H AA Signals and systems Fall 2003 Lecture #21 25 November 2003 Feedback a) Root locus Tracking c) Disturbance rejection d) The Inverted Pendulum 2. Introduction to the z-transform
Signals and Systems Fall 2003 Lecture #21 25 November 2003 1. Feedback a) Root Locus b) Tracking c) Disturbance Rejection d) The Inverted Pendulum 2. Introduction to the Z-Transform
The concept of a root locus x()(+) C(s) H(s) G(s Q(s) C(SH(s 1+C(SG(sH(S) C(s), G(s)-Designed with one or more free parameters Question: How do the closed-loop poles move as we vary these parameters?-Root locus of 1+C(SG(sH(S)
The Concept of a Root Locus • C(s), G ( s) — Designed with one or more free parameters • Question: How do the closed-loop poles move as we vary these paramet ers? — Root locus of 1+ C(s) G ( s ) H( s )
The“ Classical” Root locus problem C(s)=K-a simple linear amplifier K H(S G(s) Q(s) KH(s 1+KH(s)G(s) Closed-loop H(s) y(t) poles are same Q(s)= H(s) 1+KH(s)G(s)
The “Classical” Root Locus Problem C ( s) = K — a simple linear amplifier Closed-loop poles are the same
A Simple example +2 G(s)=1 K K (a)Q( +k s+K+ (b)Q(s) 1+ +K+2 In either case. pole is at s=-2-K Sketch where gm pole moves 9x as k increases. K>0 2 Becomes more stable Becomes less stable
A Simple Example Becomes more stable Becomes less stable Sketch where pole moves as |K| increases... In either case, pole is at so = -2 - K
What Happens More generally For simplicity, suppose there is no pole-zero cancellation in G(s)H(s h( Q()=1+KG(s)H() Closed-loop poles are the solutions of 1+KGSH(S That is G(SH(s Difficult to solve explicitly for solutions given any specific value of K, unless G(s)H(s)is second-order or lower Much easier to plot the root locus the values of s that are solutions for some value of k because 1)It is easier to find the roots in the limiting cases for K=0.± 2) There are rules on how to connect between these limiting points
What Happens More Generally ? • For simplicity, suppose there is no pole-zero cancellation in G ( s ) H( s ) — Difficult to solve explicitly for solutions given any specific value of K, unless G ( s ) H( s) is second-order or lower. That is Closed-loop poles are the solutions of — Much easier to plot the root l ocus, the values of s that are solutions for some value of K, because: 1) It is easier to find the roots in the limiting cases for K = 0, ± ∞. 2) There are rules on how to connect between these limiting points