Research Article
Design of Gain Scheduling Control Using State
Derivative Feedback

Lázaro Ismael Hardy Llins,1 Edvaldo Assunção,1 Marcelo C. M. Teixeira,1 Rodrigo Cardim,1

Mario R. R. Cadalso,1 Diogo R. de Oliveira,2 and Emerson R. P. da Silva3

1Department of Electrical Engineering, Ilha Solteira School of Engineering, São Paulo State University (UNESP),
Control Research Laboratory, José Carlos Rossi Ave. 1370, 15385-000 Ilha Solteira, SP, Brazil
2Federal Institute of Education, Science and Technology of Mato Grosso do Sul (IFMS), Campus of Três Lagoas,
79.641-162 Três Lagoas, MS, Brazil
3Academic Department of Electrical Engineering, Federal Technological University of Paraná (UTFPR),
Alberto Carazzai Ave. 1640, 86300-000 Cornélio Procópio, PR, Brazil

Correspondence should be addressed to Lázaro Ismael Hardy Llins; hardyllins@gmail.com

Received 2 July 2017; Revised 10 November 2017; Accepted 19 November 2017; Published 17 December 2017

Academic Editor: Haranath Kar

Copyright © 2017 Lázaro Ismael Hardy Llins et al. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.

In recent years, the study of systems subject to time-varying parameters has awakened the interest of many researchers. The gain
scheduling control strategy guarantees a good performance for systems of this type and also is considered as the simplest to deal with
problems of this nature. Moreover, the class of systems in which the state derivative signals are easier to obtain than the state signals,
such as in the control for reducing vibrations in a mechanical system, has gained an important hole in control theory. Considering
those ideas, we propose sufficient conditions via LMI for designing a gain scheduling controller using state derivative feedback.
TheD-stability methodology was used for improving the performance of the transitory response. Practical implementation in an
active suspension system and comparison with other methods validates the efficiency of the proposed strategy.

1. Introduction

Since the emergence of dynamic systems control theory, the
techniques most used to feedback systems are the output
feedback or state feedback, but the use of accelerometers
allowed opening a path for the study of derivative feedback
of the state vector, due to the easy reconstruction of the
derived signals to the signals themselves. For example, from
the acceleration, velocity determination is a simple security
integral, which presents no greater problem of experimental
order [1].

The use of feedback from the state vector derivative
(derivative feedback) for linear systems has been explored
in recent years. Some researchers have sought to develop
methods similar to those existing for the state vector feed-
back; for example, [1] developed a formula similar to the
widespread Ackerman for linear systems (SISO) through
derivative feedback. In [2], a new formulation was presented

for the stabilization of linear multivariable systems under
derivative feedback states. In [3], an analysis was presented
of linear systems of observability and stability through the
state vector derived and a study on the disturbance rejection
with state derivative feedback. In [4], a geometric theory for
dynamical systems with state derivative feedback for singular
systems was shown. In [5], the theory was presented to the
design of a less conservative controller for linear systems with
polytopic uncertainties via derivative feedback ensuring the
stability and robustness of the system. An approach to the
stability and robustness with derivative feedback, including
the weakness, can be seen in [6]. Still, in the literature, it
is possible to find papers that include the use of derivative
feedback in linear systems subject to uncertainties in the
plant among others, using techniques based on linear matrix
inequalities (LMIs).

The gain scheduling controller has motivated several
studies in the field of control engineering. This control

Hindawi
Mathematical Problems in Engineering
Volume 2017, Article ID 1512679, 11 pages
https://doi.org/10.1155/2017/1512679

https://doi.org/10.1155/2017/1512679


2 Mathematical Problems in Engineering

strategy is very popular for linear application in multiple
fields such as aeronautics, military, and civil. The origin of
the gain scheduling controllers took place in the 60s with the
classic theory called gain scheduling based on linearization of
a system on their equilibrium points [7–9].

The classical gain scheduling efficiency depends on the
dynamic characteristics of the nonlinear system. These can
be described as a combination of linear systems, which was
associated with a linear controller for each operation point
[10]. The controller is designed taking into account only
the dynamics of the plant locally around an equilibrium
point [10]. Classical gain scheduling controllers were applied
intensely but had limitations. Working only in the area of the
neighborhood of operating equilibrium points represented
a technical deficiency. The gain scheduling controller does
reduce the error but there is still a periodic part in the error
signal of the system [11].

It can design the controller using methods based on
standard L2 ensuring robustness and nominal stability of
the system and improving the project gain scheduling [8].
Robust gain scheduler PI controllers are presented in [12]. It
presented the procedure for obtaining the solutions via BMI.
In [13] a methodology was described to generalize robust
gain-scheduled PID controller design for affine LPV systems
with polytopic uncertainty via BMI.

Fuzzy gain scheduling overcomes the disadvantages of
classical gain scheduling, considering the stability restriction
and performance in both the local and the global behav-
ior. Technical fuzzy gain scheduling may involve classical
scheduling gain as well as the LPV techniques [14]. Reference
[15] presents a fuzzy gain scheduling of proportional integral
derivative (FGS-PID) controller for guaranteeing stability of
a multimachine power system. The bee colony optimization
method was used to determine control rules of the FGS-PID.
Reference [16] developed a methodology to the nonfragile
H∞ control problem for a class of discrete-time Takagi-
Sugeno fuzzy systems with both randomly occurring gain
variations and channel fading via LMI.

An important consideration in the design of a linear
controller for a closed-loop system with uncertainty is the
robustness and performance. When uncertainties are invari-
ant or slow variations, the problem can be solved by using
robust control techniques [17]. Sometimes, one can have
considerable variations in the parameters; in this case, a
robust controller does not guarantee a good performance
and the stabilization of the plant by a controller designed
assumes that the same is time invariant system. Whereas
the variations of parameters can be measured during system
operation, the gain scheduling strategy can provide more
efficient solutions. The gain scheduling controllers are a
function of a variant parameter of the plant. The controllers
can be adjusted according to the variations in the dynamics of
the plant. Therefore, in many applications such drivers’ gain
scheduling is more feasible than the robust controllers. The
combination of both techniques has been studied in [18].

Themainmotivation of the work was to develop a control
strategy using gain scheduling and state derivative feedback;
both techniques are used for systems with time variant
parameters. In recent years, the LMIs are used to solve a lot

of control theory problems. It was interesting to propose a
methodology for solving this class of problems of this type.

According to the authors’ knowledge, there is no record
of studies that consider the union of both control strategies.
Ourmain challenge and objective were to develop and design
a gain scheduling type control using derivative feedback that
considers time-varying parameters to ensure stability and
good system performance.

