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1 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.451 Dynamic Systems – System Representation

System Model Representation

Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell



System models can be developed in a variety of forms. Once the governing equations are developed, then the system characteristics and response can be determined.

Consider a simple mechanical system subjected to initial conditions

and oo xx & m

c

k

x(t)

f(t)



The governing differential equation (chap 4) describing the motion is

Note that there is only one generalized coordinate to describe the motion – x

00 x)0(x;x)0(x && ==0kxxcxm =++ &&& with

Note if a force is applied instead of an initial condition, there is still only one generalized equation

)t(fkxxcxm =++ &&&


State-Space Representation

Another approach commonly used is to represent the system as a larger system of equations – The second order ODE with two variables to describe the “state” of the system rather than one variable

The number of state variable depends on the number of possible initial conditions

For the 2nd order ODE

00 x&xIC)t(fkxxcxm &&&& =++

has two state variables –displacement and velocity


Example

00 x&xIC)t(fkxxcxm &&&& =++

The system is described by

The two mutual conditions of are required to completely describe the “state” of the system. Hence there are two state variables.

00 & xx &

m

c

k

x(t)

f(t)


Example – State Space

In order to solve this, there are three mutual conditions required namely )0(x),0(x),0(x 121 &

Assume 0xxxx 2111 =−++ &&&

0xx2x 212 =++&


State Space – General Formulation

State-variable equations

Where x is a state variableu is an inputf is some function

)t;u...u;x...,x(fx m1n122 =&

)t;u...u;x...,x(fx m1n111 =&

)t;u...u;x...,x(fx m1n1nn =&M


System Outputs

State variables can be expressed as

System variables can be expressed as

)t;u...u;x...,x(hy m1n12 =

)t;u...u;x...,x(hy m1n11 =

M

Note that f and h can be general nonlinear functions

)t,u,x(fx =&

)t,u,x(hy =



For a linear system, this simplifies to

Linear State Variable

mm1111nn11n1 ub...ubxa...xax ++++=&

M

mm2121nn21212 ub...ubxa...xax ++++=&

mnm11nnnn11nn ub...ubxa...xax ++++=&

Linear Outputs

mm1111nn11111 ud...udxc...xcy +++++=

Mmm2121nn21212 ud...udxc...xcy +++++=

mpm11pnpn11pn ud...udxc...xcy +++++=



=

n

2

1

x

xx

xM

State Vector Input VectorOutput Vector

=

p

2

1

y

yy

yM

=

m

2

1

u

uu

uM



→

↓→

↓=

nn1n

n111

aa

aaA

State Matrix

Output Matrix

Input Matrix

Transmission Matrix

→

↓→

↓=

mm1m

m111

bb

bbB

→

↓→

↓=

nn1n

n111

cc

ccC

→

→↓=

mm1m

m111

dd

ddD

State Equation

Output Equation

{ } [ ]{ } [ ]{ }uBxAx +=&

{ } [ ]{ } [ ]{ }uDxCy +=




Governing Equation

)t(fkxxcxm =++ &&&

The state variables are

In order to rewrite the governing equation as 2 first order ODE, the following relationship exists

xx&xx 21 &==

21 xx =&

m

c

k

x(t)

f(t)


Substituting into the governing equation21 xx =&

)t(fkxcxxm 122 =++&

[ ])t(fkxcxm1x 122 +−−=&

which is

Now the state variables equations are:

[ ])t(fkxcxm1x

xx

122

21

+−−=

=

&

&



This can be written in matrix form as:

)t(f0

xx10

xx

m1

2

1

mc

mk

2

1

+

−−

=

&

&

[ ] u0xx

01y2

1 ⋅+

=

which is:



{ } [ ]{ } [ ]{ }uBxAx +=&

)t(f0

xx10

xx

m1

2

1

mc

mk

2

1

+

−−

=

&

&

{ } [ ]{ } [ ]{ }uDxCy +=

[ ] u0xx

01y2

1 ⋅+

=



Example – Electrical Circuit

Source: Dynamic Systems – Vu & Esfandiari


Input-Output EquationsAnother common form is the input-output equation. A single Diff Eq with time derivatives

( ) ( ) yaya...yay n1n1n

1n ++++ −

− &

( ) ( ) ubub...ubub m1m1m

1m

0 ++++= −− &

Note: y(n) is the nth derivativea and b are constant coefficientsy is system outputu is system input

These equations are normally cumbersome and a Laplace Transform is used resulting in a set of algebraic equations.


