Multi-loop Decentralized PID Control Based on

Covariance Control Criteria: an LMI Approach

Xin Huang
1
, Biao Huang
1
1
Department of Chemical & Materials Engineering
University of Alberta
Edmonton, Alberta, Canada T6G2G6
Abstract
PID control is well known and widely applied in industry and many design algorithms are
readily available in the literature. However, systematic design of multi-loop or decentralized
PID control for multivariable processes to meet certain objectives simultaneously is still a
challenging task. Designing multi-loop PID controllers such that the process variables satisfy
the generalized covariance constraints is studied in this paper. A convergent computational
algorithm is proposed to calculate the multi-loop PID controller for a process with stable
disturbances. This algorithm is then extended to a process with random-walk disturbances.
The feasibility of the proposed algorithm is veried by applying it to several simulation
examples.
Keywords: PID; Covariance control; LMI; Semidenite programming;

Corresponding author. Email Biao.Huang@ualberta.ca, tel:(780)492-9016, Fax:(780)492-2881

1
1 Introduction
The proportional-integral-derivative (PID) controller is extensively used in industry and is
well documented in the literature since the classic Ziegler-Nichols method [27] was presented.
This is because PID controller is simple, robust and well understood. To accommodate the
high performance requirement of the modern industry, optimization of the PID parameters is
also extensively studied, and many dierent tuning criteria and procedures have been proposed,
for example, decay ratio method [5], gain and phase margin method [2] and the internal model
control (IMC) based PID tuning method [17, 21]. Recently, with the popularity of the interior
point algorithm several PID design methods based on Linear Matrix Inequality (LMI ) were
proposed for the continuous-time systems [3, 7, 9]. However, none of the above mentioned
PID design methods are presented to directly achieve variance specication on all outputs for
multivariable systems.
The signicance of reducing the process variation is well appreciated in the manufacture
industry [23]; however, there are only a few papers on optimizing PID parameters in order to
reduce the process variances directly. A stochastic predictive PID controller was presented in
[13, 18] by equating a discrete PID control law with the linear form of the Generalized Predictive
Control (GPC) with steady state weighting; several self-tuning PID controllers were proposed
in [16, 22, 26] for discrete-time Linear Time-invariant (LTI ) systems by approximating the PID
controller to the generalized minimum variance control (GMVC). The philosophy behind these
methods [13, 18, 16, 22, 26] is to design the PID parameters by approximation of a PID controller
to a controller designed by other advanced methods. In such a way, it is expected that the PID
controller has similar property as the other advanced controllers. However, there is no theoretical
guarantee about how close such approximation can be; furthermore, these approximation
methods so far are available only for the single-input-single-output (SISO) systems and the
extension of the PID design from the SISO system to the multi-input-multi-output (MIMO)
system is nontrivial.
In this paper a state space approach to designing multi-loop PID controllers is proposed such
that closed-loop satises the generalized covariance constraints. One of the main advantages of
the proposed method is that the controller parameters are calculated directly according to the
covariance constraints on process variables instead of approximating other controllers. A conver-
gent computational algorithm, in which a sequence of semi-denite programming problems are
solved using the LMItool [8], is proposed to calculate the multi-loop PID controller parameters.
The proposed algorithm initially intends for the process with stable disturbances. The algorithm
is then extended to the process with unstable disturbance (random walk disturbance). The pro-
posed multi-loop PID controller design method is for the purpose of controlling the variation
of process variables. However, the proposed approach can also be applied to design multi-loop
PID controller for other performance indices, such as H
2
or H

