Abstract

The main goal of this work is to study the stability properties of an aircraft with nonlinear behavior, controlled using a gain scheduled approach. An output feedback is proposed which is able to guarantee asymptotical stability of the task-coordinates origin and safety of the operation in the entire flight envelope. The results are derived using theory of hybrid and singular perturbed systems. It is demonstrated that both body velocity and orientation asymptotic tracking can be obtained in spite of nonlinearities and uncertainty. The results are illustrated using numerical simulations in F16 jet.

1. Introduction

Gain scheduled control (GSC) is widely employed in flight control applications. The principle of operation of such a controller relies on the use of a piece-wise linear approximation of aircraft dynamics to conclude stability via direct or indirect arguments relating the linear approximation and the original nonlinear dynamics [1]. Usually, stability of every linear system is analyzed, and the design of an interpolation controller to guarantee stable switching is performed as a separated control design task [2–4].

Since the flight dynamics have nonlinear nature, the linear approximation of the system leads to controllers with stability and performance limitations. To overcome such limitations there exist in the literature a variety of control methodologies which provide important advances in GSC and hybrid and nonlinear approaches [5–8]; see also GSC surveys [1, 9–19].

Robustness of controllers is crucial not only due to the high nonlinear behavior of the system, but also because aeronautical systems are subjected to external perturbations and they must have some fault tolerant capabilities. There exist in the literature a variety of works leading the control of these systems from a robust or adaptive point of view. In [14] a multi-input, multioutput nonlinear model predictive controller is proposed that makes use of a neurofuzzy predictor model. The authors analyze the robustness of the proposed control in presence of uncertainties in the physical parameters. In [15] attitude stabilization with external disturbance, unknown inertia parameters, and actuator uncertainties is analyzed. An adaptive control is proposed and input-to-state stability is derived. Reference [16] studies the spacecraft attitude tracking problem; the conditions that guarantee that the attitude and body-rate error are ultimately bounded are derived and bounds for the steady-state tracking errors are also provided. Reference [17] proposes an observer-controller control scheme to solve the output feedback attitude control of a rigid body with bounded input.

In [10] a robust sum-of-squares (SOS) LMI (linear matrix inequality) method to design a nonlinear controller for longitudinal dynamics with parametric uncertainties is proposed. The control design problem is formulated as a robust SOS problem and then an LMI representation is derived using an algorithm that is solved via a stochastic ellipsoid method. In [11] an output feedback to stabilize a planar nonlinear model of a vertical take-off and landing (PVTOL) aircraft is proposed. To address the lack of full state measurement, a finite time convergent observer is designed. Reference [12] addresses the nonlinear tracking problem of a roll-rate-sensorless vertical take-off and landing aircraft in the presence of roll-angle measurement delay. The authors develop an extended attitude observer and a tracking controller based on the observer by using the first-order Pade approximation signal of the delayed roll angle, two global coordinate transformations, and an extension of the backstepping technique.

In many of the works above, it is not clear if the controller can deal with structural uncertainty that may arise from a failure, or from the nonlinearities of the system. Moreover, some of them make use of observers to estimate the entire state, which usually leads to high-order controllers. Some of these drawbacks have made researches investigate improved GSC strategies that can ensure the stability of the system in spite of nonlinearities but still with simple controllers. Many of these works make use of recent advances in switching and hybrid systems.

Recently, switched LPV (linear parameter varying) systems have gained the attention of researchers due to the fact that stabilization can also be derived using a multiple Lyapunov function approach. In [20] it is studied the GSC using a switched polytopic approach. In such work, it was proved that the switched polytopic system is input-to-state bounded if the subsystems share a common Lyapunov function, and average dwell time is no smaller than a given constant. On the other hand, [21] proposed a hybrid tracking control for aircraft described by discrete-time stochastic linear hybrid systems; the flight logic is described by discrete dynamics governed by Markov chains. Conditions for convergence of a model adaptive estimation algorithm are proposed, and a new residual-mean interacting multiple model algorithm is proposed.

