Switched linear parameter-varying modeling and tracking control for flexible hypersonic vehicle

Aerospace Science and Technology 95 (2019) 105445

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Switched linear parameter-varying modeling and tracking control for flexible hypersonic vehicle Lixian Zhang ∗ , Liang Nie, Bo Cai, Shuai Yuan, Dongzhe Wang Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, No. 92, West Dazhi Street, 150001 Harbin, China

a r t i c l e

i n f o

Article history: Received 10 January 2019 Received in revised form 27 July 2019 Accepted 27 September 2019 Available online 1 October 2019 Keywords: Flexible hypersonic vehicle H ∞ control Persistent dwell time Switched linear parameter-varying system

a b s t r a c t In this paper, the reference tracking problem for a class of flexible hypersonic vehicles, whose dynamics vary significantly as flight states changes, is investigated via a novel switched linear parameter-varying (LPV) framework. Specifically, ranges of velocity and altitude are divided into several partitions, and the flexible hypersonic vehicle dynamics is modeled as a LPV system in each partition. Then the switched LPV model for the underlying system is obtained over a wide range of operating conditions. Switched LPV tracking controllers based on the developed model are established under mode independent/dependent persistent dwell time (PDT/MPDT) switching signals, respectively. Compared with the conventional mode independent/dependent average dwell time (ADT/MADT) switching signals, the proposed PDT/MPDT switching signals can address some more general flight variations, such as slow maneuvers and frequent rapid maneuvers coexist in the same flight mission. Furthermore, stringent transient and steady-state performance and a guaranteed non-weighted H ∞ performance are achieved with the aid of multiple Lyapunov-like functions. A simulation is used to illustrate the designed controller via two proposed switching signals. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction As a type of potential near-space vehicles, air-breathing hypersonic vehicle has attracted extensive research due to its fast speed and large flight envelope [1–4]. Tracking control is one of the crucial research topics about hypersonic vehicles [5]. Owing to stringent performance requirements and dramatic parameter variations of the hypersonic vehicle, a robust and high-performance tracking controller for hypersonic vehicle is required. However, since the dynamics of hypersonic vehicle depends on many complicated factors, e.g., modeling uncertainty and coupling effects between elasticity, propulsion and aerodynamics, it is difficult to design a satisfactory controller for hypersonic vehicles. To this end, the nonlinear longitudinal models of hypersonic vehicles are introduced to reduce the complexity of controller design [6,7], for which numerous tracking control methods are proposed. In [8], the nonlinear dynamic inversion control method for the hypersonic vehicle tracking control in the presence of disturbances has been extensively studied. To address unknown time-varying aerodynamics, an adaptive output feedback control scheme is proposed for the hypersonic

*

Corresponding author. E-mail addresses: [email protected] (L. Zhang), [email protected] (L. Nie), [email protected] (B. Cai), [email protected] (S. Yuan), [email protected] (D. Wang). https://doi.org/10.1016/j.ast.2019.105445 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.

vehicle attitude tracking in [9]. Although high performance has been achieved by these approaches, the nonlinearity of the model increases the difficulty of stability and robustness analysis, which restricts their application in practice. The linear parameter-varying (LPV) approach, which adopts linearized system models, has attracted considerable attention [10, 11]. Due to fast and wide parameter variations during hypersonic vehicles operation, the LPV technique is well suitable for controller design of hypersonic vehicles. In view of this, a class of LPV modeling and control methodologies for hypersonic vehicles have been proposed to date. Specifically, the H 2 performance for hypersonic vehicle based on a polytopic LPV model is presented in [12]. To reduce computational complexity, the LPV method based on the linear fractional transformation (LFT) model is also applied to the tracking control system design for hypersonic vehicles [13]. In addition, the LPV method is also combined with other control methods to improve the performance of the hypersonic vehicle control system. For example, the dynamic decoupling control and receding horizon control combining with LPV modeling are investigated in [14] and [15], respectively. In fact, these modeling and control methods are based on the strict condition that the hypersonic vehicle system dynamics and corresponding controllers need to be smooth. In addition, the LPV controller is also designed by using smooth parameter-dependent or constant Lyapunov functions, which is restrictive and could introduce some conservatism as well.

2

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

To overcome the aforementioned drawbacks of LPV methods, the switched LPV approach is proposed [16]. It adopts a more general class of Lyapunov functions [17], and employs a switching signal to describe switching among local controllers [18]. It is widely applied to engineering systems, such as F-16 aircraft [19], missile autopilot [20], and aircraft engines [21,17]. H ∞ performance as a very important topic of the switched LPV approach, can effectively deal with performance degradation or system instability caused by turbulence, gusts and modeling uncertainty. Increasing attention has been drawn on switched LPV controllers with weighted performance for hypersonic vehicles in recent years. Most of them are usually designed under mode independent/dependent average dwell time (ADT/MADT) switching signals [22,23]. It is well established that, an upper bound of switching times is strictly limited by chatter bound and length of the average switching interval of ADT/MADT switching [24]. This property prevents the ADT/MADT from being applied to the hypersonic vehicles with high maneuverability, such as slow maneuvers and frequent rapid maneuvers coexist in the same flight mission. The mode independent/dependent persistent dwell time (PDT/MPDT) switching signals are more general switching signals. On the other hand, this switching strategy leads to a larger class of switching signals than ADT/MADT, which can describe slow switching and fast switching occurring in one switched system simultaneously [25]. However, to the best of authors’ knowledge, the PDT/MPDT switching signals are not introduced to deal with the problem of controller design for the complex maneuvers of hypersonic vehicles. In addition, the nonweighted H ∞ performance1 for switched LPV hypersonic vehicle system with PDT/MPDT, which is hard to obtain under existing ADT/MADT, has not been achieved. In fact, we usually need to quantitatively measure the effectiveness of disturbance suppression and the quality of reference tracking in practice. The nonweighted H ∞ performance index is very convenient for measuring this performance of hypersonic vehicle control system. All of the above motivates the research work in this paper. Compared with previous research, this paper mainly addresses controller design for the flexible hypersonic vehicle under more general PDT/MPDT, which includes a more applicable non-weighted H ∞ performance index. For controller design purpose, the nonlinear model of flexible hypersonic vehicle is first converted to a switched LPV system. Then, switched sate-feedback LPV controllers are designed, which results in performance and robustness against parameter perturbations. The main contribution of this paper is in two aspects: on the one hand, flight envelope is divided into several partitions according to velocity and altitude variations, for which a new switched LPV framework under PDT/MPDT is introduced to tracking controller design for hypersonic vehicles; on the other hand, more practical non-weighted H ∞ disturbance attenuation performance is obtained for hypersonic vehicle control system. The rest of the paper is organized as follows. Section 2 introduces the detailed switched LPV model of a class of flexible hypersonic vehicles. Section 3 proposes a stability criterion and H ∞ controller design approach under the PDT/MPDT framework, which is expressed in the form of linear matrix inequalities (LMIs). In Section 4, a numerical example is presented to illustrate the effectiveness and performance of the proposed controllers. The paper is concluded in Section 5.

