# Stability of Gain Scheduling Control for Aircraft with Highly Nonlinear Behavior

Authors: Fany Mendez-Vergara; Ilse Cervantes; Angelica Mendoza-Torres
Journal: Mathematical Problems in Engineering
 (2014)

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## 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.


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## Body

## 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. Section3 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 j be valid in the domain Ωj. 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 (Φ¯j(x)) 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 Ωj. In this way, the airplane movement description and orientation is given by
(1)x˙=Ajx(t)+Bju(t)+Φ¯j(x),ϑ˙=T(ϑ)ωw,
where x=[VT,β,α,P,Q,R]T are deviation variables with respect to an operation point, ϑ=[η,ϵ]T, ϵ=[ϵ1,ϵ2,ϵ3]T, and ωw=[P,Q,R]T. Moreover,
(2)T(ϑ)≜[-12ϵT12[ηI-S(ϵ)]],
where
(3)S(ϵ)=[0-ϵ3ϵ2ϵ30-ϵ1-ϵ2ϵ10],
with Aj, Bj, and Φ¯j(x) given by(4)Aj=m[XV+XTVcos⁡(αe+αT)0Xα0000Yβ0Yp0Yr-VTeZV-XTV(αe+αT)0Zα0VTe+Zq00μLβ+σ1Nβ0μLp+σ1Np0μLr+σ1NrMV+MTV0Mα+MTα0Mq00μNβ+σ2Lβ0μNp+σ2Lp0μNr+σ2Lr],Bj=[1m∂Da∂δT1m∂Da∂δe1m∂Da∂δa1m∂Da∂δr1mVT∂Ca∂δT1mVT∂Ca∂δe1mVT∂Ca∂δa1mVT∂Ca∂δr1mVTcos⁡β∂L∂δT1mVTcos⁡β∂L∂δe1mVTcos⁡β∂L∂δa1mVTcos⁡β∂L∂δrJw-1∂lw∂δTJw-11Ix′∂lw∂δeJw-11Ix′∂lw∂δaJw-11Ix′∂lw∂δrJw-1∂mw∂δTJw-11Ix′∂mw∂δeJw-11Ix′∂mw∂δaJw-11Ix′∂mw∂δrJw-1∂nw∂δTJw-11Ix′∂nw∂δeJw-11Ix′∂nw∂δaJw-11Ix′∂nw∂δr],(5)Φ¯j(x)=[1m{∂L∂δeDae+∑‍Da(VT,β,α,P,Q,R)-gcos⁡αcos⁡βsinθ+gsinβsinϕcos⁡θ+gsinαcos⁡ϕcos⁡θ}+φ1¯{1mVT}{∂L∂δemVTR+Cae+∑‍Ca(VT,β,α,P,Q,R)+gcos⁡αsinβsinθ+gcos⁡βsinϕcos⁡θ-gsinαsinβcos⁡ϕcos⁡θ∂L∂δe}+φ2¯{1mVTcos⁡β}{∂L∂δemVT{Qcos⁡β-Psinβ}+Le+∑‍L(VT,β,α,P,Q,R)+gsinαsinθ+gcos⁡αcos⁡ϕcos⁡θ+∂L∂δeδe+∂L∂δrδr}+φ3¯Jw-1{PQIxz′+QRIy′+RQIz′+lwe+∑‍lw(VT,β,α,P,Q,R)∂L∂δe}+φ4¯Jw-1{-P{RIz′+PIxz′}-R{PIxz′+PIz′}+mwe+∑‍mw(VT,β,α,P,Q,R)∂L∂δe}+φ5¯Jw-1{PQIz′-QPIy′+RQIxz′+nwe+∑‍nw(VT,β,α,P,Q,R)∂L∂δe}+φ6¯],where φi¯i=1,…,6 account for state-dependent uncertainties, and ∑‍Da(VT,β,α,P,Q,R), ∑‍Ca(VT,β,α,P,Q,R), and so forth, are terms accounting for higher order nonlinear terms as shown in the following example:
(6)∑‍Da(VT,β,α,P,Q,R)={∂2Da∂VT2VT22!+∂3VT∂VT3VT33!+∂4VT∂VT4VT44!+⋯}+{∂2Da∂α2α22!+∂3Da∂α3α33!+∂4Da∂α4α44!+⋯}+{∂2Da∂β2β22!+∂3Da∂β3β33!+∂4Da∂β4β44!+⋯}+{∂2Da∂P2P22!+∂3Da∂P3P33!+∂4Da∂P4P44!+⋯}+{∂2Da∂Q2Q22!+∂3Da∂Q3Q33!+∂4Da∂Q4Q44!+⋯}+{∂2Da∂R2R22!+∂3Da∂R3R33!+∂4Da∂R4R44!+⋯}.Moreover the output and input of the system arey=[VT,P,Q,R]TuT=[δTδeδaδr] where δT is the throttle position and δe, δa, and δr are the control surface positions. The following is assumed:(A.1)
Φ¯j is a bounded, continuous differentiable function of x in Ωj.The assumption (A.1) implies that Φ¯j is Lipschitz with respect to x in Ωj. 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
(7)lim⁡t→∞[VTPQRηϵ]=[VTref000ηrefϵref],
for every x∈D, where D=∪Ωj and VTref, ηref, and ϵref 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:
(8)ey=y-yref,(9)eϑ=ϑ-ϑref.Notice that (8) leads to the following implicit definition of the state error Cjex=ey. By substituting (8), (9) in (1) the regulation error dynamics is obtained as follows:
(10)e˙x=Ajex+Bju+Φj1(ex),ey=Cjex,e˙ϑ=T(eϑ)ωe,
with
(11)ωe≜ωw-ωref,
where ωw=[P,Q,R]T and
(12)Φj1(ex)≜Φ¯j(ex+xref)+Ajxref.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:
(13)u=u1+u2,
where
(14)u1=-K¯jp,1ey-εj1KjI,1∫‍ey,(15)u2=-K¯jp,2eϑ-εj2KjI,2∫‍eϑ,
with j=1,…,s where s is the number of modes defined by the design.K¯jp,1, KjI,1∈R4×4 are the control gains for the body stabilization loop for every mode j.K¯jp,2, KjI,2∈R4×4 are the control gains for the orientation control loop for every mode j.εj1 and εj2∈R 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:(A.2)
the triplet(Aj,Bj,Cj) is controllable and observable for every operation mode;(A.3)
the operation domain or flight envelopeD satisfies D=∪Ωj with int⁡Ωj⋂‍int⁡Ωi=∅ for i≠j and Ωj≠∅.Figure 1
Controller structure.


