Disturbance observer assisted robust control of wing rock motion based on contraction theory

Abstract

A nonlinear robust control strategy is proposed in this paper to deal with wing rock motion. The method is based on the disturbance observer enforced contraction theory results. The disturbance observer calculates and robustly cancels system uncertainties and input disturbances. The information regarding the known part of the nonlinear plant dynamics is used by the controller, while the uncertainties and disturbances are treated as an additional input to the system and dealt with by the disturbance observer. The algorithm demonstrates good performance in damping the oscillations while rejecting the disturbances. Simulations are provided to demonstrate the effectiveness of the proposed method via an application to an experimentally derived delta wing rock model operating in both nominal and uncertain environments.

Keywords

wing rock contraction theory disturbance observer robust control nonlinear systems

1. Introduction

Several modern high-performance aircraft exhibit lightly damped or undamped rolling oscillations around the longitudinal axis at moderate to high angle of attack (AOA), commonly referred to as wing rock (WR). Such dynamics possesses either limit cycles, which become stable after a transient phase, or divergence. WR can have a deteriorating effect on an aircraft performance in addition to distracting the pilot. WR can be anything from an early warning of imminent departure or spin entry to inertial and kinematic coupling, leading to AOA excursions and loss of control. If WR is left uncontrolled, handling qualities are compromised in addition to a degradation of maneuvering capabilities in terms of the maximum achievable AOA.^1,2

The first type of WR, usually associated with high-AOA, low-airspeed flight in blowy conditions, is characterized by unsteady lateral motions at relatively high AOA. These motions demonstrate small-amplitude intermittent non-periodic roll oscillations, and since these are assumed to be a function of pilot–vehicle interaction, flight procedures are typically changed to avoid this type of WR without significantly compromising mission accomplishment. The WR of the second type is characterized by large roll angle variations and normally associated with high-AOA maneuvering. In cases where a combat aircraft is incapable of tracking a target due to WR, severe performance degradation is obvious. Moreover, the presence of WR during the approach or landing phase may have acute influence on the aircraft operational safety. From the stability point of view, WR is created by a nonlinear aerodynamic mechanism and associated with the nonlinear trend of roll damping derivatives, causing hysteresis and sign changes of the stability parameters while increasing the AOA or airspeed during aircraft maneuvers.^3,4 Suppressing WR is accomplished via appropriate ailerons deflection (Figure 1) and is a task that is important to both aircraft designers and control engineers.

Figure 1.

The influence of aileron deflection on the roll motion (modified from http://www.grc.nasa.gov/WWW/K-12/airplane/bga.html).

The problem of controlling WR motion has attracted quite a few researchers during the last two decades. For example, adaptive feedback linearization approaches were proposed by Monahemi and Krstic⁵ and Jain,⁶ and suboptimal and optimal algorithms were discussed by Xin and Balakrishnan⁷ and Abdulwahab and Hongquan,⁸, respectively. Kalman-filter-based control has shown promising results in Shue and Agarwal.⁹ Fuzzy,^10,11 fuzzy adaptive,¹² and fuzzy neural¹³ approaches has recently gained popularity. Neural-network-based control^14,15 and wavelet adaptive backstepping approaches¹⁶ were also considered. Moreover, several nonlinear control algorithms has proven to be suitable for dealing with WR motion.^17–19 Recently, adaptive control using the contraction theory (CT) has been successfully applied to a WR system in uncertain environment.²⁰

CT is a tool for analyzing the convergence properties of nonlinear systems in the state space form.^21,22 A nonlinear dynamic system is called contracting if trajectories of perturbed system exponentially return to their nominal behavior. CT requires no specific knowledge of a specific attractor, opposed to the Lyapunov stability-based analysis. The CT may be relatively easy applied to the controller synthesis of nonlinear system in case all the parameters are known. In an uncertain environment, an estimator must be employed to ensure acceptable performance. Sharma and Kar²⁰ applied an adaptive backstepping approach to calculate the uncertain parameter estimates.

