Space Systems, Modeling and Identification

Systems and signals

Signals; input-output systems, definition and examples; dynamical systems; continuous-time and discrete-time systems; time-invariant and time-varying systems; linear and nonlinear systems; deterministic and stochastic systems.

State space description

Input-output mapping systems; memoryless and dynamical systems; anticipative and nonanticipative systems; difference and differential systems; time invariance and linearity; initial conditions; state space description, notion of state; differential and difference state space systems; linearization process.

Linear systems characteristics

Time characteristics (impulse and step responses); frequency characteristics (Bode, Nyquist).

Identification

Obtaining a model of the system from the data; identification experiments; ARX model; Least Squares methods and modifications.

Solution of differential (difference) state space equations

Response to an initial state, response to an input signal, uniqueness of the solution; modes of system; relation between continuous-time and discrete-time model, sampling of continuous-time models; synchronous and asynchronous sampling.

Realization theory, System poles and zeros

System poles (resonance), systems zeros (antiresonance), system order; transfer function description and its relation to the state space description; transfer function poles and zeros, Smith-McMillan canonical form, degree of transfer function; state space canonical realizations (reachability canonical form, observability canonical form, Jordan canonical form) transfer function matrix of Multi Input Multi Output (MIMO) systems.

Stability

Lyapunov and asymptotic stability; Lyapunov equations for continuous-time and discrete-time systems; characteristic polynomial.

Reachability, stabilizability

Reachability, stabilizability, reachability matrix, reachability Gramian, reachability criteria for continuous and discrete-time systems.

Observability, detectability

Observability, detectability, observability matrix, observability Gramian, observability criteria for continuous-time and discrete-time systems.

Block Diagrams

Parallel, series and feedback connection of systems.

Transformations of state

Regular transformation of state space; controllability and observability in the new basis; Kalman canonical realization; minimal realization. Transformation of controlability Gramian, observability Gramian; Hankel singular values, balanced realization, model reduction by balanced truncation method.

State feedback

Change of dynamical properties of the system by state feedback; relation to reachability, stabilization.

State observer

Change of dynamical properties of the system by output injection; relation to observability; state observer, identical observer, reduced observer; separation principle; stabilization.