Step disturbances

One-dimensional representation of the propagation of a concentration step-disturbance in a packed bed reactor (5). 

The response of a system to a step disturbance is called the step response or the transient response. 

Therefore the time-dependent response of C to the step disturbance in feed concentration is... 

We put in a step disturbance m, and record the output variable x, as a function of time, as illustrated in Fig. 14.1. The quick-and-dirty engineering approach is to simply look at the shape of the x, curve and find some approximate transfer function Gjj, that would give the same type of step response. 

This pulse transfer function has a zero at z = a and a pole at z = +1. It cannot produce any phase-angle advance since the pole lies to the right of the zero (a is less than 1). The pole at -I-1 is equivalent to integration (pole at s = 0 in continuous systems) which drives the system to zero steadystate error for step disturbances. 

Exercise 4. Prom the loop 1 and loop 2 of the block diagram in Fig. 10 determine the response to step disturbances in Fjit ) and Fo t)... 

Figure 7.5 shows typical responses of a controlled variable to a step disturbance in load for a simple control loop of the type illustrated in Figs. 7.1 and 7.3. The effects of different control actions are summarised in the following sections. 

Fig. 7.5. Response of controlled variable to step disturbance in load using different...

Figure 25. Simulated response of third reactor of a continuous vinyl acetate polymerization to a step disturbance at high emulsifier feed concentration (0.06 mol/L H,0) and manipulation of initiator flow rate to the third reactor at 50°C ((-------j optimum PID) (--------) Z transform (XXX) self-tuning algorithm)...

If a step disturbance reduces the feed rate F, the region of exchange and the temperature front are forced to move upward the column (Figure 4). The state variables of all other regions are only slightly altered. 

This optimization can be trivially transformed to linear programming. A step disturbance equal to -0.05 and a step setpoint change equal to 0.05 enter the closed loop at time k = Q. For move suppression coefficient values to = ti < 0.5, the resulting closed-loop response is shown in Fig. 9. The closed loop is clearly unstable. If the move suppression coefficients take values ro = ri > 0.5 (to penalize the move suppression coefficients even more) the closed loop remains unstable, as shown in the Fig. 10. The instability is due to the nonminimum phase characteristics of the process. Although a longer optimization horizon length, p, might easily solve the... 

Figure 2. Comparison of the SMB outlet purities in the controlled and uncontrolled case. The controller is switched on after reaching steady state and a step disturbance in Henry s constants takes place at cycle 60. (AHa=+10%, a Hb=+15%).

If the time between two step disturbances is an element of v, then the worst case is not necessarily at either bound of this variable. 

For example, optimizing control moves for disturbance rejection in a blending system subject to step disturbances of variable magnitude would give perfect disturbance rejection for a maxmin/perfect-knowledge formulation and a poor... 

Simplified Analysis for Series CSTRs. Although general problems require optimization of a nonlinear dynamic model as discussed above, the analysis can be greatly simplified for some special cases. The case of particular interest for the problems considered later is that of continuous-flow stirred-tank reactors (CSTRs) in series. In this case, it is desired to add reagent so as to keep variations in the net concentration of effluent and reagent, cnet, at the exit of the last tank below a certain level, 8 ei, in the face of step disturbances in the inlet concentration of magnitude A,.,. This objective can be expressed as a required disturbance attenuation, 5,., where... 

Similar expressions can be derived for responses to ramp disturbances. If the disturbance is a decaying exponential, then this is equivalent to an extra lag on a step disturbance, and the response can be evaluated by using the formulas for non-equal-sized tanks. The improvement in achievable attenuation from replacing a step by an exponential disturbance of time constant r or a ramp reaching a new steady value after r is approximately t J n + 1)t. ... 

Up to the present time it has not been possible to demonstrate the ultimate reliability of characterizations based on random disturbances. However, the use of random disturbances offers great potential advantage in studying existing process control systems where upsets like step disturbances cannot be tolerated. Because of the extensive calculation required to reduce the random operating records to statistical-correlation functions, high speed digital computation is essential in this treatment. 

PI is the most used controller. The tuning consists of a Pl-controller the finding the dynamic characteristics called ultimate gain (/C ) and ultimate period (Py). A step disturbance is introduced, and the gain of controller in P mode is slowly increased. The value where sustained bounded oscillations appear designates the ultimate gain K. The ultimate period can be measured directly. 

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