Control theory

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Control theory in control system engineering offerings with dynamic system controls that operate continuously in process and engineered machines. The goal is to develop a control model to control the system using optimal control measures without delay or overshoot and ensuring stability control.

To do this, controllers with required corrective behavior is required. This controller monitors process controlled variables (PV), and compares them with references or set points (SP). The difference between the actual and desired value of the process variable, called the signal error , or the SP-PV error, is applied as feedback to generate the control action to bring the controlled process variable to the same value as the set point. Another aspect that is also studied is control and observation. It's based on a kind of advanced automation that revolutionizes manufacturing, aircraft, communications, and other industries. This is a control feedback , which is usually continuous and involves taking measurements using sensors and making adjustments countable to keep variables measured within the specified range by means of "final control of elements" such as control valves.

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

Control of the theoretical date from the 19th century, when the theoretical basis for the governor's operation was first described by James Clerk Maxwell. Control theory was subsequently put forward by Edward Routh in 1874, Charles Sturm and in 1895, Adolf Hurwitz, all of whom contributed to the formation of the criterion of stability control; and from 1922 onwards, the development of PID control theory by Nicolas Minorsky. Although the main application of control theory is in control system engineering, which deals with the design of process control systems for industry, other applications are far beyond this. As a general theory of feedback systems, control theory is useful wherever feedback occurs. Some examples in physiology, electronics, climate modeling, machine design, ecosystems, navigation, artificial neural networks, predator interactions, gene expression, and production theory.

Video Control theory

History

Although the control system of various types dates back to antiquity, a more formal analysis of the field begins with the dynamics analysis of the centrifugal governor, conducted by the physicist James Clerk Maxwell in 1868, entitled On Governors . It describes and analyzes the phenomenon of self-oscillation, where slowness in the system can lead to overcompensation and unstable behavior. This caused confusion in the topic, in which Maxwell's classmate, Edward John Routh, abstracted Maxwell's results for a general class of linear systems. Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877, yielding what is now known as Routh-Hurwitz's theorem.

Prominent dynamic control applications are in the field of manned flight. The Wright brothers performed their first successful test flight on December 17, 1903 and were distinguished by their ability to control their flights for substantial periods (more than the ability to generate elevators from airfoils, known). Continuous and reliable aircraft control is required for flights that last longer than a few seconds.

In World War II, control theory became an important field of research. Irmgard FlÃÆ'¼gge-Lotz develops the theory of automatic discontinuous control systems, and applies the bang-bang principle for the development of automatic flight control equipment for aircraft. Other areas of application for interrupted control include fire control systems, guidance and electronic systems.

A centrifugal governor is used to regulate the speed of a windmill.

Sometimes, mechanical methods are used to improve system stability. For example, a ship's stabilizer is a fins mounted below the water surface and appears laterally. In contemporary vessels, they may be active girroscopic fins, which have the capacity to change the angle of their attacks to fight windings caused by wind or waves working on the ship.

The Space Race also relies on accurate spacecraft control, and control theory has also seen increased use in such areas as the economy.

Maps Control theory

Control open loop and closed loop (feedback )

Basically, there are two types of control loops: open-loop control and closed-loop control (feedback). Control theory deals only with closed loop control.

In the open-loop control, the control action of the controller does not depend on the "process output" (or "process variables controlled" - PV). A good example of this is a central heating boiler controlled only by a timer, so heat is applied for a constant time, regardless of the temperature of the building. Control action is a timed boiler switch/activation, process variable is the building temperature, but not linked.

In closed loop control, the control action of the controller depends on the feedback of the process in the form of a process variable value (PV). In the case of boiler analogy, the closed loop will include a thermostat to compare the building temperature (PV) with the temperature set on the thermostat (set point - SP). This produces a controller output to keep the building at the desired temperature by turning on and off the boiler. The closed-loop control, therefore, has a feedback loop that ensures the controller performs a control action to manipulate the process variable into the same as "Reference Input" or "set point". For this reason, closed-loop controllers are also called feedback controllers.

The definition of a closed-loop control system according to the British Standard Institution is "a control system having monitoring feedback, the deviation signal formed as a result of this feedback is used to control the action of the final control element in such a way that it tends to reduce the deviation to zero."

Too; "A Feedback Control System is a system that tends to maintain a defined relationship from one system variable to another by comparing the functions of these variables and using differences as a control tool."

Another example

An example of a control system is a car's cruise control, which is a device designed to maintain the vehicle speed at the desired speed of desired or the reference provided by the driver. Controllers are shipping controls, factories are cars, and systems are automobiles and shipping controls. The system output is the speed of the car, and the control itself is the engine throttle position that determines how much power is given by the machine.

