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-springRobust control applied to a cart-spring pendulum system with uncertainty Dan Dai Advisor: Professor Roy Smith University of California, Santa Barbara II. ROBUST C ONTROL STATEMENT The objective of this project is to design a controller that meets the specified robust performance criteria. When the cart-spring pendulum system is considered, these criteria reflects on robustness to outside disturbance and plant uncertainty. To get this controller, it is necessary to set up this problem in a very systematic way. The cart-spring pendulum system is a complex system and it has a few important properties to study. For the purpose of deriving a model, the experimental system will be considered to be composed of a massless spring attached to a frictionless cart from which a slender rod freely hangs. The output of the system is the position p of the cart, in meters, relative to the spring’s equilibrium point and the angular position θ of the pendulum, in radians, relative to the vertical; both positions are measured with optical encoders. The physical inputs of the system are the voltage u applied to the armature of the dc motor, in Volts, and a disturbance force w, in Newtons. The operating range of D/A converter, is [-5,5] Volts. The disturbance w is a force in the plane of motion orthogonal to the pendulum of length 2l and acts at a disturbance of (4/3)l from the cart-pendulum hinge. A nonlinear model of the system can be derived by applying standard Euler-Lagrange techniques, see [1]. Moreover, consider the plant as  ˙  xp = Ap xp + Bp,u u + Bp,w w y = Cp,y xp + Dp,yu u + Dp,yw w  z =C x +D p,z p p,zu u + Dp,zw w

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.

P

(1)

Fig. 1.

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

(2)

        

 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)

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