Control Theory: Feedback and the Pole-Shifting Theorem Joseph Downs - - PowerPoint PPT Presentation

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Control Theory: Feedback and the Pole-Shifting Theorem Joseph Downs - - PowerPoint PPT Presentation

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Control Theory: Feedback and the Pole-Shifting Theorem Joseph Downs September 3, 2013 Introduction (+ examples) What is Control Theory? Examples State-Space What is a State? Notation (State) Feedback Feedback Mechanisms LTI Systems

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Control Theory: Feedback and the Pole-Shifting Theorem

Joseph Downs September 3, 2013

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Introduction (+ examples) What is Control Theory? Examples State-Space What is a State? Notation (State) Feedback Feedback Mechanisms LTI Systems Linearity and Stationarity How to Apply the PST The Pole-Shifting Theorem Statement of the Pole-Shifting Theorem Consequences of the Pole-Shifting Theorem Conclusion

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What is Control Theory?

◮ steer physical quantities to desired values

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What is Control Theory?

◮ steer physical quantities to desired values ◮ mathematical description of engineering process

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What is Control Theory?

◮ steer physical quantities to desired values ◮ mathematical description of engineering process ◮ application of dynamical systems theory

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What is Control Theory?

◮ steer physical quantities to desired values ◮ mathematical description of engineering process ◮ application of dynamical systems theory ◮ introduce a control, u

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Examples

◮ cruise control

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Examples

◮ cruise control ◮ precision amplification (lasers + circuits)

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Examples

◮ cruise control ◮ precision amplification (lasers + circuits) ◮ biological motor control systems

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The Handstand Problem: Setup

◮ Iα = τi

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The Handstand Problem: Setup

◮ Iα = τi ◮ mL2¨

θ = mgL sin θ − u

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The Handstand Problem: Setup

◮ Iα = τi ◮ mL2¨

θ = mgL sin θ − u

◮ ¨

θ = g sin θ

L

−

u mL2

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What is a State?

◮ contains ”sufficient information”

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What is a State?

◮ contains ”sufficient information” ◮ describes system past

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What is a State?

◮ contains ”sufficient information” ◮ describes system past ◮ useful for deterministic systems

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What is a State?

◮ contains ”sufficient information” ◮ describes system past ◮ useful for deterministic systems ◮ state variable, x

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What is a State?

◮ contains ”sufficient information” ◮ describes system past ◮ useful for deterministic systems ◮ state variable, x ◮ allows us to write ˙

x = φ(t, x, u)

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Notation

◮ convenient to ”vectorize”

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Notation

◮ convenient to ”vectorize” ◮

˙ x1 = f1(x1, x2, ..., xn) ˙ x2 = f2(x1, x2, ..., xn) ... ˙ xn = fn(x1, x2, ..., xn)

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Notation

◮ convenient to ”vectorize” ◮

˙ x1 = f1(x1, x2, ..., xn) ˙ x2 = f2(x1, x2, ..., xn) ... ˙ xn = fn(x1, x2, ..., xn)

◮ →

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Notation

◮ convenient to ”vectorize” ◮

˙ x1 = f1(x1, x2, ..., xn) ˙ x2 = f2(x1, x2, ..., xn) ... ˙ xn = fn(x1, x2, ..., xn)

◮ → ◮ ˙

x = f(x)

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The Handstand Problem: Notation

◮ x1 = θ

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The Handstand Problem: Notation

◮ x1 = θ ◮ x2 = ˙

θ

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The Handstand Problem: Notation

◮ x1 = θ ◮ x2 = ˙

θ

◮ ˙

x := ˙ x1 ˙ x2

  • =
  • x2

g sin x1 L

−

u mL2

  • =: f(x, u)
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Feedback Mechanisms

◮ plug calculated values in as control

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Feedback Mechanisms

◮ plug calculated values in as control ◮ u = ψ(t, x)

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Feedback Mechanisms

◮ plug calculated values in as control ◮ u = ψ(t, x) ◮ achieved by measuring physical quantities

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Feedback Mechanisms

◮ plug calculated values in as control ◮ u = ψ(t, x) ◮ achieved by measuring physical quantities ◮ state vs. output feedback

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Linearity and Time-Invariance

◮ linear: ˙

x = A(t)x + B(t)u

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Linearity and Time-Invariance

◮ linear: ˙

x = A(t)x + B(t)u

◮ time-invariant: ˙

x = φ(t, x, u) = f (x, u)

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Linearity and Time-Invariance

◮ linear: ˙

x = A(t)x + B(t)u

◮ time-invariant: ˙

x = φ(t, x, u) = f (x, u)

◮ much simpler mathematically

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How to Apply the Pole-Shifting Theorem

