Stabilizing two-dimensional stochastic systems through sliding mode control

Accepted Manuscript

Stabilizing two-dimensional stochastic systems through sliding mode control Xinghua Liu, Alessandro N. Vargas, Xinghuo Yu, Long Xu PII: DOI: Reference:

S0016-0032(17)30329-0 10.1016/j.jfranklin.2017.07.015 FI 3054

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

23 January 2017 14 June 2017 1 July 2017

Please cite this article as: Xinghua Liu, Alessandro N. Vargas, Xinghuo Yu, Long Xu, Stabilizing twodimensional stochastic systems through sliding mode control, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.07.015

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Stabilizing two-dimensional stochastic systems through sliding mode control ✩ Xinghua Liua , Alessandro N. Vargasb,c,∗, Xinghuo Yud , Long Xud School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798 b Universidade Tecnol´ ogica Federal do Paran´ a, UTFPR, Av. Alberto Carazzai 1640, 86300-000 Cornelio Proc´ opio-PR, Brazil. c Universitat Polit`ecnica de Catalunya Barcelona Tech, Escola d’Enginyeria de Barcelona Est, CoDAlab (Control, Dynamics and Applications), Carrer d’Eduard Maristany, 10-14, 08930 Barcelona, Spain. d School of Engineering, RMIT University, Melbourne VIC 3001, Australia

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Abstract

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Stability of stochastic differential equations has been investigated intensively during the past decades, showing a rich literature, see [2, 10, 17, 28, 29] for a brief account. Despite the progress made, stability results rely on general conditions based on n-dimensional systems, having n as either finite [17] or infinite [12]. That general conditions, though, do not apply in some practical circumstances—the case in which real-time control devices are modeled as two-dimensional systems (see an example in Section 3). This paper contributes towards the stability of two-dimensional stochastic systems. To clarify the contribution, some notations are now introduced. Given a complete filtered probability space (Ω, F , {Ft }t≥0 , P), consider the next two-dimensional stochastic

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1. Introduction

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This paper presents a stabilizing control for two-dimensional stochastic differential equations. The stability concept in this study is the stability in probability. To ensure such a stability, the control is designed based on the sliding mode technique, and applied to account stochastic systems. This finding has a practical implication—the proposed control can be used to stabilize a real-time automotive electronic throttle valve. The proposed approach is verified by data collected from experiments. Keywords: stochastic systems, stability in probability, sliding mode control, control applications, automotive applications.

✩ Research supported in part by the National Key Scientific Research Project of China (61233003) and by the Brazilian agencies FAPESP Grants 03/06736-7; CNPq Grant 304856/2007-0; and CAPES Grant Programa PVE 88881.030423/2013-01. ∗ Corresponding author Email address: [email protected] (Alessandro N. Vargas)

Preprint submitted to Elsevier

July 20, 2017

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differential equation. dx1 (t) = x2 (t)dt dx2 (t) = [a(x(t)) + b(x(t))u(t)]dt + c(x(t))dW (t),

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Definition 1.1. ([17, p. 110]). The system (1) is called stable in probability if for every ε ∈ (0, 1) and r > 0, there exists a δ = δ(ε, t0 , r) > 0 such that   Pr sup |x(t; t0 , x0 )| < r ≥ 1 − ε, t≥t0

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The feedback law f : R2 7→ R that stabilizes (1) takes the form known as sliding mode control. Sliding mode control has received attention in the literature, mostly for deterministic systems [4, 5, 24, 30], but contributions for stochastic systems exist as well, see [11, 13, 14, 15, 16]. These contributions, though, assume that c(·) is a continuous function with c(0) = 0—this assumption means that the influence of the additive noise W (t) in (1) vanishes as long as x(t) approaches the origin (this assumption can also be found in stochastic systems with time-delay [3, 9] and in its discrete-time counterpart [7, 8]). In constrast, our result accounts c(x) 6= 0 for all x ∈ R2 , a condition that considers stochastic disturbances acting on (1) for all t ≥ 0 and all x0 ∈ R2 . Our condition thus represents a step toward understanding the effects of stochastic disturbances on two-dimensional systems. The main result of the paper expands the deterministic result of [4, Thm. 2.2] for stochastic systems. Namely, our result recovers the one in [4, Thm. 2.2] when c(x(t)) ≡ 0 in (1) (see Remark 2.4). In fact, whereas [4, Thm. 2.2] shows stability in the deterministic sense, our result shows stability in probability, a weaker concept. This sets the main theoretical contribution of the paper. This paper shares motivation with [21] on finding conditions to assure stability of nonlinear stochastic systems. The results in [21] account the µ-zone mean square stability; our results, in contrast, account the stability in probability—a weaker stability concept. As for applications, sliding mode control has been used in practice, such as in robotics [6, 30], and in mechanics [1]. Here, sliding mode control finds a novel application. This paper also has a contribution for real-time applications. Indeed, our theoretical result was deployed to control an electronic automotive valve in a laboratory testbed. The data collected in the laboratory demonstrated the effectiveness of the proposed theory in real-time applications.

