LMIs in Control/pages/Stabilizability LMI
A system is stabilizable if all unstable modes of the system are controllable. This implies that if the system is controllable, it will also be stabilizable. Thus, stabilizability is a essentially a weaker version of the controllability condition. The LMI condition for stabilizability of pair is shown below.
The System[edit | edit source]
where , , at any .
The Data[edit | edit source]
The matrices necessary for this LMI are and . There is no restriction on the stability of A.
The LMI: Stabilizability LMI[edit | edit source]
is stabilizable if and only if there exists such that
where the stabilizing controller is given by
Conclusion:[edit | edit source]
If we are able to find such that the above LMI holds it means the matrix pair is stabilizable. In words, a system pair is stabilizable if for any initial state an appropriate input can be found so that the state asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to approach as whereas controllability requires that the state must reach the origin in a finite time.
Implementation[edit | edit source]
This implementation requires Yalmip and Sedumi.
Related LMIs[edit | edit source]
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.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.