HRRE materiály ke zkoušce

Inverse controller

Compensation of all poles and zeros of the system by controller zeros and poles which leads to following open loop transfer function

where ωˇr is cut-off frequency.

This form is suitable for step changes in required value or in disturbance since it leads to zero steady state error (for step change in disturbance it is necessary to have integrator in a controller), MP = 90◦ a MG = ∞ (F0(p) never crosses negative real axes).

Computation of the controller transfer function is done using formula

Designed controller often lacks to causality condition. This limitation can be solved using realization time constants on higher frequencies.

ZIEGLER-NICHOLS METHOD (critical gain a critical period – co to je, účel)

ZN method can be summarized in following steps.

1. Integral and derivative parts of PID controller are excluded and only P component is active (TD = 0 a TI = ∞).

2. Proportional gain KR is successively increased until stability limit is achieved. Reached value of KR for which undamped oscillations are visible is called critical gain - Kkrit. Period of oscillations is called critical period - Tkrit.

3. Measured critical parameters Kkrit a Tkrit can be used in tabulated formulas for computing parameters of selected controller.

This test on stability limit can be dangerous and can lead to system destruction. Therefore we can

• compute critical parameters from step response,

• use model to find critical parameters from simulation results,

• use model for the computation of critical parameters, ,

• use relay test which guarantee maximum magnitude of oscillations.

ZN method - closed loop circuit on stability limit

Formulas for controller parameters computation using Ziegler-Nichol method

Following table assumes PID controller described using following transfer function

Normalized step response and its analysis

I is called inflection point. By drawing a tangent line at the inflection point of the S-shaped curve and by finding the intersections of the tangent line with the time axis and the steady-state level line we obtain delay time Tu and rise time Tn respectively. Td is time (transport) delay.

How to determine critical parameters from step response?

Delay time Tu and rise time Tn can be measured from step response which is not oscillating. Approximate formulas then hold for critical parameters computation

Such obtained critical parameters can be used in formulas on previous slide to compute parameters of selected controller.

MATLAB

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and proprietary programming language developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages, including C, C++, C#, Java, Fortran and Python.

Although MATLAB is intended primarily for numerical computing, an optional toolbox uses the MuPADsymbolic engine, allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems.