Ultimate gain

Starting with a small value, Kp can be increased until the response is unstable and oscillatory. This value is called the ultimate gain Kpp. 

A preview We can derive the ultimate gain and ultimate period (or frequency) with stability analyses. In Chapter 7, we use the substitution s = jco in the closed-loop characteristic equation. In Chapter 8, we make use of what is called the Nyquist stability criterion and Bode plots. 

B. Tuning relations based on closed-loop testing and the Ziegler-Nichols ultimate-gain (cycle) method with given ultimate proportional gain Kcu and ultimate period Tu. 

Ziegler-Nichols Continuous Cycling (empirical tuning with closed loop test) Increase proportional gain of only a proportional controller until system sustains oscillation. Measure ultimate gain and ultimate period. Apply empirical design relations. 

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. 

If we have chosen the other possibility of u = 0, meaning that the closed-loop poles are on the real axis, the ultimate gain is Kc u = -6, which is consistent with the other limit obtained with the Routh criterion. 

Example 7.3A Repeat Example 7.3 to find the condition for the ultimate gain. 

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. 

Example 7.2B Do the root locus plot and find the ultimate gain of Example 7.2 (p. 7-5). The closed-loop equation from that example is ... 

After entering the riocf ind () command, MATLAB will prompt us to click a point on the root locus plot. In this problem, we select the intersection between the root locus and the imaginary axis for the ultimate gain. 

We should find that for values of Xj > 0.5, the system stays stable. For = 0.5, the system may become unstable, but only at infinitely large Kc. The system may become unstable for x < 0.5 if Kc is too large. Finally, for the choice of x = 0.1, we should find with the MATLAB function riocf ind that the ultimate gain is roughly 0.25, the same answer from Example 7.3. How close you get depends on how accurate you can click the axis crossover point. 

The very first step is to find the ultimate gain. With the given third order process transfer function, we use the following MATLAB commands,... 

If we want to increase the margin, we either have to reduce the value ofKc or increase One possibility is to keep = 1.58 min and repeat the Bode plot calculation to find a new Kc which may provide a gain margin of, say, 2 (6 dB), as in the case of using only the proportional controller. To do so, we first need to find the new ultimate gain using the PI controller ... 

First, we need MATLAB to find the ultimate gain ... 

We may want to find the ultimate gain when the loci cross the imaginary axis. Again there are many ways to do it. The easiest method is to estimate with the MATLAB function rlocf ind (), which we will introduce next. 

Ultimate gain and ultimate period (Pu = 2tt/(0u) that can be used in the Ziegler-Nichols continuous cycling relations. Result on ultimate gain is consistent with the Routh array analysis. Limited to relatively simple systems. 

Operate with proportional control only and a step change in flowrate (set rj very high, controller 1 stepflow=l steptemp=0). Increase KP until oscillations in the response occur at KP0. Use this oscillation frequency, f0, to set the controller according to the Ultimate Gain Method (KP=0.45 KP0, rj = 1/(1.2 f0)), where f0 is the frequency of the oscillations at KP = KP0 (see Sec. 2.3.3). How high is EINT2 ... 

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