No abstract

No abstract

An upper bound on the feedback capacity of unifilar finite-state channels (FSCs) is derived. A new technique, called the Q-contexts, is based on a construction of a directed graph that is used to quantize recursively the receiver's output sequences to a finite set of contexts. For any choice of Q-graph, the feedback capacity is bounded by a singleletter expression, Cfb ≤ sup I(X, S; Y |Q), where the supremum is over P X|S,Q and the distribution of (S, Q) is their stationary distribution. It is shown that the bound is tight for all unifilar FSCs where feedback capacity is known: channels where the state is a function of the outputs, the trapdoor channel, Ising channels, the no-consecutive-ones input-constrained erasure channel and for the memoryless channel. Its efficiency is also demonstrated by deriving a new capacity result for the dicode erasure channel (DEC); the upper bound is obtained directly from the above expression and its tightness is concluded with a general sufficient condition on the optimality of the upper bound. This sufficient condition is based on a fixed point principle of the BCJR equation and, indeed, formulated as a simple lower bound on feedback capacity of unifilar FSCs for arbitrary Q-graphs. This upper bound indicates that a single-letter expression might exist for the capacity of finite-state channels with or without feedback based on a construction of auxiliary random variable with specified structure, such as Q-graph, and not with i.i.d distribution. The upper bound also serves as a non-trivial bound on the capacity of channels without feedback, a problem that is still open. Index TermsConverse, dicode erasure channel, feedback capacity, finite state channels, trapdoor channel, unifilar channels, upper bound.

The capacity of unifilar finite-state channels in the presence of feedback is investigated. We derive a new evaluation method to extract graph-based encoders with their achievable rates, and to compute upper bounds to examine their performance. The evaluation method is built upon a recent methodology to derive simple bounds on the capacity using auxiliary directed graphs. While it is not clear whether the upper bound is convex, we manage to formulate it as a convex optimization problem using transformation of the argument with proper constraints. The lower bound is formulated as a non-convex optimization problem, yet, any feasible point to the optimization problem induces a graph-based encoders. In all examples, the numerical results show near-tight upper and lower bounds that can be easily converted to analytic results. For the non-symmetric Trapdoor channel and binary fading channels (BFCs), new capacity results are eastablished by computing the corresponding bounds. For all other instances, including the Ising channel, the near-tightness of the achievable rates is shown via a comparison with corresponding upper bounds. Finally, we show that any graph-based encoder implies a simple coding scheme that is based on the posterior matching principle and achieves the lower bound.where the joint distribution is π S,Q P X|S,Q P Y |X,S , and π S,Q denotes a stationary distribution. For DRAFT

We consider the infinite-horizon, discrete-time fullinformation control problem. Motivated by learning theory, as a criterion for controller design we focus on regret, defined as the difference between the LQR cost of a causal controller (that has only access to past and current disturbances) and the LQR cost of a clairvoyant one (that has also access to future disturbances). In the full-information setting, there is a unique optimal non-causal controller that in terms of LQR cost dominates all other controllers, and we focus on the regret compared to this particular controller. Since the regret itself is a function of the disturbances, we consider the worst-case regret over all possible bounded energy disturbances, and propose to find a causal controller that minimizes this worst-case regret. The resulting controller has the interpretation of guaranteeing the smallest possible regret compared to the best non-causal controller that can see the future, no matter what the disturbances are. We show that the regretoptimal control problem can be reduced to a Nehari extension problem, i.e., to approximate an anticausal operator with a causal one in the operator norm. In the state-space setting we obtain explicit formulas for the optimal regret and for the regret-optimal controller (in both the causal and the strictly causal settings). The regret-optimal controller is the sum of the classical H2 state-feedback law and an n-th order controller (where n is the state dimension of the plant) obtained from the Nehari problem. The controller construction simply requires the solution to the standard LQR Riccati equation, in addition to two Lyapunov equations. Simulations over a range of plants demonstrates that the regret-optimal controller interpolates nicely between the H2 and the H∞ optimal controllers, and generally has H2 and H∞ costs that are simultaneously close to their optimal values. The regret-optimal controller thus presents itself as a viable option for control system design.