Pole placement controllers for linear time-delay systems with commensurate point delays

Abstract

We investigate the exact and approximate spectrum assignment properties associated with realizable output-feedback pole-placement-type controllers for single-input singleoutput linear time-invariant time-delay systems with commensurate point delays. The controller synthesis problem is discussed through the solvability of a set of coupled Diophantine equations of polynomials. An extra complexity is incorporated in the above design to cancel extra unsuitable dynamics being generated when solving the above Diophantine equations. Thus, the complete controller tracks any arbitrary prefixed (either finite or delay-dependent) closed-loop spectrum. However, if the controller is simplified by deleting the above-mentioned extra complexity, then robust stability and approximated spectrum assignment are still achievable for a certain sufficiently small amount of delayed dynamics. Finally, the approximate spectrum assignment and robust stability problems are revisited under plant disturbances if the nominal controller is maintained. In the current approach, the finite spectrum assignment is only considered as a particular case of the designer's choice of a (delay-dependent) arbitrary spectrum assignment objective.

About K-Positivity Properties of Time-Invariant Linear Systems Subject to Point Delays

“…The strong evolution operator can be calculated explicitly via (2.6) in the approach of this paper and through the Lambert matrix functions and associate coefficients in the approach of [28]. Since the solution is unique under the given weak conditions, the three expressions of the solution lead in fact to the same solution for all time.…”

“…The properties of the above second kind of systems are more difficult to investigate because of their infinite-dimensional nature [21,[25][26][27][28][29][30][31][32][33][34][35][36] although they are very important in some control applications like, for instance, the synthesis of sliding-mode controllers under delays [21,25,26]. The analytic problem becomes more difficult when delays are distributed or time varying [30,31,33,36].…”

“…The use of disturbing signals on the nominal dynamics is also common in control theory problems involving the use of backstepping techniques or the synthesis of reduced-order controllers (see, e.g., [40,41]). However, a direct inclusion of the delay effect on the dynamics leads, in general, to tighter calculus of the solution trajectories, [21,[25][26][27][28][29][30][31][32][33][34][35][36].…”

On K-Positivity Properties of Time-Invariant Linear Systems Subject to Discrete Point Lags

“…The use of disturbing signals on the nominal dynamics is also common in control theory problems involving the use of backstepping techniques or the synthesis of reduced order controllers (see, for instance, [21,22], [36]). However, a direct inclusion of the delay effect on the dynamics leads, in general, to tighter calculus of the solution trajectories, [23,24,25,26,27,28,29,30,31,32,33,34,35]. The main objective of this paper is to study the nonnegativity/positivity properties of time-invariant continuous-time dynamic systems under constant point delays.…”

“…in the state. The properties of that second kind of systems are more difficult to investigate because of their infinite-dimensional nature, [23,24,25,26,27,28,29,30,31,32,33,34,35] although they are very important in some control applications like, for instance, the synthesis of sliding-mode controllers under delays, [23,24,25]. The analytic problems becomes more difficult when delays are distributed or time-varying, [29,30], [32], [35].…”