# Methods of Pre-Identification of TITO Systems

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Decentralized Control

#### 2.2. Recursive Identification Using Approximation Polynomials

#### 2.3. Least Squares Method with Exponential Forgetting

#### 2.4. Self-Tuning Controller

#### 2.5. Suboptimal Linear Quadratic Tracking Controller

#### 2.6. Calculation of Derivatives Using Approximation Functions

#### 2.7. Recursive Instrumental Variable Method

## 3. Assumptions

#### 3.1. Decentralized Controllability

_{ij}(t) of the transformed system S(s). The transformed system S(s) je is decentrally controllable only when its main diagonal is dominant.

#### 3.2. System Model and Shape of Reference Signal

_{i}(t) of the model M(t)of the transformed system S(s). This assumes minimal impact of extra-diagonal transmissions, which is important because of the deployment of a decentralized controller. Simplification of the N-dimensional system to N-dimensional systems is simplified.

_{i}(s), i = 1, 2..., N is the i-th Laplace reference signal of Laplace transformation of the reference signal vector r(s) and has the form

_{i}, t is the time ${t}_{il}\in {R}^{+},i=1,2,\dots ,N,l=1,2,\dots ,p$, the l-th moment of the i-th in portions of the constant function h

_{i}. This means that each non-zero element of the matrix M(s) has exactly one non-zero element of the vector r(s), i.e., that each partial transmission of the overall system model has a reference signal defined for it. As for the form of the reference signal, it is a constant function in parts. This function is approximated from an arbitrary but predetermined number of p segments of a different but concise value, i.e., it varies over time.

## 4. Pre-Identification

- The controller is not connected in the closed circuit. The values of the vector of difference of output quantities and reference signals E(t) are sent to the input of the system S(t). The values of the reference signals are the same and at the same time as those that will be used during regulation.
- If switching control is considered, each time interval of the control of the system S(s) at which all reference signals have a constant value is identified separately, in so-called Identification Elements (IE).
- Each identification element is identified several times, each time by a different identification algorithm, and the obtained model can be verified by comparison with the measured data. The obtained model, which is most consistent with the measured data, is then used for control. Let us call this method of Identification More Than One Method (IMTOM).

## 5. Results

#### 5.1. Simulation Results

#### 5.2. Results in Real-Time at Laboratory Model

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**System pre-identification using recursive intrumental variable—1st subsystem (

**left**), 2nd (

**right**).

**Figure 11.**System pre-identification using recursive instrumental variable—1st subsystem (

**left**), 2nd (

**right**).

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**MDPI and ACS Style**

Saga, M.; Perutka, K.; Kuric, I.; Zajačko, I.; Bulej, V.; Tlach, V.; Bezák, M.
Methods of Pre-Identification of TITO Systems. *Appl. Sci.* **2021**, *11*, 6954.
https://doi.org/10.3390/app11156954

**AMA Style**

Saga M, Perutka K, Kuric I, Zajačko I, Bulej V, Tlach V, Bezák M.
Methods of Pre-Identification of TITO Systems. *Applied Sciences*. 2021; 11(15):6954.
https://doi.org/10.3390/app11156954

**Chicago/Turabian Style**

Saga, Milan, Karel Perutka, Ivan Kuric, Ivan Zajačko, Vladimír Bulej, Vladimír Tlach, and Martin Bezák.
2021. "Methods of Pre-Identification of TITO Systems" *Applied Sciences* 11, no. 15: 6954.
https://doi.org/10.3390/app11156954