Trustable UAV for higher level control architectures

Accepted Manuscript Trustable UAV for Higher Level Control Architectures

Chimpalthradi R. Ashokkumar, George WP York, Scott Gruber

PII: DOI: Reference:

S1270-9638(17)30827-1 http://dx.doi.org/10.1016/j.ast.2017.05.013 AESCTE 4026

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Aerospace Science and Technology

Received date: Revised date: Accepted date:

9 February 2016 12 October 2016 4 May 2017

Please cite this article in press as: C.R. Ashokkumar et al., Trustable UAV for Higher Level Control Architectures, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.05.013

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Trustable UAV for Higher Level Control Architectures Chimpalthradi R. Ashokkumar1, George WP York2 and Scott Gruber3

Abstract – Future unmanned air vehicles (UAVs) are expected to operate under the supervision of higher level control architectures that give instructions to engage and direct the UAV to perform a certain task. These instructions allow the UAV to make decisions along any point of its trajectory and then modify its flight path by using a sequence of reconfigurable controllers at the decision points. Assume that the UAV is flying with a transient and with a steady state contributing to a flight control mode (FCM) such as an altitude hold, an ascent and a descent mode when only longitudinal aircraft dynamics is considered. At the time of a decision, if the UAV bifurcates from its original (or parent) FCM in an effort to acquire a new (or a child) FCM and comply with a higher level instruction, then the UAV is said to be trustable. Mathematically, at the time of bifurcation where controller reconfiguration takes place, trustable UAV with the child trajectory must originate from a region where the stability regions of the parent and child trajectories intersect. In this paper, a procedure to reconfigure such FCMs and their controllers that develop the trustable UAV are presented. A three degree of freedom UAV is considered to illustrate the trustable UAV. Keywords: Flight control modes, UAVs, transients, steady states, higher level instructions, switching, stability.

1

NRC Research Associate, Dept. of Electrical and Computer Engineering, USAF Academy, Colorado, USA. Director and Associate Professor, USAF Academy Center for Unmanned Aircraft Systems Research, Dept. of Electrical and Computer Engineering, USAF Academy, Colorado, USA. 3 Program Manager, USAF Academy Center for Unmanned Aircraft Systems Research, Dept. of Electrical and Computer Engineering, USAF Academy, Colorado, USA. 2

1

1. Introduction Future unmanned air vehicles (UAVs), both aerial and combat air vehicles, have a major role to mitigate threats involving the life of ground personnel and pilots who are assigned to perform complex mission operations. In these operations, although human decisions to remotely operate an UAV are far superior, the UAVs are expected to be operative under the supervision of a higher level control architecture to carry out a task. The instructions are such that they prompt the UAV to make a decision and change the course of its flight path. At the decision points, if the UAV is able to reconfigure its flight path with stability, then the UAV is said to be trustable. Otherwise, the UAV at that decision point is not trustable. In this paper, a mathematical exposition using the linearized model based controllers of the nonlinear UAV that develops a trustable UAV is presented. The trusted UAV is assumed autonomous. The technical factors governing the human operated UAV will be considered at a later stage to define the trust. Nevertheless, pilot errors [1] in manned or unmanned aircraft leading to catastrophes require special attention to define trust. The aircraft simulation in these cases must be carried out like the way the pilot with an inner loop controller switching criterion operates the aircraft. In unmanned aircraft, there is a requirement to perform nonlinear aircraft simulations using switched linear model based controllers that guarantee trust, especially when higher level control architecture’s instructions [2,3] decide to direct an extremely uncertain lower level UAV [4,5] to perform a certain task. Typical uncertainties influencing the trustable UAV are higher level decision points at which the UAVs are instructed, denied airspace, friendly resources, enemy states, etc. Such uncertainties constantly drive the aircraft to adopt an unplanned flight path. One option available to validate a trustable UAV in these circumstances is to develop and automate a capability to switch flight control modes (FCMs) such as an ascent, descent, and altitude hold,

2

etc., compatible to higher level instructions so that the flight bifurcates from its original (or parent) FCM to a new (or a child) FCMs as shown in Figure 1.

Fig. 1. A trustable UAV with parent and child FCMs

The parent FCM shown in solid line (ABCD) has a transient and a steady state. The child FCM shown in dotted lines can originate from any part of the parent’s transient or the steady state depending upon the time at which a higher level instruction is directed. The solid and dotted line combination depict the trustable UAV, which takes such instructions at nodes and makes decisions to bifurcate instantaneously from the parent FCM with a FCM reconfiguration for the child. In this paper, a procedure to develop these trustable UAVs is presented. In particular, controller reconfiguration to generate parent and child FCMs in extended stability regions is presented. That is, the trustable UAV is developed by using controller switching as in gain scheduling algorithm; however, these controllers are developed to originate from a particular trim point of the flight envelope. Following the control and simulation procedure, the trustable UAV possesses the inner loop controller switching capability and guarantee stability of the reconfigurable FCMs that the parent and child generate. For a given command or pilot input at the outer loop, the rules to determine the sequence of controllers to be switched as in gain scheduling algorithms [6-11] consistent with flight envelope points are beyond the scope of the present work. Hence, in this paper, trustable UAVs are investigated with respect to the initial

