Polytopic linear parameter varying model-based tube model predictive control for hypersonic vehicles

Abstract

This article investigates a tube model predictive control scheme ensuring robustness and constraints fulfillment for hypersonic vehicles with bounded external disturbances. The polytopic linear parameter varying model is established using tensor-product modeling method, and tube-model predictive control controller is designed to satisfy all vertices of the polytopic linear parameter varying model. Firstly, the nominal model predictive control law is constructed for the nominal system without disturbances. Secondly, tube invariant set strategy is introduced for the unknown bounded disturbances, and the local robust feedback control law is presented to restrain disturbances. The proposed tube-model predictive control method not only steers the nominal state to track a reference trajectory but also keeps the real states within a tube centered with the nominal trajectory. Simulation results show the effectiveness of the proposed method for the hypersonic vehicle with bounded disturbances.

Keywords

Hypersonic vehicle polytopic LPV model bounded disturbances tube invariant set model predictive control

Introduction

Hypersonic vehicles have attracted more and more interests due to special application value in both military and civilian areas, such as horizontal landing space vehicle, hypersonic missile, hypersonic aircraft, and trans-aerosphere vehicle.¹ Since hypersonic vehicles use scramjet,² they have high speed and can carry more payload than other ordinary flight vehicles. In recent years, hypersonic vehicles have been studied extensively in the world. Moreover, the United States has attempted some experimental vehicles, such as NASA’s X-43A³ and Air Force’s X-51.⁴ Although the hypersonic vehicle is a promising technology in aeronautic and aerospace engineering, it is still a challenge to design a good controller due to its high nonlinearity and strong coupling and the existences of external disturbances.⁵

Hypersonic vehicles have a large flight envelop, where aerodynamic characteristics and flight dynamics will change dramatically during flight. Especially in rapid flight, parameters are very sensitive to the change of flight condition, and any change would possibly lead to a vehicle unstable. Therefore, there are a higher requirement for flight control system. That is, controller must be adapted to the larger parameter variation range and has strong robustness.⁶ For example, Xu et al.⁷ proposed indirect and direct global neural controllers combining dynamic surface design for hypersonic vehicle with unknown dynamics. As we know, linear parameter varying (LPV) model can fully describe the time-varying and inherent nonlinear characteristics, where the advanced linear control theory can be used instead of complex nonlinear approaches,⁸ for example, linear matrix inequalities (LMIs). Fortunately, the hypersonic vehicle system can be represented as an LPV model. Qin et al.⁹ used polytopic LPV model to design a robust variable gain controller for hypersonic vehicles. In the study by Wang et al.,¹⁰ a variable gain state feedback H ∞ controller with LPV model is proposed for the hypersonic vehicle.

In complex flight environment, in addition to parameter uncertainty, there always exist external disturbances, for example, wind, noise, and electromagnetic disturbance.¹¹ Thereby, it is very important and realistic for hypersonic vehicles to design a controller with strong anti-disturbance ability. Thus, robust or adaptive control becomes the main strategy for the hypersonic vehicle. In the study by Hu et al.,¹² robust adaptive sliding mode controller is designed for the hypersonic vehicle with parameter uncertainty and external disturbances, where a robust sliding mode surface is predesigned and then the adaptive sliding mode controller based on the error model is designed. In addition, the disturbance observer is often used to handle disturbances. For example, Xu¹³ presented dynamic surface control for longitudinal dynamics of aircraft and designed a disturbance observer to compensate model parameter uncertainty and external disturbances. Guan et al.¹⁴ proposed a fuzzy predictive control method based on disturbance observer. The disturbance observer is constructed to estimate the size of unknown disturbances, and the adaptive fuzzy system is used to approximate disturbances. However, the disturbance observer often assumes the derivative of disturbances being zero or approximating to zero, which does not meet the actual flight.

