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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 form.

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
  • addition,
  • subtraction,
  • multiplication,
  • equation,
  • norm construction,
  • tracing,
  • transposition,
  • scaling,
  • juxtaposition, and
  • construction of null and identity matrices.
The analysis routines provide for the following computations:
  • the eigenvalues and eigenvectors of real matrices;
  • the relative stability of a given matrix;
  • 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.
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.

For the 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 model.
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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