.
The rest of the paper is organized as follows: the generalized covariance constraints problem
and the preliminary results are stated in Section 2. The state space realization of the multi-loop
PID controller is given in Section 3. An algorithm is presented to calculate the multi-loop PID
2
controller parameters in Section 4, where the disturbance model is assumed to be stable. The
multi-loop PID controller design for the random walk disturbance is addressed in Section 5.
Numerical examples are presented in Section 6 followed by concluding remarks in Section 7.
The notation of this paper is quite standard: R denotes the set of real number; R
n
denotes
the n-dimensional real vector space; I
n
means the unit element in linear space R
nn
; A
T
stands
for the transpose of matrix A; A
1
means inverse of matrix A; X > Y (X Y ) means that XY
is positive denite (semi-denite); E is the expectation operator; var (y) denotes variance of y;
(k) is the discrete-time Dirac function: (k) =
_
1 k = 0
0 k = 0
; z
1
is the back shift operator:
z
1
y
k
= y
k1
; = 1 z
1
is the dierence operator: y
k
= y
k
y
k1
.
2 Problem statement
Consider a nite-dimensional discrete-time LTI system P, given as follows:
x
k+1
= Ax
k
+ Bu
k
+ G
k
y
k
= Cx
k
+ F
k
z
i
k
= C
i
x
k
+ D
i
u
k
, i = 1...l
(1)
where x
k
R
n
is the state vector, u
k
R
m
is the input, y
k
R
m
is the measurement,
k
R
g
is the external disturbance and measurement noise, and z
i
k
R
p
i
is the i-th controlled vector.
A, B, G, C, F, C
i
and D
i
are matrices with appropriate dimensions. The disturbance
k
is
unmeasurable, but it is assumed that some of its statistical properties are known:
E (
k
) = 0
E
_

T
j
_
= (i j)
(2)
It is assumed that the state space representation is a minimal realization.
To achieve better product quality, controlling the variation of the process variables is well
accepted in industry [23], and dierent control strategies have been presented, such as minimum
variance control [1], the generalized minimum variance control (GMVC) [26], linear quadratic
Gaussian (LQG) control [12, 14, 15] and the covariance control [24, 25]. For multivariable
systems controlling the plants covariance is one of the main objectives [24, 25]. The covariance
of x
k
is dened as:
= lim
k
E
_
x
k
x
T
k
_
(3)
However, as it is pointed out in [10], there is often no physical interpretations of the covariance
of the states. Therefore, it is more desirable to control covariance of process output rather
than that of states. The generalized covariance constrained control (GCC) problem is stated as
follows:
Problem 2.1 For the continuous-time LTI system (1), nd a controller such that the closed-
loop system is internally stable and the covariance of the controlled variable z
i
k
(i = 1...l) satises

i
= lim
k
E
_
z
i
k
z
i
k
T
_
<

i
(4)
3
where

i
(i = 1...l) is some pre-specied positive denite matrix.
If there exists a multi-loop PID controller such that the closed-loop system is internally stable
and the generalized covariance constraints (4) are satised, then the GCC problem is feasible
via a multi-loop PID controller. It has been shown [10] that the feasibility of GCC is equiv-
alent to feasibility of some linear matrix inequalities (LMIs) if a full order dynamic controller
is considered. Unfortunately, this conclusion does not hold for the xed-order controller and
decentralized controller. The interior point algorithm can be used to solve LMI problems quite
eciently [4, 8], but it can not be used here to solve the multi-loop PID design with the general-
ized covariance constraints. However, an iterative algorithm, in which a sequence of optimization
problems are solved, is proposed in section 5 to calculate the multi-loop PID controller param-
eters. To state the computational algorithm, the well known Schur complement Lemma will be
used:
Lemma 2.1 (Schur complement lemma) The following statements are equivalent:
1)
_
A B
B
T
C
_
> 0 (5)
2) C > 0 and ABC
1
B
T
> 0.
3 Multi-loop PID controller
Multi-loop PID controllers have certain advantages over the complex multivariable control
systems. Multi-loop PID controllers are easier to implement on DCS and requires less train-
ing compared to the multivariable controller. However, optimization of the PID parameters
to reduce the eect of disturbance is not a trivial task due to the non-convex nature of the
optimization problem. The presented approach is based on the state space representation and
a state realization of the multi-loop PID controller is given in this section.
A discrete-time single-loop PID controller can be described as:
u
k
= k
1
e
k
+ k
2
k

