Inter national J our nal of P o wer Electr onics and Dri v e Systems (IJPEDS) V ol. 12, No. 1, May 2021, pp. 567 ∼ 575 ISSN: 2088-8694, DOI: 10.11591/ijpeds.v12.i1.pp567-575 ❒ 567 Finite fr equency H ∞ contr ol design f or nonlinear systems Zineb Lahlou, Abderrahim El − Amrani, Ismail Boumhidi LISA C Laboratory , Sidi Mohamed Ben Abdellah Uni v ersity , Fes, Morocco Article Inf o Article history: Recei v ed Feb 17, 2020 Re vised Jan 18, 2021 Accepted Feb 7, 2021 K eyw ords: Finite frequenc y LMIs Nonlinear systems T -S Model ABSTRA CT The w ork deals nite frequenc y H ∞ control design for continuous time nonlinear systems, we pro vide suf cient conditions, ensuring that the closed-loop model is stable. Simulations will be gifted to sho w le v el of attenuation that a H ∞ lo wer can be by our method obtained de v eloped where further comparison. This is an open access article under the CC BY -SA license . Corresponding A uthor: Abderrahim El − Amrani LISA C Laboratory Sidi Mohamed Ben Abdellah Uni v ersity Fes 30050, Morocco Email: abderrahim.elamrani@usmba.ac.ma 1. INTR ODUCTION Fuzzy models [1] it generated widespread interest from engineers, mainly for reno wned T -S syst ems my actually approach great cate gory for non linear models. Then, the T -S systems is its uni v ersal approximation of a smooth non linear function by a f amily of IF and THEN non linear rules that represent t he output/ input relationships of the models [2]-[11]. The interest of the literature mentioned abo v e the H ∞ control design in the FF range. whereas, in such cases, standard design methods of full frequenc y range can pro vide conserv atism. Ne v ertheless, in an actual application, the design characteristics are generally gi v en in selector Frequenc y domains (see, [12]-[21]). In this w ork, we de v elop ne w our method concerning FF design of nonlinears continuous systems. Us- ing theadequate conditions are de v eloped, ensuring that the closed loop system is stable. Numerical e xamples are pro vides to pro v e the ef fecti v eness of FF propose method. Notations : • ∗ : F orm symmetry • Q > 0 : F orm positi v e • sy m ( M ) > 0 : M + M ∗ • I : form Identity • diag { .. } : Block diagonal form 2. T -S MODELS Let’ s the continuous model is gi v en by ˙ x ( t ) = P n r =1 σ r ( t )( A r x ( t ) + L r u ( t ) + B r v ( t )) P n r =1 σ r ( t ) , y ( t ) = P n r =1 σ r ( t )( C r x ( t ) + E r u ( t ) + D r v ( t )) P n r =1 σ r ( t ) (1) J ournal homepage: http://ijpeds.