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In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, [citation needed] 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 ...
The examples thus far have shown continuous time systems and control solutions. In fact, as optimal control solutions are now often implemented digitally , contemporary control theory is now primarily concerned with discrete time systems and solutions.
In control engineering and system identification, a state-space representation is a mathematical model of a physical system specified as a set of input, output, and variables related by first-order differential equations or difference equations.
A Carathéodory-π solution can be applied towards the practical stabilization of a control system. [ 6 ] [ 7 ] It has been used to stabilize an inverted pendulum, [ 6 ] control and optimize the motion of robots, [ 7 ] [ 8 ] slew and control the NPSAT1 spacecraft [ 3 ] and produce guidance commands for low-thrust space missions.
A water heater that maintains desired temperature by turning the applied power on and off (as opposed to continuously varying electrical voltage or current) based on temperature feedback is an example application of bang–bang control. Although the applied power switches from one discrete state to another, the water temperature will remain ...
The optimization of portfolios is an example of multi-objective optimization in economics. Since the 1970s, economists have modeled dynamic decisions over time using control theory. [14] For example, dynamic search models are used to study labor-market behavior. [15] A crucial distinction is between deterministic and stochastic models. [16]
The phrase H ∞ control comes from the name of the mathematical space over which the optimization takes place: H ∞ is the Hardy space of matrix-valued functions that are analytic and bounded in the open right-half of the complex plane defined by Re(s) > 0; the H ∞ norm is the supremum singular value of the matrix over that space.
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]