Unformatted text preview: Standard Forms for System Models
Standard Forms for System Models
• State Space Model Representation
State Space Model Representation
– State Variables
– Example • Input/Output Model Representation
– General Form
– Example • Comments on the Difference between State Space
and Input/Output Model Representations
and Input/Output Model Representations School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  1 State Space Model Representation
State Space Model Representation
• State Variables
The smallest set of variables {q1, q2, …, qn} such that the knowledge of these
smallest set of variables
such that the knowledge of these
variables
variables at time t = t0 , together with the knowledge of the input for t ≥ t0
completely
completely determines the behavior (the values of the state variables) of the
system for time t ≥ t0 .
Example
Example:
x
K
EOM:
M
f(t)
B Q: What information about the mass do we need to know to be able to solve for x(t)
for
for t ≥ t0 ? School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  2 State Space Model Representation
State Space Model Representation
• State Space Representation
– Two parts:
parts:
• A set of first order ODEs that represents the derivative of each state
variable qi as an algebraic function of the set of state variables {qi} and
the inputs {ui}.
&
⎧ q1
⎪q
⎪ &2
⎨
⎪
⎪ qn
⎩& = f 1 (q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m ) = f 2 (q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m )
M = f n (q1 , q 2 , q 3 , K , q n , u1 , u 2 , u 3 , K , u m ) • A set of equations that represents the output variables as algebraic
functions of the set of state variables {qi} and the inputs {ui}.
⎧ y1
⎪
⎪ y2
⎨
⎪
⎪
⎩ yp = g 1 ( q1 , q 2 , q 3 , K , q n , u 1 , u 2 , u 3 , K , u m ) = g 2 ( q1 , q 2 , q 3 , K , q n , u 1 , u 2 , u 3 , K , u m )
M = g p ( q1 , q 2 , q 3 , K , q n , u 1 , u 2 , u 3 , K , u m ) School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  3 State Space Model Representation
State Space Model Representation
• Example
EOM &&
&
M x + B x + K x = f (t ) x K
M f(t) B State Variables:
Output:
State Space Representation: School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  4 State Space Model Representation
State Space Model Representation
• Obtaining State Space Representation
– Identify State Variables
• Rule of Thumb:
– Nth order ODE requires N state variables.
– Position and velocity are natural state variables for translational
mechanical systems. – Eliminate all algebraic equations written in the modeling process.
– Express the resulting differential equations in terms of state variables and
inputs in coupled first order ODEs
inputs in coupled first order ODEs.
– Express the output variables as algebraic functions of the state variables
and inputs.
– For linear systems, put the equations in matrix form.
linear systems put the equations in matrix form &
x = A⋅ x
{ State Variables
in vector form y
{ Outputs in
vector form +B⋅ u
{ Inputs in
vector form = C ⋅x + D⋅ u
School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  5 State Space Model Representation
State Space Model Representation
• Exercise
Represent the 2 DOF suspension system
in a state space representation. Let the system output be the
position of mass M1.
&
&
M 1&&1 + B1 x1 − B1 x2 + K1 x1 − K1 x2 = 0
x x1 M1
g
K1 B1
x2 &
&
M 2 &&2 − B1 x1 + B1 x2 − K1 x1 + ( K1 + K 2 ) x2 = K 2 xr
x M2 State Variables:
K2 Output: xr State Space Representation: &
⎡ q1 ⎤ ⎡
⎤ ⎡ q1 ⎤ ⎡ ⎤
⎢q ⎥ ⎢
⎥ ⎢q ⎥ ⎢ ⎥
&2
⎢ ⎥=⎢
⎥ ⎢ 2 ⎥ + ⎢ ⎥ xr
&
⎢ q3 ⎥ ⎢
⎥ ⎢ q3 ⎥ ⎢ ⎥
⎢& ⎥ ⎢
⎥⎢ ⎥ ⎢ ⎥
⎦ ⎣ q4 ⎦ {
⎣ q4 ⎦ ⎣
{ 144 2444 { ⎣ ⎦
4
3
&
B
x
x
A , School of Mechanical Engineering
Purdue University ⎡ q1 ⎤
⎢q ⎥
y =[ ⎢ q2 ⎥
14 244 ⎢ 3 ⎥
4
3
C
⎢⎥
⎣ q4 ⎦
{
x
ME375 Standard Forms of Equation  6 Input/Output Representation
Input/Output Representation
• Input/Output Model
Uses one nth order ODE to represent the relationship between the input
Uses one nth order ODE to represent the relationship between the input
variable,
variable, u(t), and the output variable, y(t), of a system.
For linear timeinvariant (LTI) systems, it can be represented by :
time &
&&
&
an y ( n ) + L + a2 && + a1 y + a0 y = bmu ( m ) + L + b2u + b1u + b0u (t )
y where y ( n ) = ( d dt ) y
n – To solve an input/output differential equation, we need to know – To obtain I/O models:
I/O
• Identify input/output variables.
• Derive equations of motion.
• Combine equations of motion by eliminating all variables except the
Combine equations of motion by eliminating all variables except the
input
input and output variables and their derivatives.
School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  7 Input/Output Representation
Input/Output Representation
• Example
z2 Vibration Absorber
EOM: M2 &
M 1&&1 + B1 z1 + ( K1 + K 2 ) z1 − K 2 z2 = f (t )
z M 2 &&2 + K 2 z2 − K 2 z1 = 0
z
– Find input/output representation between
betwee
input f(t) and output z2. K2
z1
M1
K1 School of Mechanical Engineering
Purdue University f(t) B1 ME375 Standard Forms of Equation  8 Input/Output Representation
In
• Vibration Absorber
z2
M2
K2
z1
M1
K1 f(t) B1 Q: Find input/output representation between input f(t) and output z1.
Find
Q: What if another damper is added between masses M1 and M2 ?
What
School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  9 Comments on Input/Output and State Space Models
Comments on Input/Output and State Space Models
• State Space Models:
– consider the internal behavior of a system
– can easily incorporate complicated output variables
– have significant computation advantage for computer
simulation
– can represent multiinput multioutput (MIMO) systems and
ou
(MIMO)
nonlinear
nonlinear systems School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  10 Comments on Input/Output and State Space Models
Comments on Input/Output and State Space Models
• Input/Output Models:
– are conceptually simple
– are easily converted to frequency domain transfer functions
that
that are more intuitive to practicing engineers
– are difficult to solve in the time domain (solution: Laplace
transformation)
transformation) School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  11 ...
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 Fall '10
 Meckle
 Input/output, School of Mechanical Engineering

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