Robust PID control of quadrotors with power reduction analysis

ISA Transactions xxx (xxxx) xxx

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Research article

Robust PID control of quadrotors with power reduction analysis✩ ∗

Roger Miranda-Colorado a , , Luis T. Aguilar b a b

CONACyT-Instituto Politécnico Nacional-CITEDI, Av. Instituto Politécnico Nacional No. 1310, Nueva Tijuana, Tijuana, Baja California, 22435, Mexico Instituto Politécnico Nacional-CITEDI, Av. Instituto Politécnico Nacional No. 1310, Nueva Tijuana, Tijuana, Baja California, 22435, Mexico

article

info

Article history: Received 28 February 2019 Received in revised form 13 August 2019 Accepted 28 August 2019 Available online xxxx Keywords: Robust proportional integral derivative control Quadrotor unmanned aerial vehicle Cuckoo search algorithm Reduction of power consumption Sliding modes Quadrotor nonlinear dynamic model

a b s t r a c t This paper presents a novel robust controller applied to a quadrotor vehicle for regulation and trajectory tracking tasks. In the proposed scheme, the quadrotor position is controlled by a proportional integral derivative (PID) controller, while the orientation control is achieved through a model-based controller. The proposed controller is combined with a power reduction methodology, which includes a controller-gains tuning stage using the cuckoo search algorithm, and a minimum jerk trajectory design stage. The performance of the new controller is assessed in a free-disturbance case and under the effect of parametric uncertainty and aero-dynamical disturbances. The new controller is compared against two linear PID controllers and a nonlinear sliding mode-based controller. Numerical simulations demonstrate the superiority of the proposed scheme as well as its robustness against different types of perturbations. Also, it is proven that the power demanded by any controller is reduced when using the power reduction methodology. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Recently, the interest on Unmanned Aerial Vehicles (UAV), also called drones, has increased. They are flying platforms comprising airships, fixed wing, or Vertical Take-off and Landing (VTOL) vehicles. Quadrotors are a type of UAV with four rotors attached to a rigid cross airframe and placed equidistant from its center of mass, as depicted in Fig. 1. These systems have three degrees of freedom (DOF) for their rotational motion, three DOF for their translational motion, and only four actuators available as control inputs. Hence, quadrotors are underactuated systems, with the coordinates x, y being the underactuated translational coordinates [1]. 1.1. Overview Quadrotors are essential research systems due to their appealing features. For instance, they have the ability for hovering and operating in places with reduced space [2]; high thrust-weight ratio and outstanding maneuverability [2]; utilization of small propellers for thrust generation, which diminishes its complexity [2]; and their ability to react and continue in flight even after actuator or sensor faults [3,4]. ✩ This work was supported by the Consejo Nacional de Ciencia y Tecnología (CONACyT), Mexico under the Cátedras Project 1537: ‘‘Análisis y Control de Sistemas Mecatrónicos Complejos’’. ∗ Corresponding author. E-mail addresses: [email protected] (R. Miranda-Colorado), [email protected] (L.T. Aguilar).

Usually, quadrotors are controlled by Proportional Integral Derivative (PID) controllers, which are inherently simple and computationally efficient. Also, PID controllers are easy to implement in onboard micro control units [5]. However, the main challenges that remain are:

• the different sources of perturbation such as aerodynamic disturbances, measurement noise, and parameter uncertainty, which may degrade the performance of the closedloop system, making it necessary to resort to a robust controller to achieve the desired performance; and • the high amount of power consumption demanded by quadrotors, which limits their autonomy in real-time applications. Based on the issues mentioned above, this paper presents a novel robust PID controller which is applied to the quadrotor nonlinear dynamic model. This controller is combined with a power reduction methodology [6]. The study presented in this work proves the robustness properties of the proposed controller. Also, we develop an extensive analysis demonstrating that we can reduce the power demands of any controller by utilizing the power reduction methodology, which increases the quadrotor’s autonomy. 1.2. Literature review In the literature, many quadrotor control schemes have been reported for regulation, trajectory tracking, and obstacle avoidance tasks. These methodologies may consider the quadrotor

https://doi.org/10.1016/j.isatra.2019.08.045 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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Finally, a fundamental drawback in quadrotor control is related to their power consumption, which may affect by lowering the quadrotor performance. For instance, the controller efficiency is directly affected by the state of charge of the battery. Hence, if the quadrotor control takes into account the power consumption, we may expect to have a better operation. Some works face the power reduction problem by shutting-down the motors [33]; computing the optimal thrust from all rotors [34]; through the use of trajectory planning stages [35]; and with the help of saturated controllers [36] or controllers that minimize a performance index depending on the control effort [37]. Also, reference [38] introduced a controller tuning methodology based on energy consumption using the quadrotor control inputs. However, as pointed out in [37], this scheme may be misleading and not effective. Thus, more works presenting alternatives for reducing the quadrotor power consumption are still required. Fig. 1. Schematic diagram of a quadrotor UAV vehicle. The inertial reference frame is represented by {I }, and the body reference frame is denoted by {B}.

linear and/or nonlinear dynamic model. However, unmodeled dynamics or complex nonlinearities in the quadrotor model as well as the presence of disturbances may limit the quadrotor performance. Besides, for model-based control schemes, the system’s operation is also affected by sensitivity to parametric uncertainties. Linear control techniques are preferable due to their simplicity and ease of implementation in a real platform. Some examples applied to quadrotors encompass proportional integral, proportional–derivative, linear quadratic regulators, and PID controllers, among others [2,7,8]. However, these methodologies are limited by their vulnerability to disturbances. To overcome the problems faced by linear quadrotor control techniques, we may resort to use nonlinear control methodologies, which have a better performance than the linear ones. However, their structure is more complicated and challenging to implement in real platforms. Some examples include backstepping control [9], H∞ control [10], robust proportional–derivative control [11], neural networks [12], adaptive methods [13], L1 adaptive control [14], fault-tolerant controllers [15], cooperative tracking control [16], hysteresis-based controllers for attitude control [17], adaptive fault-tolerant control [18], fractional control [19], and terminal sliding modes [6,20,21]. Also, some interesting works reporting the design of proportional integral [22], output feedback indirect dynamic inversion [23], as well as PID passivity-based [24] controllers for nonaffine-in-control and portHamiltonian systems have been studied, and the structure of these controllers may be used to control the quadrotor nonlinear dynamic model. It can be verified that the system performance can be affected by the structure of the commanded reference signal and controller-gains tuning [6]. Improper design of the reference signal can affect by decreasing the overall system’s performance [25]. Also, a wrong selection of the controller gains may yield unsatisfactory performance. Hence, the stages devoted to trajectory design and controller tuning deserve to be investigated. Some works dealing with trajectory design employ optimization-based methods, learning strategies, convex optimization, minimumtime trajectory generation, among others [26–28]. The controller tuning stage can be carried out through the use of empirical techniques such as the Ziegler–Nichols and the Cohen–Coon methods [29]. Also, we may resort to a numerical method such as genetic algorithms [30], particle swarm optimization [31], and cuckoo search algorithm (CSA) [32] for tuning the controller gains.

1.3. Contribution of the paper The previous literature review shows that, although there exist many control techniques for quadrotor control, these schemes are not widely adopted in real-time applications due to their structural complexity. Also, the robustness of the controller is related to its complexity, which may produce excessive computational burden, complicating its implementation in a real platform. Besides, the PID controller is commonly utilized for quadrotor control because of its appealing features such as ease of implementation and their computational efficiency in onboard micro control units [1,4,5,29,30,38–43]. However, most of the PID quadrotor control methodologies presented in the literature are based on a linearized model and/or are not supported by rigorous closed-loop stability analysis. Also, the quadrotor operation is directly affected by adverse effects such as communications delays, changing dynamics, external disturbances, and power consumption. Finally, some works provide guidelines for tuning the proposed controller, others include a procedure for designing the reference trajectory, and only a few develop a methodology for reducing the power consumption. Besides, some works on quadrotor control focus on either regulation (point-to-point) or trajectory tracking tasks. However, both tasks are essential for all the applications involving a quadrotor system, such as surveillance, mapping, photography, farming, among others. Hence, by taking into account the issues mentioned above, this work aims to contribute by 1. designing a novel robust PID controller for regulation and trajectory tracking tasks, with a simple structure, which is applied directly to the quadrotor nonlinear model, and is robust against parametric uncertainties, state estimation error, measurement noise, as well as external disturbances; 2. including a meticulous stability analysis utilizing a Lyapunov-based approach; 3. presenting an extensive power consumption study, which is applied to three different PID controllers and a sliding mode-based controller with a PID-like sliding surface; 4. providing guidelines for implementing the proposed controller as well as the power reduction methodology. The proposed controller consists on a PID position control and a model-based orientation control. Also, the controller structure is simple, which simplifies its implementation in a real platform. The controller performance is assessed by comparing it against two other linear PID control schemes as well as a robust sliding mode-based controller with a PID-like sliding surface. Vast numerical simulations indicate that the proposed controller outperforms the other PID standard schemes when affected by aerodynamic disturbances, state estimation error, measurement

