SIDE NOTES
CLASSICAL AND MODERN CONTROL THEORIES.
Some differences between classical and modern control theories are highlighted below.
Classical Theory Modern Theory Transfer function (TF)
I/O based principally on TF. Used only to obtain corresponding D.E. Differential equations (D.E.) Linearized when encountered. Constraints req'd. for useful I/O relationship. Direct use of D.E. Input Impulse, step, sinusoidal. Initial conditions must obtained. Indifferent to type of input. Controllers Limited to linear controller. Non-linear controller (optimal) can be obtained. Design process Synthesis Trial-and-error Modeled systems limited to SISO SISO/MIMO capability ____________
We have discussed at great length on the subject of control actions in class. Here, some highlights of the three main controllers, namely proportional, integral, and derivative controllers are re-visited.
The block diagram of a typical control system is presented in Figure 1.
Figure 1.
Proportional Control Action
This type of control action involves a proportional constant, namely Kp, which is defined as
,
where U(s) is the output signal and E(s) is the error signal, measured in frequency domain.
You may think of Kp as a volume control knob on your car radio. It is essentially an adjustable amplifier that helps reducing the effects of disturbances. This relatively simple type of controller is also useful in reducing the system sensitivity to changes in the plant's parameter. However, if the proportional gain is set too high, system instability (closed-loop) and signal distortion may result. Proportional control action is generally not useful for controlling steady state errors. This shortcoming of proportional control action gives rise to the need of another type of control action: the integral control action.
Integral Control Action
The relationship of the integral control action with the output and error signals, in frequency domain, is defined as
.
(2)
where KI is the integral gain, U(s) is output, and E(s) relates to the error signal. In the time domain, the rate of change of the output is proportional to the error signal by the constant KI. This means when the error changes, the rate at which the output signal change depends on the constant KI. If error is zero, the output signal remains unchanged (slope = 0). This control action is also known as the reset control.
Derivative Control Action
In this control action, the magnitude of the output of the controller U(s) is proportional to the rate of change of error signal E(s) by the constant KD. This action is described by Eq (3).
.
(3)
In a linear system, derivative controller measures the instantaneous slope of the error signal, predicts or anticipates the large overshoot, and makes proper adjustment before the overshoot actually happens. Because of this inherent characteristic, derivative controller is classified as anticipatory type of controller. Noted that if the steady-state error of a system is unchanged in the time domain, the derivative control has no effect since the time derivative is zero in this case. Because derivative control action only works in transient mode, it should never be used alone!
Proportional-plus-Integral Control Action (PI)
Combining Eq (1) and Eq (2), we have
.
(4)
Rearranging (4) we obtain
.
(5)
where TI = KP/KI is known as the integral time or reset rate (times per minute), the rate at which KP is repeated (duplicated). Note that KP affects both proportional and integral parts of the controller. For a system of type 1 (see lecture notes) with a ramp input, KP controls the ramp-error constant and consequently affects the magnitude of the non-zero steady-state error. However, as demonstrated in lab, if you crank up KP too high in a type 1 system, the system could become very unstable (remember the roaring noise?). For a type 0 system, the value of KI affects the ramp-error constant while KP affects phase margin, gain margin, resonant peak, and bandwidth. In general, a PI control action can improve steady-state error, but at the expense of stability. In addition, if the location of the zero of the controller is correctly placed, you can also improve the damping as well. In the design of a PI controller, it's good to keep the zero close to the origin and far away from the most significant poles, while keeping KP and KI small. When properly designed, a PI controller can improve damping but extend rise time. In frequency domain, it can reduce bandwidth and improve gain and phase margins as well as resonant peak.
Proportional-plus-Derivative Control Action (PD)
The combination of proportional and derivative control actions forms
.
(6)
where TD = KD/KP is called the derivative time, during which interval the proportional control action takes effect.
The anticipatory characteristic of derivative control action is found in PD control action. This means, in transient mode, PD can anticipate the direction of the error in making adjustments before excessive overshoot occurs. During the stead-state mode, PD has an effect on the stead-state error only when the error changes with respect to time. In the design of a PD controller, we want to place the controller's corner frequency ω = 1/TP so that the phase margin is improved with the new gain-crossover frequency. The effects of a PD controller in frequency domain include the increase of bandwidth and an accentuation of high-frequency noise, as PD controller acts like a high-pass filter. In the time domain, PD improves damping while reduces maximum overshoot, rise time, and settling time. PD is generally not ideal for lightly damped systems.
Proportional-plus-Intergal-plus-Derivative Control Action (PID)
As the name suggests, this type of controller represents the combined action of Eq. (5) and Eq. (6) or
.
(7)
This type of controller contains the inherent characteristics of PD and PI controllers. With PI controller acts like a low-pass filter (phase-lag controller) and PD acts like a high-pass filter (phase-lead controller), PID controller functions as a bandpass filter. The phase-lag part of the controller is used to boost the damping of the system while the phase-lead part is used to reduce the rise time.
The general effects of a properly designed controller are summarized below:
Max. overshoot Damping Rise time Settling time Steady-state error Phase margin Gain margin Resonant peak Bandwidth filter 1- the observed effects based on unit-step response of a third-order system with KP held constant.
2- If KDis set too high, max overshoot and settling time may increase significantly. In the frequency domain, the excessive gain of KD can cause the phase margin to decrease and resonant peak to increase.
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Handle is Matlab's way of identifying an object. You may assign a handle to a Matlab function for subsequent changes of properties or execution of the function. A function handle is assigned with the use of the "@" sign.
Example:
We will use the handle name "banana" as an identifier in the plotting of a natural log function.
Actual Matlab input Comments >> x=(0.1:0.1:10);
setting an array >> banana = @log; function_handle 'banana' is defined >> juice=plot(x,banana(x)) handle 'juice' is defined >> set(findobj(juice,'type','line',...
'color','blue'),'color',...
'yellow','linewidth',2)Find the specified properties of handle 'juice' (the plot) and change its properties such as colour and line width. *Note: the three dots ("...") used in the command lines above are continuation designator telling Matlab that the lines are to be interpreted one string of command.
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RECOMMENDED READINGS
Hale F., Introduction to Control System Analysis and Design, 2nd. ed., Prentice-Hall,1988.
Kou B. C., Automatic Control Systems, 7th. ed. John Wiley & Son, 1995.
Ogata K., Modern Control Engineering, 2nd., Prentice Hall, 1990.
Phelan R. M., Automatic Control System, Cornell University Press, 1977.
Raven F. H., Automatic Control Engineering, 3rd ed., McGraw-Hill, 1978.
© MMIV. T. Nguyen.