Asymptotic Tracking Control for Mismatched Uncertain Systems with Active Disturbance Rejection

Abstract

By introducing a set of exact disturbance estimators, a continuously tracking controller for a class of mismatched uncertain systems with exogenous disturbances will be proposed. The most appealing superiority is that the proposed exact disturbance estimators can not only estimate the external disturbances but also achieve an asymptotic estimation performance. Furthermore, with the help of a set of first-order asymptotic filters and an auxiliary system, the developed control algorithm is able to compensate for these total disturbances feedforwardly. Consequently, the whole closed-loop stability with an asymptotic tracking performance is strictly analyzed, and meanwhile applications are conducted to indicate the effectiveness of the proposed controller.

1. Introduction

Disturbances extensively exist in all practical systems, which may cause critical control performance degradation and even instability in developing high-performance closed-loop controllers [1,2,3]. Over the past decades, many advanced control algorithms such as adaptive robust control [4], robust adaptive control [5,6], sliding mode control [7,8] and so on have been proposed for various nonlinear systems to cope with modeling uncertainties. Additionally, many studies focus on rejecting disturbances by combining with disturbance observers.

Currently, there has been a growing interest in disturbance-observer-based control strategies with an active disturbance rejection ability for uncertain nonlinear systems [9,10,11,12,13,14,15,16,17,18]. And the main concept of these control strategies is to estimate the disturbances via different disturbance observers and thus compensate for them feedforwardly in developing the closed-loop controllers. Typically, Chen et al., proposed a nonlinear disturbance observer (NDOB) for nonlinear systems with disturbances governed by the exogenous system to estimate the total disturbances in an exponentially convergent rate [19]. It is worth noting that NDOB has been successfully applied to various practical systems [20,21]. Moreover, Won et al., have proposed a high-gain-disturbance-observer-based controller for hydraulic systems to improve the output tracking performance and meanwhile constrain the output tracking error [14]. Furthermore, Han has developed an active disturbance rejection controller (ADRC) for uncertain systems [22]. And the main support for ADRC is the extended state observer (ESO) [12,23,24]. Moreover, there are still some other disturbance observers [25,26,27] have been proposed. Notably, the aforementioned disturbance observers can only achieve a bounded estimation performance. How to develop exact disturbance estimators and meanwhile acquire an asymptotic tracking performance, especially for mismatched nonlinear systems, is extremely important and challenging in designing high-performance closed-loop controllers.

Inspired by the above discussions, we will propose an asymptotic tracking controller for systems with matched and mismatched exogenous disturbances, which is of great significance both in theory and practice. Especially, the main contributions of this paper are shown as follows:

(1)

A set of exact disturbance estimators (EDEs) with optimized design parameters which can acquire an asymptotic estimation performance is proposed;

(2)

Both mismatched and matched exogenous disturbances can be effectively compensated, and meanwhile an asymptotic tracking performance can be acquired.

Notation: 

$\widehat{•}$ is the estimate of $•$ with $\widetilde{•}=•-\widehat{•}$ being the estimation error; and sgn(•) is the signum function. In addition, the variable i = 1,…, n with n being the system order; the variable j = 1,…, n–1; and the variable θ = 2,…, n–1.

2. Problem Formulation

A class of nonlinear systems is employed as

$$\left\{\begin{array}{l}{\dot{\phi }}_j={\phi }_{j+1}+{\psi }_j({\overline{\phi }}_j)+f_j(t)\\ {\dot{\phi }}_n=g({\overline{\phi }}_n)u+{\psi }_n({\overline{\phi }}_n)+f_n(t)\\ y={\phi }_1\end{array}\right.$$

(1)

where ${\overline{\phi }}_i$ = [φ₁,…, φ_i]^T∈Rⁱ with φ_i being the system states; $g({\overline{\phi }}_n)$ and ${\psi }_i({\overline{\phi }}_i)$ are known nonlinear functions, especially, $g({\overline{\phi }}_n)>0$; $f_i(t)$ denote unknown time-varying functions which can describe external disturbances; moreover, u and y are the control input and output, respectively.

Given a desired trajectory φ_1d, the control objective is to design a continuously disturbance-compensation-based control law u for (1) so that y = φ₁ can asymptotically track φ_1d.

Assumption 1. 

The desired trajectory φ_1d∈Ϲ¹.

