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Journal of Control Science and Engineering
Volume 2012 (2012), Article ID 269346, 9 pages
http://dx.doi.org/10.1155/2012/269346
Research Article

Adaptive Control for Nonlinear Systems with Time-Varying Control Gain

1Programa de Ingeniería Ambiental, Facultad de Ingeniería y Arquitectura, Universidad Católica de Manizales, Carrena 23 No. 60-30, Manizales 170002, Colombia
2Departamento de Ingeniería Eléctrica, Facultad de Ingeniería y Arquitectura, Universidad Nacional de Colombia, Sede Manizales, Electrónica y Computación, Percepción y Control Inteligente, Bloque Q, Campus La Nubia, Manizales 170003, Colombia

Received 21 November 2011; Accepted 11 April 2012

Academic Editor: Chengyu Cao

Copyright © 2012 Alejandro Rincon and Fabiola Angulo. 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

We propose a scheme for nonlinear plants with time-varying control gain and time-varying plant coefficients, on the basis of a plant model consisting of a Brunovsky-type model with polynomials as approximators. We develop an adaptive robust control scheme for this plant, under the following assumptions: (i) the plant terms involve time-varying but bounded coefficients, being its upper bound unknown; (ii) the control gain is unknown, not necessarily bounded, and only its signum is known. To achieve robustness, we use a combination of robustifying control inputs and dead zone-type update laws. We apply this methodology to the speed control of a permanent magnet synchronous motor (PMSM), and we achieve proper tracking results.

1. Introduction

Nonlinear behavior is difficult to model accurately, rendering controller design cumbersome. An approach to handle this is to use linear models, which have local validity in the state space, that is, different values of the parameter set would be required for each region in the state space. A second approach is to use a plant model in either the Brunovsky form defined in [1], or the parametric-pure feedback form defined in [2], using the so-called “function approximation techniques’’, such as locally weighted statistical learning [3], fuzzy sets [4], Fourier series [5], orthonormal functions, [6] or neural networks [7]. In the last case, state and time dependent terms, known as “basis functions,’’ are used to represent the nonlinear and time-varying behaviour. Some identification or learning methods are used to perform the approximation, resulting in a negligible modelling error if there are a sufficient number of basis functions [3]. In some cases, a Brunovsky form model of the plant is used, assuming that there is full-state measurement and that the basis functions are multiplied by unknown constants. Then, a sliding surface model reference adaptive control (SSMRAC) is devised, handling the residual approximation error by means of auxiliary robustifying inputs [7].

Common drawbacks of the above-mentioned schemes are(i)the convergence of the tracking error to some small residual set is not ensured in a strict sense and depends on the value of the approximation error [7];(ii)upper or lower bounds of the plant coefficients are required to be known [8];(iii)discontinuous auxiliary inputs are used, which may lead to chattering [5];(iv)high enough gains are used, which require excessive values [6].

Due to environment changes, the coefficients of the plant model may experience time-varying but bounded behavior [9, 10]. Both, the control gain and other coefficients may exhibit this behavior. The time-varying behavior of coefficients different from the control gain may be handled by means of robustness techniques. See for instance the robust MRAC scheme in [10], whose drawback is that projection- or 𝜎-type update laws are used, and hence upper bounds of the plant coefficients are required to be known, in order to achieve the convergence of the tracking error to some residual set of user-defined size. The time-varying control gain may be handled by means of robustness techniques [11] or Nussbaum gain technique [12], a projection-type update law is used, such that a lower bound of the control gain is required to be known. In [11], a 𝜎 type update law is used, such that bounds of the plant coefficients are required to be known. In summary, the main drawback of using robustness techniques to handle varying control gain is that upper or lower bounds of some plant coefficients are required to be known. On the other hand, the Nussbaum gain technique relaxes this requirement. The main drawback is that the upper bound of the transient behavior of the state 𝑆 is significantly altered with respect to that of the disturbance-free case: the value of this bound depends on time integrals of the terms that comprise the Nussbaum function. Another drawback is that the control gain is usually required to be upper-bounded by some constant.

The scheme of [13] achieves adequate robustness properties. Mainly, upper bounds of the plant coefficients are not required to be known. The essential element of the technique is to define the quadratic forms related to the sliding surface 𝒮 in terms of a dead zone function of 𝒮 rather than in terms of 𝒮. This scheme has the following benefits.(Ri) The tracking error converges to a residual set whose size is of the user’s choice;(Rii) known upper bounds of the plant coefficients are not required, such that high enough gains are not used;(Riii) all the closed-loop signals are bounded (parameter drifting is avoided);(Riv) auxiliary control signals are not discontinuous in terms of both the tracking error and the sliding surface, hence input chattering is avoided;(Rv) upper bounds of time-varying but bounded coefficients are not required to be known.

The disadvantages of [13] are (i) the design is only developed for systems with hysteresis nonlinearities in the actuator; (ii) the time-varying control gain is tackled by means of the Nussbaum gain method, which entails higher complexity of the Lyapunov analysis; (iii) the control gain is assumed to be time-varying but bounded.

