Mathematical Theories and Applications for Nonlinear Control SystemsView this Special Issue
A New Robust Nonfragile Controller Design Scheme for a Class of Hybrid Systems through Piecewise Affine Models
This paper investigates the robust 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 performance . 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.
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 [1–6]. 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 [7–9]. 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 [12–22]. In , the stability of continuous-time PWA systems was analyzed by utilizing discontinuous piecewise Lyapunov functions (PLFs). On the basis of the idea in , some algorithms for stability and performance analysis of discrete-time PWA systems were proposed in  and the continuity of the PLFs was not required for the discrete-time case. The authors in  designed a static output-feedback controller for discrete-time PWA systems such that the resulting closed-loop system was asymptotically stable with robust performance . The authors in  developed a reconfigurable control approach for continuous-time PWA systems with actuator and sensor faults, and the output tracking problem was investigated simultaneously. In , the problem of delay-dependent fixed-order memory piecewise affine output-feedback control for a class of nonlinear systems with time-varying delay was considered. The authors in  have proposed a novel system-augmentation approach to the delay-dependent reliable piecewise affine static output-feedback control for nonlinear systems with time-varying delay and sensor faults in the piecewise-Markovian-Lyapunov-functional-based framework.
On the other hand, since 1980s, the robust 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 . Numbers of literatures have already shown that relatively small perturbations in the controller parameters can even result in instability of the closed-loop system [28–30]. 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 controller design problem [31–40]. For instance, a nonfragile controller with multiplicative gain perturbations was proposed for linear time invariant (LTI) systems in . The author in  developed a decentralized robust state-feedback nonfragile control method for uncertain discrete-time large-scale systems with time-delays and controller gain variations. In , 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  considered the nonfragile robust control problem for fuzzy systems based on output information. In spite of the attractive features in piecewise affine systems, the authors in  proposed a state-feedback nonfragile robust controller design method for continuous-time PWA systems. The results in  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 indicates being positive-definite (positive-semi-definite). is the shorthand notation for .
2. Problem Formulation
2.1. Discrete-Time Piecewise Affine Systems
A discrete-time piecewise affine system is shown as where is the system state; is the control input; is the regulated output; is the external nonlinear disturbance belonging to ; is a polyhedral partition from the state-space and is the index set of these polyhedral regions; , , and are the uncertainty terms of the -th local affine model satisfyingwith , , , and being known real constant matrices. 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 is involved. One can easily conclude that this type of models is more accurate for approximation to nonlinear systems .
Following the idea given in , define the region index , where denotes the index set of subspaces without the origin and refers to the index set of subspaces containing the origin.
For future use, a new set is introduced to depict all possible region transitionsWhen and , the state is involved in the same region at the time . Otherwise, the state trajectories will jump from the region to at that time.
In this paper, it is assumed that each polyhedral regions can be outer approximated by an ellipsoid , that is to say, there exist matrices and such that
When are slab regions, the above covering is useful, and then the parameters and are guaranteed to exist, and and . If the polyhedral regions are slabs withwhere , , then each slab can be represented by a degenerate ellipsoid as in (6) with
For each ellipsoid region, we have the following relationship:
2.2. Piecewise Affine Nonfragile Controller Design
Borrowing the idea from , define . 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 aswhere and are the controller gains to be determined, and for . and are unknown matrices standing for the gain perturbations satisfyingwhere , , and are known real constant matrices of appropriate dimensions. The real-valued matrix functions satisfy
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 performance asunder zero initial conditions for all nonzero .
Before ending this section, the following lemmas will be employed to prove the main results in this paper.
Lemma 2 (see ). Real matrices , , , and are regular matrices with appropriate dimensions. is time-varying satisfying . The inequality holds if and only if for some
3. Piecewise Affine Nonfragile Controller Analysis and Design
On the basis of piecewise Lyapunov functions (PLFs), some new approaches to robust piecewise affine nonfragile controller synthesis will be proposed in this section.
Theorem 3. For a given constant scalar , the closed-loop system (12) is asymptotically stable with disturbance attenuation level if there exist matrices , , , , , , and and two sets of scalars , and , such that the following LMIs hold:whereand is an arbitrary matrix.
