Optimal Control. Lecture 1. Prof. Daniela Iacoviello

Transcription

1 Optimal Control Lecture 1 Prof. Daniela Iacoviello

2 Prof. Daniela Iacoviello Department of computer, control and management engineering Antonio Ruberti Office: A19 Via Ariosto 5 1/9/18 Pagina

3 Mailing list: Send an to: Subject: Optimal Control 18 1/9/18 Pagina 3

4 Grading Project+ oral eam Eample of project (1-3 Students): -Read a paper on an optimal control problem -Study: background, motivations, model, optimal control, solution, results -Simulations -Conclusions -References You must give me, before the date of the eam: -A.doc document -A power point presentation -Matlab simulation files Oral eam: Discussion of the project AND on the topics of the lectures 1/9/18 Pagina 4

5 Some projects proposed in 14-15, 15-16, Application Of Optimal Control To Malaria: Strategies And Simulations Performance Compare Between Lqr And Pid Control Of Dc Motor Optimal Low-thrust Leo (Low-earth Orbit) To Geo (Geosynchronous-earth Orbit) Circular Orbit Transfer Controllo Ottimo Di Una Turbina Eolica A Velocità Variabile Attraverso Il Metodo dell'inseguimento Ottimo A Regime Permanente Optimalcontrol In Dielectrophoresis On The Design Of P.I.D. Controllers Using Optimal Linear Regulator Theory Rocket Railroad Car Optimal Control Of Quadrotor Altitude Using Linear Quadratic Regulator Optimal Control Of An Inverted Pendulum Glucose Optimal Control System In Diabetes Treatment 1/9/18 Pagina 5

6 Some projects proposed in 14-15, 15-16, Optimal Control Of Shell And Tube Heat Echanger Optimal Control Analysis Of A Mathematical Model For Unemployment Time optimal control of an automatic Cableway Glucose Optimal Control System In Diabetes Treatment Optimal Control Of Shell And Tube Heat Echanger Optimal Control Analysis Of A Mathematical Model For Unemployment Time Optimal Control Of An Automatic Cableway Optimal Control Project On Arduino Managed Module For Automatic Ventilation Of Vehicle Interiors Optimal Control For A Suspension Of A Quarter Car Model 1/9/18 Pagina 6

7 THESE SLIDES ARE NOT SUFFICIENT FOR THE EXAM: YOU MUST STUDY ON THE BOOKS Part of the slides has been taken from the References indicated below 1/9/18 Pagina 7

8 References B.D.O.Anderson, J.B.Moore, Optimal control, Prentice Hall, 1989 E.R.Pinch, Optimal control and the calculus of variations, Oford science publications, 1993 C.Bruni, G. Di Pillo, Metodi Variazionali per il controllo ottimo, Masson, 1993 A.Calogero, Notes on optimal control theory, 17 L. Evans, An introduction to mathematical optimal control theory, 1983 H.Kwakernaak, R.Sivan, Linear Optimal Control Systems, Wiley Interscience, 197 D. E. Kirk, "Optimal Control Theory: An Introduction, New York, NY: Dover, 4 D. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton University Press, 11 How, Jonathan, Principles of optimal control, Spring 8. (MIT OpenCourseWare: Massachusetts Institute of Technology). License: Creative Commons BY-NC-SA... 1/9/18 Pagina 8

9 Course outline Introduction to optimal control Nonlinear optimization Dynamic programming Calculus of variations Calculus of variations and optimal control LQ problem Minimum time problem 1/9/18 Pagina 9

10 Background First course on linear systems (free evolution, transition matri, gramian matri, ) 1/9/18 Pagina 1

11 Notations ( t) n R State variable u( t) R p Control variable f : R n p R R : R Function C k (function with k-th derivative continuous a.e.) 1/9/18 Pagina 11

12 Introduction Optimal control is one particular branch of modern control that sets out to provide analytical designs of a special appealing type. The system, that is the end result of an optimal design, is supposed to be the best possible system of a particular type A cost inde is introduced 1/9/18 Pagina 1

13 Introduction Linear optimal control is a special sort of optimal control: the plant that is controlled is assumed linear the controller is constrained to be linear Linear controllers are achieved by working with quadratic cost indices 1/9/18 Pagina 13

14 Introduction Advantages of linear optimal control Linear optimal control may be applied to nonlinear systems Nearly all linear optimal control problems have computational solutions The computational procedures required for linear optimal design may often be carried over to nonlinear optimal problems 1/9/18 Pagina 14

15 VARIATIONAL METHODS Early Greeks Ma. area/perimeter Hero of Aleandria Equal angles of incidence /reflection Fermat - least time principle (Early 17 th Century) Newton and Leibniz Calculus (Mid 17th Century) Johann Bernoulli - brachistochrone problem (1696) Euler - calculus of variations (1744) Joseph-Louis Lagrange Euler-Lagrange Equations (??) William Hamilton Hamilton's Principle (1835) Raleigh-Ritz method VA for linear eigenvalue problems (late 19 th Century) Quantum Mechanics - Computational methods (early th Century) Morse & Feshbach technology of variational methods (1953) Solid State Physics, Chemistry, Engineering (mid-late th Century) Personal computers new computational power (198 s) Technology of variational methods essentially lost (198-) D. Anderson VA for perturbations of solitons (1979) Malomed, Kaup VA for solitary wave solutions (1994 present) 1/9/18 Pagina 15

