# A co-design method of online learning SMC law via an input-mappping strategy

Yaru Yu<EMAIL_ADDRESS>Dewei Li<EMAIL_ADDRESS>Dongya Zhao
<EMAIL_ADDRESS>Yugeng Xi<EMAIL_ADDRESS>Department of Automation,
Shanghai Jiao Tong University, Key Laboratory of System Control and
Information Processing, Ministry of Education of China, Shanghai, China,
200240  College of New Energy, China University of Petroleum (East China),
Qingdao, China, 266580

###### Abstract

The research on sliding mode control strategy is generally based on the robust
approach. The larger parameter space consideration will inevitably sacrifice
part of the performance. Recently, the data-driven sliding mode control method
attracts much attention and shows excellent benefits in the fact that data is
introduced to compensate the controller. Nevertheless, most of the research on
data-driven sliding mode control relied on identification techniques, which
limits its online applications due to the special requirements of the data. In
this paper, an input-mapping technique is inserted into the design framework
of sliding mode control to compensate for the influence generated by the
unknown dynamic of the system. The major novelty of the proposed input-mapping
sliding mode control strategy lies in that the sliding mode surface and the
sliding mode controller are co-designed through online learning from
historical input-output data to minimize an objective function. Then, the
convergence rate of the system is improved significantly based on the method
designed in this work. Finally, some simulations are provided to show the
effectiveness and superiority of the proposed methods.

###### keywords:

Sliding mode control, input-mapping, co-design.

††thanks: This paper was not presented at any IFAC meeting. Corresponding
author: Dewei Li (dwli@sjtu.edu.cn).

, , ,

##  1 Introduction

It is well known that sliding mode control (SMC) is a typical nonlinear
control method and it shows strong robustness to uncertain systems [1, 2, 3,
4]. So far, abundant theoretical and applied achievements have been presented
due to the characteristics of simplicity and flexibility of SMC strategy [5,
6, 7, 8].

With the development and application of computer technology, considerable
attention has been taken into account for SMC of discrete-time systems. The
most noteworthy one is that the system trajectories of the discrete-time
systems move in the vicinity of the ideal sliding mode surface. The region
where the system state trajectory is ultimately maintained is the so-called
quasi-sliding mode band (QSMB) [9, 10]. In fact, such a chattering phenomenon
is unfavorable to the implementation of the sliding mode controller.
Therefore, many efforts focus on the research of eliminating or weakening
chattering. Such as the reaching law method [11], the boundary layer technique
[12], and others [13]. Where the reaching law method proposed by Gao et al
[11] has attracted wide attention due to its superiority in simplicity and
robustness. Then, Qu et al proposed a disturbance dynamic compensation
reaching law method in [14] to compensate the sliding mode controller by
introducing the superposition sum of the deviation between ideal reaching law
and real reaching law. In this way, the chattering phenomenon is significantly
weakened and the dynamic performance of the system is improved. Therefore, the
proposed disturbance dynamic compensation method has enriched the theoretical
achievements in the field of SMC. However, the robust design may lead to the
loss of control performance owing to the considered large space of unknown
dynamic.

For the past few decades, offline and online data is beginning to play a
significant role in controller design. In [15, 16, 17, 18, 19, 20, 21], the
data-driven SMC methods were presented to compensate for the influence of
unknown dynamic by identifying an approximate model according to data. It is
undeniable that the usage of data indeed offsets the deficiency of SMC
strategy based on robust methods. Nevertheless, the existing works for data-
driven model-free adaptive control usually need to identify a pseudo-partial
derivative model and the identification has special requirements on data which
limits the parameters close to the actual value. This is difficult for the
actual application.

Note that the model information of the system is implicated in its input-
output data. Therefore, an input-mapping technique was proposed in [22, 23] to
online compensate for the impact of unknown dynamic by a linear combination of
past data. Inspired by the input-mapping strategy, this paper develops an
input-mapping sliding mode control (IMSMC) strategy for the system with
unknown dynamic. Especially, the SMC law and the sliding variable are co-
designed to minimize an objective function according to the past information
and the superposition parameters are obtained by optimizing in the meantime.
For the co-design method, the SMC law and the sliding variable are designed
simultaneously to guarantee satisfactory system performance.

Compared with the robust sliding mode control method in [14], the input-
mapping technique is inserted into the design framework of the SMC strategy to
accelerate the convergence rate of the system trajectory by online input-
output data. On the other hand, the design method proposed in this paper
directly predicts the future system dynamic by historical data generated
online. This avoids the need to adjust parameters by a trial-and-error method
in [15] and the special requirements on the data used for model identification
in [16]. Moreover, the stability of the addressed system is analyzed and
demonstrated by the Lyapunov method. Some comparisons with existing methods
are given in the simulation to show the superiority of the algorithm.

The rest of this paper is organized as follows. The system description and
problem formulation are given in Section II. The specific input-mapping-based
online learning SMC strategy is introduced in Section III. In section IV, two
examples are presented to illustrate the effectiveness and advantages of the
methods proposed in this paper. Section V shows some important conclusions.

Notations: For a matrix $A$, $A^{T}$ denote its transpose.
$\|x\|:=\sqrt{x^{T}x}$ stands for its 2-norm; $\mathbb{R}^{n}$ denotes the
$n$-dimensional Euclidean space; $\mathbb{N}$ represents positive integers;
$\mathrm{sgn}(\cdot)$ satisfies $\mathrm{sgn}(s(k))=1$ if $s(k)>0$ and
$\mathrm{sgn}(s(k))=-1$ if $s(k)\leq 0$ ; $\sup_{k\in\mathbb{N}}()$ is the
maximum value of a parameter within $k\in\mathbb{N}$;
$x:=\mathrm{col}(x_{1},\ldots,x_{n})=[x_{1}^{T},\ldots,x_{n}^{T}]^{T}$.

##  2  System description and problem formulation

###  2.1 System description

In this paper, we consider a class of systems suffer from unknown-but-bounded
unknown dynamic described by

$\displaystyle\varsigma(k+1)=[\tilde{A}+\triangle\tilde{A}]\varsigma(k)+\tilde{B}u(k),$
(1)

where $\varsigma(k)\in\mathbb{R}^{n_{\zeta}}$ is the state and
$u(k)\in\mathbb{R}^{n_{u}}$ is the control input at $k\in\mathbb{N}$.
$\tilde{A}$, $\bigtriangleup\tilde{A}$ and $\tilde{B}$ represent the system
matrix, the system unknown dynamic and the input matrix, respectively. Then,
some assumptions are given for the subsequent derivation.

######  Assumption 1.

The unknown dynamic $\triangle\tilde{A}$ satisfies $\triangle\tilde{A}=D\Delta
E$, where $D$ and $E$ are known matrices. $\Delta$ is unknown and it satisfies
$\Delta^{T}\Delta\leq I$.

######  Assumption 2.

The pair $(\tilde{A},\tilde{B})$ is controllable and the matrix $\tilde{B}$ is
column full rank.

Now, the input matrix $\tilde{B}$ can be partitioned as
$\tilde{B}=[\tilde{B}^{T}_{1}~{}\tilde{B}^{T}_{2}]^{T}$ from the Assumption 2
and $\tilde{B}_{2}$ is invertible. Then, by implementing a coordinate
transformation $x(k)=T_{c}\varsigma(k)\in\mathbb{R}^{n_{x}}$
$(n_{x}=n_{\varsigma})$ with

$\displaystyle T_{c}\in\mathbb{R}^{n_{x}\times
n_{x}}=\left[\begin{array}[]{cc}I_{(n_{x}-n_{u})\times(n_{x}-n_{u})}&-\tilde{B}_{1}\tilde{B}^{-1}_{2}\\\
0_{n_{u}\times(n_{x}-n_{u})}&I_{n_{u}\times n_{u}}\end{array}\right].$

Then, the system (1) can be reshaped as

$\displaystyle x(k+1)=(A+\triangle A)x(k)+Bu(k).$ (3)

Specifically, $T_{c}$ is an invertible matrix. Matrices $A$, $B$ and
$\triangle A$ satisfy $A=T_{c}\tilde{A}T^{-1}_{c}$, $B=T_{c}\tilde{B}$ and
$\triangle A=T_{c}\triangle\tilde{A}T^{-1}_{c}$. Then, we have (3) is
equivalent to

$\displaystyle\left[\begin{array}[]{cc}x_{1}(k+1)\\\
x_{2}(k+1)\end{array}\right]=\left(\left[\begin{array}[]{cc}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{array}\right]+\left[\begin{array}[]{cc}\bigtriangleup
A_{11}&\bigtriangleup A_{12}\\\ \bigtriangleup A_{21}&\bigtriangleup
A_{22}\end{array}\right]\right)$ (10)
$\displaystyle\times\left[\begin{array}[]{cc}x_{1}(k)\\\
x_{2}(k)\end{array}\right]+\left[\begin{array}[]{cc}0_{(n_{x}-n_{u})\times
n_{u}}\\\ B_{1}\end{array}\right]u(k),$ (15)

and the unknown dynamic is supposed to be represented by

$\displaystyle\left[\begin{array}[]{cc}\bigtriangleup A_{11}&\bigtriangleup
A_{12}\\\ \bigtriangleup A_{21}&\bigtriangleup
A_{22}\end{array}\right]=\bar{D}\Delta\bar{E}=\left[\begin{array}[]{cc}\bar{D}_{1}\\\
\bar{D}_{2}\end{array}\right]\Delta[\begin{array}[]{cc}\bar{E}_{1}&\bar{E}_{2}\end{array}].$
(21)

######  Remark 1.

It can be observed from (15) that the state component $x_{1}(k+1)$ is
independent of control input $u(k)$, and only the state $x_{2}(k+1)$ is
related to $u(k)$. The system model in this form is the so-called “regular
form” as in [24] for the convenience of designing a SMC strategy.

