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Linear-quadratic regulatorFrom Wikipedia, the free encyclopedia

The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where

the system dynamics are described by a set of linear differential equations and the cost is described by a

quadratic functional which is called the LQ problem. One of the main results in the theory is that the solution isprovided by the linear-quadratic regulator (LQR), a feedback controller whose equations are given below.

The LQR is an important part of the solution to the LQG problem. Like the LQR problem itself, the LQG

problem is one of the most fundamental problems in control theory.

Contents

1 General description

2 Finite-horizon, continuous-time LQR3 Infinite-horizon, continuous-time LQR

4 Finite-horizon, discrete-time LQR

5 Infinite-horizon, discrete-time LQR

6 References

7 External links

General description

This means that the settings of a (regulating) controller governing either a machine or process (like an airplane or

chemical reactor) are foundby using a mathematical algorithm that minimizes a cost function with weighting

factors supplied by a human (engineer). The "cost" (function) is often defined as a sum of the deviations of key

measurements from their desired values. In effect this algorithm finds those controller settings that minimize the

undesired deviations, like deviations from desired altitude or process temperature. Often the magnitude of the

control action itself is included in this sum so as to keep the energy expended by the control action itself limited.

In effect, the LQR algorithm takes care of the tedious work done by the control systems engineer in optimizing

the controller. However, the engineer still needs to specify the weighting factors and compare the results with the

specified design goals. Often this means that controller synthesis will still be an iterative process where the

engineer judges the produced "optimal" controllers through simulation and then adjusts the weighting factors to

get a controller more in line with the specified design goals.

The LQR algorithm is, at its core, just an automated way of finding an appropriate state-feedback controller. As

such it is not uncommon to find that control engineers prefer alternative methods like full state feedback (also

known as pole placement) to find a controller over the use of the LQR algorithm. With these the engineer has a

much clearer linkage between adjusted parameters and the resulting changes in controller behavior. Difficulty in

finding the right weighting factors limits the application of the LQR based controller synthesis.

Finite-horizon, continuous-time LQR
http://en.wikipedia.org/wiki/Full_state_feedbackhttp://en.wikipedia.org/wiki/Control_theoryhttp://en.wikipedia.org/wiki/Quadratic_polynomialhttp://en.wikipedia.org/wiki/Functional_(mathematics)http://en.wikipedia.org/wiki/Optimal_controlhttp://en.wikipedia.org/wiki/Dynamic_systemhttp://en.wikipedia.org/wiki/Full_state_feedbackhttp://en.wikipedia.org/wiki/State_space_(controls)http://en.wikipedia.org/wiki/Control_theoryhttp://en.wikipedia.org/wiki/Linear-quadratic-Gaussian_controlhttp://en.wikipedia.org/wiki/Functional_(mathematics)http://en.wikipedia.org/wiki/Quadratic_polynomialhttp://en.wikipedia.org/wiki/Linear_differential_equationhttp://en.wikipedia.org/wiki/Dynamic_systemhttp://en.wikipedia.org/wiki/Optimal_control


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For a continuous-time linear system, defined on , described by

with a quadratic cost function defined as

the feedback control law that minimizes the value of the cost is

where is given by

and is found by solving the continuous time Riccati differential equation.

with the boundary condition

The first order conditions for Jmin are

(i) State equation

(ii) Co-state equation

(iii) Stationary equation

(iv) Boundary conditions

and

Infinite-horizon, continuous-time LQR

For a continuous-time linear system described by
http://en.wikipedia.org/wiki/Costate_equationshttp://en.wikipedia.org/wiki/Riccati_differential_equation


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the optimal control sequence minimizing the performance index is given by

where

and is the unique positive definite solution to the discrete time algebraic Riccati equation (DARE)

.

Note that one way to solve this equation is by iterating the dynamic Riccati equation of the finite-horizon case

until it converges.

References

1. ^ Chow, Gregory C. (1986). Analysis and Control of Dynamic Economic Systems. Krieger Publ. Co. ISBN 0-

89874-969-7.

Kwakernaak, Huibert and Sivan, Raphael (1972).Linear Optimal Control Systems. First

Edition. Wiley-Interscience. ISBN 0-471-51110-2.

Sontag, Eduardo (1998).Mathematical Control Theory: Deterministic Finite DimensionalSystems. Second Edition. Springer. ISBN 0-387-98489-5.

External links

MATLAB function for Linear Quadratic Regulator design

(http://www.mathworks.com/help/toolbox/control/ref/lqr.html)

Mathematica function for Linear Quadratic Regulator design

(http://reference.wolfram.com/mathematica/ref/LQRegulatorGains.html)

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Categories: Optimal control

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