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IEEE TRANSACTIONS MANUSCRIPT 1

Finite-Time Trajectory Tracking Control of a Class

of Nonlinear Discrete-Time Systems

Zhuo Wang, Member, IEEE, Renquan Lu, Member, IEEE, Furong Gao, and Hong Wang, Senior Member, IEEE

Abstract—This paper studies how to control a class of nonlin-

ear discrete-time systems, to completely track any given bounded

expected trajectories in ﬁnite time. For this problem, we develop

some constructive control methods for both the total output case

as well as the partial output case. Some of these control methods

can design the tracking instant when the trajectory tracking is

just accomplished, but cannot guarantee the monotonic decrease

of the norm of the tracking error before that instant. The other

control methods not only can determine the tracking instant after

which the (partial) output trajectory coincides with the expected

trajectory, but also can make the norm of the tracking error

decrease monotonically before that instant. In the mean time,

for the partial output case, the rest part of the system output is

bounded for all the time. Then, we give some simulation examples

to demonstrate the effectiveness of these control methods. Among

these examples, a permanent magnet linear motor system is

employed to illustrate the practicability.

Index Terms—Bounded expected trajectories, ﬁnite-time tra-

jectory tracking, nonlinear discrete-time systems, partial output

trajectory, total output trajectory.

I. INTRODUCTION

N control theory and control engineering, trajectory track-

ing is one of the most important subjects to be extensively

studied for both linear/nonlinear continuous-time systems and

discrete-time systems [1], [2], [3]. The objective of these

studies is to make the system state/output track a prescribed

trajectory. The trajectory tracking has so far been investigated

can be divided into two categories: the asymptotic trajectory

tracking [4], [5], [6] and the ﬁnite-time trajectory tracking [7],

[8], [9]. The asymptotic trajectory tracking is generally in the

Lyapunov’s sense [5], [6], [10], [11], that the norm of the

tracking error decreases monotonically as time increases, and

the system state/output will converge to the expected trajectory

as time tends to inﬁnity. Besides, it is often required that

the system model has smooth dynamic characteristics [12],

[13], such as the continuous partial derivatives. However, it is

impractical to wait for an inﬁnitely long time, thus people need

to introduce a prescribed accuracy (usually referred to as the

maximum tolerable tracking error), and consider the trajectory

This work was supported in part by Guangzhou Scientiﬁc and Technological

Project under Grant 12190007, NSFC-On-Line Quality Measurement Instru-

ment for Injection Molding under Grant 61227005, and Guangdong Scientiﬁc

and Technological Project under Grant 2012B090500010.

Z. Wang and F. Gao are with Fok Ying Tung Graduate School, Hong Kong

University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

(e-mail: zwang8381@foxmail.com, Kefgao@ust.hk).

R. Lu is with Institute of Information and Automation, Hangzhou Dianzi

University, Hangzhou 310000, China (e-mail: rqlu@hdu.edu.cn).

H. Wang is with the Control Systems Centre, University of Manchester,

Manchester M60 1QD, U.K. (e-mail: hong.wang@manchester.ac.uk).

tracking is accomplished when the accuracy requirement is

satisﬁed.

The ﬁnite-time trajectory tracking, on the other hand, is

quite a different problem. It is concerned with how to design

a tracking instant when the trajectory tracking is accomplished,

and how to make the tracking error keep zero from that instant

on. In this sense, the ﬁnite-time trajectory tracking is superior

to the asymptotic trajectory tracking regardless of the system

requirements, since it has higher tracking precision, faster

convergence rate, and can determine the exact time when the

trajectory tracking is accomplished. In engineering practice,

there are some precise tracking tasks need to be done in

ﬁnite time, due to the resource limitation or the standard of

workmanship, etc. For example, in the areas of shipbuilding

and car manufacturing, it has become very common that the

industrial robot precisely cuts materials along the preset edge

or welds workpieces along the preset welding seam [14], [15].

When launching aircraft, the aircraft is supposed to enter

the predesigned trajectory in ﬁnite time, and subsequently,

ﬂies along this trajectory precisely [16], [17]. In astronomical

observation area, the attitude precision and the ﬂying precision

along the predesigned trajectory of spacecrafts (such as Hubble

Space Telescope), will directly affect the observation of the

far-away stars [18], [19]. In this circumstance, the study and

investigation of ﬁnite-time trajectory tracking problems have

important signiﬁcance in theory and practice.

Up to now, most of the existing works on ﬁnite-time

trajectory tracking are about the control design problem, and

some typical control schemes have been proposed for it. Such

as the backstepping methods [10], [19], [20], the sliding mode

methods [18], [21], [22], and the neural network methods [23],

[24]. Despite the merits of these control methods, they have

some shortcomings: they can only deal with some particular

systems with speciﬁc structures or dynamic characteristics, but

cannot control a general class of systems; they can only drive

the system state/output to track some particular trajectories,

but not to track arbitrary bounded trajectories; they can only

accomplish the trajectory tracking within a ﬁnite time horizon,

but cannot determine the exact tracking instant when the

state/output trajectory begins to coincide with the expected

trajectory. In this situation, further researches need to be

conducted. The study on these problems is necessary to reveal

the essential role of ﬁnite-time trajectory tracking control.

In this paper, we will conduct research on the ﬁnite-time

output trajectory tracking control problem for a class of

nonlinear discrete-time systems. We will develop control laws

in a constructive way, for both the total output case and the

partial output case of these systems. Our control methods

IEEE TRANSACTIONS MANUSCRIPT 2

have some advantages: ﬁrst, they are not limited to particular

systems, but can control a general class of nonlinear discrete-

time systems; second, they do not require special structure

or smooth dynamic characteristics (like continuous partial

derivatives) of the system model; third, they are applicable

to any bounded expected trajectories; and ﬁnally, they can

arbitrarily design the tracking instant when the trajectory

tracking is accomplished.

The remainder of this paper is organized as follows. Section

II studies how to control the system output to completely

track the given bounded expected trajectory in ﬁnite time,

and how to make the norm of the tracking error decrease

monotonically before the trajectory tracking is accomplished.

Section III studies how to control a part of the system output

to completely track the given bounded expected trajectory in

ﬁnite time while keeping the other part bounded, and also

studies the conditions of monotonic decrease of the norm

of the tracking error. Section IV presents some simulation

examples to demonstrate the effectiveness of these control

methods. In particular, a permanent magnet linear motor sys-

tem is employed to illustrate the practicability in engineering

applications. Finally, Section V draws the conclusion of this

paper and looks forward to the prospect of future researches.

II. FINITE-TIME TOTAL OUTPUT TRAJECTORY TRACKING

CONTROL

Consider the following nonlinear discrete-time system:

y(k + 1) = f



y(k)



+ B



y(k)



u(k), (1)

where u(k), y(k) ∈ R

are the input and the output, respec-

tively. Integer k ≥ 0 is the tracking instant. f : R

→ R

is piecewise continuous with respect to y(k), and f(0) = 0.



y(k)



∈ R

n×n

is the continuous input gain matrix.

In this section, we study how to control the total output of

system (1) to completely track any given bounded trajectory

in ﬁnite time. Deﬁne the set of the trajectories to be tracked:



g(k)



k ≥ 0, g(k) ∈ R



g(k)



≤ α < ∞



. (2)

In this paper, k·k denotes the spectral norm. g(t) is continuous

with respect to t ∈ R

and g(k) is the discrete value of g(t)

at t = kT , where T > 0 is the sampling period. In this way,

0 < α < ∞ is the boundary of any g(k) ∈ S

. Moreover, we

suppose that each given g(k) ∈ S

is known for all k ≥ 0.

Assumption 1: For system (1), assume that ∀y(k) ∈ R

Rank





y(k)



= n.

When Assumption 1 is satisﬁed, system (1) is controllable.

To facilitate discussion, we suppose that



y(0)



≤ α for any

given y(0); otherwise, we may ﬁrst control the output to satisfy

this initial condition, and then discuss the tracking problem.

Theorem 1: Suppose that system (1) satisﬁes Assumption

1. Then, for any given y(0) and g(k) ∈ S

, the output y(k)

of system (1) can completely track g(k) in ﬁnite time.

Proof: If y(0) = g(0), we design the control input as

u(k) = B

−1



y(k)



g(k + 1) − f



y(k)



(k ≥ 0). (3)

From (1) and (3), we can see that y(k) = g(k) (k ≥ 0). Thus,

the system output y(k) completely tracks g(k) instantaneously

by using control law (3).

If y(0) 6= g(0), preset a tracking instant M ≥ 1 when the

trajectory tracking is just accomplished. Then, we design the

control input as

u(k) =











−1



y(k)





k + 1

g(k + 1) − f



y(k)





(0 ≤ k ≤ M −1);

−1



y(k)



g(k + 1) − f



y(k)



(k ≥ M).

(4)

From (1) and (4), we can obtain

y(k + 1) =

(

k + 1

g(k + 1), (0 ≤ k ≤ M − 1);

g(k + 1), (k ≥ M).

(5)

When k = M −1, y(k+1) = y(M) =

k + 1

g(k+1) = g(M).

When k ≥ M , y(k + 1) = g(k + 1). Therefore, the system

output y(k ) completely tracks g(k) from k = M on by using

control law (4).

In summary, the output y(k) of system (1) can completely

track g(k) in ﬁnite time.

From the above proof, we can see that when y(0) = g(0),



y(k)



g(k)



≤ α for all k ≥ 0. When y(0) 6= g(0),



y(k)



(



g(k)



≤

α, (1 ≤ k ≤ M);



g(k)



≤ α, (k ≥ M + 1).

Since we have supposed that



y(0)



≤ α for any given y(0),

then



y(k)



≤ α for all k ≥ 0. In summary, the system output

also has a boundary of 0 < α < ∞.

In the above discussion, though not speciﬁcally mentioned,

system (1) is usually supposed to be globally asymptotically

stable. Otherwise, according to Assumption 1, system (1) is

stabilizable, we can stabilize it ﬁrst and then study the tracking

problem. Note that whether control inputs (3) and (4) are

bounded, is not discussed. But, the control input should always

be constrained in engineering applications, and we will discuss

this issue below.

Suppose that sup

y(k)∈R





−1



y(k)







= M

< ∞. Then if

y(0) = g(0), from (2) and (3), the control input should satisfy



u(k)



≤



−1



y(k)







kg(k + 1)k +





y(k)







≤ M



α +





y(k)







When system (1) is globally asymptotically stable, it is easy

to see that





y(k)





y(k)



. With the above conclusion

that



y(k)



g(k)



≤ α for all k ≥ 0, we can obtain



u(k)



< M



α +



y(k)





≤ 2αM

< ∞ for all k ≥ 0,

which indicates the boundary of control input (3) for ﬁnite-

time trajectory tracking. For the case that y(0) 6= g(0), we can

still get the same result, which is omitted here.

It is worth mentioning that although control law (4) can

design when the system output y(k) starts to completely track

g(k) by choosing proper tracking instant M ≥ 1, which is

its contribution, we cannot guarantee that the norm of the

tracking error decreases for all 0 ≤ k ≤ (M − 1). As a

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