3

conditions and various controllers, all originating from a single linear time invariant model (or the trim point). It is possible to develop a trustable UAV whose FCMs are derived by considering various flight conditions on the flight envelope. Here, the linearized model with respect to its equilibrium or trim point varies for each flight condition. Controllers are determined for each linear model and they are scheduled as per the norms of the gain scheduling algorithm. Note that controller switching needs to take place at a point where the stability regions (regions of attraction) of the respective controllers intersect [12]. In current gain scheduling algorithms, the stability regions are so intact and the switching criteria are therefore taking place within a region where stability regions intersect. Alternative approach to avoid switching rules is simultaneous stabilization [13,14]. Here a single controller attempts to stabilize as many linear models as possible; however, such controllers are difficult to determine for more than two linear models [15,16]. While there is a need to understand the selection of flight conditions in accordance with a flight trajectory, in this case FCMs, it is also possible to override controller variations with altitude (one of the coordinates of the flight envelope) where their linear models simultaneously vary. For instance, a sequence of controllers pertaining to a single linear model can generate ascent and descent modes as against the gain scheduling algorithm where controllers and their linear models are assumed to vary with altitude. For UAVs, these procedures are open for implementation. Along these lines, presently, sophisticated six degree of freedom nonlinear aircraft and its equations for navigation is not considered. This paper assumes a single longitudinal linear model to generate parent and child FCMs for a trustable UAV. The child FCMs are reconfigured by using controllers resulting from various closed loop pole locations. Presently, the linear model corresponding to steady level flight of a longitudinal micro air

4

vehicle model [17] is considered. Generally, FCM variations are possible with thrust and elevator as control inputs. Hence the linear model becomes a two input system. FCMs of the nonlinear aircraft and eigenvalue and eigenvector (eigenstructure) options for the linear model based controller design are strongly connected; however, eigenvalue selection or the choice of eigenvector elements for a particular ascent or a descent mode is difficult to establish. In fact, in aircraft applications, the choice of eigenvector elements for eigenmode decoupling [18], minimum norm gains for control input constraints [19], etc., does not apply when the controller is implemented for the nonlinear aircraft. For each set of desired eigenvalue locations, eigenvector options are indeed infinite. Accordingly, FCM options become infinite. A procedure to determine such FCMs in state feedback format [20], in output feedback format [21], and in observer based feedback format [22] is presented. Potential steady state trajectory transcriptions for stability of reconfigurable controllers in an integrated FCM to depict a typical reconfigurable aircraft control are discussed. In this paper, in order to develop a trustable UAV, both transient trajectory transcriptions (TTTs, for higher level instructions at transients) as well as steady state trajectory transcriptions (SSTTs, for higher level instructions at steady states) using controllers resulting from different closed loop eigenvalue locations are considered. Eigenvalue variations in a way depict variations in eigenvectors. Each of these choices will have a unique steady state value called auxiliary equilibrium point to define the FCM and several transients depending upon the initial condition one picks up from the stability region. If a parent and its associated child FCM is required to be configured, initial condition (or transient) for the parent FCM becomes instrumental to have acceptable higher level decision points where the child FCMs emerge. Such conclusions are also derivable by using the controllers resulting from robust pole assignment [23], linear quadratic regulator [24], etc. The basic objective behind a

5

trustable UAV is that the bifurcation from parent to child FCM is stable and it is a property associated with the variations in eigenstructure of the linear model. Eigenstructure variations can be pursued through linear functional controllers (LFCs) [20-22] (the eigenvalues are retained at fixed locations but their eigenvectors are varied through a multiple input state variable feedback controllers). In this paper, the basic requirement on eigenvector variations for FCM options is obtained through eigenvalue variations. The rest of the paper is organized as follows. The problem description is presented in Section 2. Trajectory transcription criteria for a trustable UAV is presented in Section 3. In Section 4, the autonomous attributes of the trustable UAV are discussed. Simulations and conclusions are provided in Section 5 and 6, respectively. 2. Problem description When a higher level instruction to engage an UAV and perform a certain task is made, it is important that the UAV need to cooperate with the instruction and make a decision to develop a child trajectory. Several scenarios exist to define the present and future state of the UAV at the time the decision is made. For instance, prior to decision, the UAV may be accelerating to a new altitude in an ascent mode and after the decision it may be required to pass through a waypoint (say a latitude, longitude and altitude point). A question that would naturally arise is the following. Can the UAV bifurcate from its present path to its future path? As in standard aircraft control problem, when the inner loop is scheduled with linear model based controllers, this paper considers the FCMs of the longitudinal aircraft dynamics and proposes parent and child FCMs for bifurcation at the decision points. As discussed in the previous section, the controllers work like the gain scheduling algorithm but the trim points are not. That is, several pole placing controllers designed for one linear model are scheduled to derive the child FCMs.

