Applications of Verified Methods for Solving Non-smooth Initial - - PowerPoint PPT Presentation

▶
applications of verified methods for solving non smooth
SMART_READER_LITE
LIVE PREVIEW

Applications of Verified Methods for Solving Non-smooth Initial - - PowerPoint PPT Presentation

Feb 06, 2024 99 likes •425 views

Motivation Non-smooth IVPs Example Applications Conclusions Applications of Verified Methods for Solving Non-smooth Initial Value Problems Ekaterina Auer, Andreas Rauh University of Duisburg-Essen, University of Rostock June 14, 2011

slide-1
SLIDE 1

Motivation Non-smooth IVPs Example Applications Conclusions

Applications of Verified Methods for Solving Non-smooth Initial Value Problems

Ekaterina Auer, Andreas Rauh

University of Duisburg-Essen, University of Rostock

June 14, 2011 (updated)

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 1

slide-2
SLIDE 2

Motivation Non-smooth IVPs Example Applications Conclusions

Non-smooth Models in Engineering

Friction Contact dynamics Besides: saturation effects, ensuring good numerical behavior, etc.

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 2

slide-3
SLIDE 3

Motivation Non-smooth IVPs Example Applications Conclusions

Implicitly Non-smooth Models: Traps in Code

Trap Example IF-THEN-ELSE Force: F ≤ 0 SWITCH Muscle activation function: 0 ≤ a(t) = A1e−c1(t−t1) + A2e−c2(t−t2) ≤ 1 |x| Hysteresis: ˙ ω (t) =ρ ·

  • v (t) − σ · |v (t)| · |ω (t)|ν−1 · ω (t)

+ (σ − 1) · v (t) · |ω (t)|ν) sgnx Friction: F(v) =sgn(v) · F + µ · v

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 3

slide-4
SLIDE 4

Motivation Non-smooth IVPs Example Applications Conclusions

Biomechanical Context: MobileBody

Chief coordination: Prof. A. Kecskem´ ethy (UDE)

Gait lab MRT X-Ray MobileBody Assistance during OP planning OP assessment rehabilitation

Our major task: Characterization of uncertain parameters Non-smoothness in: Muscle models, stabilization of stance

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 4

slide-5
SLIDE 5

Motivation Non-smooth IVPs Example Applications Conclusions

Fuel Cells Context: Development of VeriCell

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 5

Cooperation: Chair of Mechatronics, Rostock Gas supply Preheater SOFC stack module (30 fuel cells) Our task: Control design, verified simulation environment Non-smoothness: Saturation effects in reaction kinetics, etc.

slide-6
SLIDE 6

Motivation Non-smooth IVPs Example Applications Conclusions

Background: Verified Methods for Non-smooth Systems

Description of a non-smooth IVP ւ ց Analytical Graph-like x′ = f+(x), h(x(t), t) < 0 f−(x), h(x(t), t) > 0 Rihm (1993), Rauh (2006), Eggers (2008), Mahmoud and Chen (2008) Nedialkov and Mohrenschildt (2002) Verified non-smooth optimization: Slopes, generalized gradients ...

Ratz (1995), Kearfott (2004), Schnurr (2007), ...

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 6

slide-7
SLIDE 7

Motivation Non-smooth IVPs Example Applications Conclusions

Problem Definition

Interval IVP: x′ = f(x), x(0) ∈ [x0] where f : D ⊂ Rn → Rn or D ⊂ IRn → IRn and is given in algorithmic representation:    τi(x) = gi(x) = xi, i = 1 . . . n τi(x) = gi(τ1(x), . . . , τi−1(x)), i = n + 1 . . . l, gi ∈ SEO ∪ SPW . SEO = {+, −, ∗, /, sin, cos, . . . } and SPW are piecewise smooth functions

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 7

slide-8
SLIDE 8

Motivation Non-smooth IVPs Example Applications Conclusions

Definition of Piecewise Functions

φi,τν,c(τi0(x), . . . , τip(x)):

           τi0(x), c−1 = −∞ < τν(x) ≤ c0, τi1(x), c0 < τν(x) ≤ c1, . . . . . . τip−1, (x) cp−2 < τν(x) ≤ cp−1, τip(x), cp−1 < τν(x) < cp = +∞

c0 c1 c2 x f(x)

An interval extension of φ over X (φ(X)): τi(X), if X ⊆ (ci−1, ci), j−1

k=i+1τk([ck−1, ck])∪τi([x, ci])∪τj([cj−1, x]),

if X ⊆ (ci−1, cj)

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 8

slide-9
SLIDE 9

Motivation Non-smooth IVPs Example Applications Conclusions

Definition of the Derivative

An interval extension of φ′ over X (φ′(X))        τ ′

i(X),

if X ⊆ (ci−1, ci), j−1

k=i+1τ ′ k([ck−1, ck])∪τ ′ i([x, ci])∪τ ′ j([cj−1, x])

∪ rest, if X ⊆ (ci−1, cj), where REST depends on: – how many switching points X contains, – whether φ is continuous, if we want the mean value theorem to hold.

