CARLETON UNIVERSITY Department of Systems and Computer Engineering SYSC 3610 - Biomedical Systems, Modeling, and Control Fall Semester, 2016 Assignment #1 Assigned: 21-September-2016 Due: in drop-box near 4499ME by Noon on 4-October-2016 Assignment is to be done individually. PROBLEM 1.1: Derive the Laplace transform for the following time function. f (t) = tu(t) where u(t) is the unit step function. Note that f (t) is the unit ramp function. PROBLEM 1.2: Determine the Laplace transform of the function y(t) shown in the following figure. Begin by writing an equation for y(t) using simple functions. Hint: y(t) can be expressed as a linear combination of two ramp functions and two step functions. 1 of 3 PROBLEM 1.3: Consider a linear time-invariant causal system with input x(t) and output y(t) x(t) - LTIC System H(s) y(t) - that is described the following differential equation d3 y d2 y dy dx d2 x + 5 + 3 +4 + y = + 7x. dt3 dt2 dt dt2 dt Find the expression for the transfer function of the system, H(s) = Y (s)/X(s). PROBLEM 1.4: Consider the following Laplace transforms: 1. F1 (s) = 2(s + 2) s(s + 1)(s + 3) 2. F2 (s) = s s2 + 2s + 10 Solve the following problems for these functions. (a) Determine the inverse Laplace transform f (t) for each function. (b) Determine the initial value f (0+ ) for each function. Use the initial-value theorem. (c) Determine the final value f () (if it exists) for each function. Use the final-value theorem. (d) Show that the results obtained for the initial values and final values in (b) and (c) are the same as those obtained by directly evaluating f (0+ ) and f () in the inverse Laplace transforms from (a). 2 of 3 PROBLEM 1.5: Consider a simplified human kinetics problem where a speed skater is skating along a straight outdoor ice track. Let us presume that the skater has a mass of M = 75 kg and that the sliding coefficient of friction for a skate on ice is bskateonice = 0.0046 and that we are dealing with Couloumb friction (linear with velocity). Let us also presume that the input force over time generated by the skater's feet isfpush (t) and that a tailwind behind the skater would push them with an additional force of fwind (t) over time. To simplify any modelling, we will assume that all forces are forward/backward without any side-to-side or up-and-down forces. Given this description, we can write the following linear constant coefficient ordinary differential equation (LCCODE) describing the velocity of the skater given the two inputs force functions. M v(t) + bskateonice v(t) = fpush (t) + fwind (t) = f (t) (1) (a) Determine the transfer function of the system V (s)/Fpush (s). Set the other inputs, in this case fwind (t), to zero. (b) Determine the transfer function of the system V (s)/Fwind (s). Set the other inputs, in this case fpush (t), to zero. (c) If f (t) = (t), where (t) is a unit impulse, solve for v(t) using Laplace transform techniques. (d) If f (t) = u(t), where u(t) is a unit step, solve for v(t) using Laplace transform techniques. (e) If the skater gives an impulsive push of 1160 N at time t = 4 s, a 935 N push at time t = 6 s, and a 708 N push at time t = 7.8 s, and ( 4.5 1 t < 10, fwind (t) = 0 otherwise, then plot f (t) and give an equation for f (t) as a mixture of impulse functions and unit step functions. (f) Given f (t) from part (e) and assuming the speed skater initially at rest, solve for v(t) so that you know the skater's velocity at any point in time. Note: This is an easy problem if you use the properties of linear time-invariant systems and the results from previous parts of this question. End of Assignment 3 of 3