1.1 Describing Systems

ECE 5800Western 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.

SystemInputx(t)

Outputz(t)

Statey(t)

environment

Zeroth order system

Properties

1. All environmental influences on a system can be reduced to a vector of m real variable varying with time.

2. All system effects can be summarized by a vector of n real variables varying with time, z

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.

Properties Continued

3. The system can be written as two algebraic equations involving the input, state, and output:

zFor suitable functions and .4. If the input signal depends dynamically on the output, there must also be system memory. The state and output equations are dynamic, and depend on time delays, advances, derivatives, and integrals.

Dynamic System

Dynamic systems have memory, delays, time advances, derivatives, and integrals.

𝑓 1

𝑓 2state

Input Output

VS(t) VR2(t)

𝑉 𝑅 2(𝑡)=𝑅2

𝑅1+𝑅2

𝑉 𝑠 (𝑡)VS(t) VR2(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: Output: State:

𝑖=𝑉 𝑠 (𝑡 )𝑅1+𝑅2

Example 1.1

Show LtSpice and MatLab Example.

Time Driven ModelsThe solution to example

𝑣𝐶 (𝑡 )=𝑣𝐶 (𝑡 0 )+ 1𝑅𝐶∫

𝑡0

𝑡

𝑣𝑆 (𝜏 )𝑒(𝜏−𝑡 ) /𝑅𝐶𝑑𝜏

Example 1.2The 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.

𝑉 𝑆=𝑉 𝑅+𝑉 𝐶

𝑉 𝑅=𝐼𝑅

𝑉 𝑅=𝐶𝑑𝑉 𝐶

𝑑𝑡×𝑅

𝑉 𝑆=𝑅𝐶𝑑𝑉 𝐶

𝑑𝑡+𝑉 𝐶

𝑉 𝑆=𝑅𝐶𝑑𝑉 𝐶

𝑑𝑡+𝑉 𝐶

VS(t) VR2(t)

MATLAB Solution%Example 1.2%ECE 5800%John Stahl clc;clear all; %% Constantspi = 3.1415926; %%n = 1000;t = 0:1/n:60e-3-1/n; %% SolutionVs = 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

Open loop controlPlant: subsystem with a relationship we want to have a prescribed output.Controller: a subsystem with alters the behavior of the plant.

Controller Plantdesired response

Reference signal

Control Systems

Closed loop controlFeedback: a signal giving the controller the response of the plant to the reference signal.Disturbance: a signal which alters the behavior of the plant.

Controller Plantdesired response

disturbance

𝑢1 (𝑡 )