1

The hypersonic vehicle system is said to have a guaranteed non-weighted H ∞

∞

e T (t ) e (t ) dt ≤

performance, if the L2 -gain holds 0

∞

γ 2 d T (t ) d (t ) dt, where e (t ) 0

is the tracking error, d (t ) is the external disturbance [25].

Notation. The notations in this paper are standard. The superscript ‘T ’ stands for vector or matrix transposition. Rn denotes the n-dimensional Euclidean space (R stands for R1 ), R[0,∞) represents the set of non-negative real numbers, and Z+ represents the set of non-negative integers. C 1 indicates the space of continuously differentiable functions. The notation · refers to the Euclidean vector norm and L2 is used to express the space of square-integrable functions. In addition, diag {· · · } indicates a block-diagonal matrix. The symbol ‘∗’ is adopted to signify an ellipsis for the terms introduced by symmetry. I is an identity matrix. The subscripts ‘max’ and ‘min’ represent the maximum and minimum value of a number, respectively. 2. Switched LPV modeling for flexible hypersonic vehicles In this section, a switched LPV model for the longitudinal dynamics of flexible air-breathing hypersonic vehicles is presented. The longitudinal model of the flexible hypersonic vehicles considered in this paper, is a model with curve-fitted approximations [1]. In particular, the earth curvature is ignored, and there is no gyroscopic effect. For simplicity, the hypersonic vehicle is modeled as a pair of cantilever beams, of which only the first two modes are considered. The interconnection between the propulsion system and aerodynamic effects is also added in this model. The longitudinal model of the flexible hypersonic vehicles can be described as follows

⎧ ˙ = V sin (θ − α ) H ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ V˙ = m ( T cos α − D ) − g sin (θ − α ) ⎪ ⎪ ⎪ 1 ⎪ α˙ = mV (− T sin α − L ) + Q + Vg cos (θ − α ) ⎪ ⎪ ⎨ θ˙ = Q ⎪ ⎪ Q˙ = M ⎪ ⎪ I yy ⎪ ⎪   ⎪ ⎪ ¨ −2ζm,1 ωm,1 , −2ζm,2 ωm,2 η˙ η = diag ⎪ ⎪ ⎪

⎪ ⎪ T 2 2 ⎩ + diag −ωm ,1 , −ωm,2 η + [N 1 , N 2 ]

(1)

where M ≈ z T T + q¯ S c¯ (C M ,α (α ) + C M ,δe (δe )), L ≈ q¯ SC L (α , δe ), 2 η = [η1 , η2 ]T , D ≈ q¯ SC D (α , δe ), N 1 ≈ N 1α α 2 + N 1α α + N 10 , N 2 ≈ 2

δ

3

2

N 2α α 2 + N 2α α + N 2e δe + N 20 , T ≈ C Tα α 3 + C Tα α 2 + C Tα α + C T0 , the rigid body states are velocity (V ), altitude (H ), angle-of-attack (α ), pitch angle (θ ), and pitch rate ( Q ). The two vibrational modes are represented by η1 and η2 . The definitions of the notations T , L, D, G, M, and N are shown in Table 1. The miscellaneous coefficient values C M ,α , C M ,δe , C L , C D are presented in [1]. Fig. 1 shows the geometry and mechanical analysis of the longitudinal model of a hypersonic vehicle. Firstly, a family of discrete linearization models are achieved by the Jacobian linearization method with respect to adequate equilibrium points. Then, due to the velocity and altitude are crucial factors during the flight, we choose the vector [V , H] as the scheduling variable vector. A continuous LPV model is achieved by the curve fitting method. Finally, we divide the scheduling variable range into several partitions as shown in Fig. 2. According to the partitions of the scheduling variable range, the switched LPV model of the hypersonic vehicle can be established as follows



x˙ δ = A σ (t ) ( V , H ) xδ + B σ (t ) ( V , H ) u δ y δ = C σ (t ) ( V , H ) xδ

(2)

where xδ = x − xeq , u δ = u − u eq , and y δ = y − y eq . Herein, (·)eq represents the values at the equilibrium points, x = [ H , V , α , θ, Q , η1 , η˙ 1 , η2 , η˙ 2 ]T is the state vector, u = [φ, δe ] is the control input, and y denotes the output. The expressions of matrices A σ (t ) ( V , H ) ∈ R9×9 , B σ (t ) ( V , H ) ∈ R9×2 and C σ (t ) ( V , H ) ∈ R2×9 are given in

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445



Table 1 The definitions of symbols in hypersonic vehicle model. Parameters

The definitions in Fig. 1

T (lbf) D (lbf) L (lbf) G (lbf) M (lbf.ft) N i (i = 1, 2) (lbf) R (ft) H (ft)

Thrust Drag Lift Graviton force Moment Generalized forces (Related to elastic forces) Earth radius Altitude of Vehicle

x˙ a ya

 =S

N 

3

 W σ (t ),n (ρn (t ))

n =1

xa ua

 + B 1 yr .

