Prescribed Performance Tracking Control of Lower-Triangular Systems with Unknown Fractional Powers

Abstract

This paper is concerned with the tracking control problem for the lower-triangular systems with unknown fractional powers and nonparametric uncertainties. A prescribed performance control approach is put forward as a means of resolving this problem. The proposed control law incorporates a set of barrier functions to guarantee error constraints. Unlike the previous works, our approach works for the cases where the fractional powers, the nonlinearities, and their bounding functions or bounds are totally unknown; no restrictive conditions on the powers, such as power order restriction, specific size limitation or homogeneous condition, are made. Moreover, neither the powers and system nonlinearities nor their bounding functions or bounds are needed. It achieves reference tracking with the preassigned tracking accuracy and convergence speed. In addition, our controller is simple, as it does not necessitate parameter identification, function approximation, derivative calculation, or adding a power integrator technique. At the end, a comparative simulation demonstrates the effectiveness and advantage of the proposed approach.

1. Introduction

Due to the theoretical challenge and practical needs, the issue of controlling uncertain nonlinear systems with odd powers has garnered significant attention. Examples of odd-power nonlinear systems in engineering include but are not limited to dynamical boiler–turbine units [1], jet engine compression systems [2], and under-actuated mechanical systems [3]. Compared with the strict feedback systems whose powers are one, odd-power nonlinear systems exhibit more general and complex behavior due to their exponential powers. It is worth claiming that such a system cannot conduct feedback linearization caused by the uncontrollability of its Jacobian matrix and is nonaffine with respect to the control input. Therefore, the control development of odd-power systems poses significant challenges and difficulties.

Various methods for the odd-power nonlinear systems have been developed, which are mainly based on adaptive control [4,5,6,7,8,9,10,11,12,13,14], neural or fuzzy control [15,16,17,18,19,20], funnel or prescribed performance control [14,17,18,19,20,21], and adding a power integrator technique [11,13,22,23,24,25,26,27,28]. The results [22,23,24] work well under the power order restriction (i.e., $p_1\ge p_i\ge \dots \ge p_n,i=1,\cdots ,n$, where $p_1,\dots ,p_n$ are positive odd powers), and they also extend to the odd-power stochastic nonlinear systems [5,8,29]. Later, this restriction was removed, but the powers need to be identical [30]. The above limitations were relaxed [3,4,16,17,20,25,31]; however, the sphere of application for the above methods is limited to integer powers. The control designs for the systems with fractional powers were performed in recent years [6,7,9,10,18,21,32,33]. Nonetheless, in all the aforementioned developments, the powers are required not less than one. In the literature [14,19], the powers can be odd numbers greater than zero and less than one. Notably, a common feature among the aforesaid findings is that the powers should be known. Nevertheless, as summarized in Ref. [27], the aging of hardening spring and diverse operating conditions may result in time-varying and unknown powers in some particular cases, such as the boiler–turbine systems [1] and the under-actuated, weakly coupled mechanical systems [3]. It is noteworthy that the technique of adding a power integrator, an effective strategy for the odd-power systems, is not suitable for the unknown power systems due to its reliance on the system’s homogeneous dominant part. To deal with this problem, numerous approaches were put forward in the literature [11,13,26,27,28], but the bounds of their powers need to be available for the control design. The prior knowledge of the bounds is eliminated, either by imposing order restriction [12] or by placing a specific size limitation on the powers [12,15]. Additionally, the system nonlinearities of the aforementioned results are either considered to be known [32] or constrained by known functions [3,5,8,22,23,24,30] or expressed as a form containing unknown parameters and known functions [11,12,13,28,29].

On the other hand, in the presence of unknown nonlinearities, the results [3,4,16,25,28,32] show only the tracking error’s boundedness, but the specific behavior (e.g., the convergence speed and the accuracy) cannot be predetermined. To overcome this problem, researchers propose the prescribed performance (PPC) method [34,35], which enables quantitatively pre-specification of both transient and steady-state reference tracking behavior. This method has been applied to the first-order systems [36,37,38], feedback linearizable systems [39,40,41], strict-feedback systems [42,43], and odd-power systems [14,17,18,19,20,21]. However, there are some restrictive conditions on the powers and nonlinearities as mentioned above. Therefore, the control development for the odd-power nonlinear systems without the aforesaid requirements still remains open.

Inspired by the above discussion, this paper introduces a PPC strategy for the lower-triangular systems with unknown fractional powers and nonparametric uncertainties. The primary contributions and advantages are outlined:

• Our approach works well when the fractional powers, the system nonlinearities, and their bounds or bounding functions are unknown, without the power order restriction [5,8,12,13,22,23,24,29], the specific size limitation [3,4,5,6,7,8,9,10,11,12,13,15,16,17,18,20,21,22,23,24,25,26,27,28,29,30,31,32,33], or the homogeneous condition [30].
• It guarantees reference tracking with the prescribed convergence speed and accuracy in addition to the boundedness of the tracking error [3,4,16,25,28,32].
• It exhibits simplicity, without function approximation [15,16,17,18,19,20,44,45], parameter identification [4,5,6,7,8,9,10,11,12,13,14], command filtering [44,45], or adding a power integrator technique [11,13,22,23,24,25,26,27,28].

This paper is structured as follows. Section 2 presents the system description and control objective. In Section 3, we state the composition of the proposed control scheme. Its feasibility is demonstrated in Section 4. The simulation results are detailed in Section 5. Finally, we draw the conclusion in Section 6.

Notations: The notations used in this paper are standard and are summarized as follows. ${\mathfrak{R}}^i$ denotes the i-dimensional Euclidean space, with ${\mathfrak{R}}^1=\mathfrak{R}$; $sgn(·)$ denotes the sign function.

2. Problem Description

2.1. System Description

Consider the lower-triangular systems with unknown fractional powers as follows:

$$\left\{\begin{array}{c}{\dot{x}}_i=f_i\left({\overline{x}}_i\right)+{x_{i+1}}^{\frac{p_i}{q_i}},\hfill \\ {\dot{x}}_n=f_n\left({\overline{x}}_n\right)+u^{\frac{p_n}{q_n}},\hfill \\ y=x_{1,}\hfill \end{array}\right.$$

(1)

where ${\overline{x}}_i={[x_1,\cdots ,x_i]}^T\in {\mathfrak{R}}^i$, $i=1,\cdots ,n$; ${\overline{x}}_n$ is composed of the system state; $u\in \mathfrak{R}$ and $y\in \mathfrak{R}$ are the input and the output, respectively; $p_i$ and $q_i$ are positive odd integers, $i=1,\cdots ,n$; $f_i(·)\in \mathfrak{R}$, $i=1,\cdots ,n$ denote the continuous nonlinear functions.

Remark 1. 

Distinct from the available findings, only the basic structural properties of (1) are required for the subsequent control design. Specifically, the nonlinear function $f_i(·)$ and the powers $p_i$ and $q_i$ are unknown, which helps to design the universal controller. In this case, the commonly used model-based technique of adding a power integrator cannot be applied to (1). Moreover, it is worth claiming that the fractional powers are allowed in this study rather than integers [3,4,5,8,16,17,20,22,23,24,25,29,30,31] and are not required to satisfy the power order restriction and specific size limitation [3,4,5,6,7,8,9,10,11,12,13,15,16,17,18,20,21,22,23,24,25,26,27,28,29,30,31,32,33].

2.2. Control Objective

The control target for (1) is let $y\left(t\right)$ follow a reference $r\left(t\right)$, which meets the following assumption [14,16,23,32].

Assumption 1. 

$r\left(t\right)$ and $\dot{r}\left(t\right)$ are bounded on $[0,\infty )$.

