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Three-dimensional trajectory design for horizontal well based on optimal switching algorithms Xiang Wu a,n, Kanjian Zhang b,c a
School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, PR China School of Automation, Southeast University, Nanjing 210096, PR China c Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing 210096, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 5 February 2015 Received in revised form 25 March 2015 Accepted 10 April 2015 This paper was recommended for publication by Jeff Pieper.
This paper considers a three-dimensional trajectory design problem for horizontal well. The problem is formulated as an optimal control problem of switched systems with continuous state inequality constraints. Since the complexity of such constraints and the switching instants is unknown, it is difficult to solve the problem by standard optimization techniques. To overcome the difficulty, by a timescaling transformation, a smoothing technique and a penalty function method, an efficient computational method is proposed for solving this problem. Convergence results show that, for a sufficiently large penalty parameter, any local optimal solution of the approximate problem is also a local optimal solution of the original problem. Two numerical examples are presented to illustrate the efficiency of the approach proposed. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Optimal control Horizontal well trajectory design Time-scaling transformation Smoothing technique Penalty function method
1. Introduction In general, a well trajectory is a three-dimensional curve, where the trajectory is described as a combination of turn sections and straight sections. The main objective of designing a horizontal well trajectory is to construct a trajectory that reaches a given target at a specified inclination and azimuth from a given starting location subject to various constraints arising from engineering specifications. Several well trajectory designing software are available commercially for finding an appropriate horizontal well trajectory. However, most of them use a trial-and-error procedure to obtain a solution. In addition to being time-consuming and depending on user-experience, these softwares are limited to designing simple well trajectories, and may not generate an optimal trajectory, since the trial-and-error search is usually user-driven. Thus, some methods have been proposed to design a three-dimensional well trajectory. In 1979, Planeix and Fox [1] for the first time considered the problem. In their work, an approach to reach a given target from a current position was obtained by controlling the azimuth and the inclination independently. Guo et al. [2] proposed a
n
Corresponding author. Tel.: þ 86 85186702059. E-mail addresses: [email protected] (X. Wu), [email protected] (K. Zhang).
constant curvature method to solve the set end (set inclination and set azimuth at the target) three-dimensional well trajectory design problem in 1992. In 1995, Ebrahim [3] obtained general expressions of the three-dimensional well trajectory calculation by vector algebra and differential geometry. In 2001, Liu and Shi [4] developed a general method where two circle arcs and a straight line are used to obtain simple three-dimensional trajectories to solve the set end three-dimensional well trajectory problem. Suppose that the three-dimensional well trajectory is a combination of several smooth turn segments, two three-dimensional well trajectory design methods are developed by heuristic or direction search methods in [5,6]. For more discussions on various literature results, the reader may refer to [7–14] and the references therein. Switched systems are a particular class of hybrid systems, which have received considerable attention in the past two decades. Most work focuses on stability and stabilization problems [15–25]. From the application perspective, however, besides stability and stabilization, performance is also crucial. Recently, optimal control problems of switched systems have been attracting researchers from various fields in science and engineering, due to their significance in theory and applications [26–33]. These works are both theoretical results and numerical methods. The available theoretical results usually extend the classical maximum principle or the dynamic programming approach to switched systems. The numerical methods take advantage of efficient
http://dx.doi.org/10.1016/j.isatra.2015.04.002 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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2
nonlinear optimization techniques and high-speed computers to develop efficient numerical methods for the optimal control of switched systems [34]. Theoretical results of switched systems are now well developed. However, for switched systems with nonlinear modes, analytical techniques are not sufficient to determine an optimal control policy. In addition, many dynamic control processes are subject to continuous-state equality or inequality constraints, such as container cranes [35] and aircraft trajectory planning [36]. Note that one continuous-time or one continuousstate constraint is equivalent to an uncountable number of conventional constraints. Thus, it is difficult to solve the problem by conventional optimization techniques. The existing three-dimensional well trajectory design approaches have always their drawbacks. For example, the approach given in [6] is computationally expensive with poor convergence properties. In addition, the continuous state inequality constraints arising from engineering specifications are ignored in [7,8]. Then some optimal solutions obtained using the algorithms given in [7,8] actually fail to satisfy the continuous state inequality constraints at some points along the well trajectory. This is clearly undesirable in practice. Furthermore, compared with nonlinear systems, switched systems have many very good properties. For instance, the analysis and control of switched systems are easier due to their structures being simple. Thus, we reconsider the three-dimensional horizontal well trajectory design problem under the framework of switched systems. The main contribution of this paper is that the three-dimensional trajectory design problem for horizontal well is formulated as an optimal control problem of switched systems with continuous state inequality constraints, and an efficient smoothed penalty approach is proposed to solve the problem. By introducing a time-scaling transformation [37] and a novel smoothing technique, all the continuous state inequality constraints are incorporated into the original objective function by the l1 penalty function, giving rise to an approximate nonlinear parameter optimization problem that can be solved using any gradient-based method. Convergence results indicate that any local optimal solution of the approximate problem is also a local optimal solution of the original problem as long as the penalty parameter is sufficiently large. Compared with the existing methods, the penalty parameter and the smoothing parameter of our proposed algorithm are not fixed artificially by experience. The rest of the paper is organized as follows. Section 2 presents the three-dimensional trajectory design problem for horizontal well. A time-scaling transformation is introduced is Section 3. Then, in Section 4, by the time-scaling transformation, the three-dimensional trajectory design problem for horizontal well is transformed into an equivalent problem with fixed switching instants. However, these constraints are very complex, which pose a challenge for standard optimization algorithms. Thus, in Section 5, by a smoothing technique and a penalty function method, the problem is transformed into a sequence of unconstrained problems. Each of these unconstrained problems can be effectively solved by any gradient-based optimization technique. The gradient formulae of the cost function and our algorithm are given in Section 6. Section 7 provides the Convergence analysis. Two numerical examples in Section 8 show that our method is effective.
2. Problem formulation As shown in Fig. 1, a three-dimensional trajectory design problem for horizontal well is considered. The trajectory design problem is described in the Cartesian coordinate system, with xaxis representing North/South (positive x being North), yaxis representing East/West (positive y being East), and the zaxis (positive z being downward) representing the true vertical depth. The arc length from the kick-off point is denoted by t, and any point P(t) on the curve is described by its inclination βðtÞ, azimuth
Fig. 1. Three-dimensional trajectory design for horizontal well.
ϕðtÞ and coordinate ðx ðt Þ; y ðt Þ; z ðt ÞÞ. In order to make the model more close to the practical problems, suppose that the following three conditions are satisfied: Assumption 1. The well trajectory is a combination of n smooth turn segments. Assumption 2. The curvature Ki and the tool-face angle γi are constants in each ith turn segment, i ¼ 1; …; n. Assumption 3. The design of the well trajectory is only for nonstraight horizontal well. Then, the well trajectory, which is required to reach a given target from the kick-off point at a specified inclination and azimuth, can be described the following switched system: x_ ðt Þ ¼ f xðt Þ; ζ i ; t A ½t i 1 ; t i Þ; i ¼ 1; …; n; ð1Þ xðt i Þ ¼ x t i ;
i ¼ 1; …; n 1;
xð0Þ ¼ x0 ;
ð2Þ ð3Þ
where t is the arc length from the kick-off point; βðt Þ and ϕðt Þ are the inclination and azimuth of any point P ðt Þ on the curve respectively; xðt Þ ¼ ðβðt Þ; ϕðt Þ; x ðt Þ; y ðt Þ; z ðt ÞÞ is the location of the point P ðt Þ on the curve; Ki is the curvature; γi is the tool-face angle which is the and position direction of the deflecting tool bowed angle; ζ i ¼ K i ; γ i ; i ¼ 1; …; n; x t i ¼ limt-ti xðt i Þ; i ¼ 1; …; n 1; x0 is a given initial state; t 0 ¼ 0 is the initial arc length; t i ; i ¼ 1; …; n 1 are the switching instants; tn is the well's whole trajectory length; and 0 1 K i cos γ i B C K i sin γ i B C sin βðt Þ C B B C i f xðt Þ; ζ ¼ B sin βðt Þ cos ϕðt Þ C: ð4Þ B C B sin βðt Þ sin ϕðt Þ C @ A cos βðt Þ Define n o Γ ¼ τ ¼ ðt 1 ; …; t n ÞT A Rn : t i t i 1 A c1 ; d1 ; i ¼ 1; …; n ;
ð5Þ
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
X. Wu, K. Zhang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
n o T Ξ ¼ ζ ¼ ζ 1 ; …; ζ n A R2n : ζ i A c2 ; d2 c3 ; d3 ; i ¼ 1; …; n ;
ð6Þ
where ci and di, i ¼ 1; 2; 3, are given real numbers such that ci o di , i ¼ 1; 2; 3. Because of engineering specifications, for any t A ½0; t n , βðt Þ and ϕðt Þ are required to satisfy the following constraints: π δ0 rβðt Þ r ; 2
ð7Þ
0 r ϕðt Þ r 2π;
ð8Þ
where δ0 is a given positive number. Lemma 1. The function f defined in (4) satisfies that (1) f is continuously differentiable. (2) For any x; ζ i A R5 c2 ; d2 c3 ; d3 , there exists a real number L 40 such that ð9Þ J f x; ζ i J r Lð1 þ J x J Þ;
3
Lemma 1 ensures that the switched system (1)–(3) has a unique solution. Our goalTis to construct a well trajectory to reach the target xf ¼ xf1 ; …; xf5 from the kick-off point x0 by choosing the parameter vectors τ and ζ such that the cost function J ðτ; ζ Þ ¼ t n þ η
5 X j¼1
xj ðt n jτ; ζ Þ xfj
2
;
ð11Þ
is minimized, where η is a given weight coefficient and the cost function (11) denotes the weighted sum of the trajectory length and the error of reaching the target, xj ðt Þ, j ¼ 1; …; 5, are the jth component of the state vector xðt Þ, i.e. x1 ðt Þ ¼ βðt Þ; x2 ðt Þ ¼ ϕðt Þ; x3 ðt Þ ¼ x ðt Þ; x4 ðt Þ ¼ y ðt Þ; x5 ðt Þ ¼ z ðt Þ, xðtjτ; ζ Þ denotes the solution of switched system (1)–(4). Now, we define our optimal control problem formally as follows. Problem 1. Given the switched system (1)–(3), choose ðτ; ζ Þ A Γ Ξ such that the cost function (11) is minimized subject to the continuous state inequality constraints (7) and (8). 3. Time-scaling transformation
where J J denotes the usual Euclidean norm.