Assumption 2. 

${\dot{f}}_i(t)$ exist and meanwhile are bounded.

Remark 1. 

Notably, not only mismatched external disturbances f_j(t) but also matched external disturbances f_n(t) are simultaneously considered in Model (1), which means that it can be applied to many practical systems, such as motor servo systems [28,29] and so on.

3. Asymptotic Tracking Controller with Active Disturbance Rejection

3.1. Exact Disturbance Estimator

Inspired by the ESO, we extend the external disturbances f_i(t) as new state variables φ_fi. According to Assumption 2, we can define ${\dot{f}}_i(t)=d_i(t)$ and then transform the system (1) as

$$\begin{array}{l}\left\{\begin{array}{l}{\dot{\phi }}_j={\phi }_{j+1}+{\psi }_j({\overline{\phi }}_j)+{\phi }_{fj}\\ {\dot{\phi }}_{fj}=d_j(t)\end{array}\right.\\ \left\{\begin{array}{l}{\dot{\phi }}_n=g({\overline{\phi }}_n)u+{\psi }_n({\overline{\phi }}_n)+{\phi }_{fn}\\ {\dot{\phi }}_{fn}=d_n(t)\end{array}\right.\end{array}$$

(2)

Stimulated by [30,31], a set of novel EDEs can be proposed as [32]

$$\begin{array}{l}\left\{\begin{array}{l}{\dot{\widehat{\phi }}}_j={\phi }_{j+1}+{\psi }_j({\overline{\phi }}_j)+{\widehat{\phi }}_{fj}+L_{j1}({\phi }_j-{\widehat{\phi }}_j)\\ {\dot{\widehat{\phi }}}_{fj}=L_{j2}({\phi }_j-{\widehat{\phi }}_j)+{\widehat{\alpha }}_j\mathrm{sgn}({\phi }_j-{\widehat{\phi }}_j)\end{array}\right.\\ \left\{\begin{array}{l}{\widehat{\phi }}_n=g({\overline{\phi }}_n)u+{\psi }_n({\overline{\phi }}_n)+{\widehat{\phi }}_{fn}+L_{n1}({\phi }_n-{\widehat{\phi }}_n)\\ {\dot{\widehat{\phi }}}_{fn}=L_{n2}({\phi }_n-{\widehat{\phi }}_n)+{\widehat{\alpha }}_n\mathrm{sgn}({\phi }_n-{\widehat{\phi }}_n)\end{array}\right.\end{array}$$

(3)

where L_i1 and L_i2 are adjustable positive design parameters; in addition, ${\widehat{\alpha }}_i$ are the estimates of α_i which satisfy (23) which will be introduced later.

As conducted in [23], we can parameterize L_i1 and L_i2 as 2β_oi and ${\beta }_{oi}^2$, respectively, with β_oi being adjustable positive design parameters. Therefore, (3) can be rearranged as

$$\begin{array}{l}\left\{\begin{array}{l}{\dot{\widehat{\phi }}}_j={\phi }_{j+1}+{\psi }_j({\overline{\phi }}_j)+{\widehat{\phi }}_{fj}+2{\beta }_{oj}({\phi }_j-{\widehat{\phi }}_j)\\ {\dot{\widehat{\phi }}}_{fj}={\beta }_{oj}^2({\phi }_j-{\widehat{\phi }}_j)+{\widehat{\alpha }}_j\mathrm{sgn}({\phi }_j-{\widehat{\phi }}_j)\end{array}\right.\\ \left\{\begin{array}{l}{\widehat{\phi }}_n=g({\overline{\phi }}_n)u+{\psi }_n({\overline{\phi }}_n)+{\widehat{\phi }}_{fn}+2{\beta }_{on}({\phi }_n-{\widehat{\phi }}_n)\\ {\dot{\widehat{\phi }}}_{fn}={\beta }_{on}^2({\phi }_n-{\widehat{\phi }}_n)+{\widehat{\alpha }}_n\mathrm{sgn}({\phi }_n-{\widehat{\phi }}_n)\end{array}\right.\end{array}$$

(4)

Especially, ${\widehat{\alpha }}_i$ can be updated via

$${\dot{\widehat{\alpha }}}_i=\frac{{\lambda }_i}{{\beta }_{oi}}{\eta }_i^T{F}_o{C}_o\mathrm{sgn}({\eta }_{i1})$$

