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If the state equation is polynomial then the problem is known as the polynomial-quadratic regulator (PQR). Again, the Al'Brekht algorithm can be applied to reduce this problem to a large linear one which can be solved with a generalization of the Bartels-Stewart algorithm; this is feasible provided that the degree of the polynomial is not too high.
This control law which is known as the LQG controller, is unique and it is simply a combination of a Kalman filter (a linear–quadratic state estimator (LQE)) together with a linear–quadratic regulator (LQR). The separation principle states that the state estimator and the state feedback can be designed independently.
Linear-quadratic regulator rapidly exploring random tree (LQR-RRT) is a sampling based algorithm for kinodynamic planning. A solver is producing random actions which are forming a funnel in the state space. The generated tree is the action sequence which fulfills the cost function.
A particular form of the LQ problem that arises in many control system problems is that of the linear quadratic regulator (LQR) where all of the matrices (i.e., , , , and ) are constant, the initial time is arbitrarily set to zero, and the terminal time is taken in the limit (this last assumption is what is known as infinite horizon). The LQR ...
The Kalman filter, the linear-quadratic regulator, and the linear–quadratic–Gaussian controller are solutions to what arguably are the most fundamental problems of control theory. In most applications, the internal state is much larger (has more degrees of freedom ) than the few "observable" parameters which are measured.
The algebraic Riccati equation determines the solution of the infinite-horizon time-invariant Linear-Quadratic Regulator problem (LQR) as well as that of the infinite horizon time-invariant Linear-Quadratic-Gaussian control problem (LQG). These are two of the most fundamental problems in control theory.
Model predictive control is a multivariable control algorithm that uses: an internal dynamic model of the process; a cost function J over the receding horizon; an optimization algorithm minimizing the cost function J using the control input u; An example of a quadratic cost function for optimization is given by:
When process and observation noise are Gaussian, the optimal solution separates into a Kalman filter and a linear-quadratic regulator. This is known as linear-quadratic-Gaussian control . More generally, under suitable conditions and when the noise is a martingale (with possible jumps), again a separation principle applies and is known as the ...