In contrast to the approaches of [3, 4, 1416], we propose that an adequate regression model for highly nonlinear systems may be obtained from a Brunovsky type model, inserting polynomials to approximate the nonlinear behavior, and considering also: (i) time-varying but bounded behavior of some plant coefficients; (ii) unknown control gain, time-varying, and not necessarily bounded. We develop a robust adaptive scheme for this plant, achieving benefits (Ri) to (Rv). If we compare the robust technique developed here with the technique developed in [13], the main differences, which are also contributions, are(Rvi) we consider unknown time-varying control gain, not necessarily bounded, not restricted to actuator nonlinearities;(Rvii) we consider time-varying but bounded behavior of some plant terms;(Rviii) we tackle the control gain by means of robustness techniques, which gives a simpler design in comparison with the Nussbaum technique.

This paper is organized as follows. The outline of the scheme is given in Section 2. The function approximation is discussed in Section 3. The plant regression model and the statement of the problem are established in Section 4. The control and update laws are derived in Section 5. The control algorithm and its stability are presented in Section 6. An application of the scheme to a PMSM is given in Section 7. Finally, the conclusions are presented in Section 8.

2. Outline of the Scheme

We propose the use of polynomials to approximate the nonlinear behavior, taking into account the fact that polynomials are universal approximators for continuous functions, according to [17]. We consider a Brunovsky-form plant model, into which we introduce the polynomial terms.

We devise a robust adaptive controller for this plant, achieving benefits (Ri) to (Rix) mentioned in the introduction. We use the SSMRAC method stated in [18], as the basic framework for designing the control and update laws. To handle the effect of modelling error and time-varying behavior of plant terms, we use a robust continuous control input, whose magnitude is adjusted, and a dead zone-type update law. The resulting controller has the following features: (i) the magnitude of the control input is adjusted to cope with the unknown upper bounds of the time-varying coefficients; (ii) the control input is proportional to some continuous function of the sliding surface 𝒮, so that chattering is avoided; (iii) an inactivation is introduced in all the update laws, which occurs when the sliding surface 𝒮 reaches a target region. If we compare this with projection-type update laws, it has the advantage of not requiring the upper bounds of the plant parameters.

For the stability analysis, we use a truncated version of the quadratic form related to the sliding surface, denoted by 𝑉𝑠. The validity of this technique, including the conditions of the Lyapunov function, is stated in [19]. It is worth noticing that the standard conditions of the Lyapunov theory for time-variable systems, shown in [18] or [20], are not satisfied because of the truncation. We prove the asymptotic convergence of the tracking error in a rigorous manner by means of Barbalat’s Lemma. To that end, we redefine the expression of ̇𝑉 as an inequality in terms of 𝑉𝑠.

3. Function Approximation Based on Polynomials

According to [17, page 116], a continuous real-valued function 𝑓(𝑥), where 𝑥𝐷, 𝐷𝑅𝑛, being 𝐷 a compact set, may be approximated by a polynomial function 𝑔 in the interval 𝑥𝐷, with some finite error, being 𝑔 defined as𝑔𝑥,𝜃𝑜=𝑘1,𝑘2,,𝑘𝑛𝜃𝑜[𝑗]𝑥𝑘11𝑥𝑘22𝑥𝑘𝑛𝑛𝜃𝑜Ω,Ω𝑅𝑝,(1) where Ω is a convex set and 𝜃𝑜[𝑗] means the 𝑗th element of the vector 𝜃𝑜. Thus, 𝑓(𝑥) can be expressed in terms of 𝑔 as follows (cf. [17] page 122):𝑓(𝑥)=𝑔𝑥,𝜃𝑎+𝜖𝑜||𝜖(𝑥),𝑜||𝜃(𝑥)<𝜖,𝑎=𝜃𝑜Ω𝜃𝑜=argmin𝜃𝑜sup𝑥𝐷|||𝑓(𝑥)𝑔𝑥,𝜃𝑜|||,(2) where 𝜖 is a positive constant, and it is known as approximation error, 𝜃𝑎 is an ideal parameter set and 𝑔(𝑥,𝜃𝑎) is an ideal representation of 𝑓(𝑥). The polynomial can be linearly parameterized with respect to its coefficients:𝑓(𝑥)=𝜙𝜃𝑎+𝜖𝑜(𝑥),(3) where 𝜙 contains polynomial terms, whereas 𝜃𝑎 contains constants. In this work, 𝑓 represents the nonlinear part of the dynamical equation. We will handle the unknown constant vector 𝜃𝑎 by means of adjustment rules and 𝜖𝑜 by means of robust techniques.

