Variable sampling time discrete sliding mode control for a flapping wing micro air vehicle using flapping frequency as the control input

Joshua Hill; Farbod Fahimi

doi:10.1515/nleng-2024-0017

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Variable sampling time discrete sliding mode control for a flapping wing micro air vehicle using flapping frequency as the control input

Joshua Hill and Farbod Fahimi

Published/Copyright: August 9, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

In this article, two main assumptions commonly used in the reported literature are relaxed. First, instead of a continuous time controller, a discrete time controller is proposed. Second, the controller uses the flapping frequency as opposed to flapping force/moments as the control input commonly assumed by other research groups. A discrete time controller is more suitable for implementation and deployment to most real flapping wing micro aerial vehicles (FWMAVs), in which a computer controls a sampled system and the flapping wings are driven by servomotors. A robust reaching law based on the discrete sliding mode control method is proposed to stabilize the vertical altitude of a simulated FWMAV. The continuous time model of the FWMAV is converted to a discrete time model, and the discrete time sliding mode controller is developed for the new form of the model. Using the variable sampling time, the robust discrete sliding mode controller developed in this article is able to control a simulated FWMAV to successfully stabilize its position in spite of a 16% difference in model parameters, as well as a 0.7 m/s wind gust. The controller was able to withstand the wind gust with negligible deviation from its desired position. These simulation results further show an input frequency range similar to that of the experimentally derived model previously reported in the literature.

Keywords: flapping wing; air vehicle; discrete sliding model control; flapping frequency

1 Introduction

1.1 Literature survey

Micro aerial vehicles (MAVs) are a class of miniature air vehicles that have a wingspan of less than 15 cm. These vehicles show significant promise in sensing and information gathering capabilities in fields such as environmental monitoring and homeland security. Inspired by nature, flapping wing micro aerial vehicles (FWMAVs) have the potential to augment the current capabilities of MAVs in the military and civilian field due to their capacity to be more aerodynamically agile and efficient, while appearing benign. There has been significant research over the years to study and create robotic representations of what is found in nature, such as the herring gull inspired Festo SmartBird, the hummingbird inspired aerovironment nano Hummingbird, and the dragonfly inspired Festo BionicOpter. The aerodynamics of these FWMAVs are complex and nonlinear, and their small size makes them highly susceptible to external disturbances such as wind gusts. To overcome these challenges, robust control methods are necessary to overcome these complexities and achieve the desired flight characteristics.

Over the past decade, there has been much attention given to FWMAVs and flight control methodologies. Simple proportional-integral-derivative (PID) controllers designed as proof of concepts for new FWMAV designs to prove out attitude control and hovering [1–3], but require the system to train in an idealized environment with the inability to compensate for disturbances or uncertainties. A nonlinear proportional-derivative (PD) controller that uses flapping angle amplitude for attitude control was explored by Wang [4]. A linear Proportional-integral (PI) controller is presented by Gayango et al. [5]. Trajectory tracking control [6] requires much knowledge of the systems states.

The disturbance observer based control [7], while it can handle limited disturbance and uncertainty, needs a nominal PID controller. Adaptive control methods are attractive in that they can balance controller parameters to provide optimal performance based on particular situations, such as reaching and settling time. Neural networks [8–10] and other iterative learning controllers [11,12] require a high computational cost and require adequate training. Active disturbance rejection controllers, such as those presented by Liang et al. [13] and Feng et al. [14] are cumbersome to implement, requiring much rigor to derive. Model-free methods have advantages over model-based strategies, in that they require less information about the FWMAV model, and are improved on with adaption methods [15]. However, they also have disadvantages, such as higher computational loads due to less transparent fuzzy logic rules [16]. Sliding mode control (SMC) is one of the best known methods for handling disturbances and uncertainties and has been successfully used with a variety of adaptation methods to further improve the performance of FWMAVs [17–19]. In addition, a discrete time model predictive control was simulated by Zheng et al. [20] and a continuous time observer-based adaptive control was simulated by Meng et al. [21].

In the study by Khosravi and Novinzadeh [22], a model-free adaptive control is augmented with a discrete sliding mode controller (DSMC) with good results. Others have also reported research on robust DSMC for tracking periodic signals and repetitive tasks [23,24], but have used different applications than the FWMAV application.

1.2 Contributions

The control modes, methods, inputs, and the test methods used in the cited literature are listed in Table 1 as a quick reference. It can be seen that the majority of the research work done on FWMAVs assume that (1) the moments can be used as the control input, and (2) the system is operating in the continuous time domain. Most work, that make the aforementioned two assumptions use simulations to demonstrate the effectiveness of their proposed method.

Table 1

Methodology comparison

Citation	Control mode	Control method	Control input	Test method
[1]	Continuous	PID	Servo speed	Experiment
[2]	Continuous	PID	Servo speed	Experiment
[3]	Continuous	PID	Moments	Experiment
[4]	Continuous	PD	Flapping amplitude	Simulation
[5]	Continuous	PI	Thrust/pitch	Simulation
[6]	Continuous	Linear state feedback	Wing angles	Simulation
[7]	Continuous	PID	Servo speed	Experiment
[8]	Continuous	NN adaptive	Servo speed	Simulation
[9]	Continuous	NN adaptive	Servo speed	Simulation
[10]	Continuous	PD	Flapping amplitude	Simulation
[11]	Discrete	Learning NN	Moments	Simulation
[12]	Continuous	LQR method	Wind angles	Simulation
		Iterative learning tuning
[13]	Continuous	State feedback	Moments	Simulation
[14]	Continuous	Dynamic inversion	Moments	Simulation
[15]	Discrete	Model free adaptive	Moments	Simulation
[16]	Continuous	Adaptive fuzzy	Flapping amplitude	Simulation
[17]	Continuous	Adaptive sliding mode	Moments	Simulation
[18]	Continuous	Adaptive sliding mode	Voltage	Experiment
[19]	Continuous	Adaptive	Voltage	Experiment
[20]	Discrete	Predictive control	Force/moments	Simulation
[21]	Continuous	Adaptive	Forces	Simulation
[22]	Discrete	Sliding mode	Amplitude/phase shift	Simulation

