What is a negative feedback loop? how is it used to keep a project in control?

Example

Determine the overall transfer function for a control system (Figure 2.26) which has a negative feedback loop with a transfer function 4 and a forward path transfer function of 2/(s + 2).

The overall transfer function of the system is:

Goverall(s)=2s+21+4×2s+2=2s+10

Example

Determine the overall transfer function for a system (Figure 2.27) which has a positive feedback loop with a transfer function 4 and a forward path transfer function of 2/(s + 2).

The overall transfer function is:

Goverall(s)=2s+21−4×2s+2=2s−6

6.1.2.1 Effects of a delayed response

Introducing a delay function to our simple negative feedback loop will produce an unexpected oscillation (a counterintuitive behavior). For example, a delay between the time a discrepancy is observed and a corrective action is taken will result in an oscillation in room temperature (see Figure 6.6).

Second, we will consider the common phenomenon known as a positive feedback loop, such as a bank account earning compound interest or a company growing annually at a fixed rate. We said that it would result in an exponential growth curve (see Figure 6.7).

The exponential growth curve resulting from a positive feedback loop assumes unlimited resources, but in reality, a resource is a universal constraint and all exponential growth curves will eventually be influenced by carrying capacity and therefore will ultimately convert to an S-shaped curve.

The Key Distinction Between Rate of O2 Supply and of O2 Consumption (Metabolic Rate)

An engineering control analogy similar to the classic negative feedback loop is useful. This has

a sensor (an input = a chemoreceptor?) that continuously measures O2 consumption somewhere, detecting O2 levels falling as metabolic rate rises and generating a proportional signal,

an integrator (the brainstem) that converts this signal into a proportionate drive to

an effector (an output = breathing = respiratory muscles) to increase O2 supply.

By definition, metabolic rate is the rate of O2 consumption. It is not quite accurate to derive metabolic rate from the rate of CO2 production. This is because CO2 production rate can be partially dissociated from metabolic rate when metabolic substrates other than carbohydrate are used, or during respiratory compensation for metabolic acidosis. The presumption has always been that chemoreceptors must form a major contribution to whatever measure metabolic rate. Nevertheless, other control systems can also be devised (see below) that do not involve chemoreceptors.

At rest, the systemic PaO2 is ∼ 100 mmHg (SpO2 is ∼ 98%), PaCO2 is ∼ 40 mmHg (PetCO2 is ∼ 43 mmHg) and pHa ∼ 7.4 (40 nM). The obsession with partial pressures (as opposed to other functions of blood gas content) arises for two reasons:

the fact that in animal studies the action potential frequency in afferent nerves from carotid chemoreceptors relates better to the PO2 or PCO2 in the fluid bathing the chemoreceptors, than to any other mathematical functions of their gas content

the partial pressure gradients govern the direction and rate of diffusion of gases across barriers and between functionally important locations (Fick’s law).

Obviously the brain ought somehow to monitor continuously the O2 supply (both arterial blood gas levels and blood flow i.e., cardiac output), since any impending failure of the arterial O2 supply is so serious, almost immediately for the brain itself and ultimately for all other tissues. And it is the fall in PaO2, not the normal PaO2 level itself, that the arterial chemoreceptors signal. This is because recordings from arterial chemoreceptors in animals also show that their action potential firing rate is negligible when arterial blood gases are normal. Fig. 1A shows how breathing is appropriately stimulated by lowering PaO2 at a constant (resting) metabolic rate (or, see Fig. 1B, by raising PaCO2).

The discovery of arterial chemoreceptors deservedly won their discoverer the 1938 Nobel prize. The trap is in believing that this monitoring of arterial O2 supply is the same as the monitoring of O2 consumption. Ever since, many have fallen into this trap. Nobody yet has looked properly for the metabolic rate chemoreceptor(s) elsewhere. Fig. 2 shows that the chemoreceptors for metabolic rate should be located ideally in skeletal muscle or somewhere along the systemic venous tree (and preferably in multiple, or all veins). Only here do O2 levels fluctuate in proportion to how much O2 is consumed by the structures drained by each vein. The brain could then integrate the signal from all structures. If the brain also knew what was cardiac output and the arterio-venous O2 difference (even knowing just the PvO2 alone might suffice), it could measure overall metabolic rate continuously using the Fick Principle (Parkes, 2017).

