Feedback Control Systems 2012/2013

Linear systems analysis

The matrix exponential; the variation of constants formula. Computation of the matrix exponential via eigenvalues and eigenvectors  and via residual matrices. Necessary and sufficient conditions for exponential stability: Routh-Hurwitz criterion. Modal analysis: mode excitation by initial conditions and by impulsive inputs; modal observability from output measurements; modes which are both excitable and observable. Popov conditions for modal excitability and observability.

Reachability and observability

Kalman reachability conditions, gramian reachability matrices and the computation of input signals to drive  the system between two given states. Kalman observability conditions, gramian observability matrices and the computation of initial conditions given input and output signals. Equivalence between Kalman and Popov conditions.Eigenvalues assignment by state feedback for reachable systems. Design of asymptotic observers and Kalman filters for state estimation of observable systems. Design of dynamic compensators to stabilize any reachable and observable system. LQG (Linear Quadratic and Gaussian) compensator design via Riccati equations.Design of regulators to reject disturbances generated by linear exosystems. Kalman decomposition for non reachable and non observable systems.

Classical feedback design

Impulse responses, step responses and steady state responses to sinusoidal inputs. Transient behaviors. Autoregressive moving average (ARMA) models and transfer functions. Bode plots.  Static gain, system gain and high frequency gain. Zero-pole cancellation. Non minimum phase systems. Nyquist plot and Nyquist criterion. Root locus analysis. Stability margins. Proportional Integral Derivative  (PID) control. Frequency domain design: lead and lag compensation. Bode’s integral formula. Robust pole placement and robust performance.

Textbooks.

K.J.Astrom, R. Murray. Feedback systems. An introduction for scientists and engineers. Princeton University Press 2008

C.M.Verrelli. La matematica elementare del feedback, II Edizione. Esculapio, 2013.