Recent!trends!in!control!synthesisfor! …...Recent!trends!in!control!synthesisfor!...
Transcript of Recent!trends!in!control!synthesisfor! …...Recent!trends!in!control!synthesisfor!...
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Recent trends in control synthesis for
hybrid systems: a personal view
Laurent Fribourg LSV – CNRS & ENS Cachan, U. Paris-‐Saclay
November 4, 2016 -‐ IRT SystemX
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Plan
I Classical Control II Hybrid systems III Set-‐based approach IV (Bi)simulaSon V MINIMATOR VI ComposiSonality VII Model reducSon VIII Conclusion
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I. Classical control
2 I. Classical control
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SchemaSc view of a control system – Plant: dynamics with a state variable x(t) governed by • dx/dt = f(x,u) (conSnuous-‐Sme form) • x(t+1) = f(x(t),u(t)) (discrete-‐Sme form)
– Sensors: gives a parSal informaSon y(t) about x(t) – Controller: computes the law u(t) as a funcSon of y(t)
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x(t)
y(t)
u(t)
I. Classical control
hXp://www.dis.uniroma1.it/~iacoviel/
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Principle of feedback
• Feedback: The actual operaSon of the control system is compared to the desired operaSon and the input u(t) to the plant is adjusted on the basis of this comparison.
• Feedback control systems are able to operate saSsfactorily despite adverse condiSons, such as disturbances and variaSons in plant properSes
4 I. Classical control
hXp://www.dis.uniroma1.it/~iacoviel/
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OpSmal control (the moon lander) • Aim: bring a spacecra] to a so] landing on the lunar surface,
using the least amount of fuel
• The state variable x is a triple (h,v,m) with: – h(t) height at Sme t – v(t) velocity (= dh/dt) – m(t) mass of spacecra]
• The control (or input) u(t) is the thrust at Sme t
5 I. Classical control
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Moon lander • Consider Newton’s law: m dv/dt = -‐ g m + u
This gives: dv/dt = -‐ g + u/m dh/dt = v dm/dt = -‐ k u
• The problem is to find u(.) in order to minimize the amout of fuel, i.e., maximize the amount remaining once we landed, i.e., maximize J defined by: J(u(.))=m(T) where T is the first Cme: h(T)=0, v(T)=0.
6 I. Classical control
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7 I. Classical control
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Limit: scalability
• The real subsystems are o]en numerous: – MulS variable – High dimension – Nonlinear – Time-‐Varying – Poorly modelled
• Thus they are o]en outside the bounds of exisSng classical theory, and/or exisSng computaSonal tools
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www.ifac-‐control.org/about/ifac-‐50-‐lectures/2-‐Anderson.pdf
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Example
• The pitch control system on a commercial aircra] (2006) has two inputs, two outputs, stochasSc disturbance, is open loop unstable. The state dimension is about 50.
• Challenge: How to design a (low order) controller
9 I. Classical control
www.ifac-‐control.org/about/ifac-‐50-‐lectures/2-‐Anderson.pdf
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New control problems
• Digital computer as a control system component:
à hybrid system (with switch control) à (new) combinatorial explosion: nb of possible switches grows exponenSal with Sme horizon
• Provable safe design • Complex networked systems
• Sensor and actuator rich systems • Autonomous distributed systems
10 I. Classical control
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II. Hybrid systems
The discrete-‐Sme dynamics of a hybrid system is
x(t+1)=f(x(t),u(t)) where u(t) is a discrete variable that takes its values on a finite domain U, eg: {0,1} (instead of a dense domain, eg: [0,1])
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The control synthesis problem consists in choosing at each t = 0,1,2,… a mode (value of u) according to the current value of x (or observaSon y) in order to meet a temporal property spec(x)
II. Hybrid systems
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A special class: switched systems
ex: DC-‐DC converter
II. Hybrid systems
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13 II. Hybrid systems
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14 II. Hybrid systems
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15 II. Hybrid systems
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16 II. Hybrid systems
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17 II. Hybrid systems
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18 II. Hybrid systems
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19 II. Hybrid systems
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20 II. Hybrid systems
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III Set-‐based approach
21 III. Set-‐based approach
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Safety constraint and invariance set
• A safe set S is a constraint (i.e. a subset of the state space) that should be always saSsfied by the state of the system.
• Safety saSsfacSon can be guaranteed for all Sme if (and only if) the iniSal state of the system is contained inside a controlled invariant set of S.
22 III. Set-‐based approach
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Maximal Controlled Invariant Set [Bertsekas-‐Rhodes 1971]
• Def: A subset X of S is a controlled invariant subset of S if, for all x0 in X,
there is a controlled trajectory issued from x0 that always stays in S
• Prop: The maximal controlled invariant subset (MCIS) M of S exists. Furthermore: x in M => f(x,u) in M for some u.
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Let X be a set of states and u a mode of U, we define the predecessor operators: Pre_u(X) = { x’ | x = f(x’,u) for some x of X } Pre(X) = U Pre_u(X) = { x’ | x = f(x’,u) for some x of X, u of U } III. Set-‐based approach
M S
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MCIS algorithm
• Algo input: S output: M maximal controlled invariant of S
IniSally: M := S while Pre(M) ≠ M
M := Pre(M) ∧ S endwhile
NB1: Algo terminates if S finite NB2: M is the greatest fixed-‐point (gfp) of Pre contained in S.
24 III. Set-‐based approach
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Fixed-‐Points of Pre
• MCIS of S = gfp(Pre) included in S = ∧_k Pre (S) • Basin of a9rac:on of S = lfp(Pre) containing S = U_k Pre (S) • Reach-‐avoid set of (S, A)
= set of iniSal points for which the controlled system reaches S while always avoiding A
= lfp(Reach-‐avoid) containing S = U_k Reach-‐avoid (S,A) with Reach-‐avoid(X,A) = { x’ | x’ in Pre_u(x) & Pre_u(x) ∧ A = ø for some control u and x of X }
25 III. Set-‐based approach
k
k
k
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ApplicaSon to collision avoidance [Mitchell-‐Bayen-‐Tomlin2004]:
determinaSon of the unsafe zone using reach-‐avoid
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A= { x1, x2 | x1 + x2 ≤ r }
unsafe zone={iniSal posiSon of red plane (relaSvely to blue plane) for which there is a risk of collision}
If the red plane is outside the unsafe zone, the blue plane is always ensured to « evade » III. Set-‐based approach
2 2 2
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Pursuer-‐evader game [Tomlin-‐Mitchell-‐Bayen-‐Oishi IEEE 2003]
27 III. Set-‐based approach
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IV. Another approach: (bi)simulaSon
• Pb: non-‐terminaSon of the fixed-‐point set calculaSon in case of infinite state systems
• Idea: find a bisimular (≈ equivalent) and finite system
28 IV. (Bi)simulaSon
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BisimulaSon
Def: An equivalence relaCon R on Σ × Σ’ is a bi-‐simulaCon if, for all (P,Q) of Σ × Σ’ with P R Q :
• Any move Q – a-‐> Q’ of Q can be matched by a move P –a-‐> P’, with P’ R Q’
• Conversely, any P – a-‐> P’ is matched by a move Q –a-‐> Q’, with P’ R Q’
Consider two transiSon systems Τ and T’ of state space Σ and Σ’ resp.
Systems T and T’ are said to be bisimular IV. (Bi)simulaSon
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Goal: To parSSon the infinite state-‐space of a system T into finitely many equivalences classes so that equivalence classes exhibit similar behaviors (è construcSon of a finite bisimilar quoSent automaton T’)
ConstrucSon of bisimular quoSent automaton
IV. (Bi)simulaSon
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hdps://www.lccc.lth.se/media/LCCC2013/WS1304/Slides/belta.pdf 31
Principle of (bi)simulaSon
original system T quoSent automaton T’
spec: « there is no trajectory from green to red » ~(green ∧ured) for all trajectories
spec false for this quoSent
IV. (Bi)simulaSon
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QuoSent refinement
spec. becomes true on T’
If the quoSent T’ is a simulaCon of T, and spec is true for T’ then spec is true for T
NB: spec restricted to universal properSes
IV. (Bi)simulaSon
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QuoSent refinement
ParSSon is refined on Σ using Pre operator:
≠ ø
IV. (Bi)simulaSon
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• If the algorithm terminates, the quoSent is finite and bisimular to the original system • The quoSent can be used in lieu of the original system for verificaSon of spec
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BisimulaSon algorithm
Pb: Unfortunately, terminaSon is rare! IV. (Bi)simulaSon
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If the algorithm terminates,
the quoSent saCsfies spec and simulates the original system (no more bisimulaSon). The original system is guaranteed to saSsfy spec.
