L4-1_statefbk

L4-1_statefbk - CDS 101: Lecture 4-1 Reachability and State...

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1 CDS 101: Lecture 4-1 Reachability and State Space Feedback Douglas G. MacMynowski 18 October 2010 Goals: Define reachability of a control system Give tests for reachability of linear systems and apply to examples Describe the design of state feedback controllers for linear systems Reading: Åström and Murray, Feedback Systems, Ch 6 Review from last week Properties of Linear Systems: • Linearity with respect to inputs and initial conditions • Stability characterized by eigenvalues • Response described by convolution uy integral • Many applications and tools • Characterizes a nonlinear system around an equilibrium point No system is linear for all input amplitudes, but linearization is a good enough model for many systems 0 5 1 0 - 1 0 1 0 5 1 0 - 1 0 1 + + 0 5 1 0 - 1 0 1 0 5 1 0 - 0. 5 0 0. 5 2 0 0 2 2 18 Oct 10 D. G. MacMynowski, Caltech CDS 2 1 0 0 5 - 2 0 5 1 0 - 2 ζ 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 time (sec) y T=2 π / ω 0
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2 Control Overview Process Controller u y Trajectory Generation Design controller so that i) System is stable ii) Performance: Keep close to equilibrium despite disturbances (disturbance rejection) 18 Oct 10 D. G. MacMynowski, Caltech CDS 3 Move system to desired state (track reference) iii) Robustness to modeling errors ~ 5.8 rad/sec ( × 2 in 120 msec) First Questions… 1 Can the input u affect the dynamics? Process Controller u y 1) Can the input Equivalent to asking whether there is a u that allows us to reach any point in the state-space Reachability (today), depends on A, B Related to the design of state feedback Can’t change x 2 e.g. 18 Oct 10 D. G. MacMynowski, Caltech CDS 4 2) Does the measurement y contain enough information about the system? Observability (next week), depends on A, C Related to the design of observers to estimate state from measurement Can’t measure x 2 e.g.
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3 Control Design Concepts System description: single input, single output system (MIMO also OK) Stability:
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This note was uploaded on 01/04/2012 for the course CDS 101 taught by Professor Murray,r during the Fall '08 term at Caltech.

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L4-1_statefbk - CDS 101: Lecture 4-1 Reachability and State...

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