A novel ADRC?based design for a kind of flexible aerocraft

Sheng Zhong · Yi Huang · Sen Chen · Lin Dai

Abstract

Keywords Attitude control for flexible aerocraft · Active disturbance rejection control (ADRC) · Extended state observer(ESO)

1 Introduction

To achieve the requirements of long range, high precision and high manoeuvrability, aerocrafts are more prone to be slender, long and light, resulting in a rather acute problem of structural flexibility [1]. On the one hand, the natural frequencies of noticeable elastic modes decrease and are close to the designed bandwidth of the system, which may lead to the degradation of the control system stability and performance [2]. On the other hand, owing to the influence of elastic modes, the system measurement is usually the combination of rigid mode and elastic modes, rather than the controlled variable, i.e., the single rigid state. Consequently,if the system measurement is directly used for feedback, the elastic modes are easy to get excited and the coupling effects may cause the instability of system. Therefore, the effective suppression of structural vibration is of vital importance for the flight safety and accuracy. Moreover, besides the flexible structure, external disturbances and uncertainties of model also inevitably exist. Hence, a satisfied control scheme should be able to overcome the coupling effect of elastic modes, various external disturbances and the model uncertainties.

The vibration reduction control of flexible aerocrafts has attracted more and more attentions. An effective approach to avoid the elastic vibration, which is introduced by the measured output, is to design notch filters to alleviate the influence of elastic vibration by filtering the elastic modes[2-6]. Specifically, in order to achieve the control of rigid state, Z. Wang and Z. P. Liu [4] proposed a time-varying notch filter-based sliding mode control, A. M. Khoshnood et al. [5] designed an adaptive notch filter-based model reference adaptive control, and M. Piao [6] put forward an adaptive notch filter-based active disturbance rejection control (ADRC) design. In these notch filter-based control designs, actively suppressing the vibration of elastic modes does not considered. Moreover, when the frequency of elastic mode becomes low and close to the working bandwidth of the craft, the notch filter-based designs may lead to the degradation of the system stability. To actively suppress the vibration, several advanced controllers are applied. In[7] and [8], H∞control was used to stabilize the system.However, the order of controller was high and the calculation process was complicated. In [9] and [10], optimal control and observer-based control were proposed to suppress the vibration of flexible mode. However, both controllers depended on the system model, without considering the uncertainties and external disturbances. In this paper, to deal with the problem of controlling the attitude of the rigid mode as well as rapidly suppressing the vibration of the elastic mode, a new extended state observer (ESO) and the corresponding active disturbance rejection control (ADRC) law,which are different from the original form of ADRC [11],are proposed. The new ESO is designed to simultaneously estimate the rigid body signal, the flexible signal and the total disturbance in the channel of the rigid mode. Moreover,in the novel control design, despite the feedback of the rigid states and a compensation for the total disturbance, an elastic damping term is added to quickly suppress the vibration.

The ADRC, whose basic idea is to online estimate and compensate for the total disturbance, which lumps the internal uncertainties and the external disturbances, by a welldesigned ESO, was originally proposed by Han in 1990s[12]. The high level of robustness and the superior transient performance turn out to be the most valuable characteristics of ADRC to make it an appealing solution in dealing with real-world control problems [13-19]. In [13], the ADRC law is proposed to realize high dynamic performance for a kind of rigid body aerocraft. Moreover, in [14, 18, 19], the ADRC laws are used for vibration suppress problems under the circumstance that the controlled variable can be directly measured. In this paper, since the measurement and the controlled variable are not the same, the framework of the classical ADRC cannot be applied directly. Furthermore, there exist external disturbances acting on both the rigid mode and the elastic modes, simultaneously. Hence, both the states and the total disturbance are, in general, unobservable, which increases the difficulty for disturbance rejection.

In this paper, the performance of the novel ADRC-based design for a kind of flexible aerocraft is analyzed. It is proved that the new ESO is able to simultaneously estimate the rigid mode, the flexible mode and the total disturbance with an acceptable precision. Furthermore, the frequency characteristic of the new ESO is studied, indicating that the new ESO has a function, similar to that of notch filter, to estimate the rigid states. Then, by comparing the new ESO with a common used notch filter, the advantages of the ESO on estimation capability and phase margin are discussed. Finally, the theoretical analysis on the stability and the steady state of the closed-loop system is given to demonstrate the validity of the proposed ADRC design.

