An approach to algorithmically synthesize control
strategies for set-to-set transitions of uncertain discrete-time
switched linear systems based on a combination of tree search
and reachable set computations in a stochastic setting is
proposed in this presentation. The initial state and disturbances
are assumed to be Gaussian distributed, and a time-variant
hybrid control law stabilizes the system towards a goal set.
The algorithmic solution computes sequences of discrete states
via tree search and the continuous controls are obtained
from solving embedded semi-definite programs (SDP). These
program taking polytopic input constraints as well as timevarying
probabilistic state constraints into account. An example
for demonstrating the principles of the solution procedure with
focus on handling the chance constraints is included.
Control of Discrete-Time Piecewise Affine Probabilistic Systems using Reachab...Leo Asselborn
This presentation proposes an algorithmic approach to
synthesize stabilizing control laws for discrete-time piecewise
affine probabilistic (PWAP) systems based on computations of
probabilistic reachable sets. The considered class of systems
contains probabilistic components (with Gaussian distribution)
modeling additive disturbances and state initialization. The
probabilistic reachable state sets contain all states that are
reachable with a given confidence level under the effect of
time-variant control laws. The control synthesis uses principles
of the ellipsoidal calculus, and it considers that the system
parametrization depends on the partition of the state space. The
proposed algorithm uses LMI-constrained semi-definite programming
(SDP) problems to compute stabilizing controllers,
while polytopic input constraints and transitions between regions
of the state space are considered. The formulation of
the SDP is adopted from a previous work in [1] for switched
systems, in which the switching of the continuous dynamics
is triggered by a discrete input variable. Here, as opposed
to [1], the switching occurs autonomously and an algorithmic
procedure is suggested to synthesis a stabilizing controller. An
example for illustration is included.
Robust Control of Uncertain Switched Linear Systems based on Stochastic Reach...Leo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of uncertain discrete-time switched linear systems based on a combination
of tree search and reachable set computations in a stochastic setting. For given Gaussian
distributions of the initial states and disturbances, state sets wich are reachable to a chosen
confidence level under the effect of time-variant hybrid control laws are computed by using
principles of the ellipsoidal calculus. The proposed algorithm iterates over sequences of the
discrete states and LMI-constrained semi-definite programming (SDP) problems to compute
stabilizing controllers, while polytopic input constraints are considered. An example for illustration is included.
Probabilistic Control of Uncertain Linear Systems Using Stochastic ReachabilityLeo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of discrete-time uncertain systems based on reachable set computations in
a stochastic setting. For given Gaussian distributions of the initial states and disturbances, state
sets wich are reachable to a chosen confidence level under the effect of time-variant control laws
are computed by using principles of the ellipsoidal calculus. The proposed algorithm iterates over
LMI-constrained semi-definite programming problems to compute probabilistically stabilizing
controllers, while ellipsoidal input constraints are considered. An example for illustration is included.
Subgradient Methods for Huge-Scale Optimization Problems - Юрий Нестеров, Cat...Yandex
We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piecewise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, the total cost of which depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions.
We show that the updating technique can be efficiently coupled with the simplest subgradient methods. Similar results can be obtained for a new non-smooth random variant of a coordinate descent scheme. We also present promising results of preliminary computational experiments.
Control of Uncertain Nonlinear Systems Using Ellipsoidal Reachability CalculusLeo Asselborn
This paper proposes an approach to algorithmically synthesize control strategies for discrete-time nonlinear uncertain systems based on reachable set computations using the ellipsoidal calculus. For given ellipsoidal initial sets and bounded ellipsoidal disturbances, the proposed algorithm iterates over conservatively approximating and LMI-constrained optimization problems to compute stabilizing controllers. The method uses
first-order Taylor approximation of the nonlinear dynamics and a conservative approximation of the Lagrange remainder.
