This document discusses various applications of optimization techniques in aerospace systems modeling, control, and design. It covers topics such as aircraft design formulation as a multidisciplinary optimization problem, parameter estimation for unmanned aerial vehicles using input-output flight test data, model order reduction via optimization, and control design problems cast as convex and sum-of-squares optimizations. Specific examples discussed include stability analysis of a jet engine model using polynomial Lyapunov functions.
In the study of probabilistic integrators for deterministic ordinary differential equations, one goal is to establish the convergence (in an appropriate topology) of the random solutions to the true deterministic solution of an initial value problem defined by some operator. The challenge is to identify the right conditions on the additive noise with which one constructs the probabilistic integrator, so that the convergence of the random solutions has the same order as the underlying deterministic integrator. In the context of ordinary differential equations, Conrad et. al. (Stat.
Comput., 2017), established the mean square convergence of the solutions for globally Lipschitz vector fields, under the assumptions of i.i.d., state-independent, mean-zero Gaussian noise. We extend their analysis by considering vector fields that need not be globally Lipschitz, and by
considering non-Gaussian, non-i.i.d. noise that can depend on the state and that can have nonzero mean. A key assumption is a uniform moment bound condition on the noise. We obtain convergence in the stronger topology of the uniform norm, and establish results that connect this topology to the regularity of the additive noise. Joint work with A. M. Stuart (Caltech), T. J. Sullivan (Free University of Berlin).
Reinforcement Learning: Hidden Theory and New Super-Fast AlgorithmsSean Meyn
A tutorial, and very new algorithms -- more details on arXiv and at NIPS 2017 https://arxiv.org/abs/1707.03770
Part of the Data Science Summer School at École Polytechnique: http://www.ds3-datascience-polytechnique.fr/program/
---------
2018 Updates:
See Zap slides from ISMP 2018 for new inverse-free optimal algorithms
Simons tutorial, March 2018 [one month before most discoveries announced at ISMP]
Part I (Basics, with focus on variance of algorithms)
https://www.youtube.com/watch?v=dhEF5pfYmvc
Part II (Zap Q-learning)
https://www.youtube.com/watch?v=Y3w8f1xIb6s
Big 2017 survey on variance in SA:
Fastest convergence for Q-learning
https://arxiv.org/abs/1707.03770
You will find the infinite-variance Q result there.
Our NIPS 2017 paper is distilled from this.
In the study of probabilistic integrators for deterministic ordinary differential equations, one goal is to establish the convergence (in an appropriate topology) of the random solutions to the true deterministic solution of an initial value problem defined by some operator. The challenge is to identify the right conditions on the additive noise with which one constructs the probabilistic integrator, so that the convergence of the random solutions has the same order as the underlying deterministic integrator. In the context of ordinary differential equations, Conrad et. al. (Stat.
Comput., 2017), established the mean square convergence of the solutions for globally Lipschitz vector fields, under the assumptions of i.i.d., state-independent, mean-zero Gaussian noise. We extend their analysis by considering vector fields that need not be globally Lipschitz, and by
considering non-Gaussian, non-i.i.d. noise that can depend on the state and that can have nonzero mean. A key assumption is a uniform moment bound condition on the noise. We obtain convergence in the stronger topology of the uniform norm, and establish results that connect this topology to the regularity of the additive noise. Joint work with A. M. Stuart (Caltech), T. J. Sullivan (Free University of Berlin).
Reinforcement Learning: Hidden Theory and New Super-Fast AlgorithmsSean Meyn
A tutorial, and very new algorithms -- more details on arXiv and at NIPS 2017 https://arxiv.org/abs/1707.03770
Part of the Data Science Summer School at École Polytechnique: http://www.ds3-datascience-polytechnique.fr/program/
---------
2018 Updates:
See Zap slides from ISMP 2018 for new inverse-free optimal algorithms
Simons tutorial, March 2018 [one month before most discoveries announced at ISMP]
Part I (Basics, with focus on variance of algorithms)
https://www.youtube.com/watch?v=dhEF5pfYmvc
Part II (Zap Q-learning)
https://www.youtube.com/watch?v=Y3w8f1xIb6s
Big 2017 survey on variance in SA:
Fastest convergence for Q-learning
https://arxiv.org/abs/1707.03770
You will find the infinite-variance Q result there.
