Semiglobal Finite-Time Trajectory Tracking Realization for Disturbed Nonlinear Systems via Higher-Order Sliding Modes

This paper investigates an alternative nonrecursive finite-time trajectory tracking control methodology for a class of nonlinear systems in the presence of general mismatched disturbances. By integrating a finite-time disturbance feedforward decoupling process via higher-order sliding modes, it is shown that, a novel nonrecursive design framework resulting a simpler controller expression and easier gain tuning mechanism is presented. A new feature is that a quasi-linear inherent nonsmooth control law could be constructed straightforwardly from the system information, which is essentially detached from the determination of a series of virtual controllers. Moreover, by proposing a less ambitious semiglobal tracking control objective, the synthesis procedure can be achieved without restrictive nonlinear growth constraints. Explicit stability analysis is given to ensure the theoretical justification. A numerical example and an application to the speed regulation of permanent magnet synchronous motor are provided to illustrate the simplicity and effectiveness of the proposed nonrecursive control design approach.


I. INTRODUCTION
In this paper, we specify the control objective as to realize the finite-time exact tracking task for the following nonlinear system    ẋi (t) = x i+1 (t) + φ i (x i (t)) + d i (t), i ∈ N 1:n−1 , ẋn (t) = u(t) + φ n (x(t)) + d n (t), where xi = (x 1 , • • • , x i ) ∈ R i is the system partial state vector with i ∈ N 1:n (N j:i := {j, j + 1, • • • , i} where j and i are integers satisfying 0 ≤ j ≤ i), x = xn is the full state vector, y is the system output, φ i (•), i ∈ N 1:n is a known smooth nonlinear function (or, at least C n−i ), d i (t), i ∈ N 1:n represents a nonvanishing mismatched disturbance item.The This work is supported in part by National Natural Science Foundation of China (Nos.61503236, 61973080 and 61973081) and in part by the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning.(Corresponding author: Jun Yang) Chuanlin Zhang is with the College of Automation Engineering, Shanghai University of Electric Power, Shanghai, China, 200090 (e-mail: clzhang@shiep.edu.cn).
Leonid Fridman is with the Department of Control, Division of Electrical Engineering, Engineering Faculty, National Autonomous University of Mexico, Mexico City, Mexico, 04510 (e-mail: lfridman@unam.mx).
output reference signal, denoted by y r , and its n−th order derivative are assumed to be piecewise continuous, known and bounded.Without loss of generality, the initial time is set as zero.
The issue of exact trajectory tracking realization for nonlinear systems in the presence of mismatched uncertainties/disturbances has aroused great efforts in control community due to its significant application demands.Existing related results found in the literature could fall into two main categories: 1) nonlinear output regulation under a common assumption that the external disturbances are governed by certain deterministic exosystems, see.e.g., [1]- [3], etc.For instance, the disturbance is mostly supposed to be a harmonic one with unknown magnitude and phase but known frequency.In the case when the exosystems are completely unknown, the internal model is inaccessible in general owing to the missing information of the disturbance model; 2) backstepping design integrated with a HOSM observation/identification process, see, e.g., [4]- [8] and references therein.However, a well known side-effect of the backstepping based design approaches is the expanding complexity of the controller expression along with the increase of system order, which could possibly cause a costly implementation process.
Compared with the blossom of exact asymptotical tracking realization methods as partially mentioned above, it is also noted that there are fewer results in the literature to address the exact tracking problem for system (1) via an inherent nonsmooth (continuously non-differentiable) design, which could result in a finite-time convergence rate and stronger robustness [9].Consider the case when system (1) is presented with d i (t) = 0, finite-time control problem is actually well understood by referring to [10], [11], etc.However, it presents a nontrivial problem when the addressed nonlinear systems are perturbed by various non-vanishing disturbances, especially in a mismatched perturbation manner.Indeed, owing to the equilibrium drift caused by the adverse effects of disturbances, even asymptotical stabilization could not be achieved for system (1).In reference [12], the finite-time control problem can be solved under the assumption that φ i + d i is treated as a bounded lumped disturbance term.Nevertheless, the inherent nonlinearity characteristics are all sacrificed.A later result in [13] proposes a feedback domination method to solve the finite-time tracking problem for system (1), while a restrictive nonlinear growth constraint is necessarily required.It is also worth pointing out that, as one main character of all the existing related literatures mentioned above, a recursive design adopting series of virtual controllers is always essential to derive the final control scheme.Not surprisingly, for high order systems, recursive design will apparently cause heavy calculations of partial derivative terms and complex mathematical magnifying or reducing steps, see e.g., [5], [10], [12], [13], etc.
Following the discussions above, this paper is aiming to propose an alternative non-recursive synthesis strategy which could yield a simplified finite-time trajectory tracking controller design procedure.More distinguishably, unlike backstepping based approaches, it will be shown that the proposed control scheme could be straightforwardly derived from system (1) with largely reduced calculation burdens, owing to the fact that the proposed controller can be constructed separately from the Lyapunov function based stability analysis.To this end, firstly, we employ the higher-order sliding mode (HOSM) observer to provide a finite-time disturbance decoupling process.Secondly, we put forward a systematic feedfoward framework to transform the original system into a stabilizable system form, which facilitates the integration of homogeneous system theory.Thirdly, in order to present a simpler controller form and ease the practical implementations, a non-recursive composite control design strategy is therefore investigated.In this paper, it is also shown that, under a less demanding but more practical control objective, namely, semiglobal instead of global control, the restrictive nonlinearity growth constraints in most of the existing related continuous finite-time control works including [13] can be fully removed.In other word, an essential smooth condition of φ i in system (1) will be sufficient to derive an finite-time exact tracking control law.Theoretically, an explicit selection guideline for the bandwidth factor is formulated in a delicate semi-global attractivity analysis and the stability could be ensured after a rigourous contradiction argument.In addition, the proposed control law can also reduce to a simple linear composite controller only by tuning the homogeneous degree to zero, which could be regarded as a specific smooth control case.
Compared with existing related results, the main contribution is twofold: • An inherent nonsmooth control law could be constructed straightforwardly from the system information in a nonrecursive manner.Hence the controller could be essentially detached from the determination of a series of virtual controllers, which is a basic design principle of backstepping based approaches.
• By employing a less ambitious semi-global tracking control objective, the proposed nonsmooth synthesis procedure can be realized without any restrictive nonlinear growth constraints, which are currently essential in existing global control results.To demonstrate the simplicity and effectiveness of the proposed control design scheme, both a numerical example and an application to the speed regulation of Permanent Magnet Synchronous Motor (PMSM) are provided.
Definitions and Notations: i) The symbol C i denotes the set of all differentiable functions whose first ith time derivatives are continuous.R + represents the set of positive real numbers.A continuous function • a is defined by ii) (Weighted Homogeneity [14]): For fixed coordinates γ, a one-parameter family of dilation is a map Δ γ : R + × R n → R n , defined by Δ γ x = ( γ1 x 1 , • • • , γn x n ) for any constant ∈ R + .For a given dilation Δ γ and a real number τ , a continuous function

II. PROBLEM FORMULATION
To begin with, the following assumption of the disturbances is essentially required.
Assumption 2.1: In most of the existing output regulation results, exosystems of the disturbances are always employed as a pre-condition in order to present exact tracking results, see e.g., [2], [3], etc.In this paper, the boundedness assumption made on the disturbances is much more general from a practical point of view.
Firstly, inspired by the higher-order sliding mode observer design in Section 5, [15], the following observer can be built to realize an accurate estimation of the mismatched disturbances where . Combining (1) and ( 2), the error dynamics gives Lemma 2.1: (Theorem 5, [15]) Assume the observer gain λ i satisfies λ i > D, i ∈ N 1:n .For all possible well defined trajectories x(t), all signals in (3) are uniformly bounded and there exists a finite-time Provided that all the disturbance terms d i s are exactly known, we are thereafter able to define an auxiliary variable χi = (χ 1 , • • • , χ i ) , i ∈ N 1:n+1 , where each element χ i is determined by the following output regulation equations Note that ( 4) is clearly unaccessible in practice.However, with the corresponding estimates from the disturbance observer (2), replacing one can therefore obtain the following implementable state trajectory reference function where z

By defining a change of coordinates
where L ≥ 1 is a scaling gain to be made precise in the semi-global attractivity analysis later on, system (1) can be transformed to the following stabilizable form where Up to now, we are able to show that, without going through a series of recursive design steps, a simple controller can be explicitly pre-built of the following form

Remark 2.2:
In order to carry out a novel non-recursive controller design strategy, this paper proposes an alternative handling procedure with existing recursive design results, such as [1], [2], [13], etc.A direct benefit is that the controller can be directly derived as a simple form of (8) without going through the determination of a series of virtual controllers.It is also worth pointing out that by setting the homogeneous τ = 0, the proposed controller (8) reduces to a conventional linear state feedback control law.
Remark 2.3: As illustrated in the design procedure presented above, the proposed control methodology provides the control engineers a more practical synthesis manner.More distinguishably, the requirement of a Hölder continuous condition (or, homogeneous growth condition) on the system nonlinearities which are always employed in finite-time control related literatures is essentially relaxed, see e.g., [10], [13], etc.In addition, the proposed non-recursive composite control design strategy could largely facilitate the practical implementations, by recalling that existing backstepping based approaches always employ exhaustive recursive calculations within nondetachable step-by-step Lyapunov stability analysis, see for details in references [5], [12], etc.

III. MAIN RESULT
The main result of this paper can be summarized by the following theorem.
Theorem 3.1: Consider the closed-loop system consisting of (1) under Assumption 2.1 and the dynamic compensator ( 2)-( 8) with a sufficiently large scaling gain L. Then for any given constant ρ ∈ R + which could be arbitrarily large, all trajectories of x(t) starting from the compact set H [−ρ, ρ] n will converge to the equilibrium point within a finite-time.
Proof: Inspired by [14], [16], construct a candidate Lyapunov function with P being a positive definite and symmetrical matrix satisfying Λ P + P Λ = −I and Λ being a companion matrix of K.The time derivative of U (ξ) along the closed-loop system ( 7)-( 8) is given by By the definition of x * i in (5) and Lemma 2.1, we know that for any well defined x(t), the signal x * i is uniformly bounded, that is, there exists a constant ρ > 0 such that max i∈N 1:n {sup t≥0 {x * i (t)}} ≤ ρ.Thereafter, for a given compact set H On the other hand, with L ≥ 1 and the relation (6) in mind, we know that ∀η(t) ∈ Ω ⇒ ξ(t) ∈ Ω.
In order to proceed, the following propositions, whose proofs are included in the Appendix, are required.
Proposition 3.1: There exist a constant α ∈ R + and a constant ς Δ γ holds for η ∈ Ω.With Lemma 2.1 and Assumption 2.1 in mind, we know that for any well defined x(t), ∀0 ≤ t < T 1 , there exists a bounded constant Γ ∈ R + , such that max obtained for constants α ∈ R + and Using Lemma A.1 again, the following relation holds with Then for any arbitrarily small tolerance δ ∈ (0, c 0 /2), now we are able to choose the scaling gain L ≥ 1 under the following guideline In what follows, we will first show the uniform boundness of trajectory ξ, and then prove that a local finite-time convergence can be achieved.
If the above statement is not true, that is, at least one trajectory of η(t) will escape Ω within a finite-time.Regarding the finite-time escaping phenomenon, two cases described in Fig. 1
In a summary of the above two cases, we can arrive at the conclusion that ∀x(0 2) Local finite-time convergence: Now it is true that the relation ( 15) also holds for t ∈ [0, T 1 ].With this relation in mind, we know that any trajectory of the closed-loop system ( 7)-( 8) will be well defined.In the case when t ≥ T 1 , it concludes from Lemma 2.1 that ε i = 0, t ≥ T 1 , i ∈ N 1:n .Based on ( 13) and (15), one can also have By Lemma A.2 and with the fact that 0 < 2 2−τ < 1 in mind, the relation (17) leads to a straightforward conclusion that there exists another time instant T 2 > T 1 > 0, such that y(t)− y r = 0, t ∈ [T 2 , ∞).This completes the proof of Theorem 3.1.
Remark 3.1: In the proof of Theorem 3.1, a delicate contradiction argument is employed to guarantee the avoidance of finite-time escaping phenomenon.Under the framework of non-recursive homogeneous domination approach, we first show that under the guideline (13), the semi-global attractivity of the level set Ω can be ensured via a contradiction argument, and then all signals in the closed-loop system will be uniformly bounded.Moreover, by Lyapunov function based analysis, the finite-time convergence property of the system states is eventually guaranteed.

