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Practical discontinuous tracking control for a permanent magnet synchronous motor


  • In this paper, the practical discontinuous control algorithm is used in the tracking controller design for a permanent magnet synchronous motor (PMSM). Although the theory of discontinuous control has been studied intensely, it is seldom applied to the actual systems, which encourages us to spread the discontinuous control algorithm to motor control. Due to the constraints of physical conditions, the input of the system is limited. Hence, we design the practical discontinuous control algorithm for PMSM with input saturation. To achieve the tracking control of PMSM, we define the error variables of the tracking control, and the sliding mode control method is introduced to complete the design of the discontinuous controller. Based on the Lyapunov stability theory, the error variables are guaranteed to converge to zero asymptotically, and the tracking control of the system is realized. Finally, the validity of the proposed control method is verified by a simulation example and the experimental platform.

    Citation: Bin Liu, Dengxiu Yu, Xing Zeng, Dianbiao Dong, Xinyi He, Xiaodi Li. Practical discontinuous tracking control for a permanent magnet synchronous motor[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 3793-3810. doi: 10.3934/mbe.2023178

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  • In this paper, the practical discontinuous control algorithm is used in the tracking controller design for a permanent magnet synchronous motor (PMSM). Although the theory of discontinuous control has been studied intensely, it is seldom applied to the actual systems, which encourages us to spread the discontinuous control algorithm to motor control. Due to the constraints of physical conditions, the input of the system is limited. Hence, we design the practical discontinuous control algorithm for PMSM with input saturation. To achieve the tracking control of PMSM, we define the error variables of the tracking control, and the sliding mode control method is introduced to complete the design of the discontinuous controller. Based on the Lyapunov stability theory, the error variables are guaranteed to converge to zero asymptotically, and the tracking control of the system is realized. Finally, the validity of the proposed control method is verified by a simulation example and the experimental platform.



    Dendrolimus is a forest pest with large occurrence and wide damage area. Among them, Dendrolimus superans is mainly distributed in Northeast China, Russia Far East, Japan, and so on. Dendrolimus superans are mainly parasitic on coniferous trees such as Larix gmelinii Kuzen and Pinus tabuliformis Carriere. Their larvae eat a large number of needles, resulting in trees being unable to carry out normal photosynthesis, eventually leading to a wide range of trees being destroyed or killed. Dendrolimus superans is extremely harmful to agriculture and forestry. In addition, the outbreak of Dendrolimus superans leads to tree death, reduces vegetation coverage, changes in plant population structure, and damages forest health and ecosystem stability. The life cycle of Dendrolimus superans consists of four stages: Egg, larva, pupa, and adult. In the life cycle of Dendrolimus superans, their development typically encompasses several key stages: Adult, egg, larva, and pupa, with each stage transitioning closely to the next. Specifically, adult insects will actively approach trees as a crucial behavior for seeking suitable environments to lay eggs. After the adults deposit their eggs on pine needles, the eggs will eventually hatch into larvae. During the larval stage, insects crawl on pine needles; this behavior is likely in search of food resources to meet the demands of their rapid growth and development. Subsequently, the larvae enter the pupal stage, during which they undergo a series of complex physiological and morphological changes. Finally, they metamorphose into adults through a process of transformation, thus completing a full life cycle. This process not only showcases the morphological and behavioral characteristics of insects at different life stages but also reflects their important strategies for adapting to the environment and reproducing. Among these stages, the larval stage is the most damaging to forests. The spread mechanism of the Dendrolimus superans is illustrated in Figure 1.

    Figure 1.  Spread mechanism of Dendrolimus superans infestation.

    There are three major methods for the prevention and control of Dendrolimus superans: Physical control, chemical control, and biological control. Among them, physical control is mainly aimed at the behavior patterns and characteristics of Dendrolimus superans. However, this method requires a lot of manpower and material resources, which is not suitable for large-scale prevention and control [1]. Although chemical control is effective for large-scale pests, the use of pesticides and other chemical reagents may cause environmental pollution. By comparison, biological control is an environmentally friendly and sustainable pest control technology. The use of natural biological control of pests reduces the use of chemical pesticides, maintains ecological balance, avoids pest resistance, and is harmless to human health. Therefore, it is suitable for long-term pest control. Dendrolimus superans mostly has two major natural enemies: Parasitic natural enemies and predatory natural enemies. Among the parasitic natural enemies, Trichogramma and Exorista civilis play an important role; in terms of predatory natural enemies, it is represented by Common Cuckoo, Cyanopica cyanus, and so on, as illustrated in Figure 2.

    Figure 2.  The variety of natural enemies that prey on Dendrolimus superans.

    When exploring the interaction between Dendrolimus superans and Cyanopica cyanus, we frequently employ the predator-prey model for analysis. This model is crucial for understanding and studying the dynamic changes in the predator-prey relationship between these two species within the realm of mathematical biology. In other studies, scholars have continuously refined and optimized these models [2,3,4,5,6,7]. The flight distance of adult female Dendrolimus superans is about 1.5 km, which shows that there are some limitations in their dispersal ability. Therefore, in the study of predator-prey model, the research environment can be regarded as relatively isolated. In such a relatively closed natural environment, scholars are concerned about the various factors that affect the system. Some biological studies have shown that habitat complexity affects population size and growth trends, indicating the important role of habitat complexity in the construction of ecological communities [8,9,10]. In addition, the construction of mixed forests provides habitat for natural enemies, increases environmental complexity, creates an ecological environment that is not conducive to the growth and development of Dendrolimus superans, and can effectively control the population of Dendrolimus superans [11,12]. Therefore, it is of great significance to study the biological control of Dendrolimus superans and their impact on forest ecosystems by using mathematical models. Based on the Lotka-Volterra model[13], Ma and Wang[14] proposed a predator-prey model with time delay and habitat complexity, that is

    {dudt=ru(1uK)c(1β)uαv1+ch(1β)uα,dvdt=ec(1β)uα(tτ)v(tτ)1+ch(1β)uα(tτ)dv, (1.1)

    where u and v describe the population of prey and predator, respectively. r is the intrinsic growth rate of prey. K is the capacity of environmental for prey. c is the attack rate of predator. β stands for the strength of habitat complexity. h is the handling time of predator. e is the conversion efficiency. τ is the gestation delay of predator. d is the death rate of predator. α is a positive real number. All parameters in model (1.1) are positive.

