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Control Synthesis for Stochastic Switched Systems using the Tamed Euler Method
Adrien Le Coënt, L Fribourg, J Vacher
To cite this version:
Adrien Le Coënt, L Fribourg, J Vacher. Control Synthesis for Stochastic Switched Systems using the Tamed Euler Method. 2017. �hal-01670579�
Control Synthesis for Stochastic Switched Systems using the Tamed Euler Method
A. Le Coënt1, L. Fribourg2, and J. Vacher1 CMLA,1LSV,2CNRS & ENS Paris-Saclay & INRIA ABSTRACT
In this paper, we explain how, under the one-sided Lipschitz (OSL) hypothesis, one can find an error bound for a variant of the Euler- Maruyama approximation method for stochastic switched systems.
We then explain how this bound can be used to control stochastic switched switched system in order to stabilize them in a given re- gion. The method is illustrated on several examples of the literature.
1 INTRODUCTION
Control synthesis for stochastic switched systems has been recently explored using the construction ofapproximately bisimilar sym- bolicmodels. This approach relies on the hypothesis ofincremental stabilityof the stochastic switched system (or existence of a com- mon/multiple Lyapunov function) [8–10], which concerns only a small part of the real systems.
Here we address the problem of control synthesis in a more general setting. We do not use the construction of approximatively bisimilar symbolic models, but the (tamed) Euler method [3] for stochastic switched systems. We thus follow the lines of previous work on control synthesis for deterministic switched systems [1].
Unlike [8–10], the Euler-based method requires neither state-space discretization nor input set discretization, thus avoiding a source of combinatorial explosion.
The correctness of these Euler-based methods does not rely on the hypothesis of incremental stability as in [8, 10], but on the hypothesis of ‘one-sided Lipschitz (OSL)’ condition with constant λP Rd (also called ‘monotonicity’/‘dissipativity’, see [7]). It can be seen that if a stochastic switched system satisfies an OSL con- dition withλ ă 0, then the functionVpx,x1q “ }x´x1}2is a common incremental Lyapunov functionin the sense of [9], from which it follows that the switched system is incrementally stable, and can be treated by approximate bisimulation. However, Euler- based methods also apply when the system is not incrementally stable, in which case the constantλis necessarily positive.
The plan of the paper is as follows: In Section 2, we give an explicit upper bound on the mean square error of the tamed Euler method for SDEs under OSL condition. We apply the result in order to ensure properties of stochastic switched systems, such as “pR,Sq- stability” (Section 3). We conclude in Section 4.
2 BOUNDING THE ERROR OF THE TAMED EULER METHOD
2.1 Assumptions
The symbol} ¨ }denotes the Euclidean norm onRd.The symbol x¨,¨ydenotes the scalar product of two vectors ofRd. Given a point x PRdand a positive realrą0, the ballBpx,rqof centrexand radiusris the settyPRd | }x´y} ďru.
Letτ P p0,8qbe a fixed real number, letpΩ,F,Pqbe a prob- ability space with normal filtrationpFtqtPr0,τs, letd,mP N:“ t1,2, . . .uletW “ pWp1q, . . . ,Wpmqq :r0,Rs ˆΩ ÑRm be an m-dimensional standardpWtqtPr0,τs-Brownian motion and letx0: ΩÑRdbe anF0{BpRdq-measurable mapping withEr}x0}ps ă 8for allp P r1,8q. Moreover, let f : Rd ÑRd be a continu- ously differentiable and globally one-sided Lipschitz continuous function whose derivative grows at most polynomially and let д“ pдi,jqiPt1, ...,du,jPt1, ...,mu :Rd ÑRdˆm be a globally Lips- chitz continuous function.
Then consider the Stochastic Differential Equations (SDE):
dXt“fpXtqdt`дpXtqdWt, X0“x0 (1) fortP r0,τs. The drift coefficientf is the infinitesimal mean of the processXand the diffusion coefficientдis the infinitesimal stan- dard deviation of the processX. Under the above assumptions, the SDE (1) is known to have a unique strong solution. More formally, there exists an adapted stochastic processX :r0,τs ˆΩ ÑRd with continuous sample paths fulfilling
Xt,x0“x0` żt
0
fpXsqds` żt
0
дpXsqdWs
for allt P r0,τsP-a.s. (see, e.g., [6]).
We denote byXt,x0the solution of Equation (1) at timetfrom initial conditionX0,x0“x0P-a.s., in whichx0is a random variable that is measurable inF0.