Pb2: Unfortunately, the computaSon of Pre is difficult!
[ChuSnan-‐Krogh 2001]
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Variant 1 (spec-‐guided refinement)
IV. (Bi)simulaSon
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If the algorithm terminates, the original system saSsfies spec. NB: Refinement of the parSSon now involves a scheme independent of Pre
(e.g., split the « bad » state into two) [Tiwari-‐Khanna 2002] [Habets-‐van Schuppen 2004][Belta-‐Habets 2006] [Kloetzer-‐Belta HSCC 2006, TIMC 2012] 36
Variant 2 (not using Pre)
IV. (Bi)simulaSon
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How to refine the quoSent using Post
Post_u(X1) Post_v(X7)
u u
u v
TransiSon X – u -‐> X’ is added to the quoSent T’ when Post_u(X) ∧ X’ ≠ ø in the original system T
Post_u(X) = { x’ | x’ = f(x,u) for some x of X }
IV. (Bi)simulaSon
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38 IV. (Bi)simulaSon
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39 IV. (Bi)simulaSon
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V. MINIMATOR hXps://bitbucket.org/ukuehne/minimator/wiki/Home
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Romain Soulat, Ulrich Kühne, Adrien Le Coënt
V. MINIMATOR
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spec(X):
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IniSally, P := [R+a] while true
if some Sle X of P violates spec(X), refine P by spli~ng X endif
endwhile
44 NB: if a=0, spec states the invariance of R (with tolerance ε)
Basic algorithm
Input: R ε K Output: a Sling P of R+a saSsfying spec (for some a ≥ 0)
spec(X) = reachability of R from X in K steps while always staying inside R+a+ε = Post_π(X) R for some paXern π of length ≤ K ∧ Post_π’(X) R+a+ε for all prefix π’ of π
ê
target tolerance Cme horizon í í
iniCal zone ê
If the algorithm terminates, we have spec(X) for each Sle X of P with:
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47 V. MINIMATOR
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VI. ComposiSonality
50 VI. ComposiSonality
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spec1(X1):
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spec2(X2):
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Hence the distributed control achieves the goal of the centralized control
If spec1(X1) and spec2(X2) are true, ie.:
and
Then spec(X) (with X=(X1,X2)) is true, ie.:
Soundness of the distributed control synthesis
Advantage: the state dimension n and the nb of modes N have split in 2
with u=(u1,u2) and v=(v1,v2)
CondiSon: requires the weak interdependency of the arguments of f
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57 VI. ComposiSonality
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58 VI. ComposiSonality
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59 VI. ComposiSonality
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60 VI. ComposiSonality
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VII. Model reducSon
61 VII. Model reducSon
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VIII. Model reducSon 62
Model order reducSon
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Reduced order control synthesis
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VII. Model reducSon 64
Guaranteed on-‐line control
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VIII. Many important issues not menSoned! • Guards of hybrid systems (PWA)
• Non linearity
• ConSnuous-‐Sme dynamics
• Data structures (eg, zonotopes [Girard 2005])
• Observability
• Robustness
• Uncertainty
• StochasScity
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IX. RecapitulaSon
• Affine switched systems (special class of hybrid systems) • Set-‐based approach
• Symbolic simulaSon using Post • ComposiSonality
à Safety-‐provable design with increasing scalability (nb of conSnuous variables: n ≈ 3 in 2004, n ≈ 5 in 2010, n ≈ 11 in 2016, …)
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References • D. Bertsekas, I. Rhodes. On the minimax reachability of target sets and target
tubes. AutomaSca, 1971
• A Bouajjani, JC Fernandez, N Halbwachs. Minimal model generaSon, CAV 1990
• I. Mitchell, A. Bayen, C. Tomlin. A Time-‐Dependent Hamilton-‐Jacobi FormulaSon of Reachable Sets for ConSnuous Dynamic Games, IEEE TAC 2004
• B. Yordanov, X. Belta. Formal analysis of discrete-‐Sme piecewise affine systems. IEEE TAC 2010
• A. Le Coënt, L. Fribourg, N. Markey, F. De Vuyst, L. Chamoin. Distributed Synthesis of State-‐Dependent Switching Control. RP'16, LNCS 9899, Springer, 2016
• A. Le Coënt, F. De Vuyst, C. Rey, L. Chamoin, L. Fribourg. Control of mechanical systems using set-‐based methods. Int. J. of Dynamics and Control, 2016
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