## Description

1. (20 Points) For each of the following systems with input u and output y, t ≥ 0, determine whether the

system is memoryless, linear, time-invariant, causal, finite-dimensional ? Justify your answers!

(a) y(t) = (sin(t))3

(b) y(t) = Rt

0

τu(τ )dτ

(c) y(t) = 2u(t) + 10

(d) y(t) = cos(t)u(t)

(e) y(t) = u(t − T)

(f) y[n] = u[k − n] (Discrete time version of the above system)

(g) Now let’s consider the following input–output dynamical system. The expression inside the blockdiagram is the transfer funtion.

x(t) y(t)

2. (20 Points) In this problem, we will review the basic properties of the convolution operation, denoted

by ∗, as well as those of the Laplace transform, denoted by L. Consider f : R 7→ R, and g : R 7→ R,

and h : R 7→ R.

(a) Show that ∗ is associative that is (f ∗ g) ∗ h = f ∗ (g ∗ h).

(b) Show that f(t − τ ) = f(t) ∗ δ(t − τ ), τ ≥ 0. This property is referred to as the sifting property of

the dirac delta function δ(t).

(c) Show that L(f ∗ g) = L(f)L(g).

(d) Show that L(f + g) = L(f) + L(g).

3. (15 Points) Compute Y (s)/U(s) for the following system

y(t) = Zt

t−T

h(t − τ )u(τ )dτ

h(t) =

t if t > 10

0 if t ≤ 0

∗This document c M. Mert Ankarali

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4. (15 Points) In this question you will analyze the the control system that is illustrated with the block

diagram topology given below. Let’s assume that M(s) = 1

s−a

, a > 0 and C(s) = K

s+1 . Find the range

of K such that the closed-loop system is stable.

5. (30 points) Consider an inverted pendulum of length L, with mass m, that is actuated by an agonist/antagonist linear actuator pair that attach a distance ` from the joint / pivot point. One can

show

¨θ −

g

L

sin θ =

1

mL2

τ (t), (1)

where τ (t) is the net moment that results from forces applied

by the linear actuators.

d d

Suppose the left and right actuators produce linear contractile forces FL and FR, respectively. If we

assume that ` d, we can have the following simplification:

τ ≈ (d cos θ)u(t) (2)

where u(t) = ∆F(t) = FL(t) − FR(t), the difference between the forces applied by the muscles.

IMPORTANT: For the subsequent problems, use Eq. (2) for the torque unless you want a nightmare

of a calculation.

(a) Combine Eq. (1) with (2), make a small-angle approximation to linearize the dynamics, and find

a proper ODE that governs the linearized equations of motion.

(b) Compute the transfer function P(s) = Θ(s)/U(s). Call this the “plant”. Find the poles. Is the

system stable or unstable and why?

(c) Let

g = 9.81 m/s2

, L = 0.3924 m , M = 1 kg , d = 0.39242m

Re-evaluate the transfer function using these quantities. Then, draw the root-locus of the plant

by hand (based on rules covered in EE302) as well as in MATLAB. Decide if the system can can

be controlled with a P controller or not.

(d) Design a “controller” (Gc(s)) so that the closed-loop “linear” system is stable and provide the

transfer function of the closed-loop system. No other performance specification is given, just the

stability condition.

(e) Draw the step and impulse response of the closed loop system using Control System Toolbox

of MATLAB. Hint: “step” and “impulse ”commands. By looking at these responses can you

comment on the stability of the closed-loop system.

(f) Plot (in MATLAB) the bode diagrams/plots of the feedforward transfer function Gc(s)∗G(s) and

find the Phase and Gain margin. Can you comment on the stability of the closed-loop system

based on these margins.

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