Interval Prediction for Continuous-Time Systems with Parametric - PowerPoint PPT Presentation

Interval Prediction for Continuous-Time Systems with Parametric Uncertainties Edouard Leurent 1 , 2 , Denis Efimov 1 , ıssi 3 , Wilfrid Perruquetti 4 Tarek Ra¨ 1 Inria, Lille, France 2 Renault Group, Guyancourt, France 3 CNAM, Paris, France 4 Centrale Lille, France

Contents 01.. Problem statement 02.. Our proposed predictor 03.. Application to autonomous driving 2 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

01 Problem statement 3 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Motivation We are interested in trajectory planning for an autonomous vehicle. 1. We need to predict the behaviours of other drivers 2. These behaviours are uncertain and non-linear In order to efficiently capture model uncertainty, we consider the modelling framework of Linear Parameter-Varying systems. 4 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

The setting Linear Parameter-Varying systems x ( t ) = A ( θ ( t )) x ( t ) + Bd ( t ) ˙ There are two sources of uncertainty: • Parametric uncertainty θ ( t ) • External perturbations d ( t ) 𝑦 𝑢, 𝜄 𝑢 , 𝑒(𝑢) 𝑦 0 5 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

The goal Interval Prediction Can we design an interval predictor [ x ( t ) , x ( t )] that verifies: • inclusion property: ∀ t , x ( t ) ≤ x ( t ) ≤ x ( t ) ; • stable dynamics? We want the predictor to be as tight as possible. 𝑦 𝑢, 𝜄 𝑢 , 𝑒(𝑢) 𝑦 𝑢 , 𝑦 𝑢 𝑦 0 6 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Assumptions Assumption (Bounded trajectories) • � x � ∞ < ∞ • x ( 0 ) ∈ [ x 0 , x 0 ] for some known x 0 , x 0 ∈ R n 7 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Assumptions Assumption (Bounded trajectories) • � x � ∞ < ∞ • x ( 0 ) ∈ [ x 0 , x 0 ] for some known x 0 , x 0 ∈ R n Assumption (Bounded parameters) • θ ( t ) ∈ Θ for some known Θ • The matrix function A ( θ ) is known 7 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Assumptions Assumption (Bounded trajectories) • � x � ∞ < ∞ • x ( 0 ) ∈ [ x 0 , x 0 ] for some known x 0 , x 0 ∈ R n Assumption (Bounded parameters) • θ ( t ) ∈ Θ for some known Θ • The matrix function A ( θ ) is known Assumption (Bounded perturbations) • d ( t ) ∈ [ d ( t ) , d ( t )] for some known signals d , d ∈ L n ∞ How to proceed? 7 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

A first idea Assume that x ( t ) ≤ x ( t ) ≤ x ( t ) , for some t ≥ 0. 8 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

A first idea Assume that x ( t ) ≤ x ( t ) ≤ x ( t ) , for some t ≥ 0. ë To propagate the interval to x ( t + dt ) , we need to bound A ( θ ( t )) x ( t ) . 8 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

A first idea Assume that x ( t ) ≤ x ( t ) ≤ x ( t ) , for some t ≥ 0. ë To propagate the interval to x ( t + dt ) , we need to bound A ( θ ( t )) x ( t ) . ë Why not use interval arithmetics? 8 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

A first idea Assume that x ( t ) ≤ x ( t ) ≤ x ( t ) , for some t ≥ 0. ë To propagate the interval to x ( t + dt ) , we need to bound A ( θ ( t )) x ( t ) . ë Why not use interval arithmetics? Lemma (Image of an interval (Efimov et al. 2012)) If A a known matrix, then A + x − A − x ≤ Ax ≤ A + x − A − x . where A + = max( A , 0 ) and A − = A − A + . 8 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

A first idea Assume that x ( t ) ≤ x ( t ) ≤ x ( t ) , for some t ≥ 0. ë To propagate the interval to x ( t + dt ) , we need to bound A ( θ ( t )) x ( t ) . ë Why not use interval arithmetics? Lemma (Product of intervals (Efimov et al. 2012)) If A is unknown but bounded A ≤ A ≤ A, A + x + − A + x − − A − x + + A − x − ≤ Ax + x + − A + x − − A − x + + A − x − . ≤ A 8 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

A first idea Assume that x ( t ) ≤ x ( t ) ≤ x ( t ) , for some t ≥ 0. ë To propagate the interval to x ( t + dt ) , we need to bound A ( θ ( t )) x ( t ) . ë Why not use interval arithmetics? Lemma (Product of intervals (Efimov et al. 2012)) If A is unknown but bounded A ≤ A ≤ A, A + x + − A + x − − A − x + + A − x − ≤ Ax + x + − A + x − − A − x + + A − x − . ≤ A � Since A ( θ ) and the set Θ are known, we can easily compute such bounds A ≤ A ( θ ( t )) ≤ A 8 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

A candidate predictor Following this result, define the predictor: + x − ( t ) − A − x + ( t ) A + x + ( t ) − A x ( t ) ˙ = − x − ( t ) + B + d ( t ) − B − d ( t ) , + A (1) − x + ( t ) + x + ( t ) − A + x − ( t ) − A ˙ x ( t ) = A + A − x − ( t ) + B + d ( t ) − B − d ( t ) , x ( 0 ) = x 0 , x ( 0 ) = x 0 , 9 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

