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The unified technique for multimode flight control systems designing
The unified technique for multimode flight control systems designing V.G. Borisov * , G.N. Nachinkina * , B.V. Pavlov * , A.M. Shevchenko * * Trapeznikov Institute of Control Sciences, Russian Academy of Sciences (RAS)
65 Profsoyuznaya Str., Moscow, 117997, Russia (Fax +7495 3349340, e-mail: [email protected]) Abstract: The purpose of this paper is elaboration of uniform technology for carrying out of all stages of flight control system designing. Proprietary components of this technology are the energy approach to flight control and the special version of modal synthesis. The classical structure of an energy control system has been modified by insertion of a correction loop. The version of modal control has been developed for calculation of optimum regulator parameters in the whole flight envelope. Modal control at an incomplete control vector is formed as the weighed sum of controls. Weighting coefficients reflect a degree of influence of any control on each controlled variable. All developments are integrated into the computerized research stands. Modeling of the maneuverable aircraft with the synthesized flight control system has shown excellent characteristics in whole operational area. Keywords: flight control, energy approach, modal method, handling qualities, modeling.
INTRODUCTION The process of designing and preflight tryout of flight control systems of flying objects (FO) consists of following conditional stages: i. Analysis of static and dynamic properties of an object; ii. Synthesis of control algorithms for all possible operation modes; iii. Modeling tests and estimation of FO performance with the closed automatic control system; iv. System tests at the flight simulator with the operator in a control contour. Each stage was usually carried out by different teams often separated territorially, with different engineering tools. Such organizational management has resulted in unjustified economic expenses and long lead time of system engineering. Analytical studies of modern and prospective engineering technological level [Advanced…(2008)] have specified conservatism and backwardness of this level. Scientific achievements could hardly find application in current projects. In our paper the complex approach to flight control systems designing is developed. This approach assumes carrying out the greatest part of researches and tryouts by means of the uniform multipurpose engineering stand. Proprietary theoretical components of the submitted technology are the modified energy approach to motion control in space and the special version of modal synthesis of regulators with required quality. This work was supported in part by Russian Foundation for Basic Research, Project 09-08-00313-a
1. NEW ALGORITHMIC TOOLS 1.1 Modified Energy Control System There is a wide variety of flight control concepts. One nonconventional concept has been developed in [Kurdjukov et al. (1994), Borisov et al (1999)]. We call it “The energy approach to flying objects control” in our works. Its essence in short is as follows. The structure of traditional flight control systems can be represented by the generalized circuit (Fig. 1). ΔX
Xref
U
Controller
X
Plant
Fig.1
Control U is formed on the basis of error ΔX of a state variable vector X. The performance index is assigned in class Qx=Qx (U, X, ΔX). The nonconventional control concept (Fig. 2) is offered instead. Here a total motional energy E is a controllable variable. The performance index is also set at the form QE=QE(U, E, ΔE).
ΔX = f ∗ (ΔE )
Eref = f ( X ) Xref
f
ΔE
f*
Controller
U
Plant
E
Fig.2
The basic quantitative relation of the energy approach is established by the energy balance equation [Borisov et al. (1998), Kurdjukov et al. (1994)]
ΔH E = ΔH EEngine − ΔH EDrag − ΔH EWind
This equation is written in the form of specific energy deviations ΔH E(.) = ΔE (.) mg . Terms in the right side are normalized engine work, aerodynamic force work, and wind work, accordingly. Integral expressions for each term have been received. The left-hand side is the increment of specific energy: t2 ⎛ V& ⎞ ΔH E = ∫ V ⎜ θ + ⎟ dt g⎠ ⎝ t1 Obviously, the first term in the right side is the work of the propulsive force, which we call the specific work of the engine: t2
ΔH EEngine = ∫ V ⋅ Pnr cos α ⋅ dt . t1
The second term expresses the expenditure of energy for overcoming the harmful drag: t2
ΔH EDrag = ∫ V ⋅ Dnr ⋅ dt . t1
The third term describes the work of wind: t2
ΔH EWind = ∫ V ⋅ f w ⋅ dt . t1