To be specific, the desired tracking performance is prescribed by

$$\begin{array}{c}\hfill |y\left(t\right)-r\left(t\right)|<{\xi }_1\left(t\right),t\ge 0,\end{array}$$

(2)

with

$$\begin{array}{c}\hfill {\xi }_1\left(t\right)=({\xi }_{10}-{\xi }_{1\infty })e^{-{\iota }_{1}t}+{\xi }_{1\infty },\end{array}$$

(3)

where ${\iota }_1>0$ and ${\xi }_{1\infty }>0$ are the convergence rate and the tracking accuracy, respectively. Both of them can be chosen by the designer according to the requirements. Moreover, ${\xi }_1\left(0\right)$ should satisfy

$$\begin{array}{c}\hfill |y\left(0\right)-r\left(0\right)|<{\xi }_1\left(0\right).\end{array}$$

(4)

We take into account the problem below.

Problem 1. 

Develop a control for the odd-power systems with unknown fractional powers in (1) to ensure the fulfillment of the performance requirement stated in (2) and guarantee the boundedness of the closed-loop system signals.

3. Control Design

We present a robust PPC approach to address Problem 1. The proposed controller design starts from

$$\begin{array}{c}\hfill e_1\left(t\right)=x_1\left(t\right)-r\left(t\right).\end{array}$$

(5)

Subsequently, a barrier function is utilized to confine $e_1\left(t\right)$:

$$\begin{array}{cc}\hfill & {\eta }_1\left(t\right)=tan\left(\frac{\pi }{2}\frac{e_1\left(t\right)}{{\xi }_1\left(t\right)}\right).\hfill \end{array}$$

(6)

The resulting first intermediate control law is obtained by

$$\begin{array}{c}\hfill {\alpha }_1\left(t\right)=-c_1{\eta }_1\left(t\right),\end{array}$$

(7)

where $c_1>0$ denotes the constant control gain. Proceed with

$$\begin{array}{cc}\hfill & e_i\left(t\right)=x_i\left(t\right)-{\alpha }_{i-1}\left(t\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & {\xi }_i\left(t\right)=({\xi }_{i0}-{\xi }_{i\infty })e^{-{\iota }_{i}t}+{\xi }_{i\infty },\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\eta }_i\left(t\right)=tan\left(\frac{\pi }{2}\frac{e_i\left(t\right)}{{\xi }_i\left(t\right)}\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & {\alpha }_i\left(t\right)=-c_i{\eta }_i\left(t\right),\hfill \end{array}$$

(11)

for $i=2,\cdots ,n$, in a recursive manner, where $c_i>0$ represents the constant control gain; ${\iota }_i>0$ and ${\xi }_{i\infty }>0$ are freely designed by the designer; ${\xi }_i\left(0\right)$ is chosen such that

$$\begin{array}{c}\hfill |e_i\left(0\right)|<{\xi }_i\left(0\right),i=2\cdots ,n.\end{array}$$

(12)

In the end, the final control is obtained as follows:

$$\begin{array}{c}\hfill u\left(t\right)={\alpha }_n\left(t\right).\end{array}$$

(13)

The block diagram of the system with the controller is given in Figure 1.

Remark 2. 

The presented design in (5)–(13) depends on neither the prior knowledge of the powers and the system nonlinearities nor their specific bounding functions or bounds. Even so, no attempt is made for parameter identification [4,5,6,7,8,9,10,11,12,13,14], function approximation [15,16,17,18,19,20,44,45], gain adaptation [46,47], or adding a power integrator technique [11,13,22,23,24,25,26,27,28]. Furthermore, the reference derivative and the intermediate control signal derivatives are not involved in the control law. Nevertheless, this is accomplished without dynamic surface control [44] or auxiliary filters [45]. Thus, the controller exhibits fewer demands and simplicity.

4. Theoretical Analysis

For ease of theoretical analysis, we first give a lemma.

Lemma 1. 

For any $ϵ>0$, ${\dot{\alpha }}_i\left(t\right)$ is bounded on $[0,ϵ)$, if

1. 

$e_i\left(t\right)$ evolves within $\left(-{\xi }_i\left(t\right),{\xi }_i\left(t\right)\right)$ but keeps away from the boundaries on $[0,ϵ)$;

2. 

${\dot{e}}_i\left(t\right)$ is bounded on $[0,ϵ)$;

with $i=1,\cdots ,n$.

Proof. 

