EE402 Mini Project 1 solution


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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
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
h(t − τ )u(τ )dτ
h(t) = 
t if t > 10
0 if t ≤ 0
∗This document c M. Mert Ankarali
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
, 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
¨θ −
sin θ =
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.