In this paper, a disturbance observer (DOB) approach is employed. The use of a DOB introduces a two-degree-of-freedom structure, where tracking and disturbance rejection are treated independently.²³ The DOB is a state or output feedback algorithm, based on the model inversion and filtering approach.²⁴ Here, an uncertainty and disturbance estimator (UDE)^25,26 is employed as a DOB, which is able to quickly estimate uncertainties and disturbances. In order to implement the UDE approach, instead of estimating each of the uncertain parameters and disturbances, they are lumped together, estimated and added to the control signal to cancel out the influence of the unknown part of the plant, providing excellent robust performance.

The rest of this paper is organized as follows. In Section 2, the main results of the CT are revisited. WR modeling is described in Section 3, followed by the application of UDE-based DOB enforced CT to the WR problem in Section 4. Conclusions are made in Section 5.

2. Contraction theory: main results

A brief review of the CT results is given in this section. Readers are referred to Lohmiller and Slotine^21,22 and Jouffroy and Slotine²⁷ for more details.

Consider a nonlinear system (either open or closed loop) described in the following state space form

\overset{\cdot}{x} (t) = h (x (t), t)

(1)

where $x = {(x_{1}, \dots, x_{n})}^{T}$ is the state vector and $h (x (t), t)$ is a continuously differentiable vector function of appropriate dimension. Denote an infinitesimal displacement over a fixed time in the state $x$ (virtual displacement) as $δ x$ . Then the first variation (virtual dynamics) of (1) is given by

δ \overset{\cdot}{x} = \frac{\partial h}{\partial x} δ x

(2)

The system is called contracting if the Jacobian matrix denoted as

J = \frac{\partial h}{\partial x}

(3)

is uniformly negative definite (UND). Hence, for the system given in (1), a region in the state space is called a contracting region if the Jacobian is UND in that region. Moreover, any trajectory starting in a ball of a constant radius, centered about a given trajectory and contained at all times in a contraction region remains in that ball and converges exponentially to a given trajectory. Further, global exponent convergence to this trajectory is guaranteed if the whole state space region is contracting. Alternatively, the system is called semi-contracting if the Jacobian matrix is uniformly negative semi-definite (UNS).

In general, let

{\overset{\cdot}{x}}_{m} (t) = h_{m} (x_{m} (t), t)

(4)

be a stable reference system, defining the desired specifications. The error between the states of the reference model and the states of the system is

e {(t) = (e_{1} (t), \dots, e_{n} (t))}^{T} = x_{m} (t) - x (t)

(5)

and the error dynamic system is

\overset{\cdot}{e} (t) = h_{1} (x, e, t)

(6)

If the Jacobian $J = \frac{\partial h_{1}}{\partial e}$ is UNS, then the system is considered semi-contracting. It was shown by Lohmiller and Slotine²¹ and Sharma and Kar²⁰ that the global and asymptotic stability can be ensured using CT results for contracting and semi-contracting systems, respectively.

3. Modeling the Wing Rock Motion

The WR motion can be described by the following one-degree-of-freedom nonlinear differential equation,^1,17 based on the Newton’s second law of angular motion:

I_{xx} \overset{\cdot\cdot}{ϕ} = T_{a} - T_{r}

(7)

where $ϕ$ is the roll angle, $I_{xx}$ is the moment of inertia about the roll axis, $T_{a}$ is the torque created by the ailerons and $T_{r}$ is the rolling torque, evaluated as

T_{r} = q \cdot S \cdot w \cdot C_{r}

(8)

where $q$ is the dynamic pressure, $S$ is the wing surface area, $w$ is the wing span and $C_{r}$ is the rolling moment coefficient. The dynamic pressure is given by

q = \frac{1}{2} ρ v^{2}

(9)

where $ρ$ and $v$ are the air density and airspeed, respectively. The following expression of the rolling moment coefficient was experimentally identified by Guglieri and Quagliotti² and is adopted in this paper,

C_{r} = b_{0} ϕ + b_{1} \overset{\cdot}{ϕ} + b_{2} | \overset{\cdot}{ϕ} | \overset{\cdot}{ϕ} + b_{3} ϕ^{3} + b_{4} \overset{\cdot}{ϕ} ϕ^{2}

(10)

where $b_{0} - b_{4}$ are nonlinear functions of the aircraft AOA and Reynolds number ( $Re$ ), which is related to the airspeed as