i=0
e
i
+ k
3
(e
k
e
k1
) (6)
where u
k
is the manipulated variable and e
k
= r
k
y
k
is the error between setpoint r
k
and the
measurement y
k
. The velocity form of the discrete-time PID controller can be obtained from
equation (6):
u
k
= (k
1
+ k
2
+ k
3
) e
k
+ (k
1
2k
3
) e
k1
+ k
3
e
k2
(7)
The transfer function of the discrete-time PID controller (6) can be obtained easily from the
velocity form:
C
_
z
1
_
=
(k
1
+k
2
+k
3
)+(k
1
2k
3
)z
1
+k
3
z
2
1z
1
(8)
4
The controllable state space realization of the PID controller (6) is obtained from the transfer
function (8):
x
s
k+1
=
_
0 1
0 1
_
x
s
k
+
_
0
1
_
e
k
u
s
k
=
_

k
3

k
2
_
x
s
k
+

k
1
e
k
(9)
where

k
1
= k
1
+k
2
+k
3
,

k
2
= k
2
k
3
,

k
3
= k
3
and x
s
k
represents the state vector of a single-loop
PID. For the multivariable systems, the multi-loop PID controller C
m
_
z
1
_
, consisted of a group
of individual PID controllers, is given as:
C
_
z
1
_
= diag
_
c
1
_
z
1
_
, c
2
_
z
1
_
, , c
m
_
z
1
__
(10)
where c
i
_
z
1
_
is the ith single-loop PID controller with the same form as (8). With this multi-
loop PID controller, the diagram of the closed-loop system is shown as:
P
C1(z )
-1
Cm(z )
-1
y1
ym
r1
rm
u1
um
Multi-loop PID
Figure 1: Closed-loop diagram
The state space representation of the multi-loop PID controller can be obtained by stacking
the state space realization of each individual PID controller (where superscript m represents
multi-loop):
x
m
k+1
= A
c
x
m
k
+ B
c
e
m
k
u
m
k
= C
c
x
m
k
+ D
c
e
m
k
(11)
where matrices A
c
, B
c
, C
c
and D
c
are dened as:
A
c
= diag
__
0 1
0 1
_
,
_
0 1
0 1
_
, ,
_
0 1
0 1
__
R
2m2m
B
c
= diag
__
0
1
_
,
_
0
1
_
, ,
_
0
1
__
R
2mm
C
c
= diag
__

k
1
3
,

k
1
2
_
,
_

k
2
3

k
2
2
_
, ,
_

k
m
3

k
m
2
_ _
R
m2m
D
c
= diag
_

k
1
1
,

k
2
1

k
m
1
_
R
mm
(12)
With the multi-loop PID controller (11) the closed-loop system can be written as (assuming
r = 0):
X
k+1
= (A
0
+ B
0
KC
0
) X
k
+ (G
0
+ B
0
KF
0
)
k
z
i
k
=
_

C
i
+

D
i
KC
0
_
X
k
+

D
i
KF
0

k
, i = 1...l
(13)
5
where X
k
=
_
x
k
x
m
k
_
and the matrices A
0
, B
0
, C
0
, G
0
, F
0
, K,

C
i
and

D
i
are composed as follows:
A
0
=
_
A 0
B
c
C A
c
_
, B
0
=
_
B
0
_
, C
0
=
_
C 0
0 I
_
, G
0
=
_
G
B
c
F
_
F
0
=
_
F
0
_
,

C
i
=
_
C
i
0
_
,

D
i
= D
i
, K =
_
D
c
C
c
_
(14)
To make the closed-loop system (13) satisfy the generalized covariance constraints we use the
following lemma from [10]:
Lemma 3.1 The closed-loop system (13) is stable and satises constraints (4) if and only if
there exists a matrix > 0 such that
(A
0
+ B
0
KC
0
) (A
0
+ B
0
KC
0
)
T
+ (G
0
+ B
0
KF
0
) (G
0
+ B
0
KF
0
)
T
< 0 (15)
_

