Flight Control System Design Optimization via Genetic Algorithm Based on High-Gain Output Feedback

Transcription

1 Journal o Computer Science & Computational Mathematics, Volume, Issue, September 0 5 Flight Control System Design Optimization via Genetic Algorithm ased on High-Gain Output Feedback Mojtaba Vahedi, Mohammad Hadad Zari and ijan Moaveni Department o Electrical and computer Engineering, Islamic Azad University, Shahrood branch, Shahrood, Iran, Department o Electrical and robotic Engineering, Shahrood university o technology, Shahrood, Iran, School o Railway Engineering, Iran university o Science and Technology, Tehran, Iran, Corresponding addresses vahedi.mojtaba@yahoo.com, mhzari@shahroodut.ac.ir, b_moaveni@iust.ac.ir Abstract: In This paper, a high-gain output eedback control structure is applied in order to be optimized with GA or a MIMO nonlinear model o a light object. In control structure, a modiied GA algorithm or obtaining a suitable measurement matri in eedback loop is proposed which minimizes the interaction between the outputs. The proposed method has two major advantages in compare to other methods; irst o all, the proposed method is independent o system degree or system compleity and secondly, in this method some o unknown high-gain method parameters such as arbitrary diagonal matri, etc are discarded. Computer simuions are carried out or showing the perormance o the designed controller against common high-gain controller. eywords: multivariable systems; light control; tracking; high-gain output eedback; Genetic Algorithm.. Introduction PID controller has widely used as a classical dynamic controller or SISO system. Various PID parameter tuning methods eist or SISO systems (e.g. Ziegler-Nichols method such as C-H-R method [], and itamori s method []. Several researches have been conducted about PID control or MIMO systems which they usually restricted to stable and/or minimum phase system. Several researchers have been used classical control theory or designing and tuning PID control parameters in MIMO system [], [4], [5]. New research studies have ocused on the modern control theory which is more eective or analysis o MIMO system [6], [7], [8], [9], [0]. Lin et al. and Zheng et al., tried to determine PID parameter matrices by solving LMI ater one ormues PID control as static output eedback or the etended system [6], [7] ut, their method cannot satisy desired perormance in light control systems because the system outputs are naturally dynamic. A method by eigenvalue assignment has been proposed in [8]. In all eigenvalue based systems, or each state, a sensor is required. A complicated system such as aircrats has a signiicant number o states that in some o them sensors cannot used directly and should replaced with observers. Shimizu and. Tamura used dynamic high-gain output eedback but the reive degree o system should be less than or equal two [0]. In order to soten these limitations, GA has been used to optimize high-gain output eedback unknown parameters. In the high-gain output eedback, the closed loop system ehibits a distinctive asymptotic structure in which there are slow and ast modes. These properties are derived by using singular perturbation method to block diagonalize the close loop plant. The slow modes are asymptotically uncontrollable or unobservable. Thereore, the output response is dominated by the ast modes. This leads to track the command input by the output quickly. The design is dependent upon the irst markov parameter that is equal to matri product C. In light systems, C does not have maimal rank and is irregular and due to this property, the PI controller is augmented with an inner-loop that provides etra measurements or control purpose. The design o the control law or tracking system may include the desirable requirement i the outputs be decoupled, which minimizes the interaction between the outputs. In order to achieve this decoupling, each o the component slow and ast transer unctions must be diagonal. It may be possible to make these transer unctions diagonal by selecting the measurement matri M but, there is no guarantee that this is practical. Ridegly et al. [] represented a method or accomplishing the selection o the measurement matri but this method cannot calcue M in light or complicated systems. In order to soten this problem, a modiied GA algorithm or obtaining a suitable M matri is proposed, which minimizes the interaction between the outputs. The proposed method has two major advantages in compare with other methods; irst o all, the proposed method is independent o system degree or system compleity. Secondly, in this method some o unknown high-gain method parameters such as arbitrary diagonal matri (Σ), etc are discarded and designers do not need to estimate them.. High-gain output eedback control The state and output equations o a MIMO plant are: ( A( u( y( C( Where the dimension o A is n n, is n m, C is l n, and the rank o is m. Suppose the number o controlled outputs y( be equal to number o controls u(.in case o regular plant in which the matri product C has ull rank; the High-gain controller implements a proportional plus integral control law, but in case o irregular plants, the PI controller is augmented with an inner-loop that provides ()

2 6 Journal o Computer Science & Computational Mathematics, Volume, Issue, September 0 etra measurements through a measurement matri M or control purpose. (See Figure ) ecause o aircrat model irregularity in this section, irregular high-gain control will be illustd. A A A A4 r A r y C C C Which the sub matries are: (7) Figure. High-gain Output eedback control or irregular plants. Using rosenbrok algorithm, the state equation may be transormed to the partitioned orm o: A A 0 u A A y [C C ] () Where is m, is m m and has rank m, C is m m and has rank m, and the remaining elements in these equations have appropriate dimensions. This design method requires that the number o controlled outputs y( be equal to the number o controls u( and because o that l=m. A high-gain controller implements a proportional plus integral (PI) control law represented by u g{ e z } () Where, g is a scalar gain. The error vector between the constant command input r( and the output y( is e( = r( - y(. The integral o the error is the vector z( which satisies the ollowing equation: t z edt z r y (4) 0 Figure shows a new output described by W y M (5) Inserting the values obtained rom equation in to equation 5 yields the new output equation, equation 6. y the proper selection o the measurement matri M, the matri F in equation 6 will have ull rank. w C C M A A [( C MA )( C MA )] [ F F ] (6) The closed-loop tracking system o Figure is represented by equations through 6. Combining these equations, the composite closed-loop state and output equations have the respective orms o: A 0 F F A 0 A A A g A g F A A g F 4 I l [ g ] 0 C 0 C C C y using singular perturbation method, the resulting block diagonalization orm is: s As 0 s s r 0 A y C C s s Taking the limit as g yields the components: A A s A F A A F F 0 g F 0 g A F s C [C F C C F F ], C C s The ast transer unction determined rom equation0: l (8) (9) (0) ( ) C F [ I gf ] gf () Thus, coeicient matri that make Γ (λ) diagonal is obtained by choosing diagonal matri Σ as: 0 F, ( F ) 0 l (), () In High-gain approach usually is proportional to as (4) 0 The value o λ 0 that products satisactory perormance is determined with simuion o the system response. Equations 5 through 4 show that F=C+MA must be non singular. For an ideal system, C F - must be diagonal. Measurement matri M should be selected properly in order to satisy the above conditions. In the mean time, by increasing system gain, close-loop poles move toward

3 Journal o Computer Science & Computational Mathematics, Volume, Issue, September 0 7 transmission zeroes and M should be able to stabilize transmission zeroes.. Genetic algorithm GA is a methodology in evolutionary computation that is commonly used or selecting properly unknown parameters (such as Measurement matri M) in order to optimize system properties. The genetic algorithm transorms a popuion o individual objects, each with an associated itness value, into a new generation o the popuion. It is based on Darwinian principle o reproduction and survival o naturally occurring genetic operations such as crossover and mutation. The genetic algorithm attempts to ind an optimum (or bes solution to the problem by genetically breeding the popuion o individuals over a series o generations. It is very simple to implement and solves problems very quickly. In this work we develop a modiied GA to ind measurement matri M or obtaining the best parameters or light control system.. General methods We can divide GA methods by three main operators: selection, crossover and mutation... Selection - Selection initial samples Randomly create initial samples in search space. Create sequential initial samples in search space. - Selection scheme in algorithm The probability to choose a certain sample is proportional to its itness. Algorithm at last is permit to select N/ samples rom N initial samples. Algorithm chooses N/ samples with better itness and discards other samples. The probability to choose a certain sample is proportional to its itness but i the sample with best itness discards, algorithm replaces this sample with one o selected samples and discards it... Cross over One-point crossover: two strings cut at a randomly chosen position and swapping the two tails. One-point crossover is a simple method or GAs. N-point crossover: Instead o only one, N breaking points are chosen randomly. Every second section is