Several control and supervision problems for hybrid systems are posed in terms of abstract information. If the reduction of themeasurement resolution leads to quantized signals, the problem to stabilize a continuous or hybrid systems by quantized feedback has to be solved. For process supervision with abstract design specifications, it is reasonable to reduce the complexity of analysis and design tasks by using abstract models that ignore the continuous movement of the hybrid systems. This chapter shows how abstract models like automata or embedded maps can be set up and used for the diagnosis and supervisory control of hybrid systems.

Quantization and model abstraction

The subject of this chapter is a class of hybrid systems that have to be analyzed and controlled in terms of their symbolic dynamics. Only the sequence of symbolic inputs and outputs associated with the hybrid state trajectory is accessible for measurement and, hence, provides the on-line information to be used in fault diagnosis and feedback control.

The way of dealing with abstract measurement and modeling information is explained by considering quantized systems that consist of a continuous-variable system whose input, state, and output are accessible only through quantizers. The hybrid character of such systems becomes obvious from the fact that internally the system behavior is described by the continuous state x(t) whereas the outside observer only sees the quantized version [x(t)] of the state and of the input and output, which jumps between discrete (symbolic) values.

The Need for Fluid Power

In applications for which large forces, torques, or both are required, often with a fast response time, it is inevitable that oil-hydraulic control systems will be called on. They may be used in environmentally difficult applications because the drive part can be designed with no electrical components, and often they are the only feasible means of obtaining the forces required, particularly for linear actuation. A particularly important feature is that they almost always have a more competitive power–weight ratio when compared with electrically actuated systems, and they are the inherent choice for mobile machines and plants. Fluid power systems also have the capability of being able to control several parameters, such as pressure, speed, and position, to a high degree of accuracy and at high power levels. The latest developments are now achieving position control to an accuracy expressed in micrometers and with high-water-content fluids. In practice, there are many exciting challenges facing the fluid power engineer, who now must preferably have skills in several of the following topics:

• Materials selection, water-based fluids, higher working pressures

• Fluid mechanics and thermodynamics studies

• Wear and lubrication

• The use of alternative fluids, given the environmental aspects of mineral oil, together with the extremely important issue of future supplies of mineral oil

• Energy efficiency

• Vibration and noise analysis

• Condition monitoring and fault diagnosis

• Component design, steady-state and dynamic

• Circuit design, steady-state and dynamic

• Machine design and its integrated hydraulics

• Sensor technologies

• Electrical–electromagnetic design

• Computer control techniques

• Signal processing and associated algorithms

• Modern control theory and artificial intelligence

Introduction to Basic Concepts, the Hydromechanical Actuator

The interconnection of components to form a closed-loop control system introduces new features, primarily the consideration of steady-state error and the speed of control. Steady-state error is a function of component design, such as the existence of a servovalve spool underlap, and the speed of control is a function of the way control is dynamically achieved combined with the dynamic characteristics of the system. It has been shown in Chapter 5 that fluid compressibility and load mass–inertia play the dominant part when characterizing system dynamics, and these aspects cannot be neglected when the dynamic stability of a closed-loop system is considered. Increasing system gain – for example, by increasing a servovalve servoamplifier current gain in a closed-loop servovalve–actuator position controller – will eventually lead to closed-loop instability. This will result in severe oscillations that could rapidly lead to component damage.

The hydromechanical actuator is one of the simplest forms of closed-loop control with applications reaching back to the very earliest days of industrial manufacturing using cast-iron components and low-pressure water as the working fluid. It incorporates a spool valve and an actuator – for example as shown in Fig. 6.1.

It will be seen that as the handle is moved to the right, the spool valve opens, allowing pressurized fluid to move the actuator in the same direction as the handle movement. In its basic form, it therefore acts in a manner similar to that of a servovalve–actuator system.

Introduction

These studies represent a variety of mathematical and simulation solutions for a range of components and systems and also include much experimental testing with some novel measurement techniques and practical limitations. They are intended to bring together the various aspects of fluid power theory introduced in earlier chapters, but in a more comprehensive manner usually required for more complex systems studies involving the integration of components and control concepts.

Performance of an Axial Piston Pump Tilted Slipper with Grooves

Introduction

This study was undertaken by Bergada, Haynes, and Watton with experimental work in the author's Fluid Power Laboratory at Cardiff University as part of a comprehensive study on losses within an axial piston pump. It was concerned with a new analytical method based on the Reynolds equation of lubrication, with experimental validation, to evaluate the leakage and pressure distribution for an axial piston pump slipper, taking into account the effect of grooves.

The analytical work was developed by JM Bergada (UPC, Terrassa, Spain) with experimental work undertaken by JM Bergada and JM Haynes. Additional CFD analysis and test-rig design was undertaken by JM Haynes and J Watton. Further CFD results by R Worthing and J Watton are also presented in this overview.

The equations consider slipper spin and tilt and are extended to be used for a slipper with any number of grooves.