Our NIPS 2017 paper is distilled from this.
In this talk, I address two new ideas in sampling geometric objects. The first is a new take on adaptive sampling with respect to the local feature size, i.e., the distance to the medial axis. We recently proved that such samples acn be viewed as uniform samples with respect to an alternative metric on the Euclidean space. The second is a generalization of Voronoi refinement sampling. There, one also achieves an adaptive sample while simultaneously "discovering" the underlying sizing function. We show how to construct such samples that are spaced uniformly with respect to the $k$th nearest neighbor distance function.
A gentle introduction to 2 classification techniques, as presented by Kriti Puniyani to the NYC Predictive Analytics group (April 14, 2011). To download the file please go here: http://www.meetup.com/NYC-Predictive-Analytics/files/
Linear Bayesian update surrogate for updating PCE coefficientsAlexander Litvinenko
This is our joint work with colleagues from TU Braunschweig. Prof. H. G. Matthies had an excellent idea to develop a Bayesian surrogate formula for updating not probability densities (like in classical Bayesian formula), but PCE coefficients of the given random variable. Bojana Rosic implemented the linear case. I (with help of Elmar Zander) implemented non-linear case. Later on Elmar significantly simplified the algorithm.
Presentation on stochastic control problem with financial applications (Merto...Asma Ben Slimene
This is an introductory to optimal stochastic control theory with two applications in finance: Merton portfolio problem and Investement/consumption problem with numerical results using finite differences approach
In this talk, I address two new ideas in sampling geometric objects. The first is a new take on adaptive sampling with respect to the local feature size, i.e., the distance to the medial axis. We recently proved that such samples acn be viewed as uniform samples with respect to an alternative metric on the Euclidean space. The second is a generalization of Voronoi refinement sampling. There, one also achieves an adaptive sample while simultaneously "discovering" the underlying sizing function. We show how to construct such samples that are spaced uniformly with respect to the $k$th nearest neighbor distance function.
A gentle introduction to 2 classification techniques, as presented by Kriti Puniyani to the NYC Predictive Analytics group (April 14, 2011). To download the file please go here: http://www.meetup.com/NYC-Predictive-Analytics/files/
Linear Bayesian update surrogate for updating PCE coefficientsAlexander Litvinenko
This is our joint work with colleagues from TU Braunschweig. Prof. H. G. Matthies had an excellent idea to develop a Bayesian surrogate formula for updating not probability densities (like in classical Bayesian formula), but PCE coefficients of the given random variable. Bojana Rosic implemented the linear case. I (with help of Elmar Zander) implemented non-linear case. Later on Elmar significantly simplified the algorithm.
Presentation on stochastic control problem with financial applications (Merto...Asma Ben Slimene
This is an introductory to optimal stochastic control theory with two applications in finance: Merton portfolio problem and Investement/consumption problem with numerical results using finite differences approach
Real Time Code Generation for Nonlinear Model Predictive ControlBehzad Samadi
This is a quick introduction to optimal control and nonlinear model predictive control. It also includes code generation for a NMPC controller. For a recorded webinar, follow this link: http://goo.gl/c5zFgN
The design and simulation of magneto-rheological damper for automobile suspen...IJRES Journal
As the application of magneto-rheological (MR) damper in automobile suspension semi-active
control, according to the character of the MR fluid and the relations between the damping force of MR damper
and the structure parameters, the new-type automobile suspension MR damper was designed. At present, the
design method of MR damper is still depended on the experience of the designer, which can't be calculated
accurately. In this paper, after a review of the features and models of MR materials and devices, geometrical
design and magnetic circuit design of MR damper were presented and discussed in detail, then the performance
of MR damper was analyzed by MATLAB. Simulation results showed that the design method of this automobile
suspension MR damper was feasible and the test performance was satisfactory, which could meet the
requirements of damping system for automobile suspension and achieve the required adjustable range of
anticipative damping force.
BEADS : filtrage asymétrique de ligne de base (tendance) et débruitage pour d...Laurent Duval
This paper jointly addresses the problems of chromatogram baseline correction and noise reduction. The proposed approach is based on modeling the series of chromatogram peaks as sparse with sparse derivatives, and on modeling the baseline as a low-pass signal. A convex optimization problem is formulated so as to encapsulate these non-parametric models. To account for the positivity of chromatogram peaks, an asymmetric penalty functions is utilized. A robust, computationally efficient, iterative algorithm is developed that is guaranteed to converge to the unique optimal solution. The approach, termed Baseline Estimation And Denoising with Sparsity (BEADS), is evaluated and compared with two state-of-the-art methods using both simulated and real chromatogram data.
