Non-Monotonic Lyapunov-Krasovskii Functional Approach to Stability Analysis and Stabilization of Discrete Time-Delay Systems

Younes Solgi, Alireza Fatehi,, and Ala Shariati

Abstract—In this paper, a novel non-monotonic Lyapunov-Kra-sovskii functional approach is proposed to deal with the stability analysis and stabilization problem of linear discrete time-delay systems. This technique is utilized to relax the monotonic requirement of the Lyapunov-Krasovskii theorem. In this regard, the Lyapunov-Krasovskii functional is allowed to increase in a few steps, while being forced to be overall decreasing. As a result, it relays on a larger class of Lyapunov-Krasovskii functionals to provide stability of a state-delay system. To this end, using the non-monotonic Lyapunov-Krasovskii theorem, new sufficient conditions are derived regarding linear matrix inequalities (LMIs)to study the global asymptotic stability of state-delay systems.Moreover, new stabilization conditions are also proposed for time-delay systems in this article. Both simulation and experimental results on a pH neutralizing process are provided to demonstrate the efficacy of the proposed method.

I. Introduction

STABILITY analysis is one of the most critical issues in control engineering. Among different challenges in practical systems, stability analysis of time-delay systems has received significant attention in recent decades [1], [2]. The extensive interest in this area is understandable because timedelays are known as a source of instability and degradation of many system performances.

Several methods have been introduced for the stability analysis of the linear time-delay systems during the past decades, such as exponential stability [3], eigenvalue perturbation [4], Razumikhin stability [5] and new conditions based on Lyapunov-Krasovskii stability approach [6]. The Lyapunov-Krasovskii approach is one of the most effective methods for analyzing the stability of time-delay systems.Stability analysis of time-delay systems based on improved Lyapunov-Krasovskii theorems were considered in many studies [7]–[10]. Leeet al. [7] introduced a modified Lyapunov functional which consists of two quadratic functions with a special structural matrix for the stability of time-varying delay. A stability condition for the stochastic time-delay systems was derived by Shiet al. [8], in which a sufficient condition is obtained in terms of a priori designed feedback matrix. New multiple summation inequalities have been presented in [9] which involve the use of discrete Jensen’s and Wirtinger’s inequalities. Moreover, a new model transformation for the stability analysis of discrete time-delay systems was applied in [10], in addition to employing a new approximation for delayed states.

Since the improved Lyapunov-Krasovskii method renders some sufficient conditions for stability, reducing the conservatism in the Lyapunov-Krasovskii method plays a helpful role in providing suitable stability conditions. Among all the studies on the Lyapunov-Krasovskii methods, most researchers have focused on the functional modification to reduce conservatism [11], [12]. These modifications usually lead to new summation terms in the way of stability analysis.Therefore, a popular issue in the stability analysis of timedelay systems is estimating a lower bound for common summation terms appearing in the forward difference of Lyapunov-Krasovskii functional [13]. Resolving this challenging issue was started with the use of the freeweighting matrix (FWM) approach [14] and the Jensen-based inequality (JBI) [15], then continued in [16], [17] by relaxing obtained inequalities. In this regard, Zhanget al. improved summation inequality in [18] and also tried to reduce conservatism by providing tighter bounds of summation terms by the Abel lemma-based finite-sum inequality [19], which led to less conservative stability criteria compared with others.On the other hand, some researchers tried to improve this stability analysis method by making structural changes in the Lyapunov functional [20]–[24].