Control of Discrete-Time Piecewise Affine Probabilistic Systems using Reachab...Leo Asselborn
This presentation proposes an algorithmic approach to
synthesize stabilizing control laws for discrete-time piecewise
affine probabilistic (PWAP) systems based on computations of
probabilistic reachable sets. The considered class of systems
contains probabilistic components (with Gaussian distribution)
modeling additive disturbances and state initialization. The
probabilistic reachable state sets contain all states that are
reachable with a given confidence level under the effect of
time-variant control laws. The control synthesis uses principles
of the ellipsoidal calculus, and it considers that the system
parametrization depends on the partition of the state space. The
proposed algorithm uses LMI-constrained semi-definite programming
(SDP) problems to compute stabilizing controllers,
while polytopic input constraints and transitions between regions
of the state space are considered. The formulation of
the SDP is adopted from a previous work in [1] for switched
systems, in which the switching of the continuous dynamics
is triggered by a discrete input variable. Here, as opposed
to [1], the switching occurs autonomously and an algorithmic
procedure is suggested to synthesis a stabilizing controller. An
example for illustration is included.
Robust Control of Uncertain Switched Linear Systems based on Stochastic Reach...Leo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of uncertain discrete-time switched linear systems based on a combination
of tree search and reachable set computations in a stochastic setting. For given Gaussian
distributions of the initial states and disturbances, state sets wich are reachable to a chosen
confidence level under the effect of time-variant hybrid control laws are computed by using
principles of the ellipsoidal calculus. The proposed algorithm iterates over sequences of the
discrete states and LMI-constrained semi-definite programming (SDP) problems to compute
stabilizing controllers, while polytopic input constraints are considered. An example for illustration is included.
Probabilistic Control of Uncertain Linear Systems Using Stochastic ReachabilityLeo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of discrete-time uncertain systems based on reachable set computations in
a stochastic setting. For given Gaussian distributions of the initial states and disturbances, state
sets wich are reachable to a chosen confidence level under the effect of time-variant control laws
are computed by using principles of the ellipsoidal calculus. The proposed algorithm iterates over
LMI-constrained semi-definite programming problems to compute probabilistically stabilizing
controllers, while ellipsoidal input constraints are considered. An example for illustration is included.
Subgradient Methods for Huge-Scale Optimization Problems - Юрий Нестеров, Cat...Yandex
We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piecewise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, the total cost of which depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions.
We show that the updating technique can be efficiently coupled with the simplest subgradient methods. Similar results can be obtained for a new non-smooth random variant of a coordinate descent scheme. We also present promising results of preliminary computational experiments.
Control of Uncertain Nonlinear Systems Using Ellipsoidal Reachability CalculusLeo Asselborn
This paper proposes an approach to algorithmically synthesize control strategies for discrete-time nonlinear uncertain systems based on reachable set computations using the ellipsoidal calculus. For given ellipsoidal initial sets and bounded ellipsoidal disturbances, the proposed algorithm iterates over conservatively approximating and LMI-constrained optimization problems to compute stabilizing controllers. The method uses
first-order Taylor approximation of the nonlinear dynamics and a conservative approximation of the Lagrange remainder.
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
The standard Galerkin formulation of the acoustic wave propagation, governed by the Helmholtz partial differential equation (PDE), is indefinite for large wavenumbers. However, the Helmholtz PDE is in general not indefinite. The lack of coercivity (indefiniteness) is one of the major difficulties for approximation and simulation of heterogeneous media wave propagation models, including application to stochastic wave propagation Quasi Monte Carlo (QMC) analysis. We will present a new class of sign-definite continuous and discrete preconditioned FEM Helmholtz wave propagation models.
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.
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
The standard Galerkin formulation of the acoustic wave propagation, governed by the Helmholtz partial differential equation (PDE), is indefinite for large wavenumbers. However, the Helmholtz PDE is in general not indefinite. The lack of coercivity (indefiniteness) is one of the major difficulties for approximation and simulation of heterogeneous media wave propagation models, including application to stochastic wave propagation Quasi Monte Carlo (QMC) analysis. We will present a new class of sign-definite continuous and discrete preconditioned FEM Helmholtz wave propagation models.