Differentiating (7) and (11) by (6) and (10), respectively, yields

$$\begin{array}{cc}\hfill & {\dot{\alpha }}_i\left(t\right)=-c_i·\frac{\pi }{2}·\frac{1}{{\xi }_i\left(t\right)}·\frac{1}{{\gamma }_i\left(t\right)}·{\psi }_i\left(t\right),i=1,\cdots ,n,\hfill \end{array}$$

(14)

with

$$\begin{array}{cc}\hfill & {\gamma }_i\left(t\right)={cos}^2\left(\frac{\pi }{2}\frac{e_i\left(t\right)}{{\xi }_i\left(t\right)}\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\psi }_i\left(t\right)={\dot{e}}_i\left(t\right)-\frac{e_i\left(t\right){\dot{\xi }}_i\left(t\right)}{{\xi }_i\left(t\right)}.\hfill \end{array}$$

(16)

According to (3) and (9), we obtain

$$\begin{array}{c}\hfill 0<{\xi }_{i\infty }\le {\xi }_i\left(t\right)\le {\xi }_{i0},i=1,\cdots ,n,\end{array}$$

(17)

and

$$\begin{array}{c}\hfill -{\rho }_i({\xi }_{i0}-{\xi }_{i\infty })\le \dot{{\xi }_i}\left(t\right)<0,i=1,\cdots ,n.\end{array}$$

(18)

It follows that the reciprocal of (15) is bounded on $[0,ϵ)$ provided that the first assumed condition is met. Note from (17) and (18) that $\left|{\psi }_i\left(t\right)\right|<\infty$, $t<ϵ$, under the assumed conditions of Lemma 1. Thereby, ${\dot{\alpha }}_i\left(t\right)$ in (14) is bounded over $[0,ϵ)$ under the same conditions. □

The result of theory is outlined next.

Theorem 1. 

Under Assumption 1, as well as the initial conditions in (4) and (12), the control scheme developed in (5)–(11) and (13) effectively resolves Problem 1.

Proof. 

The argument starts from positing the claim below.

$$\begin{array}{cc}\hfill & |e_i\left(t\right)|<{\xi }_i\left(t\right),i=1,\cdots ,n,t\ge 0,\hfill \end{array}$$

(19)

This claim is demonstrated through the method of proof by contradiction. From (4) and (12), (19) is met at $t=0$. Note that ${\overline{x}}_n\left(t\right)$ is continuous on $[0,\infty )$. The same holds for $r\left(t\right)$ under Assumption 1. Hence, $e_1\left(t\right)$ in (5) is continuous on $[0,\infty )$. This combined with the continuity of ${\xi }_1\left(t\right)$ in (3) guarantees that ${\eta }_1\left(t\right)$ in (6) and ${\alpha }_1\left(t\right)$ in (7) are continuous if $|e_1\left(t\right)|<{\xi }_1\left(t\right)$. Further, $e_2\left(t\right)$ in (8) does so under the same condition. Proceed with analyzing $e_3\left(t\right),\cdots ,e_n\left(t\right)$, recursively. Then, we deduce that each $e_i\left(t\right)$ is continuous so long as $|e_{\tau }\left(t\right)|<{\xi }_{\tau }\left(t\right)$, where $\tau$ varies from 1 to $i-1$. This information indicates that the existence of $t^{*}>0$ is a necessary condition for the breach of (19) so that

$$\begin{array}{cc}\hfill & \underset{t\to t^{*}}{lim}\left|e_{\tau }\left(t\right)\right|={\xi }_{\tau }\left(t^{*}\right),\tau \in \{1,\cdots ,n\},\hfill \end{array}$$

(20)

and

$$\begin{array}{c}\hfill \begin{array}{c}\left|e_i\left(t\right)\right|<{\xi }_i\left(t\right),i=1,\cdots ,n,t<t^{*}.\hfill \end{array}\end{array}$$

(21)

Next, we posit (20) with (21) and proceed to examine each individual case defined in (20). To maintain concision, the subsequent discussion may omit some functions’ dependence on time or state.

