Research article

A new design method to global asymptotic stabilization of strict-feedforward nonlinear systems with state and input delays

  • This paper studies the global asymptotic stabilization problem of strict-feedforward nonlinear systems with state and input delays. We will first transform the considered system into an equivalent system by constructing the novel parameter-dependent state feedback controller and introducing the appropriate coordinate transformation. After that, the global asymptotic stability of the closed system is proved by giving the proper Lyapunov-Krasovskii functional and using the stability criterion of time-delay system.

    Citation: Mengmeng Jiang, Xiao Niu. A new design method to global asymptotic stabilization of strict-feedforward nonlinear systems with state and input delays[J]. AIMS Mathematics, 2024, 9(4): 9494-9507. doi: 10.3934/math.2024463

    Related Papers:

    [1] M. Mossa Al-Sawalha, Saima Noor, Saleh Alshammari, Abdul Hamid Ganie, Ahmad Shafee . Analytical insights into solitary wave solutions of the fractional Estevez-Mansfield-Clarkson equation. AIMS Mathematics, 2024, 9(6): 13589-13606. doi: 10.3934/math.2024663
    [2] Naher Mohammed A. Alsafri . Solitonic behaviors in the coupled Drinfeld-Sokolov-Wilson system with fractional dynamics. AIMS Mathematics, 2025, 10(3): 4747-4774. doi: 10.3934/math.2025218
    [3] Mohammad Alqudah, Safyan Mukhtar, Albandari W. Alrowaily, Sherif. M. E. Ismaeel, S. A. El-Tantawy, Fazal Ghani . Breather patterns and other soliton dynamics in (2+1)-dimensional conformable Broer-Kaup-Kupershmit system. AIMS Mathematics, 2024, 9(6): 13712-13749. doi: 10.3934/math.2024669
    [4] Mahmoud Soliman, Hamdy M. Ahmed, Niveen Badra, Taher A. Nofal, Islam Samir . Highly dispersive gap solitons for conformable fractional model in optical fibers with dispersive reflectivity solutions using the modified extended direct algebraic method. AIMS Mathematics, 2024, 9(9): 25205-25222. doi: 10.3934/math.20241229
    [5] Chun Huang, Zhao Li . New soliton solutions of the conformal time derivative generalized q-deformed sinh-Gordon equation. AIMS Mathematics, 2024, 9(2): 4194-4204. doi: 10.3934/math.2024206
    [6] Yanlin Li, Mohd Danish Siddiqi, Meraj Ali Khan, Ibrahim Al-Dayel, Maged Zakaria Youssef . Solitonic effect on relativistic string cloud spacetime attached with strange quark matter. AIMS Mathematics, 2024, 9(6): 14487-14503. doi: 10.3934/math.2024704
    [7] Gulnur Yel, Haci Mehmet Baskonus, Wei Gao . New dark-bright soliton in the shallow water wave model. AIMS Mathematics, 2020, 5(4): 4027-4044. doi: 10.3934/math.2020259
    [8] M. Mossa Al-Sawalha, Safyan Mukhtar, Azzh Saad Alshehry, Mohammad Alqudah, Musaad S. Aldhabani . Kink soliton phenomena of fractional conformable Kairat equations. AIMS Mathematics, 2025, 10(2): 2808-2828. doi: 10.3934/math.2025131
    [9] Yanlin Li, Dipen Ganguly, Santu Dey, Arindam Bhattacharyya . Conformal η-Ricci solitons within the framework of indefinite Kenmotsu manifolds. AIMS Mathematics, 2022, 7(4): 5408-5430. doi: 10.3934/math.2022300
    [10] Chun Huang, Zhao Li . Soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation. AIMS Mathematics, 2022, 7(8): 14460-14473. doi: 10.3934/math.2022797
  • This paper studies the global asymptotic stabilization problem of strict-feedforward nonlinear systems with state and input delays. We will first transform the considered system into an equivalent system by constructing the novel parameter-dependent state feedback controller and introducing the appropriate coordinate transformation. After that, the global asymptotic stability of the closed system is proved by giving the proper Lyapunov-Krasovskii functional and using the stability criterion of time-delay system.



    It is known that circular reflections on the unit circle leads to the computation of the digits of the trigonometric constant π [1,2]. A similar analysis shows that hyperbolic reflections on the unit hyperbola leads to the computation of the digits of the logarithmic constant ln(2) [3]. Looking at those two situations, we obtain a unified way to present an unusual alternating reflection method on conics to evaluate inverse trigonometric and hyperbolic functions. A lot of research has been done on the efficient evaluation of elementary functions, see for example [4,5,6]. The method we obtain is much more a curiosity than an efficient practical method, but since it uses only an elementary geometric operation (reflection), its presentation can be interesting.

