Mar 2, 2018 - The program adaptation of the controller for the flight control system (FCS) of an ... Linearized flight dynamic models depend mainly on the true ...
DOI 10.1007/s10778-018-0876-0
International Applied Mechanics, Vol. 54, No. 2, March, 2018
A FLIGHT CONTROL SYSTEM FOR SMALL UNMANNED AERIAL VEHICLE A. A. Tunik1 and O. I. Nadsadnaya2
The program adaptation of the controller for the flight control system (FCS) of an unmanned aerial vehicle (UAV) is considered. Linearized flight dynamic models depend mainly on the true airspeed of the UAV, which is measured by the onboard air data system. This enables its use for program adaptation of the FCS over the full range of altitudes and velocities, which define the flight operating range. FCS with program adaptation, based on static feedback (SF), is selected. The SF parameters for every sub-range of the true airspeed are determined using the linear matrix inequality approach in the case of discrete systems for synthesis of a suboptimal robust H ¥ -controller. The use of the Lagrange interpolation between true airspeed sub-ranges provides continuous adaptation. The efficiency of the proposed approach is shown against an example of the heading stabilization system. Keywords: flight control, static feedback, adaptation, H ¥ -controller Introduction. The wide operational range of an unmanned aerial vehicle (UAV) wheither civil or military-oriented results in considerable changes in the parameters of the dynamic model of an UAV when performing a flight mission in the entire range of altitudes and velocities. This circumstance often makes it impossible to solve the flight control problem based only robust control theory [12]. In this case, methods of adaptive control with a closed adaptation circuit are used [12], which leads to considerable complication of control laws. The simple method of direct program adaptation (gain scheduling) depending on the variation of the parameters of the mathematical model under parametric excitation has long been known (see [21 – 23] and the references therein). Application of this method was restricted by the necessity of synthesis of a considerable number of control laws for rather narrow ranges of variation of parametric excitation [23]. However, with the advent of robust control methods, it became possible to provide robust stability and quality for much wider ranges of variation of the parametric excitation, which made it possible to reduce the number of control laws and simplify the system structure. In such a system, there is no complex and slow-acting closed loop of adaptation, and the parameters of the controller change instantly upon changes in parametric excitation. Since cost, power, size, and weight restrictions are imposed on the solvers of the control systems of UAVs, the controller should be sufficiently simple and easy to implement. It is convenient to use static output feedback [16, 25] based on the g-optimal approach, where g means the excitation suppression factor (L2 -gain [5, 10]), synthesized by the method of linear matrix inequalities (LMI [5, 7, 8, 10]. In the most cases of using LMIs for the synthesis of control law for autopilot, continuous models are considered [10]. Since at the present time only digital autopilots are used, we will synthesize the flight control system in discrete time. The efficiency of the approach will be shown by considering an example of flight modeling of Aerosonde UAV in the mode of assigned track stabilization. The Aerosonde model is supported by Aerosim Matlab® [26] and is a benchmark model for study of UAV flight dynamics. 1. Problem Statement. Basic Equations. Let the dynamics of the vehicle be defined in continuous time in space by the system of equations