The rest of the paper is organized as follows. The detailed problem description is presented in Sect. 2. Section 3 proposes a new ESO design and the corresponding ADRC law.Then, Sect. 4 analyzes the features as well as the advantages of the new ESO and gives the theoretical analysis on the stability and performance of the proposed ADRC-based closed-loop system. Some simulation verifications are given in Sect. 5. Finally, Sect. 6 is the conclusion.

2 Problem formulation

In the paper, the following dynamic model of pitch angle error with the first elastic mode and disturbances is considered [9, 10]:

where Δφ(t)∈? is the pitch angle error,δφ(t)∈? is elevator deflection angle, i.e., the control input,q1(t)∈? is the first generalized elastic coordinate,ω1andξ1are natural frequency and damping ratio for the elastic modeq1, respectively. In (1),D31denotes the control gain for the elastic modeq1,b11,b21,D11andD21are the coupling coefficients,αwis wind attack angle,MbZandMq1are the disturbance moments, acting on the rigid mode Δφand the elastic modeq1, respectively. Owing to the influence of the elastic modeq1, the measurements of system (1) are

wherec1is the measurement coefficient. Measurement equation (2) implies that the controlled variable Δφcannot be measured directly. The detailed notations of the key coefficients and the range of their values are summarized in Table 1.

The control objective is to keep the pitch angle error Δφbe close to zero and quickly suppress the vibration of the elastic mode despite various uncertainties of the system parameters and the external disturbances.

Remark 1For the problem considered in the paper, although there are two outputs, they are not the controlled variables.The pitch angle is the only controlled variable, which cannot be measured directly. Quickly suppressing the elastic vibration is an additional requirement of stabilizing the system.Therefore, this is not a underactuated control problem.

Remark 2Since the flexible aerocraft is open-loop unstable,the controller with a high bandwidth is required to stabilize the plant [6]. Therefore, the first elastic mode is the most likely one to get excited and cause the system to be unstable. On the other hand, since the frequencies of other higher elastic modes are far from the designed bandwidth, they can be easily eliminated by classical filters [10]. Hence, in the paper, the system with the first elastic mode will be analyzed and a novel design based on ADRC will be proposed.

In the next section, it will be shown that since the controlled variable cannot be measured directly, neither simple PD feedback of the output nor the ADRC in common use can be directly applied. Then, a new ADRC-based control strategy is proposed.

3 ADRC with a new ESO design

In this section, firstly, a possible stabilization strategy is discussed. Then, the observability of the states as well as the total disturbance on the rigid mode is quantitatively analyzed. Finally, a new ESO design and the corresponding ADRC law are proposed.

3.1 Stabilization strategies

First, the difficulty of stabilizing the system, caused by the measurements, is discussed.

wherekp ＞0 andkd ＞0 are feedback coefficients. According to the Routh criterion, there exist positive constantskp0andkd0. Whenkp ＞kp0,kd ＞kd0, the closed-loop system is stable.However, no matter how the PD feedback parameterskpandkdare tuned, the characteristic polynomial of the closed-loop system always exists a pair of conjugated roots with almost fixed and small real part, which depends on the termξ1ω1, indicating that the flexible mode is easy to be unstable or converge slowly.

Thus, to guarantee the stability of the closed-loop system as well as to quickly suppress the vibration of the flexile mode,it is crucial to increase the damping of the vibration. Then,consider the control law as follows:

Table 1 Nomenclatures and values of the coefficients

Remark 3Since the feedbacks of the states Δφ,Δ˙φand˙q1are enough to assign the poles to ideal positions, there is no need to design full-state feedback.

Based on the above analysis, how to obtain the system states Δφ,Δ˙φand˙q1from the measured outputs Δφsand Δ˙φsis extremely important for this control problem. In engineering practice, notch filter is a popular tool to filter the elastic signal from the measured output.In this paper, since the function of disturbance rejection should also be added to the control law, the idea of ESO to simultaneously obtain the rigid mode Δφ, the elastic modeq1, their derivatives and the disturbance are considered.

In plant (1), there exist external disturbances acting on the rigid mode as well as the elastic mode. Moreover, the system parameters have uncertainties. According to the idea of ADRC [11], to realize the control objective, the total disturbance, which influences the controlled variable Δφ, should be estimated and compensated for in real time.However, since the measurement is Δφsrather than Δφand there also exists the external disturbanceMq1acting on the elastic mode, the observability of the state Δφ,Δ˙φ,˙q1and the total disturbance on the rigid mode needs to be analyzed in advance.

3.2 Observability of the states and the total disturbance

Modify system (1) and measurements (2) by the following form:

Due to the existence of the disturbancesfandfq, both the states [Δφ,q1,Δ˙φ,˙q1] and the total disturbancefare unobservable [20]. Fortunately, according to the following proposition, which gives the quantitative result for the estimation error ofXcaused by the unobservability, the error of the case considered in this paper may be small enough to be acceptable in engineering practice.

Proposition 1If the disturbance fq is bounded and the disturbances f and fq have the bounded first-and second-order derivatives, then the real value of X(t)will be

From Proposition 1,M(t) is an inherent error for estimatingX(t), which is caused by the unobservability and cannot be eliminated by any algorithm. Fortunately, from Table 1,M(t)of system (1) only depends on the parametersξ,andc1,which can be quantitatively calculated as follows:

(10) indicates that, although there are two disturbances (i.e.,fandfq) acting on the system, their influences on the estimation of the states [Δφ,q1,Δ˙φ,˙q1] and the total disturbancefmay be small if a suitable observer can be designed.

In the next section, a new extended state observer (ESO)will be proposed for estimating the states [Δφ,q1,Δ˙φ,˙q1]and the total disturbancef. Then, a corresponding ADRC law is developed.

3.3 A novel ADRC?based control design

To estimate the states Δφ,Δ˙φ,˙q1and the total disturbancef, a new ESO, which fully utilizes the measurementsy1andy2, is designed as follows:

whereL∈?5×2is the observer parameter to be tuned, andz=[z1,z2,z3,z4,z5]T, [z1,z2,z3,z4]Tandz5are expected to be the estimations of the states and the total disturbancef,respectively.

Remark 4In order to estimate elastic signal, which will be used for quick suppressing the vibration of the elastic mode,the nominal model of the elastic mode is used in new ESO(11). Moreover, besides estimating the rigid mode and the elastic mode, the new ESO simultaneously estimates the“total disturbance” acting on the rigid mode, which lumps the uncertainty of the rigid mode, the coupling effect of the elastic mode and the external disturbance in the channel of the pitch angle. Since the controlled variable cannot be measured directly, the way to get the total disturbance in this ESO design is different from the original one of estimating total disturbance, discussed in [11], which can be directly obtained from the differential signal of the measurement.

Then, according to the discussion in Sect. 3.1, a novel ADRC-based control law to control the angle of the rigid mode as well as rapidly suppress the vibration can be designed as:

where parametersωcandξccharacterize the poles of the rigid mode in an ideal integrator cascade canonical form,and parameterωqcharacterizes the damping of the elastic equation in the absence of the coupling effect. Moreover, in order to make the dynamic response smooth, it is suggested to setξc ＞1.

Remark 5In the original frame of ADRC [11], where the controlled variable is the measured output, the total disturbance is observable. In this paper, since the controlled variable cannot be measured directly, ESO (11) is not the original form as in most literature about ADRC. Based on Proposition 1 and (10), the estimation error of new ESO design (11) can be small enough to be acceptable. This property as well as the features of new designed ESO (11) will be further analyzed in Sect. 4.1.

Remark 6The two outputs/measurements of the system,which contain the information of the elastic mode, are fully used in new ESO (11) to extract the information of the elastic mode with a higher precision such that a better suppression of the elastic mode can be achieved. If only the measurementy1is used, theny1-based ESO has the following form:

Fig. 1 Block diagram of ADRC law (12)

whereL1∈?5is the observer parameter to be tuned. The advantage of ESO (11) for estimating˙q1, compared to ESO(14), and its advantage of phase property, compared to a common used notch filter, will be discussed in Sect. 4.2.

Remark 7ADRC law (12) is different from the original form in the existing literature. The theoretical analysis on the stability of the closed-loop system will be given in Sect. 4.3.

4 Performances of the closed?loop system

In this section, the features and advantages of new ESO (11)are analyzed. Moreover, the theoretical analysis on the stability and performances of the proposed ADRC-based closed-loop system is given.

4.1 Features of new ESO (11)

First, a lemma is proposed to demonstrate that the estimation error of new ESO (11) can be small enough to be acceptable by tuning the parameters of the ESO.

Before introducing the lemma, some mathematical notations are denoted. The notation C-represents the left plane in the complex plane. The set of the eigenvalues of matrixEis denoted asλ(E).

Lemma 1Consider uncertain system(1),(2)and ESO(11).Assume the disturbance fq is bounded and the disturbances fq and f have the bounded first3rd-order derivatives. Then,for any symmetric set Λ={λ1,…,λ5}?C-and β ＞0,there exists L such that λ(-)=βΛ and

From (17), the following proposition can be obtained.

Proposition 2For any parameter L,which satisfies the conditions in Lemma1,the transfer functions Gi(s)(i=1,…,4)have the following properties:

From Proposition 2 (1), it can be seen that bothG1(s) andG3(s) have a function similar to notch filter for filtering the elastic mode to estimate the rigid states.

At the same time, Proposition 2 (2) implies thatG2(s) andG4(s) have the ability to filter the rigid mode and extract the elastic mode, whose frequency is close to1, from measurements (2).

To verify Proposition 2, the bode plots of the transfer functionsGi(s) (i=1,…,4) are given withλ(-)=10{-0.9,-1,-0.95,-1.05,-1}. Figures 2 and 3 are the bode plots ofG1(s) andG3(s) , respectively,which show thatG1(s) andG3(s) have the function of extracting Δφand Δ˙φfrom the measurementsy1andy2,respectively. Figures 4 and 5 are the bode plots ofc1G2(s)andc1G4(s) , respectively, which show that at the frequency of1=30 rad/s, the magnitudes ofc1Gi(i1) (i=2,4) are near 1 and the phases ofGi(i1)(i=2,4) are around 0?.Thus, ESO (11) can simultaneously extract the rigid mode and the elastic mode from the measurements.

Fig. 2 The bode plot of G1(s)

Fig. 4 The bode plot of c1G2(s)

Fig. 3 The bode plot of G3(s)

Fig. 5 The bode plot of c1G4(s)

4.2 Advantages of ESO (11)

In this section, by comparing with ESO (14), and with a popular-used notch filter, the advantages of ESO (11) are discussed.

To compare with ESO (14), which only one measurement signaly1is utilized, denote4(s) as the open-loop transfer function from the derivative of the measurementy1(t) to the output(t) of ESO (14).n

whereξfis the parameter of the filter.

Define

Fig. 6 The bode plot of c1G4(s) and c1(s)

Fig. 7 The bode plots of G1(s) and Gn1(s)

Figure 7 is the bode plots of the transfer functionsG1(s) andGn1(s) . Figure 8 is the bode plots of the transfer functionsG3(s) andGn2(s) . From Figs. 7 and 8, it can be seen that,compared with the notch filerGn1(s) andGn2(s) , both the phase lags ofG1(s) andG3(s) in the low-frequency range are smaller, while the abilities of filtering the elastic mode,whose frequency is around 24-36 rad/s, are almost the same.This feature indicates that ESO (11) can obtain the signal of the rigid mode more accurately.

In the next section, the stability of the closed-loop system will be analyzed. Moreover, compared with notch filter (18),the advantage of ESO (11) in relative stability will be studied.

4.3 Stability of the closed?loop system

Theorem 1 is proposed to show that ADRC (11) and (12)can stabilize uncertain system (1) and (2) with satisfied performance.errors of the rigid modeΔφ and the estimation z of ESO(11)will satisfy

Fig. 8 The bode plots of G3(s) and Gn2(s)

The proof of Theorem 1 is given in Appendix.

Theorem 1 shows that despite various uncertainties, the closed-loop system is stable. Moreover, if the limitations of the external disturbancesαw,MbZandMq1exist, the steadystate error of the rigid mode Δφcan be controlled consistently small via proposed ADRC (11) and (12). The steadystate estimation errors ofz1andz2can be restricted by a small and acceptable boundary and those ofzi(i=3,4,5)are zero.

It is well known that stability is a fundamental but not sole index for evaluating the quality of a controller. Next,the relative stability, especially the phase margin, will be analyzed and it will be shown that the phase margin of ADRC (11) and (12) controlled system is larger than that of notch filter (18)-based system.

