# Adaptive Neural Control and Modeling for Continuous Stirred Tank Reactor with Delays and Full State Constraints

Authors: Dongjuan Li; Dongxing Wang; Ying Gao
Journal: Complexity
 (2021)

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## Abstract

In this paper, an adaptive neural network control method is described to stabilize a continuous stirred tank reactor (CSTR) subject to unknown time-varying delays and full state constraints. The unknown time delay and state constraints problem of the concentration in the reactor seriously affect the input-output ratio and stability of the entire system. Therefore, the design difficulty of this control scheme is how to debar the effect of time delay in CSTR systems. To deal with time-varying delays, Lyapunov–Krasovskii functionals (LKFs) are utilized in the adaptive controller design. The convergence of the tracking error to a small compact set without violating the constraints can be identified by the time-varying logarithm barrier Lyapunov function (LBLF). Finally, the simulation results on CSTR are shown to reveal the validity of the developed control strategy.


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## Body

## 1. Introduction

To eliminate the effect of nonlinearities and uncertainties appearing in nonlinear systems, the fuzzy logic systems (FLSs) and neural networks (NNs) [1, 2] are useful tools, which avoid the requirement that nonlinearities must be known or can be linearly parameterized. Therefore, adaptive neural or fuzzy control methods are usually applied for the nonlinear single-input-single-output (SISO) systems in [3–8] and multi-input-multi-output (MIMO) systems in [9–12]. In recent years, more and more scholars have paid attention to the practical field. By employing neural networks or fuzzy logic systems, many stable control strategies for CSTR (a common chemical engineering reactor, usually with multiple reactors connected to each other for material conversion) have been caused widespread concern in [13–19] currently. A robust adaptive control method for the nonlinear CSTR system was proposed to ensure that the entire system remains stable in [20]. A novel controller design method was reported in [21] for a CSTR system. In practical CSTR systems, the concentration or temperature must be considered and controlled within certain limits to ensure the stability and safety of the system [22]. Hence, it is meaningful to consider the constraint problem in CSTRs. It is noteworthy that these above methods neglect the constraint limitations of operating spaces and safety specifications on system states.Constraints could be seen everywhere in practical systems, for instance, car suspension [23, 24], motor servo systems [25, 26], mobile manipulators [27], and chemical systems [28]; the violation of constraints is a main source of reducing control quality and even causing instability. Logarithm barrier Lyapunov functions were first proposed for constant output constraints in [29] to ensure that system output stays within the prescribed barrier functions. Afterwards, several constraint control methods, such as Logarithm-BLFs, Tangent-BLFs, and Integral-BLFs, were presented for output constraint in [30] and full state constraints in [31–34]. In recent years, novel state-dependent nonlinear-transformation-based state-constrained control schemes have been developed, see, for example, [31, 32]. The new solution that completely removes the feasibility conditions for virtual controllers is first proposed in [31] by constructing a nonlinear state-dependent function. Besides, the authors in [32] propose a unified barrier function upon the constrained states, with which the original constrained nonlinear system is transformed into an equivalent “nonconstrained” system. From a practical point of view, comparing with constant constraint, the time-varying constraint is more general. Hence, based on time-varying BLFs, some novel adaptive constraint controllers were designed, such as nonlinear strict-feedback systems and robotic manipulator with time-varying output constraint in [35, 36] and active suspension systems and nonlinear input saturated systems with time-varying state constraints in [37]. However, a major obstacle was that these control methods were proposed for delay-free systems.The time-delay control problem always attracts the attention of the researchers. Some relative strategies were developed for the treatment of several categories of nonlinear time delayed systems. Some new impulsive delay control methods were proposed for nonlinear impulsive delay differential systems in [38–40], and in order to achieve the system stability, the change rates of states and impulses were imposed. Employing appropriate LKFs, the robust adaptive backstepping control strategies in [41–43] were reported for nonlinear strict-feedback systems to debar time delays. Afterwards, based on [41–43], some adaptive NN and fuzzy controllers in [44–47] were further constructed to debar the effect of unknown time-varying delays appearing in several classes of nonlinear systems. In practical application, it must take times to transfer materials from one reactor to the next in the CSTR system, which creates time delay. In [48, 49], two adaptive state feedback controllers were designed for the CSTR system with unknown time delays.According to the above discussion, this study develops an adaptive NN control strategy for CSTRs with both time-varying delays and full time-varying state constraints. As the best of our knowledge, there are only few results solving the stability problem of such CSTRs in the existing literatures. The primary contributions are as follows:(1)
Under the adaptive control framework, the problem of time-varying delays and time-varying full state constraints is considered for CSTRs simultaneously, which is more in line with the needs of engineering systems. By using the time-varying barrier Lyapunov functions, all the states never violate the prescribed time-varying ranges.(2)
It is inevitable that there will be time delays in the transfer of substances in the reactor, resulting in the corresponding delay of the reaction time in the next stage. The time delayed functions are decomposed into a number of positive continuous functions based on the separation technique. With the help of the Lyapunov–Krasovskii functionals, these positive continuous functions with delayed states are eliminated.Finally, the construct integrity of the closed-loop nonlinear system is obtained by Lyapunov theory and the effectiveness of the proposed control scheme is proved by giving the simulation results.The paper is divided into the following parts. Certain preliminaries are assumed during Section2. Then, the adaptive NN control and stability analysis which are designed by backstepping technique are recommended during Section 3. Section 4 demonstrates certain experimental results. In Section 5, the conclusion is presented.


