Abstract— This paper describes the application of robust control theory extended to a cart-spring pendulum system with uncertainties and disturbances. Weighting functions are chosen such that the system could meet the performance requirements. The design of the H∞ controller is done with µ-synthesis in Matlab. Simulation of the H∞ controller was done on the linearized plant, and a worst case simulation was done.
keywords: robust control, D-K iteration, µ synthesis I. I NTRODUCTION
The cart-spring pendulum system consists of a cart restricted to motion on a straight and level track which is attached via a spring to a ﬁx wall. A pendulum is suspended from the cart by a hinge so as to be constrained to the vertical plane deﬁned by the track. The cart is equipped with a DC motor that exerts a torque to a small toothed wheel which, in turn, applies a force on the cart. The system will be disturbed by a sharp tap on the pendulum that comes from a human hand. Thus, it is important to look at issues as disturbance rejection and the robustness of the controller due to uncertainties in the system description. The D-K iteration method and µ synthesis are used to design a H∞ controller. The H∞ controller was tested on the linearized plant as well as a perturbed plant, and a worst case simulation is shown.
The cart-spring pendulum system
The paper is organized as follows: Section II sets up the robust control problem and gives the cart-spring pendulum model description. In section III the choice of weighting functions are described. The D-K iteration and Hinf controller design is shown in section IV. Section V contains the simulation study and section VI gives the conclusion.
where xp ∈ Rnp is the plant state, u ∈ Rnu is the control input subject to saturation, w ∈ Rnw is the exogenous input (possibly containing disturbance, reference and measurement noise), y ∈ Rny is the measurement output and z ∈ Rnz is the performance output. ˙ T, Deﬁning the plant state as xp = p p θ θ ˙ linearized model around the origin is given by (1) and Ap Cp,y Cp,z Bp,u Dp,yu Dp,zu Bp,w Dp,yw = Dp,zw
0 1 0 0 0 0 −330.46 −12.15 −2.44 0 2.71762 0 0 0 0 0 0 1 −812.61 −29.87 −30.10 0 6.68268 15.61 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 (3)