Creative Writing

38 views

Model reduction for unstable LPV systems based on coprime factorizations and singular perturbation

Model reduction for unstable LPV systems based on coprime factorizations and singular perturbation
of 8

Please download to get full document.

View again

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Previous:

26

Next:

WF PO

Share
Tags
Transcript
  Model Reduction for Unstable LPV Systems Based onCoprime Factorizations and Singular Perturbation Widowati 1 ,  R. Bambang † ,  R. Saragih ‡ , and  S. M. Nababan  1 Mathematics DepartmentFMIPA Bandung Institute of Technology ∗ e-mail: wiwied@dns.math.itb.ac.id † Electrical Engineering DepartmentFTI Bandung Institute of Technology ∗ e-mail: briyanto@lskk-svr1.ee.itb.ac.id ‡ ,  Mathematics DepartmentFMIPA Bandung Institute of Technology ∗ e-mail:roberd@dns.math.itb.ac.id Abstract This paper proposes a reduction method of unstablelinear parameter varying systems. We generalize asingular perturbation method for linear time invari-ant systems by approximating the contractive right co-prime factorizations to reduce parameter varying reali-zations which are not quadratically stable. Based onthe reduced-order model the low-order parameter vary-ing controllers are designed. Effectiveness of the pro-posed model reduction is verified by applying it to LPVcontrol of an aircraft model. The closed-loop perfor-mance with the low-order LPV controller found by pro-posed method is compatred with balanced truncationmethod of Wood. 1 Introduction Model reduction via coprime factorizations of LinearTime Invariant (LTI) systems has been published byLi Li and Paganini [10]. They derived linear matrixinequality characterization of expansive and contrac-tive coprime factorizations that maintain structure,and use this to construct a method for structuredmodel reduction. Several authors [9, 12] have consid- ered coprime factorizations of linear time invariantand time varying systems. The generalization of Balanced Truncation (BT) to reduce the order of unstable Linear Parameter Varying (LPV) systems byapproximating contractive coprime factorizations hasbeen studied by some authors [6, 14]. In this paper we propose a method to reduce the unstable LPV systems 1 Permanent address : Department of Mathematics, FMIPA,Diponegoro University, Kamp. Tembalang, Semarang, Indonesia ∗ Jl. Ganesha 10, Bandung 40132, Indonesia by generalizing Singular Perturbation Approximation(SPA) for LTI systems [11] via Contractive RightCoprime Factorizations (CRCF). The LPV systemsare not assumed to be quadratically stable, but mustbe quadratically stabilizable and detectable [6, 14]. If  the unstable LPV system is stabilizable via a parame-ter varying state feedback, then we can constructa quadratically stable right coprime factorizationand reduce the coprime factor realization by using ageneralization of the SPA method.In comparison with method proposed in [14], thispaper uses SPA method, whereas in [14] BT methodwas used to reduce a high order-plant. To verify thevalidity of the proposed method it is applied to modelreduction for the aircraft model having 20th-order.The closed-loop performance of the low-order LPVcontrollers found by