# A Sliding Mode LCO Regulation Strategy for Dual-Parallel Underactuated UAV Systems Using Synthetic Jet Actuators

Authors: N. Ramos-Pedroza; W. MacKunis; M. Reyhanoglu
Journal: International Journal of Aerospace Engineering
 (2015)

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## Abstract

A sliding mode control- (SMC-) based limit cycle oscillation (LCO) regulation method is presented, which achieves asymptotic LCO suppression for UAVs using synthetic jet actuators (SJAs). With a focus on applications involving
small UAVs with limited onboard computational resources, the controller is designed with a simplistic structure, requiring no adaptive laws, function approximators, or complex calculations in the control loop. The control law is rigorously proven to achieve asymptotic regulation of both pitching and plunging displacements for a class of systems in a dual-parallel underactuated form, where a single scalar control signal simultaneously affects two states. Since dual-parallel underactuated systems cannot be expressed in a strict feedback or cascade form, standard backstepping-based control techniques cannot be applied. This difficulty is mitigated through
careful algebraic manipulation in the regulation error system development, along with innovative design of the sliding surface. A detailed model of the UAV LCO dynamics is utilized, and a rigorous analysis is provided to prove asymptotic regulation of the pitching and plunging displacements. Numerical simulation results are provided to demonstrate the performance of the control law.


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## Body

## 1. Introduction

Limit cycle oscillation (LCO) (flutter) is a self-excited aeroelastic phenomenon that can create difficulties in aircraft tracking control and could potentially result in structural damage and even catastrophic failures. Even in the low Reynolds number (low-Re) regimes, characteristic of small unmanned aerial vehicle (UAV) systems, LCO can be adversarial in flight control systems and can lead to unsafe UAV operating conditions [1]. Motivated by these challenges, automatic control methods for LCO regulation systems have been widely investigated in recent controls literature [2–5]. LCO suppression control systems are usually designed using moving deflection surfaces (e.g., elevators, rudders, and ailerons) to deliver the required control force or moment. However, for applications involving smaller, lighter weight UAVs, the use of heavy mechanical actuators might not be practical. As these practical considerations motivate the need for smaller, low power-consumptive control actuators, synthetic jet actuators (SJAs) have emerged as a popular alternative to mechanical deflection surfaces for UAV control applications [1, 6–14].SJAs are practical tools for UAV LCO suppression control systems due to their low cost, small size, and low power consumption properties. SJAs utilize a vibrating diaphragm to create trains of vortices (or jets) of air through the periodic ejection and suction of air through a small orifice (see Figure1). The resulting trains of air vortices impart linear momentum to a flow system, and this momentum transfer enables SJAs to deliver an equivalent control force or moment when implemented in UAV wings. Since SJAs only use the surrounding air of the flow system to generate the vortices, they can deliver a control force or moment with zero net mass injection across the flow boundary. This is a key benefit in small UAV applications: SJAs do not require space for a fuel supply. The boundary layer flow field near the surface of a UAV wing can be altered by using the trains of air vortices generated by the SJAs. By utilizing this SJA-commanded modification of the boundary layer flow field, SJAs can be employed to achieve automatic regulation control of LCO in UAV wings. SJA-based control design is complicated, however, due to the inherent nonlinearity in the equations governing the SJA dynamics. Specifically, the virtual surface deflection delivered by the SJAs is a nonlinear (nonaffine) function of the voltage signal commanded to the SJAs. An additional challenge in control design using SJAs is that the nonlinear actuator dynamic model includes parametric uncertainty. This SJA nonlinear SJA dynamic model was determined empirically through repeated experiments, and it is well-accepted in SJA-based control literature (e.g., see [10, 12, 14]).Figure 1
