1 Introduction

Robustness and tracking performance are vital indexes of controller design. Tracking performance is the system properties shown in the steady-state and transient processes, for example, convergence rate, maximum overshoot and steady-state tracking error. Robustness is the maintenance ability of tracking performance in the presence of uncertainties, such as external disturbances, system parameter variations, un-modeled dynamics, and so on. In particular, for the unknown nonlinear systems, how to guarantee these performances comprehensively is still an open and significant problem [1]. During the past several decades, back-stepping technique has been recognized as a powerful tool to design controller for a larger class of uncertain nonlinear system, and the robustness and tracking problem for unknown nonlinear systems has been intensively solved based on back-stepping technique [2,3,4,5,6,7]. A main limitation in these works is that the progress can only be applied to systems in the affine form. Compared with this progress, relatively fewer results are available for control of pure-feedback systems, which represents a more general class of triangular systems. The difficulty associated with the control design of pure feedback systems arises from the lack of the appropriate variables to be used as virtual and/or actual control in the recursive design procedure. Therefore, control synthesis and stability analysis of pure-feedback systems are challenging issues and have attracted considerable research efforts [8,9,10,11,12,13,14,15,16,17,18,19]. To enhance the robustness of uncertain nonlinearities or completely unknown nonlinearities, most of these back-stepping-based schemes were developed based on neural networks (NNs) or fuzzy logical systems (FLSs). By utilizing the mean value theorem [8,9,10,11,12,13,14,15], the auxiliary integral method [16] and the contraction mapping method [17,18,19], the original pure-feedback system was transformed into an equivalent model with quasistrict-feedback form. Subsequently, the back-stepping-based control scheme was developed by employing the NN [8,9,10,11,12,13, 16,17,18] and FLS [14, 15, 19] to approximate the unknown nonlinearities of the equivalent strict-feedback form models. However, the repeated differentiation calculations of virtual controller in back-stepping may lead to the problem of “explosion of complexity”, which results in tremendous calculation burden and undesirable numerical noise in practice. To eliminate this problem, the dynamic surface control (DSC) was developed in [20] and extended to unknown pure-feedback systems by utilizing a low-pass filter for the synthetic input at each step of back-stepping procedure. By borrowing the features that universal approximator (i.e., NN and FLS) can approximate arbitrary nonlinear continuous function to a given accuracy, an adaptive NN-DSC approach was formulated for a class of unknown pure-feedback systems, where the mean value theorem [21,22,23,24] and contraction mapping method [25, 26] were employed to transformed the original systems into strict-feedback form, respectively. In [27, 28], the pure-feedback system was transformed into pseudostrict-feedback form by adopting the contraction mapping approach, then, an adaptive NN output feedback DSC design [27] and an adaptive fuzzy output feedback DSC design [28] were presented for unknown multi-input and multi-output (MIMO) and single-input and single-output (SISO) pure-feedback systems, respectively. Unfortunately, all aforementioned works guarantee convergence of the steady-state tracking error to a residual set, whose size depends on explicit design parameters and some unknown bounded terms, which makes a priori selection of the design parameters satisfying certain steady-state behavior practically impossible. Furthermore, the analytical relationship between transient behavior (i.e., convergence rate and maximum overshoot) and design parameters is difficult to analyze by using mathematical tool. To formulate the relationship between performance indexes and design parameters as specific analytical functions, an alternative approach, named prescribed performance control (PPC) to guarantee the transient and steady-state performance was first proposed in [29], where prescribed performance bound (PPB) can characterize the convergence rate, maximum overshoot and maximum steady-state error of the tracking errors. With the appropriate performance function and error transformation, the tracking errors can converge to a predefined small residual set with a convergence rate no less than a predefined value and maximum overshoot less than a sufficiently small specified constant. Subsequently, PPC methodology was employed to design adaptive controller for various classes of unknown nonlinear systems by using NN [30, 31] and FLS [32]. Only a few results are available in the literature for the PPC of unknown pure-feedback systems. Two low-complexity global approximation-free control scheme with prescribed performance for unknown pure-feedback systems were proposed in [33, 34]. An adaptive robust control with prescribed performance for a class of unknown pure-feedback system was studied in [35]. However, a major obstacle in the application scope is that the output and state constraints are not considered in the above-mentioned results about unknown pure-feedback systems.

In recent years, adaptive control of nonlinear systems with output or state constraints has received much attention. Many significant control schemes have been developed by utilizing the back-stepping and DSC approach [36,37,38,39,40,41,42,43,44,45]. Since the barrier Lyapunov function (BLF) candidate was originally proposed in [46], the BLF-based control scheme has been widely used for the nonlinear system with state and output constraints. By employing the BLF, an adaptive control scheme was developed to tackle the problem of control output constraints in [36, 37] and time-varying output constraints in [38, 39]. An adaptive fuzzy control approach for a category of uncertain nonlinear systems with output constraint was developed in [40], and the problem of output constraint was handled by utilizing a BLF. A BLF-based adaptive control was developed for a class of strict-feedback nonlinear systems with full-state constraints in [41, 42] and partial-state constraints in [43]. But in the aforementioned works, the research results are obtained under the condition that the considered systems are strict-feedback nonlinear systems, there are very few existing research results for unknown pure-feedback systems with state and output constraints. In [44], for a class of pure-feedback systems with output constraints, a dynamic surface design approach based on an appropriate integral BLF was presented to design an adaptive controller to ensure both the constraint satisfaction and the desired tracking ability. In [45], for a class of uncertain pure-feedback parametric systems, an adaptive recursive design procedure was constructed to remove the difficulties for avoiding nonaffine terms and guarantee that the full-state constraints are not violated by introducing BLF with the error variables.

Despite the efforts made to unknown pure-feedback systems recently, certain issues still remain open. Firstly, all aforementioned works about unknown pure-feedback systems have resorted to universal approximation theorem to deal with the unknown nonlinear dynamics of the system. Unfortunately, this universal approximation-based (i.e., NN and FLS) approach inherently introduces certain issues affecting controller complexity, closed-loop stability and robustness [33]. Secondly, the assumption is always very stringent, for example, the signs of control gain are compelled to be known [8,9,10,11,12,13,14, 16,17,18,19, 21,22,28, 33, 34, 45] and the unknown nonlinear functions satisfy linear-in-the-parameters (LIP) condition [45]. Finally, there are very few results about the constrained or prescribed performance control problem of pure-feedback system. In particular, to the authors’ best knowledge, in the literature, there are no results reported on the integrated control design for prescribed performance and state/output constraints, and the previous works for all the state/output constraints are very conservative due to the need for a priori knowledge of control direction.

Based on the above discussions, an adaptive DSC scheme is proposed for a class of unknown pure-feedback systems with prescribe performance and full-state constraints. The main contributions of the proposed approach are that:

  1. (1)

    Reduced design complexity. Different from the results in [8,9,10,11,12,13,14, 16,17,18,19, 21,22,28] which focus on approximation-based techniques to tackle the unknown dynamics, this study frames a novel adaptive bounded estimation approach to deal with the unknown dynamics by combining the mean value theorem, the supremum norm theory and DSC, which is approximation-free and can avoid unnecessary repeated differentiation calculations. Namely, the proposed control scheme is low-complexity.

  2. (2)

    Reduced system prior knowledge and conservatism. Compared with results in [8,9,10,11,12,13,14, 16,17,18,19, 21,22,28, 33, 34, 45], the knowledge of the sign of control gain and the LIP condition of unknown nonlinear functions are not required by employing Nussbaum gain technique and supremum norm theory. Meanwhile, the conservatism of traditional BLF can be removed in this study.

  3. (3)

    Prescribed performance-barrier Lyapunov function (PP-BLF). The proposed control scheme constitutes a first approach to solve the problem of state constraints and prescribed performance tracking integratedly, which can guarantee that all the state constraints are not violated, and the tracking error is preserved within a specified prescribed performance bound at all times, simultaneously. Moreover, the proposed control scheme also constitutes a first approach toward the solution of the prescribed performance tracking problem and state-constrained problem for pure feedback systems.

2 Problem formulation and preliminaries

Consider a class of unknown pure-feedback systems [8] with full-state constraints as follows

$$\begin{aligned} \left\{ \begin{array}{lll} {{\dot{x}}_i} = {f_i}({x_1}, \ldots , {x_{i + 1}}),{} {} {} {} {} {} {} {} {} {} i = 1, \ldots , n - 1\\ {{\dot{x}}_n} = {f_n}(x,u)\\ {{y} =} {x_1} \end{array} \right. \end{aligned}$$
(1)

where \(x = {[{x_1}, \ldots , {x_n}]^\mathrm{T}} \in {R^n},y \in R\) and \(u \in R\) are the states, the output, and the input of system, respectively; \({{{\bar{x}}}_i} = {[{x_1}, \ldots , {x_i}]^\mathrm{T}} \in {R^i},{f_i}({{{\bar{x}}}_{i + 1}}),i = 1, \ldots , n - 1\) and \({f_n}(x,u)\) are unknown nonaffine nonlinear smooth functions. In this study, all the states are constrained in the compact sets, i.e., \(\left| {{x_i}} \right| < {k_{{\mathrm{c}_i}}}\) with \({k_{{\mathrm{c}_i}}}\) being a known positive constant. To facilitate design, let \({x_{n + 1}} = u\).

The control objective in this study is to design an adaptive state feedback controller u such that: (1) All the signals in the closed-loop system are bounded; (2) the tracking error \(z_1=y-y_\mathrm{r}\) achieves prescribed transient and steady-state performance; (3) the full-state constraints are not violated, i.e., \(\left| {{x_i}} \right| < {k_{{\mathrm{c}_i}}}\), where \({y_\mathrm{r}} \in R\) is the reference signal.

To achieve the control goal, the following assumptions, lemmas, and definitions are required in the design.

Assumption 1

For the unknown pure-feedback system (1), the function \(f_i\) is continuously differentiable and there exists unknown positive constant \({{\bar{\beta }} _{i,j}},i = 1, \ldots , n,j = 1, \ldots , i + 1\) such that

$$\begin{aligned}&\left| {\frac{{\partial {f_i}({x_1}, \ldots , {x_i},{x_{i + 1}})}}{{\partial {x_j}}}} \right| \le {{\bar{\beta }} _{i,j}},\\&\qquad \quad i = 1, \ldots ,n,{} {} {} {} j = 1, \ldots , i\\&0 < \left| {\frac{{\partial {f_i}({x_1}, \ldots , {x_i},{x_{i + 1}})}}{{\partial {x_{i + 1}}}}} \right| \le {{\bar{\beta }}_{i,i + 1}},\\&\qquad \qquad i = 1, \ldots , n \end{aligned}$$

for all \(({x_1}, \ldots , {x_n},u) \in {R^{n + 1}}\).