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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noise, and parametric uncertainties, which demonstrates its robustness properties. Also, the numerical simulations indicate that the power demanded by any of the controllers under study is reduced, and their corresponding performance is enhanced when using the power reduction methodology. Finally, note that this work is mainly focused on developing a control methodology to be applied to quadrotor vehicles, which overcomes most of the main issues own by the existing control methodologies. This methodology encompasses the controller structural simplicity, the detailed procedure for controller tuning, the design of the reference signal, the inclusion of a power reduction methodology, and an extensive numerical analysis demonstrating the robustness of the proposed control scheme against a large number of internal and external disturbances. All the previous features are only partially reported in previous works. Hence, this work is self-contained and a useful alternative for robust control of quadrotor vehicles. 1.4. Organization The rest of the paper is organized as follows. Section 2 presents the main features of a quadrotor UAV, its dynamic equations, and the problem formulation. The synthesis of the proposed PID controller is given in Section 3. The description of the power reduction methodology is analyzed in Section 4. The performance analysis of the proposed PID controller is studied in Section 5. Some concluding remarks wrap up this work in Section 6. Finally, three appendices presenting the structure of the inertia and Coriolis matrices, a stability proof, as well as the structure of the controllers used for comparison, are given at the end of the paper.

3

angular position, with φ, θ, ψ denoting the roll, pitch, and yaw angles, respectively. The dynamic equations of a quadrotor vehicle, expressed in {I }, are given by [6,11] mr¨ + mgez = R(η)ez F ,

(1)

Mo (η)η¨ + Co (η, η˙ )η˙ = τ o ,

(2)

where

τ o = W T τ¯ o , Mo (η) = W T IW , ˙, Co (η, η˙ ) = −W T S(W η˙ )IW − W T IW

(3) (4) (5)

ez = [0, 0, 1] , and S(W η˙ ) ∈ R a skew-symmetric matrix. The explicit expressions for matrices Mo and Co are given in Appendix A. Also, F is the total thrust and τ¯ o = [τ2 , τ3 , τ4 ]T ∈ R3 is the vector of external torques applied to the quadrotor. Matrix R ∈ SO(3) describes the orientation of {B} with respect to {I }, and is obtained through the xyz Euler angles [6]. Besides, η˙ (t) and ζ (t) are related by ζ = W (η)η˙ . The explicit expression for R(η) and W (η) is T

[ R(η) =

[ W (η) =

3×3

cψ cθ − sφ sψ sθ cθ sψ + cψ sφ sθ −cφ sθ cθ 0 sθ

0 1 0

−cφ sθ sφ cφ cθ

−cφ sψ cφ cψ sφ

cψ sθ + cθ sφ sψ sψ sθ − cθ sφ cψ cφ cθ

] ,

(6)

] ,

(7)

where cα , sβ represent the functions cos(α ) and sin(β ), respectively.

1.5. Notation In this work, R denotes the set of real numbers, ∥·∥ denotes the Euclidean norm of some vector or matrix, and |·| represents the absolute value. For a time-varying signal f (t), the terms f˙ (t), f¨ (t), f (n) (t) constitute the first, second and nth time derivative of f (t). In general, vectors are represented by bold lowercase letters, and matrices by uppercase letters; 0n denotes the zero matrices of order n × n, and In the identity matrix or order n × n. A matrix of the form A = diag {a1 , . . . , an } ∈ Rn×n represents a diagonal matrix with elements a1 , . . . , an on the main diagonal. For a given square matrix, λmin (·), λmax (·) represent its minimum and maximum eigenvalues, respectively. When required, the square brackets [·] are used to indicate the units of a given quantity. Finally, if no confusion is present, the arguments of functions will be omitted.

2.1. Problem formulation In this work, we design a novel PID controller for quadrotor control. This controller permits achieving point-to-point and trajectory tracking tasks. Also, we develop a methodology for reducing the quadrotor power consumption. Thus, we can state the control problem as follows. Control Objective: Let the quadrotor dynamic model (1)–(5). Design a PID controller ensuring that the quadrotor position and orientation error signals are uniformly ultimately bounded [44], i.e., for any initial time t0 ≥ 0, and for every a ∈ (0, c), c > 0, and a positive constant b > 0, there is a time T (a, b) ≥ 0 such that max r˜ (t0 ) , η˜ (t0 ) ≤ a H⇒ max r˜ (t) , η˜ (t) ≤ b,

{

 

}

{

 

}

(8)

2. Quadrotor dynamics A diagram representing the structure of a quadrotor vehicle is shown in Fig. 1. This system possesses four identical rotors and propellers. Rotors 1 and 3 rotate in the counter-clockwise direction (around vector −b3 ), while the others rotate in the opposite direction. The quadrotor position is described by the inertial frame {I } = {i1 , i2 , i3 }; the body frame {B} = {b1 , b1 , b3 }, attached to its center of mass, describes the quadrotor orientation. Parameter m ∈ R denotes the quadrotor mass, L the arm length, and I ∈ R3×3 the inertia tensor. By assuming a symmetric mass distribution, I = diag {Ixx , Iyy , Izz } in {B}, where each diagonal element refers to the corresponding principal inertia moment. Finally, the quadrotor angular velocity in {B} is represented by ζ = [p, q, r ]T ∈ R3 ; r , v ∈ R3 denote the quadrotor translational position and velocity in {I }, and η = [φ, θ, ψ]T represents the

where r˜ (t) = rd (t) − r(t), η˜ (t) = ηd (t) − η(t), rd (t) is the desired position, and ηd (t) the orientation reference signal. Also, verify that the power consumption can be reduced through a proper design of the reference signal and a controller gains tuning stage. We end up this Section by presenting some properties related to matrices Mo (η) and Co (η, η˙ ), which are essential in the stability proof of the proposed PID controller. Property 1. Matrix Mo (η) is symmetric and positive definite.

˙ o (η) − 2Co (η, η˙ ) is skewProperty 2. The matrix SMC (η, η˙ ) = M symmetric, i.e., for any vector x ∈ R3 , the equality ˙ o (η) − 2Co (η, η˙ ) x = 0, xT SMC (η, η˙ )x = xT M

[

]

(9)

holds.

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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3. Synthesis of the controller This Section describes the steps required for designing the proposed robust PID controller, as well as a simple tuning procedure. For simplicity, the controller synthesis is divided into two parts: Position control and orientation control.

Let the input vector f p (η) = [fpx , fpy , fpz ] f p (ηd ) ∈ R3 be given by

3

∈ R and vector

f p (η) = R(η)ez F ,

(10)

f p (ηd ) = R(ηd )ez F ,

(11)

where ηd = [φd , θd , ψd ]T ∈ R3 is the desired orientation vector in {I }, with φd , θd , ψd the desired roll, pitch and yaw angles; and r˜ (t) = [rd (t) − r(t)] ∈ R3 , η˜ (t) = [ηd (t) − η(t)] ∈ R3 are the position and orientation error vectors, where rd (t) = [xd , yd , zd ]T ∈ R3 is the desired position vector in {I }, with xd , yd , zd denoting the desired positions along the i1 , i2 and i3 axes, respectively. The next development considers the next assumption. Assumption 1. The angles φ (t), θ (t) fulfill the small angle condition

π

2

.

(12)

The small angle condition is realistic for many quadrotor’s applications which do not involve acrobatic or extreme maneuvers [21,39], which is the case in this work. For instance, in hovering and tracking of smooth references, the roll and pitch angles regularly remain within the set described by (12). Also, this property is strengthened when a trajectory design stage is used, because no sudden changes in position or orientation are demanded to the quadrotor vehicle. From the quadrotor position dynamics (1), Eq. (10), and the definition of the position error signal r˜ (t), we obtain the next position error dynamics mr¨˜ = mr¨d + mgez − f p (ηd ) + f p (ηd ) − f p (η) .

[

]

(13)

3

Let rc (t) ∈ R be a commanded acceleration vector satisfying the next equation [r¨d − r¨c ] + Kp r˜ + Kd r˙˜ + Ki ξ = 0,

(14)

ξ˙ = r˜ ,

(15)

with Kp = diag {kp1 , kp2 , kp3 }, Ki = diag {ki1 , ki2 , ki3 }, Kd = diag {kd1 , kd2 , kd3 } ∈ R3×3 being positive definite diagonal matrices, and ξ (t) = [ξx , ξy , ξz ]T ∈ R3 . The desired acceleration r¨d (t) and the commanded acceleration r¨c (t) are different. Vector rc (t) is designed to make the position error r˜ (t) and its time derivative r˙˜ (t) to converge to zero. Hence, the commanded acceleration is the required quadrotor acceleration that ensures that the control objective is achieved. Besides, Eq. (14) shows that the commanded acceleration also depends on r¨d (t). If perfect knowledge of the quadrotor parameters is available, the input vector f p (η) with the structure f p (η) = mgez + mr¨d + Kp r˜ + Kd r˙˜ + Ki ξ,

(16)

accomplishes the objective (14). Nevertheless, in practice, only an estimate of the system parameters is at disposal. Thus, the position control is designed as follows

ˆ z +m ˆ r¨d + Kp r˜ + Kd r˙˜ + Ki ξ, f p (η) = mge

(17)

(18)

with the orthogonal matrix A and vector b defined as A=

T

t ≥0

AT f p = bF ,

[

3.1. Position control

max {|φ (t)| , |θ (t)|} <

ˆ being a constant value representing an estimate of the with m quadrotor mass. From Eqs. (6) and (10), the next relation follows

cψ sψ 0

sψ −cψ 0

0 0 1

]

[ ,b=

sθ sφ cθ cφ cθ

] .

(19)

Therefore, the input signal F and the desired roll φd (t) and pitch θd (t) angles can be computed from (18) as F =

fpz cφ cθ

,

φd = tan−1

(20)

[

sψd fpx − cψd fpy fpz

]

,

(21)

⎡

⎤

θd = tan−1 ⎣ √[

cψd fpx + sψd fpy

sψd fpx − cψd fpy

]2

⎦.

(22)

+ fpz2

From Eqs. (10) and (17), the input signal F can be upper bounded as follows

⏐ ⏐ ⏐ ⏐ ˆ +m ˆ z¨d + kp3 r˜z + kd3 r˙˜z + ki3 ξz ⏐ |F | ≤ c¯L1 ⏐mg    r˙˜     ≤ c¯1 + c¯2   r˜  ,  ξ 

(23)

where c¯L1 , c¯1 , c¯2 are positive constants. Recall that the assumption on angles φ (t) and θ (t) stated in Eq. (12) prevents the input signal F in (20) to go to infinity, which makes feasible the upper bound given in Eq. (23). ˜ = m ˆ − m denote the parametric mass error. Finally, let m Hence, by using Eqs. (1) and (17), the position error dynamics can be expressed as

˜ [r¨d + gez ] + f p (ηd ) − f p (η) . (24) mr¨˜ = −Kp r˜ − Kd r˙˜ − Ki ξ − m

[

]

3.2. Orientation control Let us define the vector s(t) ∈ R3 as follows s = η˙˜ + Λη˜ ,

(25)

where Λ = diag {λ1 , λ2 , λ3 } ∈ R3×3 is a positive definite matrix. Note that if s(t) converges to zero, so do the orientation error η˜ (t). By using Eq. (2), the time derivative of s(t) is given by s˙ = η¨ d + Λη˙˜ − Mo−1 (η) [τ o − Co (η, η˙ )η˙ ] .

(26)

We assume that only an estimate of the system parameters is available. Hence, the orientation control signal τ o (t) is designed as follows

[ ] [ ] ˆ o (η) η¨ d + Λη˙˜ + Cˆ o (η, η˙ ) η˙ d + Λη˜ + K s, τo = M

(27)

where K = diag {k1 , k2 , k3 } ∈ R3×3 is a positive definite diagonal ˆ o , Cˆ o correspond to the known values of matrices matrix, and M Mo and Co , respectively. The property of linearity in the parameters [45] allows rewriting equation (27) as

ˆ + K s, τ o = Φ (η, η˙ , ηd , η˙ d , η¨ d )Θ

(28)

ˆ ∈ Rp is a constant vector comprising the estimated where Θ parameters of the system, and Φ ∈ R3×p is the so called regression matrix.

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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Finally, from Eqs. (26) and (28), the dynamics of s(t) is written as follows

where

˜, Mo s˙ = −K s − Co s − Φ Θ

ϵ2 λmin (Kp ) − ϵ1 λmax (Kd ) − λmax (Ki ), ϵ1 λmin (Ki )}, [ ] λmin (K ) − ϵ23 Q4 = , ϵ3 ϵ3 λmin (Λ) −2 [ ] q11 q12 Q5 = ,

(29)

˜ = Θ ˆ − Θ , Θ ∈ Rp denotes the vector of the actual where Θ ˜ ∈ Rp is the parametric error vector. system parameters, and Θ 3.3. Main result

d r˜ = r˙˜ , dt d ˜ [r¨d + gez ] m r˜˙ = −Kp r˜ − Kd r˙˜ − Ki ξ − m dt [ ] + f p (ηd ) − f p (η) , d dt d dt d

(30)

(31)

ξ = r˜ ,

(32)

η˜ = s − Λη˜ ,

(33)

˜. s = −K s − Co (η, η˙ ) − Φ Θ (34) dt The previous development shows how to design the position and orientation controllers. Now, we summarize these results in the next theorem. The proof is given in Appendix B.

Mo (η)

Theorem 1. Let us consider the quadrotor dynamic equations (1)–(2). Then, the altitude controller F and the attitude controller τ¯ o (t) = [τ2 , τ3 , τ4 ]T described by F =

fpz cφ cθ

τ¯ o = W T

[

,

]−1

Q3 = diag {λmin (Kd ) − mϵ2 ,

q12

From Eqs. (24) and (29), the quadrotor closed-loop dynamic equations can be written as the next set of first order differential equations

(35)

τo,

(36)

with f p = [fpx , fpy , fpz ] , W defined in (7), and T

[ ] [ ] ˆ o (η) η¨ d + Λη˙˜ + Cˆ o (η, η˙ ) η˙ d + Λη˜ + K s, τo = M ˆ z +m ˆ r¨d + Kp r˜ + Kd r˙˜ + Ki ξ, f p = mge ] [ −1 sψd fpx − cψd fpy , φd = tan fpz ⎡ ⎤ c f + s f ψ px ψ py d d ⎦, θd = tan−1 ⎣ √[ ]2 2 sψd fpx − cψd fpy + fpz

(37) (38) (39)

(40)

ensures that trajectories r˜ , r˙˜ , ξ , η˜ , s are uniformly ultimately bounded for some compact set of initial conditions, provided that matrices Kp , Kd , Ki , Λ, K , Q3 , Q4 and Q5 fulfill the next inequalities

{

[ ] λmin (Kp ) > max m ϵ1 + ϵ22 − ϵ2 λmin (Kd ), } ϵ2 λmin (Ki ) ϵ1 λmax (Kd ) + λmax (Ki ) , mϵ1 − , ϵ2 ϵ1 { } mϵ1 λmin (Kp ) nnum λmin (Kd ) > max + − + mϵ2 , mϵ2 , nden ϵ2 ϵ2 ϵ3 λmin (K ) > , 4λmin (Λ) [ ] α1 c¯3 1 + ϵ12 + ϵ22  ] , λmin (Q3 ) > [ ˜ 4α2 λmin (Q4 ) − d2 Θ   ˜  > 0, λmin (Q4 ) − d2 Θ λmin (Q5 ) > a1 > 0,

(41) (42) (43) (44) (45) (46)

5

q22

(47) (48) (49)

q11 = α1 λmin (Q3 ), √ 1 q12 = − α1 c¯3 1 + ϵ12 + ϵ22 , 2   ˜ , q22 = α2 λmin (Q4 ) − α2 d2 Θ

ϵ1 , ϵ2 , ϵ3 , a1 , d2 , c3 , α1 , α2 , c¯3 , d2 are arbitrary positive numbers, and nnum = [ϵ1 λmax (Kd ) + λmax (Ki ) + mϵ1 ϵ2 ]2 , nden = ϵ2 [ϵ1 λmin (Kp ) + ϵ2 λmin (Ki ) − mϵ12 ]. Remark 1. From the stability proof of Theorem 1 given in Appendix B, we observe that the time derivative of the Lyapunov function V defined in Eq. (B.1) is negative definite outside the set

⎧  ⎪  ⎨  σ1   Ω1 = σ1 , σ2 :   σ2  ⎪ ⎩ { }    [ ⏐ ⏐√ ] ⎫ ˜  , α2 Θ ˜  d3 + d4 rσ2 ⎪ ⎪ ˜ ⏐ 1 + ϵ12 + ϵ22 , α2 d1 Θ max α1 c¯4 ⏐m ⎬ , ≤ ⎪ λmin (Q5 ) − a1 ⎪ ⎭ (50) where α 1 , α2 , c¯4 , ϵ1 , ϵ2 , d1, d3 , d4 , a1 are design constants; and

  T   T   ˙  η˜ ]  r˜  , ∥ξ∥] , σ2 =  ˜ ∥ [∥ s , [ σ1 =  r ,  . Hence, if no     parametric error is present, the error signals r˜ (t), η˜ (t) converge to zero asymptotically. If this is not the case, the error signals are ultimately bounded, with the ultimate bound given in (50). Also, the set Ω1 can be made arbitrarily small by increasing the smallest eigenvalue of matrix Q5 . This is achieved by increasing the gains of the controller matrices Kp , Kd , Ki , K , and Λ.

