55 research outputs found
Control of fluid flows and other systems governed by partial differential-algebraic equations
The motion of fluids, such as air or water, is central to many engineering systems of significant
economic and environmental importance. Examples range from air/fuel mixing in combustion engines
to turbulence induced noise and fatigue on aircraft. Recent advances in novel sensor/actuator
technologies have raised the intriguing prospect of actively sensing and manipulating the motion
of the fluid within these systems, making them ripe for feedback control, provided a suitable control
model exists. Unfortunately, the models for many of these systems are described by nonlinear,
partial differential-algebraic equations for which few, if any, controller synthesis techniques exist.
In stark contrast, the majority of established control theory assumes plant models of finite (and
typically small) state dimension, expressed as a linear system of ordinary differential equations.
Therefore, this thesis explores the problem of how to apply the mainstream tools of control theory
to the class of systems described by partial differential-algebraic equations, that are either linear,
or for which a linear approximation is valid.
The problems of control system design for infinite-dimensional and algebraically constrained
systems are treated separately in this thesis. With respect to the former, a new method is presented
that enables the computation of a bound on the n-gap between a discretisation of a spatially distributed
plant, and the plant itself, by exploiting the convergence rate of the v-gap metric between
low-order models of successively finer spatial resolution. This bound informs the design, on loworder
models, of H[infinity] loop-shaping controllers that are guaranteed to robustly stabilise the actual
plant. An example is presented on a one-dimensional heat equation.
Controller/estimator synthesis is then discussed for finite-dimensional systems containing algebraic,
as well as differential equations. In the case of fluid flows, algebraic constraints typically
arise from incompressibility and the application of boundary conditions. A numerical algorithm is
presented, suitable for the semi-discrete linearised Navier-Stokes equations, that decouples the differential
and algebraic parts of the system, enabling application of standard control theory without
the need for velocity-vorticity type methods. This algorithm is demonstrated firstly on a simple
electrical circuit, and secondly on the highly non-trivial problem of flow-field estimation in the
transient growth region of a flat-plate boundary layer, using only wall shear measurements.
These separate strands are woven together in the penultimate chapter, where a transient energy
controller is designed for a channel-flow system, using wall mounted sensors and actuators
Modelling for Robust Feedback Control of Fluid Flows
This paper addresses the problem of obtaining low-order models of fluid flows for the purpose of designing robust feedback controllers. This is challenging since whilst many flows are governed by a set of nonlinear, partial differential-algebraic equations (the Navier-Stokes equations), the majority of established control theory assumes models of much greater simplicity, in that they are firstly: linear, secondly: described by ordinary differential equations, and thirdly: finite-dimensional. Linearisation, where appropriate, overcomes the first disparity, but attempts to reconcile the remaining two have proved difficult. This paper addresses these two problems as follows. Firstly, a numerical approach is used to project the governing equations onto a divergence-free basis, thus converting a system of differential-algebraic equations into one of ordinary differential equations. This dispenses with the need for analytical velocity-vorticity transformations, and thus simplifies the modelling of boundary sensing and actuation. Secondly, this paper presents a novel and straightforward approach for obtaining suitable low-order models of fluid flows, from which robust feedback controllers can be synthesised that provide~\emph{a~priori} guarantees of robust performance when connected to the (infinite-dimensional) linearised flow system. This approach overcomes many of the problems inherent in approaches that rely upon model-reduction. To illustrate these methods, a perturbation shear stress controller is designed and applied to plane channel flow, assuming arrays of wall mounted shear-stress sensors and transpiration actuators. DNS results demonstrate robust attenuation of the perturbation shear-stresses across a wide range of Reynolds numbers with a single, linear controller
Modelling for robust feedback control of fluid flows
This paper addresses the problem of designing low-order and linear robust feedback controllers that provide a priori guarantees with respect to stability and performance when applied to a fluid flow. This is challenging, since whilst many flows are governed by a set of nonlinear, partial differentialâalgebraic equations (the NavierâStokes equations), the majority of established control system design assumes models of much greater simplicity, in that they are: firstly, linear; secondly, described by ordinary differential equations (ODEs); and thirdly, finite-dimensional. With this in mind, we present a set of techniques that enables the disparity between such models and the underlying flow system to be quantified in a fashion that informs the subsequent design of feedback flow controllers, specifically those based on the Hâ loop-shaping approach. Highlights include the application