Assumption 2

For the unknown pure-feedback system (1), the \({f_i}({{\bar{x}}}_{i + 1}^0)\) is always bounded; that is, there exists unknown positive constant \({\Delta _i}\) such that \(\left| {{f_i}({{\bar{x}}}_{i + 1}^0)} \right| {\le } \, {\Delta _i}\), where \(x_j^0 = {x_j}(0),j = 1, \ldots ,i + 1\) and \({{\bar{x}}}_{i + 1}^0 = {[x_1^0, \ldots , x_{i + 1}^0]^\mathrm{T}}\) denote the system initial conditions.

Assumption 3

[27] For the unknown pure-feedback system (1), there exists positive constants \(A_0\) and \(B_0\) such that the desired trajectory \(y_\mathrm{r}\), its first-order derivative \({{\dot{y}}_\mathrm{r}}\) and its second-order derivative \({{\ddot{y}}_\mathrm{r}}\) satisfy \(\left| {{y_\mathrm{r}}} \right| \le {A_0} < {k_{{\mathrm{c}_1}}}\) and \({\Omega _0}: = \{ [{y_\mathrm{r}},{{\dot{y}}_\mathrm{r}},{{\ddot{y}}_\mathrm{r}}]:{\left| {{y_\mathrm{r}}} \right| ^2} + {\left| {{{{\dot{y}}}_\mathrm{r}}} \right| ^2} + {\left| {{{{\ddot{y}}}_\mathrm{r}}} \right| ^2} \le {B_0}\} \in {R^3},{} {} \forall t \ge 0\).

Remark 1

\(\textcircled {{\varvec{a}}}\): Without loss of generality, the stability results will be valid as long as the states remain within some compact sets, so the global Lipschitz condition on \(f_i\) can be relaxed to a local one in the Assumption 1. Such as \(f_i = 0.1x_i^2 - {x_{i+1}} + 0.2\sin ({x_i}{x_{i+1}})\) also satisfies Assumption 1. \(\textcircled {{\varvec{{b}}}}\): From a practical view, the energy and change rate of nonlinear dynamic are limited. Thus, Assumption 1–3 is reasonable. \(\textcircled {{\varvec{{c}}}}\): In [8,9,10,11,12,13,14, 16,17,18,19, 21,22,28, 33, 34], the adaptive control design is required to confirm the signs of control gain, and the unknown nonlinear function \({f_i}({{{\bar{x}}}_{i + 1}})\) satisfies \({f_i}({{\bar{x}}_{i + 1}}) = \theta _i^\mathrm{T}{\xi _i}({{\bar{x}}_{i + 1}})\) in [45], These are conservative for practical application and the problem of state constraints. In this study, the signs of control gain and the function \(f_i\) do not require to be known.

Definition 1

[47] Any continuous even function \(N(\zeta )\) is called Nussbaum-type function when there are the following properties:

$$\begin{aligned} \mathop {\lim }\limits _{s \rightarrow \infty } \sup \frac{1}{s}\int _0^s N(\zeta ){\hbox {d}}\zeta= & {} + \infty \end{aligned}$$
(2)
$$\begin{aligned} \mathop {\lim }\limits _{s \rightarrow \infty } \inf \frac{1}{s}\int _0^s N(\zeta ){\hbox {d}}\zeta= & {} - \infty \end{aligned}$$
(3)

There are many functions to be viewed as a Nussbaum-type function such as \({\mathrm{e}^{{\zeta ^2}}}\cos ((\pi /2)\zeta )\) and \({\zeta ^2}\cos (\zeta )\). In this study, \(N(\zeta ) = {\zeta ^2}\cos (\zeta )\) is used.

Lemma 1

[48] Let V(t) and \(\zeta (t)\) be smooth functions defined on \([0,{t_f})\) with \(V(t) \ge 0\), \(\forall t \in [0,{t_f})\), and \(N(\zeta )\) is a smooth Nussbaum-type function. If the following inequality holds

$$\begin{aligned} V(t) \le c + {\mathrm{e}^{ - \mu t}}\int _0^t {\left( {{\Gamma } \left( {x(\tau )} \right) N(\zeta ) + 1} \right) {{\dot{\zeta }}} {\mathrm{e}^{\mu \tau }}{\hbox {d}}\tau }\nonumber \\ \end{aligned}$$
(4)

where \(\mu > 0,{} {} {\Gamma }\left( {x(\tau )} \right) \) is a time-varying parameter which takes values in the unknown closed intervals \(I{:}{} {} = [{l^ - },{l^ + }]\) with \(0 \notin I\), and c represents some suitable constant, then \(V(t),\zeta (\tau )\) and \(\int _0^t {{\Gamma }\left( {x(\tau )} \right) N(\zeta ){{\dot{\zeta }}} {\hbox {d}}\tau } {} \) must be bounded on \([0,{t_f})\).

Lemma 2

[49] Suppose \(0 \le {t_{f}} \le \infty \) and that \(x:[0,{t_f}] \rightarrow {R^N}\) is a solution of the closed-loop system. If x is a bounded solution, then \({t_{f}} = \infty \).

Lemma 3

(Youngs inequality [15]) For \(\forall (m,n) \in {R^2}\), the following inequality holds:

$$\begin{aligned} mn \le \frac{{{\varsigma ^p}}}{p}{\left| m \right| ^p} + \frac{1}{{q{\varsigma ^q}}}{\left| n \right| ^q} \end{aligned}$$
(5)

where \(\varsigma> 0,{} {} {} p> 1,{} {} {} q > 1\) and \((p - 1)(q - 1) = 1\).

Lemma 4

For the unknown nonlinear function \({f_i}({{\bar{x}}_{i + 1}}),i = 1, \ldots , n\) of system (1), there exists unknown constant \({{\xi } _{i,j}} \in [x_j^0,{x_j}],{} {} i = 1, \ldots ,n,{} {} j = 1, \ldots , i + 1\), such that

$$\begin{aligned} {f_i}({{{\bar{x}}}_{i + 1}})= & {} \sum \limits _{j = 1}^i {{\beta _{i,j}}({x_j}} - x_j^0) - {\beta _{i,i + 1}}x_{i + 1}^0 \nonumber \\&+\,{\beta _{i,i + 1}}{x_{i + 1}} + {f_i}({\bar{x}}_{i + 1}^0) \end{aligned}$$
(6)

where \({\beta _{i,j}} = (\partial {f_i}/\partial {x_j}){|_{(0, \ldots , {{\xi } _{i,j}}, \ldots , {x_i},{x_{i + 1}})}}\).

Proof

By adding and subtracting the term \({f_i}({\bar{x}}_{i + 1}^0)\), then the \({f_i}({{\bar{x}}_{i + 1}})\) can be described as:

$$\begin{aligned} {f_i}({{{\bar{x}}}_{i + 1}})&= {f_i}({x_1}, \ldots , {x_{i + 1}}) - {f_i}(x_1^0,{x_2}, \ldots , {x_{i + 1}}) \nonumber \\&\quad +{f_i}(x_1^0,{x_2}, \ldots , {x_{i + 1}})\nonumber \\&\qquad - {f_i}(x_1^0,x_2^0,{x_3}, \ldots , {x_{i + 1}}) \nonumber \\&\quad + \cdots + {f_i}(x_1^0, \ldots x_i^0,{x_{i + 1}})\nonumber \\&\qquad - {f_i}({\bar{x}}_{i + 1}^0) + {f_i}({\bar{x}}_{i + 1}^0) \end{aligned}$$
(7)

Subsequently, since the \({f_i}({{\bar{x}}_{i + 1}})\) is continuous and differentiable for all \(({x_1}, \ldots , {x_i},{x_{i + 1}}) \in {R^{i + 1}}\) and the initial value \({f_i}({\bar{x}}_{i + 1}^0)\) are bounded, the \({f_i}({{\bar{x}}_{i + 1}})\) can be expressed as the following form with \({{\xi } _{i,j}} \in [x_j^0,{x_j}],{} {} i = 1, \ldots , n,{} {} j = 1, \ldots , i + 1\) ultimately by using mean value theorem.

$$\begin{aligned} {f_i}({{{\bar{x}}}_{i + 1}})= & {} \left( {{{\left. {\frac{{\partial {f_i}}}{{\partial {x_1}}}} \right| }_{({{\xi } _{i,1}},{x_2} \ldots ,{x_i},{x_{i + 1}})}}} \right) * ({x_1} - x_1^0) \nonumber \\&+\,\left( {{{\left. {\frac{{\partial {f_i}}}{{\partial {x_2}}}} \right| }_{(x_1^0,{{\xi } _{i,2}},{x_3} \ldots , {x_i},{x_{i + 1}})}}} \right) \nonumber \\&\quad *\, ({x_2} - x_2^0) + \cdots \nonumber \\&+\,\left( {{{\left. {\frac{{\partial {f_i}}}{{\partial {x_{i + 1}}}}} \right| }_{(x_1^0, \ldots , x_i^0,{{\xi } _{i,i + 1}})}}} \right) \nonumber \\&\quad \quad *\, ({x_{i + 1}} - x_{i + 1}^0) + {f_i}({\bar{x}}_{i + 1}^0) \end{aligned}$$
(8)

which together with the \({\beta _{i,j}} = (\partial {f_i}/\partial {x_j}){|_{(0, \ldots , {{\xi } _{i,j}}, \ldots , {x_i},{x_{i + 1}})}}\), \(i = 1, \ldots , n,j = 1, \ldots , i + 1\) gives the Eq. (6).

This concludes the proof.\(\square \)

3 Prescribed performance-Barrier Lyapunov function

3.1 Prescribed performance function

PPC was first proposed in [29]. PPC is achieved if the tracking error \({z_1} = y - {y_\mathrm{r}}\) evolves strictly within a predefined region that is bounded by a decaying function of time, i.e., prescribed performance function (PPF). The definition of PPF is given as follows.