Reference [18] uses a piecewise linear description of the system and variations in operation points modeled as switches between constituent linearized system dynamics to analyze the control tracking problem. An adaptive control scheme is proposed that can achieve asymptotic tracking performance if the reference input is sufficiently rich and the switches are sufficiently slow. Reference [19] develops a hybrid control strategy that provides autonomous transition between hovered and leveled flights to a model-scale fixed-wing aircraft. The controller, together with an appropriate choice of reference maneuvers, ensures that a transition from hovered flight to level flight, or vice versa, is achieved. In [22], a sensor-based backstepping method is implemented on a fixed-wing aircraft with its focus on handling structural changes caused by damages. The controller combines nonlinear dynamic inversion, which is highly dependent on the knowledge of the system structure, with a sensor-based backstepping control approach.

In this work, we are focused on the velocity regulation and orientation of nonlinear aircraft. It is known (see [23]) that the more general description of the flight dynamics is piece-wise nonlinear since forces and torques involved may vary dramatically with altitude, angle of attack, side-slip angle, and mach number, among others. That is, in the best situation, multiple nonlinear description of the aircraft constitutes the more reliable description that can be obtained. Under this premise, we depart from a six degree of freedom, nonlinear piece-wise description of the aircraft to study the stability of the system under a simple linear dynamic output feedback (as in GSC). Using theory of switched hybrid systems and singular perturbation systems, we demonstrate that the origin of the closed-loop piecewise nonlinear description of the aircraft is asymptotically stable, even in presence of uncertainties that can be either structural or parametric. This result constitutes the main contribution of this paper.

The tracking control strategy proposed here achieves a desired linear velocity and attitude, while ensuring body rate stabilization in spite of nonlinearities and model uncertainty. As in GSC, the proposed controller is able to adapt its operation based on the suitable definition of the switching criterion. Stability conditions for every operation mode and the overall switching system are derived; such piece-wise nonlinear analysis has not been performed before, to the authors’ knowledge.

Based on the aircraft movement restrictions, a cascade control structure is proposed to solve the underactuated task of solving simultaneously body rate stabilization and attitude regulation. The attitude problem is formulated using the kinematic restrictions relating angular body velocity and quaternions, avoiding in this way the drawback of multiple singularities in a given operation condition. It is shown that the proposed controller is able to robustly stabilize the origin of the controlled nonlinear dynamics of the aircraft in spite of parametric or structural uncertainties, as long as they satisfy being bounded and continuous Lipschitz functions of their arguments in the operation domain. Numerical simulations of an F16 aircraft are used to illustrate advantages of the proposed control.

The paper begins with the nonlinear piecewise description of the aircraft and the control problem formulation. Section 3 introduces some preliminary results to establish the stability and the equilibrium uniqueness of some auxiliary systems, while Section 4 is devoted to deriving the main result of the paper. Illustrations of the results are given in Section 5 using a fixed-wing aircraft. Finally, Section 6 summarizes the main contribution of the proposed structure and presents some conclusions.

2. Problem Statement and Dynamic Description of the Aircraft

In this work, we depart from a nonlinear description of the aircraft using wind axis. As it is known, the nonlinear behavior of the aircraft depends crucially on the movement task performed. Typically the model parameters are obtained about certain operation conditions. Due to this fact, the nonlinear behavior of the aircraft is better described using piecewise continuous models. In [23], it is stated that the nonlinear description of the aircraft depends on variables such as dynamic pressure, altitude, angle of attack, and sideslip angle among the most important. Let the nonlinear model be valid in the domain . Departing form this premise, the piecewise nonlinear description of the system can be written without loss of generality as constituted by a linear function and a nonlinear function clustering high order terms and uncertainty, depending exclusively on system’s states. In this work, we assume that such description is valid on a domain . In this way, the airplane movement description and orientation is given by where are deviation variables with respect to an operation point, , , and . Moreover, where with , , and given bywhere account for state-dependent uncertainties, and , , and so forth, are terms accounting for higher order nonlinear terms as shown in the following example:

Moreover the output and input of the system are where is the throttle position and , , and are the control surface positions. The following is assumed: is a bounded, continuous differentiable function of in .The assumption (A.1) implies that is Lipschitz with respect to in . Notice that the Euler angles in (5) are related to the quaternions (2)-(3).

