Introduction and Review of Controls

Question 1

Dynamic Systems

Answer 1

x’(t) = f(x(t),u(t),t): systems that change or evolve over time according to a fixed rule. Rule is governed by some law of physics.

Question 2

State Vector

Answer 2

x(t): variables representing the configuration of the system at time t

Question 3

Input Vectors

Answer 3

u(t): external inputs to the system at time t

Question 4

System Order

Answer 4

Dimensionality of state-space model. Corresponds to the number of independent energy storage elements in the system

Question 5

Linear Time Invariant

Answer 5

system that is obeys superposition (linear: f(ax,by) = af(x)+bf(y)) and is time-independent (time-invariant: f(x(t+T)) = f(x(t)).

Question 6

State Space Model for Continuous LTI systems

Answer 6

x' = Ax + Bu
y = Cx + Du
A: system matrix
B: input matrix
C: output matrix
D: feed-foward matrix

Question 7

Purpose of State Space Models

Answer 7

1. Handle MIMO systems
2. Solve systems with non-zero I.C.
3. Solve non-linear systems

Question 8

Frequency Response of LTI Systems

Answer 8

Magnitude and phase difference as a function of frequency for output of LTI given a sinusoidal input

Question 9

Purpose of Laplace Transform of LTI Systems

Answer 9

Calculate transfer function which represents input/output relation in frequency domain

Question 10

Laplace Transform

Answer 10

F(s) = L(f(t)) = integral(exp(-s*t)*f(t)*dt) [0, inf]
s = sigma + j*omega

Question 11

Laplace Transform of nth Derivative

Answer 11

L(d^nf/dt^n) = s^nF(s) - s^(n-1)f(0) - s^n-2*f’(0) - … - f’…‘(0)

Question 12

Frequency-Domain Methods

Answer 12

Used for analyzing LTI SISO systems

1. a_n y’…’ + … + a_1y’ + a_0y = b_mu’…’ +…+ b_1u’ + b_0u
2. a_ns^nY(s) + …+ a_1sY(s) + a_0Y(s) = b_ms^mU(s) + … + b_1sU(s) + b_0*U(s)
3. Transfer Function: G(s) = N(s)/D(s) = K*(s-z1)(s-z2)…(s-zm)/(s-p1)s-p2)…(s-pn)
4. System Gain: K = bm/an
5. Transfer Function from State Space Model: C(sI-A)^-1*B + D

Question 13

Sum of Forces

Answer 13

Sum(Forces) = m*a

Question 14

Governing Equation of Mass Spring

Answer 14

F(t) - bx’ - kx = m*x’’
k: spring constant
b: viscous damping force
x’ = [x’;x’’] = [0 1; -k/m -b/m][x;x’] + [0; 1/m] * F(t)

Question 15

Transfer Function of Mass-Spring Model

Answer 15

H(s) = 1/(ms^2+bs+k)

Question 16

KCL

Answer 16

Sum of currents entering a node is equal to sum of currents exiting a node

Question 17

KVL

Answer 17

Sum of voltages in a closed loop is zero

Question 18

ODE for Series RLC Circuit

Answer 18

V(t) - Ri -Ldi/dt - int(i*dt)/C = 0

Question 19

Relation of RLC Circuit to Spring Mass Model

Answer 19

Charge: displacement
Inductance: mass
Resistance: viscous damping
Capacitance: spring stiffness

Question 20

State Space of Series RLC Circuit

Answer 20

x' = [q; i] = [0 1; -1/LC -R/L][q; i] + [0; 1/L]*V(t)
y = [0 1][q; i]

Question 21

Transfer Function of Series RLC Circuit I(s)/V(s)

Answer 21

I(s)/V(s) = s/(Ls^2 + Rs + 1/C)