Invariant Sets of Hybrid Autonomous Systems with Disturbance

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.

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. Section 2 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 system𝑥̇=𝑓_𝑖(𝑥(𝑡)),𝑖=1,2,…,𝑚. (2.1) It can be modelled by a hybrid automaton [16].

Definition 2.1 (Hybrid Automaton). A Hybrid automaton 𝐻 is the collection 𝐻=(𝑄,𝑋,𝑈,𝑓,Init,𝐷,𝐸,𝐺,𝑅), where 𝑄 is a discrete state space, 𝑋∶𝑋⊂𝑅^𝑛 is a continuous state space, 𝑈∶𝑈⊂𝑅^𝜆 is a set of inputs, 𝑓∶Q×𝑋→𝑅^𝑛 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,𝑞_𝑖), the discrete state will change to 𝑞_𝑖. At the same time the states will reset to 𝑅(𝑞₀,𝑞_𝑖,𝑥) if the system has the reset condition. The discrete state 𝑞_𝑖 is called mode in hybrid system. So in the hybrid automaton, the system starts from initial mode 𝑞₀ to any mode 𝑞_𝑖. The process can be repeated.

By the hybrid automaton, a temperature controller can be described as follows.

In mode 𝑞₀, 𝑢_𝑑 and 𝑢 are the inputs, the temperature 𝑥 flows according to 𝑓(𝑞₀,𝑢_𝑑,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,𝑥(0)∈𝑀⟶𝑥(𝑡)∈𝑀,𝑡>0, (3.1) 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 system𝑥̇(𝑡)=𝑓(𝑥(𝑡)), (3.2) 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 set 𝜅⊂ℝ^𝑛, the tangent cone to 𝜅 in 𝑥₀ is the set 𝜅_𝑥0=𝑧∈ℝ^𝑛∶liminf_ℎ→0dist𝑥₀+ℎ𝑧,𝜅ℎ=0. (3.3)

The definition is due to Bouligand. As shown in Figure 2, if 𝑥₀∈int{𝜅}, the interior of 𝜅, then 𝜅_𝑥0=ℝ^𝑛; if 𝑥₀∉int{𝜅}, then 𝜅_𝑥0=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,𝑓(𝑥)∈𝜅_𝑥. (3.4)

Figure 2 

Tangent cone.

Every point on the boundary is directed into the set. This can be expressed as follows:𝑛_𝜅(𝑥)^𝑇𝑓(𝑥)≤0,∀𝑥∈𝜅, (3.5) where 𝑛_𝑘(𝑥) denotes a normal to 𝜕𝜅 at 𝑥. The invariant set is described by an inequality𝜅=𝑥∈ℝ^𝑛∣𝑉(𝑥)≤𝑏. (3.6)

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 system 𝑥̇=𝐴𝑥, 𝑥∈ℝ^𝑛, 𝐴∈ℝ^𝑛, 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 Lemma 3.2, 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]:𝛿=𝑥∶𝐹𝑥≤1, (3.7) where F is a 𝑟×𝑛 matrix and 1∈𝑅^𝑟 denotes vector of the form1=[11⋯1]^𝑇. (3.8)

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: 𝛿_𝑘+1={𝑥∈𝛿∶∃𝑢(𝑥)∈𝑈∶𝐴𝑥+𝐵𝑢(𝑥)+𝐸𝑤∈𝛿_𝑘,forall𝑤∈𝑊}     𝛿_∞=̇^∞_𝑖=0𝛿_𝑖 (3.9)Until 𝛿_𝑖+1=𝛿_𝑖, then 𝛿_∞=𝛿_𝑖.

At each step, the algorithm computes the set of states for which all solutions of the system stay in the 𝛿_𝑘+1. The invariant set of the system is the intersection of 𝛿_𝑖 until 𝛿_𝑖+1=𝛿_𝑖.

As an invariant ellipsoid, the invariant set can be represented as the following ellipsoid:𝛿=𝑥∈ℝ^𝑛∣𝑥−𝑥_𝑎^𝑇𝑄⁻¹𝑥−𝑥_𝑎≤1, (3.10) where 𝑄^𝑇=𝑄∈ℝ^𝑛 and 𝑄≻0. Given an initial set, the maximal invariant set in it can be computed as a convex optimization problem. 𝑥_𝑎 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:𝛿₀=𝑥∈ℝ^𝑛∣𝑣^𝑇_𝑘𝑥≤1,𝑘=1,…,𝑞. (3.11)

Then this invariant set can be computed as a convex problem (CP), and it is more convenient to compute by LMI toolbox:Minimizelogdet𝑄⁻¹Subjectto𝑄𝐴^𝑇𝑣+𝐴𝑄≤0,𝑃≻0,_𝑇𝑘𝑄⁻¹𝑣_𝑘≤1. (3.12)

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 system 𝑞_𝑖, 𝑓_𝑖(𝑥)=𝐴_𝑖𝑥 exists as invariant sets by arbitrarily transition if a common Lyapunov function exists in each subsystem.

