Tracking Control of Linear Systems

Appendix I

Proof of Theorem 116

Proof. [148]. a) Left, right, Laplace transform ℒ∓{.} of the left-hand side of (2.1), i.e., (2.9) (Section 2.1), i.e., of (7.20) (Subsection 7.2.1), yields the following:

$$\begin{array}{r}{\mathcal{L}}^{\mp }\{\sum _{k=0}^{k=\nu }{A}_k{\mathbf{Y}}^{(k)}(t)\}={\mathcal{L}}^{\mp }\{A^{(\nu )}{\mathbf{Y}}^{\nu }(t)\}=\stackrel{A^{(\nu )}{S}_N^{(\nu )}(s)}{\overbrace{(A_0{s}^0+A_1{s}^1+\dots +A_{\nu }{s}^{\nu })}}{\mathbf{Y}}^{\mp }(s)-\\ -A_0{O}_N-\\ -A_1\left(s^{1-1}{\mathbf{Y}}^{(1-1)}\left(0^{\mp }\right)\right)-\\ -A_2(s^{2-1}{\mathbf{Y}}^{(1-1)}\left(0^{\mp }\right)+s^{2-2}{\mathbf{Y}}^{(2-1)}\left(0^{\mp }\right))-\\ -A_k\left(\begin{array}{c}{s}^{k-1}{\mathbf{Y}}^{(1-1)}\left(0^{\mp }\right)+s^{k-2}{\mathbf{Y}}^{(2-1)}\left(0^{\mp }\right)+.\\ +s^{k-i}{\mathbf{Y}}^{(i-1)}\left(0^{\mp }\right)+..+s^{k-k}{\mathbf{Y}}^{(k-1)}\left(0^{\mp }\right)\end{array}\right)-\\ -A_{\nu }(s^{\nu -1}{\mathbf{Y}}^{(1-1)}\left(0^{\mp }\right)+s^{\nu -2}{\mathbf{Y}}^{(2-1)}\left(0^{\mp }\right)+\dots +s^{\nu -\nu }{\mathbf{Y}}^{(\nu -1)}\left(0^{\mp }\right))\\ =A^{(\nu )}{S}_N^{(\nu )}(s){\mathbf{Y}}^{\mp }(s)-\stackrel{A^{(\nu )},(7.23)}{\overbrace{[A_0⋮A_1⋮\dots \dots A_k⋮\dots ⋮A_{\nu }]}}\text{.}\\ \begin{array}{c}{O}_N\\ s^{1-1}{\mathbf{Y}}^{(1-1)}\left(0^{\mp }\right)\\ s^{k-2}{\mathbf{Y}}^{(2-1)}\left(0^{\mp }\right)+\dots +s^{k-k}{\mathbf{Y}}^{(k-1)}\left(0^{\mp }\right)\\ s^{\nu -2}{\mathbf{Y}}^{(2-1)}\left(0^{\mp }\right)+\dots +s^{\nu -\nu }{\mathbf{Y}}^{(\nu -1)}\left(0^{\mp }\right)\end{array}]\end{array}$$

$$\begin{array}{r}=[A_0⋮A_1⋮\dots .A_k⋮\dots \dots A_{\nu }]\bullet \\ \underbrace{\bullet \left[\begin{array}{cccccc}{O}_N& O_N& \dots \dots \dots & O_N& \dots \dots \dots & O_N\\ s^{1-1}{I}_N& O_N& \dots \dots \dots & O_N& \dots \dots \dots & O_N\\ \dots \dots \dots & \dots \dots \dots & \dots \dots \dots & O_N& \dots \dots \dots & O_N\\ s^{k-1}{I}_N& s^{k-2}{I}_N& \dots \dots \dots & s^{k-k}{I}_N& \dots \dots \dots & O_N\\ \dots \dots \dots & \dots \dots \dots & \dots \dots \dots & \dots \dots \dots & \dots \dots \dots & \dots \dots \dots \\ s^{\nu -1}{I}_N& s^{\nu -2}{I}_N& \dots \dots \dots & s^{\nu -2}{I}_N& \dots \dots \dots & s^{\nu -\nu }{I}_N\end{array}\right]}\\ \cdot \underset{}{\underset{}{[\begin{array}{c}{\mathbf{Y}}^{(1-1)}\left(0^{\mp }\right)\\ {\mathbf{Y}}^{}\end{array}}}\end{array}$$