# Analytical solutions, duality and symmetry in constrained control and estimation

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 unconstrained systems, it is well known that the optimal quadratic regulator and the Kalman filter share a duality relationship, where the different system and objective function parameters can be interchanged according to well defined relations. This duality relationship was established by R. Kalman and collaborators in the 1960s, and one important implication is that it allows for an exchange of solutions between estimation and control problems. However, the relationships between control and estimation, in the constrained cases, are—despite their importance in practical applications— not as well understood. The context of this talk will be that of Model Predictive Control (MPC) and Moving Horizon Estimation (MHE), arguably the most popular methodologies for dealing with constrained problems. We will 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—exhibit a remarkable symmetry in the light of this duality relationship. The second result is concerned with the optimal solution to both constrained problems, which will be derived analytically by using dynamic programming. The optimal solution is given by a piece-wise affine function of the data (or 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 programming will allow us to derive the solutions—at least for simple constrained problems— in an entirely analytical way, obtaining recursive equations that can be interpreted as the constrained versions of the Riccati equation. Finally, we will revisit the connection between constrained control and estimation problems. We will show that, from the analytical solutions to both problems (obtained with dynamic programming), a clear symmetry relationship is exposed between them, which is different from the Lagrangian 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.

This talk is part of the CUED Control Group Seminars series.