The Linear Quadratic Regulator

The Linear Quadratic Regulator (LQR) provides an optimal control law for a linear system with a quadratic performance index. 

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In the sixties and seventies, in contrast to literature references to the constrained on-line optimization performed by MFC, which were only sporadic, there was an already vast and growing literature on a related problem, the linear-quadratic regulator (LQR) either in deterministic or stochastic settings. Simply stated, the LQR problem is... 

The aim, is to design a controller so that the base displacements in the x and y directions are minimized. This is established by making use of the linear quadratic regulator formulation in which the cost function to be minimized is as follows... 

One of the classical strategies developed in control science and technology that has deserved attention in the field of structural control is the Linear Quadratic Regulator strategy (LQR), in which the external control force u t) is expressed by... 

Many state of the art publications on control applications are published. These publications provide a review of base isolation systems (Kelly, 1986), active control (Soong, 1988, 1990, 1994, Datta, 2003), structural control concepts and strategies (Housner, 1997, Spencer, 2003), etc. Inmost cases active control systems are designed based on the linear quadratic regulator (LQR) theory or... 

The proposed strategies for stabilization of gas-lifted oil wells are offline methods which are unable to track online dynamic changes of the system. However, system parameters such as flow rate of injected gas and also noise characteristic are not constant with respect to time. An adaptive Linear Quadratic Gaussian (LQG) approach is presented in this paper in which the state estimation is performed using an Adaptive Unscented Kalman Filter (AUKF) to deal with unknown time-varying noise statistics. State-feedback gain is adaptively calculated based on Linear Quadratic Regulator (LQR). Finally, the proposed control scheme is evaluated on a simulation case study. 

Over the last two decades, there has been increasing interest in probabilistic, or stochastic, robust control theory. Monte Carlo simulation methods have been used to synthesize and analyze controllers for uncertain systems [170,255], First- and second-order reliability methods were incorporated to compute the probable performance of linear-quadratic-regulator... 

Chapter 5 considers optimal regulator control problems. The Kalman linear quadratic regulator (LQR) problem is developed, and this optimal multivariable proportional controller is shown to be easily computable using the Riccati matrix differential equation. The regulator problem with unmeasurable... 

Chapter 12 considers the combination of optimal control with state and parameter estimation. The separation principle is developed, which states that the design of a control problem with measurement and model uncertainty can be treated by first performing a Kalman filter estimate of the states and then developing the optimal control law based upon the estimated states. For linear regulator problems, the problem is known as the linear quadratic Gaussian (LQG) problem. The inclusion of model parameter identification results in adaptive control algorithms. 

An LQR (linear quadratic regulator) is considered as the first-stage or primary controller. LQR is designed to obtain the optimal force required to minimize the cost function defined as... 

The state feedback controller is designed using Linear Quadratic Regulator (LQR) optimal design to minimize the performance variables in s without using excessive control input. There are two controller options. The first one minimizes the Kqz deviation from desired while the second minimizes deviations of both Kqz and net power. The goal for each is formulated with a suitable cost function. The cost fimction for the first is given. 

The feedback matrix can be calculated based on optimal control theory like linear quadratic regulator, LQG. Optimal control methods are based on the concept of minimizing a cost criterion. The criterion of cost, represented by J, has a common format ... 

To calculate state feedback gain K k, discrete linear-quadratic (LQ) regulator is used. The sate feedback law m[A ] = -K[k y k rninitnizes the quadratic cost function [15] ... 

Since the linear model consistently predicts the highest response frequency per unit dose in very low dose range, it is usually the most conservative or least likely to underestimate human health risk. It is often recommended by regulating agencies to determine the risk of known or suspected human carcinogens. For non-carcinogens, the quadratic model that predicts a threshold dose at which there is no effect is often used. 

There is considerable controversy over the shape of the dose-response curve at the chronic low dose levels important for environmental contamination. Proposed models include linear models, nonlinear (quadratic) models, and threshold models. Because risks at low dose must be extrapolated from available data at high doses, the shape of the dose-response curve has important implications for the environmental regulations used to protect the general public. Detailed description of dosimetry models can be found in Cember (1996), BEIR IV (1988), and Harley (2001). 

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