Notation. Throughout the text, the following notations are
used: K𝑟 = {1, 2, . . . , 𝑟}, 𝑟 ∈ N; diag{𝑀1,𝑀2, . . . ,𝑀𝑟}
represents a block-diagonal matrix in which the diagonal
elements are 𝑀1, 𝑀2, . . . ,𝑀𝑟; 𝑀 > 0 is used to represent
positive definite matrices and, equivalently,𝑀 < 0 is used to
represent negative definite matrices; (⋅ ⋅ ⋅ )𝑇 denotes a vector
or matrix transposition; (⋅ ⋅ ⋅ )−𝑇 denotes the transposition of
an inverted matrix; (∗) denotes the symmetric matrix.

Property 1 (see [19]). For all nonsymmetric𝑀 matrix (𝑀 ̸=𝑀𝑇), if𝑀+𝑀𝑇 < 0, then𝑀 is invertible.

Property 2. One has

𝐹𝑧 (𝛼 (𝑡)) = 𝑁∑
𝑖=1

𝛼𝑖 (𝑡) 𝐹𝑖,
𝐹𝑧𝑧 (𝛼 (𝑡)) = 𝑁∑

𝑖=1

𝑁∑
𝑗=1

𝛼𝑖 (𝑡) 𝛼𝑗 (𝑡) 𝐹𝑖𝑗,

Λ𝑁 = {𝛼 (𝑡) ∈ R
𝑁 : 𝑁∑
𝑖=1

𝛼𝑖 (𝑡) = 1, 𝛼𝑖 (𝑡) ≥ 0, 𝑖
= 1, . . . , 𝑁} ,

(1)

such that 𝛼(𝑡) ∈ Λ𝑁 is a time-varying parameter. The
condition ∑𝑁𝑖=1 𝛼𝑖(𝑡) = 1 was used to obtain the convexity of
the set and it was described by polytope vertices.The variable𝛼𝑖(𝑡) from 𝑖 = 1, . . . , 𝑁 represents a convex combination
of the vertices. In this case we can write the set using the
minimum and maximum vertex values. This description
in the unit simplex form provides the design of the gain
scheduling controller using LMIs. For simplicity,𝐹𝑧(𝛼(𝑡))will
be denoted by 𝐹𝑧, similarly 𝐹𝑧𝑧(𝛼(𝑡)) by 𝐹𝑧𝑧.
Property 3 (see [20]). If the following LMIs, Υ𝑖𝑖 < 0 for 𝑖 =1, 2, . . . , 𝑁 and Υ𝑖𝑗 + Υ𝑗𝑖 < 0 for 1 ≤ 𝑖 < 𝑗 ≤ 𝑁, are feasible,
then the inequality ∑𝑁𝑖=1∑𝑁𝑗=1 𝛼𝑖𝛼𝑗Υ𝑖𝑗 = Υ𝑧𝑧 < 0 holds, where𝛼(𝑡) ∈ Λ𝑁.
2. Design of Gain Scheduling Controller Using
State Derivative Feedback

In this section, a solution for linear systems with time-
varying parameters is proposed, using the strategy of gain
scheduling control using state derivative feedback. In this



Mathematical Problems in Engineering 3

section, Finsler’s Lemma is used to avoid cross product of
uncertain matrices.

2.1. Finsler’s Lemma, an LMI Formulation. Consider a linear
system with time variant uncertainties, described by

�̇� (𝑡) = 𝑁∑
𝑖=1

𝛼𝑖 (𝑡) 𝐴 𝑖𝑥 (𝑡) + 𝑁∑
𝑖=1

𝛼𝑖 (𝑡) 𝐵𝑖𝑢 (𝑡)
= 𝐴𝑧 (𝛼 (𝑡)) 𝑥 (𝑡) + 𝐵𝑧 (𝛼 (𝑡)) 𝑢 (𝑡) ,

(2)

where 𝐴𝑧(𝛼(𝑡)) ∈ R𝑛×𝑛 and 𝐵𝑧(𝛼(𝑡)) ∈ R𝑛×𝑚 are matrices
that represent the time variant systems dynamics, 𝑥(𝑡) ∈ R𝑛

is the state vector, and 𝑢(𝑡) ∈ R𝑚 is the control input vector.
Using the compact notation of Property 2, (2) can be

rewritten as

�̇� (𝑡) = 𝐴𝑧𝑥 (𝑡) + 𝐵𝑧𝑢 (𝑡) . (3)

The objective is to find a controller 𝐾𝑧, such that the
feedback system with the control input is

𝑢 (𝑡) = − 𝑁∑
𝑗=1

𝛼𝑗 (𝑡) 𝐾𝑗�̇� (𝑡) = −𝐾𝑧�̇� (𝑡) . (4)

In this case, the closed-loop system of (3) and (4) is
obtained:

�̇� (𝑡) = 𝐴𝑧𝑥 (𝑡) − 𝐵𝑧𝐾𝑧�̇� (𝑡) . (5)

or

�̇� (𝑡) = (𝐼 + 𝐵𝑧𝐾𝑧)−1 𝐴𝑧𝑥 (𝑡) . (6)

If𝐴𝑧 has null eigenvalue, then the feedback system, given
by (𝐼+𝐵𝑧𝐾𝑧)−1𝐴𝑧, has null eigenvalue.The controller𝐾𝑧 can
not modify the null eigenvalue of 𝐴𝑧, in this case rank[(𝐼 +𝐵𝑧𝐾𝑧)−1𝐴𝑧] < 𝑛.

For applying Finsler’s Lemma, it was necessary to start
from equality (7), like the result of the transformation (5):

0 = 𝐴𝑧𝑥 (𝑡) − (𝐼 + 𝐵𝑧𝐾𝑧) �̇� (𝑡) . (7)

To obtain the LMIs for the design of the controllers,
Lemma 1 is used. Using the notation of Property 2, consider
thatB𝑧 = B𝑧(𝛼(𝑡)).
Lemma 1 (Finsler’s Lemma). Considering thatW ∈ R2𝑛,D ∈
R2𝑛×2𝑛 andB𝑧 ∈ R𝑛×2𝑛 with rank(B𝑧) < 𝑛 andB⊥𝑧 is a base
for the null space ofB𝑧 (i.e.,B𝑧B⊥𝑧 = 0).

So the next conditions are equivalent:

(i) W𝑇DW < 0, ∀W ̸= 0, B𝑧W = 0,
(ii) B⊥

𝑇

𝑧 DB⊥𝑧 < 0,
(iii) ∃𝜌 ∈ R : D − 𝜌B𝑇𝑧B𝑧 < 0,
(iv) ∃Q ∈ R2𝑛×𝑛 : D + QB𝑧 +B𝑇𝑧Q

𝑇 < 0,
where 𝜌 and Q are additional variables (or multipliers).

