Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon,...
Transcript of Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon,...
![Page 1: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/1.jpg)
Optimal Control
Lecture
Prof. Daniela Iacoviello
Department of Computer, Control, and Management Engineering Antonio Ruberti
Sapienza University of Rome
![Page 2: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/2.jpg)
13/10/2015 Controllo nei sistemi biologici
Lecture 1
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
![Page 3: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/3.jpg)
Grading
Project + oral exam
The exam must be concluded before the second
part of Identification that will be held by Prof. Battilotti
![Page 4: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/4.jpg)
Grading
Project+ oral exam
Example of project:
- Read a paper on an optimal control problem
- Study: background, motivations, model, optimal control, solution,
results
- Simulations
You must give me, before the date of the exam:
- A .doc document
- A power point presentation
- Matlab simulation files
The exam must be concluded before the second
part of Identification that will be held by Prof. Battilotti
![Page 5: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/5.jpg)
Some projects studied in 2014-15
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
Optimal Control in Dielectrophoresis
On the Design of P.I.D. Controllers Using Optimal Linear Regulator Theory
Rocket Railroad Car
………
![Page 6: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/6.jpg)
THESE SLIDES ARE NOT SUFFICIENT
FOR THE EXAM: YOU MUST STUDY ON THE BOOKS
Prof.Daniela Iacoviello- Optimal Control
Part of the slides has been taken from the References indicated below
![Page 7: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/7.jpg)
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 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.
References
Prof.Daniela Iacoviello- Optimal Control
![Page 8: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/8.jpg)
Course outline
• Introduction to optimal control
• Nonlinear optimization
• Dynamic programming
• Calculus of variations
• Calculus of variations and optimal control
Prof.Daniela Iacoviello- Optimal Control
![Page 9: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/9.jpg)
Course outline
• Introduction to optimal control
• Nonlinear optimization
• Dynamic programming
• Calculus of variations
• Calculus of variations and optimal control
Prof.Daniela Iacoviello- Optimal Control
![Page 10: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/10.jpg)
First course on linear systems (free evolution, transition matrix, gramian matrix,…)
Background
Prof.Daniela Iacoviello- Optimal Control
![Page 11: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/11.jpg)
Notations
nRtx )( State variable
pRtu )( Control variable
RRRRf pn ::
Function (function with second derivative continuous a.e.)
2C
Prof.Daniela Iacoviello- Optimal Control
![Page 12: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/12.jpg)
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 Prof.Daniela Iacoviello- Optimal Control
![Page 13: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/13.jpg)
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
Prof.Daniela Iacoviello- Optimal Control
![Page 14: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/14.jpg)
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
Prof.Daniela Iacoviello- Optimal Control
![Page 15: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/15.jpg)
History
Prof.Daniela Iacoviello- Optimal Control
![Page 16: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/16.jpg)
History
In 1696 Bernoulli posed the
Brachystochrone problem to his
contemporaries: “it seems to be the
First problem which explicitely dealt
with optimally controlling the path or or the behaviour of a dynamical
system”.
Prof.Daniela Iacoviello- Optimal Control
![Page 17: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/17.jpg)
Motivations
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
TProf.Daniela Iacoviello- Optimal Control
![Page 18: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/18.jpg)
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 tuProf.Daniela Iacoviello- Optimal Control
![Page 19: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/19.jpg)
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
![Page 20: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/20.jpg)
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
Prof.Daniela Iacoviello- Optimal Control
)(th
)(tu
1)(0 tu
![Page 21: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/21.jpg)
Considere 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
Prof.Daniela Iacoviello- Optimal Control
ugmthm )(
0)(0)()()(
)()(
)(
)()(
tmthtkutm
tvth
tm
tugtv
)())(( muJ
0)(0)( vh
![Page 22: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/22.jpg)
Analysis of linear control systems
Essential components of a control system
The plant
One or more sensors
The controller
Controller Plant
Sensor
Prof.Daniela Iacoviello- Optimal Control
![Page 23: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/23.jpg)
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
Prof.Daniela Iacoviello- Optimal Control
![Page 24: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/24.jpg)
Course outline
• Introduction to optimal control
• Nonlinear optimization
• Dynamic programming
• Calculus of variations
• Calculus of variations and optimal control
Prof.Daniela Iacoviello- Optimal Control
![Page 25: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/25.jpg)
Definitions
Consider a function
And
denotes the Eucledian norm
A point is a local minimum of over
If
Prof.Daniela Iacoviello- Optimal Control
RRf n ::
nRD
Dx *f
nRD
)()(
0
*
*
xfxf
xxsatisfyingDxallforthatsuch
![Page 26: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/26.jpg)
Definitions
Consider a function
And
denotes the Eucledian norm
A point is a strict local minimum of over
If
Prof.Daniela Iacoviello- Optimal Control
RRf n ::
nRD
Dx * fnRD
**
*
),()(
0
xxxfxf
xxsatisfyingDxallforthatsuch
![Page 27: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/27.jpg)
Definitions
Consider a function
And
denotes the Eucledian norm
A point is a global minimum of over
If
Prof.Daniela Iacoviello- Optimal Control
RRf n ::
nRD
Dx *f
nRD
)()( * xfxf
Dxallfor
![Page 28: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/28.jpg)
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
Prof.Daniela Iacoviello- Optimal Control
![Page 29: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/29.jpg)
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
Prof.Daniela Iacoviello- Optimal Control
x
nRd
DnR
1Cf *x
)(:)( * dxfg
0* toenoughcloseRDdx
*x
![Page 30: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/30.jpg)
First order Taylor expansion of g around
Proof: assume
For these values of
Prof.Daniela Iacoviello- Optimal Control
0
0)(
lim),()0(')0()(0
ooggg
0)0(' g
0)0(' g
)0(')(
0
gofor
thatsoenoughsmall
)0(')0(')0()( gggg
Unconstrained optimization - first order necessary conditions
![Page 31: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/31.jpg)
If we restrict to have the opposite sign to
contradiction.
