Coining (metalworking)

Coining is a form of precision stamping in which a workpiece is subjected to a sufficiently high stress to induce plastic flow on the surface of the material. A beneficial feature is that in some metals, the plastic flow reduces surface grain size, and work hardens the surface, while the material deeper in the part retains its toughness and ductility. The term comes from the initial use of the process: manufacturing of coins.
Coining is used to manufacture parts for all industries and is commonly used when high relief or very fine features are required. For example, it is used to produce coins, badges, buttons, precision-energy springs and precision parts with small or polished surface features.
Coining is a cold working process similar in other respects to forging, which takes place at elevated temperature; it uses a great deal of force to elastically deform a workpiece, so that it conforms to a die. Coining can be done using a gear driven press, a mechanical press, or more commonly, a hydraulically actuated press. Coining typically requires higher tonnage presses than stamping, because the workpiece is elastically deformed and not actually cut, as in some other forms of stamping. The coining process is preferred when there is a high tonnage.
Coining in electronic industry.
In soldering of electronic components, bumps are formed on bonding pads to enhance adhesion, which are further flattened by the coining process. Unlike typical coining applications, in this case the goal of coining is to create a flat, rather than patterned, surface.


Computer-aided manufacturing

Computer-aided manufacturing (CAM) also known as computer-aided modeling or computer-aided machining is the use of software to control machine tools in the manufacturing of work pieces. This is not the only definition for CAM, but it is the most common. It may also refer to the use of a computer to assist in all operations of a manufacturing plant, including planning, management, transportation and storage. Its primary purpose is to create a faster production process and components and tooling with more precise dimensions and material consistency, which in some cases, uses only the required amount of raw material (thus minimizing waste), while simultaneously reducing energy consumption.
CAM is now a system used in schools and lower educational purposes.
CAM is a subsequent computer-aided process after computer-aided design (CAD) and sometimes computer-aided engineering (CAE), as the model generated in CAD and verified in CAE can be input into CAM software, which then controls the machine tool. CAM is used in many schools alongside computer-aided design (CAD) to create objects.
Overview.
Traditionally, CAM has been numerical control (NC) programming tool, wherein two-dimensional (2-D) or three-dimensional (3-D) models of components are generated in CAD. As with other "computer-aided" technologies, CAM does not eliminate the need for skilled professionals such as manufacturing engineers, NC programmers, or machinists. CAM leverages both the value of the most skilled manufacturing professionals through advanced productivity tools, while building the skills of new professionals through visualization, simulation and optimization tools.
A CAM tool generally converts a model to a language the target machine in question understands, typically G-Code. The numerical control can be applied to machining tools, or more recently to 3D printers.
History.
Early commercial applications of CAM were in large companies in the automotive and aerospace industries; for example, Pierre Béziers work developing the CAD/CAM application UNISURF in the 1960s for car body design and tooling at Renault. Alexander Hammer at DeLaval Steam Turbine Company invented a technique to progressively drill turbine blades out of a solid metal block of metal with the drill controlled by a punch card reader in 1950.
Historically, CAM software was seen to have several shortcomings that necessitated an overly high level of involvement by skilled CNC machinists. Fallows created the first CAD software but this had severe shortcomings and was promptly taken back into the developing stage. CAM software would output code for the least capable machine, as each machine tool control added on to the standard G-code set for increased flexibility. In some cases, such as improperly set up CAM software or specific tools, the CNC machine required manual editing before the program will run properly. None of these issues were so insurmountable that a thoughtful engineer or skilled machine operator could not overcome for prototyping or small production runs; G-Code is a simple language. In high production or high precision shops, a different set of problems were encountered where an experienced CNC machinist must both hand-code programs and run CAM software.
The integration of CAD with other components of CAD/CAM/CAE Product lifecycle management (PLM) environment requires an effective CAD data exchange. Usually it had been necessary to force the CAD operator to export the data in one of the common data formats, such as IGES or STL or Parasolid formats that are supported by a wide variety of software.
The output from the CAM software is usually a simple text file of G-code/M-codes, sometimes many thousands of commands long, that is then transferred to a machine tool using a direct numerical control (DNC) program or in modern Controllers using a common USB Storage Device.
CAM packages could not, and still cannot, reason as a machinist can. They could not optimize toolpaths to the extent required of mass production. Users would select the type of tool, machining process and paths to be used. While an engineer may have a working knowledge of G-code programming, small optimization and wear issues compound over time. Mass-produced items that require machining are often initially created through casting or some other non-machine method. This enables hand-written, short, and highly optimized G-code that could not be produced in a CAM package.
At least in the United States, there is a shortage of young, skilled machinists entering the workforce able to perform at the extremes of manufacturing; high precision and mass production. As CAM software and machines become more complicated, the skills required of a machinist or machine operator advance to approach that of a computer programmer and engineer rather than eliminating the CNC machinist from the workforce.
Overcoming historical shortcomings.
Over time, the historical shortcomings of CAM are being attenuated, both by providers of niche solutions and by providers of high-end solutions. This is occurring primarily in three arenas:
Machining process.
Most machining progresses through many stages, each of which is implemented by a variety of basic and sophisticated strategies, depending on the part design, material, and software available.


Self-tuning

In control theory a self-tuning system is capable of optimizing its own internal running parameters in order to maximize or minimize the fulfilment of an objective function; typically the maximization of efficiency or error minimization.
Self-tuning and auto-tuning often refer to the same concept. Many software research groups consider auto-tuning the proper nomenclature.
Self-tuning systems typically exhibit non-linear adaptive control. Self-tuning systems have been a hallmark of the aerospace industry for decades, as this sort of feedback is necessary to generate optimal multi-variable control for non-linear processes. In the telecommunications industry, adaptive communications are often used to dynamically modify operational system parameters to maximize efficiency and robustness.
Examples.
Examples of self-tuning systems in computing include:
Performance benefits can be substantial. Professor Jack Dongarra, an American computer scientist, claims self-tuning boosts performance, often on the order of 300%.
Digital self-tuning controllers are an example of self-tuning systems at the hardware level.
Architecture.
Self-tuning systems are typically composed of four components: expectations, measurement, analysis, and actions. The expectations describe how the system should behave given exogenous conditions.
Measurements gather data about the conditions and behaviour. Analysis helps determine whether the expectations are being met- and which subsequent actions should be performed. Common actions are gathering more data and performing dynamic reconfiguration of the system.
Self-tuning (self-adapting) systems of automatic control are systems whereby adaptation to randomly changing conditions is performed by means of automatically changing parameters or via automatically determining their optimum configuration. In any non-self-tuning automatic control system there are parameters which have an influence on system stability and control quality and which can be tuned. If these parameters remain constant whilst operating conditions (such as input signals or different characteristics of controlled objects) are substantially varying, control can degrade or even become unstable. Manual tuning is often cumbersome and sometimes impossible. In such cases, not only is using self-tuning systems technically and economically worthwhile, but it could be the only means of robust control. Self-tuning systems can be with or without parameter determination.
In systems with parameter determination the required level of control quality is achieved by automatically searching for an optimum (in some sense) set of parameter values. Control quality is described by a generalised characteristic which is usually a complex and not completely known or stable function of the primary parameters. This characteristic is either measured directly or computed based on the primary parameter values. The parameters are then tentatively varied. An analysis of the control quality characteristic oscillations caused by the varying of the parameters makes it possible to figure out if the parameters have optimum values, i.e.. if those values deliver extreme (minimum or maximum) values of the control quality characteristic. If the characteristic values deviate from an extremum, the parameters need to be varied until optimum values are found. Self-tuning systems with parameter determination can reliably operate in environments characterised by wide variations of exogenous conditions.
In practice systems with parameter determination require considerable time to find an optimum tuning, i.e. time necessary for self-tuning in such systems is bounded from below. Self-tuning systems without parameter determination do not have this disadvantage. In such systems, some characteristic of control quality is used (e.g., the first time derivative of a controlled parameter). Automatic tuning makes sure that this characteristic is kept within given bounds. Different self-tuning systems without parameter determination exist that are based on controlling transitional processes, frequency characteristics, etc. All of those are examples of closed-circuit self-tuning systems, whereby parameters are automatically corrected every time the quality characteristic value falls outside the allowable bounds. In contrast, open-circuit self-tuning systems are systems with para-metrical compensation, whereby input signal itself is controlled and system parameters are changed according to a specified procedure. This type of self-tuning can be close to instantaneous. However, in order to realise such self-tuning one needs to control the environment in which the system operates and a good enough understanding of how the environment influences the controlled system is required.
In practice self-tuning is done through the use of specialised hardware or adaptive software algorithms. Giving software the ability to self-tune (adapt):


Iteratively reweighted least squares

The method of iteratively reweighted least squares (IRLS) is used to solve certain optimization problems with objective functions of the form of a "p"-norm:
formula_1
by an iterative method in which each step involves solving a weighted least squares problem of the form:
formula_2
IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression to find an M-estimator, as a way of mitigating the influence of outliers in an otherwise normally-distributed data set, for example, by minimizing the least absolute errors rather than the least square errors.
One of the advantages of IRLS over linear programming and convex programming is that it can be used with Gauss–Newton and Levenberg–Marquardt numerical algorithms.
Examples.
"L"1 minimization for sparse recovery.
IRLS can be used for "ℓ"1 minimization and smoothed "ℓ"p minimization, "p" compressed sensing problems. It has been proved that the algorithm has a linear rate of convergence for "ℓ"1 norm and superlinear for "ℓ""t" with "t" restricted isometry property, which is generally a sufficient condition for sparse solutions. However, in most practical situations, the restricted isometry property is not satisfied.
"Lp" norm linear regression.
To find the parameters β = ("β"1, …,"β""k")T which minimize the "Lp" norm for the linear regression problem,
formula_3
the IRLS algorithm at step "t" + 1 involves solving the weighted linear least squares problem:
formula_4
where "W"("t") is the diagonal matrix of weights, usually with all elements set initially to:
formula_5
and updated after each iteration to:
formula_6
In the case "p" = 1, this corresponds to least absolute deviation regression (in this case, the problem would be better approached by use of linear programming methods, so the result would be exact) and the formula is:
formula_7
To avoid dividing by zero, regularization must be done, so in practice the formula is:
formula_8
where formula_9 is some small value, like 0.0001. Note the use of formula_9 in the weighting function is equivalent to the Huber loss function in robust estimation.


