Some examples of time-delay systems I. Fluid flow model for a - PowerPoint PPT Presentation

Some examples of time-delay systems

I. Fluid flow model for a congested router in TCP/AQM controlled network Model of collision-avoidance type: Hollot et al., IEEE TAC 2002 − W t W t R t 1 1 ( ) ( ( )) ɺ = − − W t p t R t W : window-size ( ) ( ( )) − R t R t R t ( ) 2 ( ( )) Q : queue length N : number of TCP sessions W t  ( ) − > N t C Q ( ) 0 R : round-trip-time  R t  ( ) ɺ C : link capacity = Q t  ( )   W t ( ) p : probability of packet mark   −  = N t C Q max ( ) , 0 , 0   Tp : propagation delay  R t  ( )   Q t ( ) = + R t T ( ) p C packet marking queue Q link c Bottleneck Sender Receiver router rtt R acknowledgement p = f ( Q ) Interpretation of AQM as a feedback control problem: We assume: - N constant, R is constant, p=K Q

Normalization of state and time − W t W t R 1 1 ( ) ( ) ɺ = − − W t K Q t R ( ) ( ) R R 2 W t  ( ) − > 4 parameters N C Q 0   R ɺ = Q t  ( ) K N R C W t , , ,   ( )  − =  N C  Q max , 0 , 0  R    Q old t ( ) = = new = w W q t ( ) , , N R 1 = − − − w ɺ t w t w t k q t ( ) 1 ( ) ( 1 ) ( 1 ) 2 parameters 2 RC − > w t c q = =  c k KN ( ) 0 , = q t ɺ  N ( ) ( ) − = w t c q  max ( ) , 0 , 0

Linearized model 1 = − − − w ɺ t w t w t k q t ( ) 1 ( ) ( 1 ) ( 1 ) 2 − >  w t c q ( ) 0 = q ɺ t  ( ) ( ) − = w t c q  max ( ) , 0 , 0 = 2 w q c * * Unique steady state solution ( , ) ( , ) kc 2 Linearization: kc 2 1 1 ɺ ɺ ɺ ɺ ~ ~ ~ ~ + + − + − = q t q t q t q t ( ) ( ) ( 1 ) ( 1 ) 0 c c 2 kc 2 1 1 − − + + + = t t e λ e λ λ 2 λ λ ( ) ( ) 0 c c 2

II. A car following system Car following model in a ring configuration Simplest model: k k-1 speed v k speed v k-1 Refinements: - taking multiple cars into account - distribution of the delay Possible choice for f: a gamma distribution with a gap 0.1 ξ τ − − f( ξ ) ∼ e T gap τ 0.05 τ T n ( , , ) three parameters: 0 0 2 4 6 8 10 ξ

Interpretation as a consensus protocol System consisting of p agents, each described by an integrator: = v t ɺ u t ( ) ( ), k k = = y t v t k p … ( ) ( ), 1, , k k Directed, time-invariant communication graph: Node set {1,…,p} ∈ ⇔ α ≠ k l E Set of vertices E: ( , ) 0 k l , �� α Weighted adjacency matrix : diagonal entries zero, non-diagonal entries k l , Strongly connected Consensus protocol : ∞ ∑ ( ) ∫ = α θ − θ − − θ θ = u t f y t y t d k p … ( ) ( ) ( ) ( ) , 1, , k k l l k , k l ∈ E ( , ) 0

III. Rotating cutting and milling machines Milling process = + −  x t ɺ A ω t x t B ω t x t τ t ( ) ( ) ( ) ( ) ( ( ))  = + Ω τ t δ f t τ  ( ) ( ) 0 � Successive passage of teeth ⇒ delay � Rotation of each tooth ⇒ periodic coefficients workpiece (fixed / translates) tool (rotates) � Successive passage of the same point Cutting process of the piece ⇒ delay � Orientation of tooth w.r.t. workpiece is fixed ⇒ constant coefficients Both cases: speed determines delay Workpiece tool (rotates) (fixed) unstable steady state chatter or oscillations of workpiece/tool irregular surface

Variable speed machines A measure to improve stability and prevent chatter: Fast modulation of rotational machine speed, N, around the nominal value (see work of Jayaram,Sexton,Stone, etc.) speed time 1 t τ ( ) ~ since N t ( ) Modulating the machine speed = modulating the delay in the model ! Stabilizing effect of delay variation !

