The document discusses PID control, which combines proportional, integral and derivative control actions. It explains each control action and its effects on the system response and stability. Proportional control reduces steady-state error but can cause oscillation. Integral control eliminates steady-state error but also causes oscillation. Derivative control reduces overshoot and damping but increases sensitivity to noise. PID controllers combine these actions to achieve desired response without oscillations or steady-state error. The document provides examples of tuning the gains of each action.

1. Control Systems I
Lecture 12: PID Control
Readings: A&M, Ch. 10, Guzzella, Chapter 11.2,
Jacopo Tani
Institute for Dynamic Systems and Control
D-MAVT
ETH Zürich
December 7, 2018
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 1 / 31

2. Tentative schedule
# Date Topic
1 Sept. 21 Introduction, Signals and Systems
2 Sept. 28 Modeling, Linearization
3 Oct. 5 Analysis 1: Time response, Stability
4 Oct. 12 Analysis 2: Diagonalization, Modal coordinates
5 Oct. 19 Transfer functions 1: Definition and properties
6 Oct. 26 Transfer functions 2: Poles and Zeros
7 Nov. 2 Analysis of feedback systems: internal stability,
root locus
8 Nov. 9 Frequency response
9 Nov. 16 Analysis of feedback systems 2: the Nyquist
condition
10 Nov. 23 Frequency response II
11 Nov. 30 Specifications for feedback systems
12 Dec. 7 PID control
13 Dec. 14 Loop Shaping
14 Dec. 21 State feedback and Luenberger observers
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 2 / 31

3. Today’s learning objectives
Learn what a PID control is and how to design one:
Proportional control: what it is, pro’s and con’s
Derivative control: what it is, pro’s and con’s
Integral control: what it is, pro’s and con’t
Tuning strategies for PID controllers.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 3 / 31

4. A nice intro to PID control
https://www.youtube.com/watch?v=4Y7zG48uHRo
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 4 / 31

5. Recall: Control Specifications
Type of the system (order of ramp to track with zero steady-state error):
Number of integrators in L(s)
Time domain specifications (max overshoot, settling time, rise time, ...):
Location of dominant closed-loop poles (damping ratio and real part)
Frequency domain specifications (command tracking, disturbance/noise
rejection, closed-loop bandwidth):
Bode obstacle course (low/high frequency)
Crossover frequency
Control synthesis: how do we choose a feedback control system that
achieves these objectives?
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 5 / 31

6. Controller design methods
What other methods do exist to design controllers C(s) that meet design
specifications? Many approaches, among them:
PID, Loop Shaping, LQR, LQG-LTR, H∞, Discrete-time optimal control,
continuous-time optimal control, model predictive control,...
Today we look at the most widely used approach for SISO systems: PID
control.
PID - control (proportional-integral-derivative control) is the most widely
applied controller design because it is able to cope well with the majority of
cases encountered in practice.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 6 / 31

7. Proportional Control
1
s+1
Transfer Fcn
Step
Scope
Band-Limited
White Noise
Step1
5
Gain
output
noise
measurement
reference error control
disturbance
The control input tries to move the system in a direction that is opposite to
the error, and is proportional to the error in magnitude.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 7 / 31

8. Proportional gain selection
0 1 2 3 4 5 6 7 8 9 10
t
0
0.2
0.4
0.6
0.8
1
1.2
y
k=2
k=5
k=10
k=50
As the proportional gain increases,
The closed-loop system remains stable;
The steady-state error decreases;
The response becomes faster;
The sensitivity to noise increases.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 8 / 31

9. Proportional gain selection
-60 -50 -40 -30 -20 -10 0 10
-3
-2
-1
0
1
2
3
60 50 40 30 20
0.88
0.975
0.989
0.995
0.997
0.999
1
1
10
0.88
0.975
0.989
0.995
0.997
0.999
1
1
Root Locus
Real Axis (seconds-1
)
Imaginary
Axis
(seconds
-1
)
Closed-loop transfer function
T(s) =
kL(s)
1 + kL(s)
=
1
s + 1 + k
,
i.e., the closed-loop pole is at s = −1 − k (see root locus above).
Steady-state error to a unit step: ess = lims→0
1
1+kL(s) = 1
1+k
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 9 / 31

10. Proportional gain selection
-40
-20
0
20
40
Magnitude
(dB)
10
-2
10
-1
10
0
10
1
10
2
-90
-45
0
Phase
(deg)
k=2
k=5
k=10
k=50
Bode Diagram
Frequency (rad/s)
As the proportional gain increases,
Phase margin remains > 90◦
;
The crossover frequency increases;
The low-frequency gain increases;
The high-frequency gain increases;
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 10 / 31