It is known that model predictive control (MPC) is an attractive control strategy for handling complex systems with diverse constraints and multiple variables effectively.^15,16 Originally, the nominal MPC can solve the open-loop optimal control problems rapidly enough; however, the robustness of MPC becomes more and more important for existence of uncertainty and disturbances. Consequently, robust MPC receives much attention. Generally, robust MPC with constraints is transformed into the LMI problem,¹⁷ and the many robust MPC control methods are proposed. For example, the robust MPC with bounded rates of parameter changes and robust MPC with saturated actuator are studied by Li and Xi¹⁸ and He et al.¹⁹ Ding et al.²⁰ and Ding²¹ designed output feedback robust MPC controllers. In flight control, robust MPC has also got development. Qin et al.²² designed a robust MPC controller for hypersonic vehicles using polytopic LPV model. A robust MPC tracking control strategy is presented for hypersonic vehicles with input constraints by Hu et al.²³ Although the open-loop min-max MPC is the feasible strategy for uncertainty and disturbances,²⁴ the control input may be too conservative and the good control performance cannot be ensured. So the feedback or closed-loop MPC becomes an attractive alternative.²⁵ However, the computation burden is often intolerable. Hence, Mayne et al. proposed the tube-MPC,²⁶ an effective technique for practical implementation of robust MPC. It can make the state of uncertain system to keep in a tube by designing a sequence of parameterized control strategy. This control action consists of a nominal MPC control law and a robust state feedback control law. Wherein the MPC control law steers the state of nominal system to track a reference trajectory, the state feedback control law maintains the actual trajectory within a tube centered along the nominal trajectory. Tube-MPC can not only deal with disturbances effectively but also improve the control performance. Meanwhile, online computation burden is reduced and the controller can be implement easily in practice. Cannon et al.²⁷ utilized successive linear approximations to develop MPC laws for nonlinear system, where tubes are used to bound the approximation errors. In addition, the tube is also used to design stochastic MPC controller.²⁸ Wang et al.²⁹ designed a tube-MPC method for constrained unmanned surface vessels based on trajectory linearization being used to translate the continuous nonlinear model to a linear time varying predictive model. In the study by Mayne et al.,³⁰ tube-MPC controller is proposed to realize robust control of nonlinear system with additive disturbances using an ancillary MPC controller to replace the local linear controller. Sun et al.³¹ proposed a method that consists of a nonlinear feedback control and a dual-mode MPC, where the nonlinear feedback control guarantees the actual trajectory being contained in a tube, and the dual-mode MPC is designed to ensure asymptotic convergence of the nominal trajectory to zero. We can see that for complicated hypersonic vehicle system, tube-MPC has the great potential to deal with disturbances and it can obtain the good control performance and stability. In the studies by Su et al.³² and Mayne et al.,³³ tube-MPC based on state observer is considered for the case of existence of unmeasured states.

Thereby in this article, the tube-MPC strategy^32,33 is introduced, and the controller based on polytopic LPV model is designed for hypersonic vehicles with bounded external disturbances. Firstly, the original nonlinear longitudinal model is linearized by Jacobian linearization, and LPV model is formulated with altitude and velocity as the scheduling parameters. Secondly, the polytopic LPV model is established using tensor-product modeling method. Then, the offline tube-MPC controller is designed based on the disturbance invariant set. The center of the tube is generated by nominal MPC optimization with tighter input constraints. The local robust feedback control is employed to weaken the effect of disturbances, that is, to restrict the size of the tube. Finally, simulations comparing with the traditional MPC show that tube-MPC is more effective, and it has lower computation complexity and strong practicability.

The rest of this article is organized as follow. In “Vehicle model” section 2, the nonlinear model of hypersonic vehicle is presented. The polytopic LPV modeling is given in “Polytopic LPV model” section. A tube-MPC controller is designed in “Design of the tube-MPC controller” section. In “Simulations” section, simulations are given. Finally, we conclude this article in the last section.

Notation: Given two subsets $P, W \in R^{n}$ , Minkowski set sum is defined as $P \oplus W : = {p + w | p \in P, w \in W}$ . Pontryagin set difference is defined as $P Θ W : = {x \in R^{n} | {x} \oplus W \subseteq P}$ . Set multiplication is defined as $K P : = {K p | p \in P}, K \in R^{m \times n}$ .

Vehicle model

In this article, the longitudinal model of hypersonic vehicle is developed by Bolender and Doman³⁴ and Parker et al.³⁵ of the air force research laboratory (AFRL), called AFRL model. The basic geometry of the hypersonic vehicle longitudinal model is given in Figure 1.³⁵

Figure 1 shows a longitudinal sketch of the hypersonic vehicle, in which $L_{v}$ is vehicle length, $M_{\infty}$ is freestream Mach number, and $x_{B}$ and $z_{B}$ are body axis coordinate frame of x- and z-direction, respectively. Only rigid dynamics of the hypersonic vehicle is considered in this article, and the equations of motion are given as follows

Figure 1.

Geometry of the hypersonic vehicle model.

\begin{array}{l} \dot{h} = V sin (θ - α) \\ \dot{V} = \frac{1}{m} (T cos α - D) - g sin (θ - α) \\ \dot{α} = \frac{1}{m V} (- T sin α - L) + q + \frac{g}{V} cos (θ - α) \\ \dot{θ} = q \\ \dot{q} = \frac{M}{I} \end{array}

where h, V, α, θ, and q denote the altitude, velocity, angle of attack, pitch angle, and pitch rate, respectively. g is the acceleration due to gravity. I is the moment of inertia. L, D, T, and M are the lift, drag, thrust, and pitching moment, respectively.

