Impulse vector

File:CGKang_fig1.jpg

For a vibratory second-order system $\backslash omega\_n^2\; /(s^2\; +\; 2\; \backslash zeta\; \backslash omega\_n\; +\; \backslash omega\_n^2\; )$ with undamped natural frequency $\backslash omega\_n$ and damping ratio $\backslash zeta$, the magnitude $I\_i$ and angle $\backslash theta\_i$ of an impulse vector (or Kang vector) $\backslash mathbf\{I\}\_i$ corresponding to an impulse function $A\_i\; \backslash delta\; (t-t\_i)$, $i\; =\; 1,2,...,n$ is defined in a 2-dimensional polar coordinate system as

::$I\_i\; =\; A\_i\; e^\{\backslash zeta\; \backslash omega\_n\; t\_i\; \}$

::$\backslash theta\_i\; =\; \backslash omega\_d\; t\_i$

where $A\_i$ implies the magnitude of an impulse function, $t\_i$ implies the time location of the impulse function, and $\backslash omega\_d$ implies damped natural frequency $\backslash omega\_n\; \backslash sqrt\{1-\backslash zeta^2\}$. 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 , 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 $\backslash theta\_i$ is the product of the impulse time and damped natural frequency. $\backslash delta(t-t\_i)$ 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, $\backslash omega\_n$ and $\backslash zeta$ 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.

File:CGKang_fig2.jpg

Consider two impulse vectors $\backslash mathbf\{I\}\_1$ and $\backslash mathbf\{I\}\_2$ in the figure on the right-hand side, in which $\backslash mathbf\{I\}\_1$ is an impulse vector with magnitude $I\_1\; (>0)$ and angle $\backslash theta\_1$ corresponding to a positive impulse with $A\_1\; >\; 0$, and $\backslash mathbf\{I\}\_2$ is an impulse vector with magnitude $I\_2\; =\; -I\_1$ and angle $\backslash theta\_2\; =\; \backslash pi\; +\; \backslash theta\_1$ corresponding to a negative impulse with . Since the two time-responses corresponding to $\backslash mathbf\{I\}\_1$ and $\backslash mathbf\{I\}\_2$ are exactly same after the final impulse time $t\_2$ as shown in the figure, the two impulse vectors $\backslash mathbf\{I\}\_1$ and $\backslash mathbf\{I\}\_2$ 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, $2\; \backslash pi$ angle, on an impulse vector diagram corresponds to one (damped) period of the corresponding impulse response.

If it is an undamped system ($\backslash zeta\; =\; 0$), the magnitude and angle of the impulse vector become $I\_i\; =\; A\_i$ and $\backslash theta\_i\; =\; \backslash omega\_n\; t\_i$.

File:CGKang_fig3.jpg

The impulse response of a second-order system corresponding to the resultant of two impulse vectors is same as the time response of the system with a two-impulse input corresponding to two impulse vectors after the final impulse time regardless of whether the system is undamped or underdamped. □

File:CGKang_fig4.jpg

If the resultant of impulse vectors is zero, the time response of a second-order system for the input of the impulse sequence corresponding to the impulse vectors becomes zero also after the final impulse time regardless of whether the system is undamped or underdamped. □

Consider an underdamped second-order system with the transfer function $4\; \backslash pi^2\; /\; (s^2\; +\; 0.4\; \backslash pi\; s\; +\; 4\; \backslash pi^2\; )$. This system has $\backslash omega\_n\; =\; 2\; \backslash pi$ and $\backslash zeta\; =\; 0.1$. For given impulse vectors $\backslash mathbf\{I\}\_1$ and $\backslash mathbf\{I\}\_2$ as shown in the figure, the resultant can be represented in two ways, $\backslash mathbf\{I\}\_\{R1\}$ and $\backslash mathbf\{I\}\_\{R2\}$, in which $\backslash mathbf\{I\}\_\{R1\}$ corresponds to a negative impulse with $A\_\{R1\}\; =\; I\_\{R1\}\; /\; e^\{\backslash zeta\; \backslash omega\_n\; t\_\{R1\}\}$ and $t\_\{R1\}\; =\; \backslash theta\_\{R1\}\; /\; \backslash omega\_d$, and $\backslash mathbf\{I\}\_\{R2\}$ corresponds to a positive impulse with $A\_\{R2\}\; =\; I\_\{R2\}\; /\; e^\{\backslash zeta\; \backslash omega\_n\; t\_\{R2\}\}$ and $t\_\{R2\}\; =\; \backslash theta\_\{R2\}\; /\; \backslash omega\_d$.

The resultants $\backslash mathbf\{I\}\_\{R1\}$, $\backslash mathbf\{I\}\_\{R2\}$ can be found as follows.

::$R\_x\; =\; I\_1\; +\; I\_2\; \backslash cos\; \backslash theta\_2\; ,\; \backslash \; \backslash \; R\_y\; =\; I\_2\; \backslash sin\; \backslash theta\_2$,

::$I\_\{R1\}\; =\; -\; \backslash sqrt\{R\_x^2\; +\; R\_y^2\},\backslash \; \backslash \; \backslash theta\_\{R1\}\; =\; \backslash pi\; +\; \backslash tan^\{-1\}\; (R\_y\; /\; R\_x)$

::$I\_\{R2\}\; =\; \backslash sqrt\{R\_x^2\; +\; R\_y^2\},\backslash \; \backslash \; \backslash theta\_\{R2\}\; =\; \backslash tan^\{-1\}\; (R\_y\; /\; R\_x)$

Note that . The impulse responses $y\_\{R1\}$ and $y\_\{R2\}$ corresponding to $\backslash mathbf\{I\}\_\{R1\}$ and $\backslash mathbf\{I\}\_\{R2\}$ 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 $\backslash mathbf\{I\}\_3$ on the impulse vector diagram to cancel the resultant $\backslash mathbf\{I\}\_1\; +\; \backslash mathbf\{I\}\_2$ as shown in the figure. The impulse vector $\backslash mathbf\{I\}\_3$ is given by

::$I\_3\; =\; \backslash sqrt\{R\_x^2\; +\; R\_y^2\},\backslash \; \backslash \; \backslash theta\_3\; =\; \backslash pi\; +\; \backslash tan^\{-1\}(R\_y\; /R\_x\; )$.

When the impulse sequence corresponding to three impulse vectors $\backslash mathbf\{I\}\_1,\; \backslash mathbf\{I\}\_2$ and $\backslash mathbf\{I\}\_3$ 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 $\backslash mathbf\{I\}^\text{'}\_3$ can exist, which is the impulse vector with the same magnitude as $\backslash mathbf\{I\}\_3$ 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 $\backslash mathbf\{I\}\_3$.

File:CGKang_fig5.jpg

Using impulse vectors, we can redesign known input shapers {{cite journal |last1=Singhose |first1=W. |title=Command shaping for flexible systems: A review of the first 50 years |journal=International Journal of Precision Engineering and Manufacturing |date=2009 |volume=10 |issue=4 |pages=153–168 |doi=10.1007/s12541-009-0084-2|s2cid=111341954 }} such as zero vibration (ZV), zero vibration and derivative (ZVD), and ZVDⁿ shapers.

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 $\backslash mathbf\{I\}\_1\; +\; \backslash mathbf\{I\}\_2\; =\; \backslash mathbf\{0\}$. Then from the impulse vector diagram of the ZV shaper on the right-hand side,

::$\backslash theta\_1\; =\; 0,\; \backslash \; \backslash \; \backslash theta\_2\; =\; \backslash pi$

::$I\_1\; =\; I\_2\; =\; I$.

Therefore, $t\_1\; =\; 0,\; \backslash \; \backslash \; t\_2\; =\; \backslash pi/\; \backslash omega\_d$.

