Trading the stability of finite zeros for global stabilization of nonlinear cascade systems

This note analyzes the stabilizability properties of nonlinear cascades in which a nonminimum phase linear system is interconnected through its output to a Stable nonlinear system. It is shown that the instability of the zeros of the linear System can be traded with the stability of the nonlinear system up to a limit fixed by the growth properties of the cascade interconnection term. Below this limit, global stabilization is achieved by smooth static-state feedback. Beyond this limit, various examples illustrate that controllability of the cascade may be lost, making it impossible to achieve large regions of attractions.

Trading the stability of finite zeros for global stabilization of nonlinear cascade systems

This paper analyzes the stabilizability properties of nonlinear cascades in which a nonminimum phase linear system is interconnected through its output to a stable nonlinear system. It is shown that the instability of the zeros of the linear system can be traded with the stability of the nonlinear system up to a limit fixed by the growth properties of the cascade interconnection term. Below this limit, global stabilization is achieved by smooth static state feedback. Beyond this limit, various examples illustrate that controllability of the cascade may be lost, making it impossible to achieve large regions of attractions.

Some remarks about periodic feedback stabilization

Periodic feedback stabilization is a very natural solution to overcome the topological obstructions which may occur when one tries to asymptotically (locally) stabilize a (locally) controllable nonlinear system around an equilibrium point. The object of this paper is to give a simple geometric interpretation of this fact, to show that one obtains a weakened form of those obstructions when periodic feedback is used, and to illustrate the success of periodic feedback stabilization on a representative system which contains a drift.

A general mass law for broadband energy harvesting

It is well known that the power absorbed by a linear oscillator when excited by white noise base acceleration depends only on the mass of the oscillator and the spectral density of the base motion. This places an upper bound on the energy that can be harvested from a linear oscillator under broadband excitation, regardless of the stiffness of the system or the damping factor. It is shown here that the same result applies to any multi-degree-of-freedom nonlinear system that is subjected to white noise base acceleration: for a given spectral density of base motion the total power absorbed is proportional to the total mass of the system. The only restriction to this result is that the internal forces are assumed to be a function of the instantaneous value of the state vector. The result is derived analytically by several different approaches, and numerical results are presented for an example two-degree-of-freedom-system with various combinations of linear and nonlinear damping and stiffness. © 2013 The Author.

Stable dynamic walking over uneven terrain

We propose a constructive control design for stabilization of non-periodic trajectories of underactuated robots. An important example of such a system is an underactuated "dynamic walking" biped robot traversing rough or uneven terrain. The stabilization problem is inherently challenging due to the nonlinearity, open-loop instability, hybrid (impact) dynamics, and target motions which are not known in advance. The proposed technique is to compute a transverse linearization about the desired motion: a linear impulsive system which locally represents "transversal" dynamics about a target trajectory. This system is then exponentially stabilized using a modified receding-horizon control design, providing exponential orbital stability of the target trajectory of the original nonlinear system. The proposed method is experimentally verified using a compass-gait walker: a two-degree-of-freedom biped with hip actuation but pointed stilt-like feet. The technique is, however, very general and can be applied to a wide variety of hybrid nonlinear systems. © The Author(s) 2011.

An extended computational model of the circulatory system for designing ventricular assist devices.

An extended computational model of the circulatory system has been developed to predict blood flow in the presence of ventricular assist devices (VADs). A novel VAD, placed in the descending aorta, intended to offload the left ventricle (LV) and augment renal perfusion is being studied. For this application, a better understanding of the global hemodynamic response of the VAD, in essence an electrically driven pump, and the cardiovascular system is necessary. To meet this need, a model has been established as a nonlinear, lumped-parameter electrical analog, and simulated results under different states [healthy, congestive heart failure (CHF), and postinsertion of VAD] are presented. The systemic circulation is separated into five compartments and the descending aorta is composed of three components to accurately yield the system response of each section before and after the insertion of the VAD. Delays in valve closing time and blood inertia in the aorta were introduced to deliver a more realistic model. Pump governing equations and optimization are based on fundamental theories of turbomachines and can serve as a practical initial design point for rotary blood pumps. The model's results closely mimic established parameters for the circulatory system and confirm the feasibility of the intra-aortic VAD concept. This computational model can be linked with models of the pump motor to provide a valuable tool for innovative VAD design.

EBSD characterisation of Y2Ba4CuUOx phase in melttextured YBCO with addition of depleted uranium oxide

Melt-textured YBCO samples processed with added Y2O3 and depleted uranium oxide (DU) contain nano-particles, which have been identified previously as Y2Ba4CuUOx (U-411). This phase has a cubic unit cell, which is clearly distinct from the orthorhombic Y-123 and Y-211 phases within the YBCO system. In samples with a high amount of DU addition (0.8 wt-% DU), U-2411 particles have sizes between 200 nm and several νm, so identification of the Kikuchi patterns of this phase becomes possible. Together with a parallel EDX analysis, the particles embedded in the Y-123 matrix can be identified unambiguously. In this way, a three-phase EBSD scan becomes possible, allowing also the identification of nanometre-sized particles in the sample microstructure. © 2006 IOP Publishing Ltd.