16 History 1/9/18 Pagina 16

17 History In 1696 Bernoulli posed the Brachistochrone problem to his contemporaries: it seems to be the first problem which eplicitely dealt with optimally controlling the path or the behaviour of a dynamical system» Suppose a particle of mass M moves from A to B along smooth wire joining the two fied points (not on the same vertical line),under the influence of gravity. What shape the wire should be in order that the bead, when released from rest at A should slide to B in the minimum time? E.R.Pinch, Optimal control and calculus of variations, Oford Science Publications, /9/18 Pagina 17

18 Motivations Eample 1 (Evans 1983) Reproductive strategies in social insects Let us consider the model describing how social insects (for eample bees) interact: w(t) represents the number of workers at time t q(t) represents the number of queens u(t) represents the fraction of colony effort devoted to increasing work force T lenght of the season 1/9/18 Pagina 18

19 Death rate Known rate at which each worker contributes to the bee economy w ( t) w() = = w( t) w + b s( t) u( t) w( t) Evolution of the worker population q ( t) = q( t) q() = q + c ( 1 u( t) ) s( t) w( t) Evolution of the Population of queens u( t) 1 Constraint for the control: 1/9/18 Pagina 19

20 The bees goal is to find the control that maimizes the number of queens at time T: ( u( t) ) q( T ) J = The solution is a bang- bang control 1/9/18 Pagina

21 Motivations Eample (Evans 1983 and everywhere!) A moon lander Aim: bring a spacecraft to a soft landing on the lunar surface, using the least amount of fuel h(t) represents the height at time t v(t) represents the velocity = h (t) m(t) u(t) represents the mass of spacecraft represents thrust at time t We assume u( t) 1 1/9/18 Pagina 1

22 Consider the Newton s law: m h ( t) = gm+ u v ( t) = g + u( t) m( t) h ( t) = v( t) m ( t) = ku( t) h( t) m( t) We want to minimize the amount of fuel that is maimize the amount remaining once we have landed J ( u( )) = m( ) where is the first time in which h( ) = v( ) = 1/9/18 Pagina

23 Analysis of linear control systems Essential components of a control system The plant One or more sensors The controller Controller Plant Sensor 1/9/18 Pagina 3

24 Analysis of linear control systems Feedback: the actual operation of the control system is compared to the desired operation and the input to the plant is adjusted on the basis of this comparison. Feedback control systems are able to operate satisfactorily despite adverse conditions, such as disturbances and variations in plant properties 1/9/18 Pagina 4

25 Definitions Consider a function f : R n : R and n D R denotes the Eucledian norm D A point is a local minimum of over f n D R If such that for all D satisfying f ( ) f ( ) 1/9/18 Pagina 5

26 Definitions Consider a function and n D R : R denotes the Eucledian norm f n : R A point D is a strict local minimum of over If f ( f ) such that f ( ) for all D satisfying n D R 1/9/18 Pagina 6

27 Definitions Consider a function and n D R : R denotes the Eucledian norm f n : R A point D is a global minimum of over f n D R If for all D f ( ) f ( ) 1/9/18 Pagina 7

28 Definitions The notions of a local/strict/global maimum are defined similarly If a point is either a maimum or a minimum is called an etremum 1/9/18 Pagina 8

29 Unconstrained optimization - first order necessary conditions All points sufficiently near in are in R n D 1 Assume f C and its local minimum. n Let R be an arbitrary vector. Being in the unconstrained case: R close Let s consider: g( ) : = f ( + ) + D enough to is a minimum of g 1/9/18 Pagina 9

30 Unconstrained optimization - first order necessary conditions First order Taylor epansion of g around g( ) = g() + g'() + o( ), g' () = lim o( ) = = Proof: assume g' () small for enough so that o( ) g'() For these values of If we restrict to have the opposite sign of g ( ) g() g'() + g'() g'() g ( ) g() g'() + g'() g( ) g() Contraddiction g' () = 1/9/18 Pagina 3

31 Unconstrained optimization - first order necessary conditions g'( ) = f ( + ) where f : = ( f f ) 1 T is the gradient of f n g'() = f ( ) = Δ is arbitrary First order necessary condition for optimality f ( ) = 1/9/18 Pagina 31

32 Unconstrained optimization - first order necessary conditions A point satisfying this condition is a stationary point 1/9/18 Pagina 3

33 Unconstrained optimization second order conditions n Assume f C and its local minimum. Let R be an arbitrary vector. Second order Taylor epansion of g around = 1 o( ) g( ) = g() + g'() + g''() + o( ), lim Since g' () = = g' '() 1/9/18 Pagina 33