Since $A\_1\; +\; A\_2\; =\; 1$ (normalization constraint) must be hold, and $A\_1\; =\; I\_1,\; \backslash \; \backslash \; A\_2\; =\; I\_2\; /\; e^\{\backslash zeta\; \backslash omega\_n\; t\_2\}$,

::$I\_1\; +\; \backslash frac\; \{I\_2\}\{e^\{\backslash zeta\; \backslash omega\_n\; t\_2\; \}\}\; =\; I\; +\; \backslash frac\; \{I\}\{K\}\; =\; 1,\; \backslash quad\; K=e^\{\backslash zeta\; \backslash pi\; /\; \backslash sqrt\; \{1-\backslash zeta^2\; \}\}$.

Therefoere, $I\; =\; K/(K+1)$.

Thus, the ZV shaper $A\_1\; \backslash delta\; (t)\; +\; A\_2\; \backslash delta\; (t-t\_2)$ 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}

.

::

$\backslash quad$

The ZVD shaper is composed of three impulse vectors, in which the first impulse vector is located at 0 rad, the second vector at $\backslash pi$ rad, and the third vector at $2\; \backslash pi$ rad, and the magnitude ratio is $I\_1\; :\; I\_2\; :\; I\_3\; =\; 1:2:1$. Then $\backslash mathbf\{I\}\_1\; +\; \backslash mathbf\{I\}\_2\; +\; \backslash mathbf\{I\}\_3\; =\; \backslash mathbf\{0\}$. From the impulse vector diagram,

::$\backslash theta\_1\; =\; 0,\; \backslash \; \backslash \; \backslash theta\_2\; =\; \backslash pi\; ,\; \backslash \; \backslash \; \backslash theta\_3\; =\; 2\; \backslash pi$.

Therefore, $t\_1\; =\; 0,\; \backslash \; t\_2\; =\; \backslash pi/\; \backslash omega\_d\; ,\; \backslash \; t\_3\; =\; 2\; \backslash pi/\; \backslash omega\_d$.

Also from the impulse vector diagram,

::$I\_1\; =\; I\_3\; =\; I,\; \backslash \; \backslash \; I\_2\; =\; 2I$.

Since $A\_1\; +\; A\_2\; +\; A\_3\; =\; 1$ must be hold,

::$I\_1\; +\; \backslash frac\; \{I\_2\}\{e^\{\backslash zeta\; \backslash omega\_n\; t\_3\}\}\; +\; \backslash frac\; \{I\_3\}\{e^\{\backslash zeta\; \backslash omega\_n\; t\_3\}\}\; =\; I+\; \backslash frac\{2I\}\{K\}+\; \backslash frac\{I\}\{K^2\}\; =\; 1,\; \backslash \; \backslash \; K=e^\{\backslash zeta\; \backslash pi\; /\; \backslash sqrt\; \{1-\backslash zeta^2\; \}\}$.

Therefore, $I=K^2\; /(K+1)^2$.

Thus, the ZVD shaper $A\_1\; \backslash delta\; (t)\; +\; A\_2\; \backslash delta\; (t-t\_2)\; +\; A\_3\; \backslash delta\; (t-\; t\_3\; )$ 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}

.

::

$\backslash quad$

The ZVD² shaper is composed of four impulse vectors, in which the first impulse vector is located at 0 rad, the second vector at $\backslash pi$ rad, the third vector at $2\; \backslash pi$ rad, and the fourth vector at $3\; \backslash pi$ rad, and the magnitude ratio is $I\_1\; :\; I\_2\; :\; I\_3\; :\; I\_4\; =\; 1:3:3:1$. Then $\backslash mathbf\{I\}\_1\; +\; \backslash mathbf\{I\}\_2\; +\; \backslash mathbf\{I\}\_3\; +\; \backslash mathbf\{I\}\_4\; =\; \backslash mathbf\{0\}$. From the impulse vector diagram,

::$\backslash theta\_1\; =\; 0,\; \backslash \; \backslash \; \backslash theta\_2\; =\; \backslash pi\; ,\; \backslash \; \backslash \; \backslash theta\_3\; =\; 2\; \backslash pi\; ,\; \backslash \; \backslash \; \backslash theta\_4\; =\; 3\; \backslash pi$.

