1 Introduction

Uncertainties, including unmodeled nonlinear dynamics, external disturbances and parametric perturbations, are ubiquitous in practice. In control science and technology, it is a central issue to ensure the normal operation of control systems despite various uncertainties [1].

Motivated by this important objective, numerous control strategies have been substantially developed, such as proportional-integral-derivative (PID) control [2], adaptive control [3], fuzzy control [4, 5], neural network-based control [6,7,8] and disturbance rejection methods [9,10,11,12], just to name a few. In [2], the capability of PID control to handle nonlinear uncertainties was proved. The reference [3] comprehensively introduced the design and application of adaptive control, which can tackle a large scope of parametric uncertainties. The design of fuzzy control for nonlinear uncertainties was reviewed in [4]. More importantly, for nonlinear systems with unknown control coefficients, a novel fuzzy controller was developed in [5]. Besides, for nonlinear lower-triangular systems, neural network-based designs were proposed in [6, 7]. In addition, various disturbance rejection designs have been proposed in the past decades [9,10,11,12], which can online estimate and compensate for uncertainties.

In recent years, disturbance rejection methods have drawn lots of attention from researches due to the simplicity in practical implementation and the superior performance to handle nonlinear uncertainties. Active disturbance rejection control (ADRC) is one of the most popular designs among various disturbance rejection methods, which has been successfully applied to flight systems [13,14,15], motion control systems [16, 17] and process control systems [18,19,20], just to name a few. In the framework of ADRC, the extended state observer (ESO) is novelly constructed to estimate the total effect of various uncertainties, named as “total disturbance” [21], and then the control input is composed of the compensation for total disturbance and the feedback of integrators chain states.

In the last two decades, the theoretical foundation of ADRC has been substantially established. For the ESO which is the vital component in ADRC, the literature [22] investigated the convergence of ESO and further showed the bound of estimating error. The closed-loop stability of ADRC was firstly presented in [23], where the considered uncertainties and their derivatives with respect to the time were assumed to be bounded. For the linear time-invariant uncertain systems, the literature [24] theoretically illustrated the satisfied tracking performance of ADRC despite a large scope of parametric variations. In [25], by assuming the existence of the Lyapunov functions related with uncertainties, the capability of ADRC to deal with nonlinear uncertainties was proved. By assuming that the uncertainties and their partial derivatives are bounded if the states are in a bounded set, the references [26, 27] rigorously studied the convergence and the transient performance of ADRC-based closed-loop system. It is remarkable that the conditions in [26, 27] can depict a wide class of nonlinear uncertainties in practice. By utilizing the concept of total disturbance, the literatures [28, 29] illuminated the capability of ADRC to handle the mismatched nonlinearities in both dynamical systems and measurement models. Besides, several successful modifications of ADRC have been made for nonlinear uncertain systems with other complicated practical factors, such as time delay [30, 31] and stochastic uncertainties [32, 33].

Up to now, the theoretical results have demonstrated the effectiveness of ADRC to tackle the time-varying disturbances and the nonlinear internal uncertainties dependent on system states. In the conventional ADRC design, it is remarkable that the information of the control coefficient is required to design the ESO and the compensation term for total disturbance [12, 34, 35]. In practical systems, the true values of control coefficients are usually unknown [24], which promotes the development of ADRC based on the nominal values or the approximative mathematical expressions of control coefficients. For the ADRC based on the nominal information of control coefficients, the literatures [24, 26, 28] quantitatively analyzed the stability region of uncertain control coefficients. Unfortunately, compared with the capability to handle the nonlinear uncertainties dependent on system states, the capability of the conventional ADRC to deal with the uncertainties of control coefficients is limited [36]. Besides, in some practical systems, it is difficult to obtain the nominal values or the approximative mathematical expressions of control coefficients [18, 19, 30]. Hence, it is significant to design a new ADRC which is featured with the strong robustness to uncertainties and does not rely on the nominal values or the approximative mathematical expressions of control coefficients.

The paper studies the control problem for a class of lower-triangular nonlinear uncertain systems. Based on the signs of control coefficients rather than the nominal values or the approximative mathematical expressions, a new ADRC design is proposed to handle the mismatched nonlinear uncertainties and the unknown values of control coefficients. The design procedure of the new ADRC is separated into three parts: (1) determining the transformation from the original states to the states of an equivalent integrators chain system, (2) designing the ESO to estimate the total disturbance and the states of the integrators chain system, and (3) forcing the actual input to track the desired input signal by designing a dynamical system. By rigorously analyzing the closed-loop properties, the bounds of tracking error, estimating error and the error between the actual and desired inputs are explicitly shown as the functions of control parameters. Based on the detailed expressions of error bounds, it is demonstrated that the satisfied closed-loop performance can be obtained despite a wide class of uncertainties by suitably enlarging the ESO’s parameter. Moreover, by meticulously studying the relationship between the original states and the integrators chain states, the paper removes the bounded hypothesis for integrators chain states, which is required in [21]. The main contributions of the paper are as follows.

  1. (1)

    In the conventional ADRC designs [12, 26, 28, 34], the nominal values or the approximative mathematical expressions of control coefficients are required. In the paper, a new ADRC based on the signs of control coefficients is proposed, which can deal with a large scope of uncertain control coefficients.

  2. (2)

    Compared with the mismatched bounded disturbances [5] and Lipschitz continuous uncertainties [7], the paper considers the mismatched uncertainties with nonlinear growth with respect to system states. Moreover, despite a wide class of nonlinear uncertainties, the satisfied tracking performance in the whole time period is proved.

  3. (3)

    The tuning principle of the control parameters is provided. Especially, the relationship between the parameter of input dynamical system and the observer parameter is explicitly shown.

The rest of the paper has the following organization. In Sect. 2, the problem formulation is given. In Sect. 3, the new ADRC based on the signs of control coefficients is proposed. In Sect. 4, the theoretical analysis of the closed-loop system is presented. The simulation studies are shown in Sect. 5. The conclusion is presented in Sect. 6.

1.1 Notations

The following notations are used throughout the paper. \(y^{(k)}(t)\) represents the k-th order derivative of y with respect to the variable t for \(k \ge 1\) and \(y^{(0)}(t) \triangleq y(t)\). The notations \(|\cdot |\) and \(\Vert \cdot \Vert \) are the absolute value of a scalar and the 2-norm of a vector or a matrix, respectively. The notation \(\text {diag}(a_1,a_2,\cdots ,a_m)\) represents a diagonal matrix with the dimension \(m\times m\), whose i-th diagonal element is \(a_i\). For a given real symmetric matrix M, the maximal eigenvalue of M is denoted as \(\lambda _\mathrm{max}(M)\) and the minimal eigenvalue of M is denoted as \(\lambda _\mathrm{min}(M)\). The following useful matrices are introduced.

$$\begin{aligned} \begin{aligned}&A=\left[ {\begin{matrix}0&{}\quad 1&{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad 0 &{}\quad \ddots &{}\quad \vdots \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \end{matrix}} \right] _{n \times n},~~ B = \left[ {\begin{matrix}0\\ \vdots \\ 0 \\ 1 \end{matrix}} \right] _{n \times 1}, \\&\quad B_f = \left[ {\begin{matrix}0\\ \vdots \\ 0 \\ 1 \end{matrix}} \right] _{(n+1) \times 1}, \qquad C = \left[ {\begin{matrix}1\\ 0\\ \vdots \\ 0 \end{matrix}} \right] _{n \times 1},\\&A_e = \begin{bmatrix} A&{}\quad B \\ 0&{}\quad 0 \end{bmatrix}_{(n+1)\times (n+1)},~~ C_e = \begin{bmatrix} C \\ 0 \end{bmatrix}_{(n+1)\times 1}. \end{aligned} \end{aligned}$$
(1)

The function \(\hbox {sgn}(\cdot )\) represents the sign function, which satisfies

$$\begin{aligned} \hbox {sgn}(a) = \left\{ \begin{aligned}&1,\quad \quad \,\, \text {if }a>0,\\&0,\quad \quad \,\, \text {if }a=0,\\&-1,\quad \text {if }a<0. \end{aligned}\right. \end{aligned}$$
(2)

2 Problem formulation

Consider the following class of lower-triangular nonlinear uncertain systems with unknown control coefficients.

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{x}}_i(t) = \theta _{i}(t) x_{i+1}(t) + \phi _i(x_1(t),\cdots ,x_i(t),t),\quad 1\le i\le n-1,\\&{\dot{x}}_n(t) = \theta _{n}(t) u(t) + \phi _n(x_1(t),\cdots ,x_n(t),t) ,\\&{\dot{z}}(t) = g(z,x,t),\\&y(t) = x_1(t), \quad t\ge t_0, \end{aligned}\right. \end{aligned}$$
(3)

where \(x_i(t)\in R\) \((1\le i \le n)\) are the system states, \(x(t)\triangleq [x_1(t)~\cdots ~x_n(t)]^T\in R^n\) is the system state vector, \(z(t)\in R^m\) is the state vector of zero dynamics, \(y(t)\in R\) is the measured output to be controlled, \(u(t)\in R\) is the control input, \(g(\cdot )\) represents the dynamics of z(t), \(\phi _i(\cdot )~(1\le i\le n)\) represent the uncertainties in various channels which might be mismatched, and \(\theta _i(t)~(1\le i\le n)\) are the unknown time-varying control coefficients. As shown in [37], the signs of the control coefficients \(\theta _i(t)\), i.e., \(\hbox {sgn}(\theta _i(t))~(1\le i \le n)\), represent the control directions of the system (3). The paper considers the situation that the control directions \(\hbox {sgn}(\theta _i(t))~(1\le i \le n)\) are known. However, both the approximative mathematical expressions and the nominal values of the control coefficients \(\theta _i(t)\) are unknown.

Based on the control directions \(\hbox {sgn}(\theta _i(t))\) rather than the nominal values or the approximative mathematical expressions of control coefficients, the control objective of the system (3) is to design the control input u(t) such that the output y(t) can track the reference signal r(t) despite the mismatched nonlinear uncertainties \(\phi _i(\cdot )~(1\le i \le n)\).

Remark 1

The system (3) can model plenty of practical processes, such as flight systems [15], motion control systems [16] and process control systems [20]. More importantly, under the strong nonlinearity of the unknown functions \(\phi _i(\cdot )\), it is challenging to achieve the output regulation task only based on the control directions \(\hbox {sgn}(\theta _i(t))\).

Before the detailed control design, the following assumptions for the reference signal r(t), the mismatched nonlinear uncertainties \(\phi _i(\cdot )~(1\le i\le n)\) and the dynamics of z(t) are introduced.

Assumption 1

There exists a positive constant \(M_r\) such that \(\sup _{t\ge t_0} |r^{(i)}(t)|\le M_r\) for \(0\le i\le n+1\).

Remark 2

Assumption 1 implies that the reference signal and its derivatives are bounded, which is rational in practice [34].

