1.1 Describing Systems
ECE 5800
Western Michigan University
Fall 2012
1.1 The nature of Systems
A System is an entity
isolated from an
environment with
entry points called
Inputs and exits into
the environment call
Outputs.
Input
x(t)
System
State
y(t)
environment
Zeroth order system
Output
z(t)
Properties
1. All environmental influences on a system can be
reduced to a vector of m real variable varying
with time. 𝑥 𝑡 = 𝑥1 𝑡 … 𝑥𝑚 𝑡 .
2. All system effects can be summarized by a
vector of n real variables varying with time,
z 𝑡 = 𝑧1 𝑡 … 𝑧𝑛 𝑡 .
3. If the output signals are algebraic functions of
only the current input, the system is said to be
zeroth order, since there can be system
dynamics. y 𝑡 = 𝑦1 𝑡 … 𝑦𝑝 𝑡 .
Properties Continued
3. The system can be written as two algebraic
equations involving the input, state, and output:
𝑦 𝑡 = 𝑓1 𝑥 𝑡
z 𝑡 = 𝑓2 𝑥 𝑡 , 𝑦 𝑡
For suitable functions 𝑓1 and 𝑓2 .
4. If the input signal depends dynamically on the
output, there must also be system memory. The
state and output equations are dynamic, 𝑓1 and 𝑓2
depend on time delays, advances, derivatives, and
integrals.
Dynamic System
Dynamic systems have
memory, delays, time
advances, derivatives,
and integrals.
Input
𝑥 𝑡
𝑓1
𝑓2
state
Output
𝑧 𝑡
Example 1.1 Zeroth Order
The source voltage, VS(t),
is the input to the resistor
network. The two
resistors form a simple
system with an output
VR2(t). The state variable is
the current.
Input: 𝑥 𝑡 = 𝑉𝑆 𝑡
Output: z 𝑡 = 𝑉𝑅2 𝑡
State: y 𝑡 = 𝑖(𝑡)
VS(t)
VS(t)
VR2(t)
𝑉𝑅2 (𝑡) =
𝑖=
𝑅2
𝑉 (𝑡)
𝑅1 + 𝑅2 𝑠
𝑉𝑠 𝑡
𝑅1 + 𝑅2
VR2(t)
Example 1.1
Show LtSpice
and MatLab
Example.
Time Driven Models
The solution to
example
𝑣𝐶 𝑡 = 𝑣𝐶 𝑡0
1
+
𝑅𝐶
𝑡
𝑣𝑆 𝜏 𝑒
𝑡0
𝜏−𝑡 𝑅𝐶 𝑑𝜏
Example 1.2
The RC circuit is
driven by a time
signal. The output is
the voltage is across
the capacitor.
The derivation of the
output voltage is
shown.
𝑉𝑆 = 𝑉𝑅 + 𝑉𝐶
𝑉𝑅 = 𝐼𝑅
𝑉𝑅 = 𝐶
𝑉𝑆 = 𝑅𝐶
VS(t)
𝑑𝑉𝐶
×𝑅
𝑑𝑡
𝑑𝑉𝐶
+ 𝑉𝐶
𝑑𝑡
𝑉𝑆 = 𝑅𝐶
𝑑𝑉𝐶
+ 𝑉𝐶
𝑑𝑡
VR2(t)
MATLAB Solution
%Example 1.2
%ECE 5800
%John Stahl
clc;
clear all;
%% Constants
pi = 3.1415926;
%%
n = 1000;
t = 0:1/n:60e-3-1/n;
%% Solution
Vs = 2 + 1*sin(2*pi*60*t);
Vo = -2*exp(-100*t)+0.0657366*sin(2*pi*60*t)+0.247821*cos(2*pi*60*t)+2;
%%
figure(1)
plot(t,Vo,'r');
title('ECE 5800 Example 1.2')
xlabel('time')
ylabel('volts')
LTSpice Solution
Control Systems
desired
𝑥(𝑡)
Controller
Plant
response
𝑧(𝑡)
Reference signal
Open loop control
Plant: subsystem with a relationship we want to
have a prescribed output.
Controller: a subsystem with alters the behavior
of the plant.
Control Systems
desired
𝑥(𝑡)
Controller
𝑢1 𝑡
disturbance
𝑢2 𝑡
Plant
response
𝑧(𝑡)
Closed loop control
Feedback: a signal giving the controller the
response of the plant to the reference signal.
Disturbance: a signal which alters the behavior of
the plant.