Feedforward–feedback-enhanced model-free adaptive iterative learning control with measurement disturbance and data dropout for an autonomous bus trajectory tracking system

Abstract

This article presents an innovative enhanced model-free adaptive iterative learning control approach suited for autonomous bus trajectory tracking systems that may experience measurement disruptions and random data dropouts. Data loss can occur independently and randomly at different times and in different iterations with varying probabilities, leading to successive data dropouts on both the time and iteration axes. The proposed enhanced model-free adaptive iterative learning control controller incorporates a data compensation mechanism to compensate for missing data, ensuring excellent control performance. This data-driven control strategy requires only input/output data for controller design. The convergence and effectiveness of the proposed approach are verified through rigorous mathematical analysis and simulation outcomes.

Keywords

Model-free adaptive iterative learning control measurement disturbance data dropout and autonomous bus

Introduction

The safety of passengers and public road traffic is significantly impacted by the control effect achieved for buses, which are the primary mode of public transport^1,2 and thus of paramount importance to the security of travelers’ lives and property. Therefore, research on trajectory tracking for autonomous buses has been a focus of scholars worldwide.

The controller design process can be greatly enhanced through the utilization of repetitive information, as bus operations exhibit certain repeatability, with fixed starting and terminal stations and daily driving routes.

In the realm of intelligent transportation, the iterative learning control (ILC) method has witnessed increasing application in recent years. ILC employs a typical finite-time repeated operation system to achieve optimal control of uncertain nonlinear systems. The literature presents various innovative approaches that provide effective solutions for speed trajectory tracking as well as the tracking of displacement–speed trajectories subject to unknown speed delays, control input saturation, and other uncertainties.

One such approach for the tracking control of high-speed trains (HSTs) is presented by Yu and Hou,³ in which an advanced ILC (AFILC) algorithm is recommended. According to Ji et al.,⁴ an adaptive ILC (AILC) approach for HSTs is introduced, which considers unknown speed delays and control input saturation to facilitate speed trajectory tracking. Similarly, an AILC method is recommended for tracking the speed and position of a subway train by Liu and Hou.⁵ Moreover, a constrained spatial AILC (CSAILC) controller is suggested by Li et al.⁶ to address displacement–speed trajectory tracking for automatic train control (ATC) systems. In contrast, a high-order pseudo-partial derivative-based model-free adaptive iterative learning controller (HOPPD-MFAILC) is proposed by Ai et al.⁷ to facilitate a fast convergence speed.

Another area in which the ILC method has seen broad application is in batch processing, for which a data-driven model-free AILC (MFAILC) method is proposed by Ren et al.⁸ to achieve precise control. A freeway traffic ramp-metering system under a non-strict repeatable pattern is addressed by Hou et al.,⁹ in which the ILC method is utilized to facilitate optimal control. Similarly, precise aircraft trajectory tracking can be achieved through ILC, as presented by Buelta et al.¹⁰

A novel approach for subway train speed tracking is presented by Zheng and Hou,¹¹ where model-free adaptive iterative learning energy-efficient control based on an extended state observer (ESO-based MFAILEEC) is introduced. Additionally, Yu et al.¹² proposed a distributed data-driven event-triggered MFAILC (ETMFAILC) method for multiple HSTs under iteration-varying topologies.

In summary, the ILC method is an effective tool in intelligent transportation systems that utilize historical data of the objects to be controlled. The ILC approach is widely applicable in tracking problems for a variety of system types, including HSTs, ATC systems, and aircraft tracking. The literature presents various innovative approaches to ILC that address system challenges and facilitate optimal control.

When an automated bus driving system is to receive the next input and output signals sent by a sensor, it may fail to receive them due to system failure or various transient factors, resulting in data loss. The normal operation of the bus will be detrimentally impacted by such data packet loss and measurement disturbances. The influence of measurement disturbances on the actual operation process will be examined in greater detail in the following section.

Numerous studies have been conducted on the resilience of ILC in the presence of data dropout and robustness issues. Upon review, it becomes apparent that most ILC techniques designed to address data dropout are exclusively applicable to linear systems^13–17 or affine nonlinear systems.^18–23 According to Bu et al.^13,14 and Shen et al.^15–17 separate ILC models are developed for linear systems. For affine nonlinear systems, Shen et al.,¹⁸ Liu and Ruan,¹⁹ Bu et al.,²⁰ and Liu and Ruan^21–23 proposed control systems combining P-type ILC with other algorithms tailored to affine nonlinear systems. This situation suggests the need for further research on ILC models capable of addressing data dropout in a broader range of systems.

Current academic research on overcoming the influence of measurement disturbances focuses on several techniques. For example, He et al.²⁴ explored the use of a flexible micro aerial vehicle (MAV) in the presence of spatiotemporally varying disturbances. Sun et al.²⁵ proposed a modified ILC strategy for a four degree-of-freedom (DOF) hybrid magnetic bearing system to reject disturbances. An innovative control method that combines active disturbance rejection control (ADRC) with current-cycle ILC (CILC) is suggested by Huang et al.²⁶ These studies showcase various approaches to addressing measurement disturbances in different contexts.

An accurate mathematical model of the control object is crucial for the above work. If the model of the object (e.g. a bus) is unknown, the above methods may not produce desirable results. For such situations, a data-driven control method provides an alternative solution.

Model-free adaptive control (MFAC) is an emerging data-driven control method that is undergoing rapid development. MFAC employs a virtual equivalent dynamic linearized data model with a pseudo-gradient (PG) at each operation point for a certain type of nonlinear discrete-time system. This approach replaces the general nonlinear discrete-time system. MFAC has proven to be an effective control method, showing promise as advancements in the field continue. The PG of the data model is estimated solely based on the online input and output from the regulated plant, and an MFAC strategy based on the established linearized data model is then formulated. The utilization of MFAC in the realm of intelligent transportation has yielded remarkable outcomes in research on the control of urban road traffic networks,²⁷ GRU networks for industrial operations,²⁸ autonomous 4 WMV parking systems,²⁹ AUV heading control,³⁰ and static synchronous compensators (STATCOMs) to achieve sub-synchronous oscillation mitigation.³¹ The MFAILC approach presented in this article achieves superior control performance due to its incorporation of the repetitive learning capabilities of ILC and the adaptive learning ability of MFAC. Unlike previous methods, the MFAILC strategy considers only the input and output data of the control object. This allows more precise adjustments and improved accuracy, making it a promising control strategy for various applications.

This article presents an enhanced MFAILC (E-MFAILC) algorithm for a self-driving bus trajectory tracking system faced with measurement disturbances and random data dropout. The E-MFAILC algorithm proposed in this study employs a compensation mechanism to overcome data loss, and it demonstrates excellent control performance. This data-driven control strategy relies only on I/O data for its design. The effectiveness and convergence of the E-MFAILC algorithm are validated through rigorous mathematical analysis and simulation results. The key contribution of this study lies in the development of a reliable trajectory-tracking system that is adaptive and iterative, making it especially suitable for real-world applications, in which precise measurements are not always guaranteed.

The article is structured as follows: The “Problem formulation” section outlines the problem formulation, “The scheme of E-MFAILC” section presents the E-MFAILC algorithm along with details on its convergence analysis, the “Simulation” section shows cases numerical examples to demonstrate the algorithm's efficacy, and finally, the “Conclusions” section concludes with a summary of the article.

Problem formulation

Currently, research on autonomous buses can be categorized into two areas: lateral angle control and longitudinal speed control. The control process can be better understood by referring to Figure 1.

Figure 1.

Longitudinal and lateral integrated control frame diagram.

Longitudinal dynamic analysis

The longitudinal speed of a bus while in motion is primarily determined by the longitudinal control forces exerted upon it. Factors that affect the vehicle during operation include various automobile principles such as air resistance, rolling resistance, and ramp resistance. These factors are depicted in Figure 2, which presents a longitudinal force analysis diagram. These factors are essential considerations for those studying or working in the field of automobile engineering.

Figure 2.

Longitudinal force diagram of the vehicle.

Newton's second law in classical mechanics states that the mechanical equation of longitudinal force can be articulated as follows:

m \overset{\cdot}{v} (t) = F_{A} (t) - F_{B} (t) - F_{C} (t) - F_{D} (t)

(1)

The variables in the above equation are the mass m, traction force

F_{A} (t)

, acceleration

\overset{\cdot}{v} (t)

, air resistance

F_{c} (t)

, rolling resistance

F_{B} (t)

, and slope resistance

F_{D} (t)

of the bus. Specifically, the traction force, which propels the vehicle via the drivetrain and tires, is expressed as

F_{A} (t) = u_{1} i_{g} i_{0} η_{t} / r

(2)

The variables involved in the above equation are the tire radius r, the transmission ratio

i_{g}

(determined by the selected gear), the output torque u of the engine, the mechanical efficiency

η_{t}

of the transmission, and the transmission ratio

i_{0}

of the main reducer.

The equation used to determine rolling resistance is

F_{B} (t) = f_{b} m g \cos α

(3)

which depends on the rolling resistance coefficient

f_{b}

of the wheels, the angle

α

between the road surface and the lateral plane, and the gravitational acceleration g. Finally, the equation for the air resistance is:

F_{C} (t) = 1 / (2 C_{D} A ρ y_{1}^{2}) .

(4)

From formulas (1) to (4), we can see that the vehicle speed is related to the traction force through a nonlinear dynamic process, which is expressed as follows:

\begin{matrix} \begin{matrix} y_{1} (k + 1) = y_{1} (k) + f_{1} (y_{1} (k), \dots, y_{1} (k - n_{y_{1}}), \\ u_{1} (k), \dots, u_{1} (k - n_{u_{1}})) \end{matrix} \end{matrix}

(5)

where

y_{1} (k + 1)

is the velocity v at moment

k + 1

and

f_{1} (\cdot)

depends on the appropriate dimensions in the longitudinal dynamic analysis.