Case 1:

At the outset, we consider

$$\begin{array}{cc}\hfill & \underset{t\to t^{*}}{lim}\left|e_1\left(t\right)\right|={\xi }_1\left(t^{*}\right).\hfill \end{array}$$

(22)

A necessary condition for (22) under (21) is

$$\begin{array}{cc}\hfill & \underset{t\to t^{*}}{lim}\frac{d\left|e_1\left(t\right)\right|}{dt}\ge \underset{t\to t^{*}}{lim}{\dot{\xi }}_1\left(t\right).\hfill \end{array}$$

(23)

Based on (1) and (5), the differential equation for $e_1$ is obtained as follows:

$$\begin{array}{c}\hfill {\dot{e}}_1=f_1+{x_2}^{\frac{p_1}{q_1}}-\dot{r}.\end{array}$$

(24)

Rewrite (8) for $i=2$ as follows:

$$\begin{array}{c}\hfill x_2=e_2+{\alpha }_1.\end{array}$$

(25)

Substituting (25) into (24) yields

$$\begin{array}{c}\hfill {\dot{e}}_1=f_1-\dot{r}+{(e_2+{\alpha }_1)}^{\frac{p_1}{q_1}}.\end{array}$$

(26)

Therefore, it holds that

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\frac{d\left|e_1\left(t\right)\right|}{dt}=\underset{t\to t^{*}}{lim}\mathrm{sgn}\left(e_1\left(t\right)\right)(f_1-\dot{r})+\underset{t\to t^{*}}{lim}\mathrm{sgn}\left(e_1\left(t\right)\right){(e_2+{\alpha }_1)}^{\frac{p_1}{q_1}}.\end{array}$$

(27)

According to (5), (21) and Assumption 1, $x_1$ and $e_2$ are bounded on $[0,t^{*})$. Further, due to the continuity of $f_1$ with respect to its arguments, it follows that $|f_1|<\infty ,t<t^{*}$. Recalling that $\dot{r}$ is bounded on $[0,\infty )$, we know

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\left|\mathrm{sgn}\left(e_1\left(t\right)\right)(f_1-\dot{r})\right|<\infty ,t<t^{*}.\end{array}$$

(28)

It follows from (6) and (22) that

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\mathrm{sgn}\left(e_1\left(t\right)\right){\eta }_1=+\infty .\end{array}$$

(29)

By (7), there is

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\mathrm{sgn}\left(e_1\left(t\right)\right){\alpha }_1=-\infty .\end{array}$$

(30)

Under (20), it holds that

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\mathrm{sgn}\left(e_1\left(t\right)\right)(e_2+{\alpha }_1)=-\infty .\end{array}$$

(31)

Noting that $p_1$ and $q_1$ are positive odd integers, we further obtain

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}{\left(\mathrm{sgn}\left(e_1\left(t\right)\right)(e_2+{\alpha }_1)\right)}^{\frac{p_1}{q_1}}=-\infty .\end{array}$$

(32)

Substituting (28) and (32) into (27) gives

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\frac{d\left|e_1\left(t\right)\right|}{dt}=-\infty .\end{array}$$

(33)

Note from (18) that

$$\begin{array}{c}\hfill -{\rho }_1({\xi }_{10}-{\xi }_{1\infty })\le \underset{t\to t^{*}}{lim}{\dot{\xi }}_1\left(t\right)<0.\end{array}$$

Apparently, (33) contradicts (23). Therefore, (22) is false. Instead, this implies the existence of a constant $o_1>0$ so that

$$\begin{array}{c}\hfill |e_1\left(t\right)|\le {\xi }_1\left(t\right)-o_1<{\xi }_1\left(t\right),t<t^{*}.\end{array}$$

(34)

As a result, ${\eta }_1$ in (6) and ${\alpha }_1$ in (7) are bounded for $t<t^{*}$. Further, by virtue of the boundedness of $f_1$, $e_2$ and $\dot{r}$ on $[0,t^{*})$, the same holds for ${\dot{e}}_1$ in (26). This above facts imply by Lemma 1 that ${\dot{\alpha }}_1$ is bounded for $t<t^{*}$, which contributes to the analysis of $e_2$ next.

