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Unformatted text preview: ME375 Handouts Standard Forms for System Models
• State Space Model Representation
– State Variables
State Variables
– Example • Input/Output Model Representation
– General Form
– Example • Comments on the Difference between State Space
and Input/Output Model Representations School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  1 State Space Model Representation
• State Variables
The smallest set of variables {q1, q2, …, qn} such that the knowledge of these
variables at time
variables at time t = t0 , together with the knowledge of the input for t ≥ t0
together with the knowledge of the input for
completely
completely determines the behavior (the values of the state variables) of the
system for time t ≥ t0 .
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)
What information about the mass do we need to know to be able to solve for
for
for t ≥ t0 ? School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  2 1 ME375 Handouts State Space Model Representation
• State Space Representation
– Two parts:
set of first order ODEs that represents the derivative of each state
• A set of first order ODEs that represents the derivative of each state
variable
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}.
th
th
⎧ y1
⎪
⎪ y2
⎨
⎪
⎪y
⎩p = 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 , u1 , u 2 , u 3 , K , u m ) School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  3 State Space Model Representation
• Example
EOM &
M && + B x + K x = f (t )
x x K
M f(t) B State Variables:
Output:
State Space Representation: School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  4 2 ME375 Handouts 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.
– Express the output variables as algebraic functions of the state variables
and inputs
and inputs.
– For linear systems, put the equations in matrix form. &
x = A⋅ x
{ +B⋅ State Variables
in vector form y
{ Outputs in
vector form u
{ Inputs in
vector form = C ⋅x + D⋅ u
School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  5 State
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 ⎦ ⎣
q
⎣
{ 144 2444 ⎦ ⎣ 4 ⎦ ⎣ ⎦
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 3 ME375 Handouts Input/Output Representation
• Input/Output Model
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.
and the output variable
of 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:
• Identify input/output variables.
• Derive equations of motion.
• Combine equations of motion by eliminating all variables except the
input and output variables and their derivatives.
School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  7 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
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 4 ME375 Handouts Comments on Input/Output and State Space Models
• State Space Models:
– consider the internal behavior of a system
the internal behavior of system
– can easily incorporate complicated output variables
– have significant computation advantage for computer
simulation
– can represent multiinput multioutput (MIMO) systems and
multimultinonlinear systems School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  9 Comments on Input/Output and State Space Models
• Input/Output Models:
– are conceptually simple
conceptually simple
– are easily converted to frequency domain transfer functions
that are more intuitive to practicing engineers
– are difficult to solve in the time domain (solution: Laplace
transformation) School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  10 5 ME375 Handouts Example: General Mechanical Systems
Rack and pinion system: School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  11 Example Continued School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  12 6 ME375 Handouts Example: General Mechanical Systems
1. Let x be measured from the unstretched state of the spring, find EOM in terms of x;
2. Let z be the motion of the system about the static equilibrium, find EOM in terms of z. School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  13 Example Continued School of Mechanical Engineering
Purdue University ME375 Standard Forms of Equation  14 7 ...
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This note was uploaded on 12/23/2011 for the course ME 375 taught by Professor Meckle during the Fall '10 term at Purdue UniversityWest Lafayette.
 Fall '10
 Meckle

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