Abstract
This paper addresses the difficulty of designing a controller for a class of multi-input multi-output uncertain nonaffine nonlinear systems governed by differential equations. We first derive the first-order characteristic model composed of a linear time-varying uncertain system for such nonaffine systems and then design an adaptive controller based on this first-order characteristic model for position tracking control. The designed controller exhibits a simple structure that can effectively avoid the controller singularity problem. The stability of the closed-loop system is analyzed using the Lyapunov method. The effectiveness of our proposed method is validated with a numerical example.
Highlights
Complexity usually utilized to produce digital control signals, which require controllers to be designed in discrete time [27]
E above discussion motivates us to use a novel method, called characteristic modeling proposed by Wu [28,29,30,31,32,33,34,35], to solve the control design problem for uncertain nonaffine systems. e key idea of the characteristic modeling is to use a lower-order discrete time-varying linear system to express an original continuous system equivalently based on both the dynamic characteristics of controlled plants and performance specifications. is discrete system is called the “characteristic model of the original system.”
In the controller design method based on the second-order characteristic model, almost all the existing control methods belong to the indirect method in adaptive control
Summary
Where ai0 ∈ (0, 1) and θi,k denotes the estimation of the parameter θi,k. Note that ai0 ∈ Based on the bounds of the characteristic parameters provided by (20), we can prove that there exist ci, ci2, and ci3 > 0, such that Ni1 ∈ (0, 1); Ni2 and Ni3 > 0. From the above results and (46), we can obtain the following result: e2i,k+1 + 1 + Ni3φ2i,k ≤ 1 − Ni1e2i,k + φ2i,k− 1 − Ni2φ2i,ky2i,k + δim. ≤ Nki e2i,1 + 1 + Ni3φ2i,0 + Nki − 1δim + Nki − 2δim + · · · + N2i δim + Niδim + δim From this inequality, both ei,k and φi,k are bounded, and we can obtain e2i,k+1 ≤ Nki e2i,1 +. If fi(x, u) is bounded, that is, there exists a constant Li > 0 such that |fi(x, u)| ≤ Li, the tracking error ei(t) xi(t) − xir of system (1) converges to the neighborhood of the origin under the conditions in eorem. For any t ∈ [kTs, (k + 1)Ts], integrating (1) from kTs to t and noting e_i(t) x_i(t), one can obtain |ei(t)| ≤ |ei(kTs)| + TsLi ≤ ρi + TsLi, where ρi is defined in (58)
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