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2012

Volume 50, Issue 3 (partial)


Real Solutions to Control, Approximation, and Factorization Problems

Kalle M. Mikkola

SIAM J. Control Optim. 50, pp. 1071-1086 (16 pages)

Online Publication Date: May 01, 2012

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During the past decades much of finite-dimensional systems theory has been generalized to infinite dimensions. However, there is one important flaw in this theory: it guarantees only complex solutions, even when the data is real. We show that the standard solutions of many classical problems with real data are also real. We call a (possibly matrix- or operator-valued) holomorphic function $G$ real (real-symmetric) if $G(\bar{z})=\overline{G(z)}$ for every $z$. We show that if such a function can be presented as $G=NM^{-1}$, where $N,M\in\mathcal{H}^\infty$, then we have $G=N_R M_R^{-1}$, where $N_R,M_R\in\mathcal{H}^\infty$ are real and weakly right coprime. Consequently, if a real function $G$ has a stabilizing compensator (i.e., a function $K$ such that $[\begin{smallmatrix}I&-K\\-G&I\end{smallmatrix}]^{-1}\in\mathcal{H}^\infty$), then $G$ has a real doubly coprime factorization and a Youla parameterization of all real stabilizing controllers. If a system of the form $\dot{x}=Ax+Bu$, $y=Cx+Du$ or of the form $x_{n+1}=Ax_n+Bu_n$, $y_n=Cx_n+Du_n$ has real (possibly unbounded, constant) coefficients $A$, $B$, $C$, and $D$, then the system is stabilizable iff it is stabilizable by a real state-feedback operator. This holds for both exponential stabilization and output stabilization. A real stabilizing state-feedback operator is then given by the standard linear quadratic regulator (LQR) feedback operator, hence the standard (complex) formulae can be used to find this real solution. Analogous results are established for other optimization, factorization, approximation, and representation problems too.

Convergence Speed of Binary Interval Consensus

Moez Draief and Milan Vojnović

SIAM J. Control Optim. 50, pp. 1087-1109 (23 pages)

Online Publication Date: May 01, 2012

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We consider the convergence time for solving the binary consensus problem using the interval consensus algorithm proposed by Bénézit, Thiran, and Vetterli [Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), IEEE Press, Piscataway, NJ, 2009, pp. 3661–3664]. In the binary consensus problem, each node initially holds one of two states and the goal for each node is to correctly decide which one of these two states was initially held by a majority of nodes. We derive an upper bound on the expected convergence time that holds for arbitrary connected graphs, which is based on the location of eigenvalues of some contact rate matrices. We instantiate our bound for particular networks of interest, including complete graphs, paths, cycles, star-shaped networks, and Erdös–Rényi random graphs; for these graphs, we compare our bound with alternative computations. We find that for all these examples our bound is tight, yielding the exact order with respect to the number of nodes. We pinpoint the fact that the expected convergence time critically depends on the voting margin defined as the difference between the fraction of nodes that initially held the majority and the minority states, respectively. The characterization of the expected convergence time yields an exact relation between the expected convergence time and the voting margin for some of these graphs, which reveals how the expected convergence time goes to infinity as the voting margin approaches zero. Our results provide insight into how the expected convergence time depends on the network topology, which can be used for performance evaluation and network design. The results are of interest in the context of networked systems, in particular, peer-to-peer networks, sensor networks, and distributed databases.

Dynamic Phasor Analysis of Pulse-Modulated Systems

Stefan Almér and Ulf T. Jönsson

SIAM J. Control Optim. 50, pp. 1110-1138 (29 pages)

Online Publication Date: May 01, 2012

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This paper considers stability and harmonic analysis of a general class of pulse-modulated systems. The systems are modeled using the dynamic phasor model, which explores the cyclic nature of the modulation functions by representing the system state as a Fourier series expansion defined over a moving time window. The contribution of the paper is to show that a special type of periodic Lyapunov function can be used to analyze the system and that the analysis conditions become tractable for computation after truncation. The approach provides a trade-off between complexity and accuracy that includes standard state space averaged models as a special case. The paper also shows how the dynamic phasor model can be used to derive a frequency domain input-to-state map which is analogous to the harmonic transfer function.

Variational Approach to Second-Order Optimality Conditions for Control Problems with Pure State Constraints

Daniel Hoehener

SIAM J. Control Optim. 50, pp. 1139-1173 (35 pages)

Online Publication Date: May 15, 2012

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For optimal control problems with set-valued control constraints and pure state constraints we propose new second-order necessary optimality conditions. In addition to the usual second-order derivative of the Hamiltonian, these conditions contain extra terms involving second-order tangents to the set of feasible trajectory-control pairs at the extremal process under consideration. The second-order necessary optimality conditions of the present work are obtained by using a variational approach. In particular, we present a new second-order variational equation. This approach allows us to make direct proofs as opposed to the classical way of obtaining second-order necessary conditions by using an abstract infinite dimensional optimization problem. No convexity assumptions on the constraints are imposed and optimal controls are required to be merely measurable.
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