of a model refinement technique as a means of obtaining low-order models with an associated bound that quantifies the closed-loop degradation incurred by using such finite-dimensional approximations of the underlying flow. In addition, we demonstrate how the influence of the nonlinearity of the flow can be attenuated by a linear feedback controller that employs high loop gain over a select frequency range, and offer an explanation for this in terms of Landahlâs theory of sheared turbulence. To illustrate the application of these techniques, an Hâ loop-shaping controller is designed and applied to the problem of reducing perturbation wall shear stress in plane channel flow. Direct numerical simulation (DNS) results demonstrate robust attenuation of the perturbation shear stresses across a wide range of Reynolds numbers with a single linear controller
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Approximate controllability and observability measures in control systems design
The selection of systems of inputs and outputs (input and output structure) forms part of early system design, which is important since it preconditions the potential for control design. Existing methodologies for input, output structure selection rely on criteria expressing distance from uncontrollability (unobservability). The thesis introduces novel measures for evaluating and estimating the distance to uncontrollability and relatively unobservability. At first, the modal measuring approach is studied in detail, providing a framework for the âbestâ structure selection. Although controllability (observability) is invariant under state feedback (output injection), the corresponding degrees expressing distance from uncontrollability (unobservability) are not. Hence, the thesis introduces new criteria for the distance problem from uncontrollability (unobservability) which is invariant under feedback transformations. The approach uses the restricted input-state (state-output) matrix pencil and then deploys exterior algebra that reduces the overall problem to the standard problem of distance of a set of polynomials from non-coprimeness. Results on the distance of the Sylvester Resultants from singularity provide the new measures. Since distance to singularity of the corresponding Sylvester matrix is the key in evaluating the distance to uncontrolability it is of the particular interest in the present work. In order to find the solution two novel methods are introduced in the thesis, namely the alternating projection algorithm and a structured singular value approach. A least-squares alternating projection algorithm, motivated by a factorisation result involving the Sylvester resultant matrix, is proposed for calculating the âbestâ approximate GCD of a coprime polynomial set. The properties of the proposed algorithm are investigated and the method is compared with alternative optimisation techniques which can be employed to solve the problem. It is also shown that the problem of an approximate GCD calculation is equivalent tothe solution of a structured singular value (”) problem arising in robust control for which numerous techniques are available. Motivated by the powerful concept of the structured singular values, the proposed method is extended to the special case of an implicit system that has a wide application in the behavioural analysis of complex systems. Moreover, ”-value approach has a potential application for the general distance problem to uncontrollability that is numerically hard to obtain. Overall, the proposed framework significantly simplifies and generalises the input-output structure selection procedure and evaluates alternative solutions for a variety of distance problems that appear in Control Theory
Optimised configuration of sensing elements for control and fault tolerance applied to an electro-magnetic suspension system
New technological advances and the requirements to increasingly abide
by new safety laws in engineering design projects highly affects industrial
products in areas such as automotive, aerospace and railway industries.
The necessity arises to design reduced-cost hi-tech products with minimal
complexity, optimal performance, effective parameter robustness properties,
and high reliability with fault tolerance. In this context the control system
design plays an important role and the impact is crucial relative to the level
of cost efficiency of a product.
Measurement of required information for the operation of the design
control system in any product is a vital issue, and in such cases a number of
sensors can be available to select from in order to achieve the desired system
properties. However, for a complex engineering system a manual procedure
to select the best sensor set subject to the desired system properties can
be very complicated, time consuming or even impossible to achieve. This is
more evident in the case of large number of sensors and the requirement to
comply with optimum performance.
The thesis describes a comprehensive study of sensor selection for control
and fault tolerance with the particular application of an ElectroMagnetic
Levitation system (being an unstable, nonlinear, safety-critical system with
non-trivial control performance requirements). The particular aim of the
presented work is to identify effective sensor selection frameworks subject to
given system properties for controlling (with a level of fault tolerance) the
MagLev suspension system. A particular objective of the work is to identify
the minimum possible sensors that can be used to cover multiple sensor faults,
while maintaining optimum performance with the remaining sensors.
