# System Theory 2

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Based on the course System Theory 1, in which continuous-time control systems with continuous-time controllers were considered, the lecture System Theory 2 imparts extended mathematical methods to analyze and synthesize discrete-time systems in the time and frequency domain. Although many technical and non-technical systems can be steered by continuous-time controllers, the use of digital computers (e.g. in the form of microcontrollers) allows much better regulation procedures to be implemented. In addition, nowadays digital controls can be implemented at low cost, which is why their use is usually preferred.

The course System Theory 2 is intended to provide the students a deep understanding of discrete-time control systems. Besides the introduction of the z-transformation, the stability analysis of discrete-time systems and the design of control algorithms for sampling controls, the model of the state space representation is considered. On the basis of the state space representation, the so-called canonical forms are derived and system properties such as controllability and observability are discussed. In addition, different discrete-time control procedures in the state space are pointed out. In the end, solutions for control systems under uncertainty are presented, such as state estimations using the Kalman filter.

## Detailed event overview

Linear discrete-time systems and sampling controls

Structure of sampling controls, sampling theory, quantization, D/A converters, discrete-time model of sampling control

Discrete-time systems in the time and frequency domain, analysis of sampling systems

System representation by differential equations, impulse response sequence, system representation by convolution sum, z-transformation, correspondence to Laplace-transformation, the transfer function of discrete-time systems, the stability of discrete-time systems, pole positions of continuous-time and discrete-time systems

Control algorithms for sampling control

Continuous-time and discrete-time PID controller, quasi-continuous sampling controls

State space representation, system description and analysis in the state space for linear continuous-time & discrete-time systems

Concept of the system states, state space model and solution of the state equations in the time domain, fundamental matrix, state-transition matrix, solution of the state equations in the frequency domain

Canonical forms for linear continuous-time & discrete-time systems

Controllable and observable canoncial forms, the duality between controllable and observable canoncial form, Jordan canonical form

Transformation of state equations to normal forms

Similarity transformation, transformation in diagonal form and Jordan canonical form, application of canonical transformations

Controllability & observability of linear continuous-time & discrete-time systems

Controllability and reachability, controllability matrix and controllability condition according to Kalman, observability, observability matrix and condition for observability, duality and dual systems

Synthesis of linear continuous-time & discrete-time control systems in the state space

Feedback of the state vector, feedback of the output vector, prefilter, controller synthesis by pole placement

State observer for linear continuous-time & discrete-time systems

State observer, gain matrix of the observer, observer synthesis by pole placement

State estimation using Kalman filter for linear systems

Probability calculations, Kalman filter, state estimation using the Kalman filter

## Questions?

In case of questions, please contact acs-teaching-sys2@eonerc.rwth-aachen.de.