Controller Design And Estimation Of Domain Of Attraction For Positive Polynomial Fuzzy Systems Under Constraints

Nonlinear positive systems are a special kind of nonlinear systems whose system states and output always remain in the positive quadrant for any nonnegative initial conditions,they are prevalent in the practical industry and life.Due to the coexistence of positive constraints and complex nonlinear characteristics in nonlinear positive systems,it is a difficult task to analyze stability and design controllers for nonlinear positive systems.In order to facilitate the analysis and controller design,the nonlinear positive systems are represented as positive polynomial fuzzy model based systems with the help of polynomial fuzzy model.Considering that the nonlinear positive system may be subject to certain constraints,such as,time delay,the unmeasurable system states,input saturation limit and the uncertain system parameters,this dissertation investigates the controller design for positive polynomial fuzzy systems under the above constrains based on the positive characteristics of systems and the information of membership functions.The main works of this dissertation include:First,when the system states of the positive polynomial fuzzy systems are unmeasurable,the polynomial fuzzy observer with mismatched premise variables is designed to estimate the unmeasurable system states.The estimated states are then employed by the polynomial fuzzy controller for the feedback positive control.In order to handle the nonconvex stability conditions and non-convex positivity conditions derived from this control strategy with imperfectly matched premises,an effective convexification method based on the matrix decoupling technique and matrix conversion technology is proposed in this dissertation,which helps to obtain the global optimal solution of the observer gains and controller gains at the same time.Finally,the simulations are provided to verify the effectiveness of the above method in expanding the stability region of the positive polynomial fuzzy systems.Second,when positive polynomial fuzzy systems suffer from input saturation limits,the polynomial fuzzy controller and polynomial fuzzy auxiliary controller are designed based on the symbolic membership function dependent method.When performing the stability analysis for the closed-loop positive polynomial fuzzy systems,the properties of membership functions,the boundary information of membership functions and premise variables are included into the stability conditions,so that the conservativeness of the controller design caused by the lack of the information of the membership function is reduced.Finally,the simulation is provided to verify the effectiveness of the above method in expanding the domain of attraction of the positive polynomial fuzzy systems.Third,considering that the above symbolic membership functions dependent method has weak ability to reduce the conservativeness of controller design,the Taylor series membership functions dependent method is used to design controller for the positive polynomial fuzzy systems subject to input saturation.Firstly,the traditional taylor series membership function dependent method is improved by local approximated errors and the threshold function which is used to distinguish the state sub-spaces.Then,the improved taylor series membership function dependent method is adopted to derive the stability condition and the estimation condition of domain of attraction for closed-loop systems,so that the conservativeness of the controller design is reduced.Finally,the simulation is provided to verify the effectiveness of the above method in expanding the domain of attraction of the positive polynomial fuzzy systems.Forth,in order to solve the hign cost and high conservatism of controller design of the positive polynomial fuzzy systems,the concept of partially matched premises is used to increase the flexibility of controller design,so that the implement cost of the controller can be reduced.Then,an analysis scheme based on the linear copositive Lyapunov function and piecewise linear membership function dependent method is proposed to perform the stability analysis and positivity analysis for positive polynomial fuzzy systems,so that the conservativeness caused by the positive constraints is reduced.And then,an effective convexification method based on the sector nonlinear technique and the membership function dependent method is proposed to handle the non-convex term of stability conditions.Finally,the simulation is provided to verify the effectiveness of the above method in expanding the stability region of the positive polynomial fuzzy systems.At last,when positive nonlinear systems suffer from input saturation limits and the uncertain parameters,they are firstly represented by interval type-2 positive polynomial fuzzy systems,so that the parameter uncertainties are handled.Then an analysis method based on the linear copositive Lyapunov function and the polyhedron invariant set is proposed,as well as a convexification method used to deal with the non-convex stability condition and the estimation condition of domain of attraction,so that the conservativeness that the positive constraint brings to the controller design and estimation of the domain of attraction is reduced.And then,an improved interval type-2 membership function dependent method is proposed by using the minimum approximation error,which further reduces the conservativeness of controller design and the estimation of the domain of attraction caused by the lack of interval type-2 membership function.Finally,the simulation is provided to verify the effectiveness of the above method in expanding the domain of attraction of the interval type-2 positive polynomial fuzzy systems.