Case 2:

Proceed with supposing

$$\begin{array}{cc}\hfill & \underset{t\to t^{*}}{lim}\left|e_2\left(t\right)\right|={\xi }_2\left(t^{*}\right),\hfill \end{array}$$

(35)

which necessitates

$$\begin{array}{cc}\hfill & \underset{t\to t^{*}}{lim}\frac{d\left|e_2\left(t\right)\right|}{dt}\ge \underset{t\to t^{*}}{lim}{\dot{\xi }}_2\left(t\right).\hfill \end{array}$$

(36)

Differentiating (8) for $i=2$ by (1) yields

$$\begin{array}{c}\hfill {\dot{e}}_2=f_2+{x_3}^{\frac{p_2}{q_2}}-\dot{{\alpha }_1}.\end{array}$$

(37)

One sees from (8) for $i=3$ that

$$\begin{array}{c}\hfill x_3=e_3+{\alpha }_2.\end{array}$$

(38)

Substituting (38) into (37) leads to

$$\begin{array}{c}\hfill {\dot{e}}_2=f_2-\dot{{\alpha }_1}+{(e_3+{\alpha }_2)}^{\frac{p_2}{q_2}}.\end{array}$$

(39)

Then, there is

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\frac{d\left|e_2\left(t\right)\right|}{dt}=\underset{t\to t^{*}}{lim}\mathrm{sgn}\left(e_2\left(t\right)\right)(f_2-\dot{{\alpha }_1})+\underset{t\to t^{*}}{lim}\mathrm{sgn}\left(e_2\left(t\right)\right){(e_3+{\alpha }_2)}^{\frac{p_2}{q_2}}.\end{array}$$

(40)

Recall the boundedness of $x_1$ and ${\alpha }_1$ on $[0,t^{*})$. From (8) and (21) for $i=2$, we have $\left|x_2\right|<\infty$, $t<t^{*}$. Due to the continuity of $f_2$ in its arguments, we know $|f_2|<\infty ,t<t^{*}$. This, along with $\left|\dot{{\alpha }_1}\right|<\infty$ over $[0,t^{*})$ established in Case 1, ensures

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\left|\mathrm{sgn}\left(e_2\left(t\right)\right)(f_2-\dot{{\alpha }_1})\right|<\infty ,t<t^{*}.\end{array}$$

(41)

Under (10), (11), and (35), it holds that

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\mathrm{sgn}\left(e_2\left(t\right)\right){\alpha }_2=-\infty .\end{array}$$

(42)

Proceed with (20), and we have

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\mathrm{sgn}\left(e_2\left(t\right)\right)(e_3+{\alpha }_2)=-\infty .\end{array}$$

(43)

Noting that $p_2$ and $q_2$ are positive odd integers, it further holds that

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}{\left(\mathrm{sgn}\left(e_2\left(t\right)\right)(e_3+{\alpha }_2)\right)}^{\frac{p_2}{q_2}}=-\infty .\end{array}$$

(44)

Inserting (41) with (44) into (40) yields

$$\begin{array}{c}\hfill \underset{t\to t^{*}}{lim}\frac{d\left|e_2\left(t\right)\right|}{dt}=-\infty ,\end{array}$$

(45)

which contradicts (36) due to (18). Hence, (35) is false. Then there exists a constant $o_2$ greater than zero so that

$$\begin{array}{c}\hfill |e_1\left(t\right)|\le {\xi }_2\left(t\right)-o_2<{\xi }_2\left(t\right),t<t^{*}.\end{array}$$

(46)

Thus, ${\eta }_2$ in (10) and ${\alpha }_2$ in (11) are bounded during $t<t^{*}$. Recalling $f_2$, $e_3$ and $\dot{{\alpha }_1}$ are bounded over $[0,t^{*})$ established above, ${\dot{e}}_2$ in (39) is bounded for $t<t^{*}$. This along with (46) ensures by Lemma 1 the boundedness of ${\alpha }_2$ on $[0,t^{*})$, which contributes to the analysis of $e_3$ next.

