Interpolation-based model reduction of nonlinear control systems - PowerPoint PPT Presentation

Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems, Trogir October 12, 2011 Interpolation-based model reduction of nonlinear control systems Tobias Breiten Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory MAX PLANCK INSTITUTE FOR DYNAMICS OF COMPLEX TECHNICAL SYSTEMS MAGDEBURG NETWORK THEORY Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 1/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Motivation Given a large-scale state-nonlinear control system of the form � x ( t ) = f ( x ( t )) + bu ( t ) , ˙ Σ : y ( t ) = cx ( t ) , x (0) = x 0 , with f : R n → R n nonlinear and b , c T ∈ R n , x ∈ R n , u , y ∈ R . Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 2/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Motivation Given a large-scale state-nonlinear control system of the form � x ( t ) = f ( x ( t )) + bu ( t ) , ˙ Σ : y ( t ) = cx ( t ) , x (0) = x 0 , with f : R n → R n nonlinear and b , c T ∈ R n , x ∈ R n , u , y ∈ R . Optimization, control and simulation cannot be done efficiently! Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 2/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Motivation Given a large-scale state-nonlinear control system of the form � x ( t ) = f ( x ( t )) + bu ( t ) , ˙ Σ : y ( t ) = cx ( t ) , x (0) = x 0 , with f : R n → R n nonlinear and b , c T ∈ R n , x ∈ R n , u , y ∈ R . Optimization, control and simulation cannot be done efficiently! MOR � ˙ x ( t ) = ˆ x ( t )) + ˆ ˆ f ( ˆ bu ( t ) , ˆ Σ : y ( t ) = ˆ x ( t ) , x (0) = ˆ ˆ cˆ ˆ x 0 , n → R ˆ n and ˆ with ˆ f : R ˆ c T ∈ R ˆ n , x ∈ R ˆ n , u ∈ R and ˆ b , ˆ y ≈ y ∈ R , ˆ n ≪ n . Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 2/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Quadratic-Bilinear DAEs State-Space Representation We will consider quadratic-bilinear SISO systems of the form x = ˙ A 1 + A 2 + xu + bu E x x ⊗ x N y = c x where E , A 1 , N ∈ R n × n , A 2 ∈ R n × n 2 (Hessian tensor) , b , c T ∈ R n . A large class of smooth nonlinear control-affine systems can be transformed into the above type of control system. The transformation is exact, but a slight increase of the state dimension has to be accepted. Input-output behavior can be characterized by generalized transfer functions. Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 3/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Quadratic-Bilinear DAEs Transformation via McCormick Relaxation [ McCormick ’76 ] Theorem [Gu’09] Assume that the state equation of a nonlinear system Σ is given by x = a 0 x + a 1 g 1 ( x ) + . . . + a k g k ( x ) + Bu , ˙ where g i ( x ) : R n → R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 4/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Quadratic-Bilinear DAEs Transformation via McCormick Relaxation [ McCormick ’76 ] Theorem [Gu’09] Assume that the state equation of a nonlinear system Σ is given by x = a 0 x + a 1 g 1 ( x ) + . . . + a k g k ( x ) + Bu , ˙ where g i ( x ) : R n → R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example � x 2 x 1 = exp( − x 2 ) · ˙ 1 + 1 , x 2 = − x 2 + u . ˙ Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 4/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Quadratic-Bilinear DAEs Transformation via McCormick Relaxation [ McCormick ’76 ] Theorem [Gu’09] Assume that the state equation of a nonlinear system Σ is given by x = a 0 x + a 1 g 1 ( x ) + . . . + a k g k ( x ) + Bu , ˙ where g i ( x ) : R n → R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example � x 2 x 1 = exp( − x 2 ) · ˙ 1 + 1 , x 2 = − x 2 + u . ˙ z 1 := exp( − x 2 ) , Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 4/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Quadratic-Bilinear DAEs Transformation via McCormick Relaxation [ McCormick ’76 ] Theorem [Gu’09] Assume that the state equation of a nonlinear system Σ is given by x = a 0 x + a 1 g 1 ( x ) + . . . + a k g k ( x ) + Bu , ˙ where g i ( x ) : R n → R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example � x 2 x 1 = exp( − x 2 ) · ˙ 1 + 1 , x 2 = − x 2 + u . ˙ � x 2 z 1 := exp( − x 2 ) , z 2 := 1 + 1 . Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 4/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Quadratic-Bilinear DAEs Transformation via McCormick Relaxation [ McCormick ’76 ] Theorem [Gu’09] Assume that the state equation of a nonlinear system Σ is given by x = a 0 x + a 1 g 1 ( x ) + . . . + a k g k ( x ) + Bu , ˙ where g i ( x ) : R n → R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example � x 2 x 1 = exp( − x 2 ) · ˙ 1 + 1 , x 2 = − x 2 + u . ˙ � x 2 z 1 := exp( − x 2 ) , z 2 := 1 + 1 . x 1 = z 1 · z 2 , ˙ Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 4/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Quadratic-Bilinear DAEs Transformation via McCormick Relaxation [ McCormick ’76 ] Theorem [Gu’09] Assume that the state equation of a nonlinear system Σ is given by x = a 0 x + a 1 g 1 ( x ) + . . . + a k g k ( x ) + Bu , ˙ where g i ( x ) : R n → R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example � x 2 x 1 = exp( − x 2 ) · ˙ 1 + 1 , x 2 = − x 2 + u . ˙ � x 2 z 1 := exp( − x 2 ) , z 2 := 1 + 1 . x 1 = z 1 · z 2 , ˙ x 2 = − x 2 + u , ˙ Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 4/9

Motivation Quadratic-Bilinear Differential-Algebraic Equations Model Reduction via Moment-Matching Quadratic-Bilinear DAEs Transformation via McCormick Relaxation [ McCormick ’76 ] Theorem [Gu’09] Assume that the state equation of a nonlinear system Σ is given by x = a 0 x + a 1 g 1 ( x ) + . . . + a k g k ( x ) + Bu , ˙ where g i ( x ) : R n → R n are compositions of uni-variable rational, exponential, logarithmic, trigonometric or root functions, respectively. Then, by iteratively taking derivatives and adding algebraic equations, respectively, Σ can be transformed into a system of QBDAEs. Example � x 2 x 1 = exp( − x 2 ) · ˙ 1 + 1 , x 2 = − x 2 + u . ˙ � x 2 z 1 := exp( − x 2 ) , z 2 := 1 + 1 . x 1 = z 1 · z 2 , ˙ x 2 = − x 2 + u , ˙ z 1 = − z 1 · ( − x 2 + u ) , ˙ Max Planck Institute Magdeburg T. Breiten, Interpolation-based model reduction of nonlinear control systems 4/9