C
i
+

D
i
KC
0
_

C
i
+

D
i
KC
0
_
T
+

D
i
KF
0
F
T
0
K
T

D
T
i
<

i
(16)
where i = 1, ..., l.
4 Computational algorithm
Inequalities (15) and (16) are dicult to solve and the diculty lies in two facts: rst,
both Inequality (15) and Inequality (16) contain cubic terms; second, the decentralized control
structure of the multi-loop PID controller makes the unknown matrix K a sparse matrix. To
solve the nonlinear matrix inequalities (15) and (16), one need change them to some equivalent
forms that can be solved. By applying the Schur complement Lemma, it can be obtained:
Proposition 4.1 The discrete-time system (1) is stabilized by a multi-loop PID controller, de-
ned in (11), and the constraints (4) are satised if and only if there exist matrices X > 0,
Y > 0 and K (the decision variable K is composed as that in (14) ) such that
_

_
X A
0
+ B
0
KC
0
G
0
+ B
0
KF
0
(A
0
+ B
0
KC
0
)
T
Y 0
(G
0
+ B
0
KF
0
)
T
0
1
_

_
< 0 (17)
_

i
_

C
i
+

D
i
KC
0
_

D
i
KF
0
_

C
i
+

D
i
KC
0
_
T
Y 0
F
T
0
K
T

D
T
i
0
1
_

_
> 0 (18)
XY = I
n+2m
(19)
where i = 1, ..., l.
The proof is straightforward and it is omitted here. Obviously the condition in the proposi-
tion (4.1) is not convex because X > 0 and Y > 0 are inverse to each other. To nd a feasible
6
solution to (17) - (19) the idea of the cone complementary linearization method [6] is adopted.
The algebraic equation (19) is relaxed with the following LMI:
_
X I
n+2m
I
n+2m
Y
_
0 (20)
and the linearized version of trace (XY ) is minimized at each step.
The algorithm to calculate the multi-loop PID controller is summarized as follows:
Algorithm 4.1 1. Set k=0. Initialize X
k
> 0 R
n+2m
, Y
k
> 0 R
n+2m
2. Find X
k+1
> 0 R
n+2m
, Y
k+1
> 0 R
n+2m
that solve the following semi-denite program-
ming problem:
minimize
X,Y,K
trace (X
k
Y + Y
k
X) subject to (17) , (18) , (20) (21)
Set t
k
= trace (X
k
Y
k+1
+ Y
k
X
k+1
).
3. Set k = k + 1. If the decrease of t
k
in last L steps is less than a small constant number

1
> 0, then the algorithm stops. If trace (X
k
Y
k
) n2m <
2
then go to step 4; otherwise, go
to step 2.
4. Find > 0 by solving LMI (15) and LMI (16) (where K is obtained from step 3). If there
is a solution, then one feasible solution is found; otherwise, go to step 2.
Remark 4.1 The above computational algorithm is an extension of the one so called cone com-
plementarity linearization algorithm presented in [6] by introducing the sparse matrix K into
inequalities (17) and (18) as decision variable. Similar to the proof of Theorem (2.1) in [6] it
can be shown that t
k
decreases with each step so that the algorithm converges.
Claim 4.1 The Algorithm (4.1) is convergent.
5 Process with random walk disturbance
In process industry random walk disturbance is often used to represent slow dynamic of
disturbances. The algorithm presented in the last section can not deal with random walk
disturbance. This is because the closed-loop system, composed by the multi-loop PID controller
and the process, is not a minimal realization. To generate a minimal realization of the closed-
loop system, one needs to change the process diagram, and the procedure is illustrated by using
a univariate feedback control example. A block diagram of a single-loop feedback system is
shown in Figure (2):
7
PID
u
y
r
e

( ) z G
( )
1 - z
z H
Figure 2: Univariate feedback block diagram
where G(z) is the process model and
H(z)
z1
is the disturbance model. Since the PID controller
block and this disturbance block both contain a pole at 1, this unstable pole is moved out of
these two blocks. The recongured block diagram is shown in Figure (3).
C(z)
y
r
e
1

