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