Introduction to Linear and Nonlinear Systems

1. System Representations
   1. State space representations (continuous-time and discrete-time);
   2. Linear and nonlinear systems;
   3. Linearization;
   4. Input-output models and transfer functions;
   5. State space realizations of transfer functions including controllable and observable realizations, minimal realizations.
2. System Behaviour and Stability
   1. Review of linear algebra including eigenvalues and the Cayley-Hamilton theorem;
   2. State transition matrix and the solution of linear state equations;
   3. Stability and boundedness;
   4. Stability tests for linear time invariant systems;
   5. Lyapunov stability theory for linear and nonlinear systems.
3. System Input-Output Properties
   1. Input-output response, impulse response, convolution integral, frequency response;
   2. Input-output stability;
   3. System performance;
   4. Observability, unobservable subspaces and unobservable modes;
   5. Controllability, controllable subspaces and controllable modes.
4. Control Design
   1. Stabilizability and state feedback, eigenvalue placement;
   2. Detectability and observers;
   3. Observer based control design;
   4. Tracking and disturbance rejection;
   5. Linear quadratic regulators (LQR) ;
   6. LQG controllers.