    The relationship between Dendrolimus superans and its natural enemies is typically characterized using the predator-prey model. The functional response, a key component in predator-prey models, is frequently employed to describe the predation ability of predators. Grounded in experimental findings, Holling[15] developed three functional responses, which are described below:

    I:mu,II:au1+mu,III:au21+mu2.

    In a specific area, we assume that the predation rate will reach saturation when the population of natural enemies is sufficiently large. This means that the predation rate will not increase further even if the number of natural enemies continues to grow. Conversely, when the population of natural enemies is low and begins to increase, the predation rate rises more rapidly than a linear function. This nonlinear increase is due to the fact that predators become more efficient at capturing prey as their numbers increase. A natural selection for such a predation rate is the Holling type-II functional response, which captures this nonlinear relationship between predator population and predation rate. In other words, the value of parameter α=1 in model (1.1).

    In a relatively closed natural environment, the spatial distribution of predators and prey significantly influences population dynamics, resource allocation, and predation efficiency. The nonlocal competition is in the interaction of predators and prey in different spatial locations, so that the system can predict the population dynamics and competition results more accurately. Peng and Zhang[16] investigated a predator-prey model incorporating collective behavior and nonlocal competition among prey. The findings indicated that nonlocal competition had a destabilizing effect on the predator-prey system.

    In addition to geographical distribution, ecological conditions, and other factors, time delay also plays an important role in the maintenance of ecological balance. In the predator-prey model, gestation delay is a key biological factor affecting the system, which is mainly reflected in the time difference between the occurrence of predation behavior and the birth of offspring. This physiological trait can significantly change the functional response pattern of predators to prey: During the breeding interval, the predator's predation frequency may decrease significantly or decrease to zero. This time delay mechanism directly affects the predator's growth rate by adjusting the number of effective predation per unit time and affects the dynamic balance of the ecosystem. The researchers in [17,18,19] developed a type of predator-prey model incorporating gestation delay and examined the dynamic characteristics of the systems.

    Inspired by the above, we incorporate the nonlocal competition and gestation delay into the model (1.1). In the ecological environment, spatial distribution is often inhomogeneous, and spatial diffusion often occurs within populations. Therefore, when we study predator-prey models, reaction-diffusion equations may be more realistic. Thus, the resulting system is given by:

    {u(x,t)t=d1Δu+ru(1˜uK)acuv1+achu,xΩ,t>0,v(x,t)t=d2Δv+gacu(x,tτ)v(x,tτ)1+ahcu(x,tτ)dv,xΩ,t>0,u(x,t)n=0,v(x,t)n=0,xΩ,t>0,u(x,t)=u0(x,t)0,v(x,t)=v0(x,t)0,(t,x)[τ,0]ׯΩ, (1.2)

    where u and v denote the density of Dendrolimus superans (prey) and Cyanopica cyanus (predator), respectively, the parameters r, K, h, c, and d have the same meanings as in model (1.1). d1 stands for the diffusion coefficients of Dendrolimus superans. d2 stands for the diffusion coefficients of Cyanopica cyanus. g denotes the efficiency with which energy is transferred from the species Dendrolimus superans to Cyanopica cyanus. a is the attack rate of predator on prey. τ is the gestation delay of Cyanopica cyanus. r, K, a, c, h, g, d, d1, d2 are a positive constant. We select Ω(0,lπ) where l>0. ˜u=1KΩG(x,y)u(y,t)dy represents the nonlocal competition, and the kernel function is given by G(x,y)=1|Ω|=1lπ, which is based on the assumption that the competition intensity among prey individuals in the habitat is uniform, meaning that the competition between any two preys is identical. The Neumann boundary condition is employed in this study, suggesting that the habitat is closed and effectively preventing any prey or predator from entering or leaving, thereby maintaining a self-contained ecosystem. We aim to investigate the dynamics of a predator-prey model, with a particular focus on the stability and dynamic properties of the system as the time delay parameter varies and serves as the bifurcation parameter.

    The structure of this paper is as follows: In Section 2, we analyze the stability of positive constant steady states and the existence of Hopf bifurcation. In Section 3, we investigate the normal form of the Hopf bifurcation. In Section 4, we present numerical simulations. Finally, conclusions are drawn in Section 5.

    System (1.2) has a boundary equilibria E0=(0,0) and a non-trivial equilibrium E1=(u,v), where

    u=d(ghd)ac,v=rg(ghd)ac(1uK).

    If the following assumption holds:

    (H0)g>hd,

    then the system (1.2) must have a positive constant steady equilibrium E1=(u,v).

    By defining U(x,t)=(u(x,t),v(x,t))T, the linearized system for Eq (1.2) can be re-expressed as a differential equation at the equilibrium E=(u0,v0),with(u0,v0)=(0,0)or(u,v):

    tU=D(Δu(x,t)Δv(x,t))+L1(u(x,t)v(x,t))+L2(u(x,tτ)v(x,tτ))+L3(˜u(x,t)˜v(x,t)),

    where

    D=(d100d2),L1=(a1a20d),L2=(00b1b2),L3=(c1000),

    with a1=r(1u0K)acv0(1+achu0)2,a2=acu01+achu0,b1=gacv0(1+achu0)2,b2=gacu01+achu0,c1=ru0K.