A. A Numerical Example
Example 4.1: Consider the following disturbed nonlinear system: where d 1 (t) and d 2 (t) are mismatched and matched disturbances, respectively.The control objective is to realize finite-time exact tracking of a given reference signal y r = 1 + √ 2 sin(t + π/4) while the disturbances are set as d 1 = 0.1 sin(t), d 2 = 1.
On one hand, the problem of finite-time trajectory tracking for system (18) presents a nontrivial task by referring to existing literature.Firstly, the existing nonsmooth control methods such as [10], [12], [17]- [19], etc. will only lead to a control result with practical stability.Secondly, the global design framework proposed in [13] cannot be applied due to the fact that the nonlinearity growth hypothesis cannot be preverified.
In this work, we show that by considering a semi-global control objective, the exact tracking control problem for Example 4.1 can now be solved by following a simple synthesis procedure depicted as follows: By skipping the pre-verifications of nonlinearity growth constraints, one can straightforwardly utilize the proposed control method with a series of pre-calculations as: y r − cos(y r )y (1) r + sin(y r )(y , the obtained exact tracking controller is depicted explicitly in Table I. Table I: Finite-time controller design for system (18) HOSM Disturbance Observer: On the other hand, in reference to existing backstepping based approaches, a clear improvement is the design simplicity under the proposed non-recursive design framework.For instance, following the backstepping based design in [6], a more complex control scheme with nested virtual controllers can be carried out via recursive design steps, as depicted sketchily in Table II.
In the simulation, by following the proposed design procedure, the gain vector K can be selected following the classical pole placement manner In the simulation, we choose K = [27, 27, 9] to place the pole of the nominal system into (−3, −3, −3).The scaling gain is selected as L = 1.5 according to the guideline (13).The designed homogeneous degree is set as τ = −0.1.The observer gains are chosen as Table II: Backstepping based controller design for system (18) Disturbance Observer: The initial values are given as For the backstepping based controller in Table II, the parameters are set as As shown in Fig. 2, the finite-time tracking objective is realized under the designed tracking scheme while the backstepping controller could only render a practical tracking result.Fig. 4 shows that under the proposed method, the states x 2 and x 3 also approach to their desired reference signal x * 2 , x * 3 within a finite-time.The time histories of two control input signals are shown in Fig. 3.In Fig. 5, the performance of the finite-time disturbance observer is demonstrated.

B. Application to Speed Regulation of PMSM
The mathematical model of the permanent magnet synchronous motor (PMSM) in the rotor reference frame is presented as follows [20]  where ω is the rotor angular velocity; i d and i q are d-and q-axis stator currents, respectively; u d and u q are d-and qaxis stator voltages, respectively; T L is the load torque; n p is the number of poles-pairs, equals 4; R is the stator resistance, equals 9.7Ω; L is the stator inductance, equals 26mH; ψ f is the magnetic flux linkage, equals 0.084Wb; J is the moment of inertia, equals 1.35 × 10 −4 kg • m 2 ; B v is the frictional coefficient, equals 7.4×10 −5 N•m•s/rad.The control object is to realize finite-time exact tracking of a given speed reference signal ω r = 100rad/s under an unknown load torque, assumed as Under a semi-global stability criterion, one can utilize the proposed exact tracking control method while several auxiliary variables are calculated as: r − Jz 2 + Ri * q + n p Lω r i * d + n p ψ f ω r ; ξ ω = ω − ω * , ξ q = (i q − i * q )/l q , v q = (u q − u * q )/l 2 q .The finite-time control scheme is then explicitly presented in Table III.
Table III: Finite-time controller design for PMSM system (19) HOSM Disturbance Observer: In the simulation, the observer parameters are set as: λ = 10 7 , l 0 = 4, l 1 = 2, l 2 = 1.The control parameters are set as: The initial values of the closed-loop system are chosen as 0.
As is clearly depicted by Fig. 6, in the presence of unknown load torque variation, the speed regulation objective can still be well achieved under the proposed finite-time controller.The response curves of i q , i d and the time histories of two control inputs u q , u d are presented in Figs.7 and 8, respectively.The response curves of disturbances and disturbance estimates are given in Fig. 9, which clearly demonstrate the effectiveness of the designed HOSM disturbance observer depicted in Table III.

V. CONCLUSIONS
In this paper, we investigate a novel non-recursive tracking control design framework under a semi-global control objective.Compared with existing related results, several distinguishable improvements can be achieved.Firstly, the proposed control scheme is presented with simpler homogenous expression and gain tuning mechanisms.Secondly, it is shown that a finite-time trajectory tracking result can also be for disturbed smooth nonlinear systems without any additional nonlinearity growth condition hypothesis.Moreover, the proposed one-step control design and stability analysis under a new non-recursive synthesis manner will largely facilitate the practical implementations, as illustrated by both a numerical example and a PMSM system application.

A. Useful Lemmas
Some useful lemmas are stated as follows.

B. Proofs of Propositions
This subsection collects the proofs of propositions used in the paper.