Transfer functions open-loop

The closed-loop transfer function is the forward-path transfer function divided by one plus the open-loop transfer function. 

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.

The position of the closed-loop poles in the. v-plane determine the nature of the transient behaviour of the system as can be seen in Figure 5.5. Also, the open-loop transfer function may be expressed as... 

The open-loop transfer function for a control system is... 

A control system has the open-loop transfer function given in Example 5.8, i.e. 

Closed-loop control systems are classified according to the number of pure integrations in the open-loop transfer function. If... 

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

A process plant has an open-loop transfer function... 

The laser guided missile shown in Figure 5.26 has an open-loop transfer function (combining the fin dynamics and missile dynamics) of... 

Nyquist plot Bode plot Nyquist plot is a frequency parametric plot of the magnitude and the argument of the open-loop transfer function in polar coordinates. Bode plot is magnitude vs. frequency and phase angle vs. frequency plotted individually. 

We can now state the problem in more general terms. Let us consider a closed-loop characteristic equation 1 + KCG0 = 0, where KCG0 is referred to as the "open-loop" transfer function, G0l- The proportional gain is Kc, and G0 is "everything" else. If we only have a proportional controller, then G0 = GmGaGp. If we have other controllers, then G0 would contain... 

Another advantage of frequency response analysis is that one can identify the process transfer function with experimental data. With either a frequency response experiment or a pulse experiment with proper Fourier transform, one can construct the Bode plot using the open-loop transfer functions and use the plot as the basis for controller design.1... 

With frequency response analysis, we can derive a general relative stability criterion. The result is apphcable to systems with dead time. The analysis of the closed-loop system can be reduced to using only the open-loop transfer functions in the computation. 

This equation, of course, contains information regarding stability, and as it is written, implies that one may match properties on the LHS with the point (-1,0) on the complex plane. The form in (7-2a) also imphes that in the process of analyzing the closed-loop stability property, the calculation procedures (or computer programs) only require the open-loop transfer functions. For complex problems, this fact eliminates unnecessary algebra. We just state the Nyquist stability criterion here.1... 

Or if we know beforehand the open-loop transfer function to be used ... 

Let say we have a simple open-loop transfer function G0 of the closed-loop characteristic equation... 

B. If the inlet liquid flowrate remains constant, prove that the open-loop transfer function for the response of y2 to a change in inlet gas composition is given by ... 

In order to determine the maximum allowable value of N, it is necessary to determine the real part of the open-loop transfer function when the imaginary part is zero. 

Note that the effect of adding a zero or a lead is to pull the root locus toward a more stable region of the s plane. The root locus starts at the poles of the open-loop transfer function. As the gain goes to infinity the two paths of the root locus go to minus infinity and to the zero of the transfer function at s = -2. We will find that this is true in general the root locus plot ends at the zeros of the openloop transfer function. 

Example 11JL Let s take the same process as studied in Example 11.7. The process open loop transfer function is... 

M i (input) and Jfr0 (output), and Gyb(s) is the product of all blocks in the whole loop—often termed the open-loop transfer function of the control system. It is possible to apply the same rule successively to simplify certain multiple loop control schemes (e.g. cascade control—Section 7.13). 

Equation 7.119 is the characteristic equation of the system shown in Fig. 7.34 and is dependent only upon the open-loop transfer function G (j)H(j) and is therefore the same for both set point and load changes (equations 7.109 and 7.110). 

The roots of the characteristic equation may be real and/or complex, depending on the form of the open-loop transfer function. Suppose at to be complex, such that ... 

Hence with Kc= 1.8 and r, = 3.5, the polar plot of the open-loop transfer function passes through the point (-1, 0). This confirms the result obtained in Example 7.6 using the Routh-Hurwitz criterion, i.e. that with these controller parameters, the response of the controlled variable is conditionally stable. Figure 7.55 shows polar plots of the open-loop transfer function Gr(i) for different values of Kc and t>. 

Fio. 7.62. Polar plot of the open-loop transfer function for the heat exchanger control system described in Example 7.6 with Kc = 1.8 and t, 2.S (a) uncompensated system (b) compensated system with rt 1 min, r2 = 0.1 min... 

Fig. 7.64. Log modulus plots of open-loop transfer function for Example 7.10 curve (i)—uncompensated system curve (ii)—compensated system...

The open loop transfer function of an IPDT IV process is the following ... 

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