Petri net

A Petri net, also known as a place/transition (PT) net, is one of several mathematical modeling languages for the description of distributed systems. It is a class of discrete event dynamic system. A Petri net is a directed bipartite graph that has two types of elements: places and transitions. Place elements are depicted as white circles and transition elements are depicted as rectangles.
A place can contain any number of tokens, depicted as black circles. A transition is enabled if all places connected to it as inputs contain at least one token. Some sources state that Petri nets were invented in August 1939 by Carl Adam Petri—at the age of 13—for the purpose of describing chemical processes.
Like industry standards such as UML activity diagrams, Business Process Model and Notation, and event-driven process chains, Petri nets offer a graphical notation for stepwise processes that include choice, iteration, and concurrent execution. Unlike these standards, Petri nets have an exact mathematical definition of their execution semantics, with a well-developed mathematical theory for process analysis.
Historical background.
The German computer scientist Carl Adam Petri, after whom such structures are named, analyzed Petri nets extensively in his 1962 dissertation .
Petri net basics.
A Petri net consists of "places", "transitions", and "arcs". Arcs run from a place to a transition or vice versa, never between places or between transitions. The places from which an arc runs to a transition are called the "input places" of the transition; the places to which arcs run from a transition are called the "output places" of the transition.
Graphically, places in a Petri net may contain a discrete number of marks called "tokens". Any distribution of tokens over the places will represent a configuration of the net called a "marking". In an abstract sense relating to a Petri net diagram, a transition of a Petri net may "fire" if it is "enabled", i.e. there are sufficient tokens in all of its input places; when the transition fires, it consumes the required input tokens, and creates tokens in its output places. A firing is atomic, i.e. a single non-interruptible step.
Unless an "execution policy" (e.g. a strict ordering of transitions, describing precedence) is defined, the execution of Petri nets is nondeterministic: when multiple transitions are enabled at the same time, they will fire in any order.
Since firing is nondeterministic, and multiple tokens may be present anywhere in the net (even in the same place), Petri nets are well suited for modeling the concurrent behavior of distributed systems.
Formal definition and basic terminology.
Petri nets are state-transition systems that extend a class of nets called elementary nets.
Definition 1. A "net" is a tuple formula_1 where
Definition 2. Given a net "N" = ("P", "T", "F"), a "configuration" is a set "C" so that "C" ⊆ "P".
Definition 3. An "elementary net" is a net of the form "EN" = ("N", "C") where
Definition 4. A "Petri net" is a net of the form "PN" = ("N", "M", "W"), which extends the elementary net so that
If a Petri net is equivalent to an elementary net, then "Z" can be the countable set {0,1} and those elements in "P" that map to 1 under "M" form a configuration. Similarly, if a Petri net is not an elementary net, then the multiset "M" can be interpreted as representing a non-singleton set of configurations. In this respect, "M" extends the concept of configuration for elementary nets to Petri nets.
In the diagram of a Petri net (see top figure right), places are conventionally depicted with circles, transitions with long narrow rectangles and arcs as one-way arrows that show connections of places to transitions or transitions to places. If the diagram were of an elementary net, then those places in a configuration would be conventionally depicted as circles, where each circle encompasses a single dot called a "token". In the given diagram of a Petri net (see right), the place circles may encompass more than one token to show the number of times a place appears in a configuration. The configuration of tokens distributed over an entire Petri net diagram is called a "marking".
In the top figure (see right), the place "p"1 is an input place of transition "t"; whereas, the place "p"2 is an output place to the same transition. Let "PN"0 (top figure) be a Petri net with a marking configured "M"0, and "PN"1 (bottom figure) be a Petri net with a marking configured "M"1. The configuration of "PN"0 "enables" transition "t" through the property that all input places have sufficient number of tokens (shown in the figures as dots) "equal to or greater" than the multiplicities on their respective arcs to "t". Once and only once a transition is enabled will the transition fire. In this example, the "firing" of transition "t" generates a map that has the marking configured "M"1 in the image of "M"0 and results in Petri net "PN"1, seen in the bottom figure. In the diagram, the firing rule for a transition can be characterised by subtracting a number of tokens from its input places equal to the multiplicity of the respective input arcs and accumulating a new number of tokens at the output places equal to the multiplicity of the respective output arcs.
Remark 1. The precise meaning of "equal to or greater" will depend on the precise algebraic properties of addition being applied on "Z" in the firing rule, where subtle variations on the algebraic properties can lead to other classes of Petri nets; for example, algebraic Petri nets.
The following formal definition is loosely based on . Many alternative definitions exist.
Syntax.
A Petri net graph (called "Petri net" by some, but see below) is a 3-tuple formula_5, where
The "flow relation" is the set of arcs: formula_7. In many textbooks, arcs can only have multiplicity 1. These texts often define Petri nets using "F" instead of "W". When using this convention, a Petri net graph is a bipartite multigraph formula_8 with node partitions "S" and "T".
The "preset" of a transition "t" is the set of its "input places": formula_9;
its "postset" is the set of its "output places": formula_10. Definitions of pre- and postsets of places are analogous.
A "marking" of a Petri net (graph) is a multiset of its places, i.e., a mapping formula_11. We say the marking assigns to each place a number of "tokens".
A Petri net (called "marked Petri net" by some, see above) is a 4-tuple formula_12, where
Execution semantics.
In words
We are generally interested in what may happen when transitions may continually fire in arbitrary order.
We say that a marking "is reachable from" a marking "in one step" if formula_18; we say that it "is reachable from " if formula_19, where formula_20 is the reflexive transitive closure of formula_21; that is, if it is reachable in 0 or more steps.
For a (marked) Petri net formula_22, we are interested in the firings that can be performed starting with the initial marking formula_14. Its set of "reachable markings" is the set
formula_24
The "reachability graph" of is the transition relation formula_21 restricted to its reachable markings formula_26. It is the state space of the net.
A "firing sequence" for a Petri net with graph and initial marking formula_14 is a sequence of transitions formula_28 such that formula_29. The set of firing sequences is denoted as formula_30.
Variations on the definition.
A common variation is to disallow arc multiplicities and replace the bag of arcs "W" with a simple set, called the "flow relation", formula_31.
This does not limit expressive power as both can represent each other.
Another common variation, e.g. in Desel and Juhás (2001), is to allow "capacities" to be defined on places. This is discussed under "extensions" below.
Formulation in terms of vectors and matrices.
The markings of a Petri net formula_12 can be regarded as vectors of non-negative integers of length formula_33.
Its transition relation can be described as a pair of formula_33 by formula_35 matrices:
Then their difference
can be used to describe the reachable markings in terms of matrix multiplication, as follows.
For any sequence of transitions , write formula_41 for the vector that maps every transition to its number of occurrences in . Then, we have
It must be required that is a firing sequence; allowing arbitrary sequences of transitions will generally produce a larger set.
formula_43
formula_44
Category-theoretic formulation.
Meseguer and Montanari considered a kind of symmetric monoidal categories known as Petri categories.
Mathematical properties of Petri nets.
One thing that makes Petri nets interesting is that they provide a balance between modeling power and analyzability: many things one would like to know about concurrent systems can be automatically determined for Petri nets, although some of those things are very expensive to determine in the general case. Several subclasses of Petri nets have been studied that can still model interesting classes of concurrent systems, while these problems become easier.
An overview of such decision problems, with decidability and complexity results for Petri nets and some subclasses, can be found in Esparza and Nielsen (1995).
Reachability.
The reachability problem for Petri nets is to decide, given a Petri net "N" and a marking "M", whether formula_45.
It is a matter of walking the reachability-graph defined above, until either the requested-marking is reached or it can no longer be found. This is harder than it may seem at first: the reachability graph is generally infinite, and it isn't easy to determine when it is safe to stop.
In fact, this problem was shown to be EXPSPACE-hard years before it was shown to be decidable at all (Mayr, 1981). Papers continue to be published on how to do it efficiently. In 2018, Czerwiński et al. improved the lower bound and showed that the problem is not ELEMENTARY. In 2021, this problem was shown to be non-primitive recursive, independently by Jerome Leroux
and by Wojciech Czerwiński and Łukasz Orlikowski. These results thus close the long-standing complexity gap.
While reachability seems to be a good tool to find erroneous states, for practical problems the constructed graph usually has far too many states to calculate. To alleviate this problem, linear temporal logic is usually used in conjunction with the tableau method to prove that such states cannot be reached. Linear temporal logic uses the semi-decision technique to find if indeed a state can be reached, by finding a set of necessary conditions for the state to be reached then proving that those conditions cannot be satisfied.
Liveness.
Petri nets can be described as having different degrees of liveness formula_46. A Petri net formula_47 is called formula_48-live if and only if all of its transitions are formula_48-live, where a transition is
Note that these are increasingly stringent requirements: formula_60-liveness implies formula_61-liveness, for formula_62.
These definitions are in accordance with Murata's overview, which additionally uses formula_63"-live" as a term for "dead".
Boundedness.
A place in a Petri net is called "k-bound" if it does not contain more than "k" tokens in all reachable markings, including the initial marking; it is said to be "safe" if it is 1-bounded; it is "bounded" if it is "k-bounded" for some "k".
A (marked) Petri net is called "k"-bounded, "safe", or "bounded" when all of its places are.
A Petri net (graph) is called "(structurally) bounded" if it is bounded for every possible initial marking.
A Petri net is bounded if and only if its reachability graph is finite.
Boundedness is decidable by looking at covering, by constructing the Karp–Miller Tree.
It can be useful to explicitly impose a bound on places in a given net.
This can be used to model limited system resources.
Some definitions of Petri nets explicitly allow this as a syntactic feature.
Formally, "Petri nets with place capacities" can be defined as tuples formula_64, where formula_12 is a Petri net, formula_66 an assignment of capacities to (some or all) places, and the transition relation is the usual one restricted to the markings in which each place with a capacity has at most that many tokens.
For example, if in the net "N", both places are assigned capacity 2, we obtain a Petri net with place capacities, say "N2"; its reachability graph is displayed on the right.
Alternatively, places can be made bounded by extending the net. To be exact,
a place can be made "k"-bounded by adding a "counter-place" with flow opposite to that of the place, and adding tokens to make the total in both places "k".
Discrete, continuous, and hybrid Petri nets.
As well as for discrete events, there are Petri nets for continuous and hybrid discrete-continuous processes that are useful in discrete, continuous and hybrid control theory, and related to discrete, continuous and hybrid automata.
Extensions.
There are many extensions to Petri nets. Some of them are completely backwards-compatible (e.g. coloured Petri nets) with the original Petri net, some add properties that cannot be modelled in the original Petri net formalism (e.g. timed Petri nets). Although backwards-compatible models do not extend the computational power of Petri nets, they may have more succinct representations and may be more convenient for modeling. Extensions that cannot be transformed into Petri nets are sometimes very powerful, but usually lack the full range of mathematical tools available to analyse ordinary Petri nets.
The term high-level Petri net is used for many Petri net formalisms that extend the basic P/T net formalism; this includes coloured Petri nets, hierarchical Petri nets such as Nets within Nets, and all other extensions sketched in this section. The term is also used specifically for the type of coloured nets supported by CPN Tools.
A short list of possible extensions follows:
There are many more extensions to Petri nets, however, it is important to keep in mind, that as the complexity of the net increases in terms of extended properties, the harder it is to use standard tools to evaluate certain properties of the net. For this reason, it is a good idea to use the most simple net type possible for a given modelling task.
Restrictions.
Instead of extending the Petri net formalism, we can also look at restricting it, and look at particular types of Petri nets, obtained by restricting the syntax in a particular way. Ordinary Petri nets are the nets where all arc weights are 1. Restricting further, the following types of ordinary Petri nets are commonly used and studied:
Workflow nets.
Workflow nets (WF-nets) are a subclass of Petri nets intending to model the workflow of process activities.
The WF-net transitions are assigned to tasks or activities, and places are assigned to the pre/post conditions.
The WF-nets have additional structural and operational requirements, mainly the addition of a single input (source) place with no previous transitions, and output place (sink) with no following transitions. Accordingly, start and termination markings can be defined that represent the process status.
WF-nets have the soundness property, indicating that a process with a start marking of "k" tokens in its source place, can reach the termination state marking with "k" tokens in its sink place (defined as "k"-sound WF-net). Additionally, all the transitions in the process could fire (i.e., for each transition there is a reachable state in which the transition is enabled).
A general sound (G-sound) WF-net is defined as being "k"-sound for every "k" > 0.
A directed path in the Petri net is defined as the sequence of nodes (places and transitions) linked by the directed arcs. An elementary path includes every node in the sequence only once.
A well-handled Petri net is a net in which there are no fully distinct elementary paths between a place and a transition (or transition and a place), i.e., if there are two paths between the pair of nodes then these paths share a node.
An acyclic well-handled WF-net is sound (G-sound).
Extended WF-net is a Petri net that is composed of a WF-net with additional transition t (feedback transition). The sink place is connected as the input place of transition t and the source place as its output place. Firing of the transition causes iteration of the process (Note, the extended WF-net is not a WF-net).
WRI (Well-handled with Regular Iteration) WF-net, is an extended acyclic well-handled WF-net.
WRI-WF-net can be built as composition of nets, i.e., replacing a transition within a WRI-WF-net with a subnet which is a WRI-WF-net. The result is also WRI-WF-net. WRI-WF-nets are G-sound, therefore by using only WRI-WF-net building blocks, one can get WF-nets that are G-sound by construction.
The design structure matrix (DSM) can model process relations, and be utilized for process planning. The DSM-nets are realization of DSM-based plans into workflow processes by Petri nets, and are equivalent to WRI-WF-nets. The DSM-net construction process ensures the soundness property of the resulting net.
Other models of concurrency.
Other ways of modelling concurrent computation have been proposed, including vector addition systems, communicating finite-state machines, Kahn process networks, process algebra, the actor model, and trace theory. Different models provide tradeoffs of concepts such as compositionality, modularity, and locality.
An approach to relating some of these models of concurrency is proposed in the chapter by Winskel and Nielsen.