    Especially, when (u0,v0)=(u,v), a1=r(1uK)hdg>0,a2=dg<0,b1=r(ghd)(1uK)>0,b2=d>0,c1=ruK<0.

    Hence, the characteristic equation of (1.2) at E=(u0,v0) is given as follows:

    λ2+Anλ+(Bnλ+Cn)eλτ+Dn=0,n=0,1,2. (2.1)

    where

    {An=(nl)2(d1+d2)+da1δnc1,Bn=b2,Cn=(nl)2b2d1+a1b2a2b1+δnb2c1,Dn=(nl)4d1d2+(nl)2d1d(nl)2a1d2a1dδnc1d,

    with

    δn={1,n=0,0,n0.

    When τ=0, Eq (2.1) for equilibrium E0=(0,0) becomes

    λ2+[n2l2(d1+d2)+dr]λ+n4l4d1d2+n2l2d1dn2l2rd2rd=0,n=0,1,2. (2.2)

    For the case of n=0 in Eq (2.2), the product of the two eigenvalues rd is negative. Therefore, the equilibrium E0=(0,0) is always unstable.

    When τ=0, Eq (2.1) for equilibrium E1=(u,v) becomes

    λ2+(An+Bn)λ+Cn+Dn=0,n=0,1,2. (2.3)

    Subsequently, we examine the conditions under which habitat complexity guarantees the stability of a positive constant steady state in the system (1.2).

    An+Bn=(nl)2(d1+d2)a1. Due to d1+d2>0, there is An+Bn>A1+B1>A0+B0=(a1+c1). Assuming A0+B0=(a1+c1)>0, there is c<hd+gahK(ghd). So we have A1+B1=d1+d2l2a1>0d1+d2l2rhdg+rhd2gk(ghd)ac>0. Thus, if rhdl2(d1+d2)g>0, that is c<rhd2l2Ka(ghd)[rhdl2(d1+d2)g]. Otherwise, if rhdl2(d1+d2)g<0, that is c>0. Define:

    c0={rhd2l2Ka(ghd)[rhdl2(d1+d2)g],rhdl2(d1+d2)g>0,hd+gahK(ghd),rhdl2(d1+d2)g<0.

    Due to a1>0,a2<0,b1>0,b2>0,C0+D0=a2b1>0 holds. When n=0, a2b1>0, a12d10 and 1=(a1d2)2+4a2b1d1d2<0 c<dk(ghd)a(14d1(ghd)rh)1c, the function Cn+Dn=(nl)4d1d2(nl)2a1d2a2b1>0 holds for any nN.

    In summary, if c<min{hd+gahK(ghd),c0,c} holds, A0B0>0,AnBn>0,Cn+Dn>0, for nN holds. Thus, we make the following hypothesis

    (H1)c<min{hd+gahK(ghd),c0,c}.

    When (H1) holds, the roots of characteristic equation (2.3) have negative real parts.

    Theorem 1. For the model (1.2) with τ=0, the stability findings for equilibria are detailed below:

    1) Equilibrium E0=(0,0) is always unstable.

    2) Equilibrium E1=(u,v) is always locally asymptotically stable when (H0) and (H1) hold.

    3) Equilibrium E1=(u,v) is always unstable when (H0) or (H1) does not hold.

    Next, we investigate the existence of bifurcating periodic solutions near the positive constant steady state E1=(u,v) for τ0.

    When τ0, the characteristic equation describing the system (2.1) is presented as:

    λ2+Anλ+(Bnλ+Cn)eλτ+Dn=0,n=0,1,2. (2.4)

    We may assume that λ=±iω (ω>0) are a pair of purely imaginary roots of Eq (2.4). By substituting λ=±iω into Eq (2.4) and separating the real and imaginary components, we derive:

    {ω2+Dn+Cncos(ωτ)Bnωsin(ωτ)=0,ωAnCnsin(ωτ)Bnωcos(ωτ)=0,

    that is,

    {cos(ωτ)=ω2(AnBn+Cn)DnCnC2n+B2nω2,sin(ωτ)=ω(AnCn+BnDn+Bnω2)C2n+B2nω2, (2.5)

    with n=0,1,2. Then, for n=0,1,2,3 and j=0,1,2,3, we get

    τ(j)n={1ωnarccos(cosωτ)+2jπ,sinωτ0,1ωn[2πarccos(cosωτ)]+2jπ,sinωτ<0. (2.6)

    From (2.5), let z=ω2, we obtain:

    h(z)=z2+(A2n2DnB2n)z+D2nC2n=0. (2.7)

    Calculating the transversality condition, we get:

    Re(dλdτ)1|τ=τ(j)n=2z+(A2n2DnB2n)C2n+B2nω2n=h(z)C2n+B2nω2n.

    Denote

    τc=min{τ(0)nnN+} (2.8)

    and

    (H2)D2nC2n<0,
    (H3)A2n2DnB2n<0,D2nC2n>0,Δn>0,

    with Δn=(A2n2DnB2n)24(D2nC2n).

    We define the following set

    G1={nkjN+|j=1,2,3},
    G2={nktN+|t=1,2,3}.

    G1 satis (H2) and G2 satis (H3), respectively.

    When (H2) holds, Eq (2.7) has the unique positive root znk for some n=nkjG1. Therefore, we can solve for ωnk and τ(j)nk, where

    ωnk=12[[A2nk2DnkB2nk])+Δnk], (2.9)

    h(znk)>0,so we haveRe(dλdτ1)1|τ=τ(j)nk>0.