Therefore, $t\_1\; =\; 0,\; \backslash \; \backslash \; t\_2\; =\; \backslash pi\; /\; \backslash omega\_d\; ,\; \backslash \; \backslash \; t\_3\; =\; 2\; \backslash pi\; /\; \backslash omega\_d\; ,\; \backslash \; \backslash \; t\_4\; =\; 3\; \backslash pi\; /\; \backslash omega\_d$.

Also, from the impulse vector diagram,

::$I\_1\; =\; I\_4\; =\; I,\; \backslash \; \backslash \; I\_2\; =\; I\_3\; =\; 3I$.

Since $A\_1\; +\; A\_2\; +\; A\_3\; +\; A\_4\; =\; 1$ must be hold,

::$I\_1\; +\; \backslash frac\; \{I\_2\}\{e^\{\backslash zeta\; \backslash omega\_n\; t\_2\}\}\; +\; \backslash frac\; \{I\_3\}\{e^\{\backslash zeta\; \backslash omega\_n\; t\_3\}\}\; +\; \backslash frac\; \{I\_4\}\{e^\{\backslash zeta\; \backslash omega\_n\; t\_4\}\}\; =\; I\; +\; \backslash frac\; \{3I\}\{K\}\; +\; \backslash frac\; \{3I\}\{K^2\}\; +\; \backslash frac\; \{I\}\{K^3\}\; =\; 1,\; \backslash \; \backslash \; \backslash \; K=e^\{\backslash zeta\; \backslash pi\; /\; \backslash sqrt\; \{1-\backslash zeta^2\; \}\}$.

Therefore, $I\; =\; K^3\; /\; (K+1)^3$.

Thus, the ZVD² shaper $A\_1\; \backslash delta\; (t)\; +\; A\_2\; \backslash delta\; (t-t\_2)\; +\; A\_3\; \backslash delta\; (t-t\_3)\; +\; A\_4\; \backslash delta\; (t-t\_4)$ 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}

.

::

$\backslash quad$

Similarly, the ZVD³ shaper with five impulse vectors can be obtained, in which the first vector is located at 0 rad, the second vector at $\backslash pi$ rad, third vector at $2\; \backslash pi$ rad, the fourth vector at $3\; \backslash pi$ rad, and the fifth vector at $4\; \backslash pi$ 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 ZVDⁿ shaper, i-th impulse vector is located at $(i-1)\; \backslash pi$ rad, and the magnitude ratio is $I\_1\; :\; I\_2\; :\; I\_3\; :\; \backslash cdots\; :\; I\_\{n+2\}\; =\; \backslash tbinom\{n+1\}\{0\}\; :\; \backslash tbinom\{n+1\}\{1\}\; :\; \backslash tbinom\{n+1\}\{2\}\; :\; \backslash cdots\; :\; \backslash tbinom\{n+1\}\{n+1\}$ where $\backslash tbinom\{m\}\{k\}$ implies a mathematical combination.

File:CGKang_fig6.jpg

Now, consider equal shaping-time and magnitudes (ETM) shapers, with the same magnitude of impulse vectors and with the same angle between impulse vectors. The ETMn shaper satisfies the conditions

::$\backslash theta\_1\; =\; 0,\; \backslash \; \backslash \; \backslash theta\_2\; =\; \backslash frac\{2\; \backslash pi\}\{n-1\},\; \backslash cdots\; ,\; \backslash theta\_\{n-1\}\; =\; \backslash frac\{(n-2)2\; \backslash pi\}\{n-1\}\backslash \; \backslash$

::$I\_2\; =\; I\_3\; =\; \backslash cdots\; =\; I\_\{n-1\}\; =\; I\_1\; +\; I\_n\; ,\; \backslash \; \backslash \; I\_n\; =\; mI\_1\; \backslash \; (m>0)$

::$\backslash sum\_\{i=1\}^\{n\}\; A\_i\; =\; 1$.

