SummaryThe approximate dynamic programming needs 2 prerequisites to be an effective optimal control method. Firstly, it must be assured to be stable and convergent before application. Secondly, the control system should mainly be a nonlinear multi-input multi-output form. Thus, this paper introduces a nonlinear multi-input multi-output approximate dynamic programming and proves that it is stable in Lyapunov sense, therefore it is convergent. Besides, the Lyapunov function design is also analyzed. These proofs are based on the Lyapunov stability theory in the form of the utility function of quadratic, square-weighted sum, and absolute value. Thereafter, 3 typical control examples of nonlinear multi-input multi-output approximate dynamic programming are offered to show their applications and verify the proofs. The proof overcomes the complex derivation, and the results contain 3 practical and systematic bounded proofs. It is for the first time that the proof focuses on nonlinear multi-input multi-output approximate dynamic programming from the view of utility function. What is more, the results can also serve as an effective analysis and guide for the utility function design and the stability criterion of nonlinear multi-input multi-output approximate dynamic programming as well.
The standard ADRC controller usually selects the canonical plant in the form of cascaded integrators. However, the condition variables of practical system do not necessarily have the cascaded integral relationship. Therefore, this paper proposes a method of total derivative of composite functions of several variables and a structure, which can convert the state space system of noncascaded integral form into the cascaded integral form. In this way, the converted system can be directly controlled with ADRC. Meanwhile, the control of Chen chaotic system is discussed in detail to show the conversion and the controller design. The control performances under different levels of complication and different strengths of disturbance are comparably researched. The converted system achieves significantly better control effects under ADRC than that of the PID. This converting method solves the control problem of some noncascaded integral systems in both theory and application and greatly expands the application scope of the standard ADRC method.
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