Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1512679, 11 pages

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

## Design of Gain Scheduling Control Using State Derivative Feedback

^{1}Department 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^{2}Federal Institute of Education, Science and Technology of Mato Grosso do Sul (IFMS), Campus of Três Lagoas, 79.641-162 Três Lagoas, MS, Brazil^{3}Academic 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

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.

#### Abstract

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. The -*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 feedback; 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 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 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 behavior. 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 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 invariant 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].

The main motivation 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. Our main 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: , ; represents a block-diagonal matrix in which the diagonal elements are , ; is used to represent positive definite matrices and, equivalently, is used to represent negative definite matrices; denotes a vector or matrix transposition; denotes the transposition of an inverted matrix; denotes the symmetric matrix.

*Property 1 (see [19]). *For all nonsymmetric matrix , if , then is invertible.

*Property 2. *One hassuch that is a time-varying parameter. The condition was used to obtain the convexity of the set and it was described by polytope vertices. The variable from 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, will be denoted by , similarly by .

*Property 3 (see [20]). *If the following LMIs, for and for , are feasible, then the inequality holds, where .

#### 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 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 bywhere and are matrices that represent the time variant systems dynamics, is the state vector, and is the control input vector.

Using the compact notation of Property 2, (2) can be rewritten as

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

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

If has null eigenvalue, then the feedback system, given by , has null eigenvalue. The controller can not modify the null eigenvalue of , in this case .

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

To obtain the LMIs for the design of the controllers, Lemma 1 is used. Using the notation of Property 2, consider that .

Lemma 1 (Finsler’s Lemma). *Considering that , and with and is a base for the null space of (i.e., ).**So the next conditions are equivalent: *(i)*,*(ii)*,*(iii)*,*(iv)*,** where and are additional variables (or multipliers).*

*Proof. *See [21].

*Finsler’s Lemma is widely used in many control applications 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:where 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 theorem was developed considering unique Lyapunov matrix , and the objective was to simplify the methodology. In this case the derivative of parameter in function of time was not included, knowing that .*

*Theorem 2. Assuming is invertible, if there exist symmetric positive definite matrices , matrices , and , with , such thatwhere is a result of adopting . Then system (5) is stabilizable and the gain matrices are given by*

*Proof. *Consider that (9) and (10) are feasible. Then, applying Property 3, observe that the following inequality holds:Replacing , , and , the following is obtained:Multiplying (14) to the left for and to the right , one obtainsSeparating in terms turnsThe multiplication of matrix can be separated in multiplications of matrices:then, inequality (17) has the form of Finsler’s Lemma:It satisfies item (i) of Finsler’s Lemma, then there is a matrix , accomplishing the Lyapunov conditions for system (2), taking into account the gain matrices (12), then system (5) is asymptotically stable.

*Figure 1 shows a set for allocating the eigenvalues of the system. It is limited for relations between the constraints that meet the parameters Decay Rate (), Ratio (), and Angle ().*