Asymptotic attenuation and rejection for a class of cascade systems with general periodic disturbances

Abstract

In this article, the problem of asymptotic attenuation and rejection is investigated for a class of cascade systems with general periodic disturbances via a non-linear disturbance observer and $H_{\infty }$ control. First, a novel disturbance observer is developed to estimate general periodic disturbances, where the disturbances are described as a linear uncertain system with a non-linear output function. Then, a composite controller is constructed by combining the disturbance estimation with the partial state feedback control law, such that, the closed-loop system is asymptotically stable and meets $H_{\infty }$ performance. With the help of the Lyapunov function theory and the linear matrix inequality method, some sufficient conditions for the desired controller and observer gains are established. Finally, numerical and application examples are presented to demonstrate the effectiveness of the proposed scheme.

Introduction

Periodic disturbances widely exist in many practical systems, for example, harmonic disturbances in power systems and non-linear vibrations in mechanical systems.^1,2 The existence of periodic disturbances always results in adverse effects on the control performance of a closed-loop system. Anti-disturbance control has been a key topic in control system fields in the past decade. At present, many elegant approaches have been proposed to attenuate and reject periodic disturbances,^3–7 including the non-linear disturbance-observer-based control (NDOBC) method,^8–13 non-linear output regulation theory^14–16 and adaptive control scheme.^17–22 The key idea of many of these methods is to use the desired feedforward input terms to cancel periodic disturbances.

In recent years, the asymptotic rejection of general periodic disturbances has received more and more attention.^23–27 In Ding, ²⁴ the asymptotic rejection of general periodic disturbances for non-linear systems is developed, in which an approach based on integrations over half of the disturbance period is proposed to estimate disturbances. An adaptive tracking control problem is presented for non-linear systems with general periodic disturbances in Xu and Xu, ²⁷ where the control scheme can only guarantee that the tracking error converges to zero in the sense of $L_2$ convergence. In Chen, ⁸ an adaptive estimation method is proposed for general periodic disturbances, and then an output feedback controller based on the disturbance estimation is presented. In Ding, ²⁶ general periodic disturbances are modelled as a linear system with a non-linear output function, and the problem of asymptotic rejection for a class of non-linear systems with general periodic disturbances via an internal model design is investigated.

In the above-mentioned literature, the systems are assumed to be subjected to just one kind of general periodic disturbance. However, in practice, the systems are always subjected to multiple disturbances, for example, external disturbances, internal noise and modelling error. In order to enhance the control accuracy of systems with multiple disturbances, composite hierarchical anti-disturbance control (CHADC) strategies are proposed in Guo and Cao. ²⁸ The main idea of CHADC is that a disturbance observer is designed to estimate disturbances generated by an exogenous system, and the disturbance estimation used in feedforward compensation plus conventional feedback control methods, such as $H_{\infty }$ control,^29,30 sliding mode control^31–33 and adaptive control,^34,35 is employed to attenuate other types of disturbances. However, the exogenous system is described by a linear system with a linear output function in the existing results of CHADC theories. However, the non-linear terms are connected to the systems in a parallel way in the above papers, which can be directly rejected by control inputs. It is worth noting that CHADC for cascade non-linear systems is much more complicated, where the non-linear item, as a non-linear subsystem, is cascaded to the other subsystems.

In this article, we investigate the problem of asymptotic attenuation and rejection for a class of cascade systems with general periodic disturbances via a non-linear disturbance observer and $H_{\infty }$ control. The disturbances are divided into two parts: one part is assumed to be norm bounded, and the other part is the general periodic disturbances. It is shown that the general periodic disturbances can be described by a linear uncertain system with a non-linear output function. A novel disturbance observer is designed to estimate the general periodic disturbances. By combining the disturbance estimation with the partial state feedback control law, a composite controller is constructed; the desired dynamic performance can be guaranteed. Finally, numerical and application examples are presented to demonstrate the effectiveness of the proposed method.

The main contributions of this work are listed as follows:

1. A CHADC strategy is proposed for a class of cascade system with multiple disturbances for the first time.

2. The systems considered in this article have two kinds of disturbances, that is, general periodic disturbances generated by a linear uncertain system with a non-linear output function and norm-bounded disturbances which can be regarded as an extension of the previous references.

3. A novel disturbance observer is designed to estimate the general periodic disturbances.

Problem formulation and preliminaries

Consider the following cascade systems $({\Sigma }_0)$

$$\stackrel{.}{\xi }(t)=g(\xi (t),x(t))$$

(1)

$$\stackrel{.}{x}(t)=Ax(t)+Ff(t,x(t))+B_2(u(t)+d_2(t))+B_1{d}_1(t)$$

(2)

$$z(t)=Cx(t)$$

(3)

where $x(t)\in {\mathbb{R}}^r$ is the system state; $\xi (t)\in {\mathbb{R}}^{n-r}$ is the remaining unmeasured state, which is referred to as the dynamic uncertainties; $u(t)\in {\mathbb{R}}^p$ and $z(t)\in {\mathbb{R}}^q$ are the control inputs and control outputs, respectively; A, F, $B_1$ , $B_2$ and C are the constant matrices of appropriate dimensions; $g(\xi (t),x(t))$ and $f(t,x(t))$ are non-linear functions; $d_1(t)\in {\mathbb{R}}^m$ is the disturbance input, which is assumed to belong to $L_2[0,\infty )$ ; and $d_2(t)\in {\mathbb{R}}^p$ represents the external disturbances, which is supposed to satisfy assumption 1.