Moreover, the controller gains for each subspace can be determined via
Proof. It is easy to see that condition (18) for is a special case of condition (19) for . Without loss of generality, in the following, the proof of the more complex case is to be considered. Take the piecewise Lyapunov function asIt is also assumed that . Then we have . The closed-loop system in (12) is asymptotically stable with robust performance , if the inequalityholds.
By substituting (22) into (23), it yields thatAlong the trajectories of closed-loop system (12), the following inequality implies (24) with ,Noticing the state-space partition (14) and utilizing S-procedure, 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 . To this end, for convenience in the controller synthesis, we make a congruence transformation to (26) by asNotice thatwhich indicates thatBased on (29), the following inequality implies (27):Notice that the controller gains are not involved in the first row of the matrices , , , and . For numerical tractability, we specify aswhere and are defined as in (20).
Based on the matrices defined as in (31), one has thatwhere , , .
Substitute (32) into (30), and use Lemma 2 to handle the uncertainty terms. Then the following inequality implies (30) for , ,whereOn 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 , 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 and 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 asFor system (35), a piecewise linear nonfragile controller in a differential form is designed aswith and defined as in (9).
Substituting the controller (36) into system (35), then the resulting closed-loop system iswhere is denoted as in (13).
Corollary 5. For a given constant scalar , the closed-loop system (37) is asymptotically stable if there exist matrices , , , , , and and a set of positive scalar , such thatMoreover, 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 aswhere .
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 asand the parameter uncertainties in controllers in terms of (10) areThe decomposed regions are given aswhere and .
From the relationship in (7), we can obtain that , for , and the parameters of the degenerate ellipsoids areThe 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 performance . Exploiting Theorem 3 with and , the controller gains can be obtained aswith robust performance .
It is worth pointing out that when , the piecewise affine nonfragile controller (9) is characterized as a piecewise linear nonfragile controller given in (36). With , 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 performance than the piecewise linear nonfragile controllers.
To verify the effectiveness of the designed controllers, simulations are carried out. With and , solving the LMIs in (18)-(19), we obtain the piecewise linear nonfragile controller gains asIn this simulation example, the initial condition is , and the external disturbance is . 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 s.
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.
Under zero initial conditions, Figure 3 presents the performance. The ratio with using piecewise affine nonfragile controller is about 1.18, which is lower than . The ratio with using piecewise linear nonfragile controller is about 1.3, which is lower than . It can also be seen that a better disturbance attenuation level can be obtained with the piecewise affine nonfragile controller.
In order to verify the advantages of piecewise affine nonfragile controller (9) over the piecewise affine state-feedback controller (40), another simulation is conducted. Solve the LMIs in (18)-(19) with , , and , . Feasible solutions can be obtained with and , and the controller gains areWith 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.
(a) State response
(b) Control input
(a) State response
(b) Control input
(a) Response of the ratio by using piecewise affine nonfragile controller (9)
(b) Response of the ratio by using piecewise affine nonfragile controller (36)
Example 2. Consider another discrete-time Chuas circuit system represented by the piecewise affine model as where the system matrices areThe decomposed regions are given as where , .
From (7), we can obtain that , for , and the parameters of the degenerate ellipsoid areIt is assumed in this example that the perturbations existing in the controller gains are set asAs 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 performance . Exploiting Theorem 3 with and , the controller gains can be obtained aswith robust performance .
To present the effectiveness of the designed controllers, simulations are carried out. With initial conditions and the external disturbance , the state trajectories of the closed-loop system are demonstrated in Figure 5(a).
Solving the LMIs in (18)-(19) with , , , , and , , then the gains of the piecewise affine state-feedback controller (40) areWith 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.
(a) Response of closed-loop system states by using piecewise affine nonfragile controller (9)
(b) Response of closed-loop system states by using piecewise affine nonfragile controller (40)
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.
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is financially supported by the National Key Research and Development Program (Grants nos. 2017YFC0306804 and 2017YFC0305700) and State Key Laboratory of Ocean Engineering (Shanghai Jiao Tong University) (Grant no. 1715).
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