International Journal of Computers, Communications & Control
Vol. II (2007), No. 2, pp. 195-204

Robust PID Decentralized Controller Design Using LMI

Danica Rosinová, Vojtech Veselý

Abstract: The new LMI based method for robust stability analysis for linear un-
certain system with PID controller is proposed. The general constrained structure of
controller matrix is considered appropriate for both output feedback and decentral-
ized control and the respective guaranteed cost control design scheme is presented.
The sufficient robust stability condition is developed for extended quadratic perfor-
mance index including first derivative of the state vector to damp oscillations. The
obtained stability condition is formulated for parameter-dependent Lyapunov func-
tion.
Keywords: Uncertain systems, Robust stability, Decentralized control, Linear matrix
inequalities (LMI), Lyapunov function

1 Introduction

Robust stability and robust control belong to fundamental problems in control theory and practice;
various approaches in this field have been proposed to cope with uncertainties that always appear in real
plant ([2];[8];[7];[5];[4]). The development of Linear Matrix Inequality (LMI) computational techniques
has brought an efficient tool to solve a large set of convex problems in polynomial time (e.g. [2]). Sig-
nificant effort has been therefore made to formulate crucial control problems in algebraic way ([12]), so
that the numerical LMI solution can be adopted. This approach is advantageously used in solving con-
trol problems for linear systems with convex (affine or polytopic) uncertainty domain. However, many
important problems in linear control design, such as decentralized control, simultaneous SOF or more
generally - structured linear control problems have been proven as NP hard ([1]). Intensive research has
been devoted to overcome nonconvexity and transform the nonconvex or NP-hard problem into convex
optimisation problem in LMI framework. Various techniques have been developed using inner or outer
convex approximation of the respective nonconvex domain. The common tool in both inner and outer
approximation is the use of linearization or convexification. In ([6]; [3]) the general convexifying algo-
rithm for the nonconvex function together with potential convexifying functions for both continuous and
discrete-time case have been proposed. Linearization approach for continuous and discrete-time system
design was independently used in ([13];[11]).
Proportional-integral-derivative (PID) controllers belong to the most popular ones in the industrial world.
The derivative part of the controller, however, causes difficulties when uncertainties are considered. In
multivariable PID control schemes using LMI developed recently ([14]) the incorporation of the deriva-
tive part requires inversion of the respective matrix, which does not allow including uncertainties. The
other way to cope with the derivative part is to assume the special case when output and its derivative are
state variables, robust PID controller for first and second order SISO systems are proposed for this case
in ([7]).
In this paper a state space approach to designing decentralized (multi-loop) PID robust controllers is
proposed for linear uncertain system with guaranteed cost with a new quadratic cost function. The major
contribution is in considering the derivative part in robust control framework. We adopt the new PID con-
trol problem formulation using LMI that is appropriate for polytopic uncertain systems. The robust PID
control scheme is proposed for structured control gain matrix, thus enabling decentralized PID control
design. In Section 2 the robust control design problem with structured control gain matrix is formulated
in general. The robust optimal control design procedure in state space with the extended cost function
is proposed in Section 3. The main result-robust PID controller design approach is provided in Section

Copyright © 2006-2007 by CCC Publications



196 Danica Rosinová, Vojtech Veselý

4, the developed approach is appropriate for decentralized control structure. In Section 5 the results are
illustrated on the examples.

2 Problem Formulation And Preliminaries

Consider a linear affine uncertain system:

δ x(t) = (A + δ A)x(t) + (B + δ B)u(t)
y(t) = Cx(t)

(1)

where
δ x(t) = ẋ(t) for continuous - time system
δ x(t) = x(t + 1) for discrete - time system

x(t) ∈ Rn, u(t) ∈ Rm, y(t) ∈ Rl are state, control and output vectors respectively; A, B,C are known con-
stant matrices of appropriate dimensions corresponding to the nominal system, δ A, δ B are matrices of
uncertainties of the respective dimensions.The uncertainties are considered to be affine of the form

δ A =
p

∑
j=1

ε jà j, δ B =
p

∑
j=1

ε jB̃ j (2)

where ε j ≤ ε j ≤ ε̄ j are unknown uncertainty parameters; Ã j, B̃ j, j = 1, 2, ..., p are constant matrices of
uncertainties of the respective dimensions and structure. The uncertain system (1), (2) can be equivalently
described by a polytopic model given by its vertices

{(A1, B1,C), (A2, B2,C), ..., (AN , BN ,C)}, N = 2p.