A bus route exhibits periodicity in a finite time domain; accordingly, introducing the iterative axis i yields

\begin{matrix} \begin{matrix} y_{1} (i, k + 1) = y_{1} (i, k) + f_{1} (y_{1} (i, k), \dots, y_{1} (i, k - n_{y_{1}}), \\ u_{1} (i, k), \dots, u_{1} (i, k - n_{u_{1}})) \end{matrix} \end{matrix}

(6)

This analysis demonstrates that when the engine output torque surpasses the tire limit, the bus is propelled forward. This finding is supported by objective observations. When the bus starts up, the driver depresses the accelerator pedal in order to accelerate, and the engine's output torque increases. At this stage, once the force exerted by the bus exceeds the combined resistance from the air and the slope of the terrain, this will result in a positive acceleration, causing the bus to move forward. However, in situations where there are pedestrians or obstacles, the bus must slow down. The driver releases the gas pedal, thereby reducing the engine output torque and the bus's traction force, ultimately leading to a decrease in speed as the external resistance becomes greater than the bus's traction force. This is an important consideration for drivers when navigating potentially hazardous situations on the road.

In summary, the longitudinal speed tracking and control of the bus will require the utilization of both the control input $u_{1} (k)$ and the control output $y_{1} (k)$ in the longitudinal controller design.

Lateral dynamic analysis

The driving wheels are typically the front wheels, while the rear wheels serve as the power steering wheels. More specifically, a bus typically has two rear wheels and two front wheels. Figure 3 displays a lateral analysis of an equivalent model.

Figure 3.

A schematic representation in which the front wheels are depicted with the same angle of rotation.

As illustrated in Figure 3, $(x, y)$ is the rear wheel axis's location, with the axis perpendicular to the wheels’ direction, and the velocity is v. The red line indicates the direction of the bus's velocity v, which lies at an angle $y_{2}$ from the x-direction. The body angle $y_{2}$ can range from 0° to 90°. The turn angle u of the front wheels is indicated, and the distance L between the front and rear wheels is also shown. Within a very short period, the bus can be approximated as moving in the direction of the vehicle body; then, the following equations are obtained:

{\begin{matrix} d z / d h = \tan α \\ \tan α = \sin α / \cos α \end{matrix}

(7)

With (7), the following equation can be introduced:

- \dot{h} \sin α + \dot{z} \cos α = 0

(8)

Assuming that the bus travels a distance of

Δ t

and that the radius of the tires is b, we have

{\begin{matrix} d s = b {u^{'}}_{2} \\ b = L / \tan y_{2} \end{matrix}

(9)

From (9), the following can be derived:

u_{2}^{'} = (\tan β_{c} / L) d s

(10)

By combining (7) to (10), the following equations are obtained as the description of the lateral dynamics, with the center point between the rear wheels serving as the reference

{\begin{matrix} \dot{h} (t) = v \cos y_{2} (t) \\ \dot{z} (t) = v \cos y_{2} (t) \\ {\dot{y}}_{2} = v \tan u_{2} (t) / L \end{matrix}

(11)

Equation (11) can be discretized as follows:

{\begin{matrix} h (k + 1) = h (k) + T v \cos (y_{2} (k)) \\ z (k + 1) = z (k) + T v \sin (y_{2} (k)) \\ y_{2} (k + 1) = y_{2} (k) + T v \tan (u_{2} (k)) / L \end{matrix}

(12)

where T is the length of the sampling period. By utilizing equation (12) as the lateral model of the vehicle, we can ascertain its position coordinates at any given moment k, given knowledge of its velocity v, and body angle

y_{2}

. Thus, the vehicle's actual current location is determined. The vehicle's trajectory can be adjusted to the correct direction by modifying the body angle

y_{2}

Because a bus route exhibits periodicity in the finite time domain, introducing the iterative axis i yields:

{\begin{matrix} h (i, k + 1) = h (i, k) + T v \cos (y_{2} (i, k)) \\ z (i, k + 1) = z (i, k) + T v \sin (y_{2} (i, k)) \\ y_{2} (i, k + 1) = y_{2} (i, k) + T v \tan (u_{2} (i, k)) / L \end{matrix}

(13)

The lateral model in (13) indicates how the front wheel angle

u_{2} (i, k)

is related to the sampling time,

k \in {0, 1, \dots, T}

, and the number of iterations,

i = 1, 2, \dots

The following nonlinear discrete input/output system can be used to represent equation (13)

\begin{matrix} \begin{matrix} y_{2} (k + 1) = y_{2} (k) + f_{2} (y_{2} (k), \dots, y_{2} (k - n_{y_{2}}), \\ u_{2} (k), \dots, u_{2} (k - n_{u_{2}})) \end{matrix} \end{matrix}

(14)

at time index

k \in {0, 1, \dots, K}

in iteration i, the system output is

y_{2} (i, k)

, and the control input vector is

y_{2} (i, k)

;

f_{2} (\cdot)

depends on the appropriate dimensions.

The complexity of the autonomous vehicle control problem is exacerbated by the need for an exact mathematical representation of the vehicle in order to utilize methods of integrated lateral and longitudinal control due to the interconnection between the longitudinal and lateral control systems.³² However, relevant investigations indicate that independent lateral and longitudinal control can also achieve a good control effect. The analysis of the control system and the utilization of full-form dynamic linearization procedures, as detailed below, are applicable to both the lateral and longitudinal control of autonomous vehicles.

To uniformly represent the entire control system, the following nonlinear discrete system is formulated:

\begin{aligned} y_{γ} (i, k + 1) = f_{γ} (y_{γ} (i, k), \dots, y_{γ} (i, k - n_{y_{γ}}) \\ \times u_{γ} (i, k), \dots, u_{γ} (i, k - n_{u_{γ}})) \end{aligned}

(15)

where the system output for either longitudinal speed control or lateral angle control is denoted by

y_{γ} (i, k) \in R

, with

γ = 1

representing the longitudinal speed and

γ = 2

representing the lateral angle;

u (i, k) \in R^{n}

denotes the system input;

f_{γ} (\cdot) \in R^{n}

is an unknown, continuously differentiable nonlinear function; N is a bounded positive integer; i is the iteration index; k is the time index; and

0 < n_{y_{γ}} < N

and

0 < n_{u_{γ}} < N

define the order of the system.

The scheme of E-MFAILC

Before introducing dynamic linearization technology, we first state that system (15) is considered to satisfy the following assumptions.

Assumption 1:

The function $f_{γ} (\cdot)$ is continuously differentiable with respect to the control input, and for all iterations k and times t, the partial derivatives are bounded, that is:

Assumption 2:

On the iteration axis, equation (15) is generalized Lipschitz, meaning that it satisfies:

Δ y_{γ} (i, k) \leq b_{1} ‖ Δ U_{γ} (i, k) ‖

(16)

Where

\begin{matrix} Δ y_{γ} (i, k + 1) = y_{γ} (i, k + 1) - y_{γ} (i - 1, k + 1), \\ Δ U_{γ} (i, k) = U_{γ} (i, k) - U_{γ} (i - 1, k) and b_{1} > 0 \end{matrix}

\begin{matrix} U_{γ} (i, k) = [y_{γ} (i, k), \dots, y_{γ} (i, k - L_{y} + 1), \\ u_{γ} (i, k), \dots, u_{γ} (i, k - L_{u} + 1)]^{T} \end{matrix}

is the vector of the controller input and output signals for the ith iteration at a time k.

U_{γ} (i, k) \in R^{L_{Y} + L_{u}}

is defined as a sliding time window consisting of all control input signals at times

[k - L_{u} + 1, k]

and all system output signals at times

[k - L_{y} + 1, k],

where

L_{y}

and

L_{u}

(0 \leq L_{y} \leq k_{y}, 1 \leq L_{u} \leq k_{u})

are constant integers representing the linearized control output length and the linearized control input length, respectively.

Assumption 3:

For a given angle $y_{λ d} (k)$ , the control input $u_{γ d} (k)$ satisfies equation (7), and $| u_{γ d} (k) \leq b_{2} |$ is assumed to be true.

Remark 1:

Generally, the assumptions above are achievable in practical applications. Among them, Assumption 1 indicates that the dynamic characteristics of the controlled system are constantly changing, which is a commonly used control assumption with strong practicability. From an energy standpoint, Assumption 2 implies that in the given circumstances, the system's rate of change in energy cannot be boundless. Assumption 3 means that the control object is controllable, which is necessary for the whole control system.

Figure 1 shows that the desired body angle $y_{γ d} (k)$ of the vehicle can be determined from the desired angles $[y_{γ} (i, k), \dots, y_{γ} (i, k - L_{y} + 1)],$ thus decreasing the tracking error as the number of iterations i nears infinity. The control goal is to identify a suitable control input.

The contents of this section are arranged as follows. In the “Full-form dynamic linearization method for the iterative domain” section, the full-format dynamic linearization analysis was performed under the iterative domain. The AFF-FFDL-MFAILC algorithm is proposed in the “AFF-FFDL-MFAILC controller design” section and the “Design of the Enhanced AFF-ILC-MFAC (EMFAILC) controller with lost data compensation” section describes the AFF-FFDL-MFAILC algorithm with data loss and then proposed the E-MFAILC algorithm with compensation mechanism.

Full-form dynamic linearization method for the iterative domain

In the iterative domain, the dynamic linearization process for the nonlinear bus dynamics system can be summarized as follows:

Lemma 1:

For the nonlinear system in (15), when $‖ Δ U_{γ} (i, k) ‖ \neq 0$ , given Assumptions 1 to 3, the system can be transformed into a full-form dynamic linearized data model in the iterative domain as follows:

Δ y_{γ} (i, k + 1) = ϕ_{f, L_{y}, L_{u}} (i, k)^{T} Δ U_{γ} (i, k)

(17)

where

\begin{matrix} U_{γ} (i, k) = [y_{γ} (i, k), \dots, y_{γ} (i, k - L_{y} + 1), \\ u_{γ} (i, k), \dots, u_{γ} (i, k - L_{u} + 1)]^{T} \end{matrix}

and

ϕ_{f, L_{y}, L_{u}} (i, k)

is the PG of the system, described as

ϕ_{f, L_{y}, L_{u}} (i, k) = [ϕ_{1} (i, k), \dots, ϕ_{L_{u}} (i, k), ϕ_{L_{y} + 1} (i, k), \dots, ϕ_{L_{y} + L_{u}} (i, k)]^{T}

(18)

Proof:

Please see Hou and Xiong.³³

According to the actual operation requirements of buses, $L_{y} = L_{u} = 1$ here; therefore, equation (17) can be rewritten as follows:

Δ y_{γ} (i, k + 1) = ϕ_{1 γ} (i, k) Δ y_{γ} (i, k) + ϕ_{2 γ} (i, k) Δ u_{γ} (i, k)

(19)

The output of a system in the real world is often detrimentally impacted by measurement disturbances. Accordingly, the measured output is

\begin{aligned} Δ y_{γ} (i, k + 1) = ϕ_{1 γ} (i, k) Δ y_{γ} (i, k) \\ + ϕ_{2 γ} (i, k) Δ u_{γ} (i, k) + d (k) \end{aligned}

(20)

The nonlinear term

d (k)

collectively represents all quantifiable disruptions of the bus system.