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.
We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution
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.
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).
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
MVPA with SpaceNet: sparse structured priorsElvis DOHMATOB
The GraphNet (aka S-Lasso), as well as other “sparsity + structure” priors like TV (Total-Variation), TV-L1, etc., are not easily applicable to brain data because of technical problems
relating to the selection of the regularization parameters. Also, in
their own right, such models lead to challenging high-dimensional optimization problems. In this manuscript, we present some heuristics for speeding up the overall optimization process: (a) Early-stopping, whereby one halts the optimization process when the test score (performance on leftout data) for the internal cross-validation for model-selection stops improving, and (b) univariate feature-screening, whereby irrelevant (non-predictive) voxels are detected and eliminated before the optimization problem is entered, thus reducing the size of the problem. Empirical results with GraphNet on real MRI (Magnetic Resonance Imaging) datasets indicate that these heuristics are a win-win strategy, as they add speed without sacrificing the quality of the predictions. We expect the proposed heuristics to work on other models like TV-L1, etc.
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.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Cosmetic shop management system project report.pdf
Probabilistic Control of Switched Linear Systems with Chance Constraints
1. Control & System
Theory
Probabilistic Control of Switched Linear Systems
with Chance Constraints
Leonhard Asselborn Olaf Stursberg
Control and System Theory
University of Kassel (Germany)
l.asselborn@uni-kassel.de
stursberg@uni-kassel.de
European Control Conference 2016
www.control.eecs.uni-kassel.de 01.07.2016
2. Introduction: Motivation in Uncertain Environment Control & System
Theory
• dynamic system with different modes (1. fast straight motion, 2. slow
curved motion, etc.)
• stochastic disturbances (wind)
• stochastic initialization (GPS coordinates)
• input constraints: uk ∈ U
• state constraints (safety critical): xk ∈ Xk cannot be guaranteed for all
disturbances → state chance constraint: Pr(xk ∈ Xk) ≥ δx
• steer the system into a terminal region T with confidence δ
x0
p
p
xN
estimated
position of
an obstacle
T
Introduction Problem Definition Method Example Conclusion Appendix 2
3. Relevant Literature (Excerpt): Control & System
Theory
Switched Linear Systems
• Liberzon [2003], Sun [2006], Sun and Ge [2011]: stability conditions for arbitrary
switching, no input constraints
Chance Constraint Optimization
• Galafiore and Campi [2005], Blackmore et al. [2010], Asselborn et al. [2012],
Prandini et al. [2015]: scenario-based approach
• van Hessem et al. [2001,2002], Ma and Borelli [2012]: set-based approach
• Prekopa [1999], Blackmore and Ono [2009], Blackmore et al [2011], Vitus and
Tomlin [2011]: Boole’s inequality for separate handling of chance constraints
