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ОРТIMAL ON-LINE CONTROL AND CLASSICAL REGULATION PROBLEMS

ОРТIMAL ON-LINE CONTROL AND CLASSICAL REGULATION PROBLEMS. Faina M. Kirillova Institute of Mathematics National Academy of Sciences of Belarus e-mail: kirill@nsys.minsk.by Minsk BELARUS. 1. Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria.

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ОРТIMAL ON-LINE CONTROL AND CLASSICAL REGULATION PROBLEMS

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  1. ОРТIMAL ON-LINE CONTROL AND CLASSICAL REGULATION PROBLEMS Faina M. Kirillova Institute of MathematicsNational Academy of Sciences of Belarus e-mail: kirill@nsys.minsk.by Minsk BELARUS 1 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  2. The final test of a theory is its capacity to solve the problems which originated itG.Dantzig OUTLINE • Basic problems of classical regulation theory • A linear optimal control problem. Optimal open-loop solutions • Optimal feedbacks to linear control systems • Stabilization by optimal control methods • Examples. An oscillating system. Damping oscillations of a string. • Stabilization of nonlinear inverted pendulums • Positional optimization of nonlinear control systems • Regulation of a crane • Realization of dynamic systems with a prescribed behavior. Synthesis of systems with prescribed limit cycles. 2 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  3. 40 – 50th A.M. Hopkin, D.W. Bushau, A. A. Feldbaum, A. Ya. Lerner Let a control object has several operating modes Regulation problem:to construct a feedback at which an object is transferred from one state to the other and stabilized at the new state. The second basic regulation problem Let a control system and a set of motions are given. It is necessary to construct a feedback at which this element is an asymptotically stable trajectory for the closed-loop system. invariance, robustness, damping, amortization V.S. Kulebakin, B.N. Petrov, N.N. Lusin, M.V. Meerov, A.I. Lurie, M.A. Aizermann, A.A. Krasovskii, A.M. Lyotov, Ya. Z. Tsypkin The Pontryagin Maximum Principle (1956):Optimal Open-loop Controls Dynamic ProgrammingFeedback Optimal Control “Curse of Dimensionality” 3 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  4. 2. A linear optimal control problem (1) (2) ( -- a quantization period) Problem (1) is equivalent to LP problem: R.Gabasov, F.M. Kirillova et al. Constructive methods of optimization: Part 1. Linear problems; Part 2. Control problems; Part 3. Transportation problems; Part 4. Convex problems; Part 5. Nonlinear problems. Minsk, University press. – 1984 – 1998. 4 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  5. Example 1. It is necessary to damp a two mass oscillating system (Fig. 1) under the minimum “fuel consumption” (3) x1, x2: deviations of masses from the equilibrium, x3, x4: velocities of the masses, u: “fuel consumption” per second. u(t), t≥0: discrete function from (2). Dimension of x: 5; complexity: amounts of total integrations of primal or adjoint systems. u x1 m C2 x2 M C1 Open-loop solutions of OC problem Fig. 1 Table 1. 5 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  6. 3. Optimal closed-loop controls (4) Let be an optimal open-loop control for (4), be a set of all possible for which problem (4) with a fixed has a solution. Definition. A function (5) s said to be an optimal positional control. • Balashevich N., Gabasov R., Kirillova F.M. Numerical methods for open-loop and closed-loop optimization of linear control systems. // Comput. Math. & Math. Phys., 40, 2000, P.137-138. • Gabasov R., Kirillova F.M. Real-time construction of optimal closable feed-backs. Proceed. of 14 IFAC Congress. San-Francisco, 1996, Vol.D, P.231-236. • Gabasov R., Kirillova F.M., Prischepova S.V. Optimal feedback control. Lecture Notes in Control and Information Sciences. (Thoma M. ed.) London-Berlin-Heidelberg: Springer. V. 207. 1995. 6 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  7. Closed-loop system under the conditions of constantly acting disturbances (6) disturbance transient (7) A device that for any current position is able to calculate a value of realization of the optimal feedback for the time which does not exceed h is said to be Optimal controller. 7 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  8. A dual method of LP adapted to dynamic problem (1). Fig. 2a, 2b disturbances: Thin line : optimal open-loop control Solid line : optimal positional control Fig. 3 : realization of the optimal feedback; Fig. 4 : values of complexity of calculation of current values of Algorithm of Acting Optimal Controller a) b) x3 x4 x1 x2 Fig. 2 Fig. 3 Fig. 4 8 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  9. 4. Stabilization of linear systems Let system (1), is not asymptotically stable, G is a vicinity of the equilibrium state . A function is said to be a bounded stabilizing feedback in G if 1) 2) 3) a zero solution of is asymptotically stable in G. R.Kalman, А.М. Lyotov (1960-1962) Positional solution of linear quadratic problems of optimal control with an infinite horizon. • Gabasov R., Kirillova F.M., Kostyukova O.I. To methods of stabilizing dynamic systems. Technical Cybernetics. 1993. № 3. P. 67-77. • Balashevich N.V., Gabasov R., Kirillova F.M. Stabilization of dynamic systems with delay in feedback loop. Automation and Remote Control. 1996. № 6. P. 31-39. 9 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  10. Auxiliary (accompanying) optimal control problem (parameter of the method). Auxiliary (accompanying) optimal control problem: (8) Let be an optimal open-loop control for z, be a set of all state z for which problem (8) has a solution. The function (9) is a bounded stabilizing feedback. The properties 1) → maximal; 2) extremal property 10 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  11. Example 2. The stabilization of an oscillating system (10) where : state of a system, u: a control function. (11) Y.J. Sussmann, E.D. Sontag, Y. Yang (1999) Accompanying optimal control problem (12) Fig.5 Fig. 5: transients of (10) with feedback (11) (Sussman et al) (line 1) and the feedback constructed for (12) (line 2). x1 11 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  12. Example 3. Let be a degree of stability: Accompanying problem (12) with the additional condition: (13) Parameters: Fig. 6: behavior of the output signal at different Curve 1: =0.1, a=0.2. Curve 2: =0.5, a=0.4. Curve 3: no constraints on . 5. Degree of stability, oscillation, monotonicity, overcontrol  Fig. 6 12 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  13. Degree of stability, oscillation, monotonicity, overcontrol Example 4. The functional The accompanying problem: (14) Fig. 7. Curve 1: transients under condition (13).  13 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  14. 6. Nonstationary systems. Systems with many-dimensional controls. Control systems with delays.Distributed parameter systems Example 5.Damping oscillations of a string. (15) Let be an open-loop control. be a vicinity of A functional (16) is said to be a damping control of feedback type if 1) problem (15) has a solution 2) at (17) 14 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  15. Results of computer experiments Parameters: Intensity of control Fig. 10 15 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  16. 7. Nonlinear systems Example 6. Stabilization of a pendulum under a large disturbances. Mathematical model: (18) Problem 1. To construct a bounded feedback ( ) stabilizing the pendulum at the upper state after the large disturbances of the initial state ( ). The transients with stabilizing feedbacks (various L). The curves 1,2 (without oscillations) The curves 3 – 6 (with the swings) Problem 2. Mathematical model (Pendulum control by horizontal movements of the suspension axis) Problem 3. Mathematical model (Control by Magnet) Fig. 11 16 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  17. 8. Classical problem of regulation (possible equilibrium state) Let be given. A function (19) is said to be a feedback solving the classical regulation problem if 1) 2) 3) closed system (20) has a solution 4) an equilibrium state of (20) is asymptotically stable in G. 17 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  18. u M 0 z x H  Example 7. The regulation problem for a crane transferring the load by rope from one equilibrium state to the vicinity of the other (Fir. 12). (21) x: deviation of the crane from the first equilibrium; φ: deviation of the rope from the vertical; I: moment of inertia. Parameters: Fig. 12 18 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  19. An accompanying optimal control problem x φ   Fig. 13. =5, 10 x φ Fig. 14   19 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  20. 9. A problem of realizing motions A set of realized motions (22) Problem. Let be given. Design a bounded stabilizing feedback such that becomes an asymptotically stable solution for the closed system Self – oscillating systems realizing given stable limit cycles. 20 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  21. Example 8. periodic solutions but not limit cycles x x x x Fig. 15 21 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  22. y y Example 9. periodic solutions x x y y x x Fig. 16 22 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  23. Example 10. no periodic solutions Limit cycle x x x x Fig. 17 23 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  24. 10. Invariance, robustness • Gabasov R., Ruzhitskaya E.A. Robust stabilization of dynamic systems by bounded controls. Appl. Math. Mech. 1998. Vol. 62. № 5. P. 778 – 785. 11. References • Gabasov R., Kirillova F.M. and Ruzhitskaya E.A. Damping and stabilization of a pendulum at the large initial disturbances // J. Comput. and Systems Sci. Intern. 2001, № 1, P. 29–38. • Gabasov R., Kirillova F.M. and Ruzhitskaya E.A. The classical regulation problem: its solution by optimal control methods // Automation and Remote Control. 2001, № 6, P. 18–19. • Gabasov R., Furuta K. et al. stabilization in Large an inverted pendulum // J. of Computer and Sciences Intern. 2005, Vol. 42, № 1. P. 13–19. • Gabasov R., Kirillova F.M. Reference governors for tracking systems with control constraints // Doklady Academii Nauk of Belarus. 2002. • Gabasov R. et al. Optimal control of nonlinear systems // Comput. Math & Math. Phys., 2002. Vol. 42, № 7, P. 931–956. Computer experiments: Dr. N. Balashevich (Minsk, Belarus), Dr. E. Ruzhitskaya (Gomel, Belarus) 24 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  25. Model Predictive Control (MPC) Industrial models Problem • To compose mathematical model to a future sampling instance using measurements of the process conducted up to the current moment • To formulate an auxiliary optimization problem (an optimal control problem) in accordance with processes under consideration and calculate the optimal program • To implement the initial part of the obtained optimal program to control up to the next sampling instance (or measurement) MPC • linear-quadratic OC problem • complete discretization of the model • use of standard methods of quadratic programming MPC approach is effective at control for sufficiently slow processes 25 Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

  26. Thank youVery much Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

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