where S ∈ R I 1 × I 2 ×...× I N × O × M is constructed from the linear  timeinvariant (LTI) vertex systems S I 1 I 2 ... I N ∈ R O × M , and denotes Kronecker product. The row vector W σ (t ),n (ρn (t )) ∈ R I n contains weighting functions σ (t ),n,in (ρn (t )), (σ (t ) ∈ I , n = 1, 2, . . . , N, and in = 1, 2, . . . , I n ). I n denotes the number of the weighting functions, and ρn (t ) represents the nth element of scheduling variable vector ρ (t ). Hence, the switched LPV system (3) can be rewritten as



x˙ a ya



where

=

n  k =1



σ (t ),k

A a,σ (t ) (ρk ) C a,σ (t ) (ρk )

B a,σ (t ) (ρk ) 0

B1 0



⎡

⎤

xa ⎣ yr ⎦ (4) ua

n 

σ (t ),k = 1, σ (t ),k ≥ 0, and σ (t ) ∈ I .

k=1

Fig. 1. The schematic diagram of mechanical analysis of hypersonic vehicle (See Table 1 for the symbols).

Fig. 2. The partitions of the scheduling parameter space.

3. Controller design under PDT/MPDT switching signals

Appendix. σ (·) : Z+ → I is the switching signal governing the switching between the subsystems. In this paper, given a reference signal y r = [V r , H r ] with bounded energy, the aim of the controller design is to eliminate the following tracking error e = y − y r . Defining a new variable xe =

t

[ y − y r ] dt, we have the aug-

0

mented system (2) as follows



x˙ a = A a,σ (t ) ( V , H ) xa + B a,σ (t ) ( V , H ) ua + B 1 y r

(3)

ya = C a,σ (t ) ( V , H ) xa where xa = [xδ , xe ] T , A a,σ (t ) ( V , H ) =

B σ (t ) ( V , H ) 0



, B1 =

0 −I



A σ (t ) ( V , H ) 0 C σ (t ) ( V , H ) 0





, B a,σ (t ) ( V , H ) =



Remark 1. In fact, the parameters uncertainties of hypersonic vehicles such as modeling error, unmodeled dynamics, and fuel consumption of scheduling parameters, can also be converted into a polytopic space such as (4). In addition, the linearized system models used in LPV techniques may result in computational problems. The proposed switched LPV model can be used to overcome these problems within an acceptable tolerance [16]. To examine the accuracy of the established polytopic LPV model, the control input u = [φ, δe ] of the flexible hypersonic vehicle, is selected as u = [0.8 + 2e − 3i , 0.25 + 1e − 3i], i = 1, 2, . . . , 50. Fig. 3 shows the rigid and flexible states responses of the established polytopic LPV model and the nonlinear model of the hypersonic vehicle. The states of rigid body and flexible modes are consistent with the nonlinear model around the equilibrium point. It is clear that the established polytopic LPV model captures the local nonlinearity of the nonlinear model, and shows high accuracy around equilibrium points. This shows that the polytopic LPV model will have sufficient accuracy in flight envelope if enough equilibrium points are selected, which indicates that the model meets the basic requirements of LPV controller design. In this paper, we aim to design a controller ensuring that the closed-loop system (3) is globally uniformly asymptotically stable, and the H ∞ norm for system (3) is less than a given γ > 0.

, C a,σ (t ) ( V , H ) = C σ (t ) ( V , H ) 0 . Based on the Tensor-Product(TP) model transformation, an approach which has the universal approximation property [26], the switched LPV system (3) can be transformed as

The switched LPV approach introduces an effective way to cope with drastic dynamic changes in hypersonic vehicle system, which provides a more accurate model and less computational effort than a single LPV controller. Moreover, the dwell time switching signals further improves the flexibility, and reduces the difficulty of switched LPV controller design. In this section, switched LPV controllers for the flexible hypersonic vehicles are designed under the framework of PDT/MPDT by solving a family number of linear matrix inequalities (LMIs). It should be pointed out that the switched LPV approach via multiple Lyapunov-like functions need no parameter variation information and avoid using gridding technique [16]. Compared to the parameter-dependent Lyapunov functions, it is a more suitable method for the hypersonic vehicle controller design. Fig. 4 is the proposed switched LPV control system structure of the flexible hypersonic vehicle. In Fig. 4, K is the control gain, u denotes the control input, ρ represents the schedule parameter, and σ is the switching signal. x, xr , w represent the practical states, desired states, and disturbances, respectively. The controller u is determined by K , x, xr and ρ = [ V , H ], while the

4

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

Fig. 3. States responses of the hypersonic vehicle model.

states x can be obtained from u, w, and the hypersonic vehicle model. 3.1. Preliminaries Definition 1 ([24]). (Function classes K, KL, K∞ ) A continuous function α : [0, ∞) → [0, ∞) is of class K if α is strictly increasing, and α (0) = 0. A continuous function β : [0, ∞) × [0, ∞) → [0, ∞) is of class KL, when β (., t ) is of class K for each fixed t ≥ 0 and β (r , t ) decreases to 0 as t → ∞ for each fixed r ≥ 0. A continuous function ξ : [0, ∞) → [0, ∞) is of class K∞ if it is strictly increasing, unbounded, and ξ (0) = 0.

Definition 2 ([27]). (Mode dependent persistent dwell time) For a sequence of switching instants t 0 , t 1 , . . . , t s , . . . , there exists an infinite number of disjoint intervals of length no smaller than τi on which subsystem σ = i is constant. The consecutive intervals with this property are separated by no more than T, where the positive constant τi and T are said to be the mode dependent persistent dwell time and the period of persistence, respectively. Remark 2. Some comments are provided for Definition 2. (i) In the period of persistence T, the switching sequence of subsystem can be arbitrary, and the running time of any activated subsystems

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

5

Fig. 3. (continued)

dwell time τa in the ADT switching, the parameter chatter bound N 0 strictly limits upper bound of switching times N [25]. Definition 3 ([24]). (Globally uniformly asymptotically stable) A switched system



x˙ = f σ (t ) (x (t ))

(5)

σ (t ) ∈ I Fig. 4. Switched LPV control system structure of flexible hypersonic vehicle.