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
568 ❒ ISSN: 2088-8694 with λ r ( t ) = p Y j =1 N r s ( µ s ( t )) and σ r = λ r ( t ) P n r =1 λ r ( t ) ; 0 ≤ λ r ≤ 1 and n X r =1 λ r = 1 (2) and σ = [ σ 1 , ..., σ r ] ∗ , the T -S system can be re written as follo ws: ˙ x ( t ) = A ( σ ) x ( t ) + L ( σ ) u ( t ) + B ( σ ) v ( t ) y ( t ) = C ( σ ) x ( t ) + E ( σ ) u ( t ) + D ( σ ) v ( t ) (3) where { A ( σ ); B ( σ ); L ( σ ); B ( σ ); C ( σ ); E ( σ ); D ( σ ) } = n X r =1 ρ r ( t ) { A r ; L r ; B r ; C r ; E r ; D r } (4) 3. PDC CONTR OLLER SCHEME The fuzzy control as follo ws: u ( t ) = n X s =1 σ s K s x ( t ) (5) then, we ha v e the closed loop model: ˙ x ( t ) = A c ( λ ) x ( t ) + B ( λ ) v ( t ) y ( t ) = C c ( λ ) x ( t ) + D ( λ ) v ( t ) (6) with A c ( σ ) = A c ( σ ) + L ( σ ) K ( σ ); c ( σ ) = C c ( σ ) + E ( σ ) K ( σ ) (7) problem formulation Gi v en: the state feedback in the form of (5) such that: Z µ ∈∇ Y ∗ ( µ ) Y ( µ ) dµ ≤ γ 2 Z µ ∈∇ V ∗ ( µ ) V ( µ ) dµ (8) with △ is gi v en in T able 1. T able 1. Dif ferent frequenc y ranges − Low f r eq u ency M iddl ef r eq uency H ig hf r eq u ency ∇ | µ | ≤ ¯ µ l ¯ µ 1 ≤ µ ≤ ¯ µ 2 | µ | ≥ ¯ µ h Π − S ( σ ) R ( σ ) R ( σ ) ¯ µ 2 l S − S ( σ ) R ( σ ) + j ¯ µ 0 S ( σ ) R ( σ ) − j ¯ µ 0 S ( σ ) − ¯ µ 1 ¯ µ 2 R S ( σ ) R ( σ ) R ( σ ) − ¯ µ 2 h S ( σ ) 4. MAIN RESUL TS 4.1. Useful lemma Lemma 4..1 T uan, H. D et al .[22] If the follo wing conditions are met: Ω r r < 0 1 ≤ r ≤ n 1 n − 1 Ω r r + 1 2 [Ω r s + Ω sr ] < 0; 1 ≤ r ̸ = s ≤ n (9) and n X r =1 n X s =1 λ r λ s Ω r s < 0 (10) Lemma 4..2 El-Amrani, A. et al . [23]. Let T ∈ R n × n and M ∈ R m × n , so that the follo wing conditions are equi v alent: Int J Po w Elec & Dri Syst, V ol. 12, No. 1, May 2021 : 567 – 575 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 569 1. M ⊥∗ T M ⊥ < 0 2. ∃N ∈ R n × m : T + sy m [ MN ] < 0 Lemma 4..3 Closed loop (6) is stable, if R ( σ ) = R ( σ ) ∗ ∈ H n , 0 < S = S ∗ ∈ H n such that A c ( σ ) B ( σ ) I 0 ∗ Π A c ( σ ) B ( σ ) I 0 + C T c ( σ ) C c ( σ ) C T c ( σ ) D ( σ ) D T ( σ ) C c ( σ ) D T ( σ ) D ( σ ) − γ 2 I < 0 (11) with Π is gi v en of T able 1. 