Interval Power Integration-Based Nonlinear Suppression Control for Uncertain Systems and Its Application to Superheated Steam Temperature Control

Abstract

The control of many industrial processes, such as superheated steam temperature control, exhibits poor robustness and degraded accuracy in the presence of model parameter uncertainties. This paper addresses this issue by developing a novel interval power integration-based nonlinear suppression scheme for a class of uncertain nonlinear systems with unknown but bounded parameters. The efficacy of this approach is specifically demonstrated for the superheated steam temperature control in thermal power plants. By integrating Lyapunov stability theory and homogeneous system theory, this method extends the traditional homogeneous degree theory to the interval domain, establishes interval boundary conditions for time-varying parameters, and constructs a Lyapunov function with interval numbers to recursively design the controller. Furthermore, the interval monotonic homogeneous degree and an admissibility index are introduced to ensure system stability under parameter uncertainties. The effectiveness of the proposed method is verified through numerical simulations of superheated steam temperature control. Simulation results demonstrate that the method effectively suppresses nonlinearities and achieves robust asymptotic stability, even when model parameters vary within bounded intervals. In the varying-exponent scenario, the proposed controller achieved an Integral of Absolute Error (IAE) of 70.78 and a convergence time of 37s for the superheated steam temperature control. This represents a performance improvement of 42.79% in IAE and 53.16% in convergence time compared to a conventional PID controller, offering a promising solution for complex thermal processes with inherent uncertainties.

1. Introduction

In the context of increasingly flexible operational demands in modern industry, the control of complex processes with significant nonlinearities and uncertainties has become a critical challenge [1,2]. The superheated steam temperature system in thermal power plants is a prime example of such a process, characterized by its complex thermal dynamics and load-dependent parameter variations. Consequently, conventional control strategies, such as the widely used PID controller, often exhibit deteriorated robustness and degraded control accuracy when faced with these complex dynamics [3]. Maintaining precise superheated steam temperature control is crucial for the safe and efficient operation of power generation units [4], typically requiring temperature deviations within $\pm 5$ °C [5]. Furthermore, enhancing the performance of the superheated steam temperature control system is a key enabler for improving the wide-load operational capabilities of thermal units [6], which is essential for effectively accommodating the intermittent renewable energies.

However, achieving high-precision robust control for the superheated steam temperature system remains a formidable challenge, attributable to its strong nonlinearities, large inertial dynamics, and significant uncertainties. In this regard, various advanced control methods have been investigated, such as fractional-order PID [7], adaptive control [8], and model predictive control [9]. Nonetheless, their reliance on accurate mathematical models or high computational complexity often impedes their practical implementation. Furthermore, while various intelligent PID controllers, such as those based on sigmoid functions [10] or neuroendocrine algorithms [11], have shown potential, they often rely on heuristic tuning rules and are not explicitly designed to handle the structured interval uncertainty of power exponents addressed in this work. Some novel control methods, such as data-driven feedback compensator [3] and active disturbance rejection control [12], have also demonstrated advantages in handling large inertia and disturbances, but they are generally not directly applicable to the control design for the aforementioned uncertain nonlinear systems. It is also noteworthy that ensuring the reliability of modern energy systems is a broad challenge, as seen in the parallel evolution of control technology, remaining useful life prediction [13], and intelligent monitoring [14].

Many thermal processes, including the superheated steam temperature system, can be represented by a specific nonlinear structure known as a P-normal form system, where the dynamics are characterized by power exponents p. The critical challenge, exacerbated by the complex thermal dynamics and load-dependent parameter variations, is that these power exponents are often uncertain and vary within a bounded interval.

From a theoretical point of view, even if the power exponent p is precisely knowable, the controller design of the nonlinear system is very difficult, because in this case the system has singularity at the equilibrium point. In the past 20 years, the control of this type system has attracted increasing attention. Among these studies, adding a power integrator is a critical contribution. It can be seen as a generalization of the traditional back-stepping method [15,16,17]. In recent years, the concept of homogeneity with monotone degrees (HWMD) was proposed in [18], and the generalized exponential integration method was proposed to solve the tracking and stabilization problems [19]. So far, these studies have been carried out under the premise that the power exponent p is a known constant. In fact, the structural parameters of thermal process models are generally determined based on the identification of running data, so the model structural parameters have the characteristics of slow time variation. Even for the operating data under the same working conditions, the structural parameters of the model tend to change due to uncertainty and deterioration of equipment performance, so the representation of the power index with appropriate boundaries is more realistic [20].

When $p_i$ is unknown but bounded, that is, $p_i\in \left[a_i,b_i\right],b_i>a_i>0$, the controlled system becomes a class of interval power indeterminate nonlinear systems, which brings great difficulty to the design of the controller. For the interval power exponential system, we need to propose a new method to design the controller. From the point of view of homogeneous systems theory, the greatest difficulty caused by the interval power exponent is that the homogeneous degree is uncertain. Corresponding to the interval power index, the concept of interval homogeneous degree is first proposed in this work. Secondly, an admissibility index is proposed to judge whether the system has interval homogeneity. Finally, we design a Lyapunov function with unknown parameters, and propose a more generalized design method to achieve global asymptotic stability of interval power exponential system. The simulation shows that when the parameters of the thermal process model change within a certain interval range, the controller designed based on the interval power integration method still has good control performance.

In summary, the contributions of this paper include the following:

• A novel interval power integration-based nonlinear suppression scheme is proposed for a class of uncertain nonlinear systems with unknown but bounded parameters.
• By integrating Lyapunov stability theory and homogeneous system theory, a novel theoretical framework is constructed to handle interval uncertainties.
• The stability of the proposed method is theoretically analyzed, and its efficacy is specifically demonstrated for the superheated steam temperature control.

The rest of this paper is arranged as follows: Section 2 reviews some basic concepts of homogeneous system theory, lists some important lemmas used in the paper, and gives a description of the controller design process based on the power integration method. In Section 3, we propose the concept of interval monotone homogeneous degree, define the admissibility index, and prove the existence of the admissibility index. In Section 4, an interval power integration method is proposed and applied to the design of state feedback controller so that the interval power index system (1) can achieve global asymptotic stability. Section 5 uses the interval power integration method to solve the control problem of a typical thermal process, i.e., superheated steam temperature system. A summary of this paper is given in Section 6.

2. Mathematic Preliminaries

The superheated steam temperature system and a broad class of nonlinear industrial processes can be described by the following P-normal form nonlinear model [21,22,23]:

$$\begin{array}{cc}\hfill & {\dot{x}}_i=x_{i+1}^{p_i}+{\varphi }_i\left(x_1,x_2,\cdots ,x_i\right),i=1,2,\dots ,n-1,\hfill \\ \hfill & {\dot{x}}_n=u+{\varphi }_n\left(x\right)\hfill \end{array}$$

(1)

where $x={\left(x_1,\cdots ,x_n\right)}^T\in {\mathbb{R}}^n$ denotes the system states, u is the control input, ${\varphi }_i(·):{\mathbb{R}}^i\to \mathbb{R}$ represents the unknown nonlinear continuous function which satisfies ${\varphi }_i\left(0\right)=0$ and can be regarded as a disturbance, and $p_i\in {\mathbb{R}}_{\mathrm{odd}}^{+},i=1,2,\dots ,n-1,p_n=1$ stands for the odd proportional numbers, i.e., positive odd numerators and denominators. When $p_1=p_2=\cdots =p_n=1$, the system (1) deforms to the widely known integral chain nonlinear system [24,25,26,27]. When there is one or more $p_i>1$, the system (1) changes to a power integral system [28,29]. Generally speaking, common single input single output thermal processes can be converted to this type of P-normal system [30].

This section provides the necessary mathematical preliminaries for the subsequent analysis. First, we introduce the fundamental concepts of homogeneous system theory. Next, we present some supporting lemmas that are critical to the development of our approach. Finally, we discuss the adding a power integrator technique, which serves as a key tool for the later derivations and stability proof.

2.1. Homogeneous System Theory

As a fundamental component of nonlinear systems theory, the homogeneous system theory provides a powerful tool for simplifying complex analysis and design problems. Its application has garnered significant research interest in recent years [31,32], especially in the domains of stability analysis and controller synthesis.

First, we provide the formal definition of a homogeneous function and its degree of homogeneity [33]:

Definition 1.

For a given function $f\left(x_1,x_2,\cdots ,x_n\right)$ and a factor ε, if the following equation holds:

$$f\left(\epsilon x_1,\epsilon x_2,\cdots ,\epsilon x_n\right)={\epsilon }^{n}f\left(x_1,x_2,\cdots ,x_n\right)$$

then the function $f\left(x_1,x_2,\cdots ,x_n\right)$ is a homogeneous one with n power, i.e., its degree of homogeneity is n.

For an autonomous system $\dot{x}=f\left(x\right),x\left(t_0\right)=x_0$, where $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ satisfies $f\left(0\right)=0$ and local Lipschitz continuity. If the system is linear, it is usually homogeneous, e.g., $\left\{\begin{array}{c}{\dot{x}}_1=3x_2,\hfill \\ {\dot{x}}_2=-x_1,\hfill \end{array}\right.$; we have $f\left(\epsilon x_1,\epsilon x_2\right)=\left[\begin{array}{c}3\epsilon x_2\hfill \\ -\epsilon x_1\hfill \end{array}\right]=\epsilon \left[\begin{array}{c}3x_2\hfill \\ -x_1\hfill \end{array}\right]=\epsilon f\left(x_1,x_2\right)$, and its degree of homogeneity is 1. For a nonlinear system, e.g., ${\dot{x}}_1=3x_1^3+2x_2^3, {\dot{x}}_2=-x_2^3$, since

$$f\left(\epsilon x_1,\epsilon x_2\right)=\left[\begin{array}{c}3{\epsilon }^3{x}_1^3+2{\epsilon }^3{x}_2^3\\ -{\epsilon }^3{x}_2^3\end{array}\right]={\epsilon }^3\left[\begin{array}{c}3x_1^3+2x_2^3\\ -x_2^3\end{array}\right]={\epsilon }^{3}f\left(x_1,x_2\right)$$

this nonlinear system is also homogeneous, with its degree of homogeneity equal to 3. While if the order of the nonlinear system is different, e.g., ${\dot{x}}_1=3x_1+2x_2^3,{\dot{x}}_2=-x_2^3$, since

$$f\left(\epsilon x_1,\epsilon x_2\right)=\left[\begin{array}{c}3\epsilon x_1^3+2{\epsilon }^3{x}_2^3\\ -{\epsilon }^3{x}_2^3\end{array}\right]\ne {\epsilon }^{3}f\left(x_1,x_2\right)\ne \epsilon f\left(x_1,x_2\right)$$

according to the Definition 1, this nonlinear system is non-homogeneous.

To describe the homogeneity of such a nonlinear system, the concept of weighted homogeneity was proposed [34]:

Definition 2.

For any positive real number $r_i>0,i=1,2,\dots ,n$ and state vector $\left(x_1,\cdots ,x_n\right)\in {\mathbb{R}}^n$, the extension is defined as ${\mathsf{\Delta }}_{\epsilon }^r\left(x\right)=\left({\epsilon }^{r_1}{x}_1,\cdots ,{\epsilon }^{r_n}{x}_n\right),\forall \epsilon >0$, and the real number $r_i$ is the extension weight coefficient of the vector, which is also called the homogeneous weight coefficient.

Definition 3.

For a continuous function $V\left(x\right):{\mathbb{R}}^n\to \mathbb{R}$, if there exist real numbers $\tau \in \mathbb{R}$ and $r_i>0,i=1,2,\dots ,n$ such that for $\forall \epsilon >0,x\in {\mathbb{R}}^n\setminus \left\{0\right\}$, and if the following equation is true,

$$V\left({\mathsf{\Delta }}_{\epsilon }^r\left(x\right)\right)={\epsilon }^{\tau }V\left(x\right)$$

the function $V\left(x\right)$ is said to be homogeneous with respect to the homogeneous weight coefficient $r_i>0,i=1,2,\dots ,n$, and the weighted homogeneous degree is τ.

Definition 4.

For a continuous vector field function $f\left(x\right)={\left(f_1\left(x\right),\cdots ,f_n\left(x\right)\right)}^T,x\in \mathbb{R}$, if there exist $\tau \in \mathbb{R}$ and $r_i>0,i=1,2,\dots ,n$, such that for $\forall \epsilon >0,x\in {\mathbb{R}}^n\setminus \left\{0\right\}$, and if the following equation is true:

$$f_i\left({\mathsf{\Delta }}_{\epsilon }^r\left(x\right)\right)={\epsilon }^{r+\tau }{f}_i\left(x\right),i=1,2,\dots ,n$$

the vector field function $f\left(x\right)={\left(f_1\left(x\right),\dots ,f_n\left(x\right)\right)}^T$ is then said to be homogeneous with respect to $r_i>0,i=1,2,\dots ,n$, and the weighted homogeneity degree is τ.

According to the Definition 4, for the system ${\dot{x}}_1=3x_1+2x_2^3,{\dot{x}}_2=-x_2^3$, if we choose $r_1=1,r_2=1/3$, then the following can be obtained

$$\begin{array}{cc}& f_1\left({\epsilon }^{r_1}{x}_1,{\epsilon }^{r_2}{x}_2\right)=3{\epsilon }^1{x}_1+2{\epsilon }^1{x}_2^3=\epsilon f_1\left(x_1,x_2\right)\hfill \\ & f_2\left({\epsilon }^{r_1}{x}_1,{\epsilon }^{r_2}{x}_2\right)=-{\epsilon }^1{x}_2^3=\epsilon f_2\left(x_1,x_2\right)\hfill \end{array}$$

indicating that the vector field function $f\left(x\right)$ satisfies the definition of the weighted homogeneity degree under the homogeneous weight coefficient $r_1=1,r_2=1/3$.

2.2. Some Important Lemmas

In this subsection, we give some important lemmas that will be used frequently in subsequent proofs in this paper.

Lemma 1

([35]). For a system of order n $\dot{x}=f\left(x\right),f\left(0\right)=0$, if there exist continuously differentiable first-order functions, $V:{\mathbb{R}}^n\to \mathbb{R}$, ${\beta }_i,i=1,2,3$ and α, which are capable of making ${\beta }_1{\parallel x\parallel }^{\alpha }\le V\left(x\right)\le {\beta }_2{\parallel x\parallel }^{\alpha },{\displaystyle \frac{\partial V}{\partial t}}+{\displaystyle \frac{\partial V}{\partial x}}f\left(x\right)\le -{\beta }_3{\parallel x\parallel }^{\alpha }$, then $x=0$ is exponentially stable.

