State space realizations and monomial equivalence for convolutional codes

In this paper, we show that a minimal state space realization in Jordan canonical form for linear multivariable continuous‐time systems described by rational transfer function matrices could be obtained in a natural and basic way by using the concept of Nerode equivalence. While the minimal state space realization is known, the contribution of this note is to provide an alternative realization procedure which is directly introduced in a simple and self‐contained manner. Both scalar and multivariable cases in the continuous‐time setting are discussed. The presentation also provides a concrete construction of rational vector function with preassigned poles to interpolate any 2‐norm bounded analytical vector function in the open left‐half of the complex plane. The basic idea of Nerode equivalence is that the state can be identified with a corresponding equivalence class of inputs. For a linear finite dimensional time‐invariant continuous‐time system, the zero state is identified with the kernel of certain Hankel operator. This characterization via Nerode equivalence class sheds light on the construction of state for general nonlinear input–output systems.

Minimal state-space realization in linear system theory: an overview

A matrix Euclidean algorithm induced by state space realization

On the realization of periodic functions

The problem of minimal state space realization of two-dimensional systems is considered. The approach followed is, initially, to derive a circuit realization of the given transfer function involving a minimum number of delay elements. To facilitate this realization, the transfer function is expanded into a continued fraction. Using this circuit realization, an algorithm is proposed which readily provides the matrices of the state space model. The simplicity in deriving this model is due to a novel state space model introduced, which is of cyclic structure. Several examples are presented to illustrate ihe proposed algorithm.

A simple technique is presented for the minimal state space realisation of product factorable two dimensional systems. The application of this method yields the matrices A, b, cT and the scalar d of the two dimensional Givone-Roesser state space model, having minimal dimension.

Given a discrete-time linear system C, a shortest basis for C is a set of linearly independent generators for C with the least possible lengths. A basis B is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J, the generators in B whose span is in J is a basis for the subsystem CJ). The dimensions of the minimal state spaces and minimal transition spaces of C are simply the numbers of generators in a shortest basis B that are active at any given state or symbol time, respectively. A minimal linear realization for C in controller canonical form follows directly from a shortest basis for C, and a minimal linear realization for C in observer canonical form follows directly from a shortest basis for the orthogonal system C⊥. This approach seems conceptually simpler than that of classical minimal realization theory.

A directional partial realization problem

Minimal realizations play a key role in system analysis and synthesis. Among a variety of realizations they are characterised by a minimal dimension, which guarantees that no pole zero cancellations occur, a very important feature for the stability analysis, and are also of importance from the numerical point of view. This paper provides a simple method for minimal realization construction for multi-linear odd rational functions an important class from the practical point of view due to strong links to so-called reactance functions.

In this paper it is shown that a natural representation of a state space is given by the predictor space, the linear space spanned by the predictors when the system is driven by a Gaussian white noise input with unit covariance matrix. A minimal realization corresponds to a selection of a basis of this predictor space. Based on this interpretation, a unifying view of hitherto proposed algorithmically defined minimal realizations is developed. A natural minimal partial realization is also obtained with the aid of this interpretation.

On minimality in the partial realization problem

After providing an intrinsic definition of the adjoint system of a linear system given in state space form, we characterize the adjoint system in terms of the system’s external behavior. Conversely we show how this external characterization allows one to recover the state space definition by constructing an isomorphism from the abstractly defined minimal state space of the adjoint system to the dual of the minimal state space of the system itself. It is shown how a coordinate expression of this isomorphism can be obtained from a modified form of the Bezoutian matrix.

An algorithm for inverting rational matrices

This paper describes an algorithm, called CQ-learning, which learns to adapt the state representation for multi-agent systems in order to coordinate with other agents. We propose a multi-level approach which builds a progressively more advanced representation of the learning problem. The idea is that agents start with a minimal single agent state space representation, which is expanded only when necessary. In cases where agents detect conflicts, they automatically expand their state to explicitly take into account the other agents. These conflict situations are then analyzed in an attempt to find an abstract representation which generalises over the problem states. Our system allows agents to learn effective policies, while avoiding the exponential state space growth typical in multi-agent environments. Furthermore, the method we introduce to generalise over conflict states allows knowledge to be transferred to unseen and possibly more complex situations. Our research departs from previous efforts in this area of multi-agent learning because our agents combine state space generalisation with an agent-centric point of view. The algorithms that we introduce can be used in robotic systems to automatically reduce the sensor information to what is essential to solve the problem at hand. This is a must when dealing with multiple agents, since learning in such environments is a cumbersome task due to the massive amount of information, much of which may be irrelevant. In our experiments we demonstrate a simulation of such environments using various gridworlds.

The state-space realization of a certain class of two-dimensional systems with applications to image processing