Dynamic Gain Adaptation in Linear Quadratic Regulators

In feedback control of dynamical systems, the choice of a higher loop gain is typically desirable to achieve a faster closed-loop dynamics, smaller tracking error, and more effective disturbance suppression. Yet, an increased loop gain requires a higher control effort, which can extend beyond the actuation capacity of the feedback system and intermittently cause actuator saturation. To benefit from the advantages of a high feedback gain and simultaneously avoid actuator saturation, this article advocates a dynamic gain adaptation technique in which the loop gain is lowered whenever necessary to prevent actuator saturation, and is raised again whenever possible. This concept is optimized for linear systems based on an optimal control formulation inspired by the notion of linear quadratic regulator (LQR). The quadratic cost functional adopted in the LQR is modified into a certain quasi-quadratic form in which the control cost is dynamically emphasized or deemphasized as a function of the system state. The optimal control law resulted from this quasi-quadratic cost functional is essentially nonlinear, but its structure resembles an LQR with an adaptable gain adjusted by the state of system, aimed to prevent actuator saturation. Moreover, under mild assumptions analogous to those of LQR, this optimal control law is stabilizing. As an illustrative example, application of this optimal control law in feedback design for dc servomotors is examined, and its performance is verified by numerical simulations.