## 2. Preliminaries and Problem Formulation

We discuss the CSTR systems with two reactorsA and B, as shown in Figure 1 [45]:(1)C˙A=−ηACA−HACA,CAt−τAθA+1−RBCBDA+ξAt,CAt−τA,C˙B=−ηBCB−HBCB,CBt−τBθB+RACAt−τADB+RBCBt−τBDB+FutDB+ξBt,CBt−τB,where Ri denotes the recycle flow rate, θi is the reactor residence time, ηi stands for the reaction constant, F is the feed rate, Di is the reactor volume, Hi is the nonlinear function representing the complex behavior of the systems, ξi is nonlinear function for describing the system uncertainties and external disturbances, Ci denotes the reactor i concentration, and τi denotes time-varying delay, i=A,B.Figure 1
Continuously stirred tank reactor (CSTR) system.The control objective is to develop an adaptive neural controller to ensure that the concentrationsCA and CB of the producing reactors A and B never overstep the correspondent barrier functions, and the boundedness of all signals in the CSTR systems are obtained.Remark 1.
Comparing with the CSRT systems with only constant time delays in [44, 45], the considered systems in this paper with unknown time-varying delays are more important and general. In addition, to improve the control performance and system stability, full time-varying state constraints’ problem is also considered in the design process. Hence, the developed control approach is more responsive to actual engineering needs.
WhenξA=0 and ξB=0, the entire CSTR system achieves dynamic equilibrium; assume HA=CA+CAt−τ1 and HB=CB2t. We can get that the equilibrium points CA∗ and CB∗ satisfy(2)2θA+ηACA∗=1−RBDACB∗,ηBCB∗+1θBCB∗2=RADBCA∗+RBDBCB∗.
Lettingx1=CAt−CA∗ and x2=CBt−CB∗, the constraints of xi,i=1,2, can be regarded as the constraints of the concentrations CA and CB; we can further have(3)x˙1=−ηAx1−x1θA−x1t−τ1θA+x21−RBDA+ξAt,CAt−τ1,x˙2=−ηBx2−x22tθB+RAx1t−τ1DB−2CB∗x2tθB+RBx2t−τ2DB+FutDB+ξBt,CBt−τ2.
Letφ1=1−RB/DA and φ2=F/DB:(4)g1x1t=−ηAx1−x1θA,g2x2t=−ηBx2−x22tθB−2CB∗x2tθB,ξ¯At,x1t−τ1=ξAt,CAt−τ1,ξ¯Bt,x2t−τ1=ξBt,CBt−τ1,h1x1,τt=−x1t−τ1θA+ξ¯At,x1t−τ1,h2x¯2,τt=RAx1t−τ1DB+ξ¯Bt,x2t−τ1+RBx2t−τ2DB.
For brevity, thus, (3) can be rewritten as(5)x˙1=φ1x2t+g1x1t+h1x1,τt,x˙2=φ2ut+g2x2t+h2x¯2,τt,yt=x1t,where xi,i=1,2, are the state variables, u∈R and yt are control input and output, respectively, x¯2=x1,x2T, gi⋅ and hi⋅ are known and unknown smooth nonlinear functions, respectively, and gi0=hi0=0, φit is the continuous function, and τit is the unknown time-varying time delay satisfying τit≤τmax and τ˙it≤τ¯≤1, with τ¯ and τmax being two known constants. Here, xi≤kcit,i=1,2, with kcit are known functions.
A continuous functionfz can be approximated by the radial basis function neural networks (RBFNN) as(6)fz=WTIz,where W=W1,W2,…,WkT∈ℝk denote the adjustable weight vector, k expresses the number of neuron, and Iz=I1z,I2z,…,IkzT signify the basis function vector. There is a smooth vector function fz∈R and ideal weights W∗; hence, the smooth function fz can be approximated by the RBFNN as follows:(7)fz=W∗TIz+ϑz,∀z∈Ω∈ℝq.
We chooseW∗ as follows:(8)W∗≜argminsupfz−WTIz,in which the error ϑz fulfills ϑz≤ε within ε>0. During the thesis, undermentioned Gaussian basis function Iiz will be employed:(9)Iiz=exp−z−ιiTz−ιiwi2,where ιi=ιi1,ιi2,…,ιiqT depicts the center of the receptive field and wi represents the width of the Gaussian function within, i=1,2,…,k.Assumption 1 (see [35]).
There exist positive constantsKij,i=1,2andj=0,1,2, such that, for t>0, the functions kcit and the jth time derivative satisfy kcit≤Ki0 and kcijt≤Kij.Assumption 2.
The smooth functionshi,i=1,2, satisfy(10)hi2xit−τit≤z12t−τita12z¯1t−τit,where a1⋅ is a positive smooth function.Assumption 3 (see [32]).
There exist a positive functionY0t:R+⟶R+ and the constants Yi>0,i=1,2, so that the desired trajectory yd and its time derivatives satisfy ydt≤Y0tandydit≤Yi.Remark 2.
Assumption1 shows that the function and the jth time derivative less than or equal to positive constants. Assumptions 1–3 are used to prove the stability of the system; all the signals in the closed-loop system are bounded.Lemma 1 (see [23]).
For allς<1, such that the following inequality holds:(11)log11−ς2≤ς21−ς2.