###  2.2 Preliminaries

####  2.2.1 Sliding mode surface design

In this section, an appropriate sliding mode surface is designed to guarantee
the stability of the states which reach the surface. Without loss of
generality, the sliding mode surface for system (3) is considered as

$\displaystyle s(k)=\bar{G}x(k)=[\begin{array}[]{cc}G&I\\\ \end{array}]x(k),$
(23)

where $G\in\mathbb{R}^{n_{u}\times(n_{x}-n_{u})}$ is the sliding variable to
be designed later. Then, the corresponding sliding manifold is assigned as

$\displaystyle\mathcal{\varrho}_{x}\triangleq\left\\{x\in\mathbb{R}^{n_{x}}|s=\bar{G}x(k)=0\right\\}.$
(24)

It is indisputable that the system trajectories arrive at sliding manifold
(24) satisfies $s(k)=0$, that is, $x_{2}(k)=-Gx_{1}(k)$. Substituting
$x_{2}(k)$ of the first term in (15) by the established relation, the sliding
mode dynamic (SMD) of the considered system (15) can be obtained as

$\displaystyle x_{1}(k+1)=A_{g}x_{1}(k),$ (25)

with $A_{g}=(A_{11}+\Delta A_{11})-(A_{12}+\Delta A_{12})G$. Apparently, the
sliding mode variable $G$ can be designed by recognizing $x_{2}(k)$ as a
virtual control input of $x_{1}(k)$. Then, a linear matrix inequality
condition is given in Lemma 1 to guarantee the robustness of the SMD (25) by
employing quadratic stability analysis.

######  Lemma 1.

[25] For a chosen Lyapunov function $V(k)=x^{T}_{1}(k)Rx_{1}(k)$, the SMD in
(25) is asymptotically stable if there exist scalar $\gamma>0$, matrix
$R_{1}=R^{-1}$ is positive definite and $R_{g}$ is a matrix with appropriate
dimension satisfy

$\displaystyle\left[\begin{array}[]{ccc}-R_{1}&R_{1}A^{T}_{11}-R_{g}A^{T}_{12}&R_{1}\bar{E}^{T}_{1}-R_{g}\bar{E}^{T}_{2}\\\
&-R_{1}+\gamma\bar{D}_{1}\bar{D}^{T}_{1}&0\\\ &*&-\gamma I\\\
\end{array}\right]<0,$ (29)

and then the variable $G$ is derived off-line as $G=R_{g}R_{1}^{-1}$.

Then, the disturbance compensation SMC law proposed in [14] is presented to
ensure the system trajectories from arbitrary initial state will converge to a
QSMB.

####  2.2.2 SMC law

In [14], a discrete-time reaching law with a disturbance compensator is
provided by utilizing the deviation between the real reaching law and the
ideal reaching law. The reaching law is shown below:

$\displaystyle s(k+1)-s(k)=-\mu Ts(k)-\xi T\mathrm{sgn}s(k)+\varpi(k)$
$\displaystyle-\sum^{k}_{i=2}\\{s(i)-[(1-\mu T)s(k-i)-\xi
T\mathrm{sgn}s(k)]\\},$ (30)

and

$\displaystyle\varpi(k)=\bar{G}\Delta Ax(k),$
$\displaystyle\varpi(k-1)=\sum^{k}_{i=2}\\{s(i)-[(1-\mu T)s(k-i)-\xi
T\mathrm{sgn}s(k)]\\},$

where $\mu$ is a converging parameter and $\xi$ is a switching parameter. $T$
is the sampling period, and $1>1-\mu T>0$. Assume that the gradient of
parameter perturbation within the system is bounded by
$\|\varpi(k)-\varpi(k-1)\|<\delta$. Then, the following result is derived to
ensure the system trajectories of system (15) can be driven onto the designed
sliding surface in finite time and stay on there all the succeeding time.

######  Lemma 2.

[14] On the basis of the sliding variable $G$ designed by Lemma 1, the
trajectories of the considered system (3) can be driven onto a QSMB in finite
time by the following controller:

$\displaystyle u(k)$ $\displaystyle=$ $\displaystyle-(\bar{G}B)^{-1}\left[\mu
Ts(k)+\xi T\mathrm{sgn}s(k)+\bar{G}Ax(k)-s(k)\right.$ (31) $\displaystyle+$
$\displaystyle\sum^{k}_{i=2}\\{s(i)-[(1-\mu T)s(k-i)-\xi
T\mathrm{sgn}s(k)]\\}].$

######  Remark 2.

The SMC strategy based on Lemmas 1-2 is designed with the idea of robustness.
Hence, the control performance sacrifices are inevitably by the derived
sliding mode controller due to the considered space of unknown dynamic being
relatively large. With the fact that the unknown dynamic of the addressed
system show the same characteristic as in the past, it is a natural idea to
collect historical information to compensate for the designed SMC strategy and
for improving the convergence rate of the system trajectory.

##  3 Input-mapping-based co-deign SMC strategy

We now aim at synthesizing a valid input-mapping-based online learning SMC law
to drive the state trajectories of the closed-loop system in (15) to
$\varrho_{x}$. Particularly, the sliding mode surface in (23) is continuously
adjusted by online learning from the past behavior of the system until it
enters the QSMB. That is, the sliding mode variable $G(k)$ and sliding mode
controller are co-designed by minimizing a given objective function, which is
different from the methods in [24, 26, 27].

In order to form an input-mapping-based online learning SMC strategy for the
addressed system, the following section is given for some preliminary
presentations.

###  3.1 Input-mapping technique

As for the system (3), we can get the following relations

$\displaystyle x(k)=(A+\triangle A)x(k-1)+Bu(k-1).$ (32)

Similar to [22, 23], at time $k$, the unknown dynamic $\triangle Ax(k-1)$ of
considered system can be calculated by

$\displaystyle\triangle Ax(k-1)=x(k)-Ax(k-1)-Bu(k-1),$ (33)

where $x(k)$, $x(k-1)$ and $u(k-1)$ are known information to exactly derive
unknown dynamic $\triangle Ax(k-1)$. Similarly, at time $k$, we have the
historical relations

$\displaystyle x(k-i+1)=(A+\triangle A)x(k-i)+Bu(k-i),~{}i\in\mathbb{N},$ (34)

where $\Delta Ax(k-i)$ can be uniquely determined by a set of known historical
data $\\{x(k-i+1),x(k-i),u(k-i)\\}$ although the item $\Delta A$ is unknown.

On the other hand, it is a fact that for a vector in $n$-dimensional linear
space, it can be uniquely linearly represented by $n$ linearly independent
basis vectors. In that way, the system state $x(k)\in\mathbb{R}^{n}$ can be
uniquely linearly represented by $n$ linearly independent basis vectors. Thus,
if the historical system states of $N$ moments near the current moment are
collected as the basis vectors of the corresponding space, the system states
$x(k)$ can be decomposed as

$\displaystyle x(k)=\sum^{N}_{i=1}l_{i}x(k-i)+\delta_{x}(k),$ (35)

where $l_{i}$ is the unknown combination parameter, $x(k-i)$ are the known
system states at time $k$, and $\delta_{x}(k)$ is the residual item to express
$x(k)$ because the linear independence of $N$ historical data may not be
guaranteed online. Analogously, the control input $u(k)$ can be decomposed as

$\displaystyle u(k)=\sum^{N}_{i=1}l_{i}u(k-i)+\delta_{u}(k),$ (36)

where $l_{i}$ is the same as in (35), $u(k-i)$ are the known control inputs at
time $k$, and $\delta_{u}(k)$ is also a residual item to represent $u(k)$.
Moreover, based on the collected $N$ historical data $x(k-i)$ and $u(k-i)$
near the current moment, (34) can be converted to

$\displaystyle\sum^{N}_{i=1}l_{i}x(k-i+1)$ $\displaystyle=(A+\triangle
A)\sum^{N}_{i=1}l_{i}x(k-i)+B\sum^{N}_{i=1}l_{i}u(k-i),$ (37)

by the homogeneity property and superposition principles of a linear system.
Based on (37), the system model in (3) can be reshaped by substituting the
expressions (35) and (36) as

$\displaystyle x(k$ $\displaystyle+$ $\displaystyle 1)=(A+\Delta
A)[\sum^{N}_{i=1}l_{i}x(k-i)+\delta_{x}(k)]$ (38)
$\displaystyle+B[\sum^{N}_{i=1}l_{i}u(k-i)+\delta_{u}(k)]$ $\displaystyle=$
$\displaystyle(A+\Delta
A)\sum^{N}_{i=1}l_{i}x(k-i)+B\sum^{N}_{i=1}l_{i}u(k-i)$
$\displaystyle+(A+\Delta A)\delta_{x}(k)+B\delta_{u}(k)$ $\displaystyle=$
$\displaystyle\sum^{N}_{i=1}l_{i}x(k-i+1)+(A+\Delta
A)\delta_{x}(k)+B\delta_{u}(k).$

For simplifying the subsequent derivation, the relations (35), (36) and (38)
are rewritten as matrix form as follows

$\displaystyle x(k)=\mathcal{X}(k-1)L(k)+\delta_{x}(k),$ (39) $\displaystyle
u(k)=\mathcal{U}(k-1)L(k)+\delta_{u}(k),$ (40) $\displaystyle
x(k+1)=\mathcal{X}(k)L(k)+(A+\Delta A)\delta_{x}(k)+B\delta_{u}(k),$ (41)

where

$\displaystyle\mathcal{X}(k)$ $\displaystyle=$
$\displaystyle\left[x(k),x(k-1),\ldots,x(k-N+1)\right]\in\mathbb{R}^{n_{x}\times
N},$ $\displaystyle\mathcal{X}(k-1)$ $\displaystyle=$
$\displaystyle\left[x(k-1),x(k-2),\ldots,x(k-N)\right]\in\mathbb{R}^{n_{x}\times
N},$ $\displaystyle\mathcal{U}(k-1)$ $\displaystyle=$
$\displaystyle\left[u(k-1),u(k-2),\ldots,u(k-N)\right]\in\mathbb{R}^{n_{u}\times
N},$ $\displaystyle L(k)$ $\displaystyle=$
$\displaystyle\left[l_{1},l_{2},\ldots,l_{N}\right]^{T}\in\mathbb{R}^{N}.$
(42)

From (41), we have vector $\mathcal{X}(k)$ is fully known historical data. The
vector variable $L(k)$ and the residual items $\delta_{x}(k)$ and
$\delta_{u}(k)$ are unknown terms need to solve. At time $k$, the control
problem of designing $u(k)$ is transformed into a problem of solving $L(k)$
and $\delta_{u}(k)$. Specially, the unknown dynamic is just related to
$\delta_{x}(k)$, which is only a part of system state $x(k)$. Therefore, the
influence caused by part of the unknown dynamic of the system can be
compensated by historical information, which can effectively improve the
system’s performance.

Furthermore, a data table needs to be created to store historical information.
The data table is updated at every moment, and the new data is stored in the
table and the data far away from the current moment is removed. Then, the
information stored in the data table is the basis for online optimization
calculations.

###  3.2 Co-design IMSMC law

In this section, a developed input-mapping-based online learning SMC law will
be constructed by a co-designed method for improving the system performance.
The sliding mode surface is online adjusted in terms of historical data to
ensure that the system state can reach the sliding surface faster. Firstly,
the sliding mode surface is rewritten as

$\displaystyle s(k)=\bar{G}(k)x(k)=[\begin{array}[]{cc}G(k)&I\\\
\end{array}]x(k),$ (44)

where the sliding variable $G(k)$ is time-dependent now. The SMD is regained
as

$\displaystyle\left\\{\begin{array}[]{ll}x_{1}(k+1)=\bar{A}_{g}x_{1}(k)\\\
\bar{A}_{g}=(A_{11}+\Delta A_{11})+(A_{12}+\Delta
A_{12})G(k)\end{array}\right..$ (47)

Then, the reaching law is given as follows

$\displaystyle s(k+1)$ $\displaystyle-$ $\displaystyle
s(k)=-\bar{\mu}(k)Ts(k)-\bar{\xi}T\mathrm{sgn}s(k)+\varpi_{1}(k)$ (48)
$\displaystyle-\sum^{k}_{i=2}\\{s(i)-[(1-\bar{\mu}(k)T)s(k-i)-\bar{\xi}T\mathrm{sgn}s(k)]\\},$

with

$\displaystyle\left\\{\begin{array}[]{llll}s(k+1)=\bar{G}(k+1)x(k+1),\\\
s(k)=\bar{G}(k)x(k),\\\ \varpi_{1}(k)=\bar{G}(k+1)\Delta
A\delta_{x}(k).\end{array}\right.$

Similar to the derivation of [14], we have
$\varpi_{1}(k-1)=\sum^{k}_{i=2}\\{s(i)-[(1-\bar{\mu}(k)T)s(k-i)-\bar{\xi}T\mathrm{sgn}s(k)]\\}$.
The convergence parameter $\bar{\mu}(k)$ satisfy $1>1-\bar{\mu}(k)T>0$.
$\bar{\xi}$ is the switching parameter.

######  Remark 3.

As the sliding mode variable is a determined value solved in advance, and then
the convergence parameter for the conventional way is a given constant that
satisfy certain conditions. However, the sliding mode surface for co-design
IMSMC strategy is online adjusted by leaning historical behavior. Thus, the
convergence parameter $\bar{\mu}(k)$ for co-design IMSMC strategy (sliding
variable is time-dependent) is assigned to time-varying due to the
reachability of the system is difficult to guarantee by a constant convergence
parameter.

On the basis of sliding mode surface (23), model (41) and reaching law (48),
the input-mapping-based SMC law is established as follows

$\displaystyle\delta_{u}(k)$ $\displaystyle=$
$\displaystyle-(\bar{G}(k+1)B)^{-1}\left[\bar{\mu}(k)Ts(k)+\bar{\xi}T\mathrm{sgn}s(k)-s(k)\right.$
(50) $\displaystyle+$
$\displaystyle\bar{G}(k+1)\mathcal{X}(k)L(k)+\bar{G}(k+1)A\delta_{x}(k)$
$\displaystyle+$
$\displaystyle\sum^{k}_{i=2}\left\\{s(i)-[(1-\bar{\mu}(k)T)s(k-i)-\bar{\xi}T\mathrm{sgn}s(k)]\right\\}],$

and $\delta_{x}(k)=x(k)-\mathcal{X}(k-1)L(k)$. Then, $u(k)$ can be obtained by
(40), that is

$\displaystyle u(k)$ $\displaystyle=$
$\displaystyle\mathcal{U}(k-1)L(k)-(\bar{G}(k+1)B)^{-1}\left[\bar{\mu}(k)Ts(k)-s(k)\right.$
(51) $\displaystyle+$
$\displaystyle\bar{\xi}T\mathrm{sgn}s(k)+\bar{G}(k+1)\mathcal{X}(k)L(k)+\bar{G}(k+1)A\delta_{x}(k)$
$\displaystyle+$
$\displaystyle\sum^{k}_{i=2}\\{s(i)-[(1-\bar{\mu}(k)T)s(k-i)-\bar{\xi}T\mathrm{sgn}s(k)]\\}],$

where $L(k)$ and $G(k+1)$ are parameters need to be designed.

######  Remark 4.

The control law in (51) is equivalent to

$\displaystyle u(k)$ $\displaystyle=$
$\displaystyle-(\bar{G}(k+1)B)^{-1}\left[\bar{\mu}Ts(k)+\bar{\xi}T\mathrm{sgn}s(k)-s(k)\right.$
$\displaystyle+$ $\displaystyle\bar{G}(k+1)Ax(k)+\bar{G}(k)\Delta
A\delta_{x}(k-1)$ $\displaystyle+$ $\displaystyle\left.\bar{G}(k+1)\Delta
A\mathcal{X}(k-1)L(k)\right],$

by implementing a transformation. In contrast to the control law (31), the co-
design IMSMC law in (51) involves the unknown dynamic of the past. Thus, the
sliding mode controller can be compensated by online learning from the past
behavior. Moreover, a certain improvement of system performance can be
obtained by compensating for the conservative processing of unknown dynamic in
the robust method in [14].

Consider $\bar{\mu}(k)T$ as a whole, i.e.
$\bar{\mu}_{0}(k)\triangleq\bar{\mu}(k)T$. Note that the optimal target for
system (3) is to control the system states of next time, that is, $x(k+1)$
reached stabilization quickly. Then, the optimal problem can be naturally
designed as follows

$\displaystyle\vspace{-1em}J_{1}$ $\displaystyle=$
$\displaystyle\min_{\phi_{0}(k)}\parallel x(k+1)\parallel^{2},$ (52)
$\displaystyle=$ $\displaystyle\min_{\phi_{0}(k)}\parallel
W(k)L(k)+\Xi(k)-B[\bar{G}(k+1)B]^{-1}$
$\displaystyle\times[\bar{\mu}(k)Ts(k)+\bar{\xi}T\mathrm{sgn}s(k)-s(k)+\varpi_{1}(k-1)]\parallel^{2},$

with

$\displaystyle\phi_{0}(k)$ $\displaystyle=$
$\displaystyle\\{L(k),G(k+1),\bar{\mu}(k)\\},$ $\displaystyle W(k)$
$\displaystyle=$
$\displaystyle\mathcal{X}(k)-B[\bar{G}(k+1)B]^{-1}\bar{G}(k+1)\mathcal{X}(k),$
$\displaystyle\Xi(k)$ $\displaystyle=$ $\displaystyle[A+\Delta
A-B[\bar{G}(k+1)B]^{-1}\bar{G}(k+1)A]\delta_{x}(k).$

Note that (52) contains unknown dynamic $\Delta A$ in $\Xi(k)$, and in view of
$\delta_{x}(k)$ is undecided. Then, the $\Xi(k)$ in (52) can be removed and
the objective function is refreshed as

$\displaystyle J$ $\displaystyle=$ $\displaystyle\min_{\phi_{0}(k)}\parallel
W(k)L(k)-B(\bar{G}(k+1)B)^{-1}[\bar{\mu}_{0}(k)s(k)$ (53)
$\displaystyle+\bar{\xi}T\mathrm{sgn}s(k)-s(k)+\varpi_{1}(k-1)]\parallel^{2}.$

Our purpose now is to minimize the objective $J$ based on the optimization
variables $L(k)$, $G(k+1)$ and $\bar{\mu}_{0}(k)$. Then, taking the derivative
with respect to $L(k)$, $G(k+1)$ and $\bar{\mu}_{0}(k)$, the analytical
solution of the optimal problem is obtained as a set of nonlinear matrix
equations as the Box. I.

$\displaystyle\left\\{\begin{array}[]{l}\frac{\partial J}{\partial
L(k)}=2W^{T}(k)\left\\{W(k)L(k)-B(\bar{G}(k+1)B)^{-1}[\bar{\mu}_{0}(k)s(k)+\bar{\xi}T\mathrm{sgn}s(k)-s(k)+\varpi_{1}(k-1)]\right\\}=0,\\\
\frac{\partial J}{\partial
G(k+1)}=2\mathcal{X}_{1}(k)L(k)L^{T}(k)\mathcal{X}^{T}_{1}(k)G^{T}(k+1)+2\mathcal{X}_{1}(k)L(k)[\bar{\mu}_{0}(k)s(k)+\bar{\xi}T\mathrm{sgn}s(k)-s(k)+\varpi_{1}(k-1)]^{T}=0,\\\
\frac{\partial
J}{\partial\bar{\mu}_{0}(k)}=2[-B(\bar{G}(k+1)B)^{-1}s(k)]^{T}\left\\{W(k)L(k)-B(\bar{G}(k+1)B)^{-1}[\bar{\mu}_{0}(k)s(k)+\bar{\xi}T\mathrm{sgn}s(k)-s(k)+\varpi_{1}(k-1)]\right\\}=0,\end{array}\right.$
(57) with
$\mathcal{X}(k)=\left[\begin{array}[]{l}\mathcal{X}_{1}(k)_{(n_{x}-n_{u})\times
N}\\\ \mathcal{X}_{2}(k)_{n_{u}\times N}\end{array}\right].$

Box I.

Next, the co-design IMSMC law constructed in (51) can be implemented directly
by means of the solution of combination coefficient $L(k)$, sliding variable
$G(k+1)$ and convergence parameter $\bar{\mu}_{0}(k)$ in (57).

######  Remark 5.

The sliding variable $G(k+1)$ in sliding mode surface (44) and the parameters
$\mu_{0}(k)$ and $L(k)$ in sliding mode controller (51) are obtained
simultaneously by optimizing. Such a co-design method is benefit to accelerate
the convergence of the system trajectory because of the sliding surface is
adjusted to reduce the time for the system state to reach it.

Moreover, the following theorem is presented for further analysis of the
proposed co-design IMSMC law.

######  Theorem 1.

Assume that the gradient of parameter perturbation within the system is
bounded by $\|\varpi_{1}(k)-\varpi_{1}(k-1)\|<\bar{\delta}$. For the sliding
manifold in (44) and the parameters $L(k)$, $G(k+1)$ and $\bar{\mu}_{0}(k)$ in
(51) derived by solving nonlinear matrix equations (57), we come to some
conclusions as follows

(1): The system trajectories of (3) can be driven onto a quasi-sliding mode
band and maintain in the QSMB all the subsequent time by IMSMC law (51). The
QSMB is bounded by

$\displaystyle\Upsilon=\\{s(k)|\parallel s(k)\parallel\leq\Omega\\},$ (58)

with

$\displaystyle\Omega=\sup_{k\in{\mathbb{N}}}\sqrt{\bar{\Phi}^{2}(k)+2\eta(k)\bar{\Phi}(k)},$
$\displaystyle\eta(k)=(1-\bar{\mu}_{0}(k))(\sqrt{n_{u}}\bar{\xi}T-\bar{\delta}),$
$\displaystyle\bar{\Phi}(k)=\frac{(\sqrt{n_{u}}\bar{\xi}T+\bar{\delta})^{2}}{2(1-\bar{\mu}_{0}(k))(\sqrt{n_{u}}\bar{\xi}T-\bar{\delta})}.$

(2): Let sliding variable $G(k+1)$ keeps the value calculated at the previous
time unchanged when the sliding mode surface enters the QSMB (58). Then the
stability of the SMD $(\ref{UBS-18-1})$ can be guaranteed by the solved
$G(k+1)$ in $(\ref{UBS-21})$.

Proof (1) For the co-designed IMSMC strategy in (51), the sliding incremental
can be obtained as

$\displaystyle\left\\{\begin{array}[]{l}s(k+1)-s(k)=-\bar{\mu}_{0}(k)s(k)-\bar{\xi}T\mathrm{sgn}s(k)+\Delta\varpi_{\delta}(k),\\\
\Delta\varpi_{\delta}=\varpi_{1}(k)-\varpi_{1}(k-1).\end{array}\right.$

Denote $V(k)=\frac{1}{2}s^{T}(k)s(k)$, and suppose that
$\|\Delta\varpi_{\delta}\|<\bar{\delta}$. Then, we have

$\displaystyle\Delta V(k)$ $\displaystyle=$ $\displaystyle s^{T}(k)\Delta
s(k)+\frac{1}{2}\Delta s^{T}(k)\Delta s(k)$ (60) $\displaystyle=$
$\displaystyle-\bar{\mu}_{0}(k)(1-\frac{1}{2}\bar{\mu}_{0}(k))\parallel
s(k)\parallel^{2}-\sqrt{n_{u}}\bar{\xi}T\parallel s(k)\parallel$
$\displaystyle+\frac{1}{2}n_{u}\bar{\xi}^{2}T^{2}+\sqrt{n_{u}}\bar{\mu}_{0}(k)\bar{\xi}T\parallel
s(k)\parallel$
$\displaystyle+(1-\mu_{0}(k))s^{T}(k)\Delta\varpi_{\delta}-\bar{\xi}T\mathrm{sgn}s(k)\Delta\varpi_{\delta}$
$\displaystyle+\frac{1}{2}\Delta\varpi_{\delta}^{T}(k)\Delta\varpi_{\delta}$
$\displaystyle<$ $\displaystyle-\sqrt{n_{u}}\bar{\xi}T\parallel
s(k)\parallel+\frac{1}{2}n_{u}\bar{\xi}^{2}T^{2}$
$\displaystyle+\sqrt{n_{u}}\bar{\mu}_{0}(k)\bar{\xi}T\parallel
s(k)\parallel+(1-\bar{\mu}_{0}(k))\parallel s(k)\parallel\bar{\delta}$
$\displaystyle+\sqrt{n_{u}}\bar{\xi}T\bar{\delta}+\frac{1}{2}\bar{\delta}^{2}$
$\displaystyle=$
$\displaystyle-(1-\bar{\mu}_{0}(k))(\sqrt{n_{u}}\bar{\xi}T-\bar{\delta})$
$\displaystyle\times\left(\parallel
s(k)\parallel-\frac{(\sqrt{n_{u}}\bar{\xi}T+\bar{\delta})^{2}}{2(1-\bar{\mu}_{0}(k))(\sqrt{n_{u}}\bar{\xi}T-\bar{\delta})}\right)$
$\displaystyle\triangleq$ $\displaystyle-\eta(k)\left(\parallel
s(k)\parallel-\bar{\Phi}(k)\right),$

and $\Delta s(k)=s(k+1)-s(k)$. It is clear that for Lyapunov function
$V(k)=\frac{1}{2}s^{T}(k)s(k)>0$, $\Delta V(k)<0$ can be ensured by choosing
parameter $\bar{\xi}T>\bar{\delta}$ if $\parallel
s(k)\parallel>\bar{\Phi}(k)$. Moreover, from $(\ref{UBS-27-1})$, the step-
forward sliding mode function satisfy $\parallel
s(k+1)\parallel\leq\sqrt{\parallel s(k)\parallel^{2}-2\eta(k)(\parallel
s(k)\parallel-\bar{\Phi}(k))}$, and then we have $\parallel
s(k+1)\parallel\leq\sqrt{\parallel s(k)\parallel^{2}+2\eta(k)\bar{\Phi}(k)}$.
It can be concluded that (i): the sliding mode function will decrease when
$\parallel s(k)\parallel>\bar{\Phi}(k)$ because $\Delta V(k)<0$; (ii):
$s(k+1)$ satisfies $\parallel
s(k+1)\parallel\leq\sqrt{\bar{\Phi}^{2}(k)+2\eta(k)\bar{\Phi}(k)}$ when
$\parallel
s(k)\parallel\leq\bar{\Phi}(k)\leq\sqrt{\bar{\Phi}^{2}(k)+2\eta(k)\bar{\Phi}(k)}$.
Therefore, the ultimate sliding trajectories maintain in the QSMB (58).

(2) Note that the reachability of the system states can be guaranteed by
selecting a large enough parameter $\bar{\xi}T$. Then, for sliding variable
$G(k+1)$ solved by nonlinear matrix equations (57), it must satisfy

$\displaystyle G(k+1)$ $\displaystyle=$
$\displaystyle[\bar{\mu}_{0}(k)s(k)+\bar{\xi}T\mathrm{sgn}s(k)-s(k)+\varpi_{1}(k-1)]$
$\displaystyle\times
L^{T}(k)\mathcal{X}^{T}_{1}(k)\\{\mathcal{X}_{1}(k)L(k)L^{T}(k)\mathcal{X}_{1}(k)\\}^{-1}.$

Denote

$\displaystyle
R_{g}\triangleq\left[\begin{array}[]{cc}\hat{R}_{g}&1\end{array}\right],~{}~{}R_{1}\triangleq\left[\begin{array}[]{cc}\hat{R}_{1}&0\\\
0&1\end{array}\right],$

with

$\displaystyle\hat{R}_{g}$ $\displaystyle\triangleq$
$\displaystyle[\bar{\mu}_{0}(k)s(k)+\bar{\xi}T\mathrm{sgn}s(k)-s(k)+\varpi_{1}(k-1)]$
$\displaystyle\times L^{T}(k)\mathcal{X}^{T}_{1}(k),$
$\displaystyle\hat{R}_{1}$ $\displaystyle\triangleq$
$\displaystyle\mathcal{X}_{1}(k)L(k)L^{T}(k)\mathcal{X}_{1}(k).$

Apparently, $R_{g}$ is a matrix with appropriate dimension and $R_{1}>0$. And
then we can conclude from Lemma 1 that the SMD in (47) will eventually
stabilize for any $G(k+1)$ derived by (57) if there exist scalar $\gamma>0$
satisfy (29) based on the above expression of $R_{g}$ and $R_{1}$. Moreover,
the stability of the SMD (47) can be guaranteed by the solved $G(k+1)$ due to
$G(k+1)$ keeps the value calculated at the previous time unchanged when
sliding mode surface enters the QSMB (58).$\hfill\blacksquare$

######  Remark 6.

For the solution of the equations (57), denote
$\omega(k)\triangleq\mathrm{col}\\{L(k)$, $G^{(1)}(k+1)$, $\ldots,$
$G^{(n_{x}-n_{u})}(k+1),\bar{\mu}_{0}(k)\\}$, where
$G^{(i)}(k+1),i\in\\{1,\ldots,n_{x}-n_{u}\\}$ is the i-column of sliding
variable $G(k+1)$. Let the initial iteration value of vector parameter
$\omega(k)$ is zero, the vector $\omega(k)$ can be derived through solving the
nonlinear matrix equations (57) by Levenberg-Marquardt algorithm, which can
find the minimum value by gradient.

Furthermore, the final implementation steps of the proposed input-mapping-
based online learning SMC strategy are shown in Algorithm. 1.

Algorithm 1 Co-design IMSMC strategy.

Step 1. Initialize $x(0)$ and $G(0)$ and assign the value to $\bar{\xi}T$. Set
a value for $N$ and let $\mathcal{X}(0)=0$, $\mathcal{U}(0)=0$.
Step 2. At the begin of $k$-th time, judge whether the condition $\parallel
s(k)\parallel\leq\Omega$ is satisfied. If meet, jump to Step 3. Otherwise,
compute $L(k)$, $G(k+1)$ and $\bar{\mu}_{0}(k)$ according to (57). Moreover,
if the obtained $\bar{\mu}_{0}(k)\geq 1$, set $\bar{\mu}_{0}(k)=0.99$. If
$\bar{\mu}_{0}(k)\leq 0$, set $\bar{\mu}_{0}(k)=0.01$. Then, regain $L(k)$ and
$G(k+1)$ by the equations composed of the first two expressions of (57) based
on the new $\bar{\mu}_{0}(k)$, and jump to Step 4.
Step 3. Let $G(k+1)$ and $\bar{\mu}_{0}(k)$ keep the value calculated at the
previous time, and solve $L(k)$ by the first relation of (57).
Step 4. Calculate remainder $\delta_{u}(k)$ of $u(k)$ by (50) based on
$\delta_{x}(k)=x(k)-\mathcal{X}(k-1)L(k)$ and the final value of $L(k)$,
$G(k+1)$ and $\bar{\mu}_{0}(k)$. Then,
$u(k)=\sum^{N}_{i=1}l_{i}u(k-i)+\delta_{u}(k)$ can be obtained. Implementing
$u(k)$ on system (3) and then measure the actual state $x(k+1)$.
Step 5. Update the data $\mathcal{X}(k)$ and $\mathcal{U}(k)$ in memory by
$x(k+1)$ and $u(k)$. Set $k=k+1$, and return to Step 2.

##  4 Illustrative Example

For demonstrating the validity and effectiveness of the proposed input-
mapping-based online learning SMC strategy for considered system (3), some
simulations are displayed in this section. Firstly, consider a numerical
example with the following parameters

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}0.1012&0.8075&1.7837\\\
-0.0529&0.0944&-0.0396\\\
0&0.1937&0.5402\end{array}\right],~{}~{}B=\left[\begin{array}[]{c}0\\\ 0\\\
0.1\end{array}\right],$ $\displaystyle\bar{D}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}0.2&0.1&0.2\end{array}\right]^{T},~{}~{}\bar{E}=\left[\begin{array}[]{ccc}0.5&0.2&0.1\end{array}\right].$

Then, based on the above system parameters, the following two examples are
given to make some comparisons for the proposed algorithm with the robust
method in [14] and the data-driven method in [15, 16].

###  4.1 Example 1

In this section, an example is given to compare the proposed co-design IMSMC
method, the robust method proposed in [14] and the adaptive method in [15] .
By solving LMI (29) with the above system parameters, the sliding mode
variable $G$ for the robust SMC method is obtained as
$G=[0.0728~{}~{}0.4562]$. Other parameters in [14] are given as
$\bar{\mu}_{0}=0.1$, $\bar{\xi}T=0.01$. For a fair comparison, the sliding
mode variable $G$ in [15] is set as robust method, and the remaining
parameters are $\delta_{0}=150$, $Tp_{0}=Tp_{1}=0.0004$,
$\alpha_{0}=\alpha_{1}=1$, $q_{0}=0.5$, $T\hat{k}_{0}(0)=0.01$ and
$T\hat{k}_{1}(0)=0$. For the proposed method in this paper, the length of the
historical data is chosen as $N=2$, and the switching parameter is set as the
robust method. In addition, a disturbance $f(k)=[0~{}0~{}1]^{T}$ is applied to
the real system at $k=50$ to $k=95$ for further comparison. For initial state
$x(0)=[-1~{}~{}1~{}-5]^{T}$, the effectiveness and superiority of the proposed
method are presented in the following two cases.

Case 1: Suppose that $\Delta=0.8$, the response curve of the closed-loop
system is depicted in Fig. 1. It can be observed from Figs. 1(a)-1(c) that the
convergence rate of system states in terms of control law (51) and the
adaptive control law in [15] are faster than that of robust SMC law. For the
adaptive in [15], the parameters needed to be tuned by trial-and-error method
to obtain a satisfactory performance, which reduces its application
efficiency. And the system state is difficult to reach the equilibrium point
if the system is disturbed during operation. Figs. 1(d)-1(e) show the
corresponding control law and sliding mode surface, from which we have the
proposed co-design IMSMC can not only keep a small chattering and improve the
performance of the system, but also better overcome the impact of the burst
disturbance.

Case 2: Suppose that $\Delta=2$, which is beyond the constraints of unknown
dynamic. The corresponding response curves of the closed-loop system are given
in Figs. 2(a)-2(c). There is no doubt that the improvement of the convergence
rate by using the co-design IMSMC method in this paper is obvious.
Furthermore, the SMC law and sliding mode surface are shown in Figs.
2(d)-2(e). It is easy to see that the robustness of the proposed co-design
IMSMC strategy is stronger.

(a) State $x_{1}(k)$.

(b) State $x_{2}(k)$.

(c) State $x_{3}(k)$.

(d) Control input $u(k)$.

(e) Sliding mode surface $s(k)$.

Figure 1: The curves when $\Delta=0.8$.

(a) State $x_{1}(k)$.

(b) State $x_{2}(k)$.

(c) State $x_{3}(k)$.

(d) Control input $u(k)$.

(e) Sliding mode surface $s(k)$.

Figure 2: The curves when $\Delta=2$.

###  4.2 Example 2

Now, the proposed co-design IMSMC law is compared with the data-driven SMC
method in [16], which uses an identification method to identify the model.
Firstly, the simulation result of the paper is reproduced for a correct
comparison. Secondly, the output of the system (15) and the tracking object
are supposed to be $y(k)=\left[\begin{array}[]{ccc}1&1&1\\\
\end{array}\right]x(k)$ and $y_{d}(k)=0$, respectively. Then, the parameters
in [16] are set as
$\mu=1.2,\eta=0.8,\beta=9,\lambda=1,\alpha=0.1,n=1,\delta=0.01,\gamma=0.5$.
However, the system output $y(k)$ is always diverging until the initial value
of the estimated pseudo gradient $\hat{\emptyset}(0)$ reaches to
$\hat{\emptyset}(0)=40$. Next, the parameters of the proposed co-design IMSMC
method are chosen as Example 1. It can be observed from the simulation result
in Figure. 3 that the proposed method can make the convergence rate of the
considered system faster. Thus, we can come to a conclusion that the model-
free adaptive method in [16] requires identifying a model, and the
identification results have high requirements on the initial value of the
model parameter. However, the co-design IMSMC method proposed in this paper is
unnecessary to identify model parameters, and the future system dynamic is
directly predicted based on the historical data. Furthermore, the convergence
rate of the addressed system is improved more significantly by the results in
this paper.

(a) System output $y(k)$. (b) Control input $u(k)$. (c) Sliding mode surface
$s(k)$.

Figure 3: The comparison of data-driven SMC and co-design IMSMC.

##  5 Conclusion

A novel online learning SMC strategy for the system is studied in this paper
via an input-mapping method. Firstly, a new form of the model is established
based on the input-mapping of historical information and a part of the model
is known accurately which is beneficial for improving the system performance.
Secondly, the input-mapping technique is introduced detailed to further
explain the motivation and principle of the method. Then, the sliding mode
surface and the IMSMC law are co-designed by minimizing an optimization
objective, and the combination coefficients of the historical information are
also derived through optimization. Next, the whole algorithm of the proposed
co-design IMSMC strategy and the stability analysis are presented. Finally,
the effectiveness of the proposed method is illustrated by a simulated system
and some given comparisons.

## References

  * [1] J. Yoneyama, ${H}_{\infty}$ filtering for fuzzy systems with immeasurable premise variables: an uncertain system approach, Fuzzy Sets and Systems 160 (12) (2009) 1738–1748.
  * [2] D. Bresch-Pietri, J. Chauvin, N. Petit, Adaptive control scheme for uncertain time-delay systems, Automatica 48 (8) (2012) 1536–1552.
  * [3] Z.-Y. Sun, L.-R. Xue, K. Zhang, A new approach to finite-time adaptive stabilization of high-order uncertain nonlinear system, Automatica 58 (2015) 60–66.
  * [4] L. Wu, P. Shi, H. Gao, State estimation and sliding-mode control of markovian jump singular systems, IEEE Transactions on Automatic Control 55 (5) (2010) 1213–1219.
  * [5] A. K. Behera, B. Bandyopadhyay, X. Yu, Periodic event-triggered sliding mode control, Automatica 96 (2018) 61–72.
  * [6] Z. Cao, Y. Niu, J. Song, Finite-time sliding-mode control of markovian jump cyber-physical systems against randomly occurring injection attacks, IEEE Transactions on Automatic Control 65 (3) (2019) 1264–1271.
  * [7] J. Hu, L. Shang, Y. He, Z. Zhu, Direct active and reactive power regulation of grid-connected DC/AC converters using sliding mode control approach, IEEE Transactions on Power Electronics 26 (1) (2010) 210–222.
  * [8] H. Li, J. Yu, C. Hilton, H. Liu, Adaptive sliding mode control for nonlinear active suspension vehicle systems using T–S fuzzy approach, IEEE Transactions on Industrial Electronics 60 (8) (2012) 3328–3338.
  * [9] H. Hou, X. Yu, L. Xu, R. Chuei, Z. Cao, Discrete-time terminal sliding-mode tracking control with alleviated chattering, IEEE/ASME Transactions on Mechatronics 24 (4) (2019) 1808–1817.
  * [10] H. Ma, J. Wu, Z. Xiong, A novel exponential reaching law of discrete-time sliding-mode control, IEEE Transactions on Industrial Electronics 64 (5) (2017) 3840–3850.
  * [11] W. Gao, Y. Wang, A. Homaifa, Discrete-time variable structure control systems, IEEE Transactions on Industrial Electronics 42 (2) (1995) 117–122.
  * [12] A. Saghafinia, H. W. Ping, M. N. Uddin, K. S. Gaeid, Adaptive fuzzy sliding-mode control into chattering-free im drive, IEEE Transactions on Industry Applications 51 (1) (2014) 692–701.
  * [13] J.-N. Li, H. Su, Y. Zhang, Z.-G. Wu, J. Chu, Chattering free sliding mode control for uncertain discrete time-delay singular systems, Asian Journal of Control 15 (1) (2013) 260–269.
  * [14] S. Qu, X. Xia, J. Zhang, Dynamics of discrete-time sliding-mode-control uncertain systems with a disturbance compensator, IEEE Transactions on Industrial Electronics 61 (7) (2013) 3502–3510.
  * [15] H. Ma, Z. Xiong, Y. Li, Z. Liu, Sliding mode control for uncertain discrete-time systems using an adaptive reaching law, IEEE Transactions on Circuits and Systems II: Express Briefs 68 (2) (2020) 722–726.
  * [16] N. Ebrahimi, S. Ozgoli, A. Ramezani, Data-driven sliding mode control: a new approach based on optimization, International Journal of Control 93 (8) (2020) 1980–1988.
  * [17] X. Wang, X. Li, J. Wang, X. Fang, X. Zhu, Data-driven model-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeleton, Information Sciences 327 (2016) 246–257.
  * [18] Y. Weng, X. Gao, Data-driven sliding mode control of unknown mimo nonlinear discrete-time systems with moving pid sliding surface, Journal of the Franklin Institute 354 (15) (2017) 6463–6502.
  * [19] M. Hou, Y. Wang, Y. Han, Data-driven discrete terminal sliding mode decoupling control method with prescribed performance, Journal of the Franklin Institute 358 (13) (2021) 6612–6633.
  * [20] M. Hou, Y. Wang, Data-driven adaptive terminal sliding mode control with prescribed performance, Asian Journal of Control 23 (2) (2021) 774–785.
  * [21] D. Liu, G.-H. Yang, Data-driven adaptive sliding mode control of nonlinear discrete-time systems with prescribed performance, IEEE Transactions on Systems, Man, and Cybernetics: Systems 49 (12) (2017) 2598–2604.
  * [22] L. Yang, J. Lu, Y. Xu, D. Li, Y. Xi, Constrained robust model predictive control embedded with a new data-driven technique, IET Control Theory & Applications 14 (16) (2020) 2395–2405.
  * [23] S. He, W. Chen, D. Li, Y. Xi, Y. Xu, P. Zheng, Iterative learning control with data-driven-based compensation, IEEE Transactions on Cybernetics (2021) 1–12.
  * [24] X. Su, X. Liu, P. Shi, Y.-D. Song, Sliding mode control of hybrid switched systems via an event-triggered mechanism, Automatica 90 (2018) 294–303.
  * [25] H. H. Choi, Variable structure output feedback control design for a class of uncertain dynamic systems, Automatica 38 (2) (2002) 335–341.
  * [26] J. Zhang, P. Shi, Y. Xia, H. Yang, Discrete-time sliding mode control with disturbance rejection, IEEE Transactions on Industrial Electronics 66 (10) (2018) 7967–7975.
  * [27] J. Wang, M. Z. Chen, L. Zhang, Observer-based discrete-time sliding mode control for systems with unmatched uncertainties, Journal of the Franklin Institute 358 (16) (2021) 8470–8484.


#  Enhancing Translation Validation of Compiler Transformations with Large
Language Models

Yanzhao Wang and Fei Xie  Department of Computer Science
Portland State University
Portland, OR 97201, USA
{wyanzhao<EMAIL_ADDRESS>
###### Abstract

This paper presents a framework that integrates Large Language Models (LLMs)
into translation validation, targeting LLVM compiler transformations where
formal verification tools fall short. Our framework first utilizes existing
formal verification tools for translation validation. In this work, we use
Alive2, a well-known tool in LLVM compiler verification, as an example. When
formal verification tools are unable to confirm a transformation’s soundness,
our framework employs fine-tuned LLMs for prediction. It then applies fuzzing
to transformations predicted as potentially unsound by the LLMs due to return
values or memory inconsistencies, aiming to find counterexamples. In cases
where transformations are unsound for other reasons or sound, or if no
counterexamples emerge, the framework directly reports these outcomes without
further fuzzing. This methodology has shown effectiveness in complex
application such as deep-learning accelerator designs, where traditional
formal verification tools struggle.

##  I Introduction

LLVM [1], a versatile open-source compiler, supports an extensive range of
programming languages and hardware design languages. The core element of LLVM
is its intermediate representation (IR). It serves as a key interface that
streamlines interactions among frontends, backends, and transformation passes
in the middle of the compiler, and therefore, is critical in integrating the
varied components of the LLVM framework.

Translation validation, as detailed in [11], has evolved into a powerful
method for verifying the correctness of compiler transformations. By
validating the semantics of the source and transformed IRs, it establishes an
efficient and robust mechanism for ensuring the reliability of compiler
transformations. Alive2 [10] is such a translation validation tool specialized
for LLVM IR. It leverages formal verification techniques, aiming to prevent
that compiler optimizations do not introduce bugs into the resultant code.

Despite their advantages, translation validation tools such as Alive2 have
significant limitations. They typically cannot manage unbounded loops and have
limited ability in handling external function calls. Moreover, the SMT
(Satisfiability modulo theories) solver [3], a fundamental component of these
tools, is often incapable of dealing with complex computations.

Our framework introduces a novel integration of fine-tuned Large Language
Models (LLMs) to overcome limitations in current compiler transformation
validation flows. It can conduct predictive analyses of the correctness of
transformations at which formal verification tools fall short. The process
begins with inputting source and target Intermediate Representations (IR), the
latter being a transformed version of the former. If the verification of this
transformation is deemed unsolvable by the SMT solver of formal tools such as
Alive2 (the example used in this paper), it is then passed to our LLM-based
transformation predictor. This predictor assesses the transformation,
categorizing it as likely sound or unsound. For transformations predicted as
unsound, the predictor also provides the reasons underlying the unsoundness.
Where transformations are unsound due to return values or memory
inconsistencies, our framework employs fuzzing to find counterexamples. If the
fuzzer finds counterexamples, it confirms the transformation’s unsoundness. In
cases where transformations are unsound for other reasons or sound, or if no
counterexamples emerge, our framework reports their predicted
soundness/unsoundness. This framework has been proven effective in practical
applications, such as in Intel’s deep-learning accelerator designs [2],
addressing the challenges posed by unbounded loops and complex computations.

Our primary contributions are as follows:

  * •

We pioneered the application LLMs in enhancing the reliability of compiler
transformations.

  * •

We seamlessly integrated Large Language Models (LLMs) with formal verification
tools, exemplified by application to Alive2, to rapidly and efficiently assess
LLVM transformation soundness. This integration proves particularly effective
in scenarios where traditional formal verification tools struggle.

The remainder of this paper is structured as follows. Section 2 introduces the
background. Section 3 presents an overview of our framework, while Section 4
dives into the specifics of our implementation. Section 5 reports our
evaluation results. Section 6 discusses limitations and future work, and
Section 7 concludes.

##  II Background

Two primary methodologies are currently widely used in the domain of formal
compiler verification:

1\. Theorem Proving: This approach formally verifies if every transformation
of a compiler preserves the semantics of the input program, e.g., CompCert [9]
is a compiler for C that is formalized and verified in Coq. However, compiler
certification using theorem proving is a highly complex and labor-intensive
process, and every compiler revision requires reproofing. These drawbacks
hinder future compiler improvements.

2\. Translation Validation: The translation validation approach was first
introduced by [11]; it entails a weaker formal technique that effectively
certifies the conformity of a compiler’s individual executions to the
compiler’s specification. This approach compares a compiler’s input and output
for its specific application.

Figure 1: Workflow of the Translation Validation Approach

As illustrated in Figure 1, the workflow for translation validation sends both
source and target programs to a validator. If the validator confirms that the
target program refines the source program, it generates a proof. Conversely,
it produces a counterexample if discrepancies are detected. Translation
Validation is practical and adaptable, efficiently ensuring compiler execution
aligns with specifications.

Figure 2: Schematic Diagram of the Translation Validation Framework

##  III Overview of The Framework

This section presents our novel translation validation framework, consisting
of three major components: the formal transformation checker, LLM-based
transformation predictor, and fuzzer. As shown in Figure 2, the framework aims
to offer an efficient approach for evaluating the soundness of compiler
transformations, particularly in scenarios where traditional formal
verification tools struggle.