11. Introducing an integrator
1
s+1
Transfer Fcn
Step
Band-Limited
White Noise
Step1
2
P Gain
simout
To Workspace
1
s
Integrator
5
I Gain
output
noise
measurement
reference error
disturbance
control
control
Integrating the error allows one to detect potential ”biases” in the system
behavior.
An integral control action tries to move the response in order to reduce the
detected biases.
PI control:
u(t) = kP e(t) + kI
Z t
0
e(τ)dτ,
C(s) = kP +
kI
s
=
kP s + kI
s
= kI
kP /kI · s + 1
s
.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 11 / 31

12. Integral gain selection
0 1 2 3 4 5 6 7 8 9 10
t
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
y
kP
=2, kI
= 2
kP
=2, kI
= 2
kP
=2, kI
= 5
kP
=10, kI
= 50
As the integral gain increases,
The steady-state error is zero (as long as kI is not zero)
The response becomes more oscillatory (warning!)
The sensitivity to noise does not change!
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 12 / 31

13. Integral gain selection
-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10
-30
-20
-10
0
10
20
30
Root Locus
Real Axis (seconds-1
)
Imaginary
Axis
(seconds
-1
)
Steady-state error to a unit step: ess = lims→0
1
1+C(s)L(s) = 0
The root locus shows us that as the integral gain increases, the closed-loop
poles go from being “slow” and overdamped to being “fast” but with low
damping!
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 13 / 31

14. Integral gain selection
-60
-40
-20
0
20
40
60
80
Magnitude
(dB)
10-2
10-1
100
101
102
103
-180
-150
-120
-90
Phase
(deg)
Bode Diagram
Frequency (rad/s)
As the integral gain increases,
Phase margin decreases;
The crossover frequency increases;
The low-frequency gain increases — but goes to infinity near 0 in all cases ;
The high-frequency gain does not change.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 14 / 31

15. Proportional Control — Higher order systems
2
s +2s+2
2
Transfer Fcn
Step
Band-Limited
White Noise
Step1
50
P Gain
simout
To Workspace
output
noise
measurement
reference error
disturbance
control
control
How do the previous consideration extend to higher-order systems, e.g., 2nd
order?
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 15 / 31

16. Proportional gain selection
0 1 2 3 4 5 6 7 8 9 10
t
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
y
As the proportional gain increases,
The closed-loop system become more oscillatory (warning!);
The steady-state error decreases;
The response becomes faster;
The sensitivity to noise increases.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 16 / 31

17. Proportional gain selection
-2.5 -2 -1.5 -1 -0.5 0 0.5
-15
-10
-5
0
5
10
15
14
12
10
8
6
4
2
14
12
10
8
6
4
2
0.5
0.26
0.17 0.115 0.085 0.056 0.036 0.016
0.5
0.26
0.17 0.115 0.085 0.056 0.036 0.016
Root Locus
Real Axis (seconds-1
)
Imaginary
Axis
(seconds
-1
)
The root locus shows that as the proportional gain increases, the closed-loop
poles have decreasing damping ratio.
Steady-state error to a unit step: ess = lims→0
1
1+kL(s) = 1
1+k
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 17 / 31

18. Proportional gain selection
-80
-60
-40
-20
0
20
40
Magnitude
(dB)
10-2
10-1
100
101
102
-180
-135
-90
-45
0
Phase
(deg)
Bode Diagram
Frequency (rad/s)
As the proportional gain increases,
Phase margin gets smaller and smaller!
The crossover frequency increases;
The low-frequency gain increases;
The high-frequency gain increases;
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 18 / 31

19. Introducing a differentiator
2
s +2s+2
2
Transfer Fcn
Step
Band-Limited
White Noise
Step1
50
P Gain
simout
To Workspace
Derivative
50
D Gain
output
noise
measurement
reference error
disturbance
control
control
Differentiating the error allows one to “predict” what the error will do in the
near future.
An derivative control action tries to avoid overshooting, hence damping the
system.
PD control:
u(t) = kP e(t) + kDė(t)
C(s) = kP + kDs.
Note that this is not a causal transfer function (not physically realizable in
general). This is typically fixed by approximating the derivative as s ≈ s
cs+1
for some small c.)
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 19 / 31

20. Derivative gain selection
0 1 2 3 4 5 6 7 8 9 10
t
0
0.2
0.4
0.6
0.8
1
1.2
1.4
y
kP
=50, kD
=2
kP
=50, kD
=5
kP
=50, kD
=10
kP
=50, kD
=50
As the derivative gain increases,
The steady-state error not affected;
The response becomes less oscillatory, but potentially slower
The sensitivity to noise increases!
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 20 / 31