This book is aimed at undergraduate students as a second-year and beyond entry stage to fluid power. There is much material that will also appeal to technicians regarding the background to fluid power and the operation of components and systems. Fluid power is often considered a specialist subject but should not be so given that the same would not be said for electrical power. In fact, there are many applications for which fluid power control is the only possibility because of force/torque/power/environmental demands. In the past 20 years, a number of groups around the world have made significant steps forward in both the understanding and the application of theory and control, complementing the R&D activity undertaken within the manufacturing industry. Details of just one organization involving many participating fluid power centers around the world are available at www.fluid.power.net. I embarked on this book ostensibly as a replacement for my first book, Fluid Power Systems – Modelling, Simulation, Analog and Microcomputer Control, published by Prentice-Hall in 1989 and now out of print. However, the result is a much different book and perhaps not surprising, given the developments in fluid power in the past 20 years. Following many constructive comments by undergraduate students, friends in industry, and academic friends who still use my first book for teaching, it was clear that a new book was needed.

Fluid Types

The preferred working fluid for most applications is mineral oil, although in some applications there is a requirement for a water-based or synthetic fluid, mainly for reasons of fire hazards and increasingly for environmental considerations. The drive toward nonmineral oil fluids has seen a renewed attitude to pure water hydraulics together with the emergence of biodegradable and vegetable-based fluids. Fire-resistant fluids in use fall under the following classifications:

1. HFA 5/95 oil-in-water emulsion, typically 5% oil and 95% water

2. HFB 60/40 water-in-oil emulsion, typically 60% oil and 40% water

3. HFC 60/40 water-in-glycol emulsion, typically 60% glycol and 40% water

4. HFD synthetic fluid containing no water

5. HFE synthetic biodegradable fluid

The use of water-based fluids has implications for component material selection – for example, the use of stainless steel, plastics, and ceramics. In addition, serious consideration of fluid properties must also be given, particularly viscosity, which can be very high at low temperatures in some cases. Fluids are being continually developed, and the following information is intended to reflect the general trend and is not considered as definitive because this would require an overview of many suppliers from many countries around the world – for example, see www.shell.com.

Type HFA 5/95 oil-in-water emulsions are fire-resistant emulsions that exhibit enhanced stability, lubrication, and antiwear characteristics and have the following important aspects:

• They have much improved stability toward variations in temperature, pressure, shear, and bacterial attack.

• The performance limitations become obvious for systems operating well above 70 bar, reliability and efficiency often being sacrificed where fire resistance is of paramount importance.

• […]

Introduction

The preceding chapters considered the steady-state behavior of common fluid power elements and systems. In reality, fluid power systems handle significant moving masses, and the combination of this with fluid compressibility results in system dynamics that usually cannot be neglected. In addition, individual components such as PRVs require a finite time to accommodate flow-rate changes. This also applies, for example, to a servovalve that again requires a finite time to change its spool position in response to a change in applied current. The combination of these issues means that the design of both open-loop and closed-loop control systems must take into account these dynamic issues. In particular, a closed-loop control system will almost certainly become unstable as system gains are increased because of such dynamic effects. Instability can lead to disastrous consequences if severe pressure oscillations occur. Instability in axial piston motor speed control systems, for example, can result in severe repetitive lifting and impact of the pistons on the swash plate.

Consider the design of a servoactuator that forms one of four to be used to provide the “road” input to the wheels of a vehicle sitting on a rig commonly called a “four-poster.” Figure 5.1 shows one of the servoactuators and a block diagram of the position control system.

Determining the dynamic performance of the position control system only is relatively straightforward once the important dynamic features have been identified.

The effects of roundoff noise in control and signal processing systems, and in numerical computation have been described in detail in the previous chapters.

There is another kind of quantization that takes place in these systems however, which has not yet been discussed, and that is coefficient quantization.

The coefficients of an equation being implemented by computer must be represented according to a given numerical scale. The representation is of course done with a finite number of bits. The same would be true for the coefficients of a digital filter or for the gains of a control system.

If a coefficient can be perfectly represented by the allowed number of bits, there would be no error in the system implementation. If the coefficient required more bits than the allowed word length, then the coefficient would need to be rounded to the nearest number on the allowed number scale. The rounding of the coefficient would result in a change in the implementation and would cause an error in the computed result. This error is distinct from and independent of quantization noise introduced by roundoff in computation. Its effect is bias–like, rather than the PQN nature of roundoff, studied previously.

If a numerical equation is being implemented or simulated by computer, quantization of the coefficients causes the implementation of a slightly different equation.

The purpose of this chapter is to provide an introduction to the basics of statistical analysis, to discuss the ideas of probability density function (PDF), characteristic function (CF), and moments. Our goal is to show how the characteristic function can be used to obtain the PDF and moments of functions of statistically related variables. This subject is useful for the study of quantization noise.

PROBABILITY DENSITY FUNCTION

Figure 3.1(a) shows an ensemble of random time functions, sampled at time instant t = t1 as indicated by the vertical dashed line. Each of the samples is quantized in amplitude. A “histogram” is shown in Fig. 3.1(b). This is a “bar graph” indicating the relative frequency of the samples falling within the given quantum box. Each bar can be constructed to have an area equal to the probability of the signal falling within the corresponding quantum box at time t = t1. The sum of the areas must total to 1. The ensemble should have an arbitrarily large number of member functions. As such, the probability will be equal to the ratio of the number of “hits” in the given quantum box divided by the number of samples. If the quantum box size is made smaller and smaller, in the limit the histogram becomes fx(x), the probability density function (PDF) of x, sketched in Fig. 3.1(c).