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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.
Inspired by—but distinct from—the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. [2] Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian. [3]
Tuning a control loop is the adjustment of its control parameters (proportional band/gain, integral gain/reset, derivative gain/rate) to the optimum values for the desired control response. Stability (no unbounded oscillation) is a basic requirement, but beyond that, different systems have different behavior, different applications have ...
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 ...
An everyday example is the cruise control on a road vehicle; where external influences such as gradients cause speed changes (PV), and the driver also alters the desired set speed (SP). The automatic control algorithm restores the actual speed to the desired speed in the optimum way, without delay or overshoot, by altering the power output of ...
Single-input single-output (SISO) – This is the simplest and most common type, in which one output is controlled by one control signal. Examples are the cruise control example above, or an audio system, in which the control input is the input audio signal and the output is the sound waves from the speaker.
In this example, the term control law refers specifically to the way in which the driver presses the accelerator and shifts the gears. The system consists of both the car and the road, and the optimality criterion is the minimization of the total traveling time. Control problems usually include ancillary constraints. For example, the amount of ...
Proportional control, in engineering and process control, is a type of linear feedback control system in which a correction is applied to the controlled variable, and the size of the correction is proportional to the difference between the desired value (setpoint, SP) and the measured value (process variable, PV).