◮ assume time-invariance: ∂φ(t,:,:) ∂t

≪ 1

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How to Apply the Pole-Shifting Theorem

◮ assume time-invariance: ∂φ(t,:,:) ∂t

≪ 1

◮ linearize about an equilibrium point (˙

x = 0)

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How to Apply the Pole-Shifting Theorem

◮ assume time-invariance: ∂φ(t,:,:) ∂t

≪ 1

◮ linearize about an equilibrium point (˙

x = 0)

◮ A =

  • ∂f1

∂x1 ∂f1 ∂x2 ∂f2 ∂x1 ∂f2 ∂x2

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How to Apply the Pole-Shifting Theorem

◮ assume time-invariance: ∂φ(t,:,:) ∂t

≪ 1

◮ linearize about an equilibrium point (˙

x = 0)

◮ A =

  • ∂f1

∂x1 ∂f1 ∂x2 ∂f2 ∂x1 ∂f2 ∂x2

  • ◮ B =

∂f1

∂u ∂f2 ∂u

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The Handstand Problem: Theorem Preparation

◮ ˙

x ≈ Ax + Bu

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The Handstand Problem: Theorem Preparation

◮ ˙

x ≈ Ax + Bu

◮ A =

1

g L

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The Handstand Problem: Theorem Preparation

◮ ˙

x ≈ Ax + Bu

◮ A =

1

g L

  • ◮ B =
  • −

1 mL2

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The Handstand Problem: Theorem Preparation

◮ ˙

x ≈ Ax + Bu

◮ A =

1

g L

  • ◮ B =
  • −

1 mL2

  • ◮ AB =

−

1 mL2

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Pole-Shifting Theorem Statement

◮ A(n × n) and B(n × m) are s.t.

rank

  • [ B

AB ... An−1B ]

  • = n
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Pole-Shifting Theorem Statement

◮ A(n × n) and B(n × m) are s.t.

rank

  • [ B

AB ... An−1B ]

  • = n

◮ →

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Pole-Shifting Theorem Statement

◮ A(n × n) and B(n × m) are s.t.

rank

  • [ B

AB ... An−1B ]

  • = n

◮ → ◮ ∃F(m × n) s.t. eigenvalues of A + BF are arbitrary

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Pole-Shifting Theorem Consequences

◮ we can make a feedback law!

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Pole-Shifting Theorem Consequences

◮ we can make a feedback law! ◮ u = Fx

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Pole-Shifting Theorem Consequences

◮ we can make a feedback law! ◮ u = Fx ◮ ˙

x = (A + BF)x

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Pole-Shifting Theorem Consequences

◮ we can make a feedback law! ◮ u = Fx ◮ ˙

x = (A + BF)x

◮ control system reduces to a dynamical system!

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The Handstand Problem: Theorem Application

◮ F =

  • f1

f2

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The Handstand Problem: Theorem Application

◮ F =

  • f1

f2

  • ◮ A + BF =
  • 1

g L − f1 mL2

− f2

mL2

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The Handstand Problem: Theorem Application

◮ F =

  • f1

f2

  • ◮ A + BF =
  • 1

g L − f1 mL2

− f2

mL2

  • ◮ χA+BF = λ2 + λ f2

mL2 + ( f1 mL2 − g L) = 0

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The Handstand Problem: Theorem Application

◮ F =

  • f1

f2

  • ◮ A + BF =
  • 1

g L − f1 mL2

− f2

mL2

  • ◮ χA+BF = λ2 + λ f2

mL2 + ( f1 mL2 − g L) = 0 ◮ λ = − f2 2mL2 ± 1 2

  • f22

m2L4 − 4 f1 mL2 + 4 g L

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The Handstand Problem: Theorem Application

◮ F =

  • f1

f2

  • ◮ A + BF =
  • 1

g L − f1 mL2

− f2

mL2

  • ◮ χA+BF = λ2 + λ f2

mL2 + ( f1 mL2 − g L) = 0 ◮ λ = − f2 2mL2 ± 1 2

  • f22

m2L4 − 4 f1 mL2 + 4 g L ◮

(f2 > 0)&(f1 > f22 4mL2 + mgL) → Re(λ±) < 0

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Conclusion

◮ control theory provides framework

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Conclusion

◮ control theory provides framework ◮ linearization simplifies problem

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Conclusion

◮ control theory provides framework ◮ linearization simplifies problem ◮ simple rank condition permits use of feedback

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Conclusion

◮ control theory provides framework ◮ linearization simplifies problem ◮ simple rank condition permits use of feedback ◮ further topics: PID control, feedback linearization, robotics,

etc.

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Conclusion

◮ control theory provides framework ◮ linearization simplifies problem ◮ simple rank condition permits use of feedback ◮ further topics: PID control, feedback linearization, robotics,

etc.

◮ Thanks for your time!