Proof. See [21].

Finsler’s Lemma is widely used in many control appli-
cations or theory of stability analysis based on LMIs. This
lemma ensures relaxation of the set of LMIs due to the
disassociation of matrices or reduction of control design of
the number of LMIs [22].

2.2. Controller Design with Stability Condition. Define the
following vectors and matrices:

W = [𝑥 (𝑡)�̇� (𝑡)] ,
D = [0 𝑃

𝑃 0] ,
Q = [𝑋𝑋] ,

B𝑧 = [𝐴𝑧 − [𝐼 + 𝐵𝑧𝐾𝑧]] ,

(8)

where𝑋 is a nonsingular matrix of appropriate dimensions.
From (8) and using items (i) and (iv) of Finsler’s Lemma,

in Theorem 2 sufficient conditions are proposed so that
system (5) is stabilizable.

The following theoremwas developed considering unique
Lyapunov matrix 𝑃, and the objective was to simplify the
methodology. In this case the derivative of parameter in
function of time was not included, knowing that ∑𝑁𝑖=1 �̇�𝑖(𝑡) =0.
Theorem2. Assuming𝐴𝑧 is invertible, if there exist symmetric
positive definite matrices 𝐺 ∈ R𝑛×𝑛, matrices 𝑍𝑖, 𝑍𝑗 ∈ R𝑚×𝑛

and 𝑄 ∈ R𝑛×𝑛, with 𝑖, 𝑗 ∈ K𝑟, such that

Υ𝑖𝑖 < 0, 𝑖 = 1, 2, . . . , 𝑁, (9)

Υ𝑖𝑗 + Υ𝑗𝑖 < 0, 1 ≤ 𝑖 < 𝑗 < 𝑁, (10)

Υ𝑖𝑗 = [𝐴 𝑖𝑄𝑇 + 𝑄𝐴𝑇𝑖 𝐺 + 𝑄𝐴𝑇𝑖 − 𝑄𝑇 − 𝐵𝑗𝑍𝑖∗ −𝑄𝑇 − 𝐵𝑖𝑍𝑗 − 𝑄 − 𝑍𝑇𝑗 𝐵𝑇𝑖 ] , (11)

where Υ𝑖𝑖 is a result of adopting 𝑗 = 𝑖. Then system (5) is
stabilizable and the gain matrices are given by

𝐾𝑧 = 𝑍𝑧𝑄−𝑇. (12)

Proof. Consider that (9) and (10) are feasible. Then, applying
Property 3, observe that the following inequality holds:

[𝐴𝑧𝑄𝑇 + 𝑄𝐴𝑇𝑧 𝐺 + 𝑄𝐴𝑇𝑧 − 𝑄𝑇 − 𝐵𝑧𝑍𝑧∗ −𝑄𝑇 − 𝐵𝑧𝑍 − 𝑄 − 𝑍𝑇𝑧𝐵𝑇𝑧] < 0. (13)

Replacing𝑄 = 𝑋−1, 𝐺 = 𝑋−1𝑃𝑋−𝑇, and 𝑍𝑧 = 𝐾𝑧𝑋−𝑇, the
following is obtained:



4 Mathematical Problems in Engineering

[
[
𝐴𝑧𝑋−𝑇 + 𝑋−1𝐴𝑇𝑧 𝑋−1𝑃𝑋−𝑇 + 𝑋−1𝐴𝑇𝑧 − 𝑋−𝑇 − 𝐵𝑧𝐾𝑧𝑋−𝑇

∗ −𝑋−𝑇 − 𝐵𝑧𝐾𝑧𝑋−𝑇 − 𝑋−1 − (𝐾𝑧𝑋−𝑇)𝑇 𝐵𝑇𝑧]] < 0. (14)

Multiplying (14) to the left for [𝑋 𝑋] and to the right[𝑋 𝑋]𝑇, one obtains
[𝑋𝐴𝑧 + 𝐴𝑇𝑧𝑋𝑇 𝑃 + 𝐴𝑇𝑧𝑋𝑇 − 𝑋 − 𝑋𝐵𝑧𝐾𝑧∗ −𝑋 − 𝑋𝐵𝑧𝐾𝑧 − 𝑋𝑇 − 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇]
< 0.

(15)

Separating in terms turns

[0 𝑃
𝑃 0] + [

𝑋𝐴𝑧 −𝑋 [𝐼 + 𝐵𝑧𝐾𝑧]𝑋𝐴𝑧 −𝑋 [𝐼 + 𝐵𝑧𝐾𝑧]]
+ [ 𝐴𝑇𝑧𝑋𝑇 𝐴𝑇𝑧𝑋𝑇− [𝐼 + 𝐾𝑇𝑧 𝐵𝑇𝑧 ]𝑋𝑇 − [𝐼 + 𝐾𝑇𝑧 𝐵𝑇𝑧 ]𝑋𝑇] < 0.

(16)

Themultiplication of matrix can be separated inmultipli-
cations of matrices:

[0 𝑃
𝑃 0] + [

𝑋
𝑋] [𝐴𝑧 − [𝐼 + 𝐵𝑧𝐾𝑧]]

+ [ 𝐴𝑇𝑧
− [𝐼 + 𝐵𝑧𝐾𝑧]𝑇] [𝑋

𝑇 𝑋𝑇] < 0,
(17)

then, inequality (17) has the form of Finsler’s Lemma:

W = [𝑥 (𝑡)�̇� (𝑡)] ,
D = [0 𝑃

𝑃 0] ,
Q = [𝑋𝑋] ,

B𝑧 = [𝐴𝑧 − [𝐼 + 𝐵𝑧𝐾𝑧]] .

(18)

It satisfies item (i) of Finsler’s Lemma, then there is a
matrix 𝑃 = 𝑃𝑇 > 0, accomplishing the Lyapunov conditions
for system (2), taking into account the gainmatrices (12), then
system (5) is asymptotically stable.

Figure 1 shows a set D for allocating the eigenvalues of
the system. It is limited for relations between the constraints
that meet the parametersDecay Rate (𝛾), Ratio (𝑟), and Angle
(𝜃).

2.3. Controller Designwith Stability Condition andDecay Rate.
Given a real constant 𝛾 > 0, one can impose a decay rate
constraint as shown in Figure 1, where 𝜆 are the eigenvalues
of closed-loop (5).

The decay rate constraint is imposed to the closed-loop
system (5), if condition (19) is satisfied for the entire trajectory𝑥(𝑡) ̸= 0 of the system, 𝑡 ≥ 0 [19].

