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Discrete time optimal control with frequency constraints for non-smooth systems

Shruti Kotpalliwar, Pradyumna Paruchuri, Debasish Chatterjee, Ravi Banavar

TL;DR

This work develops a discrete-time Pontryagin maximum principle for nonlinear, possibly nonsmooth systems subject to pointwise state and control constraints as well as frequency constraints on both state and control trajectories over a finite horizon $N$. A lifting strategy converts the problem to a high-dimensional, differentiable-at-first-glance formulation, enabling Clarke-based first-order necessary conditions to be derived, including adjoint dynamics, transversality, and a frequency-aware Hamiltonian maximization. The results accommodate nonsmooth dynamics via Clarke cones and include special cases with smooth dynamics or linear dynamics where the conditions simplify to standard PMP forms. Numerical examples on an inverted pendulum, a nonsmooth 2D system, and a Buck converter demonstrate the practical impact of jointly enforcing time-domain and spectral constraints and illustrate the methodology's applicability to constrained motion planning in digital control contexts.

Abstract

We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) frequency constraints on the control and the state trajectories, and (c) nonsmooth dynamical systems. Pointwise constraints on the states and the control actions represent desired and/or physical limitations on the states and the control values; such constraints are important and are widely present in the optimal control literature. Constraints of the type (b), while less standard in the literature, effectively serve the purpose of describing important spectral properties of inertial actuators and systems. The conjunction of constraints of the type (a) and (b) is a relatively new phenomenon in optimal control but are important for the synthesis control trajectories with a high degree of fidelity. The maximum principle established here provides first order necessary conditions for optimality that serve as a starting point for the synthesis of control trajectories corresponding to a large class of constrained motion planning problems that have high accuracy in a computationally tractable fashion. Moreover, the ability to handle a reasonably large class of nonsmooth dynamical systems that arise in practice ensures broad applicability our theory, and we include several illustrations of our results on standard problems.

Discrete time optimal control with frequency constraints for non-smooth systems

TL;DR

This work develops a discrete-time Pontryagin maximum principle for nonlinear, possibly nonsmooth systems subject to pointwise state and control constraints as well as frequency constraints on both state and control trajectories over a finite horizon . A lifting strategy converts the problem to a high-dimensional, differentiable-at-first-glance formulation, enabling Clarke-based first-order necessary conditions to be derived, including adjoint dynamics, transversality, and a frequency-aware Hamiltonian maximization. The results accommodate nonsmooth dynamics via Clarke cones and include special cases with smooth dynamics or linear dynamics where the conditions simplify to standard PMP forms. Numerical examples on an inverted pendulum, a nonsmooth 2D system, and a Buck converter demonstrate the practical impact of jointly enforcing time-domain and spectral constraints and illustrate the methodology's applicability to constrained motion planning in digital control contexts.

Abstract

We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) frequency constraints on the control and the state trajectories, and (c) nonsmooth dynamical systems. Pointwise constraints on the states and the control actions represent desired and/or physical limitations on the states and the control values; such constraints are important and are widely present in the optimal control literature. Constraints of the type (b), while less standard in the literature, effectively serve the purpose of describing important spectral properties of inertial actuators and systems. The conjunction of constraints of the type (a) and (b) is a relatively new phenomenon in optimal control but are important for the synthesis control trajectories with a high degree of fidelity. The maximum principle established here provides first order necessary conditions for optimality that serve as a starting point for the synthesis of control trajectories corresponding to a large class of constrained motion planning problems that have high accuracy in a computationally tractable fashion. Moreover, the ability to handle a reasonably large class of nonsmooth dynamical systems that arise in practice ensures broad applicability our theory, and we include several illustrations of our results on standard problems.

Paper Structure

This paper contains 14 sections, 11 theorems, 44 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Preliminaries
  4. Main result
  5. Proof of Theorem \ref{['th:nonsmooth pmp']}
  6. \ref{['step1:Constructing optimization problem']}
  7. \ref{['step2:First order necessary condition']}
  8. \ref{['step3: projection to original problem']}
  9. Numerical Experiments
  10. Example 1
  11. Example 2
  12. Example 3. Buck Converter:
  13. Nonsmooth Optimization and Convex Cones
  14. Auxiliary Lemmas

Key Result

Theorem 4.2

Consider the problem e:DTOCPNSD along with its associated data, and suppose that Assumption assum:regularity of dynamics holds. Let $(u_{t}^{\ast})_{t=0}^{N-1}$ be a control trajectory that solves the optimal optimal control problem e:DTOCPNSD, and let $x_{}^{\ast} \coloneqq (x_{t}^{\ast})_{t=0}^{N} Then there exist satisfying

Figures (3)

  • Figure 1: Tangent cone to the sets $S_{1}, S_{2}$ at $o=(0,0)$
  • Figure 2: Flowchart of a proof
  • Figure 3: Time and frequency domain profiles of the optimal control $u_{\mathrm{p}}$ (obtained using the proposed necessary conditions) (Fig. 3A), and $u_{\mathrm{f}}$ (the filtered optimal control) (Fig. 3B). Phase portraits of the system for the initial condition $\bar{x}$ corresponding to the control inputs: $u_{\mathrm{p}}$ (in blue) and $u_{\mathrm{f}}$ (in red).

Theorems & Definitions (25)

  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Theorem 4.2
  • Remark 4.3
  • Remark 4.4
  • Remark 4.5
  • Remark 4.6
  • ...and 15 more