Evolutionary Computing Based Modular Control Design for Aircraft with Redundant Effectors

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Procedia Engineering

ProcediaProcedia Engineering 00 (2011) 000–000 Engineering 29 (2012) 110 – 117 www.elsevier.com/locate/procedia

2012 International Workshop on Information and Electronics Engineering (IWIEE)

Evolutionary Computing Based Modular Control Design for Aircraft with Redundant Effectors Arsalan H.Khan a*, Zhang Weiguoa, Zeashan.H.Khanb, Shi Jingpinga a

School of automation, Northwestern polytechnical university, Xi'an 710072, PR China b Comwave institute of Science and Technology, Islamabad 44000, Pakistan

Abstract In this paper, we present a genetic algorithm based modular reconfigurable control strategy for an over-actuated ADMIRE nonlinear aircraft system. The control law was based on multi-input multi-output (MIMO) linear quadratic regulator (LQR) strategy to produce virtual command signals. A pseudo-inverse based allocation method was used for effective distribution of commands produced by controller to redundant control surfaces in normal and fault condition. Actuator position limits can be considered for reconfiguration of virtual command signals. Simulation results show that satisfactory and improved performance even in fault scenario can be achieved quickly by using natural evolution based optimization technique in modular control design.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology. Keywords: Genetic algorithm; Control allocation; Linear quadratic regulator, Pseudo-inverse method; Non-linear aircraft model

1. Introduction Due to the advancement in aircraft technology and increasing demands of aircraft performance, safety and reliability, new improved multiple redundant control surfaces are introduced. In substance modern aircrafts have more actuating surfaces to control the same three rotational degrees of freedom (pitch, roll and yaw). So, the traditional approach of using optimal controller to shape the closed loop flight dynamics is no longer as persuasive as modular approach, in which separate control allocator is

* Corresponding author. Tel.:+86-18220503391. E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.12.678

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introduced for efficient control command distribution among the actuators. This modular approach has gain a lot of attention in practical safety-critical applications [1, 2, 3]. An optimized modular design approach was applied in this research work where an optimal controller is designed separately to the allocation algorithm (see Fig. 1). There is no specific methodology available for LQR design [8], except laboriously selection of state feedback gain matrix through trial and error. Here, we proposed fast approach based on genetic algorithm for determining best possible values of LQR controller for shaping the closed-loop dynamic response of the aircraft. With the addition of control allocation (CA) module in modular flight control loop, a designer has further control freedom by exploiting the redundant control surfaces in re-distribution of control commands between the remaining healthy actuators in case of aircraft actuators faults/failures without modifying the base-line controller parameters. This complex problem of desired moment commands distribution between a large number of control surfaces requires a fast, efficient and optimal solution. There are several optimization methods available for control allocation with varying performance indexes like computational requirement, allocation efficiency, constraints handling and design simplicity [4]. In this paper, an allocation strategy whose volume of attainable subset ( VΨ ) is optimized through genetic algorithm (GA) and compared with direct control allocation strategy proposed by Durham [5, 6] is used. The control allocation problem without consideration of optimized base-line controller and reallocation in faulty case for flight path control has been discussed in following work of Bordignon [7].The control surface deflection to angular acceleration relationship is treated as linear and presented in control effectiveness matrix estimated at trim conditions. Dominating performance of the modular design approach because of control allocation module is also presented in fault conditions. The objective of this paper is to design and implement a control law and a CA strategy using evolutionary algorithm for nonlinear model of an generic aircraft (ADMIRE).In view of simplicity, robustness and reasonable performance, we have used LQR as a base-line controller with control allocation approach based on pseudo-inverse method [4]. The paper is structured as follows: The baseline LQR control law is presented in subsection 2.1. Control allocation concept with selected control allocation technique for optimization is described in subsection 2.2. GA based control loop shaping is presented in Section 3. Brief description of generic aircraft model (ADMIRE) with Matlab/Simulink simulation development and results in Section 4. Lastly, conclusions with future improvements are given in Section 5. 2. Modular Control 2.1 Base-line Control Law (LQR) Consider the linearized aircraft dynamics at a trim condition in state-space form as = x& (t ) Ax(t ) + Bu u (t ) y (t ) = Cx(t )

(1)

Where A , is the n × n state matrix, Bu is the n × m input control matrix, C is the p × n output matrix, x ∈ R n is the system state vector; u ∈ R m is the control input vector and y ∈ R p is the system output vector to be controlled without excessive expense of control “effort”. Here, assume that all states are measurable and the system is full-state feedback system. In optimal control theory, Linear Quadratic Regulator (LQR) has to bring the state x to zero while minimizing the following objective function ∞

= J

∫ ( x Qx + u T

0

T

Ru )dt

(2)

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Fig. 1. Structure of modular flight control

Where, Q (positive semi-definite) and R (positive definite) are weighting matrices. The optimal control law that minimizes the cost function Eq. (2) is given as u = − Lx −1

(3)

Where, L = R B S and S is an unique positive semi-definite and symmetric matrix solution to T

AT S + SA − SBu R −1 BuT S + Q = 0

(4)

The above expression is the famous algebraic Riccatic equation (ARE).Now, the optimized closed-loop poles are the eigenvalues of A − Bu L . The best LQR controller performance can be achieved by proper selection of Q and R weighting matrices. There are several methods available for determining weighting matrices, with closed loop poles placement in complex left half plane. The new poles placement improves the stability index and minimizes the control effort. The selection of Q and R weighting matrices was done intuitively [9].Whereas, different poles locations, because of weighting matrices and gains correspond to varying system performance. Thus employing intelligent optimization techniques for searching Q and R is more impressive. 2. 2 Control Allocation (re-allocation) Control allocation is the process to determine the constrained control command vector u , in response of virtual command v . Control allocation is normally used for over-actuated systems, where the control devices are greater than the variables to be controlled (len(v ) < len(u )) . Let assume that rank ( Bu )= k ≤ m in Eq. (1). So, Bu can be factorized as Bu = Bv Be n× m

(5) n× k

k ×m

Where Bu ∈ R , Bv ∈ R and Be ∈ R are respectively, the control, virtual control and control effectiveness matrices. The alternate state equation form of Eq. (1) can be given as = x& (t ) Ax(t ) + Bv v(t )

(6)

v(t ) = Be u (t )

(7)

Where v ∈ R k is the total control effort produced by the actuators and commanded by the base-line controller. Here, we consider that the number of virtual control ( v ) equals the number of outputs to be controlled ( y ), ( k = p ). Normally, the actuator dynamics are much faster than the aircraft dynamics. So, the control allocation process has a linear relationship between constrained control command ( u ) and virtual command ( v ) in Eq. (5). The control u (t ) is limited by

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umin ≤ u ≤ umax

(8)

Where umin and umax are the lower and upper position deflection limits of physical actuators. The modular structure of flight control system (FCS) with control allocator is shown in Fig. 1. Typically, the vector v consists of, roll (Cl ) , pitch (Cm ) , and yaw (Cn ) moments and u represents the commanded actuator positions. 2.2.1Redistributed Pseudo-Inverse (RPI) Method Generalized inverse (GI) based solutions are very popular in control allocation. Where a control mixing matrix is used to satisfy Be P = I k , where I k is k × k identity matrix and P ∈ R m× k .The redistributed pseudo-inverse [7, 10] is an optimization base linear control allocation method which exploits the use of generalized inverse, where the cost function is: = u arg min

umin ≤ u ≤ umax

Wu (u − ud )

(9)

2

Subject to Beu = v and having the closed form solution: .

= u Fud + Gv, F= I − GBe , G = Wu −1 ( BeWu −1 )

†

(10)

Where ‘†’ is the pseudo-inverse operator. The RPI allocator efficiency depends on the optimized pseudoinverse matrix G , where Wu is the diagonal weighting matrix which allows the designer to prioritize between the virtual control commands. The procedure of the RPI can be described as follows. Initially, a pseudo inverse is computed which distribute the controls given in response of desired moments from base-line controller. If the control inputs exceed the respective position limits the pseudo-inverse solution is limited to its respective maximum or minimum value and removed from the optimization. Then, again the pseudo-inverse based control allocation is performed for remaining unsaturated controls to achieve the desired moments. Keep, this process continues until the desired response is achieved.

