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Nonsmooth Systems
Ricardo SanfeliceDepartment Electrical and Computer EngineeringUniversity of California
CoE Review @ Duke University - October 15, 2019
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Motivation and Approach
Common features in AFOSR applications:
! Variables changing continuously (e.g., physical quantities) anddiscretely (e.g., logic variables, resetting timers).
! Abrupt changes in the dynamics (changes in the environment,
control decisions, communication events, or failures).
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Motivation and Approach
Common features in AFOSR applications:
! Variables changing continuously (e.g., physical quantities) anddiscretely (e.g., logic variables, resetting timers).
! Abrupt changes in the dynamics (changes in the environment,
control decisions, communication events, or failures).
Driving Question:
How can we systematically design such systems featuringswitching and intermittency of information with provablerobustness to uncertainties arising in real-world environments?
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Motivation and Approach
Common features in AFOSR applications:
! Variables changing continuously (e.g., physical quantities) anddiscretely (e.g., logic variables, resetting timers).
! Abrupt changes in the dynamics (changes in the environment,
control decisions, communication events, or failures).
Driving Question:
How can we systematically design such systems featuringswitching and intermittency of information with provablerobustness to uncertainties arising in real-world environments?
Approach:
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Motivation and Approach
Common features in AFOSR applications:
! Variables changing continuously (e.g., physical quantities) anddiscretely (e.g., logic variables, resetting timers).
! Abrupt changes in the dynamics (changes in the environment,
control decisions, communication events, or failures).
Driving Question:
How can we systematically design such systems featuringswitching and intermittency of information with provablerobustness to uncertainties arising in real-world environments?
Approach:! Model continuous and discrete behavior using dynamical
models that are hybrid.
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Motivation and Approach
Common features in AFOSR applications:
! Variables changing continuously (e.g., physical quantities) anddiscretely (e.g., logic variables, resetting timers).
! Abrupt changes in the dynamics (changes in the environment,
control decisions, communication events, or failures).
Driving Question:
How can we systematically design such systems featuringswitching and intermittency of information with provablerobustness to uncertainties arising in real-world environments?
Approach:! Model continuous and discrete behavior using dynamical
models that are hybrid.! Develop systematic control theoretical tools for stability,
invariance, safety, and temporal logic, with robustness.
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Modeling Hybrid Dynamical Systems
Hybrid dynamical systems include a wide range of systems
Switched systems
z = fσ(t)(z)
σ switching signal
Impulsive systems
z(t) = f(z(t))
z(t+) = g(z(t)) t = t1, t2, . . .
Differential-algebraicequations
z = f(z, w)
0 = η(z, w)
w algebraic variables
Hybrid automata
q = 1 q = 2
q = 3
z = f1(z) z = f2(z)
z = f3(z)
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Prevalent Network Control Applications
Multi-agent Systems with Limited
Information [Automatica 16, TAC 18]
Control of Groups of Neurons
[ACC 14, TCNS 16]
Coordination of
Underactuated Vehicles
[Automatica 15, TAC 16]
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Prevalent Network Control Applications
Multi-agent Systems with Limited
Information [Automatica 16, TAC 18]
Control of Groups of Neurons
[ACC 14, TCNS 16]
Coordination of
Underactuated Vehicles
[Automatica 15, TAC 16]
Key Features:
! Nonlinearities
! Fast time scales / events
! Limited information
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Outline of Recent Results Relevant to the CoE
1. Optimization! High Performance Optimization via Uniting Control
ACC19, ACC20 (submitted), + CoE collab. (M. Hale)
! Model Predictive Control for Hybrid SystemsACC19, CDC19, ACC20 (submitted), CDC19 Workshop
+ collab. w/ AFRL/RV (S. Phillips and C. Petersen)
2. Tools to Satisfy High-level Specifications! Solution-independent Conditions for Invariance and
Finite-time Attractivity Automatica 19, TAC 19, NAHS and
IFAC WC20 (submitted) + Collab. w/ NASA (A. Mavridou)
! (Necessary and Sufficient) Safety CertificatesHSCC19, ACC19, and ACC20 (submitted)
3. Hybrid Control! Global Robust Stabilization on Manifolds
Automatica 19, TAC19, and ACC19 + CoE collab. (W. Dixon)
! Synchronization over Networks w/ IntermittentInformation Automatica 19, ACC19, and ACC20 (submitted)
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Safety Certificates
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Basic Setting
Consider the system
x = f(x) x ∈ X
and the sets
Xo ⊂ X the initial set,
Xu ⊂ X\Xo the unsafe set.
