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Control systems that include some sensing of the results they are trying to achieve are making use of feedback and can adapt to varying circumstances to some extent. Open-loop control systems do not make use of feedback, and run only in pre-arranged ways. Closed-loop controllers have the following advantages over open-loop controllers:
improved rectification of random fluctuations [2] 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. A basic feedback loop
Lur'e problem block diagram. An early nonlinear feedback system analysis problem was formulated by A. I. Lur'e.Control systems described by the Lur'e problem have a forward path that is linear and time-invariant, and a feedback path that contains a memory-less, possibly time-varying, static nonlinearity.
The associated more difficult control problem leads to a similar optimal controller of which only the controller parameters are different. [5] It is possible to compute the expected value of the cost function for the optimal gains, as well as any other set of stable gains. [12] The LQG controller is also used to control perturbed non-linear ...
The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem. One of the main results in the theory is that the solution is provided by the linear–quadratic regulator (LQR), a feedback controller whose equations are given below.
The definition of a closed loop control system according to the British Standards Institution is "a control system possessing monitoring feedback, the deviation signal formed as a result of this feedback being used to control the action of a final control element in such a way as to tend to reduce the deviation to zero." [2]
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The goal of feedback linearization is to produce a transformed system whose states are the output and its first () derivatives. To understand the structure of this target system, we use the Lie derivative. Consider the time derivative of (2), which can be computed using the chain rule,