1Table of Symbols

Ts	system time constant
T1	system time constant
T2	system time constant
y	response of the system
u	input to the system
k	the(DC) gain of the system
ξ	damping ratio
t	(continuous) time
x	state variable “column vector”
z	output (measured) variable “column vector”
z	complex variable
s	complex variable
f	sampled function
f*	sampled function value
fn	approximation of fover the nth period of sampling
ui	value of u over the ith interval of sampling
ui0	initial value of u (at the beginning of the interval i)
uif	final value of u (at the end of the interval i)
	Laplace transform operator
	inverse Laplace transform operator
Z	Z- transform operator
a, b…	polynomials in z-1
a, b…	scalar values

2Basic Premises

	(2.1)
	(2.2)
	(2.3)
	(2.4)
	(2.5)
, n order of the system	(2.6)
, n order of the system	(2.7)
	(2.8)
	(2.9)
	(2.10)

3ZOH and FOH Definitions

3.1Zero Order Hold

(3.1.1)

 

(3.1.2)

3.2First Order Hold

(3.2.1)

 

(3.2.2)

4Closed Loop Structure and Controller Synthesis

Let’s consider a closed-loop structure with sampling as follows:

Fig.1

where the continuous part of the structure is represented by controlled system S with (continuous time) transfer function F_s and Zero- or First Order Hold H with the transfer function H₀ or H₁. The whole continuous-time part has then the transfer function

(4.1)

We suppose that the input is applied periodically and the output y is also observed periodically at the same discrete instants Δt at which the input is adjusted (the displacement of the samples is insignificant in comparison with the sampling period Δt ).

4.1Discrete-time transfer function and controller synthesis

(4.1.1)

We suppose that F_c has the form:

(4.1.2)

where a and b are polynomials in z^-1 and the greatest common divisor of polynomials a and b is (a,b) = 1.

Let the reference input (setpoint) w has the form:

(4.1.3)

where q and p are polynomials in z^-1 and (q,p) = 1 is the greatest common divisor of polynomials q and p.

We define the polynomial a₀ :

(4.1.4)

For quadratic optimal control we solve the linear Diophantine equation in polynomials:

(4.1.5)

and for the finite-time-optimal control:

(4.1.6)

for unknown polynomials x and y. For the quadratic optimal control it is necessary to find a solution which satisfies the condition that ∂y < ∂a where the symbol ∂ denotes the degree of a polynomial. For the finite time-optimal control it is necessary to find a solution which minimizes degree ∂y of the polynomial y.

Superscripts ^"+" and ^"–" denote factoring of a polynomial, tilde denotes reciprocal polynomial and superscript ^"*" denotes the reflection of a polynomial.

Then, for quadratic optimal control we have a discrete transfer function:

(4.1.7)

and for finite-time-optimal control:

(4.1.8)

5Algorithm

The discrete transfer function R can be transformed into the form:

(5.1)

The relation between the transfer function and the recursive expression of the algorithm is given by:

(5.2)

The index i of the error signal e_i and of the input to the system (controller output) u_i denotes the size and sign in the previous steps of sampling (0 - i) related to the immediate value of a variable which has the index i, thus:

(5.3)