4. Plant Model and Problem Statement

We assume that the dynamical nonlinear system can be represented by a Brunovsky type model, as defined in [1], with polynomial functions:𝑦(𝑛)=𝑐𝑛1𝑦(𝑛1)𝑐1̇𝑦𝑐𝑜𝑦+𝑓(𝑥)+𝑏𝑢,(4)𝑥=𝑦,̇𝑦,,𝑦(𝑛1),(5) where 𝑦(𝑡)1 is the system output 𝑢(𝑡)1 is the input and the coefficients 𝑐𝑛1,, and 𝑐𝑜 are unknown and time-varying but bounded. The relative degree 𝑛 may be established by means of previous step response analysis. Inserting the function (3) into (4) gives𝑦(𝑛)=𝑏𝑢+𝜙𝜃𝑎+𝑑,(6)𝑑=𝜖𝑜(𝑥)𝑐𝑛1𝑦(𝑛1)𝑐1̇𝑦𝑐𝑜||𝑐𝑦,𝑛1||𝜇𝑛1||𝑐,,1||𝜇1,||𝑐𝑜||𝜇𝑛,||𝜖𝑜||(𝑥)𝜇𝑛+1=𝜖.(7) We make the following assumptions:(Ai) the entries of 𝜃𝑎 and the terms 𝜇1,, and 𝜇𝑛+1 are constant and unknown;(Aii) the control gain 𝑏 varies with respect to the state 𝑥, or time, so that it satisfies the following:(i)0<𝑏𝑚||𝑏||𝑓𝑏,𝑡𝑡𝑜,(8)(ii)sign(𝑏)constant,𝑡𝑡𝑜,(9) where the value of sign(𝑏) is known, 𝑏𝑚 is an unknown positive constant, and 𝑓𝑏 is an unknown positive function of time or 𝑥. Notice that condition (8) implies that 𝑏 is always different from zero;(Aiii) the entries of the vector 𝑥 are available for measurement;(Aiv) the entries of the vector 𝜙 are known functions of 𝑥;(Av) the value of the desired trajectory 𝑦𝑑(𝑡) and its derivatives 𝑦𝑑(𝑛1),,̇𝑦𝑑 is bounded.

The desired output 𝑦𝑑 is specified in terms of a bounded external command 𝑟(𝑡) as follows:𝑦𝑑(𝑛)+𝑎𝑚,𝑛1𝑦𝑑(𝑛1)++𝑎𝑚,𝑜𝑦𝑑=𝑎𝑚,𝑜𝑟,(10) where 𝑎𝑚,𝑛1,,𝑎𝑚,𝑜 are constant coefficients prespecified by the user, such that the polynomial 𝐾(𝑝) is Hurwitz with at least one real root, being 𝐾(𝑝) defined as 𝐾(𝑝)=𝑝(𝑛)+𝑎𝑚,𝑛1𝑝(𝑛1)++𝑎𝑚,𝑜. The external reference signal 𝑟(𝑡) must be bounded. The objective of the MRAC design is to formulate a control law 𝑢(𝑡) such that the tracking error 𝑒(𝑡)=𝑦(𝑡)𝑦𝑑(𝑡) asymptotically converges to the residual set 𝐷𝑒, defined as follows:𝐷𝑒=𝑒|𝑒|𝐶𝑏𝑒,(11) where 𝐶𝑏𝑒 is a user-defined bound.

