Impulse-to-Peak-Output Norm Optimal State-Feedback Control of Linear PDEs
It provides a novel method for I2P optimal control of PDEs, a problem previously unsolved due to lack of suitable state-space representation, enabling performance guarantees for PDE systems.
The paper extends impulse-to-peak (I2P) optimal control from ODEs to linear PDEs using a partial integral equation (PIE) framework, formulating the problem as a convex optimization and providing a constructive state-feedback control method with provable bounds.
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.