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Modal array signal processing, as treated in this book, comprises waveform estimation tasks, i.e. beamforming, as well as parameter estimation tasks, i.e. source localization and the estimation of the number of active sources. Modal array signal processing has important applications in the areas of teleconferencing, surveillance, and seamless acoustic human-machine interfaces. [ABSTRACT FROM AUTHOR]

In this chapter, the concepts of classical acoustics and wavefield decomposition as derived in Chapters 2 and 3 on the one hand, and classical array signal processing, as discussed in Chapter 4, on the other hand, are combined to yield novel solutions for sensor array signal processing tasks. As before, the signal processing tasks considered are beamforming, i.e. waveform estimation (WE), and the localization of possibly multiple acoustic sources, i.e. parameter estimation (PE). In addition, the problem of how to determine the number of active sources in a wavefield is addressed. One of the central observations to be made here, which makes the combination of the two paradigms possible, is that the individual circular or spherical harmonics resulting from the wavefield decomposition step can be regarded as individual sensors in the classical sensor array processing framework. In the following, the circular and spherical harmonics are jointly denoted as eigenbeams, a concept first applied to acoustic signal processing by Elko et al. in [EKM03] and [ME04]. [ABSTRACT FROM AUTHOR]

In this chapter, the path of classical acoustics, illuminated by the previous two chapters, is abandoned for introducing the concepts of the second pillar necessary for understanding the fruitful combination of classical acoustics with classical array signal processing to be presented in Chapter 5. Array signal processing is the science of signal processing where more than one sensor is used to obtain an additional dimension of freedom for the solution of signal processing tasks. In most cases, this additional dimension is the spatial dimension. Since this book deals with acoustic wavefields propagating in air, the sensors utilized are microphones. However, most techniques developed here are also applicable to array signal processing in the fields of radar (antennas), sonar (hydrophones), and seismic imaging (accelerometers). Indeed, most of the techniques presented in this chapter have been originally developed in the radar and sonar community. As a tribute to this development, the more general term ‘sensor' is used in most of the following discussions. [ABSTRACT FROM AUTHOR]

This chapter deals with wavefield decomposition making extensive use of the theoretical foundations of acoustic wavefields laid out in Chapter 2. Wavefield decomposition (WFD), as treated in this book, is a technique that decomposes a wavefield into spatially orthogonal eigen-solutions of the acoustic wave equation in a coordinate system that best suits the geometry of the aperture under consideration. Here, apertures are spatially distributed observation devices for analyzing the characteristics of the spatio-temporal nature exhibited by wavefields. This aperture may be a continuous processor in threedimensional space. Alternatively, this continuous aperture may be sampled by discrete points in space. The resulting processor is called an array. [ABSTRACT FROM AUTHOR]

This chapter describes a practical real-time capable system designed to verify the claim developed throughout the previous chapters that acoustic scene analysis tasks can be tackled by wavefield decomposition methods using microphone arrays. In this chapter and in Appendix F, it will be shown that a circular microphone array mounted into a rigid cylindrical baffle is capable of performing waveform estimation and parameter estimation in the sense of Chapter 5 in real acoustic environments. The remainder of this chapter is organized as follows. Section 6.1 discusses hardware, software, and algorithmic details utilized throughout the evaluation process. The measurements to be presented in Section 6.2 were performed in a real room allowing for varying degrees of reverberation. [ABSTRACT FROM AUTHOR]

This chapter is concerned with acoustic radiators and scatterers of cylindrical and spherical shape which forms the basis for the discussion of wavefield decomposition to be detailed in Chapter 3. Acoustic wavefields in cylindrical and spherical coordinate systems are considered that travel through a fluid, most notably air, and that interact with objects and structures resulting in scattering phenomena. In order to arrive at tractable mathematical descriptions of problems related to acoustic radiation and scattering a few basic assumptions are introduced here [JF93]: 1.All fluids and materials considered are assumed to obey linear equations. This restriction limits the following discussions to small-signal disturbances of the transmitting medium.2.The media the acoustic wavefields interact with are assumed to be homogeneous.3.Steady-state conditions are assumed which means that initial transient effects can be neglected.4.For scattering problems, only local interactions between a wavefield propagating in a homogeneous medium and bodies within this medium are assumed. It should be noted that all of the above assumptions are reasonably standard in many textbooks on linear acoustics, such as [MI68], [JF93], [Wil99] and [Bla00]. [ABSTRACT FROM AUTHOR]

