There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

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Linear Matrix Inequalities in System and Control Theory

This paper will very briefly review the history of the relationship between modern optimal control and robust control. The latter is commonly viewed as having arisen in reaction to certain perceived inadequacies of the former. More recently, the distinction has effectively disappeared. Once-controversial notions of robust control have become thoroughly mainstream, and optimal control methods permeate robust control theory. This has been especially true in H-infinity theory, the primary focus of this paper.

http://www.gbv.de/dms/goettingen/18606599X.pdf

Robust and Optimal Control

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Simple state-space formulas are derived for all controllers solving the following standard H/sub infinity / problem: For a given number gamma >0, find all controllers such that the H/sub infinity / norm of the closed-loop transfer function is (strictly) less than gamma . It is known that a controller exists if and only if the unique stabilizing solutions to two algebraic Riccati equations are positive definite and the spectral radius of their product is less than gamma /sup 2/. Under these conditions, a parameterization of all controllers solving the problem is given as a linear fractional transformation (LFT) on a contractive, stable, free parameter. The state dimension of the coefficient matrix for the LFT, constructed using the two Riccati solutions, equals that of the plant and has a separation structure reminiscent of classical LQG (i.e. H/sub 2/) theory. This paper is intended to be of tutorial value, so a standard H/sub 2/ solution is developed in parallel. >

State-space solutions to standard H/sub 2/ and H/sub infinity / control problems

This paper presents methods to design edge weights of a distributed control network, and to estimate a positively invariant set for a general multirate Kuramoto oscillator network to achieve bounded phase synchronization. The oscillators' phases can be ultimately bounded by any given value if the edge weights of the control network satisfy proposed conditions. And the transient boundedness on the multirate Kuramoto network's dynamics is guaranteed if its initial values are located in the estimated positively invariant set. The presented conditions on edge weights can be obtained by using the graph spectral properties, or alternatively by analysing a path set of a graph. We further improve the estimate of the positively invariant set for the multirate Kuramoto network if its parameters satisfy certain assumptions. The effectiveness and conservativeness of the proposed methods are demonstrated on some simulation studies.

Distributed control network design for multirate kuramoto networks to achieve bounded synchronization

In this paper, we extend nonlinear negative imaginary (NI) systems theory to switched systems. Switched nonlinear NI systems and switched nonlinear output strictly negative imaginary (OSNI) systems are defined. We show that the interconnection of two switched nonlinear NI systems is still switched nonlinear NI. The interconnection of a switched nonlinear NI system and a switched nonlinear OSNI system is asymptotically stable under some assumptions. This stability result is then illustrated using a numerical example.

Nonlinear Negative Imaginary Systems with Switching

In this paper, we propose the Whiplash Inertial Gradient dynamics, a closed-loop optimization method that utilises gradient information, to ﬁnd the minima of a cost function in ﬁnite-dimensional settings. We introduce the symplectic asymptotic convergence analysis for the Whiplash system for convex functions. We also introduce relaxation sequences to explain the non-classical nature of the algorithm and an exploring heuristic variant of the Whiplash algorithm to escape saddle points, deterministically. We study the algorithm’s performance for various costs and provide a practical methodology for analyzing convergence rates using integral constraint bounds and a novel Lyapunov rate method. Our results demonstrate polynomial and exponential rates of convergence for quadratic cost functions

https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/asjc.3153

Analysis of the Whiplash gradient descent dynamics

This paper is concerned with linear stochastic systems whose output is a stationary Gaussian random process related by an integral operator to a standard Wiener process at the input. We consider a performance criterion which involves the trace of an analytic function of the spectral density of the output process. This class of “covariance-analytic” cost functionals includes the usual mean square and risk-sensitive criteria as particular cases. Due to the presence of the “cost-shaping” analytic function, the performance criterion is related to higher-order Hardy-Schatten norms of the system transfer function. These norms have links with the asymptotic properties of cumulants of ﬁnite-horizon quadratic functionals of the system output and satisfy variational inequalities pertaining to system robustness to statistically uncertain inputs. In the case of strictly proper ﬁnite-dimensional systems, governed in state space by linear stochastic differential equations, we develop a method for recursively computing the Hardy-Schatten norms through a recently proposed technique of rearranging cascaded linear systems, which resembles the Wick ordering of annihilation and creation operators in quantum mechanics. The resulting computational procedure involves a recurrence sequence of solutions to algebraic Lyapunov equations and represents the covariance-analytic cost as the squared H 2 -norm of an auxiliary cascaded system. These results are also compared with an alternative approach which uses higher-order derivatives of stabilising solutions of parameter-dependent algebraic Riccati equations.

Covariance-analytic performance criteria, Hardy-Schatten norms and Wick-like ordering of cascaded systems

Systematic approaches to the design of quantum optical systems will become more and more important as their complexity grows. One such systematic technique is the Linear Quadratic Gaussian (LQG) methodology. Here we describe the design and implementation of a LQG feedback controller in a quantum optical experiment—frequency‐locking an optical cavity to a laser. The successful implementation of the LQG design procedure to this particular problem lays the groundwork for the application of other modern control techniques to quantum optical systems. Our results are promising for future, more complex stabilisation problems in quantum optics, as the described approach is inherently multi‐variable and naturally incorporates multiple sensors and actuators as well as nested loops.

Ian R. Petersen

Papers

Distributed control network design for multirate kuramoto networks to achieve bounded synchronization

Nonlinear Negative Imaginary Systems with Switching

Analysis of the Whiplash gradient descent dynamics

Covariance-analytic performance criteria, Hardy-Schatten norms and Wick-like ordering of cascaded systems

Laser Frequency Locking to an Optical Cavity using LQG Control