## 2.1. Problem Formulation

Let the airplane model be given by (1)–(6); find an output dependent piecewise linear control such that
(7)lim⁡t→∞[VTPQRηϵ]=[VTref000ηrefϵref],
for every x∈D, where D=∪Ωj and VTref, ηref, and ϵref 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:
(8)ey=y-yref,(9)eϑ=ϑ-ϑref.Notice that (8) leads to the following implicit definition of the state error Cjex=ey. By substituting (8), (9) in (1) the regulation error dynamics is obtained as follows:
(10)e˙x=Ajex+Bju+Φj1(ex),ey=Cjex,e˙ϑ=T(eϑ)ωe,
with
(11)ωe≜ωw-ωref,
where ωw=[P,Q,R]T and
(12)Φj1(ex)≜Φ¯j(ex+xref)+Ajxref.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:
(13)u=u1+u2,
where
(14)u1=-K¯jp,1ey-εj1KjI,1∫‍ey,(15)u2=-K¯jp,2eϑ-εj2KjI,2∫‍eϑ,
with j=1,…,s where s is the number of modes defined by the design.K¯jp,1, KjI,1∈R4×4 are the control gains for the body stabilization loop for every mode j.K¯jp,2, KjI,2∈R4×4 are the control gains for the orientation control loop for every mode j.εj1 and εj2∈R 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:(A.2)
the triplet(Aj,Bj,Cj) is controllable and observable for every operation mode;(A.3)
the operation domain or flight envelopeD satisfies D=∪Ωj with int⁡Ωj⋂‍int⁡Ωi=∅ for i≠j and Ωj≠∅.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
(16)e˙=[Aj1I-εj1BjKjI,1Cj06×6]e+[Φj1(ex)0],
where ex, es are the state and quaternion tracking error, respectively, e=[ex,es]T∈R12 is a virtual state of the system associated with the integral action, and
(17)Aj1≜Aj-BjK¯jp,1Cj∈R6×6.The equilibrium point of (16) is given by
(18)ex,eq=-jA1-1[bI+Φj1(ex,eq)]≜g(bI,ex,eq),
where e˙s=0⇒es=bI and bI is a constant vector. Regarding the properties of the system at equilibrium the following propositions are stated.Proposition 1.
There exists an explicit functionex,eq=h(bI) derived from (18) if
(19)∥Aj1-1∥<1Ljd1,
where Ljd1 is the Lipschitz constant of Φj1(ex).Proof.
Notice that(20)∥-Aj1-1∇ex,eqΦj1(ex,eq)∥<∥Aj1-1∥∥∇ex,eqΦj1(ex,eq)∥<∥Aj1-1∥Ljd1.
From the hypothesis, we have that∥Aj1-1∥Ljd1<1; therefore ∥-Aj1-1∇ex,eqΦj1(ex,eq)∥<1. Furthermore since Φj1(ex,eq) is differentiable, from (18) we have
(21)∥∇ex,eqg(ex,eq)∥=∥-Aj1-1∇ex,eqΦj1(ex,eq)∥<1.
Using the contraction mapping theorem [24], there exists a unique solution of (18); hence there exists an explicit map ex,eq=h(bI).
In other words, ifK¯jp,1≜γjKjp,1, where γj>0, hence (20) can be rewritten as
(22)∥γj(Ajγj-BjKjp,1Cj)∥≤|γj|∥(Ajγj-BjKjp,1Cj)∥,Ljd1<γj∥Ajγj-BjKjp,1Cj∥.
From (22) it can be seen that if γj→∞, then γj∥(Aj/γj)-BjKjp,1Cj∥→∞. Such results imply the existence of a sufficiently large γj such that condition of this proposition is satisfied.Proposition 2.
Considering the map(23)es=-Aj1ex-Φj1(ex),
there exist real negative eigenvalues of Aj1, such that [∇exes]-1 is positive definite.Proof.
Computing theex-derivative of (23), we have
(24)∇exes=-Aj1-∇exΦj1(ex).
According to Theorem  3 in [25] and Assumption (A.2), the eigenvalues of (17) can be arbitrarily assigned (by virtue of gain K¯jp,1). Therefore, they are chosen for Aj1 to be negative definite. Notice that in view of this fact Aj1 is Hurwitz and -Aj1 is positive definite. Moreover, given (17), if K¯jp,1=γjKjp,1, there exists a sufficiently large value of γj such that the term Aj1 prevails over ∇exΦj1(ex), which by virtue of Assumption (A.1) is bounded in Ωj. These arguments lead us to deduce the existence of controller gains K¯jp,1 such that ∇exes is positive definite, which implies that [∇exes]-1 is positive definite.


