verification and synthesis of hybrid systems
DESCRIPTION
Verification and Synthesis of Hybrid Systems. Thao Dang October 10, 2000. Plan. 1- Algorithmic Verification of Hybrid Systems 2- Reachability Analysis of Continuous Systems 3- Safety Verification of Hybrid Systems 4- Safety Controller Synthesis for Hybrid Systems 5- Implementation. - PowerPoint PPT PresentationTRANSCRIPT
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Verification and Synthesis of Hybrid Systems
Thao Dang
October 10, 2000
2
Plan
1- Algorithmic Verification of Hybrid Systems
2- Reachability Analysis of Continuous Systems
3- Safety Verification of Hybrid Systems
4- Safety Controller Synthesis for Hybrid Systems
5- Implementation
3
Plan
1- Algorithmic Verification of Hybrid Systems
2- Reachability Analysis of Continuous Systems
3- Safety Verification of Hybrid Systems
4- Safety Controller Synthesis for Hybrid Systems
5- Implementation
4
Hybrid systems
• Hybrid systems: systems which combine continuous-time dynamics and discrete-event dynamics
Continuous processes Digital controllers,
switches, gears..(e.g., chemical reactions)
• Arisen virtually everywhere (due to the increasing use of computers)
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Analysis of Hybrid Systems
• Formal verification: prove that the system satisfies a given property• Controller synthesis: design controllers so that the controlled system satisfies a desired property
• We concentrate on invariance properties: all trajectories of the system stay in a subset of the state space
• Hybrid systems are difficult to analyze No existing general method
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Illustrative Example: A Thermostat
on
x x
off
max x
4 x x
min x
• Verification problem: prove that the temperature x[a,b]
• Characterize all behaviors Reachability Analysis
7
The Thermostat Example (cont’d)
• Two-phase behavior
• Non-deterministic behavior
• Set of initial states
x
t
max
min
0
0
How to characterize and represent “tubes” of trajectories of continuous dynamics in order to treat discrete transitions??
How to characterize and represent “tubes” of trajectories of continuous dynamics in order to treat discrete transitions??
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Algorithmic Analysis of Hybrid Systems
• Exact symbolic methods applicable for restricted classes of hybrid systems
• Our objective: verification method for general hybrid systems in any dimension
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Algorithmic Verification of Hybrid Systems
approximate reachability techniques represent reachable sets by orthogonal polyhedra
What do we need?? a reachability technique which
is applicable for arbitrary continuous systems can be extended to hybrid systems
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Approximations by Orthogonal Polyhedra
Non-convex orthogonal polyhedra (unions of hyperrectangles)
Motivations canonical representation, efficient manipulation in any dimension easy extension to hybrid systems termination can be guaranteed
Over-approximation Under-approximation
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Plan
1- Approach to Algorithmic Verification of Hybrid Systems
2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems
3- Safety Verification of Hybrid Systems
4- Safety Controller Synthesis for Hybrid Systems
5- Implementation
12
Plan
1- Approach to Algorithmic Verification of Hybrid Systems
2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems
3- Safety Verification of Hybrid Systems
4- Safety Controller Synthesis for Hybrid Systems
5- Implementation
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Reachability Analysis of Continuous Systems
Problem
Find an orthogonal polyhedron over-approximating the reachable set from F
x(0)F, set of initial states
Lipschitzisf);(fsystemcontinuousA xx
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[0,r](F)
Successor Operator
r(F)
F
Reachable set from F: (F) = [0,)(F)
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Abstract Algorithm for Calculating (F)
P0 := F ;repeat k = 0, 1, 2 .. Pk+1 := Pk [0,r](Pk) ;until Pk+1 = Pk
Use orthogonal polyhedra to
• represent Pk
• approximate [0,r]
r : time step
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Plan
1- Algorithmic Verification of Hybrid Systems
2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems
3- Safety Verification of Hybrid Systems
4- Safety Controller Synthesis for Hybrid Systems
5- Implementation
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Reachability of Linear Continuous Systems
;AsystemlinearA xx
F is a convex polyhedron: F = conv{v1,..,vm}
r(F) = eArF
F
vir(vi)=eArvi
F is the set of initial states
r(F) = conv{r(v1),.., r(vm)}
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Over-Approximating the Reachable Set
[0,2r] (F) P2 = G1G2
X2
P2
[0,r](F) G1
P1=G1
[r,2r](F) G2
X1
X2
G2
X0=F
r(v2)
X1= r(X0)
v1
v2
r(v1) X1X1
X0
C1=conv{X1,X0}
C1Cb1
Extension to under-approximationsExtension to under-approximations
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Example
5.00.00.0
0.00.10.4
0.00.40.1
A
]1.0,05.0[]15.0,1.0[]05.0,025.0[F,Axx
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Extension to Linear Systems with Uncertain Input
setcompactandconvexa,Uinput);t()t(A)t( uuxx
Computation of r(F) [Varaiya 98]
U),t(λmaxarg)t( i*i uuu
i(r)i
F yi*(r)yi
r(F)
Bloating amount
u1
u2
(Maximum Principle)
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Example
]005.0,005.0[]5.0,5.0[]005.0,005.0[]5.0,5.0[UsetInput
]1,1[]2,0[]1,1[]2,0[FsetInitial
0400
1000
0008
0010
A,A
uxx [Kurzhanski and Valyi 97]
Advantage: time-efficiency Advantage: time-efficiency
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Plan
1- Algorithmic Verification of Hybrid Systems
2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems
3- Safety Verification of Hybrid Systems
4- Safety Controller Synthesis for Hybrid Systems
5- Implementation
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Principle of the Reachability Technique
yF
x
Lipschitzisf);(fsystemcontinuousA xx
‘Face lifting’ technique, inspired by [Greenstreet 96]
x(0)F, set of initial states
Continuity of trajectories compute from the boundary of F
The initial set F is a convex polyhedron
The boundary of F: union of its faces
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N(e)
H(e)
Over-Approximating [0,r](F)
Step 1: rough approximation N(F)
F
e
fe : projection of f on the outward normal to face e
ef̂ : maximum of fe over the neighborhood N(e) of e
ef̂
H’(e)
r
e1N(F)
Step 2: more accurate approximation
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Computation Procedure
• Decompose F into non-overlapping hyper-rectangles
• Apply the lifting operation to each hyper-rectangle (faces on the boundary of F)
• Make the union of the new hyper-rectangles
F
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Example: Airplane Safety [Lygeros et al. 98]
)anglepitch(,u);thrust(T,Tu
um
cxa
x
xcosg
m
)cx1(xax
m
uxsing
m
xax
anglepathflight:x;velocity:x
maxmin2maxmin1
21L
1
221L2
12
21D
1
21
P = [Vmin,Vmax][min,max]
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Plan
1- Algorithmic Verification of Hybrid Systems
2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems
3- Safety Verification of Hybrid Systems
4- Safety Controller Synthesis for Hybrid Systems
5- Implementation
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Hybrid Systems
Hybrid automata• continuous dynamics: linear with uncertain input, non-linear• staying and switching conditions: convex polyhedra• reset functions : affine of the form Rqq’ (x) = Dqq’x + Jqq’
q0 u x x 1 A
q1
) ( R : / G01 01x x x
0 Hx 1 Hx) ( f0x x
) ( R : / G10 10x x x
switching conditionreset function
discrete state
staying condition
continuous dynamics
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Reachability of Hybrid Automata
The state (q, x) of the system can change in two ways:• continuous evolution: q remains constant, and x changes continuously according to the diff. eq. at q• discrete evolution (by making a transition): q changes, and x changes according to the reset function.
