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A 2D-DFT based method to compute the Bezoutian and a link to Lyapunov equations Chayan Bhawal, Debasattam Pal and Madhu N. Belur Control and Computing Department of Electrical Engineering IIT Bombay Indian Control Conference, Guwahati January 6, 2017 Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

Introduction Introduction p, q are polynomials: p ( x ) q ( y ) + p ( y ) q ( x ) p ( x ) q ( y ) − p ( y ) q ( x ) =: ˜ B ( x, y ) =: b ( x, y ) x + y x − y Stability analysis Riccati equation solutions Bezoutian Storage functions Polynomial coprimeness Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

Introduction Introduction p, q are polynomials: p ( x ) q ( y ) + p ( y ) q ( x ) p ( x ) q ( y ) − p ( y ) q ( x ) =: ˜ B ( x, y ) =: b ( x, y ) x + y x − y Stability analysis Riccati equation solutions Bezoutian Storage functions Polynomial coprimeness Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

Introduction Introduction p, q are polynomials: p ( x ) q ( y ) + p ( y ) q ( x ) p ( x ) q ( y ) − p ( y ) q ( x ) =: ˜ B ( x, y ) =: b ( x, y ) x + y x − y Stability analysis Riccati equation solutions Bezoutian Storage functions Polynomial coprimeness Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

Introduction Bezoutian b ( x, y ) p, q are polynomials: p ( x ) q ( y ) + p ( y ) q ( x ) p ( x ) q ( y ) − p ( y ) q ( x ) =: b ( x, y ) =: b ( x, y ) x + y x − y Stability analysis Riccati equation solutions Bezoutian Storage functions Polynomial coprimeness Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

Introduction Bezoutian b ( x, y ) p, q are polynomials: φ ( x, y ) p ( x ) q ( y ) − p ( y ) q ( x ) x + y =: b ( x, y ) =: b ( x, y ) x − y Stability analysis Riccati equation solutions Bezoutian Storage functions Polynomial coprimeness Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

Introduction Bezoutian b ( x, y ) p, q are polynomials: p ( x ) q ( y ) − p ( y ) q ( x ) φ ( x, y ) = ( x + y ) b ( x, y ) =: b ( x, y ) x − y Stability analysis Riccati equation solutions Bezoutian Storage functions Polynomial coprimeness Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

Outline Objective and Outline p, q are polynomials: φ ( x, y ) = ( x + y ) b ( x, y ) OBJECTIVE: To compute the Bezoutian b ( x, y ). OUTLINE 1 2D-DFT based method to compute Bezoutian. 2 Bezoutian and link to Lyapunov equation. 3 Lyapunov equation and its link to two variable polynomials. Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

Outline Objective and Outline p, q are polynomials: φ ( x, y ) = ( x + y ) b ( x, y ) OBJECTIVE: To compute the Bezoutian b ( x, y ). OUTLINE 1 2D-DFT based method to compute Bezoutian. 2 Bezoutian and link to Lyapunov equation. 3 Lyapunov equation and its link to two variable polynomials. Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian The Algorithm     1 1 x y Bezoutian:      x 2   y 2  Let X : = , Y : =      .   .  ( x + y ) b ( x, y ) = φ ( x, y ) . .     . .     x N − 1 y N − 1 φ ( x, y )= X T Φ Y , Φ ∈ R N × N � J � � 0 � 0 1 Y =: X T R Y , J = ( x + y ) = X T ∈ R 2 × 2 0 0 1 0 � ˜ � 0 B Y =: X T B Y , ˜ B ∈ R ( N − 1 ) × ( N − 1 ) b ( x, y ) = X T 0 0 Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian The Algorithm     1 1 x y Bezoutian:      x 2   y 2  Let X : = , Y : =      .   .  ( x + y ) b ( x, y ) = φ ( x, y ) . .     . .     x N − 1 y N − 1 φ ( x, y ) = X T Φ Y , Φ ∈ R N × N � J � � 0 � 0 1 Y =: X T R Y , J = ( x + y ) = X T ∈ R 2 × 2 0 0 1 0 � ˜ � 0 B Y =: X T B Y , ˜ B ∈ R ( N − 1 ) × ( N − 1 ) b ( x, y ) = X T 0 0 Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian The Algorithm     1 1 x y Bezoutian:      x 2   y 2  Let X : = , Y : =      .   .  ( x + y ) b ( x, y ) = φ ( x, y ) . .     . .     x N − 1 y N − 1 φ ( x, y ) = X T Φ Y , Φ ∈ R N × N � J � � 0 � 0 1 Y =: X T R Y , J = ( x + y ) = X T ∈ R 2 × 2 0 0 1 0 � ˜ � 0 B Y =: X T B Y , ˜ B ∈ R ( N − 1 ) × ( N − 1 ) b ( x, y ) = X T 0 0 Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian The Algorithm     1 1 x y Bezoutian:      x 2   y 2  Let X : = , Y : =      .   .  ( x + y ) b ( x, y ) = φ ( x, y ) . .     . .     x N − 1 y N − 1 φ ( x, y ) = X T Φ Y , Φ ∈ R N × N � J � � 0 � 0 1 Y =: X T R Y , J = ( x + y ) = X T ∈ R 2 × 2 0 0 1 0 � ˜ � 0 B Y =: X T B Y , ˜ B ∈ R ( N − 1 ) × ( N − 1 ) b ( x, y ) = X T 0 0 Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian The Algorithm X T RY X T B Y X T Φ Y � � � � � � ( x + y ) b ( x, y ) = φ ( x, y ) = ⇒ = # Objective: Find B . # Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F ( R ) ⊗ F ( B ) = F (Φ) where ⊗ means elementwise multiplication. Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian The Algorithm X T RY X T B Y X T Φ Y � � � � � � ( x + y ) b ( x, y ) = φ ( x, y ) = ⇒ = # Objective: Find B . # Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F ( R ) ⊗ F ( B ) = F (Φ) where ⊗ means elementwise multiplication. Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian The Algorithm X T RY X T B Y X T Φ Y � � � � � � ( x + y ) b ( x, y ) = φ ( x, y ) = ⇒ = # Objective: Find B . # Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F ( R ) ⊗ F ( B ) = F (Φ) where ⊗ means elementwise multiplication. Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian The Algorithm X T RY X T B Y X T Φ Y � � � � � � ( x + y ) b ( x, y ) = φ ( x, y ) = ⇒ = # Objective: Find B . # Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F ( R ) ⊗ F ( B ) = F (Φ) where ⊗ means elementwise multiplication. Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian The Algorithm X T RY X T B Y X T Φ Y � � � � � � ( x + y ) b ( x, y ) = φ ( x, y ) = ⇒ = # Objective: Find B . # Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F ( R ) ⊗ F ( B ) = F (Φ) where ⊗ means elementwise multiplication. Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12

2D-DFT and Bezoutian B = F − 1 � � F (Φ) ./ F ( R ) . The algorithm fails when any element of F ( R ) is zero. ( k, ℓ ) th element of F ( R ) is e − j 2 π N k + e − j 2 π N ℓ i.e. pairwise sum of roots of unity. -1 1 1 N is even N is odd N even case: Zero padding required in R and Φ to make them odd. Indian Control Conference, GuwahatiJan Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. / 12