An LMI-based robust state-feedback controller design for the position control of a knee rehabilitation exoskeleton robot: Comparative analysis

Abstract

Rehabilitation exoskeleton robots play a crucial role in restoring functional lower limb movements for individuals with locomotor disorders. Numerous research studies have concentrated on adapting the control of these rehabilitation robotic systems. In this study, we investigate an affine state-feedback control law for robust position control of a knee exoskeleton robot, taking into account its nonlinear dynamic model that includes solid and viscous frictions. To ensure robust stabilization, we employ the Lyapunov approach and propose three methods to establish stability conditions using the Schur complement, the Young inequality, the matrix inversion lemma, and the S-procedure lemma. These conditions are formulated as Linear Matrix Inequalities (LMIs). Furthermore, we conduct a comprehensive comparison among these methods to determine the most efficient approach. At the end of this work, we present simulation results to validate the developed LMI conditions and demonstrate the effectiveness of the adopted control law in achieving robust position control of the knee exoskeleton robot.

Introduction

The world is undergoing a profound demographic shift characterized by the remarkable rise in the aging population over the past few decades. This demographic transformation, often referred to as population aging, carries profound implications for various facets of society, including healthcare, social welfare, labor markets, and overall economic productivity. ¹ A key domain influenced by population aging is the mobility and functional abilities of older individuals, who experience natural declines in physical capabilities, such as reductions in muscular strength, motor neurons, muscle fibers, and aerobic capacity.^2–5 As a response to these demographic changes and the pressing need to enhance the mobility and independence of older individuals, exoskeletons have emerged as a promising field of robotics and assistive technology. These wearable robotic devices are designed to augment or restore human physical capabilities by providing mechanical support, strength amplification, and motion assistance, and they have garnered significant attention and research interest due to their potential applications in rehabilitation, physical therapy, and assistance for the elderly and individuals with mobility impairments.^6–8

Within the realm of exoskeletons, a specific focus has been placed on the development of exoskeletons tailored for rehabilitation purposes.^9–11 These rehabilitation exoskeletons play a pivotal role in bridging the gap between therapeutic interventions and assistive technologies, as they provide both rehabilitative exercise training and support for daily activities. This convergence of rehabilitation and robotics has ushered in new possibilities for enhancing the lives of individuals with mobility impairments, enabling them to regain independence and actively participate in daily tasks. A critical determinant of the effectiveness of rehabilitation exoskeletons lies in their control systems.^8,12–14 Control algorithms are instrumental in ensuring the safe and effective operation of these exoskeleton devices, facilitating seamless interaction between the user and the robotic system. The control system’s primary responsibility is to generate and coordinate the motions of the exoskeleton’s actuators to assist, amplify, or restore human movements with precision and efficiency. The success of rehabilitation exoskeletons hinges on the control algorithm’s ability to mimic and enhance the wearer’s natural movements while providing necessary support and assistance.^15–19

Developing effective control strategies for rehabilitation exoskeletons is a multifaceted research endeavor, combining principles from robotics, biomechanics, and rehabilitation science.^13,20–22 The intricate nature of human-robot interaction and the variability in patients’ physical conditions pose significant challenges in the development of robust and adaptive control strategies.^23,24 Researchers and engineers are continuously exploring various control techniques and strategies to optimize the functionality and performance of rehabilitation exoskeletons. Personalized and adaptive control strategies have emerged as focal points in this field, aiming to provide efficient and precise control that seamlessly integrates the exoskeleton with the user’s body, thereby enhancing their mobility and quality of life. Furthermore, the integration of advanced sensing technologies, such as wearable sensors and inertial measurement units, has enriched the potential for developing sophisticated control strategies.^17,23,25 These sensors offer valuable insights into the wearer’s gait, posture, and muscle activities, enabling the control system to adapt in real time based on the user’s intentions and needs. The incorporation of sensor data into the control algorithm fosters a more natural and coordinated human-robot interaction, facilitating more intuitive and fluid movements.

The control of rehabilitation exoskeletons represents a captivating multidisciplinary endeavor that harmoniously integrates principles from the diverse domains of robotics, biomechanics, and rehabilitation science. This harmonious fusion paves the way for continuous research and innovation in this ever-evolving field. The relentless pursuit of resilient and adaptable control strategies serves as the driving force propelling advancements in this dynamic arena. Hence, numerous studies have delved into diverse control strategies with a specific focus on improving walking assistance and gait rehabilitation.^{12,14,25–30} These endeavors contribute to advancing the field by offering insights into the multifaceted domain of lower limb exoskeleton control. These groundbreaking exoskeletons not only promise heightened stability during the rehabilitation process but also hold the potential to bestow individuals with mobility impairments a newfound sense of independence and an enriched quality of life. The paramount focus of this paper is dedicated to the formidable task of attaining meticulous position control within a powered lower limb exoskeleton, with a specific spotlight on the pivotal knee joint.^31–35 Undoubtedly, the knee joint stands as a linchpin in fundamental activities such as walking and bearing weight, accentuating its pivotal role in the blueprint of exoskeletons designed for rehabilitation and assistance. Our primary mission revolves around the conception and deployment of a robust affine state-feedback controller, meticulously crafted to effectively preside over and stabilize the knee exoskeleton robotic system. The mastery of knee joint control stands as a linchpin in ensuring the precision and seamlessness of movements, ultimately bestowing users with heightened mobility and an amplified sense of independence.^10,24,32,33

To accomplish our goal, we delve into the intricacies of the robotic system’s nonlinear dynamic model, which encompasses factors like solid and viscous frictions. These frictions wield substantial influence over the knee exoskeleton’s performance, rendering their inclusion in the control design process imperative. Employing an affine state-feedback control approach and harnessing the principles of Lyapunov stability theory, we introduce three distinct design methods. These methods allow us to establish a set of Linear Matrix Inequality (LMI) conditions governing the feedback gain of the proposed control law.^31,36–38 These LMI conditions serve as robust safeguards for the knee exoskeleton system, ensuring stability and resilience even in the face of uncertainties and disturbances. In addition to theoretical analysis and design, this paper offers comprehensive simulation results validating the effectiveness of the LMI conditions and the proposed affine state-feedback controller. The simulations furnish compelling evidence of the controller’s capacity to maintain stability and robustness, reinforcing the credibility of our approach.

The main contributions of this paper can be succinctly outlined like so:

• A robust affine state-feedback controller is meticulously crafted to ensure the stabilization of the knee exoskeleton robot. This controller is tailored to address the intricate nonlinear dynamics inherent in the system, taking into account pivotal factors such as solid and viscous frictions, uncertain parameters, and external disturbances affecting the knee-joint exoskeleton robotic system. The amalgamation of parametric uncertainties and external disturbances is encapsulated within a unified variable, termed the lumped disturbance.

• Utilization of the Lyapunov approach, the S-procedure Lemma, the Young inequality, the Schur complement, the matrix inversion lemma, and judicious congruence transformations to formulate stability conditions in terms of LMIs via three different methods used to establish quadratic conditions on the boundedness of the nonlinearity presented in the dynamical model of the knee exoskeleton.

• Enhancement of the stability conditions in LMI form through the introduction of optimization problems. These problems permit the maximization or minimization of two critical parameters: the maximum amplitude of the lumped disturbance and the parameter incorporated into the controller constraint.

• Presentation of a series of numerical simulations validating the derived mathematical results. These simulations serve to illustrate the controller’s efficacy in compensating for the impact of the lumped disturbance on the knee exoskeleton system.

The subsequent sections of this paper are organized as follows: Section 2 delves into the description of the nonlinear dynamics of the knee exoskeleton and the formulation of the problem. Section 3 introduces the controlled dynamics and utilized affine state-feedback controller. Section 4 explores the design of LMI stability conditions through three distinct methodologies. Section 5 presents the results of the simulations, providing empirical validation of our approach. Finally, Section 6 concludes the paper while discussing potential avenues for future research.

Nonlinear dynamics of the knee exoskeleton and problem formulation

Description of knee exoskeleton system

In this comprehensive study, we embark on an illuminating exploration of the intricate nuances embodied by the knee exoskeleton, masterfully unveiled in the artfully crafted schematic showcased in Figure 1. ³² This knee exoskeleton epitomizes a design ethos meticulously honed around geometric precision, a testament to its engineering excellence in the quest for optimal functionality. Taking the form of a seated structure, it is a thoughtful creation engineered to grant unrestricted shank mobility, orchestrating the graceful symphony of knee joint extension and flexion. At $\theta =0°$ , the knee joint unfurls its full splendor in extension, while it achieves the pinnacle of flexion at $\theta =90°$ .

Figure 1.

Geometric representation of the knee exoskeleton robot for the rehabilitation purposes (adopted from Rifaï et al. ³² ).

This ingenious exoskeleton springs to life through the fusion of two seamlessly interlinked segments, astutely poised at the knee joint’s epicenter, and seamlessly entwined through a pivotal rotational axis. The upper segment, an eloquent representation of the thigh, plays the role of a vital citadel, housing the actuator that powers the exoskeleton’s fluid motion. The lower segment, with painstaking precision, mirrors the shank’s distinctive attributes. To secure an impeccable fit, both segments clasp firmly onto the user’s lower limb, courtesy of adjustable straps. This meticulously crafted configuration breathes life into the knee exoskeleton, endowing it with the remarkable ability to faithfully replicate the knee joint’s natural sagittal plane rotational ballet.

In our unwavering pursuit of simplicity and astute motion management for the knee exoskeleton, we made a deliberate choice: the adoption of a single-degree-of-freedom robotic knee joint system. This strategic decision empowers us to channel our research endeavors toward the cultivation and rigorous scrutiny of robust control strategies. Our paramount aspiration finds its expression in the enhancement of stability and precision within the realm of lower limb rehabilitation, culminating in the provision of unwaveringly dependable and profoundly comfortable assistance to users. Our overarching mission unfurls the banner of a promising era, one marked by elevated mobility and an all-encompassing sense of well-being, where effective lower limb rehabilitation and assistance shine brightly on the horizon.^39,40

Nonlinear dynamics of the exoskeleton robot

Taking into account multiple factors influencing the dynamic behavior of the exoskeleton robotic system, we can articulate its nonlinear dynamics as follows, incorporating the influences of both solid and viscous frictions:

$${\nu }_{★}\stackrel{··}{\theta }=\mathcal{U}-f_v^{★}\stackrel{·}{\theta }+a_{★}\cos (\theta )-f_s^{★}\mathrm{sgn}(\stackrel{·}{\theta })+{\mathrm{\Delta }}_t$$

(1)

In this context, the scalar value $a_{★}$ is defined as $a_{★}=g{m}_{★}{l}_{★}$ , where $m_{★}$ signifies the mass of the system, $g$ represents the acceleration due to gravity, and $l_{★}$ denotes a characteristic length. The system’s inertia is denoted as ${\nu }_{★}$ , and the applied torque is represented by $\mathcal{U}$ . The coefficient for viscous friction is indicated as $f_v^{★}$ , while $f_s^{★}$ corresponds to the coefficient of solid friction. Additionally, ${\mathrm{\Delta }}_t$ represents the term accounting for total or aggregated disturbance, which can be decomposed as follows:

$${\mathrm{\Delta }}_t={\mathrm{\Delta }}_{\mathrm{pu}}+{\mathrm{\Delta }}_{\mathrm{ed}}+{\mathrm{\Delta }}_{\mathrm{ud}}+{\mathrm{\Delta }}_{\mathrm{hl}}$$

(2)

where each component within the disturbance quantity ${\mathrm{\Delta }}_t$ is explained as follows:

• Δ _pu corresponds to the contribution stemming from parametric uncertainties.

• Δ _ed accounts for the influence of external disturbances.

• Δ _ud encompasses the impact of unmodeled dynamics.

• Δ _hl signifies the contribution arising from the force exerted by the human leg.

Moving forward, we will make the following assumption regarding the total disturbing term ${\mathrm{\Delta }}_t$

in the nonlinear dynamics (1).

• Assumption 1. It is assumed that the total disturbance ${\mathrm{\Delta }}_t$ in the nonlinear dynamics (1) is bounded and therefore satisfies the following condition:

$$\left|{\mathrm{\Delta }}_t\right|\le \overline{\delta },\overline{\delta }>0$$

(3)

This implies that the absolute magnitude of the total disturbance ${\mathrm{\Delta }}_t$ is bounded by the positive constant $\overline{\delta }$ . In other words, the total disturbance remains within a bounded range defined by this constant. This assumption is crucial for subsequent analyses and designs, providing a foundation to assess and control the impact of the disturbance on the knee exoskeleton robotic system.

This comprehensive representation (1) of the exoskeleton’s dynamics allows us to consider and address various factors affecting its behavior and control.