The decentralized feedback control law is considered in the form

u(t) = FCx(t) (3)

where F is a matrix corresponding to decentralized controller. The uncertain closed-loop polytopic
system is then

δ x(t) = AC(α)x(t) (4)

where

AC(α) ∈
{

N
∑

i=1
αiACi,

N
∑

i=1
αi = 1, αi ≥ 0

}
,

ACi = Ai + BiFC.
(5)

To assess the performance quality a quadratic cost function known from LQ theory is often used. How-
ever, in practice the response rate or overshoot are often limited. Therefore we include into the cost
function the additional derivative term for state variable to open the possibility to damp the oscillations
and limit the response rate.

Jc =
∞∫

0

[x(t)T Qx(t) + u(t)T Ru(t) + δ x(t)T Sδ x(t)]dt

for a continuous-time and

Jd =
∞

∑
k=0

[x(t)T Qx(t) + u(t)T Ru(t) + δ x(t)T Sδ x(t)] (6)



Robust PID Decentralized Controller Design Using LMI 197

for a discrete-time system,
where Q, S ∈ Rn×n, R ∈ Rm×m are symmetric positive definite matrices. The concept of guaranteed cost
control is used in a standard way: Let there exist a feedback gain matrix F0 and a constant J0 such that

J ≤ J0 (7)
holds for the closed loop system (4), (5). Then the respective control (3) is called the guaranteed cost
control and the value of J0 is the guaranteed cost. The main aim of this paper is to develop a decentralized
PID control algorithm that stabilizes the uncertain system (1), with guaranteed cost with respect to the
cost function (6).
We start with basic notions concerning Lyapunov stability and convexifying functions. In the following
we use D−stability concept ([4]) to receive the respective stability conditions in more general form.
Definition 1. (D-stability) Consider the D-domain in the complex plain defined as

D = {s is complex number :
[

1
s

]∗ [
r11 r12
r∗12 r22

] [
1
s

]
< 0}.

The considered linear system (1) is D-stable if all its poles lie in the D-domain.

(To simplify the reading of formulas we use in the Definition 1 scalar values of the parameters ri j,
in general the stability domain can be defined using matrix values of parameters ri j with the respective
dimensions.) The standard choice of ri j is r11 = 0, r12 = 1, r22= 0 for a continuous-time system;
r11 = −1, r12 = 0, r22= 1 for a discrete-time system. The quadratic D-stability is equivalent to the
existence of one Lyapunov function for the whole set that describes the uncertain system model.

Definition 2. (Quadratic D− stability)
The uncertain system (4) is quadratically D−stable if and only if there exists a symmetric positive definite
matrix P such that

r12PAC(α) + r∗12A
T
C (α)P + r11P + r22A

T
C (α)PAC(α) < 0 (8)

Instead of quadratic stability, a robust stability notion is considered based on the parameter dependent
Lyapunov function (PDLF) defined as

P(α) =
N

∑
i=1

aiPi where Pi = P
T
i > 0 (9)

to obtain less conservative results than using quadratic stability with unique Lyapunov function.

Definition 3. ([5]) System (4) is robustly D-stable in the convex uncertainty domain (5) with parameter-
dependent Lyapunov function (9) if and only if there exists a matrix P(α) = P(α)T > 0 such that

r12P(α)AC(α) + r∗12A
T
C (α)P(α) + r11P(α)+

+r22ATC (α)P(α)AC(α) < 0
(10)

for all α such that AC (α) is given by (5).

The sufficient robust D-stability condition which can be considered as not too conservative has been
proposed in ([9]), recalled in the following lemma.

Lemma 4. If there exist matrices E ∈ Rnxn, G ∈ Rnxn and N symmetric positive definite matrices Pi ∈ Rnxn
such that for all i= 1,. . . ,N:

[
r11Pi + ATCiE

T + EACi r12Pi −E + ATCiG
r∗12Pi −E T + GT ACi r22Pi −(G + GT )

]
< 0 (11)

then system (4) is robustly D−stable.



198 Danica Rosinová, Vojtech Veselý

Note that matrices E and G are not restricted to any special form; they were included to relax the
conservatism of the sufficient condition. To transform nonconvex problem of structured control (decen-
tralized control in our case) into convex form, the convexifying (linearizing) function can be used ([6];
[3];[11];[13]). The respective potential convexifying function for X−1 and XWX has been proposed in
the linearizing form:
- The linearization of X−1 ∈ Rnxn about the value Xk > 0 is

Φ(X−1, Xk) = X−1k −X−1k (X −Xk)X−1k (12)

- The linearization of XW X ∈ Rnxn about Xk is

Ψ(XW X , Xk) = −XkW Xk + XW Xk + XkW X (13)

Both functions defined in (12) and (13) meets one of the basic requirements on convexifying function:
to be equal to the original nonconvex term if and only if Xk = X . However, the question how to choose
the appropriate nice convexifying function remains still open.
In the sequel, X > 0 denotes positive definite matrix; * in matrices denotes the respective transposed
term to make the matrix symmetric; I denotes identity matrix and 0 denotes zero matrix of the respective
dimensions.