Pathogenesis

The underlying mechanism of CSR is instability of the negative feedback loop controlling ventilation during sleep. This negative feedback loop aims to maintain a constant partial pressure of arterial carbon dioxide and oxygen. It can become unstable if the gain of the system (“loop gain”) becomes greater than 1 (Khoo et al., 1982). The gain is the magnitude of a system’s response divided by the magnitude of the initial disturbance which caused that response. In other words, instability occurs when the ventilatory control system is hyperresponsive to changes in arterial partial pressure of carbon dioxide. This hyperresponsiveness is exaggerated when there is a delay in circulatory time resulting in a phasic mismatch of the arterial carbon dioxide pressure at the central chemoreceptors compared to pulmonary vasculature. It is this instability which leads to oscillating ventilation which swings between hyperventilation and hypoventilation leading to eventual apnea. The pattern of the oscillations in CSR are directly related to the magnitude of the loop gain of the system. As loop gain increases, the proportion of each cycle spent in apnea increases and can thus loop gain can be estimated using data from the standard polysomnogram (Sands et al., 2011). In heart failure, patients have an increased chemoreceptor response (Solin et al., 2000), a prolonged circulatory time and reduced lung volumes, all of which act to increase loop gain and thus ventilatory instability.

The proposed mechanisms responsible for the underlying heightened ventilatory response and associated hypocapnia with hyperventilation are fourfold: elevated sympathetic activation, loss of nitric oxide within the carotid body, pulmonary congestion with stimulation of vagal pulmonary afferents and relatively increased dead space. Studies have also suggested that rostral fluid shift from edematous lower limbs into the lungs can exacerbate CSR severity (Yumino et al., 2010). The volume of fluid moving from the lower limbs correlated strongly with central apnea frequency. The authors suggested that this fluid was moving into the lungs resulting in stimulation of pulmonary vagal afferents as well as reducing lung volume.

CSR occurs most frequently during Stage1 and 2 sleep due to the ventilation instability associated with sleep-wake transition. In addition, CSR is particularly rare during rapid eye movement sleep as ventilatory control is less dependent on autonomic control.

Summary

Traditional enterprise data warehousing projects easily fall into a negative feedback loop where fear of failure drives companies to instill so many checks and controls on the development process that delivery of value to business stakeholders slows to a crawl. To some extent these process bottlenecks can be corrected by switching to generic incremental programming methods such as Scrum and Kanban once those starter methods have been adapted for the additional complexity that data integration adds to a software development project. In order to deliver at maximum speed and with minimum risk, development teams will also need agile adaptations for the remaining components of the application development life cycle that wrap around the work of programming data transforms and front-end modules. Whereas my earlier books focused upon accelerating the work of programming business intelligence applications, this volume provides detailed guidance for fast and incremental approaches to the three remaining engineering disciplines that every EDW team must master: requirements management, database design, and quality assurance. It also describes how the latest productivity tools for data analytics, such as data virtualization, data warehouse automation, and big data management system, offer teams a new type of application development value cycle that dramatically reduces the amount of labor needed to design, build, and deploy each incremental version of an enterprise data warehouse. By following the suggestions provided in the chapters ahead, EDW project leaders such as solution architects, data modelers, and system testers can accelerate their team's delivery pace by a factor of three. Moreover, by incorporating the new breeds of productivity tools on top of those process improvements, EDW project leaders can triple again their team's delivery speed.

