8. Synopsis
1. Basic notions of control theory (3 hours of lectures): The principle of control. Presentation of control structures. Principles and differences of open and closed loop control. Functional diagrams, dataflow diagrams, conventions and standard signals and their nomenclature in a control loop. Static and dynamic characteristics of control loops, integrals of the error function. Classification of control systems. The steps of control system synthesis. Main trends in control theory including a historical review. Some important services of Matlab, Simulink, and the Control System toolbox.

2. Modeling of dynamical systems (3 hours of lectures): Dynamical systems. State, and state space. Solution of the state equation of a continuous time, linear, time varying (LTV) system, the properties of the fundamental matrix. Solution of the state equation of a continuous time, linear, time invariant (LTI) system, the exponential matrix, the transfer function, poles, and zeros. The consequences of a (invertible linear) coordinate transformation in the state space, LTV system invariants. Possibilities to solve the state

equations of a continuous time nonlinear system using numerical methods, linearization around a setpoint. Models of some classes of physical systems including mechanical and thermal processes using energy preservation laws of physics. Model establishment based on measures available on the real process.

3. Analysis of continuous time linear control systems (6 hours of lectures): Descriptions of single variable (SISO) linear transfers: ordinary differential equation, transfer function, Bode-plot, impulse response, step response, state equation. Transformations between descriptions. Fundamental interconnections of elements, open and closed loops. Elementary transfers. First and second order systems: characteristics in time and

frequency domains. Relation between the dominant pole(s) and the dynamical characteristics of a transfer. Properties of the amplitude and phase plots of a general open loop transfer function, the calculation of the crossover frequency. Steady state responses in linear control loops and consequences on reference tracking and disturbance rejection. Stability criteria: Hurwitz criterion, Nyquist criterion, Bode criterion, phase margin and crossover frequency. The description of the stability margin by the phase margin.

4. Synthesis of continuous time linear control systems (9 hours of lectures): The class of PID compensators, the PID compensator with filtered D term, Bode plots and pole zero distribution of the compensators. Properties of the compensators to be used. Setting the compensator parameters for a desired phase margin and steady state behavior. Examples for compensation with P, PD, PI, and PID controllers. Feedback compensation. Controller design for minimal error square integral. Root locus methods. Compensation of systems with time lag: compensation of an ideal time lag with an integrator, compensation of time lags using Smith predictor. Setting the controller parameters for bounded controller signals. Experimental setting of controller parameters using the Ziegler-Nichols method.

5. Analysis of discrete time linear control systems (3 hours of lectures): The Shannon law. Properties of hold elements. Signal propagation in a discrete time system in frequency domain and using state space description. Discrete time equivalent of a continuous time plant using a zero order hold element. Discrete time implementation of continuous time compensators: discrete time realization of integral and derivative operators (approximations), step response equivalence. Hardware and software realization of a PID controller using integrator anti-windup techniques. Nyquist and Bode stability criteria for discrete time control systems.

6. Synthesis of discrete time linear control systems (3 hours of lectures): Realization of a simple direct digital control scheme. Design of a discrete time controller using the bilinear (Tustin) transform: the effect of the transformation to transmission poles and zeros, the main steps of the compensator design, setting the parameters for a given phase margin and crossover frequency using the technique already presented for the continuous time case. Design of two-degree-of-freedom controllers: the choice of the observer polynomial and the transfer function of the reference model, the steps of the design procedure to arrive to a Diophantine equation. Illustration with an example. Robustness of the two-degree-of-freedom controller scheme against parameter uncertainties. Compensation of a plant with time lag, the realization of the Smith predictor.

7. Control loop synthesis in state space (6 hours of lectures): Controllability and observability in continuous time linear systems. Conditions of full state controllability and observability. Staircase forms, stabilizability and detectability. Kalman decomposition of LTV systems. Pole placement using state feedback, the Ackermann formula. Design of full state observers, algebraic equivalence to the pole placement problem. Controllability

and observability of discrete time systems. Pole placement and actual observer design for discrete time systems. Integral control and load estimator design.

8. Discrete time system models and parameter identification methods (3 hours of lectures): Autoregressive (AR) and moving average (MA) processes, ARX and ARMAX models. Parameter identification of ARX models using LS methods. Parameter identification of the ARMAX model using numerical optimization and the quasi-Newton method. The services of the Identification toolbox of Matlab. The recursive LS problem and its solution with application possibilities in control engineering and signal processing.

9. Elementary stability theory of nonlinear systems, further topics (3 hours of lectures): Equilibria and limit cycles of nonlinear systems, their stability in Lyapunov’s sense. Uniform and asymptotic stability. Positive and negative definite functions. Lyapunov’s direct and indirect methods. Relation between the classical and Lyapunov stability for LTV systems. Invariant set, LaSalle’s invariance theorem. Examples for stability analysis of nonlinear control systems. Short introduction into further topics: new trends in contemporary control theory, new tools, rapid control prototyping, case studies to present up-to-date development tools.