The tools employed combine modern control strategies and multiobjective
constraint optimisation (for tuning purposes) methods. An important part
of the work is the design and construction of a 25kg MagLev suspension
to be used for experimental verification of the proposed sensor selection
frameworks
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Robust stabilisation of multivariable systems: A super-optimisation approach
The work aims to derive extended robust stability results for the case of unstructured uncertainty models of multivariable systems. More specifically, throughout the thesis, additive and coprime unstructured perturbation models are considered. Refined robust stabilisation problems of MIMO systems are defined and maximally robust controllers are synthesised in a state-space form.
Unstructured perturbations which destabilise the feedback system for every optimal (maximally robust) controller are identified on the boundary of the optimal ball, i.e. the set of all admissible perturbations with norm equal to the maximum robust stability radius. Boundary perturbations are termed "uniformly destabilising" if they destabilise the closed-loop system for every optimal controller and it is shown that they all share a common characteristic, i.e. a projection of magnitude equal to the maximal robust stability radius, along a fixed direction defined by a pair of maximising vectors (scaled Schmidt pair) of a Hankel operator related to the problem. By imposing a directionality constraint it is shown that it is possible to increase the robust stability radius in every other direction by a subset of all optimal controllers.
In order to solve this problem, super-optimisation techniques are developed. Independently a natural extension of Hankel norm approximations, the so-called super optimisation problem is posed and solved explicitly for the case of one-block problems in a state-space setting. It is thus shown that a subset of all maximally robust controllers, namely the class of super-optimal controllers, stabilises all perturbed plants within an extended stability radius 11,*(b), subject to a directionality constraint.
In addition, the work is related to robust stabilisation subject to structured perturbations. The notions of structured robust stabilisation problem, and structured set approximation are defined in connection with the maximised set of permissible perturbations. It is further shown that ”*(J) can serve as an upper bound the structured robust stabilisation problem.
The effect of ”*(J) as an upper bound depends on the compatibility between the two structures, the true structure and the artificial structure of the extended permissible set.
[Look inside the thesis' abstract for an exact version of formulas and equations
Consistent aeroelastic linearisation and reduced-order modelling in the dynamics of manoeuvring flexible aircraft
This work proposes a novel reduced-order modelling approach in time domain for the coupled flight dynamics and aeroelastic response of manoeuvring very flexible aircraft. The
starting point is the coupling of a displacement-based, geometrically-nonlinear flexible-body
dynamics formulation with a 3-D unsteady vortex-lattice method. This is followed
by a linearisation of the structural degrees of freedom, which are assumed to be small in
a body- fixed reference frame. The translations and rotations of that reference frame and
their time derivatives, which describe the vehicle flight dynamics, can be arbitrarily large.
As a result, all couplings between the rigid and elastic motions are introduced without
the a priori assumptions of the mean axes approximation, traditionally used to decouple
the equations in flexible-aircraft dynamics. The resulting system can be projected onto
a few vibration modes of the unconstrained aircraft with geometrically-nonlinear static deflections at a trim condition. Equally, the unsteady aerodynamics are approximated
on a fixed lattice defined by the deformed static geometry. The resulting high-order
aerodynamic system, which defines the mapping between the small number of generalised
coordinates and unsteady aerodynamic loads, is then reduced through balanced truncation.
This unified description of the flexible aircraft dynamics provides a hierarchy of aeroelastic
model fidelities, which will be illustrated on a representative high-altitude, long-endurance
aircraft to identify the importance of geometrically-nonlinear wing deformations on the
vehicle dynamics. Application of the reduced-order modelling approach further shows a
very substantial reduction in model size that leads to model orders (and computational
cost) similar to those in conventional frequency-based methods but with higher modelling fidelity to compute manoeuvre loads. Closed-loop results for the Goland wing finally demonstrate the application of this approach in the synthesis of a robust flutter suppression
controller.Open Acces
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