Controllers transfer functions

For the rest of the control loop, Gc is obviously the controller transfer function. The measuring device (or transducer) function is Gm. While it is not shown in the block diagram, the steady state gain of Gm is Km. The key is that the summing point can only compare quantities with the same units. Hence we need to introduce Km on the reference signal, which should have the same units as C. The use of Km, in a way, performs unit conversion between what we dial in and what the controller actually uses in comparative tests. 2... 

The gain and phase margins are used in the next section for controller design. Before that, let s plot different controller transfer functions and infer their properties in frequency response analysis. Generally speaking, any function that introduces additional phase lag or magnitude tends to be destabilizing, and the effect is frequency dependent. 

X Example 8.12. Derive the magnitude and phase lag of the controller transfer function... 

We may look at the controller transfer function in the time constant form ... 

In this chapter we will demonstrate the signiScant computational and nota-tional advantages of LaplaTce transforms. The techniques involve finding the transfer function of the openloop process, specifying the desired performance of the closedloop system (process plus controller) and finding the feedback controller transfer function that is.required to do the job. 

Example 10.1. The closedloop transfer functions for the two-heated-lank process can be calculated from the openloop process transfer functions and the feedback controller transfer function. We will choose a proportional controller, so = K,. Note that the dimensions of the gain of the controller are mA/mA, that is, the gain is dimensionless. The controller looks at a milliampere signal (PM) and puts out a milliampere signal (CO). 

This equation shows that closedloop dynamics depend on the process openloop transfer functions (G, Gv, and Gj) and on the feedback controller transfer function (fl). Equation (10.10) applies for simple single-input-single-output systems. We will derive closedloop characteristic equations for other systems in later chapters. 

Setting equal to zero and solving for the relationship between and L, give the feedforward controller transfer function... 

Now, knowing the process model and having specified the desired closedloop servo transfer funclion, we can solve for the feedback controller transfer function. We define the closedloop servo transfer function as. ... 

Equation (11.64) contains only one unknown (i.e., the feedback controller transfer function Solving for 0, in terms of the known values of and Su, gives... 

So if we cannot attain perfect control, what do we do From the IMC perspective we simply break up the controller transfer function C( ) into two parts. The first part is the inverse of. The second part, which Morari calls a filter, is chosen to make the total physically leahzable. As we will show below, this second part turns out to be the closedloop servo transfer function that we defined as S(,j in Eq. (11.64). 

Eletive the feedforward-controller transfer function that will keep the process output constant with load changes L, y 11.2. Repeat Prob. 11.1 with... 

Find the feedforward-controller transfer functions that will keep Xj> constant, by manipulating R, despite changes in z and F. For what values of parameters are these feedforward controllers physically realizable ... 

Substituting these quantities and Eq. (20.63) (with k = 1) into Eq. (20.15) gives the controller transfer function. 

EXftMPLE 8.4. Let us start with the simplest of all processes, a first-order lag. We choose a proportional controller. The system and controller transfer functions are... 

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