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Candidiasis

Candidiasis is a fungal infection due to any species of the genus Candida (a yeast). When it affects the mouth, in some countries it is commonly called thrush. Signs and symptoms include white patches on the tongue or other areas of the mouth and throat. Other symptoms may include soreness and problems swallowing. When it affects the vagina, it may be referred to as a yeast infection or thrush. Signs and symptoms include genital itching, burning, and sometimes a white "cottage cheese-like" discharge from the vagina. Yeast infections of the penis are less common and typically present with an itchy rash. Very rarely, yeast infections may become invasive, spreading to other parts of the body. This may result in fevers, among other symptoms.

More than 20 types of Candida may cause infection with Candida albicans being the most common. Infections of the mouth are most common among children less than one month old, the elderly, and those with weak immune systems. Conditions that result in a weak immune system include HIV/AIDS, the medications used after organ transplantation, diabetes, and the use of corticosteroids. Other risk factors include during breastfeeding, following antibiotic therapy, and the wearing of dentures. Vaginal infections occur more commonly during pregnancy, in those with weak immune systems, and following antibiotic therapy. Individuals at risk for invasive candidiasis include low birth weight babies, people recovering from surgery, people admitted to intensive care units, and those with an otherwise compromised immune system.

Efforts to prevent infections of the mouth include the use of chlorhexidine mouthwash in those with poor immune function and washing out the mouth following the use of inhaled steroids. Little evidence supports probiotics for either prevention or treatment, even among those with frequent vaginal infections. For infections of the mouth, treatment with topical clotrimazole or nystatin is usually effective. Oral or intravenous fluconazole, itraconazole, or amphotericin B may be used if these do not work. A number of topical antifungal medications may be used for vaginal infections, including clotrimazole. In those with widespread disease, an echinocandin such as caspofungin or micafungin is used. A number of weeks of intravenous amphotericin B may be used as an alternative. In certain groups at very high risk, antifungal medications may be used preventively, and concomitantly with medications known to precipitate infections.

Infections of the mouth occur in about 6% of babies less than a month old. About 20% of those receiving chemotherapy for cancer and 20% of those with AIDS also develop the disease. About three-quarters of women have at least one yeast infection at some time during their lives. Widespread disease is rare except in those who have risk factors.

Signs and symptoms
Signs and symptoms of candidiasis vary depending on the area affected. Most candidal infections result in minimal complications such as redness, itching, and discomfort, though complications may be severe or even fatal if left untreated in certain populations. In healthy (immunocompetent) persons, candidiasis is usually a localized infection of the skin, fingernails or toenails (onychomycosis), or mucosal membranes, including the oral cavity and pharynx (thrush), esophagus, and the sex organs (vagina, penis, etc.); less commonly in healthy individuals, the gastrointestinal tract,   urinary tract, and respiratory tract are sites of candida infection.

In immunocompromised individuals, Candida infections in the esophagus occur more frequently than in healthy individuals and have a higher potential of becoming systemic, causing a much more serious condition, a fungemia called candidemia. Symptoms of esophageal candidiasis include difficulty swallowing, painful swallowing, abdominal pain, nausea, and vomiting.

Mouth
Infection in the mouth is characterized by white discolorations in the tongue, around the mouth, and in the throat. Irritation may also occur, causing discomfort when swallowing.

Thrush is commonly seen in infants. It is not considered abnormal in infants unless it lasts longer than a few weeks.

Genitals
Infection of the vagina or vulva may cause severe itching, burning, soreness, irritation, and a whitish or whitish-gray cottage cheese-like discharge. Symptoms of infection of the male genitalia (balanitis thrush) include red skin around the head of the penis, swelling, irritation, itchiness and soreness of the head of the penis, thick, lumpy discharge under the foreskin, unpleasant odour, difficulty retracting the foreskin (phimosis), and pain when passing urine or during sex.

Skin
Signs and symptoms of candidiasis in the skin include itching, irritation, and chafing or broken skin.

Invasive infection
Common symptoms of gastrointestinal candidiasis in healthy individuals are anal itching, belching, bloating, indigestion, nausea, diarrhea, gas, intestinal cramps, vomiting, and gastric ulcers. Perianal candidiasis can cause anal itching; the lesion can be red, papular, or ulcerative in appearance, and it is not considered to be a sexually transmitted infection. Abnormal proliferation of the candida in the gut may lead to dysbiosis. While it is not yet clear, this alteration may be the source of symptoms generally described as the irritable bowel syndrome, and other gastrointestinal diseases.

Neurological symptoms
Systemic candidiasis can affect the central nervous system causing a variety of neurological symptoms, with a presentation similar to meningitis.

Causes
Candida yeasts are generally present in healthy humans, frequently part of the human body's normal oral and intestinal flora, and particularly on the skin; however, their growth is normally limited by the human immune system and by competition of other microorganisms, such as bacteria occupying the same locations in the human body. Candida requires moisture for growth, notably on the skin. For example, wearing wet swimwear for long periods of time is believed to be a risk factor. Candida can also cause diaper rashes in babies. In extreme cases, superficial infections of the skin or mucous membranes may enter the bloodstream and cause systemic Candida infections.

Factors that increase the risk of candidiasis include HIV/AIDS, mononucleosis, cancer treatments, steroids, stress, antibiotic therapy, diabetes, and nutrient deficiency. Hormone replacement therapy and infertility treatments may also be predisposing factors. Use of inhaled corticosteroids increases risk of candidiasis of the mouth. Inhaled corticosteroids with other risk factors such as antibiotics, oral glucocorticoids, not rinsing mouth after use of inhaled corticosteroids or high dose of inhaled corticosteroids put people at even higher risk. Treatment with antibiotics can lead to eliminating the yeast's natural competitors for resources in the oral and intestinal flora, thereby increasing the severity of the condition. A weakened or undeveloped immune system or metabolic illnesses are significant predisposing factors of candidiasis. Almost 15% of people with weakened immune systems develop a systemic illness caused by Candida species. Diets high in simple carbohydrates have been found to affect rates of oral candidiases.

C. albicans was isolated from the vaginas of 19% of apparently healthy women, i.e., those who experienced few or no symptoms of infection. External use of detergents or douches or internal disturbances (hormonal or physiological) can perturb the normal vaginal flora, consisting of lactic acid bacteria, such as lactobacilli, and result in an overgrowth of Candida cells, causing symptoms of infection, such as local inflammation. Pregnancy and the use of oral contraceptives have been reported as risk factors. Diabetes mellitus and the use of antibiotics are also linked to increased rates of yeast infections.

In penile candidiasis, the causes include sexual intercourse with an infected individual, low immunity, antibiotics, and diabetes. Male genital yeast infections are less common, but a yeast infection on the penis caused from direct contact via sexual intercourse with an infected partner is not uncommon.

Breast-feeding mothers may also develop candidiasis on and around the nipple as a result of moisture created by excessive milk-production.