AFF-FFDL-MFAILC controller design

The controller searches for a suitable control input to determine the vehicle control signal based on the disparity between the desired body angle $y_{d} (k)$ and the actual output signal.

The control input criterion function is

\begin{aligned} J (u_{γ} (i, k)) = | y_{γ d} (k + 1) - y_{γ p} (i, k + 1) |^{2} \\ + λ | u_{γ} (i, k) - u_{γ} (i - 1, k) |^{2} \end{aligned}

(21)

By substituting the linearized model in (20) into the criterion function (21), computing the derivative of the control quantity

u_{γ} (i, k)

, and setting it to zero, a linearized dynamic iterative control law can be obtained:

\begin{aligned} u_{γ b} (i, k) = u_{γ b} (i - 1, k) + \frac{ρ_{1} ϕ_{γ} (i, k)}{λ + {| ϕ_{p} (i, k) |}^{2}} \\ \cdot y_{γ d} (k + 1) - y_{γ p} (i, k + 1) \end{aligned}

(22)

where the step size factor

ρ_{1} (0, 1]

is used to allow more flexibility in the control algorithm.

A high-order adaptive feedforward iterative learning law is implemented to bolster the system's steadiness

\begin{aligned} u_{γ f} (i, k) = χ_{1} u_{γ f} (i - 1, k) + χ_{2} u_{f} (i - 2, k) \\ + δ_{1 e γ} (i - 1, k + 1) + δ_{2 e γ} (i - 2, k + 1) \end{aligned}

(23)

where

χ_{1}

χ_{2}

δ_{1}

, and

δ_{2}

are adaptive learning factors that can be adjusted.

Finally, the entire input control law $u_{γ} (i, k)$ for the system can be described as follows:

u_{γ} (i, k) = u_{γ f} (i, k) + u_{γ b} (i, k)

(24)

The PG vector in the control law given in (22) is an unknown parameter, and it thus needs to be estimated. In this article, an alternating algorithm is utilized to calculate this parameter. First, the index function is specified as follows:

\begin{aligned} J (ϕ_{γ} (i, k)) = | Δ y_{γ} (i - 1, k + 1) - ϕ_{γ}^{T} (i, k) Δ U_{γ} (i - 1, k) |^{2} \\ + μ ‖ ϕ_{γ} (i, k) - {\hat{ϕ}}_{γ} (i - 1, k) ‖^{2} \end{aligned}

(25)

Let

\partial J (ϕ_{γ} (i, k)) / \partial ϕ_{γ} (i, k) = 0

; by employing the “inverse matrix” lemma, we have

\begin{aligned} {\hat{ϕ}}_{γ} (i, k) = {\hat{ϕ}}_{γ} (i - 1, k) + \frac{η Δ U_{γ} (i - 1, k)}{μ + {‖ Δ U_{γ} (i - 1, k) ‖}^{2}} \\ \cdot (Δ y_{γ} (i - 1, k) - {\hat{ϕ}}_{γ}^{T} (i - 1, k) Δ U_{γ} (i - 1, k)) \end{aligned}

(26)

where

μ \in (0, 2]

is a step size factor, which increases the flexibility of the control algorithm, and

{\hat{ϕ}}_{γ} (i, k)

is the estimate of

ϕ_{γ} (i, k)

The parameter vector $ϕ_{γ} (i, k)$ is varying in time and is of great importance in the system, as demonstrated by the control law. To bolster the stability and time-varying parameter tracking capability of the system, the following parameter resetting algorithm is applied:

\begin{aligned} {\hat{ϕ}}_{γ} (i, k) = {\hat{ϕ}}_{γ} (i, 1), sign ({\hat{ϕ}}_{2 γ} (i, k)) \neq sign ({\hat{ϕ}}_{2 γ} (i, 1)), \\ or ‖ {\hat{ϕ}}_{γ} (i, k) ‖ \leq ε, or ‖ Δ U_{γ} (i, k) ‖ \leq ε \end{aligned}

(27)

Remark 2:

Equation (27) is defined as is the reset mechanism of the PG parameter ${\hat{ϕ}}_{γ} (i, k)$ . During the control process, the ${\hat{ϕ}}_{γ} (i, k)$ may update abnormally due to the system mutation or interference by external factors. By triggering the reset mechanism described via equation (27), the algorithm becomes more flexible and the situation of abnormal updates of ${\hat{ϕ}}_{γ} (i, k)$ can also be prevented.

Upon combining the control algorithm in (22), the high-order adaptive feedforward iterative learning law in (23), the entire input control law in (24), the previously obtained PG estimation algorithm in (26), and the resetting algorithm in (27), the corresponding AFF-FFDL-MFAILC algorithm is expressed as follows:

{\begin{matrix} u_{γ b} (i, k) = u_{γ b} (i - 1, k) + \frac{ρ_{1 ϕ γ} (i, k)}{λ + {| ϕ_{p} (i, k) |}^{2}} \\ \cdot (y_{γ d} (k + 1) - y_{γ p} (i, k + 1)) \\ u_{γ f} (i, k) = χ_{1} u_{γ f} (i - 1, k) + χ_{2} u_{γ f} (i - 2, k) \\ + δ_{1} e_{γ} (i - 1, k) + δ_{2} e_{γ} (i - 2, k) \\ u_{γ} (i, k) = u_{f γ} (i, k) + u_{b γ} (i, k) \\ {\hat{ϕ}}_{γ} (i, k) = {\hat{ϕ}}_{γ} (i - 1, k) + \frac{η Δ U_{γ} (i - 1, k)}{μ + {‖ Δ u_{γ} (i - 1, k) ‖}^{2}} \\ \cdot [Δ y_{γ} (i - 1, k) - {\hat{ϕ}}_{γ}^{T} (i - 1, k) Δ U_{γ} (i - 1, k)] \\ {\hat{ϕ}}_{γ} (i, k) = {\hat{ϕ}}_{γ} (i, 1), sign ({\hat{ϕ}}_{2 γ} (i, k)) \neq sign ({\hat{ϕ}}_{2 γ} (i, 1)) \\ or ‖ {\hat{ϕ}}_{γ} (i, k) ‖ \leq ε, or ‖ Δ U_{γ} (i, k) ‖ \leq ε \end{matrix}

(28)

where

ε

is a sufficiently small positive number;

η

μ

ρ

, and

λ

are the parameters of the controller to be selected;

{\hat{ϕ}}_{γ} (i, k)

is the estimate of the parameter matrix; and

e_{γ} (i - 1, k + 1) = y_{γ d} (k + 1) - y_{γ} (i - 1, k + 1)

{\hat{ϕ}}_{γ} (0, k)

is the initial parameter matrix estimate, that is,

{\hat{ϕ}}_{γ} (i, k)

in the initial iteration.

Design of the enhanced AFF-ILC-MFAC (E-MFAILC) controller with lost data compensation

The current main concern in the field of ILC is ensuring the robustness of the regulated system (15) in cases of data failure on both the controller and actuator sides. This issue must be addressed to ensure the system's repeatability and reliability.

Describe the data dropout

The control aim is to identify the most suitable control input $u_{γ} (i, k)$ that will enable the tracking error $e_{γ} (i, k + 1) = y_{γ d} (k + 1) - y_{γ} (i, k + 1)$ to reach iterative convergence, even in the presence of packet dropout, where $y_{γ d} (k + 1)$ denotes the desired trajectory for all time intervals $k \in {0, \dots, N}$ . Bernoulli random variables $β (k)$ are used to describe the random dropout of input and output data. A value of 1 denotes regular data transmission, whereas a value of 0 denotes transmission failure.

According to equation (28), the output $y_{γ} (i, k)$ of the system is needed in the process of updating the pseudo-partial derivative (PPD) estimate ${\hat{ϕ}}_{γ} (i, k)$ and the control quantity $u_{γ b} (i, k)$ . According to Figure 4, $y_{γ} (i, k)$ may be lost due to network transmission failure or sensor failure. Suppose that the controller can detect whether the output data point $y_{γ} (i, k)$ of the system is lost; when $y_{γ} (i, k)$ is not lost, ${\hat{ϕ}}_{γ} (i, k)$ and $u_{γ b} (i, k)$ are updated normally in accordance with formula (28), and when $y_{γ} (i, k)$ is lost, the values of ${\hat{ϕ}}_{γ} (i, k)$ and $u_{γ b} (i, k)$ remain unchanged from the previous time. That is

\begin{aligned} {\hat{ϕ}}_{γ} (i, k) \\ = {\begin{matrix} \begin{matrix} {\hat{ϕ}}_{γ} (i, k - 1) + \frac{η Δ u_{γ b} (i, k - 1)}{μ + Δ u_{γ b} {(i, k - 1)}^{2}} \\ (Δ y_{γ} (i, k) - {\hat{ϕ}}_{γ} (i, k - 1) Δ U_{γ} (i, k - 1)), \end{matrix} & \begin{matrix} if y_{γ} (i, k) is not missed \end{matrix} \\ {\hat{ϕ}}_{γ} (i, k - 1), & if y_{γ} (i, k) is missed \end{matrix} \end{aligned}

(29)

\begin{aligned} u_{γ b} (i, k) \\ = {\begin{matrix} \begin{matrix} u_{γ b} (i, k - 1) + \frac{ρ {\hat{ϕ}}_{γ} (i, k)}{λ + | {\hat{ϕ}}_{γ} (i, k) |^{2}} \\ (y_{γ d} (k + 1) - y_{γ} (i, k)), \end{matrix} & if y_{γ} (k) is not missed \\ u_{γ} (i, k - 1), & if y_{γ} (k) is missed \end{matrix} \end{aligned}

(30)

Figure 4.

