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Computing Solutions to the Polynomial-Polynomial Regulator Problem

Nicholas A. Corbin, Boris Kramer

Abstract

We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, resulting in lower control costs, and 2) for some problems they can provide a larger region of stabilization. We derive explicit formulas -- and a scalable, general purpose software implementation -- for computing the polynomial approximation to the value function that solves the optimal control problem. The method is illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control. Then we demonstrate the scalability of the approach on a semidiscretization of dimension $n=129$ of a partial differential equation, for which the PPR control reduces the control cost by approximately 75% compared to LQR for the initial condition of interest.

Computing Solutions to the Polynomial-Polynomial Regulator Problem

Abstract

We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, resulting in lower control costs, and 2) for some problems they can provide a larger region of stabilization. We derive explicit formulas -- and a scalable, general purpose software implementation -- for computing the polynomial approximation to the value function that solves the optimal control problem. The method is illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control. Then we demonstrate the scalability of the approach on a semidiscretization of dimension of a partial differential equation, for which the PPR control reduces the control cost by approximately 75% compared to LQR for the initial condition of interest.

Paper Structure

This paper contains 12 sections, 5 theorems, 39 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimal Control Theory
  4. Al'brekht's Method
  5. Kronecker Product Definitions and Notation
  6. Kronecker Polynomial-Based Value Function Computations
  7. Numerical Examples
  8. 3D Example: Aircraft Stall Stabilization
  9. Allen-Cahn Equation
  10. Conclusions And Future Work
  11. Kronecker Product Identities
  12. Proof of \ref{['thm:wiPoly']}

Key Result

Theorem 1

Assume that the cost eq:performanceIndex is continuously differentiable in all of its arguments and is strictly convex in $\mathbf{u}$. Then the value function is the solution to the HJB PDE Furthermore, the optimal control $\mathbf{u}_*$ is given in feedback form by the gradient of the value function as

Figures (4)

  • Figure 1: Angle of attack response for initial conditions (from left to right) $\alpha_0 = 25^{\circ}, 27^{\circ}, 30^{\circ}, 35^{\circ}$. The PPR controllers stabilize the aircraft faster than LQR, and they are able to recover the aircraft from stall for larger initial angles of attack.
  • Figure 2: Spatial domain and control locations for the Allen-Cahn PDE discretized with 129 Chebychev nodes, which are denser near the boundaries.
  • Figure 3: Open-loop behavior of the Allen-Cahn example for $\epsilon = 0.01$, as shown in Trefethen2000.
  • Figure 4: Allen-Cahn example closed-loop simulations for $\epsilon = 0.01$

Theorems & Definitions (10)

  • Definition 1
  • Theorem 1: e.g. Kalman1963Lewis2012
  • Theorem 2: Al'brekht's method Albrekht1961Lukes1969
  • Definition 2: Symmetric coefficients
  • Proposition 1: Permutation of symmetric coefficients
  • Theorem 3: Value function coefficients
  • Theorem 4: Existence and uniqueness of solutions
  • proof
  • Remark 1
  • Remark 2