The workflow starts with a source IR and a target IR transformed from the
source IR. In this paper, we use LLVM IR as an example, as demonstrated in
Figure 3. The IRs are first processed by the formal transformation checker,
utilizing the capabilities of established translation validation tools to
analyze and verify IR transformations. In this paper, we use Alive2 as the
example. The formal transformation checker compares the memory states and
return values of the transformed code, ensuring no new undefined behavior
arises. This involves encoding the source and target IR programs into SMT
encodings and submitting these to the SMT solver [3] for verification.

Figure 3: Example of LLVM programs

If the SMT solver is able to verify the transformation, our framework reports
the transformation validation results (indicating soundness or unsoundness)
directly. However, if the SMT solver fails to verify the transformation, due
to factors such as complex SMT expressions or unbounded loops, our framework
forwards the IR program pair to the transformation predictor (Step 1 in Figure
2). After processing through the prompt encode/decoder, the input is sent to
the fine-tuned LLM models, which generate the prediction results (Steps 2 and
3). Our transformation predictor currently supports the fine-tuned models:
Mistral-7B [5], Llama2-7B [12], and GPT-3.5 [4]. The fine-tuning process of
these LLM models is elaborated in a subsequent section.

Then, for transformations predicted as unsound due to inconsistent memory or
return values, our framework sends both the program pair and the reasons for
unsoundness to the fuzzer to identify counterexamples (Step 4). If the fuzzer
finds any counterexamples, the transformation is deemed unsound (Step 5 and
6). For transformations predicted as sound, and for unsound transformations
where no counterexamples are found or those caused by other reasons, our
framework directly reports their predicted soundness/unsoundness (Step 4).

##  IV Implementations

###  IV-A Formal Transformation Checker

The formal transformation checker is crucial for analyzing compiler
transformations. It focuses on verifying memory states, return values, and
preventing the introduction of new undefined behaviors (UBs), thereby ensuring
transformation correctness and reliability.

####  IV-A1 Checking Memory States

The checker uses existing formal verification tools for a detailed comparison
of memory states between the source IR $P_{\text{src}}$ and the target IR
$P_{\text{tgt}}$. It verifies that memory operations in $P_{\text{src}}$ and
$P_{\text{tgt}}$ are equivalent, denoted as:

$\forall\sigma_{\text{src}},\sigma_{\text{tgt}}:\text{Memory}(P_{\text{src}},\sigma_{\text{src}})\equiv\text{Memory}(P_{\text{tgt}},\sigma_{\text{tgt}})$

####  IV-A2 Checking Return Values

Similar to memory state verification, the checker ensures consistent return
values post-transformation, preserving the program’s functional behavior. It
encodes return values from both source and target programs and employs a SMT
solver, such as Z3 [3], to compare these SMT encodings, flagging any
discrepancies.

####  IV-A3 Detecting Introduction of New Undefined Behaviors

The checker detects new undefined behaviors (UBs) in the transformed program,
ensuring no new UBs are introduced. This involves checking against division by
zero, null pointer dereferencing, and other unpredictable or unsafe
operations, thereby maintaining the program’s correctness and reliability.

###  IV-B LLM-Based Transformation Predictor

This subsection introduces our primary innovation: fine-tuning Large Language
Models (LLMs) in enhancing the robustness of the compiler transformations.

Currently, our training data originates from the verification outcomes of
llvm-project [7] transformations and unit tests conducted by Alive2, while
similar data from other tools could also be applied. After eliminating
duplicate records, our dataset comprised 32,850 sound transformations and 405
unsound transformations. We randomly selected 40 pairs from these to form our
test dataset and allocated the remainder to the training set. For each unsound
transformation, we included reasons for the unsoundness, such as memory
issues, return values, or the introduction of new undefined behaviors (UB). We
then formatted this data into prompts for LLM fine-tuning, utilizing the
prompt encode/decoder within the transformation predictor. We defined the
fine-tuning prompt structure as follows:

  * •

System Content: ”Your task involves analyzing the given IR transformations. On
receiving a transformation in the ”Transformation: X $\rightarrow$ Y” format,
analyze and respond in the ”Status: A Reason: B” format. ”A” denotes the
transformation’s correctness as CORRECT or UNSOUND. For UNSOUND
transformations, list reasons ”B”. Your analysis involves a two-step approach:

    * –

Special Value Injection: Inject specific values into both original and
transformed IR to observe and compare behaviors.

    * –

Step-by-Step Computation and Analysis: Execute detailed computations for both
IR versions to pinpoint discrepancies.

Base your analysis on the following to assess soundness:

    * –

Undefined Behavior Consistency: The target should only trigger UB if the
source does. New UB in the target renders the transformation unsound.

    * –

Return Domain Consistency: The target’s return domain must align with the
source’s, except when the source triggers UB. A mismatched return domain
without source UB suggests unsoundness.

    * –

Poison Value Propagation: The target’s return value should indicate poison
only if the source’s does. Any additional poison in the target signals
unsoundness.

    * –

Undefined Value Handling: The target’s return value should be Undefined only
if the source’s is Undefined or poison. Introduction of Undefined values by
the target without source justification is unsound.

    * –

Return Value Consistency: The return values of both the source and target
should match when the source is clear of Undefined or poison. Variances under
a well-defined source indicate unsoundness.

    * –

Memory State Refinement: Verify that the memory state after target execution
refines that of the source’s. Memory state inconsistencies suggest
unsoundness.

  * •

User Content: “{Source IR} $\rightarrow$ {Target IR}”

  * •

Assistant Content: ”Transformation status: {SOUND/UNSOUND} Reason: {UNSOUND
REASON}”

Choosing the appropriate training data is crucial. Our dataset is imbalanced,
with far more sound than unsound transformations. Using the dataset directly
could make the model biased. To alleviate the potential bias issue, for sound
transformations, we removed 1,874 instances where the source and target code
were exactly the same. Then we sampled the sound and unsound data at ratios of
1:1, 2:1, 4:1, and 8:1 to evaluate the performance.

After fine-tuning, when our framework identifies transformations that the SMT
solver cannot prove, it forwards the transformation pair to the transformation
predictor. First, this data is structured using the prompt encoder, adhering
to the fine-tuning prompt structure. Next, the formatted data is sent to the
LLMs. The LLMs then predict the soundness/unsoundness of the transformations.
Finally, the prompt decoder converts the models’ output into a human-readable
text format, completing the transformation prediction process.

###  IV-C Fuzzer

The fuzzer module is a key part of our framework. It aims to find concrete
counterexamples that confirm the unsoundness due to inconsistent memory or
return values predicted by the transformation predictor. For LLVM
transformations, our framework uses LibFuzzer [7].

The fuzzer takes program pairs and their LLM-predicted unsound reasons given
by the framework as inputs. The input format is:

$\text{Input: }\\{\text{Source IR},\allowbreak\text{Target
IR},\allowbreak\text{Unsoundness Reasons}\\}$

Based on the predicted reasons for unsoundness, the fuzzer employs different
strategies to fuzz the predictions:

  * •

Handling Return Value Inconsistencies: If the unsoundness is caused by return
value issues, the fuzzer focuses on the return logic of the program pair. It
tests both programs under various inputs to spot any output differences.

  * •

Handling Memory Inconsistencies: For unsoundness caused by memory, the fuzzer
examines memory operations in both source and target IRs. It aims to find any
differences in memory state after execution, checking that the target IR does
not introduce new memory behaviors or access issues not in the source IR.

##  V Evaluation Results

Figure 4: Evaluation Results on LLVM Transformations

In this section, we present the evaluation results of our framework, focusing
on assessing the accuracy of LLMs in determining the soundness of compiler
transformations. We selected 80 transformations from the llvm-project and
Alive2’s unit tests, maintaining a balanced ratio of sound to unsound results.
For locally deployed models such as Llama2-7B and Mistral-7B, we conducted our
runs on an Apple M3 Max 128GB platform. For GPT-3.5, we accessed the model via
the OpenAI API. Additionally, we assessed the baseline performance of the
original GPT-4 and GPT-3.5 models without fine-tuning.

We conducted comparison of the fine-tuned Llama2-7B, Mistral-7B, and GPT-3.5
models, as well as the GPT-4 and GPT-3.5 models without fine-tuning, using the
same test data. These models were evaluated based on their accuracy,
considering various ratios of SOUND to UNSOUND data in the training set,
specifically 1:1, 2:1, 4:1, and 8:1.

As depicted in Figure 4, the fine-tuned GPT-3.5 model achieved an accuracy of
up to 88%, while locally deployable models, like the fine-tuned Llama2-7B,
reached an accuracy of up to 86%, and the Mistral-7B reached an accuracy of up
to 79%. The GPT-4 model, without fine-tuning, attained only an accuracy of
64%, and the GPT-3.5 without fine-tuning achieved only 50% accuracy. These
results demonstrate that LLMs, when fine-tuned with an appropriate dataset,
can effectively provide rapid assessments of compiler transformations.
Moreover, it shows that properly fine-tuned smaller models, such as Llama2-7B
and GPT-3.5, can outperform larger models like GPT-4. This evaluation
highlights the potential of LLMs in assessing the soundness of compiler
transformations.

###  V-A Evaluation of Deep-learning Accelerator Designs in LLVM IR

Figure 5: load_2d module from hVTA

We employed our framework to assess Intel deep-learning accelerator designs
[2] specified using HeteroCL [6], which features a Python-based domain-
specific language (DSL) for specifying hardware designs at a high abstraction
level. It then compiles the hardware from this DSL to a lower-level design
language such as LLVM, and that of high-level synthesis. In this evaluation,
we utilized one source deep-learning design (named hVTA), and two transformed
designs from the source design (named sVTA and uVTA) as examples. These
designs adhere to the VTA open-source accelerator
specifications111github.com/apache/tvm/blob/v0.6/vta/include/vta/hw_spec.h and
consist of four modules: load, ALU, GEMM, and store.