The term ‘acoustic scene analysis' (ASA) describes the task of extracting information contained in an acoustic wavefield, such as the waveform itself or parameter describing the source of the wavefield. Since acoustic wavefields are processes spread-out in space and time it follows quite naturally that ASA is predominantly performed by evaluating the signals captured by a number of spatially distinct microphones, i.e. microphone arrays. A standard and widely applied vehicle for evaluating the microphone array signals is built upon classical array signal processing techniques [JD93, Tre02]. In this context, the term ‘classical' is used to denote signal processing algorithms, to be introduced below, that are applied directly to the individual microphones comprising the array. In contrast, the algorithms to be derived in this book are applied to signals that are obtained by transforming the microphone signals into a domain defined by the eigen-solutions of the acoustic wave equation in two- and three spatial dimensions. [ABSTRACT FROM AUTHOR]

In human walking, the swing leg moves backward just prior to ground contact, i.e. the relative angle between the thighs is decreasing. We hypothesize that this swing leg retraction may have a positive effect on gait stability, because similar effects have been reported in passive dynamic walking models, in running models, and in robot juggling. For this study, we use a simple inverted pendulum model for the stance leg. The swing leg is assumed to accurately follow a time-based trajectory. The model walks down a shallow slope for energy input which is balanced by the impact losses at heel strike. With this model we show that a mild retraction speed indeed improves stability, while gaits without a retraction phase (the swing leg keeps moving forward) are consistently unstable. By walking with shorter steps or on a steeper slope, the range of stable retraction speeds increases, suggesting a better robustness. An optimization of the swing leg trajectory of a more realistic model also consistently comes up with a retraction phase, and indeed our prototype demonstrates a retraction phase as well. The conclusions of this paper are twofold; (1) use a mild swing leg retraction speed for better stability, and (2) walking faster is easier. [ABSTRACT FROM AUTHOR]

Walking, running or jumping are special cases of articulated motions which rely heavily on contact forces for their accomplishment. This central role of the contact forces is widely recognized now, but it is rarely connected to the structure of the dynamics of articulated motion. Indeed, this dynamics is generally considered as a complex nonlinear black-box without any specific structure, or its structure is only partly uncovered. We propose here to precise this structure and show in details how it shapes the movements that an articulated system might realize. Some propositions are made then to improve the design of control laws for walking, running, jumping or free-floating motions. [ABSTRACT FROM AUTHOR]

Mechanical properties of complex biological systems are non-linear, e.g. the force-velocity-length relation of muscles, activation dynamics, and the geometric arrangement of antagonistic pair of muscles. The control of such systems is a highly demanding task. Therefore, the question arises whether these mechanical properties of a muscle-skeletal system itself are able to support or guarantee for the stability of a desired movement, indicating self-stability. Self-stability of single joint biological systems were studied based on eigenvalues of the equation of motions and the basins of attraction were analysed using Lyapunov functions. In general, we found selfstability in single muscle contractions (e.g. frog, rat, cui), in human arm and leg movements, the human spine and even in the co-ordination of complex movements such as tennis or basketball. It seems that self-stability may be a general design criterion not only for the mechanical properties of biological systems but also for motor control. [ABSTRACT FROM AUTHOR]

Stability of biped walking is an important characteristic of legged locomotion. Whereas clinical investigations often relate increased variability to decreased stability, there are only few studies examining stability aspects directly. On the other hand, various papers from the field of robotics are dedicated to the question: how can the stability of legged locomotor systems be quantified? Particularly, when it comes to realizing fast motions in robots, the question of maintaining dynamic stability is of utmost importance. The current paper presents a theoretical comparison of several measures for dynamic stability — namely Floquet multipliers and Local Divergence Exponents (LDE). The sensitivity of these parameters to changes in speed of human treadmill locomotion is investigated. Experimental results show that two different types of stability with respect to speed dependence seem to exist. Short term LDE and Floquet multipliers consider the stability over a period of one stride, which seems to be optimal at intermediate walking speeds. Long term LDE quantify stability of movement trajectories over multiple strides. This type of stability decreases with speed and may be one reason for changing gaits from walking to running at a certain speed value. [ABSTRACT FROM AUTHOR]

