Optimal output feedback for linear time-periodic systems

Abstract

An approach is developed for applying optimal output feedback control theory to the design of fixed gain controllers for time-periodic systems. Constant feedback gains based on plant outputs are calculated by minimizing a linear quadratic performance cost functional. A solution algorithm is developed using Floquet- Lyapunov theory and a generalized harmonic expansion technique. The theory is applied to the control of a helicopter rotor blade in forward flight through individual blade control. It is shown that constant gain output feedback can be used to augment the stability of the resulting time-periodic system, with the potential for greatly reducing the implementation complexity. HE primary purpose of this paper is to present a general framework for applying output feedback techniques to linear time-periodic systems. The motivation for such an ap- proach is, in part, to reduce the control implementation re- quirements. The use of constant gain optimal output feedback in designing active feedback controllers provides many advan- tages over existing linear quadratic Gaussian (LQG) design methods. In particular, an on-line observer is not required, and the method can be extended to include compensator de- sign wherein the order of the compensator may be prespecified during the design process. Unneeded feedback channels may also be eliminated. 1 The number of sensors may be reduced while eliminating the need to estimate all of the unmeasurable states in real time. Recently, efficient compensator design formulations have been developed for linear time-invariant systems including robustness considerations for both unstruc- tured uncertainty and parametric uncertainty.2'3 The potential advantages of designing a fixed-order linear time-invariant controller are especially appealing for systems whose equations of motion are characterized by linear peri- odic coefficient dynamics. Aerospace applications include he- licopter forward flight stability and control and satellite or- bital stability problems. For helicopter aeromechanical and aeroelastic stability and control applications, the number of states required to model the system accurately continues to increase as research in structural elasticity, dynamic inflow, and aeroelasticity progresses. However, the feasibility and cost effectiveness of measuring/est imating all of the higher frequency structural elastic and dynamic inflow positions and rates for helicopter rotors in real time in an operational envi- ronment is questionable. This also drives the need for poten- tial alternatives to the full state feedback solution. An output feedback approach provides additional flexibility in that higher fidelity state space physical models may be used for the