Immersion and invariance based adaptive command filtered backstepping control design for a class of SISO uncertain nonlinear systems in strict-feedback form

Abstract

In this article, an immersion and invariance based adaptive command filtered backstepping control method (I&I ACFBC) is proposed for a class of single-input and single-output (SISO) uncertain nonlinear systems in strict-feedback form. The problem of the explosion of complexity inherent to the traditional adaptive backstepping control design is solved by applying the proposed adaptive control technique, and the compensating signals are constructed to reduce the effect of the known filtering errors produced by the command filters. The adaptive laws are achieved by an immersion and invariance method. The Lyapunov stability theory is utilized to ensure the boundedness of all the signals in the closed-loop system. The feasibility and validity of the proposed adaptive control scheme are demonstrated by the simulation and experimental results on a DC-DC buck converter system.

Introduction

Over the past few years, the tracking control issue of nonlinear systems has attracted considerable attention. The adaptive control schemes for a class of uncertain nonlinear systems are one of the challenging fields of nonlinear control engineering problems (Kanellakopoulos et al., 1991; Soukkou et al., 2023a, 2023b; Tee et al., 2009). Most adaptive control techniques are based on the parameter estimation. It is challenging to develop efficient control strategies that can deal with the unknown parameters of nonlinear systems and achieve a better tracking performance. The adaptive backstepping control scheme has been widely investigated as an effective technique for a class of single-input and single-output (SISO) uncertain nonlinear systems in strict-feedback form (Ciliz, 2007; Kanellakopoulos et al., 1991; Krstić et al., 1995; Soukkou et al., 2023a; Soukkou and Labiod, 2015; Wang et al., 2016; Zhang et al., 2015; Zhou and Wen, 2008), and the stability of the closed-loop system has been proved by using the Lyapunov stability theory. The technique has been attracting considerable research attention recently. However, an obvious shortcoming existing in the adaptive backstepping control technique is the explosion of complexity problem induced by the repetitive differentiation of virtual control inputs. Recently, to overcome this problem, an adaptive dynamic surface control approach has been developed for strict-feedback nonlinear systems by introducing a first order filters (Soukkou et al., 2018, 2021, 2022; Soukkou and Labiod, 2017; Wang and Huang, 2005; Yip and Hedrick, 1998; Yu et al., 2015a). On the other hand, the adaptive command filtered backstepping control (ACFBC) approach has been introduced to solve the problem of the explosion of complexity and compensating signals have been designed to overcome the effect of the known filtering errors produced by command filters (Dong et al., 2010, 2012; Soukkou et al., 2019; Yu et al., 2015b, Yu et al., 2018a). Besides, adaptive finite time command filtered backstepping control scheme has been also proposed in (Li, 2019; Liu et al., 2021; Yang et al., 2019; Yu et al., 2018b, 2021).

The I&I based adaptive control method for nonlinear systems has been proposed as a novel design for the adaptive control and stabilization of nonlinear systems (Astolfi et al., 2003), which has received much attention (Astolfi et al., 2008; Han et al., 2018; Karagiannis and Astolfi, 2004, 2007, 2008; Zhang et al., 2013; Zhao et al., 2015). The design procedure consisted of two steps, the first step deals with the design of an estimator, and the second consists in designing a control law. Various nonlinear adaptive control strategies via I&I have been used for uncertain nonlinear systems in strict-feedback form such as adaptive backstepping control (Karagiannis and Astolfi, 2008; Monfared et al., 2013; Navabi and Soleymanpour, 2017), ACFBC (Hu and Zhang, 2013; Sonneveldt et al., 2010; Yu et al., 2021) and adaptive dynamic surface control (Fujimto and Yokoyama, 2012; Soukkou et al., 2021).

Motivated by the above discussions, an I&I ACFBC method for a class of uncertain nonlinear systems in strict-feedback form is developed in this paper. The proposed adaptive control scheme is employed to solve the problem of the explosion of complexity that exists in the conventional adaptive backstepping control design, and the compensation signals are designed to reduce the effect of known filtering errors caused by the command filters. Stability of the closed-loop system is proven under the Lyapunov stability analysis theorem. The effectiveness of the proposed adaptive control scheme is illustrated by simulation and experimental results for a DC-DC buck converter system. Compared with the most existing results, the main contributions of this paper are concluded as follows

1. By applying the proposed adaptive control scheme, we can deal with the problem of the explosion of complexity due to repeated derivatives of the virtual control signals existing in the adaptive backstepping control method, and the compensating signals are designed to reduce the effect of the errors attributed to the command filters. Furthermore, based on the proposed adaptive control technique, the boundedness of all the signals in the closed-loop system is achieved in contrast to the works in (Soukkou et al., 2021; Yu et al., 2021) that all the signals in the closed-loop system are proven to be uniformly ultimately bounded (UUB).

2. Compared with the results in (Dong et al., 2010, 2012; Soukkou et al., 2019; Yu et al., 2015b, 2018a), this paper uses an I&I scheme for adaptive parameter estimation.

3. Compared with the I&I based adaptive backstepping control (I&I ABC) method in (Karagiannis and Astolfi, 2008; Monfared et al., 2013; Navabi and Soleymanpour, 2017), the I&I ACFBC method in (Hu and Zhang, 2013; Sonneveldt et al., 2010) and the I&I based adaptive dynamic surface control method in (Fujimto and Yokoyama, 2012; Soukkou et al., 2021)), this paper further considers that the issue of the explosion of complexity caused by the repeated differentiation of the virtual control signals is avoided by introducing a command filter technique with a compensation mechanism.

