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first_order_notes - MAE107 Introduction to Modeling and...

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MAE107 Introduction to Modeling and Analysis of Dynamic Systems Prof. R.T. M’Closkey, Mechanical and Aerospace Engineering, UCLA Subject outline: 1. Intro to first order ODEs: ode, ivp, solutions, convolution, transfer function, frequency response, stability 2. Intro to second order ODEs: same subjects 3. Intro to n th order ODEs 4. Poles, zeros, coprime assumptions, block diagram manipulation with transfer functions vs. ODEs 5. Frequency response asymptotic approximations with examples. 6. The unit impulse function (various realizations), unit step function, impulse response and step response, rela- tions between the two, definitions of 0 - and 0 + 7. Superposition=convolution, properties of convolution 8. Fourier series and response to periodic inputs; energy spectral density; identification with periodic inputs; discrete-time Fourier series for periodic sequences; aliasing 9. Fourier transform; relation between frequency response and FT of impulse response 10. Correlation functions and their transforms for signals with finite energy; use in identification; extension to power signals including power spectral density functions (also give block diagram interpretation) 11. Bilateral Laplace, ROC, numerical examples 12. Unilateral Laplace and application to solving IVPs 13. State-space representation theory. Do not distribute these notes 1 Prof. R.T. M’Closkey, UCLA

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First Order Systems Do not distribute these notes 2 Prof. R.T. M’Closkey, UCLA
The Starting Point: First Order Linear Linear Differential Equations The first order linear ODE is given by: d dt ( y ( t )) = ay ( t ) + bu ( t ) (1) where y = dependent variable that is determined by “solving" the ODE t = time, the independent variable, specified on some interval I u = the “forcing" or “nonhomogeneous" term that is known or specified for t ∈ I a, b = coefficients that are assumed to be constant in these notes . This ODE is linear because y and its time derivative appear linearly in the expression. Another shorthand way of writing this expression is: ˙ y = ay + bu. In terms of using this model to predict the response of a physical system, the following initial value problem is often of interest: Initial Value Problem. The time interval of interest is I = [ t 0 , t 1 ] and an initial condition, denoted y 0 , is specified at time t 0 (in other words, y ( t 0 ) = y 0 is specified) and the continuous forcing function u is known for all t ∈ I , then, find y for all t ∈ I . Notes: the “starting time" t 0 is often 0 t 1 may be this scenario assumes the system modeled by this ODE is causal (future inputs do not affect the current value of the dependent variable) Do not distribute these notes 3 Prof. R.T. M’Closkey, UCLA

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Block diagrams are a useful way to pictorially represent relationships between systems and signals. To derive a block diagram for (7) note that the unique solution of the IVP also satisfies y ( t ) = y 0 + Z t t 0 ( ay ( τ ) + bu ( τ )) dτ.
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first_order_notes - MAE107 Introduction to Modeling and...

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