Proceeings of the 17th Worl Congress The International Feeration of Automatic Control An algebraic expression of stable inversion for nonminimum phase systems an its applications Takuya Sogo Chubu University, Aichi 487-8501, Japan (Tel: +81-568-51-9771; e-mail: sogo@isc.chubu.ac.jp). Abstract: This paper proposes an algebraic expression of noncausal stable inversion base on the two-sie Laplace transform, which is a classic mathematical tool but has not been use very much in the fiel of control engineering. This expression brings an avantage that computing of stable inversion is reuce to simulation of the response of the plant an the reverse of time horizon without solving the bounary value problem of state-space equations as the conventional efinition of stable inversion. An illustrative example emonstrates that this approach is useful to reuce the loa of the programming for a search algorithm to etermine the shape of a transition trajectory uner the constraint of input saturation. As another application of the propose expression, evelope is a metho of iterative learning control to obtain stable inversion for infinite imensional systems. An experiment to apply the iterative learning control to tip control of a flexible arm is reporte to emonstrate the effectiveness of the metho. Keywors: Stable inversion, Input esign, Iterative learning control, Dynamic inversion, Nonminimum phase systems, Feeforwar control, Preview control, Input shaping, Laplace transforms 1. INTRODUCTION System inversion plays crucial roles in many control applications such as perfect tracking, transient response shaping, isturbance attenuation, an noise cancellation. For example, consier shaping the transient response of a plant G(s). Then, if 1/G(s) is stable, one can employ F (s) = M(s)/G(s) as a prefilter of G(s), where M(s) is a moel that has a esire response. If G(s) is a nonminimum phase system or equivalently 1/G(s) is unstable, F (s) = M(s)/G(s) cannot be use as a prefilter. However it was propose to substitute the optimal function of the moel matching problem min G(s)F (s) M(s) F (s) RH (1) for a prefilter instea of the unstable F (s) = M(s)/G(s). This approach has an avantage that both prefilters an feeback controllers can be esigne in the same framework of H optimization(limebeer et al. (1993)). It is recognize that feeback controller esign base on the transfer function is very effective from the viewpoint of robustness or sensitivity esign. However, effectiveness of the prefilter esigne by the aforementione approach is controversial from the viewpoint of response shaping. On the other han, noncausal stable inversion technique was propose in orer to calculate a boune input which achieves perfect tracking even if 1/G(s) is unstable, equivalently, M(s)/G(s) / RH. (Devasia et al. (1996); Zou an This work was supporte in part by a Grant-in-Ai for Scientific Research (C) No.18560444 from the Japan Society for the Promotion of Science an a grant from the High-Tech Research Center Establishment Project from the Ministry of Eucation, Culture, Sports, Science an Technology, Japan. Devasia (1999); Hunt et al. (1996)). This technique utilizes noncausal or preview information of esire trajectories in orer to generate boune input profiles. In most of the work reporte so far, the calculation of stable inversion is base on solving the bounary value problem of the inverte state-space equations. This means that transfer functions are necessarily transforme into state-space representations to esign stable inversion even if feeback controller esign is base on transfer functions which are common in control engineering. In orer to improve this inconvenience, this paper introuces the two-sie Laplace transform to express stable inversion an its calculation simply as unstable transfer function 1/G(s) an