the proposed method is comparedwith that found by Wood method [14]. From thesimulation results the order of the model found byWood method can be reduced up to 11, whereas inthe proposed method the same model can be reducedas low as 6 while providing the same level perfor-mance of the closed-loop system with the full-orderLPV controller for all allowable parameter trajectories.The paper is organized as follows. Section 2describes parameter varying systems and some no-tations. Results concerning CRCF are presented inSection 3. In Section 4 we generalize the SPA methodfor LTI systems to reduce the order of unstable LPVsystems based on CRCF. Computational algorithmfor calculating reduced order model of unstable LPVsystems is presented in Section 5. In Section 6 thevalidity of the proposed method is demonstrated for  a lateral-directional motion of aircraft model. Finally,conclusion is drawn in Section 7. 2 Parameter Varying Systems For a compact subset P ⊂ R s , the parameter variationset  F  ρ  denotes the set of all piecewise continuous map-ping R (time) into P   with a finite number of discontinu-ity in any interval.  F  ρ  :=  { ρ ( t ) :  R → P  , ρ i min  ≤  ρ i  ≤ ρ i max , i  = 1 , 2 ,...,s } . A compact set  P ⊂ R s , alongwith continuous functions  A  :  R s → R n × n , B  :  R s →R n × m , C   :  R s → R  p × n , D  :  R s → R  p × m represent n th-order LPV systems,  G ρ , whose dynamics evolve as˙ x ( t ) =  A ( ρ ( t )) x ( t ) + B ( ρ ( t )) u ( t ) ,y ( t ) =  C  ( ρ ( t )) x ( t ) + D ( ρ ( t )) u ( t ) , ρ ( t )  ∈ F  ρ . (1)The LPV systems  G ρ  is quadratically stable [3, 4] if there exits a real positive-definite matrix  P   =  P  T  >  0such that A T  ( ρ ( t )) P   + PA ( ρ ( t ))  <  0 , ∀ ρ ( t )  ∈ F  ρ .  (2)The LPV systems  G ρ  is quadratically stabilizable if there exists a continuous matrix function  F  ( ρ ) :  R s →R m × n , and a constant positive-definite matrix  P  , suchthat  ∀ ρ  ∈ F  ρ , (for brevity, t is omitted)( A ( ρ )+ B ( ρ ) F  ( ρ )) T  P   + P  ( A ( ρ )+ B ( ρ ) F  ( ρ ))  <  0 .  (3)The LPV systems  G ρ  is quadratically detectable if there exists a continuous matrix function  L ( ρ ) :  R s →R n ×  p , and a constant positive-definite matrix  P  , suchthat  ∀ ρ  ∈ F  ρ ,( A ( ρ )+ L ( ρ ) C  ( ρ )) P   + P  ( A ( ρ )+ L ( ρ ) C  ( ρ )) T  <  0 .  (4)The induced  L 2  norm of a quadratically stable LPVsystems,  G ρ , with zero initial conditions, is defined as[3]  G ρ  i, 2  := sup ρ ( t ) ∈F  ρ sup u  =0 ,u ∈L 2  y  2  u  2 .  (5) 3 Contractive Right CoprimeFactorizations In this section, we discuss contractive right coprimefactorizations of LPV systems. It is the generaliza-tion of contractive right coprime factorizations of LTIsystems. The characteristics of the Contractive RightCoprime Factorization (CRCF) of an LPV systems isdefined as follows. Definition 3.1  [14]  Let   S  F   denotes the ring of all causal, quadratically stable, finite-dimensional LPV systems defined on the underlying feasible parameter set   F  ρ . Denote by   S  − F   the elements in   S  F   that have causal inverses. Let   N  ρ  ∈ S  F   and   M  ρ  ∈ S  − F   have the same number of columns. The ordered pair   [ N  ρ , M  ρ ] represents a contractive right coprime factorization of  G ρ  over   S  F   if 1.  