An enhanced method able to perform accurate stability of constrained uncertain systems is presented. The main objective of this method is to compute a sequence of feedback control laws which stabilizes the closed-loop system. The proposed approach is based on robust model predictive control (RMPC) and enhanced maximized sets algorithm (EMSA), which are applied to improve the performance of the closed-loop system and achieve less conservative results. In fact, the proposed approach is split into two parts. The first is a method of enhanced maximized ellipsoidal invariant sets (EMES) based on a semidefinite programming problem. The second is an enhanced maximized polyhedral set (EMPS) which consists of appending new vertices to their convex hull to minimize the distance between each new vertex and the polyhedral set vertices to ensure state constraints. Simulation results on two examples, an uncertain nonisothermal CSTR and an angular positioning system, demonstrate the effectiveness of the proposed methodology when compared to other works related to a similar subject. According to the performance evaluation, we recorded higher feedback gain provided by smallest maximized invariant sets compared to recently studied methods, which shows the best region of stability. Therefore, the proposed algorithm can achieve less conservative results.
Model predictive control (MPC) is a main concern for control design applied in different systems such as linear or nonlinear [
For constrained control problems processing, robust MPC is an effectual stabilization algorithm. This technique employs a specific model procedure based on input and output constraints, for each sampling time, in order to optimize system behavior through the prediction horizon. The controller implements merely the initial calculated input and reproduces these computations at the next sampling time, despite the fact that more than one input shift is calculated [
At each time phase, the convex problem is considered as an optimization problem including linear matrix inequalities (LMI). The main common current algorithm for RMPC is demonstrated to guarantee robust stability. But, due to the fact that the optimization problem is truly settled at each sampling time, it needs high computational time in online implementation. On the other hand, such problems rise appreciably with the size of the polytopic uncertainty set [
Many efforts have been made to design the state feedback control law which minimizes the worst-case performance cost. However, in future RMPC research [
Many researchers [
In this work, a new approach for maximizing ellipsoidal and polyhedral invariant sets associated with the determination of the corresponding state feedback control laws is developed. The contributions of this paper are twofold: firstly, to highlight the robust control of states, an RMPC algorithm [
So, in summary, using this proposed approach, we recorded these two contributions: Maximization of the invariant ellipsoidal and polyhedral sets in order to increase the region of stability Providing less conservative results and efficient system performance in terms of computational time
This paper is organized as follows. Section
Preliminaries and notation.
Notation | Signification |
---|---|
Capital letters | Real matrices |
Transpose of matrix | |
Determinant of matrix | |
Symmetric matrix | |
Symmetric matrix A is positive and definite | |
The | |
The convex hull of |
Schur’s Lemma 1 (see [
In this work, robust model predictive control (RMPC) analysis is the employed procedure to emphasize stability and effectively improve the performance of the uncertain discrete-time linear systems. RMPC method is a typical scheme for minimizing the worst-case performance cost in order to determine the state feedback control law. This technique consists of two tasks: (i) offline part is introduced to search the feedback gain where where Equation ( subject to where subject to where
As illustrated in Figure
Flowchart of the proposed methodology.
Subsequent to the RMPC problem resolution and the feedback gains determination, an invariant ellipsoidal sets sequence is built.
Let the following inequalities be
To maximize the ellipsoidal region
Let us consider the quadratic Lyapunov function
Condition (
Using Schur’s lemma, the following condition with
A natural objective enables increasing the ellipsoid volume which is proportional to
Given the state feedback gains subject to where Once the problem is solved, an optimal vertex Relations (
An uncertain nonisothermal CSTR [
Let
The two parameters
By manipulating
The cost function is given by (
The sequence of the chosen states is
These sequences are used to compute six offline feedback gains
Focused on the EMSA method, the maximized ellipsoidal and polyhedral invariant sets are larger compared to invariant sets [
Resulting maximized ellipsoidal invariant sets (in red).
Examples of maximized polyhedral invariant sets from (a) six imprecated sets and (b) first set compared to [
For both techniques, the invariant sets, ellipsoidal (Figure
The maximized polyhedral invariant sets enable us to obtain an appreciably larger domain of stability compared to the polyhedral invariant ones in [
The concentration of
The reactor temperature of the regulated output obtained with EMPS algorithm.
Performance comparison with previous works.
Methods | Years | Stabilizable region | Stabilization domain | Invariant sets number | Maximization methods | Computational time (s) | ||
---|---|---|---|---|---|---|---|---|
Wan and Kothare [ | 2003 | Ellipsoidal invariant sets | 6 | 3.672 | ||||
Bumroongsri and Kheawhom [ | 2012 | Polyhedral invariant sets | 6 | 4.372 | ||||
The proposed method | Maximized invariant sets | 6 | Semidefine and linear programming | 4.951 |
Cumulative cost in Example 1.
Methods | Cumulative cost | Cumulative equation |
---|---|---|
Wan and Kothare [ | 20.48 | |
Bumroongsri and Kheawhom [ | 19, 02 | |
Proposed approach | 17, 9 |
We consider the angular positioning system described by the following discrete-time equation [
Let
Angular positioning system.
System (
The input constraint is
The weighting matrices
Let us choose the following seven states sequence:
In this example, the sequence of seven states
(a) Maximized ellipsoidal invariant sets compared to [
Compared to the invariant set [
The regulated output obtained with EMPS approach.
The control input obtained with EMPS approach.
Performance comparison with previous works.
Methods | Years | Stabilizable region | Stabilization domain | Invariant sets number | Maximization methods | Computational time (s) | ||
---|---|---|---|---|---|---|---|---|
Different points | ||||||||
Wan and Kothare [ | 2003 | Ellipsoidal invariant sets | 7 | 2.831 | ||||
Bumroongsri and Kheawhom [ | 2012 | Polyhedral invariant sets | 7 | 3.541 | ||||
The proposed method | Maximized invariant sets | 7 | Semidefine and linear programming | 4.183 |
Cumulative cost in Example 2.
Methods | Cumulative cost | Cumulative equation |
---|---|---|
Wan and Kothare [ | 0.19 | |
Bumroongsri and Kheawhom [ | 0.12 | |
Proposed approach | 0.9 |
In this paper, we described an enhanced method which can be used for constrained uncertain discrete-time linear systems stabilization. A useful RMPC technique was applied to emphasize the robust control and improve the state stabilization. The proposed procedure gives appropriate optimization and notable precision when compared to existing model predictive control results. Then, we have suggested the combined RMPC method and maximized invariant sets process that can accurately progress the performance of the closed-loop system. The included methods are used to enlarge ellipsoidal and polyhedral invariant sets constructed by the RMPC algorithm. An online implementation for the obtained feedback control laws has been made. The proposed method has been compared with some existing algorithms in order to enlarge stability domain. Experiment results demonstrate that the proposed method can permanently control system states having a larger stabilizable region. Therefore, the performance of the proposed strategy furnishes a rigid basis in support of solving the control problem. As future works, we propose to use deep learning to obtain flexible models for nonlinear model predictive control (MPC).
No data were used to support this study.
The authors declare that they have no conflicts of interest.