( ) z G
1
( ) z H
1
1
- z
Figure 3: Univariate feedback block diagram after reconguration
where G
1
(z) = G(z) z and C (z) =
(k
1
+k
2
+k
3
)z
2
+(k
1
2k
2
)z+k
3
z
2
. The PID controller borrows a
pole (located at the origin) from the process in order to preserve its properness. The zero order
hold ensures that the process model G(z) is strictly proper, in other words, G(z) can always
have an extra pole that is lent to the controller block. After this block diagram reconguration,
a state space representation of the controller C (z) is:
x
k+1
=
_
0 1
0 0
_
x
k
+
_
0
1
_
e
k
u
k
=
_
k
3
k
1
2k
2
_
x
k
+ (k
1
+ k
2
+ k
3
) e
k
(22)
Following the same procedure as in Section 3, we can build the controller and the closed-loop
system in state space for the MIMO case, and then use the algorithm (4.1) to calculate the
multi-loop PID controller parameters.
Remark 5.1 Changing the block diagram preserves the transfer function from the disturbance
to the output so that only the variances for the output can be specied. The variance for the
manipulated variables can not be specied because the PID controller output is not a stationary
signal.
8
6 Simulation results
6.1 Example 1
The rst example [20] is to design a single-loop PID controller for a rst order plus time-delay
process subject to unstable disturbance containing an integrator. The process is as follows:
y
k
=
z
6
1 0.8z
1
u
k
+
1 + 0.6z
1
(1 0.6z
1
) (1 + 0.7z
1
) (1 0.5z
1
)

k
(23)
where the series {
k
} is white noise with zero mean and unit variance. The known minimal
output variance achieved by a PID controller is 123.54 [20]. However, our algorithm shows that
the variance of the output can be further reduced by optimizing the PID parameters. Using the
algorithm presented in the last section, we obtain the optimal PID controller as:
C
_
z
1
_
=
0.8021 1.3402z
1
+ 0.5788z
2
1 z
1
(24)
The corresponding output variance is 87.85. Compared with the known output variance 123.54,
the variance of the output is reduced by 30% by using the PID controller (24).
If let disturbance
k
= 0, the process (23) is actually obtained by sampling a rst order plus
time-delay system with sampling period 1:
G(s) =
5
4.48s + 1
e
5s
(25)
PID tuning for the rst order plus time-delay systems is well studied and there are many tuning
rules available. The PID controllers, obtained using these tuning algorithms, and the corre-
sponding output variance are listed in Table 1. The PID controllers are calculated according to
the Tables 15.2, 15.3, 15.4 and 15.6 in [19].
Tuning methods PID controller output variance
Ziegler-Nichols method
0.77421.2903z
1
+0.5376z
2
1z
1
105.87
Cohen-Coon
0.7591.163z
1
+0.437z
2
1z
1
112.22
IMC ( = 0.2)
0.63930.9794z
1
+0.3734z
2
1z
1
93.93
IMC ( = 0.4)
0.5480.839z
1
+0.320z
2
1z
1
98.29
IMC ( = 0.6)
0.4780.735z
1
+0.280z
2
1z
1
107.31
Integral Time-weighted Square Error (ITAE)
0.7531.176z
1
+0.466z
2
1z
1
93.63
Our algorithm
0.80211.3402z
1
+0.5788z
2
1z
1
87.85
Table 1: PID controllers and the corresponding output variances
The PID design algorithm in this paper is proposed for MIMO systems, but this example
shows that it can be used for SISO systems. The obtained PID controller (24) has the best
performance among all the PID controllers.
9
6.2 Example 2
The second example is from [11]. The process is described as follows:
y
k
+ A
1
y
k1
= B
0
u
k2
+
k
(26)
where A
1
=
_
0.99101 8.80512 10
3
0.80610 0.77089
_
, B
0
=
_
0.89889 0.409329
0.56 0.88052
_
and the series {
k
} is assumed to be zero mean white noise and its covariance matrix: E
_

