Practical Prescribed Time Tracking Control With Bounded Time-Varying Gain Under Non-Vanishing Uncertainties

Dahui Luo , Yujuan Wang , and Yongduan Song ,,

Abstract—This paper investigates the prescribed-time control(PTC) problem for a class of strict-feedback systems subject to non-vanishing uncertainties.The coexistence of mismatched uncertainties and non-vanishing disturbances makes PTC synthesis nontrivial.In this work, a control method that does not involve infinite time-varying gain is proposed, leading to a practical and global prescribed time tracking control solution for the strict-feedback systems, in spite of both the mismatched and nonvanishing uncertainties.Different from methods based on control switching to avoid the issue of infinite control gain that involves control discontinuity at the switching point, in our method a softening unit is exclusively included to ensure the continuity of the control action.Furthermore, in contrast to most existing prescribed-time control works where the control scheme is only valid on a finite time interval, in this work, the proposed control scheme is valid on the entire time interval.In addition, the prior information on the upper or lower bound of gi is not in need,enlarging the applicability of the proposed method.Both the theoretical analysis and numerical simulation confirm the effectiveness of the proposed control algorithm.

I.INTRODUCTION

SINCE the original work on prescribed time control (PTC)by Songet al.[1] for high order nonlinear systems, PTC has attracted considerable attention from the control community during the past few years [2]–[16].The salient feature of PTC lies in its ability to achieve closed loop system stability within finite time that is independent of system initial conditions and thus can be pre-specified arbitrarily [1].As the convergence rate is one of the most important factors for any control system, PTC is of particular interest for time-critical systems, e.g., emergency braking [17], missile interception [18],spaceship docking [19], and so forth.Compared with traditional finite-time control [20]–[28] and fixed time control[29]–[31] methods where the settling time is not at user’s disposal, PTC has its superiority, and thus has motivated numerous following up studies and extensions since its introduction,including cooperative prescribed-time control for networked multi-agent systems under local communication condition [2],[3], prescribed-time stabilization for strict feedback-like systems with mismatched uncertainties [4], prescribed-time tracking control for nonlinear multi-input and multi-output(MIMO) systems [5], and the inverse prescribed-time optimality control for stochastic strict-feedback nonlinear systems [6],etc.

The prescribed-time control method is systematically proposed by Songet al.[1] upon using infinity time-varying gain that diverges to being unbounded as the time approaches the user-defined terminal time.Then, this methodology is extended to various systems or control problems ([1], [2],[4]–[6]).However, numerical problems during the implementation of the controller may be encountered due to any disturbance/noise that is amplified, rendering the PTC method somewhat unpractical.Recently, there have been some efforts devoted to bounded time-varying gain (BTG) based PTC control methods, such as [32]–[35].Orlovet al.[32], [33] constructed a prescribed-time robust differentiator and observer upon using BTG.In [34], a predefined-time method based on BTG is proposed for arbitrary-order differentiators.However,it is rather difficult to extend the methods [32]–[34] to more general systems with unknown and time-varying control gain.More recently, a novel approach based on BTG is proposed in[35] that naturally links finite time control with prescribed time control, avoiding the numerical implementation problem associated with the current PTC method.Nevertheless, the transition time before the prescribed time still depends on the systems’ initial states and parameters, which is the first motivation of this work that integrates BTG into the controller design, allowing it to be implemented without the need for control switching.

Also, it is worth noting that most existing PTC methods based on state or time transformation are invalid beyond the prescribed time interval.Efforts have been made in allowing PTC to be functional beyond the prescribed time.In [7], the prescribed-time consensus and containment control problem of multi-agent systems is addressed upon using piecewise continuous time-varying scaling function.This idea has been extended to consensus tracking control of nonlinear multiagent systems satisfying the time-varying Lipschitz condition in [8].In addition, [9] addresses the prescribed-time cooperative guidance control problem with input saturation.In the aforementioned works [7]–[9], although the methods are valid on the whole time interval, the switching mechanism is involved.This is the second motivation of the work that establishes a new method allowing the system to continue operating beyond the prescribed time interval.

On the other hand, a precise system model is always hard to obtain under unpredictable internal/external environment,which also brings about unmatched uncertainties in the model.Although prescribed time control methods have been developed for strict-feedback systems with unmatched uncertainties [11], [12], those methods are based on vanishing uncertainties/disturbances.Given the presence of non-vanishing uncertainties, it is hard to ensure that the states of the practical system converge to the point of origin after a prescribed time [11], [12].In practice, system uncertainties and external disturbances (which are possibly non-vanishing) are inevitable, and it is highly desirable to investigate the control problem for such systems subject to non-vanishing uncertainties with high control precision and fast convergence speed.In fact, there have been a lot of efforts devoted to the control problem for systems subject to the non-vanishing uncertainties [5], [36]–[38].However, the control results for the systems subject to non-vanishing uncertainties with high control accuracy are scare.Further, it is nontrivial to extend the aforementioned stabilization methods to tracking control for general nonlinear systems with mismatched yet non-vanishing uncertainties.This is mainly because among those infinite time-varying gain based system transformation methods, it is not easy to ensure the boundedness of virtual controllers that involve the differential product of infinite time-varying gain function and non-vanishing terms through the backstepping design process.This is the third motivation in the work,where, instead of using state transformation, we introduce BTG directly into controller design and Lyapunov function based analysis.

The above analysis indicates that although rich results on PTC have been reported during the past few years, at least three major issues have not been adequately addressed: 1)practicality of PTC; 2) accommodation of gains and rejection of mismatched yet non-vanishing uncertainties; and 3) operational capability beyond the settling time.In this paper, we present a method aimed at addressing those issues simultaneously.The main contributions of this paper can be summarized as follow:

1) Different from current PTC which normally relies on time-varying feedback control gain growing limitlessly with time and becoming infinite at equilibrium, the proposed method, with the aid of a time-varying scaling function that grows monotonically with time and maintains boundedness at and beyond the settling time, does not involve infinite control gain anytime during system operation making PTC practical and linking PTC with its practical version analytically;

2) Without using any prior information on the upper or lower bound of system control gain, the proposed control is able to settle the tracking error in the neighborhood of the origin within the prescribed time; and

3) The developed solution is truly global, allowing the system to operate on the entire time interval, yet is able to deal with non-vanishing and mismatched uncertainties, leading to practical and global prescribed time tracking control solution for a larger class of nonlinear systems.In addition, a compensation mechanism based on a computable softening unit is introduced to guarantee the continuity of the control action,which solves the infinite control gain problem without involving discontinuity at the switching point.

Notations: Throughout this paper,t0denotes the initial time;Rdenotes the set of real numbers;L∞:={χ(t)|χ:R+→R,supt∈R+|χ(t)|<+∞}.We denote by χ(i)theith derivative of χ,and by χitheith power of χ.

II.PROBLEM FORMULATION AND PRELIMINARIES

A. Problem Formulation

In this paper, we consider a class of strict-feedback systems with non-vanishing uncertainties, which is modeled by

wherexi(t)∈R denotes the state withxˉi=[x1,...,xi]Tbeing the state vector,u(t)∈R denotes the control input andyis the output,fi(xˉi,t)∈R denotes the lumped uncertainty, which is unknown non-vanishing smooth function, andgi∈R is the unknown time-varying control gain.The planar systems (1)are generally employed to describe the dynamics of practical systems such as the “wing-rock” unstable motion of some high-performance aircraft [39], electromechanical system[40], etc.

The objective here is to develop a control strategy for the system (1) such that the output signaly(t) closely synchronizes withxr(t), i.e., the tracking errore(t)=y(t)-xr(t) converges to a small residual set containing origin within prescribed time and maintains synchronization thereafter.

To this end, we impose the following assumptions on system model (1) and the desired trajectoryxr(t), respectively.

Remark 1: With Assumption 1, the resultant control scheme becomes more practical and more elegant because it does not require the upper bound or lower bound of the virtual or actual control gaingi(i=1,...,n), in contrast to most of existing schemes that normally demand certain prior bound information on the virtual and/or the actual control gains ([1], [16],[37]).

Assumption 2: For the non-vanishing uncertain termfi(,t)(i=1,...,n), there exists an unknown constant νi>0 and a known continuously differentiable scalar function?i()such that

In addition, ?i() is either bounded unconditionally for anyor bounded only ifis bounded.

Remark 2: In Assumption 2, ?i() (i=1,...,n) denotes a computable scalar function carrying “core” information of the system that is independent of system parameters.

Assumption 3: For allt∈[t0,+∞), the desired trajectoryxr(t)∈R and itspth (p=1,...,n-1) order derivatives are known, bounded, and piecewise continuous.

Remark 3:Assumption 3 is a commonly required condition in addressing the tracking control problem (see [5], [13],[37]).

B. The Preliminaries

Inspired by [1], we put forward a time-varying function defined on the whole time interval as follows:

fort∈[t0,T?+t0-?), it holds that

2) if

fort∈[T?+t0-?,+∞), it holds that

Remark 4: Lemma 1 allows us to leverage prescribed time control to its practical (executable) version, making it possible to address the tracking control (not just regulation) problem for strict feedback systems with unknown time varying control gains and mismatched yet non-vanishing uncertainties.

Lemma 2: Let ?(t) be an unknown bound function and//?//[t0,t]=supτ∈[t0,t]|?(τ)|.If a continuously differentiable function χ (t) satisfies

withk>0, then

1) Fort∈[t0,T?+t0-?),

2) Fort∈[T?+t0-?,+∞),

Proof: 1) The solution of the differential (13) ont∈[t0,T?+t0-?)is derived as

2) Upon using (13) and (3), we can obtain that

which yields

It is noted that χ (t) is bounded from (15).

Lemma 3[13]: Consider the following one-order differential equation:

where χ2(t) is a nonnegative function,a>0,b>0.Then, for any given positive initial state χ1(t0)≥0, the associated solution χ1(t)≥0 holds for ?t≥t0.

Lemma 4[8]: For any given vectors η1, η2∈Rm, the following inequality (i.e., Young’s Inequality) holds:

where μ is defined as in (3) and γ>0 is a user-design constant.

Remark 5: Lemma 2 is crucial in guaranteeing the boundedness of the updated parameter utilized in the proposed adaptive control scheme.With the aid of Lemma 3, such an estimated parameter is also ensured to be non-negative.In developing Lemma 4, we purposely introduce the design variable μ through Young’s inequality, which plays an important role in our later stability analysis.

III.MAIN RESULT

A. Controller Design

The error surfacezi(i=1,...,n) is introduced as

where αi-1(t)∈R is the virtual control.LetT=T?+t0-?.The main procedure is divided into two stages:t∈[t0,T) andt∈[T,+∞).

Stage 1:t∈[t0,T).

Step 1: From the first equation of (1) and the expression of α1in (17), we can obtain that

Then, the time derivative ofalong (17) is

With the help of Lemma 4, it is not difficult to get that

where γ1>0 andr1>0 are constants.

By inserting (20)-(22) into (19), it follows that:

where ω1=max{ν21} is the unknown virtual constant,ψ1=?21and ?1=|x˙r|2are the computable bounded functions.

The Lyapunov function is chosen as

Thevirtual control α is designed as

with the adaptive law

By combining (28) and (29), (25) can be rewritten as

Step 2: The time derivative ofz2=x2-α1is

The second Lyapunov function is chosen as

Similar to the analysis procedures (20)-(22), we have

The virtual controller α2is designed as

with the adaptive law

By inserting (35) in (33), we get that

Substituting (34)-(37) into (33), we obtain

Step i(i=3,...,n): Theith Lyapunov function is chosen as

with the adaptive law

By inserting the virtual and actual control inputs (40) as well as the adaptive law (41) into the derivative of the Lyapunov function, we then arrive at

Following the similar design procedure as in Stage 1, here we only give the expression form of virtual/actual controllers and adaptive laws.

The virtual controller α1is consistent with (26) as in the first step.It is noted that in the conventional design of the backstepping method, repeated derivatives of the virtual controller are involved.In such cases, the functionμdefined in (3) is a constant.The derivative ofμis zero so that ?i(i=2,...,n) is not uniform, which affects the design of virtual control strategies.Now, we introduce the following calculable control action softening unit:

to remedy the discontinuity of the control input att=T.To this end, the virtual/actual controller laws are designed as

Correspondingly, the parameter update laws are designed as

B. Theoretical Analysis

The main result is given in the following theorem.

Theorem 1: Consider system (1) under Assumptions 1-3.If the control strategies αi, u and the adaptive laws ︿ωiin Section III-A are applied, then the following objectives are achieved.

3) All the internal signals, including the virtual control αi,the control input u andziare continuous and remain uniformly bounded for allt∈[t0,+∞).

Proof:

Stage 1:t∈[t0,T)

We first prove that the output signaly(t) can track the desired reference trajectoryxr(t) within the prescribed timeT?.The Lyapunov function is chosen as

whose time derivative along (30), (33), and (42) is

According to Lemma 1, we derive from (49) that

In light of (48), we can obtain that

and

from which we see that αi∈L∞, (i=2,...,n) after which the control inputuis bounded.

Stage 2:t∈[T,+∞)

We prove that the output error remains bounded beyond the finite timeT.

The control strategies are proved to be continuous att=T.In light of the definition of the control action softening unit δi(T)in (43), we see that

At the same time, αi(i=1,...,n-1) anducan also be ensured to be continuous at the breakpointt=T.It is easily concluded that the control action softening unit δi(T) is a finite constant.Adding such a term to the virtual control αito ensure the continuity of the control action does not impact the subsequent time derivative of the virtual control in stepi+1 because this term is a constant value.

In the following, we analyze the system convergence on[T,+∞).

Remark 6: In both virtual controller αi(i=1,...,n-1) and the actual controlleruin Section III-A, only the finite timevarying gain μ(t) and finite constant gain are involved.Thus,the proposed control scheme does not involve infinite control gain.However, the control gaingi(i=1,...,n) involved in the system dynamic model (1) is supposed to be bounded in Assumption 1 to ensure the controllability of the system,which is commonly needed in any controllable system.

Remark 7: The virtual and actual controllers as given in Section III-A (αi,i=1,...,n) are continuous everywhere including“T=T?+t0-?”becaus(e itdepends ont)hecalculable control action softeningunitδi(T),i=1,...,n.Theidea of adding such a softening unit originates from [41] and is verified in (53).This is based on the fact that the derivative of time-varying function (3) is piecewise continuous.With this compensation, the potential discontinuity of the control at the time instancet=T?+t0-? is avoided gracefully.

Remark 8: Three salient features of the proposed method are worth mentioning: 1) The control scheme is valid on the whole time interval, making it different from existing PTC results (such as [1], [4]–[6], [15]) where the control gains grow unbounded as time approaches the terminal time and,consequently, the control schemes are only valid on [t0,T?)rather than on [t0,+∞).Although the work by [11] addresses the PTC problem on the whole time interval, it only considers regulation rather than tracking, and furthermore, the nonlinear function “f” is assumed to be known and vanishing; 2) The control scheme does not need either the upper or lower bound information of the virtual or actual control gaingi(i=1,...,n), while most existing PTC works ([1], [16], [37]) depend on availability of the bound of gains; and 3) different from the existing PTC works ([11], [13], [15]) where the control scheme is discontinuous at the prescribed timeT?, the proposed scheme remains continuous and bounded throughout the whole time interval.

Remark 9: It is noted that although most PTC methods are robust against model uncertainties, [11] is the first adaptive PTC for state regulation of strict feedback systems with known (unit) control gains and vanishing uncertainties,wherein control is switched off and the corresponding adaptive law stops updating at and beyond the terminal time.Whereas the proposed method is a practical robust and adaptive solution capable of dealing with output tracking for systems with unknown time varying gains and mismatched yet nonvanishing uncertainties.It is interesting to note that no priori information on virtual and actual control gains is required in building the control scheme and the feedback gain remains bounded anywhere during system operation without the need to switch off the control action.

IV.NUMERICAL SIMULATION

To verify the effectiveness of the proposed control method,two simulation examples are given.

A. Example 1 (Application Example)

To compare the performance of the proposed control strategy with that developed by [37], we employ the same model as that in [37] for the numerical simulation

The simulation results are shown in Fig.1.From Fig.1(a),we can see that the system states can track the desired trajectory within the finite time point 0.9swhich is less than the prescribed timeT?=1.2 s under different initial condition.In addition, precise tracking can be maintained after the prescribed-time and the system can remain operating beyondT?,distinguishing itself from the method in [1].From Fig.2, it is observed that the control signal is continuous throughout the time interval.From Fig.3, it is shown that the parameter estimate ︿ω is also bounded for any initial state.

Fig.1.The performances of state x under different initial conditions.

The simulation results compared with that in [37] (all parameters are consistent) are shown in Figs.4 and 5.The performance ofx1(t) is shown in Fig.4, from which we can seethat the outputx1can trackthedesired trajectoryxrwithin theprescribed timeT?=0.8s.Theerrorbetween statex1andxris shown in Fig.5(a) and the error betweenx2andx˙ris shown in Fig.5(b).This is the same as that in [37], and further, the precise tracking can be maintained after the prescribed timeT?.From the simulation results in Fig.5, we can see that an excellent tracking control performance is guaranteed under the proposed control scheme, where the tracking error is on the verge of zero att=0.6 s before prescribed timeT?, distinguishing itself from that in [37].

Fig.2.The practical control input signal of u under different initial conditions.

Fig.3.The performances of ω? with different initial conditions.

Fig.4.Compared with [37], the performances of x1,([x1(t0), x2(t0)]=[0.5,0.3],?=0.2, T?=0.8 s, t end=1.5 s.

B. Example 2 (Numerical Example)

We conduct the simulation on the following strict-feedback nonlinear systems from [42] under Assumptions 1 and 2:

Fig.5.The tracking error e under different control schemes.

V.CONCLUSION

Fig.6.The performance of x1 under different initial conditions in Example 2.

Fig.7.The tracking error y-xr under different initial conditions in Example 2.

Fig.8.The performances of u under different initial conditions in Example 2.

In this paper, a practical prescribed time tracking control method is proposed for a class of strict-feedback systems with mismatched yet non-vanishing uncertainties.The method proposed here, by means of BTG, avoids the numerical problem during the implementation of the controller, which makes PTC practical and bridges PTC and its executable version analytically.It is shown that, without the need for any prior control gain information of system, the tracking error between the output of systems and desired trajectory settles in the neighborhood of origin within the pre-assigned time regardless of the initial condition and other design parameters and particularly, and the neighborhood of origin can be pre-given arbitrarily by simply adjusting the control parameters if the bounds ofgare known.The developed solution is truly practical and global, allowing the systems to operate throughout the whole time interval and the states to be started at any initial state.The extension of the proposed PTC method to cooperative control of multi-agent systems under directed topology or to the consensus tracking control of heterogeneous multiagent systems represents two interesting topics for future research.

ig.9.The performances of adaptive parameter ω? under different initial conditions.