Define a function tðsÞ : ½0; n-R by the following differential equation [37]:
Proof. (1) Obviously, differentiable. f is continuously (2) For any x; ζ i A R5 c2 ; d2 c3 ; d3 , we have 0 11=2 5 X 2 i i A @ J f x; ζ J ¼ f j x; ζ
t_ ðsÞ ¼
¼
2
cos γ i þ
K 2i sin 2 γ i sin 2 βðt Þ
2
þ sin βðt Þ cos ϕðt Þ
!1=2 2 2 2 K 2i cos 2 γ i sin βðt Þ þK 2i sin γ i þ sin βðt Þ 2
2
2
2
d2 1 sin βðt Þ þ d2 1 þ sin βðt Þ
r
!1=2
2
where L ¼ x1 ðt Þ is the first component of the state vector xðt Þ, i.e. x1 ðt Þ ¼ βðt Þ. The proof is complete.□
0 o c 1 r θ i ¼ t i t i 1 r d1 ;
i ¼ 1; …; n;
ð15Þ
Let θ ¼ ½θ1 ; …; θn and let Θ be the set containing all such θ. Integrating (12) with initial condition (13) yields that, for any s A k 1; kÞ, k ¼ 1; …; n,
0
d2 þ 1 , sin 2 δ0
The above differential transformation (12)–(14) is called a timescaling transformation. Clearly, θi satisfies T
2
sin δ0 11=2 2 2 d2 þ 1 sin 2 βðt Þ þ d2 @ A ¼ 2 sin δ0 11=2 0 2 2 2 d2 þ 1 sin βðt Þ þ d2 þ 1 A o@ sin 2 δ0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1=2 d2 þ 1 2 ¼ sin βðt Þ þ 1 2 sin δ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 2 1=2 d2 þ 1 2 β ðt Þ þ 1 r 2 sin δ0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d2 þ 1 βðt Þ þ 1 r sin 2 δ0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d2 þ 1 ¼ x1 ðt Þ þ 1 sin 2 δ0 rffiffiffiffiffiffiffiffiffiffiffi Lð J x J þ 1Þ; r
ð13Þ
where θi is the duration of the ith turn segment in the original arc length horizon, and for a given interval I ½0; n, the characteristic function χ I ðsÞ is defined by
1 if s A I; ð14Þ χ I ðsÞ ¼ 0 otherwise:
2
sin 2 βðt Þ
ð12Þ
t ð0Þ ¼ 0;
1=2 2 2 þ sin βðt Þ sin ϕðt Þ þ cos 2 βðt Þ !1=2 K 2 sin 2 γ i þ1 ¼ K 2i cos 2 γ i þ i 2 sin βðt Þ ¼
θi χ ½i 1;iÞ ðsÞ;
i¼1
j¼1
K 2i
n X
t ðsÞ ¼
kX 1
θi þ θk ðs k þ1Þ:
ð16Þ
i¼1
Thus, for each i ¼ 1; …; n, we have t ðiÞ ¼
i X
θk ¼
k¼1
i X
ðt k t k 1 Þ ¼ t i :
ð17Þ
k¼1
In particular, t ðnÞ ¼ t n .
4. Problem transformation
ð10Þ
Since the state in Problem 1 depends on the switching instants, it is difficult to integrate the switched system (1)–(3) numerically. To overcome the difficulty, a more tractable equivalent problem will be derived. Let x~ ðsÞ ¼ ðxðt ðsÞÞ; t ðsÞÞT and f~ x~ ðsÞ; θ; ζ i ¼ ððθi f x~ ðsÞ; ζ i ÞT ; θi ÞT . Then, applying (12) and (13) to (1)–(3), (7), (8) and (11) yields x~ ðsÞ ¼ f~ x~ ðsÞ; θ; ζ i ; s A ½i 1; iÞ; ð18Þ x~ ðiÞ ¼ x~ ði Þ;
i ¼ 1; …; n 1;
ð19Þ
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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4
h iT x~ ð0Þ ¼ x~ 0 ¼ ðx0 ÞT ; 0 ;
ð20Þ
Problem 3. Given the switched system ðθ; ζ Þ A Θ Ξ such that the cost function
π δ0 r x~ 1 ðsj θ; ζ Þ r ; 2
ð21Þ
J^ ðθ; ζ Þ ¼ x~ 6 ðnj θÞ þ η
for any s A ½0; n;
5 2 X x~ j ðnj θ; ζ Þ xfj
j¼1
0 r x~ 2 ðsj θ; ζ Þ r 2π;
for any s A ½0; n;
ð22Þ
þρ
4 Z X j¼1
J~ ðθ; ζ Þ ¼ x~ 6 ðnj θÞ þ η
5 X j¼1
x~ j ðnj θ; ζ Þ xfj
2
ð23Þ
where x~ j ðsj θ; ζ Þ, j ¼ 1; …; 6, are the jth component of the unique solution vector of the switched system (18)–(20). Then, a new optimal control problem is defined as follows. Problem 2. Given the switched system (18)–(20), choose ðθ; ζ Þ A Θ Ξ such that the cost function (23) is minimized subject to the continuous state inequality constraints (21) and (22). Clearly, Problems 1 and 2 are equivalent.
n 0
n o max g j ðx~ ðsj θ; ζ ÞÞ; 0 ds;
ð30Þ
is minimized, where ρ4 0 is the penalty parameter. n o Note that max g j ðx~ ðsj θ; ζ ÞÞ; 0 , j ¼ 1; …; 4, are non-smooth functions in θ and ζ. Thus, any gradient-based optimization algorithm would have difficulty in dealing with Problem 3. To overcome this difficulty, we introduce a novel osmooth function n given in (31) to approximate max g j ðx~ ðsj θ; ζ ÞÞ; 0 : 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð31Þ ψ ðω; αÞ ¼ ω2 þ 4α2 þ ω ; 2 where α 40 is an adjustable parameter. Lemma 2. For any ω A R and α 4 0, ψ ðω; αÞ has the following properties:
5. Smoothing technique and penalty function method To begin, for any s A ½0; n, the inequality constraints (21) and (22) are written as g j ðx~ ðsj θ; ζ ÞÞ r0;
(18)–(20), choose
j ¼ 1; 2; 3; 4;
ð24Þ
g 1 ðx~ ðsj θ; ζ ÞÞ ¼ x~ 1 ðsj θ; ζ Þ þ δ0 ;
ð25Þ
π g 2 ðx~ ðsj θ; ζ ÞÞ ¼ x~ 1 ðsj θ; ζ Þ ; 2
ð26Þ
g 3 ðx~ ðsj θ; ζ ÞÞ ¼ x~ 2 ðsj θ; ζ Þ;
ð27Þ
g 4 ðx~ ðsj θ; ζ ÞÞ ¼ x~ 2 ðsj θ; ζ Þ 2π:
ð28Þ
(1) limα-0 þ ψ ðω; αÞ ¼ maxfω; 0g, (2) ψ ðω; αÞ 4 0, ω (3) 0 oψ 0 ðω; αÞ ¼ 12 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 o 1, 2 2 ω þ 4α
(4) 0 oψ ðω; αÞ maxfω; 0g oα.
where
Obviously, the constraint (24) is equivalent to the following equality constraint: 4 Z X j¼1
n 0
Proof. See Appendix A. The approximation property of ψ ðω; αÞ is shown in Fig. 2, which shows that the approximation level can be artificially controlled by adjusting parameter α. Applying the smoothing technique to the cost function (30), a new optimal control problem is defined as follows. Problem 4. Given the switched system ðθ; ζ Þ A Θ Ξ such that the cost function J ðθ; ζ; ρÞ ¼ x~ 6 ðnj θÞ þ η
ð29Þ
þρ
4 Z X j¼1
By the idea of l1 penalty given in [38], the inequality constraint (29) is appended to the cost function (23) to form an augmented cost function. Then, we can define the following optimal control problem.
0.8
5 2 X x~ j ðnj θ; ζ Þ xfj
j¼1
n o max g j ðx~ ðsj θ; ζ ÞÞ; 0 ds ¼ 0:
1
(18)–(20), choose
alpha=0.3 alpha=0.1 alpha=0.025 max[omega,0]
0
n
n o ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α ds;
ð32Þ
is minimized. Problem 4 is an optimal parameter selection problem with simple bounds on the variables (recall the definition of Θ and Ξ). Such problem can be solved efficiently using any gradient-based optimization technique.
6. Gradient formulae and algorithm Define the Hamiltonian function
0.6
ψ
H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ¼ ρ
4 n o X ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α j¼1
þ λT ðsÞf~ x~ ðsj θ; ζ Þ; θ; ζ i ;
0.4
where λðsÞ is the costate that satisfies the following system:
0.2
0 −1
ð33Þ
∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ T ; λ_ ðsÞ ¼ ∂x~ ðsj θ; ζ Þ −0.5
0
0.5
ω Fig. 2. The approximation property of ψ ðω; αÞ.
s A ½0; n;
ð34Þ
1
T λðnÞ ¼ 2η x~ 1 ðnj θ; ζ Þ xf1 ; …; x~ 5 ðnj θ; ζ Þ xf5 ; 1 :
ð35Þ
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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Then, the gradient formulae of the cost function (32) are given in the following theorem. Theorem 1. The gradient formulae of the cost function (32) with respect to θ, ζ and ρ are given by Z n ∂J ðθ; ζ; ρ; αÞ ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ¼ ds; ð36Þ ∂θ ∂θ 0 ∂J ðθ; ζ; ρ; αÞ ¼ ∂ζ ∂J ðθ; ζ; ρ; αÞ ¼ ∂ρ
Z
n 0
Z
n 0
∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ds; ∂ζ
ð37Þ
∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ds: ∂ρ
ð38Þ
5
Then, for any ρ 4 0, we obtain 4 Z n n o X 0oρ ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α ds j¼1
ρ
0
4 Z X
j¼1
n 0
n o max g j ðx~ ðsj θ; ζ ÞÞ; 0 ds r 4nαρ;
ð43Þ
which implies that 0 o J ðθ; ζ; ρ; αÞ J^ ðθ; ζ; ρÞ r 4nρα:
ð44Þ
The proof is complete.□
Proof. See Appendix B. Based on the above discussions, the following algorithm for solving Problem 1 is proposed: Algorithm 1. Step 1: Set initial penalty parameter ρ, initial parameter α, increase coefficient aða 4 1Þ, decrease coefficient bð0 o b o 1Þ, tolerance ϵ0 for penalty. ð0Step 2: Input the initial condition x0 and the initial pair θ Þ ; ζ ð0Þ , set the iteration number k ¼0. Step 3: Solve Problem 4 by any gradient-based optimization technique (the are given in Theorem 1), a new pair gradients θðk þ 1Þ ; ζ ðk þ 1Þ and J θðk þ 1Þ ; ζ ðk þ 1Þ ; ρ; α are obtained. 4: the convergence criterion Step Check J θðk þ 1Þ ; ζ ðk þ 1Þ ; ρ; α J θðkÞ ; ζ ðkÞ ; ρ; α oε. If satisfied then go to step 5, otherwise goto step 6. Step 5: Stop. Let θðkþ 1Þ ; ζðk þ 1Þ be the solution for Problem2. Construct the solution τn ; ζ n of Problem 1 from θðk þ 1Þ ; ζ ðk þ 1Þ . Step 6: Set k≔k þ 1, ρ≔aρ, α≔bα, then go back to step 3.
