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SUMMARY:Analytical solutions\, duality and symmetry in constrained control
  and estimation - Dr Jose De Dona (The University of Newcastle\, Australia
 --on study leave during 2008/2009 at Ecole des Mines de Paris\, France)
DTSTART:20090220T140000Z
DTEND:20090220T150000Z
UID:TALK14436@talks.cam.ac.uk
CONTACT:Dr Guy-Bart Stan
DESCRIPTION:In this talk we will explore the interplay between estimation 
 and control problems for linear systems with constraints. We will present 
 results that extend\, to the constrained case\, the well-known connections
  that exist in the absence of constraints. For example\, for linear uncons
 trained systems\, it is well known that the optimal quadratic regulator an
 d the Kalman filter share a duality relationship\, where the different sys
 tem and objective function parameters can be interchanged according to wel
 l defined relations. This duality relationship was established by R. Kalma
 n and collaborators in the 1960s\, and one important implication is that i
 t allows for an exchange of solutions between estimation and control probl
 ems. However\, the relationships between control and estimation\, in the c
 onstrained cases\, are—despite their importance in practical application
 s— not as well understood. The context of this talk will be that of Mode
 l Predictive Control (MPC) and Moving Horizon Estimation (MHE)\, arguably 
 the most popular methodologies for dealing with constrained problems. We w
 ill first establish a Lagrangian duality relationship between constrained 
 state estimation and control\, and show that the well-known unconstrained 
 duality relationship is a special case of our constrained result.  We will
  also see that both problems—constrained estimation and control—exhibi
 t a remarkable symmetry in the light of this duality relationship. The sec
 ond result is concerned with the optimal solution to both constrained prob
 lems\, which will be derived analytically by using dynamic programming. Th
 e optimal solution is given by a piece-wise affine function of the data (o
 r parameter). This optimal solution— of course— coincides with the one
  obtained by other existing methods belonging to what is usually referred 
 to as explicit solutions in MPC and MHE. However\, the use of dynamic prog
 ramming will allow us to derive the solutions—at least for simple constr
 ained problems— in an entirely analytical way\, obtaining recursive equa
 tions that can be interpreted as the constrained versions of the Riccati e
 quation. Finally\, we will revisit the connection between constrained cont
 rol and estimation problems. We will show that\, from the analytical solut
 ions to both problems (obtained with dynamic programming)\, a clear symmet
 ry relationship is exposed between them\, which is different from the Lagr
 angian duality relationship. This novel symmetry is summarized by means of
  a translation table that gives a complete correspondence of all variables
  of one problem into the variables of the other.\n
LOCATION:Cambridge University Engineering Department\, Lecture Room 3B
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