As indicated in Fig. 2, we may distinguish an inner loop constituted by the altitude controller (35) and the attitude controller (36). Also, we may discern an outer loop, composed by (39)–(40), which allows handling angles roll and pitch in order to control the non-actuated coordinates x and y. A general block diagram describing this PID structure is given in Fig. 2. The accomplishment of inequalities (41)–(46) is required for a correct tuning of the proposed controller. In the next, we provide some tuning guidelines. To this end, we can follow the next steps: 1. Select small values for λmin (Ki ) and λmax (Ki ). A possible selection is λmin (Ki ) = 0.001 and λmax (Ki ) = 0.5. 2. Select some test values for λmin (Kp ), λmax (Kp ), ϵ1 , ϵ2 . We can select ϵ1 = 0.1, ϵ2 = 1, λmin (Kp ) = 0.1, and some large value for the maximum eigenvalue of matrix Kp such as λmax (Kp ) = 20. Then, use inequality (42) to obtain a set of allowable values for λmin (Kd ). Also, it can be verified that by increasing the values of the gains in Kp we also increase the set of feasible gains for Kd . Besides, this provides a minimum value for λmin (Kp ). 3. Use inequality (41) to obtain the allowable values for λmin (Kp ). 4. Select some small value for ϵ3 , for instance, ϵ3 = 0.1. Then, select some desired value for the gains of Λ. We can select unitary values for all the elements of matrix Λ. Then, by increasing these gains, we can obtain a faster response.

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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Fig. 2. General block diagram describing the structure of the proposed PID controller (35)–(40). The inner loop generates the altitude and attitude control signals, while the outer loop produces the roll and pitch angles to handle the x and y coordinates.

5. Use inequality (43) to obtain a set of allowable values for the gains of matrix K . Note that small values for the gains of K usually yield a good response. Also, by increasing the value of the gains of K , we obtain a faster response. 6. Verify that the selected values for matrices Kp , Kd , Ki , K , and Λ satisfy inequalities (44)–(46). After completing this procedure, we obtain a set of feasible gains for the proposed controller. Also, note that the previous procedure provides guidelines for obtaining the smallest eigenvalues of the controller matrices. Then, we can continue the tuning procedure by increasing the gains up to obtain the desired performance. Theorem 1 ensures that the quadrotor position and orientation error signals are UUB, i.e., the control objective (8) is achieved. Then, it is left to introduce power reduction methodology, and demonstrate that its combination with the proposed controller allows reducing the quadrotor’s power consumption. Remark 2. For the proposed controller (35)–(40), the altitude controller F and the desired roll (φd (t)) and pitch (θd (t)) angles are generated through the PID controller f p given in Eq. (38), while the control signal τ¯ o is obtained by means of the model-based controller (37), which can be viewed as a proportional–derivative controller with a nonlinear compensation term. Also, the numerical simulation results presented in Section 5 show that the proposed controller is robust against endogenous and exogenous disturbances. Hence, for simplicity, we decided to name the whole structure as a robust PID controller. 4. Power reduction methodology The block-diagram describing the power reduction methodology is depicted in Fig. 3, and consists of the next two stages: Stage 1. Design the reference signal employing an optimization method. Stage 2. Tuning the controller gains by utilizing an evolutionary meta-heuristic algorithm. In Stage 1, the reference signal was computed by minimizing the jerk of the desired trajectory. This procedure generates smooth trajectories, which demand a lower control effort. Explicitly, the minimum jerk trajectories are computed through the next polynomial rdi (t) = ci5 t 5 + ci4 t 4 + ci3 t 3 + ci2 t 2 + ci1 t + ci0 ,

(51)

where cij , j = 0, . . . , 5 are design constants depending on the boundary conditions. Further details can be found in [6]. To implement the trajectory design stage, we follow the next steps. First, we define the boundary conditions on position, velocity, and acceleration for each segment of the trajectory. Then,

employ equation (51) together with its first and second timederivatives to obtain a set of algebraic equations of the form At k = yt , with k = [ci5 , . . . , ci0 ]T . After solving this system of equations, we obtain the coefficients k for computing (51). The numerical simulations included in this work consider point-to-point and a trajectory tracking tasks. For each case, we design the corresponding trajectory through the methodology mentioned above. Specifically, the point-to-point task is mathematically described by

[rd (t), ψd ] =

] [ π 1, 1, 1, 10 [ ] π 0, 1, 1, 10 [ ] π 0, 0, 1, − 10 [ ] π 1, 0, 1, − 10 [1, 0, 0, 0]

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

, , , , ,

0≤t≤5 5 < t ≤ 10 10 < t ≤ 15 [s],

(52)

15 < t ≤ 20 20 < t ≤ 25

and the reference signal for the trajectory tracking task, which is a helix followed by a set of points, is given by

⎧ π ⎪ ⎨ [[f d (t), 10 ] ] π 0, 0, 8k¯π , 10 [rd (t), ψd ] = 3 ⎪ ] ⎩ [ π 0, 0, 0, − 10 ] [ f d (t) = f1 (t), f2 (t), t /k¯ 3 ,

, 0 ≤ t ≤ 8π , 8π < t ≤ 27 [s],

(53)

, 27 < t ≤ 30

f1 (t) = k¯ 1 cos(k¯ 2 π t), f2 (t) = k¯ 1 sin(k¯ 2 π t), where k¯ i , i = 1, 2, 3, are positive constants. In this work, the values k¯ 1 = 0.4, k¯ 2 = 0.3, and k¯ 3 = 30 were used. Regarding Stage 2, we utilized the CSA for tuning the controller gains as described in [6]. The CSA requires to define a performance index for computing the optimal controller gains. Unlike the performance index used in [6], in this work we employ the next function L(r˜ , τ ) =

Ts

1

∫

Ts

√0

+

t r˜ T (t)r˜ (t) dt

[

1

Ts

]

√

Ts

∫

2 (t)dt Pm 0

+

1 Ts

Ts

∫

Pe (t)dt ,

(54)

0

where Pm = [Rez F ]T r˙ + τ To ζ is the instantaneous mechanical power, Pe (t) = i2 R the instantaneous electrical power, and Ts the time-length of the simulation. The first term in (54) penalizes large tracking errors and duration of oscillations. The next two terms are the RMS (root mean square) values of the instantaneous mechanical and electrical power, which provide a measure of the power demanded by the system. To implement this second stage, we select some controller, a large initial value for (54), and a set of allowable values for the

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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Fig. 3. Methodology for reduction of power consumption, which includes a stage devoted to trajectory design (top) and another stage for tuning the controller gains using the CSA (bottom).

controller gains. Then, after running the CSA, we obtain a new set of gains, which permit to obtain a new value of (54). We continue this procedure up to reaching a pre-specified threshold value of the performance index. Remark 3. Note that the proposed controller (35)–(40), as well as the other controllers used in this work for comparison, are model-based. This feature implies the necessity of knowing the quadrotor system parameters to implement them. Specifically, the required parameters are the mass and the principal inertia moments. These parameters are not difficult to be obtained directly from the manufacturer or can be computed utilizing computer-aided design software. Hence, we can assume that an estimate of the quadrotor parameters is available. Thus, in order to reduce the computational burden in a real-time application, the trajectory design stage and the controller gains tuning stages are implemented offline. Remark 4. In this work, we only considered the trajectory design and controller gains tuning stages for reduction of power consumption. Nevertheless, there are also other research areas that permit to reduce the quadrotor power demands. For instance, we may consider analyzing the gains used in cruise flight; the transition between hovering and cruise flight (transition maneuver); change the quadrotor mechanical structure; or improving the power-to-weight ratio of the motors [46]. Regarding the transition maneuver, some works solve this problem by using a different controller for each flight mode and a maneuver for switching from one flight mode to the other [47]. Other works design a controller that permits a soft transition between hovering and cruise flight [48]. However, in any case, the control action increases significantly when the transition begins. Other ways used to face the transition maneuver consists of tilting the rotors, which also provides advantages encompassing reduction of power consumption [49]. 5. Performance evaluation Now, we perform the numerical simulations validating the theoretical results presented in Sections 3 and 4. First, we show