All aforementioned research has referred to modifying the Lyapunov function in the stability analysis procedure. An alternative method to reduce the conservatism in the Lyapunov-based stability approach is the non-monotonic Lyapunov technique [25], which is also known as finite-step Lyapunov method [26]. The non-monotonic Lyapunov function is utilized to relax the monotonically decreasing condition of the Lyapunov function. As a result, it adapts a larger class of functions for stability analysis. The principle of this approach is that the Lyapunov function decreases every few steps; however, it can increase in between those steps.Few works are addressing non-monotonic approach on stability analysis of discrete time systems. In [26], Aeyels and Peuteman modified some conditions to establish global asymptotic stability and named it the finite-step Lyapunov method. Then, the same concepts called the non-quadratic stabilization conditions, were used for stabilizing a Takagi-Sugeno fuzzy models [27]. Ahmadi and Parrilo in [25]replaced some conditions and established global asymptotic stability in discrete time, and labeled their work nonmonotonic Lyapunov stability. Similar to [26], Derakhshan and Fatehi introduced discrete non-monotonic Lyapunov method in [28], which was used to extract the stability condition for fuzzy control systems [28]–[30]. They reduced the conservatism in the existing results by relaxing the monotonicity requirement of Lyapunov’s theorem. Xieet al.in [31] developed a non-monotonic Lyapunov function approach for discrete-time switching linear systems to improve the stability criterion and achieve betterH∞performance. Then they presented a robustH∞control for switched systems using this method and called it theN-step ahead Lyapunov function approach [32]. Thereupon, this technique was considered in the design of guaranteed cost controller [33], robust output feedback controller [34] and robust state feedback controller [35], [36] for discrete time Takagi-Sugeno fuzzy systems. Also, non-monotonic stability technique was used in [37] to investigate the problem of robustH∞control for a class of discrete-time nonhomogenous Markovian jump linear systems. Although the aforementioned research about non-monotonic Lyapunov method have some small differences, all of them try to relax the monotonicity requirement of Lyapunov’s theorem by allowing the function to increase over few steps while overall continuing to decrease.

As previously mentioned, the stability of time-delay systems is analyzed by Lyapunov-Krasovskii method;however, all of them are based on monotonic decreasing stability criteria. In this paper, due to the high potential of non-monotonic decreasing criteria in reducing conservatism, a novel non-monotonic Lyapunov-Krasovskii (NMLK) theorem is proposed for the stability analysis of the linear discrete state-delay systems. The proposed theorems in this article give the maximum upper bound of delay. The proposed theorems can investigate the stability of systems with “constant and known” or “constant and unknown” delays. State space representation of time delay systems contains the statex(k–d).While extracting the stability conditions using the nonmonotonic technique, previous states betweenx(k) andx(k–d)will appear, which are complicated to deal with. Another issue in using this method involves the summation terms which contain previous states. These problems are solved in this paper. Developing this method, we stabilize the systems with input/output delay, in which the closed-loop equations are in the form of a state-delayed type systems with constant known/unknown delay. The proposed theorems are utilized to relax the monotonic approach mostly in twofold:

1) Enlarging the search space of solutions by allowing the Lyapunov-Krasovskii functional to reduce non-monotonically.

2) Reducing the conservatism by using Abel lemma-based finite-sum inequality [19].

Finally, the proposed theorems are verified first via numerical examples, followed by experimental implementation on a pH neutralization process pilot plant.

This paper is organized as follows: Section II provides preliminary information of the research presented in this paper. The main results, which contain obtaining both 2 andm-step NMLK based stability and stabilization conditions, are presented in Section III. In Section IV, the proposed method is evaluated using numerical examples. Simulation and experimental results on a pH neutralization process are provided in Section V. Finally, the concluding remarks are given in Section VI.

Nomenclature:

II. Preliminaries

A. Non-Monotonic Lyapunov Stability Method

Consider the discrete time system (1) with the function

whereFor the stability studies, without loss of generality, the equilibrium point is assumed at the origin [30]. The conventional monotonic stability condition states that if there exists a continuous scalar functionV(x(k)) satisfying

1)V(x(k)) is a positive definite function (PDF);

2)V(x(k))→∞ asx→∞;

3)V(x(k+1)) –V(x(k)) < 0 for

then the equilibrium statex(k) = 0 is globally asymptotically stable, andV(x(k)) is a Lyapunov function.

To reduce the conservatism in the Lyapunov stability theorem, monotonically decreasingV(x(k)) is replaced with non-monotonically decreasing criteria in [26], [28]. In this technique, it is allowed that the Lyapunov function increases over a few steps. Before presenting the related definition, the globally Lipschitz function must be defined.

Definition 1:f(x) is a globally Lipschitz function if there existsL> 0 such that

The Lipschitz condition is used to replace the monotonicity requirement of the Lyapunov theorem with a more relaxed condition which is summarized in the following proposition.