�̇� (𝑥 (𝑡)) < −2𝛾𝑉 (𝑥 (𝑡)) , (19)

then we have

�̇� (𝑡)𝑇 𝑃𝑥 (𝑡) + 𝑥 (𝑡)𝑇 𝑃�̇� (𝑡) < −2𝛾𝑥 (𝑡)𝑇 𝑃𝑥 (𝑡) . (20)

From (5),

�̇� (𝑡) = (𝐼 + 𝐵𝑧𝐾𝑧)−1 𝐴𝑧𝑥 (𝑡) . (21)

Replacing (21) in (20), the consideration of the decay rate
is equivalent to solution of

𝐴𝑇𝑧 (𝐼 + 𝐵𝑧𝐾𝑧)−𝑇 𝑃 + 𝑃 (𝐼 + 𝐵𝑧𝐾𝑧)−1 𝐴𝑧 < −2𝛾𝑃,
𝑃 > 0. (22)

Considering Lemma 1, sufficient conditions for system (5)
to be stabilizable with restrictions on the decay rate 𝛾 > 0, and
defining the following vectors and matrices,

W = [𝑥 (𝑡)�̇� (𝑡)] ,
D = [2𝛾𝑃 𝑃

𝑃 0] ,
Q = [𝑋𝑋] ,

B𝑧 = [𝐴𝑧 − [𝐼 + 𝐵𝑧𝐾𝑧]] ,

(23)

where𝑋 is a nonsingular matrix of appropriate dimensions.
Using the definitions given in (23), inTheorem3 sufficient

conditions are proposed for system (5) to be stabilizable with
decay rate 𝛾 > 0.
Theorem3. Assuming𝐴𝑧 is invertible, if there exist symmetric
positive definite matrices 𝐺 ∈ R𝑛×𝑛, matrices 𝑍𝑖, 𝑍𝑗 ∈ R𝑚×𝑛

and 𝑄 ∈ R𝑛×𝑛, with 𝑖, 𝑗 ∈ K𝑟, such that

Υ𝛾𝑖𝑖 < 0, 𝑖 = 1, 2, . . . , 𝑁,
Υ𝛾𝑖𝑗 + Υ𝛾𝑗𝑖 < 0, 1 ≤ 𝑖 < 𝑗 < 𝑁,



Mathematical Problems in Engineering 5

r



Im()

Re()





Figure 1: RegionD-stability for pole placement [5].

Υ𝛾𝑖𝑗
= [2𝛾𝐺 + 𝐴 𝑖𝑄𝑇 + 𝑄𝐴𝑇𝑖 𝐺 + 𝑄𝐴𝑇𝑖 − 𝑄𝑇 − 𝐵𝑖𝑍𝑗∗ −𝑄𝑇 − 𝐵𝑖𝑍𝑗 − 𝑄 − 𝑍𝑇𝑗 𝐵𝑇𝑖 ] ,

(24)

where Υ𝛾𝑖𝑖 is a result of adopting 𝑗 = 𝑖. Then system (5) is
stabilizable, with decay rate greater than or equal to 𝛾, and the
matrices of controller are given by

𝐾𝑧 = 𝑍𝑧𝑄−𝑇. (25)

Proof. The demonstration follows steps similar to the proof
of Theorem 2, considering the condition of stability with
restriction of decay rate (22).

2.4. Controller Design with Stability Condition, Decay Rate,
and Less Conservative Solutions (Using 𝜅). In the previ-
ous subsection the controller design using Finsler’s Lemma
considering the decay rate presents solutions that can be
improved; given the empirical results, the value of 𝛾 is limited.
Applying a similar strategy, but incorporating a scalar value𝜅 > 0 into the inequalities, one can obtain less conservative
solutions, obviously with an adequate value of 𝜅 [5].

Considering Lemma 1 (Finsler’s Lemma), sufficient con-
ditions for system (2) to be stabilized with restrictions on the
decay rates 𝛾 > 0, 𝜅 > 0, and defining the following vectors
and matrices,

W = [𝑥 (𝑡)�̇� (𝑡)] ,
D = [2𝛾𝑃 𝑃

𝑃 0] ,
Q = [ 𝑋𝜅𝑋] ,

B𝑧 = [𝐴𝑧 − [𝐼 + 𝐵𝑧𝐾𝑧]] ,

(26)

where𝑋 is a nonsingular matrix of appropriate dimensions.

From the definitions given in (26), inTheorem4 sufficient
conditions are proposed for system (5) to be stabilizable with
decay rates 𝛾 > 0 and 𝜅 > 0.
Theorem 4. Assuming 𝐴𝑧 is invertible, for given 𝛾 and 𝜅, if
there exists a positive definite symmetric matrix 𝐺 ∈ R𝑛×𝑛,
matrices 𝑍𝑖, 𝑍𝑗 ∈ R𝑚×𝑛 and 𝑄 ∈ R𝑛×𝑛 such that

Υ𝛾𝜅𝑖𝑖 < 0, 𝑖 = 1, 2, . . . , 𝑁,
Υ𝛾𝜅𝑖𝑗 + Υ𝛾𝑗𝑖 < 0, 1 ≤ 𝑖 < 𝑗 < 𝑁,
Υ𝛾𝜅𝑖𝑗
= [
[
2𝛾𝐺 + 𝐴 𝑖𝑄𝑇 + 𝑄𝐴𝑇𝑖 𝐺 + 𝜅𝑄𝐴𝑇𝑖 − 𝑄𝑇 − 𝐵𝑗𝑍𝑖∗ −𝜅 (𝑄𝑇 + 𝐵𝑖𝑍𝑗 + 𝑄 + 𝑍𝑇𝑗 𝐵𝑇𝑖 )]] ,

(27)

where Υ𝛾𝜅𝑖𝑖 is a result of adopting 𝑗 = 𝑖. Then system (5) is
stabilizable, with decay rate greater than or equal to 𝛾, and the
matrices of controller are given by

𝐾𝑧 = 𝑍𝑧𝑄−𝑇. (28)

Proof. It is similar to the proof of Theorem 2.

2.5. Controller Design with Stability Condition
and Radius Restriction 𝑟

Theorem 5. Assuming 𝐴𝑧 is invertible, for a given 𝑟, if there
exists a symmetric positive defined matrix 𝐺 ∈ R𝑛×𝑛, matrices𝑍𝑖, 𝑍𝑗 ∈ R𝑚×𝑛 and𝑊 ∈ R𝑛×𝑛 such that

Υ𝑟𝑖𝑖 < 0, 𝑖 = 1, 2, . . . , 𝑁, (29)

Υ𝑟𝑖𝑗 + Υ𝑟𝑗𝑖 < 0, 1 ≤ 𝑖 < 𝑗 < 𝑁, (30)

Υ𝑟𝑖𝑗 =
[[[[[[[
[

(−𝑟𝐺 + 𝐴 𝑖𝑊𝑇+𝑊𝐴𝑇𝑖 ) (−𝑊𝑇 − 𝐵𝑖𝑍𝑗+𝑊𝐴𝑇𝑖 ) 0𝑛×𝑛
∗ (−𝑊𝑇 − 𝐵𝑖𝑍𝑗−𝑊 − 𝑍𝑇𝑗 𝐵𝑇𝑖 ) 𝐺
∗ ∗ −𝑟𝐺

]]]]]]]
]
, (31)

where Υ𝑟𝑖𝑖 is a result of adopting 𝑗 = 𝑖. Then system (5) is
stabilizable with eigenvalues inside the circle of radius 𝑟, for all𝛼(𝑡). The gain matrices of controller are given:

𝐾𝑧 = 𝑍𝑧𝑊−𝑇. (32)

Proof. From Property 3, if the LMIs (29)-(30) are feasible,
then the inequality Υ𝑟𝑧𝑧 < 0 holds. Replacing 𝐺 = 𝑋−1𝑃𝑋−𝑇,𝑊 = 𝑋−1, and𝑍𝑧 = 𝐾𝑧𝑋−𝑇 andmultiplyingΥ𝑟𝑧𝑧 to the left for
diag (𝑋,𝑋,𝑋,𝑋) and to the right for diag (𝑋𝑇, 𝑋𝑇, 𝑋𝑇, 𝑋𝑇),
the following is obtained:

[[[[[[[
[

(−𝑟𝑃 + 𝑋𝐴𝑧+𝐴𝑇𝑧𝑋𝑇 ) (−𝑋 + 𝐴𝑇𝑧𝑋𝑇−𝑋𝐵𝑧𝐾𝑧 ) 0𝑛×𝑛
∗ ( −𝑋 − 𝑋𝐵𝑧𝐾𝑧−𝑋𝑇 − 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇) 𝑃
∗ ∗ −𝑟𝑃

]]]]]]]
]
< 0. (33)



6 Mathematical Problems in Engineering

From −𝑋−𝑋𝑇−𝑋𝐵𝑧𝐾𝑧−𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇 < 0 ⇔ 𝑋(𝐼+𝐵𝑧𝐾𝑧)+(𝐼+𝐵𝑧𝐾𝑧)𝑇𝑋𝑇 > 0 and according to Property 1, conclude that𝑋(𝐼+𝐵𝑧𝐾𝑧) is invertible, then (𝐼+𝐵𝑧𝐾𝑧) and𝑋 are invertible.
From item (iv) of Lemma 1, inequality (33) can be rewritten
asD + QB𝑧 +B𝑇𝑧Q

𝑇 < 0, where

D = [[
[
−𝑟𝑃 0𝑛×𝑛 0𝑛×𝑛∗ 0𝑛×𝑛 𝑃
∗ ∗ −𝑟𝑃

]]
]
,

Q = [[
[
𝑋
𝑋
0𝑛×𝑛

]]
]

B𝑧 = [𝐴𝑧 − (𝐼 + 𝐵𝑧𝐾𝑧) 0𝑛×𝑛] .

(34)

Now consider that

B
⊥
𝑧 = [[[

[
𝐴−1𝑧 0𝑛×𝑛

(𝐼 + 𝐵𝑧𝐾𝑧)−1 0𝑛×𝑛0𝑛×𝑛 𝐼
]]]
]
. (35)

Note thatB𝑧B
⊥
𝑧 = 0𝑛×2𝑛. Then, from (34)-(35) and con-

sidering item (ii) of Lemma 1, the equivalence is established:

B
⊥𝑇

𝑧 DB
⊥
𝑧 = [

[
−𝐴−𝑇𝑧 𝑟𝑃𝐴−1𝑧 (𝐼 + 𝐵𝑧𝐾𝑧)−𝑇 𝑃

𝑃 (𝐼 + 𝐵𝑧𝐾𝑧)−1 −𝑟𝑃 ]
]

< 0,
(36)

multiplying (36) to the left for diag (𝐴𝑇𝑧, 𝐼) and to the right for
diag (𝐴𝑧, 𝐼), the following is obtained:

[ 𝑟𝑃 𝐴𝑇𝑁𝑃𝑃𝐴𝑁 𝑟𝑃 ] < 0, (37)

where

𝐴𝑁 = (𝐼 + 𝐵𝑧𝐾𝑧)−1 𝐴𝑧, (38)

and the proof is completed.

In the next section, the LMIs conditions are proposed for
inclusion of the restriction of the angle ofD-stability.

2.6. Controller Design with Stability Condition
and Angle Restriction 𝜃

Theorem 6. Assuming 𝐴𝑧 is invertible, for a given 𝜃, if there
exists a symmetric positive defined matrix 𝐺 ∈ R𝑛×𝑛, matrices𝑍𝑖, 𝑍𝑗 ∈ R𝑚×𝑛 and𝑊 ∈ R𝑛×𝑛 such that

Υ𝜃𝑖𝑖 < 0, 𝑖 = 1, 2, . . . , 𝑁, (39)

Υ𝜃𝑖𝑗 + Υ𝜃𝑗𝑖 < 0, 1 ≤ 𝑖 < 𝑗 < 𝑁, (40)

Υ𝜃𝑖𝑗 = [𝑀𝑖𝑗11 𝑀𝑖𝑗12 𝑀𝑖𝑗13 𝑀𝑖𝑗14] ,

𝑀𝑖𝑗11 =
[[[[[[[
[

sin (𝜃) (𝐴 𝑖𝑊𝑇 +𝑊𝐴𝑇𝑖 )
sin (𝜃) (𝐺 −𝑊 + 𝐴 𝑖𝑊𝑇 − 𝑍𝑇𝑗 𝐵𝑇𝑖 )
cos (𝜃) (𝐺 + 𝐴 𝑖𝑊𝑇 +𝑊 + 𝑍𝑇𝑗 𝐵𝑇𝑖 )

cos (𝜃) (𝐴 𝑖𝑊𝑇 −𝑊𝐴𝑇𝑖 )

]]]]]]]
]
,

𝑀𝑖𝑗12 =
[[[[[[[
[

sin (𝜃) (𝐺 −𝑊𝑇 +𝑊𝐴𝑇𝑖 − 𝐵𝑖𝑍𝑗)
sin (𝜃) (−𝑊𝑇 − 𝐵𝑖𝑍𝑗 −𝑊 − 𝑍𝑇𝑗 𝐵𝑇𝑖 )
cos (𝜃) (−𝑊𝑇 − 𝐵𝑖𝑍𝑗 +𝑊 + 𝑍𝑇𝑗 𝐵𝑇𝑖 )
cos (𝜃) (−𝐺 −𝑊𝑇 − 𝐵𝑖𝑍𝑗 −𝑊𝐴𝑇𝑖 )