Reachability analysis• continuous-successors • discrete-successors approximations by orthogonal polyhedra
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Over-approximating Continuous-Successors
• Use the reachability algorithms for continuous systems• Take into account the staying conditions
Hq
F[0,r](F)P
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Fg FGqq’
Over-approximating Discrete-Successors
Rqq’(b)
Hq’
F
qq’(q, F) = (q’, Rqq’(F Gqq’) Hq’)
b Gqq’
Fg
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q0
15 . 0 x1
q1
02 . 0 x1
q0
15 . 0 x1 Example
2 3
3 2A ;
0 3
6. 0 0A1 0q0 x x1 A
q1
15 . 0 x1
15 . 0 x1 02 . 0 x1 x x0 A
02 . 0 x1
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Plan
1- Algorithmic Verification of Hybrid Systems
2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems
3- Safety Verification of Hybrid Systems
4- Safety Controller Synthesis for Hybrid Systems
5- Implementation
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Switching Controller Synthesis: Introduction
q1 q2
q3
f1
f2
f3
q x
Mode selection
Plant
Discrete Switching Controller
q3
12 Gx
21 Gx
31 Gx23 Gx
3 Hx
2 Hx1 Hx) ( f1x x ) ( f2x x
) ( f3x x
q1 q2
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The Safety Synthesis Problem
Given a hybrid automaton A and a set F How to restrict the guards and the staying conditions of A so that all trajectories of the resulting automaton A* stay in F
Solution: Compute the maximal invariant set (set of ‘winning’ states)
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Operator
Given F={(q, Fq) | qQ}, (F) consists of states from which all trajectories
• stay indefinitely in F without switching OR
• stay in F for some time and then make a transition to another discrete state and still in F
Gqq’Fq’
Fq
x1
x2
x3
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Calculation of the Maximal Invariant Set
P0 := F ; repeat k = 1, 2, .. Pk+1 := Pk (Pk) ; until Pk+1 = Pk
P* = Pk ;
P* : maximal invariant setA* : H* =H P*, G* =G P*
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Effective Approximate Synthesis Algorithm
• Use our reachability techniques for hybrid automata to approximate (F)
• Under-approximations
Effective approximate synthesis algorithm for hybrid systems with linear continuous dynamics
To approximate the maximal invariant set:
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F0 F1
G10
G01
05.00.2
5.005.0A0
05.05.0
0.205.0A1
68.0,35.035.0,65.0F
G10F0F1F0
G01F1
G01=[-0.2,-0.01]
[-0.2,-0.01]
G10=[0.01,0.32]
[-0.01,0.1]
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Plan
1- Approach to Algorithmic Verification of Hybrid Systems
2- Reachability Analysis of Continuous Systems Abstract Reachability Algorithm Algorithm for Linear Continuous Systems Algorithm for Non-Linear Continuous Systems
3- Safety Verification of Hybrid Systems
4- Safety Controller Synthesis for Hybrid Systems
5- Implementation
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The tool d/dt
Three types of automatic analysis for hybrid systems with linear differential inclusions
Reachability Analysis: compute an over-approximation of the reachable set from a given initial set
Safety Verification: check whether the system reaches a set of bad states
Safety Controller Synthesis: synthesize a switching controller so that the controlled system always remains inside a given set
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Implementation
OpenGL LEDA
Interface Verification AlgorithmsController Synthesis Algorithms
Numerical IntegrationCVODE
Geometric Algorithms
Qhull, Polka,Cubes
Orthogonal Approximations
d/dt
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The tool d/dt
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Conclusions
Generality of Systems Complexity of continuous and discrete dynamics High dimensional systems
Variety of Problems Safety Verification and Synthesis
Applications collision avoidance (4 continuous variables, 1 discrete state) double pendulum (3 continuous variables, 7 discrete states) freezing system (6 continuous variables, 9 discrete states)
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Perspectives
• More efficient analysis techniques- Combining with analytic/qualitative methods- Adapting existing techniques for discrete/timed systems
• More classes of problems - more properties to verify, more synthesis criteria - controller synthesis for more general systems, e.g linear diff. games vuxx CBA
• Tool - more interactive analysis, simulation features - experimentation: real-life problems
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Related Work
Reachability Analysis• Polygonal Projections [Greenstreet and Mitchell 99]• Ellipsoidal Techniques [Kurzhanski and Varaiya 00]• Approximations via Parallelotopes [Kostoukova 99]
Verification• CheckMate [Chutinan and Krogh 99]• HyperTech [Henzinger et al. 00]• VeriShift [Botchkarev and Tripakis 00]• Symbolic Method [Lafferriere, Pappas, and Yovine 99]
Synthesis• Synthesis for timed automata [Asarin, Maler, Pnueli, and Sifakis 98]• Hamilton Jacobi Partial Diff. Eq. [Lygeros, Tomlin, and Sastry 98]• Computer Algebra [Shakernia, Pappas, and Sastry 00]
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FinMerci