Table 1 Parameters of the quadrotor UAV, the actuators, and the disturbance model. Parameter

Value

Parameter

Value

m [kg] g [m/s2 ] Ixx [kg · m2 ] Iyy [kg · m2 ] Izz [kg · m2 ] Ra [] k3e

0.285 9.807 5.136 × 10−3 5.136 × 10−3 5.136 × 10−3 0.67 0.03

Ir [kg · m2 ] kd [Nms2 ] km [Nm/A] ki [Nm/A] k1e k2e k4e

6 × 10−5 7.5 × 10−7 4.3 × 10−3 10−9 0.13 0.03 0.005

the fulfillment of the control objective (8). Then, we focus on demonstrating that the power consumption reduction methodology allows reducing the amount of power demanded by the quadrotor for whatever controller being used. The numerical simulations were implemented utilizing MatlabSimulink R2012b, a sampling time of 1 ms, and the ode3 Bogacki– Shampine method. Table 1 provides the quadrotor inertial parameters and the actuator parameters. Also, we include the effect of aerodynamic disturbances through the disturbance model presented in Section 5.3 of reference [6]. Besides, the proposed controller is compared against the PID controllers proposed by Wang et al. [50], Mellinger et al. [26], and Sumantri et al. [51]. These three controllers are referred in the following as the WC, MC, and SC controllers, respectively. The proposed controller is termed as PC controller. The structure of the WC, MC, and SC controllers is given in Appendix C. We considered numerical simulations when the quadrotor is not affected by any disturbance, and when the system is affected by aerodynamic disturbances, measurement noise, and parametric uncertainties. The parameters used for implementing the aerodynamical disturbances are included in Table 1. Also, the parameter uncertainty was implemented by considering an increment of 20% in all the quadrotor inertial parameters. Finally, the measurement noise was added to the position and orientation signals by corrupting them with Gaussian noise with zero mean and variance of 1 × 10−6 . The numerical simulations considered point-to-point and trajectory tracking tasks. The trajectories employed for each case

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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Fig. 4. Reference signals used in numerical simulations. The small arrows on each trajectory indicate the direction of motion. The circle and triangle show the beginning and the end of each reference, respectively.

are mathematically described by Eqs. (52) and (53), respectively. Each segment was designed through the method described in Section 4, with boundary conditions on velocity and acceleration equal to zero. The corresponding reference signals are depicted in Fig. 4. On the left-hand side, we depict the reference signal used for regulation, which corresponds to Eq. (52). The plot on the right represents the reference signal used for trajectory tracking, which is described by (53). The gains of each controller were tuned by using the methodology described in Section 4, employing the CSA with 15 nests and a probability p = 0.5. The CSA was implemented to make all the controllers to achieve the same value of the performance index (54). Fig. 5 shows the behavior of the performance index when applying the controller gains tuning procedure with each controller. The gains that provided the lowest performance index values were selected as the final gains for each controller. The numerical values are given in Appendix C. To analyze the quadrotor’s performance, we considered four scenarios: (1) The case NTNG, which does not include the power reduction methodology; (2) case TG, when the power reduction methodology is employed; (3–4) and cases NTNG-P and TG-P, which are the same as the first and second cases respectively, but including the effect of parametric uncertainties, measurement noise, as well as aerodynamical disturbances. Fig. 6 depicts the time evolution of the performance index L(r˜ , τ ) given in Eq. (54) for all controllers when using the pointto-point reference signal (52). The corresponding RMS numerical values are given in Table 2, where the lowest values are marked in bold. Fig. 7 depicts the time evolution of the position and orientation error signals from each controller for the TG-P case. The RMS numerical values of the error signals corresponding to the NTNG, TG, NTNG-P, and TG-P cases are given in Table 3. Fig. 8 depicts the time evolution of L(r˜ , τ ) for all controllers in the trajectory tracking task, i.e. when using the reference signal (53). The corresponding RMS numerical values are given in Table 4. The lowest values are marked in bold. Fig. 9 depicts the position and orientation errors of each controller for the TGP case. The position and orientation RMS error numerical values for the NTNG, TG, NTNG-P, and TG-P cases are given in Table 5. For the point-to-point task, the results from Table 3 indicate that in general, the SC controller has the lowest orientation errors for both, the unperturbed and the perturbed cases. Also, its performance is enhanced when using the power reduction

Fig. 5. The behavior of the performance index when applying the controller gains tuning stage with the CSA to the WC, MC, SC, and the PC controllers. Table 2 Point-to-Point task: RMS performance index values from the WC, MC, SC and PC controllers when using the reference trajectory (52): Without using the power reduction methodology in the free disturbance (NTNG) and disturbed (NTNG-P) cases; and when using the power reduction methodology in the free-disturbance (TG) and disturbed (TG-P) cases. Controller

NTNG

TG

NTNG-P

TG-P

WC MC SC PC

256.9482 27.8209 30.5920 11.9236

16.3695 8.2632 9.2658 5.8652

282.3965 28.1971 131.2491 16.9680

17.0687 9.7557 18.9498 9.5682

methodology. Regarding the coordinates x and z, the PC controller exhibits the best performance in almost all cases. Also, its performance in the y coordinate is close to the best performance obtained with the other methodologies. Besides, the information

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Fig. 6. Point-to-Point task: Behavior of the performance index (54) for the WC, MC, SC, and PC controllers: (1) Without using the power reduction methodology in the free disturbance (NTNG) and disturbed (NTNG-P) cases; and when using the power reduction methodology in the free-disturbance (TG) and disturbed (TG-P) cases. Table 3 Point-to-Point task: Position and orientation error RMS values of the WC, MC, SC, and PC controllers for the NTNG, TG, NTNG-P and TG-P cases. NTNG

WC MC SC PC

TG ex

ey

ez

eφ

eθ

eψ

ex

ey

ez

eφ

eθ

eψ

0.180 0.355 0.224 0.175

0.148 0.295 0.198 0.179

0.143 0.232 0.196 0.138

0.132 0.023 0.025 0.058

0.157 0.030 0.037 0.092

0.030 0.066 0.028 0.035

0.166 0.142 0.128 0.119

0.142 0.148 0.136 0.157

0.115 0.157 0.130 0.101

0.042 0.019 0.006 0.004

0.039 0.027 0.007 0.015

0.025 0.069 0.007 0.034

ex

ey

ez

eφ

eθ

eψ

ex

ey

ez

eφ

eθ

eψ

0.205 0.445 0.297 0.201

0.219 0.531 0.521 0.306

0.176 0.241 0.275 0.145

0.123 0.031 0.025 0.051

0.138 0.033 0.035 0.079

0.030 0.066 0.029 0.036

0.259 0.158 0.250 0.154

0.354 0.148 0.441 0.200

0.155 0.164 0.183 0.114

0.055 0.019 0.006 0.009

0.052 0.024 0.007 0.016

0.025 0.068 0.028 0.034

NTNG-P

WC MC SC PC

TG-P

provided by Table 2 indicates that the PC controller demands the lowest RMS values of the performance index, which indicates that it requires a lower amount of power to perform the required tasks. In the trajectory tracking task, from the information given in Table 5, again the SC has the lowest orientation error values, and the PC controller the lowest position error values. Also, we verify that the magnitude of this error is reduced when using the power reduction methodology. Finally, from the data given in Table 4, we verify again that the PC controller demands the lowest amount of power. Regarding the use of the power reduction methodology, the information provided by Tables 2 and 4 indicates that in general, the amount of power demanded by any controller is reduced when the power reduction methodology is applied. For instance, in the regulation task, the PC controller reduces its power demands in 50.8% for the unperturbed case, and 43.6% in the perturbed case. This performance improvement is also better for the trajectory tracking task, where the PC controller reduces its

Table 4 Trajectory tracking task: RMS performance index values from the WC, MC, SC, and PC controllers when using the reference trajectory (53): Without using the power reduction methodology in the free disturbance (NTNG) and disturbed (NTNG-P) cases; and when using the power reduction methodology in the free-disturbance (TG) and disturbed (TG-P) cases. Controller

NTNG

TG

NTNG-P

TG-P

WC MC SC PC

36.9384 3.3582 5.5301 2.2252

10.0962 0.6263 3.2358 0.3331

40.6464 6.8752 2802 2.2280

11.0572 2.4862 7.9421 0.6015

power demands in 85% for the unperturbed case, and 73% in the perturbed case. The same conclusion can be drawn for the other controllers. This fact verifies that the power reduction methodology contributes to reducing the amount of power demanded by each controller, i.e., the last part of the control objective is proved. Note that the PC controller remains to have a good performance even when the system is being affected by disturbances,

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Fig. 7. Point-to-Point task: Position and orientation error signals for the WC, MC, SC, and PC controllers when using the power reduction methodology and the quadrotor is being affected by parametric and aerodynamic disturbances (TG-P case).

