# LMIs in Control/pages/Positive Orthant Stability

**Positive Orthant Stabilizability**

The positive orthant stability of a linear system refers to the property of the system states being real and positive for all and decaying down to zero over time. In this section, the feasibility problem for systems to be positive orthant stable, and the stabilizability conditions to make the system positive orthant stable will be covered.

**The System**[edit | edit source]

Consider a linear state-space representation of a system as:

where and are the system state and the input vector respectively. *A* and *B* are system coefficient matrices of appropriate dimensions.

**The Data**[edit | edit source]

Number of states *n* and number of control inputs *r* need to be known. Moreover, the system matrices *A,B* are also required to be known.

**The Feasibility LMI**[edit | edit source]

An LTI system is positive orthant stable if implies that . Moreover, as , . This is possible if and only if the following conditions hold:

The above LMI feasibility is the positive orthant stability criteria. To convert it into a *positive orthant stabilizability* check, the problem can be modified so that we check if is positive orthant stable. As is also a design variable here, the second inequality in the above LMI will result in bilinearity. A simple change of variables can overcome that to result in the following LMI feasibility problem for checking positive orthant stabilizability of the LTI system:

If the above LMI is feasible, the LTI system is stabilizable with controller .

**Conclusion:**[edit | edit source]

The feasibility of the above LMIs guarantees that the system is positive orthant stable if the first LMI is feasible or stabilizable with a controller if the second LMI holds.

**Implementation**[edit | edit source]

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:

https://github.com/smhassaan/LMI-Examples/blob/master/Positive_Orthant_LMI.m

## External Links[edit | edit source]

A list of references documenting and validating the LMI.

- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.