Here V is the airspeed; Pnr is the normalised thrust; Dnr is the normalised drag force. W& Wy is called the wind factor; Wx and Wy are fw ≅ x − g V horizontal and vertical wind components. The energy balance equation reflects the interconnections of all energy sources and consumers in the system «aircraft — power-plant —environment». In the given paper baseline control laws are modified to correct essential object nonlinearities. The well-known PI-algorithms for engine and elevator control [Lambreghts (1983)] are also received from the energy balance equation:
⎛ V& h& ⎞ ⎛ ΔV& Δh& ⎞ ΔP = kP ⎜ + ⎟ + kI ∫ ⎜ + ⎟dt, mg V ⎠ ⎝g V⎠ ⎝ g ⎛ V& h& ⎞ ⎛ ΔV& Δh& ⎞ Δδ e = k P ⎜ − ⎟ + k I ∫ ⎜ − ⎟dt V ⎠ ⎝g V⎠ ⎝ g where ΔV& and Δh& are longitudinal acceleration and vertical velocity errors. ΔPnr =
The energy control system (ECS) has shown excellent handling qualities of transport aircrafts at landing approach in situation of strong atmospheric perturbation, engine failures, abrupt glide slopes, etc. [Pavlov et al. (2003)]. In all situations the structure and coefficients of the ECS remained fixed for aircrafts of different classes. Similar achievements of foreign researchers are also known [Akmeliawati R., Mareels I. (2002), Voth Ch., Uy-Loi Ly.(1991)]. These control laws are valid under the assumption of small angular movements and during steady flight. In fact, the FO characteristic are different from the balancing ones at trajectory evolutions. First of all it concerns a drag D. To compensate this change we have introduced the normalized corrective term ΔDnr = ΔD into the thrust mg control law: ΔPnrcor = ΔPnr + ΔDnr Additive correction was found as follows. It is known: D = CD qS , where CD is a drag coefficient, q = 0,5ρV 2 is a dynamic pressure, S is a wing planform area. In general CD depends on a complete state vector: CD = CD (V ,α ,h,t,α& ,P,δ e ) . Keeping the most essential dependence CD = CD (V ,α ) , let's express CD variations in terms of partial derivatives CVD and
CDα :
(
)
Δ D = C VD Δ V + C Dα Δ α qS
and then the modified thrust control law becomes Δ Pnrcor = Δ Pnr + ( C DV Δ V + C Dα Δ α ) qS mg .
nx =
V& g
θ =
Vy V
∫
ΔD = ( C ΔV + C Δα )qS / mg nr
V
α
D
D
Fig.3. The structure of modified energy control system.
There is another way of corrective adjustment. Since ΔDnr variation depends on disturbances both in airspeed and angle of attack, it is advisable to add a correction signal to the elevator through cross-channel with some coefficient kcor. Studies show that the portion of elevator correction must be approximately ≈50% of the total signal ΔDnr . Figure 3 depicts a generalized block diagram of the modified energy control system (MECS). The correction unit calculates additive signals for aerodynamics nonlinearity compensation. The advantages of the MECS were demonstrated during landing approach through the zone of severe wind disturbances. For this purpose the windshear that caused the crash of the L-1011 jet at Dallas airport (USA) in 1985 was simulated. The MECS counteracted catastrophic wind disturbances and confidently steered the aircraft to the runway. However, the system with fixed parameters could not provide acceptable control quality for maneuverable high-speed aircrafts over the whole flight envelope. Therefore, adjustment of ECS parameters was needed. In this paper the version of a modal synthesis method is developed for solving adjustable optimum coefficients. 1.2 Proprietary version of modal synthesis Modal methods became widespread in the field of designing of control systems meeting the specific quality and stability requirements. Methods of synthesis of single-loop control systems are the most developed and covered in the literature [Doll et al.(1997), Tutikov et al.(2004), Dylevskii et al. (2003)] In the field of multi-loop systems the papers [Ackermann (1985a), Akunov et al. (2003)] are well known. The problem solving is ambiguous in the most general case at incomplete control vector. In this paper the control weighting matrix is proposed for elimination of solution multiplicity. Quality and stability of the closed control system are defined by its eigenvalues. Suppose that the desirable set of eigenvalues is known a priori. The only required though insufficient condition is stability of the closed-loop system. The multivariable control system is considered as
x& ( t ) = Ax ( t ) + Bu ( t )
The desirable spectrum Λ∗ = {λ1∗ ,..., λn∗} of the closed-loop system matrix A+BK should not contain multiple eigenvalues. The desirable spectrum Λ ∗ is achieved by a modal method by constructing the required matrix K. Let V = {V1 ,...,Vn } be a set of eigenvectors of the closed-loop system matrix A+BK. The matrix K in (1) is not unique for m> 1 as the spectrum correction can be achieved by various components of a control vector ui or their linear combinations. The control as the sum of weighted modal controls is offered [Borisov et al.(1990)] n
u = ∑ qi Ki x . i =1
In other words the structure of matrix K becomes as follows n
K = ∑ qi Ki .
