Different Affine Decomposition of the Model of the TORA System by TP model transformation Zolt´ an Petres, P´ eter L. V´ arkonyi, P´ eter Baranyi, and P´ eter Korondi Computer and Automation Research Institute of the Hungarian Academy of Sciences, H–1111 Budapest, Kende utca 13–17, Hungary, E-mail: petres@tmit.bme.hu Abstract — The Tensor Product (TP) model transfor- mation is a recently proposed technique for transform- ing given Linear Parameter Varying (LPV) models into affine model form, namely, to parameter varying con- vex combination of Linear Time Invariant (LTI) models. The main advantage of the TP model transformation is that the Linear Matrix Inequality (LMI) based control design frameworks can immediately be applied to the re- sulting affine models to yield controllers with tractable and guaranteed performance. The effectiveness of the LMI design depends on the LTI models of the convex combination. Therefore, the main objective of this paper is to study how the TP model transformation is capable of determining different types of convex hulls of the LTI models. This paper shows a case study of the TORA sys- tem. The theory and the definitions of the affine decom- position is discussed in the paper “Different Affine De- composition of the Model of the Prototypical Aeroelastic Wing Section by TP model transformation, PART I” of this proceedings. I CASE STUDY OF THE TORA SYSTEM The Translational Oscillations with a Rotational Actuator (TORA) system 1 was developed as a simplified model of a dual-spin spacecraft [13]. Later, Bernstein and his col- leagues at the University of Michigan, Ann Arbor, turned it into a benchmark problem for nonlinear control [1, 2, 3]. The system shown in Fig. 1 represents a translational os- cillator with an eccentric rotational proof-mass actuator. The oscillator consists of a cart of mass M connected to a fixed wall by a linear spring of stiffness k. The cart is constrained to have one-dimensional travel. The proof-mass actuator at- tached to the cart has mass m and moment of inertia I about its center of mass, which is located at distance e from the point about which the proof mass rotates. The motion occurs 1 Also referred to as the rotational/translational proof-mass actuator (RTAC) system. k m I θ e N M F Fig. 1: TORA system in a horizontal plane, so that no gravitational forces need to be considered. In Fig. 1, N denotes the control torque ap- plied to the proof mass, and F is the disturbance force on the cart. Let q and ˙ q denote the translational position and veloc- ity of the cart, and let θ and ˙ θ denote the angular position and velocity of the rotational proof mass, where θ = 0 deg is perpendicular to the motion of the cart, and θ = 90 deg is aligned with the positive q direction. The equations of motion are given by (M + m) ¨ q + kq = -me( ¨ θ cos θ - ˙ θ 2 sin θ)+ F (I + me 2 ) ¨ θ = -me ¨ q cos θ + N With the normalization ξ   M+m I +me 2 q, τ   k M+m t , u  M+m k(I +me 2 ) N, w  1 k  M+m I +me 2 F, the equation of motion become ¨ ξ + ξ = ε ( ˙ θ 2 sin θ - ¨ θ cos θ ) + w ¨ θ = -ε ¨ ξ cos θ + u where ξ is the normalized cart position, and w and u rep- resent the dimensionless disturbance and control torque, re- spectively. In the normalized equations, the symbol (·) rep- resents differentiation with respect to the normalized time τ. The coupling between the translational and rotational mo- tions is represented by the parameter ε which is defined by ε  me  (I + me 2 )(M + m) Letting x = ( x 1 x 2 x 3 x 4 ) T = ( ξ ˙ ξ θ ˙ θ ) T , the dimensionless equations of motion in first-order form are given by ˙ x = f(x)+ g(x)u + d(x)w , (1) where f(x)=      0 1 0 0 -1 1-ε 2 cos 2 x 3 0 0 εx 4 sin x 3 1-ε 2 cos 2 x 3 0 0 0 1 ε cos x 3 1-ε 2 cos 2 x 3 0 0 -εx 4 sin x 3 1-ε 2 cos 2 x 3      , g(x)=      0 -ε cos x 3 1-ε 2 cos 2 x 3 0 1 1-ε 2 cos 2 x 3      , d(x)=      0 1 1-ε 2 cos 2 x 3 0 -ε cos x 3 1-ε 2 cos 2 x 3      ,