# Stability Results of Thermal Control System with Time-Dependent Delays and Perturbations of Nonlinearity

Authors: Dagne Asefa Gemechu; K. Prabakaran; Lalithamani N; S. Nagarajan; S. Umamaheswaran; Umamageshwari. S; Meseret Leta Feyisa
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The stability of a controlled network temperature control system (heat exchanger system) is examined in this study paper. A heat exchanger system keeps the end temperature of a liquid within defined parameters. The study proposes a linearized model of a temperature control system based on a delay-dependent state equation to accomplish this job. For a temperature management system in a network having time-varying delays, the LK functional (Lyapunov–Krasovskii) method is coupled with the reciprocal convex lemma. To get less conservative stability requirements, a new LK functional was assumed in the stability analysis, and the time-dependent of the (LK) functional was taken via the reciprocal convex combination approach. Finally, under the LMI (linear matrix inequalities) paradigm, the suggested stability analysis leads to a stability criterion. The suggested results establish a new stability criterion for a more accurate operating form for a present temperature control system based on a theoretically obtained temperature management system.


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## Body

## 1. Introduction

The stability of a controlled network of heat exchanger systems (temperature control systems) is assessed in this article. Closed-loop control is extremely important when using a heat exchanger for heat transfer between one or more fluids. Fluid separation may be done using a solid wall or by direct contact [1]. Heat exchangers are used in a variety of industrial and commercial contexts, including heat pumps, freezers, air conditioning, electricity production, chemical production, petroleum industry, sewage treatment, and a variety of other applications [2]. When the heat thermal system is operated from a distant master regulator center, the closed loop feedback method is completed via computer networks [3]. A heat exchanger, detector, valve, and regulator at the remote-control center are all part of a conventional controlled network temperature monitoring system. In the case of an unexpected shift in the output variable, the temperature control mechanism will be called into action.When the temperature is severely fluctuating, the temperature device detects the temperature gain. This feedback indication is conveyed over a communication network to the controller unit (as digitally processed data packets). Proportional-integral (PI) control method computes and processes the fault among the standard transmission and actual transmission (deferred by the network of communication). To return the disturbed heat exchanger terminal temperature to its equilibrium state, the PI controller delivers an appropriate input signal to the closing control component (the valve). The PI controller, the heart of the heat exchanger management method, assumes zero network delays. Time delays in the feedback channel are introduced, however, when using data signal transmission in the feedback-loop system. The result of this is that an individual will have optimal performance while using the time-delayed system [4]. Temperature control systems use delay-bound stability conditions to figure the stable margin. According to the best of the author's understanding, a heat exchanger [5] has delay-dependent stability. In distributed control systems, communication channels include inevitable delays while the data (control or measurement) are buffered, processed, and propagated among the nodes [6]. The network introduced two additive delays, both of which have different properties due to the network [7].In the majority of situations, these network-induced delays are limiting the system's overall performance and, in other cases, may even be causing the temperature control system to become unstable. This results in the requirement for a steady delay boundary (i.e., the margin that the system with a controlled network cannot exceed before asymptotically stable) for a temperature regulator system, which is used to fine-tune regulator parameters for specific subsets of constraint values (proportional gain KP and integral gain KI). It has not been documented in the literature yet that a system under a networked control environment exists. The condition proposed in this work describes a generalized criterion (sufficient condition) for determining the time-varying loop delay-dependent stability of temperature control systems. Following that, a linear delayed delay-differential continuous-time equation is constructed for the statistical method of the PI monitored thermal control system with time-varying feedback controller delays. The generalized modeling approach described here is applicable to dynamic systems with temporal delays. Subsequently, LMI is used to demonstrate a time-delay dependent stability criteria, along with the Lyapunov–Krasovskii functional method and the Reciprocal convex combination [8], to determine the closed-loop heat exchanger's maximum value bound for time delay in the sense of Lyapunov. After this article presents simulation data to verify the systematic results on delay-bound stability, the study summarizes the simulation results and analyses the results. As with other convex optimization problems, it is possible to respond to the proposed stability requirements as a series of LMIs that would be solved by recasting the issue as a convex optimization problem [9].The following are stability requirements stated as a collection of solved LMI conditions, which are indicated as a pair of solvable LMI conditions (Stability criteria) [10]. The newer delay time-bound conditions of stability, which favor long-term characteristics, mostly depend on the kind of LK function employed in the study; they avoid the restrictive commutative condition and choose a Lyapunov function of a more general form, and a new design algorithm, which takes into account the relation between the feedback control gains and the observer and improved EID estimator gains, is developed for the nonlinear system and clear relation between the developed estimator and the GESO is also clarified [11]. Novel iterative algorithms based on optimization are developed to solve the continuous-time and discrete-time Sylvester matrix equations [12]. Convergence rates of the proposed algorithms can be markedly improved by choosing appropriate tuning parameters, efficient numerical methods are presented. However, the information on delay-dependent temperature regulation is scattered among the best knowledge, in the literature.