Proposition 1 (Non-Monotonic Lyapunov Stability) [28]:Consider a discrete time system described by (1), wherex(k) ∈Rnandf: Rn→ Rnis globally Lipschitz function with the Lipschitz constantLthat satisfiesf(0) = 0. If there exists a continuous scalar functionV(x(k)) satisfying

1)V(x(k)) is a positive definite function (PDF);

2)V(x(k))→∞ asx→∞;

3)V(x(k+m)) –V(x(k)) < 0 for

wheremdefines the number of steps that the Lyapunov function can be increasing, then the origin is stable.

B. Basic Lyapunov-Krasovskii Stability Method

then the solutionx= 0 of the system is asymptotically stable.

Finally, the following lemma will be used for obtaining the NMLK based stability criterion in this paper.

Lemma 1 (Abel lemma-Based Finite-Sum Inequality) [19]:For a constant positive definite matrixZ∈ Rn×nwithZ=ZT> 0 and integersr1andr2withr2–r1> 1, the following inequality holds

III. Main Results

This section is concerned with the conservatism reduction of monotonic Lyapunov-Krasovskii stability analysis condition.First, non-monotonic Lyapunov-Krasovskii (NMLK) stability theorem is proposed for the state-delay systems in terms of linear matrix inequalities (LMIs) in Section III-A. In the second subsection, the stabilization conditions of time-delay systems are extracted based on the proposed NMLK approach.

A. Non-Monotonic Lyapunov-Krasovskii Stability Analysis

In the non-monotonic Lyapunov stability technique, the Lyapunov functional candidate is allowed to increase provisionally, but is decreasing in general. We define the mstep difference of a functional as follows:

wheremis the number of steps that the Lyapunov-Krasovskii functional can be incremental.

Remark 1:Delays can be considered in several forms in discrete time delay systems: 1) constant and known; 2)constant and unknown; 3) time-varying and bounded. While time-varying delay is the most general form, in some industrial applications like process control systems, delay is most likely to be constant [38], [39]. Although the stability analysis and controller synthesis of the time-varying delay form is the most general, it also imposes more conservative conditions. So, it is important to improve the analysis and synthesis methods for of all forms, and then select the proper method according to the specifications of the problem. The proposed theorems in this article can investigate the stability of state-delay systems with “constant and known” or “constant and unknown” delay forms.

Assume the state-space representation of the discrete statedelay system with constant unknown delay is expressed as follows:

We consider Lyapunov-Krasovskii candidate in the form of

whereV1,V2andV3are given in (11)–(13)

where ℓ1=d−1,ℓ2=ℓ1−1. Now, we can present the main results of this paper. The following gives the delay-dependent sufficient condition for the stability of the discrete time-delay system (7) based on NMLK method.

Theorem 1: Linear discrete time-delay system (7) with constant unknown delay and given initial condition andd> 2 is 2-step NMLK stable if there exist positive definite matricesandZ∈Rn×nsuch that

where

with definitions

whereei(i= 1, 2, 3, 4) aren ×4nblock-row vectors of then×nidentity matrix for (i= 1, 2, 3, 4) so that

Proof: In the way of stability analysis, first, we calculate each term of Δ2V= Δ2V1+Δ2V2+Δ2V3in the following steps

1) CalculatingΔ2V1

using (8) in (19a)

using the state-space representation (7),x(k+2) is calculated as follows:

where

On the other hand, (19b) can be written as

using the state-space representation (7) and substituting (20)in (22), (23) is reached after some manipulations

alsoΞ1andΞ2are defined in (18).

2) CalculatingΔ2V2

Equation (24) can be written as

using model dynamics (7) in (25), we have

then (26) can be rewritten as (27) after some manipulation

The stability condition can be derived since (Δ2V1+Δ2V2) <0; however it is too conservative. To reduce the conservatism,a third term,V3, is added to (10). Thus, we calculate Δ2V3in the next subsection and try to derive the stability condition using (∆2V1+∆2V2+∆2V3)<0 to enlarge the space and reduce conservatism.