Theorem 2. If ðθn ; ζ n Þ is the solution of Problem 3 and ðθαn ; ζ αn Þ is the solution of Problem 4, we have 0 o J θn ; ζ n ; ρ; α J^ θαn ; ζ αn ; ρ r 4nρα: ð45Þ Proof. By Lemma 3, we have 0 o J θn ; ζ n ; ρ; α J^ θn ; ζ n ; ρ r 4nρα;
0 o J θαn ; ζ αn ; ρ; α J^ θαn ; ζ αn ; ρ r 4nρα; Since θn ; ζ n is the solution of Problem 3, we obtain J^ θαn ; ζ αn ; ρ Z J^ θn ; ζ n ; ρ ;
ð46Þ ð47Þ
ð48Þ
which implies that J θαn ; ζ αn ; ρ; α J^ θαn ; ζ αn ; ρ r J θαn ; ζ αn ; ρ; α J^ θn ; ζ n ; ρ : Since θαn ; ζ αn is the solution of Problem 4, we obtain J θαn ; ζ αn ; ρ; α r J θn ; ζ n ; ρ; α ;
ð50Þ
which implies that J θαn ; ζ αn ; ρ; α J^ θn ; ζ n ; ρ r J θn ; ζ n ; ρ; α J^ θn ; ζ n ; ρ :
ð51Þ
Numerical results in Section 8 will show that Algorithm 1 is a very effective method for solving Problem 1, however, it should be pointed out that generally speaking, the solution obtained by Algorithm 1 is a local minimizer.
By (46), (47), (49) and (51), we have 0 o J θαn ; ζ αn ; ρ; α J^ θαn ; ζ αn ; ρ r J θαn ; ζ αn ; ρ; α J^ θn ; ζ n ; ρ r J θn ; ζ n ; ρ; α J^ θn ; ζ n ; ρ r 4nρα:
ð52Þ
7. Convergence analysis
That is, 0 o J θαn ; ζ αn ; ρ; α J^ θn ; ζ n ; ρ r 4nρα:
ð53Þ
Fiacco and McCormick proved that the solution of Problem 2 can be obtained by solving Problem 3 by increasing the values of ρ [39]. We further prove that if the smoothing parameter α is sufficiently small, the solution of Problem 4 is an approximate solution of Problem 3. Definition 1. A pair ðθα ; ζ α Þ is α-feasible to Problem 2, if g j ðx~ ðsj θα ; ζ α ÞÞ r α;
j ¼ 1; 2; 3; 4:
ð39Þ
Lemma 3. For any ρ 40 and α 4 0, we have 0 o J ðθ; ζ; ρ; αÞ J^ ðθ; ζ; ρÞ r4nρα:
ð40Þ
The proof is complete.□ Remark 1. Theorem 2 implies that if the parameter α is sufficiently small, the solution of Problem 4 is an approximate solution of Problem 3. Theorem 3. Let ðθn ; ζ n Þ be the solution of Problem 3 and ðθαn ; ζ αn Þ be the solution of Problem 4. If ðθn ; ζ n Þ is feasible to Problem 2 and ðθαn ; ζ αn Þ is α-feasible to Problem 2, we have pffiffiffi ð54Þ 0 o J~ θαn ; ζ αn J~ θn ; ζ n r 2 3 þ 5 nρα: Proof. Since θn ; ζ n is feasible to Problem 2, we have 4 X
Proof. From the property (4) of the smooth function ψ ðω; αÞ, for any α 4 0, we have n o n o 0 o ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α max g j ðx~ ðsj θ; ζ ÞÞ; 0 rα; j ¼ 1; 2; 3; 4:
ð41Þ
That is, 0o
4 n o n o X ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α max g j ðx~ ðsj θ; ζ ÞÞ; 0 r 4α: j¼1
ð42Þ
ð49Þ
n o max g j ðx~ ðsj θα ; ζ α ÞÞ; 0 ¼ 0:
ð55Þ
j¼1
Since θαn ; ζ αn is α-feasible to Problem 2, from the property (2) of the smooth function ψ ðω; αÞ, we obtain 1 þ pffiffiffi 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 α2 þ 4α2 þ α ¼ α; ð56Þ 0 o ψ g j x~ sj θαn ; ζ αn ; α r 2 2 Z 0oρ
n 0
4 X pffiffiffi ψ g j x~ sj θαn ; ζ αn ; α ds r 2 1 þ 5 nρα:
ð57Þ
j¼1
By (55) and Theorem 2, we have
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6
Remark 2. If ρ is greater than the threshold value ρn , the solution of Problem 3 is the exact solution of Problems 1 and 2 [40]. That is, if ρ 4 ρn , Theorem 3 gives an error estimation between the solutions of Problem 4 and Problems 1 and 2. Thus, the approximate solution Problems 1 and 2 can be obtained by solving Problem 4. In essence, it can also be regarded as an extension of the sequential unconstrained minimization technique [39].
10 Our algorithm [7] [8]
y
0 −10 −20
8. Numerical results
−30 1030 1020 1010 1000 990 980
z
−50
−40
−30
−20
−10
In this section, the effectiveness of the proposed algorithm is shown using two numerical examples.
0
Example 1. Consider a three-dimensional trajectory design problem for the well Jin45-12-Cp26 (in an oil field of Liaohe, PR China) as discussed in [6]. Under the framework of switched systems, the nonlinear dynamical system given in [6] is formulated as the following switched dynamical system: 8 > x_ ðt Þ ¼ f xðt Þ; ζ i ; t A ½t i 1 ; t i Þ; i ¼ 1; 2; 3; > > > > < xðt i Þ ¼ x t i ; i ¼ 1; …; n 1; ð60Þ T > > 13π 18815π > > > ; ; 1:86; 0:77; 984:86 ; x ð 0 Þ ¼ : 200 18000
x
Fig. 3. The optimal trajectory. 1.6 Our algorithm [7] [8]
1.4
Inclination (rad)
1.2 1
with the inequality constraints: 8 0:1 r x1 ðt Þ r 0:5π > > > > > > < 0 r x2 ðt Þ r 2π 50 r K i r50; i ¼ 1; 2; 3 > > > 20 r γ i r60; i ¼ 1; 2; 3 > > > : 10 rt t r 80; i ¼ 1; 2; 3
0.8 0.6 0.4 0.2
i1
i
where
0 0
10
20
30
40
50
60
70
0
Arc length (m)
Fig. 4. The optimal inclination.
f xðt Þ; ζ i 7 Our algorithm [7] [8]
6
Azimuth (rad)
K i cos γ i
1
B C K i sin γ i B C sin x1 ðt Þ B C B C ¼ B sin x1 ðt Þ cos x2 ðt Þ C: B C B sin x1 ðt Þ sin x2 ðt Þ C @ A cos x1 ðt Þ
The cost function is given by
5
J ðτ; ζ Þ ¼ t 3 þ10
5 2 X xj ðt 3 j τ; ζ Þ xfj ;
ð62Þ
j¼1
4
where xf ¼ 3 2 1 0
ð61Þ
0
10
20
30
40
50
60
70
Arc length (m) Fig. 5. The optimal azimuth.
0 o J~ θαn ; ζ αn J~ θn ; ζ n þ ρ
Z
n 0
4 X ψ g j x~ sj θαn ; ζ αn ; α ds r 4nρα: j¼1
ð58Þ From (57) and (58), we have pffiffiffi 0 o J~ θαn ; ζ αn J~ θn ; ζ n r2 3 þ 5 nρα: The proof is complete.□
ð59Þ
873π
2119π 1800; 1800 ;
T 43:00; 29:00; 1022:00 .
Given the dynamical system (60) and the inequality constraint (61), our goal is to choose ðτ; ζ Þ A Γ Ξ such that the cost function (62) is minimized. Suppose that the initial value of ρ and the initial value of α are 1 and 0.5 respectively. Let a¼10, b¼0.1, ε ¼ 10 4 . Then the algorithm proposed by us is applied to solve the three-dimensional trajectory design for the well Jin45-12-Cp26 by Matlab 2010a on an Pentium Dual-core PC with 2.60 GB of RAM. Then the optimal control parameter vector, the optimal switching instant vector, the length of the optimal trajectory, the target error, and the corresponding optimal cost function value are ζ n ¼ ðð40:1270; 14:8491Þ; ð46:7655; 41:5060Þ; ð47:5853; 42:3112ÞÞ, τn ¼ ð28:0046; 43:3510; 60:1656ÞT , t n3 ¼ 60:1656, μn ¼ J n t n3 ¼ 1:8146 and J n ¼ 61:9802 respectively. To compare the performance of our algorithm to existing methods, we continue to perform the method given in [7] and the approach given in [8] to solve the three-dimensional trajectory design for the well Jin45-12-Cp26 under the same condition. Then the optimal control parameter vector, the optimal switching instant vector, the length of the optimal trajectory, the target error, and the corresponding optimal cost function value
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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7 Our algorithm [7] [8]
3
6
2.5
5
Azimuth (rad)
The i−th smooth turn segment
7
2 1.5 1
4 3 2
0.5
Our algorithm [7] [8]
1
0 0
10
20
30
40
50
60
0
70
0
10
20
30
Arc length (m) Fig. 6. The optimal switching law.
y
−170 −180 −190
Our algorithm [7] [8]
−200 1720 1700
1660
60
80
70
90
100
110
x
Fig. 7. The optimal trajectory.
with the inequality constraints: 8 0:1 r x1 ðt Þ r0:5π > > > > > > < 0 r x2 ðt Þ r 2π 50 r K i r 50; i ¼ 1; 2; 3 > > > 40 rγ i r 60; i ¼ 1; 2; 3 > > > : 10 rt t r 100; i ¼ 1; 2; 3 i
1.6
where
0.8 0.6
The cost function is given by
Inclination (rad)
1.2 1
0.4 0.2
where xf ¼ 30
40
50
60
70
5 X j¼1
0 20
80
80
Arc length (m) Fig. 8. The optimal inclination.
are, respectively, ζnn ¼((41.7513, 21.1622), (49.8147, 89.6557), (60.4387, −26.3517)), τnn = (16.0971, 51.3371, 68.1450)T, tnn 3 = 68.0782, μnn = Jnn − tnn 3 = 7.9655, Jnn = 76.0437 and ζnnn = ((54.9134, −4.6324), (51.0975, 22.2785), (42.5469, −47.9575)), τnnn = (30.0344, 48.2238, 66.1112)T, tnnn 3 = 66.4733, μnnn = Jnnn − tnnn 3 = 6.2760, Jnnn = 72.7493. The optimal trajectory, the optimal inclination, the optimal azimuth and the optimal switching law are presented in Figs. 3–6 respectively.
ð64Þ
0
J ðτ; ζ Þ ¼ t 3 þ 10
Our algorithm [7] [8]
10
70
i1
1 K i cos γ i B C K i sin γ i B C sin x1 ðt Þ C B B C i f xðt Þ; ζ ¼ B sin x1 ðt Þ cos x2 ðt Þ C: B C B sin x1 ðt Þ sin x2 ðt Þ C @ A cos x1 ðt Þ
1.4
0
60
Example 2. Consider a three-dimensional trajectory design problem for the well Ci-16-Cp146 (in an oil field of Liaohe, PR China) as discussed in [7,8]. Under the framework of switched systems, the nonlinear dynamical system given in [7,8] is formulated as the following switched dynamical system: 8 > x_ ðt Þ ¼ f xðt Þ; ζ i ; t A ½t i 1 ; t i Þ; i ¼ 1; 2; 3; > > > > < xðt i Þ ¼ x t i ; i ¼ 1; …; n 1; ð63Þ T > > 26π 3803π > > > : xð0Þ ¼ 450 ; 3000 ; 102:69; 156:39; 1673:15 ;
−160
z
50
Fig. 9. The optimal azimuth.
−150
1680
40
Arc length (m)
179π
xj ðt 3 jτ; ζ Þ xfj
411π 360 ; 360 ; 62:50;
2
;
ð65Þ
T 192:90; 1718:00 .
Given the dynamical system (63) and the inequality constraint (64), our goal is to choose ðτ; ζ Þ A Γ Ξ such that the cost function (65) is minimized. Suppose that the initial value of ρ and the initial value of α are 1 and 0.5 respectively. Let a¼10, b¼ 0.1, ε ¼ 10 4 . Then the algorithm proposed by us is applied to solve the three-dimensional trajectory design for the well Ci-16-Cp146 by Matlab 2010a on an Pentium Dualcore PC with 2.60 GB of RAM. Then the optimal control parameter vector, the optimal switching instant vector, the length of the optimal trajectory, the target error, and the corresponding optimal cost function value are ζ n ¼ ðð39:4427; 6:5067Þ; ð58:8530; 47:3816Þ; ð61:2551; 40:6160ÞÞ , τn ¼ ð26:0497; 59:0844; 68:0760ÞT , t n3 ¼ 68:3171, μn ¼ J n t n3 ¼ 2:3549 and J n ¼ 70:6720 respectively. To compare the
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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Acknowledgements
3.5
The i−th smooth turn segment
3
The authors express their sincere gratitude to Professor A.B. Rad, Q.G. Wang, J. Pieper, the editor and the anonymous reviewers for their constructive comments in improving the presentation and quality of this manuscript. This work was supposed by the Chinese National Natural Science Foundation under Grant no. 61374006.
2.5 2 1.5
Appendix A. Proof of Lemma 2
1 Our algorithm [7] [8]
0.5 0
0
10
20
30
40
50
60
70
80
Arc length (m) Fig. 10. The optimal switching law.
performance of our algorithm to existing methods, we continue to perform the method given in [7] and the approach given in [8] to solve the three-dimensional trajectory design for the well Ci-16-Cp146 under the same condition. Then the optimal control parameter vector, the optimal switching instant vector, the length of the optimal trajectory, the target error, and the corresponding optimal cost function value are, respectively, ζ nn ¼ ðð59:5853; 3:4909Þ; ð48:1190; 16:0540Þ; ð56:2638; nn 24:2435ÞÞ, τnn ¼ ð28:1419; 44:2858; 76:0357ÞT , t nn ¼ 3 ¼ 76:1493, μ nn nn nnn nn J t 3 ¼ 9:0759, J ¼ 85:2252 and ζ ¼ ðð51:3404; 2:3038Þ; ð60:4893; 45:1818Þ; ð43:1386; 36:2769ÞÞ, τnnn ¼ ð24:0119; 39:1299; nnn 74:1839ÞT , t nnn ¼ J nnn t nnn ¼ 7:1576, J nnn ¼ 81:4087. 3 ¼ 74:2511, μ 3 The optimal trajectory, the optimal inclination, the optimal azimuth and the optimal switching law are presented in Figs. 7–10 respectively. Above numerical results show that our algorithm is superior to the existing methods in the following three aspects. First, the length of the trajectory of Jin45-12-Cp26 and Ci-16-Cp146 is decreased by around 8–9 percent, which results in the cost reduction of the drilling the two horizontal wells. Next, the precision of reaching the target is higher, which indicates that the optimal three-dimensional well trajectory reaches the target at the desired inclination, azimuth and coordinate much more accurately. Finally, the continuous state inequality constraints arising from engineering specifications are fulfilled for the whole arc length.
(1)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi limα-0 þ ψ ðω; αÞ ¼ limα-0 þ 12 ω2 þ4α2 þ ω ¼ 12ðjωj þ ωÞ ¼
ω
if ωZ 0
0
if ωo 0
¼ maxfω; 0g. pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 þ 4α2 þω Z 12 4α2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi obtain ψ ðω; αÞ ¼ 12 ω2 þ 4α2 þ ω
(2) If ωZ 0, we have ψ ðω; αÞ ¼ 12
α 4 0; if ωo 0, we pffiffiffiffiffiffi 4 12 ω2 þ ω ¼ 12ðjωjþ ωÞ ¼ 12ð ωþ ωÞ ¼ 0. Thus, for any ω A R, ψ ðω; αÞ 40 holds. ω (3) If ω4 0, we have 0 o ψ 0 ðω; αÞ ¼ 12 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ1 o 2 2 ω þ 4α 1 pω ffiffiffiffi2 þ 1 ¼ 1; if ω ¼ 0, we obtain 0 oψ 0 ðω; αÞ ¼ 2 ω 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ω þ 1 ¼ 12 o 1; if ω o 0, we have 0 ¼ 12 ωω þ 1 ¼ 2 2 2 ω þ 4α 1 pω 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ω ffiffiffiffi þ 1 o ψ ð ω; α Þ ¼ þ1 o 12 o 1. 2 2 2 2 2 ω
ω þ 4α
ω Thus, for any ω A R, 0 o ψ 0 ðω; αÞ ¼ 12 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 o 1 holds. 2 2 pffiffiffiffiffiffi
ω þ 4α
(4) If ω Z0, we have 0 ¼ 12 ω2 þ ω ω o ψ ðω; αÞ maxfω; 0g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ¼ 2 ω2 þ 4α2 þω ωr 12 ω2 þ 4α2 þ ω ω ¼ α; if ωo 0, we
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi obtain 0 ¼ 12 ω2 þω o ψ ðω; αÞ maxfω; 0g ¼ 12 ω2 þ 4α2 þ ω r pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ω2 þ 4α2 þ ω ¼ α. Thus, for any ω A R, 0 o ψ ðω; αÞ 2 maxfω; 0g r α holds. The proof is complete.
Appendix B. Proof of Theorem 1 By (18) and (33), we have J ðθ; ζ; ρ; αÞ ¼ x~ 6 ðnj θÞ þη
5 2 Z X x~ j ðnj θ; ζ Þ xfj þ
j¼1
9. Conclusion In this paper, a three-dimensional trajectory design problem for horizontal well is formulated as an optimal control problem of switched systems with continuous state inequality constraints, and an efficient smoothed penalty approach is proposed to solve the problem. By introducing a time-scaling transformation and a novel smoothing technique, all the continuous state inequality constraints are incorporated into the original objective function by the l1 penalty function, giving rise to an approximate nonlinear parameter optimization problem that can be solved using any gradient-based method. Convergence results indicate that any local optimal solution of the approximate problem is also a local optimal solution of the original problem as long as the penalty parameter is sufficiently large. Numerical results show that our method is superior to existing algorithms, regardless of the length of the trajectory, the precision of reaching the target or obeying the continuous state inequality constraints.
n 0
ðH ðx~ ðsj θ; ζ Þ; θ; ζ;
λðsÞÞ λT ðsÞf~ x~ ðsj θ; ζ Þ; θ; ζ i Þ ds: Integrating by parts the term λT ðsÞx~ ðsj θ; ζ Þ yields J ðθ; ζ; ρ; αÞ ¼ x~ 6 ðnj θÞ þ η
5 X j¼1
Z
n
þ 0
x~ j ðnj θ; ζ Þ xfj
2
λT ðnÞx~ ðnj θ; ζ Þ þ λT ð0Þx~ 0
T H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ þ λ_ ðsÞx~ ðsj θ; ζ Þ ds:
ð67Þ
Then, the first order variation of (67) is given by δJ ðθ; ζ; ρ; αÞ ¼ ð0; 0; 0; 0; 0; 1Þ þ 2η x~ 1 ðnj θ; ζ Þ xf1 ; 0; 0; 0; 0; 0 þ 2η 0; x~ 2 ðnj θ; ζ Þ xf2 ; 0; 0; 0; 0 þ 2η 0; 0; x~ 3 ðnj θ; ζ Þ xf3 ; 0; 0; 0 þ 2η 0; 0; 0; x~ 4 ðnj θ; ζ Þ xf4 ; 0 þ 2η 0; 0; 0; x~ 5 ðnj θ; ζ Þ xf5 ; 0 λT ðnÞ δx~ ðnj θ; ζ Þ þ λT ð0Þδx~ 0
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δx~ ðsj θ; ζ Þ ∂x~ ðsj θ; ζ Þ 0 ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δθ þ ∂θ ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δζ þ δρ þ ∂ζ ∂ρ T þ λ_ ðsÞδx~ ðsj θ; ζ Þ ds: Z
þ
n
ð68Þ
Collecting terms in (68) gives δJ ðθ; ζ; ρ; αÞ ¼ 2η x~ 1 ðnj θ; ζ Þ xf1 ; ⋯; x~ 5 ðnj θ; ζ Þ xf5 ; 1 λT ðnÞ δx~ ðnj θ; ζ Þ Z
∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δθ ds ∂θ Z n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δζ ds þ ∂ζ 0 Z n ~ ∂H ðx ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δρ ds þ ∂ρ 0 Z n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ _ T þ λ ðsÞ δx~ ðsj θ; ζ Þ ds: þ ∂x~ ðsj θ; ζ Þ 0 þ
n
0
ð69Þ By (34) and (35), from (69), we have Z n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ds δθ δJ ðθ; ζ; ρ; αÞ ¼ ∂θ 0 Z n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ þ ds δζ ∂ζ Z0 n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ds δρ: þ ∂ρ 0
ð70Þ
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Three-dimensional trajectory design for horizontal well based on optimal switching algorithms Xiang Wu a,n, Kanjian Zhang b,c a
School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, PR China School of Automation, Southeast University, Nanjing 210096, PR China c Key Laboratory of Measurement and Control of CSE, Ministry of Education, Southeast University, Nanjing 210096, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 5 February 2015 Received in revised form 25 March 2015 Accepted 10 April 2015 This paper was recommended for publication by Jeff Pieper.