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 9

slide-10
SLIDE 10

Motivation Non-smooth IVPs Example Applications Conclusions

Suppose we have a single switching point (IF-THEN-ELSE) φ(x) =

  • φ0(x),

x < c0, φ1(x), x > c0. then REST is (φ′

0([x, c0]) + φ′ 1([c0, x])) · [0, 1] if φ is continuous,

φ1(c0) − φ0(c0) [c0, x] − x0 + (φ′

0([x, c0]) + φ′ 1([c0, x])) · [0, 1]

  • ∪

φ0(c0) − φ1(c0) [x, c0] − x0 + (φ′

0([x, c0]) + φ′ 1([c0, x])) · [0, 1]

  • if φ is discontinuous.

Problem: We need x0 to avoid enclosures containing ∞ afap

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 10

slide-11
SLIDE 11

Motivation Non-smooth IVPs Example Applications Conclusions

Properties of φ′(X)

1 If the derivative of φ exists for x ∈ X, then φ′(x) ∈ φ′(X) 2 The slope δφ(X, x0) ⊆ φ′(X) 3 The mean value theorem holds:

φ(x) = φ(x0) + φ′(ξ)(x − x0) ∈ φ(x0) + φ′(X)(X − x0)

4 If φ is continuous (τij(cj) = τij+1(cj), 0 ≤ j < p), then

f(x) is continuous.

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 11

slide-12
SLIDE 12

Motivation Non-smooth IVPs Example Applications Conclusions

Solution Definitions

Two situations: ւ ց (a) f is discontinuous only in t (b) f is discontinuous in t, x τν(x) = t or τij(cj) = τij+1(cj) Solution: x(t) = x0 +

t

  • f(x(s))ds, x0 ∈ [x0]

Depends on the application

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 12

slide-13
SLIDE 13

Motivation Non-smooth IVPs Example Applications Conclusions

ValEncIA-IVP1 For Non-smooth IVPs

General approach in ValEncIA: A posteriori x(t) ∈ [x (t)]

verified state enclosure

:= xapp (t)

non-verified approximation

+ [R (t)]

error bounds

Conditions for the right side:

1 continuous 2 Lipschitz 1VALidation of state ENClosures using Interval Arithmetic for Initial V alue

Problems

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 13

slide-14
SLIDE 14

Motivation Non-smooth IVPs Example Applications Conclusions

ValEncIA-IVP For Non-smooth IVPs (Cont.)

The algorithm for 0 ≤ t ≤ T:

1 Start with [x(0)], xapp(t), [R(0)] 2 k = 1 . . . kmax or while [ ˙

R(k+1)([0, T])]!=[ ˙ R(k)([0, T])] Compute [ ˙ R(k+1)([0, T])] := ˙ xapp+f([x(k)]), (MVT) where [x(k)] := [x(k)([0, T])] If [ ˙ R(k+1)([0, T])] ⊆ [ ˙ R(k)([0, T])] then

  • R(k+1)([0, T])
  • :=

[R(0)] + [ ˙ R(k+1)([0, T])][0, T]

  • x(k+1)([0, T])
  • :=

xapp + [R(k+1)([0, T])] Differences ((non-)smooth): Derivative definition, the fixed point theorem To-do-list: Discontinuities in x for the right side

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 14

slide-15
SLIDE 15

Motivation Non-smooth IVPs Example Applications Conclusions

Implementation Issues: Class pwFunc

Remarks on f(x) f′(X) is obtained with pwFunc pwFunc uses FADBAD++ and

  • verloads hull, d()

f′(X) encloses both left and right derivatives pwFunc is plugged into ValEncIA Class declaration

template<class T> class pwFunc{ public: typedef T (*ptrFct)(const T& x); pwFunc(const vector<interval>& p, const vector<ptrFct>& f); T operator()(const T& x) { return getValueAtX(x);} private: vector< ptrFct > functions; vector<interval> points; vector<T> subintervals; T getValueAtX(const T& x); void generateSubintervals(const T& x); };

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 15

slide-16
SLIDE 16

Motivation Non-smooth IVPs Example Applications Conclusions

Implementation Example: A Discontinuous Function

template <class T> T f1(const T& x){ return -1+x;} template <class T> T f2(const T& x){ return 1+x;} template<class T> T ff(const T& a){ vector<INTERVAL> p; p.push back(0); vector<pwFunc<T>::ptrFct> functions; functions.push back(&f1<T>);functions.push back(&f2<T>); pwFunc<T> fp(p, functions); return fp(a); } ff([-1,2]);