The structure diagram of the system is given in Figure 1.

Figure 1.

The structure diagram of the system.

Assumption 1

The disturbance $d_2(t)$ in the control path is generated by the following exogenous system

$$\stackrel{.}{\omega }(t)=W\omega (t)+H\delta (t),d_2(t)=V\omega (t)+h(\omega (t))$$

(4)

where $W\in {\mathbb{R}}^{o\times o},$ $V\in {\mathbb{R}}^{p\times o}$ and $H\in {\mathbb{R}}^{o\times l}$ are known matrices; $\omega (t)\in {\mathbb{R}}^o$ is the system states; and $\delta (t)\in {\mathbb{R}}^l$ represents the additional disturbances. It is also supposed that $\delta (t)\in L_2[0,\infty )$ . The non-linear function $h(\omega (t))\in {\mathbb{R}}^p$ is a Lipschitz non-linear function with a Lipschitz constant ${\gamma }_1>0$ , that is, $||h({\omega }_1(t))-h({\omega }_2(t))||\le {\gamma }_1||{\omega }_1(t)-{\omega }_2(t)||$ .

Remark 1

In assumption 1, the disturbance is modelled as a linear system with a non-linear output function, which can represent many kinds of disturbances, for example, constant disturbances, harmonic disturbances and non-harmonic disturbances.

Remark 2

Many disturbance-observer-based control methods are proposed to reject the periodic disturbances described by a linear system with a linear output function.^8,9,29,32 However, some disturbance signals cannot be modelled by a linear system with a linear output function but can be described as system (4), such as a trapezoidal wave and triangular wave. In order to extend the application of the disturbance-observer-based control method, we design a novel disturbance observer to estimate the disturbance in equation (4) in this article.

Remark 3

An internal model design method has been developed to reject the disturbances described by a linear system with a non-linear output function in Ding, ²⁶ in which the disturbance model does not contain any uncertainties. In this article, we consider the disturbances with an unmodelled error and system perturbations in the disturbance model (4).

In this article, we make the following assumptions to analyse the stability of the closed-loop system.

Assumption 2

The non-linear function $f(t,x(t))$ satisfies the following conditions ⁹

$$\begin{array}{l}(1)f(t,0)=0\\ (2)\left|\right|f(t,x_1(t))-f(t,x_2(t))\left|\right|\le \left|\right|U(x_1(t)-x_2(t))\left|\right|\end{array}$$

where U is a given constant weighting matrix.

Assumption 3

$g(\xi (t),x(t))$ satisfies the global Lipschitz condition, that is, there exists a constant $T>0$ such that

$$\begin{array}{l}\left|\right|g(\xi (t),x_1(t))-g(\xi (t),\\ x_2(t))||\le T||x_1(t)-x_2(t)||,\forall x_1(t),x_2(t)\end{array}$$

Assumption 4

There exists a smooth positive-definite function $\mathcal{V}(\xi (t))$ with $\mathcal{V}(0)=0$ and constants ${\sigma }_i>0,i=1,2,$ $\alpha >0$ and $\beta >0$ such that

$$\begin{array}{l}{\sigma }_1\left|\right|\xi (t)|{|}^2\le \mathcal{V}(\xi (t))\le {\sigma }_2||\xi (t)|{|}^2\\ \frac{d\mathcal{V}}{d\xi (t)}g(\xi (t),0)\le -\alpha \left|\right|\xi (t)|{|}^2\\ \frac{d\mathcal{V}}{d\xi (t)}\le \beta \left|\right|\xi (t)\left|\right|\end{array}$$

Assumption 5

$(W,B_{2}V)$ is observable.^9,29

Composite anti-disturbance control for the case with a known non-linear function $f(t,x(t))$

In this section, we first assume that the non-linear function $f(t,x(t))$ is known, and then a novel non-linear disturbance observer is designed as

$$\begin{array}{l}\stackrel{.}{\nu }(t)=(W+L{B}_{2}V)(\mathbf{\nu }(t)-Lx(t))+L(Ax(t)\\ +Ff(t,x(t))+B_{2}u(t))+L{B}_{2}h(\widehat{\omega }(t))\\ \widehat{\omega }(t)=\nu (t)-Lx(t),{\widehat{d}}_2(t)=V\widehat{\omega }(t)+h(\widehat{\omega }(t))\end{array}$$

(5)

where L is the observer gain, and $\hat{\omega }(t)$ is the estimation of $\omega (t)$ .

The composite controller is designed as

$$u(t)=-{\widehat{d}}_2\left(t\right)+Kx(t)$$

(6)

where ${\hat{d}}_2(t)$ is the estimation of the disturbances $d_2(t)$ , which is obtained by the disturbance observer.

Remark 4

The composite controller (6) includes two parts: one part is the negative of the output of the disturbance observer, and the other part is the conventional state feedback control law. It is obvious that the aim of the term $-{\hat{d}}_2(t)$ in controller (6) is to compensate for the disturbances $d_2(t)$ generated by the exogenous system, whereas the term $\mathrm{Kx}(t)$ plays an important role in guaranteeing the closed-loop system stability and obtaining the desired control performance.

