Sampled-data active disturbance rejection output feedback control for systems with mismatched uncertainties

Abstract

This article investigates the sampled-data disturbance rejection control problem for a class of non-integral-chain systems with mismatched uncertainties. Aiming to reject the adverse effects caused by general mismatched uncertainties via digital control strategy, a new generalized discrete-time extended state observer is first proposed to estimate the lumped disturbances in the sampling point. A disturbance rejection control law is then constructed in a sampled-data form, which will lead to easier implementation in practices. By carefully selecting the control gains and a sampling period sufficiently small to restrain the state growth under a zero-order-holder input, the bounded-input bounded-output stability of the hybrid closed-loop system and the disturbance rejection ability are delicately proved even the controller is dormant within two neighbor sampling points. Numerical simulation results demonstrate the feasibility and efficacy of the proposed method.

Introduction

In this article, we consider a class of single-input single-output (SISO) system with mismatched uncertainties of the form

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{x}(t)=\mathrm{Ax}(t)+B_{u}u(t)+B_{d}f(x(t),\omega (t),t)}\\ \multicolumn{1}{c}{y_m(t)=C_{m}x(t)}\\ \multicolumn{1}{c}{y_0(t)=C_{0}x(t)}\end{array}$$

(1)

where $x(t)\in {\mathbb{R}}^n$ is the system state vector, $u(t)\in \mathbb{R}$ is the control input, $\omega (t)\in \mathbb{R}$ is the external disturbance, $y_m(t)\in {\mathbb{R}}^r$ is the measurable outputs, $y_0\in \mathbb{R}$ is the controlled output, and $A\in {\mathbb{R}}^{n\times n},B_u\in {\mathbb{R}}^{n\times 1}$ , $B_d\in {\mathbb{R}}^{n\times 1}$ , $C_m\in {\mathbb{R}}^{r\times n}$ , $C_0\in {\mathbb{R}}^{1\times n}$ are system matrices. $f(x(t),\omega (t),t)$ is an uncertain function representing the lumped disturbance in a general way which possibly includes external disturbances, unmodeled dynamics, parameter variations, and complex nonlinear dynamics.^1–8

In modern control practices, it is a trend that sampled-data controllers with a zero-order holder (ZOH) are being digitally implemented into real-life plants with the rapid development of computational hardware technology. It is common that conventional control law design is always devoted to continuous-time design for mathematically modeled continuous-time systems due to the convenience of direct stability analysis, typically based on the continuously differentiable Lyapunov function analysis. In most practical implementation processes, continuous-time controllers can be discretized directly which exists on the fact that the closed-loop system performance can always be guaranteed while the sampling frequency is fast enough. However, in most of the times, this is done without a theoretical support, and the formulated restrictions of the sampling frequency and how it affects the controlled system performance are actually unknown. Moreover, only digital sensors are available for some real-life plants, for instance, Global Positioning System (GPS) is a discrete-time sensor and a radar is more naturally represented using a discrete-time model. ⁹ Hence, it is always a challenging but crucial issue to address the problem of designing sampled-data controllers which pursue smaller steady-state error, faster dynamical response, and milder noise susceptibility for system perturbed by matching or mismatched uncertainties.

In the literature, one of the most popular methods among those existing approaches developed to design sampled-data controllers for nonlinear plants is the discretization method.^10–12 It employs a discrete-time approximation model of the plant (typically use Euler approximation) to design discrete-time controllers since it is almost impossible to obtain an exact discrete-time model of the continuous-time nonlinear plant. Hence, the results regarding this method are always achieved within a local or semi-global control goal. The emulation method, as a global control method, designs a continuous-time controller based on the continuous-time plant, followed by a discretization process to yield sampled-data controllers which will assure the continuous-time nonlinear plant by choosing an appropriate sampling period. One can refer to papers^13–19 and the references therein.

