Semi-definite programming and functional inequalities for Distributed Parameter Systems

We study one-dimensional integral inequalities, with quadratic integrands, on bounded domains. Conditions for these inequalities to hold are formulated in terms of function matrix inequalities which must hold in the domain of integration. For the case of polynomial function matrices, sufficient conditions for positivity of the matrix inequality and, therefore, for the integral inequalities are cast as semi-definite programs. The inequalities are used to study stability of linear partial differential equations.Comment: 8 pages, 5 figure

Exploiting sparsity for neural network verification

The problem of verifying the properties of a neural network has never been more important. This task is often done by bounding the activation functions in the network. Some approaches are more conservative than others and in general there is a trade-off between complexity and conservativeness. There has been significant progress to improve the efficiency and the accuracy of these methods. We investigate the sparsity that arises in a recently proposed semi-definite programming framework to verify a fully connected feed-forward neural network. We show that due to the intrinsic cascading structure of the neural network the constraint matrices in the semi-definite program form a block-arrow pattern and satisfy conditions for chordal sparsity. We reformulate and implement the optimisation problem, showing a significant speed-up in computation, without sacrificing solution accuracy

Fast ADMM for sum-of-squares programs using partial orthogonality

IEEE When sum-of-squares (SOS) programs are recast as semidefinite programs (SDPs) using the standard monomial basis, the constraint matrices in the SDP possess a structural property that we call partial orthogonality. In this paper, we leverage partial orthogonality to develop a fast first-order method, based on the alternating direction method of multipliers (ADMM), for the solution of the homogeneous self-dual embedding of SDPs describing SOS programs. Precisely, we show how a “diagonal plus low rank” structure implied by partial orthogonality can be exploited to project efficiently the iterates of a recent ADMM algorithm for generic conic programs onto the set defined by the affine constraints of the SDP. The resulting algorithm, implemented as a new package in the solver CDCS, is tested on a range of large-scale SOS programs arising from constrained polynomial optimization problems and from Lyapunov stability analysis of polynomial dynamical systems. These numerical experiments demonstrate the effectiveness of our approach compared to common state-of-the-art solvers

Exploiting Sparsity in the Coefficient Matching Conditions in Sum-of-Squares Programming Using ADMM

This letter introduces an efficient first-order method based on the alternating direction method of multipliers (ADMM) to solve semidefinite programs arising from sum-of-squares (SOS) programming. We exploit the sparsity of the coefficient matching conditions when SOS programs are formulated in the usual monomial basis to reduce the computational cost of the ADMM algorithm. Each iteration of our algorithm requires one projection onto the positive semidefinite cone and the solution of multiple quadratic programs with closed-form solutions free of any matrix inversion. Our techniques are implemented in the open-source MATLAB solver SOSADMM. Numerical experiments on SOS problems arising from unconstrained polynomial minimization and from Lyapunov stability analysis for polynomial systems show speed-ups compared to the interior-point solver SeDuMi, and the first-order solver CDCS

Structural Identifiability Analysis via Extended Observability and Decomposition

7 páginasStructural identifiability analysis of nonlinear dynamic models requires symbolic manipulations, whose computational cost rises very fast with problem size. This hampers the application of these techniques to the large models which are increasingly common in systems biology. Here we present a method to assess parametric identifiability based on the framework of nonlinear observability. Essentially, our method considers model parameters as particular cases of state variables with zero dynamics, and evaluates structural identifiability by calculating the rank of a generalized observability-identifiability matrix. If a model is unidentifiable as a whole, the method determines the identifiability of its individual parameters. For models whose size or complexity prevents the direct application of this procedure, an optimization approach is used to decompose them into tractable subsystems. We demonstrate the feasibility of this approach by applying it to three well-known case studiesPeer reviewe

TESTING USED ROLLER BEARINGS FOR QUALITY AND SERVICE LIFE

Synthetic biology is a rapidly expanding field at the interface of the engineering and biological sciences which aims to apply rational design principles in biological contexts. Many natural processes utilise regulatory architectures that parallel those found in control and electrical engineering, which has motivated their implementation as part of synthetic biological constructs. Tools based upon control theoretical concepts can be used to design such systems, as well as to guide their experimental realisation. In this paper we provide examples of biological implementations of negative feedback systems, and discuss progress made toward realisation of other feedback and control architectures. We then outline major challenges posed by the design of such systems, particularly focusing on those which are specific to biological contexts and on which feedback control can have a significant impact. We explore future directions for work in the field, including new approaches for theoretical design of biological control systems, the utilisation of novel components for their implementation, and the potential for application of automation and machine-learning approaches to accelerate synthetic biological research

A game theoretic approach for safe and distributed control of unmanned aerial vehicles

This paper presents a distributed methodology to produce collision-free control laws for an Unmanned Aerial Vehicle (UAV) fleet. We use a game theoretic framework, where UAVs accommodate for individual and fleet goals, while respecting safety requirements. The method combines Control Barrier Functions (CBFs) and a primal-dual algorithm for Nash equilibrium (NE) seeking in generalized games. Feedback is introduced by Model Predictive Control (MPC) and we analyze its stability properties. The combination of these tools allows for a distributed, collision-free pointwise equilibrium solution, despite the agents' coupling, due to common target tracking and the collision avoidance constraints. Our algorithmic results are supported theoretically and our method's efficacy is demonstrated via extensive numerical simulations

Analysis of aircraft pitch axis stability augmentation system using sum of squares optimization

In this paper, we use SOS (sum of squares) programming approaches to analyze the stability and robustness properties of the controlled pitch axis (6 state system) of a nonlinear model of an aircraft. The controller is a LTI dynamic inversion based control law designed for the short period dynamics of the aircraft. The closed loop system is tested for its robustness to uncertainty in the location of center of gravity along the body x-axis. Results in the form of stability regions about a trim point are computed and verified using simulations

Convergence rate analysis of a subgradient averaging algorithm for distributed optimisation with different constraint sets

We consider a multi-agent setting with agents exchanging information over a network to solve a convex constrained optimisation problem in a distributed manner. We analyse a new algorithm based on local subgradient exchange under undirected time-varying communication. First, we prove asymptotic convergence of the iterates to a minimum of the given optimisation problem for time-varying step-sizes of the form c(k) = rac{eta }{{k + 1}}, for some \u3b7 > 0. We then restrict attention to step-size choices c(k) = rac{eta }{{sqrt {k + 1} }},eta > 0, and establish a convergence of mathcal{O}left( {rac{{ln (k)}}{{sqrt k }}} ight) in objective value. Our algorithm extends currently available distributed subgradient/proximal methods by: (i) accounting for different constraint sets at each node, and (ii) enhancing the convergence speed thanks to a subgradient averaging step performed by the agents. A numerical example demonstrates the efficacy of the proposed algorithm

Fast ADMM for Semidefinite Programs with Chordal Sparsity

Many problems in control theory can be formulated as semidefinite programs (SDPs). For large-scale SDPs, it is important to exploit the inherent sparsity to improve the scalability. This paper develops efficient first-order methods to solve SDPs with chordal sparsity based on the alternating direction method of multipliers (ADMM). We show that chordal decomposition can be applied to either the primal or the dual standard form of a sparse SDP, resulting in scaled versions of ADMM algorithms with the same computational cost. Each iteration of our algorithms consists of a projection on the product of small positive semidefinite cones, followed by a projection on an affine set, both of which can be carried out efficiently. Our techniques are implemented in CDCS, an open source add-on to MATLAB. Numerical experiments on large-scale sparse problems in SDPLIB and random SDPs with block-arrow sparse patterns show speedups compared to some common state-of-the-art software packages