In-cell charge control

In-Cell Charge Control or I-C3 is a method for very rapid charging of a Nickel-metal hydride battery, patented by Rayovac. Batteries using this technology are commonly sold as "15-minute rechargeables".
The charge control consists of a pressure switch built into the cell, which disconnects the charging current when the internal cell pressure rises above a certain limit; usually to . This prevents overcharging and damage to the cell.


Pocketdelta robot

The PocketDelta Robot is a microrobot based on a parallel structure called "Delta robot". It has been designed to perform micro-assembly tasks where high-speed and high-precision are needed in a reduced working space. The robot's size is 120×120×200 mm offering a workspace diameter up to 150mm × 30mm.
Design.
Prerequisites to high-speed and high-precision are a light, but stiff mechanical structure. Thus stiff and lightweight materials build up the moving part of the robot, whereas the heavy motors are attached to the supporting frame. The motors are attached directly to the robot arm (direct drive), eliminating backlash, friction and elasticity.
A high-level of integration reduces the size of the whole system. The mechanics, motion drives, control electronics and computing are all integrated into one compact unit. This innovative configuration simplifies its use and increases the flexibility of the robot.
The controller is based on PC architecture with interface boards for communication with the robot hardware. The realtime software enables the generation of trajectories in the Cartesian space using a direct and inverse kinematics model of the robot.
The robot integrates a web server which enables an easy communication through an Ethernet network using HTTP, TCP or UDP protocol.
Applications.
Typical application's domains for the PocketDelta robot are the following:
External links.
Companies and universities
Videos
Publications and articles
__notoc__


Fault-tolerant messaging

Fault Tolerant Messaging in the context of computer systems and networks, refers to a design approach and set of techniques aimed at ensuring reliable and continuous communication between components or nodes even in the presence of errors or failures. This concept is especially critical in distributed systems, where components may be geographically dispersed and interconnected through networks, making them susceptible to various potential points of failure.
The primary objective of fault-tolerant messaging is to maintain the integrity and availability of information exchange among system components, even when some components or communication channels encounter disruptions or errors. These errors may arise from hardware failures, network outages, software bugs, or other unexpected events.
Key characteristics and mechanisms commonly employed in fault-tolerant messaging include:
Several common protocols and technologies are employed to provide fault-tolerant messaging in distributed systems. These protocols are designed to ensure reliable communication, error detection and correction, and seamless failover mechanisms. Some of the most widely used protocols for fault-tolerant messaging include:


Consumption map

A consumption map or efficiency map shows the brake-specific fuel consumption in g per kWh over mean effective pressure per rotational speed of an internal combustion engine.
The x-axis shows the rotational speed range. The y-axis represents the load on the engine. The contour lines show the specific fuel consumption, indicating the areas of the speed/load regime where the engine is more or less efficient.
The map contains each possible condition, combining rotational speed and mean effective pressure. It shows the result of specific fuel consumption. A typical rotation power output P (linear to formula_1 ) is reached on several locations on the map but differing in the amount of fuel consumption. Automatic transmissions, are designed to keep the engine at the speed with the lowest possible fuel consumption, given the power demand.
The map also shows the efficiency of the engine. Depending on the fuel type, diesel and gasoline engines reach up to 210 g/kWh and about 40% efficiency. Using natural gas this efficiency is reached at 200 g/kWh.
Average values are 160–180 g/kWh for slow moving two stroke diesel cargoship engines using fuel oil, reaching up to 55% efficiency at 300 rpm. 195–210 g/kWh at cooled and pre-charged diesel engines for passenger cars, trucks 195–225 g/kWh. Non-charged Otto cycle gasoline engines for passenger cars 250–350 g/kWh.


Differential evolution

In evolutionary computation, differential evolution (DE) is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Such methods are commonly known as metaheuristics as they make few or no assumptions about the optimized problem and can search very large spaces of candidate solutions. However, metaheuristics such as DE do not guarantee an optimal solution is ever found.
DE is used for multidimensional real-valued functions but does not use the gradient of the problem being optimized, which means DE does not require the optimization problem to be differentiable, as is required by classic optimization methods such as gradient descent and quasi-newton methods. DE can therefore also be used on optimization problems that are not even continuous, are noisy, change over time, etc.
DE optimizes a problem by maintaining a population of candidate solutions and creating new candidate solutions by combining existing ones according to its simple formulae, and then keeping whichever candidate solution has the best score or fitness on the optimization problem at hand. In this way, the optimization problem is treated as a black box that merely provides a measure of quality given a candidate solution and the gradient is therefore not needed.
Storn and Price introduced DE in the 1990s. Books have been published on theoretical and practical aspects of using DE in parallel computing, multiobjective optimization, constrained optimization, and the books also contain surveys of application areas. Surveys on the multi-faceted research aspects of DE can be found in journal articles .
Algorithm.
A basic variant of the DE algorithm works by having a population of candidate solutions (called agents). These agents are moved around in the search-space by using simple mathematical formulae to combine the positions of existing agents from the population. If the new position of an agent is an improvement then it is accepted and forms part of the population, otherwise the new position is simply discarded. The process is repeated and by doing so it is hoped, but not guaranteed, that a satisfactory solution will eventually be discovered.
Formally, let formula_1 be the fitness function which must be minimized (note that maximization can be performed by considering the function formula_2 instead). The function takes a candidate solution as argument in the form of a vector of real numbers. It produces a real number as output which indicates the fitness of the given candidate solution. The gradient of formula_3 is not known. The goal is to find a solution formula_4 for which formula_5 for all formula_6 in the search-space, which means that formula_4 is the global minimum.
Let formula_8 designate a candidate solution (agent) in the population. The basic DE algorithm can then be described as follows:
Parameter selection.
The choice of DE parameters formula_37, formula_38 and formula_39 can have a large impact on optimization performance. Selecting the DE parameters that yield good performance has therefore been the subject of much research. Rules of thumb for parameter selection were devised by Storn et al. and Liu and Lampinen. Mathematical convergence analysis regarding parameter selection was done by Zaharie.
Variants.
Variants of the DE algorithm are continually being developed in an effort to improve optimization performance. Many different schemes for performing crossover and mutation of agents are possible in the basic algorithm given above, see e.g.


The Reference Broadcast Infrastructure Synchronization (RBIS) protocol is a master/slave synchronization protocol. RBIS, as the Reference Broadcast Time Synchronization (RBS), is a receiver/receiver synchronization protocol, as a consequence timestamps used for clock regulation are acquired only on the receiving of synchronization events. RBIS is specifically tailored to be used in IEEE 802.11 Wi-Fi networks configured in infrastructure mode. Such a kind of networks are based on an access point that coordinates the communication between the wireless nodes (i.e., the STAs), and they are very common.
The advantages of RBIS are that it can be directly used with common access points, no modification is required to the STAs (or minor modifications to improve synchronization performance) and a very small overhead is added to the wireless channel (typically a message every 1 second). Moreover, it allows an easily synchronization with an external time source, because it is a master/slave protocol. Its major drawback is that it does not compensate the propagation delay. This fact limits the achievable synchronization quality to a couple of microsecond, which is typically enough for the very majority of the applications, especially for home automation. An example is the connection of wireless speakers to a television.