Statement of stochastic programming problemsSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 1.
More info at http://summerschool.ssa.org.ua
We study an elliptic eigenvalue problem, with a random coefficient that can be parametrised by infinitely-many stochastic parameters. The physical motivation is the criticality problem for a nuclear reactor: in steady state the fission reaction can be modeled by an elliptic eigenvalue
problem, and the smallest eigenvalue provides a measure of how close the reaction is to equilibrium -- in terms of production/absorption of neutrons. The coefficients are allowed to be random to model the uncertainty of the composition of materials inside the reactor, e.g., the
control rods, reactor structure, fuel rods etc.
The randomness in the coefficient also results in randomness in the eigenvalues and corresponding eigenfunctions. As such, our quantity of interest is the expected value, with
respect to the stochastic parameters, of the smallest eigenvalue, which we formulate as an integral over the infinite-dimensional parameter domain. Our approximation involves three steps: truncating the stochastic dimension, discretizing the spatial domain using finite elements and approximating the now finite but still high-dimensional integral.
To approximate the high-dimensional integral we use quasi-Monte Carlo (QMC) methods. These are deterministic or quasi-random quadrature rules that can be proven to be very efficient for the numerical integration of certain classes of high-dimensional functions. QMC methods have previously been applied to linear functionals of the solution of a similar elliptic source problem; however, because of the nonlinearity of eigenvalues the existing analysis of the integration error
does not hold in our case.
We show that the minimal eigenvalue belongs to the spaces required for QMC theory, outline the approximation algorithm and provide numerical results.
Distributed solution of stochastic optimal control problem on GPUsPantelis Sopasakis
Stochastic optimal control problems arise in many
applications and are, in principle,
large-scale involving up to millions of decision variables. Their
applicability in control applications is often limited by the
availability of algorithms that can solve them efficiently and within
the sampling time of the controlled system.
In this paper we propose a dual accelerated proximal
gradient algorithm which is amenable to parallelization and
demonstrate that its GPU implementation affords high speed-up
values (with respect to a CPU implementation) and greatly outperforms
well-established commercial optimizers such as Gurobi.
Research internship on optimal stochastic theory with financial application u...Asma Ben Slimene
This is a presntation of my second year intership on optimal stochastic theory and how we can apply it on some financial application then how we can solve such problems using finite differences methods!
Enjoy it !
* ML in HEP
* classification and regression
* knn classification and regression
* ROC curve
* optimal bayesian classifier
* Fisher's QDA
* intro to Logistic Regression
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
Modeling, Control and Optimization for Aerospace Systems
1. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Modeling, Control, and Optimization
for Aerospace Systems
HYCONS Lab, Concordia University
Behzad Samadi
HYCONS Lab, Concordia University
American Control Conference
Montreal, Canada
June 2012
2. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Outline
Motivation
Aircraft design
Parameter estimation
Model order reduction
Model based control design
Convex Optimization
Sum of Squares
Lyapunov Analysis
Controller Synthesis
Safety Verification
Polynomial Controller Synthesis
Gain Scheduling
Piecewise Smooth Systems
References
3. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Motivation
There are many problems that can be formulated as optimization
problems:
Aircraft design
Modeling: Parameter estimation
Modeling: Model order reduction
Model based control design (Landing gear semi active
suspension)
4. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Aircraft Design
The aircraft designer wants to:
maximize range
minimize weight
maximize lift to drag ratio
minimize cost
minimize noise
subject to physical, geometrical, environmental, budget and safety
constraints
Multidisciplinary Optimization (MDO) problem: aerodynamics,
structure, aeroelasticity, propulsion, noise and vibration, dynamics,
stability and control, manufacturing
5. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Parameter Estimation
Unmanned Rotorcraft Technology Demonstrator ARTIS at DLR
(German Aerospace Center)
7. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Parameter Estimation
Discrete time linear model:
x(k + 1) = A(𝜃)x(k) + B(𝜃)u(k)
y(k) = C(𝜃)x(k) + D(𝜃)u(k)
where x is the state vector, u denotes the input vector and y is the
measurement vector.