## 2. Problem Statement

To evaluate the system dynamics and develop a controller with the thermal control system is modeled using a straight line [13]. Figure 1 depicts the sequence layout of a temperature control system with platform time delays. The controller is simulated by assuming that every element, such as the thermal exchanger, monitor, and injectors, is represented by a first-order transfer function [14].(1)GVs=KVTVs+1;GHs=KHTHs+1;GFs=KFTFs+1.Figure 1
Illustration of a temperature closed-loop control system with various time delays.The gains of an injector, thermal exchanger, and monitor areKV, KH, and KF, respectively, while the related time constants are TV, TH, and TF. The PI controller's transfer function is as follows:(2)GCs=KP+KIs.The proportional gain regulates the rate of temperature increase after the first transients, whileKP and KI are the PI controller gains, respectively. By placing a pole at the source and boosting the system type by one, the steady-state inaccuracy is reduced at the integral controller gain [15].The cumulative impact of the PI monitor improvements definitely shapes the thermal controller's response, resulting in less overshoot and a shorter settling time. τ1(t) represents sensor-controller delay (measured delay) and τ2(t) represents controller-actuator delay as shown in Figure 1 (processing delay).


### 2.1. System Framework

The additional state-delayed system's generic state-space framework is provided by(3)x˙t=Axt+Adxt−τ1t−τ2t,xt=∅t,t∈−τ¯1+τ¯2,0,wherein xt∈Rn×1 known as scale parameter, A∈Rn×n denotes matrices of the system, and ∅(t) is the system's starting precondition for, t∈−τ¯1+τ¯2,0.Consider system matrices (3):(4)A=−1TV000KHTH−1TH000KFTF−1TF00010;Ad=00−KPKVTV−KIKVTV000000000000.τ1(t) and τ2(t) are time-varying additive delays that meet the following criteria:(5)0≤τ1t≤τ¯1,0≤τ2t≤τ¯2,(6)τ˙1t≤μ1≤∞,τ˙2t≤μ2≤∞,here τ̅1 and τ̅2 are the higher bounds of the time-dependent additive delays τ1(t) and τ2(t), respectively, μ1 and μ2 are the delay's derivatives. Using the LK functional method and the reciprocal convex combination lemma, construct a robust stability condition in the LMI basis to learn about the delay-bound system stability (3) based on the conditions fulfilling (5) and (6). The preceding lemmas are critical for supporting the new stability criteria.Lemma 1.
For symmetric constant (positive) of any matrix N∈ℝn×n, scalars η1 and η2 sustaining η1 < η2, a vector valued function ω: [η1, η2] ⟶ℝn. If the integration in question is properly specified, then the following inequality holds [16].(7)∫η1η2ωsdsTN∫η1η2ωsds≤η12∫η1η2ωTsNωsds,where η12 = η2−η1.Lemma 2.
When a continuous function (x) and its derivative f′(x) are both defined on [a, b],(8)∫abf′xdx=fb−fa.Lemma 3.
For any vectors ɛ1, ɛ2, matrices P>0, Q and real numbers α1 > 0, α2 > 0 with α1+α2 = 1 satisfying [17](9)PQQTP≥0.
Subsequent inequality condition holds(10)−1α1ε1TPε1−1α2ε2TPε2≤−ε1ε2TPQQTPε1ε2.Proof.
The condition (9) indicates(11)α2α1ε1−α2α1ε2PQQTPα2α1ε1−α2α1ε2≥0,which gives(12)α2α1ε1TPε1+α1α2ε2TPε2≥ε1TQε2+ε2TQε1.
So may derive the inequality condition by utilizing the α1+α2 = 1 connection on (12) and rearranging the variables (10).


