Analysis is about building the right system. Design is about building the system right. Preliminary design is an intermediate step between analysis and design. Preliminary design explicitly recognizes something that many people recognize implicitly:

You usually can’t fully understand the requirements that you’re dealing with unless you do some exploratory design.

The following activity diagram (which follows on from this one) shows the preliminary design steps.

The steps in the diagram are described in detail (with lots of examples and exercises) in Chapters 5, 6 and 7 of Use Case Driven Object Modeling with UML.
Analysis and Preliminary Design