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Safety control with performance guarantees of cooperative systems - - PowerPoint PPT Presentation

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Cooperative systems Centralized symbolic control Compositional approach Safety control with performance guarantees of cooperative systems using compositional abstractions Pierre-Jean Meyer Antoine Girard Emmanuel Witrant Universit e

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Cooperative systems Centralized symbolic control Compositional approach

Safety control with performance guarantees of cooperative systems using compositional abstractions

Pierre-Jean Meyer Antoine Girard Emmanuel Witrant

Universit´ e Grenoble-Alpes

ADHS’15, October 16th 2015

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 1 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Outline

1

Cooperative control system

2

Centralized symbolic control

3

Compositional approach

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 2 / 25

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Cooperative systems Centralized symbolic control Compositional approach

System description

Nonlinear control system: ˙ x = f (x, u, w) Trajectories: x = Φ(·, x0, u, w) x: state u: control input w: disturbance input x, u, w: time functions

x t x0 x = Φ(·, x0, u, w) x(t) f(x(t), u(t), w(t))

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 3 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Cooperative system

Definition (Cooperativeness)

The system is cooperative if Φ preserves the componentwise inequality: u ≥ u′, w ≥ w′, x0 ≥ x′

0 ⇒ ∀t ≥ 0, Φ(t, x, u, w) ≥ Φ(t, x′, u′, w′)

u, w t u u′ x t x′ Φ(·, x0, u, w)

⇒

w w′ x0 Φ(·, x′

0, u′, w′)

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 4 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Bounded inputs

Control and disturbance inputs bounded in intervals: ∀t ≥ 0, u(t) ∈ [u, u], w(t) ∈ [w, w] = ⇒ ∀t ≥ 0, Φ(t, x0, u, w) ∈ [Φ(t, x0, u, w), Φ(t, x0, u, w)]

x t x0 Φ(·, x0, u, w) Φ(·, x0, u, w) Φ(·, x0, u, w)

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 5 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Outline

1

Cooperative control system

2

Centralized symbolic control

3

Compositional approach

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 6 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Abstraction-based synthesis

x+ = f(x, u, w) x+ = f(x, u, w) Controller w x u Uncontrolled Controlled Continuous state Synthesis

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 7 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Abstraction-based synthesis

u1 u1 u2 u2 x+ = f(x, u, w) Discrete state Uncontrolled Controlled Continuous state Abstraction

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 7 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Abstraction-based synthesis

u1 u1 u2 u2 x+ = f(x, u, w) Discrete state Uncontrolled Controlled Continuous state u1 u1 Abstraction Synthesis

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 7 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Abstraction-based synthesis

u1 u1 u2 u2 x+ = f(x, u, w) x+ = f(x, u, w) Controller w x u Discrete state Uncontrolled Controlled Continuous state u1 u1 Abstraction Synthesis Refining

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 7 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Transition systems

S = (X, U, − →) Set of states X Set of inputs U Transition relation − → Trajectories: x1

u1

− → x2

u2

− → x3

u3

− → . . .

u x x′ u′

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 8 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Transition systems

S = (X, U, − →) Set of states X Set of inputs U Transition relation − → Trajectories: x1

u1

− → x2

u2

− → x3

u3

− → . . .

u x x′ u′

Sampled dynamics (sampling τ) X = Rn U = [u, u] x

u

− → x′ ⇐ ⇒ ∃w : [0, τ] → [w, w] | x′ = Φ(τ, x, u, w) Safety specification in [x, x] ⊆ Rn

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 8 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Abstraction

Discretization of the control space [u, u] Partition P of the interval [x, x] into symbols

s

s s x x

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 9 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Abstraction

Discretization of the control space [u, u] Partition P of the interval [x, x] into symbols Over-approximation of the reachable set (cooperativeness) Φ(τ, s, u, w) Φ(τ, s, u, w)

s

s s x x

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 9 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Abstraction

Discretization of the control space [u, u] Partition P of the interval [x, x] into symbols Over-approximation of the reachable set (cooperativeness) Intersection with the partition

s u u

Obtain a finite abstraction Sa = (Xa, Ua, − →

a )

