Multirate Feedforward Control with State Trajectory Generation based on Time Axis Reversal for Plant with Continuous Time Unstable Zeros

Transcription

1 Multirate Feeforwar Control with State Trajectory Generation base on Time Axis Reversal for with Continuous Time Unstable Zeros Wataru Ohnishi, Hiroshi Fujimoto Abstract with unstable zeros is known as ifficult to be controlle because of initial unershoot of step response an unstable poles of its inversion system. There are two reasons why plant has unstable zeros in iscrete time omain: ) non-collocation of actuators an sensors, 2) iscretization by zero-orer-hol. Problem 2) has been solve by multirate feeforwar control propose by our research group. This paper extens this metho to solve problem ) by the state trajectory generation base on time axis reversal. The valiity of the propose methos is emonstrate by simulations. I. Introuction with unstable zeros is known as ifficult to be controlle because of unstable poles of its inversion system for a feeforwar controller an initial unershoot of step response shown in Fig.. Zeros of the iscretize transfer function can be classifie as two types [][2]: ) intrinsic zeros, an 2) iscretization zeros [3]. Intrinsic zeros correspon to zeros of its continuous time transfer function. The others are calle iscretization zeros. The poles of the continuous time omain transfer function are etermine by the plant ynamics such as the mass, amping, an stiffness. On the other han, zeros of the continuous time omain are etermine by not only ynamics but also the characteristics of the actuators an sensors [4]. Therefore, integrate esign of mechanism an control is conucte to place zeros to esire position [5][6][7][8]. However, in most cases, it is ifficult to change the characteristics of the actuators an sensors to allocate zeros at the esire position. For instance, zeros of wafer stages are position epenent [9] because wafer stages are nee to be controlle for the lens coorinate. Other examples are har isk rives [], atomic force microscopes [], bust converters [2], an interior permanent magnet synchronous motors [3]. In practice, controllers are nee to be iscretize for implementation. In single rate control scheme, perfect tracking [4] is not achievable because the iscretization zero(s) are unstable when the plant relative orer is greater than two even without intrinsic unstable zero(s). To eal with this problem, the approximate inverse base feeforwar control approaches such as nonminimum-phase zeros ignore (NPZI) [5], zero-phase-error tracking controller () [4], an zero-magnitue-error tracking controller () [6] are propose. Other approaches are FIR filter base combine input shaping an feeforwar methos [7][8][9]. These methos eal with the problem ) an 2) in the same time The University of Tokyo, 5--5 Kashiwanoha, Kashiwa, Chiba, , Japan, ohnishi@hflab.k.u-tokyo.ac.jp, fujimoto@k.u-tokyo.ac.jp y (t) P = s+ P 2 = (s) (s+) 2.2 P 3 = (s)2 (s+) 3 P 4 = (s)3 (s+) Fig.. Step response comparison. P is st orer transfer function without unstable zero. P 2, P 3, P 4 have one, two, three unstable zero(s) as shown in the legen of the figure. Step responses of the system with unstable zero(s) make unershoot. because these controllers are esigne by iscretize transfer functions. Unstable intrinsic an iscretization zeros compensation methos by the preactuation an preview are propose in [][2]. These methos also compensate for intrinsic an iscretization zeros in the same time. Continuous time omain approach is propose in [2]. This metho solves the ifferential equation in continuous time omain. In this metho, however, the reference trajectory has to be efine by one equation in the positive time omain. This paper proposes Preactuation Perfect Tracking Control metho base on multirate feeforwar an state trajectory generation by time axis reversal. This metho solves problem ) an 2) separately. The unstable zeros in the continuous time transfer function are manage by a state trajectory generation base on time axis reversal an preactuation commans. This metho can be applie for any kin of reference position trajectory as long as n th erivative of the trajectory is given. Here, n enotes the orer of the plant in the continuous time transfer function. Next, the plant iscretization problem is solve by the multirate feeforwar control [22]. Finally, the relationship between the pre/post actuation an the continuous time omain unstable/stable zero(s) is clearly escribe. The effectiveness of the propose metho is verifie by simulations. II. Perfect Tracking Control metho base on multirate feeforwar (conventional) There are two types of zeros in iscretize transfer functions calle intrinsic zeros an iscretization zeros [][2].

2 In the case of the plant with stable zeros in the continuous time omain, the intrinsic zeros are stable in the iscretize omain. However, the iscretize zeros are unstable when the relative egree of the continuous time plant is greater than two [3]. Because of this reason, in the single rate control scheme, the perfect tracking control (PTC) efine in [4] is impossible even without moeling error an isturbance. In the multirate control scheme, the perfect tracking control can be achieve [22] for the plant with unstable iscretize zeros. Perfect tracking control metho base on multirate feeforwar has been applie for high-precision stages [23][24], har isk rives [25], atomic force microscopes [26], machining tools [27] etc. However, this metho cannot be applie for the plant with continuous time omain unstable zeros without approximation because the state trajectory iverges. This problem is solve by the state trajectory generation base on time axis reversal propose in the section III. A. efinition Nominal plant in continuous time omain is efine as a control canonical form: P c (s) = B(s) A(s) = b ms m + b m s m + + b s n + a n s n + a () ẋ(t) = A c x(t) + b c u(t) (2) y(t) = c c x(t) (3) x = x x 2. x n, A c =... a a a 2 a n b c = [ ] T c c = [ b b b m ] B(s) an A(s) enotes the irreucible numerator an enominator of P c (s). n an m(< n) enotes the nominal plant orer an the number of the zeros, respectively. Discretize plant of (2) an (3) by zero-orer-hol with sampling time T u is efine as (4) x[k + ] = A s x[k] + b s u[k] (5) A s = e A ct u, b s = y[k] = c s x[k], (6) Tu B. State trajectory x generation e A cτ b c τ, c s = c c. (7) Accoring to (3), in orer to track the reference position trajectory r(t), the esire state trajectory x shoul satisfy r(t) = c c x (t). (8) In plant without continuous time zeros cases, the state trajectory x = [x x 2 x n ] T becomes b [r sr s n r] T Fig. 2. Multirate sampling perio. consiering c c = [b ] in (8). For instance, in the case of a secon orer rigi boy plant, the state trajectory coincies the position an velocity references. Generally, in the plant with continuous time zeros case, the state trajectory is obtaine by [25] x (t) = t f (t τ)r(τ)τ, (9) [ ] f (t) = L, () B(s) r(t) = [ r (t) r 2 (t) r n (t) ] T = [ s s n] T r(t). () Here, L enotes one-sie Laplace transform. In the case of plant with continuous time unstable zeros, x (t) in (9) iverges because B(s) is unstable. This problem is solve by the time axis reversal propose in the section III. C. Feeforwar output u o generation from x Effect of unstable iscretization zeros can be avoie by multirate feeforwar control [22]. Here, as shown in Fig. 2, there are three time perios T y, T u, an T r which enote the perio for y(t), u(t), an r(t). These perios are set as T r = nt u = nt y. The multirate system of (5) an (6) are written as x[i + ] = Ax[i] + Bu[i], y[i] = cx[i], (2) A = A n s, B = [ A n s b s A n 2 s b s A s b s ] b s c = c s, x[i] = x(it r ) (3) by calculating the state transition from t = it r = kt u to t = (i + )T r = (k + n)t u. Here, the input vector u[i] is efine in the lifting form: u[i] = [ u [i] u 2 [i] u n [i] ] T = [ u(kt u ) u((k + )T u ) u((k + n )T u ) ] T (4) Accoring to (2), the feeforwar output u o [i] is obtaine from the previewe state trajectory x [i + ] as following z enotes e st r. u o [i] = B (I z A)x [i + ], (5)

