## The utility SYSTPROP

SYSTPROP is a simple utility that provides a quick overview of some important properties of a linear system:

         1. time constant                             tau
         2. natural frequency of undamped system      w0
         3. eigenfrequency of the system              wn
         4. period                                    P
         5. damping factor                            zeta
         6. percentage overshoot                      PO
         7. peak-time                                 Tpeak
         8. settling time to a 2% band                Tset
         9. halve-time                                Thalve

SYSTPROP also provides conclusions about the stability of the system and possible oscillatory behaviour, and it lists a short summary of the physical meaning of some system properties. For state-space models, information about the controllability and observability of the linear system will also be given, and for transfer function models, information about possible non-minimum-phase behaviour will be presented.

Some definitions:

- The time constant 'tau' is the time in seconds for the decaying exponential transient (in case of non-oscillatory behavior) or amplitude of oscillation (in case of oscillatory behavior) to be reduced to e-1 = 0.368 of its initial value.

The results are displayed on screen and also saved to a user-specified diary-file SYSTPROP.LOG. The default location to store this file will normally be the FDC data directory; if SYSTPROP is not able to locate that directory, it will start with the current working directory. The destination directory and filename can of course be changed in the save dialog. To cancel the creation of the diary-file altogether, select 'Cancel' or click on the Close button in the save dialog.

### Usage

`[X,Y] = systprop(A,B,C,D)` will return the system properties in the matrix X and the eigenvalues of A in the columnvector Y. In addition, this call yields information about the controllability and observability of the system. The columns of the matrix X correspond with the system properties 1 through 9 in the table above, i.e.:

         X = [tau, w0, wn, P, zeta, PO, Tpeak, Tset, Thalve]

`[X,Y] = systprop(A,B,C)` does the same, except all elements of the D matrix will be assumed to equal zero.

`[X,Y] = systprop(A)` returns the system properties in the matrix X and the eigenvalues of A in the columvector Y; see the definitions above.

`[X,Y] = systprop(num,den)` will evaluate the equivalent transfer function representation.

`systprop(Y)` returns the matrix X if Y contains the eigenvalues of a system in a columnvector. If Y is a scalar, it will be treated as a single eigenvalue, instead of a (1x1) A matrix. To analyze a system consisting of a (1x1) A matrix, use the call `[X,Y] = systprop(A,0,0)` instead.

Note: this function requires the Control System Toolbox! In addition, be aware that SYSTPROP uses the routine NUM2STR2 for output formatting.

### Some general notes about system performance

1. The system is stable (i.e. the transient solution will decay to zero) if all poles are located in the left half plane of the s-plane
2. To avoid excessive overshoot and unduly oscillator behaviour, the damping ratio must be adequate, thus the angle phi (the angle between the real-axis of the s-plane and the line through the origin and the pole) must not be too close to zero.
3. The time constant and settling time can be reduced (response speed increased) by increasing the negative real part of the poles.
4. Undamped frequency = distance from pole to origin; moving the poles out radially (i.e. with constant damping factor) increases the speed of the response, thus reducing settling time, peak time and rise time, while the percentage overshoot will remain constant.
5. Natural frequency = resonant frequency or damped natural frequency. This is the frequency of transient oscillations, which equals the imaginary part of the pole positions.
6. Peak time and rise time are reduced by increasing the imaginary part of the pole locations.

For quick reference, the above notes will be included in the SYSTPROP.LOG file too. For more details, it is recommended to read the source-code of SYSTPROP.M and to refer to J. van de Vegte, 'Feedback Control Systems', Prentice Hall International Editions, London, 2nd edition, 1990.