In practical engineering, the control performance of modern industrial systems is inevitably affected by various uncertainties including parameter variations, unmodeled dynamics, and external disturbances.^20–22 Active anti-disturbance technique is generally required in the controller design to access high-precision control performance.^1,23–25 The extended state observer–based control (ESOBC) was originally proposed by Han,^26,27 and it is made practical by the tuning method which simplifies its implementation and makes the design transparent to engineers. ¹ Su et al. ²⁸ presented the relationship between time-domain and frequency-domain disturbance observers and its applications for further information. However, most of existing disturbance rejection methods, including ESOBC, are concerned with continuous control approaches, which lack sound justification since most control approaches are digitally implemented in a sampled-data manner. To this end, the development of an active disturbance rejection method for system (1) with mismatched uncertainties will be of interest for both theoretical and industrial communities. Moreover, the mismatched uncertainties, rather than the so-called matching conditions^4,20 are concerned to discover a generalized sampled-data disturbance rejection control law for system (1). Mismatched uncertainties may not act via the same channel with the control input and are regarded as a more general case concerned in uncertainty attenuation problems. As an example, the lumped disturbance torques in flight control systems always affect the states directly rather than through the input channels.^23,29

This article presents a generalized sampled-data control law design based on a discrete-time extended state observer (ESO), which estimates the unmeasurable states and the lumped disturbance information in the sampling point. Explicit formulas to select the control gains and the tunable sampling period based on a detailed stability analysis for the hybrid closed-loop system are presented. Numerical simulations are shown to demonstrate the effectiveness of the proposed method. The proposed method will be a helpful guideline for direct digital implementation.

Main results

In this section, we present a step-by-step procedure to design a discrete-time ESO-based sampled-data control law to solve the global stabilization problem for system (1).

With an added extended variable

$$x_{n+1}(t)=d=f(x(t),\omega (t),t)$$

system (1) can be extended to the following form

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\overline{x}}(t)=\overline{A}\overline{x}(t)+{\overline{B}}_{u}u(t)+\mathrm{Eh}(t)}\\ \multicolumn{1}{c}{y_m(t)={\overline{C}}_m\overline{x}(t)}\end{array}$$

(2)

with the denotation of

$$\begin{array}{cc}\multicolumn{1}{c}{\overline{x}(t)}& ={[x^T(t),x_{n+1}(t)]}^T\hfill \\ \multicolumn{1}{c}{h(t)}& =\frac{\mathrm{df}(x(t),\omega (t),t)}{\mathrm{dt}}\hfill \\ \multicolumn{1}{c}{\overline{A}}& =\left(\begin{array}{cc}A\hfill & B_d\hfill \\ \multicolumn{1}{c}{0_{1\times n}}& 0\hfill \end{array}\right)\in {\mathbb{R}}^{(n+1)\times (n+1)}\hfill \\ \multicolumn{1}{c}{{\overline{B}}_u}& =\left(\begin{array}{c}{B}_u\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)\in {\mathbb{R}}^{(n+1)\times 1}\hfill \\ \multicolumn{1}{c}{E}& =[0,\dots ,0,1{]}^T\in {\mathbb{R}}^{(n+1)\times 1}\hfill \\ \multicolumn{1}{c}{{\overline{C}}_m}& =[C_m,0_{r\times 1}]\in {\mathbb{R}}^{r\times (n+1)}\hfill \end{array}$$

Assumption 1

$(A,B_u)$ is controllable and $(\overline{A},{\overline{C}}_m)$ is observable.

Assumption 2

The lumped disturbances satisfy the following conditions: ²⁰

• $d(t)$ and $h(t)$ are bounded, that is, there exist two positive constants $\overline{d},\overline{h}$ such that $|d(t)|\le \overline{d},|h(t)|\le \overline{h}$ , respectively

$$\underset{t\to \infty }{\lim }\stackrel{·}{d}(t)=\underset{t\to \infty }{\lim }h(t)=0$$

In the article, the Assumption 2 is made on the disturbances $d(t)$ that its derivative is close to zero. However, this assumption can be relaxed using unknown observer theory.^30,31

Construction of discrete-time ESO

In what follows, an ESO for system (1) will be built using the sampled-data information $y_m(t_k)$ $(t_k=\mathrm{kT},k=0,1,2,\dots )$ , where T is the sampling period. The continuous-time state $\hat{\overline{x}}(t)\stackrel{\mathrm{\Delta }}{=}[{\hat{x}}^T,{\hat{x}}_{n+1}{]}^T$ defined in the time region $t\in [t_k,t_{k+1})$

$$\stackrel{·}{\hat{\overline{x}}}(t)=\overline{A}\hat{\overline{x}}(t)+{\overline{B}}_{u}u(t)-L({\hat{y}}_m(t)-y_m(t_k)),t\in [t_k,t_{k+1})$$

(3)

where ${\hat{y}}_m(t)={\overline{C}}_m\hat{\overline{x}}(t)$ and $L\in {\mathbb{R}}^{(n+1)\times r}$ is the observer gain matrix to be assigned later.