qi ∈ R m×1 , i = 1, n is the assigned weight vector of a λ ∗ K ∈ R1×n , i = 1, m is the required coefficient mode i ; i Here
vector of a mode
λi∗ which the additional conditions K jVi = 0 for i ≠ j
(3)
providing independence of mode control are imposed on. After implementing numerous matrix transformations the compact formula for calculation of the feedback coefficients matrix is received [Borisov et al. (2008)]: K = QV −1 ,
here, Q = (q1 K qn ) , Q ∈ R m × n is a weight vector matrix Elements qi of the matrix Q are assigned so that their relative values would reflect a degree of influence of any control on each of plant controllable coordinate. As can be seen this algorithm gives solution for feedback matrix in the explicit form per one computing cycle. The control vector is more convenient for writing partly through derivatives. In our publications control u has been found in form u = KP11 x& + KP12 , where matrixes P11 P12 are connected with −1
where A ∈ R n×n is the plant matrix, x ∈ R n ×1 is the state vector, B ∈ R n×m is the control matrix, u ∈ R m ×1 is the control vector. Let the pair of matrix (A B) be controllable. The problem is to find the control low of the form
⎡ P11 P12 ⎤ ⎡ A B ⎤ system matrixes as follows: ⎢ ⎥=⎢ ⎥ . Here ⎣ P21 P22 ⎦ ⎣C 0 ⎦ y = Cx. Generalized MECS structure with a modal correction loop is shown in fig.4. V&
∫
u = Kx that the closed-loop system
x& = ( A + BK ) x
(2)
i =1
⎡ I1_V ⎤ ⎢ I1_y ⎥ g⎦ ⎣ & ⎣⎡X⎦⎤ & ⎣⎡X⎦⎤
∫
(1)
has a desirable set of eigenvalues. Here K ∈ R required feedback matrix.
m ×n
is the
& ⎣⎡ K1_X ⎦⎤ & ⎣⎡ K2_X ⎦⎤
⎡ I2_V ⎤ ⎢ I2_y ⎥ g⎦ ⎣
h&
Fig.4. The structure of modified energy control system with modal correction.
⎡V ⎤ ⎢ ⎥ ⎢V y ⎥ X = ⎢ϑ ⎥ ⎢ ⎥ ⎢ω ⎥ ⎢y ⎥ ⎣⎢ g ⎦⎥
2. DESIGNING OF THE OPTIMUM FLIGHT CONTROL SYSTEM. The approval of our proper modal method has been carried out during designing the flight control system of the hypothetical high-speed maneuverable aircraft. The problem to provide equally high handling qualities over the whole operational area has been set. An aperiodic type reaction with duration no more than 30s was assumed as airspeed and height reference transient. Formally these system requirements were set by values of low-frequency roots.
20,0
15,0
Altitude H, km
The linearized model contained only three low-frequency
The feedback matrix in our case is K ∈ R 2×12 . To establish the range of feedback coefficients Kij their values at discrete set of flight modes were calculated over the whole operational area. The flight envelope and some specific trajectories are shown in fig. 5 in H − M coordinates. Meaningful coefficient manifolds have been reduced by excluding coefficients that did not need any adjustment or considering their smallness. For the remaining coefficients analytical approximations have been found. For example, the pairs of exact and approximated coefficients in thrust channel (i=1) and elevator channel (i=2) are shown in fig. 6. As can be seen, these dependences agree fairly well.
3
3. INTEGRATED COMPUTERIZED STAND The developed theoretical methods of synthesis have formed methodological base for designing of control systems for several types of planes.
10,0
2 5,0
1 0,0
roots. Values -0.5,-0.6, and -0.7 were assigned to them.
Mach number М 0,5
1,0
1,5
2,0
At the Russian Institute of control sciences a series of research stands has been created. The new methods have been included in its structure as major intellectual components.
2,5
Fig.5. Investigated flight envelope. In thrust channel K 15 ( H , M )
Kˆ 15 ( H , M ) = −0,0007
1 M
4
In elevator channel K 22 ( QM )
1 Kˆ 22 = 117,5497 QM
The most complete multipurpose seminatural stand contains a wide set of algorithmic and software aids for analysis and synthesis of high quality systems. The stand has a convenient friendly user-system interface. As a result an operator doesn’t need to be skilled in programming of theoretical methods. The stand equipment includes the personal computer, the monitor and models of real control handles. Peripheral interface with real actuators and real on-board computer are also provided. The short list of the functional modules used on each stage of control system creation is shown below.
Fig.6. Approximation of discrete functions by analytical ones.
Flying object research
Control algorithms synthesis
Automatic control modes modeling
Bench tests with pilots in the loop
Synthesis of basic control algorithms
Balancing in spatial maneuver
Construction of Bode plots and locus diagram
Modal method calculation of algorithm parameters
Modeling of critical modes
parameter optimization in whole flight envelope Modeling instrument flights and visual flights
Fig.7. The program modules of multifunctional stand.
