CONTROL AND IDENTIFICATION OF CONTROLLED AUTO-REGRESSIVE MOVING AVERAGE (CARMA) FORM OF AN INTRODUCED SINGLE-INPUT SINGLE-OUTPUT TUMOR MODEL

The article investigates the parameter estimation for controlled auto-regressive moving average models with gradient based iterative approach and two-stage gradient based iterative approach. Since deriving a new model for tumor model is substantial, introduced system identification algorithms are used in order to estimate parameters of a specific nonlinear tumor model. Besides, in order to estimate tumor model a collection of output and input data is taken from the nonlinear system. Apart from that, effectiveness of the identification algorithms such as convergence rate and estimation error is depicted through various tables and figures. Finally, it is shown that the two stage approach has higher identification efficacy


SYSTEM MODEL: CARMA SYSTEMS
Take the introduced below CARMA system into consideration: ()y(t) = B(q)u(t) + C(q)ν(t) (1) Here u(t) is the succession of input of the system, y(t) is the succession of output of the system and ν(t) is a succession of white noise with zero mean and variance σ 2 .Also A(q), B(q) and c(q) are multinomial in the monad backward variation agent [i.e. −1 u(t) = u(t − 1)].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 n-dimensional column which all components are 1.The superscript T indicates the transpose of a matrix; the matrix norm is described by || 2 = (  ).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: () ≔ 1 +  1  −1 +  2  −2 + ⋯ +       , () ≔  1  −1 +  2  −2 + ⋯ +       , () ≔ 1 +  1  −1 +  2  −2 + ⋯ +       .
In a generic way, it is presumed that y(t) = 0, u(t) = 0 and ν(t).=0 for t ≪ 0. Take  ≔   +   +   , Consider the system parameter vectors: and the corresponding information vectors: Type equation here.
Beyond that λ  [𝑋𝑋] 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 Φ() like: 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  1 (Θ) regarding Θ, we attain this iterative relation: Here, μ > 0 is a convergence factor or an iterative step-size.To make sure about convergence of Θ �  , all the eigenvalues of   − μΦ  ()Φ() should be in the monad circle, so −  ≤   − μΦ  ()Φ() ≤    0 ≤   − μΦ  ()Φ() ≤ 2  therefore as suitable conservative form of μ 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: The steps of calculating Θ �  from equation ( 4)-( 10) summarized as below: 1) Regarding  ≤ 0 set every variable to zero.Assume k = 1, take the data length L (L≫ ) and take the primary amounts, Θ �  = 1   0 ,  0 = 10 6 and the system identification precision ε. 2) Gather all the input u(t) and output y(t) for t=1,2,…,L.

TWO-STAGE GRADIENT BASED ITERATIVE ALGORITHMS (2S-GI)
Consider the CARMA model described in equation (\ref{eq.2}).First, we define these two imaginary output variables: Afterwards by these definitions we have: Take $L$ as data length.According to equation ( 11) and ( 12), we define these two static criterion functions: Consider the vector of stacked output Y(L), vectors of the stacked imaginary outputs  1 () and  2 (), and the matrices of stacked information ϕ() and ψ() 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 �   +  − μ 1 ϕ  ()ϕ()� and �   − μ 2 ψ  ()ψ()�, should be in the unit circle, so we have: Therefore, similar to GI algorithm as a conservative choice, we have the following relation for μ 1 and μ 2 : In brief, we have the following set of equations for 2S-GI algorithm: The steps of attaining θ �  and  ̂ included in the 2S-GI approach from equation ( 15)-( 25) are brought up as follows: 1) Regarding  ≤ 0, 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 ε.

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: 9 Here   is the controller error.The feedback form of control law is: Where  0 ,  1   2 respectively are: 0 And we have:   = 0.6  ,   = 0.5  ,   = 0.125  .And      are ultimate period and ultimate gain respectively.

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   = 2,   = 2and   = 2.In simulations,  = 2,  0 =1 and  0 =1.\subsection{Estimation of T(t)} The CARMA model of T(t) as the output and u(t) as the input is:

ESTIMATION OF I(T)
The CARMA model of $I(t)$ as the output and u(t) as the input is: Control and Identification of Controlled Auto-Regressive Moving Average (Carma) Form of an Introduced Single-Input Single-Output Tumor Model International Journal of Engineering Technologies and Management Research 4 Based on the above definitions and equation (\ref{eq.1}),we attain the the below parameter estimation configuration: Control and Identification of Controlled Auto-Regressive Moving Average (Carma) Form of an Introduced Single-Input Single-Output Tumor Model International Journal of Engineering Technologies and Management Research 6
of  1 and  2 for CARMA System with Variance  2 = 2.00 2 and Number of Data L=1000 with GI Algorithm Estimation of  1   2 for CARMA System with Variance  2 = 2.00 2 and Number of Data L=1000 with 2S-GI Algorithm 12 Estimation of  1 for CARMA System with Variance  2 = 2.00 2 and Number of Data L=1000 Estimation Error for CARMA System with Variance  2 = 2.00 2 and Number of Data L=1000 8782 −2 () = 1 − 0.1586 −1 + 0.0987 −2 7697 -0.0238 0.0200 1.7570 0.6264 -0.3459Control and Identification of Controlled Auto-Regressive Moving Average (Carma) Form of an Introduced Single-Input Single-Output Tumor Model International Journal of Engineering Technologies and Management Research