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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 205939, 17 pages
http://dx.doi.org/10.1155/2011/205939
Research Article

Analysis of Exact Linearization and Aproximate Feedback Linearization Techniques

1Universidade Federal do Recôncavo da Bahia, Rua Rui Barbosa, No. 710, CEP 44.380-000—Cruz das Almas, BA, Brazil
2Post Graduate Program in Mechatronics, Universidade Federal da Bahia, Rua Aristides Novis No. 2, Federação, CEP 40.210—630—Salvador, BA, Brazil

Received 19 October 2010; Accepted 3 March 2011

Academic Editor: Bin Liu

Copyright © 2011 Gildeberto S. Cardoso and Leizer Schnitman. 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

This paper presents a study of linear control systems based on exact feedback linearization and approximate feedback linearization. As exact feedback linearization is applied, a linear controller can perform the control objectives. The approximate feedback linearization is required when a nonlinear system presents a noninvolutive property. It uses a Taylor series expansion in order to compute a nonlinear transformation of coordinates to satisfy the involutivity conditions.

1. Introduction

The control theory of nonlinear system has been receiving increasing attention in recent years, both for its technical importance as well as for its impact in various fields of application. In strategic areas, such as aerospace, chemical and petrochemical industries, bioengineering, and robotics, a new practical application for this tool appears every day.

System nonlinearity is characterized when at least one component or subsystem is nonlinear [1]. The classic methods used in the study of nonlinear systems, particularly frequency analysis, are not usually applied to the nonlinear systems. It is necessary to use other methods to study the control of these systems.

When the nonlinear effects of systems become significant, nonlinear control techniques generally fail to produce the desired performance. In these cases, there are several nonlinear control techniques that have proven to supply better results.

Among the possible techniques used to deal with the nonlinear control problem, this paper studies the approximate feedback linearization technique for controlling nonlinear systems. The applications of these techniques are illustrated through some examples.

2. Analysis of Necessary and Sufficient Conditons for Exact Feedback Linearization

The objective of this section is to detail the necessary and sufficient conditions for input-state linearization. These conditions are limiting to the implementation of the exact feedback linearization technique. However, as a solution to this problem, the approximate linearization technique will be described.

Initially, we consider a single input 𝑢(𝑡) for a nonlinear system, given bẏ𝐱=𝐟(𝐱)+𝐠(𝐱)𝑢.(2.1) This system is known as linear in the control and linear in the input, among other denominations. The functions 𝐟(𝐱) and 𝐠(𝐱) are 𝐶 vectors with field in ̇𝐱𝑅𝑛.

The system in the form (2.1) is input-state linearizable if there is a region Ω in 𝑅𝑛 that meets the following conditions:(i) the vector field {𝐠,𝑎𝑑𝐟𝐠,,𝐚𝐝𝐟𝑛1𝐠} is linearly independents in Ω,(ii) the set of vectors [𝑎𝑑𝑖𝐟𝐠,𝑎𝑑𝑗𝐟𝐠]=span{𝐠,𝑎𝑑𝐟𝐠,,𝑎𝑑𝐟𝑛2𝐠} for 𝑖,𝑗=0,1,2,,𝑛2 is involutive,

where [𝑎𝑑𝑖𝐟𝐠,𝑎𝑑𝑗𝐟𝐠] indicates the Lie bracket and 𝑎𝑑𝑖𝐟𝐠 is defined as [2]:𝑎𝑑𝑖𝐟𝐠=𝐠,𝑎𝑑𝐟𝑖1𝐠.(2.2) The first condition indicates the system controllability in which the vector fields {𝐠,𝑎𝑑𝐟𝐠,,𝑎𝑑𝐟𝑛1𝐠} are equivalent to the controllability matrix for linear systems [𝐛𝐀𝐛𝐀2𝐛𝐀𝑛1𝐛] [1].

The second involutivity condition enables to find out a new vector of linear state through the states feedback.

