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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 is called the LQ problem. One of the main results in the theory is that

the solution is provided by the linear-quadratic regulator (LQR), a feedback controller whoseequations 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 LQR

■ 3 Infinite-horizon, continuous-time LQR

■ 4 Finite-horizon, discrete-time LQR■ 5 Infinite-horizon, discrete-time LQR

■ 6 References

■ 7 External links

General description

In layman's terms this means that the settings of a (regulating) controller governing either a machine or

process (like an airplane or chemical reactor) are found by 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 therefore 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. And 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 behaviour. Difficulty in finding the right weighting factors limits the application of

the LQR based controller synthesis.

Finite-horizon, continuous-time LQR

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

of 4Linear-quadratic regulator - Wikipedia, the free encyclopedia

02-11-2011http://en.wikipedia.org/wiki/Linear-quadratic_regulator



with a quadratic cost function defined as

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

where K is given by

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

The first order conditions for Jmin are

(i) State equation

(ii) Co-state equation

(iii) Stationary equation

0 = Ru + BT λ

(iv) Boundary conditions

x(t 0) = x0

and λ (t 1) = F (t 1) x(t 1)

Infinite-horizon, continuous-time LQR

For a continuous-time linear system described by

with a cost functional defined as

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

where K is given by





and P is found by solving the continuous time algebraic Riccati equation

Finite-horizon, discrete-time LQR

For a discrete-time linear system described by[1]

with a performance index defined as

the optimal control sequence minimizing the performance index is given by

where

and Pk is found iteratively backwards in time by the dynamic Riccati equation

from initial condition P N = Q.

Infinite-horizon, discrete-time LQR

For a discrete-time linear system described by

with a performance index defined as

the optimal control sequence minimizing the performance index is given by

where

and P 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-

898-749697

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

Edition. Wiley-Interscience. ISBN 0-471-511102.

■ Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional

Systems. Second Edition. Springer. ISBN 0-387-984895.

External links

■ Linear Quadratic Regulator

(http://documents.wolfram.com/applications/control/OptimalControlSystemsDesign/10.1.html)

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Categories: Optimal control

■ This page was last modified on 12 October 2011 at 14:43.

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