Reachability sets for stochastic hybrid systems
• Hu et al. [2000], Blom and Lygeros [2006], Cassandras and Lygeros [2006],
Kamgarpour et al. [2013], Abate et al. [2008]: control design
Previous own work
• Controller synthesis for nonlinear systems: [NOLCOS, 2013]
• Synthesis for stochastic discrete-time linear systems: [ROCOND, 2015]
• Synthesis for stochastic discrete-time switched linear systems: [ADHS, 2015]
Introduction Problem Definition Method Example Conclusion Appendix 3
4. Contribution Control & System
Theory
Controller synthesis based on probabilistc reachability computation for
discrete-time stochastic hybrid systems with state chance constraints
Solution Approach:
• forward propagation of ellipsoidal reachable sets Xδ
k with confidence δ
Xδ
0
Xk
Xδ
N
T
• offline controller synthesis by semi-definite programming (SDP) and tree
search
• approximation of chance constraints via Boole’s inequality
Introduction Problem Definition Method Example Conclusion Appendix 4
5. Sets, Distributions and Dynamic System (1) Control & System
Theory
Set representation:
• Ellipsoid: E := ε(q, Q) = x ∈ Rn
| (x − q)T
Q−1
(x − q) ≤ 1
with q ∈ Rn
, Q ∈ Rn×n
• Polytope: P := {x ∈ Rn
| Rx ≤ b} with R ∈ Rnp×n
, b ∈ Rnp
Multivariate Normal Distribution:
ξ ∼ N(µ, Ω)
The sum of two Gaussian variables ξ1 ∼ N(µ1, Ω1) and ξ2 ∼ N(µ2, Ω2) is
again a Gaussian variable:
ξ1 + ξ2 ∼ N(µ1 + µ2, Ω1 + Ω2)
Introduction Problem Definition Method Example Conclusion Appendix 5
6. Sets, Distributions and Dynamic System (2) Control & System
Theory
Probabilistic Switched Affine Systems (PSAS):
xk+1 = Azk xk + Bzk uk + Gzk vk, k ∈ {0, 1, 2, . . .}
x0 ∼ N(q0, Q0), xk ∈ Rn
Xk = {xk | Rx,kxk ≤ bx,k}
vk ∼ N(0, Σ), vk ∈ Rn
, iid
uk ∈ U = {uk | Ruuk ≤ bu} ⊆ Rm
zk ∈ Z = {1, 2, . . . , nz}
Feasible system execution for k ∈ N0:
1. select zk ∈ Z to determine the tuple (Azk , Bzk , Gzk )
2. sample the disturbance vk ∼ N(0, Σ)
3. choose a suitable input uk ∈ U
4. evaluate the continuous dynamics to get the new state xk+1
Introduction Problem Definition Method Example Conclusion Appendix 6
7. Probabilistic Reachable Sets with Confidence δ Control & System
Theory
• Surfaces of equal density for ξ ∼ N(µ, Ω)
(Krzanowski and Marriott [1994]):
(ξ − µ)T
Ω−1
(ξ − µ) = c
with χ2
-distributed random variable and:
δ := Pr (ξ ∈ ε(µ, Ωc)) = Fχ2 (c, n)
cumulative distribution function
• Initial state confidence ellipsoid:
Xδ
0 := ε(q0, Q0c) with Pr(x0 ∈ Xδ
0 ) = δ contour of pdf
samples of ξ ∼ N (µ, Ω)
ξ2
ξ1
• Evolution of the state distribution:
qk+1 = Azk qk + Bzk uk, Qk+1 = Azk QkAT
zk
+ Gzk ΣGT
zk
Xδ
k+1 := ε(qk+1, Qk+1c
=:Qδ
k+1
)
Xδ
k+1 is the confidence ellipsoid for xk+1 with confidence δ.
Introduction Problem Definition Method Example Conclusion Appendix 7
8. Problem Definition Control & System
Theory
Problem
Given PSAS, determine a hybrid control law κk = (λk(xk), νk) for which it
holds that:
• uk = λk(xk) ∈ U, zk = νk ∈ Z and xk ∈ Xδ
k
∀ k ∈ {0, 1, . . . , N − 1}, N ∈ N
• the chance constraint Pr(xk ∈ Xk) ≥ δx is satisfied ∀k ∈ {0, . . . , N − 1}
• Xδ
N ⊆ T for a finite N ≤ Nmax.
Thus, any initial state x0 ∈ Xδ
0 has to be transferred into the terminal set T
with probability δ after N steps.
