Peter M. Dower

Extremum Seeking Control of Cascaded Raman Optical Amplifiers

This paper considers the control of a particular type of optical amplifier that finds application in long-haul wavelength division multiplexed optical communications systems. The objective of this consideration is to demonstrate an application of extremum seeking to the regulation of amplifier output signal power across a range of signal wavelengths, where limited control authority is available. Although such amplifiers are nonlinear and distributed parameter devices, an extremum seeking design is demonstrated to be a promising approach for achieving the stated amplifier control objectives.

A unified approach to controller design for achieving ISS and related properties

A unified approach to the design of controllers achieving various specified input-to-state stability (ISS) like properties is presented. Both full state and measurement feedback cases are considered. Synthesis procedures based on dynamic programming are given using the recently developed results on controller synthesis to achieve uniform l/sup /spl infin// bound. Our results provide a link between the ISS literature and the nonlinear H/sup /spl infin// design literature.

The Principle of Least Action and Fundamental Solutions of Mass-Spring and N-Body Two-Point Boundary Value Problems

Two-point boundary value problems for conservative systems are studied in the context of the least action principle. One obtains a fundamental solution, whereby two-point boundary value problems are converted to initial value problems via an idempotent convolution of the fundamental solution with a cost function related to the terminal data. The classical mass-spring problem is included as a simple example. The

On linear control of backward pumped Raman amplifiers

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A dynamic programming approach to the approximation of nonlinear L 2 -gain

-body problem under gravitation is also studied. There, the least action principle optimal control problem is converted to a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. Solutions are obtained as indexed sets of solutions of Riccati equations.

A Max-plus Dual Space Fundamental Solution for a Class of Operator Differential Riccati Equations

Abstract This paper considers the modelling and control of a particular type of optical amplifier that finds application in long-haul wavelength division multiplexed optical communications systems. The objective of this consideration is to design and demonstrate a simple (and potentially cheap) linear control strategy for regulating amplifier output signal power across a range of signal wavelengths in the presence of upstream signal power uncertainty, where limited control authority is available. Although such amplifiers are highly nonlinear and distributed devices, the straight-forward application of linearization, model reduction and ℋ ∞ loopshaping design is demonstrated to be a promising approach for achieving the stated amplifier control objectives.

A max-plus based fundamental solution for a class of infinite dimensional Riccati equations

A generalization of the L2-gain inequality based on nonlinear gains is considered. Using optimization and dynamic programming to characterize lower bounds for the minimal gain function for which this nonlinear L2-gain inequality holds, a technique for computation of nonlinear L2-gain bounds is proposed. Some simple illustrative examples are explored.

Analysis of input to state stability for discrete time nonlinear systems via dynamic programming

A new fundamental solution semigroup for operator differential Riccati equations is developed. This fundamental solution semigroup is constructed via an auxiliary finite horizon optimal control problem whose value functional growth with respect to time horizon is determined by a particular solution of the operator differential Riccati equation of interest. By exploiting semiconvexity of this value functional, and the attendant max-plus linearity and semigroup properties of the associated dynamic programming evolution operator, a semigroup of max-plus integral operators is constructed in a dual space defined via the Legendre--Fenchel transform. It is demonstrated that this semigroup of max-plus integral operators can be used to propagate all solutions of the operator differential Riccati equation that are initialized from a specified class of initial conditions. As this semigroup of max-plus integral operators can be identified with a semigroup of quadratic kernels, an explicit recipe for the aforementione...

Nonlinear ℒ 2 -gain analysis via a cascade

A new fundamental solution for a specific class of infinite dimensional Riccati equations is developed. This fundamental solution is based on the max-plus dual of the dynamic programming solution operator (or semigroup) of an associated control problem. By taking the max-plus dual of this semigroup operator, the kernel of a dual-space integral operator may be obtained. This kernel is the dual-space Riccati solution propagation operator. Specific initial conditions for the Riccati equation correspond to the associated growth rates of the control problem terminal payoffs. Propagation of the solution of the Riccati equation from these initial conditions proceeds in the dual-space, via a max-plus convolution operation utilizing the aforementioned Riccati solution propagation operator.

Solving Two-Point Boundary Value Problems for a Wave Equation via the Principle of Stationary Action and Optimal Control

This paper presents novel analysis results for input-to-state stability (ISS) that utilise dynamic programming techniques to characterise minimal ISS gains and transient bounds. These characterisations naturally lead to computable necessary and sufficient conditions for ISS. Our results make a connection between ISS and optimisation problems in nonlinear dissipative systems theory (including L/sub 2/-gain analysis and nonlinear H/sub /spl infin// theory).