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whenever |x0 | ≤ δ.

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(1)

where a(x), b(x), c(x) : R2 7→ R are appropriate functions, x(t) = [x1 (t) x2 (t)]0 denote the system state evolving on R2 , u(t) denotes a control input on R, W (t) represents the standard Brownian motion on R, and x(t0 ) = x0 ∈ R2 defines an initial condition. The main contribution of the paper is to show a feedback law f : R2 7→ R such that the control u(t) = f (x(t)) turns the system (1) stable in probability. Now recall such a stability concept—for this purpose, consider x(t; t0 , x0 ) as the solution of (1).

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In summary, this paper brings benefits for both fronts, theory and applications. The remainder of this paper is organized as follows. Section 2 contains notation, definitions, assumptions, and main result. Section 3 presents data collected from real-time experiments, and finally, Section 4 presents some concluding remarks.

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2. Notation, definitions, and main result

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Assumption 2.1. The functions a(·), b(·), c(·) : R2 7→ R in (1) are chosen in such a manner that there exists a solution x(t) to the system (1). Remark 2.1. Assumption 2.1 is satisfied if the coefficients a(·), b(·), and c(·) are selected to satisfy both the global Lipschitz and linear growth conditions [17, Thm. 3.1, p. 51]. Assumption 2.1 can be guaranteed even if a(·), b(·), and c(·) present discontinuous points, see [23, p. 1221]. Assumption 2.2. There exist a constant L > 0 and a piecewise continuous function ϕ : R2 → R such that the function a : R2 → R in (1) satisfies |a(x) − ϕ(x)| ≤ L, for all x ∈ R2 . Remark 2.2. Assumption 2.2 allows the function a : R2 → R in (1) to be discontinuous, but the discontinuous points cannot exceed a distance L > 0 from some other piecewise continuous function ϕ. 2.1. Stabilizing control via sliding mode control The aim of this section is to present a feedback law f : R2 7→ R, taking the format u(t) = f (x(t)), such that the system (1) is stable in probability (recall Definition 1.1). This feedback law f is constructed upon a strategy called fast-terminal sliding mode manifold [4], as detailed next. Define

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The following notations are applied: Rn denotes the n-dimensional Euclidean space; R+ represents the set of nonnegative numbers; | · | represents the absolute value; sgn(·) denotes the signum function; 11{·} stands for the Dirac measure; and tr{·} denotes the trace operator on real-valued matrices. E[·] stands for the usual mathematical expectation. In (1), the term W (t) on R denotes a standard Brownian process. Let us consider the next assumptions.

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s(x(t)) = x2 (t) + αx1 (t) + β|x1 (t)|ξ sgn(x1 (t)),

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where α > 0, β > 0, and 0 < ξ < 1 are given constants.

Assumption 2.3. There exist two nonnegative constants K1 , K2 such that c2 (x) ≤ K1 + K2 s2 (x),

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∀x ∈ R2 .

Remark 2.3. The condition in Assumption 2.3 resembles the linear growth condition [17, Thm. 3.1, p. 51], which is necessary to show the main result. 3

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Let ε > 0, and consider the function hε : R2 7→ R+ defined as   ε + K2 hε (x(t)) = L + |s(x(t))|, ∀t ≥ t0 . 2 Now, we are able to present the main result of this paper. Theorem 2.1. Let the function f : R2 7→ R be defined as


f (x) = b−1 (x) −ϕ(x) − αx2 − hε (x) sgn(s(x)) − βξ|x1 |ξ−1 x2 ,

If the control u(t) in (1) satisfies u(t) = f (x(t)), then

E[s2 (x(t))] ≤ exp(−ε(t − t0 ))s2 (x0 ) +

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In addition, the system (1) is stable in probability.

The proof of Theorem 2.1 is presented in Section 2.2 below.

