Fuzzy Control Of Inverted Pendulum System Based On T-S Model

The inverted pendulum is a very popular experiment used for educationalpurposes in modern control theory. It is basically a pole which has a pivot on a cartthat can be moved horizontally. The pole moves freely around the cart, and thecontrol objective is to bring the pole to the upper unstable equilibrium position bymoving the cart on the horizontal plane. Since the angular acceleration of the polecannot be controlled directly, the inverted pendulum is an underactuatedmechanical system. The system has strong nonlinearity and inherent instability,therefore, the standard nonlinear control techniques are ineffective to control it.Takagi-Sugeno fuzzy model developed in the recent years has presented a new ideato the inverted pendulum system design.The design of the inverted pendulum has mostly focused on two cases. One isswinging the pole to its upper equilibrium position while the cart displacement isbrought to zero. The other is keeping the pole to the upper equilibrium while thecart is at its equilibrium position, with the external disturbance. In this paper, thestability controller has been designed.After analyzed the pendulum system which can move freely on the horizontalorbit, it can be found that the control design is mostly divided into two parts. One isto minimize the performance of the transfer function from the disturbance to theoutputs. The other is to confirm the control variable and the output variable withinthe constraints. The control strategy based on this analysis which divides the stateexit into performance state exit and restricted state exit is proposed. Then thecontrol problem of the pendulum system reduced to the disturbance restrainedproblem which is satisfied the constraints.In this paper , we adopt H_∞ norm of the closed-loop system as the criterion ofoptimization. Traditional H_∞ control does not consider the constraints of systemalthough the constraints of inputs and state exit in a common control system. Underthe base of H_∞ control, H_∞ control of constrained system is formed. It solvesthe problem of constraints by transforming the constraints of the input or state intothe constraints of optimized parameters, taking the performance of H∞ asoptimized objection and resolving the optimization problem of constrained system.In this paper, we proposed a new design method for Takagi-Sugeno fuzzycontroller with constraint H∞ control, the solution of the controller utilize an LMIapproach. The control design includes two parts, first we present theTakagi-Sugeno fuzzy control with constraint H∞ based on the state feedbackcontrol strategy. This method restricts the achievable set of the state space withinthe ellipse region, which is proposed by the H∞ norm of the optimized channels.The H∞ norm is used as the performance index. The control synthesis problemsare formulated as linear matrix inequality (LMI) problems. We get the parametersof the local controllers by solving the optimized LMIs problem. After that , theseparameters are used to form a global controller by so called parallel distributedcompensation. The global controller is the blending of each local controller. Itconfirms the control system asymptotically stable. Since some state variable is notmeasurable, considering about the feasibility of the engineering, we proposedanother method—dynamic parallel distributed compensation for Takagi-Sugenofuzzy systems. The control synthesis problems are formulated as linear matrixinequality (LMI) problems. We get the parameters of the local controllers bysolving the optimized LMIs problem. Then, these parameters are used to form aglobal controller by so called dynamic parallel distributed compensation (DPDC).It confirms the control system asymptotically stable. The DPDC controller isessentially nonlinear dynamic feedback controller. Applying the methods into thedesign of the pendulum control systems, It can be found from the response curvesof the closed loop system , the designed fuzzy controller satisfies the stability andthe performance of the controlled system.