Observer-based linear parameter varying H ∞ tracking control for hypersonic vehicles

Abstract

This article aims to develop observer-based linear parameter varying output feedback H_∞ tracking controller for hypersonic vehicles. Due to the complexity of an original nonlinear model of the hypersonic vehicle dynamics, a slow–fast loop linear parameter varying polytopic model is introduced for system stability analysis and controller design. Then, a state observer is developed by linear parameter varying technique in order to estimate the unmeasured attitude angular for slow loop system. Also, based on the designed linear parameter varying state observer, a kind of attitude tracking controller is presented to reduce tracking errors for all bounded reference attitude angular inputs. The closed-loop linear parameter varying system is proved to be quadratically stable by Lypapunov function technique. Finally, simulation results show that the developed linear parameter varying H_∞ controller has good tracking capability for reference commands.

Keywords

Linear parameter varying state observer tracking control hypersonic vehicles

Introduction

National Aeronautics and Space Administration and the US Air Force have focused on the research of hypersonic vehicles for many years because of their military and commercial applications. Among various hypersonic vehicle research programs, the air-breathing hypersonic vehicle (AHV) is a kind of aircraft which is widely studied in the world. It can fly with the speed of 5 Ma to 10 Ma for long times and distances.¹ Especially, the scramjet-powered AHV is also proposed as a possible way of making space access more affordable and delivering a quick response to global threats, because the scramjet is a more efficient propulsion system for flight within the atmosphere than a rocket.² However, the research of srcamjet-powered vehicles was stopped and continued in the early 1990s with Hyper-X project within the National Aerospace Plane.³

The critical problem existing in the hypersonic vehicle is guidance, navigation, and various control methods.⁴ However, the control design problem for the scramjet-powered AHV is still a challenging task due to the inherent strong couplings and nonlinear disturbance. Conventionally, linear control techniques are used to design linear controller for many hypersonic vehicles, for example, guaranteed cost control,⁵ fault-tolerant tracking control,⁶ linear quadratic regulator control,⁷ and so on. In these results, the first step need to do is to develop a linear model for the complex nonlinear hypersonic vehicle system by linearized method. However, the obtained linear model cannot capture the nonlinear dynamics of hypersonic vehicle, hence, linear control technique may not result in desirable controller when aircrafts encounter disturbances and large parameter uncertainties. In later studies on this subject, some typical nonlinear control techniques, such as sliding mode control,⁸ back-stepping control,⁹ adaptive control,¹⁰ composite predictive control,¹¹ and nonlinear disturbance observer-based robust control,¹² have been widely investigated.

Recently, linear parameter varying (LPV) system is a bridge between nonlinear systems and linear systems, which has important research significance. LPV control method is applied to all kinds of complex nonlinear systems.^13

–16 Also, many important LPV control techniques have been acquired for hypersonic vehicles. In the study by Lind,¹⁷ an LPV framework is used to represent structural nonlinear dynamics of a kind of hypersonic vehicle. Then, a robust gain-scheduled controller is synthesized to guarantee closed-loop H_∞ control performance. In the study by Cai et al.,¹⁸ the authors developed an LPV model for the AHV. Within LPV framework, a self-scheduled controller is designed in order to realize stable tracking. Especially, under considering flexibility effects, a gain scheduling controller is developed for a kind of flexible AHV in the study by Ge et al.¹⁹ However, the state information cannot be obtained directly in many practical nonlinear complex systems. Therefore, a kind of robust output feedback control method is presented for a nonlinear model of an AHV in the study by Sigthorsson et al.²⁰ Using only limited state information, the developed controller can realize excellent tracking responses. In the study by Li et al.,²¹ an output feedback controller combining with neural network theory and high gain observer method is developed for a generic hypersonic vehicle. More recently, in the study by Zong et al.,²² a generic hypersonic vehicle tracking control problem is investigated via dynamic output feedback control technique, which consists of a nonlinear observer and a back-stepping controller.

In this article, the objective is to develop an LPV H_∞ tracking controller for the hypersonic vehicle system based on LPV state observer. First, a polytopic simplified LPV model is adapted to design attitude tracking controller. Next, based on the polytopic LPV model, a slow loop LPV H_∞ output feedback tracking controller is constructed. The tracking controller gain is given by Linear Matrix Inequality (LMI) constraints which can be efficiently solved by standard software packages. The LPV H_∞ controller can guarantee that LPV closed-loop hypersonic vehicle system is globally asymptotically stable. Finally, the control performances of the presented attitude tracking controller is illustrated by numerical simulation.

LPV model of the hypersonic vehicle

The hypersonic vehicle nonlinear attitude control model in entry phase can be described as follows²³

\dot{x} = V \cos ϒ \cos χ

\dot{y} = V \cos ϒ sin χ

\dot{z} = - V sin ϒ = - \dot{H}

\dot{V} = \frac{1}{M} (- D + ϒ sin β - M g sin ϒ + T_{x} \cos β \cos α + T_{y} sin β + T_{z} \cos β sin α)

\dot{χ} = \frac{1}{M V \cos ϒ} [\begin{array}{l} L sin μ + ϒ \cos μ \cos β + T_{x} (sin μ sin α - \cos μ sin β \cos α) + T_{y} \cos μ \cos β \\ - T_{z} \times (sin μ \cos α + \cos μ sin β sin α) \end{array}]

\dot{ϒ} = \frac{\begin{array}{l} L \cos μ - ϒ sin μ \cos β - M g \cos ϒ + T_{x} (\cos μ sin α + sin μ sin β \cos α) \\ - T_{z} (\cos μ \cos α - sin μ sin β sin α) - T_{y} sin μ \cos β \end{array}}{(M V)}

\dot{α} = q - tan β (p \cos α + r sin α) + \frac{1}{M V \cos β} (- L + M g \cos ϒ \cos μ - T_{x} sin α + T_{z} \cos α)

\dot{β} = p sin α - r cos α + \frac{1}{M V} (ϒ \cos β + M g \cos ϒ sin μ T_{r x} sin β \cos α + T_{y} \cos β - T_{z} sin β sin α)

\dot{μ} = \frac{\begin{array}{l} sec β (p \cos α + r sin α) + [L tan ϒ sin μ + L tan β - M g \cos ϒ \cos μ tan β + (T_{r x} sin α - T_{r z} \cos α) \\ \times (tan ϒ sin μ + tan β) - (T_{x} \cos α + T_{z} sin α) tan ϒ \cos μ sin β + (ϒ + T_{y}) tan ϒ \cos μ \cos β] \end{array}}{M V}

\dot{p} = \frac{I_{y y} - I_{z z}}{I_{x x}} q r + \frac{1}{I_{x x}} (- {\dot{I}}_{x x} p + l_{A} + l_{T})

\dot{q} = \frac{I_{z z} - I_{x x}}{I_{y y}} p r + \frac{1}{I_{y y}} (- {\dot{I}}_{x x} q + m_{A} + m_{T})

\dot{r} = \frac{I_{y y} - I_{x x}}{I_{z z}} p q + \frac{1}{I_{z z}} (- {\dot{I}}_{z z} q + n_{A} + n_{T})

In equations (1) to (12), x, y, and z are positions of the hypersonic vehicle in earth coordinate system; V is velocity; χ; is flight path azimuth angle; ϒ; is flight path angle; α; is angle of attack; β; is sideslip angle; μ; is bank angle; and p, q, and r are pith, roll, and yaw rate, respectively. For more details about the nonlinear model, we can refer to the study by Huang et al.²⁴

The hypersonic vehicle attitude control model can be written as the follow slow loop and fast loop polytopic LPV systems.^24
–26

{\begin{array}{l} {\dot{x}}_{s} (t) = \sum_{i = 1}^{N} α_{s_{i}} (M_{a}, \hat{q}) (A_{s_{i}} x_{s} (t) + B_{s_{i}} u_{s}) + w (t) \\ {\dot{x}}_{f} (t) = \sum_{i = 1}^{N} α_{f_{i}} (M_{a}, \hat{q}) (A_{f_{i}} x_{s} (t) + B_{f_{i}} u_{s}) + w (t) \end{array}

where slow loop state vector $x_{s} = {[\begin{matrix} α & β & μ \end{matrix}]}^{T}$ and fast loop state vector $x_{f} = {[\begin{matrix} p & q & r \end{matrix}]}^{T}$ .