Remark 2.4. Theorem 2.1 retrieves the deterministic result of [4, Thm. 2.2] when α = K1 = K2 = 0 in (4)–(6); these null values impose no stochastic disturbances on (1) (cf., Assumption 2.3), so that (1) retrieves the deterministic case. This equivalence reveals that Theorem 2.1 expands the usefulness of the result from [4, Thm. 2.2] for the stochastic scenario. Remark 2.5. Theorem 2.1 affirms the stability in probability for the stochastic system (1), a condition weaker than the deterministic, finite-time stability to the origin shown in [4, Thm. 2.2]. Interestingly, the state x(t) in (1) never reaches exactly the origin when the Brownian motion W (t) keeps actuating (i.e., c(x(t)) 6= 0), see [17, Lem. 3.2, p. 120]. Hence, the result of [4] does not apply when (1) is driven by the Brownian motion W (t). 2.2. Proof of Theorem 2.1 The proof of Theorem 2.1 is divided into two parts. In the first part, we show (5) and (6); in the second part, we show that (5) and (6) suffice to the stability in probability for (1).

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    ε + K 2 E[s(x(t))] − exp − ≤ 4L , (t − t ) s(x ) 0 0 ε + K2 2

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Proof. (Part 1). Define the function V : R 7→ R+ as 1 V (s(x)) = s2 (x), 2

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Applying the Itˆo formula in (7) (see [17, p. 110]), we obtain

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From the assumption that {W (t)} is a standard Brownian process, we have that W (t) and x(t) are independent random variables for each t ≥ t0 ; as a result,        ∂V (s(x(t))) ∂V (s(x(t))) 0 0 dW (t) = E E [dW (t)] = 0, ∀t ≥ t0 , E c(x(t)) c(x(t)) ∂x ∂x

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∀t ≥ t0 . (8)

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  ∂V (s(x(t))) x2 (t) dV (s(x(t))) = dt a(x(t)) + b(x(t))u(t) ∂x    1 ∂ 2 V (s(x(t))) 0 + tr [0 c(x(t))] dt c(x(t)) 2 ∂ 2x   ∂V (s(x(t))) 0 + dW (t), c(x(t)) ∂x

where the righmost equality follows from the fact that E [dW (t)] = 0, see [17, p. 22]. Using this identity, and passing the expected value operator on both sides of (8), we obtain the differential equation "

Applying Lemma 4.2 (Appendix) in (9), we have that the right-hand side of (9) equals 
 E 2s(x(t)) αx2 (t) + βξ|x1 (t)|ξ−1 x2 (t) + a(x(t)) + b(x(t))u(t) + c2 (x(t)) dt. (10)

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  ∂V (s(x(t))) x2 (t) dE[V (s(x(t)))] = E a(x(t)) + b(x(t))u(t) ∂x    # 1 ∂ 2 V (s(x(t))) 0 + tr [0 c(x(t))] dt. (9) c(x(t)) 2 ∂ 2x

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  d E[V (s(x(t)))] = E 2s(x(t)) (a(x(t)) − ϕ(x(t)) − hε (x(t)) sgn(s(x))) + c2 (x(t)) . dt

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Substituting the control u(t) from (4) into (10) yields   E 2s(x(t)) [a(x(t)) − ϕ(x(t)) − hε (x(t)) sgn(s(x))] + c2 (x(t)) dt.

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From Assumptions 2.2 and 2.3, we have

  d E[V (s(x(t)))] ≤ 2E [|s(x(t))| (L − hε (x(t)))] + K1 + E K2 s2 (x(t)) . dt

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Now, by substituting the expression of hε (·) from (3) into (11), we obtain

E[−(ε + K2 )s2 (x(t)) + K2 s2 (x(t))] + K1 = −εE[V (s(x(t)))] + K1 5

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which is an upper bound for (11). Consequently, d E[V (s(x(t)))] ≤ −εE[V (s(x(t)))] + K1 , dt

which shows the first assertion. To show the second assertion, we proceed a similar analysis for s(·). Indeed, by expressing the dynamics of s(·) in the Itˆo formula, we have

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∀t ≥ t0 . (13)

Using Lemma 4.1 in (13) after passing the expected value on both sides of the resulting expression, we obtain   d E[s(x(t))] = E αx2 (t) + βξ|x1 (t)|ξ−1 x2 (t) + a(x(t)) + b(x(t))u(t) . dt

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The control u(t) as in (4) into (14) gives

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  ∂s(x(t)) x2 (t) dt ds(x(t)) = a(x(t)) + b(x(t))u(t) ∂x    1 ∂ 2 s(x(t)) 0 + tr [0 c(x(t))] dt c(x(t)) 2 ∂ 2x   ∂s(x(t)) 0 + dW (t), c(x(t)) ∂x

d E[s(x(t))] = E [a(x(t)) − ϕ(x(t)) − hε (x(t)) sgn(s(x))] . dt

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s(x(t)).

