Lec 9. Root Locus Analysis I From last lecture, the locations of the closed loop poles have important implication in –Stability –Transient behavior –Steady state error When adjusting some parameter, how will the closed loop pole locations be affected? Reading: TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

How does + closed-loop transfer function The closed loop transfer function of the unity-feedback system is with DC gain H(0)=1. Question: if we only know the closed loop transfer function H(s) with H(0)=1, can we derive the steady state error for unit ramp input directly from H(s)?

Truxal’s Formula Given a system with (closed loop) transfer function: satisfying H(0)=1. The steady state error for tracking unit ramp input is

Derivation of Truxal’s Formula When the input is the unit ramp function Hence the tracking error is The steady state error is (by L’Hopital’s rule) (steady state velocity error is equal to the negative slope of H(s) at s=0)

Derivation of Druxal’s Formula (cont.) Note that So Evaluating at s=0: From last slide,

Application to the Previous Example + closed-loop transfer function Two closed loop poles and no closed loop zeros.

Implication Suppose that the closed loop zeros are fixed. Then the steady state velocity error e ss decreases as the closed loop poles move away from the origin on LHP Similar formulas can be derived for steady state position error and acceleration error

Another Example Suppose that a third order type 1 system H(s) has closed loop poles at and only one closed loop zero. Where should the zero be if K v =10 is desired? +

Characteristic Equation Characteristic equation is the equation whose roots are closed loop poles For the feedback system below, the characteristic equation is + Example:

Characteristic Equation with Parameter The transfer functions may depend on a parameter K The characteristic equation and the closed loop poles depend on K. + Example 1: + Example 2:

Another Example The adjustable parameter K could appear elsewhere +

In General + For a feedback control system dependent on a parameter K, the characteristic equation is equivalent to for some transfer function Therefore, the closed loop poles are solutions of + In terms of closed loop poles (not closed loop transfer function!)

Root Locus Root locus of a feedback control system with parameter K is the plot of all closed loop poles as K varies from 0 to infinity. We will focus on the following special type of feedback system Have the same root locus

A Simple Example + Closed loop poles for different K: If Two open loop poles: 0, -4 No open loop zeros

Root Locus Changing K, the two closed loop poles are represented by two moving points (Breakaway point)

Another Example + Two open loop poles: j and -j One open loop zero: 0 Closed loop poles for different K:

Root Locus There are two branches that start from the open loop poles at K=0. As K increases to 1, one branch converges to the open loop zero and the other diverges. (Breakin point)

In General The root locus of a characteristic equation consists of n branches –Start from the n open loop poles p 1,…, p n at K=0 –m of them converge to the m open loop zeros z 1,…,z m –The other n-m diverge to 1 along certain asymptotic lines Points of interest –Point where root locus crosses the imaginary axis –Breakin/Breakaway points –Diverging asymptotic lines Can we find the root locus without solving the characteristic equation?

Characterizing Points on the Root Locus + How to determine if a point s on the complex plane belongs to the root locus?

Angle Condition A point s on the complex plane belongs to the root locus if and only if s is on the root locus if and only if Example:

Review on Complex Analysis Given a complex number z=a+bj Its modulus (or norm) is Its (phase) angle is Product of two complex numbers Quotient of two complex numbers

Previous Examples

Root Locus on the Real Axis A point on the real axis belongs to the root locus if and only there are odd number of open loop zeros/poles to its right Why? Use the angle condition Example: characteristic equation

Asymptotic Behaviors of Root Locus Root locus of the feedback system consists of n branches At K=0, the n branches start from the n open loop poles Why? +

Example How does the other n-m branches diverge to infinity? Example: Matlab code: The three branches diverge to infinity along three evenly distributed rays centered at -1, which is the center of the three open loop poles

Asymptotic Behaviors of Root Locus with angles Why? is approximated by As K approaches 1, m branches converge to the m open loop zeros, and the other n-m branches diverge to infinity along n-m rays (asymptotes) centered at:

Examples of Asymptotes of Root Locus