### 2.2. Feasibility Checking Methodology

Figure2 illustrates the possibility of a temperature network controller with lags based on the linear matrix inequality (LMI) condition.Figure 2
The LMI condition-based determination of the viability of a temperature network control system with delays.


## 2.1. System Framework

The additional state-delayed system's generic state-space framework is provided by(3)x˙t=Axt+Adxt−τ1t−τ2t,xt=∅t,t∈−τ¯1+τ¯2,0,wherein xt∈Rn×1 known as scale parameter, A∈Rn×n denotes matrices of the system, and ∅(t) is the system's starting precondition for, t∈−τ¯1+τ¯2,0.Consider system matrices (3):(4)A=−1TV000KHTH−1TH000KFTF−1TF00010;Ad=00−KPKVTV−KIKVTV000000000000.τ1(t) and τ2(t) are time-varying additive delays that meet the following criteria:(5)0≤τ1t≤τ¯1,0≤τ2t≤τ¯2,(6)τ˙1t≤μ1≤∞,τ˙2t≤μ2≤∞,here τ̅1 and τ̅2 are the higher bounds of the time-dependent additive delays τ1(t) and τ2(t), respectively, μ1 and μ2 are the delay's derivatives. Using the LK functional method and the reciprocal convex combination lemma, construct a robust stability condition in the LMI basis to learn about the delay-bound system stability (3) based on the conditions fulfilling (5) and (6). The preceding lemmas are critical for supporting the new stability criteria.Lemma 1.
For symmetric constant (positive) of any matrix N∈ℝn×n, scalars η1 and η2 sustaining η1 < η2, a vector valued function ω: [η1, η2] ⟶ℝn. If the integration in question is properly specified, then the following inequality holds [16].(7)∫η1η2ωsdsTN∫η1η2ωsds≤η12∫η1η2ωTsNωsds,where η12 = η2−η1.Lemma 2.
When a continuous function (x) and its derivative f′(x) are both defined on [a, b],(8)∫abf′xdx=fb−fa.Lemma 3.
For any vectors ɛ1, ɛ2, matrices P>0, Q and real numbers α1 > 0, α2 > 0 with α1+α2 = 1 satisfying [17](9)PQQTP≥0.
Subsequent inequality condition holds(10)−1α1ε1TPε1−1α2ε2TPε2≤−ε1ε2TPQQTPε1ε2.Proof.
The condition (9) indicates(11)α2α1ε1−α2α1ε2PQQTPα2α1ε1−α2α1ε2≥0,which gives(12)α2α1ε1TPε1+α1α2ε2TPε2≥ε1TQε2+ε2TQε1.
So may derive the inequality condition by utilizing the α1+α2 = 1 connection on (12) and rearranging the variables (10).


## 2.2. Feasibility Checking Methodology

Figure2 illustrates the possibility of a temperature network controller with lags based on the linear matrix inequality (LMI) condition.Figure 2
The LMI condition-based determination of the viability of a temperature network control system with delays.


