# LECTURE 27: FEEDBACK CONTROL - PowerPoint PPT Presentation

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LECTURE 27: FEEDBACK CONTROL. Objectives: Typical Feedback System Feedback Example Feedback as Compensation Proportional Feedback Applications

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LECTURE 27: FEEDBACK CONTROL

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LECTURE 27: FEEDBACK CONTROL

• Objectives:Typical Feedback SystemFeedback ExampleFeedback as CompensationProportional FeedbackApplications

• Resources:MIT 6.003: Lecture 20MIT 6.003: Lecture 21Wiki: Control SystemsBrit: Feedback ControlJC: Crash CourseWiki: Root LocusWiki: Inverted Pendulum CJC: Inverted Pendulum

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System Function For A Closed-Loop System

• The transfer function of thissystem can be derived usingprinciples we learned inChapter 6:

• Black’s Formula: Closed-loop transfer function is given by:

• Forward Gain: total gain of the forward path from the inputto the output, where the gain of a summer is 1.

• Loop Gain: total gain along the closed loop shared by all systems.

Loop

The Use Of Feedback As Compensation

• Assume the open loopgain is very large(e.g., op amp):

 Independent of P(s)

• 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.

Stabilization of an Unstable System

• If P(s) is unstable, can westabilize the system byinserting controllers?

• Design C(s) and G(s) so thatthe poles of Q(s) are in the LHP:

• Example: Proportional Feedback (C(s) = K)

• The overall system gain is:

• The transfer function is stable for K > 2.

• Hence, we can adjust K until the system is stable.

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

• Using damping and feedback, we have stabilized a second-order unstable system.

The Concept of a Root Locus

• Recall our simple control systemwith transfer function:

• The controllers C(s) and G(s) can bedesigned 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:

Closed-loop poles are the same.

Example: First-Order System

• Consider a simple first-order system:

• The pole is at s0 = -(2+K). Vary Kfrom 0 to :

• Observation: improper adjustment of the gain can cause the overall system to become unstable.

Becomes less stable

Becomes more stable

Example: Second-Order System With Proportional Control

• Using Black’s Formula:

• How does the step responsevary as a function of the gain, K?

• Note that as K increases, thesystem goes from too little gainto too much gain.

How Do The Poles Move?

Desired Response

• Can we generalize this analysis to systems of arbitrary complexity?

• Fortunately, MATLAB has support for generation of the root locus:

• num = [1];

• den = [1 101 101]; (assuming K = 1)

• P = tf(num, den);

• rlocus(P);

Example

Feedback System – Implementation

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.

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) =   s0are the poles of G(s)H(s).

• At K= , G(s0)H(s0) = 0 s0are 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.

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).

Feedback System – Use Proportional Derivative Control

• Equations describing the physics:

• The poles of the system are inherentlyunstable.

• Feedback control can be used to stabilize both the angle and position.

• Other approaches involve oscillatingthe support up and down.