These two assumptions break down for testing a majority of real FWMAVs. The reason for the breakdown of these assumptions is as follows. The means of the real world control input to most FWMAVs is changing the flapping frequency of the wings. There is no real FWMAV in which the flapping force and moment can be specified in continuous time as the control input because that is practically impossible.

Most FWMAV hardware platforms designed for size and weight reduction are incapable of controlling moments, but can control the flapping frequency by manipulating the speed (or input voltage) of a servomotor. This fact is supported by studying Table 1. It can be seen that the majority of the experimental work use servo speed or voltage (which changes the servo speed) as the control input.

In this work, first, a discrete time SMC is proposed using a linear plant. Second, the flapping frequency is proposed as the input to the control system via a virtual control that uses the nonlinear conversion to update the sampling time. The combination of the aforementioned two features is the core of the contribution of this work.

The proposed method in this publication eliminates the need for the aforementioned assumptions (i.e., using force/moment as control input and using a continuous time domain for control). First, we use the flapping frequency as the control input, which can be easily specified by changing the speed of the motor that generates the flapping motion. Second, we keep the frequency of flapping constant for a full flapping cycle. This allows for the accurate use of averaging the force/moment generated by a full cycle flapping with a constant frequency. The state sampling is done at the beginning of a flapping cycle. The controller determines the flapping frequency using a discrete-time feedback. This frequency is applied for a full flapping cycle, which constitutes a sampling duration. Then, at the end of the flapping cycle, the states and control are updated again, and the process continues. Third, since the frequency is kept constant during the complete cycle, we have to use a discrete-time controller, and a continuous-time controller is rendered useless!

Using a continuous time system means that the flapping frequency (related to servo speed) can change in the middle of a flapping cycle. As a consequence, the commonly used “averaging of the aerodynamic forces/moments” in FWMAV modeling (which is normally done for a full cycle) breaks down. With our proposed discrete time control method, the flapping frequency will change at the start of each cycle. Once a cycle starts, it will complete with a constant frequency. Then, the next cycle will have a new frequency determined by the feedback control. As a consequence, the assumption of “averaging the aerodynamic forces/moments” holds, which reduces the uncertainty terms in the model.

It should be noted that once the flapping frequency for the next step is determined by the discrete time SMC, the duration of the full flapping cycle is determined by T = 1 ∕ f , where T is the full cycle duration and f is the flapping frequency. Then, the sampling time for the current step is T , which varies as the controller calculates the input frequency through time. That is how a variable sampling time system is implemented.

In Section 2, the detailed derivation of the proposed discrete time SMC is presented. In Section 3, the implementation of the DSMC on the 2D flight of a FWMAV is demonstrated. In Section 4, the robustness in the face of system uncertainties and disturbances are discussed. In Section 5, some simulations results are presented. Finally, in Section 6, the article is concluded.

2 Derivation of the DSMC

The discrete time SMC is derived using the reaching law approach presented in [25]. Consider a discrete plant:

(1) X ( k + 1 ) = A X ( k ) + b u ( k ) ,

where X is a vector of length n , u is a scalar representing the control input, A is a n × n state matrix, and b is a n × 1 control parameter vector.

2.1 Reaching law design

The following are necessary for designing a reaching law that achieves a stable error behavior around the discrete sliding surface:

Requirement 1: The system trajectory shall move monotonically toward a switching plane s and cross it in finite time, regardless of the initial state. Requirement 2: After the system has crossed the switching plane, it shall cross the plane in each successive sampling step, resulting in a zigzagging motion centered on the switching plane. Requirement 3: The size of each successive zigzagging step shall not increase and shall remain within a bounded region around the switching plane.

For a discrete time system, the reaching law from Gao et al. [25] is considered

(2) s ( k + 1 ) − s ( k ) = − q T s ( k ) − ε T sgn ( s ( k ) ) ,

where 0 < 1 − q T < 1 , ε > 0 , and T > 0 is the sampling time.

This equation can be broken into its two parts. The first, − ε T sgn ( s ( k ) ) , ensures that the switching variable, s ( k ) , reaches the switching manifold at a constant rate given by − ε T . Additionally, due to the signum function, sgn , it will reverse, or switch, the direction on crossing the switching manifold. The second part, − q T s ( k ) , is a proportional term that will force the state to approach the switching manifold faster when s ( k ) is larger. As long as the inequality 0 < 1 − q T < 1 holds, it guarantees that the first requirement is satisfied. The latter two requirements are guaranteed by the signum properties in the discrete time domain.

Until now, the reaching law (2) guarantees that the surface variable s ( k ) approaches the surface and stays in its vicinity as time passes. While this is an excellent way to reach the surface, there must be guarantees that the states slide to the desired point after the surface has been reached. The next section discusses how this can be achieved.

2.2 Switching surface design

Consider a linear switching surface.

(3) s ( k ) = c T X ( k ) = 0 ,

where c is a n × 1 vector and c T b ≠ 0 . The surface Eq. (3) is selected linear in terms of the control gains c i . This is because a linear surface Eq. (3) leads to a linear Eq. (9) in terms of gains c i , which is easier to solve in terms of the gains.

As mentioned in the previous section, the reaching law (2) guarantees that the surface variable stays in the vicinity around the surface, which is referred to as the quasi-sliding mode (QSM). The ideal QSM satisfies the following condition:

(4) s ( k + 1 ) = s ( k ) = 0 , k = 0 , 1 , 2 … .

This condition means that if s ( k ) is zero, then s ( k + 1 ) must also be zero, ensuring that s ( k ) remains zero through time. Merging (3) and (4), and noting (1), one can obtain:

(5) s ( k + 1 ) = c T X ( k + 1 ) = c T [ A X ( k ) + b u e ( k ) ] = 0 .

Here, u e ( k ) , the equivalent control, is used instead of u ( k ) , indicating the control effort needed at the ideal QSM.

Solving (5) for the equivalent control u e can be determined as follows:

(6) u e ( k ) = − ( c T b ) − 1 c T A X ( k ) .

The dynamic equation while on the ideal sliding mode can be found by plugging in the equivalent control (6) into the following dynamic Eq. (1):

(7) X ( k + 1 ) = A ˆ X ( k ) ,

where

(8) A ˆ = [ I − b ( c T b ) − 1 c T ] A .

The vector c needs to be determined to guarantee the stability of the ideal QSM. Note that the desired behavior of the system requires that the system states error be decreasing. This can be achieved by ensuring that the roots of the characteristic polynomial of the system in (7) are between 0 and 1. The characteristic polynomial of the system in (7) is found as follows:

(9) ∣ λ I − [ I − b ( c T b ) − 1 c T ] A ∣ = 0 ,

where the gain c is designed such that the roots of the above polynomial are between 0 and 1.

2.3 Demonstration of the achievement of the QSM

The switching law is defined in Eq. (3) in Section 2.1. However, it is important to know the bounds of s ( k ) , or the width of the stabilization band achieved by the switching law. In this section, the bounds of s ( k ) or width of the stabilization band is determined.

Theorem 1

Consider the sliding mode reaching law (3). After the sliding variable s ( k ) crosses 0, it crosses 0 repeatedly in all future sampling times (defined as QSM), while it remains bounded within a band with a width equal to Δ = 2 ε T 1 − q T around 0.

Proof

According to the definition of the QSM, the sign of s ( k + 1 ) must be opposite to that of s ( k ) , indicating the s trajectory crosses 0 at each sampling time. The band around the surface s ( k ) = 0 , where every state s ( k ) satisfies this condition, makes up the QSM band. The control law (21) is now substituted into the original system dynamics (1), and both sides of the resulting equation is multiplied by c T , giving the response of the discrete variable structure control (VSC):

(10) c T X ( k + 1 ) = c T [ A X ( k ) − b ( c T b ) − 1 [ c T A X ( k ) − c T X ( k ) + q T c T X ( k ) + ε T sgn ( c T X ( k ) ) ] ] .

Noting that s ( k ) = c T X ( k ) , (10) becomes:

(11) s ( k + 1 ) = ( 1 − q T ) s ( k ) − ε T sgn ( s ( k ) ) , where 1 − q T > 0 .

With the response of the controlled system found, the width of the QSM band needs to be determined. To determine the width of this band, consider the case, where s ( k ) > 0 , and the s trajectory crosses the surface s = 0 . It follows that s ( k + 1 ) < 0 . Using (11) and noting that sgn ( s ( k ) ) = 1 :

(12) ( 1 − q T ) s ( k ) − ε T < 0 .

Solving the aforementioned for s ( k ) gives:

(13) 0 < s ( k ) < ε T 1 − q T .

Similarly, consider the case where s ( k ) < 0 , and the s trajectory crosses the surface s = 0 . It follows that s ( k + 1 ) > 0 . Using (11) and noting that sgn ( s ( k ) ) = − 1 :

(14) ( 1 − q T ) s ( k ) + ε T > 0 .

Solving the aforementioned for s ( k ) gives:

(15) − ε T 1 − q T < s ( k ) < 0 .

Finally, combining the results (13) and (15), the QSM region can be defined as follows:

(16) − ε T 1 − q T < s ( k ) < ε T 1 − q T

(17) ∣ s ( k ) ∣ < ε T 1 − q T .

The aforementioned equation proves the bounds of s ( k ) below and above zero. It can be seen that the width of this band around 0 is:

(18)□ Δ = ε T 1 − q T − − ε T 1 − q T = 2 ε T 1 − q T .

The QSM bandwidth shown in (18) is the size of the region around the surface, where the s ( k ) dynamics will remain and chatter. Note that the size of the QSM band Δ can be arbitrarily reduced by choosing a very small ε , q , and sampling time T , such that Δ ≈ 0 . Since s ( k ) remains within Δ ≈ 0 , then:

(19) s ( k ) ≈ 0 .

Thus far, it has been shown that the chosen reaching law is stable around the discrete sliding surface and is bounded within a specific QSM band. The stability of the whole control law must be studied, which is discussed in the next section.

2.4 VSC design

In this section, the reaching law (2) and the surface stabilized with the specially designed c in Section 2.2 will be used to achieve an effective controller. The time step change of the sliding surface is used as a start point in the derivations:

(20) s ( k + 1 ) − s ( k ) = c T X ( k + 1 ) − c T X ( k ) .