The primitive way to apply cruise control is to simply lock the throttle position when the driver cruises the control. However, if the cruise control moves on a flat stretch of road, then the car will run slower to climb and faster when it will decrease. This type of controller is called open loop control because there is no feedback; no measurement of system output (car speed) is used to change the control (throttle position.) As a result, the controller can not compensate for changes that work on the car, such as a change in the slope of the road.

In a closed-loop control system, data from sensors that monitor the speed of the car (the system output) enters a continuous controller comparing the number representing velocity with the reference quantity representing the desired speed. The difference, called an error, determines the position of the throttle (control). The result is to match the speed of the car to the reference speed (maintaining the desired system output). Now, as the car goes uphill, the difference between the input (perceived speed) and the reference constantly determines the throttle position. When the perceived speed drops below the reference, the difference increases, the throttle opens, and the engine power increases, accelerating the vehicle. In this way, the controller dynamically negates changes to the speed of the car. The central idea of ​​this control system is loop feedback , the controller affects the system output, which in turn is measured and fed back to the controller.

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Classic control theory

To overcome the limitations of open-loop controllers, control theory introduces feedback. Closed loop controllers use feedback to control the status or output of dynamic systems. Its name is derived from the information path in the system: the input process (eg, the applied voltage to the electric motor) has an effect on the process output (eg, speed or motor torque), as measured by the sensor and processed by the controller; the result (control signal) is "feedback" as input to the process, closing the loop.

Closed loop controllers have the following advantages over open-loop controllers:

• denial of interruption (like a hill in the cruise control example above)
• guaranteed performance even with model uncertainty, when model structure does not match real process and model parameter is not exact
• Unstable processes can be stabilized
• reduces sensitivity to parameter variations
• improve the performance of reference tracking

In some systems, closed loops and open-loop controls are used simultaneously. In such systems, open-loop controls are called feedforward and serve to further improve the performance of reference tracking.

The common closed loop control architecture is the PID controller.

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Closed-loop transfer function

The output of the system y (t) is fed back through the sensor measurement F to the comparison with the reference value r (t) . Controller C then takes the error e (difference) between the reference and the output to change the input u to the system under P . This is shown in the figure. This type of controller is a closed-loop controller or feedback controller.

This is called single single-input-output control system ( SISO ); MIMO (that is, Multi-Input-Multi-Output System), with more than one input/output, is common. In such cases, variables are represented by a vector rather than a simple scalar value. For some distributed parameter systems, the vector may be of dimensionless (usually functional) dimension.

Jika kita menganggap pengontrol C , pabrik P , dan sensor F adalah linier dan waktu-invariant (yaitu elemen fungsi transfernya) C , P (s) , dan F (s) tidak bergantung pada waktu), sistem di atas dapat dianalisis menggunakan Transformasi Laplace pada variabel. Ini memberikan hubungan berikut:

${\ displaystyle Y (s) = P (s) U (s)}  ÂÂ$

${\ displaystyle U (s) = C (s) E (s)}  ÂÂ$

${\ displaystyle E (s) = R (s) -F (s) Y (s).}  ÂÂ$

Memecahkan untuk Y ( s ) dalam hal R ( s ) memberi

${\ displaystyle Y (s) = \ kiri ({\ frac {P (s) C (s)} {1 F (s) P (s) C (s))}} \ right) R (s) = H (s) R (s).} Â Â$

Express ${\ displaystyle H (s) = {\ frac {P (s) C (s)} {1 F (s) P (s) C (s) )}}} Â Â$ disebut sebagai fungsi transfer loop tertutup dari sistem. Pembilang adalah gain maju (loop terbuka) dari r ke y , dan penyebutnya adalah satu plus keuntungan dalam mengelilingi putaran umpan balik, yang disebut loop mendapatkan. Jika ${\ displaystyle | P (s) C (s) | \ gg 1} Â Â$ , yaitu memiliki norma besar dengan setiap nilai s , dan jika ${\ displaystyle | F (s) | \ approx 1} Â$ , maka Y (s) kira-kira same dengan R (s) give output sangat melacak masukan reference.

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Kontrol umpan balik PID

A derivative control integral ( PID Controller ) is a control loop control control technique that is widely used in control systems.

The PID controller continues to calculate the error value ${\ displaystyle e (t)}$ as the difference between the desired setpoint and the measured process variable and apply the correction based on the proportional, integral, and derived terms. PID is the initialism for Proportional-Integral-Derivative , referring to three terms that operate on the error signal to generate a control signal.