5. The Control and Update Laws

Let 𝒮 the dynamics imposes over the tracking error given by𝒮=(𝑝+𝜆)𝑛1𝑒=𝑝𝑛1𝑒+𝜆𝑛2𝑝𝑛2𝑒++𝜆1̇𝑒+𝜆𝑜𝑒,(12) where 𝜆 is a positive constant chosen by the user. Having defined 𝒮, we establish the dynamic equation for 𝒮 by differentiating with respect to time:̇𝒮=𝑝𝑛𝑒+𝜆𝑛2𝑝𝑛1𝑒++𝜆1̈𝑒+𝜆𝑜̇̇𝑒,(13)𝒮=𝑦(𝑛)𝑦𝑑(𝑛)+𝜆𝑛2𝑝𝑛1𝑒++𝜆1̈𝑒+𝜆𝑜̇̇𝑒,(14)𝒮=𝑦(𝑛)+𝜑𝑎𝜑,(15)𝑎=𝑦𝑑(𝑛)+𝜆𝑛2𝑝𝑛1𝑒++𝜆1̈𝑒+𝜆𝑜̇𝑒.(16) We insert the expression for 𝑦(𝑛) of (6) into (15):̇𝒮=𝑏𝑢+𝜃𝑎𝜙+𝑑+𝜑𝑎.(17) Define𝑢=𝜃𝑎𝜙𝜑𝑎𝑎𝑚𝒮=𝜑𝜃,𝜃=𝜃𝑎,1,(18)𝜑=𝜙,𝜑𝑎𝑎𝑚𝒮,(19) where 𝜑𝑎 is defined in (16), 𝑎𝑚 is a positive constant. We express (17) in terms of 𝑢 as follows:̇𝒮=𝑎𝑚𝒮+𝑏𝑢+𝑑𝑢=𝑎𝑚𝒮+𝑏𝑢+𝑑𝜑𝜃.(20) Let𝑐𝑙=𝑛+1,(21)=2+𝑙2𝐶𝑏𝑣𝑠12𝑏𝑚𝑛2𝐶(22)𝑏𝑣𝑠12𝜆𝑛1𝐶𝑏𝑒2𝑓,(23)1=||||̇𝑦,,𝑓𝑙2=||𝑦(𝑛1)||,𝑓𝑙1=||𝑦||,𝑓𝑙𝜇=1,1||𝑐=max1||,,𝜇𝑙2||𝑐=max𝑛1||,𝜇𝑙1||𝑐=max𝑜||,𝜇𝑙=𝜖.(24) Now, we multiply (20) by 𝒮 and add and subtract the term 𝑐𝑆2:𝒮̇𝒮=𝑎𝑚𝒮2+𝑏𝒮𝑢+𝒮𝑑𝒮𝜑𝜃+𝑐𝑆2𝑐𝑆2,(25) where the terms 𝑐𝒮2𝑐𝒮2, 𝒮𝜑𝜃, and 𝒮𝑑 can be expressed in terms of adjustment errors:(i)𝑐𝒮2𝑐𝒮2=𝑐𝒮2̃𝑐𝒮2+̂𝑐𝒮2,̃𝑐=̂𝑐𝑐;(26)(ii)𝒮𝜑𝜃=𝑆𝜑̃𝜃𝒮𝜑̂̃̂𝜃,𝜃=𝜃𝜃;(27)(iii)𝒮𝑑(1)̃𝑐1||𝒮||𝑓1+̂𝑐1||𝒮||𝑓1++(1)̃𝑐𝑙||𝒮||𝑓𝑙+̂𝑐𝑙||𝒮||𝑓𝑙,(28)̃𝑐1=̂𝑐1𝜇1,,̃𝑐𝑙=̂𝑐𝑙𝜇𝑙,(29) where ̂̂𝑐,𝜃,̂𝑐1,,̂𝑐𝑙 are adjusted parameters whose update laws will be defined later. Inserting the above properties into (25) gives𝒮̇𝒮𝑎𝑚𝒮2̃𝑐𝒮2+𝒮𝜑̃𝜃+(1)̃𝑐1||𝒮||𝑓1++(1)̃𝑐𝑙||𝒮||𝑓𝑙+̂𝑐𝒮2𝒮𝜑̂𝜃+̂𝑐1||𝒮||𝑓1++̂𝑐𝑙||𝒮||𝑓𝑙+𝑏𝒮𝑢𝑐𝒮2.(30) If 𝑏 were constant, we would choose the control 𝑢 so as to cancel the terms involving updated parameters ̂̂𝑐,𝜃,̂𝑐1,,̂𝑐𝑙. Since 𝑏 is varying, we chose 𝑢 to attenuate the effect of adjusted parameters, being the remaining error rejected by 𝑐𝒮2:𝑢=(1)sgn(𝑏)̂𝑐2𝒮3𝜑+(1)sgn(𝑏)̂𝜃2𝒮+(1)sgn(𝑏)̂𝑐21𝑓21𝒮++(1)sgn(𝑏)̂𝑐2𝑙𝑓2𝑙𝒮,(31) where ̂̂𝑐,𝜃,̂𝑐1,,̂𝑐𝑙 are adjusted parameters whose update laws will be defined later. Replacing the above control law into (20) and (30) giveṡ𝒮=𝑎𝑚||𝑏||𝒮̂𝑐2𝒮3||𝑏||𝜑̂𝜃2𝒮||𝑏||̂𝑐21𝑓21||𝑏||𝒮+̂𝑐2𝑙𝑓2𝑙𝒮+𝑑𝜑𝜃,𝒮̇(32)𝒮𝑎𝑚𝒮2̃𝑐𝒮2+𝒮𝜑̃𝜃+(1)̃𝑐1||𝒮||𝑓1++(1)̃𝑐𝑙||𝒮||𝑓𝑙+̂𝑐𝒮2𝒮𝜑̂𝜃+̂𝑐1||𝒮||𝑓1++̂𝑐𝑙||𝒮||𝑓𝑙||𝑏||̂𝑐2𝒮4||𝑏||𝜑̂𝜃2𝒮2||𝑏||̂𝑐21𝑓21𝒮2||𝑏||̂𝑐2𝑙𝑓2𝑙𝒮2𝑐𝒮2,(33) where the last terms of the above equation satisfy the following inequality (see the proof in Appendix A):̂𝑐𝒮2𝒮𝜑̂𝜃+̂𝑐1||𝒮||𝑓1++̂𝑐𝑙||𝒮||𝑓𝑙||𝑏||̂𝑐2𝒮4||𝑏||𝜑̂𝜃2𝒮2||𝑏||̂𝑐21𝑓21𝒮2||𝑏||+̂𝑐2𝑙𝑓2𝑙𝒮2𝑐𝒮20if𝒮22𝐶𝑏𝑣𝑠.(34) Equation (34) expresses the attenuation of the effect of the adjusted parameters. Substituting (34) into (33) gives𝒮̇𝒮𝑎𝑚𝒮2̃𝑐𝒮2+𝒮𝜑̃𝜃̃𝑐1||𝒮||𝑓1̃𝑐𝑙||𝒮||𝑓𝑙if𝒮22𝐶𝑏𝑣𝑠,(35) or equivalently,𝒮̇𝒮𝑎𝑚𝒮2̃𝑐𝒮2+𝒮𝜑̃||𝒮||𝐶𝜃𝑑𝑓if𝒮22𝐶𝑏𝑣𝑠,𝐶(36)𝑑=̃𝑐1,,̃𝑐𝑙𝑓,𝑓=1,,𝑓𝑙.(37) We choose the following update laws:𝛾̇̂𝑐=𝑐𝒮2if𝒮22𝐶𝑏𝑣𝑠,̇𝐶0otherwise,𝑑=Γ𝑑𝑓||𝒮||if𝒮22𝐶𝑏𝑣𝑠,0otherwise,̇̂𝜃=Γ𝜑𝒮if𝒮22𝐶𝑏𝑣𝑠,0otherwise,(38) where 𝒮 is defined in (12), 𝐶𝑏𝑣𝑠 in (23), whereas 𝛾𝑐, and the diagonal entries of Γ, Γ𝑑 are positive constants chosen by the user, being Γ and Γ𝑑 diagonal matrices.