noncausal convolution, respectively. It will be shown that the propose transfer-function expression of inversion is actually equivalent to the conventional stable inversion expresse by state-space representations; this simplifies computing the input profile for stable inversion. It will be emonstrate that stable inversion base on the transfer-function expression can be applie to infinite imensional systems. This will be illustrate by an example of iterative learning control applie to tip positioning of a flexible arm. In this paper, we use following notations: For f(t) : (, + ) R, norms of f(t) are efine as f(t) := ess sup t (,+ ) f(t), f(t) 1 := + f(t) t an f(t) 2 := + f(t) 2 t. Spaces of functions which have boune 1, 2 an norms are enote by L 1, L 2 an L, respectively. For F (s) : C C, norms of F (s) 978-1-1234-7890-2/08/$20.00 2008 IFAC 2820 10.3182/20080706-5-KR-1001.1405
are efine as F (s) = ess sup ω (,+ ) F (jω) an F (s) 2 := + F (jω) 2 ω. 2. STABLE INVERSION FOR LTI SYSTEMS Consier a plant c 0 s m + + c m 1 s + c m G(s) = s n + a 1 s n 1 (2) + + a n 1 s + a n with the controllable canonical form of the state-space representation (A, B, C). We assume that all poles of G(s) or eigenvalues of A are in the left half plane. Then Silverman s inversion (Silverman (1969)) of the system is expresse as ẋ n m+1 x n m+1 0. ẋ = Φ. n 1 x +. n 1 0 y(n m) (3) ẋ n x n 1/c 0 u = Γ x n m+1. x n 1 x n + Λ where y (k) = k y/t k, equivalently y. y (n m 1) y (n m) 1/G(s) = ( s n m + ā 1 s n m 1 + + ā n m ) /c0 (4) + ān m+1s m 1 + + ā n c 0 s m + c 1 s m 1 + + c m (5) If G(s) is a nonminimum phase system, Φ has eigenvalues in the right half plane an the solution of the initial value problem of (3) is unboune. This means that the input to achieve perfect tracking to an output trajectory y is unboune in the causal framework. However, if the constraint of causality is not impose on (3), there exists an input to achieve perfect tracking for a class of output trajectories. Proposition 1. [Devasia et al. (1996); Hunt et al. (1996)] Assume that y (i) L 1 L (i = 0, 1,, n m) (6) an G(s) has no zero on the imaginary axis. Then, there exist boune x (t) an u (t) such that an ẋ = Ax + Bu (7) y = Cx (8) u (t) 0, x (t) 0 as t ± (9) Functions x (t) an u (t) given in Proposition 1 are obtaine by solving the ifferential equation (3) uner the bounary conition (9). Calculating the boune input u (t) efine by Proposition 1 requires the following steps: (1) transform of the transfer function into the state-space equation (2) inversion of the state-space equation (3) solving of the bounary value problem In orer to reuce these steps, we evelop a simple metho base on transfer functions in the next section. 3. ALGEBRAIC EXPRESSION OF STABLE INVERSION Since all elements in feeback control systems are online an causal, the one-sie Laplace transform is wiely use to analyze control systems algebraically. In orer to import noncausal elements into the same framework, we introuce the two-sie Laplace transform(van er Pol an Bremmer (1987); Papoulis (1962)), which is a classic mathematical tool but has not been very common in the fiel of control engineering 1. For a function g(t) efine on the infinite time horizon, the two-sie Laplace transform is efine as L[g(t)](s) = G(s) := + e st g(t)t (10) where the region of convergence is the strip {s; γ 1 < Re(s) < γ 2 } (Van er Pol an Bremmer (1987); Papoulis (1962)). It shoul be note that [ t L g(τ)τ [ + L g(t τ)f(τ)τ L[g (t)](s) = sg(s) (11) ] (s) = G(s) (12) s ] (s) = G(s)F (s) (13) where F (s) = L[f(t)](s). The inverse transform is expresse by α+j L 1 [G(s)](t) = g(t) = 1 e st G(s)s 2πj α j Res(e st G(s), p n ) t 0 Re(p = n )<α Res( e st (14) G(s), p m ) t < 0 Re(p m )>α where {p n } an {p m } enote the sets of poles of G(s) which are in the left an right half plane of the vertical line s = α, respectively. The next theorem shows that inversion of the transfer function base on the two-sie Laplace transform is actually equivalent to stable inversion efine in Proposition 1. Theorem 2. Assume that y (i) L 1 L (i = 0, 1,, n m) an G(s) has no zero on the imaginary axis. Functions u an x satisfying (7), (8) an (9) are expresse by u (t) = L 1 [1/G(s) Y (s)] (15) x (t) = L 1 [(si A) 1 BU (s)] (16) where Y (s) = L[y ] an U (s) = L[u ]. Proof: From the assumption on y, we have L 1 [(s n m + ā 1 s n m 1 + + ā n m )Y (s)] L 1 L (17) Since G(s) has no zero on the imaginary axis, α of (14) for the fraction term of (5) can be chosen as 0. This implies L 1 [ ān m+1 s m 1 + + ā n c 0 s m + c 1 s m 1 + + c m ] L 1 L (18) 1 As far as the author knows, an application to ientification of systems with elay was reporte in the fiel of control engineering(kachanov an Khrolovich (1993)) 2821
moreover [ L 1 ān m+1 s m 1 ] + + ā n c 0 s m + c 1 s m 1 Y (s) + + c m L 1 L (19) From (17) an (19), u efine by (15) satisfies u (t) L 1 L. Since all eigenvalues of A are in the left half plane, u (t) L 1 L implies x (t) L 1 L which leas to (9). From the efinition of G(s) an (A, B, C), u an x efine by (15) an (16) satisfies (7) an (8). This completes the proof. Remark 3. It shoul be note that the input profile (15) is efine even though y (i) / L 1 L. (See Example 1 below) This implies that the inversion base on the twosie Laplace transform generalizes the stable inversion efine by Proposition 1. The next corollary shows that the noncausal convolution for (15) can be substitute by the orinary causal convolution for two stable systems with the reverse of the time horizon. In the following iscussion, L enotes the conventional one-sie Laplace transform, namely L[g(t)](s) = G(s) := + 0 e st g(t)t (20) Corollary 4. Assume that y (i) L 1 L (i = 0, 1,, n m) an G(s) has no zero on the imaginary axis. Let 1/G(s) = ( s n m + ā 1 s n m 1 + + ā n m ) /c0 + F l (s) F r (s) (21) where F l (s) an F r (s) are proper transfer functions all poles of which are in the left an right half plane, respectively. Then u (t) efine by (15) is expresse by u (t) =(y n m v(t) = where + t σ (t) + ā 1 y n m 1 (t) + + ā n m y (t))/c 0 f l (t τ)v(τ)τ (22) f r (σ τ)y ( τ)τ σ= t (23) f l (t) = L 1 [F l (s)] (24) f r (t) = L 1 [F r ( s)] (25) Proof: From (14) with α = 0, we have Res(e st F l (s), p n ) if t 0 L 1 [F l (s)] = n an (26) = L 1 [F l (s)] = f l (t) (27) Res(e st F r ( s), p m ) if t 0 L 1 [F r ( s)] = m (28) = L 1 [F r ( s)] = f r (t) (29) j L 1 [F r (s)] = 1 e st F r ( s)s (30) 2πj +j 0 if t 0 = Res( e st F r ( s), p m ) if t < 0 m (31) = f r ( t) (32) These equalities with (11) an (13) imply (22) an v(t) = which leas to (23). + t f r ( (t τ))y (τ)τ (33) From Corollary 4, one can obtain the input profile base on the stable inversion by the following steps: (1) ecomposition of the proper part of the inverte transfer function into the cascae connection of the stable an antistable part (2) computation of the causal convolution or the solution of the initial value problem with reversing the time horizon Example 1. Consier a nonminimum phase system G(s) = (s + 4)(3 s) s 3 + 2s 2 + 3s + 4 an a function e 1 t/3 y (t) = if 0 t 3 e 1 t/3 + e 1 1 t/3 1 if t > 3 (34) (35) as the esire output trajectory of (34); the function (35) is a C function which is monotonously increasing from 0 to 1 an Y (s) = L[y (t)] exists for {s; 0 < Re(s)}. To calculate the input that achieves perfect tracking to (35), express 1/G(s) as 1 4s + 16 = s 1 + G(s) s + 4 1 3 s (36) then one can see that L 1 [1/G(s) Y (s)] is well-efine an boune since y (t) C an L 1 [(4s + 16)/{(s + 4)(3 s)}y (s)] L with {s; 0 < Re(s) < 3}. The input profile u (t) = L 1 [1/G(s) Y (s)] is compute by (23) an (22) with F l (s) = 4s + 16 s + 