3. Exact Feedback Linearization

If the aforementioned conditions were satisfied, then it is possible to find a scalar function 𝜙(𝐱) such that𝜕𝜙(𝐱)𝜕𝐱𝑎𝑑𝑖𝐟𝐠=0,𝑖=0,1,2,,𝑛2,(3.1)𝜕𝜙(𝐱)𝜕𝐱𝑎𝑑𝐟𝑛1𝐠0,(3.2) where 𝜙(𝐱) defines a state transformation 𝐳 given by [1]𝐿𝐳=𝑇(𝐱)=0𝐟𝜙(𝐱)𝐿1𝐟𝜙(𝐱)𝐿𝐟𝑛1𝜙(𝐱).(3.3) A control input is then obtained through𝑢=𝛼(𝐱)+𝛽(𝐱)𝑣=𝐿𝑛𝐟𝜙(𝐱)𝐿𝐠𝐿𝐟𝑛1+1𝜙(𝐱)𝐿𝐠𝐿𝐟𝑛1𝜙(𝐱)𝑣,(3.4) such that the closed-loop system has a new set of variables in linear statė𝐳=𝐀𝐳+𝐛𝑣,(3.5) where0001𝐀=0100001000010000,𝐛=.(3.6)𝐀 and 𝐛 are in the canonical form of Brunovski however, there is no loss of generality; as every linear system may be written in the companion form by a state transformation [1].

3.1. Inverted Pendulum Linearization

Figure 1 shows an overview of a single pendulum. The pendulum may be described as a fixed end connected to a sphere through an inflexible rod and can be swung when stimulated by a force 𝑢 or has a nonzero initial condition.

205939.fig.001
Figure 1: Simple pendulum.

The model of a pendulum is given bẏ𝑥1=𝑥2,̇𝑥2𝑔=𝑙𝑥sin1𝑏𝑚𝑥2+1𝑚𝑙𝑢,(3.7) where 𝑔, 𝑙, 𝑚, 𝑏, 𝑥1, and  𝑥2  are, respectively, gravitational acceleration, length of rod, mass of the sphere, friction coefficient, rotation angle, and angular speed.

3.1.1. Calculation of Controllability and Involutivity of the Simple Pendulum

As it is a second-order system, the vector field for verifying the controllability will be formed by{𝐠,𝑎𝑑𝐟𝐠}, where,01𝐠=,𝑚𝑙𝑎𝑑𝐟𝑥𝐠=00002𝑔𝑙𝑥sin1𝑏𝑚𝑥2𝑔01𝑙𝑥cos1𝑏𝑚01=1𝑚𝑙𝑏𝑚𝑙𝑚2𝑙.(3.8) Hence, the vector field {𝐠,𝑎𝑑𝐟𝐠} will be given by𝐠,𝑎𝑑𝐟𝐠=101𝑚𝑙𝑏𝑚𝑙𝑚2𝑙.(3.9) The vector fields are linearly independent; therefore, the simple pendulum is controllable.

In order to calculate the involutivity of the system, it is necessary to calculate the Lie bracket of {𝐠,𝑎𝑑𝐟𝐠} and verify if it may be written as a linear combination of 𝐠 and 𝑎𝑑𝐟𝐠𝐠,𝑎𝑑𝐟𝐠=0000.(3.10) The Lie bracket of {𝐠,𝑎𝑑𝐟𝐠} is equal to zero, as the column vectors of the matrix are constant. Therefore, the system is involutive, which satisfy the conditions to apply input-state linearization.

3.1.2. Calculation of the New State Vector and the Input Variable Using Differential Geometry

The first component of the state vector 𝑇1 should be obtained through the expression (3.1). As the order of the system is equal to 2, equation 𝜕𝑇1 becomes𝜕𝑇1𝜕𝑥1𝜕𝑇1𝜕𝑥201𝑚𝑙=0,(3.11) resulting in1𝑚𝑙𝜕𝑇1𝜕𝑥2=0.(3.12) Therefore,𝜕𝑇1𝜕𝑥2=0.(3.13) From (3.2),𝜕𝑇1𝜕𝑥1𝜕𝑇1𝜕𝑥21𝑏𝑚𝑙𝑚2𝑙0,(3.14) solving1𝑚𝑙𝜕𝑇1𝜕𝑥10.(3.15) Therefore,𝜕𝑇1𝜕𝑥10.(3.16) In order to satisfy the last equation, 𝑇1=𝑥1 has to be chosen.

We use (3.3) to compute the second component of the new state vector 𝑇2𝑇2=𝐿𝐟𝑇1=𝜕𝑇1𝜕𝑥1𝑥2𝐠𝑙𝑥sin1𝑏𝑚𝑥2=𝑥102𝐠𝑙𝑥sin1𝑏𝑚𝑥2=𝑥2.(3.17) This gives the new state vector𝑇𝐓=1𝑇2=𝑥1𝑥2.(3.18) The new input variable is calculated from (3.4), where𝑥𝛼(𝐱)=(𝑔/𝑙)sin1(𝑏/𝑚)𝑥2𝑥1/𝑚𝑙=𝑚𝑔sin1+𝑙𝑏𝑥2,1𝛽(𝐱)=1/𝑚𝑙=𝑚𝑙.(3.19) Therefore,𝑥𝑢=𝑚𝑔sin1+𝑙𝑏𝑥2𝑚𝑙𝑣.(3.20) This results in a linear systeṁ𝑇1̇𝑇2=𝑇2𝑣.(3.21) The system block diagram may be represented as shown in Figure 2.