Increased Loop Gain

The term loop gain is used to describe the negative feedback loop that controls ventilation to regulate blood gas levels within narrow limits (Younes et al., 2007). This feedback loop incorporates pulmonary, circulatory and central responses, and works to counteract any change in blood gas tension (increase or decrease) with a corresponding change in ventilation. Loop gain has three key gain components. Controller gain, equivalent to central plus peripheral chemosensitivity, governs hypercapnic and hypoxic ventilatory responses to changes in blood gases (controller gain = Δventilation/ΔPCO2) (Wellman et al., 2011). Plant gain is the “pulmonary filter” that processes blood gas variations in the lungs to a sudden change in ventilation. Thus, plant gain is an estimation of the effectiveness of the lungs to alter blood gases (plant gain = ΔPCO2/ventilation) and is influenced by factors such as changes in lung volume. Finally, mixing gain, a secondary “smoothing filter,” further regulates blood gas tensions when the blood passes from the pulmonary capillary to the larger chest vessels. Circulation delay times (i.e. lung to chemoreceptors) are also crucial components of loop gain and the resultant time course of unstable breathing patterns.

In unstable systems, an elevated dynamic loop gain (Hudgel et al., 1998) reacts to breathing disturbances such as hypopneas or apneas with an exaggerated response (such that the response/disturbance ratio is ≥ 1). This heightened response itself then becomes a new disturbance and propagates ventilatory instability at a rate determined by the characteristics of the plant (i.e. components of the lung). The amplitude of the response is determined by the controller (i.e. chemoreceptors). Thus, loop gain = plant × controller gain. While high loop gain underpins unstable breathing, low loop gain acts to progressively dampen the original disturbance. Thus, the subsequent, smaller ventilatory response is lower. This then forms the next cycle such that the ventilatory response/disturbance ratio is < 1 (Khoo et al., 1982).

High loop gain occurs when there is an alteration in one or more gain components. The common denominator is the increase in ventilatory drive, which can ultimately contribute to oscillations in PCO2 to below the apnea threshold. Given that PCO2 is the main driver of breathing during sleep, if PCO2 falls below a critical threshold during sleep, breathing ceases (Dempsey, 2005). These factors promote OSA through several mechanisms. First, continuous oscillations in ventilatory drive to the respiratory pump muscles lead to breathing instability with concomitant intermittent periods of reduced “mechanical drive” to the upper airway muscles. This can result in a mismatch between drive to the pump versus pharyngeal dilator muscles and further contribute to cycles of upper airway closure-reopening (Eckert and Wellman, 2015). Another contributor to unstable breathing is a heightened ventilatory response to arousal (Jordan et al., 2003). This can perpetuate transient blood gas disturbances and unstable breathing during sleep (Eckert and Younes, 2014). Third, large negative inspiratory pressure swings to small increases in CO2 can cause “negative effort dependence” (Eckert and Wellman, 2015; Genta et al., 2014b). This is where the upper airway is effectively “sucked closed” during inspiration.

Elevated loop gain contributes to OSA (Wellman et al., 2011; Eckert et al., 2013; Deacon et al., 2018; Salloum et al., 2010) in at least one third of patients (Eckert et al., 2013).

Targeted therapy for an elevated loop gain includes oxygen (Sands et al., 2018a; Wellman et al., 2008), CO2 (Xie et al., 2013) and carbonic anhydrase inhibitors (Edwards et al., 2012, 2013) (Fig. 3). Strategies to increase lung volume reduce plant gain and can therefore reduce loop gain. These include: CPAP (Messineo et al., 2018; Edwards et al., 2009), positional therapy (Joosten et al., 2015), weight loss (Parameswaran et al., 2006) and lung inflation in restrictive (Kolilekas et al., 2013) or obstructive (Krachman et al., 2016) lung diseases.

