Global prescribed-time control of a class of uncertain nonholonomic systems by smooth time-varying feedback
Kang-Kang Zhang Bin Zhou Chenchen Fan James Lam

1.0


Global prescribed-time control of a class of uncertain nonholonomic systems by smooth
time-varying feedback
    Kang-Kang Zhang, Bin Zhou, Chenchen Fan, James Lam,  Fellow, IEEE



This work was supported by the National Science Found for
Distinguished Young Scholars (62125303), the Science Center Program of
National Natural Science Foundation of China (62188101), the
Fundamental Research Funds for the Central Universities
(HIT.BRET.2021008), and HKU CRCG (2302101740). (Corresponding authors: Bin Zhou)
Kang-Kang Zhang is with the Department of Mechanical Engineering, University of Hong Kong,
Hong Kong, China, and the Department of Computer Science, KU Leuven, B-3001 Heverlee, Belgium; James Lam is with the Department of Mechanical Engineering, University of Hong Kong,
Hong Kong, China;
Bin Zhou is with the Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, 150001, China; Chenchen Fan is with the Department of Rehabilitation Sciences, The Hong Kong Polytechnic University, Hong Kong, China  (email: kang-kang.zhang@kuleuven.be, binzhou@hit.edu.cn, chenchen.fan@polyu.edu.hk, james.lam@hku.hk).


    Received ?? ; accepted ??
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§ ABSTRACT 
This paper investigates the prescribed-time smooth control problem for a class of
uncertain nonholonomic systems.  With a novel smooth time-varying state transformation, the uncertain chained nonholonomic system is reformulated as an uncertain linear time-varying system.
By fully utilizing the properties of a class of parametric Lyapunov equations and constructing time-varying Lyapunov-like functions,
smooth time-varying high-gain state and output feedback controllers are designed. The states and controllers are proven to converge to zero at any prescribed time. The proposed smooth time-varying method combines the advantage of a
time-varying high-gain function, which enhances control performance, and a
smooth time-varying function that can drive the states to zero at the prescribed time. The
effectiveness of the proposed methods is verified by a numerical example.


Shell
et al.: Bare Demo of IEEEtran.cls for IEEE Journals


Nonholonomic systems;
Uncertain nonholonomic systems;
Prescribed-time control;
Smooth time-varying high-gain feedback.






§ INTRODUCTION


Since nonholonomic systems have a wide range of applications, such as wheeled
mobile robots and surface vessels, the control problem of nonholonomic systems
has received widespread attention
<cit.>. However, nonholonomic systems
do not satisfy the necessary condition for the existence of a smooth
time-invariant feedback for a nonlinear system, called Brockett's necessary
conditions <cit.>. Therefore researchers can only turn their
attention to discontinuous time-invariant control/hybrid control and smooth
time-varying control. For discontinuous time-invariant control, a general strategy for designing discontinuous control laws that can achieve asymptotic convergence for a class of nonholonomic control systems was proposed in <cit.>. For smooth
time-varying control, a control law which globally asymptotically stabilizes
nonholonomic system in chained/power form was presented in <cit.>. In order to improve the convergence rate,
by using a time-varying state transformation, a smooth time-varying feedback
control law that can achieve exponential stability was proposed in
<cit.>. However, the above control laws can only ensure
that the states converge to zero asymptotically (exponentially), which
sometimes cannot meet the requirement in practical engineering.



Compared with traditional asymptotic stability (or exponential stability),
finite-/fixed-time control has higher convergence speed, faster
convergence accuracy and stronger robustness to external disturbances
<cit.>. However, for both finite-time control and
fixed-time control, the actual convergence time (rather than the upper bound)
depends on the initial condition <cit.>. Recently, a so-called prescribed-time control method, whose convergence time does
not depend on the initial state, have received renewed attention
<cit.>. By using
time-varying high-gain feedback, the prescribed-time control problem of
normal-form nonlinear systems was solved in <cit.>. Such time-varying high-gain methods have been utilized to address various
problems, such as prescribed-time stabilization <cit.>,
prescribed-time observer <cit.>, adaptive prescribed-time
control <cit.>, prescribed-time output feedback
<cit.>, and prescribed-time tracking
control <cit.>. With the aid of properties of a class of
parametric Lyapunov equations (PLEs) and the scalarization technique
<cit.>, the finite-time and prescribed-time control problems for
linear and a class of nonlinear systems were addressed in
<cit.> based on time-varying
high-gain feedback. Very recently, a new approach called time-space
deformation methods has been proposed in <cit.> to address
numerical singular problems that may arise in time-varying high-gain feedback.



To the best of the authors' knowledge, there are no smooth time-varying
feedback results for the prescribed-time control of nonholonomic systems.

In this paper, the smooth prescribed-time control problem for a class of
uncertain nonholonomic systems will be investigated. Firstly, with a novel smooth time-varying state
transformation, uncertain chained nonholonomic systems are reformulated as
uncertain linear time-varying systems. Secondly, by fully utilizing the
properties of a class of parametric Lyapunov equations and constructing
time-varying Lyapunov-like functions, smooth time-varying high-gain feedback
controllers are designed. It is proved that the states and controllers
converge to zero at any prescribed time. Both state feedback and
observer-based output feedback are considered.

The contributions of this paper are twofold. On the one hand, different from
the traditional discontinuous control methods
<cit.>
, which may cause singularity or chattering problems, the proposed methods in
this paper are smooth. On the other hand, different from
<cit.>, where only the state
feedback controllers were constructed to ensure that the state converges to zero
exponentially, the proposed controllers are not only based on state feedback
but also on output feedback, and can guarantee that the state converges to zero
at any prescribed time.

Notation: For a (symmetric) matrix M, let M^T,
‖ M‖, λ_max(M), and λ_min(M)
denote its transpose, 2-norm, maximal eigenvalue, and minimal eigenvalue,
respectively. In addition, diag{A_1,…, A_n} denotes a
diagonal matrix whose ith diagonal element is A_i .



§ PROBLEM INTRODUCTION AND PRELIMINARIES




 §.§ Problem Introduction


Consider the following uncertain nonholonomic system

    {[                       ẋ_0                     =u_0,;                       ẋ_1   =u_0x_2+ϕ_1(  t,u,x)  ,;                                                   ⋮;                     ẋ_n-1 =u_0x_n+ϕ_n-1(  t,u,x)  ,;                       ẋ_n        =u+ϕ_n(  t,u,x)  ,;                         y             =[x_0,x_1]^T, ].

where [x_0,x^T]^T=[x_0,x_1,…,x_n
]^T∈𝐑^n+1 is the state vector, u_0∈𝐑
and u∈𝐑 are two control inputs, y∈𝐑^2 is the
output vector, and ϕ_i(t,u,x), i=1,2,…,n are some unknown continuous
functions satisfying the following assumption.


The nonlinear functions ϕ_i(t,u,x), i=1,2,…,n, satisfy

    |ϕ_i(  t,u,x)  |≤ c_i1|
    x_1| +c_i2| x_2| +⋯+c_ii|
    x_i|,

where c_ij, j=1,2,…,i are some known constants.


In this paper, regarding system (<ref>), we are interested in the
following problem.


 Suppose that Assumption <ref> holds true. Let T>0 be a
prescribed time. Design a smooth time-varying state feedback controllers
u_0(t) and u(t) such that, for any initial condition, the states and
controls converge to zero at the prescribed time T.


Since not all states are actually measurable, it is necessary to study
observer-based output feedback control strategies.


Suppose that Assumption <ref> holds true. Let T>0 be a
prescribed time. Design a smooth time-varying output feedback

    {[                               ξ̇(  t) =Fξ(  t)  +Gu(  t)
    +L(  t)  y(  t)  ,;                                u(  t)                    =H(  t)  ξ(  t)  , ].

such that, for any initial condition, the states and controls converge to zero
at the prescribed time T.




 §.§ Preliminaries




  §.§.§ Some properties of ϕ(t,u,x)


Consider the time-varying state transformation

    z(  t)  =L_n(  1/T-t)  x(  t)
    ,   t∈0,T),

where

    L_n(  γ)  =diag{γ^n-1,γ
    ^n-2,…,1}  .

For later use, denote, for i=1,2,…,n,

    φ_i(t,u,x)       =(  T-t)  ^i-nϕ_i(t,u,x),  
    ψ_i(t,u,x)       =φ_i(t,u,x)+n-i/T-tz_i.

Therefore, by using Assumption <ref> and (<ref>), it can be
obtained that, for i=1,2,…,n,

    |φ_i(t,u,x)| =     (  T-t)
    ^i-n|ϕ_i(t,u,x)|
    ≤   (  T-t)  ^i-n∑_j=1^ic_ij|
    x_j|
    
    =     (  T-t)  ^i-n∑_j=1^ic_ij|(
    T-t)  ^n-jz_j|
    
    =     ∑_j=1^ic_ij|(  T-t)  ^i-jz_j| ,

and further

    |ψ_i(t,u,z)|≤   ∑_j=1^ic_ij|(T-t)^i-jz_j|+n-i/T-t|z_i|
    
    =     1/T-t∑_j=1^ic_ij|(T-t)^i-j+1z_j|+n-i/T-t|z_i|
    ≤   1/T-t∑_j=1^ig_ij| z_j| ,

where g_ij=c_ijT^i+1-j, j=1,2,…,i-1, i=2,3,…,n,
andg_ii=c_iiT+n-i, i=1,2,…,n.

With above preparations, we can state the following result by referring Lemma
1 in <cit.>.


Suppose that Assumption <ref> holds true. Then, for any
z∈𝐑^n and any t∈0,T), there holds

    Φ^2(  t,u,z)  ≜(  L_nψ)
    ^T(  L_nψ)  ≤ d^2γ^2(
    L_nz)  ^T(  L_nz)  ,

where γ=1/T-t, γ_0=1/T, L_n=L_n(γ),
and

    d^2=max{∑_i=1^ng_i1^2i/γ_0^2(  i-1)  ,⋯,
    
    
    ∑_i=n^ng_in^2i/γ_0^2(  i-n)  }  .





It follows from (<ref>) that

    ψ_i^2(  t,u,z)      ≤1/(  T-t)  ^2(
    
    
    ∑_j=1^i
    
    g_ij| z_j|)  ^2
       ≤i/(  T-t)  ^2∑_j=1^i
    
    g_ij^2| z_j| ^2.

Then in view of (<ref>) and (<ref>), we have

    Φ^2(  t,u,z)  =(  L_nψ)
    ^T(  L_nψ) 
         =
    
    
    ∑_i=1^nγ^2(  n-i)  ψ_i^2(  t,u,z) 
       ≤1/(  T-t)  ^2∑_i=1^nγ^2(  n-i)  i
    
    
    ∑_j=1^i
    
    g_ij^2| z_j| ^2
       ≤ d^2γ^2(  |γ^n-1z_1|
    ^2+|γ^n-2z_2| ^2+⋯+|γ
    ^0z_n| ^2)  .

The proof is finished.




  §.§.§ Some Properties of the PLE


We now introduce some properties of the parametric Lyapunov equations (PLEs)
that will be used in this paper. Denote

    A=[
    [ 0 1      ; ⋮   ⋱    ; 0     1  ; 0 0 ⋯ 0   ]]  ,   b=[
    [ 0; ⋮; 0; 1 ]]  ,   c=[
    [ 1; 0; ⋮; 0 ]]  ^T.

Consider the following PLE <cit.>

    A^TP+PA-Pbb^TP=-γ P,

and its dual form

    AQ+QA^T-Qc^TcQ=-γ Q,

where γ>0 is a parameter to be designed.


<cit.> Let (A,b)∈(𝐑^n× n
,𝐑^n×1) be given by (<ref>) and γ>0.



  * The PLE (<ref>) has a unique positive definite solution

    P(γ)=γ L_nP_nL_n,

where P_n=P(1) and L_n=L_n(γ).


  * The solution P(γ) satisfies

    dP(γ)/dγ>0,   b^T
    P(γ)b=nγ.




  * There holds

    P(γ)/nγ≤dP(γ)/dγ≤δ_cP(γ)/nγ,  ∀γ>0,

where δ_c=n(1+λ_max(E_n+P_nE_nP_n^-1))
with E_n=diag{n-1,n-2,…,1,0}.


  * There holds

    A^TPA≤3n^2γ^2P.






<cit.> Let ( A,c) ∈( 𝐑^n× n
,𝐑^1× n) be given by (<ref>) and γ>0.



  * The unique positive definite solution Q(γ) is given by

    Q(γ)=γ^2n-1L_n^-1Q_nL_n^-1,

where Q_n=Q(1) and L_n=L_n(γ).


  * There holds

    Q( γ) /nγ≤dQ( γ) /dγ≤δ_oQ( γ) /nγ,  ∀γ>0,

with δ_o is a constant.


  * The matrices P_n,Q_n,P_n^-1, and Q_n^-1 are similar to
each other and thus

    λ_max(P_n)       =λ_max(Q_n)=λ_min^-1
    (P_n)=λ_max(Q_n^-1)
         =λ_min^-1(Q_n)=λ_min^-1(Q_n^-1)≜Λ.




  * There holds

    cQ(γ)c^T=nγ.







§ STATE FEEDBACK


A solution to Problem <ref> can be stated as follows.


Let T>0 be a prescribed time, β=2n+δ
_c/n+2dΛ, and γ_0=1/T. Design the
smooth time-varying state feedback

    {[                                  u_0(  t)                 =-3/T-tx_0(  t)-β/2(T-t),;                                    u(  t) =-β b^TP(  γ (t)
    )  L_n(  1/T-t)  x(  t),;                                   γ (  t)                                =T/T-tγ_0. ].

Then, for any initial condition and all t∈0,T), the states and
controls of the closed-loop system consisting of (<ref>) and
(<ref>) satisfy

    | x_0(t)|   ≤ (  T-t)  ^2(
    T-t/T^3| x_0(0)| +β t/2T)
    ,
    | u_0(t)|   ≤(T-t)(  3(  T-t)
    (  | x_0(0) |/T^3+β/2T)
    +β)  ,
    ‖ x(  t)  ‖   ≤υ_1(
    T-t)  ^3/2e^υ_10| x_0
    (0)| t‖ x(  0)  ‖ ,
    ‖ u(t) ‖   ≤υ_2(
    T-t)  ^1/2e^υ_20| x_0
    (0)| t‖ x(  0)  ‖ ,

where υ_1, υ_10, υ_2 and υ_20 are
some positive constants, namely, Problem <ref> is solved.



Consider the x_0-subsystem in system (<ref>). The closed-loop system
consisting of (<ref>) and the first equation in (<ref>) can be
written as

    ẋ_0=-3/T-tx_0-β/2(T-t),

whose solution can be given as

    x_0(t)=     exp(  -∫_0^t3/T-sd
    s)  x_0(0)
         -β/2∫_0^texp(  -∫_s^t3/T-τdτ)  (T-s)ds
    
    =     (  T-t/T)  ^3x_0(0)-β/2
    (T-t)^2+β/2T(T-t)^3
    
    =       (T-t)^2(  T-t/T^3x_0(0)-β t/2T)  ,

which proves (<ref>). Then the control u_0 can be written as the
open-loop form

    u_0(t)=       -3/T-tx_0-β/2(T-t)
    
    =       -3/T-t(  (  T-t)  ^2(  T-t/T^3x_0(0)-β t/2T)  ) 
         -β/2(T-t)
    
    =       (T-t)(  3(  T-t)  (  -x_0(0)/T^3
    -β/2T)  +β) 
    ≜     (T-t)(  θ(t)+β)  ,

which proves (<ref>).

Consider the x-subsystem in system (<ref>). By using (<ref>),
system (<ref>) can be written as

    {[ ż_i=n-i/T-tz_i+(  β+θ(t))
    z_i+1+(  T-t)  ^i-nϕ_i,;                                        ż_n=u+ϕ_n, ].

where i=1,2,…,n-1, which can be re-expressed as

    ż=1/T-tA_0z+θ(t)Az+β Az+φ+bu,

where φ=φ(t,u,x)=[φ_1(t,u,x),φ_2(t,u,x),…
,φ_n (t,u,x)]^T, and

    A_0=diag{n-1,n-2,…,2,1}.

Such a system can be further expressed as

    ż=β(  A-bb^TP)  z+θ(t)Az+ψ
    ,

where ψ=ψ(t,u,z)=[ψ_1(t,u,z),ψ_2(t,u,z),…,ψ_n
(t,u,z)]^T. Choose the Lyapunov-like function

    V(  t,z(t))  =γ(t) z^T(t)P(γ(t))z(t),

whose time-derivative along the trajectory of the closed-loop system
(<ref>) can be written as

    V̇(  t,z)  =    γ̇z^TPz+γγ̇z^TdP/dγz
         +βγ z^T(  A^TP+PA-2Pbb^TP)  z
         +2γθ z^TPAz+2γ z^TPψ
    ≤   γ̇z^TPz+δ_c/nγ̇z^TPz-βγ^2z^TPz
         +2γθ z^TPAz+2γ z^TPψ,

where we have used (<ref>) and (<ref>). By using the Young's
inequalities with k_0>0 and k_1>0, we have

    2z^TPAz    ≤ k_0γ z^TPz+1/k_0γz^TA^TPAz,
    
    2z^TPψ   ≤ k_1γ z^TPz+1/k_1γψ^TPψ.

In addition, it can be obtained from Lemma <ref>, (<ref>),
and (<ref>) that

    ψ^TPψ     =ψ^Tγ L_nP_nL_nψ≤Λγ(  L_nψ)  ^T(
    L_nψ)  
         =d^2Λγ^3(  L_nz)  ^T(
    L_nz)  ≤ d^2Λ^2γ^2z^TPz.

With this and (<ref>) in Lemma <ref>, (<ref>) can be continued as

    V̇(  t,z)  ≤   γ̇z^TPz+δ_c/nγ̇z^TPz-βγ
    ^2z^TPz
         +k_0|θ|γ^2z^TPz+|θ|/k_0z^TA^TPAz
         +k_1γ^2z^TPz+1/k_1ψ^TPψ
    ≤   γ̇z^TPz+δ_c/nγ̇z^TPz-βγ^2z^TPz
         +k_0|θ|γ^2z^TPz+3n^2|θ|/k_0γ^2z^TPz
         +k_1γ^2z^TPz+1/k_1d^2Λ
    ^2γ^2z^TPz
    
    =    (  γ̇+δ_c/nγ̇
    -βγ^2+k_1γ^2+1/k_1d^2Λ
    ^2γ^2)  z^TPz
         +(  k_0|θ|γ^2+3n^2/k_0|θ|γ^2)  z^TPz
    
    =    (  1+δ_c/n)  μ_zz^TPz+2|θ|√(3)nγ^2z^TPz
    
    =      -γ^2z^TPz+θ_0γ z^TPz,

where

    θ_0     =6(  | x_0(0)|/T^3
    +β/2T)  √(3)n,
    μ_z     =γ̇-n/n+δ_cβγ^2
    +2ndΛ/n+δ_cγ^2=-n/n+δ_cγ^2,

and we have taken k_0=√(3)n and k_1=dΛ. By using
the comparison lemma in <cit.>, V(t,z(t)) satisfies

    V(t,z(t))     ≤exp(  -∫_0^tγ(s)ds)
    e^θ_0tV(0,z(0))
        =(  T-t)  e^θ_0tV(0,z(0)),   t∈0,T).

It follows from (<ref>) that

    V(t,z(t))=γ z^TPz≥γ^2λ_min(L_n(γ
    _0)P_nL_n(γ_0))‖ z(t)‖^2.

Therefore, we have

    ‖ z(  t)  ‖ ^2≤(  T-t)
    ^3e^θ_0tV(  0,z(  0)  )  /λ_min(  L_n(  γ_0)  P_nL_n(
    γ_0)  )  ,

which, together with (<ref>), indicates that

    ‖ x(  t)  ‖   ≤‖ L_n(
    T-t)  ‖‖ z(  t)  ‖
       ≤‖ L_n(  T)  ‖(  T-t)
    ^3/2e^1/2θ_0t√(V(  0,z(
    0)  )  )/√(λ_min(  L_n(  γ
    _0)  P_nL_n(  γ_0)  )  ).

In addition, it can be obtained from (<ref>) and (<ref>) that

    V(  0,z(  0)  )  =γ_0z^T(0)P(
    γ_0)  z(0)
       ≤γ_0^2λ_max(  L_n(  γ_0)
    P_nL_n(  γ_0)  )  ‖ z(0)‖
    ^2
       ≤γ_0^2λ_max(  L_n(  γ_0)
    P_nL_n(  γ_0)  )  ‖ L_n(  1/T)  ‖ ^2‖ x(0)‖ ^2,

combining which with (<ref>), indicates that

    ‖ x(  t)  ‖   ≤‖ L_n(
    T)  ‖(  T-t)  ^3/2e
    ^1/2θ_0t√(V(  0,z(  0)  )  )/√(λ_min(  L_n(  γ_0)  P_nL_n(
    γ_0)  )  )
    ≤   γ_0√(ω_1)‖ L_n(  T)
    ‖(  T-t)  ^3/2e^1/2θ_0t‖ L_n(  1/T)  ‖‖
    x(0)‖ ,

where ω_1=λ_max(L_n(γ_0)P_nL_n(γ
_0))/λ_min(L_n(γ_0)P_nL_n (γ_0)). Thus we
have proven (<ref>).

Finally, it follows from (<ref>), (<ref>), (<ref>) that

    ‖ u(t)‖^2     =β^2z^TP(γ(t))bb^TP(γ(t))z
       ≤β^2b^TP(γ(t))bz^TP(γ(t))z
         =β^2nγ z^TP(γ(t))z
     =β^2nV(t,z(t))
       ≤β^2n(T-t)e^θ_0tV(0,z(0))
         =β^2n(T-t)e^θ_0tλ_max(L_n(γ
    _0)P_nL_n(γ_0))
       ×‖ L_n(  1/T)  ‖ ^2‖
    x(0)‖^2,

which proves (<ref>). The proof is finished.


If the x_0-subsystem in system (<ref>) is of the following form
<cit.>

    ẋ_0=u_0+c_0x_0,

where c_0 is a known constant, we can get the following result.


Let T>0 be a prescribed time, β=(2n+δ_c)e
^| c_0| T/n+2dΛe^|
c_0| T, and γ_0=1/T. Then, for any initial condition
and all t∈0,T), the state and control of the closed-loop system
consisting of (<ref>) with x_0-subsystem be defined in (<ref>) and
the following smooth time-varying feedback

    {[                                   u_0(  t)                =-3/T-tx_0-β/2e
    ^c_0t(T-t),;                                     u(  t) =-βe^c_0tb^TP(
    γ(  t)  )  L_n(  1/T-t)  x,;                                     γ(  t)                                 =T/T-tγ_0, ].

satisfy

    | x_0(t)|   ≤(  T-t)  ^2e
    ^c_0t(  (  T-t)  | x_0(0)|/T^3+β t/2T)  ,
    | u_0(t)|   ≤(  T-t)  e^c_0
    t(  3(  T-t)  (  | x_0(0)|/T^3+β/2T)  +β)  ,
    ‖ x(  t)  ‖   ≤υ_11(
    T-t)  ^3/2e^υ_12| x_0
    (0)| t‖ x(  0)  ‖ ,
    ‖ u(  t)  ‖   ≤υ_21(T-t)^1/2e^υ_22| x_0(0)| t‖
    x(  0)  ‖ ,

where υ_11, υ_12, υ_21 and υ_22
are some positive constants.



The closed-loop system consisting of (<ref>) and u_0(  t)
in the first equation in (<ref>) can be written as

    ẋ_0(t)=-3/T-tx_0(t)+c_0x_0(t)-β/2e^c_0
    t(T-t),

whose solution can be given as

    x_0(t)=     e^c_0texp(  -∫_0^t3/T-sds)  x_0(0)
         -β/2∫_0^te^c_0texp(  -∫_s
    ^t3/T-τdτ)  (T-s)ds
    
    =     (  T-t/T)  ^3x_0(0)e^c_0t
    -β/2e^c_0t(  T-t)  ^2
         +βe^c_0t(  T-t)  ^3/2T
    
    =     (  T-t)  ^2(  (  T-t)  x_0
    (0)/T^3e^c_0t-tβ/2Te^c_0t)  ,

which proves the expression of | x_0(t)| in this
proposition. Then the control u_0 can be written as the open-loop form

    u_0(t)=-3/T-tx_0-β/2e^c_0t(T-t)
    
      =   -3/T-t(  (  T-t)  ^2(  (
    T-t)  x_0(0)/T^3e^c_0t-tβ/2Te^c_0t)  ) 
         -β/2e^c_0t(T-t)
    
      =   (  T-t)  (  -3(  T-t)  (  x_0
    (0)/T^3e^c_0t+βe^c_0t/2T)
    +βe^c_0t) 
       ≜(T-t)(  θ(t)+βe^c_0t)  ,

which proves the expression of | u_0(t)| in this
proposition. The rest of the proof is similar as the proof of Theorem
<ref>, and is omitted for brevity.


When c_0=0, the above theorem reduces to Theorem <ref>. In addition,
similar to Remark 4 in <cit.>, the proposed method can also
handle the nonholonomic system (<ref>) with the nonlinear functions
satisfying

    |ϕ_i(  t,u,x)  |≤φ
    (x_0)(| x_1| +| x_2|
    +⋯+| x_i| )

in which φ(x_0) are some known positive continuous functions. Due to
space limitations, the details are not given here.



§ OBSERVED-BASED OUTPUT FEEDBACK


To state a solution to Problem <ref>, we first denote the parameter
β as

    β=max{β_1,β_2}  ,

where

    β_1     =8√(2)ndΛcQ_nP_nQ_nc^T+2(  2+δ_c/n)  ,
    β_2     =4√(2)dΛ+2(  2n+2-1/n)  ,

in which d is defined in (<ref>).


Let T>0 be a prescribed time and γ_0=1/T. Design the smooth
time-varying observed-based output feedback

    {[ ξ̇(t)=β Aξ(t)+bu(t)+β Q(γ)c^T(
    γ^n-1y_2-cξ(t))  ,;                              u(t)=-β b^TP(γ)ξ(t),;                                  γ=γ(t)=T/T-tγ_0. ].

Then, for any initial condition and all t∈0,T), the states
x_0(t), x(t), ξ(  t), and controls u_0(t),
u(  t) of the closed-loop system consisting of (<ref>) and
(<ref>) satisfy, (<ref>), (<ref>) and

    ‖ x(  t)  ‖   ≤υ_3e
    ^υ_30| x_0| t(  T-t)  ^3/2(  ‖ x(  0)  ‖ +‖ξ(
    0)  ‖)  ,
    ‖ξ(  t)  ‖   ≤υ_4e^υ_40| x_0| t(  T-t)
    ^3/2(  ‖ x(  0)  ‖ +‖ξ(  0)  ‖)  ,
    ‖ u(  t)  ‖   ≤υ_5e
    ^υ_50| x_0| t(  T-t)  ^1/2(  ‖ x(  0)  ‖ +‖ξ(
    0)  ‖)  ,

where υ_3, υ_30, υ_4, υ_40,
υ_5 and υ_50 are some positive constants, namely,
Problem <ref> is solved.



In this proof, if not specified, we drop the dependence of variables on t,
for example, we use P=P(γ(t)) and Q=Q(γ(t)). Consider the
x-subsystem in (<ref>). Denote e=z-ξ. By using (<ref>), the
closed-loop system can be written as

    ξ̇     =β (  A-bb^TP)  ξ+β Qc^Tce,
    ė     =β Ae+θ Az+ψ-β Qc^Tce.

Choose the Lyapunov-like function

    V_ξ(  t,ξ(t))  =γ(t) ξ^T(t)P(γ(t))ξ(t),

whose time-derivative along system (<ref>) can be written as

    V̇_ξ(  t,ξ )  =    γ̇ξ^T
    Pξ+γγ̇ξ^TdP/dγξ+2βγξ^TPQc^Tce
         +βγξ^T(  A^TP+PA-2Pbb^TP)  ξ
    ≤   γ̇ξ^TPξ+δ_cγ̇/nξ^TPξ
         -βγ^2ξ^TPξ+2βγξ^TPQc^Tce,

where we have used (<ref>) and (<ref>). By using Young's
inequality with k_2>0, it follows that

    2βξ^TPQc^Tce≤ k_2βγξ^TPξ+β/k_2γe^Tc^TcQPQc^Tce.

In addition, according to Theorem 2 in <cit.>, we have
c^TcQPQc^Tc≤γ^2n+2nk_3Q^-1,
where k_3=cQ_nP_n Q_nc^T.

Therefore, (<ref>) can be continued as

    V̇_ξ(  t,ξ)  ≤   γ̇ξ^T
    Pξ+δ_cγ̇/nξ^TPξ-βγ^2ξ^TPξ+k_2βγ^2ξ^T
    Pξ
         +β/k_2γ^2n+2nk_3e^TQ^-1e
    
    =     (  γ̇/γ+δ_cγ̇/nγ-βγ+k_2βγ)  γξ^TPξ
         +β/k_2nk_3γ^2n+2e^TQ^-1e.

Take another Lyapunov-like function

    V_e(  t,e(t))  =γ^2n+1(t)e^T(t)Q^-1(γ(t))e(t),

whose time-derivative along system (<ref>) can be written as

    V̇_e(t,e)=       (2n+1)γ^2nγ̇e^T
    Q^-1e+γ^2n+1γ̇e^TdQ^-1/dγe
         +γ^2n+1(β Ae+θ Az+ψ-β Qc^Tce)^TQ^-1e
         +γ^2n+1e^TQ^-1(β Ae+θ Az+ψ-Qc^Tce)
    ≤     (2n+1)γ^2nγ̇e^TQ^-1e-γ̇/nγ^2ne^TQ^-1e
         +γ^2n+1(β Ae+θ Az+ψ-β Qc^Tce)^TQ^-1e
         +γ^2n+1e^TQ^-1(β Ae+θ Az+ψ-β
    Qc^Tce)
    ≤   (  2n+1-1/n)  γ^2nγ̇e^TQ^-1e+2γ^2n+1e^TQ^-1ψ
         +γ^2n+1β e^TQ^-1(QA^T+AQ-2Qc^TcQ)Q^-1e
         +2θγ^2n+1e^TQ^-1Az,

where we have used (<ref>). By using Young's inequalities with k_4>0
and k_5>0, we can get

    2z^TA^TQ^-1e     ≤k_4/γ
    z^TA^TQ^-1Az+γ/k_4e^T
    Q^-1e,
    
    2e^TQ^-1ψ   ≤k_5/γψ^T
    Q^-1ψ+γ/k_5e^TQ^-1e.

On the one hand, it follows from Lemma <ref>, (<ref>) and (<ref>)
that

    ψ^TQ^-1ψ     =ψ^T(  γ^2n-1
    L_n^-1Q_nL_n^-1)  ^-1ψ
       ≤Λ/γ^2n-1(  L_nψ)
    ^T(  L_nψ) 
       ≤d^2Λ/γ^2n-3(  L_nz)
    ^T(  L_nz)  .

On the other hand, in light of z=ξ+e, (<ref>), (<ref>) and
(<ref>), we can get

    (  L_nz)  ^T(  L_nz)        =(
    L_n(  ξ+e)  )  ^T(  L_n(
    ξ+e)  ) 
       ≤2(  L_nξ)  ^T(  L_nξ)
    +2(  L_ne)  ^T(  L_ne) 
       ≤2Λ(  ξ^TPξ/γ
    +γ^2n-1e^TQ^-1e)  .

Therefore, we have

    ψ^TQ^-1ψ   ≤2d^2Λ^2/γ^2n-3(  ξ^TPξ/γ+γ
    ^2n-1e^TQ^-1e) 
         =2d^2Λ^2/γ^2n-2ξ^T
    Pξ+2d^2Λ^2γ^2e^TQ^-1e.

In view of L_nA=AL_nγ, (<ref>) and (<ref>), it yields that

    A^TQ^-1A       =A^T1/γ^2n-1L_n
    Q_n^^-1L_nA
         =1/γ^2n-3L_nA^TQ_n^^-1AL_n
       ≤λ_max(  A^TQ_n^^-1A)  1/γ^2n-3L_nL_n
         =λ_max(  A^TQ_n^^-1A)  Λ1/γ^2n-3L_nΛ^-1L_n
       ≤λ_max(  A^TQ_n^^-1A)
    Λ1/γ^2n-3L_nQ_n^^-1L_n
         =λ_max(  A^TQ_n^^-1A)  Λγ^2Q^-1≜ k_8γ^2Q^-1.

Again in view of z=ξ+e, (<ref>), and (<ref>), it follows that

    z^TQ^-1z       =(  ξ+e)  ^TQ^-1(
    ξ+e) 
       ≤2ξ^TQ^-1ξ+2e^TQ^-1e
       ≤2/γ^2n-1ξ^TL_nQ_n^^-1L_nξ+2e^TQ^-1e
       ≤2Λ/γ^2n-1ξ^TL_nL_nξ+2e^TQ^-1e
       ≤2Λ^2/γ^2n-1ξ^TL_n
    P_nL_nξ+2e^TQ^-1e
         =2Λ^2/γ^2nξ^TPξ
    +2e^TQ^-1e.

Then we have

    z^TA^TQ^-1Az     ≤ k_8γ^2z^TQ^-1z
       ≤2Λ^2k_8/γ^2n-2ξ^T
    Pξ+2k_8γ^2e^TQ^-1e.

With the above results and (<ref>), (<ref>) can be written as

    V̇_e(  t,e)  ≤   (  2n+1-1/n)
    γ^2nγ̇e^TQ^-1e
         -γ^2n+2β e^TQ^-1e+|θ|γ^2nk_4z^TA^TQ^-1Az
         +|θ|1/k_4γ^2n+2e^TQ^-1e+γ^2nk_5ψ^TQ^-1ψ
         +1/k_5γ^2n+2e^TQ^-1e
    ≤   (  (  2n+1-1/n)  γ̇/γ-βγ)  γ^2n+1e^TQ^-1e
         +(  2k_5d^2Λ^2+1/k_5)
    γ^2n+2e^TQ^-1e
         +2|θ|Λ^2k_4k_8γ^2ξ^TPξ
         +|θ|(  2k_4k_8+1/k_4)
    γ^2n+2e^TQ^-1e
         +2k_5d^2Λ^2γ^2ξ^TPξ.

Take the total Lyapunov-like function

    𝒱=𝒱(  t,ξ,e)  =V_ξ(  t,ξ)
    +k_7V_e(  t,e)  ,

where k_7 is a positive constant to be designed later. It can be calculated from
(<ref>) and (<ref>) that

    𝒱̇=     V̇_ξ(  t,ξ)  +k_7V̇
    _e(  t,e) 
    ≤   (  γ̇+δ_cγ̇/n
    -βγ^2+k_2βγ^2)  ξ^TPξ
         +β/k_2γ^2n+2nk_3e^TQ^-1e
         +k_7(  (  2n+1-1/n)  γ̇-βγ
    ^2)  γ^2ne^TQ^-1e
         +k_7(  2d^2Λ^2k_5+1/k_5)
    γ^2n+2e^TQ^-1e
         +2|θ|Λ^2k_4k_7k_8γ^2ξ^TPξ
         +|θ|(  2k_4k_8+1/k_4)
    k_7γ^2n+2e^TQ^-1e
         +2d^2Λ^2k_5k_7γ^2ξ^TPξ
    
    =     μ_ξξ^TPξ+μ_ek_7γ^2ne^TQ^-1e+2|θ|Λ^2k_7k_4
    k_8γ^2ξ^TPξ
         +|θ|/k_4k_7γ^2n+2
    e^TQ^-1e+2k_4k_8|θ| k_7γ^2n+2e^TQ^-1e,

where

    μ_ξ=     γ̇+δ_cγ̇/n
    -βγ^2+k_2βγ^2+2d^2Λ^2k_5
    k_7γ^2
    μ_e=     (  2n+1-1/n)  γ̇-βγ
    ^2+1/k_5γ^2
         +2d^2Λ^2k_5γ^2+β nk_3/k_2k_7γ^2.

Take the parameters as k_2=1/2, k_5=1/(√(2)dΛ) and
k_7=4nk_3. Then it follows from (<ref>) that

    μ_ξ=     (  1+δ_c/n)  γ̇-(  1-k_2)  βγ^2+2d^2Λ^2k_5
    k_7γ^2
    ≤   (  1+δ_c/n)  γ̇-1/2β_1γ^2+4√(2)dΛcQ_nP_nQ_n
    c^Tγ^2
    
    =     (  1+δ_c/n)  γ̇-(
    1+δ_c/n)  γ^2-γ^2 = -γ^2,
    μ_e=     (  2n+1-1/n)  γ̇-(
    1-nk_3/k_2k_7)  βγ^2+1/k_5γ
    ^2
         +2d^2Λ^2k_5γ^2
    ≤   (  2n+1-1/n)  γ̇-1/2β
    _2γ^2+2√(2)dΛγ^2
    
    =     (  2n+1-1/n)  γ̇-(  2n+1-1/n)  γ^2-γ^2
    =   -γ^2.

Therefore, we have

    𝒱̇(  t,ξ,e)  ≤     -γ𝒱(
    t,ξ,e)  +2|θ|Λ^2k_7
    k_4k_8γ^2ξ^TPξ
         +|θ|1/k_4k_7γ^2n+2
    e^TQ^-1e
         +2|θ| k_4k_8k_7γ^2n+2e^TQ^-1e
    
    =       -γ𝒱(  t,ξ,e)  +6|
    x_0(0)|/T^2Λ^2k_7k_4k_8γξ^TPξ
         +3βΛ^2k_7k_4k_8γξ^TPξ
         +3(  | x_0(0)|/T^2+β/2)  1/k_4k_7γ^2n+1e^TQ^-1e
         +3(  2| x_0(0)|/T^2+β)
    k_4k_8k_7γ^2n+1e^TQ^-1e
    ≤     -γ𝒱(  t,ξ,e)  +λ𝒱(
    t,ξ,e)  ,

where

    λ=       3(  2| x_0(0)|/T^2
    +β)  Λ^2k_7k_4k_8+3(  |
    x_0(0)|/T^2+β/2)  1/k_4
         +3(  2| x_0(0)|/T^2+β)
    k_4k_8,

which, together with the comparison lemma in <cit.>, indicates

    𝒱(t,ξ(t),e(t))≤(T-t)e^λ t𝒱(0,ξ
    (0),e(0)),   t∈0,T).

It can be obtained from (<ref>) and (<ref>) that

    V_ξ(  t,ξ )        =γξ^TP(
    γ)  ξ≥γ^2ξ^TL_n(  γ_0)  P_n
    L_n(  γ_0)  ξ
       ≥γ^2λ_min(  L_n(  γ_0)
    P_nL_n(  γ_0) )  ‖ξ‖
    ^2,
    
    V_e(  t,e)        =γ^2n+1e^TQ^-1(
    γ)  e
    ≥γ^2e^TL_n(  γ_0)  Q_n^^-1L_n(  γ_0)  e
       ≥γ^2λ_min(  L_n(  γ_0)
    Q_n^^-1L_n(  γ_0)  )  ‖ e‖
    ^2,

which indicates that

    𝒱(  t,ξ,e)  =V_ξ(  t,ξ)  +k_7
    V_e(  t,e)  ≥γ^2ω_1(  γ_0)
    ‖χ‖ ^2,

where χ=[ξ^T,e^T]^T and ω
_1(γ_0)=min{λ_min(L_n(γ_0)P_n L_n
(γ_0)),λ_min(L_n(γ_0)Q_n^^-1L_n(γ
_0))}. Therefore, it yields that

    ‖χ (  t) ‖ ^2   ≤𝒱
    (  t,ξ(t),e(t))  /γ^2(  t)  ω_1(
    γ_0)  ≤(  T-t)  ^3e^λ t𝒱(
    0,ξ(0),e(0))  /ω_1(  γ_0)  
       ≤(  T-t)  ^3ω_2(  γ_0)
    /T^2ω_1(  γ_0)  e^λ t‖χ(  0) ‖ ^2,

where ω_2(γ_0)=max{λ_max(L_n(γ_0
)P_nL_n(γ_0)),λ_max(L_n (γ_0)Q_n^^-1
L_n(γ_0))}. Thus (<ref>) is proven. Moreover, according to
(<ref>) and e=z-ξ, it follows that

    ‖ x(t)  ‖   ≤‖ L_n(
    T-t)  ‖‖ z(t)  ‖
       ≤‖ L_n(  T-t)  ‖(  ‖
    e(  0)  ‖ +‖ξ (0)  ‖) 
       ≤2‖ L_n(T)  ‖‖χ(
    t)  ‖
       ≤2‖ L_n(  T)  ‖(  T-t)
    ^3/2√(ω_2(  γ_0)  )/T√(ω_1(  γ_0)  )e^λ t/2‖χ(  0)  ‖
       ≤2‖ L_n(  T)  ‖(  T-t)
    ^3/2√(ω_2(  γ_0)  )/T√(ω_1(  γ_0)  )
       ×e^λ t/2(  ‖ L_n(
    1/T)  ‖‖ x(  0)  ‖
    +2‖ξ(  0)  ‖)  ,

which proves (<ref>).

Finally, by using (<ref>), (<ref>) and (<ref>), we can obtain
that

    ‖ u(  t)  ‖ ^2     =β^2ξ^TP(  γ )  bb^TP(  γ)  ξ
       ≤β^2b^TP(  γ )  bξ^TP(  γ )  ξ
         =β^2nγξ^TP(  γ)  ξ
      =β^2nV_ξ(  t,ξ) 
       ≤β^2n𝒱(  t,ξ,e) 
       ≤β^2n(  T-t)  e^λ t𝒱(
    0,ξ(0),e(0)) 
       ≤β^2nγ_0^2ω_2(  γ_0) (
    T-t)  e^λ t‖χ (  0)  ‖
    ^2
       ≤β^2nγ_0^2ω_2(  γ_0) (
    T-t)  e^λ t
       ×(  ‖ L_n(  1/T)  ‖‖ x(  0)  ‖ +2‖ξ(  0)
    ‖)  ,

which prove (<ref>). The proof is finished.


For nonholonomic systems, compared to time-invariant discontinuous feedback,
such as <cit.> and their citations,
there are few related results on smooth time-varying feedback. To the best of
the authors' knowledge, there are no smooth time-varying feedback results for
prescribed-time control of nonholonomic systems available in the literature.

In this paper, on the one hand, different from the traditional discontinuous
control methods <cit.>, the
proposed methods are smooth, time-varying and even linear. On the other hand,
different from smooth time-varying methods
<cit.>, where only the state
feedback controllers were considered and drive the state to converge to zero
exponentially, the proposed controllers are not only based on state feedback
but also on output feedback, and can drive the state to converge to zero
globally at any prescribed time. In addition, different from
<cit.>, the parameters in our controllers do not depend
on the initial state of the x_0-subsystem.



§ A NUMERICAL EXAMPLE


In this part, we consider the uncertain bilinear model
<cit.>

    ẋ_0=(  1-ε^2/2)  v,  ż_1
    =z_2v,  ż_2=u+φ(  z_1)  ,

where ε is the bias in orientation and is assumed to be much
smaller than one, v and u are the two controls, and φ(
z_1)  =z(  1+θ_1^2) with θ_1=θ
_1(t) being an uncertain function.

Denote the output y=[x_0,z_1]^T and

    u_0     =(  1-ε^2/2)  v,   x_1
    =z_1,   x_2=2/2-ε^2z_2,
    
    u_1     =2/2-ε^2u,  ϕ (  x_1)
    =2/2-ε^2φ (  x_1)  ,

the system (<ref>) can be expressed as

    ẋ_0=u_0,  ẋ_1=x_2u_0,  ẋ_2=u_1+ϕ (
    x_1)  ,   y=[x_0,x_1]^T.


For this system, the existing results are all based on the σ process (see
<cit.>), that is, x_0(t) appears on the denominator (see
<cit.>). When x_0(t) is close to zero, this may cause
singular problems. For the situation when x_0(0)=0, most of the existing
methods rely on using switching controls (see
<cit.>), and the control strategy proposed in this paper
is workable for all initial states of x_0(t), thus having greater potential for application.






In the simulation, we take ε=0.1, [x_0(0),z_1(0),z_2
(0)]=[0,-1,1], [ξ_1(0),ξ_2(0)]=[0,0], θ_1=sin(t),
T=2.5s and β=100. The states and controllers are plotted in Fig.
<ref>, from which it can be observed that the proposed method can
indeed drive the states and controllers to converge to zero at the prescribed
time T=2.5s.





§ CONCLUSION


The smooth prescribed-time control problem for uncertain chained nonholonomic
systems was solved in this paper. With a novel smooth time-varying state
transformation, uncertain chained nonholonomic systems were reformulated as
uncertain linear time-varying systems. By fully utilizing the properties of a
class of parametric Lyapunov equations and constructing time-varying
Lyapunov-like functions, smooth time-varying high-gain controllers were
constructed. It was proved that the states and controllers can converge to zero at
the prescribed time. Both state feedback and observer-based output feedback
were considered. The effectiveness of the proposed methods was verified by a
numerical example.

99                                                                                               


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