A controllability counterexample

Abstract

ISR develops, applies and teaches advanced methodologies of design and analysis to AbstractSimulation by time discretizations can be qualitatively misleading; as counterexamples a class of non-controllable single-input two-dimensional bilinear control systems is presented whose Euler discretizations are controllable on the punctured plane.

Study of a Class of Almost‐controllable Discrete‐time Bilinear Systems

“…One illustrates that a discrete bilinear system is almost everywhere on R n and is nearly controllable, although the system itself is not controllable. The other indicates that the result in [20] is a particular case of our results and that the controllability counterexample in [21] can be extended to n-dimensional bilinear systems.…”

On controllability of discrete-time bilinear systems

“…According to Theorem 1, (2) is a controllability counterexample if B in system (1) has a non-real pair of complex conjugate eigenvalues except pure imaginary eigenvalues since (1) is uncontrollable for any B 2 R 2Â2 . Indeed, the counterexample presented in [1] is the case of a=(2/3)p in our theorem.…”

“…Note that in the existing literature [23][24][25][26][27], mainly, [24,27], either the sufficient condition of [24] or the necessary and sufficient condition of [27] is not able to discuss the controllability of (2). Secondly, we show that, in comparison with the controllability criterion of this paper, the counterexample presented by [1] is a special case of the proposed necessary and sufficient conditions. Finally, by noting that system (1) is uncontrollable in any finite dimension [4,14,22], it is shown that the discretization counterpart of (1), say system (2), is also uncontrollable if its dimension is greater than two.…”

“…Since a discrete-time approximation to a bilinear system is essential for computer simulation or digital control purpose [28][29][30], a natural question is that whether the discretization changes the system controllability. Recently, a counterexample was given by [1] for a class of uncontrollable singleinput two-dimensional bilinear systems: where xðtÞ 2 R n is the state variable with n=2, uðtÞ 2 R is a single-input, and B 2 R nÂn . With a specially chosen matrix B, it is shown that the Euler discretization of (1)…”

“…As [1] pointed out, numerical methods must be watched with care. However, what is the essence of the controllability counterexample?…”