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State Feedback Optimal L2-Induced Control of Nonlinear Systems Utilizing Universal Approximation.
Adrian-Mihail Stoica1, Isaac Yaesh2
1Faculty of Aerospace Engineering, National University of Science and Technology POLITEHNICA of Bucharest, 011061 Bucharest, Romania.
Entropy (Basel, Switzerland)
|May 26, 2026
Summary
This study introduces an optimal L2-induced control method for systems with multiple nonlinearities. The approach uses linear matrix inequalities (LMIs) to ensure system stability and performance.
Area of Science:
- Control Theory
- Nonlinear Systems Analysis
- Optimization Techniques
Background:
- Systems with multiple sector-bounded nonlinearities pose challenges for traditional control design.
- Ensuring boundedness of the L2-induced norm is crucial for stability and performance guarantees.
- Existing methods often struggle to handle complex nonlinearities effectively.
Purpose of the Study:
- To develop an optimal L2-induced control strategy for systems featuring multiple sector-bounded nonlinearities.
- To derive sufficient conditions for L2-induced norm boundedness using linear matrix inequalities (LMIs).
- To formulate and solve the optimal state feedback control problem for this class of systems.
Main Methods:
- Derivation of sufficient boundedness conditions for the L2-induced norm using a system of LMIs.
- Formulation and solution of an optimal state feedback control problem based on the derived LMI conditions.
- Development of a procedure to reduce the conservatism of the obtained conditions.
- Application of the universal approximation theorem to represent broader classes of nonlinearities using sector-bounded sigmoid functions.
Main Results:
- Sufficient conditions for L2-induced norm boundedness were established using LMIs.
- An optimal state feedback controller was designed and validated for systems with multiple sector-bounded nonlinearities.
- A method to reduce conservatism in the derived conditions was successfully implemented.
- The proposed formulation demonstrated applicability to a wider range of nonlinearities, including those approximated by sigmoid functions.
Conclusions:
- The presented optimal L2-induced control approach effectively addresses systems with multiple sector-bounded nonlinearities.
- The use of LMIs provides a systematic way to ensure stability and performance.
- The method's flexibility, enhanced by the universal approximation theorem, broadens its applicability to complex nonlinear systems.
- Numerical validation on a van der Pol oscillator confirms the theoretical developments.
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