Designing A Feedback Control System For An Elastic Motor Transmission System

Design Specifications

Instructions

The following assignment is an individualised task. You should consult the accompanying ”individual parameters” document to find the parameters a₁, a₂, b₁, b₂, ω₁ and ω₂ which correspond to your Curtin student ID number. Your assignment should begin with a clear statement of your ID number and the plant transfer function and design criteria for your individualised task.

This is a design task, which should be documented in report form. Your report should include a clear description of the task which needs to be accomplished, and the steps undertaken to address the problem. All graphs should be clearly labelled, and all design steps should include some justification. MATLAB or a similar computational package may be used for any of the calculations or graphs required. The report should be able to be read and understood on its own without reference to this document.

Design task

                       

                                                          Figure 1: Feedback control system

 A feedback control system (illustrated in Figure 1) needs to be designed for an elastic motor transmission system with plant transfer function

                                                            G(s) =      a1s + a2       .

                                                                     s(s2 + b₁s + b₂)

You should design a controller C(s), such that the closed-loop system is asymptotically stable and such that the following design criteria are met:

1. the gain crossover frequency ω_c should be approximately ω₁ (±10%);
2.  the steady-state error should be zero in response to a unit ramp reference;
3. the phase margin should be at least 60^o;
4. the effect of measurement noise on the output should be attenuated by at least ¹ at noise frequencies ω > ω₂

If the four performance criteria are met, further iteration of the controller may be undertaken to minimise the settling time of the step response from r(t) to y(t). If you cannot meet any of the design criteria, get as close as you can while ensuring closed-loop stability.

This task will be approached incrementally, beginning with a proportional controller and finishing with a filtered PID controller.

1. Proportional controller

Choose a proportional controller C(s) = k such that design criterion 1 is met. Graph the Bode plot for L(s) = C(s)G(s) to show that this is the case.

Calculate the transfer function from the reference r to the error e, and graph the ramp response of this function to demonstrate whether design criterion 2 is met.

Use your Bode plot or a margin plot for L(s) to demonstrate whether design criterion 3 is met.

Calculate the transfer function from the measurement noise n to the output y. Find the frequency response of this function at a frequency of ω₂ to demonstrate whether design criterion 4 is met.

2. Proportional plus integral controller

Consider a controller of the form

                                                              

Choose T_i and k such that design criteria 1 and 2 are both met, and the system remains stable. (Hint: choose T_i first to eliminate steady-state error, then select k to set the gain crossover frequency). With this choice of controller, demonstrate whether design criteria 3 and 4 are met.

3. Proportional plus integral plus derivative controller

 Consider a controller of the form

                                                      

Choose T_i, T_d and k such that design criteria 1, 2 and 3 are all met, and the system remains stable. Note 1: the derivative term in the controller can be used to provide additional phase-lead around the desired cross- over frequency, hence increasing the phase margin. Note 2: the numerator of the PID controller contains a second-order term, which can be used to cancel the second-order term in the denominator of the plant, leaving k to set the crossover frequency.

With this choice of controller, demonstrate whether design criterion 4 is met.

4. Filtered proportional plus integral plus derivative controller

Consider a controller of the form

                                                             

Beginning with the parameters k, T_i and T_d used for your PID controller, choose the set point of the filter term T_f such that design criterion 4 is met. In other words, set the corner frequency of the filter such that noise signals above ω₂ are attenuated sufficiently. Note that at frequencies where |C( jω)G( jω)| is small,

                                                                     

and therefore we can use the loop transfer function attenuation to set the closed-loop attenuation.

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.