6

At the inner loop, LFCs presented in [20-22] are particularly attractive to understand FCM transcriptions as a function of condition numbers for the closed loop modal matrices. Here, the respective eigenvalues in all the FCMs are fixed in the open left half plane of the complex plane. However, the eigenvectors are varied infinitely to have variations in condition numbers. In this paper, the eigenvalue locations are varied and the procedure to design these LFCs for variations in condition numbers is retained. Hence the FCM options are multiplied for every set of desired eigenvalue locations. Further, the FCM transcriptions and hierarchical controllers that assign distinct set of eigenvalue locations which transcript FCMs and configure parent and child FCMs are unknown and it is the study objective of this paper. Such an assignment can also take the controllers resulting from robust pole assignment [23] and linear quadratic regulators [24]. In order to define a FCMs and their transcriptions for a trustable UAV, the nonlinear aircraft confining to the pitch plane [17] is taken in the form, x

a( x)  b( x)u(t ) .

(1)

Assume the control input u(t) is not a scalar. The n-component state vector x(t) requires the compatible vector valued functions a(x) and b(x) to be smooth and differentiable with respect to x(t). The time (t) derivative of the state variables is denoted by a dot over the state vector x(t). Let the kinematics of the aircraft in NED (North-East-Down) coordinate frame as, X i

va (t ) cos(T (t )  D (t ))

(2)

Zi

va (t ) sin(T (t )  D (t ))

(3)

where va (t ) is the velocity of the aircraft in m/s, T (t ) is the pitch angle in radians and D (t ) is the angle of attack in radians. The flight paths are determined by X i and Z i , which are the coordinates of the inertial positions such as North-Down or East-Down positions. Other state (trajectory) and control variables that govern the dynamics of the aircraft in pitch plane are pitch 7

rate ( T Q (t), rad/sec), elevator deflection ( G e (t ) , rad) and thrust coefficient CT (t ) . The state vector is identified as follows.

>va

xc(t )

D T Q@

(4)

The control variables are identified as, u1 (t ) CT (t ) and u2 (t ) G e (t ) .

(5)

The three degree of freedom micro aerial vehicle model and its aerodynamic data is given in [17]. An equilibrium point (in Eq. (1), points where x 0 ) corresponding to a steady level flight is computed. The nonlinear UAV from [17] is given by, S S S f LD (CL0  CLD D )  g sin (T  D )  q CT cos D  q CDG G e . e m m m

va (t )

q

D (t )

1 S Sc S g [Q  q CL0  q 2 CLQ Q  q CLD D  cos(T  D )] v m v m v k 2 va m a a a S S q CT sinD  q CL G e k va m k va m Ge

T(t ) q

Sc c Sc [Cm0  CmD D  CmQ T]  q CmG G e e I yy 2v a I yy

(6a)

(6b)

(6c)

where, f LD (M ) a1M 4  a2M 3  a3M 2  a4M  a5 , M a1

0.1723; a 2 0.3161; a3

k

1 q

q

1 2

Sc CL , 2va2m D

Uva2 ,

0.2397; a4

CL0  CLD D 0.0624; a5

0.0194

(6d)

(6e) (6f)

Here dot over a variable refers the time derivative of the variable. U is the air density kg/m3, S is the surface area of the wing in m2, m is the mass of the aircraft in kg, c is the mean aerodynamic chord in meters, Iyy is the pitch moment of inertia in kg-m2 and other parameters are the aerodynamic stability and control derivatives. The data for these coefficients are given in [17]. 8

To compute the trim condition, special techniques are required to solve a set of nonlinear algebraic equation with six unknowns and four equations (for six degree of freedom aircraft, see [25]).

Fig. 2. Potential parent red) and child (blue) FCMs of an UAV.

In order to explain parent and child FCMs, consider the transcriptions (FCMs in blue transcripting FCMs in red) as in Figure 2. Each slope is a representation of the inner loop controller. The transient is a representation of the initial condition belonging to its stability region. Further, the transients are confined to admissible state variable bounds which in turn determine bounds on the control inputs. The points where the color codes change and trajectories in blue and red intersect are the decision points where controller reconfiguration takes place to oblige the higher level instructions. Note that the decision points are not observed on transient 9

part of the trajectories. Suppose the higher level instructions come in at any point of the trajectory. When a particular trajectory in red (referred as parent trajectory or FCM) with which the trajectories in blue (referred as child trajectories or FCMs) transcript at many points of the parent trajectory, the UAV becomes trustable. Depending upon the controller for parent trajectory, the child FCMs are defined. For instance, an extremely shallow parent will not have transient effects and hence the child FCMs are derivable at almost every point of the parent trajectory. The parent trajectory in this case is a property associated with the initial condition search in the stability region of its controller. Given this controller, a procedure to generate the parent trajectory using a linear time invariant model based controller is well known in the literature. Usually, the aircraft in Eq. (1) is linearized with respect to an equilibrium point. The linear model is completely controllable and is given by, 'x