## 3. Results

The following theorem presents a fresh criterion of stability for a thermal controller with additional time-dependent lagging in this part.Theorem 1.
Consider symmetric matrices Z, with real, positive, R, Qi,i = 1,2,3; free matrices S12, S13, S23 of suitable dimensions, the system (1) for nonnegative scalars τ̅1, τ̅2, μ1, μ2 fulfilling the criteria (4) and (5) is asymptotically stable.(13)ZS12S13∗ZS23∗∗Z≥0,Z>0,Q1,Q2,Q3>0,R>0;Ξ<zero,where(14)Ξ=e1Pe5T+e5Pe1T+e1Q1+Q2+Q3e1T+e2−Q11−μ1e2T+e3−Q21−μe3T+e4−Q3e4T+e5Ue5T−e1−e2Te2−e3Te3−e4TTZS12S13∗ZS23∗∗Z.
Withe1=I000T,e2=0I00T,e3=00I0T,e4=000IT,e5=Ae1T+Ade3TT,U=τ¯2Zτ¯=τ¯1+τ¯2 and μ=μ1+μ2.Proof.
Consider the LK functional V(t) = Σ Vi(t) 3i=1 with [18], [19](15)V1t=xTtPxt,V2t=∫t−τ1ttxTsQ1xsds+∫t−τttxTsQ2xsds+∫t−τ¯txTsQ3xsds,V3t=τ¯∫−τ¯0∫t+θtx˙TsZx˙sdsdθwhere P, Qi, i = 1, 2, 3, and definite real matrices Z(symmetric) with positive, and τ(t) = (τ1(t) + τ2(t)). Define ηt=xTtxTt−τ1txTt−τtxTt−τ¯T function of time-derivative (t), i = 1, 2, 3, is figured along by the trajectory of (3) as follows: The time-function V ̇1(t) is expressed as(16)V˙1t=x˙TtPxt+xTtPx˙t,V˙1t=ηTte1Pe5T+e5Pe1Tηt.
The time-derivative of V ̇2(t) is(17)V˙2t=xTtQ1+Q2+Q3xt−1−τ˙1txTt−τ1tQ1xt−τ1t−1−τ˙txTt−τtQ2xt−τt−xTt−τ¯Q3xt−τ¯.
Since in NCS’s, τ̇1(t) ≤ μ1 < 1 and τ̇ (t) ≤ μ < 1, by means of Equation (17) the following inequality may be expressed:(18)V˙2t≤xTtQ1+Q2+Q3xt−1−μ1xTt−τ1tQ1xt−τ1t−1−μxTt−τtQ2xt−τt−xTt−τ¯Q3xt−τ¯.
To put it another way, Equation (18) may be deduced from (t) as follows:(19)V˙2t≤ηTte1Q1+Q2+Q3e1T+e2−1−μ1Q1e2T+e3−1−μQ2e3T+e4−Q3e4Tηt.
V̇3(t)-time-derivative is given by(20)V3t=x˙Ttτ¯2Zx˙t−τ¯∫t−τ¯tx˙TsZx˙sds
By extending the integral component, Equation (20) may be expressed as follows:(21)V˙3t=ηTte5Ue5Tηt−τ¯∫t−τ¯t−τtx˙TsZx˙sds−τ¯∫t−τtt−τ1tx˙TsZx˙sds−τ¯∫t−τ1tx˙TsZx˙sds.
Which, when Lemma1 is reformulated on Equation (21), gives us(22)V˙3t≤ηTte5Ue5Tηt−τ¯τ¯−τtτ¯−τt∫t−τ¯t−τtx˙TsZx˙sds−τ¯τt−τ1tτt−τ1t∫t−τtt−τ1tx˙TsZx˙sds−τ¯τ1tτ1t∫t−τ1ttx˙TsZx˙sds.
Applying Lemma1 to Equation (22), we now have(23)V˙3t≤ηTte5Ue5Tηt−1δ1∫t−τ¯t−τtx˙sdsTZ∫t−τ¯t−τtx˙sds−1δ2∫t−τtt−τ1tx˙sdsTZ∫t−τtt−τ1tx˙sds−1δ3∫t−τ1ttx˙sdsTZ∫t−τ1ttx˙sds.Here δ1=τ¯−τt/τ,δ2=τt−τ1t/τ¯,δ3=τ1t/τ¯.
Equation (23) may now be expressed as follows:(24)V˙3t≤ηTte5Ue5Tηt−1δ1ηTte3−e4TZe3−e4ηt−1δ2ηTte2−e3)Ze2−e3ηt−1δ3ηTte1−e2TZe1−e2ηt.
Since δ1+δ2+δ3 = 1, ∀t, Lemma 3 is used to deal with the reciprocal convex combination method of equation (24):(25)V˙3t≤ηTte5Ue5Tηt−e1−e2Te2−e3Te3−e4TTZS12S13∗ZS23∗∗Ze1−e2Te2−e3Te3−e4T,provided(26)ZS12S13∗ZS23∗∗Z≥0,holds with S12, S13, and S23 being free matrices of any dimensions.
We get the following quadratic bounding relation by totaling equations from (16), (19), and (25) of V̇1(t), V̇2(t), and V̇3(t) correspondingly.(27)V˙t≤ηTtΞηt.
The additive system (3) with time-lagging fulfills the conditions (5) and (6) is asymptotically stable if condition Ξ<0 and relation (28) hold concurrently.