This paper considers a three-dimensional trajectory design problem for horizontal well. The problem is formulated as an optimal control problem of switched systems with continuous state inequality constraints. Since the complexity of such constraints and the switching instants is unknown, it is difficult to solve the problem by standard optimization techniques. To overcome the difficulty, by a timescaling transformation, a smoothing technique and a penalty function method, an efficient computational method is proposed for solving this problem. Convergence results show that, for a sufficiently large penalty parameter, any local optimal solution of the approximate problem is also a local optimal solution of the original problem. Two numerical examples are presented to illustrate the efficiency of the approach proposed. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Optimal control Horizontal well trajectory design Time-scaling transformation Smoothing technique Penalty function method
1. Introduction In general, a well trajectory is a three-dimensional curve, where the trajectory is described as a combination of turn sections and straight sections. The main objective of designing a horizontal well trajectory is to construct a trajectory that reaches a given target at a specified inclination and azimuth from a given starting location subject to various constraints arising from engineering specifications. Several well trajectory designing software are available commercially for finding an appropriate horizontal well trajectory. However, most of them use a trial-and-error procedure to obtain a solution. In addition to being time-consuming and depending on user-experience, these softwares are limited to designing simple well trajectories, and may not generate an optimal trajectory, since the trial-and-error search is usually user-driven. Thus, some methods have been proposed to design a three-dimensional well trajectory. In 1979, Planeix and Fox [1] for the first time considered the problem. In their work, an approach to reach a given target from a current position was obtained by controlling the azimuth and the inclination independently. Guo et al. [2] proposed a
n
Corresponding author. Tel.: þ 86 85186702059. E-mail addresses: [email protected] (X. Wu), [email protected] (K. Zhang).
constant curvature method to solve the set end (set inclination and set azimuth at the target) three-dimensional well trajectory design problem in 1992. In 1995, Ebrahim [3] obtained general expressions of the three-dimensional well trajectory calculation by vector algebra and differential geometry. In 2001, Liu and Shi [4] developed a general method where two circle arcs and a straight line are used to obtain simple three-dimensional trajectories to solve the set end three-dimensional well trajectory problem. Suppose that the three-dimensional well trajectory is a combination of several smooth turn segments, two three-dimensional well trajectory design methods are developed by heuristic or direction search methods in [5,6]. For more discussions on various literature results, the reader may refer to [7–14] and the references therein. Switched systems are a particular class of hybrid systems, which have received considerable attention in the past two decades. Most work focuses on stability and stabilization problems [15–25]. From the application perspective, however, besides stability and stabilization, performance is also crucial. Recently, optimal control problems of switched systems have been attracting researchers from various fields in science and engineering, due to their significance in theory and applications [26–33]. These works are both theoretical results and numerical methods. The available theoretical results usually extend the classical maximum principle or the dynamic programming approach to switched systems. The numerical methods take advantage of efficient
http://dx.doi.org/10.1016/j.isatra.2015.04.002 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
X. Wu, K. Zhang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
nonlinear optimization techniques and high-speed computers to develop efficient numerical methods for the optimal control of switched systems [34]. Theoretical results of switched systems are now well developed. However, for switched systems with nonlinear modes, analytical techniques are not sufficient to determine an optimal control policy. In addition, many dynamic control processes are subject to continuous-state equality or inequality constraints, such as container cranes [35] and aircraft trajectory planning [36]. Note that one continuous-time or one continuousstate constraint is equivalent to an uncountable number of conventional constraints. Thus, it is difficult to solve the problem by conventional optimization techniques. The existing three-dimensional well trajectory design approaches have always their drawbacks. For example, the approach given in [6] is computationally expensive with poor convergence properties. In addition, the continuous state inequality constraints arising from engineering specifications are ignored in [7,8]. Then some optimal solutions obtained using the algorithms given in [7,8] actually fail to satisfy the continuous state inequality constraints at some points along the well trajectory. This is clearly undesirable in practice. Furthermore, compared with nonlinear systems, switched systems have many very good properties. For instance, the analysis and control of switched systems are easier due to their structures being simple. Thus, we reconsider the three-dimensional horizontal well trajectory design problem under the framework of switched systems. The main contribution of this paper is that the three-dimensional trajectory design problem for horizontal well is formulated as an optimal control problem of switched systems with continuous state inequality constraints, and an efficient smoothed penalty approach is proposed to solve the problem. By introducing a time-scaling transformation [37] and a novel smoothing technique, all the continuous state inequality constraints are incorporated into the original objective function by the l1 penalty function, giving rise to an approximate nonlinear parameter optimization problem that can be solved using any gradient-based method. Convergence results indicate that any local optimal solution of the approximate problem is also a local optimal solution of the original problem as long as the penalty parameter is sufficiently large. Compared with the existing methods, the penalty parameter and the smoothing parameter of our proposed algorithm are not fixed artificially by experience. The rest of the paper is organized as follows. Section 2 presents the three-dimensional trajectory design problem for horizontal well. A time-scaling transformation is introduced is Section 3. Then, in Section 4, by the time-scaling transformation, the three-dimensional trajectory design problem for horizontal well is transformed into an equivalent problem with fixed switching instants. However, these constraints are very complex, which pose a challenge for standard optimization algorithms. Thus, in Section 5, by a smoothing technique and a penalty function method, the problem is transformed into a sequence of unconstrained problems. Each of these unconstrained problems can be effectively solved by any gradient-based optimization technique. The gradient formulae of the cost function and our algorithm are given in Section 6. Section 7 provides the Convergence analysis. Two numerical examples in Section 8 show that our method is effective.
2. Problem formulation As shown in Fig. 1, a three-dimensional trajectory design problem for horizontal well is considered. The trajectory design problem is described in the Cartesian coordinate system, with xaxis representing North/South (positive x being North), yaxis representing East/West (positive y being East), and the zaxis (positive z being downward) representing the true vertical depth. The arc length from the kick-off point is denoted by t, and any point P(t) on the curve is described by its inclination βðtÞ, azimuth
Fig. 1. Three-dimensional trajectory design for horizontal well.
ϕðtÞ and coordinate ðx ðt Þ; y ðt Þ; z ðt ÞÞ. In order to make the model more close to the practical problems, suppose that the following three conditions are satisfied: Assumption 1. The well trajectory is a combination of n smooth turn segments. Assumption 2. The curvature Ki and the tool-face angle γi are constants in each ith turn segment, i ¼ 1; …; n. Assumption 3. The design of the well trajectory is only for nonstraight horizontal well. Then, the well trajectory, which is required to reach a given target from the kick-off point at a specified inclination and azimuth, can be described the following switched system: x_ ðt Þ ¼ f xðt Þ; ζ i ; t A ½t i 1 ; t i Þ; i ¼ 1; …; n; ð1Þ xðt i Þ ¼ x t i ;
i ¼ 1; …; n 1;
xð0Þ ¼ x0 ;
ð2Þ ð3Þ
where t is the arc length from the kick-off point; βðt Þ and ϕðt Þ are the inclination and azimuth of any point P ðt Þ on the curve respectively; xðt Þ ¼ ðβðt Þ; ϕðt Þ; x ðt Þ; y ðt Þ; z ðt ÞÞ is the location of the point P ðt Þ on the curve; Ki is the curvature; γi is the tool-face angle which is the and position direction of the deflecting tool bowed angle; ζ i ¼ K i ; γ i ; i ¼ 1; …; n; x t i ¼ limt-ti xðt i Þ; i ¼ 1; …; n 1; x0 is a given initial state; t 0 ¼ 0 is the initial arc length; t i ; i ¼ 1; …; n 1 are the switching instants; tn is the well's whole trajectory length; and 0 1 K i cos γ i B C K i sin γ i B C sin βðt Þ C B B C i f xðt Þ; ζ ¼ B sin βðt Þ cos ϕðt Þ C: ð4Þ B C B sin βðt Þ sin ϕðt Þ C @ A cos βðt Þ Define n o Γ ¼ τ ¼ ðt 1 ; …; t n ÞT A Rn : t i t i 1 A c1 ; d1 ; i ¼ 1; …; n ;
ð5Þ
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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n o T Ξ ¼ ζ ¼ ζ 1 ; …; ζ n A R2n : ζ i A c2 ; d2 c3 ; d3 ; i ¼ 1; …; n ;
ð6Þ
where ci and di, i ¼ 1; 2; 3, are given real numbers such that ci o di , i ¼ 1; 2; 3. Because of engineering specifications, for any t A ½0; t n , βðt Þ and ϕðt Þ are required to satisfy the following constraints: π δ0 rβðt Þ r ; 2
ð7Þ
0 r ϕðt Þ r 2π;
ð8Þ
where δ0 is a given positive number. Lemma 1. The function f defined in (4) satisfies that (1) f is continuously differentiable. (2) For any x; ζ i A R5 c2 ; d2 c3 ; d3 , there exists a real number L 40 such that ð9Þ J f x; ζ i J r Lð1 þ J x J Þ;
3
Lemma 1 ensures that the switched system (1)–(3) has a unique solution. Our goalTis to construct a well trajectory to reach the target xf ¼ xf1 ; …; xf5 from the kick-off point x0 by choosing the parameter vectors τ and ζ such that the cost function J ðτ; ζ Þ ¼ t n þ η
5 X j¼1
xj ðt n jτ; ζ Þ xfj
2
;
ð11Þ
is minimized, where η is a given weight coefficient and the cost function (11) denotes the weighted sum of the trajectory length and the error of reaching the target, xj ðt Þ, j ¼ 1; …; 5, are the jth component of the state vector xðt Þ, i.e. x1 ðt Þ ¼ βðt Þ; x2 ðt Þ ¼ ϕðt Þ; x3 ðt Þ ¼ x ðt Þ; x4 ðt Þ ¼ y ðt Þ; x5 ðt Þ ¼ z ðt Þ, xðtjτ; ζ Þ denotes the solution of switched system (1)–(4). Now, we define our optimal control problem formally as follows. Problem 1. Given the switched system (1)–(3), choose ðτ; ζ Þ A Γ Ξ such that the cost function (11) is minimized subject to the continuous state inequality constraints (7) and (8). 3. Time-scaling transformation
where J J denotes the usual Euclidean norm.