1
National Aviation University, 1 Komarova Av., Kyiv, Ukraine 03058; aatunik@nau.edu.ua. 2State Enterprise “Antonov”, Tupoleva St., 1, Kyiv, Ukraine 03062; nadsadna@ukr.net. Translated from Prikladnaya Mekhanika, Vol. 54, No. 2, pp. 134–144, March–April, 2018. Original article submitted May 11, 2017. 1063-7095/18/5402-0239 ©2018 Springer Science+Business Media, LLC
239
ì dX = A ( q ) X ( t ) + B ( q )U ( t ) + B w ( q )W ( t ), ï í dt ïî Y ( t ) = CX ( t ),
(1.1)
where the matrix elements A ( q ) Î R n ´n , B ( q ) Î R n ´m , B w ( q ) Î R n ´q depend on the velocity pressure (VP) q = (rVt2 ) / 2, where r is the air density at the flight altitude; Vt is the true air speed (TAS) [6, 19]. This parameter defines the parametric excitation since it depends uniquely on the altitude and velocity and changes within the range of altitudes and velocities typical for this type of UAVs. Note also that in the general case, the air density r depends on the flight altitude; however, this parameter can be assumed constant for altitudes typical for UAVs [6]. Even the flight altitude of UAVs is determined assuming the constancy of r [6], and the error due to this assumption is neglected not only in measurement but also in the integration of measurements in the navigation system [15]. Therefore, it is possible to consider all elements of the matrices in (1.1) as functions ofVt alone, and a small deviation of the air density r from its true value above the sea level as one of the sources of uncertainty in the UAV model. The TAS is measured by the air data system of every aircraft; therefore, the flight control system does not need supplementary equipment to measure this speed. Note also that altitude and velocity are long-period variables [6, 11]; therefore, the elements of the matrices A (Vt ), B (Vt ) in (1.1) vary quite slowly. Moreover, all variables of the space of states cannot be measured; therefore, the observation matrix C belongs to the set C Î R l ´n , l < n, where C has full rank of its rows. Changing over to discrete time in a standard way and assuming constancy of r, we obtain the system of equations ì X ( k + 1) = A (V )Y ( k ) + B (V )U ( k ) + B w (V )W ( k ), d t d t d t í ( ) = × ( k ), Y k C X î
(1.2)
where X Î R n ´1 ,U Î R m ´1 , W Î R q ´1 , and Y ( k ) Î R p ´1 are the vectors of state, control and external excitation (turbulent wind) and output, respectively, A d Î R n ´n , B d Î R n ´m , B dw Î R n ´q , Ñ Î R l ´n are the dynamic matrices. Here it is assumed the pair ( A d (Vt ), B d (Vt )) is stabilizable, and the pair (A d (Vt ), C) is detectable over the entire range of variation inVt . Let us introduce a quadratic quality function: ¥
J=
å ( X Tk QX k + U Tk RU k ).
(1.3)
k=0
Define the vector of desired output variables Z ( k ) [1, 10] and supplement the system of equations (1.2) as ì X ( k + 1) = A (V )Y ( k ) + B (V )U ( k ) + B w (V )W ( k ), d t d t d t ï í Z ( k ) = C z X ( k ) + D zU ( k ), ï Y ( k ) = C × X ( k ), î
(1.4)
where the matrices Ñ z Î R n ´n , D z Î R n ´m are determined according to (1.3) and [1, 2, 10] as C z = Q,
D z = [ 0( n - m ) ´m , R ]T .
(1.5)
Define also the excitation suppression factor (L2 -gain) [1, 5, 7, 10, 18] and bound it from above by g < 1: | | Z ( k )| |2 | |W ( k )| |2
=
| |C z X ( k ) + D zU ( k )| |2 | |W ( k )| |2
£ g,
(1.6)
where ||× ||2 denotes the H 2 -norm. According to [8, 10, 13], condition (1.6) guarantees robust H ¥ -control, which is sometimes called g-optimal [1, 2]. Let static output feedback be defined by the vector L (Vt ). Then the autopilot control law is U ( k ) = -L(Vt )Y ( k ).
