Step-change disturbance

All the above changes are easily implementable in dynamic simulations, using ISIM and other digital simulation languages. The forms of response obtained differ in form, depending upon the system characteristics and can be demonstrated in the various ISIM simulation examples. The response characteristics of real systems are, however, more complex. In order to be able to explain such phenomena, it is necessary to first examine the responses of simple systems, using the concept of the simple, step-change disturbance. 

First-Order Response to an Input Step-Change Disturbance... 

Figure 2.2. First-order exponential response to an imposed step-change disturbance.

Hence the value of the equation time constant, x, is simply determined as the time at which the response achieves sixty three per cent of its eventual steady-state value, when following a step change disturbance to the system. 

Often an instrument response measurement can be fitted empirically to a first-order lag model, especially if the pure instrument response to a step change disturbance has the general shape of a first-order exponential. 

Assuming that the instrument response is first order, then as shown in Sec. 2.1.1.1, the instrument time constant Xm is then given by the value of time at the 63% point (response to a step-change disturbance), where... 

Figure 2.19. Higher order response to a step change disturbance.

FIG. 8 26 Resp onse for a step change in disturbance with tuned P, PI, and PID controllers and with no control. 

In principle, the step-response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-43. The step response coefficients are simply the values of the output variable at the samphng instants, after the initial value y(0) has been subtracted. Theoretically, they can be determined from a single-step response, but, in practice, a number of bump tests are required to compensate for unanticipated disturbances, process nonhnearities, and noisy measurements. 

These differential equations are readily solved, as shown by Luyben (op. cit.), by simple Euler numerical integration, starling from an initial steady state, as determined, e.g., by the McCabe-Thiele method, followed by some prescribed disturbance such as a step change in feed composition. Typical results for the initial steady-state conditions, fixed conditions, controller and hydraulic parameters, and disturbance given in Table 13-32 are listed in Table 13-33. 

In testing process systems, standard input disturbances such as the unit-step change, unit pulse, unit impulse, unit ramp, sinusoidal, and various randomised changes can be employed. 

DISTURBANCE IS A STEP CHANGE IN FEED CONCENTRATION AT TIME EQUAL ZERO PROM O.S TO 1.8 KG-MOLES OF A/CURIC METER. 

DISTURBANCE (CAD) IS A STEP CHANGE AT TIME EQUAL ZERO FROM 0 TO 0.2. 

Figure 5.3 shows results for a step change in the disturbance of 0.2 at time equal zero. An integration step size of 0.1 min is used. We will return to this simple system later in this book to discuss the selection of values for and r, that is, how we tune the controller. . 

DISTURBANCE IS STEP CHANGE IN FEED COMPOSITION AT TIME ZERO FROM 0.50 TO 0.55. 

J0i Use Laplace transforms to prove mathematically that a P controller produces steadystate ofiMt and that a PI controller does not. The disturbance is a step change in the load variable. The process openloop transfer functions, Gm and G[, are both liist-order lags with dUTerent gains but identical time constants. 

Figure 10.9 shows the time-domain performance of these PI and PID controllers. The disturbance is a step change in Qp. Note the improved dynamic performance of the PID controllers. 

The chromatograph deadtime is assumed equal to the sampling period, so D, = T, and k = 1. The disturbance is again a unit step change in load so (GlL),, is the same as used in Example 18.8. However, we must find the new (G ... 

Before we leave this example, let s see what happens if we use this controller, which has been designed for step changes in load, but the real disturbance is a step change in setpoint. 

The application of the SVD technique provides a measure of the controllability properties of a given d mamic system. More than a quantitative measure, SVD should provide a suitable basis for the comparison of the theoretical control properties among the thermally coupled sequences under consideration. To prepare the information needed for such test, each of the product streams of each of the thermally coupled systems was disturbed with a step change in product composition and the corresponding d3mamic responses were obtained. A transfer function matrix relating the product compositions to the intended manipulated variables was then constructed for each case. The transfer function matrix can be subjected to SVD ... 

For start-up and disturbance simulations, the linearized model does predict an eventual return to the steady state around which the system was linearized. However, for step-input changes where the final steady state differs from the original, some minimal loss in accuracy is apparent in the final steady state reached using dynamic simulations of the linear model from the original steady state. This difficulty can easily be circumvented in the case of step changes by relinearizing about the new final steady-state conditions somewhere during the simulation. 

The normal function of any control system is to ensure that the controlled variable attains its desired value as rapidly as possible after a disturbance has occurred, with the minimum of oscillation. Determination of the response of a system to a given forcing function will show what final value the controlled variable will attain and the manner in which it will arrive at that value. This latter is a function of the stability of the response. For example, in considering the response of a second order system to a step change, it can be seen that oscillation increases... 

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