We suppose thatf behaves polynomially andдis Lipschitz, i.e.:
there exist constantsDPRě0,qPNandLдPRě0such that, for allx,yPRd
}fpxq ´fpyq}2ďD}x´y}2p1` }x}q` }y}qq (H1) }дpxq ´дpyq} ďLд}x´y} (H2) We also assume that the SDE (1) satisfies the following one-sided Lipschitz (OSL) condition with constantλPR:
DλPR@x,yPRd: xfpyq ´fpxq,y´xy ďλ}y´x}2 (H3) Remark 1. Constantsλ,LдandDcan be computed using (con- strained) optimization algorithms (see [1]).
2.2 Tamed Euler approximation
The standard way to extend the classical Euler method for ordi- nary differential equations to the SDE (1) is the Euler-Maruyama scheme [4]. More precisely, givenz : Ω Ñ Rd anF0{BpRdq- measurable mapping withEr}z}ps ă 8for allp P r1,8q, the explicit Euler-Maruyama (EM) method for the SDE (1) is given by the mappingsYn,zN : Ω Ñ Rd,n P t0,1, . . . ,Nu, which satisfy Y0N,z “zand
Yn`N1,z“Yn,zN ` τ
N ¨fpYn,zN q `дpYn,zN qpWpn`1qτ{N ´Wnτ{Nq
for allnP t0,1, . . . ,N´1uand allNPN. See [4]. Unfortunately, the convergence results for the EM scheme does not hold when the drift functionf of the SDE (1) behaves polynomially (and not linearly). For the sake of generality, we will now adopt a refined scheme, which has been proposed recently in order to overcome this difficulty [3]. LetXn,zN :ΩÑRd,
Xn`N 1,z“Xn,zN `
Nτ ¨fpXNn,zq
1`Nτ ¨ }fpXn,zN q}`дpXn,zN qpWpn`1qτ
N ´Wnτ
Nq (2) for allnP t0,1, . . . ,N´1uand allN PN. We refer to the numerical method (2) as atamed Euler scheme[3]. In this method the drift termτn ¨fpXNn,zqis “tamed” by the factor 1{p1`Nτ ¨ }fpXNn,zq}q fornP t0,1, . . . ,N´1uandNPNin (2).
A time continuous interpolation of the time discrete numerical approximations (2) is also introduced in [3] as follows. LetX˜zN : r0,τs ˆΩ ÑRd,N PN, be a sequence of stochastic processes given by
X˜t,zN “X˜n,zN `pt´nτ{Nq ¨fpX˜n,zN q
1`τ{N¨ }fpX˜n,zN q} `дpX˜n,zN qpWt´WnτNq for alltP rnτN,pn`N1qτs,nP t0,1. . . ,N´1uand allN PN. Note thatX˜t,zN :r0,τs ˆΩÑRdis an adapted stochastic process with continuous sample paths for everyNPN.
Let us defineXNt,zby
XNt,z :“Xn,zN fortP rnτ
N,pn`1qτ N q.
Note thatX˜t,zN “Xt,zN “Xn,zN at timet“nτN fornP t0,1, . . . ,Nu. The following theorem is proven in [3]:
Theorem 1. [3] SupposepH1qpH2qpH3q. Let the setting in this section be fulfilled, andz :ΩÑRdbe anF0{BpRdq-measurable mapping withEr}z}ps ă 8for allp P r1,8q. Then, for allp P r1,8q
sup
NPN
sup
nPt0,1, ...,NuEr}XNn,z}ps ă 8
For the sake of simplicity, the numberNof subsampling steps is now left implicit. From Theorem 1, it follows (cf. Lemma 4.3, [2]):
Lemma 2. SupposepH1qpH2qpH3q. Let the setting in this section be fulfilled, andz:ΩÑRdbe anF0{BpRdq-measurable mapping withEr}z}ps ă 8for allpP r1,8q. Then, for any even integerrě2, there exist two constantsEr,zandFr,zsuch that
sup
0ďtďτE}Xt,z´X˜t,z}r ď p∆tqr2pEr,zp∆tqr2`Fr,zdq.
with∆t“τ{Nand:
Er,z“2rp}fp0q}r `D2
r`1 2
p1`Esup
0ďtďτ}Xt,z}qrq
1 2pEsup
0ďtďτ}Xt,z}2rq
1 2q, Fr,z “2rp}дp0q}2r`LrдEsup
0ďtďτ}Xt,z}r2q.
Proof. see Appendix.
□
Remark 2. ConstantsEr,zandFr,zare computed using the con- stantsλandLд(see Remark 1), and the expected values ofXt,zat each timet “0,∆t,2∆t, . . . ,N∆t. These expected values are com- puted using a Monte Carlo method (by averaging here the value of104 samplings).
