L3-1_stability

# L3-1_stability - CDS 101: Lecture 3.1 Stability and...

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CDS 101: Lecture 3.1 Stability and Performance Richard M. Murray 11 October 2004 Goals: y Describe different types of stability for an equilibrium point y Explain the difference between local/global stability, and related concepts y Describe performance measures for (controlled) systems, including transients and steady state response Reading: y Åström and Murray, Analysis and Design of Feedback Systems, Ch 3

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CDS 101, 11 Oct 04 R. M. Murray, Caltech CDS 2 Lecture 2.1: System Modeling Model = state, inputs, outputs, dynamics Principle : Choice of model depends on the questions you want to answer (,) () dx fxu dt yh x = = b k 3 m 1 m 2 q 1 u(t) q 2 k 2 k 1 function dydt = f(t,y, k1, k2, k3, m1, m2, b, omega) u = 0.00315*cos(omega*t); dydt = [ y(3); y(4); -(k1+k2)/m1*y(1) + k2/m1*y(2); k2/m2*y(1) - (k2+k3)/m2*y(2) - b/m2*y(4) + k3/m2*u ]; 1 11 (, ) kk k xf xu x + ++ = = Review from Last Week
CDS 101, 11 Oct 04 R. M. Murray, Caltech CDS 3 Today: Stability and Performance Goal #1: Stability y Check if closed loop response is stable Goal #2: Performance y Look at ability to track changes in reference and reject disturbances Goal #3: Robustness (later) (,(,) , ) () xf x k x r d yh x = = & reference measured output disturbance (,) x u u k x == & system input control law Sense Vehicle Speed Compute Control “Law” Actuate Gas Pedal

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CDS 101, 11 Oct 04 R. M. Murray, Caltech CDS 4 Phase Portraits (2D systems only) Phase plane plots show 2D dynamics as vector fields & stream functions y Plot f ( x ) as a vector on the plane; stream lines follow the flow of the arrows -1 0 1 -1 -0.5 0 0.5 1 x 1 x 2 12 21 2 xx d x dt ⎡⎤ = ⎢⎥ −− ⎣⎦ phaseplot(‘dosc’, . ..
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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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L3-1_stability - CDS 101: Lecture 3.1 Stability and...

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