Mikael Johansson

Piecewise quadratic stability of fuzzy systems

Presents an approach to stability analysis of fuzzy systems. The analysis is based on Lyapunov functions that are continuous and piecewise quadratic. The approach exploits the gain-scheduling nature of fuzzy systems and results in stability conditions that can be verified via convex optimization over linear matrix inequalities. Examples demonstrate the many improvements over analysis based on a single quadratic Lyapunov function. Special attention is given to the computational aspects of the approach and several methods to improve the computational efficiency are described.

Piecewise linear quadratic optimal control

The use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control. Lower bounds on the optimal control cost are obtained by semidefinite programming based on the Bellman inequality. This also gives an approximation to the optimal control law. An upper bound to the optimal cost is obtained by another convex optimization problem using the given control law. A compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy.

Piecewise Linear Control Systems

This thesis treats analysis and design of piecewise linear control systems. Piecewise linear systems capture many of the most common nonlinearities in engineering systems, and they can also be used for approximation of other nonlinear systems. Several aspects of linear systems with quadratic constraints are generalized to piecewise linear systems with piecewise quadratic constraints. It is shown how uncertainty models for linear systems can be extended to piecewise linear systems, and how these extensions give insight into the classical trade-offs between fidelity and complexity of a model. Stability of piecewise linear systems is investigated using piecewise quadratic Lyapunov functions. Piecewise quadratic Lyapunov functions are much more powerful than the commonly used quadratic Lyapunov functions. It is shown how piecewise quadratic Lyapunov functions can be computed via convex optimization in terms of linear matrix inequalities. The computations are based on a compact parameterization of continuous piecewise quadratic functions and conditional analysis using the S-procedure. A unifying framework for computation of a variety of Lyapunov functions via convex optimization is established based on this parameterization. Systems with attractive sliding modes and systems with bounded regions of attraction are also treated. Dissipativity analysis and optimal control problems with piecewise quadratic cost functions are solved via convex optimization. The basic results are extended to fuzzy systems, hybrid systems and smooth nonlinear systems. It is shown how Lyapunov functions with a discontinuous dependence on the discrete state can be computed via convex optimization. An automated procedure for increasing the flexibility of the Lyapunov function candidate is suggested based on linear programming duality. A Matlab toolbox that implements several of the results derived in the thesis is presented.

Subgradient methods and consensus algorithms for solving convex optimization problems

In this paper we propose a subgradient method for solving coupled optimization problems in a distributed way given restrictions on the communication topology. The iterative procedure maintains local variables at each node and relies on local subgradient updates in combination with a consensus process. The local subgradient steps are applied simultaneously as opposed to the standard sequential or cyclic procedure. We study convergence properties of the proposed scheme using results from consensus theory and approximate subgradient methods. The framework is illustrated on an optimal distributed finite-time rendezvous problem.

A Randomized Incremental Subgradient Method for Distributed Optimization in Networked Systems

We present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to Nedic and Bertsekas [SIAM J. Optim., 12 (2001), pp. 109-138]. Our novel algorithm is particularly suitable for distributed implementation and execution, and possible applications include distributed optimization, e.g., parameter estimation in networks of tiny wireless sensors. The stochastic component in the algorithm is described by a Markov chain, which can be constructed in a distributed fashion using only local information. We provide a detailed convergence analysis of the proposed algorithm and compare it with existing, both deterministic and randomized, incremental subgradient methods.

Cross-layer optimization of wireless networks using nonlinear column generation

We consider the problem of finding the jointly optimal end-to-end communication rates, routing, power allocation and transmission scheduling for wireless networks. In particular, we focus on finding the resource allocation that achieves fair end-to-end communication rates. Using realistic models of several rate and power adaption schemes, we show how this cross-layer optimization problem can be formulated as a nonlinear mathematical program. We develop a specialized solution method, based on a nonlinear column generation technique, and prove that it converges to the globally optimal solution. We present computational results from a large set of networks and discuss the insight that can be gained about the influence of power control, spatial reuse, routing strategies and variable transmission rates on network performance.

Optimal parameter selection for the alternating direction method of multipliers (ADMM) : quadratic problems

The alternating direction method of multipliers (ADMM) has emerged as a powerful technique for large-scale structured optimization. Despite many recent results on the convergence properties of ADMM, a quantitative characterization of the impact of the algorithm parameters on the convergence times of the method is still lacking. In this paper we find the optimal algorithm parameters that minimize the convergence factor of the ADMM iterates in the context of ℓ2-regularized minimization and constrained quadratic programming. Numerical examples show that our parameter selection rules significantly outperform existing alternatives in the literature.

Traffic matrix estimation on a large IP backbone: a comparison on real data

This paper considers the problem of estimating the point-to-point traffic matrix in an operational IP backbone. Contrary to previous studies, that have used a partial traffic matrix or demands estimated from aggregated Netflow traces, we use a unique data set of complete traffic matrices from a global IP network measured over five-minute intervals. This allows us to do an accurate data analysis on the time-scale of typical link-load measurements and enables us to make a balanced evaluation of different traffic matrix estimation techniques. We describe the data collection infrastructure, present spatial and temporal demand distributions, investigate the stability of fan-out factors, and analyze the mean-variance relationships between demands. We perform a critical evaluation of existing and novel methods for traffic matrix estimation, including recursive fanout estimation, worst-case bounds, regularized estimation techniques, and methods that rely on mean variance relationships. We discuss the weaknesses and strengths of the various methods, and highlight differences in the results for the European and American subnetworks.

Low power, low delay: opportunistic routing meets duty cycling

Traditionally, routing in wireless sensor networks consists of two steps: First, the routing protocol selects a next hop, and, second, the MAC protocol waits for the intended destination to wake up and receive the data. This design makes it difficult to adapt to link dynamics and introduces delays while waiting for the next hop to wake up. In this paper we introduce ORW, a practical opportunistic routing scheme for wireless sensor networks. In a duty-cycled setting, packets are addressed to sets of potential receivers and forwarded by the neighbor that wakes up first and successfully receives the packet. This reduces delay and energy consumption by utilizing all neighbors as potential forwarders. Furthermore, this increases resilience to wireless link dynamics by exploiting spatial diversity. Our results show that ORW reduces radio duty-cycles on average by 50% (up to 90% on individual nodes) and delays by 30% to 90% when compared to the state of the art.

Interactive tools for education in automatic control

Experiments have shown that the time is now ripe for a new generation of interactive learning tools for control. The tools are based on objects which admit direct graphical manipulation. During manipulations, objects are updated instantaneously, so that relations between objects are maintained all the time. The tools are natural complements to traditional education, and allow students to quickly gain insight and motivation. A high degree of interactivity has been found to be a key issue in the design. Together with a high bandwidth in the man-machine interaction, this enhances learning significantly. Another nice feature is the possibility to hide minor issues and focus on the essentials. It is not easy to describe the power of these tools adequately in text. The best way to appreciate them is simply to use them. We believe that there is a strong pedagogical potential for the type of tools that we have described. We are also of the opinion that we are only at the very beginning in the development of learning tools of this type. The addition of sound and animation are interesting avenues that should be pursued.