A block diagram of the proposed control system.

Therefore, the AFF-FFDL-MFAILC algorithm with data dropout can be described as follows:

\begin{aligned} {\hat{ϕ}}_{γ} (i, k) = {\hat{ϕ}}_{γ} (i - 1, k) + β (k) \cdot \frac{η Δ u_{γ b} (i - 1, k)}{μ + {‖ Δ u_{γ b} (i - 1, k) ‖}^{2}} \\ \cdot [Δ y_{γ} (i - 1, k) - {\hat{ϕ}}_{γ}^{T} (i - 1, k) Δ u_{γ b} (i - 1, k)] \end{aligned}

(31)

\begin{aligned} u_{γ b} (i, k) = u_{γ b} (i - 1, k) + β (k) \cdot \frac{ρ_{1} ϕ_{γ} (i, k)}{λ + {| ϕ_{γ} (i, k) |}^{2}} \\ \cdot (y_{γ d} (k + 1) - y_{γ} (i, k + 1)) \end{aligned}

(32)

The value of

β (k)

is 0 or 1 (i.e.

β (k) \in (0, 1)

). Due to the randomness of data dropout,

β (k)

takes a random value of 0 or 1. If

β (k) = 1

, then the output data point

y_{γ} (i, k)

is not lost at the current time. If

β (k) = 0

, then the output data point

y_{γ} (i, k)

is lost at the current time. Obviously,

β (k)

is independent of

u_{γ b} (i, k)

y_{γ} (i, k)

, and

{\hat{ϕ}}_{γ} (i, k)

. Suppose that

β (k)

satisfies:

\begin{aligned} p r o b {β (k) = 1} = E {β (k)} = \bar{β} \\ p r o b {β (k) = 0} = 1 - E {β (k)} = 1 - \bar{β} \end{aligned}

(33)

where

p r o b {\cdot}

represents the probability factor,

\bar{β}

represents the successful data transmission rate, and

1 - \bar{β}

represents the data dropout rate and satisfies

0 \leq \bar{β} \leq 1

. In the analysis in this chapter, it is assumed that

\bar{β}

is known, that is, the dropout rate of the output data is known.

Design of the data compensation mechanism

According to the above analysis, the full-format linearized version of equation (1) for the SISO nonlinear discrete-time system is as follows:

\begin{aligned} y_{γ} (i, k) = y_{γ} (i, k - 1) + ϕ_{γ} (i, k - 1)^{T} \\ \cdot Δ U_{γ} (i, k - 1) + d (k) \end{aligned}

(34)

Equation (34) can be regarded as the one-step advance prediction of the output

y_{γ} (i, k)

. Therefore, if the output measurement

y_{γ} (i, k)

at time k is lost,

y_{γ} (i, k)

is estimated using

{\hat{ϕ}}_{γ} (i, k - 1)

and

Δ u_{γ} (i, k - 1)

from time

k - 1

. The estimated value of

y_{γ} (i, k)

is:

{\hat{y}}_{p} (i, k) = y (i, k - 1) + \hat{ϕ} (i, k - 1)^{T} Δ U (i, k - 1) + d (k)

(35)

when the output data point

y_{p} (i, k)

is not lost, the AFF-FFDL-MFAILC algorithm performs the update process in accordance with formula (28). When the output data point

y_{γ} (i, k)

is lost, the control algorithm performs the update using the estimate

{\hat{y}}_{γ} (i, k)

instead of

y_{γ} (i, k)

. At this time, whether the output data are received by the controller can be expressed by

β (k)

, and we have the following:

{\bar{y}}_{γ} (i, k) = {\begin{matrix} y_{γ} (i, k), & if \begin{matrix} β (k) = 1 \end{matrix} \\ {\hat{y}}_{γ} (i, k), & if \begin{matrix} β (k) = 0 \end{matrix} \end{matrix}

(36)

Therefore, the AFF-ILC-MFAILC algorithm for the SISO nonlinear discrete-time system with the addition of the data dropout compensation mechanism, also called the E-MFAILC algorithm, can be expressed as follows:

{\begin{matrix} u_{γ b} (i, k) = u_{γ b} (i - 1, k) + \frac{ρ_{1 ϕ γ} (i, k)}{λ + {| ϕ_{γ} (i, k) |}^{2}} \\ \cdot (y_{γ d} (k + 1) - {\bar{y}}_{γ} (i, k + 1)) \\ u_{f} (i, k) = χ_{1} u_{f} (i - 1, k) + χ_{2} u_{f} (i - 2, k) \\ + δ_{1} e_{p} (i - 1, k) + δ_{2} e_{p} (i - 2, k) \\ u_{γ} (i, k) = u_{γ f} (i, k) + u_{γ b} (i, k) \\ {\hat{ϕ}}_{γ} (i, k) = {\hat{ϕ}}_{γ} (i - 1, k) + \frac{η Δ U_{γ} (i - 1, k)}{μ + {‖ Δ u_{γ b} (i - 1, k) ‖}^{2}} \\ \cdot [Δ {\bar{y}}_{γ} (i - 1, k) - {\hat{ϕ}}_{γ}^{T} (i - 1, k) Δ U_{γ} (i - 1, k)] \\ {\hat{ϕ}}_{γ} (i, k) = {\hat{ϕ}}_{γ} (i, 1), sign ({\hat{ϕ}}_{2 γ} (i, k)) \neq sign ({\hat{ϕ}}_{2 γ} (i, 1)), \\ or ‖ {\hat{ϕ}}_{γ} (i, k) ‖ \leq ε, or ‖ Δ U_{γ} (i, k) ‖ \leq ε \end{matrix}

(37)

A block diagram of the proposed E-MFAILC system with data dropout compensation is shown in Figure 4.

Remark 3:

It is evident from Figure 4 that the controller can be iteratively taught based on the actual control input $u_{γ} (i, k)$ , thus rendering E-MFAILC with data dropout compensation feasible. The control equations (37) consider both input and output data dropout and provide the appropriate compensation. For the dropout of measurement data, that is, the loss of an output measurement $y_{γ} (i, k + 1)$ , the lost measurement is replaced with the predicted output value ${\hat{y}}_{γ} (i, k + 1)$ for compensation. For the control input, $u_{γ} (i - 1, k)$ is used to compensate for the loss of $u_{γ} (i, k)$ .

The proposed E-MFAILC algorithm in this article is applied to a class of SISO systems. For general multiple-input and multiple-output (MIMO) systems, the E-MFAILC algorithm can also be applied by using the dynamical linearization technique for MIMO systems.

Convergence analysis

Before the convergence analysis, the following two lemmas are given

Lemma 2:

Let

A = {[\begin{matrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ a_{1} & a_{2} & a_{3} & \dots & a_{k} \end{matrix}]}_{k \times k};

if $\sum_{i = 1}^{k} | a_{i} | < 1$ , then the spectral radius of the matrix diameter satisfies $s (A) < 1$ .

Lemma 3:

Let $A \in R^{k \times k}$ , and let $s (A)$ denote the spectral radius of the matrix A. Then, $\forall δ > 0$ , there will always exist a suitable matrix parametrization $‖ A ‖_{v}$ such that $‖ A (i, k) ‖_{v} \leq s (A) + δ$ .

Theorem 1:

The plant described by (15), which satisfies Assumptions 1 to 4, is controlled by the E-MFAILC scheme expressed in (37) for the regulator $y_{γ d} (k + 1) = c o n s t$ . Then, there exists a $λ > 0$ such that if $λ > λ_{min}$ , the following statements hold:

$\forall k$ , ${\hat{ϕ}}_{γ} (i, k)$ is bounded.

When $k \to \infty$ , $‖ e (i, k + 1) ‖$ can converge to a bounded range, that is, the bounded convergence of $lim_{k \to \infty} ‖ e (i, k + 1) ‖ \leq \frac{2 L_{σ}}{1 - d_{8} (i, k)} + | (1 - α (i, k + 1)) | L_{σ}$ is guaranteed.

Proof:

See the appendix.

Simulation

Construction of the simulation environment

The selected experimental bus's equipment specifications and the necessary speed it must maintain on the road lead to a scenario that solely focuses on the operation between two stations. To accurately simulate this scenario, a simulation duration of 300 s is set, equivalent to the 5-minute bus journey from one station to the next. The dynamic model used is based on data collected from the Beijing Foton BJ6105EVCA-49 bus, as detailed in Table 1, combined with the bus kinematic process analysis in the “Problem formulation” section. This represents an objective and precise approach to modeling the behavior of autonomous buses.

Table 1.

Data related to the BJ6105EVCA-49 bus.

	Test items		Set value
1	Parameters of mass (kg)	Quality of preparation	$11, 400 kg$
		Front axle mass	$3700 kg$
		Rear axle mass	$7700 kg$
		Maximum total mass	$18, 000 kg$
		Front axle mass	$7000 kg$
2	Mechanical structure parameters	Main reducer transmission ratio $i_{0}$	$4.33$
		Transmission ratio $i_{g}$	$0.90$
		Mechanical efficiency of the transmission $η_{t}$	$0.90$
		Wheel rolling resistance coefficient $f_{b}$	$17100$
		Radius of bus tire $r$	$0.9 m$
		Windward area of vehicles $A$	$1 m^{2}$
3	Other dynamics-related parameters	Density of air $ρ$	$1.29 kg / m^{3}$
		Time of sampling $p$	$0.1 s$
		Acceleration of gravity $g$	$9.8 N / m^{2}$
		Drag coefficient of air $C_{D}$	$0.4$
		Maximum traction power $L_{max}$	$46.8 kW$

A dynamic model with the following input‒output relationship is used for simulation:

\begin{aligned} y_{1} (k + 1) - y_{1} (k) = \frac{u_{1} (k) i_{g} i_{0} η_{k} p}{m r} - p f_{b} g \cos α \\ - \frac{p C_{D} A ρ y_{1} {(k)}^{2}}{2 m} - p g \sin α \end{aligned}

(38)

By substituting the relevant parameters from Table 1 into equation (38), a nonlinear relationship regarding the longitudinal motion of the bus is achieved as follows:

\begin{aligned} y_{1} (k + 1) - y_{1} (k) \\ = \frac{u_{1} (k) i_{g} i_{0} η_{k} p}{m r} - p f_{b} g \cos α \\ - \frac{p C_{D} A ρ y_{1} {(k)}^{2}}{2 m} - p g \sin α \\ = 2.165 \times 10^{- 5} u_{1} (k) - 1.65 \times 10^{5} - 1.43 \times 10^{- 6} y_{1}^{2} (k) \end{aligned}

(39)

Remark 4:

Notably, the model in equation (39) is employed only to create the independent input‒output data mapping of the bus; it is not involved in the controller design from the perspective of the model structure or parameter data.

