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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.

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 [

Common drawbacks of the above-mentioned schemes are

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 [

upper or lower bounds of the plant coefficients are required to be known [

discontinuous auxiliary inputs are used, which may lead to chattering [

high enough gains are used, which require excessive values [

Due to environment changes, the coefficients of the plant model may experience time-varying but bounded behavior [

The scheme of [

The tracking error converges to a residual set whose size is of the user’s choice;

known upper bounds of the plant coefficients are not required, such that high enough gains are not used;

all the closed-loop signals are bounded (parameter drifting is avoided);

auxiliary control signals are not discontinuous in terms of both the tracking error and the sliding surface, hence input chattering is avoided;

upper bounds of time-varying but bounded coefficients are not required to be known.

The disadvantages of [

In contrast to the approaches of [

we consider unknown time-varying control gain, not necessarily bounded, not restricted to actuator nonlinearities;

we consider time-varying but bounded behavior of some plant terms;

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

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 [

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 [

For the stability analysis, we use a truncated version of the quadratic form related to the sliding surface, denoted by

According to [

We assume that the dynamical nonlinear system can be represented by a Brunovsky type model, as defined in [

the entries of

the control gain

the entries of the vector

the entries of the vector

the value of the desired trajectory

The desired output

Let

Now we recall the equations corresponding to the controller and establish the tracking convergence theorem. The control law is given by (

If the controller designed in Section

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

The closed-loop dynamics is given by (

A PMSM is a kind of highly efficient and high-powered motor. The benefits of the PMSM are discussed in [

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 [

The results are shown in Figure

Transient behavior of the output and control input.

Performance of the tracking error and control input varying the resistance value.

In addition, we consider the variation of the damping constant

Performance of the tracking error and control input varying the damping constant value.

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

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

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

As the first step, we factorize several summands of (

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