Diagrams closed loops

Figure 1. Artificial pancreas block diagram. Closed-loop control (—) uses the sensor ana programmers to direct flow controller. In open-loop control (—), augmentation is initiated by the diabetic, without the sensor. Typical design criteria are indicated on the right-hand side of the drawing.

Open-Loop versus Closed-Loop Dynamics It is common in industry to manipulate coolant in a jacketed reacdor in order to control conditions in the reacdor itself. A simplified schematic diagram of such a reactor control system is shown in Fig. 8-2. Assume that the reacdor temperature is adjusted by a controller that increases the coolant flow in proportion to the difference between the desired reactor temperature and the temperature that is measured. The proportionality constant is K. If a small change in the temperature of the inlet stream occurs, then depending on the value or K, one might observe the reactor temperature responses shown in Fig. 8-3. The top plot shows the case for no control (K = 0), which is called the open loop, or the normal dynamic response of the process by itself. As increases, several effects can be noted. First, the reactor temperature responds faster and faster. Second, for the initial increases in K, the maximum deviation in the reactor temperature becomes smaller. Both of these effects are desirable so that disturbances from normal operation have... 

Figure 6.6 Typical block diagram of a W/control scheme with open- or closed-loop control scheme...

The elements of a closed-loop control system are represented in block diagram form using the transfer function approach. The general form of such a system is shown in Figure 4.1. 

Fig. 4.1 Block diagram of a closed-loop control system. R s) = Laplace transform of reference input r(t) C(s) = Laplace transform of controlled output c(t) B s) = Primary feedback signal, of value H(s)C(s) E s) = Actuating or error signal, of value R s) - B s), G s) = Product of all transfer functions along the forward path H s) = Product of all transfer functions along the feedback path G s)H s) = Open-loop transfer function = summing point symbol, used to denote algebraic summation = Signal take-off point Direction of information flow.

Fig. 4.39 Block diagrams for closed-loop systems. Amplifier...

The root locus method provides a very powerful tool for control system design. The objective is to shape the loci so that closed-loop poles can be placed in the. v-plane at positions that produce a transient response that meets a given performance specification. It should be noted that a root locus diagram does not provide information relating to steady-state response, so that steady-state errors may go undetected, unless checked by other means, i.e. time response. 

In practice, only the frequencies lu = 0 to+oo are of interest and since in the frequency domain. v = jtu, a simplified Nyquist stability criterion, as shown in Figure 6.18 is A closed-loop system is stable if, and only if, the locus of the G(iLu)H(iuj) function does not enclose the (—l,j0) point as lu is varied from zero to infinity. Enclosing the (—1, jO) point may be interpreted as passing to the left of the point . The G(iLu)H(iLu) locus is referred to as the Nyquist Diagram. 

The M and N circles can be superimposed on a Nyquist diagram (called a Hall chart) to directly obtain closed-loop frequency response information. 

Alternatively, the closed-loop frequency response can be obtained from a Nyquist diagram using the direct construction method shown in Figure 6.25. From equation (6.73)... 

Fig. 6.25 Closed-loop frequency response from Nyquist diagram using the direct construction method.

The Nichols chart shown in Figure 6.26 is a rectangular plot of open-loop phase on the x-axis against open-loop modulus (dB) on the jr-axis. M and N contours are superimposed so that open-loop and closed-loop frequency response characteristics can be evaluated simultaneously. Like the Bode diagram, the effect of increasing the open-loop gain constant K is to move the open-loop frequency response locus in the y-direction. The Nichols chart is one of the most useful tools in frequency domain analysis. 

Using MATLAB to design a system, it is possible to superimpose lines of constant ( and ajn on the root locus diagram. It is also possible, using a cursor in the graphics window, to select a point on the locus, and return values for open-loop gain K and closed-loop poles using the command... 

Running script file fig629.m will produce the closed-loop frequency response gain diagrams shown in Figure 6.29 for Example 6.4 when K = 3.8 and 3.2 (value of K for best flatband response). 