2.3 Mean square error bounding
The following Theorem holds for SDE (1). This corresponds to a stochastic version of Theorem 1 of [1], showing that a similar result holds on average, using the tamed Euler method of [3]. It is an adaptation of Theorem 4.4 in [2].
Theorem 3. Given the SDE system(1)satisfying (H1)-(H2)-(H3). Letδ0PRě0. Suppose thatzis a random variable onRd such that
Er}x0´z}2s ďδ02. Then, we have, for allτě0:
Er sup
0ďtďτ}Xt,x0´X˜t,z}2s ďδτ,δ2
0,
withδτ,δ2
0
:“βpτqeγ τ, where:
γ “2p?
∆t`2λ`L2д`128L4дq, and βpτq “2δ02
`2τ∆tL2дp1`128L2дqpF2,zd`E2,z∆tq
`4τa
∆tDpF4,zd`E4,z∆2tq
1 2
p1`4E sup
0ďtďτ}Xt,z}2q`4E sup
0ďtďτ}X˜t,z}2qq
1 2. (3) with∆t “τ{N.
Proof. The proof closely follows the proof of Theorem 4.4 in [2].
Letet “Xt,x0´X˜t,z. We have, for all 0ďtďτ:
det “ pfpXt,x0q ´fpzqqdt` pдpXt,x0q ´дpzqqdWt. (4) Then, by using Equation (4) and the integral version of Itô formula applied to functionx ÞÑ }x}2we obtain
}et}2“ }e0}2`
żt
0
2xes, fpXs,x0q ´fpXs,zqyds
` żt
0
}дpXs,x0q ´дpXs,zq}2ds`Mptq,
(5)
wheree0“x0´z, and Mptq “
żt
0
2xes,дpXs,x0q ´дpXs,zqydWs. So we have using (H2):
2
}et}2ď }e0}2`
żt
0
2xes, fpXs,x0q ´fpX˜s,zqyds
`L2д żt
0
}Xs,x0´Xs,z}2ds
` żt
0
2xes,fpX˜s,zq ´fpXs,zqyds`Mptq.
(6)
So we have using (H3) and Young’s inequality:
}et}2ď }e0}2`
żt
0
p2λ}es}2`L2д}es}2qds
`L2д żt
0
}Xs,z´X˜s,z}2ds
` żt
0
p 1
?∆t}fpX˜s,zq ´fpXs,zq}2`a
∆t}es}2qds
`Mptq.
(7) So we have using (H1), for all 0ďtďτ:
}et}2ď
}e0}2` pa
∆t`2λ`L2дq żt
0
}es}2ds
`L2д żt
0
}Xs,z´X˜s,z}2ds
` D
?∆t
żt
0
p1` }Xs,z}q` }X˜s,z}qq}Xs,z´X˜s,z}2ds
`Mptq.
(8) It follows using Lemma 2 forr “2, and Cauchy-Schwarz in- equality:
Er sup
0ďsďt}es}2s ď
E}e0}2` pa
∆t`2λ`L2дq żt
0
E}es}2ds
`L2дτ∆tpE2,z∆t`F2,zdq
` D
?∆t żt
0
pEp1` }Xs,z}q` }X˜s,z}qq2q
1
2pE}Xs,z´X˜s,z}4q
1 2ds
`mptq,
(9) where
mptq “Er sup
0ďsďt}Mpsq}s.
Hence, using using Lemma 2 forr“4, and inequalitypa`bqr ď 2rpar `brq:
Er sup
0ďsďt}es}2s ď
E}e0}2` pa
∆t`2λ`L2дqq żt
0
E}es}2ds
`L2дτ∆tpE2,z∆t`F2,zdq
`2Dτa
∆tpE4,z∆2t`F4,zdq12 p1`4E sup
0ďtďτ}Xt,z}2q`4E sup
0ďtďτ}X˜t,z}2qq
1 2
`mptq.
(10) On the other hand, from the Burkholder-Davis-Gundy inequality, we get:
mptq ď16Er żt
0
}es}2}дpXs,x0q ´дpXs,zq}2dss12 Hence, using (H2):
mptq ď16L2дEr sup
0ďsďt}es}2 żt
0
}Xs,x0´Xs,z}2dss12 Then, using Young’s inequality (for anyαą0):
mptq ď8L2дpαEr sup
0ďsďt}es}2s ` 1 αEr
żt
0
}Xs,x0´Xs,z}2dssq.