Define a function tðsÞ : ½0; n-R by the following differential equation [37]:
Proof. (1) Obviously, differentiable. f is continuously (2) For any x; ζ i A R5 c2 ; d2 c3 ; d3 , we have 0 11=2 5 X 2 i i A @ J f x; ζ J ¼ f j x; ζ
t_ ðsÞ ¼
¼
2
cos γ i þ
K 2i sin 2 γ i sin 2 βðt Þ
2
þ sin βðt Þ cos ϕðt Þ
!1=2 2 2 2 K 2i cos 2 γ i sin βðt Þ þK 2i sin γ i þ sin βðt Þ 2
2
2
2
d2 1 sin βðt Þ þ d2 1 þ sin βðt Þ
r
!1=2
2
where L ¼ x1 ðt Þ is the first component of the state vector xðt Þ, i.e. x1 ðt Þ ¼ βðt Þ. The proof is complete.□
0 o c 1 r θ i ¼ t i t i 1 r d1 ;
i ¼ 1; …; n;
ð15Þ
Let θ ¼ ½θ1 ; …; θn and let Θ be the set containing all such θ. Integrating (12) with initial condition (13) yields that, for any s A k 1; kÞ, k ¼ 1; …; n,
0
d2 þ 1 , sin 2 δ0
The above differential transformation (12)–(14) is called a timescaling transformation. Clearly, θi satisfies T
2
sin δ0 11=2 2 2 d2 þ 1 sin 2 βðt Þ þ d2 @ A ¼ 2 sin δ0 11=2 0 2 2 2 d2 þ 1 sin βðt Þ þ d2 þ 1 A o@ sin 2 δ0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1=2 d2 þ 1 2 ¼ sin βðt Þ þ 1 2 sin δ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0 2 1=2 d2 þ 1 2 β ðt Þ þ 1 r 2 sin δ0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d2 þ 1 βðt Þ þ 1 r sin 2 δ0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 d2 þ 1 ¼ x1 ðt Þ þ 1 sin 2 δ0 rffiffiffiffiffiffiffiffiffiffiffi Lð J x J þ 1Þ; r
ð13Þ
where θi is the duration of the ith turn segment in the original arc length horizon, and for a given interval I ½0; n, the characteristic function χ I ðsÞ is defined by
1 if s A I; ð14Þ χ I ðsÞ ¼ 0 otherwise:
2
sin 2 βðt Þ
ð12Þ
t ð0Þ ¼ 0;
1=2 2 2 þ sin βðt Þ sin ϕðt Þ þ cos 2 βðt Þ !1=2 K 2 sin 2 γ i þ1 ¼ K 2i cos 2 γ i þ i 2 sin βðt Þ ¼
θi χ ½i 1;iÞ ðsÞ;
i¼1
j¼1
K 2i
n X
t ðsÞ ¼
kX 1
θi þ θk ðs k þ1Þ:
ð16Þ
i¼1
Thus, for each i ¼ 1; …; n, we have t ðiÞ ¼
i X
θk ¼
k¼1
i X
ðt k t k 1 Þ ¼ t i :
ð17Þ
k¼1
In particular, t ðnÞ ¼ t n .
4. Problem transformation
ð10Þ
Since the state in Problem 1 depends on the switching instants, it is difficult to integrate the switched system (1)–(3) numerically. To overcome the difficulty, a more tractable equivalent problem will be derived. Let x~ ðsÞ ¼ ðxðt ðsÞÞ; t ðsÞÞT and f~ x~ ðsÞ; θ; ζ i ¼ ððθi f x~ ðsÞ; ζ i ÞT ; θi ÞT . Then, applying (12) and (13) to (1)–(3), (7), (8) and (11) yields x~ ðsÞ ¼ f~ x~ ðsÞ; θ; ζ i ; s A ½i 1; iÞ; ð18Þ x~ ðiÞ ¼ x~ ði Þ;
i ¼ 1; …; n 1;
ð19Þ
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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4
h iT x~ ð0Þ ¼ x~ 0 ¼ ðx0 ÞT ; 0 ;
ð20Þ
Problem 3. Given the switched system ðθ; ζ Þ A Θ Ξ such that the cost function
π δ0 r x~ 1 ðsj θ; ζ Þ r ; 2
ð21Þ
J^ ðθ; ζ Þ ¼ x~ 6 ðnj θÞ þ η
for any s A ½0; n;
5 2 X x~ j ðnj θ; ζ Þ xfj
j¼1
0 r x~ 2 ðsj θ; ζ Þ r 2π;
for any s A ½0; n;
ð22Þ
þρ
4 Z X j¼1
J~ ðθ; ζ Þ ¼ x~ 6 ðnj θÞ þ η
5 X j¼1
x~ j ðnj θ; ζ Þ xfj
2
ð23Þ
where x~ j ðsj θ; ζ Þ, j ¼ 1; …; 6, are the jth component of the unique solution vector of the switched system (18)–(20). Then, a new optimal control problem is defined as follows. Problem 2. Given the switched system (18)–(20), choose ðθ; ζ Þ A Θ Ξ such that the cost function (23) is minimized subject to the continuous state inequality constraints (21) and (22). Clearly, Problems 1 and 2 are equivalent.
n 0
n o max g j ðx~ ðsj θ; ζ ÞÞ; 0 ds;
ð30Þ
is minimized, where ρ4 0 is the penalty parameter. n o Note that max g j ðx~ ðsj θ; ζ ÞÞ; 0 , j ¼ 1; …; 4, are non-smooth functions in θ and ζ. Thus, any gradient-based optimization algorithm would have difficulty in dealing with Problem 3. To overcome this difficulty, we introduce a novel osmooth function n given in (31) to approximate max g j ðx~ ðsj θ; ζ ÞÞ; 0 : 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð31Þ ψ ðω; αÞ ¼ ω2 þ 4α2 þ ω ; 2 where α 40 is an adjustable parameter. Lemma 2. For any ω A R and α 4 0, ψ ðω; αÞ has the following properties:
5. Smoothing technique and penalty function method To begin, for any s A ½0; n, the inequality constraints (21) and (22) are written as g j ðx~ ðsj θ; ζ ÞÞ r0;
(18)–(20), choose
j ¼ 1; 2; 3; 4;
ð24Þ
g 1 ðx~ ðsj θ; ζ ÞÞ ¼ x~ 1 ðsj θ; ζ Þ þ δ0 ;
ð25Þ
π g 2 ðx~ ðsj θ; ζ ÞÞ ¼ x~ 1 ðsj θ; ζ Þ ; 2
ð26Þ
g 3 ðx~ ðsj θ; ζ ÞÞ ¼ x~ 2 ðsj θ; ζ Þ;
ð27Þ
g 4 ðx~ ðsj θ; ζ ÞÞ ¼ x~ 2 ðsj θ; ζ Þ 2π:
ð28Þ
(1) limα-0 þ ψ ðω; αÞ ¼ maxfω; 0g, (2) ψ ðω; αÞ 4 0, ω (3) 0 oψ 0 ðω; αÞ ¼ 12 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 o 1, 2 2 ω þ 4α
(4) 0 oψ ðω; αÞ maxfω; 0g oα.
where
Obviously, the constraint (24) is equivalent to the following equality constraint: 4 Z X j¼1
n 0
Proof. See Appendix A. The approximation property of ψ ðω; αÞ is shown in Fig. 2, which shows that the approximation level can be artificially controlled by adjusting parameter α. Applying the smoothing technique to the cost function (30), a new optimal control problem is defined as follows. Problem 4. Given the switched system ðθ; ζ Þ A Θ Ξ such that the cost function J ðθ; ζ; ρÞ ¼ x~ 6 ðnj θÞ þ η
ð29Þ
þρ
4 Z X j¼1
By the idea of l1 penalty given in [38], the inequality constraint (29) is appended to the cost function (23) to form an augmented cost function. Then, we can define the following optimal control problem.
0.8
5 2 X x~ j ðnj θ; ζ Þ xfj
j¼1
n o max g j ðx~ ðsj θ; ζ ÞÞ; 0 ds ¼ 0:
1
(18)–(20), choose
alpha=0.3 alpha=0.1 alpha=0.025 max[omega,0]
0
n
n o ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α ds;
ð32Þ
is minimized. Problem 4 is an optimal parameter selection problem with simple bounds on the variables (recall the definition of Θ and Ξ). Such problem can be solved efficiently using any gradient-based optimization technique.
6. Gradient formulae and algorithm Define the Hamiltonian function
0.6
ψ
H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ¼ ρ
4 n o X ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α j¼1
þ λT ðsÞf~ x~ ðsj θ; ζ Þ; θ; ζ i ;
0.4
where λðsÞ is the costate that satisfies the following system:
0.2
0 −1
ð33Þ
∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ T ; λ_ ðsÞ ¼ ∂x~ ðsj θ; ζ Þ −0.5
0
0.5
ω Fig. 2. The approximation property of ψ ðω; αÞ.
s A ½0; n;
ð34Þ
1
T λðnÞ ¼ 2η x~ 1 ðnj θ; ζ Þ xf1 ; …; x~ 5 ðnj θ; ζ Þ xf5 ; 1 :
ð35Þ
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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Then, the gradient formulae of the cost function (32) are given in the following theorem. Theorem 1. The gradient formulae of the cost function (32) with respect to θ, ζ and ρ are given by Z n ∂J ðθ; ζ; ρ; αÞ ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ¼ ds; ð36Þ ∂θ ∂θ 0 ∂J ðθ; ζ; ρ; αÞ ¼ ∂ζ ∂J ðθ; ζ; ρ; αÞ ¼ ∂ρ
Z
n 0
Z
n 0
∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ds; ∂ζ
ð37Þ
∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ds: ∂ρ
ð38Þ
5
Then, for any ρ 4 0, we obtain 4 Z n n o X 0oρ ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α ds j¼1
ρ
0
4 Z X
j¼1
n 0
n o max g j ðx~ ðsj θ; ζ ÞÞ; 0 ds r 4nαρ;
ð43Þ
which implies that 0 o J ðθ; ζ; ρ; αÞ J^ ðθ; ζ; ρÞ r 4nρα:
ð44Þ
The proof is complete.□
Proof. See Appendix B. Based on the above discussions, the following algorithm for solving Problem 1 is proposed: Algorithm 1. Step 1: Set initial penalty parameter ρ, initial parameter α, increase coefficient aða 4 1Þ, decrease coefficient bð0 o b o 1Þ, tolerance ϵ0 for penalty. ð0Step 2: Input the initial condition x0 and the initial pair θ Þ ; ζ ð0Þ , set the iteration number k ¼0. Step 3: Solve Problem 4 by any gradient-based optimization technique (the are given in Theorem 1), a new pair gradients θðk þ 1Þ ; ζ ðk þ 1Þ and J θðk þ 1Þ ; ζ ðk þ 1Þ ; ρ; α are obtained. 4: the convergence criterion Step Check J θðk þ 1Þ ; ζ ðk þ 1Þ ; ρ; α J θðkÞ ; ζ ðkÞ ; ρ; α oε. If satisfied then go to step 5, otherwise goto step 6. Step 5: Stop. Let θðkþ 1Þ ; ζðk þ 1Þ be the solution for Problem2. Construct the solution τn ; ζ n of Problem 1 from θðk þ 1Þ ; ζ ðk þ 1Þ . Step 6: Set k≔k þ 1, ρ≔aρ, α≔bα, then go back to step 3.
