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

The System

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+\sum _{i=1}^{L}A_{i}x(t-\tau _{i}),\\\end{aligned}}}

where ${\displaystyle x(t)\in \mathbb {R} ^{n}}$  and ${\displaystyle \tau _{i}>0}$ .

The Data

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

The LMI:

Solve the following LMIP

{\displaystyle {\begin{aligned}&{\text{Find}}\{P\succ 0,P_{1}\succ 0,\dots ,P_{L}\succ 0\}:\\&\quad s.t.{\begin{bmatrix}A^{\top }P+PA+\sum _{i=1}^{L}P_{i}&PA_{1}&\dots &PA_{L}\\A_{1}^{\top }P&-P_{1}&\dots &0\\\vdots &\vdots &\ddots &\vdots \\A_{L}^{\top }P&0&\dots &-P_{L}\end{bmatrix}}\prec 0,P_{1}\succ 0,\dots ,P_{L}\succ 0.\end{aligned}}}

Conclusion

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

Remark

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