T
j
_
=
(i j) 0.01 I
2
.
The process is controlled by the conventional GMVC [11]:
u
k
=
_
1.8989 + 1.9957z
1
0.4093 0.4134z
1
0.56 + 0.2929z
1
1.8805 + 1.4488z
1
_ _
0.975 0.0155
1.4203 0.5872
_
y
k
(27)
With the controller (27) implemented on the process, the covariance for the output y
k
is
_
0.0621 0.0782
0.0782 0.1411
_
and the covariance for the input u
k
is
_
0.0169 0.0255
0.0255 0.0413
_
.
To use the algorithm in Section 4, the state space model is rst generated from (26):
x
k+1
=
_
A
1
B
0
0 0
_
x
k
+
_
0
I
2
_
u
k
+
_
A
1
0
_

k
y
k
=
_
I
2
0
_
x
k
+
k
(28)
where x
k
=
_
y
k

k
u
k1
_
.
The obtained multi-loop PID control is as follows:
C
_
z
1
_
=
_
_
0.56470.5387z
1
0.02z
2
z
1
(1z
1
)
0.38150.5476z
1
+0.1678z
2
z
1
(1z
1
)
_
_
(29)
With the multi-loop PID controller (29) implemented on the process, the covariance for the sim-
ulated y
k
is
_
0.0249 0.0224
0.0224 0.1158
_
and the covariance for the simulated u
k
is
_
0.0084 0.0041
0.0041 0.0075
_
.
It can be seen that by using the multi-loop PID controller the variances of the process input and
output are smaller than those by using the GMVC. The performance comparison of multi-loop
PID and GMVC is shown in Figure (4).
6.3 Example 3
The third example is a dry rotary cement kiln with capacity 1000 tons/day [15]. The kiln is 105
meters long and 5 meters in diameter. After two pre-heaters, where the dry homogenized raw
material is heated to 800
o
C, then goes to the kiln. The dry homogenized raw material enters
the kiln and then passes it. The nal temperature of the material is around 1450
o
C. Coal is
burned in the lower front end of the kiln in order to produce the high temperature, which is
required to start the chemical reactions taking place in the raw materials. The product of the
10
0 50 100 150 200 250 300
0.5
0
0.5
0 50 100 150 200 250 300
1
0
1
0 50 100 150 200 250 300
1
0
1
0 50 100 150 200 250 300
2
0
2
Time
U
1

U
2

Y
1

Y
2

Figure 4: Simulation results: the dashed line is from the GMVC; the solid line is the from the
multi-loop PID controller
reaction is called clinker that is cooled in a planetary cooler before it leaves the process. The kiln
process has two controlled variables: the combustion gas temperature and the kiln drive power.
The latter is chosen as a controlled variable because it correlates to the burning temperature
and clinker quality and the clinker quality can only be analyzed every two hours. The two
manipulated variables of the kiln process are the kiln exhaust fan speed and raw material feed
rate. The process is exposed to random disturbance. The sampling period is 5 minutes. The
original process model is shown as follows:
y
k+1
+ A
0
y
k
= B
0
u
k
+
k+1
+ C
0

k
(30)
where
A
0
=
_
0.917 0.0846
0.132 0.915
_
, B
0
=
_
2.06 0.0746
0.108 0.0192
_
C
0
=
_
0.0449 0.216
0.0256 0.841
_
, E
_