Multidimensional scaling

Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset. MDS is used to translate "information about the pairwise 'distances' among a set of formula_1 objects or individuals" into a configuration of formula_1 points mapped into an abstract Cartesian space.
More technically, MDS refers to a set of related ordination techniques used in information visualization, in particular to display the information contained in a distance matrix. It is a form of non-linear dimensionality reduction.
Given a distance matrix with the distances between each pair of objects in a set, and a chosen number of dimensions, "N", an MDS algorithm places each object into "N"-dimensional space (a lower-dimensional representation) such that the between-object distances are preserved as well as possible. For "N" = 1, 2, and 3, the resulting points can be visualized on a scatter plot.
Core theoretical contributions to MDS were made by James O. Ramsay of McGill University, who is also regarded as the founder of functional data analysis.
Types.
MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix:
Classical multidimensional scaling.
It is also known as Principal Coordinates Analysis (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called "strain", which is given by
formula_3
where formula_4 denote vectors in "N"-dimensional space, formula_5 denotes the scalar product between formula_4 and formula_7, and formula_8 are the elements of the matrix formula_9 defined on step 2 of the following algorithm, which are computed from the distances.
Metric multidimensional scaling (mMDS).
It is a superset of classical MDS that generalizes the optimization procedure to a variety of loss functions and input matrices of known distances with weights and so on. A useful loss function in this context is called "stress", which is often minimized using a procedure called stress majorization. Metric MDS minimizes the cost function called “stress” which is a residual sum of squares:formula_33
Metric scaling uses a power transformation with a user-controlled exponent formula_34: formula_35 and formula_36 for distance. In classical scaling formula_37 Non-metric scaling is defined by the use of isotonic regression to nonparametrically estimate a transformation of the dissimilarities.
Non-metric multidimensional scaling (NMDS).
In contrast to metric MDS, non-metric MDS finds both a non-parametric monotonic relationship between the dissimilarities in the item-item matrix and the Euclidean distances between items, and the location of each item in the low-dimensional space. For NMDS, it is unnecessary to
Let formula_38 be the dissimilarity between points formula_39. Let formula_40 be the Euclidean distance between embedded points formula_41.
Now, for each choice of the embedded points formula_42 and is a monotonically increasing function formula_43, define the "stress" function:
The factor of formula_45 in the denominator is necessary to prevent a "collapse". Suppose we define instead formula_46, then it can be trivially minimized by setting formula_47, then collapse every point to the same point.
A few variants of this cost function exist. MDS programs automatically minimize stress in order to obtain the MDS solution.
NMDS algorithm is a twofold optimization. The optimal monotonic transformation of the proximities has to be found. The points of a configuration have to be optimally arranged, so that their distances match the scaled proximities as closely as possible. This is usually done iteratively:
Louis Guttman's smallest space analysis (SSA) is an example of a non-metric MDS procedure.
Generalized multidimensional scaling (GMD).
An extension of metric multidimensional scaling, in which the target space is an arbitrary smooth non-Euclidean space. In cases where the dissimilarities are distances on a surface and the target space is another surface, GMDS allows finding the minimum-distortion embedding of one surface into another.
Details.
The data to be analyzed is a collection of formula_54 objects (colors, faces, stocks, . . .) on which a "distance function" is defined,
These distances are the entries of the "dissimilarity matrix"
The goal of MDS is, given formula_59, to find formula_54 vectors
formula_61 such that
where formula_64 is a vector norm. In classical MDS, this norm is the Euclidean distance, but, in a broader sense, it may be a metric or arbitrary distance function.
In other words, MDS attempts to find a mapping from the formula_54 objects into formula_66 such that distances are preserved. If the dimension formula_67 is chosen to be 2 or 3, we may plot the vectors formula_42 to obtain a visualization of the similarities between the formula_54 objects. Note that the vectors formula_42 are not unique: With the Euclidean distance, they may be arbitrarily translated, rotated, and reflected, since these transformations do not change the pairwise distances formula_71.
There are various approaches to determining the vectors formula_42. Usually, MDS is formulated as an optimization problem, where formula_78 is found as a minimizer of some cost function, for example,
A solution may then be found by numerical optimization techniques. For some particularly chosen cost functions, minimizers can be stated analytically in terms of matrix eigendecompositions.
Procedure.
There are several steps in conducting MDS research:


List of SysML tools

This article compares SysML tools. SysML tools are software applications which support some functions of the Systems Modeling Language.


Track technology

Depending on the supplier, track technology has been variously termed a smart conveyance system, intelligent track system, industrial transport system, independent cart technology, smart carriage technology, linear or extended or flexible transport system, or simply a conveyor or conveyance platform. They are also referred to as linear motors or long stator linear motors, reflecting the underlying technology of the track (stator) and shuttles (platen, equivalent to the rotor in a conventional rotary electric motor).  Shuttles have also been called carriers, movers, platforms and pallets.
List of commercially available track systems.
The following is a list of commercially available track systems by product name:
Areas of application.
Track technology is – among other technologies like machine vision and robotics – one of the key enablers for the adaptive machine.
The concept of the adaptive machine also goes beyond track technology to achieve their high levels of flexibility.  One complementary technology is the industrial robot, which by definition possesses the same programmable flexibility.  Of particular interest is the ability of both robots and track systems to operate safely along with humans in a collaborative environment. This recent development allows for a combination of manual and automated assembly tasks, maintenance and materials replenishment without stopping production.
Machine vision can play a pivotal role when integrated into an adaptive machine. Vision can identify individual shuttles and their contents in order to guide them to the appropriate workstations. Vision has long been used to automate robot guidance, inspection, orientation and related tasks.
Given the adaptive machine's flexibility to respond to consumer demand generation, Internet of Things and e-commerce technologies are complementary, providing the connection between internal production resources and commercial systems in a manufacturer's digital business model.


Artefaktur

Artefaktur Component Development Kit (ACDK) is a platform-independent library for generating distributed server-based components and applications. Services are provided by a C++ framework.
Artefaktur is free software, distributed under the terms of the GNU Lesser General Public License.


Siconos

SICONOS is an Open Source scientific software primarily targeted at
modeling and simulating non-smooth dynamical systems (NSDS):
Other applications are found in Systems and Control (hybrid systems, differential inclusions, optimal control with state constraints), Optimization (Complementarity problem and Variational inequality) Biology Gene regulatory network, Fluid Mechanics and Computer graphics, etc.
Components.
The software is based on 3 main components
Performance.
According to peer reviewed studies published by its developers, Siconos was approximately five times faster than Ngspice or ELDO (a commercial SPICE by Mentor Graphics) and 250 times faster than PLECS when solving a buck converter.


PivotPoint Technology Corporation

PivotPoint Technology Corporation is a software and systems engineering services company headquartered in Fallbrook, California. PivotPoint was founded in 2003 by Cris Kobryn, a noted expert in visual modeling languages and model-driven development technologies. PivotPoint is best known for its model-driven development consulting and training services, the latter which feature UML, SysML, BPMN and DoDAF workshops. PivotPoint is a founding member and a major contributor to the SysML Partners, the group of software tool vendors and industry leaders that convened in 2003 to create a UML dialect for systems engineering called SysML (Systems Modeling Language). In June 2007 the SysML Partners were named a winner in the "Modeling" category of the SD Times 100, which recognizes the leaders and innovators of the software development industry.


Separation principle

In control theory, a separation principle, more formally known as a principle of separation of estimation and control, states that under some assumptions the problem of designing an optimal feedback controller for a stochastic system can be solved by designing an optimal observer for the state of the system, which feeds into an optimal deterministic controller for the system. Thus the problem can be broken into two separate parts, which facilitates the design.
The first instance of such a principle is in the setting of deterministic linear systems, namely that if a stable observer and a stable state feedback are designed for a linear time-invariant system (LTI system hereafter), then the combined observer and feedback is stable. The separation principle does not hold in general for nonlinear systems.
Another instance of the separation principle arises in the setting of linear stochastic systems, namely that state estimation (possibly nonlinear) together with an optimal state feedback controller designed to minimize a quadratic cost, is optimal for the stochastic control problem with output measurements. When process and observation noise are Gaussian, the optimal solution separates into a Kalman filter and a linear-quadratic regulator. This is known as linear-quadratic-Gaussian control. More generally, under suitable conditions and when the noise is a martingale (with possible jumps), again a separation principle applies and is known as the separation principle in stochastic control.
The separation principle also holds for high gain observers used for state estimation of a class of nonlinear systems and control of quantum systems.
Proof of separation principle for deterministic LTI systems.
Consider a deterministic LTI system:
where
We can design an observer of the form
and state feedback
Define the error "e":
Then
Now we can write the closed-loop dynamics as
Since this is a triangular matrix, the eigenvalues are just those of "A" − "BK" together with those of "A" − "LC". Thus the stability of the observer and feedback are independent.


Hamilton–Jacobi–Bellman equation

The Hamilton-Jacobi-Bellman (HJB) equation is a nonlinear partial differential equation that provides necessary and sufficient conditions for optimality of a control with respect to a loss function. Its solution is the value function of the optimal control problem which, once known, can be used to obtain the optimal control by taking the maximizer (or minimizer) of the Hamiltonian involved in the HJB equation.
The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The connection to the Hamilton–Jacobi equation from classical physics was first drawn by Rudolf Kálmán. In discrete-time problems, the analogous difference equation is usually referred to as the Bellman equation.
While classical variational problems, such as the brachistochrone problem, can be solved using the Hamilton–Jacobi–Bellman equation, the method can be applied to a broader spectrum of problems. Further it can be generalized to stochastic systems, in which case the HJB equation is a second-order elliptic partial differential equation. A major drawback, however, is that the HJB equation admits classical solutions only for a sufficiently smooth value function, which is not guaranteed in most situations. Instead, the notion of a viscosity solution is required, in which conventional derivatives are replaced by (set-valued) subderivatives.
Optimal-control-problems.
Consider the following problem in deterministic optimal control over the time period formula_1:
where formula_3 is the scalar cost rate function and formula_4 is a function that gives the bequest value at the final state, formula_5 is the system state vector, formula_6 is assumed given, and formula_7 for formula_8 is the control vector that we are trying to find. Thus, formula_9 is the value function.
The system must also be subject to
where formula_11 gives the vector determining physical evolution of the state vector over time.
The partial differential equation.
For this simple system, the Hamilton–Jacobi–Bellman partial differential equation is
subject to the terminal condition
As before, the unknown scalar function formula_9 in the above partial differential equation is the Bellman value function, which represents the cost incurred from starting in state formula_15 at time formula_16 and controlling the system optimally from then until time formula_17.
Deriving the equation.
Intuitively, the HJB equation can be derived as follows. If formula_18 is the optimal cost-to-go function (also called the 'value function'), then by Richard Bellman's principle of optimality, going from time "t" to "t" + "dt", we have
Note that the Taylor expansion of the first term on the right-hand side is
where formula_21 denotes the terms in the Taylor expansion of higher order than one in little-"o" notation. Then if we subtract formula_18 from both sides, divide by "dt", and take the limit as "dt" approaches zero, we obtain the HJB equation defined above.
Solving the equation.
The HJB equation is usually solved backwards in time, starting from formula_23 and ending at formula_24.
When solved over the whole of state space and formula_25 is continuously differentiable, the HJB equation is a necessary and sufficient condition for an optimum when the terminal state is unconstrained. If we can solve for formula_26 then we can find from it a control formula_27 that achieves the minimum cost.
In general case, the HJB equation does not have a classical (smooth) solution. Several notions of generalized solutions have been developed to cover such situations, including viscosity solution (Pierre-Louis Lions and Michael Crandall), minimax solution , and others.
Approximate dynamic programming has been introduced by D. P. Bertsekas and J. N. Tsitsiklis with the use of artificial neural networks (multilayer perceptrons) for approximating the Bellman function in general. This is an effective mitigation strategy for reducing the impact of dimensionality by replacing the memorization of the complete function mapping for the whole space domain with the memorization of the sole neural network parameters. In particular, for continuous-time systems, an approximate dynamic programming approach that combines both policy iterations with neural networks was introduced. In discrete-time, an approach to solve the HJB equation combining value iterations and neural networks was introduced.
Alternatively, it has been shown that sum-of-squares optimization can yield an approximate polynomial solution to the Hamilton–Jacobi–Bellman equation arbitrarily well with respect to the formula_28 norm.
Extension to stochastic problems.
The idea of solving a control problem by applying Bellman's principle of optimality and then working out backwards in time an optimizing strategy can be generalized to stochastic control problems. Consider similar as above
now with formula_30 the stochastic process to optimize and formula_31 the steering. By first using Bellman and then expanding formula_32 with Itô's rule, one finds the stochastic HJB equation
where formula_34 represents the stochastic differentiation operator, and subject to the terminal condition
Note that the randomness has disappeared. In this case a solution formula_36 of the latter does not necessarily solve the primal problem, it is a candidate only and a further verifying argument is required. This technique is widely used in Financial Mathematics to determine optimal investment strategies in the market (see for example Merton's portfolio problem).
Application to LQG-Control.
As an example, we can look at a system with linear stochastic dynamics and quadratic cost. If the system dynamics is given by
and the cost accumulates at rate formula_38, the HJB equation is given by
with optimal action given by
Assuming a quadratic form for the value function, we obtain the usual Riccati equation for the Hessian of the value function as is usual for Linear-quadratic-Gaussian control.