Creates closed-loop Bode Gain Diagrams for K=3.8 and 3.2 %Prints in Command Window Mp,k,wp and bandwidth 0 If... 

The command cloop is used to find the closed-loop transfer function. The command max is used to find the maximum value of 20 logio (mag), i.e. Mp and the frequency at which it occurs i.e. tUp = uj k). A while loop is used to find the —3 dB point and hence bandwidth = ca (n). Thus, in addition to plotting the closed-loop frequency response gain diagrams,/ gd29.7 will print in the command window ... 

FIG. 4 Qualitative phase diagram close to a first-order irreversible phase transition. The solid line shows the dependence of the coverage of A species ( a) on the partial pressure (Ta). Just at the critical point F2a one has a discontinuity in (dashed line) which indicates coexistence between a reactive state with no large A clusters and an A rich phase (hkely a large A cluster). The dotted fine shows a metastability loop where Fas and F s are the upper and lower spinodal points, respectively. Between F2A and Fas the reactive state is unstable and is displaced by the A rich phase. In contrast, between F s and F2A the reactive state displaces the A rich phase. 

Example 2.16. Derive the closed-loop transfer function X,/U for the block diagram in Fig. E2.16a. We will see this one again in Chapter 4 on state space models. With the integrator 1/s, X2 is the Laplace transform of the time derivative of x,(t), and X3 is the second order derivative of x,(t). 

We can check with MATLAB that the model matrix A has eigenvalues -0.29, -0.69, and -10.02. They are identical with the closed-loop poles. Given a block diagram, MATLAB can put the state space model together for us easily. To do that, we need to learn some closed-loop MATLAB functions, and we will defer this illustration to MATLAB Session 5. 

Figure 5.4. Block diagram of a simple SISO closed-loop system.

We first establish the closed-loop transfer functions of a fairly general SISO system. After that, we ll walk through the diagram block by block to gather the thoughts that we must have in synthesizing and designing a control system. An important detail is the units of the physical properties. 

We first need to derive the closed-loop functions for the system. Based on the block diagram, the error is... 

The time delay effect is canceled out, and this equation at the summing point is equivalent to a system without dead time (where the forward path is C = GCGE). With simple block diagram algebra, we can also show that the closed-loop characteristic polynomial with the Smith predictor... 

In this section, we add the so-called decoupler functions to a 2 x 2 system. Our starting point is Fig. 10.12. The closed-loop system equations can be written in matrix form, virtually by visual observation of the block diagram, as... 

Fig. 10 Phase diagrams of dPS/PnPMA blends, o UCST at ambient pressure, LCST at ambient pressure. Closed-loop phase behaviours are observed at higher pressure A 97 bar 117 bar 138 bar 166 bar 186 bar. Dashed line Prediction of Tg, blend by Fox equation at ambient pressure. From [59]. Copyright 2004 American Chemical Society... 

Below are several terms associated with the closed-loop block diagram. 

Figure 5.26. Iron binary alloys. Examples of the effects produced by the addition of different metals on the stability of the yFe (cF4-Cu type) field are shown. In the Fe-Ge and Fe-Cr systems the 7 field forms a closed loop surrounded by the a-j two-phase field and, around it, by the a field. Notice in the Fe-Cr diagram a minimum in the a-7 transformation temperature. The iron-rich region of the Fe-Ru diagram shows a different behaviour the 7 field is bounded by several, mutually intersecting, two (and three) phase equilibria. The Fe-Ir alloys are characterized, in certain temperature ranges, by the formation of a continuous fee solid solution between Ir and yFe. Compare with Fig. 5.27 where an indication is given of the effects produced by the different elements of the Periodic Table on the stability and extension of the yFe field.

Fig. 2. Block diagram for the complete closed-loop behavior. Here, the block corresponding to the refrigeration system is neglected.

Fig. 3. Bode diagram of the closed-loop under the estimation/compensation control approach. Closed-loop system is expected to be non-sensitive to high frequency signais. Arrows indicate how the frequency response of the ciosed-ioop as Kc is increased.

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