G ρ  =  N  ρ M  − 1 ρ  ;2. there exist   U  ρ , V   ρ  ∈ S  F   ∋  U  ρ N  ρ  + V   ρ M  ρ  =  I  ;3.  [ N  T ρ  , M  T ρ  ] T  is a contractive in the following sense  sup ρ ∈F  ρ sup { u ∈ L +2  :  u  2 ≤ 1 }  N  ρ M  ρ  u  ≤  1 .  (6)Now define the Contractive Right Graph Symbol(CRGS)  G  ρ  :  L +2  →  L +2  ⊗  L +2  of LPV systems  G ρ ,as  G  ρ  :=  N  ρ M  ρ  ,  where [ N  ρ , M  ρ ] is CRCF of   G ρ . Theabove definition indicates that  G  ρ  generates the set of all stable input-output pairs of the LPV systems  G ρ by allowing  G  ρ  to act on the whole of   L +2  . Theorem 3.1  [14]  Let   G ρ  have a continuous,quadratically stabilizable, quadratically detectable realizations, then CRGS of   G ρ  is given by  G  ρ  :=  A ( ρ ) + B ( ρ ) F  ( ρ )  B ( ρ ) S  − 1 / 2 ( ρ ) C  ( ρ ) + D ( ρ ) F  ( ρ )  D ( ρ ) S  − 1 / 2 ( ρ ) F  ( ρ )  S  − 1 / 2 ( ρ )  ,  (7) where   F  ( ρ ) =  − S  − 1 ( ρ )( B T  ( ρ ) X  + D T  ( ρ ) C  ( ρ )) , S  ( ρ ) = I   +  D T  ( ρ ) D ( ρ ) , R ( ρ ) =  I   +  D ( ρ ) D T  ( ρ ) , dan   X   = X  T  >  0  is a constant solution of the Generalized Con-trol Riccati Inequality (GCRI) ( A ( ρ ) − B ( ρ ) S  − 1 ( ρ ) D T  ( ρ ) C  ( ρ )) T  X  + X  ( A ( ρ ) − B ( ρ ) S  − 1 ( ρ ) D T  ( ρ ) C  ( ρ )) − XB ( ρ ) S  − 1 ( ρ ) B T  ( ρ ) X   + C  T  ( ρ ) R − 1 ( ρ ) C  ( ρ )  <  0 , ∀ ρ  ∈ F  ρ . (8)Consider Generalized Filtering Riccati Inequality(GFRI)( A ( ρ ) − B ( ρ ) D T  ( ρ ) R − 1 ( ρ ) C  ( ρ )) Y  + Y  ( A ( ρ ) − B ( ρ ) D T  ( ρ ) R − 1 ( ρ ) C  ( ρ )) T  − YC  T  ( ρ ) R − 1 ( ρ ) C  ( ρ ) Y   + B ( ρ ) S  − 1 ( ρ ) B T  ( ρ )  <  0 , ∀ ρ  ∈ F  ρ . (9)The connection between the generalized Gramians of CRGS and solutions of Riccati inequality is given inthe following result.  Lemma 3.1  [14]  Let   G ρ  have a continuous, quadrati-cally stabilizable, quadratically detectable realizations,and let the ordered pair   [ N  ρ , M  ρ ]  represents the CRCF of   G ρ ,  X   =  X  T  >  0  and   Y   =  Y  T  >  0  solve the GCRI and GFRI, then  Q  =  X, P   = ( I   + YX  ) − 1 Y   (10) are generalized observability and controllability Grami-ans for   G  ρ . By Schur complement and changing variable ¯ X   =  X  − 1 and ¯ Y   =  Y  − 1 , the GCRI and GFRI are equivalent tothe following LMIs.  ˜ A ( ρ ) T  ¯ Y   + ¯ Y   ˜ A ( ρ ) − C  T  ( ρ ) R − 1 ( ρ ) C  ( ρ ) ¯ YB ( ρ ) B T  ( ρ )¯ Y   − S  ( ρ )   <  0 , (11)  ˘ A ( ρ ) ¯ X   + ¯ X   ˘ A T  ( ρ ) − B ( ρ ) S  − 1 ( ρ ) B T  ( ρ ) ¯ XC  T  ( ρ ) C  ( ρ ) ¯ X   − R ( ρ )   <  0 , (12) where ˜ A ( ρ ) =  A ( ρ )  −  B ( ρ ) S  − 1 ( ρ ) D T  ( ρ ) C  ( ρ ) and˘ A ( ρ ) =  A ( ρ ) − B ( ρ ) D T  ( ρ ) R − 1 ( ρ ) C  ( ρ ).The stabilizing solutions of GCRI (8) dan GFRI(9) can be obtained by taking the inversion of the fea-sible solution of the two above LMIs. From Theorem3.1, (12) guarantees that the parameter varying statefeedback  F  ( ρ ) =  − S  − 1 ( ρ )( B T  ( ρ ) X   +  D T  ( ρ ) C  ( ρ ))will make  G  ρ  contractive, hence we can construct aquadratically stable right coprime factorization. 