IV. Heating system ( PhD Thesis Vyhlidal, CTU Prague, 2003) setpoint temperature to be controlled Linear system of dimension 6, 5 delays,, Goal of feedback: achieving asymptotic stability, and maximizing response time

System = − − η + − τ + − τ  T x t ɺ x t K x t K x t ( ) ( ) ( ) ( ) h h h h b a b u h set u ,  + −  q q  1 1  = − + − τ + − − − τ T x t ɺ x t x t K x t x t x t ( ) ( ) ( )  ( ) ( ) ( )   a a a c e a h a c e   2 2  = − + − τ T x ɺ t x t K x t  ( ) ( ) ( ) d d d d a d  = − − η + − τ T x t ɺ x t K x t ( ) ( ) ( )  c c c c c d c = −  x t ɺ x t x t ( ) ( ) ( ) e c set c ,   Control law (PI+ state feedback) T   = x K x x x x x   h set h a d c e ,

Computation of characteristic roots and stability regions

Operators associated to a delay equation ∑ m = + − τ ∈ n τ = τ x t ɺ A x t A x t x t ℝ ( ) ( ) ( ), ( ) , max i i i ϕ ϕ ϕ ϕ 0 = max i 1 i [ ] ϕ ∈ − τ n ℝ � Initial condition is a function segment ( ,0 , ), max − τ 0 max [ ] be the forward solution with initial condition ϕ and let ∈ − τ ∞ → ϕ t x t Let max , ( )( ) [ ] ϕ = + θ θ ∈ − τ x x t ( ) ( ), ,0 t max [ ] − τ n ℝ � ( ,0 , ), Reformulation of the DDE over max mapping abstract ‘ODE’ = ≥ = ∈ x t x t x x x � � � � ( ) , 0 , ( ) t t t 0 0 � (t) : solution (time-integration) � : infinitesimal generator of � (t) operator over interval t { ( ) = ϕ ∈ − τ ϕ − τ ɺ � � � ( ) ([ ,0]): continuous on ,0 and t ϕ θ = � max max ( ) ( )  m ∑ ϕ + θ + θ ≤ t t  ϕ = ϕ + ϕ − τ ( ), 0, ɺ A A  (0) (0) ( ) , i i i   + θ i = t 1  m ∑ ∫ ϕ + ϕ + ϕ − τ + θ > A s A s ds t � � ϕ = ϕ ɺ ϕ ∈ (0) ( ( ) )(0) ( ( ) )( ) , 0 � � �  , ( ). i i 0  i = 1 0

Spectral properties λ is a characteristic root if and only if it satisfies the characteristic equation   m ∑ Ae λτ − λ = λ = λ − − H H I A ( ) 0, ( ) : det ,  i  i 0   i = 1 or equivalently { } ∃ ∈ n n × v ℂ \ 0 : finite-dimensional nonlinear   m ∑ λ − − − λτ = I A Ae v  )  0 i eigenvalue problem i 0   = i 1 Properties H λ = ⇔ λ ∈ σ � ( ) 0 ( ) ve λθ θ [ ] σ = σ P A ∈ − τ � ( ) ( ), , ,0 eigenfunction max ( σ (.): spectrum, P σ (.): point-spectrum) infinite-dimensional linear eigenvalue problems for � and � (t) ( ) σ = σ t t � � ( ( )) exp ( ) ve λθ θ [ ] ∈ − τ λ ∈ σ ⇒ λ t ∈ σ e P t � � , ,0 ( ) ( ( )), eigenfunction max

100 1 50 Imaginary axis Imaginary axis 0 0 −50 −1 −100 −3 −2 −1 0 1 −1 0 1 1.5 Real axis Real axis exp(.) Eigenvalues of � (1) Characteristic roots, eigenvalues of � Mapping is not one-to-one But: characteristic roots can be obtained from σ ( � (t)) by computing also the corresponding eigenfunction

Two -stage approach to compute characteristic roots 1a. Discretize � or � (t) , with t fixed, into a matrix Discretizing � (t) - linear multi-step methods (Engelborghs et al.) - subspace iteration (Engelborghs at al) - spectral collocation (Verheyden et al.) - Chebychev expansion (Butcher, Bühler et al.) - semi-discretization (Stepan et al.) Discretizing � (Breda et al) 1b. Compute the (rightmost or dominant) eigenvalues of this matrix 2 . Correct the approximate characteristic roots with Newton iterations on the characteristic equation, up to the desired accuracy

Routine in the Matlab package DDE -BIFTOOL - Linear multi-step method to discretize � (h), combined with Lagrange interpolation to evaluate delayed terms - Newton correction - Automatic choice of discretization steplength h, to capture all the characteristic roots in a given half plane, possible + uncorrected roots o corrected roots

Pseudospectra and stability radii of nonlinear eigenvalue problems, with application to time -delay systems

Overview � Pseudospectra � Approaches to exploit structure of nonlinear eigenvalue problems � via structured matrix perturbations � by redefining pseudospectra Emphasis on computable expressions � Numerical examples � Concluding remarks

Pseudospectra d x  ε -pseudospectrum of an operator � = x �  (or system dt    1 Λ = Σ ∪ λ ∈ � λ > Σ ⋅ � � � �   ( ) ( ) : ( , ) , ( ) : spectrum ε ε   − λ = λ − A I � � 1 ( , ) ( ) : { } resolvent Λ = λ ∈Σ + δ = δ δ < ε � � � � � ( ) ( ) 0,forsome with ε computable as level sets of resolvent norm 50 50 (a) (b) ℑ ( λ ) ℑ ( λ ) 0 0 −50 −50 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 ℜ ( λ ) ℜ ( λ ) spectrum pseudospectra