    When (H3) holds, Eq (2.7) has two positive root znt,i for some n=nktG2. Therefore, we can solve for ωnt,i and τ(j)nt,i,i=1,2, where

    ωnt,i=12[(A2nt,i2Dnt,iB2nt,)Δnt,i]. (2.10)

    Due to ωnt,1<ωnt,2, we have h(znt,1)<0 and h(znt,2)>0. Thus Re(dλdτ)1|τ=τ(j)nt,1<0, Re(dλdτ)1|τ=τ(j)nt,2>0 hold.

    In summary, we can get the following conclusions.

    Theorem 2. Under the framework of model (1.2) with τ0, when (H0) and (H1) are valid and τ(j)n is defined by Eq (2.6), the subsequent conclusions hold.

    1) Under the conditions G1= and G2=,Eq (2.7) has no positive roots, and the positive constant steady state E1 retains local asymptotic stability for all τ0.

    2) Under the conditions G1 and G2=,Eq (2.7) admits a unique positive root znk for some nkG1, the positive constant steady state E1 of system (1.2) exhibits asymptotic stability for 0<τ1<τc and unstable for τ1>τc, where τc=min{τ(0)nknkG1}. Additionally, system (1.2) undergoes nkmode Hopf bifurcation near E1 when τ=τ(j)nk(nkG1,j=0,1,2,3).

    3) Under the conditions G1= and G2,Eq (2.7) admits two positive roots znt,i(i=1,2) for some ntG2, the positive constant steady state E1 of system (1.2) exhibits asymptotic stability for 0<τ1<τc, but a stability switch may occur for τ1>τc, where τc=min{τ(0)ntntG1}. Furthermore, ntmode Hopf bifurcation emerges near E1 at τ=τ(j)nt(ntG1,j=0,1,2,3).

    4) Under the conditions G1 and G2,Eq (2.7) admits positive roots znk and znt,i(i=1,2) and letting np correspond to the smallest critical time delay τc, where npG1 or npG2, the positive constant steady state E1 of model (1.2) remains asymptotically stable for 0<τ1<τc, but a stability switch may arise for τ1>τc, where τc=min{τ(0)npnpG1ornpG2}. Furthermore, npmode Hopf bifurcation occurs near E1 at τ=τ(j)np(npG1ornpG2,j=0,1,2,3).

    In this section, we use the multiple time scales method to derive the normal form of the Hopf bifurcation for system (1.2).

    Suppose that the characteristic equation (2.4) has a pair of pure imaginary roots λ=±iω for τ=τc Then, system (1.2) undergoes n-mode Hopf bifurcation near the equilibrium point E1. Then, we derive the normal form of Hopf bifurcation of system (1.2) by the multiple time scales method. We choose time delay τ as a bifurcation parameter, where τ=τc+εμ. The parameter ε is a dimensionless scaling factor, μ is a perturbation parameter, and τc is given in Eq (2.8).

    We let u(x,t)u(x,t)u, v(x,t)v(x,t)v, α=ac, then model (1.2) can be rewritten as

    {u(x,t)t=d1Δu+u(rf1uK)ruK˜uf2v12f11u2f12uv16f111u312f112u2v,v(x,t)t=d2Δu+gf2v(x,tτ)+gf1u(x,tτ)+g2f11u2(x,tτ)+gf12u(x,tτ)v(x,tτ)+g6f111u3(x,tτ)+g2f112u2(x,tτ)v(x,tτ)dv, (3.1)

    where

    f1=αv(1+αhu)2,f2=αu1+αhu,f11=2α2hv(1+αhu)3+rK,f12=α(1+αhu)2,f111=6α3h2v(1+αhu)4,f112=2α2h(1+αhu)3.

    Let h=(h11,h12)T be the the eigenvector of the linear operator of system (3.1) corresponding to the eigenvalue λ=iω, and let h=(h11,h12)T be the normalized eigenvector of the adjoint operator of the linear operator of system (3.1) corresponding to the eigenvalue iω satisfying the inner product <h, h>=¯hTh=1. By a simple calculation, we get

    h=(h11,h12)T=(1,iω(nl)2d1+rf1uKuKδnf2),h=s(h21,h22)T=s(f2iω(nl)2d1+rf1uKuKδn,1), (3.2)

    with s=(¯h11h21+h22¯h12)1, and

    δn={1,n=0,0,n0.

    Suppose the solution of Eq (3.1) is

    U(x,t)=U(x,T0,T1,T2,)=+k=1εkUk(x,T0,T1,T2,), (3.3)

    where

    U(x,T0,T1,T2,)=(u(x,T0,T1,T2,),v(x,T0,T1,T2,))T,Uk(x,T0,T1,T2,)=(uk(x,T0,T1,T2,),vk(x,T0,T1,T2,))T.

    The derivation with respect to t is

    ddt=T0+εT1+ε2T2+=D0+εD1+ε2D2+,

    where the differential operator Di=Ti,i=0,1,2,.

    We obtain

    U(x,t)t=εD0U1+ε2D0U2+ε2D1U1+ε3D0U3+ε3D1U2+ε3D2U1+,ΔU(x,t)=εΔU1(x,t)+ε2ΔU2(x,t)+ε3ΔU3(x,t)+. (3.4)

    Denote

    uj=uj(x,T0,T1,T2,),vj=vj(x,T0,T1,T2,),uj,τc=uj(x,T0τc,T1,T2,),vj,τc=vj(x,T0τc,T1,T2,),

    with j=1,2,3,.

    We take perturbations as τ=τc+εμ to deal with the delayed terms. By expand U(x,tτ) at U(x,T0τc,T1,T2,···), respectively, that is,

    {u(x,tτ)=εu1,τc+ε2u2,τc+ε3u3,τcε2μD0u1,τcε3μD0u2,τcε2τcD1u1,τcε3μD1u1,τcε3τcD2u1,τcε3τcD1u2,τc+,v(x,tτ)=εv1,τc+ε2v2,τc+ε3v3,τcε2μD0v1,τcε3μD0v2,τcε2τcD1v1,τcε3μD1v1,τcε3τcD2v1,τcε3τcD1v2,τc+. (3.5)

    By substituting Eqs (3.3)–(3.5) into Eq (3.1), we obtain the expression for ε.