B Multiple sources for oscillations in the circadian regulatory network

The genetic regulatory network underlying circadian rhythms contains intertwined positive and negative feedback loops. In view of the complexity of these regulatory interactions, it should not be a surprise that more than one mechanism in the network may give rise to sustained oscillations. Evidence pointing to the existence of a second oscillatory mechanism (Leloup and Goldbeter 2003, 2004) stems from the fact that sustained oscillations generally disappear in the absence of PER protein (Figure 13.17a). However, even in such conditions sustained oscillations may occur with a period that is not necessarily circadian (Figure 13.17b). This second oscillator is based on the negative autoregulation exerted by BMAL1 on the expression of its gene, via the Rev-Erbα gene (see Figure 13.15).

Experimental observations so far suggest that if a second oscillator exists in the circadian regulatory network it does not manifest itself in producing rhythmic behavior. Thus, mPer1/mPer2 (Zheng et al. 2001) or mCry1/mCry2 (Van der Horst et al. 1999) double-knockout mice are arrhythmic. In some conditions, however, an extended light pulse can restore rhythmic behavior in a low proportion of mPer1/mPer2 double-knockout mice (K. Bae and D. Weaver, personal communication).

In the absence of the negative feedback exerted by BMAL1 on the expression of its gene, oscillations can still originate from the PER—CRY negative feedback loop involving BMAL1. This result holds with the observation that circadian oscillations occur in the absence of REV-ERBa in mice (Preitner et al. 2002). Preventing altogether the synthesis of BMAL1 suppresses oscillations, because BMAL1 is involved in the mechanism of the two oscillators described previously.

1 Mammalian circadian genetic control mechanism

Molecular clocks reside within suprachiasmatic nucleus cells. Each molecular circadian clock is a negative feedback loop of the gene transcription and its translation into protein. The loop includes several genes and their protein products. In the case of mammals, three period genes (Per1, Per2, and Per3) and two Cryptochrome genes (Cry1 and Cry2) constitute the negative limb, whereas Clock and Bmal1 (Bmal) genes constitute the positive limb of the feedback loop in the molecular circadian clock. To simplify the model and gain the insight of each interaction path, we deal with two groups of genes—Per1, Per2, and Per3 genes and Cry1 and Cry2 genes—as Per and Cry, respectively.

The mammalian circadian genetic control mechanism consists of two interlocked negative feedback loops. PER and CRY proteins collaborate in the regulation of their own expression, assembling in PER/CRY complexes that permit nuclear translocation and inactivation of Per and Cry transcription in a cycling negative feedback loop. At the same time, the PER/CRY complex inactivates the expression of the Rev-Erb gene. Proteins of Bmal and Clock form heterodimers that activate Per, Cry, and Rev-Erb transcriptions. The Bmal gene is inactivated by the REV-ERB protein in the nucleus. Except for the gene Clock, the genes are rhythmically expressed in about 24 hours according to these molecular interactions of genes and their products.

1 Systems Theory

Systems or control theory originated from practical, engineering-related questions concerning regulation processes. Its aim is to analyze the structure of systems and to provide adequate, preferably quantitative, models for them. The basic unit of systems control is a negative feedback loop which consists of the following elements: a detector which provides the system with information about the environment (input), a reference value which indicates what the input ideally should be, and an output function that causes the system in case of a discrepancy between input and reference value to ‘behave’ in a way that affects the environment in order to reduce the discrepancy (for details see Bischof 1985, 1995).

The notion that processes such as these not only occur in machines, but also can be found in living systems such as animals or human beings was first introduced into the life sciences by Wiener (1948). Almost at the same time, it was also formulated in German by von Holst and Mittelstaedt (1971). Since then, this idea has appeared in different theories. The models based on systems theory, however, vary considerably in their degree of quantification, their complexity, and their generalizability.

One simple, general, and well-known example of a negative feedback loop used to characterize human behavior is the TOTE unit described by Miller et al. (1960). A more complex attempt to explain the general principles of goal directed behavior in terms of systems theory stems from Carver and Scheier (1998). Examples of mathematically formulated control systems and their experimental verification can be found in Powers (1978).