Lemma 2

([36]). Given normal numbers $c,d$, for any function with real value $\eta (x,y)>0$, there is

$${\left|x\right|}^c{\left|y\right|}^d\le {\displaystyle \frac{c}{c+d}}{\eta (x,y)\left|x\right|}^{c+d}+{\displaystyle \frac{d}{c+d}}{\eta }^{-{\displaystyle \frac{c}{d}}}(x,y){\left|y\right|}^{c+d}$$

Lemma 3

([37]). For any $x,y\in \mathbb{R}$ and any $\epsilon >0$, if the constant $p>1,q>1$ satisfies $1/q+1/q=1$, then we have:

$$xy\le {\displaystyle \frac{{\epsilon }^p}{p}}{\left|x\right|}^p+{\displaystyle \frac{1}{q{\epsilon }^p}}{\left|y\right|}^q$$

Lemma 4

([28,36]). For any $x,y\in \mathbb{R}$ and a constant $p\ge 1$, there exists the following inequality:

$$\begin{array}{c}{|x+y|}^p\le 2^{p-1}\left|x^p+y^p\right|\\ \left(\right|x|+{\left|y\right|)}^{1/p}\le {\left|x\right|}^{1/p}+{\left|y\right|}^{1/p}\le 2^{{\displaystyle \frac{p-1}{p}}}\left(\right|x|+{\left|y\right|)}^{1/p},\end{array}$$

and if $p\ge 1$ is odd, we have

$$\begin{array}{cc}& {|x-y|}^p\le 2^{p-1}\left|x^p-y^p\right|\hfill \\ & \left|x^{1/p}-y^{1/p}\right|\le 2^{(p-1)/p}{|x-y|}^{1/p}.\hfill \end{array}$$

Lemma 5

([38]). If the continuous function $f:[a,b]\to \mathbb{R}(a\le b)$ is monotonic and satisfies $f\left(a\right)=0$, then the following holds:

$${\int }_a^{b}f\left(x\right)dx\le \left|f\left(b\right)\right||b-a|.$$

2.3. Adding a Power Integrator Technique

The technique involving adding a power integrator was proposed by Lin and Qian [39,40]. This method introduces a virtual controller for iterative design. In each iteration of the virtual controller design, it suppresses the system’s nonlinearity by designing a larger nonlinear function based on the idea of nonlinear suppression, rather than eliminating the system’s nonlinearity. As a result, the designed controller has good robustness and has been widely applied [19,36].

Next, we use a simple second-order nonlinear system as an example to introduce how to design a controller based on the addition of a power integrator technique.

Example 1.

Design a state feedback controller to ensure global asymptotic stability for the following second-order nonlinear system.

$${\dot{x}}_1=x_2^3+x_1,{\dot{x}}_2=u.$$

Step 1: Define the following state variables ${\xi }_1=x_1,{\xi }_2=x_2^3-x_2^{\ast 3}$, where $x_2^{\ast }$ is the virtual controller to be designed.

Step 2: Define the Lyapunov function $V_1={\xi }_1^2/2$, with its derivative provided by the following:

$${\dot{V}}_1={\xi }_1{\dot{\xi }}_1={\xi }_1{\dot{x}}_1={\xi }_1\left(x_2^3+x_1\right)={\xi }_1{\xi }_2+{\xi }_1{x}_2^{\ast 3}+{\xi }_1^2,$$

If a virtual controller $x_2^{\ast }=-3^{1/3}{\xi }_1^{1/3}$ is designed, there exists the relation ${\dot{V}}_1=-2{\xi }_1^2+{\xi }_1{\xi }_2$.

Step 3: Define a Lyapunov function $V_2=V_1+W_2=V_1+{\int }_{x_2^{\ast }}^{x_2}{\left(s^3-x_2^{\ast 3}\right)}^{5/3}ds$; according to the Lemma 4, as $x_2\ge x_2^{\ast }$, we have

$$W_2={\int }_{x_2^{\ast }}^{x_2}{\left(s^3-x_2^{\ast 3}\right)}^{5/3}ds\ge c_1{\int }_{x_2^{\ast }}^{x_2}{\left(s-x_2^{\ast }\right)}^{5}ds>0$$

while $x_2<x_2^{\ast }$ is as follows:

$$\begin{array}{cc}\hfill W_2& ={\int }_{x_2^{\ast }}^{x_2}{\left(s^3-x_2^{\ast 3}\right)}^{5/3}ds={\int }_{x_2}^{x_2^{\ast }}-{\left(s^3-x_2^{\ast 3}\right)}^{5/3}ds\hfill \\ & ={\int }_{x_2}^{x_2^{\ast }}{\left(x_2^{\ast 3}-s^3\right)}^{5/3}ds\ge c_2{\int }_{x_2}^{x_2^{\ast }}{\left(x_2^{\ast }-s\right)}^{5}ds>0.\hfill \end{array}$$

Therefore, $V_2$ holds positive. By taking the derivative of $V_2$, we get the following:

$${\dot{V}}_2={\dot{V}}_1+{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1+{\displaystyle \frac{\partial W_2}{\partial x_2}}=-2{\xi }_1^2+{\xi }_1{\xi }_2+{\displaystyle \frac{\partial W_2}{\partial x_1}}\left(x_2^3+x_1\right)+{\xi }_2^{5/3}{\dot{x}}_2$$

Since $x_2^{\ast }=-3^{1/3}{\xi }_1^{1/3}$, the following equation holds:

$${\displaystyle \frac{\partial W_2}{\partial x_1}}={\displaystyle \frac{\partial {\int }_{x_2^{\ast }}^{x_2}{\left(s^3-x_2^{\ast 3}\right)}^{5/3}ds}{\partial x_1}}=-{\displaystyle \frac{5}{3}}{\displaystyle \frac{\partial x_2^{\ast 3}}{\partial x_1}}{\int }_{x_2^{\prime }}^{x_2}{\left(s^3-x_2^{\ast 3}\right)}^{2/3}ds=5{\int }_{x_2^{\ast }}^{x_2}{\left(s^3-x_2^{\ast 3}\right)}^{2/3}ds$$

According to Lemma 5, we can obtain the following:

$${\displaystyle \frac{\partial W_2}{\partial x_1}}=5{\int }_{x_2^{\ast }}^{x_2}{\left(s^3-x_2^{\ast 3}\right)}^{2/3}ds\le 5{\left(x_2^3-x_2^{\ast 3}\right)}^{2/3}\left(x_2-x_2^{\ast }\right)$$

According to Lemma 4, we have $\left|x_2-x_2^{\ast }\right|<2^{2/3}{\left|x_2^3-x_2^{\ast 3}\right|}^{1/3}$ and ${\xi }_1=x_1,{\xi }_2=x_2^3-x_2^{\ast 3}$, which yields the following:

$$\begin{array}{cc}& {\displaystyle \frac{\partial W_2}{\partial x_1}}\le 5{\left(x_2^3-x_2^{\ast 3}\right)}^{2/3}\left(x_2-x_2^{\ast }\right)\le 5\left|{\left(x_2^3-x_2^{\ast 3}\right)}^{2/3}\right|\left|\left(x_2-x_2^{\ast }\right)\right|\hfill \\ & \le 5^{\ast }{2}^{2/3}\left|{\left(x_2^3-x_2^{\ast 3}\right)}^{2/3}\right|{\left|\left(x_2^3-x_2^{\ast 3}\right)\right|}^{1/3}=5\ast 2^{2/3}\left|{\xi }_2\right|,\hfill \\ & {\displaystyle \frac{\partial W_2}{\partial x_1}}\left(x_2^3+x_1\right)={\displaystyle \frac{\partial W_2}{\partial x_1}}\left({\xi }_2+x_2^{\ast 3}+{\xi }_1\right)={\displaystyle \frac{\partial W_2}{\partial x_1}}\left({\xi }_2-2{\xi }_1\right)\hfill \\ & \le {\displaystyle \frac{\partial W_2}{\partial x_1}}\left(\left|{\xi }_2\right|+2\left|{\xi }_1\right|\right)\le 5^{\ast }{2}^{2/3}\left|{\xi }_2\right|\left(\left|{\xi }_2\right|+2\left|{\xi }_1\right|\right),\hfill \\ & {\xi }_1{\xi }_2+{\displaystyle \frac{\partial W_2}{\partial x_1}}\left(x_2^3+x_1\right)\le \left|{\xi }_1\right|\left|{\xi }_2\right|+5^{\ast }{2}^{2/3}\left|{\xi }_2\right|\left(\left|{\xi }_2\right|+2\left|{\xi }_1\right|\right)\hfill \\ & \le \left|{\xi }_1\right|\left|{\xi }_2\right|\left(1+10\ast 2^{2/3}\right)+5^{\ast }{2}^{2/3}{\left|{\xi }_2\right|}^2.\hfill \end{array}$$

According to Lemma 3, and through defining $p=2,q=2$, we have $\left|{\xi }_1\right|\left|{\xi }_2\right|\le {\displaystyle \frac{{\epsilon }^2}{2}}{\left|{\xi }_1\right|}^2+{\displaystyle \frac{1}{2{\epsilon }^2}}{\left|{\xi }_2\right|}^2$, where ${\displaystyle \frac{{\epsilon }^2}{2}}=1/\left(1+10\times 2^{2/3}\right)$. Substituting the inequality relation into the above equation, we can obtain the following:

$$\begin{array}{c}\left|{\xi }_1\right|\left|{\xi }_2\right|\left(1+{10}^2{2}^{2/3}\right)\le {\left|{\xi }_1\right|}^2+{\displaystyle \frac{{\left(1+10\ast 2^{2/3}\right)}^2}{4}}{\left|{\xi }_2\right|}^2\\ {\xi }_1{\xi }_2+{\displaystyle \frac{\partial W_2}{\partial x_1}}\left(x_2^3+x_1\right)\le {\left|{\xi }_1\right|}^2+79.2{\left|{\xi }_2\right|}^2\end{array}$$

Thus, the controller can be designed as $u=-80.2{\xi }_2^{1/3}$. In this way, we have

$$\begin{array}{cc}\hfill {\dot{V}}_2& =-2{\xi }_1^2+{\xi }_1{\xi }_2+{\displaystyle \frac{\partial W_2}{\partial x_1}}\left(x_2^3+x_1\right)+{\xi }_2^{5/3}{\dot{x}}_2\hfill \\ & \le -2{\xi }_1^2-80.2{\xi }_2^{1/3}{\xi }_2^{5/3}+{\left|{\xi }_1\right|}^2+79.2{\left|{\xi }_2\right|}^2=-{\left|\xi \right|}_1^2-{\left|{\xi }_2\right|}^2\hfill \end{array}$$

According to the Lyapunov theorem, the system states $x_1,x_2$ converge globally to the origin.

3. Interval Monotonic Homogeneity

The weighted homogeneity extends the definition of a homogeneous system, but is still unable to deal with cases where the power exponent p is uncertain, just as described in the introduction. To overcome this limitation, we introduce the concept of interval monotone homogeneous degree as follows:

Definition 5.

For a continuous vector function $f\left(x\right)={\left(f_1\left(x\right),\cdots ,f_n\left(x\right)\right)}^T,x\in \mathbb{R}$, if there are positive real numbers $r_i>0,i=1,2,\dots ,n$ and a series of intervals $\left[{\underline{\tau }}_i,{\overline{\tau }}_i\right],i=1,\cdots ,n$ that satisfy ${\underline{\tau }}_i\ge {\overline{\tau }}_{i+1},i=1,\cdots ,n-1$, this makes the following formula true:

$$f_i\left({\epsilon }^{r_1}{x}_1,\cdots ,{\epsilon }^{r_n}{x}_n\right)={\epsilon }^{r_i+{\tau }_i}{f}_i\left(x\right),\forall \epsilon >0,{\tau }_i\in \left[{\underline{\tau }}_i,{\overline{\tau }}_i\right],x\in {\mathbb{R}}^n\setminus \left\{0\right\},$$

the continuous vector function $f\left(x\right)$ is said to have successively decreasing homogeneity degrees ${\tau }_1,{\tau }_2,\cdots ,{\tau }_n$, which are referred to as interval monotonic homogeneity degrees.

Combined with the interval homogeneity degree, an admissibility index is defined below to judge whether the interval power exponential nonlinear system can find the homogeneous weight coefficient and expansion coefficient that meet the definition requirements of the interval monotonic homogeneity degree.

Definition 6.

For an interval power exponential nonlinear system with $p_i\in \left[a_i,b_i\right],i=1,2,\dots ,n-1$, the admissibility index is recursively calculated as follows:

$$\begin{array}{cc}\hfill & D_0:=1\hfill \\ \hfill & D_1:=a_{n-1}+1,\hfill \\ \hfill & D_k:=\left(a_{n-k}+1\right)D_{k-1}-b_{n-k+1}{D}_{k-2},k=2,3,\cdots ,n-1\hfill \end{array}$$

(2)

According to Definition 6, the admissibility index is only related to the interval power index. The following two lemmas give some properties of the admissibility index.

Lemma 6.

There exists a nonlinear system (1) with an interval power exponent $p_i\in \left[a_i,b_i\right]$ satisfying $D_k>0,k=0,1,\cdots n-1$, especially when $p_i$ precisely known, e.g., $a_i\equiv b_i\equiv p_i$, $D_k>0$ always holds.

The proof of Lemma 6 is provided in the Appendix A.

Lemma 7.

As $p_i\in \overline{p}[1-\overline{\delta },1+\overline{\delta }]$, the exact center value $\overline{p}$ and scale range $\overline{\delta }$ are given to satisfy

$$\overline{\delta }<{\delta }^{\ast }\left(n\right):={\displaystyle \frac{1+\overline{p}+2{cos}^2\left(\frac{\pi }{n}\right)-2cos\left(\frac{\pi }{n}\right)\sqrt{1+2\overline{p}+{cos}^2\left(\frac{\pi }{n}\right)}}{\overline{p}}},$$

then, we have $D_k\left(\overline{\delta }\right)>0,k=0,1,\cdots ,n-1$

The proof of Lemma 7 is similar to Lemma 2.1 in [41], and is therefore omitted here.

In the case of $\overline{p}=1$, the value of ${\delta }^{\ast }$ is related to the system dimensionality according to the above lemmas, with the detailed results shown in the

Table 1. Upper bounds corresponding to different dimensionalities.

n	2	3	4	5	6	⋯
${\delta }^{\ast }\left(n\right)$	2	0.6972	0.3542	0.2159	0.1459	⋯

.

Lemmas 6 and 7 indicate that there exists a nonlinear system (1) satisfying $D_k>0$. On the other hand, for the interval power exponential nonlinear system (1), a sufficient condition is provided to determine whether the homogeneous weight coefficient $r_i$ can be found to guarantee the existence of the interval HWMD.

4. Stabilization Analysis of Interval Power Exponential Nonlinear Systems

In this section, we consider the global stabilization of a nonlinear system (1) under the condition of unknown but bounded powers, i.e., $p_i\in \left[a_i,b_i\right],i=1,2,\cdots n-1$. The overall control structure is depicted in Figure 1.

Specifically, we first give some key assumptions and lemmas that are fundamental to the controller design. Then, we present the detailed recursive design of the controller and provides a rigorous proof of the closed-loop system’s global asymptotic stability. Finally, the effectiveness of the proposed method is validated through two illustrative simulation examples.

4.1. Some Key Assumptions and Lemmas

Considering that the selected homogeneous weight may not have a positive odd denominator in some cases, the index value of the function is expressed in the following form: ${[·]}^{\alpha }=sign(·){|·|}^{\alpha }$, with the system definition becoming as follows:

$$\begin{array}{cc}\hfill & {\dot{x}}_i=sign\left(x_{i+1}\right){\left|x_{i+1}\left(t\right)\right|}^{p_i}+{\varphi }_i\left(x_1,x_2,\cdots ,x_i\right),i=1,2,\dots ,n-1,\hfill \\ \hfill & {\dot{x}}_n=u+{\varphi }_n\left(x\right).\hfill \end{array}$$

(3)

Assumption 1.

The admissibility index $D_k$ of an interval power exponential nonlinear system (3) meets the following requirements: $D_k>0,k=0,1,\cdots ,n-1$.