By the common Lyapunov function, if there exists 𝐴^𝑇_𝑖𝑃𝐴_𝑖≤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 𝑞_𝑖, 𝑓_𝑖, and define 𝑣(𝑡)∶=𝑉(𝑥(𝑡))𝑣̇=𝜕𝑣(𝑥)̇𝜕𝑥𝑥=𝜕𝑣𝑓𝜕𝑥^𝑖≤0. (4.1)
(2) There is 𝑣(𝑡)∶=𝑉(𝑥(𝑡))≤𝑣(0)∶=𝑉(𝑥(0)). (4.2)
So, if the hybrid system is given an initial set which can be found as 𝑉(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 Theorem 4.1 exists. If the systems satisfy the following condition, as matrices 𝑅_𝑖, 𝑅_𝑖≻0 can be found to satisfy _{𝑞𝑖∈𝑄}𝐴^𝑇_𝑖𝑅_𝑖+𝑅_𝑖𝐴_𝑖>0, (4.3) then there is not a solution 𝑃=𝑃^𝑇 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 Lemma 3.2. It can be shown as the example below.

Example 4.2. One has the following:  𝑞₁∶𝑓₁(𝑥)=𝐴₁𝑥,𝑞₂∶𝑓₂(𝑥)=𝐴₂𝐴𝑥,₁=−0.11−10−0.1,𝐴₂=.−0.110−1−0.1 (4.4)

This is a simple transition system. In mode 𝑞₁, 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.

(a)

(b)

Figure 3 

Trajectory of system 𝑥̇=𝑓1(𝑥),𝑥̇=𝑓2(𝑥).

However, after arbitrary transitions between the two modes, the states of the system may not shrink to an invariant set. As Figure 4 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 Section 4.2 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 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐸_𝑖(𝑡)𝑥(𝑡),𝐸_𝑖(𝑡)≤𝜂_𝑖,𝑖∈𝑄 (4.5) if there exist a positive matrix 𝑃(𝑃=𝑃^𝑇) and positive number 𝛽 satisfying the matrix inequalities below: ^𝑚_𝑖=1𝛿_𝑖𝐴_𝑖^𝑇𝑃+𝑃^𝑚_𝑖=1𝛿_𝑖𝐴_𝑖+𝑃²+𝛽𝐼≤0,^𝑚_𝑖=1𝛿_𝑖=1,𝜂≤𝛽2. (4.6)(4.7) Then, there exist invariant sets in this hybrid system by efficient transition.

In Theorem 4.3, (∑^𝑚_𝑖=1𝛿_𝑖)=1, 𝐸_𝑖(𝑡)𝑥(𝑡) is the disturbance of each system𝑖, and ||𝐸_𝑖(𝑡)||≤𝜂_𝑖, 𝜂_𝑖is a constant, 𝜂=max(𝜂_𝑖). The domain of each system can be computed as follows:𝐷_𝑖=𝑥(𝑡)∣𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖+𝑃²+𝛽𝐼𝑥(𝑡)≤0. (4.8)