We discovered that individual modules generated LLVM code exceeding the LLMs’
4096 token limit. Therefore, we further divided the modules into smaller,
functionally discrete units such as store_2d, load_data, load_uop, and others,
totaling 19 modules. Figure 5 shows an example of the store_2d module’s code
from hVTA, illustrating how it performs tiling according to VTA’s SRAM shape
and writes data into DRAM. Out of these 19 modules, 12 were not successfully
processed by Alive2. For each of these 12 modules from hVTA, sVTA, and uVTA,
we used hVTA’s modules’ LLVM IR as the source program and the respective
modules’ LLVM IR from sVTA and uVTA as the targets. We employed a fine-tuned
GPT-3.5 model, with a SOUND: UNSOUND data ratio of 2:1, to predict the
soundness of their LLVM IR. The model indicated UNSOUND results for the
pad_top modules in hVTA and uVTA with inconsistent memory. The fuzzer
generated a counterexample with is_min_pad_value set to true. As shown in
Figure 6, we found that uVTA altered hVTA’s behavior. While the hVTA module
directly wrote 0 to SRAM, uVTA’s value written to SRAM depended on pad_val.

Figure 6: load_2d module from hVTA-uVTA

##  VI Limitations and Future Work

This section discusses current limitations of our approach and outlines
potential directions for future work.

Limitations: Our framework, despite promising results, has limitations. One
primary constraint is the limited diversity of our dataset. We managed to
collect only 405 unsound transformations, which may not sufficiently represent
the wide spectrum of potential unsound transformations in the wild. This
limitation could affect the generalizability of our model’s predictions.

Moreover, our fine-tuning efforts were constrained by platform capabilities.
We only conducted the fine-tuning of the Llama2-7B model and Mistral-7B
locally. Larger models may yield better performance.

Additionally, the inherent limitations of the models’ context window restrict
our framework’s ability to handle larger programs. This limitation can prevent
the processing of more extensive and complex program structures, posing a
challenge to the scope of transformations that our framework can accurately
evaluate.

Future Work: To address these limitations and enhance the robustness of our
framework, we plan to undertake the following initiatives in our future work:

  * •

Synthetic Data Generation: We are considering developing rules for synthetic
data generation. This approach could potentially enable us to generate a
larger number of training samples, thereby enhancing the training process and
potentially improving the model’s predictive accuracy. A larger and more
diverse dataset will likely improve the model’s ability to generalize and
accurately predict the correctness of LLVM transformations.

  * •

Exploring Larger Models: We intend to experiment with fine-tuning larger LLM
models. Larger models have a greater capacity for learning and might
demonstrate superior performance in predicting transformation correctness.

  * •

Exploring Other IRs: While our method is currently using LLVM IR and Alive2 as
examples, it is not inherently limited to them. In our future work, we also
aim to explore its applicability to other IRs, such as MLIR [8], broadening
the scope and versatility of our framework.

##  VII Summary

In this paper, we have presented a Translation Validation framework integrated
with LLMs. We utilized verification data from an existing formal verification
tool’s processing of LLVM transformations as the training set to fine-tune
models such as Llama2-7B, Mistral-7B, and GPT-3.5. This approach aimed to
provide a predictive mechanism for transformations that prove challenging for
traditional formal verification tools. The evaluation results underscore the
potential of LLMs in enhancing the robustness of compiler transformations.

## References

  * [1] The llvm compiler infrastructure.  llvm.org.
  * [2] Vta implementations in heterocl.
  * [3] L. De Moura and N. Bjørner.  Z3: An efficient smt solver.  In Tools and Algorithms for the Construction and Analysis of Systems: 14th International Conference, TACAS 2008, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2008, Budapest, Hungary, March 29-April 6, 2008. Proceedings 14.
  * [4] L. Floridi and M. Chiriatti.  Gpt-3: Its nature, scope, limits, and consequences.  Minds and Machines, 30:681–694, 2020.
  * [5] A. Q. Jiang, A. Sablayrolles, A. Mensch, C. Bamford, D. S. Chaplot, D. d. l. Casas, F. Bressand, G. Lengyel, G. Lample, L. Saulnier, et al.  Mistral 7b.  arXiv preprint arXiv:2310.06825, 2023.
  * [6] Y.-H. Lai, Y. Chi, Y. Hu, J. Wang, C. H. Yu, Y. Zhou, J. Cong, and Z. Zhang.  Heterocl: A multi-paradigm programming infrastructure for software-defined reconfigurable computing.  In Proceedings of the 2019 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays.
  * [7] C. Lattner and V. Adve.  Llvm: A compilation framework for lifelong program analysis & transformation.  In International Symposium on Code Generation and Optimization, 2004\. CGO 2004., 2004.
  * [8] C. Lattner, M. Amini, U. Bondhugula, A. Cohen, A. Davis, J. Pienaar, R. Riddle, T. Shpeisman, N. Vasilache, and O. Zinenko.  Mlir: Scaling compiler infrastructure for domain specific computation.  In 2021 IEEE/ACM International Symposium on Code Generation and Optimization (CGO).
  * [9] X. Leroy, S. Blazy, D. Kästner, B. Schommer, M. Pister, and C. Ferdinand.  Compcert-a formally verified optimizing compiler.  In ERTS 2016: Embedded Real Time Software and Systems, 8th European Congress.
  * [10] N. P. Lopes, J. Lee, C.-K. Hur, Z. Liu, and J. Regehr.  Alive2: bounded translation validation for llvm.  In Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation.
  * [11] A. Pnueli, M. Siegel, and E. Singerman.  Translation validation.  In International Conference on Tools and Algorithms for the Construction and Analysis of Systems.
  * [12] H. Touvron, L. Martin, K. Stone, P. Albert, A. Almahairi, Y. Babaei, N. Bashlykov, S. Batra, P. Bhargava, S. Bhosale, et al.  Llama 2: Open foundation and fine-tuned chat models.  arXiv preprint arXiv:2307.09288, 2023.


# Experimental results on $t\bar{t}+W/Z/\gamma$ and SM top couplings from the
Tevatron and the LHC

Tamara Vázquez Schröder for the ATLAS  CDF  CMS and D0 Collaborations  II.
Physikalisches Institut, Georg-August Universität Göttingen, Göttingen,
Germany<EMAIL_ADDRESS>
###### Abstract

Experimental results from the CDF and D0 Collaborations at the Tevatron and
the ATLAS and CMS Collaborations at the LHC on the processes related to
probing top quark couplings are presented. Evidence of both $t\bar{t}Z$ and
$t\bar{t}W$ processes is reported. All measurements are in agreement with the
SM expectations.

##  1 Motivation

The top quark was discovered in 1995 by the CDF and D0 Collaborations
[D0:topdisc, CDF:topdisc]. It couples to the Standard Model fields through its
gauge and Yukawa interactions. Some of these couplings have been investigated
at the Tevatron, through studies of the $Wtb$ vertex and the $t\bar{t}\gamma$
production, while others, such as the $t\bar{t}Z$ and $t\bar{t}H$ production,
are becoming accessible only with the high statistics top quark sample at the
LHC, also called for this reason a ‘top quark factory’. At hadron colliders,
the first evidence of the coupling of the top quark to the $\gamma,Z,$ and $H$
boson will come from the production rate, while constraints on the coupling of
the top quark with the $W$ boson come from both the top quark decay and the
single top production. Given its large mass, the top quark may play a special
role in the electroweak symmetry breaking (EWSB) and therefore, new physics
related to EWSB may be found first in top quark precision measurements.
Possible new physics signals would cause deviations of the top quark couplings
$tZ$, $t\gamma$, and $Wtb$, from the SM prediction.

##  2 Wtb coupling

Information on the coupling of the top quark to the $W$ boson can be obtained
from the top quark decay and electroweak single top production.

###  2.1 W-helicity measurements

Since the top quark decays almost exclusively as $t\rightarrow W^{+}b$, the
measurement of the $W$ boson helicity in top decays probes the structure of
the $Wtb$ vertex, which in the Standard Model (SM) is V-A. Since $W$ bosons
are produced as on-shell particles in top quark decays, their polarisation can
be longitudinal, left-handed or right-handed. The fractions with a certain
polarisation, $F_{0},F_{L}$ and $F_{R}$, can be extracted from measurements of
the angular distribution of the decay products of the top quark, given by:

$\frac{1}{\sigma}\frac{d\sigma}{d\cos{\theta^{*}}}=\frac{3}{4}(1-\cos^{2}{\theta^{*}})F_{0}+\frac{3}{8}(1-\cos{\theta^{*}})^{2}F_{L}+\frac{3}{8}(1+\cos{\theta^{*}})^{2}F_{R},$
(1)

where $\theta^{*}$ is defined as the angle between the $W$ boson momentum in
the top quark rest frame and the momentum of the down-type decay fermion in
the rest frame of the $W$ boson. The next-to-next-to-leading-order (NNLO) QCD
prediction for the helicity fractions in the SM, for a top quark mass
$m_{t}=172.8$ GeV and a $b$-quark mass $m_{b}=4.8$ GeV, are $F_{0}=0.687\pm
0.005$, $F_{L}=0.311\pm 0.005$ and $F_{R}=0.0017\pm 0.0001$
[TheoryNNLO:Whelicity]. Measurements of the $W$ boson helicity fractions have
been performed by both, CDF and D0 experiments [CDF:Whelicity, D0:Whelicity,
CDFD0:WhelicityComb]. The CDF measurement is performed with the full dataset
of 8.7