control theory: feedback and the pole-shifting...
TRANSCRIPT
![Page 1: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/1.jpg)
Control Theory:Feedback and the Pole-Shifting Theorem
Joseph Downs
September 3, 2013
![Page 2: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/2.jpg)
Introduction (+ examples)What is Control Theory?Examples
State-SpaceWhat is a State?Notation
(State) FeedbackFeedback Mechanisms
LTI SystemsLinearity and StationarityHow to Apply the PST
The Pole-Shifting TheoremStatement of the Pole-Shifting TheoremConsequences of the Pole-Shifting Theorem
Conclusion
![Page 3: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/3.jpg)
What is Control Theory?
I steer physical quantities to desired values
I mathematical description of engineering process
I application of dynamical systems theory
I introduce a control, u
![Page 4: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/4.jpg)
What is Control Theory?
I steer physical quantities to desired values
I mathematical description of engineering process
I application of dynamical systems theory
I introduce a control, u
![Page 5: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/5.jpg)
What is Control Theory?
I steer physical quantities to desired values
I mathematical description of engineering process
I application of dynamical systems theory
I introduce a control, u
![Page 6: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/6.jpg)
What is Control Theory?
I steer physical quantities to desired values
I mathematical description of engineering process
I application of dynamical systems theory
I introduce a control, u
![Page 7: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/7.jpg)
Examples
I cruise control
I precision amplification (lasers + circuits)
I biological motor control systems
![Page 8: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/8.jpg)
Examples
I cruise control
I precision amplification (lasers + circuits)
I biological motor control systems
![Page 9: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/9.jpg)
Examples
I cruise control
I precision amplification (lasers + circuits)
I biological motor control systems
![Page 10: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/10.jpg)
The Handstand Problem: Setup
I Iα =∑τi
I mL2θ = mgL sin θ − u
I θ = g sin θL − u
mL2
![Page 11: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/11.jpg)
The Handstand Problem: Setup
I Iα =∑τi
I mL2θ = mgL sin θ − u
I θ = g sin θL − u
mL2
![Page 12: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/12.jpg)
The Handstand Problem: Setup
I Iα =∑τi
I mL2θ = mgL sin θ − u
I θ = g sin θL − u
mL2
![Page 13: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/13.jpg)
What is a State?
I contains ”sufficient information”
I describes system past
I useful for deterministic systems
I state variable, x
I allows us to write x = φ(t, x , u)
![Page 14: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/14.jpg)
What is a State?
I contains ”sufficient information”
I describes system past
I useful for deterministic systems
I state variable, x
I allows us to write x = φ(t, x , u)
![Page 15: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/15.jpg)
What is a State?
I contains ”sufficient information”
I describes system past
I useful for deterministic systems
I state variable, x
I allows us to write x = φ(t, x , u)
![Page 16: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/16.jpg)
What is a State?
I contains ”sufficient information”
I describes system past
I useful for deterministic systems
I state variable, x
I allows us to write x = φ(t, x , u)
![Page 17: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/17.jpg)
What is a State?
I contains ”sufficient information”
I describes system past
I useful for deterministic systems
I state variable, x
I allows us to write x = φ(t, x , u)
![Page 18: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/18.jpg)
Notation
I convenient to ”vectorize”
I x1 = f1(x1, x2, ..., xn)x2 = f2(x1, x2, ..., xn)...xn = fn(x1, x2, ..., xn)
I →I x = f(x)
![Page 19: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/19.jpg)
Notation
I convenient to ”vectorize”
I x1 = f1(x1, x2, ..., xn)x2 = f2(x1, x2, ..., xn)...xn = fn(x1, x2, ..., xn)
I →I x = f(x)
![Page 20: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/20.jpg)
Notation
I convenient to ”vectorize”
I x1 = f1(x1, x2, ..., xn)x2 = f2(x1, x2, ..., xn)...xn = fn(x1, x2, ..., xn)
I →
I x = f(x)
![Page 21: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/21.jpg)
Notation
I convenient to ”vectorize”
I x1 = f1(x1, x2, ..., xn)x2 = f2(x1, x2, ..., xn)...xn = fn(x1, x2, ..., xn)
