A unified framework of rapid exponential stability and optimal feedback control for nonlinear systems

Abstract

A novel framework of rapid exponential stability and optimal feedback control is investigated and analyzed for a class of nonlinear systems through a variant of continuous Lyapunov functions and Hamilton–Jacobi–Bellman equation. Rapid exponential stability means that the trajectories of nonlinear systems converge to equilibrium states in accelerated time. The sufficient conditions of rapid exponential stability are developed using continuous Lyapunov functions for nonlinear systems. Furthermore, according to a variant of continuous Lyapunov functions, a rapid exponential stability is guaranteed which satisfies some canonical conditions and Hamilton–Jacobi–Bellman equation for controlled nonlinear systems. It is can be seen that the solution of Hamilton–Jacobi–Bellman equation is a continuous Lyapunov function, and, therefore, rapid exponential stability and optimality are guaranteed for nonlinear systems. Last, the main result of this article is investigated via a nonlinear model of a spacecraft with one axis of symmetry through simulations and is used to check rapid exponential stability. Moreover, for the disturbance problem of initial point, a rapid exponential stable controller can reject the large-scale disturbances for controlled nonlinear systems. In addition, the proposed optimal feedback controller is applied to the tracking trajectories of 2-degree-of-freedom manipulator, and the numerical results have illustrated high efficiency and robustness in real time. The simulation results demonstrate the use of the rapid exponential stability and optimal feedback approach for real-time nonlinear systems.

Introduction

Nonlinear systems have many applications in natural science and practical engineering. According to the fact that stabilities are frequently encountered in science and engineering systems, for instance, physical system, mechanical system, biological system, chemical system, robotic system, and so on.^1–5

In the study of nonlinear systems’ applications, people have researched the stability of such systems and developed various conditions for asymptotic stability or exponential stability. Moreover, exponential stabilities of nonlinear systems have been investigated during the past few years and the references therein.^6–8 Particularly, more and more people have paid increasing attention to the problem of rapid exponential stability of nonlinear systems in recent years.^9,10 For nonlinear systems, a classical Lyapunov method to exponential stability of an equilibrium point requires that there exists a Lyapunov function with some basic properties. Indeed, the exponential stability of an equilibrium point is equivalent to the existence of a Lyapunov function for nonlinear systems. In Aeyels and Peuteman, ¹¹ the exponential stability has been proved for nonlinear time-varying systems; what’s more, the negative definiteness of the derivative of Lyapunov function was not required. However, in order to get such stability of an equilibrium point for nonlinear time-varying systems, a sequence of time should be satisfied with the descent property of the positive definite functions. Furthermore, the bounded positive definite function should be given. Ngoc ¹² proposed several explicit criteria for the global exponential stability of nonlinear systems with time-varying delay. Moreover, the advantaged criteria can be easy to investigate and analyze the exponential stability of equilibriums for the complex nonlinear systems. According to the multiple local quadratic Lyapunov functions, Pettersson and Lennartson ¹³ have developed an exponential stability for nonlinear autonomous systems. Indeed, the sufficient conditions of stability were derived by linear matrix inequalities (LMIs). Furthermore, the overall Lyapunov functions were allowed to be discontinuous at the states of nonlinear autonomous systems where the trajectories traversed from one local domain to another. However, according to the uncertainty or noise, the discontinuous property of the states of nonlinear autonomous systems might lead to chattering phenomena for actual applications. To cope with this difficulty, Shen and Wang ¹⁴ developed results on the robustness of global exponential stability of nonlinear systems with random disturbances and time delays. Nevertheless, the controlled nonlinear systems were not considered in Shen and Wang. ¹⁴ Recently, problems of necessary and sufficient conditions of nonlinear systems have attracted much attention from researchers.^15–17 Bernstein ¹⁸ developed sufficient conditions for asymptotic stability and optimality of nonlinear systems. Combined Hamilton–Jacobi–Bellman equation (HJBE) and Lyapunov theorem, the asymptotic stability and optimality were guaranteed for nonlinear systems. According to the basic idea of the results of Bernstein, ¹⁸ the finite-time stable and optimal feedback controllers were proposed and investigated for nonlinear systems.^19–21 The most important features of HJBE are as follows. First, the theoretical solution of the HJBE is obviously showed to be a Lyapunov-like function for the closed-loop system which satisfies a differential inequality involving an accelerator, and hence, guaranteeing both rapid exponential stability and optimality. Second, to overcome the complexity in solving the theoretical solution of HJBE, a family of rapidly exponentially stabilizing controllers are parameterized so as to minimize some cost functional. Therefore, owing to different parameters in the Lyapunov-like function and the cost functional, the developed framework could be used to achieve a class of rapidly exponentially stabilizing controllers that could satisfy closed-loop system. However, to the best of our knowledge, there are not many sufficient conditions for rapid exponential stability of nonlinear dynamical systems through HJBE and rapid exponential Lyapunov theorem. In general, it is difficult to construct optimal feedback controllers for nonlinear systems. Recently, Sui et al.^22–25 developed, investigated, and analyzed the non-triangular stochastic nonlinear systems via adaptive fuzzy control theory. However, the nonlinear system is considered and investigated in this article and the uncertain stochastic nonlinear systems with non-triangular form is investigated and analyzed in Sui et al.^22–24 Furthermore, it is a keynote step which is the criterion of stochastically semiglobal finite-time stable for the uncertain stochastic nonlinear systems with non-triangular form. In addition, the sufficient conditions of rapidly exponentially stabilizing are developed and analyzed using a variant of continuous Lyapunov functions and HJBE for nonlinear systems in this article. To reduce the complexity of calculation by choosing the quadratic Lyapunov function, the proposed control method is used solve the tracking problem in non-triangular stochastic nonlinear system under a reasonable assumption in Sui et al. ²² However, how to construct the finite-time controller is a main difficulty based on the semiglobal finite-time stable in probability.

The rest of this article is organized as follows: we will extend the framework proposed in Bernstein ¹⁸ and Haddad and L’Afflitto ¹⁹ to achieve the problem of rapid exponential stability for nonlinear systems. Section “Notations and definitions” reviews the relative notations and definitions of nonlinear systems. Moreover, the sufficient conditions of rapid exponential stability have been developed using continuous differential Lyapunov function with performance functional for nonlinear systems. Section “A new rapid exponential stability for controlled nonlinear system” solves the optimal controller of controlled nonlinear systems via HJBE. Sufficient conditions for optimality are developed for the controlled nonlinear systems. Furthermore, according to the rapid exponential Lyapunov function, the rapid exponential stability of controlled nonlinear systems is proposed in this section. In section “A rapid exponential stability for controlled nonlinear affine system,” an optimal exponential stability of nonlinear affine systems is considered and sufficient conditions based on the HJBE and Lyapunov theorem are utilized to obtain an optimal feedback control law. Due to the HJBE and Lyapunov theorem, a rapid exponential Lyapunov function guarantees both rapid exponential stability and optimality for the nonlinear controlled affine systems. The spacecraft model with one axis of symmetry and 2-degree-of-freedom manipulator will be illustrated and investigated in section “Simulations.” Finally, section “Conclusion and future works” describes the conclusions and future works for nonlinear systems. Furthermore, the proposed framework may be extended to address rapid exponential controllers for manipulator systems, ²⁶ path planning,^27,28 bipedal walking systems,^29,30 and mobile robot systems. ³¹