Schematic layout of a synthetic jet actuator.In addition to the control design challenges involved in compensating for actuator uncertainties, significant challenges arise in situations where multiple actuators lose control effectiveness. When the number of control actuators is less than the number of degrees of freedom to be controlled, the system becomes underactuated, and control design for underactuated systems presents a nontrivial challenge. The integrator backstepping technique is often utilized to address the challenge of control design for underactuated systems [15]. However, the backstepping-based control design approach can only be applied to systems in strict feedback or cascade form. Backstepping techniques cannot be utilized for systems in a dual-parallel underactuated form, where a single scalar control input simultaneously affects two states. Several open problems remain in control design for dual-parallel underactuated systems.To compensate for the nonlinearity and parametric uncertainty inherent in the SJA dynamic model, recent approaches typically employ adaptive parameter estimation, neural networks (NN), fuzzy logic rule sets, or complex fluid dynamics computations in the feedback loop (e.g., see [10, 14, 16]). These types of SJA-based control design approaches have been shown to achieve good closed-loop performance; however, control designs involving complex calculations or function approximation methods can incur an increased computational requirement, which might not be practical for applications involving small UAVs. An adaptive inverse control method is presented in [10, 14], which achieves asymptotic trajectory tracking for an aircraft dynamic model equipped with SJAs. To achieve the results in [10, 14], a series of adaptive parameter estimation laws are utilized along with rigorous Lyapunov-based stability analyses. Inspired by these results and motivated by the desire to investigate a more computationally minimal control strategy, our previous result in [17] was the first SJA-based tracking control approach to achieve asymptotic trajectory tracking for an aircraft using a simple (single-loop) feedback control law without the use of parameter adaptation or function approximators in the control law. A question that remains to be answered is as follows: Can a computationally minimal nonlinear SJA-based control law achieve asymptotic regulation of LCO for UAV systems in a dual-parallel underactuated form?The contribution of this paper is the development and rigorous analysis of a sliding mode control law, which achieves asymptotic regulation of both pitching and plunging LCO in UAV wings using a single, scalar SJA as the control input (i.e., a dual-parallel underactuated system). To the best of the authors’ knowledge, this is the first SJA-based nonlinear control result to rigorously prove asymptotic regulation of both pitching and plunging LCO states in a dual-parallel underactuated UAV system. Moreover, to address a practical UAV scenario where onboard computational resources are limited, the result presented here compensates for the inherent SJA actuator nonlinearity and parametric uncertainty without the use of adaptive laws or function approximators. To achieve the result, a sliding mode control strategy is utilized, which employs a periodic switching law. A detailed mathematical model of the UAV dynamics is utilized to develop the regulation error dynamics, and a rigorous Lyapunov-based stability analysis is presented to prove asymptotic regulation of the pitching and plunging displacements. Numerical simulation results are also provided to complement the theoretical development.


## 2. Dynamic Model and Properties

The equation describing dynamics of LCO in an airfoil can be expressed as [18](1)Msp¨+Csp˙+Fpp=-LM,where the coefficients Ms,Cs∈R2×2 denote the structural mass and damping matrices and F(p)∈R2×2 is a nonlinear stiffness matrix. In (1), pt≜htαtT∈R2, where ht,αt∈R denote the plunging and pitching displacements, respectively. Figure 2 illustrates the pitching and plunging displacements in a standard airfoil.Figure 2
Illustration of pitch (α) and plunge (h) in an airfoil, including lift (L) and moment (M) definitions and the nonlinear stiffness coefficient (Kα).Also in (1), the structural linear mass matrix Ms is defined as [18](2)Ms=mmxαbmxαbIα,where the parameter xα∈R denotes the nondimensional distance measured from the elastic axis to the center of mass, b∈R is the semichord of the wing [m], m∈R is the mass of the wing section, and Iα∈R is the mass moment of inertia of the wing about the elastic axis (see Figures 2 and 3).Figure 3
Diagram of the wing section, showing geometric parameters and the deflection angleβ [18].The structural linear damping matrix is described as(3)Cs=Ch00Cα,where the parameters Ch,Cα∈R are the structural damping coefficient in plunge due to viscous damping kg/s and structural damping coefficient in pitch due to viscous damping [(Kg·m2)/s], respectively. The nonlinear stiffness matrix utilized in this study is(4)Fp=Kh00Kα,where Kh∈R is the structural spring constant in plunge N/m and Kα∈R in [(N·m)/rad] is the torsion stiffness coefficient described in terms of a polynomial as(5)Kα=2.821-22.1α+1315.5α2-8580α3+17289.7α4.The right hand side of (1) is explicitly given by [18](6)L=ρU2spbclαα+h˙b+12-abα˙U+ρU2spbclββ,M=ρU2spb2cmαα+h˙b+12-abα˙U+ρU2spb2cmββ,where U∈R is the velocity m/s, ρ is the density of air kg/m3, sp is the wing span [m], clα is the lift coefficient per angle of attack, cmα is the moment coefficient per control surface deflection, clβ is the lift coefficient per control surface deflection, cmβ is the moment coefficient per control surface deflection, and a is the nondimensional distance from the midchord to elastic axis. The term β denotes the surface deflection angle of the wing (see Figure 3). By rearranging (1), the dynamics equations can be expressed as(7)x˙=Axx+Bβ,where xt≜x1tx2tx3tx4tT∈R4 is the state vector with x1t≜ht, x2t≜αt, x3t≜h˙t, and x4t≜α˙t. In (7), A(x)∈R4×4 is a nonlinear state matrix (containing nonlinearities due to the torsion stiffness coefficient introduced in (5)), and B∈R4×1 is the control input gain matrix. By separating the constant elements from the state-dependent elements of the matrix Ax, the dynamic model (7) can be rewritten as(8)x˙=00100001a1a2a3a4c1c2c3c4x+Fx2+00b1b2u,where a1,…,a4∈R and c1,…,c4∈R denote known constant parameters, which are dependent on the physical parameters of the wing section. The explicit definitions of the constant parameters are unwieldy and are omitted here for brevity. In (8), ut∈R represents the virtual surface deflection delivered by the SJA (or SJA array) (i.e., the term β introduced in (7)). Also in (8), the nonlinear vector function F(x2) is defined in terms of the stiffness coefficient Kα as(9)
F
x
2
=
0
0
-
K
α
m
x
a
b
-
I
α
/
m
x
a
b
1
/
x
a
b
K
α
m
x
a
b
-
I
α
/
m
x
a
b
α
.The constant control input gain terms b1,b2∈R introduced in (8) are explicitly defined as(10)b1=ρv2b2cmβsp+Iα/mxabρv2bclβspmxab-Iα/mxab,b2=-ρv2bclβsp-1/xabρv2b2cmβspmxab-Iα/mxab.


## 3. SJA Dynamics

The dynamics of the SJA are nonlinear and they contain parametric uncertainty. Figure1 represents the basic structure of a SJA [10, 14]. This SJA has a piezoelectrically driven diaphragm in its cavity that generates time varying pressure gradients across a small orifice. The periodic ingestion and expulsion of the air through the small orifice results in the formation of vortex rings which formulates a steady turbulent jet of air.Based on empirical data, the dynamics of a SJA can be expressed as [10, 14](11)u=θ2∗-θ1∗v,where u(t)∈R denotes the virtual airfoil surface deflection (i.e., the control input), v(t)=Appi2(t)∈R denotes the peak-to-peak SJA voltage, and θ1∗,θ2∗∈R are uncertain physical parameters.To compensate for the SJA nonlinearity and parametric uncertainty in (11), a robust inverse control design structure will be utilized for the voltage input signal v(t) [17]. The robust inverse controller can be expressed as(12)vt=θ^1θ^2-ud,where θ^1,θ^2∈R denote constant feedforward estimates of the uncertain parameters θ1∗ and θ2∗; and ud(t)∈R is a subsequently defined auxiliary control signal.


## 4. Control Development

The objective is to design the scalar control signaludt to asymptotically regulate the plunging and pitching dynamics (i.e., h(t) and α(t)) to zero. By leveraging the result in [19], udt will be designed using a sliding mode control law with a periodic switching function as(13)ud=M0tanh⁡sin⁡πϵst+λ∫0ttanh⁡sτdτ.Based on the dynamic equations in (8) and the subsequent stability analysis, the sliding surface st∈R in (13) is designed as(14)sx=-Kαdx2+k1x1+k3x3+∑i=14aixi,i=1,…,4,where k1,k3∈R are positive constant control gains and a1,…,a4 are introduced in (8).