The expressions of L, D, T, and M are modeled as

\begin{array}{l} L = \frac{1}{2} ρ V^{2} S C_{L} (α, δ_{e}) \\ D = \frac{1}{2} ρ V^{2} S C_{D} (α, δ_{e}) \\ T = C_{T}^{α^{3}} α^{3} + C_{T}^{α^{2}} α^{2} + C_{T}^{α} α + C_{T}^{0} \\ M = z_{T} T + \frac{1}{2} ρ V^{2} S \bar{c} (C_{M, α} (α) + C_{M, δ_{e}} (δ_{e})) \end{array}

where S is reference area of the vehicle and ρ is the air density. The lift, drag, moment, and thrust coefficients are given as follows

\begin{array}{l} C_{L} = C_{L}^{α} α + C_{L}^{δ_{e}} δ_{e} + C_{L}^{0} \\ C_{D} = C_{D}^{α^{2}} α^{2} + C_{D}^{α} α + C_{D}^{δ_{e}^{2}} δ_{e}^{2} + C_{D}^{δ_{e}} δ_{e} + C_{D}^{0} \\ C_{M, α} = C_{M, α}^{α^{2}} α^{2} + C_{M, α}^{α} α + C_{M, α}^{0} \\ C_{M, δ_{e}} = c_{e} δ_{e} \\ C_{T}^{α^{3}} = β_{1} (h, \bar{q}) Φ + β_{2} (h, \bar{q}) \\ C_{T}^{α^{2}} = β_{3} (h, \bar{q}) Φ + β_{4} (h, \bar{q}) \\ C_{T}^{α} = β_{5} (h, \bar{q}) Φ + β_{6} (h, \bar{q}) \\ C_{T}^{0} = β_{7} (h, \bar{q}) Φ + β_{8} (h, \bar{q}) \end{array}

where $δ_{e}$ is the elevator deflection, Φ is the fuel-to-air ratio, and $\bar{q} = \frac{1}{2} ρ V^{2}$ is the dynamic pressure.

The engine dynamics can be seen as a second-order system

\begin{array}{l} \ddot{Φ} = - 2 ζ ω \dot{Φ} - ω^{2} Φ + ω^{2} Φ_{c} \end{array}

where $Φ_{c}$ is the commanded value of Φ.

Polytopic LPV model

In this section, the polytopic LPV model of hypersonic vehicle is constructed by the Jacobian linearization and tensor-product modeling method. Firstly, Jacobian linearization is used to develop the LPV model. The scheduling variables are altitude and velocity, that is, $p (t) = [h, V]^{T} \in Γ$ , and Γ is the flight envelope. Here, we choose $h \in [85000 ft,87000 ft]$ and $V \in [7500 ft/s,8500 ft/s]$ . The state and input variables of the hypersonic vehicle are defined as $x_{s} = [h, V, α, θ, q, Φ, \dot{Φ}]^{T}$ and $u_{s} = [δ_{e}, Φ_{c}]^{T}$ , respectively, and the outputs are $y_{s} = [h, V]^{T}$ . In order to get equilibrium points of the hypersonic vehicle, we define

\begin{array}{l} f_{1} = V sin (θ - α) \\ f_{2} = \frac{1}{m} (T cos α - D) - g sin (θ - α) \\ f_{3} = \frac{1}{m V} (- T sin α - L) + q + \frac{g}{V} cos (θ - α) \\ f_{4} = q \\ f_{5} = \frac{M}{I} \\ f_{6} = \dot{Φ} \\ f_{7} = - 2 ζ ω \dot{Φ} - ω^{2} Φ + ω^{2} Φ_{c} \end{array}

By solving equations $f_{i} = 0, i = 1, \dots,7$ , and fixing scheduling variables, we can get the equilibrium points and control inputs of the hypersonic vehicle as follows

\begin{array}{l} x_{r} = {[h_{r}, V_{r}, α_{r} (h, V), θ_{r} (h, V), q_{r} (h, V), Φ_{r} (h, V), {\dot{Φ}}_{r} (h, V)]}^{T} \\ u_{r} = {[δ_{e r} (h, V), Φ_{c r} (h, V)]}^{T} \end{array}

Then, AFTR model is linearized by Jacobian linearization at different equilibrium points in the flight envelope Γ. Corresponding to these equilibrium points, the LPV model of the hypersonic vehicle is described as

\dot{x} = A (p (t)) x + B (p (t)) u

where $x = x_{s} - x_{r} = [Δ h, Δ V, Δ α, Δ θ, Δ q, Δ Φ, Δ \dot{Φ}]^{T}$ and $u = u_{s} - u_{r} = [Δ δ_{e}, Δ Φ_{c} {)]}^{T}$ . By means of curve fitting, the Jacobian varying parameter matrices $A (p (t))$ and $B (p (t))$ are presented as follows

