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

# Under these conditions t he only way t o describe or

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Unformatted text preview: when t he s ystem is u ncontrollable a nd/or u nobservable. Figures 1.35a a nd 1.35b show a s tructural r epresentation of simple uncontrollable a nd unobservable systems respectively. In Fig. 1.35a we n ote t hat p art of t he s ystem (subsystem 82) inside t he box cannot be controlled by t he i nput f (t). I n Fig. 1.35b some of t he s ystem o utputs ( those in subsystem 82) c annot be observed from t he o utput terminals. I f we t ry t o describe either of these systems by applying an external i nput f (t) a nd t hen measuring t he o utput y(t), t he m easurement will not characterize t he complete system b ut only t he p art of t he s ystem (here 8 Il t hat is b oth controllable a nd observable (linked t o b oth t he i nput a nd o utput). Such systems are undesirable in practice a nd should be avoided in any system design. T he s ystem in Fig. 1.35a can be shown t o be neither controllable nor observable. I t can be represented structurally as a combination of t he s ystems in Figs. 1.35a a nd 1.35b. . ................................... _-_ . ..... _. ........ - .... -.... ; I tt) ~ S, ~ • .• y (t) . ~ ~ / (t) I • s, 1 i tt) = 5 f(t) (a) Also, because y (t) = 2 x ( i/2) = i , (b) F ig. 1.35 Structures of uncontrollable and unobservable systems. y(t) = 1 y( ( 1) + (1.63) 30 .&quot; 20 f 5 + i &quot;2 = 1 f it) = 7 (t) =K rf(t) which can be expressed as i 30 (1.62) where K r is a constant of the motor. This torque drives a mechanical load whose freebody diagram is illustrated in Fig. 1.33b. The viscous damping (with coefficient B ), which is proportional t o the angular velocity 0, dissipates a torque BiJ(t). I f J is the moment of inertia of the load (including the rotor of the motor), then the net torque 7 (t) - BiJ(t) available must equal to JU(t); 93 1.9 Summary 1 Introduction to Signals a nd Systems 5f (t) (1.66) y ( t) 94 1 .9 1 I ntroduction t o Signals a nd Systems Summary A signal is a s et of information or d ata. A s ystem processes input signals t o modify them or extract additional information from them to produce o utput signals (response). A system may be made up of physical components (hardware realization) o r m ay be a n a lgorithm t hat computes a n o utput signal from a n i nput signal (software realization). A convenient measure of the size of a signal is its energy if it is finite. I f t he signal energy is infinite, t he a ppropriate measure is its power, if it exists. T he signal power is t he t ime average of its energy (averaged over t he entire time interval from - 00 t o ( 0). F or p eriodic signals t he t ime averaging need be performed only over one period in view of t he periodic repetition of the signal. Signal power is also equal t o t he m ean s quared value of t he signal (averaged over t he entire time interval from t = - 00 t o (0). Signals can be classified in several ways as follows: 1. A c ontinuous-time signal is specified for a continuum of values of t he independent variable (such as time t). A discrete-time signal is specified only a t a finite or a countable s et of time ins...
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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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