Proof. Construct the Lyapunov function as follows: 𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝛾^𝑡₀𝑥^𝑇(𝜏)𝑥(𝜏)𝑑(𝜏),𝛾>0. (4.9) so 𝑉̇(𝑥(𝑡))=𝑥^𝑇(𝑡)𝐴^𝑇𝑃+𝑃𝐴𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐸(𝑡)𝑥(𝑡)+𝛾𝑥^𝑇(𝑡)𝑥(𝑡)≤𝑥^𝑇(𝑡)𝐴^𝑇𝑃+𝑃𝐴𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃²𝑥(𝑡)+𝑥^𝑇(𝑡)𝐸^𝑇(𝑡)𝐸(𝑡)𝑥(𝑡)+𝛾𝑥^𝑇(𝑡)𝑥(𝑡)=𝑥^𝑇(𝑡)𝐴^𝑇𝑃+𝑃𝐴+𝑃²+2𝛾𝐼𝑥(𝑡)+𝑥^𝑇(𝑡)𝜂²−𝛾𝑥(𝑡). (4.10)
From (4.3), 𝛿₁𝐴^𝑇₁𝑃+𝑃𝐴₁+𝛿₂𝐴^𝑇₂𝑃+𝑃𝐴₂+⋯+𝛿_𝑚𝐴^𝑇_𝑚𝑃+𝑃𝐴_𝑚+𝑃²+𝛽𝐼≤0. (4.11) Considering the condition 𝛿₁+𝛿₂+⋯+𝛿_𝑚=1, the inequality can also be written as 𝛿₁𝐴^𝑇₁𝑃+𝑃𝐴₁+𝑃²+𝛽𝐼+𝛿₂𝐴^𝑇₂𝑃+𝑃𝐴₂+𝑃²+𝛽𝐼+⋯+𝛿_𝑚𝐴^𝑇_𝑚𝑃+𝑃𝐴_𝑚+𝑃²+𝛽𝐼≤0. (4.12) Suppose that 𝐷_𝑖=𝑥(𝑡)∣𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖+𝑃²+𝛽𝐼𝑥(𝑡)≤0. (4.13) Taking 𝛽=2𝛾, when 𝑥(𝑡)∈𝐷_𝑖, 𝑉̇(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖+𝑃²+𝛽𝐼𝑥(𝑡)+𝑥^𝑇(𝑡)𝜂²−𝛽2𝑥(𝑡). (4.14)
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 Theorem 4.3, the computation for the efficient transition can be concluded as follows.(1)Compute whether there exists common Lyapunov function by Theorem 4.1 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)Compute 𝛽_max and matrix 𝑃 in (4.6), and inequality (4.6) is equivalent to the linear matrix inequalities (LMI) as below: 𝑃>0,𝛽>0,^𝑚_𝑖=1𝛿_𝑖𝐴_𝑖^𝑇𝑃+𝑃^𝑚_𝑖=1𝛿_𝑖𝐴_𝑖+𝑃²+𝛽𝐼𝑃𝑃−𝐼<0. (4.15)𝛽_max and 𝑃 can be computed by LMI toolbox in Matlab. (3)Compute whether 𝜂≤𝛽_max/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 in ℝ² 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:  𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐸_𝑖(𝑡)𝑥(𝑡),𝐴₁=−30−22,𝐸₁=,𝐴0.6sin𝑡00−0.3cos𝑡₂=160−6,𝐸₂=.−0.1sin𝑡000.5sin𝑡 (5.1)(5.2) For the system, we know that 𝜂=max𝜂₁,𝜂₂=0.6. (5.3) 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, maxRe𝜆𝐴₁𝜆𝐴=2,maxRe₂=1. (5.4) 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 𝛽_max and matrix 𝑃. Fist, 𝛿_𝑖 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: 𝛽_max=0.993,𝑃=0.61740.79780.79782.8776. (5.5) (3) Compute the maximum range of the disturbance, 𝛽_max/2=0.7069, 𝜂≤𝛽_max/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)

𝐷₁=𝑥(𝑡)∣𝑥^𝑇(𝑡)𝐴^𝑇₁𝑃+𝑃𝐴₁+𝑃²𝑥(𝑡)=𝑥(𝑡)+𝛽𝐼<0∣𝑥^𝑇=𝑥−4.8786−3.7647−3.764721.4268𝑥<0₁,𝑥₂^𝑇∣−4.8786𝑥²₁−7.5294𝑥₁𝑥₂+21.4268𝑥²₂,𝐷<0₂=𝑥(𝑡)∣𝑥^𝑇(𝑡)𝐴^𝑇₂𝑃+𝑃𝐴₂+𝑃²𝑥(𝑡)=𝑥(𝑡)+𝛼𝐼<0∣𝑥^𝑇=𝑥3.25182.50372.5037−15.0412𝑥<0₁,𝑥₂^𝑇∣3.2518𝑥²₁−5.0074𝑥₁𝑥₂−15.0412𝑥²₂<0. (5.6)

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 Figure 5, 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.

Acknowledgments

The authors gratefully acknowledge the contribution of the National Science Foundation of China [61001185][60903114][61003271], the Specialized Research Fund for the Doctoral Program of Higher Education [20040561031], and the National Natural Science Foundation of Guangdong Province [031407].