4. MODELLING THE CONTROL SYSTEM WITH ADJUSTED COEFFICIENTS The mentioned hypothetical maneuverable aircraft model equipped with the designed adaptive control system has been tested in various flight modes. Test signals were simultaneous step commands on height and airspeed. On the plots in fig. 8-a,b the transients on height d_H and airspeed d_V in five points near to the second trajectory are shown. In fig. 8-c,d corresponding reactions of an elevator d_Dle and engine thrust d_P are represented. Flight
conditions in these points are resulted in the same figure. These records show that in the diversified flight conditions the aircraft carries out maneuvers practically equally. Only the controls vary. Actions of the system look very rational and economic. This system property is specified by the energy approach. And the property of behavior invariance with respect to aircraft parameters variation is provided by adjustment of control system coefficients. CONCLUSIONS The technique for designing of high-quality flight control systems has been developed. The basic structural concept is the energy approach. On its base the rational structure of the energy control system (ECS) has been received. Moreover, the modified energy control system (MECS) for aircrafts with essentially nonlinear aerodynamics has been developed. The MECS has shown excellent handling quality of transport aircrafts at landing approach in diversified conditions. However, for maneuverable high-speed aircrafts the system with constant coefficients could not provide acceptable control quality over the operational area. Therefore adjustment of the MECS parameters was required for its use over the whole flight envelope. The proprietary version of the modal method supplying the required performance of dynamic system has been developed. In case of multivariable systems with an incomplete control vector the required feedback matrix was ambiguous. For unique solution the control on each mode as the sum of available controls with weight coefficients is offered. Synthesis of the flight control system for a high-speed maneuverable aircraft has been carried out. Optimum feedback coefficients in discrete points of all operational area have been found. The coefficients adjustment circuit depending on flight conditions was introduced into the system structure. All new theoretical methods have been included in the structure of several research stands as major intellectual components. The most complete multipurpose seminatural stand contains a wide set of algorithmic and software aids for analysis and synthesis of high quality systems. The stand has a convenient friendly operator-system interface. Modeling of the ECS with adjusted parameters has shown that transients had been invariant in various flight conditions and had been practically identical at speed from 100 up to 500 m/s and height from 0 up to 10,000m. The developed technology was used in several aviation firms for creating control systems of liners.
Mode No
V (m/s) H (m) M
1
2
3
4
5
100
200
300
400
500
0
2500
5000
7500
10000
0.294
0.605
0.936
1.29
1.67
Fig. 8.Transitions records at various flight modes:
REFERENCES Ackermann,J (1985a). Multi-model approaches to robust control system design. // Multi-model approaches to robust control system design.- Lecture Notes in Control and Information Sciences. v.70.-. pp. 108-130.
Doll, C, Magni, J.F. and Gorrec, Y. Le. A Modal MultiModel Approach. // Robust Flight Control : A design challenge. - Lecture Notes in Control & Information Sciences. v. 224. -1997.
Ackermann,J. (1985b). Sampled-Data Control Systems. – Berlin, Germany: Springer-Verlag, 1985.
Dylevskii, A.V., Lozgachev ,G.I. (2003). Sintez lineinyh sistem upravleniya s zadannym harakteristicheskim mnogochlenom. // Izv. RAN. Teoriya i sistemy upravleniya. № 4. s. 17-20. In Russian
Advanced Guidance and Control—Operational and Safety Benefits. Final report DOT/FAA/AR-08/27 Air Traffic Organization Operations Planning Office of Aviation Research and Development Washington, DC 20591. June 2008. Akmeliawati R., Mareels I. (2002) Nonlinear energy-based control method for landing autopilot // Proc. 15th Triennial World Congr. of IFAC (b'02). Barcelona, Akunov, T.A., Ushakov, A.V. (2003). Sintez sistem garantirovannoi modal'noi stabil'nosti. // Izv. RAN. Teoriya i sistemy upravleniya. № 4. s. 9 16. In Russian Borisov, V.G., Diligenskii, S.N., Efremov, A.Yu. (1990) Synthesis of Invariant Control Systems Using Eigenstructures. Automation and Remote Control, vol.51, No.7, Part.1, July, 1990. Borisov, V.G., Nachinkina, G.N., Shevchenko, A.M. (1999) Energy approach to flight control. Automation and remote control. Vol.60, №6, pp.805-813 Borisov, V.G., Nachinkina, G.N., Shevchenko, A.M. (2008). Modal method of optimization of the energy flight control system of the flying objects.//Problemy upravleniya. №6, pp. 74-80. In Russian
Kurdjukov, A.P., Natchinkina G.N., Shevtchenko A.M. (1998). Energy Approach to Flight Control. AIAA Guidance Navigation and Control Conf., AIAA Paper 98-4211, Boston, MA, Lambregts, A.A.(1983). Vertical Flight Path and Speed Control Autopilot Design Using Total Energy Principles. AIAA Paper 83-2239CP. Pavlov, B.V., Shevchenko, A.M., Nachinkina ,G.N. (2003). Energy Approach and its Usage for Flight Control System Design. Actual Problems of Aviation and Aerospace, vol.8, 2(16), pp. 24-43. Tutikov, V.V., Tararykin, S.V., Varkov, E.A. Obespechenie staticheskoi tochnosti sistem modal'nogo upravleniya s polinomial'nymi regulyatorami. // Pribory i sistemy. 2004. № 2. s. 1-4. In Russian Voth Ch., Uy-Loi Ly. (1991). Design of a total energy control autopilot using constrained parameter optimization. J. of Guidance, Control, and Dynamics., Vol. 14, No.5 pp. 927-935.