205939.fig.002
Figure 2: Control structure.

The control structure, according to Figure 2, has two feedbacks. The first feedback is responsible for system linearization, eliminating the existing linearity. The second feedback is the controller project based on the state feedback.

3.2. The Need for Differential Geometry to Calculate the Exact Linearization

The linearization result shown previously might be obtained directly from1𝑢=𝐠(𝐱)(𝐟(𝐱)+𝑣).(3.22) When this procedure is applied, the linearization calculation would be significantly less. The problem occurs when the control law obtained by this means does not permit the exact linearization.

As an example of the aforementioned problem, consider a nonlinear system given by𝐟(𝐱)=3𝑥1+2𝑥2+𝑒𝑥1𝑥2𝑒𝑥10𝑥,(3.23)𝐠(𝐱)=1.(3.24) Applying (3.22), we will obtain:1𝑢=𝑥1𝑥2𝑒𝑥1+𝑣.(3.25) This result does not make equation (3.23) a linear system. It requires a variable transformation by inference or by differential geometry. An approach through differential geometry is more generic and can be implemented in software through simple programming.

3.2.1. The Use of Differential Geometry to Linearize the Previous Example

The first component of the state vector 𝑇(𝐱) is obtained through𝑇1(𝐱)=𝑎𝑑0𝐟(𝐱)𝐠(𝐱)=0.(3.26) As 𝑎𝑑0𝐟(𝐱)𝐠(𝐱)=0𝑥1, we will have𝜕𝑇1(𝐱)𝜕𝑥1𝜕𝑇1(𝐱)𝜕𝑥20𝑥1=0.(3.27) Therefore,𝜕𝑇1(𝐱)𝜕𝑥2=0.(3.28) If 𝑎𝑑𝐟(𝐱)𝐠(𝐱)=2𝑥13𝑥1+2𝑥2+𝑒𝑥1+𝑥1𝑒𝑥1, obtained through the expression (3.2), then𝜕𝑇1(𝐱)𝜕𝑥1𝜕𝑇1(𝐱)𝜕𝑥22𝑥13𝑥1+2𝑥2+𝑒𝑥1+𝑥1𝑒𝑥10.(3.29) Substituting the result of (3.28) in the matrix 𝑇1(𝐱), we obtain𝜕𝑇1(𝐱)𝜕𝑥102𝑥13𝑥1+2𝑥2+𝑒𝑥1+𝑥1𝑒𝑥10.(3.30) Consequently, 2𝑥1𝜕𝑇1(𝐱)𝜕𝑥10.(3.31) This expression indicates that 𝜕𝑇1(𝐱)/𝜕𝑥10; therefore, we should choose the new state 𝑇1(𝐱) in such way that it satisfies this condition.

Choosing 𝑇1(𝐱)=𝑥1, we obtain𝑇2(𝐱)=𝐿𝐟𝑇1(𝐱)=𝑇1(𝐱)3𝑥1+2𝑥2+𝑒𝑥1𝑥2𝑒𝑥1=103𝑥1+2𝑥2+𝑒𝑥1𝑥2𝑒𝑥1=3𝑥1+2𝑥2+𝑒𝑥1.(3.32) With the new state space given by 𝐳=𝐓(𝐱),𝑇𝐳=𝐓(𝐱)=1𝑇(𝐱)2=𝑥(𝐱)13𝑥1+2𝑥2+𝑒𝑥1.(3.33)

4. Approximate Feedback Linearization

There are situations in the design of controllers where the nonlinear terms are significant to the extent that failure to consider these terms in the system model could cause it to perform badly or to act restricted to a narrow operation band.

For systems where the nonlinear terms cannot be ignored, however, a physical model very close to reality can be made. The solution is to use exact feedback linearization. In this case, it is possible to linearize the system and apply linear control methods. Unfortunately exact linearization cannot be used for noninvolutive systems, as they do not satisfy the aforementioned conditions.

To solve this problem, the approximate feedback linearization, formulated by Krener [3], has been adopted using differential geometry techniques.