d was arbitrary
First order necessary condition for
optimality
Prof.Daniela Iacoviello- Optimal Control
)0(')0(')0()( gggg
)0('g
0)0()( gg
fofgradienttheisfffwhereddxfgT
xx n
1:)()(' *
0)()0(' * dxfg
0)( * xf
Unconstrained optimization - first order necessary conditions
![Page 32: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/32.jpg)
A point satisfying this condition is a stationary point
Prof.Daniela Iacoviello- Optimal Control
*x
Unconstrained optimization - first order necessary conditions
![Page 33: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/33.jpg)
Unconstrained optimization- second order conditions
Assume and its local minimum. Let an
arbitrary vector.
Second order Taylor expansion of g around
Since
Proof: suppose
Prof.Daniela Iacoviello- Optimal Control
nRd2Cf *x
0
0)(
lim),()0(''2
1)0(')0()(
2
2
0
2
oogggg
0)0(' g
0)0('' g
0)0('' g
22 )0(''2
1)(
0
gofor
thatsoenoughsmall
![Page 34: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/34.jpg)
For these values of
contradiction
By differentiating both sides with respect to
Unconstrained optimization- second order conditions
Prof.Daniela Iacoviello- Optimal Control
0)0()( gg
n
i
ix ddxfgi
1
* )()('
dxfdddxfg
dddxfg
T
j
n
ji
ixx
j
n
ji
ixx
ji
ji
)()()0(''
)()(''
*2
1,
*
1,
*
nnn
n
xxxx
xxxx
ff
ff
f
1
111
2
Hessian matrix
![Page 35: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/35.jpg)
Unconstrained optimization- second order conditions
Second order necessary condition for
optimality
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
Prof.Daniela Iacoviello- Optimal Control
0)( *2 xf
![Page 36: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/36.jpg)
Unconstrained optimization- second order conditions
Let and
is a strict local minimum of f
Prof.Daniela Iacoviello- Optimal Control
0)( * xf2Cf 0)( *2 xf
*x
![Page 37: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/37.jpg)
Definitions- Remarks
A vector is a feasible direction at if
If D in not the entire then D is the constraint set over
which f is being minimized
Prof.Daniela Iacoviello- Optimal Control
*xnRd
0* enoughsmallforDdx
nR
![Page 38: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/38.jpg)
Global minimum
Weierstrass Theorem
Let f be a continuous function and D a compact set
there exist a global minimum of f over D
Prof.Daniela Iacoviello- Optimal Control
![Page 39: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/39.jpg)
Let
Equality constraints
Inequality constraints
Regularity condition:
Lagrangian function
If the stationary point is called normal and we can assume .
Constrained optimization
Prof.Daniela Iacoviello- Optimal Control
1, CfRD n 1,:,0)( ChRRhxh p
1,:,0)( CgRRgxg q
a
a
a
x
a
qg
ofconstrainactivethearegwhere
qpx
ghrank
dimensionwith
,
*
)()()(,,, 00 xgxhxfxL TT
*x00
10
![Page 40: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/40.jpg)
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
Prof.Daniela Iacoviello- Optimal Control
Constrained optimization
i
10
)()()(,, xgxhxfxL TT
![Page 41: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/41.jpg)
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!!!
Constrained optimization
Prof.Daniela Iacoviello- Optimal Control
i
ixg
x
L
i
ii
T
T
x
0
,0)(
0
*
*
Dx * 1,, Cghf
*x
![Page 42: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/42.jpg)
Constrained optimization
Second order sufficient conditions for constrained optimality:
Let and and assume the conditions
is a strict constrained local minimum if
Prof.Daniela Iacoviello- Optimal Control
ixgx
Liii
T
T
x
0,0)(0 *
*
Dx * 2,, Cghf
*x
piddx
xdhthatsuchdd
x
Ld
x
i
x
T ,...,1,0)(
0**
2
2
![Page 43: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/43.jpg)
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
![Page 44: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/44.jpg)
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
![Page 45: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/45.jpg)
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:
Prof.Daniela Iacoviello- Optimal Control
RVJz
:2
and
)()()()()( 222 oJJzJzJ zz
0)(*
2 z
J
Az * VA
Function spaces
![Page 46: Linear Optimal control - uniroma1.itiacoviel/Materiale/OptimalControl-Lecture1.pdfD. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton](https://reader033.fdocuments.us/reader033/viewer/2022050105/5f4354a22175b52181020cfb/html5/thumbnails/46.jpg)
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
Prof.Daniela Iacoviello- Optimal Control
VA
Function spaces