Adaptive nonlinear control with partial overparametrization

Abstract

The main purpose of this paper is to show that overparametrization in adaptive nonlinear control is not always a bad thing and can be intentionally introduced to achieve similar performance with less control effort. To this end, an adaptive controller with partial overparametrization is first designed to solve the regulation problem of parametric-strict-feedback form nonlinear systems by combining the design procedure in Krstic et al. (System Control Lett. 19 (1992) 177) with that in Kanellakopoulos et al. (IEEE Trans. Automat. Control 36 (1991) 1241). Then, to reduce the control effort needed by the above controller further, a modified adaptive controller is designed by introducing a reference signal and reformulating the regulation problem as a tracking problem. The simulation results show that, compared with the adaptive controller without overparametrization, the adaptive controller with partial overparametrization can achieve similar performance with less control effort, and the modified one can do the same thing with even less control effort.

Introduction

In this paper, we will study again the control problem of the following system considered in [4]:

$$\text{x}\text{̇}{}_{\mathrm{i}}\text{=x}{}_{\mathrm{i+1}}\text{+θ}{}^{\mathrm{<mtext>T</mtext>}}\text{φ}{}_{\mathrm{i}}\text{(x}{}_1\text{,…,x}{}_{\mathrm{i}}\text{),}\text{1⩽i⩽n−1,}\text{x}\text{̇}{}_{\mathrm{n}}\text{=φ}{}_0\text{(x)+θ}{}^{\mathrm{<mtext>T</mtext>}}\text{φ}{}_{\mathrm{n}}\text{(x)+β}{}_0\text{(x)u,}$$

where θ=(θ₁,…,θ_p)^T∈R^p is the vector of unknown constant parameters, φ₀,β₀, and the components of φ_i, 1⩽i⩽n are known smooth nonlinear functions in Rⁿ, and β₀(x)≠0, for all x∈Rⁿ.

For parametric-strict-feedback nonlinear system (1), by using a powerful design tool – back-stepping, two kinds of nonlinear adaptive controllers have been designed. One is without overparametrization (see [4]); the other is with overparametrization (see [3], [2]). The first adaptive controller with full overparametrization designed by applying back-stepping was presented in [3], which employed np estimates for p unknown parameters. (We use “full overparametrization” to describe the situation.) Its main drawback is that the dynamic order of the resulting adaptive controller may be too high. The number of estimates was reduced in half in [2], and overparametrization was completely removed in [4].

In the development of adaptive nonlinear control, overparametrization is always regarded as a bad thing. Therefore, the adaptive controller in [4] seems to be perfect since it needs no overparametrization and possesses stronger stability properties than those using overparametrization. However, it is shown by simulations in [1] that the adaptive controller without overparametrization may require too large control signals, which is also a drawback. We also find in our simulations that, compared with the adaptive controller without overparametrization, the one with full overparametrization requires much smaller control signals to achieve similar performance under the same initial state conditions, which implies that overparametrization is not always bad.

As we have already known, the drawback of the adaptive controller with full overparametrization is that it employs too many estimates for one parameter, while the drawback of the one without overparametrization is that it requires too large control signals. The main idea of this paper is to make a tradeoff between them, that is, we will design adaptive controllers with partial overparametrization (here “partial overparametrization” means that mp, 1<m<n estimates are employed for p unknown parameters) which neither employs too many estimates for one parameter nor requires too large control signals. Here, instead of completely removing overparametrization, some overparametrization is intentionally introduced to reduce the control effort needed by the adaptive controller without overparametrization.

As in [4], we will only study the regulation problem. The control objective is also the same as in [4].

The rest of this paper is organized as follows: In Section 2, an adaptive controller with partial overparametrization is designed for system (1) to solve the regulation problem. The controller is then modified to achieve similar performance with even less control effort by introducing a reference signal and reformulating the regulation problem as a tracking problem in Section 3. The simulation results are presented in Section 4 for a third-order nonlinear system. Conclusions are made at last.