Control and identification of controlled autoregressive moving average (CARMA) form of an introduced singleinput singleoutput tumor model Kiavash Hossein Sadeghi ^{1}, Abolhassan Razminia ^{2}, Abolfazl Simorgh ^{3} ^{1}^{ }Department
of Electrical Engineering, Faculty of Intelligent Systems Engineering and Data
Science, Persian Gulf University, Bushehr 75169, Iran ^{2}^{ }Department
of Electrical Engineering, Faculty of Intelligent Systems Engineering and Data
Science, Persian Gulf University, Bushehr 75169, Iran ^{3}^{ }Department
of Aerospace Engineering, Universidad Carlos III de Madrid, 28911 Leganés, Spain
1. INTRODUCTION The iterative and recursive algorithms could be used to solve matrix equations Wang (2007), Ding (2005), Xie (2010), parameter estimation problems Li (2018), Li (2018), Liu (2010) and filtering issues Ma (2020). In parameter estimation approaches which are recursive, the estimation of parameters can to be calculated in an online framework Du (2017), Wei (2017). On the other hand, the primary notion of the hierarchical algorithms is to update estimation of the parameters by applying a set of data Ding (2018), Ding (2019), Sadeghi (2023). The hierarchical parameter estimation approaches make adequate use of all output and input Data Li (2020), Wang (2020), and could enhance the accuracy of estimation of parameters Li (2020), Ding (2020) and convergence rate of parameters Li (2021), Chen (2020). Twostage
algorithms have an enormous usage in the realm of parameter identification Sadeghi (2023), Sadeghi (2023) developed a twostage
stepwise system identification approach for a class of nonlinear dynamic
systems Li
et al. (2006). In Raja
(2015), twostage least mean
square adaptive methods relying on process of fractional signal were fostered
regarding CARMA systems. A twostage neural network
algorithms related to ARMA model estimation by the use of a
simple mean called extended sample autocorrelation function is presented
Lee
(1994). In Bin
(2012), a twostage method is
introduced regarding the system identification of an ARMAX model which
identifies ARX and MA part separately by biaseliminated least squares
method and another basic method respectively. Also in Ding
(2020), a new twostage
algorithm for estimating parameter of system is brought up but in this article
as a novelty, a CARMA system is discussed. Having a suitable model for tumor system has become an integral issue since the death rate of cancer has become considerable. Accessing a suitable polynomial model for tumor can make the designing of a controller for system much easier. In Pillis (2020), a four population model is presented which contains tumor cells, host cells, drug interaction, immune cells and a controller based on optimization, which is used to satisfy the specific desire. In Sweilam & ALMekhlafi (2018), an updated nonlinear mathematical format of a general tumor beneath immune suppression is discussed. The brought up model in this paper is ruled by a fractional differential equations system. Lobato (2016) presented another model for tumor and in their works they aim to reach a protocol of optimization for injection of drug to sick individuals having cancer, by the making both of the cells having cancer and the drug concentration which has been prescribed minimum Lobato (2016). Tumor model presented in this last research is the basis of our study throughout the rest of the paper. Controlling a CARMA or ARMAX model system has been the subject of a few papers and not much work has been done in this field. For instance, In Chen & Guo (1987), an optimal adaptive control for ARMAX systems using a quadratic loss function is introduced. In Li (2021), abrupt faults in ARMAX models have been taken into consideration and reliable control problem has been studied. Multivariable system control is discussed in OsorioArteaga (2020) where a robust adaptive control is applied to ARMA and ARMAX structures of an electric arc model. Furthermore, linear neural networks was set as a study tool for adpative control of CARMA systems Watanabe (1992). In the following section, a nuance characteristic of the system configuration regarding the CARMA configuration is brought up. Also, section section 3 includes the mathematics of two novel GI algorithm. Section 4 describes a specific tumor model. In section 5, all the necessary simulations for showing the effectiveness of new algorithms are illustrated by identifying a tumor model. Eventually, in the last section, all the outcomes were derived. 2. System model: Carma systems Take the introduced below CARMA system into consideration: Here u(t) is the succession of input of the system, y(t) is the succession of output of the system and is a succession of white noise with zero mean and variance Also A(q), B(q) and c(q) are multinomial in the monad backward variation agent [i.e. For simplicity in the rest of the paper, we have the following notations: A =: X describes A is described as X; The indication I () is an identity matrix with suitable dimensions ($1_{n}$ indicates a vector of ndimensional column which all components are 1. The superscript T indicates the transpose of a matrix; the matrix norm is described by .
