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On moment relaxations for linear state feedback controller synthesis with non-convex quadratic costs and constraints

Dennis Gramlich, Sheng Gao, Hao Zhang, Carsten W. Scherer, Christian Ebenbauer

Abstract

We present a simple and effective way to account for non-convex costs and constraints~in~state feedback synthesis, and an interpretation for the variables in which state feedback synthesis is typically convex. We achieve this by deriving the controller design using moment matrices of state and input. It turns out that this approach allows the consideration of non-convex constraints by relaxing them as expectation constraints, and that the variables in which state feedback synthesis is typically convexified can be identified with blocks of these moment matrices.

On moment relaxations for linear state feedback controller synthesis with non-convex quadratic costs and constraints

Abstract

We present a simple and effective way to account for non-convex costs and constraints~in~state feedback synthesis, and an interpretation for the variables in which state feedback synthesis is typically convex. We achieve this by deriving the controller design using moment matrices of state and input. It turns out that this approach allows the consideration of non-convex constraints by relaxing them as expectation constraints, and that the variables in which state feedback synthesis is typically convexified can be identified with blocks of these moment matrices.
Paper Structure (10 sections, 3 theorems, 34 equations, 4 figures)

This paper contains 10 sections, 3 theorems, 34 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem statement
  3. Convexification over moment matrices
  4. The time-invariant case
  5. The moment matrix perspective and duality
  6. Example: Non-convex constraints
  7. Test 1
  8. Test 2
  9. Example: Fixed-point escape
  10. Conclusion

Key Result

Theorem III.1

Let $\Sigma_t = \Sigma_t^\top \in \mathbb{R}^{(1+n+m) \times (1+n+m)}$ be a sequence of matrices satisfying the initial condition eq:initialMoments. Then there exists a policy $(\mathbb{P}_{u_t})$ generating the sequence of moments $(\Sigma_t)$ satisfying eq:momentEquation if and only if hold for $t = 0,\ldots,N$. Moreover, if eq:cond holds, then there exist controller parameters $K_t = $ and $\S

Figures (4)

  • Figure 1: Vizualization of trajectories generated by the random policy \ref{['eq:controllerReconstruction']} and two black areas to be avoided.
  • Figure 2: Vizualization of trajectories generated by the random policy \ref{['eq:controllerReconstruction']} for an obstacle avoidance problem with one obstacle.
  • Figure 3: Two closed loop trajectories of the angle of the nonlinear inverted pendulum model \ref{['eq:invertedPendulum']} with a swing up controller. Where the color of the trajectories is orange or blue, an escape controller for the lower equilibrium is used. Where the color of the trajectories is red, a stabilizing controller for the upper equilibrium is used.
  • Figure 4: Two closed loop trajectories of the cart position of the nonlinear inverted pendulum model \ref{['eq:invertedPendulum']} with a swing up controller. Where the color of the trajectories is orange or blue, an escape controller for the lower equilibrium is used. Where the color of the trajectories is red, a stabilizing controller for the upper equilibrium is used.

Theorems & Definitions (9)

  • Theorem III.1
  • proof
  • Remark III.2
  • Remark III.3
  • Lemma IV.1
  • proof
  • Remark IV.2
  • Remark IV.3
  • Lemma V.1: derived from Lemma 10.2, scherer2000robust