3. Problem statement

navigation facilities. The deviations contain altitude error zδ, velocity error v<sup>δ</sup> and flight path

Denote the compensatory bank angle by σδ. In order to minimize the deviations and keep the aerial vehicle tracking nominal profile properly, the optimal feedback gain of guidance com-

Algorithm 1. The actual guidance signal in trajectory tracking law based on LQR can be

Step 1. Based on perturbation theory, one establishes the linear equations of motion by taking

Step 3. The weighting matrices Q and R in Eq. (20) are determined by Bryson principle. Since altitude z and velocity v are main factors in profile-following, q3, which is the weighting element of path angle γ, can be ignored. Using Eq. (18), the other elements of weighting

where zδmax and vδmax are anticipated maximum deviations between the actual profile and the nominal profile, and σδmax is the maximum allowable modification of guidance signal σ. Based

Step 4. In order to minimize the index J in Eq. (20), one calculates the Riccati Eqs. (11) and (12) to obtain the optimal feedback gain K(t). Then, the compensatory signal can be obtained as

Step 5. The actual guidance signal σ(t) which consists of guidance reference signal u(t) and

<sup>v</sup><sup>δ</sup>max<sup>2</sup> , r<sup>1</sup> <sup>¼</sup> <sup>1</sup>

ðÞ¼ t A tð Þδx tð Þþ B tð Þδu tð Þ,

<sup>δ</sup>xTð Þ<sup>τ</sup> <sup>Q</sup>δxð Þþ <sup>τ</sup> <sup>δ</sup>uð Þ<sup>τ</sup> <sup>R</sup>δuð Þ<sup>τ</sup> � �dτ: (20)

σδmax<sup>2</sup> , (21)

� � , R <sup>¼</sup> <sup>r</sup>1: (22)

δu tðÞ¼�K tð Þδx tð Þ: (23)

σðÞ¼ t u tð Þþ δu tðÞ¼ σref � K tð Þδx tð Þ: (24)

(19)

pensatory signal σδ can be calculated by LQR in the following algorithm.

δx<sup>0</sup>

ðtf t

�

Step 2. Construct the quadratic performance index as follows:

<sup>q</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup>

on Eq. (21), one can get the weighting matrices:

J t; tf � � <sup>¼</sup>

y tðÞ¼ C tð Þδx tð Þ:

<sup>z</sup><sup>δ</sup>max<sup>2</sup> , q<sup>2</sup> <sup>¼</sup> <sup>1</sup>

Q ¼ diag q1; q<sup>2</sup>

guidance compensatory signal δu(t) can be obtained. It can be shown that

angle error γδ.

determined in the following procedure.

138 Advances in Some Hypersonic Vehicles Technologies

where δx(t)=[zδ(t), vδ(t), γδ(t)]<sup>T</sup> and δu(t) = σδ(t).

the deviations as state parameters.

matrices can be obtained as

In this section, the problem of HSV profile-following using trajectory tracking law based on LQR in Section 2 is presented.

The traditional reference guidance is not suitable for hypersonic vehicles because of its particular characteristics, including strong nonlinear, large flight envelope and complex entry environment. In the process of entry flight, it is difficult to constrain the deviation between real profile and nominal profile into a proper scope. Furthermore, strict terminal accuracy requirement demands that hypersonic vehicles track nominal profile precisely, that is, deviations in the terminal stage must be smaller.

Consequently, in the reference profile-following of HSV based on LQR, new problems occur in the selection of weighting matrices. In the initial flight stage, it is assumed that the deviations of altitude and velocity are zδ<sup>0</sup> and vδ0, respectively, which are chosen to be

$$\begin{cases} z\_{\delta0} = 3 \,\text{km}, \\ v\_{\delta0} = 200 \,\text{m/s}. \end{cases} \tag{25}$$

Let zδ<sup>1</sup> and vδ<sup>1</sup> be the anticipated maximum deviations accuracy in the terminal stage, expressed as

$$\begin{cases} z\_{\delta1} = 0.5 \text{ km}, \\ v\_{\delta1} = 20 \text{m/s.} \end{cases} \tag{26}$$

Substituting zδ0, vδ0, zδ<sup>1</sup> and vδ<sup>1</sup> into Eq. (21), one can get the weighting matrix Q<sup>0</sup> and Q<sup>1</sup> as

$$\mathbf{Q}\_0 = \begin{bmatrix} \frac{1}{\mathbf{z}\_{\delta 0}} & \mathbf{0} \\ \mathbf{0} & \frac{1}{\mathbf{z}\_{\delta 0}} \end{bmatrix}, \quad \mathbf{Q}\_1 = \begin{bmatrix} \frac{1}{\mathbf{z}\_{\delta 1}} & \mathbf{0} \\ \mathbf{0} & \frac{1}{\mathbf{z}\_{\delta 1}} \end{bmatrix}. \tag{27}$$

From Eqs. (25) and (26), one sees that zδ<sup>0</sup> and vδ<sup>0</sup> are bigger than zδ<sup>1</sup> and vδ1, respectively. The weighting matrix Q0, which is determined by zδ<sup>0</sup> and vδ0, can effectively eliminate the large initial stage deviations between the real and nominal profiles. Nevertheless, the capacity of Q<sup>0</sup> for resisting disturbance in the process of flight is not strong enough to satisfy the terminal accuracy requirement. On the contrary, the weighting matrix Q<sup>1</sup> constructed by zδ<sup>1</sup> and vδ<sup>1</sup> can eliminate the disturbance in the process of flight effectively. However, facing the existence of large deviations in the initial entry flight, it is difficult to keep HSV tracking the nominal profile properly, which will further influence the terminal accuracy.