## 4. Main Results

In this section, the main contributions of this work are stated in Theorems3 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 Ωj, let KjI,i∈R4×4, i∈{1,2}, j∈{1,…,s}, and let K¯jp,1 be as in Propositions 1 and 2; then there exist constants 0<jε1<εmax⁡ and 0<jε2<εmax⁡, 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 Proposition1, a unique isolated equilibrium exists; let us denote such equilibrium values as ex,eq and es,eq. In view of this fact, the following coordinates are introduced to shift such equilibrium to the origin
(25)z≜ex-ex,eq,w≜es-es,eq.
At this point, let us use variableεj1>0 as a perturbation parameter; hence the closed-loop system (10)–(13) can be rewritten as a standard singularly perturbed system given by
(26)[jε1z′w′]=[Aj1I-BjKjI,1Cj04×4][zw]+[Φj2(z)0],
where Φj2(z)≜jΦ1(z+ex,eq), t′=ϵj1t, z≜z(t′), z′≜dz/dt′, w≜w(t′), and w′≜dw/dt′.
The boundary-layer and reduced systems associated with (26) are given, respectively, by
(27)z˙=Aj1z+Φj2(z),(28)0=Aj1z+w+Φj2(z),(29)w′=-BjKjI,1Cjz.
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 z and w that satisfies z=-Aj1-1[w+Φj2(z)], which has the same structure of (18); therefore Proposition 1 can be used; hence, for a suitable proportional gain (i.e., sufficiently large γj), there exists an explicit function z≜h(w) such that
(30)w′=-BjKjI,1Cjh(w).
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 largeγj, the origin of (27) is exponentially stable. To this end, let us consider the following scalar function:
(31)Vj1=zTPj1z,
where Pj1∈R6×6 is positive definite. Since Aj1 is Hurwitz, the time derivative of (31) along (27) is given by
(32)V˙j1=-zTQj1z+2zTPj1Φj2(z),
with Qj1 positive definite, since Φj2(z) is Lipschitz; hence
(33)V˙j1≤-∥z∥2∥Qj1∥+2∥z∥2∥Pj1∥L1,V˙j1≤(-∥Qj1∥+2L1∥Pj1∥)∥z∥2,
where L1 is the Lipschitz constant of Φj2(z). To ensure that (-∥Qj1∥+2L1∥Pj1∥) is negative definite, it is required that 2L1∥Pj1∥≤∥Qj1∥, since ∥Qj1∥=∥jA1TPj1+Pj1Aj1∥ and K¯jp,1=γjKjp,1 with γj>0; hence
(34)∥Qj1∥=∥jA1TPj1+Pj1Aj1∥=∥γj[Ajγj-BjKjp,1Cj]TPj1+γjPj1[Ajγj-BjKjp,1Cj][Ajγj-BjKjp,1Cj]T∥=∥γj∥∥[Ajγj-BjKjp,1Cj]TPj1+Pj1[Ajγj-BjKjp,1Cj][Ajγj-BjKjp,1Cj]T∥.
From (34) it can be observed that lim⁡γj→∞∥Qj1∥=∞, which implies that, for a sufficiently large γj, the origin of (27) is exponentially stable.
On the other hand, to study stability of the reduced system (30), consider the scalar function Vj2=(1/2)wTw. Let E≜BjKjI,1Cj; hence the time derivative of Vj2 along (27) is V˙j2=-wTEh(w).