Similar to control laws (12), notch filter (18)-based control lawδφn(t) can be designed as follows:

whereUn(s) is the Laplace transform ofδφn(t) ,kpn,kdnandkqnare the controller parameters to be designed. The loop transfer functionGon(s) of notch filter (18)-based system can be obtained as follows:

The controller parameters for the two controllers are chosen as

- notch filter-based control (21):ξf1=16π,ξf2=64π,kpn=6,kdn=5.6 andkqn=0.24,

such that both the closed-loop systems are stable and the crossover frequencies ofGo(s) andGon(s) are almost the same.

Figure 9 is the bode plots of the loop transfer functionsGo(s) andGon(s) by setting the system parametersb1,b11,b21,D11andD21being their upper bounds andb2,b3,D31andω1being their nominal values, respectively.Figure 10 is the local amplification of Fig. 9 in the lowfrequency range. It can be seen that the phase margins ofGo(s) andGon(s) for the rigid mode are 33?and 27?, respectively, that is, the phase margin ofGo(s) for the rigid mode is 6?larger than that ofGon(s) . Moreover, Fig. 9 shows that the phase margin ofGo(s) for the elastic mode is 40?larger than that ofGon(s).

This analysis demonstrates that new ADRC law (12) has a better relative stability than the classical design. On the other hand, notch filter-based control law (21) does not have the function of compensating for the uncertainties and disturbances. Therefore, new ADRC law (12) can timely estimate and compensate for the total disturbancefto guarantee the control precision without loss of phase margin.

Fig. 9 The bode plots of Go(s) and Gon(s)

Fig. 10 The local amplification of Fig. 9 in the low-frequency range

5 Simulation

In this section, the simulation results of the proposed ADRC control law are presented to verify the efficiency of the controller.

Figure 11 is the response curve of the rigid mode under 16 different cases. It is shown that under novel control law(11) and (12), the pitch angle error can be quickly controlled converging to zero. Moreover, the control law is robust to the uncertainties of the system.

The key of ADRC law (11) and (12) to suppress the vibration is by adding the elastic damping feedbackkqz4. To show the efficiency of elastic suppression, Fig. 12 is the comparison on the response curves of the elastic mode for the nominal case, which are based on ADRC law (12) and ADRC law(12) without elastic damping feedback, i.e.,kq=0 , respectively. It can be seen that by ADRC law (12), the vibration of the elastic mode can indeed be promptly suppressed.

Fig. 11 Response curves of the pitch angle error Δφ under 16 cases

In order to verify the estimation capability of ESO (11),Figs. 13, 14, 15 give the simulation results of the estimations and the estimation errors of the nominal case. It is shown that the estimation errors of the states and the total disturbance are small, which coincide with that of the theoretical analysis.

Fig. 12 Comparison on the response curves of the elastic mode q1

Fig. 13 Estimations of the system states

6 Conclusions

Fig. 14 Estimation of the total disturbance f

Fig. 15 Estimation error of ESO (11)

In this paper, a novel control design based on the ADRC is proposed for a kind of uncertain flexible aerocraft. What makes the problem more meaningful and challenging is that because of the existence of elastic mode, the controlled variable cannot be directly measured. First, a novel ADRC-based control design with special forms of ESO and feedback law is proposed in order to achieve the desired performance of the rigid mode as well as vibration suppression. Then, it is proved that new ESO (11) can estimate the system states and the total disturbancefwith an acceptable precision,although there exist errors, which are caused by the unobservability and cannot be eliminated by any algorithm. Furthermore, it is demonstrated that ESO (11) has the advantage of extracting the elastic mode signal with strong robustness to the uncertainty in the frequency of the elastic mode. On the other hand, compared with a popular-used notch filter,ESO (11) also shows the advantage of extracting the rigid mode with higher accuracy. Finally, Theorem 1 is given to illustrate the performance of the closed-loop system. The relative stability of the closed-loop system is also discussed.

Appendix

where ΔΦ(s),Q1(s),F(s) andFq(s) are the Laplace transforms of Δφ(t),q1(t),f(t) andfq(t) , respectively.

Denote the estimation error asez=z-X. From (a15), it can be obtained that, when the closed-loop system reaches steady state, there is

Table a1 Ranges of the characteristic roots

AcknowledgementsThis work was supported by the National Key R&D Program of China (No. 2018YFA0703800) and the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.