Considering the relationship $\varphi =\theta -{\theta }_d$ , we can equate $\stackrel{·}{\theta }$ with $\stackrel{·}{\varphi }$ and $\stackrel{··}{\theta }$ with $\stackrel{··}{\varphi }$ . Here, ${\theta }_d$ stands for the target state at which the knee exoskeleton robotic system is intended to be controlled. Consequently, the expression (1) transforms into:

$${\nu }_{★}\stackrel{··}{\varphi }=\mathcal{U}-f_v^{★}\stackrel{·}{\varphi }+a_{★}\cos (\varphi +{\theta }_d)-f_s^{★}\mathrm{sgn}(\stackrel{·}{\varphi })+{\mathrm{\Delta }}_t$$

(4)

Desired controller at the equilibrium

After successfully positioning the knee exoskeleton robotic system at the desired state ${\theta }_d$ and maintaining zero angular velocity ${\stackrel{·}{\theta }}_d=0$ using the controller $\mathcal{U}$ , a customized control effort, denoted as ${\mathcal{U}}_d$ , is required to sustain this equilibrium state. Utilizing the nonlinear dynamic model (4), we can derive the following expression:

$$0={\mathcal{U}}_d+a_{★}\cos ({\theta }_d)$$

(5)

Hence, in a state of equilibrium, we can express the desired control input ${\mathcal{U}}_d$ through the following equation:

$${\mathcal{U}}_d=-a_{★}\cos ({\theta }_d)$$

(6)

State space representation

Let’s define $\mathcal{X}=\left[\begin{array}{c}\varphi \hfill \\ \multicolumn{1}{c}{\stackrel{·}{\varphi }}\end{array}\right]$ as the state vector and consider $\mathcal{U}=\mathrm{\Delta }\mathcal{U}+{\mathcal{U}}_d$ . Consequently, the nonlinear dynamics (4) of the robotic system can be expressed in the state-space representation as follows:

$$\stackrel{·}{\mathcal{X}}=\mathcal{A}\mathcal{X}+\mathcal{B}\mathrm{\Delta }\mathcal{U}+{\mathcal{D}}_1{\mathrm{\Phi }}_1(\mathcal{X})+{\mathcal{D}}_2{\mathrm{\Phi }}_2(\mathcal{X})+\mathcal{B}{\mathrm{\Delta }}_t$$

(7)

In this model (7), the functions ${\mathrm{\Phi }}_1(\mathcal{X})$ and ${\mathrm{\Phi }}_2(\mathcal{X})$ are described as follows:

$${\mathrm{\Phi }}_1(\mathcal{X})=\mathrm{sgn}(\stackrel{·}{\varphi })$$

(8a)

$${\mathrm{\Phi }}_2(\mathcal{X})=\cos (\varphi +{\theta }_d)-\cos ({\theta }_d)$$

(8b)

The following expressions represent the matrices $\mathcal{A}$ , $\mathcal{B}$ , ${\mathcal{D}}_1$ , and ${\mathcal{D}}_2$ presented in the nonlinear dynamics (7):

$$\mathcal{A}=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& -\frac{f_v^{★}}{{\nu }_{★}}\hfill \end{array}\right]\mathcal{B}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\frac{1}{{\nu }_{★}}}\end{array}\right]$$

$${\mathcal{D}}_1=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-\frac{f_s^{★}}{{\nu }_{★}}}\end{array}\right]{\mathcal{D}}_2=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\frac{a_{★}}{{\nu }_{★}}}\end{array}\right]$$

It is important to note that in the dynamical model (7), the non-linearities of the exoskeleton robot’s dynamic model are grouped into two nonlinear terms, ${\mathrm{\Phi }}_1(\mathcal{X})$ and ${\mathrm{\Phi }}_2(\mathcal{X})$ . These two nonlinear terms, or more precisely their squared quantities, are assumed to be bounded by either a constant, a linear function, or an order-2 quadratic nonlinear function, allowing us to convert them to or handle them as linear components within the framework of our research using the LMI approach.

Problem formulation

Individuals facing mobility challenges stand to gain significant advantages from lower limb exoskeleton orthoses, potentially rekindling their mobility and enhancing lower limb functionality.^14,41,42 This study places a specific emphasis on an actuated knee joint orthosis, chosen for its straightforward implementation and limited degrees of freedom. This orthosis model, capable of simulating flexion and extension motions of the knee joint in the sagittal plane concurrently, serves as the focal point of our investigation.

Recent years have witnessed a growing interest in the field of controlling lower limb exoskeletons, a domain that has increasingly captivated the attention of robotics researchers. ⁴³ Our primary objective in this research is to develop advanced control strategies tailored for lower extremity exoskeleton devices designed to assist elderly individuals with essential standing and walking tasks.^26,44,45

This research unfolds with a distinct emphasis on the control of a knee exoskeleton robot, specifically one equipped with a single degree of freedom, meticulously illustrated in Figure 1. Furthermore, in our unique approach, we amalgamate the rod and the foot into a unified rotational system at the knee joint level, treating them as a singular entity.

Hence, our primary mission embarks on the establishment of an effective control and a stabilization mechanism for the knee exoskeleton robotic system presented in Figure 1, and governed by its intricate nonlinear dynamics (1) or equivalently (7). To achieve this goal, we will use the Lyapunov approach and we will propose three methods to establish LMI stability conditions, empowering us to wield robust control over robotic systems, capitalizing on the insights gleaned from prior research in this dynamic arena. ⁴⁶ In the first approach to designing the LMI conditions, we assume that the squared quantity of the function ${\mathrm{\Phi }}_2(\mathcal{X})$ defined by (8b) remains bounded by a constant value. In the second LMI design approach, we will operate under the assumption that the squared quantity of the function ${\mathrm{\Phi }}_2(\mathcal{X})$ is bounded by a linear equation constraint. However, in the third LMI design approach, we will proceed with the assumption that the squared quantity of the function ${\mathrm{\Phi }}_2(\mathcal{X})$ (8b) is bounded by a second-degree quadratic nonlinear function.

Moreover, in order to handle with these nonlinear functions ${\mathrm{\Phi }}_1(\mathcal{X})$ and ${\mathrm{\Phi }}_2(\mathcal{X})$ , and the lumped disturbance term ${\mathrm{\Delta }}_t$ presented in the nonlinear dynamical system (7) of the knee exoskeleton robot, and therefore to design the LMI stability conditions, the idea is to employ some technical lemma, namely the S-procedure lemma, the matrix inversion lemma, the Schur complement and the Young inequality, and also some judicious congruence transformations. All these tools will be used to convert the nonlinear matrix inequality conditions into LMIs, which are traceable conditions that can be solved numerically compared to the nonlinear inequality conditions.

Adopted affine state-feedback controller and controlled dynamics

Utilized affine state-feedback controller

Our primary goal in this study is to achieve precise position control of the knee exoskeleton robotic system. To efficiently control the robotic system with its nonlinear dynamics (1), we propose an affine state-feedback control law.

This envisioned control law is defined by the following expression:

$$\mathcal{U}={\mathcal{U}}_d+\mathrm{\Delta }\mathcal{U}$$

(9)

with $\mathrm{\Delta }\mathcal{U}$ is expressed like so:

$$\mathrm{\Delta }\mathcal{U}=\{\begin{array}{ccc}\mathcal{K}\mathcal{X}\hfill & \mathrm{if}\hfill & \parallel \mathcal{X}{\parallel }^2\ge {ϵ}^2\hfill \\ \multicolumn{1}{c}{0}& \mathrm{if}\hfill & \parallel \mathcal{X}{\parallel }^2<{ϵ}^2\hfill \end{array}$$

where here $ϵ>0$ is a small constant that will be optimized/minimized later.

Controlled dynamic model

The previously suggested control law governs the nonlinear dynamic model of the knee exoskeleton robotic system, which can be formulated as follows:

$$\begin{array}{c}\stackrel{·}{\mathcal{X}}=(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{X}+{\mathcal{D}}_1{\mathrm{\Phi }}_1(\mathcal{X})+{\mathcal{D}}_2{\mathrm{\Phi }}_2(\mathcal{X})+\mathcal{B}{\mathrm{\Delta }}_t\hfill \\ \mathrm{where}\parallel \mathcal{X}{\parallel }^2\ge {ϵ}^2\hfill \end{array}$$

(10)

• Remark 1. The adopted algorithm functions as a crucial component in mediating the interaction between the exoskeleton and the user, playing a pivotal role in customizing the assistance provided. The adoption of this control algorithm is suggested to address uncertainties within the system and bolster the robustness of position control in the robotic system. It’s worth noting that the chosen control algorithm offers several advantages over both linear and nonlinear controllers. While linear control laws, like the Linear Quadratic Regulator (LQR), are conventionally applied to linear systems, our methodology takes a distinct approach by leveraging a nonlinear dynamic model to elucidate the movement of the knee exoskeleton robotic system. Furthermore, the approach we have chosen showcases superior applicability in comparison to the classical linear Proportional-Derivative (PD) controller or also the nonlinear PD with gravity compensation controller, despite the latter being grounded in a nonlinear dynamic model. ⁴⁶ In practical scenarios, the nonlinear PD-based controllers often necessitate a substantial computation time, and the selection of controller gains is confined by constraints, leading to a constrained and somewhat specialized choice of control parameters. Notably, the PD nonlinear controllers rely on the gravity matrix (potential energy), and the matrices of gains associated with the position and velocity of different joints in the robotic system are typically mandated to be diagonal. ⁴⁶ Our utilization of the LMI approach adeptly overcomes these challenges, providing a solution that is not only more efficient but also more adaptable for controlling knee exoskeleton robotic systems. Importantly, there are no restrictions on the format of the matrix gain $\mathcal{K}$ . Furthermore, the LMI approach allows for the consideration of additional conditions, facilitating the reduction of the feedback gain $\mathcal{K}$ and, consequently, the realization of a simpler and more feasible controller applicable in real-world settings.

Design of LMI stability conditions

In this section, we focus on formulating an LMI condition for the feedback gain of the proposed controller (9). This controller plays a dual role in both controlling and stabilizing the knee exoskeleton robot. Our approach involves utilizing the Lyapunov technique to achieve this goal. Accordingly, we will adopt the subsequent candidate Lyapunov function:

$$\mathcal{V}(\mathcal{X})={\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{X}\mathrm{such}\mathrm{that}\parallel \mathcal{X}{\parallel }^2\ge {ϵ}^2$$

(11)

Subsequently, the time derivative of the candidate Lyapunov function can be expressed as follows:

$$\stackrel{·}{\mathcal{V}}(\mathcal{X})=2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}\stackrel{·}{\mathcal{X}}\mathrm{such}\mathrm{that}\parallel \mathcal{X}{\parallel }^2\ge {ϵ}^2$$

(12)

Considering the controlled nonlinear dynamics (10), the expression (12) can be rephrased in the following manner:

$$\begin{array}{c}\stackrel{·}{\mathcal{V}}(\mathcal{X})=2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{X}+2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_1{\mathrm{\Phi }}_1(\mathcal{X})\hfill \\ +2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathrm{\Phi }}_2(\mathcal{X})+2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{B}{\mathrm{\Delta }}_t\hfill \end{array}$$

(13)

To further enhance the efficacy of the control outcomes, we introduce the subsequent criterion for achieving exponential stability:

$$\stackrel{·}{\mathcal{V}}(\mathcal{X})+\alpha \mathcal{V}(\mathcal{X})<0\mathrm{such}\mathrm{that}\parallel \mathcal{X}{\parallel }^2\ge {ϵ}^2$$

(14)

with $\alpha >0$ is a fixed positive scalar.

Furthermore, it is worth acknowledging that the constraint $\parallel \mathcal{X}{\parallel }^2\ge {ϵ}^2$ on the candidate Lyapunov function $\mathcal{V}(\mathcal{X})$ , as indicated in equation (11), can be restated in the subsequent manner:

$$\mathcal{N}(\mathcal{X})={\left[\begin{array}{c}\mathcal{X}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}\mathcal{I}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& -{ϵ}^2\hfill \end{array}\right]\left[\begin{array}{c}\mathcal{X}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]\ge 0$$

(15)

where here and in the following, $\mathcal{I}$ and $\mathcal{O}$ respectively represent the identity and zero matrices with appropriate dimensions.

Subsequently, we will proceed to develop three distinct approaches for designing LMI-based stability conditions for the feedback gain $\mathcal{K}$ using the foundation provided by expression (13).