6. The Control Algorithm

Now we recall the equations corresponding to the controller and establish the tracking convergence theorem. The control law is given by (31); the update laws are given by (38); the terms 𝜑 and 𝒮 are defined in (19) and (12); 𝜑𝑎 is defined in (16); 𝐶𝑏𝑣𝑠 is defined in (23) and 𝑓 is defined in ((37)). The adjusted parameters ̂𝑐1,,̂𝑐𝑙, required to compute 𝑢, are the entries of the vector 𝐶𝑑, that is, 𝐶𝑑=[̂𝑐1,,̂𝑐𝑙].

6.1. Theorem: Boundedness and Tracking Convergence

If the controller designed in Section 5 is applied to the plant (6), then the tracking error 𝑒(𝑡) converges to 𝐷𝑒 asymptotically, and the closed-loop signals remain bounded.

Proof. Now we proceed to analyze the stability of the controlled system using the direct Lyapunov method and the Barbalat’s Lemma. First, we establish the boundedness of the closed-loop signals on the basis of the time derivative of the Lyapunov function. Then, we establish the convergence of the tracking error to the target region 𝐷𝑒 defined in (11), on the basis of the Barbalat’s Lemma. The validity of the procedure can be stated using the theory developed and applied in [18, 19, 21, 22].
The closed-loop dynamics is given by (32), and (38). We define the following truncated quadratic form, on the basis of the truncation presented in [19]: 𝑉𝑠=𝑉𝑠if𝑉𝑠𝐶𝑏𝑣𝑠,𝐶𝑏𝑣𝑠if𝑉𝑠<𝐶𝑏𝑣𝑠,𝑉(39)𝑠12𝒮2,(40) where 𝐶𝑏𝑣𝑠 is defined in (23). The form 𝑉𝑠 has the following properties: 𝑉𝑠<𝑉𝑠+𝐶𝑏𝑣𝑠<0if𝑉𝑠>𝐶𝑏𝑣𝑠,𝑉𝑠<𝑉𝑠+𝐶𝑏𝑣𝑠=0if𝑉𝑠=𝐶𝑏𝑣𝑠,𝑉𝑠0=𝑉𝑠+𝐶𝑏𝑣𝑠=0if𝑉𝑠<𝐶𝑏𝑣𝑠.(41) The states of the closed-loop system may be grouped in the following vector: 𝐶𝑥=𝒮,̃𝑐,𝑑,̃𝜃,(42) where ̃𝑐, 𝐶𝑑, and ̃𝜃 are defined in (26), (37), (27). We define the following function: 𝑉=𝑥(𝑡)𝑉𝑠+𝑉𝜃+𝑉𝑐+𝑉𝑑,𝑉𝜃=12̃𝜃Γ1̃𝜃,𝑉𝑐=12𝛾𝑐1̃𝑐2,𝑉𝑑=12𝐶𝑑Γ𝑑1𝐶𝑑.(43) Notice that 𝑉(𝑥) does not satisfy standard conditions of Lyapunov theory for time-variable systems, shown in [18]. It satisfies the conditions of [19] instead, which we use to prove the boundedness of the closed-loop signals and the convergence of the tracking error to the residual set 𝐷𝑒 (11). The time derivative of 𝑉𝑠 along the trajectory (32) is ̇𝑉𝑠̇=𝒮𝒮=𝑎𝑚𝒮2||𝑏||̂𝑐2𝒮4||𝑏||𝜑̂𝜃2𝒮2||𝑏||+(1)̂𝑐21𝑓21𝒮2||𝑏||++(1)̂𝑐2𝑙𝑓2𝑙𝒮2+𝒮𝑑𝜑𝜃𝑎𝑚𝒮2̃𝑐𝒮2+𝒮𝜑̃𝜃||𝒮||𝐶+(1)𝑑𝑓if𝑉𝑠𝐶𝑏𝑣𝑠,(44) where the last inequality is obtained from (36). The time derivative of 𝑉𝑠 can be derived from (39) and expressed on the basis of the above equation: ̇𝑉𝑠=̇𝑉𝑠𝑎𝑚𝒮2̃𝑐𝒮2+𝒮𝜑̃||𝒮||𝐶𝜃+(1)𝑑𝑓if𝑉𝑠𝐶𝑏𝑣𝑠,0if𝑉𝑠<𝐶𝑏𝑣𝑠.