A New Robust Nonfragile Controller Design Scheme for a Class of Hybrid Systems through Piecewise Affine Models

This paper investigates the robustH∞ nonfragile control problem for a class of discrete-time hybrid systems based on piecewise affine models. The objective is to develop an admissible piecewise affine nonfragile controller such that the resulting closed-loop system is asymptotically stable with robust H∞ performance γ. By employing a state-control augmentation methodology, some new sufficient conditions for the controller synthesis are formulated based on piecewise Lyapunov functions (PLFs).The controller gains can be obtained via solving a set of linear matrix inequalities. Simulation examples are finally presented to demonstrate the feasibility and effectiveness of the proposed approaches.


Introduction
Over the past few decades, hybrid systems have drawn tremendous attention from the control community, as they contain the competence to model the interaction between logic components and continuous dynamics [1][2][3][4][5][6].Piecewise affine (PWA) systems are a rich class of hybrid systems, which can provide a useful modeling approach for the stability analysis and controller synthesis of hybrid systems [7][8][9].In addition, the PWA models can approximate smooth nonlinear plants with arbitrary degrees of accuracy [10,11].
In recent years, many valuable references on the systematic analysis and synthesis for the PWA systems have been published [12][13][14][15][16][17][18][19][20][21][22].In [12], the stability of continuous-time PWA systems was analyzed by utilizing discontinuous piecewise Lyapunov functions (PLFs).On the basis of the idea in [12], some algorithms for stability and performance analysis of discrete-time PWA systems were proposed in [14] and the continuity of the PLFs was not required for the discrete-time case.The authors in [20] designed a static output-feedback controller for discrete-time PWA systems such that the resulting closed-loop system was asymptotically stable with robust H ∞ performance .The authors in [21] developed a reconfigurable control approach for continuous-time PWA systems with actuator and sensor faults, and the output tracking problem was investigated simultaneously.In [23], the problem of delay-dependent fixed-order memory piecewise affine H ∞ output-feedback control for a class of nonlinear systems with time-varying delay was considered.The authors in [24] have proposed a novel system-augmentation approach to the delay-dependent reliable piecewise affine H ∞ static output-feedback control for nonlinear systems with timevarying delay and sensor faults in the piecewise-Markovian-Lyapunov-functional-based framework.
On the other hand, since 1980s, the robust H ∞ control problem has drawn much attention as it is robust against the system parameter uncertainties, unmodeled dynamics, and external disturbances [25,26].It is implicitly assumed in this control approach that the controllers are implemented precisely.In fact, controllers gains inevitably possess some degrees of errors or parameter variations due to physical limitations such as the inherent inaccuracy in analog systems, finite word length in arbitrary digital systems, and round-off errors in process of numerical computations [27].
Numbers of literatures have already shown that relatively small perturbations in the controller parameters can even result in instability of the closed-loop system [28][29][30].For this case, it is realistic to synthesize a controller for a given uncertain system such that the controller is insensitive to some predefined admissible variations with respect to its gains.More recently, many efforts have been devoted to the nonfragile robust H ∞ controller design problem [31][32][33][34][35][36][37][38][39][40].For instance, a nonfragile H ∞ controller with multiplicative gain perturbations was proposed for linear time invariant (LTI) systems in [31].The author in [32] developed a decentralized robust state-feedback nonfragile control method for uncertain discrete-time large-scale systems with time-delays and controller gain variations.In [34], a nonfragile robust controller with parameter uncertainties was constructed for a class of nonlinear systems based on T-S fuzzy neural models.The authors in [40] considered the nonfragile robust H ∞ control problem for fuzzy systems based on output information.In spite of the attractive features in piecewise affine systems, the authors in [41] proposed a state-feedback nonfragile robust controller design method for continuoustime PWA systems.The results in [41] were derived based on a common Lyapunov function (CLF), and the matrices   are not empowered to have uncertainties.To the best of the authors' knowledge, few attempts have been devoted to the robust nonfragile controller design for discrete-time piecewise affine systems based on PLFs, which inspires us for this study.
In this paper, we will consider the piecewise affine nonfragile control problem for a class of discrete-time hybrid systems based on piecewise affine models.Via employing a state-control augmentation approach, a new nonfragile controller synthesis method is proposed based on piecewise Lyapunov functions (PLFs), which removes the restrictive condition that the input matrices do not have uncertainties.This feature enables one to design the nonfragile controller for a broader class of industrial systems.It is shown that the resulting closed-loop PWA system is asymptotically stable with a prescribed disturbance attenuation level , and the controller gains can be acquired via solving a set of linear matrix inequalities (LMIs).Finally, two simulation examples are presented to show the effectiveness and amenability of the proposed approaches.The main contributions of this paper are listed as follows: (1) The controller is designed based on PLFs and piecewise affine models and the conservation can be reduced.(2) A new robust piecewise affine nonfragile controller is designed to deal with the situations that perturbations exist in the controller gains.(3) The input matrices are empowered to have uncertainties, which is more applicable for some practical requirements.
The rest of this paper is organized as follows.The piecewise affine model description and nonfragile controller design are given in Section 2. The asymptotic stability analysis for resulting closed-loop system is shown in Section 3. Simulation examples are shown in Section 4 to verify the effectiveness of the proposed methods.Section 5 gives the conclusions.
Notations.The notations employed in this paper are standard.A real symmetric matrix  > 0(≥ 0) indicates  being positive-definite (positive-semi-definite).sym{} is the shorthand notation for  T + .