Introduction Problem Definition Method Example Conclusion Appendix 8
9. Main Idea Control & System
Theory
Solution procedure:
1. tree search algorithm (Asselborn and Stursberg [2015]) to determine the
sequence of discrete control inputs νk ∈ Z
2. solution of an SDP provides continuous control law:
uk = λ(xk) = −Kkxk + dk
with closed-loop dynamics:
xk+1 = Azk xk + Bzk uk + Gzk vk
= (Aνk − Bνk Kk)
:=Acl,k,νk
xk + Bνk dk + Gνk vk
Main Challenge:
• state chance constraint with multi-dimensional integral:
Pr(xk ∈ Xk) =
w∈Xk
N (qx,k, Qx,k) dw ≥ δx
• no closed form solution available
Idea: separate consideration of each half-space of Xk
Introduction Problem Definition Method Example Conclusion Appendix 9
10. Approximation of State Chance Constraints Control & System
Theory
• projection of the random variables onto each hyperplane
rx,k,1
rx,k,2
Rx,k
xk ≤
bx,k,1
bx,k,2
bx,k
yk,i := rx,k,i · xk ∼ N(qy,k,i, Qy,k,i)
qx,k
rx,k,1
rx,k,2
• yk,1 is a univariate random variable (Blackmore and Ono [2009])
qy,k,i = rx,k,iqx,k,i, Qy,k,i = rx,k,iQx,k,irT
x,k,i
• consideration of the violation of the chance constraint: Pr(yk,i > bx,k,i)
• with Boole’s inequality Pr nx
i=1 yk,i > bx,k,i ≤ nx
i=1 Pr(yk,i > bx,k,i)
the following Lemma holds:
Lemma
The chance constraint Pr(xk ∈ Xk) ≥ δx is satisfied, if it holds for
i ∈ {1, 2, . . . , nx} that:
ǫi := Pr(yk,i > bx,k,i),
nx
i=1
ǫi < 1 − δx
Introduction Problem Definition Method Example Conclusion Appendix 10
11. Approximation of State Chance Constraint (2) Control & System
Theory
Transformation into a standard normal distribution
yk,i ∼ N(qy,k,i, Qy,k,i), bx,k,i → yk,i ∼ N(0, 1),
bx,k,i − qy,k,i
Qy,k,i
Evaluation of the probability with the cumulative distribution function:
Pr(yk,i > bx,k,i) = 1 − cdf
bx,k,i − qy,k,i
Qy,k,i
with:
cdf(x) =
1
√
2π
x
−∞
e− z2
2 dz, no closed form solution available
In Soranzo [2014]:
cdf(x) ≈ fcdf (x) = 2−221−41x/10
with max error: 1.27e − 4 for x ≥ 0
Introduction Problem Definition Method Example Conclusion Appendix 11
12. Solution based on SDP (1) Control & System
Theory
Linearization of fcdf (x):
¯fcdf (qy,k,i, Qy,k,i, bx,k,i) :=fcdf (¯qy, ¯Qy, bx,k,i) + . . .
∇qy fcdf (qy,k,i − ¯qy) + ∇Qy fcdf (Qy,k,i − ¯Qy)
Approximation of Pr(xk ∈ Xk) ≥ δx by:
ǫi = 1 − ¯fcdf (qy,k,i, Qy,k,i, bx,k,i) ,
nx
i=1
ǫi < 1 − δx, ∀i = {1, 2, . . . nx}
Introduction Problem Definition Method Example Conclusion Appendix 12
13. Solution based on SDP (2) Control & System
Theory
• Convergence of the covariance matrix of N(qx,k+1, Qx,k+1):
Sk+1 ≥ Qx,k+1 = Acl,k,zk
Qx,kAT
cl,k,zk
+ GΣGT
or with Schur complement:
Sk+1 Acl,k,zk
Qx,k GΣ
Qx,kAT
cl,k,zk
Qx,k 0
ΣGT
0 Σ
≥ 0
• Convergence of the expected value qx,k
use of flexible Lyapunov functions. (Lazar et al. [2009] ) suitable for
switched dynamics
V
k
Introduction Problem Definition Method Example Conclusion Appendix 13
14. Solution based on SDP (3) Control & System
Theory
Proposition
The input constraint uk = −Kkxk + dk ∈ U holds for Kk, dk and all xk ∈ Xδ
k
if:
(bu,i − ru,idk)In −ru,iKk(Qδ
x,k)− 1
2
−(Qδ
x,k)− 1
2 KT
k rT
u,i bu,i − ru,idk
≥ 0 ∀i = {1, . . . , nu}.