4.2. Finite fr equency analysis Theor em 4..4 The fuzzy model (6) is stable, if R ( σ ) ∈ H n , 0 < S ∈ H n , 0 < W ( σ ) ∈ H n , Z ( σ ) ∈ H n , H ( σ ) ∈ H n such that − sy m [ Z ( σ )] W ( σ ) + Z ( σ ) A ( σ ) − H ∗ ( σ ) ⋆ sy m [ H ( σ ) A c ( σ )] < 0 (12) Ψ 11 ( σ ) Ψ 12 ( σ ) + Z ( σ ) A c ( σ ) − H ∗ ( σ ) Z ( σ ) B ( σ ) 0 ⋆ Ψ 22 ( σ ) + sy m [ H ( σ ) A c ( σ )] H ( σ ) B ( σ ) C ∗ c ( σ ) ⋆ ⋆ − γ 2 I D ∗ ( σ ) ⋆ ⋆ ⋆ − I < 0 (13) • Lo w frequenc y (LF) range: Ψ 11 ( σ ) = − S ( σ ) − Z ( σ ) − Z ∗ ( σ ); Ψ 12 ( σ ) = R ( σ ); Ψ 22 ( σ ) = ¯ µ 2 l S ( σ ) (14) • Middle frequenc y range (MF) range: Ψ 11 = − S ( σ ) − Z ( σ ) − Z ∗ ( σ ); Ψ 12 = R ( σ ) + j ¯ µ 0 S ( σ ); Ψ 22 = − ¯ µ 1 ¯ µ 2 S ( σ ) (15) • High frequenc y (HF) range: Ψ 11 ( σ ) = S ( σ ) − Z ( σ ) − Z ∗ ( σ ); Ψ 12 ( σ ) = R ( σ ); Ψ 22 ( σ ) = − ¯ µ 2 h S ( σ ) Pr oof 4..5 Let ¯ A ( σ ) , W ( σ ) = W ( σ ) ∗ > 0 such that A c ( σ ) I ∗ 0 W ( σ ) W ( σ ) 0 A c ( σ ) I < 0 (16) dene: T = 0 W ( σ ) W ( σ ) 0 ; N = Z ( σ ) H ( σ ) ; M = − I A c ( σ ) ; M ⊥ = A c ( σ ) I (17) let lemma 4.1., (16) and (17) are equi v alent to: 0 W ( σ ) W ( σ ) 0 + Z ( σ ) H ( σ ) − I A c ( σ ) + − I A c ( σ ) ∗ Z ( σ ) H ( σ ) ∗ < 0 (18) which is nothing b ut (12), let LF case : T = − S R ( σ ) 0 ⋆ ¯ µ 2 l S + C ∗ c ( σ ) C c ( σ ) C ∗ c ( σ ) D ( σ ) ⋆ ⋆ − γ 2 I + D ∗ ( σ ) D ( σ ) ; M ⊥ = A c ( σ ) B ( σ ) I 0 0 I ; M = − I A c ( σ ) B ( σ ) ; N = Z ( σ ) T H ( σ ) T 0 T (19) we ha v e T + sy m ( N M ) < 0 (20) using Lemma 4.1., we obtain (11). F inite fr equency H ∞ contr ol design for nonlinear systems (Zineb Lahlou) Evaluation Warning : The document was created with Spire.PDF for Python.