For the comparative experiments in this article, two iterative learning algorithms are employed.

The conventional MFAILC algorithm by Ren et al.³⁴ is shown as follows:

\begin{aligned} u_{r} (i, k) = u_{r} (i - 1, k) + \frac{ρ \hat{φ} (i, k)}{λ + {| \hat{φ} (i, k) |}^{2}} \cdot [y (i - 1, k + 1) - y_{d} (k + 1)] \\ {\hat{φ}}_{p} (i, k) = {\hat{φ}}_{p} (i - 1, k) + \frac{η Δ u_{r} (i - 1, k)}{μ + {| Δ u_{r} (i - 1, k) |}^{2}} \\ \begin{matrix}  \end{matrix} \cdot [Δ \bar{y} (i - 1, k + 1) - {\hat{φ}}_{p} (i - 1, k) Δ u_{p} (i - 1, k)] \\ \begin{matrix} {\hat{φ}}_{p} (i, k) = {\hat{φ}}_{p} (i, 1), {\hat{φ}}_{p} (i, k) \leq ε & or \end{matrix} \begin{matrix} sgn ({\hat{φ}}_{p} (i, k)) \neq sgn ({\hat{φ}}_{p} (i, 1)) \end{matrix} \end{aligned}

(40)

According to Ji et al.,³⁵ a PD-type ILC algorithm is proposed, as follows:

u_{r} (i, k) = u_{r} (i - 1, k) + α_{d} Δ e_{c} (i, k - 1) + α_{p} e_{c} (i - 1, k)

(41)

Table 2 displays the parameters of the two algorithms considered for comparison, where the gain coefficient of the differential term is

α_{d}

, the error difference with respect to the previous moment is

Δ e_{c} (i, k - 1)

e_{c} (i, k - 1) = β_{c p} (k - 1) - β_{c} (i, k - 1)

, and the gain coefficient of the proportional term is

α_{p}

Table 2.

Parameter settings of the two control methods are considered for comparison.

Algorithm	Parameter settings
MFAILC	$λ_{γ} = 1$ , $μ_{γ} = 2$ , $ρ_{γ} = 0.2$ , $η_{γ} = 1$ , $ϕ_{γ} (i, 1) = 0$
PD-ILC	$α_{d_{γ}} = 0.6, α_{p_{γ}} = 0.6$ (longitudinal control)	$α_{d_{γ}} = 0.6, α_{p_{γ}} = 0.6$ (lateral control)

MFAILC: model-free adaptive iterative learning control; PD-ILC: partial derivative-iterative learning control.

Simulation analysis of longitudinal control

Consider the SISO nonlinear longitudinal control system of an autonomous bus as given in (15), which is contaminated by system noise $d_{1} (k)$ satisfying Assumption 1 with $σ = 0.005$ :

d_{1} (k + 1) = 0.4 \times r a n d n (1, k)

(42)

The desired output is

y_{1 d} (k) = {\begin{matrix} 0.28 \times k & if & k \leq 50 \\ 14 & if & 50 < k \leq 100 \\ 0.3 \times k - 16 & if & 100 < k \leq 120 \\ 20 & if & 120 < k \leq 180 \\ 18 & if & 180 < k \leq 230 \\ - 0.257 \times k + 77.11 & if & 230 < k \leq 300 \end{matrix}

(43)

In the simulation, the initial control input signals in each iteration were set to zero,

u_{1} (i, 1) = 0

and

y_{1 d} (0) = 0

. The PPD estimates were set to

{\hat{ϕ}}_{1} (i, 1) = 2

and

{\hat{ϕ}}_{1} (i, 2) = 0.5

. The other controller parameters were set to

ε_{γ} = 10^{- 5}

ρ_{1} = 0.5

λ_{1} = 0.3

η_{1} = 0.5

, and

μ_{1} = 1

. The successful data transfer rate was set to

β = 0.9

(

10 %

dropout). The simulation results of E-MFAILC in various iterations are depicted in Figure 5. As the number of iterations increases, the performance of the E-MFAILC method improves. In the sixth iteration, safe tracking of the expected trajectory is realized.

Figure 5.

Tracking the performance of enhanced model-free adaptive iterative learning control (E-MFAILC) with 10% data dropout.

The curves of the tracking effects produced by the E-MFAILC, MFAILC, and PD-ILC algorithms are shown in Figures 5 to 7. The tracking effect of the E-MFAILC algorithm is superior to those of the other two ILC algorithms, as evidenced by the up-and-down oscillation of its tracking curve in the 4th iteration. In contrast, the MFAILC algorithm is unable to accurately track the desired velocity beyond 100–180 s. Similarly, the PD-ILC algorithm cannot fully track the required velocity throughout the whole process. These findings indicate that the adaptation enabled by the data compensation mechanism ability makes the proposed approach perform significantly better than the algorithms considered for comparison.

Figure 6.

Tracking the performance of model-free adaptive iterative learning control (MFAILC) with 10% data dropout.

Figure 7.

Tracking the performance of partial derivative-iterative learning control (PD-ILC) with 10% data dropout.

As can be seen in Figures 8 to 11, both E-MFAILC and MFAILC algorithms can track the expected trajectory after four to six iterations, while the PD-ILC algorithm cannot meet the tracking requirements in terms of control inputs.

Figure 8.

The control input of enhanced model-free adaptive iterative learning control (E-MFAILC) with 10% data dropout.

Figure 9.

The control input of enhanced model-free adaptive iterative learning control (E-MFAILC) with 10% data dropout.

Figure 10.

The control input of partial derivative-iterative learning control (PD-ILC) with 10% data dropout.

Figure 11.

Errors of E-MFAILC, MFAILC, and PD-ILC with 10% data dropout.

Considering the error convergence rate and initial error value, the adaptability and anti-disturbance capability of the E-MFAILC algorithm are thus verified. The mean absolute error (MAE) index for error evaluation is expressed as shown in the following formula:

M A E = \sum_{i = 1}^{n} | e_{c p} (i, k) | / n

(44)

where

e_{c p} (i, k)

represents the absolute error derived from iteration i.

Table 3 lists the calculated errors of the three algorithms (E-MFAILC, MFAILC, and PD-ILC) after various numbers of iterations. These error statistics demonstrate that E-MFAILC's error values are lower than those of the other two methods, indicating both a faster error convergence speed (the MAE of E-MFAILC in the sixth iteration is 3.268, that of PD-ILC is 18.967, and that of MFAILC is 7.489) and a lower final stable error value (the MAEs in the 100th iteration for E-MFAILC, PD-ILC, and MFAILC are 0.0001315, 3.085, and 0.00013, respectively). From these two evaluations, it is not difficult to find that the E-MFAILC algorithm has better data compensation and adaptation capabilities than the other two algorithms.

Table 3.

The MAEs of the three algorithms in various iterations.

Number of iterations	Types of algorithms
Number of iterations	E-MFAILC	PD-ILC	MFAILC
3	20.000	20.000	20.000
6	5.268	18.967	7.489
20	0.001	5.23	0.223
100	0.00013	3.085	0.015

MAEs: mean absolute error; E-MFAILC: enhanced model-free adaptive iterative learning control; MFAILC: model-free adaptive iterative learning control; PD-ILC: partial derivative-iterative learning control.

From the error analysis in Figures 5 to 8 and Table 2, it can be seen that the tracking effects of the three algorithms are different. The E-MFAILC algorithm is validated in terms of its adaptability and data compensation ability in comparison to the MFAILC algorithm and PD-ILC.

Simulation analysis of lateral control

Consider the following SISO nonlinear lateral control system for an autonomous bus:

{\begin{matrix} h (i, k + 1) = h (i, k) + T v \cos (y_{2} (i, k)) \\ z (i, k + 1) = z (i, k) + T v \sin (y_{2} (i, k)) \\ y_{2} (i, k + 1) = y_{2} (i, k) + T v \tan (u_{2} (i, k)) / L \end{matrix}

(45)

where

T = \begin{matrix} 0.01 & s \end{matrix}

and

L = \begin{matrix} 20 & s \end{matrix}

; additionally, in the ideal case, the constant tracking velocity is 10 km/h. The desired body angle of the bus during curve tracking is described as follows:

y_{2} d (k) = {\begin{matrix} - 10, & 0 < k \leq 50 \\ 30 \times \sin (k π / 450), & 50 < k \leq 300 \end{matrix}

(46)

In each iteration, the initial simulation state was set to zero

(u_{2} (i, 0) = 0, y_{2 d} (0) = 0)

. The PPD estimates were set to

{\hat{ϕ}}_{2} (i, 1) = 1

and

{\hat{ϕ}}_{2} (i, 2) = 0.5

. The other controller parameters were set to

ε_{γ} = 10^{- 5}

ρ_{2} = 0.5

λ_{2} = 0.3

η_{2} = 0.5

, and

μ_{2} = 1

The system noise $d_{2} (k)$ , satisfying Assumption 1 with $σ = 0.001$ , is described as follows:

d_{2} (k + 1) = 0.4 \times r a n d n (1, k)

(47)

Under the same conditions, the control performance achieved using the E-MFAILC, MFAILC, and PD-ILC algorithms in the fourth to sixth iterations is shown in Figures 12 to 14. The E-MFAILC algorithm, in comparison to the other two ILC algorithms, exhibits a superior tracking effect. This is particularly evident in relation to the MFAILC algorithm, which displays a more visible up-and-down oscillation of the tracking curve in the fourth iteration; however, the MFAILC algorithm is unable to fully track the necessary angle after the 80-s mark. Similarly, the PD-ILC algorithm cannot fully track the required angle throughout the whole process.

Figure 12.

Tracking the performance of enhanced model-free adaptive iterative learning control (E-MFAILC) with 10% data dropout.

Figure 13.

Tracking the performance of MFAILC with 10% data dropout.