First approach for formulating LMI conditions

In the initial approach to designing the LMI conditions, we make the assumption that the function ${\mathrm{\Phi }}_2(\mathcal{X})$ remains bounded by a constant value to be calculated. Therefore, our focus is on guaranteeing the stability of the controlled knee exoskeleton robotic system, utilizing the extended expression (13). Consequently, we will proceed to establish a dedicated Linear Matrix Inequality (LMI) constraint on the gain matrix $\mathcal{K}$ .

By employing the Young inequality lemma^37,47–49 on equation (13), it becomes apparent that:

$$2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_1{\mathrm{\Phi }}_1(\mathcal{X})\le {\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_1\mathcal{P}\mathcal{X}+{\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1{\mathrm{\Phi }}_1(\mathcal{X}{)}^2$$

where ${\mathcal{M}}_1={\mathcal{M}}_1^{\mathrm{T}}>0$ , and recall that ${\mathrm{\Phi }}_1(\mathcal{X})$ denotes a scalar function.

Moreover, relying on expression (8a), considering that ${\mathrm{\Phi }}_1(\mathcal{X})=\mathrm{sgn}(\stackrel{·}{\varphi })=\pm 1$ , it becomes apparent that ${\mathrm{\Phi }}_1(\mathcal{X}{)}^2=1$ . Substituting this outcome into the previously established inequality, we arrive at the subsequent expression:

$$2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_1{\mathrm{\Phi }}_1(\mathcal{X})\le {\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_1\mathcal{P}\mathcal{X}+{\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1$$

(16)

Consequently, given that ${\mathrm{\Phi }}_2(\mathcal{X})$ is a scalar function, we can employ the Young inequality lemma to equation (13), leading the subsequent equation:

$$2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathrm{\Phi }}_2(\mathcal{X})\le {\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_2\mathcal{P}\mathcal{X}+{\mathcal{D}}_2^{\mathrm{T}}{\mathcal{M}}_2^{-1}{\mathcal{D}}_2{\mathrm{\Phi }}_2(\mathcal{X}{)}^2$$

where ${\mathcal{M}}_2={\mathcal{M}}_2^{\mathrm{T}}>0$ .

As ${\mathrm{\Phi }}_2(\mathcal{X})=\cos (\varphi +{\theta }_d)-\cos ({\theta }_d)=\cos (\theta )-\cos ({\theta }_d)$ , it follows that:

$${\mathrm{\Phi }}_2(\mathcal{X}{)}^2={\cos }^2(\theta )-2\cos ({\theta }_d)\cos (\theta )+{\cos }^2({\theta }_d)$$

(17)

Moreover, we can demonstrate the subsequent set of inequalities:

$${\cos }^2(\theta )\le 1-2\cos (\theta )\le 0$$

Therefore, by extending the previous expressions, it is straightforward to derive the following inequality:

$${\mathrm{\Phi }}_2(\mathcal{X}{)}^2\le {\mu }_{11}$$

(18)

with ${\mu }_{11}$ is expressed like so:

$${\mu }_{11}=1+{\cos }^2({\theta }_d)$$

This leads to rewriting of the previously developed inequality as follows:

$$2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathrm{\Phi }}_2(\mathcal{X})\le {\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_2\mathcal{P}\mathcal{X}+{\mu }_{11}{\mathcal{D}}_2^{\mathrm{T}}{\mathcal{M}}_2^{-1}{\mathcal{D}}_2$$

(19)

Furthermore, given that ${\mathrm{\Delta }}_t$ is a scalar function, we can employ the Young inequality lemma within equation (14) to deduce the subsequent equation:

$$2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{B}{\mathrm{\Delta }}_t\le {\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_3\mathcal{P}\mathcal{X}+{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}{\mathrm{\Delta }}_t^2$$

where ${\mathcal{M}}_3={\mathcal{M}}_3^{\mathrm{T}}>0$ .

Considering the constraint on ${\mathrm{\Delta }}_t$ in Assumption 1, it can be deduced that ${\mathrm{\Delta }}_t^2\le {\overline{\delta }}^2$ , consequently, this allows for the following reformulation:

$$2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{B}{\mathrm{\Delta }}_t\le {\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_3\mathcal{P}\mathcal{X}+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}$$

(20)

By replacing constraints (16), (19), and (20) into the resultant equation (13) and considering constraint (14), the following inequality may be constructed:

$$\begin{array}{c}\stackrel{·}{\mathcal{V}}(\mathcal{X})+\alpha \mathcal{V}(\mathcal{X})\le 2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{X}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_1\mathcal{P}\mathcal{X}\hfill \\ +{\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_2\mathcal{P}\mathcal{X}+{\mu }_{11}{\mathcal{D}}_2^{\mathrm{T}}{\mathcal{M}}_2^{-1}{\mathcal{D}}_2\hfill \\ +{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_3\mathcal{P}\mathcal{X}+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}+\alpha {\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{X}\hfill \end{array}$$

Let consider $\mathcal{Z}(\mathcal{X})=2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{X}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_1\mathcal{P}\mathcal{X}+{\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_2\mathcal{P}\mathcal{X}$ $+{\mu }_{11}{\mathcal{D}}_2^{\mathrm{T}}{\mathcal{M}}_2^{-1}{\mathcal{D}}_2+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_3\mathcal{P}\mathcal{X}+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}+\alpha {\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{X}$ Therefore, we have $\stackrel{·}{\mathcal{V}}(\mathcal{X})+\alpha \mathcal{V}(\mathcal{X})\le \mathcal{Z}(\mathcal{X})$ , and hence we should satisfy the following condition:

$$\mathcal{Z}(\mathcal{X})={\left[\begin{array}{c}\mathcal{X}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\Sigma }_{11}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{★}& {\Sigma }_{22}\hfill \end{array}\right]\left[\begin{array}{c}\mathcal{X}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]<0$$

(21)

with

$$\begin{array}{c}{\Sigma }_{11}=(\mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})+{(\mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})}^{\mathrm{T}}+\mathcal{P}{\mathcal{M}}_1\mathcal{P}\hfill \\ +\mathcal{P}{\mathcal{M}}_2\mathcal{P}+\mathcal{P}{\mathcal{M}}_3\mathcal{P}+\alpha \mathcal{P}\hfill \end{array}$$

(22a)

$${\Sigma }_{22}={\mathcal{D}}_1^{\mathrm{T}}{ℳ}_1^{-1}{\mathcal{D}}_1+{\mu }_{11}{\mathcal{D}}_2^{\mathrm{T}}{ℳ}_2^{-1}{\mathcal{D}}_2+{\overline{\delta }}^2{ℬ}^{\mathrm{T}}{ℳ}_3^{-1}ℬ$$

(22b)

Employing the S-procedure lemma^37,47–49 and incorporating the expressions (21) and (15), we formulate the subsequent stability condition:

$$\left[\begin{array}{cc}{\Sigma }_{11}+{\eta }_1{\mathcal{I}}_n\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{★}& {\Sigma }_{22}-{\eta }_1{ϵ}^2\hfill \end{array}\right]<0$$

(23)

where ${\eta }_1>0$ .

Derived from the matrix inequality in (23), we can demonstrate the ensuing pair of inequalities:

$$\begin{array}{c}(\mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})+{(\mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})}^{\mathrm{T}}+\mathcal{P}{\mathcal{M}}_1\mathcal{P}+\mathcal{P}{\mathcal{M}}_2\mathcal{P}\hfill \\ +\mathcal{P}{\mathcal{M}}_3\mathcal{P}+\alpha \mathcal{P}+{\eta }_1{\mathcal{I}}_n<0\hfill \\ {\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\mu }_{11}{\mathcal{D}}_2^{\mathrm{T}}{\mathcal{M}}_2^{-1}{\mathcal{D}}_2+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}-{\eta }_1{ϵ}^2<0\hfill \end{array}$$

(24)

Subsequently, once the first extended condition in (24) has been multiplied on both the right and left sides with $\mathcal{S}={\mathcal{P}}^{-1}$ , the resulting inequality is as follows:

$$\begin{array}{c}(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\mathcal{M}}_2+{\mathcal{M}}_3\hfill \\ +\alpha \mathcal{S}+{\eta }_1{\mathcal{S}}^2<0\hfill \end{array}$$

with

$$\mathcal{R}=\mathcal{K}\mathcal{S}$$

(25)

Setting ${\gamma }_1={\eta }_1^{-1}$ and employing the Schur complement,^37,47–49 we derive the subsequent LMI stability condition:

$$\left[\begin{array}{cc}\mathrm{\Lambda }\hfill & \mathcal{S}\hfill \\ \multicolumn{1}{c}{\mathcal{S}}& -{\gamma }_1{\mathcal{I}}_n\hfill \end{array}\right]<0$$

(26)

with

$$\begin{array}{c}\mathrm{\Lambda }=(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}\hfill \\ +{\mathcal{M}}_1+{\mathcal{M}}_2+{\mathcal{M}}_3+\alpha \mathcal{S}\hfill \end{array}$$

Additionally, by multiplying the second developed stability condition in (24) by ${\gamma }_1^2$ , we arrive at the subsequent inequality:

$$\begin{array}{c}{\gamma }_1^2{\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\gamma }_1^2{\mu }_{11}{\mathcal{D}}_2^{\mathrm{T}}{\mathcal{M}}_2^{-1}{\mathcal{D}}_2\hfill \\ +{\gamma }_1^2{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}-{\gamma }_1{ϵ}^2<0\hfill \end{array}$$

Applying the Schur complement and introducing the following new change of variables ${\tau }_1={\gamma }_1{ϵ}^2$ and ${\kappa }_1={\gamma }_1\overline{\delta }$ , the previously extended condition can be recast as follows:

$$\left[\begin{array}{cccc}{\tau }_1\hfill & {\kappa }_1{\mathcal{B}}^{\mathrm{T}}\hfill & {\gamma }_1{\mathcal{D}}_2^{\mathrm{T}}\hfill & {\gamma }_1{\mathcal{D}}_1^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{{\kappa }_1\mathcal{B}}& {\mathcal{M}}_3\hfill & \mathcal{O}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{{\gamma }_1{\mathcal{D}}_2}& \mathcal{O}\hfill & \frac{1}{{\mu }_{11}}{\mathcal{M}}_2\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{{\gamma }_1{\mathcal{D}}_1}& \mathcal{O}\hfill & \mathcal{O}\hfill & {\mathcal{M}}_1\hfill \end{array}\right]>0$$

(27)

It is crucial to highlight that ${\tau }_1={\gamma }_1{ϵ}^2$ , leading to the relation $ϵ=\sqrt{\frac{{\tau }_1}{{\gamma }_1}}$ . To minimize $ϵ$ , we strive to minimize $({\tau }_1+{\gamma }_1^{-1})$ . Then, let’s consider the subsequent condition:

$${\tau }_1+{\gamma }_1^{-1}<{\nabla }_1$$

Therefore, the strategy to minimize $ϵ$ entails minimizing the parameter ${\nabla }_1$ .

Evidently, the previous expression can be rephrased in the subsequent manner:

$${\nabla }_1-{\tau }_1-{\gamma }_1^{-1}>0$$

Using the previous expression and applying the Schur complement lemma, we derive the ensuing LMI stability condition:

$$\left[\begin{array}{cc}{\nabla }_1-{\tau }_1\hfill & 1\hfill \\ \multicolumn{1}{c}{1}& {\gamma }_1\hfill \end{array}\right]>0$$

(28)

where ${\tau }_1>0$ .

Furthermore, it is important to highlight that ${\kappa }_1={\gamma }_1\overline{\delta }$ . This relation implies $\overline{\delta }=\frac{{\kappa }_1}{{\gamma }_1}$ . Additionally, let take $\Omega ={\overline{\delta }}^{-1}=\frac{{\gamma }_1}{{\kappa }_1}$ . To maximize $\overline{\delta }$ , it is necessary to minimize $\Omega$ , and consequently minimize ${\gamma }_1+{\kappa }_1^{-1}$ . Thus, we introduce the subsequent constraint:

$${\gamma }_1+{\kappa }_1^{-1}<{\nabla }_2$$

Hence, the objective to maximize $\overline{\delta }$ necessitates minimizing ${\nabla }_2$ . Consequently, it becomes evident that the previous inequality can be reformulated as follows:

$${\nabla }_2-{\gamma }_1-{\kappa }_1^{-1}>0$$

By utilizing the previous inequality and applying the Schur complement lemma, we can formulate the following LMI stability condition:

$$\left[\begin{array}{cc}{\nabla }_2-{\gamma }_1\hfill & 1\hfill \\ \multicolumn{1}{c}{1}& {\kappa }_1\hfill \end{array}\right]>0$$

(29)

where ${\gamma }_1>0$ .