(1.7)
The problem consists in the syntheses of the control law (1.7) that satisfies inequality (1.6) in the whole range of Vt . 240
2. Synthesis of Discrete Static Output Feedback. Let us synthesize discrete static output feedback for incomplete measurement of components of the state vector and for a fixed value of the parameter Vt by the method of linear matrix inequalities (LMI) [1, 2, 5, 7, 9, 10, 16, 18]. To this end, it is convenient to use the results from [5]. Since the value ofVt is fixed, this parameter is avoided in Eqs. (1.2), (1.4), (1.7). If the matrix C has full row rank, then there is a matrix T such that CT -1 = [ I p 0 p ´( n - p ) ] [5]. In the case of SC models, it is assumed that the matrix C consists of zeros and units [19]. Arranging Eqs. (1.1), (1.2) in an order such that the measurable variables go first, followed by the non-measurable ones, it is possible to represent the matrix C as C = [ I p 0 p ´( n - p ) ] ,
(2.1)
where I p and 0 p ´( n - p ) are unit and zero matrices. Here the matrix T transformed to the form (2.1) turns out to be a unit matrix, which allows avoiding to reduce the state variables in (1.1), (1.2) to a new basis, used in [5], and, thus, to simplify further derivations. The closed system (1.2), (1.7) is known to be stable if the matrix of the closed system A cl = A d + B d LC is of Hurwitz type. On the other hand, according to the Lyapunov theorem, the closed system remains stable if there exists a positive definite T - P < 0. According to [5], the matrix P can be represented as é P11 P12 ù for P > 0 and symmetric matrix P such that A cl PA cl ú êP 11 ë 21 P22 û -1 P P22 - P12 P11 12 > 0. Let
P1 = P11 ,
-1 N = P11 P12 ,
T -1 P2 = P22 - P12 P11 P12
(2.2)
and é P P = ê T1 ëN P1
ù = T N Pdg T NT P2 + N T P1 N úû P1 N
æ é I çç Pdg = diag ( P1 , P2 ); T N = ê T ëN è
0ù ö ÷, I úû ÷ø
(2.3)
(2.4)
where T N is a non-singular matrix. The algorithm of synthesis is based on two theorems proved in [5]. Theorem 1 [5]. System (1.2), (1.7) is stabilizable by static output feedback if and only if there exist matrixes T P1 = P1 Î R p ´ p , P2 = P2T Î R ( n - p ) ´( n - p ) , G Î R n ´ p , N Î R p ´( n - p ) such that the following inequality holds: é Pdg ê T T T T 1 -1 T êëPdg (T N A d T N ) + C G (T N B d )
(T N-1 A d T N )Pdg + (T N-1 B d )GC ù ú > 0, Pdg úû
(2.5)
where Pdg and T N are defined by (2.3), (2.4), and inequality (2.5) holds for a controller L = GP1-1 , where the matrix P1 is defined by (2.2). This theorem enables determining a stabilizing controller for system (1.2) and is the first stage in the synthesis of discrete output feedback. To determine the g-optimal controller in the sense of (1.6) for system (1.4), we need the second theorem. Theorem 2 [5]. The closed system (1.4), (1.7) is stabilizable by static output feedback and its H ¥ -norm is less than the given level g if there exist a symmetric matrix P Î R n ´n and a scalar s Î R such that the following inequalities hold: é-P + sB B T d d ê T PA d ê ê sD B T z d ê w T B ( êë d )
Ad P
sB d DTz
-P
PC Tz
CzP
-gI + sD z DTz
0 q ´n
0 q ´n
B dw ù ú 0 n ´q ú < 0, 0 n ´q ú ú -gI q úû
(2.6)
241
é -P ê ê 0 n ´n ê( B w )T ë d
0 n ´n -gI n 0 q ´n
B dw ù é A d ù ú ê ú é 0 p´ p 0 n ´q ú + ê C z ú ê ë0( n - p ) ´ p -gI q úû êë0q ´n úû
éA ù 0 p ´( n - p ) ù ê d ú C P22 úû ê z ú ê0q ´n ú ë û
T
< 0.