σ = i will be less than τi . While in the τi -portion, one subsystem will be active staying at least τi . (ii) In the MPDT switching, the interval including one τi -portion and the adjacent period of persistence (T-portion) can be treated as a stage of switching. In the p-th 1 stage, t sp and t sp is denoted as the initial instant of τi -portion and T-portion, respectively. The actual running time of T-portion is de1 . Additionally, the Zeno behavior is excluded for noted as t sp +1 − t sp arbitrary switching sequence, therefore an upper bound of switching times within T-portion denoted as Q exists, which is assumed to be known a priori. (iii) When τi = τ , MPDT is reduced to PDT, and σ (·): Z+ → P are all permissible PDT switching signals. Fig. 5 shows a scenario of PDT and ADT switching. Note that PDT represents a larger class of switching signals than ADT, since there is no requirement on the frequency of switching during the period of persistence T. While within an interval of length less than

is said to be globally uniformly asymptotically stable (GUAS) if there exists a KL function β(·) such that for some switching signals σ (·): Z+ → P and for any initial condition x(0) the inequality is satisfied: x (t ) ≤ β (x (0) , t ) , ∀t ≥ 0. Definition 4 ([25]). (Non-weighted L2 gain) The switched system

⎧ ⎨ x˙ = f σ (t ) (x (t ) , d (t )) e = mσ (t ) (x (t ) , d (t )) ⎩ σ (t ) ∈ I

(6)

is said to have a non-weighted L2 gain γ > 0 (i.e. non-weighted H ∞ performance), if under zero initial conditions, the following inequality holds

∞

∞ T

e (t ) e (t ) dt ≤ 0

γ

2

d T (t ) d (t ) dt 0

for all t ≥ 0, and all d (t ) ∈ L2 .

6

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

Remark 5. Most of the existing switched or switched LPV methods for hypersonic vehicles adopt switching signals similar to [17,18, 28,29], i.e., DT and ADT/MADT switching signals [23], which are restrictive when PDT/MPDT signals are concerned [25]. Lemma 2. Let μi > 1, αi > 0 be given constants. Suppose that there   exist positive definite C 1 functions V σ (t ) : Rn , R → R, σ (t ) = i ∈ I , two class K∞ functions κ1 , κ2 and a scalar γ0 > 0, such that ∀t ≥ 0, satisfying inequalities (7) and (9), and

V˙ σ (t ) (x (t ) , t )  −αi V σ (t ) (x (t ) , t ) −  (t )

(11)

where  (t ) = e (t ) e (t ) − γ0 d (t ) d (t ). Then, the switched system (6) is GUAS and has a non-weighted L2 -gain no greater than γ (i.e. non-weighted H ∞ performance) for any T

T

MPDT switching signals satisfying τi >

   β =

Fig. 5. A scenario of PDT (on the top) and ADT (on the bottom) switching.

Lemma 1. Let μi > 1, αi > 0 be given constants. Suppose that there   exist positive definite C 1 functions V σ (t ) : Rn , R → R, σ (t ) = i ∈ I , and two class K∞ functions κ1 and κ2 , such that ∀t ≥ 0

κ1 (x (t )) ≤ V σ (t ) (x (t )) ≤ κ2 (x (t ))

(7)

V˙ σ (t ) (x (t )) ≤ −αi V σ (t ) (x (t ))     and ∀ σ (t ) = i , σ t − = j ∈ I × I , i = j

(8)

V i (x (t )) ≤ μ j V j (x (t )) .

(9)

τi >

Q¯ max ln μmax

αi

(10)

where Q¯ = Q + 1. Proof. For conditions (7)–(10), we will prove that the Lyapunov function decreases to zero in infinite time. The proof of Lemma 1 can be found in Appendix B. 2 Remark 3. It needs to be pointed out that: the above conclusion is applicable to 0 < T < τmin Q . In addition, from the derivation process, let μmax = μ, αmin = α , it is straightforward that the system (5) is GUAS for any switching signal with PDT satisfies

τ>

Q¯ max ln μ

α

.

Remark 4. The Lyapunov functions used in this paper can be discontinuous and increasing at the switching instant. This meets the different local performance requirements of hypersonic vehicles in a broad flight envelope. Compared to the nonquadratic Lyapunov function which is considered in [11], the multiple Lyapunov-like scheme used in this paper is more convenient for controller design, and with less computational cost. It also can deal with the nonsmooth systems, and with a more accurate approximation of the nonlinear hypersonic vehicle dynamics. In fact, from a practical consideration, this multiple Lyapunov-like scheme might be more suitable for the flight vehicle autopilots design [16,20]. Also, compared to the multiple discontinuous convex or parameterdependent Lyapunov function approach in [17,18], the Lyapunov functions used in this paper can reduce computational cost while guaranteeing desired performance.

1 ¯ αmin − τmin Q max ln μmax

αmin

, where γ = β γ0 and

!.

(12)

2

Proof. The proof of Lemma 2 can be found in Appendix B.

Remark 6. From the derivation process, let μmax = μ, and αmax = α . It can be concluded that the switched system (6) is GUAS. And it has a non-weighted L2 -gain no greater than γ (i.e. nonweighted H ∞ performance) for any PDT switching signals satisfy" ing

Then, the system (5) is GUAS for any switching signal with MPDT satisfies

αmax e2 Q¯ max ln μmax

Q¯ max ln μmax

τ>

Q¯ max ln μ

α

, where

α e 2 Q¯ max ln μ ! . α − τ1 Q¯ max ln μ

γ = β γ0 and β =

Remark 7. Most of the existing ADT switching scheme only give a weighted H ∞ control performance index, which is inappropriate to measure the disturbances rejection capability desired in the practical because the output is exponentially attenuated [23]. For various complex disturbances encountered by hypersonic vehicles in flight, a guaranteed non-weighted disturbance attenuation performance is obtained, which is of explicit physical sense and less conservatism than weighted one. 3.2. Controller design Considering the switched LPV system (3), the switched controller is constructed as



u i = K i (ρ ) x

(13)

i ∈ I.