2.1. Problem Formulation

Let the airplane model be given by (1)–(6); find an output dependent piecewise linear control such that for every , where and , , and are the desired values of the aircraft velocity magnitude and quaternion orientation.

2.2. Control Structure

To accomplish both the movement and orientation control defined in (7), it is necessary to define the regulation error as follows:

Notice that (8) leads to the following implicit definition of the state error . By substituting (8), (9) in (1) the regulation error dynamics is obtained as follows: with where and

To achieve (7) two feedback loops can be used: one that depends on the velocity regulation error (body stabilization task) and another that performs the orientation regulation. In this work the following gain scheduling control structure is studied: where with where is the number of modes defined by the design. , are the control gains for the body stabilization loop for every mode . , are the control gains for the orientation control loop for every mode .  and are constant.The approach proposed in this work is to use the integral parameters epsilon 1 and epsilon 2 as perturbation parameters in order to establish stability conditions for the nonlinear closed-loop system. The schematic diagram of the controller (13)–(15) is given in Figure 1. To solve the regulation problem, consider the following additional assumptions:the triplet is controllable and observable for every operation mode;the operation domain or flight envelope satisfies with for and .

Figure 1 

Controller structure.

3. Preliminaries

In this section, important results to deduce the main contribution of this paper are introduced. To this end, let us analyze the aircraft dynamics (10) under the action of the control law (14), which is given by where , are the state and quaternion tracking error, respectively, is a virtual state of the system associated with the integral action, and

The equilibrium point of (16) is given by where and is a constant vector. Regarding the properties of the system at equilibrium the following propositions are stated.

Proposition 1. There exists an explicit function derived from (18) if where is the Lipschitz constant of .

Proof. Notice that
From the hypothesis, we have that ; therefore . Furthermore since is differentiable, from (18) we have
Using the contraction mapping theorem [24], there exists a unique solution of (18); hence there exists an explicit map .
In other words, if , where , hence (20) can be rewritten as
From (22) it can be seen that if , then . Such results imply the existence of a sufficiently large such that condition of this proposition is satisfied.

Proposition 2. Considering the map there exist real negative eigenvalues of , such that is positive definite.

Proof. Computing the -derivative of (23), we have
According to Theorem  3 in [25] and Assumption (A.2), the eigenvalues of (17) can be arbitrarily assigned (by virtue of gain ). Therefore, they are chosen for to be negative definite. Notice that in view of this fact is Hurwitz and is positive definite. Moreover, given (17), if , there exists a sufficiently large value of such that the term prevails over , which by virtue of Assumption (A.1) is bounded in . These arguments lead us to deduce the existence of controller gains such that is positive definite, which implies that is positive definite.

4. Main Results

In this section, the main contributions of this work are stated in Theorems 3 and 4. These theorems contain results regarding the stability of every operation mode (i.e., stability of a two-loop controller acting in a given operation region) and the stability of the entire operation region or flight envelope conditions.

4.1. Stability of Every Operation Mode

This stability analysis is performed in two parts; firstly the stability of the body stabilization loop is analyzed and afterwards the orientation stabilization loop. The stability analysis makes use of singularly perturbed systems [26], since the idea is to use integral control gains as perturbation parameters. In other words, since the integral actions constitute slow actions of the controller, the closed-loop system can be described naturally using two systems: a fast system given by the aircraft dynamics under a pure proportional action and a slow system given by the integral dynamics of the regulation error. The singular perturbed system arises as the integral gains approach zero. The interested reader is referenced to [26, 27] for a more detailed explanation of singularly perturbed systems.