(2.7)
This controller synthesis procedure is easily programmed and is convenient from the point of view of its multiple use for linearized models corresponding to typical nodal values of Vt within its admissible range. The synthesis of the controller parameters for every nodal value of Vt consists of the following stages: 1) the whole range of admissible values Vt for UAV is divided into a number of subranges, inside which the mathematical model is linearized; 2) for the obtained family of the linearized models, the family of local controllers is synthesized; 2.1) the scalars s > 0, g > 0 appearing in (2.6), (2.7) and the weighting matrices Q , R in (1.3) are specified; 2.2) the solutions of LMI (2.6), (2.7) for P are found; 2.3) the matrix P is represented in the form (2.4); 2.3) the solution of inequality (2.5) for G is found; 2.4) the matrix of parameters of the controller L = GP1-1 is calculated; 3) for every subrange Vt , we determine the mechanism of program adaptation of the controller (PAC) is determined. 3. Program Adaptation of the Controller. Two ways of implementation of the PAC mechanism are known, namely, switching from one synthesized control law to another upon changes in the external conditions (flight velocity in our case) [18] and linear interpolation of the controller coefficients as functions of parametric excitation [22]. Here in both cases, for simplification of the PAC structure, one tries to reduce the number of table controllers (in the first case) and the number of interpolation nodes (in the second case). The use of controller switching is associated with the requirement of smooth control transfer from one controller to another (bumpless transfer [11]), which considerably complicates the system. From this point of view, the interpolation method is preferable. For minimization of the number of interpolation nodes, it is proposed to use sensitivity theory [4]. Since the output variables of the closed system have different sensitivity to variations of the controller parameters, then on interpolation of the functional dependence of a definite controller coefficient on TAS, it is possible to reduce the number of interpolation nodes if the norm of the corresponding sensitivity function will be small. The Lyapunov stability of systems with a program-adapted controller in a wide range of variation of parametric excitation was detailed in [7]. The corresponding results for airplanes were considered in [14]. Let us now consider the procedure of PAC synthesis by the method of linear matrix inequalities for an Aerosonde UAV [26], whose mathematical models can be determine in MATLAB software. The vector of state X of the mathematical model (1.1) has the form X = [b, p , r, j, y ]T , where b is the slip angle; p is the angular roll rate; r is the angular yaw rate; j is the roll angle; y is the yaw angle. Control is performed by deflecting an aileron d a and rudder d r : u = [ d a , d r ] T . Since the slip angle bis not measured for UAVs in most cases [6], the observation vector has the form y = [ p , r, j, y ] T . According to [19, 24], the matrices A (Vt ) and B (Vt ) in (1.1), (1.2) depend on Vt such that the elements of the matrix A (Vt ) are partial derivatives of the lateral force Y with respect to the state variables b, p, r, j, y, i.e., Yb , Y p , Y , … and so on, as well as partial derivatives of moments along the longitudinal l and vertical N axes with respect to the same variables, i.e., Lb , L p , Lr , … and N b , N p , N r , … . The elements of the matrix B (Vt )are also partial derivatives with respect to the control variables d a , d r for both the lateral forces Y and the moments along the longitudinal L and vertical N axes, i.e., Y da , Y dr , Lda , Ldr and N da , N dr . Some of these partial derivatives are expressed as follows [19, 24]: Yp = Lda =
qSb C , 2mVt Y p
qSb C , J X lda
Y da = Np =
qS C , m Yda
qSb C , J Z 2V t N p
Lp =
qSb C , J X 2V t l p
N dr =
qSb , C J Z N dr
