Linear active disturbance rejection control for oscillatory systems with large time-delays

Abstract

Oscillatory systems with time delays exist widely in actual industrial process. This paper discusses the design and tuning of linear active disturbance rejection controller (LADRC) for the oscillatory systems with large time delays. First, internal model controllers (IMC) are designed for the oscillatory systems to compensate the time-delay and cancel the complex poles; then they are implemented with the general LADRC structures and approximated with observer-bandwidth-based LADRCs. Afterwards, the third-order LADRC tuning formulas for oscillatory systems are derived from the IMC controllers. Simulation examples and load frequency control(LFC) in power system with communication delay are used to test the applicability of the proposed tuning formula.

Introduction

It is well-known that a stable process can be oscillatory or non-oscillatory. In practice, the non-oscillatory process is often modeled as a first-order process with dead-time (FOPDT) model, and the oscillatory process is modeled as an underdamped second-order process with dead-time (SOPDT) model. Some SOPDT systems in practical application can be found in Ghorai et al. [1], Huey et al. [2], Fu and Tan [3]. The first one has been studied thoroughly [4], [5], [6], [7]. But there are few studies focusing on the second one, especially the underdamped systems with time delay.

The SOPDT system has rich dynamic characteristics, including underdamped, critically damped and overdamped dynamics. Critically damped and overdamped are non-oscillatory systems, so they can be treated as FOPDT systems. Because of the existence of conjugate complex poles for the underdamped systems, parameter tuning becomes particularly difficult. So far, almost all tuning formulas for SOPDT underdamped systems are based on PID. For example, a method of dominant pole assignment of the proportional integral differential (PID) for the oscillatory SOPDT is proposed in Das et al. [8]; A fuzzy neural network control method for stabilizing underdamped plants is proposed in Shen [9]; A new control structure based on Posicast control and PID control is proposed for underdamped second-order system to achieve minimum overshoot in Oliveira and Vrančić [10]; A PID controller with cascaded delay filters is used for the control of SOPDT system in Nath et al. [11]; And a model free PID controller is designed to stabilize SOPDT system in Siddiqui et al. [12].

However, PID is an error feedback control method which will lead to some deviations in the output of the systems [13]. For oscillatory systems, PID may not perform well. To overcome the limit of PID controllers, Han proposed a new control technology which is simple, practical and independent of system model, namely active disturbance rejection control(ADRC) [14]. ADRC has strong disturbance rejection ability [15], [16], [17]. However, the application of ADRC is limited by its complex structure and numerous tuning parameters, so Gao proposed linear ADRC(LADRC) [18], [19]. The structure and parameter tuning of LADRC are greatly simplified, and it has become a potential technology to replace the traditional PID control. Tian showed that LADRC can achieve good performance through frequency domain analysis [20]. Zheng proved the stability of LADRC [21]. Zhao and Zheng applied LARDC into delayed systems [22], [23]. A complete survey of ADRC method can be found in Feng and Guo [24], Huang and Xue [25]. In addition, LADRC has been widely applied in several practical processes [26], [27], [28], [29], [30].

Consider a $n\text{th}$-order controlled plant model:

$$y^{\left(n\right)}\left(t\right)=b_{0}u\left(t\right)+f(y\left(t\right),u\left(t\right),d\left(t\right)).$$

where $b_0$ is the high frequency gain of controlled plant,$f\left(y\right(t),u(t),d(t\left)\right)$ contains the internal uncertainties of the plant and the external disturbances, that is, the ‘total disturbance’.

Let $z={[z_1{z}_2\dots z_n{z}_{n+1}]}^T,z_1=y,z_2=\dot{y},\dots ,z_n=y^{(n-1)},z_{n+1}=f$. Then model Eq. (1) can be presented as:

$$\{\begin{array}{c}\dot{z}=A_{e}z+B_{e}u+E_e\dot{f},\hfill \\ y=C_{e}z.\hfill \end{array}$$

where

$$\begin{array}{ccc}& & A_e={\left[\begin{array}{ccccc}0& 1& 0& \dots & 0\\ 0& 0& 1& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & 1\\ 0& 0& 0& \dots & 0\end{array}\right]}_{(n+1)\times (n+1)},\hfill \\ & & B_e={\left[\begin{array}{c}0\\ 0\\ ⋮\\ b_0\\ 0\end{array}\right]}_{(n+1)\times 1},E_e={\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ 1\end{array}\right]}_{(n+1)\times 1},\hfill \\ & & C_e={\left[\begin{array}{ccccc}1& 0& 0& \dots & 0\end{array}\right]}_{1\times (n+1)}.\hfill \end{array}$$

Design a full-order linear extended state observer (ESO):

$$\dot{\widehat{z}}=A_e\widehat{z}+B_{e}u+L_o(y-C_e\widehat{z})$$

where $\widehat{z}$ represent the estimation of $z$, $L_o$ represents the observer gain:

$$L_o={[{\beta }_1{\beta }_2\dots {\beta }_n{\beta }_{n+1}]}^T.$$

When $A_e-L_o{C}_e$ is asymptotically stable, ${\widehat{z}}_1$ is close to $y\left(t\right)$, ${\widehat{z}}_2$ is close to $\dot{y\left(t\right)},\dots$, and ${\widehat{z}}_{n+1}$ is close to $f$, thus, the total disturbance can be estimated. In order to eliminate this disturbance $f$, choose:

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{k_1(r\left(t\right)-{\widehat{z}}_1\left(t\right))+k_2(\dot{r}\left(t\right)-{\widehat{z}}_2\left(t\right))+\dots +k_n(r^{(n-1)}\left(t\right)-{\widehat{z}}_n\left(t\right))-{\widehat{z}}_{n+1}\left(t\right)}{b_0}\hfill \\ \hfill & =& K_o(\widehat{r}\left(t\right)-\widehat{z}\left(t\right)).\hfill \end{array}$$

where $K_o$ represents the feedback control gain and $\widehat{r}\left(t\right)$ represents the generalized reference input signal:

$$K_{\mathrm{o}}=[k_1{k}_2\dots k_{n}1]/b_0.$$

$$\widehat{r}\left(t\right)={[r\left(t\right)\dot{r}\left(t\right)\dots r^{(n-1)}\left(t\right)0]}^T.$$

The state-space form of LADRC is shown as follows and the structure is shown in Fig. 1, where $P$ is the controlled plant.

$$\{\begin{array}{c}\dot{\widehat{z}}=(A_e-L_o{C}_e)\widehat{z}+B_{e}u+L_{o}y,\hfill \\ u=K_o(\widehat{r}-\widehat{z}).\hfill \end{array}$$

Apply Laplace transform to Eq. (9) and assume that the initial conditions are zero, we get the transfer function of LADRC from $y$ to $u$:

$$K\left(s\right)=\frac{u\left(s\right)}{y\left(s\right)}=-K_o{(sI-A_e+B_e{K}_o+L_o{C}_e)}^{-1}{L}_o.$$

Although LADRC has become popular in many fields, two major difficulties remain to be solved:

• Selection of the order of LADRC. Theoretically, the order of LADRC should be the same as the relative degree of the controlled plant [31], however, for time-delayed or high-order systems, will high-order LADRCs improve the control performance?

• Tuning of the parameters of LADRC. The conventional bandwidth idea reduces the tuning of LADRC parameters to 3 [19], i.e., the high-frequency gain $b_0$, the controller bandwidth ${\omega }_c$, and the observer bandwidth ${\omega }_o$. In this case, the elements of the controller gain can be obtained from ${\omega }_c$ as

  $$k_i=C_n^{i-1}{\omega }_c^{n-i+1},i=1,\dots ,n.$$

  And the elements of the observer gain can be obtained from ${\omega }_o$ as

  $${\beta }_i=C_{n+1}^i{\omega }_o^i,i=1,\dots ,n+1.$$

  However, as discussed in Zhou and Tan [32], the performance of bandwidth-based LADRCs may be restricted for some systems, and general LADRCs should be used to obtain better control effect. In this case, the number of a general LADRC parameters increases dramatically compared with the conventional bandwidth-based LADRC. For example, for a 2nd-order general LADRC, there are totally 6 parameters to be tuned, and for a 3rd-order LADRC, the number is then 8.