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    ε + K2 (t − t0 ) s(x0 ) E[s(x(t))] = exp − 2     Z t ε + K2 + exp − (t − τ ) E [a(x(τ )) − ϕ(x(τ )) − L sgn(s(x(τ )))] dτ. (16) 2 t0

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This argument completes the proof. (Part 2). Now, we show that (5) and (6) implies in the stability in probability for (1). Indeed, by substituting dx1 (t) = x2 (t)dt into the expression of s(x(t)) (see (2)), we have
dx1 (t) = −αx1 (t) + s(x(t)) − β|x1 (t)|ξ sgn(x1 (t)) dt. (17) 6

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It means that s(x(t)) is uniformly bounded (with respect to time) by the constant Mε for most of the trajectories x(t) from (1). That applied in (17) yields dx1 (t) ξ ≤ Mε . + αx (t) + β|x (t)| sgn(x (t)) (19) 1 1 1 dt

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On the other hand, it follows from (5) that supt≥t0 E[s2 (x(t))] is finite, and according to the Markov’s inequality, there exists some constant Mε = M (ε), for any ε > 0, such that Pr [ω : |s(x(t, ω))| ≤ Mε , ∀t ≥ t0 ] ≥ 1 − ε. (18)

Now, we deploy (19) for showing that x1 (t) is uniformly bounded. To show this result, we employ a contradiction argument; namely, we assume that |x1 (t)| diverges to infinity and prove that it is an absurd. The contradiction argument follows through two cases. Case 1: Suppose that supt≥t0 x1 (t) = +∞. In this case, we can choose a sequence of instants t1 < t2 < · · · < tk < · · · such that x(tk ) > k for all k ≥ 1. Since the inequality

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The rightmost inequality allows us to write x1 (tk ) ≤ exp(−α(tk − t0 ))x1 (t0 ) + Mε /α, which represents an absurd because it means that x1 (tk ) does not diverge to infinity. This argument proves that x1 (t) is bounded from above for all t ≥ t0 . Case 2: Suppose that supt≥t0 x1 (t) = −∞, and take t1 < t2 < · · · < tk < · · · such that x(tk ) < −k for all k ≥ 1. It follows that −β|x1 (tk )|ξ sgn(x1 (tk )) > 0 for all k ≥ 1. Besides, we can use this inequality and (19) to write

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The absurd then arises because this inequality assures that x1 (tk ) is bounded from below by exp(−α(tk − t0 ))x1 (t0 ) − Mε /α, which contradicts our initial assumption. In summary, the argument described in Cases 1 and 2 proves that the trajectory x1 (t, ω) is uniformly bounded (with respect to time) whenever it corresponds to the probability level in (18). Finally, to complete the proof, one can check that x2 (t, ω) is uniformly bounded provided that x1 (t, ω) is uniformly bounded due to (19).

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3. Applying the control of Theorem 2.1 to stabilize an automotive electronic throttle valve This section aims to illustrate the usefulness of Theorem 2.1 in practice. Experiments were performed so as to check the control u(t) from (4) in practice. The experimental data were collected in the laboratory described in [26]—the devices, as well as the automotive throttle valve, were the same as the ones described in [26]. 7

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3.1. Automotive throttle valve: modeling Recall that the automotive electronic throttle valve is a mechatronic device used in the automotive industry to control the power produced by spark-ignition engines [19, 20, 22, 26, 31]. The automotive throttle valve has been modelled successfully by a three-dimensional system, which accounts (i) the angular position of the throttle valve; (ii) the corresponding angular velocity; and (iii) the consumed electrical current. The system input corresponds to the voltage applied in the equipment terminals. The authors of [18, 26, 27] suggest piecewise linear systems to model the throttle device. In particular, [26] presents a discrete-time model accounting additive noise input modeled as a Gaussian process; here, the Gaussian process is converted into a Brownian motion because we are interested in the continuous-time counterpart of [26]. According to [26], three main regions of operation should be used to represent the throttle position, i.e., Θ1 = [0◦ , 8◦ ], Θ2 = (8◦ , 16◦ ], and Θ3 = (16◦ , 90◦ ]. With x(t) = [x1 (t) x2 (t) x3 (t)]0 ∈ R3 representing the position, velocity, and electrical current, respectively, we obtain