Re = \frac{ρ vL}{μ}

(11)

with $L$ and $μ$ being the wing root chord and air viscosity, respectively. Substituting (8), (9), (10) and (11) into (7), the WR motion can be described as

\overset{\cdot\cdot}{ϕ} + a_{0} ϕ + a_{1} \overset{\cdot}{ϕ} + a_{2} | \overset{\cdot}{ϕ} | \overset{\cdot}{ϕ} + a_{3} ϕ^{3} + a_{4} \overset{\cdot}{ϕ} ϕ^{2} = \frac{1}{I_{xx}} T_{a}

(12)

with

a_{i} = \frac{q \cdot S \cdot w \cdot b_{i}}{I_{xx}}, i = 0, \dots, 4

which are nonlinearly dependent on AOA and $Re$ as well as on the aircraft dimensions and environmental conditions and are normally determined experimentally.

Assuming an additive input disturbance (such as wind gusts) of the form $d = (\begin{matrix} 0 \\ d \end{matrix})$ and model parameters uncertainty of the form $a_{i} = a_{in} + Δ a_{i}, i = 0, \dots, 4$ and $b = b_{n} + Δ b$ with $b_{n} = I_{xx}^{- 1}$ , the WR dynamics (12) can be reformulated into the form of system

\overset{\cdot}{x} (t) = g (x (t), t) + b (x (t), t) u (t) + f (x (t), u (t), d (t), t)

(13)

where $x = {(\begin{matrix} ϕ & p \end{matrix})}^{T}$ , $p = \overset{\cdot}{ϕ}$ is the roll rate and

\begin{matrix} g = (\begin{matrix} p \\ g_{2} \end{matrix}) \\ b = (\begin{matrix} 0 \\ b_{n} \end{matrix}) \\ u = (\begin{matrix} 0 \\ T_{a} \end{matrix}) \\ f = (\begin{matrix} 0 \\ f_{2} \end{matrix}) \end{matrix}

(14)

where $g_{2} = - a_{0 n} ϕ - a_{1 n} p - a_{2 n} | p | p - a_{3 n} ϕ^{3} - a_{4 n} p ϕ^{2}$ is a known nonlinear function of the state vector and $f_{2} = - Δ a_{0} ϕ - Δ a_{1} p - Δ a_{2} | p | p - Δ a_{3} ϕ^{3} - Δ a_{4} p ϕ^{2} + Δ b \cdot T_{a} + d$ is an unknown nonlinear function of the state vector, the control input and the unpredictable disturbances.

Consider an aircraft with an $80^{\circ}$ delta wing with $L = 479$ mm, $b = 169$ mm and $I_{xx} = 1.0117 \times 10^{- 3}$ kg m² from Guglieri and Quagliotti.² According to the free-to-roll experiments described by Guglieri and Quagliotti² on the delta wing for $AOA = 25^{\circ}$ – $45^{\circ}$ and $v = 15$ –40 m s⁻¹, corresponding to $Re = 486, 000$ –1,290,000, the coefficients $a_{0}$ – $a_{4}$ were determined and are shown in Figure 2. Apparently, these parameters are non-constant and highly dependent on the operating conditions.

Figure 2.

The coefficients (a) $a_{0}$ , (b) $a_{1}$ , (c) $a_{2}$ , (d) $a_{3}$ and (e) $a_{4}$ corresponding to different AOA and Reynolds number $Re$ Guglieri and Quagliotti.²

The Simulink plant model is shown in Figure 3. The uncertain parameters $a_{0}$ – $a_{4}$ are realized via two-dimensional look-up tables with $Re$ and AOA as inputs. The dynamics inputs are the aileron created torque $T_{a}$ as the control input and the unknown signal $d$ as the disturbance input. An additional input $rst$ allows the system states to be reset to initial conditions. The nonlinear dynamics of (12) is realized in Matlab using a $Matlab$ $Function$ block.

Figure 3.

Simulink plant model.