Assumption 2

The functions \(\phi _i(\cdot )\) and \(\theta _i(\cdot )\) are \((n+1-i)\)-th order differentiable with respect to their variables for \(1\le i \le n\). There exist positive constants \({\bar{M}}_{\theta ,i}\) and \({\underline{M}}_{\theta ,i}\) and continuous functions \(\psi _{\phi ,i}(x_1,\cdots ,x_i)\) such that

$$\begin{aligned}&\sup _{t\ge t_0, 0\le j\le n+1-i}|\theta _i^{(j)}(t)| \le {\bar{M}}_{\theta ,i},\nonumber \\&\quad \inf _{t\ge t_0}|\theta _i(t)| \ge {\underline{M}}_{\theta ,i}>0, \end{aligned}$$
(4)
$$\begin{aligned}&\sup _{t\ge t_0} \left| \frac{\partial ^{ \Sigma _{k=1}^{i+1} j_k } \phi _i(x_1,\cdots ,x_i,t)}{\partial x_1^{j_1} \cdots \partial x_i^{j_i} \partial t^{j_{i+1}}} \right| \nonumber \\&\quad \le \psi _{\phi ,i}(x_1,\cdots ,x_i), \end{aligned}$$
(5)

for \(\Sigma _{k=1}^{i+1} j_k \le n+i-1\), \(1\le i \le n\) and \(j_{p}\ge 0~(1\le p \le i+1)\).

Remark 3

Assumption 2 describes a large scope of internal nonlinear uncertainties and external disturbances in practice, which is more general than the assumptions in [5, 7]. From (4), the control coefficients and their derivatives are assumed to be bounded. Since the practical systems are controllable, the control coefficients \(\theta _i(t)\) are assumed to be nonzero for \(t\ge t_0\), as shown in (4) [38]. Hence, (4) further implies that the control directions \(\hbox {sgn}(\theta _i(t))\) will not change. We remark that the assumption for \(\theta _i(t)\) is a common one [5, 37]. As for the nonlinear uncertainties \(\phi _i(\cdot )\), (5) implies that the uncertainties and their partial derivatives are bounded by some continuous functions \(\psi _{\phi ,i}(\cdot )\) dependent on system states. Due to (5), the uncertainties \(\phi _i(\cdot )\) and their partial derivatives are assumed to be bounded if the system states are bounded, which allows the uncertainties to grow nonlinearly with respect to system states.

Assumption 3

There exists a radially unbounded positive-definite function \(V_z(z)\) such that

$$\begin{aligned}&{\dot{V}}_z(z(t)) = \frac{\hbox {d} V_z}{\hbox {d} z} g(z,x,t)\le 0, \quad \nonumber \\&\forall V_z(z) \ge r_z(\Vert x\Vert ),\quad t\ge t_0, \end{aligned}$$
(6)

where \(r_z(\cdot )\) is a nonnegative continuous increasing function.

Remark 4

By regarding x(t) as the input of the z-subsystem, Assumption 3 implies that the z-subsystem is uniformly input-state-stable [26]. For the system (3) being a linear time-invariant system, Assumption 3 is equivalent to that the system (3) is a minimum phase plant.

Remark 5

As shown in the conventional ADRC design [12, 26, 28, 34], the nominal values or the approximative mathematical expressions of control coefficients are required, which play important roles in the design of ESO and compensation term. However, it is sometimes hard to acquire the nominal values or the approximative mathematical expressions in practical systems [18, 19], which leads to the ruleless tuning of nominal control coefficients. In the other hand, due to physical mechanism, the control directions, i.e., the signs of the control coefficients, can be easily determined. Hence, the paper aims to develop a new ADRC based on the control directions rather than the nominal values or the approximative mathematical expressions of control coefficients.

In the next section, an ADRC design based on the control directions \(\hbox {sgn}(\theta _i(t))\) is proposed to achieve the output regulation task despite multiple uncertainties.

3 ADRC based on control directions

In this section, a new ADRC based on control directions is proposed. The corresponding control diagram is shown in Fig. 1. The rest of this section consists of the following three parts.

  1. 1.

    The equivalent integrators chain form for the system (3) is investigated, which shows the essential relationship between the input and the output. Moreover, the output regulation task is transmitted to the state tracking task.

  2. 2.

    Based on the equivalent integrators chain system, an ESO is presented to estimate the derivatives of output and the unknown term named as “total disturbance.”

  3. 3.

    Via the estimations from ESO, the control input is generated by a dynamical system, which forces the input to track the desired input signal.

Fig. 1
figure 1

Control diagram for the proposed ADRC

Full size image

3.1 Analysis for the equivalent integrators chain form

In this subsection, the equivalent integrators chain form for the system (3) is investigated.

Denote the new state vector \({\tilde{x}}(t)= [{\tilde{x}}_1(t)~\cdots ~{\tilde{x}}_n(t)]^T \in R^n\) as follows.

$$\begin{aligned} \left\{ \begin{aligned} {\tilde{x}}_1(t)&= x_1 (t),\\ {\tilde{x}}_{i}(t)&= \dot{{\tilde{x}}}_{i-1}(t) \\&= \left( \mathop {\Pi }\limits _{j=1}^{i-1} \theta _j(t)\right) x_i (t) + \mathop {\Sigma }\limits _{j=1}^{i-2} \frac{\hbox {d}^{j-1}}{\hbox {d}t^{j-1}} \left( \frac{\hbox {d}\left( \mathop {\Pi }\limits _{k=1}^{i-j-1}\theta _k(t)\right) }{\hbox {d}t} x_{i-j}(t) \right) \\&\quad + \mathop {\Sigma }\limits _{j=1}^{i-1} \frac{\hbox {d}^{j-1}}{\hbox {d}t^{j-1}} \left( \left( \mathop {\Pi }\limits _{k=0}^{i-j-1} \theta _k(t)\right) \phi _{i-j}(x_1,\cdots ,x_{i-j},t) \right) ,~~ 2 \le i\le n, \end{aligned}\right. \end{aligned}$$
(7)

where \(\theta _0(t) = 1\) for \(t\ge t_0\).

The following proposition illustrates the relationship between the state vectors x and \({\tilde{x}}\).

Proposition 1

Consider the transformation (7) under Assumption 2. Then, there exist two continuous mappings satisfying

$$\begin{aligned} \gamma (x,t)\triangleq & {} \begin{bmatrix} \gamma _1(x,t) \\ \vdots \\ \gamma _{n+1}(x,t) \end{bmatrix} = \begin{bmatrix} {\tilde{x}}_1 \\ \vdots \\ {\tilde{x}}_n \\ t \end{bmatrix},\quad \nonumber \\ \varphi ({\tilde{x}},t)\triangleq & {} \begin{bmatrix} \varphi _1({\tilde{x}},t) \\ \vdots \\ \varphi _{n+1}({\tilde{x}},t) \end{bmatrix} = \begin{bmatrix} x_1 \\ \vdots \\ x_n \\ t \end{bmatrix}, \end{aligned}$$
(8)

and

$$\begin{aligned}&\sup _{t\ge t_0,\Vert x\Vert \le \varrho _x }\left\{ |\gamma _i(x,t)|, \left\| \frac{\partial \gamma _i(x,t)}{\partial x} \right\| , \left| \frac{\partial \gamma _i(x,t)}{\partial t} \right| \right\} \nonumber \\&\quad \le \psi _{\gamma }(\varrho _x),\quad \forall \varrho _x\ge 0,~ \forall 1\le i\le n, \end{aligned}$$
(9)
$$\begin{aligned}&\sup _{t\ge t_0,\Vert {\tilde{x}}\Vert \le {\tilde{\varrho }}_x }\left\{ |\varphi _i({\tilde{x}},t)|, \left\| \frac{\partial \varphi _i({\tilde{x}},t)}{\partial {\tilde{x}}} \right\| , \left| \frac{\partial \varphi _i({\tilde{x}},t)}{\partial t} \right| \right\} \nonumber \\&\quad \le \psi _{\varphi }({\tilde{\varrho }}_x),~~ \forall {\tilde{\varrho }}_x\ge 0,~ \forall 1\le i\le n, \end{aligned}$$
(10)

where \(\psi _{\gamma }(\cdot )\) and \(\psi _{\varphi }(\cdot )\) are nonnegative continuous increasing functions dependent on \({\bar{M}}_{\theta ,i}\), \({\underline{M}}_{\theta ,i}\) and \(\psi _{\phi ,i}\) for \(1\le i \le n\).

The proof of Proposition 1 is given in Appendix.

Remark 6

In the existing studies [21, 28], the bounds of the mappings \(\gamma _i(\cdot )\) and \(\varphi _i(\cdot )\) are provided in additional assumptions. In this paper, by rigorously analyzing the detailed expressions of the mappings \(\gamma _i(\cdot )\) and \(\varphi _i(\cdot )\), we prove that \(\gamma _i(\cdot )\) and \(\varphi _i(\cdot )\) and their partial derivatives satisfy (9)–(10). Hence, the assumptions for the bounds of \(\gamma _i(\cdot )\) and \(\varphi _i(\cdot )\) are removed in the paper.

Based on the transformation (7) and Proposition 1, the system (3) can be rewritten as the following integrators chain system.

$$\begin{aligned} \left\{ \begin{aligned}&\dot{{\tilde{x}}}_i(t) = {\tilde{x}}_{i+1}(t) ,\quad 1\le i\le n-1,\\&\dot{{\tilde{x}}}_n(t) = b(t) u(t) + f({\tilde{x}}(t),t) ,\\&{\dot{z}}(t) = g(z,\varphi _1({\tilde{x}},t),\cdots ,\varphi _n({\tilde{x}},t),t),\\&y(t) = {\tilde{x}}_1(t), \quad t\ge t_0, \end{aligned} \right. \end{aligned}$$
(11)

with the initial condition \({\tilde{x}}(t_0) = [\gamma _1(x(t_0),t_0)~\cdots \gamma _n(x(t_0),t_0)]^T\). The control coefficient b(t) and the uncertainty \(f({\tilde{x}}(t),t)\) have the following form.

$$\begin{aligned} \left\{ \begin{aligned}&b(t) = \Pi _{i=1}^{n} \theta _i(t),\\&f({\tilde{x}},t) = \Sigma _{j=1}^{n-1} \frac{\hbox {d}^{j-1}}{\hbox {d}t^{j-1}} \left( \frac{\hbox {d} \left( \Pi _{k=1}^{n-j}\theta _k(t)\right) }{\hbox {d}t} \varphi _{n+1-j} ({\tilde{x}},t) \right) \\&\quad \quad \quad \quad \quad + \Sigma _{j=1}^{n} \frac{\hbox {d}^{j-1}}{\hbox {d}t^{j-1}} \left( \left( \Pi _{k=0}^{n-j} \theta _k(t)\right) {\tilde{\phi }}_{n+1-j} ({\tilde{x}},t) \right) , \end{aligned} \right. \nonumber \\ \end{aligned}$$
(12)

where \({\tilde{\phi }}_{i}({\tilde{x}},t) = \phi _i(\varphi _1({\tilde{x}},t),\cdots ,\varphi _i({\tilde{x}},t),t)\) for \(1 \le i \le n\).