T
j
_
=
_
0.0639 0.00188
0.00188 0.0233
_
(i j)
One can obtain the following state space model from the kiln process (30):
x
k+1
=
_
0.917 0.0846
0.132 0.915
_
x
k
+
_
2.06 0.0746
0.108 0.0192
_
u
k
+
_
0.8721 0.1314
0.1064 1.7560
_

k
y
k
= x
k
+
k
(31)
It is desired to control the variances of both the output and input so that the controlled variables
11
are chosen as:
z
1
k
=
_
1 0
_
y
k
z
2
k
=
_
0 1
_
y
k
z
3
k
=
_
1 0
_
u
k
z
4
k
=
_
0 1
_
u
k
(32)
It is shown in [15] that a reasonable control criterion is to minimize the joint variation of the
controlled variables:
J = lim
k
Ey
T
k
y
k
(33)
However, the variances of the input variables become unacceptable if the minimum variance
control law is implemented: lim
k
E
_
z
3
k
z
3
k
_
= 0.148 and lim
k
E
_
z
4
k
z
4
k
_
= 108. As it is pointed
out in [15] it is appropriate to restrict the variances of the input variables according to:
lim
k
E
_
z
3
k
z
3
k
_
< 0.004
lim
k
E
_
z
4
k
z
4
k
_
< 1.5
(34)
Minimization of the (33) subject to the variance constraints (34) by using a full order dynamic
controller leads to the output variances [10, 15]:
lim
k
E
_
z
1
k
z
1
k
_
= 0.0939
lim
k
E
_
z
2
k
z
2
k
_
= 0.189
(35)
The proposed algorithm can not nd a multi-loop PID controller if the variance constraints
for the input variables are chosen as (34) and the variance constraints for the output variables
are specied as 0.939 and 0.189 respectively. This is not surprising because the decentralized
controller structure adds performance limit compared to the full order centralized controller. If
the variance constraint for z
2
k
is relaxed to 0.345, by using the algorithm in Section 4 a multi-loop
PID controller can be obtained as follows:
C
_
z
1
_
=
_
_
0.17430.1612z
1
+0.0064z
2
(1z
1
)z
1
1.8252+1.8139z
1
0.0207z
2
(1z
1
)z
1
_
_
(36)
The simulation results are shown in Figure (5):
With the multi-loop PID controller (36) implemented, the variances for the controlled vari-
ables, calculated from the simulated data, are:
_
_
_
lim
k
Ez
1
k
z
1
k
= 0.0923
lim
k
Ez
2
k
z
2
k
= 0.3419
,
_
_
_
lim
k
Ez
3
k
z
3
k
= 0.0034
lim
k
Ez
4
k
z
4
k
= 1.1198
It is shown that the variance for the rst output is even slightly better than that achieved by
the constrained LQG controller in [15]; the variances of the input variables are better than
those achieved by the constrained LQG controller. However, the variance of z
2
k
is larger than
that achieved by the constrained LQG controller in [15] but still satises the design specication.
Simulation also shows that if the variance bound for the second output is smaller than 0.33, then
12
0 20 40 60 80 100 120 140 160 180 200
1
0
1
0 20 40 60 80 100 120 140 160 180 200
2
0
2
0 20 40 60 80 100 120 140 160 180 200
0.2
0
0.2
0 20 40 60 80 100 120 140 160 180 200
5
0
5
Time
y
1
y
2

u
1

u
2

Figure 5: Simulation results for the kiln process
the algorithm can not nd a solution. This may be caused by two reasons: 1) the multi-loop-PID-
controller structure is far simpler than that of the full-order multivariable controller, and this
simplicity will reduce the closed-loop performance compared to the full-order optimal constrained
LQG controller in [15]; 2) the proposed algorithm may not be globally convergent, which means
that if there exists a multi-loop PID controller such that the variance z
2
k
is smaller than 0.33,
the proposed algorithm may not nd it. There is no strict proof of the global convergency of the
proposed algorithm in Section 4; however, our simulation shows the algorithm always converges
to one value no matter what is the initial condition. It is worth a investigation of the global
convergent properties of the proposed algorithm as a future research topic.
7 Conclusion
The multi-loop or decentralized PID controller design based on the generalized covariance
constraints has been considered in this paper and an iterative LMI approach is proposed to solve
the problem. The algorithm is shown to be convergent. This algorithm is originally derived for
the process with stable disturbances; after the reconguration of the process block diagram, it
can also be applied to the process with unstable random-walk disturbances. Several simulation
results are used to illustrate the eectiveness of the proposed method.
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15