Optimality conditions and duality for a non-linear time-delay control problem

1997 ◽  
Vol 18 (5) ◽  
pp. 327-340 ◽  
Author(s):  
Richard C. H. Lee ◽  
S. P. Yung
Keyword(s):  
Time Delay ◽  
Control Problem ◽  
Optimality Conditions ◽  
Linear Time ◽  
Time Delay Control ◽  
Delay Control ◽  
Non Linear

Stability of Uncertain Linear Systems With Time Delay

1991 ◽  
Vol 113 (4) ◽  
pp. 558-567 ◽  
Author(s):  
K. Youcef-Toumi ◽  
J. Bobbett
Keyword(s):  
Time Delay ◽  
Linear Time ◽  
Sufficient Condition ◽  
Single Input Single Output ◽  
Time Delay Control ◽  
Linear Time Invariant ◽  
Single Output ◽  
Uncertain Dynamics ◽  
Delay Control ◽  
The Stability

The control of systems with uncertain dynamics and unpredictable disturbances has raised some challenging problems. This is particularly important when high system performance is to be guaranteed at all times. Recently, Time Delay Control has been suggested as an alternative control scheme. The proposed control system does not require an explicit plant model nor does it depend on the estimation of specific plant parameters. Rather, it combines adaptation with past observations to directly estimate the effect of the plant dynamics. This paper outlines the Time Delay Control law for a class of linear dynamic systems and then presents a sufficient condition for stability of linear uncertain systems with time delay. The ideas of Nyquist and Kharitonov are used in the development of a sufficient condition, which does not resort to using approximations for time delay. Like Nyquist, the condition depends on maps of the Nyquist path and, like Kharitonov, stability depends on four functions each yielding a stable system. In this paper we combine these ideas to determine the stability of systems where the Time Delay Controller is applied to single input single output, linear time-invariant plants whose coefficients are known to vary within certain defined intervals. The development is carried out in the context of Time Delay Control but it can be applied in more general cases. Two examples will illustrate the approach and the usefulness of the technique.

Improving Stability Margins via Time Delay Control

2013 ◽  
Author(s):  
A. Galip Ulsoy
Keyword(s):  
Time Delay ◽  
Time Delays ◽  
Linear Time ◽  
Mechanical Vibration ◽  
Stability Margins ◽  
Single Input Single Output ◽  
Time Delay Control ◽  
Linear Time Invariant ◽  
Single Output ◽  
Delay Control

While time delays typically lead to poor control performance, and even instability, previous research has shown that introduction of time delays in controlling a dynamic system can, in some cases, be beneficial. This paper presents a new benefit of time delay control for single-input single-output linear time invariant systems: it can be used to improve robustness, as measured by increased stability margins. The proposed method utilizes time delays to approximate state-derivative feedback, which can be used, together with state feedback, to reduce sensitivity and improve robustness. Additional sensors are not required since the state-derivatives are approximated using available measurements and time delays. The method is introduced using a scalar example, then applied to a single degree-of-freedom mechanical vibration control problem in simulations to demonstrate excellent performance with improved stability margins.

Analysis of Linear Time Invariant Systems With Time Delay

1992 ◽  
Vol 114 (4) ◽  
pp. 544-555 ◽  
Author(s):  
K. Youcef-Toumi ◽  
S. Reddy
Keyword(s):  
Time Delay ◽  
Continuous Time ◽  
Control Algorithm ◽  
Closed Loop ◽  
Linear Time ◽  
Necessary Condition ◽  
Closed Loop System ◽  
Time Delay Control ◽  
Delay Control ◽  
Continuous Time Delay

Time Delay Control has recently been suggested as an alternative scheme for control of systems with unknown dynamics and unpredictable disturbances. The proposed control algorithm does not require an explicit plant model nor does it depend on the estimation of specific plant parameters. Rather, it uses information in the recent past to directly estimate the unknown dynamics at any given instant, through time delay. In earlier papers, analysis and implementation of Time Delay Controller for nonlinear systems were discussed. This paper analyzes the continuous Time Delay Controller for a class of linear systems and presents necessary and sufficient conditions for control system stability. A necessary condition for stability is derived using the properties of linear time-delayed systems. This condition involves only a few of the system and controller parameters and facilitates design of the Time Delay Controller. It is proved that this necessary condition is also sufficient if the delay time is chosen to be infinitesimally small. The convergence of closed loop system error to zero for certain classes of inputs and disturbances when the system is stable is also established. It is also shown that certain approximations in the control algorithm and certain additional unmodeled dynamics render the closed loop system under continuous Time Delay Control to be not exponentially stable due to the controller poles on the imaginary axis at infinitely high frequencies. However, in digital implementation, all the signals are prefiltered by anti-aliasing filters prior to sampling. Hence, the highest frequency component is automatically limited and the issue of exponential instability is not encountered. A discussion is presented comparing Time Delay Control with Repetitive Control. It is indicated that the Time Delay Controller can perform the functions of a repetitive controller with the delay time replaced by the period of the reference input while the repetitive controller can perform the functions of Time Delay Controller for sufficiently small “period” for a certain class of linear systems. Furthermore, examples are included to illustrate the results.

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