Using the result in Proposition2, ∇wh(w)=[∇exes]-1 is positive definite, which in turn implies that V˙j2 is positive semidefinite. Using the Lasalle invariance principle and provided that V˙j2=0 only in z=0, w=0, 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 εmax⁡>0 such that, for every 0<εj1<εmax⁡, the origin of (16) is asymptotically stable, with a scalar function
(35)Vj3=(Vj1+Vj2)εj1.
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:
(36)0=f(σeq)-Λ(eϑ,eq;Kjp,2)+B¯(ρeq),(37)0=T(eϑ,eq)[P,Q,R]T,(38)0=εj2KjI,2eϑ,eq,
where the subindex eq is the value of the variable under equilibrium conditions, σ=[z,w]T, f(σ) is the right-hand side of (26), ρ is the virtual state associated with the integral action, B¯(ρ)≜[Bjρ,0]T, and
(39)Λ(e;Kjp,2)≜[BjKjp,2eϑ0].
From (38) it is clear that eϑ,eq=0, since KjI,2 is full rank. Observe that T(0)=0, Λ(0;Kjp,2)=0, and B¯(ρeq) is a constant vector, so (36) can be rewritten as
(40)0=f(σeq)+B¯(ρeq).
Observe that (40) has the same structure as (18), except for the translation term B¯(ρeq). 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:
(41)εj2σ~′=f(σ~′)-Λ(e~ϑ′;Kjp,2)+B¯(ρ~′),εj2e~ϑ′=T(e~ϑ′)[P,Q,R]T,ρ~′=-KjI,2e~ϑ′,
with σ~=[z~,w~]≜σ-σeq=[z-zeq,w-weq], ρ~≜ρ-ρeq, τ′=ϵj2t is the new time scale, and e~ϑ≜eϑ-eϑ,eq, σ~′≜dσ~/dτ′, ρ~′≜dρ~/dτ′, and e~ϑ′≜de~ϑ/dτ′. Observe that (41) has an equilibrium point in (σ~,e~ϑ,ρ~)=(0,0,0). Moreover, observe that (σ~,e~ϑ,ρ~)→(0,0,0), as τ′→∞ implies that (σ~,e~ϑ,ρ~)→(0,0,0) as t→∞.
At this point, let us compute the reduced and boundary layer systems associated with (41), which are given, respectively, by
(42)ρ~′=-KjI,2Kjp,2-1ρ~,(43)σ~˙=f(σ~).
Notice that (43) has been studied before; the origin of such system is asymptotically stable for 0<εj1<εmax⁡. On the other hand, the stability of the reduced system (42) can be readily established by suitable gains -KjI,2jKp,2 that make the matrix Hurwitz, hence, the existence of a Lyapunov function Vj4=ρ~TPj4ρ~ associated with (42) can be derived such that V˙j4=-ρ~TQj4ρ~<0.
Given the arguments above and the Lipschitz nature ofΦj2(z~), Theorem  2.1 in [26] implies the existence of a parameter εmax⁡>0 such that for all 0<εj2<εmax⁡ the origin of (41) is asymptotically stable.So far, it has been proven that a linear control is able to stabilize both velocity and orientation of the nonlinear aircraft; however, such result has been derived from the hypothesis that the system description is continuous. To relax such assumption, in the following we will study the stability of a family of piecewise continuous functions, which describes the dynamic of the aircraft at a family of desired operation points.


