LMIs in Control/pages/Stability of Linear Delayed Differential Equations

The System

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+\sum _{i=1}^L{A}_{i}x(t-{\tau }_i),\\ \end{array}$

where $x(t)\in {ℝ}^n$ and ${\tau }_i>0$.

The Data

The matrices $A,\{A_i.{\tau }_i{\}}_{i=1}^L$.

The LMI:

Solve the following LMIP

$\begin{array}{cc}& \text{Find}\{P\succ 0,P_1\succ 0,\dots ,P_L\succ 0\}:\\ & s.t.\left[\begin{array}{cccc}{A}^{\mathrm{\top }}P+PA+\sum _{i=1}^L{P}_i& P{A}_1& \dots & P{A}_L\\ A_1^{\mathrm{\top }}P& -P_1& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ A_L^{\mathrm{\top }}P& 0& \dots & -P_L\end{array}\right]\prec 0,P_1\succ 0,\dots ,P_L\succ 0.\end{array}$

Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/50fc71737b69f2cf57d15634f2f19d091bf37d02

Conclusion

The stability of the above linear delayed differential equation is proved, using Lyapunov functionals of the form $V(x,t)=x(t{)}^{\mathrm{\top }}Px(t)+\sum _{i=1}^L\underset{0}{\overset{L}{\int }}{x}^{\mathrm{\top }}(t-s)P_{i}x(t-s)ds$, if the provided LMIP is feasible.

Remark

The techniques for proving stability of norm-bound LDIs [Boyd, ch.5] can also be used.

External Links

• 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.