I →I x = f(x)
![Page 22: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/22.jpg)
The Handstand Problem: Notation
I x1 = θ
I x2 = θ
I x :=
(x1x2
)=
(x2
g sin x1L − u
mL2
)=: f(x,u)
![Page 23: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/23.jpg)
The Handstand Problem: Notation
I x1 = θ
I x2 = θ
I x :=
(x1x2
)=
(x2
g sin x1L − u
mL2
)=: f(x,u)
![Page 24: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/24.jpg)
The Handstand Problem: Notation
I x1 = θ
I x2 = θ
I x :=
(x1x2
)=
(x2
g sin x1L − u
mL2
)=: f(x,u)
![Page 25: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/25.jpg)
Feedback Mechanisms
I plug calculated values in as control
I u = ψ(t, x)
I achieved by measuring physical quantities
I state vs. output feedback
![Page 26: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/26.jpg)
Feedback Mechanisms
I plug calculated values in as control
I u = ψ(t, x)
I achieved by measuring physical quantities
I state vs. output feedback
![Page 27: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/27.jpg)
Feedback Mechanisms
I plug calculated values in as control
I u = ψ(t, x)
I achieved by measuring physical quantities
I state vs. output feedback
![Page 28: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/28.jpg)
Feedback Mechanisms
I plug calculated values in as control
I u = ψ(t, x)
I achieved by measuring physical quantities
I state vs. output feedback
![Page 29: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/29.jpg)
Linearity and Time-Invariance
I linear: x = A(t)x + B(t)u
I time-invariant: x = φ(t, x,u) = f (x,u)
I much simpler mathematically
![Page 30: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/30.jpg)
Linearity and Time-Invariance
I linear: x = A(t)x + B(t)u
I time-invariant: x = φ(t, x,u) = f (x,u)
I much simpler mathematically
![Page 31: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/31.jpg)
Linearity and Time-Invariance
I linear: x = A(t)x + B(t)u
I time-invariant: x = φ(t, x,u) = f (x,u)
I much simpler mathematically
![Page 32: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/32.jpg)
How to Apply the Pole-Shifting Theorem
I assume time-invariance: ∂φ(t,:,:)∂t � 1
I linearize about an equilibrium point (x = 0)
I A =
(∂f1∂x1
∂f1∂x2
∂f2∂x1
∂f2∂x2
)
I B =
(∂f1∂u∂f2∂u
)
![Page 33: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/33.jpg)
How to Apply the Pole-Shifting Theorem
I assume time-invariance: ∂φ(t,:,:)∂t � 1
I linearize about an equilibrium point (x = 0)
I A =
(∂f1∂x1
∂f1∂x2
∂f2∂x1
∂f2∂x2
)
I B =
(∂f1∂u∂f2∂u
)
![Page 34: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/34.jpg)
How to Apply the Pole-Shifting Theorem
I assume time-invariance: ∂φ(t,:,:)∂t � 1
I linearize about an equilibrium point (x = 0)
I A =
(∂f1∂x1
∂f1∂x2
∂f2∂x1
∂f2∂x2
)
I B =
(∂f1∂u∂f2∂u
)
![Page 35: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/35.jpg)
How to Apply the Pole-Shifting Theorem
I assume time-invariance: ∂φ(t,:,:)∂t � 1
I linearize about an equilibrium point (x = 0)
I A =
(∂f1∂x1
∂f1∂x2
∂f2∂x1
∂f2∂x2
)
I B =
(∂f1∂u∂f2∂u
)
![Page 36: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/36.jpg)
The Handstand Problem: Theorem Preparation
I x ≈ Ax + Bu
I A =
(0 1gL 0
)I B =
(0− 1
mL2
)I AB =
(− 1
mL2
0
)
![Page 37: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/37.jpg)
The Handstand Problem: Theorem Preparation
I x ≈ Ax + Bu
I A =
(0 1gL 0
)
I B =
(0− 1
mL2
)I AB =
(− 1
mL2
0
)
![Page 38: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/38.jpg)
The Handstand Problem: Theorem Preparation
I x ≈ Ax + Bu
I A =
(0 1gL 0
)I B =
(0− 1
mL2
)
I AB =
(− 1
mL2
0
)
![Page 39: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/39.jpg)
The Handstand Problem: Theorem Preparation
I x ≈ Ax + Bu
I A =
(0 1gL 0
)I B =
(0− 1
mL2
)I AB =
(− 1
mL2
0
)
![Page 40: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/40.jpg)
Pole-Shifting Theorem Statement
I A(n × n) and B(n ×m) are s.t.rank
([ B AB ... An−1B ]
)= n
I →I ∃F (m × n) s.t. eigenvalues of A + BF are arbitrary
![Page 41: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/41.jpg)
Pole-Shifting Theorem Statement
I A(n × n) and B(n ×m) are s.t.rank
([ B AB ... An−1B ]
)= n
I →
I ∃F (m × n) s.t. eigenvalues of A + BF are arbitrary
![Page 42: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/42.jpg)
Pole-Shifting Theorem Statement
I A(n × n) and B(n ×m) are s.t.rank
([ B AB ... An−1B ]
)= n
I →I ∃F (m × n) s.t. eigenvalues of A + BF are arbitrary
![Page 43: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/43.jpg)
Pole-Shifting Theorem Consequences
I we can make a feedback law!
