The Robust Control of a Nonsmooth or Switched Control-Affine Uncertain Nonlinear System Using an Auxiliary Robust Integral of the Sign of the Error (ARISE) Controller ^†

Abstract

To deal with uncertainties in a dynamic system, many nonlinear control approaches have been considered. Unique challenges arise from uncertainties that are bounded by constants, which has led to the development of both continuous and discontinuous control methods. However, these methods either are limited to classes of smooth nonlinear models or have a tendency to result in chattering during practical applications. In this work, a novel auxiliary robust integral of the sign of the error (ARISE) controller is proposed to prevent chattering and deal with uncertainties (even those bounded by constants) for general, switched, and nonsmooth control-affine nonlinear systems. The ARISE control system includes a unique auxiliary error that is designed to inject a sliding mode (SM) term directly into the error system without including an SM term in the controller itself. In fact, the ARISE control law includes an integral SM term that is continuous. Consequently, the ARISE control law minimizes the chattering effect that results from discontinuous SM terms. The proposed ARISE control system is augmented with an adaptive update law to deal with the unknown control effectiveness matrix in the dynamic model. To prove the effectiveness of the ARISE controller, a nonlinear stability analysis was conducted and resulted in semi-global exponential tracking towards an ultimate bound. Furthermore, the performance of the proposed controller was evaluated and compared against a traditional SM controller through simulations using a switched Van der Pol oscillator model. It was concluded that the proposed ARISE controller performs better for a switched system than an SM controller. The improved performance of the ARISE controller was consistent across different dynamic parameters and disturbances.

1. Introduction

Over many years, research has been ongoing in the field of control design for complicated, uncertain nonlinear dynamic systems [1,2,3,4,5,6]. Given that most real-world systems exhibit nonlinear behavior, nonlinear control design is a topic of practical significance [7]. To obtain the best stability outcomes in the presence of system uncertainty and nonlinearities, various control approaches have been developed by numerous researchers [8,9,10,11]. For example, high-gain control is a common robust control approach that is used to deal with model uncertainty [12]. However, robust high-gain controllers do not provide the best control and error convergence results when uncertain or unstructured terms within the dynamics can only be upper-bounded by constants. To enhance the performance of the controller and obtain a better stability outcome, robust controllers have been combined with feedforward adaptive terms to compensate for the uncertain parameters [13,14,15,16]. To elaborate, adaptive terms in a control law are capable of reducing the amount of control effort needed for the system to converge since some of the uncertainty has been isolated and approximated by feedforward terms. Thus, a feedforward adaptive term incorporates some dynamic information in the control law, thereby enhancing controller performance.

Another approach to compensating for dynamic uncertainties that are bounded by constants is to augment robust or adaptive controllers with discontinuous terms. For example, sliding mode (SM) controllers incorporate a discontinuous signum function within the control law to compensate for these uncertain terms [10,17,18,19]. Theoretically, SM control immensely enhances stability and yields a favorable transient performance; however, the discontinuous signum function within the SM controller architecture introduces abrupt switching within the control input. Though SM controllers are known for their robustness to uncertainties, this rapid switching between differing control actions requires infinite control frequency and introduces chattering, all of which hinder the practical application of SM control [13,20,21].

Robust integral of the sign of the error (RISE) control, which is a high-gain feedback control approach, was developed to address the shortcomings of SM control while still exploiting its advantages and robustness. Unlike conventional SM control development methods where a signum function is included directly in the control input, the signum function is instead included in the time derivative of an auxiliary error system in the RISE-based approach. Incorporating the auxiliary error in the RISE control input introduces an integral SM term in the control law, rather than an SM term itself, making the control input continuous. The continuous RISE controller eliminates the SM controller’s requirement for infinite control frequency and reduces chattering, bolstering the RISE controller’s utility in real-world systems. In fact, RISE controllers have been implemented in numerous systems; for example, autonomous flight, delayed systems, underwater vehicles, and many more [13,22,23,24,25,26]. Additionally, many researchers have demonstrated that RISE-based methods have produced excellent results in a variety of applications, including estimation, control design, and optimization [23,27,28,29]. Across a broad class of smooth systems, RISE-based control laws typically produce an asymptotic tracking outcome [22,30,31,32,33,34,35,36,37,38], and after some modification, researchers have also been successful in achieving an exponential error convergence [39]. Though RISE controllers are highly robust and leverage the benefits of SM all while mitigating its downsides, the RISE-based method requires the dynamic model including the control input to be continuously differentiable, which precludes its application in switched/discontinuous systems.

To counter the limitations of RISE control, the authors developed a novel auxiliary RISE (ARISE) control approach, upon which this work is built, for a general, nonsmooth, and uncertain control-affine nonlinear dynamic system with the goal of developing a chatter-mitigating controller that can be applied to a wide range of uncertain nonlinear systems [40]. Similar to [40], this work developed a filtered error signal to inject an SM term into the closed-loop error system, whereas the controller itself includes an integral SM term to mitigate chattering and to reduce the need for an infinite control frequency. The control effectiveness used in the dynamics in this work is considered to be unknown and potentially switched or nonsmooth. To account for the uncertain control effectiveness matrix, an adaptive estimate is proposed. A nonsmooth Lyapunov-like stability analysis is performed to confirm the efficacy of the proposed adaptive ARISE control system. The stability result shows that, if the gain and initial state requirements are satisfied, the designed control system produces semi-global exponential trajectory tracking towards an ultimate bound. In contrast to our work in [40], this work includes a series of numerical simulations using a modified two-state Van der Pol oscillator as the dynamic model to validate the performance of the proposed control system. The oscillator was specifically modified by including control-affine control inputs and by enabling the model parameters to be switched. The performance of the proposed ARISE controller was compared with a traditional SM controller to determine their relative performances when implemented on a switched system. The root mean square (RMS) error was calculated for the trajectory tracking errors for both the ARISE and SM controllers across different disturbances and dynamic parameters to compare both approaches robustness. The comparative results demonstrated that the RMS tracking errors of the ARISE controller were less than those of the SM controller across every tested scenario, which indicates that the ARISE controller is more robust than the SM controller for the considered switched system. Overall, the proposed ARISE controller can reduce chattering, handle model uncertainties, and account for dynamic parameter switching. Because of this, the controller can be used with a wide range of real-world engineering systems, including robotic manipulators, spaceflight systems with switched dynamics, mobile robots, and many more. To illustrate the practicality of ARISE control, an additional simulation was conducted on a soft robotic end-effector.

2. Materials and Methods

2.1. Dynamics

Consider a nonsmooth, uncertain, and control-affine nonlinear dynamic system

$$\dot{x}\left(t\right)=f_{\sigma }\left(x\left(t\right)\right)+d_{\sigma }\left(t\right)+\underset{{\tau }_{\varphi }\left(x\left(t\right),t\right)}{\underbrace{g_{\varphi }\left(x\left(t\right),t\right)u\left(t\right)},}$$

(1)

where $x:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n$ represents the generalized states, $f_{\sigma }:{\mathbb{R}}^n\to {\mathbb{R}}^n$ represents the uncertain, nonlinear, and nonsmooth drift dynamics, and $d_{\sigma }:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n$ represents the nonsmooth and unknown disturbances. The signal $\sigma \left(x\left(t\right),t\right)$, denoted by $\sigma :{\mathbb{R}}^n\times {\mathbb{R}}_{\ge 0}\to \mathcal{P}\triangleq \left\{1,2,\dots ,p\right\}$, represents an uncertain switching signal that indicates the index of $f_{\sigma }\left(x\left(t\right)\right)$ and $d_{\sigma }\left(t\right)$, where the number of unique forms for $f_{\sigma }\left(x\left(t\right)\right)$ and $d_{\sigma }\left(t\right)$ is represented by $p\in \mathbb{N}$. The generalized input torque or force in (1) is represented as ${\tau }_{\varphi }:{\mathbb{R}}^n\times {\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n$ and is defined as the product of the control input $u:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^m$ and the unknown and nonsmooth control effectiveness matrix $g_{\varphi }:{\mathbb{R}}^n\times {\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^{n\times m}$. The signal $\varphi \left(x\left(t\right),t\right)$, denoted by $\varphi :{\mathbb{R}}^n\times {\mathbb{R}}_{\ge 0}\to \mathcal{S}\triangleq \left\{1,2,\dots ,s\right\}$, represents a known switching signal that indicates the index of ${\tau }_{\varphi }\left(x\left(t\right),t\right)$ and $g_{\varphi }\left(x\left(t\right),t\right)$, where the number of unique forms for ${\tau }_{\varphi }\left(x\left(t\right),t\right)$ and $g_{\varphi }\left(x\left(t\right)\right)$ is represented by $s\in \mathbb{N}$.

Remark 1.

Every unique form of $f_{\sigma }\left(x\left(t\right)\right)$ and $d_{\sigma }\left(t\right)$ is represented by the signal $\sigma \in \mathcal{P}$, which switches values whenever $f_{\sigma }\left(x\left(t\right)\right)$ and $d_{\sigma }\left(t\right)$ have a discrete jump. Likewise, every unique form of ${\tau }_{\varphi }\left(x\left(t\right),t\right)$ and $g_{\varphi }\left(x\left(t\right),t\right)$ is represented by the signal $\varphi \in \mathcal{S}$, which switches values whenever ${\tau }_{\varphi }\left(x\left(t\right),t\right)$ and $g_{\varphi }\left(x\left(t\right),t\right)$ have a discrete jump. Whenever $\sigma \in \mathcal{P}$ and $\varphi \in \mathcal{S}$ are constant, the dynamics in (1) are continuous (i.e., $f_{\sigma }\left(x\left(t\right)\right)$, $d_{\sigma }\left(t\right)$, $g_{\varphi }\left(x\left(t\right),t\right)$, and ${\tau }_{\varphi }\left(x\left(t\right),t\right)$ are continuous for constant σ and ϕ).

The following assumptions must be satisfied to facilitate the control development:

Assumption 1.

The drift function $f_{\sigma }\left(x\left(t\right)\right)$ is upper-bounded by a known constant $c_F\in {\mathbb{R}}_{>0}$ such that $∥f_{\sigma }\left(x\left(t\right)\right)∥\le c_F,$ $\forall t\ge t_0$, and $\forall \sigma \in \mathcal{P}$ whenever $∥x\left(t\right)∥\in {\mathcal{L}}_{\infty }$, where the term $t_0\in {\mathbb{R}}_{\ge 0}$ represents the initial time and $∥·∥$ represents the standard Euclidean norm.

Assumption 2.

The disturbance $d_{\sigma }\left(t\right)$ is upper-bounded by a known constant $c_D\in {\mathbb{R}}_{>0}$ such that $∥d_{\sigma }\left(t\right)∥\le c_D,$ $\forall t\ge t_0$, and $\forall \sigma \in \mathcal{P}$.

Assumption 3.

The control effectiveness matrix $g_{\varphi }\left(x\left(t\right),t\right)$ is a full-row rank matrix for all $t\ge t_0$.

Assumption 4.

The generalized torque/force input ${\tau }_{\varphi }\left(x\left(t\right),t\right)=g_{\varphi }\left(x\left(t\right),t\right)u\left(t\right)$ can be linearly parameterized for all $t\ge t_0$ such that

$$g_{\varphi }\left(x\left(t\right),t\right)u\left(t\right)=Y_{\varphi }\left(x\left(t\right),t,u\left(t\right)\right)\theta ,$$

(2)

where $\theta \in {\mathbb{R}}^q$ represents a vector of the unknown constants in $g_{\varphi }\left(x\left(t\right),t\right)$ and $q\in \mathbb{N}$ represents the number of uncertain constants in $g_{\varphi }\left(x\left(t\right),t\right)$. The matrix $Y_{\varphi }:{\mathbb{R}}^n\times {\mathbb{R}}_{\ge 0}\times {\mathbb{R}}^m\to {\mathbb{R}}^{n\times q}$ represents a measurable regression matrix. Also, the portions of $g_{\varphi }\left(x\left(t\right),t\right)$ that vary explicitly with time are assumed to be bounded such that $g_{\varphi }\left(x\left(t\right),t\right)\in {\mathcal{L}}_{\infty }$ whenever $x\left(t\right)\in {\mathcal{L}}_{\infty },\forall t\ge t_0$. The $i$th parameter in θ, denoted by ${\theta }_i$, can be upper- and lower-bounded by ${\overline{\theta }}_i\in \mathbb{R}$ and ${\underline{\theta }}_i\in \mathbb{R}$, respectively, such that $\theta \in {\mathcal{L}}_{\infty }$ and

$${\underline{\theta }}_i\le {\theta }_i\le {\overline{\theta }}_i,\forall i\in \left\{1,\dots ,q\right\}.$$

(3)

2.2. Controller Design

The control objective in this work is to ensure that the generalized states in the dynamics in (1) track a sufficiently smooth desired trajectory denoted by $x_d:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n$. The desired trajectory $x_d$ and its time derivative ${\dot{x}}_d$ are known and bounded such that $x_d,{\dot{x}}_d\in {\mathcal{L}}_{\infty }$. A known tracking error, denoted as $e:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n$, is defined as

$$e\left(t\right)\triangleq x\left(t\right)-x_d\left(t\right).$$

(4)

Another objective of this work is to modify the error system to inject an SM term into the open-loop error system (i.e., the SM term is not in the controller). This task is accomplished through the design of an auxiliary tracking error, denoted by $r:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n,$ that is defined as

$$r\left(t\right)\triangleq e\left(t\right)+\alpha e_f\left(t\right),$$

(5)

where $e_f:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n$ represents an auxiliary filtered error signal and $\alpha \in {\mathbb{R}}_{>0}$ represents a user-defined control gain. The time derivative of the auxiliary filtered error is defined as

$${\dot{e}}_f\left(t\right)\triangleq -k_1\mathrm{sgn}\left(r\left(t\right)\right)-k_{2}r\left(t\right)-\beta e_f\left(t\right),$$

(6)

where $k_1,k_2,\beta \in {\mathbb{R}}_{>0}$ are user-defined control gains and $\mathrm{s}\mathrm{g}\mathrm{n}\left(·\right)$ represents the signum function. Notice that the filter dynamics in (6) can be solved to determine $e_f\left(t\right)$.

Taking the time derivative of (5) and using (1), (4) and (6) yields

$$\begin{array}{ccc}\hfill \dot{r}\left(t\right)& =& f_{\sigma }\left(x\left(t\right)\right)+d_{\sigma }\left(t\right)+g_{\varphi }\left(x\left(t\right),t\right)u\left(t\right)-{\dot{x}}_d\left(t\right)\hfill \\ \hfill & & -\alpha k_1\mathrm{sgn}\left(r\left(t\right)\right)-\alpha k_{2}r\left(t\right)-\alpha \beta e_f\left(t\right).\hfill \end{array}$$

(7)

For notational brevity, an auxiliary term ${\chi }_{\sigma }:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n$ is defined as

$${\chi }_{\sigma }\left(t\right)\triangleq f_{\sigma }\left(x\left(t\right)\right)+d_{\sigma }\left(t\right).$$

(8)

The auxiliary term can be upper-bounded using Assumptions 1 and 2 and Equations (4) and (5) such that

$$∥{\chi }_{\sigma }\left(t\right)∥\le \Phi +\rho \left(∥z\left(t\right)∥\right)∥z\left(t\right)∥,$$

(9)

where $\rho \left(·\right):\mathbb{R}\to {\mathbb{R}}_{>0}$ is a known, radially unbounded, and strictly increasing positive function, $\Phi \in {\mathbb{R}}_{>0}$ is a known constant, and $z:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^{2n}$ is a composite error vector defined as

$$z\left(t\right)\triangleq {\left[\begin{array}{cc}{r}^T\left(t\right)& e_f^T\left(t\right)\end{array}\right]}^T.$$

(10)

Substituting (8) into (7) yields an open-loop error function as

$$\begin{array}{ccc}\hfill \dot{r}\left(t\right)& =& {\chi }_{\sigma }\left(t\right)+g_{\varphi }\left(x\left(t\right),t\right)u\left(t\right)-{\dot{x}}_d\left(t\right)\hfill \\ \hfill & & -\alpha k_1\mathrm{sgn}\left(r\left(t\right)\right)-\alpha k_{2}r\left(t\right)-\alpha \beta e_f\left(t\right).\hfill \end{array}$$

(11)

Since the control effectiveness matrix $g_{\varphi }\left(x\left(t\right),t\right)$ is unknown, the matrix must be estimated using an adaptive approach. In this work, the estimate of the generalized torque/force input is expressed as

$${\widehat{g}}_{\varphi }\left(x\left(t\right),t\right)u\left(t\right)=Y_{\varphi }\left(x\left(t\right),t,u\left(t\right)\right)\widehat{\theta }\left(t\right),$$

(12)

where ${\widehat{g}}_{\varphi }:{\mathbb{R}}^n\times {\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^{n\times m}$ is an estimate of $g_{\varphi }\left(x\left(t\right),t\right)$ and $\widehat{\theta }:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^q$ is an estimate of the unknown vector of constants $\theta$. The difference between the uncertain parameters and their estimates results in a parameter estimation error, denoted as $\tilde{\theta }:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^q$, that is defined as

$$\tilde{\theta }\left(t\right)\triangleq \theta -\widehat{\theta }\left(t\right).$$

(13)

Assumption 5.

The estimate of the control effectiveness matrix ${\widehat{g}}_{\varphi }\left(x\left(t\right),t\right)$, $\forall t\ge t_0$, is a full-row rank matrix. The right pseudo inverse of ${\widehat{g}}_{\varphi }\left(x\left(t\right),t\right)$ is denoted as ${\widehat{g}}_{\varphi }^{+}:{\mathbb{R}}^n\times {\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^{m\times n}$, where ${\widehat{g}}_{\varphi }^{+}\left(·\right)\triangleq {\widehat{g}}_{\varphi }^T\left(·\right){\left({\widehat{g}}_{\varphi }\left(·\right){\widehat{g}}_{\varphi }^T\left(·\right)\right)}^{-1}.$ Also, ${\widehat{g}}_{\varphi }\left(x\left(t\right),t\right)$ is bounded such that ${\widehat{g}}_{\varphi }\left(x\left(t\right),t\right)\in {\mathcal{L}}_{\infty }$ provided that $\widehat{\theta }\left(t\right)\mathrm{and}x\left(t\right)$ are bounded $\left(\mathrm{i}.\mathrm{e}.,\widehat{\theta }\left(t\right),x\left(t\right)\in {\mathcal{L}}_{\infty }\right)$.

Adding and subtracting $Y\left(x\left(t\right),t,u\left(t\right)\right)\widehat{\theta }\left(t\right)$ in (11) and using (2), (12) and (13) yields a modified open-loop error system as

$$\begin{array}{ccc}\hfill \dot{r}\left(t\right)& =& {\chi }_{\sigma }\left(t\right)+Y_{\varphi }\left(x\left(t\right),t,u\left(t\right)\right)\tilde{\theta }\left(t\right)+{\widehat{g}}_{\varphi }\left(x\left(t\right),t\right)u\left(t\right)\hfill \\ \hfill & & -{\dot{x}}_d\left(t\right)-\alpha k_1\mathrm{sgn}\left(r\left(t\right)\right)-\alpha k_{2}r\left(t\right)-\alpha \beta e_f\left(t\right).\hfill \end{array}$$

(14)

Based on the open-loop error dynamics in (14), the control input is designed as

$$u\left(t\right)\triangleq {\widehat{g}}_{\varphi }^{+}\left(x\left(t\right),t\right)\left[-k_{3}r\left(t\right)+\left(k_2+\alpha \beta \right)e_f\left(t\right)+{\dot{x}}_d\left(t\right)\right],$$

(15)

where $k_3\in {\mathbb{R}}_{>0}$ is a user-defined control gain. Similarly, the parameter estimate law is designed as

$$\dot{\widehat{\theta }}\left(t\right)\triangleq \mathrm{Proj}\left(\Gamma Y_{\varphi }^T\left(x\left(t\right),t,u\left(t\right)\right)r\left(t\right)\right),$$

(16)

where $\Gamma \in {\mathbb{R}}^{q\times q}$ is a positive definite, diagonal gain matrix and $\mathrm{Proj}\left(·\right)$ denotes the smooth projection algorithm defined in Cai et al. [41], which ensures that the estimate $\widehat{\theta }\left(t\right)$ stays within bounds defined in (3) for $\theta \left(t\right)$. The controller from (15) is substituted into the open-loop error system in (14) to obtain a closed-loop error system as

$$\begin{array}{ccc}\hfill \dot{r}\left(t\right)& =& Y_{\varphi }\left(x\left(t\right),t,u\left(t\right)\right)\tilde{\theta }\left(t\right)-\left(k_3+\alpha k_2\right)r\left(t\right)\hfill \\ \hfill & & -\alpha k_1\mathrm{sgn}\left(r\left(t\right)\right)+k_2{e}_f\left(t\right)+{\chi }_{\sigma }\left(t\right).\hfill \end{array}$$

(17)

The closed-loop error system in (17) has an SM term due to the implementation of the auxiliary error term $e_f$. The SM term compensates for uncertain terms upper-bounded by constants in the dynamics. Due to the controller implementing the integral of the signum function, chattering is effectively reduced.

2.3. Stability Analysis

The $i^{th}$ time when $\varphi$ switches to $w\in \mathcal{S}$ can be represented by $\left\{t_i^{w,\mathrm{on}}\right\},$$\forall i\in \mathbb{N}$, and the $i^{th}$ time when $\varphi$ switches from $w\in \mathcal{S}$ can be represented by $\left\{t_i^{w,\mathrm{off}}\right\}$, $\forall i\in \mathbb{N}$. Let $V_L:\mathcal{D}\times {\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$ be a positive definite common Lyapunov-like candidate function that is continuously differentiable in y, where y is a concatenated state vector that is defined as

$$y\triangleq {\left[\begin{array}{cc}{z}^T& {\tilde{\theta }}^T\end{array}\right]}^T.$$

(18)

The Lyapunov candidate function $V_L$ is defined within the domain $\mathcal{D}\subseteq {\mathbb{R}}^{2n+q}$ as

$$V_L\left(y,t\right)\triangleq {\displaystyle \frac{1}{2}}{r}^{T}r+{\displaystyle \frac{1}{2}}{e}_f^T{e}_f+{\displaystyle \frac{1}{2}}{\tilde{\theta }}^T{\Gamma }^{-1}\tilde{\theta },$$

(19)

where $\mathcal{D}\triangleq \left\{y\in {\mathbb{R}}^{2n+q}\left|∥y∥\le \gamma \right.\right\}$, $\gamma \in {\mathbb{R}}_{>0}$ is a known constant that satisfies the inequality $\gamma \le inf\left\{{\rho }^{-1}\left(\left(\sqrt{2\alpha k_2\delta },\infty \right)\right)\right\}$, $\delta \in \mathbb{R}$ is a known constant defined as $\delta \triangleq min\left(k_3,\beta -{\displaystyle \frac{\epsilon k_1}{2}}\right),$ and $\epsilon \in {\mathbb{R}}_{>0}$ is a selectable constant. Using the concatenated state vector y defined in (18), the Lyapunov-like function $V_L$ can be upper- and lower-bounded as

$${\lambda }_1{∥y∥}^2\le V_L\le {\lambda }_2{∥y∥}^2,$$

(20)

where the constants ${\lambda }_1,{\lambda }_2\in {\mathbb{R}}_{\ge 0}$ are defined as ${\lambda }_1\triangleq min\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}{\lambda }_{\min }\left\{{\Gamma }^{-1}\right\}\right)$ and ${\lambda }_2\triangleq max\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}{\lambda }_{\max }\left\{{\Gamma }^{-1}\right\}\right)$, ${\lambda }_{\min }$ is defined as the minimum eigenvalue $\left\{·\right\}$, and ${\lambda }_{\max }$ is defined as the maximum eigenvalue $\left\{·\right\}$. Additionally, based on Assumption 4 and (18), $V_L$ can be further bounded as

$${\displaystyle \frac{1}{2}}{∥z∥}^2\le V_L\le {\displaystyle \frac{1}{2}}{∥z∥}^2+v_1.$$

(21)

In (21), the function $v_1:{\mathbb{R}}^q\to {\mathbb{R}}_{\ge 0}$ is defined as

$$v_1\triangleq {\displaystyle \frac{1}{2}}{\lambda }_{\max }\left\{{\Gamma }^{-1}\right\}{∥\tilde{\theta }∥}_{\max }^2,$$

(22)

where ${∥\tilde{\theta }∥}_{\max }\in {\mathbb{R}}_{>0}$ is the maximum value of $∥\tilde{\theta }\left(t\right)∥$ such that $∥\tilde{\theta }\left(t\right)∥\le {∥\tilde{\theta }∥}_{max},\forall t\ge t_0,$ resulting from the projection algorithm in (16) and the bounds defined in Assumption 4. Moreover, let ${\mathcal{S}}_{\mathcal{D}}\subseteq {\mathbb{R}}^{2n+q}$ be a set of allowable initial conditions that maintain system stability, defined as

$${\mathcal{S}}_{\mathcal{D}}\triangleq \left\{y\in {\mathbb{R}}^{2n+q}\left|∥y∥\le \sqrt{{\displaystyle \frac{{\lambda }_1}{{\lambda }_2}}}\gamma \right.\right\}.$$

(23)

Theorem 1.

Consider the uncertain, nonsmooth, and nonlinear dynamics in (1) that satisfy Assumptions 1–5. The control input in (15) and the parameter update law in (16) ensure semi-global exponential trajectory tracking to an ultimate bound in the sense that

$$\begin{array}{ccc}\hfill {∥y\left(t\right)∥}^2& \le & {\displaystyle \frac{{\lambda }_2}{{\lambda }_1}}{∥y\left(t_0\right)∥}^{2}exp\left(-\delta \left(t-t_0\right)\right)+{\displaystyle \frac{v}{{\lambda }_1\delta }}\left(1-exp\left(-\delta \left(t-t_0\right)\right)\right),\hfill \end{array}$$

(24)

for all $t\in \left[t_0,\infty \right)$, provided that $y\left(t_0\right)\in {\mathcal{S}}_{\mathcal{D}},$ where $v\triangleq \delta v_1+{\displaystyle \frac{n{k}_1}{2\epsilon }}$, and provided that the following gain conditions are satisfied:

$$\beta >{\displaystyle \frac{\epsilon k_1}{2}},\alpha k_1\ge \Phi ,\sqrt{{\displaystyle \frac{v}{{\lambda }_1\delta }}}\le \gamma .$$

(25)

Proof. 

Since the closed-loop error system in (17) contains discontinuous terms, the time derivative of (19) exists almost everywhere (a.e.) for all $t\in \left[t_0,\infty \right).$ Let $y\left(t\right)$ be a Filippov solution to the differential inclusion $\dot{y}\in K\left[h\right]\left(y\right)$ for all $t\in \left[t_0,\infty \right)$, where $K[·]$ is defined in [42] and $h:{\mathbb{R}}^{2n+q}\to {\mathbb{R}}^{2n+q}$ is defined as $h\left(y\right)\triangleq {\left[\begin{array}{ccc}{\dot{r}}^T,& {\dot{e}}_f^T,& {\dot{\tilde{\theta }}}^T\end{array}\right]}^T.$

Consider times when $\varphi =w$ for any arbitrary $w\in \mathcal{S}$ (i.e., $t\in \left[t_i^{w,\mathrm{on}},t_i^{w,\mathrm{off}}\right)$ for some $i\in \mathbb{N}$). During these times, the dynamic functions $Y_{\varphi }$, $g_{\varphi }$, and ${\widehat{g}}_{\varphi }$ are continuous; thus, $K\left[{\widehat{g}}_{\varphi }\right]={\widehat{g}}_{\varphi }$ and $K\left[Y_{\varphi }\right]=Y_{\varphi }$. Taking the generalized time derivative of (19), using the calculus of $K\left[.\right]$ from [43], and using (6), (13), (16) and (17) yields

$$\begin{array}{ccc}{\dot{\tilde{V}}}_L& \subseteq & r^T\left(K\left[{\chi }_{\sigma }\right]+Y_{\varphi }\tilde{\theta }-\left(k_3+\alpha k_2\right)r-\alpha k_{1}K\left[\mathrm{sgn}\left(r\right)\right]+k_2{e}_f\right)\hfill \\ & & +e_f^T\left(-k_{1}K\left[\mathrm{sgn}\left(r\right)\right]-k_{2}r-\beta e_f\right)-{\tilde{\theta }}^T{\Gamma }^{-1}\left(\Gamma Y_{\varphi }^{T}r\right).\hfill \end{array}$$

(26)

Implementing (9) and (25) into (26), applying Young’s Inequality, completing the squares, canceling common terms, and using the facts that ${\dot{V}}_L\left(y\right)\stackrel{\mathrm{a}.\mathrm{e}.}{\in }{\dot{\tilde{V}}}_L\left(y\right)$, $∥K\left[\mathrm{sgn}\left(r\right)\right]∥\le \sqrt{n}$, and $∥y∥\ge ∥z∥$ yields

$$\begin{array}{ccc}\hfill {\dot{V}}_L& \stackrel{\mathrm{a}.\mathrm{e}.}{\le }& -k_3{∥r∥}^2-\left(\beta -{\displaystyle \frac{\epsilon k_1}{2}}\right){∥e_f∥}^2+{\displaystyle \frac{1}{4\alpha k_2}}{\rho }^2\left(∥y∥\right){∥z∥}^2+{\displaystyle \frac{n{k}_1}{2\epsilon }}.\hfill \end{array}$$

(27)

