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Worst-Case Analysis of Digital Control Loops with Uncertain Input/Output Timing (Benchmark Proposal) Maximilian Gaukler and Peter Ulbrich Friedrich-Alexander-Universitt Erlangen-Nrnberg (FAU) ARCH19, Montral, Canada April 15, 2019 M.

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Worst-Case Analysis of Digital Control Loops with Uncertain Input/Output Timing

(Benchmark Proposal) Maximilian Gaukler and Peter Ulbrich

Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)

ARCH’19, Montréal, Canada April 15, 2019

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Motivation

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Motivation Plant Controller

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Motivation Plant Controller Controller Design: input/output assumed periodic

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Motivation

Real-time computing system

Plant Controller Other applications and controllers Controller Design: input/output assumed periodic

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Motivation

Real-time computing system

Plant Controller Other applications and controllers Controller Design: input/output assumed periodic

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Motivation

Real-time computing system

Plant Controller Other applications and controllers Controller Design: input/output assumed periodic Modern Real-Time Systems:

  • Network / bus systems
  • Smart sensors
  • High complexity
  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Motivation

Real-time computing system

Plant Controller Other applications and controllers Controller Design: input/output assumed periodic Modern Real-Time Systems:

  • Network / bus systems
  • Smart sensors
  • High complexity

Strict timing is difficult!

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Motivation

Real-time computing system

Plant Controller Other applications and controllers

Input timing Output timing

Controller Design: input/output assumed periodic Modern Real-Time Systems:

  • Network / bus systems
  • Smart sensors
  • High complexity

Strict timing is difficult! Desired Alternative: Allow some timing deviation Formal safety guarantees?

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Timing Model: Periodic Timing Windows kT (k − 1)T (k + 1)T t dataflow

Sample y[k − 1] Compute u[k] Actuate u[k] Sample y[k] Compute u[k + 1] Actuate u[k +1] Sample y[k + 1] Actuate u[k −1]

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Timing Model: Periodic Timing Windows kT (k − 1)T (k + 1)T t dataflow

Sample y[k − 1] Compute u[k] Actuate u[k] Sample y[k] Compute u[k + 1] Actuate u[k +1] Sample y[k + 1] Actuate u[k −1]

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Timing Model: Periodic Timing Windows kT (k − 1)T (k + 1)T t dataflow

Sample y[k − 1] Compute u[k] Actuate u[k] Sample y[k] Compute u[k + 1] Actuate u[k +1] Sample y[k + 1] Actuate u[k −1]

∆t

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Timing Model: Periodic Timing Windows kT (k − 1)T (k + 1)T t dataflow

Sample y[k − 1] Compute u[k] Actuate u[k] Sample y[k] Compute u[k + 1] Actuate u[k +1] Sample y[k + 1] Actuate u[k −1]

∆t ∆t ∆t

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Formalization: Hybrid Automata

clock (τ, startOfCycle) Controller and Clock . . . Sample-and-Hold

with timing uncertainty

Sample-and-Hold

with timing uncertainty

Plant . . . Sample-and-Hold

with timing uncertainty

Sample-and-Hold

with timing uncertainty

unext,1 unext,m u1 um y1 yp yd,1 yd,p

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Formalization: Hybrid Automata

clock (τ, startOfCycle) Controller and Clock . . . Sample-and-Hold

with timing uncertainty

Sample-and-Hold

with timing uncertainty

Plant . . . Sample-and-Hold

with timing uncertainty

Sample-and-Hold

with timing uncertainty

unext,1 unext,m u1 um y1 yp yd,1 yd,p always ˙ τ = 1, ˙ xd = 0 −T/2 ≤ τ ≤ T/2 startOfCycle τ = T/2 τ ′ = −T/2, x′

d = fd(xd, yd)

T 2T

T 2

− T

2

t Clock (τ) Controller (xd) Controller output: unext = gd(xd)

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Formalization: Hybrid Automata

clock (τ, startOfCycle) Controller and Clock . . . Sample-and-Hold

with timing uncertainty

Sample-and-Hold

with timing uncertainty

Plant . . . Sample-and-Hold

with timing uncertainty

Sample-and-Hold

with timing uncertainty

unext,1 unext,m u1 um y1 yp yd,1 yd,p ˙ xp(t) = fp(xp(t), u(t), d(t)) y(t) = gp(xp(t), d(t)) d(t) ∈ D

  • Multiple inputs and outputs
  • Bounded disturbance and measurement noise
  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Formalization: Hybrid Automata

clock (τ, startOfCycle) Controller and Clock . . . Sample-and-Hold

with timing uncertainty

Sample-and-Hold

with timing uncertainty

Plant . . . Sample-and-Hold

with timing uncertainty

Sample-and-Hold

with timing uncertainty

unext,1 unext,m u1 um y1 yp yd,1 yd,p wait ˙

  • ut = 0

τ ≤ ∆t done ˙

  • ut = 0

∆t ≤ τ ≤ ∆t

  • ut′

= in startOfCycle

(T + ∆t) T (T + ∆t) 2T

t

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Benchmark Setup

1 System model: network of hybrid automata

  • Machine-readable, unambiguous
  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Benchmark Setup

1 System model: network of hybrid automata

  • Machine-readable, unambiguous

2 Verification goal: tight worst-case bounds (infinite-time reachable set)

  • Metric: bloating factor
  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Benchmark Setup

1 System model: network of hybrid automata

  • Machine-readable, unambiguous

2 Verification goal: tight worst-case bounds (infinite-time reachable set)

  • Metric: bloating factor

x2 x1 simulation analysis a b K = b a = “ upper bound from analysis worst observed in simulation ”

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Benchmark Setup

1 System model: network of hybrid automata

  • Machine-readable, unambiguous

2 Verification goal: tight worst-case bounds (infinite-time reachable set)

  • Metric: bloating factor

3 Example systems

  • Linear, no disturbance
  • From 1D examples to a simplified 3-axis quadrocopter controller
  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Benchmark Setup

1 System model: network of hybrid automata

  • Machine-readable, unambiguous

2 Verification goal: tight worst-case bounds (infinite-time reachable set)

  • Metric: bloating factor

3 Example systems

  • Linear, no disturbance
  • From 1D examples to a simplified 3-axis quadrocopter controller

4 Experiments with SpaceEx: Success only for trivial examples

Reachable set over time: 1D, small uncertainty 1D, large uncertainty × 3D, perfect timing (!)

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Summary and Outlook

  • Problem: Timing uncertainties in digital control
  • Hard to avoid
  • Verification is challenging, but of high practical relevance
  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Summary and Outlook

  • Problem: Timing uncertainties in digital control
  • Hard to avoid
  • Verification is challenging, but of high practical relevance
  • Is a pure hybrid-automata approach suitable here?
  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Summary and Outlook

  • Problem: Timing uncertainties in digital control
  • Hard to avoid
  • Verification is challenging, but of high practical relevance
  • Is a pure hybrid-automata approach suitable here?
  • Future work: “non-hybrid” alternatives
  • Continuous-time abstraction: continuization
  • Discrete-time: LMI-based robust stability
  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Summary and Outlook

  • Problem: Timing uncertainties in digital control
  • Hard to avoid
  • Verification is challenging, but of high practical relevance
  • Is a pure hybrid-automata approach suitable here?
  • Future work: “non-hybrid” alternatives
  • Continuous-time abstraction: continuization
  • Discrete-time: LMI-based robust stability

Can your tool solve the benchmark? http://qronos.de → Files and code (GPLv3)

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Appendix

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Experimental Results 1D example np nd m p timing SpaceEx tSE KSE A2 1 1 1 1 varying (negligible)

  • 1 s

1.001 A1 1 1 1 1 varying (small)

  • 1 s

1.010 A3 1 1 1 1 varying (medium)

  • 2 s

1.059 A4 1 1 1 1 varying (large) × error (GLPK) — — A5 2 2 2 2 varying (like A3) × crash (GLPK) — — 3D, trivial (stable, negligible controller) B1 3 2 2 1 varying

  • 16 s

1.097 1-axis quadrotor angular rate control C1 1 2 1 1 constant × timeout — — C2 1 2 1 1 varying × crash — — 3-axis quadrotor angular rate control D1 3 6 4 3 constant × diverging — — D2 3 6 4 3 varying × crash — —

  • M. Gaukler: Digital Control Loops with Uncertain Input/Output Timing

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Experiments: Discussion

  • Structural problem: Failures even with ∆t = 0

Tools optimized for mostly-continuous systems? = here: many discrete transitions, “unstable” inbetween

  • Dimension problematic: 2(#inputs+#outputs) discrete states
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