the work of peter crouch the control theorist* conference on decision and control december 11, 2011...
TRANSCRIPT
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.
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!
Peter Crouch at the Center: From the Web
Some Lie Theoretic, Least Squares, State Transfer Problems involving Z2 Graded Lie Algebras
The first two have finite Volterra series
Recall
What about regulator versions of these systems?
What it Approximates
Our Quadratic Regulator Problem
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
This is from the zeroth order term.
This is from the first order term.
Formula for Z
Factoring the Euler-Lagrange Equation
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
An Example
These solutions are stable for all $a$ and generate a Z displacement.
A Further Elaboration
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”.
As for the Cost---
This is not a dead end—Many more possibilities
Peter---
Congratulations on a distinguished career based on talent, hard work, discipline, service to the community.