where J X , J Z are the corresponding moments of inertia of UAVC; m is the mass; S , b are the wing area and wingspan, respectively; CY , CY , C l , C l , C N , C N are partial derivatives of the coefficients of the lifting force, moments of roll and p
242
da
p
da
p
dr
dr
A
UAV da
A
yspec
y j p r b
W WF (z)
ey
Ly (n t ) Lj (n t )
nt
Lp (n t )
ADS
Lr (n t )
Fig. 3.1
yaw with respect to the state and control variables. The rest of partial derivatives (elements of the matrices A (Vt )and B (Vt )) have a similar form [19, 24]; therefore, their expressions can be omitted. Due to the capabilities of the UAV measurement system represented by the state vector y = [ p , r, j, y ]T , the vector of controller parameters has the form
[
]
L(Vt ) = L p (Vt ) Lr (Vt ) Lj (Vt ) Ly (Vt ) ,
(3.1)
where L p , Lr , Lj , Ly are the gains for the respective output variables. Note that the control object should include the actuators of the rudder and aileron with the transfer functions WÈÏ ( p ) =
1 , 0.25 p + 1
(3.2)
as well as the local angular yaw rate feedback connected to the input of the rudder actuator and used as a yaw damper and for partial compensation of the slip angle during a bank-to-turn maneuver [19]. This feedback represents a wash-out filter [19] whose transfer function in discrete form is WWF ( z ) = LWF
z -1 , z -d
(3.3)
æ T ö where LWF is the gain; d = expçç - s ÷÷ (Ts is the discreteness interval (sec); t is the time constant of the wash-out filter (sec)). è t ø Starting from the conditions of bank-to-turn maneuver [17] we choose LWF = 7. The discreteness interval is chosen to be 0.02 sec due to the frequency of the navigation system (50 Hz). With (3.2) and (3.3), the vector of state of the extended control object has the form X = [b, p , r, j, y , ¶r, ¶a, x w ]T , where x w is the variable of state of wash-out filter (3.3). A structural diagram of the heading stabilization system is shown in Fig. 3.1, where A denotes the actuators of the rudder and ailerons, ADS is the air data system. Nonlinear models of Aerosonde were linearized by procedures supported by the MATLAB software and presented in [26]. The admissible range of variation in TAS (21–32 m/sec) was divided into nine subranges, inside which model (1.1) can be considered as a system with constant coefficients. Note that the program of determination of the linearized models presented in [26] accounts for the variation in the air density r with the altitude, which corresponds to the real conditions of flight. After the standard procedure of discretization of continuous models, nine discrete models (1.2) were obtained. The matrices A d , B d of these models for the minimum (Vt = 21 m/sec, height 400 m) and the maximum (Vt = 32 m/sec, height 1000 m) TASs have the following form: 243
A dmin
é 2.44 0.45 -2.63 1.63 ê 1.56 -0.07 0.72 -0.21 ê ê-0.23 -0.06 0.49 0.13 ê 0.34 0.04 0.12 0.98 =ê ê 0.01 -0.01 0.15 0.01 ê-0.36 0 0 0 ê 0 0 0 ê 0 êë 1 0 0 0
0 -2.33 -0.66 3.67 ù 0 -2.08 -1.68 -1.02ú ú 0 0.44 0.14 -1.46ú 0 -0.45 -0.15 -0.1 ú ú, 1 0.03 0.01 -0.19ú 0 0.45 0 0.36 ú ú 0 0 0.45 0 ú 0 0 0 0 úû
A dmax
0.46 -3.22 1.37 é 4.67 ê 2.04 -0.1 0.74 -0.18 ê ê-0.75 -0.02 0.04 0.19 ê 0.6 0.02 0.12 0.98 =ê 0.07 0.014 0.12 0.01 ê ê-0.36 0 0 0 ê 0 0 0 ê 0 êë 1 0 0 0
0 -5.56 0 -2.58 0 1.14 0 -0.77 1 0.12 0 0.45 0 0
0 0
12.77 -4.45ù -3.21 -2.06ú ú -2.55 0.91 ú -0.29 -0.62ú ú, -0.41 0.1 ú 0 0.36 ú ú 0.45 0 ú 0 0 úû
B dmin
. é -187 ê -167 . ê ê 0.36 ê-0.36 =ê ê 0.03 ê 0.36 ê ê 0 êë 0
2.94 ù -0.82ú ú -117 . ú -0.08ú ú, -0.15ú 0 ú ú 0.36 ú 0 úû
B dmax
é-4.47 ê -2.06 ê ê 0.91 ê -0.62 =ê ê 0.09 ê 0.36 ê ê 0 êë 0
10.21ù -2.57ú ú -2.04ú -0.23ú ú. -0.33ú 0 ú ú 0.36 ú 0 úû