Letting $e_{\omega }(t)=\omega (t)-\hat{\omega }(t)$ leads to

$$\begin{array}{l}{\stackrel{.}{e}}_{\omega }(t)=(W+L{B}_{2}V)e_{\omega }(t)+L{B}_1{d}_1(t)+H\delta (t)\\ +L{B}_2(h(\omega (t))-h(\widehat{\omega }(t)))\end{array}$$

(7)

By applying the controller (6) to the system $({\Sigma }_0)$ and combining with (7), the following augmented closed-loop system $({\Sigma }_{\mathrm{cl}})$ can be obtained

$$\stackrel{·}{\xi }(t)=g(\xi (t),x(t))$$

(8)

$$\begin{array}{l}\stackrel{.}{x}(t)=(A+B_{2}K)x(t)+B_{2}V{e}_{\omega }(t)\\ +{\overline{B}}_1\overline{d}(t)+B_2(h(\omega (t))-h(\widehat{\omega }(t)))+Ff(t,x(t))\end{array}$$

(9)

$$\begin{array}{l}{\stackrel{.}{e}}_{\omega }(t)=(W+L{B}_{2}V)e_{\omega }(t)+\overline{H}\overline{d}(t)\\ +L{B}_2(h(\omega (t))-h(\widehat{\omega }(t)))\end{array}$$

(10)

$$z(t)=\mathrm{Cx}(t)$$

(11)

where

$$\overline{d}(t)=\left[\begin{array}{c}{d}_1(t)\hfill \\ \multicolumn{1}{c}{\delta (t)}\end{array}\right],\overline{H}=[L{B}_{1}H],{\overline{B}}_1=[B_{1}0]$$

Remark 5

The disturbance $\overline{d}(t)\in L_2[0,\infty )$ because of $\delta (t)\in L_2[0,\infty )$ and $d_1(t)\in L_2[0,\infty )$ . In this situation, the system $({\Sigma }_{\mathrm{cl}})$ becomes a non-linear system with norm-bounded disturbances. Thus, it is a good choice to design a controller for the system $({\Sigma }_{\mathrm{cl}})$ with the $H_{\infty }$ control scheme.

Remark 6

In the composite control method, the conventional $H_{\infty }$ control scheme generally obtains the attenuation performance with respect to the disturbances belonging to $L_2[0,\infty )$ , while disturbance-observer-based control (DOBC) is employed to reject the influence of the disturbance with some known information.

Before presenting the main results, a definition is formulated.

Definition 1

Given a scalar $\gamma >0$ , the system $({\Sigma }_{\mathrm{cl}})$ is asymptotically stable with an $H_{\infty }$ performance index $\gamma$ if the following conditions are satisfied:

1. The system $({\Sigma }_{\mathrm{cl}})$ is asymptotically stable when $\overline{d}(t)=0$ ;

2. Under zero initial conditions, there exists $||z(t)|{|}_2\le \gamma ||\overline{d}(t)|{|}_2$ for all non-zero $\overline{d}(t)$ , where $\gamma$ is a given disturbance attenuation level.

To analyse the stability of the closed-loop system $({\Sigma }_{\mathrm{cl}})$ , we develop the following theorem.

Theorem 1

Consider the system ( ${\Sigma }_0$ ) under assumptions 1–5. Given scalars $\gamma >0$ , ${\gamma }_1>0$ , $\alpha >0$ , $T>0,$ $\beta >0$ and $\lambda >0,$ the closed-loop system $({\Sigma }_{\mathrm{cl}})$ is asymptotically stable with $H_{\infty }$ performance if there exist matrices $X>0$ , $Q>0$ , $\overline{K}$ and $\overline{L}$ , such that the following inequality holds

$${\mathrm{\Pi }}_1=\left[\begin{array}{cccccccc}{\mathrm{\Pi }}_{11}\hfill & F\hfill & B_{2}V\hfill & {\overline{B}}_1\hfill & X{C}^T\hfill & X{U}^T\hfill & X^T\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& -\frac{1}{{\lambda }^2}I\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathrm{\Pi }}_{33}\hfill & Q\overline{H}\hfill & 0\hfill & 0\hfill & 0\hfill & \overline{L}{B}_2\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -{\gamma }^{2}I\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -I\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -{\lambda }^{2}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & -\frac{2\alpha }{{\beta }^2{T}^2}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & -I\hfill \end{array}\right]<0$$

(12)

where

$$\begin{array}{c}{\mathrm{\Pi }}_{11}=\mathrm{AX}+B_2\overline{K}+X{A}^T+{\overline{K}}^T{B}_2^T+B_2{B}_2^T\hfill \\ {\mathrm{\Pi }}_{33}=\mathrm{QW}+\overline{L}{B}_{2}V+W^{T}Q+(B_{2}V{)}^T{\overline{L}}^T+2{\gamma }_1^{2}I\hfill \end{array}$$

(13)

Moreover, the desired controller and observer gains are as follows

$$K=\overline{K}{X}^{-1},L=Q^{-1}\overline{L}$$

Proof

Choose the Lyapunov function as

$$\begin{array}{l}V(t)=\mathcal{V}(\xi (t))+x^T(t)Px(t)\\ +\frac{1}{{\lambda }^2}{\displaystyle \underset{0}{\overset{t}{\int }}(||Ux(\tau )|{|}^2-\left|\right|f(\tau ,x(\tau ))|{|}^2)d\tau +e_{\omega }^T(t)Q{e}_{\omega }(t)}\end{array}$$