65 Profsoyuznaya Str., Moscow, 117997, Russia (Fax +7495 3349340, e-mail: [email protected]) Abstract: The purpose of this paper is elaboration of uniform technology for carrying out of all stages of flight control system designing. Proprietary components of this technology are the energy approach to flight control and the special version of modal synthesis. The classical structure of an energy control system has been modified by insertion of a correction loop. The version of modal control has been developed for calculation of optimum regulator parameters in the whole flight envelope. Modal control at an incomplete control vector is formed as the weighed sum of controls. Weighting coefficients reflect a degree of influence of any control on each controlled variable. All developments are integrated into the computerized research stands. Modeling of the maneuverable aircraft with the synthesized flight control system has shown excellent characteristics in whole operational area. Keywords: flight control, energy approach, modal method, handling qualities, modeling.
INTRODUCTION The process of designing and preflight tryout of flight control systems of flying objects (FO) consists of following conditional stages: i. Analysis of static and dynamic properties of an object; ii. Synthesis of control algorithms for all possible operation modes; iii. Modeling tests and estimation of FO performance with the closed automatic control system; iv. System tests at the flight simulator with the operator in a control contour. Each stage was usually carried out by different teams often separated territorially, with different engineering tools. Such organizational management has resulted in unjustified economic expenses and long lead time of system engineering. Analytical studies of modern and prospective engineering technological level [Advanced…(2008)] have specified conservatism and backwardness of this level. Scientific achievements could hardly find application in current projects. In our paper the complex approach to flight control systems designing is developed. This approach assumes carrying out the greatest part of researches and tryouts by means of the uniform multipurpose engineering stand. Proprietary theoretical components of the submitted technology are the modified energy approach to motion control in space and the special version of modal synthesis of regulators with required quality. This work was supported in part by Russian Foundation for Basic Research, Project 09-08-00313-a
1. NEW ALGORITHMIC TOOLS 1.1 Modified Energy Control System There is a wide variety of flight control concepts. One nonconventional concept has been developed in [Kurdjukov et al. (1994), Borisov et al (1999)]. We call it “The energy approach to flying objects control” in our works. Its essence in short is as follows. The structure of traditional flight control systems can be represented by the generalized circuit (Fig. 1). ΔX
Xref
U
Controller
X
Plant
Fig.1
Control U is formed on the basis of error ΔX of a state variable vector X. The performance index is assigned in class Qx=Qx (U, X, ΔX). The nonconventional control concept (Fig. 2) is offered instead. Here a total motional energy E is a controllable variable. The performance index is also set at the form QE=QE(U, E, ΔE).
ΔX = f ∗ (ΔE )
Eref = f ( X ) Xref
f
ΔE
f*
Controller
U
Plant
E
Fig.2
The basic quantitative relation of the energy approach is established by the energy balance equation [Borisov et al. (1998), Kurdjukov et al. (1994)]
ΔH E = ΔH EEngine − ΔH EDrag − ΔH EWind
This equation is written in the form of specific energy deviations ΔH E(.) = ΔE (.) mg . Terms in the right side are normalized engine work, aerodynamic force work, and wind work, accordingly. Integral expressions for each term have been received. The left-hand side is the increment of specific energy: t2 ⎛ V& ⎞ ΔH E = ∫ V ⎜ θ + ⎟ dt g⎠ ⎝ t1 Obviously, the first term in the right side is the work of the propulsive force, which we call the specific work of the engine: t2
ΔH EEngine = ∫ V ⋅ Pnr cos α ⋅ dt . t1
The second term expresses the expenditure of energy for overcoming the harmful drag: t2
ΔH EDrag = ∫ V ⋅ Dnr ⋅ dt . t1
The third term describes the work of wind: t2
ΔH EWind = ∫ V ⋅ f w ⋅ dt . t1
Here V is the airspeed; Pnr is the normalised thrust; Dnr is the normalised drag force. W& Wy is called the wind factor; Wx and Wy are fw ≅ x − g V horizontal and vertical wind components. The energy balance equation reflects the interconnections of all energy sources and consumers in the system «aircraft — power-plant —environment». In the given paper baseline control laws are modified to correct essential object nonlinearities. The well-known PI-algorithms for engine and elevator control [Lambreghts (1983)] are also received from the energy balance equation:
⎛ V& h& ⎞ ⎛ ΔV& Δh& ⎞ ΔP = kP ⎜ + ⎟ + kI ∫ ⎜ + ⎟dt, mg V ⎠ ⎝g V⎠ ⎝ g ⎛ V& h& ⎞ ⎛ ΔV& Δh& ⎞ Δδ e = k P ⎜ − ⎟ + k I ∫ ⎜ − ⎟dt V ⎠ ⎝g V⎠ ⎝ g where ΔV& and Δh& are longitudinal acceleration and vertical velocity errors. ΔPnr =
The energy control system (ECS) has shown excellent handling qualities of transport aircrafts at landing approach in situation of strong atmospheric perturbation, engine failures, abrupt glide slopes, etc. [Pavlov et al. (2003)]. In all situations the structure and coefficients of the ECS remained fixed for aircrafts of different classes. Similar achievements of foreign researchers are also known [Akmeliawati R., Mareels I. (2002), Voth Ch., Uy-Loi Ly.(1991)]. These control laws are valid under the assumption of small angular movements and during steady flight. In fact, the FO characteristic are different from the balancing ones at trajectory evolutions. First of all it concerns a drag D. To compensate this change we have introduced the normalized corrective term ΔDnr = ΔD into the thrust mg control law: ΔPnrcor = ΔPnr + ΔDnr Additive correction was found as follows. It is known: D = CD qS , where CD is a drag coefficient, q = 0,5ρV 2 is a dynamic pressure, S is a wing planform area. In general CD depends on a complete state vector: CD = CD (V ,α ,h,t,α& ,P,δ e ) . Keeping the most essential dependence CD = CD (V ,α ) , let's express CD variations in terms of partial derivatives CVD and
CDα :
(
)
Δ D = C VD Δ V + C Dα Δ α qS
and then the modified thrust control law becomes Δ Pnrcor = Δ Pnr + ( C DV Δ V + C Dα Δ α ) qS mg .
nx =
V& g
θ =
Vy V
∫
ΔD = ( C ΔV + C Δα )qS / mg nr
V
α
D
D
Fig.3. The structure of modified energy control system.
There is another way of corrective adjustment. Since ΔDnr variation depends on disturbances both in airspeed and angle of attack, it is advisable to add a correction signal to the elevator through cross-channel with some coefficient kcor. Studies show that the portion of elevator correction must be approximately ≈50% of the total signal ΔDnr . Figure 3 depicts a generalized block diagram of the modified energy control system (MECS). The correction unit calculates additive signals for aerodynamics nonlinearity compensation. The advantages of the MECS were demonstrated during landing approach through the zone of severe wind disturbances. For this purpose the windshear that caused the crash of the L-1011 jet at Dallas airport (USA) in 1985 was simulated. The MECS counteracted catastrophic wind disturbances and confidently steered the aircraft to the runway. However, the system with fixed parameters could not provide acceptable control quality for maneuverable high-speed aircrafts over the whole flight envelope. Therefore, adjustment of ECS parameters was needed. In this paper the version of a modal synthesis method is developed for solving adjustable optimum coefficients. 1.2 Proprietary version of modal synthesis Modal methods became widespread in the field of designing of control systems meeting the specific quality and stability requirements. Methods of synthesis of single-loop control systems are the most developed and covered in the literature [Doll et al.(1997), Tutikov et al.(2004), Dylevskii et al. (2003)] In the field of multi-loop systems the papers [Ackermann (1985a), Akunov et al. (2003)] are well known. The problem solving is ambiguous in the most general case at incomplete control vector. In this paper the control weighting matrix is proposed for elimination of solution multiplicity. Quality and stability of the closed control system are defined by its eigenvalues. Suppose that the desirable set of eigenvalues is known a priori. The only required though insufficient condition is stability of the closed-loop system. The multivariable control system is considered as
x& ( t ) = Ax ( t ) + Bu ( t )
The desirable spectrum Λ∗ = {λ1∗ ,..., λn∗} of the closed-loop system matrix A+BK should not contain multiple eigenvalues. The desirable spectrum Λ ∗ is achieved by a modal method by constructing the required matrix K. Let V = {V1 ,...,Vn } be a set of eigenvectors of the closed-loop system matrix A+BK. The matrix K in (1) is not unique for m> 1 as the spectrum correction can be achieved by various components of a control vector ui or their linear combinations. The control as the sum of weighted modal controls is offered [Borisov et al.(1990)] n
u = ∑ qi Ki x . i =1
In other words the structure of matrix K becomes as follows n
K = ∑ qi Ki .
qi ∈ R m×1 , i = 1, n is the assigned weight vector of a λ ∗ K ∈ R1×n , i = 1, m is the required coefficient mode i ; i Here
vector of a mode
λi∗ which the additional conditions K jVi = 0 for i ≠ j
(3)
providing independence of mode control are imposed on. After implementing numerous matrix transformations the compact formula for calculation of the feedback coefficients matrix is received [Borisov et al. (2008)]: K = QV −1 ,
here, Q = (q1 K qn ) , Q ∈ R m × n is a weight vector matrix Elements qi of the matrix Q are assigned so that their relative values would reflect a degree of influence of any control on each of plant controllable coordinate. As can be seen this algorithm gives solution for feedback matrix in the explicit form per one computing cycle. The control vector is more convenient for writing partly through derivatives. In our publications control u has been found in form u = KP11 x& + KP12 , where matrixes P11 P12 are connected with −1
where A ∈ R n×n is the plant matrix, x ∈ R n ×1 is the state vector, B ∈ R n×m is the control matrix, u ∈ R m ×1 is the control vector. Let the pair of matrix (A B) be controllable. The problem is to find the control low of the form
⎡ P11 P12 ⎤ ⎡ A B ⎤ system matrixes as follows: ⎢ ⎥=⎢ ⎥ . Here ⎣ P21 P22 ⎦ ⎣C 0 ⎦ y = Cx. Generalized MECS structure with a modal correction loop is shown in fig.4. V&
∫
u = Kx that the closed-loop system
x& = ( A + BK ) x
(2)
i =1
⎡ I1_V ⎤ ⎢ I1_y ⎥ g⎦ ⎣ & ⎣⎡X⎦⎤ & ⎣⎡X⎦⎤
∫
(1)
has a desirable set of eigenvalues. Here K ∈ R required feedback matrix.
m ×n
is the
& ⎣⎡ K1_X ⎦⎤ & ⎣⎡ K2_X ⎦⎤
⎡ I2_V ⎤ ⎢ I2_y ⎥ g⎦ ⎣
h&
Fig.4. The structure of modified energy control system with modal correction.
⎡V ⎤ ⎢ ⎥ ⎢V y ⎥ X = ⎢ϑ ⎥ ⎢ ⎥ ⎢ω ⎥ ⎢y ⎥ ⎣⎢ g ⎦⎥
2. DESIGNING OF THE OPTIMUM FLIGHT CONTROL SYSTEM. The approval of our proper modal method has been carried out during designing the flight control system of the hypothetical high-speed maneuverable aircraft. The problem to provide equally high handling qualities over the whole operational area has been set. An aperiodic type reaction with duration no more than 30s was assumed as airspeed and height reference transient. Formally these system requirements were set by values of low-frequency roots.
20,0
15,0
Altitude H, km
The linearized model contained only three low-frequency
The feedback matrix in our case is K ∈ R 2×12 . To establish the range of feedback coefficients Kij their values at discrete set of flight modes were calculated over the whole operational area. The flight envelope and some specific trajectories are shown in fig. 5 in H − M coordinates. Meaningful coefficient manifolds have been reduced by excluding coefficients that did not need any adjustment or considering their smallness. For the remaining coefficients analytical approximations have been found. For example, the pairs of exact and approximated coefficients in thrust channel (i=1) and elevator channel (i=2) are shown in fig. 6. As can be seen, these dependences agree fairly well.
3
3. INTEGRATED COMPUTERIZED STAND The developed theoretical methods of synthesis have formed methodological base for designing of control systems for several types of planes.
10,0
2 5,0
1 0,0
roots. Values -0.5,-0.6, and -0.7 were assigned to them.
Mach number М 0,5
1,0
1,5
2,0
At the Russian Institute of control sciences a series of research stands has been created. The new methods have been included in its structure as major intellectual components.
2,5
Fig.5. Investigated flight envelope. In thrust channel K 15 ( H , M )
Kˆ 15 ( H , M ) = −0,0007
1 M
4
In elevator channel K 22 ( QM )
1 Kˆ 22 = 117,5497 QM
The most complete multipurpose seminatural stand contains a wide set of algorithmic and software aids for analysis and synthesis of high quality systems. The stand has a convenient friendly user-system interface. As a result an operator doesn’t need to be skilled in programming of theoretical methods. The stand equipment includes the personal computer, the monitor and models of real control handles. Peripheral interface with real actuators and real on-board computer are also provided. The short list of the functional modules used on each stage of control system creation is shown below.
Fig.6. Approximation of discrete functions by analytical ones.
Flying object research
Control algorithms synthesis
Automatic control modes modeling
Bench tests with pilots in the loop
Synthesis of basic control algorithms
Balancing in spatial maneuver
Construction of Bode plots and locus diagram
Modal method calculation of algorithm parameters
Modeling of critical modes
parameter optimization in whole flight envelope Modeling instrument flights and visual flights
Fig.7. The program modules of multifunctional stand.