3)CalculatingΔ2V3

The third term in the Δ2Vis

Thus, by some manipulations

then

Using (9), η(k)=Ae1+Ade2−e1=L1−L0and similarly,η(k+1)=L2−L1, η(k−d)=e3−e2, Δ2V3can be represented as

The summation term in (30) can easily be neglected because it is negative definite. However, this increases the conservatism. Instead of removing the summation term, we try to calculate an upper bound for it, which helps to lessen its conservatism. For this purpose, Lemma 1 (Abel lemma) can be used. Thus, following inequalities are obtained for the last term of (30).

where

Thus, the following inequality is provided using (21) and block-row vectorsei

The above inequality can be represented in the following form

whereφ2andΠ2are defined in (18).

It follows from (23), (27), (30), (31) and definition of Δ2Vat(10) that

The stability is achieved if Δ2V< 0. Therefore, we obtain the following inequality

with some manipulation, the recent sufficient stability conditions can be written as (33) considering definitions (15)–(17)

It is obvious that ifΨ1+Ψ2+Ψ3< 0, then (33) is satisfied, and therefore, the LMI (14) is obtained.

Remark 2: Abel lemma is used in this paper instead of a common lemma in the stability analysis of time-delay systems called discrete Jensen’s inequality. This lemma reduces the conservatism by providing a tighter lower bound [19].

Theorem 1 can be generalized to them-step NMLK. First, to simplify the representation, let

and

wherei= 1,...,(m+2).

Theorem 2: Linear discrete time-delay system (7) with constant unknown delay and given initial condition ism-step NMLK stable with 1 ≤m

where

where

Proof: The Lyapunov-Krasovskii functional is assumed as(10), where

Then, the proof is similar to the proof of Theorem 1.

Remark 3:In the proposed non-monotonic Lyapunov-Krasovskii method, the step of non-monotonicity is demonstrated bym. Therefore, a system is calledm-step nonmonotonic stable if it is stable based on NMLK theorem. As the non-monotonicity stepmincreases, the conservatism reduces at the cost of increasing the calculations. Thus,choosingmis a trade-off between reducing the conservatism and increasing the computational cost.

Remark 4:Ifm= 1, then a stability criteria similar to [19] is obtained. Please notice that, in the case ofm= 1, the upper bound of the second summation term inΨ3will be less than the lower bound which is called empty-sum. By convention,the empty sum is equal to zero.

B. Stabilization Based on Non-Monotonic Lyapunov-Krasovskii Functional

In this section, the objective is to design a state feedback controller for a time-delay system by the use of the NMLK stability theorem. Assume the following discrete time-delay system

The state feedback controller is assumed as

By applying this controller, the resulting closed-loop system is derived as a state-delay system shown in (45)

Theorem 3:The closed-loop time-delay system (45) is globally asymptotically 2-step non-monotonic stabilizable if there exist a controller gainFand positive definite matricesandZ∈Rn×nsuch that

where

and s ystem coefficient matrices areand

Proof:The proof can be carried out similarly to the proof of Theorem 1 for the closed loop time-delay system described by(45). Finally,Ψ1+Ψ2+Ψ3< 0 is attained for stabilization in whichΨ1,Ψ2andΨ3are defined in (15)–(17) where the parameters are as (47). The stabilization conditionΨ1+Ψ2+Ψ3< 0 isnot an LMI because of the termwhich contains the controller gainFmultiplied by the unknown variablesP,QorZ. Considering the conditionΨ1+Ψ2+Ψ3<0 and using Schur complement, the LMI (46) is obtained.

Remark 5:Having different units in a large-scale process causes different parallel streams of materials. As a result, a state can affect the output through different paths with different time delays. This kind of output-delayed systems,which is modeled by (48), can be called output-state-delay system

The pH neutralization pilot plant discussed in Section V is an example of this kind of systems. Applying the output feedbacku(k)=−Fy(k)+r(k) on the plant (48), provides the following state-delay closed loop system.

Corollary 1 summarizes the stabilization condition of such systems.

Corollary 1:The closed-loop time-delay system (49) is globally asymptotically 2-step non-monotonic stabilizable if there exist a controller gainFand positive definite matricesP∈R2n×2n,Q∈Rn×nandZ∈Rn×nsuch that (46) holds,whereand

Proof:In the state-delay system (49),and. Theorem 3 can be applied to this system.