## 4. Time-Dependent Delays

The efficacy of the stability criteria for thermal control systems through added time-dependent delays is shown in this discussion. Using PI range controller gains, the maximum permissible delayed margin by adding time-varying delays of τ̅1and τ̅2 is calculated for stability. The temperature control gains and time constants utilized in the study are as follows in Table 1 [20].Table 1
Temperature control system Parameter values.
SystemGainTime-constantTemperature ControllerKH = 33TH = 30ValveKV = 1.25TV = 3SensorKF = 0.07TF = 2The controller capacity curve may be readily produced for the aforementioned system characteristics in a delay-free situation; it is shown in Figure3. The closed-loop system is asymptotically stable for all values of KP and KI below the controller capability curve, as shown in Figure 3. If KP=2 and KI = 0.15 per second, from the chart, the feedback loop will converge to equilibrium asymptotically, as shown in Figure 4 for a single perturbation step in the heat exchange objective system function.Figure 3
KI determined values for various KP values.Figure 4
Progression of Δ(t) less than zero delay condition.The system (3) is modified when the network delay τ is zero.(28)x˙t=A+Adxt.The state matrix eigenvalues of the network and eigenvalues of the input matrix (A+Ad) are now found to be dependent on the controller parameters KP and KI. Providing the highest K(1/S) values for a static number of KP, the thermal system control is on the brink of instability, with unit complex conjugate pair on the jω axis eigenvalues in Table 2. The propensity for the curve to progressively diminish with the rise of KI can be seen in Figure 4.Table 2
For a givenP, the maximum integral controller gain.
KPKIEigenvalues of (A + Ad)00.06−0.432 ± 0.009j; −0.0003; −0.0810.25−0.000 ± 0.16j; −0.580; −0.2920.39−0.0001 ± 0.23j; −0.641; −0.2330.47−0.0000 ± 0.26j; −0.689; −0.1840.5−0.0000 ± 0.30j; −0.732; −0.1450.49−0.0000 ± 0.3j; −0.763; −0.1060.42−0.0001 ± 0.37j; −0.801; −0.0770.2−0.0000 ± 0.4j; −0.823; −0.04380.13−0.0000 ± 0.42j; −0.851; −0.02The thermal system control drive now stays exposed near additional system-driven time-dependent delays τ1(t)and τ2(t) when this PI controller works in a networked environment. Let τ̇1(t) = 0.1 and τ̇2(t) = 0.2 be the additive time-varying delays representing time derivatives. Let us say the controller parameters are KP = 2 and KI = 0.151/S, and in the existence of network-generated delays, the temperature regulation is based on the stability criteria given in the statement.If network-generated delays are raised much above this amount, τ1 = 1 second and τ2 = 1.3305 seconds, the system output becomes unbounded evolution. As a result, the effect of network-induced delays on system efficiency and stability is explicitly shown in this study. Let μ1 = 0.1, and μ2 = 0.2 be the values. The maximum permissible delay limit τ̅2 for a given τ̅1 is provided in Tables 3–7 for various values of KP and KI derived from the capacity curve. On the other hand, Tables 8–12 show the determined allowed delay margin τ̅1 to an applied τ̅2. The greatest delay limit that the system can tolerate without affecting stability falls as the different values of KP and KI increase. The comparable stability margin in Figure 5 clearly shows a decrease.Table 3
For a given τ̅1 − 1, the highest acceptable delay bound τ̅2.
