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Unformatted text preview: ME 575 Handouts Modeling
Physical Description
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Schematic
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Differential Equations
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Transfer Function
State Space Model ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 1 Some Terminology
• Nominal Model
Simplified approximate model used for control design • Calibration (or Simulation) Model
Model that captures all pertinent aspects of plant behavior
to be used for controller validation • Model Error
The difference between the nominal and calibration model ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 2 1 ME 575 Handouts Building Models
Three approaches:
• From physical principles
• From experimental data (blackbox approach)
• Combination of the above Always use the simplest model that captures
the essential aspects of the process for control
design.
ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 3 Example: Disk Drive Head
Objective:
to precisely locate the
head over the disk ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 4 2 ME 575 Handouts Model of Head Disk Assembly
Schematic: V =L di
+ Ri + Vb
dt Tm = K Ti
Vb = Kbω J dω
+ Bω = Tm
dt School of Mechanical Engineering
Purdue University ME575 Session 4 – Modeling Slide 5 Model of Head Disk Assembly
Block Diagram:
+
− ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 6 3 ME 575 Handouts Model of Head Disk Assembly Transfer Function: Input/Output Equation: ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 7 State Space Representation ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 8 4 ME 575 Handouts In Matrix Form ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 9 Some Definitions
• State  the smallest set of n variables (state
variables) such that knowledge of these n
variables at t = t0, together with knowledge of
the input for t ≥ t0, completely and uniquely
determines system behavior for t ≥ t0.
• State vector  nth order vector whose
components are the state variables
• State space  ndimensional space whose
coordinate axes consist of the x1 axis, x2 axis,
etc.
ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 10 5 ME 575 Handouts More Definitions
• State trajectory  path produced in the state space
by the state vector as it changes over time
• General state space equations:
&
x(t) = f(x,u, t)
y(t) = g(x,u,t) ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 11 Linear Systems
&
x(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t)
&
x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 12 6 ME 575 Handouts Input/Output vs. State Space Models
• Input/Output Models:
– are 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) • 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
multimultinonlinear systems ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 13 Modeling Errors
• Additive Modeling Error (AME):
Let yo be the nominal system output and y be the
actual system output, then AME = y − yo
• Multiplicative Modeling Error (MME): ME575 Session 4 – Modeling School of Mechanical Engineering
Purdue University Slide 14 7 ...
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 Fall '10
 Meckl
 Mechanical Engineering

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