]]]]]]]
]
,

𝑀𝑖𝑗13 =
[[[[[[[
[

cos (𝜃) (𝐺 +𝑊𝑇 +𝑊𝐴𝑇𝑖 + 𝐵𝑖𝑍𝑗)
cos (𝜃) (𝑊𝑇 + 𝐵𝑖𝑍𝑗 −𝑊 − 𝑍𝑇𝑗 𝐵𝑇𝑖 )
sin (𝜃) (−𝑊𝑇 − 𝐵𝑖𝑍𝑗 −𝑊 − 𝑍𝑇𝑗 𝐵𝑇𝑖 )
sin (𝜃) (𝐺 −𝑊𝑇 − 𝐵𝑖𝑍𝑗 +𝑊𝐴𝑇𝑖 )

]]]]]]]
]
,

𝑀𝑖𝑗14 =
[[[[[[[
[

cos (𝜃) (−𝐴 𝑖𝑊𝑇 +𝑊𝐴𝑇𝑖 )
cos (𝜃) (−𝐺 − 𝐴 𝑖𝑊𝑇 −𝑊 − 𝑍𝑇𝑗 𝐵𝑇𝑖 )
sin (𝜃) (𝐺 + 𝐴 𝑖𝑊𝑇 −𝑊 − 𝑍𝑇𝑗 𝐵𝑇𝑖 )

sin (𝜃) (𝐴 𝑖𝑊𝑇 +𝑊𝐴𝑇𝑖 )

]]]]]]]
]
,

(41)

where Υ𝑟𝑖𝑖 is a result of adopting 𝑗 = 𝑖. Then system (5) is
stabilizable with eigenvalues, inside of angle 𝜃, for all 𝛼(𝑡). The
gain matrices of controller are given by

𝐾𝑧 = 𝑍𝑧𝑊−𝑇. (42)

Proof. From Property 3, if the LMIs (39)-(40) are feasible,
then the inequality Υ𝜃𝑧𝑧 < 0 holds. Replacing 𝐺 = 𝑋−1𝑃𝑋−𝑇,𝑊 = 𝑋−1, and𝑍𝑧 = 𝐾𝑧𝑋−𝑇 andmultiplyingΥ𝜃𝑧𝑧 to the left for
diag(𝑋,𝑋,𝑋,𝑋) and to the right for diag(𝑋𝑇, 𝑋𝑇, 𝑋𝑇, 𝑋𝑇),
the following is obtained:

[𝑀21 𝑀22 𝑀23 𝑀24] < 0, (43)

where

𝑀21 =
[[[[[[[
[

sin (𝜃) (𝑋𝐴𝑧 + 𝐴𝑇𝑧𝑋𝑇)
sin (𝜃) (𝑃 + 𝑋𝐴𝑧 − 𝑋𝑇 − 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇)
cos (𝜃) (𝑃 + 𝑋𝐴𝑧 + 𝑋𝑇 + 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇)

cos (𝜃) (𝑋𝐴𝑧 − 𝐴𝑇𝑧𝑋𝑇)

]]]]]]]
]
,



Mathematical Problems in Engineering 7

𝑀22 =
[[[[[[
[

sin (𝜃) (𝑃 − 𝑋 + 𝐴𝑇𝑧𝑋𝑇 − 𝑋𝐵𝑧𝐾𝑧)
sin (𝜃) (−𝑋 − 𝑋𝐵𝑧𝐾𝑧 − 𝑋𝑇 − 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇)
cos (𝜃) (−𝑋 − 𝑋𝐵𝑧𝐾𝑧 + 𝑋𝑇 + 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇)
cos (𝜃) (−𝑃 − 𝑋 − 𝑋𝐵𝑧𝐾𝑧 − 𝐴𝑇𝑧𝑋𝑇)

]]]]]]
]
,

𝑀23 =
[[[[[[
[

cos (𝜃) (𝑃 + 𝑋 + 𝐴𝑇𝑧𝑋𝑇 + 𝑋𝐵𝑧𝐾𝑧)
cos (𝜃) (𝑋 + 𝑋𝐵𝑧𝐾𝑧 − 𝑋𝑇 − 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇)
sin (𝜃) (−𝑋 − 𝑋𝐵𝑧𝐾𝑧 − 𝑋𝑇 − 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇)

sin (𝜃) (𝑃 − 𝑋 − 𝑋𝐵𝑧𝐾𝑧 + 𝐴𝑇𝑧𝑋𝑇)

]]]]]]
]
,

𝑀24 =
[[[[[[
[

cos (𝜃) (−𝑋𝐴𝑧 + 𝐴𝑇𝑧𝑋𝑇)
cos (𝜃) (−𝑃 − 𝑋𝐴𝑧 − 𝑋𝑇 − 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇)
sin (𝜃) (𝑃 + 𝑋𝐴𝑧 − 𝑋𝑇 − 𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇)

sin (𝜃) (𝑋𝐴𝑧 + 𝐴𝑇𝑧𝑋𝑇)

]]]]]]
]
.

(44)

From −𝑋−X𝑇−𝑋𝐵𝑧𝐾𝑧 −𝐾𝑇𝑧 𝐵𝑇𝑧𝑋𝑇 < 0 ⇔ 𝑋(𝐼+𝐵𝑧𝐾𝑧)+(𝐼+𝐵𝑧𝐾𝑧)𝑇𝑋𝑇 > 0 and according to Property 1, conclude that𝑋(𝐼+𝐵𝑧𝐾𝑧) is invertible, then (𝐼+𝐵𝑧𝐾𝑧) and𝑋 are invertible.
From item (iv) of Lemma 1, inequality (43) can be rewritten
asD + QB𝑧 +B𝑇𝑧Q

𝑇 < 0, where

D = [[[[[
[

0𝑛×𝑛 sin (𝜃) 𝑃 cos (𝜃) 𝑃 0𝑛×𝑛∗ 0𝑛×𝑛 0𝑛×𝑛 − cos (𝜃) 𝑃
∗ ∗ 0𝑛×𝑛 sin (𝜃) 𝑃
∗ ∗ ∗ 0𝑛×𝑛

]]]]]
]
,

Q = [[[[[
[

sin (𝜃)𝑋 − cos (𝜃)𝑋
sin (𝜃)𝑋 − cos (𝜃)𝑋
cos (𝜃)𝑋 sin (𝜃)𝑋
cos (𝜃)𝑋 sin (𝜃)𝑋

]]]]]
]
,

B𝑧 = [ 𝐴𝑧 − (𝐼 + 𝐵𝑧𝐾𝑧) 0𝑛×𝑛 0𝑛×𝑛0𝑛×𝑛 0𝑛×𝑛 − (𝐼 + 𝐵𝑧𝐾𝑧) 𝐴𝑧 ] .