Hence, by using Lemma 2 forr“2:
mptq ď8αL2дEr sup
0ďsďt}es}2s
` 8L2д
α żt
0
Er sup
0ďrďs}er}2sds
` 8L2д
α τ∆tpE2,z∆t`F2,zdq.
(11) Hence, lettingα“161L2
д
, we have by replacing in (10):
1 2Er sup
0ďsďt}es}2s ď δ02
` pa
∆t`2λ`L2д`128L4дq żt
0
Er sup
0ďrďs}er}2sds
`τpL2д`128L4дq∆tpE2,z∆t`F2,zdq
`τ2Da
∆tpE4,z∆2t`F4,zdq12 p1`4E sup
0ďtďτ}Xt,z}2q`4E sup
0ďtďτ}X˜t,z}2qq
1 2. (12) It results from Gronwall’s inequality:
Er sup
0ďtďτ}et}2s “βpτqeγ τ, with
3
γ“2p?
∆t`2λ`L2д`128L4дq, and βpτq “2δ02
`2τp∆tL2дp1`128L2дqpF2,zd`E2,z∆tq
`4τa
∆tDpF4,zd`E4,z∆2tq
1 2
p1`4E sup
0ďtďτ}Xt,z}2q`4E sup
0ďtďτ}X˜t,z}2qq
1 2. (13)
□
It follows from Theorem 3 and Jensen’s inequality:
Proposition 1. Consider two pointsx0andzofRd,and a positive real numberδ0. Suppose thatx0PBpz,δ0q(i.e.}x0´z} ďδ0). Then EXt,x0PBpX˜t,z,δt,δ0qfor allt P r0,τs.
It also follows from Theorem 3:
Proposition 2. In the setting of Theorem 3, the expressionδτ,δ0
tends to
δ0?
2e2λτ`Lд2`128Lд4 when∆t tends to 0 (i.e., whenNtends to8).
2.4 Implementation
This method has been implemented in the interpreted language Octave, and the experiments performed on a 2.80 GHz Intel Core i7-4810MQ CP U with 8 GB of memory. The implementation is an adaptation of the program described in [1] for controlling determin- istic switched systems, but makes use of the tamed Euler scheme for SDEs (with the error functionδgiven in Theorem 3) instead of the classical Euler scheme.
Example 1. Consider the following system, corresponding to the example in Section 6.2 of [8] (cf. [9]) for modeu“1:
dx1“ p´0.25x1`x2`0.25qdt`0.05x1dWt1 dx2“ p´2x1´0.25x2´2qdt`0.05x2dWt2
The program gives (forτ “ 1,∆t “ τ{104):q “ 0,D “ 1.36, Lд“0.05,λ“0.25; and forz“ p´4,´3.8q:E2,z “893.3,E4,z“ 2.14¨105,F2,z“0.002,F4,z“4.9¨10´6.
Consider now the system corresponding to the example of [8] for modeu“2:
dx1“ p´0.25x1`2x2´0.25qdt`0.05x1dWt1 dx2“ p´x1´0.25x2`1qdt`0.05x2dWt2
The program gives (forτ“1,∆t “τ{104):q“0,D“1.36,Lд“ 0.05,λ“0.25, and, forz“ p0,3q:E2,z “543.2,E4,z“7.94¨104, F2,z“0.0442,F4,z“0.00178.
Both computations take less than 10 s. of CPU time. Simulations of the two systems are given in Figure 1 for modeu“1and starting pointz “ p´4,3.8q, and in Figure 2 for modeu “2and starting pointz“ p0,3q. On each figure, the initial ball (t “0) is depicted in black, the final ball (t “τ) in red, and 200 random sampling trajectories in blue fortP r0,τs.1
1Note that, in the figures, all the end points (att“τ) of the sampling trajectories lie in the final ball, but this is not true in general; we only know by Proposition 1 that, for all starting pointx0of the initial ball, theexpected valueof the end point lie in the final ball.
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2
-6 -4 -2 0 2 4
X1
X2
Figure 1: Example 1 with modeu “ 1,τ “ 1,∆t “ 10´4, initial ballBpz,δ0qwithz“ p´4,3.8qandδ0“0.5, final ball Bpz1,δτ,δ0qwithz1“ p´3.6,2.56qandδτ,δ0“1.17
-1 0 1 2 3 4 5 6
-1 0 1 2 3 4
X1
X2
Figure 2: Example 1 with modeu “ 2,τ “ 1,∆t “ 10´4, initial ballBpz,δ0qwithz “ p0,3qandδ0 “ 0.5, final ball Bpz1,δτ,δ0qwithz1“ p0.79,´0.63qandδτ,δ0“1.17
4