Equation:

Ff (v) = −1.0 + x x < 0 +1.0 + x x > 0

Result:

[-2,3]([1,6])

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 16

slide-17
SLIDE 17

Motivation Non-smooth IVPs Example Applications Conclusions

A Mechanical System with Friction and Hysteresis

˙ x (t) = 1

  • x (t) +
  • 1

m (Fa (t) − Ff (x2))

  • x =
  • x1

x2 T Friction force: Ff (x2) = − [Fs] + [µ] · x2 for S1 = true

  • r

S2 = true + [Fs] + [µ] · x2 for S4 = true

  • r

S5 = true with the static friction Ff (x2) ∈ [F max

s

] :=

  • −F s ; F s
  • for

S3 = true

S1 = {x < 0, ω ≥ 0}, S2 = {x < 0, ω < 0}, S3 = {x = 0}, S4 = {x > 0, ω ≥ 0}, S5 = {x < 0, ω > 0}

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 17

slide-18
SLIDE 18

Motivation Non-smooth IVPs Example Applications Conclusions

A Mechanical System with Friction and Hysteresis (Cont.)

Accelerating force: Fa (t) := u (t) − φ (x1 (t) , ω (t)) control variable u (t) provided by an actuator restoring spring force φ (x1 (t) , ω (t)) = κxx1 + κωω Restoring spring force with hysteresis (the Bouc-Wen model): ˙ ω (t) =ρ ·

  • x2 (t) − σ · |x2 (t)| · |ω (t)|ν−1 · ω (t)

+ (σ − 1) · x2 (t) · |ω (t)|ν)

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 18

slide-19
SLIDE 19

Motivation Non-smooth IVPs Example Applications Conclusions

Automaton-Based Method: Rauh et al.

sliding friction forward motion S4 sliding friction forward motion S5 sliding friction backward motion S2 static friction no motion S3 sliding friction backward motion S1 T 1

4 ∪ T 2 4 ∪ T 1 5 ∪ T 2 5

state transitions of the T 1

1 ∪ T 2 1 ∪ T 1 2 ∪ T 2 2

T 1

1

T 2

2

T 2

1

T 1

2

T 4

1 ∪ T 5 1 ∪ T 4 2 ∪ T 5 2

T 1

3 ∪ T 2 3

T 3

1 ∪ T 3 2

T 4

4

T 5

5

T 4

5

T 5

4

T 4

4 ∪ T 5 4 ∪ T 4 5 ∪ T 5 5

T 3

3

T 3

4 ∪ T 3 5

Bouc-Wen hysteresis model T 4

3 ∪ T 5 3

  • A. Rauh, Ch. Siebert, H. Aschemann, Verified Simulation and Optimization of

Dynamic Systems with Friction and Hysteresis, ENOC 2011, Rome, Italy, 2011

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 19

slide-20
SLIDE 20

Motivation Non-smooth IVPs Example Applications Conclusions

Idea of the Algorithm

1 Detection of all possible points of time at which transition

conditions T j

i (x, u) are activated

= ⇒ Validated enclosures of switching times as well as state variables

2 Detection of deactivation of model states

= ⇒ Computation of tight enclosures of state variables

3 Algorithm relying on Taylor-series method with a priori

enclosures determined by the Picard iteration to detect activation of switching conditions

  • A. Rauh, M. Kletting, et al.: Interval Methods for Simulation of Dynamical

Systems with State-Dependent Switching Characteristics, Proc. of the IEEE CCA 2006, pp. 355–360

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 20

slide-21
SLIDE 21

Motivation Non-smooth IVPs Example Applications Conclusions

Comparison: pwFunc+ValEncIA vs. Rauh et al.

κx = x1(0) = x2(0) = −0.1 ω(0) = −0.001 u(t) = 0.01 κx = 0.001 x1(0) = x2(0) = ω(0) = 0.001 u(t) = 2 sin (3t)

  • 0.2
  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 0.2 0.25 0.3 2 4 6 8 10 Velocity (m/s) Time (s) point (lower bound) point (upper bound) ValEncIA+pwFunc Matlab simulation

  • 1
  • 0.5

0.5 1 1.5 2 2 4 6 8 10 Velocity (m/s) Time (s) ValEncIA+pwFunc Matlab simulation

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 21

slide-22
SLIDE 22

Motivation Non-smooth IVPs Example Applications Conclusions

The PT Muscle Model

Activation function: 0 ≤ a(t) = A1e−c1(t−t1) + A2e−c2(t−t2) ≤ 1 PT muscle force : F ≤ 0 Thigh length: p = 0.45 ± 0.1% Dynamics in SmartMOBILE with ValEncIA-IVP with and without piecewise functions:

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 knee angle enclosures (rad) time (s) PT model without pwFunc PT model with pwFunc

Improvement in break down times!