First, we give the following lemma to determine the homogeneous weight coefficient $r_i$.

Lemma 8.

With the above assumption, for any real $r_1>0$ and $r_{n+1}>0$, there exists a set of homogeneous weights

$$r_i={\displaystyle \frac{1}{D_{n-i+1}}}\left(D_{n-i}{r}_{i-1}+b_i\cdots b_{n-1}{b}_n{r}_{n+1}\right),i=2,3,\cdots ,n$$

(4)

which can ensure that system (3) has interval monotone homogeneous degree ${\tau }_i\in \left[{\underline{\tau }}_i,{\overline{\tau }}_i\right]$, where ${\tau }_i=a_i{r}_{i+1}-r_i,{\overline{\tau }}_i=b_i{r}_{i+1}-r_i,i=1,2,\cdots ,n-1$.

The proof of Lemma 8 is provided in the Appendix A.

When we know exactly that $p_i\ge 1$, with initializing $r_1=1,r_{n+1}={\displaystyle \frac{1}{p_1\cdots p_n}}$, we can recursively obtain according to (4)

$$r_2={\displaystyle \frac{1}{p_1}},r_3={\displaystyle \frac{1}{p_1{p}_2}},\cdots ,r_n={\displaystyle \frac{1}{p_1\cdots p_{n-1}}}.$$

(5)

According to the definition of Lemma (8), and $p_i=a_i=b_i$, the following can be obtained

$$\begin{array}{cc}\hfill & {\underline{\tau }}_i=a_i{r}_{i+1}-r_i=p_i{\displaystyle \frac{1}{p_1\cdots p_i}}-{\displaystyle \frac{1}{p_1\cdots p_{i-1}}}=0\hfill \\ \hfill & {\overline{\tau }}_i=b_i{r}_{i+1}-r_i=p_i{\displaystyle \frac{1}{p_1\cdots p_i}}-{\displaystyle \frac{1}{p_1\cdots p_{i-1}}}=0\hfill \end{array}$$

(6)

Thus, we get ${\tau }_i=0,i=1,2,\cdots ,n$, indicating that the positive real number $r_i$ can be found to satisfy the definition of homogeneous degree. The above results are consistent with those in [28].

Assumption 2.

For an interval power exponential nonlinear system (3), when ${\varphi }_i(·)\ne 0,i=1,\cdots ,n$, the homogeneous weight coefficients defined by (4) simultaneously satisfy $b_i{r}_{i+1}\le r_j,j=1,2,\cdots ,i$.

Lemma 9.

Under the above assumption, we can obtain the following:

$$\left|{\varphi }_i(·)\right|\le {\tilde{\eta }}_i\left(x_i\right)\left({\left|{\xi }_1\right|}^{{\displaystyle \frac{r_i+{\tau }_i}{\sigma }}}+{\left|{\xi }_2\right|}^{{\displaystyle \frac{r_i+{\tau }_i}{\sigma }}}+\cdots +{\left|{\xi }_i\right|}^{{\displaystyle \frac{r_i+{\tau }_i}{\sigma }}}\right)$$

where ${\tilde{\eta }}_i\left(x_i\right)$ is a positive smooth function, the definition of ${\xi }_i$ is as in (9), and $\sigma \ge {max}_{1\le i\le n}\left\{r_i\right\}$.

The proof of Lemma 9 is also provided in the Appendix A.

4.2. Stability Analysis

In this subsection, a Lyapunov function with unknown parameters is constructed by using the interval monotone homogeneous degree and Lemmas 8 and 9, and a state feedback controller is designed based on the power integration method.

Theorem 1.

Under the Assumptions 1 and 2, the following state feedback controller can be designed:

$$u=-{\beta }_n(·){\left({\left[x_n\right]}^{{\displaystyle \frac{\sigma }{{\sigma }_n}}}+{\beta }_{n-1}^{{\displaystyle \frac{\sigma }{n_n}}}(·)\left({\left[x_{n-1}\right]}^{{\displaystyle \frac{\sigma }{r_{n-1}}}}+\cdots +{\beta }_2^{{\displaystyle \frac{\sigma }{r_3}}}\left({\overline{x}}_2\right)\left({\left[x_2\right]}^{{\displaystyle \frac{\sigma }{\frac{\sigma }{2}}}}+{\beta }_1^{{\displaystyle \frac{\sigma }{r_1}}}\left({\overline{x}}_1\right){\left[x_1\right]}^{{\displaystyle \frac{\sigma }{r_1}}}\right)\right)\right)}^{{\displaystyle \frac{r_{n+1}}{\sigma }}}$$

which makes the interval power exponential nonlinear system (3) achieve global asymptotic stability, where the homogeneous weight coefficient $r_i$ is defined according to Lemma 8, and $\sigma \ge {max}_{1\le i\le n}\left\{r_i\right\},{\beta }_i\left(x_1,x_2,\cdots ,x_i\right)$ are the undetermined positive smooth functions.

Proof of Theorem 1. 

The proof is conducted with the following steps:

Step 1: We select a constant, $\rho \ge {max}_{1\le i\le n}\left\{r_i+{\overline{\tau }}_i,\sigma \right\}$, where ${\tau }_i$ is the upper boundary of the interval monotonically homogeneous degree. We then establish a Lyapunov function: $V_1\left(x_1\right)={\int }_0^{x_1}{\left({\left[s\right]}^{{\displaystyle \frac{\sigma }{r_1}}}-0\right)}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}$ and the derivative of $V_1$ along the system (3) is as follows:

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill & {\dot{V}}_1={\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\dot{x}}_1\right)\hfill \\ \hfill & ={\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\left[x_2^{\ast }\right]}^{p_1}+{\varphi }_1\right)+{\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)\hfill \end{array}\hfill \\ \hfill & \begin{array}{cc}\hfill & \le {\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}{\left[x_2^{\ast }\right]}^{p_1}+{\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)+{\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}{\tilde{\eta }}_1\left(x_1\right)\left({\left|{\xi }_1\right|}^{{\displaystyle \frac{r_1+{\tau }_1}{\sigma }}}\right)\hfill \\ \hfill & ={\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}{\left[x_2^{\ast }\right]}^{p_1}+{\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)+{\gamma }_1\left({\overline{x}}_1\right){\xi }_1^{{\displaystyle \frac{2\rho }{\sigma }}},\hfill \end{array}\hfill \end{array}$$

(7)

By constructing the virtual controller $x_2^{\ast }:=-{\beta }_1\left(x_1\right){\left[{\xi }_1\right]}^{r_2/\sigma }$, the above formula becomes

$$\begin{array}{cc}\hfill {\dot{V}}_1& \le {\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}{\left[-{\beta }_1\left(x_1\right){\left[{\xi }_1\right]}^{{\displaystyle \frac{p_2}{\sigma }}}\right]}^{p_1}+{\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)+{\gamma }_1\left({\overline{x}}_1\right){\xi }_1^2\hfill \\ & ={\left[-{\beta }_1\left(x_1\right)\right]}^{p_1}{\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho }{\sigma }}}+{\gamma }_1\left({\overline{x}}_1\right){\xi }_1^{{\displaystyle \frac{2\rho }{\sigma }}}+{\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)\hfill \\ & =\left({\left[-{\beta }_1\left(x_1\right)\right]}^{p_1}+{\gamma }_1\left({\overline{x}}_1\right)\right){\xi }_1^{{\displaystyle \frac{2\rho }{\sigma }}}+{\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right),\hfill \end{array}$$

Suppose that ${\beta }_1(·)={\left(n+{\alpha }_1\left(x_1\right)+{\gamma }_1\left({\overline{x}}_1\right)\right)}^{1/a_1}$, where ${\alpha }_1\left(x_1\right)>0$, and since $n+{\alpha }_1\left(x_1\right)+{\gamma }_1\left({\overline{x}}_1\right)>1,p_1{\ge }_1$, the above formula becomes

$$\begin{array}{cc}\hfill {\dot{V}}_1& \le \left({\left[-{\beta }_1\left(x_1\right)\right]}^{p_1}+{\gamma }_1\left({\overline{x}}_1\right)\right){\xi }_1^{{\displaystyle \frac{2\rho }{\sigma }}}+{\left[{\xi }_1\right]}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)\hfill \\ \hfill & \le -\left(n+{\alpha }_1\left(x_1\right)\right){\xi }_1^{{\displaystyle \frac{2\rho }{\sigma }}}+x_1^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right).\hfill \end{array}$$

(8)

Step 2: Suppose that at step $k_1(k<n)$ there exists a continuous and positive definite Lyapunov function of first order: $V_{k-1}:{\mathbb{R}}^{k-1}\to \mathbb{R}$, and the continuous virtual controller $x_1^{\ast },x_2^{\ast },\cdots ,x_k^{\ast }$ is defined as follows:

$$\begin{array}{cc}\hfill & x_1^{\ast }:=0,{\xi }_1={\left[x_1\right]}^{{\displaystyle \frac{\sigma }{r_1}}}-{\left[x_1^{\ast }\right]}^{{\displaystyle \frac{\sigma }{r_1}}},\hfill \\ \hfill & x_2^{\ast }:=-{\beta }_1\left({\overline{x}}_1\right){\left[{\xi }_1\right]}^{{\displaystyle \frac{r_2}{\sigma }}},{\xi }_2={\left[x_2\right]}^{{\displaystyle \frac{\sigma }{r_2}}}-{\left[x_2^{\ast }\right]}^{{\displaystyle \frac{\sigma }{r_2}}},{\beta }_1\left({\overline{x}}_1\right)={\left(n+{\alpha }_1\left({\overline{x}}_1\right)\right)}^{{\displaystyle \frac{1}{{\alpha }_1}}},\hfill \\ \hfill & x_k^{\ast }:=-{\beta }_{k-1}\left({\overline{x}}_{k-1}\right){\left[{\xi }_{k-1}\right]}^{{\displaystyle \frac{r_k}{\sigma }}},{\xi }_k={\left[x_k\right]}^{{\displaystyle \frac{\sigma }{r_k}}}-{\left[x_k^{\ast }\right]}^{{\displaystyle \frac{\sigma }{\frac{\sigma }{r_k}}}},{\beta }_{k-1}\left({\overline{x}}_{k-1}\right)={\left(n+{\alpha }_{k-1}\left({\overline{x}}_{k-1}\right)\right)}^{{\displaystyle \frac{1}{{\alpha }_{k-1}}}},\hfill \end{array}$$

(9)

where ${\overline{x}}_i:=\left(x_1,\cdots ,x_i\right)$, smooth function ${\alpha }_i\left({\overline{x}}_i\right)>0,{\beta }_i\left({\overline{x}}_i\right)>0,i=1,\cdots ,k-1$, satisfying:

$${\dot{V}}_{k-1}\le -\sum _{i=1}^{k-1}\left(n-k+2+{\alpha }_i\left({\overline{x}}_i\right)\right){\xi }_i^{{\displaystyle \frac{2\rho }{\sigma }}}+{\left|{\xi }_{k-1}\right|}^{{\displaystyle \frac{2\rho -r_{k-1}-{\tau }_{k-1}}{\sigma }}}\left|{\left[x_k\right]}^{p_{k-1}}-{\left[x_k^{\ast }\right]}^{p_{k-1}}\right|,$$

(10)

It is evident that Equation (10) is equivalent to Equation (8) as $k=2$.

Step 3: The Equation (10) is to proved to remain true at step k $(k=2,\cdots ,\mathrm{n}-1)$. First we define

$$V_k\left({\overline{x}}_k\right)=V_{k-1}\left({\overline{x}}_{k-1}\right)+W_k\left({\overline{x}}_k\right),$$

(11)

where $W_k$ is a continuous function of first order and is defined as

$$W_k={\int }_{x_k^{\ast }}^{x_k}{\left[{\left[s\right]}^{{\displaystyle \frac{\sigma }{r_k}}}-{\left[x_k^{\ast }\right]}^{{\displaystyle \frac{\sigma }{{]}_k}}}\right]}^{{\displaystyle \frac{2\rho -r_k-{\tau }_k}{\sigma }}}ds$$

(12)

According to the proof in [28], it is clear that $V_k$ defined by (11) is positive definite and regular, with its derivation given by the following:

$$\begin{array}{cc}\hfill {\dot{V}}_k=& {\dot{V}}_{k-1}+\sum _{i=1}^{k-1}{\displaystyle \frac{\partial W_k}{\partial x_i}}{\dot{x}}_i+{\left[{\xi }_k\right]}^{{\displaystyle \frac{2\rho -r_k-{\tau }_k}{\sigma }}}{\dot{x}}_k\hfill \\ \hfill =& {\dot{V}}_{k-1}+\sum _{i=1}^{k-1}{\displaystyle \frac{\partial W_k}{\partial x_i}}\left({\left[x_{i+1}\right]}^{p_i}+{\varphi }_i\right)+{\left[{\xi }_k\right]}^{{\displaystyle \frac{2\rho -r_k-{\tau }_k}{\sigma }}}{\dot{x}}_k\hfill \\ \hfill =& {\dot{V}}_{k-1}+\sum _{i=1}^{k-1}{\displaystyle \frac{\partial W_k}{\partial x_i}}{\left[x_{i+1}\right]}^{p_i}+\sum _{i=1}^{k-1}{\displaystyle \frac{\partial W_k}{\partial x_i}}{\varphi }_i+{\left[{\xi }_k\right]}^{{\displaystyle \frac{2\rho -r_k-{\tau }_k}{\sigma }}}{\dot{x}}_k\hfill \\ \hfill \le & -\sum _{i=1}^{k-1}\left(n-k+2+{\alpha }_i\left({\overline{x}}_i\right)\right){\xi }_i^{{\displaystyle \frac{2\rho }{\sigma }}}+{\left|{\xi }_{k-1}\right|}^{{\displaystyle \frac{2\rho -r_{k-1}-{\tau }_{k-1}}{\sigma }}}\left|{\left[x_k\right]}^{p_{k-1}}-{\left[x_k^{\ast }\right]}^{p_{k-1}}\right|\hfill \\ \hfill & +{\left[{\xi }_k\right]}^{{\displaystyle \frac{2\rho -r_k-{\tau }_k}{\sigma }}}{\left[x_{k+1}^{\ast }\right]}^{p_k}+{\left[{\xi }_k\right]}^{{\displaystyle \frac{2\rho -r_k-{\tau }_k}{\sigma }}}\left|{\left[x_{k+1}\right]}^{p_k}-{\left[x_{k+1}^{\ast }\right]}^{p_k}\right|\hfill \\ \hfill & +\sum _{i=1}^{k-1}{\displaystyle \frac{\partial W_k}{\partial x_i}}{\left[x_{i+1}\right]}^{p_i}+\sum _{i=1}^k{\displaystyle \frac{\partial W_k}{\partial x_i}}{\varphi }_i.\hfill \end{array}$$

(13)

The three terms on the right of the above inequality can be obtained:

$${\left|{\xi }_{k-1}\right|}^{{\displaystyle \frac{2\rho -r_{k-1}-{\tau }_{k-1}}{\sigma }}}\left|{\left[x_k\right]}^{p_{k-1}}-{\left[x_k^{\ast }\right]}^{p_{k-1}}\right|\le {\displaystyle \frac{1}{4}}{\xi }_{k-1}^{{\displaystyle \frac{2\rho }{\sigma }}}+c_k\left({\overline{x}}_k\right){\xi }_k^{{\displaystyle \frac{2\rho }{\sigma }}},$$