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 9 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Alternating simulation

Definition (Alternating simulation relation)

H : X → Xa is an alternating simulation relation from Sa to S if: ∀ua ∈ Ua, ∃u ∈ U | x

u

− → x′ in S = ⇒ H(x)

ua

− →

a

H(x′) in Sa

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 10 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Alternating simulation

Definition (Alternating simulation relation)

H : X → Xa is an alternating simulation relation from Sa to S if: ∀ua ∈ Ua, ∃u ∈ U | x

u

− → x′ in S = ⇒ H(x)

ua

− →

a

H(x′) in Sa

Proposition

The map H : X → Xa defined by H(x) = s ⇐ ⇒ x ∈ s is an alternating simulation relation from Sa to S: ∀ua ∈ Ua ⊆ U | x

ua

− → x′ in S = ⇒ H(x)

ua

− →

a

H(x′) in Sa

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 10 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Safety synthesis

Specification: safety of Sa in P (the partition of the interval [x, x]) FP(Z) = {s ∈ Z ∩ P | ∃ u, ∀ s

u

− →

a

s′, s′ ∈ Z}

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 11 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Safety synthesis

Specification: safety of Sa in P (the partition of the interval [x, x]) FP(Z) = {s ∈ Z ∩ P | ∃ u, ∀ s

u

− →

a

s′, s′ ∈ Z} Fixed-point Za of FP reached in finite time Za is the maximal safe set for Sa, associated with the safety controller: Ca(s) = {u | ∀ s

u

− →

a

s′, s′ ∈ Za}

Theorem

Ca is a safety controller for S in Za.

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 11 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Performance criterion

Minimize on a trajectory (x0, u0, x1, u1, . . . ) of S:

+∞

  • k=0

λkg(xk, uk) with a cost function g and a discount factor λ ∈ (0, 1)

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 12 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Performance criterion

Minimize on a trajectory (x0, u0, x1, u1, . . . ) of S:

+∞

  • k=0

λkg(xk, uk) with a cost function g and a discount factor λ ∈ (0, 1) Cost function on Sa: ga(s, u) = max

x∈s g(x, u)

Focus the optimization on a finite horizon of N sampling periods Accurate approximation if λN+1 ≪ 1

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 12 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Optimization

Dynamic programming algorithm: JN

a (s) =

min

u∈Ca(s) ga(s, u)

Jk

a (s) =

min

u∈Ca(s)

 ga(s, u) + λ max

s

u

− →

a s′ Jk+1

a

(s′)   , ∀k < N J0

a(s) is the worst-case minimization of N

  • k=0

λkga(sk, uk)

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 13 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Optimization

Dynamic programming algorithm: JN

a (s) =

min

u∈Ca(s) ga(s, u)

Jk

a (s) =

min

u∈Ca(s)

 ga(s, u) + λ max

s

u

− →

a s′ Jk+1

a

(s′)   , ∀k < N J0

a(s) is the worst-case minimization of N

  • k=0

λkga(sk, uk) Receding horizon controller: C ∗

a (s) = arg min u∈Ca(s)

 ga(s, u) + λ max

s

u

− →

a s′ J1

a(s′)

 

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 13 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Performance guarantees

Theorem

Let (x0, u0, x1, u1, . . . ) be a trajectory of S controlled with C ∗

a .

Let s0, s1, . . . such that xk ∈ sk, for all k ∈ N. Then,

+∞

  • k=0

λkg(xk, uk) ≤

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 14 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Performance guarantees

Theorem

Let (x0, u0, x1, u1, . . . ) be a trajectory of S controlled with C ∗

a .

Let s0, s1, . . . such that xk ∈ sk, for all k ∈ N. Then,

+∞

  • k=0

λkg(xk, uk) ≤ J0

a(s0) +

Worst-case minimization on finite horizon:

. . . . . . 1 N N + 1

N

  • k=0

λkga(sk, uk) ≤ J0

a(s0)

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 14 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Performance guarantees

Theorem

Let (x0, u0, x1, u1, . . . ) be a trajectory of S controlled with C ∗

a .