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Basic Setting
Consider the system
x = f(x) x ∈ X
and the sets
Xo ⊂ X the initial set,
Xu ⊂ X\Xo the unsafe set.
Safety with respect to (Xo,Xu) ⇔ reach(Xo) ∩Xu = ∅
reach(Xo) := {x ∈ Rn : x = φ(t;xo),with φ a solution from xo ∈ Xo
and any t ∈ domφ} – namely, the infinite reach set
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Sufficient Conditions for Safety when X = Rn
Let B be continuous such that
B(x) > 0 ∀x ∈ Xu
B(x) ≤ 0 ∀x ∈ Xo
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Sufficient Conditions for Safety when X = Rn
Let B be continuous such that
B(x) > 0 ∀x ∈ Xu
B(x) ≤ 0 ∀x ∈ Xo
and for each solution φ from xo ∈ Rn\Ke
t (→ B(φ(t;xo)) is nonincreasing
where Ke :=!
x ∈ R2 : B(x) ≤ 0"
– the zero-sublevel set of B
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Sufficient Conditions for Safety when X = Rn
Let B be continuous such that
B(x) > 0 ∀x ∈ Xu
B(x) ≤ 0 ∀x ∈ Xo
and for each solution φ from xo ∈ Rn\Ke
t (→ B(φ(t;xo)) is nonincreasing
where Ke :=!
x ∈ R2 : B(x) ≤ 0"
– the zero-sublevel set of B
It follows that the system x = f(x) is safe w.r.t. (Xo,Xu)
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Converse Safety Problem
Given a safe system x = f(x) w.r.t. (Xo,Xu), find a scalarfunction B : Rn → R (at least continuous) such that
B(x) > 0 ∀x ∈ Xu, B(x) ≤ 0 ∀x ∈ Xo
and t (→ B(φ(t;xo)) is nonincreasing on Rn\Ke
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Converse Safety Problem
Given a safe system x = f(x) w.r.t. (Xo,Xu), find a scalarfunction B : Rn → R (at least continuous) such that
B(x) > 0 ∀x ∈ Xu, B(x) ≤ 0 ∀x ∈ Xo
and t (→ B(φ(t;xo)) is nonincreasing on Rn\Ke
Two solutions to the converse safety problem in the literature are
[Prajna & Rantzer 05] when
1. f ∈ C1
2. ∃V ∈ C1 s.t.⟨∇V (x), f(x)⟩ < 0 ∀x ∈ X
3. (X,Xo,Xu) are compact
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Converse Safety Problem
Given a safe system x = f(x) w.r.t. (Xo,Xu), find a scalarfunction B : Rn → R (at least continuous) such that
B(x) > 0 ∀x ∈ Xu, B(x) ≤ 0 ∀x ∈ Xo
and t (→ B(φ(t;xo)) is nonincreasing on Rn\Ke
Two solutions to the converse safety problem in the literature are
[Prajna & Rantzer 05] when
1. f ∈ C1
2. ∃V ∈ C1 s.t.⟨∇V (x), f(x)⟩ < 0 ∀x ∈ X
3. (X,Xo,Xu) are compact
[Wisniewski & Sloth 17] when
1. f ∈ C1
2. ∃V smooth s.t.⟨∇V (x), f(x)⟩ < 0 on Xexcept at critical points of V+ geometric conditions
3. (X,Xo,Xu) are compactmanifolds
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Safe System but No Continuous Barrier Exists
Consider the planar continuous-time system [Krasovskii 63]
x1 = −x2 + rx1 sin2(1/r)
x2 = x1 + rx2 sin2(1/r)
wherer = |x|
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Safe System but No Continuous Barrier Exists
Consider the planar continuous-time system [Krasovskii 63]
x1 = −x2 + rx1 sin2(1/r)
x2 = x1 + rx2 sin2(1/r)
wherer = |x|
This system can be rewritten in polar coordinates (r, θ) as
r = r2 sin2(1/r), θ = 1
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Safe System but No Continuous Barrier Exists
Consider the planar continuous-time system [Krasovskii 63]
x1 = −x2 + rx1 sin2(1/r)
x2 = x1 + rx2 sin2(1/r)
wherer = |x|
This system can be rewritten in polar coordinates (r, θ) as
r = r2 sin2(1/r), θ = 1
Consider the initial and unsafe sets
Xo := {0} , Xu := R2\δB with δ ≥ 0
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Safe System but No Continuous Barrier Exists
Consider the planar continuous-time system [Krasovskii 63]
x1 = −x2 + rx1 sin2(1/r)
x2 = x1 + rx2 sin2(1/r)
wherer = |x|
This system can be rewritten in polar coordinates (r, θ) as
r = r2 sin2(1/r), θ = 1
Consider the initial and unsafe sets
Xo := {0} , Xu := R2\δB with δ ≥ 0
It is easy to check that the system is safe w.r.t. (Xo,Xu)due to Xo being forward invariant
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Safe System w/o C0 State-Dependent Barrier
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Safe System w/o C0 State-Dependent Barrier
There is no continuous barrier function with B(0) = 0, B(x) > 0forall x ∈ Xu, and t (→ B(φ(t;xo)) nonincreasing on Rn \Xo
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Modeling Hybrid Systems: Closed Loop
Hybrid closed-loop systems are given by hybrid inclusions
H
#
x = F (x) x ∈ Cx+ = G(x) x ∈ D
where x is the state
! C is the flow set
! F is the flow map
! D is the jump set
! G is the jump map
Solutions are functions parameterized by hybrid time (t, j):! Flows parameterized by t ∈ R≥0 := [0,+∞)! Jumps parameterized by j ∈ N≥0 := {0, 1, 2, . . .}
Then, solutions to H are given by hybrid arcs x defined on
([0, t1]× {0}) ∪ ([t1, t2]× {1}) ∪ . . . ([tj, tj+1]× {j}) ∪ . . .
The state x can have logic, memory, and timer components.The hybrid system H satisfies the hybrid basic conditions ifC, D are closed and F , G are “continuous”
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Modeling Hybrid Systems: Closed Loop
Hybrid closed-loop systems are given by hybrid inclusions
H
#
x ∈ F (x) x ∈ Cx+ ∈ G(x) x ∈ D
where x is the state
! C is the flow set
! F is the flow map
! D is the jump set
! G is the jump map
Solutions are functions parameterized by hybrid time (t, j):! Flows parameterized by t ∈ R≥0 := [0,+∞)! Jumps parameterized by j ∈ N≥0 := {0, 1, 2, . . .}
Then, solutions to H are given by hybrid arcs x defined on
([0, t1]× {0}) ∪ ([t1, t2]× {1}) ∪ . . . ([tj, tj+1]× {j}) ∪ . . .
The state x can have logic, memory, and timer components.The hybrid system H satisfies the hybrid basic conditions ifC, D are closed and F , G are “continuous”
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Modeling Hybrid Systems: Closed Loop
Hybrid closed-loop systems are given by hybrid inclusions
H
#
x = F (x) x ∈ Cx+ = G(x) x ∈ D
For this class of systems, we have the following:
! An initial set Xo ⊂ C ∪D and an unsafe set Xu ⊂ Rn.
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Modeling Hybrid Systems: Closed Loop
Hybrid closed-loop systems are given by hybrid inclusions
H
#
x ∈ F (x) x ∈ Cx+ ∈ G(x) x ∈ D
For this class of systems, we have the following:
! An initial set Xo ⊂ C ∪D and an unsafe set Xu ⊂ Rn.
! Assume that Rn\(C ∪D) ⊂ Xu.
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Modeling Hybrid Systems: Closed Loop
Hybrid closed-loop systems are given by hybrid inclusions
H
#
x ∈ F (x) x ∈ Cx+ ∈ G(x) x ∈ D
For this class of systems, we have the following:
! An initial set Xo ⊂ C ∪D and an unsafe set Xu ⊂ Rn.
! Assume that Rn\(C ∪D) ⊂ Xu.
! A barrier function candidate B : Rn → R is defined asB(x) > 0 ∀x ∈ Xu ∩ (C ∪D) and B(x) ≤ 0 ∀x ∈ Xo.
! A barrier function candidate B defines the set
K := {x ∈ C ∪D : B(x) ≤ 0}
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Sufficient Conditions for Safety
Theorem: The system H is safe w.r.t. (Xo,Xu) if a barriercandidate B exists such that K = {x ∈ C ∪D : B(x) ≤ 0} isclosed and forward pre-invariant for H.
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Sufficient Conditions for Safety
Theorem: The system H is safe w.r.t. (Xo,Xu) if a barriercandidate B exists such that K = {x ∈ C ∪D : B(x) ≤ 0} isclosed and forward pre-invariant for H.
The closed set K is forward pre-invariant if
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Sufficient Conditions for Safety
Theorem: The system H is safe w.r.t. (Xo,Xu) if a barriercandidate B exists such that K = {x ∈ C ∪D : B(x) ≤ 0} isclosed and forward pre-invariant for H.
The closed set K is forward pre-invariant if
1. B(η) ≤ 0 ∀η ∈ G(x) ∀x ∈ D ∩K
2. G(D ∩K) ⊂ C ∪D
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Sufficient Conditions for Safety
Theorem: The system H is safe w.r.t. (Xo,Xu) if a barriercandidate B exists such that K = {x ∈ C ∪D : B(x) ≤ 0} isclosed and forward pre-invariant for H.
The closed set K is forward pre-invariant if
1. B(η) ≤ 0 ∀η ∈ G(x) ∀x ∈ D ∩K
2. G(D ∩K) ⊂ C ∪D
3. t (→ B(φ(t, 0)) is nonincreasing for flowing solutionst (→ φ(t, 0) in (U(∂K)\K) ∩ C
where U(S) is any neighborhood around the set S
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Monotone Behavior Along the Flows
Proposition:
! When B is C1, 3. is satisfied if⟨∇B(x), η⟩ ≤ 0 ∀x ∈ (U(∂Ke)\Ke) ∩C ∀η ∈ F (x) ∩ TC(x).
! When B is loc. Lip., we replace ∇B by ∂B.
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Monotone Behavior Along the Flows
Proposition:
! When B is C1, 3. is satisfied if⟨∇B(x), η⟩ ≤ 0 ∀x ∈ (U(∂Ke)\Ke) ∩C ∀η ∈ F (x) ∩ TC(x).
! When B is loc. Lip., we replace ∇B by ∂B.
! When B is lower semicontinuous and F locally bounded wereplace ∇B by ∂pB and (U(∂Ke)\Ke) ∩ C byU(Ke ∩C)\Ke, Ke := {x ∈ Rn : B(x) ≤ 0}.
(∂B, ∂pB,TC) are the generalized gradient, the proximalsubdifferential, and the contingent cone w.r.t. C [Clarke & al 08].
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Safety for Bouncing Ball
Consider a hybrid system H = (C,F ,D,G) modeling a ballbouncing vertically on the ground, with x = (x1, x2) ∈ R2 given by
F (x) :=
$
x2−γ
%
∀x ∈ C, γ > 0
G(x) :=
$
0−λx2
%
∀x ∈ D, λ ∈ [0, 1]
C := {x ∈ R2 : x1 ≥ 0}
D := {x ∈ R2 : x1 = 0, x2 ≤ 0}
x1
x2
D
C
F (x)
x
❅❅G(x′)
x′
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Safety for Bouncing Ball
Consider a hybrid system H = (C,F ,D,G) modeling a ballbouncing vertically on the ground, with x = (x1, x2) ∈ R2 given by
F (x) :=
$
x2−γ
%
∀x ∈ C, γ > 0
G(x) :=
$
0−λx2
%
∀x ∈ D, λ ∈ [0, 1]
C := {x ∈ R2 : x1 ≥ 0}
D := {x ∈ R2 : x1 = 0, x2 ≤ 0}
x1
x2
D
C
F (x)
x
❅❅G(x′)
x′
! Let Xo := {x ∈ C : |x| ≤ 1/(4γ)} andXu := {x ∈ C : x1 > 1/γ, x2 = 0} .
! Consider the barrier candidate B(x) := 2γx1 + (x2 − 1)(x2 + 1).
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Safety for Bouncing Ball
Consider a hybrid system H = (C,F ,D,G) modeling a ballbouncing vertically on the ground, with x = (x1, x2) ∈ R2 given by
F (x) :=
$
x2−γ
%
∀x ∈ C, γ > 0
G(x) :=
$
0−λx2
%
∀x ∈ D, λ ∈ [0, 1]
C := {x ∈ R2 : x1 ≥ 0}
D := {x ∈ R2 : x1 = 0, x2 ≤ 0}
x1
x2
D
C
F (x)
x
❅❅G(x′)
x′
! Let Xo := {x ∈ C : |x| ≤ 1/(4γ)} andXu := {x ∈ C : x1 > 1/γ, x2 = 0} .
! Consider the barrier candidate B(x) := 2γx1 + (x2 − 1)(x2 + 1).
! Condition 1) holds sinceB(G(x)) = 2γx1 + λ2x2
2 − 1 ≤ 2γx1 + x22 − 1 ≤ 0 ∀x ∈ K ∩D.
! Condition 2) holds since G(D) = {0}× R≥0 ⊂ C ∪D.
! Condition 3) holds since ⟨∇B(x), F (x)⟩ = 0 ∀x ∈ C.
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Safety for Bouncing Ball
Consider a hybrid system H = (C,F ,D,G) modeling a ballbouncing vertically on the ground, with x = (x1, x2) ∈ R2 given by
F (x) :=
$
x2−γ
%
∀x ∈ C, γ > 0
G(x) :=
$
0−λx2
%
∀x ∈ D, λ ∈ [0, 1]
C := {x ∈ R2 : x1 ≥ 0}
D := {x ∈ R2 : x1 = 0, x2 ≤ 0}
x1
x2
D
C
F (x)
x
❅❅G(x′)
x′
! Let Xo := {x ∈ C : |x| ≤ 1/(4γ)} andXu := {x ∈ C : x1 > 1/γ, x2 = 0} .
! Consider the barrier candidate B(x) := 2γx1 + (x2 − 1)(x2 + 1).
! Condition 1) holds sinceB(G(x)) = 2γx1 + λ2x2
2 − 1 ≤ 2γx1 + x22 − 1 ≤ 0 ∀x ∈ K ∩D.
! Condition 2) holds since G(D) = {0}× R≥0 ⊂ C ∪D.
! Condition 3) holds since ⟨∇B(x), F (x)⟩ = 0 ∀x ∈ C.The system H is safe with respect to (Xo, Xu)
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Safety for System with Discrete States
Consider a hybrid system H = (C,F ,D,G) modeling a thermostatsystem, with the state x = (q, z) ∈ X := {0, 1} × R given by
F (x) :=
$
0−z + z0 + z∆q
%
∀x ∈ C
G(x) :=
$
1− qz
%
∀x ∈ D
C := ({0} × C0) ∪ ({1} × C1),C0 := {z ∈ R : z ≥ zmin}C1 := {z ∈ R : z ≤ zmax}
D := ({0} ×D0) ∪ ({1} ×D1).D0 := {z ∈ R : z ≤ zmin}D1 := {z ∈ R : z ≥ zmax}
! z is the room temperature, zo the room temperature when theheater is OFF
! z∆ the capacity of the heater to raise the temperature
! q the state of the heater 1 (ON) or 0 (OFF)
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Safety for System with Discrete States
Consider a hybrid system H = (C,F ,D,G) modeling a thermostatsystem, with the state x = (q, z) ∈ X := {0, 1} × R given by
F (x) :=
$
0−z + z0 + z∆q
%
∀x ∈ C
G(x) :=
$
1− qz
%
∀x ∈ D
C := ({0} × C0) ∪ ({1} × C1),C0 := {z ∈ R : z ≥ zmin}C1 := {z ∈ R : z ≤ zmax}
D := ({0} ×D0) ∪ ({1} ×D1).D0 := {z ∈ R : z ≤ zmin}D1 := {z ∈ R : z ≥ zmax}
! z stays between zmin and zmax satisfyingzo < zmin < zmax < zo + z∆.
! Let Xo := {(q, z) ∈ X : z ∈ [zmin/2, zmax/2]}.! Let Xu := {(q, z) ∈ X : z ∈ (−∞, zmin) ∪ (zmax,+∞)}.! Consider the barrier candidate B(x) := (z − zmin)(z − zmax)
and let Ke := {x ∈ R2 : B(x) ≤ 0}.
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Safety for System with Discrete States
Consider a hybrid system H = (C,F ,D,G) modeling a thermostatsystem, with the state x = (q, z) ∈ X := {0, 1} × R given by
F (x) :=
$
0−z + z0 + z∆q
%
∀x ∈ C
G(x) :=
$
1− qz
%
∀x ∈ D
C := ({0} × C0) ∪ ({1} × C1),C0 := {z ∈ R : z ≥ zmin}C1 := {z ∈ R : z ≤ zmax}
D := ({0} ×D0) ∪ ({1} ×D1).D0 := {z ∈ R : z ≤ zmin}D1 := {z ∈ R : z ≥ zmax}
! C ∪D = {0, 1} × R; hence, condition 2) holds sinceG(x) = [1− q z]⊤ ∈ C ∪D ∀x ∈ C ∪D.
! Condition 1) holds sinceB(G(x)) = B([(1− q) z]⊤) = B(x) ≤ 0 ∀x ∈ Ke ∩D.
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Safety for System with Discrete States
Consider a hybrid system H = (C,F ,D,G) modeling a thermostatsystem, with the state x = (q, z) ∈ X := {0, 1} × R given by
F (x) :=
$
0−z + z0 + z∆q
%
∀x ∈ C
G(x) :=
$
1− qz
%
∀x ∈ D
C := ({0} × C0) ∪ ({1} × C1),C0 := {z ∈ R : z ≥ zmin}C1 := {z ∈ R : z ≤ zmax}
D := ({0} ×D0) ∪ ({1} ×D1).D0 := {z ∈ R : z ≤ zmin}D1 := {z ∈ R : z ≥ zmax}
! Ke = R× [zmin, zmax].! Furthermore, for some ϵ > 0, (U(Ke)\Ke) ∩C =
({0}× (zmax, zmax + ϵ)) ∪ ({1}× (zmin, zmin − ϵ)). Hence,condition 3) holds since
⟨∇B(x), F (x)⟩ = (zmin + zmax − 2z)(z − zo − z∆q) ≤ 0
for all x ∈ (U(Ke)\Ke) ∩ C.
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Safety for System with Discrete States
Consider a hybrid system H = (C,F ,D,G) modeling a thermostatsystem, with the state x = (q, z) ∈ X := {0, 1} × R given by
F (x) :=
$
0−z + z0 + z∆q
%
∀x ∈ C
G(x) :=
$
1− qz
%
∀x ∈ D
C := ({0} × C0) ∪ ({1} × C1),C0 := {z ∈ R : z ≥ zmin}C1 := {z ∈ R : z ≤ zmax}
D := ({0} ×D0) ∪ ({1} ×D1).D0 := {z ∈ R : z ≤ zmin}D1 := {z ∈ R : z ≥ zmax}
! Ke = R× [zmin, zmax].! Furthermore, for some ϵ > 0, (U(Ke)\Ke) ∩C =
({0}× (zmax, zmax + ϵ)) ∪ ({1}× (zmin, zmin − ϵ)). Hence,condition 3) holds since
⟨∇B(x), F (x)⟩ = (zmin + zmax − 2z)(z − zo − z∆q) ≤ 0
for all x ∈ (U(K )\K ) ∩ C.
The system H is safe with respect to (Xo,Xu)
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Safety-Based Control for Agile Evasion
Presented at 2019 CASE Robotics Conference – Best Paper Award Finalist
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Hybrid Control for Aggresive Maneuvering
coning
recoveryandintercept
launch
glide
glide trackingset
vehicle
obst
acl
eAchieving non-parabolic ballistic trajectories for a guided
munition using aggressive maneuvers using multiple controllers.
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists a lower semicontinuous barrier function
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists B : R≥0 ×N× Rn → R such that
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists B : R≥0 ×N× Rn → R such that
(τ, x) (→ B(τ, k, x) is lower semicontinuous (uniformly in k)
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists B : R≥0 ×N× Rn → R such that
(τ, x) (→ B(τ, k, x) is lower semicontinuous (uniformly in k)B is nonincreasing along the flows of H,
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists B : R≥0 ×N× Rn → R such that
(τ, x) (→ B(τ, k, x) is lower semicontinuous (uniformly in k)B is nonincreasing along the flows of H,B(τ, k, x) ≤ 0 ∀(τ, k, x) ∈ R≥0 × N×Xo
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists B : R≥0 ×N× Rn → R such that
(τ, x) (→ B(τ, k, x) is lower semicontinuous (uniformly in k)B is nonincreasing along the flows of H,B(τ, k, x) ≤ 0 ∀(τ, k, x) ∈ R≥0 × N×Xo
B(τ, k, x) > 0 ∀(τ, k, x) ∈ R≥0 × N× (Xu ∩ (C ∪D))
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists B : R≥0 ×N× Rn → R such that
(τ, x) (→ B(τ, k, x) is lower semicontinuous (uniformly in k)B is nonincreasing along the flows of H,B(τ, k, x) ≤ 0 ∀(τ, k, x) ∈ R≥0 × N×Xo
B(τ, k, x) > 0 ∀(τ, k, x) ∈ R≥0 × N× (Xu ∩ (C ∪D))B(τ, k + 1, η) ≤ 0 ∀η ∈ G(x), ∀(τ, k, x) ∈ Ke ∩ (R≥0 ×N×D)
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists B : R≥0 ×N× Rn → R such that
(τ, x) (→ B(τ, k, x) is lower semicontinuous (uniformly in k)B is nonincreasing along the flows of H,B(τ, k, x) ≤ 0 ∀(τ, k, x) ∈ R≥0 × N×Xo
B(τ, k, x) > 0 ∀(τ, k, x) ∈ R≥0 × N× (Xu ∩ (C ∪D))B(τ, k + 1, η) ≤ 0 ∀η ∈ G(x), ∀(τ, k, x) ∈ Ke ∩ (R≥0 ×N×D)G(x) ⊂ C ∪D ∀x ∈ D : (τ, k, x) ∈ Ke, (τ, k) ∈ R≥0 × N
where Ke := {(τ, k, x) ∈ R≥0 × N× Rn : B(τ, k, x) ≤ 0}
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists B : R≥0 ×N× Rn → R such that
(τ, x) (→ B(τ, k, x) is lower semicontinuous (uniformly in k)B is nonincreasing along the flows of H,B(τ, k, x) ≤ 0 ∀(τ, k, x) ∈ R≥0 × N×Xo
B(τ, k, x) > 0 ∀(τ, k, x) ∈ R≥0 × N× (Xu ∩ (C ∪D))B(τ, k + 1, η) ≤ 0 ∀η ∈ G(x), ∀(τ, k, x) ∈ Ke ∩ (R≥0 ×N×D)G(x) ⊂ C ∪D ∀x ∈ D : (τ, k, x) ∈ Ke, (τ, k) ∈ R≥0 × N
where Ke := {(τ, k, x) ∈ R≥0 × N× Rn : B(τ, k, x) ≤ 0} Moreover,! If Xo is compact, then the pre-completeness condition on
the backward solutions is not needed.
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Necessary and Sufficient Conditions for Safety
Theorem: Assume that H is satisfies the hybrid basic conditions*and that the backward solutions are either bounded or complete.The hybrid system H is safe w.r.t. (Xo,Xu)
if and only if
there exists B : R≥0 ×N× Rn → R such that
(τ, x) (→ B(τ, k, x) is lower semicontinuous (uniformly in k)B is nonincreasing along the flows of H,B(τ, k, x) ≤ 0 ∀(τ, k, x) ∈ R≥0 × N×Xo
B(τ, k, x) > 0 ∀(τ, k, x) ∈ R≥0 × N× (Xu ∩ (C ∪D))B(τ, k + 1, η) ≤ 0 ∀η ∈ G(x), ∀(τ, k, x) ∈ Ke ∩ (R≥0 ×N×D)G(x) ⊂ C ∪D ∀x ∈ D : (τ, k, x) ∈ Ke, (τ, k) ∈ R≥0 × N
where Ke := {(τ, k, x) ∈ R≥0 × N× Rn : B(τ, k, x) ≤ 0} Moreover,! If Xo is compact, then the pre-completeness condition on
the backward solutions is not needed.! If the solutions to H are not Zeno, then the result holds with
B independent of k.
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Summary:! Overview of Recent Results
! Introduction to Safety
! Sufficient Conditions
! Necessary and Sufficient Conditions
Next steps:
! Estimation
! Reachability
! Approximations
! Robustness
References at hybrid.soe.ucsc.edu
Acknowledgments: Partially supported by NSF, AFOSR,
AFRL and by CITRIS and the Banatao Institute at the
University of California
IEEE 2009 Princeton U. Press
2012
CRC Press 2015