### 4.1. Stability Analysis

Theorem 1.
The robust control law in (13) ensures asymptotic convergence to the sliding manifold s(x)=0.Proof.
Proof of Theorem1 can be found in [19] and is omitted here to avoid distraction from the main contribution of the current result.Theorem 2 (main result).
Convergence to the sliding manifoldsx=0 results in asymptotic regulation of both pitching and plunging displacements in the sense that(15)sx⟶0⟹ht,αt⟶0as  t→∞.Proof (plunging regulation).
It follows directly from the definition of the sliding surfacesx in (14) and the LCO dynamic equations in (8) that(16)sx⟶0⟹x˙3⟶-k1x1-k3x3.By using the state definition x3t≜x˙1t, the expression in (16) can be used to show that, on the sliding manifold sx=0, the x1t dynamics are governed by(17)x¨1+k3x˙1+k1x1=0,where k1, k3 are introduced in (14). Since k1,k3>0, the ODE in (17) is Hurwitz, and (17) can be used to prove that x1t≜ht→0 and x3t≜h˙t→0. Thus, convergence to the sliding manifold sx=0 directly results in asymptotic (exponential) regulation of the plunging displacement ht to zero.Proof (pitching regulation).
By substituting (14) into (8) and using the fact that x1t,x3t→0, it can be shown that convergence to the sliding manifold sx=0 results in the pitching dynamics(18)x˙4-c2x2-c4x4-d2dKαx2=0,where c2 and c4 are known constant parameters introduced in (8); d2≜1/(xab), d≜mxab-Iα/(mxab); and Kα is defined in (5). By using the state definition x4t≜x˙2t, the expression in (18) can be rewritten as the second-order nonlinear ODE(19)x¨2-c4x˙2-c2x2-d2dKαx2=0.
After utilizing the definition ofKα given in (5), the ODE in (19) can be expressed in the form [20](20)x¨2+bx˙2+cx2=0.Noting that c4<0, the auxiliary functions bx˙2,cx2∈R in (20) are explicitly defined as(21)bx˙2≜c4x˙2,(22)cx2≜-d2d17289.7x25+d2d8580x24-d2d1315.5x23+d2d22.1x22-d2d2.82+c2x2.
Note. It can be shown that cx2=0⇒x2t=0. The expressions in (21) and (22) satisfy the following properties:(23)x˙2bx˙2>0forx˙2∈R-0,(24)x2cx2>0forx2∈R-0.To prove asymptotic regulation of the pitching displacement x2t to zero, consider the positive definition function (i.e., Lyapunov function candidate) [20](25)V=12x˙22+∫0x2cξdξ.After taking the time derivative of (25) and utilizing (20), V˙t can be expressed as(26)V˙=x˙2-bx˙2-cx2+cx2x˙2=-x˙2bx˙2.The Lyapunov derivative in (26) is negative semidefinite based on inequality (23). We now use LaSalle’s invariance principle to state that V˙t→0 as t→∞⇒x˙2t→0 as t→∞, and thus, x¨2t→0 as t→∞. It further follows that bx˙2→0 as t→∞ from (21). Since bx˙2,x¨2t→0 as t→∞, (20) can be used along with property (24) to prove that x2t→0 as t→∞.


### 4.2. Results

In this section, a numerical simulation was created to demonstrate the performance of the control law developed in (13) and (14). The simulation is based on the dynamic model given in (1). The dynamic parameters utilized in the simulation are summarized in Table 2 and were obtained from [18].The control gainsks, ϵ, M0, and λ were manually selected as described in Table 1. The actual SJA parameters θ2∗ and θ1∗ and estimates θ^2 and θ^1 were selected as 32.9 and 14.7, respectively.Table 1
Manually selected gains for robust control.
k
s
=
9.0
ϵ
=
100
M
0
=
1.0
λ
=
17.0Table 2
Constant parameters.
ρ = 1.225 kg/m3
a
=
-
0.6
c
m
α
=
-
0.635
m = 12.387 kg
b = 0.125 m
v = 13 m/s
C
α = 0.036 (kg⋅m2)/s
c
l
β = 3.358
s
p = 0.6 m
I
α = 0.065 kg⋅m
C
h = 27.43 kg/s
c
l
α
=
6.28
K
h = 2844.4 N/m
c
m
β = −0.635
x
a
=
0.2847The control law was simulated for three different cases of initial conditions: Case 1:x0=0.020.200T (Figures 4 and 5), Case 2: x0=000.050.5T (Figures 6 and 7), and Case 3: x0=0.020.20.030.3T (Figures 8 and 9). Figures 4, 6, and 8 show the time evolution of the pitching and plunging displacements αt and ht during closed-loop controller operation for Cases 1, 2, and 3, demonstrating the rapid convergence of the pitching and plunging displacements to zero. In addition, Figures 5, 7, and 9 show that the pitching and plunging velocities α˙t and h˙t converge to zero rapidly during closed-loop operation. Further, the top plots in Figures 4, 6, and 8 show the control effort u(t) used during closed-loop controller for Cases 1, 2, and 3.Figure 4
Case 1:x0=0.02,0.2,0,0T; time evolution of the control input, ut, regulation error for plunging, ht in [m], and regulation error for pitching, αt in [rad].Figure 5
Case 1:x0=0.02,0.2,0,0T; convergence of the regulation error rate for plunging, h˙(t), and pitching, α˙(t).Figure 6
Case 2:x0=0,0,0.05,0.5T; time evolution of the control input, ut, regulation error for plunging, h(t) in [m], and regulation error for pitching, α(t) in [rad].Figure 7
Case 2:x0=0,0,0.05,0.5T; convergence of the regulation error rate for plunging, h˙t, and pitching, α˙t.Figure 8
Case 3:x0=0.02,0.2,0.03,0.3T; time evolution of the control input, ut, regulation error for plunging, ht in [m], and regulation error for pitching, αt in [rad].Figure 9
Case 3:x0=0.02,0.2,0.03,0.3T; convergence of the regulation error rate for plunging, h˙t, and pitching, α˙t.