The remainder of this paper is arranged as follows. The Problem formulation and preliminaries are given in “Problem formulation and preliminaries” section. In “I&I ACFBC design” section, an I&I ACFBC scheme and stability analysis are designed for uncertain nonlinear systems in strict-feedback form. In “Simulation and experimental results on the DC-DC buck converter system” section, simulation and experimental results on a DC-DC buck converter system are provided to show the validity of the proposed adaptive control method. The conclusions are presented in “Conclusion” section.

Problem formulation and preliminaries

Consider the SISO uncertain nonlinear system in strict-feedback form, given by

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=g_1\left(x_1\right)x_2+{\phi }_1^T\left(x_1\right)\theta +{\psi }_1\left(x_1\right)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_i=g_i\left({\overline{x}}_i\right)x_{i+1}+{\phi }_i^T\left({\overline{x}}_i\right)\theta +{\psi }_i\left({\overline{x}}_i\right),i=2,\cdots ,n-1}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_n=g_n\left(x\right)u+{\phi }_n^T\left(x\right)\theta +{\psi }_n\left(x\right)}\\ \multicolumn{1}{c}{y=x_1}\end{array}$$

(1)

where $x=[\begin{array}{cccc}{x}_1\hfill & x_2\hfill & \dots \hfill & x_n\hfill \end{array}{]}^T\in {\mathbb{R}}^n$ is the system states, $u\in \mathbb{R}$ and $y\in \mathbb{R}$ denote the control input and the system output, respectively. ${\overline{x}}_i=[\begin{array}{cccc}{x}_1\hfill & x_2\hfill & \dots \hfill & x_i\hfill \end{array}{]}^T\in {\mathbb{R}}^i$ and $\theta \in {\mathbb{R}}^p$ is the unknown constant parameter vector. The nonlinear functions ${\phi }_i:{\mathbb{R}}^i\to {\mathbb{R}}^p$ , ${\psi }_i:{\mathbb{R}}^i\to \mathbb{R}$ and $g_i:{\mathbb{R}}^i\to \mathbb{R}$ with $g_i\ne 0$ are known and continuous. The control objective in this paper is to construct an adaptive controller $u$ such that the system output $x_1$ tracks the reference signal $x_{1d}$ , while guaranteeing the boundedness of all the signals in the closed-loop system.

Throughout this article, to achieve the control objective, the assumptions of the system (equation (1)) are required.

Assumption 1. There exists a known positive constants $\underset{¯}{g}$ and $\overline{g}$ such that $\underset{¯}{g}\le \left|g_i\right|\le \overline{g}$ , $i=1,\dots ,n$ . Without loss of generality, it is further assumed that $0<\underset{¯}{g}\le g_i$ .

Assumption 2. The reference signal $x_{1d}$ and its first derivative ${\stackrel{·}{x}}_{1d}$ are known, smooth, and bounded.

Remark 1. The above assumptions are necessary and reasonable. For the conventional adaptive backstepping control design, the information of $x_{1d}^{\left(i\right)}$ $(i=0,\dots ,n)$ is required, and the adaptive dynamic surface control design requires $x_{1d}$ and both its first and second derivatives ${\stackrel{·}{x}}_{1d}$ and ${\stackrel{··}{x}}_{1d}$ , whereas our proposed adaptive control approach only requires $x_{1d}$ and its first derivative ${\stackrel{·}{x}}_{1d}$ , which is less stringent. Therefore, it is more appropriate for some practical applications.

I&I ACFBC design

The detailed design procedures of the proposed adaptive controller with stability analysis for the nonlinear system (equation (1)) are presented as follows.

Estimator design

Defining the estimation errors as

$${\tilde{\theta }}_i={\hat{\theta }}_i-\theta +{\beta }_i\left({\overline{x}}_i\right)$$

(2)

where ${\hat{\theta }}_i$ are the estimator states and ${\beta }_i:{\mathbb{R}}^i\to {\mathbb{R}}^p$ are functions to defined later. Then, the dynamics of ${\tilde{\theta }}_i$ are given by (Karagiannis and Astolfi, 2008)

$$\begin{array}{c}{\stackrel{·}{\stackrel{~}{\theta }}}_i={\stackrel{·}{\hat{\theta }}}_i+\sum _{k=1}^i\frac{\partial {\beta }_i}{\partial x_k}{\stackrel{·}{x}}_k\hfill \\ ={\stackrel{·}{\hat{\theta }}}_i+\sum _{k=1}^i\frac{\partial {\beta }_i}{\partial x_k}(g_k{x}_{k+1}+{\phi }_k^T({\hat{\theta }}_i+{\beta }_i-{\tilde{\theta }}_i)+{\psi }_k)\hfill \end{array}$$

(3)

where $x_{n+1}=u$ . The adaptive laws for ${\hat{\theta }}_i$ can be selected as

$${\stackrel{·}{\hat{\theta }}}_i=-\sum _{k=1}^i\frac{\partial {\beta }_i}{\partial x_k}(g_k{x}_{k+1}+{\phi }_k^T({\hat{\theta }}_i+{\beta }_i)+{\psi }_k)$$

(4)

To cancel all the known parts of the ${\tilde{\theta }}_i$ dynamics, resulting in

$${\stackrel{·}{\stackrel{~}{\theta }}}_i=-\left[\sum _{k=1}^i\frac{\partial {\beta }_i}{\partial x_k}{\phi }_k^T\right]{\tilde{\theta }}_i$$

(5)

The functions ${\beta }_i$ are defined as (Karagiannis and Astolfi, 2008)

$${\beta }_i={\mathrm{\Gamma }}_i\underset{0}{\overset{x_i}{\int }}{\phi }_i({\overline{x}}_{i-1},\chi )d\chi +{\epsilon }_i\left(x_i\right)$$

(6)

where ${\mathrm{\Gamma }}_i>0$ are constants and ${\epsilon }_i$ are continuously differentiable functions with ${\epsilon }_1=0$ .

