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Reachability-Analysis-Based PID Control for Practical Stabilizability
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DOI：10.1109/tase.2026.3727086.png)
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Abstract
En 中文
In this paper, we investigate the practical stabilizability problem (i.e., the reach-and-stay control problem) for a class of $n$ -dimensional control systems and PID control is introduced to achieve this objective. First, we construct a $2n$ -dimensional system, such that the projection of its reachable set onto the first $n$ -dimensional space is the reachable set of the original $n$ -dimensional system with pre-assumed PID control. After that, we introduce the evolution function for exactly describing the reachable set of the $2n$ -dimensional system, and find a series representation of this evolution function, which can be regarded as a series function with respect to the parameters $P$ , $I$ , and $D$ . Moreover, we introduce zonotope representation and multi-start local optimization to approximate the evolution function with arbitrary precision, improving the efficiency of the algorithm described in She and Li (2021) and Hu et al. (2025). Then, barrier certificates based method is used for reach-and-stay verification. Especially, due to the use of PID control, we can present a specific barrier certificate template together with a polar coordinate transformation based verification method. Finally, a particle swarm optimization (PSO) based approach is proposed to iteratively compute PID control parameters together with as small a stabilization time as possible, achieving the practical stabilizability of the $n$ -dimensional system. We implement our approach and test it on some benchmarks with comparisons. The comparison results demonstrate the performance of our approach. It is worth mentioning that the comparison results also show the advantage of PID control over P control, PI control, and PD control, respectively. Note to Practitioners—This paper studies the practical stabilizability of a class of $n$ -dimensional systems. Practical stabilizability is an important criterion for ensuring system reliability and safety. We propose a novel approach for achieving the practical stabilizability using traditional PID control through reachable set approximation and the construction of a barrier certificate template. Experimental results show that our method can handle systems that cannot be addressed by traditional Lyapunov theory. To further enhance performance, a PSO algorithm is employed to obtain a stabilization time as small as possible. Practical examples demonstrate that our method can handle a variety of complex systems, and comparisons with different optimization methods and alternative control highlight the advantages of using the PSO algorithm and PID control. In future work, we will consider the robust practical stabilizability of systems with disturbances.
Keywords:
Practical stabilizability
PID control
reachability
evolution function
barrier certificate
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