Polynomial-input-type final-state control taking account of input saturation

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1 Milano (Italy) August 28 - September 2, 211 Polynomial-input-type final-state control taking account of input saturation Mitsuo Hirata Fujimaru Ueno Graduate School of Engineering, Utsunomiya University, Yoto, Utsunomiya, Tochigi , Japan Graduate School of Engineering, Utsunomiya University, Yoto, Utsunomiya, Tochigi , Japan Abstract: A final-state control method based on a polynomial input with a time-symmetry assumptionwasproposedhowever,themethodcannotconsiderthesaturationofcontrolinput When a positioning time is shorten, it is inevitable to consider the input saturation to achieve a faster positioning time In this study, we propose a polynomial-input-type final-state control taking account of input saturation The effectiveness is shown by simulations Keywords: Polynomial-input-type final-state control, input saturation, linear matrix ineuality 1 INTRODUCTION For information devices such as hard disk drives, it is important to achieve fast and accurate positioning of the recording head Since the system has mechanical vibration modes at high freuency, the feedforward input must be obtained so that the mechanical vibration modes are not excited while minimizing access time Thus, various attempts have been made to design feedforward inputs (Yamaguchi et al, 27) We developed a design method for feedforward inputs that does not excite high freuency resonance modes based on freuency-shaped final-state control () method, and the effectiveness of the method has been shown by simulation and experiment by applying it to track-seeking problem of HDD (Hirata et al, 22) Further, the proposed method has been applied to any other mechatronic systems and the effectiveness has been shown (Hirose et al, 28) Inthemethod,thefeedforwardinputmustbestored in memory since it is obtained as a time-series data Therefore, a larger memory size is reuired if the step number of input is increased To overcome the problem, we proposed a method based on a polynomial input (Hirata and Ueno, 29) Since the polynomialinput-type (P) method only reuires storing the coefficients, the memory size is drastically reduced Moreover, we proposed a P method that assumes a time symmetry of input to reduced the order of the polynomial (Hirata and Ueno, 21) In practical control applications, the magnitude of the inputislimitedthus,thecontrolinputmustbegenerated soasnottosaturateespecially,whenthepositioningtime is shorten, it is inevitable to consider the input saturation to achieve a faster positioning time This paper proposes a P method that can take into account the input saturation In particular, we show that the time-symmetry assumption is effective for the P method with input saturation The effectiveness of the proposed method is shown by simulations 2 FINAL-STATE CONTROL TAKING ACCOUNT OF INPUT SATURATION For a controllable discrete-time system of m-th order xk +1=Axk+Buk (1) yk=cxk, (2) let us consider the problem of determining an N-step (N m) control input uk that drives an initial state x to a final state xn By substituting k =,1,,N 1 into E(1) recursively we have Y =ΣU (3) Σ= A N 1 BA N 2 B B (4) U =u u1 un 1 T (5) Y =xn A N x (6) It is obvious that the input vector U satisfying the constraint E(3) is not uniue, then the following performance index is introduced: J =U T QU, Q > (7) The minimization problem of E(7) subject to the constraint E(3) can be formulated by linear-matrix ineualities (LMIs) Since Σ is guaranteed to be full rank by the controllability of (A,B), Σ R N (N m) and Σ R N m satisfying ΣΣ = O and ΣΣ = I can be introduced Then Ũ satisfying U = Σ Σ Ũ (8) is introduced Now substituting E(8) into E(3), we have Y =IOŨ Therefore, Ũ can be represented by Y and a free parameter R (N m) 1 as follows: Copyright by the International Federation of Automatic Control (IFAC) 461