### 4.2. Stability of the Switching Structure

Conditions in Theorem3 guarantee the stability in a local domain Ωj. However, they do not constitute sufficient evidence to guarantee the stability of the entire flight envelope. To obtain such evidence, the piecewise continuous system under the action of discrete event switching has to be studied.Since the origin of every continuous description of the system can be defined using different operation points (reference), the switching among modes can be described using discrete event switching depending on time, system’s state, or both.In view of this fact, it is necessary to study the requirements on the discrete event actions that ensure asymptotic stability of the closed-loop hybrid system. The following theorem establishes such conditions.Theorem 4.
Assume that conditions of Theorem3 are satisfied and let
(44)Vjtot(π)=zTPj1z+12wTw+ρ~TPj4ρ~≜πTP^jπ,
with πT=[z,w,ρ~]. The origin of the hybrid system (10)–(13) is asymptotically stable for the entire flight envelope using switching criteria guaranteeing an average dwell time for the jth mode, TAD>0, such that
(45)Vjtot(π(tj+l))≥Vjtot(π(tj-l+1)),
with tj+l, tj-l are the initial time and final time when the jth mode becomes active for the lth time, respectively.Proof.
The proof will be performed using Theorem  7 in [28]. According to the result in [28], the existence of a differentiable and radially unbounded scalar function Vjtot>0 must be shown such that V˙jtot≤0 along trajectories of (41). Let us consider the following quadratic positive definite scalar function:
(46)Vjtot(π)=zTPj1z+12wTw+ρ~TPj4ρ~.
From Theorem3,
(47)α1(∥π∥)≤Vjtot(π)≤α2(∥π∥),
withα2(∥π∥)=3max⁡j{λmax⁡(Pj1)/εj1εj2,1/εj1εj2,λmax⁡(Pj4)/εj2}∥π∥2, and α1(∥π∥)=3min⁡j{λmin⁡(Pj1)/εj1εj2,1/εj1εj2,λmin⁡(Pj4)/εj2}∥π∥2. On the other hand, the time derivative of (44) along (10)–(13) is given by
(48)V˙jtot(π)≤-{{-∥Qj1∥+2L1∥Pj1∥}∥z∥2+wTEh(w)+ρ~TQj4ρ~}≜-Wjtot(π).
From Theorem3 in this work, the origin of every operation mode j is asymptotically stable. Using this property and Remark  6 in [28] it is possible to conclude the satisfaction of Conditions 1 and 2 of Theorem  7 in [28], that is, the existence of a class K∞ function (κ) such that |π(t)|≤κ(∥Wjtot(π)∥[t,t+τ]).
Finally, Remark 7 and Lemma  8 in [28] let us conclude that if a minimum average dwell time TAD>0,
(49)Vjtot(π(tj+l))≥Vjtot(π(tj-l+1)),
withtj-l, tj+l are the initial time and final time when the jth mode becomes active for the lth time, respectively, the origin of the hybrid system is asymptotically stable in D, where D=∪Ωj.Remark 5.
Theorem4 provide us with guidelines to design switching laws for the hybrid system. According to such theorem, the gain scheduling control must use a supervisory action that decides whether a given mode is active using switching laws that can be either time or state dependent, as long as a minimum average dwell time is satisfied for every mode. This average time can be arbitrary small as long as TAD>0. The stability result in Theorem 4 actually establishes that the system will be stable if chattering does not occur. The phenomenon of chattering occurs when infinite switching actions occur in finite time and it may lead to finite time destabilization.Remark 6.
The results discussed in Theorems3 and 4 are in fact quite general, and can be used to other systems that admit the same description, including aircrafts with some actuator failures. That is, the same result can be applied to other highly nonlinear systems as long as Assumptions (A.1)–(A.3) are satisfied and the piecewise description of the system fits the model in (10)–(13).