A (p (t)) = [\begin{matrix} 0 & 0 & - V & V & 0 & 0 & 0 \\ A_{21} (h, V) & A_{22} (h, V) & A_{23} (h, V) & - g & 0 & A_{26} (h, V) & 0 \\ A_{31} (h, V) & A_{32} (h, V) & A_{33} (h, V) & 0 & 1 & A_{36} (h, V) & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ A_{51} (h, V) & A_{52} (h, V) & A_{53} (h, V) & 0 & 0 & A_{56} (h, V) & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & - ω^{2} & - 2 ζ ω \end{matrix}]

B (p (t)) = [\begin{matrix} 0 & 0 \\ B_{21} (h, V) & 0 \\ B_{31} (h, V) & 0 \\ 0 & 0 \\ B_{51} (h, V) & 0 \\ 0 & 0 \\ 0 & ω^{2} \end{matrix}]

Then, the continuous LPV models (2) and (3) are transformed into polytopic formation using tensor-product approach.³⁶ Moreover, the high-order singular value decomposition method³⁷ is used to simplify the complicated tensor. In the flight envelope Γ, choosing a certain number of sampling points, the continuous LPV model is discretized, and the tensor is established to store this discrete system. Next, we extract the linear time invariant (LTI) vertex system by high-order singular value decomposition and construct the weight function of the LTI vertex system. In this article, sampling density is 21 × 21, and four big singular values are remained for h and five big singular values for V by high-order singular value decomposition. Therefore, the polytopic convex system with 20 vertices is constructed. Consequently, time-varying matrix $S (p (t)) = [A (p (t)), B (p (t))]$ can be defined as $S (p (t)) \underset{ζ}{\approx} Σ_{n = 1}^{20} τ_{n} (p (t)) S_{n}$ , where $S_{n}$ is vertex system, $τ_{n}$ is weighting coefficient, $Σ_{n = 1}^{20} τ_{n} = 1$ , and ζ denotes the approximation error between the polytopic LPV model and the original system.

Generally, the hypersonic vehicle is affected by external disturbances, such as strong wind in aerospace, which degrades the performance of control system. Therefore, the following discrete polytopic LPV model with external disturbances is formulated

\begin{array}{l} x (k + 1) = A (p (k)) x (k) + B (p (k)) u (k) + w (k) \\ y (k) = C x (k) \end{array}

where $x (k) \in R^{7}$ is the state vector, $u (k) \in R^{2}$ is control input vector, and $y (k) \in R^{2}$ is output vector. $w (k) \in R^{7}$ is the bounded external disturbances, that is, $w (k) \in W$ , where W is compact and contains the origin. The matrices $A (p (k)), B (p (k)) \in Ω$ and $Ω = C o {[A_{1}, B_{1}], [A_{2}, B_{2}],... [A_{L}, B_{L}]}$ . L is the number of vertices. Here, L = 20. $[A_{i}, B_{i}], i = 1, \dots,20$ are the vertices of the convex hull. $A (p (k)), B (p (k))$ is the linear combination of $[A_{i}, B_{i}]$ , that is, $[A (p (k)), B (p (k))] = \sum_{i = 1}^{L} p_{i} (k) [A_{i}, B_{i}]$ , where $p_{i} (k)$ is the weight of each vertex; it is the function of $h, V$ , and $0 \leq p_{i} (k) \leq 1$ and $\sum_{i = 1}^{L} p_{i} (k) = 1$ .

The system (4) is subject to the input constraints

u \in U : = {u \in R^{2} | | u (k) | \leq u_{max}}

where $U \subset R^{2}$ is compact containing the origin.

Design of the tube-MPC controller

In this part, a tube-MPC control approach is presented for polytopic LPV model (4) of the hypersonic vehicle. Firstly, the nominal polytopic LPV system (without disturbances) is defined as

x' (k + 1) = A (p (k)) x' (k) + B (p (k)) u' (k)

where $x' (k) \in R^{7}$ and $u' (k) \in R^{2}$ are the state and control input of the nominal system (6), respectively.

The purpose of the proposed algorithm is to keep the state of polytopic LPV system with disturbances (4) within a tube, which centered with the nominal trajectory. Meanwhile, the control input constraint (5) is satisfied. That is $x_{i} = {x'}_{i} \oplus Z$ , where ${x'}_{i}$ and $x_{i}$ are the nominal and actual states, respectively. Z is a disturbance invariant set used to bound $x_{i} - {x'}_{i}$ .

Define the following error equation by subtracting equation (6) from equation (4)

x (k + 1) - x' (k + 1) = A_{p} (x (k) - x' (k)) + B_{p} (u (k) - u' (k)) + w (k)

where $A_{p} and B_{p}$ are abbreviations of $A (p (k)) and B (p (k))$ , respectively. Let $e (k) = x (k) - x' (k)$ and $u_{e} (k) = u (k) - u' (k)$ , the system (7) is rewritten as

e (k + 1) = A_{p} e (k) + B_{p} u_{e} (k) + w (k)

The overall tube-MPC control scheme design is shown in Figure 2.

Figure 2.