2 Milano (Italy) August 28 - September 2, 211 Y Ũ = Thus we have U = Σ Σ Y By introducing M as (Σ M = ) T (Σ ) T = M11 M 12, M 21 M 22 Q Σ Σ (9) the performance index J can be described as follows: J =U T QU =Y T M 11 Y +2Y T M 12 + T M 22 The ineuality J<γfor a given γ can be represented by a LMI as follows: γ 2Y T M 12 Y T M 11 Y T M 1 22 > (1) In order to account for input saturation, we consider the minimization problem of E(7) subject to the constraint z min <zk <z max, (k =,,N 1) (11) zk is defined by zk=c z xk+d z uk (12) It is assumed that z max and z min are a positive and a negative values, respectively In E(12), matrices C z and D z are design parameters to define zkfromxkanduk By introducing the following matrices for simple notation Z =z z1 zn 1 T (13) C z C z A Φ z = C z A N 1 D z Ω z = C z B D z, C z A N 2 B C z BD z Z can be described as an affine function of as Z()=Φ z x+ω z U (14) =Φ z x+ω z Σ Σ Y (15) By defining a row vector δ i whose i-th element is one and the other elements are zero, E(11) is represented by N LMIs as z δ i Z() z (δ i Z()) T z z > (16) i =1,,N and z =(z max +z min )/2, z =(z max z min )/2 The control input that drives x to xn minimizing J subject to the constraint E(11) can be obtained by minimizing γ subject to E(1) and E(16) This method is referred to as a conventional method 3 POLYNOMIAL-INPUT-TYPE FSC TAKING ACCOUNT OF INPUT SATURATION 31 Proposed method 1 In this section, the polynomial-input-type FSC method proposed in (Hirata and Ueno, 29) is extended to treat input saturation It is assumed that the FSC input uk is generated by a polynomial as follows: uk=α +α 1 k 1 + +α p 1 k p 1 (17) p is a number of terms, α i (i =,,p 1) are coefficients By substituting E(17) into U of E(5) we have U =Γα (18) 1 1 p p 1 Γ= 1(N 2) 1 (N 2) p 1 1(N 1) 1 (N 1) p 1 (19) α=α α 1 α p 1 T (2) It is easy to show that Γ is full rank By substituting E(18) into E(3) we have Y = Σα, Σ=ΣΓ (21) On the other hand, the performance index J of E(7) becomes Q =Γ T QΓand Q> J p =U T QU = α T Qα (22) Since E(21) and E(22) correspond to E(3) and E(7), respectively, this problem can also be formulated by LMIs Let us define Σ R p (p m) and Σ R p m that satisfy Σ Σ = O and Σ Σ = I for the full-rank matrix Σ Thus the coefficient vector α can be described by a free parameter R (p m) 1 as α= Y Σ Σ (23) It is obvious that E(23) corresponds to E(9) if α is regarded as U Thus we have the following LMI to satisfy J p <γ: γ 2Y T M12 Y T M11 Y T M 1 22 > (24) 462

3 Milano (Italy) August 28 - September 2, 211 u max A B C D E Time Fig 2 Augmented system for FSC design u min Fig 1 Saturated control input M11 M12 = ( Σ ) T Q M 21 M22 ( Σ ) T Σ Σ On the other hand, Z of E(13) can be described as an affine function of by substituting E(18) and E(23) into E(14) as follows: Z( )=Φ z x+ω z (Γα) =Φ z x+ Ω z Σ Σ Y (25) Ω z =Ω z Γ Thus we have N LMIs for the constraint of E(11) as z δ i Z( ) z (δ i Z( )) T > (26) z z i =1,,N and Z( ) is defined by E(25) Hence,apolynomial-input-typeFSCconsideringtheinput saturation can be formulated by LMIs of E(24) and E(26) This method is referred to as a proposed method 1 32 Proposed method 2 When the input is saturated, it resides between u min and u max and the intervals A B and D E become almost flat asshowninfig1infscmethods,anactualinputu c kis augmented by an integrator as shown in Fig2 to evaluate its derivative for smooth feedforward input Therefore, the input to the augmented system becomes almost zero at the intervals A B and D E as shown in Fig3 However, it is difficult for a polynomial to generate such a flat input Thus, local minima and