Finite-Time Boundedness and H ∞ Control for Affine Switched Systems

For affine switched systems, the existence of multiple equilibria is related to subsystems owing to the affine terms, which makes asymptotic and finite-time stability analysis nontrivial. In this paper, the problems of finite-time boundedness (FTB) analysis and stabilization are addressed for affine switched systems, and several definitions and sufficient conditions are proposed to study FTB andH∞ performance. At first, the definition of FTB for affine switched systems is improved concerning the affine terms andmultiple equilibria. Based on the FTB definition, sufficient conditions ensuring finite-time boundedness for affine switched systems under a prespecified state boundary are given. Then the results are extended to solve H∞ finite-time boundedness problem, in which theH∞ controllers are designed to guarantee the finite-time boundedness of affine switched system withH∞ performance. In our investigation, average dwell-time approach is employed to study the time-dependent constrained switching case. Finally, several numerical examples are given to illustrate the effectiveness of the proposed results.


Introduction
Switched systems are distinctive subclass of hybrid systems.They are composed of a family of continuous-time or discrete-time subsystems with a criterion that rules the switching among them.This switching rule can be classified as time-dependent, state-dependent, or time-statedependent [1].Since many physical processes possess switching nature, and many real-world applications resort to switching strategy to improve the control performance, the theory and application of switched systems have received a great attention during the recent decades.For more details on the recent results about the basic problems in stability and stabilization for switched systems, readers are referred to surveys [2][3][4] and books [1,5] and the references cited therein.
The issue of stability analysis and stabilization is an important topic for switched dynamical systems [6][7][8][9][10].Finding sufficient conditions ensuring the Lyapunov asymptotic stability dealing with infinite time interval has been the major concern for switched systems.Numerous published results discussed the asymptotic stability analysis and stabilization employing different variations of Lyapunov function [7,11,12].Average dwell-time approach [13,14] and Lie-algebraic condition technique [15,16] are effective tools for analysis of switched systems.On the contrary, the finite-time behavior of dynamical systems is also of interest in many practical applications.It concerns that the states do not exceed a certain bound during a fixed time interval, e.g., to avoid saturations or excitation.The theory of finite-time stability (FTS) and finite-time boundedness (FTB) focuses on the transient response of dynamical systems over a finite-time interval, while asymptotic behavior is for infinite time.In the survey of recent development of this innovative theory, some necessary and sufficient conditions for finite-time stability and stabilization of continuous-time systems or discretetime systems have been provided in [17,18].Based upon it, necessary and sufficient conditions for finite-time stability of systems with impulsive effects were obtained in [19,20].The authors [21,22] applied FTS/FTB conceptions to switched systems and compared the conservativeness among different conditions.In [23], the mixed  ∞ /finite-time stability control problem was discussed.For quadratic input-output finite-time stability with an  ∞ bound, [23] provided a necessary and sufficient condition.Then the method was extended to robust  ∞ controller and filter design for switched system with exogenous noise [24,25].It should be noted that finite-time stability and Lyapunov asymptotic stability are independent concepts: a Lyapunov asymptotic stable system may not fulfill FTS/FTB criteria since the transient response of a system may exceed the bound, and vice versa [26].In many practical applications, switching is likely to occur in some short-time intervals, whereas for remaining long time no switching occurs.Since Lyapunov stability concerns with infinite time, it may not be influenced by such short-time switching.However, the boundedness of state may be affected by the switching.Hence, FTB criteria are needed to be considered for designing controller and switching laws during such applications.
Most of the existing literatures on stability issues of switched systems are based on the premise that all subsystems share a common equilibrium (typically the origin).On the other hand, for affine switched system, subsystems have different equilibria, so complex and interesting phenomena emerge.Almost all the practical hybrid systems can be modeled as affine switched systems.Many results like [27][28][29][30] analyzed interesting behaviors similar to those of asymptotically stable systems near an equilibrium for affine switched systems and depicted their real-world applications.Many extensions of the conventional stability concepts have been obtained for affine switched systems.S-Procedure method with the extensional state vector has been proposed in [31,32] to analyze the asymptotic stability for continuous affine switched system.The relative results were extended to discrete affine switched systems in [33].In [34], a method for designing switching rules driving the state of affine switched system to a desired equilibrium was investigated.Almost all the existing literatures on stability analysis of affine switched systems focused on the asymptotic stability.However, the boundedness of state for affine switched systems under constrained dwell-time switching is also of significant interest for affine switched systems.In FTB analysis, we also need to deal with affine terms leading to multiple equilibria for affine switched systems, but the investigation of this problem lacks researchers' interest previously.Potential of affine switched systems theory and importance of finitetime transient behavior from the perspective of real-world applications are the major motivations for this investigation presented in this paper.
The main objective in this paper is to find sufficient conditions ensuring the FTB of affine switched systems by switching signal and feedback controllers design and to drive the state of affine switched system to the prescribed neighborhood of a desired equilibrium during a finite-time interval.Taking into account the influence of affine terms on FTB for affine switched system, we propose an innovative FTB concept.Based on this definition, sufficient conditions ensuring the affine switched system finite-time bounded are proposed.Specifically, with the prespecified state boundary, average dwell time and state-feedback controllers for each subsystem are determined to guarantee the finite-time boundedness.The paper [22] points out that the more information about switching signal we know, the less conservative results can be derived.We extend this idea to switched affine systems to further reduce the conservatism.Classifying subsystems into asymptotically stable and unstable systems, we get the less conservative results of finite-time boundedness for affine switched system with the help of additional information of switching signal.Then, results are extended to solve the FTB problem for  ∞ controller design.
The rest of this paper is organized as follows.In Section 2, definitions of finite-time boundedness and  ∞ finite-time boundedness for affine switched system are revisited.Based on these definitions, finite-time boundedness analysis and finite-time stabilization are presented in Section 3. Then in presence of exogenous signals,  ∞ finite-time boundedness and the controllers design are investigated in Section 4. In Section 5, several numerical examples are presented to validate the proposed results.Conclusions are given in Section 6.

Preliminaries and Problem Formulation
For our investigation, we consider continuous-time affine switched system described as where () ∈ R  is the system state, () ∈ R  is the control input, () ∈ R  is the measurement output,   ,   , and   are system matrices with appropriate dimensions, constants   are affine terms, and () : R + →  = {1, ⋅ ⋅ ⋅ , } is switching signal.For notational simplicity, we use  in place of ().
Matrix variables   ,   , and   give rise to an equilibrium (stable or unstable) for each subsystem; assuming all   to be nonsingular, we consider a given reference   as the required equilibrium for the whole system, referred to as switched equilibrium.Without loss of generality, it is assumed that the desirable equilibrium is different from all the equilibria of subsystems.Now although the asymptotic stability of affine switched system may be achieved by other types of switching strategy such as min-switching and sliding method, the state will not exactly converge to   under dwell-time constrained switching.The reason is that there always exist time interval (dwell time is always greater than zero) in which state must diverge from   .In our FTB investigation, we provide solution for boundedness of error state under dwelltime switching, which depicts the importance and innovation of our approach.
Here first we will extend the FTS and FTB concepts for affine switched systems keeping in view prescribed equilibrium   .In absence of control input, system (1) can be stated as Definition 1. Autonomous affine switched system (2) is said to be finite-time bounded with respect to (  ,   , ,   ,   , ) if the following inequalities hold: where  max = argmax =1,⋅⋅⋅ , {       },   =     +   , 0 ≤   < ,   ≥ 0,   > 0,   > 0, and  ∈ R + .Remark 2. Given equilibrium   and system (2), its tracking error system can be written as where In other words, it guarantees the error state () tracking the origin in finite-time interval.Therefore, our study about FTB of affine switched systems can be turned into analyzing its corresponding tracking error system.Moreover, it is worth noting that FTB theory for general switched systems is related to initial state  0 [35,36]; whereas for affine switched systems, we are concerned with  0 as well as the desired equilibrium   and affine terms   .Thus, in the Definition 1, the premise constraint conditions are extended to both initial state  0 and   to analyze the FTB of affine switched systems, where   is related to the desired equilibrium   and affine terms   .Remark 3.With the state-feedback controller () =   (),  ∈ , affine switched system (1) can be rewritten into the following closed-loop system: where   =   +     and the FTB analysis method can be used directly.Similar to the significant impact of switching laws on asymptotic stability, the switching signals affect the finite-time boundedness of affine switched systems property significantly.Therefore, both switching signals and robust controllers should be designed during the FTB analysis of affine switched systems.
Definition 4. For affine switched system (7), considering state-feedback controller () =   () and H ∞ performance index  > 0, if the following two conditions are satisfied: (1) the closed-loop error tracking switched system ( 8) is finite-time bounded; (2) under zero-initial condition, the controlled output  satisfies the inequality where Assuming () = 0,   = 0 system (7) is expressed as Now Definition 4 can be reduced to the following form.Switched system ( 10) is said to be H ∞ finite-time bounded with performance index , if (1) the error tracking switched system (10) is FTB; (2) under zero-initial condition, the controlled output  satisfies Based upon the above preliminaries we will focus on how to find sufficient conditions to ensure the finite-time boundedness of affine switched systems and address the  ∞ analysis and synthesis of piecewise linear state-feedback controllers resorting to LMI-based algorithms.The main problems we concern in this paper can be stated as follows.
Problem 7 ( ∞ performance and controller design).Given affine switched system (8), analyze the  ∞ performance and design set of  ∞ controllers defined in Definition 4 to ensure the finite-time boundedness with respect to (  ,   , ,   ,   , ) and reduce the effect of the exogenous signal  and   on the controlled output  to a prescribed level .

Finite-Time Boundedness and State-Feedback Stabilization
In this section, Problems 5 and 6 are taken into consideration.
Remark 9. When other parameters are fixed, condition (12c) can be described by average dwell time as [37] where  *  = / [0,] .In other words, the average dwell time   should be chosen large enough to ensure that inequality (22) is satisfied, which is necessary to guarantee the finitetime boundedness of affine switched system (2).Moreover, assuming   = , from (12a) and ( 19) we deduce When   → ∞,   → ∞ and the term √ (/)  [0,]  2   (/) on the right side of (23) will become infinite, which explains that the affine switched system (2) is not ultimately bounded, which illustrates FTB and ultimately boundedness are independent concepts.Remark 10.Once the state bound  is not ascertained, the minimum value  min is of interest, which can be found through optimization problem min(/)  [0,]  (/) ( 2  +  2  ) −1 subject to (12a) and (12b).If we fix the parameter  and let  = 1,  = , the optimization problem becomes min Then  min = √  [0,]   ( 2  +  2  ) can be derived with the optimized value .
It is evident that smaller value of  gives rise to less conservative FTB conditions.In Theorem 8, the parameter  indicates the asymptotic stability property of each subsystem.It is well known that when  = 0 in condition (12b), this condition can be regarded as Lyapunov function condition which ensures each subsystem to be asymptotic stable; whereas when  > 0, the condition that V() must be negative is relaxed in FTB sense, and V() just should be no greater than V  () <  −1   +  2  to guarantee the boundedness of state in finite-time interval [0, ].The parameter  ≥ 0 in condition (12b) covers both the asymptotic stable and unstable subsystems.Now let subsystems  1 , ⋅ ⋅ ⋅ ,   be asymptotic stable and  +1 , ⋅ ⋅ ⋅ ,   are unstable, and  − ,  + denote the total activation time for stable and unstable subsystems during [0, ].Then the less conservative results about FTB of affine switched system can be obtained in the following corollary.
Remark 12. Similar to the optimization problem of state bound  described in Remark 10, the optimal value  min can be found according to min(/)  [0,]   +  −1  + ( 2  +  +  +  2  ) −1 subject to (25a) and (25b).We fix the parameter  + and let  = 1,  = , the optimization problem becomes min Then the minimum  min = √  [0,]   +  + ( 2  +  +  +  2  ) can be derived with the optimized value .Since  + ≤ , comparing the value of the optimal state bound  min in Theorem 8 and Corollary 11, we know that, by classifying subsystems into asymptotically stable and unstable, the FTB conditions derived in Corollary 11 are less conservative than that in Theorem 8.
Constituting state-feedback controller of the form () =   (), affine switched system (1) can be transformed into the closed-loop form of ( 5) and Definition 1 of FTB can be used directly.Now we will consider problem-2 to provide sufficient conditions for finite-time state-feedback stabilization.
Proof.Assume   is the switched equilibrium point of affine switched system (1).Applying coordinate transformation we can get its corresponding error tracking switched system as ė () =    () +    () +   (33) where   =     +   and () = () −   .Then under the state-feedback controllers () =   (), and the closed-loop error system can be written as where   =   +     .From Remark 2, we know that FTB analysis and finite-time control can be realized employing tracking error system.Hence, we consider the closed-loop error system (34) here to design the controllers stabilizing the system (1) in finite-time interval.

𝐻 ∞ Performance Analysis and Controller Design of Affine Switched Systems
Based upon FTB investigation of previous section, our main aim now is to design a set of  ∞ controllers to solve Problem 7. As stated in Remark 2, finite-time  ∞ control can be realized through tracking error system, and this will be the main focus in this section.For the sake of simplicity, we firstly consider the autonomous error switched system in the form of (10) assuming that   = 0, (t) = 0 and the corresponding theorem is stated as follows; then we will show how to remove the assumption and extend the results to the general affine switched system with exogenous signal input.