It has long been the dream to build robots which could walk and run with ease. To date, the stance phase of walking robots has been characterized by the use of either straight, rigid legs, as is the case of passive walkers, or by the use of articulated, kinematically-driven legs. In contrast, the design of most hopping or running robots is based on compliant legs which exhibit quite natural behavior during locomotion. [ABSTRACT FROM AUTHOR]

The spring loaded inverted pendulum model (SLIP) has been shown to accurately model sagittal plane locomotion for a variety of legged animals. Tuned appropriately, the model exhibits passively stable periodic gaits using either fixed leg touchdown angle or swing-leg retraction protocols. In this work, we investigate the relevance of the model in insect locomotion and develop a simple feedback control law to enlarge the basin of stability and produce stable periodic gaits for both the point mass and rigid body models. Control is applied once per stance phase through appropriate choice of the leg touchdown angle. The control law is unique in that stabilization is achieved solely through direct observation of the leg angle and body orientation, rather than through feedback of system positions, velocities, and orientation. [ABSTRACT FROM AUTHOR]

The paper considers the use of sum of squares techniques in nonlinear model predictive control. To be more precise, sum of squares techniques are used to solve at each sampling instant a finite horizon optimal control problem which arises in nonlinear model predictive control for discrete time polynomial systems. The combination of nonlinear model predictive control and sum of squares techniques is motivated by the successful application of semidefinite programming in linear model predictive control. The advantages and disadvantages of applying sum of squares techniques to nonlinear model predictive control are illustrated on a small example. [ABSTRACT FROM AUTHOR]

We present velocity-based stability margins for fast bipedal walking that are sufficient conditions for stability, allow comparison between different walking algorithms, are measurable and computable, and are meaningful. While not completely necessary conditions, they are tighter necessary conditions than several previously proposed stability margins. The stability margins we present take into consideration a biped's Center of Mass position and velocity, the reachable region of its swing leg, the time required to swing its swing leg, and the amount of internal angular momentum available for capturing balance. They predict the opportunity for the biped to place its swing leg in such a way that it can continue walking without falling down. We present methods for estimating these stability margins by using simple models of walking such as an inverted pendulum model and the Linear Inverted Pendulum model. We show that by considering the Center of Mass location with respect to the Center of Pressure on the foot, these estimates are easily computable. Finally, we show through simulation experiments on a 12 degree-of-freedom distributed-mass lower-body biped that these estimates are useful for analyzing and controlling bipedal walking. [ABSTRACT FROM AUTHOR]

This paper develops a class of bipedal running controllers based on the hybrid zero dynamics (HZD) framework and discusses related experiments conducted in September 2004 in Grenoble, France. In these experiments, RABBIT, a five-link, four-actuator, planar bipedal robot, executed six consecutive running steps. The observed gait was remarkably human-like, having long stride lengths (approx. 50 cm or 36% of body length), flight phases of significant duration (approx. 100 ms or 25% of step duration), an upright posture, and an average forward rate of 0.6 m/s. A video is available at [7, 17]. In the time allotted for experiments, stability of the gait could not be validated. To put the results into context, background information on hybrid robot modeling, control philosophy, and gait optimization techniques accompany final experimental observations. An additional discussion about some unmodeled dynamic and geometric effects that contributed to implementation difficulties is given. [ABSTRACT FROM AUTHOR]

For fast motions in biomechanics and robotics, stability and robustness against perturbations are critical issues. The faster a motion the more important it is to exploit the system's natural stability properties for control. The stability of a periodic motion can be measured in terms of the spectral radius of the monodromy matrix. We optimize this stability criterion for a given robot topology, using special purpose optimization methods and leaving the model parameters, actuator inputs, trajectory start values and cycle time free to be determined by the optimization. This approach allows us to create simulations of robots that can move stably without any feedback. In order to analyze the robustness of a resulting periodic motion, we propose two methods, the first of which relies on forward simulations using perturbed start data and parameters while the second is based on the pseudospectra of the matrix. As a new example for a fast open-loop stable motion that has been produced by stability optimization, we present a biped gymnastics robot performing repetitive flip-flops (i.e. back handsprings). A similar model has previously been shown capable of performing open-loop stable running motions and repetitive somersaults. [ABSTRACT FROM AUTHOR]