3 State variable reference generation State variable reference generation Multirate feeforwar (stable inversion Multirate for unstable feeforwar iscretization zeros) (stable inversion for unstable iscretization zeros) Fig. 3. Perfect Tracking Control metho base on multirate feeforwar propose in reference [22]. S, H, H M enote a sampler, a holer, an a multirate holer, respectively. z an z s enote e str an e stu, respectively. State trajectory generation State with trajectory time axis generation reversal (stable inversion with time for axis unstable reversal intrinsic zeros) (stable inversion for unstable intrinsic zeros) Stable part state trajectory generation Stable part state trajectory generation Multirate feeforwar (stable inversion Multirate for unstable feeforwar iscretization zeros) (stable inversion for unstable iscretization zeros) Unstable part state trajectory generation Unstable part state trajectory generation Fig. 4. Propose Preactuation Perfect Tracking Control metho base on multirate feeforwar an state trajectory generation by time axis reversal. III. Preactuation Perfect Tracking Control metho base on multirate feeforwar (propose) A. Design steps In the case of plant with continuous time unstable zeros, the state trajectory iverges accoring to (9). In this section, a stable variable reference generation metho for the plant with continuous time omain is formulate. ) Separation to a stable an unstable part: B(s) = F st (s) + F ust (s) (6) f st (t) = L [ F st (s) ], f ust (t) = L [ () l F ust ( s) ] (7) F st (s) an F ust (s) have the stable an unstable pole(s), respectively. l enotes the orer of F ust (s). Note that F ust ( s) is stable. 2) Stable part state trajectory generation: As for the stable part, the esire state trajectory (t) is generate forwarly. (t) = [ (t) = t xst 2 (t) xst n (t)] T f st (t τ)r(τ)τ (8) 3) Unstable part state trajectory generation: As for the unstable part, the esire state trajectory (t) is generate by (t) = [ (t) xust 2 (t) xust n (t)] T t = f ust ( t τ)r( τ) τ t= t (9) (t) is calculate by two steps. First, a convolution of the time reverse reference position trajectory r( t) an the stable signal f ust ( t) is calculate. Secon, time axis is reverse. This metho is base on the two-sie Laplace transform [28][29]. 4) State trajectory generation: 5) Multirate feeforwar: x (t) = (t) + xust (t) (2) u o [i] = B (I z A)x [i + ] (2) B. Preactuation an postactuation The propose metho shown in the section III-A clearly shows the relationship between the pre/post actuation an the continuous time omain unstable/stable zero(s). The stable zeros in the continuous time omain result in postactuation accoring to (8). In contrast, the unstable zeros in the continuous time omain result in the preactuation accoring to (9). On the other han, the iscretization zeros are compensate by the multirate feeforwar with preview formulate in (2). A. Simulation conition IV. Simulation results In this section, simulations are performe by the moel illustrate in Fig. 5(b). This moel assumes a high-precision stage shown in Fig. 5(a) with a current feeback. Here, the continuous time omain transfer function from the current

4 Linear encoer Carriage Table Linear motor Linear encoer (a) High-precision stage. Air guie (measure) (b) Moel of (a). Fig. 5. Experimental high-precision stage an its moel for simulation[24], [6], [3]. reference of x axis actuator which generates force f x to the measure stage position x is efine as P c (s) = 3.48 (.228 L m )s s s(s + )(s +.846)(s s ), (22) L m enotes the height of the measurement point illustrate in Fig. 5(b). From (22), in the case of.228 < L m, P c (s) has a continuous time unstable zero. In this paper, L m =.3 is consiere: P c (s) = 599(s 4.2)(s ) s(s + )(s +.846)(s s ) (23) Discretize transfer function of (23) by zero-orer-hol with µs sampling is obtaine as P s [z s ] = 2.2 (z s )(z s.4)(z s.9862)(z s +.245), (24) (z s )(z s.9998)(z s.3679)(z 2 s.999z s ) z s enotes e st u. Boe iagram of P c (s) an polezero map of P c (s) an P s [z s ] are shown in Fig. 6 an 7. In the continuous time omain, P c (s) has one stable zero (s = 38.9) an one unstable zero (s = +4.2). In the iscrete time omain, P s [z s ] has the intrinsic zeros at z s = +.4 an the iscretize zeros at z s = 2.97,.245. Here, P s [z s ] has one unstable intrinsic zero an one unstable iscretize zero. Step target trajectory r(t) is esigne as Fig. (a) by 9th orer polynomial uring step motion. Step time is set as.2 [s]. In Fig. 4 an 8 configurations, the feeback controller C f b [z s ] generates force only if moeling error or isturbance exist. Simulation step is set as µs. Simulations are conucte between. [s] < t <. [s]. B. Simulation results of the propose metho Simulation results are shown in Fig. 9 an. The reference position trajectory r(t) an the generate state trajectory x (t), (t), xust (t) are shown in Fig. 9. The continuous time omain unstable zero generates the state trajectory in the negative time omain by (8). Fig. (a) an (c) inicate that the output trajectory y(t) can track the reference position trajectory r(t) without any unershoot or overshoot. Fig. (c) emonstrates the perfect tracking is achieve for every T r = 5 µs by the propose metho. Fig. (e) shows the preactuation (t < ) an postactuation (.2 < t). Fig. 9 an Fig. (e) inicate that the preactuation an postactuation are cause Magnitue [B] Phase [eg] P c(s) Frequency [Hz] Fig (a) P c (s) (c) P s [z s ]. Approximate plant inverse Boe iagram of P c (s) (b) P c (s) zoom () P s [z s ] zoom. Fig. 7. Pole-zero map of P c (s) an P s [z s ]. Fig. 8. Approximate plant inverse feeforwar control configuration. Simulation block iagram for NPZI,, methos. by the (t) an xst (t), respectively. Note that the unstable iscretization zero (z s = 2.97) is compensate by the multirate feeforwar scheme introuce in the section II. From the above, the effectiveness of the propose Preactuation Perfect Tracking Control metho base on multirate feeforwar an state trajectory is verifie. C. Comparison of approximate inverse methos Simulation results of the NPZI metho [5], the metho [4], an the metho [6] are also shown in Fig.. Block iagram shown in Fig. 8 is use for these methos. These three methos are esigne by sampling

