# LMIs in Control/Stability Analysis/Continuous Time/Output Energy Bound for Discrete-Time Autonomous LTI Systems

**Output Energy Bound for Discrete-Time Autonomous LTI Systems**

Autonomous systems are initialized under a given set of initial conditions, and then run without any additional inputs. It is useful to know ahead of time the bounds such a system will operate within. This analysis can be used to determine the upper bound on the output of a given autonomous LTI system operating in discrete time.

**The System**

[edit | edit source]Consider the discrete-time, LTI autonomous system with state space representation

,

,

where and

**Determining an Upper Bound**

[edit | edit source]The output of this system will satisfy

if there exists some matrix and scalar such that

,

,

.

**Conclusion**

[edit | edit source]Given an autonomous system operating in discrete-time conditions, the parameter can be used to determine the largest feasible bound on the output of that system.

**External Links**

[edit | edit source]A list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.