Additionally, using (10) and (21), $v\triangleq \delta v_1+{\displaystyle \frac{n{k}_1}{2\epsilon }}$, and $\delta \triangleq min\left(k_3,\beta -{\displaystyle \frac{\epsilon k_1}{2}}\right)$ with (27) yields the following bound:

$$\begin{array}{ccc}\hfill {\dot{V}}_L& \stackrel{\mathrm{a}.\mathrm{e}.}{\le }& -\delta V_L+v,\hfill \end{array}$$

(28)

∀$t\in \left[t_i^{w,\mathrm{on}},t_i^{w,\mathrm{off}}\right)$ provided that $y\left(t\right)\in \mathcal{D}$, ∀$t\in \left[t_i^{w,\mathrm{on}},t_i^{w,\mathrm{off}}\right)$. Since ${\dot{V}}_L$ is bounded for any arbitrary $\varphi \in \mathcal{S}$, it can likewise be bounded for all $\varphi \in \mathcal{S}$. Hence, (19) is a common Lyapunov-like function across all $t\in \left[t_0,\infty \right)$, which means that (28) holds true for all $t\in \left[t_0,\infty \right)$, provided that $y\left(t\right)\in \mathcal{D}$, $\forall t\in \left[t_0,\infty \right)$. The differential equation in (28) can be further solved to yield

$$V_L\left(t\right)\le \left(V_L\left(t_0\right)-{\displaystyle \frac{v}{\delta }}\right)exp\left(-\delta \left(t-t_0\right)\right)+{\displaystyle \frac{v}{\delta }},$$

(29)

$\forall t\in \left[t_0,\infty \right)$, provided that $y\left(t\right)\in \mathcal{D}$, $\forall t\in \left[t_0,\infty \right)$. The solution in (24) in the theorem statement can be determined using (20) and (29). Provided that $y\left(t_0\right)$ lies within ${\mathcal{S}}_D$ and the gain condition $\sqrt{{\displaystyle \frac{v}{{\lambda }_1\delta }}}\le \gamma$ is satisfied, it can be proven that $y\left(t\right)\in \mathcal{D}$, $\forall t\in \left[t_0,\infty \right)$. It can be observed from (24) in the theorem statement that as $t\to \infty$, the first term in (24) converges exponentially to 0 and the second term in (24) converges to ${\displaystyle \frac{v}{{\lambda }_1\delta }}$. Therefore, the state vector $y\left(t\right)$ converges to the ultimate bound $\sqrt{{\displaystyle \frac{v}{{\lambda }_1\delta }}}$, which establishes a neighborhood about the origin defined by $\{y\mid ∥y∥\le \sqrt{{\displaystyle \frac{v}{{\lambda }_1\delta }}}\}$ that y will ultimately reside within. Moving on, the proper selection of control gains $\alpha$, $k_1$, $k_2$, $k_3$, $\beta$, and $\Gamma$ and the selectable constant $\epsilon$ can guarantee that $y\left(t_0\right)$ lies within ${\mathcal{S}}_D$. The common Lyapunov-like function in (19) can be used with (29) to prove that $r,e_f,\tilde{\theta }\in {\mathcal{L}}_{\infty }$. Noting that $x_d$ is user-defined and is bounded and using (4) and (5), it can be shown that $e,x\in {\mathcal{L}}_{\infty }$. Using (3) and (13) and the fact that $\tilde{\theta }\in {\mathcal{L}}_{\infty }$ implies that $\widehat{\theta }\in {\mathcal{L}}_{\infty }$. Additionally, utilizing Assumption 5 and (16), it can be proven that ${\widehat{g}}_{\varphi },{\widehat{g}}_{\varphi }^{+}\in {\mathcal{L}}_{\infty }$. Therefore, since ${\dot{x}}_d\in {\mathcal{L}}_{\infty }$ and from (16), it is clear that $u\in {\mathcal{L}}_{\infty }$. □

2.4. Simulation

The ARISE control framework is presented in Figure 1 to provide a better understanding of the simulation flow required by the control architecture. This flowchart outlines the error computation, update law, auxiliary filter, ARISE controller, and plant, as implemented in the subsequent simulations. A series of simulations were performed to demonstrate the effectiveness of the proposed ARISE controller. To demonstrate ARISE’s performance compared to SM control, simulations were performed using the developed ARISE controller and a traditional SM controller. The dynamics considered during the simulations were similar to a two-state Van der Pol oscillator; however, the dynamics were modified to be switched, to include disturbances, and to have a control-affine control input. Specifically, the unknown drift dynamics used in the simulations were

$$f_{\sigma }\left(x\right)=\left[\begin{array}{c}\mu \sigma \left(x_1-{\displaystyle \frac{1}{3}}{x}_1^3-x_2\right)\\ {\displaystyle \frac{1}{\mu \sigma }}{x}_1\end{array}\right],$$

where $\mu \in {\mathbb{R}}_{>0}$ is an unknown system parameter, $x={\left[x_1,x_2\right]}^T$, and

$$\sigma =\left\{\begin{array}{cc}1,& x_1\ge 0\\ 2,& \mathrm{otherwise}\end{array}\right.$$

(30)

is an unknown state-dependent switching signal that indicates the index of the drift dynamics. The disturbances used during the simulation were designed as

$$d\left(t\right)=\left[\begin{array}{c}{d}_1\\ d_2\end{array}\right],$$

where $d_1,d_2\in {\mathbb{R}}_{>0}$ are uncertain parameters. The control effective matrix used in the simulations was

$$g_{\varphi }\left(x,t\right)=\left[\begin{array}{cc}{c}_1{\varphi }_1\left(x\right)+c_2{\varphi }_2\left(x\right)& 0\\ 0& c_3{\varphi }_1\left(x\right)+c_4{\varphi }_2\left(x\right)\end{array}\right],$$

where $c_1,c_2,c_3,c_4\in \mathbb{R}$ are unknown, constant, and non-zero parameters and

$${\varphi }_1\left(x\right)\triangleq \left\{\begin{array}{cc}1,& x_2\ge 0\\ 0,& \mathrm{otherwise}\end{array}\right.\mathrm{and}$$