A 'x  B 'u(t )

'x

x (t )  x e

'u

u( t )  ue

(7)

In the present case, a LFC G defines a control law of the form, (8)

'u(t ) G 'x(t )

For each of these controllers ( G i ), the trajectories x(t ) in Figure 2 are generated from the following differential equations (for brevity, the superscript i is ignored), x

a( x)  b( x)[ue  G ( x  xe )] ,

x(t

0) z 0 .

(9)

Note that the auxiliary equilibrium points xai ,e at FCM transcriptions are the points that the numerical integration scheme employed in navigation and control algorithm satisfy, a( xai ,e )  b( xa(i,)e )[ue  G i ( xai ,e  xe )] 0 ,

i 1, 2 

(10)

Also, when the auxiliary equilibrium point is too close to the equilibrium point, transcription is favored by the fact that it will be in a region where stability regions of the LFCs intersect, a condition investigated in detail for a second order nonlinear system [12]. A procedure for such 10

transcriptions is presented in sequel and it holds for state feedback based LFCs when the controllers are originating from the model in Eq. (7). These LFCs for state feedback case is presented in [20]. In addition, the state feedback controllers resulting from robust pole assignment and linear quadratic regulators are also considered which primarily distinguishes the slopes of the FCMs from other techniques. Prior to presenting transient trajectory transcriptions (TTTs) and steady state trajectory transcriptions (SSTTs) by using a combination of stability regions pertaining to the LFCs, the stability region is defined as follows: Definition 1 (Stability Region): SGj

^x

0

 R 4 : If x(0)

x0 then limtof x(t )

`

xaj,,e , j represents the linear functional controller G j .

Note that the classical stability region definition takes the equilibrium ( xe ) and auxiliary equilibrium ( xaj,e ) points as xe xaj,e 0 . In aircraft, these non-zero points are crucial to explain trustable UAV, where the controllers are switched to define child FCMs from a parent FCM. Consider now one of these trustable UAVs and denote its controller for a parent FCM by G p and its controllers for child FCMs by G d , d 1,2 . The TTT is defined as follows: Definition 2 (Transient trajectory transcriptions TTTs): Consider a stability region S Gp for a parent controller G p . Denote a trajectory in the stability region S Gp by x p (t ) . Similarly, consider a stability region S Gd for a child controller G d . Let the decision point on the parent trajectory be t d , which is the time at which the UAV with a child trajectory begins to bifurcate from the parent trajectory. Then the UAV with a TTT is contained in the child stability region S Gd with the following criterion. S Gd

^x

d p

 R 4 : If x p (t

td )

x dp then lim t t t d of x(t )

`

x ad,e , G represents a child controller G d .

Similarly, the SSTTs for decision points on auxiliary equilibrium points are defined.

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Definition 3 (Steady state trajectory transcriptions SSTTs): Consider a stability region S Gp for a parent controller G p . Denote the auxiliary equilibrium point of the trajectory x p (t ) in the stability region S Gp by xap,e . Similarly, consider a stability region S Gd for a child controller G d . Let the decision point on the parent trajectory be t d , which is the time at which the UAV with a child trajectory begins to bifurcate from the parent trajectory. Then the UAV with a SSTT is contained in the child stability region S Gd with the following criterion. S Gd

^x

p a ,e

 R 4 : If x p (t

td )

x ap,e then limt tt d of x(t )

`

x ad,e , G represents a child controller G d .

One of the implications of the definitions provided for TTTs and SSTTs is that the stability regions S Gp and S Gd must intersect so that x dp and x(t d ) (parent and child trajectories at the decision point) reside at the intersected region. These definitions are used to define the FCM and the trusted UAV as follows: ௣

Definition 4 (Flight Control Mode): In the stability region ܵீ (or ܵீௗ ), consider an initial condition ‫ݔ‬଴ (or ‫ݔ‬ሺ‫ݐ‬ௗ ሻ) such that the trajectories contained in their respective stability regions ௣ ௗ (or ‫ݔ‬௔ǡ௘ ). The FCM with an ascent (+) or descent (െ) will be approach the steady state ‫ݔ‬௔ǡ௘ ௣ ௣ ௗ ௗ േ–ƒሺߠ௔ǡ௘ െ ߙ௔ǡ௘ )) (or േ–ƒሺߠ௔ǡ௘ െ ߙ௔ǡ௘ )).

