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Time-optimal steering of an object moving in a viscous medium to a desired phase state
Journal of Applied Mathematics and Mechanics 75 (2011) 534–538
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Time-optimal steering of an object moving in a viscous medium to a desired phase state夽 L.D. Akulenko Moscow, Russia
a r t i c l e
i n f o
Article history: Received 16 August 2010
a b s t r a c t The two-point problem of the time-optimal attainment of a desired phase state by a multidimensional dynamic object is investigated. The motion occurs in a viscous medium by means of a limited force. The open-loop and/or feedback control laws constructed by numerical-analytical methods for arbitrary initial data. An asymptotically approximate solution of the maximum principle boundary-value problem is presented for short and long time intervals. The singularities of the optimal trajectory are established for the initial and final parts of the motion. The solution obtained of the two-point problem of the optimal control of the motion of a dynamic object in a homogeneous viscous medium by means of a force of bounded modulus is compared with the known solutions in special formulations. © 2011 Elsevier Ltd. All rights reserved.
1. Formulation of the problem The problem of the time-optimal control of a multidimensional dynamic object in a homogeneous medium with viscous friction is investigated. It is required to steer a dynamical system (a point mass) to a specified phase state by means of a force of bounded modulus. The n-dimensional model under consideration is described by the relations
(1.1) Here, t0 is the initial instant, tf is the final instant of the time t, m is the mass, k is the friction coefficient and u is the control. The solution of problem (1.1) and its analysis is of interest in the case of an arbitrary dimensionality n ≥ 2. Classical results are available for n = 1.1,2 Cases of a problem of the form (1.1) have been investigated3 for n = 1 when there are perturbations of an arbitrary nature. A special class of feedback control problems when there is no dissipation (k = 0) is of interest in the mechanics of flight4–8 (see below). The solution of the class of problems (1.1) when there are no constraints on the terminal velocity vector x˙ f 1,9–12 is rather significant for control theory and its applications. The discontinuities in the analogue of the Bellman function that make the use of the method of dynamic programming difficult1,2 have to be regarded as the important features of the investigation of such problems. ˙ t according to the It follows from the structure of expressions (1.1) that, with the introduction of the dimensionless variables x, x, formulae
(1.2) = xf
= 0. In the limit case when k = 0, change the optimal control problem reduces to relations (1.1) in which m = k = u0 = 1, that is, |u| ≤ 1 and t0 of variables (1.2) degenerates and the dimensional parameters l and v are related by the single expression lv2 = u0 /m. Such a problem has been studied in detail using numerical-analytical methods4,6–10 (see below).
夽 Prikl. Mat. Mekh. Vol. 75, No. 5, pp. 763–770, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.11.007
L.D. Akulenko / Journal of Applied Mathematics and Mechanics 75 (2011) 534–538
535
The property of the central symmetry of the optimal control problem (1.1) is of considerable significance. It enables one to simplify its overall solution and analysis by means of a smaller number of governing parameters (self-similar variables). The formulation of the time-optimal problem can be modified by taking account of known perturbing factors in the equation of motion and also by generalizing the initial and terminal conditions. Similar formulations of the control problems with other expressions for the performance index that also possess symmetry properties are possible. Thus, the multidimensional time-optimal control problem (1.1), taking account of change of variables (1.2), is investigated in what follows. As is customary,1–7 it is required to construct an open-loop optimal control u∗ (t, x0 , x˙ 0 ) and the phase trajectories ˙ = u∗ (0, x, x) ˙ x∗ (t, x0 , x˙ 0 ), x˙ ∗ (t, x0 , x˙ 0 ), to calculate the response time tf∗ (x0 , x˙ 0 ) and also to determine the control in the feedback form us (x, x) ˙ t) = tf∗ (x, x). ˙ It is of interest to analyse the solution for arbitrary values of the parameters and the analogue of the Bellman function (x, x,
x0 , x˙ 0 , x˙ f (including asymptotically small and large values). Note that the quantity |x˙ f | has an upper limit of unity. A solution of the time-optimal control problem in the class of piecewise-continuous functions of time exists and it is unique. The admissible controls are quite simple to construct. In particular, such a control can consist of three stages. At the initial stage, deceleration to a complete stop is produced. Then the system is driven from the state as rest to an arbitrary point of the line (for example, to the nearest point or to the origin of coordinates) containing the final velocity vector, the velocity at the end of this stage being equal to zero at the final stage, the motion occurs along the line to the origin of coordinates with the desired value of the velocity, taking account of the constraint due to friction. Such as these three stages can be performed by time-optimal controls but, naturally, this control mode will not be globally optimal. Obviously, the solution of the optimal control problem can be considerably simplified when the directions, in particular, the values, of the initial and final velocity vectors coincide.6 It is also considerably simpler to construct a numerical-analytical solution when the initial and final positions coincide.7 In the general case of initial and terminal conditions, the use of complex software and laborious calculations are required (see Section 4). 2. Solution of the time-optimal problem using the conditions of the maximum principle We will now write out the truncated Hamiltonian H of problem (1.1) and maximize it with respect to u:
(2.1) ˙ They are explicitly defined by the formulae Here, p and q are n-vectors of the variables conjugate to x and ≡ x.
(2.2) The values of the parameters pf , g (or qf ) and tf are unknown and to be determined from the final conditions for x and . This boundaryvalue problem causes the major computational difficulty in solving the optimal control problem. In order to find the unknown parameters, the expression u*(t), defined by relations (2.1) and (2.2), is substituted into Eq. (1.1) (taking account of substitutions (1.2)), and the Cauchy problem
(2.3) is solved. We next use the terminal conditions (when t = tf ) and, as result, obtain a system of 2n governing transcendental equations in the n-vectors p and g and the unknown tf (everywhere henceforth integration with respect to t is carried out from 0 to tf )
(2.4) Adding the two equalities of (2.4), we obtain the apparently simpler equation
(2.5)
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L.D. Akulenko / Journal of Applied Mathematics and Mechanics 75 (2011) 534–538
Then, in relations (2.3) - (2.5), it is convenient to introduce the argument z = et , where 1≤ z ≤ zf < ∞ and to normalize the vectors p and g by |p| = / 0 or |g| = / 0. The argument z, dz = zdt is introduced with the aim of simplifying the analytical calculation of the quadratures, and the numerator and denominator in the expression for u* (2.5) are transformed
(2.6) The expression for |q| (2.6) contains three unknown parameters: |p|, zf (when |g| = 1) or |g|, zf (when |p| = 1) and , the cosine of the angle between the vectors p and g, and q2 is a second-degree polynomial in z, which enables us to obtain analytical expressions for the quadratures (2.4) and (2.5) of the form
(2.7) z = et ,
= etf ,
Henceforth the argument that varies within the limits from 1 to zf is introduced instead of the argument t. We substitute expressions (2.5) and (2.6) for u* into formulae (2.7) and normalize the vectors p and g by |p| = / 0 (to be specific) and obtain
(2.8) The coefficients Ij (j = − 1, 0, 1) have the
form13
(2.9) Transcendental expressions (2.7)- (2.9) are substituted into any two of relations (2.4) and (2.5). The total number of unknown parameters of the maximum principle boundary-value problem is equal to 2n + 4; the n-vectors and and the scalar quantities zf , tf , , . We have the above-mentioned number of equations for finding them. In addition to the equations (2.4), (2.5), (2.7) and (2.8) (a total number of 2n) that are “linear” in and , four relations are taken into account: the expressions for their moduli || = 1, || = (2.8), for the scalar product (, ) = and for zf = etf . 3. Investigation of the modified boundary-value problem An analysis of the transcendental system of 2n + 4 equations suggests that its straight forward numerical solution and investigation are difficult, including the case when n = 2. The computational difficulties are aggravated in view of the high dimensionality when n ≥ 2 and increase because of the wide range of variation in the governing parameters x0 , 0 , f . We recall that, for n = 1, the solution can be constructed comparatively easily analytically.1–3 When f = 0, the case of the plane problem holds and the optimal motion and the control vector are located in the plane formed by the vectors x0 and 0 . The special case of the collinearity of these vectors corresponds to the classical situation when n = 1. If the vectors x0 , 0 , f are non-collinear, then the case is general and equivalent to the three-dimensional (n = 3) case, and the spatial trihedron, in particular, is constructed by means of the three above-mentioned vectors when n ≥ 3. In the generic case system (2.4), (2.5) can be reduced to three scalar equations in the parameters zf(tf = ln zf > ) and by scalar multiplication.4,6–12 To be specific, we take the first equations of (2.4) and (2.5):
(3.1) I02
System (3.1) can be readily solved for the vectors and since it is a block-diagonal system with a non-degenerate matrix = / I−1 I1 . If the quantities zf , and are determined for given x0 , 0 , f , then the optimal control u*(t) or u˜ ∗ (z), z = et , 0 ≤ t ≤ tf is found using of the expressions for and given by relations (2.5)-(2.9). In order to obtain the required system of three transcendental equations in zf , , from system (3.1), it is necessary to square each vector relation for ␣ and  and form the scalar product of the vectors ␣ and 
(3.2)
L.D. Akulenko / Journal of Applied Mathematics and Mechanics 75 (2011) 534–538
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The right-hand sides of Eqs (3.2) only contain the unknown parameters zf , and , and the left-hand sides are defined in terms of the known quantities x0 , 0 , f and the unknown quantity zf . It is difficult to determine the roots as functions of the known vectors, and this requires the development of a software. Note that the left-hand sides contain the moduli |x0 |, |0 |, |f | and scalar products (x0 , 0 ), (x0 , f ), (0 , f ), that is the moduli and the cosines of the angles between the above-mentioned vectors. The limit case when k = 0 (f = / 0 and when there is no dissipation and f = 0) has been investigated earlier.4,8 In the generic case, a solution of (3.2) exists and is unique. The Jacobian of the equation is non-zero, which enables us to use the classical numerical procedures of the methods of successive approximation including the algorithm for the method of accelerated convergence (the method of tangents4,6–8 ).
