View
4
Download
0
Category
Preview:
Citation preview
Optimal Control
Prof. Daniela Iacoviello
Lecture 1
25/09/2017 Pagina 2
Prof. Daniela Iacoviello
Department of computer, control and management
engineering Antonio Ruberti
Office: A219 Via Ariosto 25
http://www.dis.uniroma1.it/~iacoviel
Prof.Daniela Iacoviello- Optimal Control Pagina 2 25/09/2017
25/09/2017 Pagina 3 Prof.Daniela Iacoviello- Optimal Control Pagina 3 25/09/2017
Mailing list:
Send an e-mail to: iacoviello@dis.uniroma1.it
Subject: Optimal Control 2017
25/09/2017 Pagina 4 Prof.Daniela Iacoviello- Optimal Control
Grading
Project+ oral exam
Example 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 exam:
-A .doc document
-A power point presentation
-Matlab simulation files
Oral exam: Discussion of the project AND on the topics of the lectures
Pagina 4 25/09/2017
25/09/2017 Pagina 5 Prof.Daniela Iacoviello- Optimal Control
Some projects studied in 2014-15, 2015-16, 2016-17
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
………
Pagina 5 25/09/2017
25/09/2017 Pagina 6 Prof.Daniela Iacoviello- Optimal Control
Some projects studied in 2014-15, 2015-16, 2016-17 2
Optimal Control Of Shell And Tube Heat Exchanger
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 Exchanger
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
………
Pagina 6 25/09/2017
25/09/2017 Pagina 7 Prof.Daniela Iacoviello- Optimal Control
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
Pagina 7 25/09/2017
25/09/2017 Pagina 8 Prof.Daniela Iacoviello- Optimal Control
References
B.D.O.Anderson, J.B.Moore, Optimal control, Prentice Hall, 1989
C.Bruni, G. Di Pillo, Metodi Variazionali per il controllo ottimo, Masson , 1993
A.Calogero, Notes on optimal control theory, 2017
L. Evans, An introduction to mathematical optimal control theory, 1983
H.Kwakernaak , R.Sivan, Linear Optimal Control Systems, Wiley Interscience, 1972
D. E. Kirk, "Optimal Control Theory: An Introduction, New York, NY: Dover, 2004
D. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise
Introduction", Princeton University Press, 2011
How, Jonathan, Principles of optimal control, Spring 2008. (MIT OpenCourseWare:
Massachusetts Institute of Technology). License: Creative Commons BY-NC-SA.
……………………………………..
Pagina 8 25/09/2017
25/09/2017 Pagina 9 Prof.Daniela Iacoviello- Optimal Control
Course outline
• Introduction to optimal control
• Nonlinear optimization
• Dynamic programming
• Calculus of variations
• Calculus of variations and optimal control
• LQ problem
• Minimum time problem
Pagina 9 25/09/2017
25/09/2017 Pagina 10 Prof.Daniela Iacoviello- Optimal Control
Background
First course on linear systems
(free evolution, transition matrix, gramian
matrix,…)
Pagina 10 25/09/2017
25/09/2017 Pagina 11 Prof.Daniela Iacoviello- Optimal Control
Notations
nRtx )(
pRtu )(
RRRRf pn ::
kC
State variable
Control variable
Function (function with k-th derivative continuous
a.e.)
Pagina 11 25/09/2017
25/09/2017 Pagina 12 Prof.Daniela Iacoviello- Optimal Control
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 index is introduced
Pagina 12 25/09/2017
25/09/2017 Pagina 13 Prof.Daniela Iacoviello- Optimal Control
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
Pagina 13 25/09/2017
25/09/2017 Pagina 14 Prof.Daniela Iacoviello- Optimal Control
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
Pagina 14 25/09/2017
25/09/2017 Pagina 15 Prof.Daniela Iacoviello- Optimal Control
Hystory
Pagina 15 25/09/2017
25/09/2017 Pagina 16 Prof.Daniela Iacoviello- Optimal Control
Hystory
In 1696 Bernoulli posed the
Brachistochrone problem to his
contemporaries: “it seems to be the first problem which
explicitely dealt with optimally controlling the path or the
behaviour of a dynamical system»
Suppose a particle of mass M moves from A to B under the
influence of gravity.
What is the shape of the wire from which the time to get
from A to B is minimized?
