1SC2 2014/2015 Department of Electronics Telecommunications and Informatics University of Aveiro Subject: Sistemas de Controlo II (Control Systems II) 2014/2015 Lecture 2 (13/Feb 2015) Petia Georgieva

Continuous (analog) control systems Chapter 1. Review of the main topics of control system theory (SC1) (chapters I-VII of the book of Prof. Melo) Chapter 2. Basic compensators (chapter VIII) Chapter 3. PID compensator (chapter VIII) Chapter 4. State-feedback control (chapters X, XI) 2

Linear mechanical translation systems (body - spring - damping) 3

Linear mechanical rotation systems (body - spring - friction) 4

Linear mechanical rotation systems with gears 5

Thermodynamic systems and laws 6

Fluid (hydraulic, pneumatic) systems and laws 7

Time (transient) response of 1 st order systems

Time (transient) response of 2 nd order systems

Typical 2 nd order system under-damped response

Linear system stability analysis (Routh-Hurwitz)

Rule 1 (necessary but not sufficient rule for system stability): In order to have all roots with a negative real part it is necessary that the coefficients of the denominator are positive. Rule 2 (necessary and sufficient rule for system stability ): The system is stable if there are no changes of the sign of the 1ª column coefficients and the table has not a line with zeros. Rule 3 : If all coefficients of the 1ª column are different from 0, then the number of changes of the sign of these coefficients = the number of roots of the denominator D(s) with positive real part. Rule 4 : If one or more coefficients of the 1ª column are equal to zero, then there are roots with non negative real part. Case 1 : If the first coefficient of one line is equal to zero but not all other coefficients of the same line are zeros, we substitute this coefficient with a small positive parameter ε and continue to build the table. After that, we compute the coefficients of the table assuming that ε tends to zero. Case 2: If the table has a complete line of zeros, the system is on the boarder of stability. Determine the auxiliary equation from the coefficients of the line above the line of zeros. Substitute the zero line with the coefficients of the first derivative of the auxiliary equation. The poles on the imaginary axes (on the border of stability, i.e. with real part=0) are the solution of the auxiliary equation. Linear system stability analysis (Routh-Hurwitz)

Exemples Determine the values of k that make the system on the figure stable, unstable, marginally stable, apply the Routh-Hurwitz criterion.