Assumption 3. There exist functions ${\epsilon }_i$ that satisfy the partial differential matrix inequality (Karagiannis and Astolfi, 2008)

$$\begin{array}{cc}{F}_i+F_i^T\ge 0,\hfill & i=2,\dots ,n\hfill \end{array}$$

(7)

where

$$F_i={\mathrm{\Gamma }}_i\sum _{k=1}^{i-1}\frac{\partial }{\partial x_k}\left(\underset{0}{\overset{x_i}{\int }}{\phi }_i({\overline{x}}_{i-1},\chi )d\chi \right){\phi }_k^T+\frac{\partial {\epsilon }_i}{\partial x_i}{\phi }_i^T$$

(8)

Remark 2. In the case that ${\phi }_i(·)$ only depends on $x_i$ , the trivial solution ${\epsilon }_i=0$ for, $i=2,\dots ,n$ satisfies the inequality (equation (7)) (Karagiannis and Astolfi, 2008).

The following lemma of the proposed estimator can be established.

Lemma 1. Consider the system (equation (5)), where the functions ${\beta }_i$ are given by equation (6) and functions ${\epsilon }_i$ exist which satisfy equation (7). From assumption 3, the system (equation (1)) is globally uniformly stable equilibrium at the origin, the states ${\tilde{\theta }}_i$ are bounded (i.e. ${\tilde{\theta }}_i\in L_{\infty }$ ) and the signals ${\phi }_i^T{\tilde{\theta }}_i$ are square integrable (i.e. ${\phi }_i^T{\tilde{\theta }}_i\in L_2$ ). If, in addition, ${\phi }_i$ and its time derivative ${\stackrel{·}{\phi }}_i$ are bounded, then ${\phi }_i^T{\tilde{\theta }}_i$ converges to zero.

Proof. Define the Lyapunov function $W$ as

$$W=\sum _{i=1}^n{\tilde{\theta }}_i^T{\mathrm{\Gamma }}_i^{-1}{\tilde{\theta }}_i$$

(9)

The time derivative of $W$ becomes

$$\begin{array}{c}\stackrel{·}{W}=\sum _{i=1}^n({\stackrel{·}{\stackrel{~}{\theta }}}_i^T{\mathrm{\Gamma }}_i^{-1}{\tilde{\theta }}_i+{\tilde{\theta }}_i^T{\mathrm{\Gamma }}_i^{-1}{\stackrel{·}{\stackrel{~}{\theta }}}_i)\hfill \\ =-\sum _{i=1}^n{\tilde{\theta }}_i^T{\mathrm{\Gamma }}_i^{-1}(2{\mathrm{\Gamma }}_i{\phi }_i{\phi }_i^T+F_i+F_i^T){\tilde{\theta }}_i\hfill \\ \le -2\sum _{i=1}^n{\left({\phi }_i^T{\tilde{\theta }}_i\right)}^2\hfill \end{array}$$

(10)

From equation (10), we establish that $W$ is non-increasing. As a result, $W\in L_{\infty }$ and ${\tilde{\theta }}_i\in L_{\infty }$ , and ${\phi }_i^T{\tilde{\theta }}_i\in L_2$ . Integrating it over $[0,\infty ]$ , it leads to

$$\underset{0}{\overset{\infty }{\int }}\sum _{i=1}^n{\left({\phi }_i^T{\tilde{\theta }}_i\right)}^{2}d\tau \le \frac{1}{2}(W\left(0\right)-W\left(\infty \right))<\infty$$

(11)

Furthermore, if ${\phi }_i$ and ${\stackrel{·}{\phi }}_i$ are bounded, then by Barbalat’s lemma (Slotine and Li, 1991), we get $\underset{t\to \infty }{lim}{\phi }_i^T{\tilde{\theta }}_i=0$ . From equation (2), an asymptotically converging estimate of each unknown term ${\phi }_i^T\theta$ in equation (1) is given by ${\phi }_i^T({\hat{\theta }}_i+{\beta }_i)$ .

Controller design

In this section, an I&I based adaptive command filtered backstepping controller is designed for uncertain strict-feedback nonlinear system described by equation (1) to guarantee the boundedness of all the signals in the closed-loop system. The controller design presented here includes command filters. Define the tracking errors as follows

$$\{\begin{array}{c}{e}_1=x_1-x_{1d}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}{e}_i=x_i-x_{\mathrm{ic}},\hfill & i=2,\dots ,n\hfill \end{array}}\end{array}$$

(12)

where $x_{\mathrm{ic}}$ is the output signal of the command filter whose input signal is ${\alpha }_{i-1}$ . For $i=2,\dots ,n$ , the command filter is designed as (Dong et al., 2010, 2012)

$${\stackrel{·}{x}}_{\mathrm{ic}}=-{\omega }_i(x_{\mathrm{ic}}-{\alpha }_{i-1}),x_{\mathrm{ic}}\left(0\right)={\alpha }_{i-1}\left(0\right)$$

(13)

where ${\omega }_i>0$ is a positive design constant.