If the nominal value of the control coefficient b(t), denoted as \({\bar{b}}(t)\), can be obtained, the conventional ADRC can be applied to the system (11) by regarding \(f({\tilde{x}},t)+(b(t)-{\bar{b}}(t))u(t)\) as the total disturbance [28, 34]. However, only the sign of the control coefficient b(t) is known in the paper, rather than the nominal value or the approximative mathematical expression.

$$\begin{aligned} \hbox {sgn}(b(t)) = \Pi _{i=1}^{n} \hbox {sgn}(\theta _i(t)). \end{aligned}$$
(13)

Next, based on the integrators chain form (11) and the sign of control coefficient b(t), an ADRC design will be proposed.

3.2 ESO design

Since the nominal value of the control coefficient b(t) is unknown, the total disturbance of the system (11) is denoted as

$$\begin{aligned} f_t({\tilde{x}},u,t) \triangleq b(t)u(t)+f({\tilde{x}},t). \end{aligned}$$
(14)

Then, the following ESO is presented to estimate \({\tilde{x}}\) and \(f_t\).

$$\begin{aligned} \begin{bmatrix} \dot{\hat{{\tilde{x}}}}(t) \\ \dot{{\hat{f}}}_t(t)\end{bmatrix} = A_e \begin{bmatrix} \hat{{\tilde{x}}}(t) \\ {\hat{f}}_t(t)\end{bmatrix} + L_e \left( y(t)-C_e^T \begin{bmatrix} \hat{{\tilde{x}}}(t) \\ {\hat{f}}_t(t)\end{bmatrix} \right) ,\nonumber \\ \end{aligned}$$
(15)

where \(\hat{{\tilde{x}}}(t) = [\hat{{\tilde{x}}}_1(t)~\cdots ~\hat{{\tilde{x}}}_{n}(t)]^T\in R^{n}\) is the estimation for the state vector \({\tilde{x}}(t)\) and \({\hat{f}}_t(t) \in R\) is the estimation for the total disturbance \(f_t({\tilde{x}},u,t)\). In addition, the constant vector \(L_e\in R^{(n+1)\times 1}\) is the tunable parameter vector of ESO such that the matrix \(A_L \triangleq A_e-L_eC_e^T\) is Hurwitz. Owing to [39], the following concise tuning method of \(L_e\) is presented.

$$\begin{aligned} L_e= & {} \begin{bmatrix}\varsigma _1 \omega _o&\varsigma _2 \omega _o^2&\cdots&\varsigma _{n+1} \omega _o^{n+1} \end{bmatrix}^{T}, \quad \nonumber \\ \varsigma _i= & {} \frac{(n+1)!}{(n+1-i)! i!}, \quad \omega _o> 0, \end{aligned}$$
(16)

which ensures that all the eigenvalues of \(A_L\) are set at \(- \omega _o\).

Next, a dynamical design of ADRC input based on the estimations from ESO will be presented.

3.3 Dynamical input design

Firstly, the desired control input is introduced. For the system (11), the desired closed-loop system satisfies the following form.

$$\begin{aligned} \left\{ \begin{aligned}&\dot{{\tilde{x}}}_i^*(t) = {\tilde{x}}_{i+1}^*(t) ,\quad 1\le i\le n-1,\\&\dot{{\tilde{x}}}_n^*(t) = -K^T ({\tilde{x}}^*(t)-{\bar{r}}(t)) + r^{(n)}(t) ,\\&{\dot{z}}^*(t) = g(z^*,\varphi _1({\tilde{x}}^*,t),\cdots ,\varphi _n({\tilde{x}}^*,t),t),\\&y^*(t) = {\tilde{x}}_1^*(t), \quad t\ge t_0, \quad {\tilde{x}}^*(t_0) = {\tilde{x}}(t_0), \end{aligned} \right. \end{aligned}$$
(17)

where \({\tilde{x}}^*(t)\triangleq [{\tilde{x}}^*_1(t)~\cdots ~{\tilde{x}}^*_n(t)]\in R^n\) is the state vector of the desired system, \(z^*(t)\in R^m\) is the state vector of the desired zero dynamics, \(y^*(t)\in R\) is the desired output, \({\bar{r}}(t) \triangleq [r(t)~\cdots ~r^{(n-1)}(t)]^T\in R^n\) and the constant vector \(K\in R^{n\times 1}\) is feedback gain vector satisfying that the matrix \(A_K\triangleq A-BK^T\) is Hurwitz.

Remark 7

For the desired system (17), the output \(y^*(t)\) can exponentially converge to the reference signal r(t) with the desired convergence rate by tuning the feedback gain vector K. Moreover, the system states \(({\tilde{x}}^*(t),z^*(t))\) are bounded for \(t\ge t_0\).

By comparing the integrators chain system (11) with the desired system (17), the desired control input can be obtained as follows.

$$\begin{aligned} u^*(t)= & {} \frac{-f({\tilde{x}},t) - K^T ({\tilde{x}}(t)-{\bar{r}}(t))+r^{(n)}(t)}{b(t)}. \end{aligned}$$
(18)

Inspired by the approximative dynamic inversion method in [40], we design the following dynamical system to generate the actual input u(t) which can approach the desired input signal \(u^*(t)\).

$$\begin{aligned} {\dot{u}}(t)= & {} -\hbox {sgn}(b(t)) \kappa (\omega _o) ({\hat{f}}_t(t) \nonumber \\&+ K^T (\hat{{\tilde{x}}}(t)-{\bar{r}}(t))-r^{(n)}(t)), \end{aligned}$$
(19)

where \(\kappa (\omega _o) >0\) is a function to be designed.

The parameter \(\kappa (\omega _o)\) should be carefully selected to ensure that the dynamics (19) is “slower” than the dynamics of the ESO (15) [41]. However, the explicit tuning law of \(\kappa (\omega _o)\) has not been provided in the existing studies. To make the proposed method more friendly to practitioners, the tuning law of \(\kappa (\omega _o)\) is depicted in the following assumption.

Assumption 4

The increasing function \(\kappa (\omega _o)>0\) for \(\omega _o>0\) and

$$\begin{aligned}&\lim _{\omega _o \rightarrow \infty } \frac{\ln \omega _o}{ \sqrt{\kappa (\omega _o)}} =0,\quad \nonumber \\&\lim _{\omega _o \rightarrow \infty }\frac{\kappa (\omega _o)}{\omega _o} =0. \end{aligned}$$
(20)

Assumption 4 describes the growth rate of the increasing function \(\kappa (\omega _o)\) which grows faster than the log function \(f_1(\omega _o) = (\ln \omega _o)^2\) and slower than the linear function \(f_2(\omega _o)=\omega _o\). It is remarkable that the function \(\kappa (\omega _o) = \omega _o^k\) with \(0<k<1\) can satisfy Assumption 4.

Remark 8

This remark provides the design ideology of the dynamical system (19). Firstly, the dynamics of u(t) is supposed to satisfy the following equation, which can force u(t) to track \(u^*(t)\).

$$\begin{aligned} {\dot{u}}(t) = -\varpi (t)(u(t)-u^*(t)), \end{aligned}$$
(21)

where \(\varpi (t)>0\) is a function to be designed. By substituting (18) into (21), we have

$$\begin{aligned} \begin{aligned} {\dot{u}}(t)&= -\frac{\varpi (t)}{b(t)}(b(t)u(t) +f({\tilde{x}},t) \\&\quad + K^T ({\tilde{x}}(t)-{\bar{r}}(t))-r^{(n)}(t))\\&= -\frac{\varpi (t)}{b(t)}(f_t({\tilde{x}},u,t) \\&\quad + K^T ({\tilde{x}}(t)-{\bar{r}}(t))-r^{(n)}(t)). \end{aligned} \end{aligned}$$
(22)

Then, by substituting the estimations \({\hat{f}}_t\) and \(\hat{{\tilde{x}}}\) into (22), the control input can be designed as follows.

$$\begin{aligned} {\dot{u}}(t)= & {} -\frac{\varpi (t)}{b(t)}({\hat{f}}_t(t) \nonumber \\&+ K^T (\hat{{\tilde{x}}}(t)-{\bar{r}}(t))-r^{(n)}(t)). \end{aligned}$$
(23)

Finally, the dynamical system of input (19) is obtained by designing \(\varpi (t) = \kappa (\omega _o) |b(t)|\).

4 Performance analysis of closed-loop system

The performance of the closed-loop system based on the ADRC design (15) and (19) is investigated in this section.

The following theorem shows the satisfactory transient performance of the closed-loop system based on the proposed ADRC despite a wide class of uncertainties.

Theorem 1

Consider the system (3) with Assumptions 1–4. Let \(u(t)=0\) for \(t\in [t_0,t_u)\) where

$$\begin{aligned} \left\{ \begin{aligned}&t_u = t_0 + 2 n c_{\varsigma 2} \frac{\max \left\{ \ln \left( \omega _o \varrho _0 \right) ,~0 \right\} }{\sqrt{\kappa (\omega _o)}},\quad \varrho _0 = \mathop {\max }\limits _{2\le i \le n}|{\tilde{x}}_{i}(t_0)-\hat{{\tilde{x}}}_{i}(t_0) |^{\frac{1}{n}},\\&c_{\varsigma 2} = \lambda _\mathrm{max}(P_\varsigma ),\quad A_\varsigma ^T P_\varsigma + P_\varsigma A_\varsigma =-I,~~ A_\varsigma = \left[ {\begin{matrix} -\varsigma _1&{}\quad 1&{}\quad 0&{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad 0 &{}\quad \ddots &{}\quad \ddots &{}\quad \vdots \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad 0 \\ -\varsigma _{n} &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 1 \\ -\varsigma _{n+1} &{}\quad 0 &{}\quad \cdots &{}\quad \cdots &{}\quad 0 \end{matrix}} \right] . \end{aligned} \right. \end{aligned}$$
(24)

For \(t\ge t_u\), u(t) is designed according to (15) and (19). Then, there exist positive constants \(\eta _i^*~(1\le i\le 5)\) and \(\omega ^*\) dependent on \((x(t_0),\hat{{\tilde{x}}}(t_0),{\hat{f}}_t(t_0),M_r, {\underline{M}}_{\theta ,i}, {\bar{M}}_{\theta ,i},\psi _{\phi i}, K)\) such that

$$\begin{aligned}&\sup \limits _{t\ge t_0}|y(t)-y^*(t)|\nonumber \\&\quad \le \eta _1^* \max \left\{ \frac{\ln \omega _o}{\sqrt{\kappa (\omega _o)}} , \frac{\kappa (\omega _o)}{\omega _o}, \frac{1}{\kappa (\omega _o)} \right\} , \end{aligned}$$
(25)
$$\begin{aligned}&\left\| \begin{bmatrix} {\tilde{x}}(t) - \hat{{\tilde{x}}}(t) \\ f_t({\tilde{x}},u,t) - {\hat{f}}_t(t) \end{bmatrix} \right\| \nonumber \\&\quad \le \eta _2^*\left( \frac{\kappa (\omega _o)}{\omega _o} + e^{-\eta _3^* \omega _o (t-t_u)} \right) ,~~\forall t\ge t_u, \end{aligned}$$
(26)
$$\begin{aligned}&|u(t)-u^*(t)|\nonumber \\&\quad \le \eta _4^*\left( \frac{\kappa (\omega _o)}{\omega _o} \!+\! \frac{1}{\kappa (\omega _o)} \!+\! e^{-\eta _5^* \kappa (\omega _o) (t-t_u)} \right) ,~~\forall t\ge t_u, \end{aligned}$$
(27)

for any \(\omega _o \ge \omega ^*\).