It can be seen that the elements of these matrices depend considerably on the TAS. Note also that for these altitudes and speeds, the air pressure vary considerably: 250–650 kg×m/sec2. For smooth heading stabilization upon TAS variation, it is expedient to use interpolation of the feedback coefficients as functions of TAS using Lagrange polynomials [3]. In this connection, the feedback coefficients were calculated by the technique described in Sec. 2 for every subrange, resulting in a family of 89 controllers. The following values were used for the linear matrix inequalities (2.6), (2.7): g = 0.85, s = 1; R = [.01], Q min = 10-3 diag([ 99 9.5 3.5 10 1 1 10 1]), Q max = 10-3 diag([ 99 1.5 20 10 1 1 10 0.1]) (to save space, the values for the minimum and maximum admissible TASs are given). Thus, we obtain nine interpolation nodes for every feedback coefficient. However, the use of polynomials of the 9th degree for the approximating the dependence of these coefficients on TAS is not expedient due to extreme complexity. To reduce the degree of interpolation polynomials, we analyzed the sensitivity of the closed system to variations of the controller coefficients. According to [4] this sensitivity can be estimated using the transfer function of the sensitivity of the closed system to the variation of the corresponding parameter Li : Wi ( z ) =
where T ( z ) =
WÐÑ ( z ) 1+ WÐÑ ( z )
¶T ( z ) , ¶Li
(3.4)
is the transfer function of the closed system, i.e., the complementary sensitivity function [13]; WÐÑ ( z )
is the transfer function of the open system for the series connection of the object and the controller. Determining the H ¥ -norm of the sensitivity function Wi ( z ) to the parameter Li (3.4), it is possible to estimate numerically the effect of change in this parameter Li on the variation of the output coordinate y of the system. For the system in Fig. 3.1, according to (3.4), the H ¥ -norm of the sensitivity function of the closed system to the variation of the ith parameter of the controller Li has the form
¶Ty ( z ) ¶Li
244
= ¥
W
di ( z )
i æ ç 1+ Li W å ç è pÎi
ö ÷ z ( ) di ÷ ø
,
2
¥
(3.5)
TABLE 1 ¶Ty
Flight conditions
¶Lp
¶Ty ¶Lj
¥
¶Ty ¶Lr
¥
¶Ty ¥
¶Ly
¥
1
0.7393
0.6432
0.1052
0.0513
2
0.3597
0.3876
0.0552
0.0293
3
0.11
0.3312
0.0196
0.0067
Lj 0.8
0.6
0.4 0.2 18
22
26
30
nt, m/sec
Fig. 3.2
where Wdi ( z ) is the transfer function from the deviation of ailerons to the corresponding output variable. Numerical values of the H ¥ -norm of the sensitivity functions to variations of controller parameters determined by formula (3.5) for three different flight conditions are presented in Table 1, namely, 1 corresponds to an altitude of 400 m, velocity of 21 m/sec, and air pressure of 259 kg×m/sec2; 2 corresponds to an altitude of 900 m, velocity of 26 m/sec, and air pressure of 379 kg×m/sec2; 3 corresponds to an altitude of 1000 m, velocity of 32 m/sec, and are pressure of 569 kg×m/sec2. As follows from Table 1, variations of the gains with respect to the angular heading velocity r and heading angle y have a much weaker effect on the output coordinate y than the variations in the gains with respect to the angular roll rate p and roll angle j . Therefore, to approximate the controller gain for y, it is expedient to Lagrange polynomials of lower degree than those for approximation of the gains for j . The values of the coefficient Lj (Vt )nine values ofVt are shown in Fig. 3.2 by x’s and the solid line is an approximation of Lj (Vt ) by a Lagrange polynomial of the 3rd degree. As is seen from Fig. 3.2, the maximal deviation of the synthesized value Lj (Vt )from the approximation by Lagrange polynomial of the 3rd degree is » 0.09, which implies that the maximal value of the deviation of the output coordinate will not exceed | | ¶Ty / ¶Lj | |¥ ×0.09 ; 0.058 rad. Guided by these considerations, it is sufficient to approximate the coefficients Ly (Vt ), Lr (Vt ) by Lagrange polynomials of the 2nd degree. Finally, the Lagrange formulas for the family of the controller coefficients take the form . × 10-3 Vt3 + 0.01025Vt2 - 0.253Vt + 2152 . , L p (Vt ) = -1358 Lr (Vt ) = 1018 . × 10-3 Vt2 - 0.05332Vt + 1015 . , . × 10-4 Vt3 + 17.372× 10-4 Vt2 - 01265 . Lj (Vt ) = -1071 Vt + 0.7358, Ly (Vt ) = 6.445× 10-3 Vt2 - 0.0418Vt + 0.5325.