Case i (i = 3, …, n):

Adopt the same method as in Case 2 to analyze $e_i$, $i=3,\dots ,n$, recursively. We can deduce that a constant $o_i$ greater than zero exists, $i=3,\cdots ,n$, so that

$$\begin{array}{cc}\hfill & \left|e_i\left(t\right)\right|\le {\xi }_i\left(t\right)-o_i<{\xi }_i\left(t\right),i=3,\cdots ,n,t<t^{*}.\hfill \end{array}$$

(47)

Now, we arrive at a contradiction between (34), (46), (47), and (20). Therefore, (20) is false, and instead

$$\begin{array}{cc}\hfill & \left|e_i\left(t\right)\right|\le {\xi }_i\left(t\right)-o_i<{\xi }_i\left(t\right),i=1,\cdots ,n,t\ge 0.\hfill \end{array}$$

(48)

Obviously, (19) is correct. This means that the controller not only guarantees the error constraints but also excludes the boundary contact. Hence, the prescribed tracking performance as described in (2) is achieved.

It remains to demonstrate that the state variables, $x_1,\cdots ,x_n$, the intermediate control law, ${\alpha }_1,\cdots ,{\alpha }_{n-1}$, and the control input, u are all bounded. From (48), ${\eta }_i\left(t\right)$, $i=1,\cdots ,n$, in (6) and (10), ${\alpha }_i\left(t\right)$, $i=1,\cdots ,n-1$, in (7) and (11) and $u\left(t\right)$ in (13) are all bounded. Further, under (48) and Assumption 1, we know that $x_i$, $i=1,\cdots ,n$ are bounded on $[0,\infty )$. □

Remark 3. 

Contrary to the classical Lyapunov stability theory, this study employs a constraint analysis based on dialectic by contradiction. It reveals the control system’s robustness against the unknown fractional powers and the unknown uncertainties. This is attributed to the infinity property of the PPC method [34,35], as shown in (31). When it extends to the nonlinear system whose powers are unknown, the infinity property is preserved as shown in (32). This means that the controller has sufficient potential to suppress the effects of the above unknown terms. However, this does not mean that such an infinity phenonmenon would occur in the control implementation. The reason has been elaborated in the related works [48,49,50].

Remark 4. 

Due to the aging of hardening spring and diverse operating conditions, the powers are not fixed but varied within a range in some particular cases, such as the boiler–turbine systems [1] and the under-actuated, weakly coupled mechanical systems [3]. The proposed approach extends to the nonlinear systems with time-varying powers [26] as follows.

$$\left\{\begin{array}{c}{\dot{x}}_i=f_i\left({\overline{x}}_i\right)+{\left[x_{i+1}\right]}^{p_i\left(t\right)},\hfill \\ {\dot{x}}_n=f_n\left({\overline{x}}_n\right)+{\left[u\right]}^{p_n\left(t\right)},\hfill \\ y=x_1,\hfill \end{array}\right.$$

(49)

where $p_i\left(t\right)$ is a time-varying continuous function, $i=1,\cdots ,n$; the power sign function ${\left[·\right]}^{\alpha }$ is defined as ${\left[·\right]}^{\alpha }=sgn\left(·\right){\left|·\right|}^{\alpha }$ for a real number $\alpha >0$. When extending the proposed approach to (49), the existence of ${\left[·\right]}^{\alpha }$ has no influence on the infinity property in (33). Therefore, the robustness of the PPC method [34,35] against the time varying powers is exploited.

Remark 5. 

In the presence of external disturbances, the predetermined transient and steady-state performance of the control system still holds, i.e., the stability of the system is still guaranteed. This is because the effect of external disturbances is finite, and it can be sufficiently counteracted by feat of the infinity property of the PPC controller. Therefore, the control system is robust against external disturbances.

5. Simulation Study

To provide the illustration of the above theoretical findings, two simulation studies are carried out.