maxima of the polynomial are repeated at the intervals As a result, a high order polynomial will be reuired to obtain a good performance In order to solve this problem, a time-symmetry assumption of input is introduced in the proposed method 1 Under this assumption the polynomial only generate an input from t = to the point C in Fig3 (Hirata and Ueno, 21), then the order of the polynomial might be reduced The input vector U with the time-symmetry assumption can be defined as: { Uo, if N =2n 1 U = (27) U e, if N =2n A B C D E Time Fig 3 Input to the augmented system when the actual input is saturated u u1 u u1 un 2 un 1 U o = un 1,U un 2 e = un 1 u1 u1 u u n =1,2, and the definition of U depends on whether N is an even or an odd number By substituting E(17) into E(27) we have U =Γ s α (28) { Γo, if N =2n 1 Γ s = Γ e, if N =2n 1 1 p p 1 1(n 2) 1 (n 2) p 1 Γ o = 1(n 1) 1 (n 1) p 1 1(n 2) 1 (n 2) p p p 1 (29) 463

4 Milano (Italy) August 28 - September 2, LMI Gain db Phase deg Freuency Hz Fig 4 Freuency responce of the plant 1 1 p p 1 Γ e = 1(n 1) 1 (n 1) p 1 1(n 1) 1 (n 1) p p p 1 Since E(28) is similar to E(18), J p <γis represented by the LMI of E(24) in which Γ is replaced by Γ s As for the constraint to zk, it is enough to impose the constrain to the first half of zk, eg, from z to zn 1, since the input is time symmetry Therefore, the LMI of E(26) is evaluated for i =,,n 1 The method considering the time symmetry of input is referred to as a proposed method 2 41 Plant model 4 SIMULATIONS In order to verify the effectiveness of the proposed method 1 and 2, a mechanical system that has a ridged-body mode and two vibration modes are assumed P(s)= k s i=1 k i s 2 +2ζ i ω i s+ω 2 i The plant parameters are defined as shown in Table 1 to have vibration modes at 1kHz and 2kHz The sampling period τ isassumedtobe25µs,andthebodeplotofthe discretized plant is shown in Fig4 Table 1 Parameters of the plant model k k k ζ ζ ω 1 2π 1 ω 2 2π 2 Spectrum db (a) Time responses of input LMI Freuency Hz (b) Spectra of input Fig 5 LMI (Conventional method) 42 Feedforward input design For feedforward input design, a ridged-mode model is used In order to take into account the smoothness and the freuency components of the input, the following performance index is introduced for the augmented system in Fig2 (Hirata et al, 22) J w = N 1 k= u 2 k+ l i Ûc(ω i ) 2 (3) Nτ Û c (ω i )= u c (t)e jωit dt Eachω i isselectedtobefreuencybetween1khz±2%and 2kHz ±2% in order to reduced the freuency components of the input around vibration modes of the plant The variation range of each resonance freuency is divided into 1freuenciestospecifyω i Theweight i for ω i isselected to be for all freuency points The performance index of E(3) can be represented by a uadratic form as J w = U T Q w U (Hirata et al, 22) i=1 Asanexample,thestepnumberofN =8andthenumber of terms of p = 7 are assumed, and the initial and final states are selected to be a zero velocity at the origin and a zero velocity at r m = 1, respectively Thefreuency-shapedfinal-statecontrolinputobtainedby using E(3) without input constraint is shown in Fig5 464

5 Milano (Italy) August 28 - September 2, LMI P 1 8 LMI P TS Spectrum db (a) Time responses of input LMI P Freuency Hz (b) Spectra of input Fig 6 LMI P (Proposed method 1) by dashed line It confirms that the spectrum around 1kHz and 2kHz is well reduced In order to constrain the actual input u c kas u c k < u max, C z and D z in E(12) are selected so that zk= u c k holds The maximum absolute value u max is determined to be 8% of the maximum absolute value of the input without input constraint, and u max =79is determined in this simulation Then, z max = u max and z min = u max are defined in E(11) to obtain feedforward inputs for all methods For simple