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   0 1 0 0 (s) (s) (s) (s) (s)  dt dx(t) = a21 a22 a23  x(t)dt + c(s) 1 sgn(x2 (t)) + c2 sgn(x1 (t) − 10) + c3 (s) (s) 0 0 a32 a33     0 0    + 0 u(t)dt + 0  dW (t), x1 (t) ∈ Θs , s = 1, 2, 3, ∀t ≥ 0, (20) b(s) f (s)

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dx1 (t) = x2 (t)dt dx2 (t) = [a(x(t)) + b(x(t))u(t)] dt + c(x(t))dW (t),

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3.2. Automotive throttle valve: experiments The aim of this section is to illustrate the benefits of Theorem 2.1 for applications. As a byproduct, the stability in probability for the throttle valve is checked, as detailed next. The control strategy u(t) = f (x(t)) from Theorem 2.1 was implemented in the laboratory, according to the scheme shown in Fig. 1. The control used online measurements from both sensors, position and velocity. To be evaluated in practice, the function f (·) in Theorem 2.1 required some parameters— in the experiments they were chosen so as to respect the constraints of the throttle model (21). The chosen values were ξ = 0.6, ε = 0.012, L = 9, α = 0.03, β = 1, L = 9, K1 = 100, and K2 = 0. The experiments followed the next procedure. The throttle valve was kept opened, fixed at three distinct positions, i.e., 6.3◦ , 54◦ , and 75◦ . Afterwards, the control u(t) = f (x(t)) entered in operation. Fig. 2 shows the experimental outcome. As can be seen, the process showed a stable behaviour, reaching a region around the origin; the origin means the throttle valve is closed. For the sake of improving the stability interpretation, it was necessary to carry out additional experiments—five hundred experiments were carried out in the laboratory for each initial position: 15◦ , 35◦ , and 54◦ . The data is summarized in Fig. 3. The curves depicted in Fig. 3 suggest that the automotive throttle valve be not only second moment stable [25], but also stable in probability—the latter property confirmed by Theorem 2.1. In conclusion, experimental evidence indicates that the control of Theorem 2.1 can be applied successfully in real-time applications—particularly in stabilizing an automotive electronic throttle valve.

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Figure 2: Experimental result for the automotive throttle valve. The cases correspond to three distinct realizations, taking distinct initial positions. The curves suggest that the control of Theorem 2.1 be able to drive the automotive valve from an opened position to a closed position, in a stable trajectory.

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Appendix

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This paper proposed a control strategy for stabilizing two-dimensional stochastic systems. Theorem 2.1, the main result of the paper, expands the result of [4, Thm. 2.2] for stochastic systems. In fact, it has been shown that the result in [4, Thm. 2.2] is a particular case of Theorem 2.1. The effectiveness of Theorem 2.1 was verified in practice. To do so, experiments have been designed for controlling an automotive electronic throttle valve in a laboratory testbed. Collected data confirmed the usefulness of Theorem 2.1 for real-time applications.

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The next two lemmas are used in the proof of Theorem 2.1.

Lemma 4.1. If s(·) satisfies (2), then there hold  ∂s(x)  = α + βξ|x1 |ξ−1 1 ∂x

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  ∂ 2 s(x) βξ (ξ − 1) x1 |x1 |ξ−2 0 = . 0 0 ∂ 2x

Lemma 4.2. Let s(·) be as in (2), and define V (s(x)) = s2 (x) for all x ∈ R2 . Then   ∂V (s(x)) = 2s(x) α + βξ|x1 |ξ−1 1 ∂x 10

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Figure 3: Three phase portraits represent the position and velocity measured from an automotive throttle device. Each curve corresponds to 500 distinct realizations. The shading of the colors stands for the statistical dispersion of the measured values around their statistical mean. Each phase portrait follows an spiral path, from outside to inside, reaching the equilibrium around the innermost circle—a behaviour that resembles an ellipsoidal limit cycle (colored in red). Such a limit cycle suggests that the automotive throttle device be stable in probability, an evidence in accordance with the result of Theorem 2.1.

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2 α + βξ|x1 |ξ−1 + βξ(ξ − 1)s(x)x1 |x1 |ξ−3 α + βξ|x1 |ξ−1 . α + βξ|x1 |ξ−1 1

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