Consequently, the LMI stability condition outlined in (26), (27), (28), and (29) are derived. These conditions are characterized in terms of the unknown variables $\mathcal{S}$ , $\mathcal{R}$ , ${\mathcal{M}}_1$ , ${\mathcal{M}}_2$ , ${\mathcal{M}}_3$ , ${\gamma }_1$ , ${\tau }_1$ , ${\nabla }_1$ , ${\nabla }_2$ , and ${\kappa }_1$ .

• Remark 2. It is important to note that the LMI conditions designed previously can result in a considerably large control gain $\mathcal{K}$ . Therefore, the objective is to reduce the size of the controller’s gain $\mathcal{K}$ . Relying on expression (25), if follows then that $\mathcal{K}=\mathcal{R}{\mathcal{S}}^{-1}$ . Hence, in order to reduce the size of $\mathcal{K}$ , the following inequality conditions are introduced:

$${\mathcal{R}}^{\mathrm{T}}\mathcal{R}<{\zeta }_1\mathcal{I},$$

(30a)

$${\mathcal{S}}^{-1}<{\zeta }_2\mathcal{I},{\zeta }_2>0$$

(30b)

where ${\zeta }_1>0$ and ${\zeta }_2>0$ some two free variables to be optimized.

As a result, the two constraints defined by (30a) and (30b) can be reformulated as follows:

$$\left[\begin{array}{cc}{\zeta }_1\mathcal{I}\hfill & {\mathcal{R}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{\mathcal{R}}& \mathcal{I}\hfill \end{array}\right]>0$$

(31a)

$$\left[\begin{array}{cc}{\zeta }_2\mathcal{I}\hfill & \mathcal{I}\hfill \\ \multicolumn{1}{c}{\mathcal{I}}& \mathcal{S}\hfill \end{array}\right]>0$$

(31b)

The two parameters ${\zeta }_1$ and ${\zeta }_2$ in the two LMIs (31a) and (31b) must be decreased to effectively reduce the size of the controller gain.

In the intended LMIs (31a) and (31b), the two parameters ${\zeta }_1$ and ${\zeta }_2$ need to be minimized to effectively reduce the controller gain’s size $\mathcal{K}$ .

• Theorem 1. Considering the availability of matrices $\mathcal{S}$ , $\mathcal{R}$ , ${\mathcal{M}}_1$ , ${\mathcal{M}}_2$ , and ${\mathcal{M}}_3$ , along with scalar parameters ${\gamma }_1$ , ${\tau }_1$ , ${\nabla }_1$ , ${\nabla }_2$ , and ${\kappa }_1$ , we can formulate the subsequent LMI-based minimization problem:

$$\begin{array}{c}\min {\nabla }_1+{\nabla }_2+{\zeta }_1+{\zeta }_2\hfill \\ s.t.\mathrm{LMIs}(26),(27),(28),(29),(31a)\mathrm{and}(31b).\hfill \end{array}$$

(32)

is feasible, Therefore, it can be inferred that the proposed controller outlined in (9) effectively stabilizes the nonlinear dynamic model (1) of the knee exoskeleton. Furthermore, the determination of the gain matrix $\mathcal{K}$ for the adopted controller can be achieved using the following expression:

$$\mathcal{K}=\mathcal{R}{\mathcal{S}}^{-1}$$

(33)

Additionally, the subsequent formula can be employed to calculate the minimum value of the parameter $ϵ$ :

$${ϵ}_{\min }=\sqrt{\frac{{\tau }_1}{{\gamma }_1}}$$

(34)

Moreover, the following expression may be used to determine the maximum value of parameter $\overline{\delta }$ :

$${\overline{\delta }}_{\max }=\frac{{\kappa }_1}{{\gamma }_1}$$

(35)

Second approach for formulating LMI conditions

In this second design approach, we make the assumption that the function ${\mathrm{\Phi }}_2(\mathcal{X})$ remains within bounded limits, as prescribed by a linear equation constraint, which will be formulated subsequently.

As mentioned previously, the constraint applied to the proposed Lyapunov function (11) can be redefined by employing the expression (15). Furthermore, we will adhere to the same formulation (13) to derive certain LMI conditions for the feedback gain matrix $\mathcal{K}$ .

It is crucial to recall that ${\mathrm{\Phi }}_1(\mathcal{X})=\mathrm{sgn}(\stackrel{·}{\varphi })=\pm 1$ , implying ${\mathrm{\Phi }}_1(\mathcal{X}{)}^2=1$ . Therefore, we can justify the expression (16).

Since ${\mathrm{\Phi }}_2(\mathcal{X})$ is a scalar function, we can apply the Young inequality in (13), enabling us to express it as follows:

$$2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathrm{\Phi }}_2(\mathcal{X})\le {\tau }_2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\mathcal{P}\mathcal{X}+{\tau }_2^{-1}{\mathrm{\Phi }}_2(\mathcal{X}{)}^2$$

(36)

where here ${\tau }_2>0$ .

Moreover, recall that the function ${\mathrm{\Phi }}_2(\mathcal{X}{)}^2$ can be rewritten under expression (17). In addition, developing the following inequalities is a straightforward task:

$${\cos }^2(\theta )\le 1-2\cos (\theta )\le a_2\theta +b_2$$

with $a_2=\frac{4}{\pi }$ and $b_2=-2$ .

Since $\theta =\varphi +{\theta }_d$ as our starting point, we can proceed to infer the following expression:

$$\begin{array}{c}-2\cos ({\theta }_d)\cos (\theta )\le a_2\cos ({\theta }_d)\varphi \hfill \\ +a_2\cos ({\theta }_d){\theta }_d+b_2\cos ({\theta }_d)\hfill \end{array}$$

Hence, considering the above expressions, we can establish the following inequality:

$${\mathrm{\Phi }}_2(\mathcal{X}{)}^2\le 2{\mu }_{21}\varphi +{\mu }_{22}$$

where here ${\mu }_{21}$ and ${\mu }_{22}$ are defined by the following expressions:

$$\begin{array}{c}{\mu }_{21}=\frac{a_2}{2}\cos ({\theta }_d)\hfill \\ {\mu }_{22}=1+a_2\cos ({\theta }_d){\theta }_d+b_2\cos ({\theta }_d)+{\cos }^2({\theta }_d)\hfill \end{array}$$

It is essential to highlight that for all values of ${\theta }_d$ within the range of 0 to $\frac{\pi }{2}$ , we consistently observe that ${\mu }_{21}>0$ .

Let consider $\varphi ={\mathcal{C}}_1\mathcal{X}$ , where ${\mathcal{C}}_1=\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right]$ . Therefore, the previous equation transforms into:

$${\mathrm{\Phi }}_2(\mathcal{X}{)}^2\le 2{\mu }_{21}{\mathcal{C}}_1\mathcal{X}+{\mu }_{22}$$

(37)

Consequently, condition (38) is recast as follows:

$$\begin{array}{c}2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathrm{\Phi }}_2(\mathcal{X})\le {\tau }_2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\mathcal{P}\mathcal{X}\hfill \\ +2{\mu }_{21}{\tau }_2^{-1}{\mathcal{C}}_1\mathcal{X}+{\mu }_{22}{\tau }_2^{-1}\hfill \end{array}$$

(38)

Moreover, taking into account that ${\mathrm{\Delta }}_t$ is a scalar function, we can apply the Young inequality lemma to the term $2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{B}{\mathrm{\Delta }}_t$ in equation (13). Assuming that ${\mathrm{\Delta }}_t$ adheres to the constraint outlined in (3), we can subsequently derive the inequalities as depicted in (20).

By substituting conditions (16), (38), and (20) into the resulting expression (13) and incorporating the constraint (14), we arrive at the following inequality:

$$\begin{array}{c}\stackrel{·}{\mathcal{V}}(\mathcal{X})+\alpha \mathcal{V}(\mathcal{X})\le 2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{X}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_1\mathcal{P}\mathcal{X}\hfill \\ +{\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\tau }_2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\mathcal{P}\mathcal{X}+2{\mu }_{21}{\tau }_2^{-1}{\mathcal{C}}_1\mathcal{X}\hfill \\ +{\mu }_{22}{\tau }_2^{-1}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_3\mathcal{P}\mathcal{X}+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}+\alpha {\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{X}\hfill \end{array}$$

(39)

Let consider $\mathcal{Z}(\mathcal{X})=2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{X}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_1\mathcal{P}\mathcal{X}+{\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\tau }_2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\mathcal{P}\mathcal{X}$ $+2{\mu }_{21}{\tau }_2^{-1}{\mathcal{C}}_1\mathcal{X}+{\mu }_{22}{\tau }_2^{-1}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_3\mathcal{P}\mathcal{X}+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}+\alpha {\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{X}$ Consequently, we obtain $\stackrel{·}{\mathcal{V}}(\mathcal{X})+\alpha \mathcal{V}(\mathcal{X})\le \mathcal{Z}(\mathcal{X})$ , and thus, to satisfy the constraint (14), we must meet the following inequality condition:

$$\mathcal{Z}(\mathcal{X})={\left[\begin{array}{c}\mathcal{X}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\mathrm{\Upsilon }}_{11}\hfill & {\mathrm{\Upsilon }}_{12}\hfill \\ \multicolumn{1}{c}{★}& {\mathrm{\Upsilon }}_{22}\hfill \end{array}\right]\left[\begin{array}{c}\mathcal{X}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]<0$$

(40)

where

$$\begin{array}{c}{\mathrm{\Upsilon }}_{11}=(\mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})+{(\mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})}^{\mathrm{T}}+\mathcal{P}{\mathcal{M}}_1\mathcal{P}\hfill \\ +{\tau }_2\mathcal{P}{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\mathcal{P}+\mathcal{P}{\mathcal{M}}_3\mathcal{P}+\alpha \mathcal{P}\hfill \\ {\mathrm{\Upsilon }}_{12}={\mu }_{21}{\tau }_2^{-1}{\mathcal{C}}_1^{\mathrm{T}}\hfill \\ {\mathrm{\Upsilon }}_{22}={\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\mu }_{22}{\tau }_2^{-1}+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}\hfill \end{array}$$

Utilizing the S-procedure and employing the expressions (40) and (15), we formulate the following stability condition:

$$\left[\begin{array}{cc}{\mathrm{\Upsilon }}_{11}+{\eta }_2{\mathcal{I}}_n\hfill & {\mathrm{\Upsilon }}_{12}\hfill \\ \multicolumn{1}{c}{★}& {\mathrm{\Upsilon }}_{22}-{\eta }_2{ϵ}^2\hfill \end{array}\right]<0$$

(41)

where ${\eta }_2>0$ .

Therefore, by multiplying the extended condition defined in (41) on the right and left sides by $\mathcal{Q}=\left[\begin{array}{cc}\mathcal{S}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& \mathcal{I}\hfill \end{array}\right]$ , where $\mathcal{S}={\mathcal{P}}^{-1}$ , we obtain:

$$\left[\begin{array}{cc}{\mathrm{\Gamma }}_{11}\hfill & {\mathrm{\Gamma }}_{12}\hfill \\ \multicolumn{1}{c}{★}& {\mathrm{\Gamma }}_{22}\hfill \end{array}\right]<0$$

(42)

where

$$\begin{array}{c}{\mathrm{\Gamma }}_{11}=(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\hfill \\ +{\mathcal{M}}_3+\alpha \mathcal{S}+{\eta }_2{\mathcal{S}}^2\hfill \\ {\mathrm{\Gamma }}_{12}={\mu }_{21}{\tau }_2^{-1}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}\hfill \\ {\mathrm{\Gamma }}_{22}={\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\mu }_{22}{\tau }_2^{-1}+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}-{\eta }_2{ϵ}^2\hfill \end{array}$$

with $\mathcal{R}$ is defined by expression (25).