Remark 6:Since the stability conditions in Corollary 1 are no longer LMI, a convex optimization algorithm cannot be directly applied. In other words, substituting calculatedandcause nonlinearity inL2and the stabilization condition will be BMI. Therefore, we define. ThenL2is obtained asChange of variables converts the BMI problem to an LMI problem with nonlinear constraints, which can be solved by following the same algorithm as explained in Algorithm 7.3.1 at [20].

Remark 7:Steady state error is a common problem in the state feedback controlled systems. An integrator can be added to the system to overcome this problem. In this regard, the integrator dynamic model is ase(k+1)=e(k)+(r(k)−y(k))wheree(k) implies the output error. By substitutingy(k) from(48) in the integrator dynamic model, we reache(k+1)=e(k)+(r(k)−Cx(k)−Cx(k−d)). The feed-back control law is assumed asu(k)=He(k)−Fy(k)+r(k), in whichFandHare output feedback controller and integrator gains, respectively.Therefore, the closed-loop system is derived as (50)

Then, state augmentation yields the following representation.

Following form Corollary 1 and Remark 7, we express Corollary 2.

Corollary 2: The closed-loop time-delay system (51) is globally asymptotically 2-step non-monotonic stabilizable if there exist a controller gainFand positive definite matricessuch that (46) holds,whereand

Proof:Considering the system coefficient matrices in (51),stabilization conditions are obtained by following the same procedure as given in Theorem 3.

Remark 8:The obtained stabilization conditions in Corollary 2 are BMIs due to the termL2, and definitions ofandTherefore, by defining the new variablesand using Algorithm 7.3.1 in[20], the nonconvex BMI problem can be converted into an LMI-based nonlinear minimization problem in which the parameterL2is as follows:

IV. Numerical Examples

Two numerical examples are provided to evaluate and demonstrate the advantages of the proposed stability analysis criterion in Theorems 1 and 2. This will be followed by an experimental example in the next section.

Example 1:Consider the following state-delay system

whereα∈R anddis the constant unknown delay. This example investigates the calculation of maximum admissible upper bounds of unknown delayd. In this plant, asαincreases,the maximum admissibleddecreases to retain stability. Stability and stabilization conditions in Theorems 1 and 2 are LMI which can be solved using YALMIP toolbox in MATLAB software [40].

The implementation result of the proposed stability analysis conditions in Theorem 2 is compared with several methods for different values of the parameterα. In Theorem 2, the step of non-monotonicity is assumedm= 1,…,4. Table I lists the obtained results for the aforementioned LMI in Theorem 2 along with those calculated by 1) Zhang and Han [19]; 2)Yoneyama and Tsuchiya in [2], and 3) Xuet al. in [41].

TABLE I Maximum Upper Bound of Constant Unknown d

Table I depicts that Theorem 2 can provide stability for a larger value of time delay compared to other methods. Also,Theorem 2 operates better for different value of parametera.Overall, the Abel lemma-based method by Zhang and Han[19] is less conservative than the methods presented in [41]and [2]. The results show that the proposed multi-step NMLK method in this paper is clearly more effective than the onestep stabilizing techniques. In the case ofm= 1, stability conditions in Theorem 2 will be the same as stability conditions in [19]. As it is shown in Table I, Theorem 2 has better performance withm= 4 among other methods and other non-monotonicity steps.

Fig. 1 illustrates a non-monotonically decreasing trend in the NMLK functional, in which the NMLK functional increases in some periods while the overall trend is decreasing. Incremental steps in Fig. 1 is less than or equal to non-monotonicity step,m= 4. This is well understood from Fig. 2 which represents the repetitions of the consecutive incremental steps.

The NMLK functional increases eleven times for four consecutive steps. Also, the repetitions of 2 and 3 consecutive incremental steps are 2 and 5 times, respectively. It is obvious that the maximum of the incremental step of the NMLK in Fig. 1, which is plotted in Fig. 2, is less than or equal to nonmonotonicity stepm= 4.

Fig. 2. The repetition number of incremental steps NMLKF of Example 1.

