# Design of Multiple Observers Based Anti-Disturbance Control

In order to mitigate the influence of periodic disturbance and wind disturbance, the MOBADC is composed of a DO and an ESO in the position loop, and only ESO based nonlinear control in the attitude loop. The proposed control scheme and data flows can be seen in Fig. 4.3.

## Control for Translational Dynamics

Firstly, the errors of position and linear velocity are defined as

where 7*d* and *i>,i* are the desired trajectory and velocity of the quadrotor UAV, and 7 and *v* are the current position and velocity.

The controller of the position loop is designed as

where a,/ is the desired control acceleration in the position loop, *K,_* and *K„ *are positive definite gain matrices, 7,/ is the desired trajectory acceleration. Note that the estimation values *d _{m}f* and

*di*/ will be given below.

FIGURE 4.3: MOBADC scheme and relevant data flows.

### DO Design

The DO is designed to estimate the periodic disturbance *d _{m}.* Based on Eq. (4.3a) and inspired by Refs. [104, 106], the DO is designed as

where £ and *d _{m}f* represent the estimates of £ and

*d*is the auxiliary state variable of the observer,

_{m}f. z*df*denotes the estimate of

*dif*which can be obtained by ESO,

*p(*7,

*is)*is an auxiliary function to be designed, and

*l(*7,

*is)*is the gain function of the observer determined by

and *G =* ^[0_{3х}з,/зхз]^{Т}, /М *=* [^^{Т},0,0, -p]^{T}.

### ESO Design

In order to facilitate the design process of ESO, we first denote *x _{p} =* 7,

*x*

_{p}2*= is.*and

*x*

_{P}3*= dif,*respectively. Subsequently, the translational dynamic in Eq. (4.3a) can be transformed to a general form [33] as

The ESO serving in the position loop is designed to estimate the lumped disturbance d/ *f.*

where *z _{p}* 1,

*z*

_{p}*2*

*,*and

*z*3 are the estimates of

_{p}*x*

_{p}. x_{p}*2*

*,*and

*х*2, and

_{рЛ}. K_{v}, K_{p}*K*

*€ IR*

_{P}3^{:!x3}are the observer gains of the ESO in the position loop and

*e*For the sake of simplification, define the stacked gain matrix

_{pl}= Xpi — Zpi.*K*3].

_{p}= {K_{pl},K_{p2},K_{p}The gains of control *(K _{7}. K„)* and the observers

*(K*/(7.

_{p}.*is))*can be chosen based on Theorem 4.1.

Remark 4.1. *The DO and ESO in this part are designed to respectively tackle the payload oscillating disturbance and disturbances with bounded derivative value that occur in the position loop. Their respective disturbance estimates are utilized by each other which can be seen from Eqs.* (4.9) *and* (4.12).

Remark 4.2. *The payload oscillating disturbance is a kind of periodic disturbance and the disturbance like wind is a kind of disturbance with bounded derivative value. Since there is a great difference between these kinds of disturbances, the proposed DO and ESO can effectively distinguish and estimate their respective disturbances.*

## Control for Rotational Dynamics

According to Eq. (4.3b). the desired input for the attitude loop is driven by

where *ft _{(}i* and

*o,i*are the desired pitch and roll angles of the quadrotor, and

*ifd*is the desired yaw angle that is set to zero in the chapter.

Define the attitude tracking error as

where *rpi = [*

•,/] and *rj* is the current Euler angular.

Hence, the controller for rotational dynamics is designed as

where *K _{4}* and

*К*are positive definite gain matrices, and

_{ш}*di*is the lumped disturbance estimate, respectively.

_{T}Remark 4.3. *The desired attitude velocity u>d is set to zero, which is applicable to the quadrotor UAV operating within a small angle.*

Denote *x _{a} = *tj,

*x*

_{a}2*=*17, and

*x*

_{a}z*= d*[

_{T}, the rotational dynamic in Eq. (4.4) can be thereby rewritten as

The ESO in the attitude loop is developed to estimate the lumped disturbance *di _{T}.*

where *z _{a}*i,

*z*2, and

_{a}*z*are the estimates of

_{a3}*x*and ж

_{a}, x_{a2},_{а}з,

*K„ , K*

_{a}*2*

*,*and

*Ka-s e*R

^{3x3}are the observer gains of ESO in the attitude loop and e„ i =

*x*and

_{a}i — z_{a}. M*C*are the estimates of

*M*and

*C*with respect to

*z*1,

_{a}*z*2, and

_{a}*z*3. For the sake of simplification, the stacked gain is denoted as

_{a}*K*

_{a}= [К_{аЛ}, K_{a}*2*

*, K*3], which can be chosen based on Theorem 4.2.

