# VU 325.055 Feedback Control

## Content

### Advanced Methods of Feedback Control

Methods of feedback control have undergone a tremendous development in the last three decades. While basic control theory is generally restriced to linear single-input-single-output systems, the automation and control of complex systems in most industries requires far more sophisticated approaches. This course provides an insight into some cutting edge control methodologies.

### The Root Locus Method

Root locus analysis is a graphical method for examining how the eigenvalues of a system change with variation of a certain system parameter, commonly a control gain value. This technique can thus determine and analyse the stability of the feedback system visually. Besides control design the root locus method can be also used to analyse how stability, damping or other essential system properties change as parameters in the control system are varied. A typical application would be parameters changes due to ageing, e.g. friction values, valve characteristics, and so on.

### State Space Control Methods

In control theory, a state-space representation is a mathematical model of a physical system, represented as a set of first order differential equations, relating input, output, and so-called state variables. The state of the system can be represented as a vector within that space and any dynamic transient of the system refers to a state trajectory. There is a number of major advantages when representing systems in the state space: First, as opposed to only looking at input to output relations the state variables reveal much deeper insight into the system behaviour. Once a state space model is available, its inputs and outputs can be changed instantly, without the need to otherwise re-model the whole system.

Even the number of inputs and outputs can be changed immediately, opening the door to multi-input-multi-output system analysis. State space control is an advanced method of control that directly builds on the structure of a state space model. Compared to classic PID control it provides much more design freedom and methods for analysis. State feedback controllers capture the full structure and interconnections of arbitrarily complex systems. Methods for the design of state feedback controllers can be easily extended to multivariable systems.

### Nonlinear Control Methods

Non-linear control methods are required when the nonlinear behaviour of a system is so pronounced that it cannot be approximated by linearisation anymore. Feedback linearization is a common approach used with nonlinear systems. It involves a transformation of the nonlinear system into an equivalent linear system through a change of state variables and a suitable transformed control input, so that subsequently linear control techniques can be applied.

Lyapunov-based control is a generic non-linear technique where the control law is derived in such a way that the closed-loop system is asymptotically stable, based on a so-called control Lyapunov-function. For nonlinear systems that have a cascaded structure, the Backstepping approach can be used to systematically create control Lyapunov-functions.

## Contact

TU Wien

Institute of Mechanics and Mechatronics

Research Unit of Control and Process Automation

Getreidemarkt 9 / E325-04 / 6th floor

1060 Vienna

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