The approximate feedback linearization may be considered a near equivalent version of the exact feedback linearization. On the other hand, it offers the possibility to control noninvolutive systems, which was not possible using exact linearization. This new approach has recently been used in several applications.

As an example to validate this technique, the inverted pendulum model (Figure 3) with its characteristic nonlinear dynamic and an unstable open loop will be used. It is a classic problem in control theory and is widely used as a benchmark for testing of control algorithms (adaptive control, robust control, fuzzy, etc.).

205939.fig.003
Figure 3: Inverted pendulum.
4.1. Approximate Linearization Calculation

The approximate feedback linearization has been applied in cases where exact linearization is not feasible or has not yielded satisfactory results [4]. This process consists of finding approximate output functions that satisfy the involutivity condition up to determined system order.

The basic idea is [5], giveṅ𝐱=𝐟(𝐱)+𝐠(𝐱)𝑢,(4.1) where 𝐱𝑅.

Considering a state transformation in the control variable, we obtain𝐳=𝑊(𝐱),𝑢=𝛼(𝐱)+𝛽(𝐱)𝑣,(4.2) so thaṫ𝐳=𝐴𝐳(𝐱)+𝑏𝑣+𝑂𝑝+1(̃𝐱,𝑢),(4.3) where 𝐴=𝜕𝐟(𝐱0)/𝜕𝐱, 𝑏=𝑔(𝐱0), ̃𝐱=𝐱𝐱0, 𝐟(𝐱0)=0.

We apply a state transformation in the control variable such as̃1𝐳(𝐱)=𝐱+2𝐳𝑥𝑥𝐱0̃𝐱2+𝑂3(̃𝐱,𝑢),(4.4)𝑣(𝐱,𝑢)=𝑢+𝑣𝑥̃1𝐱+2𝑣𝑥𝑥̃𝐱2+𝑣𝑥𝑢̃𝐱𝑢+𝑂3(̃𝐱,𝑢),(4.5) where𝐳𝑥𝑥=𝜕2𝐳𝐱0𝜕𝐱2||||𝑥=𝑥0,𝑣𝑥𝑥=𝜕2𝑣(𝐱,𝑢)𝜕𝐱2||||𝑥=𝑥0,𝑣𝑥𝑢=𝜕2𝑣(𝐱,𝑢)||||𝜕𝐱𝜕𝑢𝑥=𝑥0.(4.6) Differentiating (4.4), we obtaiṅ̇𝐳(𝐱)=𝐱+𝐳𝑥𝑥̃𝐱̇𝐱.(4.7) Substituting (4.1) in (4.7),̇𝐳(𝐱)=𝐟(𝐱)+𝐠(𝐱)𝑢+𝐟(𝐱)𝐳𝑥𝑥̃𝐱+𝐠(𝐱)𝐳𝑥𝑥̃𝐱𝑢.(4.8) Expanding in a Taylor series,̇̃𝐱𝐳(𝐱)=𝐴𝐱+𝐠0𝐱𝑢+𝜕𝐟0𝐳𝜕𝐱𝑥𝑥+12𝜕2𝐟𝐱0𝜕𝐱2̃𝐱2+𝐱𝜕𝐠0𝐱𝜕𝐱+𝐠0𝐳𝑥𝑥̃𝐱𝑢+𝑂3(̃𝐱,𝑢).(4.9) Substituting (4.4) and (4.5) in (4.3), we obtaiṅ̃𝐱𝐳(𝐱)=𝐴𝐱+𝐠01𝑢+2𝐱𝜕𝐟0𝐳𝜕𝐱𝑥𝑥𝐱+𝐠0𝑣𝑥𝑥̃𝐱2𝐱+𝐠0𝑣𝑥𝑢̃𝐱𝑢+𝑂3(̃𝐱,𝑢).(4.10) Equalizing the second order term of (4.9) and (4.10) so that the unknown constants (𝐳𝑥𝑥,𝑣𝑥𝑥,𝑣𝑥𝑢) cancel out the second-order terms (̃𝐱2,̃𝐱𝑢), we obtain𝐱𝜕𝐟0𝐳𝜕𝐱𝑥𝑥+𝜕2𝐟𝐱0𝜕𝐱2𝐱𝐠0𝑣𝑥𝑥𝐱=0,𝜕𝐠0𝐱𝜕𝐱+𝐠0𝐳𝑥𝑥𝐱𝐠0𝑣𝑥𝑢=0.(4.11) Thus, there is an approximate output 𝑊1(𝐱), such as the set of vectors {𝐠,𝑎𝑑𝐟𝐠,,𝑎𝑑n𝐟2𝐠} are involutive until certain order 𝑝 [6]. In other words, the gradient 𝑊1(𝐱) multiplied by the Lie bracket of 𝐟 and 𝐠 is equal to the superior terms 𝑝+1. The following equation illustrates this:𝑊1(𝐱)𝑎𝑑𝑖𝐟𝐠=𝑂𝑝+1(̃𝐱),𝑖=0,1,2,,𝑛2.(4.12) In order to find vector 𝑊1, it is necessary to expand the involutive distribution 𝐷𝐼={𝐠,𝑎𝑑𝐟𝐠,,𝑎𝑑𝐟𝑛2𝐠} in a Taylor series on an equilibrium point 𝐱0. The expansion of the involutive distribution in 𝐱0 can be represented through the following equation:𝐷𝐼(𝐱)=𝐷𝐼𝐱0+𝑛𝑘1𝜕𝐷𝐼𝐱0𝜕𝐱𝑘̃𝐱𝑘+12𝑛𝑘,𝑖1𝜕2𝐷𝐼𝐱0𝜕𝐱𝑘𝜕𝐱𝑖̃𝐱𝑘̃𝐱𝑖+O3(̃𝐱),(4.13) where ̃𝐱𝑘=𝐱𝑘𝐱0 and ̃𝐱𝑖=𝐱𝑖𝐱0

