Prakties 2 - Electrical, Electronic and Computer Engineering

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Figure 1. Reaction curve (m.s.t = maximum slope tangent. Reproduced from Goodwin et al., (2001)) The values for , and are read off from the reaction curve where is the time at which the step was applied, is the time at which the system starts to react and is the time where the maximum slope tangent (drawn from ) intercepts the steady state output value. Once , and have been determined from the graph, the dead time, approximate time constant and gain are calculated. The dead time of the system is calculated as . The approximate time constant is and the gain is calculated as: where is the output and is the input. These values are then substituted into the equations from the following table to determine the PID parameters. [ ] ( ) Table 1: Controller settings for the Cohen-Coon process reaction curve method The standard form for the PID controller is given by: ( ) ( Note that the values in Table 1 (and Table 2) are given for the standard PID controller form. 2.2 Ziegler-Nichols oscillation method This method is described in Seborg et al., (2004) as follows: 1. After the process has reached steady state, eliminate the integral and derivative action (i.e. implement proportional only control). 2. Set the controller gain to a small non-zero value. 3. Introduce a small momentary set-point change and observe the output. Gradually increase the controller gain and repeat the introduction of the momentary set-point change until sustained oscillation is seen at the output. The oscillation at the output should have a constant amplitude (i.e. it should not die out over time). The numerical value of the proportional controller gain that produces the continuous oscillation is called the ultimate gain . The period of the oscillation is called the ultimate period . 4. Calculate the PID controller settings from the Ziegler-Nichols tuning settings table. Ziegler-Nichols Method PID Table 2: Controller settings based on the Ziegler-Nichols oscillation method Control Systems (EBB 320) Practical 2 Guide 2012 Page 2 )
The numeric value for the ultimate controller gain may be calculated from the method of Goodwin et al., (2001) (p.162) or from the Bode stability criterion (in Seborg p.317, (2004)). 2.3 Bode stability criterion The Bode stability criterion is a method to evaluate the stability of a system containing time delays. There is a host of applications for this method but all we are interested in is the calculation of the ultimate controller gain that leads to sustained oscillation. This gain is achieved at the point where the system is marginally stable. From the Bode stability criterion this is the point where or equivalently | | and . Here is the open loop transfer function. In our case it is simply the product of the proportional controller and the plant transfer function (first order plus time delay). Rewrite the open loop transfer function into magnitude and phase form. Determine the frequency at which the phase becomes . At this frequency the magnitude must equal 1 for marginal stability. Substitute the calculated frequency back into the magnitude equation and solve for the ultimate controller gain. 3. Additional circuits The controller ( ) can be written in the following form: ( ) ( ) ( ) [( Note that you will need to rewrite the standard PID controller form, given in section 2.1, into this form to calculate the component values. Figure 1: Realization of controller ( ( )). Control Systems (EBB 320) Practical 2 Guide 2012 Page 3 ) ⁄ ]
Page 1: Departement Elektriese, Elektronies
Page 5 and 6: Figure 5: The implementation of the

Prakties 2 - Electrical, Electronic and Computer Engineering

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