Remark 15.The parameter  is  ∞ performance index and its minimum value  min is often of interest from practical viewpoint; hence, we can state the optimization problem as min  2 ..( Similarly, fulfilling FTB criteria, minimum value of state bound  min is also desired, which can be found as the optimization problem: min(/)  [0,]  (/) ( 2  +  2  +  2  2 ) −1 subject to (39a) and (39b).If we fix the parameter  and let  = 1,  = , then we can state optimization problem as min and ) is derived with the optimized value of .We can adopt a convex combination of  min and  min as () =  2 min + (1 − ) 2 min , 0 ≤  ≤ 1 and a more general convex optimization problem can be stated as min  () Now we will extend the results to design the  ∞ controllers, ensuring FTB of the closed-loop affine switched system (8).Different equilibria for subsystems exist because of the affine terms   , and hence stability analysis and  ∞ control are not trivial.To solve this problem, a few results are available proposing extended state space method in [12,31].However, this approach seems conservative for system synthesis because the eigenvalues of the extended state matrices   related to the affine terms are not exactly the same as for the original state matrices   .For state-dependent affine switched system, S-procedure method can be used to reduce the conservatism.However, for time-dependent affine switched systems, there are only few effective results.In our investigation, we redefine exogenous signal () as Hence, the closed-loop switched system (8) can be rewritten as ė () =    () + G ω () , (0) =  0  () =    () + D2 ω () , ω (0) = ω0 where Employing (60a) and using Schur complement formula, Following the proof guidelines of Theorem 14, condition (60c) which guarantees the FTB of robust affine switched system can be obtained.Now we need to prove condition (9) for  ∞ performance under zero-initial conditions.From (60b), Applying integration and iterations, and setting ( 0 ) = 0 under zero-initial conditions, we get Then setting  = ,  0 = 0, we obtain that which illustrates that condition (9) is satisfied.We conclude that the affine switched system (59), and hence closed-loop affine switched system (8), is FTB with H ∞ performance .
Remark 17.Unlike the normal switched system, the existence of multiple equilibria for affine switched systems is related to subsystems owing to the affine terms   , which makes asymptotic and finite-time stability analysis nontrivial.As for the finite-time  ∞ controller design, concerning with both the external disturbance () and the affine terms   , we redefined the conception of  ∞ controller for affine switched system in Definition 4, based on which the results in this section are obtained.It is worth noting that the  ∞ performance of affine switched system reduces to normal  ∞ performance when assuming   = 0.
It is easy to see in Figure 1 that subject system is FTB with conditions (12a), (12b), and (12c) satisfied.Moreover, assuming  + =  = 1 and using optimization process (24) and (31), the optimal value  min = 1.0058,  1 = [ 0.8425 0.1287 0.1287 1.2053 ],  2 = [  We observe in Figure 2 that the system is FTB, and the  ∞ performance satisfies Thus, according to Definition 4, the autonomous robust error switched system can be regarded as finite-time  ∞ bounded.Moreover, using optimization procedure (56) we get  min = 1.932,  1,1 = [ Substitute controller gains into system (8), the closed-loop error switched system can be written as State responses under state-feedback controllers and switching signal  are shown in Figure 3.We can observe that the closed-loop system is FTB, and the  ∞ performance satisfies Thus, according to Definition 4, the given affine switched system can be regarded as finite-time  ∞ bounded under designed  ∞ controller gains.

Conclusion
In this paper, the problem of finite-time boundedness and finite-time  ∞ control for affine switched systems has been investigated.Several definitions and sufficient conditions for FTB and  ∞ performance are proposed.Based on the average dwell-time method, the FTB conditions of affine switched linear system with known state boundary are derived first in this investigation.To reduce the conservatism of FTB conditions, by classifying subsystems into asymptotically stable and unstable systems, we get the improved FTB conditions for affine switched system presented in Corollary 11.The conservatism of conditions under the two situations is compared.Then applying the finite-time boundedness analysis results, finite-time  ∞ performance is discussed.Finite-time  ∞ controllers are designed to ensure the corresponding closedloop switched system FTB with  ∞ performance.Numerical examples are finally provided to validate our theoretical results.Many real-world systems concern with finite-time and transient behavior; meanwhile, many engineering applications can be modeled as affine switched systems.Therefore, our theoretical results about finite-time boundedness of affine switched systems are supposed to have great potential in the application of practical switched systems.Furthermore, the proposed results in this paper can be extended to the nonlinear affine switched systems which will be considered in future work.