Theoretical understanding and date of application from the 1920s, and they are implemented in almost all analog control systems; initially in a mechanical controller, and then using discrete electronics and the latter in industrial process computers. The PID controller is probably the most commonly used feedback control design.

Jika u (t) adalah sinyal control yang dikirim that sistem, y (t) adalah output terukur dan r (t) adalah output yang diinginkan , dan ${\ displaystyle e (t) = r (t) -t (t)} Â$ adalah kesalahan pelacakan, control PID to memble bentuk umum

${\ displaystyle u (t) = K_ {P} e (t) K_ {I} \ int e (\ tau) {\ text {d}} \ tau K_ {D} {\ frac {\ text {de (t)}} {{\ text {d}} t}}.} Â Â$

Dinamika loop tertutup yang diinginkan diperoleh dengan menyesuaikan tiga parameter ${\ displaystyle K_ {P}}  ÂÂ$ , ${\ displaystyle K_ {I}}  ÂÂ$ dan ${\ displaystyle K_ {D}}  ÂÂ$ , sering kali secara iteratif dengan "tuning" dan tanpa pengetahuan khusus tentang model tanaman. Stabilitas sering dapat dipastikan hanya menggunakan istilah proporsional. Istilah integral memungkinkan penolakan gangguan langkah (sering spesifikasi mencolok dalam pengendalian proses). Istilah derivatif digunakan untuk memberikan redaman atau pembentukan respon. Kontrol PID adalah kelas sistem kontrol yang paling mapan: namun, mereka tidak dapat digunakan dalam beberapa kasus yang lebih rumit, terutama jika sistem MIMO dipertimbangkan.

Menerapkan transformasi Laplace menghasilkan persamaan pengendali PID yang diubah

${\ displaystyle u (s) = K_ {P} e (s) K_ {I} {\ frac {1} {s}} e (s) K_ {D} se (s)} Â Â$

${\ displaystyle u (s) = \ kiri (K_ {P} K_ {I} {\ frac {1} {s}} K_ {D} s \ right) e (s)} Â Â$

deny fungal transfer transfer control PID

${\ displaystyle C (s) = \ kiri (K_ {P} K_ {I} {\ frac {1} {s}} K_ {D} s \ right).} Â Â$

Sebagai contoh penyetelan kontroler PID dalam sistem loop tertutup ${\ displaystyle H (s)}  ÂÂ$ , pertimbangkan tanaman pesanan pertama yang diberikan oleh

${\ displaystyle P (s) = {\ frac {A} {1 sT_ {P}}}}  ÂÂ$

di mana ${\ displaystyle A} Â$ dan ${\ displaystyle T_ {P}} Â$ adalah beberapa constant. Output tanaman diberi makan kembali

${\ displaystyle F (s) = {\ frac {1} {1 sT_ {F}}}} Â Â$

Memasukkan ${\ displaystyle P (s)}  ÂÂ$ , ${\ displaystyle F (s)}  ÂÂ$ , dan ${\ displaystyle C (s)}  ÂÂ$ ke dalam fungsi transfer loop tertutup ${\ displaystyle H (s)}  ÂÂ$ , kami menemukannya dengan pengaturan

${\ displaystyle K = {\ frac {1} {A}}, T_ {I} = T_ {F}, T_ {D} = T_ {P}}  ÂÂ$

${\ displaystyle H (s) = 1} Â Â$ . Deny penyetelan ini dalam contoh ini, output system mengikuti input references dengan tepat.

However, in practice, the pure differentiator can not be realized physically or desirably because of noise amplification and resonance mode in the system. Therefore, a major-phase compensator-type approach or differentiator with a low-pass roll-off is used instead.

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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 that adhere to the superposition principle, which means roughly that the output is proportional to the input. They are governed by linear differential equations. The main subclass is a system that in addition has parameters that do not change over time, is called linear time invariant (LTI) systems. These systems accept strong frequency domain mathematical techniques of large generalities, such as Laplace transform, Fourier transform, Z transform, Bode plot, locus root, and Nyquist stability criteria. This leads to system descriptions using terms such as bandwidth, frequency response, eigenvalues, reinforcement, resonance frequency, zero and poles, which provide solutions for system responses and design techniques for most systems of interest.
• Nonlinear control theory - This includes a broader system class that does not adhere to the principle of superposition, and applies to real-world systems because all real control systems are nonlinear. This system is often governed by nonlinear differential equations. Some of the mathematical techniques that have been developed to deal with them are more difficult and much more common, often applicable only to narrow category systems. These include cycle boundary theory, PoincarÃÆ'Â © maps, Lyapunov stability theorem, and describe functions. Nonlinear systems are often analyzed using numerical methods on computers, for example by simulating their operations using a simulated language. If only solutions close to an attractive stable point, nonlinear systems can often be linearized by estimating them with a linear system using perturbation theory, and linear techniques can be used.

