Objectives:

1. LECTURE 39: APPLICATIONS OF STATE VARIABLES • Objectives: • Review of State EquationsExample – Two-Car MotionTime-Domain ResponseTransfer FunctionCourse Evaluations • Resources:WikiBooks: State VariablesWC: State Variables Audio: URL:

2. Discrete-Time Systems • The discrete-time equivalents of the state equations are: • The process is completely analogous to CT systems, with one exception: these equations are easy to implement and solve using difference equations  • For example, note that if v[n] = 0 for n ≥ 0, • and An is the state-transition matrix for the discrete-time case. • The output equation can be derived following the procedure for the CT case: • The impulse response is given by:

3. Solutions Using the z-Transform • Just as we solved the CT equations with the Laplace transform, we can solve the DT case with the z-transform: • Note that historically the CT state equations were solved numerically using a discrete-time approximation for the derivatives (see Section 11.6), and this provided a bridge between the CT and DT solutions. • Example:

4. State Variable Equations and Solutions – CT Systems • Recall our state equations for CT systems: • the time-domain solutions: • and the frequency domain solutions: • An equivalent formulation exists for discrete-time (DT) systems.

5. Example – Coupled Two-Car System

6. Example (Cont.)

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14. Summary • Introduced the discrete-time versions of the state equations. • Reviewed an example in the textbook (Kamen and Heck) demonstrating the use of state variables on a real application. • Demonstrated how to solve this system in MATLAB. • The end…