This is a parametric model, based on physical principles. In order to
have a virtual model of the UAV, we need to find the best
parameter vector using input-output data of a few flight tests.
8. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Parameter Estimation
Assume that we are given a set of flight test data:
𝒟N = {(uft(k), yft(k)) |k = 0, . . . , N}
The parameter estimation problem can be formulated as:
minimize
𝜃,x(0)
ΣN
i=1
‖y(tk) − yft(tk)‖2
2
subject to x(k + 1) = A(𝜃)x(k) + B(𝜃)uft(k) for k = 0, . . . , N − 1
y(k) = C(𝜃)x(k) + D(𝜃)uft(k) for k = 1, . . . , N
9. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Model Order Reduction
After estimating the parameter vector, we have a high-order
linear model.
To design a controller for the pitch dynamics, we don’t need
all the degrees of freedom.
If G(s) is the transfer function of the original model, we need
to compute ˆG(s) such that it captures the main characteristics
of the pitch dynamics.
Model order reductoion, in this case, can be formulated as the
following optimization problem:
minimize
^G(s)
‖G(s) − ˆG(s)‖∞
10. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Model Based Control Design
Design a semi-active landing gear to:
maximize stability on the ground
maximize stability during taxi
minimize noise
minimize cost
minimize weight
subject to physical, geometrical, environmental, budget and
safety constraints
11. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Convex optimization problems
have extensive, useful theory
have a unique optimal answer
occur often in engineering problems
often go unrecognized
[cvxbook]
12. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Convex optimization problem
minimize f (x)
subject to x ∈ C
where f is a convex function and C is a convex set.
13. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Convex set
C ⊆ Rn is convex if
x, y ∈ C, 𝜃 ∈ [0, 1] =⇒ 𝜃x + (1 − 𝜃)y ∈ C
[cvxbook]
14. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Convex function
f : Rn −→ R is convex if
x, y ∈ Rn, 𝜃 ∈ [0, 1]
⇓
f (𝜃x + (1 − 𝜃)y) ≤ 𝜃f (x) + (1 − 𝜃)f (y)
[cvxbook]
15. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Linear programming
minimize aT
0
x
subject to aT
i x ≤ bi, i = 1, . . . , m
[cvxbook]
16. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Semidefinite programming
minimize cTx
subject to x1F1 + · · · + xnFn + G ⪯ 0
Ax = b,
where P ⪯ 0 for a matrix P ∈ Rn×n means that for any vector
v ∈ Rn, we have:
vT
Pv ≤ 0
This is equivalent to all the eigenvalues of P being nonpositive. P
is called negative semidefinite.
17. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Why Convex Optimization?
In fact the great watershed in optimization isn’t between
linearity and nonlinearity, but convexity and nonconvexity (R.
Tyrrell Rockafellar, in SIAM Review, 1993).
Convex optimization problems can be solved almost as quickly
and reliably as linear programming problems.
18. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Nonnegativity of polynomials
Polynomials of degree d in n variables:
p(x) p(x1, x2, . . . , xn) =
∑︁
k1+k2+···+kn≤d
ak1k2...kn xk1
1
xk2
2
· · · xkn
n
How to check if a given p(x) (of even order) is globally
nonnegative?
p(x) ≥ 0, ∀x ∈ Rn
For d = 2, easy (check eigenvalues). What happens in
general?
Decidable, but NP-hard when d ≥ 4.
“Low complexity” is desired at the cost of possibly being
conservative.
[Parrilo]
19. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
A sufficient condition
A “simple” sufficient condition: a sum of squares (SOS)
decomposition:
p(x) =
m∑︁
i=1
f 2
i (x)
If p(x) can be written as above, for some polynomials fi, then
p(x) ≥ 0.
p(x) is an SOS if and only if a positive semidefinite matrix Q
exists such that
p(x) = ZT(x)QZ(x)
where Z(x) is the vector of monomials of degree less than or
equal to deg(p)/2
[Parrilo]
21. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Sum of squares programming
A sum of squares program is a convex optimization program of the
following form:
Minimize
J∑︁
j=1
wj 𝛼j
subject to fi,0 +
J∑︁
j=1
𝛼jfi,j(x) is SOS, for i = 1, . . . , I
where the 𝛼j’s are the scalar real decision variables, the wj’s are
some given real numbers, and the fi,j are some given multivariate
polynomials.
