# Damping of Oscillations of a Rotary Pendulum System
## 2 Input Shaping  
Input shaping with time delays is a method of reducing residual vibrations by convolving a sequence of impulses with a baseline reference command to compensate for the undesired oscillatory couple of system poles by the dominant couple of shaper zeros, which ensures a finite settling time for system modes with a known natural frequency and damping ratio. This method can be divided into two categories. In off-line shaping, the trajectory is defined in advance and is constantly repeated during the motion cycle, such as in motor drives used in a printer, hard disks, etc. Suppose we know the trajectory in advance (or at least any initial part), which can be updated in regular cycles. In that case, it is possible to use different forward control (MPC) techniques, which require detailed information about the system and knowledge of the input signal [25; 26]. On the other hand, online shaping is a method of preshaping a profile to generate a previously unknown trajectory in which the system is controlled by a human operator or a superior system that acts as a trajectory generator, which means that the input signal needs to be modified in real-time to eliminate spectral components from around the natural frequency.  
To demonstrate the basic principle of filtering, we consider the response of a dynamic oscillatory system, which is usually expressed simply by a slightly damped second-order linear transfer function consisting of the estimated values of natural frequency and damping to a series of two pulses, as shown in Figure 1 (red shows the response of the system to the pulses if each was applied individually, and blue shows the combined response of the system resulting from superposition).  
This example illustrates that pulses of suitable amplitudes and times can be applied to a system so that the resonant modes of the system are destructively combined, resulting in the minimization of the resulting vector component pulses. The FIR shaper structure can be expressed by a series of Dirac pulses, where \(D_{t}\) represents the time course of the Dirac pulse. Each of the pulses is assigned the corresponding amplitude \(A_{t}\) for the time delay \(T_{Dn}\) of the (\(n\)th) Dirac pulse relative to the first of the series, for which we assume a delay with a zero value (Figure 2).  
From a mathematical point of view, the transition of the signal through a parallel combination of transport delays can therefore be replaced by the operation of convolutional multiplication of the reference signal with a sequence of Dirac pulses, which means that the control signal will be decomposed into several time-shifted components, which will individually excite sub-oscillations. However, these oscillations are attenuated in sum due to appropriately set phase shifts and amplitudes. This process requires one multiplication and one addition operation for each pulse. The basic block diagram of the molding principle is shown in Figure 3.  
Since the input shaper must be placed directly in front of the time-varying controlled system, there is no feedback to help set the control signal correctly [5,27]. By identifying the output of the controlled system, it is possible to correctly configure the input shaper and then modify the control signal in the desired way.