Speeds and feeds

The phrase speeds and feeds or feeds and speeds refers to two separate velocities in machine tool practice, cutting speed and feed rate. They are often considered as a pair because of their combined effect on the cutting process. Each, however, can also be considered and analyzed in its own right.
"Cutting speed" (also called "surface speed" or simply "speed") is the speed difference (relative velocity) between the cutting tool and the surface of the workpiece it is operating on. It is expressed in units of distance across the workpiece surface per unit of time, typically surface feet per minute (sfm) or meters per minute (m/min). "Feed rate" (also often styled as a solid compound, "feedrate", or called simply "feed") is the relative velocity at which the cutter is advanced along the workpiece; its vector is perpendicular to the vector of cutting speed. Feed rate units depend on the motion of the tool and workpiece; when the workpiece rotates ("e.g.", in turning and boring), the units are almost always distance per spindle revolution (inches per revolution [in/rev or ipr] or millimeters per revolution [mm/rev]). When the workpiece does not rotate ("e.g.", in milling), the units are typically distance per time (inches per minute [in/min or ipm] or millimeters per minute [mm/min]), although distance per revolution or per cutter tooth are also sometimes used.
If variables such as cutter geometry and the rigidity of the machine tool and its tooling setup could be ideally maximized (and reduced to negligible constants), then only a lack of power (that is, kilowatts or horsepower) available to the spindle would prevent the use of the maximum possible speeds and feeds for any given workpiece material and cutter material. Of course, in reality those other variables are dynamic and not negligible, but there is still a correlation between power available and feeds and speeds employed. In practice, lack of rigidity is usually the limiting constraint.
The phrases "speeds and feeds" or "feeds and speeds" have sometimes been used metaphorically to refer to the execution details of a plan, which only skilled technicians (as opposed to designers or managers) would know.
Cutting speed.
Cutting speed may be defined as the rate at the workpiece surface, irrespective of the machining operation used. A cutting speed for mild steel of 100 ft/min is the same whether it is the speed of the cutter passing over the workpiece, such as in a turning operation, or the speed of the cutter moving past a workpiece, such as in a milling operation. The cutting conditions will affect the value of this surface speed for mild steel.
Schematically, speed at the workpiece surface can be thought of as the tangential speed at the tool-cutter interface, that is, how fast the material moves past the cutting edge of the tool, although "which surface to focus on" is a topic with several valid answers. In drilling and milling, the outside diameter of the tool is the widely agreed surface. In turning and boring, the surface can be defined on either side of the depth of cut, that is, either the starting surface or the ending surface, with neither definition being "wrong" as long as the people involved understand the difference. An experienced machinist summed this up succinctly as "the diameter I am turning from" versus "the diameter I am turning to." He uses the "from", not the "to", and explains why, while acknowledging that some others do not. The logic of focusing on the largest diameter involved (OD of drill or end mill, starting diameter of turned workpiece) is that this is where the highest tangential speed is, with the most heat generation, which is the main driver of tool wear.
There will be an optimum cutting speed for each material and set of machining conditions, and the spindle speed (RPM) can be calculated from this speed. Factors affecting the calculation of cutting speed are:
Cutting speeds are calculated on the assumption that optimum cutting conditions exist. These include:
The cutting "speed" is given as a set of constants that are available from the material manufacturer or supplier. The most common materials are available in reference books or charts, but will always be subject to adjustment depending on the cutting conditions. The following table gives the cutting speeds for a selection of common materials under one set of conditions. The conditions are a tool life of 1 hour, dry cutting (no coolant), and at medium feeds, so they may appear to be incorrect depending on circumstances. These cutting speeds may change if, for instance, adequate coolant is available or an improved grade of HSS is used (such as one that includes [cobalt]).
Machinability rating.
The machinability rating of a material attempts to quantify the machinability of various materials. It is expressed as a percentage or a normalized value. The American Iron and Steel Institute (AISI) determined machinability ratings for a wide variety of materials by running turning tests at 180 surface feet per minute (sfpm). It then arbitrarily assigned 160 Brinell B1112 steel a machinability rating of 100%. The machinability rating is determined by measuring the weighed averages of the normal cutting speed, surface finish, and tool life for each material. Note that a material with a machinability rating less than 100% would be more difficult to machine than B1112 and material and a value more than 100% would be easier.
Machinability ratings can be used in conjunction with the Taylor tool life equation, in order to determine cutting speeds or tool life. It is known that B1112 has a tool life of 60 minutes at a cutting speed of 100 sfpm. If a material has a machinability rating of 70%, it can be determined, with the above knowns, that in order to maintain the same tool life (60 minutes), the cutting speed must be 70 sfpm (assuming the same tooling is used).
When calculating for copper alloys, the machine rating is arrived at by assuming the 100 rating of 600 SFM. For example, phosphorus bronze (grades A–D) has a machinability rating of 20. This means that phosphor bronze runs at 20% the speed of 600 SFM or 120 SFM. However, 165 SFM is generally accepted as the basic 100% rating for "grading steels".
Formula
Cutting Speed (V)= [πDN]/1000 m/min
Where
D=Diameter of Workpiece in meter or millimeter
N=Spindle Speed in rpm
Spindle speed.
The spindle speed is the rotational frequency of the spindle of the machine, measured in revolutions per minute (RPM). The preferred speed is determined by working backward from the desired surface speed (sfm or m/min) and incorporating the diameter (of workpiece or cutter).
The spindle may hold the:
Excessive spindle speed will cause premature tool wear, breakages, and can cause tool chatter, all of which can lead to potentially dangerous conditions. Using the correct spindle speed for the material and tools will greatly enhance tool life and the quality of the surface finish.
For a given machining operation, the cutting speed will remain constant for most situations; therefore the spindle speed will also remain constant. However, facing, forming, parting off, and recess operations on a lathe or screw machine involve the machining of a constantly changing diameter. Ideally, this means changing the spindle speed as the cut advances across the face of the workpiece, producing constant surface speed (CSS). Mechanical arrangements to effect CSS have existed for centuries, but they were never applied commonly to machine tool control. In the pre-CNC era, the ideal of CSS was ignored for most work. For unusual work that demanded it, special pains were taken to achieve it. The introduction of CNC-controlled lathes has provided a practical, everyday solution via automated CSS Machining Process Monitoring and Control. By means of the machine's software and variable speed electric motors, the lathe can increase the RPM of the spindle as the cutter gets closer to the center of the part.
Grinding wheels are designed to be run at a maximum safe speed, the spindle speed of the grinding machine may be variable but this should only be changed with due attention to the safe working speed of the wheel. As a wheel wears it will decrease in diameter, and its effective cutting speed will be reduced. Some grinders have the provision to increase the spindle speed, which corrects for this loss of cutting ability; however, increasing the speed beyond the wheels rating will destroy the wheel and create a serious hazard to life and limb.
Generally speaking, spindle speeds and feed rates are less critical in woodworking than metalworking. Most woodworking machines including power saws such as circular saws and band saws, jointers, Thickness planers rotate at a fixed RPM. In those machines, cutting speed is regulated through the feed rate. The required feed rate can be extremely variable depending on the power of the motor, the hardness of the wood or other material being machined, and the sharpness of the cutting tool.
In woodworking, the ideal feed rate is one that is slow enough not to bog down the motor, yet fast enough to avoid burning the material. Certain woods, such as black cherry and maple are more prone to burning than others. The right feed rate is usually obtained by "feel" if the material is hand fed, or by trial and error if a power feeder is used. In thicknessers (planers), the wood is usually fed automatically through rubber or corrugated steel rollers. Some of these machines allow varying the feed rate, usually by changing pulleys. A slower feed rate usually results in a finer surface as more cuts are made for any length of wood.