KPKIτ̅1 = 2.3τ̅1 = 2.2τ̅1 = 2τ̅1 = 1.8τ̅1 = 1.6τ̅1 = 1.4τ̅1 = 1.2τ̅1 = 1.0τ̅1 = 0.8τ̅1 = 0.6τ̅1 = 0.4τ̅1 = 0.2τ̅1 = 020.150.000.130.330.530.730.931.131.331.531.731.932.132.33Table 4
For a given τ̅1 − 2, the highest acceptable delay bound τ̅2.
KPKIτ̅1 = 1.1408τ̅1 = 1.0τ̅1 = 0.8τ̅1 = 0.6τ̅1 = 0.4τ̅1 = 0.2τ̅1 = 0.1τ̅1 = 0.08τ̅1 = 0.06τ̅1 = 0.04τ̅1 = 0.02τ̅1 = 020.2500.120.320.520.720.921.021.041.061.11.121.14Table 5
For a given τ̅1 − 3, the highest acceptable delay bound τ̅2.
KPKIτ̅1 = 1.0983τ̅1 = 1.0τ̅1 = 0.8τ̅1 = 0.6τ̅1 = 0.4τ̅1 = 0.2τ̅1 = 0.1τ̅1 = 0.08τ̅1 = 0.06τ̅1 = 0.04τ̅1 = 0.02τ̅1 = 040.100.10.30.510.710.911.011.021.041.061.081.1Table 6
For a given τ̅1 − 4, the highest acceptable delay bound τ̅2.
KPKIτ̅1 = 0.2611τ̅1 = 0.2τ̅1 = 0.18τ̅1 = 0.16τ̅1 = 0.14τ̅1 = 0.12τ̅1 = 0.1τ̅1 = 0.08τ̅1=0.06τ̅1 = 0.04τ̅1 = 0.02τ̅1 = 060.100.060.080.10.120.140.160.180.200.220.240.26Table 7
For a given τ̅1 − 5, the highest acceptable delay bound τ̅2.
KPKIτ̅1 = 0.1644τ̅1 = 0.16τ̅1 = 0.14τ̅1 = 0.12τ̅1 = 0.1τ̅1 = 0.08τ̅1 = 0.06τ̅1 = 0.04τ̅1 = 0.02τ̅1 = 02.50.40.000.000.020.040.060.080.100.120.140.16Table 8
For a given τ̅2 − 1, the highest acceptable delay bound τ̅1.
KPKIτ̅2 = 2.331τ̅2 = 2.2τ̅2 = 2τ̅2=1.8τ̅2 = 1.6τ̅2 = 1.4τ̅2 = 1.2τ̅2 = 1.0τ̅2 = 0.8τ̅2 = 0.6τ̅2 = 0.4τ̅2 = 0.2τ̅2 = 020.150.000.130.330.530.730.931.131.331.531.731.932.132.33Table 9
For a given τ̅2 − 2, the highest acceptable delay bound τ̅1.
KPKIτ̅2 = 1.0983τ̅2 = 1.0τ̅2 = 0.8τ̅2 = 0.6τ̅2 = 0.4τ̅2 = 0.2τ̅2 = 0.1τ̅2 = 0.08τ̅2 = 0.06τ̅2 = 0.04τ̅2 = 0.02τ̅2 = 040.10.000.100.300.500.700.901.001.021.041.061.081.10Table 10
For a given τ̅2-3, the highest acceptable delay bound τ̅1.
KPKIτ̅2 = 1.1408τ̅2 = 1.0τ̅2 = 0.8τ̅2 = 0.6τ̅2 = 0.4τ̅2 = 0.2τ̅2 = 0.1τ̅2 = 0.08τ̅2 = 0.06τ̅2 = 0.04τ̅2 = 0.02τ̅2 = 020.250.000.140.340.540.740.941.041.061.081.101.121.14Table 11
For a given τ̅2–4, the highest acceptable delay bound τ̅1.
KPKIτ̅2 = 0.26τ̅2 = 0.2τ̅2 = 0.18τ̅2 = 0.16τ̅2 = 0.14τ̅2 = 0.12τ̅2 = 0.1τ̅2 = 0.08τ̅2 = 0.06τ̅2 = 0.04τ̅2 = 0.02τ̅2 = 060.10.000.060.080.100.120.140.160.180.200.220.240.26Table 12
For a given τ̅2–5, the highest acceptable delay bound τ̅1.
KPKIτ̅2 = 0.1644τ̅2 = 0.16τ̅2 = 0.14τ̅2 = 0.12τ̅2 = 0.1τ̅2 = 0.08τ̅2 = 0.06τ̅2 = 0.04τ̅2 = 0.02τ̅2 = 02.50.40.0000.0040.020.040.060.080.120.140.140.16Figure 5
Heat exchanger thermal control resilience limit.The sensitivity analysis of the closed-loop thermal system control carrying network delay as zero is limited to the stability as illustrated in Figure4; the problem is simple to deduce the effect of network stability delays on the system. The temperature system control responsiveness becomes deformed as network delays grow, and the system closed-loop becomes longer unpredictable delays.


## 5. Conclusion

The issue of delay-dependent temperature control system stability has indeed been solved using additive time-varying delay, and this study investigates the method. The temperature control system is expressed mathematically as a system model in the stability research, and inverse convex pairing is utilized in the investigation. In order to achieve the desired results, this benchmark heat exchanger includes time-varying additive lags. The operational conditions of a real-time temperature control system have been deduced using the findings of this study.



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