[Prajna]
22. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Numerical Solvers
SOSTOOLS handles the general SOS programming.
MATLAB toolbox, freely available.
Requires SeDuMi (a freely available SDP solver).
Natural syntax, efficient implementation
Developed by S. Prajna, A. Papachristodoulou and P. Parrilo
Includes customized functions for several problems
[Parrilo]
23. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Global optimization
Consider for example:
min
x,y
F(x, y)
with F(x, y) = 4x2
− 21
10
x4
+ 1
3
x6
+ xy − 4y2
+ 4y4
[Parrilo]
24. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Global optimization
Not convex, many local minima. NP-Hard in general.
Find the largest 𝛾 s.t.
F(x, y) − 𝛾 is SOS.
A semidefinite program (convex!).
If exact, can recover optimal solution.
Surprisingly effective.
Solving, the maximum value is −1.0316. Exact value.
Many more details in Parrilio and Strumfels, 2001
[Parrilo]
25. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Lyapunov stability analysis
To prove asymptotic stability of ˙x = f (x),
V(x) > 0, x ̸= 0, ˙V(x) =
(︂
𝜕V
𝜕x
)︂T
f (x) < 0, x ̸= 0
For linear systems ˙x = Ax, quadratic Lyapunov functions
V(x) = xTPx
P > 0, ATP + PA < 0
With an affine family of candidate Lyapunov functions V, ˙V is
also affine.
Instead of checking nonnegativity, use an SOS condition
[Parrilo]
26. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Lyapunov stability - Jet Engine Example
A jet engine model (derived from Moore-Greitzer), with controller:
˙x = −y +
3
2
x2
−
1
2
x3
˙y = 3x − y
Try a generic 4th order polynomial Lyapunov function.
Find a V(x, y) that satisfies the conditions:
V(x, y) is SOS.
− ˙V(x, y) is SOS.
Can easily do this using SOS/SDP techniques...
[Parrilo]
27. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Lyapunov stability - Jet Engine Example
After solving the SDPs, we obtain a Lyapunov function.
[Parrilo]
28. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Lyapunov stability - Jet Engine Example
Consider the nonlinear system
˙x1 = −x3
1
− x1x2
3
˙x2 = −x2 − x2
1
x2
˙x3 = −x3 −
3x3
x2
3
+ 1
+ 3x2
1
x3
Looking for a quadratic Lyapunov function s.t.
V − (x2
1
+ x2
2
+ x2
3
) is SOS,
(x2
3
+ 1)(−
𝜕V
𝜕x1
˙x1 −
𝜕V
𝜕x2
˙x2 −
𝜕V
𝜕x3
˙x3) is SOS,
we have V(x) = 5.5489x2
1
+ 4.1068x2
2
+ 1.7945x2
3
.
[sostools]
29. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Parametric robustness analysis - Example
Consider the following linear system
d
dt
⎡
⎣
cx1
x2
x3
⎤
⎦ =
⎡
⎣
−p1 1 −1
2 − p2 2 −1
3 1 −p1p2
⎤
⎦
⎡
⎣
cx1
x2
x3
⎤
⎦
where p1 ∈ [p1, p1] and p2 ∈ [p2, p2] are parameters.
Parameter set can be captured by
a1(p) (p1 − p1)(p1 − p1) ≤ 0
a2(p) (p2 − p2)(p2 − p2) ≤ 0
[sostools]
30. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Parametric robustness analysis - Example
Find V(x; p) and qi,j(x; p), such that
V(x; p) − ‖x‖2
+
∑︀2
j=1
q1,j(x; p)ai(p) is SOS,
− ˙V(x; p) − ‖x‖2
+
∑︀2
j=1
q2,j(x; p)ai(p) is SOS, qi,j(x; p) is
SOS, for i, j = 1, 2.
[sostools]
32. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Safety verification - Example
Barrier certificate B(x)
B(x) < 0, ∀x ∈ 𝒳0
B(x) > 0, ∀x ∈ 𝒳u
𝜕B
𝜕x1
˙x1 + 𝜕B
𝜕x2
˙x2 ≤ 0
SOS program: Find B(x) and 𝜎i(x)
−B(x) − 0.1 + 𝜎1(x)g0(x) is SOS,
B(x) − 0.1 + 𝜎2(x)gu(x) is SOS,
− 𝜕B
𝜕x1
˙x1 − 𝜕B
𝜕x2
˙x2 is SOS
𝜎1(x) and 𝜎2(x) are SOS
[sostools]
33. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Safety verification - Example
[sostools]
34. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Nonlinear control synthesis
Consider the system
˙x = f (x) + g(x)u
State dependent linear-like representation
˙x = A(x)Z(x) + B(x)u
where Z(x) = 0 ⇔ x = 0
Consider the following Lyapunov function and control input
V(x) = ZT(x)P−1
Z(x)
u(x) = K(x)P−1
Z(x)
[Prajna]
35. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Nonlinear control synthesis
For the system ˙x = A(x)Z(x) + B(x)u, suppose there exist a
constant matrix P, a polynomial matrix K(x), a constant 𝜖1 and a
sum of squares 𝜖2(x), such that:
vT(P − 𝜖1I)v is SOS,
−vT(PAT(x)MT(x) + M(x)A(x)P + KT(x)BT(x)MT(x) +
M(x)B(x)K(x) + 𝜖2(x)I) is SOS,
where v ∈ RN and Mij(x) = 𝜕Zi
𝜕xj
(x). Then a controller that
stabilizes the system is given by:
u(x) = K(x)P−1
Z(x)
Furthermore, if 𝜖2(x) > 0 for x ̸= 0, then the zero equilibrium is
globally asymptotically stable.
[Prajna]
36. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Nonlinear control synthesis - Example
Consider a tunnel diode circuit:
˙x1 = 0.5(−h(x1) + x2)
˙x2 = 0.2(−x1 − 1.5x2 + u)
where the diode characteristic:
h(x1) = 17.76x1 − 103.79x2
1
+ 229.62x3
1
− 226.31x4
1
+ 83.72x5
1
[Prajna]
37. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Nonlinear control synthesis - Example
[Prajna]
38. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
How conservative is SOS?
It is proven by Hilbert that “nonnegativity” and “sum of
squares” are equivalent in the following cases.
Univariate polynomials, any (even) degree
Quadratic polynomials, in any number of variables
Quartic polynomials in two variables
When the degree is larger than two it follows that
There are signitcantly more nonnegative polynomials than
sums of squares.
There are signitcantly more sums of squares than sums of even
powers of linear forms.
[soscvx]
39. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Flutter Phenomenon
Mechanism of Flutter
Inertial Forces
Aerodynamic Forces (∝ V2) (exciting the
motion)
Elastic Forces (damping the motion)
Flutter Facts
Flutter is self-excited
Two or more modes of motion (e.g. flexural and torsional)
exist simultaneously
Critical Flutter Speed, largely depends on torsional and flexural
stiffnesses of the structure
[flutter96]
41. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Flutter Phenomenon
State Space Equations:
M
[︂
¨h
¨𝛼
]︂
+ (C0 + C 𝜇)
[︂
˙h
˙𝛼
]︂
+ (K0 + K 𝜇)
[︂
h
𝛼
]︂
+
[︂
0
𝛼K 𝛼(𝛼)
]︂
= B 𝛽o
State variables: plunge deflection (h), pitch angle ( 𝛼), and
their derivatives ( ˙h and ˙𝛼)
Inputs: angular deflection of the flaps ( 𝛽o ∈ R2
)
Constraints: on states and actuators
[flutter07] [flutter98]
42. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Active Flutter Suppression
Bombardier Q400
HYCONS Lab, Concordia University
43. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
LQR Controller
Very large control inputs
R = 10I, Q = 10
4
I
44. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
LQR Controller
Divergence: the effect of actuator saturation
maximum admissible flap angles: 15 deg
45. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
LQR Controller
Region of attraction: plung deflection - pitch angle plane
46. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
LQR Controller
Region of attraction: plung deflection - plung deflection rate plane
47. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
LQR Controller
Region of attraction: pitch angle - pitch rate plane
50. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Polynomial Controller
Consider x3 and x4 as inputs of the following system:
˙x1 = x3
˙x2 = x4
Consider the controller
[︂
x3
x4
]︂
= −10
[︂
x1
x2
]︂
for the above system.