This expansion is carried out in order to obtain an involutivity until a certain order 𝑝, where 𝑝 indicates the order in which the involutive distribution is expanded in a Taylor series to satisfy the involutivity condition. Therefore, the calculation of the approximate output function 𝑊1 is done through the equation𝐷0+𝐷1++𝐷𝑝𝑇𝑊1=0𝑛1x1,(4.14) where,𝐷0=𝐷𝐼𝐱0,𝐷1=𝑛𝑘=1𝜕𝐷𝐼𝐱0𝜕𝐱𝑘̃𝐱𝑘,𝐃𝑝=1𝑝!𝑛𝑘,𝑙,,𝑧=1𝜕𝑝𝐃𝐼𝐱0𝜕𝐱𝑘𝜕𝐱𝑙𝜕𝐱𝑧̃𝐱𝑘̃𝐱𝑙̃𝐱𝑧.(4.15) Once 𝑊1(𝐱) is known, the equation of the remaining state is given by𝜆𝑘(𝐱)=𝐿𝐟𝑘1𝑊1(𝐱).(4.16) And the input transformation𝑣=𝐿𝑘𝐟𝑊1(𝐱)+𝐿𝐠𝐿𝐟𝑘1𝑊1(𝐱)𝑢.(4.17) The state equations are represented aṡ𝑊𝑖=𝑊𝑖+1+O𝑝+1(̃̇𝑊𝐱,𝑢),𝑛=𝑣+O𝑝+1(̃𝐱,𝑢).(4.18) It is important to emphasize that the choice for the representation of the linearized system was based on the Brunovski canonical form.

4.2. Inverted Pendulum Linearization

The proposed inverted pendulum consists of a rigid rod of negligible weight, attached to a mobile base called car and with a fixed mass at the other end. The inferior part of the pendulum rod is in direct contact with the car, which aims to keep the rod in the upright position according to the movement performance of the car control position. In the context of this paper, a pendulum that performs its movement in only one plane was conidered. In other words, it has only one degree of freedom.

We consider the following nonlinear function:̇𝐱=𝐟(𝐱)+𝐠(𝐱)𝑢,(4.19) where 𝑥1=𝑥, 𝑥2=̇𝑥1, 𝑥3=𝜃, and 𝑥4=̇𝜃, and the movement equation may be described as [7]𝑥𝐟(𝐱)=2𝑥sin3𝑥24𝑥𝑔sin3𝑥cos3𝑥2cos32𝑥4𝑥sin3𝑥cos3𝑥24𝑥+2𝑔sin3𝑥2cos32,01𝐠(𝐱)=𝑥2cos320𝑥cos3𝑥2cos32.(4.20) For the involutive order 1, there should be a 𝑊1, not trivial, that satisfies the equation below [8]:𝑊1𝐃0=0,(4.21) where𝐃0=01010𝑔010102𝑔,(4.22) resulting in𝑊1=1010𝑇,(4.23) integrating𝑊1=𝑥1+𝑥3.(4.24) Once 𝑊1 has been calculated, the remaining states and the control variable are obtained from [6].𝜕𝑊𝜕𝑡1𝑊𝑛1𝑊𝑛=𝐿𝐟+𝐠𝑢𝑊1𝐿𝐟𝑛2𝑊1𝐿𝐟𝑛1𝑊1=𝜆2𝜆𝑛𝐿𝐟𝑊+𝐿𝐠𝑊1𝐿𝐠𝐿𝐟𝑛2𝑊1𝐿𝐠𝐿𝐟𝑛1𝑊1𝑢.(4.25) The last column on the right side represents the liberalized system neglected terms.