Since systems theory not only is especially apt for the description of purposive behavior, but also allows for the description of short-term as well as long-term regulation processes, models or theories dealing with more specific topics can be found in the fields of motivational as well as developmental psychology. Examples in the area of motivational psychology are the control theory model of work motivation by Klein (1989) and the control system model of organizational motivation by Lord and Hanges (1987). The former describes individual work motivation as a function of feedback and several cognitive processes. The latter also deals with task performance, but can be applied to individual motivation as well as interpersonal activities in organizations. A more comprehensive model, which is at least partly based on a systemic approach, is presented by Dörner (1994, 1998), who intends nothing less than to discover the ‘construction plan’ of the human soul. In contrast to these authors, Hyland (1988) uses his motivational control theory as a meta-theoretical framework to integrate different, already existing, motivational theories which—according to him—reflect different aspects of a single underlying process which can again be described with a negative feedback loop.

In developmental psychology, a dynamic systems approach is used, for example, by Fogel and Thelen (1987) in their theory of motor development, even if they do not put it into terms of systems theory. The models of Burke (1991) and Bell (1971) are more explicitly based on systems theory. The former describes the process of identity formation as a negative feedback loop. The latter develops a control system to explain reciprocal socialization effects between children and their caregivers. A much broader model which is concerned with the general principles of human development is the developmental systems theory by Ford and Lerner (1992). The best known example for a systems approach in developmental psychology, however, is probably Bowlby's (1969) theory of infant attachment, a phenomenon which he viewed in feedback terms, although he never ventured to formulate it in a mathematical model. This was eventually performed by Bischof (1975, 1985, 1993), who provided a completely quantified model of social motivation. This is referred to as the Zurich Model of Social Motivation (or Zurich Model) and will be described in the following section.

1.7 Open Systems and the Environment

An isolated physical system does not exchange matter, energy, or information with its environment; thus, an isolated system has a limited supply of energy. A closed physical system exchanges energy, but not matter with the environment.

In contrast, an open system interacts with its environment transferring, matter, energy, and/or information through the boundary separating it from its environment. An open system is receptive to new information. The theory of open systems was formulated in the mid-1950s by the biologist Ludwig von Bertalanffy [36].

Living systems are open to exchange matter, energy, and information, while in thermodynamically closed systems the entropy only increases. Complex systems interact with their environment, therefore, they are open systems. For example, living systems have to exchange matter and energy with the environment in order to survive.

Open systems are analyzed in [132]. The author stresses that open systems evolve as cycles of events and have several defining properties:

they import energy from the environment;

they transform the resources available to them;

they export some resources to the environment;

they generate negative entropy;

they have a negative feedback loop to maintain a steady-state;

they dynamically achieve homeostasis;

they strive to achieve differentiation and specialization;

they include mechanisms for integration and coordination; and

they enjoy equifinality.

Homeostasis describes the successful survival of an organism, and equifinality means that a given end state can be reached by many potential means [36].

Open systems are classified in several categories in the order of their complexity [40]:

1.

systems comprising static structures, e.g., crystals;

simple dynamic systems with predetermined motions, e.g., clocks or the solar system;

systems capable of self-regulation with an externally prescribed target, the cybernetic systems, e.g., a thermostat;

systems capable of self-maintenance through exchange of resources with the environment, e.g., a cell;

blueprint-growth systems, e.g., systems reproducing through seeds or eggs;

systems with a detailed awareness of their environment, e.g., animals;

self-consciousness systems, e.g., humans;

social systems groups sharing a common order and culture; and

transcendental systems—“absolutes and inescapable unknowables.”

The concept of open systems has many applications, not only in natural sciences, but also in social sciences and economics. In computing, an open system supports a combination of interoperability, portability, and open software standards. The idea of open computer systems, initially referred to systems based on Unix, in contrast to the proprietary systems of vendors such as IBM, HP, or Microsoft.

What is a negative feedback loop and how does it work?

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What is negative feedback and where it is used?

What is a feedback loop for projects?