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This paper investigates the robust

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 [

In recent years, many valuable references on the systematic analysis and synthesis for the PWA systems have been published [

On the other hand, since 1980s, the robust

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

The rest of this paper is organized as follows. The piecewise affine model description and nonfragile controller design are given in Section

A discrete-time piecewise affine system is shown as

The system model in (

Following the idea given in [

For future use, a new set

In this paper, it is assumed that each polyhedral regions

When

For each ellipsoid region, we have the following relationship:

Borrowing the idea from [

The parameter uncertainties in (

By applying the piecewise affine nonfragile controller (

Based on the augment vector

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 (

Before ending this section, the following lemmas will be employed to prove the main results in this paper.

Real matrices

On the basis of piecewise Lyapunov functions (PLFs), some new approaches to robust

For a given constant scalar

Moreover, the controller gains for each subspace can be determined via

It is easy to see that condition (

It is also assumed that

By substituting (

Along the trajectories of closed-loop system (

Noticing the state-space partition (

It can be concluded that the Lyapunov matrices

Notice that

Based on (

Notice that the controller gains are not involved in the first row of the matrices

Based on the matrices defined as in (

Substitute (

On the basis of Schur complement, it is easy to conclude that (

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 [

If the uncertainty terms, affine terms, and external disturbance do not exist, system (

For system (

Substituting the controller (

For a given constant scalar

Moreover, the controller gain can be obtained by

The derivation procedures are similar to that of Theorem

Note that when the uncertainty terms in the form of (

In this section, two simulation examples are shown to illustrate the effectiveness of the proposed approaches.

Consider a discrete-time piecewise affine system in the form of (

The decomposed regions are given as

From the relationship in (

The objective is to design a piecewise affine nonfragile controller in the form of (

It is worth pointing out that when

It can be seen from Table

To verify the effectiveness of the designed controllers, simulations are carried out. With

In this simulation example, the initial condition is

With the same initial conditions and external disturbance, use the piecewise linear nonfragile controller as in (

Compare Figure

Under zero initial conditions, Figure

In order to verify the advantages of piecewise affine nonfragile controller (

With the same initial conditions, disturbance, and parameter variations in the controller gains, apply controller (

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.

Comparison of robust

Controller form | | | |
---|---|---|---|

piecewise affine controller | 1.4222 | 1.6797 | 4.6871 |

piecewise linear controller | 1.5924 | 1.7997 | 4.7362 |

Simulation results in Example

State response

Control input

Simulation results in Example

State response

Control input

Response of the ratio

Response of the ratio by using piecewise affine nonfragile controller (

Response of the ratio by using piecewise affine nonfragile controller (

State responses in Example

Consider another discrete-time Chuas circuit system represented by the piecewise affine model as

The decomposed regions are given as

From (

It is assumed in this example that the perturbations existing in the controller gains are set as

As the controller gains contain parameter perturbations, we will design a piecewise affine nonfragile controller in (

To present the effectiveness of the designed controllers, simulations are carried out. With initial conditions

Solving the LMIs in (

With the same initial conditions and disturbance, we use controller (

Response of the system states in Example

Response of closed-loop system states by using piecewise affine nonfragile controller (

Response of closed-loop system states by using piecewise affine nonfragile controller (

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.

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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