Example 2:Consider the following time-delay system

In this example, the feasible space provided by the proposed method is studied. Table II compares maximum upper bound of the delaydobtained by Theorem 2 with different nonmonotonicity steps (m= 1,…,58) in this article, stabilization criteria derived using the Lyapunov functional in [15], a finite sum inequality based approach in [42], utilization of zero equalities in [43] and Abel lemma based method in [19], for different values ofβ. It can be observed that forβ= 0, the proposed NMLK algorithm is less conservative than those in[15], [42], [43]. Using NMLK functional and applying Theorem 2 withm= 57, the stability of the system (54) isguaranteed ford≤ 58, in the case ofβ= 0.056; while the methods in [15], 42], [43] cannot prove the stability for this value ofβ. Zhang and Han method [19] can also guarantee stability ford≤ 57. In the case ofm= 1, stability conditions in Theorem 2 will be the same as the stability conditions in [19].

TABLE II Maximum Upper Bound of Constant Unknown d

Fig. 3. Proposed NMLKF of Example 2.

Fig. 4 illustrates the repetitions of the incremental steps. For instance, the assumed NMLKF is incremented 5 times for 57 consecutive steps.

Also, there are 31 repetitions in which NMLK functional increases for 54 steps. As expected, the maximum of incremental steps in the proposed non-monotonic Lyapunov-Krasovskii functional is less than or equal to the nonmonotonicity stepm= 57.

Fig. 4. The repetition number of incremental steps NMLKF of Example 2.

V. Stabilization of the PH Neutralization Process

In this section, a controller is designed based on the proposed NMLK stabilization theorem for a pH neutralizing pilot process. Then, simulation and experimental implementation are presented. First, the pH neutralizing process plant is introduced, and then the simulation and experimental results are discussed.

A. pH Neutralization Process

The pH neutralizing process contains a continuous stirred tank reactor (CSTR), where the pH of the content is controlled by sodium hydroxide flow rate as the manipulated variable.The flow rate of the acetic acid (CH3COOH) is considered as a disturbance. Also, tap water flow is used to control the tank level. Acid, base, and water are injected into the CSTR by corresponding dosing pumps [44], [45]. There is a motorized mixer inside the CSTR in order to have a well-mixed tank.The effluent stream flow rate is pumped out using an outlet pump, which its flow rate is kept constant. As shown in Fig. 5,the output of the reactor tank can be transferred through two different pipes with different delay times; a direct path and a delayed path. The delayed path affects transportation delay and the time constant of the process. A pH sensor is located after the combination of direct and delayed paths to measure the pH of the effluent stream. If both paths are opened, the process converts to the output- state-delay system of (48). Fig. 5 depicts this process while Fig. 6 illustrates its schematic diagram. A Siemens SIMATIC® S7-300 PLC is installed for data acquisition which communicates with the computer through a PROFIBUS Fieldbus network. Some of the tunable parameters of the pH neutralization process plant are shown in Table III. The control objective is to achieve the desired pH values by manipulating the base flow rate in the presence of changes in the acid stream flow rate as a disturbance to evaluate the proposed control strategy.

Besides the advanced pH controller, there is a level controller to control the CSTR level by manipulating the buffer input flow. The block diagram of the system is illustrated in Fig. 7.

Fig. 5. Neutralization pH process plant.

Fig. 6. Schematic diagram of the pH neutralization process.

TABLE III Tunable Parameters of pH Process Pilot Plant

Fig. 7. The control loops in pH neutralization process.

In this article, the focus is on designing a controller to control pH value. Therefore, a PI controller with transfer functionCl(s)=0.2+20/sis used as the level controller to maintain the CSTR level ath= 11 cm [45].

B. Simulation Results

In this section, the stabilization strategy described in Corollary 2 is applied to the model of the pH neutralization process. A first-order dynamic process is constructed for the process using the sequential step responses method [46].Considering the combination of both direct and delayed output paths, the output-state-delay state-space model of the plant is as follows:

whereA= 0.97,B= 0.02,C= 1.

An integral term is added to the control loop to remove the steady-state error. The dynamic model of the augmented system is calculated similarly as stated in Remark 7. The state space representation of the closed-loop controlled system is reached as in (51) where system coefficient matrices are denoted in (55). Fig. 8 depicts the whole control structure.