Substituting (13) into (3), and control inputs u passing through a filter [30], we have the following closed-loop system

⎧ ⎪ ⎨ x˙ cl = A cl,i xcl + B cl,i d (t ) z = C cl,i xcl ⎪ ⎩ i∈I

(14)

¯ a,i (ρ ) + B¯ a,i (ρ ) K i (ρ ), B cl,i  B¯ 1 , C cl,i  C¯ a,i (ρ ), where A cl, i  A  

 ¯C a,i (ρ ) = C a,i (ρ ) 0 , A¯ a,i (ρ ) = Aa,i (ρ ) B a,i (ρ )C u , B¯ a,i (ρ ) = 0 , B 0 A Au =

0

0 1 −144 −19.2

0



, Bu =

01 10



, B¯ 1 =

u

I 0 0 0 0 0 −I 0

T

, Cu =

10 01



u

, xcl is

the state. Remark 8. In (14), d (t ) consists of two parts, i.e., the reference signal y r and external disturbances, which are both assumed to be a kind of signal with bounded energy.

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

⎡

Remark 9. A single parameter-dependent controller is usually conservative, and it is difficult to obtain for hypersonic vehicles with drastic dynamic changes [16,31]. The proposed switched LPV method uses several LPV polytopic controllers to overcome these shortcomings, and improves the performance of the control system. The following theorem presents the synthesis conditions and construction algorithms for the switched LPV controller, which asymptotically stabilizes the switched plant (14) with guaranteed non-weighted H ∞ performance. Theorem 1. (MPDT case) Let μi > 1, αi > 0 be given constants. Suppose that there exist symmetry positive definite matrices X i , rectangular matrices W i ,k , ∀σ (t ) = i , j ∈ I , i = j , X j < μ j X i , and a scalar γ0 > 0 such that ⎡ ⎤ ¯ a,i (ρk ) X i + B¯ a,i (ρk ) W i ,k  T A ¯ ¯ B C ρ X ( ) 1 a , i i k ⎢ ⎥ + ( A¯ a,i (ρk ) X i + B¯ a,i (ρk ) W i ,k )T + αi X i ⎢ ⎥ ⎣ ⎦ ∗ −γ02 I 0 ∗ ∗ −I

< 0.

(15)

Then, the system (14) is GUAS with a non-weighted L2 -gain no greater than γ (i.e. non-weighted H ∞ performance) for any switching signal satisfying in the following

K i (ρ ) =

τi >

Q¯ max ln μmax

αmin

. Moreover, the controller is obtained

n 

k W i,k X i−1 .

(16)

k =1

Proof. The main idea is to transform the condition (11) into LMI condition by means of Schur complement [32], and the detail of proof can be found in Appendix B. 2 Remark 10. With aid of LMI toolbox in Matlab, the controller gain K i can be obtained by solving (15) and (16) (see reference [23]). ¯ a,i (ρk ) , B¯ a,i (ρk ) , C¯ a,i (ρk )), we can get a For each system vertex ( A K i ,k . With the same weight functions can be achieved.

k in (4), K i =

n %

k=1

k K i,k

Corollary 1. (PDT case) Let μ > 1, α > 0 be given constants. Suppose that there exist symmetry positive definite matrices X i , rectangular matrices W i ,k , ∀σ (t ) = i , j ∈ I , i = j , X j < μ X i , and a scalar γ0 > 0 such that

⎢ ⎢ ⎣

7

¯ a,i (ρk ) X i + B¯ a,i (ρk ) W i ,k A + ( A¯ a,i (ρk ) X i + B¯ a,i (ρk ) W i ,k )T + α X i ∗ ∗

< 0.

B¯ 1

−γ02 I ∗



C¯ a,i (ρk ) X i 0 −I

T

⎤ ⎥ ⎥ ⎦

(17)

Then, the system (14) is GUAS with an non-weighted L2 -gain no greater than γ (i.e. non-weighted H ∞ performance) for any switching signal satisfying the following

K i (ρ ) =

τ>

Q¯ max ln μ

n 

α

k W i,k X i−1 .

. Moreover, the controller is obtained in

(18)

k =1

Proof. The proof of PDT case is similar to that of Theorem 1 and that is omitted here. 2 Remark 11. With regard to the various disturbances and uncertainties in hypersonic vehicles, the proposed switched LPV controllers, could achieve guaranteed disturbance attenuation and robustness performances when the system parameters changed within (4). 4. Numerical example In this section, based on the presented switched LPV model, the effectiveness of the proposed theorems is verified by simulations. The considered flight envelope covers a velocity of V ∈ [8200 9800] f t /s and altitude H ∈ [85000 90000] f t. The velocity and altitude ranges are divided into 3 partitions as in Fig. 2. For simplicity, external disturbances are chosen as w 1 = 0.1e −0.001t sin(0.1t + π ) and w 2 = 0.2e −0.001t sin(0.1t + π ) [33], respectively. The distribution of open-loop poles is displayed in Fig. 6(a). It can be seen that the open-loop hypersonic vehicle system is not stable, and the poles distribution is quite different from those of conventional aircrafts. In controller synthesis, to guarantee the stability of closed-loop system and to avoid the numerical issue caused by ill-conditioning matrices, the closedloop poles are constrained on the left half plane as shown in Fig. 6(b) via the technique in [34]. For the PDT framework, let α = 0.005, μ = 1.65 and Q max = 3. For the MPDT framework, let α = [0.005 0.007 0.009] , μ = [1.65 1.65 1.65] and Q max = 3. By solving the LMIs (15) and (17) to find the optimal γ0 , the desired controller gains are obtained. Fig. 7 shows the output responses to reference command inputs of the hypersonic vehicle under PDT/MPDT switching signals, arbitrary switching [16] and LPV method [30]. In particular, the

Fig. 6. Poles of the hypersonic vehicle system.

8

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

Fig. 7. The tracking responses of the flexible hypersonic vehicle.

