Comparison between an auto-tuned PI controller,
a predictive controller and a predictive functional
controller in elementary dynamic systems
G. Valencia-Palomo∗ and J.A. Rossiter.
Automatic Control and Systems Engineering, University of Sheffield, UK.
e-mail: ∗ g.valencia-palomo@sheffield.ac.uk
Abstract— In this paper three controllers are compared: a
novel auto-tuned PI, a Model Predictive Controller (MPC) and
a Predictive Functional Controller (PFC) in order to illustrate
the similarities and differences as well as their advantages
and disadvantages. These controllers are tested in different
elementary systems and the tuning is made following specific
rules. The results of these tests shows the benefits of use simple
algorithms to control different plants that exhibit low order
systems behaviors.
I.
I NTRODUCTION
In the process industry most of the controllers are PI(D)
mainly due to its price and ease of tuning. These controllers
solves well most of the monovariable control tasks (e.g.
flow, pressure, temperature and so on). However, in more
complex systems that exhibits dead time, non-minimum
phase or when the system is subject to constraints or
constant perturbations, the so-called conventional control
structures do not always give good results and are difficult
to maintain. Therefore, in the practice, the process is often
regulated manually by an operator.
On the other hand, a modern control algorithm that has
had success in the process industry is indeed the predictive
control (MPC) (Maciejowski, 2002; Rossiter, 2003; Camacho and Bordons, 2004). Model predictive control is
a common name for computer control algorithms that use
explicit process model to predict the future plant response.
According to this prediction the algorithm optimices the
manipulated variable to obtain an optimal plant response.
But while there are currently many academic predictive
controller developments that solves a large variety of complex problems, the majority of these have not been tested
experimentally nor exploited commercially.
The different linear MPC approaches involves the resolution of a quadratic problem (QP) in order to obtain the
optimal input to the process. The computational load associated to solve an on-line QP can be heavy and my require
a standalone computer. In the last decade a predictive
functional control (PFC) has been pioneered by Richalet
(1993). The advantage of PFC compared to different MPC
configurations is its flexibility to transform a QP problem
into a square system of equations, witch allows for an
easy implementation in practice. But despite its industrial
success does not enjoy wide acceptance in the academic
world because of the inconsistency of its mathematical
formulation.
This paper aims to make a comparison between these
three control techniques: PI(D), MPC and PFC in order to
illustrate the similarities and differences as well as their
advantages and disadvantages when are used to control
different types of systems.
II. T HE CONTROLLERS
In this section the background of the controllers are
exposed. The PI controller is adjusted by a state of the art
tuning rule while the algorithms of the predictive controllers
MPC and PFC are formulated in state-space.
II-A. Auto-tuned PI(D)
A novel auto-tuned PI(D) controller described in (Clarke,
2006; Gyöngy and Clarke, 2006) is used. A schematic
diagram of the system is shown in Fig. 1. The objetive is
to adapt the controller so as to achieve a carefully chosen
design point on the Nyquist diagram.
The key components are phase/frecuancy and plant gain
estimators (PFE, GE), described in detail in (Clarke, 2002).
In essence a PFE injects a test sinewave into a system and
continuously adapts its frequency ω1 until its phase shift
attains a desired value θd (in this case the design point).
Also forming important part of the tuner, but not shown
in Fig. 1, are variable band-pass filters (VBPF) at the inputs
of the PFE and GE. These are second-order filters centered
on the current value of the test frequency. They are used to
isolate the probing signal from the other signals circulating
on the loop (such as noise, set-point changes and load
disturbances).
The algorithm is initialized using a first-order/dead-time
(FODT) approximation Ga (s) for the plant, obtained from a
simple step test. The initialization involves the computation
of suitable values for the parameters associated with the
GE, PFE and the controller.
II-B. Model predictive control (MPC)
Consider the discrete-time system with no disturbing and
no measurement errors:
(
x (k + 1) = Ax (k) + Bu (k)
(1)
y (k) = Cx (k)
Test frecuency
(from FPE)
Set-point r
1
+
+
where:
1
Design
GE
Controller
C(s)
u
Control
signal
u = ∆UHC ;

IHC ×HC
 −IHC ×HC


T
R=

−T

 GH P C
−GHP C
Output y
Plant
G(s)
PFE
Fig. 1. Schematic diagram of the autotuning PI(D).
where x(k), y(k) and u(k) are the state vector, the measured
output and the process input respectively. x(k) ∈ Rn , in
the SISO case: y(k), u(k) ∈ R. A MIMO process has the
same description but with y(k) ∈ Rl and u(k) ∈ Rm . By
simplicity in the notation only the first case is considered,
the extension to MIMO case is straightforward.
By iterating the model, the output prediction with a
prediction horizon HP and a control horizon HC is given
by:
ŶHP = FHP + GHP C ∆UHC
(2)
where matrices ŶHP , ∆UHC , FHP and GHP C are defined
as follows:
ŶHP =
£
ŷ (k + 1|k)
...
ŷ (k + HP |k)
¤T








;c = 






1m ∆umax
1m ∆umin
1m Umax − 1m u(k − 1)
−1m Umin + 1m u(k − 1)
1n ymax − FHP
−1n ymin + FHP








1m is an (HC × m) × m matrix formed by HC m × m
identity matrices; 1n is an (HP × n) × n matrix formed by
HP n × n identity matrices; and T its a lower triangular
block matrix whose nonnull block entries are HC × HC
identity matrices.
The set of control increments is calculated by minimizing
an objective function for a prediction horizon. The objective
is to minimize the error between the predicted output
ŷ(k) and the reference trajectory w(k) along the prediction
horizon:
J=
HP
X
j=1
HC
X
2
2
[w (k + j|k) − ŷ (k + j|k)] +
λ [∆u (k + j|k)]
j=1
(4)
where the second term in the Eq. (4) is the control effort and
λ is the weighting sequence factor. The reference trajectory
w(k), is the desired output in closed loop of the system and
is given by:
w (k + i|k) = s (k + i) − αi [s (k) − y (k)] ; 1 ≤ i ≤ HP
∆UHC = ∆ u (k|k) . . . ∆u (k + HC |k)
(5)




where
s(k)
is
the
set-point
and
α
determines
the
smoothness
CB
CA


..
of the approach
from the output to s(k).
..


£
¤




.
.
 P

w
(k
+ 1|k), . . . , w (k + HP |k) , then, the
If
W
=


H
−1


i
C
H
 CA C 
 Pi=0 CA B 

FHP = 
 u (k − 1)objective function (4) can be expressed in terms of the
HC
 CAHC +1  x (k) + 
CAi B 

predicted output (2):
i=0







..
..






.
.
 mı́n J (u) = 1 uT Hu + fT u + b
PHP −1
i
(6)
2
CAHP
i=0 CA B
 subject to


Ru ≤ c
CB
···
0
³
´
..
..
..


where: H = 2 GTHP C GHP C + λI ,
fT =


.
.
.


T
 PHC −1

..
2(FHP − W)T GHP C and b = (FHP − W) (FHP − W) .


.
CAi B
CB
GHP C =  Pi=0

Therefore the solution takes the form of a standard
HC
i


···
CAB + CB
i=0 CA B


Quadratic
Programming (QP) formulation. The objective


..
..
..


.
.
.
function (6) is convex since it is quadratic with positive
PHP −1
PHP −HC
definite Hessian (H > 0). Also the constraints are linear.
CAi B · · ·
CAi B
i=0
i=0
Given these conditions, it is a well known result that a local
minimum, if it exists, is also a global minimum (Bazaraa
The considered constraints are: limits in the control et al., 1993). In this work the active set method proposed
signal (Umin ,Umax ), limits in the slew rate of the ac- by Fletcher (1987) is used to solve the QP problem.
tuator (∆umin ,∆umax ) and limits in the output signals
(ymin ,ymax ). Then for a process with m inputs, n outputs, II-C. Predictive functional control (PFC)
This control algorithm has two main characteristics that
the constraints along the horizons HP and HC can be
distinguish it from the rest of the predictive controllers: the
expressed as:
use of coincidence points and basis functions.
The concept of coincidence points is used to simplify
Ru ≤ c
(3) the calculations considering only a subset of points on the
£
¤T
TABLE I
S YSTEMS TO BE CONTROLLED .
TABLE II
M ODELS FOR PREDICTION .
Sys
Transfer function
Umin
Umax
∆umin
∆umax
Sys
Transfer function
1
1
s+1
e2
G(s) = s+1
1
G(s) = 2s2 +2s+1
−s+1
G(s) = s2 +1.5s+0.5
1
G(s) = s−1
s−1
G(s) = s2 −s−2
3
-3
1.5
-1.5
1
1
2
1
2
3
-1
-2
-1
-2
-3
0.4
0.5
0.3
0.5
0.5
-0.4
-0.5
-0.3
-0.5
-0.5
2
3
4
5
6
0.99
s+0.9
2
G(s) = 0.99e
s+0.9
0.9
G(s) = 1.8s2 +1.8s+0.9
−s+0.9
G(s) = s2 +1.35s+0.45
0.9
G(s) = s−0.9
s−0.9
G(s) = s2 −1.8s−1.8
2
3
4
5
6
G(s) =
prediction horizon. The desired output and predicted output
must match in these points, not in all the horizon.
The other characteristic is the representation of the manipulated variable as a sum of a set of pre-determined basis
functions:
u (k + i|k) =
nB
X
µj (k) B (i)
(7)
j=1
They are chosen according to the set-point profile and
the expected disturbances (in this case a first order model response). Minimizing the objective function requires
therefore less calculation because only the optimal set of
weighting factors µj has to be found. Thus, the concept of
control horizon, does not exist in this algorithm because it
is replaced by the number of basis functions used.
The function to be minimized is:
J=
HP
X
2
[ŷ (k + nCj |k) − w (k + nCj |k)]
(8)
j=1
where nCj corresponds to the chosen coincidence points
(one coincidence point in this case). Notice that the coincidence points, must be, at least equal to the number of basis
functions used.
III.
T HE SYSTEMS TO BE CONTROLLED
For the simulations six elementary systems to be controlled are chosen in order to test different aspects of the
controllers. These systems with its constraints are shown in
Table I. Each one of these systems has a characteristic type
of response: (1) first order system, (2) first order system
with delay, (3) second order system, (4) non-minimum
phase system, (5) instable system, (6) instable system with
instable zero.
IV.
C ONTROLLER DESIGN
IV-A. PI
The controller design is based in choose a design point in
the Nyquist diagram. This design point is chosen to obtain
the desired closed loop behavior, i.e. rise time, damping
value, settling time. In this case, the desired damping
value of 0.5 for all the systems is chosen. From this
desired damping value, the variables for all the auto-tuning
process are obtained as is shown in (Clarke, 2006; Gyöngy
and Clarke, 2006). Unfortunately this controller can not
G(s) =
be useful for unstable systems since it only works with
stable systems. The final tuning parameter values for the PI
controller in the diverse cases studied are shown in Table III.
For this controller the constraints are not used to compute
the control signal. It will be shown in the numerical
simulations how this controller will act in the system when
the constraints do not exist; this controlled will be labelled
as PI unconst. And also there is going to be shown the same
controller acting in the same system when this constrains
do exist, and for this controller (labelled as PI) in case that
one constraint is violated the variable is saturated to their
maximum or minimum value.
IV-B. MPC and PFC
The predictive control design and tuning procedure is
generally described as follows (DMC Corp., 1994):
1 Variable selection: From the control objectives, define
the size of the problem, and determine relevant control
variables and manipulated variables. For these systems
the ‘real’ models are known and since they are SISO is
straightforward determine the controlled and manipulated
variables.
2 Plant test: Test the plant systematically by varying
manipulated variables and capturing data showing how
the controlled variables respond. This step is mainly to
determine the move sizes of the manipulated variables
(already showed in Table I).
3 Model for prediction: This step is the most important
fact in this type of controllers since the quality of the
model will determine the accuracy of the predictions and the
complexity of the optimization. The considered prediction
models in this work are slightly different from the ‘real’
models of the system, since they are derived only with
the information of the dead time, rise time and overshoot
to build a first or second order system. The models for
prediction are shown in Table II. In order to introduce these
models for prediction to the controller, they are discretized
with a sampling time of Ts and converted to a state-space
representation.
4 Tuning parameters for the controller: Configure the controller and enter initial tuning parameters. For the MPC the
following parameters are required to be tuned: prediction
horizon HP , control horizon HC and control weighting
sequence factor λ. For the PFC are required to be tuned:
prediction horizon HP , coincidence point nC and the rise
I
7.05
0.30
0.50
0.16
—
—
HP
5
6
10
10
10
20
MPC
HC
3
4
3
2
5
5
λ
0.3
1
1
2
1
1
HP
5
6
10
10
10
—
PFC
nC
4
5
4
6
5
—
α
0.7
0.4
0.7
0.6
0.7
—
time α of the first order model used as reference trajectory.
For both cases (MPC and PFC) HP is chosen so that HP is
equal to the settling time τs of the system that is going to be
controlled taken into account the sample time. In case that
the system is unstable, HP should be a big value, typically
10 ≤ HP ≤ 20 depending on the sample time 1 . For the
MPC HC and λ are chosen in certain way to establish
a compromise between the control effort and the time to
reach the set-point, and always with 1 ≤ HC < HP and
0.1 ≤ λ ≤ 10. In the same way, α (0.1 × τs ≤ α ≤ τs ) and
nC (1 ≤ nC < HP ) are chosen for the PFC. This is done
doing a ‘global search’ by varying the parameters within its
permitted values and choosing the best pair that minimizes
τs + Σ∆u2 .
5 Simulation: Test the controller off-line using closed-loop
simulation to verify the controller performance.
V.
reader is reminded that, for unstable systems, if HP
the values of the matrices of the form Ai could be high,
computational ill-conditions and therefore to bad results as
(Rossiter et al., 1998). One alternative proposed is to stabilize
before trying to control it.
is too big
leading to
noticed in
the system
Controlled output (y)
1.5
0.8
0.6
0.4
1
0.5
PI
PI unconst
MPC
PFC
ref
0.2
0
0
2
4
0
6
0
2
Time (sec)
Input increments (∆
∆u)
N UMERICAL SIMULATIONS
The controllers designed in the Section IV will be applied
to systems of the Section III. The simulations were carried
out in Matlabr . The results can be seen in Figs. 2 to 7.
Besides the qualitative results for each case, some quantitative values are shown for each simulation: the settling time
τs , the overshoot Mp and the control effort Σ∆u2 . These
quantitative values are going to be the performance indices
of the controllers.
For the first simulation, in Fig 2 can be appraised that
the PI controller for the unconstrained system works as its
expected according to the parameters for what it was tuned.
However, when this controller is applied to the system with
constraints the performance is very poor since the first
control move because the slew rate needs to be saturated
for the constraint violation. This constraint violation affects
the controller behavior throughout the transient, despite that
this violation is almost negligible.
Also in the same figure is shown the MPC and the
PFC controllers. Due to their tune, the MPC and PFC
controllers have no need to operate near the constraints of
the system and the settling times are almost the same as
the PI controllers with a smoother input. The performance
indices of this simulation are shown in Table IV.
1 The
Step response of G(s)
1
Amplitude
P
1.58
0.56
0.71
0.34
—
—
Amplitude
Ts
0.5 s
0.5 s
0.5 s
1s
0.1 s
0.3 s
3
1
2.5
0.5
0
-0.5
PID
PID unconst
MPC
PFC
-1
-1.5
0
2
4
Time (sec)
6
Inputs (u)
1.5
4
Time (sec)
Amplitude
PI
Sys
1
2
3
4
5
6
For the second simulation, in Fig. 3 can be seen that, as in
the fist case, the PI controller for the unconstrained system
works well, but when its applied to the constrained system
its performance decrease because of the violation of the
constraints. The system response of the constrained system
with the PI controller is almost the same as with the MPC
controller and the PFC controller, being the plots practically
overlayed. The performance indices of this simulation are
shown in Table V and shows for the PI, MPC and PFC very
similar values.
Amplitude
TABLE III
T UNING PARAMETERS .
2
1.5
1
0.5
0
6
PID
PID unconst
MPC
PFC
0
2
4
Time (sec)
Fig. 2. System 1.
TABLE IV
P ERFORMANCE INDICES FOR THE SYSTEM 1.
PI
PI unconst
MPC
PFC
τs
5.1 s
3.9 s
3.8 s
3.7 s
Mp
36 %
22 %
2%
—
Σ∆u2
8.31
3.88
1.31
2.32
6
Step response of G(s)
Controlled output (y)
Step response of G(s)
1
0.4
0.2
0.6
0.4
PI
PI unconst
MPC
PFC
ref
0.2
0
2
4
Time (sec)
6
0
8
0
5
10
Amplitude
0.6
1
1
0.8
Amplitude
Amplitude
Amplitude
1.2
1
0.8
0
Controlled output (y)
1.4
0.8
0.6
0.4
0
0
5
Time (sec)
Input increments (∆
∆u)
10
0
15
PI
PI unconst
MPC
PFC
ref
0
5
Input increments (∆
∆u)
Inputs (u)
0.5
10
15
Time (sec)
Inputs (u)
0.8
1.5
0.6
0.2
0.8
0.6
0.1
0.4
0
0.2
0
5
10
15
0
PID
PID unconst
MPC
PFC
0
Time (sec)
5
10
Time (sec)
0.4
PID
PID unconst
MPC
PFC
0.2
Amplitude
Amplitude
PID
PID unconst
MPC
PFC
0.3
Amplitude
1
0.4
Amplitude
0.4
Time (sec)
1.2
-0.1
0.6
0.2
0.2
15
0.8
0
1
0.5
PID
PID unconst
MPC
PFC
-0.2
15
-0.4
0
5
10
15
0
0
Time (sec)
5
10
Fig. 3. System 2.
Fig. 4. System 3.
TABLE V
P ERFORMANCE INDICES FOR THE SYSTEM 2.
P ERFORMANCE INDICES FOR THE SYSTEM 3.
PI
PI unconst
MPC
PFC
τs
7.4 s
12.5 s
7.6 s
8.0 s
Mp
7.1 %
—
—
—
Σ∆u2
0.30
0.43
0.29
0.33
15
Time (sec)
TABLE VI
PI
PI unconst
MPC
PFC
τs
12.2 s
12 s
8.3 s
6.4 s
Mp
12.6 %
9.5 %
4.0 %
—
Σ∆u2
0.50
0.59
0.49
0.95
In the third simulation, the system to be controlled is a
second order system. The PI controller for the unconstrained
system and the PI controller for the constrained system
performance are almost the same despite the slew rate
constraint violation. And for the MPC and PFC the system
have almost the same performance with the tuning, reaching the set-point within less time than the PI controllers.
The performance indices of this simulation are shown in
Table VI.
For the fourth simulation, as in the previous case, the PI
controller for the unconstrained system and the PI controller
for the constrained system performance are almost the same.
The system regulated by MPC and the PFC controller shows
a less inverse response because the control effort es less
than the PIs and for that the system input is smother.
The performance indices of this simulation are shown in
