(pp. 131-140)

In this chapter we specialize the results of Chapter 3 to the case of large-scale dynamical systems with linear energy exchange between subsystems, that is,*w(E) = WE*and*d(E) = DE*, where$W\in{\mathbb{R}^{q\timesq}}$and\[D\in{\mathbb{R}^{q\timesq}}\]In this case, the vector form of the energy balance equation (3.1), with${t_0}=0$, is given by

$E(T)=E(0)+\int_0^T{WE(t){\rm{d}}t}-\int_0^T{DE(t){\rm{d}}t+\int_0^T{S(t){\rm{d}}t},T\ge0,}$

or, in power balance form,

$\dotE(t)=WE(t)-DE(t)+S(t),E(0)={E_0},t\ge0.$

Next, let the net energy flow from the*j*th subsystem${{\calG}_j}$to the*i*th subsystem${{\calG}_i}$be parameterized as${\phi_{ij}}(E)=\Phi_{ij}^{\rm{T}}E$, where${\Phi_{ij}}\in{\mathbb{R}^q}$and$E\in\bar\mathbb{R}_+^q$. In this case, since${w_i}(E)=\sum\nolimits_{i=1,j\nei}^q{{\phi_{ij}}(E)}$, it follows that

$W\; = \;{\left[ {\begin{array}{*{20}{c}} {\sum\limits_{j\, = \,2}^q {{\Phi _{1j,\; \ldots \,,}}} } & {\sum\limits_{j\, = \,1,\;j \ne \,i}^q {{\Phi _{ij,\, \ldots ,\;}}} } & {\sum\limits_{j\, = \,1}^{q\, - \,1} {{\Phi _{{q_j}}}} } \\\end{array}} \right]^{\rm{T}}}$.

Since${\phi _{ij}}(E)\; = - {\phi _{ji}}(E),\;i,\;j\; = \;1,\; \ldots ,\;q,\;i\; \ne j$and$E\; \in \;\bar \mathbb{R}_ + ^q$it follows...