Ultimate frequency

Direct substitution Substitute s = jto in characteristic polynomial and solve for closed-loop poles on /m-axis. The Im and Re parts of the equation allow the ultimate gain and ultimate frequency to be solved. 

The closed-loop poles may lie on the imaginary axis at the moment a system becomes unstable. We can substitute s = jco in the closed-loop characteristic equation to find the proportional gain that corresponds to this stability limit (which may be called marginal unstable). The value of this specific proportional gain is called the critical or ultimate gain. The corresponding frequency is called the crossover or ultimate frequency. 

From the imaginary part equation, the ultimate frequency is u = Vl 1. Substituting this value in the real part equation leads to the ultimate gain Kc u = 60, which is consistent with the result of the Routh criterion. 

One may question whether direct substitution is a better method. There is no clear-cut winner here. By and large, we are less prone to making algebraic errors when we apply the Routh-Hurwitz recipe, and the interpretation of the results is more straightforward. With direct substitution, we do not have to remember ary formulas, and we can find the ultimate frequency, which however, can be obtained with a root locus plot or frequency response analysis—techniques that we will cover later. 

When the system has dead time, we must make an approximation, such as the Pade approximation, on the exponential dead time function before we can apply the Routh-Hurwitz criterion. The result is hence only an estimate. Direct substitution allows us to solve for the ultimate gain and ultimate frequency exactly. The next example illustrates this point. 

The solution of this equation is the ultimate frequency cou = 0.895, and from the real part equation, the corresponding ultimate proportional gain is Kc u = 5.73. Thus the more accurate range of Kc that provides system stability is 0 < Kc < 5.73. 

Note 2 The iterative solution in solving the ultimate frequency is tricky. The equation has poor numerical properties—arising from the fact that tan9 "jumps" from infinity at 9 = (ir/2) to negative infinity at 9 = (ir/2)+. To better see why, use MATLAB to make a plot of the function (LHS of the equation) with 9 < co < 1. With MATLAB, we can solve the equation with the f zero () function. Create an M-file named f. m, and enter these two statements in it ... 

To find the ultimate gain K, and ultimate frequency qj, we substitute w> for s. 

When a proportional feedback controller is used, this process has an ultimate gain of 64 and an ultimate frequency of... 

To solve for the ultimate gain and ultimate frequency, wc substitute i[Pg.380]

E3umple 11.S. Let us take the same third-order process analyzed in Example 11.4. For a t = 5 minutes and a proportional controller, the ultimate gain was 6.17 and the ultimate frequency was 1.18 radians per minute. 

Determine the ultimate gain and ultimate frequency of the process from either the transfer function or experimentally. For our three-CSTR example, these values are K, = 64 and to —. ... 

The Bode plot of is given in Fig. 13.20 for D = 0.5. The ultimate gain is 3.9 (11.6 dB), and the ultimate frequency is 3.7 radians per minute. The ZN controller settings for P and PI controllers and the corresponding phase and gain margins and log moduli are shown in Table 13.2 for several values of deadtime D. Also shown are the values for a proportional controller that give +2-dB maximum closedloop log modulus. 

Astrbm and Hagglund (Proceedings of the 1983 IF AC Conference, San Francisco) suggested an autotune procedure that is a very attractive technique for determining the ultimate frequency and ultimate gain. We call this method ATV (autotune variation. The acronym also stands for all-terrain vehicle which makes it easy to remember and is not completely inappropriate since ATV does provide a useful tool for the rough and rocky road of process identification. 

The period of the limit cycle is the ultimate period (PJ for the transfer function relating the controlled variable x and the manipulated variable m. So the ultimate frequency is... 

Once the test has been performed and the ultimate gain and ultimate frequency have been determined, we may simply use it to calculate Ziegler-Nichols settings. Alternatively, it is possible to use this information, along with other easily determined data, to calculate approximate transfer functions. The idea is to pick some simple forms of transfer functions (gains, deadtime, first- or second-order lags) and find the parameter values that fit the ATV results. 

Both of these transfer functions pass through exactly the same zero frequency and ultimate frequency points. So we could use either... 

The important feature of the ATV method is that it gives transfer function models that fit the frequency-response data very well near the important frequencies of zero (steadystate gains) and the ultimate frequency (which determines closedloop stability). 

Figures 6.14 and 6.15 give dynamic responses of the tray temperatures, reboiler heat input, and bottoms product impurity. The temperature loops were tuned using the TL (Tyreus-Luyben) tuning rules after the ultimate gain and ultimate frequency had been determined using a relay-feedback test. Two 0.5-minute first-order lags are used in the temperature loop. Temperature transmitter spans are 100T. The ultimate gain and period for the tray 6 temperature loop are 4.2 and 2.7 minutes, and for the tray 14 loop are 12.7 and 2.5 minutes. These results reflect the fact that the process gain is higher when tray 6 is...

The method (B. D. Tyreus and W. L. Luyben, Z EC Research 31 2625, 1992) uses the ultimate gain and the ultimate frequency The formulas for P[ and PID controllers are given in Table 3.1, and the TLC settings for the three-heated-tank process are given in Table 3.2. The performance of these settings is shown in Fig. 3.16. 

The value of the gain at the limit of stability is 64. It is called the ultimate ain K . The (O at this limit is the value of the imaginary part of. Y when the roots lie right on the imaginary axis. Since the real part of s is zero, the system will show a sustained oscillation with this frequency called the ultimate frequency, in radians per time. The period of the oscillation is exactly the same as the ultimate period that we defined in Chapter 2 in the Ziegler-Nichols tuning method... 

Ultimate period is in "hours ultimate frequency is in radians/hour) pult-2 3.1416/wult ... 

Figure 8.8 gives the root locus plot for a proportional controller. The three curves start at - 10 on the real axis (only one complex root is shown). Two of the loci go off at 60 angles and cross the imaginary axis at 17.3 (the ultimate frequency) vdien the gain is 6 (the ultimate gain). For a closedloop damping coefficient of 0.3 (the radial line), the closed-loop time constant is about 0.085 hours. 

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