In this artitio, reference model (14) is considered:

{\dot{x}}_{r} (t) = A_{r} x_{r} (t) + r (t)

where $r (t)$ is bounded reference input, $x_{r} (t)$ is reference system state, and $A_{r} (t)$ is a asymptotically stable matrix.

For system tracking error $x (t) - x_{r} (t)$ , we consider the following H_∞ tracking performance.

\int_{0}^{t_{f}} {[x (t) - x_{r} (t {)]}^{T} Q [x (t) - x_{r} (t)]} d t \leq γ^{2} \int_{0}^{t_{f}} w^{T} (t) w (t) d (t)

where x(t) is state vector, Q is positive definite weighting matrix, γ; is prescribed attenuation level, and $w (t)$ is external disturbance.

Now, we design the following LPV observer in order to estimate unmeasured state of the slow loop system (13).

{\begin{array}{l} \dot{\hat{x}} (t) = \sum_{i = 1}^{N} α_{s_{i}} [A_{i} \hat{x} (t) + B_{i} u (t) + L_{i} (y (t) - \hat{y} (t))] \\ \dot{\hat{y}} (t) = \sum_{i = 1}^{N} α_{s_{i}} C_{i} \hat{x} (t) \end{array}

Define the observer error as follows

e (t) = x (t) - \hat{x} (t)

By differentiating equation (17), we have

\begin{matrix} \dot{e} (t) = \dot{x} (t) - \dot{\hat{x}} (t) \\ = \sum_{i = 1}^{N} \sum_{j = 1}^{N} α_{s}_{_{i}} (M_{a}, \hat{q}) α_{s}_{_{j}} (M_{a}, \hat{q}) [A_{i} x (t) + B_{i} u (t) + ω (t)] - [A_{i} \hat{x} (t) + B_{i} u (t) + L_{i} C_{j} (x (t) - \hat{x} (t))] \\ = \sum_{i = 1}^{N} \sum_{j = 1}^{N} α_{s}_{_{i}} (M_{a}, \hat{q}) α_{s}_{_{j}} (M_{a}, \hat{q}) [(A_{i} - L_{i} C_{j}) e (t)] + ω (t) \end{matrix}

In this article, the following slow loop LPV controller is considered.

u (t) = \sum_{j = 1}^{N} α_{s}_{_{j}} (M_{a}, \hat{q}) K_{j} (\hat{x} (t) - x_{r} (t))

Therefore, the augmented system can be described as

\dot{\tilde{x}} (t) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} α_{s}_{_{i}} (M_{a}, \hat{q}) α_{s}_{_{j}} (M_{a}, \hat{q}) [{\tilde{A}}_{i j} \tilde{x} (t) + {\tilde{B}}_{i} \tilde{w} (t)]

where ${\tilde{A}}_{i j} = [\begin{matrix} A_{i} - L_{i} C_{j} & 0 & 0 \\ - B_{i} K_{j} & A_{i} + B_{i} K_{j} & - B_{i} K_{j} \\ 0 & 0 & A_{r} \end{matrix}], {\tilde{A}}_{i j} = [\begin{matrix} A_{i} - L_{i} C_{j} & 0 & 0 \\ - B_{i} K_{j} & A_{i} + B_{i} K_{j} & - B_{i} K_{j} \\ 0 & 0 & A_{r} \end{matrix}], \tilde{x} (t) = [\begin{matrix} e (t) \\ x (t) \\ x_{r} (t) \end{matrix}], \tilde{w} = [\begin{matrix} 0 \\ w (t) \\ r (t) \end{matrix}], {\tilde{B}}_{i} = [\begin{matrix} - L_{i} & I & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix}]$

Based on eqaution (20), the modified slow loop H_∞ performance can be described as follows

\int_{0}^{t_{f}} {(x (t) - x_{r} (t {))}^{T} Q (x (t) - x_{r} (t))} d t = \int_{0}^{t_{f}} {\tilde{x}}^{T} (t) \dot{\tilde{Q}} \tilde{x} (t) d t \leq {\tilde{x}}^{T} (0) \tilde{P} \tilde{x} (0) + γ^{2} \int_{0}^{t_{f}} {\tilde{w}}^{T} (t) \tilde{w} (t) d t

where $\tilde{Q} = [\begin{matrix} 0 & 0 & 0 \\ 0 & Q & - Q \\ 0 & - Q & Q \end{matrix}]$ and $\tilde{P}$ is a symmetric positive definite weighting matrix.