notation, the results of the conventional method,theproposedmethod1,andtheproposedmethod 2 are referred to as LMI, LMI P (LMI Polynomial-input-type ), and LMI P-TS (LMI P with Time Symmetry), respectively Figs5-7 show the time responses and their spectra of LMI, LMI P, and LMI P-TS inputs In these figures, the result of without input constraint is shown by dashed line for comparison Fig5 shows that the LMI method satisfies the constraint of u c k < 79 and the spectrum around 1kHz and 2kHz is reduced In Table 2 Performance degradation Methods J RE % LMI (Conventional method) 258 LMI P (Proposed method 1) 525 LMI P-TS (Proposed method 2) 295 Spectrum db (a) Time responses of input LMI P TS Freuency Hz (b) Spectra of input Fig 7 LMI P-TS (Proposed method 2) the LMI P method, as shown in Fig6, the input constraint is satisfied while the spectrum is not well reduced both around 1kHz and 2kHz On the other hand, Fig7 shows that the LMI P-TS method satisfies input constraint while minimizing the spectrum around 1kHz and 2kHz The value of the performance index J p of the LMI P and LMI P-TS will be larger than that of J of LMI because the class of input is restricted to polynomialtherefore,therelativeerrorj RE isintroduced to evaluate the gap between J p and J: J RE = J p J 1 % J The value of J RE foreachmethodisshownintable2 The LMI P method shows significant performance degradation of J RE = 525% while the LMI P- TS achieves a small performance degradation of J RE = 295 Since J RE of the LMI method is 258, the performance degradation of the LMI P-TS method is kept to be minimum 43 Simulations Residual vibrations are evaluated when the obtained feedforward inputs are imposed to the plant The results are shown in Fig8 For comparison, the result of the without input constraint is also shown by dashed line FromFig8(b),theresultofLMIPshowslargeresidualvibrationThisisbecausethatthespectrumoftheLMI 465

6 Milano (Italy) August 28 - September 2, 211 Output (a) LMI (Conventional method) LMI LMI P Hirata, M and Ueno, F (21) Final-state control using a time-symmetrical polynomial input In Proc of Advanced Motion Control (AMC21), Hirose, N, Terachi, Y, Kawafuku, M, Iwasaki, M, and Hirai, H (28) Real-time compensation for positioning peformance using on-line parameter identification and initial value compensation IEEJ Transactions on Industry Applications (in Japanese), 128(6), Yamaguchi, T, Hirata, M, and Fujimoto, H (27) Nano-scale servo control TDU Press Output (b) LMI P (Proposed method 1) LMI P TS Output (c) LMI P-TS (Proposed method 2) Fig 8 Time responses of output P input is not reduced around 1kHz and 2kHz On the other hand, the result of the LMI P-TS achieves a small residual vibration, and the performance is almost the same that of the LMI Since the feedforward input of the LMI P-TS is generated by a polynomial with p = 7, the reuired memory size is reduced 5 CONCLUSION In this paper, we have proposed two P methods considering input saturation The effectiveness has been evaluated by applying them to the plant having a ridgedbody mode and two mechanical vibration modes From simulation results, it has been confirmed that the timesymmetry assumption in the proposed method 2 plays an important role to achieve a small residual vibration The proposed method can easily apply to various positioning systems such as HDD and Galvano scanner REFERENCES Hirata, M, Hasegawa, T, and Nonami, K (22) Seek control of hard disk drives based on final-state control taking account of the freuency components and the magnitudeofcontrolinput InProc of Advanced Motion Control (AMC22), 4 45 Hirata, M and Ueno, F (29) Final-state control using a sampled-data polynomial for hard disk drives In Proc Micromechatronics for Information and Precision Euipment (MIPE29),