4. Numerical-analytical solution of the boundary-value problem The method of successive approximations (or the accelerated convergence method) combined with the continuation with respect to the known parameters x0 , 0 , f can be used for the calculations. In the general case, the coefficients Ij (2.9) are strictly positive, which follows from expressions (2.7). Preliminary calculations and analysis suggested4,6–8 that the solutions of the maximum-principle boundary-value problem (2.4), (2.5) as well as of the modified problem (3.1), (3.2) exist and are unique. The time-optimality of the solution of the initial control problem (1.1) is thereby established. For illustration, we will first present a brief analysis of the solution of the boundary-value problem and optimal motion in special cases. The investigation of the system of 2n transcendental equations (2.4) in the unknown parameters = p|p|−1 , = g|p|−1 and tf with the additional conditions || = 1 and zf = etf = 1 corresponds to the situation k → 0 (the absence of a viscous medium). A fundamental property of the solution in the case when n = 2 (motion in a plane, that is, the vectors x0 , 0 , f are collinear)8 involves the above-mentioned collinearity of the optimal control vector. The velocity vector (t) of the optimal trajectory in the general case in the final part of the motion t = tf − t, that is, when t/tf 1, is then coaxial to the final velocity vector f with a relative error O((t/tf )2 ), that is, the motion takes place along a coaxial line. The final part of the optimal motion trajectory is located in a phase plane corresponding to unidimensional motion.7 Four types of trajectories are observed in the final stage: two “hill” trajectories (from above and from below), a “serpentine curve” and a “loop” (Ref. 8, Figs 1 and 2). The case of the time-optimal steering of a dynamic object to the initial (starting) point x(0) = x(tf ) = 0 of a geometric plane with the required value of the final velocity vector is of interest in the class of plane optimal-control problems: the vectors (0) = 0 and (tf ) = f are specified (Ref. 7, Figs 1 and 2). The time-optimal plane trajectories passing through a given point when the initial and final velocities are equal possess the interesting properties of considerable final rectification (Ref. 6, Figs 4 and 5). In the case of zero final velocity (x˙ f = 0, a “soft” encounter or landing4 ), the optimal control problem appears to be a plane problem: the initial position x0 , velocity 0 and control u* vectors will be coplanar. The final part of the optimal trajectory x(t) also satisfies the above-mentioned rectification property, but the final part of the uniformly retarded motion can begin at an arbitrary geometric point of a fairly small neighbourhood of the origin of coordinates x = 0; the change in the velocity vector (t) has a similar character. Hence, the fact that the trajectories are rectified at the final stage of the time-optimal control of the motion is established, which leads to the degeneration of the problem. The solution of applied problems of guidance, interception, docking, mooring, etc. is at the same time complicated because of the effect of perturbing factors leading to considerable errors near the terminal point, that is, to instability of the control process including the feedback control. Great computational difficulties arise under self-guidance conditions when the measuring instruments are located on the controlled object. The existence of a large number of methods for controlling moving objects is explained by these circumstances.14 Similar properties of the trajectories also hold when there is viscous friction. However, the difficulties in carrying out exact calculations are aggravated by the great complexity of the resulting expressions for the maximum-principle boundary-value problem (2.4), (2.5) and (3.1), (3.2). They can be carried out by numerical methods (the method of successive approximations, accelerated convergence, etc.) on the basis of approximate solutions corresponding to limit situations. For example, when tf 1 (we recall that tf → vtf (1.2)), that is, in the case of a small dissipation effect, the quantity zf ≈ 1, and the motion of the object in the first approximation in tf corresponds to the case k = 0. This is established by taking the limit in expressions (2.4), (2.5) and (3.2) as k → 0 (zf = etf ). The generating solutions and their properties are described above. In the case of small values of the quantity tf , recurrence procedures are applicable which rapidly converge to the desired exact solution. The standard procedure of continuation with respect to the parameter k is then used. The quantities x0 , 0 , f can vary over wide ranges. Another (“inverse”) limit case corresponds to the strong inequality zf 1 (tf = ln zf ); then the quantity 0 /zf in Eqs (2.4) can be neglected. The basic control stage then involves reaching the origin of coordinates x = 0 in a minimum time with subsequent adjustment of the motion to the state x = 0, = f for a certain t = tf . The characteristics of the motion in the first approximation are determined by the control u(1) = − x0 /|x0 |, with the velocity vector being approximated as follows: ≈ u(1) . In particular, when f = 0, the initial approximation according to relations (2.4) and (3.1) leads to collinearity of the and n-vectors of the form = I0−1 I1 , and the value of the parameter = 1. The relation between the unknown parameters zf and (4.1) follows from this. The functions I0 and I1 have the form (2.9) when = 1. The first and third relations of (3.2) are identically satisfied and the second relation together with the relation obtained above serve to find the required values of zf and : (4.2)
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The approximate solution of the problem of the time-optimal steering of a dynamic object to the origin of coordinates with zero velocity taking account of relations (4.1) and (4.2) is found in explicit form. The open-loop control up and the time T have the form
(4.3) x0
+ 0 ).