    {D0u1d1Δu1(rf1uK)u1+uK~u1+f2v1=0,D0v1d2Δv1gf2v1,τcgf1u1,τc+dv1=0. (3.6)

    The solution of Eq (3.6) can be written as follows:

    {u1=GeiωτcT0h11cos(nx)+c.c.,v1=GeiωτcT0h12cos(nx)+c.c., (3.7)

    where h11 and h12 are given in Eq (3.2), and c.c. means the complex conjugate of the preceding terms.

    For the ε2, we have

    {D0u2d1u2(rf1uK)u2+uK~u2+f2v2=D1u1~u1v1K12f11u21f12u1v1,D0v2d2v2gf2v2,τcgf1u2,τc+dv2=D1v1+gf2(τcD1v1,τc+μD0v1,τc)+gf1(τcD1u1,τc+μD0u1,τc)+g2f11u21,τc+gf12u1,τcv1,τc. (3.8)

    Substituting Eq (3.7) into the right side of Eq (3.8), we denote the coefficient vector of eiωT0 as m1, from solvability condition <h,(m1,cosnlx)>=0, we obtain

    GT1=μMG, (3.9)

    with

    M=iω(gf2h12+gf1h11)[iωn2l2d1+rf1uK(1δn)]f2h11+(h12+τcgf2h12+τcgf1h11)[iωn2l2d1+rf1uK(1δn)].

    Assume the solution of Eq (3.8) to be as follows:

    {u2=+k=0(η0kG¯G+η1kG2e2iωT0+¯η1k¯G2e2iωT0)cos(kxl),v2=+k=0(ς0kG¯G+ς1kG2e2iωT0+¯ς1k¯G2e2iωT0)cos(kxl). (3.10)

    Denote

    {ck=lπ0cos(nxl)cos(kxl)dx={lπ,k=n0,lπ2,k=n=0,0,kn.fk=lπ0cos2(nxl)cos(kxl)dx={lπ2,k=0,lπ4,k=2n0,0,k2n0.

    Substituting solutions Eqs (3.7) and (3.10) into Eq (3.8), we obtain

    η0k=X0kD0kY0kB0kA0kD0kC0kB0k,ζ0k=A0kY0kC0kX0kA0kD0kC0kB0k,η1k=X1kD1kY1kB1kA1kD1kC1kB1k,ζ1k=A1kY1kC1kX1kA1kD1kC1kB1k,

    where

    {A0k=[(kl)2d1(rf1uK)]lπ0cos2(klx)dx+uklπ[lπ0cos2(klx)dx]2,B0k=f2lπ0cos2(klx)dx,C0k=gf1lπ0cos2(klx)dx,D0k=(d1+d2gf2)lπ0cos2(klx)dx,X0k=[(h11¯h12+h12¯h11)f2+f11h11¯h11+(h11¯h12+h11h12)]fk,Y0k=g[f11h11¯h11+f12(h12¯h11+h11¯h12)]fk,A1k=[2iω+(kl)2d1(rf1uK)]lπ0cos2(klx)dx+uklπ[lπ0cos2(klx)dx]2,B1k=f2lπ0cos2(klx)dx,C1k=gf1lπ0cos2(klx)dx,D1k=[2iω+d2(kl)2gf2+d1]lπ0cos2(klx)dx,X1k=h11h12ck(12f11h211+f2h11h12)fk,Y1k=(g2f11h211+gf12h12h11)fk.

    For the ε3, we obtain

    {D0u3d1u3(rf1u3K)u3+u3K~u3+f2v3+16f111u33=D2u1D1u2~u1v2+~u2v1Kf11(u1u2+u2v1)16f111u3112f112u21v1+o(μ),D0v3d2v3+gf2v3,τc+dv3=D1u2D2u1gf2τc(D2v1,τc+D1v2,τc)gf1τc(D2u1,τc+D1u2,τc)+gf11(u2,τcu1,τcu1D1u1,τcτc)++g6f11u31,τc+g2f11u21,τ1v1,τc+gf12[u1,τcv2,τc+u2,τcv1,τcτc(D1u1,τcv1,τc+D1v1,τcv1,τc)]+o(μ). (3.11)

    Substituting Eqs (3.7) and (3.10) into the right side of Eq (3.11), we denote the coefficient vector of eiωT0 as m2, by solvability condition <h,(m2,cosnlx)>=0, we obtain

    GT2=XG2¯G, (3.12)

    where

    X=σφ, (3.13)

    with

    {σ=k=01klπ(η0kh11+η1k¯h11)fk+k=0(f11η0kh11+f11η1k¯h11+f12ς0kh12+f12ς1k¯h12)ck12[(1g)f111h211¯h11+(f112gf111)(h211¯h12+2h11¯h11h12)]lπ0cos3(nlx)dx+k=0g[f11(η0kh11+η1k¯h11)+f12(ς0kh11+ς1k¯h11+η0kh12+η1k¯h12)]ck,φ=[2h11+gτc(f2h12+f1h11)]lπ0cos2(nlx)dx.

    According to the above analysis, the normal form of Hopf bifurcation for system (1.2) reduced on the center manifold is

    ˙G=MμG+XG2¯G, (3.14)

    where M and X are given by Eqs (3.9) and (3.13), respectively.

    Let G=reiθ and substitute it into Eq (3.14), and we obtain the Hopf bifurcation normal form in polar coordinates:

    {˙r=Re(M)μr+Re(X)r3,˙θ=Im(M)μ+Im(X)r2. (3.15)

    Therefore, we arrive at the following theorem.