(3.6)
245
y, deg.
y, deg.
40
8
20
4
0
50
100
t, sec
0
50
Fig. 4.1
100
t, sec
Fig. 4.2 y, deg.
da , deg. 0
40 –1 20
–2
–3 0
20
40
Fig. 4.3
60
t, sec
0
50
100
150
t, sec
Fig. 4.4
4. Modeling the Heading Stabilization System Using PAC. For estimating the efficiency of PAC, we modeling of the system for the above three flight conditions using SIMULINK software for drive signal y dr = 60 deg. The results of this modeling are presented in Figs. 4.1–4.4, where solid lines correspond to flight conditions 1, dashed lines correspond to flight conditions 2, dash-and-dot lines correspond to flight conditions 3. Figure 4.1 represents the heading angle; Fig. 4.2, the roll angle; and Fig. 4.3, aileron deviations. Figure 4.4 compares the transient processes upon change in the heading angle for flight conditions 3 in the presence (solid line) and absence (dashed line) of PAC. Here the values of the gain in the second case (without PAC) were computed by the above technique for flight conditions 1 as the most typical for UAV. As is seen from Fig. 4.1, with PAC, the transient processes slightly differ from one another upon considerable changes in the flight conditions, which is achieved by considerable variations in the controlling input (deviations of ailerons, Fig. 4.3) and the roll angle (Fig. 4.2). The absence of PAC increases at least twice the time of the transient processes with respect to the heading, which has an adverse effect on the maneuverability of the UAV at high altitudes (Fig. 4.4). Conclusions. 1. The use of PAC enables maintaining the flight-technical characteristics of the UAV at a high level with significant changes in flight conditions. When output feedback is used in the controller, PAC is implemented with minimum complication of the autopilot algorithm. 2. It is expedient to provide the synthesis of static output feedback for digital autopilot using the method of linear matrix inequalities for a family of controllers providing the g-optimal control under definite flight conditions, which are characterized by the true airspeed of UAV. 3. In order to simplify the digital autopilot software, it is convenient to use Lagrange interpolation polynomials for the approximation of the dependences of the static output feedback coefficients on the TAS. To minimize the degree of these polynomials, it is expedient to analyze the sensitivity of the output coordinate of the system to variations of the static output feedback coefficients. 4. The conducted modeling of the heading stabilization system of a definite UAV with PAC confirms the efficiency of the approach used in the present article. 246
Similar results were also obtained for control at flight altitude in the continuous case [20]. Moreover, considering the simplicity of software implementation of PAC, it is expedient to use it for control of other types of UAVs, such as quadrocopters [17].