(14)

where $P>0,$ $Q>0,$ and $\lambda >0$ . Computing the derivative of equation (14) along the trajectories of system $({\Sigma }_{\mathrm{cl}})$ leads to

$$\begin{array}{l}\stackrel{.}{V}(t)=\frac{d\mathcal{V}}{d\xi (t)}\stackrel{.}{\xi }(t)+2x^{T}P\stackrel{.}{x}\\ +\frac{1}{{\lambda }^2}(||Ux(t)|{|}^2-\left|\right|f(t,x(t))|{|}^2)+2e_{\omega }^{T}Q{\stackrel{.}{e}}_{\omega }\end{array}$$

(15)

From assumptions 3 and 4 and equation (8), we have

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d\mathcal{V}}{d\xi (t)}\stackrel{·}{\xi }(t)=\frac{d\mathcal{V}}{d\xi (t)}g(\xi (t),x)}\\ =\frac{d\mathcal{V}}{d\xi (t)}g(\xi (t),0)+\frac{d\mathcal{V}}{d\xi (t)}(g(\xi (t),x)-g(\xi (t),0))\hfill \\ \le -\alpha ||\xi (t)|{|}^2+\beta T||\xi (t)||||x(t)||\hfill \\ \le -\alpha ||\xi (t)|{|}^2+\frac{\alpha }{2}||\xi (t)|{|}^2+\frac{{\beta }^2{T}^2}{2\alpha }||x(t)|{|}^2\hfill \\ =-\frac{\alpha }{2}||\xi (t)|{|}^2+\frac{{\beta }^2{T}^2}{2\alpha }||x(t)|{|}^2\hfill \end{array}$$

(16)

Based on equations (9) and (10), we can obtain

$$\begin{array}{c}2x^{T}P\stackrel{·}{x}+2e_{\omega }^{T}Q{\stackrel{·}{e}}_{\omega }\hfill \\ =2x^T(t)P((A+B_{2}K)x(t)+B_{2}V{e}_{\omega }(t)\hfill \\ +{\overline{B}}_1\overline{d}(t)+B_2(h(\omega (t))-h(\hat{\omega }(t)))+\mathrm{Ff}(t,x(t)))\hfill \\ +2e_{\omega }^T(t)Q((W+L{B}_{2}V)e_{\omega }(t)\hfill \\ +\overline{H}\overline{d}(t)+L{B}_2(h(\omega (t))-h(\hat{\omega }(t))))\hfill \\ =x^T(t)(P(A+B_{2}K)+(A+B_{2}K{)}^{T}P)x(t)\hfill \\ +2x^T(t)P{B}_{2}V{e}_{\omega }(t)+2x^T(t)\mathrm{PFf}(t,x(t))\hfill \\ +2x^T(t)P{B}_2(h(\omega (t))-h(\hat{\omega }(t)))\hfill \\ +2x^T(t)P{\overline{B}}_1\overline{d}(t)+e_{\omega }^T(t)(Q(W+L{B}_{2}V)\hfill \\ +(W+L{B}_{2}V{)}^{T}Q)e_{\omega }(t)\hfill \\ +2e_{\omega }^T(t)\mathrm{QL}{B}_2(h(\omega (t))-h(\hat{\omega }(t)))+2e_{\omega }^T(t)Q\overline{H}\overline{d}(t)\hfill \end{array}$$

(17)

Then, one obtains

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(t)\le -\frac{\alpha }{2}||\xi (t)|{|}^2+\frac{{\beta }^2{T}^2}{2\alpha }||x(t)|{|}^2}\\ +x^T(t)(P(A+B_{2}K)+(A+B_{2}K{)}^{T}P)x(t)\hfill \\ +2x^T(t)\mathrm{PFf}(t,x(t))\hfill \\ +2x^T(t)P{B}_{2}V{e}_{\omega }(t)+2x^T(t)P{B}_2(h(\omega (t))\hfill \\ -h(\hat{\omega }(t)))+2x^T(t)P{\overline{B}}_1\overline{d}(t)\hfill \\ +\frac{1}{{\lambda }^2}(||\mathrm{Ux}(t)|{|}^2-||f(t,x(t))|{|}^2)\hfill \\ +e_{\omega }^T(t)(Q(W+L{B}_{2}V)+(W+L{B}_{2}V{)}^{T}Q)e_{\omega }(t)\hfill \\ +2e_{\omega }^T(t)\mathrm{QL}{B}_2(h(\omega (t))-h(\hat{\omega }(t)))+2e_{\omega }^T(t)Q\overline{H}\overline{d}(t)\hfill \end{array}$$

(18)

According to the characteristics of the non-linear function $h(·)$ , we obtain

$$\begin{array}{c}2x^T(t)P{B}_2(h(\omega (t))-h(\hat{\omega }(t)))\hfill \\ \le 2||x^T(t)P{B}_2||||h(\omega (t))-h(\hat{\omega }(t))||\hfill \\ \le 2{\gamma }_1||x^T(t)P{B}_2||||e_{\omega }(t)||\hfill \\ \le x^T(t)P{B}_2{B}_2^T\mathrm{Px}(t)+{\gamma }_1^2{e}_{\omega }^T(t)e_{\omega }(t)\hfill \end{array}$$