Example 1

Consider the following heavily oscillatory (damping ratio 0.1) plants with small time-delay $\left(G_1\right)$ and large time-delay $\left(G_2\right)$.

$$G_1\left(s\right)=\frac{e^{-0.5s}}{s^2+0.2s+1}.$$

$$G_2\left(s\right)=\frac{e^{-2s}}{s^2+0.2s+1}.$$

Second-order LADRCs and third-order LADRCs are optimized with the genetic algorithm to minimize the integral of the time squared error(ITSE) index with the limitation of robustness measure ($ϵ$) less than 3 for these two systems following the idea in Zhang et al. [4]. Table 1 shows the optimal parameters of both general LADRCs and conventional bandwidth-based LADRCs.

A unit step set-point reference and a unit step input disturbance are inserted at 0s and suitable time respectively for every simulation in this paper. Fig. 2 shows the control effects of the second-order LADRCs and the third-order LADRCs. Fig. 2(a) and (b) are the responses of $G_1$ and $G_2$ under general LADRCs, the parameters are shown in Table 1(a). Fig. 2(c) and (d) are the responses of $G_1$ and $G_2$ under bandwidth-based LADRCs, and Table 1(b) shows the parameters of bandwidth-based LADRCs. The general LADRCs can achieve better control effects than the bandwidth-based LADRCs for both the second-order and the third-order cases from the observation of Fig. 2.

It is also observed that the second-order LADRC can achieve similar control effects as the third-order LADRC for $G_1$, but the third-order LADRC has better performance in both tracking and disturbance rejection for $G_2$. Because second-order LADRC treats SOPDT model as a second-order system, the delay (high-order) dynamics of SOPDT is ignored. When the delay is small, ignoring it will not have much effect on the controller design. But when the delay is large, ignoring it will have a great influence on the controller. Thus for the delay-dominant SOPDT plants, the second-order LADRC may not work well, and the third-order LADRC is a better choice. Furthermore, it is also noted that the general LADRCs can achieve better performance than the bandwidth-based LADRCs, therefore, it is necessary to discuss the general LADRCs for oscillatory systems with large delays.

Up to now, there are several papers discussing the parameter setting of LADRC. A tuning formula under the limitation of robustness measure between 2.4 and 2.5 for FOPDT system is proposed in Zhang et al. [4]. The selection of the order of LADRC is discussed in Zhang et al. [33]. Chen proposed a tuning method for LADRC by the expected settling time [34]. Fu added model information to the design of the observer which can lighten the burden of ESO [35]. However, the researches of the tuning of underdamped SOPDT systems for LADRC are few, especially for those with large time-delay.

This paper will discuss the design and tuning of the third-order LADRC for oscillatory SOPDT systems with large delay. The contributions of this paper are shown below:

• The proposed approach can obtain satisfactory control effect for oscillatory systems with large time delays. The robustness and disturbance rejection performance are considered in the derivation of the tuning formula.

• Compared with the general LADRC where the controller gain and the observer gain can be arbitrary, the proposed method is observer-bandwidth-based, which reduces the tuning parameters and facilitates engineering application.

• Compared with the conventional LADRC where the controller gain and the observer gain are tuned with two bandwidths, the proposed method assumes that the controller gain can be arbitrary. The control performance is improved with the extra tuning parameters in the controller gain.

The tuning formulas are verified by several examples and the load frequency control(LFC) with communication delay in power system is also used to verify the effect of the proposed formula.

The rest of this paper is arranged as follows: In the Section 2, a third-order LADRC tuning method is designed for SOPDT system with large delay based on IMC, several examples are given to illustrate the design process. Section 3 introduces the derivation of the tuning formulas of third-order LADRC. Section 4 verifies the effect of the proposed formulas by simulation examples. Section 5 applies the proposed formula into LFC with communication delay to verify the effect. Section 6 summarizes the conclusion and future work.