Definition 2

[29]. A continuous function \(\rho (t)\) is called PPF if

  1. (i)

    \(\rho (t)\) is positive and strictly decreasing;

  2. (ii)

    \(\mathop {\lim }\nolimits _{t \rightarrow \infty } \rho (t) = {\rho _\infty }\). In this paper, the following exponential function is chosen as PPF

    $$\begin{aligned} \rho (t) = ({k_{{\mathrm{b}_1}}} - {\rho _\infty }){\mathrm{e}^{ - \ell t}} + {\rho _\infty } \end{aligned}$$
    (9)

    where \({k_{{\mathrm{b}_1}}} = {k_{{\mathrm{c}_1}}} - {{A_0}} > {\rho _\infty }\) is the boundedness of tracking error \({z_1},{\rho _\infty } > 0\) and \(\ell > 0\) are appropriately prescribed scalars. Then, the aforementioned second objective is obviously equivalent to the following mathematical inequality:

    $$\begin{aligned} - \rho (t)< {z_1} < \rho (t),{} {} {} {} {} {} {} \forall t > 0 \end{aligned}$$
    (10)

    The constant \(\rho _\infty \) represents the maximum allowable size of the tracking error \(z_1\) at the steady state, which may even be set arbitrarily small, thus accomplishing practical convergence of \(z_1\) to zero. Moreover, the decreasing rate of \(\rho (t)\), which is determined by the constant \(\ell \), imposes a lower bound on the required speed of convergence of \(z_1\). Therefore, the appropriate selection of the PPF \(\rho (t)\) imposes performance characteristics on the tracking error \(z_1\).

Remark 2

To integrate PPF with BLF, the mathematical expression of prescribed performance is given as (9) and (10) which introduce parameter \({k_{{\mathrm{b}_1}}}\). Different from the results in [29,30,31,32] which employ an error transformation technique to transform the constrained tracking error (10) into an unconstrained one, this study integrates the constrained tracking error (10) into BLF (i.e., PP-BLF) to achieve PPC and state-constrained control, simultaneously.

3.2 BLF and PP-BLF

To avoid the violation of output constraints, we employ a BLF with the following definitions.

Definition 3

[36] V(x) as a scalar function defined with respect to the system \(\dot{x} = f(x)\) on an open region \({\mathcal {D}}\) including the origin, is a BLF if it has the following properties.

  1. (1)

    V(x) is continuous, positively definite, and has continuous first-order partial derivatives at every point of \({\mathcal {D}}\).

  2. (2)

    \(V(x) \rightarrow \infty \) as x approaches the boundary of \({\mathcal {D}}\).

  3. (3)

    \(V(x) \le {d},{} \forall t \ge 0\) along the solution of \(\dot{x} = f(x)\) with \(x(0) \in {\mathcal {D}}\), where d is some positive constant.

To prevent the states or outputs of system from violating their constraints, one kind of traditional BLF is always defined in the following form with a compact set \({\Omega _z} = \{ {z_i}:\left| {{z_i}} \right| < {k_{{\mathrm{b}_i}}},i = 1, \ldots , n\} \):

$$\begin{aligned} {V_{{\mathrm{b}_i}}} = \frac{1}{2}\ln \frac{{k_{{\mathrm{b}_i}}^2}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}},{} {} {} {} {} {} {} i = 1, \ldots , n \end{aligned}$$
(11)

where \({k_{{\mathrm{b}_1}}} > 0\) is the bounded of \(z_1\), \({k_{{\mathrm{b}_i}}} > 0,i = 2, \ldots , n\) is a design constant, and the \({k_{{\mathrm{b}_i}}}\) and \(z_i\) will be determined later on. However, the traditional BLF is incapable of the requirement of prescribed performance. In order to guarantee that the tracking error \(z_1\) satisfies predefined inequality (10), the PP-BLF is developed in this study, which can be defined as follows with a compact set \({\Omega _{{z_1}}} = \{ {z_1}:\left| {{z_1}/\rho (t)} \right| < 1\}\).

$$\begin{aligned} {V_{{\mathrm{b}_1}}} = \frac{1}{2}\ln \frac{1}{{1 - {{\left( {{z_1}/\rho (t)} \right) }^2}}} \end{aligned}$$
(12)

The following lemma formalizes the result for (11) and (12) barrier functions and is used in the control design and analysis for pure-feedback system (1) to guarantee that state constraints and prescribed performance scalars are not violated.

Remark 3

In fact, the time-varying BLF in [38] can solve the problem of prescribed performance, but the relationship between BLF and PPB cannot be expressed clearly. Moreover, only strict-feedback system is considered with LIP condition in [38].

Lemma 5

[36, 38] For any \({\chi _1}\), any positive constant \({k_{{\mathrm{b}_i}}}\) and any \({z_i} \in R\) satisfying \(\left| {{\chi _1}} \right| < 1\) and \(\left| {{z_i}} \right| < {k_{{\mathrm{b}_i}}},i = 1,2, \ldots , n\), we have

$$\begin{aligned} \ln \frac{{k_{{\mathrm{b}_i}}^2}}{{k_{{\mathrm{b}_i}}^2 -{ z_i^2}}} \le \frac{{z_i^2}}{{k_{{\mathrm{b}_i}}^2 - {z_i^2}}},{} {} {} {} {} {} {} {} {\ln \frac{1}{{1 - \chi _1^2}} \le \frac{{\chi _1^2}}{{1 - \chi _1^2}}} \end{aligned}$$
(13)

Proof

The inequality (13) can be verified easily based on the Lemma 2 in [36, 38]. The proof is omitted here due to the limited space.\(\square \)

Lemma 6

[37] For any positive constant , let and \({\varvec{N}} = {{\varvec{R}}^l} \times {\varvec{S}} \subset {{\varvec{R}}^{l + 1}}\) be open sets. Consider the system

$$\begin{aligned} {{\dot{\eta }}} = h(t,\eta ) \end{aligned}$$
(14)

where \(\eta = {[\omega , s]^\mathrm{T}} \in {\varvec{N}}\) and \(h:{{\varvec{R}}^ + } \times {\varvec{N}} \rightarrow {{\varvec{R}}^{l + 1}}\) is piecewise continuous in t and locally Lipschitz in \(\eta \) , uniformly in t, on \({R^ + } \times N\). Suppose that there exists functions \( U:{{\varvec{R}}^l} \times {{\varvec{R}}^ + } \rightarrow {R^ + }\) and \({V_\mathrm{b}}:{\varvec{S}} \rightarrow {{\varvec{R}}^ + }\), continuously differentiable and positive definite in their respective domains, such that

(15)
$$\begin{aligned} {\Upsilon _1}(\left\| \omega \right\| ) \le U(\omega , t) \le {\Upsilon _2}(\left\| \omega \right\| ) \end{aligned}$$
(16)

where \(\Upsilon _1\) and \(\Upsilon _2\) are class \({K_\infty }\) functions. Let \(V(\eta ) = {V_\mathrm{b}}(s) + U(\omega , t)\) and \(s(0) \in {\varvec{S}}\). If the inequality holds:

$$\begin{aligned} {\dot{V}} = \frac{{\partial V}}{{\partial \eta }}h \le - \mu V + \sigma \end{aligned}$$
(17)

where \(\mu \) and \(\sigma \) are positive constants, then \(s(t) \in {\varvec{S}},\forall t \in [0,\infty )\).

Proof

The proof is omitted here due to the limited space. Interested readers can follow the similar procedures of the proof of Lemma 1 in [36]. \(\square \)

4 Main results

4.1 Control scheme

In this section, the approximation-free PP-BLF-based control scheme for systems (1) is designed step-by-step in the presence of unknown dynamics and full-state constraints.

Step 1 Define the tracking error as \({z_1} = {x_1} - {y_\mathrm{r}}\) and invoking (6), it has

$$\begin{aligned} {{{\dot{z}}}_1}= & {} {{\dot{x}}_1} - {{{\dot{y}}}_\mathrm{r}} = {\beta _{1,1}}({x_1} - x_1^0) \nonumber \\&- {\beta _{1,2}}x_2^0 + {\beta _{1,2}}{x_2} + {f_1}({\bar{x}}_2^0) - {{{\dot{y}}}_\mathrm{r}}\nonumber \\= & {} \varphi _1^\mathrm{T}{\psi _1} + {\beta _{1,2}}{x_2} - {{{\dot{y}}}_\mathrm{r}} \end{aligned}$$
(18)

where \({\varphi _1} = {[{\beta _{1,1}},{\beta _{1,2}},{f_1}({\bar{x}}_2^0)]^\mathrm{T}}\) and \({\psi _1} = [{x_1} - x_1^0, - x_2^0,1]^\mathrm{T}\). Since the \({\beta _{1,1}},{\beta _{1,2}}\) and \({f_1}({\bar{x}}_2^0)\) are bounded by Assumption 1 and 2, the \(\left\| {{\varphi _1}} \right\| \) is also bounded. With the supremum norm theory in mild, let

$$\begin{aligned} {\vartheta _1} = \mathop {\sup }\limits _{t \ge 0} {} {\left\| {{\varphi _1}} \right\| ^2} \end{aligned}$$
(19)

Design the virtual control \(\alpha _1\) as

$$\begin{aligned} {\alpha _1}= & {} N({\zeta _1}){\varpi _1} \end{aligned}$$
(20)
$$\begin{aligned} {\varpi _1}= & {} {\kappa _1}{\chi _1} - {{\dot{y}}_\mathrm{r}} - {\rho ^{ - 1}}{\dot{\rho }} {z_1} + {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}}\nonumber \\&+{{\left( {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}{{\hat{\vartheta }}_1}{{\left\| {{\psi _1}} \right\| }^2}} \right) } \big / {(2\delta _1^2)}} \end{aligned}$$
(21)

where \({\chi _1} = {z_1}/\rho (t)\). The parameter updating laws are designed as

$$\begin{aligned} {{{\dot{\zeta }}} _1}= & {} {\rho ^{ - 1}}{{{\chi _1}{\varpi _1}} \big / {(1 - \chi _1^2)}} \end{aligned}$$
(22)
$$\begin{aligned} {\dot{\hat{\vartheta }}} _{1}= & {} {\gamma _1}\left[ { - {{{\hat{\vartheta }} }_1} + {{{{\left( {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}} \right) }^2}{{\left\| {{\psi _1}} \right\| }^2}} \big / {(2\delta _1^2)}}} \right] \end{aligned}$$
(23)

where \({\kappa _1},{} {} {\delta _1}\) and \({\gamma _1}\) are designed positive constants, \({{\hat{\vartheta }} _1}\) denotes the estimation of \(\vartheta _1\).