Theorem 1 demonstrates that the tracking error between the actual and desired outputs, the estimating error of the ESO (15) and the error between the actual and desired inputs are bounded. Furthermore, (25)–(27) explicitly show the bounds of the tracking error, estimating error and the error between the actual and desired inputs. More importantly, as shown in (25), the tracking error between the actual and ideal outputs can be sufficiently small for \(t\ge t_0\) by tuning \(\omega _o\) to be suitably large, which ensures the satisfied transient performance despite various uncertainties. In addition, (26)–(27) imply that both the estimating error and the error between the actual and desired inputs can converge into a neighborhood with a small boundary by suitably enlarging \(\omega _o\).

Remark 9

In the proposed design (15) and (19), the adjustable parameters are \(\kappa \), K and \(\omega _o\). According to Assumption 4, the parameter \(\kappa \) can be designed as a function with respect to \(\omega _o\), such as \(\kappa =\omega _o^k\) for a constant k satisfying \(0<k<1\). The feedback gain K should satisfy that the matrix \(A_K\) is Hurwitz, which determines the convergence rate of the desired output \(y^*\) generated by the system (17). Based on Theorem 1, the observer parameter \(\omega _o\) should be suitably large to ensure the small tracking error and estimating error.

Remark 10

The design (24) is a common way to avoid the peaking phenomenon of ESO [26, 29]. Moreover, since plenty of practical systems satisfy the initial condition that \({\tilde{x}}(t_0) = [0~\cdots ~0]^T\), by designing the initial values of ESO as \(\hat{{\tilde{x}}}(t_0)=[0~\cdots ~0]^T\), then it can be obtained that \(t_u=t_0\). In addition, for the initial values satisfying \(\varrho _0\ge \frac{1}{\omega _o}\), it can be deduced from (24) and Assumption 4 that \(t_u\) can be close to \(t_0\) by suitably enlarging \(\omega _o\).

To simplify the proof of Theorem 1, Propositions 2–3 are introduced. Proposition 2 describes the bounds of the uncertainty \(f({\tilde{x}},t)\), the control coefficient b(t) and their derivatives. Proposition 3 provides the closed-loop form and the bounds of the uncertain terms in closed-loop system. The proofs of Propositions 2–3 are given in Appendix.

Proposition 2

Let Assumption 2 holds. For any given positive constant \({\tilde{\varrho }}_x\), the functions b(t) and \(f({\tilde{x}},t)\) in (12) satisfy the following equations:

$$\begin{aligned}&\sup _{ t\ge t_0 } \left\{ |(b(t))^{-1}|, |b(t)|, |{\dot{b}}(t)| \right\} \le \psi _b, \end{aligned}$$
(28)
$$\begin{aligned}&\sup _{t\ge t_0, \Vert {\tilde{x}} \Vert \le {\tilde{\varrho }}_x} \left\{ |f({\tilde{x}},t)|, \left\| \frac{\partial f({\tilde{x}},t)}{\partial {\tilde{x}}}\right\| ,\left| \frac{\partial f({\tilde{x}},t)}{\partial t}\right| \right\} \nonumber \\&\quad \le \psi _{f}({\tilde{\varrho }}_x), \end{aligned}$$
(29)

where the positive constant \(\psi _b\) and the non-decreasing function \(\psi _{f}(\cdot )\) are dependent on \({\bar{M}}_{\theta ,i}\), \({\underline{M}}_{\theta ,i}\) and \(\psi _{\phi ,i}\) for \(1\le i \le n\).

Denote the tracking error vector, estimating error vector and the error between the actual and desired inputs as follows.

$$\begin{aligned} e(t)= & {} {\tilde{x}}(t)-{\tilde{x}}^*(t),\quad \nonumber \\ \zeta (t)= & {} T_1^{-1} \begin{bmatrix} {\tilde{x}}(t) - \hat{{\tilde{x}}}(t) \\ f_t({\tilde{x}},u,t) - {\hat{f}}_t(t) \end{bmatrix},\quad \delta _u(t) = u(t) - u^*(t),\nonumber \\ \end{aligned}$$
(30)

where \(T_1=\text {diag}(\omega _o^{-n},\cdots ,\omega _o^{-1},1)\). Then, the following proposition presents the closed-loop system and further analyzes the properties of the uncertain terms in closed-loop system.

Proposition 3

Let Assumptions 1–3 hold. Design \(u(t)=0\) for \(t\in [t_0,t_u)\), and design u(t) by (15) and (19) for \(t\in [t_u,\infty )\). Then, the closed-loop system is shown as follow.

$$\begin{aligned}&\left\{ \begin{aligned}&{\dot{e}}(t) = A e(t) +B \Delta _{e0}(e,t),\\&{\dot{z}}(t) = g(z,\varphi _1({\tilde{x}}^*+e,t),\cdots ,\varphi _n({\tilde{x}}^*+e,t),t),\\&{\dot{\zeta }}(t) = \omega _o A_\varsigma \zeta (t) +B_f \Delta _{\zeta 0}(e,t),\\&{\dot{\delta }}_u(t) = -|b(t)| \kappa \delta _u(t) + \Delta _{\delta _u 0}(e,\zeta ,\omega _o,\kappa ,t), \end{aligned} \right. \quad t\in [t_0,t_u),\nonumber \\ \end{aligned}$$
(31)
$$\begin{aligned}&\left\{ \begin{aligned}&{\dot{e}}(t) = A_K e(t) +B \Delta _{e1}(\delta _u,t),\\&{\dot{z}}(t) = g(z,\varphi _1({\tilde{x}}^*+e,t),\cdots ,\varphi _n({\tilde{x}}^*+e,t),t),\\&{\dot{\zeta }}(t) = \omega _o A_\varsigma \zeta (t) +B_f \Delta _{\zeta 1}(e,\zeta ,\delta _u,\omega _o,\kappa ,t),\\&{\dot{\delta }}_u(t) = -|b(t)| \kappa \delta _u(t) + \Delta _{\delta _u 1}(e,\zeta ,\delta _u,\omega _o,\kappa ,t), \end{aligned} \right. \quad t\in [t_u,\infty ).\nonumber \\ \end{aligned}$$
(32)

Moreover, the uncertain terms \(\Delta _{e0}\), \(\Delta _{\zeta 0}\), \(\Delta _{\delta _u 0}\), \(\Delta _{e1}\), \(\Delta _{\zeta 1}\) and \(\Delta _{\delta _u 1}\) have the following bounds.

$$\begin{aligned} \left\{ \begin{aligned}&|\Delta _{e0}| \le \pi _{e0}(\varrho _e),\quad |\Delta _{e1}| \le \psi _b |\delta _u|,\\&|\Delta _{\zeta 0}| \le \pi _{\zeta 0}(\varrho _e),\quad |\Delta _{\zeta 1}| \le \pi _{\zeta 1}(\varrho _e) + \pi _{ \omega }(\omega _o^*) \kappa \Vert \zeta \Vert + (\kappa +1) \pi _{\delta _u}(\varrho _e,\varrho _u) ,\\&|\Delta _{\delta _u 0}| \le \pi _{\delta _u 0}(\varrho _e)+\pi _{ \omega }(\omega _o^*)\kappa \Vert \zeta \Vert ,\quad |\Delta _{\delta _u 1}| \le \pi _{\delta _u 1}(\varrho _e , \varrho _u) + \pi _{ \omega }(\omega _o^*)\kappa \Vert \zeta \Vert , \end{aligned} \right. \end{aligned}$$
(33)

for \(e\in \{e|~\Vert e\Vert \le \varrho _e \}\), \(\zeta \in \{\zeta |~\Vert \zeta \Vert \le \varrho _\zeta \}\), \(\delta _u \in \{\delta _u|~|\delta _u|\le \varrho _u \}\) and \(\omega _o\in \{\omega _o|~\omega _o\ge \omega _o^* \}\) with any given positives \(\varrho _e\), \(\varrho _\zeta \), \(\varrho _u\) and \(\omega _o^*\). The functions \(\pi _{e0}(\cdot )\), \(\pi _{\zeta 0}(\cdot )\), \(\pi _{\zeta 1}(\cdot )\), \(\pi _{\delta _u}(\cdot )\), \(\pi _{\delta _u 0}(\cdot )\) and \(\pi _{\delta _u 1}(\cdot )\) are non-decreasing and are dependent on K, \(M_r\), \({\bar{M}}_{\theta ,i}\), \({\underline{M}}_{\theta ,i}\) and \(\psi _{\phi ,i}\) for \(1\le i \le n\). The function \(\pi _\omega (\cdot )\) is non-increasing and is dependent on \({\bar{M}}_{\theta ,i}\), \({\underline{M}}_{\theta ,i}\) and K.

Based on Propositions 2–3, the proof of Theorem 1 is presented as follows.

Proof of Theorem 1

With the mapping from (x, t) to \(({\tilde{x}},t)\) presented in Proposition 1 and the discussions in Sect. 3.1, the plant (3) can be rewritten as the integrators chain system (11). Owing to Propositions 2–3, the closed-loop system is formulated as (31)–(32).

Since the matrices \(A_K\) and \(A_\varsigma \) are Hurwitz, there exist positive definite matrices \(P_K\) and \(P_\varsigma \) such that \(A_K^T P_K+ P_K A_K=-I\) and \(A_\varsigma ^T P_\varsigma + P_\varsigma A_\varsigma =-I\). Then, the following Lyapunov functions are introduced.

$$\begin{aligned} V_K(t)= & {} e^T(t) P_K e(t),\quad V_\varsigma (t) = \zeta ^T(t) P_\varsigma \zeta (t),\quad \nonumber \\ V_u(t)= & {} \frac{\delta _u^2(t)}{2}. \end{aligned}$$
(34)

Let \(c_{k1}\) and \(c_{k2}\) be the minimal and maximal eigenvalues of \(P_K\) and denote \(c_{\varsigma 1}\) and \(c_{\varsigma 2}\) as the minimal and maximal eigenvalues of \(P_\varsigma \). Then, the following inequalities hold.

$$\begin{aligned}&c_{k1} \Vert e(t)\Vert ^2 \le V_K(t) \le c_{k2}\Vert e(t)\Vert ^2,\quad \nonumber \\&c_{\varsigma 1} \Vert \zeta (t)\Vert ^2 \le V_\varsigma (t) \le c_{\varsigma 2}\Vert \zeta (t)\Vert ^2. \end{aligned}$$
(35)

Next, we analyze the properties of the closed-loop system for \(t\in [t_0,t_u)\) and \(t\in [t_u,\infty )\).

Part 1: The analysis for the closed-loop system for \(t\in [t_0,t_u)\).

Owing to (11) and (17), the initial condition satisfies that \(e(t_0) = 0\). The dynamical systems (31)–(32) imply the continuity of e(t). Hence, for a sufficiently small positive constant \(\Delta t\), there exists a positive constant \(\eta _{e1}\) such that

$$\begin{aligned} \Vert e(t)\Vert \le \eta _{e1},\quad \forall t\in [t_0,t_0+\Delta t]. \end{aligned}$$
(36)

According to Assumption 4, it can be verified that \(\lim _{\omega _o \rightarrow \infty } \frac{\ln \omega _o}{\sqrt{\kappa (\omega _o)} } = 0\). In addition, it can be verified from (24) that \(\lim _{\omega _o \rightarrow \infty } t_u = t_0\). Hence, there exists a positive constant \(\omega _1\) such that \(t_u- t_0 \le \Delta t\) for \(\omega _o \ge \omega _1\). Combined with the statement (36), it can be concluded that \(\Vert e(t)\Vert \le \eta _{e1}\) for \(t\in [t_0,t_u)\) and \(\omega _o \ge \omega _1\).

Due to Proposition 1, the bound of x(t) for \(t\in [t_0,t_u)\) satisfies that \(\sup _{t_0\le t\le t_u} \Vert x(t)\Vert \le n \psi _\varphi (\eta _{e1}+M_{x^*})\). Combined with Assumption 3, the following equation is satisfied.

$$\begin{aligned}&{\dot{V}}_z (z(t)) \le 0,\quad \forall V_z(z) \ge r_z(n \psi _\varphi (\eta _{e1}+M_{x^*})),\nonumber \\&\quad t_0\le t\le t_u, \end{aligned}$$
(37)

which further implies that

$$\begin{aligned} \sup _{t_0\le t\le t_u}V_z(z(t))\le & {} \eta _{V_z 1} \nonumber \\\triangleq & {} r_z(n \psi _\varphi (\eta _{e1}+M_{x^*})). \end{aligned}$$
(38)

Based on the bound of e(t) for \(t\in [t_0,t_u)\), the dynamics (31) implies that

$$\begin{aligned} \begin{aligned} \sup _{t_0\le t\le t_u}\Vert e(t)\Vert&\le (t_u-t_0)\cdot \sup _{t_0\le t\le t_u}\{ \Vert A\Vert \Vert e(t)\Vert \\&\quad + \Vert B\Vert |\Delta _{e0}(e,t)|\} \\&\le 2n c_{\varsigma 2} (\Vert A\Vert \eta _{e1} \\&\quad + \pi _{e0}(\eta _{e1})) \frac{\ln \omega _o+|\ln \rho _0|}{\sqrt{\kappa }}. \end{aligned}\nonumber \\ \end{aligned}$$
(39)

Due to (31) and (33), the following dynamics of \(\sqrt{V_\varsigma (t)}\) and \(\sqrt{V_u(t)}\) hold for \(t\in [t_0,t_u]\).

$$\begin{aligned} \left\{ \begin{aligned}&\frac{d \sqrt{V_\varsigma (t)}}{dt} = \frac{-\omega _o \Vert \zeta \Vert ^2 + 2\zeta ^T P_\varsigma B_f \Delta _{\zeta 0}}{2\sqrt{V_\varsigma }}\\&\quad \quad \quad \quad \quad \le -\frac{\omega _o}{2c_{\varsigma 2}} \sqrt{V_\varsigma (t)} + \frac{\Vert P_\varsigma \Vert \pi _{\zeta 0} (\eta _{e1})}{\sqrt{c_{\varsigma 1}}},\\&\frac{d \sqrt{V_u(t)}}{dt} = \frac{-|b|\kappa |\delta _u|^2+\delta _u \Delta _{\delta _u 0}}{2\sqrt{V_u}} \\&\quad \quad \quad \quad \quad \le - \psi _b^{-1}\kappa \sqrt{V_u(t)} + \frac{\sqrt{2}\pi _{\delta _u 0}(\eta _{e1}) +\sqrt{2}\pi _{\omega }(\omega _1)\kappa \Vert \zeta (t)\Vert }{2}, \end{aligned} \right. \nonumber \\ \end{aligned}$$
(40)

for \(\omega _o \ge \omega _1\). With the help of Gronwall lemma, it can be deduced from (40) that \(\sqrt{V_\varsigma (t)}\) has the following bound.

$$\begin{aligned} \sqrt{V_\varsigma (t)}\le & {} \frac{2c_{\varsigma 2} \Vert P_\varsigma \Vert \pi _{\zeta 0 }(\eta _{e1}) }{\sqrt{c_{\varsigma 1}}\omega _o} \nonumber \\&+ \sqrt{V_\varsigma (t_0)} e^{-\frac{\omega _o(t-t_0)}{2c_{ \varsigma 2}}},\quad t\in [t_0,t_u]. \end{aligned}$$
(41)

Based on Gronwall lemma and (40)–(41), the bound of \(\sqrt{V_u(t)}\) is obtained as follows.

$$\begin{aligned} \begin{aligned} \sqrt{V_u(t)}&\le \frac{\sqrt{2}\pi _{\delta _u 0}(\eta _{e1}) }{2\psi _b^{-1}\kappa } + \sqrt{V_u(t_0)} e^{-\psi _b^{-1}\kappa (t-t_0)}\\&\quad +\int _{t_0}^{t} e^{-\psi _b^{-1}\kappa (t-s)} \sqrt{2}\pi _{\omega }\kappa \Vert \zeta (s)\Vert ds\\&\le \frac{\sqrt{2}\pi _{\delta _u 0}(\eta _{e1}) }{2\psi _b^{-1}\kappa } \\&\quad +\frac{2\sqrt{2}\pi _{\omega } \kappa c_{\varsigma 2} \Vert P_\varsigma \Vert \pi _{\zeta 0}(\eta _{e1}) }{c_{\varsigma 1}\psi _b^{-1}\omega _o}\\&\quad + \sqrt{V_u(t_0)} e^{-\psi _b^{-1}\kappa (t-t_0)}\\&\quad + \frac{\sqrt{2}\pi _{\omega }\kappa \sqrt{V_\varsigma (t_0)} }{\sqrt{c_{\varsigma 1}} \left( \frac{\omega _o}{2c_{\varsigma 2}}-\psi _b^{-1}\kappa \right) } \\&\qquad e^{-\psi _b^{-1}\kappa (t-t_0)}(1 - e^{-(\frac{\omega _o}{2c_{\varsigma 2}}-\psi _b^{-1}\kappa )(t-t_0)} ). \end{aligned} \end{aligned}$$

Next, the bounds of \(\sqrt{V_\varsigma (t_u)}\) and \(\sqrt{V_u(t_u)}\) will be analyzed.

Based on the initial condition \(\varrho _0> \frac{1}{\omega _o}\), the initial value of \(\sqrt{V_\varsigma }\) satisfies that \(\sqrt{V_\varsigma (t_0)} \le n\sqrt{c_{\varsigma 2}}(\omega _o^{n-1}\varrho _0^{n}+\eta _{f0})\) where \(\eta _{f0} = |{\hat{f}}_t(t_0)-f_t({\tilde{x}}(t_0),u(t_0),t_0)|\). Owing to the definition of \(t_u\) (24), the following inequalities hold for \(\frac{\omega _o}{\kappa }\ge 1\) and \(\kappa \ge \max \{1, \frac{1}{4 \psi _b^{-2} c_{\varsigma 2}^2} \}\).

$$\begin{aligned} \left\{ \begin{aligned}&\sqrt{V_\varsigma (t_0)} e^{-\frac{\omega _o(t_u-t_0)}{2c_{\varsigma 2}}} = \sqrt{V_\varsigma (t_0)} e^{-\frac{n\omega _o}{\sqrt{\kappa }} \ln (\omega _o \varrho _0) } \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \le \frac{n\sqrt{c_{\varsigma 2}}}{\omega _o } +\frac{n\sqrt{c_{\varsigma 2}}\eta _{f0} }{\omega _o^n \varrho _0^n},\\&e^{-\psi _b^{-1}\kappa (t_u-t_0)} = e^{- 2nc_{\varsigma 2}\psi _b^{-1}\sqrt{\kappa } \ln (\omega _o \varrho _0) } \le \frac{1}{\varrho _0^n \omega _o^n}. \end{aligned} \right. \end{aligned}$$
(42)

Owing to (41)–(42), the following inequalities are satisfied for any \(\omega _o\ge \omega _2\), \(\kappa \ge \kappa _2\) and \(\frac{\omega _o}{\kappa }\ge \tau _2\), where \(\omega _2=\max \{\kappa _2\tau _2,\omega _1\}\), \(\kappa _2=\max \{1,\frac{1}{4 \psi _b^{-2} c_{\varsigma 2}^2}\}\) and \(\tau _2 =\max \{1,4\psi _b^{-1}c_{\varsigma 2}\} \).

$$\begin{aligned} \left\{ \begin{aligned}&\sqrt{V_\varsigma (t_u)} \le \eta _{V_\varsigma 1}(\omega _2)\triangleq \frac{2c_{\varsigma 2} \Vert P_\varsigma \Vert \pi _{\zeta 0 }(\eta _{e1}) }{\sqrt{c_{\varsigma 1}}\omega _2}+ \frac{n\sqrt{c_{ \varsigma 2}}}{\omega _2} +\frac{n\sqrt{c_{\varsigma 2}}\eta _{f0} }{\omega _2^n \varrho _0^n},\\&\sqrt{V_u(t_u)} \le \eta _{V_u 1}(\omega _2,\kappa _2) \triangleq \frac{\sqrt{2}\pi _{\delta _u 0}(\eta _{e1}) }{2\psi _b^{-1}\kappa _2} +\frac{2\sqrt{2}\pi _{\omega }(\omega _2) c_{\varsigma 2} \Vert P_\varsigma \Vert \pi _{\zeta 0}(\eta _{e1}) \tau _2}{c_{\varsigma 1}\psi _b^{-1}} \\&\quad \quad \quad \quad \quad \quad + \frac{ \sqrt{V_u(t_0)} }{\omega _2^n \varrho _0^n} + \frac{\sqrt{2}\pi _{\omega }(\omega _2) n \sqrt{c_{\varsigma 2}} }{\sqrt{c_{\varsigma 1}} \psi _b^{-1} \omega _2 } + \frac{\sqrt{2}\pi _{\omega }(\omega _2) n \sqrt{c_{\varsigma 2}} \eta _{f0} }{\sqrt{c_{\varsigma 1}} \psi _b^{-1} \omega _2^n \varrho _0^n }. \end{aligned} \right. \end{aligned}$$
(43)

Considering the initial condition satisfying \(\varrho _0 \le \frac{1}{\omega _o}\), (24) implies that \(t_u=t_0\). It can be directly proved that \(\Vert e(t_u)\Vert \), \(V_z(z(t_u))\), \(\sqrt{V_\varsigma (t_u)}\) and \(\sqrt{V_u(t_u)}\) are bounded for \( \omega _o\ge \omega _2\). Without loss of generality, the bounds of \(\Vert e(t_u)\Vert \), \(V_z(z(t_u))\), \(\sqrt{V_\varsigma (t_u)}\) and \(\sqrt{V_u(t_u)}\) can be also denoted as \(\eta _{e1}\), \(\eta _{V_z 1}\), \(\eta _{V_\varsigma 1}\) and \(\eta _{V_u 1}\), respectively.

