Pure delay

Figure 10.15 Section of a stiff string where the allpass delay elements are consolidated at two points, and a sample of pure delay is extracted from each allpass chain.

Pure delays in dynamic response impose fundamental bounds on achievable disturbance rejection, as discussed in Section II.B.2. Uncertainty imposes additional limitations. For example, if the measurement lag were accurately known, it could be canceled and would not limit control performance. But the measurement response is more complex than this and it is variable over time thus it cannot readily be canceled and combines with the delay to limit controller performance (see Sections V.A.3 and V.A.5). Limits on the reagent addition rate prevent the ideal delay-limited control bound from being achieved in some cases, even in the absence of uncertainty, as infinite actuator range is required (see Section V.A.5). However, these limits are not observed to have a major effect on PI control performance as the degree of transient overshoot is moderate compared to the steady-state control output change required. 

Section II.B.2 suggested that when minor lags are present in addition to pure delays, a heuristic effective delay could be computed based on the natural period of the control loop divided by 4. Numerical optimization suggested that Eq. (52) remained approximately valid, although the optimized performance was slightly improved compared to the pure delay case. 

In Section V.A.4, we showed that optimal volume is proportional to t j and that there is a factor of about (1.5n ) " between the optimal volume with PI control (without uncertainty) and the volume required, assuming the performance bound is reached. The PI performance is governed by the effective delay it d — tJ4) rather than the actual pure delay. The maximum effective delay, assuming a mixing delay of 10 s, a probe lag of up to 30 s, and a CSTR with > td. , is 28.7 s. The measurement lag may therefore imply almost a threefold increase in the required volume, compared to that required with an instantaneous measurement response. 

The experimental curve in Figure 3 demonstrates overshoot in the tissue oxygen response. It was determined previously (22) that a term representing pure delay along with the steady-state blood flow vs. arterial oxygen tension data would cause overshoot. In this investigation it was found that a first-order time constant delay would also produce overshoot. Therefore, since exact controller mechanisms are not being postulated, the flow controlled dynamics used in this study include pure delay and time constant lag. To consider the problem of sensor location, feedback and feedforward control loops were superimposed on the capillary-tissue model. 

Figure 3 demonstrates the simulated tissue overshoot with a pure delay of 20 sec and time constant lag of 10 sec. Manipulation of the two variables yields overshoot peaks of various amplitudes and shapes. 

Autoregulatory action helps to reduce nerve cell destruction resulting from brain tissue anoxia. Two possible mechanisms include flow controller dynamics in the form of pure delays and time constant lags and oxygen consumption control with Michaelis-Menten behavior. Response curves also suggest the possibility of facilitated or active transport of oxygen in tissue and resistance to the diffusion of oxygen from the tissue into the blood stream. 

The oversimplified picture given above is contrary to our physical experience, which dictates that whenever an input variable of a system changes, there is a time interval (short or long) during which no effect is observed on the outputs of the system. This time interval is called dead time, or transportation lag, or pure delay, or distance-velocity lag. 

Pure delay or dead time usually occurs due to transportation lag, such as flow through a pipe. For a velocity V and distance L,9 = L / v. Another case is a measurement delay, such as for a gas or Uquid chromatograph. 

Individual linear control loops The bandwidth of each loop is restricted to 0.5 times the modulus of the smallest RHP zero of the transfer function or the reciprocal value of the pure delay (dead time) in the channel, respectively. This corresponds to rise times not smaller than the dead time or the inverse of the zero. If these bounds are not respected, the sensitivity transfer function has a large peak and the control loop is highly sensitive to model errors and therefore the controller is not applicable in practice. 

A pure delayed hypersensitivity without simultaneous production of humoral antibody is not easily induced in mice. The experimental model chosen was that described by Asherson and Ptak where the delayed hypersensitivity reaction is measured by the increase in ear thickness 24 hours after a local application of picryl chloride in mice pre-sensitized 6-7 days previously by skin painting with this antigen. 

One may proceed in a similar fashion to obtain solutions for more and more tanks in series with equal time constants, with the constraint that the total time constant is equal to the sum of the identical individual time constants so that ti = T2 = = %n = %/n, creating systems of higher and higher order. These results are plotted in Figure 12.20 for the cases with the numbers of tanks in series n = 1,2,5,10, CO with M = 1 m min KpiKp2. .. Kp = 1 min m , t = 1 min. As the number of tanks increases, the response approaches that of a pure delay or dead time, 8, equal to the total time constant of the infinite tanks, 0 = z. 

Dead time is a characteristic of a physical system that causes an input disturbance to be delayed in time, but unaffected in form. Whereas capacitance changes the form of the input disturbance (i.e. a step is filtered into a typical first-order curve), dead time is a pure delay of the input disturbance. Dead time is also referred to as transport lag, or distance velocity lag. A typical example process is the continuous weighing system shown in Figure 3.21. 

Dead time imposes a pure delay on disturbances, effectively hiding the disturbance from the process, the measurements and the controls until it is well into the system. Dead time deteriorates controllability, especially if it is large relative to the amount of capacitance in the system with which it is associated. Dead time should generally be minimized as far as possible. 

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