# On Stability for State Constrained Hybrid Systems via Barrier Lyapunov Functions

Authors: Liangliang Li; Wenlin Jiang
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, the stability of familiar state constrained hybrid systems is considered. In the first, we prove the invariant set stability for the state constrained impulsive hybrid systems. Specifically, a robust control feedback method is applied for state constrained uncertain impulsive hybrid systems. With the auxiliary matrix assistance, some convergence criteria are derived to guarantee robust stability for state constrained uncertain hybrid systems with output disturbance by constructing the symmetric and asymmetric barrier Lyapunov functions (BLF), respectively. Finally, two comparative examples with simulations show that the proposed results are effective and superior.


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## Body

## 1. Introduction

A hybrid system consists of many aspects, such as computation, communication, control, continuous dynamics, and discrete behavior [1, 2]. Hybrid system is a complex system, and its modeling methods mainly consist of two kinds: hybrid state machine and differential equation. Within each step, the state of the system is constantly evolving according to the dynamic law, until a transition occurs. Some transient phenomena of state changes (impulsive system) are one of the most important branches of hybrid systems and are essentially a nonlinear system. The theory of stability or finite-time stability of impulsive systems or hybrid systems has also obtained rapid development [3–7] in recent years. For the actual engineering system, all the actuators have the constraints of human participation or the inherent physical constraints of the dynamical system itself. For example, the valve has jammed shut while driving to a certain extent. The saturation nonlinearity of this actuator makes it necessary for us to consider controlling the influence of input saturation or constrained nonlinearity on the dynamical system when designing the control system. In [8], Bemporad et al. considered the problem of explicit linear quadratic regulator for constrained systems. In [9], Li et al. studied the input delayed nonlinear state constrained systems via adaptive tracking control. There are alternatives to solve the state constraint problems that are effective [10–17], such as basing on the invariant set principle and model predictive control. Recently, BLF is presented to consider the problem of constraint dynamical system. In [14], control design for partial participation constraints nonlinear systems is considered. In [15], Tee et al. present control design for delayed and output constraint nonlinear systems. In [16], Liu et al. develop a state adaptive controller for state constraint stochastic nonlinear systems with some parameters. While impulsive systems have been studied comprehensively, state constrained impulsive nonlinear systems require further analysis.To our knowledge, many interesting results on state constraint systems have been reported in the literature [13–18]. But, those results are not suitable for the hybrid systems due to its particularity and complexity. That is, the control problem for state constrain impulsive dynamic systems has not been thoroughly studied, many challenging works still exist. In this paper, we propose a special type of feedback controller, which can be addressed by the robust control for uncertain state constraint impulsive hybrid systems with input disturbance. The matrix theory and inequality technique show that the control impulsive nonlinear dynamic systems are stable. Finally, some comparative examples with simulations show that the results in this paper are viable.The thesis consists of the following: Section2 is devoted to the model and some theorems are presented. In Section 3, the robust control applications are established for the main model. Examples are reported in Section 4. In Section 5, the conclusion is drawn.Notations 1.
ℝn:n-dimensional Euclidean space. ℝ+=0,+∞. u denotes the vector norm of the vector u∈ℝn. ∂S is the boundary of set S. ℝn×m: n×m matrices. 0<X∈ℝn×n0≤X: a real symmetric and positive definite (positive semidefinite) matrix.