I u = Fx
I x = (A + BF )x
I control system reduces to a dynamical system!
![Page 44: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/44.jpg)
Pole-Shifting Theorem Consequences
I we can make a feedback law!
I u = Fx
I x = (A + BF )x
I control system reduces to a dynamical system!
![Page 45: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/45.jpg)
Pole-Shifting Theorem Consequences
I we can make a feedback law!
I u = Fx
I x = (A + BF )x
I control system reduces to a dynamical system!
![Page 46: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/46.jpg)
Pole-Shifting Theorem Consequences
I we can make a feedback law!
I u = Fx
I x = (A + BF )x
I control system reduces to a dynamical system!
![Page 47: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/47.jpg)
The Handstand Problem: Theorem Application
I F =(f1 f2
)
I A + BF =
(0 1
gL −
f1mL2
− f2mL2
)I χA+BF = λ2 + λ f2
mL2+ ( f1
mL2− g
L ) = 0
I λ = − f22mL2
± 12
√f2
2
m2L4− 4 f1
mL2+ 4g
L
I
(f2 > 0)&(f1 >f2
2
4mL2+ mgL)
→
Re(λ±) < 0
![Page 48: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/48.jpg)
The Handstand Problem: Theorem Application
I F =(f1 f2
)I A + BF =
(0 1
gL −
f1mL2
− f2mL2
)
I χA+BF = λ2 + λ f2mL2
+ ( f1mL2− g
L ) = 0
I λ = − f22mL2
± 12
√f2
2
m2L4− 4 f1
mL2+ 4g
L
I
(f2 > 0)&(f1 >f2
2
4mL2+ mgL)
→
Re(λ±) < 0
![Page 49: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/49.jpg)
The Handstand Problem: Theorem Application
I F =(f1 f2
)I A + BF =
(0 1
gL −
f1mL2
− f2mL2
)I χA+BF = λ2 + λ f2
mL2+ ( f1
mL2− g
L ) = 0
I λ = − f22mL2
± 12
√f2
2
m2L4− 4 f1
mL2+ 4g
L
I
(f2 > 0)&(f1 >f2
2
4mL2+ mgL)
→
Re(λ±) < 0
![Page 50: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/50.jpg)
The Handstand Problem: Theorem Application
I F =(f1 f2
)I A + BF =
(0 1
gL −
f1mL2
− f2mL2
)I χA+BF = λ2 + λ f2
mL2+ ( f1
mL2− g
L ) = 0
I λ = − f22mL2
± 12
√f2
2
m2L4− 4 f1
mL2+ 4g
L
I
(f2 > 0)&(f1 >f2
2
4mL2+ mgL)
→
Re(λ±) < 0
![Page 51: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/51.jpg)
The Handstand Problem: Theorem Application
I F =(f1 f2
)I A + BF =
(0 1
gL −
f1mL2
− f2mL2
)I χA+BF = λ2 + λ f2
mL2+ ( f1
mL2− g
L ) = 0
I λ = − f22mL2
± 12
√f2
2
m2L4− 4 f1
mL2+ 4g
L
I
(f2 > 0)&(f1 >f2
2
4mL2+ mgL)
→
Re(λ±) < 0
![Page 52: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/52.jpg)
Conclusion
I control theory provides framework
I linearization simplifies problem
I simple rank condition permits use of feedback
I further topics: PID control, feedback linearization, robotics,etc.
I Thanks for your time!
![Page 53: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/53.jpg)
Conclusion
I control theory provides framework
I linearization simplifies problem
I simple rank condition permits use of feedback
I further topics: PID control, feedback linearization, robotics,etc.
I Thanks for your time!
![Page 54: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/54.jpg)
Conclusion
I control theory provides framework
I linearization simplifies problem
I simple rank condition permits use of feedback
I further topics: PID control, feedback linearization, robotics,etc.
I Thanks for your time!
![Page 55: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/55.jpg)
Conclusion
I control theory provides framework
I linearization simplifies problem
I simple rank condition permits use of feedback
I further topics: PID control, feedback linearization, robotics,etc.
I Thanks for your time!
![Page 56: Control Theory: Feedback and the Pole-Shifting Theoremdrp.math.umd.edu/Project-Slides/DownsSummer13.pdfI cruise control I precision ampli cation (lasers + circuits) I biological motor](https://reader033.vdocuments.us/reader033/viewer/2022053119/609f66cf12064e0a4402ad36/html5/thumbnails/56.jpg)
Conclusion
I control theory provides framework
I linearization simplifies problem
I simple rank condition permits use of feedback
I further topics: PID control, feedback linearization, robotics,etc.
I Thanks for your time!