    Theorem 3. For system (3.15), if Re(M)μRe(X)<0 holds, model (1.2) exists periodic solutions near equilibrium E1=(u,v).

    1) If Re(M)μ<0 the bifurcating periodic solutions reduced on the center manifold are unstable, and the direction of bifurcation is forward (backward) for μ>0(μ<0).

    2) If Re(M)μ>0 the bifurcating periodic solutions reduced on the center manifold are stable, and the direction of bifurcation is forward (backward) for μ>0(μ<0).

    In this section, we perform numerical simulations to verify the correctness of the theoretical analysis. In summary, we choose d=0.3,g=0.6,a=0.6,d1=0.01,d2=0.2,c=0.55, r=0.41, K=5, h=0.41, and l=1.

    Based on the data above, (H0) and (H1) always hold, and system (1.2) has one unstable boundary equilibrium E0=(0,0), and one nontrivial equilibrium E1=(u,v)=(1.90585,0.96710). What matters most to us is the stability of the nontrivial equilibrium E1.

    According to Theorem 1, then E1 is local asymptotically stable. This means that if system (1.2) is without delay, although Cyanopica cyanus and Dendrolimus superans can coexist at this time, natural predators are capable of suppressing the reproduction of the pests.

    When τ0, by a simple calculation, (H2) only holds for n=1 there does not exist any n such that (H3) holds. Thus, Eq (2.9) has a unique positive root ω=0.1616, which corresponds to the critical delay τ(0)1=2.5897. From the definition of τc, we obtain τc=2.5897, theorem 2.2 supports the conclusion that the positive constant steady state E1 achieves local asymptotic stability when τ[0,2.5897), and instability when τ(2.5897,+). We select τ=1.5[0,2.5897) to conduct the simulation. See Figure 3.

    Figure 3.  When τ=1.5, the simulation of system (1.2) reveals that the equilibrium E1 is locally asymptotically stable.

    We choose τ=2.8>τc=2.5879, from Eqs (3.9) and (3.12), we obtain Re(M)>0, Re(X)<0. Thus, according to Theorem 3, system (1.2) will generate inhomogeneous periodic solutions near the positive constant steady state E1 of model (1.2), and bifurcating periodic solutions are stable and forward (see Figure 4).

    Figure 4.  When τ=2.8, system (1.2) produces stable inhomogeneous forward periodic solutions near E1.

    By analyzing Figures 3 and 4, we can draw the following conclusions.

    1) When the gestation delay of the predator is less than the critical value, the predator can quickly control the number of pests and reach a stable state by preying on pests.

    2) When the gestation delay of the predator is slightly more than the critical value, the predator has a certain control effect on the number of pests. This will cause the number of pest populations to erupt periodically, and the number of natural enemy populations will fluctuate periodically.

    3) Comparing Figures 3 and 4, when the gestation delay of insectivorous birds is long, so it is necessary to artificially intervene in the population of pests to prevent the outbreak of insect pests. By releasing insectivorous birds to increase their number, we can indirectly reduce the increase in the number of pests and the outbreak of pests due to the prolonged pregnancy of predators, so that the population of pests can return to a controllable stable state.

    In this paper, aiming at the population density control problem of the pine caterpillar, considering the Holling-II type functional response, we developed a pest control model with gestation delay and nonlocal competition. For our analysis, we focused on two major aspects: first, the existence and stability of the positive constant steady state, and second, the conditions under which Hopf bifurcations emerge near this steady state. In the part of numerical simulation, we selected a set of suitable parameters for numerical simulation. It provided an explanation for effectively controlling the population density of the pine caterpillar and maintained the stability between natural enemies and the pine caterpillar species, so as to realize effective environmental protection and true green pest control. Therefore, gestation delay has a significant impact on the stability of the population. When the gestation delay is less than the critical delay, the predator can effectively control the pest population. When the gestation delay exceeds the critical delay, the pest population experiences periodic outbreaks.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This study was funded by the Heilongjiang Provincial Natural Science Foundation of China (Grant No. LH2024A001), and the College Students Innovations Special Project funded by Northeast Forestry University (No. DCLXY-2025011).

    The authors declare there are no conflicts of interest.



