5. Simulation results

In this section, a numerical simulation is given to demonstrate the effectiveness of the proposed method in the previous section.

Q<sup>0</sup> and Q<sup>1</sup> are defined in Section 3, and the time-varying weighting matrix is denoted by Q. The simulations for hypersonic vehicle following reference profile with Q are shown in Figures 5–7.

Figure 5. The profile-following of HSV entry guidance with initial altitude deviation by Q<sup>0</sup> and Q.

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multiplying feedback gain K. Then one can input the compensatory signal and the reference guidance signal into HSV guidance loop. In this way, actual flight profile of the next step is obtained. The calculation of the feedback gain K by LQR involves four matrices. As shown in the Figure 4, the construction of system matrices A and B needs actual state parameters. Weighting matrices Q and R need to be determined and downloaded into the onboard com-

Instead of obtaining the specific elements in traditional method, the LQR design method using time-varying weighting matrix substitutes the flight state deviations z<sup>δ</sup> and v<sup>δ</sup> into the calculation of Q. The main idea of this method can be explained as the dashed line in Figure 4. With the help of Bryson principle, the calculation of elements in weighting matrix Q involves two parameters zδmax and vδmax. These two parameters represent maximal allowable deviations in altitude and velocity between actual and reference profiles, respectively. In the time-varying optimization method, one can make a comparison between the actual real-time profile and the relevant reference profile, and get the current deviations zδ(t) and vδ(t). Then substitute them

Substituting zδmax and vδmax into Eq. (21), the weighting matrix Q can be obtained. The following algorithm is proposed to determine the actual guidance signal σ(t) with time-

Algorithm 2. The actual guidance signal in trajectory tracking law based on LQR using time-

Step 1 Measure the actual current flight profile which contains altitude z and velocity v.

Step 2 Substitute z, v, and γ into system, and calculate the linear system matrices A(t) and B(t). Step 3 Construct the weighting matrices Q(t) and R(t) by substituting zδ, v<sup>δ</sup> and maximal allowable adjustment of guidance signal σδmax into Eqs. (28), (21) and (22).

Step 5 The compensatory guidance signal δu can be calculated by K(t) in Eq. (23). Then one

In this section, a numerical simulation is given to demonstrate the effectiveness of the pro-

Q<sup>0</sup> and Q<sup>1</sup> are defined in Section 3, and the time-varying weighting matrix is denoted by Q. The simulations for hypersonic vehicle following reference profile with Q are shown in

Step 4 Calculate the feedback gain K(t) by A(t), B(t), Q(t), R(t) in Eqs. (11) and (12).

can get the actual guidance signal in Eq. (24), namely bank angle σ(t).

Compare them with the relevant reference altitude zref and velocity vref, the current

varying weighting matrix can be designed in the following procedure.

deviations z<sup>δ</sup> and v<sup>δ</sup> can be obtained, respectively.

zδmax ¼ zδð Þt , vδmax ¼ vδð Þt : (28)

puter before starting entry guidance of HSV.

144 Advances in Some Hypersonic Vehicles Technologies

into zδmax and vδmax, that is,

varying weighting matrix in LQR.

5. Simulation results

Figures 5–7.

posed method in the previous section.

Figure 5. The profile-following of HSV entry guidance with initial altitude deviation by Q<sup>0</sup> and Q.

Figure 6. The profile-following of HSV entry guidance with initial path angle deviation by Q<sup>0</sup> and Q.

Figure 7. The profile-following of HSV entry guidance with 20% aerodynamic parameter error by Q<sup>1</sup> and Q.

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Figure 7. The profile-following of HSV entry guidance with 20% aerodynamic parameter error by Q<sup>1</sup> and Q.

Figure 6. The profile-following of HSV entry guidance with initial path angle deviation by Q<sup>0</sup> and Q.

146 Advances in Some Hypersonic Vehicles Technologies

For initial deviations of altitude and path angle, it's clear that the profile-following performances of Q<sup>0</sup> are better than Q<sup>1</sup> from Figures 1 and 2. Consequently, for the same deviations, Figures 5 and 6 choose Q<sup>0</sup> to compare with Q. Since Q<sup>1</sup> performs better than Q<sup>0</sup> in Figure 3 under the aerodynamic parameter error, one can choose Q<sup>1</sup> to compare with Q under the same disturbance in Figure 7.

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From Figures 5–7, it can be shown that the time-varying matrix Q has better performance than Q<sup>0</sup> and Q<sup>1</sup> in the application of LQR on HSV following reference profile.