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Analysis technique - frequency domain and time domain

Mathematical techniques for analyzing and designing control systems are divided into two distinct categories:

• Frequency domains - In this type the state variable value, the mathematical variable representing input, output, and system feedback is represented as a function of frequency. Input signals and system transfer functions are converted from time-to-function functions by transformations such as the Fourier transform, laplace transform, or Z transform. The advantage of this technique is that it results in a simplified math; the differential equation representing the system is replaced by the algebraic equation in the frequency domain which is much easier to solve. However, frequency domain techniques can only be used with linear systems, as mentioned above.
• Time-domain status representation - In this type the status variable value is represented as a function of time. With this model, the analyzed system is represented by one or more differential equations. Because frequency domain techniques are limited to linear systems, time domains are widely used to analyze real-world nonlinear systems. Although this is more difficult to solve, modern computer simulation techniques such as simulation languages ​​have made routine analysis.

Unlike the frequency domain analysis of classical control theory, modern control theory uses time-domain status-domain representations, mathematical models of physical systems as a set of input, output and status variables associated with first-order differential equations. To abstract from the number of inputs, outputs, and states, variables are expressed as vectors and differential and algebraic equations are written in matrix form (the latter only becomes possible when dynamic systems are linear). Spatial representation (also known as "time domain approximation") provides an easy and concise way to model and analyze systems with multiple inputs and outputs. With input and output, we should instead write Laplace transforms to encode all information about a system. Unlike frequency domain approach, the use of space-state representation is not limited to systems with linear components and zero initial conditions. "Space state" refers to the space where the ax is a status variable. The state of the system can be represented as a point in the space.

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System Interface - SISO & amp; 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 a single control signal. An example is an example of a roaming control above, or an audio system, in which the control input is an audio input signal and the output is the sound wave from the speaker.
• Multiple-input multiple-output (MIMO) - This is found in more complicated systems. For example, modern large telescopes such as Keck and MMT have mirrors consisting of many separate segments each controlled by the actuator. The shape of the entire mirror is constantly adjusted by the active optical control system MIMO uses input from multiple sensors in the focal plane, to compensate for changes in the shape of the mirror due to thermal expansion, contraction, pressure due to rotation and distortion of the wavefront due to turbulence in the atmosphere. Complex systems such as nuclear reactors and human cells are simulated by computers as large MIMO control systems.

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Topics in control theory

Stability

The stability of a general dynamic system without input can be explained by the Lyapunov stability criterion.

• A linear system called bounded-input bounded-output (BIBO) is stable if its output will remain limited for a restricted input.
• The stability for nonlinear systems that take input is the input-to-state stability (ISS), which combines Lyapunov stability and ideas similar to BIBO stability.

For simplicity, the following description focuses on the linear system of time-continuous and discrete-time linear linear .

Mathematically, this means that for a stable causal linear system all poles of the transfer function must have a real negative value, ie the real part of each pole must be less than zero. Practically, stability requires that a complex pole displacement function resides

• in the open left of the complex field for continuous time, when the Laplace transform is used to obtain the transfer function.
• inside the unit loop for discrete time, when Z-transform is used.

The difference between the two cases is simply because the traditional method of planning a continuous time versus a discrete time transfer function. Continuous Laplace transform is in Cartesian coordinate where ${\ displaystyle x}$ is the real axis and the discrete Z-transform is in a circular coordinate where ${\ displaystyle \ rho}$ axis is the actual axis.

When the above conditions are met, a system is said to be asymptotically stable; the asymptotic stable control system variable always decreases from the initial value and does not indicate permanent oscillation. Permanent oscillations occur when a pole has a real part exactly equal to zero (in case of continuous time) or a modulus equal to one (in the case of discrete time). If a stable system response does not decay or grow over time, and has no oscillation, it is slightly stable; in this case the system transfer function has a non-repetitive pole on the origin of the complex (ie the real component and the zero complex in case of continuous time). Oscillation exists when the pole with the real part equal to zero has an imaginary part that is not equal to zero.

Jika sistem yang dimaksud memiliki respon impuls

${\ displaystyle \ x [n] = 0,5 ^ n [n]} Â Â$

kemudian Z-transform (lihat contoh ini), diberikan oleh

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

yang memiliki tiang di ${\ displaystyle z = 0.5} Â$ (bagian imajiner nol). Sistem ini BIBO (asimtotik) stabil karena kutub adalah di dalam lingkaran unit.

Namun, jika respons impuls itu

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Source of the article : Wikipedia