Standard Forms for System Models

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School of Mechanical Engineering

Purdue University ME375 Standard Forms of Equation - 1

Standard Forms for System Models Standard Forms for System Models

• • State Space Model Representation State Space Model Representation

– – State Variables State Variables – – Example Example

• • Input/Output Model Representation Input/Output Model Representation

–

– General Form General Form – – Example Example

• • Comments on the Difference between State Space Comments on the Difference between State Space and Input/Output Model Representations

and Input/Output Model Representations

• • State Variables State Variables

The smallest set of variables {

The smallest set of variables {qq_1,_1,qq_2,_2,……_,_,qq_n_n} such that the knowledge of these } such that the knowledge of these variables at time

variables at time t = tt = t₀₀, together with the knowledge of the input for t , together with the knowledge of the input for t ≥ tt₀₀ completely determines the behavior (the values of the state vari

completely determines the behavior (the values of the state variables) of the ables) of the system for time

system for time t t ≥ tt₀₀.. Example

Example:: EOM:EOM:

State Space Model Representation State Space Model Representation

K x

M B

f(t)

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• • State Space Representation State Space Representation

–

– Two parts:Two parts:

•

• A set of A set of first order ODEsfirst order ODEsthat represents the derivative of each state that represents the derivative of each state variable

variable qq_i_ias an algebraic function of the set of state variables {as an algebraic function of the set of state variables {qq_i_i} and } and the inputs {

the inputs {uu_i_i}.}.

•• A set of equations that represents the output variables as algebA set of equations that represents the output variables as algebraic raic functions of the set of state variables {

functions of the set of state variables {qq_i_i} and the inputs {u} and the inputs {u_i_i}.}.

State Space Model Representation State Space Model Representation

( )











=

m n

n n

m n

u u u u q q q q f q

, , , , , , , , ,

3 2 1 3

2 1

3 2 1 3

2 1 2 2

3 2 1 3

2 1 1 1

…

( )

1 1 1 2 3 1 2 3

2 2 1 2 3 1 2 3

1 2 3 1 2 3

, , , , , , , , ,

n m

p p n m

y g q q q q u u u u

y g q q q q u u u u

=



 =



 =



… …

• • Example Example

EOM EOM

State Variables:

Output:

State Space Representation:

State Space Model Representation State Space Model Representation

K x

M B

M x+B x+K x= f t( ) f(t)

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• • Obtaining State Space Representation Obtaining State Space Representation

–– Identify State VariablesIdentify State Variables

•

• Rule of Thumb:Rule of Thumb:

–

– Nth order ODE requires N state variables.Nth order ODE requires N state variables.

–

– Position and velocity are natural state variables for translational Position and velocity are natural state variables for translational mechanical systems.

mechanical systems.

–

– Eliminate all algebraic equations written in the modeling procesEliminate all algebraic equations written in the modeling process.s.

–– Express the resulting differential equations in terms of state vExpress the resulting differential equations in terms of state variables and ariables and inputs in coupled first order ODEs.

inputs in coupled first order ODEs.

–– Express the output variables as algebraic functions of the stateExpress the output variables as algebraic functions of the statevariables variables and inputs.

and inputs.

–– For linear systems, put the equations in matrix form.For linear systems, put the equations in matrix form.

State Space Model Representation State Space Model Representation

State Variables Inputs in in vector form vector form

Outputs in vector form

= ⋅ + ⋅

x A x B u

y C x D u

•• ExerciseExercise

Represent the 2 DOF suspension system Represent the 2 DOF suspension system

in a state space representation. Let the system output be the in a state space representation. Let the system output be the position of mass

position of mass MM₁₁..

State Variables:

Output:

State Space Model Representation State Space Model Representation

K₁ M₁

B1

M₂

K₂

x₁

x₂

x_r g

( )

1 1 1 1 1 2 1 1 1 2

2 2 1 1 1 2 1 1 1 2 2 2

0

r

M x B x B x K x K x

M x B x B x K x K K x K x

+ − + − =

− + − + + =

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• • Input/Output Model Input/Output Model

Uses one nth order ODE to represent the relationship between the Uses one nth order ODE to represent the relationship between theinput input variable,

variable, uu((tt), and the output variable, ), and the output variable, yy((tt), of a system.), of a system.

For linear time

For linear time--invariant (LTI) systems, it can be represented by :invariant (LTI) systems, it can be represented by :

where where

–– To solve an input/output differential equation, we need to knowTo solve an input/output differential equation, we need to know

–

– To obtain I/O models:To obtain I/O models:

•

• Identify input/output variables.Identify input/output variables.

•

• Derive equations of motion.Derive equations of motion.

•• Combine equations of motion by eliminating all variables except Combine equations of motion by eliminating all variables except the the input and output variables and their derivatives.

input and output variables and their derivatives.

Input/Output Representation Input/Output Representation

( ) ( )

2 1 0 2 1 0

( )

n m

a y + + a y + a y + a y = b u + + b u b u + + b u t

( )

( )n d n

y =

dt

y

Input/Output Representation Input/Output Representation

•

• ExampleExample Vibration Absorber Vibration Absorber EOM:EOM:

–– Find input/output representation betweenFind input/output representation between input

input ff((tt) and output ) and output zz₂₂..

M₂

K₂ z₂

K1

M₁

B₁ z₁

f(t)

( )

1 1 1 1 1 2 1 2 2

2 2 2 2 2 1

( ) 0

M z B z K K z K z f t

M z K z K z

+ + + − =

+ − =

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Input/Output Representation Input/Output Representation

•• Vibration AbsorberVibration Absorber

Q

Q: : Find input/output representation between input f(t) and output zFind input/output representation between input f(t) and output z₁1.. QQ: : What if another damper is added between masses MWhat if another damper is added between masses M₁₁and Mand M₂₂??

M₂

K₂ z₂

K1

M₁

B₁ z₁

f(t)

Comments on Input/Output and State Space Models Comments on Input/Output and State Space Models

• • State Space Models: State Space Models:

– – consider the internal behavior of a system consider the internal behavior of a system

– – can easily incorporate complicated output variables can easily incorporate complicated output variables – – have significant computation advantage for computer have significant computation advantage for computer

simulation simulation

– – can represent multi can represent multi- -input multi input multi -output (MIMO) systems and - output (MIMO) systems and nonlinear systems

nonlinear systems

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Comments on Input/Output and State Space Models Comments on Input/Output and State Space Models

• • Input/Output Models: Input/Output Models:

– – are conceptually simple are conceptually simple

– – are easily converted to frequency domain transfer functions are easily converted to frequency domain transfer functions that are more intuitive to practicing engineers

Standard Forms for System Models