## Switching in Systems and Control (Systems & Control: Foundations & Applications)

A proportional—integral—derivative controller PID controller is a control loop feedback mechanism control technique widely used in control systems. PID is an initialism for Proportional-Integral-Derivative , referring to the three terms operating on the error signal to produce a control signal.

The theoretical understanding and application dates from the s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and latterly in industrial process computers. The PID controller is probably the most-used feedback control design.

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. 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.

Applying Laplace transformation results in the transformed PID controller equation. The plant output is fed back through.

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With this tuning in this example, the system output follows the reference input exactly. However, in practice, a pure differentiator is neither physically realizable nor desirable  due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach or a differentiator with low-pass roll-off are used instead.

Mathematical techniques for analyzing and designing control systems fall into two different categories:. In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.

To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form the latter only being possible when the dynamical system is linear. The state space representation also known as the "time-domain approach" provides a convenient and compact way to model and analyze systems with multiple inputs and outputs.

With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions.

The state of the system can be represented as a point within that space. Control systems can be divided into different categories depending on the number of inputs and outputs. The stability of a general dynamical system with no input can be described with Lyapunov stability criteria.

## Systems & Control: Foundations & Applications

For simplicity, the following descriptions focus on continuous-time and discrete-time linear systems. Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must have negative-real values, i. Practically speaking, stability requires that the transfer function complex poles reside. The difference between the two cases is simply due to the traditional method of plotting continuous time versus discrete time transfer functions.

When the appropriate conditions above are satisfied a system is said to be asymptotically stable ; the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero in the continuous time case or a modulus equal to one in the discrete time case.

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If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable ; in this case the system transfer function has non-repeated poles at the complex plane origin i. Oscillations are present when poles with real part equal to zero have an imaginary part not equal to zero. If a system in question has an impulse response of. This system is BIBO asymptotically stable since the pole is inside the unit circle. Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the root locus , Bode plots or the Nyquist plots.

Mechanical changes can make equipment and control systems more stable. Sailors add ballast to improve the stability of ships. Cruise ships use antiroll fins that extend transversely from the side of the ship for perhaps 30 feet 10 m and are continuously rotated about their axes to develop forces that oppose the roll. Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system.

Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are stable, then the state is termed stabilizable. Observability instead is related to the possibility of observing , through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the behavior of an unobservable state and hence cannot use it to stabilize the system.

However, similar to the stabilizability condition above, if a state cannot be observed it might still be detectable. From a geometrical point of view, looking at the states of each variable of the system to be controlled, every "bad" state of these variables must be controllable and observable to ensure a good behavior in the closed-loop system.

## Switching in Systems and Control : Daniel Liberzon :

That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis. Solutions to problems of an uncontrollable or unobservable system include adding actuators and sensors.

Several different control strategies have been devised in the past years.

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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.

## Industrial Automation & Control

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. Another typical specification is the rejection of a step disturbance; including an integrator in the open-loop chain i. Other classes of disturbances need different types of sub-systems to be included. Other "classical" control theory specifications regard the time-response of the closed-loop system.

These include the rise time the time needed by the control system to reach the desired value after a perturbation , peak overshoot the highest value reached by the response before reaching the desired value and others settling time , quarter-decay. Frequency domain specifications are usually related to robustness see after. A control system must always have some robustness property. A robust controller is such that its properties do not change much if applied to a system slightly different from the mathematical one used for its synthesis. This requirement is important, as no real physical system truly behaves like the series of differential equations used to represent it mathematically.

Typically a simpler mathematical model is chosen in order to simplify calculations, otherwise, the true system dynamics can be so complicated that a complete model is impossible. The process of determining the equations that govern the model's dynamics is called system identification. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its transfer function or matrix. Such identification from the output, however, cannot take account of unobservable dynamics. Even assuming that a "complete" model is used in designing the controller, all the parameters included in these equations called "nominal parameters" are never known with absolute precision; the control system will have to behave correctly even when connected to a physical system with true parameter values away from nominal.

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