Apparatus and method for rapid identification of laser communication beacons

Abstract

An apparatus and method for tracking a laser communication signal of interest incident on a focal plane array (FPA) identifies a plurality of hotspots in a scene of interest, and aligns a hotspot that is a signal of interest (SOI) or beacon component thereof with the FPA. Hotspot centroids can be estimated within a fraction of an FPA pixel by considering squares of four pixels and comparing their signal amplitudes. A multi-spot calculation is used to improve the position estimates of all of the hotspots by applying a Kalman filter to the hotspot position data and assuming that the relative hotspots positions are fixed. The calculation is periodically repeated to enable tracking of the SOI. Amplitude variability of the hotspots is accommodated by weighting the hotspot contributions according to their intensities. In embodiments, estimation error of the SOI centroid is less than a smallest dimension of the FPA pixels.

Claims

1. A method of identifying and tracking a laser communication signal of interest, the method comprising: directing incident light from a scene onto a focal plane array (FPA); designating a plurality of light intensity maxima detected by the FPA as hotspots belonging to a group of hotspots; identifying one of the hotspots in the group of hotspots as the laser communication signal of interest (SOI), and designating the other hotspots in the group of hotspots as competing hotspots; adjusting an incident direction of the incident light from the scene so as to align the laser communication SOI with a laser data receiver; measuring location and amplitude data for each of the hotspots in the group of hotspots; performing a multi-spot estimation of a centroid of the laser communication SOI under an assumption that relative locations of all of the hotspots in the group of hotspots are constant, said multi-spot estimation comprising applying a Kalman filter to the location data measured for the hotspots, the location data for each of said hotspots being weighted according to a magnitude of the amplitude data measured for said hotspot, so that the result of the multi-spot estimation is more strongly influenced by hotspots for which the measured amplitude data is stronger, and less strongly influenced by hotspots for which the measured amplitude data is weaker; and adjusting the incident direction of the light from the scene as needed to maintain alignment of the laser communication SOI with the laser data receiver.

2. The method of claim 1, wherein aligning the laser communication SOI with the laser data receiver includes positioning the laser communication SOI at a tracking location on the FPA.

3. The method of claim 1, wherein designating the plurality of light intensity maxima detected by the FPA as hotspots includes considering pixels of the FPA in a sliding window that defines square pixel groups of six by six pixels, and designating pixels within the pixel groups having output amplitudes that are local maxima as hotspots.

4. The method of claim 1, wherein for each of the hotspots, measuring the location information for the hotspot comprises: identifying a quad-cell of the FPA within which the hotspot is located, the quad-cell being a group of four adjacent pixels of the FPA arranged in a square; and estimating a centroid of the hotspot according to relative output amplitudes of the four pixels of the quad-cell.

5. The method of claim 1, wherein the laser communication SOI comprises a signal component and a beacon component that differ from each other in wavelength, said components being overlapping and parallel.

6. The method of claim 5, further comprising separating beacon light having the beacon component wavelength from the incident light, and directing only the beacon light onto the FPA.

7. The method of claim 1, wherein the multi-spot estimation of the centroid of the laser communication SOI has an estimation error that is less than a smallest dimension of each of the pixels of the FPA.

8. The method of claim 1, further comprising repeating the step of designating the plurality of light intensity maxima as hotspots, whereby newly appearing hotspots are added to the group of hotspots and previously designated hotspots that are not detected are deleted from the group of hotspots.

9. An apparatus for tracking a laser communication signal of interest, the apparatus comprising: a beam-directing device configured to direct incident light from a scene onto a focal plane array (FPA); a hotspot identifier configured to receive light intensity data from the FPA, to designate a plurality of maxima of the FPA light intensity data as hotspots belonging to a group of hotspots, and to determine amplitude and location information pertaining to the hotspots; a signal identifier configured to identify one of the hotspots as the laser communication signal of interest (SOI); a laser data receiver; and a controller configured to accept data pertaining to the incident light from the FPA, hotspot identifier, and signal identifier, the controller being further configured to: estimate a location of a centroid of the laser communication SOI; according to the estimated location of the SOI centroid, cause the beam-directing device to adjust an incident direction of the incident light from the scene so as to align the laser communication SOI with the laser data receiver; perform a multi-spot estimation of the centroid location of the laser communication SOI under an assumption that relative locations of all of the hotspots in the group of hotspots are constant, said multi-spot calculation comprising applying a Kalman filter to the location data pertaining to the hotspots, the location data for each of said hotspots being weighted according to a magnitude of the amplitude data pertaining to said hotspot, so that the result of the multi-spot calculation is more strongly influenced by hotspots having stronger amplitude data, and less strongly influenced by hotspots having weaker amplitude data; and cause the beam-directing device to adjust the incident direction of the light from the scene as needed to maintain alignment of the laser communication SOI with the laser data receiver, so that the laser data receiver is able to receive wireless data from the laser communication signal of interest.

