1. Lec 9. Root Locus Analysis I
Locations of closed-loop-poles matter a lot
Stability
Transient behavior
Steady state error
When adjusting some parameter, how will the closed-loop pole locations be affected?
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2. Steady State Error from Closed-Loop TF
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)?
closed-loop transfer function +

3. 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
Suppose closed-loop zeros are fixed. Then the steady state velocity error ess 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

4. Application to the Previous Example
Two closed-loop poles
and no closed-loop zeros.
By the Druxal’s formula
closed-loop transfer function +

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

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

7. Another Example
The adjustable parameter K could appear elsewhere +

8. 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!)

9. 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

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

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

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

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

14. In General The root locus of a characteristic equation
consists of n branches
Start from the n open-loop poles p1,…, pn at K=0
m of them converge to the m open-loop zeros z1,…,zm
The other n-m diverge to ∞ 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?

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

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

17. Review on Complex Analysis
Given a complex number z=a+bj
Its modulus (or norm) is
Its (phase) angle is
∠𝑧=𝑎𝑡𝑎𝑛2(𝑏,𝑎)
Product of two complex numbers
Quotient of two complex numbers

18. Previous Examples

19. 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

20. 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? +

21. 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

22. Asymptotic Behaviors of Root Locus
As K approaches ∞, 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:
with angles
Why?
is approximated by

23. Examples of Asymptotes of Root Locus