Sample-time control algorithm

The variable sample time control algorithm was tested experimentally and the results compared with computer simulations. Tests were made with and without modeling error (parameter shift) for set point and load changes. 

SIGN is used to determine whether the carriage position is incrementing or decremented. TIME is the sampling time of the proportional control algorithm. 

In a digital computer-control system, the feedback controller has a pulse transfer function. What we need is an equation or algorithm that can be programmed into the digital computer. At the sampling time for a given loop, the computer looks at the current process output x, compares it to a setpoint, and calculates a current value of the error. This error, plus some old values of the error and old values of the controller output or manipulated variable that have been stored in computer memory, are then used to calculate a new value of the controller output m,. 

Leemans described a sampling scheme based on these algorithms that considers sampling frequency, sampling time, dead time and accuracy of the method of analysis to obtain optimal information yield or maximal profit when controlling a factory. 

Variable Sample Time Algorithm for Microcomputer Control of a Heat Exchanger... 

The purpose of the present study is to present a simply derived and implemented DDC algorithm for the flow-forced heat exchanger which does not require storage of previous values of the manipulated variable or error. The algorithm requires that Steady-state exist at the sampling instants which means that the sample time is variable. However, the algorithm is shown to control for the constant sample time case. 

Because the control algorithm was derived from the steady-state portion of Equation 1 it may be applied, strictly, only after the system has reached steady-state following a change in velocity, that is, after one residence time. This means that the minimum sampling time, Tg, is given by... 

Set Point Changes, Constant Sample Time. When the constant twenty seconds was added to the variable residence time, the resulting variable sample times were in the 22-26 second range. Since this variable sample time is similar to a long constant sample time, the performance of the control algorithm was tested with various constant sample times. The constant sample times... 

Results for an intermediate constant sample time of 10 seconds are shown in Figure 5. The control in this case and also for the 15 second sample time case, not shown, is improved compared with the constant 25 and 5 second sample times. This suggests that an optimal constant sample time exists for application of the variable sample time algorithm. The optimum constant sample time appears to be on the order of 2 to 3 thermocouple time constants plus the average fluid residence time. 

Cohen-Coon settings (see Section 16.5) From the process reaction curve we can estimate the process static gain K, the dominant time constant r, and the process dead time td Then, from eqs. (16.9) through (16.11c), we can compute the parameters Kc, r/, and rD of a P, PI, or PID control algorithm. The effect of the sampling period T has been accounted for by the nature of the experiment itself, because the reaction curve has been determined using sampled-values of the process output. 

Design the deadbeat control algorithms for a first-order process with dead time equpl to 3T. Can we have a physically realizable controller if we require that the response exhibit zero error at all sampling instants after the first ... 

Equation (29.3) yields the discrete transfer function of the velocity PID control algorithm. For given changes in the sampled values of the error signal, the resulting discrete-time control action can be found from the inverse z-transform ... 

Consider the block diagram of a direct digital feedback control loop shown in Figure 29.9. Such loops contain both continuous- and discrete-time signals and dynamic elements. Three samplers are present to indicate the discrete-time nature of the set point j/Sp( ), control command c(z), and sampled process output y(z). The continuous signals are denoted by their Laplace transforms [i.e., y(s), Jn(s), and d(s)]. Furthermore, the continuous dynamic elements (e.g., hold, process, disturbance element) are denoted by their continuous transfer functions, H(s), Gp(s), and GAs), respectively. For the control algorithm, which is the only discrete element, we have used its discrete transfer function, D(z). 

Because the digital computer calculates a new value of output for a given loop only once each cycle, it cannot solve differential equations. Instead, digital control algorithms are difference equations, whose time base is the sampling interval. The differential equation for an ideal noninteracting analog controller is... 

Microprocessor controls usually have a sampling time of a fraction of a second. Although slightly slower than analog controls, their performance can generally be approximated by analog control algorithms. 

The nonlinear simulation was used to illustrate the closed-loop response of the controlled variable X2 following a 30 percent increase in feed composition. The results are shown in Figure 21.4b with the feedback-only dual and PID algorithms. Control is immensely improved with the feedforward action. The slight deviation in X2 with feedforward control is due to inaccuracies in the linear model and the long sampling time relative to the process dead time. The... 

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