Adaptive Iterative Learning Control for Nonlinear Nonsquare Systems Subject to Unknown Control Gain Matrices With Applications to PMSM

This work presents a novel adaptive iterative learning control (AILC) approach for a class of nonlinear nonsquare systems with unknown control gain matrices. The proposed strategy avoids explicitly incorporating the control gain matrices into the control algorithms and does not require them to be invertible, thereby significantly broadening the applicability of AILC. In the proposed AILC method, a newly designed virtual control gain matrix is introduced, enabling the transformation of the unknown control gain matrix into a form of norm-bounded system uncertainty while accounting for input saturation. Building on this reformulation, a structurally simple AILC scheme is developed, incorporating an input-dependent auxiliary system to mitigate the effects of input constraints. Moreover, the proposed method is extended to nonaffine nonlinear systems by integrating neural network techniques. The convergence of the proposed AILC laws is rigorously analyzed within the composite energy function (CEF) framework, and its effectiveness is demonstrated through implementation on a permanent magnet synchronous motor (PMSM) and a numerical example of a nonaffine system. Note to Practitioners—In control engineering, knowledge of the control gain matrix is crucial for controller design. However, in practical applications, this information is often unavailable, posing significant challenges to control system development. While this issue has traditionally been addressed using Nussbaum-type function techniques, such methods require the control gain matrix to be either positive or negative definite. When the gain matrix is indefinite, the control design problem becomes substantially more difficult. Moreover, many engineering systems, such as underactuated or overactuated systems, feature nonsquare control gain matrices, rendering the existing Nussbaum-type function approach inapplicable. The proposed AILC method can effectively address all of these challenges. By introducing a virtual designer-specified control gain matrix, the originally unknown gain matrix can be modeled as a norm-bounded uncertainty, which is then compensated through adaptive learning mechanisms. Additionally, the proposed method is applicable to both square and nonsquare systems, enhancing its suitability for real-world applications. Detailed guidelines are provided for selecting controller parameters and their impacts on control performance are analyzed for practical applications.