Approximate state-space and transfer function models for \(2 \times 2\) linear hyperbolic systems with collocated boundary inputs (Q2023598)

The paper deals with the hyperbolic system \[ \displaystyle{\partial_t x + \Lambda\partial_l = Kx(l,t)\ ;\ x=col\{x_1,x_2\}} \] with \(\Lambda=diag\{\lambda_1,\lambda_2\}\), \(\lambda_i>0\). The boundary conditions are controlled and given by \[ \displaystyle{x_i(0,t)=u_i(t)\ ,\ i=1,2} \] while the observed output is given by \[ \displaystyle{y_i(t) = x_i(L,t)\ ,\ i=1,2} \] In this way an input-state-output system (in the sense of control theory) has been defined, displaying unbounded input and output operators. The paper deals with a finite dimensional approximation theory for this system, expressed in the input-state-output representation, transfer function and steady state versus transient response. An engineering application is given, including simulation results.