Decay rate estimations for linear quadratic optimal regulators

Let $u(t)=-Fx(t)$ be the optimal control of the open-loop system $x'(t)=Ax(t)+Bu(t)$ in a linear quadratic optimization problem. By using different complex variable arguments, we give several lower and upper estimates of the exponential decay rate of the closed-loop system $x'(t)=(A-BF)x(t)$. Main attention is given to the case of a skew-Hermitian matrix $A$. Given an operator $A$, for a class of cases, we find a matrix $B$ that provides an almost optimal decay rate. We show how our results can be applied to the problem of optimizing the decay rate for a large finite collection of control systems $(A, B_j)$, $j=1, \dots, N$, and illustrate this on an example of a concrete mechanical system. At the end of the article, we pose several questions concerning the decay rates in the context of linear quadratic optimization and in a more general context of the pole placement problem.Comment: 25 pages, 1 figur

Traffic-responsive urban network control using multivariable regulators

The paper presents the philosophy, the aim, the development, the advantages, and the potential shortcomings of the TUC (Traffic-responsive Urban Control) strategy. Based on a store-and-forward modeling approach and using well-known methods of the Automatic Control Theory, the approach followed by TUC designs (off-line) and employs (on-line) a multivariable regulator for traffic-responsive co-ordinated network-wide signal control. Simulation investigations are used to demonstrate the efficiency of the proposed approach. Based on the presented investigations, summarising conclusions are drawn and future work is outlined

Covariant Pauli-Villars Regularization of Quantum Gravity at the One Loop Order

We study a regularization of the Pauli-Villars kind of the one loop gravitational divergences in any dimension. The Pauli-Villars fields are massive particles coupled to gravity in a covariant and nonminimal way, namely one real tensor and one complex vector. The gauge is fixed by means of the unusual gauge-fixing that gives the same effective action as in the context of the background field method. Indeed, with the background field method it is simple to see that the regularization effectively works. On the other hand, we show that in the usual formalism (non background) the regularization cannot work with each gauge-fixing.In particular, it does not work with the usual one. Moreover, we show that, under a suitable choice of the Pauli-Villars coefficients, the terms divergent in the Pauli-Villars masses can be corrected by the Pauli-Villars fields themselves. In dimension four, there is no need to add counterterms quadratic in the curvature tensor to the Einstein action (which would be equivalent to the introduction of new coupling constants). The technique also works when matter is coupled to gravity. We discuss the possible consequences of this approach, in particular the renormalization of Newton's coupling constant and the appearance of two parameters in the effective action, that seem to have physical implications.Comment: 26 pages, LaTeX, SISSA/ISAS 73/93/E

Robust ℋ2 Performance: Guaranteeing Margins for LQG Regulators

This paper shows that ℋ2 (LQG) performance specifications can be combined with structured uncertainty in the system, yielding robustness analysis conditions of the same nature and computational complexity as the corresponding conditions for ℋ∞ performance. These conditions are convex feasibility tests in terms of Linear Matrix Inequalities, and can be proven to be necessary and sufficient under the same conditions as in the ℋ∞ case. With these results, the tools of robust control can be viewed as coming full circle to treat the problem where it all began: guaranteeing margins for LQG regulators

Relaxing Lorentz invariance in general perturbative anomalies

We analyze the role of Lorentz symmetry in the perturbative non-gravitational anomalies for a single family of fermions. The theory is assumed to be translational invariant, power-counting renormalizable and based on a local action, but is allowed to have general Lorentz violating operators. We study the conservation of global and gauge currents associate with general internal symmetry groups and find, by using a perturbative approach, that Lorentz symmetry does not participate in the clash of symmetries that leads to the anomalies. We first analyze the triangle graphs and prove that there are regulators for which the anomalous part of the Ward identities exactly reproduces the Lorentz invariant case. Then we show, by means of a regulator independent argument, that the anomaly cancellation conditions derived in Lorentz invariant theories remain necessary ingredients for anomaly freedom.Comment: 18 pages, 1 figure. Few comments added. Article published in Physical Review

Corporate Control, Portfolio Choice, and the Decline of Banking

The authors focus on the persistence of bank unprofitability during the 1980s. A large literature in banking, following Merton (1977), concentrates on the incentives of shareholders to maximize the value of the (fixed rate) deposit insurance subsidy provided by the government by taking on risk inefficiently, so called moral hazard' risk. This paper takes issue with this moral hazard explanation for the performance of the banking industry. The moral hazard view assumes that shareholders make the lending decisions and can take on risk to maximize the value of insurance if they desire. The authors assume bank managers, who may own a fraction of the bank, make the lending decisions. If managers have different objectives than outside shareholders and disciplining in managers is costly, then managerial decisions may be at odds with the decisions outside shareholders would like them to take. When investment opportunities are declining, managers behave differently than in healthy' industries. This is particularly true in banking, where asymmetric information and deposit insurance allow banks resources to invest even if there are few good lending opportunities. The risk-avoiding behavior of managers stressed in the corporate finance literature presumes that conservative behavior is sufficient for job and perquisite preservation. When bad managers predominate, conservative behavior may not allow most managers to keep their jobs and perquisites. These managers may find it optimal to take excessively risky actions. The paper sets out a game between a bank manager and shareholders and solves for a sequential Nash equilibrium. A bank manager chooses either risky or safe loans based on the quality of the loan opportunities available to the manager (the manager s type). The choice of loan portfolios is observed by shareholders, but the manager s type is not. If the manager is fired, shareholders decide whether to invest in new bank assets (hire a new manager) or move their capital out of banking (liquidate capital). In any period that they are employed, managers receive a private benefit. Using data on the equity ownership structure of large bank holding companies, the authors test the predictions of the corporate control model of banking against an alternative model based on moral hazard problems between banks and regulators. With respect to the choice of loans made, the authors findings are consistent with corporate control problems playing an important role, but are inconsistent with moral hazard playing a dominant role in banking. None of the results are what a moral hazard model would predict. However, the analysis is done for adequately-capitalized banks. Thus, if the value of bank equity is low enough, the interests of inside and outside owners are aligned, so there are no corporate control problems of the sort modeled by the authors. It may be accurate to say that, for large U.S. banks, corporate control problems have been the cause of the conditions of which moral hazard may be an accurate characterization. The presence of agency costs suggests that the underlying trends that reduced profitability in the 1980s may persist, despite high bank earnings in the early 1990s. That banking is regulated does not appear to be a sufficient countervailing force. To the extent that chartered banks must transform themselves into nonbanks as they seek nonlending and deposit-taking activities which are profitable, the authors suggest that banking' is in decline. Their conclusions concern the difficulties that outside equityholders face during the transition period.

Robustness results in LQG based multivariable control designs

The robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria. Results are derived under a common framework in which the minimum singular value of the return difference transfer matrix is the key quantity. In particular, the LQ and LQG robustness results are discussed