Now look at the CARMA system shown in Figure \ref{fig.1}. We define A(q), B(q) and C(q) as polynomials of known orders as follows: In a generic way, it is presumed that y(t) = 0, u(t) = 0 and = 0 for t 0. Take , Consider the system parameter vectors: and the corresponding information vectors:
Based on the above definitions and equation (\ref{eq.1}), we attain the the below parameter estimation configuration: =, y(t)= + + , (2) y(t)= + , (3) 3. Theory of identification and
control algorithms 3.1. Gradient based iterative
algorithms(GI) We consider k=1,2,3,… as an
hierarchical variable and
as
the hierarchical identification of and
while k iteration has established. Beyond that is the biggest eigenvalue of the matrix of
symmetric format X. Now we take an array of data with length L which works with the model introduced in. Here, we consider the vector of stacked output data Y(L) and matrix of the stacked data like: Y(L):= := Now we define the static criterion function as follows: which can be equally described as:
By taking advantage of negative gradient probe, calculating the partial derivative of regarding , we attain this iterative relation:
=
Here, is a convergence factor or an iterative stepsize. To make sure about convergence of , all the eigenvalues of should be in the monad circle, so therefore as suitable conservative form of we have: As to eschew calculating the intricate eigenvalues of a matrix which is square and to decrease evaluation expense, the trace of matrix is taken advantage of and capitalized on a different manner for picking up the convergence rate:
Now it is possible to attain the gradient based iterative method for CARMA system presented in equation (1) with the following set of equations: (4) (5) (6) (7) (8) (9) (10) The steps of calculating from equation (4)(10) summarized as below: 1) Regarding set every variable to zero. Assume k = 1, take the data length L (Land take the primary amounts, and the system identification precision . 2) Gather all the input u(t) and output y(t) for t=1,2,…,L. 3) Attain the vectors of information by equation (9), by equation (10) and by equation (8). 4) Form the vector of stacked output Y(L) regarding equation (6) and the matrix of stacked information regarding equation (7), also pick up a large based on equation (5). 5) Upgrade the parameter estimation vector $\hat{\Theta}{k}$ by equation (\ref{eq.4}). 6) Contrast with . If extend k in unit order and start from step 5. In all other respects, attain iteration k and the system identification vector . 3.2. Twostage Gradient based iterative algorithms (2SGI) Consider the CARMA model described in equation (\ref{eq.2}). First, we define these two imaginary output variables:
Afterwards by these definitions we have: (11) (12) Take $L$ as data length. According to equation (11) and (12), we define these two static criterion functions: (13)
Consider the vector of stacked output Y(L), vectors of the stacked imaginary outputs and , and the matrices of stacked information and are as follows:
Equations (13) and (14) can be equivalently written as:
By taking advantage of the search of negative gradient to make the criterion functions above minimum, we have:
=
To make sure about convergence of and all the eigenvalues of and , should be in the unit circle, so we have: Therefore, similar to GI algorithm as a conservative choice, we have
the following relation for and In brief, we have the following set of equations for 2SGI algorithm: (15) (16)
(20) (22) (24) (25) The steps of attaining and included in the 2SGI approach from equation (15)–(25) are brought up as follows: 1) Regarding , put every parameter to 0. Imagine k=1 take the length of data as L ( and set the initial values as: and the parameter estimation accuracy 2) Gather all the input u(t) and output y(t) for t=1,2,…,L. Attain the information vectors (22) by equation (22) and (t) by equation (23). 3) Build the vector of stacked output Y(L) by (19) and the matrices of stacked information and by (20) and (21), calculate the convergence factor and regarding (16) and (18). 4) Update the vectors of parameter approximation by (15) and (17). 5) Compare with and with :If +> , extend k by $1$ and start from step 4. In all other respects attain iteration k and the vectors of estimation of parameters and . 4. Control theory In this part of the paper, theory of a ziegler nichols PID controller for third order processes introduced in (Bobal, 2006) is brought up. The control law which we took advantage of is: (26) Here is the controller error. The feedback form of control law is: (28) Where respectively are: And we have: . And are ultimate period and ultimate gain respectively.
5. Tumor model I indicate the immune cells number at time t, T denotes the tumor cells number at time t, N describes the normal (host) cells number at time t, and u is the plan of control.\begin{figure}[h] \centering \includegraphics[width=.5\linewidth]{TIN.eps} \caption{Random tumor and immune cells interactions.}
,
. Values of known parameters in above equations are listed below Lobato (2016)
Therefore, we yield: \begin{equation*} \begin{split} , , . 6. Simulations 6.1. Estimation of T(t) In this paper, we aim to identify T(t) as the quantity of tumor cells at time t and I(t) as the quantity of immune cells at time t, by presenting novel parameter estimation method. In simulations assume , and . In simulations,, =1 and =1. \subsection{Estimation of T(t)} The CARMA model of T(t) as the output and u(t) as the input is:
Table 1
Table 2
