Feedback Control and Automation - Research Paper Example

Running Head: FEEDBACK Feedback Control and Automation Question Research at least two methods of design compensators, which are used to cause a practical unstable system to run as a stable system under the desired operating frequency range. Write a comparison of relative merits of each type. Answer: Although the prime objective of a control system is to have optimum performance yet without any adjustment it is rare indeed. Usually we have to compromise among many conflicting and damaging specifications and have to adjust the system parameters to achieve a suitable performance although it is not possible to obtain all desired optimum specifications. The control system adjustment or stabilization and alteration is carried out by introducing a compensating block within the control loop that will not allow the steady state error to exceed to a certain limit. This compensating block is called compensator or filter and physically it may be an electric network or mechanical equipment containing levers, springs, dashpots, etc. The compensator may be placed cascade with forward transfer function Gx(s) or in feed back path as shown in fig.1.1. The selection of location for inserting the compensator largely depends on the control system, the necessary physical modifications and results desired (D'Azzo, 1995, pp. 319-320). The available techniques to achieve improvement in the response of feedback control system are time domain, frequency domain, root locus and disturbance rejection technique. In next section we will examine cascade and feed back path compensators using frequency domain technique and a comparison is made relative to merits of each design. Cascade Compensator Let we consider a system as shown in fig. 1.2, then forward transfer function in terms of frequency domain with a compensator be, G(j) = Gc(j) Gx(j) or, Gc(j) = .(1) Now for lag compensators, the equation (1) in terms of frequency domain becomes (D'Azzo, 1995, pp. 333), Lag Compensator Gc(j) = .(2) Also for Lead Compensator Gc(j) = . .(3) Fig. 1.1 Block Diagrams of Control Systems with Feedback Compensators Fig. 1.2 Feedback System with Compensators The polar plots of equations (2) and (3) can be plotted as shown in fig. 1.3. From graph we observe that the cascade lag compensator results in a clockwise rotation of G(j) and can produce a large increase in gain (Km) with a small decrease in m where as cascade lead compensator results in the counter clockwise rotation of G(j) and produce a large increase in m with small increase in Km where both Km and m determine the transit response of a feedback system. Fig. 1.3 Polar graphs of Cascade Compensator Also from equation (1), Gc(j) = Gc(j) Solving for ', we have (Dorf, 1989, p. 266), (D'Azzo, 1995, pp. 370-371). = tan-1T - tan-1T = tan-1 . (3) or T + T + = 0 . (4) Where for lag compensator > 1 and < 1 for lead compensator. Differentiating equation (3) w.r.t. and setting it equal to '0, we have, m = . (5) Substituting the value in (3) we have the max. phase shift, () max = sin-1 . (6) From equation (6) we can calculate typical shift values as given in table (a) which are very useful in design application of lead compensator as we will see in design examples. 0 0.1 0.2 0.3 0.4 () max 90 54.9 41.8 32.6 19.5 Table. a Design Example of Phase - lag Compensator Network Let we consider an uncompensated transfer function give as, GH(j) = GH(j) = (where Kv = K/20) Let we compensate this network to Kv = 20 with phase margin of 45. If we observe the uncompensated Bode diagram as given in fig. 1.4 we see it has as Fig. 1.4 Bode Diagram shown in solid lines then we see that uncompensated system has a phase margin of 20. Allowing 5 for an increase in phase -lag compensator, we relocate the frequency, (where = -130) which is to be our crossover frequency c. From Bode diagram c = 1.5 which allow a small margin of safety. The attenuation necessary to cause c to be new crossover frequency is equal to 2odb, accounting for a 2-db difference between the actual and asymptotic curves. Therefore, 20db = 20log or = 10. Hence zero is one decade below the crossover and z = c/10 =0.15 and pole p = z/10 = 0.015. Hence the compensation system is then (Dorf, 1989, pp459-460). GH(j) = The frequency response of system is shown in fig. 1.5. Fig. 1.5 Feedback Compensator Let we consider the block diagram of a control system with feed back compensator H(j) as given in fig 1.6. The forward transfer function of this system be, G1(j) = . (7) Compare to cascade compensator, in feedback compensator we have to consider the gain and frequency characteristics of minor feedback compensator. Now if we consider the functions, Gc(j)H(j)> 1, then equation (7) will reduced to G1(j) Gx(j) for Gc(j)H(j)>1. . (9) Also Gc(j)H(j) 1 is undefined. In above approximation procedure we neglect the undefined condition (D'Azzo, 1995, pp. 391) and using approximation (8) and (9) satisfactory design results are obtained. Comparison of Merits of Cascade and Feedback Compensator The design producers for cascade compensator Gc(j) is more direct then those for Hc(j). The design of feedback compensator is more laborious then cascade. The physical form of control system determines the ease of implementing a practical cascade or feedback compensator. The economics, size, weight and cost of components and amplifier are important. In the forward path the signal goes from a low to a high energy level where as the reverse is true in feedback loop. Hence generally an amplifier is not implemented in feedback path while cascade path generally requires an amplifier for gain and/or isolation. The environmental conditions also affect the accuracy and stability of controlled quantity. The problem of noise within a control may determine the choice factor. This is accentuated in the situations in which a large amplifier gain is required with a forward compensator than by the use of feedbacks. Compared to cascade a faster time response is achieved by implementing a feedback compensator. Except these the designer's experience and preference greatly influence the choice between a cascade and feedback compensator for achieving the desired performance (D'Azzo, 1995, pp.320-321). Question 2) The diagram below illustrates a multi-loop control system. By reducing the diagram to its Canonical form, establish the overall transfer function for the system. Explain the process this involves. Solution: Let we consider the given block diagram (fig. 1), now to reduce it in required Canonical form let we precede with following steps. Fig. 1 Block diagram of multi-loop control system. Step 1: As any finite block in series can be algebraically combined by multiplication (DiStefano, 1998, p.113), therefore combining all cascade blocks we have following block diagram as in figure 2. The original block diagram will reduce to block Fig. 2 (Step 1 reduction) diagram as in fig. 3. Step 2: Now from fig. 3, G2G3 and H1 parameter blocks are parallel, hence reducing parallel blocks into control ratio we have reduced block diagram as in fig. 4. (Dorf, 1989, p. 60). The positive sign refers to -ve feedback. Step 3: From fig. 5 as we again have cascade situation, hence the required reduced form is shown in fig. 6. Fig. 3 Fig. 4 (Step 2 reduction) Fig. 5 Fig. 6 (Step 3 reduction) Step 4: Repeating step 2 procedure for H2 and G1 () blocks with +ve feedback we have fig. 7 reduced form. Fig. 7 (Step 4 reduction) After simplification of the expression in block diagram of fig. 7 the control ratio turn to be . Step 5: Finally we have unity + ve feedback, hence considering a parallel block with unity value we have over all transfer function, Fig. 8 Overall transfer function block diagram. Block diagram is actually shorthand graphical representation of a physical system. It illustrates the functional relationships among its components. The block diagram letter feature permits evaluation of the contributions of the individual elements to the overall performance of the system. References D'Azzo, J. J. & Constantine, H. H. (1995). Linear Control System Analysis and Design Conventional and Modern. Singapore: McGraw-Hill Book Co. DiStefano, J. J., Allen, R. S. & Ivan, J. W. (1998). Schaum's Outline of Theory and Problems of Feedback and Control Systems. Singapore: McGraw-Hill Book Co. Dorf, R. C. (1989). Modern Control Systems. 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