570 ❒ ISSN: 2088-8694 4.3. Finite fr equency design Theor em 4..6 The fuzzy model (6) is stable, if ˜ R ( σ ) ∈ H n , 0 < ˜ S ∈ H n , 0 < ˜ W ( σ ) ∈ H n , G ( σ ) , ¯ Z ( σ ) such that: − ¯ Z ∗ ( σ ) − ¯ Z ( σ ) ˜ W ( σ ) + A ( σ ) ¯ Z ∗ ( σ ) + B ( σ ) G ∗ ( σ ) − β ¯ Z ( σ ) ⋆ sy m [ β ( A ( σ ) ¯ Z ∗ ( σ ) + B ( σ ) G ∗ ( σ ))] < 0 (21) ¯ Ψ 11 ( σ ) ¯ Ψ 12 ( σ ) B ( σ ) 0 ⋆ ¯ Ψ 22 ( σ ) β B ( σ ) ¯ Z ( σ ) C ∗ ( σ ) + G ( σ ) E ∗ ( σ ) ⋆ ⋆ − γ 2 I D ∗ ( σ ) ⋆ ⋆ ⋆ − I < 0 (22) • | µ | ≤ ¯ µ l ¯ Ψ 11 ( σ ) = − ˜ S ( σ ) − ¯ Z ∗ ( σ ) − ¯ Z ( σ ); ¯ Ψ 22 ( σ ) = ¯ µ 2 l ˜ S ( σ ) + sy m [ β ( A ( σ ) ¯ Z ∗ ( σ ) + B ( σ ) G ∗ ( σ ))]; ¯ Ψ 12 ( σ ) = ˜ R ( σ ) − β ¯ Z ( σ ) + A ( σ ) ¯ Z ∗ ( σ ) + B ( σ ) G ∗ ( σ ) . • ¯ µ 1 ≤ µ ≤ ¯ µ 2 ¯ Ψ 11 ( σ ) = − ˜ S ( σ ) − ¯ Z ∗ ( σ ) − ¯ Z ( σ ); ¯ Ψ 22 ( σ ) = − ¯ µ 1 ¯ µ 2 ˜ S ( σ ) + sy m [ β ( A ( σ ) ¯ Z ∗ ( σ ) + B ( σ ) G ∗ ( σ ))]; ¯ Ψ 12 ( σ ) = ˜ R ( σ ) + j ¯ µ 0 ˜ S ( σ ) − β ¯ U ( σ ) + A ( σ ) ¯ Z ∗ ( σ ) + B ( σ ) G ∗ ( σ ) • | µ | ≥ ¯ µ h ¯ Ψ 11 ( σ ) = ˜ S ( σ ) − ¯ Z ∗ ( σ ) − ¯ Z ( σ ); ¯ Ψ 12 ( σ ) = ˜ R ( σ ) − β ¯ Z ( σ ) + A ( σ ) ¯ Z ∗ ( σ ) + B ( σ ) G ∗ ( σ ); ¯ Ψ 22 ( σ ) = − ¯ µ 2 h ˜ S ( σ ) + sy m [ β ( A ( σ ) ¯ Z ∗ ( σ ) + B ( σ ) G ∗ ( σ ))] . therefore : K ( σ ) = ( ¯ Z − 1 ( σ ) G ( σ )) ∗ (23) Pr oof 4..7 First, for matrix v ariable H ( σ ) in theorem 4.2., let H ( σ ) = β Z ( σ ) , after that by replacing (7) into (12) and (13), respecti v ely , moreo v er , let, ¯ Z ( σ ) = Z − 1 ( σ ); G ( σ ) = ¯ Z ( σ ) K ∗ ; ˜ S ( σ ) = Z − 1 ( σ ) S ( σ ) Z −∗ ( σ ); ˜ R ( σ ) = Z − 1 ( σ ) R ( σ ) Z −∗ ( σ ); ˜ W ( σ ) = Z − 1 ( σ ) W ( σ ) Z −∗ ( σ ) Multiplying (12) by diag { Z − 1 ( σ ) , Z − 1 ( σ ) } , and (13) by diag { Z − 1 ( σ ) , Z − 1 ( σ ) , I , I } , we ha v e (12) and (13) are equi v alent (21) and (22), respecti v ely . Theor em 4..8 The fuzzy model (6) is stable. If R r ∈ H n , 0 < S ∈ H n , 0 < W r ∈ H n , ¯ Z s , G s such that ˜ Ψ r r < 0; ˜ Υ r r < 0; 1 ≤ r ≤ n (24) 1 r − 1 ˜ Ψ r r + 1 2 [ ˜ Ψ r s + ˜ Ψ sr ] < 0; 1 ≤ r ̸ = s ≤ n (25) 1 r − 1 ˜ Υ r r + 1 2 [ ˜ Υ r s + ˜ Υ sr ] < 0; 1 ≤ r ̸ = s ≤ n (26) where ˜ Ψ r s = ˜ Ψ 11 r s ˜ Ψ 12 r s B r 0 ⋆ ˜ Ψ 22 r s β B r ¯ F s C ∗ r + G s E ∗ r ⋆ ⋆ − γ 2 I D ∗ r ⋆ ⋆ ⋆ − I ; Int J Po w Elec & Dri Syst, V ol. 12, No. 1, May 2021 : 567 – 575 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 571 ˜ Υ r s = − ¯ Z ∗ s − ¯ Z s W r + A r ¯ Z ∗ s + B r G ∗ s − β ¯ Z s ⋆ sy m [ β ( A r ¯ Z ∗ s + B r G ∗ s )] • | µ | ≤ ¯ µ l ˜ Ψ 11 