(45) The time derivative ̇𝑉𝜃+̇𝑉𝑐+̇𝑉𝑑 along trajectories (38) is ̇𝑉𝜃+̇𝑉𝑐+̇𝑉𝑑=̃𝜃(1)𝒮𝜑+̃𝑐𝒮2+𝐶𝑑𝑓||𝒮||if𝑉𝑠𝐶𝑏𝑣𝑠,0otherwise.(46) Therefore, ̇𝑉 is given by ̇̇𝑉=𝑉𝑠+̇𝑉𝜃+̇𝑉𝑐+̇𝑉𝑑=̇𝑉𝑠+̇𝑉𝜃+̇𝑉𝑐+̇𝑉𝑑𝑎𝑚𝒮2if𝑉𝑠𝐶𝑏𝑣𝑠,0if𝑉𝑠<𝐶𝑏𝑣𝑠.(47) The above equation can be rewritten in terms of 𝑉𝑠 defined in (40): ̇𝑉2𝑎𝑚𝑉𝑠if𝑉𝑠𝐶𝑏𝑣𝑠,̇𝑉=0if𝑉𝑠<𝐶𝑏𝑣𝑠.(48) This implies that ̇𝑉0 for all 𝑡𝑡𝑜. Thus, according to [19], all the closed-loop signals are bounded, that is, (𝒮, ̃𝜃, ̃𝑐, 𝐶𝑑)𝐿, or equivalently, 𝑥𝐿, where 𝑥 is defined in (42). As a consequence, (𝑉,  𝑉𝑠, 𝑉𝑠, 𝑉𝜃, 𝑉𝑐, 𝑉𝑑)𝐿, according to the definitions in (39) and (43). The boundedness of 𝑥 implies (𝑒, , 𝑒(𝑛1))𝐿 according to (12); hence, ̇𝒮𝐿 [22], and ̇𝑉𝑠𝐿. To establish the convergence of the tracking error, we begin by expressing the equation ahead in terms of 𝑉𝑠, according to the definition of 𝑉𝑠 in (39) and the properties of (41): ̇𝑉2𝑎𝑚𝑉𝑠<2𝑎𝑚𝑉𝑠+2𝑎𝑚𝐶𝑏𝑣𝑠0if𝑉𝑠𝐶𝑏𝑣𝑠,̇𝑉=0=2𝑎𝑚𝑉𝑠+𝐶𝑏𝑣𝑠=0if𝑉𝑠<𝐶𝑏𝑣𝑠,̇𝑉2𝑎𝑚𝑉𝑠+2𝑎𝑚𝐶𝑏𝑣𝑠0.(49) We reorganize the above equation and integrate as follows: 2𝑎𝑚𝑡𝑡𝑜𝑉𝑠𝐶𝑏𝑣𝑠𝑡𝑑𝜏+𝑉(𝑡)𝑉𝑜.(50) Thus, (𝑉𝑠𝐶𝑏𝑣𝑠)𝐿1. Recall that (𝑉𝑠,̇𝑉𝑠)𝐿. Thus, (𝑉𝑠𝐶𝑏𝑣𝑠)𝐿𝐿1, (𝑑/𝑑𝑡)(𝑉𝑠𝐶𝑏𝑣𝑠)𝐿. By invoking the Barbalat Lemma [23], we obtain (𝑉𝑠𝐶𝑏𝑣𝑠)0 as 𝑡. In turn, this implies that 𝑉𝑠𝐷𝑣𝑠, being 𝐷𝑣𝑠 defined as 𝐷𝑣𝑠=𝑉𝑠𝑉𝑠𝐶𝑏𝑣𝑠,(51) where 𝐶𝑏𝑣𝑠 is defined in (23). 𝒮 can be expressed in terms of 𝑉𝑠, on the basis of the definition of 𝑉𝑠 in (40): 𝒮=2𝑉𝑠.(52) In the following, we analyze the convergence of 𝒮. Taking into account the definition of 𝐶𝑏𝑣𝑠 in (23), the fact that 𝑉𝑠 converges to 𝐷𝑣𝑠 defined in (51), and (52) the convergence of 𝒮 is: lim𝑡𝒮=𝐷𝑠,𝐷𝑠=||𝒮||𝒮𝐶𝑏𝑠,𝐶𝑏𝑠=2𝐶𝑏𝑣𝑠=𝜆𝑛1𝐶𝑏𝑒.(53) To analyze the convergence of 𝑒, we begin by expressing 𝑒 in terms of 𝒮, according to (12): 1𝑒=(𝑝+𝜆)𝑛1𝒮.(54) The convergence of the tracking error may be established on the basis of (53) and (54) [18, 21]: lim𝑡𝑒(𝑡)=𝐷𝑒,𝐷𝑒=𝑒|𝑒|𝐶𝑏𝑒,(55) which means that the tracking error 𝑒 converges to a residual set whose size is of the user’s choice.