Figure 14.

Tracking the performance of partial derivative-iterative learning control (PD-ILC) with 10% data dropout.

As can be seen in Figures 15 to 17, both E-MFAILC and MFAILC algorithms can track the expected trajectory after four to six iterations, while the PD-ILC algorithm cannot meet the tracking requirements in terms of control inputs.

Figure 15.

The control input of enhanced model-free adaptive iterative learning control (E-MFAILC) with 10% data dropout.

Figure 16.

The control input of model-free adaptive iterative learning control (MFAILC) with 10% data dropout.

Figure 17.

The control input of partial derivative-iterative learning control (PD-ILC) with 10% data dropout.

Figure 18 shows the tracking error curves obtained using the three algorithms with the MAE as the error evaluation index. The E-MFAILC algorithm is clearly superior to the other two algorithms in terms of its initial error value and error convergence, particularly after 100 iterations. PD-ILC has difficulty maintaining a consistent final error value, and the error convergence rate of MFAILC is evidently inferior to that of E-MFAILC. The MFAILC algorithm's error convergence rate is faster than that of PD-ILC, but its initial error value is too high. Through comprehensive consideration of the error convergence rate and initial error value, the adaptability and anti-disturbance capability of the E-MFAILC algorithm are further verified.

Figure 18.

MAEs of E-MFAILC, MFAILC, and PD-ILC.

Table 4 lists the errors of the three algorithms (E-MFAILC, MFAILC, and PD-ILC) after various numbers of iterations. The error statistics demonstrate that E-MFAILC's error values are lower than those of the other two methods, with E-MFAILC's MAE in the sixth iteration being 7.358, that of PD-ILC being 30.367, and that of MFAILC being 7.489 and the final stable error values in the 100th iteration being 0.00024 for E-MFAILC, 43.085 for PD-ILC, and 0.015 for MFAILC. From these two evaluations, it is not difficult to find that the E-MFAILC algorithm has better data compensation and adaptation capabilities than the other two algorithms.

Table 4.

The MAEs of the three algorithms in various iterations.

Number of iterations	Types of algorithms
Number of iterations	E-MFAILC	PD-ILC	MFAILC
3	30.400	30.400	30.400
6	7.358	30.367	7.489
20	0.223	30.023	0.373
100	0.00024	43.085	0.015

With the increase in the data packet loss rate, the control performance will gradually weaken. After a large number of tests, when the data packet loss rate is greater than or equal to 20%, the control performance of the proposed controller in this article will significantly deteriorate. In practical applications, with the current sensing accuracy, the data packet loss rate is less than 5% in most cases. It has been measured by many numerical simulation experiments, when the data packet loss rate is 10%, the desired tracking effect can still be achieved, which can be satisfied in practical application.

Conclusions

This academic work presents the development of the novel E-MFAILC algorithm for a class of SISO systems that experience data dropout and measurement disturbances. To address the challenges presented by an unknown nonlinear system, an iterative learning method based on full-form dynamic linearization is proposed, which utilizes only the system input and output information. An E-MFAILC controller is then formulated. The use of data compensation allows the controller to maintain excellent performance in the event of packet loss. This approach is a data-driven control strategy that relies solely on I/O data for controller design. The effectiveness and convergence of the proposed approach are validated through rigorous mathematical analysis and simulation results.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by Beijing Municipal Natural Science Foundation under Grants 4212035, 4222045, National Natural Science Foundation (NNSF) of China under Grants 62073025, North China University of Technology Scientific Research Foundation, North China University of Technology YuYou Talent Training Program. R&D Program of Beijing Municipal Education Commission (KM202310009010, KM202210009011)). North China University of Technology Unveils Project Support (2023YZZKY03).

ORCID iD

Shida Liu

Author biographies

Shida Liu received his PhD degree at the Advanced Control Systems Laboratory from Beijing Jiaotong University in 2017. He is currently an associate professor at School of Automation, North China University of Technology University. His research interests include learning control, data-driven control, complex industrial process, and autonomous car.

Wei Huang received his bachelor's degree from Henan Institute of Technology, Xinxiang, China, in 2017 and is currently pursuing a master's degree from North China University of Technology Beijing, China. His current research interests include iterative learning control, adaptive control, and data loss.

Ye Ren received the bachelor's degree and PhD degree from Beijing Jiaotong University, Beijing, China, in 2013 and 2020, respectively. In 2018, he has visited École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland. He is currently a lecturer with North China University of Technology, Beijing, China. He is a member of the Beijing Key Laboratory on Intelligent Control Technology for Urban Road Traffic, and the youth member of the 3rd Transport Planning Technical Committee of World Transport Convention. He has authored over 20 papers in peer-reviewed journals and prestigious conference proceedings. His current research interests include data-driven control, multidimensional traffic systems control, and multi-agent systems control.

Li Wang, PhD supervisor, vice president of North China University of Technology, director of Beijing Key Laboratory of Intelligent Control Technology for Urban Road Traffic, Beijing Great Wall Scholar, Beijing Science and Technology Rising Star. He has presided over more than 30 projects, such as the National Natural Science Foundation of China, the National Key R&D Program, the Beijing Science and Technology Program, the Beijing Science and Technology Rising Star Program, the 2011 Scientific Research and Development Plan of the Beijing Municipal Education Commission, and the horizontal commissioned projects of enterprises, won nine provincial and ministerial science and technology awards, published 41 papers as the first author or corresponding author (23 papers retrieved by SCI/EI), and authorized 15 invention patents. He has six software copyrights and four published works. In 2017, he was rated as “Outstanding Communist Party Member of Beijing Universities”, and in 2020, he was rated as “Beijing Advanced Worker”. In 2015, the scientific research team led by him was awarded the “Annual Scientific Research and Innovation Team of the Chinese Society of Automation”. In terms of teaching and educating people, he has always adhered to front-line classroom teaching. In 2020, as the person in charge, he was approved as the “‘Control and Intelligence’ Beijing Excellent Undergraduate Education Team”, and in 2021, he was shortlisted for the first prize of Beijing Teaching Achievement Award, and his lecture on “Urban Road Traffic Control Theory and Technology” was rated as a demonstration course of ideology and politics in the postgraduate education course of Beijing universities.

Honghai Ji received his PhD degree at the Advanced Control Systems Laboratory from Beijing Jiaotong University in 2017. He is currently an associate professor at School of Automation, North China University of Technology University. His research interests include learning control, data-driven control, complex industrial process, and autonomous car.

Appendix

The following inequality can be easily obtained by taking the norms of both sides: (49)

\begin{matrix} \begin{matrix} ‖ {\tilde{ϕ}}_{γ} (i, k) ‖ \leq ‖ {\tilde{ϕ}}_{γ} (i - 1, k) (I_{1} - α (i, k + 1) \end{matrix} \end{matrix} \cdot \frac{η_{γ} Δ U_{γ} (i - 1, k) Δ U_{γ}^{T} (i - 1, k)}{μ_{γ} + {‖ Δ U_{γ} (i - 1, k) ‖}^{2}}) ‖

Furthermore, taking the square of the right-hand side of inequality (49) yields the following result: (50)

\begin{matrix} \begin{matrix} {‖ {\tilde{ϕ}}_{γ} (i - 1, k) (I_{1} - \frac{α (i, k + 1) η_{γ} Δ U_{γ} (i - 1, k) Δ U_{γ}^{T} (i - 1, k)}{μ_{γ} + {‖ Δ U_{γ} (i - 1, k) ‖}^{2}}) ‖}^{2} \\ = ‖ {\tilde{ϕ}}_{γ} (i - 1, k) ‖^{2} + (- 2 + \frac{α (i, k + 1) η_{γ} {‖ Δ U_{γ} (i - 1, k) ‖}^{2}}{μ_{γ} + {‖ Δ U_{γ} (i - 1, k) ‖}^{2}}) \\ \frac{α (i, k + 1) η_{γ} {‖ {\tilde{ϕ}}_{γ} (i - 1, k) Δ U_{γ} (i - 1, k) ‖}^{2}}{μ_{γ} + {‖ Δ U_{γ} (i - 1, k) ‖}^{2}} \end{matrix} \end{matrix}

Since the parameters satisfy

0 < η_{γ} < 2

μ_{γ} > 0

, and

α (i, k + 1) \in {0, 1}

, a constant

0 < d_{1} < 1

can be found such that the following inequality holds: (51)

‖ {\tilde{ϕ}}_{γ} (i, k) ‖ \leq d_{1} ‖ {\hat{ϕ}}_{γ} (i - 1, k) ‖ \leq \dots \leq d_{1}^{k} ‖ {\tilde{ϕ}}_{γ} (0, k) ‖

and since

{\tilde{ϕ}}_{γ} (0, k)

is bounded, it is clear that

lim_{k \to \infty} ‖ {\tilde{ϕ}}_{γ} (i, k) ‖ = 0

holds. Thus,

{\tilde{ϕ}}_{γ} (i, k)

and

ϕ_{γ} (k)

are bounded. The proof of property (a) in Theorem 4.1 is complete.

By solving equation (1.13), we can obtain: (53)

σ_{γ} (i, k + 1) = \sum_{i = 0}^{k} {\tilde{ϕ}}_{γ} (i, k) Δ U_{γ} (i, k)

where

{\tilde{ϕ}}_{γ i} = [{\tilde{ϕ}}_{γ} (0, k), {\tilde{ϕ}}_{γ} (1, k), \dots, {\tilde{ϕ}}_{γ} (i, k)] \in R^{i \times (i * (k + 1) * (i + 1))}

; when

k < 0

, the output estimation error

σ_{γ} (i, k + 1)

is set to zero; and

Δ U_{γ i} = [Δ U_{γ}^{T} (1, k), Δ U_{γ}^{T} (2, k), \dots, Δ U_{γ}^{T} (i, k)]^{T} \in R^{i * (k + 1) * (i + 1)}

From inequality (21), it can be deduced that when $k \to \infty$ , $‖ {\tilde{ϕ}}_{γ} (i, k) ‖$ converges to zero. It is obvious that we can further obtain $lim_{k \to \infty} ‖ {\tilde{ϕ}}_{γ} (i, k) Δ U (i, k) ‖ = 0$ . Therefore, $lim_{k \to \infty} ‖ \sum_{j = 0}^{k} {\tilde{ϕ}}_{γ} (i, k) Δ U_{γ} (i, k) ‖$ is bounded. In turn, it can be shown that $σ_{γ} (i, k + 1)$ is bounded. In the next analysis, we assume that $‖ σ_{γ} (i, k + 1) ‖ \leq L_{σ_{γ}}$ , where $L_{σ_{γ}}$ is a bounded positive constant.