Definition 5 (Trustable UAV) : If the UAV trajectory ‫ݔ‬ሺ‫ݐ‬ሻ comprising of parent and child trajectories is contained within the union of the stability regions ܵீ௣೛ ‫ீܵ ׫‬ௗ೏ such that the point ‫ݔ‬ሺ‫ ݐ‬ൌ ‫ݐ‬ௗ ሻ ൌ ‫ݔ‬௣ௗ resides at the intersected region ܵீ௣೛ ‫ீܵ ת‬ௗ೏ , then the UAV is said to be trustable. Note that the parent and child controllers ‫ ܩ‬௣ and ‫ ܩ‬ௗ , respectively, are reconfigurable at ‫ݔ‬௣ௗ . In order to know if the SSTT and TTT criteria for a trustable UAV are satisfied, it becomes necessary to interpret Lyapunov stability criterion for nonlinear autonomous system using eigenvalues of the Jacobian matrix. This technique also accommodates the LFCs G d and G p that 12

are designed for eigenvector variations which are required to have options in auxiliary equilibrium points xad,e that in turn decide ascent and descent modes of the child FCMs. Some of these principles are discussed in sequel. 3. Transcription criterion in a trustable UAV The nonlinear aircraft in Eq. (9) is indeed the nonlinear autonomous system of the form, x

f (x ) ,

x(t0 )

(11)

x0 z 0

and one of its non-zero equilibrium points xe such that (12)

f ( xe ) 0

As in [26,27], choose the Krasovskii’s Lyapunov function V ( x) ! 0 such that 1 2

V ( x)

f cP f , P ! 0

where prime denotes the transpose. The time derivative of the Lyapunov function is, f c ^Ac( x) P  PA( x)` f

V ( x)

, A(x ) is Jacobian

wf ( x ) . wx

(13)

Clearly, for stability, V ( x)  0 implies that A(x ) is asymptotically stable along the trajectory x(t ) . In [26,27] it has been shown that Eq. (13) has a Lyapunov equation connection [28]. Ac( x) P  PA( x)

Q ,

P ! 0 and Q t 0 .

(14)

Theorem 1: The equilibrium point x e is asymptotically stable if A(x ) is in Ф, where Ф is the set of all matrices whose eigenvalues have negative real parts. In order to connect condition numbers and transcriptions, Theorem 1 is used. Consider the LFCs G (i ) and define the closed loop system matrices A0(i ) such that, A0(i )

A  BG (i ) , i 1, 2,

(15)

Theorem 1 immediately defines unstructured error matrices E (i ) ( x) such that, A(i ) ( x)

A0(i )  E (i ) ( x) , i 1, 2,

(16) 13

The eigenvalue perturbations of A(i ) ( x ) from the nominal eigenvalues of A0(i ) are connected as follows. Theorem 2: Let the nominal matrix A0(i ) is diagonalizable such that A0(i ) V (i ) /V (i )1 where V (i )1 is the inverse of V (i ) and / diag (O1 On ) . If an error matrix that perturbs the nominal matrix is E (i ) ( x) and if P (ip ) is the eigenvalue of A(i ) ( x) A0(i )  E (i ) ( x) , then

A0(i )

min P (pi )  Oq d N (V (i ) ) V max ( E (i ) ( x)) .

1d q dn

(17)

Theorem 2 is well known when x(t ) and i are fixed [29,30]. Here N (.) refers the condition number of the matrix (.). Similarly, V max (.) refers to the maximum singular value of the matrix (.). Note that in Eq. (17) the property of the LFC for eigenvector variations when eigenvalues are fixed or eigenvalue variations is employed. In order to present the transcription criterion, few inferences of Theorem 2 are presented as follows. A. Smaller the condition number larger the stability region.

Consider the left hand side of Eq. (17). The perturbed eigenvalue P q(i ) will be within a circle of radius ‘R’ with center at Oq if [30], V max ( E (i ) ( x )) d

R N (V (i ) )

(18)

Clearly, smaller the condition number larger the error size E (i ) ( x) . That is, larger the trajectory dispersions (that is, x(t) – dispersions) from the equilibrium point xe at which E (i ) ( xe ) 0 . Hence stability region is larger when condition number for the modal matrix of the nominal matrix A0(i ) is smaller. Note that the circles with radius ‘R’ and with centers at the eigenvalues of the nominal matrix are contained in the open left half plane of the complex plane. The transcription criterion in a trustable UAV is immediately inferred as follows. B. Transcription criterion.

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It is observed that all the controllers for parent and child FCMs originate from the linear model in Eq. (7) whose equilibrium point is given by ( xe , u e ). The Jacobian matrix A(i ) ( x) in Eq.(16) at this equilibrium point is indeed the nominal matrix A0(i ) for which the error matrix E (i ) ( xe ) is a zero matrix. Hence when the Jacobian matrix at all x(t ) is asymptotically stable, the error matrix E (i ) ( x)

measures the relative position of x(t ) with respect to the equilibrium state xe . If one takes

the 2-norm of the error matrix E (i ) ( x) denoted by V max ( E (i ) ( x)) and take this norm for the parent and child trajectories at the decision point t d , transcription criterion is inferred as follows. For the child trajectory x(t d ) to transcript with parent trajectory x dp , a qualitative requirement is that, V max ( E ( p) ( x dp ))  V max ( E (d ) ( x(t d ))

(19)

Eq. (19) suggests that at the decision point, the stability region of the child controller most likely contains the stability region of the parent controller enabling the potential TTT. The same principle applies to SSTT where for the child FCM to most likely transcript the parent FCM, the condition is established as follows. V max ( E ( p) ( xap,e ))  V max ( E (d ) ( x(t d ))

(20)

These qualitative rules of transcription criterion assist to accommodate the decision points of the higher level control architecture while an UAV is directed to perform a certain task. Eqs (19) and (20) further suggest (qualitatively) that when a decision point is active, the distance of the child trajectory point from the equilibrium point is more compared to the distance of the parent trajectory point from the same equilibrium point (as in Figure 3). These qualitative statements are used for a transcription to occur.

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Fig. 3a. Child controller selection criterion.

Fig. 3b. Stable child trajectories for higher level decisions. C. LFC effects on transcriptions.

LFCs fundamentally adopt eigenstructure variations to modify transients and steady states in their respective stability regions. Such variations may be established by designing a controller whose closed loop eigenvalues are either fixed or varied in the open left half plane of the complex plane. The freedom available to design these controllers via eigenstructure options offers flexibility to access any decision point of the higher level control architecture; however, identifying a controller for child FCM that satisfies the transcription criterion (Eq. (19) or Eq. (20)) becomes difficult. In this framework, it can be claimed that all decision points of the higher level control architecture are accessible through appropriate child controllers. Suppose the child controller is fixed. Again it is observed that the initial condition options suggest different

16

decision points at transients. Hence it becomes important to highlight the significance of the point x(t d ) on the child trajectory. Theoretically, the UAV at the decision point t d begins to acquire a child trajectory with an initial condition x dp on the parent trajectory. While it is known that x dp is in the stability region S Gp for stable switching of the controller G d from G p . In order to check if x dp is in the stability region S Gd

of the controller G d , the following hypothesis test is applied. Let x0 be an initial condition

that generates trajectories x p (t ) in S Gp and x(t ) in S Gd . Clearly if x(t d ) satisfies the trajectory transcription criterion in Eq. (19), it is likely that x dp is in S Gd . The hypothesis test is a procedure to choose potential controllers for child FCMs. The entire procedure selects the initial condition x dp for the child FCM generated by a controller option G d . It is anticipated that the Lyapunov

stability criterion in the form of an asymptotically stable Jacobian at Eqs. (13) and (14) will be satisfied for an f (x) satisfying, x

f (x) , x(t d )

f ( x)

x dp , where

(21)

a( x)  b( x) [ue  G d ( x  xe )]

In Figure 4, a detailed description of a trustable UAV is presented. 4. Autonomous attributes of a trustable UAV Before illustrating the autonomous attributes of a trustable UAV, the current practices in UAV operations by a remote operator are reviewed. Figure 5 depicts the typical control architecture with a remote operator at the outer loop. The main feature of the architecture is that at the inner loop a fixed order linear control structure with constant gains is adopted. If it is a PID controller, the gain tuning is performed usually by trial and error method. If a FCM such as an ascent is desired, the remote operator intuitively gives a command input for which the UAV ascends with 17

Fig. 4. Trustable UAV complying the higher level instructions.

a constant thrust. Note that from an affixed order inner loop controller, these are the limited maneuver capabilities from an UAV without the thrust management. However, human decisions significantly contribute to the feasible mission objectives of UAV. In contrast, the control architecture of an autonomous trustable UAV for higher level decisions is depicted in Figure 6. These architectures are programmable if the number of parent and child controllers is known. The decision points may be appropriately directed; however, the difficulty prevails for decision points at transients. In this regard, LFC may be designed such that, V max ( E p ( x p (t ))  V max ( E d ( x(t ))

(22)

Here, the higher level decisions can be made at any part of the parent trajectory for all of which the transcription criterion will be met to generate child FCMs. In the next section, the

18

autonomous features of a trustable UAV presented in Figure 5 are simulated by using the parent and child FCMs.

Fig. 5. FCM in a human operated UAV.

Fig. 6. Autonomous parent and child FCMs in a trustable UAV.

