Jacob Hammer

On the control of asynchronous machines with races

The problem of eliminating the effects of critical races on asynchronous machines is considered in a control theoretic context. State feedback controllers that eliminate the effects of critical races are developed. The results include necessary and sufficient conditions for the existence of such controllers and algorithms for their design. When the controllers exist, they eliminate the race effects and control the machine to match a given race-free model.

Non-linear systems: stability and rationality

The problem of when a non-linear system can be represented as a quotient of two stable non-linear systems is considered. Attention is mainly directed toward nonlinear discrete-time recursive systems, where recursive means that the relationship between an input sequence and the corresponding output sequence can be expressed in terms of a finite number of recursive equations. Necessary and sufficient conditions are derived for the existence of a fraction representation of a recursive system, where the numerator and the denominator are stable recursive systems. The explicit construction of such a fraction representation is described.

Input/output control of asynchronous sequential machines

The problem of controlling a finite-state asynchronous sequential machine is examined. Main consideration is given to input/output control, where access to the state of the machine is not available. The objective is to use output feedback to control the machine so as to match a prescribed model. It is shown that necessary and sufficient conditions for the existence of appropriate controllers can be stated in terms of a simple comparison of two numerical matrices. Whenever controllers exist, algorithms for their design are outlined.

Fraction representations of non-linear systems: a simplified approach

Abstract A non-linear system Σ has a right fraction representation if it can be represented as Σ = PQ−1, where P and Q are stable systems, and it has a left fraction representation if it can be represented as Σ = G −1 T, where G and T are stable systems. We develop here a theory of right and of left fraction representations for discrete-time non-linear systems with bounded input sequences. We indicate the connection between fraction representations and the stabilization problem for non-linear systems.

On the control of asynchronous sequential machines with infinite cycles

The problem of eliminating the effects of infinite cycles on asynchronous sequential machines is considered in a control theoretic context. The main objective is to develop state feedback controllers that stop infinite cycles in an existing asynchronous machine, while controlling the machine to match a prescribed model. Necessary and sufficient conditions for the existence of such controllers are derived in terms of an inequality condition between two numerical matrices. The results include an algorithm for the characterization of all infinite cycles of a given machine as well as an algorithm for the construction of appropriate controllers, whenever they exist.

Causal Factorization and Linear Feedback

An algebraic framework for the investigation of linear dynamic output feedback is introduced. Pivotal in the present theory is the problem of causal factorization, i.e. the problem of factoring two systems over each other through a causal factor. The basic issues are resolved with the aid of the new concept of latency kernels.

State feedback control of asynchronous sequential machines with adversarial inputs

The problem of controlling an asynchronous sequential machine in the presence of adversarial inputs is considered. Here, an adversarial input is an unknown and unauthorised input that attempts to interfere with the operation of the machine. The objective is to develop automatic state feedback controllers that counteract the effects of the adversarial input and restore desirable behaviour to the controlled machine. Necessary and sufficient conditions for the existence of such controllers are presented in terms of an inequality condition between two numerical matrices. Whenever a controller exists, an algorithm for its design is provided.

Input/output control of asynchronous sequential machines with races

The design of output feedback controllers that eliminate uncertainties caused by critical races in asynchronous sequential machines is considered. The objective is to build controllers that drive a race-afflicted machine so as to match a prescribed deterministic model. Necessary and sufficient conditions for the existence of such controllers are presented in terms of a numerical matrix derived from the given machine. When controllers exist, an algorithm for their construction is also provided. The discussion depends on the novel notion of ‘generalised state’, which helps represent the uncertainty created by critical races and facilitates the construction of controllers.

Stabilization of non-linear systems†

The problem of stabilizing a discrete-time non-linear system is considered. For a rather large class of common stabilizable non-linear systems, a procedure leading to the stabilization of a given non-linear system Σ belonging to that class is derived. In this procedure, a pair of compensators is constructed, consisting of a precompensator and an output feedback compensator, which, when connected in closed loop around the system Σ, yield a closed-loop system that is internally stable for bounded input sequences. The procedure allows the construction of infinitely many different pairs of such compensators, thus facilitating the choice of a convenient one.

On the corrective control of sequential machines

The paper deals with the design of controllers that correct faulty behaviour of sequential machines caused by corrupted inputs. Alternatively, the results can be interpreted as the design of controllers that steer a sequential machine from an unknown initial condition to a prescribed steady-state course. In these terms, the paper characterizes the uncertainties about the initial condition under which the prescribed steady-state course can be achieved. The paper is written within the input/output framework of nonlinear control, and is motivated in part by potential applications to molecular biology.