EE 510: Lumped Systems Theory - Model Representations of Systems, Lecture notes of Teaching method

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EE 510: Lumped Systems Theory

Fall 2016

Handout 2:

Model Representations of Systems

Prof. Mohamed Zribi

Updated 2 October 2016

Outline of the Handout

1. Introduction to Mathematical Models of Physical Systems

2. Conversion between the Different Model Representations

3. Linking State Space Representations And Transfer Functions

4. Equivalent State Space Representations

5. Solutions of State Space Equations

6. Modal Decomposition of the Solution of

7. Modal Decomposition of the Solution of

8. Linearization of Nonlinear Systems

x t ^ ( ) = Ax t ( )

x t ^ ( ) = Ax t ( ) + Bu t ( )

1. Introduction to Mathematical

Models of Physical Systems

Control systems give desired output by controlling the

input. Therefore, control systems and mathematical modeling are inter-linked.

Mathematical Models of Physical Systems

Why are mathematical models of physical systems needed?

 Design of engineering systems can be completed by 1)

trial and error, or 2) by using mathematical models.

 Mathematical model gives the mathematical relationships

relating the output of a system to its input.

 Physical laws such as Newton’s second law of motion

result in a mathematical model.

Modeling

 Models are used to describe the operation of the plant, including sensors and actuators

 Models capture how variables relate to each other  The designer needs to pay close attention to how input(s) affects output(s)

 The designer needs to use appropriate level of abstraction vs details  Note that many types of physical systems share the same math model  focus on models

Modeling Guidelines

 Focus on important variables  Use reasonable approximations

 Write mathematical equations from physical laws, don’t invent your own  Eliminate intermediate variables

 Obtain o.d.e. involving input/output variables I/O model  Or, obtain 1 st^ order o.d.e.s state space model

 Or, obtain I/O transfer function

Classical Approach (Transfer Function)

 The classical approach or frequency domain technique is based on converting a system’s differential equation to a transfer function.

 It relates a representation of the output to a representation of the input.

 It can only be applied to linear, time-invariant systems.

 It rapidly provides stability and transient response information.

EE 510 Lumped Systems Theory Dr. Mohamed Zribi (^) 10

Modern Approach (State-space)

 It can be used to model and analyze nonlinear

(backlash, saturation), time-varying (missiles with varying fuel levels), multi-input multi-output systems (i.e. an airplane) with nonzero initial conditions.

 But it is not as intuitive as the classical approach. The

designer has to engage in several calculations before the physical interpretation of the model is apparent.

EE 510 Lumped Systems Theory Dr. Mohamed Zribi (^) 11

Type 2: Transfer function model:

Using the I/O o.d.e. model, take the Laplace transform

(assuming zero IC): [ ]

[ ]

(^22) 2

( ) ( ), ( ) ( ),

( ) ( ), ( ) ( ), ( ) ( )

y t Y s d y t sY s dt d (^) y t s Y s dt u t U s d u t sU s dt

→   →     →   →   →  





L L

L L L

Then, the I/O model in the s-domain becomes:

This is the T.F. model of the system.

or

i.e. the Laplace transform of the output is equal to the Laplace transform input L.T. with gain G(s).

( ) ( ) ( )

( ) ( ) ( ) ( ) 1 0

1 0 1 1 b s U s b sU s b U s

s Y s a s Y s a sY s a Y s m m

n n n

= + + +

• (^) − − + + + 



1 0

1 1

1 0 ( )

( ) s a s a s a

b s b s b U s

Y s n n

n

m m

= (^) − − 



denote = G s ( )

( ) ( ) ( )

G s Y s U s

= Y s ( )^^ = G s U s ( )^ ( )

Remark: Rational Transfer Function

For linear lumped system, is a rational function. We define that is proper

is strictly proper

is improper

G s N s

D s

G s ( )

⇔ deg D s ( ) ≥ deg N s ( )

⇔ deg D s ( ) > deg N s ( )

⇔ deg D s ( ) < deg N s ( )

G s ( )

G s ( )

Input equation Output equation

Type 3: State Space Model: The state space representation of

a a linear system is such that:

State Space Model

A state space model is a representation of the

dynamics of an Nth^ -order system as a first-order

equation in an N-vector, which is called the state.

Convert an Nth-order differential equation that

governs the dynamics of the system into N first-

order differential equations.

State-Space Model

The model of a system is described by a set of first-order

differential equations written in terms of the state variables.