(45)

Now consider that

B
⊥𝑇

𝑧 = [
[
𝐴−1𝑧 (𝐼 + 𝐵𝑧𝐾𝑧)−1 0𝑛×𝑛 0𝑛×𝑛
0𝑛×𝑛 0𝑛×𝑛 (𝐼 + 𝐵𝑧𝐾𝑧)−1 𝐴−1𝑧 ]] . (46)

Note that B𝑧B
⊥
𝑧 = 02𝑛×2𝑛. Then, from (45)-(46)

and considering item (ii) of Lemma 1, the equivalence is
established:

B
⊥𝑇

𝑧 DB
⊥
𝑧 =

[[[[[[
[

( sin (𝜃) (𝐼 + 𝐵𝑧𝐾𝑧)−𝑇 𝑃𝐴−1𝑧
+sin (𝜃) 𝐴−𝑇𝑧 𝑃 (𝐼 + 𝐵𝑧𝐾𝑧)−1) ( cos (𝜃) 𝐴−𝑇𝑧 𝑃 (𝐼 + 𝐵𝑧𝐾𝑧)−1

−cos (𝜃) (𝐼 + 𝐵𝑧𝐾𝑧)−𝑇 𝑃𝐴−1𝑧 )
( cos (𝜃) 𝐴−𝑇𝑧 𝑃 (𝐼 + 𝐵𝑧𝐾𝑧)−1
+cos (𝜃) (𝐼 + 𝐵𝑧𝐾𝑧)−𝑇 𝑃𝐴−1𝑧 ) ( sin (𝜃) 𝐴−𝑇𝑧 𝑃 (𝐼 + 𝐵𝑧𝐾𝑧)−1

+sin (𝜃) (𝐼 + 𝐵𝑧𝐾𝑧)−𝑇 𝑃𝐴−1𝑧 )
]]]]]]
]
< 0. (47)

Multiplying (47) to the left for diag (𝐴𝑇𝑧, 𝐴𝑇𝑧) and to the
right for diag (𝐴𝑧, 𝐴𝑧), the following is obtained:

[
[
sin (𝜃) (𝐴𝑇𝑁𝑃 + 𝑃𝐴𝑁) cos (𝜃) (𝑃𝐴𝑁 − 𝐴𝑇𝑁𝑃)
cos (𝜃) (𝐴𝑇𝑁𝑃 − 𝑃𝐴𝑁) sin (𝜃) (𝑃𝐴𝑁 + 𝐴𝑇𝑁𝑃)]]
< 0;

(48)

where

𝐴𝑁 = (𝐼 + 𝐵𝑧𝐾𝑧)−1 𝐴𝑧, (49)

then the proof is completed.

3. Active Suspension System of a Vehicle

The laboratory of active suspension, built by Quanser, can
be represented in Figure 2. The schematic model of active
suspension system is presented in Figure 3.

The system has a set consisting of two masses, called𝑀𝑠
and𝑀𝑢𝑠.Themass𝑀𝑠 represents 1/4 of the total vehicle body
and is supported by the spring 𝑘𝑠 and the damper 𝑏𝑠.Themass𝑀𝑢𝑠 corresponds to the mass of the vehicle and tire assembly
is supported by the spring 𝑘𝑢𝑠 and by the damper 𝑏𝑢𝑠.

In order to diminish the vibrations caused by irregu-
larities in the track the active suspension system is used,
represented by a motor (actuator) connected between the
masses𝑀𝑠 and𝑀𝑢𝑠 and controlled by force 𝐹𝑐. In Table 1 the
values of parameters are represented.

The schematic model [23] can be represented in state
space, like what is shown in

�̇� (𝑡) =
[[[[[[[[[
[

0 1 0 −1−𝑘𝑠𝑀𝑠
−𝑏𝑠𝑀𝑠 0 𝑏𝑠𝑀𝑠0 0 0 1

−𝑘𝑠𝑀𝑢𝑠
𝑏𝑠𝑀𝑢𝑠

−𝑘𝑢𝑠𝑀𝑢𝑠
− (𝑏𝑠 + 𝑏𝑢𝑠)𝑀𝑢𝑠

]]]]]]]]]
]
𝑥 (𝑡)

+
[[[[[[[[
[

01𝑀𝑠0−1𝑀𝑢𝑠

]]]]]]]]
]
𝑢 (𝑡) ,



8 Mathematical Problems in Engineering

Table 1: Parameters of the active suspension.

Symbol Value𝑀𝑠 2.45 (kg)𝑀𝑢𝑠 1 (kg)𝐾𝑠 900 (N/m)𝑘𝑢𝑠 2500 (N/m)𝑏𝑠 7.5 (Ns/m)𝑏𝑢𝑠 5 (Ns/m)

Figure 2: Active suspension system of the Quanser, belonging to
LPC-FEIS-UNESP.

𝑥 (𝑡) = [[[[[
[

𝑧𝑠 (𝑡) − 𝑧𝑢𝑠 (𝑡)�̇�𝑠 (𝑡)𝑧𝑢𝑠 (𝑡) − 𝑧𝑟 (𝑡)�̇�𝑢𝑠 (𝑡)
]]]]]
]
,

𝑦 (𝑡) = [1 0 0 0
0 0 1 0] 𝑥 (𝑡) ,

(50)

where 𝑥2(𝑡) = �̇�𝑠(𝑡) and 𝑥4(𝑡) = �̇�𝑢𝑠(𝑡), representing the
respective velocities of each mass.

For accomplishing the control design of active suspen-
sion, it was considered that the system has the amplified
control signal, producing a variation on the input vector
matrix. This variation is given taking into account the fault
of the 40% of the amplified control signal. An actuator fault
can be described as politopic uncertainties.

Therefore, to perform the control project, the following
vertices of the polytope were considered:

𝐴1 = 𝐴2 =
[[[[[
[

0 1 0 −1
−367.347 −3.061 0 3.061

0 0 0 1
900 7.5 −2500 −12.5

]]]]]
]
,

𝐵1 = [0 −0.408 0 −1]𝑇 ,
𝐵2 = [0 −0.244 0 −0.6]𝑇 .

(51)

(Road)

Active suspension

Tire

zs(t)

zus(t)

Accelerometer (⇒z̈s(t))

kus

ks

bus

zr(t)

Ms →

bs

Accelerometer (⇒ z̈us(t))

Mus → tire assembly mass

Fc

1/4 the vehicle mass

Figure 3: Schematic model of active suspension system.

