Signal Processing and Linear Systems-B.P.Lathi copy

T t his behavior is exactly opposite of what was m

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: find t he s tep a nd r amp r esponse for a given value o f K , we first c reate a n m-fiIe c64a.m as r. a nd e o (s) -- o + 4)2 4 +4 - + t + ~e-8t cos (8t + 36.87 tTypical percentage values used are 2% to 5% for t •. 433 T his response, sketched in Fig. 6.36d, shows t hat t here is a s teady-state error e r = 0.1 radian. In many cases such a small steady-state error may be tolerable. If~ however, a zero steady-state error to a r amp i nput is reauired, t his system in its present form is unsatisfactory. We m ust add some form of compensator t o t he system. 16 16 e o(s) = s (s2 6.7 Application t o Feedback a nd Control S tate E quations 0 )] u(t) ( c64a.m) To plot t he s tep r esponse (as in Fig. 6.36c), we c reate a nother file c64b.m as: K =7;c64a; h old o n, K =16;c64aj K =80jc64a T he u nit r amp r esponse o f t his s ystem is t he s ame a s t he u nit s tep r esponse o f a s ystem with transfer function T (s)/s. Hence, we c an use t he file c64a t o find t he r amp r esponse. To plot t he r amp r esponse, for K = 80, we c reate c64c.m file as follows: K =80j c 64aj N umTFr=NumTFj D enTFr=conv([O 1 O J,DenTF)j p rintsys(NumTFr,DenTFr); s tep(NumTFr,DenTFr) 0 Design Specifications T he above discussion has given t he r eader some idea o f t he various specifications a control system might require. In general we m ay be required to design a control system t o m eet some o r all o f t he following specifications: 1 . T ransient R esponse (a) Specified overshoot t o s tep input. (b) Specified rise time t r a nd lor delay time t d. (c) Specified settling time t s. 2. S teady-State E rror Specified ~te.ady-state e rror t o c ertain expected inputs such as step, ramp, o r p~rabohc Inputs. ~n control systems, t he t ransient response is generally specIfied for t he s tep Input. T he reason is t hat t he s tep i nput represents a sudden j ump discontinuity. Hence if a system has a n a cceptable transient response for s tep i nput, i t is likely t o have acceptable transient response for most o.f th~ p ractical inputs. Steady-state errors, however, must be specified for tYPIcal I nputs o f t he system. For t he given system one must determine what kind of i nput ( step, ramp, etc) are likely to occur, a nd t hen specify acceptable steady-state requirements for these inputs. 434 6 C ontinuous-Time S ystem Analysis Using t he Laplace Transform 3. S ensitivity T he s ystem s hould satisfy a specified sensitivity specifications t o some system p arameter v ariations, or t o c ertain d isturbances. Sensitivity analysis will n ot b e considered here. 6.7 Application t o Feedback a nd Control S tate E quations 435 . T he n ature of this response for t he u nderdamped case ( ( < 1) is i llustrated in FIg. 6.38. T he response decays exponentially as e -(wnt. Hence, t he t ime constant o~ response is 1 /(w n . I t takes four time constants for an exponential t o decay t o slIghtly less t han 2% of its initial value. Hence, t he s ettling Lime t . is given by 6.7-2 Analysis o f a Second-Order System T he t ransient...
View Full Document

Ask a homework question - tutors are online