Therefore, compared with Q0, the weighting matrix Q<sup>1</sup> is not applicable to initial deviations, and has good robustness to deal with disturbance in the process of entry flight. The

Figure 1. The profile-following of HSV entry guidance with initial 3 km altitude deviation by Q<sup>0</sup> and Q1.

Figure 2. The profile-following of HSV entry guidance with initial path angle deviation by Q<sup>0</sup> and Q1.

Hypersonic Vehicles Profile-Following Based on LQR Design Using Time-Varying Weighting Matrices

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Hypersonic Vehicles Profile-Following Based on LQR Design Using Time-Varying Weighting Matrices http://dx.doi.org/10.5772/intechopen.70659 141

Figure 2. The profile-following of HSV entry guidance with initial path angle deviation by Q<sup>0</sup> and Q1.

Figure 1. The profile-following of HSV entry guidance with initial 3 km altitude deviation by Q<sup>0</sup> and Q1.

140 Advances in Some Hypersonic Vehicles Technologies

simulations for hypersonic vehicles profile-following with Q<sup>0</sup> and Q<sup>1</sup> under different disturbances are shown in Figures 1–3. The flight profiles are expressed in altitude and velocity

Hypersonic Vehicles Profile-Following Based on LQR Design Using Time-Varying Weighting Matrices

http://dx.doi.org/10.5772/intechopen.70659

143

The reference profile described in this study is similar to the shuttle entry reference profile. The

Figure 1 shows that hypersonic vehicle tracks nominal profile with initial altitude deviation of positive 3 km, where circle line, solid line, dashed line indicate nominal profile, actual profile with Q0, actual profile with Q1, respectively. From Figure 1, one sees that the performance of Q<sup>0</sup> tracking nominal profile is better than Q1. Figures 2 and 3 are in respect to HSV profile-following with initial deviation of path angle and process disturbance of positive 20% aerodynamic parameter error. It can be seen that the performance of Q<sup>0</sup> tracking nominal profile is better than Q<sup>1</sup> in Figure 2, and Q<sup>1</sup> is better than Q<sup>0</sup> in Figure 3. Based on Figures 1–3, it can be obtained that the LQR with weighting matrices constructed by Bryson principle hasn't strong robustness to different disturbances in HSV

In order to solve above problem, it is required that LQR cannot only minimize the initial deviations, but also enhance the capability that resists the process disturbance effectually. Therefore, it is necessary to develop an algorithm to determine a proper weighting matrix in LQR. With the help of Bryson principle, an approach to determine the weighting matrix in

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

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

initial altitude of simulation is 55 km, and the initial velocity is 6 km/s.

LQR with current flight states is proposed in the following section.

4. LQR with time-varying weighting matrices

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

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

plane.

profile-following.

in Figure 4.

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

simulations for hypersonic vehicles profile-following with Q<sup>0</sup> and Q<sup>1</sup> under different disturbances are shown in Figures 1–3. The flight profiles are expressed in altitude and velocity plane.

The reference profile described in this study is similar to the shuttle entry reference profile. The initial altitude of simulation is 55 km, and the initial velocity is 6 km/s.

Figure 1 shows that hypersonic vehicle tracks nominal profile with initial altitude deviation of positive 3 km, where circle line, solid line, dashed line indicate nominal profile, actual profile with Q0, actual profile with Q1, respectively. From Figure 1, one sees that the performance of Q<sup>0</sup> tracking nominal profile is better than Q1. Figures 2 and 3 are in respect to HSV profile-following with initial deviation of path angle and process disturbance of positive 20% aerodynamic parameter error. It can be seen that the performance of Q<sup>0</sup> tracking nominal profile is better than Q<sup>1</sup> in Figure 2, and Q<sup>1</sup> is better than Q<sup>0</sup> in Figure 3. Based on Figures 1–3, it can be obtained that the LQR with weighting matrices constructed by Bryson principle hasn't strong robustness to different disturbances in HSV profile-following.

In order to solve above problem, it is required that LQR cannot only minimize the initial deviations, but also enhance the capability that resists the process disturbance effectually. Therefore, it is necessary to develop an algorithm to determine a proper weighting matrix in LQR. With the help of Bryson principle, an approach to determine the weighting matrix in LQR with current flight states is proposed in the following section.