−10
−8
−6
−4
−2

0
2
4
6
8

10

El
em

en
ts 

of
K

z
(

(t
))

13 14 15 16 17 18 19 20 2112
Time (s)

K1((t))/100

K2((t))

K3((t))/100

K4((t))

Figure 4: Elements of 𝐾𝑧(𝛼(𝑡)) as function of time.

4. Practical Implementation

In this section, the practical implementation and results will
be presented, with which it is possible to make a detailed
analysis of the technique implemented.

For designing the values of the gain matrix, it was
necessary to solve the LMI presented inTheorem 2:

𝐾𝑑1 = [381.9988 5.6279 −955.6624 −3.5189] ,
𝐾𝑑2 = [526.7000 7.8000 −1320.0000 −4.9000] , (52)

and the value of 𝐾𝑧(𝛼(𝑡)) shown in Figure 4 is given by the
sum of the products of two sinusoidal functions 𝛼1(𝑡) shown
in Figure 5 and 𝛼2(𝑡) which is the complement of the unit
simplex.

The first practical implementation of the reference signal𝑧𝑟 was chosen to reproduce a square wave 0.02m amplitude,
frequency 1/3Hzpulse width 50%. In the range of 0 to 12 s the
system is open loop and in the range of 12.01 to 21 s the system
is closed-loop. It is highlighted that in closed-loop used the
control law is given by 𝑢 = 𝐾𝑧(𝛼(𝑡))�̇�.

Figure 4 illustrates𝐾𝑧(𝛼(𝑡)) along time.



Mathematical Problems in Engineering 9

Over a period
Closed-loop system

20 40 60 80 1000
Time (s)

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

1


1
(t
)

Figure 5: Sinusoidal function 𝛼1(𝑡).

Road
Chassis
Seat

−0.02

−0.01

0

0.01

0.02

0.03

0.04

D
isp

la
ce

m
en

t (
m

)

2 4 6 8 10 12 14 16 18 200
Time (s)

Figure 6: Transient response practice open loop (0–12 s) and closed-
loop (12.01–21 s).

In this case the controller is given by

𝐾𝑧 (𝛼 (𝑡)) = 𝛼1 (𝑡) 𝐾𝑑1 + (1 − 𝛼1 (𝑡))𝐾𝑑2 , (53)

where the variation of 𝛼1(𝑡) is illustrated in Figure 5, which
highlights the period in which the system is closed-loop. As
shown in Figure 6, the total time of the suspension drive was
21 s, so it is possible to make the data acquisition to results
analysis and display in a single graph.

Figure 10 shows how the open loop system tends to sta-
bility, although it has fluctuations that may affect the driver’s
comfort. As estimated, the closed-loop system ensures a good
performance to the system, reducing the oscillations and the
amplitude of the driver’s seat displacement.

4.1. Derivate State Feedback Using D-Stability Constraints.
The values of parameters 𝛾, 𝑟, and 𝜃 were obtained consider-
ing the normof controller gains. It was given for the allocation
of eigenvalues.

K1((t))/100

K2((t))

K3((t))/100

K4((t))

14 16 18 20 22 24 26 28 3012
Time (s)

−10

−5

0

5

10

15

20

25

30

35

40

El
em

en
ts 

of
K

z
(

(t
))

Figure 7: Elements of 𝐾𝑧(𝛼(𝑡)) as a function of time.

−50
−40
−30
−20
−10

0
10
20
30
40
50

Im
ag

in
ar

y 
ax

is 
(

)

−60 −50 −40−70 −20 −10−30 0
Real axis ()

Figure 8: Placement of system eigenvalue cloud.

Using the set of LMIs of Theorems 4, 5, and 6 with 𝛾 =1.11, 𝜅 = 0.9, 𝑟 = 70, 𝜃 = 45∘, and parameter 𝛼(𝑡) varying
with frequency of 0.01Hz, the calculated gains were

𝐾𝑑1 = [292.7180 24.5552 −466.9545 1.5677] ,
𝐾𝑑2 = [486.0575 40.8744 −787.9186 2.4722] . (54)

The frequency of𝛼(𝑡)was adopted equal to 0.01Hz; in this
case the time evolution of the elements of the controller is
shown in Figure 7 to 30 seconds.

Figure 8 shows that the eigenvalues are insideD-stability
set for variations of 0 ≤ 𝛼(𝑡) ≤ 1.

In Figure 9, it is noticed that the system presents a good
dynamic performancewhich decreases overshoot of transient
response, showing the efficiency of the new methodology.

In [24], there is no limitation of the gain of feedback
which makes its practical implementation difficult; therefore
this new proposal is more comprehensive.

4.2. Comparison Analysis. Figure 10 represents the transitory
response considering the methodology proposed in [5].



10 Mathematical Problems in Engineering

Road
Chassis
Seat

5 10 15 20 25 300
Time (s)

−0.02

−0.01

0

0.01

0.02

0.03

0.04

D
isp

la
ce

m
en

t (
m

)

Figure 9: Transient response of open loop (0–12 s) and closed-loop
(12.01–30 s).

5 10 15 20 25 300
Time (s)

−0.02

−0.01

0

0.01

0.02

0.03

0.04

D
isp

la
ce

m
en

t (
m

)

Figure 10: Transient response practice open loop (0–12 s) and
closed-loop (12.01–30 s).

The system was subject to the same conditions showed in
the practical implementation. We can observe that the red
response belonging to the chassis is affected by influence of
the parametric variable in time function 𝛼(𝑡). The controller
does not get to eliminate this oscillations; however the
designed controller has a good performance for any moment
time.We can conclude that, for high frequency, the controller
will have difficulty for guaranteeing good performance of
system.

5. Conclusions

In this work a methodology was proposed to design a state
derivative feedback robust gain scheduling controller that
stabilizes an LTI system. In the practical implementation, the
efficiency of the new technique presented was verified. The
formulations of LMI with D-stability constraints design the
controller without having to invert a matrix dependent on
parameter 𝛼(𝑡); it is an advantage. The presented technique

represented a useful tool for linear systems with derivative
feedback subject to faults. The method D-stability for allo-
cating the eigenvalues allowed concluding the antagonistic
nature of the system between the state derivative and state
feedback showing how the value of the controller gains can
decrease or increase, depending on how near or far from the
imaginary axis the eigenvalues are.

Conflicts of Interest

The authors declare that there are no conflicts of interest
regarding the publication of this paper.

Acknowledgments

The authors would like to thank the Brazilian agencies
CAPES, CNPq, and FAPESP (no. 2011/17610-0) which have
supported this research.

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