This paper deals with a methodology to design optimal reference trajectories for walking gaits. This methodology consists of two steps: (i) design a parameterized family of motions, and (ii) determine the optimal parameters giving the motion that minimizes a criterion and satisfies some constraints within this family. This approach is applied to a five link biped, the prototype Rabbit. It has point feet and four actuators which are located in each knee and haunch. Rabbit is underactuated in single support since it has no actuated feet and is overactuated in double support. To take into account this under-actuation, a characteristic of the family of motions considered is that the four actuated joints are prescribed as polynomials in function of the absolute orientation of the stance ankle. There is no impact. The chosen criterion is the integral of the square of torques. Different technological and physical constraints are taken into account to obtain a walking motion. Optimal process is solved considering an order of treatment of constraints, according to their importance on the feasibility of the walking gait. Numerical simulations of walking gaits are presented to illustrate this methodology. [ABSTRACT FROM AUTHOR]

In the next few years considerable effort will be expended to make humanoid robots that can do true dynamic walking, or even running. One may numerically compute a desired gait, e.g. one that has been optimized to be asymptotically stable without feedback. One would normally give the gait as commands to the controllers for the robot joints. However, control system outputs generally differ from the command given, and the faster the command changes with time, the more deviation there is. Iterative learning control (ILC) and repetitive control (RC) aim to fix this problem in situations where a command is repeating or periodic. Since gaits are periodic motions, it is natural to ask whether ILC/RC can be of use in implementing gaits in hardware. These control concepts are no substitutes for feedback control but work in combination with them by adjusting the commands to the feedback controllers from a higher level perspective. It is shown that the gait problem does not precisely fit either the ILC or the RC problem statements. Gait problems are necessarily divided into phases defined by foot strike times, and furthermore the durations of the phases are not the same from cycle to cycle during the learning process. Several methods are suggested to address these issues, and four repetitive control laws are studied numerically. The laws that include both position and velocity error in the updates are seen to be the most effective. It appears that with appropriate refinement, such generalized RC laws could be very helpful in getting hardware to execute desired gaits. [ABSTRACT FROM AUTHOR]

Dynamic walking with two-legged robots is still an unsolved problem of todays robotics research. Beside finding mathematical models for the walking process, suitable mechanical designs and control methods must be found. This paper presents concepts for the latter two points. As biological walking makes use of the elastic properties of e.g. tendons and muscles, a joint design using a pneumatic rotational spring with adjustable stiffness is proposed. Equations to model the spring's dynamics as well as the supporting sensor systems and electronics are presented. For controlling the robot a behaviour-based approach is suggested. [ABSTRACT FROM AUTHOR]

The Lateral Leg Spring model (LLS) was developed by Schmitt and Holmes to model the horizontal-plane dynamics of a running cockroach. The model captures several salient features of real insect locomotion, and demonstrates that horizontal plane locomotion can be passively stabilized by a well-tuned mechanical system, thus requiring minimal neural reflexes. We propose two enhancements to the LLS model. First, we derive the dynamical equations for a more flexible placement of the center of pressure (COP), which enables the model to capture the phase relationship between the body orientation and center-of-mass (COM) heading in a simpler manner than previously possible. Second, we propose a reduced LLS "plant model" and biologically inspired control law that enables the model to follow along a virtual wall, much like antenna-based wall following in cockroaches. [ABSTRACT FROM AUTHOR]

Movements of humans are achieved by muscle contractions. Humans are able to perform coordinated movements even in the presence of perturbations from the environment or of the muscles themselves. But which properties of the muscles and the geometry of the joints are responsible for the stability? Does the stability depend on the joint angle? How large are the perturbations, the muscle-skeletal system can cope with before reflexes or controls by the brain are necessary? To answer these questions, we will derive a mathematical model of the muscle-skeletal system without reflexes. We present different mathematical methods to analyze these systems with respect to the stability of movements and thus provide the mathematical tools to answer the above questions. This paper is a companion paper to [13] where the biological applications of the mathematical methods presented in this paper are discussed in more detail. [ABSTRACT FROM AUTHOR]