Part 2: The analysis for the closed-loop system for \(t\in [t_u,\infty )\).

According to the control design (19) and the closed-loop system (31)–(32), the variables e,\(\zeta \),z and u are continuous at \(t_u\). To simplify the mathematical expressions in this part, we introduce the following notations.

$$\begin{aligned} \left\{ \begin{aligned}&\eta _{V_K2} \triangleq \max \{\sqrt{c_{k2}}\eta _{e1}, 4\Vert P_K\Vert \psi _b \eta _{\delta _u 2} \},\quad \eta _{V_\varsigma 2} \triangleq \eta _{V_\varsigma 1},\\&\eta _{V_u 2} \triangleq \max \{\eta _{V_u 1}, {2\pi _{\omega }(\omega _2)\eta _{\zeta 2} \psi _b} \},\\&\eta _{V_z 2} \triangleq r_z ( n \psi _\varphi (\eta _{e2}+M_{x^*}) ),\\&\eta _{e2} \triangleq \frac{\eta _{V_K1}}{\sqrt{c_{k1}}},~~~~ \eta _{\zeta 2} \triangleq \frac{\eta _{V_\zeta 2}}{\sqrt{c_{\zeta 1}}},~~~~ \eta _{\delta _u 2} \triangleq \sqrt{2}\eta _{V_u 2}. \end{aligned} \right. \nonumber \\ \end{aligned}$$
(44)

Then, we proceed to prove that there exist positive constants \((\omega _3,\kappa _3,\tau _3)\) such that \((e(t),z(t),\zeta (t),\delta _u(t))\) stay in a bounded set \(\Omega _1\triangleq \{(e,z,\zeta ,\delta _u)~|~ \sqrt{V_K}\le \eta _{V_K2},~ \sqrt{V_z}\le \eta _{V_z 2},~\sqrt{V_\varsigma }\le \eta _{V_\varsigma 2},~\sqrt{V_u} \le \eta _{V_u 2} \}\) for \(\omega _o\ge \omega _3\), \(\kappa \ge \kappa _3\), \(\frac{\omega _o}{\kappa }\ge \tau _3\) and \(t\ge t_u\). The proof consists of the following four steps:

Step 1 We assume that there exists \(t^*\in [t_u,\infty )\) such that \(\sqrt{V_\varsigma (t^*)}= \eta _{V_\varsigma 2}\). Besides, \(\sqrt{V_\varsigma (t)}\le \eta _{V_\varsigma 2}\), \( \sqrt{V_K(t)}\le \eta _{V_K2}\), \( V_z(t)\le \eta _{V_z2}\) and \(\sqrt{V_{\delta _u}(t)} \le \eta _{V_u 2}\) for \(t\in [t_u,t^*]\). Then, it can be deduced that \(\Vert e(t)\Vert \le \eta _{e2}\), \( \Vert \zeta (t)\Vert \le \eta _{\zeta 2}\) and \(|\delta _u(t)|\le \eta _{\delta _u 2}\) for \(t\in [t_u,t^*]\). Owing to the dynamics (32) and the bound of \(\Delta _{\zeta 1}\) (33), the derivative of \(\sqrt{V_\varsigma (t^*)}\) satisfies

$$\begin{aligned} \frac{d \sqrt{V_\varsigma (t^*)}}{d t}\le & {} -\frac{ \omega _o \eta _{V_\varsigma 2} }{2 c_{\varsigma 2} } \nonumber \\&+ \frac{\Vert P_\varsigma \Vert (\pi _{\zeta 1} (\eta _{e2}) +(\kappa +1)\pi _{\delta _u}(\eta _{e2},\eta _{\delta _u 2}) + \pi _{\omega }(\omega _o) \kappa \eta _{\zeta 2} ) }{\sqrt{c_{\varsigma 1}}}. \end{aligned}$$
(45)

By selecting \(\omega _3 = \max \{\omega _2 ,6 c_{\varsigma 2} \Vert P_\varsigma \Vert (\pi _{\zeta 1}(\eta _{e2}) +\pi _{\delta _u} (\eta _{e2},\eta _{\delta _u 2}) ) / (\eta _{V_\varsigma 2}\sqrt{c_{\varsigma 1}}) \}\) and \(\tau _3 = 6 c_{\varsigma 2} \Vert P_\varsigma \Vert ( \pi _{\omega } (\omega _2) \eta _{\zeta 2} + \pi _{\delta _u} (\eta _{e2} , \eta _{\delta _u 2}) )/( \eta _{V_\varsigma 2}\sqrt{c_{\varsigma 1}} )\), (45) directly implies that \(\frac{d \sqrt{V_\varsigma (t^*)}}{dt} < 0\) for the \((\omega _o,\kappa )\) satisfying \(\omega _o\ge \omega _3\) and \(\frac{\omega _o}{\kappa }\ge \tau _3\).

Step 2 We assume that there exists \(t^*\in [t_u,\infty )\) such that \(V_z(t^*)= \eta _{V_z 2}\). Besides, \(\sqrt{V_u(t)} \le \eta _{V_u 2}\), \(\sqrt{V_K(t)} \le \eta _{V_K2}\), \( V_z(t)\le \eta _{V_z2}\) and \(\sqrt{V_\varsigma (t)}\le \eta _{V_\varsigma 2}\) for \(t\in [t_u,t^*]\). Proposition 1, the bound of x(t) satisfies that

$$\begin{aligned} \sup _{t_u\le t\le t^*}\Vert x(t)\Vert\le & {} n\psi _\varphi \left( \sup _{t_u\le t\le t^*} \Vert {\tilde{x}}(t)\Vert \right) \nonumber \\\le & {} n\psi _\varphi (\eta _{e2}+M_{x^*}),\quad \forall t\in [t_u,t^*] .\nonumber \\ \end{aligned}$$
(46)

The definition of \(\eta _{V_z 2}\) (44) implies that \(\eta _{V_z 2} \ge r_z(\sup _{t_u \le t\le t^*}\Vert x(t)\Vert ) \ge r_z(\Vert x(t^*)\Vert ).\) Due to Assumption 3, it can be verified that \({\dot{V}}_z(z(t^*)) \le 0.\)

Step 3 We assume that there exists \(t^*\in [t_u,\infty )\) such that \(\sqrt{V_u(t^*)} = \eta _{V_u 2}\). Besides, \(\sqrt{V_u(t)} \le \eta _{V_u 2}\), \(\sqrt{V_K(t)} \le \eta _{V_K2}\), \( V_z(t)\le \eta _{V_z2}\) and \(\sqrt{V_\varsigma (t)}\le \eta _{V_\varsigma 2}\) for \(t\in [t_u,t^*]\). Hence, \(\Vert e(t)\Vert \le \eta _{e2}\), \(\Vert \zeta (t)\Vert \le \eta _{\zeta 2}\), \(|\delta _u (t)|\le \eta _{\delta _u 2}\) and \(|(b(t))^{-1}|\ge \psi _b^{-1}\) for \(t\in [t_u,t^*]\). Based on (32)–(33), the derivative of \(\sqrt{V_u(t^*)}\) satisfies

$$\begin{aligned} \frac{\hbox {d} \sqrt{V_u(t^*)}}{\hbox {d}t}\le & {} -\psi _b^{-1} \eta _{V_u 2} \kappa \nonumber \\&+ \frac{\sqrt{2}}{2} (\pi _{\delta _u 1}(\eta _{e2},\eta _{\delta _u 2})+\pi _{\omega }\kappa \eta _{\zeta 2}). \end{aligned}$$
(47)

Owing to the definition of \(\eta _{V_u2}\) (44), it can be verified that \(-\frac{\psi _b^{-1} \eta _{V_u 2}}{2}+ \pi _{\omega }(\omega _o)\eta _{\zeta 2} = -\pi _{\omega }(\omega _2)\eta _{\zeta 2} \le 0\) for any \( \omega _o \ge \omega _3\). By choosing \(\kappa _3 = \frac{2\sqrt{2} \pi _{\delta _u 1} (\eta _{e2},\eta _{\delta _u 2}) }{\psi _b^{-1} \eta _{V_u 2}}\), the following inequality holds for \(\kappa \ge \kappa _3\).

$$\begin{aligned} -\frac{\psi _b^{-1} \eta _{V_u 2}}{2} \kappa +\frac{\sqrt{2} \pi _{\delta _u 1} (\eta _{e2},\eta _{\delta _u 2})}{2} <0. \end{aligned}$$
(48)

Hence, \(\frac{d \sqrt{V_u(t^*)}}{dt} < 0 \) for \(\omega _o \ge \omega _3\) and \(\kappa \ge \kappa _3\).

Step 4 We assume that there exists \(t^*\in [t_u,\infty )\) such that \(\sqrt{V_K(t^*)} = \eta _{V_K2}\). Besides, \(\sqrt{V_K(t)} \le \eta _{V_K2}\), \(\sqrt{V_\varsigma (t)}\le \eta _{V_\varsigma 2}\), \( V_z(t)\le \eta _{V_z2}\) and \(\sqrt{V_u(t)} \le \eta _{V_u 2}\) for \(t\in [t_u,t^*]\). Hence, \(\Vert e(t)\Vert \le \eta _{e2}\), \(\Vert \zeta (t)\Vert \le \eta _{\zeta 2}\) and \(|\delta _u(t)|\le \eta _{\delta _u 2}\) for \(t\in [t_u,t^*]\). According to (32), (33) and (44), the derivative of \(\sqrt{V_K(t^*)}\) satisfies

$$\begin{aligned} \frac{\hbox {d} \sqrt{V_K(t^*)}}{\hbox {d}t}\le & {} -\frac{\Vert e(t^*)\Vert }{2 \sqrt{V_K(t^*)}} (\Vert e(t^*)\Vert \nonumber \\&- 2\Vert P_K\Vert \psi _b \eta _{\delta _u 2} ) <0. \end{aligned}$$
(49)

Due to Steps 1–4, it can be concluded that the variables e(t), z(t), \(\zeta (t)\) and \(\delta _u(t)\) stay in \(\Omega _1\) for \(t\in [t_u,\infty )\) if the parameters satisfy \(\omega _o\ge \omega _3\), \(\kappa \ge \kappa _3\) and \(\frac{\omega _o}{\kappa }\ge \tau _3\). Next, we will analyze the bounds of \(\sqrt{V_K(t)}\), \(\sqrt{V_\varsigma (t)}\) and \(\sqrt{V_u(t)}\) for \(t\ge t_u\).

