Ads
related to: linear quadratic regulator example equation with solution of 1
Search results
Results from the WOW.Com Content Network
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. LQR controllers possess inherent robustness with guaranteed gain and phase margin, [1] and they also are part of the solution to the LQG (linear–quadratic–Gaussian) problem.
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.
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.
The control theory is using differential equations to describe complex physical systems like an inverted pendulum. [1] A set of differential equations forms a physics engine which maps the control input to the state space of the system. The forward model is able to simulate the given domain.
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 substitution that is needed to solve this Bernoulli equation is = Substituting = + directly into the Riccati equation yields the linear equation ′ + (+) = A set of solutions to the Riccati equation is then given by = + where z is the general solution to the aforementioned linear equation.
where is the gain of the optimal linear-quadratic regulator obtained by taking = = and () deterministic, and where is the Kalman gain. There is also a non-Gaussian version of this problem (to be discussed below) where the Wiener process w {\displaystyle w} is replaced by a more general square-integrable martingale with possible jumps. [ 1 ]
Ads
related to: linear quadratic regulator example equation with solution of 1