3 Robust Optimal Controller Design

In this section the new design algorithm for optimal control with guaranteed cost is developed us-
ing parameter dependent Lyapunov function and convexifying approach employing iterative procedure.
The proposed control design approach uses sufficient stability condition inspired by the result of ([9]).
The next theorem provides the new form of robust stability condition for linear uncertain system with
guaranteed cost.

Theorem 5. Consider uncertain linear system (1), (2) with static output feedback (3) and cost function
(6). The following statements are equivalent:

i) Closed loop system (4) is robustly D-stable with PDLF (9) and guaranteed cost with respect to cost
function (6): J ≤ J0 = xT (0)P(α)x(0).

ii) There exist matrices P(α) > 0 defined by (9) such that

r12P(α)AC(α) + r∗12A
T
C (α)P(α) + r22A

T
C (α)P(α)AC(α)+

+r11P(α) + Q + CT F T RFC + ATC (α)SAC(α) < 0
(14)

iii) There exist matrices P(α) > 0 defined by (9) and H, G and F of the respective dimensions such that
[

r11P(α) + ATC (α)H
T + HAC(α) + Q + CT F T RFC

r∗12P(α)−H T + GT AC(α)
]
.

[
.

∗
r22P(α)−(G + GT ) + S

]
< 0

(15)

ACi = (Ai + BiFC) denotes the i−th closed loop system vertex. Matrix F is the guaranteed cost decen-
tralized control gain for the uncertain system (4), (5).



Robust PID Decentralized Controller Design Using LMI 199

Proof. For brevity the detail steps of the proof are omitted where standard tools are applied. (i) ⇔(ii): the
proof is analogous to that in ([10]). The (ii) ⇒(i) is shown by taking V (t) = x(t)P(α)x(t) as a candidate
Lyapunov function for (4) and writing δV (t), where

δV (t) = V̇ (t) for continuous - time system
δV (t) = V (t + 1)−V (t) for discrete - time system

δV (t) = r∗12δ x(t)
T P(α)x(t) + r12x(t)T P(α)δ x(t) + r11x(t)T P(α)x(t) + r22δ x(t)T P(α)δ x(t) (16)

Substituting for δ x from (4) to (16) and comparing with (14) provides D−stability of the considered
system when the latter inequality holds. The guaranteed cost can be proved by summing or integrating
both sides of the following inequality for t from 0 to ∞:

δV (t) < −x(t)T [Q + CT F T RFC + ATC (α)SAC(α)]x(t)
The (i) ⇒(ii) can be proved by contradiction. (ii)⇔(iii): The proof follows the same steps to the proof
of Lemma 1: (iii) ⇒(ii) is proved in standard way multiplying both sides of (15) by the full rank matrix:

[
I ATC (α)

]
{l.h.s.(15)}

[
I

AC(α)

]
< 0.

(ii) ⇒ (iii) follows from applying a Schur complement to (14) rewritten as
r12P(α)AC(α) + r∗12A

T
C (α)P(α) + Q + C

T F T RFC+
+r11P(α) + ATC (α)[r22P(α) + S]AC(α) < 0

Therefore
[

X11 X12
X T12 X22

]
< 0 where

X11 = r11P(α) + r12P(α)AC(α) + r∗12A
T
C (α)P(α) + Q+

+CT F T RFC
X12 = ATC (α)[r22P(α) + S]
X22 = −[r22P(α) + S]

which for H = r12P(α), G = [r22P(α) + S] gives (15).

The guaranteed cost control design is based on the robust stability condition (15). Since the matrix
inequality (15) is not LMI we use the inner approximation for the continuous time system applying
linearization formula (13) together with using the respective quadratic forms to obtain LMI formulation,
which is then solved by iterative procedure.