## 2. Preliminaries

The following nonlinear impulsive hybrid dynamic systems are described:(1)yt=gt,yt,t≥0,yt∈S,△yt=Eyt,yt∈M⊂∂ℝnS,y0=y0∈S,where yt∈ℝn denotes the state of system (1); S is a bounded ellipsoid in ℝn. gt,yt∈Cℝ+×S,ℝn. If yt∈M, the impulsive point set M⊂∂ℝn/S. It is denoted that △ytk=ytk+−ytk−, ytk+=ytk, and ytk−=limt⟶tk−yt. Eyt is a continuous impulsive function. For dynamic systems (1), the initial conditions were y0=y0∈S.A setT is called an invariant set [19] of system (1), if xt,x0∈T for any x0∈T, t≥0. In the following, some sufficient conditions for the stability of system (1) will be given.Theorem 1.
LetS be a bounded ellipsoid in ℝn, for 0<μ≤1. If there exists a positive definite function Vt,yt such that the Vt,0=0 for all t and(2)D+Vt,y|1≤0,y∈S,Vt,y+Ey≤μVt,y,y∈M.Then, the stateyt of system (1) will remain in the bounded ellipsoid S.proof.
There are two possible solutions to the system (1).Case 1.
The solution never hits the boundary ofS. That is, yy0,t∉M, for any time t. From the condition of Theorem 1, we have Vy≤Vy0. This implies that the conclusion is clearly valid.Case 2.
The solution will hit the boundary ofS; assume that the impulsive set tky0, k=1,2,… (tk+1y0>tky0 and tky0⟶+∞ as k⟶+∞).
Fort∈[0,t1y0), and D+Vt,y≤0, then Vyt≤Vy0 for t∈[0,t1y0).
Combining with (2), when t=t1y0, we have Vt1y0+≤μVt1y0−≤Vy0. By deduction method, for t∈[tky0,tk+1y0), from (2), we have Vtky0+≤μVtky0−≤Vy0.
Then,Vt≤Vy0 for ∈[tkx0,tk+1y0).
Then, the stateyt will stay in ellipsoid S.Remark 1.
From Theorem1, if y0∈S, the trajectory of system (1) will remain in the set S. In addition, in this paper, we assume that there are no impulses at initial value.Remark 2.
From Theorem1, it can be used to solve the bouncing ball system [20,21] which is typical state constrained systems or state-dependent impulsive systems. Here, we consider that the system state is limited to a bounded region, and when the trajectory of systems reaches the boundary, it will trigger the impulsive input. Then, the state returns to the internal regionExample 1.
In Figure1, let yt be altitude of the oscillating platform, xt be altitude of the ball, ϖ1t denote the relative displacement between altitude of the ball and altitude of the oscillating platform, and ϖ2t denote the velocity. The mass of the ball is ignored and all collisions are instantaneous, and g denotes the gravitational acceleration. Then, dynamic model simulates can be written as(3)ϖ˙1t=ϖ2t,ϖ˙2t=−g−y¨t,ϖ1t,ϖ2t∈S;ϖ2t+=−μϖ2t−,ϖ1t,ϖ2t∈M=∂S,where S=ϖ1t,ϖ2t∈ℝ2|0≤ϖ1t≤h; M=ϖ1t,ϖ2t∈ℝ2|ϖ1t=0orh. Assume Vϖt=1/2ϖ22t+gϖ1t, while ϖ1t,ϖ2t∉M. Such collision can lead to energy degradation, then V˙ϖt≤0. When ϖ1t,ϖ2t∈M, and μ≤1, then Vϖt+=1/2ϖ22t++gϖ1t+=1/2μ2ϖ22t−+gϖ1t−≤1/22ϖ22t−+gϖ1t−=Vϖt−. All the conditions are met to Theorem 1, then the ball and the platform finally moved steadily together (see Figure 2).
Recently, the BLF is proposed to solve the state-constrained system [13, 15, 16, 22]. In the following, we consider the uncertain state constraint impulsive systems.Figure 1
Bouncing ball dynamics (model).Figure 2
Bouncing ball dynamics (trajectory).