To avoid repeatedly differentiating \(\alpha _1\), which leads to the so-called “explosion of complexity”, the DSC technique in [20] is employed here. Namely, introduce a new variable \({{\overline{\alpha }}}_1\) and let \(\alpha _1\) pass through a first-order filter with time constant \(g_1\) to gain \({\overline{\alpha }} _1\)

$$\begin{aligned} {g_1}{\dot{{{\overline{\alpha }}} _1}} + {{{\overline{\alpha }}} _1} = {\alpha _1},{} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {{{\overline{\alpha }}} _1}(0) = {\alpha _1}(0) \end{aligned}$$
(24)

By defining the output error \(\phi _1\) of this filter and tracking error \(z_2\) as

$$\begin{aligned} {\phi _1}= & {} {{{{\overline{\alpha }}}} _1} - {\alpha _1} \end{aligned}$$
(25)
$$\begin{aligned} {z_2}= & {} {x_2} - {{{{\overline{\alpha }}}} _1} \end{aligned}$$
(26)

Differentiating \(\phi _1\) yields

$$\begin{aligned} {{{{\dot{\phi }}} }_1}= & {} {{\dot{{{{\overline{\alpha }}}} }}_1} - {{{\dot{\alpha }} }_1} = - \frac{{{\phi _1}}}{{{g_1}}} - {{{\dot{\alpha }} }_1}\nonumber \\= & {} - \frac{{{\phi _1}}}{{{g_1}}} + \left[ - \frac{{\partial N({\zeta _1})}}{{\partial {\zeta _1}}}{{{\dot{\zeta }} }_1}{\varpi _1} - N({\zeta _1})\left( \frac{{\partial {\varpi _1}}}{{\partial {x_1}}}{{\dot{x}}_1} \right. \right. \nonumber \\&\quad \left. \left. + \frac{{\partial {\varpi _1}}}{{\partial {z_1}}}{{{\dot{z}}}_1} + \frac{{\partial {\varpi _1}}}{{\partial {\rho _1}}}{{{\dot{\rho }} }_1} - {{{\ddot{y}}}_\mathrm{r}} + \frac{{\partial {\varpi _1}}}{{\partial {{{\dot{\rho }} }_1}}}{{\ddot{\rho }}_1} + \frac{{\partial {\varpi _1}}}{{\partial {{{\hat{\vartheta }} }_1}}}{{\dot{{\hat{\vartheta }} }}_1} \right) \right] \nonumber \\= & {} - \frac{{{\phi _1}}}{{{g_1}}} + {{T}_1}({z_1},{z_2},{\phi _1},{\rho _1},{{{\hat{\vartheta }} }_1},{y_\mathrm{r}},{{{\dot{y}}}_\mathrm{r}},{{{\ddot{y}}}_\mathrm{r}}) \end{aligned}$$
(27)

All of variables of the function \({{T}_1}\) are from compact sets and \({{T}_1}\) is a smooth function, so the \(\left| {{{T}_1}} \right| \) has its maximum value \({{\overline{{T}}}_1}\), i.e., \(\left| {{{T}_1}} \right| \le {{\overline{{T}}}_1}\) with \({{\overline{{T}}}_1}\) being an unknown constant (Please refer to [20] for details).

Then, consider the PP-BLF in (12) and choose the following Lyapunov function candidate

$$\begin{aligned} {V_1} = {V_{{\mathrm{b}_1}}} + \frac{1}{{2{\gamma _1}}}{{\tilde{\vartheta }}} _1^2 + {\frac{1}{2}\phi _1^2} \end{aligned}$$
(28)

where \({\gamma _1}\) is a designed positive constant, \({{\tilde{\vartheta }} _1} = {{\hat{\vartheta }} _1} - {\vartheta _1}\) is the estimation error. Define a set \({\Omega _{{\chi _1}}} = \{ {\chi _1}:\left| {{\chi _1}} \right| < 1\}\). In the set \({\Omega _{{\chi _1}}} = \{ {\chi _1}:\left| {{\chi _1}} \right| < 1\}\), \(V_1\) is continuous. Then, in the view of (12), (18), (20), (21), (22), (25), and (26), differentiating (28) yields

$$\begin{aligned} {{{\dot{V}}}_1}= & {} \frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}\left[ \varphi _1^\mathrm{T}{\psi _1} + {\beta _{1,2}}({z_2} + {\phi _1}) + {\beta _{1,2}}N({\zeta _1}){\varpi _1} \right. \nonumber \\&\left. +\, {\varpi _1} - {\varpi _1} - {{{\dot{y}}}_\mathrm{r}} - {\rho ^{ - 1}}{\dot{\rho }} {z_1} \right] \nonumber \\&+\,\frac{{{{{\tilde{\vartheta }} }_1}{{\dot{{\hat{\vartheta }}} }_1}}}{{{\gamma _1}}} + {{\phi _1}{{{\dot{\phi }} }_1}}\nonumber \\= & {} -\, {\kappa _1}\frac{{{\rho ^{ - 1}}\chi _1^2}}{{1 - \chi _1^2}} - {\left( {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}} \right) ^2}\nonumber \\&+\, \frac{{{\beta _{1,2}}{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}({z_2} + {\phi _1}) + \left[ {{\beta _{1,2}}N({\zeta _1}) + 1} \right] {{{\dot{\zeta }} }_1} \nonumber \\&-\, {{{{\left( {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}} \right) }^2}{{{\hat{\vartheta }} }_1}{{\left\| {{\psi _1}} \right\| }^2}} \big / {(2\delta _1^2)}} + \frac{{{{{\tilde{\vartheta }} }_1}{{\dot{{\hat{\vartheta }} }}_1}}}{{{\gamma _1}}}\nonumber \\&+\, \frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}\varphi _1^\mathrm{T}{\psi _1} + {{\phi _1}{{{\dot{\phi }} }_1}} \end{aligned}$$
(29)

where \(\rho \) is the abbreviation for \(\rho (t)\). Consider (19), (27) and Assumption 1, using the Youngs inequality, we have

$$\begin{aligned} \frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}\varphi _1^\mathrm{T}{\psi _1}\le & {} {{{{\left( {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}} \right) }^2}{\vartheta _1}{{\left\| {{\psi _1}} \right\| }^2}} \big / {(2\delta _1^2) + }}\frac{{\delta _1^2}}{2} \end{aligned}$$
(30)
$$\begin{aligned} \frac{{{\beta _{1,2}}{\rho ^{ - 1}}{\chi _1}{z_2}}}{{1 - \chi _1^2}}\le & {} \frac{1}{2}{\left( {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}} \right) ^2} + \frac{1}{2}{\bar{\beta }} _{1,2}^2z_2^2 \end{aligned}$$
(31)
$$\begin{aligned} \frac{{{\beta _{1,2}}{\rho ^{ - 1}}{\chi _1}{\phi _1}}}{{1 - \chi _1^2}}\le & {} \frac{1}{2}{\left( {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}} \right) ^2} + \frac{1}{2}{\bar{\beta }} _{1,2}^2\phi _1^2 \end{aligned}$$
(32)
$$\begin{aligned} {\phi _1}{{\dot{\phi }} _1}= & {} {\phi _1}\left( { - \frac{{{\phi _1}}}{{{g_1}}} - {{{\dot{\alpha }} }_1}} \right) \nonumber \\\le & {} -\, \frac{{\phi _1^2}}{{{g_1}}} + \frac{{\phi _1^2}}{2} + \frac{{{T}_1^2}}{2}\nonumber \\&\le -\, \frac{{\phi _1^2}}{{{g_1}}} + \frac{{\phi _1^2}}{2} + \frac{{{\overline{{T}}}_1^2}}{2} \end{aligned}$$
(33)

Substituting (30), (31), (32) and (33) into (29), we have

$$\begin{aligned} {{{\dot{V}}}_1}= & {} -\, {\kappa _1}\frac{{{\rho ^{ - 1}}\chi _1^2}}{{1 - \chi _1^2}} - {\left( \frac{1}{{{g_1}}} - \frac{1}{2} - \frac{{{\bar{\beta }} _{1,2}^2}}{2}\right) \phi _1^2} \nonumber \\&+\, \frac{1}{2}{\bar{\beta }} _{1,2}^2z_2^2 + \left[ {{\beta _{1,2}}N({\zeta _1}) + 1} \right] {{{\dot{\zeta }} }_1} \nonumber \\&-\,{{{{\left( {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}} \right) }^2}{{{\tilde{\vartheta }} }_1}{{\left\| {{\psi _1}} \right\| }^2}} \big / {(2\delta _1^2)}} \nonumber \\&+\, \frac{{{{{\tilde{\vartheta }} }_1}{{\dot{{\hat{\vartheta }} }}_1}}}{{{\gamma _1}}} + \frac{{\delta _1^2}}{2} + {\frac{{{\overline{{T}}}_1^2}}{2}} \end{aligned}$$
(34)

According to (23), it is easy to obtain

$$\begin{aligned}&- \,{{{{\left( {\frac{{{\rho ^{ - 1}}{\chi _1}}}{{1 - \chi _1^2}}} \right) }^2}{{{\tilde{\vartheta }} }_1}{{\left\| {{\psi _1}} \right\| }^2}} \big / {(2\delta _1^2)}} + \frac{{{{{\tilde{\vartheta }} }_1}{{\dot{\hat{\vartheta }}}_1}}}{{{\gamma _1}}} = - {\tilde{\vartheta }} _{1}{{\hat{\vartheta }} _1} \nonumber \\&\quad = -\, {\tilde{\vartheta }} _1^2 - {{\tilde{\vartheta }} _1}{\vartheta _1} \le - \frac{1}{2}{\tilde{\vartheta }} _1^2 + \frac{1}{2}\vartheta _1^2 \end{aligned}$$
(35)