According to relations (43), the quantity t = T − corresponds to the instant of the switching of the control up (t, As established above, the parameter = 1 and the unknown = e . An additional time interval 2 is required in order to reduce the magnitude of the velocity to approximately zero ( T). Expressions (4.3) can be reduced to the feedback form when |x + | 1: (4.4) where is the time remaining until the end of the process. When approaching the final phase point (x, ) = (0, 0), the motion occurs along a line and the steering is performed in accordance with a one-dimensional optimal feedback control.1,2 The method discussed for sinthesizing the feedback control can be applied to the problem of the time-optimal “soft landing” on a convex manifold such as, for example, on a fixed sphere in an n-dimensional (n = 3) space.15 When there is a phase constraint (the penetration of the sphere is prohibited) the problem of the deviation from a sphere can be solved16 with subsequent control of the “landing”.15 The existence of viscosity makes the solution more difficult because of the considerable increase in the complexity of the maximum-principle boundary value problem. However, its character is preserved and is analogous to that discussed above in the case of a terminal point. Acknowledgements This research was financed by the Russian Foundation for Basic Research (11-01-00472 and 11-01-12110), the Programme for the Support of Leading Scientific Schools (NSh-64817.2010.1) and Programme No. 15 of the Department of Power Engineering, Machine Building, Mechanics and Control Processes of the Russian Academy of Sciences. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF. Mathematical Theory of Optimal Processes. New York: Wiley; 1962. Boltyanskii VG. Mathematical Methods of Optimal Control. New York: Holt, Rinehart and Winston; 1971. Akulenko LD. Asymptotic Methods of Optimal Control. Moscow:Nauka;1987. Akulenko LD, Shmatkov AM. Feedback control in the problem of the time-optimal steering of a point mass to a specified position with zero velocity. Prikl Mat Mekh 1998;62(1):129–38. Grozdovskii GL, Ivanov YuN, Tokarev VV. The Mechanics of Space Flight. Optimization Problems. Moscow: Nauka; 1975. Akulenko LD, Koshelev AP. Time-optimal steering of a dynamic object to a specified position in the case of the equality of the initial and terminal velocities. Izv Ross Akad Nauk Theory and Control Systems 2003;6:98–105. Akulenko LD, Koshelev AP. Time-optimal return of a dynamic object with the required velocity. Dokl Ross Akad Nauk 2005;403(5):614–8. Akulenko LD, Koshelev AP. Time-optimal steering of a point mass to a specified position with the required velocity. Prikl Mat Mekh 2007;71(2):228–36. Akulenko LD. The perturbed time-optimal problem of controlling the final position of a point mass by means of a limited force. Prikl Mat Mekh 1994;58(2):12–21. Akulenko LD. Feedback control in the problem of the time-optimal intersection of a sphere. Prikl Mat Mekh 1996;60(5):724–35. Vnuchkov DV. The time-optimal steering of a dynamical system with linear dissipation to a specified terminal position. Izv Ross Akad Nauk Theory and Control Systems 1998;3:56–61. Akulenko LD, Shmatkov AM. Time-optimal intersection of a sphere in a viscous medium. Izv Ross Akad Nauk Theory and Control Systems 2007;1:19–26. Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. New York: Academic Press; 2000. Gutkin LS, Borisov YuP, Valuyev AA. Radiocontrol of Rocket Missiles and Spacecraft. Moscow: Sovetokoye Radio; 1968. Akulenko LD, Shmatkov AM. Time-optimal steering of a point mass onto the surface of a sphere at zero velocity. Prikl Mat Mekh 2002;66(1):10–23. Akulenko LD, Shmatkov AM. Evasion of a fixed sphere by a dynamic object driven by a bounded force. Prikl Mat Mekh 2003;67(2):179–90.