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 22

slide-23
SLIDE 23

Motivation Non-smooth IVPs Example Applications Conclusions

Stabilization of Stance∗

Human skeleton ⇓ Mass parameters Foot contact Hunt-Crossley contact ⇓ Contact parameters Stance stabilizer PID controller Q = Kp · ϕ+ Kd · dϕ

dt +

Ki

  • ϕdt

⇓ Force parameters

∗ Modeling: X. Liu, H. Albassam (UDE)

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 23

slide-24
SLIDE 24

Motivation Non-smooth IVPs Example Applications Conclusions

Stance Stabilization: Equations of Motion

M(q; t)q′′ + b(q, q′; t) = Q(q, q′; t)

  • f

, dof=26 Parameters of interest: mpelvis, px, Fx, mrfemur = 10.34kg

[f1 f2 f4 f6] = [[0, 200] [−940.00, −595.69] [−31.89, 31.89] [−50.17, 45.49]]

Sensitivity (interval/nominal) mpelvis px mrfemur Fx f1 0.0 0.0 0.0 1 f2

  • 9.8

0.0

  • 9.8

0.0 f4 [-0.5,0.5]/0 0.0 0.7848 0.0 f6 [-9.81,0.5]/-0.25 [-637.66,-343.34]/-490.5 0.5 0.0

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 24

slide-25
SLIDE 25

Motivation Non-smooth IVPs Example Applications Conclusions

Stance Stabilization: To-Do-List

Major challenge: Foot contact problem Advantages: e.g. contact area modeling with intervals Necessary: A solver similar to Rihm’s for dynamics A solver similar to Chen’s for equilibria

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 25

slide-26
SLIDE 26

Motivation Non-smooth IVPs Example Applications Conclusions

Outlook: Control-Oriented Modeling and Simulation for SOFC Fuel Cells

Three subprocesses: ϑFC pH 2, pH 2O, pO2 ˙ mH 2, ˙ mH 2O, ˙ mN 2, ϑAG ˙ mCG ,ϑCG ϑFC U FC I Thermo- dynamics Fluid mechanics Electro- chemistry Elec. Load

  • A. Rauh, Th. D¨
  • tschel, H. Aschemann: Experimental Parameter Identification

for a Control-Oriented Model of the Thermal Behavior of High-Temperature Fuel Cells MMAR 2011, accepted.

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 26

slide-27
SLIDE 27

Motivation Non-smooth IVPs Example Applications Conclusions

Outlook: Switchings in SOFC Modeling

Hysteresis in the electro-chemical subsystem

Activation losses Concentration losses Storage of charge carriers in the double layer

Detection of predominant influence factors

(Partial) Pressures of fuel gas and air Temperatur Electric load conditions

Modeling of disturbances, especially electric load drop-off

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 27

slide-28
SLIDE 28

Motivation Non-smooth IVPs Example Applications Conclusions

Outlook: Switchings in SOFC Modeling (Cont.)

Locally restricted validity of system models Piecewise (polynomial) approximation of

(Specific) Heat capacity Heat conductivity Thermal resistances Enthalpy

Physical limitation of the range of state variables

Non-negativity of partial pressures Temperatures Volume and mass flow rates

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 28

slide-29
SLIDE 29

Motivation Non-smooth IVPs Example Applications Conclusions

Outlook: Switchings in SOFC Modeling (Cont.)

Saturation in actuator dynamics

Gas preheaters for fuel gas and air Mass flow controllers for fuel gas and air

Introduction of hysteresis as an additional degree of freedom in controller design to reduce actuator Goal: Reliable detections of possible limit cycles Goal: Reliable prevention of limit cycles

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 29

slide-30
SLIDE 30

Motivation Non-smooth IVPs Example Applications Conclusions

Outlook: Switchings in SOFC Modeling (Cont.)

General variable structure control and estimation approaches Nonlinear control design

Model predictive control with active avoidance of limit cacles Sliding mode controllers Sliding mode observers

Discrete-time implementation with piecewise constant signals (control and estimated values) Necessity to compute derivatives of at least first order during sensitivity analysis

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 30

slide-31
SLIDE 31

Motivation Non-smooth IVPs Example Applications Conclusions

Conclusions

Results: – Implementation of a simple extension for ValEncIA to work with non-smooth functions – A proof for continuous functions – Simulation for a system with friction and hysteresis – Results similar to Rauh Future work: – Application to stance stabilization – Application to SOFC

  • E. Auer, A. Rauh

University of Duisburg-Essen Applications of Verified Methods for Non-smooth IVPs 31