Case 1: Take account of the subsequent second-order lower-triangular systems with time varying powers

$$\left\{\begin{array}{c}{\dot{x}}_1=-x_1-{x_1}^3+{\left[x_2\right]}^{1.2+cos\left(t\right)},\hfill \\ {\dot{x}}_2=x_1{x_2}^2+u^{\frac{7}{5}},\hfill \\ y=x_1.\hfill \end{array}\right.$$

(50)

In the simulation, let $x_1\left(0\right)=-1$, $x_2\left(0\right)=1.5$. The control target for (50) is let $y\left(t\right)$ track $r\left(t\right)=0.7sin\left(t\right)$ with

$$\begin{array}{c}\hfill |y\left(t\right)-r\left(t\right)|<{\xi }_1\left(t\right)=(1.5-0.01)e^{-0.5t}+0.01.\end{array}$$

(51)

Following Theorem 1, a model-free controller is obtained with $c_1=8$, $c_2=10$ and

$$\begin{array}{cc}\hfill & {\xi }_2\left(t\right)=(17-0.15)e^{-t}+0.15.\hfill \end{array}$$

(52)

Applying the above control scheme to (50), the simulation results are exhibited in Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6. Figure 2 displays that the output varies along with the reference. The tracking error, plotted in Figure 3, is inside the predefined performance funnel. Hence, the performance requirement in (51) is fulfilled. Likewise, Figure 4 shows that the prescribed specification of the intermediate error in (52) is also met. Lastly, Figure 5 and Figure 6 depict the boundedness of the state variable, the intermediate control law and the input. Thus, our approach is effective.

To perform a comparative study, another controller employing backstepping design method is applied to (50). This is executed with the same control goal and under the same simulation condition. The controller is designed in the case where the nonlinear functions are known but the fractional powers are unknown. The simulation results are displayed in Figure 7 and Figure 8. Figure 7 depicts a large basis between the output and the reference. It is demonstrated by Figure 8 that the tracking error violates the performance constraint in both the transient and steady-state phrases. Therefore, the control target in (51) fails to be achieved. Accordingly, the comparative findings show the advantages of our approach.

Case 2: Consider the following three-order lower-triangular systems with positive powers:

$$\left\{\begin{array}{c}{\dot{x}}_1=-x_1-{x_1}^2+{x_2}^{\frac{9}{7}},\hfill \\ {\dot{x}}_2=-{x_2}^2+x_3,\hfill \\ {\dot{x}}_3={x_3}^3+u^{\frac{7}{5}},\hfill \\ y=x_1.\hfill \end{array}\right.$$

(53)

The control goal for (53) is steering its output to track $r\left(t\right)=0.5sin\left(0.2t\right)$ and satisfy

$$\begin{array}{c}\hfill |y\left(t\right)-r\left(t\right)|<{\xi }_1\left(t\right)=(1.6-0.05)e^{-0.48t}+0.05.\end{array}$$

(54)

The performance functions are chosen as ${\xi }_2\left(t\right)=(9-0.45)e^{-0.5t}+0.45$ and ${\xi }_3\left(t\right)=(13-0.2)e^{-0.8t}+0.2$. According to the design procedure in (5)–(13), we can obtain a model-free controller. In the simulation, let $x_1\left(0\right)=0.5$, $x_2\left(0\right)=0.5$ and $x_3\left(0\right)=0$. Applying the designed controller to (53), the simulation results are displayed in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. Figure 9 shows that the output nearly tracks the reference after $t=10$ s. The tracking error, plotted in Figure 10, evolves within the prescribed performance envelope, and thus (54) is satisfied. Figure 11 and Figure 12 exhibit that the intermediate tracking errors are also inside the performance funnel. Finally, Figure 13 and Figure 14 show that the state variables, the intermediate control law, and the control input are all bounded. Accordingly, the above results verify the effectiveness of our approach.

6. Conclusions

An approach for prescribed performance tracking control is put forward in this paper. It is capable of handling unknown fractional powers and unknown nonlinearities. It achieves the reference tracking with the arbitrarily preassigned accuracy and speed. It eliminates the power order restriction, the specific size limitation, and the homogeneous condition. Additionally, the powers, the system nonlinearities, and their bounds or bounding functions are totally unknown. The proposed control is simple in the sense that it does not involve derivative calculation, parameter identification, function approximation, or adding a power integrator technique. The simulation results validate the theoretical findings.