Translated by E.L.S.
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Time-optimal steering of an object moving in a viscous medium to a desired phase state夽 L.D. Akulenko Moscow, Russia
a r t i c l e
i n f o
Article history: Received 16 August 2010
a b s t r a c t The two-point problem of the time-optimal attainment of a desired phase state by a multidimensional dynamic object is investigated. The motion occurs in a viscous medium by means of a limited force. The open-loop and/or feedback control laws constructed by numerical-analytical methods for arbitrary initial data. An asymptotically approximate solution of the maximum principle boundary-value problem is presented for short and long time intervals. The singularities of the optimal trajectory are established for the initial and final parts of the motion. The solution obtained of the two-point problem of the optimal control of the motion of a dynamic object in a homogeneous viscous medium by means of a force of bounded modulus is compared with the known solutions in special formulations. © 2011 Elsevier Ltd. All rights reserved.
1. Formulation of the problem The problem of the time-optimal control of a multidimensional dynamic object in a homogeneous medium with viscous friction is investigated. It is required to steer a dynamical system (a point mass) to a specified phase state by means of a force of bounded modulus. The n-dimensional model under consideration is described by the relations
(1.1) Here, t0 is the initial instant, tf is the final instant of the time t, m is the mass, k is the friction coefficient and u is the control. The solution of problem (1.1) and its analysis is of interest in the case of an arbitrary dimensionality n ≥ 2. Classical results are available for n = 1.1,2 Cases of a problem of the form (1.1) have been investigated3 for n = 1 when there are perturbations of an arbitrary nature. A special class of feedback control problems when there is no dissipation (k = 0) is of interest in the mechanics of flight4–8 (see below). The solution of the class of problems (1.1) when there are no constraints on the terminal velocity vector x˙ f 1,9–12 is rather significant for control theory and its applications. The discontinuities in the analogue of the Bellman function that make the use of the method of dynamic programming difficult1,2 have to be regarded as the important features of the investigation of such problems. ˙ t according to the It follows from the structure of expressions (1.1) that, with the introduction of the dimensionless variables x, x, formulae
(1.2) = xf
= 0. In the limit case when k = 0, change the optimal control problem reduces to relations (1.1) in which m = k = u0 = 1, that is, |u| ≤ 1 and t0 of variables (1.2) degenerates and the dimensional parameters l and v are related by the single expression lv2 = u0 /m. Such a problem has been studied in detail using numerical-analytical methods4,6–10 (see below).
夽 Prikl. Mat. Mekh. Vol. 75, No. 5, pp. 763–770, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.11.007
L.D. Akulenko / Journal of Applied Mathematics and Mechanics 75 (2011) 534–538
535
The property of the central symmetry of the optimal control problem (1.1) is of considerable significance. It enables one to simplify its overall solution and analysis by means of a smaller number of governing parameters (self-similar variables). The formulation of the time-optimal problem can be modified by taking account of known perturbing factors in the equation of motion and also by generalizing the initial and terminal conditions. Similar formulations of the control problems with other expressions for the performance index that also possess symmetry properties are possible. Thus, the multidimensional time-optimal control problem (1.1), taking account of change of variables (1.2), is investigated in what follows. As is customary,1–7 it is required to construct an open-loop optimal control u∗ (t, x0 , x˙ 0 ) and the phase trajectories ˙ = u∗ (0, x, x) ˙ x∗ (t, x0 , x˙ 0 ), x˙ ∗ (t, x0 , x˙ 0 ), to calculate the response time tf∗ (x0 , x˙ 0 ) and also to determine the control in the feedback form us (x, x) ˙ t) = tf∗ (x, x). ˙ It is of interest to analyse the solution for arbitrary values of the parameters and the analogue of the Bellman function (x, x,
x0 , x˙ 0 , x˙ f (including asymptotically small and large values). Note that the quantity |x˙ f | has an upper limit of unity. A solution of the time-optimal control problem in the class of piecewise-continuous functions of time exists and it is unique. The admissible controls are quite simple to construct. In particular, such a control can consist of three stages. At the initial stage, deceleration to a complete stop is produced. Then the system is driven from the state as rest to an arbitrary point of the line (for example, to the nearest point or to the origin of coordinates) containing the final velocity vector, the velocity at the end of this stage being equal to zero at the final stage, the motion occurs along the line to the origin of coordinates with the desired value of the velocity, taking account of the constraint due to friction. Such as these three stages can be performed by time-optimal controls but, naturally, this control mode will not be globally optimal. Obviously, the solution of the optimal control problem can be considerably simplified when the directions, in particular, the values, of the initial and final velocity vectors coincide.6 It is also considerably simpler to construct a numerical-analytical solution when the initial and final positions coincide.7 In the general case of initial and terminal conditions, the use of complex software and laborious calculations are required (see Section 4). 2. Solution of the time-optimal problem using the conditions of the maximum principle We will now write out the truncated Hamiltonian H of problem (1.1) and maximize it with respect to u:
(2.1) ˙ They are explicitly defined by the formulae Here, p and q are n-vectors of the variables conjugate to x and ≡ x.
(2.2) The values of the parameters pf , g (or qf ) and tf are unknown and to be determined from the final conditions for x and . This boundaryvalue problem causes the major computational difficulty in solving the optimal control problem. In order to find the unknown parameters, the expression u*(t), defined by relations (2.1) and (2.2), is substituted into Eq. (1.1) (taking account of substitutions (1.2)), and the Cauchy problem
(2.3) is solved. We next use the terminal conditions (when t = tf ) and, as result, obtain a system of 2n governing transcendental equations in the n-vectors p and g and the unknown tf (everywhere henceforth integration with respect to t is carried out from 0 to tf )
(2.4) Adding the two equalities of (2.4), we obtain the apparently simpler equation
(2.5)
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L.D. Akulenko / Journal of Applied Mathematics and Mechanics 75 (2011) 534–538
Then, in relations (2.3) - (2.5), it is convenient to introduce the argument z = et , where 1≤ z ≤ zf < ∞ and to normalize the vectors p and g by |p| = / 0 or |g| = / 0. The argument z, dz = zdt is introduced with the aim of simplifying the analytical calculation of the quadratures, and the numerator and denominator in the expression for u* (2.5) are transformed
(2.6) The expression for |q| (2.6) contains three unknown parameters: |p|, zf (when |g| = 1) or |g|, zf (when |p| = 1) and , the cosine of the angle between the vectors p and g, and q2 is a second-degree polynomial in z, which enables us to obtain analytical expressions for the quadratures (2.4) and (2.5) of the form
(2.7) z = et ,
= etf ,
Henceforth the argument that varies within the limits from 1 to zf is introduced instead of the argument t. We substitute expressions (2.5) and (2.6) for u* into formulae (2.7) and normalize the vectors p and g by |p| = / 0 (to be specific) and obtain
(2.8) The coefficients Ij (j = − 1, 0, 1) have the
form13
(2.9) Transcendental expressions (2.7)- (2.9) are substituted into any two of relations (2.4) and (2.5). The total number of unknown parameters of the maximum principle boundary-value problem is equal to 2n + 4; the n-vectors and and the scalar quantities zf , tf , , . We have the above-mentioned number of equations for finding them. In addition to the equations (2.4), (2.5), (2.7) and (2.8) (a total number of 2n) that are “linear” in and , four relations are taken into account: the expressions for their moduli || = 1, || = (2.8), for the scalar product (, ) = and for zf = etf . 3. Investigation of the modified boundary-value problem An analysis of the transcendental system of 2n + 4 equations suggests that its straight forward numerical solution and investigation are difficult, including the case when n = 2. The computational difficulties are aggravated in view of the high dimensionality when n ≥ 2 and increase because of the wide range of variation in the governing parameters x0 , 0 , f . We recall that, for n = 1, the solution can be constructed comparatively easily analytically.1–3 When f = 0, the case of the plane problem holds and the optimal motion and the control vector are located in the plane formed by the vectors x0 and 0 . The special case of the collinearity of these vectors corresponds to the classical situation when n = 1. If the vectors x0 , 0 , f are non-collinear, then the case is general and equivalent to the three-dimensional (n = 3) case, and the spatial trihedron, in particular, is constructed by means of the three above-mentioned vectors when n ≥ 3. In the generic case system (2.4), (2.5) can be reduced to three scalar equations in the parameters zf(tf = ln zf > ) and by scalar multiplication.4,6–12 To be specific, we take the first equations of (2.4) and (2.5):
(3.1) I02
System (3.1) can be readily solved for the vectors and since it is a block-diagonal system with a non-degenerate matrix = / I−1 I1 . If the quantities zf , and are determined for given x0 , 0 , f , then the optimal control u*(t) or u˜ ∗ (z), z = et , 0 ≤ t ≤ tf is found using of the expressions for and given by relations (2.5)-(2.9). In order to obtain the required system of three transcendental equations in zf , , from system (3.1), it is necessary to square each vector relation for ␣ and  and form the scalar product of the vectors ␣ and 
(3.2)
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The right-hand sides of Eqs (3.2) only contain the unknown parameters zf , and , and the left-hand sides are defined in terms of the known quantities x0 , 0 , f and the unknown quantity zf . It is difficult to determine the roots as functions of the known vectors, and this requires the development of a software. Note that the left-hand sides contain the moduli |x0 |, |0 |, |f | and scalar products (x0 , 0 ), (x0 , f ), (0 , f ), that is the moduli and the cosines of the angles between the above-mentioned vectors. The limit case when k = 0 (f = / 0 and when there is no dissipation and f = 0) has been investigated earlier.4,8 In the generic case, a solution of (3.2) exists and is unique. The Jacobian of the equation is non-zero, which enables us to use the classical numerical procedures of the methods of successive approximation including the algorithm for the method of accelerated convergence (the method of tangents4,6–8 ).