Theorem 3. Let the dynamics of the aircraft be described by (10)–(15) in the set , let , , , and let be as in Propositions 1 and 2; then there exist constants and , such that the origin of (10) under the action of control (13) is asymptotically stable.

Proof. As a first step, consider the aircraft under the body stabilization loop. From Proposition 1, a unique isolated equilibrium exists; let us denote such equilibrium values as and . In view of this fact, the following coordinates are introduced to shift such equilibrium to the origin
At this point, let us use variable as a perturbation parameter; hence the closed-loop system (10)–(13) can be rewritten as a standard singularly perturbed system given by where , , , , , and .
The boundary-layer and reduced systems associated with (26) are given, respectively, by Observe that (27) can be seen as the aircraft system under pure proportional action. On the other hand, notice that (28) is a steady-state relationship between and that satisfies , which has the same structure of (18); therefore Proposition 1 can be used; hence, for a suitable proportional gain (i.e., sufficiently large ), there exists an explicit function such that Firstly, we will show that the linear output feedback (14) is able to track asymptotically the velocity of the nonlinear aircraft. Observe that (30) can be seen as the aircraft system under pure integral action. To derive the stability properties of (26), we can use systems (27) and (30).
We start by showing that, for a sufficiently large , the origin of (27) is exponentially stable. To this end, let us consider the following scalar function: where is positive definite. Since is Hurwitz, the time derivative of (31) along (27) is given by with positive definite, since is Lipschitz; hence where is the Lipschitz constant of . To ensure that is negative definite, it is required that , since and with ; hence From (34) it can be observed that , which implies that, for a sufficiently large , the origin of (27) is exponentially stable.
On the other hand, to study stability of the reduced system (30), consider the scalar function . Let ; hence the time derivative of along (27) is .
Using the result in Proposition 2, is positive definite, which in turn implies that is positive semidefinite. Using the Lasalle invariance principle and provided that only in , , it is possible to conclude that the origin of (27) is asymptotically stable.
At this point, asymptotic stability of (16) can be concluded using Theorem  2.1 in [26]. The stability properties of the reduced and boundary layer system imply the existence of a parameter such that, for every , the origin of (16) is asymptotically stable, with a scalar function So far, it has been shown that the linear output feedback (14) is able to asymptotically stabilize the velocity of the nonlinear aircraft.
To show that the second linear loop (15) is able to regulate the orientation, consider the following: let us compute the closed-loop system (10)–(15) under steady-state conditions: where the subindex is the value of the variable under equilibrium conditions, , is the right-hand side of (26), is the virtual state associated with the integral action, , and From (38) it is clear that , since is full rank. Observe that , , and is a constant vector, so (36) can be rewritten as Observe that (40) has the same structure as (18), except for the translation term . Hence Proposition 1 can be used to guarantee the existence and uniqueness of the equilibrium point. That implies that in steady-state conditions for every value of the state one and only one integral action corresponds. As before, let us consider system (26) under the action of controller (15) as a singularly perturbed system, with the gain of the integral action of the second loop as a new perturbation parameter. In this way, such closed-loop system can be rewritten as: with , , is the new time scale, and , , , and . Observe that (41) has an equilibrium point in . Moreover, observe that , as implies that as .
At this point, let us compute the reduced and boundary layer systems associated with (41), which are given, respectively, by Notice that (43) has been studied before; the origin of such system is asymptotically stable for . On the other hand, the stability of the reduced system (42) can be readily established by suitable gains that make the matrix Hurwitz, hence, the existence of a Lyapunov function associated with (42) can be derived such that .
Given the arguments above and the Lipschitz nature of , Theorem  2.1 in [26] implies the existence of a parameter such that for all the origin of (41) is asymptotically stable.