4 CRCF Reduction of Unstable LPVsystems In this section, we propose results regarding thegeneralization of singular perturbation method forLTI systems to reduce the order of unstable parametervarying systems by approximating contractive rightcoprime factorizations. Consider the CRCF of the n th-order LPV systems  G  ρ  in equation (7). By usinga balancing state transformation matrix we obtainthe transformed controllability and observabilityGramians˜ P   = ˜ Q  = Σ =  diag (Σ 1 , Σ 2 ), with ˜ P   = T  − 1 PT  − T  , ˜ Q  =  T  T  QT,  Σ 1  =  diag ( σ 1 , ···  ,σ r ),Σ 2  =  diag ( σ r +1 , ···  ,σ n ),  σ r  > σ r +1 , and  σ j  =   λ j ( PQ ) , σ j  ≥  σ j +1 , j  = 1 , 2 , ···  ,r,r  + 1 , ···  ,n .A balanced parameter varying CRCF of   G ρ  canbe expressed by G  ρ  :=  ¯ A ( ρ ) + ¯ B ( ρ ) ¯ F  ( ρ ) ¯ B ( ρ ) S  − 1 / 2 ( ρ )¯ C  ( ρ ) + D ( ρ ) ¯ F  ( ρ )  D ( ρ ) S  − 1 / 2 ( ρ )¯ F  ( ρ )  S  − 1 / 2 ( ρ )  ,  (13)where¯ A ( ρ ) =  T  − 1 A ( ρ ) T,  ¯ B ( ρ ) =  T  − 1 B ( ρ ) ,  ¯ C  ( ρ ) = C  ( ρ ) T,  ¯ F  ( ρ ) =  − S  − 1 ( ρ )( ¯ B T  ( ρ )Σ + D T  ( ρ ) ¯ C  ( ρ )) . Partition the balanced parameter varying CRCFconformably with Σ =  diag (Σ 1 , Σ 2 ) is described asfollows G ρ  :=  ¯ A 11 ( ρ ) + ¯ B 1 ( ρ ) ¯ F  1 ( ρ ) ¯ A 12 ( ρ ) + ¯ B 1 ( ρ ) ¯ F  2 ( ρ )¯ A 21 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  1 ( ρ ) ¯ A 22 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  2 ( ρ )¯ C  1 ( ρ ) + D ( ρ ) ¯ F  1 ( ρ ) ¯ C  2 ( ρ ) + D ( ρ ) ¯ F  2 ( ρ )¯ F  1 ( ρ ) ¯ F  2 ( ρ )¯ B 1 ( ρ ) S  − 1 / 2 ( ρ )¯ B 2 ( ρ ) S  − 1 / 2 ( ρ ) D ( ρ ) S  − 1 / 2 ( ρ ) S  − 1 / 2 ( ρ )  (14)with ¯ A 11  ∈ R r × r ,  ¯ A 12  ∈ R r × ( n − r ) ,  ¯ A 21  ∈R ( n − r ) × r ,  ¯ A 22  ∈ R ( n − r ) × ( n − r ) ,  ¯ B 1  ∈ R r × m ,  ¯ B 2  ∈R ( n − r ) × m ,  ¯ C  1  ∈ R  p × r ,  ¯ C  2  ∈ R  p × ( n − r ) ,  ¯ F  1  ∈R m × r ,  ¯ F  2  ∈ R m × ( n − r ) .By using the concept of the SPA method[11], weset to zero the derivative of all states correspondingto Σ 2 . Furthermore, the generalized SPA method canbe applied to approximate the realization (14).ˆ G  ρ r  :=  N  ρ r M  ρ r   =  A s ( ρ )  B s ( ρ ) C  ns ( ρ )  D ns ( ρ ) C  ms ( ρ )  D ms ( ρ )  ,  (15)where A s ( ρ ) = ¯ A 11 ( ρ ) + ¯ B 1 ( ρ ) ¯ F  1 ( ρ ) − ( ¯ A 12 ( ρ ) + ¯ B 1 ( ρ ) ¯ F  2 ( ρ )) × ( ¯ A 22 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  2 ( ρ )) − 1 ( ¯ A 21 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  1 ( ρ )) ,B s ( ρ ) = ¯ B 1 ( ρ ) S  − 1 / 2 ( ρ ) − ( ¯ A 12 ( ρ ) + ¯ B 1 ( ρ ) ¯ F  2 ( ρ )) × ( ¯ A 22 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  2 ( ρ )) − 1 ( ¯ B 2 ( ρ ) S  − 1 / 2 ( ρ )) ,C  ns ( ρ ) = ¯ C  1 ( ρ ) + D ( ρ ) ¯ F  1 ( ρ ) − ( ¯ C  2 ( ρ ) + D ( ρ ) ¯ F  2 ( ρ )) × ( ¯ A 22 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  2 ( ρ )) − 1 ( ¯ A 21 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  1 ( ρ )) ,C  ms ( ρ ) = ¯ F  1 ( ρ ) −  ¯ F  2 ( ρ )( ¯ A 22 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  2 ( ρ )) − 1 × ( ¯ A 21 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  1 ( ρ )) ,D ns ( ρ ) =  D ( ρ ) S  − 1 / 2 ( ρ ) − ( ¯ C  2 ( ρ ) + D ( ρ ) ¯ F  2 ( ρ )) × ( ¯ A 22 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  2 ( ρ )) − 1 ¯ B 2 ( ρ ) S  − 1 / 2 ( ρ )) ,D ms ( ρ ) =  S  − 1 / 2 ( ρ ) −  ¯ F  2 ( ρ )( ¯ A 22 ( ρ ) + ¯ B 2 ( ρ ) ¯ F  2 ( ρ )) − 1 × ( ¯ B 2 ( ρ ) S  − 1 / 2 ( ρ )) , by assuming ( ¯ A 22 ( ρ )+ ¯ B 2 ( ρ ) ¯ F  2 ( ρ )) invertible  ∀ ρ  ∈ F  ρ .  