34 Unconstrained optimization second order conditions Proof: suppose g' '() For these values of small for g( ) g() enough o( ) so that 1 g''() Contraddiction g' '() 1/9/18 Pagina 34

35 By differentiating both sides with respect to Second order necessary condition for optimality = + = n i i f g i 1 ) ( ) ( ' ) ( ) ( ''() ) ( ) ''( 1, 1, f f g f g T j n j i i j n j i i j i j i = = + = = = = n n n n f f f f f Hessian matri ( ) f 1/9/18 Pagina 35 Unconstrained optimization second order conditions

36 Unconstrained optimization second order conditions Remark: The second order condition distinguishes minima from maima: At a local maimum the Hessian must be negative semidefinite At a local minimum the Hessian must be positive semidefinite 1/9/18 Pagina 36

37 Unconstrained optimization- second order conditions Let f and f ( ) = f ( ) C is a strict local minimum of f 1/9/18 Pagina 37

38 Global minimum Eistence result Weierstrass Theorem Let f be a continuous function and D a compact set there eist a global minimum of f over D 1/9/18 Pagina 38

39 Constrained optimization Let D R n, f C 1 Equality constraints p h( ) =, h : R R, hc 1 Inequality constraints q g( ), g : R R, g C 1 Regularity condition: rank where g g with ( h, g ) a a are dimension = p + qa the active constraint q a of 1/9/18 Pagina 39

40 Constrained optimization Lagrangian function L ( ) T T,, = f ( ) + h( ) + g( ), If the stationary point is called normal and we can assume =1 1/9/18 Pagina 4

41 Constrained optimization From now on =1 and therefore the Lagrangian is L T T (,, ) = f ( ) + h( ) + g( ) If there are only equality constraints the Lagrange multipliers i are called The inequality multipliers are called Kuhn Tucker multipliers 1/9/18 Pagina 41

42 Constrained optimization First order necessary conditions for constrained optimality: Let D and f, h, gc 1 The necessary conditions for to be a constrained local minimum are L g i ( i i T i = T ) =, i If the functions f and g are conve and the functions h are linear these conditions are necessary and sufficient 1/9/18 Pagina 4

43 Constrained optimization Second order sufficient conditions for constrained optimality: D f, h, gc Let and and assume the conditions L T = T g i i ( ) =, i is a strict constrained local minimum if i T L such that dhi ( ) d =, i = 1,..., p 1/9/18 Pagina 43

44 Definition: 1/9/18 Pagina 44 Constrained optimization j i T j i h L h Bordered Hessian A point in which and is called a non-degenerate critical point of the constrained problem. L = det j i T j i h L h

45 Constrained optimization Theorem: A necessary and sufficient condition for the non-degenerate critical point to minimize the cost f subjects to the constraints h ( ) =, j = 1,,..., m j is that T L for all non-zero tangent vector E.R.Pinch, Optimal control and calculus of variations, Oford Science Publications, /9/18 Pagina 45

46 A geometrical interpretation Let us consider the problem of minimizing function with one constraint h( 1, ) = c f (, ) 1 The previous theorem implies that we should look for such that: ( ) f + h = at f = h at and = ( 1, ) If f h the normal to the constraint curve h( 1, ) = c at = ( 1, ) normal parallel to f ( ) has its the level surface of f ( 1, ) that passes through = ( 1, ) has the same normal direction as the constraint curve at = (, ) 1 In R this means that the two curves touch at = ( 1, ) 1/9/18 Pagina 46

47 A geometrical interpretation - Eample Minimize the function subject to 1 = 1 f (, ) h(, ) = 1+ = Introduce a Lagrange multiplier and consider: ( ) 1 1 ( f + h) = 1 + ( 1 + ) = + = + = 1 1 There are three unknowns and we need another equation: 1 1+ = 1 =, = 1, = and 1 = 1, = 1/, = 1 1/9/18 Pagina 47

48 A geometrical interpretation - Eample The minimum is at You can find it evaluating =, = = 1 f (, ) 1 in the two points =, = 1, = =, = 1/, = 1 Note that the level curves of f are centered in the origin O and f decreases as you move away from the origin. 1/9/18 Pagina 48

49 Function spaces Functional J : V R Vector space V, A V z A is a local minimum of J over A if there eists an such that for all z A satisfying z z J( z ) J( z) 1/9/18 Pagina 49

50 Function spaces Consider function in V of the form z +, V, R The first variation of J at z is the linear function such that and J z : V R J( z + ) = J( z) + J z ( ) + o( ) First order necessary condition for optimality: For all admissible perturbation we must have: J z = 1/9/18 Pagina 5

51 Function spaces A quadratic form J z : V R is the second variation of J at z if and we have: J( z + ) = J( z) + J ( ) + J ( ) z z + o( ) second order necessary condition for optimality: If z A A V is a local minimum of J over for all admissible perturbation we must have: J z ( ) 1/9/18 Pagina 51

52 Function spaces The Weierstrass Theorem is still valid If J is a conve functional and A V is a conve set a local minimum is automatically a global one and the first order condition are necessary and sufficient condition for a minimum 1/9/18 Pagina 5