Ahmed-ElTahan/Determinist​ic-Indirect-Self-Tuning-Regulator-Two-Degree-Controller-2nd-Method

# Deterministic-Indirect-Self-Tuning-Regulator-Two-Degree-Controller-2nd-Method
It's intended to apply the self-tuning regulator for a given system
such as
y z^(-d) Bsys
Gp = ------ = ----------------------
u Asys

the controller is given in the form of

T S
u = ------ uc - ------ y = L1 - L2
R R

the closed loop transfer function
y z^(-d)BsysT z^(-d)BsysT z^(-d)BsysT
------ = ---------------------------------- = ------------------- = -------------------
uc AsysR + z^(-d)BsysS Am A0 alpha

where
-- y : output of the system
-- u : control action (input to the system)
-- uc : required output (closed loop input-reference, command signal)
-- err = error between the required and the output --> = uc - y
-- Asys = 1 + a_1 z^-1 + a_2 z^-1 + ... + a_na z^(-na)
-- Bsys = b_0 + b_1 z^-1 + b_2 z^-1 + ... + b_nb z^(-nb)
-- R = 1 + r_1 z^-1 + r_2 z^-1 + ... + r_nr z^(-nr) --> [1, r_1, r_2, r_3, ..., r_nr]
-- S = s_0 + s_1 z^-1 + s_2 z^-1 + ... + s_ns z^(-ns) --> [s_0, s_1, s _2, s_3, ..., s_ns]
-- T : another choice that to affect the close loop zeros and it's determined based
on several ways. Here use T = z^(-n)/B, n >=d, choose n = d
-- d : delay in the system. Notice that this form of the Diaphontaing solution
is available for systems with d>=1
-- Am = required polynomial of the model = 1+m_1 z^-1 + m_2 z^-1 + ... + m_nm z^(-m_nm)
-- A0 = observer polynomail for compensation of the order = 1 + o_1 z^-1 + o_2 z^-1 + ... + o_no z^(-no)
-- alpha:required characteristic polynomial = Am A0 = 1 + alpha1 z^-1 + alpha2 z^-1 + ... + alpha_(nalpha z)^(-nalpha)

Steps of solution:
1- initialization of the some parameters (theta0, P, Asys, Bsys, S, R, T, y, u, err, dc_gain).
2- assume at first the controllers are unity. Get u, y of the system
3- RLS and get A, B estimated for the system.
4- Solve the Diophantine equation using A, B and the specified "alpha = AmA0" and get S, R of the controller.
5- choose T = z^(-n)/B, n >=d, choose n = d
5- find "u" due to this new controller and then "y"

T S
u = ------ uc - ------ y
R R

6- repeat from 3 till the system converges.

Function Inputs and Outputs
Inputs
uc: command signal (column vector)
Asys = [1, a_1, a_2, a_3, ..., a_na] ----> size(1, na)
Bsys = [b_0, b_1, b _2, b_3, ..., a_nb]----> size(1, nb)
d : delay in the system (d>=1)
Ts : sample time (sec.)
Am = [1, m_1, m_2, m_3, ..., m_nm]---> size(1, nm)
A0 = [1, o_1, o_2, o_3, ..., o_no]---> size(1, no)

Outputs
Theta_final : final estimated parameters
Gz_estm : estimated pulse transfer function
Gc1: first controller S/R
Gc2: second controller T/R
Gcl = closed loop transfer function

Note: in order to acheive the dc gain which is the y_ss/uc_ss we may use
here T = T/dc_gain