Remark 3. To eliminate the influence of the known filtering errors $(x_{(i+1)c}-{\alpha }_i)$ produced by the command filters, the compensating signals ${\xi }_i$ are introduced

$$\{\begin{array}{c}{\stackrel{·}{\xi }}_1=-k_1{\xi }_1+g_1{\xi }_2+g_1(x_{2c}-{\alpha }_1)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\xi }}_i=-k_i{\xi }_i+g_i{\xi }_{i+1}-g_{i-1}{\xi }_{i-1}+g_i(x_{(i+1)c}-{\alpha }_i)}\\ \multicolumn{1}{c}{{\stackrel{·}{\xi }}_n=-k_n{\xi }_n-g_{n-1}{\xi }_{n-1}}\end{array}$$

(14)

with ${\xi }_i(0)=0$ . The compensated tracking errors ${\upsilon }_i$ are defined as

$$\begin{array}{cc}{\upsilon }_i=e_i-{\xi }_i,\hfill & i=1,\dots ,n\hfill \end{array}$$

(15)

Construct the virtual control signals ${\alpha }_i$ and actual control input $u$ as follows

$${\alpha }_1=\frac{1}{g_1}(-k_1{e}_1-{\phi }_1^T({\hat{\theta }}_1+{\beta }_1)-{\psi }_1+{\stackrel{·}{x}}_{1d})$$

(16)

$${\alpha }_i=\frac{1}{g_i}(-k_i{e}_i-{\phi }_i^T({\hat{\theta }}_i+{\beta }_i)-{\psi }_i+{\stackrel{·}{x}}_{\mathrm{ic}}-g_{i-1}{e}_{i-1})$$

(17)

$$u=\frac{1}{g_n}(-k_n{e}_n-{\phi }_n^T({\hat{\theta }}_n+{\beta }_n)-{\psi }_n+{\stackrel{·}{x}}_{\mathrm{nc}}-g_{n-1}{e}_{n-1})$$

(18)

where $k_i>0$ are positive design parameters.

Now, the detailed design procedures of the adaptive controller are given as follows.

• ▪ Step $1(i=1)$ : The derivative of $e_1$ is obtained as

$${\stackrel{·}{e}}_1=g_1{e}_2+g_1{x}_{2c}+{\phi }_1^T\theta +{\psi }_1-{\stackrel{·}{x}}_{1d}$$

(19)

Choose the Lyapunov function $V_1$ as

$$V_1=\frac{1}{2}{\upsilon }_1^2$$

(20)

The time derivative of $V_1$ is

$$\begin{array}{c}{\stackrel{·}{V}}_1={\upsilon }_1{\stackrel{·}{\upsilon }}_1={\upsilon }_1({\stackrel{·}{e}}_1-{\stackrel{·}{\xi }}_1)\hfill \\ ={\upsilon }_1(g_1{\upsilon }_2+g_1{\alpha }_1+{\phi }_1^T\theta +{\psi }_1-{\stackrel{·}{x}}_{1d}+k_1{\xi }_1)\hfill \end{array}$$

(21)

Substituting equation (16) into equation (21), we have

$${\stackrel{·}{V}}_1=-k_1{\upsilon }_1^2+g_1{\upsilon }_1{\upsilon }_2-{\upsilon }_1{\phi }_1^T{\tilde{\theta }}_1$$

(22)

• ▪ Step $i(i=2,\dots ,n-1)$ : The time derivative of $e_i$ is obtained as

$${\stackrel{·}{e}}_i=g_i{e}_{i+1}+g_i{x}_{(i+1)c}+{\phi }_i^T\theta +{\psi }_i-{\stackrel{·}{x}}_{\mathrm{ic}}$$

(23)

Consider the Lyapunov function $V_i$ as

$$V_i=V_{i-1}+\frac{1}{2}{\upsilon }_i^2$$

(24)

The time derivative of $V_i$ is obtained as

$${\stackrel{·}{V}}_i={\stackrel{·}{V}}_{i-1}+{\upsilon }_i(g_i{\upsilon }_{i+1}+g_i{\alpha }_i+{\phi }_i^T\theta +{\psi }_i-{\stackrel{·}{x}}_{\mathrm{ic}}+k_i{\xi }_i+g_{i-1}{\xi }_{i-1})$$

(25)

Substituting equation (17) into equation (25) yields

$${\stackrel{·}{V}}_i=-\sum _{j=1}^i{k}_j{\upsilon }_j^2+g_i{\upsilon }_i{\upsilon }_{i+1}-\sum _{j=1}^i{\upsilon }_j{\phi }_j^T{\tilde{\theta }}_j$$

(26)

• ▪ Step $n$ : The time derivative of $e_n$ is obtained as

$${\stackrel{·}{e}}_n=g_{n}u+{\phi }_n^T\theta +{\psi }_n-{\stackrel{·}{x}}_{\mathrm{nc}}$$

(27)

Choose the Lyapunov function $V=V_n$ as

$$V=\frac{1}{2}\sum _{i=1}^n{\upsilon }_i^2$$

(28)

The derivative of $V$ is expressed as

$$\stackrel{·}{V}={\stackrel{·}{V}}_{n-1}+{\upsilon }_n(g_{n}u+{\phi }_n^T\theta +{\psi }_n-{\stackrel{·}{x}}_{\mathrm{nc}}+k_n{\xi }_n+g_{n-1}{\xi }_{n-1})$$

(29)

Substituting equation (18) into equation (29), we can obtain

$$\stackrel{·}{V}=-\sum _{i=1}^n{k}_i{\upsilon }_i^2-\sum _{i=1}^n{\upsilon }_i{\phi }_i^T{\tilde{\theta }}_i$$

(30)

Based on Young’s inequality, we have

$$-{\upsilon }_i{\phi }_i^T{\tilde{\theta }}_i\le \frac{1}{4}{\upsilon }_i^2+{\left({\phi }_i^T{\tilde{\theta }}_i\right)}^2$$

(31)

The time derivative of $V$ becomes

$$\stackrel{·}{V}\le -\sum _{i=1}^n(k_i-\frac{1}{4}){\upsilon }_i^2+\sum _{i=1}^n{\left({\phi }_i^T{\tilde{\theta }}_i\right)}^2$$

(32)

For the stability analysis, the following theorem is established.

Theorem 1. Consider the uncertain nonlinear system in strict-feedback form (equation (1)) under assumptions 1 and 2. By constructing the virtual controllers (equations (16) and (17)), the actual controller (equation (18)), and the adaptive laws (equation (4)), the proposed adaptive control method can guarantee that all the signals in the closed-loop system are bounded.

Proof. Construct the Lyapunov function $V_c$ as

$$V_c=V+W$$

(33)

where

$$V=\frac{1}{2}\sum _{i=1}^n{\upsilon }_i^2,W=\sum _{i=1}^n{\tilde{\theta }}_i^T{\mathrm{\Gamma }}_i^{-1}{\tilde{\theta }}_i$$

(34)

According to equations (10) and (32), the derivative of $V_c$ is given by

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_c=\stackrel{·}{V}+\stackrel{·}{W}}\\ \multicolumn{1}{c}{\le -\sum _{i=1}^n(k_i-\frac{1}{4}){\upsilon }_i^2+\sum _{i=1}^n{\left({\phi }_i^T{\tilde{\theta }}_i\right)}^2-2\sum _{i=1}^n{\left({\phi }_i^T{\tilde{\theta }}_i\right)}^2}\\ \multicolumn{1}{c}{\le -\sum _{i=1}^n(k_i-\frac{1}{4}){\upsilon }_i^2-\sum _{i=1}^n{\left({\phi }_i^T{\tilde{\theta }}_i\right)}^2}\end{array}$$

(35)

Based on equation (35), by choosing $k_i>1/4$ , we can conclude that $V_c$ , ${\upsilon }_i$ and ${\phi }_i^T{\tilde{\theta }}_i$ are bounded. Because $e_i={\upsilon }_i+{\xi }_i$ and ${\xi }_i$ are bounded, the signals $e_i$ are bounded too. Furthermore, all the signals in the closed-loop system, that is., $x_{1d}$ , ${\stackrel{·}{x}}_{1d}$ , $x_{\mathrm{ic}}$ , ${\stackrel{·}{x}}_{\mathrm{ic}}$ $(i=2,\dots ,n)$ , ${\alpha }_i$ , $u$ , and ${\hat{\theta }}_i+{\beta }_i$ $(i=1,\dots ,n)$ are also bounded. This completes the proof of theorem 1.

The following Lyapunov function is established to guarantee the boundedness of ${\xi }_i$ , given by

$$V_{\xi }=\frac{1}{2}\sum _{i=1}^n{\xi }_i^2$$

(36)

Taking the time derivative of $V_{\xi }$ yields

$${\stackrel{·}{V}}_{\xi }=\sum _{i=1}^n{\xi }_i{\stackrel{·}{\xi }}_i=-\sum _{i=1}^n{k}_i{\xi }_i^2+\sum _{i=1}^{n-1}{g}_i{\xi }_i(x_{(i+1)c}-{\alpha }_i)$$

(37)

By using lemma 2 in (Dong et al., 2012; Li, 2019; Yu et al., 2018c), we have $|x_{(i+1)c}-{\alpha }_i|\le \overline{\mu }$ , then, we can obtain the results as follows

$$\begin{array}{c}{\stackrel{·}{V}}_{\xi }=-\sum _{i=1}^n{k}_i{\xi }_i^2+\sum _{i=1}^{n-1}\left|g_i\right|\left|{\xi }_i\right||x_{(i+1)c}-{\alpha }_i|\hfill \\ \le -\sum _{i=1}^n{k}_i{\xi }_i^2+\overline{\mu }\overline{g}\sum _{i=1}^{n-1}\left|{\xi }_i\right|\hfill \\ \le -k_0{V}_{\xi }+\sqrt{2n}\overline{\mu }\overline{g}{V}_{\xi }^{\frac{1}{2}}\hfill \end{array}$$

(38)

where $k_0=\underset{1\le i\le n}{min}\{2k_i\}$ , $\overline{g}=\underset{1\le i\le n-1}{max}\{g_i\}$ and $\overline{\mu }>0$ . From equations (36) and (38), we can get

$$\underset{t\to \infty }{lim}\left|{\xi }_i\right|\le \frac{\overline{\mu }\overline{g}}{2k_0}$$

(39)

Remark 4. Distinctly from the adaptive laws presented by an I&I method in (Karagiannis and Astolfi, 2008), several techniques of robustifying the adaptive laws have been suggested in literature over the years such as σ-modification and projection (Farrell and Polycarpou, 2006; Ioannou and Fidan, 2007; Ioannou and Sun, 1996; Polycarpou and Ioannou, 1996; Soukkou et al., 2021; Yu et al., 2021), an overview is given as follows

• ▪ Projection-based adaptive laws

Projection properties: We assume that $\theta$ is estimated by ${\overline{\mathrm{\Theta }}}_i={\hat{\theta }}_i+{\beta }_i$ , where ${\overline{\mathrm{\Theta }}}_i$ are the estimates of $\theta$ . Terms ${\hat{\theta }}_i$ and ${\beta }_i$ are two parts of the estimates need to be judiciously designed.

The estimation errors are defined as

$${\tilde{\theta }}_i={\overline{\mathrm{\Theta }}}_i-\theta ={\hat{\theta }}_i+{\beta }_i-\theta$$

(40)

The projection operator is given by

$$\mathrm{Pro}{\mathrm{j}}_{{\overline{\mathrm{\Theta }}}_i}\left({•}_i\right)=\{\begin{array}{c}\begin{array}{c}{•}_i\mathrm{if}{\overline{\mathrm{\Theta }}}_i^T{\overline{\mathrm{\Theta }}}_i<{\overline{M}}_i^2\mathrm{or}\mathrm{if}({\overline{\mathrm{\Theta }}}_i^T{\overline{\mathrm{\Theta }}}_i={\overline{M}}_i^2\mathrm{and}{•}_i{\overline{\mathrm{\Theta }}}_i\le 0),\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & \mathrm{otherwise}.\hfill \end{array}}\end{array}$$

(41)

where “ ${•}_i$ ”represents any reasonable adaptation function. The projection algorithm ensures that the estimated parameters ${\overline{\mathrm{\Theta }}}_i$ of $\theta$ remain bounded and satisfy that $||{\overline{\mathrm{\Theta }}}_i||\le {\overline{M}}_i$ . The projection mapping used in equation (41) ensures that

$$-{\tilde{\theta }}_i^T\mathrm{Pro}{\mathrm{j}}_{{\overline{\mathrm{\Theta }}}_i}\left({•}_i\right)\le -{\tilde{\theta }}_i^T{•}_i$$

(42)

Applying the projection algorithm, the standard adaptive laws (equation (4)) become

$${\stackrel{·}{\hat{\theta }}}_i=\mathrm{pro}{\mathrm{j}}_{{\hat{\theta }}_i}(-\sum _{k=1}^i\frac{\partial {\beta }_i}{\partial x_k}(g_k{x}_{k+1}+{\phi }_k^T({\hat{\theta }}_i+{\beta }_i)+{\psi }_k))$$

(43)

• ▪ σ-modification-based adaptive laws

The adaptive laws are achieved by fusing an I&I method and σ-modification technique to robustify the adaptive parameters (Soukkou et al., 2021; Yu et al., 2021). Modifying the adaptive laws (equation (4)) using the σ-modification technique results in

$${\stackrel{·}{\hat{\theta }}}_i=-\sum _{k=1}^i\frac{\partial {\beta }_i}{\partial x_k}(g_k{x}_{k+1}+{\phi }_k^T({\hat{\theta }}_i+{\beta }_i)+{\psi }_k)-{\mathrm{\Gamma }}_i{\sigma }_{{\theta }_i}({\hat{\theta }}_i+{\beta }_i)$$

(44)

where ${\sigma }_{{\theta }_i}>0$ are small constants.

Remark 5. The command filters are also designed as follows (Farrell et al., 2008, 2009)

$$\{\begin{array}{c}{\stackrel{·}{\phi }}_{i,1}={\omega }_n{\phi }_{i,2}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\phi }}_{i,2}=-2\zeta {\omega }_n{\phi }_{i,2}-{\omega }_n({\phi }_{i,1}-{\alpha }_i)}\end{array}$$

(45)

with $i=1,\dots ,n-1$ , $x_{(i+1)c}={\phi }_{i,1}$ , ${\phi }_{i,1}(0)={\alpha }_i(0)$ and ${\phi }_{i,2}(0)=0$ . There exist ${\omega }_n>0$ and $\zeta \in (0,1]$ such that $|{\phi }_{i,1}-{\alpha }_i|\le \mu$ for any $\mu >0$ .

Remark 6. Compared with the existing results (Dong et al., 2010, 2012; Soukkou et al., 2019; Yu et al., 2015b, 2018a), a developed ACFBC method is proposed by fusing an I&I approach and ACFBC technique to improve parameter estimation.

Remark 7. Due to the introduction of the command filter, the repeated differentiations of the virtual control signals in the traditional adaptive backstepping control method are obviated. Accordingly, the computational burden is effectively reduced. Moreover, the compensating signals are introduced to overcome the effect of the errors caused by command filter.

Remark 8. The parameter selection guideline in the proposed adaptive control design is presented as follows

• ▪ Step 1: Choose suitable parameter so that $k_1>0$ , and determine the virtual control signal ${\alpha }_1$ in equation (16).

• ▪ Step 2: Choose suitable parameters so that $k_i>0$ , and determine the virtual control signals ${\alpha }_i$ in equation (17), $i=2,\dots ,n-1$ .

• ▪ Step 3: Choose suitable parameters so that $k_n>0$ , and determine the actual control input $u$ in equation (18), the adaptive laws ${\stackrel{·}{\hat{\theta }}}_i$ in equation (4).

Simulation and experimental results on the DC-DC buck converter system

In this section, the availability of the designed adaptive control method is demonstrated by a typical pulse width modulation (PWM) based DC-DC buck converter system. The DC-DC buck converter system comprises a DC input voltage source $(V_{\mathrm{in}})$ , a PWM gate drive controlled switch $(V_T)$ , a diode $(V_D)$ , a filter inductor $(L)$ , a filter capacitor $(C)$ , and a load resistor $(R)$ . The circuit diagram on the DC-DC buck converter system is shown in Figure 1.

Figure 1.

Schematic diagram on the DC-DC buck converter system.

To show the effectiveness of the proposed adaptive control method, simulation and experimental results on a DC-DC buck converter system are validated.

Simulation results

Considering the following state variables $x_1=v_o$ and $x_2=i_L$ , the dynamic model on the overall DC-DC buck converter system is constructed by the following form as (Guo et al., 2019; Jia et al., 2019; Nizami and Mahanta, 2016; Wang et al., 2018; Wen et al., 2021)

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=\frac{1}{C}{x}_2-\frac{1}{\mathrm{CR}}{x}_1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=\frac{1}{L}{V}_{\mathrm{in}}u-\frac{1}{L}{x}_1}\end{array}$$

(46)

where $v_o$ and $i_L$ are the output voltage and the inductance current, respectively. The duty ratio function $u\in [0,1]$ represents the control signal of PWM. The unknown parameters ${\theta }_1$ and ${\theta }_2$ are assumed as ${\theta }_1=1/\mathrm{CR}$ and ${\theta }_2=1/\mathrm{L}$ , where $\theta =[\begin{array}{cc}{\theta }_1\hfill & {\theta }_2\hfill \end{array}{]}^T$ . The control objective is to follow the reference output voltage $V_d=x_{1d}$ . The nominal values of the real parameters on the DC-DC buck converter system are listed in Table 1.

Table 1.

Simulation and experimental on the DC-DC buck converter system real parameters.

Parameters	Values
Input voltage	Vin = 100 V
Inductance	L = 10 mH
Capacitance	C = 1000 µF
Load resistance	R = 82 Ω
Switching frequency	fs = 4.05 kHz

The signal $x_{2c}$ and the actual controller $u$ are selected as follows

$${\stackrel{·}{x}}_{2c}=-{\omega }_2(x_{2c}-C(-k_1{e}_1+x_1({\hat{\theta }}_1+{\beta }_1)+{\stackrel{·}{x}}_{1d}))$$

(47)

$$u=\frac{L}{V_{\mathrm{in}}}(-k_2{e}_2-\frac{1}{C}{e}_1+x_1({\hat{\theta }}_2+{\beta }_2)+{\stackrel{·}{x}}_{2c})$$

(48)

The adaptive laws for ${\hat{\theta }}_1$ and ${\hat{\theta }}_2$ are defined as

$${\stackrel{·}{\hat{\theta }}}_1={\mathrm{\Gamma }}_1{x}_1(\frac{1}{C}{x}_2-x_1({\hat{\theta }}_1+{\beta }_1))$$

(49)

$${\stackrel{·}{\hat{\theta }}}_2={\mathrm{\Gamma }}_2{x}_2(\frac{1}{C}{x}_2-x_1({\hat{\theta }}_2+{\beta }_2))+{\mathrm{\Gamma }}_2{x}_1(\frac{1}{L}{V}_{\mathrm{in}}u-x_1({\hat{\theta }}_2+{\beta }_2))$$

(50)

where ${\beta }_1=-0.5{\mathrm{\Gamma }}_1{x}_1^2$ and ${\beta }_2=-{\mathrm{\Gamma }}_2{x}_1{x}_2$ . The initial conditions are chosen as $x\left(0\right)=[\begin{array}{cc}0\hfill & 0\hfill \end{array}{]}^T$ , ${\hat{\theta }}_1(0)={\hat{\theta }}_2(0)=0$ and $x_{2c}(0)=0$ . The control parameters are chosen as $k_1=250$ , $k_2=9000$ , ${\mathrm{\Gamma }}_1=3\times {10}^{-3}$ , ${\mathrm{\Gamma }}_2=5\times {10}^{-3}$ , and ${\omega }_2=5$ .

The DC-DC buck converter system is studied for the following simulation cases:

• ▪ Case 1: The reference output voltage $V_d=50\mathrm{V}$ .

• ▪ Case 2: Robustness against reference output voltage change from $50$ to $70\mathrm{V}$ and vice versa.

The simulation results obtained by using the proposed adaptive control technique are compared against the simulation results of the conventional adaptive command filtered backstepping control method (CACFBC) proposed in (Dong et al., 2012) and the simulation results of the I&I ABC proposed in (Karagiannis and Astolfi, 2008). The simulation results of the DC-DC buck converter system with reference output voltage and reference output voltage change are shown in Figures 2–4, respectively.

• ▪ Case 1: The reference output voltage is set to $V_d=50\mathrm{V}$ .

Figure 2.

Output voltage.

Figure 3.

Output voltage tracking error.

Figure 4.

Output voltage.

The response curves of the output voltage and the output voltage tracking error under the above three adaptive control approaches (I&I ABC, CACFBC, and I&I ACFBC) are displayed in Figures 2 and 3.

The simulation results are shown by Figures 2 and 3. Figure 2 shows the responses of output voltage and the reference output voltage, from which it is observed that the tracking performance is achieved. Figure 3 indicates the response of output voltage tracking error, from which we can see that the tracking error converges to zero. It is observed from the result that the proposed adaptive control scheme has better control performance and higher tracking accuracy in our developed adaptive control scheme than the I&I ABC approach in (Karagiannis and Astolfi, 2008) and CACFBC approach in (Dong et al., 2012).

• ▪ Case 2: Robustness against reference output voltage change is set from $50$ to $70\mathrm{V}$ and vice versa.

The reference output voltage is tending to vary suddenly from $50$ to $70\mathrm{V}$ and vice versa. The response curve of the output voltage under the above three adaptive controller approaches is hence shown in Figure 4.

