Steady-state gain constant

Equations (3.42) and (3.43) are the standard forms of transfer functions for a second-order system, where K = steady-state gain constant, Wn = undamped natural frequency (rad/s) and ( = damping ratio. The meaning of the parameters Wn and ( are explained in sections 3.6.4 and 3.6.3. 

The principal limitation to using these rules is that the true process parameters are often unknown. Steady-state gain K can be calculated from a process model or determined from the steady-state results of a step test as Ac/Au, as shown in Fig. 8-28. The test will not be viable, however, if the time constant of the process is longer than a few... 

The constant in the numerator can always be chosen to preserve the steady state gain of the transfer function. As suggested by Luus (1980) the 5 unknown parameters can be obtained by minimizing the following quadratic objective function... 

In addition to transfer functions, we make extensive use of steady state gain and time constants in our analysis. 

The steady state gain is the ratio of the two constant coefficients. Take note that the steady state gain value is based on the transfer function only. From Eqs. (2-31) and (2-32), one easy way to "spot" the steady state gain is to look at a transfer function in the time constant form. 

In this rearrangement, xp is the process time constant, and Kd and Kp are the steady state gains.2 The denominators of the transfer functions are identical, they both are from the LHS of the differential equation—the characteristic polynomial that governs the inherent dynamic characteristic of the process. 

K and KH in (2-49a) are referred to as gains, but not the steady state gains. The process time constant is also called a first-order lag or linear lag. 

After this exercise, let s hope that we have a better appreciation of the different forms of a transfer function. With one, it is easier to identify the pole positions. With the other, it is easier to extract the steady state gain and time constants. It is veiy important for us to leam how to interpret qualitatively the dynamic response from the pole positions, and to make physical interpretation with the help of quantities like steady state gains, and time constants. 

With the time constant defined as x = V/Qm s, the steady state gain for the transfer function for the inlet flow rate is (Cin s - Cs)/Qin s, and it is 1 for the inlet concentration transfer function. 

In the event that we are modeling a process, we would use a subscript p (x = xp, K = Kp). Similarly, the parameters would be the system time constant and system steady state gain when we analyze a control system. To avoid confusion, we may use a different subscript for a system. 

Figure 3.1. Properties of a first order transfer function in time domain. Left panel y/MK effect of changing the time constant plotted with x = 0.25, 0.5, 1, and 2 [arbitrary time unit]. Right panel y/M effect of changing the steady state gain all curves have x = 1.5.

From that same section, we know that the steady state gain and the time constant are dependent on the values of flow rate, liquid density, heat capacity, heat transfer coefficient, and so on. For the sake of illustration, we are skipping the heat transfer analysis. Let s presume that we have done our homework, substituted in numerical values, and we found Kp = 0.85 °C/°C, and xp = 20 min. 

When the steady state gains of all three assumptions are lumped together, we may arrive at a valve gain Kv with the units of °C/mV. For this illustration, let s say the valve gain is 0.6 °C/mV and the time constant is 0.2 min. The actuator controller function would appear as... 

Rearrange the expressions such that we can redefine the parameters as time constants and steady state gains for the closed-loop system. 

The system steady state gain is the same as that with proportional control in Example 5.1. We, of course, expect the same offset with PD control too. The system time constant depends on various parameters. Again, we defer this analysis to when we discuss root locus. 

Example 7.7 Consider installing a PI controller in a system with a first order process such that we have no offset. The process function has a steady state gain of 0.5 and a time constant of 2 min. Take Ga = Gm = 1. The system has the simple closed-loop characteristic equation ... 

Note The factor k is 6 here, and in the MATLAB manual, it is referred to as the "gain." This factor is really the ratio of the leading coefficients of the two polynomials q(s) and p(s). Make sure you understand that the "k" here is not the steady state gain—which is the ratio of the last constant coefficients. (In this example, the steady state gain is -12/-4 = 3.) MATLAB actually has a function named degain to do this. 

This is first order with a time constant of 1.5 min. and a steady-state gain of 4.8. With the AR plot,... 

The transmitter and thermocouple have a combined steady-state gain of 0.5 units and negligible time constants. Assuming the solenoid switch to act as a standard on-off element determine the limit of the disturbance in output gas temperature that the system can tolerate. 

A process consists of two transfer functions in series. The fint, relates the manipulated variable M to the variable and is a steady state gain of 1 and two filSt-order lags in series with equal time constants of 1 minute. 

G( ) is the transfer function relating 0O and 0X. It can be seen from equation 7.18 that the use of deviation variables is not only physically relevant but also eliminates the necessity of considering initial conditions. Equation 7.19 is typical of transfer functions of first order systems in that the numerator consists of a constant and the denominator a first order polynomial in the Laplace transform parameter s. The numerator represents the steady-state relationship between the input 0O and the output 0 of the system and is termed the system steady-state gain. In this case the steady-state gain is unity as, in the steady state, the input and output are the same both physically and dimensionally (equation 7.16h). Note that the constant term in the denominator of G( ) must be written as unity in order to identify the coefficient of s as the system time constant and the numerator as the system... 

Equation 7.52 is the standard form of a second-order transfer function arising from the second-order differential equation representing the model of the process. Note that two parameters are now necessary to define the system, viz. r (the time constant) and (the damping coefficient). The steady-state gain KMT represents the steady-state relationship between the input to the system AP and the output of the system z (cf. equation 7.50). 

There are distinct similarities between second order systems and two first-order systems in series. However, in the latter case, it is possible physically to separate the two lags involved. This is not so with a true second order system and the mathematical representation of the latter always contains an acceleration term (i.e. a second-order differential of displacement with respect to time). A second-order transfer function can be separated theoretically into two first-order lags having the same time constant by factorising the denominator of the transfer function e.g. from equation 7.52, for a system with unit steady-state gain ... 

A measuring instrument consists basically of a sensor and a transducer. The sensor transmits a signal x to the transducer every second and the transducer responds as a first-order system with a time constant of Ss and a steady-state gain of 2 units. The output yt of the transducer drives a transmitter which also approximates to first-order behaviour with a time constant of 4s and a steady-state gain of 5. 

K is the steady-state gain t is the time constant td is the dead time... 

The output of GFu, is y, which also recycles back through a second transfer function GRs- in the recycle path. This recycle transfer function also consists of a steady-state gain and a time constant. 

We want tight control of these important quantities for economic and operational reasons. Hence we should select manipulated variables such that the dynamic relationships between the controlled and manipulated variables feature small time constants and deadtimes and large steady-state gains. The former gives small closed-loop time constants and the latter prevents problems with the rangeability of the manipulated variable (control valve saturation t... 

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