Pagina 16 25/09/2017
Example 1 (Evans 1983)
Reproductive strategies in social insects
Let us consider the model describing how social
insects (for example bees) interact:
represents the number of workers at time t
represents the number of queens
represents the fraction of colony effort
devoted to increasing work force
lenght of the season
)(tw)(tq
)(tu
T
Motivations
Prof.Daniela Iacoviello- Optimal Control Pagina 17 25/09/2017
0)0(
)()()()()(
ww
twtutsbtwtw
Evolution of the
worker population
Death rate Known rate at which each worker
contributes to the bee economy
0)0(
)()()(1)()(
twtstuctqtq
Evolution of the
Population
of queens
Constraint for the control: 1)(0 tu
Prof.Daniela Iacoviello- Optimal Control Pagina 18 25/09/2017
The bees goal is to find the control that
maximizes the number of queens at time T:
The solution is a bang- bang control
)()( TqtuJ
Prof.Daniela Iacoviello- Optimal Control Pagina 19 25/09/2017
Motivations
Example 2 (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
represents the height at time t
represents the velocity =
represents the mass of spacecraft
represents thrust at time t
We assume
)(th
)(tv
)(tm
)(th
)(tu
1)(0 tu
Prof.Daniela Iacoviello- Optimal Control Pagina 20 25/09/2017
Consider the Newton’s law:
We want to minimize the amount of fuel
that is maximize the amount remaining once we have
landed
where
is the first time in which
ugmthm )(
0)(0)()()(
)()(
)(
)()(
tmthtkutm
tvth
tm
tugtv
)())(( muJ
0)(0)( vh
Pagina 21 Prof.Daniela Iacoviello- Optimal Control 25/09/2017
25/09/2017 Pagina 22 Prof.Daniela Iacoviello- Optimal Control Pagina 22 25/09/2017
Analysis of linear control systems
Essential components of a control system
The plant
One or more sensors
The controller
Controller Plant
Sensor
25/09/2017 Pagina 23 Prof.Daniela Iacoviello- Optimal Control Pagina 23 25/09/2017
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
25/09/2017 Pagina 24 Prof.Daniela Iacoviello- Optimal Control
Course outline
• Introduction to optimal control
• Nonlinear optimization
• Dynamic programming
• Calculus of variations
• Calculus of variations and optimal control
• LQ problem
• Minimum time problem
Pagina 24 25/09/2017
25/09/2017 Pagina 25 Prof.Daniela Iacoviello- Optimal Control Pagina 25 25/09/2017
Definitions
Consider a function
And
denotes the Eucledian norm
A point is a local minimum of over
If
RRf n ::
nRD
Dx *f
nRD
)()(
0
*
*
xfxf
xxsatisfyingDxallforthatsuch
25/09/2017 Pagina 26 Prof.Daniela Iacoviello- Optimal Control Pagina 26 25/09/2017
Definitions
Consider a function
And
denotes the Eucledian norm
A point is a strict local minimum of over
If
RRf n ::
nRD
Dx *f
nRD
*)()(
0
*
*
xxxfxf
xxsatisfyingDxallforthatsuch
25/09/2017 Pagina 27 Prof.Daniela Iacoviello- Optimal Control Pagina 27 25/09/2017
Definitions
Consider a function
And
denotes the Eucledian norm
A point is a global minimum of over
If
RRf n ::
nRD
Dx *f
nRD
)()( * xfxf
Dxallfor
25/09/2017 Pagina 28 Prof.Daniela Iacoviello- Optimal Control Pagina 28 25/09/2017
Definitions
The notions of a local/strict/global maximum are defined
similarly
If a point is either a maximum or a minimum
is called an extremum
Unconstrained optimization - first order necessary conditions
All points sufficiently near in are in
Assume and its local minimum. Let
an arbitrary vector.
Being in the unconstrained case:
Let’s consider:
0 is a minimum of g
x
nR
DnR
1Cf *x
)(:)( * xfg
0
*
toenoughcloseR
Dx
*x
Prof.Daniela Iacoviello- Optimal Control 25/09/2017 Pagina 29
First order Taylor expansion of g around
Proof: assume
For these values of
If we restrict to have the opposite sign of
Contraddiction
0
0)(
lim),()0(')0()(0
ooggg
0)0(' g
25/09/2017 Pagina 30 Prof.Daniela Iacoviello- Optimal Control
Unconstrained optimization - first order necessary conditions
0)0(' g
)0(')(
0
gofor
thatsoenoughsmall
)0(')0(')0()( gggg
)0('g
)0(')0(')0()( gggg 0)0()( gg
0)0(' g
δ is arbitrary
First order necessary condition for optimality
fofgradienttheis
fffwherexfgT
xx n
1:)()(' *
0)()0(' * xfg
0)( * xf
Prof.Daniela Iacoviello- Optimal Control 25/09/2017 Pagina 31
Unconstrained optimization - first order necessary conditions
25/09/2017 Pagina 32 Prof.Daniela Iacoviello- Optimal Control Pagina 32 25/09/2017
A point satisfying this condition is a stationary point *x
Unconstrained optimization - first order necessary conditions
Assume and its local minimum. Let
an arbitrary vector.