Since ${\hat{ϕ}}_{γ} (i, k)$ is bounded and $β (k) \in {0, 1}$ , there exists a $λ_{γ min} > 0$ such that the following inequalities hold when $λ_{γ} > λ_{γ min}$ : (54)

\begin{matrix} \begin{matrix} ‖ \frac{β (k) {\hat{ϕ}}_{γ} (i, k)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} ‖ \leq \frac{β (k) ‖ {\hat{ϕ}}_{γ} (i, k) ‖}{2 \sqrt{λ_{γ}} ‖ {\hat{ϕ}}_{γ} (i, k) ‖} \\ \leq \frac{β (k)}{2 \sqrt{λ_{γ min}}} = d_{2} (i, k) < \frac{0.5}{L_{u}} \end{matrix} \end{matrix}

(55)

\begin{matrix} \begin{matrix} 0 < ‖ \frac{β (k) {\hat{ϕ}}_{γ} (i, k) {\hat{ϕ}}_{γ} (t, k)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} ‖ \leq L_{u} \frac{β (k) ‖ {\hat{ϕ}}_{γ} (i, k) ‖}{2 \sqrt{λ_{γ}} ‖ {\hat{ϕ}}_{γ} (i, k) ‖} \\ < \frac{β (k) L_{u}}{2 \sqrt{λ_{min}}} = d_{3} (i, k) < 0.5 \end{matrix} \end{matrix}

(56)

\begin{matrix} {(\sum_{t = 0}^{k - 1} ‖ \frac{β (i, k) {\hat{ϕ}}_{γ} (i, k) {\hat{ϕ}}_{γ} (t, k)}{λ + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} ‖)}^{\frac{1}{t}} \leq d_{4} (i, k) \end{matrix} .

According to equations (22) and (23), the following result can be obtained: (57)

\begin{matrix} \begin{matrix} u (i, k) = β (k) (u (i - 1, k) + \frac{ρ {\hat{ϕ}}_{γ}^{T} (i, k)}{λ_{γ} + {‖ \hat{φ} (i, k) ‖}^{2}} (\tilde{e} (i - 1, k + 1) \\ - \sum_{i = 0}^{k - 1} {\hat{ϕ}}_{γ} (i, k) Δ u (i, k))) + (1 - β (k)) u (i - 1, k) \\ = u (i - 1, k) + \frac{ρ β (k) {\hat{ϕ}}_{γ}^{T} (i, k)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} (\tilde{e} (i - 1, k + 1) \\ - \sum_{i = 0}^{k - 1} {\hat{ϕ}}_{γ} (i, k) Δ u (i, k)) \end{matrix} \end{matrix}

Let (58)

A (i, k) = [\begin{matrix} 0 & I_{2} & 0 & \dots & 0 \\ 0 & 0 & I_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & I_{2} \\ 0 & - \frac{ρ_{γ} β (k) {\hat{ϕ}}_{γ} (i, k) {\hat{ϕ}}_{γ} (i, 0)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} & - \frac{ρ_{γ} β (k) {\hat{ϕ}}_{γ} (i, k) {\hat{ϕ}}_{γ} (i, 1)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} & \dots & - \frac{ρ_{γ} β (i, k) {\hat{ϕ}}_{γ} (i, k) {\hat{ϕ}}_{γ} (i, k - 1)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} \end{matrix}]

and

C = [0, \dots, 0, I_{2}]^{T} \in R^{(i * (k + 1)) \times i}

, where

0 \in R^{i \times i}

is a zero matrix and

I_{2} \in R^{k \times k}

is a unit moment array. This leads to the following expression: (59)

\begin{matrix} \begin{matrix} Δ {\bar{U}}_{γ} (i, k) = A (i, k) Δ {\bar{U}}_{γ} (i, k - 1) \\ + \frac{ρ_{γ} β (k) C {\hat{ϕ}}_{γ}^{T} (t, k)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} \tilde{e} (i - 1, k + 1) \end{matrix} \end{matrix}

where

Δ {\bar{U}}_{γ} (i, k) = [Δ u_{γ}^{T} (i, 0), \dots, Δ u_{γ}^{T} (i, k)]^{T} \in R^{(i * (k + 1)) \times 1}

denotes an iterative difference vector with a fixed number of dimensional control inputs. For example,

Δ {\bar{U}}_{γ} (i, k - 1) = [o^{T}, Δ u_{γ}^{T} (i, 0), \dots, Δ u_{γ}^{T} (i, k - 1)]^{T} \in R^{(i * (k + 1)) \times 1}

, where

o = [0, \dots, 0]^{T} \in R^{i}

and when

i < 0

Δ u (i, k) = o

The characteristic equation of the matrix $A (i, k)$ is (60)

\begin{matrix} \begin{matrix} det (z^{i + 1} I_{2} + \frac{ρ_{γ} β (k) {\hat{ϕ}}_{γ}^{T} (i, k) {\hat{ϕ}}_{γ} (i, k - 1)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} z^{t} \\ + \dots + \frac{ρ_{γ} β (k) {\hat{ϕ}}_{γ}^{T} (i, k - 1) {\hat{ϕ}}_{γ} (i, k)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (i, k) ‖}^{2}} z) = 0 \end{matrix} \end{matrix}

Based on inequality (60) , we can find a suitable parameter

ρ_{γ}

such that (61)

ρ_{γ} \sum_{i = 0}^{k - 1} ‖ \frac{β (k) {\hat{ϕ}}_{γ} (i, k) {\hat{ϕ}}_{γ} (t, k)}{λ + ‖ {\hat{ϕ}}_{γ} (i, k) ‖} ‖ | z |^{k - i} \leq ρ d_{4}^{t} (i, k) < 1

Then, by Lemma 2, it follows that: (62)

\begin{matrix} \begin{matrix} | z |^{t} \leq ρ_{γ} \sum_{i = 0}^{k - 1} ‖ \frac{β (k) {\hat{ϕ}}_{γ} (i, k) {\hat{ϕ}}_{γ} (t, k)}{λ_{γ} + ‖ {\hat{ϕ}}_{γ} (i, k) ‖} ‖ | z |^{k - i} \\ \leq ρ_{γ} \sum_{i = 0}^{k - 1} ‖ \frac{β (k) {\hat{ϕ}}_{γ} (i, k) {\hat{ϕ}}_{γ} (t, k)}{λ_{γ} + ‖ \hat{φ} (i, k) ‖} ‖ | z |^{k - i} \leq ρ d_{4}^{k} (i, k) \end{matrix} \end{matrix}

This means

| z |^{t} \leq ρ_{γ} d_{4}^{t} (i, k) < 1

. Therefore, we can find the appropriate controller parameters

ρ_{γ}

and

λ

ensure that

| z | \leq^{t} \sqrt{ρ γ} d_{4} (i, k) < 0.5

. According to Lemma 2, we can obtain that: (63)

‖ A (i, k) ‖ \leq^{t} \sqrt{ρ_{γ}} d_{4} (i, k) + δ \leq d_{5} (i, k) < 0.5

where

‖ \cdot ‖_{v}

denotes a suitable matrix parametrization and

0 < d_{5} (i, k) < 0.5

is a constant.The following inequality can be derived by taking the norms of both sides of equation (63): (64)

\begin{matrix} \begin{matrix} ‖ Δ {\bar{U}}_{γ} (i, k) ‖ \leq ‖ A (i, k) ‖_{v} ‖ Δ {\bar{U}}_{γ} (i, k - 1) ‖_{v} \\ + ρ_{γ} ‖ \frac{β (k) {\hat{ϕ}}_{γ} (t, k}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (t, k ‖}^{2}} ‖ ‖ {\tilde{e}}_{γ} (i - 1, k + 1) ‖ \end{matrix} \end{matrix} .

Then, according to inequalities (62) to (64), the following results can be deduced: (65)

‖ Δ {\bar{U}}_{γ} (i, k) ‖_{v} \leq ρ_{γ} d_{2} (i, k) \sum_{i = 0}^{t} d_{5}^{t - i} (i, k) ‖ {\tilde{e}}_{γ} (i - 1, k + 1) ‖

where

{\bar{U}}_{γ} (i, k)

is set to zero at

t < 0

. From the definition of

{\tilde{e}}_{γ} (i, k + 1)

, we can obtain the following result: (66)

\begin{matrix} \begin{matrix} {\tilde{e}}_{γ} (i, k + 1) = y_{d} (k + 1) - α (i, k + 1) y_{γ} (i, k + 1) \\ - (1 - α (i, k + 1)) {\hat{y}}_{γ} (i, k + 1) \\ = y_{γ d} (k + 1) - α (i - 1, k + 1) y_{γ} (i - 1, k + 1) \\ - (1 - α (i - 1, k + 1)) {\hat{y}}_{γ} (i - 1, k + 1) \\ + α (i - 1, k + 1) y_{γ} (i - 1, k + 1) \\ - (1 - α (i - 1, k + 1)) {\hat{y}}_{γ} (i - 1, k + 1) \\ - α (i, k + 1) y_{γ} (i, k + 1) - (1 - α (i, k + 1)) {\hat{y}}_{γ} (i, k + 1) \\ = {\tilde{e}}_{γ} (i - 1, k + 1) + α (i - 1, k + 1) σ_{γ} (i - 1, k + 1) \\ - α (i, k + 1) σ_{γ} (i, k + 1) - {\hat{ϕ}}_{γ} (i, k) Δ U_{γ} (i, k) \end{matrix} \end{matrix}

According to equation (63), equation (66) can be rewritten as (67)

\begin{matrix} \begin{matrix} {\tilde{e}}_{γ} (i, k + 1) \\ = (I_{2} - \frac{ρ_{γ} β (i, k) {\hat{ϕ}}^{T} (i, k) \hat{ϕ} (t, k)}{λ + {‖ \hat{ϕ} (i, k) ‖}^{2}}) {\tilde{e}}_{γ} (i - 1, k + 1) \\ + α (i - 1, k + 1) σ_{γ} (i - 1, k + 1) \\ - α (i, k + 1) σ_{γ} (i, k + 1) - \hat{ϕ} (i, k) A (i, k) Δ {\bar{U}}_{γ} (i, k - 1) \end{matrix} \end{matrix}

