The work of Peter Crouch the control theorist* Conference on Decision and Control December 11, 2011...
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![Page 1: The work of Peter Crouch the control theorist* Conference on Decision and Control December 11, 2011 Canonical Geometrical Control Problems: New and Old.](https://reader035.fdocuments.in/reader035/viewer/2022062712/56649c9a5503460f94957fc3/html5/thumbnails/1.jpg)
The work of Peter Crouch the control theorist*Conference on Decision and Control
December 11, 2011
Canonical Geometrical Control Problems: New and Old
Roger Brockett
Engineering and Applied Sciences
Harvard University
*Not to be confused with the bad-boy English footballer of
Tottenham Hotspur, Stoke City, Abigail Clancy, etc., etc.
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Some of my early interactions:
The London NATO meeting– September, 1973
Student at Harvard, 1974-1977: Thesis: “Dynamical Realizations of Finite Volterra Series”
It showed that the natural state space for a finite Volterra series is diffeomorphic to Rn
Cohort included P. S. Krishnaprassad and Joseph Ja’ Ja”
Sabbatical at Harvard in 1982
Peter Crouch: The reason we are here!
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Peter Crouch at the Center: From the Web
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Some Lie Theoretic, Least Squares, State Transfer Problems involving Z2 Graded Lie Algebras
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The first two have finite Volterra series
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Recall
![Page 7: The work of Peter Crouch the control theorist* Conference on Decision and Control December 11, 2011 Canonical Geometrical Control Problems: New and Old.](https://reader035.fdocuments.in/reader035/viewer/2022062712/56649c9a5503460f94957fc3/html5/thumbnails/7.jpg)
What about regulator versions of these systems?
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What it Approximates
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Our Quadratic Regulator Problem
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The Euler-Lagrange Equations
We need to factor the linear operator into a stable and unstable factors.
The value of x(0) is given. Its derivative is to be determinedso as to put x on the right submanifold
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This is from the zeroth order term.
This is from the first order term.
Formula for Z
Factoring the Euler-Lagrange Equation
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Relating Properties of x and Z through Q
It is important that we are now dealing with initial values
Theta and Q are functions of x(0) and Z(0).
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Here we first define the optimal trajectory using initial conditionsgiving an open loop control.
Actually it is true at all times and states!
If considered as a “gain”
From the perspective of achieving the correct homogeneity, this is quite remarkable, even miraculous.
is homogeneous of degree zero
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An Example
These solutions are stable for all $a$ and generate a Z displacement.
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A Further Elaboration
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A Further Elaboration
As x(0) approaches 0 the cost is upper bounded by the cost of the u-only optimal trajectory. However, this cost is not differentiable on the “Z axis”.
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As for the Cost---
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This is not a dead end—Many more possibilities
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Peter---
Congratulations on a distinguished career based on talent, hard work, discipline, service to the community.