Adaptive nonlinear control with constrained parallel parameter estimates

Abstract

The state feedback regulation of nonlinear systems of order $n$ in parametric strict-feedback form is considered. A simple, easy to tune, adaptive control with projected parameter estimates is proposed. The uncertain parameter vector is estimated by $n$ parallel vector estimates whose differences are constrained to tend asymptotically to zero. When the uncertain parameter vector is identifiable, the closed loop system is globally asymptotically and locally exponentially stable. Three comparative examples illustrate the advantages of the proposed simple adaptive control over the tuning functions or extended matching approaches.

Introduction

A basic and desirable property for any adaptive control design is to guarantee converging parameter estimation errors, when the parameter vector is identifiable either in certain operating conditions or under persistently exciting exogenous signals. In these cases exponential stability and, consequently, robustness for the closed loop system hold; in addition the performance of the known parameter controller can be recovered. Hence, it is preferable to parametrize the adaptive control in terms of the minimum number of parameter estimates so that overparametrization is avoided. There are, however, cases in which an overparametrized control has some advantages [1], [2]. A second order example in parametric strict-feedback form is presented in [1] showing that the overparametrized adaptive control proposed in [3] gives better transient performance than the non-overparametrized control in [4]. In [2] an alternative adaptive control employing overparametrization is presented which has the advantage of improved performance with respect to the tuning functions approach in [5]. The overparametrization problem will be addressed for feedback linearizable nonlinear systems, depending linearly on an unknown parameter vector. They have been widely studied (see [6], [7]) with the aim of designing adaptive controls under matching [8] or extended matching [4] conditions. Adaptive controls for the more general case of systems in parametric strict-feedback form were designed in [3] via adaptive backstepping and overparametrization. Tuning functions were introduced in [5] to avoid overparametrization. A robust adaptive design is proposed in [9] following [5]: it allows for parameter projections and improved transient performance at the expense of introducing nonlinear high-gain damping feedback terms. In the presence of unknown nonlinear functions, smooth projections and sliding mode controls are used in [10] to design adaptive robust controls. Additive disturbances are dealt with in [11] by incorporating robust control design techniques which guarantee asymptotic reference tracking, provided that the control gains are sufficiently large with respect to the system initial conditions. Dynamic uncertainties are addressed in [12] by robust modifications. Actuator saturations are considered in [13] for bounded input-bounded state systems. If the nonlinearities are either unknown or not linearly parametrizable, robust approaches such as [14] may be followed. However, the robust approaches in [10], [12], [14] do not guarantee asymptotic reference tracking.

In this paper it is shown how the adaptive backstepping technique in [3], which requires an overparametrization, can be simply equipped by constrained parallel estimates so that the same stability results presented in [5] via the tuning functions design can be reobtained. The key idea is to constrain the parallel estimates differences to tend asymptotically to zero. This alternative approach has the advantage of allowing for simple parameter projection algorithms as in [15], without resorting to complex smooth projection operators (see [10], [11], [16], [17]), when a priori parameter bounds are available. However, it has the drawback of requiring $n$ parallel estimates for each unknown parameter. The same third order system reported in [5] is considered in Section 3 as a comparative example showing that the proposed constrained parallel estimation approach is simpler to design. Numerical simulations are carried out with a nonlinearity consisting of a quadratic term. Better transient performance are obtained with respect to the controller designed following [5], since the tuning functions approach leads to high frequency oscillations and higher amplitude of the control signal. A simpler second order example illustrates that the extended matching approach in [4] leads to a controller amplitude which is one order of magnitude larger than the amplitude of the controller proposed in this paper. Finally, a third order linear system is considered for which the approaches followed in [5] and in this paper lead to linear closed loop dynamics whose eigenvalues can be easily compared.