Subset of moments attainable using RPI Π 1.5 1

C

n

0.5 0 -0.5 -1 -1.5

6

Attaiable moments subset Φ

4 2

10

0 -2

0

-4 -6 Cl

-10 Cm

Fig. 2. Subset of moments attainable using optimized RPI approach

The redistributed pseudo-inverse method is simple and computationally efficient. The allocation efficiency of optimized RPI allocator is 46% .The subset of attainable moments of RPI is shown in Fig. 2. Where, it does not ensure full utilization of redundant control effectors capabilities.

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3. Optimization using GA Optimization algorithms are widely used in designing of close-loop control systems e.g. [15, 16]. For an appropriate design of closed loop flight control system, where we want to achieve the desired response output vector Yd = [yd ……yid]T, whereas the actual response of the system, is Y = [y1,………..yi]T. So, it is necessary to use some intelligent optimization technique for closed loop system performance improvement by varying Y as close to desired output Yd, through closed loop gain tuning and poles placement.The optimization problem is to minimize the difference between desired output vector Yd and real output Y. Considering the 2-norm of the difference vector, which sometimes referred to as minimum norm solution, we can relate the optimal index with the desired vector (Yd) as described: min Yd − Y

(11)

2

In modular flight control, one of the methods for improving control allocation efficiency for overactuated systems is to maximize the volume of attainable moment subset of allocation scheme (Π) as compare to the volume of attainable moment subset (AMS) Φ through optimization. Here, by searching the best possible generalized inverse solution improve the efficiency of redistributed pseudo-inverse based allocator. Optimization objective for control allocator also involves finding a vector of control variable in the AMS that produce moment of same magnitude and direction as that of desired moment.  

Best individuals (SOLUTION)

Yes D E F & S E L E C T I O N

P O P U L A T I O N G E N

O B J E C T I V E E V A L U

Termination criteria met?

No

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M U T A T I O N

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R E I N S E R T I O N

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C O M P E T I T I O N

Fig. 3 Multi-population GA algorithm execution (modified from [12])

Genetic algorithm is one of the most widely employed stochastic search and optimization technique for evolutionary computation which is a developing area of artificial intelligence. The GA starts with randomly initialized population of chromosomes and evolves towards the objective by utilizing evolution operators occurring in nature. The GA has been used in search, machine learning and optimal control [11]. Several advantages of GA includes simultaneous searching, large number of variables handling and providing multiple optimum solutions where traditional optimization techniques fail. In this paper we utilize the GA to find best weighting matrices for closed loop poles placement. The GA was introduced and developed by John Holland and proved by one of his students, David Goldberg, as a strong method for optimizing large complex control systems. GA is a searching technique based on the process of natural genetics, selection, recombination and mutation. GA operates on the population of chromosomes based on the principle of fitness to produce best possible solution and selects chromosomes for crossover and mutation. After definition of fitness function and selection of GA parameters, the algorithm proceeds as shown in Fig. 3. (1) Select an initial, random population of chromosomes (elements of diagonal weighting matrices) of specified size. (2) Evaluate these chromosomes for Continuous Algebraic Riccati Eq. (4) for controlling gain L.