(19)

$$\begin{array}{c}2e_{\omega }^T(t)\mathrm{QL}{B}_2(h(\omega (t))-h(\hat{\omega }(t)))\hfill \\ \le 2||e_{\omega }^T(t)\mathrm{QL}{B}_2||||h(\omega (t))-h(\hat{\omega }(t))||\hfill \\ \le {\gamma }_1||e_{\omega }^T(t)\mathrm{QL}{B}_2||||e_{\omega }(t)||\hfill \\ \le e_{\omega }^T(t)\mathrm{QL}{B}_2{B}_2^T{L}^{T}Q{e}_{\omega }(t)+{\gamma }_1^2{e}_{\omega }^T(t)e_{\omega }(t)\hfill \end{array}$$

(20)

From equations (15)–(20), we have

$$\begin{array}{c}\stackrel{·}{V}\le -\frac{\alpha }{2}||\xi (t)|{|}^2+{\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{f(t,x(t))}\\ \multicolumn{1}{c}{e_{\omega }(t)}\\ \multicolumn{1}{c}{\overline{d}(t)}\end{array}\right]}^T\hfill \\ \left[\begin{array}{cccc}{\mathrm{\Phi }}_{11}\hfill & \mathrm{PF}\hfill & P{B}_{2}V\hfill & P{\overline{B}}_1\hfill \\ \multicolumn{1}{c}{*}& -\frac{1}{{\lambda }^2}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathrm{\Phi }}_{33}\hfill & Q\overline{H}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -{\gamma }^{2}I\hfill \end{array}\right]\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{f(t,x(t))}\\ \multicolumn{1}{c}{e_{\omega }(t)}\\ \multicolumn{1}{c}{\overline{d}(t)}\end{array}\right]\hfill \\ -x^T(t)C^T\mathrm{Cx}(t)+{\gamma }^2||\overline{d}(t)|{|}^2\hfill \end{array}$$

(21)

where

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\Phi }}_{11}=P(A+B_{2}K)+(A+B_{2}K{)}^{T}P}\\ +P{B}_2{B}_2^{T}P+C^{T}C+\frac{1}{{\lambda }^2}{U}^{T}U+\frac{{\beta }^2{T}^2}{2\alpha }I\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{33}=Q(W+L{B}_{2}V)+(W+L{B}_{2}V{)}^{T}Q}\\ +\mathrm{QL}{B}_2{B}_2^T{L}^{T}Q+2{\gamma }_1^{2}I\hfill \end{array}$$

Consider the following performance index

$$J_{\overline{T}}=\underset{0}{\overset{\overline{T}}{\int }}(z^T(t)z(t)-{\gamma }^2{\overline{d}}^T(t)\overline{d}(t))\mathrm{dt}$$

(22)

Under zero initial conditions, we have

$$\begin{array}{c}{J}_{\overline{T}}={\int }_0^{\overline{T}}(z^T(t)z(t)-{\gamma }^2{\overline{d}}^T(t)\overline{d}(t))\mathrm{dt}\hfill \\ ={\int }_0^{\overline{T}}(z^T(t)z(t)-{\gamma }^2{\overline{d}}^T(t)\overline{d}(t)+\stackrel{·}{V})\mathrm{dt}-V(\overline{T})\hfill \end{array}$$

(23)

According to equation (21), we obtain

$$\begin{array}{c}{z}^T(t)z(t)-{\gamma }^2{\overline{d}}^T(t)\overline{d}(t)+\stackrel{·}{V}\hfill \\ \le {\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{f(t,x(t))}\\ \multicolumn{1}{c}{e_{\omega }(t)}\\ \multicolumn{1}{c}{\overline{d}(t)}\end{array}\right]}^T\left[\begin{array}{cccc}{\mathrm{\Phi }}_{11}\hfill & \mathrm{PF}\hfill & P{B}_{2}V\hfill & P{\overline{B}}_1\hfill \\ \multicolumn{1}{c}{*}& -\frac{1}{{\lambda }^2}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathrm{\Phi }}_{33}\hfill & Q\overline{H}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -{\gamma }^{2}I\hfill \end{array}\right]\hfill \\ \left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{f(t,x(t))}\\ \multicolumn{1}{c}{e_{\omega }(t)}\\ \multicolumn{1}{c}{\overline{d}(t)}\end{array}\right]\hfill \end{array}$$

(24)

The combination of equation (12) and the Schur complement formula yields

$$\overline{\mathrm{\Pi }}=\left[\begin{array}{cccc}{\overline{\mathrm{\Pi }}}_{11}\hfill & F\hfill & B_{2}V\hfill & {\overline{B}}_1\hfill \\ \multicolumn{1}{c}{*}& -\frac{1}{{\lambda }^2}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\overline{\mathrm{\Pi }}}_{33}\hfill & Q\overline{H}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -{\gamma }^{2}I\hfill \end{array}\right]<0$$

(25)

where

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{\mathrm{\Pi }}}_{11}=\mathrm{AX}+B_2\overline{K}+X{A}^T+{\overline{K}}^T{B}_2^T+B_2{B}_2^T}\\ +X{C}^T\mathrm{CX}+\frac{1}{{\lambda }^2}X{U}^T\mathrm{UX}+\frac{{\beta }^2{T}^2}{2\alpha }{X}^{T}X\hfill \\ \multicolumn{1}{c}{{\overline{\mathrm{\Pi }}}_{33}=\mathrm{QW}+\overline{L}{B}_{2}V+W^{T}Q+(B_{2}V{)}^T{\overline{L}}^T}\\ +2{\gamma }_1^{2}I+\overline{L}{B}_2{B}_2^T{\overline{L}}^T\hfill \\ \multicolumn{1}{c}{\overline{K}=\mathrm{KX},\overline{L}=\mathrm{QL}}\end{array}$$