r s = − ˜ S s − ¯ Z ∗ s − ¯ Z s ; ˜ Ψ 12 r s = ˜ R r − β ¯ Z s + A r ¯ Z ∗ s + B r G ∗ s ; ˜ Ψ 22 r s = ¯ µ 2 l ˜ S s + sy m [ β ( A r ¯ Z ∗ s + B r G ∗ s )] . • ¯ µ 1 ≤ µ ≤ ¯ µ 2 ˜ Ψ 11 r s = − ˜ S s − ¯ Z ∗ s − ¯ Z s ; ˜ Ψ 12 r s = ˜ R r + j ¯ µ 0 ˜ S s − β ¯ U s + A r ¯ Z ∗ s + B r G ∗ s ; ˜ Ψ 22 r s = − ¯ µ 1 ¯ µ 2 ˜ S s + sy m [ β ( A r ¯ Z ∗ s + B r G ∗ s )] . • | µ | ≥ ¯ µ h ˜ Ψ 11 r s = ˜ S s − ¯ Z ∗ s − ¯ Z s ; ˜ Ψ 12 r s = ˜ R r − β ¯ Z s + A r ¯ Z ∗ s + B r G ∗ s ; ˜ Ψ 22 r s = − ¯ µ 2 h ˜ S s + sy m [ β ( A r ¯ Z ∗ s + B r G ∗ s )] . The matrices g ains are obtained by: K s = ( ¯ Z − 1 s G s ) ∗ , 1 ≤ s ≤ n (27) Pr oof 4..9 by applying the Lemma 4.1., we ha v e Theorem 4.3. 5. SIMULA TIONS 5.1. Example 1 Consider fuzzy system (3) with tw o rules [24]: A 1 = 0 1 17 . 2941 0 ; A 2 = 0 1 12 . 6305 0 ; E 1 = 0 − 0 . 1765 ; L 1 = L 2 = 0 . 1 0 . 1 ; E 2 = 0 − 0 . 0779 ; B 1 = B 2 = 0 . 1; C 1 = C 2 = 1 1 ; D 1 = D 2 = 0 (28) and N 2 ( x 1 ) = ( 1 1 + exp( − 7( x 1 − π 4 )) )( 1 1 + exp( − 7( x 1 + π 4 )) ); N 1 ( x 1 ) = 1 − N 2 ( x 1 ) . (29) let v ( t ) = 2 2 ≤ t ≤ 3 2 5 ≤ t ≤ 6 0 other s (30) W e propose in table 2 sho ws the v alues of γ obtained in dif ferent frequenc y ranges. By Theorem 4.3., the controller g ains are gi v en by: • Lo w frequenc y (LF) range (with β 1 = 0 . 0502 and γ = 0 . 2507 ): K 1 = 168 . 2205 22 . 9439 ; K 2 = 353 . 5002 115 . 2592 (31) F inite fr equency H ∞ contr ol design for nonlinear systems (Zineb Lahlou) Evaluation Warning : The document was created with Spire.PDF for Python.
572 ❒ ISSN: 2088-8694 • Middle frequenc y (MF) range (with β 1 = 0 . 5025 and γ = 0 . 8355 ): K 1 = 186 . 8988 36 . 1978 ; K 2 = 378 . 9459 109 . 7698 (32) • High frequenc y (HF) range (with β 1 = 0 . 2478 and γ = 0 . 5702 ): K 1 = 203 . 4092 43 . 1392 ; K 2 = 356 . 0428 101 . 8833 (33) T able 2. Obtained γ by dif ferent domains F r eq uency r ang es methods γ EF ( 0 ≤ µ ≤ ∞ ) [16] Infeasible EF ( 0 ≤ µ ≤ ∞ ) Theorem 4 . 3 . ( ˜ S s = 0 ) 1 . 1789 LF ( | µ | ≤ 0 . 7 ) T heor em 2 in [16] 1.3598 LF ( | µ | ≤ 0 . 7 ) Theorem 4 . 3 . 0 . 2507 MF ( 1 ≤ µ ≤ 5 ) C or ol l ar y 1 in [16] 1.3010 MF ( 1 ≤ µ ≤ 5 ) Theorem 4 . 3 . 0 . 8355 HF ( | µ | ≥ 6 ) C or ol l ar y 2 in [16] - HF ( | µ | ≥ 6 ) Theorem 4 . 3 . 0 . 5702 MF ( 628 ≤ µ ≤ 6283 ) C or ol l ar y 1 in [16] 1.5786 MF ( 628 ≤ µ ≤ 6283 ) Theorem 4 . 3 . 0 . 9245 HF ( | µ | ≥ 6283 ) C or ol l ar y 2 in [16] - HF ( | µ | ≥ 6283 ) Theorem 4 . 3 . 0 . 2102 Figure 1. T rajectories of x i ( t ) , i = 1 , 2 , u ( t ) and y ( t ) for LF | µ | ≤ 0 . 7 range, (a) state x 1 ( t ) , (b) state x 2 ( t ) , (c) estimation controlled output y ( t ) , (d) estimation controlled output u ( t ) Int J Po w Elec & Dri Syst, V ol. 12, No. 1, May 2021 : 567 – 575 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 573 5.2. Example 2 Let the fuzzy system (3) [25], where: A 1 = − 1 − 1 . 155 1 0 ; A 2 = − 1 − 1 . 155 1 0 ; L 1 = 1 . 4387 0 ; L 2 = 0 . 5613 0 ; B 1 = B 2 = 1 0 ; C 1 = C 2 = 0 2 0 0 ; E 1 = E 2 = 0 2 ; D 1 = D 2 = 0 2 (34) and N 2 ( x 1 ( t )) = 0 . 5 − x 3 1 ( t ) 6 . 75 ; N 1 ( x 1 ( t )) = 1 − N 2 ( x 1 ( t )); x 1 ( t ) ∈ 1 . 5 1 . 5 (35) Figure 2. T rajectories of x i ( t ) , i = 1 , 2 , u ( t ) and y ( t ) for MF 1 ≤ | µ | ≤ 5 range, (a) state x 1 ( t ) , (b) state x 2 ( t ) , (c) estimation controlled output y ( t ) , (d) estimation controlled output u ( t ) F inite fr equency H ∞ contr ol design for nonlinear systems (Zineb Lahlou) Evaluation Warning : The document was created with Spire.PDF for Python.
574 ❒ ISSN: 2088-8694 T able 3. H ∞ performance bounds γ by dif ferent domains F r eq uency r ang es methods γ 0 ≤ µ ≤ ∞ [16] Infeasible 0 ≤ µ ≤ ∞ Theorem 4 . 3 . ( ˜ S s = 0 ) 1 . 4517 | µ | ≤ 628 [16] 1.2215 | µ | ≤ 628 Theorem 4 . 3 . 0 . 7514 628 ≤ µ ≤ 6283 [16] 1.025 628 ≤ µ ≤ 6283 Theorem 4 . 3 . 0 . 5274 | µ | ≥ 6283 [16] - | µ | ≥ 6283 Theorem 4 . 3 . 0 . 3854 The FF case of u ( t ) is assumed to satisfy 100 Hz; [100 1000] Hz and 1000 Hz, i.e.,. | µ | ≤ 628 rad/s; 628 ≤ µ ≤ 6283 rad/s; and | µ | ≥ 6283 rad/s, respecti v ely for u ( t ) and β = 1 . 6. CONCLUSION W e sent the FF state feedback design. T o reduce the closed-loop system and establish less conserv ati v e results, we ha v e considered tw o practical e xamples has been pro vides to sho w the feasibility of tuning FF H ∞ fuzzy control design method. 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