Closed forms of stabilizing sets are generally only available for linearized systems. An innovative numerical strategy to estimate stabilizing sets of PI or PID controllers tackling (uncertain) nonlinear systems is proposed. The stability of the closed-loop system is characterized by the sign of the largest Lyapunov exponent (LLE). In this framework, the bottleneck is the computational cost associated with the solution of the system, particularly including uncertainties. To overcome this issue, an adaptive surrogate algorithm, the Monte Carlo intersite Voronoi (MiVor) scheme, is adopted to pertinently explore the domain of the controller parameters and classify it into stable/unstable regions from a low number of nonlinear estimations. The result of the random analysis is a stochastic set providing probability information regarding the capabilities of PI or PID controllers to stabilize the nonlinear system and the risk of instabilities. The minimum of the LLE is proposed as tuning rule of the controller parameters. It is expected that using a tuning rule like this results in PID controllers producing the highest closed-loop convergence rate, thus being robust against model parametric uncertainties and capable of avoiding large fluctuating behavior. The capabilities of the innovative approach are demonstrated by estimating robust stabilizing sets for the blood glucose regulation problem in type 1 diabetes patients.

稳定集的封闭形式通常仅适用于线性化系统。提出了一种用于估计处理（不确定）非线性系统的 PI 或 PID 控制器稳定集的创新数值策略。闭环系统的稳定性以最大李雅普诺夫指数 (LLE) 的符号为特征。在这个框架中，瓶颈是与系统解决方案相关的计算成本，特别是包括不确定性。为了克服这个问题，采用了一种自适应代理算法，即蒙特卡罗站点间 Voronoi (MiVor) 方案，有针对性地探索控制器参数的域，并将其从少量非线性估计中分类为稳定/不稳定区域。随机分析的结果是一个随机集，提供关于 PI 或 PID 控制器稳定非线性系统的能力和不稳定性风险的概率信息。建议 LLE 的最小值作为控制器参数的调整规则。预计使用这样的调整规则会导致 PID 控制器产生最高的闭环收敛速度，从而对模型参数不确定性具有鲁棒性并能够避免大的波动行为。通过为 1 型糖尿病患者的血糖调节问题估计稳健的稳定集，证明了创新方法的能力。建议 LLE 的最小值作为控制器参数的调整规则。预计使用这样的调整规则会导致 PID 控制器产生最高的闭环收敛速度，从而对模型参数不确定性具有鲁棒性并能够避免大的波动行为。通过为 1 型糖尿病患者的血糖调节问题估计稳健的稳定集，证明了创新方法的能力。建议 LLE 的最小值作为控制器参数的调整规则。预计使用这样的调整规则会导致 PID 控制器产生最高的闭环收敛速度，从而对模型参数不确定性具有鲁棒性并能够避免大的波动行为。通过为 1 型糖尿病患者的血糖调节问题估计稳健的稳定集，证明了创新方法的能力。