Design Specifications
Questions :-
INTRODUCTION
Controllers in linear time-invariant systems are used to ensure that the intended output is yielded despite the errors and disturbances in the surrounding. The control systems are used in industrial applications as they tend to improve the manufacturing processes, the efficiency of energy usage, advanced automobile control which includes the rapid transport alongside other merits (Dorf & Bishop, 2008). The controllers use an iterative approach to get the accurate output. The iterative approach is achieved using the feedback loop which enables the summer to compute the error by comparing the actual output to the input. Some systems have very complex plants which need to be controlled hence the controller should be designed appropriately.
The controller seeks to ensure that the intended output of the process, G(s), is obtained despite the effects of the noise signal at the feedback loop. The variables that need to be controlled are the gain crossover frequency, steady state error, phase margin, and the noise measure. A closed loop system has a feedback loop that aims at enabling modification of the system behavior (Kuo, 2001). The impact of the feedback loop is to ensure that the desired output is obtained from the system. The loop takes back the actual output to the summer to check for errors and the output signal from the summer is taken into a controller for appropriate adjustment. The feedback loop performs the measurement and signal transmission of the system.
DESIGN SPEFICATIONS
i. The gain crossover frequency should be,
ii. The steady state error ought to be zero in response to a unit ramp reference
iii. The phase margin should be at least 600
iv. The effect of measurement noise on the output should be attenuated by at least 1/100 at noise frequencies,
RESULTS AND DISCUSSION
i. Proportional controller
It is applied to first order process systems that have a solitary energy storage to stabilize the unstable process. The system is set up to reduce the steady state error. An increase in the proportional constant decreases the steady state error of the system. Unfortunately, the controller reduces but does not remove the steady state error. Some of the merits associated with the proportional constant gain are small signal amplitudes and phase margins, the dynamic attributes of the system have a wider frequency bands and larger sensitivity to noise (Karl, 2002). It decreases the signal rise time up to a certain value. Adding the value of Kp after that could lead to overshoot of the system response and amplifies the process noise.
ii. Proportional plus integral controller
The P-I controller seeks to eliminate the steady state error that results from using proportional gain constant. Unfortunately, the PI controller affects the system stability and decreases the speed of response. It does not predict future errors in the system and cannot decrease the rise time nor eliminate the oscillations.
iii.Proportional plus integral plus derivative controller
It provides the ultimate control of the system dynamics to achieve a reduced steady state error to negligible magnitudes with short rise time. There are no oscillations and as a result, a higher stability of the system is achieved. The derivative additional component eliminates the system response overshoot as well as the cycles that occur during the output response of the system.
iv. Filtered proportional plus integral plus derivative controller
DISCUSSION
A transfer function is the minimum phase were all the pole and zeros are reflecting system stability and the non-minimum phase when the poles and zeros depict an unstable system. The gain margin is the detachment on the bode magnitude plot from the amplitude at the phase crossover frequency up to the 0dB point. As demonstrated from the output responses, the proportional and integral components ensure that the response has no offset and it provides a better dynamic response than reset alone. It improves from proportional controller which is experiences offset at steady state. The PI controller introduces instability due to the introduced lag. Adding the derivative component restores the system stability and reduces the lags which are more rapid responses. The filtered version of PID requires proper tuning using Nichol-Ziegler methods which is the most prevalent method in controller design (Tehrani & Augustin , n.d.).
i. Gain crossover frequency design,
To get the gain crossover at unity magnitude,
Finding the magnitude of the equation above,
Using a bode plot, one can obtain the gain crossover frequency directly.
% Design of a control system using the ramp input
clc
clear
close all
format short
% Controller design with 4 design criterions to meet
a1=1;
a2=5;
b1=20;
b2=2500;
w1=100;
w2=1500;
%% elastic motor system
s=tf(‘s’);
num=[a1 a2];
den=[1 b1 b2 0];
Gs=tf(num,den)
t = 0:100;
u = t;
[y,x] = lsim(Gs,u,t);
figure(1)
plot(t,y,t,u)
xlabel(‘Time(secs)’)
ylabel(‘Amplitude’)
title(‘The proportional Controller Ramp Response’)
k=2.5; % proportional constant, Kp (Design criterion 1)
Cs=k;
Ls=Cs*Gs
% To meet Design criterion 2: wc=w1=4 (tolerance of about 10%)
figure(2)
bodeplot(Ls)
grid on
[Gm, Pm,Wcg,Wcp]=margin(Ls)
It is obtained as,
Wcg= 4.5673 rads/sec
ii.Steady state error design using ramp input,
Computing the steady state error for the unit ramp input system response,
In our case study the disturbance is at the feedback loop, it is factored in while determining the systems feedback.
CONCLUSION
In a nutshell, a good control system aims at generating a response quickly and without oscillation such that there is a good transient response. The system should have a low error once it settles hence it is considered to have a good steady-state response.
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