We obtain the reduced-order model,  r th-order,as follows.ˆ G ρ r  :=  N  ρ r M  − 1 ρ r =   A s ( ρ ) − B s ( ρ )D − 1 ms ( ρ )C ms ( ρ )  B s ( ρ )D − 1 ms ( ρ )C ns ( ρ ) − D ns ( ρ )D − 1 ms ( ρ )C ms ( ρ )  D ns ( ρ )D − 1 ms ( ρ )  . (16) 5 Computational Algorithm Based on the preceding discussions, the procedure tocompute the reduced-order model of unstable LPV sys-tems via parameter varying CRCF is described as fol-lows.1. Find  X   =  X  T  >  0 and  Y   =  Y  T  >  0 satisfyingLMIs (11) and (12).2. Set a controllability and an observability Grami-ans  Q  =  X, P   = ( I   + YX  ) − 1 Y. 3. Compute F  ( ρ ) =  − S  − 1 ( ρ )( B T  ( ρ ) X   + D T  ( ρ ) C  ( ρ )) , ∀ ρ ( t )  ∈ F  ρ .4. Construct a parameter varying CRCF, G ρ  =  N  ρ M  − 1 ρ  , with the form (7).5. Use a similarity transformation to find a balan-ced CRCF realization and partition the balancedCRCF corresponding to Σ =  diag (Σ 1 , Σ 2 ).6. Apply the generalization of the SPA method toobtain ˆ G  ρ  =  N  ρ r M  ρ r  .7. Form the reduced-order model ˆ G ρ  =  N  ρ r M  − 1 ρ r , r th-order,  r < n,  with state space realization inequation (16).The constrains given by LMIs (11) and (12) areparameter dependent i.e., there is an infinite set of LMIs, one for every value of the parameter. Thesetypes of feasibility problems are known to be hard tosolve. To be able to solve these LMIs without gridingthe parameter space [5], we assume that system matri-ces associated with the plant have affine dependence onthe parameters and that the parameters are boundedin a hypercube. Hence, it is equivalent to check LMIs inthe  L  = 2 s corners of this hypercube, where the systemmatrices are convex combinations of the vertices ma-trices, for example, ˜ A ( ρ ) = L  i =1 ˜ A i α i , L  i =1 α i  = 1 , α i  ∈ [0 ,  1]. The convex feasibility problem on LMI formcan be solved numerically using LMI Control toolboxfor MATLAB [8]. 6 Simulation Results In this section, the theoretical results in previoussection are used to reduce unstable LPV lateral-directional dynamic of the aircraft [2]. The Aircraftprimary control surface consists of aileron and rudderfor lateral-directional motion and elevator for longi-tudinal motion. Secondary control surface consistsof spoiler and flaps. Only lateral-directional dyna-mics are considered in this paper, the longitudinaldynamics are assumed to remain at equilibrium.Lateral-directional dynamic of the aircraft can beexpressed as the LPV system with the state-spacemodel as follows.  ˙ v ˙  p ˙ r ˙ φ ˙ ψ  =  Y  ′ v ( V,f  )  Y  ′ p ( V,f  )  Y  ′ r ( V,f  )  gcosθ  0 L ′ v ( V,f  )  L ′ p ( V,f  )  L ′ r ( V,f  ) 0 0 N  ′ v ( V,f  )  N  ′ p ( V,f  )  N  ′ r ( V,f  ) 0 00  I Tanθ  0 00 0  Secθ  0 0  v prφψ  +  Y  ′ da ( V,f  )  Y  ′ dr ( V,f  ) L ′ da ( V,f  )  L ′ dr ( V,f  ) N  ′ da ( V,f  )  N  ′ dr ( V,f  )0 00 0  δ  A δ  R  ,  β  prφψa y  =  57 . 3 /V   0 0 0 00 57 . 3 0 0 00 0 57 . 3 0 00 0 0 57 . 3 00 0 0 0 57 . 