4, F r( s) = 1 s + 3 (37) which can be easily compute by a stanar tool for numerical simulation (e.g. lsim comman in MATLAB). It shoul be note that the convolution over the infinite time horizon for (35) can be approximate by a convolution over a sufficiently long time interval with truncation. Fig. 1 shows the compute u an its response. Remark 5. For MIMO systems, algebraic expression of stable inversion correspons to the inversion of transfer function matrix. The time-omain calculation propose in Corollary 4 can be applie to each element of the transfer function matrix. 2822
Fig. 1. The input profile an its response for Example 1 Fig. 2. The input profile with T = 1 an its response for Example 2 4. APPLICATIONS OF TRANSFER FUNCTION EXPRESSION OF STABLE INVERSION In the preceing section, suggeste was that the algebraic expression of stable inversion has avantages in computing an the range of applications. This section presents two practical examples. 4.1 Application to optimization of a transition trajectory uner input constraint In many control applications, of importance is shaping a trajectory that achieves a fast monotonous transition between two constant values uner input saturation: u(t) u max for t (, + ) (38) Example 2. Consier (34) with a C 1 function 3! 1 ( 1) 1 i y (t) = T 3 i!(1 i)!(3 i) T i t 3 i if 0 t T i=0 1 if t > T (39) as a esire trajectory where T is the time parameter of transition that affects the peak value of the input(piazzi an Visioli (2005)). Fig. 2 shows the input profile u (t) that achieves y(t) = y (t) with T = 1. It was proven that if u max 1 G(0) (40) Fig. 3. The input profile satisfying the constraint an its response for Example 2 the feasible set {T } for (38) is nonempty an there exists the minimum transition time T min uner the constraint(piazzi an Visioli (2005)). One can fin T min practically by a bisection search algorithm with respect to T. It shoul be note that Corollary 4 reuces the loa of coing of the search algorithm since no explicit expression u (t) for each y (t) is require. For u max = 0.5 (41) a value T = 1.4609 was foun by a bisection search starting from the interval [1, 2] for T. Fig. 3 shows the foun input u (t) an its response y(t). 4.2 Application to an infinite imensional system The algebraic expression of stable inversion presente in Section 3 suggests that stable inversion is essentially applicable to infinite imensional systems. It is, however, practically ifficult to execute the computation given in Corollary 4 because of the ifficulty of ientification of the infinite-imensional moel. To circumvent this, we evelop an iterative metho to obtain the esire input profile without the ecomposition of 1/G(s). Theorem 6. Assume that there exists L 1 [1/G(s) Y (s)] L 1 L (42) an G(s) has no zeros on the imaginary axis an G(jω) 0 as ω. Let y (t) be a esire output trajectory an Y (s) = L 1 [y (t)]. Then, the sequence of input functions {U k (s); k = 0, 1, } efine by U 0 (s) 0 (43) U k+1 (s) = U k (s) αg( s){g(s)u k (s) Y (s)} (44) satisfies U k (s) 1/G(s) Y (s) 2 0 as k (45) where α is a constant satisfying 0 < α < 1/ G(s) 2. Proof: Let E k (s) = U k (s) 1/G(s) Y (s) Then we have E k+1 (s) = (1 αg( s)g(s))e k (s) which leas to E k+1 (jω) = (1 α G(jω) 2 )E k (jω) moreover E k (jω) 2 = (1 α G(jω) 2 ) 2k E 0 (jω) 2 Since 1 α G(jω) 2 < 1 for ω (, + ), we have +Ω E k (s) 2 2 Ω + (1 α G(jω) 2 ) 2k E 0 (jω) 2 ω Ω< ω E 0 (jω) 2 ω (46) 2823