Theorem 2. If ðθn ; ζ n Þ is the solution of Problem 3 and ðθαn ; ζ αn Þ is the solution of Problem 4, we have 0 o J θn ; ζ n ; ρ; α J^ θαn ; ζ αn ; ρ r 4nρα: ð45Þ Proof. By Lemma 3, we have 0 o J θn ; ζ n ; ρ; α J^ θn ; ζ n ; ρ r 4nρα;
0 o J θαn ; ζ αn ; ρ; α J^ θαn ; ζ αn ; ρ r 4nρα; Since θn ; ζ n is the solution of Problem 3, we obtain J^ θαn ; ζ αn ; ρ Z J^ θn ; ζ n ; ρ ;
ð46Þ ð47Þ
ð48Þ
which implies that J θαn ; ζ αn ; ρ; α J^ θαn ; ζ αn ; ρ r J θαn ; ζ αn ; ρ; α J^ θn ; ζ n ; ρ : Since θαn ; ζ αn is the solution of Problem 4, we obtain J θαn ; ζ αn ; ρ; α r J θn ; ζ n ; ρ; α ;
ð50Þ
which implies that J θαn ; ζ αn ; ρ; α J^ θn ; ζ n ; ρ r J θn ; ζ n ; ρ; α J^ θn ; ζ n ; ρ :
ð51Þ
Numerical results in Section 8 will show that Algorithm 1 is a very effective method for solving Problem 1, however, it should be pointed out that generally speaking, the solution obtained by Algorithm 1 is a local minimizer.
By (46), (47), (49) and (51), we have 0 o J θαn ; ζ αn ; ρ; α J^ θαn ; ζ αn ; ρ r J θαn ; ζ αn ; ρ; α J^ θn ; ζ n ; ρ r J θn ; ζ n ; ρ; α J^ θn ; ζ n ; ρ r 4nρα:
ð52Þ
7. Convergence analysis
That is, 0 o J θαn ; ζ αn ; ρ; α J^ θn ; ζ n ; ρ r 4nρα:
ð53Þ
Fiacco and McCormick proved that the solution of Problem 2 can be obtained by solving Problem 3 by increasing the values of ρ [39]. We further prove that if the smoothing parameter α is sufficiently small, the solution of Problem 4 is an approximate solution of Problem 3. Definition 1. A pair ðθα ; ζ α Þ is α-feasible to Problem 2, if g j ðx~ ðsj θα ; ζ α ÞÞ r α;
j ¼ 1; 2; 3; 4:
ð39Þ
Lemma 3. For any ρ 40 and α 4 0, we have 0 o J ðθ; ζ; ρ; αÞ J^ ðθ; ζ; ρÞ r4nρα:
ð40Þ
The proof is complete.□ Remark 1. Theorem 2 implies that if the parameter α is sufficiently small, the solution of Problem 4 is an approximate solution of Problem 3. Theorem 3. Let ðθn ; ζ n Þ be the solution of Problem 3 and ðθαn ; ζ αn Þ be the solution of Problem 4. If ðθn ; ζ n Þ is feasible to Problem 2 and ðθαn ; ζ αn Þ is α-feasible to Problem 2, we have pffiffiffi ð54Þ 0 o J~ θαn ; ζ αn J~ θn ; ζ n r 2 3 þ 5 nρα: Proof. Since θn ; ζ n is feasible to Problem 2, we have 4 X
Proof. From the property (4) of the smooth function ψ ðω; αÞ, for any α 4 0, we have n o n o 0 o ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α max g j ðx~ ðsj θ; ζ ÞÞ; 0 rα; j ¼ 1; 2; 3; 4:
ð41Þ
That is, 0o
4 n o n o X ψ max g j ðx~ ðsj θ; ζ ÞÞ; 0 ; α max g j ðx~ ðsj θ; ζ ÞÞ; 0 r 4α: j¼1
ð42Þ
ð49Þ
n o max g j ðx~ ðsj θα ; ζ α ÞÞ; 0 ¼ 0:
ð55Þ
j¼1
Since θαn ; ζ αn is α-feasible to Problem 2, from the property (2) of the smooth function ψ ðω; αÞ, we obtain 1 þ pffiffiffi 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 α2 þ 4α2 þ α ¼ α; ð56Þ 0 o ψ g j x~ sj θαn ; ζ αn ; α r 2 2 Z 0oρ
n 0
4 X pffiffiffi ψ g j x~ sj θαn ; ζ αn ; α ds r 2 1 þ 5 nρα:
ð57Þ
j¼1
By (55) and Theorem 2, we have
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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6
Remark 2. If ρ is greater than the threshold value ρn , the solution of Problem 3 is the exact solution of Problems 1 and 2 [40]. That is, if ρ 4 ρn , Theorem 3 gives an error estimation between the solutions of Problem 4 and Problems 1 and 2. Thus, the approximate solution Problems 1 and 2 can be obtained by solving Problem 4. In essence, it can also be regarded as an extension of the sequential unconstrained minimization technique [39].
10 Our algorithm [7] [8]
y
0 −10 −20
8. Numerical results
−30 1030 1020 1010 1000 990 980
z
−50
−40
−30
−20
−10
In this section, the effectiveness of the proposed algorithm is shown using two numerical examples.
0
Example 1. Consider a three-dimensional trajectory design problem for the well Jin45-12-Cp26 (in an oil field of Liaohe, PR China) as discussed in [6]. Under the framework of switched systems, the nonlinear dynamical system given in [6] is formulated as the following switched dynamical system: 8 > x_ ðt Þ ¼ f xðt Þ; ζ i ; t A ½t i 1 ; t i Þ; i ¼ 1; 2; 3; > > > > < xðt i Þ ¼ x t i ; i ¼ 1; …; n 1; ð60Þ T > > 13π 18815π > > > ; ; 1:86; 0:77; 984:86 ; x ð 0 Þ ¼ : 200 18000
x
Fig. 3. The optimal trajectory. 1.6 Our algorithm [7] [8]
1.4
Inclination (rad)
1.2 1
with the inequality constraints: 8 0:1 r x1 ðt Þ r 0:5π > > > > > > < 0 r x2 ðt Þ r 2π 50 r K i r50; i ¼ 1; 2; 3 > > > 20 r γ i r60; i ¼ 1; 2; 3 > > > : 10 rt t r 80; i ¼ 1; 2; 3
0.8 0.6 0.4 0.2
i1
i
where
0 0
10
20
30
40
50
60
70
0
Arc length (m)
Fig. 4. The optimal inclination.
f xðt Þ; ζ i 7 Our algorithm [7] [8]
6
Azimuth (rad)
K i cos γ i
1
B C K i sin γ i B C sin x1 ðt Þ B C B C ¼ B sin x1 ðt Þ cos x2 ðt Þ C: B C B sin x1 ðt Þ sin x2 ðt Þ C @ A cos x1 ðt Þ
The cost function is given by
5
J ðτ; ζ Þ ¼ t 3 þ10
5 2 X xj ðt 3 j τ; ζ Þ xfj ;
ð62Þ
j¼1
4
where xf ¼ 3 2 1 0
ð61Þ
0
10
20
30
40
50
60
70
Arc length (m) Fig. 5. The optimal azimuth.
0 o J~ θαn ; ζ αn J~ θn ; ζ n þ ρ
Z
n 0
4 X ψ g j x~ sj θαn ; ζ αn ; α ds r 4nρα: j¼1
ð58Þ From (57) and (58), we have pffiffiffi 0 o J~ θαn ; ζ αn J~ θn ; ζ n r2 3 þ 5 nρα: The proof is complete.□
ð59Þ
873π
2119π 1800; 1800 ;
T 43:00; 29:00; 1022:00 .
Given the dynamical system (60) and the inequality constraint (61), our goal is to choose ðτ; ζ Þ A Γ Ξ such that the cost function (62) is minimized. Suppose that the initial value of ρ and the initial value of α are 1 and 0.5 respectively. Let a¼10, b¼0.1, ε ¼ 10 4 . Then the algorithm proposed by us is applied to solve the three-dimensional trajectory design for the well Jin45-12-Cp26 by Matlab 2010a on an Pentium Dual-core PC with 2.60 GB of RAM. Then the optimal control parameter vector, the optimal switching instant vector, the length of the optimal trajectory, the target error, and the corresponding optimal cost function value are ζ n ¼ ðð40:1270; 14:8491Þ; ð46:7655; 41:5060Þ; ð47:5853; 42:3112ÞÞ, τn ¼ ð28:0046; 43:3510; 60:1656ÞT , t n3 ¼ 60:1656, μn ¼ J n t n3 ¼ 1:8146 and J n ¼ 61:9802 respectively. To compare the performance of our algorithm to existing methods, we continue to perform the method given in [7] and the approach given in [8] to solve the three-dimensional trajectory design for the well Jin45-12-Cp26 under the same condition. Then the optimal control parameter vector, the optimal switching instant vector, the length of the optimal trajectory, the target error, and the corresponding optimal cost function value
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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7 Our algorithm [7] [8]
3
6
2.5
5
Azimuth (rad)
The i−th smooth turn segment
7
2 1.5 1
4 3 2
0.5
Our algorithm [7] [8]
1
0 0
10
20
30
40
50
60
0
70
0
10
20
30
Arc length (m) Fig. 6. The optimal switching law.
y
−170 −180 −190
Our algorithm [7] [8]
−200 1720 1700
1660
60
80
70
90
100
110
x
Fig. 7. The optimal trajectory.
with the inequality constraints: 8 0:1 r x1 ðt Þ r0:5π > > > > > > < 0 r x2 ðt Þ r 2π 50 r K i r 50; i ¼ 1; 2; 3 > > > 40 rγ i r 60; i ¼ 1; 2; 3 > > > : 10 rt t r 100; i ¼ 1; 2; 3 i
1.6
where
0.8 0.6
The cost function is given by
Inclination (rad)
1.2 1
0.4 0.2
where xf ¼ 30
40
50
60
70
5 X j¼1
0 20
80
80
Arc length (m) Fig. 8. The optimal inclination.
are, respectively, ζnn ¼((41.7513, 21.1622), (49.8147, 89.6557), (60.4387, −26.3517)), τnn = (16.0971, 51.3371, 68.1450)T, tnn 3 = 68.0782, μnn = Jnn − tnn 3 = 7.9655, Jnn = 76.0437 and ζnnn = ((54.9134, −4.6324), (51.0975, 22.2785), (42.5469, −47.9575)), τnnn = (30.0344, 48.2238, 66.1112)T, tnnn 3 = 66.4733, μnnn = Jnnn − tnnn 3 = 6.2760, Jnnn = 72.7493. The optimal trajectory, the optimal inclination, the optimal azimuth and the optimal switching law are presented in Figs. 3–6 respectively.