The objective of this study is to analyze the stability of two control strategies for a planar biped robot. The unexpected rotation of the supporting foot is avoided via the control of the center of pressure or CoP. For the simultaneous control of the joints and of the CoP, the system is under-actuated in the sense that the number of inputs is less than the number of outputs. Thus a control strategy developed for planar robot without actuated ankles can be used in this context. The control law is defined in such a way that only the geometric evolution of the biped configuration is controlled, but not the temporal evolution. The temporal evolution during the geometric tracking is completely defined and can be analyzed through the study of a model with one degree of freedom. Simple conditions, which guarantee the existence of a cyclic motion and the convergence toward this motion, are deduced. These results are illustrated with some simulation results. In the first control strategy, the position of the CoP is tracked precisely, in the second one, only the limits on the CoP position are used to speed-up the convergence to the cyclic motion. [ABSTRACT FROM AUTHOR]

This paper first introduces a multi-locomotion robot with high mobility and then proposes Passive Dynamic Autonomous Control (PDAC) for the comprehensive control method of multiple types of locomotion. PDAC is the method to take advantage of the robot inherent dynamics and to realize natural dynamic motion. We apply PDAC to a biped walk control. On the assumption that the sagittal and lateral motion can be separated and controlled individually, each motion is designed based on the given desired step-length and period. In order to stabilize walking, the landing position control according to the status is designed. In addition, a coupling method between these motions, which makes the period of each motion identical, is proposed. Finally, we show that the multi-locomotion robot realizes the 3-dimensional dynamic walking using the PDAC control. [ABSTRACT FROM AUTHOR]

In this overview paper, we first survey numerical approaches to solve nonlinear optimal control problems, and second, we present our most recent algorithmic developments for real-time optimization in nonlinear model predictive control. [ABSTRACT FROM AUTHOR]

A dynamic model of running-the spring-loaded inverted pendulum (SLIP)-has proven effective in describing the force patterns found in a wide variety of animals and in designing and constructing a number of terrestrial running robots. Climbing or vertical locomotion has, on the other hand, lacked such a simple and powerful model. Climbing robots to date have all been quasi-static in their operation. This paper introduces a one degree of freedom model of a climbing robot used to investigate the power constraints involved with climbing in a dynamic manner. Particular attention is paid to understanding how springs and body dynamics can be exploited to help relieve a limited power/weight ratio and achieve dynamic running and climbing. [ABSTRACT FROM AUTHOR]

An important technique for computing motions for robot systems is to conduct a numerical search for a trajectory that minimizes a physical criteria like energy, control effort, jerk, or time. In this paper, we provide example solutions of these types of optimal control problems, and develop a framework to solve these problems reliably. Our approach uses an efficient solver for both inverse and forward dynamics along with the sensitivity of these quantities used to compute gradients, and a reliable optimal control solver. We give an overview of our algorithms for these elements in this paper. The optimal control solver has been the primary focus of our recent work. This algorithm creates optimal motions in a numerically stable and efficient manner. Similar to sequential quadratic programming for solving finite-dimensional optimization problems, our approach solves the infinite-dimensional problem using a sequence of linear-quadratic optimal control subproblems. Each subproblem is solved efficiently and reliably using the Riccati differential equation. [ABSTRACT FROM AUTHOR]

In this paper, a constrained nonlinear predictive control scheme is proposed for a class of under-actuated nonholonomic systems. The scheme is based on fast generation of steering trajectories that inherently fulfill the contraints while showing a "translatability" property which is generally needed to derive stability results in receding-horizon schemes. The corresponding open-loop optimization problem can be solved very efficiently making possible a real-time implementation on fast systems (The resulting optimization problem is roughly scalar). The whole framework is shown to hold for the well known challenging problem of a snakeboard constrained stabilization. Illustrative simulations are proposed to assess the efficiency of the proposed solution under saturation constraints and model uncertainties. [ABSTRACT FROM AUTHOR]

