(235e) Generic Semi-Infinite Stochastic Programming Formulation and Algorithm for Control-Lyapunov Function Design

Lyapunov stability analysis is the classical approach for establishing the stability of nonlinear systems and enforcing closed-loop stability in nonlinear control [1]. For example, the “universal” formula given by Lin and Sontag [2] specifies an explicit analytical feedback law that guarantees the descent of any given control-Lyapunov function, and the Lyapunov-based model predictive control (LMPC) [3] incorporates the Lyapunov descent as a constraint in the receding horizon optimization problem. Evidently, the design of the control-Lyapunov function has a fundamental impact on the shapes of the closed-loop trajectories and also on the region of attraction (RoA).

However, the systematic design of control-Lyapunov functions is a challenging problem. The most common approach, namely sum-of-squares (SOS) programming based on algebraic geometry [4], applies only to polynomial systems and can optimize objectives that are simple, such as the volume of an estimated RoA. For general nonlinear systems, the problem can be expressed as an optimization problem over measures (called Lyapunov measures) [5], which is computationally intractable except for very low-dimensional systems.

In this talk, we introduce a generic formulation of the control-Lyapunov function design problem in the semi-infinite stochastic programming form, namely as an optimization problem whose objective function is defined as the expectation of a cost involving random variables and the constraints are infinite. The formulation is derived based on the following considerations.

1. The control-Lyapunov function is represented by a linear parameterization, thus formulating the parameters as the variables to be optimized. This can accommodate parameterizations of different complexity, such as quadratic forms or SOS polynomials, and is more scalable than the grid-point discretization of the state space (e.g., used in [5]).

2. The stability requirement is formulated as an inequality that captures Lyapunov descent or bounded disturbance effect. The inequality should hold for relevant state-dependent functions and is thus parameterized by an infinite number of states. The user may specify the functions involved, e.g., the decreasing rate of the control-Lyapunov function or the gain function from disturbances.

3. The objective function (cost) is evaluated on a set of simulation scenarios that reflect typical operating conditions or concerns, for which random variables are assigned. Unlike the classical quadratic cost in optimal control that only considers the magnitudes of the states and inputs, the cost can account for various characteristics of dynamic behaviors, such as smoothness and overshooting.

To solve the semi-infinite stochastic program, we introduce and propose a modification of the stochastic proximal primal-dual algorithm from [6]. The algorithm is illustrated with a reactor case study. We also highlight the conceptual connection between our proposed approach for Lyapunov function design and flexibility analysis in the process design literature [7].

References

1. Khalil, H. K.(2002). Nonlinear systems, 3^rd ed. Prentice Hall.

2. Lin, Y., & Sontag, D. (1991). A universal formula for stabilization with bounded controls. Syst. Control Lett., 16, 393–397.

3. Mhaskar, P., El-Farra, N. H., & Christofides, P. D. (2006). Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control. Syst. Control Lett., 55, 650–659.

4. Souaiby, M., Tanwani, A., & Henrion, D. (2020). Computation of Lyapunov functions under state constraints using semidefinite programming hierarchies. IFAC-PapersOnLine, 53(2), 6281–6286.

5. Raghunathan, A. & Vaidya, U. (2013). Optimal stabilization using Lyapunov measures. IEEE Trans. Autom. Control, 59, 1316–1321.

6. Boob, D., Deng, Q., & Lan, G.(2022). Stochastic first-order methods for convex and nonconvex functional constrained optimization. Math. Program., in press, doi:10.1007/s10107-021-01742-y.

7. Grossmann, I. E., Calfa, B. A., & Garcia-Herreros, P. (2014). Evolution of concepts and models for quantifying resiliency and flexibility of chemical processes. Comput. Chem. Eng., 70, 22-34.