ð64Þ
0
J ðτ; ζ Þ ¼ t 3 þ 10
Our algorithm [7] [8]
10
70
i1
1 K i cos γ i B C K i sin γ i B C sin x1 ðt Þ C B B C i f xðt Þ; ζ ¼ B sin x1 ðt Þ cos x2 ðt Þ C: B C B sin x1 ðt Þ sin x2 ðt Þ C @ A cos x1 ðt Þ
1.4
0
60
Example 2. Consider a three-dimensional trajectory design problem for the well Ci-16-Cp146 (in an oil field of Liaohe, PR China) as discussed in [7,8]. Under the framework of switched systems, the nonlinear dynamical system given in [7,8] is formulated as the following switched dynamical system: 8 > x_ ðt Þ ¼ f xðt Þ; ζ i ; t A ½t i 1 ; t i Þ; i ¼ 1; 2; 3; > > > > < xðt i Þ ¼ x t i ; i ¼ 1; …; n 1; ð63Þ T > > 26π 3803π > > > : xð0Þ ¼ 450 ; 3000 ; 102:69; 156:39; 1673:15 ;
−160
z
50
Fig. 9. The optimal azimuth.
−150
1680
40
Arc length (m)
179π
xj ðt 3 jτ; ζ Þ xfj
411π 360 ; 360 ; 62:50;
2
;
ð65Þ
T 192:90; 1718:00 .
Given the dynamical system (63) and the inequality constraint (64), our goal is to choose ðτ; ζ Þ A Γ Ξ such that the cost function (65) is minimized. Suppose that the initial value of ρ and the initial value of α are 1 and 0.5 respectively. Let a¼10, b¼ 0.1, ε ¼ 10 4 . Then the algorithm proposed by us is applied to solve the three-dimensional trajectory design for the well Ci-16-Cp146 by Matlab 2010a on an Pentium Dualcore PC with 2.60 GB of RAM. Then the optimal control parameter vector, the optimal switching instant vector, the length of the optimal trajectory, the target error, and the corresponding optimal cost function value are ζ n ¼ ðð39:4427; 6:5067Þ; ð58:8530; 47:3816Þ; ð61:2551; 40:6160ÞÞ , τn ¼ ð26:0497; 59:0844; 68:0760ÞT , t n3 ¼ 68:3171, μn ¼ J n t n3 ¼ 2:3549 and J n ¼ 70:6720 respectively. To compare the
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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Acknowledgements
3.5
The i−th smooth turn segment
3
The authors express their sincere gratitude to Professor A.B. Rad, Q.G. Wang, J. Pieper, the editor and the anonymous reviewers for their constructive comments in improving the presentation and quality of this manuscript. This work was supposed by the Chinese National Natural Science Foundation under Grant no. 61374006.
2.5 2 1.5
Appendix A. Proof of Lemma 2
1 Our algorithm [7] [8]
0.5 0
0
10
20
30
40
50
60
70
80
Arc length (m) Fig. 10. The optimal switching law.
performance of our algorithm to existing methods, we continue to perform the method given in [7] and the approach given in [8] to solve the three-dimensional trajectory design for the well Ci-16-Cp146 under the same condition. Then the optimal control parameter vector, the optimal switching instant vector, the length of the optimal trajectory, the target error, and the corresponding optimal cost function value are, respectively, ζ nn ¼ ðð59:5853; 3:4909Þ; ð48:1190; 16:0540Þ; ð56:2638; nn 24:2435ÞÞ, τnn ¼ ð28:1419; 44:2858; 76:0357ÞT , t nn ¼ 3 ¼ 76:1493, μ nn nn nnn nn J t 3 ¼ 9:0759, J ¼ 85:2252 and ζ ¼ ðð51:3404; 2:3038Þ; ð60:4893; 45:1818Þ; ð43:1386; 36:2769ÞÞ, τnnn ¼ ð24:0119; 39:1299; nnn 74:1839ÞT , t nnn ¼ J nnn t nnn ¼ 7:1576, J nnn ¼ 81:4087. 3 ¼ 74:2511, μ 3 The optimal trajectory, the optimal inclination, the optimal azimuth and the optimal switching law are presented in Figs. 7–10 respectively. Above numerical results show that our algorithm is superior to the existing methods in the following three aspects. First, the length of the trajectory of Jin45-12-Cp26 and Ci-16-Cp146 is decreased by around 8–9 percent, which results in the cost reduction of the drilling the two horizontal wells. Next, the precision of reaching the target is higher, which indicates that the optimal three-dimensional well trajectory reaches the target at the desired inclination, azimuth and coordinate much more accurately. Finally, the continuous state inequality constraints arising from engineering specifications are fulfilled for the whole arc length.
(1)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi limα-0 þ ψ ðω; αÞ ¼ limα-0 þ 12 ω2 þ4α2 þ ω ¼ 12ðjωj þ ωÞ ¼
ω
if ωZ 0
0
if ωo 0
¼ maxfω; 0g. pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 þ 4α2 þω Z 12 4α2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi obtain ψ ðω; αÞ ¼ 12 ω2 þ 4α2 þ ω
(2) If ωZ 0, we have ψ ðω; αÞ ¼ 12
α 4 0; if ωo 0, we pffiffiffiffiffiffi 4 12 ω2 þ ω ¼ 12ðjωjþ ωÞ ¼ 12ð ωþ ωÞ ¼ 0. Thus, for any ω A R, ψ ðω; αÞ 40 holds. ω (3) If ω4 0, we have 0 o ψ 0 ðω; αÞ ¼ 12 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ1 o 2 2 ω þ 4α 1 pω ffiffiffiffi2 þ 1 ¼ 1; if ω ¼ 0, we obtain 0 oψ 0 ðω; αÞ ¼ 2 ω 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ω þ 1 ¼ 12 o 1; if ω o 0, we have 0 ¼ 12 ωω þ 1 ¼ 2 2 2 ω þ 4α 1 pω 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ω ffiffiffiffi þ 1 o ψ ð ω; α Þ ¼ þ1 o 12 o 1. 2 2 2 2 2 ω
ω þ 4α
ω Thus, for any ω A R, 0 o ψ 0 ðω; αÞ ¼ 12 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 o 1 holds. 2 2 pffiffiffiffiffiffi
ω þ 4α
(4) If ω Z0, we have 0 ¼ 12 ω2 þ ω ω o ψ ðω; αÞ maxfω; 0g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ¼ 2 ω2 þ 4α2 þω ωr 12 ω2 þ 4α2 þ ω ω ¼ α; if ωo 0, we
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi obtain 0 ¼ 12 ω2 þω o ψ ðω; αÞ maxfω; 0g ¼ 12 ω2 þ 4α2 þ ω r pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ω2 þ 4α2 þ ω ¼ α. Thus, for any ω A R, 0 o ψ ðω; αÞ 2 maxfω; 0g r α holds. The proof is complete.
Appendix B. Proof of Theorem 1 By (18) and (33), we have J ðθ; ζ; ρ; αÞ ¼ x~ 6 ðnj θÞ þη
5 2 Z X x~ j ðnj θ; ζ Þ xfj þ
j¼1
9. Conclusion In this paper, a three-dimensional trajectory design problem for horizontal well is formulated as an optimal control problem of switched systems with continuous state inequality constraints, and an efficient smoothed penalty approach is proposed to solve the problem. By introducing a time-scaling transformation and a novel smoothing technique, all the continuous state inequality constraints are incorporated into the original objective function by the l1 penalty function, giving rise to an approximate nonlinear parameter optimization problem that can be solved using any gradient-based method. Convergence results indicate that any local optimal solution of the approximate problem is also a local optimal solution of the original problem as long as the penalty parameter is sufficiently large. Numerical results show that our method is superior to existing algorithms, regardless of the length of the trajectory, the precision of reaching the target or obeying the continuous state inequality constraints.
n 0
ðH ðx~ ðsj θ; ζ Þ; θ; ζ;
λðsÞÞ λT ðsÞf~ x~ ðsj θ; ζ Þ; θ; ζ i Þ ds: Integrating by parts the term λT ðsÞx~ ðsj θ; ζ Þ yields J ðθ; ζ; ρ; αÞ ¼ x~ 6 ðnj θÞ þ η
5 X j¼1
Z
n
þ 0
x~ j ðnj θ; ζ Þ xfj
2
λT ðnÞx~ ðnj θ; ζ Þ þ λT ð0Þx~ 0
T H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ þ λ_ ðsÞx~ ðsj θ; ζ Þ ds:
ð67Þ
Then, the first order variation of (67) is given by δJ ðθ; ζ; ρ; αÞ ¼ ð0; 0; 0; 0; 0; 1Þ þ 2η x~ 1 ðnj θ; ζ Þ xf1 ; 0; 0; 0; 0; 0 þ 2η 0; x~ 2 ðnj θ; ζ Þ xf2 ; 0; 0; 0; 0 þ 2η 0; 0; x~ 3 ðnj θ; ζ Þ xf3 ; 0; 0; 0 þ 2η 0; 0; 0; x~ 4 ðnj θ; ζ Þ xf4 ; 0 þ 2η 0; 0; 0; x~ 5 ðnj θ; ζ Þ xf5 ; 0 λT ðnÞ δx~ ðnj θ; ζ Þ þ λT ð0Þδx~ 0
Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
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∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δx~ ðsj θ; ζ Þ ∂x~ ðsj θ; ζ Þ 0 ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δθ þ ∂θ ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δζ þ δρ þ ∂ζ ∂ρ T þ λ_ ðsÞδx~ ðsj θ; ζ Þ ds: Z
þ
n
ð68Þ
Collecting terms in (68) gives δJ ðθ; ζ; ρ; αÞ ¼ 2η x~ 1 ðnj θ; ζ Þ xf1 ; ⋯; x~ 5 ðnj θ; ζ Þ xf5 ; 1 λT ðnÞ δx~ ðnj θ; ζ Þ Z
∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δθ ds ∂θ Z n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δζ ds þ ∂ζ 0 Z n ~ ∂H ðx ðsj θ; ζ Þ; θ; ζ; λðsÞÞ δρ ds þ ∂ρ 0 Z n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ _ T þ λ ðsÞ δx~ ðsj θ; ζ Þ ds: þ ∂x~ ðsj θ; ζ Þ 0 þ
n
0
ð69Þ By (34) and (35), from (69), we have Z n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ds δθ δJ ðθ; ζ; ρ; αÞ ¼ ∂θ 0 Z n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ þ ds δζ ∂ζ Z0 n ∂H ðx~ ðsj θ; ζ Þ; θ; ζ; λðsÞÞ ds δρ: þ ∂ρ 0
ð70Þ
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Please cite this article as: Wu X, Zhang K. Three-dimensional trajectory design for horizontal well based on optimal switching algorithms. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.002i
