Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. In Control theory, a controller is a device which monitors and affects the operational conditions of a given Dynamical system.

The concept of the feedback loop to control the dynamic behavior of the reference: this is negative feedback because the sensed value is subtracted from the desired value to create the error signal which is amplified by the controller.

Overview

Control theory is

• theory, that deals with influencing the behavior of dynamical systems
• interdisciplinary subfield of science which originated in engineering and mathematics, and evolved into use by the social sciences, like psychology, sociology and criminology. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Psychology (from Greek grc ψῡχή psȳkhē, "breath life soul" and grc -λογία -logia) is an Academic and Control theory, as an extension to the field of Psychoanalysis, postulates human behaviors driven by the therapeutic function of taming the threatening otherness of Schools of thought In the mid-18th century criminology arose as social philosophers gave thought to crime and concepts of law

An example

Consider an automobile's cruise control, which is a device designed to maintain a constant vehicle speed; the desired or reference speed, provided by the driver. Cruise control (sometimes known as speed control or autocruise) is a system that automatically controls the rate of motion of a Motor vehicle. The system in this case is the vehicle. The system output is the vehicle speed, and the control variable is the engine's throttle position which influences engine torque output. A throttle is the mechanism by which the flow of a fluid is managed by constriction or obstruction A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about

A simple way to implement cruise control is to lock the throttle position when the driver engages cruise control. However, on hilly terrain, the vehicle will slow down going uphill and accelerate going downhill. In fact, any parameter different than what was assumed at design time will translate into a proportional error in the output velocity, including exact mass of the vehicle, wind resistance, and tire pressure. This type of controller is called an open-loop controller because there is no direct connection between the output of the system (the engine torque) and the actual conditions encountered; that is to say, the system does not and can not compensate for unexpected forces. An open-loop controller, also called a non-feedback controller, is a type of controller which computes its input into a system using only the current state

In a closed-loop control system, a sensor monitors the output (the vehicle's speed) and feeds the data to a computer which continuously adjusts the control input (the throttle) as necessary to keep the control error to a minimum (to maintain the desired speed). Feedback on how the system is actually performing allows the controller (vehicle's on board computer) to dynamically compensate for disturbances to the system, such as changes in slope of the ground or wind speed. An ideal feedback control system cancels out all errors, effectively mitigating the effects of any forces that may or may not arise during operation and producing a response in the system that perfectly matches the user's wishes.

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 is a specific type of governor that controls the Speed of an Engine by regulating the amount of Fuel (or Working James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. Year 1868 ( MDCCCLXVIII) was a Leap year starting on Wednesday (link will display the full calendar of the Gregorian Calendar (or a Leap ^[1] This described and analyzed the phenomenon of "hunting", in which lags in the system can lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate Edward John Routh generalized the results of Maxwell for the general class of linear systems. Edward John Routh FRS (20 January 1831–7 June 1907 was an English Mathematician, noted as the outstanding coach of students preparing for the Mathematical ^[2] Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877. Adolf Hurwitz ( 26 March 1859 - 18 November 1919) (ˈadɒlf ˈhurvits was a German mathematician and was described by Jean-Pierre Year 1877 ( MDCCCLXXVII) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a Common This result is called the Routh-Hurwitz theorem. In Mathematics, Routh–Hurwitz theorem gives a test to determine whether a given Polynomial is Hurwitz- stable. ^[3][4]

A notable application of dynamic control was in the area of manned 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). WikipediaWikiProject Aircraft. Please see WikipediaWikiProject Aircraft/page content for recommended layout Events 546 - Gothic War (535–554: The Ostrogoths of King Totila Year 1903 ( MCMIII) was a Common year starting on Thursday (link will display calendar of the Gregorian calendar or a Common year starting Control of the airplane was necessary for safe flight.

By World War II, control theory was an important part of fire-control systems, guidance systems and electronics. World War II, or the Second World War, (often abbreviated WWII) was a global military conflict which involved a majority of the world's nations, including Note the term " fire control " may also refer to means of stopping a fire such as sprinkler systems A fire-control system A guidance system is a device or group of devices used to navigate a Ship, Aircraft, Missile, Rocket, Satellite, or other Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical The Space Race also depended on accurate spacecraft control. The Space Race was a competition of space exploration between the Soviet Union and the United States, which lasted roughly from 1957 to 1975 However, control theory also saw an increasing use in fields such as economics . Economics is the social science that studies the production distribution, and consumption of goods and services.

People in systems and control

Main article: People in systems and control

Many active and historical figures made significant contribution to control theory, for example:

• Alexander Lyapunov (1857-1918) in the 1890s marks the beginning of stability theory. People in systems and control is an alphabetical list (in two parts of people who have made significant contributions in the fields of System analysis and Control theory Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов ( June 6 1857 &ndash November 3 1918, all In Mathematics, stability theory deals with the stability of solutions (or sets of solutions for Differential equations and Dynamical systems
• Harold S. Black (1898-1983), invented the negative feedback amplifier in the 1930s. Harold Stephen Black (1898-1983 was an American Electrical engineer, who revolutionized the field of applied electronics by inventing the Negative feedback amplifier When a fraction of the output of an amplifier is combined with the input Feedback exists if the feedback opposes the original signal it is negative feedback and if it increases
• Harry Nyquist (1889-1976), developed the Nyquist stability criterion for feedback systems in the 1930s. Harry Nyquist ( né Harry Theodor Nyqvist pron, not as often pronounced ( February 7, 1889 – April 4, 1976) was an important The Nyquist stability criterion, named after Harry Nyquist, provides a simple test for Stability of a Closed-loop Control system by examining
• Richard Bellman (1920-1984), developed dynamic programming since the 1940s. Richard Ernest Bellman ( August 26, 1920 – March 19, 1984) was an applied mathematician, celebrated for his invention of In Mathematics and Computer science, dynamic programming is a method of solving problems exhibiting the properties of Overlapping subproblems and
• Norbert Wiener (1894-1964) coined the term Cybernetics in the 1940s. Norbert Wiener ( November 26, 1894, Columbia Missouri – March 18, 1964, Stockholm, Sweden) was an American Cybernetics is the interdisciplinary study of the Structure of Complex systems especially Communication processes control mechanisms and Feedback
• John R. Ragazzini (1912-1988) introduced digital control and the z-transform in the 1950s. John Ralph Ragazzini (1912 &ndash November 22, 1988) was an American Electrical engineer and a professor of Electrical Engineering Digital control is a branch of Control theory that uses Digital Computers to act as a system In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real

Classical control theory

To avoid the problems of the open-loop controller, control theory introduces feedback. Feedback is a circular causal Process whereby some proportion of a system's output is returned (fed back to the Input. A closed-loop controller uses feedback to control states or outputs of a dynamical system. In Control theory, a controller is a device which monitors and affects the operational conditions of a given Dynamical system. In Control theory, states are what characterize a System. With Linear systems states are not unique but can be transformed into equivalent states Output is the term denoting either an exit or changes which exit a System and which activate/modify a Process. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position Its name comes from the information path in the system: process inputs (e. g. voltage applied to an electric motor) have an effect on the process outputs (e. Electrical tension (or voltage after its SI unit, the Volt) is the difference of electrical potential between two points of an electrical An electric motor uses Electrical energy to produce Mechanical energy. g. velocity or torque of the motor), which is measured with sensors and processed by the controller; the result (the control signal) is used as input to the process, closing the loop. A sensor is a device that measures a physical quantity and converts it into a signal which can be read by an observer or by an instrument

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

• disturbance rejection (such as unmeasured friction in a motor)
• guaranteed performance even with model uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
• unstable processes can be stabilized
• reduced sensitivity to parameter variations
• improved reference tracking performance

In some systems, closed-loop and open-loop control are used simultaneously. An open-loop controller, also called a non-feedback controller, is a type of controller which computes its input into a system using only the current state Note The term model has a different meaning in Model theory, a branch of Mathematical logic. Instability in systems is generally characterized by some of the Outputs or internal states growing without Bounds. In such systems, the open-loop control is termed feedforward and serves to further improve reference tracking performance. Feed-forward is a term describing a kind of System which reacts to changes in its environment usually to maintain some desired state of the system

A common closed-loop controller architecture is the PID controller. A proportional–integral–derivative controller (PID controller is a generic Control loop Feedback mechanism widely used in industrial Control systems

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

This is called a single-input-single-output (SISO) control system; MIMO (i. e. Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. In Linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or equivalently as an In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication For some distributed parameter systems the vectors may be infinite-dimensional (typically functions). A distributed parameter system (as opposed to a lumped parameter system) is a System whose State space is infinite- dimensional. In Mathematics, the dimension of a Vector space V is the cardinality (i

If we assume the controller C and the plant P are linear and time-invariant (i. The word linear comes from the Latin word linearis, which means created by lines. e. : elements of their transfer function C(s) and P(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. A transfer function is a mathematical representation in terms of spatial or temporal frequency of the relation between the input and output of a ( linear time-invariant) In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic This gives the following relations:

Solving for Y(s) in terms of R(s) gives:

The term is referred to as the transfer function of the system. The numerator is the forward gain from r to y, and the denominator is one plus the loop gain of the feedback loop. If , i. e. it has a large norm with each value of s, then Y(s) is approximately equal to R(s). In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length This means simply setting the reference controls the output.

