has been computed. The invariant set here is easy to visualize, as shown in Figure 5; the invariant set is a part of polygon shadow. The invariant set also can be computed as an invariant ellipsoid in (5.1) by LMI toolbox.


## 6. Conclusions

The concepts of hybrid systems and invariant set have been introduced in this paper, and the computation of invariant set is also introduced. The concept is extended to invariant set in hybrid systems in this paper. It is shown that the existence of invariant sets by arbitrary transition in hybrid systems is determined by whether there exists a common Lyapunov function in the systems. Based on the Lyapunov function, an efficient transition method is proposed to ensure the existence of an invariant set. Given an initial set, an algorithm is proposed to compute the transition mode, and the invariant set is also computed as an invariant ellipsoid.Since the invariant sets and the efficient transition mode can be computed efficiently, the proposed invariant sets make it possible to model predictive control, protection, and decision for mode transitions. An example of hybrid systems is given in this paper. The applications to the realistic problem are currently being studied.After the efficient transition in hybrid system, the computation for complex systems may be difficult as it has large event sets. More efficient methods for the invariant set are investigated in the next step. Controller design is important in the invariant set. By the effect of the control, the system stays in the invariant set, and the application of the invariant set in hybrid systems will be studied in the future. The method can be used to the realistic application such as unmanned helicopter control [20] and power system.



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