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 4.3 Step, Impulse, and Other Responses Using the general time-domain response functions defined in this chapter, it is easy to investigate typical problems such as step, impulse, and ramp responses. This section presents some examples. Let us create the second-order system with the natural frequency and damping ratio . In[37]:= Out[37]= Here is the symbolic form of the step response for the critically damped case. In[38]:= Out[38]= This is the analog simulation of the step response for the system with the natural frequency equal to unity and a particular value of from the underdamped case region. In[39]:= This plots the response for various values of the damping ratio . We can see that the response changes from pure oscillation for = 0 (undamped case) to the exponential for = 1.5 (overdamped case). In[40]:= Here is a third-order system. In the next inputs, we compute the step response for a few values of the parameter defined as for the particular case of . In[41]:= Out[41]= For , the system degenerates to the second-order one. In[42]:= Out[42]= This is its step response. In[43]:= Out[43]= This is the step response for . In[44]:= Out[44]= Finally, here is the step response for . In[45]:= Out[45]= This combines the three previous results. The solid, dashed, and dashed-dotted lines represent the respective curves for , , and . In[46]:= This is the simulation of a ramp response of the third-order system for a particular set of parameters. To visualize the steady-state error, we include the dashed straight line. In[47]:= Consider a two-input, second-order system. In[48]:= Out[48]= This applies the impulse function to all inputs in turn and simplifies the result. In[49]:= Out[49]= Here is the plot of the impulse responses. Note that we have joined the lists in the previous result to plot all curves in one graph. The impulse responses for the first and second inputs are shown as a solid and a dashed line, respectively. In[50]:= This is the simulated (rather than computed symbolically) impulse response for the first input. As mentioned in Section 4.2, SimulationPlot performs the discrete simulation for the input signal in a form of the Dirac delta function. In[51]:= To perform the analog simulation, we can introduce an approximation of the Dirac delta function by an impulse with a finite width. In myDiracDelta, the parameter determines the offset of the impulse along the time axis and determines the width of the impulse. In[52]:= In[53]:=