Stability of Feedback Control System Notes - Electrical Engineering (EE)

Electrical Engineering (EE): Stability of Feedback Control System Notes - Electrical Engineering (EE)

The document Stability of Feedback Control System Notes - Electrical Engineering (EE) is a part of Electrical Engineering (EE) category.
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 Stability of feedback control system
Frequency response analysis is an useful tool for designing feedback controllers. It enables the designer to study the stability characteristics of a closed loop system using Bode or Nyquist plots and also to select the appropriate design values of controller parameters.

 Bode Diagram of closed loop process
 Bode diagram of PID controler

It is also worth to study the Bode diagram of PID controller in the following figure.
Stability of Feedback Control System Notes - Electrical Engineering (EE)

Stability of Feedback Control System Notes - Electrical Engineering (EE)

The PID controllers have the following characteristics:
Stability of Feedback Control System Notes - Electrical Engineering (EE)

And

Stability of Feedback Control System Notes - Electrical Engineering (EE)

It is left to the reader to exercise how to arrive at above eqs. Note that the PID controllers ideally have three asymptotes.

Stability of Feedback Control System Notes - Electrical Engineering (EE)  with a (-1) slope 

•  Stability of Feedback Control System Notes - Electrical Engineering (EE)  with a slope (1+)

•   Stability of Feedback Control System Notes - Electrical Engineering (EE)Stability of Feedback Control System Notes - Electrical Engineering (EE)Stability of Feedback Control System Notes - Electrical Engineering (EE)  with a zero slope.

Bode diagram of Processes in series
Let us have a control loop with two components, viz ., one PI controller and one first order plus dead time model.
Stability of Feedback Control System Notes - Electrical Engineering (EE)

The open loop transfer function is
Stability of Feedback Control System Notes - Electrical Engineering (EE)

The above open loop transfer function is a combination of four individual processes in series, viz., pure gain, pure dead time, first order system and PI controller. Two time constants are observed in the series that would yield the location of corner frequencies viz .,  Stability of Feedback Control System Notes - Electrical Engineering (EE)  or  Stability of Feedback Control System Notes - Electrical Engineering (EE)

Stability of Feedback Control System Notes - Electrical Engineering (EE)

The above figure shows the Bode plot of the open loop process indicated in this example. AR and phase shift for all individual transfer functions as well as the overall transfer function have been indicated along with the location of corner frequencies.

 Bode Stability Criterion
Consider a simple first order plus dead time process to be controlled by a proportional controller:
Stability of Feedback Control System Notes - Electrical Engineering (EE)Stability of Feedback Control System Notes - Electrical Engineering (EE)

Example of first order system for studying Bode stability criterion

The open-loop transfer function for this system is given by
Stability of Feedback Control System Notes - Electrical Engineering (EE)          71

The Bode plot of the above open loop transfer function is given by the following figure.

Stability of Feedback Control System Notes - Electrical Engineering (EE)

We are interested to know the frequency where the phase shift is -1800. Numerically it can be solved by the equation
Stability of Feedback Control System Notes - Electrical Engineering (EE)         72

The frequency is  Stability of Feedback Control System Notes - Electrical Engineering (EE)  and the value of AR/2Kc at  Stability of Feedback Control System Notes - Electrical Engineering (EE)  is observed to be 0.0463 which can also be found numerically by,
Stability of Feedback Control System Notes - Electrical Engineering (EE)           73

The above exercise indicates that in order to obtain AR = 1 at this frequency  Stability of Feedback Control System Notes - Electrical Engineering (EE)  one needs to set the value of Kc as,
Stability of Feedback Control System Notes - Electrical Engineering (EE)     74

At this juncture, one needs to perform a thought experiment in order to understand the Bode stability criterion. Let us set the value of controller gain, Kc = 10.8 and let us “open up” the feedback loop as indicated in the figure before. Suppose, we vary the setpoint as a sinusoidal function  Stability of Feedback Control System Notes - Electrical Engineering (EE) . As the loop is open, the error will be equal to the setpoint  Stability of Feedback Control System Notes - Electrical Engineering (EE)  and thereby yield an output,

Stability of Feedback Control System Notes - Electrical Engineering (EE)              75

Now, suppose two events occur simultaneously

•  The setpoint perturbation is stopped

•  The feedback loop is reconnected

Then, the error signal will remain as  Stability of Feedback Control System Notes - Electrical Engineering (EE) . In other words, the response of the system will continue to oscillate with constant amplitude even when the setpoint signal is withdrawn.

Alternatively, if we choose the value of controller gain less than 10.8 , (say Kc = 8) then

Stability of Feedback Control System Notes - Electrical Engineering (EE)                     76

If we repeat the above thought experiment, the output signal will take the form
Stability of Feedback Control System Notes - Electrical Engineering (EE)           77

Upon closing the loop and withdrawing the setpoint perturbation, the new value for the error for the next cycle will be  Stability of Feedback Control System Notes - Electrical Engineering (EE)  that will eventually yield an output response of  Stability of Feedback Control System Notes - Electrical Engineering (EE)  and so on. It is evident that the amplitude of the error signal would diminish at every cycle and eventually lead to zero.

In case we choose the value of controller gain greater than 10.8 , (say Kc = 12) then

Stability of Feedback Control System Notes - Electrical Engineering (EE)             78

The same thought experiment would lead to ever increasing error signal because the amplitude ratio is greater than 1.

Hence the above thought experiment indicates that we have been able to find a combination of frequency  Stability of Feedback Control System Notes - Electrical Engineering (EE)  and controller gain  Stability of Feedback Control System Notes - Electrical Engineering (EE)  such that the AR of the process becomes 1 and phase shift becomes -1800 simultaneously at that combination  Stability of Feedback Control System Notes - Electrical Engineering (EE) . The output response shows a sustained oscillation with a time period  Stability of Feedback Control System Notes - Electrical Engineering (EE)  at this combination. Any frequency,  Stability of Feedback Control System Notes - Electrical Engineering (EE) , will lead to oscillation with increasing amplitude and eventually will lead to instability. Hence, the frequency  Stability of Feedback Control System Notes - Electrical Engineering (EE)  is termed as the crossover frequency , the gain value Kis termed as ultimate gain and Pu is called the ultimate period of oscillation of the closed loop system.

The conclusion drawn from the above thought experiment is the Bode Stability Criterion and can be stated as follows -A feedback control system is unstable if the amplitude ratio of the corresponding open loop transfer function is greater than one at the crossover frequency. The value of controller gain is the decisive factor in order to ensure its stability.

It is further understood from eq. (72) that large dead time leads to smaller value crossover frequency. In other words, even a low frequency signal will be able to destabilize such process.

 

 

The document Stability of Feedback Control System Notes - Electrical Engineering (EE) is a part of Electrical Engineering (EE) category.
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)

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