## 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 Ωj, let KjI,i∈R4×4, i∈{1,2}, j∈{1,…,s}, and let K¯jp,1 be as in Propositions 1 and 2; then there exist constants 0<jε1<εmax⁡ and 0<jε2<εmax⁡, 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 Proposition1, a unique isolated equilibrium exists; let us denote such equilibrium values as ex,eq and es,eq. In view of this fact, the following coordinates are introduced to shift such equilibrium to the origin
(25)z≜ex-ex,eq,w≜es-es,eq.
At this point, let us use variableεj1>0 as a perturbation parameter; hence the closed-loop system (10)–(13) can be rewritten as a standard singularly perturbed system given by
(26)[jε1z′w′]=[Aj1I-BjKjI,1Cj04×4][zw]+[Φj2(z)0],
where Φj2(z)≜jΦ1(z+ex,eq), t′=ϵj1t, z≜z(t′), z′≜dz/dt′, w≜w(t′), and w′≜dw/dt′.
The boundary-layer and reduced systems associated with (26) are given, respectively, by
(27)z˙=Aj1z+Φj2(z),(28)0=Aj1z+w+Φj2(z),(29)w′=-BjKjI,1Cjz.
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 z and w that satisfies z=-Aj1-1[w+Φj2(z)], which has the same structure of (18); therefore Proposition 1 can be used; hence, for a suitable proportional gain (i.e., sufficiently large γj), there exists an explicit function z≜h(w) such that
(30)w′=-BjKjI,1Cjh(w).
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 largeγj, the origin of (27) is exponentially stable. To this end, let us consider the following scalar function:
(31)Vj1=zTPj1z,
where Pj1∈R6×6 is positive definite. Since Aj1 is Hurwitz, the time derivative of (31) along (27) is given by
(32)V˙j1=-zTQj1z+2zTPj1Φj2(z),
with Qj1 positive definite, since Φj2(z) is Lipschitz; hence
(33)V˙j1≤-∥z∥2∥Qj1∥+2∥z∥2∥Pj1∥L1,V˙j1≤(-∥Qj1∥+2L1∥Pj1∥)∥z∥2,
where L1 is the Lipschitz constant of Φj2(z). To ensure that (-∥Qj1∥+2L1∥Pj1∥) is negative definite, it is required that 2L1∥Pj1∥≤∥Qj1∥, since ∥Qj1∥=∥jA1TPj1+Pj1Aj1∥ and K¯jp,1=γjKjp,1 with γj>0; hence
(34)∥Qj1∥=∥jA1TPj1+Pj1Aj1∥=∥γj[Ajγj-BjKjp,1Cj]TPj1+γjPj1[Ajγj-BjKjp,1Cj][Ajγj-BjKjp,1Cj]T∥=∥γj∥∥[Ajγj-BjKjp,1Cj]TPj1+Pj1[Ajγj-BjKjp,1Cj][Ajγj-BjKjp,1Cj]T∥.
From (34) it can be observed that lim⁡γj→∞∥Qj1∥=∞, which implies that, for a sufficiently large γj, the origin of (27) is exponentially stable.
On the other hand, to study stability of the reduced system (30), consider the scalar function Vj2=(1/2)wTw. Let E≜BjKjI,1Cj; hence the time derivative of Vj2 along (27) is V˙j2=-wTEh(w).
Using the result in Proposition2, ∇wh(w)=[∇exes]-1 is positive definite, which in turn implies that V˙j2 is positive semidefinite. Using the Lasalle invariance principle and provided that V˙j2=0 only in z=0, w=0, 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 εmax⁡>0 such that, for every 0<εj1<εmax⁡, the origin of (16) is asymptotically stable, with a scalar function
(35)Vj3=(Vj1+Vj2)εj1.
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:
(36)0=f(σeq)-Λ(eϑ,eq;Kjp,2)+B¯(ρeq),(37)0=T(eϑ,eq)[P,Q,R]T,(38)0=εj2KjI,2eϑ,eq,
where the subindex eq is the value of the variable under equilibrium conditions, σ=[z,w]T, f(σ) is the right-hand side of (26), ρ is the virtual state associated with the integral action, B¯(ρ)≜[Bjρ,0]T, and
(39)Λ(e;Kjp,2)≜[BjKjp,2eϑ0].
From (38) it is clear that eϑ,eq=0, since KjI,2 is full rank. Observe that T(0)=0, Λ(0;Kjp,2)=0, and B¯(ρeq) is a constant vector, so (36) can be rewritten as
(40)0=f(σeq)+B¯(ρeq).
Observe that (40) has the same structure as (18), except for the translation term B¯(ρeq). 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:
(41)εj2σ~′=f(σ~′)-Λ(e~ϑ′;Kjp,2)+B¯(ρ~′),εj2e~ϑ′=T(e~ϑ′)[P,Q,R]T,ρ~′=-KjI,2e~ϑ′,
with σ~=[z~,w~]≜σ-σeq=[z-zeq,w-weq], ρ~≜ρ-ρeq, τ′=ϵj2t is the new time scale, and e~ϑ≜eϑ-eϑ,eq, σ~′≜dσ~/dτ′, ρ~′≜dρ~/dτ′, and e~ϑ′≜de~ϑ/dτ′. Observe that (41) has an equilibrium point in (σ~,e~ϑ,ρ~)=(0,0,0). Moreover, observe that (σ~,e~ϑ,ρ~)→(0,0,0), as τ′→∞ implies that (σ~,e~ϑ,ρ~)→(0,0,0) as t→∞.
At this point, let us compute the reduced and boundary layer systems associated with (41), which are given, respectively, by
(42)ρ~′=-KjI,2Kjp,2-1ρ~,(43)σ~˙=f(σ~).
Notice that (43) has been studied before; the origin of such system is asymptotically stable for 0<εj1<εmax⁡. On the other hand, the stability of the reduced system (42) can be readily established by suitable gains -KjI,2jKp,2 that make the matrix Hurwitz, hence, the existence of a Lyapunov function Vj4=ρ~TPj4ρ~ associated with (42) can be derived such that V˙j4=-ρ~TQj4ρ~<0.
Given the arguments above and the Lipschitz nature ofΦj2(z~), Theorem  2.1 in [26] implies the existence of a parameter εmax⁡>0 such that for all 0<εj2<εmax⁡ the origin of (41) is asymptotically stable.So far, it has been proven that a linear control is able to stabilize both velocity and orientation of the nonlinear aircraft; however, such result has been derived from the hypothesis that the system description is continuous. To relax such assumption, in the following we will study the stability of a family of piecewise continuous functions, which describes the dynamic of the aircraft at a family of desired operation points.