## 3. Robust Control for Uncertain State Constraint Impulsive Systems

Consider the system in the following:(4)z˙t=A+△Azt+B+△Bfzt+Ut,zt∈Θ,△zt=Ezt,zt∈M⊂∂ℝΘ,φt=zt+ξt,z0=zt0,ξ0=ξ0,with A,B∈ℝn×n, where state vector zt∈ℝn is constrained in a ellipsoid Θ∈ℝn. The uncertainties △A, △B are time-varying matrices. φt∈ℝn is the measurable output. Ezt denotes the impulsive effects. Ut is the control input, fz is the nonlinear continuous function, and fz∈ℒ.(5)ℒ=f⋅|f∈CΘ,ℝn,∃Li>0,0<zifizi≤Lizi2,,for zi∈ℝ. Let L=diagL1,L2,…,Ln, where ξt∈ℝn is partial unmeasurable, which is bounded for the symmetric matrix M and a constant η>0 such that(6)ξTMξ≤η.For the ellipsoidΘ, it must exist as 0<R∈ℝn×n such that Θ≔z|zTRz≤η. In this paper, design the controller.(7)Ut=Kφt,where K is a control gain. Also, we take the general assumption for the impulsive effects.Assumption 1.
There exists a positive constantμ≤1 such that(8)ztk++Eztk+Tztk++Eztk+≤μzTtkztk.Lemma 1.
∀ϵ>0,a1,a2∈ℝn; the inequality 2a1Tb≤ϵa1Ta1+ϵ−1a2Ta2 holds.We will present the stability for the state constrained uncertain hybrid systems (4).Theorem 2.
Letγ, ε1, ε2∈ℝ+. If there exists 0<P,R∈ℝn×n such that(9)Ξ+2γPR−P+A+△ATR+KTRKTP∗−2R+ε2RB+△BB+△BTRR∗∗−γM≤0.Then, the state systems (4) start in the ellipsoid Θ and remain in set Θ, and it converges to zero, where Ξ=−γR+PA+△A+A+△ATP+2PK+ε1PB+△BB+△BTP+ε1−1L2I+ε2−1L2I.proof.
We choose the BLF as(10)Vz=Vzt=logηη−zTRz.
With the help of matrixP, add the term(11)Φz,z˙=1η−zTRz2zTP+2z˙TRA+△Az+B+△Bfz+Ut−z˙.
Design the control input asUt=Kφ=Kz+ξ. When z∈Θ, calculating time derivative of Vz along the trajectory of system (4), one may obtain that(12)V˙z|3=2zTRz˙η−zTRz=2zTRz˙+2zTP+2z˙TRA+△Az+B+△Bfz+Ut−z˙η−zTRz=1η−zTRz2zTRz˙+2zTPA+△Az+2zTPB+△Bfz−2zPz˙t+2zTPKz+2zTPKξt+2z˙TtRA+△Az+2z˙TRB+△Bfz+2z˙TtRKz+2z˙TtRξt−2z˙TtRz˙≤1η−zTRz2zTRz˙t+2zTPA+△Az+ε1zTPB+△BB+△BTPz+ε1−1fTzfz−2zTPz˙t+2zTPKz+2zTPKξt+2z˙TtRA+△Az+ε2z˙TtRB+△BB+△BTRz˙Tt+ε2−1fTzfz+2z˙TtRKz+2z˙TtRξt−2z˙TtRz˙t.
Let2zRz˙t+2zPA+△Az+ε1zPB+△BB+△BTPz+ε1−1L2zTz−2zPz˙t+2zPKz+2zPKξt+2z˙TtRA+△Az+ε2z˙TtRB+△BB+△BTRz˙Tt+ε2−1L2zTz+2z˙TtRKz+2z˙TtRξt−2z˙TtRz˙≤−γη−zTRz hold. That is to say, one can get that V˙z≤−γη, if δTΩ0δ+γη≤0, where(13)Ω0=ΞR−P+A+△ATR+KTRKTP∗−2R+ε2RB+△BB+△BTRR∗∗0.
Denote thatδ=zTt,z˙Tt,ξTtT. We have(14)δTΩδ+γη+ξTtMξt−2zTtPzt≤0.
We can finally write that(15)δTΩδ+2γη−zTtPzt≤0.
That is, under the condition that (9) and (15) hold, then the inequality (14) implies that V˙z≤−γη<0.
Whenzt∈M, combining with Assumption 1,(16)Vztk+=logηη−ztk−+Eztk−TRztk−+Eztk−≤logηη−μzTtk−Rztk−≤Vtk−.
From Theorem1, then the state systems (4) start in the ellipsoid Θ and remain in set Θ, and it converges to zero.Remark 3.
As we have known, most of the existing literature only considered the general impulsive system and the cases of state constrained are not included in Theorem1, the BLF is applied for the state constrained impulsive systems. In the following, we assume that z1t∈−d2,d1, in fact, in Theorem 2, d1=d2, the bound of state can be as a special suprasphere yt<ω. So, there exist some positive constants η1, η2 such that(17)zTRz≤η1,∀z1t∈[0,d1).zTRz≤η2,∀z1t∈−d2,0.
For the asymmetrical case, we will consider the asymmetric BLF [15, 16].Theorem 3.
