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Modeling in the Time
Domain
• How to find a mathematical model, called a state-space
representation, for a linear, time-invariant system

• How to convert between transfer function and state-space
models

• How to linearize a state-space representation

Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.

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Introduction

Classical (frequency-domain) technique:
• Modeling of systems using linear, time-invariant differential equations
and subsequent transfer functions

State-space (time-domain) technique:
• More powerful tool than classical techniques
• Can be used to represent nonlinear systems, systems with nonzero
initial conditions, time-varying systems, multiple-input multiple-
output systems


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State-Space Approach
1. We select a particular subset of all possible system variables and call the
variables in this subset state variables.

2. For an nth-order system, we write n simultaneous, first-order differential
equations in terms of the state variables. We call this system of simultaneous
differential equations state equations.

3. If we know the initial condition of all of the state variables at t0 as well as the
system input for t≥t0, we can solve the simultaneous differential equations for
the state variables for t≥t0.

4. We algebraically combine the state variables with the system's input and find
all of the other system variables for t≥t0. We call this algebraic equation the
output equation.

5. We consider the state equations and the output equations a viable representation
of the system. We call this representation of the system a state-space
representation.


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Example:

We select i(t) and q(t), as the two state variables.

We transform this equation into 2 first order equations in terms of i(t) and q(t)


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From these two state variables, we can solve for all other network variables.

… output equation

state-space representation.

!!! State variables must be linearly independent which means that no
state variable can be chosen if it can be expressed as a linear combination
of the other state variables.


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The state equations

can be written in matrix form as

where

The output equation can be written in matrix form as

where


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The General State-Space Representation

A system is represented in state space by the following equations:
•
x = Ax + Bu
y = Cx + Du


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Example of a second order system:

•
x = Ax + Bu
y = Cx + Du


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Applying the State-Space Representation

State-vector selection:

1. A minimum number of state variables must be selected as components
of the state vector. This minimum number of state variables is sufficient
to describe completely the state of the system.

(Note: Typically, the minimum number equals the order of the
differential equation describing the system or to the number of
independent energy-storage elements in the system)

2. The components of the state vector must be linearly independent.

(Note: Variables related by derivative are linearly independent)


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Problem Given the electrical network, find a state-space representation
if the output is the current through the resistor.

Solution
Select the state variables by writing the derivative equation for all energy-
storage elements, that is, the inductor and the capacitor

next step is to express ic and vL as linear combinations of the state variables
vc and iL, and the input, v(t).


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State variables

After substitution we obtain:


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The output equation:

Final state-space representation:


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Problem Find the state equations for the translational mechanical system

Solution
• It is more convenient when working with mechanical systems to obtain the
state equations directly from the equations of motion rather than from the
energy-storage elements.

• State variables will be the position and velocity of each point of linearly
independent motion.


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We select x1, x2, v1, and v2 as state variables
dx1 / dt = v1
dx2 / dt = v2
dv1 / dt = d 2 x1 / dt 2
dv2 / dt = d 2 x2 / dt 2

State equations:


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Converting a Transfer Function to State Space
• We will show how to convert a transfer function to a state-space representation

• First. we select a set of state variables, called phase variables, where each
subsequent state variable is defined to be the derivative of the previous state
variable.

Consider:

- we choose output y(t) and its (n-1) derivatives as the state variables


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Converting a transfer function with constant term in numerator


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Converting a transfer function with polynomial in numerator


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Example: Converting a transfer function with polynomial in numerator


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Converting from State Space to a Transfer
Function

Taking the Laplace transform of :

Taking the Laplace transform :


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Linearization

• state –space representation can be used to represent systems with
nonlinearities

• state –space representation can be linearized for small perturbations
about an equilibrium point


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Example

Let


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In order to find the transfer function we must linearize the nonlinear state-space
representation

Let equilibrium point x1 = 0, x2 = 0


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