Example – I/O Equation



System Transfer FunctionOnce again, consider the linear time-invariant differential equation as:

( ) ( ) yaya...yay n1n1n

1n ++++ −

− &

( ) ( ) ubub...ubub m1m1m

1m

0 ++++= −− &

Assuming the initial conditions are zero, the Laplace Transform yields:

( ) )s(yasa...sas n1n1n

1n ++++ −

−

( ) )s(Ubsb...sbsb m1m1m

1m

0 ++++= −−


System Transfer Function

So that the system transfer function is

n1n1n

1n

m1m1m

1m

0

asa...sasbsb...sbsb

)s(U)s(Y)s(G

++++++++

==−

−−

−

This is the system transfer function for a single input-single output (SISO). If there are other inputs and outputs to consider, then the problem becomes a multiple input-multiple output (MIMO)


Relationship between state space and system transfer function

State space

Transfer function

DuCxyBuAxx

+=+=&

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

Laplace Transform of State equation (w/IC = 0) )s(DU)s(CX)s(Y

)s(BU)s(AX)s(sX+=+=

( ))s(DU)s(CX)s(Y)s(BU)s(XAsI

+==−


State space and system transfer function

( ))s(DU)s(CX)s(Y)s(BUAsI)s(X 1

+=−= −

Then

( ) )s(DU)s(BUAsIC)s(Y 1 +−= −

( )[ ] )s(UDBAsIC 1 +−= −

( ) DBAsIC)s(U)s(Y)s(G 1 +−==∴ −


State space and system transfer function

Realizing that ( ) ( )( )AsIdet

AsIadjAsI 1

−−

⇒− −

So that( )( ) D

AsIdetBAsICadj)s(G +

−−

=


Example – State Space Representation

The SDOF state equation:DuCxyBuAxx

+=+=&

=

−−

=

=

m1

mc

mk

2

1

0B

10A

xx

x

[ ]

0D

01C

)t(fu

=

=

=


Now

( )

+−

=−mc

mk s

1sAsI

( ) ( )

+−

++=− −

mc

mk

mk

mc

1s1s

ss1AsI

Then

[ ] ( )

+−

++=

m1

mc

mk

mk

mc

0s1s

ss101)s(G

( ) [ ] ( )( )

+++=

mc

m1

m1

mk

mc s

01ss1



which is the system transfer function. State space representation can also be obtained from the input-output equation.

kcsms1)s(G 2 ++

=⇒

( ) m1

ss1

mk

mc ⋅++

=



LinearizationMany times the equations describing a system may be non-linear. Often, these equations can be “linearized” to determine an acceptable solution. The basic approach is to perform a Taylor Series Expansion about some “operating” point to find an equivalent linear set of equations with the assumption that the system will operate around this point.


Graphical InterpretationThe function can be evaluated in the vicinity of and causes to result.

Considering a small variation of ∆x results in ∆f. In this region, the slope is approximated

by

x f

xxdxdfm ==

Thus xmf)xx(mff ∆=∆⇒−=−


Taylor Series ExpansionThe analytical counterpart to the graphical representation is performed.The Taylor Series Expansion about the pointfor a non-linear function

x)x(f

K+−′′+−′+= ==2

xxxx )xx()x(f!21)xx()x(f)x(f)x(f

In the event that is small then higher order terms can be negligible. Thus

)xx( −

)xx()x(f)x(f)x(f xx −′+≈ =

x)x(fyy xx ∆′+≈ =

OR as done graphically:


Example - Linearization



Mount Stiffness

Example

Car Suspension


Example – Nonlinear Spring


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