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Impulse-to-Peak-Output Norm Optimal State-Feedback Control of Linear PDEs

Tristan Thomas, Sachin Shivakumar, Javad Mohammadpour Velni

Abstract

Impulse-to-peak response (I2P) analysis for state-space ordinary differential equation (ODE) systems is a well-studied classical problem. However, the techniques employed for I2P optimal control of ODEs have not been extended to partial differential equation (PDE) systems due to the lack of a universal transfer function and state-space representation. Recently, however, partial integral equation (PIE) representation was proposed as the desired state-space representation of a PDE, and Lyapunov stability theory was used to solve various control problems, such as stability and optimal ${H}_\infty$ control. In this work, we utilize this PIE framework, and associated Lyapunov techniques, to formulate the I2P response analysis problem as a solvable convex optimization and obtain provable bounds for the I2P-norm of linear PDEs. Moreover, by establishing strong duality between primal and dual formulations of the optimization problem, we develop a constructive method for I2P optimal state-feedback control of PDEs and demonstrate the effectiveness of the method on various examples.

Impulse-to-Peak-Output Norm Optimal State-Feedback Control of Linear PDEs

Abstract

Impulse-to-peak response (I2P) analysis for state-space ordinary differential equation (ODE) systems is a well-studied classical problem. However, the techniques employed for I2P optimal control of ODEs have not been extended to partial differential equation (PDE) systems due to the lack of a universal transfer function and state-space representation. Recently, however, partial integral equation (PIE) representation was proposed as the desired state-space representation of a PDE, and Lyapunov stability theory was used to solve various control problems, such as stability and optimal control. In this work, we utilize this PIE framework, and associated Lyapunov techniques, to formulate the I2P response analysis problem as a solvable convex optimization and obtain provable bounds for the I2P-norm of linear PDEs. Moreover, by establishing strong duality between primal and dual formulations of the optimization problem, we develop a constructive method for I2P optimal state-feedback control of PDEs and demonstrate the effectiveness of the method on various examples.

Paper Structure

This paper contains 15 sections, 4 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. The Class of Partial Integral Operators
  4. The Class of Partial Integral Equations
  5. The Class of Partial Differential Equations
  6. Solving LPIs and PI Operator Inversion
  7. Duality in PIEs
  8. Impulse-to-Peak Norm Calculation of a PIE
  9. Equivalence in Impulse-to-peak Norm
  10. Linear PI Inequality Formulations
  11. Optimal State-feedback Controller Synthesis
  12. Numerical Results
  13. Impulse-to-peak Norm Upper-bounding
  14. Impulse-to-peak Norm Optimal State Feedback Control
  15. Conclusion

Key Result

Theorem 1

Given $\mathcal{T,A}, \mathcal{B}, \mathcal{C} \in \Pi_4$, if the PIE Eq. eqn:PIEaux and its dual have well-posed solutions, then $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure E1: Evolution of the PDE state for the reaction-diffusion equation under state-feedback controller
  • Figure E2: The regulated output, $\int_0^1 x(t,s) ds$, for the reaction-diffusion equation. The control stabilizes the system and satisfies peak output bounds.
  • Figure E3: Evolution of the PDE state for the transport equation under state-feedback controller
  • Figure E4: The regulated output, $\int_0^1 x(t,s)ds$, for the controlled ($\textcolor{red}{--}$) and uncontrolled ($\textcolor{blue}{-}$) transport equation. The control reduces the peak response under impulsive input in comparison to uncontrolled system.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • proof
  • Theorem 2
  • proof : Proof of (i)
  • Theorem 3
  • ...and 7 more