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L3-1_linsys

# L3-1_linsys - CDS 101/110a Lecture 3.1 Linear Systems...

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1 CDS 101/110a: Lecture 3.1 Linear Systems Douglas G. MacMynowski 11 October 2010 Goals: Describe linear system models: properties, examples, and tools Characterize stability and performance of linear systems in terms of eigenvalues Compute linearization of a nonlinear systems around an equilibrium point Reading: Åström and Murray, Analysis and Design of Feedback Systems, Ch 5 Review from Last Week Key topics Stability of equilibrium points Eigenvalues determine stability for linear systems Local versus global behavior Note on eigenvalues: If eigenvectors are unique then: 11 Oct 10 D. G. MacMynowski, Caltech CDS 2 Choose: then

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2 Linear Systems Input/output linearity at x(0) = 0 Linear systems are linear in initial condition and input need to use x(0) = 0 to add outputs together For different initial conditions, you uy need to be more careful Linear system step response and frequency response scale with input amplitude 2X input 2X output Allows us to use ratios and percentages in step/freq response. These are independent of input u 1 0 5 10 -1 0 1 0 5 10 -1 0 1 y 1 ++ u 2 -1 0 1 -05 0 0.5 y 2 11 Oct 10 D. G. MacMynowski, Caltech CDS 3 These are independent of input amplitude Limitation: input saturation only holds up to certain input amplitude 0 5 10 051 0 0.5 2 10 0 5 -2 0 y 1 + y 2 0 5 10 -2 0 2 u + u Many important examples Electronic circuits Why are Linear Systems Important? Many important tools Frequency response, step response, etc Traditional tools of control theory Developed in 1930’s at Bell Labs Especially true after feedback Frequency response is key performance specification Many mechanical systems u(t) q Developed in 1930 s at Bell Labs Classical control design toolbox Nyquist plots, gain/phase margin Loop shaping Optimal control and estimators Linear quadratic regulators Kalman estimator CDS 101/ 110a CDS 110b 11 Oct 10 D. G. MacMynowski, Caltech CDS 4 Quantum mechanics, Markov chains c k 3 m 1 m 2 q 1 2 k 2 k 1 Kalman estimators Robust control design H control design analysis for structured uncertainty CDS 212
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L3-1_linsys - CDS 101/110a Lecture 3.1 Linear Systems...

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