Taking the norms of both sides of equation (67) yields: (68)

\begin{matrix} \begin{matrix} ‖ {\tilde{e}}_{γ} (i, k + 1) ‖ \leq ‖ (I_{2} - \frac{ρ_{γ} β (k) {\hat{ϕ}}_{γ}^{T} (i, k) {\hat{ϕ}}_{γ} (t, k)}{λ_{γ} + {‖ {\hat{ϕ}}_{γ} (t, k ‖}^{2}}) ‖ \\ ‖ {\tilde{e}}_{γ} (i - 1, k + 1) ‖ \\ + ‖ α (i - 1, k + 1) σ_{γ} (i - 1, k + 1) ‖ \\ + ‖ α (i, k + 1) σ_{γ} (i, k + 1) ‖ \\ + ‖ {\hat{ϕ}}_{γ} (i, k) ‖_{v} ‖ Δ {\bar{U}}_{γ} (i, k - 1) ‖_{v} \end{matrix} \end{matrix}

It is clear that (69)

\begin{matrix} \begin{matrix} {‖ (I_{2} - \frac{ρ_{γ} β_{γ} (k) {\hat{ϕ}}^{T} (i, k) \hat{ϕ} (t, k)}{λ γ + {‖ \hat{ϕ} (i, k) ‖}^{2}}) ‖}^{2} \\ = ‖ I_{2} ‖^{2} + (- 2 + \frac{ρ_{γ} β (k) ‖ {\hat{ϕ}}^{T} (i, k) \hat{ϕ} (t, k) ‖}{λ + {‖ \hat{ϕ} (i, k) ‖}^{2}}) \\ \frac{ρ_{γ} β (k) ‖ {\hat{ϕ}}^{T} (i, k) \hat{ϕ} (t, k) ‖}{λ + {‖ \hat{ϕ} (i, k) ‖}^{2}} \end{matrix} \end{matrix} .

Based on equality (69), we can get equation (70) in the following form: (70)

\begin{matrix} \begin{matrix} {‖ (I_{2} - \frac{ρ_{γ} β (k) {\hat{ϕ}}^{T} (i, k) \hat{ϕ} (t, k)}{λ + {‖ \hat{ϕ} (i, k) ‖}^{2}}) ‖}^{2} \\ = 1 + (- 2 + ρ_{γ} d_{3} (i, k)) \\ = (1 - ρ_{γ} d_{3} (i, k))^{2} \end{matrix} \end{matrix}

Similarly, we can discover suitable controller parameters such that the following inequality is valid (71)

0 < ‖ (I_{2} - \frac{ρ_{γ} β (k) {\hat{ϕ}}^{T} (i, k) \hat{ϕ} (i, k)}{λ + {‖ \hat{ϕ} (i, k) ‖}^{2}}) ‖ = d_{6} (i, k) < 1

where

0 < d_{6} (i, k) = 1 - ρ_{γ} d_{3} (i, k) < 1

is a constant.From the previous analysis of (70) and (71), it follows that

\forall i

and

\forall k

α (i, k + 1) \in {0, 1}

. Therefore,

‖ α (i - 1, k + 1) σ_{γ} (i - 1, k + 1) ‖ + ‖ α (i, k + 1) σ_{γ} (i, k + 1) ‖ \leq 2 L_{σ_{γ}}

is bounded. Thus, according to inequality (71), we can obtain (72)

\begin{matrix} \begin{matrix} ‖ {\tilde{e}}_{γ} (i, k + 1) ‖ \\ \leq d_{6} (i, k) ‖ {\tilde{e}}_{γ} (i - 1, k + 1) ‖ \\ + ρ L_{u_{γ}} d_{2} (i, k) \sum_{i = 0}^{t - 1} d_{5}^{t - i} (i, k) \\ \cdot ‖ {\tilde{e}}_{γ} (i - 1, k + 1) ‖ + 2 L_{σ_{γ}} \end{matrix} \end{matrix}

Let us define

‖ e_{max, k} ‖ = max {‖ e (i, k + 1) ‖}

. Suppose that

‖ e_{max, k} ‖ = ‖ e_{k} (τ) ‖

, where

τ

is some sampling moment in

{1, \dots, N}

. According to inequality (72), we can obtain (73)

\begin{matrix} \begin{matrix} ‖ {\tilde{e}}_{max, k} ‖ = ‖ {\tilde{e}}_{k} (τ) ‖ \leq d_{6} (i, k) ‖ {\tilde{e}}_{k - 1} (τ) ‖ \\ + ρ_{γ} d_{3} (i, k) \sum_{i = 1}^{τ} d_{5} (τ - i, k) ‖ {\tilde{e}}_{γ} (i - 1, k) ‖ + 2 L_{σ_{γ}} \end{matrix} \end{matrix}

Let

d_{7} (i, k) = ρ_{γ} d_{3} (i, k)

; then, we can obtain (74)

\begin{matrix} \begin{matrix} ‖ {\tilde{e}}_{max, k} ‖ = ‖ {\tilde{e}}_{k} (τ) ‖ \leq d_{6} (i, k) ‖ {\tilde{e}}_{k - 1} (τ) ‖ \\ + ρ_{γ} d_{7} (i, k) \sum_{i = 1}^{τ} d_{5} (τ - i, k) ‖ {\tilde{e}}_{γ} (i - 1, k) ‖ + 2 L_{σ_{γ}} \\ \leq (d_{6} (i, k) + d_{7} (i, k) \frac{d_{5} (i, k)}{1 - d_{5} (i, k)}) ‖ {\tilde{e}}_{max, k - 1} ‖ + 2 L_{σ_{γ}} \end{matrix} \end{matrix}

From equations (73) and (74), it is clear that

0 < d_{7} (i, k) = 1 - d_{6} (i, k) < 1

, and thus, we can obtain the following result: (75)

\begin{matrix} \begin{matrix} d_{6} (i, k) + d_{7} (i, k) \frac{d_{5} (i, k)}{1 - d_{5} (i, k)} \\ = 1 - d_{7} (i, k) + d_{7} (i, k) \frac{d_{5} (i, k)}{1 - d_{5} (i, k)} \\ = 1 + \frac{(2 d_{5} (i, k) - 1) d_{7} (i, k)}{1 - d_{5} (i, k)} \end{matrix} \end{matrix}

From inequality (75), it is known that

0 < d_{5} (i, k) < 0.5

and

d_{7} (i, k)

lie on the interval

(0, 1)

, and it is clear that: (76)

0 < 1 + \frac{(2 d_{5} (i, k) - 1) d_{7} (i, k)}{1 - d_{5} (i, k)} = d_{8} (i, k) < 1

Thus, inequality (1.34) can be reformulated as (77)

\begin{matrix} \begin{matrix} ‖ {\tilde{e}}_{max, k} ‖ \leq d_{8} (i, k) ‖ {\tilde{e}}_{max, k - 1} ‖ + 2 L_{σ_{γ}} \\ \leq \dots \leq d_{8}^{k} (i, k) ‖ {\tilde{e}}_{max, ()} ‖ + \frac{2 L_{σ_{γ}}}{1 - d_{8} (i, k)} \end{matrix} \end{matrix}

Because

‖ {\tilde{e}}_{max, ()} ‖

and

\frac{2 L_{σ_{γ}}}{1 - d_{8} (i, k)}

are bounded, it is clear that

lim_{k \to \infty} ‖ {\tilde{e}}_{max, k} ‖ \leq \frac{2 L_{σ_{γ}}}{1 - d_{8} (i, k)}

also converges to a small, bounded range.In accordance with the definition of the error

{\tilde{e}}_{γ} (i, k + 1)

, we can obtain the relationship between the actual output error

e_{γ} (i, k + 1)

and

{\tilde{e}}_{γ} (i, k + 1)

as follows: (78)

\begin{matrix} \begin{matrix} {\tilde{e}}_{γ} (i, k + 1) = y_{d} (k + 1) - α (i, k + 1) y_{γ} (i, k + 1) \\ - (1 - α (i, k + 1)) {\hat{y}}_{γ} (i, k + 1) \\ = α (i, k + 1) y_{γ d} (k + 1) - α (i, k + 1) y_{γ} (i, k + 1) \\ + (1 - α (i, k + 1)) y_{γ d} (k + 1) \\ - (1 - α (i, k + 1)) {\hat{y}}_{γ} (i, k + 1) \\ = α (i, k + 1) e_{γ} (i, k + 1) \\ + (1 - α (i, k + 1)) (y_{γ d} (k + 1) - {\hat{y}}_{γ} (i, k + 1)) \\ = α (i, k + 1) e_{γ} (i, k + 1) \\ + (1 - α (i, k + 1)) (y_{γ d} (k + 1) - {\hat{y}}_{γ} (i, k + 1)) \\ + y_{γ} (i, k + 1) - {\hat{y}}_{γ} (i, k + 1) \\ = e_{γ} (i, k + 1) + (1 - α (i, k + 1)) σ_{γ} (i, k + 1) \end{matrix} \end{matrix}

After some simplification and manipulation, equation (78) can be transformed into the following form: (79)

e_{γ} (i, k + 1) = {\tilde{e}}_{γ} (i, k + 1) - (1 - α (i, k + 1)) σ_{γ} (i, k + 1)

Taking the norms of both sides of equation (79) yields the following: Since

lim_{k \to \infty} ‖ {\tilde{e}}_{γ} (i, k + 1) ‖

and

lim_{k \to \infty} ‖ σ_{γ} (i, k + 1) ‖

are bound, we can thus conclude that when

k \to \infty

‖ e_{γ} (i, k + 1) ‖

converges to a bounded range. Thus, the bounded convergence of

lim_{k \to \infty} ‖ e_{γ} (i, k + 1) ‖ \leq \frac{2 L_{σ_{γ}}}{1 - d_{8} (i, k)} + | (1 - α (i, k + 1)) | L_{σ_{γ}}

is proven, and conclusion (b) of Theorem 1 is proven.

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