## 4.1. Stability Analysis

Theorem 1.
The robust control law in (13) ensures asymptotic convergence to the sliding manifold s(x)=0.Proof.
Proof of Theorem1 can be found in [19] and is omitted here to avoid distraction from the main contribution of the current result.Theorem 2 (main result).
Convergence to the sliding manifoldsx=0 results in asymptotic regulation of both pitching and plunging displacements in the sense that(15)sx⟶0⟹ht,αt⟶0as  t→∞.Proof (plunging regulation).
It follows directly from the definition of the sliding surfacesx in (14) and the LCO dynamic equations in (8) that(16)sx⟶0⟹x˙3⟶-k1x1-k3x3.By using the state definition x3t≜x˙1t, the expression in (16) can be used to show that, on the sliding manifold sx=0, the x1t dynamics are governed by(17)x¨1+k3x˙1+k1x1=0,where k1, k3 are introduced in (14). Since k1,k3>0, the ODE in (17) is Hurwitz, and (17) can be used to prove that x1t≜ht→0 and x3t≜h˙t→0. Thus, convergence to the sliding manifold sx=0 directly results in asymptotic (exponential) regulation of the plunging displacement ht to zero.Proof (pitching regulation).
By substituting (14) into (8) and using the fact that x1t,x3t→0, it can be shown that convergence to the sliding manifold sx=0 results in the pitching dynamics(18)x˙4-c2x2-c4x4-d2dKαx2=0,where c2 and c4 are known constant parameters introduced in (8); d2≜1/(xab), d≜mxab-Iα/(mxab); and Kα is defined in (5). By using the state definition x4t≜x˙2t, the expression in (18) can be rewritten as the second-order nonlinear ODE(19)x¨2-c4x˙2-c2x2-d2dKαx2=0.
After utilizing the definition ofKα given in (5), the ODE in (19) can be expressed in the form [20](20)x¨2+bx˙2+cx2=0.Noting that c4<0, the auxiliary functions bx˙2,cx2∈R in (20) are explicitly defined as(21)bx˙2≜c4x˙2,(22)cx2≜-d2d17289.7x25+d2d8580x24-d2d1315.5x23+d2d22.1x22-d2d2.82+c2x2.
Note. It can be shown that cx2=0⇒x2t=0. The expressions in (21) and (22) satisfy the following properties:(23)x˙2bx˙2>0forx˙2∈R-0,(24)x2cx2>0forx2∈R-0.To prove asymptotic regulation of the pitching displacement x2t to zero, consider the positive definition function (i.e., Lyapunov function candidate) [20](25)V=12x˙22+∫0x2cξdξ.After taking the time derivative of (25) and utilizing (20), V˙t can be expressed as(26)V˙=x˙2-bx˙2-cx2+cx2x˙2=-x˙2bx˙2.The Lyapunov derivative in (26) is negative semidefinite based on inequality (23). We now use LaSalle’s invariance principle to state that V˙t→0 as t→∞⇒x˙2t→0 as t→∞, and thus, x¨2t→0 as t→∞. It further follows that bx˙2→0 as t→∞ from (21). Since bx˙2,x¨2t→0 as t→∞, (20) can be used along with property (24) to prove that x2t→0 as t→∞.


## 4.2. Results

In this section, a numerical simulation was created to demonstrate the performance of the control law developed in (13) and (14). The simulation is based on the dynamic model given in (1). The dynamic parameters utilized in the simulation are summarized in Table 2 and were obtained from [18].The control gainsks, ϵ, M0, and λ were manually selected as described in Table 1. The actual SJA parameters θ2∗ and θ1∗ and estimates θ^2 and θ^1 were selected as 32.9 and 14.7, respectively.Table 1
Manually selected gains for robust control.
k
s
=
9.0
ϵ
=
100
M
0
=
1.0
λ
=
17.0Table 2
Constant parameters.
ρ = 1.225 kg/m3
a
=
-
0.6
c
m
α
=
-
0.635
m = 12.387 kg
b = 0.125 m
v = 13 m/s
C
α = 0.036 (kg⋅m2)/s
c
l
β = 3.358
s
p = 0.6 m
I
α = 0.065 kg⋅m
C
h = 27.43 kg/s
c
l
α
=
6.28
K
h = 2844.4 N/m
c
m
β = −0.635
x
a
=
0.2847The control law was simulated for three different cases of initial conditions: Case 1:x0=0.020.200T (Figures 4 and 5), Case 2: x0=000.050.5T (Figures 6 and 7), and Case 3: x0=0.020.20.030.3T (Figures 8 and 9). Figures 4, 6, and 8 show the time evolution of the pitching and plunging displacements αt and ht during closed-loop controller operation for Cases 1, 2, and 3, demonstrating the rapid convergence of the pitching and plunging displacements to zero. In addition, Figures 5, 7, and 9 show that the pitching and plunging velocities α˙t and h˙t converge to zero rapidly during closed-loop operation. Further, the top plots in Figures 4, 6, and 8 show the control effort u(t) used during closed-loop controller for Cases 1, 2, and 3.Figure 4
Case 1:x0=0.02,0.2,0,0T; time evolution of the control input, ut, regulation error for plunging, ht in [m], and regulation error for pitching, αt in [rad].Figure 5
Case 1:x0=0.02,0.2,0,0T; convergence of the regulation error rate for plunging, h˙(t), and pitching, α˙(t).Figure 6
Case 2:x0=0,0,0.05,0.5T; time evolution of the control input, ut, regulation error for plunging, h(t) in [m], and regulation error for pitching, α(t) in [rad].Figure 7
Case 2:x0=0,0,0.05,0.5T; convergence of the regulation error rate for plunging, h˙t, and pitching, α˙t.Figure 8
Case 3:x0=0.02,0.2,0.03,0.3T; time evolution of the control input, ut, regulation error for plunging, ht in [m], and regulation error for pitching, αt in [rad].Figure 9
Case 3:x0=0.02,0.2,0.03,0.3T; convergence of the regulation error rate for plunging, h˙t, and pitching, α˙t.


## 5. Conclusion

In this paper, a sliding mode control law is presented, which achieves asymptotic LCO regulation in UAV wings using SJAs. Moreover, the control law achieves asymptotic regulation of both the pitching and plunging displacements for UAV LCO dynamics in a dual-parallel underactuated form, where a single scalar control signal simultaneously affects both displacements. To achieve the result, a sliding mode control strategy is utilized, which employs a periodic switching function and a novel sliding surface, which is derived based on the nonlinear dynamics of the UAV LCO system. The inherent nonlinearity and parametric uncertainty in the SJA dynamic model is addressed by means of a robust inverse control structure, which utilizes constant feedforward estimates of the uncertain SJA parameters. The use of constant estimates as opposed to time varying adaptive parameter estimates is motivated by the desire to develop a practical LCO regulation method that can be implemented on small UAVs with limited onboard computational resources. A rigorous analysis is presented to prove the theoretical result, and numerical simulation results are provided to complement the theoretical development.



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*Source: 795348-2015-10-27.xml*