Title: The Computation of the Inverse of a Square Polynomial Matrix
Abstract: An approach to calculate the inverse of a square polynomial
matrix is suggested. The approach consists of two similar algorithms:
One calculates the determinant polynomial and the other calculates
the adjoint polynomial. The algorithm to calculate the determinant
polynomial gives the coefficients of the polynomial in a recursive
manner from a recurrence formula. Similarly, the algorithm to
calculate the adjoint polynomial also gives the coefficient matrices
recursively from a similar recurrence formula. Together, they give
the inverse as a ratio of the adjoint polynomial and the determinant
Title: Four Essential Optimal Discrete Controllers
for Control Applications
Abstract: Four optimal discrete controllers for a single-input-single-output (SISO), state space model,
stochastic regulating control system are presented. They are the one step optimal or myopic controller,
the $N$ steps optimal controller, the pseudo-infinite steps optimal controller and the infinite steps optimal controller.
Each controller has different characteristics. Depending on the application, a particular controller might be stronger
than the others and be the most suitable controller for the application.
Title: A Comparison of Some LQG Discrete Control Algorithms
Abstract: The paper presents a new discrete linear quadratic Gaussian (LQG) control
algorithm and compares it with three existing optimal discrete LQG control
algorithms. The control system is a single-input-single-output (SISO)
regulating control system, and the model is an ARMAX model. The new
algorithm gives the same controller as that of the three existing algorithms.
However, the new control algorithm is easy to understand and more versatile.
It can be used to design other stochastic control models such as the Box-Jenkins
and innovation state space models and also deterministic linear quadratic
Title: ISO, MPC and PID: The Good, The Bad and The Ugly Discrete
Abstract: In this paper, three well-known discrete controllers - Infinite Steps Optimal (ISO),
Model Predictive Control (MPC) and Proportional Integral Derivative (PID) - are
discussed and compared. The ISO controller is the perfect controller; it should
be the industry standard. The MPC controller is a heterodox controller, which
should be scrutinized before application. The PID controller is acceptable
for low-order control systems - a fact known for a long time by control engineers.
Title: Discrete H-infinity Control Theory for Chemical Process Control
Abstract: In this paper, the discrete H-infinity least sensitivity controller,
which minimizes the infinity norm of a weighted sensitivity function is
presented. The minimal weighted sensitivity function, which can be interpreted
as a Box-Jenkins model for a stochastic regulating control system with the
ARIMA time series disturbance model as the weighting function, has a flat spectrum.
The controller is obtained by searching for an embedding polynomial and solving
for the minimum variance controller. When there is no pure dead time, the least
sensitivity controller is the same as the minimum variance controller of the
Title: An Assessment of a Linear Quadratic Stochastic Control
Abstract: The existing N finite steps optimal control algorithm of
a discrete state space model, stochastic regulating control
system is under review and compared with a new
algorithm. The new algorithm is derived by the method
of dynamic programming. The two algorithms give the
same value for each controller of the N steps. The algorithms
are physically implementable and must be used
for applications with a small number of control steps.
For a large or an infinite number of steps, a steady state
controller, obtained by convergence of the algorithms,
can be used. In this case, the controller has the physical
meaning of a pseudo infinite steps optimal controller.
Title: The Sample Variance Formula:
A Detailed Study of an Old Controversy
Abstract: The two biased and unbiased formulae for the sample variance of a random variable
are under scrutiny. New research result proves that the formula with a smaller divisor is
unbiased and the formula with a larger divisor is biased. This fact agrees with the current
belief in statistics literature. Many mathematical proofs for both formulae contain errors:
This is the reason for the controversy and this new research. The final verdict comes from