Our control objective is to develop slow loop LPV output controller (19), which guarantees the slow loop LPV system (22) to be asymptotically stable and also satisfies the modified H_∞ performance (21).

\dot{\tilde{x}} (t) = \sum_{i = 1}^{N} a_{s_{i}} (M_{a}, \hat{q}) \sum_{j = 1}^{N} a_{s_{j}} (M_{a}, \hat{q}) {\tilde{A}}_{i j} \tilde{x} (t)

Observer-based LPV H_∞ controller design

In this session, first, in order to design the LPV H_∞ controller, we give the following useful result.

Theorem 1

If there exists a symmetric positive definite matrix $\tilde{P}$ such that inequality (22) holds, then the closed-loop LPV system (22) is quadratically stable and H_∞ performance (21) is satisfied for a prescribed γ².

{\tilde{A}}_{i j}^{T} \tilde{P} + \tilde{P} {\tilde{A}}_{i j} + \frac{1}{γ^{2}} \tilde{P} {\tilde{B}}_{i} {\tilde{B}}_{i}^{T} \tilde{P} + \tilde{Q} < 0

Proof

For the the modified slow loop H_∞ performance (21), we have

\begin{matrix} \int_{0}^{t_{f}} {(x (t) - x_{r} (t {))}^{T} Q (x (t) - x_{r} (t))} d t \\ = \int_{0}^{t_{f}} {\tilde{x}}^{T} (t) \tilde{Q} \tilde{x} (t) d t \\ = {\tilde{x}}^{T} (0) \tilde{P} \tilde{x} (0) - {\tilde{x}}^{T} (t_{f}) \tilde{P} \tilde{x} (t_{f}) + \int_{0}^{t_{f}} {{\tilde{x}}^{T} (t) \tilde{Q} \tilde{x} (t) + \frac{d}{d t} ({\tilde{x}}^{T} (t) \tilde{P} \tilde{x} (t))}dt \\ \leq {\tilde{x}}^{T} (0) \tilde{P} \tilde{x} (0) + \int_{0}^{t_{f}} {{\tilde{x}}^{T} (t) \tilde{Q} \tilde{x} (t) + \sum_{i = 1}^{N} α_{s}_{_{i}} (M_{a}, \hat{q}) \sum_{j = 1}^{N} α_{s}_{_{j}} (M_{a}, \hat{q}) [({\tilde{A}}_{i j} {\tilde{x}}^{T} (t) \tilde{P} \tilde{x} (t) \\ + {\tilde{x}}^{T} (t) \tilde{P} {\tilde{A}}_{i j} \tilde{x} (t) + [({\tilde{B}}_{i} \tilde{w} (t {))}^{T} \tilde{P} \tilde{x} (t) + {\tilde{x}}^{T} (t) \tilde{P} {\tilde{B}}_{i} \tilde{w} (t) - γ^{2} w^{T} (t) \tilde{w} (t) + \frac{1}{γ^{2}} {\tilde{x}}^{T} (t) \tilde{P} {\tilde{B}}_{i} {\tilde{B}}_{i}^{T} \tilde{P} \tilde{x} (t) + γ^{2} w^{T} (t) \tilde{w} (t)]} d t \\ = {\tilde{x}}^{T} (0) \tilde{P} \tilde{x} (0) + \int_{0}^{t_{f}} {{\tilde{x}}^{T} (t) \tilde{Q} \tilde{x} (t) + \sum_{i = 1}^{N} α_{s}_{_{i}} (M_{a}, \hat{q}) \sum_{j = 1}^{N} α_{s}_{_{j}} (M_{a}, \hat{q}) [({\tilde{A}}_{i j} \tilde{x} (t {))}^{T} \tilde{P} \tilde{x} (t) \\ + {\tilde{x}}^{T} (t) \tilde{P} {\tilde{A}}_{i j} \tilde{x} (t) - {[\frac{1}{γ} {\tilde{B}}_{i}^{T} \tilde{P} \tilde{x} (t) - γ \tilde{w} (t)]}^{T} [\frac{1}{γ} {\tilde{B}}_{i}^{T} \tilde{P} \tilde{x} (t) - γ \tilde{w} (t)] + \frac{1}{γ^{2}} {\tilde{x}}^{T} (t) \tilde{P} {\tilde{B}}_{i} {\tilde{B}}_{i}^{T} \tilde{P} \tilde{x} (t)] + γ^{2} \tilde{w} {(t)}^{T} \tilde{w} (t)]} d t \\ \leq {\tilde{x}}^{T} (0) \tilde{P} \tilde{x} (0) + \int_{0}^{t_{f}} {\sum_{i = 1}^{N} \sum_{i = 1}^{N} α_{s}_{_{i}} (M_{a}, \hat{q}) α_{s}_{_{j}} (M_{a}, \hat{q}) {\tilde{x}}^{T} (t) [\tilde{Q} + {\tilde{A}}_{i j}^{T} \tilde{P} + \tilde{P} {\tilde{A}}_{i j} + \frac{1}{γ^{2}} \tilde{P} {\tilde{B}}_{i} {\tilde{B}}_{i}^{T} \tilde{P}] \tilde{x} (t) + γ^{2} w^{T} (t) \tilde{w} (t)} d t \end{matrix}

