## Linear functional state observer for time-delay systems.(English)Zbl 0953.93012

The design of a low-order functional state observer for a system of the form \begin{aligned} \dot x(t) & = Ax(t)+A_d(t-\tau)+Bu(t)\\ x(t) & = g(t),-\tau\leq t\leq 0,\\ y(t) & = Cx(t)\end{aligned} is addressed. Under the assumption that a memoryless linear state feedback controller $u(t)=Fx(t)$ can be designed, it is shown that a generalisation of Luenberger’s observer can be obtained in the form of $$m$$ dynamic systems $\dot z_i(t)= E_iz_i(t)+ T_iBu(t)+ G_iy(t)+M_i y(t-\tau)$ provided \begin{aligned} A_d & = LC\\ F_i & = K_i T_i+W_i C\text{ (for some }K_i,T_i,W_i)\\ G_iC-T_iA+E_iT_i & = 0\\ M_i & = T_iL\\ p & \geq(n-r)/r\end{aligned} where $$r$$ is the output dimension and $$p$$ is the column dimension of $$K_i$$.

### MSC:

 93B07 Observability 93C23 Control/observation systems governed by functional-differential equations
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