Before ending this introductory section, we highlight the main contributions of this article in the following facts. There are three important contributions to this article. The first one is a rigorous proof of rapid exponential stability for nonlinear systems. Moreover, some simply sufficient conditions are developed for controlled nonlinear systems through HJBE and Lyapunov theorem. The second contribution is an optimal feedback controller for the nonlinear controlled affine systems based on a novel rapid exponential Lyapunov function with performance functional. In addition, a nonlinear controlled affine system illustrates and verifies the proposed methodology. Last, the researches on the aforementioned problems for the spacecraft model with one axis of symmetry and 2-degree-of-freedom manipulator via HJBE and Lyapunov theorem are scarce. Therefore, it motivates our work in this study.

Notations and definitions

To develop the results in this context, it is necessary to review the notations and definitions for rapid exponential stabilization of nonlinear systems. $\mathbb{R}$ represents the set of real numbers, ${\mathbb{R}}_{+}$ means the set of positive real numbers, ${\mathbb{R}}_{+0}={\mathbb{R}}_{+}\cup \{0\}$ denotes the set of nonnegative real numbers. ${\mathbb{R}}^n$ represents the set of real column vectors, ${\mathbb{R}}^{n\times m}$ denotes the set of real matrices with the dimension is $n\times m$ , and $\stackrel{·}{V}(\chi )=(\partial V/\partial \chi )\stackrel{·}{\chi }$ for the Fréchet derivative of $V$ at $\chi$ , ∥·∥ means the Euclidean vector norm, $A^{\mathrm{T}}$ denotes the transpose of the matrix $A$ , and $I$ denotes the identity matrix.

Consider the following nonlinear systems

$$\stackrel{·}{\chi }(\tau )=F(\chi (\tau )),\chi (0)={\chi }_0,\tau \ge 0$$

(1)

where $\chi (\tau )\in \mathbb{D}\subseteq {\mathbb{R}}^n$ is a state vector and $F(·)$ is a continuous differential function on $\mathbb{D}$ . $\tau \in [0,+\infty )$ is the maximal interval of existence of a solution $\chi (\tau )$ of the nonlinear system (1). Moreover, $\mathbb{D}$ is an open set with $0\in \mathbb{D},$ in addition, let $F(0)=0$ .

Definition 1 (The solution of nonlinear systems)

A continuously differentiable function $\chi :[0,+\infty )\to \mathbb{D}$ is said to be the solution of nonlinear system (1) on the internal $[0,+\infty )$ if the continuously differentiable function $\chi (·)$ satisfies nonlinear systems (1) for all $\tau \in [0,+\infty )$ . ²²

Definition 2 (Rapidly exponential stable for nonlinear systems)

Consider nonlinear system (1), the zero solution $\chi (\tau )=0$ is rapidly exponentially stable for nonlinear systems (1) if there exist three positive constants as follows ²²

$$\delta >0,M>0,\lambda >0,0<ϵ<1$$

such that the corresponding solution $\chi (\tau )$ of equation (1) satisfies

$$\parallel \chi (\tau )\parallel \le M{e}^{-\frac{\lambda \tau }{ϵ}}\parallel \chi ({\tau }_0)\parallel ,\forall \tau \ge {\tau }_0$$

where $\chi ({\tau }_0)={\chi }_0,\forall {\tau }_0\in {\mathbb{R}}_{+0}$ , and $\parallel {\chi }_0\parallel \le \delta$ .

Definition 3 (The continuous-time part of hybrid systems)

For the nonlinear system (1), a continuously differentiable function $V_{ϵ}:\mathbb{D}\to \mathbb{R}$ with parameter $ϵ$ is a rapidly exponential stable Lyapunov function if there exist three positive constants ⁹

$$c_1>0,c_2>0,c_3>0$$

and satisfy $0<ϵ<1$ and $\forall \chi \in \mathbb{D}$ and $\chi \ne 0$

$$c_1\parallel \chi {\parallel }^2\le V_{ϵ}(\chi )\le \frac{c_2}{{ϵ}^2}\parallel \chi {\parallel }^2$$

(2)

$${\stackrel{·}{V}}_{ϵ}(\chi )f(\chi )\le -\frac{c_3}{ϵ}{V}_{ϵ}(\chi )<0$$

(3)

According to Definitions 1–3, the sufficient conditions will be proposed for rapid exponential stability of nonlinear system (1).

Theorem 1

Consider nonlinear system

$$\stackrel{·}{\chi }(\tau )=F(\chi (\tau )),\chi (0)={\chi }_0,\tau \ge 0$$

(4)

with performance functional

$$\mathcal{J}({\chi }_0)=\underset{0}{\overset{\infty }{\int }}\mathrm{\Phi }(\chi (\tau ))d\tau$$

Moreover, given that there exists a continuously differentiable function $V_{ϵ}:\mathbb{D}\to \mathbb{R}$ satisfies

$$V_{ϵ}(0)=0$$

(5)

$$c_1\parallel \chi {\parallel }^2\le V_{ϵ}(\chi )\le \frac{c_2}{{ϵ}^2}\parallel \chi {\parallel }^2,\chi \in \mathbb{D},\chi \ne 0$$

(6)

$${\stackrel{·}{V}}_{ϵ}(\chi )F(\chi )\le -\frac{c_3}{ϵ}{V}_{ϵ}(\chi )<0,\chi \in \mathbb{D},\chi \ne 0$$

(7)

$$\mathrm{\Phi }(\chi )+{\stackrel{·}{V}}_{ϵ}(\chi )F(\chi )=0,\chi \in \mathbb{D}$$

(8)

where $c_1>0,c_2>0,c_3>0$ are three positive real numbers, $\mathbb{D}\subseteq {\mathbb{R}}^n$ is a neighborhood of the origin; therefore, the zero solution $\chi (\tau )=0,\tau \ge 0$ of nonlinear systems is rapidly exponentially stabilizing. Furthermore

$$\mathcal{J}({\chi }_0)=V_{ϵ}({\chi }_0),{\chi }_0\in \mathbb{D}$$

(9)

Proof

Due to Definitions 1–3 and the assumptions, let $\chi (\tau ),\tau \ge 0$ satisfy equation (4), then

$${\stackrel{·}{V}}_{ϵ}(\chi (\tau ))=\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }={\stackrel{·}{V}}_{ϵ}(\chi (\tau ))F(\chi (\tau )),\tau \ge 0$$

Consequently, it follows from equation (7) that

$${\stackrel{·}{V}}_{ϵ}(\chi (\tau ))\le -\frac{c_3}{ϵ}{V}_{ϵ}(\chi (\tau ))<0,\tau \ge 0,\chi (\tau )\ne 0$$

(10)

Hence, combining equations (5), (6), and (10), it can be seen that $V_{ϵ}(·)$ is a rapidly exponential stable Lyapunov function for the nonlinear system (4), which proves exponential stability of the solution $\chi (\tau )=0,\tau \ge 0$ . As a matter of the fact, according to the inequality (10), the inequality is generalized as $V_{ϵ}(\chi (\tau ))\le e^{-\frac{c_3\tau }{ϵ}}{V}_{ϵ}(\chi (0)).$ Furthermore, since the following condition is

$$c_1\parallel \chi {\parallel }^2\le V_{ϵ}(\chi )\le \frac{c_2}{{ϵ}^2}\parallel \chi {\parallel }^2$$

the following inequalities can be computed as follows

$$c_1\parallel \chi (\tau ){\parallel }^2\le V_{ϵ}(\chi (\tau ))\le e^{-\frac{c_3\tau }{ϵ}}\parallel \chi (0){\parallel }^2$$

that is

$$\parallel \chi (\tau )\parallel \le \frac{1}{ϵ}\sqrt{\frac{c_2}{c_1}}{e}^{-\frac{c_3\tau }{2ϵ}}\parallel \chi (0)\parallel$$

Therefore, $\forall {\chi }_0\in \mathbb{D}$ , let $\tau \to \infty$ , then $\chi (\tau )\to 0$ is rapidly exponentially stable.

Moreover, the following equation can be obtained as

$$-{\stackrel{·}{V}}_{ϵ}(\chi (\tau ))+{\stackrel{·}{V}}_{ϵ}(\chi (\tau ))F(\chi (\tau ))=0,\tau \ge 0$$

and based on equation (8), the following equation can be described as $\mathrm{\Phi }(\chi (\tau ))=-{\stackrel{·}{V}}_{\epsilon }(\chi (\tau )).$

Integrating two sides of the above equation over the interval $[0,\tau )$ leads to

$$\underset{0}{\overset{\tau }{\int }}\mathrm{\Phi }(\chi (s))\mathrm{ds}\le -e^{-\frac{c_3\tau }{ϵ}}{V}_{ϵ}({\chi }_0)+V_{ϵ}({\chi }_0)$$

Noting that $-e^{-\frac{c_3\tau }{ϵ}}{V}_{ϵ}(\chi (0))\to 0$ as $\tau \to \infty$ , ${\chi }_0\in \mathbb{D}$ . Therefore, $\mathcal{J}({\chi }_0)=V_{ϵ}({\chi }_0),{\chi }_0\in \mathbb{D}.$ The conclusion is true.

A new rapid exponential stability for controlled nonlinear system

The objective of this subsection is to address sufficient conditions for establishing a novel rapid exponential stability for a controlled nonlinear system. Furthermore, combined HJBE and rapid exponential Lyapunov function, the sufficient conditions of optimality are proposed for controlled nonlinear system (11).

Considering the following controlled nonlinear system

$$\stackrel{·}{\chi }(\tau )=F(\chi (\tau ),\mu (\tau )),\chi (0)={\chi }_0,\tau \ge 0$$

(11)

where, for all $\tau \ge 0,\chi (\tau )\in \mathbb{D}\subseteq {\mathbb{R}}^n$ is a controlled state, and $\mathbb{D}$ is an open set with $0\in \mathbb{D},\mu (\tau )\in \mathbb{U}\subseteq {\mathbb{R}}^m$ is a control signal with $0\in \mathbb{U}$ , and $\mathbb{U}$ is the set of admissible control values for $\mu$ . That is to say, the definition of admissible control input $\mu (•)$ should satisfy the admissible set $\mathbb{U}\subseteq {\mathbb{R}}^m$ , that is, $\stackrel{·}{\chi }(\tau )=F(\chi (\tau ),\mu (\tau ))$ . $F:\mathbb{D}\times \mathbb{U}\to {\mathbb{R}}^n$ is assumed to be continuously differential in $\chi$ and $\mu$ . In addition, given that $F(0,0)=0$ . Figure 1 is the schematic diagram of closed-loop controlled nonlinear systems.

Figure 1.

The schematic diagram for closed-loop controlled nonlinear systems.

Definition 4

Consider the controlled nonlinear system (11), the feedback controller $\mu =\varphi (\chi )$ is rapidly exponential stabilizing if the following closed-loop system (12) is rapidly exponential stable

$$\stackrel{·}{\chi }(\tau )=F(\chi (\tau ),\varphi (\chi (\tau ))),\chi (0)={\chi }_0,\tau \ge 0$$

(12)

Theorem 2

Consider the controlled nonlinear system (11) with the following performance functional

$$\mathcal{J}({\chi }_0,\mu (·))=\underset{0}{\overset{\infty }{\int }}\mathrm{\Phi }(\chi (\tau ),\mu (\tau ))d\tau$$

(13)

where $\mu (·)$ is a control signal. Furthermore, given that there exists a continuously differentiable function $V_{ϵ}:\mathbb{D}\to \mathbb{R}$ and a continuous controller $\varphi :\mathbb{D}\to \mathbb{U}$ such that

$$\varphi (0)=0$$

(14)

$$V_{ϵ}(0)=0$$

(15)

$$c_1\parallel \chi {\parallel }^2\le V_{ϵ}(\chi )\le \frac{c_2}{{ϵ}^2}\parallel \chi {\parallel }^2,\chi \in \mathbb{D},\chi \ne 0$$

(16)

$${\stackrel{·}{V}}_{ϵ}(\chi )F(\chi ,\varphi (\chi ))\le -\frac{c_3}{ϵ}{V}_{ϵ}(\chi )<0,\chi \in \mathbb{D},\chi \ne 0$$

(17)

$$\mathrm{\Phi }(\chi ,\varphi (\chi ))+{\stackrel{·}{V}}_{ϵ}(\chi )F(\chi ,\varphi (\chi ))=0,\chi \in \mathbb{D}$$

(18)

where $c_1>0,c_2>0,c_3>0$ are three positive real numbers, $\mathbb{D}\subseteq {\mathbb{R}}^n$ is a neighborhood of the origin. Therefore, the zero solution of controlled nonlinear system (11) is rapidly exponentially stable with the optimal feedback controller $\mu =\varphi (\chi )$ . Furthermore

$$\mathcal{J}({\chi }_0,\varphi (\chi (·)))=V_{ϵ}({\chi }_0),{\chi }_0\in \mathbb{D}$$

Proof

According to Definition 4 and the assumptions, let $\chi (\tau ),\tau \ge 0$ satisfy (11), then

$$\frac{d{V}_{ϵ}(\chi (t))}{d\tau }={\stackrel{·}{V}}_{ϵ}(\chi (\tau ))F(x(\tau ),\varphi (\chi (\tau ))),\tau \ge 0$$

therefore, it follows from equation (17), the following inequality can be described as

$${\stackrel{·}{V}}_{ϵ}(\chi (\tau ))F(\chi (\tau ),\varphi (\chi (\tau )))\le -\frac{c_3}{ϵ}{V}_{ϵ}(\chi (\tau )),\chi \ne 0,\tau \ge 0$$

Hence

$$V_{ϵ}(\chi (\tau ))\le e^{-\frac{c_3}{ϵ}\tau }{V}_{ϵ}(\chi (0))$$

Based on condition (16), the following inequalities can be obtained as

$$c_1\parallel \chi (\tau ){\parallel }^2\le V_{ϵ}(\chi (\tau ))\le e^{-\frac{c_3}{ϵ}\tau }\frac{c_2}{{ϵ}^2}\parallel \chi (0){\parallel }^2$$

then

$$\parallel \chi (\tau ){\parallel }^2\le e^{-\frac{c_3}{ϵ}\tau }\frac{c_2}{c_1{ϵ}^2}\parallel \chi (0){\parallel }^2$$

Thereby

$$\parallel \chi (\tau )\parallel \le \frac{1}{ϵ}\sqrt{\frac{c_2}{c_1}}{e}^{-\frac{c_3}{2ϵ}\tau }\parallel \chi (0)\parallel$$

The fact that $\underset{\tau \to \infty }{lim}\chi (\tau )=0,$ that is, the zero solution is rapidly exponentially stable to (11).

In addition, given $\mu (·)=\varphi (\chi (·))$ and $\chi (\tau ),\tau \ge 0$ satisfies equation (11). Owing to the following equation

$$0=-\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }+{\stackrel{·}{V}}_{ϵ}(\chi (\tau ))F(\chi (\tau ),\varphi (\chi (\tau ))),\tau \ge 0$$

It follows from equation (18) that it is easy to compute the following equation and can be generalized as follows

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Phi }(\chi (\tau ),\varphi (\chi (\tau ))=-\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }+\mathrm{\Phi }(\chi (\tau ),\varphi (\chi (\tau ))}\\ \multicolumn{1}{c}{+{\stackrel{·}{V}}_{ϵ}(\chi (\tau ))F(\chi (\tau ),\varphi (\chi (\tau ))}\\ =-\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau },\tau \ge 0\hfill \end{array}$$

(19)

Now, integrating equation (19) over the interval $[0,\tau ]$ yields

$$\begin{array}{c}\multicolumn{1}{c}{\underset{0}{\overset{\tau }{\int }}\mathrm{\Phi }(\chi (s),\varphi (\chi (s)))\mathrm{ds}=\underset{0}{\overset{\tau }{\int }}-{\stackrel{·}{V}}_{ϵ}(\chi (s))\mathrm{ds}}\\ =-V_{ϵ}(\chi (\tau ))+V_{ϵ}({\chi }_0)\hfill \\ \le -e^{-\frac{c_3}{ϵ}\tau }{V}_{ϵ}({\chi }_0)+V_{ϵ}({\chi }_0)\hfill \end{array}$$

(20)

Noting equation (15) and assuming $\tau \to \infty ,$ it can be directly computed that

$$\underset{0}{\overset{\infty }{\int }}\mathrm{\Phi }(\chi (s),\varphi (\chi (s)))\mathrm{ds}\le V_{ϵ}({\chi }_0)$$

Therefore

$$\mathcal{J}({\chi }_0,\varphi (\chi (·))=V_{ϵ}({\chi }_0),{\chi }_0\in \mathbb{D}$$

The conclusion is true.

Definition 5

The set of rapidly exponential stabilizing controllers $\psi ({\chi }_0)$ for each initial condition ${\chi }_0\in D$ , that is

$$\psi ({\chi }_0)=\{\mu (·):\mu (·)\mathrm{is}\mathrm{admissible}\mathrm{control}\mathrm{inputs}\}$$

(21)

Theorem 3

Consider the controlled nonlinear system (11) with performance functional

$$\mathcal{J}({\chi }_0,\mu (·))=\underset{0}{\overset{\infty }{\int }}\mathrm{\Phi }(\chi (\tau ),\mu (\tau ))\mathrm{dt}$$

Moreover, assume that there exists a continuous differential function $V_{ϵ}:\mathbb{D}\to \mathbb{R}$ and a continuous controller $\varphi :\mathbb{D}\to \mathbb{U}$ such that

$$\varphi (0)=0$$

(22)

$$V_{ϵ}(0)=0$$

(23)

$$c_1\parallel \chi {\parallel }^2\le V_{ϵ}(\chi )\le \frac{c_2}{{ϵ}^2}\parallel \chi {\parallel }^2,\chi \in \mathbb{D},\chi \ne 0$$

(24)

$${\stackrel{·}{V}}_{ϵ}(\chi )F(\chi ,\varphi (\chi ))\le -\frac{c_3}{ϵ}{V}_{ϵ}(\chi )<0,\chi \in \mathbb{D},\chi \ne 0$$

(25)

$$\mathrm{\Phi }(\chi ,\varphi (\chi ))+{\stackrel{·}{V}}_{ϵ}(\chi )(\chi )F(\chi ,\varphi (\chi ))=0,\chi \in \mathbb{D}$$

(26)

$$\mathrm{\Phi }(\chi ,\mu )+{\stackrel{·}{V}}_{ϵ}(\chi )F(\chi ,\mu )\ge 0,(\chi ,\mu )\in \mathbb{D}\times \mathbb{U}$$

(27)

Furthermore, if the initial state ${\chi }_0\in \mathbb{D}$ , the optimal feedback controller $\mu (·)=\varphi (\chi (·))$ minimizes $\mathcal{J}({\chi }_0,\mu (·))$ in the sense that

$$\mathcal{J}({\chi }_0,\varphi (·))=\underset{\mu (·)\in \psi ({\chi }_0)}{min}\mathcal{J}({\chi }_0,\mu (·))$$

(28)

Proof

To prove equation (28), let $\mu (·)\in \psi ({\chi }_0)$ and $\chi (·)$ be the solution of controlled nonlinear system (11), the following equation can be obtained as

$$\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }={\stackrel{·}{V}}_{ϵ}(\chi (\tau ))F(\chi (\tau ),\mu (\tau ))$$

therefore

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Phi }(\chi (\tau ),\mu (\tau ))={\stackrel{·}{V}}_{ϵ}(\chi (\tau ))F(\chi (\tau ),\mu (\tau ))-\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }}\\ \multicolumn{1}{c}{+\mathrm{\Phi }(\chi (\tau ),\mu (\tau ))}\end{array}$$

(29)

Integrating equation (29) over the interval $[0,\infty )$ yields

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{J}({\chi }_0,\mu (·))=\underset{0}{\overset{\infty }{\int }}\mathrm{\Phi }(\chi (\tau ),\mu (\tau ))d\tau }\\ =\underset{0}{\overset{\infty }{\int }}[-\frac{{\mathrm{dV}}_{ϵ}(\chi (\tau ))}{d\tau }d\tau +{\stackrel{·}{V}}_{ϵ}(\chi (\tau ))F(\chi (\tau ),\hfill \\ \multicolumn{1}{c}{\mu (\tau ))+\mathrm{\Phi }(\chi (\tau ),\mu (\tau ))]d\tau }\end{array}$$

According to equation (27) and Theorem 2

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{J}({\chi }_0,\mu (·))\ge \underset{0}{\overset{\infty }{\int }}-\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }d\tau }\\ =-V_{ϵ}(\chi (\tau )){|}_{\tau \to \infty }+V_{ϵ}({\chi }_0)\hfill \\ =V_{ϵ}({\chi }_0)\hfill \end{array}$$

Therefore, $\mathcal{J}({\chi }_0,\mu (·))\ge \mathcal{J}({\chi }_0,\varphi (·))$ which yields equation (28).

