Geometry of adaptive control, part II: optimization and geodesics

Two incompatible topologies appear in the study of adaptive systems: the graph topology in control design, and the coefficient topology in system identification. Their incompatibility is manifest in the stabilization problem of adaptive control. We argue that this problem can be approached by changing the geometry of the sets of control systems under consideration: estimating np parameters in an np-dimensional manifold whose points all correspond to stabilizable systems. One way to accomplish this is using the properties of the algebraic Riccati equation. Parameter estimation in such a manifold can be approached as an optimal control problem akin to the deterministic Kalman filter, leading to algorithms that can be used in conjunction with standard observers and controllers to construct stable adaptive systems. 1 Topologies in adaptive control Two topologies appear in the study of adaptive systems. Relevant for feedback control is the graph topology, induced by both the gap and graph metrics; it is the coarsest topology on sets of linear systems for which feedback stability is a robust property [1, 13]. System identification, on the other hand, makes implicit use of the topology induced by the metric in which the distance between two systems is given by the Euclidean distance between the coefficients of their transfer functions. Even on sets of linear systems with dimension not exceeding a given n, on which both are defined, these topologies are not compatible: one is neither finer nor coarser than the other, that is, a set open in the graph topology may not be an open set in the coefficient topology, and vice-versa. Adaptive controllers are characterized by a double feedback loop: the control and the adaptation loops. The incompatibility between the topologies underlying the design of the loops manifests itself in the form of the stabilization problem. In fact, the parameter values for which the design model, upon which certainty-equivalence control laws are designed, loses stabilizability, are exactly those for which the operations of addition and multiplication of transfer functions are discontinuous under the graph topology. Myriad adaptive algorithms in the literature start by designing certainty-equivalence controllers, and jury-rig alternative feedback signals to be used when the certainty-equivalence