## Dissipative control for linear discrete-time systems.(English)Zbl 0949.93068

The paper considers the “equation + index” system $x_{k+1}= Ax_k+B_1u_k$
$z_k=C_1x_k+ D_{11}u_k$
$E(u,z,N)= \sum^N_0(z^*_k Qz_k+2x_k Su_k+ u^*_kRu_k).$ A review of dissipativity theory is given for such systems. Then one considers the two-input/two-output system $x_{k+1}= Ax_k+B_1 \omega_k+B_2u_k$
$z_k=C_1x_k+ D_{11}\omega_k +D_{12}u_k$
$y_k=C_2 x_k+D_{21} \omega_k.$ A Linear Matrix Inequalities (LMI) approach is used to obtain the feedback control $$u_k=Kx_k$$ so that the resulting system should be dissipative and exponentially stable. An example is given.

### MSC:

 93D15 Stabilization of systems by feedback 15A39 Linear inequalities of matrices 93D10 Popov-type stability of feedback systems 93C55 Discrete-time control/observation systems
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