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Optimal damping of forced oscillations based on prescribed system output. (English. Russian original) Zbl 0847.93033
Phys.-Dokl. 39, No. 7, 477-481 (1994); translation from Dokl. Akad. Nauk 337, No. 3, 323-327 (1994).
The following optimal damping problem of forced oscillations is considered: ${{dx(t)} \over {dt}}= Ax(t)+ bu(t)+ f^0 \varphi (t), \quad y(t)= c^* x(t), \quad \Phi= \varlimsup_{T\to \infty} {1\over T} \int^T_0 {\mathcal G} [x(t), u(t) ]dt\to \min,$ where $${\mathcal G} (x,u)$$ is a quadratic form of the state variable $$x= x(t)\in \mathbb{R}^n$$ and the control function $$u= u(y (t))\in \mathbb{R}^m$$. The external forcing $$\varphi (t)= \varphi_1 e^{i\omega_1 t}+ \cdots+ \varphi_N e^{i\omega_N t}$$ is an unknown harmonic function with given spectrum $$\{ \omega_j \}$$. The problem is to construct a controller $$u$$ which guarantees the minimum of the functional $$\Phi$$ for a $$\varphi (t)$$ of the type cited above. The problem under consideration has been studied by many authors under various assumptions. The main result of the paper is that under some “natural” assumptions, there exists a “universal” controller which solves the problem for all the indicated $$\varphi (t)$$ simultaneously. It is physically realizable, satisfies a stability condition and can be described by relatively simple formulas.

##### MSC:
 93C41 Control/observation systems with incomplete information 93D10 Popov-type stability of feedback systems 93D09 Robust stability 49N10 Linear-quadratic optimal control problems 93C73 Perturbations in control/observation systems
##### Keywords:
optimal damping; harmonic