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

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 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

What is Control Theory? ◮ steer physical quantities to desired values

What is Control Theory? ◮ steer physical quantities to desired values ◮ mathematical description of engineering process

What is Control Theory? ◮ steer physical quantities to desired values ◮ mathematical description of engineering process ◮ application of dynamical systems theory

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

Examples ◮ cruise control

Examples ◮ cruise control ◮ precision amplification (lasers + circuits)

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

The Handstand Problem: Setup ◮ I α = � τ i

The Handstand Problem: Setup ◮ I α = � τ i ◮ mL 2 ¨ θ = mgL sin θ − u

The Handstand Problem: Setup ◮ I α = � τ i ◮ mL 2 ¨ θ = mgL sin θ − u ◮ ¨ θ = g sin θ u − mL 2 L

What is a State? ◮ contains ”sufficient information”

What is a State? ◮ contains ”sufficient information” ◮ describes system past

What is a State? ◮ contains ”sufficient information” ◮ describes system past ◮ useful for deterministic systems

What is a State? ◮ contains ”sufficient information” ◮ describes system past ◮ useful for deterministic systems ◮ state variable, x

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 )

Notation ◮ convenient to ”vectorize”

Notation ◮ convenient to ”vectorize” x 1 = f 1 ( x 1 , x 2 , ..., x n ) ˙ ◮ x 2 = f 2 ( x 1 , x 2 , ..., x n ) ˙ ... x n = f n ( x 1 , x 2 , ..., x n ) ˙

Notation ◮ convenient to ”vectorize” x 1 = f 1 ( x 1 , x 2 , ..., x n ) ˙ ◮ x 2 = f 2 ( x 1 , x 2 , ..., x n ) ˙ ... x n = f n ( x 1 , x 2 , ..., x n ) ˙ ◮ →

Notation ◮ convenient to ”vectorize” x 1 = f 1 ( x 1 , x 2 , ..., x n ) ˙ ◮ x 2 = f 2 ( x 1 , x 2 , ..., x n ) ˙ ... x n = f n ( x 1 , x 2 , ..., x n ) ˙ ◮ → ◮ ˙ x = f ( x )

The Handstand Problem: Notation ◮ x 1 = θ

The Handstand Problem: Notation ◮ x 1 = θ ◮ x 2 = ˙ θ

The Handstand Problem: Notation ◮ x 1 = θ ◮ x 2 = ˙ θ � ˙ � � � x 1 x 2 ◮ ˙ x := = =: f ( x , u ) g sin x 1 u ˙ x 2 − mL 2 L

Feedback Mechanisms ◮ plug calculated values in as control

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

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

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

Linearity and Time-Invariance ◮ linear: ˙ x = A ( t ) x + B ( t ) u

Linearity and Time-Invariance ◮ linear: ˙ x = A ( t ) x + B ( t ) u ◮ time-invariant: ˙ x = φ ( t , x , u ) = f ( x , u )

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

How to Apply the Pole-Shifting Theorem ◮ assume time-invariance: ∂φ ( t , : , :) ≪ 1 ∂ t

How to Apply the Pole-Shifting Theorem ◮ assume time-invariance: ∂φ ( t , : , :) ≪ 1 ∂ t ◮ linearize about an equilibrium point (˙ x = 0)

How to Apply the Pole-Shifting Theorem ◮ assume time-invariance: ∂φ ( t , : , :) ≪ 1 ∂ t ◮ linearize about an equilibrium point (˙ x = 0) � � ∂ f 1 ∂ f 1 ∂ x 1 ∂ x 2 ◮ A = ∂ f 2 ∂ f 2 ∂ x 1 ∂ x 2

How to Apply the Pole-Shifting Theorem ◮ assume time-invariance: ∂φ ( t , : , :) ≪ 1 ∂ t ◮ linearize about an equilibrium point (˙ x = 0) � � ∂ f 1 ∂ f 1 ∂ x 1 ∂ x 2 ◮ A = ∂ f 2 ∂ f 2 ∂ x 1 ∂ x 2 � ∂ f 1 � ◮ B = ∂ u ∂ f 2 ∂ u

The Handstand Problem: Theorem Preparation ◮ ˙ x ≈ A x + B u

The Handstand Problem: Theorem Preparation ◮ ˙ x ≈ A x + B u � 0 � 1 ◮ A = g 0 L

The Handstand Problem: Theorem Preparation ◮ ˙ x ≈ A x + B u � 0 � 1 ◮ A = g 0 L � 0 � ◮ B = 1 − mL 2

The Handstand Problem: Theorem Preparation ◮ ˙ x ≈ A x + B u � 0 � 1 ◮ A = g 0 L � 0 � ◮ B = 1 − mL 2 � − 1 � mL 2 ◮ AB = 0

Pole-Shifting Theorem Statement ◮ A ( n × n ) and B ( n × m ) are s.t. � A n − 1 B ] � [ B = n rank AB ...

Pole-Shifting Theorem Statement ◮ A ( n × n ) and B ( n × m ) are s.t. � A n − 1 B ] � [ B = n rank AB ... ◮ →

Pole-Shifting Theorem Statement ◮ A ( n × n ) and B ( n × m ) are s.t. � A n − 1 B ] � [ B = n rank AB ... ◮ → ◮ ∃ F ( m × n ) s.t. eigenvalues of A + BF are arbitrary

Pole-Shifting Theorem Consequences ◮ we can make a feedback law!

Pole-Shifting Theorem Consequences ◮ we can make a feedback law! ◮ u = F x

Pole-Shifting Theorem Consequences ◮ we can make a feedback law! ◮ u = F x ◮ ˙ x = ( A + BF ) x

Pole-Shifting Theorem Consequences ◮ we can make a feedback law! ◮ u = F x ◮ ˙ x = ( A + BF ) x ◮ control system reduces to a dynamical system!

The Handstand Problem: Theorem Application ◮ F = � � f 1 f 2

The Handstand Problem: Theorem Application ◮ F = � � f 1 f 2 � 0 1 � ◮ A + BF = g f 1 − f 2 L − mL 2 mL 2

The Handstand Problem: Theorem Application ◮ F = � � f 1 f 2 � 0 1 � ◮ A + BF = g f 1 − f 2 L − mL 2 mL 2 ◮ χ A + BF = λ 2 + λ f 2 mL 2 − g mL 2 + ( f 1 L ) = 0

The Handstand Problem: Theorem Application ◮ F = � � f 1 f 2 � 0 1 � ◮ A + BF = g f 1 − f 2 L − mL 2 mL 2 ◮ χ A + BF = λ 2 + λ f 2 mL 2 − g mL 2 + ( f 1 L ) = 0 � f 22 2 mL 2 ± 1 f 2 m 2 L 4 − 4 f 1 mL 2 + 4 g ◮ λ = − 2 L

The Handstand Problem: Theorem Application ◮ F = � � f 1 f 2 � 0 1 � ◮ A + BF = g f 1 − f 2 L − mL 2 mL 2 ◮ χ A + BF = λ 2 + λ f 2 mL 2 − g mL 2 + ( f 1 L ) = 0 � f 22 2 mL 2 ± 1 f 2 m 2 L 4 − 4 f 1 mL 2 + 4 g ◮ λ = − 2 L ◮ f 22 ( f 2 > 0)&( f 1 > 4 mL 2 + mgL ) → Re ( λ ± ) < 0

Conclusion ◮ control theory provides framework

Conclusion ◮ control theory provides framework ◮ linearization simplifies problem

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

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

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!