By the end of this section, you should be able to

1. Put 2nd-order systems into standard form

2. Identify defining parameters of system

   • Classify stability of system in TF or SS form

3. Sketch response of system due to step input

4. Modify sketch when extra poles or zeros are added

This section builds your core intuition for second-order systems, which show up everywhere in engineering dynamics, vibration, and instrumentation. You will learn how to rewrite a second-order transfer function in standard form, identify the defining parameters that control response, and classify stability from either transfer function form or state-space form by looking at pole locations. From there, the focus shifts to step response sketching. You will practice drawing a clean first-pass sketch, then refining it as you incorporate the key time-domain performance measures used in analysis and design, including 10 percent to 20 percent rise time, peak time, maximum overshoot, and 1 percent settling time. You will also learn how to adjust your sketch and your expectations when the model is no longer a pure second-order system, for example when extra poles or zeros are added. The section includes worked examples that move from qualitative reasoning to more accurate predictions, plus a dedicated connection between state-space representation and the familiar second-order response shape. Use the table of contents below to jump between the introduction, examples, state-space analysis, the extra poles and zeros material, and the lecture code so you can practice the workflow repeatedly until it is automatic.