An Analysis and Comparison of Frequency-domain and Time-domain Input Shaping*

The technique of input shaping has been successfully applied to the problem of maneuvering flexible structures without excessive residual vibration. With input shaping, non-negative impulse shapers are often desired because they can be used with any arbitrary (unshaped) commands and not cause actuator saturation (if these original unshaped commands do not cause actuator saturation). We outline conditions when non-negative amplitude shapers will result when using frequency-domain methods of input shaping, and we draw comparisons with timedomain input shaping in terms of shaper length (speed) and number of impulses (ease of implementation). 1.0 INTRODUCTION Input shaping is a feedforward technique used to reduce residual vibration in flexible structures. The system parameters (frequency and damping) are used to design an input shaper, which is an impulse sequence that is convolved with the input. This shaped input is the new input to the system. If the parameters used to design the shaper are accurately known, all or most of the residual vibration would disappear after the time of the last impulse. This time, of the last impulse, is referred to as the length or duration of a shaper. Favorable characteristics result if the following two constraints are satisfied: (1) All impulse amplitudes are nonnegative. (2) The sum of the impulse amplitudes is one. If the amplitudes sum to one, the final set point resulting from the shaped input will be the same as that resulting from the original unshaped input. In addition, all non-negative amplitudes prevent undue actuator saturation. A number of researchers have investigated command shaping methods in the frequency domain [1,5,6]. The purpose of this paper is to establish some constraints that guarantee the existence of a non-negative impulse shaper solution when using a frequency-domain (FD) zero-placement method [6] and to compare the performance of this frequency-domain method with conventional time-domain (TD) input shapers. Section 2 reviews TD and FD input shaping. Section 3 establishes conditions which guarantee that FD input shaping will yield non-negative impulse shapers. Section 4 determines the shortest length shapers that can be achieved for two-mode, zero damping, flexible systems using FD zero-placement input shaping. Finally, concluding remarks are given in section 5. 2.0 INPUT SHAPING TECHNIQUES Singer and Seering [4] developed several TD input-shaping methods that only require knowledge of the natural frequency and damping of each flexible mode of a system. The simplest type of shaper is one that only guarantees zero residual vibration (a ZV shaper) at the modeling frequencies and damping ratios. For a multi-mode system, a one-mode shaper is computed for each mode, then the multi-mode shaper is obtained by convolving all single-mode shapers [4]. The length of a multi-mode ZV shaper is 1 /(2 f i ) i ∑ where fi is the frequency of the ith mode. Tuttle and Seering [6] established a FD method of placing zeros to cancel system vibration where the unwanted system poles are calculated in the z-plane. A discrete shaper is then formed by placing a zero on each pole (or more to add robustness). In order to keep the shaper causal, for each zero placed, an additional pole is placed (at the origin so that no new vibration is added to the system). Once the discrete transfer function is established, it is transformed from the z-plane to the splane by the mapping z = exp(sT) , where T is the impulse spacing. Taking the inverse Laplace transform yields a sequence of impulses. Sequences of impulses are solved for a range of T’s, and the smallest T that yields all non-negative amplitudes is chosen as the desired impulse spacing since it leads to the shortest shaper with all non-negative amplitudes. Using this zero-placement method, the general solution to a single mode shaper is: C(A0z (t)+ A1z(t −T ) + A2z (t − 2T )) where: A0 = 1, A1 = ) exp( ) 1 cos( 2 2 T T z w z w − − − , and A2 = exp(−zwT ) . A0 , A1 , and A2 are the impulse amplitudes, w and z are the natural frequency and damping of the flexible mode of the system, T is the impulse spacing, and C is a scaling constant. Since the first impulse is at t = 0 , the length of this shaper is 2T. A FD designed shaper for n modes is just the convolution of n single mode shapers, and the n-mode shaper is of length 2nT . Placing two zeros on one mode for added robustness is just a special case of a two-mode shaper where wa = wb and za = zb . The number of impulses for TD ZV multi-mode shapers, consisting of the convolution of n single-mode shapers is 2 ; as the number of modes increases, this quickly leads to a large number of impulses. The FD ZV zero-placement method results in 2n +1 impulses. Because the conventional TD method yields spacing between the impulses that is not constant, one might solve for all the impulse amplitudes simultaneously to decrease the number of impulses to 2n +1 , where the impulses satisfy 2n nonlinear plus one linear constraint equations [2,3]. Numerical routines are generally required to obtain shaper designs, and convergence to a solution is not always guaranteed. The FD zero*This work was supported in part under a National Science Foundation Early Faculty CAREER Development Award (Grant CMS-9625086), a University of Colorado Junior Faculty Development Award, and an Office of Naval Research Young Investigator Award (Grant N00147-97-1-0642). 0-7803-4530-4/98 $10.00 © 1998 AACC placement method provides a good alternative for multi-mode systems and can often be easier to implement. The number of impulses increases linearly with the number of modes, not exponentially; the impulse spacing is constant; and solving for the amplitudes does not involve complex numerical optimization packages. 3.0 EXISTENCE OF A POSITIVE SOLUTION It is not clear that a non-negative shaper solution always exists when using the zero-placement method. A sufficiency condition to ensure that the final multi-mode shaper has all nonnegative amplitudes is to ensure that each single mode shaper has all non-negative amplitudes. A0 (which equals one) is always positive. Knowing w > 0 , z ≥ 0 , and T ≥ 0 , A2 = exp(−2zwT ) will always be positive. The factor −2 exp(−2zwT ) in A1 is always negative, leaving the remaining factor, cos(wdT ) , as the only one that can change sign, where z w w 2 1− = d . If one can show that cos(wd T ) ≤ 0 , then all the amplitudes will be nonnegative. For a multi-mode zero-placement shaper, if one can show that there exists a T such that the shaper amplitudes for each mode are non-negative, then there indeed exists a T for the resultant convolved shaper such that all its amplitudes are nonnegative. Is it possible to demonstrate that for a number of flexible modes, wdi , that there exists a T such that cos(w di T ) ≤ 0 for all i ? A simple example can be used to show that there does not always exist a T such that these constraints are met for any number of frequencies. Pick wda ; wdb = 2wda ; wdc = 3wda ; wdd = 4w da . For any four frequencies meeting these conditions; there is no T at which all four signals cos(wdiT ) , i = a, b,c,d , are non-positive. It can be shown, however, that a T always exists if there are two frequencies. THEOREM Given A(t) = cos(2pfat + f a ) and B(t) = cos(2pfa t + f a) where f a = f b = 0 then ∃ t * s.t. A(t*) ≤ 0, B(t*) ≤ 0 ∀ fa and fb , where fb ≥ fa . Proof: Let ta and tb be the period of A(t) and B(t) , respectively, that is ta =1/fa and tb =1/fb . If fa ≤ fb ≤ 3 fa then tb /4 ≤ ta / 4 ≤3tb / 4 . We know that B(t) ≤ 0 ∀ t s.t. tb / 4 ≤ t ≤ 3tb / 4 and that A(ta /4) = 0 . ∴ ∃ a t * s.t. A(t*) ≤ 0, B( t*) ≤ 0 when fa ≤ fb ≤ 3 fa . If fb >3 fa , let ∆T be the interval: [ta/ 4,3ta / 4] , A(t) ≤ 0 ∀ t ∈∆T . For fb > 3fa , tb < ta /3 and the length(∆T ) = 3ta / 4 − ta / 4 = t a / 2 , and tb < ta / 3 < ta / 2 = length(∆T ) . Therefore B(t) must complete at least one cycle within ∆T and ∃ a t* ∈∆T s.t. B(t*) ≤ 0 . ∴ ∃ a t * s.t. A(t*) ≤ 0, B(t*) ≤ 0 when fb >3 fa . ∴ ∃ a t * s.t. A(t*) ≤ 0, B(t*) ≤ 0 when fb ≥ fa .