## 4.2. Stability of the Switching Structure

Conditions in Theorem3 guarantee the stability in a local domain Ωj. However, they do not constitute sufficient evidence to guarantee the stability of the entire flight envelope. To obtain such evidence, the piecewise continuous system under the action of discrete event switching has to be studied.Since the origin of every continuous description of the system can be defined using different operation points (reference), the switching among modes can be described using discrete event switching depending on time, system’s state, or both.In view of this fact, it is necessary to study the requirements on the discrete event actions that ensure asymptotic stability of the closed-loop hybrid system. The following theorem establishes such conditions.Theorem 4.
Assume that conditions of Theorem3 are satisfied and let
(44)Vjtot(π)=zTPj1z+12wTw+ρ~TPj4ρ~≜πTP^jπ,
with πT=[z,w,ρ~]. The origin of the hybrid system (10)–(13) is asymptotically stable for the entire flight envelope using switching criteria guaranteeing an average dwell time for the jth mode, TAD>0, such that
(45)Vjtot(π(tj+l))≥Vjtot(π(tj-l+1)),
with tj+l, tj-l are the initial time and final time when the jth mode becomes active for the lth time, respectively.Proof.
The proof will be performed using Theorem  7 in [28]. According to the result in [28], the existence of a differentiable and radially unbounded scalar function Vjtot>0 must be shown such that V˙jtot≤0 along trajectories of (41). Let us consider the following quadratic positive definite scalar function:
(46)Vjtot(π)=zTPj1z+12wTw+ρ~TPj4ρ~.
From Theorem3,
(47)α1(∥π∥)≤Vjtot(π)≤α2(∥π∥),
withα2(∥π∥)=3max⁡j{λmax⁡(Pj1)/εj1εj2,1/εj1εj2,λmax⁡(Pj4)/εj2}∥π∥2, and α1(∥π∥)=3min⁡j{λmin⁡(Pj1)/εj1εj2,1/εj1εj2,λmin⁡(Pj4)/εj2}∥π∥2. On the other hand, the time derivative of (44) along (10)–(13) is given by
(48)V˙jtot(π)≤-{{-∥Qj1∥+2L1∥Pj1∥}∥z∥2+wTEh(w)+ρ~TQj4ρ~}≜-Wjtot(π).
From Theorem3 in this work, the origin of every operation mode j is asymptotically stable. Using this property and Remark  6 in [28] it is possible to conclude the satisfaction of Conditions 1 and 2 of Theorem  7 in [28], that is, the existence of a class K∞ function (κ) such that |π(t)|≤κ(∥Wjtot(π)∥[t,t+τ]).
Finally, Remark 7 and Lemma  8 in [28] let us conclude that if a minimum average dwell time TAD>0,
(49)Vjtot(π(tj+l))≥Vjtot(π(tj-l+1)),
withtj-l, tj+l are the initial time and final time when the jth mode becomes active for the lth time, respectively, the origin of the hybrid system is asymptotically stable in D, where D=∪Ωj.Remark 5.
Theorem4 provide us with guidelines to design switching laws for the hybrid system. According to such theorem, the gain scheduling control must use a supervisory action that decides whether a given mode is active using switching laws that can be either time or state dependent, as long as a minimum average dwell time is satisfied for every mode. This average time can be arbitrary small as long as TAD>0. The stability result in Theorem 4 actually establishes that the system will be stable if chattering does not occur. The phenomenon of chattering occurs when infinite switching actions occur in finite time and it may lead to finite time destabilization.Remark 6.
The results discussed in Theorems3 and 4 are in fact quite general, and can be used to other systems that admit the same description, including aircrafts with some actuator failures. That is, the same result can be applied to other highly nonlinear systems as long as Assumptions (A.1)–(A.3) are satisfied and the piecewise description of the system fits the model in (10)–(13).