swapped. Segmented crossover: Similar to N-point crossover with the dierence that the number o breaking points can vary. Uniorm crossover: For each position, it is decided randomly i the positions are swapped. Shule crossover: First a randomly chosen permutation is applied to the two parents and then N-point crossover is applied to the shuled parents... Mutation Inversion o single bits: With probability P, one randomly chosen bit is negated. itwise inversion: The whole string is inverted bit by bit with probability P Random selection: With probability replaced by a randomly chosen one. P, the string is Any combination o these operator types makes a GA method. In practice, a desired GA method rapidly and eectively optimizes comple, highly nonlinear, multidimensional systems. A desired GA method should be aster than other methods and more precise. Among these operators, deining mutation is more crucial than others because o its uncertain nature. Setting this probability higher than critical value, lead to high answer accuracy. The drawback is increasing the numbers o iterations. I this value is assumed smaller than critical value, answer accuracy will be poor and number o iterations will be low. There is a narrow band or this parameter that guarantee answer accuracy with low iteration. ecause o certain nature o other parameters in compare with mutation, they are not as important as mutation.. Proposed method In this method, mutation operator has been changed or improving GA parameters. Mutation only occurs in positions where bit value o all samples at that position is the same. It is obvious that the mutation in proposed method occurs only in deined bits while general methods apply mutation in all bits. This modiied mutation point selection lead to better system perormance. Assume one point crossover occur in group and group at deined positions. As can be seen rom Figure, ater mutation or samples and 4 the bit value in mutation point is negated. At the net step i child with mutation remain in selection process, the mutation is said to be good and algorithm continues without any change. On the other hand i these children do not remain in selection process, mutation is not appropriate. This indicates increasing the probability o appearing zeroed bit at this location in the inal answer. This means that probability that deined bit (at mutation poin be zero at inal answer is big. Facing this conditions lead us to inding a way that decrease the probability o mutation in deined bit. In the above eample the irst assumed P is 0.5 because the mutation occurred in hal o child. As a solution we can assume at net step this probability is 0.5 as a result deined bits in one o our children is negated. In this method we deine dierent probability o mutation or each position (In Figure eight sepa P ). This probability P is similar or all bits comprising the mutation point. Mutation point Crossover point Roulette wheel Ideal selection 00 0 group group 00 00

4 8 Journal o Computer Science & Computational Mathematics, Volume, Issue, September Figure. Parents and child in an eample o proposed algorithm. In order to evaluate the probability changes in a better way we deine P instead o P. P is the probability o changing bits in mutation point at irst mutation which is 0.5 in the above eample. P is assumed to be equal in all bit positions. Another major parameter in the proposed method is decrease which is deined De. De describes the o decreasing P between two successive mutation steps. In above eample P is assumed 0.5 and 0.5 or irst and second steps, respectively. De is deined by division o these two probabilities which is 0.5. Following this procedure, the third P computed or above eample is 0.5 (see 5). P 0.5 P P De 0.5 P P De 0.5 It is obvious that Crossover point 4 Mutation (5) De value would be bigger than and should be positive. Setting this parameter close to may result in low convergence (very high number o iterations). Setting this parameter to a big value leads to a general mutation system with low P. 4. Controller Design Procedure Crossover High-gain output eedback control is a linear controller. First o all, nonlinear aircrat model is linearized by Jacobian method over accumuion point and then, obtained linear model is used or controller design. The aircrat model should be included both itudinal and itudinal channels. The states used to deine itudinal channel are velocity (v), angle o attack (α), pitch angle (θ), pitch (q) and or itudinal channel are sideslip angle (β), roll angle (φ), roll (P), yaw angle (ψ), and yaw (r). The states velocity, pitch angle, yaw angle and sideslip are considered as outputs. Throttle setting (δ T ), elevator delection (δ e ), aileron (δ a ), and rudder delection (δ Y ) are selected as control inputs. The irst and second input- outputs are itudinal parameters and others are itudinal which satisies equality between number o input and output parameters. 4 The nonlinear state-equations o the light-system are as ollowed: w qu pv qcos( )cos( ) rmqs. czt V ( uu vv ww) / v t ( uw wu )/( u w ) ( vv vv )cos( sin( ) 4 p ( qsin( ) r cos( )) (6) cos( ) 5 qcos( ) r sin( ) 6 ( qsin( ) r cos( ))/cos( ) 7 p ( c p cr c4he) q qsb( c. clt c4cnt ) q c p c he) r c ( r p ) qscc. cmt 8 ( r ( c8 p cr c9he) q qsb( c4. clt c9. 9 cnt ) And in this model the parameters are: q 0.5 rhov t rho g., mass 06, rm / mass, b 7.4 s 400, he 0, c c c ,c 4.59e - 005,c c 5.89e , c ) ( e - 006,c 5 h) e-007,c , 0.97 (7) The purpose is to design a PI controller with ast and accu command tracking and to make the sideslip angle approimately zero. This system was in the orm o () or light condition o 0000 t altitude and V 500 t sec.the respective state matries or itudinal and itudinal channels ater linearization are: A , C, D A Decrease Rate

5 Journal o Computer Science & Computational Mathematics, Volume, Issue, September , C, D (8) And their eigenvalues are: Eigen values { i, i } (9) Eigen values {0,.0, 6.074, i, i} The eigenvalues show system instability in both o the subsystems. This instability is usual in high maneuverability light systems. The eect o interaction should be analyzed in MIMO systems such as aircrats. The transer unction values or two itudinal and itudinal subsystems at F=0Hz are: G _ (0) G _ (0) (0) This values show linear model is not diagonal dominance and a central multivariable approach should be used. ecause o system irregularity and compleity, in Highgain output eedback method the measurement matri is highly important and should be chosen properly. In the mean time, this matri cannot be obtained by classical methods thereore, in this paper the proposed GA is used or choosing the best measurement matri rom the total set o acceptable measurement matri. The parameters used in GA are as ollows: Variables are the elements o itudinal and itudinal measurement matri. Number o variables are 0 (si or itudinal and 4 or itudinal matri). All the variable spaces are the same and are equal to [-00 00]. Number o bits in each variable is selected as 0 so that the total length o each popuion is 00. The number o initial popuions is assumed to be 5. In the mean time, the popuions that can not satisy transmission zeros stability or make F =C +MA non ull rank are regened. The number o cross points is assumed to be 9 according to Alavi gharahbagh []. According to Alavi gharahbagh [] the parameters P Start =0.5 and De =. are applied to guarantee answer accuracy. Answer accuracy is a rational actor or breaking computation process. The best answer the worst answer Answer Accuracy 00 The best answer This value is assumed 0.0%. () The cost unction is the maimum settling time o system outputs which include velocity, pitch, yaw, sideslip angle that should be minimized. ased on equations, or computing the PI coeicients ( and matries), the diagonal matri Σ should be determined. In our solution, this matri is assumed as I (identity matri), according to equation ; is proportional to and at irst is equal to. y using GA with above conditions, the best measurement matries can be determined as ollows: M M () ased on equations and the matries are obtained: () At the end o simuion and when is determined, equality o and is not necessary and optimum. Thereore, based on some simuions, λ 0 =0.57, λ 0 =0.57 are assumed or obtaining in order to achieve a better perormance. With the above assumption (4) Finally, by attention to application details such as control eorts the values o gain matries are selected which are g =I, g =6I. The closed-loop eigenvalues or this system are : Eigenvalues Eigenvalues { 0.05, 0.05, 0.94, 6.66, 7, } { 0.05, 0.57, 0.57, 0.5,, 4, 4.66} These eigenvalues emphasize system stability. 5. Simuion result 5. Situations out o accumuion point (5) In this Simuion, a nonlinear model o a light object with 6 degrees o reedom is considered. This system outputs should be tracked velocity, pitch and yaw angle commands. In the mean time, sideslip angle must be approimately zero. System is simued over a wide range o commands around accumuion point. The close-loop system response or v=600, θ=, ψ=0 is illustd in Figure. The proposed system tracks input commands very good and sideslip angle is approimately zero. In Figure 4, the amplitude o control eort or v=600, θ=, ψ=0 is shown. I the system designed poorly, the amplitude o these signals would be very large and does not work properly or actuators, but in proposed system these values are in the acceptable range. For illustrating designed controller reliability, a time variant input or pitch and yaw angles is applied to system and the results are shown in Figures 5, 6.the results illust the good reliability o proposed system.

6 0 Journal o Computer Science & Computational Mathematics, Volume, Issue, September 0 In many practical maneuvers, the light object needs to roll and change yaw angle simultaneously. Figure 7 shows that the proposed system is compatible with this condition. 5. Comparison between designed PI controller and common PI high-gain controller In this section, proposed controller is compared with common high-gain controller. In the design o common highgain controller an acceptable measurement matri according to the work by [] is selected. This matri make F non singular, C F - diagonal, and transmission zeroes stable. These measurement matries are as ollows: M , M (6) ased on equations and the matries are obtained as ollows: (7) From the comparison results, it is obvious that designed controller based on GA is much better than common PI highgain controller which is shown in Figure Conclusion In this paper a central high-gain output eedback controller or a multivariable light object optimized by genetic algorithm. ecause o linearized plant irregularity, this controller required a suitable measurement matri. To ulill this requirement, GA is used. The designed controller was tested on a nonlinear 6 degrees o reedom light model. All simuion results showed system reliability and stability in practical situations. In addition, simuion results showed that the time response o designed controller based on GA is much better than common PI high-gain controller. Moreover, the proposed controller has a good perormance in a wide range o varieties over accumuion point in compare to other controllers. Figure. Nonlinear close-loop system response or v=600, θ=, ψ=0 Figure 4. Control eorts or v=600, θ=, ψ=0 Figure 5. Nonlinear system response or pitch command.

7 Journal o Computer Science & Computational Mathematics, Volume, Issue, September 0 Figure 6. Nonlinear system response or yaw command Figure 7. Simultaneous change o roll and yaw angles Reerences [] N.Suda, "PID Control," Asakura Pub, 99. [] T.itamori, "A Method o Control System Design ased upon Partial nowledge about Controlled Processes," SICE, Vol.5, No.4, pp , 979. [].J. A stro m,.h.johansson, and Q.G.Wang: "Design o Decoupled PID Controllers or MIMO Systems," Proceedings o the American Control Conerence Arlington, pp , 00. [4] W.. Ho and Wen Xu, "Multivariable PID Controller Design ased on the Direct Nyquist ArrayMethod," Proc. American Control Conerence, Pennsylvania, pp , 998. [5] Wang, Q.G., Zou,., Lee, T.H. and i, Q.: "Autotuning o multivariable PID controllers rom decentralized relay eedback," Automatica, Vol., No., pp. 9-0, 997. [6] C. Lin, Q-G. Wang and T. H. Lee, "An improvement on multivariable PID controller design via iterative LMI Approach," Automatica, Vol.40, pp , 004. [7] F. Zheng, Q-G. Wang and T. H. Lee, "On the Design o Multivariable PID Controllers via LMI Approach," Automatica, Vol. 8, pp , 00. [8]. Tamura, and. Shimizu, "Eigenvalue Assignment Method by PIDControl or MIMO system," ISCIE, Vol. 9, No. 5, pp. 9-0, 006. [9]. Shimizu, and. Tamura, "Epanded PID Control o MIMO system- Stabilization ased on Minimum Phase Property and High-Gain Feedback," SICE, Vol.4, No. 9, pp , 005. [0]. Shimizu, and. Tamura, "P.quasi-I.D Control or MIMO Systems," American Control ConerenceWestin Seattle Hotel, Seattle, Washington, USA, June -, 008. [] Ridegly. D., S.S.anda, and J.J.DAzzo, Decoupling o High-Gain Multivariable Tracking systems, AIAA J.Guidance, control, Dynamics, vol.8, pp , 985. [] Alavi gharahbagh. Abdorreza, and Abolghasemi. vahid. "A Novel Accu Genetic Algorithm or Multivariable Systems," World Applied Sciences Journal 5 (), pp. 7-4, Figure 8. Designed controller based on GA responses in compare with common PI high-gain controller.