Thus, the resultant of the impulse vectors of the ETMn shaper becomes always zero for all $n\; \backslash ge\; 2$. One merit of the ETMn shaper is that, unlike the ZVDⁿ or extra insensitive (EI) shapers,{{cite journal |last1=Singhose |first1=W. |last2=Derezinski |first2=S. |last3=Singer |first3=N. |title=Extra-insensitive input shapers for controlling flexible spacecraft |journal=Journal of Guidance, Control and Dynamics |date=1996 |volume=19 |issue=2 |pages=385–391 |doi=10.2514/3.21630 |bibcode=1996JGCD...19..385S }} 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}

.

::$I=\backslash frac\{(1+m)K^2\}\{K^2\; +\; (1+m)(K^\{4/3\}\; +\; K^\{2/3\})+m\}\; ,\; \backslash quad\; K=e^\{\backslash zeta\; \backslash pi\; /\; \backslash sqrt\{1-\backslash zeta^2\}\}$.

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}

.

::$I=\backslash frac\{(1+m)K^2\}\{K^2\; +\; (1+m)(K^\{3/2\}\; +\; K\; +\; K^\{1/2\})+m\}\; ,\backslash quad\; K=e^\{\backslash zeta\; \backslash pi\; /\; \backslash sqrt\{1-\backslash zeta^2\}\}$.

In the same way, the ETMn shaper with $n\; \backslash ge\; 6$ can be obtained easily. In general, ETM shapers are less sensitive to modeling errors than ZVDⁿ shapers in a large positive error range. Note that the ZVD shaper is an ETM3 shaper with $m=1$.

File:CGKang_fig7.jpg

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 $I\_1\; =\; I\; (>0),\; \backslash \; I\_2\; =\; -I,\backslash \; I\_3\; =\; I$, and the angles are $\backslash theta\_1\; =\; 0,\; \backslash \; \backslash theta\_2\; =\; \backslash pi\; /3,\backslash \; \backslash theta\_3\; =\; 2\; \backslash pi\; /3$. 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 $t\_2\; =\; (\backslash pi\; /3)/\; \backslash omega\_d,\; \backslash \; t\_3\; =\; (2\; \backslash pi\; /3)/\; \backslash omega\_d$, and impulse magnitudes are obtained easily by solving the simultaneous equations

::$A\_1\; =\; I,\; A\_2\; =\; -I/e^\{\backslash zeta\; \backslash omega\_n\; t\_2\},\; \backslash \; \backslash \; A\_3\; =\; I/e^\{\backslash zeta\; \backslash omega\_n\; t\_3\}$

::$A\_1\; +\; A\_2\; +\; A\_3\; =\; 1$.

File:CGKang_fig8.jpg

The resulting NMe shaper $A\_1\; \backslash delta\; (t)\; +\; A\_2\; \backslash delta\; (t-t\_2)\; +\; A\_3\; \backslash delta\; (t-t\_3)$ 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}

.

::$I\; =\; K/(K-K^\{2/3\}\; +\; K^\{1/3\}),\; \backslash \; \backslash \; \backslash \; K=e^\{\backslash zeta\; \backslash pi\; /\; \backslash sqrt\{1-\; \backslash zeta^2\}\}$.

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{{cite journal |last1=Singhose |first1=W. E. |last2=Seering |first2=W. P. |last3=Singer |first3=N. C. |title=Time-optimal negative input shapers |journal=Journal of Dynamic Systems, Measurement, and Control |date=1997 |volume=119 |issue=2 |pages=198–205 |doi=10.1115/1.2801233 }} if the system is undamped ($\backslash zeta\; =\; 0$).

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 $\backslash omega\_n$ and $\backslash zeta$.

{{clear}}

{{Reflist}}

Category:Dynamics (mechanics)

Category:Control theory

Category:Mechanical vibrations