Four Algorithms for Boundary Control with Breaking in Space and Time

Typically, active control systems either have a priori complete information about the boundary-value problem and damped waves before switching on, or get it during the measurement process or accumulate and update information online (identification process in adaptive systems). In this case, the boundary problem is completely imprinted in the information arrays of the control system. However, very often complete information about a boun-dary-value problem is not available in principle or this info is changing in time faster than the process of its accumulation. The article considers examples of boundary control algorithms based almost without any information. The algorithms presented in the article cannot be obtained within the frame-work of the harmonic representation of the problem by complex amplitudes. And these algorithms carry out fast control in microstructured boundary problems. It is shown that in some cases it is possible to find simple solutions if we remove restrictions: 1) on the spatio-temporal resolution of controlling elements of a boundary-value problem; 2) on the high-frequency radiation of the controlled boundary.


Introduction
With constant (in time) parameters (or without frequency conversion) devices for wideband non-resonant sound suppression (or other types of waves) should have large wave sizes (thickness D  , see Figure 1

Algorithm of Half-Return of the Boundary (AHRB)
The goal of the algorithm is to suppress reflections from boundary b x . We consider a semi-infinite ( b x x ≤ < ∞ ) elastic rod, with longitudinal impedance Z, Figure 1. General statement of the problem and goals of approach: black area means suppressing layer of thickness D  (traditional thick wideband suppressing layer with parameters constant in time (a) and thin wideband suppressing layer with high frequency operations (b)); white-black wave (b) means reflected high frequency waves conversed from incident one (gray wave means incident wave); (c) small size muffler for gas stream with long sound waves. Journal of Applied Mathematics and Physics sound speed c of waves and the field ( , ) U x t of longitudinal displacement of particles ( Figure 2(a)). The boundary condition at the end b the cross-section of the rod, ε is Young's modulus. A smooth incident wave The boundary-value problem can be represented as the sum of two partial linear problems: (a1) reflection of the incident wave (IW) W 0 U ≠ from the free (at b 0 F = ) the end of the rod; (a2) wave generation by force b 0 F ≠ in the absence of an incident wave (at W 0 U = ). The "breaker" jumps ((a1) ↔ (a2)) in accordance with the control algorithm: measurement in (a1), action in (a2). We assume that the force b F has a compact support: . For a clear distinction between the causes and consequences in the work of AHRB, it is extremely important that after the termina- x caused by this force is saved indefinitely long [1] after its switching off. Now we will directly consider AHRB, which is a sequence of time cycles (see The peak hf P and average hf P (on the cycle ~T) power of the high-frequency radiation, generated by the impact force b F , and time averaged power flux W P in the low frequency IW are satisfying to the following relations impedance: Z is unknown and can slowly change in time ( ) Z t . About impact force: b ( ) F t can be of arbitrary pulse shape, but with constant sign and at any moment of impact satisfies the condition b the returning path scale is

Algorithm of Maximum Instant Power Absorbed (AMIP)
The goal of the algorithm AMIP is to maximize the instantaneous power absorbed by the boundary b x . Consider above rod problem: some electric drive (as above "breaker") can ensure any constant velocity b V of edge b x independently of any incident wave (IW). Wave problem can be represented as the sum of two linear problems: (a1) reflection of IW from a fixed x ; (a2) the radiation of waves by a boundary b x at a given velocity b V in the absence of IW. The breaker jumps: where T is the period of velocity switching (and measuring between switching).
( ) E x ct + are longitudinal particle velocity and stress in IW in infinite rod.
AMIP is expressed by the iterative (recurrent) relation n n 1 If at the previous step the velocity increase causes the decrease ( 0 W < ) of the absorbed power, at the next step the velocity increase will change its sign and will not change it in the opposite case. In above one dimensional statement of the problem absorption maximum corresponds to the minimum of reflection and radiation too. AMIP does not need to know either rod impedance Z and IW. AMIP effectively traces IW if the following conditions are satisfied: , AMIP resembles the algorithm of random search, considered in [2]. Weak boundary radiation on frequencies f n/ T = ( n 1, 2,3,...

Algorithm for Boundary Condition Modulation (ABCM)
The goal of the algorithm is to suppress reflections from boundary b x in the As a result of such control (see Figure 4(c)), we obtain an oscillogram b ( ) V t of the velocity of the boundary b ( ) V t , which on average (over a period T) tends to the velocity of particles in the incident wave (i.e., in an infinite rod without ref- , as was required above. An experimental verification of the ABCM algorithm is presented below in Section 6. In this case, the boundary b x converts the low-frequency (with a spatial scale W cτ ) IW  x , where D D (f) =   is the frequency dependent dissipative attenuation length or equivalent damping device size (see Figure 1(b)).
Thus, having fulfilled the condition possible to ensure the smallness of the attenuation length and smallness of effect of dissipation on the above boundary condition.

Algorithm for Sound Blocking in the Gas Stream (ASBG)
The goal of ASBG is to block the sound propagation in a gas stream with average Here 2T is the period of states repetition, and T is the period of states change, respectively. In even time intervals (of duration T) between two sections at a distance L from each other, rigid thin flat walls arise instantaneously and simultaneously (state   -dimension of muffler, L-length of waveguides, Λ -wideness of blades, d-cross dimension of waveguide, g-gap between blades and waveguides, δ -mean free path of gas molecules. Running blades do not produce high frequency sound (on the frequencies ~/ V d ). Above described parametric system equally blocks the propagation of sound from left to right, and from right to left.

Experimental Testing of the Algorithm ABCM
The goal is to reduce the ringingness of a tank (as a resonator for surface water waves) without increasing the viscosity of waveguiding media (water). In the traditional case of time constant parameters (Section 1, Figure 1(a)) of wave-suppressing devices, their dimensions are not less than basin length  . The algorithm ABCM (Section 4) was experimentally tested [3] in application to surface water waves in a tank with a length 1.5 =  m and a filling depth 0.22 h = m (see Figure 6(a)). The setup was conceived as an attempt to simulate the above-described one-dimensional boundary acoustic problem for a boundary with a modulated reflection coefficient, despite the two-dimensionality and dispersion of surface water waves.

Description of the Experimental Setup
On the left edge of the tank (Figure 6 ing waves is shown in Figure 6(b).

Pulse Drive Excitation
There two experiments with pulsed excitation (Figure 7

Sinusoidal Excitation of the Tank
The wavemaker on the right wall R (Figure 7(c)) turns on at the moment 0 t = and produces a sinusoidal force ( )