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Module 5 Slides
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School of Electrical and Computer Engineering
Control of Mobile Robots
Dr. Magnus Egerstedt Professor School of Electrical and Computer Engineering
Module 5 Hybrid Systems
How make mobile robots move in effective, safe, predictable, and collaborative ways using modern control theory?
So far, the models stay the same over time
We have a designed one-size-fits-all controllers
Lecture 5.1 Switches Everywhere
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Switches by Necessity
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Switches by Design
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Switches by Design
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Models? Stability and Performance? Compositionality? Traps?
Issues
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
We need to be able to describe systems that contain both the continuous dynamics and the discrete switch logic
Hybrid Automata = Finite state machines (discrete logic) on steroids (continuous dynamics)
Lecture 5.2 Hybrid Automata
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Let, as before, the (continuous) state of the system be x As we will be switching between different modes of operation,
lets add an additional discrete state q Dynamics:
The transitions between different discrete modes can be encoded in a state machine:
Modes, Transitions, Guards, and Resets
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
The conditions under which a transition occurs are called guard conditions, i.e., a transition occurs from q to q if
As a final component, we would like to allow for abrupt changes in the continuous state as the transitions occur, which we will call resets:
Modes, Transitions, Guards, and Resets
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Putting all of this together yields a very rich model known as a hybrid automata (HA) model:
The Hybrid Automata Model
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
HA Example 1 - Thermostat
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
HA Example 2 Gear Shift
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
HA Example 3 Behaviors
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
What can possibly go wrong when you start switching between different controllers?
Lecture 5.3 A Counter Example
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Two modes:
Both modes are asymptotically stable!
A Simple 2-Mode System
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Mode 1
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Mode 2
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
HA 1
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
HA 2
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
By combining stable modes, the resulting hybrid system may be unstable!
By combining unstable modes, the resulting hybrid system may be stable!
Design stable modes but be aware that this is a risk one may face!
Punchlines
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Stable subsystems do not guarantee a stable hybrid system
Lecture 5.4 Danger, Beware!
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
We saw last time that it was possible to destabilize stable subsystems by an unfortunate series of switches
Ignoring resets, we can write the hybrid system as a switched system:
The switch signal dictates which discrete mode the system is in
Switched Systems
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Given a switched system universal, asymptotic stability:
existential, asymptotic stability:
If the switch signal is generated by an underlying hybrid automaton: hybrid, asymptotic stability:
Different Kinds of Stability
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Some Results
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Practically Speaking
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Lets model a ball bouncing on a surface:
Equations of motion in-between bounces:
Bounces:
Lecture 5.5 The Bouncing Ball
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
The Ball HA
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
The Ball HA
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Solving for the Output
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Solving for the Output
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Time In-Between Bounces?
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Accumulated Bounce Times
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
So What?
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Problem with the bouncing ball: Infinitely many switches in finite time
This is bad: Simulations crash Model is not accurate System behavior fundamentally ill-defined beyond Zeno
point
Lecture 5.6 The Zeno Phenomenon
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
The Zeno Phenomenon
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
The Zeno Phenomenon
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
The Zeno Phenomenon
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
The Zeno Phenomenon
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Example
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Super-Zeno?
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Zeno is a problem Type 1 is not only detectable, but one can deal with it in a
rather straightforward manner Type 2 is overall hard to handle!
Good News and Bad News
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
It is clear what should happen! How do we make that mathematically sound? Sliding Mode Control
Lecture 5.7 Sliding Mode Control
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Both vector fields point inwards = bad! We should keep sliding along the switching surface Sliding Mode Control
Switching Surfaces
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins 0
g(x) < 0
g(x) = 0
switching surface
Sliding?
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins 0
g(x) < 0
g(x) = 0
g
x
T
f1
f2
Sliding occurs if g
xf1 < 0 AND
g
xf2 > 0
derivative of g in direction f = Lfg = Lie derivative
Sliding?
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins 0
g(x) < 0
g(x) = 0
g
x
T
f1
f2
Sliding occurs if
derivative of g in direction f = Lfg
Lf1g < 0 AND Lf2g > 0
= Lie derivative
Next time: But what happens beyond the Zeno point?
A Test For Type 1 Zeno
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins 0
How do we move beyond the Zeno point?
Lecture 5.8 Regularizations
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins 0
g(x) < 0
g(x) = 0
f2
f1
x = f1(x)
x = f2(x)
g(x) < 0g(x) 0
Sliding occurs if
Lf1g < 0 AND Lf2g > 0
The Sliding Mode
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Back to the Example
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
The Induced Mode
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
Regularizations of Type 1 Zeno HA
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins 0
Regularizations of Type 1 Zeno HA
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins 0
x =1
Lf2g Lf1gLf2gf1 Lf1gf2
g < 0g > 0
g = 0 and Lf1g < 0 and Lf2g > 0
g = 0 and Lf1g < 0 and Lf2g > 0
We have Models Stability Awareness Zeno Regularizations
Next Module: Back to ROBOTICS!
Hybrid Systems: In Summary
Magnus Egerstedt, Control of Mobile Robots, Georgia Ins
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