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Direct Data-Driven Discrete-time Bilinear Biquadratic Regulator

Shanelle G. Clarke, Omanshu Thapliyal, Inseok Hwang

TL;DR

A novel direct data-driven algorithm is presented that learns an optimal control policy for the Bilinear Biquadratic Regulator for an unknown bilinear system through the adroit use of the Hamiltonian and Pontryagin's Minimum Principle.

Abstract

We present a novel direct data-driven algorithm that learns an optimal control policy for the Bilinear Biquadratic Regulator (BBR) for an unknown bilinear system. The BBR is difficult to solve owing to the presence of the nonlinear biquadratic performance index and the bilinear cross-term in the dynamics. To address these difficulties, we apply several transformations on the state decision variables to obtain a nonlinear optimization problem with a linear performance index and affine (in the parameterized control) state-dependent equality. The adroit use of the Hamiltonian and Pontryagin's Minimum Principle allows us to derive a pair of first-order necessary conditions that, at each point in time, are easily solvable linear matrix equalities (LMEs) which give the optimal state-dependent control law. We then use the marginal sample autocorrelation of the collected data to obtain a direct data-driven equivalent of these LMEs. We demonstrate the performance of the proposed algorithm via illustrative numerical examples.

Direct Data-Driven Discrete-time Bilinear Biquadratic Regulator

TL;DR

A novel direct data-driven algorithm is presented that learns an optimal control policy for the Bilinear Biquadratic Regulator for an unknown bilinear system through the adroit use of the Hamiltonian and Pontryagin's Minimum Principle.

Abstract

We present a novel direct data-driven algorithm that learns an optimal control policy for the Bilinear Biquadratic Regulator (BBR) for an unknown bilinear system. The BBR is difficult to solve owing to the presence of the nonlinear biquadratic performance index and the bilinear cross-term in the dynamics. To address these difficulties, we apply several transformations on the state decision variables to obtain a nonlinear optimization problem with a linear performance index and affine (in the parameterized control) state-dependent equality. The adroit use of the Hamiltonian and Pontryagin's Minimum Principle allows us to derive a pair of first-order necessary conditions that, at each point in time, are easily solvable linear matrix equalities (LMEs) which give the optimal state-dependent control law. We then use the marginal sample autocorrelation of the collected data to obtain a direct data-driven equivalent of these LMEs. We demonstrate the performance of the proposed algorithm via illustrative numerical examples.
Paper Structure (9 sections, 3 theorems, 16 equations, 3 figures, 1 algorithm)

This paper contains 9 sections, 3 theorems, 16 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results and Algorithm
  4. Reparametrization of P$_\text{BBR}$.
  5. The Hamiltonian and Pontryagin's Minimum Principle (PMP).
  6. Direct Data-Driven Representation.
  7. Numerical Simulations
  8. Discussion of Results:
  9. Conclusions

Key Result

lemma 1

Let $(K^\star_{\text{P}_\text{BBR}}(x_t),J_t^{t \star})$ and $(S_t^\star, K^\star_{\text{P}_1}(x_t),\hat{J}_t^{t \star})$ denote the respective optimal solution tuples to P$_\text{BBR}$ and P$_1$ at time step $t$. Given $u_t = K(x_t) x_t$, P$_1$ is equivalent to P$_\text{BBR}$ in the sense that $(K^

Figures (3)

  • Figure 1: Control of a bilinear DC motor using wang2018free with quadratic performance costs: $Q = \mathbb I_2$, $R = 1$.
  • Figure 2: Performance of Algorithm \ref{['alg:onlineLQ']} for a Nuclear Fission Reactor system Mohler1970.
  • Figure 3: Performance of Algorithm \ref{['alg:onlineLQ']} for a 2-motor Hydraulic system shaker2013interaction.

Theorems & Definitions (9)

  • lemma 1
  • proof
  • proposition 1
  • proof
  • proposition 2
  • proof
  • remark 1
  • remark 2
  • remark 3