Control knob

A control knob is a rotary device used to provide manual input adjustments to a mechanical/electrical system when grasped and turned by a human operator, so that differing extent of knob rotation corresponds to different desired input. Control knobs are a simpler type of input hardware and one of the most common components in control systems, and are found on all sorts of devices from taps and gas stoves to optical microscopes, potentiometers, radio tuners and digital cameras, as well as in aircraft cockpits.
Operation.
A control knob works by turning a shaft which connects to the component which produces the actual input. Common control components used include potentiometers, variable capacitors, and rotary switches. An example where the knob does not produce a variation in an electrical signal may be found in many toasters, where the darkness knob moves the thermostat in such a way as to change the temperature at which it opens and releases the cooked toast. Some similar controls produce similar inputs using different geometry; for example, the knob may be replaced by a lever which is moved through an angle. Another example is the sliding controls which frequently replace knobs as level controls in audio equipment.
Feedback.
The use of knobs is an important aspect of the design of user interfaces in these devices. Particular attention needs to be paid to the feedback to the operator from the adjustments being made. The use of a pointer on the knob in conjunction with a scale assists in producing repeatable settings; in other cases there may be a dial or other indicator which is either mechanically linked the knob's rotation (as in many older radio tuners) or which reports the behavior being controlled.


Wahba's problem

In applied mathematics, Wahba's problem, first posed by Grace Wahba in 1965, seeks to find a rotation matrix (special orthogonal matrix) between two coordinate systems from a set of (weighted) vector observations. Solutions to Wahba's problem are often used in satellite attitude determination utilising sensors such as magnetometers and multi-antenna GPS receivers. The cost function that Wahba's problem seeks to minimise is as follows:
where formula_3 is the "k"-th 3-vector measurement in the reference frame, formula_4 is the corresponding "k"-th 3-vector measurement in the body frame and formula_5 is a 3 by 3 rotation matrix between the coordinate frames.
formula_6 is an optional set of weights for each observation.
A number of solutions to the problem have appeared in literature, notably Davenport's q-method, QUEST and methods based on the singular value decomposition (SVD). Several methods for solving Wahba's problem are discussed by Markley and Mortari.
This is an alternative formulation of the Orthogonal Procrustes problem (consider all the vectors multiplied by the square-roots of the corresponding weights as columns of two matrices with "N" columns to obtain the alternative formulation). An elegant derivation of the solution on one and a half page can be found in.
Solution via SVD.
One solution can be found using a singular value decomposition (SVD).
1. Obtain a matrix formula_7 as follows:
2. Find the singular value decomposition of formula_7
3. The rotation matrix is simply:
where formula_12


Moving magnet actuator

A moving magnet actuator is a type of electromagnetic linear actuator. It typically consists of an arrangement of a mobile permanent magnet and fixed coil, arranged so that currents in the coil generate a pair of equal and opposite forces between the coil and magnet.
A voice coil actuator, also called a voice coil motor (VCM), is an electromagnetic linear actuator where the magnet is fixed and the coil is mobile. In this configuration the coil is common called a voice coil.


Cross Gramian

In control theory, the cross Gramian (formula_1, also referred to by formula_2) is a Gramian matrix used to determine how controllable and observable a linear system is.
For the stable time-invariant linear system
the cross Gramian is defined as:
and thus also given by the solution to the Sylvester equation:
This means the cross Gramian is not strictly a Gramian matrix, since it is generally neither positive semi-definite nor symmetric.
The triple formula_7 is controllable and observable, and hence minimal, if and only if the matrix formula_1 is nonsingular, (i.e. formula_1 has full rank, for any formula_10).
If the associated system formula_7 is furthermore symmetric, such that there exists a transformation formula_12 with
then the absolute value of the eigenvalues of the cross Gramian equal Hankel singular values:
Thus the direct truncation of the Eigendecomposition of the cross Gramian allows model order reduction (see ) without a balancing procedure as opposed to balanced truncation.
The cross Gramian has also applications in decentralized control, sensitivity analysis, and the inverse scattering transform.


Controlled invariant subspace

In control theory, a controlled invariant subspace of the state space representation of some system is a subspace such that, if the state of the system is initially in the subspace, it is possible to control the system so that the state is in the subspace at all times. This concept was introduced by Giuseppe Basile and Giovanni Marro .
Definition.
Consider a linear system described by the differential equation
Here, x("t") ∈ R"n" denotes the state of the system and u("t") ∈ R"p" is the input. The matrices "A" and "B" have size "n" × "n" and "n" × "p" respectively.
A subspace "V" ⊂ R"n" is a "controlled invariant subspace" if for any x(0) ∈ "V", there is an input u("t") such that x("t") ∈ "V" for all nonnegative "t".
Properties.
A subspace "V" ⊂ R"n" is a controlled invariant subspace if and only if "AV" ⊂ "V" + Im "B". If "V" is a controlled invariant subspace, then there exists a matrix "K" such that the input u("t") = "K"x("t") keeps the state within "V"; this is a simple feedback control .


Pole–zero plot

In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as:
A pole-zero plot shows the location in the complex plane of the poles and zeros of the transfer function of a dynamic system, such as a controller, compensator, sensor, equalizer, filter, or communications channel. By convention, the poles of the system are indicated in the plot by an X while the zeros are indicated by a circle or O.
A pole-zero plot can represent either a continuous-time (CT) or a discrete-time (DT) system. For a CT system, the plane in which the poles and zeros appear is the s plane of the Laplace transform. In this context, the parameter "s" represents the complex angular frequency, which is the domain of the CT transfer function. For a DT system, the plane is the z plane, where "z" represents the domain of the Z-transform.
Continuous-time systems.
In general, a rational transfer function for a continuous-time LTI system has the form:
formula_1
where
Either M or N or both may be zero, but in real systems, it should be the case that formula_9; otherwise the gain would be unbounded at high frequencies.
Region of convergence.
The region of convergence (ROC) for a given CT transfer function is a half-plane or vertical strip, either of which contains no poles. In general, the ROC is not unique, and the particular ROC in any given case depends on whether the system is causal or anti-causal.
The ROC is usually chosen to include the imaginary axis since it is important for most practical systems to have BIBO stability.
Example.
formula_14
This system has no (finite) zeros and two poles:
formula_15
and
formula_16
The pole-zero plot would be:
Notice that these two poles are complex conjugates, which is the necessary and sufficient condition to have real-valued coefficients in the differential equation representing the system.
Discrete-time systems.
In general, a rational transfer function for a discrete-time LTI system has the form:
formula_17
where
Either M or N or both may be zero.
Region of convergence.
The region of convergence (ROC) for a given DT transfer function is a disk or annulus which contains no poles. In general, the ROC is not unique, and the particular ROC in any given case depends on whether the system is causal or anti-causal.
The ROC is usually chosen to include the unit circle since it is important for most practical systems to have BIBO stability.
Example.
If formula_26 and formula_27 are completely factored, their solution can be easily plotted in the z-plane. For example, given the following transfer function:
formula_28
The only (finite) zero is located at: formula_29, and the two poles are located at: formula_30, where "j" is the imaginary unit.
The pole–zero plot would be:


Data warehouse automation

Data warehouse automation (DWA) refers to the process of accelerating and automating the data warehouse development cycles, while assuring quality and consistency. DWA is believed to provide automation of the entire lifecycle of a data warehouse, from source system analysis to testing to documentation. It helps improve productivity, reduce cost, and improve overall quality.
General.
Data warehouse automation primarily focuses on automation of each and every step involved in the lifecycle of a data warehouse, thus reducing the efforts required in managing it.
Data warehouse automation works on the principles of design patterns. It comprises a central repository of design patterns, which encapsulate architectural standards as well as best practices for data design, data management, data integration, and data usage.
In November 2015, an analyst firm has published a guide "Which Data Warehouse Automation Tool is Right for You?" covering four of the leading products in the DWA space. In November 2015, an international software and technology services company engaged in developing ‘agile tools’ for the data integration industry, was named by CIO Review as one of the 20 most promising productivity tools solution providers 2015
Benefits.
Data warehouse automation can provide advantages like source data exploration, warehouse data models, ETL generation, test automation, metadata management, managed deployment, scheduling, change impact analysis and easier maintenance and modification of the data warehouse.
More important than the technical features of DWA tools, however, is the ability to deliver projects faster and with less resources.


Comparison function

In applied mathematics, comparison functions are several classes of continuous functions, which are used in stability theory to characterize the stability properties of control systems as Lyapunov stability, uniform asymptotic stability etc. 1 + 1 equals 2, which can be used in comparison functions.
Let formula_1 be a space of continuous functions acting from formula_2 to formula_3. The most important classes of comparison functions are:
Functions of class formula_5 are also called "positive-definite functions".
One of the most important properties of comparison functions is given by Sontag’s formula_6-Lemma, named after Eduardo Sontag. It says that for each formula_7 and any formula_8 there exist formula_9:
Many further useful properties of comparison functions can be found in.
Comparison functions are primarily used to obtain quantitative restatements of stability properties as Lyapunov stability, uniform asymptotic stability, etc. These restatements are often more useful than the qualitative definitions of stability properties given in formula_10 language.
As an example, consider an ordinary differential equation
where formula_11 is locally Lipschitz. Then:
The comparison-functions formalism is widely used in input-to-state stability theory.


GestureTek

GestureTek is an American-based interactive technology company headquartered in Silicon Valley, California, with offices in Toronto and Ottawa, Ontario and Asia.
Founding.
Founded in 1986 by Canadians Vincent John Vincent and Francis MacDougall, this privately held company develops and licenses gesture recognition software based on computer vision techniques. The partners invented video gesture control in 1986 and received their base patent in 1996 for the GestPoint video gesture control system. GestPoint technology is a camera-enabled video tracking software system that translates hand and body movement into computer control. The system enables users to navigate and control interactive multi-media and menu-based content, engage in virtual reality game play, experience immersion in an augmented reality environment or interact with a consumer device (such a television, mobile phone or set top box) without using touch-based peripherals. Similar companies include gesture recognition specialist LM3LABS based in Tokyo, Japan.
Technology.
GestureTek's gesture interface applications include multi-touch and 3D camera tracking. GestureTek's multi-touch technology powers the multi-touch table in Melbourne's Eureka Tower. A GestureTek multi-touch table with object recognition is found at the New York City Visitors Center. Telefónica has a multi-touch window with technology from GestureTek. GestureTek's 3D tracking technology is used in a 3D television prototype from Hitachi and various digital signage and display solutions based on 3D interaction.
Patents.
GestureTek currently has 8 patents awarded, including: 5,534,917 (Video Gesture Control Motion Detection); 7,058,204 (Multiple Camera Control System, Point to Control Base Patent); 7,421,093 (Multiple Camera Tracking System for Interfacing With an Application); 7,227,526 (Stereo Camera Control, 3D-Vision Image Control System); 7,379,563 (Two-Handed Movement Tracker Tracking Bi-Manual Movements); 7,379,566 (Optical Flow-Based Tilt Sensor For Phone Tilt Control); 7,389,591 (Phone Tilt for Typing & Menus/Orientation-Sensitive Signal Output); 7,430,312 (Five Camera 3D Face Capture).
GestureTek's software and patents have been licensed by Microsoft for the Xbox 360, Sony for the EyeToy, NTT DoCoMo for their mobile phones and Hasbro for the ION Educational Gaming System. In addition to software provision, GestureTek also fabricates interactive gesture control display systems with natural user interface for interactive advertising, games and presentations.
In addition, GestureTek's natural user interface virtual reality system has been the subject of research by universities and hospitals for its application in both physical therapy and physical rehabilitation.
In 2008, GestureTek received the Mobile Innovation Global Award from the GSMA for its software-based, gesture-controlled user interface for mobile games and applications. The technology is used by Java platform integration providers and mobile developers. Katamari Damacy is one example of a gesture control mobile game powered by GestureTek software.
Competitors.
Other companies in the industry of interactive projections for marketing and retail experiences include Po-motion Inc., Touchmagix and LM3LABS.


Polished plate glass

Polished plate is a type of hand-made glass. It is produced by casting glass onto a table and then subsequently grinding and polishing the glass. This was originally done by hand, and then later by machine. It was an expensive process requiring a large capital investment.
Other methods of producing hand-blown window glass included: broad sheet, blown plate, crown glass and cylinder blown sheet. These methods of manufacture lasted at least until the end of the 19th century. The early 20th century marks the move away from hand-blown to machine manufactured glass such as rolled plate, machine drawn cylinder sheet, flat drawn sheet, single and twin ground polished plate, and float glass.
The Frenchman, Louis Lucas de Nehou, in 1688, in conjunction with Abraham Thevart, succeeded in perfecting the process of casting plate-glass. Mirror plates previous to the invention had been made from blown "sheet" glass, and were consequently very limited in size. De Nehou's process of rolling molten glass poured on an iron table rendered the manufacture of very large plates possible.
In 1773 English polished plate (by the French process) was produced at Ravenhead.
By 1800 a steam engine was used to carry out the grinding and polishing of the cast glass.


Outline of control engineering

The following outline is provided as an overview of and topical guide to control engineering:
Control engineering – engineering discipline that applies control theory to design systems with desired behaviors. The practice uses sensors to measure the output performance of the device being controlled and those measurements can be used to give feedback to the input actuators that can make corrections toward desired performance. When a device is designed to perform without the need of human inputs for correction it is called automatic control (such as cruise control for regulating a car's speed).


