L4-1_linsys - CDS 101: Lecture 4.1 Linear Systems Richard...

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CDS 101: Lecture 4.1 Linear Systems Richard M. Murray 18 October 2004 Goals: y Describe linear system models: properties, examples, and tools y Characterize stability and performance of linear systems in terms of eigenvalues y Compute linearization of a nonlinear systems around an equilibrium point Reading: y Åström and Murray, Analysis and Design of Feedback Systems, Ch 4 y Packard, Poola and Horowitz, Dynamic Systems and Feedback, Sections 19, 20, 22 (available via course web page)
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18 Oct 04 R. M. Murray, Caltech CDS 2 Lecture 3.1: Stability and Performance Key topics for this lecture y Stability of equilibrium points y Local versus global behavior y Performance specification via step and frequency response -2 π 0 2 π -2 0 2 x 1 x 2 0 5 10 15 20 25 0 0.5 1 1.5 0 5 10 15 20 u y Review from Last Week
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18 Oct 04 R. M. Murray, Caltech CDS 3 What is a Linear System? Linearity of functions: y Zero at the origin: y Addition: y Scaling: Linearity of systems : sums of solutions : nm f ± ± (0) 0 f = () ( ) ( ) f x yf x fy += + ( ) f x f x α = fx y f x β αβ + = + f xA x = Canonical example: x = & x B u y Cx Du = + =+ & 10 20 12 xx x xt x t →= + 10 1 = 20 2 = 1 1 0, ( ) ( ) xu t u t y t y t = = 2 2 0, ( ) ( ) t u t y t y t = = 0, ( ) ( ) ( ) t u t u t y t y t y t = + Dynamical system Control system
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18 Oct 04 R. M. Murray, Caltech CDS 4 Linear Systems Input/output linearity at x(0) = 0 y Linear systems are linear in initial condition and input need to use x(0) = 0 to add outputs together y For different initial conditions, you need to be more careful (sounds like a good midterm question) Linear system step response and frequency response scale with input amplitude y 2X input 2X output y Allows us to use ratios and percen-tages in step/freq response. These are independent of input amplitude y Limitation: input saturation only holds up to certain input amplitude xA x B u y Cx Du =+ & u 1 0 5 10 -1 0 1 uy + + 0 5 10 -2 0 2 y 1 + y 2 (0) 0 x = 0 5 10 -1 0 1 y 1 u 2 0 5 10 -1 0 1 0 5 10 -0.5 0 0.5 y 2 0 5 10 -2 0 2 u + u
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18 Oct 04 R. M. Murray, Caltech CDS 5 b k 3 m 1 m 2 q 1 u (t ) q 2 k 2 k 1 Why are Linear Systems Important? Many important tools Frequency response, step response, etc y Traditional tools of control theory y Developed in 1930’s at Bell Labs; intercontinental telecom Classical control design toolbox y Nyquist plots, gain/phase margin y Loop shaping Optimal control and estimators y Linear quadratic regulators y Kalman estimators Robust control design y H control design y μ analysis for structured uncertainty Many important examples
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This note was uploaded on 09/09/2009 for the course EE 4233 taught by Professor Georgio during the Spring '09 term at Minnesota Colleges.

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L4-1_linsys - CDS 101: Lecture 4.1 Linear Systems Richard...

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