As mentioned in Section 1, an uncontrolled WR motion results either in limit cycle oscillations or in unstable divergence, depending on the initial condition.¹³ This is shown in Figure 4 for two different initial conditions. In the case of initial conditions of $x_{0} = (\begin{matrix} 35 \times 10^{- 3} rad \\ 0 rad s^{- 1} \end{matrix})$ , the WR motion results in limit cycle oscillations, as shown in Figure 4(a). Step changes of $Re$ and AOA affect the oscillation characteristics but do not cause divergence. In the case of initial conditions of $x_{0} = (\begin{matrix} 35 \times 10^{- 3} rad \\ 52 \times 10^{- 3} rad s^{- 1} \end{matrix}),$ the WR motion results in roll angle and rate divergence, as shown in Figure 4(b). Hence, appropriate control strategies should be developed to avoid these undesirable behaviors.

Figure 4.

Uncontrolled WR dynamics: (a) limit cycle oscillations; (b) divergence.

4. Control of WR motion in nominal and uncertain environments

The desired closed-loop system behavior is defined by a reference model, chosen here as the following second-order system,

(\begin{matrix} {\overset{\cdot}{ϕ}}_{m} \\ {\overset{\cdot}{p}}_{m} \end{matrix}) = (\begin{matrix} 0 & 1 \\ - ω^{2} & - 2 ξ ω \end{matrix}) (\begin{matrix} ϕ_{m} \\ p_{m} \end{matrix})

(15)

Note that since the control objective is to suppress the WR dynamics, the desired steady-state operating point for the system states $(\begin{matrix} ϕ_{m} \\ p_{m} \end{matrix})$ is $(\begin{matrix} 0 \\ 0 \end{matrix})$ . Combining (13) and (15), the error dynamics is given by

\begin{matrix} {\overset{\cdot}{e}}_{1} & = {\overset{\cdot}{ϕ}}_{m} - \overset{\cdot}{ϕ} = p \\ {\overset{\cdot}{e}}_{2} & = {\overset{\cdot}{p}}_{m} - \overset{\cdot}{p} = - ω^{2} ϕ_{m} - 2 ξ ω p_{m} - g_{2} - f_{2} + b_{n} u \end{matrix}

(16)

When all of the parameters are known and there are no uncertainty and external disturbances, the controller

\begin{matrix} u (t) = b_{n}^{- 1} (- (ω^{2} - 2) e_{1} (t) - (2 ξ ω - 2) e_{2} (t) + (a_{0} - ω^{2}) x_{1} (t) + (a_{1} - 2 ξ ω) x_{2} (t) + a_{2} | x_{2} (t) | x_{2} (t) + a_{3} x_{1}^{3} (t) + a_{4} x_{1}^{2} (t) x_{2} (t)) \end{matrix}

(17)

is proposed, according to Lohmiller and Slotine^21,22 and Jouffroy and Slotine.²⁷ Substituting (17) into (13) and rearranging the results, then the error dynamics is

(\begin{matrix} {\overset{\cdot}{e}}_{1} \\ {\overset{\cdot}{z}}_{1} \end{matrix}) = (\begin{matrix} - 1 & 1 \\ - 1 & - 1 \end{matrix}) (\begin{matrix} e_{1} \\ z_{1} \end{matrix})

(18)

with $z_{1} = e_{1} + e_{2} .$ The Jacobian of the system (18) is UND so the error dynamics is contracting and the trajectories of the WR system converge to the reference model trajectories.

When the coefficients $a_{0}$ – $a_{4}$ are unknown, their nominal values can be chosen as $a_{0 n} = 7 \times 10^{- 3}$ , $a_{1 n} = - 0.02$ , $a_{2 n} = 0.25$ , $a_{3 n} = - 0.01$ and $a_{4 n} = 0.05$ . In addition, the moment of inertia about the roll axis was measured incorrectly as $I_{xx} = 2 \times 10^{- 3}$ kg·m² and an external input disturbance $d (t) = 2.5 \times 10^{- 4} \underset{k = 0}{\sum^{\infty}} (1 (t - 100 k) - 1 (t - 50 - 100 k))$ is applied to the system. In such a case the controller is designed as

\begin{matrix} u (t) = b_{n}^{- 1} (- (ω^{2} - 2) e_{1} (t) - (2 ξ ω - 2) e_{2} (t) + (a_{0 n} - ω^{2}) x_{1} (t) + (a_{1 n} - 2 ξ ω) x_{2} (t) + a_{2 n} | x_{2} (t) | x_{2} (t) + a_{3 n} x_{1}^{3} (t) + a_{4 n} x_{1}^{2} (t) x_{2} (t)) - {\hat{f}}_{2} \end{matrix}