Tube-MPC control scheme. MPC: model predictive control.

As shown in Figure 2, tube-MPC controller can be regarded as a feedback MPC with two degrees of freedom. Wherein, the nominal MPC controller is employed in an outer loop to provide a suitable central path, and the controller is only based on the knowledge of the nominal system without disturbances. The local robust feedback controller is separately designed in an inner loop to attenuate the effect of disturbances. The control input for the hypersonic vehicle with bounded disturbances is obtained as the sum of two components, that is, $u = K (x - x') + u'$ . Where the component $u'$ is generated by the outer loop controller, while K is generated by the inner loop controller to keep the actual state x close to the nominal state $x'$ . Since K is fixed, this method has the same complexity as conventional nominal MPC.

Due to $u_{e} = u - u' = K (x - x') = K e$ , equation (8) can be rewritten as

e (k + 1) = (A_{p} + B_{p} K) e (k) + w (k)

If the auxiliary feedback gain K, such that $A_{p} + B_{p} K$ , is Hurwitz, then the evolution of $e (k)$ is bounded and an admissible disturbance invariant set Z exists.³⁸ It can be checked by the following theorem.³²

Theorem 1

Consider the polytopic LPV error system (9), if there exist matrices Y and $Q > 0$ , such that

[\begin{matrix} Q & {(A_{p} Q + B_{p} Y)}^{T} \\ A_{p} Q + B_{p} Y & Q \end{matrix}] > 0, A_{p}, B_{p} \in Ω, p = 1, \dots,20

Then, feedback controller $K = Y Q^{- 1}$ , and the dynamic of $e (k)$ is stable.

Proof

It is known that error system (9) is linear, $w (k) \in W$ is bounded, and we will consider the influence of the bounded disturbances when calculating the disturbance invariant set Z. Therefore, the stability can be analyzed on the nominal error system without disturbances

e (k + 1) = (A_{p} + B_{p} K) e (k)

It is clear that if there are a positive-definite function $V (k) = e {(k)}^{T} P e (k), P > 0$ , and a feedback gain K such that $V ((A_{p} + B_{p} K) e (k)) - V (e (k)) < 0$ for $\forall e (k) \in R^{n}$ and $\forall (A_{p}, B_{p}) \in Ω$ , then $e (k)$ is stable. The quadratic stability condition is equivalent to

P - {(A_{p} + B_{p} K)}^{T} P (A_{p} + B_{p} K) > 0

Let $P = Q^{- 1}$ and $K = Y Q^{- 1}$ , inequality (12) can be rewritten as

Q^{- 1} - {(A_{p} + B_{p} Y Q^{- 1})}^{T} Q^{- 1} (A_{p} + B_{p} Y Q^{- 1}) > 0

Multiplying equation (13) from the left and right by Q, respectively, then there is

Q - {(A_{p} Q + B_{p} Y)}^{T} Q^{- 1} (A_{p} Q + B_{p} Y) > 0

Applying Schur complement⁸ to equation (14), then there is

[\begin{matrix} Q & {(A_{p} Q + B_{p} Y)}^{T} \\ A_{p} Q + B_{p} Y & Q \end{matrix}] > 0

Next, we should bound $x - x'$ by a disturbance invariant set Z. We always desire that Z be as small as possible. Generally, we choose the minimum robust positive invariant (mRPI) set as the disturbance invariant set. The definition of RPI and mRPI is as follows³⁹

Definition 1 (RPI set)

The set $S \subset R^{n}$ is an RPI set of the polytopic LPV model (4), if $(A_{p} + B_{p} K) x (k) + w (k) \in S$ for all $x (k) \in S$ , $w (k) \in W$ , and $\forall (A_{p}, B_{p}) \in Ω$ , that is, if and only if $(A_{p} + B_{p} K) S \oplus W \subseteq S$ .

Definition 2 (mRPI set)

The RPI set S is the mRPI set of the system (4), if and only if S is contained in every closed RPI set of the system (4).

We can calculate the mRPI set by the approach in the study by Raković et al.³⁹ and Tahir.⁴⁰ The mRPI set in the study by Raković et al.³⁹ is given by $S_{\infty} = \oplus_{i = 0}^{\infty} {(A_{p} + B_{p} K)}^{i} W$ . Since $S_{\infty}$ involves Minkowski set sum of infinite terms, it is impossible to calculate the mRPI set unless the dynamic of system is nilpotent.⁴¹ Moreover, this algorithm requires a lot of iterative computations, and it will induce a considerably large calculation for the high-dimensional multivariable system. Therefore, it is not suitable for hypersonic vehicles. As a consequence, in this article, we adopt the algorithm in the study by Tahir.⁴⁰ The invariant set is given by $Z : = {x \in R^{n} : - z \leq x \leq z}, z > 0$ . Different from the iterative algorithm in the study by Raković et al.,³⁹ the mRPI set is computed in the study by Tahir⁴⁰ by solving a linear program. By avoiding the set iterative multiplication or addition computation, this algorithm is more fast and efficient.