(14)

$$\left|\sum _{i=1}^{k-1}{\displaystyle \frac{\partial W_k}{\partial x_i}}{\left[x_{i+1}\right]}^{p_i}\right|\le {\displaystyle \frac{1}{2}}\sum _{i=1}^{k-2}{\xi }_i^{{\displaystyle \frac{2\rho }{\sigma }}}+{\displaystyle \frac{1}{4}}{\xi }_{k-1}^{{\displaystyle \frac{2\rho }{\sigma }}}+d_k\left({\overline{x}}_k\right){\xi }_k^{{\displaystyle \frac{2\rho }{\sigma }}},$$

(15)

$$\left|\sum _{i=1}^k{\displaystyle \frac{\partial W_k}{\partial x_i}}{\varphi }_i\right|\le {\displaystyle \frac{1}{2}}\sum _{i=1}^{k-1}{\xi }_i^{{\displaystyle \frac{2\rho }{\sigma }}}+{\gamma }_k\left({\overline{x}}_k\right){\xi }_k^{{\displaystyle \frac{2\rho }{\sigma }}},$$

(16)

where $c_k\left({\overline{x}}_k\right),d_k\left({\overline{x}}_k\right),{\gamma }_k\left({\overline{x}}_k\right)$ are positive smooth functions. Taking Equations (14)–(16) to (13) yields the following:

$$\begin{array}{cc}\hfill {\dot{V}}_k\le & -\sum _{i=1}^{k-1}\left(n-k+1+{\alpha }_i\left({\overline{x}}_i\right)\right){\xi }_i^{{\displaystyle \frac{2\rho }{\sigma }}}+\left(c_k\left({\overline{x}}_k\right)+d_k\left({\overline{x}}_k\right)+{\gamma }_k\left({\overline{x}}_k\right)\right){\xi }_k^{{\displaystyle \frac{2\rho }{\sigma }}}\hfill \\ \hfill & +{\left[{\xi }_k\right]}^{{\displaystyle \frac{2\rho -r_k-{\tau }_k}{\sigma }}}{\left[x_{k+1}^{\ast }\right]}^{p_k}+{\left[{\xi }_k\right]}^{{\displaystyle \frac{2\rho -l_k-{\tau }_k}{\sigma }}}\left|{\left[x_{k+1}\right]}^{p_k}-{\left[x_{k+1}^{\ast }\right]}^{p_k}\right|,\hfill \end{array}$$

(17)

Therefore, the virtual controller $x_{k+1}^{\ast }$ can be designed as follows:

$$x_{k+1}^{\ast }:=-{\beta }_k\left({\overline{x}}_k\right){\left[{\xi }_k\right]}^{{\displaystyle \frac{r_{k+1}}{\sigma }}},$$

(18)

where the positive smooth function is

$${\beta }_k\left({\overline{x}}_k\right)={\left[n-k+1+{\alpha }_k\left({\overline{x}}_k\right)+c_k\left({\overline{x}}_k\right)+d_k\left({\overline{x}}_k\right)+{\gamma }_k\left({\overline{x}}_k\right)\right]}^{1/a_k}.$$

(19)

Since $p_k\ge a_k$ and $n-k+1+{\alpha }_k\left({\overline{x}}_k\right)+c_k\left({\overline{x}}_k\right)+d_k\left({\overline{x}}_k\right)+{\gamma }_k\left({\overline{x}}_k\right)\ge 1$, the following can be obtained:

$$\begin{array}{cc}\hfill {\left[{\xi }_k\right]}^{{\displaystyle \frac{2\rho -r_k-{\tau }_k}{\sigma }}}{\left[x_{k+1}^{\ast }\right]}^{p_k}& \le -{\left(n-k+1+{\alpha }_k\left({\overline{x}}_k\right)+c_k\left({\overline{x}}_k\right)+d_k\left({\overline{x}}_k\right)+{\gamma }_k\left({\overline{x}}_k\right)\right)}^{p_k/a_k}{\xi }_k^{{\displaystyle \frac{2\rho }{\sigma }}}\hfill \\ \hfill & \le -\left(n-k+1+{\alpha }_k\left({\overline{x}}_k\right)+c_k\left({\overline{x}}_k\right)+d_k\left({\overline{x}}_k\right)+{\gamma }_k\left({\overline{x}}_k\right)\right){\xi }_k^{{\displaystyle \frac{2\rho }{\sigma }}}\hfill \end{array}$$

(20)

Taking the above equation to (18) yields

$${\dot{V}}_k\le -\sum _{i=1}^k\left(n-k+1+{\alpha }_i\left({\overline{x}}_i\right)\right){\xi }_i^{{\displaystyle \frac{2\rho }{\sigma }}}+{\left[{\xi }_k\right]}^{{\displaystyle \frac{2\rho -r_k-{\tau }_k}{\sigma }}}\left({\left[x_{k+1}\right]}^{p_k}-{\left[x_{k+1}^{\ast }\right]}^{p_k}\right),$$

(21)

Thus, we can get the recurrence formula ${\dot{V}}_k$.

Step 4: The recursive proof shows that the Lyapunov function can be defined when $k=n$, and with the action of the virtual controller, ${\dot{V}}_k$ satisfies (13).

$$V_n\left({\overline{x}}_n\right)=V_{n-1}\left({\overline{x}}_{n-1}\right)+W_n\left({\overline{x}}_n\right),W_n={\int }_{x_n^{\ast }}^{x_n}{\left[{\left[s\right]}^{{\displaystyle \frac{\sigma }{n_n}}}-{\left[x_n^{\ast }\right]}^{{\displaystyle \frac{\sigma }{{\sigma }_n}}}\right]}^{{\displaystyle \frac{2\rho -r_n-{\tau }_n}{\sigma }}}ds,$$

(22)

In the last step, select the last virtual controller:

$$\begin{array}{cc}\hfill u& =x_{n+1}=x_{n+1}^{\ast }:=-{\beta }_n\left({\overline{x}}_n\right){\left[{\xi }_n\right]}^{{\displaystyle \frac{r_{n+1}}{\sigma }}}\hfill \\ \hfill & =-{\beta }_n(·){\left({\left[x_n\right]}^{{\displaystyle \frac{\sigma }{r_n}}}-{\left[x_n^{\ast }\right]}^{{\displaystyle \frac{\sigma }{r_n}}}\right)}^{{\displaystyle \frac{r_{n+1}}{\sigma }}}\hfill \\ \hfill & =-{\beta }_n(·){\left({\left[x_n\right]}^{{\displaystyle \frac{\sigma }{r_n}}}-{\left[-{\beta }_{n-1}\left({\overline{x}}_{n-1}\right){\left[{\xi }_{n-1}\right]}^{{\displaystyle \frac{r_n}{\sigma }}}\right]}^{{\displaystyle \frac{\sigma }{r_n}}}\right)}^{{\displaystyle \frac{r_{n+1}}{\sigma }}}\hfill \\ \hfill & =-{\beta }_n(·){\left({\left[x_n\right]}^{{\displaystyle \frac{\sigma }{r_n}}}+{\beta }_{n-1}^{{\displaystyle \frac{\sigma }{r_n}}}(·)\left({\left[x_{n-1}\right]}^{{\displaystyle \frac{\sigma }{r_{n-1}}}}+\cdots +{\beta }_2^{{\displaystyle \frac{\sigma }{r_3}}}\left({\overline{x}}_2\right)\left({\left[x_2\right]}^{{\displaystyle \frac{\sigma }{r_2}}}+{\beta }_1^{{\displaystyle \frac{\sigma }{r_2}}}\left({\overline{x}}_1\right){\left[x_1\right]}^{{\displaystyle \frac{\sigma }{r_1}}}\right)\right)\right)}^{{\displaystyle \frac{r_{n+1}}{\sigma }}},\hfill \end{array}$$

(23)

where the positive smooth function ${\beta }_n(·)=1+{\alpha }_n\left({\overline{x}}_n\right)+c_n\left({\overline{x}}_n\right)+d_n\left({\overline{x}}_n\right)+{\gamma }_n\left({\overline{x}}_n\right)$, and $+{\alpha }_n\left({\overline{x}}_n\right)$ is also a positive smooth function. Thus we have

$${\dot{V}}_n\le -\sum _{i=1}^n\left(1+{\alpha }_i\left({\overline{x}}_i\right)\right){\xi }_i^{{\displaystyle \frac{2\rho }{\sigma }}}\le -\sum _{i=1}^n{\xi }_i^{{\displaystyle \frac{2\rho }{\sigma }}}$$

(24)

It can be seen that under the action of the virtual controller (23) ${\dot{V}}_n<0$, and the Lyapunov function $V_n$ defined according to (22) is positively definite regular [28]. According to Lemma 1, it can be concluded that the closed-loop system (3) is globally asymptotically stable under the control action (23). □

In Theorem 1, an unknown power does not need to satisfy any restrictions on power exponents, and its boundaries can be the same $\left(\left[a_i,b_i\right]:=[a,b],i=1,2,\cdots ,n-1\right)$, piecewise crossed $\left(a_{i-1}<b_i,i=2,3,\cdots n\right)$ or otherwise. In the case of the same boundary, such as the special case in Lemma 6, there is a permitted upper boundary ${\delta }^{\ast }$ for any $\overline{\delta }<{\delta }^{\ast }$, which guarantees $D_k>0,k=0,1,\cdots n-1$; Lemma 8 guarantees the existence of the homogeneous weight coefficients defined by Equation (4). Therefore, a state feedback controller can be obtained by this method to realize the global asymptotic stability of the system.

From the above proof, it can be seen that the proof procedure in Theorem 1 is more advanced than the generalized exponents integration method proposed in [36], and can deal more effectively with P-normal form nonlinear systems where the unknown exponents drift in a more general interval, rather than the special form in [41].

4.3. Simulation Examples

In this subsection, we present two illustrative examples, a second-order and a third-order nonlinear system, to demonstrate the design procedure and validate the effectiveness of the proposed control design method.

4.3.1. Example 1: Stabilization Analysis of a Second-Order Nonlinear System

Consider the following second-order nonlinear system [36]:

$${\dot{x}}_1=x_2^{p_1}+x_1,x_2=u,$$

(25)

where $p_1\in \left[a_1,b_1\right]:=\left[{\displaystyle \frac{13}{5}},{\displaystyle \frac{17}{5}}\right],p_2\in \left[a_2,b_2\right]:=[1,1]$.

According to the Definition 6, it can be obtained that $D_0=1,D_1=a_1+1={\displaystyle \frac{18}{5}}$, suggesting that the system has an interval monotonically homogeneous degree. Based on the Lemma 8, by initializing $r_1=17,r_3=1$ and calculating $r_2={\displaystyle \frac{1}{D_1}}\left(D_0{r}_1+b_2{r}_3\right)={\displaystyle \frac{5}{18}}\left(r_1+1\right)=5$, a set of homogeneous weight coefficients $r_1=17,r_2=5,r_3=1$ can be obtained, and the homogeneous degree ${\tau }_1\in [-4,0),{\tau }_2=-4,\sigma =17,\rho =17$ can be calculated. At the same time, because of $\left|{\varphi }_1\left(x_1\right)\right|\le \left|x_1\right|$ and $b_1{r}_2={\displaystyle \frac{17}{5}}\times 5\le r_1,b_2{r}_3=1\le r_{j}j=1,2$, Assumption 2 can be satisfied.

Defining

$$V_1\left(x_1\right)={\int }_0^{x_1}{\left[s\right]}^{1-{\displaystyle \frac{{\tau }_1}{17}}}ds={\displaystyle \frac{1}{2-\frac{{\tau }_1}{17}}}{x}_1^{2-{\displaystyle \frac{{\tau }_1}{17}}},$$

and taking the derivative of the above formula gives

$${\dot{V}}_1={\xi }_1^{1-{\displaystyle \frac{{\tau }_1}{17}}}{\dot{x}}_1={\xi }_1^{1-{\displaystyle \frac{{\tau }_1}{17}}}{\left[x_2^{\ast }\right]}^{p_1}+{\xi }_1^{1-{\displaystyle \frac{{\tau }_1}{17}}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)+{\xi }_1^{1-{\displaystyle \frac{{\tau }_1}{17}}}{x}_1,$$

by further constructing a virtual controller

$$x_2^{\ast }=-{\beta }_1(·){\xi }_1^{{\displaystyle \frac{r_2}{\sigma }}}=-{\beta }_1(·){\xi }_1^{{\displaystyle \frac{5}{17}}},$$

where ${\beta }_1(·)={\left({\displaystyle \frac{5}{2}}+{\alpha }_1\left(x_1\right)\right)}^{1/a_1},{\alpha }_1\left(x_1\right):=2+x_1^2$; then, we can obtain the following relation based on $p_1{r}_2-{\tau }_1=r_1=1$:

$${\dot{V}}_1\le -\left({\displaystyle \frac{5}{2}}+{\alpha }_1\left(x_1\right)\right){\xi }_1^2+{\left|{\xi }_1\right|}^{\left|-{\displaystyle \frac{{\tau }_1}{17}}\right.}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)+{\left|{\xi }_1\right|}^{1-{\displaystyle \frac{{\tau }_1}{17}}}\left|x_1\right|$$

Defining

$$V_2=V_1+W_2,W_2={\int }_{x_2^{\ast }}^{x_2}{\left({\left[s\right]}^{17/5}-{\left[x_2^{\ast }\right]}^{17/5}\right)}^{33/17}ds$$

and calculating the derivation of above $V_2$, we can obtain the following:

$$\begin{array}{cc}\hfill {\dot{V}}_2=& {\dot{V}}_1+{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1+{\xi }_2^{{\displaystyle \frac{33}{17}}}{\dot{x}}_2\le -\left({\displaystyle \frac{5}{2}}+{\alpha }_1\left(x_1\right)\right){\xi }_1^2+{\left|{\xi }_1\right|}^{1-{\displaystyle \frac{{\tau }_1}{17}}}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)\hfill \\ & +{\left|{\xi }_1\right|}^{1-{\displaystyle \frac{{\tau }_1}{17}}}\left|x_1\right|+\left|{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1\right|+{\xi }_2^{{\displaystyle \frac{33}{17}}}u,\hfill \end{array}$$

The three items on the right side of the above formula are estimated, respectively, as follows:

(a) Since ${\xi }_1={\left[x_1\right]}^{{\displaystyle \frac{\sigma }{{\gamma }_1}}}=x_1,{\tau }_1\in [-4,0]$, ${\left|{\xi }_1\right|}^{1-{\displaystyle \frac{{\tau }_1}{17}}}\left|x_1\right|$ can be estimated as follows:

$${\left|{\xi }_1\right|}^{1-{\displaystyle \frac{{\tau }_1}{17}}}\left|x_1\right|={\left|{\xi }_1\right|}^{2-{\displaystyle \frac{{\tau }_1}{\sigma }}}={\left|{\xi }_1\right|}^{-{\displaystyle \frac{{\tau }_1}{\sigma }}}{\xi }_1^2\le \gamma \left({\overline{x}}_1\right){\xi }_1^2$$

where the positive smooth function satisfies $\gamma \left({\overline{x}}_1\right):=\left(1+x_1^2\right)=\left(1+{\xi }_1^2\right)\ge {\left|{\xi }_1\right|}^{-{\displaystyle \frac{{\tau }_1}{\sigma }}}$.

