4 - modeling4 filled

4 - modeling4 filled - ME 575 Handouts Modeling Physical...

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Unformatted text preview: ME 575 Handouts Modeling Physical Description ⇓ Schematic ⇓ Differential Equations ⇓ ⇓ 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 (black-box 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 - n-dimensional 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 multi-input multi-output (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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