Similar to backstepping approach, we construct the following
Lyapunov function:
V(x) =
1
2
{︀
x2
1
+ x2
2
+ (x3 + 10x1)2
+ (x4 + 10x2)2
}︀
Find a polynomial u(x) such that −∇V.f (x) − V(x) is SOS
where f (x) is the vector field of the closed loop system.
51. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Polynomial Controller
smaller control inputs
52. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Polynomial Controller
Divergence: the effect of actuator saturation
maximum admissible flap angles: 15 deg
53. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Polynomial Controller
Future work:
To construct a nonlinear model of Q400
To design a nonlinear controller in order to enlarge the region
of convergence in the presence of input saturation
54. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Gain Scheduling
Design an autopilot to:
minimize steady state tracking error
maximize robustness to wind gust
subject to varying flight conditions
For controller design, consider the following issues:
Theory of Linear Systems is very rich in terms of analysis and
synthesis methods and computational tools.
Real world systems, however, are usually nonlinear.
What can be done to extend the good properties of linear
systems theory to nonlinear systems?
55. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Gain Scheduling
Gain scheduling is an attempt to address this issue
Divide and conquer
Approximating nonlinear systems by a combination of local
linear systems
Designing local linear controllers and combining them
Started in 1960s, very popular in a variety of fields from
aerospace to process control
Problem: proof of stability!
Problem: By switching between two stable linear system, you
can create an unstable system.
56. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Piecewise Smooth Systems
The dynamics of a piecewise smooth smooth (PWS) is defined as:
˙x = fi(x), x ∈ ℛi
where x ∈ 𝒳 is the state vector. A subset of the state space 𝒳 is
partitioned into M regions, ℛi, i = 1, . . . , M such that:
∪M
i=1
¯ℛi = 𝒳, ℛi ∩ ℛj = ∅, i ̸= j
where ¯ℛi denotes the closure of ℛi.
57. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
Conclusion
Sum of squares, conservative but much more tractable than
nonnegativity
Many applications in control theory
Try your problem!
58. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
References I
[cvxbook] Convex optimmization, Stephen Boyd and Lieven
Vandenberghe, http://www.stanford.edu/~boyd/cvxbook
[Parrilo] Certificates, convex optimization, and their
applications, Pablo A. Parrilo, Swiss Federal Institute of
Technology Zurich, http:
//www.mat.univie.ac.at/~neum/glopt/mss/Par04.pdf
[Prajna] Nonlinear control synthesis by sum of squares
optimization: a Lyapunov-based approach, Stephen Prajna et
al, the 5th Asian Control Conference, 2004
[sostools] SOSTOOLS: control applications and new
developments, Stephen Prajna et al, IEEE Conference on
Computer Aided Control Systems Design, 2004
59. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
References II
[soscvx] A convex polynomial that is not sos-convex, Amir Ali
Ahmadi and Pablo A. Parrilo,
http://arxiv.org/pdf/0903.1287.pdf
[yalmip] YALMIP, A Toolbox for Modeling and Optimization in
MATLAB, J. Löfberg. In Proceedings of the CACSD
Conference, Taipei, Taiwan, 2004,
http://users.isy.se/johanl/yalmip
[sos] Pre- and post-processing sum-of-squares programs in
practice. J. Löfberg. IEEE Transactions on Automatic Control,
54(5):1007-1011, 2009.
[dual] Dualize it: software for automatic primal and dual
conversions of conic programs. J. Löfberg. Optimization
Methods and Software, 24:313 - 325, 2009.
60. Motivation Convex Optimization SOS Flutter Suppression Gain Scheduling PWS Conclusion
References III
[sedumi] SeDuMi, a MATLAB toolbox for optimization over
symmetric cones, http://sedumi.ie.lehigh.edu
[flutter96] Modeling the benchmark active control technology
windtunnel model for application to flutter suppression, M. R.
Waszak, AIAA 96 - 3437, http://www.mathworks.com/
matlabcentral/fileexchange/3938
[flutter98] Stability and control of a structurally nonlinear
aeroelastic system, Jeonghwan Ko and Thomas W. Strganacy,
Journal of Guidance, Control, and Dynamics, 21 , 718-725.
[flutter07] Nonlinear control design of an airfoil with active
flutter suppression in the presence of disturbance, S. Afkhami
and H. Alighanbari, IET Control Theory Appl., vol. 1 ,
1638-1649.