Second order Taylor expansion of g around
Since
nR2Cf *x
0
0)(
lim),()0(''2
1)0(')0()(
2
2
0
22
oogggg
0)0(' g
0)0('' g
Unconstrained optimization – second order conditions
25/09/2017 Pagina 33 Prof.Daniela Iacoviello- Optimal Control
Proof: suppose
For these values of
Contraddiction
0)0('' g
22 )0(''2
1)(
0
gofor
thatsoenoughsmall
0)0()( gg
25/09/2017 Pagina 34 Prof.Daniela Iacoviello- Optimal Control
Unconstrained optimization – second order conditions
0)0('' g
By differentiating both sides with respect to
Second order necessary condition for optimality
n
i
ix xfgi
1
* )()('
)()()0(''
)()(''
*2
1,
*
1,
*
xfxfg
xfg
T
j
n
ji
ixx
j
n
ji
ixx
ji
ji
nnn
n
xxxx
xxxx
ff
ff
f
1
111
2
Hessian matrix
0)( *2 xf
25/09/2017 Pagina 35
Unconstrained optimization – second order conditions
Prof.Daniela Iacoviello- Optimal Control
Remark:
The second order condition distinguishes minima from
maxima:
At a local maximum the Hessian must be negative semidefinite
At a local minimum the Hessian must be positive semidefinite
25/09/2017 Pagina 36
Unconstrained optimization – second order conditions
Prof.Daniela Iacoviello- Optimal Control
Unconstrained optimization- second order conditions
Let and
is a strict local minimum of f
0)( * xf2Cf 0)( *2 xf
*x
25/09/2017 Pagina 37 Prof.Daniela Iacoviello- Optimal Control
Global minimum – Existence result
Weierstrass Theorem
Let f be a continuous function and D a compact set
there exist a global minimum of f over D
25/09/2017 Pagina 38 Prof.Daniela Iacoviello- Optimal Control
Constrained optimization
Let
Equality constraints
Inequality constraints
Regularity condition:
1, CfRD n
1,:,0)( ChRRhxh p
1,:,0)( CgRRgxg q
a
a
x
a
qg
qpx
ghrank
dimensionwith
ofconstraintactivethearegwhere
,
a
*
25/09/2017 Pagina 39 Prof.Daniela Iacoviello- Optimal Control
Constrained optimization
Lagrangian function
If the stationary point is called normal
and we can assume
Prof.Daniela Iacoviello- Optimal Control
)()()(,,, 00 xgxhxfxL TT
*x00
10
25/09/2017 Pagina 40 Prof.Daniela Iacoviello- Optimal Control
From now on and therefore the Lagrangian is
If there are only equality constraints the are called
Lagrange multipliers
The inequality multipliers are called Kuhn – Tucker
multipliers
i
10
)()()(,, xgxhxfxL TT
25/09/2017 Pagina 41
Constrained optimization
Prof.Daniela Iacoviello- Optimal Control
First order necessary conditions for constrained optimality:
Let and
The necessary conditions for to be a constrained local
minimum are
If the functions f and g are convex and the functions h are
linear these conditions are necessary and sufficient
i
ixg
x
L
i
ii
T
T
x
0
,0)(
0
*
*
Dx * 1,, Cghf *x
25/09/2017 Pagina 42 Prof.Daniela Iacoviello- Optimal Control
Constrained optimization
Second order sufficient conditions for constrained optimality:
Let and and assume the conditions
is a strict constrained local minimum if
ixgx
Liii
T
T
x
0,0)(0 *
*
Dx * 2,, Cghf
*x
pidx
xdhthatsuch
x
L
x
i
x
T ,...,1,0)(
0**
2
2
25/09/2017 Pagina 43 Prof.Daniela Iacoviello- Optimal Control
Constrained optimization
Function spaces
Functional
Vector space V,
is a local minimum of J over A if there exists an
Prof.Daniela Iacoviello- Optimal Control
RVJ :VA
Az *
)()(
0
*
*
zJzJ
zzsatisfyingAzallforthatsuch
25/09/2017 Pagina 44 Prof.Daniela Iacoviello- Optimal Control
Consider function in V of the form
The first variation of J at z is the linear function
such that
First order necessary condition for optimality:
For all admissible perturbation we must have:
Prof.Daniela Iacoviello- Optimal Control
RVz ,,
RVJz
:
and
)()()()( oJzJzJ z
0* z
J
Function spaces
25/09/2017 Pagina 45 Prof.Daniela Iacoviello- Optimal Control
A quadratic form is the second variation
of J at z if
we have:
second order necessary condition for optimality:
If is a local minimum of J over
for all admissible perturbation we must have:
RVJz
:2
and
)()()()()( 222 oJJzJzJ zz
0)(*
2 z
J
Az * VA
Function spaces
25/09/2017 Pagina 46 Prof.Daniela Iacoviello- Optimal Control
The Weierstrass Theorem is still valid
If J is a convex functional and is a convex set
A local minimum is automatically a global one and the first
order condition are necessary and sufficient condition for
a minimum
VA
Function spaces
25/09/2017 Pagina 47 Prof.Daniela Iacoviello- Optimal Control
Recommended