Discrete-time Bilinear Systems Research Articles

The proof of Goka's conjecture

If the system X_{k+1} = (A + u_{k}B)X_{k}, k = 0, 1, ..., is controllable with |u_{k}| for all \delta > 0 , then the eigenvalues of A lie on the unit circle. This is Goka's conjecture. Through algebraic transformation and discussion of invariant proper subsets, this note gives a proof of the conjecture, and shows that for more general discrete-time bilinear systems, the conjecture is still true.

Stability of discrete-time bilinear systems with constant inputs

In this paper discrete-time bilinear systems are considered. Necessary and sufficient conditions are developed which ensure that for any constant input in an unbounded control domain, the trajectories of the system tend to the origin as time increases.

On discrete stochastic bilinear systems stability

Mean square stability conditions for discrete-time bilinear systems operating in a stochastic environment are given in this paper. Only independence and wide sense stationarity are required for the second order disturbance sequences involved, thus dismissing ergodicity and zero-mean assumptions. Stochastic stability conditions are derived by using a deterministic stability result for a class of separable nonlinear dynamical systems evolving in a Banach space.

Invertibility of Bilinear Discrete-Time Systems

This paper gives necessary and sufficient conditions for the (left) invertibility of discrete time single-input single-output bilinear systems extending previous work on the continuous time systems. For invertible systems the (left) inverse system is constructed. The class of sequences which can appear as outputs of a given system is described are the inverse system is used to generate the required control.

Identification of discrete-time stochastic bilinear systems

This paper considers the problem of identifying unknown parameters appearing in discrete time-varying bilinear models. The system is assumed to operate in a stochastic environment, where the input disturbance and the noise observation have unknown probability distribution. Identifiability conditions are investigated in the light of a state covariance analysis for stochastic bilinear systems. By assuming that the only accessible process is supplied by the noisy observation sequence, a recursive identification procedure is proposed which is suitable for on-line applications. Stochastic approximation algorithms are used for identifying time-invariant bilinear systems.

Controllability of two-input, discrete time bilinear systems

In this paper, the controllability of discrete time bilinear systems with two inputs is examined. Two cases are studied and some sufficient conditions for controllability are obtained. The extension to more than two inputs is also briefly discussed.

Second-order correlation method for bilinear system identification

An algorithm is developed to estimate parameters of a class of discrete-time bilinear systems using second-order correlations. It is shown that for pseudorandom binary and ternary input signals the computations in the estimation algorithm can be simplified. The method is compared with least-squares algorithm for a model of nuclear fission. The proposed algorithm gives a computationally simple and effective way of parameter estimation.

Dead-beat control of discrete-time bilinear systems †

The subject, of this paper is the dead-beat control of discrete-time bilinear systems with independent additive and multiplicative controls. The first, issue is the proof that controllability to the origin is a necessary and sufficient existence condition, and that the dead-beat controller can be chosen, without loss of generality, with a two-layer structure. Then, the controllability to the origin of the system is shown to be equivalent to that of homogeneous subsystems, obtained through successive decompositions of it. Thus, the problem of finding a dead-beat controller is reduced to that of zeroing the state of these subsystems. Lastly, some conditions for the controllability to the origin of a special type of homogeneous bilinear system are given.

Controllability of a discrete-time bilinear system

In this note we examine the controllability of a discrete-time bilinear system characterized by an nth order difference equation.

Deterministic and stochastic control of discrete-time bilinear systems

Optimal control of a class of time invariant single-input, discrete bilinear systems is investigated in this paper. Both deterministic and stochastic problems are considered. In the deterministic problem, for the initial state in a certain set ∑ 0, the solution is the same as the solution to the associated linear system. The optimal path may be a regular path or a singular path. The stochastic control problem is considered with perfect state observation, and additive and multiplicative noise in the state equation. It is demonstrated that the presence of noise simplifies the analysis compared to that in the determinstic case.

An observable canonical form of discrete-time bilinear systems

A canonical form of discrete-time bilinear systems is investigated. A new concept of sequential observability is introduced and it is shown that the discrete bilinear system is sequentially observable if and only if it can be transformed into some canonical form. One of the distinctions between discrete-time and continuous-time bilinear systems is discussed from uniform observability and sequential observability.

Comments on "Controllability of a class of discrete time bilinear systems"

This note simplifies some of the conditions of the above paper. The form of the systems is characterized in an explicit way.

Controllability of discrete time inhomogeneous bilinear systems

In this paper we derive a set of conditions which are both necessary and sufficient for complete controllability of a class of inhomogeneous discrete time bilinear systems.

Controllability of a class of discrete time bilinear systems

In this paper the controllability of a class of discrete time bilinear systems is discussed. Necessary and sufficient conditions for complete controllability are derived. In addition a complete characterization of the controllable regions of this class of systems, when not completely controllable, is made.

Controllability of discrete bilinear systems with bounded control

Controllability of time-invariant discrete-time bilinear systems is discussed. Bilinear systems are classified into two categories: homogeneous and inhomogeneous. Sufficient conditions which ensure the global controllability of discrete-time bilinear systems are obtained by localized analysis in control variables.