In this chapter we deal with the discrete-time linear uncertain system in the form [ABSTRACT FROM AUTHOR]

In this section we consider an uncertain system in the form [ABSTRACT FROM AUTHOR]

As stated in Theorem 3.1, quadratic stability of system (3.1) guarantees exponential stability for all time behaviors of parameters which are of interest in the practise (see also Remark 3.2); in particular, exponential stability is guaranteed for discontinuous parameters which exhibit an unbounded rate of variation at the discontinuity points. [ABSTRACT FROM AUTHOR]

Most part of this chapter deals with the Lyapunov stability analysis of a linear system subject to parametric uncertainties as given by [ABSTRACT FROM AUTHOR]

In this chapter we consider the qualitative behavior of solutions of the system of linear differential equations [ABSTRACT FROM AUTHOR]

According to the celebrated paper [68], linear systems uncertainties can be divided into two big families: dynamical input-output uncertainties and state space uncertainties. [ABSTRACT FROM AUTHOR]

This chapter summarises the results from the previous three chapters, and shows how they can be used to analyse and solve the reconfiguration problem. Two new algorithms are presented here: a structural test for reconfigurability and an algorithm for reconfiguration. Since the structural test cannot detect parameter cancellation, only the second algorithm can definitively determine the solvability of the problem. The applicability of structural analysis to specific aspects of the reconfiguration problem has already been shown in studies by Izadi-Zamanabadi et al. [1998] and Lorentzen et al. [2003], but this chapter tries to present a complete approach. [ABSTRACT FROM AUTHOR]

It is possible to attribute properties to a system structure. These properties are called structural properties, and they hold for almost all systems of the studied structure. There are also strong structural properties which hold for every system of the given structure (without exception). [ABSTRACT FROM AUTHOR]

A two-degrees-of-freedom helicopter experiment (see Fig. 17.1) is used as a more complex reconfiguration example. As demonstrated by Lunze et al. [2003], this system lends itself for control reconfiguration after actuator faults. It consists of a main lever that can rotate in two dimensions around its centre. Strong rotors are positioned at both ends which can create an upward force by blowing air down. The speed of the rotors can be controlled, and they can be rotated around the axis of the lever to change the direction of the force. There is a second set of smaller rotors that can only blow air sidewards (tangentially). A sketch of the system is shown in Fig. 17.2. A similar system without the redundant lateral rotors is studied by López-Martínez and Rubio [2003]. [ABSTRACT FROM AUTHOR]

The 3-Tank experiment is shown in Fig. 16.1. It consists of three tanks, which are connected via the valves u2, u3 and u4. Pumps can bring water into the left and the right tank, but not into the middle tank. The control objective is to maintain a certain level in the middle tank, such that the outflow z is constant. The relevant components of the system are shown in Fig. 16.2. The system is very similar to the one defined in the original publication of the 3-Tank Benchmark Problem by Heiming and Lunze [1999]. However, because a real experiment is used here, the identified parameters have to be used. [ABSTRACT FROM AUTHOR]

This chapter develops a novel algorithm which solves the disturbance decoupling problem by repeatedly decoupling one output variable at a time. The central routine decouples a single output variable by assigning the necessary value to one control input variable, such that all influences on the chosen output variable are cancelled out. The important aspect is that the output variable is decoupled not only from the disturbance, but also from all other control input variables. Therefore, this routine can be applied repeatedly until the system is completely decoupled (or there are no control inputs left). [ABSTRACT FROM AUTHOR]

The strong reconfiguration problem leads to a disturbance decoupling problem as stated in Sect. 10.4. There are known structural tests for the solvability of this problem, which will be presented in this chapter. The next chapter will develop a novel solution inspired by the test presented here. In both cases, the structural approach to the disturbance decoupling problem has advantages over the classical approach. It is simple to implement, and it is well suited for the specific problems typical for reconfiguration. [ABSTRACT FROM AUTHOR]

The goal of structural modelling is to define all signal paths in a system.1 So for every possible signal path, the structural model has to determine whether a signal path is present or not. The structural model is, therefore, a collection of binary values. The parameters of the system (the amplification of the signal paths) are not stored in the structural model. There are two main types of structural models developed by different communities: the structural matrix approach and the graph approach. [ABSTRACT FROM AUTHOR]