For the sake of simplicity, let define ${\beta }^{-1}={\eta }_2{ϵ}^2$ , $\mathcal{J}=\left[\begin{array}{ccc}\mathcal{O}\hfill & {\mu }_{21}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}\hfill & \mathcal{O}\hfill \end{array}\right]$ , $\mathcal{D}=\left[\begin{array}{c}{\mathcal{D}}_1\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{\mathcal{B}}\end{array}\right]$ , ${\mathcal{M}}^{-1}=\left[\begin{array}{ccc}{\mathcal{M}}_1^{-1}\hfill & \mathcal{O}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& {\tau }_2^{-1}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& \mathcal{O}\hfill & {\mathcal{M}}_3^{-1}\hfill \end{array}\right]$ , ${\mathcal{Q}}^{-1}=\left[\begin{array}{ccc}\mathcal{I}\hfill & \mathcal{O}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& {\mu }_{22}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& \mathcal{O}\hfill & {\overline{\delta }}^2\mathcal{I}\hfill \end{array}\right]$ . As a result, inequality (42) can be rewritten as:

$$\left[\begin{array}{cc}{\mathrm{\Gamma }}_{11}\hfill & \mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}\hfill \\ \multicolumn{1}{c}{★}& -{\beta }^{-1}+{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{Q}}^{-1}\mathcal{D}\hfill \end{array}\right]<0$$

(43)

Applying the Schur complement lemma, the matrix inequality (43) can be reformulated as follows:

$$-{\beta }^{-1}+{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{Q}}^{-1}\mathcal{D}<0$$

(44a)

$$\begin{array}{c}{\mathrm{\Gamma }}_{11}+\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}{({\beta }^{-1}-{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{Q}}^{-1}\mathcal{D})}^{-1}\times \hfill \\ {\left(\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}\right)}^{\mathrm{T}}<0\hfill \end{array}$$

(44b)

Using the matrix inversion lemma for the term on the left-hand side of the equation (44a), we obtain

$${({\beta }^{-1}-{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{Q}}^{-1}\mathcal{D})}^{-1}=\beta -{\beta }^2{\mathcal{D}}^{\mathrm{T}}{\mathcal{H}}^{-1}\mathcal{D}$$

(45)

with

$$\mathcal{H}=\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M}$$

(46)

In accordance with equation (45), we can express inequality (44b) as equivalent to:

$${\mathrm{\Gamma }}_{11}+\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}(\beta -{\beta }^2{\mathcal{D}}^{\mathrm{T}}{\mathcal{H}}^{-1}\mathcal{D}){\left(\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}\right)}^{\mathrm{T}}<0$$

(47)

Let posing

$$\begin{array}{c}\Psi =\beta \mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}\hfill \\ -{\beta }^2\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}{\mathcal{D}}^{\mathrm{T}}{\mathcal{H}}^{-1}\mathcal{D}{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}\hfill \end{array}$$

(48)

Since the matrix $\mathcal{H}$ is defined by expression (46), it is straightforward to illustrate the following expressions:

$$\begin{array}{c}\beta \mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}=\mathcal{J}{\mathcal{M}}^{-1}\mathcal{H}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}\hfill \\ +\mathcal{J}{\mathcal{M}}^{-1}\mathcal{Q}\mathcal{M}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}\hfill \end{array}$$

(49a)

$$\begin{array}{c}{\beta }^2\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}{\mathcal{D}}^{\mathrm{T}}{\mathcal{H}}^{-1}\mathcal{D}{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}=\mathcal{J}{\mathcal{M}}^{-1}\mathcal{H}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}\hfill \\ +2\mathcal{J}{\mathcal{M}}^{-1}\mathcal{Q}\mathcal{M}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}+\mathcal{J}{\mathcal{M}}^{-1}\mathcal{Q}\mathcal{M}{\mathcal{H}}^{-1}\mathcal{Q}\mathcal{M}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}\hfill \end{array}$$

(49b)

Utilizing the previously mentioned equations, we can simplify expression (48) as follows:

$$\Psi =-\mathcal{J}\mathcal{Q}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}-\mathcal{J}\mathcal{Q}{\mathcal{H}}^{-1}\mathcal{Q}{\mathcal{J}}^{\mathrm{T}}$$

(50)

By employing (50) in (47), we derive:

$$\begin{array}{c}(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}+{\mathcal{M}}_3\hfill \\ +\alpha \mathcal{S}+{\eta }_2{\mathcal{S}}^2-\mathcal{J}\mathcal{Q}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}-\mathcal{J}\mathcal{Q}{\mathcal{H}}^{-1}\mathcal{Q}{\mathcal{J}}^{\mathrm{T}}<0\hfill \end{array}$$

Given that $\mathcal{Q}\mathcal{M}>0$ , we can rewrite the previous inequality as follows:

$$\begin{array}{c}(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}+{\mathcal{M}}_3\hfill \\ +\alpha \mathcal{S}+{\eta }_2{\mathcal{S}}^2-\mathcal{J}\mathcal{Q}{(\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M})}^{-1}\mathcal{Q}{\mathcal{J}}^{\mathrm{T}}<0\hfill \end{array}$$

(51)

Furthermore, based on inequality (44a), and considering that $\beta >0$ and $\mathcal{M}>0$ , we can apply the Schur complement to express:

$$\left[\begin{array}{cc}-{\beta }^{-1}\hfill & {\mathcal{D}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{★}& -\mathcal{M}\mathcal{Q}\hfill \end{array}\right]<0$$

(52)

Therefore, by utilizing the Schur complement, we can establish that $\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M}<0$ . Consequently, we have $-{(\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M})}^{-1}>0$ . Therefore, introducing ${\gamma }_2={\eta }_2^{-1}$ and employing the Schur complement, we expand as follows:

$$\left[\begin{array}{ccc}{\Pi }_1\hfill & \mathcal{S}\hfill & \mathcal{J}\mathcal{Q}\hfill \\ \multicolumn{1}{c}{★}& -{\gamma }_2{\mathcal{I}}_n\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & \beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M}\hfill \end{array}\right]<0$$

(53)

with

$$\begin{array}{c}{\Pi }_1=(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1\hfill \\ +{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}+{\mathcal{M}}_3+\alpha \mathcal{S}\hfill \end{array}$$

Additionally, using the following new change of variables ${\kappa }_2=\frac{{\gamma }_2}{{ϵ}^2}$ and ${\lambda }_2=\frac{m_3}{{\overline{\delta }}^2}$ , and assuming that ${\mathcal{M}}_3=m_3\mathcal{I}$ , we can reformulate expression (53) as an LMI stability condition, which can be expressed as follows:

$$\begin{array}{c}\left[\begin{array}{ccccccc}{\Pi }_2\hfill & \mathcal{S}\hfill & \mathcal{O}\hfill & \cdots \hfill & \cdots \hfill & \frac{{\mu }_{21}}{{\mu }_{22}}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{★}& -{\gamma }_2{\mathcal{I}}_n\hfill & \mathcal{O}\hfill & \cdots \hfill & \cdots \hfill & \mathcal{O}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & {\kappa }_2{\mathcal{D}}_1{\mathcal{D}}_1^{\mathrm{T}}-{\mathcal{M}}_1\hfill & \cdots \hfill & \cdots \hfill & {\kappa }_2{\mathcal{D}}_1\hfill & {\kappa }_2{\mathcal{D}}_1{\mathcal{B}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & ★\hfill & \cdots \hfill & \cdots \hfill & {\kappa }_2-\frac{1}{{\mu }_{22}}{\tau }_2\hfill & {\kappa }_2{\mathcal{B}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & ★\hfill & \cdots \hfill & \cdots \hfill & ★\hfill & {\kappa }_2\mathcal{B}{\mathcal{B}}^{\mathrm{T}}-{\lambda }_2\mathcal{I}\hfill \end{array}\right]<0\hfill \end{array}$$

(54)

where

$$\begin{array}{c}{\Pi }_2=(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1\hfill \\ +{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}+m_3\mathcal{I}+\alpha \mathcal{S}\hfill \end{array}$$

It is worth emphasizing that ${\kappa }_2=\frac{{\gamma }_2}{{\epsilon }^2}$ , which implies $ϵ=\sqrt{\frac{{\gamma }_2}{{\kappa }_2}}$ . To minimize $ϵ$ , we aim to minimize $({\gamma }_2+{\kappa }_2^{-1})$ . Therefore, let’s consider the following condition:

$${\gamma }_2+{\kappa }_2^{-1}<{\nabla }_3$$

(55)

Thus, we can conclude that to minimize $ϵ$ , our goal should be to minimize ${\nabla }_3$ . It is evident that inequality (55) can be restated as follows:

$${\nabla }_3-{\gamma }_2-{\kappa }_2^{-1}>0$$

(56)

By employing this expression and relying on the Schur complement lemma, we can derive the following LMI stability condition:

$$\left[\begin{array}{cc}{\nabla }_3-{\gamma }_2\hfill & 1\hfill \\ \multicolumn{1}{c}{1}& {\kappa }_2\hfill \end{array}\right]>0$$

(57)

where ${\gamma }_2>0$ .

Additionally, it is important to observe that ${\lambda }_2=\frac{m_3}{{\overline{\delta }}^2}$ . Thus, $\overline{\delta }=\sqrt{\frac{m_3}{{\lambda }_2}}$ . Furthermore, let take $\Omega ={\overline{\delta }}^{-1}=\sqrt{\frac{{\lambda }_2}{m_3}}$ . To maximize $\overline{\delta }$ , our objective is to minimize $\Omega$ , and consequently, minimize the quantity ${\lambda }_2+m_3^{-1}$ . Therefore, let’s impose the following constraint:

$${\lambda }_2+m_3^{-1}<{\nabla }_4$$

(58)

Hence, to maximize $\overline{\delta }$ , our aim should be to minimize ${\nabla }_4$ . It is evident that inequality (58) can be reformulated as follows:

$${\nabla }_4-{\lambda }_2-m_3^{-1}>0$$

(59)

By utilizing this expression and applying the Schur complement lemma, we can formulate the following LMI stability condition:

$$\left[\begin{array}{cc}{\nabla }_4-{\lambda }_2\hfill & 1\hfill \\ \multicolumn{1}{c}{1}& m_3\hfill \end{array}\right]>0$$

(60)

where ${\lambda }_2>0$ .

As a result, we have derived the LMI stability conditions specified by (76), (57), and (60). These conditions are formulated in terms of the unknown variables $\mathcal{S}$ , $\mathcal{R}$ , ${\mathcal{M}}_1$ , ${\tau }_2$ , $m_3$ , ${\gamma }_2$ , ${\lambda }_2$ , ${\nabla }_3$ , ${\nabla }_4$ , and ${\kappa }_2$ .

• Theorem 2. Assuming the existence of matrices $\mathcal{S}$ , $\mathcal{R}$ , and ${\mathcal{M}}_1$ , along with scalar values ${\tau }_2$ , $m_3$ , ${\gamma }_2$ , ${\lambda }_2$ , ${\nabla }_3$ , ${\nabla }_4$ , and ${\kappa }_2$ , the following LMI-based optimization problem

$$\begin{array}{c}\min {\nabla }_3+{\nabla }_4+{\zeta }_1+{\zeta }_2\hfill \\ s.t.\mathrm{LMIs}(54),(57),(60),(31a)\mathrm{and}(31b).\hfill \end{array}$$

(61)

is feasible, then, it can be deduced that the proposed affine state-feedback controller defined in (9) stabilizes the nonlinear dynamics (1) of the knee exoskeleton robot. Additionally, the gain matrix $\mathcal{K}$ of the adopted control law can be computed using the following expression (33).

Moreover, the minimum value of the parameter $ϵ$ can be calculated like so:

$${ϵ}_{\min }=\sqrt{\frac{{\gamma }_2}{{\kappa }_2}}$$

(62)

Furthermore, the maximum value of the parameter $\overline{\delta }$ can be determined as follows:

$${\overline{\delta }}_{\max }=\sqrt{\frac{m_3}{{\lambda }_2}}$$

(63)

Third design approach of the LMI condition

In this third approach, let’s proceed with the assumption that the function ${\mathrm{\Phi }}_2(\mathcal{X})$ can be bounded by a second-degree equation, to be designed subsequently.

As previously highlighted, the constraint applied to the candidate Lyapunov function (11) can be redefined by employing the expression (15). Additionally, we will retain the same formulation (13) to derive specific LMI conditions related to the feedback gain matrix $\mathcal{K}$ .

Furthermore, let’s reconsider the condition (16) and (20), along with the relation (17). It is straightforward to demonstrate the following inequalities:

$${\cos }^2(\theta )\le a_3{\theta }^2+1-2\cos (\theta )\le 0$$

with $a_3=-\frac{4}{{\pi }^2}$ .

Recall that $\theta =\varphi +{\theta }_d$ . Then, we can move forward to derive the following expression:

$${\cos }^2(\theta )\le a_3{\varphi }^2+2a_3{\theta }_d\varphi +a_3{\theta }_d^2+1$$

Hence, based on the evaluations above, we can establish the following inequality:

$${\mathrm{\Phi }}_2(\mathcal{X}{)}^2\le {\mu }_{31}{\varphi }^2+2{\mu }_{32}\varphi +{\mu }_{33}$$

(64)

where the scalars ${\mu }_{31}$ , ${\mu }_{32}$ and ${\mu }_{33}$ are defined by the following expressions:

$$\begin{array}{c}{\mu }_{31}=a_3\hfill \\ {\mu }_{32}=a_3{\theta }_d\hfill \\ {\mu }_{33}=1+a_3{\theta }_d^2+{\cos }^2({\theta }_d)\hfill \end{array}$$

An important point to highlight is that, within the range of ${\theta }_d$ values spanning from 0 to $\frac{\pi }{2}$ , we have ${\mu }_{31}<0$ and ${\mu }_{32}<0$ .