The term X ( k + 1 ) is expanded using the plant’s dynamics (1). Then, the right-hand side of the resulting equation is replaced by the reaching law (2). Finally, solving for the control u ( k ) the following variable structure control law is synthesized:

(21) u ( k ) = − ( c T b ) − 1 [ c T A X ( k ) − c T X ( k ) + q T s ( k ) + ε T sgn ( s ( k ) ) ] .

The VSC law presented in (21) stabilizes the system (1) in the QSM domain around the surface. This is proven in the following theorem.

Theorem 2

Consider a dynamic system represented by (1). If the switching law (3) is used and if c is designed such that the system in (7) is stable, the control law (21) stabilizes system (1) in the QSM domain around the surface.

Proof

Substituting control law (3) in system (21), we obtain:

X ( k + 1 ) = A X ( k ) + b ( − c T b ) − 1 [ c T A X ( k ) − c T X ( k ) + q T s ( k ) + ε T sgn ( s ( k ) ) ] .

Multiplying both sides of the aforementioned equation by c T and rearranging some terms, we have:

c T X ( k + 1 ) = c T A X ( k ) − c T [ b ( c T b ) − 1 c T A X ( k ) ] + c T X ( k ) − q T s ( k ) − ε T sgn ( s ( k ) ) .

By using the definition of s ( k ) from (3), we write:

c T X ( k + 1 ) = c T [ I − b ( c T b ) − 1 c T ] A X ( k ) + s ( k ) − q T s ( k ) − ε T sgn ( s ( k ) ) .

According to the result of Theorem 2 shown in (18), presented in section 2.3, the steady-state value of s ( k ) can be arbitrarily reduced very close to zero by the choice of a very small ε , q , and T (i.e., s ( k ) ≈ 0 ). The aforementioned equation reduces to:

X ( k + 1 ) = [ I − b ( c T b ) − 1 c T ] A X ( k ) .

If c is designed correctly such that the aforementioned linear system is stable, X ( k + 1 ) → 0 , when k → ∞ . The correct design of c is discussed later in Section 4.3. This concludes the proof.□

The aforementioned results are valid only for an ideal system that is perfectly known, which is unreasonable in an actual system. Model parameter uncertainties and disturbances would cause the idealized controller to become unstable, and thus, a robust controller must take these factors into consideration to provide the desired system performance.

3 DSMC for robust control

A system with parameter perturbations and external disturbances will now be considered. Assume the previously described nominal system (1) but introduce system parameter uncertainty and a disturbance term:

(22) X ( k + 1 ) = A X ( k ) + Δ A X ( k ) + b u ( k ) + Δ b u ( k ) + g ( k ) .

Here, Δ A and Δ b represent the system parameter uncertainty, and g ( k ) is the disturbance. The terms Δ b u ( k ) and g ( k ) can be combined into a single uncertainty term f ( k ) , which gives:

(23) X ( k + 1 ) = A X ( k ) + Δ A X ( k ) + b u ( k ) + f ( k ) .

For simplicity, notations A ˜ and f ˜ are defined such that:

(24) Δ A = b A ˜ , f = b f ˜ ,

where A ˜ is a row vector and f ˜ is a scalar. From (24), (22) becomes:

(25) X ( k + 1 ) = A X ( k ) + b [ A ˜ X ( k ) + u ( k ) + f ˜ ( k ) ] .

The equation of the ideal QSM is unchanged from the nominal case despite the presence of perturbations and the disturbance. The control law from (21) is amended to cancel out the uncertain terms A ˜ X ( k ) + f ˜ ( k ) :

(26) u ( k ) = − ( c T b ) − 1 [ c T A X ( k ) − c T X ( k ) + q T c T X ( k ) + ε T sgn ( c T X ( k ) ) ] − A ˜ X ( k ) − f ˜ ( k ) .

Here, A ˜ and f ˜ are not known a priori. Therefore, the aforementioned control (26) cannot be implemented in its current form. To create a usable control, assume that A ˜ and f ˜ can be replaced with sufficiently conservative terms A c and f c to maintain the discrete reaching condition. Eq. (26) will thus become:

(27) u ( k ) = − ( c T b ) − 1 [ c T A X ( k ) − s ( k ) + q T s ( k ) + ε T sgn ( s ( k ) ) ] − A c X ( k ) − f c ( k ) .

The effect of the controller (27) with conservative uncertain terms on the system (25) with the actual uncertain terms is studied next.

Consider the surface:

(28) s ( k + 1 ) = c T X ( k + 1 ) .

We substitute X ( k + 1 ) using (25) while using the control u ( k ) determined from (27). This yields:

(29) s ( k + 1 ) = s ( k ) − q T s ( k ) − ε T sgn ( s ( k ) ) + c T b [ − A c X ( k ) − f c ( k ) + A ˜ X ( k ) + f ˜ ( k ) ] .

To capture the unknown term f ˜ and A ˜ and their conservative counterparts f c and A c in single terms, the following notations are defined:

(30) S ˜ ( k ) = c T b A ˜ X ( k ) S c ( k ) = c T b A c X ( k ) , F ˜ ( k ) = c T b f ˜ ( k ) F c ( k ) = c T b f c ( k ) .

Eq. (29) can now be represented succinctly as follows:

(31) s ( k + 1 ) − s ( k ) = − q T s ( k ) − ε T sgn ( s ( k ) ) + S ˜ ( k ) − S c ( k ) + F ˜ ( k ) − F c ( k ) .

It is reasonable to assume that the unknown terms S ˜ and F ˜ are upper and lower bounded such that:

(32) S L ≤ S ˜ ≤ S U , F L ≤ F ˜ ≤ F U .

The choice of S c and F c is done in such a way to ensure that the incremental change in s ( k ) (i.e., s ( k + 1 ) − s ( k ) ) has the opposite sign as that of s ( k ) . A practical choice for this condition is:

(33) if s ( k ) > 0 , set S c = S U and F c = F U if s ( k ) < 0 , set S c = S L and F c = F L .

To streamline the use of Eq. (33) and avoid “if clauses” in the control law, the following new notations are defined:

(34) S 1 = S U + S L 2 S 2 = S U − S L 2 F 1 = F U + F L 2 F 2 = F U − F L 2 .

Now, instead of using Eq. (33) to determine S c and F c at any given control sampling time, the following equations can be used, which have the same effect as Eq. (33).

(35) S c = S 1 + S 2 sgn ( s ( k ) ) , F c = F 1 + F 2 sgn ( s ( k ) ) .

Finally, substituting (35) into the control (27), the robust control can be expressed as follows:

(36) u ( k ) = − ( c T b ) − 1 [ c T A X ( k ) − c T X ( k ) + q T s ( k ) + ε T sgn ( s ( k ) ) + S 1 + F 1 + ( S 2 + F 2 ) sgn ( s ( k ) ) ] .

4 DSMC design for the FWMAV

4.1 The FWMAV continuous time dynamic model

A theoretical FWMAV based on an experimentally derived dynamic model of a butterfly [26,27] is used to demonstrate the derived DSMC. A brief derivation of the FWMAV system and the conversion to discrete time will provide context for the results.

The free-body diagram of the FWMAV is shown in Figure 1. The x – z coordinates are the body axes of the FWMAV. In Figure 1, U is the forward velocity, L is the lift generated by the flapping wings, m g is the weight of the FWMAV, and θ is the angle between the lift and the weight forces.

Figure 1

Forces acting on a FWMAV in a climbing trajectory.

According to Figure 1, the balance of forces along the z axis is written as follows:

(37) m z ¨ = L − m g cos ( θ ) ,

where g is the gravitational constant, z ¨ is the acceleration of the body, m is the mass of the butterfly, and θ is the climb angle with respect to the horizon. The lift, L , is modeled with Theodorsen’s lift equation in the QSM, and is expanded as follows:

(38) L = 1 2 ρ S r ˆ 2 2 C L = L = 1 2 ρ S r 2 ˆ 2 × π c m 2 U 2 ( z ¨ − h ¨ ) + 2 π U ( z ˙ − h ˙ ) + C L , α .

This lift term shows the coupling between the body’s velocity z ˙ and acceleration z ¨ , and the wing’s velocity h ˙ and acceleration h ¨ . It includes, as constants, the fluid density, ρ , mean velocity magnitude, U , area of the left and right forewings, S , nondimensional second moment of inertia of the wing area, r ˆ 2 2 , mean chord c m , and the lift coefficient due to a mean angle of attack, C L , α .

The motion of the wing is given as follows:

(39) h ( t ) = − h a cos ( 2 π f t ) ,

where h a is the wing plunge amplitude, f is the flapping frequency, and t is time. For the sake of simplicity, the constant terms are consolidated, and the lift equation now becomes:

(40) L = C 1 ( C 2 ( z ¨ − h ¨ ) + C 3 ( z ˙ − h ˙ ) + C L , α ) ,

where

C 1 = 1 2 ρ U 2 S r ˆ 2 2 , C 2 = π c m 2 U 2 , C 3 = 2 π U ,

Substituting the lift equation back into the force balance equation and solving for z ¨ yields

(41) z ¨ = C 1 C 3 m − C 1 C 2 z ˙ − C 1 C 2 m − C 1 C 2 h ¨ − C 1 C 3 m − C 1 C 2 h ˙ + C 1 C L , α − m g cos ( θ ) m − C 1 C 2 .

There is a lift component C L , α ρ U 2 S r ˆ 2 2 ∕ 2 that always offsets the weight component, m g cos θ . This lift component is modeled as C L , α = 2 π α , where α is the mean angle of attack, and is equal to α = m g cos θ ∕ ( ρ U 2 S r ˆ 2 2 ) [27]. Substituting the derivatives of (39) into the aforementioned force balance equation gives:

(42) z ¨ = C 5 z ˙ + 4 W 1 π 2 f 2 h a cos ( 2 π f t ) + 2 W 2 π f h a sin ( 2 π f t ) + C 8 + C 9 ,

where

C 5 = C 1 C 3 m − C 1 C 2 .

W 1 = − C 1 C 2 m − C 1 C 2 , W 2 = − C 1 C 3 m − C 1 C 2 .

C 8 = C 1 C L , α m − C 1 C 2 , C 9 = − m g cos θ m − C 1 C 2 .

For simplicity, the frequency control terms are grouped into a single variable, w ( t ) .

(43) w ( t ) = 4 W 1 π 2 f 2 h a cos ( 2 π f t ) + 2 W 2 π f h a sin ( 2 π f t ) + C 9 .

This produces the continuous time linear plant that serves as the starting point for discrete time conversion.

(44) z ¨ = C 5 z ˙ + w ( t ) + C 8 .

To simplify the notation, a virtual control v ( t ) = w ( t ) + C 8 is introduced. Eq. (44) becomes:

(45) z ¨ = C 5 z ˙ + v ( t ) .

4.2 Discretizing the FWMAV’s dynamic model

The linear plant (45) is discretized using the Euler method. The position, velocity, and acceleration terms are discretized as follows:

(46) z ( k ) − z d ( k ) = x 2 ( k ) z ˙ ( k ) − z ˙ d ( k ) = x 2 ( k ) − x 2 ( k − 1 ) T z ¨ ( k ) − z ¨ d ( k ) = 1 T x 2 ( k + 1 ) − x 2 ( k ) T − x 2 ( k ) − x 2 ( k − 1 ) T ,

where z d ( k ) is the desired position of the FWMAV at instance k . Plugging equations (46) into the system (42) and solving for x 2 ( k + 1 ) yields:

(47) x 2 ( k + 1 ) = ( 2 + T C 5 ) x 2 ( k ) − ( 1 + T C 5 ) x 2 ( k − 1 ) + T 2 u ( k ) ,

where

(48) u ( k ) = v ( k ) + C 5 z ˙ d ( k ) − z ¨ d ( k ) .

Set

(49) x 1 ( k ) = x 2 ( k − 1 ) .

The aforementioned equation is substituted into (47), and the result is written in the matrix form:

(50) x 1 ( k + 1 ) x 2 ( k + 1 ) = 0 1 − ( 1 + T C 5 ) ( 2 + T C 5 ) × x 1 ( k ) x 2 ( k ) + 0 T 2 u ( k ) ,

which is the standard form of

(51) X ( k + 1 ) = A X ( k ) + b u ( k ) .

4.3 Designing the surface gain vector c

Following Section 2.2, the vector c needs to be determined to guarantee the stability of the ideal QSM. Recall the dynamic Eq. (7) while on the ideal QSM. The vector c can be determined by first defining the plant’s parameters on the ideal QSM as A ˆ , as in (7), but with the FWMAV system as follows:

(52) A ˆ = I − 0 T 2 c 1 c 2 0 T 2 − 1 c 1 c 2 0 1 − ( 1 + T C 5 ) ( 2 + T C 5 ) .

With A ˆ defined, the dynamics of the FWMAV on the ideal sliding mode can be described as follows:

(53) X ( k + 1 ) = A ˆ X ( k ) = 0 1 0 − c 1 c 2 X ( k ) .

The characteristic equation of the system to-be designed is determined.

(54) λ I − 0 1 0 − c 1 c 2 ⇒ λ − 1 0 λ + c 1 c 2 ⇒ λ 2 + c 1 c 2 λ + 0 .

To ensure that the characteristic equation in (54) results in a stable system (53), we design the elements of the c matrix as follows:

(55) c 1 = − L 1 and c 2 = 1 ,

where 0 < L 1 < 1 . The gains c 1 and c 2 in (55) have been designed to form a stable sliding mode for the FWMAV system. The stability of the system with the aforementioned gains is proven in the following theorem.

Theorem 3

Consider the dynamics of the FWMAV in the ideal sliding mode, as described in (53). If the gains c 1 and c 2 are determined by (55), where 0 < L 1 < 1 , the vector X ( k + 1 ) approaches a zero vector from any initial condition X ( 0 ) as k increases.

Proof

Substituting (55) in (53) results in:

X ( k + 1 ) = 0 1 0 L 1 X ( k ) .

Substituting X ( k ) = [ x 1 ( k ) , x 2 ( k ) ] T in the aforementioned equation yields:

(56) x 1 ( k + 1 ) = x 2 ( k ) x 2 ( k + 1 ) = L 1 x 2 ( k ) .

The second equation in (56) ensures that all x 2 ( k + 1 ) are smaller than x 2 ( k ) , because 0 < L 1 < 1 . This means that x 2 is reduced at any step and approaches zeros as k increases. The first equation in (56) shows that x 1 ( k + 1 ) is reduced to zero as x 2 approaches zero. So, both x 1 and x 2 approach zero, and consequently, the vector X ( k ) approaches a zero vector as k increases.□

Finally, according to Theorem 3, the coefficients of c 1 and c 2 have been designed to form a stable sliding mode for the FWMAV system.

4.4 Bounds of the FWMAV system uncertainty and disturbance

Next, the control parameters that sufficiently bound the state parameter uncertainty and disturbance terms need to be determined. In Section 3, state parameter uncertainty and a disturbance term were incorporated in (23):

(57) X ( k + 1 ) = A X ( k ) + Δ A X ( k ) + b u ( k ) + f ( k ) .

For the FWMAV system, the previous equation is explicitly:

(58) x 2 ( k + 1 ) = ( 2 + T ( C 5 + Δ C 5 ) ) x 2 ( k ) − ( 1 + T ( C 5 + Δ C 5 ) ) x 1 ( k ) + T 2 u ( k ) + d ( k ) ,

which can be re-written as follows:

(59) x 2 ( k + 1 ) = ( 2 + T C 5 ) x 2 ( k ) − ( 1 + T C 5 ) x 1 ( k ) + T 2 [ A 1 ˜ x 1 ( k ) + A 2 ˜ x 2 ( k ) + u ( k ) + d ˜ ( k ) ] ,

with

− T Δ C 5 = T 2 A 1 ˜ , T Δ C 5 = T 2 A 2 ˜ , d ( k ) = T 2 d ˜ ( k ) .