It can be seen from Figure 4 that the output voltage can track the reference output voltage. As shown by Figure 4, the proposed adaptive control approach further improves transient and static performance in the presence of reference output voltage variation compared with the I&I ABC and CACFBC methods.

Experimental results

To investigate the efficiency of the proposed adaptive control method, experiments for the DC-DC buck converter system are performed. The experimental test bench is depicted in Figure 5, which consists of a host PC with MATLAB/Simulink, DC-DC buck converter (SEMIKRON inverter, inductance, capacitance, and load resistance), interface card, current and voltage sensors, DC power source, dSPACE DS1103, and digital oscilloscope.

Figure 5.

Experimental test setup on DC-DC buck converter system.

In the experimental tests, the validity of the proposed adaptive control method is constructed in the two cases via the following results compared with the CACFBC method. One case is for robustness against the load resistance variation and the other is concerned with the robustness against the reference output voltage variation. The experimental results of the DC-DC buck converter system with load resistance change and reference output voltage change are shown in Figures 6–8, respectively.

• ▪ Case 1: Robustness against load resistance change is set from $82$ to $9.72\mathrm{\Omega }$ and vice versa.

Figure 6.

Experiment results for the output voltage and inductance current under different adaptive control algorithms and load resistance variation. (a) I&I ACFBC. (b) CACFBC.

Figure 7.

Experiment results for the output voltage and inductance current under different adaptive control algorithms and reference output voltage variation. (a) I&I ACFBC. (b) CACFBC.

Figure 8.

Experiment results for the output voltage and inductance current under different adaptive control algorithms and reference output voltage variation. (a) I&I ACFBC. (b) CACFBC.

In this case, the output voltage and inductance current are compared among the different adaptive control methods (I&I ACFBC and CACFBC) by changing suddenly the load resistance from $82$ to $9.72\mathrm{\Omega }$ and vice versa.

The experimental results of DC-DC buck converter system with load variation are plotted in Figure 6(a) and 6(b), respectively. The robustness against load resistance is demonstrated when the load resistance changes suddenly from $82$ to $9.72\mathrm{\Omega }$ and vice versa. It is observed in Figure 6(a) and 6(b) that the output voltage and inductance current are changed when the load resistance changes and returned to a steady state. As shown in Figure 6(a) and 6(b), the output voltage and inductance current reveal the best performances in the I&I ACFBC method compared with the CACFBC method.

Figure 6(a) and 6(b) shows the experimental output voltage and inductance current when the load resistance drops from $82$ to $9.72\mathrm{\Omega }$ and rises from $9.72$ to $82\mathrm{\Omega }$ . The reference output voltage $V_d$ and the signal $x_{2c}$ under the proposed adaptive control scheme can be tracked by the output voltage and the inductance current faster than the CACFBC method. The results show that the proposed I&I ACFBC scheme has satisfactory robustness and has attained much better transient performance when the load resistance changes from $82$ to $9.72\mathrm{\Omega }$ and vice versa.

• ▪ Case 2: Robustness against reference output voltage change is set from $50$ to $70\mathrm{V}$ and vice versa.

In this case, the load resistance is kept unchanged, and the reference output voltage is changed suddenly from $50$ to $70\mathrm{V}$ and vice versa. The output voltage and inductance current responses are compared for the different adaptive control methods (I&I ACFBC and CACFBC).

The experimental reference output voltage is set to change suddenly from $50$ to $70\mathrm{V}$ and vice versa, and the experimental comparison results are displayed in Figures 7(a), 7(b), 8(a) and 8(b). It is shown that the output voltage and inductance current can track the desired values when the reference output voltage changes suddenly from $50$ to $70\mathrm{V}$ and vice versa. It is observed that the proposed adaptive control scheme has attained much better transient performance and faster convergence rate as compared with the CACFBC algorithm when the reference output voltage is changed suddenly from $50$ to $70\mathrm{V}$ and vice versa.

Figures 7(a), 7(b), 8(a), and 8(b) show the experimental output voltage and inductance current when the reference output voltage drops from $50$ to $70\mathrm{V}$ and rises from $70$ to $50\mathrm{V}$ . It can be observed that the proposed I&I ACFBC provides a better transient response than the CACFBC method in tracking the reference output voltage.

Figure 6(a) and 6(b) is the experimental output voltage and inductance current when the load resistance is changed from $82$ to $9.72\mathrm{\Omega }$ and vice versa, and Figures 7(a), 7(b), 8(a), and 8(b) are the experimental output voltage and inductance current when the reference output voltage is changed from $50$ to $70\mathrm{V}$ and vice versa. Compared with the CACFBC method, it can be seen from the experimental results that the proposed adaptive control scheme can show fastest transient response. In conclusion, the effectiveness of the designed adaptive control strategy is demonstrated by the above experimental results.

Conclusion

In this paper, an I&I ACFBC strategy is proposed for a class of uncertain nonlinear systems in strict-feedback form. By adopting the proposed adaptive control technique, the problem of the explosion of complexity existing in the adaptive backstepping control procedure is solved. The parameter estimations are driven by an I&I approach. Moreover, according to the Lyapunov stability theory, it is proved that all the signals in the closed-loop system are bounded. Simulation and experimental results of the DC-DC buck converter system have been utilized to illustrate the efficiency of the proposed adaptive control scheme. In our future work, we will consider the I&I based adaptive finite time command filtered backstepping control design for a class of uncertain nonlinear systems in strict-feedback form.