Vaginal candidiasis can cause congenital candidiasis in newborns.

Diagnosis
In oral candidiasis, simply inspecting the person's mouth for white patches and irritation may make the diagnosis. A sample of the infected area may also be taken to determine what organism is causing the infection.

Symptoms of vaginal candidiasis are also present in the more common bacterial vaginosis; aerobic vaginitis is distinct and should be excluded in the differential diagnosis. In a 2002 study, only 33% of women who were self-treating for a yeast infection were found to have such an infection, while most had either bacterial vaginosis or a mixed-type infection.

Diagnosis of a yeast infection is confirmed either via microscopic examination or culturing. For identification by light microscopy, a scraping or swab of the affected area is placed on a microscope slide. A single drop of 10% potassium hydroxide (KOH) solution is then added to the specimen. The KOH dissolves the skin cells, but leaves the Candida cells intact, permitting visualization of pseudohyphae and budding yeast cells typical of many Candida species.

For the culturing method, a sterile swab is rubbed on the infected skin surface. The swab is then streaked on a culture medium. The culture is incubated at 37 °C (98.6 °F) for several days, to allow development of yeast or bacterial colonies. The characteristics (such as morphology and colour) of the colonies may allow initial diagnosis of the organism causing disease symptoms. Respiratory, gastrointestinal, and esophageal candidiasis require an endoscopy to diagnose. For gastrointestinal candidiasis, it is necessary to obtain a 3–5 milliliter sample of fluid from the duodenum for fungal culture. The diagnosis of gastrointestinal candidiasis is based upon the culture containing in excess of 1,000 colony-forming units per milliliter.

Classification
Candidiasis may be divided into these types:
 * Mucosal candidiasis
 * Oral candidiasis (thrush, oropharyngeal candidiasis)
 * Pseudomembranous candidiasis
 * Erythematous candidiasis
 * Hyperplastic candidiasis
 * Denture-related stomatitis — Candida organisms are involved in about 90% of cases
 * Angular cheilitis — Candida species are responsible for about 20% of cases, mixed infection of C. albicans and Staphylococcus aureus for about 60% of cases.
 * Median rhomboid glossitis
 * Candidal vulvovaginitis (vaginal yeast infection)
 * Candidal balanitis — infection of the glans penis, almost exclusively occurring in uncircumcised males
 * Esophageal candidiasis (candidal esophagitis)
 * Gastrointestinal candidiasis
 * Respiratory candidiasis
 * Cutaneous candidiasis
 * Candidal folliculitis
 * Candidal intertrigo
 * Candidal paronychia
 * Perianal candidiasis, may present as pruritus ani
 * Candidid
 * Chronic mucocutaneous candidiasis
 * Congenital cutaneous candidiasis
 * Diaper candidiasis: an infection of a child's diaper area
 * Erosio interdigitalis blastomycetica
 * Candidal onychomycosis (nail infection) caused by Candida
 * Systemic candidiasis
 * Candidemia, a form of fungemia which may lead to sepsis
 * Invasive candidiasis (disseminated candidiasis) — organ infection by Candida
 * Chronic systemic candidiasis (hepatosplenic candidiasis) — sometimes arises during recovery from neutropenia
 * Antibiotic candidiasis (iatrogenic candidiasis)

Prevention
A diet that supports the immune system and is not high in simple carbohydrates contributes to a healthy balance of the oral and intestinal flora. While yeast infections are associated with diabetes, the level of blood sugar control may not affect the risk. Wearing cotton underwear may help to reduce the risk of developing skin and vaginal yeast infections, along with not wearing wet clothes for long periods of time. For women who experience recurrent yeast infections, there is limited evidence that oral or intravaginal probiotics help to prevent future infections. This includes either as pills or as yogurt.

Oral hygiene can help prevent oral candidiasis when people have a weakened immune system. For people undergoing cancer treatment, chlorhexidine mouthwash can prevent or reduce thrush. People who use inhaled corticosteroids can reduce the risk of developing oral candidiasis by rinsing the mouth with water or mouthwash after using the inhaler. People with dentures should also disinfect their dentures regularly to prevent oral candidiasis.

Treatment
Candidiasis is treated with antifungal medications; these include clotrimazole, nystatin, fluconazole, voriconazole, amphotericin B, and echinocandins. Intravenous fluconazole or an intravenous echinocandin such as caspofungin are commonly used to treat immunocompromised or critically ill individuals.

The 2016 revision of the clinical practice guideline for the management of candidiasis lists a large number of specific treatment regimens for Candida infections that involve different Candida species, forms of antifungal drug resistance, immune statuses, and infection localization and severity. Gastrointestinal candidiasis in immunocompetent individuals is treated with 100–200 mg fluconazole per day for 2–3 weeks.

Localized infection
Mouth and throat candidiasis are treated with antifungal medication. Oral candidiasis usually responds to topical treatments; otherwise, systemic antifungal medication may be needed for oral infections. Candidal skin infections in the skin folds (candidal intertrigo) typically respond well to topical antifungal treatments (e.g., nystatin or miconazole). For breastfeeding mothers topical miconazole is the most effective treatment for treating candidiasis on the breasts. Gentian violet can be used for thrush in breastfeeding babies. Systemic treatment with antifungals by mouth is reserved for severe cases or if treatment with topical therapy is unsuccessful. Candida esophagitis may be treated orally or intravenously; for severe or azole-resistant esophageal candidiasis, treatment with amphotericin B may be necessary.

Vaginal yeast infections are typically treated with topical antifungal agents. Penile yeast infections are also treated with antifungal agents, but while an internal treatment may be used (such as a pessary) for vaginal yeast infections, only external treatments – such as a cream – can be recommended for penile treatment. A one-time dose of fluconazole by mouth is 90% effective in treating a vaginal yeast infection. For severe nonrecurring cases, several doses of fluconazole is recommended. Local treatment may include vaginal suppositories or medicated douches. Other types of yeast infections require different dosing. C. albicans can develop resistance to fluconazole, this being more of an issue in those with HIV/AIDS who are often treated with multiple courses of fluconazole for recurrent oral infections.

For vaginal yeast infection in pregnancy, topical imidazole or triazole antifungals are considered the therapy of choice owing to available safety data. Systemic absorption of these topical formulations is minimal, posing little risk of transplacental transfer. In vaginal yeast infection in pregnancy, treatment with topical azole antifungals is recommended for seven days instead of a shorter duration.

For vaginal yeast infections, many complementary treatments are proposed, however a number have side effects. No benefit from probiotics has been found for active infections.

Blood-borne infection
Candidemia occurs when any Candida species infects the blood. Its treatment typically consists of oral or intravenous antifungal medications. Examples include intravenous fluconazole or an echinocandin such as caspofungin may be used. Amphotericin B is another option.

Prognosis
In hospitalized patients who develop candidemia, age is an important prognostic factor. Mortality following candidemia is 50% in patients aged ≥75 years and 24% in patients aged <75 years. Among individuals being treated in intensive care units, the mortality rate is about 30–50% when systemic candidiasis develops.

Epidemiology
Oral candidiasis is the most common fungal infection of the mouth, and it also represents the most common opportunistic oral infection in humans. Infections of the mouth occur in about 6% of babies less than a month old. About 20% of those receiving chemotherapy for cancer and 20% of those with AIDS also develop the disease.

It is estimated that 20% of women may be asymptomatically colonized by vaginal yeast. In the United States there are approximately 1.4 million doctor office visits every year for candidiasis. About three-quarters of women have at least one yeast infection at some time during their lives.

Esophageal candidiasis is the most common esophageal infection in persons with AIDS and accounts for about 50% of all esophageal infections, often coexisting with other esophageal diseases. About two-thirds of people with AIDS and esophageal candidiasis also have oral candidiasis.

Candidal sepsis is rare. Candida is the fourth most common cause of bloodstream infections among hospital patients in the United States. The incidence of bloodstream candida in intensive care units varies widely between countries.

History
Descriptions of what sounds like oral thrush go back to the time of Hippocrates circa 460–370 BCE.

The first description of a fungus as the causative agent of an oropharyngeal and oesophageal candidosis was by Bernhard von Langenbeck in 1839.

Vulvovaginal candidiasis was first described in 1849 by Wilkinson. In 1875, Haussmann demonstrated the causative organism in both vulvovaginal and oral candidiasis is the same.

With the advent of antibiotics following World War II, the rates of candidiasis increased. The rates then decreased in the 1950s following the development of nystatin.

The colloquial term "thrush" is of unknown origin but may stem from an unrecorded Old English word *þrusc or from a Scandinavian root. The term is not related to the bird of the same name. The term candidosis is largely used in British English, and candidiasis in American English. Candida is also pronounced differently; in American English, the stress is on the "i", whereas in British English the stress is on the first syllable.

The genus Candida and species C. albicans were described by botanist Christine Marie Berkhout in her doctoral thesis at the University of Utrecht in 1923. Over the years, the classification of the genera and species has evolved. Obsolete names for this genus include Mycotorula and Torulopsis. The species has also been known in the past as Monilia albicans and Oidium albicans. The current classification is nomen conservandum, which means the name is authorized for use by the International Botanical Congress (IBC).

The genus Candida includes about 150 different species. However, only a few are known to cause human infections. C. albicans is the most significant pathogenic species. Other species pathogenic in humans include C. auris, C. tropicalis, C. parapsilosis, C. dubliniensis, and C. lusitaniae.

The name Candida was proposed by Berkhout. It is from the Latin word toga candida, referring to the white toga (robe) worn by candidates for the Senate of the ancient Roman republic. The specific epithet albicans also comes from Latin, albicare meaning "to whiten". These names refer to the generally white appearance of Candida species when cultured.

Alternative medicine
A 2005 publication noted that "a large pseudoscientific cult" has developed around the topic of Candida, with claims stating that up to one in three people are affected by yeast-related illness, particularly a condition called "Candidiasis hypersensitivity". Some practitioners of alternative medicine have promoted these purported conditions and sold dietary supplements as supposed cures; a number of them have been prosecuted. In 1990, alternative health vendor Nature's Way signed an FTC consent agreement not to misrepresent in advertising any self-diagnostic test concerning yeast conditions or to make any unsubstantiated representation concerning any food or supplement's ability to control yeast conditions, with a fine of $30,000 payable to the National Institutes of Health for research in genuine candidiasis.

Research
High level Candida colonization is linked to several diseases of the gastrointestinal tract including Crohn's disease.

There has been an increase in resistance to antifungals worldwide over the past 30–40 years.

title: Candidiasis


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Control theory

Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality.

To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the error signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. Control theory is used in control system engineering to design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as robotics.

Extensive use is usually made of a diagrammatic style known as the block diagram. In it the transfer function, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the differential equations describing the system.

Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by James Clerk Maxwell. Control theory was further advanced by Edward Routh in 1874, Charles Sturm and in 1895, Adolf Hurwitz, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of PID control theory by Nicolas Minorsky. Although a major application of mathematical control theory is in control systems engineering, which deals with the design of process control systems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs - thus control theory also has applications in life sciences, computer engineering, sociology and operations research.

History

Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the centrifugal governor, conducted by the physicist James Clerk Maxwell in 1868, entitled On Governors. A centrifugal governor was already used to regulate the velocity of windmills. Maxwell described and analyzed the phenomenon of self-oscillation, in which lags in the system may lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate, Edward John Routh, abstracted Maxwell's results for the general class of linear systems. Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877, resulting in what is now known as the Routh–Hurwitz theorem.

A notable application of dynamic control was in the area of crewed flight. The Wright brothers made their first successful test flights on December 17, 1903, and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Continuous, reliable control of the airplane was necessary for flights lasting longer than a few seconds.

By World War II, control theory was becoming an important area of research. Irmgard Flügge-Lotz developed the theory of discontinuous automatic control systems, and applied the bang-bang principle to the development of automatic flight control equipment for aircraft. Other areas of application for discontinuous controls included fire-control systems, guidance systems and electronics.

Sometimes, mechanical methods are used to improve the stability of systems. For example, ship stabilizers are fins mounted beneath the waterline and emerging laterally. In contemporary vessels, they may be gyroscopically controlled active fins, which have the capacity to change their angle of attack to counteract roll caused by wind or waves acting on the ship.

The Space Race also depended on accurate spacecraft control, and control theory has also seen an increasing use in fields such as economics and artificial intelligence. Here, one might say that the goal is to find an internal model that obeys the good regulator theorem. So, for example, in economics, the more accurately a (stock or commodities) trading model represents the actions of the market, the more easily it can control that market (and extract "useful work" (profits) from it). In AI, an example might be a chatbot modelling the discourse state of humans: the more accurately it can model the human state (e.g. on a telephone voice-support hotline), the better it can manipulate the human (e.g. into performing the corrective actions to resolve the problem that caused the phone call to the help-line). These last two examples take the narrow historical interpretation of control theory as a set of differential equations modeling and regulating kinetic motion, and broaden it into a vast generalization of a regulator interacting with a plant.

Linear and nonlinear control theory
The field of control theory can be divided into two branches:
 * Linear control theory – This applies to systems made of devices which obey the superposition principle, which means roughly that the output is proportional to the input. They are governed by linear differential equations.  A major subclass is systems which in addition have parameters which do not change with time, called linear time invariant (LTI) systems.   These systems are amenable to powerful frequency domain mathematical techniques of great generality, such as the Laplace transform, Fourier transform, Z transform, Bode plot, root locus, and Nyquist stability criterion.  These lead to a description of the system using terms like bandwidth, frequency response, eigenvalues, gain, resonant frequencies, zeros and poles, which give solutions for system response and design techniques for most systems of interest.
 * Nonlinear control theory – This covers a wider class of systems that do not obey the superposition principle, and applies to more real-world systems because all real control systems are nonlinear. These systems are often governed by nonlinear differential equations.  The few mathematical techniques which have been developed to handle them are more difficult and much less general, often applying only to narrow categories of systems.  These include limit cycle theory, Poincaré maps, Lyapunov stability theorem, and describing functions.  Nonlinear systems are often analyzed using numerical methods on computers, for example by simulating their operation using a simulation language.  If only solutions near a stable point are of interest, nonlinear systems can often be linearized by approximating them by a linear system using perturbation theory, and linear techniques can be used.

Analysis techniques - frequency domain and time domain
Mathematical techniques for analyzing and designing control systems fall into two different categories: In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a point within that space.
 * Frequency domain – In this type the values of the state variables, the mathematical variables representing the system's input, output and feedback are represented as functions of frequency. The input signal and the system's transfer function are converted from time functions to functions of frequency by a transform such as the Fourier transform, Laplace transform, or Z transform.  The advantage of this technique is that it results in a simplification of the mathematics; the differential equations that represent the system are replaced by algebraic equations in the frequency domain which is much simpler to solve.  However, frequency domain techniques can only be used with linear systems, as mentioned above.
 * Time-domain state space representation – In this type the values of the state variables are represented as functions of time. With this model, the system being analyzed is represented by one or more differential equations.  Since frequency domain techniques are limited to linear systems, time domain is widely used to analyze real-world nonlinear systems.   Although these are more difficult to solve, modern computer simulation techniques such as simulation languages have made their analysis routine.

System interfacing - SISO & MIMO
Control systems can be divided into different categories depending on the number of inputs and outputs.
 * Single-input single-output (SISO) – This is the simplest and most common type, in which one output is controlled by one control signal. Examples are the cruise control example above, or an audio system, in which the control input is the input audio signal and the output is the sound waves from the speaker.
 * Multiple-input multiple-output (MIMO) – These are found in more complicated systems. For example, modern large telescopes such as the Keck and MMT have mirrors composed of many separate segments each controlled by an actuator.  The shape of the entire mirror is constantly adjusted by a MIMO active optics control system using input from multiple sensors at the focal plane, to compensate for changes in the mirror shape due to thermal expansion, contraction, stresses as it is rotated and distortion of the wavefront due to turbulence in the atmosphere.  Complicated systems such as nuclear reactors and human cells are simulated by a computer as large MIMO control systems.

Classical SISO system design
The scope of classical control theory is limited to single-input and single-output (SISO) system design, except when analyzing for disturbance rejection using a second input. The system analysis is carried out in the time domain using differential equations, in the complex-s domain with the Laplace transform, or in the frequency domain by transforming from the complex-s domain. Many systems may be assumed to have a second order and single variable system response in the time domain. A controller designed using classical theory often requires on-site tuning due to incorrect design approximations. Yet, due to the easier physical implementation of classical controller designs as compared to systems designed using modern control theory, these controllers are preferred in most industrial applications. The most common controllers designed using classical control theory are PID controllers. A less common implementation may include either or both a Lead or Lag filter. The ultimate end goal is to meet requirements typically provided in the time-domain called the step response, or at times in the frequency domain called the open-loop response. The step response characteristics applied in a specification are typically percent overshoot, settling time, etc. The open-loop response characteristics applied in a specification are typically Gain and Phase margin and bandwidth. These characteristics may be evaluated through simulation including a dynamic model of the system under control coupled with the compensation model.

Modern MIMO system design
Modern control theory is carried out in the state space, and can deal with multiple-input and multiple-output (MIMO) systems. This overcomes the limitations of classical control theory in more sophisticated design problems, such as fighter aircraft control, with the limitation that no frequency domain analysis is possible. In modern design, a system is represented to the greatest advantage as a set of decoupled first order differential equations defined using state variables. Nonlinear, multivariable, adaptive and robust control theories come under this division. Matrix methods are significantly limited for MIMO systems where linear independence cannot be assured in the relationship between inputs and outputs. Being fairly new, modern control theory has many areas yet to be explored. Scholars like Rudolf E. Kálmán and Aleksandr Lyapunov are well known among the people who have shaped modern control theory.

Stability
The stability of a general dynamical system with no input can be described with Lyapunov stability criteria.
 * A linear system is called bounded-input bounded-output (BIBO) stable if its output will stay bounded for any bounded input.
 * Stability for nonlinear systems that take an input is input-to-state stability (ISS), which combines Lyapunov stability and a notion similar to BIBO stability.

For simplicity, the following descriptions focus on continuous-time and discrete-time linear systems.

Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must have negative-real values, i.e. the real part of each pole must be less than zero. Practically speaking, stability requires that the transfer function complex poles reside The difference between the two cases is simply due to the traditional method of plotting continuous time versus discrete time transfer functions. The continuous Laplace transform is in Cartesian coordinates where the $$x$$ axis is the real axis and the discrete Z-transform is in circular coordinates where the $$\rho$$ axis is the real axis.
 * in the open left half of the complex plane for continuous time, when the Laplace transform is used to obtain the transfer function.
 * inside the unit circle for discrete time, when the Z-transform is used.

When the appropriate conditions above are satisfied a system is said to be asymptotically stable; the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a modulus equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable; in this case the system transfer function has non-repeated poles at the complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are present when poles with real part equal to zero have an imaginary part not equal to zero.

If a system in question has an impulse response of

 * $$\ x[n] = 0.5^n u[n]$$

then the Z-transform (see this example), is given by

 * $$\ X(z) = \frac{1}{1 - 0.5z^{-1}}$$

which has a pole in $$z = 0.5$$ (zero imaginary part). This system is BIBO (asymptotically) stable since the pole is inside the unit circle.

However, if the impulse response was

 * $$\ x[n] = 1.5^n u[n]$$

then the Z-transform is

 * $$\ X(z) = \frac{1}{1 - 1.5z^{-1}}$$

which has a pole at $$z = 1.5$$ and is not BIBO stable since the pole has a modulus strictly greater than one.

Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the root locus, Bode plots or the Nyquist plots.

Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the stability of ships. Cruise ships use antiroll fins that extend transversely from the side of the ship for perhaps 30 feet (10 m) and are continuously rotated about their axes to develop forces that oppose the roll.

Controllability and observability
Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are stable, then the state is termed stabilizable. Observability instead is related to the possibility of observing, through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the behavior of an unobservable state and hence cannot use it to stabilize the system. However, similar to the stabilizability condition above, if a state cannot be observed it might still be detectable.

From a geometrical point of view, looking at the states of each variable of the system to be controlled, every "bad" state of these variables must be controllable and observable to ensure a good behavior in the closed-loop system. That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis.

Solutions to problems of an uncontrollable or unobservable system include adding actuators and sensors.

Control specification
Several different control strategies have been devised in the past years. These vary from extremely general ones (PID controller), to others devoted to very particular classes of systems (especially robotics or aircraft cruise control).

A control problem can have several specifications. Stability, of course, is always present. The controller must ensure that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even worsen the stability of the open-loop system, which must normally be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have $$Re[\lambda] < -\overline{\lambda}$$, where $$\overline{\lambda}$$ is a fixed value strictly greater than zero, instead of simply asking that $$Re[\lambda]<0$$.

Another typical specification is the rejection of a step disturbance; including an integrator in the open-loop chain (i.e. directly before the system under control) easily achieves this. Other classes of disturbances need different types of sub-systems to be included.

Other "classical" control theory specifications regard the time-response of the closed-loop system. These include the rise time (the time needed by the control system to reach the desired value after a perturbation), peak overshoot (the highest value reached by the response before reaching the desired value) and others (settling time, quarter-decay). Frequency domain specifications are usually related to robustness (see after).

Modern performance assessments use some variation of integrated tracking error (IAE, ISA, CQI).

Model identification and robustness
A control system must always have some robustness property. A robust controller is such that its properties do not change much if applied to a system slightly different from the mathematical one used for its synthesis. This requirement is important, as no real physical system truly behaves like the series of differential equations used to represent it mathematically. Typically a simpler mathematical model is chosen in order to simplify calculations, otherwise, the true system dynamics can be so complicated that a complete model is impossible.

 * System identification

The process of determining the equations that govern the model's dynamics is called system identification. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its transfer function or matrix. Such identification from the output, however, cannot take account of unobservable dynamics. Sometimes the model is built directly starting from known physical equations, for example, in the case of a mass-spring-damper system we know that $$ m \ddot(t) = - K x(t) - \Beta \dot{x}(t)$$. Even assuming that a "complete" model is used in designing the controller, all the parameters included in these equations (called "nominal parameters") are never known with absolute precision; the control system will have to behave correctly even when connected to a physical system with true parameter values away from nominal.

Some advanced control techniques include an "on-line" identification process (see later). The parameters of the model are calculated ("identified") while the controller itself is running. In this way, if a drastic variation of the parameters ensues, for example, if the robot's arm releases a weight, the controller will adjust itself consequently in order to ensure the correct performance.

Analysis of the robustness of a SISO (single input single output) control system can be performed in the frequency domain, considering the system's transfer function and using Nyquist and Bode diagrams. Topics include gain and phase margin and amplitude margin. For MIMO (multi-input multi output) and, in general, more complicated control systems, one must consider the theoretical results devised for each control technique (see next section). I.e., if particular robustness qualities are needed, the engineer must shift their attention to a control technique by including these qualities in its properties.
 * Analysis

A particular robustness issue is the requirement for a control system to perform properly in the presence of input and state constraints. In the physical world every signal is limited. It could happen that a controller will send control signals that cannot be followed by the physical system, for example, trying to rotate a valve at excessive speed. This can produce undesired behavior of the closed-loop system, or even damage or break actuators or other subsystems. Specific control techniques are available to solve the problem: model predictive control (see later), and anti-wind up systems. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold.
 * Constraints

Linear systems control
For MIMO systems, pole placement can be performed mathematically using a state space representation of the open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems this can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design.

Nonlinear systems control
Processes in industries like robotics and the aerospace industry typically have strong nonlinear dynamics. In control theory it is sometimes possible to linearize such classes of systems and apply linear techniques, but in many cases it can be necessary to devise from scratch theories permitting control of nonlinear systems. These, e.g., feedback linearization, backstepping, sliding mode control, trajectory linearization control normally take advantage of results based on Lyapunov's theory. Differential geometry has been widely used as a tool for generalizing well-known linear control concepts to the nonlinear case, as well as showing the subtleties that make it a more challenging problem. Control theory has also been used to decipher the neural mechanism that directs cognitive states.

Decentralized systems control
When the system is controlled by multiple controllers, the problem is one of decentralized control. Decentralization is helpful in many ways, for instance, it helps control systems to operate over a larger geographical area. The agents in decentralized control systems can interact using communication channels and coordinate their actions.

Deterministic and stochastic systems control
A stochastic control problem is one in which the evolution of the state variables is subjected to random shocks from outside the system. A deterministic control problem is not subject to external random shocks.

Main control strategies
Every control system must guarantee first the stability of the closed-loop behavior. For linear systems, this can be obtained by directly placing the poles. Nonlinear control systems use specific theories (normally based on Aleksandr Lyapunov's Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to fulfill different specifications varies from the model considered and the control strategy chosen.

 * List of the main control techniques

 * Optimal control is a particular control technique in which the control signal optimizes a certain "cost index": for example, in the case of a satellite, the jet thrusts needed to bring it to desired trajectory that consume the least amount of fuel. Two optimal control design methods have been widely used in industrial applications, as it has been shown they can guarantee closed-loop stability. These are Model Predictive Control (MPC) and linear-quadratic-Gaussian control (LQG). The first can more explicitly take into account constraints on the signals in the system, which is an important feature in many industrial processes. However, the "optimal control" structure in MPC is only a means to achieve such a result, as it does not optimize a true performance index of the closed-loop control system. Together with PID controllers, MPC systems are the most widely used control technique in process control.
 * Robust control deals explicitly with uncertainty in its approach to controller design. Controllers designed using robust control methods tend to be able to cope with small differences between the true system and the nominal model used for design. The early methods of Bode and others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness.  Examples of modern robust control techniques include H-infinity loop-shaping developed by Duncan McFarlane and Keith Glover, Sliding mode control (SMC) developed by Vadim Utkin, and safe protocols designed for control of large heterogeneous populations of electric loads in Smart Power Grid applications.  Robust methods aim to achieve robust performance and/or stability in the presence of small modeling errors.
 * Stochastic control deals with control design with uncertainty in the model. In typical stochastic control problems, it is assumed that there exist random noise and disturbances in the model and the controller, and the control design must take into account these random deviations.
 * Adaptive control uses on-line identification of the process parameters, or modification of controller gains, thereby obtaining strong robustness properties. Adaptive controls were applied for the first time in the aerospace industry in the 1950s, and have found particular success in that field.
 * A hierarchical control system is a type of control system in which a set of devices and governing software is arranged in a hierarchical tree. When the links in the tree are implemented by a computer network, then that hierarchical control system is also a form of networked control system.
 * Intelligent control uses various AI computing approaches like artificial neural networks, Bayesian probability, fuzzy logic, machine learning, evolutionary computation and genetic algorithms or a combination of these methods, such as neuro-fuzzy algorithms, to control a dynamic system.
 * Self-organized criticality control may be defined as attempts to interfere in the processes by which the self-organized system dissipates energy.

People in systems and control
Many active and historical figures made significant contribution to control theory including
 * Pierre-Simon Laplace invented the Z-transform in his work on probability theory, now used to solve discrete-time control theory problems. The Z-transform is a discrete-time equivalent of the Laplace transform which is named after him.
 * Irmgard Flugge-Lotz developed the theory of discontinuous automatic control and applied it to automatic aircraft control systems.
 * Alexander Lyapunov in the 1890s marks the beginning of stability theory.
 * Harold S. Black invented the concept of negative feedback amplifiers in 1927. He managed to develop stable negative feedback amplifiers in the 1930s.
 * Harry Nyquist developed the Nyquist stability criterion for feedback systems in the 1930s.
 * Richard Bellman developed dynamic programming in the 1940s.
 * Warren E. Dixon, control theorist and a professor
 * Kyriakos G. Vamvoudakis, developed synchronous reinforcement learning algorithms to solve optimal control and game theoretic problems
 * Andrey Kolmogorov co-developed the Wiener–Kolmogorov filter in 1941.
 * Norbert Wiener co-developed the Wiener–Kolmogorov filter and coined the term cybernetics in the 1940s.
 * John R. Ragazzini introduced digital control and the use of Z-transform in control theory (invented by Laplace) in the 1950s.
 * Lev Pontryagin introduced the maximum principle and the bang-bang principle.
 * Pierre-Louis Lions developed viscosity solutions into stochastic control and optimal control methods.
 * Rudolf E. Kálmán pioneered the state-space approach to systems and control. Introduced the notions of controllability and observability. Developed the Kalman filter for linear estimation.
 * Ali H. Nayfeh who was one of the main contributors to nonlinear control theory and published many books on perturbation methods
 * Jan C. Willems Introduced the concept of dissipativity, as a generalization of Lyapunov function to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called, led to the study of the linear matrix inequality (LMI) in control theory. He pioneered the behavioral approach to mathematical systems theory.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Control theory


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Joint cracking

Joint cracking is the manipulation of joints to produce a sound and related "popping" sensation. It is sometimes performed by physical therapists, chiropractors, osteopaths pursuing a variety of outcomes.

The cracking of joints, especially knuckles, was long believed to lead to arthritis and other joint problems. However, this is not supported by medical research.

The cracking mechanism and the resulting sound is caused by dissolved gas (nitrogen gas) cavitation bubbles suddenly collapsing inside the joints. This happens when the joint cavity is stretched beyond its normal size. The pressure inside the joint cavity drops and the dissolved gas suddenly comes out of solution and takes gaseous form which makes a distinct popping noise. To be able to crack the same knuckle again requires waiting about 20 minutes before the bubbles dissolve back into the synovial fluid and will be able to form again.

It is possible for voluntary joint cracking by an individual to be considered as part of the obsessive–compulsive disorders spectrum.

Causes

For many decades, the physical mechanism that causes the cracking sound as a result of bending, twisting, or compressing joints was uncertain. Suggested causes included:

 * Cavitation within the joint—small cavities of partial vacuum form in the synovial fluid and then rapidly collapse, producing a sharp sound.
 * Rapid stretching of ligaments.
 * Intra-articular (within-joint) adhesions being broken.
 * Formation of bubbles of joint air as the joint is expanded.

There were several hypotheses to explain the cracking of joints. Synovial fluid cavitation has some evidence to support it. When a spinal manipulation is performed, the applied force separates the articular surfaces of a fully encapsulated synovial joint, which in turn creates a reduction in pressure within the joint cavity. In this low-pressure environment, some of the gases that are dissolved in the synovial fluid (which are naturally found in all bodily fluids) leave the solution, making a bubble, or cavity (tribonucleation), which rapidly collapses upon itself, resulting in a "clicking" sound. The contents of the resultant gas bubble are thought to be mainly carbon dioxide, oxygen and nitrogen. The effects of this process will remain for a period of time known as the "refractory period", during which the joint cannot be "re-cracked", which lasts about 20 minutes, while the gases are slowly reabsorbed into the synovial fluid. There is some evidence that ligament laxity may be associated with an increased tendency to cavitate.

In 2015, research showed that bubbles remained in the fluid after cracking, suggesting that the cracking sound was produced when the bubble within the joint was formed, not when it collapsed. In 2018, a team in France created a mathematical simulation of what happens in a joint just before it cracks. The team concluded that the sound is caused by bubbles' collapse, and bubbles observed in the fluid are the result of a partial collapse. Due to the theoretical basis and lack of physical experimentation, the scientific community is still not fully convinced of this conclusion.

The snapping of tendons or scar tissue over a prominence (as in snapping hip syndrome) can also generate a loud snapping or popping sound.

Relation to arthritis
The common old wives' tale that cracking one's knuckles causes arthritis is without scientific evidence. A study published in 2011 examined the hand radiographs of 215 people (aged 50 to 89). It compared the joints of those who regularly cracked their knuckles to those who did not. The study concluded that knuckle-cracking did not cause hand osteoarthritis, no matter how many years or how often a person cracked their knuckles. This early study has been criticized for not taking into consideration the possibility of confounding factors, such as whether the ability to crack one's knuckles is associated with impaired hand functioning rather than being a cause of it.

The medical doctor Donald Unger cracked the knuckles of his left hand every day for more than sixty years, but he did not crack the knuckles of his right hand. No arthritis or other ailments formed in either hand, and for this, he was awarded 2009's satirical Ig Nobel Prize in Medicine.

title: Joint cracking


text:
Chemical formula

A chemical formula is a way of presenting information about the chemical proportions of atoms that constitute a particular chemical compound or molecule, using chemical element symbols, numbers, and sometimes also other symbols, such as parentheses, dashes, brackets, commas and plus (+) and minus (−) signs. These are limited to a single typographic line of symbols, which may include subscripts and superscripts. A chemical formula is not a chemical name since it does not contain any words. Although a chemical formula may imply certain simple chemical structures, it is not the same as a full chemical structural formula. Chemical formulae can fully specify the structure of only the simplest of molecules and chemical substances, and are generally more limited in power than chemical names and structural formulae.

The simplest types of chemical formulae are called empirical formulae, which use letters and numbers indicating the numerical proportions of atoms of each type. Molecular formulae indicate the simple numbers of each type of atom in a molecule, with no information on structure. For example, the empirical formula for glucose is CH2O (twice as many hydrogen atoms as carbon and oxygen), while its molecular formula is C6H12O6 (12 hydrogen atoms, six carbon and oxygen atoms).

Sometimes a chemical formula is complicated by being written as a condensed formula (or condensed molecular formula, occasionally called a "semi-structural formula"), which conveys additional information about the particular ways in which the atoms are chemically bonded together, either in covalent bonds, ionic bonds, or various combinations of these types. This is possible if the relevant bonding is easy to show in one dimension. An example is the condensed molecular/chemical formula for ethanol, which is CH3\sCH2\sOH or CH3CH2OH. However, even a condensed chemical formula is necessarily limited in its ability to show complex bonding relationships between atoms, especially atoms that have bonds to four or more different substituents.

Since a chemical formula must be expressed as a single line of chemical element symbols, it often cannot be as informative as a true structural formula, which is a graphical representation of the spatial relationship between atoms in chemical compounds (see for example the figure for butane structural and chemical formulae, at right). For reasons of structural complexity, a single condensed chemical formula (or semi-structural formula) may correspond to different molecules, known as isomers. For example, glucose shares its molecular formula C6H12O6 with a number of other sugars, including fructose, galactose and mannose. Linear equivalent chemical names exist that can and do specify uniquely any complex structural formula (see chemical nomenclature), but such names must use many terms (words), rather than the simple element symbols, numbers, and simple typographical symbols that define a chemical formula.

Chemical formulae may be used in chemical equations to describe chemical reactions and other chemical transformations, such as the dissolving of ionic compounds into solution. While, as noted, chemical formulae do not have the full power of structural formulae to show chemical relationships between atoms, they are sufficient to keep track of numbers of atoms and numbers of electrical charges in chemical reactions, thus balancing chemical equations so that these equations can be used in chemical problems involving conservation of atoms, and conservation of electric charge.

Overview
A chemical formula identifies each constituent element by its chemical symbol and indicates the proportionate number of atoms of each element. In empirical formulae, these proportions begin with a key element and then assign numbers of atoms of the other elements in the compound, by ratios to the key element. For molecular compounds, these ratio numbers can all be expressed as whole numbers. For example, the empirical formula of ethanol may be written C2H6O because the molecules of ethanol all contain two carbon atoms, six hydrogen atoms, and one oxygen atom. Some types of ionic compounds, however, cannot be written with entirely whole-number empirical formulae. An example is boron carbide, whose formula of CB_{n} is a variable non-whole number ratio with n ranging from over 4 to more than 6.5.

When the chemical compound of the formula consists of simple molecules, chemical formulae often employ ways to suggest the structure of the molecule. These types of formulae are variously known as molecular formulae and condensed formulae. A molecular formula enumerates the number of atoms to reflect those in the molecule, so that the molecular formula for glucose is C6H12O6 rather than the glucose empirical formula, which is CH2O. However, except for very simple substances, molecular chemical formulae lack needed structural information, and are ambiguous.

For simple molecules, a condensed (or semi-structural) formula is a type of chemical formula that may fully imply a correct structural formula. For example, ethanol may be represented by the condensed chemical formula CH3CH2OH, and dimethyl ether by the condensed formula CH3OCH3. These two molecules have the same empirical and molecular formulae (C2H6O), but may be differentiated by the condensed formulae shown, which are sufficient to represent the full structure of these simple organic compounds.

Condensed chemical formulae may also be used to represent ionic compounds that do not exist as discrete molecules, but nonetheless do contain covalently bound clusters within them. These polyatomic ions are groups of atoms that are covalently bound together and have an overall ionic charge, such as the sulfate [SO4]^{2-} ion. Each polyatomic ion in a compound is written individually in order to illustrate the separate groupings. For example, the compound dichlorine hexoxide has an empirical formula ClO3, and molecular formula Cl2O6, but in liquid or solid forms, this compound is more correctly shown by an ionic condensed formula [ClO2]+[ClO4]-, which illustrates that this compound consists of [ClO2]+ ions and [ClO4]- ions. In such cases, the condensed formula only need be complex enough to show at least one of each ionic species.

Chemical formulae as described here are distinct from the far more complex chemical systematic names that are used in various systems of chemical nomenclature. For example, one systematic name for glucose is (2R,3S,4R,5R)-2,3,4,5,6-pentahydroxyhexanal. This name, interpreted by the rules behind it, fully specifies glucose's structural formula, but the name is not a chemical formula as usually understood, and uses terms and words not used in chemical formulae. Such names, unlike basic formulae, may be able to represent full structural formulae without graphs.

Empirical formula
In chemistry, the empirical formula of a chemical is a simple expression of the relative number of each type of atom or ratio of the elements in the compound. Empirical formulae are the standard for ionic compounds, such as CaCl2, and for macromolecules, such as SiO2. An empirical formula makes no reference to isomerism, structure, or absolute number of atoms. The term empirical refers to the process of elemental analysis, a technique of analytical chemistry used to determine the relative percent composition of a pure chemical substance by element.

For example, hexane has a molecular formula of C6H14, and (for one of its isomers, n-hexane) a structural formula CH3CH2CH2CH2CH2CH3, implying that it has a chain structure of 6 carbon atoms, and 14 hydrogen atoms. However, the empirical formula for hexane is C3H7. Likewise the empirical formula for hydrogen peroxide, H2O2, is simply HO, expressing the 1:1 ratio of component elements. Formaldehyde and acetic acid have the same empirical formula, CH2O. This is also the molecular formula for formaldehyde, but acetic acid has double the number of atoms.

Like the other formula types detailed below, an empirical formula shows the number of elements in a molecule, and determines whether it is a binary compound, ternary compound, quaternary compound, or has even more elements.

Molecular formula

Molecular formulae simply indicate the numbers of each type of atom in a molecule of a molecular substance. They are the same as empirical formulae for molecules that only have one atom of a particular type, but otherwise may have larger numbers. An example of the difference is the empirical formula for glucose, which is CH2O (ratio 1:2:1), while its molecular formula is C6H12O6 (number of atoms 6:12:6). For water, both formulae are H2O. A molecular formula provides more information about a molecule than its empirical formula, but is more difficult to establish.

Structural formula
In addition to indicating the number of atoms of each elementa molecule, a structural formula indicates how the atoms are organized, and shows (or implies) the chemical bonds between the atoms. There are multiple types of structural formulas focused on different aspects of the molecular structure.

The two diagrams show two molecules which are structural isomers of each other, since they both have the same molecular formula C4H10, but they have different structural formulas as shown.

Condensed formula
The connectivity of a molecule often has a strong influence on its physical and chemical properties and behavior. Two molecules composed of the same numbers of the same types of atoms (i.e. a pair of isomers) might have completely different chemical and/or physical properties if the atoms are connected differently or in different positions. In such cases, a structural formula is useful, as it illustrates which atoms are bonded to which other ones. From the connectivity, it is often possible to deduce the approximate shape of the molecule.

A condensed (or semi-structural) formula may represent the types and spatial arrangement of bonds in a simple chemical substance, though it does not necessarily specify isomers or complex structures. For example, ethane consists of two carbon atoms single-bonded to each other, with each carbon atom having three hydrogen atoms bonded to it. Its chemical formula can be rendered as CH3CH3. In ethylene there is a double bond between the carbon atoms (and thus each carbon only has two hydrogens), therefore the chemical formula may be written: CH2CH2, and the fact that there is a double bond between the carbons is implicit because carbon has a valence of four. However, a more explicit method is to write H2C\dCH2 or less commonly H2C::CH2. The two lines (or two pairs of dots) indicate that a double bond connects the atoms on either side of them.

A triple bond may be expressed with three lines (HC\tCH) or three pairs of dots (HC:::CH), and if there may be ambiguity, a single line or pair of dots may be used to indicate a single bond.

Molecules with multiple functional groups that are the same may be expressed by enclosing the repeated group in round brackets. For example, isobutane may be written (CH3)3CH. This condensed structural formula implies a different connectivity from other molecules that can be formed using the same atoms in the same proportions (isomers). The formula (CH3)3CH implies a central carbon atom connected to one hydrogen atom and three methyl groups (CH3). The same number of atoms of each element (10 hydrogens and 4 carbons, or C4H10) may be used to make a straight chain molecule, n-butane: CH3CH2CH2CH3.

Chemical names in answer to limitations of chemical formulae
The alkene called but-2-ene has two isomers, which the chemical formula CH3CH\dCHCH3 does not identify. The relative position of the two methyl groups must be indicated by additional notation denoting whether the methyl groups are on the same side of the double bond (cis or Z) or on the opposite sides from each other (trans or E).

As noted above, in order to represent the full structural formulae of many complex organic and inorganic compounds, chemical nomenclature may be needed which goes well beyond the available resources used above in simple condensed formulae. See IUPAC nomenclature of organic chemistry and IUPAC nomenclature of inorganic chemistry 2005 for examples. In addition, linear naming systems such as International Chemical Identifier (InChI) allow a computer to construct a structural formula, and simplified molecular-input line-entry system (SMILES) allows a more human-readable ASCII input. However, all these nomenclature systems go beyond the standards of chemical formulae, and technically are chemical naming systems, not formula systems.

Polymers in condensed formulae
For polymers in condensed chemical formulae, parentheses are placed around the repeating unit. For example, a hydrocarbon molecule that is described as CH3(CH2)50CH3, is a molecule with fifty repeating units. If the number of repeating units is unknown or variable, the letter n may be used to indicate this formula: CH3(CH2)_{n}CH3.

Ions in condensed formulae
For ions, the charge on a particular atom may be denoted with a right-hand superscript. For example, Na+, or Cu(2+). The total charge on a charged molecule or a polyatomic ion may also be shown in this way, such as for hydronium, H3O+, or sulfate, SO4(2-). Here + and − are used in place of +1 and −1, respectively.

For more complex ions, brackets [ ] are often used to enclose the ionic formula, as in [B12H12](2-), which is found in compounds such as caesium dodecaborate, Cs2[B12H12]. Parentheses can be nested inside brackets to indicate a repeating unit, as in Hexamminecobalt(III) chloride, [Co(NH3)6](3+)Cl3-. Here, (NH3)6 indicates that the ion contains six ammine groups (NH3) bonded to cobalt, and [ ] encloses the entire formula of the ion with charge +3.

This is strictly optional; a chemical formula is valid with or without ionization information, and Hexamminecobalt(III) chloride may be written as [Co(NH3)6](3+)Cl3- or [Co(NH3)6]Cl3. Brackets, like parentheses, behave in chemistry as they do in mathematics, grouping terms together – they are not specifically employed only for ionization states. In the latter case here, the parentheses indicate 6 groups all of the same shape, bonded to another group of size 1 (the cobalt atom), and then the entire bundle, as a group, is bonded to 3 chlorine atoms. In the former case, it is clearer that the bond connecting the chlorines is ionic, rather than covalent.

Isotopes
Although isotopes are more relevant to nuclear chemistry or stable isotope chemistry than to conventional chemistry, different isotopes may be indicated with a prefixed superscript in a chemical formula. For example, the phosphate ion containing radioactive phosphorus-32 is [^{32}PO4]^{3-}. Also a study involving stable isotope ratios might include the molecule ^{18}O^{16}O.

A left-hand subscript is sometimes used redundantly to indicate the atomic number. For example, _{8}O2 for dioxygen, and  for the most abundant isotopic species of dioxygen. This is convenient when writing equations for nuclear reactions, in order to show the balance of charge more clearly.

Trapped atoms

The @ symbol (at sign) indicates an atom or molecule trapped inside a cage but not chemically bound to it. For example, a buckminsterfullerene (C60) with an atom (M) would simply be represented as MC60 regardless of whether M was inside the fullerene without chemical bonding or outside, bound to one of the carbon atoms. Using the @ symbol, this would be denoted M@C60 if M was inside the carbon network. A non-fullerene example is [As@Ni12As20](3-), an ion in which one arsenic (As) atom is trapped in a cage formed by the other 32 atoms.

This notation was proposed in 1991 with the discovery of fullerene cages (endohedral fullerenes), which can trap atoms such as La to form, for example, La@C60 or La@C82. The choice of the symbol has been explained by the authors as being concise, readily printed and transmitted electronically (the at sign is included in ASCII, which most modern character encoding schemes are based on), and the visual aspects suggesting the structure of an endohedral fullerene.

Non-stoichiometric chemical formulae
Chemical formulae most often use integers for each element. However, there is a class of compounds, called non-stoichiometric compounds, that cannot be represented by small integers. Such a formula might be written using decimal fractions, as in Fe0.95O, or it might include a variable part represented by a letter, as in Fe_{1–x}O, where x is normally much less than 1.

General forms for organic compounds
A chemical formula used for a series of compounds that differ from each other by a constant unit is called a general formula. It generates a homologous series of chemical formulae. For example, alcohols may be represented by the formula C_{n}H_{2n + 1}OH (n ≥ 1), giving the homologs methanol, ethanol, propanol for 1 ≤ n ≤ 3.

Hill system
The Hill system (or Hill notation) is a system of writing empirical chemical formulae, molecular chemical formulae and components of a condensed formula such that the number of carbon atoms in a molecule is indicated first, the number of hydrogen atoms next, and then the number of all other chemical elements subsequently, in alphabetical order of the chemical symbols. When the formula contains no carbon, all the elements, including hydrogen, are listed alphabetically.

By sorting formulae according to the number of atoms of each element present in the formula according to these rules, with differences in earlier elements or numbers being treated as more significant than differences in any later element or number—like sorting text strings into lexicographical order—it is possible to collate chemical formulae into what is known as Hill system order.

The Hill system was first published by Edwin A. Hill of the United States Patent and Trademark Office in 1900. It is the most commonly used system in chemical databases and printed indexes to sort lists of compounds.

A list of formulae in Hill system order is arranged alphabetically, as above, with single-letter elements coming before two-letter symbols when the symbols begin with the same letter (so "B" comes before "Be", which comes before "Br").

The following example formulae are written using the Hill system, and listed in Hill order:

 * BrClH2Si
 * BrI
 * CCl4
 * CH3I
 * C2H5Br
 * H2O4S

title: Chemical formula