Study On Closed-Form Dynamics Modeling Of Mechanical Systems And Theory Of Optimal Robust Control With Fuzzy Uncertainty

Model uncertainty is widespread in the control systems. A main research contentof modern control theory is to keep the system maintaining the desired performanceunder the condition of model imprecise and/or other uncertainties. Model-based con-trol design is an important way in developing control theory, and the control designrequires the closed-form (i.e., analytical form) of a dynamical model. The current dy-namics modeling approaches, in obtaining the model of a complex multi-body mechanicalsystem, is either difcult to manipulate, or can not obtain the dynamics model ana-lytically, thus can not serve control design well. Among the approaches of controllingsystem uncertainty in control area, the uncertainty-boundary-based robust control andthe stochastic-uncertainty-based control are the most successful two so far. However,the uncertainty-boundary-based robust control does not work in the optimal state; thetheory of stochastic-uncertainty-based control although is rather mature, the validity ofusing the stochastic theory describes the real-world uncertainties has been questioned.The fuzzy set theory expressed uncertainty of the system has many advantages.For the problems above, this paper systematically studied new approaches for closed-form dynamics modeling of mechanical systems and new theories for fuzzy uncertaintysystems control. The main contents and novelties are as follows:1. This paper proposes a cascading approach for multi-body system dynamics mod-eling, which provides a convenient way for closed-form dynamics modeling of complexmulti-body systems under ideal constraints. Based on the Udwadia-Kalaba equation,the approach is the frst to propose a way of hierarchically establishing closed-form dy-namics equations of multi-body systems. Compared with the current existing modelingapproaches, the new approach is rather simple, do not use any auxiliary variable, andcan get the closed-form equations, which efectively simplifes the closed-form dynamicsmodeling of complex multi-body systems.2. This paper proposes a closed-form dynamics modeling method for the contactnormal force in mechanical system with friction, which, from the dynamics point of view,solves the modeling of closed-form friction force for mechanical systems under non-idealconstraints. This paper correctly gived the closed-form expressions of friction force andefectively enhanced the modeling accuracy of the mechanical systems, which compensatesthe drawbacks caused by the assumption of constant normal force while modeling friction.3. This paper creatively uses fuzzy set theory to express uncertainty of system parameters, and proposes the theory of optimal robust control with fuzzy uncertainty for(frst order) fuzzy dynamical systems. Nonlinear uncertainties can be presented in thefuzzy dynamical systems. Unlike the previous IF-THEN rules-based fuzzy control, theproposed control theory uses a deterministic control design to drive the system to geta deterministic performance. Fuzzy information of the system uncertainties is used forthe optimal design of the control gain. The paper, based on the proposed fuzzy systemperformance index, transforms the optimal control design into a constrained optimizationproblem. It is proven that the global solution to this problem always exists and is unique.The closed-form solution and the closed-form minimum cost are derived.4. This paper systematically studies the properties of mechanical systems, proposesthat the inertia matrix of mechanical systems may not be positive defnite and proposesthe upper polynomial boundary property for the inertia matrix. This study clarifesthe misconceptions of the lower and upper boundary properties of mechanical systeminertia matrix. The upper polynomial boundary property for the inertia matrix plays animportant role in building a valid Lyapunov function candidate.5. This paper uses fuzzy set theory to express uncertainties in (second order) me-chanical systems (i.e., fuzzy mechanical systems), and proposes the control theory ofoptimal robust control with fuzzy uncertainty for mechanical systems. The paper studiesthe optimal robust trajectory following control and optimal robust constraint followingcontrol for mechanical systems. The study takes full advantage of the inherent char-acteristics of mechanical systems, takes the following error as the control variable, andmakes the fuzzy mechanical system to have a deterministic following performance whileperforming optimal control design based on fuzzy uncertainties. The study also creativelycombines fuzzy uncertainty and the Udwadia-Kalaba dynamics equation in doing robustcontrol design of constraint following. This study is an extension and deepening of theprevious (frst order) fuzzy dynamical system control.The above contents and novelties systematically propose the closed-form dynamicsmodeling approaches for mechanical systems and the theory of optimal robust controldesign with fuzzy uncertainty. The paper provides illustrative examples for every newtheories and approaches to show their validity. This study provides new approachesfor closed-form dynamics modeling of mechanical systems, and provides new ways ofcontrolling system uncertainties. As a result, the paper is with rather good academicmerits and practical value.