4. LQR with time-varying weighting matrices

In this section, first, the flow chart of HSV profile-following is presented. Then, LQR design method using time-varying weighting matrix for HSV reentry trajectory tracking law is derived.

Based on LQR, here is the flow chart of HSV tracking reference profile shown as the solid lines in Figure 4.

The work flow of HSV profile-following is explained as follows:

Comparing the actual flight profile with the reference profile, one can get the state deviations containing z<sup>δ</sup> and vδ. With these deviations, the compensatory signal u<sup>δ</sup> can be calculated by

Figure 4. The flow chart of HSV profile-following based on LQR.

Figure 3. The profile-following of HSV entry guidance with 20% aerodynamic parameter error by Q<sup>0</sup> and Q1.

142 Advances in Some Hypersonic Vehicles Technologies

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 computer before starting entry guidance of HSV.

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 into zδmax and vδmax, that is,

$$
\omega\_{\delta \text{max}} = \underline{z}\_{\delta}(t) \; , \; \underline{v}\_{\delta \text{max}} = \underline{v}\_{\delta}(t) \; . \tag{28}
$$

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 timevarying weighting matrix in LQR.

Algorithm 2. The actual guidance signal in trajectory tracking law based on LQR using timevarying weighting matrix can be designed in the following procedure.