Mitsubishi S-AWC

S-AWC (Super All Wheel Control) is the brand name of an advanced full-time four-wheel drive system developed by Mitsubishi Motors. The technology, specifically developed for the new 2007 Lancer Evolution, the 2010 Outlander (if equipped), the 2014 Outlander (if equipped), the Outlander PHEV and the Eclipse Cross have an advanced version of Mitsubishi Motors' AWC system. Mitsubishi Motors first exhibited S-AWC integration control technology in the "Concept-X" model at the 39th Tokyo Motor Show in 2005. According to Mitsubishi Motors, "the ultimate embodiment of the company's AWC philosophy is the S-AWC system, a 4WD-based integrated vehicle dynamics control system".
It integrates management of its "Active Center Differential" (ACD), "Active Yaw Control" (AYC), "Active Stability Control" (ASC), and "Sports ABS" components, while adding braking force control to Mitsubishi Motors' own AYC system, allowing regulation of torque and braking force at each wheel. S-AWC employs yaw rate feedback control, a direct yaw moment control technology that affects left-right torque vectoring (this technology forms the core of S-AWC system) and controls cornering maneuvers as desired during acceleration, steady state driving, and deceleration. Mitsubishi Motors claims the result is elevated drive power, cornering performance, and vehicle stability regardless of driving conditions.
Components.
Active Center Differential (ACD).
"Active Center Differential" incorporates an electronically-controlled hydraulic multi-plate clutch. The system optimizes clutch cover clamp load for different driving conditions, regulating the differential limiting action between free and locked states to optimize front/rear wheel torque split and thereby producing the best balance between traction and steering response.
Active Yaw Control (AYC).
"Active Yaw Control" uses a torque transfer mechanism in the rear differential to control rear wheel torque differential for different driving conditions and so limit the yaw moment that acts on the vehicle body and enhance cornering performance. AYC also acts like a limited slip differential by suppressing rear wheel slip to improve traction. In its latest form, AYC now features yaw rate feedback control using a yaw rate sensor and also gains braking force control. Accurately determining the cornering dynamics on a realtime basis, the system operates to control vehicle behavior through corners and realize vehicle behavior that more closely mirrors driver intent.
Active Stability Control (ASC).
Active Stability Control stabilizes vehicle attitude while maintaining optimum traction by regulating engine power and the braking force at each wheel. Taking a step beyond the previous generation Lancer Evolution, the fitting of a brake pressure sensor at each wheel allows more precise and positive control of braking force. ASC improves traction under acceleration by preventing the driving wheels from spinning on slippery surfaces. It also elevates vehicle stability by suppressing skidding in an emergency evasive maneuver or the result of other sudden steering inputs.
Sport ABS.
The Sports ABS system supports braking when entering into a corner by controlling power to all tires depending on handling characteristics. Braking can be controlled to obtain optimal damping at each tyre based on information from four wheel-speed sensors and steering wheel angle sensor. The addition of yaw rate sensors and brake pressure sensors to the Sport ABS system has improved braking performance through corners compared to the Lancer Evolution IX.
Concept components for 2007 Lancer Evolution.
The prototype system also featured two additional components controlling suspensions and steering, which failed to make the production version of S-AWC system:
Active Steering System.
Active Steering System realizes handling with more linear response by adaptively controlling front wheel turn angle according to steering input and vehicle speed. At slower vehicle speeds the system improves response by shifting to a quicker steering gear ratio, while at higher speeds it substantially improves stability by moving to a slower gear ratio. For rapid steering inputs, S-AWC momentarily increases front wheel turn angle and Super AYC control to realize sharper response. In countersteer situations, S-AWC increases responsiveness further to assist the driver with steering precision.
Roll Control Suspension (RCS).
RCS effectively reduces body roll and pitching by hydraulically connecting all the shock absorbers together and regulating their damping pressures as necessary. Able to control both roll and pitching stiffness separately, RCS can operate in a variety of ways. It can, for example, reduce roll only when required during turn in or in other situations while being set up on the soft side to prioritize tire contact and ride comfort. Since the system controls roll stiffness hydraulically, it eliminates the need for stabilizer bars. In the integrated control of its component systems, S-AWC employs information from RCS's hydraulic system to estimate the tire load at each wheel.
Control system.
The use of engine torque and brake pressure information in the regulation of the ACD and AYC components allows the S-AWC system to determine more quickly whether the vehicle is accelerating or decelerating. S-AWC also employs yaw rate feedback for the first time. The system helps the driver follow his chosen line more closely by comparing how the car is running, as determined from data from the yaw rate sensors, and how the driver wants it to behave, as determined from steering inputs, and operates accordingly to correct any divergence. The addition of braking force regulation to AYC's main role of transferring torque between the right and left wheels allows S-AWC to exert more control over vehicle behavior in on-the-limit driving situations. Increasing braking force on the inside wheel during understeer and on the outer wheel during oversteer situations, AYC's new braking force control feature works in conjunction with torque transfer regulation to realize higher levels of cornering performance and vehicle stability.
Using integrated management of the ASC and ABS systems allows S-AWC to effectively and seamlessly control vehicle dynamics when accelerating, decelerating or cornering under all driving conditions. S-AWC offers three operating modes:
When the driver selects the mode best suited to current road surface conditions S-AWC operates to control vehicle behavior accordingly and allow the driver to extract the maximum dynamic performance from his vehicle.
ECU integration.
Two electronic control units (ECU) regulate vehicle motion. One is an ECU developed by Mitsubishi Electric to control ACD and AYC. The other is an ECU developed by Continental Automotive Systems of Germany that controls ASC and ABS. The two ECUs can communicate with other ECUs through a CAN, an in-vehicle LAN interface standard. In addition, the two ECUs are communicating with each other through a dedicated CAN, enabling vehicle motion to be controlled more quickly. The cable and communication standard for the dedicated CAN are the same as those for other CANs.
A longitudinal acceleration sensor, lateral acceleration sensor and yaw rate sensor are installed as one module near the gravity center of a vehicle, which is located between the driver's and passenger's seats. Other sensors, such as a wheel-speed sensor and steering-angle sensor, are installed in different places. However, no vertical acceleration sensor is used.
Also, when the vehicle is equipped with Mitsubishi's "Twin Clutch SST" transmission, S-AWC analyzes the behavior of the turning vehicle and if it judges that it is safer not to shift gears, it sends a signal to tell Twin Clutch SST that the gear must not be changed. However, S-AWC does not control vehicle motion by using control information from Twin Clutch SST. The co-operation is a one-way communication.
The control algorithms of vehicle motion were developed by Mitsubishi in-house, with MATLAB and Simulink: control system modeling tools. Mitsubishi adopted model-based method, which combines an algorithm and physical model of a vehicle to run a simulation. The physical model of a vehicle was constructed with CarSim, a simulation-package software developed by Mechanical Simulation Corporation of the United States. The algorithms were developed for each function such as ACD and AYC, not for each vehicle type. Therefore, the algorithms can be employed by various types of vehicles.
Concept components for 2010 Outlander.
The 2010MY Outlander adopts a new S-AWC (Super All Wheel Control) that has added and refined an active front differential that controls the differential limiting force of the left and right front wheels based on an electronically controlled 4WD that distributes drive force to the rear wheels and integrates this Active Stability Control (ASC) and ABS. The result is greater turning performance, stability and drive performance while maintaining fuel economy equal to traditional electronically-controlled 4WD.
Structure.
The S-AWC ECU calculates the amount of control according to drive condition and vehicle behavior based on sensor and switch data and ECU operation data. Control instructions are sent to the active front diff and electronic control couplings.
Active control differential.
Electronically-controlled couplings used in electronic –control 4WD are located in the transfer case to limit differential between the front left and right wheels and control drive force distribution on either side.
Electronic control coupling.
An electronic control coupling within the rear differential distributes drive force to the rear wheels according to driving conditions. This is the same as used for 4WD electronic control in the 2009 model Outlander.
S-AWC ECU.
The optimal amount of drive force control is calculated from sensor information obtained from CAN communications etc. to control the active front diff and the electronically-controlled coupling. Compared with the 2009 Outlander, Microcomputer performance has been enhanced and calculation speed and accuracy have been improved.
Sensor information.
Compared with electronically-controlled 4WD, sensor information has been significantly augmented to accurately assess vehicle driving conditions and realize highly-responsive, finely tuned control.
S-AWC control mode switch.
S-AWC in the 2010 model Outlander has three selectable modes of control (NORMAL/SNOW/OFFROAD) that have been tuned to suit the road surface. Making the switch according to road surface conditions enables proper control.
Indicator.
S-AWC control information will be constantly displayed on the upper level of the multi-information display. A dedicated screen has been provided to display S-AWC operation information. The center displays traction control condition while yaw movement control conditions are displayed on either side.
Control.
Changes to the 2009 Outlander's electronically-controlled 4WD.
1) Addition of integrated control with the active front differential
In addition to front and rear drive force distribution, enabling integrated control of drive force distribution to both front wheels delivers a higher level of driving on all fronts (turning performance, stability and road performance) compared with the 2009 Outlander.
2) Introduction of a yaw rate feedback control
Vehicle behavior faithful to drive input is realized by precise assessment of vehicle turning movement based on yaw rate sensor data and the provision of achieve close to target vehicle behavior obtained from speed and steering angle.
3) Evolution of coordinated ASC/ABS control
Properly controlling active front differential and electronically controlled coupling according to the operating status of ASC and ABS, improves turning performance and stability.
Concept components for 2014 Outlander.
The following functions have been recently added.
Brake control.
When the under steer condition, the beginning of turning response by steering operation
is dramatically improved by adding the brake force to the inner wheel.
In addition, the wheel slippage is reduced during start moving.
EPS control.
Suppress the steering wheel movement which generated by the slippery road.
As a result, the traction performance improves because the amount of the control of Active Front Differential (AFD) can be increased.
Synchronized with ECO MODE.
By selecting the ECO MODE, Engine and climate control are controlled as an "ECO
mode." Likewise, S-AWC control also turn to AWC ECO.
At the result of this control, the driver can easily engage "ECO mode."
Control.
S-AWC Control Mode
By pushing S-AWC Control switch, the control mode can be changed.
Concept components for Outlander PHEV.
Fail-safe function.
Fault detection
The ECU performs the following checks at the appropriate moment. The ECU determines that a fault has occurred when the fault detection conditions are met. Then the ECU stores the diagnosis code and ensures that the vehicle can still be driven. When the failure resume conditions are met, ECU determines the status is normal, and resumes the system. Start-up (Initial check immediately after the power supply mode of the electric motor switch is turned on.)
• CPU check
• Performs the ROM and RAM checks.
Always (while the power supply mode of the electric motor switch is turned on except during initial check)
1. CPU check
• Performs CAN communication and interactive check between CPUs.
2. Power supply check
• Monitors the CPU supply voltage and checks if the voltage is within specifications.
3. External wire connection check
• Checks if the input and output of each external wire connection is open or shorted.
4WD lock switch.
The 4WD lock switch is located on the floor console. When the 4WD lock switch is pressed with the electric motor switch ON, "4WD LOCK" will be turned on and off. When the 4WD lock switch is turned on with the drive mode at ECO, or the ECO mode switch is turned on with the drive mode at 4WD lock, the drive mode will be switched to "ECO MODE/4WD LOCK". The driver can obtain better ground-covering ability by choosing the drive mode between "4WD LOCK" and "ECO MODE/4WD LOCK". When the ECO mode switch is turned off, the drive mode will return from "ECO MODE/4WD LOCK" to "4WD LOCK."
Cornering Performance.
Enhancement of the cornering stability.
It is optimization of the torque distribution ratio between front and rear wheels when cornering. In order to keep the cornering stability against the direction of steering wheel on the slippery road.
Enhancement of the vehicle maneuverability.
The optimization of the control value for the AYC (Active Yaw Control) with braking, in order to enhance the vehicle maneuverability.
Traction performance.
Launching performance on the icy slope is enhanced.
Concept components for Eclipse Cross.
S-AWC (Super All Wheel Control) is an integration of vehicle dynamics control systems whose design goals include safety and comfort.
S-AWC of NEW ECLIPSE CROSS adopted the integration system that controlled with Active Stability Control (ASC) and ABS based on Electronically Controlled 4WD that distributes driving torque to rear wheel and Active Yaw Control (AYC) that controlled drive/braking torque between right and left wheel. The goal of the design is to prevent loss of control while excessive braking or accelerating on slippery roads. AYC of ECLIPSE CROSS controls drive/braking torque between right and left wheel by additional brake force. There are three modes of operation:
• AUTO This mode achieves adequate 4WD performance on various conditions.
• SNOW This mode enhances stability on the slippery road surface.
• GRAVEL This mode excels at rough road driving and escape from stuck conditions.
Electronically controlled 4WD.
An electronically-controlled coupling integrated within the rear differential assembly distributes optimum driving forces between the front and rear axles, thus improving acceleration and driving stability.
Brake AYC.
The AWC-ECU is a computer that uses the inputs from various sensors to assess the state of vehicle stability and, if necessary, compensates for an instability by controlling the braking forces of the left and right wheels to generate a yaw moment.
AWC-ECU Function.
The main functions of AWC-ECU are as follows:
1. Communication function
• CAN communication with other ECUs (Engine-ECU, CVT-ECU, ASC-ECU, ETACS, EPS-ECU).
• Communication with drive mode selector: The signal from the drive mode selector changes the drive mode.
• Combination meter display: Drive mode is displayed.
2. Coupling control function
• Current output: Differential control function of the electronic control coupling according to the vehicle conditions.
3. ECU self-diagnosis function
• Initial check: ROM check, relay check, etc.
• Recording function of diagnostic trouble codes and freeze frame data in case of failure.
• If a fault occurs, the system will be disabled and a warning icon will be displayed.
• Normal control: Malfunction of CPU power supply, relay check, open or short circuit of the I/O signal, abnormal CAN communication.
External links.
"S-AWC schematics"
"ACD/AYC programming information"
'Tangime'