By theorem 1, if inequality (21) holds, we get

\int_{0}^{t_{f}} {(x (t) - x_{r} (t {))}^{T} Q (x (t) - x_{r} (t))} d t \leq {\tilde{x}}^{T} (0) \tilde{P} \tilde{x} (0) + γ^{2} \int_{0}^{t_{f}} {\tilde{w}}^{T} (t) \tilde{w} (t) d t

Then, we give the Lyapunov function candidate (26).

V (t) = {\tilde{x}}^{T} (t) \tilde{P} \tilde{x} (t)

By differentiating equation (26), we have

\dot{V} (t) = {\dot{\tilde{x}}}^{T} (t) \tilde{P} \tilde{x} (t) + {\tilde{x}}^{T} (t) \tilde{P} \dot{\tilde{x}} (t) = \sum_{i = 1}^{N} \sum_{j = 1}^{N} a_{s_{i}} (M_{a}, \hat{q}) a_{s_{j}} (M_{a}, \hat{q}) {\tilde{x}}^{T} (t) [{\tilde{A}}_{i j}^{T} \tilde{P} + \tilde{P} {\tilde{A}}_{i j}] \tilde{x} (t)

Therefore, we can obtain $\dot{V} (t) < 0$ using theorem 1. The proof is completed.

Theorem 2

For a prescribed attenuation level γ², there exists an observer-based LPV H_∞ tracking controller, which makes the closed-loop LPV system (22) H_∞ tracking control performance (21) is guaranteed, if there exists a symmetric positive definite matrix U and W_j such that LMIs (28) hold

[\begin{matrix} U A_{i}^{T} + A_{i} U + B_{i} W_{j} + W_{j}^{T} B_{i}^{T} + \frac{1}{γ^{2}} I & U \\ U & - Q^{- 1} \end{matrix}] < 0

Then, the observer-based LPV controller H_∞ tracking controller can be obtain by the following form

K_{j} = W_{j} U^{- 1} (j = 1, 2, \dots, N)

Proof

Define

\tilde{P} = [\begin{matrix} {\tilde{P}}_{11} & 0 & 0 \\ 0 & {\tilde{P}}_{22} \\ 0 & 0 & {\tilde{P}}_{33} \end{matrix}]

where ${\tilde{P}}_{11}, {\tilde{P}}_{22}$ , and ${\tilde{P}}_{33}$ are positive definite matrices.