Letγ, ε1, ε2∈ℝ+. If there exists 0<P,R∈ℝn×n such that(18)Ξ1+2γ1PΞ3KTP∗Ξ5R∗∗−γ1M≤0,Ξ2+2γ2PΞ4KTP∗Ξ5R∗∗−γ2M≤0.Then, the state of systems (4) remains in set Θ, and it converges to zero, where(19)Ξ1=−γ1R+PA+△A+A+△ATP+2PK+ε1PB+△BB+△BTP+ε1−1L2I+ε2−1L2I,Ξ2=−γ2R+PA+△A+A+△ATP+2PK+ε1PB+△BB+△BTP+ε1−1L2I+ε2−1L2I,Ξ3=ϱx1R−P+A+△ATR+KTR,Ξ4=1−ϱx1R−P+A+△ATR+KTR,Ξ5=−2R+ε2RB+△BB+△BTR.proof.
Consider the BLF candidate as(20)Vz=ϱz1logη1η1−zTRz+1−ϱz1logη2η2−zTRz,where(21)ϱz1=1,ifz1≥0,0,ifz1<0.
One can rewriteVz as(22)Vz=logη1η1−zTRz,if0≤z1<d1,logη2η2−zTRz,if−d2<z1<0.
With the help of matrixP, add the term(23)1η1−zTRz2zTP+2z˙TRA+△Az+B+△Bfz+Ut−z˙,1η2−zTRz2zTP+2z˙TRA+△Az+B+△Bfz+Ut−z˙.
Design the control input asUt=Kφ. When zt∈Θ, calculating time derivative of Vz, one may obtain that V˙z|3(24)=ϱz12zTRz˙η1−zTRz+1−ϱz12zTRz˙η2−zTRz=2ϱz1zTRz˙+2zTP+2z˙TRA+△Az+B+△Bfz+Ut−z˙η1−zTRz+21−ϱz1zTRz˙+2zTP+2z˙TRA+△Az+B+△Bfz+Ut−z˙η2−zTRz≤Υ1η1−zTRz+Υ2η2−zTRz.
LetΥ1=2ϱz1zRz˙t+2zPA+△Az+ε1zPB+△BB+△BTPz+ε1−1L2zTz−2zPz˙t+2zPKz+2zPKξt+2z˙TtRA+△Az+ε2z˙TtRB+△BB+△BTRz˙Tt+ε2−1L2zTz+2z˙TtRKz+2z˙TtRξt−2z˙TtRz˙≤−γ1η1−zTRz and Υ2=21−ϱz1zRz˙t+2zPA+△Az+ε1zPB+△BB+△BTPz+ε1−1L2zTz−2zPz˙t+2zPKz+2zPKξt+2z˙TtRA+△Az+ε2z˙TtRB+△BB+△BTRz˙Tt+ε2−1L2zTz+2z˙TtRKz+2z˙TtRξt−2z˙TtRz˙≤−γ2η2−zTRz hold. That is, we can get that V˙z≤−γ1η1−γ2η2, if δTΩ1δ+δTΩ2δ+γ1η1+γ2η2≤0, where(25)Ω1=Ξ1Ξ3KTP∗Ξ5R∗∗0,Ω2=Ξ2Ξ4KTP∗Ξ5R∗∗0.
Note thatδ=zT,z˙T,ξTT. Then, we have(26)δTΩ1δ+δTΩ2δ+γ1η1+ξTMξ−2zTPz+γ2η2+ξTMξ−2zTPz≤0.
We can finally write that(27)δTΩ1δ+δTΩ2δ+2γ1η1−zTPz+2γ2η2−zTPz≤0.
That is, if (27) holds, then the inequality (26) implies in the subset of ℝn described by Θ, then V˙z≤−γ1η1−γ2η2<0 for zt∉M.
Whenzt∈M, under the condition Assumption 1, we can get that Vztk+(28)=ϱz1tk+logη1η1−zTtk+Rztk++1−ϱz1tk+logη2η2−zTtk+Rztk+≤ϱz1tk+logη1η1−μzTtk−Rztk−+1−ϱz1tk+logη2η2−μzTtk−Rztk−≤Vtk−.
Combining with Theorem1, then the state systems (4) start in the ellipsoid Θ and remain in set Θ, and it converges to zero. This completes the proof.Remark 4.
Without loss of generality, letη=1, R=I, the following corollary can be drawn.Corollary 1.
Assume that there existγ, ε1, ε2∈ℝ+. 0<P∈ℝn×n such that(29)Ξ+2γPI−P+A+△AT+KTIKTP∗−2R+ε2B+△BB+△BTI∗∗−γM≤0.Then, the state of systems (4) remains in set Θ, and it converges to zero, where Ξ=−γI+PA+△A+A+△ATP+2PK+ε1PB+△BB+△BTP+ε1−1L2I+ε2−1L2I.proof.
Consider the BLF that(30)Vz=log11−zTz.
Similar to the discussion of Theorem2, more detailed proof is omitted here.
For model (4), if there is no external disturbance, that is, ξt=0, △A=△B=0. From the above, we can get a corollary that the state constrained system stability can be ensured when relative variables are used as feedback variables and appropriate gains have been chosen.Corollary 2.
Letγ, ε1, ε2∈ℝ+. If there exists 0<P∈ℝn×n such that(31)Ξ+2γPR−P+ATR+KTR∗−2R+ε2RBBTR≤0.Then, the state of systems (4) (with ξt=0, △A=△B=0) remains in set Θ, and it converges to zero, where Ξ=−γR+PA+ATP+2PK+ε1PBBTP+ε1−1L2I+ε2−1L2I.Remark 5.
It could tell that it is difficult to calculate theoretically for BLF [23, 24] which is different from the traditional Lyapunov function. In this paper, introducing the descriptor method and adding an additional term, some linear matrix inequality conditions are obtained.Remark 6.
For the uncertain△A, △B, also use the assumptions, such as (1) ∃EA, EB such that △A≤EA, △B≤EB. (2) ∃FA, FB such that △AT△A≤FATFA, △BT△B≤FBTFB. Some useful conclusions can also be drawn. For the convenience of calculation, we usually use one of the above assumptions in numerical validation.


