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Modeling in the Time Domain

Fourth Academic Year

Electrical Engineering Department

College of Engineering

Salahaddin University

November - 2015

Why time-domain representation?

Two approaches are available for analysis and design of feedback control systems:

1. Classical or frequency-domain

2. Modern or time-domain, also known as a state-space approach

The first approach is based on converting a system’s differential equation to a

transfer function. Thus generating a mathematical model of the system that

algebraically relates a representation of the output to a representation of the

input.

Replacing a differential equation with an algebraic equation not only simplifies

the representation of individual subsystems but also simplifies modeling

interconnected subsystems.

The main disadvantage of the classical approach is its limited applicability. It can

be applied only a linear, time-invariant systems or systems that can be

approximated as such.

On the other hand, the major advantage of this technique is that they rapidly

provide stability and transient response information. That is, the effect of varying

system parameters can be seen immediately until an acceptable design is met.

Control Engineering Salahaddin University/College of Engineering2 of 21

Why time-domain representation?

With the presence of space exploration, modeling systems using classical

approaches are inadequate.

The state-space approach is a unified method for modeling, analyzing, and

designing a wide range of systems.

For example, state-space approach can be used to represent nonlinear systems

that have backlash, saturation, and dead zone. Missile is one example that can

well represented.

Many systems do not have just a single input and a single output.

Multiple-input, multiple-output, such as a vehicle with input direction and input

velocity yielding an output direction and an output velocity, can be compactly

represented in state-space with a model similar in for and complexity to that used

for single-input, single-output systems.


General State-Space Representation

The general form of the state-space model is



System variable: Any variable that responds to an input or initial conditions in a

system.

State variable: The smallest set of linearly independent system variables such that

the values of the members of the set at time t0 along with known forcing functions

completely determine the value of all system variable for all t t0.

State vector: A vector whose elements are the state variables.

State space: The n-dimensional space whose axes are the state variables.

State equation: A set of n simultaneous, first-order differential equations with n

variable, where the n variables to be solved are the state variables.

Output equation: The algebraic equation that expresses the output variables of a

system as linear combinations of the state variables and the inputs.



For example, for a linear, time-invariant, second-order system with a single input

v(t), the state equations could be as the following form:

Where x1 and x2 are the state variables.

If there is a single output, the equation could be as:

𝑥 =

𝑑𝑥1

𝑑𝑡𝑑𝑥2

𝑑𝑡

=

𝑥1

𝑥2

𝑥 =

𝑥1

𝑥2

A=𝑎11 𝑎12

𝑎21 𝑎22

B=

𝑏1

𝑏2

C= 𝑐1 𝑐2 D= 𝑑1 𝑢 = 𝑣(𝑡)


State-Space Modeling

Example: Find a state-space representation of the following electrical circuit if the

output is the current through the resistor.

Answer



Example: Find the space-state representation of the following circuit for the output v0(t).

Answer



Example: Find the state-space representation for the following mechanical system.

Answer



Assignment: Chapter 3 Part One

HW3.1 Find a state-space representation for the following electrical circuit, where vo(t) is

the output.

HW3.2 Consider the dc servomotor and load shown in the following figure, represent the

system in state space, where the state variables are the armature current, ia, load

displacement, L, and load angular velocity, wL. Assume that the output is the angular

displacement of the armature. Do not neglect armature inductance.


Converting a Transfer Function to the State Space

Consider the differential equation

A convenient way to choose state variable is to choose the output, y(t), and its (n-1)

derivatives as the state variables. This is called the phase-variable technique.

Let the state variable denoted by xi; thus



Differentiating both sides, yields

Note: the dot above the x signifies differentiation with respect to time.



Now, for both state variables and their derivatives:

In matrix form,



Finally, since the solution to the differential equation is y(t) or x1, therefore the output

equation in matrix for is:



Example: Find the state-space model of the transfer function for the following system block

diagram.

Answer



Example: Find the state-space representation of the transfer function shown in following

block diagram.

Answer


Converting from State Space to a Transfer Function

Given the state and output equations

Solving for X(s)

Taking the Laplace transform assuming zero initial conditions:

or

where I is the identity matrix. Substituting the above equation in the output equation of Y(s):



𝑪 𝒔𝑰 − 𝑨 −𝟏𝑩+ 𝑫 called a transfer function matrix. Since it releases the output vector,

Y(s), to the input vector, U(s). However, if U(s)=U(s) and Y(s) = Y(s) are scalars, then the

transfer function will be



Example: Given the system defined by the following state-space model, find the transfer

function, T(s)=Y(s)/U(s), where U(s) is the input and Y(s) is the output.

Answer


State Space Model

Assignments: Chapter 3 Part Two

HW 3.3

Find the state equations and the output equation for the following transfer function:

𝑮 𝒔 =𝟒𝒔 + 𝟐

𝟐𝒔𝟐 + 𝟖𝒔 + 𝟏𝟎

HW 3.4

Convert the following state-space model to a transfer function:

𝒙 =𝟓 −𝟑. 𝟓𝟐 𝟖

𝒙 +𝟎𝟐

u(t)

𝒚 = 𝟏 𝟒 𝒙


End of Chapter Three!


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