The observer (3) can be rewritten as follows

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\hat{\overline{x}}}(t)=\overline{A}\hat{\overline{x}}(t)+{\overline{B}}_{u}u(t_k)-L({\hat{y}}_m(t)-y_m(t_k))}\\ \multicolumn{1}{c}{=(\overline{A}-L{\overline{C}}_m)\hat{\overline{x}}(t)+{\overline{B}}_{u}u(t_k)+L{y}_m(t_k),t\in [t_k,t_{k+1})}\end{array}$$

(4)

Integrating the continuous-time observer (4) from $t_k$ to $t_{k+1}$ , it can be concluded that $\hat{\overline{x}}(t_{k+1})$ can also be generated by the following discrete-time ESO

$$\begin{array}{c}\widehat{\overline{x}}(t_{k+1})=e^{\left(\overline{A}-L{\overline{C}}_m\right)T}\widehat{\overline{x}}(t_k)+{\displaystyle \underset{0}{\overset{T}{\int }}{e}^{\left(\overline{A}-L{\overline{C}}_m\right)s}ds}\left({\overline{B}}_{u}u(t_k)+L{y}_m(t_k)\right)\\ \triangleq F\widehat{\overline{x}}(t_k)+Gu(t_k)+N{y}_m(t_k)\end{array}$$

(5)

with the denotation of

$$\begin{array}{c}F=e^{(\overline{A}-L{\overline{C}}_m)T}\hfill \\ G=\underset{0}{\overset{T}{\int }}{e}^{(\overline{A}-L{\overline{C}}_m)s}\mathrm{ds}{\overline{B}}_u\hfill \\ N=\underset{0}{\overset{T}{\int }}{e}^{(\overline{A}-L{\overline{C}}_m)s}\mathrm{dsL}\hfill \end{array}$$

The state and disturbances errors are defined as $e(t)=[e_x^T(t),e_d(t){]}^T$ where $e_x(t)=\hat{x}(t)-x(t)$ , $e_d(t)=\hat{d}(t)-d(t)$ , respectively. With equations (2) and (3), the error dynamics are

$$\stackrel{·}{e}(t)=(\overline{A}-L{\overline{C}}_m)e(t)-\mathrm{Eh}(t)-L{\overline{C}}_m(\overline{x}(t)-\overline{x}(t_k)),t\in [t_k,t_{k+1})$$

(6)

Sampled-data disturbance rejection law design

The sampled-data disturbance rejection law can be designed as

$$u(t)=u(t_k)=K\hat{\overline{x}}(t_k)=K_x\hat{x}(t_k)+K_d\hat{d}(t_k),t\in [t_k,t_{k+1})$$

(7)

where $K=[K_x,K_d]$ . $K_x$ is the control gain to be designed. $K_d$ is the disturbance compensation gain. Motivated by the results, ²⁰ $K_d$ is assigned by the following equation

$$K_d=-{\left[C_0{(A+B_u{K}_x)}^{-1}{B}_u\right]}^{-1}{C}_0{(A+B_u{K}_x)}^{-1}{B}_d$$

Remark 1

With equation (7) in mind, the designed discrete-time extended observer can also be presented as

$$\hat{\overline{x}}(t_{k+1})=M\hat{\overline{x}}(t_k)+N{y}_m(t_k)$$

(8)

where $M=F-\mathrm{GK}\in {\mathbb{R}}^{(n+1)\times (n+1)}$ and N are two matrices dependent of the sampling period T.

Hybrid closed-loop system stability analysis

Combing equations (1), (6), and (7) together, one can obtain the hybrid closed-loop system as

$$\begin{array}{c}\left[\begin{array}{c}\stackrel{·}{x}(t)\hfill \\ \multicolumn{1}{c}{\stackrel{·}{e}(t)}\end{array}\right]=\left[\begin{array}{cc}A+B_u{K}_x\hfill & B_{u}K\hfill \\ \multicolumn{1}{c}{0}& \overline{A}-L{\overline{C}}_m\hfill \end{array}\right]\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{e(t)}\end{array}\right]\hfill \\ \multicolumn{1}{c}{+\left[\begin{array}{cc}0\hfill & B_u{K}_d+B_d\hfill \\ \multicolumn{1}{c}{-E}& 0\hfill \end{array}\right]\left[\begin{array}{c}h(t)\hfill \\ \multicolumn{1}{c}{d(t)}\end{array}\right]+\left[\begin{array}{cc}-B_{u}K\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -L{\overline{C}}_m\hfill \end{array}\right]}\\ \multicolumn{1}{c}{\left[\begin{array}{c}\hat{\overline{x}}(t)-\hat{\overline{x}}(t_k)\hfill \\ \multicolumn{1}{c}{\overline{x}(t)-\overline{x}(t_k)}\end{array}\right],t\in [t_k,t_{k+1})}\end{array}$$

(9)

If we choose the observer gain L and the feedback gain $K_x$ such that $A+B_u{K}_x$ and $\overline{A}-L{\overline{C}}_m$ are Hurwitz matrices, it is easy to verify that the following matrix

$$\left[\begin{array}{cc}A+B_u{K}_x\hfill & B_{u}K\hfill \\ \multicolumn{1}{c}{0}& \overline{A}-L{\overline{C}}_m\hfill \end{array}\right]\stackrel{\mathrm{\Delta }}{=}\mathrm{\Lambda }$$

is also a Hurwitz one. Hence, there exists a positive definite matrix $P=P^T\in {\mathbb{R}}^{(2n+1)\times (2n+1)}$ such that

$${\mathrm{\Lambda }}^{T}P+P\mathrm{\Lambda }=-\alpha I$$

where $\alpha >0$ is a constant will be determined later.

Construct a positive definite and proper Lyapunov function $V(Z(t))=Z^T(t)\mathrm{PZ}(t)$ where $Z(t)=[x^T(t),e^T(t){]}^T$ . The derivative of $V(Z(t))$ along system (9) is given as follows

$$\begin{array}{c}\stackrel{·}{V}(Z(t))=-\alpha \parallel Z(t){\parallel }^2+2Z(t{)}^T\hfill \\ \multicolumn{1}{c}{P\left[\begin{array}{cc}0\hfill & B_u{K}_d+B_d\hfill \\ \multicolumn{1}{c}{-E}& 0\hfill \end{array}\right]\left[\begin{array}{c}h(t)\hfill \\ \multicolumn{1}{c}{d(t)}\end{array}\right]}\\ +2Z(t{)}^{T}P\left[\begin{array}{cc}-B_{u}K\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -L{\overline{C}}_m\hfill \end{array}\right]\left[\begin{array}{c}\hat{\overline{x}}(t)-\hat{\overline{x}}(t_k)\hfill \\ \multicolumn{1}{c}{\overline{x}(t)-\overline{x}(t_k)}\end{array}\right],\hfill \\ \multicolumn{1}{c}{t\in [t_k,t_{k+1})}\end{array}$$

(10)

Now, we will estimate the items in the right hand side of equation (10). First, with $|h(t)|\le \overline{h}$ , $\parallel P\parallel ={\lambda }_{\max }(P),\parallel E\parallel =1$ in mind, we have

$$\begin{array}{c}\multicolumn{1}{c}{2\left|Z{(t)}^{T}P\left[\begin{array}{cc}0\hfill & B_u{K}_d+B_d\hfill \\ \multicolumn{1}{c}{-E}& 0\hfill \end{array}\right]\left[\begin{array}{c}h(t)\hfill \\ \multicolumn{1}{c}{d(t)}\end{array}\right]\right|}\\ \multicolumn{1}{c}{\le 2{\overline{c}}_1\parallel P\parallel (\overline{h}+\overline{d})\parallel Z(t)\parallel }\\ \multicolumn{1}{c}{=c_1(\overline{h}+\overline{d})\parallel Z(t)\parallel }\end{array}$$

(11)

where ${\overline{c}}_1>0$ and $c_1=2{\overline{c}}_1{\lambda }_{\max }(P)$ are constants.

Since $K,L$ are set, one can conclude that

$$\Vert \left[\begin{array}{cc}-B_{u}K\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -L{\overline{C}}_m\hfill \end{array}\right]\Vert \le {\overline{c}}_2$$

(12)

where ${\overline{c}}_2>0$ is a constant dependent on the value of $K,L$ .