## 4. Examples

Two comparative examples with simulations show that the proposed results are effective.Example 2.
Consider the state constrained nonlinear impulsive system.(32)y˜˙1=−1.45y˜1+0.23y˜2+1.2tanhy˜1+0.9tanhy˜2+13sin−y˜1,y˜∈S,y˜˙2=0.35y˜1t−1.5y˜2+1.3tanhy˜2+13cos−y˜2,y˜∈S,△y˜1=−0.34y˜1,y˜t∈M⊂∂S,△y˜2=−0.45y˜2,y˜t∈M⊂∂S,y˜10=−0.49,y˜20=0.85,where S=y˜t∈ℝ2‖y˜1|<1.2,y˜2<1.2, M=y˜∈ℝ2‖y˜1|=1.2,y˜2=1.2. Let Vt=1/2y˜Tty˜t; combining with Theorem 1, we can verify that D+Vy˜≤0, y∈S, Vy˜+Ey˜≤Vy˜ hold. From Theorem 1, the trajectory of system (32) remains in one region. Figure 3 shows the trajectory of states y˜t of systems (32).
From Figure3, we can find that when y˜1=±1.2, or y˜2=±1.2, it will trigger the impulsive mechanism. The impulses can also be an effective strategy which can stabilize the system. It can keep its solution remain in one constraint region.Figure 3
State trajectoriesyt for system (32).Example 3.
Consider the uncertain nonlinear impulsive state constrained dynamic systems (4) with Θ=zt∈ℝ2‖z1t|<0.13,z2t<0.2, fs=s+1+s−1/2,(33)A=2.10.30.51.8,△A=15sint0015cost,B=1.51.201.54,△B=15sint0015cost,M=z∈ℝ2‖z1|=0.13,z2=0.2.
△z1=−0.74z1, △z2=−0.65z2. The disturbing function ξt=1/5te−t,te−tT with ξ0=0,0T. Here, z0=−0.05,0.05T.
DesignUt=Kφt=−5zt+ξt. Choose ε1=0.92, ε2=0.91, R=I. By MATLAB tool, take(34)P=0.87620.65320.65320.7937.
From Theorem2, then the state constrained impulsive hybrid systems (4) is stable. Figures 4 and 5 show the trajectory of states z1t and z2t of (4).Figure 4
State trajectorieszt for system (4) without controller.Figure 5
State trajectorieszt for system (4) with controller.


## 5. Conclusions

In this text, the stability of state constrained hybrid systems has been considered. By constructor barrier Lyapunov method, the stabilization for a class of uncertain hybrid dynamical systems has been derived. Furthermore, using the examples, the availability of the proposed main results has been demonstrated. The main results may extend to consider the synchronization of state constrains impulsive hybrid dynamic systems by barrier Lyapunov functions. In the future, studying the stability of delayed state constrained impulsive hybrid system will be another interesting subject.



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