Optimal Regulator Algorithms for the Control of Linear Systems
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This control theory design package, called Optimal Regulator Algorithms
for the Control of Linear Systems (ORACLS), was developed to aid in
the design of controllers and optimal filters for systems which can be modeled
by linear, time-invariant differential and difference equations.
Optimal linear quadratic regulator theory, currently referred to as the
Linear-Quadratic-Gaussian (LQG) problem, has become the most widely accepted
method of determining optimal control policy. Within this theory,
the infinite duration time-invariant problems, which lead to constant
gain feedback control laws and constant Kalman-Bucy filter gains for reconstruction
of the system state, exhibit high tractability and potential
ease of implementation. A variety of new and efficient methods in the field
of numerical linear algebra have been combined into the ORACLS program,
which provides for the solution to time-invariant continuous or discrete
LQG problems. The ORACLS package is particularly attractive to the control
system designer because it provides a rigorous tool for dealing with
multi-input and multi-output dynamic systems in both continuous and discrete
The ORACLS programming system is a collection of subroutines which
can be used to formulate, manipulate, and solve various LQG design problems.
The ORACLS program is constructed in a manner which permits the user
to maintain considerable flexibility at each operational state. This flexibility
is accomplished by providing primary operations, analysis of linear
time-invariant systems, and control synthesis based on LQG methodology.
The input-output routines handle the reading and writing of numerical
matrices, printing heading information, and accumulating output information.
The basic vector-matrix operations include
The analysis routines provide for the following computations:
- norm construction,
- juxtaposition, and
- construction of null and identity matrices.
The control law design routines of ORACLS implement some of the more
common techniques of time-invariant LQG methodology. For the finiteduration
optimal linear regulator problem with noise-free measurements,
continuous dynamics, and integral performance index, a routine is provided
which implements the negative exponential method for finding both the transient
and steady-state solutions to the matrix Riccati equation.
- the eigenvalues and eigenvectors of real matrices;
- the relative stability of a
- matrix factorization;
- the solution of linear constant coefficient
vector-matrix algebraic equations;
- the controllability properties
of a linear time-invariant system;
- the steady-state covariance matrix of an
open-loop stable system forced by white noise;
- and the transient response
of continuous linear time-invariant systems.
discrete version of this problem, the method of backwards differencing is
applied to find the solutions to the discrete Riccati equation. A routine
is also included to solve the steady-state Riccati equation by the Newton
algorithms described by Klein, for continuous problems, and by Hewer, for
discrete problems. Another routine calculates the prefilter gain to eliminate
control state cross-product terms in the quadratic performance index
and the weighting matrices for the sampled data optimal linear regulator
problem. For cases with measurement noise, duality theory and optimal
regulator algorithms are used to calculate solutions to the continuous and
discrete Kalman-Bucy filter problems.
Finally, routines are included to
implement the continuous and discrete forms of the explicit (model-in-thesystem)
and implicit (model-in-the-performance-index) model following
theory. These routines generate linear control laws which cause the output
of a dynamic time-invariant system to track the output of a prescribed
ORACLS carries the NASA case numbers LAR-12953, LAR-12313 and GSC-13067. It was originally released as part of the NASA COSMIC collection.
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