## 3. The Controller Design and Stability Analysis

An adaptive NN control approach will be investigated in this section. It is used to solve the control problem of time delays and state constraints. The time variablet is omitted except for delay terms to improve the readability of this paper.Step 1: define the errorz1=x1 and a compact set Ωz=zi<kbi,i=1,2 with kbi,i=1,2, are known positive constants, and we have(12)z˙1=x˙1=φ1x2+g1x1+h1x1,τt.Consider the time-varying BLF candidate:(13)VD1=12logkb12kb12−z12+12W˜1TΓ1−1W˜1,whereΓ1 is a positive constant matrix satisfying Γ1=Γ1−1>0, W˜1=W^1−W1∗ represents the estimate error, and W^1 is the estimation weight of the optimal weight W1∗. We define(14)ψbi=zikbi,μi=1kbi2−zi2,i=1,2.Combining (13) and (14), one can obtain(15)V˙D1=ψb1kb11−ψb12z˙1−k˙b1ψb12kb11−ψb12+W˜1TΓ1−1W^˙1.Based on (12) and (15), we obtain(16)V˙D1=ψb1kb11−ψb12φ1x2+g1x1+W˜1TΓ1−1W^˙1+ψb1kb11−ψb12h1x1,τt−k˙b1ψb12kb11−ψb12.Using Young’s inequality, one has(17)ψb1kb11−ψb12h1x1,τt≤expτmax2cψb1kb11−ψb122+cexp−τmax2h12x1,τt,wherec=1−d¯.Define the unknown functionG1Z1 as(18)G1Z1=g1x1+1μ1H1+expτmax2cz1μ1,whereH1=z1a12z1+z1ψ211z1+ψ212z1/2 with a1⋅, ψ211⋅, and ψ212⋅ being continuous smooth functions and defined later.By substituting (17) and (18) into (16), meanwhile, x2=z2+α1, we can get the following inequality:(19)V˙D1≤μ1z1G1Z1+φ1z2+α1−k˙b1kb1z1−z1H1+W˜1TΓ1−1W^˙1+cexp−τmax2h12x1,τt.By using the NN approximation property, the continuous functionG1Z1 can be approximated as(20)G1Z1=W1∗TI1Z1+ϑ1Z1,whereI1Z1∈R indicates the Gaussian basis function and ϑ1Z1 indicates inherent approximation error of the NNs satisfying ϑ1Z1≤ϑ¯1 with ϑ¯1 being a positive constant; Z1=x1.The middle controller and the adaptive law are designed as(21)α1=1φ1−k1z1−K1z1−W^1TI1Z1−12γ1μ1z1−12φ1z1,(22)W^˙1=Γ1μ1z1I1Z1−σ1W^1,wherek1, γ1, and σ1 are the positive constants.DenoteKi=supk˙bi/kbi2+εi with εi>0 being a small constant.According to (20)–(22), (19) becomes(23)V˙D1≤μ1z1−k1z1−12γμ1z1+φ1z2−12φ1z1+ϑ1Z1−z1H1+cexp−τmax2h12x1,τt−σ1W˜1TW^1.By employing Young’s inequality, we can obtain(24)μ1z1ϑ1Z1≤12γ1μ12z12+γ12ϑ¯12,(25)μ1φ1z1z2≤μ1φ112z12+z22.Based on (24) and (25), (23) can be written as(26)V˙D1≤12μ1φ1z22−k1μ1z12+γ12ϑ¯12−σ1W˜1TW^1+cexp−τmax2h12x1,τt−z1H1.Step 2
: sincez2=x2−α1, its time derivative is given as follows:(27)z˙2=x˙2−α˙1=φ2u+g2x¯2+h2x¯2,τt−α˙1,where(28)α˙1=∂α1∂x1φ1x2+g1x1+h1x1,τt+∂α1∂W^1W^˙1+∑i=01∂α1∂kb12kb1i+1.The BLF candidate is given as(29)VD2=VD1+12logkb22kb22−z22+12W˜2TΓ2−1W˜2,where Γ2=Γ2−1>0 is a constant matrix and W˜2=W^2−W2∗ denotes the estimate error with W^2 being the estimation weight of W2∗.Based on (27), differentiating VD2, one