Fig. 8. The control inputs of the flexible hypersonic vehicle.

command inputs are chosen to be similar as in [23]. The trajectory tracking performances of the hypersonic vehicle system using four different methods, are both meet the control system requirements over the entire time range. It is noticed that the output responses speed under PDT switching signal is better than arbitrary switching and LPV method, while the responses under MPDT switching signal is faster than PDT. The reasons for this result consists in two points: one is that the proposed SLPV approach adopts a more general class of Lyapunov functions, and the other one is the MPDT switching signal takes into account the specific properties of each subsystem while to PDT. The fuel rate and pitch control surface deflection stay in allowed ranges over the entire time range, which are shown in Fig. 8(a) and Fig. 8(b), respectively. In particular, if τa in the ADT case is the same as the τ in PDT case and the chatter bound N 0 ≤ 1, then the ADT switching signals can’t switch two times within an interval of length less than τa [24]. This means that ADT is not applicable in above controller design for hypersonic vehicle. Table 2 presents the influence of system parameters μ, Q max , and τ on the robust performance γ . It is observed that if τ and Q max are fixed at 25.86 s and 3, respectively, while μ varies from 1.22 to 1.1, the robust performance γ is improved with the decrease of μ. On the other hand, the robust performance γ has been improved with the decrease of parameter Q max and the increase of parameter τ .

Table 2 Effects of system parameters on the performance

γ.

μ

Q max

τ (s)

γ

1.22 1.15 1.10 1.10 1.10 1.10 1.10

3 3 3 4 5 5 5

25.86 25.86 25.86 25.86 25.86 33.77 40.12

2.80 2.28 1.94 2.17 2.43 2.40 2.38

5. Conclusion This paper presents the controller design problem of a generic flexible hypersonic vehicle based on switched LPV control approach. For the hypersonic vehicle with a broad flight envelope, with the aid of Jacobian linearization and curve fitting methods, a switched LPV model is derived. Then, state-feedback controllers are given under PDT and MPDT switching signals based on the established model, respectively. The nonlinear simulations show that the proposed controllers are more applicable than arbitrary switching and LPV method, while MPDT signal is more flexible than the PDT. Also, the switched LPV control approach proposed in this paper, can be extended to applications in other fields, such as mobile robot [35], morphing aircraft [36], and spacecraft [37]. In our future work, it is a promising way to incorporate nonquadratic Lyapunov function [11] into the proposed switched

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

LPV control scheme, which can further reduce conservatism and deal with a class of more general dynamic systems. Since eventtriggered control is a powerful tool to improve the resource utilization of systems [21,38], it is also a interesting point to extend the application scope of the proposed method by event-triggered control.

9

a93 = 5.4043e3 + 0.0063V − 1.4049e − 5V 2 − 0.0528H

+ 3.753e − 6V H , b12 = −8.4028 − 1.1909e − 04V + 1.6472e − 7V 2 + 6.2042e

− 4H − 4.3694e − 8V H , b22 = 110.51 − 0.0246V − 4.0854e − 7V 2 − 0.0015H

Declaration of competing interest

+ 2.9589e − 7V H ,

The authors declared that they have no conflicts of interest to this work.

b13 = 0.0012 + 1.8044e − 09V − 9.5212e − 12V 2 − 2.6527e

− 8H + 2.2882e − 12V H , Acknowledgements

b23 = 1.1024e − 7 − 6.047e − 06V + 3.2755e − 16V 2

The work was supported in part by Foundation of State Key Laboratory of Automotive Simulation and Control (Grant no. 20160101), Natural Science Foundation of Heilongjiang Province (Grant no. JC2015015), the Fundamental Research Funds for Central Universities, China (Grant nos. HIT.MKSTISP.2016.32, HIT.BRETIII.201211 and HIT.BRETIV.201306), Self-Planned Task (Grant no. SKLRS201617B), Foundation of State Key Laboratory of Robotics and System (HIT) (Grant no. SKLRS2016KF03), and Natural Sciences and Engineering Research Council of Canada (Grant no. RGPIN-2016-05386), Cultivation Plan of Major Research Program of Harbin Institute of Technology (Grant ZDXMPY20180101).

− 9.3102e − 13H + 5.4263e − 11V H , b15 = −0.0448 − 4.7503e − 07V + 8.3781e − 10V 2

+ 3.1552e − 6H − 2.2321e − 10V H , b25 = 6.7981 − 0.0015V − 2.007e − 8V 2 − 7.5535e

− 5H + 1.6857e − 8V H .  0 b12 b13 0 b15 0 0 0 B σ (t ) ( V , H ) =  C σ (t ) ( V , H ) =

Appendix A The expressions of matrices A σ (t ) ( V , H ), B σ (t ) ( V , H ) and C σ (t ) ( V , H ) in system (2) are as follows

a21 = −0.0014 + 1.4985e − 07V + 6.2174e − 12V 2 + 2.3325e

− 8H − 2.8086e − 12V H ,

⎡

=

− 7H + 1.2413e − 11V H ,

0 0

0 0

0 0

0 0

0 0

0 0



0 ⎢ a21 ⎢ ⎢ a31 ⎢ ⎢ 0 ⎢ ⎢ a51 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

0 a22 a32 0 a52 0 0 0 0

−V a23 a33 0 a53 0 a73 0 a93

V a24 0 0 0 0 0 0 0

0 0 1 1 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0 0 0

−ω12 −2ζ1 ω1 0 0

0 0

⎤

0 0 0 0 0 0 0 1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−ω22 −2ζ2 ω2





Proof of Lemma 1. Without loss of generality, for t ∈ t nsp , t nsp+1 , n ≥ 1 ∈ I , according to (8) and (9), one has

a24 = −32.2171 − 1.1574e − 11V + 1.5019e − 16V 2

+ 3.0433e − 6H − 9.8309e − 17V H , 2



  μn V

a32 = −4.8794e − 6 + 7.1280e − 10V − 2.9041e − 14V 2

×e

2

a33 = 0.0019 − 3.7193e − 05V − 2.4197e − 11V − 5.7995e a51 = −7.01e − 6 + 7.5283e − 10V + 3.1305e − 14V 2

+ 2.3325e − 10H − 2.8086e − 12V H , 2

a53 = −9.5943 + 0.0023V + 3.003e − 8V + 1.1295e − 4H

− 2.5707e − 8V H ,

!