• ru,i and bu,i denote the i−th row of Ru and bu, respectively.
• Xδ
k is mapped into a unit ball by a suitable coordinate transformation
h(xk)
• the Euclidean norm ||h(xk)||2 ≤ 1 can be expressed as LMI, which results
in the above formulation
• complete proof can be found in Asselborn et al. [2013]
Introduction Problem Definition Method Example Conclusion Appendix 14
15. Determination of the Continuous Controller Control & System
Theory
Semidefinite program to be solved for chosen zk ∈ Z:
min
Sk+1,Kk,dk
Jk,zk
center point convergence:
qT
x,k+1,zk
Lqx,k+1,zk
− ρqT
x,kLqx,k ≤ αk
qx,k+1,zk
= (Azk − Bzk Kk)qx,k + Bzk dk
αk ≤ maxl∈{1,...,k} ωl
αk−l
ellipsoidal shape convergence:
Sk+1 Acl,k,zk
Qx,k Gzk Σ
Qx,kAT
cl,k,zk
Qx,k 0
ΣGT
zk
0 Σ
≥ 0
trace(Sk+1) ≤ trace(Qk)
input constraint:
(bu,i − ru,idk)In −ru,iKk(Qδ
x,k)− 1
2
−(Qδ
x,k)− 1
2 KT
k rT
u,i bu,i − ru,idk
≥ 0,
∀i = {1, . . . , nu}
state chance constraint:
ǫi = 1 − ¯fcdf (qy,k+1,i, Qy,k+1,i, bx,k+1,i) ,
nx
i=1 ǫi < 1 − δx, ∀i = {1, 2, . . . nx}
Introduction Problem Definition Method Example Conclusion Appendix 15
16. Determination of the Hybrid Controller Control & System
Theory
Probabilistic Ellipsoidal Control Algorithm (PECA)
Given: x0 ∼ N(qx,0, Qx,0), vk ∼ N(0, Σ), U = {uk | Ruuk ≤ bu},
Xk = {xk | Rx,kxk ≤ bx,k}, T, δ, δx, γmin, ω, ρ, α0
Define: k := 0, γ0 = γmin, O0 = ∅
while Xδ
k T and γk ≥ γmin do
(1) for i = 1, . . . , nz do
◮ compute Xδ
k, and solve the SDP problem for z = i and Xδ
k
◮ if solution exists do Ok := Ok ∪ {i} else Ok := Ok end
end
(2) if Ok = ∅ do choose the tuple (Kk, dk, νk) with best performance
else
if k = 0 do Termination without success
else k = k − 1, Ok := Ok {νk}, go to step (2) end
end
(3) compute (qx,k+1, Qx,k+1) with the selected controller (Kk, dk, νk)
(4) check state chance constraint:
if nx
j=1 1 − fcdf (qy,k+1,j, Qy,k+1,j, bx,k,j ) > 1 − δx do mark νk = i as
infeasible solution, go to step 2
(5) compute γk+1 = qk+1 − qk , k := k + 1, end while
Introduction Problem Definition Method Example Conclusion Appendix 16
17. Termination with success Control & System
Theory
Lemma
The control problem with a confidence δ, an initialization x0 ∼ N(qx,0, Qx,0),
vk ∼ N(0, Σ), (λk(xk), νk) ∈ U × Z ∀ k, and Pr(xk ∈ Xk) ≥ δx is
successfully solved with selected parameters γmin, ω, ρ and α0, if PECA
terminates in N steps with Xδ
N ⊆ T.
