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Primary controller
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FIG 6.1. Cascade control resolves the process into two parts, each within a closed loop.
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FIG 6.2. The primary controller sees a closed loop as a part of the process.
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It must be recognized, however, that cascade control cannot be employed unless a suitable intermediate variable can be measured. Many processes are so arranged that they cannot be readily broken apart in this way.
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Properties of the Inner Loop
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The secondary or inner loop confronts the primary controller as a new type of dynamic element. The inner loop can be represented as a single block, the diagram of Fig. 6.1 being resolved into the simpler configuration shown in Fig. 6.2. Heretofore the dynamic properties of a closed loop were of little concern. The controller was simply adjusted for a damping which satisfied certain transient response specifications. Moreover there was only one period of oscillation to be considered. But each loop has its own natural period and, as may be expected, the period of the primary loop is to a great extent determined by that of the secondary. Consequently the gain and phase of the secondary loop, whose natural period will be designated T,~, must be known for any value of the primary period 701, since the latter is dependent on the former. The dynamic properties of the open secondary loop can be converted into its closed-loop characteristics by solving for the response of CP with respect to 1 2. Refer to the block diagram in Fig. 6.3. Let g, and g, be vectors representing gain and phase of the process and the controller, respectively. Then
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c2 = mg, - c2)gcgp g&J = rzgcg, c2 = (r2 c2u +
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FIG 6.3. The input to the secondary loop is r2, its output is 0.
156 1Multiple-loop Systems
The vector gain of the closed secondary loop will be designated g02: it is the ratio of output c2 to input r2. The vector consists of a scalar gain Go2 and a phase angle c#+,~.
c2 go2 = T, = 1 EC&P + g,g,
(6.1)
The product g,g, is the open-loop vector. If the inner loop has been adjusted for f/4-amplitude damping, its open-loop gain will be 0.5 at the period of oscillation. But the phase lag at the period of oscillation is 180 , which makes the gain vector 0.5, L-180 , or -0.5. The closedloop vector go2 at the natural period is then g02 = -0.5 = -1.0 1.0 - 0.5 or
Go2 = 1.0
= -180
If the open-loop gain were 1.0 at the natural period, undamped oscillation would result: -1.0 go2 = 1.0 _ 1.0 = - Oc This indicates that an infinitesimal change in 1 2 would change c2 enough so that it would never return to equilibrium, and indeed this is the case. To find the gain and phase characteristics of a loop away from its natural period, the vector equation for the inner loop must be solved for various values of input period 701. This entails first finding the gain and phase of the open loop, g,g,. This vector must then be added to the vector 1.0, LO to form the denominator of the equation. Then the closed-loop gain is the quotient of the magnitude of the two vectors, and its phase is the difference between their phase angles.
example 6.1
A typical example is that of a closed loop cont,aining dead time, an integrating capacity, and a proportional controller adjusted for >i-amplitude damping. The natural period is known to he 7,,2 = 4rd2. The open-loop gain is 0.5 at 701 = 70C and varies directly as TV,. The open-loop I)hase is -90 for the integrating element, with an additional -360r&r01 or -907,,JTo1 for the dead time. Then g,g, = 0.5 2, L -90 - 90 rs J 0 701 From this information, closed-loop gain and phase arc plotted in Fig. 6.4. The primary loop will contain certain eIements of the process in addition t o the secondary loop. These elements can be expected to contribute phase lag of 90 or more. Therefore the area of greatest interest
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