Substituting (35) into (34), we have

$$\begin{aligned} {{{\dot{V}}}_1}\le & {} -\, {\kappa _1}\frac{{{\rho ^{ - 1}}\chi _1^2}}{{1 - \chi _1^2}} - \frac{1}{2}{\tilde{\vartheta }} _1^2 - \left( \frac{1}{{{g_1}}} - \frac{1}{2} - \frac{{{\bar{\beta }} _{1,2}^2}}{2}\right) \phi _1^2 \nonumber \\&+\, \left[ {{\beta _{1,2}}N({\zeta _1}) + 1} \right] {{{\dot{\zeta }} }_1} + \frac{1}{2}{\bar{\beta }} _{1,2}^2z_2^2\nonumber \\&+ \frac{1}{2}\vartheta _1^2 + \frac{{\delta _1^2}}{2} + {\frac{{{\overline{{T}}}_1^2}}{2}} \end{aligned}$$
(36)

From Lemma 5, it is the fact that \(\ln \frac{1}{{1 - \chi _1^2}} \le \frac{{\chi _1^2}}{{1 - \chi _1^2}}\) in the interval \(\left| {{\chi _1}} \right| < 1\), and the fact \({\rho ^{ - 1}} \in [k_{{\mathrm{b}_1}}^{ - 1},\rho _\infty ^{ - 1})\), we have

$$\begin{aligned} {{\dot{V}}_1} \le - {\mu _1}{V_1} + {\varepsilon _1} + \left[ {{\beta _{1,2}}N({\zeta _1}) + 1} \right] {{\dot{\zeta }} _1} + 0.5{{\bar{\beta }} _{1,2}^2z_2^2}\nonumber \\ \end{aligned}$$
(37)

where \({\mu _1} = \min \{ 2{\kappa _1}k_{{\mathrm{b}_1}}^{ - 1},{\gamma _1},{2g_1^{ - 1} - 1 - {\bar{\beta }} _{1,2}^2\} > 0}\) and \({\varepsilon _1} = 0.5\vartheta _1^2 + 0.5\delta _1^2 + { 0.5{\overline{{T}}}_1^2}\).

Step\(i,{} {} {} {} {} i = 2, \ldots , n - 1\). Define the variable \( {\varphi _i} = {[{\beta _{i,1}}, \ldots , {\beta _{i,i}},{\beta _{i,i + 1}},{f_i}({\bar{x}}_{i + 1}^0)]^\mathrm{T}},{} {} {} {} {\psi _i} = [{x_1} - x_1^0, \ldots , {x_i} - x_i^0, - x_{i + 1}^0,1]^\mathrm{T}, {z_i} = {x_i} - {{{{{\overline{\alpha }}}} _{i - 1}}},\) and invoking (6), it also has

$$\begin{aligned} {{\dot{z}}_i} = {\dot{x}_i} -{ {\dot{{{{\overline{\alpha }}}}} _{i - 1}}} = \varphi _i^\mathrm{T}{\psi _i} + {\beta _{i,i + 1}}{x_{i + 1}} - {{\dot{{{{\overline{\alpha }}}}} _{i - 1}}} \end{aligned}$$
(38)

Similarly, the \({\beta _{i,j}},j = 1, \ldots , i + 1\) and \({f_i}({\bar{x}}_{i + 1}^0)\) are bounded, so the \(\left\| {{\varphi _i}} \right\| \) is also bounded. With the supremum norm theory in mild again, let

$$\begin{aligned} {\vartheta _i} = \mathop {\sup }\limits _{t \ge 0} {} {\left\| {{\varphi _i}} \right\| ^2} \end{aligned}$$
(39)

Design the virtual control \(\alpha _{i}\) as

$$\begin{aligned} {\alpha _i}= & {} N({\zeta _{i}}){\varpi _i} \end{aligned}$$
(40)
$$\begin{aligned} {\varpi _{i}}= & {} {\kappa _i}{z_i} -{ {\dot{{{{\overline{\alpha }}}}} _{i - 1}} + \frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}} \nonumber \\&+\, {{\left( {\frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}{{{\hat{\vartheta }} }_i}{{\Vert {{\psi _i}} \Vert }^2}} \right) } \big / {(2\delta _i^2)}} \end{aligned}$$
(41)

where \({\kappa _i}\) and \({\delta _i}\) are designed positive constants. The parameter updating laws are designed as

$$\begin{aligned} {{\dot{\zeta }} _i}= & {} {{{z_i}{\varpi _i}} \big / {(k_{{\mathrm{b}_i}}^2 - z_i^2)}} \end{aligned}$$
(42)
$$\begin{aligned} {\dot{{\hat{\vartheta }}} _i}= & {} {\gamma _i}\left[ { - {{{\hat{\vartheta }} }_i} + {{{{\left( {\frac{{{z_i}}}{{k_{{\mathrm{b}_1}}^2 - z_i^2}}} \right) }^2}{{\left\| {{\psi _i}} \right\| }^2}} \big / {(2\delta _i^2)}}} \right] \end{aligned}$$
(43)

Similar to Step 1, to avoid repeatedly differentiating \(\alpha _i\), we introduce a new variable \({{{{\overline{\alpha }}}} _i}\) and let \(\alpha _i\) pass through a first-order filter with time constant \(g_i\) to gain \({{{{\overline{\alpha }}}} _i}\).

$$\begin{aligned} {g_i}{\dot{{{{\overline{\alpha }}}}} _i} + {{{\overline{\alpha }}} _i} = {\alpha _i},{} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {{{\overline{\alpha }}} _i}(0) = {\alpha _i}(0) \end{aligned}$$
(44)

By defining the output error of this filter and tracking error \(z_{i+1}\) as

$$\begin{aligned} {\phi _i}= & {} {{\bar{\alpha }} _i} - {\alpha _i} \end{aligned}$$
(45)
$$\begin{aligned} {z_{i + 1}}= & {} {x_{i + 1}} - {{\bar{\alpha }} _i} \end{aligned}$$
(46)

Similar to Step 1, we can obtain that

$$\begin{aligned} {{\dot{\phi }} _i}= & {} - \frac{{{\phi _i}}}{{{g_i}}}+ {{T}_i}({z_1}, \ldots , {z_i},{\phi _1}, \ldots , {\phi _i}, \nonumber \\&{\rho _1},{{\hat{\vartheta }} _1}, \ldots , {{\hat{\vartheta }} _i},{y_\mathrm{r}},{{\dot{y}}_\mathrm{r}},{{\ddot{y}}_\mathrm{r}}) \end{aligned}$$
(47)

and \(\left| {{{T}_i}} \right| \) has its maximum value \({\overline{{T}}_i}\), i.e., \(\left| {{{T}_i}} \right| \le {{\overline{{T}}}_i}\) with \({\overline{{T}}}_i\) being an unknown constant.

Then, consider the BLF in (11) and choose the following Lyapunov function candidate

$$\begin{aligned} {V_i} = {V_{{\mathrm{b}_i}}} + \frac{1}{{2{\gamma _i}}}{\tilde{\vartheta }} _i^2 + {\frac{1}{2}\phi _i^2} \end{aligned}$$
(48)

where \({\gamma _i}\) is a designed positive constant, \({{{\tilde{\vartheta }} _i}} = {{\hat{\vartheta }} _i} - {\vartheta _i}\) and \({k_{{\mathrm{b}_i}}}\) is specified later on. In the set \({\Omega _{{z_i}}} = \{ {z_i}:\left| {{z_i}} \right| < {k_{{\mathrm{b}_i}}}\} \), \(V_i\) is continuous. Then, we obtain \({{\dot{V}}_i}\) from (11), (38), (40), (41), (42), (45), and (46) as

$$\begin{aligned} {{{\dot{V}}}_i}= & {} -\, {\kappa _i}\frac{{z_i^2}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}} - {\left( {\frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}} \right) ^2}\nonumber \\&+\, \frac{{{\beta _{i,i + 1}}{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}({z_{i + 1}} + {\phi _i}) \nonumber \\&+\, \left[ {{\beta _{i,i + 1}}N({\zeta _i}) + 1} \right] {{{\dot{\zeta }} }_i} \nonumber \\&-\,{{{{\left( {\frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}} \right) }^2}{{{\hat{\vartheta }} }_i}{{\left\| {{\psi _i}} \right\| }^2}} \big / {(2\delta _i^2)}} \nonumber \\&+\, \frac{{{{{\tilde{\vartheta }} }_i}{{\dot{{\hat{\vartheta }} }}_i}}}{{{\gamma _i}}} + \frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}\varphi _i^\mathrm{T}{\psi _i} + {{\phi _i}{{{\dot{\phi }} }_i}} \end{aligned}$$
(49)

Consider (39), (47) and Assumption 1, using the Youngs inequality, we have

$$\begin{aligned}&\frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}\varphi _i^\mathrm{T}{\psi _i} \le {{{{\left( {\frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}} \right) }^2}{\vartheta _i}{{\left\| {{\psi _i}} \right\| }^2}} \big / {(2\delta _i^2) + }}\frac{{\delta _i^2}}{2} \nonumber \\\end{aligned}$$
(50)
$$\begin{aligned}&{} \frac{{{\beta _{i,i + 1}}{z_i}{z_{i + 1}}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}} \le \frac{1}{2}{\left( {\frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}} \right) ^2} + \frac{1}{2}{\bar{\beta }} _{i,i + 1}^2z_{i + 1}^2 \end{aligned}$$
(51)
$$\begin{aligned}&{{} \frac{{{\beta _{i,i + 1}}{z_i}{\phi _i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}} \le \frac{1}{2}{\left( {\frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}} \right) ^2} + \frac{1}{2}{\bar{\beta }} _{i,i + 1}^2\phi _i^2} \end{aligned}$$
(52)
$$\begin{aligned}&{{\phi _i}{{\dot{\phi }} _i} = {\phi _i}\left( { - \frac{{{\phi _i}}}{{{g_i}}} - {{{\dot{\alpha }} }_i}} \right) \le - \frac{{\phi _i^2}}{{{g_i}}} + \frac{{\phi _i^2}}{2} + \frac{{{\overline{{T}}}_i^2}}{2}} \end{aligned}$$
(53)

Substituting (50), (51), (52) and (53) into (49), we have

$$\begin{aligned} {{{\dot{V}}}_i}\le & {} -\, {\kappa _i}\frac{{z_i^2}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}} - { \left( {\frac{1}{{{g_i}}} - \frac{1}{2} - \frac{{{\bar{\beta }} _{i,i + 1}^2}}{2}} \right) \phi _i^2} \nonumber \\&+\, \frac{1}{2}{\bar{\beta }} _{i,i + 1}^2z_{i + 1}^2 + \left[ {{\beta _{i,i + 1}}N({\zeta _i}) + 1} \right] {{{\dot{\zeta }} }_i} \nonumber \\&-\, {{{{\left( {\frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}} \right) }^2}{{{\tilde{\vartheta }} }_i}{{\left\| {{\psi _i}} \right\| }^2}} \big / {(2\delta _i^2)}}\nonumber \\&+\, \frac{{{{{\tilde{\vartheta }} }_i}{{\dot{{\hat{\vartheta }} }}_i}}}{{{\gamma _i}}} + \frac{{\delta _i^2}}{2} + {\frac{{{\overline{{T}}}_i^2}}{2}} \end{aligned}$$
(54)

According to (43), it is easy to obtain

$$\begin{aligned}&{{ - {{\left( {\frac{{{z_i}}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}} \right) }^2}{{{\tilde{\vartheta }} }_i}{{\left\| {{\psi _i}} \right\| }^2}} \big / {(2\delta _i^2)}} + \frac{{{{{\tilde{\vartheta }} }_i}{{\dot{{\hat{\vartheta }} }}_i}}}{{{\gamma _i}}}\nonumber \\&\quad \le - \frac{1}{2} {{\tilde{\vartheta }} _i^2} + \frac{1}{2} {\vartheta _i^2} \end{aligned}$$
(55)

Substituting (55) into (54), we have

$$\begin{aligned} {{{\dot{V}}}_i}\le & {} - {\kappa _i}\frac{{z_i^2}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}} - {\left( {\frac{1}{{{g_i}}} - \frac{1}{2} - \frac{{{\bar{\beta }} _{i,i + 1}^2}}{2}} \right) \phi _i^2 }- \frac{1}{2}{\tilde{\vartheta }} _i^2 \nonumber \\&+ \left[ {{\beta _{i,i + 1}}N({\zeta _i}) + 1} \right] {{{\dot{\zeta }} }_i} + \frac{1}{2}{\bar{\beta }} _{i,i + 1}^2z_{i + 1}^2\nonumber \\&+ \frac{1}{2}\vartheta _i^2 + \frac{{\delta _i^2}}{2} + {\frac{{{\overline{{T}}}_i^2}}{2}} \end{aligned}$$
(56)

From Lemma 5, it is the fact that \(\ln \frac{{k_{{\mathrm{b}_i}}^2}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}} \le \frac{{z_i^2}}{{k_{{\mathrm{b}_i}}^2 - z_i^2}}\) in the interval \(\left| {{z_i}} \right| < {k_{{\mathrm{b}_i}}}\), we have

$$\begin{aligned} {{\dot{V}}_i}\le & {} - {\mu _i}{V_i} + {\varepsilon _i} \nonumber \\&+ \left[ {{\beta _{i,i + 1}}N({\zeta _i}) + 1} \right] {{\dot{\zeta }} _i} + 0.5{\bar{\beta }} _{i,i + 1}^2z_{i + 1}^2 \end{aligned}$$
(57)

where \({\mu _i} = \min \{ 2{\kappa _i},{\gamma _i},{2g_i^{ - 1} - 1 - {\bar{\beta }} _{i,i+1}^2\} > 0}\) and \({\varepsilon _i} = 0.5\vartheta _i^2 + 0.5\delta _i^2 + {0.5{\overline{{T}}}_i^2}\).

Stepn Define the variable \({z_n} = {x_n} - {{{{\overline{\alpha }}} _{n - 1}}},{} {} {\varphi _n} = {[{\beta _{n,1}}, \ldots , {\beta _{n,n}},{\beta _{n,n + 1}},{f_n}({\bar{x}}_{n + 1}^0)]^\mathrm{T}}\), \({\psi _n} = [{x_1} - x_1^0, \ldots , {x_n} - x_n^0, - {u^0},1]^\mathrm{T}\) with \({u^0} = u(0)\), and invoking (6) with \(u = {x_{n + 1}}\), it has

$$\begin{aligned} {{\dot{z}}_n} = {\dot{x}_n} - {{\dot{{{\overline{\alpha }}}} _{n - 1}}} = \varphi _n^\mathrm{T}{\psi _n} + {\beta _{n,n + 1}}u - {{\dot{{{\overline{\alpha }}}} _{n - 1}}} \end{aligned}$$
(58)

Similarly, the \(\left\| {{\varphi _n}} \right\| \) is also bounded. With the supremum norm theory in mild again, let

$$\begin{aligned} {\vartheta _n} = \mathop {\sup }\limits _{t \ge 0} {\left\| {{\varphi _n}} \right\| ^2} \end{aligned}$$
(59)

Then, consider the BLF in (11) again and choose the following Lyapunov function candidate

$$\begin{aligned} {V_n} = {V_{{\mathrm{b}_n}}} + \frac{1}{{2{\gamma _n}}}{\tilde{\vartheta }} _n^2 \end{aligned}$$
(60)

where \({\gamma _n}\) is a designed positive constant, \({{\tilde{\vartheta }} _n} = {{\hat{\vartheta }} _n} - {\vartheta _n}\), \({\hat{\vartheta }} _n\) denotes the estimation of \({\vartheta _n}\) and \({k_{{\mathrm{b}_n}}}\) is specified later on. In the set \({\Omega _{{z_n}}} = \{ {z_n}:\left| {{z_n}} \right| < {k_{{\mathrm{b}_n}}}\} \), \(V_n\) is continuous. Then, we obtain \({\dot{V}}_{n}\) from (11) and (58) as

$$\begin{aligned} {\dot{V}}_{n} = \frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}\left( {\varphi _n^\mathrm{T}{\psi _n} + {\beta _{n,n + 1}}u - {{{\dot{{{\overline{\alpha }}}} }_{n - 1}}}} \right) + \frac{{{{{\tilde{\vartheta }} }_n}{{\dot{\hat{\vartheta }} }_n}}}{{{\gamma _n}}}\nonumber \\ \end{aligned}$$
(61)

Design the actual control u as

$$\begin{aligned} {u}= & {} N({\zeta _n}){\varpi _n} \end{aligned}$$
(62)
$$\begin{aligned} {\varpi _n}= & {} {\kappa _n}{z_n} - {{\dot{{{\overline{\alpha }}}} _{n - 1}}} + \frac{1}{2}\frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}} \nonumber \\&+ {{\left( {\frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}{{{\hat{\vartheta }} }_n}{{\left\| {{\psi _n}} \right\| }^2}} \right) } \big / {(2\delta _n^2)}} \end{aligned}$$
(63)

where \({\kappa _n}\) and \({\delta _n}\) are designed positive constants. The parameter updating laws are designed as

$$\begin{aligned} {{\dot{\zeta }} _n}= & {} {{{z_n}{\varpi _n}} \big / {(k_{{\mathrm{b}_n}}^2 - z_n^2)}} \end{aligned}$$
(64)
$$\begin{aligned} {\dot{{\hat{\vartheta }}} _n}= & {} {\gamma _n}\left[ { - {{\hat{\vartheta }}_n} + {{{{\left( {\frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}} \right) }^2}{{\left\| {{\psi _n}} \right\| }^2}} \big / {(2\delta _n^2)}}} \right] \end{aligned}$$
(65)

Using (62), (63) and (64), the (61) can be rewritten as

$$\begin{aligned} {{\dot{V}}}_n= & {} -\, {\kappa _n}\frac{{z_n^2}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}} - \frac{1}{2}{\left( {\frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}} \right) ^2} \nonumber \\&-\, {{{{\left( {\frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}} \right) }^2}{{{\hat{\vartheta }} }_n}{{\left\| {{\psi _n}} \right\| }^2}} \big / {(2\delta _n^2)}} \nonumber \\&+\, \frac{{{{{\tilde{\vartheta }} }_n}{{\dot{\hat{\vartheta }}}_n}}}{{{\gamma _n}}} + \frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}\varphi _n^\mathrm{T}{\psi _n} \nonumber \\&+\, \left[ {{\beta _{n,n + 1}}N({\zeta _n}) + 1} \right] {{{\dot{\zeta }} }_n} \end{aligned}$$
(66)

Consider (59) and Assumption 1, using the Youngs inequality, we have

$$\begin{aligned} \frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}\varphi _n^\mathrm{T}{\psi _n} \le {{{{\left( {\frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}} \right) }^2}{\vartheta _n}{{\left\| {{\psi _n}} \right\| }^2}} \big / {(2\delta _n^2) + }}\frac{{\delta _n^2}}{2}\nonumber \\ \end{aligned}$$
(67)

Substituting (67) into (66), we have

$$\begin{aligned} {{{\dot{V}}}_n}\le & {} -\, {\kappa _n}\frac{{z_n^2}}{{k_{{\mathrm{b}_i}}^2 - z_n^2}} + \frac{{\delta _n^2}}{2}\nonumber \\&-\, {{{{\left( {\frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}} \right) }^2}{{{\tilde{\vartheta }} }_n}{{\left\| {{\psi _n}} \right\| }^2}} \big / {(2\delta _n^2)}} \nonumber \\&+\,\frac{{{{{\tilde{\vartheta }} }_n}{{\dot{\hat{\vartheta }}}_n}}}{{{\gamma _n}}} + \left[ {{\beta _{n,n + 1}}N({\zeta _n}) + 1} \right] {{{\dot{\zeta }} }_n} \end{aligned}$$
(68)

According to (65), it is easy to obtain

$$\begin{aligned}&- {{{{\left( {\frac{{{z_n}}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}} \right) }^2}{{{\tilde{\vartheta }} }_n}{{\left\| {{\psi _n}} \right\| }^2}} \big / {(2\delta _n^2)}} + \frac{{{{{\tilde{\vartheta }} }_n}{{\dot{{\hat{\vartheta }}} }_n}}}{{{\gamma _n}}} \nonumber \\&\quad \le - \frac{1}{2}{\tilde{\vartheta }} _n^2 + \frac{1}{2}\vartheta _n^2 \end{aligned}$$
(69)

Substituting (69) into (68), we have

$$\begin{aligned} {\dot{V}}_{n}\le & {} - {\kappa _n}\frac{{z_n^2}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}} - \frac{1}{2}{\tilde{\vartheta }} _n^2 \nonumber \\&+ \left[ {{\beta _{n,n + 1}}N({\zeta _n}) + 1} \right] {{\dot{\zeta }} _n} + \frac{1}{2}\vartheta _n^2 + \frac{{\delta _n^2}}{2} \end{aligned}$$
(70)

From Lemma 5, it is the fact that \(\ln \frac{{k_{{\mathrm{b}_n}}^2}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}} \le \frac{{z_n^2}}{{k_{{\mathrm{b}_n}}^2 - z_n^2}}\) in the interval \(\left| {{z_n}} \right| < {k_{{\mathrm{b}_n}}}\), we have

$$\begin{aligned} {\dot{V}}_{n} \le - {\mu _n}{V_n} + {\varepsilon _n} + \left[ {{\beta _{n,n + 1}}N({\zeta _n}) + 1} \right] {{\dot{\zeta }} _n} \end{aligned}$$
(71)

where \({\mu _n} = \min \{ 2{\kappa _n},{\gamma _n}\} \) and \({\varepsilon _n} = 0.5\vartheta _n^2 + 0.5\delta _n^2\).

4.2 Performance analysis of the closed-loop system

The main results of this study are summarized as the following theorem where it is proved that the aforementioned control scheme solves the control problem of the system (1).

Theorem 1

Consider the closed-loop system consisting of the system (1) obeying Assumptions 1–3, the virtual controllers \({\alpha _i},i = 1,2, \ldots , n - 1\) in (20), (21), (40) and (41), the actual controller u in (62) and (63), and the update laws in (22), (23), (42), (43), (64) and (65) are constructed. If the following conditions hold:

(C1) Choosing appropriate design parameters \({\kappa _i},{\delta _i},{\gamma _i},i = 1, \ldots , n\) and \({g_i},i = 1, \ldots , n - 1\) such that

$$\begin{aligned} {k_{{\mathrm{c}_1}}}> {k_{{\mathrm{b}_1}}} + {{\bar{y}}_\mathrm{r}}, {} {} {} {k_{{\mathrm{c}_{i + 1}}}} > {k_{{\mathrm{b}_{i + 1}}}} + {{\overline{\alpha }}} _i^m, {} {} {} i = 1, \ldots , n - 1\nonumber \\ \end{aligned}$$
(72)

where \({{\bar{y}}_\mathrm{r}} = \max (\left| {{y_\mathrm{r}}} \right| )\) and \({{\overline{\alpha }}} _i^m = \max (\left| {{{{{\overline{\alpha }}} }_i}} \right| )\).

(C2) the initial state \(x_i^0, i = 1, \ldots , n - 1\) satisfies:

$$\begin{aligned} \left| {{z_1}(0)} \right|< {k_{{\mathrm{b}_1}}}, {} {} \left| {{z_i}(0)} \right| < {k_{{\mathrm{b}_i}}}, {} {} i = 2, \ldots , n \end{aligned}$$
(73)

where \({z_1}(0) = x_1^0 - {y_\mathrm{r}}(0)\) and \({z_i}(0) = x_i^0 - {\alpha _i}(0)\).

Then, the following properties hold:

  1. (i)

    All the signals in the closed-loop system are bounded;

  2. (ii)

    The tracking error \({z_1} = y - {y_\mathrm{r}}\) is preserved within a specified prescribed performance bound at all times, i.e., \( - \rho (t)< {z_1} < \rho (t), {} {} {} \forall t > 0\).

  3. (iii)

    the full-state constraints are not violated, i.e., \(\left| {{x_i}} \right| < {k_{{\mathrm{c}_i}},i = 1, \ldots , n}\).

Remark 4

According to (20), (21), (24), (40), (41), (44) and the properties of low-pass filter, we known that \({{{{\overline{\alpha }}} _i}},i = 1, \ldots , n - 1\) is a continuous function of \({{\hat{\vartheta }} _i},{{\bar{x}}_i},\rho , {\dot{\rho }}, {y_\mathrm{r}},{{\dot{y}}_\mathrm{r}}\) and \({{\ddot{y}}_\mathrm{r}}\). Because the boundedness of \({{\hat{\vartheta }} _i},{{\bar{x}}_i},\rho , {\dot{\rho }}, {y_\mathrm{r}},{{\dot{y}}_\mathrm{r}}\) and \({{\ddot{y}}_\mathrm{r}}\), the \({{{\overline{\alpha }}} _i}\) is bounded due to the continuous function properties and moreover assumed to be \(\left| {{{{{\overline{\alpha }}} }_i}} \right| \le \alpha _i^m\) with a positive constant \(\alpha _i^m\). We also can prove the boundedness of \({{\overline{\alpha }}} _i\) by employing Lyapunov stability theory later on.

Table 1 Comparison of proposed controller of existing controller
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Remark 5

The given constrained constants \({k_{{\mathrm{c}_i}}} i = 1, \ldots , n\) need to satisfy feasibility condition (72), we can choose \({k_{{\mathrm{b}_1}}} = {k_{{\mathrm{c}_1}}} - {A_0}\), then it is obvious that \({k_{{\mathrm{c}_1}}} > {k_{{\mathrm{b}_1}}} + {{\bar{y}}_\mathrm{r}}\) according to the Assumption 3. From the (20), (21), (40) and (41), we known that \({\alpha _i},i = 1, \ldots , n - 1\) can be derived step-by-step in back-stepping design, and then we also can gain \({{{\overline{\alpha }}} _i}\) step-by-step. Subsequently, similar to [43], using the Matlab routine function \( \left\langle {fmincon.m} \right\rangle \), we can compute the maximum solution \({k_{{\mathrm{b}_i}}} > 0,i = 2, \ldots ,n\) which satisfy inequality \({k_{{\mathrm{c}_{i + 1}}}} > {k_{{\mathrm{b}_{i + 1}}}} + {{\overline{\alpha }}} _i^m, {} {} i = 1, \ldots , n - 1\) in (72). The computational process is similar to that in [43] and is omitted here due to the limited space.

Proof

Multiplying (71) by \({\mathrm{e}^{{\mu _n}t}}\) and integrating over [0, t], we have

$$\begin{aligned} {V_n} \le {\mathrm{c}_n} + {\mathrm{e}^{ - {\mu _n}t}}\int _0^t {\left[ {{\beta _{n,n + 1}}N({\zeta _n}) + 1} \right] {{{\dot{\zeta }} }_n}{\mathrm{e}^{ - {\mu _n}\tau }}} {\hbox {d}}\iota \nonumber \\ \end{aligned}$$
(74)

where \({\mathrm{c}_n} = {V_n}(0) + {\varepsilon _n}/{\mu _n}\). With the aid of Lemma 1, we can known that \(V_n\) and \({\zeta _n}\) are bounded, and \({z_n}, {} {{\hat{\vartheta }} _n}\) and \(\int _0^t {\left[ {{\beta _{n,n + 1}}N({\zeta _n}) + 1} \right] {{{\dot{\zeta }} }_n}} {\hbox {d}}\iota \) are bounded.

Suppose that there exists unknown positive constant \({{\bar{z}}_n}\) such that \(\left| {{z_n}} \right| \le {{\bar{z}}_n}\), and defining \({{\bar{\varepsilon }} _{n - 1}} = {\varepsilon _{n - 1}} + 0.5{\bar{\beta }} _{n - 1,n}^2{\bar{z}}_n^2\). Consider step \(n-1\) and let in \(i=n-1\) in (57), multiplying both sides by \({\mathrm{e}^{{\mu _{n - 1}}t}}\), we have

$$\begin{aligned} {V_{n - 1}}&\le {\mathrm{c}_{n - 1}}+ {\mathrm{e}^{ - {\mu _{n - 1}}t}} \nonumber \\&\quad \int _0^t {\left[ {{\beta _{n - 1,n}}N({\zeta _{n - 1}}) + 1} \right] {{\dot{\zeta }}_{n - 1}}{\mathrm{e}^{ - {\mu _{n - 1}}\tau }}} {\hbox {d}}\iota \end{aligned}$$
(75)

where \({\mathrm{c}_{n - 1}} = {V_{n - 1}}(0) + {{\bar{\varepsilon }} _{n - 1}}/{\mu _{n - 1}}\). Thus, we can conclude from (75) that \({V_{n - 1}}, {} {\zeta _{n - 1}}, {} {z_{n - 1}}\), \({{\hat{\vartheta }} _{n - 1}}\) and \(\int _0^t {\left[ {{\beta _{n - 1,n}}N({\zeta _{n - 1}}) + 1} \right] {{{\dot{\zeta }} }_{n - 1}}} {\hbox {d}}\iota \) are bounded by using Lemma 1 again. Similarly, repeating \(n-1\) times backwardly, we can prove in turn that \({V_i}, {} {} {\zeta _i}, {} {} {z_i}, {} {{\hat{\vartheta }} _i}\) and \(\int _0^t {\left[ {{\beta _{i,i + 1}}N({\zeta _i}) + 1} \right] {{{\dot{\zeta }} }_i}} {\hbox {d}}\iota , {} {} i = 1, \ldots , n - 2\) are bounded. From (20), (21), (25), (40), (41), (45), (62) and (63), the virtual controllers \({\alpha _i},{{{{\overline{\alpha }}} _i}}, {} {} i = 1, \ldots , n - 1\) and actual controller u are bounded, namely, we have proved property (i) All the signals in the closed-loop system are bounded.

Fig. 1
figure 1

The output y and reference signal \(y_\mathrm{r}\)

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Fig. 2
figure 2

The state \(x_2\)

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Fig. 3
figure 3

The tracking error \(z_1\) and PPB

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Fig. 4
figure 4

The adaptive parameters \({\zeta _1}\) and \({\zeta _2}\)

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Let \({\upsilon _n}\) be the upper bound of \(\left| {\left[ {{\beta _{n,n + 1}}N({\zeta _n}) + 1} \right] {{{\dot{\zeta }} }_n}} \right| \), and from (71), we have

$$\begin{aligned} {\dot{V}}_{n} \le - {\mu _n}{V_n} + {{\sigma _n}} \end{aligned}$$
(76)

where \({\sigma _n} = {\varepsilon _n} + {\upsilon _n}\). According to Lemma 6, we can obtain that \(\left| {{z_n}} \right| < {k_{{\mathrm{b}_n}}}\). Similarly, Let \({\upsilon _i}\) be the upper bound of \(\left| {\left[ {{\beta _{i,i + 1}}N({\zeta _i}) + 1} \right] {{\dot{\zeta }}_i}} \right| \), and form (37) and (57), we have

$$\begin{aligned} {{\dot{V}}_i} \le - {\mu _i}{V_i} + {\sigma _i}, {} {} {} i = 1, \ldots , n - 1 \end{aligned}$$
(77)

where \({\sigma _i} = {\varepsilon _i} + {\upsilon _i} + 0.5{\bar{\beta }} _{i,i + 1}^2{\bar{z}}_{i + 1}^2\) with \({{\bar{z}}_{i\mathrm{{ + }}1}}\) being the upper bound of \(\left| {{z_{i + 1}}} \right| ,i = 1, \ldots , n - 1\). According to Lemma 6 again and again, we can obtain that \(\left| {{z_1}/\rho (t)} \right| < 1\) and \(\left| {{z_i}} \right| < {k_{{\mathrm{b}_i}}}, {} i = 2, \ldots ,n\) from (12) and (11), i.e., (ii) The tracking error \({z_1} = y - {y_\mathrm{r}}\) is preserved within \( - \rho (t)< {z_1} < \rho (t),{} {} {\forall t > 0}\). \(\square \)

Remark 6

The cyclic argument may arise from the Nussbaum-type function for unknown control direction and BLF for constraints, so the Lemmas 1, 2 and 6 are employed to handle the problem of cyclic argument and extend the solution of the closed-loop system to infinity.

Based on the fact that \(\rho (t) = ({k_{{\mathrm{b}_1}}} - {\rho _\infty }){\mathrm{e}^{ - \ell t}} + {\rho _\infty }\) is positive and strictly decreasing and \(- \rho (t)< {z_1} < \rho (t)\), we have \(- {k_{{\mathrm{b}_1}}}< {z_1} < {k_{{\mathrm{b}_1}}}\). Then from \({x_1} = {z_1} + {y_\mathrm{r}}\) and \(\left| {{y_\mathrm{r}}} \right| < {A_0}\), we have \( - {k_{{\mathrm{b}_1}}} - {A_0}< {x_1} < {k_{{\mathrm{b}_1}}} + {A_0}\). Thus, we can obtain that \( - {k_{{\mathrm{c}_1}}}< {x_1} < {k_{{\mathrm{c}_1}}}\) according to the fact \({k_{{\mathrm{b}_1}}} = {k_{{\mathrm{c}_1}}} - {A_0}\). From the definition of \({z_i},i = 2, \ldots , n\), we known that \({x_i} = {z_i} + {{{{\overline{\alpha }}} _{i - 1}}}\), then based on \(\left| {{z_i}} \right| < {k_{{\mathrm{b}_i}}}, {} {\left| {{{{{\overline{\alpha }}} }_{i-1}}} \right| \le {{\overline{\alpha }}} _{i-1}^m}\) and condition (72): \({k_{{\mathrm{c}_i}}} > {k_{{\mathrm{b}_i}}} + {{\overline{\alpha }}} _{i - 1}^m\), we can obtain \(\left| {{x_i}} \right| < {k_{{\mathrm{c}_i}}}\), i.e., we have proved property (iii).

Fig. 5
figure 5

The adaptive parameters \({\vartheta _1}\) and \({\vartheta _2}\)

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5 Simulation result

To clarify and verify the performance of the proposed approximation-free PP-BLF-based adaptive control scheme, we present some simulation studies in this section. In the simulation, the unknown systems are assumed that (i) Numerical example and (ii) Single-link robot.

5.1 Numerical example

Consider the following pure-feedback nonlinear systems

$$\begin{aligned} \left\{ \begin{array}{lll} {{\dot{x}}_1} = 0.1x_1^2 - {x_2} + 0.2\sin ({x_1}{x_2})\\ {{\dot{x}}_2} = 0.2{x_1}{x_2} + {x_1} + 2u + 0.01{u^3}\\ y = {x_1} \end{array} \right. \end{aligned}$$
(78)

where the states are constrained in \(\left| {{x_1}} \right| < {k_{{\mathrm{c}_1}}} = 0.8\) and \(\left| {{x_2}} \right| < {k_{{\mathrm{c}_2}}} = 2.5\). The initial values are \(x_1^0 = 0.6,x_2^0 = - 0.1,u(0) = 0,{\zeta _1}(0) = 0.3,{\zeta _2}(0) = 0.1,{\vartheta _1}(0) = 0.1\) and \({\vartheta _2}(0) = 0.1\), the reference signal is \({y_\mathrm{r}} = 0.5\cos (t/5)\) with \({A_0} = 0.5\) and \({k_{{\mathrm{b}_1}}} = {k_{{\mathrm{c}_1}}} - {A_0} = 0.3\), it is easy to know that the \(z_1(0)\) satisfies \(\left| {{z_1}(0)} \right| < {k_{{\mathrm{b}_1}}}\). For the control performance requirements, we allow steady-state error no more than 0.005, minimum speed of convergence as obtained by the exponential \({\mathrm{e}^{ - 6t}}\). Then, the performance function can be given as \(\rho (t) = ({k_{{\mathrm{b}_1}}} - 0.005){\mathrm{e}^{ - 6t}} + 0.005\).

In the simulation, to show the superiority, the investigated PP-BLF-based controller is compared with a non-BLF/PPC-based control scheme and a BLF-based control scheme proposed in [45]. The concrete controller and design parameters are shown in Table 1. Similar to [43], the parameter \({k_{{\mathrm{b}_2}}}\) is computed by utilizing the Matlab function \( \left\langle {fmincon.m} \right\rangle \) to satisfy feasibility condition (72) for the proposed PP-BLF-based control scheme and satisfy feasibility condition in [45] (Please refer toTheorem 1in [45] for details) for BLF-based control scheme.

Remark 7

Compared with results in [45], the proposed PP-BLF-based control scheme can solve the problem of state constraints, prescribed performance tracking and “explosion of complexity” integratedly, and the knowledge of the sign of control gain and the LIP condition of unknown nonlinear functions are not required from the Table 1.

Figures 1, 2, 3, 4, 5 and 6 show the simulation results. Figures 1 and 2 show that the output can accurately track the desired signals. Simultaneously, the PP-BLF-based and BLF-based adaptive control scheme can guarantee that the full-state constraints are not violated. However, the non-BLF/PPC-based control scheme cannot guarantee that full-state constraints are not violated (State\(x_2\)breaks the constraint in Fig. 2). Figure 3 shows the system output tracking error and it can be seen from the figure that the tracking error with prescribed performance is achieved all the time by using the proposed PP-BLF-based controller, and the non-BLF/PPC-based and BLF-based in [45] control scheme cannot achieve the prescribed performance all the time. The estimation value of \({\zeta _1},{} {} {\zeta _2},{} {} {} {\vartheta _1}\) and \({\vartheta _2}\) is shown in Figs. 4 and 5, which are easily asymptotic with respect to the zero point. The control signals are show in Fig. 6.

Fig. 6
figure 6

The control signal u

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Fig. 7
figure 7

The output q and reference signal \(y_\mathrm{r}\)

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Fig. 8
figure 8

The state \(\dot{q} \)

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5.2 Single-link robot

The single-link robot dynamic equations are

$$\begin{aligned} \left\{ \begin{array}{l} M{\ddot{q}} + \frac{1}{2}mgl\sin q = u\\ y = q \end{array} \right. \end{aligned}$$
(79)

where M is the moment of inertia, q is the angle, u is the input torque, g is the gravity acceleration, m and l are the mass and the length of the link. The robot parameters are \(m = l = 1,M = 0.5\) and \(g=0.8\). Let \({x_1} = q,{x_2} = \dot{q}\), we can gain that \(f({x_1},{x_2}) = {x_2},{} f({x_1},{x_2},u) = (u - \frac{1}{2}mgl\sin {x_1})/M\) and \(y=x_1\). The states are constrained in \(\left| {{x_1}} \right| < {k_{{\mathrm{c}_1}}} = 1\) and \(\left| {{x_2}} \right| < {k_{{\mathrm{c}_2}}} = 1.5\). The initial values are \(x_1^0 = 0.5,x_2^0 = 0.1,u(0) = 0,{\zeta _1}(0) = 0.3,{\zeta _2}(0) = 0.1,{\vartheta _1}(0) = 0.1\) and \({\vartheta _2}(0) = 0.1\), the reference signal is \({y_\mathrm{r}} = 0.5\cos (t/5)\) with \({A_0} = 0.5\) and \({k_{{\mathrm{b}_1}}} = {k_{{\mathrm{c}_1}}} - {A_0} = 0.5\), it is easy to know that the \(z_1(0)\) satisfies \(\left| {{z_1}(0)} \right| < {k_{{\mathrm{b}_1}}}\). Then, the PPF is selected as \(\rho (t) = ({k_{{\mathrm{b}_1}}} - 0.02){\mathrm{e}^{ - 4t}} + 0.02{} \).

The design parameters are chosen as \({\kappa _1} = 2.05, {\kappa _2} = 5,{\gamma _1} = 2,{\gamma _1} = 2,{\sigma _1} = 1,{\sigma _2} = 1, g_1=0.001\). Similar to [43], we can obtain \({k_{{\mathrm{b}_2}}} = 1.103\) by using Matlab routine. The simulation results are obtained in Figs. 7, 8, 9, 10 and 11. From these figures, we can see that the prescribed performance and state constraints requirements can be guaranteed with the investigated PP-BLF-based adaptive controller. However, the non-BLF/PPC-based and BLF-based controller cannot achieve these two indexes together.

Fig. 9
figure 9

The tracking error \(z_1\) and PPB

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Fig. 10
figure 10

The adaptive parameters \({\zeta _1}\) and \({\zeta _2}\)

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Fig. 11
figure 11

The adaptive parameters \({\vartheta _1}\) and \({\vartheta _2}\)

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6 Conclusions

In this paper, a novel approximation-free PP-BLF-based adaptive control technique has been proposed for unknown pure-feedback nonlinear systems with full-state constraints. By employing the mean value theorem, the systems are transformed into linear structure form. Then, a novel back-stepping design is developed with the aid of use BLF, PP-BLF, Nussbaum-type function and supremum norm theory to eliminate the difficult problems of unknown dynamics, full-state constraints and prescribed performance requirements. We have proved that all the signals in the closed-loop system are uniformly bounded and the tracking error is preserved within a specified prescribed performance bound without violating the constraints. Finally, simulation results have demonstrated the feasibility and validity of the proposed control scheme.