Let s0, s1, . . . such that xk ∈ sk, for all k ∈ N. Then,

+∞

  • k=0

λkg(xk, uk) ≤ J0

a(s0) + λN+1

1 − λMa Worst-case minimization of each remaining steps (receding horizon):

. . . . . . 1 N N + 1 ga(sk, uk) ≤ max

s∈Za

min

u∈Ca(s) ga(s, u) = Ma

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 14 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Outline

1

Cooperative control system

2

Centralized symbolic control

3

Compositional approach

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 15 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Compositional synthesis

x+ = f(x, u, w) x+ = f(x, u, w) Controller w x u Subsystems Uncontrolled Controlled Whole system Decomposition

z+ 1 = g1(z1, v1, d1) z+ 2 = g2(z2, v2, d2) z+ 3 = g3(z3, v3, d3)

Controller composition Abstraction and synthesis

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 16 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Decomposition

Decomposition into m subsystems: Partition (I1, . . . , Im) of the state dimensions {1, . . . , n}

1 2 3 4 5 . . . n − 1 n I1 I2 I3 Im

Partition (J1, . . . , Jm) of the input dimensions {1, . . . , p}

1 2 3 4 5 . . . p − 1 p J1 J2 J3 Jm

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 17 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Decomposition

Decomposition into m subsystems: Partition (I1, . . . , Im) of the state dimensions {1, . . . , n}

1 2 3 4 5 . . . n − 1 n I1 I2 I3 Im K1

Partition (J1, . . . , Jm) of the input dimensions {1, . . . , p}

1 2 3 4 5 . . . p − 1 p J1 J2 J3 Jm L1

Control the states xI1 using the inputs uJ1 with disturbances xK1 and uL1

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 17 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Abstraction

Symbolic abstraction Si = (Xi, Ui, − →

i ) of subsystem i ∈ {1, . . . , m}:

Classical method, but with an assume-guarantee obligation:

A/G Obligation (Ki)

Unobserved states: xKi ∈ [xKi, xKi]

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 18 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Abstraction

Symbolic abstraction Si = (Xi, Ui, − →

i ) of subsystem i ∈ {1, . . . , m}:

Classical method, but with an assume-guarantee obligation:

A/G Obligation (Ki)

Unobserved states: xKi ∈ [xKi, xKi] xK1 ∈ [xK1, xK1] xKm ∈ [xKm, xKm] xI1 ∈ [xI1, xI1] xIm ∈ [xIm, xIm] . . . . . . Assumptions Composition x ∈ [x, x] Guarantees Abstractions and safety syntheses

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 18 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Synthesis

Safety synthesis in the partition of [xIi, xIi]: maximal safe set: Zi ⊆ Xi safety controller: Ci : Zi → 2Ui Performances optimization: cost function gi(sIi, uJi), with ga(s, u) ≤

m

  • i=1

gi(sIi, uJi) deterministic controller: C ∗

i : Zi → Ui

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 19 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Safety

Composition of safe sets and safety controllers: Zc = Z1 × · · · × Zm ∀s ∈ Zc, Cc(s) = C1(sI1) × · · · × Cm(sIm)

Theorem

Cc is a safety controller for S in Zc.

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 20 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Safety

Composition of safe sets and safety controllers: Zc = Z1 × · · · × Zm ∀s ∈ Zc, Cc(s) = C1(sI1) × · · · × Cm(sIm)

Theorem

Cc is a safety controller for S in Zc.

Proposition (Safety comparison)

Zc ⊆ Za.

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 20 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Performance guarantees

∀s ∈ Zc, C ∗

c (s) = (C ∗ 1 (sI1), . . . , C ∗ m(sIm))

Let Mi = max

si∈Zi

min

ui∈Ci(si) gi(si, ui)

Theorem

Let (x0, u0, x1, u1, . . . ) be a trajectory of S controlled with C ∗

c .

Let s0, s1, . . . such that xk ∈ sk, for all k ∈ N. Then,

+∞

  • k=0

λkg(xk, uk) ≤

m

  • i=1

J0

i (s0 Ii ) + λN+1

1 − λ

m

  • i=1

Mi

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 21 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Performance guarantees

∀s ∈ Zc, C ∗

c (s) = (C ∗ 1 (sI1), . . . , C ∗ m(sIm))

Let Mi = max

si∈Zi

min

ui∈Ci(si) gi(si, ui)

Theorem

Let (x0, u0, x1, u1, . . . ) be a trajectory of S controlled with C ∗

c .

Let s0, s1, . . . such that xk ∈ sk, for all k ∈ N. Then,

+∞

  • k=0

λkg(xk, uk) ≤

m

  • i=1

J0

i (s0 Ii ) + λN+1

1 − λ

m

  • i=1

Mi

Proposition (Guarantees comparison)

∀s ∈ Zc, J0

a(s) + λN+1

1 − λMa ≤

m

  • i=1

J0

i (sIi) + λN+1

1 − λ

m

  • i=1

Mi

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 21 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Complexity

n: state space dimension p: control space dimension αx ∈ N: number of symbols per dimension in the state partition αu ∈ N: number of controls per dimension in the input discretization | · |: cardinality of a set Method Centralized Compositional Complexity αn

xαp u m

  • i=1

α|Ii|

x α|Ji| u

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 22 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Complexity example

Application to temperature control 4-room building Each room equipped with one fan n = 4 states p = 4 control inputs Centralized (4D) Compositional (4 ∗ 1D) Precisions of abstraction αx = 10 αx = 20 αu = 4 αu = 9 Computation time > 2 days 1.1 second

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 23 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Conclusions and perspectives

The compositional approach provides: Similar safety and performance results to the centralized method, although weaker due to the loss of information The possibility of a significant complexity reduction = ⇒ Tradeoff between the accuracy and the complexity reduction

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 24 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Conclusions and perspectives

The compositional approach provides: Similar safety and performance results to the centralized method, although weaker due to the loss of information The possibility of a significant complexity reduction = ⇒ Tradeoff between the accuracy and the complexity reduction Perspectives Extension of the symbolic compositional approach

to non-cooperative systems to other specifications than safety

Adaptive symbolic control framework:

measure the disturbance; tight estimation of its future bounds synthesize compositional controller on the more accurate abstraction apply controller until the next measure

= ⇒ increased precision and robustness, local cooperativeness

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 24 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Safety control with performance guarantees of cooperative systems using compositional abstractions

Pierre-Jean Meyer Antoine Girard Emmanuel Witrant

Universit´ e Grenoble-Alpes

ADHS’15, October 16th 2015

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 25 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Symbolic abstraction

State partition P of [x, x] ⊆ Rn into αx identical intervals per dimension P =

  • s, s + x − x

αx

  • | s ∈
  • x + x − x

αx ∗ Zn

  • ∩ [x, x]
  • Input discretization Ua of [u, u] ⊆ Rp into αu ≥ 2 values per dimension

Ua =

  • u + u − u

αu − 1 ∗ Zp

  • ∩ [u, u]

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 26 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Sampling period

Guidelines for the viability kernel 1 (maximal invariant set): 2Lτ 2 sup

x∈[x,x]

f (x, u, w) ≥ x − x αx τ: sampling period x − x αx : step of the state partition L: Lipschitz constant sup

x∈[x,x]

f (x, u, w): supremum of the vector field

  • 1P. Saint-Pierre. Approximation of the viability kernel. Applied Mathematics and

Optimization, 29(2):187–209, 1994.

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 27 / 25

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Cooperative systems Centralized symbolic control Compositional approach

Complexity

n: state space dimension p: control space dimension αx ∈ N: number of symbols per dimension in the state partition αu ∈ N: number of controls per dimension in the input discretization | · |: cardinality of a set Method Centralized Compositional Abstraction (successors computed) 2αn

xαp u m

  • i=1

2α|Ii|

x α|Ji| u

Dynamic programming (max iterations) Nα2n

x αp u m

  • i=1

Nα2|Ii|

x

α|Ji|

u

Pierre-Jean Meyer (Grenoble) Compositional abstractions October 16th 2015 28 / 25