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

The paper considers the “equation + index” system

${x}_{k+1}=A{x}_{k}+{B}_{1}{u}_{k}$
${z}_{k}={C}_{1}{x}_{k}+{D}_{11}{u}_{k}$
$E\left(u,z,N\right)=\sum _{0}^{N}\left({z}_{k}^{*}Q{z}_{k}+2{x}_{k}S{u}_{k}+{u}_{k}^{*}R{u}_{k}\right)·$

A review of dissipativity theory is given for such systems. Then one considers the two-input/two-output system

${x}_{k+1}=A{x}_{k}+{B}_{1}{\omega }_{k}+{B}_{2}{u}_{k}$
${z}_{k}={C}_{1}{x}_{k}+{D}_{11}{\omega }_{k}+{D}_{12}{u}_{k}$
${y}_{k}={C}_{2}{x}_{k}+{D}_{21}{\omega }_{k}·$

A Linear Matrix Inequalities (LMI) approach is used to obtain the feedback control ${u}_{k}=K{x}_{k}$ 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 systems