This chapter generalizes the concepts introduced in Chap. 9 to multi-input multi-output systems. It begins with the Linear Quadratic Regulator (LQR), deterministic and stochastic. A procedure is described which allows to prescribe a minimum degree of stability (alpha-shift). Next, the state reconstruction is discussed through the full state observer and the Kalman Filter, leading to the Linear Quadratic Gaussian (LQG). The duality between the LQR and the Kalman Filter is stressed. A discussion of the spillover follows; this phenomenon is typical of large lightly damped systems: the modes which are not included in the controller (residual modes) and which have a small stability margin (due to natural damping) tend to be destabilized by the state feedback controller. Finally, the integral control and more generally the frequency shaping of the LQG is discussed. The chapter concludes with a short list of references and a set of problems.
KeywordsLinear Quadratic Regulator (LQR) Lyapunov equation Riccati equation Linear Quadratic Gaussian (LQG) State observer Kalman filter Spillover Integral control Frequency shaping
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