Here, we present a lemma about the disturbance invariant set:

Lemma 1

Assume that Z is a disturbance invariant set of system (9), equation (4) is the real system with bounded disturbances, and equation (6) is the corresponding nominal system.⁴² If $x (k) \in x' (k) \oplus Z$ and $u = K (x - x') + u'$ , then $x (k + 1) \in x' (k + 1) \oplus Z$ for all $w (k) \in W$ and $\forall (A_{p}, B_{p}) \in Ω$ .

As can be seen from lemma 1, the control law $u = K (x - x') + u'$ keeps the actual state x close to the nominal state $x'$ . If we can steer the nominal state $x'$ to track the reference trajectory, then the actual state x must be regulated to a tube Z, which centered with the nominal trajectory. It requires the initial state $x (0) = x' (0)$ , that is $e (0) = 0$ , then ensure $e (k) \in Z, k > 0$ .

A significant requirement for the tube-MPC controller is to get a suitable nominal trajectory. Thereby, we employ the algorithm in the study by Kothare et al.¹⁷ and make some changes. The nominal control law is parameterized as $u' = F x'$ for the nominal system (6). We can know from the characteristics of polytopic LPV model, if the every vertex of the convex hull can be stably controlled, then the whole system is stable. Therefore, the control law is designed to satisfy all vertices of the polytopic LPV nominal system (6). That is we adopt the same control law at every vertex.

Consider the following optimization problem 1 for the nominal system (6).

Optimization problem 1

\begin{array}{l} min_{F} max_{[A_{p}, B_{p}] \in Ω} J_{\infty}' (k) = \sum_{i = 0}^{\infty} [{‖ x' (k + i | k) ‖}_{L}^{2} + {‖ F x' (k + i | k) ‖}_{R}^{2}] \\ s.t. \\ {‖ x' (k) ‖}_{P^{'}}^{2} \leq γ, x' (k) = x' (k | k) \\ {‖ x' (k + i + 1 | k) ‖}_{P^{'}}^{2} - {‖ x' (k + i | k) ‖}_{P^{'}}^{2} \leq - {‖ x' (k + i | k) ‖}_{L}^{2} \\ - {‖ F x' (k + i | k) ‖}_{R}^{2} \\ u' (k) \in U Θ K Z \end{array}

where $x' (k + i | k)$ is the predictive state not corrupted by disturbances, $L > 0$ and $R > 0$ are suitable weighting matrices, and $P^{'} > 0$ is a positive-definite matrix. The optimization problem 1 ensures the Lyapunov function of nominal system decays with time and then guarantees the stability of the system. Different from Kothare et al.,¹⁷ the tighter input constraints for the nominal system (6) are employed to satisfy the actual control input constraints, that is, $u' (k) \in U Θ K Z$ . Here, the Pontryagin set difference computation is involved. In order to avoid the occurrence of an empty set, the following assumption needs to be given.

Assumption 1

Considering the input constraint U and disturbance invariant set Z, we assume that $K Z \subset U$ , that is, $U Θ K Z \neq \emptyset$ , where ∅ denotes an empty set.

In the robust control, such an assumption is not uncommon. Because if disturbance set W is too large, the actuator constraints are unlikely to meet. Meanwhile, the corresponding disturbance invariant set Z gets bigger, and the control performance of system will be difficult to guarantee. Subsequently, optimization problem 1 is transformed into the following LMI formulation. Given a feasible initial state $x' (k)$ , if there exist a symmetric positive matrix $Q' \in R^{7 \times 7}$ , a matrix $Y^{'} \in R^{2 \times 7}$ , and a scalar $γ > 0$ suitable for all vertices of the nominal polytopic LPV model (6) and satisfy the following LMIs, then the feedback control law $u' = F x'$ could guarantee the stability of the nominal system.

Optimization problem 2

\begin{array}{l} min_{γ, Y, Q} γ \\ \begin{array}{l} s.t. \\ [\begin{matrix} 1 & * \\ x' (k) & Q' \end{matrix}] \geq 0 \\ [\begin{matrix} Q' & * & * & * \\ A_{p} Q' + B_{p} Y^{'} & Q' & * & * \\ L^{\frac{1}{2}} Q' & 0 & γ I & * \\ R^{\frac{1}{2}} Y^{'} & 0 & 0 & γ I \end{matrix}] \geq 0, p = 1, \dots, 20 \\ [\begin{matrix} χ & * \\ {Y^{'}}^{T} & Q' \end{matrix}] \geq 0, χ \leq {u'}^{2}_{max}, u' \in U Θ K Z \end{array} \end{array}

where $\forall (A_{p}, B_{p}) \in Ω$ , $F = Y^{'} {Q'}^{- 1}$ , and $P^{'} = γ {Q'}^{- 1}$ .

