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

Speed Control Based on ESO for the Pitching Axis of Satellite Cameras

The pitching axis is the main axis of a satellite camera and is used to control the pitch posture of satellite cameras. A control strategy based on extended state observer (ESO) is designed to obtain a fast response speed and highly accurate pitching axis control system and eliminate disturbances during the adjustment of pitch posture. First, a sufficient condition of stabilization for ESO is obtained by analyzing the steady-state error of the system under step input. Parameter tuning and disturbance compensation are performed by ESO. Second, the ESO of speed loop is designed by the speed equation of the pitching axis of satellite cameras. The ESO parameters are obtained by pole assignment. By ESO, the original state variable observes the motor angular speed and the extended state variable observes the load torque. Therefore, the external load disturbances of the control system are estimated in real time. Finally, simulation experiments are performed for the system on the cases of nonload starting, adding external disturbances on the system suddenly, and the load of system changing suddenly. Simulation results show that the control strategy based on ESO has better stability, adaptability, and robustness than the PI control strategy

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 differential–algebraic equations (the Navier–Stokes 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 H∞ 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 Landahl’s theory of sheared turbulence. To illustrate the application of these techniques, an H∞ 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

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

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

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 µ*(J) can serve as an upper bound the structured robust stabilisation problem. The effect of µ*(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

Application of robust control in unmanned vehicle flight control system design

The robust loop-shaping control methodology is applied in the flight control system design of the Cranfield A3 Observer unmanned, unstable, catapult launched air vehicle. Detailed linear models for the full operational flight envelope of the air vehicle are developed. The nominal and worst-case models are determined using the v-gap metric. The effect of neglecting subsystems such as actuators and/or computation delays on modelling uncertainty is determined using the v-gap metric and shown to be significant. Detailed designs for the longitudinal, lateral, and the combined full dynamics TDF controllers were carried out. The Hanus command signal conditioning technique is also implemented to overcome actuator saturation and windup. The robust control system is then successfully evaluated in the high fidelity 6DOF non-linear simulation to assess its capability of launch stabilization in extreme cross-wind conditions, control effectiveness in climb, and navigation precision through the prescribed 3D flight path in level cruise. Robust performance and stability of the single-point non-scheduled control law is also demonstrated throughout the full operational flight envelope the air vehicle is capable of and for all flight phases and beyond, to severe launch conditions, such as 33knots crosswind and exaggerated CG shifts. The robust TDF control law is finally compared with the classical PMC law where the actual number of variables to be manipulated manually in the design process are shown to be much less, due to the scheduling process elimination, although the size of the final controller was much higher. The robust control law performance superiority is demonstrated in the non-linear simulation for the full flight envelope and in extreme flight conditions

Robust polynomial controller design

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The work presented in this thesis was motivated by the desire to establish an alternative approach to the design of robust polynomial controllers. The procedure of pole-placement forms the basis of the design and for polynomial systems this generally involves the solution of a diophantine equation. This equation has many possible solutions which leads directly to the idea of determining the most appropriate solution for improved performance robustness. A thorough review of many of the aspects of the diophantine equation is presented, which helps to gain an understanding of this extremely important equation. A basic investigation into selecting a more robust solution is carried out but it is shown that, in the polynomial framework, it is difficult to relate decisions in the design procedure to the effect on performance robustness. This leads to the approach of using a state space based design and transforming the resulting output feedback controller to polynomial form. The state space design is centred around parametric output feedback which explicitly represents a set of possible feedback controllers in terms of arbitrary free parameters. The aim is then to select these free parameters such that the closed-loop system has improved performance robustness. Two parametric methods are considered and compared, one being well established and the other a recently proposed scheme. Although the well established method performs slightly better for general systems it is shown to fail when applied to this type of problem. For performance robustness, the shape of the transient response in the presence of model uncertainty is of interest. It is well known that the eigenvalues and eigenvectors play an important role in determining the transient behaviour and as such the sensitivities of these factors to model uncertainty forms the basis on which the free parameters are selected. Numerical optimisation is used to select the free parameters such that the sensitivities are at a minimum. It is shown both in a simple example and in a more realistic application that a significant improvement in the transient behaviour in the presence of model uncertainty can be achieved using the proposed design procedure.UK Science and Engineering Research Counci