Spindle speed becomes important in the operation of routers, spindle moulders or shapers, and drills. Older and smaller routers often rotate at a fixed spindle speed, usually between 20,000 and 25,000 rpm. While these speeds are fine for small router bits, using larger bits, say more than or 25 millimeters in diameter, can be dangerous and can lead to chatter. Larger routers now have variable speeds and larger bits require slower speed. Drilling wood generally uses higher spindle speeds than metal, and the speed is not as critical. However, larger diameter drill bits do require slower speeds to avoid burning.
Cutting feeds and speeds, and the spindle speeds that are derived from them, are the "ideal" cutting conditions for a tool. If the conditions are less than ideal then adjustments are made to the spindle's speed, this adjustment is usually a reduction in RPM to the closest available speed, or one that is deemed (through knowledge and experience) to be correct.
Some materials, such as machinable wax, can be cut at a wide variety of spindle speeds, while others, such as stainless steel require much more careful control as the cutting speed is critical, to avoid overheating both the cutter and workpiece. Stainless steel is one material that hardens very easily under cold working, therefore insufficient feed rate or incorrect spindle speed can lead to less than ideal cutting conditions as the work piece will quickly harden and resist the tool's cutting action. The liberal application of cutting fluid can improve these cutting conditions; however, the correct selection of speeds is the critical factor.
Spindle speed calculations.
Most metalworking books have nomograms or tables of spindle speeds and feed rates for different cutters and workpiece materials; similar tables are also likely available from the manufacturer of the cutter used.
The spindle speeds may be calculated for all machining operations once the SFM or MPM is known. In most cases, we are dealing with a cylindrical object such as a milling cutter or a workpiece turning in a lathe so we need to determine the speed at the periphery of this round object. This speed at the periphery (of a point on the circumference, moving past a stationary point) will depend on the rotational speed (RPM) and diameter of the object.
One analogy would be a skateboard rider and a bicycle rider travelling side by side along the road. For a given surface speed (the speed of this pair along the road) the rotational speed (RPM) of their wheels (large for the skater and small for the bicycle rider) will be different. This rotational speed (RPM) is what we are calculating, given a fixed surface speed (speed along the road) and known values for their wheel sizes (cutter or workpiece).
The following formulae may be used to estimate this value.
Approximation.
The exact RPM is not always needed, a close approximation will work (using 3 for the value of formula_1).
e.g. for a cutting speed of 100 ft/min (a plain HSS steel cutter on mild steel) and diameter of 10 inches (the cutter or the work piece)
and, for an example using metric values, where the cutting speed is 30 m/min and a diameter of 10 mm (0.01 m),
Accuracy.
However, for more accurate calculations, and at the expense of simplicity, this formula can be used:
and using the same example
and using the same example as above
where:
Feed rate.
Feed rate is the velocity at which the cutter is fed, that is, advanced against the workpiece. It is expressed in units of distance per revolution for turning and boring (typically "inches per revolution" ["ipr"] or "millimeters per revolution"). It can be expressed thus for milling also, but it is often expressed in units of distance per time for milling (typically "inches per minute" ["ipm"] or "millimeters per minute"), with considerations of how many teeth (or flutes) the cutter has then determined what that means for each tooth.
Feed rate is dependent on the:
When deciding what feed rate to use for a certain cutting operation, the calculation is fairly straightforward for single-point cutting tools, because all of the cutting work is done at one point (done by "one tooth", as it were). With a milling machine or jointer, where multi-tipped/multi-fluted cutting tools are involved, then the desired feed rate becomes dependent on the number of teeth on the cutter, as well as the desired amount of material per tooth to cut (expressed as chip load). The greater the number of cutting edges, the higher the feed rate permissible: for a cutting edge to work efficiently it must remove sufficient material to cut rather than rub; it also must do its fair share of work.
The ratio of the spindle speed and the feed rate controls how aggressive the cut is, and the nature of the swarf formed.
Formula to determine feed rate.
This formula can be used to figure out the feed rate that the cutter travels into or around the work. This would apply to cutters on a milling machine, drill press and a number of other machine tools. This is not to be used on the lathe for turning operations, as the feed rate on a lathe is given as "feed per revolution."
formula_8
Where:
Depth of cut.
Cutting speed and feed rate come together with "depth of cut" to determine the "material removal rate", which is the volume of workpiece material (metal, wood, plastic, etc.) that can be removed per time unit.
Interrelationship of theory and practice.
Speed-and-feed selection is analogous to other examples of applied science, such as meteorology or pharmacology, in that the theoretical modeling is necessary and useful but can never fully predict the reality of specific cases because of the massively multivariate environment. Just as weather forecasts or drug dosages can be modeled with fair accuracy, but never with complete certainty, machinists can predict with charts and formulas the approximate speed and feed values that will work best on a particular job, but cannot know the exact optimal values until running the job. In CNC machining, usually the programmer programs speeds and feedrates that are as maximally tuned as calculations and general guidelines can supply. The operator then fine-tunes the values while running the machine, based on sights, sounds, smells, temperatures, tolerance holding, and tool tip lifespan. Under proper management, the revised values are captured for future use, so that when a program is run again later, this work need not be duplicated.
As with meteorology and pharmacology, however, the interrelationship of theory and practice has been developing over decades as the theory part of the balance becomes more advanced thanks to information technology. For example, an effort called the Machine Tool Genome Project is working toward providing the computer modeling (simulation) needed to predict optimal speed-and-feed combinations for particular setups in any internet-connected shop with less local experimentation and testing. Instead of the only option being the measuring and testing of the behavior of its own equipment, it will benefit from others' experience and simulation; in a sense, rather than 'reinventing a wheel', it will be able to 'make better use of existing wheels already developed by others in remote locations'.
Academic research examples.
Speeds and feeds have been studied scientifically since at least the 1890s. The work is typically done in engineering laboratories, with the funding coming from three basic roots: corporations, governments (including their militaries), and universities. All three types of institution have invested large amounts of money in the cause, often in collaborative partnerships. Examples of such work are highlighted below.
In the 1890s through 1910s, Frederick Winslow Taylor performed turning experiments that became famous (and seminal). He developed Taylor's Equation for Tool Life Expectancy.
Scientific study by Holz and De Leeuw of the Cincinnati Milling Machine Company did for milling cutters what F. W. Taylor had done for single-point cutters.
"Following World War II, many new alloys were developed. New standards were needed to increase [U.S.] American productivity. Metcut Research Associates, with technical support from the Air Force Materials Laboratory and the Army Science and Technology Laboratory, published the first Machining Data Handbook in 1966. The recommended speeds and feeds provided in this book were the result of extensive testing to determine optimum tool life under controlled conditions for every material of the day, operation and hardness."
A study on the effect of the variation of cutting parameters in the surface integrity in turning of an AISI 304 stainless steel revealed that the feed rate has the greatest impairing effect on the quality of the surface, and that besides the achievement of the desired roughness profile, it is necessary to analyze the effect of speed and feed on the creation of micropits and microdefects on the machined surface. Moreover, they found that the conventional empirical relation that relates feed rate to roughness value does not fit adequately for low cutting speeds.


Switching Kalman filter

The switching Kalman filtering (SKF) method is a variant of the Kalman filter. In its generalised form, it is often attributed to Kevin P. Murphy, but related switching state-space models have been in use.
Applications.
Applications of the switching Kalman filter include: Brain–computer interfaces and neural decoding, real-time decoding for continuous neural-prosthetic control, and sensorimotor learning in humans.
It also has application in econometrics, signal processing, tracking, computer vision, etc. It is an alternative to the Kalman filter when the system's state has a discrete component. The additional error when using a Kalman filter instead of a Switching Kalman filter may be quantified in terms of the switching system's parameters. For example, when an industrial plant has "multiple discrete modes of behaviour, each of which having a linear (Gaussian) dynamics".
Model.
There are several variants of SKF discussed in.
Special case.
In the simpler case, switching state-space models are defined based on a switching variable which evolves independent of the hidden variable. The probabilistic model of such variant of SKF is as the following:
[This section is badly written: It does not explain the notation used below.]
The hidden variables include not only the continuous formula_2, but also a discrete *switch* (or switching) variable formula_3. The dynamics of the switch variable are defined by the term formula_4. The probability model of formula_2 and formula_6 can depend on formula_3.
The switch variable can take its values from a set formula_8. This changes the joint distribution formula_9 which is a separate multivariate Gaussian distribution in case of each value of formula_3.
General case.
In more generalised variants, the switch variable affects the dynamics of formula_11, e.g. through formula_12.
The filtering and smoothing procedure for general cases is discussed in.


Optimal projection equations

In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller.
The linear-quadratic-Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, incomplete state information (i.e. not all the state variables are measured and available for feedback) also disturbed by additive white Gaussian noise and quadratic costs. Moreover, the solution is unique and constitutes a linear dynamic feedback control law that is easily computed and implemented. Finally the LQG controller is also fundamental to the optimal perturbation control of non-linear systems.
The LQG controller itself is a dynamic system like the system it controls. Both systems have the same state dimension. Therefore, implementing the LQG controller may be problematic if the dimension of the system state is large. The reduced-order LQG problem (fixed-order LQG problem) overcomes this by fixing a-priori the number of states of the LQG controller. This problem is more difficult to solve because it is no longer separable. Also the solution is no longer unique. Despite these facts numerical algorithms are available to solve the associated optimal projection equations.
Mathematical problem formulation and solution.
Continuous-time.
The reduced-order LQG control problem is almost identical to the conventional full-order LQG control problem. Let formula_1 represent the state of the reduced-order LQG controller. Then the only difference is that the state dimension formula_2 of the LQG controller is a-priori fixed to be smaller than formula_3, the state dimension of the controlled system.
The reduced-order LQG controller is represented by the following equations:
These equations are deliberately stated in a format that equals that of the conventional full-order LQG controller. For the reduced-order LQG control problem it is convenient to rewrite them as
where
The matrices formula_9 and formula_10 of the reduced-order LQG controller are determined by the so-called optimal projection equations (OPE).
The square optimal projection matrix formula_11 with dimension formula_12 is central to the OPE. The rank of this matrix is almost everywhere equal to formula_13 The associated projection is an oblique projection: formula_14 The OPE constitute four matrix differential equations. The first two equations listed below are generalizations of the matrix Riccati differential equations associated to the conventional full-order LQG controller. In these equations formula_15 denotes formula_16 where formula_17 is the identity matrix of dimension formula_18.
If the dimension of the LQG controller is not reduced, that is if formula_21, then formula_22 and the two equations above become the uncoupled matrix Riccati differential equations associated to the conventional full-order LQG controller. If formula_23 the two equations are coupled by the oblique projection formula_24 This reveals why the reduced-order LQG problem is not separable. The oblique projection formula_11 is determined from two additional matrix differential equations which involve rank conditions. Together with the previous two matrix differential equations these are the OPE. To state the additional two matrix differential equations it is convenient to introduce the following two matrices:
Then the two additional matrix differential equations that complete the OPE are as follows:
with
Here * denotes the group generalized inverse or Drazin inverse that is unique and given by
where + denotes the Moore–Penrose pseudoinverse.
The matrices formula_34 must all be nonnegative symmetric. Then they constitute a solution of the OPE that determines the reduced-order LQG controller matrices formula_35 and formula_10:
In the equations above the matrices formula_41 are two matrices with the following properties:
They can be obtained from a projective factorization of formula_43.
The OPE can be stated in many different ways that are all equivalent. To identify the equivalent representations the following identities are especially useful:
Using these identities one may for instance rewrite the first two of the optimal projection equations as follows:
This representation is both relatively simple and suitable for numerical computations.
If all the matrices in the reduced-order LQG problem formulation are time-invariant and if the horizon formula_49 tends to infinity, the optimal reduced-order LQG controller becomes time-invariant and so do the OPE. In that case the derivatives on the left hand side of the OPE are zero.
Discrete-time.
Similar to the continuous-time case, in the discrete-time case the difference with the conventional discrete-time full-order LQG problem is the a-priori fixed reduced-order formula_50 of the LQG controller state dimension. As in continuous-time, to state the discrete-time OPE it is convenient to introduce the following two matrices:
Then the discrete-time OPE is
The oblique projection matrix is given by
The nonnegative symmetric matrices formula_60 that solve the discrete-time OPE determine the reduced-order LQG controller matrices formula_61 and formula_62:
In the equations above the matrices formula_67 are two matrices with the following properties:
They can be obtained from a projective factorization of formula_69. To identify equivalent representations of the discrete-time OPE the following identities are especially useful:
As in the continuous-time case if all the matrices in the problem formulation are time-invariant and if the horizon formula_71 tends to infinity the reduced-order LQG controller becomes time-invariant. Then the discrete-time OPE converge to a steady state solution that determines the time-invariant reduced-order LQG controller.
The discrete-time OPE apply also to discrete-time systems with variable state, input and output dimensions (discrete-time systems with time-varying dimensions). Such systems arise in the case of digital controller design if the sampling occurs asynchronously.


Future Immersive Training Environment

The Future Immersive Training Environment (FITE) Joint Capability Technology Demonstration (JCTD) was a three-year $36-million Department of Defense initiative to demonstrate the value of advanced small unit immersive infantry training systems. It demonstrated infantry applications of virtual reality, mixed reality, and augmented reality.


Hinge loss

In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs).
For an intended output and a classifier score , the hinge loss of the prediction is defined as
Note that formula_2 should be the "raw" output of the classifier's decision function, not the predicted class label. For instance, in linear SVMs, formula_3, where formula_4 are the parameters of the hyperplane and formula_5 is the input variable(s).
When and have the same sign (meaning predicts the right class) and formula_6, the hinge loss formula_7. When they have opposite signs, formula_8 increases linearly with , and similarly if formula_9, even if it has the same sign (correct prediction, but not by enough margin).
Extensions.
While binary SVMs are commonly extended to multiclass classification in a one-vs.-all or one-vs.-one fashion,
it is also possible to extend the hinge loss itself for such an end. Several different variations of multiclass hinge loss have been proposed. For example, Crammer and Singer
defined it for a linear classifier as
Where formula_11 is the target label, formula_12 and formula_13 are the model parameters.
Weston and Watkins provided a similar definition, but with a sum rather than a max:
In structured prediction, the hinge loss can be further extended to structured output spaces. Structured SVMs with margin rescaling use the following variant, where denotes the SVM's parameters, the SVM's predictions, the joint feature function, and the Hamming loss:
Optimization.
The hinge loss is a convex function, so many of the usual convex optimizers used in machine learning can work with it. It is not differentiable, but has a subgradient with respect to model parameters of a linear SVM with score function formula_16 that is given by
However, since the derivative of the hinge loss at formula_18 is undefined, smoothed versions may be preferred for optimization, such as Rennie and Srebro's
or the quadratically smoothed
suggested by Zhang. The modified Huber loss formula_21 is a special case of this loss function with formula_22, specifically formula_23.


Progressive stamping

Progressive Die is a metalworking method that can encompass punching, coining, bending and several other ways of modifying metal raw material, combined with an automatic feeding system.
The feeding system pushes a strip of metal (as it unrolls from a coil) through all of the stations of a progressive stamping die. Each station performs one or more operations until a finished part is made. The final station is a cutoff operation, which separates the finished part from the carrying web. The carrying web, along with metal that is punched away in previous operations, is treated as scrap metal. Both are cut away, knocked down (or out of the dies) and then ejected from the die set, and in mass production are often transferred to scrap bins via underground scrap material conveyor belts.
The progressive stamping die is placed into a reciprocating stamping press. As the press moves up, the top die moves with it, which allows the material to feed. When the press moves down, the die closes and performs the stamping operation. With each stroke of the press, a completed part is removed from the die.
Since additional work is done in each "station" of the die, it is important that the strip be advanced very precisely so that it aligns within a few thousandths of an inch as it moves from station to station. Bullet shaped or conical "pilots" enter previously pierced round holes in the strip to assure this alignment since the feeding mechanism usually cannot provide the necessary precision in feed length.
Progressive stamping can also be produced on transfer presses. These are presses that transfer the components from one station to the next with the use of mechanical "fingers". For mass production of stamped parts which do require complicated in-press operations, it is always advisable to use a progressive press. One of the advantages of this type of press is the production cycle time. Depending upon the part, productions can easily run well over 800 parts/minute. One of the disadvantages of this type of press is that it is not suitable for high precision deep drawing which is when the depth of the stamping exceeds the diameter of the part. When necessary, this process is performed upon a transfer press, which run at slower speeds, and rely on the mechanical fingers to hold the component in place during the entire forming cycle. In the case of the progressive press, only part of the forming cycle can be guided by spring-loaded sleeves or similar, which result in concentricity and ovality issues and non uniform material thickness.
Other disadvantages of progressive presses compared to transfer presses are: increased raw material input required to transfer parts, tools are much more expensive because they are made in blocks with very little independent regulation per station; impossibility to perform processes in the press that require the part leave the strip (example beading, necking, flange curling, thread rolling, rotary stamping etc.).
The dies are usually made of tool steel to withstand the high shock loading involved, retain the necessary sharp cutting edge, and resist the abrasive forces involved.
The cost is determined by the number of features, which determine what tooling will need to be used. Engineers keep the features as simple as possible to keep the cost of tooling to a minimum. Features that are close together produce a problem because it may not provide enough clearance for the punch, which could result in another station. It can also be problematic to have narrow cuts and protrusions.
Applications.
A representative example of the product of a progressive die is the lid of a beverage can. The pull tab is made in one progressive stamping process and the lid & assembly is made in another, the pull tab simultaneously feeding at a right angle into the lid & assembly process. Also various car brake calipers have plates that are bent into shape, possibly cut too using these methods.


Iterative learning control

Iterative Learning Control (ILC) is a method of tracking control for systems that work in a repetitive mode. Examples of systems that operate in a repetitive manner include robot arm manipulators, chemical batch processes and reliability testing rigs. In each of these tasks the system is required to perform the same action over and over again with high precision. This action is represented by the objective of accurately tracking a chosen reference signal formula_1 on a finite time interval.
Repetition allows the system to improve tracking accuracy from repetition to repetition, in effect learning the required input needed to track the reference exactly. The learning process uses information from previous repetitions to improve the control signal ultimately enabling a suitable control action can be found iteratively. The internal model principle yields conditions under which perfect tracking can be achieved but the design of the control algorithm still leaves many decisions to be made to suit the application. A typical, simple control law is of the form:
where formula_3 is the input to the system during the pth repetition, formula_4 is the tracking error during the pth repetition and K is a design parameter representing operations on formula_4. Achieving perfect tracking through iteration is represented by the mathematical requirement of convergence of the input signals as formula_6 becomes large whilst the rate of this convergence represents the desirable practical need for the learning process to be rapid. There is also the need to ensure good algorithm performance even in the presence of uncertainty about the details of process dynamics. The operation formula_7 is crucial to achieving design objectives and ranges from simple scalar gains to sophisticated optimization computations.


Controllability

Controllability is an important property of a control system and plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
Controllability and observability are dual aspects of the same problem.
Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.
The following are examples of variations of controllability notions which have been introduced in the systems and control literature:
State controllability.
The state of a deterministic system, which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any given time. In particular, no information on the past of a system is needed to help in predicting the future, if the states at the present time are known and all current and future values of the control variables (those whose values can be chosen) are known.
"Complete state controllability" (or simply "controllability" if no other context is given) describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any final state in a finite time interval.
That is, we can informally define controllability as follows:
If for any initial state formula_1 and any final state formula_2 there exists an input sequence to transfer the system state from formula_1 to formula_2 in a finite time interval, then the system modeled by the state-space representation is controllable. For the simplest example of a continuous, LTI system, the row dimension of the state space expression formula_5 determines the interval; each row contributes a vector in the state space of the system. If there are not enough such vectors to span the state space of formula_6, then the system cannot achieve controllability. It may be necessary to modify formula_7 and formula_8 to better approximate the underlying differential relationships it estimates to achieve controllability.
Controllability does not mean that a reached state can be maintained, merely that any state can be reached.
Controllability does not mean that arbitrary paths can be made through state space, only that there exists a path within the prescribed finite time interval.
Continuous linear systems.
Consider the continuous linear system
There exists a control formula_11 from state formula_12 at time formula_13 to state formula_14 at time formula_15 if and only if formula_16 is in the column space of
where formula_18 is the state-transition matrix, and formula_19 is the Controllability Gramian.
In fact, if formula_20 is a solution to formula_21 then a control given by formula_22 would make the desired transfer.
Note that the matrix formula_23 defined as above has the following properties:
Rank condition for controllability.
The Controllability Gramian involves integration of the state-transition matrix of a system. A simpler condition for controllability is a rank condition analogous to the Kalman rank condition for time-invariant systems.
Consider a continuous-time linear system formula_31 smoothly varying in an interval formula_32 of formula_33:
The state-transition matrix formula_18 is also smooth. Introduce the n x m matrix-valued function formula_37 and define
Consider the matrix of matrix-valued functions obtained by listing all the columns of the formula_40, formula_41:
formula_42.
If there exists a formula_43 and a nonnegative integer k such that formula_44, then formula_31 is controllable.
If formula_31 is also analytically varying in an interval formula_32, then formula_31 is controllable on every nontrivial subinterval of formula_32 if and only if there exists a formula_43 and a nonnegative integer k such that formula_51.
The above methods can still be complex to check, since it involves the computation of the state-transition matrix formula_18. Another equivalent condition is defined as follow. Let formula_53, and for each formula_54, define
In this case, each formula_57 is obtained directly from the data formula_58 The system is controllable if there exists a formula_43 and a nonnegative integer formula_60 such that formula_61.
Example.
Consider a system varying analytically in formula_62 and matrices
formula_63, formula_64
Then
formula_65 and since this matrix has rank 3, the system is controllable on every nontrivial interval of formula_33.
Continuous linear time-invariant (LTI) systems.
Consider the continuous linear time-invariant system
where
The formula_83 controllability matrix is given by
The system is controllable if the controllability matrix has full row rank (i.e. formula_85).
Discrete linear time-invariant (LTI) systems.
For a discrete-time linear state-space system (i.e. time variable formula_86) the state equation is
where formula_75 is an formula_76 matrix and formula_77 is a formula_78 matrix (i.e. formula_73 is formula_93 inputs collected in a formula_74 vector). The test for controllability is that the formula_83 matrix
has full row rank (i.e., formula_97). That is, if the system is controllable, formula_98 will have formula_99 columns that are linearly independent; if formula_99 columns of formula_98 are linearly independent, each of the formula_99 states is reachable by giving the system proper inputs through the variable formula_103.
Derivation.
Given the state formula_104 at an initial time, arbitrarily denoted as "k"=0, the state equation gives formula_105 then formula_106 and so on with repeated back-substitutions of the state variable, eventually yielding
or equivalently
Imposing any desired value of the state vector formula_109 on the left side, this can always be solved for the stacked vector of control vectors if and only if the matrix of matrices at the beginning of the right side has full row rank.
Example.
For example, consider the case when formula_110 and formula_111 (i.e. only one control input). Thus, formula_77 and formula_113 are formula_114 vectors. If formula_115 has rank 2 (full rank), and so formula_77 and formula_117 are linearly independent and span the entire plane. If the rank is 1, then formula_77 and formula_117 are collinear and do not span the plane.
Assume that the initial state is zero.
At time formula_120: formula_121
At time formula_122: formula_123
At time formula_120 all of the reachable states are on the line formed by the vector formula_77.
At time formula_122 all of the reachable states are linear combinations of formula_117 and formula_77.
If the system is controllable then these two vectors can span the entire plane and can be done so for time formula_129.
The assumption made that the initial state is zero is merely for convenience.
Clearly if all states can be reached from the origin then any state can be reached from another state (merely a shift in coordinates).
This example holds for all positive formula_99, but the case of formula_110 is easier to visualize.
Analogy for example of "n" = 2.
Consider an analogy to the previous example system.
You are sitting in your car on an infinite, flat plane and facing north.
The goal is to reach any point in the plane by driving a distance in a straight line, come to a full stop, turn, and driving another distance, again, in a straight line.
If your car has no steering then you can only drive straight, which means you can only drive on a line (in this case the north-south line since you started facing north).
The lack of steering case would be analogous to when the rank of formula_79 is 1 (the two distances you drove are on the same line).
Now, if your car did have steering then you could easily drive to any point in the plane and this would be the analogous case to when the rank of formula_79 is 2.
If you change this example to formula_134 then the analogy would be flying in space to reach any position in 3D space (ignoring the orientation of the aircraft).
You are allowed to:
Although the 3-dimensional case is harder to visualize, the concept of controllability is still analogous.
Nonlinear systems.
Nonlinear systems in the control-affine form
are locally accessible about formula_12 if the accessibility distribution formula_137 spans formula_99 space, when formula_99 equals the rank of formula_140 and R is given by:
Here, formula_142 is the repeated Lie bracket operation defined by
The controllability matrix for linear systems in the previous section can in fact be derived from this equation.
Null Controllability.
If a discrete control system is null-controllable, it means that there exists a controllable formula_103 so that formula_145 for some initial state formula_146. In other words, it is equivalent to the condition that there exists a matrix formula_147 such that formula_148 is nilpotent.
This can be easily shown by controllable-uncontrollable decomposition.
Output controllability.
"Output controllability" is the related notion for the output of the system (denoted "y" in the previous equations); the output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. It is not necessary that there is any relationship between state controllability and output controllability. In particular:
For a linear continuous-time system, like the example above, described by matrices formula_75, formula_77, formula_79, and formula_81, the formula_153 "output controllability matrix"
has full row rank (i.e. rank formula_155) if and only if the system is output controllable.
Controllability under input constraints.
In systems with limited control authority, it is often no longer possible to move any initial state to any final state inside the controllable subspace. This phenomenon is caused by constraints on the input that could be inherent to the system (e.g. due to saturating actuator) or imposed on the system for other reasons (e.g. due to safety-related concerns). The controllability of systems with input and state constraints is studied in the context of reachability and viability theory.
Controllability in the behavioral framework.
In the so-called behavioral system theoretic approach due to Willems (see people in systems and control), models considered do not directly define an input–output structure. In this framework systems are described by admissible trajectories of a collection of variables, some of which might be interpreted as inputs or outputs.
A system is then defined to be controllable in this setting, if any past part of a behavior (trajectory of the external variables) can be concatenated with any future trajectory of the behavior in such a way that the concatenation is contained in the behavior, i.e. is part of the admissible system behavior.
Stabilizability.
A slightly weaker notion than controllability is that of stabilizability. A system is said to be stabilizable when all uncontrollable state variables can be made to have stable dynamics. Thus, even though some of the state variables cannot be controlled (as determined by the controllability test above) all the state variables will still remain bounded during the system's behavior.
Reachable set.
Let T ∈ Т and x ∈ "X" (where X is the set of all possible states and Т is an interval of time). The reachable set from x in time T is defined as:
formula_156, where xz denotes that there exists a state transition from x to z in time T.
For autonomous systems the reachable set is given by :
where R is the controllability matrix.
In terms of the reachable set, the system is controllable if and only if formula_158.
Proof
We have the following equalities:
Considering that the system is controllable, the columns of R should be linearly independent. So:
A related set to the reachable set is the controllable set, defined by:
The relation between reachability and controllability is presented by Sontag:
(a) An n-dimensional discrete linear system is controllable if and only if:
(b) A continuous-time linear system is controllable if and only if:
if and only if formula_168 for all e>0.
Example
Let the system be an n dimensional discrete-time-invariant system from the formula:
It follows that the future state is in formula_170 ⇔ it is in the image of the linear map:
which maps,
When formula_173 and formula_174 we identify R(A,B) with a n by nm matrix whose columns are the columns of formula_175 in that order. If the system is controllable the rank of formula_171 is n. If this is truth, the image of the linear map R is all of X. Based on that, we have:


List of people in systems and control

This is an alphabetical list of people who have made significant contributions in the fields of system analysis and control theory.
Eminent researchers.
The eminent researchers (born after 1920) include the winners of at least one award of the IEEE Control Systems Award, the Giorgio Quazza Medal, the Hendrik W. Bode Lecture Prize, the Richard E. Bellman Control Heritage Award, the Rufus Oldenburger Medal, or higher awards such as the IEEE Medal of Honor and the National Medal of Science. The earlier pioneers such as Nicolas Minorsky (1885–1970), Harry Nyquist (1889–1976), Harold Locke Hazen (1901–1980), Charles Stark Draper (1901–1987), Hendrik Wade Bode (1905–1982), Gordon S. Brown (1907–1996), John F. Coales (1907–1999), Rufus Oldenburger (1908–1969), John R. Ragazzini (1912–1988), Nathaniel B. Nichols (1914–1997), John Zaborszky (1914–2008) and Harold Chestnut (1917–2001) are not included.
Historical figures in systems and control.
These people have made outstanding historical contributions to systems and control.


Gun drill

Gun drills (through coolant drill) are straight fluted drills which allow cutting fluid (either compressed air or a suitable liquid) to be injected through the drill's hollow body to the cutting face. They are used for deep hole drilling—a depth-to-diameter ratio of 300:1 or more is possible. Gun barrels are the obvious example; hence the name. Other uses include moldmaking, diemaking, and the manufacture of combustion engine parts such as crankcase, cylinder head, and woodwind musical instruments, such as uilleann pipes, as gun drills can drill long straight holes in metal, wood, and some plastics. The coolant provides lubrication and cooling to the cutting edges and removes the swarf or chips from the hole. Modern gun drills use carbide tips to prolong life and reduce total cost when compared with steel tips. Speed of drilling depends on the material being drilled, rotational speed, and the drill diameter; a high speed drill can cut a hole in P20 steel at 30 inches per minute.
Gun drilling can be done on several kinds of machine tools. On lathes, it is generally practical with hole depths of less than 50 diameters. There are also purpose-built gun drilling machines, where longer aspect ratios can be drilled.
Requirement.
With a standard twist drill, it is difficult to drill a straight and accurately sized hole of a depth more than about 5 times the diameter. This is a problem in many manufacturing processes, especially the firearms industry: the barrel of a gun must be very straight and accurately sized. Gun barrels are far longer than their inside diameter; as an example, the caliber barrel of the M16 rifle is long, nearly 90 times the diameter of the bore. The gun drill was developed to drill such long, straight holes.
Gun drilling is possible over a range of depths and diameters. For diameters between , gun drilling can be performed successfully with special equipment. It is a common process between in diameter. It is also possible for the range, however less efficient than BTA deep hole drilling.
Types.
There are three basic types of deep hole drilling. Processes are categorized by how the cutting coolant flushes heat and chips from the cutting face. The three types of deep drilling are:


Control reconfiguration

Control reconfiguration is an active approach in control theory to achieve fault-tolerant control for dynamic systems. It is used when severe faults, such as actuator or sensor outages, cause a break-up of the control loop, which must be restructured to prevent failure at the system level. In addition to loop restructuring, the controller parameters must be adjusted to accommodate changed plant dynamics. Control reconfiguration is a building block toward increasing the dependability of systems under feedback control.
Reconfiguration problem.
Fault modelling.
The figure to the right shows a plant controlled by a controller in a standard control loop.
The nominal linear model of the plant is
formula_1
The plant subject to a fault (indicated by a red arrow in the figure) is modelled in general by
formula_2
where the subscript formula_3 indicates that the system is faulty. This approach models multiplicative faults by modified system matrices. Specifically, actuator faults are represented by the new input matrix formula_4, sensor faults are represented by the output map formula_5, and internal plant faults are represented by the system matrix formula_6.
The upper part of the figure shows a supervisory loop consisting of "fault detection and isolation" (FDI) and "reconfiguration" which changes the loop by
To this end, the vectors of inputs and outputs contain "all available signals", not just those used by the controller in fault-free operation.
Alternative scenarios can model faults as an additive external signal formula_9 influencing the state derivatives and outputs as follows:
formula_10
Reconfiguration goals.
The goal of reconfiguration is to keep the reconfigured control-loop performance sufficient for preventing plant shutdown. The following goals are distinguished:
Internal stability of the reconfigured closed loop is usually the minimum requirement. The equilibrium recovery goal (also referred to as weak goal) refers to the steady-state output equilibrium which the reconfigured loop reaches after a given constant input. This equilibrium must equal the nominal equilibrium under the same input (as time tends to infinity). This goal ensures steady-state reference tracking after reconfiguration. The output trajectory recovery goal (also referred to as strong goal) is even stricter. It requires that the dynamic response to an input must equal the nominal response at all times. Further restrictions are imposed by the state trajectory recovery goal, which requires that the state trajectory be restored to the nominal case by the reconfiguration under any input.
Usually a combination of goals is pursued in practice, such as the equilibrium-recovery goal with stability.
The question whether or not these or similar goals can be reached for specific faults is addressed by reconfigurability analysis.
Reconfiguration approaches.
Fault hiding.
This paradigm aims at keeping the nominal controller in the loop. To this end, a reconfiguration block can be placed between the faulty plant and the nominal controller. Together with the faulty plant, it forms the reconfigured plant. The reconfiguration block has to fulfill the requirement that the behaviour of the reconfigured plant matches the behaviour of the nominal, that is fault-free plant.
Linear model following.
In linear model following, a formal feature of the nominal closed loop is attempted to be recovered. In the classical pseudo-inverse method, the closed loop system matrix formula_11 of a state-feedback control structure is used. The new controller formula_12 is found to approximate formula_13 in the sense of an induced matrix norm.
In perfect model following, a dynamic compensator is introduced to allow for the exact recovery of the complete loop behaviour under certain conditions.
In eigenstructure assignment, the nominal closed loop eigenvalues and eigenvectors (the eigenstructure) is recovered to the nominal case after a fault.
Optimisation-based control schemes.
Optimisation control schemes include: linear-quadratic regulator design (LQR), model predictive control (MPC) and eigenstructure assignment methods.
Probabilistic approaches.
Some probabilistic approaches have been developed.
Learning control.
There are learning automata, neural networks, etc.
Mathematical tools and frameworks.
The methods by which reconfiguration is achieved differ considerably. The following list gives an overview of mathematical approaches that are commonly used.
See also.
Prior to control reconfiguration, it must be at least determined whether a fault has occurred (fault detection) and if so, which components are affected (fault isolation). Preferably, a model of the faulty plant should be provided (fault identification). These questions are addressed by fault diagnosis methods.
Fault accommodation is another common approach to achieve fault tolerance. In contrast to control reconfiguration, accommodation is limited to internal controller changes. The sets of signals manipulated and measured by the controller are fixed, which means that the loop cannot be restructured.


Stabilized soil mixing plant

A stabilized soil mixing plant is a combination of kinds of machines used for mixing stabilized soil, which is used for highway construction, municipal road projects, and fertile airport areas. The plant produces stabilized soil with different gradings in a continuous way. Such a plant usually contains a cement silo, measuring and conveying system, and mixing devices.
Stabilized soil.
Stabilized soil is a mixture of lime, cement, coal ash, soil, sand, and other aggregates.
Classification.
Stabilized soil mixing plants are of two kinds: the portable stabilized soil mixing plant and the stationary stabilized soil mixing plant. The portable stabilized soil mixing plant has wheels on each part and can be driven by a trailer, but has low productivity. The stationary plant has larger productivity but is less flexible, and needs a firm groundwork.
Operating principle.
All aggregates like lime, sand, soil, coal ash, and other materials are loaded into batching hoppers by a loading machine. After measuring, the belt feeder transports the aggregates into a mixing device. Meanwhile, stabilizing powders like lime or cement are transferred from a powder material warehouse to the batch hopper by a spiral conveyor, and then moved to the belt feeder by a powder material feeder. All ingredients then go into the mixing device for final processing. Finally, the feeding belt conveyor takes the final product and delivers it to the storage warehouse.


Roll bonding

Roll bonding is a solid state, cold welding process, obtained through flat rolling of sheet metals. In roll bonding, two or more layers of different metals are passed through a pair of flat rollers under sufficient pressure to bond the layers. The pressure is high enough to deform the metals and reduce the combined thickness of the clad material. The mating surfaces must be previously prepared (scratched, cleaned, degreased) in order to increase their friction coefficient and remove any oxide layers.
The process can be performed at room temperature or at warm conditions. In warm roll bonding, heat is applied to pre-heat the sheets just before rolling, in order to increase their ductility and improve the strength of the weld. The strength of the rolled bonds depends on the main process parameters, including the rolling conditions (entry temperature of the sheets, amount of thickness reduction, rolling speed, etc.), the pre-rolling treatment conditions (annealing temperature and time, surface preparation techniques, etc.) and the post-rolling heat treatments.
Applications.
The applications of roll bonding can be used for cladding of metal sheets, or as a sub-step of the accumulative roll bonding. Bonding of the sheets can be controlled by painting a pattern on one sheet; only the bare metal surfaces bond, and the un-bonded portion can be inflated if the sheet is heated and the coating vaporizes. This is used to make heat exchangers for refrigeration equipment.


Augmented marked graph

An augmented marked graph is basically a Petri net with a specific set of places called resource places.
If removing these resource places and their associated arcs, it will become a marked graph where every cycle is marked. For each resource place, there are pairs of outgoing and incoming transitions connected by elementary paths.
Application.
Augmented marked graphs are often used for modelling systems with shared resources, such as manufacturing systems. Based on the special properties of augmented marked graphs, the properties of the modelled systems, such as liveness, boundedness and reversibility, can be effectively analyzed.


Clinching

In metalworking, clinching or press-joining is a bulk sheet metal forming process aimed at joining thin metal sheets without additional components, using special tools to plastically form an interlock between two or more sheets. The process is generally performed at room temperature, but in some special cases the sheets can be pre-heated to improve the material ductility and thereby avoid the formation of cracks during the process. Clinching is characterized by a series of advantages over competitive technologies:
Tools.
Because the process involves relatively low forces (ranging from 5 to 50 kN depending on the material to join, type of tools and sheet thicknesses), clinching generally involves reduced size (often portable) machines. The tools typically consist of a punch and a die. Different tools have been developed so far, which can be classified in round and rectangular tools. Round clinching tools include: fixed grooved dies, split dies (with 2–4 movable sectors) and flat dies. Such tools produce round joints which show almost identical mechanical behaviors in all plane directions. When round tools are adopted, the integrity of the sheet in the joint must be guaranteed in order to preserve a good mechanical behavior of the joints.
On the other hand, rectangular clinched joints exhibit behaviors which depend on the loading direction and both sheets are intentionally sheared along the "long direction" in order to produce the interlock. The choice of the tools is highly influenced by:
In addition, the choice of the clinching tools highly affects the joining strength and the absorbed energy of a clinched connection other than the joining force. Rectangular tools, for example, require lower joining forces than round tools since the material shearing, while among the round clinching tools split dies require the minimum joining force and the largest interlock.
One benefit of clinching is the capability to join prepainted sheet metal commonly used in the appliance industry without damaging the painted surface. Clinching is an important means of fastening aluminum panels, such as hoods and decklids, in the automotive industry, due to the difficulty of spot welding of aluminum.
Main advantages as compared to welding.
Clinching is used primarily in the automotive, appliance and electronic industries, where it often replaces spot welding. Clinching does not require electricity or cooling of the electrodes commonly associated with spot welding. Being a mechanical joining process, clinching can be used to join materials showing no electrical conductivity such as polymers or plastic-metal composites. In addition, it does not require a substrate preparation such as pre-cleaning of surfaces which is required for welding processes. This fact contributes to reduce the joining costs and the environmental impact (since chemical cleaning is not required).
Clinching does not generate sparks or fumes. The strength of a clinched joint can be tested non-destructively using a simple measuring instrument to measure the remaining thickness at the bottom of the joint, of the diameter of the produced button depending on the type of tools employed. Life expectancy for clinching tools is in the hundreds of thousands of cycles, making it an economical process. Clinched connections performed on aluminum sheets have higher fatigue life as compared to spot welding.
Main advantages as compared to adhesive joining.
Clinching does not require a pre-cleaning of the surfaces, which is needed before applying adhesives. Clinching is almost an instant joining process (the required joining time is lower than a second) while adhesive joining often requires a much longer time mainly owing to the curing of the joint (up to many hours). Clinched joints are less affected by environmental agents and effect of aging.
Main limitations.
Because it is based on the plastic deformation of the sheets, clinching is limited by the sheet material formability (ductility). Metal ductility increases with temperature, so heat assisted clinching processes have been developed, extending the clinching "joinability". Increasing the joining temperature reduces the material's yield stress, so that less joining force is required. Different heating systems are used to heat the sheets before clinching:
Prolonged heating can increase the grain size or cause metallurgical changes in alloys, which can alter the mechanical behavior of the material at the joint site.
Materials.
Clinching has been widely employed for joining ductile metals, including the following:
It has recently extended to other metals, such as:
It has also extended to non-metallic materials, such as:


Negative imaginary systems

Negative imaginary (NI) systems theory was introduced by Lanzon and Petersen in. A generalization of the theory was presented in
In the single-input single-output (SISO) case, such systems are defined by considering the properties of the imaginary part of the frequency response G(jω) and require the system to have no poles in the right half plane and formula_1 > 0 for all ω in (0, ∞). This means that a system is Negative imaginary if it is both stable and a nyquist plot will have a phase lag between [-π 0] for all ω > 0.
Negative Imaginary Definition.
A square transfer function matrix formula_2 is NI if the following conditions are satisfied:
These conditions can be summarized as:
Negative Imaginary Lemma.
Let formula_18 be a minimal realization of the transfer function matrix formula_13. Then, formula_13 is NI if and only if formula_21 and there exists a matrix
formula_22 such that the following LMI is satisfied:
formula_23
This result comes from positive real theory after converting the negative imaginary system to a positive real system for analysis.


Current reality tree (theory of constraints)

One of the thinking processes in the theory of constraints, a current reality tree (CRT) is a tool to analyze many systems or organizational problems at once. By identifying root causes common to most or all of the problems, a CRT can greatly aid focused improvement of the system. A current reality tree is a directed graph.
Simplified explanation.
A CRT is a focusing procedure formulated by Eliyahu Goldratt, developer of the theory of constraints. This process is intended to help leaders gain understanding of cause and effect in a situation they want to improve. It treats multiple problems in a system as symptoms arising from one or a few ultimate root causes or systemic core problems. It describes, in a visual (cause-and-effect network) diagram, the main perceived symptoms (along with secondary or hidden ones that lead up to the perceived symptoms) of a problem scenario and ultimately the apparent root causes or core conflict. The benefit of building a CRT is that it identifies the connections or dependencies between perceived symptoms (effects) and root causes (core problems or conflicts) explicitly. If core problems are identified, prioritized, and tackled well, multiple undesirable effects in the system will disappear. Leaders may then focus on solving the few core problems which would cause the biggest positive systemic changes.
Contextual explanation.
A CRT is a statement of an underlying core problem and the symptoms that arise from it. It maps out a sequence of cause and effect from the core problem to the symptoms. Most of the symptoms will arise from the one core problem or a core conflict. Removing the core problem may well lead to removing each of the symptoms as well. Operationally working backwards from the apparent undesirable effects or symptoms to uncover or discover the underlying core cause.
Example.
A CRT begins with a list of problems, known as undesirable effects (UDEs.) These are assumed to be symptoms of a deeper common cause. To take a somewhat frivolous example, a car owner may have the following UDEs:
The CRT depicts a chain of cause-and-effect reasoning (if, and, then) in graphical form, where ellipses or circles represent an "and". The graphic is constructed by:
This approach tends to converge on a single root cause. In the illustrated case, the root cause of the above UDEs is seen as being a faulty handbrake.


Computer Aided Verification

In computer science, the International Conference on Computer-Aided Verification (CAV) is an annual academic conference on the theory and practice of computer-aided formal analysis of software and hardware systems, broadly known as formal methods. It is one of the highest-ranked conferences in computer science. Among the important results originally published in CAV are breakthrough techniques in model checking, such as Counterexample-Guided Abstraction Refinement (CEGAR) and partial order reduction.
The first CAV was held in 1989 in Grenoble, France. The CAV proceedings (1989-present) are published by Springer Science+Business Media and are open access.