## 5. Illustrative Results

The aim of this section is to illustrate the theoretical results above. To this end, the nonlinear model of F16 (see [29]) is simulated using Simulink. In order to illustrate the effect of the uncertainty, the simulations are conducted as follows: firstly, a set of linear approximations are proposed at different operation points, each of these approximations is associated with a domain Ωj and the linear approximation is used only to perform the controller tuning at the corresponding modes. Secondly, the gains are used to control a piecewise nonlinear model of the system whose modes coincide with the modes of the linear counterpart. For illustration purposes only two flight modes are used, namely, steady-state fight at different cruise velocities.Firstly, the effect of the integral action is evaluated tracking a sinusoidal function. The switching criterion is chosen to be velocity dependent and at the surfacev=325 ft/s, so to perform the task the aircraft must switch consecutively between Mode 1 and Mode 2. The results are displayed in Figures 2, 3, and 4.Figure 2
Time evolution of aircraft cruise velocity for three different integral gains (ε1,a>ε1,b>ε1,c).Figure 3
Time evolution of the aircraft attitude for three different integral gains (ε1,a>ε1,b>ε1,c).Figure 4
Time evolution of the angular velocity for three different integral gains (ε2,a>ε2,b>ε2,c).Figure2 shows the time evolution of the cruise velocity for three different integral gains, which, according to Theorem 3, are parameterized using jε1. Figure 2 shows that for simplicity ε11=ε21=ε1 have been chosen and the three integral parameters are denoted as ε1,a, ε1,b, ε1,c. Theorem 3 establishes the existence of a maximum integral gain such that the system is asymptotically stable. For ε1,a<ε1,b, the velocity reference can be followed; however, for ε1,c≫ε1,b, the system leads to instability.In Figure3 it can be observed that for ε1,a the body stabilization task is able to avoid rotation, still with a short transient period. Such transient response is reduced with ε1,b and finally with ε1,c the aircraft is unstable. That is, moderate values of the integral action result in a successful velocity and orientation tracking and good performance. However since the maximum value of the integral gain that can be used is upper bounded, a performance criterion to choose the integral gains can be used to define properly the operation domains Ωj.Similar observations can be performed for the orientation loop (see Figure4), which integral gains can be parameterized using εj2. For simplicity, the integral parameters are chosen such that ε12=ε22=ε2. Three different gains are evaluated ε2,a>ε2,b>ε2,c. For small integral gains, the orientation can be successfully tracked (ε2,a); however, higher gains lead firstly to a performance degradation (ε2,b) and secondly to instability (ε2,c).Finally, to illustrate the result in Theorem4, fast switching actions are induced by following a high frequency sinusoidal velocity reference (see Figures 5 and 6). The switching conditions are velocity dependent and set at a cruise velocity of 35 ft/sec. In Figure 6 it is possible to observe the corresponding behavior of the angular velocity and the aircraft attitude. Notice that even if the model is switching continuously, the evolution of the system remains stable and, for the controller gains in the simulations, they are able to track correctly the desired reference. Such result agrees with theoretical results in Theorem 4.Figure 5
Time evolution of aircraft cruise velocity under fast switching conditions.Time evolution of (a) aircraft attitude and (b) angular velocity under fast switching conditions.
(a)
(b)


## 6. Conclusions

In this work, the stability properties of an aircraft with nonlinear behavior controlled with a gain scheduled approach are studied. It is shown that the inclusion of a linear integral action to an output feedback linear controller makes it possible to guarantee asymptotical tracking of velocity and orientation in spite of nonlinearities and uncertainties. This result is interesting since even if the behavior of the aircraft system is highly nonlinear, the controller gains can be tuned using linear approximations of the system in given operation points and the performance of the system can be set using a trade-off between the size of the integral gains and the size of the operating domainΩj. The switching criteria can be either time or state dependent as long as a minimum residence time (TAD>0) is used in every mode and it can be designed using a supervisory structure which evaluates the satisfaction of condition (45). Due to this fact the control scheme is flexible and physical or user-defined restrictions can be taken into account.



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*Source: 906367-2014-05-21.xml*