Substituting equation (30) and ${\tilde{A}}_{i j}$ into equation (23), we obtain

[\begin{matrix} E_{11} & E_{12} & 0 \\ E_{21} & E_{22} & E_{23} \\ 0 & E_{32} & E_{33} \end{matrix}] < 0

where

E_{11} = (A_{i} - L_{i} C_{j})^{T} {\tilde{P}}_{11} + {\tilde{P}}_{11} (A_{i} - L_{i} C_{j}) + \frac{1}{γ^{2}} {\tilde{P}}_{11} (L_{i} L_{i}^{T} + I) {\tilde{P}}_{11}

E_{12} = E_{21}^{T} = - {\tilde{P}}_{22} B_{i} K_{j} + \frac{1}{γ^{2}} {\tilde{P}}_{11} {\tilde{P}}_{22}

E_{22} = (A_{i} + B_{i} K_{j})^{T} {\tilde{P}}_{22} + {\tilde{P}}_{22} (A_{i} + B_{i} K_{j}) + \frac{1}{γ^{2}} {\tilde{P}}_{22} {\tilde{P}}_{22} + Q

E_{23} = E_{32}^{T} = - {\tilde{P}}_{22} B_{i} K_{j} - Q

E_{33} = A_{r}^{T} {\tilde{P}}_{33} + {\tilde{P}}_{33} A_{r} + \frac{1}{γ^{2}} {\tilde{P}}_{33} {\tilde{P}}_{33} + Q

With $T_{i} = {\tilde{P}}_{11} L_{i}$ and Schur complement is applied, we can obtain inequality (32).

[\begin{matrix} N_{11} & {\tilde{P}}_{11} & T_{i} & N_{41}^{T} & 0 & 0 \\ {\tilde{P}}_{11} & - ρ^{2} I & 0 & 0 & 0 & 0 \\ T_{i}^{T} & 0 & - ρ^{2} I & 0 & 0 & 0 \\ N_{41} & 0 & 0 & N_{44} & N_{45} & 0 \\ 0 & 0 & 0 & N_{45}^{T} & N_{55} & {\tilde{P}}_{33} \\ 0 & 0 & 0 & 0 & {\tilde{P}}_{33} & - ρ^{2} I \end{matrix}] < 0

where $N_{11} = A_{i}^{T} {\tilde{P}}_{11} + {\tilde{P}}_{11} A_{i} - Z_{i} C_{j} - {(Z_{i} C_{j})}^{T}$

N_{41} = - {(B_{i} K_{j})}^{T} {\tilde{P}}_{22} + \frac{1}{γ^{2}} {\tilde{P}}_{22} {\tilde{P}}_{11}

N_{44} = (A_{i} + B_{i} K_{j})^{T} {\tilde{P}}_{22} + {\tilde{P}}_{22} (A_{i} + B_{i} K_{j}) + \frac{1}{γ^{2}} {\tilde{P}}_{22} {\tilde{P}}_{22} + Q

N_{45} = - {\tilde{P}}_{22} B_{i} K_{j} - Q

N_{55} = A_{r}^{T} {\tilde{P}}_{33} + {\tilde{P}}_{33} A_{r} + Q

By formula (32), it implies that

N_{44} = (A_{i} + B_{i} K_{j})^{T} {\tilde{P}}_{22} + {\tilde{P}}_{22} (A_{i} + B_{i} K_{j}) + \frac{1}{γ^{2}} {\tilde{P}}_{22} {\tilde{P}}_{22} + Q < 0

Now, pre-multiplying and post-multiplying the inequality (33) by ${\tilde{P}}_{22}^{- 1}$ , we get the follow inequality.

{\tilde{P}}_{22}^{- 1} A_{i}^{T} + {\tilde{P}}_{22}^{- 1} K_{j}^{T} B_{i}^{T} + A_{i} {\tilde{P}}_{22}^{- 1} + B_{i} K_{j} {\tilde{P}}_{22}^{- 1} + \frac{1}{γ^{2}} I + {\tilde{P}}_{22}^{- 1} Q {\tilde{P}}_{22}^{- 1} < 0

Inequality (34) is equivalent to linear matrix inequality (35) using Schur complement.

