Structural analysis, design and optimization of nonlinear control systems using the linear algebraic equivalence of nonlinear controllers

Designing a nonlinear control scheme addressing major design issues in a unified way is a difficult task. However, structural information of a system/controller allows one to diagnose and identify performance characteristics at the design stage, to select better solution methods, and identify and eliminate potential problems, thereby improving the design of the controllers. In this research, structural analysis, design and optimization of nonlinear control systems are presented using the linear algebraic equivalence of nonlinear controllers (LAENC) to address the design issues of nonlinear control systems in a systematic way. It is shown that feedback linearization and sliding mode control possess the very useful structural feature, the linear algebraic equivalence, on which many well-developed linear algebraic solution techniques can be applied directly. Using this LAENC, input-constrained nonlinear optimal controllers are proposed. Investigation of the nonlinear control structure by applying the singular value decomposition to the LAENC shows that control inputs/outputs are composed of a linear combination of finite number of mode shapes and that the linear combination coefficients for the control input determine the final distribution of the control inputs. It is also shown that the contribution of each output mode towards the target vector is determined by colinearity, and colinearity can be used as a criterion for mode truncation optimization. The proposed algorithms are applied to the temperature control of an enclosed and radiation-dominant thermal system. The pure nonlinear controllers generate input-constraint-violating and large-oscillating solutions due to the ill-conditionedness, hence regularization methods that reduce the effects of ill-conditionedness are embedded into the nonlinear controller designs based on the LAENC. Also, the proposed input-constrained nonlinear optimal controller is applied and successful results are obtained. Using structural analysis, control input modes that cause input-constraint-violation and that contribute little to the total performance but use large control energy are truncated. As a result, input-constraint-satisfying solutions with a reduction in control effort are obtained.