Let us consider $\varphi ={\mathcal{C}}_1\mathcal{X}$ , where ${\mathcal{C}}_1=\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right]$ , then equation (64) transforms into:

$${\mathrm{\Phi }}_2(\mathcal{X}{)}^2\le {\mu }_{31}{\mathcal{X}}^{\mathrm{T}}{\mathcal{C}}_1^{\mathrm{T}}{\mathcal{C}}_1\mathcal{X}+2{\mu }_{32}{\mathcal{C}}_1\mathcal{X}+{\mu }_{33}$$

(65)

Consequently, condition (36) is recast like so:

$$\begin{array}{c}2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathrm{\Phi }}_2(\mathcal{X})\le {\tau }_2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\mathcal{P}\mathcal{X}+{\mu }_{31}{\tau }_2^{-1}{\mathcal{X}}^{\mathrm{T}}{\mathcal{C}}_1^{\mathrm{T}}{\mathcal{C}}_1\mathcal{X}\hfill \\ +2{\mu }_{32}{\tau }_2^{-1}{\mathcal{C}}_1\mathcal{X}+{\mu }_{33}{\tau }_2^{-1}\hfill \end{array}$$

(66)

By substituting the constraints (16), (66), and (20) into the resulting expression (13) and by considering the constraint (14), we arrive at the following inequality:

$$\begin{array}{c}\stackrel{·}{\mathcal{V}}(\mathcal{X})+\alpha \mathcal{V}(\mathcal{X})\le 2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}(\mathcal{A}+\mathcal{B}\mathcal{K})\mathcal{X}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_1\mathcal{P}\mathcal{X}\hfill \\ +{\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\tau }_2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\mathcal{P}\mathcal{X}+2{\mu }_{32}{\tau }_2^{-1}{\mathcal{C}}_1\mathcal{X}\hfill \\ +{\mu }_{31}{\tau }_2^{-1}{\mathcal{X}}^{\mathrm{T}}{\mathcal{C}}_1^{\mathrm{T}}{\mathcal{C}}_1\mathcal{X}+{\mu }_{33}{\tau }_2^{-1}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{M}}_3\mathcal{P}\mathcal{X}\hfill \\ +{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}+\alpha {\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{X}\hfill \end{array}$$

$\mathrm{Let}\mathrm{us}\mathrm{take}\mathcal{Z}(\mathcal{X})=2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}\left(\mathcal{A}+ℬ\mathcal{K}\right)\mathcal{X}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{ℳ}_1\mathcal{P}\mathcal{X}+{\mathcal{D}}_1^{\mathrm{T}}{ℳ}_1^{-1}{\mathcal{D}}_1+{\tau }_2{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\mathcal{P}\mathcal{X}+{\mu }_{31}{\tau }_2^{-1}{\mathcal{X}}^{\mathrm{T}}{\mathcal{C}}_1^{\mathrm{T}}{\mathcal{C}}_1\mathcal{X}+2{\mu }_{32}{\tau }_2^{-1}{\mathcal{C}}_1\mathcal{X}+{\mu }_{33}{\tau }_2^{-1}+{\mathcal{X}}^{\mathrm{T}}\mathcal{P}{ℳ}_3\mathcal{P}\mathcal{X}+{\overline{\delta }}^2{ℬ}^{\mathrm{T}}{ℳ}_3^{-1}ℬ+\alpha {\mathcal{X}}^{\mathrm{T}}\mathcal{P}\mathcal{X}.$

As a result, we obtain $\stackrel{·}{\mathcal{V}}(\mathcal{X})+\alpha \mathcal{V}(\mathcal{X})\le \mathcal{Z}(\mathcal{X})$ . Therefore, to meet the constraint (14), we must satisfy the following inequality condition:

$$\mathcal{Z}(\mathcal{X})={\left[\begin{array}{c}\mathcal{X}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]}^{\mathrm{T}}\left[\begin{array}{cc}{\mathrm{\Theta }}_{11}\hfill & {\mathrm{\Theta }}_{12}\hfill \\ \multicolumn{1}{c}{★}& {\mathrm{\Theta }}_{22}\hfill \end{array}\right]\left[\begin{array}{c}\mathcal{X}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]<0$$

(67)

with

$$\begin{array}{c}{\mathrm{\Theta }}_{11}=(\mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})+{(\mathcal{P}\mathcal{A}+\mathcal{P}\mathcal{B}\mathcal{K})}^{\mathrm{T}}+\mathcal{P}{\mathcal{M}}_1\mathcal{P}\hfill \\ +{\tau }_2\mathcal{P}{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\mathcal{P}+{\mu }_{31}{\tau }_2^{-1}{\mathcal{C}}_1^{\mathrm{T}}{\mathcal{C}}_1+\mathcal{P}{\mathcal{M}}_3\mathcal{P}+\alpha \mathcal{P}\hfill \\ {\mathrm{\Theta }}_{12}={\mu }_{32}{\tau }_2^{-1}{\mathcal{C}}_1^{\mathrm{T}}\hfill \\ {\mathrm{\Theta }}_{22}={\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\mu }_{33}{\tau }_2^{-1}+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}\hfill \end{array}$$

Utilizing the S-procedure and employing the expressions (67) and (15), we formulate the following stability condition:

$$\left[\begin{array}{cc}{\mathrm{\Theta }}_{11}+{\eta }_2{\mathcal{I}}_n\hfill & {\mathrm{\Theta }}_{12}\hfill \\ \multicolumn{1}{c}{★}& {\mathrm{\Theta }}_{22}-{\eta }_2{ϵ}^2\hfill \end{array}\right]<0$$

(68)

where ${\eta }_2>0$ .

Consequently, by multiplying the extended condition defined by (68) on both the right and left sides with $\mathcal{Q}=\left[\begin{array}{cc}\mathcal{S}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& \mathcal{I}\hfill \end{array}\right]$ , where $\mathcal{S}={\mathcal{P}}^{-1}$ , we arrive at:

$$\left[\begin{array}{cc}{\chi }_{11}\hfill & {\chi }_{12}\hfill \\ \multicolumn{1}{c}{★}& {\chi }_{22}\hfill \end{array}\right]<0$$

(69)

with

$$\begin{array}{c}{\chi }_{11}=(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\hfill \\ +{\mu }_{31}{\tau }_2^{-1}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}{\mathcal{C}}_1\mathcal{S}+{\mathcal{M}}_3+\alpha \mathcal{S}+{\eta }_2{\mathcal{S}}^2\hfill \\ {\chi }_{12}={\mu }_{32}{\tau }_2^{-1}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}\hfill \\ {\chi }_{22}={\mathcal{D}}_1^{\mathrm{T}}{\mathcal{M}}_1^{-1}{\mathcal{D}}_1+{\mu }_{33}{\tau }_2^{-1}+{\overline{\delta }}^2{\mathcal{B}}^{\mathrm{T}}{\mathcal{M}}_3^{-1}\mathcal{B}-{\eta }_2{ϵ}^2\hfill \end{array}$$

where the matrix $\mathcal{R}$ is defined by expression (25).

To simplify further, let’s define: ${\beta }^{-1}={\eta }_2{ϵ}^2$ , $\mathcal{J}=\left[\begin{array}{ccc}\mathcal{O}\hfill & {\mu }_{32}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}\hfill & \mathcal{O}\hfill \end{array}\right]$ , $\mathcal{D}=\left[\begin{array}{c}{\mathcal{D}}_1\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{\mathcal{B}}\end{array}\right]$ , ${\mathcal{M}}^{-1}=\left[\begin{array}{ccc}{\mathcal{M}}_1^{-1}\hfill & \mathcal{O}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& {\tau }_2^{-1}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& \mathcal{O}\hfill & {\mathcal{M}}_3^{-1}\hfill \end{array}\right]$ , ${\mathcal{Q}}^{-1}=\left[\begin{array}{ccc}\mathcal{I}\hfill & \mathcal{O}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& {\mu }_{33}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{\mathcal{O}}& \mathcal{O}\hfill & {\overline{\delta }}^2\mathcal{I}\hfill \end{array}\right]$ . As a result, inequality (69) transforms into:

$$\left[\begin{array}{cc}{\chi }_{11}\hfill & \mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}\hfill \\ \multicolumn{1}{c}{★}& -{\beta }^{-1}+{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{Q}}^{-1}\mathcal{D}\hfill \end{array}\right]<0$$

(70)

According to the Schur complement, we can reformulate the matrix inequality (70) as follows:

$$\begin{array}{c}-{\beta }^{-1}+{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{Q}}^{-1}\mathcal{D}<0\hfill \\ {\chi }_{11}+\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}{({\beta }^{-1}-{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{Q}}^{-1}\mathcal{D})}^{-1}\times \hfill \end{array}$$

(71a)

$${\left(\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}\right)}^{\mathrm{T}}<0$$

(71b)

Using the matrix inversion lemma to the term on the left-hand side of expression (71a), we obtain the same expression (45).

Subsequently, based on equation (45), we can establish that inequality (71b) is equivalent to:

$${\chi }_{11}+\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}(\beta -{\beta }^2{\mathcal{D}}^{\mathrm{T}}{\mathcal{H}}^{-1}\mathcal{D}){\left(\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}\right)}^{\mathrm{T}}<0$$

(72)

Posing

$$\begin{array}{c}\Psi =\beta \mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}\hfill \\ -{\beta }^2\mathcal{J}{\mathcal{M}}^{-1}\mathcal{D}{\mathcal{D}}^{\mathrm{T}}{\mathcal{H}}^{-1}\mathcal{D}{\mathcal{D}}^{\mathrm{T}}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}\hfill \end{array}$$

Given that $\mathcal{H}=\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M}$ , it is straightforward to demonstrate the expressions (50).

Then, by substituting (50) into (72), we obtain:

$$\begin{array}{c}(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\hfill \\ +{\mu }_{31}{\tau }_2^{-1}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}{\mathcal{C}}_1\mathcal{S}+{\mathcal{M}}_3+\alpha \mathcal{S}+{\eta }_2{\mathcal{S}}^2\hfill \\ -\mathcal{J}\mathcal{Q}{\mathcal{M}}^{-1}{\mathcal{J}}^{\mathrm{T}}-\mathcal{J}\mathcal{Q}{\mathcal{H}}^{-1}\mathcal{Q}{\mathcal{J}}^{\mathrm{T}}<0\hfill \end{array}$$

Given that $\mathcal{Q}\mathcal{M}>0$ , we can reformulate this inequality like so:

$$\begin{array}{c}(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\hfill \\ +{\mu }_{31}{\tau }_2^{-1}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}{\mathcal{C}}_1\mathcal{S}+{\mathcal{M}}_3+\alpha \mathcal{S}+{\eta }_2{\mathcal{S}}^2\hfill \\ -\mathcal{J}\mathcal{Q}{(\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M})}^{-1}\mathcal{Q}{\mathcal{J}}^{\mathrm{T}}<0\hfill \end{array}$$

Therefore, when we consider ${\overline{\mathcal{C}}}_1=\left[\begin{array}{c}{\mathcal{C}}_1\hfill \\ \multicolumn{1}{c}{\mathcal{O}}\end{array}\right]$ , it becomes evident that:

$${\mu }_{31}{\tau }_2^{-1}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}{\mathcal{C}}_1\mathcal{S}={\mu }_{31}{\tau }_2^{-1}\mathcal{S}{\overline{\mathcal{C}}}_1^{\mathrm{T}}{\overline{\mathcal{C}}}_1\mathcal{S}$$

Hence, by employing this expression, we can represent inequality (51) as:

$$\begin{array}{c}(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\hfill \\ +{\mu }_{31}{\tau }_2^{-1}\mathcal{S}{\overline{\mathcal{C}}}_1^{\mathrm{T}}{\overline{\mathcal{C}}}_1\mathcal{S}+{\mathcal{M}}_3+\alpha \mathcal{S}+{\eta }_2{\mathcal{S}}^2\hfill \\ -\mathcal{J}\mathcal{Q}{(\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M})}^{-1}\mathcal{Q}{\mathcal{J}}^{\mathrm{T}}<0\hfill \end{array}$$

(73)

It is important to note that ${\mu }_{31}<0$ and ${\tau }_2^{-1}>0$ . Therefore, ${\mu }_{31}{\tau }_2^{-1}\mathcal{S}{\overline{\mathcal{C}}}_1^{\mathrm{T}}{\overline{\mathcal{C}}}_1\mathcal{S}<0$ . Consequently, by applying the Young inequality, we obtain:

$${\mu }_{31}{\tau }_2^{-1}\mathcal{S}{\overline{\mathcal{C}}}_1^{\mathrm{T}}{\overline{\mathcal{C}}}_1\mathcal{S}\le -\frac{1}{{\mu }_{31}}{\tau }_2\mathcal{I}-{\overline{\mathcal{C}}}_1\mathcal{S}-\mathcal{S}{\overline{\mathcal{C}}}_1^{\mathrm{T}}$$