Proof: by construction
If no success: adjust δ, γmin, ω, ρ, tree search strategy
Introduction Problem Definition Method Example Conclusion Appendix 17
18. Numerical Example (1) Control & System
Theory
Initial distribution and disturbance:
x0 ∼ N(qx,0, Qx,0) with qx,0 =
−10
50
, Qx,0 =
1 0
0 1
vk ∼ N(0, Σ) with Σ =
0.02 0.01
0.01 0.02
.
Discrete input set: Z = {1, 2, 3}
The continuous dynamic is specified by the following system matrices:
A1 =
1.35 −0.06
0.11 0.95
, A2 =
0.82 0.05
−0.14 1.10
, A3 =
0.86 0.05
−0.09 0.99
B1 =
0.58 −0.03
0.03 0.97
, B2 =
0.48 0.01
1.01 0.53
, B3 =
0.49 0.01
0.98 0.50
G1 = G2 = G3 =
0.1 0.05
0.08 0.2
Note that all three subsystems are chosen to have unstable state matrices.
Introduction Problem Definition Method Example Conclusion Appendix 18
19. Numerical Example (2) Control & System
Theory
Input constraints:
uk ∈ U =
u ∈ R2
|
1 0
0 1
−1 0
0 −1
u ≤
3
3
3
3
,
Target set:
T = ε 0,
0.96 0.64
0.64 0.8
Cost function:
Jk = trace
Sk+1 0
0 0.8 qx,k+1
State chance constraint:
Pr(xk ∈ Xk) ≥ δx = 0.95, Xk = {x ∈ Rn
| Rx,kxk ≤ bx,k}
with:
Rx,k =
1 0 −1 0 − 2√
2
3
0 1 0 −1 − 1√
2
1
T
bx,k = 5 60 20.5 5 2.5 + 0.2k 40
T
Parameters: δ = 0.95, γmin = 0.01, α0 = 10−4
, ω = 0.8 and ρ = 0.98
Introduction Problem Definition Method Example Conclusion Appendix 19
20. Numerical Example (3) Control & System
Theory
x2
x1
Xδ
0
T
−20 −15 −10 −5 0 5
0
10
20
30
40
50
60
• Termination with N = 38 steps
in 80s using 2.8 Ghz Quad-Core
CPU
• Implementation with Matlab
7.12.0, YALMIP 3.0, SeDuMi
1.3, and ellipsoidal toolbox ET
(Kurzhanskiy and Varaiya
[2006]).
Xδ
27
Xδ
26
Xδ
25
x2
x1
−8 −7.5 −7 −6.5 −6
2
3
4
5
6
7
Introduction Problem Definition Method Example Conclusion Appendix 20
21. Conclusion and Outlook Control & System
Theory
Summary:
• Algorithm for control of PSAS with state chance constraints
• Offline control law synthesis based on a combination of probabilistic
reachability analysis and tree search
• Explicit consideration of input constraints
• Tight approximation of state chance constraints
Future work:
• Consideration of piecewise-affine systems with autonomous switching
• Explore measures to reduce the computational complexity
Introduction Problem Definition Method Example Conclusion Appendix 21
22. End Control & System
Theory
Thank you for your attention!
Introduction Problem Definition Method Example Conclusion Appendix 22
23. Attractivity and Stochastic Stability Control & System
Theory
Stability with confidence δ
The switched uncertain linear system is called attractive with confidence δ on a
bounded time domain [0, N], if for any initial condition x0 ∈ Xδ
0 and any
vk ∈ ε(0, Σc), finite parameters ¯q ∈ Rn
and ¯Q ∈ Rn×n
exist such that:
||qN || ≤ ||¯q||, ||QN || ≤ || ¯Q||.
The system is said stable with confidence δ on a bounded time domain [0, N] if
in addition
||qk+1|| ≤ ||qk||, ||Qk+1|| ≤ ||Qk||.
holds for any 0 ≤ k ≤ N − 1.
Interpretation:
• qk converges to a finite neighborhood of the origin
• covariance matrix Qk converges, such that the confidence ellipsoid is of
decreasing size over k (while rotation is still possible).
Introduction Problem Definition Method Example Conclusion Appendix 23
24. References Control & System
Theory
Liberzon, D.
Switching in Systems and Control
Birkhaeuser, 2003
Sun, Z.
Switched Linear Systems: Control and Design
Springer, 2006
Sun, Z. and Ge, S.S.
Stability theory of switched dynamical systems;
Springer. 2011
Blackmore, L. and Ono, M.
Convex chance constrained predictive control without sampling
Proceedings of the AIAA Guidance, Navigation and Control Conference, 2009
Blackmore, L., Ono, M., William, B.C.
Chance-constrained optimal path planning with obstacles
IEEE Transactions on Robotics, 2011
Vitus, M.P. and Tomlin, C.J.
Closed-loop belief space planning for linear gaussian systems
IEEE Conference on Robotics and Automation, 2011
Calafiore, G. and Campi, M.C.
Uncertain convex programs: randomized solutions and confidence levels
Mathematical Programming, 2005
Blackmore, L., Ono, M., Bektassov, A., Williams, B.C.
A probabilistic particle-control approximation of chance-constrained stochastic predictive control
IEEE Transactions on Robotics, 2010
Introduction Problem Definition Method Example Conclusion Appendix 24
25. References Control & System
Theory
Asselborn, L., Jilg, M., and Stursberg, O.
Control of uncertain hybrid nonlinear systems using particle filters
In 4th IFAC Conf. on Analysis and Design of Hybrid Systems, 2012
Prandini, M., Garatti, S., Vignali, R.
Performance assessment and design of abstracted models for stochastic hybrid systems through a randomized approach
Autmatica, vol. 50, 2014
Hu, J., Lygeros, J., and Sastry, S.
Towards a theory of stochastic hybrid systems;
In Hybrid systems: Computation and Control, volume 1790, 160-173. Springer. 2000
Blom, H.A. and Lygeros, J.
Stochastic hybrid systems: theory and safety critical applications;
volume 337. Springer. 2006
Cassandras, C.G. and Lygeros, J.
Stochastic hybrid systems;
CRC Press.2006
Kamgarpour, M., Summers, S., and Lygeros, J.
Control design for specifications on stochastic hybrid systems;
In Hybrid systems: computation and control, 303-312. ACM. 2013
Abate, A., Prandini, M., Lygeros, J., and Sastry, S.
Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems;
Automatica,44(11), 2724-2734. 2008
Asselborn, L., Gross, D., and Stursberg, O.
Control of uncertain nonlinear systems using ellipsoidal reachability calculus;
In 9th IFAC Symp. on Nonlinear Control Systems, 50-55. 2013
Introduction Problem Definition Method Example Conclusion Appendix 25
26. References Control & System
Theory
Asselborn, L. and Stursberg, O.
Probabilistic control of uncertain linear systems using stochastic reachability;
In 8th IFAC Symp. on Robust Control Design. 2015
Asselborn, L. and Stursberg, O.
Robust Control of Uncertain Switched Linear Systems based on Stochastic Reachability
In 5th IFAC Conf. on Analysis and Design of Hybrid Systems, 2015
Soranzo, E.E.A.
Very simple explicit invertible approximation of normal cumulative and normal quantile function
Applied Mathematical Science, 2014
Boyd, S.P., El Ghaoui, L., Feron, E., and Balakrishnan, V.
Linear matrix inequalities in system and control theory;
volume 15. SIAM. 1994
Lazar, M.
Flexible control lyapunov functions;
In American Control Conf., 102-107.2009
Kurzhanski, A. and Varaiya, P.:
Ellipsoidal Calculus for Estimation and Control;
Birkh¨auser, 1996.
Introduction Problem Definition Method Example Conclusion Appendix 26