The controller gainF, integrator gainHand stabilizing matricesP,QandZare obtained using Corollary 2 and Remark 8, where the non-monotonicity step ism= 2.

Fig. 8. pH neutralizing system controlled by output feedback with integrator controller.

In the mentioned iterative LMI-based nonlinear minimization algorithm, controller gainFand integrator gainHconverges in 30 and 5 iterations, respectively. The convergence procedure ofFandHare indicated in Figs. 9 and 10.

Fig. 9. Convergence procedure of the controller gain (F).

Fig. 10. The convergence procedure of the integrator gain (H).

The closed-loop responses using the designed controller are shown in Figs. 11 and 12. A zero-mean Gaussian white noise with variance 0.01 is added to the output of the process. Fig. 11 illustrates the reference tracking of the control system. The reference signal consists of 3 different steps around pH =4.5–5.5. The output of the system, which is the pH of the CSTR, follows the reference precisely. Fig. 12 demonstrates the control signal which is the base feed rate.

Fig. 11. Reference tracking in controlled pH neutralizing system.

Fig. 12. The control signal (Base feed rate).

The trend of the Lyapunov-Krasovskii functional candidate is depicted in Fig. 13, which shows the non-monotonic decreasing of the functional. Magnified part of the Fig. 13 emphasizes that incremental steps in the functional are less than non-monotonicity stepm= 2; it increases at step 8 and decreases afterward.

Fig. 13. NMLKF.

C. Experimental Results

The control of pH neutralizing process is a challenging problem that is fascinating to researchers in process control systems. It is extremely nonlinear, such that the static gain changes by a ratio of more than 10, even for a small change in pH value. To keep the process maintained in a less nonlinear operating area, the reference tracking is in pH value 4.5–5.5[45]. It also has a wide source of disturbances and uncertainty which makes it very difficult to control. A buffer is usually added to the process to control the level of the reactor. But, tap water is used in our process instead of buffer, which adds some disturbance to the control loop. One of the main sources of disturbances is the effect of the minerals in the tap water. While the pH of the water is usually considered to be around 7 in the models, it is measured to be around 7.8 during the experiment.In addition, acid and base solution are prepared manually. So,the concentration of materials inadvertently changes every time that they are prepared. Furthermore, acid neutralization will occur in the source tank in connection with air over time, and changes in room temperature also affect the reaction speed.Besides, the pH sensor has some measurement noise which affects the controller performance. These practical facts in a pH neutralization process make this apparatus one of the most challenging problems in the process control study.

In this subsection, the pH neutralization process is controlled using the proposed NMLK based stabilization method. The designed output feedback controller in the previous subsection using system model (55) is used for this plant. The output feedback controller gain and integrator gain are given in (57). The applied setpoint consists of 3 different steps around pH = 4.5–5.5. The measured pH value is the pH of the effluent stream in the CSTR, which is the combination of direct and delayed paths. As it is depicted in Fig. 14, the pH of the CSTR follows the reference. Although there are some random changes in the output response, which is due to noise and disturbances in the process, the designed controller using the proposed NMLK method controls the process with a practically overdamped response. Fig. 15 illustrates the control signal which is the base feed rate.

Fig. 14. Reference tracking in controlled pH neutralizing process.

Fig. 15. Control signal.

VI. Conclusion

A non-monotonic Lyapunov-Krasovskii theorem was proposed for stability analysis of state-delay time-delay systems. The proposed theorem is based on the principle that it is not necessary for Lyapunov-Krasovskii functional to be strictly decreasing all the times. Instead, Lyapunov-Krasovskii functional can increase over a few steps, while decreasing overall. In other words, the Lyapunov-Krasovskii functional is non-monotonically decreasing. In this regard, stability and stabilization theorems were presented. In the proposed NMLK based theorems, a non-monotonicity step is defined, which is the upper bound for admissible incremental steps in the NMLK functional. Increasing the non-monotonicity step can directly lessen conservatism at the cost of increasing computational cost. Numerical examples show that conservatism is reduced compared with other methods and the feasible space is enlarged. Stabilization theorem was used to design a stable output feedback controller for a pH neutralizing process. Experimental results illustrate the efficacy of the applied controller.