5 x [m] 2 x 3 r (t) (t) (t) 8 x (t) (a) r (t), x st (t), xust (t), x (t)..5 x 5 x2 [m/s] r 2 (t) 2 (t) 2 (t) x 2 (t) (b) r 2 (t), x2 st (t), xust 2 (t), x 2(t). 4 x 7 r 5 (t) 5 (t) 5 2 (t) x 5 (t) x3 [m/s 2 ] 3 2 r 3 (t) 2 3 (t) 3 (t) x 3(t) (c) r 3 (t), x3 st (t), xust 3 (t), x 3(t). x4 [m/s 3 ] x5 [m/s 4 ].5 r 4 (t) 4 (t) 4 (t) x 4(t) () r 4 (t), x4 st (t), xust 4 (t), x 4(t) (e) r 5 (t), x5 st (t), xust 5 (t), x 5(t). Fig. 9. Generate state trajectory by time axis reversal (see equations (8), (9), an (2)). Position [m] r(t) NPZI (a) Position. Tracking error [m] NPZI (b) Tracking error. Tracking error [m].5 x 9.5 NPZI x 3 (c) Tracking error (zoom). Fig.. Force [N].5 x 4.5 NPZI () Force. Force [N] (e) Force (). Simulation results of the plant with continuous time unstable zero. From (c), the perfect tracking is achieve for every T r = 5 µs.

6 perio T u. In the, preview is use to achieve the zero phase error characteristics. These methos create the unershoot an/or overshoot to compensate for the unstable intrinsic zero (z s = +.4) an the unstable iscretization zero (z s = 2.97). From the above, without the preactuation, there are trae-offs between the unershoot an/or overshoot amplitue an settling time. V. Conclusion This paper proposes the Preactuation Perfect Tracking Control metho base on multirate feeforwar an state trajectory generation by time axis reversal. In the iscretize omain, there are two types of zeros: ) the intrinsic zeros which have counterparts in the continuous time omain, 2) the iscretization zeros generate by iscretization. In the non-collocate system, the iscretize plant has unstable intrinsic zeros. On the other han, the iscretize zeros become unstable when the relative egree of the plant in the continuous time omain is greater than two. Propose metho eals with problem ) an 2) separately. The unstable intrinsic zeros are compensate by the preactuation. The reference of the preactuation is generate by the negative time omain state trajectory calculate by the time axis reversal proceure. Next, the unstable iscretization zeros are compensate by the multirate feeforwar scheme with preview. The effectiveness of propose metho is emonstrate by simulations. Comparison with other preview/preactuation base methos will be performe. Acknowlegment This work is supporte by JSPS KAKENHI Grant Number 5J8488. References [] T. Hagiwara, T. Yuasa, an M. Araki, Stability of the limiting zeros of sample-ata systems with zero-an first-orer hols, International Journal of Control, vol. 58, no. 6, pp , 993. [2] T. Hagiwara, Analytic stuy on the intrinsic zeros of sample-ata systems, IEEE Transactions on Automatic Control, vol. 4, no. 2, pp , 996. [3] K. Åström, P. Haganer, an J. Sternby, Zeros of sample systems, Automatica, vol. 2, no., pp. 3 38, 984. [4] J. Hoagg an D. 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