  −α  μn V σ tnsp  t nsp e n ! n −1

n   ! t n−1 e −αn−1 t sp −t sp n −1 sp σ t sp

e

t −t nsp

−αn t −t nsp

!

!

! ! ! 1 −αi t sp −t sp −...−αn−1 t nsp −t nsp−1 −αn t −t nsp

  Q¯ max  μmax V σ t sp  t sp e −αi τi

− 8H + 3.396e − 10V H ,

a52 = 1.7812e − 5 − 1.2672e − 09V − 2.11314e − 12V 2

−αn t −t nsp

   . . .  μn μn−1 . . . μi V σ t sp  t sp

+ 1.7817e − 12H − 1.589e − 16V H ,

+ 1.1743e − 10H − 1.4129e − 14V H ,



V σ (t ) (t )  V σ tnsp  t nsp e

− 2.9661e − 13H + 2.4873e − 17V H ,

=e



Q¯ max ln μmax −αi τi







V σ t sp  t sp .

Let λi = − Q¯ max ln μmax + αi τi . If λi > 0, then Hence, it follows that

τi >

Q¯ max ln μmax

αi

.

 

V σ (t ) (t )  e −λ p V σ t p  t p  . . .  e −λ p −...−λ1 V σ (t1 ) (t 1 ) . Then, the system (5) is GUAS for any switching signal MPDT satisfying

a73 = 3.8648e3 − 0.0018V + 3.9195e − 6V 2 + 0.0147H

− 1.0471e − 6V H ,

0 0

0 b25

Appendix B

− 5.4828e − 7V H ,

a31 = 5.1542e − 7 − 5.5967e − 11V + 1.9736e − 15V

1 0 0 1

b23

A σ (t ) ( V , H )

a22 = 0.0035 − 2.5161e − 07V − 4.197e − 11V 2 − 1.0292e a23 = −264 + 0.0444V + 7.8961e − 7V 2 + 0.003H

0 b22

T

0 0 0 0 1247

τi >

Q¯ max ln μmax

αi

.

2

σ (t ) with

10

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

Proof of Lemma 2. First of all, for d (t ) = 0, from Lemma 1, it can be derived that if (7), (9) and (11) hold, the GUAS of system (6) can be guaranteed.   Consider d (t ) = 0, for t ∈ t nsp , t nsp+1 , n ≥ 1 ∈ I . It has

!



t −t sp



1

e −αmin (t −θ ) d T (θ) d (θ) dθ + . . .

+ γ02nμmax t sp

  −α  V σ tnsp  t nsp e n

t −t nsp

t

!

−

n

t sp + γ02 μmax

e −αn (t −θ )  (θ) dθ

  −α t −tnsp  μn V σ tnsp  t nsp e n

t

t

!

−

e

−αn (t −θ )

 μn V

σ t nsp−1

 n −1

t sp

e

−αn−1 t nsp −t nsp−1

t nsp

 − μn

e

−αn−1 t nsp −θ

!

 (θ) dθ e

+ γ02

 (θ) dθ

!

e

−αn t −t nsp

    ¯ ≤ V σ t sp  t sp e −αmin t −t sp + Q max ln μmax

!

t

−αn t −t nsp

+ γ02

!

which implies that

  ¯ V σ (t ) (t ) ≤ V σ t sp  t sp e −αmin t −t sp + Q max ln μmax 

t e

−αn (t −θ )

 (θ) dθ

t

t nsp

 ... 

n &





μi V σ t sp  t sp e

−

n% −1 i =0

!

i +1 i αi t sp −t sp −αn t −t nsp

+γ

!

n &

1 t sp



μi e

i =1

1 −α0 t sp −θ

!

 (θ) dθ e

−

n% −1 i =1

!

i +1 i αi t sp −t sp −αn t −t nsp

n

− . . . − μn

e

−αn−1 t nsp −θ

!

 (θ) dθ e

−

!

−αn t −t nsp

!

V σ (t ) (t )

    ¯  V σ t sp−1  t sp −1 e −αmin t −t sp−1 e 2 Q max ln μmax

t sp

e −αn (t −θ )  (θ) dθ

2 0

t + γ02

= F (− (θ)) . ! E (θ) = F e (θ) e (θ)

μi e−αmax (t −θ ) e T (θ) e (θ) dθ + . . . t sp



t sp



t −

¯

e −αmax



t sp −θ







¯ e T (θ) e (θ) dθ e −αmin t −t sp + Q max ln μmax

e −αmax (t −θ ) e T (θ) e (θ) dθ

e

e (θ) e (θ) dθ

≤ . . . ≤ V σ (t s1 ) (t s1 ) e −αmin (t −t s1 ) e p Q max ln μmax t

¯

e −αmin (t −θ )+( p −i +1) Q max ln μmax d T (θ) d (θ) dθ

t nsp−1

+ γ02

e −αmax (t −θ ) e T (θ) e (θ) dθ

t − e −αmax (t −θ ) e T (θ) e (θ) dθ.

t s1

t s1

e



¯

−αmax (t −θ ) T

t nsp

≥



¯ d T (θ) d (θ) dθ e −αmin t −t sp +2 Q max ln μmax

t sp

t nsp

t



t sp −1

1

t sp

t

t sp −θ

e −αmin (t −θ )+ Q max ln μmax d T (θ) d (θ) dθ

t sp

−

T

+



t sp

For  (θ) = e T (θ) e (θ) − γ02 d T (θ) d (θ), it follows that

+ μn

e −αmin

t sp −1

t nsp

i =1

e −αmax (t −θ ) e T (θ) e (θ) dθ.

t sp

+γ

n &

¯

e −αmin (t −θ )+ Q max ln μmax d T (θ) d (θ) dθ

t sp

t nsp−1

−



Therefore, we can obtain the following inequality

t sp

t sp

2 0

t

i =1

t

¯

e −αmin (t −θ )+ Q max ln μmax d T (θ) d (θ) dθ.

