Sensitivity Equations for the Design of Control Systems

Systematic strategies for optimal actuator and sensor locations require £nding extrema of control performance measures. When the control is designed for a distributed parameter system, these performance measures frequently involve the kernel of the Riccati operator or that of the feedback operator. For example, measuring the optimal linear quadratic regulator (LQR) cost over a range of initial data involves the Riccati operator. To aid in the design process, we consider sensitivity equations for Riccati and Chandrasekhar equations. The latter is well-suited for computing feedback kernels when there are a small number of control inputs and control outputs. As we demonstrate, the sensitivity of these kernels to actuator positions can lead to ef£cient computation of gradients for optimization algorithms. Numerical examples corresponding to placing an actuator in the heat equation are provided.