The PID controller is probably the most-used feedback control design, being the simplest one. A proportional–integral–derivative controller (PID controller is a generic Control loop Feedback mechanism widely used in industrial Control systems "PID" means Proportional-Integral-Derivative, referring to the three terms operating on the error signal to produce a control signal. If u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and tracking error e(t) = r(t) − y(t), a PID controller has the general form

The desired closed loop dynamics is obtained by adjusting the three parameters K_P, K_I and K_D, often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in process control). Process control is a Statistics and Engineering discipline that deals with Architectures mechanisms and Algorithms for controlling The derivative term is used to provide damping or shaping of the response. PID controllers are the most well established class of control systems: however, they cannot be used in several more complicated cases, especially if MIMO systems are considered.

Modern control theory

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. In Control engineering, a state space representation is a mathematical model of a physical system as a set of input output and state variables related by first-order Differential 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 last one can be done when the dynamical system is linear and time invariant). 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 vector within that space.

Topics in control theory

Stability

Stability (in control theory) often means that for any bounded input over any amount of time, the output will also be bounded. This is known as BIBO stability (see also Lyapunov stability). Bibo redirects here For the Egyptian football player nicknamed Bibo see Mahmoud El-Khateeb. In Mathematics, the notion of Lyapunov stability occurs in the study of Dynamical systems In simple terms if all solutions of the dynamical system that start out If a system is BIBO stable then the output cannot "blow up" (i. e. , become infinite) if the input remains finite.

Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must satisfy some criteria depending on whether a continuous or discrete time analysis is used:

• In continuous time, the Laplace transform is used to obtain the transfer function. In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 A transfer function is a mathematical representation in terms of spatial or temporal frequency of the relation between the input and output of a ( linear time-invariant) In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic A system is stable if the poles of this transfer function lie strictly in the closed left half of the complex plane (i. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis e. the real part of all the poles is less than zero).

OR

• In discrete time the Z-transform is used. In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real A system is stable if the poles of this transfer function lie strictly inside the unit circle. In Mathematics, a unit circle is i. e. the magnitude of the poles is less than one).

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. See also Lyapunov stability for an alternate definition used in Dynamical systems. 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 complex plane origin (i. In the theory of Dynamical systems, and Control theory, a continuous linear Time-invariant system is marginally stable If and only if 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.

Differences between the two cases are not a contradiction. The Laplace transform is in Cartesian coordinates and the Z-transform is in circular coordinates, and it can be shown that

• the negative-real part in the Laplace domain can map onto the interior of the unit circle
• the positive-real part in the Laplace domain can map onto the exterior of the unit circle

If a system in question has an impulse response of

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

which has a pole in z = 0. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by The impulse response of a system is its output when presented with a very brief input signal an impulse In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real In Mathematics and Signal processing, the Z-transform converts a discrete Time-domain signal which is a Sequence of real 5 (zero imaginary part). Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane This system is BIBO (asymptotically) stable since the pole is inside the unit circle.

However, if the impulse response was

then the Z-transform is

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. In Control theory, the root locus is the locus of the poles and zeros of a Transfer function as the System gain K is varied A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot A Bode magnitude plot is a graph of log A Nyquist plot is used in automatic control and Signal processing for assessing the stability of a system with Feedback.

Controllability and observability

Main articles: 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 an important property of a Control system, and the controllability property plays a crucial role in many control problems such as stabilization of unstable Observability in Control theory is a Measure for how well internal states of a System can be inferred by Knowledge of its external Outputs Controllability is an important property of a Control system, and the controllability property plays a crucial role in many control problems such as stabilization of unstable Observability in Control theory is a Measure for how well internal states of a System can be inferred by Knowledge of its external Outputs 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 it 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 behaviour 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 behaviour 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. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes 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 uncontrollable or unobservable system include adding actuators and sensors.

Control specifications

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 proportional–integral–derivative controller (PID controller is a generic Control loop Feedback mechanism widely used in industrial Control systems See also Robot Robotics is the science and technology of Robots and their design manufacture and application

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 , where is a fixed value strictly greater than zero, instead of simply ask that Re[λ] < 0.

Another typical specification is the rejection of a step disturbance; including an integrator in the open-loop chain (i. An integrator is a device to perform the mathematical operation known as integration, a fundamental operation in Calculus. 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). In Electronics, when describing a Voltage or current Step function, rise time (also risetime) refers to the time required for a signal The term overshoot has the following meanings Aviation In Aviation, an overshoot is an aborted landing The settling time of an Amplifier or other output device is the time elapsed from the application of an ideal instantaneous step input to the time at which the amplifier Frequency domain specifications are usually related to robustness (see after). Robustness is the quality of being able to withstand stresses pressures or changes in procedure or circumstance

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

Model identification and robustness

Main article: System identification

A control system must always have some robustness property. System identification is a general term to describe mathematical Tools and Algorithms that build dynamical models from measured data 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. Robust control is a branch of Control theory that explicitly deals with uncertainty in its approach to controller design This specification is important: 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. System identification is a general term to describe mathematical Tools and Algorithms that build dynamical models from measured data 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. A transfer function is a mathematical representation in terms of spatial or temporal frequency of the relation between the input and output of a ( linear time-invariant) 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 . Damping is any effect either deliberately engendered or inherent to a system that tends to reduce the amplitude of Oscillations of an oscillatory system 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 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

Analysis of the robustness of a SISO control system can be performed in the frequency domain, considering the system's transfer function and using Nyquist and Bode diagrams. A Nyquist plot is used in automatic control and Signal processing for assessing the stability of a system with Feedback. A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot A Bode magnitude plot is a graph of log Topics include gain and phase margin and amplitude margin. A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot A Bode magnitude plot is a graph of log For MIMO 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 his attention to a control technique including them in its properties.

Constraints

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 break actuators or other subsystems. Specific control techniques are available to solve the problem: model predictive control (see later), and anti-wind up systems. Model Predictive Control, or MPC is an advanced method of Process control that has been in use in the process industries such as Chemical plants and The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold.

System classifications

Linear control

Main article: State space (controls)

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 Control engineering, a state space representation is a mathematical model of a physical system as a set of input output and state variables related by first-order Differential In Control engineering, a state space representation is a mathematical model of a physical system as a set of input output and state variables related by first-order Differential 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 control

Main article: Nonlinear control

Processes in industries like robotics and the aerospace industry typically have strong nonlinear dynamics. Nonlinear control is a sub-division of Control engineering which deals with the control of nonlinear systems See also Robot Robotics is the science and technology of Robots and their design manufacture and application This article is about the field of research and industry for the corporation see The Aerospace Corporation Aerospace comprises the 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. Feedback linearization is a common approach used in controlling Nonlinear systems. In Control theory backstepping is a technique for designing controls for nonlinear systems developed around 1990 by Petar V In Control theory, sliding mode control is a type of Variable structure control where the dynamics of a nonlinear system is altered via application of a high-frequency Lyapunov theory is a collection of results regarding stability of Dynamical systems named after a Russian Mathematician Aleksandr Mikhailovich Lyapunov. Differential geometry has been widely used as a tool for generalizing well-known linear control concepts to the non-linear case, as well as showing the subtleties that make it a more challenging problem. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry

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. A linear system is a mathematical model of a System based on the use of a Linear operator. Non-linear control systems use specific theories (normally based on Aleksandr Lyapunov's Theory) to ensure stability without regard to the inner dynamics of the system. Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов ( June 6 1857 &ndash November 3 1918, all The possibility to fulfill different specifications varies from the model considered and the control strategy chosen. Here a summary list of the main control techniques is shown:

Adaptive control

Main article: Adaptive control

Adaptive control uses on-line identification of the process parameters, or modification of controller gains, thereby obtaining strong robustness properties. Adaptive control involves modifying the control law used by a controller to cope with the fact that the parameters of the system being controlled are slowly time-varying or uncertain Adaptive controls were applied for the first time in the aerospace industry in the 1950s, and have found particular success in that field. This article is about the field of research and industry for the corporation see The Aerospace Corporation Aerospace comprises the The 1950s Decade refers to the years of 1950 to 1959 inclusive

Hierarchical control

Main article: Hierarchical control system

A Hierarchical control system is a form of Control System in which a set of devices and governing software is arranged in a hierarchical tree. A Hierarchical control system is a form of Control System in which a set of devices and governing software is arranged in a Hierarchical tree. A control system is a device or set of devices to manage command direct or regulate the behavior of other devices or systems @@@ main@@@ - title Hierarchy@@@ keywords structure; sociology; information@@@ review@@@ - In Computer science, a tree is a widely-used Data structure that emulates a Tree structure with a set of linked nodes 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. A computer network is a group of interconnected Computers. Networks may be classified according to a wide variety of characteristics A Networked Control System ( NCS) is a Control system wherein the control loops are closed through a real-time network.

Intelligent control

Main article: Intelligent control

Intelligent control use various AI computing approaches like neural networks, Bayesian probability, fuzzy logic, machine learning, evolutionary computation and genetic algorithms to control a dynamic system

Optimal control

Main article: Optimal control

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. Intelligent control is a class of Control techniques that use various AI computing approaches like Neural networks, Bayesian probability, Fuzzy logic Traditionally the term neural network had been used to refer to a network or circuit of biological neurons. Bayesian probability interprets the concept of Probability as 'a measure of a state of knowledge'. Fuzzy logic is a form of Multi-valued logic derived from Fuzzy set theory to deal with Reasoning that is approximate rather than precise Machine learning is a subfield of Artificial intelligence that is concerned with the design and development of Algorithms and techniques that allow computers to "learn" In Computer science evolutionary computation is a subfield of Artificial intelligence (more particularly Computational intelligence) that involves A genetic algorithm (GA is a Search technique used in Computing to find exact or Approximate solutions to optimization and Search The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position Optimal control theory, a modern extension of the Calculus of variations, is a mathematical optimization method for deriving control policies. 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). Model Predictive Control, or MPC is an advanced method of Process control that has been in use in the process industries such as Chemical plants and In control, the Linear-Quadratic-Gaussian (LQG control problem is probably the most fundamental Optimal control problem 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. Process control is a Statistics and Engineering discipline that deals with Architectures mechanisms and Algorithms for controlling

Robust control

Main article: Robust control

Robust control deals explicitly with uncertainty in its approach to controller design. Robust control is a branch of Control theory that explicitly deals 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 1970's were sometimes found to lack robustness. A modern example of a robust control technique is H-infinity loop-shaping developed by Duncan McFarlane and Keith Glover of Cambridge University. H-infinity loop-shaping is a design methodology in modern Control theory. The University of Cambridge (often Cambridge University) located in Cambridge, England, is the second-oldest university in the Robust methods aim to achieve robust performance and/or stability in the presence of small modelling errors. In Mathematics, stability theory deals with the stability of solutions (or sets of solutions for Differential equations and Dynamical systems

Stochastic control

Main article: Stochastic control

Stochastic control deals with control design with uncertainty in the model. Stochastic control is a subfield of Control theory which addresses the design of a control methodology to address the probability of uncertainty in the data In typical stochastic control problems, it is assume that there exist random noise and disturbances in the model and the controller, and the control design must take into account these random deviations.

See also

References

Literature

• Christopher Kilian (2005). Modern Control Technology. Thompson Delmar Learning. ISBN 1-4018-5806-6.  
• Vannevar Bush (1929). Operational Circuit Analysis. John Wiley and Sons, Inc. .  
• Robert F. Stengel (1994). Optimal Control and Estimation. Dover Publications. ISBN 0-486-68200-5, ISBN-13: 978-0-486-68200-6.  
• Franklin et al. (2002). Feedback Control of Dynamic Systems, 4, New Jersey: Prentice Hall. ISBN 0-13-032393-4.  
• Joseph L. Hellerstein, Dawn M. Tilbury, and Sujay Parekh (2004). Feedback Control of Computing Systems. John Wiley and Sons. ISBN 0-47-126637-X, ISBN-13: 978-0-471-26637-2.  
• Diederich Hinrichsen and Anthony J. Diederich Hinrichsen (born 17 February 1939 in Nürnberg) is a German Mathematician, whom together with Hans W Pritchard (2005). Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness. Springer. ISBN 0-978-3-540-44125-0.  
• Sontag, Eduardo (1998). Eduardo Sontag ( 1951 in Buenos Aires, Argentina is an American/Argentine Mathematician, and Professor at the State University of New Jersey Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition. Springer. ISBN 0-387-984895.  

Further reading

• Andrei, Neculai (2005). "Modern Control Theory - A historical Perspective". Retrieved on 2007-10-10. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 680 - Battle of Karbala: Shia Imam Husayn bin Ali, the grandson of the Prophet Muhammad, is decapitated