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.
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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. 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.
Time Response of Second-Order Control System
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.
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. 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. 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 of the robustness of a SISO single input single output control system can be performed in the frequency domain, considering the system's transfer function and using Nyquist and Bode diagrams.
Topics include gain and phase margin and amplitude margin. For MIMO multi-input multi output and, in general, more complicated control systems, one must consider the theoretical results devised for each control technique see next section. 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 damage or break actuators or other subsystems. Specific control techniques are available to solve the problem: model predictive control see later , and anti-wind up systems. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold.
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 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.
Processes in industries like robotics and the aerospace industry typically have strong nonlinear dynamics. 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. 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.
Control theory has also been used to decipher the neural mechanism that directs cognitive states. When the system is controlled by multiple controllers, the problem is one of decentralized control. Decentralization is helpful in many ways, for instance, it helps control systems to operate over a larger geographical area. The agents in decentralized control systems can interact using communication channels and coordinate their actions. A stochastic control problem is one in which the evolution of the state variables is subjected to random shocks from outside the system.
A deterministic control problem is not subject to external random shocks.
Time control - Wikipedia
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. 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.
The possibility to fulfill different specifications varies from the model considered and the control strategy chosen. From Wikipedia, the free encyclopedia. This article is about control theory in engineering. For control theory in linguistics, see control linguistics. For control theory in psychology and sociology, see control theory sociology and Perceptual control theory. Main article: Classical control theory. Further information: closed-loop transfer function. Main article: PID Controller.