# barcode reader integration with asp.net [(s + l)(s + 7)+ Kl(s + 5)][(s + l)(s + 7) + 2K2(s + 3)] - 16K1K2 = 0 in Software Generator Code 128A in Software [(s + l)(s + 7)+ Kl(s + 5)][(s + l)(s + 7) + 2K2(s + 3)] - 16K1K2 = 0

[(s + l)(s + 7)+ Kl(s + 5)][(s + l)(s + 7) + 2K2(s + 3)] - 16K1K2 = 0
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For given values of K1 and K2, this expression can be expanded into a fourth order polynomial equation of the form s4 + crs3 + ,Gs2 + ys + A = 0 where (Y, p , y, and A will include the gains K 1 and K2. (30.34)
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466 STATE-SPACE METHODS The Routh test can be applied to Eq. (30.34) to determine whether or not the system is stable. From this simple example, the reader can appreciate the algebraic tedium that may be needed to determine the stability of a multivariable system. One way to express the stability of this system is to plot the stability boundaries on a graph of K1 versus K2. The region within the boundaries gives the combinations of values of K1 and K-J for which the system is stable. Since the details of stability boundaries is beyond the scope of this chapter, the reader may consult Seborg, Edgar, and Mellichamp (1989) for examples of stability boundaries for multivariable systems.
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Most of the systems encountered are multiple-input multiple-output (MIMO) systems. Such systems have several inputs and several outputs that are often interacting, meaning that a disturbance at any input causes a response in some or all of the outputs. This interaction in an MIMO system makes control and stability analysis of the system very complicated compared to a single-input single-output (SISO) system. A convenient way to describe an MIMO system is by means of a block diagram in which each block contains a matrix of transfer functions that relates an input vector to an output vector. It is often desirable to have a control system decoupled so that certain outputs can be controlled independently of other outputs. A systematic procedure was described for decoupling a control system by including cross-controllers along with the principal controllers. This approach to decoupling requires an accurate model of the system; the number of controllers (principal controllers and crosscontrollers) increases rapidly with the number of inputs and outputs. A system represented by two inputs and two outputs requires as many as four controllers; a system of three inputs and three outputs requires as many as nine controllers, and so on. The characteristic equation for a multivariable control system, from which one can determine stability by examining its roots, can be of high order for a relatively simple system. Expressing stability boundaries in terms of controller parameters becomes complex because of the large number of controller parameters that can be adjusted.
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30.1. For the liquid-level system shown in Fig. P30.1 determine the cross-controller transfer functions that will decouple the system. Fill in each block of the diagram shown in Fig. 30.5 with a tmnsfer function obtained from an analysis of the control system. The transfer function for each feedback measuring element is unity. The following data apply:
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A1 = 1, A2 = 0.5, Resl = 0.5, Res2 = 213, G,-11 = Kl, Gc22 = K2
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The resistance on the outlet of a tank has been denoted by Res to avoid confusion with the symbol for set point (R).
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Res 2
FIGURE P30-1
30.2. (a) For the interacting liquid-level system shown in Fig. P30.2, draw very neatly a block diagram that corresponds to Fig. 30.4. Each block should contain a transfer function obtained from an analysis of the liquid-level system. There are no cross-controllers in this system. The transfer function for each feedback element is unity. The following data apply: Al = l,A2 = 1/2, Rest = 1/2, Res;! = 2, Res.3 = 1 (b) Obtain the characteristic equation of this system in the form s + (yp-1 + pp-2 + . . . = 0 Obtain expressions for CY, p, etc. in terms of K2, (Kl = 1) (c) How would you determine stability limits for this interacting control system