Pure time delay

As shown in Fig. 2.14, the input signal from the process is transmitted through the sample pipe until it arrives at the measuring instrument at a delay time tL. In all other respects, however, the signal arriving at the measurement point is identical to the response of the actual system. 

In Fig. 2.15, the pointers are moved one element around the array at every communication interval CINT and thus with N elements separating the two pointers, a total time delay of N CINT is achieved. Provision must be made to zero the array initially and also to reset the pointers back to 1 as each pointer passes through the final element of the array M. 

Alternatively as shown in Sec. 2.1.2.1, as more and more tanks are connected in series, the obtained response approximates more and more to that of plug flow. Hence a very simple but approximate presentation of a time delay is that of a cascade of tanks in series, as shown in Fig. 2.16. 

In this procedure, it is necessary to write the balance equations for each tank, where for tank n 

Most simulation languages include a standard time delay function, which is pre-programmed into the language structure. This facility is also available in Madonna and is implemented in several of the simulation examples. 

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This is recognisable as a series combination of two unequal size CSTRs together with a pure time delay. 

However, the T-distribution permits an extension of the plate theory, which is also usable in case of asymmetric peaks. The chromatogram (1 component) is considered to be the result of a pure time delay and a T-distribution response. The procedure implies the fitting of a function f(t) given in Eq. (15) to the chromatographic peak. The asymmetry of the peak determines the new plate number n, decreasing with increasing asymmetry. 

Using the transfer function concept, Koppel (1967) derived the optimal control policy for a heat exchanger system described by hyperbolic partial differential equations using the lumped system approach. Koppel and Shih (1968) also presented a feedback interior control for a class of hyperbolic differential equations with distributed control. In an earlier paper Koppel e/ al. (1968) discussed the necessary conditions for the system with linear hyperbolic partial differential equations having a control which is independent of spatial coordinates. The optimal feedback-feedforward control law for linear hyperbolic systems, whose dynamical response to input variations is characterized by an initial pure time delay, was derived by Denn... 

The most important fact about piston flow is that disturbances at the inlet are propagated down the tube with no dissipation due to mixing. They arrive at the outlet t seconds later. This pure time delay is known as dead time. Systems with substantial amounts of dead time oscillate when feedback control is attempted. This is caused by the controller responding to an output caused by an input t seconds ago. The current input may be completely different. Feedforward control represents a theoretically sound approach to controlling systems with appreciable dead time. Sensors are installed at the inlet to the reactor to measure fluctuating inputs. The... 

Equation (22.35) may be implemented in a simulation by a delay function, incorporating a pure time delay equal to the controller timestep, Xc. ... 

W.H. Ray and M.A. Soliman. The optimal control of processes containing pure time delays — I. Necessary conditions for an optimum. Chem. Eng. Sci., 25 1911-1925, 1970. 

The structure is thus entirely defined by the three integers na, nb, and nk. na is equal to the number of poles and nb- is the number of zeros, while nk is the pure time-delay (the dead-time) in the system. For a system under sampled-data control, typically nk is equal to 1... 

In order to be able to establish confidence intervals, it also assumed that Ck is Gaussian distributed. The ARX model (1) is characterized by 3 numbers Ua, the auto-regressive order ny, the exogeneous order and rik, the pure time delay between input and output. A regression model is an ARXOlO model (with [ua, ny, n-k] = [0,1,0]). 

After some iterations, a six-ordered model with a pure time delay of 2 ms primarily due to the data aequisition system has fitted well to the frequeney response data of an aetuator with dimensions of 10 x 2 x 0.17 mm, as shown in Fig. 13 (John et al. 2008b). The identified model whieh is a blaek box model deseribing the relationship between the input voltage and the tip displaeement is obtained as... 

The term is the process gain, the term indicates a pure time delay. When the concentration changes at the beginning of the reactor, it takes Tr time units before the change reaches the end of the reactor. If the residence time Zr = 10 s and k = 0.2 s then k- TR = 2, hence the process gain is 0.135, i.e. a unit change in the inlet concentration of 1.0 results in a change in the outlet concentration of 0.135. The step response for the outlet concentration is shown in Fig. 13.2. 

Figmes 14.4-14.6 show the step responses of the transfer functions. As can be seen from Fig. 14.4, the model between fluid outlet temperature changes and fluid inlet temperature changes is a pure time delay of 12 seconds. This can be expected since the fluid has to move through the pipe, before the inlet change reaches the outlet. 

Add tip the pure time delays and the effective time delays from small time constants for one pass around the control loop to get the total dead time. 

If the time delay is much larger than the time constant (Tj To), it can be shown that Equation 8.10s reduces to the ultimate gain being the inverse of the open loop gain. This relationship can also be realized from the amplitude ratio being 1 for a pure time delay. 

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