0. LECTURE 27: FEEDBACK CONTROL
Objectives: Typical Feedback System Feedback Example Feedback as Compensation Proportional Feedback Applications
Resources: MIT 6.003: Lecture 20 MIT 6.003: Lecture 21 Wiki: Control Systems Brit: Feedback Control JC: Crash Course Wiki: Root Locus Wiki: Inverted Pendulum CJC: Inverted Pendulum
MS Equation 3.0 was used with settings of: 18, 12, 8, 18, 12.
Audio:
URL:

1. System Function For A Closed-Loop System
The transfer function of this system can be derived using principles we learned in Chapter 6:
Black’s Formula: Closed-loop transfer function is given by:
Forward Gain: total gain of the forward path from the input to the output, where the gain of a summer is 1.
Loop Gain: total gain along the closed loop shared by all systems.
Loop
]i=k,

2. The Use Of Feedback As Compensation
Assume the open loop gain is very large (e.g., op amp):
 Independent of P(s)
]i=k,
The closed-loop gain depends only on the passive components (R1 and R2) and is independent of the open-loop gain of the op amp.

3. Stabilization of an Unstable System
If P(s) is unstable, can we stabilize the system by inserting controllers?
Design C(s) and G(s) so that the poles of Q(s) are in the LHP:
Example: Proportional Feedback (C(s) = K)
]i=k,
The overall system gain is:
The transfer function is stable for K > 2.
Hence, we can adjust K until the system is stable.

4. Second-Order Unstable System
Try proportional feedback:
One of the poles is at
Unstable for all values of K.
Try damping, a term proportional to :
This system is stable as long as:
K2 > 0: sufficient damping force
K1 > 4: sufficient gain
]i=k,
Using damping and feedback, we have stabilized a second-order unstable system.

5. The Concept of a Root Locus
Recall our simple control system with transfer function:
The controllers C(s) and G(s) can be designed to stabilize the system, but that could involve a multidimensional optimization. Instead, we would like a simpler, more intuitive approach to understand the behavior of this system.
Recall the stability of the system depends on the poles of 1 + C(s)G(s)P(s).
A root locus, in its most general form, is simply a plot of how the poles of our transfer function vary as the parameters of C(s) and G(s) are varied.
The classic root locus problem involves a simplified system:
]i=k,
Closed-loop poles are the same.

6. Example: First-Order System
Consider a simple first-order system:
The pole is at s0 = -(2+K). Vary K from 0 to  :
Observation: improper adjustment of the gain can cause the overall system to become unstable.
]i=k,
Becomes more stable
Becomes less stable

7. Example: Second-Order System With Proportional Control
Using Black’s Formula:
How does the step response vary as a function of the gain, K?
Note that as K increases, the system goes from too little gain to too much gain.
]i=k,

8. How Do The Poles Move? Desired Response
]i=k,
Can we generalize this analysis to systems of arbitrary complexity?
Fortunately, MATLAB has support for generation of the root locus:
num = [1];
den = [ ]; (assuming K = 1)
P = tf(num, den);
rlocus(P);

9. Example
]i=k,

10. Feedback System – Implementation
]i=k,

11. Summary Introduced the concept of system control using feedback.
Demonstrated how we can stabilize first-order systems using simple proportional feedback, and second-order systems using damping (derivative proportional feedback).
Why did we not simply cancel the poles?
In real systems we never know the exact locations of the poles. Slight errors in predicting these values can be fatal.
Disturbances between the two systems can cause instability.
There are many ways we can use feedback to control systems including feedback that adapts over time to changes in the system or environment.
Discussed an application of feedback control involving stabilization of an inverted pendulum.

12. More General Case Assume no pole/zero cancellation in G(s)H(s):
Closed-loop poles are the roots of:
It is much easier to plot the root locus for high-order polynomials because we can usually determine critical points of the plot from limiting cases (e.g., K = 0, ), and then connect the critical points using some simple rules.
The root locus is defined as traces of s for unity gain:
Some general rules:
At K = 0, G(s0)H(s0) =   s0 are the poles of G(s)H(s).
At K = , G(s0)H(s0) = 0  s0 are the zeroes of G(s)H(s).
Rule #1: start at a pole at K = 0 and end at a zero at K = .
Rule #2: (K  0) number of zeroes and poles to the right of the locus point must be odd.
]i=k,

13. Inverted Pendulum Pendulum which has its mass above its pivot point.
It is often implemented with the pivot point mounted on a cart that can move horizontally.
A normal pendulum is stable when hanging downwards, an inverted pendulum is inherently unstable.
Must be actively balanced in order to remain upright, either by applying a torque at the pivot point or by moving the pivot point horizontally (Wiki).
]i=k,

14. Feedback System – Use Proportional Derivative Control
Equations describing the physics:
The poles of the system are inherently unstable.
Feedback control can be used to stabilize both the angle and position.
Other approaches involve oscillating the support up and down.
]i=k,