Proofs

Abstract

The forward difference of the Lyapunov function (2.14) for (2.29) is

$$\begin{aligned} \Delta V(k)&=x^T(k+1)Px(k+1)-x^T(k)Px(k) \nonumber \\&= \big (A_{dK}x(k)+B_dKe(k)+w(k)\big )^TP\big (A_{dK}x(k)+BKe(k)+w(k)\big )-x^T(k)Px(k) \nonumber \\&= -x^T(k)Qx(k)+2e^T(k)(B_dK)^TPA_{dK}x(k)+e^T(k)(B_dK)^TPB_dKe(k) \nonumber \\&\quad +2w^T(k)PA_{dK}x(k)+2w^T(k)PB_dKe(k)+w^T(k)Pw(k) \nonumber \\&\le -\lambda _{min}(Q)\Vert x(k)\Vert ^2+2\Vert (BK)^TPA_{dK}\Vert \Vert e(k)\Vert \Vert x(k)\Vert +\Vert (BK)^TPBK\Vert \Vert e(k)\Vert ^2 \nonumber \\&\quad +2\Vert PB_dK\Vert \Vert w(k)\Vert \Vert e(k)\Vert +2\Vert PA_{dK}\Vert \Vert w(k)\Vert \Vert x(k)\Vert +\lambda _{max}(P)\Vert w(k)\Vert ^2. \end{aligned}$$