By performing a congruence transformation of equation (25) by $\mathrm{diag}\{P,I,I,I\}$ , we obtain

$$\left[\begin{array}{cccc}{\mathrm{\Phi }}_{11}\hfill & \mathrm{PF}\hfill & P{B}_{2}V\hfill & P{\overline{B}}_1\hfill \\ \multicolumn{1}{c}{*}& -\frac{1}{{\lambda }^2}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathrm{\Phi }}_{33}\hfill & Q\overline{H}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -{\gamma }^{2}I\hfill \end{array}\right]<0$$

(26)

Therefore, we have $J_{\overline{T}}\le 0$ . As $\overline{T}\to \infty ,$ yields $||z(t)||\le \gamma ||\overline{d}(t)||$ .

In addition, when $\overline{d}(t)=0$ , equation (21) is rewritten as

$$\begin{array}{l}\stackrel{.}{V}\le -\frac{\alpha }{2}\left|\right|\xi (t)|{|}^2+{\left[\begin{array}{c}x(t)\\ f(t,x(t))\\ e_{\omega }(t)\end{array}\right]}^T\\ \left[\begin{array}{ccc}{\mathrm{\Phi }}_{11}& PF& P{B}_{2}V\\ \ast & -\frac{1}{{\lambda }^2}I& 0\\ \ast & \ast & {\mathrm{\Phi }}_{33}\end{array}\right]\left[\begin{array}{c}x(t)\\ f(t,x(t))\\ e_{\omega }(t)\end{array}\right]-x^T{C}^{T}Cx\end{array}$$

(27)

According to equation (26), we have

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le -\frac{\alpha }{2}||\xi (t)|{|}^2+{\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{f(t,x(t))}\\ \multicolumn{1}{c}{e_{\omega }(t)}\end{array}\right]}^T}\\ \left[\begin{array}{ccc}{\mathrm{\Phi }}_{11}\hfill & \mathrm{PF}\hfill & P{B}_{2}V\hfill \\ \multicolumn{1}{c}{*}& -\frac{1}{{\lambda }^2}I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathrm{\Phi }}_{33}\hfill \end{array}\right]\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{f(t,x(t))}\\ \multicolumn{1}{c}{e_{\omega }(t)}\end{array}\right]-x^T(t)C^T\mathrm{Cx}(t)\hfill \\ \le -\frac{\alpha }{2}||\xi (t)|{|}^2+{\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{f(t,x(t))}\\ \multicolumn{1}{c}{e_{\omega }(t)}\end{array}\right]}^T\hfill \\ \left[\begin{array}{ccc}{\mathrm{\Phi }}_{11}\hfill & \mathrm{PF}\hfill & P{B}_{2}V\hfill \\ \multicolumn{1}{c}{*}& -\frac{1}{{\lambda }^2}I\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathrm{\Phi }}_{33}\hfill \end{array}\right]\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{f(t,x(t))}\\ \multicolumn{1}{c}{e_{\omega }(t)}\end{array}\right]<0\hfill \end{array}$$

(28)

Then, the closed-loop system is asymptotically stable when $\overline{d}(t)=0.$ According to the above analysis, the closed-loop system is asymptotically stable with $H_{\infty }$ performance.

Composite anti-disturbance control for the case with an unknown non-linear function $f(t,x(t))$

In this section, we consider that the non-linear function $f(t,x(t))$ is unknown. In contrast to section ‘Composite anti-disturbance control for the case with a known non-linear function $f(t,x(t))$ ’, the non-linear function $f(t,x(t))$ is unavailable in the disturbance observer design.

The non-linear disturbance observer is developed as follows

$$\begin{array}{c}\stackrel{·}{\nu }(t)=(W+\mathrm{LV})(\nu (t)-\mathrm{Lx}(t))\hfill \\ +L(\mathrm{Ax}(t)+B_{2}u(t))+L{B}_{2}h(\hat{\omega }(t))\hfill \\ \hat{\omega }(t)=\nu (t)-\mathrm{Lx}(t),{\hat{d}}_2(t)=V\hat{\omega }(t)+h(\hat{\omega }(t))\hfill \end{array}$$

(29)

where L is the observer gain and $\hat{\omega }(t)$ is the estimation of $\omega (t)$ .

The disturbance estimation error system is described as

$$\begin{array}{l}{\stackrel{.}{e}}_{\omega }(t)=(W+L{B}_{2}V)e_{\omega }(t)+L{B}_1{d}_1(t)\\ +LFf(t,x(t))+H\delta (t)+L{B}_2(h(\omega (t))-h(\widehat{\omega }(t)))\end{array}$$

(30)

By applying the controller (6) to the system $({\Sigma }_0)$ and combining with (30), we obtain the following augmented closed-loop system $({\Sigma }_{\mathrm{clu}})$

$$\stackrel{·}{\xi }(t)=g(\xi (t),x(t))$$

(31)

$$\begin{array}{l}\stackrel{.}{x}(t)=(A+B_{2}K)x(t)+B_{2}V{e}_{\omega }(t)+{\overline{B}}_1\overline{d}(t)\\ +B_2(h(\omega (t))-h(\widehat{\omega }(t)))+Ff(t,x(t))\end{array}$$