(31)

$${\varphi }_2\left(x\right)\triangleq \left(1-{\varphi }_1\left(x\right)\right)$$

(32)

are known state-dependent switching signals. The index of the control effectiveness matrix is

$$\varphi =\left\{\begin{array}{cc}1,& {\varphi }_1=1\\ 2,& {\varphi }_1=0\end{array}\right.,$$

based on the switching signals defined in (31) and (32). Additionally, satisfying Assumption 4, the product $g_{\varphi }u$ can be linearly parameterized as

$$g_{\varphi }u=\underset{Y}{\underbrace{\left[\begin{array}{cccc}{\varphi }_1{u}_1& \left(1-{\varphi }_1\right)u_1& 0& 0\\ 0& 0& {\varphi }_1{u}_2& \left(1-{\varphi }_1\right)u_2\end{array}\right]}}\underset{\theta }{\underbrace{\left[\begin{array}{c}{c}_1\\ c_2\\ c_3\\ c_4\end{array}\right]}},$$

where $u={\left[u_1,u_2\right]}^T$.

The initial conditions of the states and the desired trajectory for the simulation were $x\left(0\right)={\left[\begin{array}{cc}3& 2\end{array}\right]}^T$ and $x_d={\left[5cos\left(t\right),5sin\left(t\right)\right]}^T$, respectively. The control gains for the ARISE controller were selected as $\alpha =0.1$, $k_1=3.5$, $k_2=100$, $k_3=20$, and $\beta =10$. The parameter estimates were initialized as $\widehat{\theta }\left(0\right)={\left[\begin{array}{cccc}5& 5& 5& 5\end{array}\right]}^T$. Two sets of unknown constants were selected for the control effectiveness matrix to analyze the change in performance of the control law across varying parameter selections. For the first set, the values of $c_1$, $c_2$, $c_3$, and $c_4$ were selected as 5, 10, 3, and 6, respectively, and the set was named GA. For the other set of unknown constants, the parameters were, respectively, set as 7, 10, 3, and 4, and the set was named GB. Similarly, the performances of the control laws were analyzed for 8 different sets of disturbances: 4 continuous disturbances and 4 constant disturbances, as presented in

Table 1. Continuous and constant disturbances used in simulation.

Disturbances	${\mathit{d}}_1$	${\mathit{d}}_2$
a	0.5cos(t)	0.5sin(t)
b	0.1cos(t)	0.1sin(t)
c	0.05cos(t)	0.05sin(t)
d	0.01cos(t)	0.01sin(t)
e	0.5	0.5
f	0.1	0.1
g	0.05	0.05
h	0.01	0.01

. The control performance was also analyzed for 7 different values of $\mu$ evenly spaced from 0.5 to 3.5. Analyzing the performance at different $\mu$ values helps to ensure the robustness of the controllers across varying dynamic parameters.

To ensure a fair comparison between the ARISE controller and the SM controller, the dynamic parameters and adaptive update law were kept the same during each simulation. The control law itself is the only difference (note that the proposed ARISE controller is derived in Section 2.2, while the SM controller is introduced below). The SM controller that is used for comparative purposes in the simulations is defined as $u={\widehat{g}}_{\varphi }^{+}\left(x\left(t\right),t\right)\left(-k_{4}e\left(t\right)-k_5\mathrm{sgn}\left(e\right)+{\dot{x}}_d\left(t\right)\right)$, where $k_4$ is a robust control gain for the SM controller and $k_5$ is an SM control gain. The SM control gains were selected as $k_4=90$ and $k_5=3.5$. The listed gains for both controllers were adjusted using an empirical-based method for a single set of dynamic parameters and disturbances (i.e., $\mu =1.5,$ control effectiveness parameters = set ‘GA’, and disturbances = set ‘a’). First, the control gains for the ARISE controller were selected and then adjusted to achieve an improved performance (i.e., minimum tracking errors). After that, the control gains for the SM controller were selected and then adjusted similar to the ARISE controller gains. While selecting the control gains for the SM controller, similar gain values were considered to ensure a fair comparison between the two controllers. The selected control gain values were then used across all other dynamic sets during the simulations. Note: The control gains for the ARISE controller could be adjusted using a variety of optimization techniques, including Particle Swarm Optimization, Genetic Algorithms, and the Nelder–Mead Simplex Method. However, a manual empirical gain tuning (i.e., trial-and-error) method was employed in this work. The use of optimization methods to tune the gains can be investigated in future research to further enhance the control performance. Similar to the development of the ARISE adaptive update law, an adaptive update law was developed for the SM controller as $\dot{\widehat{\theta }}\left(t\right)\triangleq \mathrm{Proj}\left(\Gamma Y_{\varphi }^T\left(x\left(t\right),t,u\left(t\right)\right)e\left(t\right)\right)$. The initial estimates of $\theta$ and the gain $\Gamma$ were selected to be the same for the ARISE and SM controllers. The RMS of the tracking errors were evaluated for both controllers to compare their effectiveness, thereby providing valuable insights into their robustness when implemented on a switched system.

To further demonstrate the practical use of the proposed ARISE controller, another simulation was carried out on a soft robotic arm. A general nonlinear dynamic model of a 2D steering robotic arm without switching is presented. The dynamic parameters for the simulation were selected as $f\left(x\right)=\left[-0.5x_1;-0.5x_2\right]$, $d\left(t\right)=\left[0.05cos\left(t\right);0.05sin\left(t\right)\right]$, $g=\left[1,0.5,-0.5;0.2,-1,0.7\right]$. Similarly, the control parameters were selected as $k_1=1.5$, $k_2=100$, $k_3=20$, $\alpha =5$, $\beta =10$, and $\Gamma =\mathrm{diag}{\left\{0.1,0.2,0.05,0.15,0.1,0.2\right\}}^{-1}$. To start the simulation, we needed to define the initial conditions for the robot’s states and the unknown parameters of the control effectiveness matrix g. These initial conditions were selected as $x\left[0\right]=\left[0.1;0.1\right]$ and $\widehat{\theta }\left(0\right)=\left[0.5;0.5;0.5;0.5;0.5;0.5\right]$. Similarly, the desired trajectories were selected as $x_d={\left[0.3sin\left(t\right),0.3cos\left(t\right)\right]}^T$. Note that the dynamic model presented above is a generic model for soft robotic end-effectors that use 3 actuators to control motion in 2 dimensions. The drift function $f\left(x\right)$ captures linear damping or elastic resistance, and the control effectiveness matrix g maps each actuator’s input directions to the end-effector’s motion.

