This paper studies the finite-time stability problem of a class of switched nonlinear systems with state constraints and control constrains. For each subsystem, optimization controller is designed by choosing the appropriate Lyapunov function to stabilize the subsystem in finite time and the estimation of the region of attraction can be prescribed. For the whole switched nonlinear system, a suitable switched law is designed to ensure the following:
Switched systems, which consist of dynamical subsystems and specific switching rule which are used to coordinate the operations of all the subsystems, belong to an important and typical class of hybrid control systems [
Finite-time stabilization of a class of continuous autonomous systems is researched in [
There are many research results about finite-time stabilization of switched nonlinear systems at present. However, most of results do not consider the optimizing of system performance and energy using in design process; that is, it does not optimize the given performance index. It is worth mentioning that dynamic optimization control method can handle systems constrains and consider fully performance index [
So far, a little attention has been paid to study finite-time stabilization for switched nonlinear systems by using finite-time optimization control method. On the basis of the above references, this paper presents a novel optimization control method to stabilize switched nonlinear systems with state constrains and control constrains in finite time. The main contributions of this paper are:
(1) In the process of designing subsystem’s controller, the optimization control method proposed in this paper cannot only consider the performance of the system sufficiently but also gives the description of the initial stability region. Finite-time optimization controller based on given objective function is designed to pull subsystem’s states into the stability region; at the same time the objective function is optimized, the system can achieve the best performance and the lowest energy consumption.
(2) A new finite-time robust controller based on selected Lyapunov function is constructed to stabilize subsystem in finite time, and the estimation of the stability region can be prescribed by Lyapunov function method. Compared with the precious robust controller [
(3) An appropriate switching law based on selected Lyapunov function is designed to stabilize the whole switched nonlinear systems in finite time.
The paper is organized as follows: Section
Consider a class of switched nonlinear systems as follows:
In this paper,
The aim of this paper is that appropriate controllers and switching signals are designed to stabilize switched nonlinear systems (
First the definition of finite-time stability is given.
Consider the autonomous system:
This section is divided into two parts. The first part is that controllers are designed for each subsystem; the finite-time optimization controller and finite-time robust controller are designed to switch according to the different states to stabilize the closed-loop subsystems in finite time. The second part is that integrated controller and appropriate switching law are designed to stabilize switched nonlinear systems in finite time.
Consider system (
First, given finite time
The controller can pull subsystem’s states into
The performance index
When subsystem’s states are out of the given area
For subsystem (
So, applying the finite-time optimization controllers (
The finite-time optimization controllers (
When states are in
The choice of Lyapunov function
Based on formulas (
When system’s states are within the given area
For system (
Based on literature [
The controller depends on the selected Lyapunov function. There are many results about controller design based on Lyapunov function and stability for nonlinear systems [
The Lyapunov function in this paper is the same as [
The given area
In order to give complete controller description, system (
For subsystem (
Let
For subsystem (
For
The proof process consists of two parts:
(1) When initial state satisfies
(2) When initial state satisfies
All in all, consider that the arbitrary initial states, the optimization controllers (
Subsystem’s state trajectory curve is shown in Figure
Subsystem’s state trajectory curve.
We assume that switching time sequence of
For each subsystem of switched nonlinear system, the controller switches reasonably between dynamic optimization controllers (
Consider the switched nonlinear system (
The proof of Theorem
(1) When
(2) When system switches before
(a) When
(b) When
In summary, we can see that switched nonlinear system (
Switched nonlinear system’s diagram is shown in Figure
Switched nonlinear system’s diagram.
The algorithm steps of Theorem
(1) Given
(2) When
(3) When
(4) When system switches before
To verify the effectiveness of the proposed finite-time optimization control method, we apply it into a continuous stirred tank reactor where an irreversible, first-order exothermic reaction of the form
Process parameters and steady-state values.
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The control objectives are using the heat input speed
Consider Lyapunov function
Stability area.
Select the initial state as
State trajectory curve.
State trajectory curve.
State trajectory curve.
Control curve.
In this paper, we consider a class of switched nonlinear systems; first, for each subsystem, optimization controller and finite-time robust controller based on Lyapunov function are designed to stabilize subsystem; then, switching laws is designed to ensure that the value of the Lyapunov function has been reduced and ultimately achieve stability in finite time. The next step of the research work is to apply optimization control method proposed in this paper to solve finite-time stability of switching nonlinear time-delay systems, switched nonlinear disturbance system, and so on.
No data were used to support this study.
The authors declare that they have no conflicts of interest.
This research was supported by the Natural Science Foundation of China (Grant nos. 61374004, 61773237, and 61473170) and the Key Research and Development Programs of Shandong Province (2017GSF18116).