4. Numerical-analytical solution of the boundary-value problem The method of successive approximations (or the accelerated convergence method) combined with the continuation with respect to the known parameters x0 , 0 , f can be used for the calculations. In the general case, the coefficients Ij (2.9) are strictly positive, which follows from expressions (2.7). Preliminary calculations and analysis suggested4,6–8 that the solutions of the maximum-principle boundary-value problem (2.4), (2.5) as well as of the modified problem (3.1), (3.2) exist and are unique. The time-optimality of the solution of the initial control problem (1.1) is thereby established. For illustration, we will first present a brief analysis of the solution of the boundary-value problem and optimal motion in special cases. The investigation of the system of 2n transcendental equations (2.4) in the unknown parameters = p|p|−1 , = g|p|−1 and tf with the additional conditions || = 1 and zf = etf = 1 corresponds to the situation k → 0 (the absence of a viscous medium). A fundamental property of the solution in the case when n = 2 (motion in a plane, that is, the vectors x0 , 0 , f are collinear)8 involves the above-mentioned collinearity of the optimal control vector. The velocity vector (t) of the optimal trajectory in the general case in the final part of the motion t = tf − t, that is, when t/tf 1, is then coaxial to the final velocity vector f with a relative error O((t/tf )2 ), that is, the motion takes place along a coaxial line. The final part of the optimal motion trajectory is located in a phase plane corresponding to unidimensional motion.7 Four types of trajectories are observed in the final stage: two “hill” trajectories (from above and from below), a “serpentine curve” and a “loop” (Ref. 8, Figs 1 and 2). The case of the time-optimal steering of a dynamic object to the initial (starting) point x(0) = x(tf ) = 0 of a geometric plane with the required value of the final velocity vector is of interest in the class of plane optimal-control problems: the vectors (0) = 0 and (tf ) = f are specified (Ref. 7, Figs 1 and 2). The time-optimal plane trajectories passing through a given point when the initial and final velocities are equal possess the interesting properties of considerable final rectification (Ref. 6, Figs 4 and 5). In the case of zero final velocity (x˙ f = 0, a “soft” encounter or landing4 ), the optimal control problem appears to be a plane problem: the initial position x0 , velocity 0 and control u* vectors will be coplanar. The final part of the optimal trajectory x(t) also satisfies the above-mentioned rectification property, but the final part of the uniformly retarded motion can begin at an arbitrary geometric point of a fairly small neighbourhood of the origin of coordinates x = 0; the change in the velocity vector (t) has a similar character. Hence, the fact that the trajectories are rectified at the final stage of the time-optimal control of the motion is established, which leads to the degeneration of the problem. The solution of applied problems of guidance, interception, docking, mooring, etc. is at the same time complicated because of the effect of perturbing factors leading to considerable errors near the terminal point, that is, to instability of the control process including the feedback control. Great computational difficulties arise under self-guidance conditions when the measuring instruments are located on the controlled object. The existence of a large number of methods for controlling moving objects is explained by these circumstances.14 Similar properties of the trajectories also hold when there is viscous friction. However, the difficulties in carrying out exact calculations are aggravated by the great complexity of the resulting expressions for the maximum-principle boundary-value problem (2.4), (2.5) and (3.1), (3.2). They can be carried out by numerical methods (the method of successive approximations, accelerated convergence, etc.) on the basis of approximate solutions corresponding to limit situations. For example, when tf 1 (we recall that tf → vtf (1.2)), that is, in the case of a small dissipation effect, the quantity zf ≈ 1, and the motion of the object in the first approximation in tf corresponds to the case k = 0. This is established by taking the limit in expressions (2.4), (2.5) and (3.2) as k → 0 (zf = etf ). The generating solutions and their properties are described above. In the case of small values of the quantity tf , recurrence procedures are applicable which rapidly converge to the desired exact solution. The standard procedure of continuation with respect to the parameter k is then used. The quantities x0 , 0 , f can vary over wide ranges. Another (“inverse”) limit case corresponds to the strong inequality zf 1 (tf = ln zf ); then the quantity 0 /zf in Eqs (2.4) can be neglected. The basic control stage then involves reaching the origin of coordinates x = 0 in a minimum time with subsequent adjustment of the motion to the state x = 0, = f for a certain t = tf . The characteristics of the motion in the first approximation are determined by the control u(1) = − x0 /|x0 |, with the velocity vector being approximated as follows: ≈ u(1) . In particular, when f = 0, the initial approximation according to relations (2.4) and (3.1) leads to collinearity of the and n-vectors of the form = I0−1 I1 , and the value of the parameter = 1. The relation between the unknown parameters zf and (4.1) follows from this. The functions I0 and I1 have the form (2.9) when = 1. The first and third relations of (3.2) are identically satisfied and the second relation together with the relation obtained above serve to find the required values of zf and : (4.2)
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The approximate solution of the problem of the time-optimal steering of a dynamic object to the origin of coordinates with zero velocity taking account of relations (4.1) and (4.2) is found in explicit form. The open-loop control up and the time T have the form
(4.3) x0
+ 0 ).