Following the procedure laid out in Section 3, the control can be ascertained:

(60) u ( k ) = 1 T 2 ( 1 − q T ) s ( k ) − ε T sgn ( s ( k ) ) − c 1 x 2 ( k ) c 2 − ( 2 + T C 5 ) x 2 ( k ) + ( 1 + T C 5 ) x 1 ( k ) ] … − A 1 c x 1 ( k ) − A 2 c x 2 ( k ) − D c ( k ) ,

where A ˜ 1 , A ˜ 2 , and d ˜ ( k ) are replaced by their conservative counterparts, A 1 c , A 2 c , and D c ( k ) . Also, the values of S ˜ ( k ) and F ˜ ( k ) are as follows:

(61) S ˜ ( k ) = c 2 T 2 ( A 1 ˜ x 1 ( k ) + A 2 ˜ x 2 ( k ) ) , F ˜ ( k ) = c 2 T 2 d ˜ ( k ) .

The aforementioned equations are used to determine the bounds of uncertainty and disturbance.

(62) S L ≤ c 2 T 2 ( A 1 ˜ x 1 ( k ) + A 2 ˜ x 2 ( k ) ) ≤ S U F L ≤ c 2 T 2 d ˜ ( k ) ≤ F U ,

where S 1 , S 2 , F 1 , and F 2 are determined from Eqs (34) and (35). Finally, the control for the FWMAV system is expressed as follows:

(63) u ( k ) = 1 T 2 c 2 [ ( 1 − q T ) s ( k ) − ε T sgn ( s ( k ) ) − c 1 x 2 ( k ) + c 2 [ ( 1 + T C 5 ) x 1 ( k ) − ( 2 + T C 5 ) x 2 ( k ) ] + ( S 1 + F 1 ) + ( S 2 + F 2 ) sgn ( s ( k ) ) ] .

As this represents a “virtual control” that does not directly translate to the flapping frequency, a conversion must be formulated to realize this physical control input.

4.5 Virtual control to physical control conversion

The nonlinear control terms are grouped as shown in (43), which is repeated here for simplicity.

(64) w ( t ) = 4 W 1 π 2 f 2 h a cos ( 2 π f t ) + 2 W 2 π f h a sin ( 2 π f t ) + C 9 .

In the discrete model, v ( k ) is constant during each sampling period, T . As such, v ( k ) must have the same effect as the right-hand side of (64) during the sampling time on the FWMAV’s body displacement, z .

As shown in (44) and (45), z ¨ is a function of v ( t ) , and therefore, a function of w ( t ) . From this, a relationship for the flapping frequency f as a function of the control v in discrete time can be surmised by twice integrating (64).

After the first integration of (64), it is noted that w ( t ) at t = 0 is C 9 , which yields:

(65) ( w − C 9 ) t = 2 W 1 π f h a sin ( 2 π f t ) − W 2 h a cos ( 2 π f t ) + C 9 t + W 2 h a .

The second integration is performed over the sampling time T :

(66) 1 2 ( w 2 − C 9 2 ) T 2 = − W 1 h a cos ( 2 π f T ) + W 1 h a − W 2 h a 2 π f sin ( 2 π f T ) + 1 2 C 9 T 2 + W 2 h a T .

Since T = 1 ∕ f , Eq. (66) can be reduced and the frequency term f can be solved for:

(67) f ( t ) = w ( t ) − 2 C 9 2 W 2 h a .

Noting that w ( t ) = v ( t ) − C 8 , Eq. (67) can finally result in the direct relation between the virtual control u and the flapping frequency f .

(68) f ( t ) = v ( t ) − C 8 − 2 C 9 2 W 2 h a .

5 Numerical simulation and results

For the simulation, the system will begin with an initial condition of x 2 ( 0 ) = 0.2 m with a variable time step. The simulated FWMAV has the nominal system parameters shown in Table 2.

Table 2

Nominal system parameters

Parameter	Value
Mass (m)	0.415 g
Wing area (A)	19 mm²
Drag coefficient ( C D )	1.28
Air density ( ρ )	1.205 kg/m³
Wing plunge amplitude ( h a )	0.0324 m
System constant C 5	10.124 s ‒ 1
System constant C 8	‒ 9.191 m/s 2
System constant C 9	9.191 m/s²
System constant W 2	‒ 10.125 s ‒ 1

The control parameters selected for the DSMC are shown in Table 3.

Table 3

DSMC parameters

Parameter	Value
L 1	0.4
L 2	0
c 1	1
c 2	‒ 0.4
ε	0.005
q	3
S U	0.0008 m
S L	‒ 0.0008 m
F U	0.002 m
F L	‒ 0.002 m

To simulate system parameter variation, an additive system perturbation of 16% above nominal system parameters is included. The simulated disturbance is the resulting acceleration due to drag on the FWMAV wings due to a gust of wind.

(69) a D = 1 2 m ρ C D A v 2 .

This gust of wind has a maximum velocity of 0.7 m/s, beginning 10 s into the simulation, and ending 5 s later, as shown in the following equation and graphically in Figure 2:

(70) v ( t ) = 0.28 t − 2.8 10 ≤ t < 12.5 − 0.28 t + 4.2 12.5 ≤ t ≤ 15 .

Figure 2

Wind gust profile.

The FWMAV has a mass of m = 0.415 g, with wing area of A = 19 mm 2 , treated as a flat plate with a drag coefficient of C D = 1.28 , in air with density ρ = 1.205 kg ∕ m 3 .

Two controllers are tested via simulations. The first one is the robust DSMC proposed in this article (control law (35)). The second one is a nonrobust (standard) DSMC (control law (21)).