(5)

where λ_i are positive design parameters and η_i = [η_i1, η_i2]^T= ${[{\tilde{\phi }}_i,{\tilde{\phi }}_{fi}/{\beta }_{oi}]}^T$ are new vectors. Due to incalculable variables existing in (5), we can obtain ${\widehat{\alpha }}_i$ via the following method:

$${\widehat{\alpha }}_i(t)={\widehat{\alpha }}_i(0)+\frac{1}{2}\frac{{\lambda }_i}{{\beta }_{oi}}\left[\frac{3}{{\beta }_{oi}}\left|{\eta }_{i1}(t)\right|-\frac{3}{{\beta }_{oi}}\left|{\eta }_{i1}(0)\right|+5{\displaystyle {\int }_0^t\left|{\eta }_{i1}(\upsilon )\right|d\upsilon }\right]$$

(6)

Noting (2) and (4), we can obtain

$$\left\{\begin{array}{l}{\dot{\tilde{\phi }}}_i={\tilde{\phi }}_{fi}-2{\beta }_{oi}{\tilde{\phi }}_i\\ {\dot{\tilde{\phi }}}_{fi}=d_i(t)-{\beta }_{oi}^2{\tilde{\phi }}_i-{\widehat{\alpha }}_i\mathrm{sgn}({\tilde{\phi }}_i)\end{array}\right.$$

(7)

After introducing new vectors as η_i = [η_i1, η_i2]^T= ${[{\tilde{\phi }}_i,{\tilde{\phi }}_{fi}/{\beta }_{oi}]}^T$, we can rewrite (7) as

$${\dot{\eta }}_i={\beta }_{oi}{B}_o{\eta }_i+\frac{1}{{\beta }_{oi}}{C}_o\left[d_i(t)-{\widehat{\alpha }}_i\mathrm{sgn}({\eta }_{i1})\right]$$

(8)

where B_o = $\left[\begin{array}{cc}-2& 1\\ -1& 0\end{array}\right]$ and C_o = [0 1]^T.

As B_o is Hurwitz, there is a positive definite matrix F_o guaranteeing $B_o^T{F}_o+F_o{B}_o=-I$ [12].

3.2. Controller Design

Firstly, we introduce a set of error variables e_i(t) and compensation signals ε_i(t) as [33]

$$\begin{array}{l}{e}_1={\phi }_1-{\phi }_{1d},e_{j+1}={\phi }_{j+1}-v_{jf}\\ {\epsilon }_1=e_1-{\zeta }_1,{\epsilon }_{j+1}=e_{j+1}-{\zeta }_{j+1}\end{array}$$

(9)

where e₁ = φ₁–φ_1d(t) is the tracking error; ζ_i represent the auxiliary variables; and v_jf indicate the filtered values of the virtual control laws v_j to be synthesized later, which can be produced via the following filters [34].

$$\begin{array}{l}{L}_{cj}{\dot{v}}_{jf}=-v_{jf}+v_j-{\xi }_{js},v_{jf}(0):=v_j(0)\\ {\xi }_{js}=-\frac{L_{cj}{\xi }_j{\widehat{\sigma }}_j^2}{\sqrt{{\widehat{\sigma }}_j^2{\xi }_j^2+{\delta }_j^2(t)}}\end{array}$$

(10)

where L_cj are the positive parameters; ξ_j = v_j –v_jf indicate the filtering errors; and δ_j(t) > 0 and satisfy ${\displaystyle {\int }_0^t{\delta }_j(\upsilon )}d\upsilon \le {\vartheta }_j<\infty$, $\forall t$ ≥ 0, with ϑ_i being some positive constants [34]; especially, σ_j represent the upper bounds of ${\dot{v}}_j$, which can be updated via

$${\dot{\widehat{\sigma }}}_j=r_j\left|{\xi }_j\right|$$

(11)

with r_j being the adjustable positive gains.

An auxiliary system is introduced as [33]

$$\begin{array}{l}{\dot{\zeta }}_j=-k_j{\zeta }_j+{\zeta }_{j+1}+(v_{jf}-v_j)\\ {\dot{\zeta }}_n=-k_n{\zeta }_n\end{array}$$

(12)

where k_i are the positive gains.