obtains(30)V˙D2=V˙D1+ψb2kb21−ψb22φ2u+g2x¯2+h2x¯2,τt−∂α1∂x1φ1x2−∂α1∂x1g1x1−∂α1∂x1h1x1,τt−∂α1∂W^1W^˙1−∑i=01∂α1∂kb12kb1i+1−k˙b2ψb22kb21−ψb22+W˜2TΓ2−1W^˙2.By using Young’s inequality, it has(31)ψb2kb21−ψb22h2x¯2,τt≤expτmax2cψb2kb21−ψb222+cexp−τmax2h22x¯2,τt,(32)−ψb2kb21−ψb22∂α1∂x1h1x1,τt≤expτmax2cψb2kb21−ψb22∂α1∂x12+cexp−τmax2h12x1,τt.The unknown functionG2Z2 is concluded as(33)G2Z2=g2x¯2−∂α1∂x1g1x1+φ1x2−∂α1∂W^1W^˙1−∑i=01∂α1∂kb12kb1i+1+1μ2H2+expτmaxμ2z22c1+∂α1∂x12,where H2=z2q22z2/2 with q2⋅ being a continuous and smooth function.By using the NNs, the continuous unknown functionG2Z2 is approximated as(34)G2Z2=W2∗TI2Z2+ϑ2Z2,where Z2=x1,x2,W^1TT, I2Z2∈R indicates the Gaussian basis function, and ϑ2Z2 indicates inherent approximation error of the NNs satisfying ϑ2Z2≤ϑ¯2 with ϑ¯2 being a positive constant.The virtual controller and the adaptive law are given as(35)u=1φ2−k2z2−K2z2−W^2TI2Z2−γ22μ2z2−μ1φ1z22μ2,(36)W^˙2=Γ2μ2z2I2Z2−σ2W^2,where k2, γ2, and σ2 are the positive constants.By using Young’s inequality, it has(37)μ2z2ϑ2Z2≤γ22μ22z22+12γ2ϑ¯22.Along with (31)–(37), we obtain(38)V˙D2≤V˙D1−12μ1φ1z22−k2μ2z22+12γ2ϑ¯22−σ2W˜2TW^2−z2H2+∑i=12cexp−τmax2hi2x¯i,τt.In Step 1, we can obtain(39)V˙D1≤−k1μ1z12+12μ1φ1z22−z1H1+12γ1ϑ¯12−σ1W˜1TW^1+cexp−τmax2h12x1,τt.Hence, (38) becomes(40)V˙D2≤−∑i=12kiμizi2−∑i=12ziHi+∑i=1212γiϑ¯i2−∑i=12σiW˜iTW^i+cexp−τmaxh12x1,τt+12h22x¯2,τt.Based on Assumption1 in [45], the delay term in (40) becomes (41)hi2x¯i,τt≤q¯izit−τit+∑j=1i−1∑l=1iψ¯ijlzjt−τlt,where q¯i and ψ¯ijl represent positive definite and sufficiently smooth known functions with q¯ix=x2qix and ψ¯mjlx=x2ψmjlx with qix and ψmjlx being continuous and smooth functions.Noting the termσiW˜iTW^i, the following inequality can be obtained:(42)−σiW˜iTW^i=−σiW˜iTW˜i+Wi∗≤−12σiW˜i2+12σiWi∗2.Thus, (40) can be rewritten as(43)V˙D2≤−∑i=12kiμizi2−∑i=12ziHi+∑i=1212γiϑ¯i2+cexp−τmax22z12t−τ1tq1z1t−τ1t+z22t−τ2tq2z2t−τ2t+12∑i2σiWi∗2+z12t−τ1tψ211z1t−τ1t−12∑i2σiW˜i2+z12t−τ2tψ212z1t−τ2t.Choose the Lyapunov function candidate as(44)V=VD2+VL,in which, LKF VL is considered as(45)VL=12∑m=12∫t−τmtt3−mexps−tzm2sqmzmsds+12∑m=12∑j=1m−1∑l=1m∫t−τlttexps−tzj2sψmjlzjsds.The time derivative ofVL is given as(46)V˙L=12∑m=123−mzm2qmzm+12∑m=12∑j=1m−1∑l=1mzj2ψmjlzj−VL−∑m=12c3−m2exp−τmtzm2t−τmtqmzmt−τmt−c2∑m=12∑j=1m−1∑l=1mexp−τltzj2t−τltψmjlzjt−τlt.Notingτi≤τmax, according to (43) and (46), the time derivative of V can be obtained as(47)V˙≤−∑i=12kiμizi2+∑i=1212γiϑ¯i2−12∑i2σiW˜i2+12∑i2σiWi∗2+Λ1+Λ2+Λ3−VL,where(48)Λ1=cexp−τmax22z12t−τ1tq1z1t−τ1t+z22t−τ2tq2z2t−τ2t−cexp−τmax2×∑m=123−mzm2t−τmtqmzmt−τmt=0,Λ2=cexp−τmax2z12t−τ1tψ211z1t−τ1t+z12t−τ2tψ212z1t−τ2t−cexp−τmax2×∑m=12∑j=1m−1∑l=1mzj2t−τltψmjlzjt−τlt=0,Λ3=12∑m=123−mzm2qmzm+12∑m=12∑j=1m−1∑l=1mzj2ψmjlzj−∑i=12ziHi=0.So,V˙ further becomes(49)V˙≤−∑i=12kiμizi2−12∑i=12σiW˜i2+12∑i=12σiWi∗2+∑i=1212γiϑ¯i2−VL.With the help of Lemma1, we have(50)−∑i=12kiμizi2≤−∑i=12logkikbi2kbi2−zi2.Then, (49) is further shown as(51)V˙≤−∑i=12logkikbi2kbi2−zi2+∑i=1212γiϑ¯i2−12∑i=12σiW˜i2+12∑i=12σiWi∗2−VL.According to (13), (29), (44), and (45), one has(52)V=12∑i=12logkbi2kbi2−zi2+12∑i=12W˜iTΓi−1W˜i+12∑m=12∫t−τmtt3−mexps−tzm2samzmsds+12∑m=12∑j=1m−1∑l=1m∫t−τlttexps−tzj2sψmjlzjsds.Based on (51) and (52), the following inequality holds:(53)V˙≤−pV+q,where p=min2ki,σiλminΓi,1,i=1,2,(54)q=12∑i=12σiWi∗2+∑i=1212γiϑ¯i2.Theorem 1.
Consider the CSTR systems (1); if Assumptions 1–3 hold, the proposed virtual controller (21), the actual controller (35), and the adaptation laws (22) and (36) are utilized to ensure the boundedness of closed-loop system signals, and all the system states remain within the corresponding constraint set.Proof.
According to (53), one can obtain(55)Vt≤V0−qpe−pt+qp≤V0+qp.
From (13), we can get that every term is positive, so, the following term satisfies(56)12log11−ψi2≤Vt≤V0e−pt+qp,(57)12W˜iTΓi−1W˜i≤V0e−pt+qp.
From (56) and (57), the error signal is bounded:(58)−Δi≤zit≤Δi,where Δi=kbit1−e−2V0e−pt−2q/p1/2.
The adaptation laws can be obtained:(59)W˜i≤2λmaxΓiV0e−pt+qp.
Then, according to Assumptions1–3, it is known that k˙b1t≤Y1+K11 and ydt−Δ1≤x1t≤ydt+Δ1, so we can obtain −kc1t≤x1t≤kc1t. Due to the definition of α1, meanwhile, the boundedness of yd, kb1t, k˙b1t, and x1; it is easy to obtain α1 is bounded and x2=z2+α1 is bounded. From the definition in (35), it can be obtained that the controller u is bounded by the same way. Therefore, all the signals in the closed-loop system are bounded. The proof completes.Remark 3.
CSTRs’ system is a complex and nonlinear system. It has time-varying delayed states. This paper adopts the mechanism modeling method; according to the kinetic equation, by conservation of mass and conservation of energy, the nonlinear model equation is derived. The Lyapunov–Krasovskii functional is used to eliminate these continuous functions with the delayed state in this paper so that the CSTRs system with the delayed state is stable.
In the case of cascade reaction between two reaction reactors, there will be time delays in the transfer of substances in the reactor, which has a strong delay characteristic. At the same time, the CSTR system is nonlinear. In the reaction process, the reactant concentration and other parameters are easy to fluctuate with time-varying, and the temperature in the chemical reaction system must be controlled within a certain safety limit since its data are constrained. Therefore, the time delay and state constraints of the CSTR system should be considered when designing the controller.