3 Y  ′′ v  ( V,f  ) 0 0 0 0  v prφψ  , where Y  ′ v  =  ̺u e S  2 m  C  y β ( V,f  ) , Y  ′  p  =  ̺u e S  4 m  C  y p ( V,f  ) ,L ′  p  =  L p I  xxs + I  xzs ∗ N  p ( V,f  ) , L ′ r  =  L r I  xxs + I  xzs ∗ N  r ( V,f  ) ,L v  =  ̺u 2 e Sb 2 I  xx C  l β ( V,f  ) , N  v  =  ̺u 2 e Sb 2 I  zz C  n β ( V,f  ) ,N  ′ v  =  N  v I  zzs + I  xzs ∗ L v ( V,f  ) , N  ′  p  =  N  p I  zzs + I  xzs ∗ L p ( V,f  ) ,Y  ′ dr  =  Y  dr m  ( V,f  ) , L da  =  L da I  xxs + I  xzs ∗ N  da ( V,f  ) ,N  ′ dr  =  N  dr I  zzs + I  xzs ∗ L dr ( V,f  ) , Y  ′′ v  =  Y  v ∗ V mg  ( V,f  ) ,Y  ′ r  =  ̺u e S  4 m  C  y r ( V,f  ) , L ′ v  =  L v I  xxs + I  xzs ∗ N  v ( V,f  ) ,L  p  =  ̺u e Sb 2 4 I  xx C  l p ( V,f  ) , L r  =  ̺u e Sb 2 4 I  xx C  l r ( V,f  ) ,N   p  =  ̺u e Sb 2 4 I  zz C  n p ( V,f  ) , N  r  =  ̺u e Sb 2 4 I  zz C  n r ( V,f  ) ,N  ′ r  =  N  r I  zzs + I  xzs ∗ L r ( V,f  ) , Y  ′ da  =  Y  da m  ( V,f  ) ,L dr  =  L dr I  xxs + I  xzs ∗ N  dr ( V,f  ) ,N  ′ da  =  N  da I  zzs + I  xzs ∗ L da ( V,f  ) , inwhich state vector consists of lateral velocity( v ),roll rate (  p ), yaw rate ( r ), roll angle( φ ), and azimuthangle( ψ ); input vector consists of aileron ( δ  A ) andrudder ( δ  R ); measurement vector consists of side-slip( β  ), roll rate (  p ), yaw rate ( r ), roll angle( φ ), azimuthangle( ψ ), and lateral acceleration( a y ),  ̺  : air density, g  : gravity,  m  : mass of the aircraft,  S   : surface area of the air craft,  b  : wing span. For the detail of symbols  refer to Etkin [7]. The dynamics of the aircraft underconsideration vary greatly as a function of speed ( V   )and flaps setting ( f  ). Signal  V    and  f   are measuredin real time. The aircraft speed ranges between 80KEAS and 320 KEAS. The flap setting varies in 0, 20,30, and 40 deg. The nominal data of aircraft is givenin Table 1. Table 1:  The nominal data of N-250 aircraftData ValuesSpeed 80-320 KEASCentral Gravity 26.7 %Aircraft Mass 20,267.5 kgAltitude 17500 feetFlap setting 0, 20, 30, 40 degThe parameter dependent state space data are ob-tained by trimming and linearizing the non linearmodel at speeds V= { 80, 85, 90, 95, 100, 105, 110, 115,120, 125, 130, 135, 140, 145, 150, 155, 160, 160, 170,175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225,230, 235, 240, 245, 250, 255, 260, 260, 270, 275, 280,285, 290, 295, 300, 305, 310, 315, 320 } KEAS (knotsequivalent air speeds) and flaps setting f= { 0, 20, 30,40 }  deg. Furthermore, we choose entries of parameterdependent state-space data as the vertices of thepolytopic plant [2]. However, choosing all the entriesas the vertices would results in large number of LMIconstrains to be checked and can cause computationalimpractical. Hence, we consider only 5 entries of state-space data which are most significantly affectedby the aircraft speed and flaps setting. By observingsingular values plots of the plants with respect tovariation of state-space data which are affected bythe true aircraft speed and flaps setting, it turns outthat entries  L ′ v ,L ′  p ,N  ′ v ,N  ′  p ,N  ′ r  provide significantchanges on the plant. The variation  V    and  f   resultin parameters range of   L ′ v  ∈  [ − 0 . 3132 , − 0 . 0181] ,L ′  p  ∈ [ − 5 . 6951 , − 1 . 2095] ,N  ′ v  ∈  [ − 0 . 0220 , 0 . 0237] ,N  ′  p  ∈ [ − 1 . 4561 , 0 . 9450] ,  and  N  ′ r  ∈  [ − 1 . 1020 , − 0 . 2057] . Control system of the aircraft is designed to meet thefollowing criteria [2, 7]: Frequency domain criteria •  The closed-loop system has bandwidth of 10rad/sec. •  The closed-loop system has low sensitivity in fre-quency  ≤  8rad/sec. •  In high frequency range the measurement noiseis attenuated around 2 dB. Time domain criteria •  Steady state errors are within 7% tolerance, over-shoot  ≤  10% and transient response in 4 − 8 sec-onds. •  Control surface magnitudes and rates do not ex-ceed actuator saturation limits. •  System response to command meets level 1 flyingquality.The purpose of the controller design is to satisfythe criteria requirements. For this purpose we selectweighting functions [2]. Based on the properties of noise where its effect is small at low frequencies andlarge at high frequencies, the noise weighting functi-ons corresponding to  β, φ, ψ, a y , p, r  are chosenas follows.  W  β  =  W  ψ  =  W  a y  = 0 . 01  s +1 s 10 +1 ,W  φ  =0 . 01 s 0 . 33 +1 s 2 +1  ,W   p  =  W  r  = 0 . 01 s 0 . 25 +1 s 0 . 5 +1  .  Weighting func-tion corresponding to pilot command are  W  βcmd  = W  rcmd  = s 2 +1 s 0 . 5 +1 .  Aileron and rudder actuator dyna-mics are modelled as first order system with time con-stant  τ   = 0 . 1 seconds. These models are describedas  G A ( s ) =  G R ( s ) =  1010+ s .  Weighting functions foraileron and rudder deflection, and their rate are cho-sen to be constants i.e.  122 ,  120 ,  150 ,  137 ,  respectively tomeet the saturation limits given in Table 2. Table 2:  Maximum deflection and rate of aileron andrudderMaximum deflection Maximum rateAileron  ±  22 deg 50 degRudder  ±  20 deg 37 degIdeal model corresponding to yaw rate is a firstorder systems with time constant  τ   = 0 . 5 seconds,the model is chosen such that settling time 2%criterion is achieved within 2 . 3 second. For idealmodel corresponding to  β  , a second order system ischosen with  ζ  D  >  0 . 08 , ω D  >  0 . 5 ,  and  ζ  D ω D  >  0 . 15 . This model is chosen to satisfy the specification forlevel 1 dutch-roll mode flying quality for class IIaircraft and flight phase B [2, 7]. In this design wechoose  ζ  D  = 0 . 7 , ω D  = 1 . 5 rad/sec. This model hassettling time  <  5 seconds and maximum overshoot5%.  W  β ideal  =  22+ s , W  r ideal  =  1 . 5 2 s 2 +2(0 . 7)(1 . 5) s +1 . 5 2 . Using the model matching technique, the actualvariables are compared to those obtained from themodel :  β  error  =  β   −  β  ideal  and  r error  =  r  −  r ideal .Weighting functions corresponding to  β  eror  and  r error are  W  β error  = 50  s +1 s 0 . 02 +1 , W  r error  = 50  s +1 s 0 . 1 +1 . Theseweighting functions are chosen to guarantee thatsteady state errors for  β   and  r  do not exceed 2%. Thehigh-order generalized plant consists of the lateral-directional dynamic and all weighting functions.We have the order of the generalized plant is 20th.Furthermore, the order of the plant(model) is reducedby using the CRCF with the algorithm in section 5.Note that the parameters  L ′ v ,L ′  p ,N  ′ v ,N  ′  p ,N  ′ r  enterthe state space matrices in an affine way and are
Advertisement
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks