Impulse vector

An impulse vector, also known as Kang vector, is a mathematical tool used to graphically design and analyze input shapers that can suppress residual vibration. The impulse vector can be applied to both undamped and underdamped systems, as well as to both positive and negative impulses in a unified manner. The impulse vector makes it easy to obtain impulse time and magnitude of the input shaper graphically. A vector concept for an input shaper was first introduced by W. Singhose for undamped systems with positive impulses. Building on this idea, C.-G. Kang introduced the impulse vector (or Kang vector) to generalize Singhose's idea to undamped and underdamped systems with positive and negative impulses.

An impulse vector, also known as Kang vector, is a mathematical tool used to graphically design and analyze input shapers that can suppress residual vibration. The impulse vector can be applied to both undamped and underdamped systems, as well as to both positive and negative impulses in a unified manner. The impulse vector makes it easy to obtain impulse time and magnitude of the input shaper graphically. A vector concept for an input shaper was first introduced by W. Singhose for undamped systems with positive impulses. Building on this idea, C.-G. Kang introduced the impulse vector (or Kang vector) to generalize Singhose's idea to undamped and underdamped systems with positive and negative impulses. For a vibratory second-order system \omega_n^2 /(s^2 + 2 \zeta \omega_n + \omega_n^2 ) with undamped natural frequency \omega_n and damping ratio \zeta , the magnitude I_i and angle \theta_i of an impulse vector (or Kang vector) \mathbf{I}_i corresponding to an impulse function A_i \delta (t-t_i) , i = 1,2,...,n is defined in a 2-dimensional polar coordinate system as I_i = A_i e^{\zeta \omega_n t_i } \theta_i = \omega_d t_i where A_i implies the magnitude of an impulse function, t_i implies the time location of the impulse function, and \omega_d implies damped natural frequency \omega_n \sqrt{1-\zeta^2} . For a positive impulse function with A_i > 0 , the initial point of the impulse vector is located at the origin of the polar coordinate system, while for a negative impulse function with A_i , the terminal point of the impulse vector is located at the origin. □ In this definition, the magnitude I_i is the product of A_i and a scaling factor for damping during time interval t_i , which represents the magnitude A_i before being damped; the angle \theta_i is the product of the impulse time and damped natural frequency. \delta(t-t_i) represents the Dirac delta function with impulse time at t=t_i . Note that an impulse function is a purely mathematical quantity, while the impulse vector includes a physical quantity (that is, \omega_n and \zeta of a second-order system) as well as a mathematical impulse function. Representing more than two impulse vectors in the same polar coordinate system makes an impulse vector diagram. The impulse vector diagram is a graphical representation of an impulse sequence. Consider two impulse vectors \mathbf{I}_1 and \mathbf{I}_2 in the figure on the right-hand side, in which \mathbf{I}_1 is an impulse vector with magnitude I_1 (>0) and angle \theta_1 corresponding to a positive impulse with A_1 > 0 , and \mathbf{I}_2 is an impulse vector with magnitude I_2 = -I_1 and angle \theta_2 = \pi + \theta_1 corresponding to a negative impulse with A_2 . Since the two time-responses corresponding to \mathbf{I}_1 and \mathbf{I}_2 are exactly same after the final impulse time t_2 as shown in the figure, the two impulse vectors \mathbf{I}_1 and \mathbf{I}_2 can be regarded as the same vector for vector addition and subtraction. Impulse vectors satisfy the commutative and associative laws, as well as the distributive law for scalar multiplication. The magnitude of the impulse vector determines the magnitude of the impulse, and the angle of the impulse vector determines the time location of the impulse. One rotation, 2 \pi angle, on an impulse vector diagram corresponds to one (damped) period of the corresponding impulse response. If it is an undamped system ( \zeta = 0 ), the magnitude and angle of the impulse vector become I_i = A_i and \theta_i = \omega_n t_i . Consider an underdamped second-order system with the transfer function 4 \pi^2 / (s^2 + 0.4 \pi s + 4 \pi^2 ) . This system has \omega_n = 2 \pi and \zeta = 0.1 . For given impulse vectors \mathbf{I}_1 and \mathbf{I}_2 as shown in the figure, the resultant can be represented in two ways, \mathbf{I}_{R1} and \mathbf{I}_{R2} , in which \mathbf{I}_{R1} corresponds to a negative impulse with A_{R1} = I_{R1} / e^{\zeta \omega_n t_{R1}} and t_{R1} = \theta_{R1} / \omega_d , and \mathbf{I}_{R2} corresponds to a positive impulse with A_{R2} = I_{R2} / e^{\zeta \omega_n t_{R2}} and t_{R2} = \theta_{R2} / \omega_d . The resultants \mathbf{I}_{R1} , \mathbf{I}_{R2} can be found as follows. R_x = I_1 + I_2 \cos \theta_2 , \ \ R_y = I_2 \sin \theta_2 , I_{R1} = - \sqrt{R_x^2 + R_y^2},\ \ \theta_{R1} = \pi + \tan^{-1} (R_y / R_x) I_{R2} = \sqrt{R_x^2 + R_y^2},\ \ \theta_{R2} = \tan^{-1} (R_y / R_x) Note that - \pi /2 . The impulse responses y_{R1} and y_{R2} corresponding to \mathbf{I}_{R1} and \mathbf{I}_{R2} are exactly same with y_1 + y_2 after each impulse time location as shown in green lines of the figure (b). Now, place an impulse vector \mathbf{I}_3 on the impulse vector diagram to cancel the resultant \mathbf{I}_1 + \mathbf{I}_2 as shown in the figure. The impulse vector \mathbf{I}_3 is given by I_3 = \sqrt{R_x^2 + R_y^2},\ \ \theta_3 = \pi + \tan^{-1}(R_y /R_x ) . When the impulse sequence corresponding to three impulse vectors \mathbf{I}_1, \mathbf{I}_2 and \mathbf{I}_3 is applied to a second-order system as an input, the resulting time response causes no residual vibration after the final impulse time t_3 as shown in the red line of the bottom figure (b). Of course, another canceling vector \mathbf{I}^'_3 can exist, which is the impulse vector with the same magnitude as \mathbf{I}_3 but with an opposite arrow direction. However, this canceling vector has a longer impulse time that can be as much as a half period compared to \mathbf{I}_3 . The ZV shaper is composed of two impulse vectors, in which the first impulse vector is located at 0°, and the second impulse vector with the same magnitude is located at 180° for \mathbf{I}_1 + \mathbf{I}_2 = \mathbf{0} . Then from the impulse vector diagram of the ZV shaper on the right-hand side, \theta_1 = 0, \ \ \theta_2 = \pi I_1 = I_2 = I . Therefore, t_1 = 0, \ \ t_2 = \pi/ \omega_d . Since A_1 + A_2 = 1 (normalization constraint) must be hold, and A_1 = I_1, \ \ A_2 = I_2 / e^{\zeta \omega_n t_2} , I_1 + \frac {I_2}{e^{\zeta \omega_n t_2 }} = I + \frac {I}{K} = 1, \quad K=e^{\zeta \pi / \sqrt {1-\zeta^2 }} . Therefoere, I = K/(K+1) . Thus, the ZV shaper A_1 \delta (t) + A_2 \delta (t-t_2) is given by \begin{bmatrix} t_i \\ A_i \end{bmatrix} = \begin{bmatrix} 0, & \pi / \omega_d \\ K/(K+1), & 1/(K+1) \end{bmatrix} . \quad The ZVD shaper is composed of three impulse vectors, in which the first impulse vector is located at 0 rad, the second vector at \pi rad, and the third vector at 2 \pi rad, and the magnitude ratio is I_1 : I_2 : I_3 = 1:2:1 . Then \mathbf{I}_1 + \mathbf{I}_2 + \mathbf{I}_3 = \mathbf{0} . From the impulse vector diagram, \theta_1 = 0, \ \ \theta_2 = \pi , \ \ \theta_3 = 2 \pi . Therefore, t_1 = 0, \ t_2 = \pi/ \omega_d , \ t_3 = 2 \pi/ \omega_d . Also from the impulse vector diagram, I_1 = I_3 = I, \ \ I_2 = 2I . Since A_1 + A_2 + A_3 = 1 must be hold, I_1 + \frac {I_2}{e^{\zeta \omega_n t_3}} + \frac {I_3}{e^{\zeta \omega_n t_3}} = I+ \frac{2I}{K}+ \frac{I}{K^2} = 1, \ \ K=e^{\zeta \pi / \sqrt {1-\zeta^2 }} . Therefore, I=K^2 /(K+1)^2 . Thus, the ZVD shaper A_1 \delta (t) + A_2 \delta (t-t_2) + A_3 \delta (t- t_3 ) is given by \begin{bmatrix} t_i \\ A_i \end{bmatrix} = \begin{bmatrix} 0, & \pi / \omega_d, & 2 \pi / \omega_d\\ K^2/(K+1)^2, & 2K/(K+1)^2, & 1/(K+1)^2 \end{bmatrix} . \quad The ZVD 2 shaper is composed of four impulse vectors, in which the first impulse vector is located at 0 rad, the second vector at \pi rad, the third vector at 2 \pi rad, and the fourth vector at 3 \pi rad, and the magnitude ratio is I_1 : I_2 : I_3 : I_4 = 1:3:3:1 . Then \mathbf{I}_1 + \mathbf{I}_2 + \mathbf{I}_3 + \mathbf{I}_4 = \mathbf{0} . From the impulse vector diagram, \theta_1 = 0, \ \ \theta_2 = \pi , \ \ \theta_3 = 2 \pi , \ \ \theta_4 = 3 \pi . Therefore, t_1 = 0, \ \ t_2 = \pi / \omega_d , \ \ t_3 = 2 \pi / \omega_d , \ \ t_4 = 3 \pi / \omega_d . Also, from the impulse vector diagram, I_1 = I_4 = I, \ \ I_2 = I_3 = 3I . Since A_1 + A_2 + A_3 + A_4 = 1 must be hold, I_1 + \frac {I_2}{e^{\zeta \omega_n t_2}} + \frac {I_3}{e^{\zeta \omega_n t_3}} + \frac {I_4}{e^{\zeta \omega_n t_4}} = I + \frac {3I}{K} + \frac {3I}{K^2} + \frac {I}{K^3} = 1, \ \ \ K=e^{\zeta \pi / \sqrt {1-\zeta^2 }} . Therefore, I = K^3 / (K+1)^3 . Thus, the ZVD 2 shaper A_1 \delta (t) + A_2 \delta (t-t_2) + A_3 \delta (t-t_3) + A_4 \delta (t-t_4) is given by \begin{bmatrix} t_i \\ A_i \end{bmatrix} = \begin{bmatrix} 0, & \pi / \omega_d, & 2 \pi / \omega_d, & 3 \pi / \omega_d \\ K^3/(K+1)^3, & 3K^2/(K+1)^3, & 3K/(K+1)^3, & 1/(K+1)^3 \end{bmatrix} . \quad Similarly, the ZVD 3 shaper with five impulse vectors can be obtained, in which the first vector is located at 0 rad, the second vector at \pi rad, third vector at 2 \pi rad, the fourth vector at 3 \pi rad, and the fifth vector at 4 \pi rad, and the magnitude ratio is I_1 : I_2 : I_3 : I_4 : I_5 = 1:4:6:4:1 . In general, for the ZVD n shaper, i-th impulse vector is located at (i-1) \pi rad, and the magnitude ratio is I_1 : I_2 : I_3 : \cdots : I_{n+2} = \tbinom{n+1}{0} : \tbinom{n+1}{1} : \tbinom{n+1}{2} : \cdots : \tbinom{n+1}{n+1} where \tbinom{m}{k} implies a mathematical combination. \theta_1 = 0, \ \ \theta_2 = \frac{2 \pi}{n-1}, \cdots , \theta_{n-1} = \frac{(n-2)2 \pi}{n-1}\ \ I_2 = I_3 = \cdots = I_{n-1} = I_1 + I_n , \ \ I_n = mI_1 \ (m>0) \sum_{i=1}^{n} A_i = 1 . Thus, the resultant of the impulse vectors of the ETMn shaper becomes always zero for all n \ge 2 . One merit of the ETMn shaper is that, unlike the ZVD n or extra insensitive (EI) shapers, the shaping time is always one (damped) period of the time response even if n increases. The ETM4 shaper with four impulse vectors is obtained from the above conditions together with impulse vector definitions as \begin{bmatrix} t_i \\ A_i \end{bmatrix} = \begin{bmatrix} 0, & (2 \pi /3) / \omega_d, & (4 \pi /3) / \omega_d, & 2 \pi / \omega_d \\ I/(1+m), & I/K^{2/3}, & I/K^{4/3}, & mI/ [(1+m)K^2] \end{bmatrix} . I=\frac{(1+m)K^2}{K^2 + (1+m)(K^{4/3} + K^{2/3})+m} , \quad K=e^{\zeta \pi / \sqrt{1-\zeta^2}} . The ETM5 shaper with five impulse vectors is obtained similarly as \begin{bmatrix} t_i \\ A_i \end{bmatrix} = \begin{bmatrix} 0, & 0.5 \pi / \omega_d, & \pi / \omega_d, & 1.5 \pi / \omega_d , & 2 \pi / \omega_d\\ I/(1+m), & I/K^{1/2}, & I/K, & I/K^{3/2}, & mI/ [(1+m)K^2] \end{bmatrix} . I=\frac{(1+m)K^2}{K^2 + (1+m)(K^{3/2} + K + K^{1/2})+m} ,\quad K=e^{\zeta \pi / \sqrt{1-\zeta^2}} . In the same way, the ETMn shaper with n \ge 6 can be obtained easily. In general, ETM shapers are less sensitive to modeling errors than ZVD n shapers in a large positive error range. Note that the ZVD shaper is an ETM3 shaper with m=1 . Moreover, impulse vectors can be applied to design input shapers with negative impulses. Consider a negative equal-magnitude (NMe) shaper, in which the magnitudes of three impulse vectors are I_1 = I (>0), \ I_2 = -I,\ I_3 = I , and the angles are \theta_1 = 0, \ \theta_2 = \pi /3,\ \theta_3 = 2 \pi /3 . Then the resultant of three impulse vectors becomes zero, and thus the residual vibration is suppressed. Impulse time t_2, t_3 of the NMe shaper are obtained as t_2 = (\pi /3)/ \omega_d, \ t_3 = (2 \pi /3)/ \omega_d , and impulse magnitudes are obtained easily by solving the simultaneous equations A_1 = I, A_2 = -I/e^{\zeta \omega_n t_2}, \ \ A_3 = I/e^{\zeta \omega_n t_3} A_1 + A_2 + A_3 = 1 . The resulting NMe shaper A_1 \delta (t) + A_2 \delta (t-t_2) + A_3 \delta (t-t_3) is \begin{bmatrix} t_i \\ A_i \end{bmatrix} = \begin{bmatrix} 0, & ( \pi /3) / \omega_d, & (2 \pi /3)/ \omega_d \\ I, & -I/K^{1/3}, & I/K^{2/3} \end{bmatrix} . I = K/(K-K^{2/3} + K^{1/3}), \ \ \ K=e^{\zeta \pi / \sqrt{1- \zeta^2}} . The NMe shaper has faster rise time than the ZVD shaper, but it is more sensitive to modeling error than the ZVD shaper. Note that the NMe shaper is the same with the UM shaper if the system is undamped ( \zeta = 0 ). Figure (a) in the right side shows a typical block diagram of an input-shaping control system, and figure (b) shows residual vibration suppressions in unit-step responses by ZV, ZVD, ETM4 and NMe shapers. Refer to the reference for sensitivity curves of the above input shapers, which represent the robustness to modeling errors in \omega_n and \zeta .

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