The analysis of the bound of \(\sqrt{V_\varsigma (t)}\). Due to (32)–(33), for \(t\ge t_u\), we have

$$\begin{aligned}&\frac{\hbox {d}\sqrt{V_\varsigma (t)}}{\hbox {d}t} \le -\frac{\omega _o}{2c_{\varsigma 2}} \sqrt{V_\varsigma (t)} \nonumber \\&\quad + \frac{\Vert P_\varsigma \Vert (\pi _{\zeta 1} (\eta _{e2}) +(\kappa +1)\pi _{\delta _u}(\eta _{e2},\eta _{\delta _u 2}) + \pi _{\omega }(\omega _o) \kappa \eta _{\zeta 2} ) }{\sqrt{c_{\varsigma 1}}}.\nonumber \\ \end{aligned}$$
(50)

Combined with Gronwall lemma, \(\sqrt{V_\varsigma (t)}\) has the following bound for \(\omega _o\ge \omega _3\), \(\kappa \ge \kappa _3\) and \(\frac{\omega _o}{\kappa }\ge \tau _3\).

$$\begin{aligned} \sqrt{V_\varsigma (t)}\le & {} \eta _{V_\varsigma 2} e^{-\frac{\omega _o(t-t_u)}{2 c_{\varsigma 2}}} \nonumber \\&+ \theta _{\varsigma 1} \frac{1}{\omega _o} + \theta _{\varsigma 2} \frac{\kappa }{\omega _o},\quad \forall t\in [t_u, \infty ), \end{aligned}$$
(51)

where \(\theta _{\varsigma 1} = (\Vert P_\varsigma \Vert (\pi _{\zeta 1}(\eta _{e2}) +\pi _{\delta _u}(\eta _{e2},\eta _{\delta _u 2}) )/\sqrt{c_{\varsigma 1}}\) and \(\theta _{\varsigma 2} = (\Vert P_\varsigma \Vert ( \pi _{\delta _u}(\eta _{e2},\eta _{\delta _u 2})+ \pi _{\omega }(\omega _3) \eta _{\zeta 2} ))/ \sqrt{c_{\varsigma 1}} \).

The analysis of the bound of \(\sqrt{V_u(t)}\). Owing to (32), (33) and (51), for \(t\ge t_u\), there is

$$\begin{aligned} \begin{aligned} \frac{\hbox {d} \sqrt{V_u(t)}}{\hbox {d}t}&\le -\psi _b^{-1}\kappa \sqrt{V_u(t)} \\&\quad + \frac{\sqrt{2}}{2} (\pi _{\delta _u 1} (\eta _{e2},\eta _{\delta _u 2})+\pi _{\omega }\kappa \Vert \zeta (t)\Vert )\\&\le -\psi _b^{-1}\kappa \sqrt{V_u(t)} \\&\quad +\frac{\sqrt{2}}{2} \left( \pi _{\delta _u 1} (\eta _{e2},\eta _{\delta _u 2})\right. \\&\left. \quad +\kappa \frac{ \pi _{\omega }\left( \eta _{V_\varsigma 2} e^{-\frac{\omega _o}{2 c_{\varsigma 2}} (t-t_u) }+ \theta _{\varsigma 1}\frac{1}{\omega _o} + \theta _{\varsigma 2}\frac{\kappa }{\omega _o} \right) }{\sqrt{c_{\varsigma 1}}} \right) . \end{aligned} \end{aligned}$$

Combined with Gronwall lemma, the bound of \(\sqrt{V_u(t)}\) for \(\omega _o \ge \omega _3\), \(\kappa \ge \kappa _3\) and \(\frac{\omega _o}{\kappa } \ge \tau _4 \triangleq \max \{\tau _3,4 c_{\varsigma 2} \psi _b^{-1} \}\) is shown as follows.

$$\begin{aligned} \begin{aligned} \sqrt{V_u(t)}&\le \eta _{V_u 2} e^{-\psi _b^{-1} \kappa (t-t_u)} + \frac{\theta _{u1} }{\omega _o} + \frac{\theta _{u2}\kappa }{\omega _o} \\&\quad + \frac{\theta _{u3} }{\kappa } + \theta _{u4} \int _{t_u}^{t} e^{-\psi _b^{-1}\kappa (t-s) } e^{-\frac{\omega _o}{2c_{\varsigma 2}} (s-t_u)} \hbox {d}s \\&\le (\eta _{V_u 2} + \frac{\theta _{u4}}{\psi _b^{-1}\kappa _3}) e^{-\psi _b^{-1} \kappa (t-t_u)} + \theta _{u1} \frac{1}{\omega _o} \\&\quad + \theta _{u2} \frac{\kappa }{\omega _o} + \theta _{u3} \frac{1}{\kappa },\quad \forall t\in [t_u, \infty ), \end{aligned} \end{aligned}$$
(52)

where \(\theta _{u1} = \frac{\sqrt{2} \pi _{\omega } (\omega _3) \theta _{\varsigma 1} }{ 2 \psi _b^{-1} \sqrt{c_{\varsigma 1}}}\), \(\theta _{u2} = \frac{\sqrt{2} \pi _{\omega } (\omega _3) \theta _{\varsigma 2} }{2 \psi _b^{-1} \sqrt{c_{\varsigma 1}}}\), \(\theta _{u3} = \frac{\sqrt{2} \pi _{\delta _u 1}(\eta _{e2}, \eta _{\delta _u 2}) }{2 \psi _b^{-1} }\) and \(\theta _{u4} = \frac{ \sqrt{2} \pi _{\omega } (\omega _3) \eta _{V_\varsigma 2} }{ 2 \psi _b^{-1} \sqrt{c_{\varsigma 1}} }\).

The analysis of the bound of \(\sqrt{V_K(t)}\). By denoting \({\tilde{t}}_u = t_u + \frac{\ln \omega _o}{ \psi _b^{-1} \sqrt{\kappa } }\), it can be deduced from (32) and (39) that

$$\begin{aligned} \left\{ \begin{aligned}&\sup _{t_u \le t\le {\tilde{t}}_u} \sqrt{V_K(t)} \le \sqrt{c_{k2}} \left( \Vert e(t_u)\Vert +(\Vert A_K\Vert \eta _{e2}+\psi _b \eta _{\delta _u 2}) \frac{\ln \omega _o}{ \psi _b^{-1} \sqrt{\kappa } } \right) \\&\quad \quad \quad \quad \quad \quad \quad \quad \le \theta _{e} \frac{\ln \omega _o}{\sqrt{\kappa }},\\&e^{-\psi _b^{-1} \kappa ({\tilde{t}}_u-t_u) } = \frac{1}{\omega _o^{\sqrt{\kappa }}} \le \frac{1}{\omega _o}, \end{aligned} \right. \end{aligned}$$
(53)

for \(\omega _o\ge \omega _3\), \(\kappa \ge \kappa _4 \triangleq \max \{ \kappa _3, 1 \}\) and \(\frac{\omega _o}{\kappa }\ge \tau _4\), where \(\theta _{e} = 2\sqrt{c_{k2}}n c_{\varsigma 2} (\Vert A\Vert \eta _{e1} + \pi _{e0}(\eta _{e1})) + \frac{\sqrt{c_{k2}}\Vert A_K\Vert \eta _{e2}+\sqrt{c_{k2}}\psi _b \eta _{\delta _u 2}}{\psi _b^{-1}}\). Then, the bound of \(\sqrt{V_K(t)}\) for \(t\ge {\tilde{t}}_u\) is analyzed. According to (32), (33) and (52)–(53), the dynamics of \(\sqrt{V_K(t)}\) satisfies the following equation for \(t\ge {\tilde{t}}_u\).

$$\begin{aligned} \begin{aligned} \frac{d \sqrt{V_K(t)}}{dt}&\le -\frac{\sqrt{V_K(t)}}{2c_{k2}} + \frac{\Vert P_K\Vert \psi _b |\delta _u(t)|}{\sqrt{c_{k1}}}\\&\le -\frac{\sqrt{V_K(t)}}{2c_{k2}} + \frac{\sqrt{2}\Vert P_K\Vert \psi _b}{\sqrt{c_{k1}}}\\&\quad \left( (\eta _{V_u 2} + \frac{\theta _{u4}}{\psi _b^{-1}\kappa }) e^{-\psi _b^{-1} \kappa (t-t_u)} \right. \\&\quad \left. + \frac{\theta _{u1}}{\omega _o} + \frac{\theta _{u2}\kappa }{\omega _o} + \frac{\theta _{u3}}{\kappa } \right) \\&\le -\frac{\sqrt{V_K(t)}}{2c_{k2}} + \frac{\sqrt{2}\Vert P_K\Vert \psi _b}{\sqrt{c_{k1}}} \left( (\eta _{V_u 2} \right. \\&\quad \left. + \frac{\theta _{u4}}{\psi _b^{-1}\kappa } + \theta _{u1} ) \frac{1}{\omega _o} + \frac{\theta _{u2}\kappa }{\omega _o} + \frac{\theta _{u3}}{\kappa } \right) \end{aligned} \end{aligned}$$

for \(\omega _o\ge \omega _3\), \(\kappa \ge \kappa _4 \) and \(\frac{\omega _o}{\kappa }\ge \tau _4\). With the help of Gronwall lemma, we get the following bound of \(\sqrt{V_K}\) for \(\omega _o\ge \omega _3\), \(\kappa \ge \kappa _4\), and \(\frac{\omega _o}{\kappa } \ge \tau _4\).

$$\begin{aligned} \begin{aligned}&\sup _{t\ge {\tilde{t}}_u} \sqrt{V_K(t)} \le \sqrt{V_K({\tilde{t}}_u)} + \frac{\theta _{e1}}{\omega _o} + \frac{\theta _{e2}\kappa }{\omega _o} + \frac{\theta _{e3}}{\kappa }\\&\quad \le \theta _{e} \frac{\ln \omega _o}{\sqrt{\kappa }} + \frac{\theta _{e1} }{\omega _o} + \frac{\theta _{e2}\kappa }{\omega _o} + \frac{\theta _{e3}}{\kappa }, \end{aligned} \end{aligned}$$
(54)

where \(\theta _{e1} = \frac{4 c_{k2} \Vert P_K\Vert \psi _b (\eta _{V_u 2} + \frac{\theta _{u4}}{\psi _b^{-1}\kappa _3} + \theta _{u1} )}{\sqrt{c_{k1}}} \), \(\theta _{e2}= \frac{4 c_{k2} \Vert P_K\Vert \psi _b \theta _{u2}}{\sqrt{c_{k1}}}\) and \(\theta _{e3} = \frac{4 c_{k2} \Vert P_K\Vert \psi _b \theta _{u3}}{\sqrt{c_{k1}}}\).