3. Results

The aforementioned simulations demonstrate the capability of the proposed ARISE control system for a switched dynamic system. Figure 2 depicts the tracking errors of the ARISE and SM controllers for a single set of dynamic parameters and disturbance inputs, highlighting the different tracking performance between the two controllers. As can be seen in Figure 2, the ARISE controller demonstrates a better tracking performance compared to the SM controller. A more thorough analysis of this performance difference is presented in Section 4. The results in Figure 2 are representative across all dynamic parameter sets and disturbances that were considered. Likewise, the RMS tracking errors for both of the controllers were analyzed across varying sets of continuous disturbances (see Table 1), and the results for a single set of dynamic parameters are presented in Figure 3. Correspondingly, Figure 4 presents the RMS tracking errors for both controllers under different control effectiveness values (i.e., ‘GA’ and ‘GB’) and a single set of dynamic parameters and disturbances. The RMS errors of the system once it reaches a steady state (SS) are also illustrated in Figure 4. Moreover, Figure 5 depicts the RMS tracking errors for both controllers when varying the parameter $\mu$.

A representative depiction of the control inputs for both controllers is presented in Figure 6 for a single set of dynamics and disturbances. The proposed ARISE controller, having reduced the tracking errors, required a similar (although slightly larger) amount of control input compared to the SM controller. Figure 7 is generated by narrowing Figure 6 to view the control inputs across a narrow subset of time. Finally, the tracking performance of the soft robotic system is presented in Figure 8. Figure 8 serves as evidence of the applicability of the proposed ARISE controller in real-world robotic systems.

4. Discussion

By referencing Figure 2, it is observed that the tracking errors decreased with increasing time for both ARISE and SM, which is likely due to the improving estimate of $\theta$ due to the adaptive update laws. However, the convergence in the proposed ARISE controller was better than in the SM controller, which is likely due to the ARISE controller implementing an integral SM term rather than a discontinuous SM term. Referencing Figure 3, it is clear that the RMS errors across different disturbance values for both controllers were consistent. However, the ARISE controller exhibited considerably smaller RMS errors than the SM controller, which indicates that the ARISE controller has a better tracking performance than SM control. When the process to generate Figure 3 was repeated using the constant disturbances in Table 1 (‘e’–‘h’), similar results were obtained for both controllers (i.e., the results for ‘a’ and ‘e’, ‘b’ and ‘f’, and so on were nearly identical), and thus the figure is omitted. Similarly, Figure 4 depicts that the RMS errors of the ARISE controller were smaller than those of the SM controller even across varying control effectiveness parameters, thereby indicating that the ARISE controller performs better on a switched system. Furthermore, there was a notable decrease in the RMS errors of the second state for both controllers once the system reached SS, as illustrated in Figure 4. This result may be due to the second state having fewer nonlinearities associated with its dynamic model. Likewise, referencing Figure 5, the RMS errors of the ARISE controller increased negligibly with increasing values of $\mu$. Comparatively, the RMS errors of the SM controller significantly increased with increasing values of $\mu$. This consistent performance of the ARISE controller despite changing the dynamic parameter suggests that ARISE is more robust than SM control when implemented on a switched system. It is also noticeable from Figure 5 that the magnitude of the RMS errors associated with ARISE was always smaller than that of the errors from the SM controller, further demonstrating the ARISE controller’s effectiveness over SM control.

Referencing Figure 7, the SM controller exhibits a rapid switching within the control input, which induces the chattering effect. Conversely, the ARISE controller displays a smoother control input, effectively mitigating the chattering effect. A key contribution of the proposed controller is that it is able to control a nonsmooth system, unlike traditional RISE controllers. Even though RISE controllers also use an integral SM term similar to ARISE, it can only be implemented on differentiable or smooth dynamic systems. The ARISE controller, on the other hand, obtains the benefits of RISE while being implementable on a wider class of dynamic systems. Referencing Figure 8, it can be seen that the proposed ARISE controller maintained a bounded tracking performance, reinforcing the applicability of the controller in a real-world scenario.

It should furthermore be noted that SM control is capable of achieving exponential tracking of the considered dynamic system, whereas ARISE control is only able to achieve uniformly ultimately bounded tracking of the same system. In particular, while using the ARISE controller, the system converges to a local neighborhood around the target trajectory, whereas an SM controller can theoretically converge to the exact trajectory. Despite the theoretical limitations of the ARISE approach, its tracking performance was superior to the SM control in simulation. It is anticipated that the improved performance of the ARISE controller compared to SM control will be more substantial in experimentation since real actuators are unable to perfectly implement SM control signals due to their requirement for infinite control frequency.

5. Conclusions

An ARISE control architecture was developed, and the effectiveness of the controller was compared with a traditional general SM controller. The ARISE controller comprises a novel auxiliary error system that is designed to compensate for uncertain terms that are upper-bounded by a constant. The chattering effect caused by SM terms is mitigated by the introduction of the novel auxiliary error system. A traditional SM controller has a discontinuous SM term directly in the controller, whereas the proposed ARISE controller implements an integral of the SM term in the controller. Unlike SM, the integral of a signum function enables a continuous controller, eliminating chattering and the requirement for infinite control frequency. Unlike RISE control, the developed ARISE controller can be applied to switched and nonsmooth systems. The performance of the proposed ARISE controller was evaluated and compared against a traditional SM controller via simulations. Results from the numerical simulations prove that ARISE control can outperform SM control. Future work will investigate mathematical approaches to eliminate the ultimate bound in the stability analysis. Furthermore, experiments will be performed to experimentally evaluate the performance of the developed ARISE control structure in real systems.