A distributed parameter control approach to sensor location for optimal feedback control of thermal processes

Functional gains are kernel functions in integral representations of feedback operators. We consider the case where these operators are computed by solving algebraic Riccati equations arising from infinite dimensional LQR/LQG control problems. When these gains exist and can be computed, one has information that provides insight into sensor location and design of low order-local dynamic compensators. Burns and King (1994) showed that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smoothes solutions of the corresponding Riccati equation. When this result is applied to problems with distributed controllers it can be established that the resulting feedback operator is also smooth. However, it is the continuity of the input operator that leads to a positive result in this case. We present numerical results that suggest that for boundary control of the 2D heat equation, the functional gains exist and have compact support near the boundary where the control is applied. Both properties are important in addressing sensor and actuator location problems and they have practical implications in the design of reduced order controllers for PDE systems. We use these results to guide the placement of discrete sensors and compare the results to full state feedback.