Individual wheel drive

Individual-wheel drive (IWD) is a wheeled vehicle with a drivetrain that allows all wheels to receive torque from several motors independent of each other. The term was coined to identify those electric vehicles whereby each wheel is driven by its own individual electric motor, as opposed to conventional differentials.
Characteristics.
These vehicles inherently have a range of characteristics built-in that are more commonly attributed to four-wheel drive vehicles or vehicles with extensive control systems. These characteristics can be:
Other features
The motors that are used in these vehicles are commonly wheel hub motors, since no transmission components are then required. Alternative layouts with inboard motors and drive shafts are also possible.
Hydraulic wheel drive.
Hydraulic wheel drives share many of the same features as an electric wheel drive. They also lack the need for a central gear box, mechanical differentials, drive shafts, and provide on the fly switching between front, rear and all-wheel drive. Hydraulic individual wheel drives are standard in various machines, such as zero-turn mowers, multi one lifts / front end loaders, and forklifts. Hydraulic drives are primarily found in machines that serve uses which benefit from the ability to "turn on a dime", i.e. with an exceptionally short turning diameter, and move between forward and reverse modes without shifting gears, such as lawn mowers and loading equipment.
Although one may be conflicted in considering such systems as a direct drive system, being that a motorized pump must drive the hydraulic system from a position other than the wheel hub. Nonetheless the drive is provided directly from the hydraulic rotary motor found in or adjacent to the wheel hub.


GNU Linear Programming Kit

The GNU Linear Programming Kit (GLPK) is a software package intended for solving large-scale linear programming (LP), mixed integer programming (MIP), and other related problems. It is a set of routines written in ANSI C and organized in the form of a callable library. The package is part of the GNU Project and is released under the GNU General Public License.
GLPK uses the revised simplex method and the primal-dual interior point method for non-integer problems and the branch-and-bound algorithm together with Gomory's mixed integer cuts for (mixed) integer problems.
History.
GLPK was developed by Andrew O. Makhorin (Андрей Олегович Махорин) of the Moscow Aviation Institute. The first public release was in October 2000.
Interfaces and wrappers.
Since version 4.0, GLPK problems can be modeled using GNU MathProg (GMPL), a subset of the AMPL modeling language used only by GLPK. However, GLPK is most commonly called from other programming languages. Wrappers exist for:


Computer-aided production engineering

Computer-aided production engineering (CAPE) is a relatively new and significant branch of engineering. Global manufacturing has changed the environment in which goods are produced. Meanwhile, the rapid development of electronics and communication technologies has required design and manufacturing to keep pace.
Description of CAPE.
CAPE is seen as a new type of computer-aided engineering environment which will improve the productivity of manufacturing/industrial engineers. This environment would be used by engineers to design and implement future manufacturing systems and subsystems. Work is currently underway at the United States National Institute of Standards and Technology (NIST) on CAPE systems. The NIST project is aimed at advancing the development of software environments and tools for the design and engineering of manufacturing systems.
CAPE and the Future of Manufacturing.
The future of manufacturing will be determined by the efficiency with which it can incorporate new technologies. The current process in engineering manufacturing systems is often ad hoc, with computerized tools being used on a limited basis. Given the costs and resources involved in the construction and operation of manufacturing systems, the engineering process must be made more efficient. New computing environments for engineering manufacturing systems could help achieve that objective.
Why is CAPE important? In much the same way that product designers need computer-aided design systems, manufacturing and industrial engineers need sophisticated computing capabilities to solve complex problems and manage the vast data associated with the design of a manufacturing system.
In order to solve these complex problems and manage design data, computerized tools must be used in the application of scientific and engineering methods to the problem of the
design and implementation of manufacturing systems. Engineers must address the entire factory as a system and the interactions of that system with its surrounding environment.
Components of a factory system include:
CAPE must not only be concerned with the initial design and engineering of the factory, it must also address enhancements over time. CAPE should support standard engineering methods and problem-solving techniques, automate mundane tasks, and provide reference data to support the decision-making process.
The environment should be designed to help engineers become more productive and effective in their work. This would be implemented on personal computers or engineering workstations which have been configured with appropriate peripheral devices. Engineering tool developers will have to integrate the functions and data used by a number of different disciplines, for example:
Many of the methods, formulas, and data associated with these technical areas currently exist only in engineering handbooks. Although some computerized tools are available, they are often very specialized, difficult to use, and do not share information or work together. Engineering tools built by different vendors must be made compatible through open systems architectures and interface standards.
What CAPE will look like.
CAPE will be based upon computer systems providing an integrated set of design and engineering tools. These software tools will be used by a company's manufacturing engineers to continuously improve its production systems. They will maintain information about manufacturing resources, enhance production capabilities, and develop new facilities and systems. Engineers working on different workstations will share information through a common database.
Using CAPE, an engineering team will prepare detailed plans and working models for an entire factory in a matter of days. Alternative solutions to production problems could be quickly developed and evaluated. This would be a significant improvement over current manual methods which may require weeks or months of intensive activity.
To achieve this goal, a new set of engineering tools are needed. Examples of functions which should be supported include:
The tools implementing these functions must be highly automated and integrated; and will need to provide quick access to a wide range of data. This data must be maintained in a format that is accessible and usable by the engineering tools. Some examples of the information that might be contained in these electronic libraries include:
These on-line libraries would allow engineers to quickly develop solutions based upon the work of others.
Another critical aspect of this engineering environment is affordability, which
can best be achieved by designing an environment that can be constructed from low cost "off-the-shelf" commercial products, rather than custombuilt computer hardware and software. The basic engineering environment must be affordable. For both cost and technical reasons, it must be designed to be able to support incremental upgrades. Incremental upgrades would allow companies to add capabilities as they are needed. Commercial software products must be easy to install and integrate with other software already in use. These capabilities exist to a limited extent in some general purpose commercial software today, e.g., word processors, databases, spreadsheets.
Technical Concerns.
Many technical issues must be considered in the design and development of new engineering tools for CAPE. These issues include:
There are three critical elements to be addressed: creating a common manufacturing systems information model; using an engineering life cycle approach; and developing a software tool integration framework.
Resolution of these elements will help ensure that independently developed systems will be able to work together. The common information model should identify the elements of the manufacturing system and their relationships to each other; the functions or processes performed by each element; the tools, materials, and information required to perform those functions; and measures of effectiveness for the model and its component elements.
There have been many efforts over the years to develop information models for different
aspects of manufacturing, but no known existing model fully meets the needs of a CAPE ernviroment. Therefore, a life cycle approach is needed to identify the different processes that a CAPE environment must support, and must define all phases of a manufacturing system or subsystem's existence. Some of the major phases which may be included in a system life cycle approach are, requirements identification; system design specification; vendor selection; system development and upgrades; installation, testing, and training; and benchmarking of production operations.
Management, coordination, and administration functions need to be performed during each phase of the life cycle. Phases may be repeated over time as a system is upgraded or re-engineered to meet changing needs or incorporate new technologies.
A software tool integration framework should specify how the tools could be independently designed and developed. The framework would define how CAPE tools would deal with common services, interact with each other and coordinate problem solving activities. Although some existing software products and standards currently address the common services issue, the problem of tool interaction remains largely unsolved. The problem of tool interaction is not limited to the domain of computer-aided manufacturing systems engineering—it is pervasive across the software industry.
CAPE's current state.
An initial CAPE environment has been established from commercial off-the-shelf (COTS) software packages. This new environment is being used to demonstrate commercially available tools to perform CAPE functions, to develop a better understanding and define functional requirements for individual engineering tools and the overall environment, and to identify the integration issues which must be addressed to implement compatible environments in the future.
Several engineering demonstrations using COTS tools are under development. These demonstrations are designed to illustrate the various types of functions that must be performed in engineering a manufacturing system.
Functions supported by the current COTS environment include: system specification/diagramming,
process flowcharting, information modeling, computer-aided design of products, plant layout, material flow analysis, ergonomic workplace design, mathematical modeling, statistical analysis, line balancing, manufacturing simulation, investment analysis, project management, knowledge-based system development, spreadsheets, document preparation, user interface development, document illustration, forms and database management.


Iterative rational Krylov algorithm

The iterative rational Krylov algorithm (IRKA), is an iterative algorithm, useful for model order reduction (MOR) of single-input single-output (SISO) linear time-invariant dynamical systems. At each iteration, IRKA does an Hermite type interpolation of the original system transfer function. Each interpolation requires solving formula_1 shifted pairs of linear systems, each of size formula_2; where formula_3 is the original system order, and formula_1 is the desired reduced model order (usually formula_5).
The algorithm was first introduced by Gugercin, Antoulas and Beattie in 2008. It is based on a first order necessary optimality condition, initially investigated by Meier and Luenberger in 1967. The first convergence proof of IRKA was given by Flagg, Beattie and Gugercin in 2012, for a particular kind of systems.
MOR as an optimization problem.
Consider a SISO linear time-invariant dynamical system, with input formula_6, and output formula_7:
Applying the Laplace transform, with zero initial conditions, we obtain the transfer function formula_9, which is a fraction of polynomials:
Assume formula_9 is stable. Given formula_12, MOR tries to approximate the transfer function formula_9, by a stable rational transfer function formula_14, of order formula_1:
A possible approximation criterion is to minimize the absolute error in formula_17 norm:
This is known as the formula_17 optimization problem. This problem has been studied extensively, and it is known to be non-convex; which implies that usually it will be difficult to find a global minimizer.
Meier–Luenberger conditions.
The following first order necessary optimality condition for the formula_17 problem, is of great importance for the IRKA algorithm.
Note that the poles formula_21 are the eigenvalues of the reduced formula_22 matrix formula_23.
Hermite interpolation.
An Hermite interpolant formula_14 of the rational function formula_9, through formula_1 distinct points formula_27, has components:
where the matrices formula_29 and formula_30 may be found by solving formula_1 dual pairs of linear systems, one for each shift [Theorem 1.1]:
IRKA algorithm.
As can be seen from the previous section, finding an Hermite interpolator formula_14 of formula_9, through formula_1 given points, is relatively easy. The difficult part is to find the correct interpolation points. IRKA tries to iteratively approximate these "optimal" interpolation points.
For this, it starts with formula_1 arbitrary interpolation points (closed under conjugation), and then, at each iteration formula_37, it imposes the first order necessary optimality condition of the formula_38 problem:
1. find the Hermite interpolant formula_14 of formula_9, through the actual formula_1 shift points: formula_42.
2. update the shifts by using the poles of the new formula_14: formula_44
The iteration is stopped when the relative change in the set of shifts of two successive iterations is less than a given tolerance. This condition may be stated as:
As already mentioned, each Hermite interpolation requires solving formula_1 shifted pairs of linear systems, each of size formula_2:
Also, updating the shifts requires finding the formula_1 poles of the new interpolant formula_14. That is, finding the formula_1 eigenvalues of the reduced formula_22 matrix formula_23.
Pseudocode.
The following is a pseudocode for the IRKA algorithm [Algorithm 4.1].
 algorithm IRKA
 input: formula_54, formula_55, formula_56 closed under conjugation
 formula_57 % Solve primal systems
 formula_58 % Solve dual systems
 while relative change in {formula_59} > tol
 formula_60 % Reduced order matrix
 formula_61 % Update shifts, using poles of formula_62
 formula_57 % Solve primal systems
 formula_64 % Solve dual systems
 end while
 return formula_65 % Reduced order model
Convergence.
A SISO linear system is said to have symmetric state space (SSS), whenever: formula_66 This type of systems appear in many important applications, such as in the analysis of RC circuits and in inverse problems involving 3D Maxwell's equations. For SSS systems with distinct poles, the following convergence result has been proven: "IRKA is a locally convergent fixed point iteration to a local minimizer of the formula_17 optimization problem."
Although there is no convergence proof for the general case, numerous experiments have shown that IRKA often converges rapidly for different kind of linear dynamical systems.
Extensions.
IRKA algorithm has been extended by the original authors to multiple-input multiple-output (MIMO) systems, and also to discrete time and differential algebraic systems [Remark 4.1].
See also.
Model order reduction


Stability (learning theory)

Stability, also known as algorithmic stability, is a notion in computational learning theory of how a machine learning algorithm output is changed with small perturbations to its inputs. A stable learning algorithm is one for which the prediction does not change much when the training data is modified slightly. For instance, consider a machine learning algorithm that is being trained to recognize handwritten letters of the alphabet, using 1000 examples of handwritten letters and their labels ("A" to "Z") as a training set. One way to modify this training set is to leave out an example, so that only 999 examples of handwritten letters and their labels are available. A stable learning algorithm would produce a similar classifier with both the 1000-element and 999-element training sets.
Stability can be studied for many types of learning problems, from language learning to inverse problems in physics and engineering, as it is a property of the learning process rather than the type of information being learned. The study of stability gained importance in computational learning theory in the 2000s when it was shown to have a connection with generalization. It was shown that for large classes of learning algorithms, notably empirical risk minimization algorithms, certain types of stability ensure good generalization.
History.
A central goal in designing a machine learning system is to guarantee that the learning algorithm will generalize, or perform accurately on new examples after being trained on a finite number of them. In the 1990s, milestones were reached in obtaining generalization bounds for supervised learning algorithms. The technique historically used to prove generalization was to show that an algorithm was consistent, using the uniform convergence properties of empirical quantities to their means. This technique was used to obtain generalization bounds for the large class of empirical risk minimization (ERM) algorithms. An ERM algorithm is one that selects a solution from a hypothesis space formula_1 in such a way to minimize the empirical error on a training set formula_2.
A general result, proved by Vladimir Vapnik for an ERM binary classification algorithms, is that for any target function and input distribution, any hypothesis space formula_1 with VC-dimension formula_4, and formula_5 training examples, the algorithm is consistent and will produce a training error that is at most formula_6 (plus logarithmic factors) from the true error. The result was later extended to almost-ERM algorithms with function classes that do not have unique minimizers.
Vapnik's work, using what became known as VC theory, established a relationship between generalization of a learning algorithm and properties of the hypothesis space formula_1 of functions being learned. However, these results could not be applied to algorithms with hypothesis spaces of unbounded VC-dimension. Put another way, these results could not be applied when the information being learned had a complexity that was too large to measure. Some of the simplest machine learning algorithms—for instance, for regression—have hypothesis spaces with unbounded VC-dimension. Another example is language learning algorithms that can produce sentences of arbitrary length.
Stability analysis was developed in the 2000s for computational learning theory and is an alternative method for obtaining generalization bounds. The stability of an algorithm is a property of the learning process, rather than a direct property of the hypothesis space formula_1, and it can be assessed in algorithms that have hypothesis spaces with unbounded or undefined VC-dimension such as nearest neighbor. A stable learning algorithm is one for which the learned function does not change much when the training set is slightly modified, for instance by leaving out an example. A measure of Leave one out error is used in a Cross Validation Leave One Out (CVloo) algorithm to evaluate a learning algorithm's stability with respect to the loss function. As such, stability analysis is the application of sensitivity analysis to machine learning.
Preliminary definitions.
We define several terms related to learning algorithms training sets, so that we can then define stability in multiple ways and present theorems from the field.
A machine learning algorithm, also known as a learning map formula_9, maps a training data set, which is a set of labeled examples formula_11, onto a function formula_12 from formula_13 to formula_14, where formula_13 and formula_14 are in the same space of the training examples. The functions formula_12 are selected from a hypothesis space of functions called formula_1.
The training set from which an algorithm learns is defined as
formula_19
and is of size formula_20 in formula_21
drawn i.i.d. from an unknown distribution D.
Thus, the learning map formula_9 is defined as a mapping from formula_23 into formula_1, mapping a training set formula_2 onto a function formula_26 from formula_13 to formula_14. Here, we consider only deterministic algorithms where formula_9 is symmetric with respect to formula_2, i.e. it does not depend on the order of the elements in the training set. Furthermore, we assume that all functions are measurable and all sets are countable.
The loss formula_31 of a hypothesis formula_12 with respect to an example formula_33 is then defined as formula_34.
The empirical error of formula_12 is formula_36.
The true error of formula_12 is formula_38
Given a training set S of size m, we will build, for all i = 1...,m, modified training sets as follows:
formula_39
formula_40
Definitions of stability.
Hypothesis Stability.
An algorithm formula_9 has hypothesis stability β with respect to the loss function V if the following holds:
formula_42
Point-wise Hypothesis Stability.
An algorithm formula_9 has point-wise hypothesis stability β with respect to the loss function V if the following holds:
formula_44
Error Stability.
An algorithm formula_9 has error stability β with respect to the loss function V if the following holds:
formula_46
Uniform Stability.
An algorithm formula_9 has uniform stability β with respect to the loss function V if the following holds:
formula_48
A probabilistic version of uniform stability β is:
formula_49
An algorithm is said to be stable, when the value of formula_50 decreases as formula_51.
Leave-one-out cross-validation (CVloo) Stability.
An algorithm formula_9 has CVloo stability β with respect to the loss function V if the following holds:
formula_53
The definition of (CVloo) Stability is equivalent to Pointwise-hypothesis stability seen earlier.
Expected-leave-one-out error (formula_54) Stability.
An algorithm formula_9 has formula_54 stability if for each n there exists a formula_57 and a formula_58 such that:
formula_59, with formula_57 and formula_58 going to zero for formula_62
Classic theorems.
From Bousquet and Elisseeff (02):
For symmetric learning algorithms with bounded loss, if the algorithm has Uniform Stability with the probabilistic definition above, then the algorithm generalizes.
Uniform Stability is a strong condition which is not met by all algorithms but is, surprisingly, met by the large and important class of Regularization algorithms.
The generalization bound is given in the article.
From Mukherjee et al. (06):
This is an important result for the foundations of learning theory, because it shows that two previously unrelated properties of an algorithm, stability and consistency, are equivalent for ERM (and certain loss functions).
The generalization bound is given in the article.
Algorithms that are stable.
This is a list of algorithms that have been shown to be stable, and the article where the associated generalization bounds are provided.