FREQUENCY RESPONSE ANALYSIS OF CONTROL SYSTEM
Description of the simulated system
The control system of a process given by a transfer function model is
simulated. You can adjust the setpoint and the disturbance. The frequency
response of the control system is shown in a Bode diagram.
The controlled process
The process model which is controlled in this simulator can be chosen
you. By default the transfer function from control variable, u, to
measurement signal, y, is a second order transfer function with
timedelay, with a disturbance, v, acting on the process at the same place
dynamically as the control variable. More specifically, the default
process model is
y(s) = H_{ps}(s)u(s) + H_{ps}(s)v(s)
where
H_{ps}(s) = [1/(s^2 + 2s + 1)]e^{0.5s})
The nominal operating point
The nominal operating point is characterized as
follows:
 The process measurement ym is 50 (in a proper unit)
 The disturbance is 10.
 The nominal control signal, u_{0}, which keeps the
process in or close to the operating point, is 40.
Controller function
The
PID controller is on serial form, i.e. its transfer function is
H_{c}(s) = K_{p }[(1+T_{i}s)(1+T_{d}s)]/[T_{i}s(T_{f}s+1)]
The aim of this simulator is to give practice in frequency response based
analysis of feedback control systems.
The frequency response is a good tool for expressing the dynamic
properties of a control system, regarding setpoint tracking and
disturbance compensation, and stability.
In the tasks below it is assumed that the process is in
the nominal operating point,
which is defined above.
 Controller tuning:
Find proper PID settings using e.g. ZieglerNichols' closedloop
method (the sustained oscillations method or the frequency response
method). Unless otherwise stated, it is assumed that you use a PID
controller with these settings in the tasks below.
 Static setpoint tracking:
 Characterize qualitatively the setpoint tracking property of the
control system as read off from both the tracking function and the
sensitivity function.
 Confirm the results in subtask a by running a simulation (apply
steps in the setpoint and the disturbance).
 Dynamic setpoint tracking:
 What is the 3dB bandwidth?
 Run a proper simulation which hopefully confirms that the
bandwidth is as found in subtask a above.
 Run simulations which demonstrate that the tracking is good
"below" the bandwidth, and poor above the bandwidth.
 Assume the setpoint tracking response time T_{r} from
the bandwidth. (The response time is gthe 63% rise time of the step
response, analog to the time constant of a first order system.) T_{r}
can be estimated as
T_{r} = 1.5/w_{b}.
Compare with the response time as found from a simulation. Are
the values similar?
 What is the bandwidth and the response time for a properly tuned
PI controller? Compare with the PID controller.
 Static disturbance compensation:
 Explain from the shape of S (sensitivity function) in a Bode
plot why the control system gives perfect static disturbance
compensation.
 Confirm the result in subtask a by running a proper simulation.
 Dynamic disturbance compensation:
 Let the setpoint be constant (50). Run the following two
simulations with a sinusoidal disturbance of frequency w_{v}
less than the bandwidth (e.g. 0.3rad/s): In the first
simulation the control loop shall be open (i.e. the controller is in
manual mode). In the second simulation the controller the control
loop is closed (i.e. the controller is in automatic modus).
Calculate from the simulations the ratio of the respective
amplitudes in the response in ym. Is this ratio in accordance with
the S(jw) plot?
 As in subtask a above, but the disturbance frequency shall now
be larger than the bandwidth (e.g. 5 rad/s).
 Stability:
 What are the stability margins (gain margin GM and phase margin
PM) with the PID controller (tuned in task 1)? Do they have proper
values?

 Increase the controller gain K_{p} until the control
system becomes marginally stable as seen from the time response.
(Keep this value K_{p} in the following subtasks.)
 What does the tracking function T and the sensivity
function S look like (at marginal stability)?
 Read off the amplitude crossover frequency and the phase
crossover frequency. Do they have equal values?
 Calculate the period of the time response from the w_{c}
value. Confirm the result by a simulation.

 Reset the PID controller to proper settings (as found in task
1).
 Increase the controller gain K_{p} until the control
system gets poor stability. How are the T and S curves changed?
(Then reset K_{p}.)
 Increase the process time delay until the control system gets
poor stability. How are the T and S curves changed?
[KYBSIM] [TechTeach]
Updated October 4, 2004.
Developed by
Finn Haugen.
Email: finn@techteach.no. 