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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
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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
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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)
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.
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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quadratic_regulator&oldid=455213827"
Categories: Optimal control
■ This page was last modified on 12 October 2011 at 14:43.
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