Control Theory

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Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. 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 (control theory) manipulates the inputs to a system to obtain the desired effect on the output of the system.

Overview Control theory is

theory, that deals with 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, control theory (sociology) and criminology.

An example Consider an automobile's cruise control, which is a device designed to maintain a constant vehicle speed; the reference speed, provided by the driver. 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 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. 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 its input (the throttle position).

In a closed-loop control system, a feedback controller monitors the output (the vehicle's speed) and adjusts the control input (the throttle) as necessary to keep the control error to a minimum (to maintain the desired speed). This feedback dynamically compensates for disturbances to the system, such as changes in slope of the ground or wind speed.

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 GovernorsJ. C. Maxwell, "On Governers," Proc. R Soc. London 16, 270-283 (1968) Reprinted. 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. Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877. This result is called the Routh-Hurwitz theorem.

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). 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. The Space Race also depended on accurate spacecraft control. However, control theory also saw an increasing use in fields such as economics and sociology.

People in systems and control A lot of active and historical figures made significant contribution to control theory, for example:

Richard Bellman (1920-1984), developed dynamic programming since the 1940s.
Harold Stephen Black (1898-1983), invented the negative feedback amplifier in the 1930s.
Alexander Lyapunov (1857-1918) in the 1890s marks the beginning of stability theory.
Harry Nyquist (1889-1976), developed the Nyquist stability criterion for feedback systems in the 1930s.
John R. Ragazzini (1912-1988) introduced digital control and the z-transform in the 1950s.
Norbert Wiener (1894-1964) coined the term Cybernetics in the 1940s.

Classical control theory: the closed-loop controller To avoid the problems of the open-loop controller, control theory introduces feedback.A closed-loop controller (control theory) uses feedback to control state (controls) or outputs of a dynamical system. 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.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.

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

disturbance rejection (such as unmeasured friction in a motor)
guaranteed performance even with mathematical model uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
instability 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. In such systems, the open-loop control is termed feedforward and serves to further improve reference tracking performance.

A common closed-loop controller architecture is the PID controller.

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 coordinate vectors instead of simple scalar (mathematics) values. For some Distributed parameter systems the vectors may be infinite-Dimension (vector space) (typically functions).

If we assume the controller C and the plant P are linear and time-invariant (i.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. This gives the following relations:

Y(s) = P(s) U(s)\,\! U(s) = C(s) E(s)\,\! E(s) = R(s) - Y(s)\,\!

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

Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)} \right) R(s)

The term \frac{P(s)C(s)}{1 + P(s)C(s)} 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 P(s)C(s) \gg 1, i.e. it has a large norm (mathematics) with each value of s, then Y(s) is approximately equal to R(s). This means simply setting the reference controls the output.

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).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 continuous-time system to be stable all of the Pole (complex analysis) of its transfer function must

lie in the closed left half of the complex plane if the Laplace transform is used (i.e. its real part is less than or equal to zero)

lie on or inside the unit circle if the Z-transform is used (i.e. its modulus is less than or equal to one)

In the two cases, if respectively the pole has a real part strictly smaller than zero or a modulus strictly smaller than one, it is asymptotic stability: the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations, which are instead present if a pole has a real part exactly equal to zero (or a modulus equal to one). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginal stability: in this case it has non-repeated poles along the vertical axis (i.e. their real and complex component is zero). 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 the system in question has an impulse response of

x = 0.5^n u

and considering the Z-transform (see Z-transform#Example 2 (causal ROC)), it yields

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

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

However, if the impulse response was

x = 1.5^n u

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.

Controllability and observability Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied. 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 stabilize the system. 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 correct the closed-loop behaviour if such a state is not desirable.

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. 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 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 < -\overline{\lambda}, where \overline{\lambda} is a fixed value strictly greater than zero, instead of simply ask that Re

Further reading

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