REFERENCES 1. D. V. Balandin and M. M. Kogan, “Linear matrix inequalities in the problem of the robust Hinf-control by output,” Dokl. RAN, 396, No. 6, 759–761 (2004). 2. D. V. Balandin and M. M. Kogan, Control Law Synthesis Based on Linear Matrix Inequalities [in Russian], Fizmatlit, Moscow (2007). 3. N. I. Bronshein and K. A. Semendyaev, Handbook of Mathematics [in Russian], Nauka, Moscow (1964). 4. R. M. Yusupov, Handbook on Theory of Automated Control [in Russian], Nauka, Moscow (1987). 5. G. I. Bara and M. Boutayeb, “Static output feedback stabilization with Hinf-performance for linear discrete-time systems,” IEEE Trans. on Automatic Control, 50, No. 2, 250–255 (2005). 6. R. W. Beard and T. W. Randal, Small Unmanned Aircraft: Theory and Practice, Princeton University Press, Princeton (2012). 7. F. Blanchini, D. Casagrande, S. Miani, and U. Viaro, “Stable LPV realization of parametric transfer functions and its application to gain-scheduling control design,” IEEE Trans. on Automatic Control., 55, No. 10, 2271–2281 (2010). 8. S. Boyd S., L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, PA SIAM, Philadelphia (1994). 9. X. Chen, J. Song, and G. K. Cai, “Robust gain-scheduled autopilot design with LPV reference model for portable missile,” in: Proc. of IEEE Aerospace Conf., Montana (2012), pp. 461–467. 10. J. Gadewadikar, F. Lewis, and M. Abu-Khalaf, “Necessary and sufficient conditions for H-infinity static output feedback control,” J. Guid. Contr. Dynam., 29, 915–921 (2006). 11. S.F. Graebe and A. L. B. Allen, “Dynamic transfer among alternative controllers and its relation to antiwindup controller design,” IEEE Trans. on Control Systems Technology, 4, No. 1, 92–99 (1996). 12. A. S. Holtsov, R. M. Farhadi, V. L. Kortunov, and A. Mohammadi, “Comparison of the UAV adaptive control with the robust control based on mu-synthesis,” in: Proc. the 4th Int. Conf. on Methods and Systems of Navigation and Motion Control (MSNMC), NAU, Kiev (2016), pp. 18–21. 13. H. Kwakernaak, “Robust Control and Hinf-Optimization,” Automatica, 29, No. 2, 255–273 (1993). 14. A. S. Khoroshun, “Stability of the horizontal flight of an aircraft,” Int. Appl. Mech., 52, No. 1, 96–103 (2016). 15. V. B. Larin, “On navigation sensor error correction,” Int. Appl. Mech., 52, No. 1, 90–95 (2016). 16. V. B. Larin, A. Al-Lawama, and A. A. Tunik, “Exogenous disturbance compensation with static output feedback,” Appl. and Comput. Math., 3, No. 2, 75–83 (2004). 17. V. B. Larin and A. A. Tunik, “On the problem of the quadrotor control system synthesis,” Int. Appl. Mech., 53, No. 2, 185–194 (2017). 18. L. Masubuchi, S. Ishii, Y. Ohta, and M. Saeki, “Gain-scheduled control via switching of LTI controllers and state reset,” Asian J. of Control, 18, No. 5, 1619–1629 (2016). 19. D. McLean, Automatic Flight Control Systems, Prentice Hall, NY (1990). 20. O. I. Nadsadna, “Static output feedback design of robust gain scheduled control system,” Electronics and Control Systems, 1, 43–49 (2016). 21. L. Ravanbod and D. Noll, “Gain-scheduled two-loop autopilot for an aircraft,” J. Dynam. Syst., Measurement, and Control, 136, No. 4 (2014). 22. J. W. Rugh and S. J. Shamma, “Research on gain scheduling,” Automatica, 36, 1401–1425 (2000). 23. S. Sastry and M. Bodson, Adaptive Control. Stability, Convergence, and Robustness, Prentice Hall: Englewood Cliffs (1989). 24. B. I. Stevens and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, Inc., NY (1992). 25. V. L. Syrmos, C. Abdallah, and Dorato, “Static output feedback: a survey,” in: Proc. 33rd IEEE Conf. on Decision and Control, Orlando, Florida (1997), pp. 837–842. 26. Unmanned Dynamics LLC, AEROSIM BLOCKSET Version 1.2 User’s Guide, 2003. 247