Similar to equation (11), there exists a constant $c_2>0$ , such that the following estimation holds

$$\begin{array}{c}2\left|Z{(t)}^{T}P\left[\begin{array}{cc}-B_{u}K& 0\\ 0& -L{\overline{C}}_m\end{array}\right]\left[\begin{array}{c}\widehat{\overline{x}}(t)-\widehat{\overline{x}}(t_k)\\ \overline{x}(t)-\overline{x}(t_k)\end{array}\right]\right|\\ \le 2{\overline{c}}_2\parallel Z(t)\parallel \cdot \parallel P\parallel \cdot \Vert \left[\begin{array}{c}e(t)-e(t_k)+\overline{x}(t)-\overline{x}(t_k)\\ \overline{x}(t)-\overline{x}(t_k)\end{array}\right]\Vert \\ \le 4{\overline{c}}_2{\lambda }_{\max }\left\{P\right\}\parallel Z(t)\parallel \left(\parallel Z(t)-Z(t_k)\parallel +\parallel d(t)-d(t_k)\parallel \right)\\ \le c_2\parallel Z(t)\parallel \parallel Z(t)-Z(t_k)\parallel +c_2\overline{d}\parallel Z(t)\parallel \mathrm{,}t\in [t_k\mathrm{,}{t}_{k+1})\end{array}$$

(13)

Substituting equations (11) and (13) into equation (10) yields

$$\begin{array}{c}\stackrel{·}{V}(Z(t))\le -\alpha \parallel Z(t){\parallel }^2+(c_1\overline{h}+c_1\overline{d}+c_2\overline{d})\parallel Z(t)\parallel \hfill \\ +c_2\parallel Z(t)\parallel \parallel Z(t)-Z(t_k)\parallel ,t\in [t_k,t_{k+1})\hfill \end{array}$$

(14)

Next, we introduce a lemma which plays a key role in the main theorem.

Lemma 1

There exist proper constants $c_3>0$ , $c_4>0$ , such that the following inequality holds

$$\parallel Z(t)-Z(t_k)\parallel \le c_4(\sqrt{V(Z(t_k))}+\overline{h}+\overline{d})(e^{c_3(t-\mathrm{kT})}-1),t\in [t_k,t_{k+1})$$

(15)

Proof 1

Denote the whole closed-loop system (9) to be $\stackrel{·}{Z}(t)=\mathrm{\Psi }(u(t_k),Z(t))$ . With the fact that $h(t)$ and $d(t)$ are bounded in mind, the following inequality holds for $s\in [t_k,t)$

$$\parallel \mathrm{\Psi }(u(t_k),Z(s))\parallel \le c_3(\parallel Z(s)-Z(t_k)\parallel +\parallel Z(t_k)\parallel +\overline{h}+\overline{d})$$

(16)

where $c_3>0$ is a constant. By equation (16), one can obtain that

$$\begin{array}{c}\multicolumn{1}{c}{\parallel Z(t)-Z(t_k)\parallel \le \underset{t_k}{\overset{t}{\int }}\parallel \mathrm{\Psi }(u(t_k),Z(s))\parallel \mathrm{ds}}\\ \multicolumn{1}{c}{\le c_3(\parallel Z(t_k)\parallel +\overline{h}+\overline{d})(t-\mathrm{kT})+}\\ \multicolumn{1}{c}{\underset{t_k}{\overset{t}{\int }}{c}_3\parallel Z(s)-Z(t_k)\parallel \mathrm{ds},t\in [t_k,t_{k+1})}\end{array}$$

(17)

Based on equation (17), applying the Gronwall inequality, we have

$$\begin{array}{c}\parallel Z(t)-Z(t_k)\parallel \le c_3(\parallel Z(t_k)\parallel +\overline{h}+\overline{d})(t-\mathrm{kT})\hfill \\ +(c_3^2\parallel Z(t_k)\parallel +c_3^2\overline{h}+c_3^2\overline{d})\underset{t_k}{\overset{t}{\int }}(s-\mathrm{kT})e^{c_3(t-s)}\mathrm{ds}\hfill \\ \le c_4(\sqrt{V(Z(t_k))}+\overline{h}+\overline{d})(e^{c_3(t-\mathrm{kT})}-1),t\in [t_k,t_{k+1})\hfill \end{array}$$

(18)

where $c_4>0$ is a constant.

With Lemma 1 in mind, equation (14) becomes

$$\begin{array}{l}\stackrel{.}{V}\left(Z(t)\right)\le -\alpha \parallel Z(t){\parallel }^2+\left(c_1\overline{h}+c_1\overline{d}+c_2\overline{d}\right)\parallel Z(t)\parallel \\ +c_2{c}_4\parallel Z(t)\parallel \left(\sqrt{V\left(Z(t_k)\right)}+\overline{h}+\overline{d}\right)\left(e^{c_3(t-kT)}-1\right)\\ \le -\left(\frac{\alpha }{\sqrt{{\lambda }_{\min }\left\{P\right\}}}-c_5\right)V\left(Z(t)\right)\\ +c_6\sqrt{V\left(Z(t)\right)}\sqrt{V\left(Z(t_k)\right)}\left(e^{c_3(t-kT)}-1\right)\\ +\left(c_5+c_6\left(e^{c_3(t-kT)}-1\right)\right)(\overline{h}+\overline{d})\sqrt{V\left(Z(t)\right)},t\in [t_k\mathrm{,}{t}_{k+1})\end{array}$$

(19)

with two positive constants $c_5,c_6>0$ .

