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

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.

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 K_P and integral gain K_I). 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].

Figure 1 

Illustration of a temperature closed-loop control system with various time delays.

The gains of an injector, thermal exchanger, and monitor are K_V, K_H, and K_F, respectively, while the related time constants are T_V, T_H, and T_F. The PI controller's transfer function is as follows:

The proportional gain regulates the rate of temperature increase after the first transients, while K_P and K_I 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 bywherein known as scale parameter, denotes matrices of the system, and ∅(t) is the system's starting precondition for, .

Consider system matrices (3):

τ₁(t) and τ₂(t) are time-varying additive delays that meet the following criteria: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∈ℝ_n×n, scalars η₁ and η₂ sustaining η₁ < η₂, a vector valued function ω: [η₁, η₂] ⟶ℝ_n. If the integration in question is properly specified, then the following inequality holds [16].where η₁₂ = η₂−η₁.

Lemma 2. When a continuous function (x) and its derivative f′(x) are both defined on [a, b],

Lemma 3. For any vectors ɛ₁, ɛ₂, matrices P>0, Q and real numbers α₁ > 0, α₂ > 0 with α₁+α₂ = 1 satisfying [17]Subsequent inequality condition holds

Proof. The condition (9) indicateswhich givesSo may derive the inequality condition by utilizing the α₁+α₂ = 1 connection on (12) and rearranging the variables (10).

2.2. Feasibility Checking Methodology

Figure 2 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.whereWith and .

Proof. Consider the LK functional V(t) = Σ Vi(t) 3i=1 with [18], [19]where P, Qi, i = 1, 2, 3, and definite real matrices Z(symmetric) with positive, and τ(t) = (τ₁(t) + τ₂(t)). Define 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 asThe time-derivative of V ̇₂(t) isSince in NCS’s, τ̇₁(t) ≤ μ₁ < 1 and τ̇ (t) ≤ μ < 1, by means of Equation (17) the following inequality may be expressed:To put it another way, Equation (18) may be deduced from (t) as follows:V̇₃(t)-time-derivative is given byBy extending the integral component, Equation (20) may be expressed as follows:Which, when Lemma 1 is reformulated on Equation (21), gives usApplying Lemma 1 to Equation (22), we now haveHere .
Equation (23) may now be expressed as follows:Since δ₁+δ₂+δ₃ = 1, ∀t, Lemma 3 is used to deal with the reciprocal convex combination method of equation (24):providedholds with S₁₂, S_13, 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.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.

The controller capacity curve may be readily produced for the aforementioned system characteristics in a delay-free situation; it is shown in Figure 3. The closed-loop system is asymptotically stable for all values of K_P and K_I below the controller capability curve, as shown in Figure 3. If K_P=2 and K_I = 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 

K_I determined values for various K_P values.

Figure 4 

Progression of Δ(t) less than zero delay condition.

The system (3) is modified when the network delay τ is zero.

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 K_P and K_I. Providing the highest K(1/S) values for a static number of K_P, 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 K_I can be seen in Figure 4.

Table 2 

For a given P, the maximum integral controller gain.

The 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 K_P = 2 and K_I = 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 K_P and K_I 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 K_P and K_I 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.

Table 4 

For a given τ̅₁ − 2, the highest acceptable delay bound τ̅₂.

Table 5 

For a given τ̅₁ − 3, the highest acceptable delay bound τ̅₂.

Table 6 

For a given τ̅₁ − 4, the highest acceptable delay bound τ̅₂.

Table 7 

For a given τ̅₁ − 5, the highest acceptable delay bound τ̅₂.

Table 8 

For a given τ̅₂ − 1, the highest acceptable delay bound τ̅₁.

Table 9 

For a given τ̅₂ − 2, the highest acceptable delay bound τ̅₁.

Table 10 

For a given τ̅₂-3, the highest acceptable delay bound τ̅₁.

Table 11 

For a given τ̅_2–4, the highest acceptable delay bound τ̅₁.

Table 12 

For a given τ̅₂–5, the highest acceptable delay bound τ̅₁.

Figure 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 Figure 4; 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.

Data Availability

The underlying data supporting the results of this study were included in the paper.

Conflicts of Interest

The authors declare that there are no conflicts of interest.