[\begin{matrix} {\tilde{P}}_{22}^{- 1} A_{i}^{T} + {\tilde{P}}_{22}^{- 1} K_{j}^{T} B_{i}^{T} + A_{i} {\tilde{P}}_{22}^{- 1} + B_{i} K_{j} {\tilde{P}}_{22}^{- 1} + \frac{1}{γ^{2}} I < 0 & {\tilde{P}}_{22}^{- 1} \\ {\tilde{P}}_{22}^{- 1} & - Q^{- 1} \end{matrix}] < 0

Denote $U = P_{22}^{- 1}$ and $W_{j} = K_{j} P_{22}^{- 1}$ , we can obtain LMIs (28). The proof of theorem 2 is completed.

Remark 2

There are no effective algorithms for solving inequality (32) but we can obtain matrices $P_{22}$ and K_j by LMI optimization software packages. Then, inequality (32) becomes standard LMIs by substituting P₂₂ and K_j into inequality (32). So, we can obtain slow loop LPV observer gain L_i. Similarly, for fast-loop system of the hypersonic vehicle, we can develop the fast-loop LPV controller by theorems 1 and 2.

Numerical simulations

The hypersonic vehicle system parameters are given as $V (0) = 7.2 km/s, H (0) = 71.1 k m, M = 65000 kg, γ (0) = 0 rad, χ (0) = 0 rad, α (0) = 0.31 rad, β (0) = 0 rad, μ (0) = 0.26 rad, p (0) = 0 rad/s, q (0) = 0 rad/s$ , and $r (0) = 0 rad/s$ .

In this article, step signals are used to output attitude angle tracking commands. Also, the step tracking commands will pass second-order filters (36).

{\begin{array}{l} H_{α} (s) = \frac{ω_{α}^{2}}{s^{2} + 2 ς_{α} ω_{α} s + ω_{α}^{2}} \\ H_{α} (s) = \frac{ω_{β}^{2}}{s^{2} + 2 ς_{β} ω_{β} s + ω_{β}^{2}} \\ H_{α} (s) = \frac{ω_{α}^{2}}{s^{2} + 2 ς_{u} ω_{u} s + ω_{u}^{2}} \end{array}

where damping ratio $ς_{α} = ς_{β} = ς_{u} = 0.5$ and natural frequency $ω_{α} = 0.2 rad/s, ω_{β} = 0.25 rad/s$ , and $ω_{u} = 0.3 rad/s$

Figures 1 to 3 illustrate attitude angle H_∞ tracking response curves for the slow loop system. Also, attitude angle tracking errors are given in Figure 4, and Figures 5 –7 present fast-loop pitch rate, roll rate, and yaw rate response curves. More specifically, control torques are shown in Figure 8. From the above simulation results, the presented observer-based LPV H_∞ control method has excellent tracking performance.

Figure 1.

Slow loop angle of attack tracking.

Figure 2.

Slow loop angle of sideslip tracking.

Figure 3.

Slow loop bank angle tracking.

Figure 4.

Slow loop attitude angle tracking error.

Figure 5.

Fast loop pitch rate response.

Figure 6.

Fast loop roll rate response.

Figure 7.

Fast loop yaw rate response.

Figure 8.

Control torques.

Conclusion

In this article, an observer-based LPV H_∞ control method has been addressed for the problem of a hypersonic vehicle attitude tracking control. A polytopic LPV model of the hypersonic vehicle is adopted for designing tracking controller due to complex couplings and nonlinear disturbance. An LPV state observer is used to estimate slow loop attitude angles due to the fact that the hypersonic vehicle state cannot be obtain directly. Applying the estimated system state, slow loop LPV H_∞ tracking controllers are designed, which satisfy the global stability and the given H_∞ tracking control performance for the closed-loop LPV system. The gain matrix of the resulting controller can be obtained by standard software packages. Simulation results illustrate that the presented control method achieves good attitude angle tracking control performance.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by the National Natural Science Foundation of China [grant number 61304127] and the Natural Science Foundation of Anhui Province [grant number 1308085QF120].

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Observer-based linear parameter varying H ∞ tracking control for hypersonic vehicles

Abstract

Keywords

Introduction

LPV model of the hypersonic vehicle

Observer-based LPV H∞ controller design

Theorem 1

Proof

Theorem 2

Proof

Remark 2

Numerical simulations

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References

Observer-based LPV H_∞ controller design