4 PID Robust Controller Design For Continuous-Time Systems

Control algorithm for PID is considered as

u(t) = KPy(t) + KI

t∫

0

y(t)dt + FdCd ẋ(t) (17)

The proportional and integral term can be included into the state vector in the common way defining

the auxiliary state z =
t∫

0
y(t), i.e. ż(t) = y(t) = Cx(t). Then the closed-loop system for PI part of the

controller is

ẋn =
[

ẋ
ż

]
=

[
A + δ A 0

C 0

] [
x
z

]
+

[
B + δ B

0

]
u(t)



200 Danica Rosinová, Vojtech Veselý

and
u(t) = FCx(t) + FdCd ẋ(t) (18)

where FCx(t) and FdCd ẋ(t) correspond respectively to the PI and D term of PID controller. The resulting
closed loop system with PID controller (17) is then

ẋn(t) = AC(α)xn(t) + B(α)
[

FdCd 0
]

ẋn(t) (19)

where the PI controller term is included inAC(α). (For brevity we omit the argument t.) To simplify the
denotation, in the following we consider PD controller (which is equivalent to the assumption, that the I
term of PID controller has been already included into the system dynamics in the above outlined way)
and the closed loop is described by

ẋ(t) = AC(α)x(t) + B(α)FdCd ẋ(t) (20)

Let us consider the following performance index

Js =
∞∫

0

[
x ẋ

]T
[

Q + CT F T RFC 0
0 S

] [
x
ẋ

]
dt (21)

which formally corresponds to (6). Then for Lyapunov function (9) we have the necessary and sufficient
condition for robust stability with guaranteed cost in the form (14), i.e. for continuous time system:

[
x ẋ

]T
[

Q + CT F T RFC P(α)
P(α) S

] [
x
ẋ

]
< 0. (22)

The main result on robust PID control stabilization is summarized in the next theorem

Theorem 6. Consider a continuous uncertain linear system (1), (2) with PID controller (17) and cost
function (21). The following statements are equivalent:

• Closed loop system (19) is robustly D-stable with PDLF (9) and guaranteed cost with respect to
cost function (21):

J ≤ J0 = xT (0)P(α)x(0).

• There exist matrices P(α) > 0 defined by (9), and H, G, F and Fd of the respective dimensions
such that [

ATCiH
T + HACi + Q + CT F T RFC

Pi −MTdiH + GT ACi
∗

−MTdiG−GT Mdi + S
]

< 0
(23)

ACi = (Ai + BiFC) denotes the i−th closed loop system vertex, Mdi includes the derivative part of the
PID controller: Mdi = I −BiFdCd .
Proof. Owing to (20) for any matrices H and G:

(
−xT H − ẋT GT

)
(ẋ−AC(α)x−B(α)FdCd ẋ)+

+(ẋ−AC(α)x−B(α)FdCd ẋ)T
(
H T x−Gẋ

)
= 0

(24)

Summing up the l.h.s of (24) and (22) and taking into consideration linearity w.r.t. α we get con-
dition (23). Theorem 6 provides the robust stability condition for the linear uncertain system with PID
controller. Notice that the derivative term does not appear in the matrix inversion and allows including
the uncertainty in control matrix B into the stability condition.



Robust PID Decentralized Controller Design Using LMI 201

Considering PID control design, there are unknown matrices H, G, F and Fd to be solved from (23).
(Recall thatACi = (Ai + BiFC), Mdi = I −BiFdCd .) Then the inequality (23) is not LMI, to cope with the
respective unknown matrices products the linearizing approach using (13) has been adopted and the PID
iterative control design algorithm based on LMI (4x4 matrix) has been proposed. The resulting closed
loop system with PD controller is

ẋ(t) = (I −BiFdCd )−1(Ai + BiFC)x(t), i = 1, ..., N (25)

The extension of the proposed algorithm to decentralized control design is straightforward since the
respective F and Fd matrices are assumed as being of the prescribed structure, therefore it is enough to
prescribe the decentralized structure for both matrices.

5 Examples

In this section the major contribution of the proposed approach: design of robust derivative feedback
is illustrated on the examples. The results obtained using the proposed new iterative algorithm based on
(23) to design the PD controller are provided and discussed. The impact of matrix S choice is studied as
well.
We consider affine models of uncertain system (1), (2) with symmetric uncertainty domain:

ε j = −q, ε j = q (26)

Example 1.
Consider the uncertain system (1), (2) where:

A =



−4.365 −0.6723 −0.3363
7.0880 −6.5570 −4.6010
−2.4100 7.5840 −14.3100


 B =




2.3740 0.7485
1.3660 3.4440
0.9461 −9.6190




C = Cd =
[

0 1 0
0 0 1

]

uncertainty parameter q=1; uncertainty matrices:

Ã1 =



−0.5608 0.8553 0.5892
0.6698 −1.3750 −0.9909
3.1917 1.7971 −2.5887


 B̃1 =



−0.1602 −0.3521
0.1162 −2.4839
−0.1106 −4.6057




Ã2 =




0.6698 −1.3750 −0.9909
−2.8963 −1.5292 10.5160
−3.5777 2.8389 1.9087


 B̃2 =




0.1562 0.1306
−0.4958 4.0379
−0.0306 0.8947




The uncertain system can be described by four vertices; the corresponding maximal eigenvalues in the
vertices of open loop system are respectively:
-4.0896 ± 2.1956i ; -3.9243 ; 1.5014; -4.9595
Notice that the open loop uncertain system is unstable (see third vertex). The stabilizing optimal PD
controller has been designed. Optimality is considered in the sense of guaranteed cost w.r.t. cost function
(21) with matrices R = I2x2,Q = 0.001∗I3x3 . The results summarized in the Tab. 1 indicate the differences
between results obtained for different values of S



202 Danica Rosinová, Vojtech Veselý

S F (proportional part)
Fd (derivative part)

Max Eig
in vertices

1e-6 *I -1.0567 -0.5643
-2.1825 -1.4969
-0.3126 -0.2243
-0.0967 0.0330

-4.8644
-2.4074
-3.8368 ± 1.1165i
-4.7436

0.1*I -1.0724 -0.5818
-2.1941 -1.4642
-0.3227 -0.2186
-0.0969 0.0340

-4.9546
-2.2211
-3.7823 ± 1.4723i
-4.7751

Table 1: Example 1, Example 2

Consider the uncertain system (1), (2) where:

A =




−2.9800 0.9300 0 −0.0340
−0.9900 −0.2100 0.0350 −0.0011

0 0 0 1
0.3900 −5.5550 0 −1.89


 ,

Ã1 =




0 1.5 0 0
0 0 0 0
0 0 0 0
0 0 0 0


 ,

B =




−0.0320
0
0

−1.6000


 , B̃1 =




0
0
0
0


 ,

C =
[

0 0 1 0
0 0 0 1

]

The results are summarized in Tab. 2 for R = 1, Q = 0.0005 ∗ I4x4 for various values of matrix S in cost
function. As indicated in Tab.2, increasing values of S slow down the response (max.eig.CL shifted to
zero) as assumed.

6 Conclusion

The new robust PID controller design method is proposed for uncertain linear system based on LMI.
The important feature of this PID design approach is that the derivative term appears in such form that
enables to consider the model uncertainties. Since the structured feedback matrix is assumed, this ap-
proach is appropriate for decentralized PID control design. The guaranteed cost control is proposed with
a new quadratic cost function including the derivative term for state vector as a tool to influence the
overshoot and response rate. The obtained results are illustrated on the examples: to show the robust PID
control design and the influence of the choice of matrix S in the extended cost function.
Acknowledgment The work has been supported by Slovak Scientific Grant Agency, Grant N 1/3841/06.



Robust PID Decentralized Controller Design Using LMI 203

S=10e-8*I4x4
qmax 1.1
max.eig.CL -0.189

S=0.1*I4x4
qmax 1.1
max.eig.CL -0.1101

S=0.2*I4x4
qmax 1.1
max.eig.CL -0.0863

S=0.29*I4x4
qmax 1.02
max.eig.CL -0.0590

Table 2: Comparison for various S - Example 2

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204 Danica Rosinová, Vojtech Veselý

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Danica Rosinová and Vojtech Veselý
Slovak University of Technology

Institute for Control and Industrial Informatics
Ilkovičova 3 81219 Bratislava , Slovakia

{danica.rosinova;vojtech.vesely}@stuba.sk
Received: December 24, 2006

Danica Rosinová was born in Bratislava, Slovak Republic in
1961. She received MSc. and PhD from Slovak University of
Technology in 1985 and 1996. Since 1985 she has been with
the Department of Automatic Control Systems, now Institute for
Control and Industrial Informatics at the Faculty of Electrical En-
gineering and Information Technology STU Bratislava. Since
2006 she has been Associate Professor. Her research interests
concentrate on robust control, large scale systems theory and op-
timization. students.

Vojtech Veselý was born in 1940. Since 1964 he has worked at
the Department of Automatic Control Systems at the Faculty of
Electrical Engineering and Information Technology, Slovak Uni-
versity of Technology in Bratislava. Since 1986 he has been Full
Professor. His research interests include the areas of power sys-
tem control, robust control, decentralized control of large-scale
systems, proces control and optimization. He is author and coau-
thor of more then 270 scientific and technical papers, he success-
ful supervised up today 19 PhD