    [1] D. Yu, J. Long, C. L. P. Chen, Z. Wang, Adaptive swarm control within saturated input based on nonlinear coupling degree, IEEE Trans. Syst. Man Cybern.: Syst., 52 (2022), 4900–4911. https://doi.org/10.1109/TSMC.2021.3102587 doi: 10.1109/TSMC.2021.3102587
    [2] H. Xu, D. Yu, S. Sui, Y. P. Zhao, C. L. P. Chen, Z. Wang, Nonsingular practical fixed-time adaptive output feedback control of mimo nonlinear systems, IEEE Trans. Neural Networks Learn. Syst., (2022), 1–13. https://doi.org/10.1109/TNNLS.2021.3139230 doi: 10.1109/TNNLS.2021.3139230
    [3] T. Li, X. Sun, G. Lei, Z. Yang, Y. Guo, J. Zhu, Finite-control-set model predictive control of permanent magnet synchronous motor drive systems–an overview, IEEE-CAA J. Autom. Sin., 9 (2022), 2087–2105. https://doi.org/10.1109/JAS.2022.105851 doi: 10.1109/JAS.2022.105851
    [4] T. Zwerger, P. Mercorelli, Using a bivariate polynomial in an ekf for state and inductance estimations in the presence of saturation effects to adaptively control a pmsm, IEEE Access, 10 (2022), 111545–111553. https://doi.org/10.1109/ACCESS.2022.3215511 doi: 10.1109/ACCESS.2022.3215511
    [5] Z. Li, T. Li, G. Feng, R. Zhao, Q. Shan, Neural network-based adaptive control for pure-feedback stochastic nonlinear systems with time-varying delays and dead-zone input, IEEE Trans. Syst. Man Cybern.: Syst., 50 (2020), 5317–5329. https://doi.org/10.1109/TSMC.2018.2872421 doi: 10.1109/TSMC.2018.2872421
    [6] Y. X. Li, G. H. Yang, Observer-based fuzzy adaptive event-triggered control codesign for a class of uncertain nonlinear systems, IEEE Trans. Fuzzy Syst., 26 (2018), 1589–1599. https://doi.org/10.1109/TFUZZ.2017.2735944 doi: 10.1109/TFUZZ.2017.2735944
    [7] X. Sun, T. Li, Z. Zhu, G. Lei, Y. Guo, J. Zhu, Speed sensorless model predictive current control based on finite position set for pmshm drives, IEEE Trans. Transp. Electrif., 7 (2021), 2743–2752. https://doi.org/10.1109/TTE.2021.3081436 doi: 10.1109/TTE.2021.3081436
    [8] T. Zwerger, P. Mercorelli, Combining a pi controller with an adaptive feedforward control in pmsm, in 2020 21th International Carpathian Control Conference (ICCC), (2020), 1–5. https://doi.org/10.1109/ICCC49264.2020.9257288
    [9] Y. Li, Y. Liu, S. Tong, Observer-based neuro-adaptive optimized control of strict-feedback nonlinear systems with state constraints, IEEE Trans. Neural Networks Learn. Syst., 33 (2022), 3131–3145. https://doi.org/10.1109/TNNLS.2021.3051030 doi: 10.1109/TNNLS.2021.3051030
    [10] X. Xie, T. Wei, X. Li, Hybrid event-triggered approach for quasi-consensus of uncertain multi-agent systems with impulsive protocols, IEEE Trans. Circuits Syst. I-Regul. Pap., 69 (2022), 872–883. https://doi.org/10.1109/TCSI.2021.3119065 doi: 10.1109/TCSI.2021.3119065
    [11] J. Wei, S. Zhang, A. Adaldo, J. Thunberg, X. Hu, K. H. Johansson, Finite-time attitude synchronization with distributed discontinuous protocols, IEEE Trans. Autom. Control, 63 (2018), 3608–3615. https://doi.org/10.1109/TAC.2018.2797179 doi: 10.1109/TAC.2018.2797179
    [12] Z. Li, L. Chen, Z. Liu, Periodic solution of a chemostat model with variable yield and impulsive state feedback control, Appl. Math. Model., 36 (2012), 1255–1266. https://doi.org/10.1016/j.apm.2011.07.069 doi: 10.1016/j.apm.2011.07.069
    [13] W. Zhu, D. Wang, L. Liu, G. Feng, Event-based impulsive control of continuous-time dynamic systems and its application to synchronization of memristive neural networks, IEEE Trans. Neural Networks Learn. Syst., 29 (2018), 3599–3609. https://doi.org/10.1109/TNNLS.2017.2731865 doi: 10.1109/TNNLS.2017.2731865
    [14] X. Li, D. Peng, J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Autom. Control, 65 (2020), 4908–4913. https://doi.org/10.1109/TAC.2020.2964558 doi: 10.1109/TAC.2020.2964558
    [15] X. Li, X. Yang, J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica, 117 (2020), 108981. https://doi.org/10.1016/j.automatica.2020.108981 doi: 10.1016/j.automatica.2020.108981
    [16] X. Tan, J. Cao, Intermittent control with double event-driven for leader-following synchronization in complex networks, Appl. Math. Model., 64 (2018), 372–385. https://doi.org/10.1016/j.apm.2018.07.040 doi: 10.1016/j.apm.2018.07.040
    [17] Z. Wang, C. Mu, S. Hu, C. Chu, X. Li, Modelling the dynamics of regret minimization in large agent populations: a master equation approach, in Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence (IJCAI-22), 23 (2022), 534–540. https://doi.org/10.24963/ijcai.2022/76
    [18] Y. Yang, Y. He, Non-fragile observer-based robust control for uncertain systems via aperiodically intermittent control, Inf. Sci., 573 (2021), 239–261. https://doi.org/10.1016/j.ins.2021.05.046 doi: 10.1016/j.ins.2021.05.046
    [19] S. Chen, G. Song, B. C. Zheng, T. Li, Finite-time synchronization of coupled reaction–diffusion neural systems via intermittent control, Automatica, 109 (2019), 108564. https://doi.org/10.1016/j.automatica.2019.108564 doi: 10.1016/j.automatica.2019.108564
    [20] Y. Wu, H. Li, W. Li, Intermittent control strategy for synchronization analysis of time-varying complex dynamical networks, IEEE Trans. Syst. Man Cybern.: Syst., 51 (2021), 3251–3262. https://doi.org/10.1109/TSMC.2019.2920451 doi: 10.1109/TSMC.2019.2920451
    [21] B. Wang, W. Chen, B. Zhang, Semi-global robust tracking consensus for multi-agent uncertain systems with input saturation via metamorphic low-gain feedback, Automatica, 103 (2019), 363–373. https://doi.org/10.1016/j.automatica.2019.02.002 doi: 10.1016/j.automatica.2019.02.002
    [22] V. T. Do, S. G. Lee, Neural integral backstepping hierarchical sliding mode control for a ridable ballbot under uncertainties and input saturation, IEEE Trans. Syst. Man Cybern.: Syst., 51 (2021), 7214–7227. https://doi.org/10.1109/TSMC.2020.2967433 doi: 10.1109/TSMC.2020.2967433
    [23] X. Yang, B. Zhou, F. Mazenc, J. Lam, Global stabilization of discrete-time linear systems subject to input saturation and time delay, IEEE Trans. Autom. Control, 66 (2021), 1345–1352. https://doi.org/10.1109/TAC.2020.2989791 doi: 10.1109/TAC.2020.2989791
    [24] Y. Su, Q. Wang, C. Sun, Self-triggered consensus control for linear multi-agent systems with input saturation, IEEE-CAA J. Autom. Sin., 7 (2020), 150–157. https://doi.org/10.1109/JAS.2019.1911837 doi: 10.1109/JAS.2019.1911837
    [25] C. Behn, K. Siedler, Adaptive pid-tracking control of muscle-like actuated compliant robotic systems with input constraints, Appl. Math. Model., 67 (2019), 9–21. https://doi.org/10.1016/j.apm.2018.10.012 doi: 10.1016/j.apm.2018.10.012
    [26] Q. Zhu, Y. Liu, G. Wen, Adaptive neural network control for time-varying state constrained nonlinear stochastic systems with input saturation, Inf. Sci., 527 (2020), 191–209. https://doi.org/10.1016/j.ins.2020.03.055 doi: 10.1016/j.ins.2020.03.055
    [27] Y. Wu, X. J. Xie, Adaptive fuzzy control for high-order nonlinear time-delay systems with full-state constraints and input saturation, IEEE Trans. Fuzzy Syst., 28 (2020), 1652–1663. https://doi.org/10.1109/TFUZZ.2019.2920808 doi: 10.1109/TFUZZ.2019.2920808
    [28] D. Yu, J. Long, C. L. P. Chen, Z. Wang, Bionic tracking-containment control based on smooth transition in communication, Inf. Sci., 587 (2022), 393–407. https://doi.org/10.1016/j.ins.2021.12.060 doi: 10.1016/j.ins.2021.12.060
    [29] H. Xu, D. Yu, S. Sui, C. L. P. Chen, An event-triggered predefined time decentralized output feedback fuzzy adaptive control method for interconnected systems, IEEE Trans. Fuzzy Syst., (2022), 1–14. https://doi.org/10.1109/TFUZZ.2022.3184834 doi: 10.1109/TFUZZ.2022.3184834
    [30] T. Zwerger, P. Mercorelli, Combining smc and mtpa using an ekf to estimate parameters and states of an interior pmsm, in 2019 20th International Carpathian Control Conference (ICCC), (2019), 1–6. https://doi.org/10.1109/CarpathianCC.2019.8766063
    [31] D. Yu, C. L. P. Chen, H. Xu, Fuzzy swarm control based on sliding-mode strategy with self-organized omnidirectional mobile robots system, IEEE Trans. Syst. Man Cybern.: Syst., 52 (2022), 2262–2274. https://doi.org/10.1109/TSMC.2020.3048733 doi: 10.1109/TSMC.2020.3048733
    [32] D. Shang, X. Li, M. Yin, F. Li, Dynamic modeling and fuzzy compensation sliding mode control for flexible manipulator servo system, Appl. Math. Model., 107 (2022), 530–556. https://doi.org/10.1016/j.apm.2022.02.035 doi: 10.1016/j.apm.2022.02.035
    [33] N. Zhang, W. Qi, G. Pang, J. Cheng, K. Shi, Observer-based sliding mode control for fuzzy stochastic switching systems with deception attacks, Appl. Math. Comput., 427 (2022), 127153. https://doi.org/10.1016/j.amc.2022.127153 doi: 10.1016/j.amc.2022.127153
    [34] W. H. Chen, X. Deng, W. X. Zheng, Sliding-mode control for linear uncertain systems with impulse effects via switching gains, IEEE Trans. Autom. Control, 67 (2022), 2044–2051. https://doi.org/10.1109/TAC.2021.3073099 doi: 10.1109/TAC.2021.3073099
    [35] L. Y. Hao, J. H. Park, D. Ye, Integral sliding mode fault-tolerant control for uncertain linear systems over networks with signals quantization, IEEE Trans. Neural Networks Learn. Syst., 28 (2017), 2088–2100. https://doi.org/10.1109/TNNLS.2016.2574905 doi: 10.1109/TNNLS.2016.2574905
    [36] A. Vahidi-Moghaddam, A. Rajaei, M. Ayati, Disturbance-observer-based fuzzy terminal sliding mode control for mimo uncertain nonlinear systems, Appl. Math. Model., 70 (2019), 109–127. https://doi.org/10.1016/j.apm.2019.01.010 doi: 10.1016/j.apm.2019.01.010
    [37] H. Xu, S. Li, D. Yu, C. Chen, T. Li., Adaptive swarm control for high-order self-organized system with unknown heterogeneous nonlinear dynamics and unmeasured states, Neurocomputing, 440 (2021), 24–35. https://doi.org/10.1016/j.neucom.2021.01.069 doi: 10.1016/j.neucom.2021.01.069
    [38] B. Jiang, H. R. Karimi, Y. Kao, C. Gao, Takagi–sugeno model based event-triggered fuzzy sliding-mode control of networked control systems with semi-markovian switchings, IEEE Trans. Fuzzy Syst., 28 (2020), 673–683. https://doi.org/10.1109/TFUZZ.2019.2914005 doi: 10.1109/TFUZZ.2019.2914005
    [39] G. Wang, J. Kuang, N. Zhao, G. Zhang, D. Xu, Rotor position estimation of pmsm in low-speed region and standstill using zero-voltage vector injection, IEEE Trans. Power Electron., 33 (2018), 7948–7958. https://doi.org/10.1109/TPEL.2017.2767294 doi: 10.1109/TPEL.2017.2767294
    [40] A. Kolli, O. Béthoux, A. D. Bernardinis, E. Labouré, G. Coquery, Space-vector pwm control synthesis for an h-bridge drive in electric vehicles, IEEE Trans. Veh. Technol., 62 (2013), 2441–2452. https://doi.org/10.1109/TVT.2013.2246202 doi: 10.1109/TVT.2013.2246202
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