The input variable is given by𝑢=𝛼(𝐱)+𝛽(𝐱)𝑣,(4.26) where𝛼(𝐱)=𝐿𝑛f𝑊1𝐿𝐠𝐿𝐟𝑛1𝑊1,1𝛽(𝐱)=𝐿𝐠𝐿𝐟𝑛1𝑊1,(4.27) resulting in system approximately linear given by𝜕𝑊𝜕𝑡1𝑊𝑛1𝑊𝑛=𝑊01000010000100001𝑊𝑛1𝑊𝑛+0001𝑣,(4.28) wherė𝑊𝑖=𝑊𝑖+1+𝑂𝑝+1(̃̇𝑊𝐱,𝑢),𝑛=𝑣+𝑂𝑝+1(̃𝐱,𝑢).(4.29)

5. Simulation Results

In the simulation, a comparison of a controller for an inverted pendulum using the Taylor linearization and the approximate feedback linearization techniques was carried out. The pole placement control technique was used, where the close loop poles in both linearizations were placed in 𝑠=1, 𝑠=2, 𝑠=1+𝑗, and 𝑠=1𝑗. It can be seen from Figures 4 and 5 that the controller based on the approximate feedback linearization performed better for the initial positions 𝑥1(0)=𝑥2(0)=𝑥4(0)=0, 𝑥3(0)=40, and 𝑥3(0)=30. From the simulation results, it can be seen that for the Taylor linearization, the controller managed to stabilize the inverted pendulum for 𝑥3(0)41, while with the approximate linearization, the controller managed the stabilization for 𝑥3(0)53, which represents an increase in the inverted pendulum operation region.

205939.fig.004
Figure 4: Inverted pendulum for 𝑥3(0)=𝟑0.
205939.fig.005
Figure 5: Inverted pendulum for 𝑥3(0)=40.

The simulation shows the effectiveness of the proposed controller. Another important factor was the response of the controller as regards its position. According to Figures 6 and 7, the displacement of the inverted pendulum was less for the controller based on the approximate feedback linearization.

205939.fig.006
Figure 6: Inverted pendulum for 𝑥3(0)=𝟑0.
205939.fig.007
Figure 7: Inverted pendulum for 𝑥3(0)=40.

Another simulation was performed increasing the weight of the fixed mass attached to the car which contains the weight five times.

6. Conclusions

This work describes feedback linearization techniques used as a tool for differential geometry calculus. The existing restrictions to the use of the exact input-state feedback linearization technique were analysed, and the impossibility of using exact linearization in involutive systems was observed. As a solution to this, we proposed approximate linearization.

To illustrate the problem of exact linearization, a simple pendulum was used, where the calculation of the input variable as well as the state transformations were performed using differential geometry as the main calculus tool.

In the approximate linearization technique, an inverted pendulum was used, as it is an example of great interest. The oscillating behaviour of the inverted pendulum rod reproduces the stabilization problem common in situations such as the trajectory of a projectile or the movement of a satellite. In addition, it is an unstable in open-loop, nonlinear, noninvolutive, and an underactuated system, which makes it attractive for application in advanced control techniques.

As a demonstration of the effectiveness of the method, a simulation was carried out to compare the Taylor linearization with the approximate feedback linearization, with the latter technique increasing the inverted pendulum operation region.

According to the results obtained with a five-kilogram weight, shown in Figure 8, on the top of the inverted pendulum flexible rod, it can be inferred that there is the possibility possess a good robustness.

205939.fig.008
Figure 8: Inverted pendulum for 𝑥3(0)=40 (𝑚=5 kg).

Noninvolutivity, an important detail that prevents the application of the exact feedback linearization technique, was solved through coordinate transformation. This new state space, in turn, is involutive, enabling, therefore, exact linearization. In other words, as this coordinate transformation generates small errors, the linearization is no longer exact, making it an approximate linearization. However, it has been noted that despite these errors, the results indicate that the technique achieved a superior performance when compared to the Taylor linearization.

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