21. Derivative gain selection
-120 -100 -80 -60 -40 -20 0 20
-30
-20
-10
0
10
20
30
Root Locus
Real Axis (seconds-1
)
Imaginary
Axis
(seconds
-1
)
Steady-state error to a unit step: ess = lims→0
1
1+C(s)L(s) = 1
1+kP L(0)
The root locus shows us that as the derivative gain increases, the closed-loop
poles are “pulled” into the left half plane!
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 21 / 31

22. Derivative gain selection
-60
-40
-20
0
20
40
Magnitude
(dB)
k_P=50, k_D=2
k_P=50, k_D=5
k_P=50, k_D=10
k_P=50, k_D=50
10-2
10-1
100
101
102
103
-180
-135
-90
-45
0
Phase
(deg)
Bode Diagram
Frequency (rad/s)
As the derivative gain increases,
Phase margin increases;
The crossover frequency increases;
The low-frequency gain does not change;
The high-frequency gain increases.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 22 / 31

23. Proportional-Integral-Derivative Control
2
s +2s+2
2
Transfer Fcn
Step
Band-Limited
White Noise
Step1
10
P Gain
simout
To Workspace
Derivative
5
D Gain
1
s
Integrator
10
D Gain1
output
noise
measurement
reference error
disturbance
control
control
One can also combine the effects of an integrator and of a differentiator with
the basic proportional controller.
PID control:
u(t) = kP e(t) + kI
Z t
0
e(τ)dτ + kDė(t),
C(s) = kP +
kI
s
+ kDs =
kDs2
+ kP s + kI
s
.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 23 / 31

24. PID Tuning
PID tuning corresponds to choosing the parameters kp, ki and kd to reach the
feedback control design specifications.
PID tuning can be done with tuning rules by hand or numerically using
MATLAB or other tools (the latter requires a system model).
There exist heuristic methods to tune a PID controller without a model of
the plant P(s), e.g. the tuning rules proposed by Ziegler and Nichols.
My recommendation: think of a PID as
C(s) = kRL
(s − z1)(s − z2)
s
i.e., as two zeros and one pole at the origin. Decide where you want these
zeros (in the complex plane, or in terms of natural frequency and damping
ratio on the Bode plot), and what you want the (root-locus) gain to be.
Finally, compute the corresponding kP , kI , kD.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 24 / 31

25. Summary
Proportional control
Decrease the steady-state error;
Increase the closed-loop bandwidth;
Increase sensitivity to noise;
Can reduce stability margins for higher-order systems (2nd order or more).
Integral control
Eliminates the steady-state error to a step (if the closed-loop is stable);
Reduces stability margins, can make a higher-order system unstable.
Derivative control
Reduce overshooting, increase damping;
Improves stability margins;
Increase sensitivity to noise.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 25 / 31

26. Today’s learning objectives
Learn what a PID control is and how to design one:
Proportional control: what it is, what it does, pro’s and con’s
Derivative control: what it is, what it does, pro’s and con’s
Integral control: what it is, what it does, pro’s and con’s
Tuning strategies for PID controllers.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 26 / 31

27. Ziegler Nichols Tuning Rules
Assumption: Plant can be approximated by the transfer function
P(s) =
k
τs + 1
e−Ts
with T/(T + τ) small.
Apply the controller C(s) = kp to the system starting at kp = 0 and increase
kp until the system is in a steady-state oscillation, then note the ”critical kp”
called k∗
p and the corresponding critical oscillation period T∗
.
Use k∗
p and T∗
to calculate the control gains:
type kp Ti Td
P 0.5 · k∗
p ∞ · T∗
0 · T∗
PI 0.45 · k∗
p 0.85 · T∗
0 · T∗
PD 0.55 · k∗
p ∞ · T∗
0.15 · T∗
PID 0.6 · k∗
p 0.5 · T∗
0.125 · T∗
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 27 / 31

28. Ziegler Nichols Tuning Rules
Graphically:
1
1(Lino Guzzella ”Analysis and Synthesis of Single-Input Single-Output Control Systems)
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 28 / 31

29. Ziegler Nichols Tuning Example
Plant: P(s) =
1
(s + 1) · (s2 + 2s + 2)
Approximation: Papprox =
0.5
0.5 · s + 1
· e−0.01s
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 29 / 31

30. Ziegler Nichols Tuning Example
Set Ti = ∞, Td = 0, τ = 0 and increase gain kp.
Critical gain k∗
p = 10 with critical oscillation period T∗
= 2π
ω∗ = 2π
2 = π
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 30 / 31

31. Ziegler Nichols Tuning Example
P, PI, PD, PID controller according to Ziegler and Nichols tuning rules.
PID controller derived with MATLAB sisotool.
Ziegler Nichols tuning rules can be useful when no model of the plant is
available but generally other tuning rules provide better results.
J. Tani, E. Frazzoli (ETH) Lecture 12: Control Systems I 7/12/2018 31 / 31