Subsequently, building upon this expression, we can express equation (73) like so:

$$\begin{array}{c}(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}+{\mathcal{M}}_3\hfill \\ +\alpha \mathcal{S}+{\eta }_2{\mathcal{S}}^2-\frac{1}{{\mu }_{31}}{\tau }_2\mathcal{I}-{\overline{\mathcal{C}}}_1\mathcal{S}-\mathcal{S}{\overline{\mathcal{C}}}_1^{\mathrm{T}}\hfill \\ -\mathcal{J}\mathcal{Q}{(\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M})}^{-1}\mathcal{Q}{\mathcal{J}}^{\mathrm{T}}<0\hfill \end{array}$$

(74)

Therefore, using expression (52), by employing the Schur complement, we can establish that $\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M}<0$ . Consequently, we can infer that $-{(\beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M})}^{-1}>0$ . Subsequently, by introducing ${\gamma }_2={\eta }_2^{-1}$ and utilizing the Schur complement, we can derive:

$$\left[\begin{array}{ccc}{\Pi }_3\hfill & \mathcal{S}\hfill & \mathcal{J}\mathcal{Q}\hfill \\ \multicolumn{1}{c}{★}& -{\gamma }_2{\mathcal{I}}_n\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & \beta \mathcal{D}{\mathcal{D}}^{\mathrm{T}}-\mathcal{Q}\mathcal{M}\hfill \end{array}\right]<0$$

(75)

where

$$\begin{array}{c}{\Pi }_3=(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\hfill \\ +{\mathcal{M}}_3+\alpha \mathcal{S}-\frac{1}{{\mu }_{31}}{\tau }_2\mathcal{I}-{\overline{\mathcal{C}}}_1\mathcal{S}-\mathcal{S}{\overline{\mathcal{C}}}_1^{\mathrm{T}}\hfill \end{array}$$

Furthermore, let us take the following new variables’ change ${\kappa }_2=\frac{{\gamma }_2}{{\epsilon }^2}$ , and ${\lambda }_2=\frac{m_3}{{\overline{\delta }}^2}$ , and also let us assume that ${\mathcal{M}}_3=m_3\mathcal{I}$ , we can rephrase expression (75) as an LMI stability condition, which can be presented in the following manner:

$$\begin{array}{c}\left[\begin{array}{ccccccc}{\Pi }_4\hfill & \mathcal{S}\hfill & \mathcal{O}\hfill & \cdots \hfill & \cdots \hfill & \frac{{\mu }_{32}}{{\mu }_{33}}\mathcal{S}{\mathcal{C}}_1^{\mathrm{T}}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{★}& -{\gamma }_2{\mathcal{I}}_n\hfill & \mathcal{O}\hfill & \cdots \hfill & \cdots \hfill & \mathcal{O}\hfill & \mathcal{O}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & {\kappa }_2{\mathcal{D}}_1{\mathcal{D}}_1^{\mathrm{T}}-{\mathcal{M}}_1\hfill & \cdots \hfill & \cdots \hfill & {\kappa }_2{\mathcal{D}}_1\hfill & {\kappa }_2{\mathcal{D}}_1{\mathcal{B}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & ★\hfill & \cdots \hfill & \cdots \hfill & {\kappa }_2-\frac{1}{{\mu }_{31}}{\tau }_2\hfill & {\kappa }_2{\mathcal{B}}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{★}& ★\hfill & ★\hfill & \cdots \hfill & \cdots \hfill & ★\hfill & {\kappa }_2\mathcal{B}{\mathcal{B}}^{\mathrm{T}}-{\lambda }_2\mathcal{I}\hfill \end{array}\right]<0\hfill \end{array}$$

(76)

with

$$\begin{array}{c}{\Pi }_4=(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})+{(\mathcal{A}\mathcal{S}+\mathcal{B}\mathcal{R})}^{\mathrm{T}}+{\mathcal{M}}_1+{\tau }_2{\mathcal{D}}_2{\mathcal{D}}_2^{\mathrm{T}}\hfill \\ +m_3\mathcal{I}+\alpha \mathcal{S}-\frac{1}{{\mu }_{31}}{\tau }_2\mathcal{I}-{\overline{\mathcal{C}}}_1\mathcal{S}-\mathcal{S}{\overline{\mathcal{C}}}_1^{\mathrm{T}}\hfill \end{array}$$

it is essential to emphasize that ${\kappa }_2=\frac{{\gamma }_2}{{ϵ}^2}$ , and this implies that $ϵ=\sqrt{\frac{{\gamma }_2}{{\kappa }_2}}$ . In order to minimize $ϵ$ , our objective is to minimize $({\gamma }_2+{\kappa }_2^{-1})$ . Therefore, we can introduce the condition (55), which we can rephrase by (56). Hence, by applying the Schur complement lemma, we can formulate the LMI stability condition (57).

Moreover, it is crucial to recognize that ${\lambda }_2=\frac{m_3}{{\overline{\delta }}^2}$ . Consequently, we can represent $\overline{\delta }$ as $\overline{\delta }=\sqrt{\frac{m_3}{{\lambda }_2}}$ . Furthermore, let introduce a variable $\Omega ={\overline{\delta }}^{-1}=\sqrt{\frac{{\lambda }_2}{m_3}}$ . To maximize $\overline{\delta }$ , we should minimize $\Omega$ , and by extension, minimize ${\lambda }_2+m_3^{-1}$ . Therefore, we can impose the constraint (58), which can be reformulated as (59). Then, we can state the LMI-based stability condition (60).

Hence, we have derived the LMI stability conditions presented in (76), (57), and (60). These conditions are formulated in relation to the unknown variables $\mathcal{S}$ , $\mathcal{R}$ , ${\mathcal{M}}_1$ , ${\tau }_2$ , $m_3$ , ${\gamma }_1$ , $\lambda$ , ${\nabla }_3$ , ${\nabla }_4$ , and $\kappa$ .

• Theorem 3. Assuming the existence of matrices $\mathcal{S}$ , $ℛ$ , and ${\mathcal{M}}_1$ , as well as scalar variables ${\tau }_2$ , $m_3$ , ${\gamma }_1$ , $\lambda$ , ${\nabla }_3$ , ${\nabla }_4$ , and $\kappa$ , such that the following LMI-based minimization problem:

$$\begin{array}{c}\min {\nabla }_3+{\nabla }_4+{\zeta }_1+{\zeta }_2\hfill \\ s.t.\mathrm{LMIs}(76),(57),(60),(31a)\mathrm{and}(31b).\hfill \end{array}$$

(77)

is feasible. Accordingly, it can be asserted that the proposed controller, as outlined in (9), successfully achieves the stabilization of the nonlinear dynamics delineated in (1) of the knee exoskeleton rehabilitation robot. Additionally, the gain matrix $\mathcal{K}$ for the applied control law can be determined using the expression (33). Moreover, the minimum value of the parameter $ϵ$ can be computed using the relation (62). Furthermore, we can determine the maximum value of the parameter $\overline{\delta }$ using the equation (63).

• Remark 3. Our primary focus in this research centers on refining precise control mechanisms for a lower limb exoskeleton, specifically honing in on the pivotal knee joint.^24,31 To attain this objective, we have introduced a unique control strategy grounded in state feedback. What distinguishes our work from previous studies^32,35,50 is the application of the LMI approach. This departure from conventional methodologies not only ensures efficient control and stabilization of the knee rehabilitation exoskeleton robot but also introduces a novel perspective to the existing body of knowledge. Our innovative approach demonstrates its prowess in handling parametric uncertainties and disturbances, ultimately augmenting the overall control performance.

• Remark 4. It is worth noting that the unique parameter to fix in the previous LMI-based minimization problems developed in the three LMI design approaches and therefore to compute the feedback gain $\mathcal{K}$ , is the decay rate $\alpha >0$ . The parameter $\alpha$ influences the behavior of both the controlled robotic system and the controller over time. By increasing the value of such parameter $\alpha$ , the size of the matrix feedback gain $\mathcal{K}$ will consequently increase and the control effort will become high. Thus, the goal is to select a value that yields the most suitable and feasible simulation results, balancing effective control with practicality.

Results of simulations

To ensure the stability of the actuated lower limb orthosis/exoskeleton at the knee joint level, we have developed three distinct LMI techniques. In the following section, we present the results of our comprehensive numerical analyses, meticulously demonstrating the effectiveness of each formulated LMI condition. To provide a comprehensive context, Table 1 compiles the values of various parameters crucial to the actuated knee exoskeleton robotic system, as described by equation (1).

Table 1.

Parameters’ value of the knee exoskeleton robot. ³⁵

Parameter	Value	Unit
${\nu }_{★}$	0.323	kg.m2/rad
$m_{★}$	0.000105	kg
$l_{★}$	0.25	M
$f_v^{★}$	0.35	N.m.s/rad
$f_s^{★}$	0.059	N.m/A
$g$	9.8	N

Continuing, we will vividly illustrate the control performance of the knee exoskeleton robotic system in response to the predefined target state ${\theta }_d$ . This target state has been intentionally set to ${\theta }_d=30°$ for our experimental evaluation.

Additionally, we will introduce a distinctive external disturbance, denoted as ${\mathrm{\Delta }}_t$ . It is crucial to note that this disturbance has been carefully designed to replicate the characteristics of a sinusoidal input:

$${\mathrm{\Delta }}_t=\overline{\delta }\times \sin ({\omega }_f\times t)$$

In this expression, ${\omega }_f$ represents the frequency of the signal. For our specific scenario, we have configured ${\omega }_f$ to be $5\pi$ . It is crucial to emphasize that this disturbance input is exclusively active within the time interval from 2 to 7 s. After this timeframe, the value of ${\mathrm{\Delta }}_t$ is reset to zero.

Results obtained using the first design approach

By applying the LMI stability condition established in Theorem 1 and by solving the associated LMI problem with a chosen parameter value of $\alpha =3$ , we obtain the resulting gain matrix $\mathcal{K}$ along with the corresponding values of ${ϵ}_{\min }$ and ${\overline{\delta }}_{\max }$ , presented below:

$$\mathcal{K}=\left[\begin{array}{cc}-2.7806\hfill & -1.1165\hfill \end{array}\right]$$

(78a)

$${ϵ}_{\min }=0.7252$$

(78b)

$${\overline{\delta }}_{\max }=0.1973$$

(78c)

By incorporating the feedback gain (78a) into the state-feedback controller (9), we can examine the graphical outcomes as illustrated in Figure 2. The results clearly demonstrate the successful adjustment of the robot to the desired position ${\theta }_d$ , which has been set to ${\theta }_d=30°$ . Additionally, once the knee exoskeleton robotic system achieves stability, the control effort $\mathcal{U}$ converges toward ${\mathcal{U}}_d$ .

Figure 2.

Graphical results of the regulated knee exoskeleton robotic system by adopting the first development approach: (a) presents the angular positions error $\varphi$ , (b) presents the angular velocities error $\stackrel{·}{\varphi }$ , (c) shows the adopted controller $\mathcal{U}$ , and (d) denotes the applied external disturbance ${\mathrm{\Delta }}_t$ .

Results obtained using the second design approach

Implementing the LMI stability condition outlined in Theorem 2 and solving the associated LMI problem with $\alpha =3$ , we have successfully computed the feedback gain matrix $\mathcal{K}$ , as well as the values of ${ϵ}_{\min }$ and ${\overline{\delta }}_{\max }$ , presented below:

$$\mathcal{K}=\left[\begin{array}{cc}-3.3902\hfill & -1.1314\hfill \end{array}\right]$$

(79a)

$${ϵ}_{\min }=3.5660$$

(79b)

$${\overline{\delta }}_{\max }=0.3362$$

(79c)

Implementing the feedback gain (79a) in the state-feedback controller (9), we can observe the graphical results illustrated in Figure 3. The robot accurately converges to the desired state ${\theta }_d$ , set in this case to ${\theta }_d=30°$ . The effectiveness of the control approach is evident as the knee exoskeleton robotic system achieves stability. Additionally, the control effort $\mathcal{U}$ gradually decreases and eventually converges to ${\mathcal{U}}_d$ .

Figure 3.

Graphical results of the regulated knee exoskeleton robotic system by adopting the second development approach: (a) presents the angular positions error $\varphi$ , (b) presents the angular velocities error $\stackrel{·}{\varphi }$ , (c) shows the adopted controller $\mathcal{U}$ , and (d) denotes the applied external disturbance ${\mathrm{\Delta }}_t$ .