Theorem 1

Under Assumptions 1 and 2, with the following selection formulas for the parameter $\alpha$ and the allowable sampling period T

$$\frac{\alpha }{\sqrt{{\lambda }_{\min }\{P\}}}\ge c_5+1;T<\frac{1}{c_3}\ln (1+\frac{1}{c_6})$$

(20)

and the observer gain L and the feedback control gain $K_x$ are selected such that $A+B_u{K}_x$ and $\overline{A}-L{\overline{C}}_m$ are Hurwitz matrices; system (1) under the proposed sampled-data control law (5)–(7) can be rendered bounded stable for any bounded $d(t),h(t)$ .

Proof

Define $\xi (t)=\sqrt{V(Z(t))}/\sqrt{V(Z(t_k))}$ . It can be concluded that equation (19) equals to the following inequality

$$\begin{array}{c}\stackrel{·}{\xi }(t)\le -\frac{1}{2}(\frac{\alpha }{\sqrt{{\lambda }_{\min }\{P\}}}-c_5)\xi (t)+\frac{c_6}{2}(e^{c_3(t-\mathrm{kT})}-1)\hfill \\ +\frac{1}{2}(c_5+c_6(e^{c_3(t-\mathrm{kT})}-1))\frac{\overline{h}+\overline{d}}{\sqrt{V(Z(t_k))}},\hfill \\ \multicolumn{1}{c}{t\in [t_k,t_{k+1})}\end{array}$$

(21)

With equation (20) in mind, the following inequality can be concluded

$$\stackrel{·}{\xi }(t)\le -\frac{1}{2}\xi (t)+\frac{c_6}{2}(e^{c_{3}T}-1)+\frac{1}{2}(c_5+c_6(e^{c_{3}T}-1))\frac{\overline{h}+\overline{d}}{\sqrt{V(Z(t_k))}},t\in [t_k,t_{k+1})$$

(22)

Solving the above inequality with $\xi (t_k)=1$ , we have

$$\xi (t_{k+1})\le \left(c_6\left(e^{c_{3}T}-1\right)+\left(c_5+c_6\left(e^{c_{3}T}-1\right)\right)\frac{\overline{h}+\overline{d}}{\sqrt{V\left(Z(t_k)\right)}}\right)\left(1-e^{\frac{-T}{2}}\right)+e^{\frac{-T}{2}}\triangleq \rho (T)$$

(23)

which leads to

$$V(Z(t_{k+1}))\le \rho (T)V(Z(t_k))$$

(24)

According to the condition (20), we have $1\ne c_6(e^{c_{3}T}-1)$ . Hence, the difference equation $\mathrm{\Delta }V(Z)=V(Z(t_{k+1}))-V(Z(t_k))\le 0$ if

$$V(Z(t_k))\ge {\left(\frac{(c_5+c_6(e^{c_{3}T}-1))(\overline{h}+\overline{d})}{1-c_6(e^{c_{3}T}-1)}\right)}^2$$

(25)

which implies that $Z(t_k)$ decreases for any $Z(t_k)$ satisfies condition (25); hence, the state $Z(t_k)$ is bounded for any bounded $d(t)$ and $h(t)$ if those parameters are properly selected. This completes the proof of Theorem 1.

Remark 2

As one can mention, the discrete-time observer (8) will generate the same sampled information $\hat{\overline{x}}(\mathrm{kT})$ at the sampling point for the same initial condition with the continuous-time observer (3). Hence, in the design, we could use the continuous-time form (3) to take fully advantages of the continuous-time design method.^16,18 As a matter of fact, we do not need to consider the signal information for $t\in (\mathrm{kT},(k+1)T)$ for the observer (3), what we need is only the sampled information $\hat{\overline{x}}(\mathrm{kT})$ which can be generated from the sampled output information $y(\mathrm{kT}),k=0,1,\dots$ . As one can observe from the proof of Theorem 1 that even the controller is dormant within two neighbor sampling points and the inter-sample information is not analyzed, the fact that the difference Lyapunov function $\mathrm{\Delta }V(Z)$ is semi-negative definite can assure the bounded-input bounded-output stability of the hybrid closed-loop system.