t sp

t nsp−1

−

e −αmin (t −θ ) d T (θ) d (θ) dθ

t nsp

t nsp

 !

e −αmin (t −θ ) d T (θ) d (θ) dθ

t nsp−1

t nsp

≥



t sp

V σ (t ) (t )

−



γ02d T (θ) d (θ)  nμmax V σ t sp  t sp e−αmin

D (θ) = F

−αmax (t −θ ) T

e (θ) e (θ) dθ.





where θ ∈ t si , t s(i +1) , i = 1, . . . , p. Substituting the initial condition V σ (t s1 ) (t s1 ) = 0, it yields that

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

t V σ (t ) (t )  γ02

e

T  + A¯ cl,i xcl (t ) + B¯ cl,i d (t ) P i xcl (t )  T + αi xclT (t ) P i xcl (t ) + C¯ cl,i xcl (t ) C¯ cl,i xcl (t ) − γ0 d T (t ) d (t ) T    ¯ cl,i + A¯ T P i + αi P i + C¯ T C¯ cl,i P i B¯ cl,i Pi A x (t ) cl,i cl,i = cl d (t ) ∗ −γ02 I   x (t ) × cl . d (t )

−αmin (t −θ )+( p −i +1) Q¯ max ln μmax T

d (θ) d (θ) dθ

t s1

t −

11

e −αmax (t −θ ) e T (θ) e (θ) dθ

t s1

Since 0  p − i  τt −θ + 1, we can obtain min

If the following inequality holds, then (11) holds

t e

−αmax (t −θ ) T



e (θ) e (θ) dθ

t s1

¯

L (t , θ)d T (θ) d (θ) dθ

t s1

−(t −θ) αmin −

Q¯ max ln μmax

1



!

< 0.

(B.1)

¯ cl,i + A¯ T P i + αi P i Pi A cl,i  −

C¯ clT ,i 0



∗

P i B¯ cl,i −γ02 I



  (− I ) C¯ cl,i 0 < 0.

(B.2)

Follow similar steps, we have the following inequality from (9)

e −αmax (t −θ ) e T (θ) e (θ) dθ dt

X j < μ j Xi

0 t s1

∞

¯

 e 2 Q max ln μmax

t

γ02 L (t , θ)d T (θ) d (θ) dθ dt . t s1

0

∞

∞ T

e (θ) e (θ) t s1

e

⎣ −αmax (t −θ )

¯

diag P i−1 , I , I

∞

d T (θ) d (θ)

t s1

L (t , θ)dtdθ θ

⎡ ⎢ ⎣

∞ T

e (θ) e (θ) dθ t s1

1

αmin − τmin

Q¯ max ln μmax

∞ "

T

d (θ) d (θ) dtdθ.

t s1

∞

e T (θ) e (θ) dθ 

Hence,

!γ

2 0

⎡ ⎢ ⎣

γ 2 d T (θ) d (θ) dθ where γ = β γ0 and

t s1

(B.4)

[32], we can get T

¯ cl,i P −1 + P −1 A¯ + P −1 α A i i i cl,i i ∗ ∗

B¯ cl,i

−γ 20 I ∗

P i−1 C¯ clT ,i 0 −I

⎤ ⎥ ⎦ < 0.

T

¯ cl,i (ρk ) P −1 + P −1 A¯ (ρk ) + P −1 α A i i i i cl,i ∗ ∗

B¯ cl,i

(B.5)

P i−1 C¯ clT ,i (ρk )

−γ 20 I ∗

0 −I

< 0.

t s1

⎤ ⎥ ⎦

(B.6)

From (4) and (14), let X i  P i−1 , W i ,k  K i ,k P i−1 , (B.6) can be rewritten as

αmax e 2 Q¯ max ln μmax ! . αmin − τ 1 Q¯ max ln μmax min

Let t s1 = 0, the proof is thus completed according to Definition 4. 2 Proof of Theorem 1. The following Lyapunov function is considered T V i (xcl (t ) , t ) = xcl (t ) P i xcl (t ) ,

∗

⎤

C¯ clT ,i 0 ⎦ < 0. −I

For vertex properties, (B.5) can be rewritten as [30]

∞

αmax e2 Q¯ max ln μmax



∗ ∗

P i B¯ cl,i −γ02 I

Performing congruence transformation on above inequality with

θ

 e 2 Q max ln μmax γ02

¯ cl,i + A¯ T P i + αi P i Pi A cl,i

dtdθ

∞

(B.3)

1 where X i = P i−1 , X j = P − . j By Schur complement [32], (B.2) can be reformulated as

⎡

That is

β=



It follows that

τmin where L (t , θ) = e . By integrating the above inequality from t = 0 to t = ∞, the following inequality is obtained

∞ t

P i B¯ cl,i −γ02 I

∗

t

 γ02 e 2 Q max ln μmax

¯ cl,i + A¯ T P i + αi P i + C¯ T C¯ cl,i Pi A cl,i cl,i

i∈I

where xcl (t ) is the state of closed-loop system. In the ith subsystem, it readily follows

V˙ i (xcl (t ) , t ) +α i V i (xcl (t ) , t ) + z T (t ) z (t ) −γ 20 d T (t ) d (t )

  = xclT (t ) P i A¯ cl,i xcl (t ) + B¯ cl,i d (t )

⎡

¯ a,i (ρk ) X i + B¯ a,i (ρk ) W i ,k A ⎢ + ( A¯ a,i (ρk ) X i + B¯ a,i (ρk ) W i ,k )T + αi X i ⎢ ⎣

∗ ∗

B¯ 1



−γ02 I ∗

< 0.

C¯ a,i (ρk ) X i 0 −I

T

⎤ ⎥ ⎥ ⎦

(B.7)

According to Lemma 2, if (15) holds, the switched system (14) is GUAS and has a non-weighted L2 -gain no greater than γ (i.e. non-weighted H ∞ performance) for any MPDT switching signals satisfying (10). 2 For more similar proof, we can refer to [31,23].

12

L. Zhang et al. / Aerospace Science and Technology 95 (2019) 105445

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