for any positive Ω. Note that (42) implies E 0 (s) = 1/G(s) Y (s) L 2. Then we have E 0 (jω) 2 ω 0 (47) Ω< ω as Ω. Hence, for any given ɛ > 0, there exist Ω an an positive integer N such that E k (s) 2 2 < ɛ/2 + ɛ/2 for all k > N. This completes the proof. The algorithm given in Theorem 6 is equivalent to a timeomain algorithm which requires only measuring of the response of the plant G(s). Corollary 7. Assume the same properties as Theorem 6 an all poles of G(s) are in the left half plane. Then the sequence of inputs {u k (t); k = 0, 1, } efine by t η k (t) = g(t τ)u k (τ)τ y (t) (48) σ u k+1 (t) = u k (t) α g(σ τ)η k ( τ)τ (49) σ= t satisfies u k (t) L 1 [1/G(s) Y (s)] 2 0 (50) as k. Proof: The assumption of poles of G(s) with (13) an (14) implies (48). Since all poles of G( s) are in the right half plane, (49) is obtaine by the same iscussion as the proof of Corollary 4. By the Parseval equality, (50) follows (45). In contrast to the metho in Corollary 4, there is no nee of the ecomposition of the inverte transfer function for the metho given in Corollary 7. Moreover, all require integrations are the convolution for G(s), which is obtaine as the response of G(s). This can be one experimentally on real plants without ientifying a moel of the plant. The iterative metho presente in Corollary 4 is summarize as the following steps. (0) k := 0; u 0 (t) 0 (1) measure the response of G(s) for the input u k (t) an recor the error η k (t) (2) measure the response of G(s) for the time-reverse error η k ( t) (3) upate the input with the time-reverse response an the last input (4) k := k + 1 an go to (1) Remark 8. The iterative metho presente in Corollary 7 is actually a generalization of iterative learning control using ajoint systems which was propose with statespace representations(kinoshita et al. (2002)). Theorem 6 extens the applicable range of iterative learning control(kinoshita et al. (2002); Markusson et al. (2001); Owens an Hätönen (2005); Moore (1993)). Example 3. Consier tip control of a experimental flexible arm epicte in Fig. 4 (The length of the arm: L = 3.0 10 1 m, the moment of inertia of the arm incluing the hub: I = 637.4 10 6 Kg m 2 ). The hub is irectly riven by a DC motor; the rotational angle θ(t) [ra] with respect to the inertial reference frame is measure by a rotary encoer embee in the motor. The eflection of the tip w(l, t) with respect to the frame fixe on the hub is measure by an optical evice on the hub, which senses the horizontal location of a light source attache to the Fig. 4. experimental setup of the flexible arm tip. Assuming that the eflection of the tip is sufficiently smaller than the length of the arm, we consier y(t) = Lθ(t) + w(l, t) [m] (51) as the position of the tip. The ynamics of the motor is expresse by T (s) = K m /R V in (s) Km/R 2 θ(s) where K m = 7.67 10 3 Nm/A an R = 2.60Ω. Observations base on the Euler-Bernoulli moel lea to transfer functions of the flexible arm as follows(cannon an Schmitz (1984)): θ(s) T (s) = 1 Is + 1 a i s I s 2 + 2ζ i=1 i ω i s + ωi 2 (52) y(s) T (s) = P (s) = L Is 2 + 1 I i=1 A PD-feeback with a reference input v k i s 2 + 2ζ i ω i s + ω 2 i (53) V in (t) = K P θ(t) K D θ(t) + v(t) (54) is applie to the aforementione experimental setup; PD gain K P an K D are experimentally chosen to make the system stable. Letting a esire trajectory of the tip position as 0 0 t 2 πl y (t) = 2 { f(t)6 + 3f(t) 4 3f(t) 2 + 1} 2 t 3 0 3 t 5 (55) where f(t) = 2(t 2.5), conucte was an experiment to apply the scheme of iterative learning control given in Corollary 7 to upate the reference input v(t) with respect to the trajectory of the error y(t) y (t) without ientifying the transfer function G(s) = L[y(t)]/L[v(t)] The length of the time horizon for the experiment is chosen sufficiently long in view of the interval of the support of y (t) an the ecay time of the impulse response of G(s) estimate by an experiment. Fig. 5 an 6 show the input v k an tip position y k with y, respectively, for the number of iteration k = 1 an k = 15. Fig. 7 shows the tip eflection w(l, t) with the tip position y k for k = 15 (Nishiki (2004)). 5. CONCLUSION This paper propose an algebraic expression of noncausal stable inversion base on the two-sie Laplace transform, which is classic but has not been use very much. It was shown that this simple expression brings an avantage that computing of stable inversion is reuce to simulation of the response of the plant an the reverse of the time 2824