4. MODELLING THE CONTROL SYSTEM WITH ADJUSTED COEFFICIENTS The mentioned hypothetical maneuverable aircraft model equipped with the designed adaptive control system has been tested in various flight modes. Test signals were simultaneous step commands on height and airspeed. On the plots in fig. 8-a,b the transients on height d_H and airspeed d_V in five points near to the second trajectory are shown. In fig. 8-c,d corresponding reactions of an elevator d_Dle and engine thrust d_P are represented. Flight
conditions in these points are resulted in the same figure. These records show that in the diversified flight conditions the aircraft carries out maneuvers practically equally. Only the controls vary. Actions of the system look very rational and economic. This system property is specified by the energy approach. And the property of behavior invariance with respect to aircraft parameters variation is provided by adjustment of control system coefficients. CONCLUSIONS The technique for designing of high-quality flight control systems has been developed. The basic structural concept is the energy approach. On its base the rational structure of the energy control system (ECS) has been received. Moreover, the modified energy control system (MECS) for aircrafts with essentially nonlinear aerodynamics has been developed. The MECS has shown excellent handling quality of transport aircrafts at landing approach in diversified conditions. However, for maneuverable high-speed aircrafts the system with constant coefficients could not provide acceptable control quality over the operational area. Therefore adjustment of the MECS parameters was required for its use over the whole flight envelope. The proprietary version of the modal method supplying the required performance of dynamic system has been developed. In case of multivariable systems with an incomplete control vector the required feedback matrix was ambiguous. For unique solution the control on each mode as the sum of available controls with weight coefficients is offered. Synthesis of the flight control system for a high-speed maneuverable aircraft has been carried out. Optimum feedback coefficients in discrete points of all operational area have been found. The coefficients adjustment circuit depending on flight conditions was introduced into the system structure. All new theoretical methods have been included in the structure of several research stands as major intellectual components. The most complete multipurpose seminatural stand contains a wide set of algorithmic and software aids for analysis and synthesis of high quality systems. The stand has a convenient friendly operator-system interface. Modeling of the ECS with adjusted parameters has shown that transients had been invariant in various flight conditions and had been practically identical at speed from 100 up to 500 m/s and height from 0 up to 10,000m. The developed technology was used in several aviation firms for creating control systems of liners.
Mode No
V (m/s) H (m) M
1
2
3
4
5
100
200
300
400
500
0
2500
5000
7500
10000
0.294
0.605
0.936
1.29
1.67
Fig. 8.Transitions records at various flight modes:
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Doll, C, Magni, J.F. and Gorrec, Y. Le. A Modal MultiModel Approach. // Robust Flight Control : A design challenge. - Lecture Notes in Control & Information Sciences. v. 224. -1997.
Ackermann,J. (1985b). Sampled-Data Control Systems. – Berlin, Germany: Springer-Verlag, 1985.
Dylevskii, A.V., Lozgachev ,G.I. (2003). Sintez lineinyh sistem upravleniya s zadannym harakteristicheskim mnogochlenom. // Izv. RAN. Teoriya i sistemy upravleniya. № 4. s. 17-20. In Russian
Advanced Guidance and Control—Operational and Safety Benefits. Final report DOT/FAA/AR-08/27 Air Traffic Organization Operations Planning Office of Aviation Research and Development Washington, DC 20591. June 2008. Akmeliawati R., Mareels I. (2002) Nonlinear energy-based control method for landing autopilot // Proc. 15th Triennial World Congr. of IFAC (b'02). Barcelona, Akunov, T.A., Ushakov, A.V. (2003). Sintez sistem garantirovannoi modal'noi stabil'nosti. // Izv. RAN. Teoriya i sistemy upravleniya. № 4. s. 9 16. In Russian Borisov, V.G., Diligenskii, S.N., Efremov, A.Yu. (1990) Synthesis of Invariant Control Systems Using Eigenstructures. Automation and Remote Control, vol.51, No.7, Part.1, July, 1990. Borisov, V.G., Nachinkina, G.N., Shevchenko, A.M. (1999) Energy approach to flight control. Automation and remote control. Vol.60, №6, pp.805-813 Borisov, V.G., Nachinkina, G.N., Shevchenko, A.M. (2008). Modal method of optimization of the energy flight control system of the flying objects.//Problemy upravleniya. №6, pp. 74-80. In Russian
Kurdjukov, A.P., Natchinkina G.N., Shevtchenko A.M. (1998). Energy Approach to Flight Control. AIAA Guidance Navigation and Control Conf., AIAA Paper 98-4211, Boston, MA, Lambregts, A.A.(1983). Vertical Flight Path and Speed Control Autopilot Design Using Total Energy Principles. AIAA Paper 83-2239CP. Pavlov, B.V., Shevchenko, A.M., Nachinkina ,G.N. (2003). Energy Approach and its Usage for Flight Control System Design. Actual Problems of Aviation and Aerospace, vol.8, 2(16), pp. 24-43. Tutikov, V.V., Tararykin, S.V., Varkov, E.A. Obespechenie staticheskoi tochnosti sistem modal'nogo upravleniya s polinomial'nymi regulyatorami. // Pribory i sistemy. 2004. № 2. s. 1-4. In Russian Voth Ch., Uy-Loi Ly. (1991). Design of a total energy control autopilot using constrained parameter optimization. J. of Guidance, Control, and Dynamics., Vol. 14, No.5 pp. 927-935.