According to Assumption 4, there exists a positive constant \(\omega _4\ge \omega _3\) such that

$$\begin{aligned} \kappa (\omega _o)\ge \kappa _4,\quad \frac{\omega _o}{\kappa (\omega _o)} \ge \tau _4, \end{aligned}$$
(55)

for any \( \omega _o\ge \omega _4\). Notice that \(\frac{1}{\omega _o} \le \frac{\kappa }{\omega _o}\) for \(\kappa \ge \kappa _4\ge 1\). With the combination of the bounds of \(\sup \limits _{t_0 \le t\le t_u}\Vert e(t)\Vert \), \(\sqrt{V_K(t)}\), \(\sqrt{V_\varsigma (t)}\) and \(\sqrt{V_u(t)}\), i.e., (39), (51), (52), (53) and (54), the equations (25)–(27) hold for \(\omega _o \ge \omega _4\). \(\square \)

Fig. 2
figure 2

Chua’s circuit

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5 Simulation

In this section, the simulation for an application example, Chua’s circuit, is presented.

Figure 2 describes Chua’s circuit, which is featured with strong nonlinearity and can generate chaotic response [42]. The mathematical model of Chua’s circuit is presented as follows [37, 42].

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{x}}_1 (t) = -\frac{1}{C_1 R} x_1(t) + \frac{1}{C_1 R} x_2(t) -\frac{1}{C_1} f_D(x_1),\\&{\dot{x}}_2 (t) = -\frac{1}{C_2 R} x_1(t) + \frac{1}{C_2 R} x_2(t) + \frac{1}{C_2} x_3(t),\\&{\dot{x}}_3 (t) = \frac{1}{L} u(t) - \frac{1}{L}x_2(t) -\frac{R_0}{L} x_3 (t), \end{aligned} \right. \nonumber \\ \end{aligned}$$
(56)

where \(x_1(t)\) and \(x_2(t)\) represent the voltages across the capacitor \(C_1\) and \(C_2\), \(x_3(t)\) is the current through the inductor L, u(t) is the input voltage, \(f_D(x_1)\) represents the nonlinear current caused by the nonlinear resistor D, and R and \(R_0\) are resistances. The units of voltage, current, capacitance, inductance and resistance are volt (V), ampere (A), farad (F), henry (H) and ohm \((\Omega )\), respectively.

The control objective is to design the input voltage u(t) such that the system states \((x_1(t),x_2(t),x_3(t))\) can track the reference signal (0, 0, 0).

In the simulation, the measurement of \(x_1\) can be obtained, while \(x_2\) and \(x_3\) cannot be measured. Besides, the detailed values of the system parameters, i.e., \(C_1\), \(C_2\), L, R and \(R_0\), are unknown for control design, whereas the signs of system parameters can be directly verified by physical mechanism:

$$\begin{aligned}&C_1>0~(F),\quad C_2>0~(F), \quad L>0~(H),\quad \nonumber \\&R>0~(\Omega ),\quad R_0\ge 0~(\Omega ). \end{aligned}$$
(57)

According to [37, 42], the unknown nonlinear function \(f_D(x_1)\) satisfies that \(f_D(0)=0\).

Remark 11

The presented control objective is a classical stabilization problem [37, 42]. However, the methods proposed in [37, 42] require the measurements for all states, i.e., \(x_1\), \(x_2\) and \(x_3\). In this paper, only the measurement of \(x_1\) is utilized for the stabilization problem. Moreover, the nominal values of system parameters, i.e., \(C_1\), \(C_2\), L, R and \(R_0\), are unknown.

Fig. 3
figure 3

The response curves of the state \(x_1\) for Cases 1–3

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Fig. 4
figure 4

The response curves of the state \(x_2\) for Cases 1–3

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Fig. 5
figure 5

The response curves of the state \(x_3\) for Cases 1–3

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Fig. 6
figure 6

The response curves of the input u for Cases 1–3

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Fig. 7
figure 7

The estimation of the state \(x_2\) via ESO for Cases 1–3

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Fig. 8
figure 8

The estimation of the state \(x_3\) via ESO for Cases 1–3

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Fig. 9
figure 9

The estimation of the total disturbance via ESO for Cases 1–3

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Next, the proposed ADRC is applied to the system (56). Firstly, we denote the following new states.

$$\begin{aligned} \left\{ \begin{aligned}&{\tilde{x}}_1(t) =x_1(t),\\&{\tilde{x}}_2(t) = -\frac{1}{C_1 R} x_1(t) + \frac{1}{C_1 R} x_2(t) -\frac{1}{C_1} f(x_1),\\&{\tilde{x}}_3(t) = \left( -\frac{1}{C_1 R}- \frac{1}{C_1} \frac{df(x_1)}{dx_1}\right) \left( -\frac{x_1(t)}{C_1 R} + \frac{x_2(t) }{C_1 R} -\frac{f(x_1)}{C_1} \right) \\&\qquad \quad \,\, +\frac{1}{C_1 R} \left( -\frac{x_1(t)}{C_2 R} + \frac{ x_2(t)}{C_2 R} + \frac{x_3(t)}{C_2} \right) . \end{aligned} \right. \nonumber \\ \end{aligned}$$
(58)

Then, the integrators chain form is obtained as follows.

$$\begin{aligned} \left\{ \begin{aligned}&\dot{{\tilde{x}}}_1 (t) = {\tilde{x}}_2(t),\\&\dot{{\tilde{x}}}_2 (t) = {\tilde{x}}_3(t),\\&\dot{{\tilde{x}}}_3 (t) = b u(t) + f, \end{aligned} \right. \end{aligned}$$
(59)

where b and f satisfy the following equation.

$$\begin{aligned} \left\{ \begin{aligned} b&= \frac{1}{C_1C_2 R L}, \\ f&= - \frac{1}{C_1} \frac{d^2 f_D(x_1)}{dx_1^2} \left( -\frac{x_1(t)}{C_1 R} + \frac{x_2(t) }{C_1 R} -\frac{f_D(x_1)}{C_1} \right) ^2 -\frac{1}{C_1C_2 R^2} \left( -\frac{x_1(t)}{C_1 R} + \frac{x_2(t) }{C_1 R} -\frac{f_D(x_1)}{C_1} \right) \\&+\frac{1}{C_1C_2 R^2} \left( -\frac{x_1(t)}{C_2 R} + \frac{ x_2(t)}{C_2 R} + \frac{x_3(t)}{C_2} \right) -\frac{1}{C_1C_2 R}\left( \frac{x_2(t)}{L} + \frac{R_0 x_3(t)}{L} \right) \\&\quad +\left( -\frac{1}{C_1 R}- \frac{1}{C_1} \frac{df_D(x_1)}{dx_1}\right) \left( -\frac{1}{C_1 R}- \frac{1}{C_1} \frac{df_D(x_1)}{dx_1}\right) \left( -\frac{x_1(t)}{C_1 R} + \frac{x_2(t) }{C_1 R} -\frac{f_D(x_1)}{C_1} \right) \\&\quad + \frac{1}{C_1 R} \left( -\frac{1}{C_1 R}- \frac{1}{C_1} \frac{df_D(x_1)}{dx_1}\right) \left( -\frac{x_1(t)}{C_2 R} + \frac{ x_2(t)}{C_2 R} + \frac{x_3(t)}{C_2} \right) . \end{aligned} \right. \end{aligned}$$

Due to the transformation (58), it can be verified that \([x_1~x_2~x_3]=[0~0~0]\) is equivalent to \([{\tilde{x}}_1~{\tilde{x}}_2~{\tilde{x}}_3]=[0~0~0]\). Hence, the stabilization problem of the system (56) can be reformulated as the stabilization problem of the system (59).

Although the nominal value of b is unknown due to the unknown system parameters \(C_1\), \(C_2\), L and R, it can be verified by (57) that \(\hbox {sgn}(b)>0\). Based on the sign of b, the proposed ADRC (15) and (19) with the following controller parameters is utilized.

$$\begin{aligned} \omega _o =100,\quad K = [8~12~6]^T,\quad \kappa (\omega _o) = \sqrt{\omega _o}. \end{aligned}$$
(60)

To investigate the capability of disturbance rejection, the following cases of uncertainties are considered, including the cubic function in [37] (Case 1).

$$\begin{aligned} \begin{aligned}&\text {Case 1 } f_D(x_1) = -x_1 + x_1^3,\\&\text {Case 2 } f_D(x_1) = -1.3x_1 + x_1^3,\\&\text {Case 3 } f_D(x_1) = -x_1 + x_1^3 - \sin (x_1). \end{aligned} \end{aligned}$$

We consider the following system parameters and initial condition, which are provided in [37].

$$\begin{aligned} \left\{ \begin{aligned}&C_1=C_2 = 1~(F),~ R_0=0~(\Omega ),~ R=1~(\Omega ),~ L=2~(H),~\\&x_1(0) = 0.01~(V),~ x_2(0) = 0.2~(V),~ x_3(0)=0.5~(A). \end{aligned} \right. \end{aligned}$$
(61)
Table 1 Performance indicators for proposed ADRC and backstepping-based funnel control
Full size table

The simulation results of the proposed ADRC and the following backstepping-based funnel control [37] are presented in Figs. 3–9.

$$\begin{aligned} \left\{ \begin{aligned}&u(t) = 7r_3(t) \cos (\pi r_3(t)) z_3(t),\\&z_1(t) = x_1(t),\\&z_2(t) = x_2(t) - 10r_1(t) \cos (\pi r_1(t))z_1(t),\\&z_3(t) = x_3(t) - 7 r_2(t) \cos (\pi r_2(t))z_2(t),\\&r_i(t) = \frac{1}{1-(e^t-1)^2 z_i^2(t)},\quad i=1,2,3. \end{aligned} \right. \end{aligned}$$
(62)

In addition, the integral of squared tracking error (ISE) and the integral of squared control input (ISCI) are presented in Table 1.

For Case 1, the satisfied closed-loop performance of the proposed ADRC and the backstepping-based funnel control is shown in Figs. 3–5. From Figs. 3–5 and ISE in Table 1, the closed-loop performance of the backstepping-based funnel control becomes poor for Cases 2–3. Especially for Case 3, the backstepping-based funnel control systems becomes unstable. Moreover, Figs. 7–9 depict the estimating performance of the proposed method, where the estimations for unmeasured integrators chain states and total disturbance are close to the real value. The satisfied estimating performance results in the highly consistent tracking performance of the proposed ADRC despite mismatched nonlinear uncertainty. Furthermore, according to the response curves of inputs shown in Fig. 6 and ISCI in Table 1, the control energy consumption of proposed ADRC is much smaller.

6 Conclusion

For a class of lower-triangular nonlinear uncertain systems, the paper proposes a new ADRC based on the control directions rather than the nominal values or the approximative mathematical expressions of control coefficients. The design ideology can be summarized as the following three parts: (1) By transforming the original states into the states of an integrators chain system, the effects from the control input and uncertainties to the controlled output are clearly shown; (2) Based on the integrators chain form, the ESO is presented to estimate the total disturbance and the integrators chain states; (3) Inspired by the approximative dynamic inversion method, a dynamical system is designed to generate the input, which can approach the desired input signal. Moreover, by associating the parameter in dynamical input design with the ESO’s parameter, the tuning method of the parameter in dynamical input design is explicitly provided. With the consideration of a large scope of mismatched nonlinear uncertainties, the transient performance of the proposed ADRC is theoretically investigated. Based on the presented theoretical results, the satisfied tracking and estimating performance can be ensured by suitably enlarging the ESO’s parameter.