Continuous-time Stochastic Grey-box Model of the Nonlinear Feedback - - PowerPoint PPT Presentation

DTU Compute 28 March 2019

Continuous-time Stochastic Grey-box Model of the Nonlinear Feedback System based on Residual Analysis Rune Grønborg Junker Rishi Relan

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Workshop on Nonlinear Benchmarks TU Eindhoven, April, 10, 2019

DTU Compute 28 March 2019

Modelling Philosophies

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Silverbox Benchmark

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m¨ x(t) + d ˙ x(t) + k1x(t) + k3x3(t) = u(t),

DTU Compute 28 March 2019

Stochastic Differential Equations

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dXt = ft(Xt, Ut, t, θ)dt | {z }

Drift

+ σt(Xt, Ut, t, θ)dWt | {z }

Diffusion

,

Ytk = gtk(Xtk, Utk, tk, θ) + etk |{z}

Measurement noise

,

θ ∈ Θ ⊂ Rp

etk ∈ N(0, Stk(Utk, tk, θ))

Xt ∈ X ⊂ Rn

Wt = (W 1

t , W 2 t , · · · , W m t )T ∈ Rm

Ytk ∈ Y ⊂ RL

DTU Compute 28 March 2019

Stochastic Differential Equations

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Ytk = gtk(Xtk, Utk, tk, θ) + etk |{z}

Measurement noise

,

θ ∈ Θ ⊂ Rp

etk ∈ N(0, Stk(Utk, tk, θ))

Xt ∈ X ⊂ Rn

Wt = (W 1

t , W 2 t , · · · , W m t )T ∈ Rm

dXt = ft(Xt, Ut, t, θ)dt | {z }

Drift

+ σt(Ut, t, θ)dWt | {z }

Diffusion

,

Ytk ∈ Y ⊂ RL

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Maximum Likelihood Estimation

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One-Step Ahead Prediction Error:

L(✓; YN) = p(YN|✓) = N Y

k=1

exp ⇣ − 1

2✏T tkR−1 tk|tk−1✏tk

⌘ q det(Rtk|tk−1)( √ 2⇡)L ! p(y0|✓) = −1 2

N

X

k=1

⇣ ✏T

tkR−1 tk|tk−1✏tk + log detR−1 tk|tk−1 + L log 2⇡

⌘

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N

X

k=1

⇣ F(xk, U, θ|x0, θ0) − yk ⌘2

Maximum Likelihood Estimation

Simulation Error:

log L =

N

X

k=1

⇣ F(xk, U, θ|xo, θ0) − yk ⌘2 σ2

k

− log YN

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Physical Model

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dXt = Vtdt + σxdW 1

t

dVt = (−a1Vt + k1Xt + k3X3

t + b1ut)dt + σvdW 2 t

Ytk = Xt + etk

Model 1:

m¨ x(t) + d ˙ x(t) + k1x(t) + k3x3(t) = u(t),

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Residual Analysis

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Residual Analysis

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dXt = (Vt + d1u1)dt + σxdW 1

t

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Residual Analysis

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Residual Analysis

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Ytk = (Xt + d1u1) + etk

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Residual Analysis

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Residual Analysis

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Residual Analysis

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dXt = (Vt + d1u1 + c3V 3)dt + σxdW 1

t

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Final Results

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dXt = (Vt + d1ut + c3V 3

t + f1pt)dt + σxdW 1 t ,

dVt = (−a1Vt + k1Xt + k3X3 + b1ut)dt + σV dW 2

t ,

dpt = g1(Vt − pt)dt + σpdW 3

t ,

Ytk = Xtk + d2utk + f2pt + f3p2

t + f4p3 t + etk.

DTU Compute 28 March 2019

Final Results

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dXt = (Vt + d1ut + c3V 3

t + f1pt)dt + σxdW 1 t ,

dVt = (−a1Vt + k1Xt + k3X3 + b1ut)dt + σV dW 2

t ,

dpt = g1(Vt − pt)dt + σpdW 3

t ,

Ytk = Xtk + d2utk + f2pt + f3p2

t + f4p3 t + etk.

Excluded terms Simulation error None 0.418 f2, f3, f4 0.448 c3, f2, f3, f4 0.457 d2, f2, f3, f4 0.625

DTU Compute 28 March 2019

Take home messages

• Flexible and easier way to include the system knowledge
• Continuous-time
• Stochastic
• Explicit estimation of process uncertainty
• Proper handling of measurement uncertainty
• Suitable for control

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