coho sinh d A sinhs cosh (1.16) inh d In other words, if we let ao, Tl,... denote the sequence of two dimensional g2(k6 p(k6) then Ek satisfies the recursion (1.18) me now that we wish to program our computer to calculate the constant ol values vk to be applied during any interval via a linear transformation of the measured values of position and velocity at the start of the interval. Here F is just a row vector (1, f2) that gives the coefficients of a linear combination of these measured values. Formally we are in a situation analogous to the PD control(1.6), except that we now assume that the measurements are being made only at discrete times and that a constant control will be applied on each intervaL. Substituting(1. 19) into the difference equation(1. 18), there results the new difference equation k+1=(A+ BF). k nce for any Tk+2=(A+BF)2ck it follows that, if one finds gains f, and f2 with the property that the matrix A+ BF is nilpotent, that is (A+BF)2=0 then we would have a controller with the property that after two sampling steps necessarily Ik+2=0. That is, both y and c vanish after these two steps, and he system remains at rest after that. This is the objective that we wanted to achieve all along. We now show that this choice of gains is always possible Consider the characteristic polynomial det(2I-A-BF)= 22+(2 cosh8-f2 sinh -fi cosh+ f1)z f1 cosh+1+f1+f2 sinh 8 It follows from the Cayley-Hamilton Theorem that condition(1. 22) will hold provided that this polynomial reduces to just z. So we need to solve for th8 1. Introduction where A =  cosh δ sinh δ sinh δ cosh δ  (1.16) and B =  cosh δ − 1 sinh δ  . (1.17) In other words, if we let x0,x1,... denote the sequence of two dimensional vectors xk :=  ϕ(kδ) ϕ˙(kδ)  , then {xk} satisfies the recursion xk+1 = Axk + Bvk . (1.18) Assume now that we wish to program our computer to calculate the constant control values vk to be applied during any interval via a linear transformation vk := Fxk (1.19) of the measured values of position and velocity at the start of the interval. Here F is just a row vector (f1,f2) that gives the coefficients of a linear combination of these measured values. Formally we are in a situation analogous to the PD control (1.6), except that we now assume that the measurements are being made only at discrete times and that a constant control will be applied on each interval. Substituting (1.19) into the difference equation (1.18), there results the new difference equation xk+1 = (A + BF)xk . (1.20) Since for any k xk+2 = (A + BF) 2xk , (1.21) it follows that, if one finds gains f1 and f2 with the property that the matrix A + BF is nilpotent, that is, (A + BF) 2 = 0 , (1.22) then we would have a controller with the property that after two sampling steps necessarily xk+2 = 0. That is, both ϕ and ˙ϕ vanish after these two steps, and the system remains at rest after that. This is the objective that we wanted to achieve all along. We now show that this choice of gains is always possible. Consider the characteristic polynomial det(zI − A − BF) = z2 + (−2cosh δ − f2 sinh δ − f1 cosh δ + f1)z − f1 cosh δ +1+ f1 + f2 sinh δ . (1.23) It follows from the Cayley-Hamilton Theorem that condition (1.22) will hold provided that this polynomial reduces to just z2. So we need to solve for the