7. Application to a Permanent Magnet Synchronous Motor

A PMSM is a kind of highly efficient and high-powered motor. The benefits of the PMSM are discussed in [24, 25]. The PMSM has the following difficulties [26]: (i) it is highly nonlinear; (ii) the parameters of the physical model experience unknown time-varying behavior, for example, the stator resistance 𝑅 and the friction coefficient 𝐵; (iii) unknown external disturbances appear, that is, the load torque disturbance 𝑇𝐿. Moreover, by varying the permanent magnet flux 𝜆𝑎𝑓 the state 𝜔 exhibits a pitchfork bifurcation [27]. All the mentioned facts lead to response deterioration of many controllers, specially for high-speed and high-precision tasks in real applications [28].

In view of the complex behavior, with parameters varying with respect to time and state plane, the scheme developed in this work is suitable. We apply the developed scheme by simulation, to a PMSM whose model and parameters are presented in [29]. Therein, the possible manipulated inputs are 𝑢𝑑 and 𝑢𝑞; and 𝑖𝑑, 𝑖𝑞, and 𝜔 are possible outputs. We choose the motor angular frequency 𝜔 as the output to be controlled and the 𝑑-axis voltage 𝑢𝑑 as the control input, that is, 𝑦=𝜔, 𝑢=𝑢𝑑, 𝜐𝑑=0. A similar choice is made in [24], where an adaptive controller is derived for a PMSM. After performing a step response analysis, the variable 𝜔 exhibits a second-order behavior when a step change is introduced in 𝑢𝑑. This second order behavior was already noticed in [29]. Thus, we use the regression model (6) with 𝑛=2. We can summarize the basic features of the plant as𝑛=2,sgn(𝑏)=+1,𝑙=3,𝑓1=||||̇𝑦,𝑓2=||𝑦||,𝑓3||||,||y||=1,𝑓=̇𝑦,1,𝑦𝜙=2,𝑦3,𝜑𝑎=̈𝑦𝑑+𝜆𝑜̇𝑒,𝜑=𝜙,𝜑𝑎𝑎𝑚𝒮.(56) We used a factorization with terms 𝑦2 and 𝑦3 to take into account the presence of the pitchfork bifurcation. According to [30] the normal form of the pitchfork bifurcation gives a description of the system behavior in a tight neighborhood of the bifurcation point. For the case of the pitchfork bifurcation, the normal form includes the terms 𝑦2 and 𝑦3. Although the system usually works in different regions far from the bifurcation point, we wanted to include the behavior in the neighborhood of the bifurcation point. Thus, we included the terms of the normal form for the pitchfork bifurcation [30], that is, 𝑦2 and 𝑦3. The terms 1 and 𝑦 are not included in 𝜙 because they are already present in 𝑑. The remaining parameters of the controller are defined on the basis of the parameters in (56):𝑒=𝑦𝑦𝑑,𝒮=(𝑝+𝜆)𝑒=̇𝑒+𝜆𝑒𝜆𝑜𝐶=𝜆,𝑏𝑣𝑠=12𝜆𝐶𝑏𝑒2,𝑢=(1)sgn(𝑏)̂𝑐2𝒮3𝜑̂𝜃+(1)sgn(𝑏)2𝒮+(1)sgn(𝑏)𝒮̂𝑐21𝑓21+̂𝑐22𝑓22+̂𝑐23𝑓23,=(1)sgn(𝑏)𝒮̂𝑐2𝒮2+𝜑̂𝜃2+̂𝑐21𝑓21+̂𝑐22𝑓22+̂𝑐23𝑓23.(57) In addition, we choose:𝜆=3,𝑎𝑚=10,𝑎𝑚,1𝑎=70,𝑚,𝑜=1225(forthereferencemodel),𝐶𝑏𝑒𝛾=0.1,𝑐=20,Γ𝑑[],[].=diag20,20,20Γ=diag0.4,0.4,0.003(58)

The results are shown in Figure 1. The simulation shows expected results: all the closed-loop signals remain bounded, and transient values of the tracking error remain in a small interval. It is worthy of note that the transient values of 𝑒 depend on its initial value and the initial values of the adjustment errors. To show the effect of the time-varying behavior of the model coefficients, we change the stator resistance 𝑅 from 1.4 to 1.7 Ω at the time instant 0.6 sec. Results are shown in Figure 2.

fig1
Figure 1: Transient behavior of the output and control input.
fig2
Figure 2: Performance of the tracking error and control input varying the resistance value.

In addition, we consider the variation of the damping constant 𝐵 from 0.00038818 to 0.00046582 Nm/(rad/s) at the instant 0.6 sec. Results are shown in Figure 3. Notice that in the three cases the final value of |𝑒| is less than 𝐶𝑏𝑒=0.1. Moreover, the control input 𝑢 belongs to the interval [200200] V. Notice in Figures 2 and 3 that the effect of the disturbance on the tracking error is almost negligible. Nevertheless, the control input experiences a large variation, as can be seen in Figure 3.

fig3
Figure 3: Performance of the tracking error and control input varying the damping constant value.

8. Conclusions

In this work, we have proposed a control scheme for highly nonlinear plants, based on a simple plant model with polynomial approximators, which provides an adequate description of transient behavior. It is worth noticing the fact that benefits Ri to Rviii (Section 2) are achieved at the same time, with minimal requirements on the knowledge of the plant. Indeed, the relative degree and the signum of the control gain can be established by a previous step response analysis. Many techniques are combined at the same time: sliding surface MRAC, dead zone-type update law, robustifying auxiliary control, approximation techniques, and truncation of the quadratic forms.

The disadvantage of polynomials as approximators is that they may be less accurate than other techniques, for example, neural networks or fuzzy sets, leading to higher approximation error. Since the approximation error is bounded, it can be handled by means of robustness techniques without requiring the upper bound to be known. Moreover, we considered the coefficients of the terms 𝑦,,𝑦(𝑛1) as time-varying but bounded, with constant and unknown bounds. Then, we handled this by means of robust control, without requiring the upper bounds to be known.

We handled the time-varying behavior of the control gain by means of robustness techniques, without using the Nussbaum gain method. The redefinition of the plant terms in terms of adjustment errors and adjustment parameters is a fundamental step. The resulting expression allows a straightforward definition of the control law. The variation of the control gain implies that the terms involving adjusted parameters cannot be cancelled. Rather, we attenuate its effect by means of squared terms and handle the residual error by means of an additional control term that is only a function of the sliding surface. The resulting expression for 𝑉, ̇𝑉, and the design is simpler in comparison with the Nussbaum technique.

Appendix

Proof of (34)

As the first step, we factorize several summands of (34) and apply the property (8):||𝑏||̂𝑐2𝒮4+̂𝑐𝒮2𝑏𝑚̂𝑐2𝒮4+̂𝑐𝒮2=𝑏𝑚̂𝑐𝒮212𝑏𝑚2+12𝑏𝑚212𝑏𝑚2.(A.1) Likewise, we obtain:||𝑏||𝜑(1)̂𝜃2𝒮2𝒮𝜑̂1𝜃2𝑏𝑚2,||𝑏||(1)̂𝑐21𝑓21𝒮2+̂𝑐1||𝒮||𝑓112𝑏𝑚2,||𝑏||(1)̂𝑐2𝑙𝑓2𝑙𝒮2+̂𝑐𝑙||𝒮||𝑓𝑙12𝑏𝑚2.(A.2) By adding (A.1) and (A.2) we obtain:||𝑏||̂𝑐2𝒮4||𝑏||𝜑+(1)̂𝜃2𝒮2||𝑏||+(1)̂𝑐21𝑓21𝒮2||𝑏||++(1)̂𝑐2𝑙𝑓2𝑙𝒮2+̂𝑐𝒮2𝒮𝜑̂𝜃+̂𝑐1||𝒮||𝑓1++̂𝑐𝑙||𝒮||𝑓𝑙1(2+𝑙)2𝑏𝑚2.(A.3) As a second step, we use the definition of 𝑐 in (22) to rewrite the term 𝑐𝒮2:𝑐𝒮2𝒮=(2+𝑙)22𝐶𝑏𝑣𝑠12𝑏𝑚21(2+𝑙)2𝑏𝑚2if𝒮22𝐶𝑏𝑣𝑠.(A.4) As the third step, we add (A.3) and (A.4):||𝑏||̂𝑐2𝒮4||𝑏||𝜑+(1)̂𝜃2𝒮2||𝑏||+(1)̂𝑐21𝑓21𝒮2||𝑏||++(1)̂𝑐2𝑙𝑓2𝑙𝒮2+̂𝑐𝒮2𝒮𝜑̂𝜃+̂𝑐1||𝒮||𝑓1++̂𝑐𝑙||𝒮||𝑓𝑙𝑐𝒮20if𝒮22𝐶𝑏𝑣𝑠,(A.5) which is (34).

Acknowledgments

This work was partially supported by Universidad Nacional de Colombia-Manizales, project 12475, Vicerrectoría de Investigación, DIMA, resolution number VR-2185.

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