10. The apparatus of claim 9, further comprising a beam dividing device configured to separate overlapping and parallel laser beam components of differing wavelengths included in the laser communication SOI, one of said components being a beacon component.

11. The apparatus of claim 10, wherein the beam dividing device comprises at least one of a diffraction grating, a prism, a beam splitter, and a bandpass filter.

12. The apparatus of claim 10, wherein the beam dividing device is configured to direct the beacon component onto the FPA, while directing at least one other of the components onto the laser data receiver.

13. The apparatus of claim 12, wherein the controller is configured to align the laser communication SOI with the laser data receiver by aligning the beacon component of the laser communication SOI with a tracking location of the FPA.

14. The apparatus of claim 9, wherein the hotspot identifies is configured to determine the location information pertaining to each of the hotspots by: identifying a quad-cell of the FPA within which the hotspot is located, the quad-cell being a group of four adjacent pixels of the FPA arranged in a square; and estimating a centroid of the hotspot according to relative output amplitudes of the four pixels of the quad-cell.

15. The apparatus of claim 9, wherein the controller is configured to perform the multi-spot estimation with an error that is less than a smallest dimension of each of the pixels of the FPA.

16. The apparatus of claim 9, wherein the hotspot identifier is further configured to periodically update the group of hotspots, whereby newly appearing hotspots are added to the group of hotspots and previously designated hotspots that are no longer detected are deleted from the group of hotspots.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 illustrates laser communication between orbiting satellites, and between the satellites and ground-based nodes;

(2) FIG. 2 is a flow diagram illustrating steps required for establishing and maintaining alignment between a laser communication receiving system and a transmitting node;

(3) FIG. 3 illustrates quad-cell estimation of a light source location;

(4) FIG. 4A is a block diagram that illustrates components of a laser communication receiving apparatus according to an embodiment of the present disclosure;

(5) FIG. 4B is a block diagram illustrating details of the structure of a controller in an embodiment of the present disclosure;

(6) FIG. 4C is a flow diagram illustrating steps performed by the controller of FIG. 4B in embodiments of the present disclosure.

(7) FIG. 5A illustrates relative positions of an SOI and two competing hotspots;

(8) FIG. 5B illustrates a mathematical formulation of the relative positions of FIG. 5A; and

(9) FIG. 6 illustrates relative positions of an SOI and a single competing hotspot.

DETAILED DESCRIPTION

(10) The present disclosure is a reliable and accurate apparatus and method for identifying, aligning, and tracking a laser beam hotspot that is carrying a laser communication signal of interest, also referred to herein as an SOI hotspot, or simply as an SOI. Embodiments of the disclosed method and apparatus provide reliable alignment and tracking of an SOI even when other, possibly stronger hotspots are present within the scene of interest that may vary in amplitude, and may even appear and disappear as a function of time.

(11) According to the present disclosure, with reference to FIGS. 3 and 4A, a frame of incident light 400 from a scene of interest is directed onto a focal plane array (FPA) 310 by a beam-directing device 404, which in the embodiment of FIG. 4A includes a mirror 402 driven by a servo motor 418 that is under control of a controller 416. Narrow beams of light arising from the scene of interest that are candidate laser communication signals give rise to spots of light within the frame 400, referred to herein as hotspots 304, 306, that are more intense than a surrounding background light intensity of the frame 400.

(12) With reference to FIG. 4B, a hotspot identifier 420, which may be physically part of the controller 416, is used to identify 200 the hotspots 304, 306 within the frame of light 400 that is incident on the FPA 310. With reference again to FIG. 3, in embodiments this step includes applying a sliding window to the FPA 310 that divides the FPA pixels into squares of pixels 308, 312, which in embodiments are 66 squares of pixels. For each of the pixel squares 308, 312, the signal intensities arising from the pixels in the square are compared. If the strongest signal intensity in a square is reported by a pixel that is not at the boundary of the square, that pixel is designated as a local maximum pixel, and the beam of light that is striking the local maximum pixel is designated as a hotspot.

(13) Once the local maximum pixels have been identified, a signal identifier 422 is used to identify a hotspot that is a signal of interest 400 (SOI). Once the SOI 400 has been identified, the controller 416 directs a beam directing device 404 to position the SOI at a tracking location on the FPA, typically at the center of the FPA. This initial positioning of the SOI 302 on the FPA 310 represents the pull in alignment step 204 of FIG. 2. The subsequent maintaining of this alignment represents the tracking step 206 of FIG. 2.

(14) In embodiments, it is assumed that location of the centroid of the SOI 304 on the FPA 310, and thereby the alignment and tracking of the SOI 304 on the FPA 310, must be determined and controlled with an accuracy that is on the order of a fraction of the size of an FPA pixel. In some of these embodiments, a quad-cell approach is employed, whereby the pixels of the FPA are divided into square groups of four adjacent cells 300, 302, referred to herein as quad-cells, which for convenience are identified by the indices of the pixels at the lower left corners of the squares. In FIG. 3 the axes at the center of the FPA 310 fall between the pixels 300 at the FPA center, as shown.

(15) In these quad-cell embodiments, after identifying the hotspots 304, 306 and verifying that one of the hotspots 304 is a laser communication signal of interest (SOI), the controller 416 adjusts the beam-directing device 404 until the SOI 304 is located within a specified tracking quad-cell 300, which is typically the quad-cell 300 at the center of the FPA 310. In some of these embodiments, this initial alignment is accomplished simply by causing the servo 418 to adjust the mirror 402 of the beam-directing device 404 until the local maximum pixel associated with the SOI 304 is one of the pixels of the tracking quad-cell 300.

(16) In the example of FIG. 3, the quad-cell 300 at the center of the FPA 310 is the tracking quad-cell, and is identified by the index that corresponds to the pixel (0,0) in the lower corner of the cell 300.

(17) Once the SOI 304 is within the tracking quad-cell 300, the controller 416 further improves the alignment of the SOI by comparing the signal amplitudes produced by the four pixels within the tracking quad-cell 300 (the pixel amplitudes), and estimating the offset of the SOI centroid within the tracking quad-cell 300 according to the differences between the pixel amplitudes. The beam-directing device 404 is then adjusted by the controller 416 until the pixel amplitudes of the tracking quad-cell 300 are all equal to each other. In some embodiments, the position of the SOI centroid within the tracking quad-cell 300 is estimated according to the pixel amplitudes by assuming that the SOI 304 is a circular spot with a sharp boundary, as illustrated in FIG. 3. In other embodiments, the SOI 304 is assumed to have a Gaussian profile, or a circular profile with edges that fall off as Gaussians. In embodiments, the calculated location of the SOI centroid is continuously updated, and is used by the controller 416 in determining the adjustments that must be made by the beam-directing device 404 to maintain alignment of the SOI centroid on the FPA 310.

(18) With continued reference to FIG. 4A, in embodiments laser communication signals of interest 400 comprise two overlapping, co-linear beam components 408, 412 that are transmitted at different wavelengths, whereby one of the two beam components 412 is used for data communication, and the other of the two beam components 408 is a beacon that is used to identify the laser communication signal source to the satellite 100 or other laser signal receiving node. In the embodiment of FIG. 4A, the incoming light 400 is directed by the beam directing device 404, which in FIG. 4A includes a steering mirror 402 driven by a servo-motor 418, onto a beam dividing device 406 that spatially separates the two components 408, 412 from each other, so that the beacon component 408 is directed to a focal plane array 310 while the signal component 412 is directed to a laser data receiver 414. In similar embodiments, the two components 408, 412 are separated by other mechanisms, which can include prisms, beam splitters, gratings, and/or bandwidth filters. The SOI beacon 408 is aligned with the FPA 301, causing the signal component 412 to be simultaneously aligned with a signal sensor 414.

(19) As is noted above, tracking of a signal (or beacon) of interest can be difficult if other, competing light sources are present in the scene of interest, especially if one or more of these competing light sources is geographically close to the node that is transmitting the SOI. The tracking becomes even more difficult if one or more of these competing hotspots is stronger than the SOI. For example, in FIG. 3, a competing hotspot 306 is also incident on the FPA 310. The competing hotspot 306 is shown as incident on a quad-cell having indices (i.sub.C, j.sub.C).

(20) In embodiments of the present disclosure, it is assumed that any competing light sources 306 are stationary relative to the SOI 304, or at least that their relative positions are changing slowly in comparison to the timescales over which data is transmitted by the SOI 304, as is generally the case, for example, when laser communication signals are received by a satellite 100. Under this assumption, rather than attempting to exclude competing hotspots 306 from the SOI centroid estimation, embodiments of the present disclosure use a multi-spot centroid estimator 424 to track the SOI and maintain its alignment with the laser data receiver 414. The multi-spot centroid estimator 424 includes the competing hotspots 306 in a multi-spot estimation, and thereby improves the SOI centroid estimation, especially in cases where the competing hotspots 306 are stronger light sources than the SOI 304.

(21) FIG. 4C is a flow diagram that illustrates steps performed by the controller 416 in embodiments of the present disclosure. After accepting data 426 from the hotspot identifier 420 and identification of the SOI by the signal recognizer 422, the controller 416 estimates the SOI centroid location 430 of the SOI, possibly using the quad-cell approach described above, and aligns 432 the SOI with the laser data receiver 414. After this initial alignment of the SOI, the controller 416 uses the multi-spot centroid estimator 424 to perform a multi-spot estimation 434 of the SOI centroid location, and then uses the beam directing device 404 to update and adjust the SOI alignment 436 according to the result of the multi-spot estimation. The controller proceeds to periodically update its data 436 regarding the centroids of the SOI and other hotspots, and repeats the multi-spot calculation and SOI realignment as needed to compensate for jitter and maintain alignment with the SOI.

(22) In some of these embodiments, a mathematical, multi-spot model is used as the basis for estimating the centroids of both the SOI 304 and the competing hotspots 306. According to this approach, in embodiments a single, multi-spot calculation is performed that results in estimates of the centroids for all of the tracked light sources 304, 306, rather than performing separate calculations for each of the tracked light sources 304, 306 in the scene.

(23) Of course, the origins of competing hotspots 306 are generally not known to the controller. They may arise from competing laser communication signals, transmitted by friendly or hostile entities. They may also arise from naturally occurring features such as locations filled with water or ice, from metallic structures or other reflective objects, or even from hostile jamming sources. Accordingly, under many circumstances it can be assumed that the intensities of competing hotspots 306 may not be constant, and may even appear and disappear over time. And of course, even for a constant light source, the apparent intensity will vary due to the atmospheric effects that can be referred to generically as scintillation.

(24) Embodiments of the present disclosure compensate for amplitude variability of the SOI 304 and competing hotspots 306 by weighting their contributions to the multispot estimation according to their intensities. In other words, the estimated locations of the SOI 304 and competing hotspots 306 are given more weight as they grow stronger, and less weight as they grow weaker. And if a competing hotspot 306 disappears entirely, it is given a weight of zero, and thereby excluded from the estimation altogether. In this way, the quality of each centroid observation is linked to the strength of that observation, and thus to its signal to noise ratio, such that when a competing hotspot disappears, observations of the centroid of that hotspot are unweighted and thereby no longer included in the multi-spot estimation. And when a new competing hotspot appears, it can be added to the estimation process and can begin to serve as an additional source of observation information and estimation accuracy.

(25) FIG. 5A presents an example where a multi-spot analysis can be used to simultaneously estimate the centroid location of an SOI 500 and two competing hotspots 502, 504. It is assumed that the SOI 500 and the two competing hotspots 502, 504 have fixed locations relative to each other, and that they move in tandem within the scene. The problem is then to estimate the location of the centroid 500 of the SOI {x.sub.s, y.sub.s} by combining centroid data measured for each of the spots 500, 502, 504. It is assumed that the accuracy of the estimate must be radially better than the size of a single pixel of the FPA 310, and that the intensities of the two competing hotspots 502, 504 can vary at any time and according to any pattern, including completely disappearing and reappearing, and that a specified SOI location accuracy must be maintained despite these variations in competing hotspot intensity.

(26) With reference to FIG. 5B, the problem can be formulated according to observations of the centroids of the competing hotspots 502, 504 as follows:

(27) TABLE-US-00001 TABLE 1 State and observation equations for multi-spot centroid estimation State Equations Observation Equations x.sub.s = 0 + w.sub.sx z.sub.xs = x.sub.s y.sub.s = 0 + w.sub.sy z.sub.x1 = x.sub.s + b.sub.1x b.sub.1x = 0 + w.sub.1x z.sub.x2 = x.sub.s + b.sub.1y b.sub.1y = 0 + w.sub.1y z.sub.ys = x.sub.s b.sub.2x = 0 + w.sub.2x z.sub.y1 = x.sub.s + b.sub.2x b.sub.2y = 0 + w.sub.2y z.sub.y2 = x.sub.s + b.sub.2y
where the w terms are process noise signals. According to this approach, the SOI centroid is modeled as a random walk process (integral of white noise) and the fixed offsets are modeled as biases that are driven by weak process noise representing the assumption that they vary slowly.

(28) This 6.sup.th order problem can be solved as 2 independent 3.sup.rd order problems, solving for x and y separately. Accordingly, the equations used to solve for x are:

(29) TABLE-US-00002 TABLE 2 State and observation equations for estimating x State Equations Observation Equations x.sub.s = 0 + w.sub.sx z.sub.xs = x.sub.s b1.sub.x = 0 + w.sub.1x z.sub.x1 = x.sub.s + b.sub.1x b2.sub.x = 0 + w.sub.2x z.sub.x2 = x.sub.s + b.sub.2x

(30) The goal is then to obtain an optimal estimate of the SOI centroid location, given the three centroid observations, and also to estimate the fixed biases as part of the solution. The problem can be expressed in vector format as x=Ax+w and y Cx+v as follows:

(31) $\begin{array}{cc}\left[\begin{array}{c}{x}_s\\ b_{1x}\\ b_{2x}\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_s\\ b_{1x}\\ b_{2x}\end{array}\right]+\left[\begin{array}{c}{w}_s\\ w_{1x}\\ w_{2x}\end{array}\right]& \left(1\right)\\ \left[\begin{array}{c}{z}_{\mathrm{xs}}\\ z_{x1}\\ z_{x2}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_s\\ b_{1x}\\ b_{2x}\end{array}\right]+\left[\begin{array}{c}{v}_s\\ v_{1x}\\ v_{2x}\end{array}\right]& \left(2\right)\end{array}$
Where the square matrices in Eqns. 1 and 2 are A and C, respectively, and where the vector v in Eqn. 2 represents the observation noise process.

(32) The Kalman filter equations can be applied to the solution of this problem as follows:

(33) $\begin{array}{cc}K={\mathrm{PC}}^{}{V}^{-1}& \left(3\right)\\ P=W-\mathrm{KCP}& \left(4\right)\\ \hat{x}=K\left(z-C\hat{x}\right)& \left(5\right)\end{array}$
where Eqn. 3 is the equation for the Kalman gain K, Eqn. 4 is the equation for the covariance matrix P, and Eqn. 5 is the state estimate equation. In Eqn. 5, z is the vector of observations. In Eqns. 3 and 4, the W and V matrices are the process and observation noise spectral density matrices, which are given by

(34) $\begin{array}{cc}W=\left[\begin{array}{ccc}{w}_s& 0& 0\\ 0& w_{b1}& 0\\ 0& 0& w_{b1}\end{array}\right]& \left(6\right)\\ V=\left[\begin{array}{ccc}{v}_s& 0& 0\\ 0& v_{b1}& 0\\ 0& 0& v_{b1}\end{array}\right]& \left(7\right)\end{array}$
where the diagonal elements in eqns. 6 and 7 are the variances of the noise terms during their respective equations.

(35) As discussed above, it is know that centroid measurements degrade with the amplitude of adjacent hotspots. Accordingly, an observation noise covariance matrix can be postulated having covariance terms that are inversely proportional to squared amplitude (A.sup.2), according to:

(36) $\begin{array}{cc}V=\left[\begin{array}{ccc}\left(\frac{1}{A_s^2}\right)& 0& 0\\ 0& \left(\frac{1}{A_1^2}\right)& 0\\ 0& 0& \left(\frac{1}{A_2^2}\right)\end{array}\right]& \left(8\right)\end{array}$
Eqn. 8 can be re-written in terms of the strongest amplitude. For example, if A.sub.1 is the strongest amplitude, then Eqn. 8 can be re-written as:

(37) $\begin{array}{cc}V=\left[\begin{array}{ccc}\left(\frac{A_1^2}{A_s^2}\right)& 0& 0\\ 0& 1& 0\\ 0& 0& \left(\frac{A_1^2}{A_2^2}\right)\end{array}\right]& \left(9\right)\end{array}$

(38) Eqn. 9 expresses the fact that the noise of an observation grows as its amplitude shrinks, i.e. as the observation worsens in quality. Formulating Eqn. 9 in this way allows the calculated Kalman filter to adapt to the amplitudes of the spots (i.e. the SOI and competing hotspot centroids), even when the amplitudes go to zero, meaning that the associated observation has infinite noise power.

(39) The inverse matrix V.sup.1 can then be formulated as follows:

(40) if A.sub.s=max(A.sub.s, A.sub.1, A.sub.2), then:

(41) $\begin{array}{cc}{V}^{-1}=\left[\begin{array}{ccc}1& 0& 0\\ 0& \left(\frac{A_1^2}{A_s^2}\right)& 0\\ 0& 0& \left(\frac{A_2^2}{A_2^2}\right)\end{array}\right]& \left(10\right)\end{array}$
if A.sub.1=max(A.sub.s, A.sub.1, A.sub.2), then:

(42) $\begin{array}{cc}{V}^{-1}=\left[\begin{array}{ccc}\left(\frac{A_s^2}{A_1^2}\right)& 0& 0\\ 0& 1& 0\\ 0& 0& \left(\frac{A_2^2}{A_1^2}\right)\end{array}\right]& \left(11\right)\end{array}$
and if A.sub.2=max(A.sub.s, A.sub.1, A.sub.2), then:

(43) $\begin{array}{cc}{V}^{-1}=\left[\begin{array}{ccc}\left(\frac{A_s^2}{A_2^2}\right)& 0& 0\\ 0& \left(\frac{A_1^2}{A_2^2}\right)& 0\\ 0& 0& 1\end{array}\right]& \left(12\right)\end{array}$
where each of the amplitudes A.sub.s, A.sub.1, A.sub.2 is obtained from a sensor of the system, for example as a pixel output of an FPA or as the sum of the four pixel outputs of a quad-cell of the FPA.

(44) With reference to FIG. 6, the equations for a 2-spot formulation, where only the SOI 600 and one competing hotspot 602 are present, are similar to the 3-spot formulation, but simplified. As before, the problem is to measure or estimate the centroid location of the SOI 600 by combining the centroid location data measured for each of the spots 600, 602 such that the accuracy of the SOI centroid estimate is better than a fraction of a pixel, even when the amplitude of the competing hotspot 602 is much greater than the amplitude of the SOI spot 600, and even when the intensity of the competing hotspot 602 can change in amplitude at any time, and can even come and go. The second order Kalman filter equations for this two-spot case are as follows:

(45) $\begin{array}{cc}P=\left[\begin{array}{cc}{p}_{11}& p_{12}\\ p_{21}& p_{22}\end{array}\right]& \left(13\right)\\ X=\left[\begin{array}{c}{x}_s\\ x_1\end{array}\right]& \left(14\right)\\ P=W-{\mathrm{PC}}^{}{V}^{-1}\mathrm{CP}& \left(15\right)\\ \hat{x}=K\left(y-C\hat{x}\right)& \left(16\right)\\ K={\mathrm{PC}}^{}{V}^{-1}& \left(17\right)\\ P\left(0\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& \left(18\right)\\ \hat{x}\left(0\right)=\left\{0,0\right\}& \left(19\right)\\ C=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]& \left(20\right)\\ W=\left[\begin{array}{cc}100& 0\\ 0& 100\end{array}\right]& \left(21\right)\\ V^{-1}=500\left[\begin{array}{cc}1& 0\\ 0& \frac{A_1^2}{A_s^2}\end{array}\right]\mathrm{if}{A}_s\max & \left(22\right)\\ V^{-1}=500\left[\begin{array}{cc}\frac{A_s^2}{A_1^2}& 0\\ 0& 1\end{array}\right]\mathrm{if}{A}_1\max & \left(23\right)\end{array}$

(46) The foregoing description of the embodiments of the invention has been presented for the purposes of illustration and description. Each and every page of this submission, and all contents thereon, however characterized, identified, or numbered, is considered a substantive part of this application for all purposes, irrespective of form or placement within the application. This specification is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of this disclosure.

(47) Although the present application is shown in a limited number of forms, the scope of the invention is not limited to just these forms, but is amenable to various changes and modifications without departing from the spirit thereof. The disclosure presented herein does not explicitly disclose all possible combinations of features that fall within the scope of the invention. The features disclosed herein for the various embodiments can generally be interchanged and combined into any combinations that are not self-contradictory without departing from the scope of the invention. In particular, the limitations presented in dependent claims below can be combined with their corresponding independent claims in any number and in any order without departing from the scope of this disclosure, unless the dependent claims are logically incompatible with each other.