Telecom network protocol analyzer

A telecom network protocol analyzer is a protocol analyzer to analyze a switching and signaling telecommunication protocol between different nodes in PSTN or Mobile telephone networks, such as 2G or 3G GSM networks, CDMA networks, WiMAX and so on.
In a mobile telecommunication network it can analyze the traffic between MSC and BSC, BSC and BTS, MSC and HLR, MSC and VLR, VLR and HLR, and so on.
Protocol analyzers are mainly used for performance measurement and troubleshooting. These devices connect to the network to calculate key performance indicators to monitor the network and speed-up troubleshooting activities.


RunBot

RunBot is a miniature bipedal robot which belongs to the class of limit cycle walkers. Instead of using a central pattern generator it uses reflexes which generate the gait. The reflexes are triggered by ground contact sensors in the feet which then activate the motors. The generation of the walking gait is straightforward: when a foot touches the ground the "other" leg is lifted upwards so that the robot falls forward. This then causes this leg to touch the ground and so forth. The walking speed can be improved by means of reinforcement learning because there are only a few parameters in this scheme. RunBot was built in 2005 by Tao Geng as part of his PhD under supervision of Prof Woergoetter and after an idea by Dr Porr to use a walking robot to benchmark reflex based reinforcement learning rules. Its movements and adaptability are based on the work of neurophysiologist Nikolai Bernstein.
Since its inception the RunBot has undergone numerous design iterations, for example where a moveable upper body mass on the robot keeps the walking pattern stable even on uneven terrain.
Design.
The locomotion system is kept simple with four motors: one on each of two knees, one on each of two hips. The sensory system is of similar simplicity, with the ability to detect the ground contact and the angles of the hips/knee motors. The motors are controlled by force and not by angle.
External links.
Runbot's creators recorded demonstrations of RunBot:


Tisean

TISEAN (acronym for Time Series Analysis) is a software package for the analysis of time series with methods based on the theory of nonlinear dynamical systems. It was developed by Rainer Hegger, Holger Kantz and Thomas Schreiber and is distributed under the GPL licence. Two highly cited scientific publications serve as an introduction to the methods addressed in the package: the article "Practical implementation of nonlinear time series methods: The TISEAN package" and the book "Nonlinear time series analysis".