According to relations (43), the quantity t = T − corresponds to the instant of the switching of the control up (t, As established above, the parameter = 1 and the unknown = e . An additional time interval 2 is required in order to reduce the magnitude of the velocity to approximately zero ( T). Expressions (4.3) can be reduced to the feedback form when |x + | 1: (4.4) where is the time remaining until the end of the process. When approaching the final phase point (x, ) = (0, 0), the motion occurs along a line and the steering is performed in accordance with a one-dimensional optimal feedback control.1,2 The method discussed for sinthesizing the feedback control can be applied to the problem of the time-optimal “soft landing” on a convex manifold such as, for example, on a fixed sphere in an n-dimensional (n = 3) space.15 When there is a phase constraint (the penetration of the sphere is prohibited) the problem of the deviation from a sphere can be solved16 with subsequent control of the “landing”.15 The existence of viscosity makes the solution more difficult because of the considerable increase in the complexity of the maximum-principle boundary value problem. However, its character is preserved and is analogous to that discussed above in the case of a terminal point. Acknowledgements This research was financed by the Russian Foundation for Basic Research (11-01-00472 and 11-01-12110), the Programme for the Support of Leading Scientific Schools (NSh-64817.2010.1) and Programme No. 15 of the Department of Power Engineering, Machine Building, Mechanics and Control Processes of the Russian Academy of Sciences. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF. Mathematical Theory of Optimal Processes. New York: Wiley; 1962. Boltyanskii VG. Mathematical Methods of Optimal Control. New York: Holt, Rinehart and Winston; 1971. Akulenko LD. Asymptotic Methods of Optimal Control. Moscow:Nauka;1987. Akulenko LD, Shmatkov AM. Feedback control in the problem of the time-optimal steering of a point mass to a specified position with zero velocity. Prikl Mat Mekh 1998;62(1):129–38. Grozdovskii GL, Ivanov YuN, Tokarev VV. The Mechanics of Space Flight. Optimization Problems. Moscow: Nauka; 1975. Akulenko LD, Koshelev AP. Time-optimal steering of a dynamic object to a specified position in the case of the equality of the initial and terminal velocities. Izv Ross Akad Nauk Theory and Control Systems 2003;6:98–105. Akulenko LD, Koshelev AP. Time-optimal return of a dynamic object with the required velocity. Dokl Ross Akad Nauk 2005;403(5):614–8. Akulenko LD, Koshelev AP. Time-optimal steering of a point mass to a specified position with the required velocity. Prikl Mat Mekh 2007;71(2):228–36. Akulenko LD. The perturbed time-optimal problem of controlling the final position of a point mass by means of a limited force. Prikl Mat Mekh 1994;58(2):12–21. Akulenko LD. Feedback control in the problem of the time-optimal intersection of a sphere. Prikl Mat Mekh 1996;60(5):724–35. Vnuchkov DV. The time-optimal steering of a dynamical system with linear dissipation to a specified terminal position. Izv Ross Akad Nauk Theory and Control Systems 1998;3:56–61. Akulenko LD, Shmatkov AM. Time-optimal intersection of a sphere in a viscous medium. Izv Ross Akad Nauk Theory and Control Systems 2007;1:19–26. Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. New York: Academic Press; 2000. Gutkin LS, Borisov YuP, Valuyev AA. Radiocontrol of Rocket Missiles and Spacecraft. Moscow: Sovetokoye Radio; 1968. Akulenko LD, Shmatkov AM. Time-optimal steering of a point mass onto the surface of a sphere at zero velocity. Prikl Mat Mekh 2002;66(1):10–23. Akulenko LD, Shmatkov AM. Evasion of a fixed sphere by a dynamic object driven by a bounded force. Prikl Mat Mekh 2003;67(2):179–90.
Translated by E.L.S.