Application of the aforementione iea to iscrete-time or sample-ata systems is straightforwar by introucing the two-sie z-transform. This helps us to clarify a relation between inversion of sample-ata an continuoustime systems, the former of which mostly has unstable zeros even if the latter has no unstable zero. In another paper, the author shows that the former with a small sample time actually approximates the latter(sogo (2008)). ACKNOWLEDGEMENTS The author is grateful to Masakazu Nishiki for his help to conuct the experiment. Fig. 5. Input profiles for k = 1 an k = 15 Fig. 6. Output for k = 1 an k = 15 with the esire trajectory Fig. 7. Output with the tip eflection for k = 15 horizon without solving the bounary value problem as the conventional efinition of stable inversion. An illustrative example emonstrates that this approach is useful for a search algorithm to choose a transition trajectory for the case where there is an input saturation. Moreover, algebraic expression extene the application class of stable inversion into infinite imensional systems. A practical iterative metho to obtain stable inversion for such systems was propose. An experiment to apply the metho to tip control of a flexible arm was presente. REFERENCES Robert H. Cannon an Eric Schmitz. Initial experiments on the en-point control of a flexible one-link robot. The International Journal of Robotics Research, 3(3):62 75, 1984. S. Devasia, D. Chen, an B. Paen. Nonlinear inversionbase output tracking. IEEE Transactions on Automatic Control, 41(7):930 942, 1996. L. R. Hunt, G. Meyer, an R. Su. Noncausal inverses for linear systems. IEEE Transactions on Automatic Control, 41(4):608 611, 1996. B. O. Kachanov an K. B. Khrolovich. Metho for ientification of ynamic systems with elays. Automation an Remote Control, 54(1):60 64, 1993. K. Kinoshita, T. Sogo, an N. Aachi. Iterative learning control using ajoint systems an stable inversion. Asian Journal of Control, 4(1):60 67, 2002. D. J. N. Limebeer, E. M. Kasenally, an J. D. Perkins. On the esign of robust two egree of freeom controllers. Automatica, 29(1):157 168, 1993. O. Markusson, H. Hjalmarsson, an M. Norrlöf. Iterative learning control of nonlinear non-minimum phase systems an its application to system an moel inversion. Proc of the 40th IEEE Conference on Decision an Control, 2001. K. L. Moore. Iterative Learning Control for Deterministic Systems. Springer-Verlag, 1993. M. Nishiki. Iterative learning control using input-output ata for linear nonminimum phase systems. Master s thesis at the grauate school of infomatics, Kyoto University, 2004. (in Japanese). D. H. Owens an J. Hätönen. Iterative learning control an optimization paraigm. Annual Reviews in Control, 29: 57 70, 2005. A. Papoulis. The Fourier Integral an Its Applications. McGraw-Hill, 1962. A. Piazzi an A. Visioli. Using stable input-output inversion for minimum-time feeforwar constraine regulation of scalar systems. Automatica, 41(2):305 313, 2005. L. M. Silverman. Inversion of multivariable linear systems. IEEE Transactions on Automatic Control, AC-14(3): 270 276, 1969. T. Sogo. Inversion of sample-ata system approximates the continuous-time counterpart in a noncausal framework. Automatica, 44(3):823 829, 2008. B. Van er Pol an H. Bremmer. Operational Calculus base on the Two-sie Laplace Integral. Chelsea Publishing Company, 1987. Q. Zou an S. Devasia. Preview-base stable-inversion for output tracking of linear systems. ASME Journal of Dynamic Systems, Measurement, an Control, 121: 625 630, 1999. 2825