8 PID Control
The relationship between the control variable and the system output is
U
(s)=
1
G(s)e
−Ts
Y(s), (27)
and since G
(s)=
ˆ
G
(s), Eq.(26) becomes
Y
f
(s)=
ˆ
G
(s)
1
G(s)e
−Ts
Y(s)=e
Ts
Y(s). (28)
This shows that the internal loop containing the plant model feeds back a signal that is a
prediction of the output, since e
Ts
represents a prediction y(t + T) in the time domain. The
closed loop transfer function of the system can be determined by using

(s)
. (32)
As can be seen, the controller can now be designed without considering the effect of the
time delay. (Hägglund, 1992; 1996) combined the properties of the Smith predictor with a
PI controller to control a first order plant with a time delay. The transfer function of the plant
is given by
G
p
(s)=
Ke
−Ts
τs + 1
, (33)
where K
> 0 is the plant gain, τ the time constant and T the time-delay of the plant. The PI
controller is given by
G
c
(s)=K
p

1
+
1
τ
i
s

, (34)
where the K

, (35)
10
Advances in PID Control
Predictive PID Control of Non-Minimum
Phase Systems 9
Fig. 5. PI with Smith predictor control structure
where G
d
(s) represents the time-delay dynamics. Let the model of the plant be given by
G
m
(s)=
ˆ
G
(s)
ˆ
G
d
(s)=
2
2s + 1
(−2s + 2)
(2s + 2)
, (36)
where
ˆ
G
d
(s) represents the Padé approximation of the time-delay. The PI control constants
are set to K

c
(s)
1 +
ˆ
G
d
(s)G
c
(s)
=
C(s)
ˆ
G
(s)
ˆ
G
d
1 + C(s)
ˆ
G
(s)
ˆ
G
d
, (39)
C
(s)=
G
c
(s)

+ 20s
3
+ 17.8s
2
+ 4.4s
. (41)
11
Predictive PID Control of Non-Minimum Phase Systems
10 PID Control
Applying model reduction techniques C(s) reduces to a PID control structure which is a
second order transfer function
C
(s)=
1.002s
2
+ 2.601s + 1.098
s(s + 4.025)
, (42)
where K
d
= 1.002, K
p
= 2.601, K
i
= 1.098 and F(s)=1/(s + 4.025). Fig. 7 shows the
time response of the system output along with the control variable. It can be seen that the
control signal acts immediately and not after the occurrence of the time-delay, demonstrating
the predictive properties of the PID controller. Fig. 8 shows the time response of the
0 5 10 15 20 25 30 35 40 45 50
−0.2


Reference
System output with T = 2 s
System output with T = 3 s
System output with T = 4 s
System output with T = 5 s
Fig. 8. Time responses of control system based on Smith predictor for different time-delays
5.1.2 Internal model control
The internal model control (IMC) design method starts with the assumption that a model
of the system is available that allows the prediction of the system output response due to a
output of the controller. In this discussion it is also assumed that the model is a "perfect"
representation of the plant. The basic structure of IMC is given in Fig. 9 (Brosilow & Joseph,
2002; Garcia & Morari, 1982). The transfer functions of the plant, the IMC controller and plant
model is given by G
p
(s, ε), G
IMC
(s) and G
m
(s) respectively. In the case when the model is not
12
Advances in PID Control
Predictive PID Control of Non-Minimum
Phase Systems 11
a perfect representation of the actual plant the tuning parameter ε is used to compensate for
modelling errors.
Fig. 9. Internal model control structure
The structure of Fig. 9 can be rearranged into a classical PID structure as shown in Fig. 10.
This allows the PID controller to have predictive properties derived from the IMC design.
Fig. 10. Classical feedback representation of the IMC structure

(s) would drive the output Y(s) of the system to track the reference
input Y
(s) instantaneously, that is
Y
(s)=R(s), (45)
and this requires that
G
IMC
(s, ε)G
p
(s)=1, (46)
G
m
(s)=G
p
(s). (47)
To have a "perfect" controller, a "perfect" model is needed. Unfortunately it is not possible to
model the dynamics of the plant perfectly. However, depending on the controller design
method, the controller can come close to show the inverse response of the plant model.
Usually the design method incorporates a tuning parameter to accommodate modelling
errors.
13
Predictive PID Control of Non-Minimum Phase Systems
12 PID Control
The plant considered is a non-minimum phase system of the following form
G
p
(s)=
N(s)
D(s)

where the zeros of N
+
(−s) are all in the left half plane and are the mirror images of the zeros of
N
+
(s). The filter constant ε is a tuning parameter that can be used to avoid noise amplification
and to accommodate modelling errors; and r is the relative order of N
(s)/D(s) (Brosilow &
Joseph, 2002).
Example
Consider the following non-minimum phase system
G
p
(s)=
2(−2s + 2)
(2s + 1)(2s + 2)
. (50)
The IMC controller can be derived by using Eq.(49), but in order to ensure zero offset for step
inputs G
p
(s) is adapted as follows
G
p
(s)=
2(−2s + 2)
2(2s + 1)(2s + 2)
. (51)
Then
G
IMC

2
+ 3s
=
s
2
+ 1.5s + 0.5
s(s + 3)
. (54)
The form of C
(s) corresponds to the form of a PID controller (Dorf & Bishop, 2011):
C
PID
(s)=
K
d
(s
2
+ as + b)
s
(55)
where a
= K
p
/K
d
and b = K
i
/K
d
. The IMC-based controller, Eq.(54), is therefore a PID

predictive control (MPC). It is an optimal control structure utilising a receding horizon
principle. This method have found wide-spread application in process industries and research
in the field is very active (Wang, 2009). In MPC the control law is computed via optimisation
of a quadratic cost function and a plant model is used to predict the future output response to
possible future control trajectories. These predictions are computed for a finite time horizons,
but only the first value of the optimal control trajectory is used at each sample instant.
Following a model predictive approach for the design of PID controllers is a challenging
task. Two routes can be followed namely a restricted model approach or a control signal matching
approach (Johnson & Moradi, 2005; Tan et al., 2000; 2002). In this section the restricted model
approach will be considered. This approach formulates the control problem in terms the
generalised predictive control (GPC) algorithm. The model used by the controller is restricted
to second order such that the predictive control law that emerges has a PID structure. The
following control algorithm is discussed in discrete-time since it offers a more natural setting
for the derivation of predictive control techniques. It also simplifies the description of the
design process and has a strong relevance to industrial applications when presented in
discrete-time (Wang, 2009).
5.2.1 The GPC-based algorithm
Augmented state space model
The main idea is to derive an MPC control law equivalent to the second order control law
of a PID controller. This can be done by developing an MPC control law, but considering
a second-order general plant (Tan et al., 2000; 2002). Consider a single-input, single-output
model of a plant described by:
X
m
(k + 1)=A
m
X
m
(k)+B
m

(k −1)) + B(u(k) − u (k − 1)). (58)
The difference of the state variables and output is given by
ΔX
m
(k + 1)=X
m
(k + 1) − X
m
(k), (59)
ΔX
m
(k)=X
m
(k) − X
m
(k −1) , (60)
Δu
(k)=u(k) −u(k − 1). (61)
The integrating effect is obtained by connecting ΔX
m
(k) to the output y(k). To do so the new
augmented state vector is chosen to be
X
(k)=

ΔX
m
(k)
T
y(k)

m
A
m
ΔX
m
(k)+C
m
B
m
Δu(k). (65)
Eqs. (63) and (64) can be written in state space form where

ΔX
m
(k + 1)
y(k + 1)

=

A
m
O
T
m
C
m
A
m
1


where O
m
=

00
···0

is a 1
× n vector, and n = 2 in the predictive PID case. This
augmented model will be used in the GPC-based predictive PID control design.
Prediction
The next step in the predictive PID control design is to predict the second order plant output
with the future control variable as the adjustable parameter. This prediction is done within
one optimisation window. Let k
> 0 be the sampling instant. Then the future control trajectory
is denoted by
Δu
(k), Δu(k + 1), ···, Δu(k + N
c
−1), (68)
where N
c
is called the control horizon. The future state variables are denoted by
X
(k + 1|k), X(k + 2|k), ···, X(k + m|k), ···, X(k + N
p
|k), (69)
where N
p
is the length of the optimisation window and X(k + m|k) is the predicted state

Δu(k)+B
m
Δu(k + 1) ,
.
.
.
X
(k + N
p
|k)=A
N
p
m
X(k)+A
N
p
−1
m
B
m
Δu(k)+A
N
p
−2
m
B
m
Δu(k + 1)
+ ··· +
A

m
B
m
Δu(k)+C
m
B
m
Δu(k + 1) ,
y
(k + 3|k)=C
m
A
3
m
X(k)+C
m
A
2
m
B
m
Δu(k)+C
m
A
m
B
m
Δu(k + 1)
+
C

−2
m
B
m
Δu(k + 1)
+ ···+
C
m
A
N
p
−N
c
m
B
m
Δu(k + N
c
−1).
The equations above can now be ordered in matrix form as
Y
= FX(k)+ΦΔU, (70)
where
Y
=

y
(k + 1|k) y(k + 2|k) y(k + 3|k) y(k + N
p
|k)

2
m
C
m
A
3
m
.
.
.
C
m
A
N
p
m
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, (73)
Φ
=
⎡
⎢
⎢

A
m
B
m
C
m
B
m
0
.
.
.
C
m
A
N
p
−1
m
B
m
C
m
A
N
p
−2
m
B
m

16 PID Control
Optimisation and control design
Let r(k) be the set-point signal at sample time k. The idea behind the predictive PID control
methodology is to drive the predicted output signal as close as possible to the set-point signal.
It is assumed that the set-point signal remains constant during the optimisation window, N
p
.
Consider the following quadratic cost function which is very similar to the one obtained by
(Tan et al., 2002)
J
=(r −y)
T
(r −y)+ΔU
T
RΔU, (75)
where the set-point information is given by
r
T
=

11 1

×r(k), (76)
and the dimension of r is N
p
×1. The cost function, Eq.(75) comprises two parts, the first part
focus on minimising the errors between the reference and the output; the second part focus
on minimising the control effort.
R is a diagonal weight matrix given by
R = r

Φ + R)ΔU = 0. (79)
Therefore, the optimal control law is given as
ΔU
=(Φ
T
Φ + R)
−1
Φ
T
(r −FX(k)) (80)
or
ΔU
=(Φ
T
Φ + R)
−1
Φ
T
e(k) (81)
where e
(k) represents the errors at sample k.
Emerging predictive control with PID structure
The discrete configuration of a PID controller has the following form (Huang et al., 2002;
Phillips & Nagle, 1995):
u
(k)=K
p
e(k)+K
i
k

d
are the proportional, integral and derivative gains, respectively, and
q
0
= K
p
+ K
i
+ K
d
, (84)
q
1
= −K
p
−2K
d
, (85)
q
2
= K
d
. (86)
By taking the difference on both sides of Eq.(82), the velocity form of the PID control law is
obtained:
Δu(k)=K
p
[e(k) − e(k −1)] + K
i
e(k)+K

y
(k −2) y(k −1) y(k)

T
(90)
e
(k)=

e
(k −2) e(k −1) e(k)

T
(91)
r
(k)=

r
(k −2) r(k −1) r(k)

T
. (92)
By equating Eq.(81) to Eq.(88 )the following is obtained
ΔU
(k)=(Φ
T
Φ + R)
−1
Φ
T
e(k)=K

−1
Φ
T
. (95)
Example
Consider the following discrete-time state space model of a non-minimum phase system
˙
X
(k)=

−0.0217 −0.3141
0.3141 0.7636

X
(k)+

0.3141
0.2364

u
(k), (96)
y
(k)=

−12

X
(k). (97)
The first step is to create the augmented model for the MPC design, and choose the values of
the prediction and control horizon. In this example the control horizon is selected to be N

.
.
.
2.1515 4.9290 1.0000
2.1516 4.9292 1.0000
2.1517 4.9294 1.0000
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, Φ
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

law (Eq. (81)) is given by
ΔU
=
⎡
⎣
0.0628 0.2602 0.2108
···−0.0144 −0.0144 −0.0145
−0.0554 −0.1681 0.0617 ··· 0.0035 0.0035 0.0035
−0.0085 −0.0976 −0.2766 ··· 0.0452 0.0453 0.0453
⎤
⎦
e
(k), (99)
where the matrix multiplied with the error vector has 3 rows and 20 columns.
Fig. 12 shows the closed loop response of the system output along with the control variable.
It can be seen that the control variable acts immediately and not after the occurrence of the
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Sampling instant
Closed loop responseSystem output

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22
Advances in PID Control
0
Adaptive PID Control System Design Based on
ASPR Property of Systems
Ikuro Mizumoto
1
and Zenta Iwai
2
1
Department of Mechanical Systems Engineering, Kumamoto University
2
Kumamoto Prefectural College of Technology
Japan
1. Introduction
PID control is one of the most common control schemes applied to many industrial processes
and mechanical systems. Because, the PID can be tuned according to the experience of
operators and can applied to uncertain system without a certain system’s model. However
in cases where there are some changes of system properties, it has been pointed out the
difficulties of maintaining the desired control performance and stability during operation,

Consider a SISO continuous-time system with a relative degree of γ.
˙x
(t)=Ax(t)+bu
f
(t)+C
d1
w
d
(t)
y(t)=c
T
x(t)+d
T
1
w
d
(t)
(1)
where x
(t) ∈ R
n
is a state vector, u(t) and y (t) ∈ R are the input and the output of the system,
respectively. w
d
(t) ∈ R
m×1
is a disturbance. The system (1) is not required to be stable and/or
minimum-phase.
Suppose that the disturbance w
d

(3)
We assume that the exosystem is stable or neutrally stable. That is, all its eigenvalues are
located on the left half-plane and/or the imaginary axis.
The objective is to design an adaptive PID controller so as to have the output y
(t) track the
reference signal r
(t).
Remark 1: The exosystem is divided into two parts for the disturbance model and the
reference signal. The part of reference signal is available so that r
(t) is known, but the part
of disturbance is just a model of the disturbance and practical signal of the disturbance is not
available, only the characteristic polynomial is known.
2.1 Transformed system
For the system (1) with a relative degree of γ, there exists a nonsingular variable
transformation:

z
(t)
η(t)

= Φx(t) (4)
such that the system (1) can be transformed into the form (Isidori, 1995):
˙z(t)=A
z
z(t)+b
z
u
f
(t)+C
z

1
w
d
(t) (7)
where
A
z
=

0 I
γ−1×γ−1
−a
0
···−a
γ−1

b
z
=

0
···0 b
z

T
C
z
=

0

Internal Model
Plant
N
IM
(s)
r(t)
+
ᯝ
G
P
(s)
)(
1
sD
IM
)(tu
)(tu
f
e(t) u(t) y(t)
Internal Model
Plant
N
IM
(s)
r(t)
+
ᯝ
G
P
(s)

where
D
IM
(s)=det(sI − A
m
) (10)
and N
IM
(s) is any stable polynomial of order m of the form:
N
IM
(s)=s
m
+ β
m−1
s
m−1
+ ···+ β
1
s + β
0
(11)
Defining the output following error by e
(t)=y(t) − r(t), consider the error system from u(t)
to e(t) as shown Fig. 1.
Define new variables X
1
(t) ∈ R
γ×1
, X

It follows from (2), (3), (5), (6) that
˙
X
1
(t)=A
z
X
1
(t)+b
z
¯
u
(t)+C
z
X
2
(t) (14)
˙
X
2
(t)=Q
η
X
2
(t)+C
η
X
1
(t) (15)
where

Next defining
E
(t)=

e
(t)
˙
e
(t) ···e
(m−1)
(t)

T
(19)
25
Adaptive PID Control System Design Based on ASPR Property of Systems
4 Will-be-set-by-IN-TECH
we have from (2), (3) and (7) that
˙
E
(t)=A
E
E(t)+C
E
X
1
(t) (20)
where
A
E

X
1
(t)
˙
X
1
(t)=A
z
X
1
(t)+b
z
¯
u
(t)+C
z
X
2
(t)
˙
X
2
(t)=Q
η
X
2
(t)+C
η
X
1

ze
(t)=Q
ze
η
ze
(t)+c
ηe
z
e1
(t)
e(t )=z
e1
(t) (23)
with
z
ze
(t)=

z
e1
(t)
.
.
.z
eγ+m
(t)

η
ze
(t)=

0
.
.
.
0
b
ze
⎤
⎥
⎥
⎥
⎦
C
ze
=

0
c
T
ze

c
ηe
=

0
1

(24)
and θ

(m−1)
(t)+···+ β
1
˙
u
(t)+β
0
u(t)
=
¯
u
(t) (25)
26
Advances in PID Control
Adaptive PID Control System Design Based on ASPR Property of Systems 5
Defining
¯z
IM
(t)=

u
(t) ,
˙
u(t) , ··· , u
(m−1)
(t)

T
(26)
we have the following system representation from

0
···−β
m−1

, b
IM
=

0
1

,¯c
T
IM
=

10
···0

(28)
Consider the following variable transformation using the state variable ¯z
IM
(t) in (27).
ξ
k
(t)=−b
ze
u
(k−γ−1)
(t)+e

j−1
C
ξγ+j−1
(i = γ + m)
(30)
Δ
i
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
β
m−1
(i = 1)
β
m−i
+
∑
m−1
j
=1
β
m−i+j
C
ξ j
(2 ≤ i ≤ m)

ξ
η
ze
(t)
˙η
ze
(t)=Q
ze
η
ze
(t)+c
ηe
e(t )
e(t )=

10

e
(t)
(32)
where
e
(t)=

e
(t),
˙
e(t ), ··· , e
(γ−1)
(t)


b
e
=

0
···0b
ze

T
, b
ze
= c
T
1
A
γ+m−1
1
B
1
∈ R (34)
27
Adaptive PID Control System Design Based on ASPR Property of Systems
6 Will-be-set-by-IN-TECH
A
IM
=

0 I
m−1×m−1

⎥
⎦
, C
ξ
=

0
c
T
ze

(35)
Note that this obtained error system with u
(t) as an input has relative degree of γ.
3. Adaptive PID control system design
Here we show an adaptive PID control system design scheme for the error system (32) based
on system’s ASPR properties.
3.1 Almost Strictly Positive Realness (ASPR-ness)
Let’s consider the following nth order SISO system:
˙x
(t)=Ax(t)+bu(t)
y(t)=c
T
x(t)
(36)
where, x
(t) ∈ R
n
is a state vector and u(t), y(t) ∈ R are the input and the output, respectively.
The ASPR-ness (almost strictly positive real-ness) of the system (36) is defined as follows:

T
(39)
is strictly positive real (SPR).
The sufficient conditions for a system to be ASPR are given as follows (Kaufman et al., 1997):
(1) The relative degree of the system is 0 or 1.
(2) The system is minimum-phase.
(3) The high frequency gain of the system is positive.
Remark 2: The system (38) with the transfer function G
c
(s)=c
T
(sI − A
cl
)
−1
b is positive real
if, for Re
(s) ≥ 0, ReG
c
(s) ≥ 0, and it is SPR if , for some ε > 0, G
c
(s − ε) is PR. Furthermore,
if the system (38) is SPR, then there exist symmetric positive definite matrices P and Q such
that the following Kalman-Yakubovich-Popov Lemma is satisfied.
A
T
cl
P + PA
cl
= −Q