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(3) Simulation is performed with fittest individual set of the population. If the termination criterion is met, then go to 9. (4) Reproduce next generation using probabilistic method. (5) Implement crossover operation on reproduced chromosomes. (6) Perform mutation operation with offspring evaluation. (7) Execute reinsertion and migration. (8) Repeat step 2 until best weighting matrices and control gain L matrix is achieved. (9) End. This intelligent and systematic approach of determining optimized Q and R matrices has great precision and advantage over the time consuming trial-error approach. Through this method we can achieve best possible closed loop response by placing poles of the system. The new gain matrix L increases the system stability to perturbations. Also, by employing the RPI allocator to efficiently distribute the input command vector among the control surfaces even in the case of partial or total failure of some actuating device improve overall system performance to faults/failures and make the whole system active fault tolerant. In this paper, we have used the Control System and GA toolboxes with modifications. The detailed description of the genetic algorithm based improved RPI method can be found in our previous publication [13]. 4. Simulation Results For evaluation of our purposed strategy we used generic aircraft model (ADMIRE) [14], developed by Swedish Defense Research Agency (FOI).It is a nonlinear, 6-DOF simulation model of a delta-canard configured, small single seated and single engine fighter aircraft with twelve control actuators. The simulation presented here shows a linear model trimmed at low speed flight condition of Mach 0.3 at an altitude of 3000m.The state vector x = [α β p q r ]T consist of α angle of attack (rad), β angle of sideslip (rad), p roll rate (rad/sec), q pitch rate (rad/sec) and r yaw rate (rad/sec). The controlled output vector is y = [α β p ]T .For controller design with control allocation strategy; we will use only seven control surfaces [δ lc , δ rc , δ roe , δ rie , δ lie , δ loe , δ r ]T . Following are the respective linearized model matrices: ⎡-0.6973 ⎢ ⎢ 0 A= ⎢ 0 ⎢ ⎢ 3.7422 ⎢⎣ 0

0.9765 0 ⎤ 0.0090 0 ⎥ −0.9816⎥ 0 -0.1680 0.1303 0.5268 ⎥ , Bu= -14.595 -1.3423 0 ⎥ 0 ⎥ -0.0053 0 −0.7045 0 −0.2992⎥⎦ 0.8291 -0.0889

⎡ 0.0003 ⎢−0.0060 ⎢ ⎢ 0.7984 ⎢ ⎢ 1.3841 ⎢⎣−0.3970

0.0003 −0.0508 −0.0813 0.0060 0.0032 0.0135 −0.7984 −4.5787 −3.9413 1.3841 −1.0906 −1.7433

−0.0813 −0.0508 0.0004 ⎤ −0.0135 −0.0032 0.0395 ⎥⎥ 3.9413 4.5787 2.6919 ⎥ ⎥ −1.7433 −1.0906 0.0046 ⎥ 0.3970 −0.2014 −0.4256 0.4256 0.2014 −1.6265⎦⎥

In this example, the actuators position limits are considered, and the approximate model with allocator can be given where: Bu = Bv Be and where Bv = [02×3 I 3×3 ] T , ⎡ 0.7984 − 0.7984 − 4.5787 = B e ⎢⎢ 1.3841 1.3841 − 1.0906 ⎢⎣ − 0.3970 0.3970 − 0.2014

− 3 .9 4 1 3 3 .9 4 1 3 4 .5 7 8 7 2 .6 9 1 9 ⎤ − 1 .7 4 3 3 − 1 .7 4 3 3 − 1 .0 9 0 6 0 .0 0 4 6 ⎥⎥ 0 .2 0 1 4 − 1 .6 2 6 5 ⎦⎥ − 0 .4 2 5 6 0 .4 2 5 6

The control limits used in the simulation were umin = {-55 - 55 - 30 - 30 - 30 - 30 - 30} (deg) and umax = {25 25 30 30 30 30 30} (deg) . The resulting virtual control for control allocator input v = Beu consist of pure moments in roll, pitch and yaw produced by the control effectors [1]. The superior performance of purposed GA based optimization as compare to manual selection of Q and R weighting matrices for closed loop feedback system is shown in Fig. 4 and parameters presented in Table 1.

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30 ref LQR1 LQR2 LQR_GA

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Fig. 4. Aircraft trajectories and control surfaces. (a) Reference and actual trajectories; (b) Control surface deflections δ Table 1. LQR controller parameters

q11

q22

q33

q44

q55

r11

r22

r33

LQR1 LQR2

5 0.01

5 0.01

10 0.01

10 0.02

10 0.02

1 1

1 1

1 1

LQR_GA

18.46

0.72

0.74

6.78

1.48

0.15

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3.50

15

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Fig. 5. Aircraft controls redistribution in fault. (a) Canard surface jammed at 1deg position; (b) Left elevon saturation

The presence of control allocation module in the suggested modular approach, actuator faults can easily be accommodated with modified control effectiveness matrix instead of modifying the base-line controller as shown in Fig. 5. Because of faults in either canards or elevons cause an overshoot in pitch variables, angle of attack (AoA) and pitch rate. But to the availability of redundancy in ADMIRE aircraft, pitch moment can be control by either the canard or elevons (left & right).We can see that in the event of faults or failures, healthy elevons can replace the damaged canard by redistribution of control effort to

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elevons as much as possible to achieve the desired pitch moment (see Fig. 5(a)) and left elevon saturation can be compensated by redistributing the lost control effect to the right elevons and canards (see Fig. 5(b)). 5. Conclusion A contribution for a GA based optimal strategy of control allocation with control law is proposed for the closed loop flight dynamics control. This strategy guarantees the optimal performance of controller with efficient distribution of the desired efforts between redundant control surfaces. Redistributed pseudoinverse method is presented with ADMIRE benchmark model. Reallocation of controlled command in actuator saturation fault condition is demonstrated through control effectiveness modification. Through adopted modular approach, actuator constraints can be considered as demonstrated in simulation results. Future works include more advance optimization based control allocation techniques (linear programming (LP) or quadratic programming (QP)), which will be considered with fault detection and diagnosis strategy for online reconfiguration of allocation scheme in the ADMIRE environment. References [1] O.Harkegard, Backstepping and Control Allocation with Applications to Flight Control. PhD thesis no. 820, Department of Electrical Engineering, Linkoping University, May 2003. [2] R.H. Shertzer, D. J. Zimpfer, and P.D. Brown. Control allocation for next generation of entry vehicles. In AIAA Guidance, Navigation, and Control Conference and Exhibit, Monterey, CA, Aug. 2002. [3] O. J. Sordalen, Optimal thrust allocation for marine vessels. Control Engineering Practice, 5(9): 1223-1231, 1997. [4] Marc Bodson, Evaluation of optimization methods for control allocation, In AIAA Guidance, Navigation, and Control Conference and Exhibit, Montreal, Canada, 6-9 August 2001. [5] W.C. Durham, Constrained control allocation. Journal of Guidance, Control, and Dynamics, 16(4): 717-725, July-Aug.1993. [6] W.C. Durham, Constrained control allocation: Three-moment problem. Journal of Guidance, Control, and Dynamics,17(2): 330-336, 1994. [7] K. A. Bordigon, Constrained control allocation for systems with redundant control effectors, Dissertation for the Doctoral Degree. Department of Aerospace and Ocean Engineering, Blackburn, Va. Virginia Polytechnic Inst. and State Univ., 1996. [8] F. L. Lewis and V. L. Syrmos, Optimal Control, 2nd Edition, John Wiley & Sons, Inc., 1995. [9] H. Ahmad, T. M. Young, D. Toal, and E. Omerdic, Design of control law and control allocation for B747-200 using a linear quadratic regulator and the active set method. Journal of Aerospace Engineering, 817-830, Nov 2009. [10] J.C. Virning and D. S. Bodden, Multivariable control allocation and control law conditioning when control effectors limit. In Proc. of AIAA Guidance, Navigation, and Control Conference, pages 572-582.Scottsdale, AZ, Aug 1994. [11] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine learning. Reading, MA: Addison-Wesley, 1989. [12] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, New York: Wiley, 2004. [13] Shi J P, Zhang W G, Li G W, et al. Research on allocation efficiency of redistributed pseudo inverse algorithm. Sci China Inf Sci, 53v, 271-277. [14] L. Forssell and U. Nilson, ADMIRE the aero-data model in a research environment version 4.0, model description.Technical report, FOI-R-1624-SE, FOI, Stockholm, Sweden, Dec. 2005. [15] Karam M. Elbayomy, Jiao Zongxia, Zhang Huaqing, PID controller optimization by GA and its performances on the electro-hydraulic servo control system. Chinese Journal of Aeronautics, 21v, 378-384, Aug 2008. [16] P. Wang, D. P. Kwok, Optimal design of PID process controllers based on genetic algorithms, Control Engineering Practice, 2v, 641-648, Aug1994.

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