(19)

where ${\hat{f}}_{2}$ is the estimate of the lumped uncertainty and disturbance function $f_{2}$ calculated by a UDE-based DOB. Substituting (19) into (13) and rearranging results, then the error dynamics is

(\begin{matrix} {\overset{\cdot}{e}}_{1} \\ {\overset{\cdot}{z}}_{1} \end{matrix}) = (\begin{matrix} - 1 & 1 \\ - 1 & - 1 \end{matrix}) (\begin{matrix} e_{1} \\ z_{1} \end{matrix}) - (\begin{matrix} 0 \\ f_{2} - {\hat{f}}_{2} \end{matrix})

(20)

In order to design the UDE-based DOB, the following procedure is proposed. First, note that the lumped uncertainty and disturbance function $f_{2}$ may be rewritten from (14) as

f_{2} = \overset{\cdot}{p} - g_{2} - b_{n} u

(21)

but it cannot be used directly in order to preserve the causality. Instead, its estimate is created after appropriate filtering

{\hat{f}}_{2} = f_{2} ☆ q (t) = (\overset{\cdot}{p} - g_{2} - b_{n} u) ☆ q (t)

(22)

where $q (t)$ is an impulse response of a so-called $Q$ -filter, whose passband contains the majority of the spectral content of $f_{2}$ and $☆$ is the convolution operator. Observing the Laplace domain of (22),

{\hat{F}}_{2} (s) = F_{2} (s) Q (s) = sQ (s) P (s) - (G_{2} (s) + b_{n} U (s)) Q (s)

(23)

it can be concluded that if the $Q$ -filter is strictly proper, there is no need of measuring or calculating the roll acceleration since $sQ (s)$ is feasible.

If the majority of the spectral content of $f_{2}$ resides within the passband of $q (t)$ or in other words $Q (s) \approx 1$ in the frequency band where $F_{2} (s) \neq 0,$ then $F_{2} (s) - {\hat{F}}_{2} (s) = F_{2} (s) (1 - Q (s)) \approx 0$ , hence $f_{2} - {\hat{f}}_{2} \approx 0$ and the Jacobian of the system (20) is UND. Then the error dynamics is contracting and the trajectories of the WR system converge to the reference model trajectories. The Simulink block diagram of the overall closed-loop system is shown in Figure 5.

Figure 5.

Simulink plant model.

In order to suppress the WR within 10 s without overshoot, $ξ = 1$ and $ω = 0.2 π$ were chosen. The $Q$ -filter was selected as $Q (s) = 1 / (10 s + 1)$ in order to include the system dynamics within the filter passband. During the simulations the controller was periodically turned on and off to demonstrate the differences between controlled and uncontrolled motions. The operating conditions were also changed during the simulations, as shown in Figure 6, to better demonstrate the performance of the controller under different operating conditions. The simulation results under nominal and uncertain conditions are shown in Figures 7 and 8, respectively. Both approaches demonstrate excellent results which are very similar since the DOB is able to estimate and quickly cancel the system uncertainty. However, the difference between two approaches is well reflected by the control effort (aileron torque) which is much higher in case of operating in an uncertain environment.

Figure 6.

Change of the operating conditions during the simulations.

Figure 7.

Simulation results: operation under nominal conditions.

Figure 8.

Simulation results: operation under uncertain conditions.

5. Conclusions

In this paper, the CT results have been applied to the problem of WR motion. In order to cancel the uncertainties and disturbances, a nonlinear state-feedback DOB based on the UDE strategy is employed. Both nominal and robust control strategies have been developed. The proposed algorithms have demonstrated excellent performance in suppressing the WR oscillations under both nominal and uncertain conditions.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Guglieri

Quagliotti

. Experimental observation and discussion of the wing rock phenomenon. Aerospace Sci Technol 1997; 2: 111–123.

Guglieri

Quagliotti

. Analytical and experimental analysis of wing rock. Nonlin Dynam 2001; 24: 129–146.

Liebst

. The dynamics, prediction, and control of wing rock in high-performance aircraft. Phil Trans R Soc A 1998; 356: 2257–2276.