Remark

If $\mathbb{D}={\mathbb{R}}^n,\mathbb{U}={\mathbb{R}}^m,V_{ϵ}(·)$ is a continuous differential function and radially unbounded Lyapunov function and conditions (22)–(27) hold on ${\mathbb{R}}^n\backslash \{0\}$ , then the controlled nonlinear system (12) is globally exponentially stable.

A rapid exponential stability for controlled nonlinear affine system

Motivated by section “A new rapid exponential stability for controlled nonlinear system,” we return to controlled nonlinear affine systems. Analogous to the case of controlled nonlinear systems in equation (11), we will specialize the results of section “A new rapid exponential stability for controlled nonlinear system” to the following controlled nonlinear affine system (30). The principal objective of this section is to develop the sufficient conditions of the rapid exponential stability for a controlled nonlinear affine system.

On one hand, considering the following controlled nonlinear affine system

$$\stackrel{·}{\chi }(\tau )=F(\chi (\tau ))+\mathrm{\Psi }(\chi (\tau ))\mu (\tau ),\chi (0)={\chi }_0,\tau \ge 0$$

(30)

where for all $\tau \ge 0,\chi (\tau )\in \mathbb{D}\subseteq {\mathbb{R}}^n$ is defined as in equation (11), $\mathbb{D}$ is an open set with $0\in \mathbb{D},\mu (\tau )\in \mathbb{U}\subseteq {\mathbb{R}}^m$ is control signal with $0\in \mathbb{U}$ . $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$ and $\mathrm{\Psi }:{\mathbb{R}}^n\to {\mathbb{R}}^{n\times m}$ are continuous differential functions in $\chi$ and $\mu$ . In addition, $F(0)=0$ . Assume that the performance functional $\mathrm{\Phi }(\chi ,\mu )$ has the form as

$$\mathrm{\Phi }(\chi ,\mu )={\chi }^{\mathrm{T}}{\mathrm{\Lambda }}_1(\chi )\chi +{\mu }^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )\mu .$$

Denoting the right hand of the function (30) as follows

$$F(\chi ,\mu )=F(\chi )+\mathrm{\Psi }(\chi )\mu (\tau )$$

where $(\chi ,\mu )\in {\mathbb{R}}^n\times {\mathbb{R}}^m$ , ${\mathrm{\Lambda }}_1(\chi )$ , and ${\mathrm{\Lambda }}_2(\chi )$ are positive matrices.

On the other hand, the performance functional of controlled nonlinear system is as follows

$$\mathcal{J}({\chi }_0,\mu (·))=\underset{0}{\overset{\infty }{\int }}{\chi }^{\mathrm{T}}{\mathrm{\Lambda }}_1(\chi )\chi +{\mu }^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )\mu d\tau$$

(31)

To obtain the rapid exponential stability of controlled nonlinear affine system, the following assumptions and definitions will be adopted in this subsection. Given $\mathrm{\Phi }:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$ and ${\stackrel{·}{V}}_{ϵ}(\chi )\in {\mathbb{R}}^n,$ and defining

$$\mathscr{H}(\chi ,\mu ,{\stackrel{.}{V}}_{ϵ}(\chi ))=\mathrm{\Phi }(\chi ,\mu )+{\stackrel{.}{V}}_{ϵ}{(\chi )}^{{\rm T}}Ϝ(\chi ,\mu )$$

(32)

Theorem 4

Consider the controlled nonlinear affine system (30) with performance integrands as follows

$$\mathcal{J}({\chi }_0,\mu (·))=\underset{0}{\overset{\infty }{\int }}{\chi }^{\mathrm{T}}{\mathrm{\Lambda }}_1(\chi )\chi +{\mu }^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )\mu d\tau$$

(33)

Moreover, assume that there exists a continuously differentiable, radially unbounded function $V_{ϵ}:{ℝ}^n\to ℝ$ and positive constants $c_1>0,c_2>0,c_3>0$ , and $0<ϵ<1$ such that

$$V_{ϵ}(0)=0$$

(34)

$$c_1\parallel \chi {\parallel }^2\le V_{ϵ}(\chi )\le \frac{c_2}{{ϵ}^2}\parallel \chi {\parallel }^2,\chi \in \mathbb{D},\chi \ne 0$$

(35)

$${\stackrel{·}{V}}_{ϵ}(\chi )[F(\chi )-\frac{1}{2}\mathrm{\Psi }(\chi ){\mathrm{\Lambda }}_2^{-1}(\chi ){\mathrm{\Psi }}^{\mathrm{T}}(\chi ){\stackrel{·}{V}}_{ϵ}{(\chi )}^{\mathrm{T}}]\le -\frac{c_3}{ϵ}{V}_{ϵ}(\chi )$$

(36)

$$A_1(\chi )=B_1(\chi )$$

(37)

where

$$A_1(\chi )={\chi }^{\mathrm{T}}{\mathrm{\Lambda }}_1(\chi )\chi +\frac{1}{4}\mathrm{\Gamma }(\chi ){\mathrm{\Lambda }}_2^{-1}(\chi )\mathrm{\Gamma }(\chi {)}^{\mathrm{T}}$$

and

$$B_1(\chi )=-{\stackrel{·}{V}}_{ϵ}(\chi )[F(\chi )-\frac{1}{2}\mathrm{\Psi }(\chi ){\mathrm{\Lambda }}_2^{-1}(\chi )\mathrm{\Gamma }{(\chi )}^{\mathrm{T}}]$$

and

$$\mathrm{\Gamma }(\chi )={\stackrel{·}{V}}_{ϵ}(\chi )\mathrm{\Psi }(\chi )$$

Furthermore, with the optimal feedback controller of this subsection

$$\mu =\varphi (\chi )=-\frac{1}{2}{\mathrm{\Lambda }}_2^{-1}(\chi )\mathrm{\Gamma }(\chi {)}^{\mathrm{T}}$$

(38)

the zero solution of controlled nonlinear affine system (30) is globally rapidly exponential stable. Moreover

$$\mathcal{J}({\chi }_0,\varphi (\chi (·)))=V_{ϵ}({\chi }_0)$$

Finally, if the initial state ${\chi }_0\in \mathbb{D}$ satisfies controlled nonlinear affine system, the feedback controller $\mu (·)=\varphi (\chi (·))$ minimizes $J({\chi }_0,\mu (·))$ in the sense that

$$\mathfrak{J}({\chi }_0,\varphi (\cdot ))=\underset{\mu (\cdot )\in \psi ({\chi }_0)}{\min }\mathcal{J}({\chi }_0,\mu (\cdot ))$$

(39)

Proof

According to the assumptions in this article, let $\chi (\tau ),\tau \ge 0$ satisfy (30) and the optimal feedback controller $\varphi (\chi )=-(1/2){\mathrm{\Lambda }}_2^{-1}(\chi )[{\stackrel{·}{V}}_{ϵ}(\chi )\mathrm{\Psi }(\chi ){]}^{\mathrm{T}},$ the following equation can be obtained as follows

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }={\stackrel{·}{V}}_{ϵ}(\chi (\tau ))[F(\chi (\tau ))+\mathrm{\Psi }(\chi (\tau ))\mu (\tau )]}\\ ={\stackrel{·}{V}}_{ϵ}(\chi (\tau ))[F(\chi (\tau ))-\frac{1}{2}{\mathrm{\Lambda }}_2^{-1}(\chi )[{\stackrel{·}{V}}_{ϵ}(\chi )\mathrm{\Psi }(\chi {)]}^{\mathrm{T}}]\hfill \\ \le -\frac{c_3}{ϵ}{V}_{ϵ}(\chi (\tau ))\hfill \end{array}$$

Due to the above inequality, the following inequality can be directly computed as follows

$$V_{ϵ}(\chi (\tau ))\le e^{\frac{c_3\tau }{ϵ}}{V}_{ϵ}(\chi (0))$$

According to equation (35), the following inequality can be generalized as follows

$$c_1\parallel \chi (\tau ){\parallel }^2\le V_{ϵ}(\chi (\tau ))\le e^{\frac{c_3\tau }{ϵ}}\frac{c_2}{{ϵ}^2}\parallel \chi (0){\parallel }^2$$

then

$$\parallel \chi (\tau )\parallel \le \frac{1}{ϵ}\sqrt{\frac{c_2}{c_1}}{e}^{-\frac{c_3\tau }{2ϵ}}\parallel \chi (0)\parallel$$

Therefore, the zero solution of controlled nonlinear affine system is rapidly exponentially stable.

Let $\varphi (\chi )=-(1/2){\mathrm{\Lambda }}_2^{-1}(\chi )\mathrm{\Gamma }(\chi {)}^{\mathrm{T}}$ and $\chi (\tau ),\tau \ge 0$ satisfy equation (30). Then, since

$$\begin{array}{c}-\frac{d{V}_{ϵ}(\chi (\tau ))}{\mathrm{dt}}+{\stackrel{·}{V}}_{ϵ}(\chi (\tau ))[F(\chi (\tau ))+\mathrm{\Psi }(\chi (\tau ))\mu ]\hfill \\ ={\stackrel{·}{V}}_{ϵ}(\chi (\tau ))[F(\chi (\tau ))-\frac{1}{2}\mathrm{\Psi }(\chi (\tau )){\mathrm{\Lambda }}_2^{-1}(\chi )\mathrm{\Gamma }{(\chi )}^{\mathrm{T}}]\hfill \\ \multicolumn{1}{c}{-\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }}\\ =0\hfill \end{array}$$

it follows from equation (37) that

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Phi }(\chi (\tau ),\varphi (\chi ))={\chi }^{\mathrm{T}}{R}_1(\chi )\chi +\frac{1}{4}\mathrm{\Gamma }(\chi ){\mathrm{\Lambda }}_2^{-1}(\chi )\mathrm{\Gamma }(\chi {)}^{\mathrm{T}}}\\ \multicolumn{1}{c}{-\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }+{\stackrel{·}{V}}_{ϵ}(\chi (\tau ))}\\ \multicolumn{1}{c}{[F(\chi (\tau ))-\frac{1}{2}\mathrm{\Psi }(\chi (\tau )){\mathrm{\Lambda }}_2^{-1}(\chi )\mathrm{\Gamma }{(\chi )}^{\mathrm{T}}]}\\ =-\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }\hfill \end{array}$$

(40)

Now, integrating equation (40) over the interval $[0,\infty )$ yields

$$\begin{array}{c}\multicolumn{1}{c}{\underset{0}{\overset{\infty }{\int }}\mathrm{\Phi }(\chi (\tau ),\varphi (\chi (\tau )))=-\underset{0}{\overset{\infty }{\int }}\frac{d{V}_{ϵ}(\chi (\tau ))}{d\tau }d\tau }\\ =-V_{ϵ}(\chi (\tau )){|}_{\tau \to \infty }+V_{ϵ}(\chi (0))\hfill \\ \le -\underset{\tau \to \infty }{lim}{e}^{-\frac{c_3\tau }{ϵ}}{V}_{ϵ}(\chi (0))+V_{ϵ}(\chi (0))\hfill \end{array}$$

Therefore

$$\mathcal{J}({\chi }_0,\varphi (\chi (·)))=V_{ϵ}({\chi }_0)$$

Furthermore, since $V_{ϵ}(·)$ is a continuously differentiable function and $V_{ϵ}(0)=0,c_1\parallel \chi {\parallel }^2\le V_{ϵ}(\chi )\le c_2\parallel \chi {\parallel }^2.$ Moreover, $V_{ϵ}(0)$ is a local minimum value of $V_{ϵ}(·)$ ; therefore, it is can be easily derived as ${\stackrel{·}{V}}_{ϵ}(0)=0$ . Due to equation (38), the equation can be generalized as follows

$$\varphi (0)=-\frac{1}{2}{\mathrm{\Lambda }}_2^{-1}(0)[{\stackrel{·}{V}}_{ϵ}(0)\mathrm{\Psi }(0){]}^{\mathrm{T}}=0$$

Finally

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Phi }(\chi ,\mu )+{\stackrel{·}{V}}_{ϵ}(\chi )[F(\chi )+\mathrm{\Psi }(\chi )\mu ]={\mu }^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )\mu +{\stackrel{·}{V}}_{ϵ}(\chi )\mathrm{\Psi }(\chi )\mu -\varphi (\chi {)}^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )\varphi (\chi )-\mathrm{\Gamma }(\chi )\varphi (\chi )}\\ ={\mu }^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )\mu -\varphi (\chi {)}^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )\varphi (\chi )+\mathrm{\Gamma }(\chi )[\mu -\varphi (\chi )]\hfill \\ ={\mu }^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )\mu -\varphi (\chi {)}^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )\varphi (\chi )-2\varphi (\chi {)}^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )[\mu -\varphi (\chi )]\hfill \\ =[\mu -\varphi (\chi ){]}^{\mathrm{T}}{\mathrm{\Lambda }}_2(\chi )[\mu -\varphi (\chi )]\hfill \\ \ge 0\hfill \end{array}$$

where $\chi \in {\mathbb{R}}^n,{\mathrm{\Lambda }}_2(\chi )>0.$ The conclusion is true.

Simulations

In this subsection, some numerical results are investigated and analyzed through a spacecraft with one axis of symmetry model^1,19 and 2-degree-of-freedom manipulator. ²⁶ The simulation results report that the optimal feedback controller which is feasible and effective for nonlinear system.

The spacecraft with one axis of symmetry model

The model of spacecraft with one axis of symmetry is as follows

$$\{\begin{array}{c}{\stackrel{·}{\nu }}_1(\tau )={\pi }_{23}{\nu }_3{\nu }_2(\tau )+{\mu }_1(\tau ),{\nu }_1(0)={\nu }_{10},\tau \ge 0\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\nu }}_2(\tau )=-{\pi }_{23}{\nu }_3{\nu }_1(\tau )+{\mu }_2(\tau ),{\nu }_2(0)={\nu }_{20}}\end{array}$$

(41)

where ${\pi }_{23}=(({\pi }_2-{\pi }_3)/{\pi }_1)$ , ${\pi }_1,{\pi }_2,{\pi }_3$ are three principal moments of inertia for the spacecraft such that

$$0<{\pi }_1={\pi }_2<{\pi }_3$$

${\nu }_1:[0,\infty )\to R,{\nu }_2:[0,\infty )\to R$ , and ${\nu }_3\in R$ represent the angular velocity vectors with respect to a given inertial reference frame, and moreover, ${\mu }_1$ and ${\mu }_2$ are the control torques for the spacecraft.

Let $\mu =[{\mu }_1{\mu }_2{]}^{\mathrm{T}}=\varphi (\chi )$ and $\chi =[{\nu }_1{\nu }_2{]}^{\mathrm{T}}$ , then the following equations can be obtained as

$$\stackrel{·}{\chi }=\left(\begin{array}{cc}{\pi }_{23}{\nu }_3\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -{\pi }_{23}{\nu }_3\hfill \end{array}\right)\left(\begin{array}{c}{\nu }_1\hfill \\ \multicolumn{1}{c}{{\nu }_2}\end{array}\right)+\left(\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right)\left(\begin{array}{c}{\mu }_1\hfill \\ \multicolumn{1}{c}{{\mu }_2}\end{array}\right)$$

that is

$$\stackrel{·}{\chi }=\left(\begin{array}{c}{\pi }_{23}{\nu }_3{\nu }_2\hfill \\ \multicolumn{1}{c}{-{\pi }_{23}{\nu }_3{\nu }_1}\end{array}\right)+\left(\begin{array}{c}{\mu }_1\hfill \\ \multicolumn{1}{c}{{\mu }_2}\end{array}\right)=F(\chi )+I\mu$$

where $I$ is an unit matrix with two dimensions. Furthermore, let the Lyapunov function $V_{ϵ}(\chi )$ and performance functional $\mathrm{\Phi }(\chi ,\mu )$ are as follows $V_{ϵ}(\chi )=(1/2ϵ)\parallel \chi {\parallel }^4$ and $\mathrm{\Phi }(\chi ,\mu )=\parallel \chi {\parallel }^2+\parallel \mu {\parallel }^2.$ According to Theorem 4, the following optimal feedback controller can be described as

$$\begin{array}{c}\multicolumn{1}{c}{\varphi (\chi )=-\frac{1}{2}{\mathrm{\Lambda }}_2^{-1}(\chi )[{\stackrel{·}{V}}_{ϵ}(\chi )\mathrm{\Psi }(\chi ){]}^{\mathrm{T}}}\\ \multicolumn{1}{c}{=-[\frac{1}{ϵ}{\nu }_1\parallel \chi {\parallel }^2\frac{1}{ϵ}{\nu }_2\parallel \chi {\parallel }^2]}\end{array}$$

then

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{ϵ}(\chi )[F(\chi )-\frac{1}{2}\mathrm{\Psi }(\chi ){\mathrm{\Lambda }}_2(\chi {)}^{-1}\mathrm{\Gamma }(\chi {)}^{\mathrm{T}}]}\\ \multicolumn{1}{c}{={\stackrel{·}{V}}_{ϵ}(\chi )[F(\chi )-\frac{1}{2}\mathrm{\Psi }(\chi ){\mathrm{\Psi }}^{\mathrm{T}}(\chi ){\stackrel{·}{V}}_{ϵ}^{\mathrm{T}}(\chi )]}\\ =[\frac{1}{ϵ}{\nu }_1\parallel \chi {\parallel }^2\frac{1}{ϵ}{\nu }_2\parallel \chi {\parallel }^2][{\pi }_{23}{\nu }_3{\nu }_2-{\pi }_{23}{\nu }_3{\nu }_1{]}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{-\frac{1}{2ϵ}[2{\nu }_1\parallel \chi {\parallel }^{2}2{\nu }_2\parallel \chi {\parallel }^2]}\\ =-\frac{2}{ϵ}({\nu }_1^2+{\nu }_2^2)\parallel \chi {\parallel }^2\hfill \\ \le -\frac{4}{ϵ}{V}_{ϵ}(\chi )\hfill \end{array}$$

Therefore, according to the rapid exponential stable conditions of this article, it can be concluded that the zero solution of spacecraft system with one axis of symmetry (41) is rapidly exponentially stable.

To achieve numerical experiments, the parameters of principal moments of inertia and angular velocity are selected as follows

$${\pi }_1={\pi }_2=4\mathrm{kg}{\mathrm{m}}^2,{\pi }_3=20\mathrm{kg}{\mathrm{m}}^2,{\nu }_1=-2\mathrm{Hz}$$

and

$${\nu }_2=2\mathrm{Hz},{\nu }_3=1\mathrm{Hz}$$

In this subsection, the trajectories versus time with parameter $ϵ=0.001$ of the spacecraft systems are shown in Figure 2. Furthermore, the phase portrait with parameter $ϵ=0.001$ is shown in Figure 3. From Figures 2 and 3, it is easy can be seen that the rapid exponential controller is feasible and effective in this article. As shown in Figures 2 and 3, it means that the rapidly exponentially stabilizing controller can be utilized to nonlinear systems in real time. Furthermore, the proposed optimal feedback control technique might open a door to the performance improvement of the related applications, which is a major breakthrough in the optimal feedback control fields as well as in the research of real-time control problem solving. When parameter $ϵ=0.05$ , it is easy can be seen that Figures 3 and 5 represent the state trajectories and the phase portrait. From Figures 3 to 5, due to the different parameters, the convergent rate of state trajectories changes significantly for the spacecraft system. The parameters of this article is compared with that of the literature. ¹⁹ Figure 6 represents the trajectories versus time of the nonlinear controlled system with finite-time controller, and the phase portrait with finite-time controller for the angular velocity vectors of nonlinear systems is shown in Figure 7. As shown in Figure 6, it can be seen that the controlled nonlinear system is finite-time stable in 4.4717 s through finite-time controller in the literature. ¹⁹ However, according to the rapidly exponentially stabilizing Lyapunov function and optimal feedback controller, the nonlinear controlled system is exponentially convergent. Particularly, when parameter $ϵ=0.001$ , the controlled nonlinear system is exponentially stable in 0.5 s. In comparison with finite-time stabilizing controller, several numerical simulations are provided and illustrated to substantiate the feasibility and superiority of the developed rapid exponential controller for nonlinear systems.

Figure 2.

The trajectories versus time with parameter $ϵ$  = 0.001.

Figure 3.

The phase portrait with parameter $ϵ=0.001$ .

Figure 4.

The trajectories versus time with parameter $ϵ=0.05$ .

Figure 5.

The phase portrait with parameter $ϵ=0.05$ .

Figure 6.

The trajectories versus time with finite-time controller. ¹⁹

Figure 7.

The phase portrait with finite-time controller. ¹⁹

Furthermore, considering the large-scale disturbance problem of initial point as follows. In this context, according to the requirements of numerical experiments, the following parameters can be chosen as

$${\nu }_1=-9\mathrm{Hz},{\nu }_2=9\mathrm{Hz},{\nu }_3=4\mathrm{Hz},ϵ=0.005$$

Figures 8 and 9 represent the closed-loop system trajectories and phase portraits by the rapid exponential stable controller. As shown in Figures 8 and 9, it is easy to say that the system is exponentially stable in 1 s. Moreover, the parameter $ϵ$ will determine the convergent rate of nonlinear systems, that is, the smaller the parameter, the faster the convergent rate. In this research, from Figures 8 and 9, a rapid exponential stable controller can reject the large-scale disturbance of initial point. In this article, in this numerical example, the benchmark system with large-scale disturbance presented will be adopted in Haddad and L’Afflitto ¹⁹ for comparison. The simulation results of rapid exponential controller and finite-time controller for solving nonlinear system of equations (14) are shown in Figures 8 and 11. From Figures 8 and 9, it reports that the rapidly exponentially stabilizing controller can be adopted to nonlinear systems with large-scale disturbance in real time. Figures 10 and 11 represent the trajectories and phase portraits through a finite-time controller. However, as shown in Figures 10 and 11, the stable time is 11 s in the literature. ¹⁹ Therefore, due to a simple experimental validation, it can be concluded the usefulness of the rapid exponential stability and optimal feedback approach for real-time control problem. In addition, this article opens a door to the research on solving nonlinear systems with large-scale disturbance problems, and eventually, based on the optimal feedback control technique, large-scale disturbance problems can be handled.

Figure 8.

The trajectories versus time with parameter $ϵ=0.005$ .

Figure 9.

The phase portrait with parameter $ϵ=0.05$ .

Figure 10.

The trajectories versus time with finite-time controller. ¹⁹

Figure 11.

The phase portrait with finite-time controller. ¹⁹

Two-degree-of-freedom manipulator

The model of 2-degree-of-freedom manipulator is as follows

$$M(q(\tau ))\stackrel{··}{q}(\tau )+C(q(\tau ),\stackrel{·}{q}(\tau ))\stackrel{·}{q}(\tau )+G(q(\tau ))=\zeta (\tau )$$

(42)

where $M(q(\tau ))\in {\mathbb{R}}^{2\times 2}$ is an inertia matrix, $C(q(\tau ),\stackrel{·}{q}(\tau ))\in {\mathbb{R}}^{2\times 2}$ is a centripetal-Coriolis matrix, $G(q(\tau ))\in {\mathbb{R}}^{2\times 1}$ is a gravity matrix, and $\zeta (\tau )\in {\mathbb{R}}^{2\times 2}$ is a torque vector, respectively. Due to the page limit, the detailed descriptions are omitted in this article but can be seen in Bouakrif and Zasadzinski. ²⁶ Equation (42) can be transformed into the following controlled nonlinear affine system

$$\stackrel{·}{\chi }=F(\chi )+\mathrm{\Psi }(\chi )\mu$$

(43)

Therefore, the simulation results of rapidly exponentially stabilizing controller and iterative learning controller for solving time-varying 2-degree-of-freedom manipulator of equation (42) are shown in Figures 12–14.

Figure 12.

The trajectories of 2-degree-of-freedom manipulator: (a) trajectory q₁ versus time t and (b) trajectory q₂ versus time t.

Figure 13.

The trajectories of 2-degree-of-freedom manipulator: ²⁶ (a) trajectory q₁ versus time t and (b) trajectory q₂ versus time t.

Figure 14.

Performance comparisons of this article and the literature ²⁶ for soling time-varying 2-degree-of-freedom manipulator: (a, c) error q₁ versus time t and (b, d) error q₂ versus time t.

As shown in Figure 12, the desired trajectories are dashed blue lines and the actual trajectories are dotted red lines. It is can be seen that the proposed approach of this article is feasible and effective. Moreover, simulation results showed that the rapidly exponentially stabilizing controller achieves the trajectory tracking for 2-degree-of-freedom manipulator system. Figure 13 shows the trajectories of 2-degree-of-freedom manipulator system versus time $t$ . The red lines represent the actual trajectories and the blue lines mean the desired trajectories. Indeed, the trajectories of 2-degree-of-freedom manipulator system track the desired trajectories less than 0.6 s. However, as shown in Figure 14(a) and (c), the error of trajectory $q_1$ converges to zero in 0.7 s, which shows that the developed approach is superior than the error of iterative learning approach, which converges to zero in 1 s. In addition, the iterative learning controller cannot converge to zero less than 1 s as shown in Figure 14(c). However, the rapidly exponentially stabilizing controller can converge to zero less than 0.8 s. Moreover, as shown in Figure 14(b) and (d), the error of trajectory $q_2$ converges to zero in 0.4 s, which shows the developed approach is superior than the iterative learning approach, which converges to zero in 0.6 s. Therefore, owing to the rapidly exponentially stabilizing controller and optimal feedback control technique, this article might open a door to the study on solving manipulator system problem in real time.

Conclusion and future works

This article developed some simple sufficient conditions for a class of optimal feedback controllers that rapidly and exponentially stabilize trajectories in nonlinear systems. According to the Lyapunov theorem, the rapid exponential stability was proposed, analyzed, and investigated for the nonlinear systems. Furthermore, the HJBE was utilized to achieve optimal feedback control for nonlinear affine systems. Moreover, numerical simulations have demonstrated that this optimal feedback control method could be very useful in real time. In the future works, the developed unified framework may be extended to address optimal feedback controllers and rapid exponential stability for nonlinear hybrid systems and nonlinear stochastic systems, for instance, bipedal walking robot systems and stochastic systems. In addition, the proposed sufficient conditions of this article may open a door to the performance improvement of the related applications, such as non-triangular stochastic nonlinear systems, ²² switched stochastic nonlinear systems, ²⁵ robot control,^26,29,30 and path planning, ²⁸ with great capacity in accelerating convergence and computing accuracy.