(32)

$$\begin{array}{l}{\stackrel{.}{e}}_{\omega }(t)=(W+L{B}_{2}V)e_{\omega }(t)+\overline{H}\overline{d}(t)+\overline{F}f(t,x(t))\\ +L{B}_2(h(\omega (t))-h(\widehat{\omega }(t)))\end{array}$$

(33)

$$z(t)=\mathrm{Cx}(t)$$

(34)

where

$$\overline{d}(t)=\left[\begin{array}{c}{d}_1(t)\hfill \\ \multicolumn{1}{c}{\delta (t)}\end{array}\right],\overline{H}=[L{B}_{1}H],{\overline{B}}_1=[B_{1}0],\overline{F}=\mathrm{LF}$$

Remark 7

Compared with the error system (10), the disturbance estimation error system (33) contains a non-linear function $f(t,x(t))$ , which is more complicated than equation (10). Fortunately, the augmented closed-loop systems $({\Sigma }_{\mathrm{clu}})$ and $({\Sigma }_{\mathrm{cl}})$ have a similar structure. Thus, the stability analysis of the system $({\Sigma }_{\mathrm{clu}})$ is similar to that of the system $({\Sigma }_{\mathrm{cl}}$ ).

Theorem 2 presents the asymptotical stability of the system $({\Sigma }_{\mathrm{clu}})$ with $H_{\infty }$ performance.

Theorem 2

Consider the system ( ${\Sigma }_0$ ) under assumptions 1–5. Given scalars $\gamma >0$ , ${\gamma }_1>0$ , $\alpha >0$ , $T>0,$ $\beta >0$ and $\lambda >0,$ the closed-loop system ( ${\Sigma }_{\mathrm{clu}}$ ) is asymptotically stable with $H_{\infty }$ performance if there exist matrices $X>0,$ $Q>0$ , $\overline{K}$ and $\overline{L}$ , such that the following inequality holds

$${\mathrm{\Pi }}_2=\left[\begin{array}{cccccccc}{\mathrm{\Pi }}_{11}\hfill & F\hfill & B_{2}V\hfill & {\overline{B}}_1\hfill & X{C}^T\hfill & X{U}^T\hfill & X\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& -\frac{1}{{\lambda }^2}I\hfill & F^T\overline{L}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathrm{\Pi }}_{33}\hfill & Q\overline{H}\hfill & 0\hfill & 0\hfill & 0\hfill & \overline{L}{B}_2\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -{\gamma }^{2}I\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -I\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -{\lambda }^{2}I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & -\frac{{\beta }^2{T}^2}{2\alpha }\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & -I\hfill \end{array}\right]<0$$

(35)

where ${\mathrm{\Pi }}_{11}$ and ${\mathrm{\Pi }}_{33}$ are defined in equation (13). Moreover, the desired controller and observer gains are

$$K=\overline{K}{X}^{-1},L=Q^{-1}\overline{L}$$

Proof

By comparing the system matrices in equations (31) and (34) with the system matrices in equations (8) and (11) and following similar arguments for the proof of theorem 1, we can readily obtain theorem 2.

Numerical and application examples

In this section, a numerical example and a practical application to the linearised model of the longitudinal dynamics of an aircraft are provided to show the effectiveness of the main results in this article.

Example 1

The system matrices in the system $({\Sigma }_0)$ are given as follows

$$\begin{array}{c}A=\left[\begin{array}{cc}-2.625\hfill & 3.270\hfill \\ \multicolumn{1}{c}{4.54}& -8.124\hfill \end{array}\right],B_2=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{2}\end{array}\right],B_1=\left[\begin{array}{c}-0.1\hfill \\ \multicolumn{1}{c}{0.5}\end{array}\right]\hfill \\ F=\left[\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0.1}\end{array}\right],C=\left[\begin{array}{cc}1\hfill & -0.8\hfill \end{array}\right]\hfill \end{array}$$

The parameters of system (4) are

$$W=\left[\begin{array}{cc}0\hfill & -2\pi \hfill \\ \multicolumn{1}{c}{2\pi }& 0\hfill \end{array}\right],H=\left[\begin{array}{c}0.1\hfill \\ \multicolumn{1}{c}{-1}\end{array}\right],V=\left[\begin{array}{cc}5\hfill & -2\hfill \end{array}\right]$$

Suppose $\gamma =0.1$ and $U=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right].$

Our aim is to design composite controllers for the case with known and unknown non-linearities such that the closed-loop system is asymptotically stable and satisfies the prescribed $H_{\infty }$ performance.

• Case 1. The non-linear function $f(t,x(t))$ is known.

By solving the linear matrix inequality (LMI) in equation (12), the gains of the controller and observer are given by

$$\begin{array}{l}K=\left[\begin{array}{cc}-13.2047& -27.4121\end{array}\right],\\ L=\left[\begin{array}{cc}-11.5441& -2.1621\\ 24.5622& 4.4922\end{array}\right]\end{array}$$

Assume that $f(t,x(t))=x_1\sin (x_2)$ . The initial value is $x(0)=[2-1{]}^T$ .