In Figures 3 and 4, it can be seen that the FWMAV begins at an initial height of 0.2 m and flies to the desired sinusoidal trajectory with amplitude 0.15 m and period of 30 s. At t = 10 s , the disturbance begins and the FWMAV using the robust DSMC in Figure 3 maintains the trajectory, with little to no trajectory deviation. This is in contrast to the FWMAV using the nonrobust DSMC in Figure 4, where the FWMAV clearly deviates from the desired trajectory before returning to the desired trajectory after the disturbance ends at t = 15 s.

Figure 3

Robust DSMC: Results for vertical displacement of the FWMAV z .

Figure 4

Nonrobust DSMC: Results for vertical displacement of the FWMAV z .

Similarly, Figures 5 and 6 show the corresponding trajectory tracking error for the robust and nonrobust DSMC controllers, respectively. In these figures, it can be seen that the maximum tracking error for the worst case disturbance for the nonrobust and robust DSMC are 0.049 m and 0.027 m, respectively. This demonstrates the superiority of the proposed robust DSMC, which has reduced the maximum tracking error by approximately 45%.

$Figure 5 Robust DSMC: Results for vertical displacement error of the FWMAV z z – z d {z}_{d} .$

Figure 5

Robust DSMC: Results for vertical displacement error of the FWMAV z – z d .

$Figure 6 Nonrobust DSMC: Results for vertical displacement error of the FWMAV z z – z d {z}_{d} .$

Figure 6

Nonrobust DSMC: Results for vertical displacement error of the FWMAV z – z d .

The error reduction is supported by observing the performance of the switching variable s in Figure 7, noting that it very quickly transitions from the reaching phase to the sliding phase and remains stable despite the presence of the disturbance. Note that, as proved in Section 2.3, the sliding variable s remains in a bound around the surface s = 0 . The sliding variable’s band is in the range of [ − 0.004 , 0.018 ] m.

Figure 7

Robust DSMC: Results for the hyperplane, s .

Looking at the control effort in Figure 8, one can see that the FWMAV control input smoothly and quickly reaches a steady-state response, increasing its relative effort to compensate for the disturbance during the disturbance presence period. The steady-state control input fluctuates within the range of [ − 6.1 , 2.5 ] m/s 2 .

Figure 8

Robust DSMC: Results for control effort, v .

Converting the control effort to flapping frequency using Eq. (68), one can show that the flapping frequency maintains a frequency range of [ 10.7 , 23.4 ] Hz. The resulting input frequency range is similar to that of the experimentally derived model by Kang et al. [27]. This is an expected result. The FWMAV dynamic model used in our simulation is adopted from system identification of a monarch butterfly presented by Kang et al. [27]. Our defined desired flight regime is defined to closely match the flights used for system identification by Kang et al. [27]. Therefore, it is expected that the frequency control effort calculated by the DSMC controller must match the open-loop flapping frequency measured by experiments by Kang et al. [27]. The matching frequencies between the two publications further verifies the validity of the proposed DSMC controller.

Note that the input chatter (flapping frequency f ) that is shown in the result of the application of the DSMC on the FWMAV (Figure 9) is misleading. In each sampling time, an input frequency f is calculated, with its duration of application T = 1 ∕ f . Then, one cycle of wing flap following Eq. (39) is applied for T seconds. The wing flap h ( t ) starts at 0 ( h ( t ) = 0 ) at the current time t , it lasts T seconds, and at the end of T seconds, it ends a 0 ( h ( t + T ) = 0 ). Then, at time t + T , the same flapping pattern repeats itself. So, there is no discontinuity in the wing flapping motion. In consequence, there is no chatter in the wing flapping motion. The wing flapping motion consists of a series of continuous curves in the harmonic shape of Eq. (39) with different frequencies.

Figure 9

Robust DSMC: Results for flapping frequency, f .

As a direct result of the changing flapping frequencies, the sampling time must change throughout the simulation. For higher flapping frequencies, the time between subsequent wing flaps naturally shortens. Conversely, for lower flapping frequencies, the time between wing flaps lengthens. The change in the sampling time T is shown in Figure 10. At the steady-state, the sampling time fluctuates in the range of [ 0.043 , 0.097 ] s. However, the sampling time trends lower as the flapping frequency increases in the duration of disturbances.

Figure 10

Robust DSMC: Results for variable sampling time, T .

6 Conclusion

A robust discrete sliding mode reaching controller was proposed to stabilize the vertical altitude of a FWMAV. The continuous time plant of the FWMAV was converted to a discrete time plant and the controller was developed for this new form. As can be seen by the results presented in this article, the FWMAV was able to successfully stabilize its position with negligible error despite a plant uncertainty of 16% and a wind gust with a maximum peak of 0.7 m/s. These trajectory and error results are compared to that of a nonrobust DSMC in Figures 3–6. The robust controller performance is further bolstered by Figure 9, which shows an input frequency range of [10.7, 23.4] Hz during the steady state, which are similar to that of the experimentally derived model by Kang et al. [27], thus validating the controller.

Funding information: This research was not funded by any funding agency.
Author contributions: The research idea and the overall methodology were formulated by J.H. The detailed derivations of the approach and the creation of the computer simulation code were performed by J.H., with suggested corrections in the derivations by F.F and suggested improvements to the simulations. The draft manuscript was authored by J.H., which went through several revisions according to feedback from F.F. All authors have accepted responsibility for the entire content of this manuscript and approved its submission. The first author was a PhD Candidate who conducted this research. The second author was the first author’s dissertation research advisor.
Conflict of interest: The authors state that there is no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-02-12

Revised: 2024-05-21

Accepted: 2024-06-07

Published Online: 2024-08-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0017

Keywords for this article

flapping wing; air vehicle; discrete sliding model control; flapping frequency

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