Results obtained using the third design approach

By solving the LMI problem associated with the LMI stability condition outlined in Theorem 3, specifically with $\alpha =3$ , the resulting feedback gain matrix $\mathcal{K}$ and the values of the two parameters ${ϵ}_{\min }$ and ${\overline{\delta }}_{\max }$ , can be expressed as follows:

$$\mathcal{K}=\left[\begin{array}{cc}-2.8223\hfill & -1.3758\hfill \end{array}\right]$$

(80a)

$${ϵ}_{\min }=6.7080$$

(80b)

$${\overline{\delta }}_{\max }=0.3606$$

(80c)

By incorporating the feedback gain (80a) into the adopted control law (9), we illustrate the graphical results shown in Figure 4. Consequently, the exoskeleton robot is successfully controlled to the intended position, specified as ${\theta }_d=30°$ . Furthermore, as the system is stabilized in the intended state, the control effort $\mathcal{U}$ converges to the desired level ${\mathcal{U}}_d$ .

Figure 4.

Graphical results of the regulated knee exoskeleton robotic system by adopting the third development approach: (a) presents the angular positions error $\varphi$ , (b) presents the angular velocities error $\stackrel{·}{\varphi }$ , (c) shows the adopted controller $\mathcal{U}$ , and (d) denotes the applied external disturbance ${\mathrm{\Delta }}_t$ .

Discussion

The proposed state-feedback control strategy plays a crucial role in effectively managing the intricate task of controlling the position of the actuated knee exoskeleton robotic system. This approach utilizes an extended feedback gain derived from three distinct methodologies, ensuring precise control over the robot’s motion. As the robot is guided toward the target state ${\theta }_d$ , the control method ensures that the angular position error $\varphi$ steadily converges to zero. These remarkable outcomes are vividly illustrated in Figures 2(a), 3(a) and 4(a).

Furthermore, the utilization of these three methodologies not only lead to a reduction in the angular position error $\varphi$ but also effectively manages the angular velocity error $\stackrel{·}{\varphi }$ , as demonstrated by the data presented in Figures 2(b), 3(b) and 4(b). The minor fluctuations observed in these plots can be attributed to the presence of an external torque ${\mathrm{\Delta }}_t$ , underscoring the control method’s capability to adeptly handle disturbances.

Figures 2(d), 3(d) and 4(d) offer a visual insight into the impact of an external disturbance, denoted as the applied load ${\mathrm{\Delta }}_t$ , on the knee exoskeleton robot. Notably, this lumped torque ${\mathrm{\Delta }}_t$ manifests as a sinusoidal signal active only between 2 and 7 s, modeled within the range of $-\overline{\delta }$ to $\overline{\delta }$ . Correspondingly, Figures 2(c), 3(c) and 4(c) portray the dynamics of the proposed controller $\mathcal{U}$ . The knee exoskeleton system demonstrates stability amid the presence of the external load ${\mathrm{\Delta }}_t$ , with observable oscillations around its equilibrium point. Once the influence of the disturbing torque diminishes, the controlled knee exoskeleton robot promptly regains stability at its desired position. It is crucial to note that, in each case, the numerical values for the matrix feedback gain $\mathcal{K}$ and the parameters ${ϵ}_{\min }$ and ${\overline{\delta }}_{\max }$ were automatically determined. Consequently, the first case yields $\overline{\delta }=0.1973$ , the second case, we obtained $\overline{\delta }=0.3362$ , whereas in the third case, we obtained $\overline{\delta }=0.3606$ . The simulation results underscore the efficacy of the designed affine state-feedback controller in mitigating the impact of the lumped disturbance, ensuring the knee exoskeleton robotic system remains closely aligned with its desired position.

Moreover, it is noteworthy that the time required for the knee exoskeleton robotic system to achieve stability varies among the three approaches. The first approach achieves stability in approximately 3 s, the second approach accomplishes this in under 2.7 s, and the third approach stabilizes the actuated knee exoskeleton in less than 2.6 s.

Significantly, the initial control effort $\mathcal{U}$ exerted on the joint at the beginning of the simulation is remarkably low for all three approaches, as indicated by the plots in Figures 2(c), 3(c), and 4(c). Therefore, the outcomes derived from the three distinct methodologies underscore the successful regulation of the robot, effectively achieving the desired state with optimal stability.

Comparison with the PID controller

In this part, the objective is to compare the performance of the adopted affine state-feedback controller to the PID controller. Such control law has the following expression:

$$\mathcal{U}={\mathcal{K}}_1\varphi +{\mathcal{K}}_2\stackrel{·}{\varphi }+{\mathcal{K}}_3{\int }_0^t\varphi (\zeta )d\zeta$$

(81)

The goal is to find the values of the gains ${\mathcal{K}}_1$ , ${\mathcal{K}}_2$ and ${\mathcal{K}}_3$ of this PID controller (81). It is possible to check several sets of the gains and verify what set can stabilize the nonlinear model (1) of the knee exoskeleton robot under the presence of the lumped disturbance ${\mathrm{\Delta }}_t$ . However, this task is very fastidious. The goal is to use the same LMI design approaches developed in the previous parts where the objective is to find the minimum value of the parameter $ϵ$ , and also the maximum value of the parameter $\overline{\delta }$ :

By making some transformation of the nonlinear model (1) under the PID controller (81), it is possible to re-obtain the same nonlinear state-space representation (7) with same nonlinear functions ${\mathrm{\Phi }}_1(\mathcal{X})$ and ${\mathrm{\Phi }}_2(\mathcal{X})$ defined by expressions (8a) and (8b). The tiny difference lies in the expression of the new matrices $\mathcal{A}$ , $\mathcal{B}$ , ${\mathcal{D}}_1$ , and ${\mathcal{D}}_2$ defining the nonlinear model (7). Moreover, the gain matrix ${\mathcal{K}}_{\mathrm{PID}}$ of the PID controller (81) becomes as follows:

$$\mathcal{K}=\left[\begin{array}{ccc}{\mathcal{K}}_1\hfill & {\mathcal{K}}_2\hfill & {\mathcal{K}}_3\hfill \end{array}\right]$$

(82)

Using then the same three approaches for the design of the LMI conditions, we reconsider the LMI-based minimization problems presented in Theorems 1–3 for the computation of the gain ${\mathcal{K}}_{\mathrm{PID}}$ and also for the calculation of the values of ${ϵ}_{\min }$ and ${\overline{\delta }}_{\max }$ .

As previously, the desired position ${\theta }_d$ is fixed to ${\theta }_d=30°$ , and the decay rate parameter $\alpha$ is fixed to the value $\alpha =3$ . The results obtained via the three design approaches for the PID controller are presented in the sequel.

Results using the first design approach for the PID controller

By adopting the previous fixed parameters ${\theta }_d$ and $\alpha$ , we obtain the following values of the feedback gain ${\mathcal{K}}_{\mathrm{PID}}$ and the associated parameters ${ϵ}_{\min }^{\mathrm{PID}}$ and ${\overline{\delta }}_{\max }^{\mathrm{PID}}$ :

$${\mathcal{K}}_{\mathrm{PID}}=\left[\begin{array}{ccc}-6.3855\hfill & -1.4331\hfill & -7.3188\hfill \end{array}\right]$$

(83a)

$${ϵ}_{\min }^{\mathrm{PID}}=0.5625$$

(83b)

$${\overline{\delta }}_{\max }^{\mathrm{PID}}=0.0528$$

(83c)

Compared to the results in (78), it is obvious that the maximum bound of the lumped disturbance (i.e. ${\overline{\delta }}_{\max }$ ) obtained by considering the PID controller is very small. In contrast, even if the value of ${\overline{\delta }}_{\max }$ is decreased for the PID controller, the size of the feedback gain ${\mathcal{K}}_{\mathrm{PID}}$ was increased. This result will allow the increase of the control effort $\mathcal{U}$ applied to the exoskeleton robot.

Results using the second design approach for the PID controller

Let us consider the second design approach of the LMI conditions and then the LMI-based minimization problem in Theorem 2. Then, by considering the case of the PID controller, we obtain the following numerical results:

$${\mathcal{K}}_{\mathrm{PID}}=\left[\begin{array}{ccc}-7.6766\hfill & -1.5722\hfill & -9.7522\hfill \end{array}\right]$$

(84a)

$${ϵ}_{\min }^{\mathrm{PID}}=6.8425$$

(84b)

$${\overline{\delta }}_{\max }^{\mathrm{PID}}=0.1662$$

(84c)

As found in the previous part, the value of ${\overline{\delta }}_{\max }$ obtained for the PID controller is smaller than that obtained using the affine state feedback controller, that is the result in (79c). Moreover, the value of ${ϵ}_{\min }$ is considerably increased using the PID controller, as found in (84b), compared to the value (79b). Furthermore, the size of the gain ${\mathcal{K}}_{\mathrm{PID}}$ is also higher than that obtained using the state-feedback controller, the feedback gain in (79a).

Results using the third design approach for the PID controller

Using the minimization problem under LMI constraints presented in Theorem 3 by considering in the present part the case of the PID controller, the following numerical results are then obtained:

$${\mathcal{K}}_{\mathrm{PID}}=\left[\begin{array}{ccc}-18.9072\hfill & -3.1094\hfill & -41.8243\hfill \end{array}\right]$$

(85a)

$${ϵ}_{\min }^{\mathrm{PID}}=40.4498$$

(85b)

$${\overline{\delta }}_{\max }^{\mathrm{PID}}=0.1149$$

(85c)

As previously, compared to the results obtained via the affine state-feedback controller, the gain ${\mathcal{K}}_{\mathrm{PID}}$ of the PID controller has a relatively big size. As noted previously, this high value of the size of the feedback gain induces an increase in the magnitude of the controller applied to the exoskeleton robot, which is a drawback since it allows an increase in energy consumption, and it can also be unrealizable in some real-life applications.

In addition, for the PID controller, the value of ${ϵ}_{\min }$ has been considerably increased compared to the two other approaches. This big value can be seen as unfeasible since such parameter $ϵ$ should be small (enough). Furthermore, as in the previous two cases, the maximum bound of the lumped disturbance ${\overline{\delta }}_{\max }$ is found to be smaller than that (value (80c)) obtained using the affine state-feedback control law.

These obtained results reveal accordingly the superiority of the adopted affine state-feedback controller against the PID controller.

Conclusion and future directions

This study employed an affine state-feedback controller to effectively manage and stabilize the position of a knee exoskeleton robotic system, considering its nonlinear dynamics encompassing frictions, parameter uncertainties, and external disturbances. The control law was meticulously designed using the Lyapunov methodology, and we introduced three distinct methods to establish stability conditions for the feedback gain, leveraging specific technical lemmas. These conditions were subsequently translated into Linear Matrix Inequalities (LMIs). Through a comparative analysis, we evaluated the efficacy of each method, utilizing the LMI approach to derive gain conditions that guarantee the stability of the controlled knee exoskeleton. The simulation results presented in this study demonstrate that the designed LMI-based controller successfully achieves excellent stabilization of the knee exoskeleton system.

Looking ahead, several promising avenues for future research emerge. Firstly, while this study specifically focuses on one scenario of external disturbance, our goal is to evaluate the robustness of the implemented control law by subjecting it to various scenarios involving different external disturbances and uncertainties in the system’s parameters. Secondly, as we assumed in the present work that the external disturbances are bounded by a constant value, we plan to design an uncertainty and disturbance estimator for tracking control in fuzzy-switched systems. This additional feature will address potential challenges and contribute to the overall robustness of our proposed approach. Furthermore, we aim to explore the integration of a filtering mechanism that could potentially reduce the amplitude or duration of oscillations in the controlled system. Additionally, our future endeavors will concentrate on designing a robust static output feedback control law using the LMI approach. ³⁸ Moreover, it’s worth noting that this work does not consider movement constraints related to the knee joint level. Hence, we are eager to incorporate movement constraints related to different lower limb articulations into the controller design. This integration aims to minimize control effort, reduce energy consumption, and ultimately enhance the overall efficiency and usability of the knee exoskeleton robotic system. In addition, the objective is to extend the LMI design approaches adopted in the present work and also the previous ideas to multi-degree-of-freedom knee-joint robotic systems.

To further advance the proposed approach, future developments could involve exploring adaptive control strategies for self-adjustment in dynamic conditions, investigating real-time implementation for practical validation, and conducting human-in-the-loop experiments to assess the system’s efficacy in rehabilitation scenarios. In terms of applications, the controller exhibits potential in rehabilitation robotics, assistive devices for mobility-impaired individuals, human-machine collaboration scenarios, and sports performance enhancement, showcasing its adaptability and robustness in diverse settings.