The response curves of the system states are shown in Figure 2, which demonstrates that the proposed method disturbance-observer-based control plus $H_{\infty }$ control (DOBCPHC) can obtain better control performance than $H_{\infty }$ control. In order to illustrate the effectiveness of the proposed disturbance observer technique, the curves of the disturbances and disturbance estimation are presented in Figure 3. From this figure, we can see that the disturbance observer can effectively estimate the unknown disturbances. The curve of the system output z is given in Figure 4, which implies that the proposed method can obtain good disturbance attenuation performance, while $H_{\infty }$ control method has a poor control performance.

• Case 2. The non-linear function $f(t,x(t))$ is unknown.

Figure 2.

Response curves of system states.

Figure 3.

Curves of disturbances $d_2$ and disturbance estimation ${\hat{d}}_2$ .

Figure 4.

Curve of output z.

We resort to the LMI toolbox to solve the LMI in equation (35); the gains of the desired controller and observer are

$$\begin{array}{l}K=\left[\begin{array}{cc}-36.7029& -36.9134\end{array}\right],\\ L=\left[\begin{array}{cc}-10.0473& -2.6010\\ 40.8917& 8.7770\end{array}\right]\end{array}$$

In contrast to case 1, we assume that the non-linear function $f(t,x(t))$ is unknown. Similar to Guo and Chen, ⁹ we suppose $f(t,x(t))=\alpha (t)x_2$ , where $\alpha (t)$ is a random input signal with an upper bound of $1$ . We can obtain $||f(t,x(t))||\le ||\mathrm{Ux}(t)||.$ The simulation results are presented as follows. Figure 5 depicts the curves of system states. From this figure, we can see that the closed-loop system has satisfactory performance. Figure 6 shows the disturbances and disturbance estimation, which implies that the proposed disturbance observer can obtain a high estimated accuracy. The curve of the system output z is given in Figure 7, which demonstrates that the proposed method can effectively attenuate external disturbances.

Figure 5.

Response curves of system states.

Figure 6.

Curves of disturbances $d_2$ and disturbance estimation ${\hat{d}}_2$ .

Figure 7.

Curve of output z.

Example 2

Consider the following linearised model of the longitudinal dynamics of an aircraft in Heck and Ferri. ³⁶ The states are the angle of attack $\alpha$ , pitch rate q and elevator angle ${\delta }_e$ , and the control input u is the command to the elevator. The system with disturbances is expressed as

$$\begin{array}{l}\left[\begin{array}{c}\stackrel{.}{\alpha }\\ \stackrel{.}{q}\\ {\stackrel{.}{\delta }}_e\end{array}\right]=\left[\begin{array}{ccc}-0.277& 1& -0.0002\\ -17.1& -0.178& -12.2\\ 0& 0& -6.67\end{array}\right]\left[\begin{array}{c}\alpha \\ q\\ {\delta }_e\end{array}\right]\\ +\left[\begin{array}{c}0\\ 0\\ 6.67\end{array}\right](u+d_2)+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]d_1\end{array}$$

(36)

$$z=\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]\left[\begin{array}{c}\alpha \hfill \\ \multicolumn{1}{c}{q}\\ \multicolumn{1}{c}{{\delta }_e}\end{array}\right]$$

(37)

Thus, we have

$$\begin{array}{l}A=\left[\begin{array}{ccc}-0.277& 1& -0.0002\\ -17.1& -0.178& -12.2\\ 0& 0& -6.67\end{array}\right]\\ B_2=\left[\begin{array}{c}0\\ 0\\ 6.67\end{array}\right],B_1=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],C=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}$$

The parameters of system (4) are

$$W=\left[\begin{array}{cc}0\hfill & 5\hfill \\ \multicolumn{1}{c}{-5}& 0\hfill \end{array}\right],H=\left[\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{-1}\end{array}\right],V=\left[\begin{array}{cc}5\hfill & 0\hfill \end{array}\right]$$

By choosing $\gamma =0.5$ and solving the inequality in equation (12), we have a set of feasible solutions of the controller and observer gains as follows

$$\begin{array}{l}K=\left[\begin{array}{ccc}-3.0656& 4.1865& -5.5773\end{array}\right],\\ L=\left[\begin{array}{ccc}0& 0& -2.0285\\ 0& 0& -1.7007\end{array}\right]\end{array}$$

Choose the initial value as $x(0)=[\begin{array}{ccc}0\hfill & 2\hfill & 0\hfill \end{array}{]}^T$ In Figure 8, the response curves of the system states x are presented. Note that the proposed method can guarantee that the system has a good control performance in the presence of disturbances, while the $H_{\infty }$ control scheme does not have a satisfactory control performance. The system input signal is depicted in Figure 9, which illustrates that the designed composite anti-disturbance controller is feasible and effective. In order to present the effectiveness of the proposed method, the curves of the disturbances and disturbance estimation are described in Figure 10, which demonstrates that the disturbance observer can effectively estimate disturbances.

Figure 8.

Response curves of system states.

Figure 9.

Curve of output z.

Figure 10.

Curves of disturbances $d_2$ and disturbance estimation ${\hat{d}}_2$ .

Conclusion

In this article, asymptotic attenuation and rejection for a class of cascade systems with general periodic disturbances has been investigated via non-linear disturbance observer technique and $H_{\infty }$ control method. The general periodic disturbances have been modelled as linear uncertain systems with a non-linear output function. A novel non-linear disturbance observer has been proposed to estimate the general periodic disturbances. Then, by combining the disturbance estimation and conventional feedback control law, a composite anti-disturbance controller has been proposed. Finally, numerical and application examples have been employed to demonstrate the effectiveness of the proposed method.