Positive Continuous-Discrete Linear Systems

General Model of Continuous-Discrete Linear Systems and Its Solution

Consider the general model of linear continuous-discrete systems described by the equations

$$ \dot{x}(t,i+1) = A_{0} x(t,i)+A_{1} \dot{x}(t,i)+A_{2} x(t,i+1)$$

$$ + B_{0} u(t,i)+B_{1} \dot{u}(t,i)+B_{2} u(t,i+1), \quad \quad (12.1a)$$

$$ y(t,i) = Cx(t,i)+Du(t,i),\quad t\in{\mathbb R}_{+} =[0,+\infty ], \quad i\in{\mathbb Z}_{+},\quad \quad (12.1a) $$

where $\dot{x}(t,i)=\frac{\partial x(t,i)}{\partial t}, x(t,i)\in{\mathbb R}^{n} , u(t,i)\in{\mathbb R}^{m} y(t,i)\in{\mathbb R}^{p}$ are the state, input and output vectors and A _k  ∈ ℝ^n×n, B _k  ∈ ℝ^n×m, k = 0,1,2; C ∈ ℝ^p×n, D ∈ ℝ^p×m .