Disturbance rejection analysis

In what follows, we will give a detailed disturbance rejection analysis to show the proposed ESO-based sampled-data disturbance rejection control law will effectively reject the lumped disturbances.

Theorem 2

Under Assumptions 1 and 2, the lumped disturbances from system (1) can be effectively rejected from the output channel in steady-state under the proposed sampled-data control law (7) provided that the observer gain L and the feedback control gain $K_x$ are selected such that $A+B_u{K}_x$ , $\overline{A}-L{\overline{C}}_m$ are Hurwitz matrices and $C_0(A+B_u{K}_x{)}^{-1}{B}_u$ is invertible.

Proof

Substituting the sampled-data controller (7) into system (1) yields

$$\begin{array}{c}\stackrel{·}{x}(t)=\mathrm{Ax}(t)+B_u{K}_x{e}_x(t_k)+B_u{K}_{x}x(t_k)\hfill \\ +B_d(d(t)-d(t_k))-B_d{e}_d(t_k),t\in [t_k,t_{k+1})\hfill \end{array}$$

(26)

By Theorem 1, the following relations hold for the hybrid closed-loop system (1)–(7)–(8)

$$\underset{t\to \infty }{\lim }\stackrel{·}{x}(t)=\underset{k\to \infty }{\lim }\stackrel{·}{x}(t_k)=0,\underset{k\to \infty }{\lim }e(t_k)=0$$

(27)

Hence, one can conclude from equations (26) and (27) that

$$\begin{array}{c}\underset{t\to \infty }{\lim }{y}_0(t)=\underset{k\to \infty }{\lim }{y}_0(t_k)=C_0{(A+B_u{K}_x)}^{-1}\stackrel{·}{x}(t_k)-C_0{(A+B_u{K}_x)}^{-1}\times (B_u{K}_x{e}_x(t_k)+B_d(d(t_k)-d(t_k))-B_d{e}_d(t_k))=0\hfill \end{array}$$

(28)

which concludes that the lumped disturbances can be effectively rejected from the output channel in steady-state.

Remark 3

In this article, we assume some states of the systems are not measurable; hence, the proposed sampled-data control strategy has to be implemented by output feedback of the form of equation (7). If we consider the case that the states are available or can be easily measured, the sampled-data state-feedback control law can be simply designed as

$$\begin{array}{c}\multicolumn{1}{c}{u(t)=u(t_k)=K_{x}x(t_k)+K_d\hat{d}(t_k),t\in [t_k,t_{k+1})}\\ \multicolumn{1}{c}{K_d=-{\left[C_0{(A+B_u{K}_x)}^{-1}{B}_u\right]}^{-1}{C}_0{(A+B_u{K}_x)}^{-1}{B}_d}\end{array}$$

(29)

where $\hat{d}(t_k)$ can be generated from the discrete-time ESO (8).

Remark 4

Figure 1 depicts the implementation structure of practical plant with the proposed ESO-based sampled-data disturbance rejection controller (7) and (8). In practical engineering, the dominated dynamics of a system has been stabilized by feedback control, and the nonlinear character of those uncertainties contained in the system are relatively weak, which means the presence of the uncertainties will not cause much damage to the closed-loop system’s stability. Hence, such uncertainties can be regarded as part of the lumped disturbances and can be reasonably handled by sampled-data control law of the form (7). This fact will support the general availability of the proposed control method.

Figure 1.

Implementation structure of extended state observer–based sampled-data controller.

Example and simulation results

Next, we use an example and numerical simulations to show how an ESOBC law is designed and the effectiveness of the proposed method.

A two-dimensional example

Consider a two-dimensional uncertain nonlinear system with mismatching condition of the form

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{x}}_1(t)=x_2(t)+f(x(t),\omega (t),t)}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2(t)=-2x_1(t)-x_2(t)+u(t)}\\ \multicolumn{1}{c}{y(t)=x_1(t)}\end{array}$$

(30)

which can also be written as the state-space form

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{x}(t)=\mathrm{Ax}(t)+B_{u}u(t)+B_{d}d(t)}\\ \multicolumn{1}{c}{y(t)=\mathrm{Cx}(t)}\end{array}$$

(31)

with the denotation of $A=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{-2}& -1\hfill \end{array}\right]$ , $B_u=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$ , $B_d=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$ , $C_m=C_0=C=[1,0]$ , and $d(t)=f(x_1(t),x_2(t),\omega (t),t)$ .