## 4. Simulation Example

In this part, the simulation example is shown to demonstrate the effectiveness of the presented method for CSTR systems (1) with both time-varying full state constraints and time-varying delays.According to the considered CSTR systems, system statesx1 and x2 are constrained by −kc1t≤x1t≤kc1t and −kc2t≤x2t≤kc2t with kc1t=0.4e−2t+0.005 and kc2t=2e−2t+0.05. The equilibrium points are CA∗=14/9 and CB∗=7/3. The system parameters are chosen as ηA=ηB=0.5, θA=θB=2, RA=RB=0.5, DA=DB=0.5, F=0.5, ξAt,CAt−τ1t=0.5sintx1t−τ1t, and ξBt,CBt−τ2t=0.5×exp0.01x2t−τ2tcost×x22t−τ2t with time-varying delays τ1t=0.25−0.2sin0.5tcos2t and τ2t=0.55−0.5sin0.5t. The initial conditions are selected as x10=0.1, x20=0, W^1=W^1=0, Γ1=0.4diagones1,30, and Γ2=0.2diagones1,30. The design parameters are selected as k1=1.1, k2=12, γ1=0.5, γ2=2, and ε1=ε2=0.01.The simulation results are shown as Figures2–5 to certify the effectiveness of the presented control method. Figure 2 illustrates the responses of system state x1 and the corresponding constrained functions. In Figure 3, it obtains the trajectories of x2 and its constrained functions. It is worth stressing that system states x1 and x2 always stay within the prescribed compact set in Figures 2 and 3. From Figures 4 and 5, the boundedness of control input and adaptation laws are obtained.Figure 2
Responses ofx1 and its constrained functions.Figure 3
Response of system statex2 and its constrained functions.Figure 4
Responses of adaptation parametersW^1 and W^2.Figure 5
Response of control inputu.


## 5. Conclusion

The time delays and constraints are often occurring in the chemical reactor system, which also are main limitation factors of system performance severely. This paper has studied the tracking problem of the continuous stirred tank reactor containing the time-varying delays and full state constraints simultaneously. By constructing appropriate LKFs and LBLF, the effects of unknown time-varying delays were eliminated and the time-varying full states are never violated. By employing the proposed control method, we can make the conclusion that the tracking error can converge into a small set of zero and ensure all the signals in the system are bounded. The simulation results are proved to the rationality and effectiveness of the scheme. The future research directions should focus on the intergral BLF-based finite-time adaptive control for a CSTR system.



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