(b) According to the Lemma 4 and $p_1{r}_2=r_1+{\tau }_1$, we can obtain the following:

$$\begin{array}{cc}\hfill \left|{\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right|& =\left|{\left({\left[x_2\right]}^{{\displaystyle \frac{17}{5}}}\right)}^{{\displaystyle \frac{5p_1}{17}}}-{\left({\left[x_2^{\ast }\right]}^{{\displaystyle \frac{17}{5}}}\right)}^{{\displaystyle \frac{5p_1}{17}}}\right|\le 2^{1-{\displaystyle \frac{5p_1}{17}}}{\left|{\left[x_2\right]}^{{\displaystyle \frac{17}{5}}}-{\left[x_2^{\ast }\right]}^{{\displaystyle \frac{17}{5}}}\right|}^{{\displaystyle \frac{5p_1}{17}}}\hfill \\ & ={\left.2^{1-{\displaystyle \frac{5p_1}{17}}}\left|{\xi }_2{\xi }^{{\displaystyle \frac{5p_1}{17}}}=2^{1-{\displaystyle \frac{5p_1}{17}}}\right|{\xi }_2\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}\hfill \end{array}.$$

Thus, $\left|\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)\right|\le 2^{1-{\displaystyle \frac{5p_1}{17}}}{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}$ can be obtained based on Lemma 4. By further selecting $\eta ={\displaystyle \frac{1}{2}}{\left(2^{1-{\displaystyle \frac{5p_1}{17}}}{\displaystyle \frac{17+{\tau }_1}{34}}\right)}^{-1}$ and Lemma 2, we have the following:

$$\begin{array}{cc}& {\left|{\xi }_1\right|}^{1-{\displaystyle \frac{{\tau }_1}{17}}}\left|\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)\right|\le 2^{1-{\displaystyle \frac{5p_1}{17}}}{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}{\left|{\xi }_1\right|}^{1-{\displaystyle \frac{{\tau }_1}{17}}}\hfill \\ & \le 2^{1-{\displaystyle \frac{5p_1}{17}}}{\displaystyle \frac{17+{\tau }_1}{34}}\eta {\xi }_1^2+2^{1-{\displaystyle \frac{5p_1}{17}}}{\displaystyle \frac{17-{\tau }_1}{34}}{\eta }^{-{\displaystyle \frac{17-{\tau }_1}{17+{\tau }_1}}}{\xi }_2^2\le {\displaystyle \frac{1}{2}}{\xi }_1^2+c{\xi }_2^2\hfill \end{array}$$

(c) We estimate the term $\left|{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1\right|$. The following relationship can be derived based on Lemma 5:

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1\right|& =\left|{\displaystyle \frac{\partial {\int }_{x_2^{\ast }}^{x_2}{\left({\left[s\right]}^{17/5}-{\left[x_2^{\ast }\right]}^{17/5}\right)}^{33/17}ds}{\partial x_1}}{\dot{x}}_1\right|\hfill \\ & =\left|{\displaystyle \frac{33}{17}}{\int }_{x_2^{\ast }}^{x_2}{\left({\left[s\right]}^{17/5}-{\left[x_2^{\ast }\right]}^{17/5}\right)}^{16/17}ds{\displaystyle \frac{\partial {\left[x_2^{\ast }\right]}^{17/5}}{x_1}}{\dot{x}}_1\right|\hfill \\ & \le \left|{\displaystyle \frac{33}{17}}{\left|{\xi }_2\right|}^{16/17}\left|x_2-x_2^{\ast }\right|{\displaystyle \frac{\partial {\left[x_2^{\ast }\right]}^{17/5}}{\partial x_1}}{\dot{x}}_1\right|\hfill \end{array}$$

According to Lemma 4, we have

$${\left|x_2-x_2^{\ast }\right|}^{{\displaystyle \frac{17}{5}}}\le 2^{{\displaystyle \frac{17}{5}}-1}\left|{\left[x_2\right]}^{{\displaystyle \frac{17}{5}}}-{\left[x_2^{\ast }\right]}^{{\displaystyle \frac{17}{5}}}\right|=2^{{\displaystyle \frac{17}{5}}-1}\left|{\xi }_2\right|,\mathrm{and}{\displaystyle \frac{33}{17}}{\left(2^{{\displaystyle \frac{17}{5}}-1}\right)}^{{\displaystyle \frac{5}{17}}}<4$$

which yields the following:

$$\left|{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1\right|<4{\left|{\xi }_2\right|}^{21/17}\left|{\displaystyle \frac{\partial {\left[x_2^{\ast }\right]}^{17/5}}{\partial x_1}}{\dot{x}}_1\right|$$

With the definition of $x_2^{\ast }$, the following can be obtained:

$${\left[x_2^{\ast }\right]}^{{\displaystyle \frac{17}{5}}}={\left[-{\beta }_1{\xi }_1^{{\displaystyle \frac{17}{5}}}\right]}^{{\displaystyle \frac{17}{5}}}=-{\beta }_1^{{\displaystyle \frac{17}{5}}}{\xi }_1:=-{\overline{\beta }}_1{\xi }_1=-{\left({\displaystyle \frac{5}{2}}+{\alpha }_1\left(x_1\right)\right)}^{{\displaystyle \frac{17}{13}}}{\xi }_1$$

With the definition of ${\dot{x}}_1$, the following can be obtained:

$$\left|{\dot{x}}_1\right|={\left|x_2\right|}^{p_1}={|{\left|x_2\right|}^{{\displaystyle \frac{17}{5}}}|}^{{\displaystyle \frac{5p_1}{17}}}={\left|{\xi }_2-{\overline{\beta }}_1{\xi }_1\right|}^{{\displaystyle \frac{5p_1}{17}}}\le {\left(\left|{\xi }_2\right|+\left|{\overline{\beta }}_1{\xi }_1\right|\right)}^{{\displaystyle \frac{5p_1}{17}}}\le {\left|{\xi }_2\right|}^{\left|+{\displaystyle \frac{{\tau }_1}{17}}\right.}+{\left|{\overline{\beta }}_1\right|}^{\left|+{\displaystyle \frac{{\tau }_1}{17}}\right.}{\left|{\xi }_1\right|}^{\left|+{\displaystyle \frac{{\tau }_1}{17}}\right.}$$

According to the above inequality, we have

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{\partial {\left[x_2^{\ast }\right]}^{17/5}}{\partial x_1}}{\dot{x}}_1\right|& \le \left|{\displaystyle \frac{\partial {\overline{\beta }}_1}{\partial x_1}}{\xi }_1+{\overline{\beta }}_1{\displaystyle \frac{\partial {\xi }_1}{\partial x_1}}\right|\left({\left|{\xi }_2\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}+{\left|{\overline{\beta }}_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}{\left|{\xi }_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}+\left|{\xi }_1\right|\right)\hfill \\ & =h\left({\overline{x}}_2\right)\left({\left|{\xi }_2\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}+{\left|{\overline{\beta }}_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}{\left|{\xi }_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}+\left|{\xi }_1\right|\right),\hfill \end{array}$$

where the positive smooth function $h\left({\overline{x}}_2\right):={\displaystyle \frac{34}{13}}{\left({\displaystyle \frac{5}{2}}+{\alpha }_1\left(x_1\right)\right)}^{{\displaystyle \frac{4}{13}}}{\xi }_1^2+{\left({\displaystyle \frac{5}{2}}+{\alpha }_1\left(x_1\right)\right)}^{{\displaystyle \frac{17}{13}}}$.

Based on the above derivations, the following can be concluded:

$$\begin{array}{cc}& \left|{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1\right|\le 4h\left({\overline{x}}_2\right){\left|{\xi }_2\right|}^{21/17}\left({\left|{\xi }_2\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}+{\left|{\overline{\beta }}_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}{\left|{\xi }_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}+\left|{\xi }_1\right|\right)\hfill \\ \hfill =& 4h\left({\overline{x}}_2\right){\left|{\xi }_2\right|}^{2+{\displaystyle \frac{{\tau }_1}{17}}}+4h\left({\overline{x}}_2\right){\left|{\overline{\beta }}_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}{\left|{\xi }_2\right|}^{{\displaystyle \frac{21}{17}}}{\left|{\xi }_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}+4h\left({\overline{x}}_2\right){\left|{\xi }_2\right|}^{{\displaystyle \frac{21}{17}}}\left|{\xi }_1\right|\hfill \\ \hfill \le & 4h\left({\overline{x}}_2\right){\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{2}{17}}}{\xi }_2^2+4h\left({\overline{x}}_2\right){\left|{\overline{\beta }}_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}{\left|{\xi }_2\right|}^{{\displaystyle \frac{21}{17}}}{\left|{\xi }_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}+4h\left({\overline{x}}_2\right){\left|{\xi }_2\right|}^{{\displaystyle \frac{21}{17}}}{\left|{\xi }_1\right|}^2.\hfill \end{array}$$

Applying Lemma 2 to the last two terms of the above equation, and selecting the appropriate function, we can obtain the following:

$$4h\left({\overline{x}}_2\right){\left|{\overline{\beta }}_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}{\left|{\xi }_2\right|}^{{\displaystyle \frac{21}{17}}}{\left|{\xi }_1\right|}^{1+{\displaystyle \frac{{\tau }_1}{17}}}\le {\displaystyle \frac{1}{2}}{\xi }_1^2+d_1\left({\overline{x}}_2\right){\xi }_2^2,4h\left({\overline{x}}_2\right){\left|{\xi }_2\right|}^{{\displaystyle \frac{21}{17}}}\left|{\xi }_1\right|\le {\displaystyle \frac{1}{2}}{\xi }_1^2+d_2\left({\overline{x}}_2\right){\xi }_2^2,$$

where

$$\begin{array}{cc}\hfill d_1\left({\overline{x}}_2\right)& :=7c_3^{{\displaystyle \frac{38}{21}}}\left({\overline{x}}_2\right){\left({\displaystyle \frac{5}{2}}+{\alpha }_1\left(x_1\right)\right)}^{{\displaystyle \frac{38}{21}}}{\left(1+{\xi }_1^2\right)}^{{\displaystyle \frac{2}{21}}}{\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{2}{17}}},\hfill \\ \hfill d_2\left({\overline{x}}_2\right)& :=7c_3^{{\displaystyle \frac{38}{21}}}\left({\overline{x}}_2\right){\left(1+{\xi }_1^2\right)}^{{\displaystyle \frac{2}{21}}}{\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{2}{17}}}.\hfill \end{array}$$

Taking the above inequality to ${\dot{V}}_2$ yields the following:

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -\left({\displaystyle \frac{5}{2}}+{\alpha }_1\left(x_1\right)\right){\xi }_1^2+\gamma \left(x_1\right){\xi }_1^2+{\displaystyle \frac{1}{2}}{\xi }_1^2+c{\xi }_2^2+4h\left({\overline{x}}_2\right){\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{2}{17}}}{\xi }_2^2\hfill \\ & +\left[{\displaystyle \frac{1}{2}}{\xi }_1^2+d_1\left({\overline{x}}_2\right){\xi }_2^2\right]+\left[{\displaystyle \frac{1}{2}}{\xi }_1^2+d_2\left({\overline{x}}_2\right){\xi }_2^2\right]+{\xi }_2^{{\displaystyle \frac{21}{17}}}u\hfill \\ \hfill \le & -\left(1+{\alpha }_1\left(x_1\right)-\gamma \left(x_1\right)\right){\xi }_1^2+\left(c+4h\left({\overline{x}}_2\right){\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{2}{17}}}+d_1\left({\overline{x}}_2\right)+d_2\left({\overline{x}}_2\right)\right)+{\xi }_2^{{\displaystyle \frac{21}{17}}}u\hfill \\ \hfill \le & -{\xi }_1^2+\left(c+4h\left({\overline{x}}_2\right){\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{2}{17}}}+d_1\left({\overline{x}}_2\right)+d_2\left({\overline{x}}_2\right)\right)+{\xi }_2^{{\displaystyle \frac{21}{17}}}u\hfill \end{array}$$

To derive ${\dot{V}}_2\le -{\xi }_1^2-{\xi }_2^2\le 0$, the following controller is designed:

$$u=-{\beta }_2(·){\xi }_2^{{\displaystyle \frac{13}{\sigma }}}={\beta }_2\left(x_1,x_2\right){\left(x_2^{{\displaystyle \frac{17}{5}}}+{\beta }_1^{{\displaystyle \frac{17}{5}}}\left(x_1\right)x_1\right)}^{{\displaystyle \frac{1}{17}}},$$

where ${\beta }_1(·)={\left({\displaystyle \frac{5}{2}}+{\alpha }_1\left(x_1\right)\right)}^{5/13},{\beta }_2(·)=1+{\alpha }_2\left({\overline{x}}_2\right)+c+4h\left({\overline{x}}_2\right){\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{2}{17}}}+d_1\left({\overline{x}}_2\right)+d_2\left({\overline{x}}_2\right)$. Then, the stabilization analysis is finished.

To verify the controller performance, a numerical simulation was conducted with the initial condition set to $x=(3,2)$. The simulation was performed using MathWorks MATLAB R2023a on a laptop equipped with an AMD Ryzen 7 5800H processor (Advanced Micro Devices, Inc., Santa Clara, CA, USA) and 32 GB of RAM (Samsung Electronics Co., Ltd., Seoul, South Korea). As depicted in Figure 2, the state trajectories are presented for the boundary values of the power exponent’s uncertain interval. The resulting state trajectories demonstrate that the controller effectively drives the system states to the origin even when the power exponent varies within the given bounded interval $p_1\in \left[{\displaystyle \frac{13}{5}},{\displaystyle \frac{17}{5}}\right]$.

For this nonlinear system, the controller is designed in [26] as follows: $u=-{\beta }_1{x}_1^{1/r_1}-{\beta }_2{x}_2^{1/r_2}$, where ${\beta }_1=10,{\beta }_2=9,r_1=1,r_2=1/3$. When $p_1\ne 3$, it cannot be restored to the initial point, as shown in Figure 3.

