Step responses

In this section we will look at the response of an RC circuit to a pulse input. We will work with the circuit below  

The time constant of the circuit is T = RC = (1 k 2)(l fiF) = 1 ms. The response of an inductive or capacitive circuit will reach 99% of its final value in 3 time constants, and will reach steady state in about 5 time constants. We will run two simulations. The first will let the capacitor charge and discharge for 3 time constants (3 ms in this case) and the second will let the capacitor charge and discharge for 5 time constants (5 ms in this example). For the first simulation we will use the waveform below  

The length of the simulation is 9 ms. The pulse will start at zero volts and remain there for 3 ms, then go to 5 V for 3 ms, and then return to 0 for another 3 ms. To create this waveform we can use either a pulsed voltage source (VPULSE) or a piecewise linear voltage source (VPWL). For this example we will use the VPWL source. To edit the attributes of the PWL 

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Scroll the window to the right until you see the time and voltage attributes  

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The step response function h(x) is the response of a system to an unit step s(x) at the input. 

The unit step function s(x) is defined as a step from 0 to /. The function s(x) is shown in fig. 1 (centre) together with an example of a step response function h(x). 

That means, the derivation of the measured step response h(x) along the path x delivers the impulse response function g(x) of the system. 

Unit impulse response Unit step response responses for input example... 

All described sensor probes scan an edge of the same material to get the characteristic step response of each system. The derivation of this curve (see eq.(4) ) causes the impulse responses. The measurement frequency is 100 kHz, the distance between sensor and structure 0. Chapter 4.2.1. and 4.2.2. compare several sensors and measurement methods and show the importance of the impulse response for the comparison. 

The step response of this transfer function is shown in Fig. 8-19. Note that all curves reach about 60 percent of their final value at t = nX. 

FIG. 8-28 If a steady state can be reached, gain K and time constant T can be estimated from a step response if not, use Tj instead. 

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

The original formulations of MPC (i.e., DMC and IDCOM) were based on empirical hnear models expressed in either step-response or impulse-response form. For simphcity, we will consider only a singleinput, single-output (SISO) model. However, the SISO model can be easily generalized to the MIMO models that are used in industrial applications. The step response model relating a single controlled variable y and a single manipiilated variable u can be expressed as... 

The step-response model is also referred to as a finite impulse response (FIR) model or a discrete convolution model. 

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. 

The step-response model in Eq. (8-63) is equivalent to the following impulse response model ... 

FIG. 8-43 Step response for u, a unit step change in the input. 

Table 3.2 Unit step response of a first-order system...

Experimental determination of system time constant using step response... 

This gives a step response function of the form shown in Figure 3.16. 

Fig. 3.18 Step response of a generalized second-order system for C < 1-Expanding equation (3.52) using partial fractions...

Fig. 3.19 Unit step response of a second-order system.

It is possible to identify the mathematieal model of an underdamped seeond-order system from its step response funetion. 

Figure 3.22 shows, in bloek diagram form, the transfer funetions for a resistanee thermometer and a valve eonneeted together. The input X[ t) is temperature and the output Xo t) is valve position. Find an expression for the unit step response funetion when there are zero initial eonditions. 

Determine the values of Wn and ( and also expressions for the unit step response for the systems represented by the following second-order transfer functions... 

Fig. 4.24 Step response of a first-order plant using proportional control.

Fig. 4.25 Step response of a first-order plant using PI control. When there are step ehanges in r t) and riit). ...

Note that the Proeess Reaetion Method eannot be used if the open-loop step response has an overshoot, or eontains a pure integrator(s). 

Fig. 4.35 Closed-loop step response of temperature control system using PID controller tuned using Zeigler-Nichols process reaction method.

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