Step 1:

Differentiating ε₁ via (1), (9) and (12), one has

$${\dot{\epsilon }}_1=v_1+k_1{\zeta }_1+{\epsilon }_2+{\psi }_1({\overline{\phi }}_1)+{\phi }_{f1}-{\dot{\phi }}_{1d}$$

(13)

Thus, v₁ can be designed as

$$v_1=-k_1{e}_1-{\psi }_1({\overline{\phi }}_1)-{\widehat{\phi }}_{f1}+{\dot{\phi }}_{1d}$$

(14)

Substituting (14) into (13) yields

$${\dot{\epsilon }}_1=-k_1{\epsilon }_1+{\epsilon }_2+{\tilde{\phi }}_{j1}$$

(15)

Step θ:

Differentiating ε_θ via (1), (9) and (12), one yields

$${\dot{\epsilon }}_{\theta }=v_{\theta }+k_{\theta }{\zeta }_{\theta }+{\epsilon }_{\theta +1}+{\psi }_{\theta }({\overline{\phi }}_{\theta })+{\phi }_{f\theta }-{\dot{\alpha }}_{(\theta -1)f}$$

(16)

Therefore, the virtual control law v_θ can be designed as

$$v_{\theta }=-k_{\theta }{e}_{\theta }-{\psi }_{\theta }({\overline{\phi }}_{\theta })-{\widehat{\phi }}_{f\theta }-{\dot{v}}_{(\theta -1)f}-{\epsilon }_{\theta -1}$$

(17)

Substituting (17) into (16) obtains

$${\dot{\epsilon }}_{\theta }=-k_{\theta }{\epsilon }_{\theta }+{\epsilon }_{\theta +1}-{\epsilon }_{\theta -1}+{\tilde{\phi }}_{f\theta }$$

(18)

Step n:

Considering (1), (9) and (12), we can acquire the time derivative of ε_n as

$${\dot{\epsilon }}_n=g({\overline{\phi }}_n)u+{\psi }_n({\overline{\phi }}_n)+{\phi }_{fn}-{\dot{v}}_{(n-1)f}+k_n{\zeta }_n$$

(19)

Finally, the actual control law u can be designed as

$$u=g^{-1}({\overline{\phi }}_n)\left[-k_n{e}_n-{\psi }_n({\overline{\phi }}_n)-{\widehat{\phi }}_{fn}-{\dot{v}}_{(n-1)f}-{\epsilon }_{n-1}\right]$$

(20)

Substituting (20) into (19) achieves

$${\dot{\epsilon }}_n=-k_n{\epsilon }_n-{\epsilon }_{n-1}+{\tilde{\phi }}_{fn}$$

(21)

3.3. Main Theoretical Results

Proposition 1. 

Define a set of auxiliary functions Q_i(t) as:

$$Q_i=\frac{1}{{\beta }_{oi}}{\eta }_i^T{F}_o{C}_o\left[d_i(t)-{\alpha }_i\mathrm{sgn}({\eta }_{i1})\right]$$

(22)

If the parameters α_i satisfy the following sufficient condition:

$${\alpha }_i>D_{i1}+\frac{3D_{i2}}{5{\beta }_{oi}}$$

(23)

where D_i1 = sup_t≥0|d_i(t)| and D_i2 = sup_t≥0|${\dot{d}}_i(t)$| are some unknown positive constants. Hence, the following inequality holds

$${\displaystyle {\int }_0^t{Q}_i(\upsilon )}d\upsilon \le {\mu }_i$$

(24)

where ${\mu }_i:=3\left[{\alpha }_i\left|{\eta }_{i1}(0)\right|-{\eta }_{i1}(0)d_i(0)\right]/\left(2{\beta }_{oi}^2\right)$.

Proof. 

See Appendix A. □

Theorem 1. 

Consider (1), the proposed controller with the designed EDEs in (4) can guarantee that all system signals are bounded as well as e₁→0 and${\tilde{\phi }}_{fi}$→0 as t→∞.

Proof. 

See Appendix B. □

Remark 2. 

Notably, some advanced control strategies [35,36] have been proposed. However, compared with these controllers, we have constructed a set of novel EDEs via the traditional ESO which can estimate the states and disturbances asymptotically. Additionally, to prevent over-parameterization of${\widehat{\alpha }}_i$and${\widehat{\sigma }}_j$, we can constrain their adaptive laws through the projection mapping function in [1].