Since the nominal control law $u' = F x'$ , then the actual control law can be rewritten as $u = K (x - x') + F x'$ . We can know that if $x (k) \in x' (k) \oplus Z$ and $u' = F x' \in U Θ K Z$ , then this control law can ensure the actuator of the hypersonic vehicle with disturbances meeting the original input constraint $u \in U$ for all $w (k) \in W$ . Since the feedback control F can robustly stabilize the nominal system, the nominal state $x'$ can stably track the reference trajectory. In addition, since $x \in x' \oplus Z$ , the real state x can converge to a tube centered with the nominal state trajectory. Therefore, the robust stability of the real system with bounded disturbances is guaranteed.

In the following, we formulate a tube-MPC algorithm for the hypersonic vehicle and then present the main results of this article.

Algorithm (tube-MPC)

Step 1. The polytopic LPV model of the hypersonic vehicle is established using Jacobian linearization and tensor-product modeling method.

Step 2. Calculate the local feedback gain K using theorem 1.

Step 3. Compute the mRPI set corresponding to the feedback gain K, that is, the disturbance invariant set Z of the hypersonic vehicle.

Step 4. Get a suitable nominal trajectory. Compute the nominal control law $u' = F x'$ for the ploytopic LPV system (6) without disturbances by solving the optimization problem 2.

Step 5. Apply the input $u = K (x - x') + F x'$ to the original nonlinear model of the hypersonic vehicle (1).

Simulations

In this part, the simulations are presented to verify the effectiveness of the proposed method. Here, the bounded external disturbances $w = [0.2 sin (t),0.3 cos (t), 0.03 sin (t), 0.03 sin (t), {0.05 sin (t),0,0]}^{T}$ are considered, which can be seen as a gust of wind in aerospace. According to lemma 1, we require that the initial values of state variable of the real system are same as the nominal system. Then, the initial values of altitude and velocity are $h_{0} = {h^{'}}_{0} = 85000 ft$ and $V_{0} = {V^{'}}_{0} = 7500 ft/s$ . The input constraints are $| δ_{e} | \leq 30$ ° and $0 \leq Φ_{c} \leq 1.5$ . We illustrate the effectiveness of the proposed method by comparing the general robust MPC method in the study by Bumroongsri.⁴³ Here, we have made the following two maneuver control simulations.

Case 1

In this case, the multistep maneuver commands for climbing and accelerating are selected with the same step changes $Δ h = 650 ft$ and $Δ V = 300 ft/s$ .

First, by solving LMI (10), we obtain the feedback control gain K as follows

K = [\begin{matrix} 0.0007 & 0.0000 & - 2.2601 & 2.3355 & 0.0112 & - 0.0013 & 0.0000 \\ - 0.0000 & - 0.0002 & 0.1403 & - 0.1392 & 0.0008 & - 0.0013 & - 0.0002 \end{matrix}] \times 10^{4}

The mRPI set corresponding to the feedback gain K is computed using the method in the study by Raković et al.³⁹ and as the disturbance invariant set Z of the hypersonic vehicle

Z : [\begin{matrix} - 4.6753 \\ - 0.3000 \\ - 0.0306 \\ - 0.0305 \\ - 0.0545 \\ - 0.0000 \\ - 0.0000 \end{matrix}] \leq x \leq [\begin{matrix} 4.6753 \\ 0.3000 \\ 0.0306 \\ 0.0305 \\ 0.0545 \\ 0.0000 \\ 0.0000 \end{matrix}]

For case 1, the tube about the altitude and velocity is written as $Z_{1}$ , as shown in Figure 3. The black line is the trajectory of altitude and velocity of the error system (9). We can see that the trajectory of the error system starts from the origin and always stays in the tube $Z_{1}$ .

Figure 3.

A tube $Z_{1}$ about altitude and velocity.

The feedback gain matrix $F_{1}$ for the nominal state $x'$ is calculated by solving the optimization problem 2

F_{1} = [\begin{matrix} 0.0002 & - 0.0002 & - 1.1677 & 4.2958 & 0.7689 & 0.0302 & 0.0009 \\ - 0.0004 & - 0.0028 & 1.5544 & - 3.1557 & - 0.3347 & - 0.0190 & - 0.0006 \end{matrix}]

The altitude and velocity tracking results together with the control inputs are presented in Figures 4 to 7. Other state variables of the hypersonic vehicle are shown in Figures 8 to 10.

Figure 4.

Altitude tracking performance.

Figure 5.

Velocity tracking performance.

Figure 6.

Values of fuel-to-air ratio.

Figure 7.

Elevator deflection.

Figure 8.

Attack angle.

Figure 9.

Pitch angle.

Figure 10.

Pitch rate.