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5. Simulations The three degree of freedom aircraft model of Langelaan [17] is considered to simulate a trustable UAV. The aircraft is actually a four-state and three-input system. In this paper, the utility of the flap as an input is not investigated. The state and other control variables defined in Eq. (4) respectively are recalled as follows.

x(t )

ªva (t )º « D (t ) » « » « T (t ) » « » ¬ Q(t ) ¼

ªForward velocity, m/sº « Angle of attack, rad » « ». « Pitch angle, rad » « » ¬ Pitch rate, rad/s ¼

u (t )

ªCT (t )º « G (t ) » ¬ e ¼

ª Thrust coefficien t º «Elevator deflection , rad » . ¬ ¼

The kinematics of the aircraft in pitch plane are given in Eqs. (2) and (3). The position of the aircraft is identified by ( X i (t ), Z i (t ) ) as in Figure 2, where X i (t ) is the latitude longitude position in meters and Z i (t ) is the altitude in meters. One of the simplest equilibrium points considered to develop the trustable UAV is computed as, xec

>21.021

0 0 0@ and uec

>0.0163 0@

The aircraft at this equilibrium point is in cruise (altitude hold) mode with D (t ) T (t ) 0 . For a cruise mode with non-zero angle of attack, generally an optimization is required to compute the associated equilibrium point and it is illustrated in [25]. For Eq. (23), the controllable linear model is computed as (all data are approximated to the fourth decimal place), 20

6.7890 ª 0 « 0.0452  7.2167 A « « 0 0 «  0 34.3897 ¬

B

ª27.1758 «0 « «0 « ¬0

 9.8100 0 0 0

º 1.0280 »» 1.0000» »  2.7163¼ 0

º 0.4806 »» » 0 » 54.8718¼ 0

Following the procedure presented in [20], the state feedback LFCs G (i ) are computed for eigenvector variations of the closed loop system matrices Ac(i )

A  BG (i ) .

For instance, when

eigenvalues of Ac(i ) are fixed at, 5 r 5.5 j  2 r 2.5 j

two LFCs out of infinite options are computed as, G1

ª0.1552 0.5727  0.7073  0.1845º «1.1461  3.2203  2.2719 0.0254» ¼ ¬

G2

ª 0.1108 0.8881  0.2090  0.0405 º «  0.8533 1.1311 0.6653 0.1191»¼ ¬

Other eigenvector variations are pursued by changing the eigenvalues. For, 4 r 7.5 j  2 r 2.5 j

,

one of the controllers assigning the above closed loop poles is, G3

ª 0.0054  0.1911  0.2758 0.1415º « 0.9719  1.5841  1.2641 0.0489» ¼ ¬

For, 3 r 7.5 j ,  2 r 2.0 j

one of the controllers assigning the above closed loop poles is,

21

G4

ª 1.3607 7.0370 2.2027 1.1524º « 0.7126 2.5202 1.2994 0.6348» . ¼ ¬

Table 1 Auxiliary equilibrium points of S Gj

xaj,e

Controller

G

1

G2

G

3

G

4

ª v aj,e º « j » «D a ,e » «T j » « aj,e » «¬Qa ,e »¼ ª 20.9683 º « 0.0014» « » « 0.0250» « » ¬ 0.0000¼ ª 21.0259 º « 0.0017 » « » « 0.0108 » « » ¬ 0.0000 ¼ ª 21.0362 º « 0.0018» « » « 0.0103» « » ¬ 0.0000 ¼ ª 21.0451 º « 0.0019» « » « 0.0177 » « » ¬ 0.0000 ¼

dZ i dX i

 tan(T aj,e  D aj,e ) FCM

0.0237

Ascent Mode

0.0125

Descent Mode

0.0085

Ascent Mode

0.0196

Descent Mode

Given these stabilizing controllers, the auxiliary equilibrium points xaj,e of the stability regions S Gj

are presented next. For any initial condition x0 in S Gj , the trajectory satisfying Lyapunov

stability criterion derived in terms of stable eigenvalues of the Jacobian matrix in Eq. (13) will determine these auxiliary equilibrium points. Such points depict the slope of the FCMs in ascent and descent modes. These details of the controllers G j are presented in Table 1. In order to accommodate the decision points of the higher level control architecture, which direct the UAV to bifurcate from its parent trajectory, the transcription criterion in Eqs. (19) or (20) is adopted. It 22

basically suggests controller switching within a region where their stability regions intersect, a rule exhaustively illustrated in [12] for a simple second order nonlinear system. It is also important to observe that there are state trajectory transcriptions as well as control input trajectory transcriptions at the decision points. With respect to a transient (initial condition) option pertaining to a stability region S Gj , the state trajectory transcription is addressed first for a fixed initial condition x0 , which is taken as,