4. Illustration Example

A one-link robot arm driven by the permanent magnet direct-current motor will be employed to verify the performance of the designed controller. Deniting state variables φ₁ = y_m, φ₂ $={\dot{y}}_m$ and φ₃ = K_mI_m/J_m, the considered system can be presented as follows

$$\left\{\begin{array}{l}{\dot{\phi }}_1={\phi }_2+{\psi }_1({\overline{\phi }}_1)+f_1(t)\\ {\dot{\phi }}_2={\phi }_3+{\psi }_2({\overline{\phi }}_2)+f_2(t)\\ {\dot{\phi }}_3=g({\overline{\phi }}_3)u+{\psi }_3({\overline{\phi }}_3)+f_3(t)\end{array}\right.$$

(25)

where y_m and I_m are the angular displacement of the load and the electric current, respectively; ${\psi }_1({\overline{\phi }}_1)$ = 0, $f_1(t)$ = 0,${\psi }_2({\overline{\phi }}_2)$ = –B_mφ₂/J_m, f₂(t) = Δ₂(t)/J_m, $g({\overline{\phi }}_3)$ = K_mK_v/(J_mL_m), ${\psi }_3({\overline{\phi }}_3)$ = –K_mK_Eφ₂/(J_mL_m)–R_m/L_m, $f_3(t)$ = K_mΔ₃(t)/(J_mL_m). The definitions and physical values of the system parameters are given in

Table 1. The physical parameters of the system.

Variable	Implication	Value (Unit)
Jm	The rotational inertia of the load	2.4 × 10−3 (kg·m2)
Bm	The viscous friction coefficient	2.26 (N·m·s/rad)
Rm	The armature resistance	3.0 (Ω)
Lm	The armature inductance	0.08 (H)
Km	The torque constant	1.85 (N·m/A)
Kv	The electrical gain	2.26
KE	The electromotive force coefficient	1.25 (V·s/rad)
Δ2(t)	External disturbance	Sin (πt)
Δ3(t)	External disturbance	15sin (πt)

.

For (25), the following four controllers are employed to track the trajectory φ_1d = 0.1sin(πt)[1–exp(–0.01t³)]rad.

(1)

C1: This is the developed controller in Section 3. And its parameters are tuned as

Table 2. The controller parameters of C1.

Controller Parameter	Value	Controller Parameter	Value
k1	300	λ2	1 × 107
k2	100	r1	150
k3	100	r2	150
βo2	300	Lc1	5 × 10−4
βo3	500	Lc2	5 × 10−4
λ1	1.5 × 105

.

(2)

C2: It is same as C1 but without compensation of the mismatched external disturbances.

(3)

C3: It is same as C1 but without compensation of the matched external disturbances.

(4)

C4: It is same as C1 but without compensation of the mismatched and matched external disturbances simultaneously.

For fairness, all design parameters of C2, C3 and C4 are chosen as same as that of C1.

The contrastive tracking errors are plotted in Figure 1. It can be clearly discovered that C1 performs the best tracking performance in terms of both transient and final tracking errors, which verifies the effectiveness of the compensation performance for mismatched and matched external disturbances. Moreover, it follows from Figure 1 that the tracking error of C1 gradually approaches zero, which demonstrates the achievable asymptotic output tracking performance. Furthermore, it also means that mismatched and matched external disturbances can be exactly estimated by the introduced observer. To support this claim, the estimation performance of the system states and modeling uncertainties with the proposed observer (5) are exhibited in Figure 2 and Figure 3, respectively. In addition, Figure 4 plots the control input of C1, which demonstrates that the resulting control law is smooth and meanwhile bounded.

5. Conclusions

A novel asymptotic tracking controller for a class of nonlinear systems with mismatched and matched exogenous disturbances has been proposed. Especially, a set of novel exact disturbance estimators with nonlinear robust terms to further reject disturbances has been creatively constructed to estimate the total disturbances in real time. Meanwhile, the exact disturbance estimations have been exploited in designing the resulting control scheme to eliminate the effects of disturbances. Especially, asymptotic tracking performance and asymptotic disturbance estimation performance have been demonstrated via strict theoretical analysis. In addition, the application on a one-link robotic arm driven by a direct-current servo motor has been conducted to verify the achievable results.