Clearly, system (30) can be seen as an example of system (1) and can satisfy Assumptions 1 and 2. Let $d(t)=x_3(t)$ , one can obtain an extended system as

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\overline{x}}(t)=\overline{A}\overline{x}(t)+\overline{B}u(t)+\mathrm{Eh}(t)}\\ \multicolumn{1}{c}{y=\overline{C}\overline{x}(t)}\end{array}$$

(32)

where $\overline{A}=\left[\begin{array}{ccc}0\hfill & 1\hfill & 1\hfill \\ \multicolumn{1}{c}{-2}& -1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]$ , ${\overline{B}}_u=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{0}\end{array}\right]$ , $\overline{C}=[1,0,0]$ , and $E=[0,0,1{]}^T$ . One can construct a discrete-time ESO-based sampled-data output feedback controller of the following form

$$\hat{\overline{x}}\left(t_{k+1}\right)=M\hat{\overline{x}}(t_k)+\mathrm{Ny}(t_k)$$

(33)

$$u(t)=K\hat{\overline{x}}(t_k),t\in [t_k,t_{k+1})$$

(34)

where $\widehat{\overline{x}}={[{\widehat{x}}_1,{\widehat{x}}_2,\widehat{d}]}^T$ , $K=[k_1,k_2,k_3]$ , $M=e^{(\overline{A}-L\overline{C})T}+{\int }_0^T{e}^{(\overline{A}-L\overline{C})s}\mathrm{ds}{\overline{B}}_{u}K$ , and $N={\int }_0^T{e}^{L(\overline{A}-L\overline{C})s}\mathrm{dsL}$ .

Numerical simulations

In what follows, we will use numerical simulations for system (30) to demonstrate the effectiveness of the proposed method.

The disturbance is considered as $d(t)=e^{x_1(t)}+\omega$ , where $\omega$ represents the external disturbance assigned as $\omega =2$ acting on the system at the time instant $t=5\mathrm{s}$ , the observer gain vector L is selected as $L={[14,-66,125]}^T$ , and the controller gain $K=[-4,-4,-5]$ based on the guideline (7). Hence, in this case, we can verify that $A+B_u{K}_x$ and $\overline{A}-L\overline{C}$ are both Hurwitz. By Theorem 1, we can choose a proper sampling period $T=0.05\mathrm{s}$ . The discrete-time observer matrices can be calculated as

$$M=\left(\begin{array}{ccc}0.5286\hfill & 1.0833\hfill & 1.0913\hfill \\ \multicolumn{1}{c}{26.3854}& 2.8041\hfill & 3.3161\hfill \\ \multicolumn{1}{c}{-5.6081}& -4.6100\hfill & -6.0125\hfill \end{array}\right),N=\left(\begin{array}{c}0.5278\hfill \\ \multicolumn{1}{c}{-437.8961}\\ \multicolumn{1}{c}{500.2623}\end{array}\right)$$

Figures 2–4 show the response curves of the system and estimated states. One can observe from Figures 2 and 3 that the system states converge to the equilibrium quickly in the presence of both internal uncertainties and external disturbances. It is illustrated by Figure 2 that the lumped disturbance can be successfully rejected in the output channel. And from Figure 4, it is shown that the discrete-time ESO works effectively and leads to high-precision observation of the disturbance $d(t)$ . The time history of the sampled-data output feedback control law (34) is shown in Figure 5.

Figure 2.

Trajectories of $x_1(t)$ and ${\hat{x}}_1(t_k)$ of the closed-loop system (30)–(33)–(34).

Figure 3.

Trajectories of $x_2(t)$ and ${\hat{x}}_2(t_k)$ of the closed-loop system (30)–(33)–(34).

Figure 4.

Trajectories of the disturbance $d(t)$ and estimated disturbance $\hat{d}(t_k)$ from the ESO (33).

Figure 5.

Time history of the sampled-data control input signal $u(t)$ .

Conclusion

A novel discrete-time ESO-based sampled-data active disturbance rejection output feedback control law has been proposed in this article to achieve high-precision control performances for a class of systems with mismatched uncertainties. With a delicate design procedure, the careful selection of the involved parameters ensures the global stability of the hybrid closed-loop system and disturbance rejection ability. Direct digital design strategy will lead to easier implementation. Numerical simulations have shown the effectiveness of the proposed method.