Table VII.
For the fifth simulation, the auto-tuned PI cannot be tuned
because this controller has the restriction to work with stable
systems. However, the MPC and the PFC can be tuned
without problems using the methodology presented in Section IV. The Fig. 6 shows the system output using this two
controllers. The constraints does not affect the controllers
behavior since they are included in the calculation of the
input signal. The performance indices of this simulation are
shown in Table VIII.
Step response of G(s)
Controlled output (y)
Step response of G(s)
2
Controlled output (y)
2.5
1
1.5
1.2
1
2
0.5
0.6
0.4
PI
PI unconst
MPC
PFC
ref
0.2
0
0
5
10
-0.2
15
0
5
Time (sec)
1
Input increments (∆
∆u)
0
0.2
0.4
0.6
0.8
Time (sec)
1
PID
PID uncost
MPC
PFC
10
15
Time (sec)
20
25
Inputs (u)
2.5
MPC
2
0.2
Amplitude
Amplitude
0.4
0.2
0
5
Input increments (∆
∆u)
0.6
0.1
0
0.6
0.4
0.2
MPC
ref
0
Inputs (u)
PID
PID uncost
MPC
PFC
0.4
-0.2
0
15
0.8
0.3
0.6
0.2
Time (sec)
0.4
Amplitude
10
1.5
0.5
Amplitude
-0.5
0
Amplitude
1
0.8
Amplitude
Amplitude
Amplitude
0.8
0
-0.2
1.5
1
0.5
0
-0.4
MPC
-0.5
-0.1
0
5
10
0
15
0
5
Time (sec)
10
15
Time (sec)
0
5
10
15
Time (sec)
20
25
0
5
10
15
Time (sec)
Fig. 5. System 4.
Fig. 7. System 6.
TABLE VII
P ERFORMANCE INDICES FOR THE SYSTEM 6.
20
25
TABLE IX
P ERFORMANCE INDICES FOR THE SYSTEM 4.
τs
12.0 s
10.4 s
7.8 s
9.7 s
PI
PI unconst
MPC
PFC
PI
PI unconst
MPC
PFC
Σ∆u2
0.17
0.18
0.16
0.18
Mp
13.4 %
8.5 %
1.0 %
—
Step response of G(s)
1
0.8
Amplitude
Amplitude
10
6
4
0
0.6
0.4
MPC
PFC
ref
0.2
2
0
0.5
1
1.5
Time (sec)
2
0
2.5
0
1
Input increments (∆
∆u)
4
5
Inputs (u)
0.6
1.5
MPC
PFC
0.4
MPC
PFC
1
0.2
Amplitude
Amplitude
2
3
Time (sec)
0
-0.2
0.5
0
-0.5
-0.4
-1
0
1
2
3
Time (sec)
4
5
0
1
2
3
Time (sec)
Fig. 6. System 5.
TABLE VIII
P ERFORMANCE INDICES FOR THE SYSTEM 5.
PI
PI unconst
MPC
PFC
τs
∞
∞
2.7 s
3.2 s
Mp
—
—
3.1 %
—
Σ∆u2
∞
∞
0.89
1.14
4
Mp
xx %
xx %
16.5 %
—
Σ∆u2
∞
∞
1.42
∞
For the sixth simulation, the system to be controlled is
also a unstable system so as in the previous case, the autotuned PI does not work. But since this is a more difficult
case, the PFC cannot be tuned either. On the other hand
the MPC controller can reach the set-point successfully.
The performance indices of this simulation are shown in
Table IX.
Controlled output (y)
12
8
τs
∞
∞
13.2 s
∞
5
VI.
C ONCLUSIONS
In this work three controllers applied to different elementary systems are presented: an auto-tuned PI controller, an
MPC controller and a PFC controller.
As it can be seen in all the simulations, the systems with
the PI controller performs really well, and the auto-tuning
makes the controller easy to implement even in the practise.
Although the operation of the controller near the constraints
might lower the performance.
For both, the MPC controller and PFC controllers the
most difficult part is the formulation of the model for
prediction as in any model based controller. In the case
of the PFC controller the best characteristic is how the
controllers deals with the optimization problem to obtain
a faster calculation.
However, the best performance indices are from the MPC,
in fact this is the only controller than could deal with all the
systems presented. But in spite of that, they are not as used
in practice as one would think. Existing MPC products are
not suitable for all applications or they do not capture all the
benefits possible when they are applied. And a high level
of expertise is required to implement and maintain a MPC
controller is an impediment to more widespread utilization.
So, provide a higher performance and make it easier for
non-experts to actually configure, tune and maintain realworld MPC applications is needed in order to bring the
MPC to the practice.
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