Characterisation and fluidisation of synthetic pit latrine sludge

Half of the world’s urban population will live in informal settlements or “slums” by 2030. Affordable urban sanitation presents a unique set of challenges as the lack of space and resources to construct new latrines makes the de-sludging of existing pits necessary and is something that is currently done manually with significant associated health risks. Therefore various mechanised technologies have been developed to facilitate pit emptying, with the majority using a vacuum system to remove material from the top of the pit. However, this results in the gradual accumulation of unpumpable sludge in the pit, which eventually fills the latrine and forces it to be abandoned. This study has developed a method for fluidising unpumpable pit latrine sludge, based on laboratory experiments using a harmless synthetic sludge. Such a sludge consisting of clay and compost was developed to replicate the physical characteristics of pit latrine sludges characterised in Botswana during the 1980s. Undrained shear strength and density are identified as the critical parameters in controlling pumpability and a method of sludge characterisation based on these parameters is reported. In a series of fluidisation tests using a one fifth scale pit emptying device the reduction in sludge shear strength was found to be caused by i) dilution, which increases water content, and ii) remoulding, which involves mechanical agitation to break down the structure of the material. The tests demonstrated that even the strongest of sludges could be rendered “pumpable” by sufficient dilution. Additionally, air injection alone produced a three-fold decrease in strength of consolidated samples as a result of remoulding at constant water content. The implications for sludge treatment and disposal are discussed, and the classification of sludges according to the equipment required to remove them from the latrine is proposed. Possible field tests to estimate sludge density and shear strength are suggested. The feasibility of using low cost vacuum cleaners to replace expensive vane pumps is demonstrated. This offers great potential for the development of affordable pit emptying technologies that can remove significantly stronger sludges than current devices through fluidising the wastes at the bottom of the pit before emptying

Modelling and control of nonlinear systems using Gaussian processes with partial model information

Gaussian processes are gaining increasing popularity among the control community, in particular for the modelling of discrete time state space systems. However, it has not been clear how to incorporate model information, in the form of known state relationships, when using a Gaussian process as a predictive model. An obvious example of known prior information is position and velocity related states. Incorporation of such information would be beneficial both computationally and for faster dynamics learning. This paper introduces a method of achieving this, yielding faster dynamics learning and a reduction in computational effort from O(Dn2) to O((D - F)n2) in the prediction stage for a system with D states, F known state relationships and n observations. The effectiveness of the method is demonstrated through its inclusion in the PILCO learning algorithm with application to the swing-up and balance of a torque-limited pendulum and the balancing of a robotic unicycle in simulation. © 2012 IEEE.

Nonlinear phenomena in thermoacoustics: A comparison between single-mode and multi-mode methods

Nonlinear analysis of thermoacoustic instability is essential for prediction of frequencies and amplitudes of limit cycles. In frequency domain analyses, a quasi-linear transfer function between acoustic velocity and heat release rate perturbations, called the flame describing function (FDF), is obtained from a flame model or experiments. The FDF is a function of the frequency and amplitude of velocity perturbations but only contains the heat release response at the forcing frequency. While the gain and phase of the FDF provide insight into the nonlinear dynamics of the system, the accuracy of its predictions remains to be verified for different types of nonlinearity. In time domain analyses, the governing equations of the fully coupled problem are solved to find the time evolution of the system. One method is to discretize the governing equations using a suitable basis, such as the natural acoustic modes of the system. The number of modes used in the discretization alters the accuracy of the solution. In our previous work we have shown that predictions using the FDF are almost exactly the same as those obtained from the time-domain using only one mode for the discretization. We call this the single-mode method. In this paper we compare results from the single-mode and multi-mode methods, applied to a thermoacoustic system of a premixed flame in a tube. For some cases, the results differ greatly in both amplitude as well as frequency content. This study shows that the contribution from higher and subharmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems. Hence multi-mode simulations are necessary, and the single-mode method or the FDF may be insufficient to capture some of the complex nonlinear behaviour in fhermoacoustics.

Characterisation and fluidisation of synthetic pit latrine sludge

Half of the world's urban population will live in informal settlements or ‘slums’ by 2030. Affordable urban sanitation presents a unique set of challenges as the lack of space and resources to construct new latrines makes the de-sludging of existing pits necessary and is something that is currently done manually with significant associated health risks. Various mechanised technologies have therefore been developed to facilitate pit emptying, with the majority using a vacuum system to remove material from the top of the pit. However, this results in the gradual accumulation of unpumpable sludge at the bottom of the pit, which eventually fills the latrine and forces it to be abandoned. This study has developed a method for fluidising unpumpable pit latrine sludge, based on laboratory experiments using a harmless synthetic sludge. The implications for sludge treatment and disposal are discussed, and the classification of sludges according to the equipment required to remove them from the latrine is proposed. Finally, further work is suggested, including the ongoing development of a device to physically characterise latrine sludge in-situ within the pit.

Shock transmission in a coupled beam system

This paper investigates the circumstances under which high peak acceleration can occur in the internal parts of a system when subjected to impulsive driving on the outside. Motivating examples include the design of packaging for transportation of fragile items. The system is modelled in an idealised form using two beams coupled with point connections. A Rayleigh-Ritz model of such coupled beams was validated against measurements on a particular beam system, then the model was used to explore the acceleration response to impulsive driving in the time, frequency and spatial domains. This study is restricted to linear vibration response and additional mechanisms for high internal acceleration due to nonlinear effects such as internal impacts are not considered. Using Monte Carlo simulation in which the indirectly driven beam was perturbed by randomly placed point masses a wide range of system behaviour was explored. This facilitates identification of vulnerable configurations that can lead to high internal acceleration. The results from the study indicate the possibility of curve veering influencing the peak acceleration amplification. The possibility of veering within an ensemble was found to be dependent on the relative coupling strength of the modes. Understanding of the mechanism may help to avoid vulnerable cases, either by design or by preparatory vibration testing. © 2013 Elsevier Ltd.