x0

ª21.0210 º « 0.0002» « ». «0.0001 » « » ¬0.0001 ¼

To identify parent and child FCMs, the norm of the error matrix in Eq. (18) is plotted for three controllers taken at a time. Figure 7 is utilized to depict transcription criterion in Eqs. (19) and (20). Controllers G1 (indicated in black color), G 3 (blue) and G 4 (red) are considered. Controller ‫ ܩ‬ଶ is not included as it will be used to illustrate the effect of controller combinations in transcriptions subsequently. It is possible to select as many controllers as one can and illustrate transcription criteria. For instance, in Figure 7, controller G 3 offers a trajectory which is likely to contain in the most parts of the stability regions S G1 and S G4 of the controllers G1 and G 4 , respectively. Hence the trajectory resulting from controller G 3 becomes a parent trajectory. The resulting FCM is referred as parent FCM. The fact that it is contained in S G1 and S G4 further enables the controllers G1 and G 4 to transcript with controller G 3 . Upon satisfying the Lyapunov stability criterion (Eq. (13) and (14)), the resulting trajectories and FCMs from these transcriptions are referred as child trajectories/FCMs. Note that in steady state and at other parts

23

of the transients, other conclusions can also be drawn from Figure 7. That is, unlike the first part

Fig. 7. Transcription criterion depicting potential parent and child FCMs.

of the transients, at steady states, G1 transcripts with G 4 (as against G 4 transcripting with G1 at the first part of the transients). This observation is useful if the child FCM becomes a parent FCM at some decision points. To illustrate the main properties of transcription, consider a decision point t d 3 seconds. From Figure 7, V max ( E1 ( x(t d ))) 0.9976

V max ( E 3 ( x(t d ))) 0.1606

(23)

V max ( E 4 ( x(t d ))) 0.6338

24

Clearly, the point x(td ) on the trajectory resulting from the controller G 3 (denoted by x dp ) is likely to contain in the stability regions S G1 and S G4 of controllers G1 and G 4 , respectively. Hence these controllers transcript with controller G 3 . That is, assuming either G1 or G 4 as a child controller G d , the trajectories resulting from the following equation, x

f (x) , x(td )

f ( x)

x dp , where

a( x)  b( x) [ue  G d ( x  xe )]

will have a Jacobian that is asymptotically stable. These parent and child trajectories for various decision points are presented in Figure 8. The ascent and descent modes are due to the trajectories generated by child controllers G1 and G 4 respectively in SG1 and SG4 .

Fig. 8. Child FCMs bifurcating from parent FCM at decision points.

25

Fig. 9. Controller options influencing transcriptions.

Fig. 10. Initial conditions influencing transcriptions.

26

As it has been observed in the example, parent and child controller selection procedures are context based. That is, they primarily depend on controller type (eigenvector variations) and initial conditions. For instance, the transcription criterion shown in Figure 7 is modified as shown in Figure 9, when controller G1 is replaced with controller G 2 . Here the transcriptions discussed similar to Eqs. (23) are preserved for all decision points across the transients and steady states. In contrast, these controllers in Figure 9 modify the transcriptions as shown in Figure 10, when initial conditions in their respective stability regions are varied. Thus these techniques in a way search whether decision points reside in the stability regions of the controllers. Given an active controller as a parent, child controllers compatible to this parent may be designed following the procedures presented in this paper. Generally, the transcription criteria for decision points across transients of the initial conditions are unpredictable. Hence decision points are favored at steady states. Lastly, the control transcriptions are presented. The thrust coefficient ( CT (t ) ) in the aircraft of Langelaan [17] must be positive. In thrust management, generally the transients are ignored but the steady state values are programmable. Linearized model based nonlinear aircraft control presented in this paper offers these steady state values for thrust management. Although positive thrust management schemes for FCM analysis is important, in this paper, parent and child FCMs with positive transients and steady states in thrust coefficient are considered to illustrate the control transcriptions. For one of the trustable UAV transcriptions in Figure 9, the control transcriptions are depicted in Figure 11 and 12, respectively. The control inputs for child FCMs are presented in black color. The parent control inputs before bifurcations are presented in blue color. In these problems, it is difficult to impose control input constraints.

27

Fig. 11. Control transcription with thrust coefficient.

Fig. 12. Control transcription with elevator inputs.

28

6. Conclusions There has been a significant interest to automate the air defense system by using higher level control architectures where the unmanned air vehicles are directed to accomplish their mission operations. These decisions to engage an unmanned air vehicle require instantaneous controller reconfiguration at inner loop so that it bifurcates from the intended path to the assigned task. While the intended path is referred as the parent flight control mode, the path adopted to perform the assigned task is referred as a child flight control mode. The decision points where the tasks are assigned must reside in a region where stability regions of the controllers intersect. Given a class of linear functional controllers for eigenvector variations, this paper establishes a search technique for controller selection so that the decision points of the higher level control architecture are naturally adaptive to stable reconfigurations of the controllers in their respective stability regions. An unmanned air vehicle exhibiting these characteristics meets a transcription criterion to define a trustable unmanned air vehicle for higher level control architectures. A three degree of freedom aircraft is considered to illustrate the trustable unmanned air vehicle. ACKNOWLEDMENT: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. References: [1]

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