Control and identification of controlled auto-regressive moving average (CARMA) form of an introduced single-input single-output tumor model Kiavash Hossein Sadeghi 1 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). Two-stage
algorithms have an enormous usage in the realm of parameter identification Sadeghi (2023), Sadeghi (2023) developed a two-stage
step-wise system identification approach for a class of nonlinear dynamic
systems Li
et al. (2006). In Raja
(2015), two-stage least mean
square adaptive methods relying on process of fractional signal were fostered
regarding CARMA systems. A two-stage 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 two-stage method is
introduced regarding the system identification of an ARMAX model which
identifies ARX and MA part separately by bias-eliminated least squares
method and another basic method respectively. Also in Ding
(2020), a new two-stage
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 & AL-Mekhlafi (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 Osorio-Arteaga (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 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 follows: In a generic way, it is presumed that y(t) = 0, u(t) = 0
and 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)= y(t)= 3. Theory of identification and
control algorithms 3.1. Gradient based iterative
algorithms(GI) We consider k=1,2,3,… as an
hierarchical variable 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
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
= Here, 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 1) Regarding
2) Gather all the input u(t) and output y(t) for t=1,2,…,L. 3) Attain
the vectors of information 4) Form
the vector of stacked output Y(L) regarding equation (6) and the matrix of
stacked information 5) Upgrade the parameter estimation vector $\hat{\Theta}{k}$ by equation (\ref{eq.4}). 6) Contrast
3.2. 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 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 Therefore, similar to GI algorithm as a conservative choice, we have
the following relation for In brief, we have the following set of equations for 2S-GI algorithm:
The steps of attaining 1) Regarding
2) Gather
all the input u(t) and output y(t) for t=1,2,…,L.
Attain the information vectors 3) Build
the vector of stacked output Y(L) by (19) and the matrices of stacked
information 4) Update
the vectors of parameter approximation (15) and (17). 5) Compare
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:
Here
Where And we have:
And
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]{T-I-N.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 The CARMA model of T(t) as the output and u(t) as the input is: Table 1
Table 2
6.2. Estimation of I(t) The CARMA model of $I(t)$ as the output and u(t) as the input is: Table 3
Table 4
7. Control of tumor models The final goal of this research is to make the amount of tumor cells minimum, therefore we
take T(t)=0 as the desired output of the system. Based on control theory
introduced in the third section and the identified polynomial model of T(t),
the ultimate period and ultimate gain is The output and input of the feedback form is depicted in the next two figures. From tables and figures above, the below results are derived: · The system identification errors of the GI and 2S-GI approaches decrease as the data length increases. · 2S-GI method, compared to GI method, produces less error and therefore is more effective at estimating parameters. · As the noise to ratio signal rises, both introduced algorithms produce a larger amount of error. · From figures, it is perceived that both introduced algorithms converge at a final point and have a competent convergence rate. · The introduced controller proved that, it is able to make the amount of tumor cells in a specific period of time minimum. 8. Conclusion In this contribution, mathematical theories and algorithms of two identification methods of GI and 2S-GI for CARMA systems were developed. GI is an old method but 2S-GI is a novel method which introduced in this paper. Furthermore, a tumor model with one input and three outputs were presented by works of other scholars. By means of introduced parameter estimation approaches, the model were identified. Above that, by taking advantage of a ziegler nichols PID controller the amount of tumor cells were controlled and it was illustrated that the controller could minimize amount of tumor cells in a specific span of time. Also, the GI and 2S-GI algorithm showed that they both are able to estimate parameter of a polynomial CARMA configuration in fast convergence rate and by producing an insignificant amount of error.
CONFLICT OF INTERESTS None. ACKNOWLEDGMENTS None. REFERENCES Bin, X. I. (2012). A Two-Stage ARMAX Identification Approach Based on Bias-Eliminated Least Squares and Parameter Relationship Between MA Process and Its Inverse. Acta Automática Sinica, 491-496. https://doi.org/10.1016/S1874-1029(11)60310-8 Bobál, V. E. (2006). Digital Self-Tuning Controllers: Algorithms, Implementation and Applications. Springer Science & Business Media. Chen, H.-F., & Guo, L. (1987). Optimal Adaptive Control and Consistent Parameter Estimates for ARMAX Model with Quadratic Cost. SIAM Journal on Control and Optimization, 845-867. https://doi.org/10.1137/0325047 Chen, J. Q. (2020). Modified Kalman Filtering Based Multi-Step-Length Gradient Iterative Algorithm for ARX Models with Random Missing Outputs. Automatica. https://doi.org/10.1016/j.automatica.2020.109034 De Pillis, L. G. (2001). A Mathematical Tumor Model with Immune Resistance and Drug Therapy: An Optimal Control Approach. Computational and Mathematical Methods in Medicine, 79-100. https://doi.org/10.1080/10273660108833067 Ding, F. A. (2005). Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations. IEEE Transactions on Automatic Control, 1216-1221. https://doi.org/10.1109/TAC.2005.852558 Ding, F. E. (2019). Gradient-Based Iterative Parameter Estimation Algorithms for Dynamical Systems from Observation Data. Mathematics. https://doi.org/10.3390/math7050428 Ding, F. E. (2020). Gradient Estimation Algorithms for the Parameter Identification of Bilinear Systems using the Auxiliary Model. Journal of Computational and Applied Mathematics. https://doi.org/10.1016/j.cam.2019.112575 Ding, F. E. (2020). Two-Stage Gradient-Based Iterative Estimation Methods for Controlled Autoregressive Systems using the Measurement Data. International Journal of Control, Automation and Systems, 886-896. https://doi.org/10.1007/s12555-019-0140-3 Ding, F. E. (2018). Iterative Parameter Identification for Pseudo-Linear Systems with ARMA Noise Using the Filtering Technique. IET Control Theory and Applications. https://doi.org/10.1049/iet-cta.2017.0821 Du, D. E. (2017). A Novel Networked Online Recursive Identification Method for Multivariable Systems with Incomplete Measurement Information. IEEE Transactions on Signal and Information Processing over Networks, 744-759. https://doi.org/10.1109/TSIPN.2017.2662621 Ji, Z. E. (2020). An Attention-Driven Two-Stage Clustering Method for Unsupervised Person Re-Identification. Computer Vision-ECCV 2020: 16th European Conference. https://doi.org/10.1007/978-3-030-58604-1_2 Lee, J. K. (1994). A Two-Stage Neural Network Approach for ARMA Model Identification with ESACF. Decision Support Systems. https://doi.org/10.1016/0167-9236(94)90019-1 Li, K., Peng, J.-X., & Bai, E.-W. (2006). A Two-Stage Algorithm for Identification of Nonlinear Dynamic Systems. Automatica, 1189-1197. https://doi.org/10.1016/j.automatica.2006.03.004 Li, L. Z. (2020). A Two-Stage Maximum a Posterior Probability Method for Blind Identification of LDPC Codes. IEEE Signal Processing Letters, 111-115. https://doi.org/10.1109/LSP.2020.3047334 Li, M. A (2018). The Least Squares Based Iterative Algorithms for Parameter Estimation of a Bilinear System with Autoregressive Noise Using the Data Filtering Technique. Signal Processing, 23-34. https://doi.org/10.1016/j.sigpro.2018.01.012 Li, M. A. (2018). Auxiliary Model Based Least Squares Iterative Algorithms for Parameter Estimation of Bilinear Systems using Interval-Varying Measurements. IEEE Access, 21518-21529. https://doi.org/10.1109/ACCESS.2018.2794396 Li, M. A. (2020). Maximum Likelihood Least Squares Based Iterative Estimation for a Class of Bilinear Systems using the Data Filtering Technique. International Journal of Control, Automation and Systems, 1581-1592. https://doi.org/10.1007/s12555-019-0191-5 Li, M. A. (2021). Maximum Likelihood Hierarchical Least Squares-Based Iterative Identification for Dual-Rate Stochastic Systems. International Journal of Adaptive Control and Signal Processing, 240-261. https://doi.org/10.1002/acs.3203 Liu, Y. D. (2010). Least Squares Based Iterative Algorithms for Identifying Box-Jenkins Models with Finite Measurement Data. 1458-1467. https://doi.org/10.1016/j.dsp.2010.01.004 Lobato, F. S. (2016). Determination of an Optimal Control Strategy for Drug Administration in Tumor Treatment using Multi-Objective Optimization Differential Evolution. Computer Methods and Programs in Biomedicine, 51-61. https://doi.org/10.1016/j.cmpb.2016.04.004 Ma, H. E. (2020). Partially-Coupled Gradient-Based Iterative Algorithms for Multivariable Output-Error-Like Systems with Autoregressive Moving Average Noises. IET Control Theory and Applications, 2613-2627. https://doi.org/10.1049/iet-cta.2019.1027 Osorio-Arteaga, F. J.-D. (2020). Robust Multivariable Adaptive Control of Time-Varying Systems. IAENG International Journal of Computer Science, 605-612. Raja, M. A. (2015). Two-Stage Fractional Least Mean Square Identification Algorithm for Parameter Estimation of CARMA Systems. Signal Processing, 327-339. https://doi.org/10.1016/j.sigpro.2014.06.015 Sadeghi, K. H. (2023). Efficient Identification Algorithm for Controlling Multivariable Tumor Models: Gradient-Based and Two-Stage Method. Advanced Mathematical Models and Applications, 8(2), 185-198. Sadeghi, K. H. (2023). Multi-Innovation Iterative Identification Algorithms for CARMA Tumor Models. International Review on Modelling and Simulation. https://doi.org/10.15866/iremos.v16i2.23270 Sadeghi, K. H. (2023). Utilizing ARMA Models for System Identification in Stirred Tank Heater: Different Approaches. Computing Open. https://doi.org/10.1142/S2972370123300030 Sweilam, N. H., & AL-Mekhlafi, S. M. (2018). Optimal Control for a Nonlinear Mathematical Model of Tumor Under Immune Suppression: A Numerical Approach. Optimal Control Applications and Methods, 1581-1596. https://doi.org/10.1002/oca.2427 Wang, L. E. (2020). Decomposition-Based Multiinnovation Gradient Identification Algorithms for a Special Bilinear System Based on its Input-Output Representation. International Journal of Robust and Nonlinear Control, 3607-3623. https://doi.org/10.1002/rnc.4959 Wang, M. X. (2007). "Iterative Algorithms for Solving the Matrix Equation AXB+ CXTD= E.". Applied Mathematics and Computation, 622-629. Watanabe, K. T. (1992). An Adaptive Control for CARMA Systems Using Linear Neural Networks. International Journal of Control, 483-497. https://doi.org/10.1080/00207179208934324 Wei, Z. E. (2017). Online Model Identification and State-of-Charge Estimate for Lithium-Ion Battery with a Recursive Total Least Squares-Based Observer. IEEE Transactions on Industrial Electronics, 1336-1346. Xie, L. Y. (2010). Gradient Based and Least Squares Based Iterative Algorithms for Matrix Equations AXB+ CXTD= F. Applied Mathematics and Computation, 2191-2199. https://doi.org/10.1016/j.amc.2010.07.019
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