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Robust Model Predictive Control for Constrained Uncertain Systems Based on Concentric Container and Varying Tube

Shibo Han, Yuhao Zhang, Xiaotong Shi, Xingwei Zhao

TL;DR

This paper tackles robust stabilization of constrained linear systems subject to both additive and multiplicative disturbances by introducing a novel CC-VT RMPC framework. It combines varying tubes with concentric containers of fixed shape to bound model-mismatch-induced disturbances more tightly and with lower online complexity than traditional homothetic-tube approaches. A container-optimization strategy (both base and augmented bounds) is developed to enlarge the region of attraction while maintaining tractable constraints, and a recursive terminal set construction guarantees robust recursive feasibility and stability. Numerical results on a representative example show that CC-VT RMPC achieves a larger region of attraction and fewer online decision variables than an HT-based RMPC, demonstrating reduced conservativeness and improved scalability for higher-dimensional systems.

Abstract

This paper proposes a novel robust model predictive control (RMPC) method for the stabilization of constrained systems subject to additive disturbance (AD) and multiplicative disturbance (MD). Concentric containers are introduced to facilitate the characterization of MD, and varying tubes are constructed to bound reachable states. By restricting states and the corresponding inputs in containers with free sizes and a fixed shape, feasible MDs, which are the products of model uncertainty with states and inputs, are restricted into polytopes with free sizes. Then, tubes with different centers and shapes are constructed based on the nominal dynamics and the knowledge of AD and MD. The free sizes of containers allow for a more accurate characterization of MD, while the fixed shape reduces online computational burden, making the proposed method less conservative and computationally efficient. Moreover, the shape of containers is optimized to further reduce conservativeness. Compared to the RMPC method using homothetic tubes, the proposed method has a larger region of attraction while involving fewer decision variables and constraints in the online optimization problem.

Robust Model Predictive Control for Constrained Uncertain Systems Based on Concentric Container and Varying Tube

TL;DR

This paper tackles robust stabilization of constrained linear systems subject to both additive and multiplicative disturbances by introducing a novel CC-VT RMPC framework. It combines varying tubes with concentric containers of fixed shape to bound model-mismatch-induced disturbances more tightly and with lower online complexity than traditional homothetic-tube approaches. A container-optimization strategy (both base and augmented bounds) is developed to enlarge the region of attraction while maintaining tractable constraints, and a recursive terminal set construction guarantees robust recursive feasibility and stability. Numerical results on a representative example show that CC-VT RMPC achieves a larger region of attraction and fewer online decision variables than an HT-based RMPC, demonstrating reduced conservativeness and improved scalability for higher-dimensional systems.

Abstract

This paper proposes a novel robust model predictive control (RMPC) method for the stabilization of constrained systems subject to additive disturbance (AD) and multiplicative disturbance (MD). Concentric containers are introduced to facilitate the characterization of MD, and varying tubes are constructed to bound reachable states. By restricting states and the corresponding inputs in containers with free sizes and a fixed shape, feasible MDs, which are the products of model uncertainty with states and inputs, are restricted into polytopes with free sizes. Then, tubes with different centers and shapes are constructed based on the nominal dynamics and the knowledge of AD and MD. The free sizes of containers allow for a more accurate characterization of MD, while the fixed shape reduces online computational burden, making the proposed method less conservative and computationally efficient. Moreover, the shape of containers is optimized to further reduce conservativeness. Compared to the RMPC method using homothetic tubes, the proposed method has a larger region of attraction while involving fewer decision variables and constraints in the online optimization problem.

Paper Structure

This paper contains 16 sections, 80 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation and preliminaries
  3. Formulation of CC-VT RMPC
  4. VT-based Structure
  5. Additional Constraints
  6. Terminal Constraints
  7. RMPC Controller
  8. Optimization of Container
  9. Optimization of $\mathbb{Z}_m^0$
  10. Optimization of $\mathbb{PZ}_m^0$
  11. Numerical Example
  12. Discussion
  13. Conclusion
  14. Appendix
  15. Proof of Theory 1
  16. ...and 1 more sections

Figures (6)

  • Figure 1: Compare of containers $\mathbb{Z}_m^0$, $\mathbb{Z}_m^1$, and $\mathbb{Z}_m^2$.
  • Figure 2: Compare of $\mathbb{PZ}_m^1$ and $\mathbb{PZ}_m^2$.
  • Figure 3: Trajectory of states, input, and cost.
  • Figure 4: Compare of regions of attraction of different methods.
  • Figure 5: State trajectory and tubes $\mathbb{X}(k|0)$ with the proposed method, $k = 1,2,3.$
  • ...and 1 more figures

Theorems & Definitions (9)

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