    In Section 2, we introduce parametrization of the conics and express reflections and rotations in terms of the parameter. The numerical method is presented in Section 3. The way to compute the digits of the solution of the problem is explained in Section 4. In Section 5 we present some examples to illustrate the unusual methods we obtained to evaluate inverse trigonometric and inverse hyperbolic functions.

    Let G=(100δ), where δ=±1, so G2=I, and

    S={(x,y)R2 | (x,y)G(xy)=x2+δy2=1, (1δ)x0 }.

    So S is the unit circle for δ=+1, and S is the right branch (x0) of the unit hyperbola for δ=1.

    Suppose we have two regular functions c() and s() defined in such a way that to every point (x,y)S there exist θ such that (x,y)=(c(θ),s(θ)), so

    c2(θ)+δs2(θ)=1.

    It is clear here that for δ=+1 we can take

    (c(θ),s(θ))=(cos(θ),sin(θ)),

    and for δ=1 we can take

    (c(θ),s(θ))=(cosh(θ),sinh(θ)).

    For a given point (c(),s())S we observe that

    (c(θ),s(θ))G(δs(θ)c(θ))=0,

    we say that (δs(θ),c(θ)) is G-orthogonal to (c(),s()). From the regularity of c() and s() we get that

    0=c(θ)c(θ)+δs(θ)s(θ)=(c(θ),s(θ))G(c(θ)s(θ)),

    and since (c(θ),s(θ)) and (δs(θ),c(θ)) are both G-orthogonal to (c(θ),s(θ)), we can suppose that

    (c(θ),s(θ))=(δs(θ),c(θ))

    then (δs(θ),c(θ)) is the direction of the tangent to S at (c(θ),s(θ)).

    Let us also suppose that

    {c(α+β)=c(α)c(β)δs(α)s(β),s(α+β)=s(α)c(β)+c(α)s(β),and{c(θ)=c(θ),s(θ)=s(θ).

    In particular (c(0),s(0))=(1,0).

    We introduce the following definitions and notations

    rot(α)=(c(α)δs(α)s(α)c(α)),

    and

    ref(α)=(c(2α)δs(2α)s(2α)c(2α)).

    As expected we have

    rot(α)(c(θ)s(θ))=(c(θ+α)s(θ+α)),

    and

    ref(α)(c(α+θ)s(α+θ))=(c(αθ)s(αθ)).

    We can verify that rot(α)rot(β)=rot(α+β), and

    {ref(α)=rot(2α)ref(0),ref(0)ref(α)=rot(2α),ref(α)rot(β)=rot(2α+β)ref(0).

    Let (c(α),s(α))S be a fixed point in the first quadrant, so c(α)>0 and s(α)>0. Let us start with an admissible x value such that there exists a point P=(x,y)S with y0. Then we look for θ such that c(θ)=x and y=s(θ)=δ(1x2)0, for an unknown θ. We apply to P=P0 a sequence of reflections. We start with ref(α) followed by ref(0), and we repeat the process. Starting with P0=PS, after k reflections we get the point Pk=(c(θk),s(θk))S. It follows that after 2n reflections P2n=(c(θ2nα),s(θ2nα)), so θ2n=θ2nα, and after 2n+1 reflections P2n+1=(c(2(n+1)αθ),s(2(n+1)αθ)), so θ2n+1=2(n+1)αθ.

    Starting with θ=θ0>α, we would like to stop when θk[0,α] for the first time. The trajectory of the points Pk (k=0,1,2,) is illustrated for δ=+1 on Figure 1 and for δ=1 on Figure 2, where Pf is the final point.

    Figure 1.  Trajectory of P on the unit circle (δ=+1).
    Figure 2.  Trajectory of P on the unit hyperbola (δ=1).

    If K is the total number of reflections (2n or 2n+1), it means that

    0<θKαα.

    Indeed, for an even number of reflections K=2n, the last reflection is with respect to the axis y=0, so θ2n cannot be 0 and we have θ2n(0,α], or 0<θ2nαα. For an odd number of reflections K=2n+1, the last reflection is with respect to the line of direction α, so θ2n+1 cannot be α, and we have θ2n+1[0,α), so 02(n+1)αθ<α or 0<θ(2n+1)αα.

    So the total number K of reflections is

    K={θα1ifθαisaninteger,θαifθαisnotaninteger.

    In the applications the coordinates of P are known but not θ. Moreover, α is not known and only some information about it is given that allow us to determine c(α) and s(α). The problem here is to find an approximation of θ from K and an approximation of α. So as a result we get a method to find θ=c1(x), i.e. an evaluation of the inverse function c1().

    The preceding result suggests a way to compute the digits of θ. Since the integer part of θ10N, noted θ10N, add the first N digits of the fractional part of θ, for an angle α10N we will be near the goal.

    To get the digits of θ we could take α=10N, then

    K={θ10N1ifθ10Nisaninteger,θ10Nifθ10Nisnotaninteger.

    Unfortunately it is not the interesting situation for the applications.

    In the applications, α is not exactly given, but there is a function T() for which the value T(α)=σ can be fixed, so α=T1(σ)σ. It is possible to take σ=10N. We will consider the next two situations using the Taylor's expansion of T1().

    (a) Let us consider T()=sin() for δ=+1, and T()=tanh() for δ=1. Taylor's expansions of their corresponding T1() are

    arcsin(σ)=+=0(2)!4(!)2σ2+1(2+1),

    and

    arctanh(σ)=+=0σ2+1(2+1).

    Thanks to the geometric series, we can consider the following lower and upper bounds for both series

    σ<T1(σ)<σ[1+σ23(11σ2)],

    for 0<σ<1. Moreover

    1+σ23(11σ2)11σ22,

    for 0<σ<1/2. So we can write

    1σσ2<1T1(σ)<1σ

    for 0<σ<ρ where 1/2<ρ=1/2<1. We set σ=10N, α=T1(10N), and multiply by θ to get

    θ10Nθ210N<θα<θ10N

    As long as θα is not an integer and that θ10Nθ210N>θ10N we have K=θ10N. In general in this case we get K=θ10N1 or K=θ10N.

    (b) Let us consider T()=tan() for δ=+1, and T()=sinh() for δ=1. Taylor's expansions of their corresponding T1() are

    arctan(σ)=+=0(1)σ2+1(2+1),

    and

    arcsinh(σ)=+=0(1)(2)!4(!)2σ2+1(2+1).

    We can consider the following lower and upper bounds for both series

    σσ33<T1(σ)<σ,

    for 0<σ<1. Moreover

    1σ23>11+σ22,

    for 0<σ<1. So we can write

    1σ<1T1(σ)<1σ+σ2

    for 0<σ<ρ where 1/2<ρ<1. We set σ=10N, α=T1(10N), and multiply by θ to get

    θ10N<θα<θ10N+θ210N

    As long as θα is not an integer and that θ10N+θ210N<θ10N+1 we have K=θ10N. In general in this case we get K=θ10N or K=θ10N+1.

    In both cases, the result K=θ10N depends of the expansion of θ. let

    {θ=a0.a1a2aNaN+1a2Na2N+1θ2=˜a0.˜a1˜a2˜aN˜aN+1˜a2N˜a2N+1

    and

    {θ10N=a0a1a2aN.aN+1a2N1a2Na2N+1θ210N=0.0     0(N1)times˜a0   ˜a1.

    A sufficient condition to get K=θ10N can be given for the two cases. The conditions are:

    (a) there exists an index n[N+1,2N1] such that an>0,

    (b) there exists an index n[N+1,2N1] such that an<9.

    For example, up to now and with modern computational facilities and methods [7], for small values of N and up to very large values of N it has not been observed sequences such that:

    (a') an=0 for n[N+1,2N1],

    (b') an=9 for n[N+1,2N1],

    in the expansion of π and ln(2). So we could claim that K=θ10N holds up to very large values of N for θ=π and θ=ln(2).

    The algorithm is as follows :

    Alternating reflection method

    Step 0. Enter P=(x,y)S with y0, and σ=T(α).

    Step 1. Determine c(α) and s(α) from the data σ.

    Step 2. The target zone is { PS | Pisbetween(1,0)and(c(α),s(α)) }.

    Step 3. Compute c(2α)=c2(α)δs2(α) and s(2α)=2s(α)c(α).

    Step 4. Set

    ref(0)=(1001)andref(α)=(c(2α)δs(2α)s(2α)c(2α))

    Step 5. With P0=P, compute Pk+1 as long as Pk is not in the target zone

    (i) for k=2n, compute Pk+1=P2n+1=ref(α)P2n=ref(α)Pk,

    (ii) for k=2n+1, compute Pk+1=P2n+2=ref(0)P2n+1=ref(0)Pk.

    We have considered the two situations δ=±1 and we got the results given in Table 1 for π and ln(2).

    Table 1.  Number of reflections needed to get θK[0,α].
    θ=π=3.14159265358979... θ=ln2=0.69314718055995...
    N σ=10N K=π10N K=ln(2)10N
    1 101 31 6
    2 102 314 69
    3 103 3141 693
    4 104 31415 6931
    5 105 314159 69314

     | Show Table
    DownLoad: CSV

    S is the unit circle, and (c(),s())=(cos(),sin()). To get θ=π, we start with x=cos(π)=1, so we consider P=(1,0). K is computed using σ=10N for N=1,2,3,4,5, for T(α)=sin(α)=σ and T(α)=tan(α)=σ. As expected, both situations generate the same value of K.

    S is the right branch (x0) of the unit hyperbola, and (c(),s())=(cosh(),sinh()). To get θ=ln(2), we start with x=cosh(ln(2))=5/4, so we consider P=(5/4,3/4). K is computed using σ=10N for N=1,2,3,4,5, for T(α)=tanh(α)=σ and T(α)=sinh(α)=σ. As expected, both situations generate the same value of K.

    We not only have an unusual method to find the digits of the trigonometric constant π and the logarithmic constant ln(2) but also to evaluate the inverse trigonometric functions (for δ=+1) and inverse hyperbolic functions (for δ=1). Indeed the method can be extended to solve for θ the equation c(θ)=x. If the given data for α allows us to say that α=10N+O(102N), we get that

    |θ(K+12)10N|=1210N+O(102N).

    The next examples illustrate that both values of K can be obtained. Numerical results are reported in Table 2 where examples were chosen to get the same figures.

    Table 2.  Values of K.
    θ=1.00000000 θ=0.99999999
    K K
    N θ10N (a) (b) θ10N (a) (b)
    1 10 9 10 9 9 10
    2 100 99 100 99 99 100
    3 1000 999 1000 999 999 1000
    4 10000 9999 10000 9999 9999 10000
    5 100000 99999 100000 99999 99999 99999

     | Show Table
    DownLoad: CSV

    We fix x and compute y=1x20 for the point P=P0. We use for (a) T(α)=sin(α), and for (b) T(α)=tan(α). We consider two situations to illustrate that we can get the two values of K. Firstly, we consider x=0.540302305...=cos(1) (and y=1x20), so θ=1=arccos(x). We get for (a) K=110N1 and for (b) K=110N. Secondly we consider x=0.540302314...=cos(0.99999999), so θ=0.99999999=arccos(x). We get for (a) K=0.9999999910N for N=1,2,3,4, and for (b) K=0.9999999910N+1 for N=1,2,3,4, but not for N5.

    We fix x and compute y=x210 for the point P=P0. We use for (a) T(α)=tanh(α), and for (b) T(α)=sinh(α). We consider two situations to illustrate the different values of K. Firstly, we consider x=1.543080635...=cosh(1), so θ=1=arccosh(x). We get for (a) K=110N1 and for (b) K=110N. Secondly, we consider x=1.543080623...=cosh(0.99999999), so θ=0.99999999=arccosh(x). We get for (a) K=0.9999999910N and for (b) K=0.9999999910N+1 for N=1,2,3,4, but not for N5.

    The computational cost of this method is quite low at each iteration. It requires 4 multiplications and 2 additions to compute P2n+1 from P2n, and only a sign change to get P2n+2 from P2n+1. Also, at each iteration a test is required to eventually terminate the process.

    We used MATLAB with single-precision computation for the numerical examples. For σ=10N, we had c(α)=1+O(102N) and s(α)=O(10N), and the matrix ref(α) looked like

    (1+O(102N)δO(10N)O(10N)1+O(102N)),

    so it explains why we stopped at N=5 in the computation. We could increase N with multi-precision computation.

    In this paper we have presented an unusual method to find θ=c1(x). We have considered the following problem:

    {Supposegivenanadmissiblevaluex,andsety=δ(1x2)0,thenfindθsuchthatθ=c1(x),so(x,y)=(c(θ),s(θ)).

    We can extend this problem to find also θ=s1(y) and θ=t1(z), where t()=s()/c(). Indeed we have θ=s1(y)=sign(y)c1(δ(1δy2)) for an appropriate y value, and θ=t1(z)=sign(z)c1(1/1+δz2) for an appropriate value of z.

    This work has been financially supported by an individual discovery grant from the Natural Sciences and Engineering Research Council of Canada.

    The author declares no conflict of interests.



    [1] A. L. Teel, A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE T. Automat. Contr., 41 (1996), 1256–1270. https://doi.org/10.1109/9.536496 doi: 10.1109/9.536496
    [2] R. Sepulchre, M. Janković, P. V. Kokotović, Constructive Nonlinear Control, Springer: London, 1997. https://doi.org/10.1007/978-1-4471-0967-9
    [3] F. Mazenc, L. Parly, Adding integrations, saturated controls, and stabilization of feedback systems, IEEE T. Automat. Contr. 41 (1996), 1559–1578. https://doi.org/10.1109/9.543995
    [4] F. Mazenc, L. Parly, Tracking trajectories of the cart-pendulum system. Automatica, 39(2003), 677–684. https://doi.org/10.1016/S0005-1098(02)00279-0
    [5] G. Kaliora, A. Astolfi, Nonlinear control of feedforward systems with bounded signals, IEEE T. Automat. Contr., 49 (2004), 1975–1990. https://doi.org/10.1109/TAC.2004.837572 doi: 10.1109/TAC.2004.837572
    [6] J. Y. Zhai, C. J. Qian, Global control of nonlinear systems with uncertain output function using homogeneous domination approach, Int. J. Robust Nonlin., 22 (2012), 1543–1561. https://doi.org/10.1002/rnc.1765 doi: 10.1002/rnc.1765
    [7] Q. Liu, Z. Liang, X. Zhang, Global state feedback stabilization of a class of feedforward systems under sampled-data control, In: Proceedings of the 33rd Chinese Control Conference, Nanjing, China, 2014, 2293–2298. https://doi.org/10.1109/ChiCC.2014.6896990
    [8] L. J. Long, Stabilization by forwarding design for switched feedforward systems with unstable modes, Int. J. Robust Nonlin., 27 (2017), 4808–4824. https://doi.org/10.1002/rnc.3832 doi: 10.1002/rnc.3832
    [9] H. W. Ye, Stabilization of uncertain feedforward nonlinear systems with ation to underactuated systems, IEEE T. Autom. Contr., 64 (2019), 3484–3491. https://doi.org/10.1109/TAC.2018.2882479 doi: 10.1109/TAC.2018.2882479
    [10] J. P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667–1694. https://doi.org/10.1016/S0005-1098(03)00167-5 doi: 10.1016/S0005-1098(03)00167-5
    [11] H. Zhang, G. Duan, L. Xie, Linear quadratic regulation for linear time-varying systems with multiple input delays, Automatica, 42 (2006), 1465–1476. https://doi.org/10.1016/j.automatica.2006.04.007 doi: 10.1016/j.automatica.2006.04.007
    [12] N. Bekiaris-Liberis, M. Krstić, Delay-adaptive feedback for linear feedforward systems, Syst. Control. Lett, 59 (2010), 277–283. https://doi.org/10.1016/j.sysconle.2010.03.001 doi: 10.1016/j.sysconle.2010.03.001
    [13] X. D. Ye, Adaptive stabilization of time-delay feedforward nonlinear systems, Automatica, 47 (2011), 950–955. https://doi.org/10.1016/j.automatica.2011.01.006 doi: 10.1016/j.automatica.2011.01.006
    [14] G. Z. Meng, K. M. Ma, Global output feedback stabilization of upper-triangular nonlinear time-delay systems, J. Control Theory Appl., 10 (2012), 533–538. https://doi.org/10.1007/s11768-012-0216-6 doi: 10.1007/s11768-012-0216-6
    [15] X. F. Zhang, L. Baron, Q. R. Liu, E. K. Boukas, Design of stabilizing controllers with a dynamic gain for feedforward nonlinear time-delay systems, IEEE T. Automat. Contr., 56 (2011), 692–697. https://doi.org/10.1109/TAC.2010.2097150 doi: 10.1109/TAC.2010.2097150
    [16] X. F. Zhang, Q. R. Liu, L. Baron, E. K. Boukas, Feedback stabilization for high order feedforward nonlinear time-delay systems, Automatica, 47 (2011), 962–967. https://doi.org/10.1016/j.automatica.2011.01.018 doi: 10.1016/j.automatica.2011.01.018
    [17] M. M. Jiang, K. M. Zhang, X. J. Xie, Output feedback stabilisation of high-order nonlinear feedforward time-delay systems, Int. J. Syst. Sci., 48 (2017), 1692–1707. https://doi.org/10.1080/00207721.2017.1280556 doi: 10.1080/00207721.2017.1280556
    [18] L. Liu, Y. Wang, Decentralized output feedback control of large-scale stochastic high-order feedforward time-delay systems, In: Proceedings of 2019 Chinese Control Conference, Guangzhou, China, 2019, 1364–1369. https://doi.org/10.23919/ChiCC.2019.8865340
    [19] W. Michiels, S. I. Niculescu, Stability and stabilization of time-delay systems: An eigenvalue-based approach, SIAM: Singapore, 1965. https://doi.org/10.1137/1.9780898718645
    [20] P. Krishnamurthy, F. Khorrami, Adaptive dynamic high-gain scaling based output-feedback control of nonlinear feedforward systems with time delays in input and state, In: Proceedings of 2010 American Control Conference, Baltimore, USA, pp.5794–5799. https://doi.org/10.1109/ACC.2010.5530429
    [21] X. D. Ye, Universal stabilization of feedforward nonlinear systems, Automatica, 39 (2003), 141–147. https://doi.org/10.1016/S0005-1098(02)00196-6 doi: 10.1016/S0005-1098(02)00196-6
    [22] X. Zhang, H. Gao, C. Zhang, Global asymptotic stabilization of feedforward nonlinear systems with a delay in the input, Int. J. Syst. Sci., 37 (2006), 141–148. https://doi.org/10.1080/00207720600566248 doi: 10.1080/00207720600566248
    [23] H. L. Choi, J. T. Lim, Stabilization of a chain of integrators with an unknown delay in the input by adaptive output feedback. IEEE T. Automat. Contr., 51 (2006), 1359–1363. https://doi.org/10.1109/TAC.2006.878742
    [24] W. T. Zha, J. Y. Zhai, S. M. Fei, Global output feedback control for a class of high-order feedforward nonlinear systems with input delay, ISA Trans., 52 (2013), 494–500. https://doi.org/10.1016/j.isatra.2013.04.001 doi: 10.1016/j.isatra.2013.04.001
    [25] C. R. Zhao, W. Lin, Memoryless linear feedback control for a class of upper-triangular systems with large delays in state and input, Syst. Control. Lett., 139 (2020), 104679. https://doi.org/10.1016/j.sysconle.2020.104679 doi: 10.1016/j.sysconle.2020.104679
    [26] C. R. Zhao, W. Lin, Global stabilization by memoryless feedback for nonlinear systems with a limited input delay and large state delays, IEEE T. Automat. Contr., 66 (2021), 3702–3709. https://doi.org/10.1109/TAC.2020.3021053 doi: 10.1109/TAC.2020.3021053
    [27] C. R. Zhao, W. Lin, Homogeneous output feedback design for time-delay nonlinear integrators and beyond: an emulation, Automatica, 146 (2022), 110641. https://doi.org/10.1016/j.automatica.2022.110641 doi: 10.1016/j.automatica.2022.110641
    [28] K. Q. Gu, V. L. Kharitonov, J. Chen, Stability of time-delay systems, Berlin: Birkhauser, 2003. https://doi.org/10.1007/978-1-4612-0039-0
    [29] L. Liu, M. Kong, A new design method to global asymptotic stabilization of strict-feedforward stochastic nonlinear time delay systems, Automatica, 151 (2023), 110932. https://doi.org/10.1016/j.automatica.2023.110932 doi: 10.1016/j.automatica.2023.110932
    [30] X. F. Zhang, E. K. Boukas, Y. G. Liu, L. Baron, Asymptotic stabilization of high-order feedforward systems with delays in the input, Int. J. Robust Nonlin., 20 (2010), 1395–1406. https://doi.org/10.1002/rnc.1521 doi: 10.1002/rnc.1521
  • This article has been cited by:

    1. Azzh Saad Alshehry, Safyan Mukhtar, Ali M. Mahnashi, Optical fractals and Hump soliton structures in integrable Kuralay-Ⅱ system, 2024, 9, 2473-6988, 28058, 10.3934/math.20241361
    2. Nauman Ahmed, Jorge E. Macías-Díaz, Shazia Umer, Muhammad Z. Baber, Muhammad Jawaz, Héctor Vargas-Rodríguez, On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time, 2024, 22, 2391-5471, 10.1515/phys-2024-0109
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(881) PDF downloads(48) Cited by(0)

Figures and Tables

Figures(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog