ОРТIMAL ON-LINE CONTROL AND CLASSICAL REGULATION PROBLEMS Faina M. Kirillova Institute of...

ОРТIMAL ON-LINE CONTROL ОРТIMAL ON-LINE CONTROL AND CLASSICAL REGULATION AND CLASSICAL REGULATION

PROBLEMSPROBLEMS Faina M. Kirillova

Institute of Mathematics National Academy of Sciences of Belarus e-mail: [email protected]

Minsk BELARUS

Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria1

The final test of a theory is its capacity The final test of a theory is its capacity to solve the problems which originated itto solve the problems which originated it

G.DantzigG.Dantzig

OUTLINE 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.


40 – 50th40 – 50th

A.M. Hopkin, D.W. Bushau, A. A. Feldbaum, A. Ya. Lerner Let a control object has several operating modes

Regulation problemRegulation 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 problemThe second basic regulation problemLet 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, amortizationV.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. TsypkinThe Pontryagin Maximum PrincipleThe Pontryagin Maximum Principle (1956): (1956): Optimal Open-loop ControlsDynamic ProgrammingDynamic Programming Feedback Optimal Control “Curse of Dimensionality”


2. A linear optimal control problem2. A linear optimal control problem

(1)

(2)

( -- a quantization period)

Problem (1) is equivalent to LP problem:

R.Gabasov, F.M. Kirillova et al.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.


max,)( * txc

,buAxx ,)0( 0xx ,)( * gtHx ,1)( tu ].,0[ *tTt

),()( khutu [,)1(,[ hkkht 1,0 Nk

Nth *

Open-loop solutions of OC problemOpen-loop solutions of OC problem Example 1.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.

m

M

C1

C2

u x1

x2

Fig. 1

N h Cost function

100 0,25 6,353339

1000 0,025 6,331252

10000 0,0025 6,330941

25000 0,001 6,330938

Table 1.


25

0

min,)( dttu

,31 xx ,42 xx ,213 uxxx ,02.11.0 214 xxx

,0)0()0( 21 xx ,2)0(3 x ,1)0(4 x,0)25()25()25()25( 4321 xxxx

],25,0[,1)(0 ttu

3. Optimal closed-loop controls3. 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.

max,)( * txc,buAxx ,)( zx ,)( * gtHx

,1)( tu ],,[)( *tTt ),(),,|(0 Ttztu X

nRz

),,|(),( 00 zuzu ,Xz ,,...,,0 * hthTu


Closed-loop system under the conditions of constantly Closed-loop system under the conditions of constantly acting disturbancesacting 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 controllerOptimal controller.

),(),(0 twxtbuAxx .)0( 0xx

),(* tw Tt),(* tx Tt

),())(,()()( **0** twtxtbutAxtx .Tt

)),(,()( *0* txtutu uTt

))(,( * x)(* u ,),(* Tttu


Algorithm of Acting Optimal ControllerAlgorithm of Acting Optimal Controller 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

,31 xx ,42 xx ,213 uxxx

)(02.11.0 *214 twxxx

[;75.9,0[,4sin3.0)(* tttw.75.9,0)(* ttw

.),(* Tttu

a) b)

x1

x3

x2

x4

)(* uFig. 2

Fig. 3 Fig. 4


4. Stabilization of linear systems4. 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.

,0,0)( ttu0x

,),( Gxxuu ;0)0( u

;,)( GxLxu

)(xbuAxx

xkxu ')(0


Auxiliary (accompanying) optimal control problemAuxiliary (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

,0,

,)(min)(0

dttuzB

,buAxx ,)0( zx ,0)( x ,)( Ltu ],,0[ Tt

.,..., 1 nbAAbbrank n

,),|(0 Ttztu )(G

),|0()( 0 xuxu ),(Gx

)(G

00

0

0

* )|()( dtxtudttu

,)(maxmin)( tuzBtu

0

2 )()( dttuzB


Example 2.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 Accompanying optimal control problem 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).

,21 xx ,312 xxx ,43 xx ,34 uxx

),,,( 4321 xxxx

)),(29(( 431291

4 xxxsatxsatu

,1)( tu .1,min)sgn()( ssssat

min,)(0

dttu

,21 xx ,312 xxx ,43 xx ,34 uxx ),()0( *

11 xx ),()0( *

22 xx ),()0( *33 xx )()0( *

44 xx ,0)( ix ,4,1i ,1)( tu ],0[ Tt

xx11


5. Degree of stability, oscillation, monotonicity, overcontrol5. Degree of stability, oscillation, monotonicity, overcontrol Example 3.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 .

*1x

Fig. 6

Faina M. Kirillova NATO ARW, October 21-24, 2006, Bulgaria

),()( 1 txty .0t

),exp()(1 tatx .0t

),exp()()exp( 1 tatxta .Tt

).1.0,1.0,1.0,1.0(,4.0,8 *0 xh

,0),()( *1 xy .,a

0),( tty

1212

Degree of stability, oscillation, monotonicity, overcontrolDegree of stability, oscillation, monotonicity, overcontrol

Example 4.Example 4. The functional

The accompanying problem:

(14)

Fig. 7. Curve 1: transients under condition (13).

)(max),(],0[

tuut

min,,21 xx ,312 xxx ,43 xx ,34 uxx

),()0( * ii xx ,0)( ix ,4,1i

),exp()()exp( 1 tatxta ,)( tu ].,0[ Tt11.0,1.0,08.0,2 ah

*1x


6. Nonstationary systems. Systems with many-dimensional 6. Nonstationary systems. Systems with many-dimensional controls. Control systems with delays.controls. Control systems with delays.

Distributed Distributed paparameter systemsrameter systems Example 5.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 if1) problem (15) has a solution 2) at (17)

,),(),(

2

22

2

2

x

xtya

t

xty

;0,0),(),()0,( tltytuty .],0[:),(),0(),(),0( lxxxxxyxxy t ,0,),,( txxty ,);(),( 21 xxzxzz ,),,()( xxtztz

;),(),,(),( 21 xtzxtzxtz ),,(),(1 xtyxtz );,(),(2 xtyxtz t xLxzLxzzG ;)(,)(: 1201 ,0z

.),(max,),(maxmax)( 21

xtzxtztz

xx

,),( Gzzu

,0,),,( txxty ,0)),(()( ttzutu0)( tz .t

.,)( GzLzu


Results of computer experimentsResults of computer experiments

Parameters:

Intensity of control

;1,1 al ),sin()(* xx .5.0)(* x

.1,3.0,6,2 Lm

Fig. 10


7. Nonlinear systems7. Nonlinear systems Example 6.Example 6. Stabilization of a pendulum under a large disturbances.

Mathematical model: (18)Problem 1.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.Problem 2. Mathematical model

(Pendulum control by horizontal movements of the suspension axis)

Problem 3.Problem 3. Mathematical model (Control by Magnet)


.sin uxx

Lxxu ),(

)0(),0( xx

Fig. 11

0cossin xuxax

16

8. Classical problem of regulation8. Classical problem of regulation

(possible equilibrium state)Let be given.A function (19)

is said to be a feedback solving the classical regulation problem if1)2) 3) closed system

(20)has a solution

4) an equilibrium state of (20) is asymptotically stable in G.


LubuAxRxX xxn ,0:0

,0, LL ,int 0Xz nRG

,),( Gxxuu z

;)( zz uzu ;,)( GxLxuz

,)0(),( 0 GxxxbuAxx z ;0,)( tGtx

,0,)( tztx

Example 7.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:


,)( umHxmM ,xmHmgHI ,0)0()0( xx ,0)0()0(

,)( ztx ,0)( tx ,0)( t ,0)( t

.27,10,3,3,7 2 mHIgHmM

Fig. 12

u

H

M

x

z0

An accompanying optimal control problemAn accompanying optimal control problem


min,)(0

dtutu z

,21 xx ,7/17/30 12 uxx ,43 xx ,21/12/100 14 uxx ,0)0()0()0()0( 4321 xxxx ,6)(1 x ,0)()()( 432 xxx

,)( Ltu ].,0[ Tt

Fig. 13. =5, 10

Fig. 14

.0,2sin1.0)(* tttw

x

x

φ

φ

9. A problem of realizing motions9. A problem of realizing motions

A set of realized motions

(22)

Problem.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.


buAxx

.0,)(,:)( tLtubuAffRtfF ffn

Ff )(*

,),( Gxxu f )(* f

.),( GxxbuAxx f


periodic solutions butperiodic solutions butnot limit cyclesnot limit cycles


,uxx

0,0)( ttu

,1)()( 22 tytx ,0t

),( xxuxx x x

x x

x x

x x

Fig. 15


periodic solutionsperiodic solutions


,uyx ,uxy

0,0)( ttu

.12)(2)(2*2* tytx

x x

x x

yy

y y

Fig. 16


no periodic solutionsno periodic solutions

Limit cycle


uxxx

,0,0)( ttu

.122 xx x x

x x

x x

x x

Fig. 17

10. Invariance, robustness10. 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. References11. 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:Computer experiments: Dr. N. Balashevich (Minsk, Belarus),

Dr. E. Ruzhitskaya (Gomel, Belarus)


Model Predictive Control (MPC)Model Predictive Control (MPC) Industrial modelsIndustrial models

ProblemProblem 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)

MPCMPC 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 processesMPC approach is effective at control for sufficiently slow processes


Thank youThank youVery muchVery much


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