Fig. 8. Trajectory tracking task: Behavior of the performance index (54) for the WC, MC, SC, and PC controllers: Without using the power reduction methodology in the free disturbance (NTNG) and disturbed (NTNG-P) cases; and when using the power reduction methodology in the free-disturbance (TG) and disturbed (TG-P) cases.

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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Table 5 Trajectory tracking task: Position and orientation error RMS values of the WC, MC, SC, and PC controllers for the NTNG, TG, NTNG-P and TG-P cases. NTNG

WC MC SC PC

TG ex

ey

ez

eφ

eθ

eψ

ex

ey

ez

eφ

eθ

eψ

0.114 0.223 0.127 0.079

0.115 0.222 0.148 0.115

0.069 0.120 0.099 0.069

0.030 0.006 0.008 0.018

0.031 0.005 0.009 0.029

0.025 0.057 0.024 0.030

0.058 0.050 0.042 0.039

0.021 0.004 0.006 0.002

0.0013 0.00057 0.00025 0.00015

0.015 0.005 0.004 0.008

0.015 0.008 0.005 0.021

0.021 0.057 0.024 0.029

ex

ey

ez

eφ

eθ

eψ

ex

ey

ez

eφ

eθ

eψ

0.161 0.371 0.178 0.110

0.129 0.331 0.163 0.195

0.096 0.141 0.107 0.073

0.033 0.009 0.008 0.018

0.038 0.016 0.010 0.036

0.025 0.057 0.024 0.030

0.263 0.191 0.270 0.079

0.109 0.156 0.118 0.080

0.069 0.064 0.103 0.034

0.026 0.006 0.004 0.012

0.040 0.009 0.005 0.021

0.021 0.057 0.024 0.029

NTNG-P

WC MC SC PC

TG-P

Fig. 9. Trajectory tracking task: Position and orientation error signals for the WC, MC, SC, and PC controllers when using the proposed power reduction methodology and the quadrotor is being affected by parametric and aerodynamic disturbances (TG-P case).

while the other controllers are more affected. This fact verifies the robustness of the PC controller to parametric uncertainties, measurement noise, state estimation error, and aerodynamic disturbances. Also, the PC controller has the lowest RMS values of the performance index for all cases in both point-to-point and trajectory tracking tasks. For instance, Table 2 indicates that the RMS values of the performance index for the WC, MC and SC controllers in the NTNG-P case are 21.5, 2.3 and 2.6 times the value demanded by the proposed PC controller. We obtain similar values for the TG, TG-P, and NTNG-P cases. Besides, the same conclusions can be drawn from Table 4 for the trajectory tracking task. Tables 3 and 5 indicate that, for all the controllers, the RMS tracking errors are also reduced for the TG and TG-P cases in comparison to the values obtained in the NTNG and NTNG-P cases, respectively. This fact points out that, by using the power reduction methodology, it is possible to enhance the performance of a given system without requiring to change the controller, and

also to reduce the power demanded by the actuators. This conclusion, together with Theorem 1, demonstrates that the control objective is achieved. Fig. 10 shows the time evolution of the current corresponding to each of the four quadrotor motors in a point-to-point task when using the proposed controller, the power reduction methodology, and the system is being affected by parametric and aerodynamic disturbances (TG-P case). Signals imi , i = 1, 2, 3, 4 correspond to the current in the ith motor. The data indicate that the required current levels are feasible for a real motor. In conclusion, the PC controller exhibits good performance and is not considerably affected by either endogenous or exogenous disturbances. Also, its power demands are small when compared to that required by the other control methodologies. This fact indicates that the PC controller is a good choice for quadrotor control in both regulation and trajectory tracking tasks. Besides, the power reduction methodology allows enhancing the controller performance and the power demands. Hence, it can be considered an additional tool for increasing the autonomy of a quadrotor vehicle.

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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Fig. 10. Time evolution of the currents corresponding to each motor for the point-to-point task, when using the proposed power-reduction methodology and the system is affected by parametric and aerodynamic disturbances, i.e., the TG-P case.

6. Conclusions

Declaration of competing interest

This paper presented a novel robust PID control methodology for controlling a quadrotor UAV and a procedure for reducing the power demanded by the controller. The stability of the proposed controller was studied using Lyapunov-based techniques. Two additional PID control methodologies existing in the literature with slight variations and a robust sliding modes-based controller were used for comparing the performance of the proposed controller. Numerical simulations were used to assess the performance of the proposed controller, including disturbances due to parametric uncertainties and aerodynamic disturbances. It was shown that the proposed controller exhibits good performance in regulation and trajectory tracking tasks, even when the system is being affected by internal or external disturbances. Also, it requires a lower amount of power than that demanded by the other methodologies. Besides, it was also shown that, by using the proposed power reduction methodology, it is possible to reduce the tracking error and power consumption of any controller, although the proposed controller remains as the one having the lowest power demand. Hence, the proposed PID-P controller is a good choice for controlling a quadrotor vehicle in real-time applications. Many aspects deserve to be studied in future work. Some of them include: Carry out real-time experiments in order to verify the performance of the proposed controller and power reduction methodology in a real platform; checking the behavior of the proposed power reduction methodology when the quadrotor battery has low energy levels; and compare the performance of the controller-gains tuning stage by using other algorithms instead of the CSA.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the Consejo Nacional de Ciencia y Tecnología (CONACyT), Mexico under the Cátedras Project 1537: ‘‘Análisis y Control de Sistemas Mecatrónicos Complejos’’. This work was supported in part by CONACyT under Grant 285279. Also, R. Miranda thanks to Lorena Ledón for her valuable and priceless support during the writing of this manuscript. Appendix A. Structure of matrices Mo and Co By using the Christoffel symbols, the explicit description of matrices Mo (η) and Co (η, η˙ ) in Eq. (2) is as follows

[ Mo (η) =

m11 0 m31

0 m22 m32

m13 m23 m33

] (A.1)

with m11 = Ixx cθ2 + Izz s2θ , m13 = [−Ixx + Izz ] cφ sθ cθ , m22 = Iyy , m23 = Iyy sφ , m33 = Ixx cφ2 s2θ + Iyy s2φ + Izz cφ2 cθ2 , and

[ Co (η, η˙ ) =

C11 C21 C31

C12 C22 C32

C13 C23 C33

] ,

(A.2)

with C11 = [−Ixx + Izz ] θ˙ sθ cθ , C12 = [−Ixx + Izz ] φ˙ sθ cθ + 21 [−Ixx +Izz ] ψ˙ cφ c2θ − 12 Iyy ψ˙ cφ , C13 = 12 [−Ixx + Izz ] θ˙ cφ c2θ − 21 Iyy θ˙ cφ + ˙ sφ cφ s2θ − Iyy ψ˙ sφ cφ + Izz ψ˙ sφ cφ cθ2 , C21 = [Ixx − Izz ] φ˙ sθ cθ + Ixx ψ

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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˙ cφ + [Ixx − Izz ] ψ˙ cφ c2θ , C22 = 0, C23 = Iyy ψ Iyy φ˙ cφ + 2 ] 2 ˙ ˙ ˙ [Ixx − Izz ] φ cφ c2θ + [−Ixx + Izz ] ψ cφ sθ cθ , C31 = [Ixx − Izz ] φ sφ sθ cθ + [ ] 1 ˙ sφ cφ s2θ + Iyy ψ˙ sφ cφ − Izz ψ˙ [−Ixx + Izz ] θ˙ c[φ c2θ + Iyy θ˙ cφ − Ixx ψ 2 ] 1 2 ˙ ˙ cφ2 sθ cθ , sφ cφ cθ , C32 = 2 Iyy φ cφ + [−Ixx + Izz ] φ˙ cφ c2θ + [Ixx − Izz ] ψ 2 2 ˙ ˙ ˙ ˙ C33 = −Ixx φ sφ cφ sθ + Iyy φ sφ cφ − Izz φ sφ cφ cθ + [Ixx − Izz ] θ cφ2 sθ cθ .

[ 1

[ 1

]

2

Appendix B. Proof of Theorem 1 Let us consider the next Lyapunov candidate function 1 1 (B.1) V (r˜ , r˙˜ , ξ, s, η˜ ) = α1 xTp Q1 xp + α2 xTo Q2 xo , 2 2

[

where xTp = r˙˜ T , r˜ T , ξ T q1,11 q1,12 q1,13

[ Q1 =

q1,12 q1,22 q1,23

]T

[

, xTo = sT , η˜

q1,13 q1,23 q1,33

T

] , Q2 =

]T

[

,

Mo 03

+

[

1 2

1 2 1

]

03

ϵ3 I3

mr˙˜ T r˙˜ + mϵ2 r˙˜ T r˜ + mϵ1 r˙˜ T ξ

r˜ T [Kp + ϵ2 Kd − mϵ1 I3 ]˜r + r˜ T [ϵ1 Kd + Ki ]ξ

2

2

2

mr˙˜ T r˙˜ ≥

1 2

 2  

m r˙˜  ,

(B.2)

     mϵ2 r˙˜ T r˜ ≥ −mϵ2 r˜  r˙˜  ,     mϵ1 r˙˜ T ξ ≥ −mϵ1 r˙˜  ∥ξ∥ , 1 2

(B.3) (B.4)

r˜ T [Kp + ϵ2 Kd − mϵ1 I3 ]˜r ≥ λa1 ,

(B.5)

r˜ T [ϵ1 Kd + Ki ]ξ ≥ λa2 ,

(B.6)

ξ T [ϵ1 Kp + ϵ2 Ki ]ξ ≥ λa3 ,

(B.7)

1 2

]  2

with λa1 = 21 λmin (Kp )+ ϵ2 λmin (Kd ) − mϵ1 r˜  , λa2 = − [ϵ1 λmax (Kd ) + λmax (Ki )] r˜  ∥ξ∥, and λa3 = 0.5[ϵ1 λmin (Kp ) + ϵ2 λmin (Ki )] ∥ξ∥2 . Inequalities (B.2)–(B.7) allow bounding V from below as follows

[

] −mϵ1 va,23 V ≥ rv rv , (B.8) va,33 [     ]T   with rv = r˙˜  r˜  ∥ξ∥ , va,22 = λmin (Kp ) + ϵ2 λmin (Kd ) − mϵ1 , va,23 = −[ϵ1 λmax (Kd ) + λmax (Ki )], va,33 = ϵ1 λmin (Kp ) + ϵ2 λmin (Ki ), and it can be verified that V is positive definite if [

T

m −mϵ2 −mϵ1

−mϵ2 va,22 va,23

inequalities (41)–(42) hold. The time derivative of V along the trajectories of (30)–(34) is

[

˜ r˙˜ T + ϵ1 ξ T + ϵ2 r˜ T [r¨d + gez ] −m [ ] ˜ . + α2 −sT K s − ϵ3 η˜ T Λη˜ + ϵ3 η˜ T s − sT Φ Θ Let us consider the inequalities

      h2 η˙ d 2   h1 η¨d +    ∥Λ∥η˜  h1 ∥Λ∥ η˙˜  + h2 η˙ d   d5 ∥s∥ + h2 η˙ d  ∥s∥  2 ∥2 η˜  h2 ∥Λ h2 ∥Λ∥ η˜  ∥s∥

d1 , d5 ∥s∥ , d2 ∥s∥ ,

≤ ≤ ≤ ≤ ≤

(B.9)

 2

 d3  η˜  , d4 η˜  ∥s∥ .

˜ is obtained Then, the next upper bound on Φ Θ (B.10)

 [  ]T   ˙ m  (B.11)  ˜ r˜ + ϵ1 ξ + ϵ2 r˜ [r¨d + gez ] ≤ n11 , T  s Φ Θ ˜  ≤ n12 , (B.12)   τ p (ηd ) − τ p (η) ≤ n13 , (B.13) [  ]T [  ]  r˙˜ + ϵ1 ξ + ϵ2 r˜ τ p (η d ) − τ p (η )  (B.14)   ≤ n14 , ⏐ ⏐     ˜ ⏐ [r˜  + ϵ1 ∥ξ∥ + ϵ2 r˜ ], n12 = [d1 + d2 ∥s∥+ with n11 = c4 ⏐m      2      ˜ , n13 = g1 ∥η∥ [c1 + c2  d3 η˜ + d4 η˜ ∥s∥] ∥s∥ Θ [r˙˜ , r˜ , ξ]], ] [      n14 = c3 r˙˜  + ϵ1 ∥ξ∥ + ϵ2 r˜  ∥η∥, where c3 ≥ g1 [c1 +     c2 [r˙˜ , r˜ , ξ]] ≥ g1 [c1 + c2 rc ], d1 , d2 , d3 , d4 , c1 , c2 , c3 , c4 , g1 are 3

Let us consider the next inequalities 1

]

with h1 , h2 , d1 , d2 , d3 , d4 and d5 being positive constants. Also, note that

,

] ξ T [ϵ1 Kp + ϵ2 Ki ]ξ 2 [ ] 1 T 1 T + α2 s Mo s + ϵ3 η˜ η˜ .

+

13

]

]        [ Φ Θ ˜ , ˜  ≤ d1 + d2 ∥s∥ + d3 η˜ 2 + d4 η˜  ∥s∥ Θ

q1,11 = mI3 , q1,12 = mϵ2 I3 , q1,13 = mϵ1 I3 , q1,22 = Kp + ϵ2 Kd − mϵ1 I3 , q1,23 = ϵ1 Kd + Ki , q1,33 = ϵ1 Kp + ϵ2 Ki , and α1 , α2 , ϵ1 , ϵ2 , ϵ3 are positive constants. Note that V = α1

[

V˙ = α1 −r˙˜ T [Kd − mϵ2 I3 ] r˙˜ − r˜ T ϵ2 Kp − ϵ1 Kd − Ki r˜

[

−ξ T [ϵ1 Ki ] ξ [ ][ ] + r˙˜ T + ϵ1 ξ T + ϵ2 r˜ T τ p (ηd ) − τ p (η)

]

positive constants, and Brc3 = {x ∈ R9 : ∥x∥ ≤ rc3 }. By using inequalities (B.11)–(B.14), the time derivative of function V can be upper bounded as follows V˙ ≤ α1 c3 r˜  + ϵ1 ∥ξ∥ + ϵ2 r˜  η˜ 

 ]   [  ⏐ ⏐ [   ] ˜ ⏐ r˜  + ϵ1 ∥ξ∥ + ϵ2 r˜  + α1 c4 ⏐m [ ]  2     ˜ + α2 d1 + d2 ∥s∥ + d3 η˜  + d4 η˜  ∥s∥ ∥s∥ Θ [ ]T [ ] ∥s∥ ∥s∥ − α1 rvT Q3 rv − α2  Q4  η˜  η˜  ,

(B.15)

where matrices Q3 , Q4 are defined in (47) and (48), respectively. Furthermore, matrices Q3 , Q4 in (B.15) are positive definite if inequalities (41), (42) and (43) hold.  

   

T

˙    Let us define the vectors σ1 =  [r˜  , r˜ , ∥ξ∥] , σ2 = 

   T   [∥s∥ , η˜ ] . Then, inequality (B.15) can be written as follows [ ]T [ ] ⏐ ⏐√ σ1 σ1 ˜ ⏐ 1 + ϵ12 + ϵ22 σ1 V˙ ≤ − Q5 + α1 c4 ⏐m σ2 σ2  [ ] ˜  d1 σ2 + [d3 + d4 ] σ23 , + α 2 Θ

where Q5 is defined in (49) and is positive definite if the inequalities BRσ of the state-space { (44)–(45) hold. Thus, for a region } BRσ = [σ1 , σ2 ] ∈ R2 : ∥[σ1 , σ2 ]∥ ≤ rσ , it follows that

     σ1 2    + ao  σ 1    σ2 σ2   [     ]  σ1       a1  σ1  + [λmin (Q5 ) − a1 ]  σ1  − ao , = −  σ2   σ2   σ2  √     ⏐ ⏐ ˜ , α 2 Θ ˜ ˜ ⏐ 1 + ϵ12 + ϵ22 , α2 d1 Θ with ao = max{α1 c4 ⏐m [ ] 2 d3 + d4 rσ }, and a1 > 0 is a constant. V˙ ≤ −λmin (Q5 ) 

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

14

R. Miranda-Colorado and L.T. Aguilar / ISA Transactions xxx (xxxx) xxx

SC controller are

If inequalities (46) and

√

   σ1  ao    σ2  > λmin (Q5 ) − a1

(B.16)

Appendix C. Controllers used for comparison

[

⎡

] ⎤

⎤ τ τ ⎥ ⎦. τ

1 Ixx 2 1 Iyy 3 1 Izz 4

(C.1)

Finally, the SC controller consists of a sliding mode controller with a PID-like sliding surface, which is applied to the quadrotor nonlinear dynamic model. The WC controller is described by the next set of equations F =

t

∫

mg cθ cφ

ez (ϑ )dϑ + kdz e˙ z ,

+ kpz ez + kiz

(C.2)

0

τ¯ o = KpW eoW − KdW η˙ , ] [ ∫ t ex (ϑ )dϑ + kdx e˙ x , θd = − kpx ex + kix 0 ∫ t ey (ϑ )dϑ + kdy e˙ y , φd = kpy ey + kiy

(C.4) (C.5)

where ex = xd − x, ey = yd − y, ez = zd − z are the position errors; eoW = [eφ , eθ , eψ ]T , eφ = φd − φ , eθ = θd − θ , eψ = ψd − ψ are orientation errors; KpW = diag {kpφ , kpθ , kpψ }, KdW = diag {kdφ , kdθ , kdψ } are positive definite matrices, and kpz , kdz , kiz , kpx , kix , kdx , kpy , kiy , kdy are positive constants. The MC controller is described by (C.6)

τ¯ o = Iη¨ + KpoM eoM + KioM

t

∫

eoM (ϑ )dϑ 0

+ KdoM eoM , ¨rcM = r¨d + KdM [r˙d − r˙ ] + KpM [rd − r] ∫ t [rd − r] dϑ, + KiM

(C.7)

(C.8)

0

φd = θd =

1[ g 1[ g

x¨ cM sψd − y¨ cM cψd ,

(C.9)

x¨ cM cψd + y¨ cM sψd ,

(C.10)

] ]

[

]

−

(C.12)

(C.13)

(C.14)

[ Ir =

Ixx cθ 0 Izz sθ

0 Iyy 0

−Ixx cφ sθ Iyy sφ Izz cφ cθ

] ,

(C.15)

γ1 = [Ixx + Iyy − Izz ]φ˙ θ˙ sθ + [−Ixx + Iyy − Izz ]φ˙ ψ˙ sφ sθ + [Ixx + Iyy − Izz ]θ˙ ψ˙ cφ cθ + [Iyy − Izz ]ψ˙ 2 sφ cφ cθ , γ2 = [−Iyy + [−Ixx + Izz ]c2θ ]φ˙ ψ˙ cφ + [−Ixx + Izz ][φ˙ 2 − ψ˙ 2 cφ2 ]sθ cθ , γ3 = [Ixx − Iyy − Izz ]φ˙ θ˙ cθ + [Ixx − Iyy + Izz ]φ˙ ψ˙ sφ cθ + [−Ixx + Iyy + Izz ]θ˙ ψ˙ cφ sθ + [−Ixx + Iyy ]ψ˙ 2 sφ cφ sθ , v = v¯ + K3 s + Q sign(s), ¯ r, v¯ = r¨d + Λr˙˜ + α˜ ∫ t ¯ r˜ + α¯ s = r˙˜ + Λ r˜ (ϑ )dϑ, 0

(C.3)

0

F = mg + mz¨cM ,

]

where γ = [γ1 , γ2 , γ3 ]T , v = [vx , vy , vz , vφ , vθ , vψ ]T ,

The WC controller consists of a PID action applied to the altitude control of the quadrotor, while the attitude is controlled using a Proportional–Derivative (PD) controller. The MC methodology is a modification of the controller developed in [26] where both, the altitude and attitude of the quadrotor are controlled using PID controllers. These two controllers are applied to the next quadrotor linearized dynamic model g [θ cψd + φ sψd ] ⎢ r¨ = ⎣ g θ sψd − φ cψd ⎦ , η¨ = ⎣ 1 F −g m

τ¯ o = Ir

vφ vθ vψ [

(C.11)

γ1 γ2 , γ3 ] v s − v x ψ y cψ −1 , φd = tan vz + g ⎡ ⎤ v c + v s x ψ y ψ ⎦, θd = tan−1 ⎣ √[ ]2 vx sψ − vy cψ + [vz + g]2

[

hold, it follows that V˙ ≤ −a1 ∥[σ1 , σ2 ]∥2 . Therefore, according to Theorem 4.18, pp. 172 in [44], we conclude that the trajectories [r˙˜ , r˜ , ξ, η˜ , s] are uniformly ultimately bounded for some compact set of initial conditions [44]. Finally, after computing the torque vector τ o , we may use matrix W from (7) together with Eq. (3) to obtain the torques τ2 , τ3 , τ4 as indicated in (36), which completes the proof. ■

⎡

F = m vx2 + vy2 + [vz + g]2 ,

where KpoM = diag {kpφ M ,pθ M ,pψ M }, KioM = diag {kiφ M ,iθ M ,iψ M }, KdoM = diag {kdφ M ,dθ M ,dψ M }, KpM = diag {kp1M , kp2M , kp3M }, KiM = diag {ki1M , ki2M , ki3M }, KdM = diag {kd1M , kd2M , kd3M } are positive definite matrices; eoM = eoW , and rcM = [xcM , ycM , zcM ]T . The SC controller was designed by considering the quadrotor nonlinear dynamic model (1)–(2). The equations describing the

K3 = diag {k31 , k32 , k33 , k34 , k35 , k36 }, Q = diag {q1 , q2 , q3 , q4 , q5 , ¯ = diag {λ¯ 1 , λ¯ 2 , λ¯ 3 , λ¯ 4 , λ¯ 5 , λ¯ 6 }, and α¯ = diag {α¯ 1 , α¯ 2 , α¯ 3 , α¯ 4 , q6 }, Λ α¯ 5 , α¯ 6 } are positive definite diagonal matrices. After tuning the PC, WC, MC, and SC controllers through the procedure described in Section 4, we obtained the next values:

• PC: kp1 = 9.3026, kp2 = 5.4638, kp3 = 13.6632, kd1 = 6.3434, kd2 = 6.3917, kd3 = 6.4503, ki1 = 0.0926, ki2 = 0.0197, ki3 = 0.0959, λ1 = 21.2403, λ2 = 19.8394, λ3 = 10.8302, k1 = 0.4723, k2 = 0.4801, k3 = 2.3441. • WC: kpx = 0.8493, kix = 0.0104, kdx = 0.8577, kpy = 0.8358, kiy = 0.0082, kdy = 1.1615, kpz = 5.4532, kiz = 0.079, kdz = 3.1281, kpφ = 37.862, kdφ = 4.4508, kpθ = 37.5587, kdθ = 5.1839, kpψ = 88.3806, kdψ = 6.5328. • MC: kp1M = 19.7999, kp2M = 9.4806, kp3M = 28.5796, ki1M = 0.1024, ki2M = 0.095, ki3M = 0.0921, kd1M = 10.7832, kd2M = 10.1723, kd3M = 32.632, kpφ M = 2018.5, kiφ M = 0.1038, kdφ M = 88.9676, kpθ M = 4733.5, kiθ M = 0.1013, kdθ M = 195.9768, kpψ M = 282.335, kiψ M = 5.0128, kdψ M = 108.8706. • SC: k31 = 3.3354, k32 = 1.7149, k33 = 1.4739, k34 = 23.7404, k35 = 48.6369, k36 = 47.6822, q1 = 0.2891, q2 = 0.3232, q3 = 0.4972, q4 = 52.6812, q5 = 51.2313, q6 = 10.5882, λ¯ 1 = 2.795, λ¯ 2 = 3.8763, λ¯ 3 = 8.5806, λ¯ 4 = 87.8798, λ¯ 5 = 78.4185, λ¯ 6 = 28.87, α¯ 1 = 0.0009, α¯ 2 = 0.0011, α¯ 3 = 0.0011, α¯ 4 = 0.0368, α¯ 5 = 0.02, α¯ 6 = 0.0583. Note that the gains of the PC controller fulfill the stability conditions given by inequalities (41)–(46). An example of this can be obtained by selecting ϵ1 = ϵ3 = 1, ϵ2 = 2, d1 = d3 = d4 = 10−5 , d2 = 0.1, α1 = 0.1, α2 = 2, c3 = 0.01, c4 = 10−5 , and a parameter variation of 50%.

Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.

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Please cite this article as: R. Miranda-Colorado and L.T. Aguilar, Robust PID control of quadrotors with power reduction analysis. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.045.