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

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

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

Simple state-space formulas are presented for a controller solving a standard H∞-problem. The controller has the same state-dimension as the plant, its computation involves only two Riccati equations, and it has a separation structure reminiscent of classical LQG (i.e., H2) theory. This paper is also intended to be of tutorial value, so a standard H2-solution is developed in parallel.

/pdf/state-space-solutions-to-standard-h-2-and-h-control-problems-54ciqky6pt.pdf

State-space solutions to standard H 2 and H ∞ control problems

This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.

Coverage control for mobile sensing networks

Project Management has long established the need for risk management techniques to be utilised in the succinct identification and mitigation of associated risks in projects. Such techniques aim at the reconciliation of countervailing project activities to reduce scope creep, increase the probability of on-time and within-budget delivery. Uncontrolled changes, regardless of size and complexity, can develop risks to projects and affect project success or even an organisation’s project delivery coherence. Ideally, a change or consequence based upon a decision should have a fairly high level of predictability and thus a low level of a potential risk materializing, which would significantly undo the decision taken. This paper proposes a novel modeling process approach; CRAM (Change Risk Assessment Model), which could significantly contribute to the missing formality of business models especially in the change risk assessment area.

/pdf/measuring-change-risk-for-organisational-decision-making-4q7523nj0t.pdf

Measuring Change Risk for Organisational Decision Making through a Hierarchical Model Process Approach

An optimisation algorithm is proposed for designing PID controllers, which minimises the asymptotic open-loop gain of a system, subject to appropriate robust stability and performance QFT constraints. The algorithm is simple and can be used to automate the loop-shaping step of the QFT design procedure. The effectiveness of the method is illustrated with an example

/pdf/optimal-design-of-pid-controllers-using-the-qft-method-59jf4qa0pw.pdf

Optimal design of PID controllers using the QFT method

This paper describes an optimization method for designing feedback systems subject to large parameter uncertainty. Following the design philosophy of the Quantitative Feedback Theory (Horowitz and Sidi 1972, 1978) the objective is to minimize the magnitude of the open loop L(jω) at high frequencies subject to: (a) low and intermediate-frequency bounds capturing the closed-loop robust-performance objectives; (b) a universal-frequency bound on L(jω) which limits the effects of disturbances; and (c) realization constraints on L(s) in the form of Bode's integral. This last relation is discretized at a number of frequencies and defines, together with (a) and (b), the overall set of linear constraints in a resulting linear programming optimization problem.

Optimal loop-shaping for systems with large parameter uncertainty via linear programming

A state-space algorithm is studied which generates the (unique) superoptimal Nehari extension of a general rational matrix $. The procedure is to use a set of all-pass transformations to sequentially minimize each frequency-dependent singular value (of the interpolating function) in a dimension peeling algorithm. These all-pass transformations are determined by the maximal Schmidt pairs of a sequence of Hankel operators. The process terminates when the original problem is reduced to one of rank one; at this stage all the available degrees of freedom have been exhausted. The work is an extension of that by Young (1986) and gives a ‘concrete’ state-space implementation of his operator-theoretic arguments. In addition, bounds are given on the minimum achievable values for s1 ∞ (E) = supωeRSi (E(jω)), i = 1, 2,..., rank (G0), and also the McMillan degree of the final superoptimal extension. Here Si.(.) denotes the ith singular value of a (frequency-dependent) matrix, and the numbering is taken to be in decrea...

State-space algorithm for the computation of superoptimal matrix interpolating functions

This paper is a continuation of our work on $\mathcal{H}^\infty $-optimal control problems which may be embedded in the linear fractional configuration of Fig. 1. In two previous articles [19], [20], a controller degree bound was established for problems in which both $P_{12} (s)$ and $P_{21} (s)$ are square (problems of the first kind). If the McMillan degree of $P(s)$ is n, it was shown that there exist $\mathcal{H}^\infty $-optimal controllers with McMillan degree no greater than $n - 1$.Here we switch our attention to problems of the second kind. That is, we allow $P_{12} (s)$ to have more rows than columns (with $P_{21} (s)$ square), or alternatively, we allow $P_{21} (s)$ to have more columns than rows (with $P_{12} (s)$ square). Our main result shows that the degree bound derived previously for problems of the first kind carries over to problems of the second kind without change. In addition to the controller degree bound, our analysis suggests a number of modifications which are easily made to cur...

George Halikias

Papers

Measuring Change Risk for Organisational Decision Making through a Hierarchical Model Process Approach

Optimal design of PID controllers using the QFT method

Optimal loop-shaping for systems with large parameter uncertainty via linear programming

State-space algorithm for the computation of superoptimal matrix interpolating functions

A controller degree bound for H ∞ 0E control problems of the second kind