Succinct game

In algorithmic game theory, a succinct game or a succinctly representable game is a game which may be represented in a size much smaller than its normal form representation. Without placing constraints on player utilities, describing a game of formula_1 players, each facing formula_2 strategies, requires listing formula_3 utility values. Even trivial algorithms are capable of finding a Nash equilibrium in a time polynomial in the length of such a large input. A succinct game is of "polynomial type" if in a game represented by a string of length "n" the number of players, as well as the number of strategies of each player, is bounded by a polynomial in "n" (a formal definition, describing succinct games as a computational problem, is given by Papadimitriou & Roughgarden 2008).
Types of succinct games.
Graphical games.
Graphical games are games in which the utilities of each player depends on the actions of very few other players. If formula_4 is the greatest number of players by whose actions any single player is affected (that is, it is the indegree of the game graph), the number of utility values needed to describe the game is formula_5, which, for a small formula_4 is a considerable improvement.
It has been shown that any normal form game is reducible to a graphical game with all degrees bounded by three and with two strategies for each player. Unlike normal form games, the problem of finding a pure Nash equilibrium in graphical games (if one exists) is NP-complete. The problem of finding a (possibly mixed) Nash equilibrium in a graphical game is PPAD-complete. Finding a correlated equilibrium of a graphical game can be done in polynomial time, and for a graph with a bounded treewidth, this is also true for finding an "optimal" correlated equilibrium.
Sparse games.
Sparse games are those where most of the utilities are zero. Graphical games may be seen as a special case of sparse games.
For a two player game, a sparse game may be defined as a game in which each row and column of the two payoff (utility) matrices has at most a constant number of non-zero entries. It has been shown that finding a Nash equilibrium in such a sparse game is PPAD-hard, and that there does not exist a fully polynomial-time approximation scheme unless PPAD is in P.
Symmetric games.
In symmetric games all players are identical, so in evaluating the utility of a combination of strategies, all that matters is how many of the formula_1 players play each of the formula_2 strategies. Thus, describing such a game requires giving only formula_9 utility values.
In a symmetric game with 2 strategies there always exists a pure Nash equilibrium – although a "symmetric" pure Nash equilibrium may not exist. The problem of finding a pure Nash equilibrium in a symmetric game (with possibly more than two players) with a constant number of actions is in AC0; however, when the number of actions grows with the number of players (even linearly) the problem is NP-complete. In any symmetric game there exists a symmetric equilibrium. Given a symmetric game of "n" players facing "k" strategies, a symmetric equilibrium may be found in polynomial time if k=formula_10. Finding a correlated equilibrium in symmetric games may be done in polynomial time.
Anonymous games.
In anonymous games, players have different utilities but do not distinguish between other players (for instance, having to choose between "go to cinema" and "go to bar" while caring only how crowded will each place be, not who'll they meet there). In such a game a player's utility again depends on how many of his peers choose which strategy, and his own, so formula_11 utility values are required.
If the number of actions grows with the number of players, finding a pure Nash equilibrium in an anonymous game is NP-hard. An optimal correlated equilibrium of an anonymous game may be found in polynomial time. When the number of strategies is 2, there is a known PTAS for finding an ε-approximate Nash equilibrium.
Polymatrix games.
In a polymatrix game (also known as a "multimatrix game"), there is a utility matrix for every pair of players "(i,j)", denoting a component of player i's utility. Player i's final utility is the sum of all such components. The number of utilities values required to represent such a game is formula_12.
Polymatrix games always have at least one mixed Nash equilibrium. The problem of finding a Nash equilibrium in a polymatrix game is PPAD-complete. Moreover, the problem of finding a constant approximate Nash equilibrium in a polymatrix game is also PPAD-complete. Finding a correlated equilibrium of a polymatrix game can be done in polynomial time. Note that even if pairwise games played between players have pure Nash equilibria, the global interaction does not necessarily admit a pure Nash equilibrium (although a mixed Nash equilibrium must exist). Checking if a pure Nash equilibrium exists is a strongly NP-complete problem.
Competitive polymatrix games with only zero-sum interactions between players are a generalization of two-player zero-sum games. The Minimax theorem originally formulated for two-player games by von Neumann generalizes to zero-sum polymatrix games.
Same as two-player zero-sum games, polymatrix zero-sum games have mixed Nash equilibria that can be computed in polynomial time and those equilibria coincide with correlated equilibria. But some other properties of two-player zero-sum games do not generalize. Notably, players need not have a unique value of the game and equilibrium strategies are not max-min strategies in a sense that worst-case payoffs of players are not maximized when using an equilibrium strategy. There exists an open source Python library for simulating competitive polymatrix games.
Polymatrix games which have coordination games on their edges are potential games and can be solved using a potential function method.
Circuit games.
The most flexible of way of representing a succinct game is by representing each player by a polynomial-time bounded Turing machine, which takes as its input the actions of all players and outputs the player's utility. Such a Turing machine is equivalent to a Boolean circuit, and it is this representation, known as circuit games, that we will consider.
Computing the value of a 2-player zero-sum circuit game is an EXP-complete problem, and approximating the value of such a game up to a multiplicative factor is known to be in PSPACE. Determining whether a pure Nash equilibrium exists is a formula_13-complete problem (see Polynomial hierarchy).
Other representations.
Many other types of succinct game exist (many having to do with allocation of resources). Examples include congestion games, network congestion games, scheduling games, local effect games, facility location games, action-graph games, hypergraphical games and more.
Summary of complexities of finding equilibria.
Below is a table of some known complexity results for finding certain classes of equilibria in several game representations. "NE" stands for "Nash equilibrium", and "CE" for "correlated equilibrium". "n" is the number of players and "s" is the number of strategies each player faces (we're assuming all players face the same number of strategies). In graphical games, "d" is the maximum indegree of the game graph. For references, see main article text.


Feedback linearization

Feedback linearization is a common strategy employed in nonlinear control to control nonlinear systems. Feedback linearization techniques may be applied to nonlinear control systems of the form
where formula_1 is the state, formula_2 are the inputs. The approach involves transforming a nonlinear control system into an equivalent linear control system through a change of variables and a suitable control input. In particular, one seeks a change of coordinates formula_3 and control input formula_4 so that the dynamics of formula_5 in the coordinates formula_6 take the form of a linear, controllable control system,
An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.
Feedback linearization of SISO systems.
Here, consider the case of feedback linearization of a single-input single-output (SISO) system. Similar results can be extended to multiple-input multiple-output (MIMO) systems. In this case, formula_7 and formula_8. The objective is to find a coordinate transformation formula_9 that transforms the system (1) into the so-called normal form which will reveal a feedback law of the form
that will render a linear input–output map from the new input formula_10 to the output formula_11. To ensure that the transformed system is an equivalent representation of the original system, the transformation must be a diffeomorphism. That is, the transformation must not only be invertible (i.e., bijective), but both the transformation and its inverse must be smooth so that differentiability in the original coordinate system is preserved in the new coordinate system. In practice, the transformation can be only locally diffeomorphic and the linearization results only hold in this smaller region.
Several tools are required to solve this problem.
Lie derivative.
The goal of feedback linearization is to produce a transformed system whose states are the output formula_11 and its first formula_13 derivatives. To understand the structure of this target system, we use the Lie derivative. Consider the time derivative of (2), which can be computed using the chain rule,
Now we can define the Lie derivative of formula_15 along formula_16 as,
and similarly, the Lie derivative of formula_15 along formula_19 as,
With this new notation, we may express formula_21 as,
Note that the notation of Lie derivatives is convenient when we take multiple derivatives with respect to either the same vector field, or a different one. For example,
and
Relative degree.
In our feedback linearized system made up of a state vector of the output formula_11 and its first formula_13 derivatives, we must understand how the input formula_27 enters the system. To do this, we introduce the notion of relative degree. Our system given by (1) and (2) is said to have relative degree formula_28 at a point formula_29 if,
Considering this definition of relative degree in light of the expression of the time derivative of the output formula_11, we can consider the relative degree of our system (1) and (2) to be the number of times we have to differentiate the output formula_11 before the input formula_27 appears explicitly. In an LTI system, the relative degree is the difference between the degree of the transfer function's denominator polynomial (i.e., number of poles) and the degree of its numerator polynomial (i.e., number of zeros).
Linearization by feedback.
For the discussion that follows, we will assume that the relative degree of the system is formula_37. In this case, after differentiating the output formula_37 times we have,
where the notation formula_40 indicates the formula_37th derivative of formula_11. Because we assumed the relative degree of the system is formula_37, the Lie derivatives of the form formula_44 for formula_45 are all zero. That is, the input formula_27 has no direct contribution to any of the first formula_13th derivatives.
The coordinate transformation formula_48 that puts the system into normal form comes from the first formula_13 derivatives. In particular,
transforms trajectories from the original formula_51 coordinate system into the new formula_52 coordinate system. So long as this transformation is a diffeomorphism, smooth trajectories in the original coordinate system will have unique counterparts in the formula_52 coordinate system that are also smooth. Those formula_52 trajectories will be described by the new system,
Hence, the feedback control law
renders a linear input–output map from formula_57 to formula_58. The resulting linearized system
is a cascade of formula_37 integrators, and an outer-loop control formula_57 may be chosen using standard linear system methodology. In particular, a state-feedback control law of
where the state vector formula_52 is the output formula_11 and its first formula_13 derivatives, results in the LTI system
with,
So, with the appropriate choice of formula_68, we can arbitrarily place the closed-loop poles of the linearized system.
Unstable zero dynamics.
Feedback linearization can be accomplished with systems that have relative degree less than formula_37. However, the normal form of the system will include zero dynamics (i.e., states that are not observable from the output of the system) that may be unstable. In practice, unstable dynamics may have deleterious effects on the system (e.g., it may be dangerous for internal states of the system to grow unbounded). These unobservable states may be controllable or at least stable, and so measures can be taken to ensure these states do not cause problems in practice. Minimum phase systems provide some insight on zero dynamics.


ULMA Handling Systems

ULMA Handling Systems is a material handling and logistics automation company, supplier of automated storage and retrieval systems, based in Oñati, Spain. The company engineers design, produce, and install material handling systems in installations, from small warehouses to complex systems.
Corporate information.
The firm has operational subsidiaries in several countries
History.
ULMA Handling Systems is one of the 8 companies which make up ULMA Group, which dates back to 1957 when six young mechanics set up a small workshop in Oñati (Guipúzcoa). In 1988 was founded ULMA Handling Systems, after a technology transfer agreement was signed with the Japanese company Daifuku for the sale and introduction of automatic material handling.
In 1997, the company broke into overseas markets installing warehouses in Brazil, France and Italy and nowadays has subsidiaries in Spain, France, Netherlands, Brazil, Chile and Peru.
Products and services.
The company designs material handling systems involving automatic movements of the products, improving the productivity rates and the efficiency of the warehouses.
The company develops order picking solutions, automated storage and retrieval systems (AS/RS), conveyor and automated guided vehicles, automated sorting solutions and end of line solutions. ULMA Handling Systems offers from logistics consulting, planning, design to after sales service.
In addition, the company provides warehouse management software which guarantees the optimization and optimum control of all the movements of the goods that are located into the warehouse.
It also offers baggage handling solutions, designing and developing integral solutions and health logistics solutions, such as storage of medication and its dispensing.


Equal channel angular extrusion

Equal channel angular extrusion (ECAE) called also equal channel angular pressing (ECAP) is one technique from the Severe Plastic Deformation (SPD) group, aimed at producing Ultra Fine Grained (UFG) material. Developed in the Soviet Union in 1973 by Segal.  However, the dates are not always consistent. In industrial metalworking, it is an extrusion process, The technique is able to refine the microstructure of metals and alloys, thereby improving their strength according to the Hall-Petch relationship. This process improves not only the strength but also other properties such as corrosion and wear resistance of alloys and compounds.
ECAE is unique because significant cold work can be accomplished without reduction in the cross sectional area of the deformed workpiece. In conventional deformation processes like rolling, forging, extrusion, and drawing, strain is introduced by reduction in the cross sectional area. ECAE produces significant deformation strain without reducing the cross sectional area. This is accomplished by extruding the work piece around a corner. For example, a square cross section bar of metal is forced through a channel with a 90° degree angle. The cross section of the channel is equal on entry and exit. The complex deformation of the metal as it flows around the corner produces very high strain. Because the cross section remains the same, a work piece can be extruded multiple times with each pass introducing additional strain.
Die design is critical because of the large forces required.
To reduce the friction of the pushed sample is lubricated with grease for example mixture of graphite and oil, and to reduce the forces, the process is sometimes carried out at elevated temperatures but then recrystallization can occur which can also leads to excessive grain growth at elevated temperature.
There are some modifications of the process e.g. incremental ECAP (I-ECAP) for the production of continuous products.
Process routes.
The process can be carried out in multiple passes. According to the rotation angle and direction between next passes, there can be four fundamental process routes named A, Ba, Bc, and C:
Finite element method in the ECAE process.
The behave during deformation and flow of the material, are analyzed by scientists and there are many articles on computer simulation, finite element method is one of the important approaches to understand the deformation occurring in the ECAE process.
See also.
Severe plastic deformation
Strengthening mechanisms of materials


Control of chaos

In lab experiments that study chaos theory, approaches designed to control chaos are based on certain observed system behaviors. Any chaotic attractor contains an infinite number of unstable, periodic orbits. Chaotic dynamics, then, consists of a motion where the system state moves in the neighborhood of one of these orbits for a while, then falls close to a different unstable, periodic orbit where it remains for a limited time and so forth. This results in a complicated and unpredictable wandering over longer periods of time.
Control of chaos is the stabilization, by means of small system perturbations, of one of these unstable periodic orbits. The result is to render an otherwise chaotic motion more stable and predictable, which is often an advantage. The perturbation must be tiny compared to the overall size of the attractor of the system to avoid significant modification of the system's natural dynamics.
Several techniques have been devised for chaos control, but most are developments of two basic approaches: the OGY (Ott, Grebogi and Yorke) method and Pyragas continuous control. Both methods require a previous determination of the unstable periodic orbits of the chaotic system before the controlling algorithm can be designed.
OGY method.
E. Ott, C. Grebogi and J. A. Yorke were the first to make the key observation that the infinite number of unstable periodic orbits typically embedded in a chaotic attractor could be taken advantage of for the purpose of achieving control by means of applying only very small perturbations. After making this general point, they illustrated it with a specific method, since called the OGY method (Ott, Grebogi and Yorke) of achieving stabilization of a chosen unstable periodic orbit. In the OGY method, small, wisely chosen, kicks are applied to the system once per cycle, to maintain it near the desired unstable periodic orbit.
To start, one obtains information about the chaotic system by analyzing a slice of the chaotic attractor. This slice is a Poincaré section. After the information about the section has been gathered, one allows the system to run and waits until it comes near a desired periodic orbit in the section. Next, the system is encouraged to remain on that orbit by perturbing the appropriate parameter. When the control parameter is actually changed, the chaotic attractor is shifted and distorted somewhat. If all goes according to plan, the new attractor encourages the system to continue on the desired trajectory. One strength of this method is that it does not require a detailed model of the chaotic system but only some information about the Poincaré section. It is for this reason that the method has been so successful in controlling a wide variety of chaotic systems.
The weaknesses of this method are in isolating the Poincaré section and in calculating the precise perturbations necessary to attain stability.
Pyragas method.
In the Pyragas method of stabilizing a periodic orbit, an appropriate continuous controlling signal is injected into the system, whose intensity is practically zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit. Both the Pyragas and OGY methods are part of a general class of methods called "closed loop" or "feedback" methods which can be applied based on knowledge of the system obtained through solely observing the behavior of the system as a whole over a suitable period of time.
Applications.
Experimental control of chaos by one or both of these methods has been achieved in a variety of systems, including turbulent fluids, oscillating chemical reactions, magneto-mechanical oscillators and cardiac tissues. attempt the control of chaotic bubbling with the OGY method and using electrostatic potential as the primary control variable.
Forcing two systems into the same state is not the only way to achieve synchronization of chaos. Both control of chaos and synchronization constitute parts of cybernetical physics, a research area on the border between physics and control theory.


Automation surprise

An automation surprise is an action that is performed by an automation system and is unexpected by the user. A mode error can be a common cause of an automation surprise. Automation surprise can be dangerous when it upsets the situational awareness of a control operator.


Tool and cutter grinder

A Tool and Cutter Grinder is used to sharpen milling cutters and tool bits along with a host of other cutting tools.
It is an extremely versatile machine used to perform a variety of grinding operations: surface, cylindrical, or complex shapes. The image shows a manually operated setup, however highly automated Computer Numerical Control (CNC) machines are becoming increasingly common due to the complexities involved in the process.
The operation of this machine (in particular, the manually operated variety) requires a high level of skill. The two main skills needed are understanding of the relationship between the grinding wheel and the metal being cut and knowledge of tool geometry. The illustrated set-up is only one of many combinations available. The huge variety in shapes and types of machining cutters requires flexibility in usage. A variety of dedicated fixtures are included that allow cylindrical grinding operations or complex angles to be ground. The vise shown can swivel in three planes.
The table moves longitudinally and laterally, the head can swivel as well as being adjustable in the horizontal plane, as visible in the first image. This flexibility in the head allows the critical clearance angles required by the various cutters to be achieved.
CNC tool and cutter grinder.
Today's tool and cutter grinder is typically a CNC machine tool, usually 5 axes, which produces endmills, drills, step tools, etc. which are widely used in the metal cutting and woodworking industries.
Modern CNC tool and cutter grinders enhance productivity by typically offering features such as automatic tool loading as well as the ability to support multiple grinding wheels. High levels of automation, as well as automatic in-machine tool measurement and compensation, allow extended periods of unmanned production. With careful process configuration and appropriate tool support, tolerances less than 5 micrometres (0.0002") can be consistently achieved even on the most complex parts.
Apart from manufacturing, in-machine tool measurement using touch-probe or laser technology allows cutting tools to be reconditioned. During normal use, cutting edges either wear and/or chip. The geometric features of cutting tools can be automatically measured within the CNC tool grinder and the tool ground to return cutting surfaces to optimal condition.
Significant software advancements have allowed CNC tool and cutter grinders to be utilized in a wide range of industries. Advanced CNC grinders feature sophisticated software that allows geometrically complex parts to be designed either parametrically or by using third party CAD/CAM software. 3D simulation of the entire grinding process and the finished part is possible as well as detection of any potential mechanical collisions and calculation of production time. Such features allow parts to be designed and verified, as well as the production process optimized, entirely within the software environment.
Tool and cutter grinders can be adapted to manufacturing precision machine components. The machine, when used for these purposes more likely would be called a CNC Grinding System.
CNC Grinding Systems are widely used to produce parts for aerospace, medical, automotive, and other industries. Extremely hard and exotic materials are generally no problem for today's grinding systems and the multi-axis machines are capable of generating quite complex geometries.
Radius grinder.
A radius grinder (or radius tool grinder) is a special grinder used for grinding the most complex tool forms, and is the historical predecessor to the CNC tool and cutter grinder. Like the CNC grinder, it may be used for other tasks where grinding spherical surfaces is necessary. The tool itself consists of three parts: The grinder head, work table, and holding fixture. The grinder head has three degrees of freedom. Vertical movement, movement into the workpiece, and tilt. These are generally set statically, and left fixed throughout operations. The work table is a T-slotted X-axis table mounted on top of a radial fixture. Mounting the X axis on top of the radius table, as opposed to the other way around, allows for complex and accurate radius grinds. The holding fixtures can be anything one can mount on a slotted table, but most commonly used is a collet or chuck fixture that indexes and has a separate Y movement to allow accurate depth setting and endmill sharpening. The dressers used on these grinders are usually quite expensive, and can dress the grinding wheel itself with a particular radius.
D-bit grinder.
The D-bit (after Deckel, the brand of the original manufacturer) grinder is a tool bit grinder designed to produce single-lip cutters for pantograph milling machines. Pantographs are a variety of milling machine used to create cavities for the dies used in the molding process; they are largely obsolete and replaced by CNC machining centers in modern industry.
With the addition of accessory holders, the single-lip grinding capability may also be applied to grinding lathe cutting bits, and simple faceted profiles on tips of drill bits or end mills. The machine is sometimes advertised as a "universal cutter-grinder", but the "universal" term refers only to the range of compound angles available, not that the machine is capable of sharpening the universe of tools. The machine is not capable of sharpening drill bits in the standard profiles, or generating any convex or spiral profiles.


Punch press

A punch press is a type of machine press used to cut holes in material. It can be small and manually operated and hold one simple die set, or be very large, CNC operated, with a multi-station turret and hold a much larger and complex die set.
Description.
Punch presses are large machines with either a 'C' type frame, or a 'portal' (bridge) type frame. The "C" type has the hydraulic ram at the top foremost part, whereas the portal frame is much akin to a complete circle with the ram being centered within the frame to stop frame deflection or distortion.
C type presses have a bed plate which is used to lock the die bottom bolster. For locking the die, T-bolts are used and so this plate contains T-slots into which T-bolts are slid in. These slots are placed diagonally and with a slot horizontal to the longer side of the plate, is the general practice. These slots run up to a central hole made in the plate, the hole being large enough to accommodate another bush with a hole, the hole being used for dropping the punched part to the bottom of the press. The top of the tool butted against a vertical sliding ram with a clamping system which accommodates only a particular diameter of a threaded cylindrical member called the "shank" of the tool. The bottom portion of the tool is locked to the bottom bed plate and the top portion of the tool is locked to the sliding ram. Top and bottom portions of the tool are generally guided by suitable pillar and bush assemblies, which gives safety to the punching elements of the tool.
Generally the tool is placed slightly above the bottom bed plate by providing two parallel blocks accurately ground to the same size. This is a necessary action since many tools, scrap (cut pieces which are a waste) is discharged through the bottom element of the tool, not necessarily in the centre of the tool. The scrap or the blank (the required portion) come out from the die at different places. These have to be taken out horizontally from between the parallels placed. Otherwise they get accumulated inside the tool itself and cause severe damage to the tool.
In very heavy presses with higher tonnage, the sliding ram has also a thick plate with T slots for locking the top plate of the tool (called the top bolster). In such cases the threaded cylinder called shank is not attached to the tool. The clamps are either mechanical (manually operated using spanners) or air operated varieties.
Turret type punch press machines have a table or bed with brushes or rollers to allow the sheet metal workpiece to traverse with low friction. Brushes are used where scratches on the workpiece must be minimized, as with brushed aluminium or high polished materials.
The punch press is characterized by parameters such as:
Punch presses are usually referred to by their tonnage and table size. In a production environment a 30-ton press is mostly the machine used today. The tonnage needed to cut and form the material is well known, so sizing tooling for a specific job is a fairly straightforward task. According to the requirement the tonnage may even go up to 2000 to 2500 ton presses.
Die set.
A die set consists of a set of punches (male) and dies (female) which, when pressed together, form a hole in a workpiece (and may also deform the workpiece in some desired manner). The punches and dies are removable, with the punch being attached to the ram during the punching process. The ram moves up and down in a vertically linear motion, forcing the punch through the material into the die.
Axis.
The main bed of most machines is called the 'X' axis, with the 'Y' axis being at right angles to that and allowed to traverse under CNC control. Dependent on the size of the machine, the beds, and the sheet metal workpiece weight, the motors required to move these axis tables will vary in size and power. Older styles of machines used DC motors; however, with advances in technology, today's machines mostly use AC brushless motors for drives.
CNC-controlled operation.
To start a cycle, the CNC controller commands the drives to move the table along the X and the Y axis to a desired position. Once in position, the control initiates the punching sequence and pushes the ram from top dead center (TDC) to bottom dead center (BDC) through the material plane. (The terms BDC and TDC go back to older presses with pneumatic or hydraulic clutches. On today's machines BDC/TDC do not actually exist but are still used for the bottom and top of a stroke.)
On its stroke from TDC to BDC, the punch enters the material, pushing it through the die, obtaining the shape determined by the design of the punch and die set. The piece of material (slug) cut from the workpiece is ejected through the die and bolster plate and collected in a scrap container. The return to TDC signals to the control to begin the next cycle.
The punch press is used for high volume production. Cycle times are often measured in milliseconds. Material yield is measured as a percentage of parts to waste per sheet processed. CAD/CAM programs maximize yield by nesting parts in the layout of the sheet.
Drive type.
Flywheel drive.
Most punch presses today are hydraulically powered. Older machines, however, have mechanically driven rams, meaning the power to the ram is provided by a heavy, constantly rotating flywheel. The flywheel drives the ram using a Pitman arm. In the 19th century, the flywheels were powered by leather drive belts attached to line shafting, which in turn ran to a steam plant. In the modern workplace, the flywheel is powered by an electric motor.
Mechanical punch press.
Mechanical punch presses fall into two distinct types, depending on the type of clutch or braking system with which they are equipped. Generally, older presses are "full revolution" presses that require a full revolution of the crankshaft for them to come to a stop. Full revolution clutch presses are known to be dangerous and outlawed in many countries unless the pinch point is fully guarded. This is because the braking mechanism depends on a set of raised keys or "dogs" to fall into matching slots to stop the ram. A full revolution clutch can only bring the ram to a stop at the same location - top dead center. Newer presses are often "part revolution" presses equipped with braking systems identical to the brakes on commercial trucks. When air is applied, a band-type brake expands and allows the crankshaft to revolve. When the stopping mechanism is applied the air is bled, causing the clutch to open and the braking system to close, stopping the ram in any part of its rotation. Modern part revolution clutch and brake units are normally combined units that operate in a fail safe mode, a dual air safety valve engages clutch and starts slide motion and brake is applied by springs.
Hydraulic punch press.
Hydraulic punch presses power the ram with a hydraulic cylinder rather than a flywheel, and are either valve controlled or valve and feedback controlled. Valve controlled machines usually allow a one stroke operation allowing the ram to stroke up and down when commanded. Controlled feedback systems allow the ram to be proportionally controlled to within fixed points as commanded.
This allows greater control over the stroke of the ram, and increases punching rates as the ram no longer has to complete the traditional full stroke up and down but can operate within a very short window of stroke.
Servo drive turret punch press.
A servo drive turret punch press uses twin AC servo drives directly coupled to the drive shaft. This drive system combines the simplicity of the original clutch and brake technology with the speed of a hydraulic ram driven system. This results in high performance, reliability, and lower operating costs. A servo drive press system has no complex hydraulics or oil-cooling chillers, thus reducing maintenance and repair costs. A turret press can be equipped with advanced technology that stores and reuses energy generated during ram deceleration, providing extended electrical power savings.


Pistonless rotary engine

A pistonless rotary engine is an internal combustion engine that does not use pistons in the way a reciprocating engine does. Designs vary widely but typically involve one or more rotors, sometimes called rotary pistons. Although many different designs have been constructed, only the Wankel engine has achieved widespread adoption.
The term rotary combustion engine has been used as a name for these engines to distinguish them from early (generally up to the early 1920s) aircraft engines and motorcycle engines also known as "rotary engines". However, both continue to be called "rotary engines" and only the context determines which type is meant, whereas the "pistonless" prefix is less ambiguous.
Pistonless rotary engines.
A pistonless rotary engine replaces the linear reciprocating motion of a piston with more complex compression/expansion motions with the objective of improving some aspect of the engine's operation, such as: higher efficiency thermodynamic cycles, lower mechanical stress, lower vibration, higher compression, or less mechanical complexity. the Wankel engine is the only successful pistonless rotary engine, but many similar concepts have been proposed and are under various stages of development. Examples of rotary engines include:


Load rejection

Load rejection in an electric power system is the condition in which there is a sudden load loss in the system which causes the generating equipment to be over-frequency.
A load rejection test is part of commissioning for power systems to confirm that the system can withstand a sudden loss of load and return to normal operating conditions using its governor. Load banks are normally used for these tests.


ION LMD

ION LMD system is one of the laser microdissection systems and a name of device that follows Gravity-Assisted Microdissection method, also known as GAM method. This non-contact laser microdissection system makes cell isolation for further genetic analysis possible. It is the first developed laser microdissection system in Asia.
History.
At first, proto type of ION LMD system was developed in 2004.
The first generation of ION LMD was developed in 2005 and then the second generation(so-called G2) was developed in 2008. At last, the third generation(so-called ION LMD Pro) was developed in 2012.
Manufacturer.
JungWoo F&B was founded in 1994, and offers various factory automation products for clients in semiconductor, consumer electronics, LCD, automotive manufacturing and ship-building industries. In 2003, the company entered the bio-mechanics business for the medical laboratory market and developed an ION LMD system which is utilized in cancer research.
Awards.
This ION LMD system has got some reliable awards.


Thin-wall injection molding

Thin wall injection molding is a specialized form of conventional injection molding that focuses on mass-producing plastic parts that are thin and light so that material cost savings can be made and cycle times can be as short as possible. Shorter cycle times means higher productivity and lower costs per part.
The definition of thin wall is really about the size of the part compared to its wall thickness. For any particular plastic part, as the wall thickness reduces the harder it is to manufacture using the injection molding process. The size of a part puts a limit on how thin the wall thickness can be. For packaging containers thin wall means wall thicknesses that are less than 0.025 inch (0.62mm) with a flow length to wall thickness greater than 200.
Markets.
The trend towards thin wall molding continues to increase in many plastic industries as plastic material and energy costs continue to rise and delivery lead times are squeezed.
The following industries make use of thin wall molding:
Examples.
Plastic resins suitable for thin-wall molding should have high-flow properties, particularly low melt viscosity. In addition, they need to be robust enough to avoid degradation from the heat generated by high shear rates (high injection speeds)
Some plastic manufacturers make plastics specifically for thin wall applications that have excellent flow properties inside the mold cavity. For example, plastic manufacturer Sabic has a polypropylene food contact grade plastic which is specifically designed for thin wall margarine containers and lids.
Another plastic manufacturer, Bayer, makes a blend of Polycarbonate (PC) and Acrylonitrile butadiene styrene (ABS) specifically designed to make thin wall mobile housings.
Equipment.
Plastic injection molding machine.
Compared to conventional injection molding, thin wall molding requires molding machines that are designed and built to withstand higher stresses and injection pressures. The molding machines computer control should also be precise in order to make quality parts. For this reason these molding machines are more expensive than general purpose machines.
Thin-wall-capable machines usually also have accumulator-assisted clamps to accommodate fast cycle times.
Regular maintenance schedules must be completed so that the machine and part quality does not suffer. These machines usually work 24/7 so they need to be well maintained.
Injection mold design.
As with the injection molding machines, injection molds need to be robust enough to withstand high stresses and pressures. Heavy mold construction with through hardened tool steels will ensure a long lasting mold.
The mold must also have a well designed cooling system so that heat can be quickly extracted from the hot plastic part allowing fast cycle times. To achieve this, cooling channels need to be designed close to the molding surface.
Cleaning the mould on a daily basis is also a critical requirement to maintain the part quality.
Robotics.
In countries where manual labour is expensive, robots are commonly used to remove the plastic parts from the mold and order them into equal stacks. These robots are fixed to the molding machine and need to be fast and reliable.
Plastic injection molding process.
The range of process parameters, which are employed for thin wall molded parts, is considerably narrower than that of conventional injection molding because thin parts are difficult for the injection unit of the machine to fill compared to thicker parts. Even with optimally designed parts and molds, it is still more difficult to produce parts with thin walls.
Consistent injection speeds and pressures are required to maintain the quality of the parts produced from thin wall molding. A properly trained molding technician who understands and operates the machine within the confines of the narrow processing window at which the molded product's production cost-effectiveness is optimized will ensure that the process produces quality parts during the production.


Disjunctive graph

In the mathematical modeling of job shop scheduling problems, disjunctive graphs are a way of modeling a system of tasks to be scheduled and timing constraints that must be respected by the schedule.
They are mixed graphs, in which vertices (representing tasks to be performed) may be connected by both directed and undirected edges (representing timing constraints between tasks). The two types of edges represent constraints of two different types:
Pairs of tasks that have no constraint on their ordering – they can be performed in either order or even simultaneously – are disconnected from each other in the graph.
A valid schedule for the disjunctive graph may be obtained by finding an acyclic orientation of the undirected edges – that is, deciding for each pair of non-simultaneous tasks which is to be first, without introducing any circular dependencies – and then ordering the resulting directed acyclic graph. In particular, suppose that all tasks have equal length and the goal is to find a schedule that minimizes the makespan, the total time until all tasks have been completed. In this case, the makespan can be computed from the longest path in the oriented graph, which can be found in polynomial time for directed acyclic graphs. However, the orientation stage of the solution is much more difficult: it is NP-hard to find the acyclic orientation that minimizes the length of the longest path. In particular, by the Gallai–Hasse–Roy–Vitaver theorem, if all edges are initially undirected, then orienting them to minimize the longest path is equivalent to finding an optimal graph coloring of the initial undirected graph.


Redmi 10 5G

The Redmi 10 5G is a 5G Android-based smartphone developed by Redmi, a sub-brand of Xiaomi Inc. Was introduced 29 March 2022 with Redmi Note 11S 5G and global version of Redmi Note 11 Pro+ 5G. Earlier in China Redmi 10 5G was released with Redmi Note 11E Pro as Redmi Note 11E. and later it was released in India with Redmi 11 Prime and Redmi A1 as Redmi 11 Prime 5G with other front camera.
Redmi 10 5G was launched under the POCO brand as Poco M4 5G (stylized and marketed as POCO M4 5G) with different designs and a primary camera. The Indian variant of Poco M4 5G has more advanced cameras.
In China global Poco M4 5G is sold as Redmi Note 11R with Ice Crystal Galaxy (silver) color option instead POCO Yellow and a bigger memory configuration.
Design.
The front is made of Gorilla Glass 3. The back is made of plastic with a wavy texture.
The design of Redmi 10 5G/Note 11E and Redmi 11 Prime 5G back is similar to Oppo smartphones, when in Poco M4 5G/Note 11R it is similar to Pixel 6. Also, all models have IP53 dust and splash protection.
On the bottom side there are USB-C port, speaker and microphone. On the top side there are additional microphone, IR blaster and 3.5mm audio jack. On the left side there is dual SIM tray with microSD slot. On the right side are a volume rocker and power button with a mounted fingerprint scanner.
Redmi 10 5G sells in 3 colors: Graphite Gray, Chrome Silver and Aurora Green.
Redmi Note 11E sells in 3 colors: Mysterious Darkness (gray), Ice Crystal Galaxy (silver) and Microbrewed Mint (green).
Redmi 11 Prime 5G sells in 3 colors: Thunder Black, Chrome Silver and Meadow Green.
Poco M4 5G sells in 3 colors: Power Black, Cool Blue and POCO Yellow.
Redmi Note 11R sells in 3 colors: Polar Blue Ocean, Mysterious Darkness (gray) and Ice Crystal Galaxy (silver).
Specifications.
Hardware.
Platform.
The smartphones have, like Redmi Note 10 5G, MediaTek Dimensity 700 with Mali-G57 MC2.
Battery.
These devices have non-removable battery with capacity 5000 mAh and 18 W fast charging.
Camera.
All models have dual rear camera with 50 MP, on Redmi 10 5G/Note 11E, Redmi 11 Prime 5G and Indian Poco M4 5G and 13 MP on global Poco M4 5G/Note 11R wide camera plus 2 MP, depth sensor. Redmi 10 5G/Note 11E and Poco M4 5G have 5 MP, Indian Poco M4 5G has 8 MP, and Redmi 11 Prime 5G has 8 MP, front camera. Both the rear and front camera can record video in 1080p@30fps.
Display.
Phones have 6.58" IPS LCD display with Full HD+ (2408 × 1080; ~401 ppi) image resolution, 90 Hz refresh rate and waterdrop notch.
Memory.
Redmi 10 5G solds with 4/64, 4/128 and 6/128 GB, Redmi Note 11E ― 4/128 and 6/128 GB, Redmi 11 Prime 5G and Poco M4 5G ― 4/64 and 6/128 GB, and Redmi Note 11E ― 4/128, 6/128 and 8/128 GB.
All models have LPDDR4X type of RAM and type of storage UFS 2.2 which could be extanded by microSD up to 1 TB in global Poco M4 5G/Redmi Note 11R and up to 512 GB in other models.
Software.
Initially, Redmi 10 5G/Note 11E, Redmi 11 Prime 5G, and Redmi Note 11R were released with MIUI 13 custom skin, and Poco M4 5G was released with MIUI 13 for POCO. Both based on Android 12. Redmi devices were updated to MIUI 14 and Poco M4 5G to MIUI 14 for POCO. Both ROMs are based on Android 13.


Time-invariant system

In control theory, a time-invariant (TI) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends "only" indirectly on the time-domain (via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".
Mathematically speaking, "time-invariance" of a system is the following property:
In the language of signal processing, this property can be satisfied if the transfer function of the system is not a direct function of time except as expressed by the input and output.
In the context of a system schematic, this property can also be stated as follows, as shown in the figure to the right:
If a time-invariant system is also linear, it is the subject of linear time-invariant theory (linear time-invariant) with direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. Nonlinear time-invariant systems lack a comprehensive, governing theory. Discrete time-invariant systems are known as shift-invariant systems. Systems which lack the time-invariant property are studied as time-variant systems.
Simple example.
To demonstrate how to determine if a system is time-invariant, consider the two systems:
Since the System Function formula_4 for system A explicitly depends on "t" outside of formula_5, it is not time-invariant because the time-dependence is not explicitly a function of the input function.
In contrast, system B's time-dependence is only a function of the time-varying input formula_5. This makes system B time-invariant.
The Formal Example below shows in more detail that while System B is a Shift-Invariant System as a function of time, "t", System A is not.
Formal example.
A more formal proof of why systems A and B above differ is now presented. To perform this proof, the second definition will be used.
More generally, the relationship between the input and output is
and its variation with time is
For time-invariant systems, the system properties remain constant with time,
Applied to Systems A and B above:
Abstract example.
We can denote the shift operator by formula_26 where formula_27 is the amount by which a vector's index set should be shifted. For example, the "advance-by-1" system
can be represented in this abstract notation by
where formula_30 is a function given by
with the system yielding the shifted output
So formula_33 is an operator that advances the input vector by 1.
Suppose we represent a system by an operator formula_34. This system is time-invariant if it commutes with the shift operator, i.e.,
If our system equation is given by
then it is time-invariant if we can apply the system operator formula_34 on formula_30 followed by the shift operator formula_26, or we can apply the shift operator formula_26 followed by the system operator formula_34, with the two computations yielding equivalent results.
Applying the system operator first gives
Applying the shift operator first gives
If the system is time-invariant, then


Time-variant system

A time-variant system is a system whose output response depends on moment of observation as well as moment of input signal application. In other words, a time delay or time advance of input not only shifts the output signal in time but also changes other parameters and behavior. Time variant systems respond differently to the same input at different times. The opposite is true for time invariant systems (TIV).
Overview.
There are many well developed techniques for dealing with the response of linear time invariant systems, such as Laplace and Fourier transforms. However, these techniques are not strictly valid for time-varying systems. A system undergoing slow time variation in comparison to its time constants can usually be considered to be time invariant: they are close to time invariant on a small scale. An example of this is the aging and wear of electronic components, which happens on a scale of years, and thus does not result in any behaviour qualitatively different from that observed in a time invariant system: day-to-day, they are effectively time invariant, though year to year, the parameters may change. Other linear time variant systems may behave more like nonlinear systems, if the system changes quickly – significantly differing between measurements.
The following things can be said about a time-variant system:
Linear time-variant systems.
Linear-time variant (LTV) systems are the ones whose parameters vary with time according to previously specified laws. Mathematically, there is a well defined dependence of the system over time and over the input parameters that change over time.
In order to solve time-variant systems, the algebraic methods initial conditions of the system i.e. whether the system is zero-input or non-zero input system.
Examples of time-variant systems.
The following time varying systems cannot be modelled by assuming that they are time invariant:


RP-570

RP-570 is a communications protocol used in industrial environments to communicate between a front-end computer and the substation to be controlled.
It is a SCADA legacy protocol and is based on the low-level protocol IEC TC57, format class 1.2.
RP-570 stands for:
"RTU Protocol based on IEC 57 part 5-1 (present IEC 870) version 0 or 1"
External links.
Details may be found here:


Stochastic variance reduction

(Stochastic) variance reduction is an algorithmic approach to minimizing functions that can be decomposed into finite sums. By exploiting the finite sum structure, variance reduction techniques are able to achieve convergence rates that are impossible to achieve with methods that treat the objective as an infinite sum, as in the classical Stochastic approximation setting.
Variance reduction approaches are widely used for training machine learning models such as logistic regression and support vector machines as these problems have finite-sum structure and uniform conditioning that make them ideal candidates for variance reduction.
Finite sum objectives.
A function formula_1 is considered to have finite sum structure if it can be decomposed into a summation or average:
where the function value and derivative of each formula_3 can be queried independently. Although variance reduction methods can be applied for any positive formula_4 and any formula_3 structure, their favorable theoretical and practical properties arise when formula_4 is large compared to the condition number of each formula_3, and when the formula_3 have similar (but not necessarily identical) Lipschitz smoothness and strong convexity constants.
The finite sum structure should be contrasted with the stochastic approximation setting which deals with functions of the form formula_9
which is the expected value of a function depending on a random variable formula_10. Any finite sum problem can be optimized using a stochastic approximation algorithm by using formula_11.
Rapid Convergence.
Stochastic variance reduced methods without acceleration are able to find a minima of formula_1 within accuracy formula_13, i.e. formula_14 in a number of steps of the order:
The number of steps depends only logarithmically on the level of accuracy required, in contrast to the stochastic approximation framework, where the number of steps formula_16 required grows proportionally to the accuracy required.
Stochastic variance reduction methods converge almost as fast as the gradient descent method's formula_17 rate, despite using only a stochastic gradient, at a formula_18 lower cost than gradient descent.
Accelerated methods in the stochastic variance reduction framework achieve even faster convergence rates, requiring only
steps to reach formula_20 accuracy, potentially formula_21 faster than non-accelerated methods. Lower complexity bounds. for the finite sum class establish that this rate is the fastest possible for smooth strongly convex problems.
Approaches.
Variance reduction approaches fall within 3 main categories: table averaging methods, full-gradient snapshot methods and dual methods. Each category contains methods designed for dealing with convex, non-smooth, and non-convex problems, each differing in hyper-parameter settings and other algorithmic details.
SAGA.
In the SAGA method, the prototypical table averaging approach, a table of size formula_4 is maintained that contains the last gradient witnessed for each formula_3 term, which we denote formula_24. At each step, an index formula_25 is sampled, and a new gradient formula_26 is computed. The iterate formula_27 is updated with:
and afterwards table entry formula_25 is updated with formula_30.
SAGA is among the most popular of the variance reduction methods due to its simplicity, easily adaptable theory, and excellent performance. It is the successor of the SAG method, improving on its flexibility and performance.
SVRG.
The stochastic variance reduced gradient method (SVRG), the prototypical snapshot method, uses a similar update except instead of using the average of a table it instead uses a full-gradient that is reevaluated at a snapshot point formula_31 at regular intervals of formula_32 iterations. The update becomes:
This approach requires two stochastic gradient evaluations per step, one to compute formula_26 and one to compute formula_35 where-as table averaging approaches need only one.
Despite the high computational cost, SVRG is popular as its simple convergence theory is highly adaptable to new optimization settings. It also has lower storage requirements than tabular averaging approaches, which make it applicable in many settings where tabular methods can not be used.
SDCA.
Exploiting the dual representation of the objective leads to another variance reduction approach that is particularly suited to finite-sums where each term has a structure that makes computing the convex conjugate formula_36 or its proximal operator tractable. The standard SDCA method considers finite sums that have additional structure compared to generic finite sum setting:
where each formula_3 is 1 dimensional and each formula_39 is a data point associated with formula_3.
SDCA solves the dual problem:
by a stochastic coordinate ascent procedure, where at each step the objective is optimized with respect to a randomly chosen coordinate formula_42, leaving all other coordinates the same. An approximate primal solution formula_43 can be recovered from the formula_44 values:
This method obtains similar theoretical rates of convergence to other stochastic variance reduced methods, while avoiding the need to specify a step-size parameter. It is fast in practice when formula_46 is large, but significantly slower than the other approaches when formula_46 is small.
Accelerated approaches.
Accelerated variance reduction methods are built upon the standard methods above. The earliest approaches make use of proximal operators to accelerate convergence, either approximately or exactly. Direct acceleration approaches have also been developed
Catalyst acceleration.
The catalyst framework uses any of the standard methods above as an inner optimizer to approximately solve a proximal operator:
after which it uses an extrapolation step to determine the next formula_49:
The catalyst method's flexibility and simplicity make it a popular baseline approach. It doesn't achieve the optimal rate of convergence among accelerated methods, it is potentially slower by up to a log factor in the hyper-parameters.
Point-SAGA.
Proximal operations may also be applied directly to the formula_3 terms to yield an accelerated method. The Point-SAGA method replaces the gradient operations in SAGA with proximal operator evaluations, result in a simple, direct acceleration method:
with the table update formula_53 performed after each step. Here formula_54 is defined as the proximal operator for the formula_55th term:
Unlike other known accelerated methods, Point-SAGA requires only a single iterate sequence formula_43 to be maintained between steps, and it has the advantage of only having a single tunable parameter formula_58. It obtains the optimal accelerated rate of convergence for strongly convex finite-sum minimization without additional log factors.


Plant (control theory)

A plant in control theory is the combination of process and actuator. A plant is often referred to with a transfer function
(commonly in the s-domain) which indicates the relation between an input signal and the output signal of a system without feedback, commonly determined by physical properties of the system. An example would be an actuator with its transfer of the input of the actuator to its physical displacement. In a system with feedback, the plant still has the same transfer function, but a control unit and a feedback loop (with their respective transfer functions) are added to the system.


Common Industrial Protocol

The Common Industrial Protocol (CIP) is an industrial protocol for industrial automation applications. It is supported by ODVA.
Previously known as Control and Information Protocol, CIP encompasses a comprehensive suite of messages and services for the collection of manufacturing automation applications – control, safety, synchronization, motion, configuration and information. It allows users to integrate these manufacturing applications with enterprise-level Ethernet networks and the Internet. It is supported by hundreds of vendors around the world, and is media-independent. CIP provides a unified communication architecture throughout the manufacturing enterprise. It is used in EtherNet/IP, DeviceNet, CompoNet and ControlNet.
ODVA is the organization that supports network technologies built on the Common Industrial Protocol (CIP). These also currently include application extensions to CIP: CIP Safety, CIP Motion and CIP Sync.


Youla–Kucera parametrization

In control theory the Youla–Kučera parametrization (also simply known as Youla parametrization) is a formula that describes all possible stabilizing feedback controllers for a given plant "P", as function of a single parameter "Q".
Details.
The YK parametrization is a general result. It is a fundamental result of control theory and launched an entirely new area of research and found application, among others, in optimal and robust control. The engineering significance of the YK formula is that if one wants to find a stabilizing controller that meets some additional criterion, one can adjust the parameter "Q" such that the desired criterion is met.
For ease of understanding and as suggested by Kučera it is best described for three increasingly general kinds of plant.
Stable SISO plant.
Let formula_1 be a transfer function of a stable single-input single-output system (SISO) system. Further, let formula_2 be a set of stable and proper functions of "formula_3". Then, the set of all proper stabilizing controllers for the plant formula_1 can be defined as
where formula_6 is an arbitrary proper and stable function of "s". It can be said, that formula_6 parametrizes all stabilizing controllers for the plant formula_1.
General SISO plant.
Consider a general plant with a transfer function formula_1. Further, the transfer function can be factorized as
Now, solve the Bézout's identity of the form
where the variables to be found formula_14 must be also proper and stable.
After proper and stable formula_15 are found, we can define one stabilizing controller that is of the form formula_16. After we have one stabilizing controller at hand, we can define all stabilizing controllers using a parameter formula_6 that is proper and stable. The set of all stabilizing controllers is defined as
General MIMO plant.
In a multiple-input multiple-output (MIMO) system, consider a transfer matrix formula_19. It can be factorized using right coprime factors formula_20 or left factors formula_21. The factors must be proper, stable and doubly coprime, which ensures that the system formula_19 is controllable and observable. This can be written by Bézout identity of the form:
After finding formula_24 that are stable and proper, we can define the set of all stabilizing controllers formula_25 using left or right factor, provided having negative feedback.
where formula_27 is an arbitrary stable and proper parameter.
Let formula_1 be the transfer function of the plant and let formula_29 be a stabilizing controller. Let their right coprime factorizations be:
then all stabilizing controllers can be written as
where formula_33 is stable and proper.


Buick Century Cruiser

The Buick Century Cruiser was a dream car (concept car) created by Buick in 1969. It was conceived as being designed for automated highways, where steering wheels would be unnecessary. The vehicle offered swivel contour seats, a refrigerator, and a TV set. The computerized car would be programmed by punch cards with predetermined routes programmed by information provided by electric highway centers. The vehicle would be monitored by a radar like device. The vehicle would also have a device to steer the vehicle manually as well as controlling speed. The canopy would slide open for easy cockpit access. It was related to the Firebird IV concept car and shares an appearance with the GM-X Stiletto.


OPEX (corporation)

OPEX Corporation is a manufacturing company based in Moorestown, New Jersey. They primarily manufacture warehouse automation equipment, high volume mailroom automation equipment, document scanners, and remittance processors. Their warehouse automation products have been implemented at retail and e-commerce companies such as: HBC, BOXED, and iHERB
OPEX employs approximately ~1600 people throughout the world with locations in Moorestown Township, New Jersey USA, Plano, Texas, Bolton, England, Villebon Sur Yvette, France, and Wiesbaden, Germany.


VCDIFF

VCDIFF is a format and an algorithm for delta encoding, described in IETF's RFC 3284. The algorithm is based on Jon Bentley and Douglas McIlroy's paper "Data Compression Using Long Common Strings" written in 1999. VCDIFF is used as one of the delta encoding algorithms in "Delta encoding in HTTP" (RFC 3229) and was employed in Google's Shared Dictionary Compression Over HTTP technology, formerly used in their Chrome browser.
Delta instructions.
VCDIFF has 3 delta instructions. ADD, COPY, and RUN. ADD adds a new sequence, COPY copies from an old sequence, and RUN adds repeated data.
Implementations.
Free software implementations include xdelta (version 3) and open-vcdiff.


Conformal cooling channel

Conformal cooling channel is a cooling passageway which follows the shape or profile of the mould core or cavity to perform rapid uniform cooling process for injection moulding or blow moulding processes.


Four-slide

A four-slide, also known as a multislide, multi-slide, or four-way, is a metalworking machine tool used in the high-volume manufacture of small stamped components from bar or wire stock. The press is most simply described as a horizontal stamping press that uses cams to control tools. The machine is used for progressive or transfer stamping operations.
Design.
A four-slide is quite different from most other presses. The key of the machine is its moving slides that have tools attached, which strike the workpiece together or in sequence to form it. These slides are driven by four shafts that outline the machine. The shafts are connected by bevel gears so that one shaft is driven by an electric motor, and then that shaft's motion drives the other three shafts. Each shaft then has cams which drive the slides, usually of a split-type. This shafting arrangement allows the workpiece to be worked for four sides, which makes this machine extremely versatile. A hole near the center of the machine is provided to expel the completed workpiece.
Advantages and disadvantages.
The greatest advantage of the four-slide machine is its ability to complete all of the operations required to form the workpiece from start to finish. Moreover, it can handle certain parts that transfer or progressive dies cannot, because it can manipulate from four axes. Due to this flexibility it reduces the cost of the finished part because it requires less machines, setups, and handling. Also, because only one machine is required, less space is required for any given workpiece. As compared to standard stamping presses the tooling is usually inexpensive, due to the simplicity of the tools. A four-slide can usually produce 20,000 to 70,000 finished parts per 16-hour shift, depending on the number of operations per part; this speed usually results in a lower cost per part.
The biggest disadvantage is its size constraints. The largest machines can handle stock up to wide, long, and thick. For wires the limit is . Other limits are the travel on the slides, which maxes out at , and the throw of the forming cams, which is between . The machine is also limited to only shearing and bending operations. Extrusion and upsetting operations are impractical because it hinders the movement of the workpiece to the next station. Drawing and stretching require too much tonnage and the mechanisms required for the operations are space prohibitive. Finally, this machine is only feasible to use on high volume parts because of the long lead time required to set up the tooling.
Materials.
The material stock used in four-slides is usually limited by its formability and not the machine capabilities. Usually the forming characteristics and bending radii are the most limiting factors. The most commonly used materials are:
Use.
Items that are commonly produced on this machine include: automotive stampings, hinges, links, clips, and razor blades.


Sanjoy K. Mitter

Sanjoy Kumar Mitter (December 9, 1933 – June 26, 2023) was a Professor in the Department of Electrical Engineering and Computer Science at MIT who was a noted control theorist.
Life and career.
Mitter was born in 1933 in Calcutta, India. He received a B.Sc. in mathematics from the University of Calcutta, and a B.Sc. in Engineering at City and Guilds of London Institute. He continued his studies in the United Kingdom and Ph.D. from Imperial College of Science and Technology, London. After graduation, he worked at Brown, Boveri & Cie, the Battelle Memorial Institute, and the Central Electricity Generating Board before joining Case Western Reserve University (CWRU) in 1965 as an assistant professor. Mitter became an associate professor at CWRU in 1967 and moved to MIT in 1969. He became a professor of electrical engineering at MIT in 1973. At MIT, he was director for both the Center for Intelligent Control and the Laboratory for Information and Decision Systems.
Mitter's research is concerned with Systems, Control and Communication. He has furnished proofs in nonlinear filtering and optimal control theory, as well as carrying out more applied work in image analysis, computation of optimal controls and reliability of electrical power systems.
Mitter lived in Cambridge, Massachusetts. He died in June 2023.
Honors and awards.
Mitter received both the Richard E. Bellman Control Heritage Award from the American Automatic Control Council (2007) and the IEEE Control Systems Award (in 2000). In 1988 he was elected a member of the National Academy of Engineering "for outstanding contributions to the theory and applications of automatic control and nonlinear filtering".


Deep drawing

Deep drawing is a sheet metal forming process in which a sheet metal blank is radially drawn into a forming die by the mechanical action of a punch. It is thus a shape transformation process with material retention. The process is considered "deep" drawing when the depth of the drawn part exceeds its diameter. This is achieved by redrawing the part through a series of dies.
The flange region (sheet metal in the die shoulder area) experiences a radial drawing stress and a tangential compressive stress due to the material retention property. These compressive stresses (hoop stresses) result in flange wrinkles (wrinkles of the first order). Wrinkles can be prevented by using a blank holder, the function of which is to facilitate controlled material flow into the die radius. Deep drawing presses, especially in the Aerospace and Medical industries, require unparalleled accuracy and precision. Sheet hydroforming presses do complex draw work. Bed size, tonnage, stroke, speed, and more can be tailored to your specific draw forming application.
Process.
The total drawing load consists of the ideal forming load and an additional component to compensate for friction in the contacting areas of the flange region and bending forces as well as unbending forces at the die radius. The forming load is transferred from the punch radius through the drawn part wall into the deformation region (sheet metal flange). In the drawn part wall, which is in contact with the punch, the hoop strain is zero whereby the plane strain condition is reached. In reality, mostly the strain condition is only approximately plane. Due to tensile forces acting in the part wall, wall thinning is prominent and results in an uneven part wall thickness, such that the part wall thickness is lowest at the point where the part wall loses contact with the punch, i.e., at the punch radius.
The thinnest part thickness determines the maximum stress that can be transferred to the deformation zone. Due to material volume constancy, the flange thickens and results in blank holder contact at the outer boundary rather than on the entire surface. The maximum stress that can be safely transferred from the punch to the blank sets a limit on the maximum blank size (initial blank diameter in the case of rotationally symmetrical blanks). An indicator of material formability is the limiting drawing ratio (LDR), defined as the ratio of the maximum blank diameter that can be safely drawn into a cup without flange to the punch diameter. Determination of the LDR for complex components is difficult and hence the part is inspected for critical areas for which an approximation is possible. During severe deep drawing the material work hardens and it may be necessary to anneal the parts in controlled atmosphere ovens to restore the original elasticity of the material.
Commercial applications of this metal shaping process often involve complex geometries with straight sides and radii. In such a case, the term stamping is used in order to distinguish between the deep drawing (radial tension-tangential compression) and stretch-and-bend (along the straight sides) components. Deep drawing is always accompanied by other forming techniques within the press. These other forming methods include:
Often components are partially deep drawn in order to create a series of diameters throughout the component (as in the image of the deep draw line). It common use to consider this process as a cost saving alternative to turned parts which require much more raw material.
The sequence of deep drawn components is referred to as a "deep draw line". The numbers of components that form the deep draw line is given by the quantity of "stations" available in the press. In the case of mechanical presses this is determined by the number of cams on the top shaft.
For high precision mass productions, it is always advisable to use a transfer press also known as eyelet press. The advantage of this type of press, in respect to conventional progressive presses, is that the parts are transferred from one die to the next by means of so-called "fingers". Not only do the fingers transfer the parts but they also guide the component during the process. This allows parts to be drawn to the deepest depths with the tightest tolerances.
Other types of presses:
Variations.
Deep drawing has been classified into "conventional" and "unconventional" deep drawing. The main aim of any unconventional deep drawing process is to extend the formability limits of the process. Some of the unconventional processes include hydromechanical deep drawing, Hydroform process, Aquadraw process, Guerin process, Marform process and the hydraulic deep drawing process to name a few.
The Marform process, for example, operates using the principle of rubber pad forming techniques. Deep-recessed parts with either vertical or sloped walls can be formed. In this type of forming, the die rig employs a rubber pad as one tool half and a solid tool half, similar to the die in a conventional die set, to form a component into its final shape. Dies are made of cast light alloys and the rubber pad is 1.5-2 times thicker than the component to be formed. For Marforming, single-action presses are equipped with die cushions and blank holders. The blank is held against the rubber pad by a blank holder, through which a punch is acting as in conventional deep drawing. It is a double-acting apparatus: at first the ram slides down, then the blank holder moves: this feature allows it to perform deep drawings (30-40% transverse dimension) with no wrinkles.
Industrial uses of deep drawing processes include automotive body and structural parts, aircraft components, utensils and white goods. Complex parts are normally formed using progressive dies in a single forming press or by using a press line.
Workpiece materials and power requirements.
Softer materials are much easier to deform and therefore require less force to draw. The following is a table demonstrating the draw force to percent reduction of commonly used materials.
Tool materials.
Punches and dies are typically made of tool steel, however cheaper (but softer) carbon steel is sometimes used in less severe applications. It is also common to see cemented carbides used where high wear and abrasive resistance is present.
Alloy steels are normally used for the ejector system to kick the part out and in durable and heat resistant blankholders.
Lubrication and cooling.
Lubricants are used to reduce friction between the working material and the punch and die. They also aid in removing the part from the punch. Some examples of lubricants used in drawing operations are heavy-duty emulsions, phosphates, white lead, and wax films. Plastic films covering both sides of the part while used with a lubricant will leave the part with a fine surface.