4.3.2. Example 2: Stabilization Analysis of a Third-Order Nonlinear System

Consider the following third-order nonlinear system

$${\dot{x}}_1=x_2,{\dot{x}}_2=sign\left(x_3\right){\left|x_3\right|}^{p_2}+{\displaystyle \frac{1}{4}}sin\left(x_1\right),{\dot{x}}_3=u$$

(26)

where $p_1\in \left[a_1,b_1\right]=[1,1],p_2\in \left[a_2,b_2\right]=\left[{\displaystyle \frac{13}{5}},3\right],p_3\in \left[a_3,b_3\right]=[1,1]$. Based on Definition 6, we have $D_0=1,D_1=a_2+1={\displaystyle \frac{18}{5}},D_2=\left(a_1+1\right)D_1-b_2{D}_0={\displaystyle \frac{21}{5}}$. Therefore, the above system satisfies Assumption 1. With Lemma 7 and initializing $r_1=5,r_4=1$, $r_2={\displaystyle \frac{1}{D_2}}\left(D_1{r}_1+b_2{b}_3{r}_4\right)=5$, $r_3={\displaystyle \frac{1}{D_1}}\left(D_0{r}_2+b_3{r}_4\right)={\displaystyle \frac{5}{3}},{\tau }_1=0,{\tau }_2\in \left[-{\displaystyle \frac{2}{3}},0\right],{\tau }_3=-{\displaystyle \frac{2}{3}},\rho =\sigma =5$ can be obtained. In addition, $\left|{\varphi }_2\left(x_1\right)\right|={\displaystyle \frac{1}{4}}\left|sin\left(x_1\right)\right|\le {\displaystyle \frac{1}{4}}{x}_1,b_2{r}_3\le 3\times {\displaystyle \frac{5}{3}}\le r_j,j=1,2$. In this way, this system also meets Assumption 2. The global stability controller of the system is derived as follows:

Defining $V_1\left(x_1\right)={\int }_0^{x_1}{\left[{\left[s\right]}^{{\displaystyle \frac{\sigma }{{\tau }_1}}}-0\right)}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}ds={\displaystyle \frac{1}{2}}{x}_1^2$, its derivation is calculated as follows:

$${\dot{V}}_1=x_1{\dot{x}}_1={\xi }_1{x}_2={\xi }_1{x}_2^{\ast }+{\xi }_1\left(x_2-x_2^{\ast }\right).$$

According to Lemma 2, by establishing a virtual controller $x_2^{\ast }=-{\beta }_1(·){\xi }_1^{{\displaystyle \frac{r_2}{\sigma }}}=-{\xi }_1$ and choosing $\eta (x,y)=1/4$, the following relationship is obtained:

$${\dot{V}}_1=-{\xi }_1^2+{\xi }_1\left(x_2-x_2^{\ast }\right)\le -{\xi }_1^2+\left|{\xi }_1\right|\left|{\xi }_2\right|\le -{\xi }_1^2+{\displaystyle \frac{1}{8}}{\xi }_1^2+2{\xi }_2^2.$$

Defining $V_2=V_1+W_2$, where $W_2={\int }_{x_2^{\ast }}^{x_2}{\left(x_2-x_2^{\ast }\right)}^{1-{\displaystyle \frac{{\tau }_2}{5}}}ds$, we have the following:

$$\begin{array}{cc}\hfill {\dot{V}}_2={\dot{V}}_1+{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1& +{\xi }_2^{1-{\displaystyle \frac{{\tau }_2}{5}}}{\dot{x}}_2\le -{\displaystyle \frac{7}{8}}{\xi }_1^2+2{\xi }_2^2+{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1+{\xi }_2^{1-{\displaystyle \frac{{\tau }_2}{5}}}{\dot{x}}_2\hfill \\ \hfill \left|{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1\right|& \le \left(1-{\displaystyle \frac{{\tau }_2}{5}}\right)\left|x_2-x_2^{\ast }\right|{\left|{\xi }_2\right|}^{-{\displaystyle \frac{{\tau }_2}{5}}}\left|{\displaystyle \frac{\partial \left[x_2^{\ast }\right]}{\partial x_1}}{\dot{x}}_1\right|\hfill \\ & =\left(1-{\displaystyle \frac{{\tau }_2}{5}}\right){\left|{\xi }_2\right|}^{1-{\displaystyle \frac{{\tau }_2}{5}}}\left|{\displaystyle \frac{\partial \left[x_2^{\ast }\right]}{\partial x_1}}{\dot{x}}_1\right|\hfill \\ & =\left(1-{\displaystyle \frac{{\tau }_2}{5}}\right){\left|{\xi }_2\right|}^{1-{\displaystyle \frac{{\tau }_2}{5}}}\left|{\xi }_2+{\xi }_1\right|\hfill \\ & \le \left(1-{\displaystyle \frac{{\tau }_2}{5}}\right){\left|{\xi }_2\right|}^{2-{\displaystyle \frac{{\tau }_2}{5}}}+\left(1-{\displaystyle \frac{{\tau }_2}{5}}\right)\left|{\xi }_1\right|{\left|{\xi }_2\right|}^{1-{\displaystyle \frac{{\tau }_2}{5}}},\hfill \end{array}$$

According to ${\left|{\xi }_2\right|}^{2-{\displaystyle \frac{{\tau }_2}{5}}}<\left(1+{\xi }_2^2\right){\xi }_2^2,{\left|{\xi }_2\right|}^{1-{\displaystyle \frac{{\tau }_2}{5}}}<\left(1+{\xi }_2^2\right){\xi }_2^1,\left(1-{\displaystyle \frac{{\tau }_2}{5}}\right)<2$ and Lemma 2, the following can be obtained:

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1\right|& \le \left(1-{\displaystyle \frac{{\tau }_2}{5}}\right){\left|{\xi }_2\right|}^{-{\displaystyle \frac{{\tau }_2}{5}}}+\left(1-{\displaystyle \frac{{\tau }_2}{5}}\right)\left|{\xi }_1\right|{\left|{\xi }_2\right|}^{1-{\displaystyle \frac{{\tau }_2}{5}}}\hfill \\ & \le 2\left(1+{\xi }_2^2\right){\xi }_2^2+2\left(1+{\xi }_2^2\right)\left|{\xi }_1\right|\left|{\xi }_2\right|\hfill \\ & \le 2\left(1+{\xi }_2^2\right){\xi }_2^2+{\displaystyle \frac{1}{8}}{\xi }_1^2+c_2\left({\overline{x}}_2\right){\xi }_2^2\hfill \end{array}$$

where $c_1\left({\overline{x}}_2\right)=2\left(1+{\xi }_2^2\right),c_2\left({\overline{x}}_2\right)=8\left(1+{\xi }_1^2+{\xi }_2^2\right)$.

Since

$${\displaystyle \frac{1}{4}}{\xi }_2^{1-{\displaystyle \frac{{\tau }_2}{5}}}sin\left(x_1\right)\le {\displaystyle \frac{1}{4}}{\left|{\xi }_2\right|}^{1-{\displaystyle \frac{{\tau }_2}{5}}}\left|{\xi }_1\right|\le {\displaystyle \frac{1}{8}}{\xi }_1^2+c_3\left({\overline{x}}_2\right){\xi }_2^2,c_3\left({\overline{x}}_2\right)=\left(1+{\xi }_1^2+{\xi }_2^2\right)$$

according to the above inequality, the relationship for ${\dot{V}}_2$ can be derived:

$$\begin{array}{cc}\hfill {\dot{V}}_2& \le -{\displaystyle \frac{7}{8}}{\xi }_1^2+2{\xi }_2^2+{\displaystyle \frac{1}{8}}{\xi }_1^2+c_1\left({\overline{x}}_2\right){\xi }_2^2+c_2\left({\overline{x}}_2\right){\xi }_2^2\hfill \\ & +{\displaystyle \frac{1}{4}}{\xi }_2^{1-{\displaystyle \frac{{\tau }_2}{5}}}sin\left(x_1\right)+{\xi }_2^{1-{\displaystyle \frac{{\tau }_2}{5}}}\left({\left[x_3\right]}^{p_2}-{\left[x_3^{\ast }\right]}^{p_2}\right)+{\xi }_2^{1-{\displaystyle \frac{{\tau }_2}{5}}}{\left[x_3^{\ast }\right]}^{p_2}\hfill \\ & \le -{\displaystyle \frac{5}{8}}{\xi }_1^2+\left(2+c_1\left({\overline{x}}_2\right)+c_2\left({\overline{x}}_2\right)+c_3\left({\overline{x}}_2\right)\right){\xi }_2^2\hfill \\ & +{\left({\xi }_2\right)}^{1-{\displaystyle \frac{{\tau }_2}{5}}}\left({\left[x_3\right]}^{p_2}-{\left[x_3^{\ast }\right]}^{p_2}\right)+{\left({\xi }_2\right)}^{1-{\displaystyle \frac{{\tau }_2}{5}}}{\left[x_3^{\ast }\right]}^{p_2},\hfill \end{array}$$

Then, the virtual controller is constructed according to the above formula

$$\begin{array}{c}{x}_3^{\ast }=-{\beta }_2(·){\left[{\xi }_2\right]}^{{\displaystyle \frac{1}{3}}}\\ {\beta }_2(·)={\left(2{\displaystyle \frac{5}{8}}+c_1\left({\overline{x}}_2\right)+c_2\left({\overline{x}}_2\right)+c_3\left({\overline{x}}_2\right)\right)}^{{\displaystyle \frac{5}{13}}}={\left(13{\displaystyle \frac{5}{8}}+9{\xi }_1^2+11{\xi }_2^2\right)}^{{\displaystyle \frac{5}{13}}}\end{array}$$

Because of $p_2{r}_3=r_2+{\tau }_2=5+{\tau }_2$, we have the following:

$${\dot{V}}_2\le -{\displaystyle \frac{5}{8}}{\xi }_1^2-{\displaystyle \frac{5}{8}}{\xi }_2^2+{\xi }_2^{1-{\displaystyle \frac{{\tau }_2}{5}}}\left({\left[x_3\right]}^{p_2}-{\left[x_3^{\ast }\right]}^{p_2}\right)$$

We define $V_3=V_2+W_3$, where $W_3={\int }_{x_3^{\ast }}^{x_3}{\left({\left[x_3\right]}^3-{\left[x_3^{\ast }\right]}^3\right)}^{{\displaystyle \frac{9}{5}}}ds$. Taking the derivative with respect to $V_3$ yields the following:

$$\begin{array}{cc}\hfill {\dot{V}}_3& ={\dot{V}}_2+\sum _{i=1}^2{\displaystyle \frac{\partial W_3}{\partial x_i}}{\dot{x}}_i+{\xi }_3^{{\displaystyle \frac{9}{5}}}{\dot{x}}_3\hfill \\ & \le -{\displaystyle \frac{5}{8}}{\xi }_1^2-{\displaystyle \frac{5}{8}}{\xi }_2^2+{\xi }_2^{1-{\displaystyle \frac{{\tau }_2}{5}}}\left({\left[x_3\right]}^{p_2}-{\left[x_3^{\ast }\right]}^{p_2}\right)+\sum _{i=1}^2{\displaystyle \frac{\partial W_3}{\partial x_i}}{\dot{x}}_i+{\xi }_3^{{\displaystyle \frac{9}{5}}}{\dot{x}}_3,\hfill \end{array}$$

According to the Lemma 2, the following inequality is obtained:

$$\begin{array}{c}{\xi }_2^{1-{\displaystyle \frac{{\tau }_2}{5}}}\left({\left[x_3\right]}^{p_2}-{\left[x_3^{\ast }\right]}^{p_2}\right)\le 2^{1-{\displaystyle \frac{p_2}{3}}}{\left|{\xi }_2\right|}^{1-{\displaystyle \frac{{\tau }_2}{5}}}{\left|{\xi }_3\right|}^{1+{\displaystyle \frac{{\tau }_2}{5}}}\le {\displaystyle \frac{1}{16}}{\xi }_2^2+9{\xi }_3^2,\\ {\displaystyle \sum _{i=1}^2}\left|{\displaystyle \frac{\partial W_3}{\partial x_i}}{\dot{x}}_i\right|\le {\displaystyle \frac{9}{5}}\left|x_3-x_3^{\ast }\right|{\left|{\xi }_3\right|}^{{\displaystyle \frac{4}{5}}}{\displaystyle \sum _{i=1}^2}\left|{\displaystyle \frac{\partial {\left[x_3^{\ast }\right]}^3}{\partial x_i}}{\dot{x}}_i\right|\le 3{\left|{\xi }_3\right|}^{{\displaystyle \frac{17}{15}}}{\displaystyle \sum _{i=1}^2}\left|{\displaystyle \frac{\partial {\left[x_3^{\ast }\right]}^3}{\partial x_i}}{\dot{x}}_i\right|.\end{array}$$

Because of ${\left(x_3^{\ast }\right)}^3=-{\beta }_2^3(.){\xi }_2=-{\left(13{\displaystyle \frac{5}{8}}+9{\xi }_1^2+11{\xi }_2^2\right)}^{{\displaystyle \frac{15}{13}}}{\xi }_2$ and ${\xi }_2=x_2+x_1$, the partial of ${\left(x_3^{\ast }\right)}^3$ with respect to $x_1$ and $x_2$ can be calculated:

$$\begin{array}{c}{\displaystyle \frac{\partial {\left(x_3^{\ast }\right)}^3}{\partial x_1}}=-{\displaystyle \frac{15}{13}}{\left(13{\displaystyle \frac{5}{8}}+9{\xi }_1^2+11{\xi }_2^2\right)}^{{\displaystyle \frac{2}{13}}}\left(18{\xi }_1+22{\xi }_2\right){\xi }_2-{\left(13{\displaystyle \frac{5}{8}}+9{\xi }_1^2+11{\xi }_2^2\right)}^{{\displaystyle \frac{15}{13}}},\\ {\displaystyle \frac{\partial {\left(x_3^{\ast }\right)}^3}{\partial x_2}}=-{\displaystyle \frac{15}{13}}{\left(13{\displaystyle \frac{5}{8}}+9{\xi }_1^2+11{\xi }_2^2\right)}^{{\displaystyle \frac{2}{13}}}22{\xi }_2^2-{\left(13{\displaystyle \frac{5}{8}}+9{\xi }_1^2+11{\xi }_2^2\right)}^{{\displaystyle \frac{15}{13}}},\end{array}$$

According to the above derivation, the following can be obtained:

$$\begin{array}{cc}\hfill \sum _{i=1}^2\left|{\displaystyle \frac{\partial W_3}{\partial x_i}}{\dot{x}}_i\right|& \le 3h\left({\overline{x}}_2\right){\left|{\xi }_3\right|}^{{\displaystyle \frac{17}{15}}}\left(\left|{\xi }_2\right|+\left|{\xi }_1\right|+\left|x_3^{p_2}\right|+{\displaystyle \frac{1}{4}}\left|{\xi }_1\right|\right)\hfill \\ & =3h\left({\overline{x}}_2\right){\left|{\xi }_3\right|}^{{\displaystyle \frac{17}{15}}}\left(\left|{\xi }_2\right|+{\displaystyle \frac{5}{4}}\left|{\xi }_1\right|+{\left|{\xi }_3\right|}^{1+{\displaystyle \frac{{\tau }_2}{17}}}+{\beta }_2^{1+{\displaystyle \frac{{\tau }_2}{5}}}(·){\left|{\xi }_2\right|}^{1+{\displaystyle \frac{{\tau }_2}{5}}}\right)\hfill \end{array}$$

where $h\left({\overline{x}}_2\right):={\displaystyle \frac{15}{13}}{\left(13{\displaystyle \frac{5}{8}}+9{\xi }_1^2+11{\xi }_2\right)}^{{\displaystyle \frac{2}{13}}}\left(9{\xi }_1^2+53{\xi }_2^2\right)+2{\left(13{\displaystyle \frac{5}{8}}+9{\xi }_1^2+11{\xi }_2^2\right)}^{{\displaystyle \frac{15}{13}}}$.

The following inequality can be derived based on Lemma 2

$$\begin{array}{c}3h\left({\overline{x}}_2\right){\left|{\xi }_3\right|}^{{\displaystyle \frac{17}{15}}}\left|{\xi }_2\right|\le {\displaystyle \frac{1}{32}}{\xi }_2^2+c_4\left({\overline{x}}_3\right){\xi }_3^2,\\ {\displaystyle \frac{4}{15}}h\left({\overline{x}}_2\right){\left|{\xi }_3\right|}^{{\displaystyle \frac{17}{15}}}\left|{\xi }_1\right|\le {\displaystyle \frac{1}{8}}{\xi }_1^2+c_5\left({\overline{x}}_3\right){\xi }_2^2,\\ 3h\left({\overline{x}}_2\right){\beta }_2^{1+{\displaystyle \frac{{\tau }_2}{5}}}(·){\left|{\xi }_3\right|}^{{\displaystyle \frac{17}{15}}}{\left|{\xi }_2\right|}^{1+{\displaystyle \frac{{\tau }_2}{5}}}\le {\displaystyle \frac{1}{32}}{\xi }_2^2+c_6\left({\overline{x}}_3\right){\xi }_3^2\\ 3h\left({\overline{x}}_2\right){\left|{\xi }_3\right|}^{{\displaystyle \frac{17}{15}}}{\left|{\xi }_3\right|}^{1+{\displaystyle \frac{{\tau }_2}{5}}}\le 3h\left({\overline{x}}_2\right)\left(1+{\xi }_3^2\right){\xi }_3^2:=c_7\left({\overline{x}}_3\right){\xi }_3^2\end{array}$$

where

$$c_4\left({\overline{x}}_3\right)=46h^{{\displaystyle \frac{32}{17}}}\left({\overline{x}}_3\right)\left(1+{\xi }_2^2+{\xi }_3^2\right),c_5\left({\overline{x}}_3\right)=21h^{{\displaystyle \frac{32}{17}}}\left({\overline{x}}_3\right)\left(1+{\xi }_1^2+{\xi }_2^2\right),$$

$$c_6\left({\overline{x}}_3\right)=29\left(1+h^2\left({\overline{x}}_3\right)\right)\left(1+{\beta }_2^2(·)\right)\left(1+{\xi }_2^2+{\xi }_3^2\right)$$

Based on the above derivation, the following results can be concluded:

$$\sum _{i=1}^2\left|{\displaystyle \frac{\partial W_3}{\partial x_i}}{\dot{x}}_i\right|\le {\displaystyle \frac{1}{8}}{\xi }_1^2+{\displaystyle \frac{1}{16}}{\xi }_2^2+\sum _{k=4}^7{c}_k(·){\xi }_3^2$$

Then, ${\dot{V}}_3$ can finally be derived:

$${\dot{V}}_3\le -{\displaystyle \frac{1}{2}}{\xi }_1^2-{\displaystyle \frac{1}{2}}{\xi }_2^2+\left(9+\sum _{k=4}^7{c}_k(·)\right){\xi }_3^2+{\xi }_3^{{\displaystyle \frac{17}{15}}}u$$

To obtain ${\dot{V}}_3\le -{\displaystyle \frac{1}{2}}{\xi }_1^2-{\displaystyle \frac{1}{2}}{\xi }_2^2-{\displaystyle \frac{1}{2}}{\xi }_3^2$, the controller can be defined as follows:

$$u=-{\beta }_3{\left(x_3^3+{\beta }_2^3\left(x_2+{\beta }_1{x}_1\right)\right)}^{1/5}$$

where

$${\beta }_1=1,{\beta }_2={\left(13{\displaystyle \frac{5}{8}}+9{\xi }_1^2+11{\xi }_2^2\right)}^{{\displaystyle \frac{5}{13}}},{\beta }_3=9.5+c_4\left({\overline{x}}_3\right)+c_5\left({\overline{x}}_3\right)+c_6\left({\overline{x}}_3\right)+c_7\left({\overline{x}}_3\right)$$

In this way, the stabilization analysis is completed.

When the initial condition of the system is $\left(x_1\left(0\right),x_2\left(0\right),x_3\left(0\right)\right)=(0.5,-1,2)$, and $p_2\in \left[{\displaystyle \frac{13}{5}},3\right]$, the control performance is verified through simulation. The resulting state trajectories are presented in Figure 4, which demonstrates that the system states successfully converge to the origin for different values of the power exponent within the uncertain interval.

The simulation results for both the second-order and third-order systems confirm that the proposed interval power integration method successfully stabilizes the system, driving the states to the origin even when the power exponent varies within its defined interval. In contrast, Figure 3 illustrates that a conventional controller may fail, leading to instability under similar uncertain conditions.

Furthermore, the simulation results suggest that the control performance can be influenced by the value of the power exponent within the uncertain interval. For the second-order system in this study, a smaller power exponent value appears to result in a slightly faster convergence rate, as shown in the comparison in Figure 2.

5. Superheated Steam Temperature Control: Control Design and Simulation Verification

When designing the control system, the thermal process is generally approximated as a common second-order inertial model near a certain working condition point. As mentioned in the introduction, such common single-input single-output thermal processes can be converted to the P-normal form structure. For example, the differential equation corresponding to a second order thermal process $G\left(s\right)={\displaystyle \frac{b_0}{a_2{s}^2+a_{1}s+a_0}}$ is $\ddot{y}={\displaystyle \frac{b_0}{a_0}}u-{\displaystyle \frac{1}{a_1}}\dot{y}-{\displaystyle \frac{1}{a_2}}y$. In this way, we can describe this system with a power $p=1$ if we let $y=x_1,\dot{y}=x_2$:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2^p,\hfill \\ {\dot{x}}_2={\displaystyle \frac{b_0}{a_0}}u+{\varphi }_2\left(x\right),\hfill \end{array}\right.$$

(27)

where ${\varphi }_2\left(x\right)=-{\displaystyle \frac{1}{a_1}}{x}_2-{\displaystyle \frac{1}{a_2}}{x}_1$.

Then, the traditional linear controller, such as PID, can be designed accordingly. However, when the working conditions or the dynamic characteristics of the model change, this method cannot obtain good control performance [42].

The interval power integration control method is well-suited for such challenges, as it is designed to maintain robust performance when the system’s dynamic characteristics vary within a bounded range. Accordingly, this section applies the proposed method to design a controller for a typical thermal process and validates its effectiveness through simulation.

5.1. Simplified Analysis

As mentioned above, most of the single input single output thermal processes, such as superheated steam temperature control, reheated steam temperature control, etc., can be expressed as [6]:

$$\begin{array}{cc}\hfill & {\dot{x}}_i=sign\left(x_{i+1}\right){\left|x_{i+1}\left(t\right)\right|}^{1+{\delta }_t},i=1,2,\dots ,n-1,\hfill \\ \hfill & {\dot{x}}_n=u+\alpha \left(x\right)\hfill \end{array}$$

(28)

According to Equation (28), it is found that the power index varies in a smaller range and varies around 1, and the nonlinear term exists only in the last term.

It is evident that Equation (28) is a special case of an interval power exponential nonlinear system. According to the Lemma 7, when p = 1, if the range of power exponents satisfies $\left|{\delta }_i\right|<\overline{\delta },i=1,2,\dots ,n-1$ and $\overline{\delta }<{\delta }^{\ast }\left(n\right)$, the relation $D_k\left(\overline{\delta }\right)>0,k=0,1,\cdots ,n-1$ holds, satisfying Assumption 1. Then, according to Lemma 8, a set of homogeneous weight coefficients can be found to make the system (28) satisfy the requirement of interval monotonically homogeneous degree. We have the following Lemma.

Lemma 10.

According to the homogeneous weight defined by Lemma 8, the system (28) can recursively obtain $r_i\ge r_{\mathrm{n}+1},i=2,\dots ,n$ when it takes $r_1\ge r_{\mathrm{n}+1}$.

The proof of Lemma 10 is provided in the Appendix A.

According to Lemma 10, we get $b_n{r}_{n+1}=r_{n+1}\le r_j,j=1,2,\cdots ,n$. Then, following the procedure in Lemma 9, we can establish that

$$\left|a\left({\overline{x}}_n\right)\right|\le {\tilde{\eta }}_n\left(x_n\right)\left({\left|{\xi }_1\right|}^{{\displaystyle \frac{r_n+{\tau }_n}{\sigma }}}+{\left|{\xi }_2\right|}^{{\displaystyle \frac{r_n+{\tau }_n}{\sigma }}}+\cdots +{\left|{\xi }_n\right|}^{{\displaystyle \frac{r_n+{\tau }_n}{\sigma }}}\right).$$

That is, the simplified system (28) satisfies the conditions of Assumptions 1 and 2. Therefore, the controller design methodology and stability analysis presented in Section 4 are directly applicable to this system. The concrete proof process follows the same procedure as that of Theorem 1 and is thus omitted for brevity.

The superheated steam temperature plays an important role in the safe and economic operation of the power plant [3,43]. A conventional control approach involves approximating the plant dynamics with a second-order inertial model at a specific operating point, upon which a traditional PID controller is then designed. While this method can achieve satisfactory performance at the design point, its effectiveness often deteriorates when the plant’s working conditions vary significantly.

The control method proposed in this paper is designed to maintain satisfying performance even as plant characteristics change. Accordingly, this section considers the leading area of the superheated steam temperature of a once through boiler as a representative case study. The procedure involves first deriving a simplified P-normal form model from the system’s transfer function. Subsequently, the proposed interval power integration control method is applied, and the resulting control performance is analyzed.

The transfer function model of the superheated steam temperature to the disturbance of the water spraying volume under a specific load point of a boiler is as follows

$$G\left(s\right)={\displaystyle \frac{-8}{225s^2+30s+1}},$$

(29)

which is a typical second-order inertia process. Let $y=x_1,\dot{y}=x_2$, Equation (29) transforms to the following:

$$\ddot{y}=-{\displaystyle \frac{8}{225}}u-{\displaystyle \frac{30}{225}}\dot{y}-{\displaystyle \frac{1}{225}}y$$

(30)

or

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2^p,\hfill \\ {\dot{x}}_2=-{\displaystyle \frac{8}{225}}u+{\varphi }_2\left(x_1,x_2\right)\hfill \end{array}\right.$$

(31)

where $p=1,{\varphi }_2\left(x_1,x_2\right)=-{\displaystyle \frac{30}{225}}{x}_2-{\displaystyle \frac{1}{225}}{x}_1$. Generally, the typical higher order model of thermal engineering can also be transformed into a class of simplified P-normal form nonlinear systems.

In practical operation, the dynamic characteristics of the thermal process are subject to change due to load variations and various on-site disturbances. Consequently, the model parameters will not remain constant. Specifically, the power exponent p and ${\varphi }_2$ may vary during operation, transforming the system into an uncertain nonlinear system.

From Definition 6, we can get $D_0=1,D_1=a_1+1$, and obviously $D_k>0$, satisfying Assumption 1. According to Assumption 2 and Lemma 8, it follows that $\left(b_1-a_1\right)\le 1$, that is, the power exponent varies within a range of less than 1. For the purpose of subsequent discussion, suppose that the center of the power index p of the controlled plant is 1 and fluctuates within the range 0.8, obtaining $p\in \left[{\displaystyle \frac{3}{5}},{\displaystyle \frac{7}{5}}\right]$.

In order to verify that the transformed nonlinear system (31) can accurately represent the controlled plant (29), the step response curves of the transfer function model (29) and the transformed model (31) at $p=1,p=4/5$ and $p=6/5$ are given respectively.

Figure 5 validates the P-normal form model (31) against the original second-order system (29). It is observed that the step response of the P-normal form model with $p=1$ closely matches that of the original system, confirming the accuracy of the transformation. Furthermore, the responses for $p=4/5$ and $p=6/5$ form an envelope around the original system’s response. This indicates that the P-normal form model (31) not only represents the nominal system well but can also effectively capture a wider range of its potential dynamic behaviors.

5.2. Controller Design

The above analysis demonstrates that the transformed nonlinear model (31) is a valid representation of the original plant dynamics (29). Since this model fits the required P-normal form, the control methodology introduced in Section 4 can be outlined here.

The specific steps are as follows:

(1) According to $p_1\in \left[a_1,b_1\right]:=\left[{\displaystyle \frac{3}{5}},{\displaystyle \frac{7}{5}}\right],p_2\in \left[a_2,b_2\right]:=[1,1]$ and definition 6, $D_0=1,D_1=a_1+1={\displaystyle \frac{8}{5}}$ can be obtained, which indicates that the system (31) satisfies the assumption. Based on the Lemma 8, and initializing $r_1=1$, we have $r_2=1,r_3=3/5,\rho =\sigma =1,{\tau }_1\in \left[-{\displaystyle \frac{2}{5}},{\displaystyle \frac{2}{5}}\right],{\tau }_2=-{\displaystyle \frac{2}{5}}$. And since the relation $\left|{\varphi }_n\left(x\right)\right|\le \left|x_1\right|+\left|x_2\right|$ and $b_1{r}_2={\displaystyle \frac{7}{5}}\times {\displaystyle \frac{3}{5}}={\displaystyle \frac{21}{25}}\le r_1,b_2{r}_3={\displaystyle \frac{3}{5}}\le r_{j}j=1,2$ both hold, the assumption is satisfied.

(2) By defining $V_1\left(x_1\right)={\int }_0^{x_1}{\left({\left[s\right]}^{{\displaystyle \frac{\sigma }{1_1}}}-0\right)}^{{\displaystyle \frac{2\rho -r_1-{\tau }_1}{\sigma }}}ds,{\xi }_1={\left[x_1\right]}^{{\displaystyle \frac{\sigma }{r_1}}}-{\left[x_1^{\ast }\right]}^{{\displaystyle \frac{\sigma }{{\frac{1}{1}}_1^2}}}=\left[x_1\right]$ and differentiating $V_1$, we have the following:

$${\dot{V}}_1={\left[x_1\right]}^{1-{\tau }_1}{\dot{x}}_1={\left[x_1\right]}^{1-{\tau }_1}{\left[x_2\right]}^{p_1}={\left[{\xi }_1\right]}^{1-{\tau }_1}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)+{\left[{\xi }_1\right]}^{1-{\tau }_1}{\left[x_2^{\ast }\right]}^{p_1},$$

Let $x_2^{\ast }=-{\beta }_1\left[{\xi }_1\right],{\xi }_2={\left[x_2\right]}^{{\displaystyle \frac{\sigma }{\frac{1}{2}}}}-{\left[x_2^{\ast }\right]}^{{\displaystyle \frac{\sigma }{{\sigma }^2}}}$ and ${\tau }_1=p_1{r}_2-r1\Rightarrow p_1-{\tau }_1=1$; the following condition can be obtained by Lemma 4.

$${\dot{V}}_1={\left[{\xi }_1\right]}^{1-{\tau }_1}\left({\left[x_2\right]}^{p_1}-{\left[x_2^{\ast }\right]}^{p_1}\right)-{\left[{\xi }_1\right]}^{1-{\tau }_1}{\beta }_1^{p_1}{\xi }_1^{p_1}\le 2^{1-p_1}{\left[{\xi }_1\right]}^{1-{\tau }_1}{\left|{\xi }_2\right|}^{p_1}-{\beta }_1^{p_1}{\xi }_1^2,\mid$$

According to Lemma 2 and letting $\eta ={\displaystyle \frac{1}{2}}{\left(2^{1-p_1}{\displaystyle \frac{1-{\tau }_1}{2}}\right)}^{-1},c_1={\displaystyle \frac{p_1}{2}}{\left(2\left(2^{1-p_1}{\displaystyle \frac{1-{\tau }_1}{2}}\right)\right)}^{{\displaystyle \frac{1-{\tau }_1}{p_1}}}$, the following can be obtained:

$$2^{1-p_1}{\left|{\xi }_1\right|}^{\mid -{\tau }_1}{\left|{\xi }_2\right|}^{p_1}\le 2^{1-p_1}\left({\displaystyle \frac{1-{\tau }_1}{2}}\eta {\left|{\xi }_1\right|}^2+{\displaystyle \frac{p_1}{2}}{\eta }^{-{\displaystyle \frac{1-{\tau }_1}{p_1}}}{\xi }_2^2\right)\le {\displaystyle \frac{1}{2}}{\xi }_1^2+c_1{\xi }_2^2$$

Letting ${\beta }_1={\left(2\right)}^{1/a_1}$, then ${\beta }_1^{p_1}={\left(2\right)}^{p_1/a_1}$. Because of $p_1\ge a_1$, the derivation of $V_1$ is calculated as follows:

$${\dot{V}}_1\le {\displaystyle \frac{1}{2}}{\xi }_1^2+c_1{\xi }_2^2-2{\xi }_1^2=-{\displaystyle \frac{3}{2}}{\xi }_1^2+c_1{\xi }_2^2$$

(3) Defining $V_2=V_1+W_2$, where $W_2={\int }_{x_2^{\ast }}^{x_2}{\left(\left[s\right]-\left[x_2^{\ast }\right]\right)}^{1-{\tau }_2}ds$, then the following condition can be derived by differentiating $W_2$:

$${\dot{W}}_2={\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1+{\displaystyle \frac{\partial W_2}{\partial x_2}}{\dot{x}}_2$$

where

$$\begin{array}{c}{\displaystyle \frac{\partial W_2}{\partial x_2}}{\dot{x}}_2={\xi }_2^{{\displaystyle \frac{7}{5}}}{\dot{x}}_2\\ {\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1={\displaystyle \frac{7}{5}}{\int }_{x_2^{\ast }}^{x_2}{\left(\left[s\right]-\left[x_2^{\ast }\right]\right)}^{{\displaystyle \frac{2}{5}}}ds{\displaystyle \frac{\partial \left[x_2^{\ast }\right]}{\partial x_1}}{\dot{x}}_1={\displaystyle \frac{7}{5}}{\left|{\xi }_2\right|}^{{\displaystyle \frac{2}{5}}}\left(x_2-x_2^{\ast }\right){\displaystyle \frac{\partial \left[x_2^{\ast }\right]}{\partial x_1}}{\dot{x}}_1={\displaystyle \frac{7}{5}}{\left|{\xi }_2\right|}^{{\displaystyle \frac{7}{5}}}\left(-{\beta }_1\right)x_2^{p_1}\end{array}$$

Letting $c_2=max\left\{n^{p-1},1\right\}$, we have $x_2^{p_1}={\left[{\xi }_2-{\beta }_1{\xi }_1\right]}^{p_1}\le c_2{\left|{\xi }_2\right|}^{p_1}+c_2{\left|{\beta }_1{\xi }_1\right|}_1^{p_1}$ we get the following condition:

$${\displaystyle \frac{\partial W_2}{\partial x_1}}{\dot{x}}_1\le {\displaystyle \frac{7}{5}}{c}_2{\beta }_1\left({\xi }_2^2{\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{4}{5}}}+{\beta }_1^{p_1}{\xi }_1^{p_1}{\left|{\xi }_2\right|}^{{\displaystyle \frac{7}{5}}}\right).$$

According to Lemma 2 and letting $\eta ={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{7}{5}}{c}_2{\beta }_1^{1+p_1}{\displaystyle \frac{p_1}{\frac{7}{5}+p_1}}{\left(1+{\xi }_1^2\right)}^{{\displaystyle \frac{4}{5}}}\right)}^{-1}$, the following can be obtained:

$${\displaystyle \frac{7}{5}}{\beta }_1^{1+p_1}{\xi }_1^{p_1}{\left|{\xi }_2\right|}^{{\displaystyle \frac{7}{5}}}\le {\displaystyle \frac{1}{2}}{\xi }_1^2+d_1{\xi }_2^2,$$

where

$$d_1={\beta }_1^{p_1}{\displaystyle \frac{\frac{7}{5}}{\frac{7}{5}+p_1}}{\left(2{\displaystyle \frac{7}{5}}{\beta }_1^{1+p_1}{\displaystyle \frac{p_1}{\frac{7}{5}+p_1}}{\left(1+{\xi }_1^2\right)}^{{\displaystyle \frac{4}{5}}}\right)}^{{\displaystyle \frac{5p_1}{7}}}\left({\xi }_2^2+1\right)$$

It further can be obtained that

$${\dot{V}}_2\le -{\displaystyle \frac{3}{2}}{\xi }_1^2+c_1{\xi }_2^2+{\displaystyle \frac{7}{5}}{\beta }_1{c}_2{\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{4}{5}}}{\xi }_2^2+\left({\displaystyle \frac{1}{2}}{\xi }_1^2+d_1{\xi }_2^2\right)+{\xi }_2^{{\displaystyle \frac{7}{5}}}\left(-{\displaystyle \frac{8}{225}}u-{\displaystyle \frac{30}{225}}{x}_2-{\displaystyle \frac{1}{225}}{x}_1\right),$$

By design, the control input

$$u=-{\displaystyle \frac{30}{8}}{x}_2-{\displaystyle \frac{1}{8}}{x}_1+{\displaystyle \frac{225}{8}}{\beta }_2{\xi }_2^{3/5}$$

(32)

where ${\beta }_2=c_1+{\displaystyle \frac{7}{5}}{\beta }_1{c}_2{\left(1+{\xi }_2^2\right)}^{{\displaystyle \frac{4}{5}}}+d_1+1$; we can obtain ${\dot{V}}_2\le -{\xi }_1^2-{\xi }_2^2$, and the closed-loop system (31) is globally asymptotically stable under the control action (32).

5.3. Simulation Verification

In this subsection, simulation-based verification is provided for the proposed control strategy. To demonstrate its robustness and superior performance, the proposed controller is compared against both a traditional PID and a state feedback controller under scenarios with different fixed and time-varying power exponents.

The interval power integration method designed in this paper is essentially a state feedback controller, suggesting that the state $x_1$ and $x_2$ of the system (31) will be used in the procedure of the controller design. The state $x_1$ represents the response of super-heater temperature to the amount of cooling water spray, and its value is measurable; $x_2$ is the derivative of the state parameter $x_1$ and cannot be measured during actual operation. Since the superheated temperature changes slowly during the actual operation of the power plant, the derivative of $x_1$ can be calculated by using backward difference in the simulation at k time: ${\dot{x}}_1\left(k\right)=\left(x_1\left(k\right)-x_1(k-1)\right)/T$. Then, we can obtain the estimate of the state parameter $x_2$ at k time: ${\widehat{x}}_2\left(k\right)={\left({\dot{x}}_1\left(k\right)\right)}^{1/p}$, further using ${\widehat{x}}_2$ as the estimate of $x_2$ for the solution and simulation of the control effect. The simulation results show that this substitution does not affect the performance of the proposed interval power integration method.

5.3.1. Simulation Example 1

To verify the control performance, the designed homogeneous domination-based controller (32) was compared with the traditional PID controller and the state feedback controller, hereinafter abbreviated as ‘hd’, ‘pid’, and ‘sfc’, respectively. In Example 1, three simulation cases are considered, i.e., $p_1=0.8,p_1=1,p_1=1.2$ and the initial condition is $\left(x_1\left(0\right),x_2\left(0\right)\right)=(3,0.5)$.

The parameters of PID are obtained by the tuning method combined with the critical proportional band and Nyquist method: $k_p=0.05,k_i=0.0001,k_d=0.0015$. The control function of the state feedback controller is $u=k_1{x}_1+k_2{x}_2$, with the control parameters obtained by combining Lyapunov stability theorem and frequency domain analysis: $k_1=0.05,k_2=0.0015$.

The simulation results are shown in Figure 6, Figure 7 and Figure 8. In these figures, $x_{1hd},x_{2hd},u_{hd},x_{1pid},$$x_{2pid},u_{pid},x_{1sfc},x_{2sfc},u_{sfc}$ represent the state $x_1,x_2$ and the control action u under the action of the proposed homogeneous domination-based controller, PID, and state feedback control methods, respectively.

To quantitatively characterize and compare the control performance, two key performance indices are adopted: the Integral of Absolute Error (IAE) and convergence time (CT). The IAE provides a measure of the cumulative control deviation, while the convergence time reflects the response speed. These metrics will be used for control performance comparison of different control strategies.

The control performance metrics for the three cases in Example 1 are summarized in

Table 2. Metrics comparison in Example 1.

	Cases	Case 1 (p = 0.8)				Case 2 (p = 1)				Case 3 (p = 1.2)
Method		${\mathbf{IAE}}_{{\mathbf{x}}_{\mathbf{1}},\mathbf{1}}$	${\mathbf{CT}}_{{\mathbf{x}}_{\mathbf{1}},\mathbf{1}}$	${\mathbf{IAE}}_{{\mathbf{x}}_{\mathbf{2}},\mathbf{1}}$	${\mathbf{CT}}_{{\mathbf{x}}_{\mathbf{2}},\mathbf{1}}$	${\mathbf{IAE}}_{{\mathbf{x}}_{\mathbf{1}},\mathbf{2}}$	${\mathbf{CT}}_{{\mathbf{x}}_{\mathbf{1}},\mathbf{2}}$	${\mathbf{IAE}}_{{\mathbf{x}}_{\mathbf{2}},\mathbf{2}}$	${\mathbf{CT}}_{{\mathbf{x}}_{\mathbf{2}},\mathbf{2}}$	${\mathbf{IAE}}_{{\mathbf{x}}_{\mathbf{1}},\mathbf{3}}$	${\mathbf{CT}}_{{\mathbf{x}}_{\mathbf{1}},\mathbf{3}}$	${\mathbf{IAE}}_{{\mathbf{x}}_{\mathbf{2}},\mathbf{3}}$	${\mathbf{CT}}_{{\mathbf{x}}_{\mathbf{2}},\mathbf{3}}$
HD (proposed)		71.59	33	4.23	33	80.29	37	5.49	47	88.94	45	7.08	60
SFC		130.64	71	5.58	80	145.24	64	6.63	81	195.58	115	9.08	120
PID		138.57	76	5.68	80	153.66	72	6.74	81	185.06	110	9.24	120
Improvements (with PID)		48.33%	56.58%	25.53%	58.75%	47.75%	48.61%	18.55%	41.98%	51.94%	59.09%	23.38%	50%

. In the table, subscripts are used to denote state variables and simulation cases. For instance, ${\mathrm{IAE}}_{x_1,1}$ represents the IAE index of state variable $x_1$ in simulation case 1. Furthermore, the performance improvement percentage of the proposed method over PID control was quantified to better demonstrate its superior performance.

Quantitative performance analysis: As shown in Table 2, the proposed ‘hd’ controller consistently outperforms both the state feedback control and PID controller across all test cases. In the nominal case (Case 2, p = 1), the ‘hd’ controller reduces the convergence time for state $x_1$ to 37 s, a significant improvement over the 64 s for SFC and 72 s for PID. This advantage is even more pronounced in the off-nominal cases. For instance, in Case 3 (p = 1.2), the IAE for $x_1$ under the ‘hd’ controller is 88.94, whereas the IAE for SFC and PID controllers are 195.58 and 185.06, respectively. The “Improvements” row in Table 2 quantifies this superiority, showing that the proposed method achieves a performance improvement of up to 59.09% over the traditional PID controller.

Qualitative response characteristics: This quantitative superiority is visually corroborated by the state trajectory curves in Figure 6, Figure 7 and Figure 8. In every scenario, the proposed ‘hd’ controller exhibits a significantly faster transient response, with less overshoot and quicker settling compared to the other methods. While the PID and SFC controllers show sluggish or oscillatory behavior, particularly as the power exponent p deviates from the nominal value of 1 (as seen in Figure 8), the ‘hd’ controller maintains a stable and well-damped response.

Robustness verification: The results strongly validate the robustness of the proposed method against parameter uncertainty. As the power exponent p changes from 0.8 to 1.2, the performance of the PID and SFC controllers degrades substantially, with convergence times and IAE values increasing by over 50%. In contrast to this, the performance metrics for the ‘hd’ controller show a much smaller degradation, demonstrating its ability to maintain effective control despite significant variations in the plant’s nonlinear dynamics. This confirms that the interval power integration approach successfully handles the defined parameter uncertainty.

5.3.2. Simulation Example 2

In order to verify the influence of p continuously changing in the specified range on the control performance, the designed controller (32) is compared with the state feedback controller and the traditional PID controller. In this simulation example, we set the power index to $p_1=1+0.2sin\left(0.1\pi t\right)$. The initial condition is $\left(x_1\left(0\right),x_2\left(0\right)\right)=(3,0.5)$. The PID controller and state feedback controller are consistent with those implemented in Section 5.3.1.

Simulation results are shown in Figure 9. The control performance metrics in Example 2 are summarized in

Table 3. Metrics comparison in Example 2.

	Cases	Example 2
Method		${\mathbf{IAE}}_{{\mathbf{x}}_{\mathbf{1}}}$	${\mathbf{CT}}_{{\mathbf{x}}_{\mathbf{1}}}$	${\mathbf{IAE}}_{{\mathbf{x}}_{\mathbf{2}}}$	${\mathbf{CT}}_{{\mathbf{x}}_{\mathbf{2}}}$
HD (proposed)		70.78	37	4.29	54
SFC		120.94	77	5.36	86
PID		123.71	79	5.44	86
Improvements (with PID)		42.79%	53.16%	21.14%	37.21%

. The data in Table 3 clearly demonstrates the superiority of the proposed controller. It achieved an IAE of 70.78 for state $x_1$, which is a 42.79% improvement over the PID controller’s IAE of 123.71. The most significant advantage is seen in the convergence time ${\mathrm{CT}}_{x_1}$, where the proposed method (37 s) is more than twice as fast as the PID (79 s) and SFC (77 s) controllers. This indicates a much faster and more efficient control action in a time-varying environment.

The graphical results in Figure 9 corroborate these quantitative findings. The proposed ‘hd’ controller exhibits a rapid and stable convergence to the origin. In contrast, both the PID and SFC controllers display a significantly more sluggish response, struggling to eliminate the steady-state error as the system dynamics continuously change. This demonstrates the proposed method’s strong robustness, as it effectively maintains performance despite the persistent variation in the power exponent.

6. Conclusions

This paper addressed the robust control problem for a class of P-normal form nonlinear systems characterized by unknown but bounded power exponents. To solve this challenge, a novel interval power integration-based nonlinear suppression control method was proposed. The theoretical foundation was laid by extending the concept of homogeneity to the interval domain, introducing the principles of interval monotonic homogeneity and an admissibility index. Based on this framework, a state feedback controller was recursively designed using a Lyapunov-based approach, with its stability rigorously established through theoretical analysis. The effectiveness and superiority of the proposed method were validated through its application to a superheated steam temperature system. Quantitative analysis demonstrated significant performance improvements over traditional PID and state feedback controllers, with the proposed method reducing key metrics like the Integral of Absolute Error and convergence time by up to 40%.

Future research should address some critical extensions of this work. First, while the current architecture assumes full-state accessibility, developing an observer-based output feedback implementation would enhance its practical utility in measurement-constrained environments. Second, extending the methodology to multivariable (MIMO) systems would broaden its industrial relevance. From an implementation perspective, successful industrial deployment will need to overcome challenges including real-time state derivative estimation with quantized measurements and robustness enhancement against sensor noise. These advancements would establish a theoretically grounded yet practically viable control framework for improving thermal power plant operational flexibility in renewable-rich power grids.