Figures 4 and 5 show the reference trajectory, nominal trajectory, and two trajectories obtained by different control strategies. The black line is the reference trajectory. The red line is the trajectory obtained by the tube-MPC controller in this article. The gray dotted line is the trajectory obtained by the robust MPC controller in the study by Bumroongsri.⁴³ And the blue line is the nominal trajectory without disturbances, that is, the center of the tube. We can see from these figures, the tube-MPC scheme achieves stable tracking of altitude and velocity. Figures 6 and 7 show the control inputs $Φ_{c}$ and $δ_{e}$ . The red and gray dotted lines are control inputs of tube-MPC and robust MPC, respectively. And we can see that both two control methods satisfy the input constraints. But, compared with the fixed value of the nominal control input, and as a result of the existence of bounded disturbances, control inputs of the actual system are not constant value. Figures 8 to 10 present the results of attack angle, pitch angle, and pitch rate, respectively. More importantly, we can see clearly from these figures, there is a smaller chattering of trajectories controlled by tube-MPC than robust MPC controller. Tube-MPC method introduces disturbance invariant set to restrict the effect of disturbances. Robust MPC method in the study by Bumroongsri⁴³ forces the predicted states stay in an invariant ellipsoid using the norm-bounding technique. However, when external disturbance is large, this robust MPC method is very difficult to guarantee the predicted states enter into the invariant ellipsoid.

Case 2

In this case, the reference value of altitude h is chosen as 85650 ft and switches to 85000 ft at 100 s. And the velocity V is chosen as 7800 ft/s and switches to 7500 ft/s at 100 s. Firstly, same as the case 1, we calculate the local robust feedback gain K that restricts the effect of disturbances, the feedback gain F that ensures the stability of nominal system, and the disturbance invariant set Z corresponding to the feedback gain K. Because the size of disturbances does not change, the values of K and Z are same as the case 1. Here, the new feedback gain $F_{2}$ is given below

F_{2} = [\begin{matrix} 0.0003 & - 0.0003 & - 2.3544 & 5.4617 & 0.7291 & 0.0283 & 0.0009 \\ - 0.0002 & - 0.0032 & 1.4792 & - 3.4889 & - 0.1635 & - 0.0137 & - 0.0005 \end{matrix}]

The trajectory of altitude and velocity of the error system (9) is given below.

Figure 11 shows the tube $Z_{2}$ about the altitude and velocity. Similarly, we can see that the trajectory of error system starts from the origin and is restricted within the tube $Z_{2}$ .

Figure 11.

A tube $Z_{2}$ about altitude and velocity.

Altitude and velocity tracking results and the control input curves are given in Figures 12 to 15. Other state variables of the hypersonic vehicle are shown in Figures 16 to 18.

Figure 12.

Altitude tracking performance.

Figure 13.

Velocity tracking performance.

Figure 14.

Values of fuel-to-air ratio.

Figure 15.

Elevator deflection.

Figure 16.

Attack angle.

Figure 17.

Pitch angle.

Figure 18.

Pitch rate.

Figures 12 and 13 show the reference trajectory, nominal trajectory, and two trajectories obtained by different control strategies. From these results, we can see that the tube-MPC scheme achieves stable tracking of altitude and velocity. As a result of the existence of bounded disturbances, the state variables of the hypersonic vehicle cannot be stable to a certain point but into a bounded set, that is, the tube $Z_{2}$ . Figures 16 to 18 present the results of attack angle, pitch angle, and pitch rate, respectively. We can see from Figures 12, 13, and 16 to 18, both methods make state variables stable within a bounded range of variation. But there is obviously a smaller chattering of trajectories controlled by tube-MPC than robust MPC controller in the study by Bumroongsri,⁴³ thus manifested the effectiveness of tube-MPC control strategy to deal with bounded disturbances. Figures 14 and 15 show the control inputs $Φ_{c}$ and $δ_{e}$ of tube-MPC and robust MPC which both satisfy the input constraints.

Conclusion

We have presented a tube-MPC strategy for hypersonic vehicles with bounded external disturbances. Firstly, the polytypic LPV model of the hypersonic vehicle is built in a given flight range using tensor-product transformation. The nominal trajectory is determined using traditional MPC with the tighter input constraints. The auxiliary robust feedback controller steers the actual state with disturbances toward the nominal trajectory. The proposed method has same computation burdens as the nominal MPC despite the uncertainty. Moreover, the robust stability and input constraints can be satisfied. The tube-MPC controller developed in this article assumes that the states of hypersonic vehicles are available for measurement. However, during the actual flight, the attack angle and pitch angle are very small, so that it is difficult to get exact measurements. Therefore, the control approach can be extended to the hypersonic vehicle subject to unmeasured states in the future work.

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 National Natural Science Foundation of China under grant (61673294) and Research Foundation of Tianjin Key Laboratory of Process Measurement and Control (TKLPMC-201613).

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