A candidate predictor Following this result, define the predictor: + x − ( t ) − A − x + ( t ) A + x + ( t ) − A x ( t ) ˙ = − x − ( t ) + B + d ( t ) − B − d ( t ) , + A (1) − x + ( t ) + x + ( t ) − A + x − ( t ) − A ˙ x ( t ) = A + A − x − ( t ) + B + d ( t ) − B − d ( t ) , x ( 0 ) = x 0 , x ( 0 ) = x 0 , Proposition (Inclusion property) � The predictor (1) satisfies x ( t ) ≤ x ( t ) ≤ x ( t )( t ) 9 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

A candidate predictor Following this result, define the predictor: + x − ( t ) − A − x + ( t ) A + x + ( t ) − A x ( t ) ˙ = − x − ( t ) + B + d ( t ) − B − d ( t ) , + A (1) − x + ( t ) + x + ( t ) − A + x − ( t ) − A ˙ x ( t ) = A + A − x − ( t ) + B + d ( t ) − B − d ( t ) , x ( 0 ) = x 0 , x ( 0 ) = x 0 , Proposition (Inclusion property) � The predictor (1) satisfies x ( t ) ≤ x ( t ) ≤ x ( t )( t ) ? But is it stable? 9 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Motivating example Consider the scalar system, for all t ≥ 0:   x ( 0 ) ∈ [ x 0 , x 0 ] = [ 1 . 0 , 1 . 1 ] ,  x ( t ) = − θ ( t ) x ( t ) + d ( t ) , where ˙ θ ( t ) ∈ Θ = [ θ, θ ] = [ 1 , 2 ] ,   d ( t ) ∈ [ d , d ] = [ − 0 . 1 , 0 . 1 ] , 1 . 2 x ( t ) 1 . 0 0 . 8 0 . 6 0 . 4 0 . 2 0 . 0 − 0 . 2 0 1 2 3 4 5 � The system is always stable 10 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Motivating example Consider the scalar system, for all t ≥ 0:   x ( 0 ) ∈ [ x 0 , x 0 ] = [ 1 . 0 , 1 . 1 ] ,  x ( t ) = − θ ( t ) x ( t ) + d ( t ) , where ˙ θ ( t ) ∈ Θ = [ θ, θ ] = [ 1 , 2 ] ,   d ( t ) ∈ [ d , d ] = [ − 0 . 1 , 0 . 1 ] , 1 . 2 x ( t ) 1 . 0 0 . 8 x ( t ) , x ( t ) 0 . 6 0 . 4 0 . 2 0 . 0 − 0 . 2 0 1 2 3 4 5 � The system is always stable ✗ The predictor (1) is unstable 10 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

02 Our proposed predictor 11 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Additional assumption Assumption (Polytopic Structure) There exist A 0 Metzler and ∆ A 0 , · · · , ∆ A N such that: N N � � A ( θ ) = A 0 + λ i ( θ )∆ A i , λ i ( θ ) = 1 ; ∀ θ ∈ Θ ���� � �� � i = 1 i = 1 Nominal ≥ 0 dynamics Δ𝐵 1 Δ𝐵 5 Δ𝐵 2 𝐵 0 𝐵(𝜄) Δ𝐵 4 Δ𝐵 3 12 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Our proposed predictor Denote N N � � ∆ A + ∆ A − ∆ A + = i , ∆ A − = i , i = 1 i = 1 We define the predictor A 0 x ( t ) − ∆ A + x − ( t ) − ∆ A − x + ( t ) x ( t ) ˙ = + B + d ( t ) − B − d ( t ) , ˙ A 0 x ( t ) + ∆ A + x + ( t ) + ∆ A − x − ( t ) x ( t ) = (2) + B + d ( t ) − B − d ( t ) , x ( 0 ) = x 0 , x ( 0 ) = x 0 Theorem (Inclusion property) The predictor (2) ensures x ( t ) ≤ x ( t ) ≤ x ( t ) . 13 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Stability Theorem (Stability) If there exist diagonal matrices P, Q, Q + , Q − , Z + , Z − , Ψ + , Ψ − , Ψ , Γ ∈ R 2 n × 2 n such that the following LMIs are satisfied: P + min { Z + , Z − } > 0 , Υ � 0 , Γ > 0 , Q + min { Q + , Q − } + 2 min { Ψ + , Ψ − } > 0 , where Υ = Υ( A 0 , ∆ A − , ∆ A + , Ψ − , Ψ + , Ψ) , then the predictor (2) is input-to-state stable with respect to the inputs d, d. 14 -Interval Prediction with Parametric Uncertainties- Edouard Leurent

Sketch of proof 1. Define the extended state vector as X = [ x ⊤ x ⊤ ] ⊤ 2. It follows the dynamics ˙ X ( t ) = A X ( t ) + R + X + ( t ) − R − X − ( t ) + δ ( t ) � A 0 � � 0 � � � 0 − ∆ A − ∆ A + 0 A = R + = , R − = 0 0 ∆ A + − ∆ A − 0 A 0 3. Consider a candidate Lyapunov function: V ( X ) = X ⊤ PX + X ⊤ Z + X + − X ⊤ Z − X − 4. V ( X ) is positive definite provided that P + min { Z + , Z − } > 0 , 5. Check on which condition we have ˙ V ( X ) ≤ 0 15 -Interval Prediction with Parametric Uncertainties- Edouard Leurent