_{a}Remark 4.4. *The ESO in the attitude loop intends to attenuate the external disturbance and model uncertainty fed into the attitude control of the quadrotor UAV. Hence, the roles that the ESO herein, and the DO and ESO in the position loop are separated.*

Remark 4.5. *The presented MOBADC scheme can be seen as a refined antidisturbance control. It is easy to tailor such that different types of control methodologies can be integrated with various disturbance observers. The developed scheme can handle the complicated case that multiple disturbances exist in different channels. Such an anti-disturbance control can also be applied to the environment where only one kind of disturbance is present. In such circumstance, the estimate of the other disturbance is negligibly small and has little effect on the control input.*

Remark 4.6. *The developed MOBADC scheme intends to reject multiple disturbances exposed on a UAV. It should be emphasized that either ESO or DO can be chosen according to the disturbance characteristics. In comparison of the studies [34, 35], multiple disturbances can be estimated and thereby rejected within the presented scheme. In contrast to the work [103], multiple observers are exploited in the developed scheme in terms of the explicit disturbance analysis and characteristics. Hence, multiple disturbances can be handled in a delicacy manner. From the aspect of anti-disturbance performance, the conservatism can be substantially reduced especially in the presence of multiple disturbances.*

## Stability Analysis

### Position Loop

Define the estimation errors of DO and ESO as

where *x _{p}* = ,

*x*J

_{2},

*xj*

_{3}]

^{T}and

*z*= [zj

_{p}_{l5}zj

_{2},2^3]

^{T}. Next, the stability

result is given.

In real applications, for convenience, the auxiliary function *p(*7, *u)* can be chosen as a linear function such that /(7.*1**/)* will be a constant matrix and rewritten as *l.*

Theorem 4.1. *The position control system with the proposed composited DO has a bounded error if the disturbance di*/ *is bounded and there exist the controller gains K _{y}, K„ and the observer gains K_{p}, l such that the eigenvalues of* S

*have negative real parts, where*2

*is given by*

*- (A-lGB) _{6x6} (—lGE)_{6xg}*

Озхб

*with E =* [Озхз, Озхз,/зхз], Yfsxis = 1 *and*

— *3x3-0 *^9x9*

*m*

Озхб

*—Kpi* /3x3 Озхз

^9x9 = *—K** _{p}2* ОзхЗ —

*I'3x3*

*m*

*—K _{P}3* Озхз Озхз

*Proof.* By taking into account Eqs. (4.3a), (4.9), and (4.10), the differential equation of Eq. (4.18a) is given by

where / := *f(i'),* and (a), (b), and (c) follow from Eqs. (4.9), (4.6), and (4.5), respectively.

According to Eqs. (4.11) and (4.12), the differential equation of Eq. (4.18b) is given by

By combining Eqs. (4.20) and (4.22), the dynamics of the estimation errors of the position loop can be presented as

Substitute Eq. (4.8) into Eq. (4.3a), and we can get

According to Eq. (4.23), together with Eq. (4.24), the error dynamics of the entire position system is obtained as

Apparently, based on Assumption 4.1, if the position controller gains and observer gains are selected appropriately to make the eigenvalues of S have negative real parts, the composited DO has a bounded error [102] and the proof is completed. □

### Attitude Loop

Define the estimation errors of ESO in the attitude loop as

where *x _{a}* = [a:I

_{1},aJ

_{2},:rT

_{3}]

^{T}and

*z„ =*[z^,zj

_{2},zJ

_{3}]

^{T}. Before moving on, it is necessary to make the following assumption.

Assumption 4.2. *The quadrotor UAV flies at a small angle and its attitude dynamics can be linearized such that M(rj) and C(t),t*/) *will become constant matrices* Mo *and Co-*

Now, we are in a position to give the following result.

Theorem 4.2. *Under the proposed control scheme in Eq.* (4.15) *and the designed ESO in Eq.* (4.17), *the attitude control system at a small angle is asymptotically stable.*

*Proof.* Based on the small-angle approximation, a linearized attitude control system is obtained. According to the so-called separation principle, design of the state observer can be separated from the controller design of the linear system under certain condition [107]. To proceed, two conditions should be satisfied: (1) the attitude loop using the control scheme in Eq. (4.15) is asymptotically stable in the absence of disturbances; (2) the ESO is stable with appropriately chosen observer gain *K _{a}.*

Obviously, in the absence of disturbances, the attitude controller Eq. (4.15) will be reduced to a proportional-differential controller and the condition (1) can be satisfied.

According to Assumption 4.2 and similar to the derivation of Eq. (4.18b), the dynamics of the estimation errors in the attitude loop can be obtained by

*—Kai ^3x3 ^3x3*

where W_{9X}9 = *~K _{a2} -M_{0}~^{[1]}C_{0}* M

_{0}

^{_1}.

*— КаЗ* ОзхЗ Озхз

According to Assumption 4.1, *K _{a}* can be appropriately selected such that the eigenvalues of W

_{)x9}have negative real parts, then the ESO in the attitude loop is stable [102] and the condition (2) is satisfied. In addition, Eq. (4.27) shows that the convergence of the observer does not depend on the variation of the state, hence the above theorem can be proved [107]. □

Remark 4.7. *In real applications, the basic control gains (K _{q}. K_{M}, K_{1} and K„ ) are determined first from the inner loop to outer loop. Then, according to Theorems 4-1 and 4-2, the gains of DO and ESOs are determined respectively.*

Remark 4.8. *Although the stability analysis of the attitude loop is based on the approximate linearization, the proposed ESO based controller still works well for its original model Eq.* (4.4) *without linearization as demonstrated in the real flight experiments.*

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