This chapter addresses the strong reconfiguration goal for plants after actuator faults. The objective is to find a reconfigured block that makes the external output of the reconfigured control loop match the output of the nominal control loop. The approach builds on the virtual actuator, designed in Chap. 8 to stabilise a plant after actuator faults. In contrast to the reconfiguration problems treated earlier, this goal does not lead to an equivalent controller design problem. Instead, the problem can be transformed into a disturbance decoupling problem with stabilisation, which belongs to a different class of problems. Known approaches can be used to find a solution. [ABSTRACT FROM AUTHOR]

This chapter treats the weak reconfiguration problem: the output equilibrium of the plant (for the same input) has to be restored (see Sect. 5.3 for the introduction of this goal). Two approaches are presented, which both build on the solution from the previous chapter. [ABSTRACT FROM AUTHOR]

This chapter develops the concept of a virtual actuator. The idea of a virtual actuator is to use the input signal meant for the nominal process and to transform it into a signal useful for the remaining actuators of the faulty plant. As shown in Chap. 6, a statical reconfiguration block is generally not sufficient, and a dynamical reconfiguration block is necessary to solve this problem. The virtual actuator for the reconfiguration after actuator faults is the dual approach to the use of a virtual sensor after sensor faults (detailed in the Chap. 7 [ABSTRACT FROM AUTHOR]

This chapter develops the concept of a virtual sensor: when a sensor is at fault, an observer is used to calculate a replacement value. This approach goes back to the 1970s when it has been studied in the context of fault detection as "dedicated (and generalised) observer scheme" (see Schröder [2003] for a modern version of this idea). However, in the literature the observer has only been used in combination with state feedback, while a general dynamical controller will be considered here. It will be shown that the virtual sensor can be used as a reconfiguration block by translating the measurements from the faulty plant into the values the controller can handle. The stability of the reconfigured loop can be guaranteed as long as the faulty plant is detectable. The results found here are the basis for the derivation of the virtual actuator in the following chapter. [ABSTRACT FROM AUTHOR]

This chapter employs a static reconfiguration block in order to solve the direct reconfiguration problem. It is shown that this approach gives excellent results, if the solvability condition is met. However, this condition is very strong, and it cannot be assumed to be satisfied for typical reconfiguration problems. The pseudo-inverse method is discussed as an approach to find an approximation if an exact solution does not exist. [ABSTRACT FROM AUTHOR]

In this chapter, the general reconfiguration problem is specialised by applying it to linear systems. The advantage is that the problem can be broken down into smaller parts by making use of superposition. This permits the analysis of how typical fault locations interact with the reconfiguration goals defined in the previous chapter. [ABSTRACT FROM AUTHOR]

In this chapter, a formal definition of the reconfiguration problem is developed. While the general idea of reconfiguration may appear obvious, there are many different ways to formulate the problem (several common approaches are listed in Patton [1997]). The problem formulation developed here is special for two reasons. On the one hand, it is kept as simple as possible. Only the minimal number of signals is used, and the definition is kept strictly within the theory of continuous (linear or nonlinear) time-invariant systems. On the other hand, as few assumptions as possible are made during the problem formulation, so as to keep it as general as possible. [ABSTRACT FROM AUTHOR]

A simple system will be used as a running example throughout the manuscript. It serves to illustrate typical faults and the resulting reconfiguration approaches. The example is a reduced version of the popular Three-Tank Benchmark Problem first described by Heiming and Lunze [1999]. The system contains two tanks connected by a valve and filled by a pump. The goal is to maintain a constant outflow of the system, which requires a constant level in the right tank. Although the system is simple (see Fig. 3.1), it is sufficient to demonstrate the relevant effects encountered with respect to reconfiguration. [ABSTRACT FROM AUTHOR]

This chapter introduces the main literature relevant for the field of reconfi-guration. During the last decade, different communities have dealt with this topic, and unfortunately a lot of incompatible terminology has been used. Several different definitions are used for the term "reconfiguration", leading to approaches that cannot be reasonably compared to each other. This overview tries to mention all approaches that are relevant for control reconfiguration in the sense defined in the introduction. [ABSTRACT FROM AUTHOR]