Problem Formulation
where () ∈ R  is the system state; () ∈ R  is the control input; () ∈ R  is the regulated output; () is the external nonlinear disturbance belonging to  2 [0, ∞); Σ  is a polyhedral partition from the state-space and Π is the index set of these polyhedral regions; Δ  , Δ  , and Δ  are the uncertainty terms of the -th local affine model satisfying with  1 ,  1 ,  2 , and  3 being known real constant matrices.Δ  () is unknown time-varying matrix satisfying Remark 1.The system model in (1) is in fact affine systems rather than linear systems as the offset term (  + Δ  ) is involved.One can easily conclude that this type of models is more accurate for approximation to nonlinear systems [41].
Following the idea given in [20], define the region index Π = Π 0 ∪ Π 1 , where Π 1 denotes the index set of subspaces without the origin and Π 0 refers to the index set of subspaces containing the origin.
For future use, a new set Ω is introduced to depict all possible region transitions When (, ) ∈ Ω and  = , the state is involved in the same region Σ  at the time .Otherwise, the state trajectories will jump from the region Σ  to Σ  at that time.
In this paper, it is assumed that each polyhedral regions Σ  can be outer approximated by an ellipsoid Ψ  , that is to say, there exist matrices   and   such that When Σ  are slab regions, the above covering is useful, and then the parameters   and   are guaranteed to exist, and Σ  ⊆ Ψ  and Ψ  ⊆ Σ  .If the polyhedral regions Σ  are slabs with where   ,   ∈ R,   ∈ R  , then each slab can be represented by a degenerate ellipsoid as in (6) with For each ellipsoid region, we have the following relationship:

Piecewise Affine Nonfragile Controller Design.
Borrowing the idea from [42], define () = [ T ()  T ()] T .As the inaccuracies or uncertainties inevitably involved in the controllers in many practical situations, for the piecewise affine system (1), a new nonfragile piecewise affine controller in differential form is designed as where   ∈ R   ×(  +  ) and   ∈ R   are the controller gains to be determined, and   = 0 for  ∈ Π 0 .Δ  and Δ  are unknown matrices standing for the gain perturbations satisfying where  2 ,  1 , and  2 are known real constant matrices of appropriate dimensions.The real-valued matrix functions Δ  () satisfy The parameter uncertainties in (1) and ( 9) are recognized to be admissible if (2)-( 3) and ( 10)-( 11) hold.
By applying the piecewise affine nonfragile controller (9) into system (1), the closed-loop system can be formulated as where A  =   +   , ΔA  = Δ  + Δ  , a  =   +   , Δa  = Δ  + Δ  , and Based on the augment vector (), the following inequality implies (8) for each ellipsoid region, where As practical control systems are always subject to external disturbance, thus, in this paper, we aim to synthesize an admissible piecewise affine nonfragile controller in the form of ( 9) such that the resulting closed-loop system is asymptotically stable with robust H ∞ performance  as under zero initial conditions for all nonzero () ∈  2 [0, ∞].Before ending this section, the following lemmas will be employed to prove the main results in this paper.

Piecewise Affine Nonfragile Controller Analysis and Design
On the basis of piecewise Lyapunov functions (PLFs), some new approaches to robust H ∞ piecewise affine nonfragile controller synthesis will be proposed in this section.
By substituting ( 22) into (23), it yields that  T ( + 1)    ( + 1) −  T ()    () +  T ()  () −  2  T ()  () < 0, (, ) ∈ Ω. ( Along the trajectories of closed-loop system (12), the following inequality implies (24) with Noticing the state-space partition ( 14) and utilizing Sprocedure, the following inequality implies (25) based on Schur complement: It can be concluded that the Lyapunov matrices   and   are coupled with the system matrices.In order to facilitate the controller design, we need to handle with the terms   and  −1  .To this end, for convenience in the controller synthesis, we make a congruence transformation to (26) by diag{  , I, I, I, I} as Notice that which indicates that Based on (29), the following inequality implies (27): Notice that the controller gains are not involved in the first row of the matrices A  , ΔA  , a  , and Δa  .For numerical tractability, we specify   as where  and  are defined as in (20).
Based on the matrices defined as in (31), one has that where Substitute (32) into (30), and use Lemma 2 to handle the uncertainty terms.Then the following inequality implies (30) for where On the basis of Schur complement, it is easy to conclude that ( 19) implies (33).The proof is thus completed.Remark 4. Note that in most existing nonfragile controller synthesis references for PWA systems, the control input matrices are not empowered to have uncertainties, which is restrictive for the control system design.In addition, it is also difficult to design a nonfragile piecewise affine controller via the approach proposed in [16], as one needs to develop a singular value decomposition to the control input matrices.However, by employing a state-control augmentation strategy for the nonfragile piecewise affine controller design, then the controller gains and the control input matrices can be decoupled from each other.As the state-control augmentation methodology is exploited, the uncertainty terms Δ  and Δ  in the controller are decoupled from the system matrices.This feature makes it possible to design a piecewise affine nonfragile controller via convexification techniques in a uniform framework.
If the uncertainty terms, affine terms, and external disturbance do not exist, system (1) reduces to a nominal piecewise system as For system (35), a piecewise linear nonfragile controller in a differential form is designed as with () and Δ  defined as in (9).Substituting the controller (36) into system (35), then the resulting closed-loop system is  ( + 1) = (  +   + Δ  )  () (37) where   is denoted as in (13).
Corollary 5.For a given constant scalar , the closed-loop system (37) is asymptotically stable if there exist matrices 0

and 𝑖 ∈ Π and a set of positive scalar 𝜀
Moreover, the controller gain can be obtained by The derivation procedures are similar to that of Theorem 3. The proof is omitted.Remark 6.Note that when the uncertainty terms in the form of (10) in the controller gains do not exist, that is to say, then the piecewise affine nonfragile controller reduces to a regular piecewise affine state-feedback controller as where () = [ T ()  T ()] T .

Simulation
In this section, two simulation examples are shown to illustrate the effectiveness of the proposed approaches.
Example 1.Consider a discrete-time piecewise affine system in the form of (1) as follows: and the system matrices are given as 3 = [ 0.9 and the parameter uncertainties in controllers in terms of (10) are The decomposed regions are given as where  = 40 and  = 400.
From the relationship in (7), we can obtain that  T  = [1 0], for  ∈ Π, and the parameters of the degenerate ellipsoids are The objective is to design a piecewise affine nonfragile controller in the form of ( 9)-( 10) such that the closed-loop system in ( 12) is asymptotically stable with a robust H ∞ performance .Exploiting Theorem 3 with  = [1 0] T and  = 1, the controller gains can be obtained as with robust H ∞ performance  min = 1.4222.
It is worth pointing out that when   ≡ 0, the piecewise affine nonfragile controller ( 9) is characterized as a piecewise linear nonfragile controller given in (36).With  = 1, Table 1 compares the robust performance between piecewise affine nonfragile controllers with piecewise linear nonfragile controllers with different matrix .
It can be seen from Table 1 that the piecewise affine nonfragile controllers have a better robust H ∞ performance than the piecewise linear nonfragile controllers.
To verify the effectiveness of the designed controllers, simulations are carried out.With  = [1 0] T and  = 1, solving the LMIs in ( 18)-( 19), we obtain the piecewise linear nonfragile controller gains as In this simulation example, the initial condition is T , and the external disturbance is () = 120 −3.6 ⋅ 8 Mathematical Problems in Engineering   cos(4).By using the piecewise affine nonfragile controller in (9), the state response of the closed-loop system is demonstrated in Figure 1(a), and the control input is shown in Figure 1(b).It can be seen from Figure 1(a) that the curves of system states converge to the zero after 20s.With the same initial conditions and external disturbance, use the piecewise linear nonfragile controller as in (36), and the time responses of the closed-loop system are demonstrated in Figures 2(a) and 2(b).Figure 2(a) shows that the curves of the closed-loop system states converge to the zero after 28s.
Compare Figure 1(a) with Figure 2(a), and one can conclude that the convergence rate of the closed-loop system states with using piecewise affine nonfragile controller is faster than those with using piecewise linear nonfragile controller.
In order to verify the advantages of piecewise affine nonfragile controller (9) over the piecewise affine statefeedback controller (40), another simulation is conducted.
Solve the LMIs in ( 18)- (19) with  2 = 0,  1 = [0 0 0], and  2 = 0,  ∈ {1, 2, 3}.Feasible solutions can be obtained with  = 1 and  = [1 0] T , and the controller gains are With the same initial conditions, disturbance, and parameter variations in the controller gains, apply controller (40) to system (41), and the trajectories of the closed-loop system states are shown in Figure 4.It can be seen from Figure 4 that the piecewise affine state-feedback controller (40) can not stabilize system (39) with parameter variations existing in the controller gains.From this case, we can see that the regular state-feedback controller can not deal with the situations that parameter variations exist in the controller gains.
This example clearly illustrates the advantages of the proposed piecewise affine nonfragile controller over the piecewise linear nonfragile controller and the piecewise affine state-feedback controller.
where the system matrices are The decomposed regions are given as where  = 1,  = 5.From ( 7), we can obtain that  T  = [1 0 0], for  ∈ Π, and the parameters of the degenerate ellipsoid are As the controller gains contain parameter perturbations, we will design a piecewise affine nonfragile controller in (9) to guarantee that the closed-loop system is asymptotically stabile with a robust H ∞ performance .Exploiting Theorem 3 with  = [1 0] T and  = 1, the controller gains can be obtained as With the same initial conditions and disturbance, we use controller (40) to system (49), and the closed-loop system states are demonstrated in Figure 5(b).It can be seen from Figure 5 that the piecewise affine state-feedback controller (40) can not stabilize system (49) with parameter variations existing in the controller gains.

Conclusion
In this paper, we have studied the nonfragile control problem for a class of discrete-time hybrid systems based on piecewise affine model.Based on PLFs, some new sufficient conditions for piecewise affine nonfragile controller synthesis are presented.By virtue of several convexification strategies, the controller gains can be attained via solving a set of linear matrix inequalities.Finally, simulation examples are carried out to demonstrate the advantages of the proposed methodologies.

𝐾 1 =Figure 5 :
Figure 5: Response of the system states in Example 2.