Katz

. Wing/vortex interactions and wing rock. Prog Aerospace Sci 1999; 35: 727–750.

Monahemi

Krstic

. Control of wing rock motion using adaptive feedback linearization. J Guidance Control Dynam 1996; 19: 905–912.

Jain

Kaul

Ananthkrishnan

. Parameter estimation of unstable, limit cycling systems using adaptive feedback linearization: example of delta wing roll dynamics. J Sound Vibration 2005; 287: 939–960.

Xin

Balakrishnan

. Control of the wing rock motion using a new suboptimal control method. J Aerospace Eng 2004; 218: 257–266.

Abdulwahab

Hongquan

. Periodic motion suppression based on control of wing rock in aircraft lateral dynamics. Aerospace Sci Technol 2008; 12: 295–301.

Shue

S-P

Agarwal

. Extension of Kalman filter theory to nonlinear systems with application to wing rock motion. Asian J Control 2000; 2: 42–49.

10.

Sreenatha

Patki

Joshi

. Fuzzy logic control for wing rock phenomenon. Mechanics Research Communications 2000; 27: 359–364.

11.

Sreenatha

Choi

J-Y

Wong

P-P

. Design and implementation of fuzzy logic controller for wing rock. Int J Control Automat Syst 2004; 2: 494–500.

12.

Liu

. Reinforcement adaptive fuzzy control of wing rock phenomena. IEE Proc Control Theory Appl 2005; 152: 615–620.

13.

Lin

C-M

Hsu

C-F

. Supervisory recurrent fuzzy neural network control of wing rock for slender delta wings. IEEE Trans Fuzzy Syst 2004; 12: 733–742.

14.

Joshi

Strenatha

Chandrasekhar

. Suppression of wing rock of slender delta wings using a single neuron controller. IEEE Trans Control Syst Technol 1998; 6: 671–677.

15.

Hsu

C-F

Lin

C-M

Chen

T-Y

. Neural-network-identification-based adaptive control of wing rock motions. IEE Proc Control Theory Appl 2005; 152: 65–71.

16.

Hsu

C-F

Lin

C-M

Lee

T-T

. Wavelet adaptive backstepping control for a class of nonlinear systems. IEEE Trans Neural Networks 2006; 17: 1175–1183.

17.

Liu

Z-L

C-Y

Svoboda

. Variable phase control of wing rock. Aerospace Sci Technol 2006; 10: 27–35.

18.

Liu

Z-L

Svoboda

. A new control scheme for nonlinear systems with disturbances. IEEE Trans Control Syst Technol 2006; 14: 176–181.

19.

Astolfi

Karagiannis

Ortega

. Towards applied nonlinear adaptive control. Annu Rev Control 2008; 32: 136–148.

20.

Sharma

Kar

. Adaptive control of wing rock system in uncertain environment using contraction theory. In Proceedings of the American Control Conference, Seattle, WA, 2008, pp. 2963–2968.

21.

Lohmiller

Slotine

JJE

. On contraction analysis for nonlinear systems. Automatica 1998; 34: 683–696.

22.

Lohmiller

Slotine

JJE

. Control system design for mechanical systems using contraction theory. IEEE Trans Automat Control 2000; 45: 884–889.

23.

Radke

Gao

. A survey of state and disturbance observers for practitioners. In Proceedings of the American Control Conference, Minnesota, MN, 2006, pp. 5183–5188.

24.

Liu

Z-L

Sloboda

. A new control scheme for nonlinear systems with disturbances. IEEE Trans Control Syst Technol 2006; 14: 176–181.

25.

Zhong

Q-C

Rees

. Control of uncertain LTI systems based on an uncertainty and disturbance estimator. J Dynam Syst Meas Conrol Trans ASME 2004; 126: 905–910.

26.

Zhong

Q-C

Kuperman

Stobart

. Design of UDE-based controllers from their two-degree-of-freedom nature. Int J Robust Nonlin Control 2011; to appear.

27.

Jouffroy

Slotine

JJE

. Methodological remarks on contraction theory. In Proceedings 43rd IEEE Conference on Decision and Control, Bahamas, 2004, pp. 2537–2543.