Asynchronous H∞ Control for Unmanned Aerial Vehicles: Switched Polytopic System Approach

IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 2, NO. 2, APRIL 2015

207

Asynchronous H∞ Control for Unmanned Aerial Vehicles: Switched Polytopic System Approach Zhaolei Wang, Qing Wang, and Chaoyang Dong

Abstract—This study is concerned with the H∞ control for the full-envelope unmanned aerial vehicles (UAVs) in the presence of missing measurements and external disturbances. With the dramatic parameter variations in large flight envelope and the locally overlapped switching laws in flight, the system dynamics is modeled as a locally overlapped switched polytopic system to reduce designing conservatism and solving complexity. Then, considering updating lags of controller0 s switching signals and the weighted coefficients of the polytopic subsystems induced by missing measurements, an asynchronous H∞ control method is proposed such that the system is stable and a desired disturbance attenuation level is satisfied. Furthermore, the sufficient existing conditions of the desired switched parameter-dependent H∞ controller are derived in the form of linear matrix inequality (LMIs) by combining the switched parameter-dependent Lyapunov function method and average dwell time method. Finally, a numerical example based on a highly maneuverable technology (HiMAT) vehicle is given to verify the validity of the proposed method. Index Terms—Switched polytopic system, asynchronous switching, robust control, unmanned aerial vehicles (UAVs), missing measurements.

I. I NTRODUCTION

I

N recent years, unmanned aerial vehicles (UAVs), such as hypersonic aircraft, highly maneuverable technology (HiMAT) vehicle and agile missile, have received considerable amounts of attention[1−4] . The control of UAVs is recognized to be extraordinarily challenging, due to the high nonlinearity and complexity of their motion equations, which stem from the strong couplings between dramatic parameter variations within the full-envelope, wide velocity range and the large unknown environmental disturbances[5−8] . A traditional approach to handle the nonlinearity of fullenvelope UAVs is by means of gain-scheduling[9−10] . Its design principle is to design local controllers for each member of a set of operating points that cover the whole flight envelope firstly and then to obtain a global controller in flight by using interpolation method according to the current value of the scheduling parameters[11−12] . Manuscript received October 10, 2013; accepted May 15, 2014. This work was supported by National Natural Science Foundation of China (61273083, 61074027). Recommended by Associate Editor Bin Xian. Citation: Zhaolei Wang, Qing Wang, Chaoyang Dong. Asynchronous H∞ control for unmanned aerial vehicles: switched polytopic system approach. IEEE/CAA Journal of Automatica Sinica, 2015, 2(2): 207−216 Zhaolei Wang is with the School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China, and also with Beijing Aerospace Automatic Control Institute, Beijing 100854, China (e-mail: [email protected]). Qing Wang is with the School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China (e-mail: [email protected]). Chaoyang Dong is with the School of Aeronautical Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China (e-mail: [email protected]).

However, some poorly developed but important theoretical issues still exist in the gain-scheduled method, such as no guarantees on the robustness, performance, or even nominal stability of the overall gain-scheduled design, especially when the flight envelope becomes much wider[13−14] . Therefore, it is quite attractive to improve the gain-scheduled method by combining other approach which has appeared in the extant literatures[15−16] . Considering the advantages of switched system theory and the computational complexity of the switched linear parameter-varying (LPV) approach[17−20] , an improved gainscheduled controller design approach is proposed in this work based on the switched polytopic systems which have been paid much attention in recent years[21−26] . To describe the full-envelope vehicle0 s nonlinear model, a locally overlapped switched polytopic system is established evolving on the locally overlapped switching laws which are constructed by considering that the flight dynamics can only switch from one region to another adjacent region, where each polytopic subsystem presents the system dynamics in a part of the full flight envelope, and its vertices describe the linear dynamics on given characteristic operating points within this part of flight envelope. For each polytopic subsystem, a gain-scheduled subcontroller is obtained by interpolating between the controllers on vertices according to the weighted coefficients, and the switched parameter-dependent controllers of the fullenvelope are synthesized using these polytopic subsystems0 gain-scheduled subcontrollers. On the other hand, the implicit assumption that the measurements of the system states always contain useful signals is not actually possible[27−31] . It can be proved that many measurements consist of noise alone, or involve data missing which can be caused by many practical reasons, such as, measurement failure, intermittent sensor failure, imperfect communication, some of the data being disturbed or coming from a very noisy environment[27] . In many engineering applications, such as flight control[29−31] , due to sensor temporal failure or transmission disturbance at certain time instants, the system measurement may only contain noise, which means the real signal is missing[28] . Moreover, because switching signals and weighted coefficients of the switched polytopic systems are dependent on the current Mach number and altitude which are measured online in flight, missing measurements will lead to the asynchronous switching problem which means there are updating lags between the system mode switching and the controller switching[32−33] . To the best of the authors0 knowledge, the UAVs0 control with missing measurements based on switched polytopic system approach is challenging, and has not been fully investigated yet. This paper investigates the H∞ controller design for UAVs

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IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 2, NO. 2, APRIL 2015

with missing measurements. Firstly, to reduce designing conservatisms and solving complexities, a locally overlapped switched polytopic system is established to describe the fullenvelope flight dynamics evolving on the locally overlapped switching laws with the aid of the proposition of a locally overlapped region division method. Secondly, to overcome the effects of the asynchronous switching problems induced by the missing measurements, an asynchronous H∞ control method is proposed to guarantee that the system is stable and satisfies a prescribed attenuation level against the external disturbances. Then, the stability and l2 performance analysis is performed by combining the switched parameter-dependent Lyapunov function method and the average dwell time method, and the sufficient existing conditions of the switched parameterdependent H∞ controller are established in the form of LMIs. Finally, a full-envelope HiMAT flight example is given to demonstrate the effectiveness of the proposed approach.

II. M ODEL DESCRIPTION The UAV investigated in this paper is a HiMAT vehicle, a full-envelope and open-loop unstable UAV, which is sponsored by NASA and the U.S. Air Force. It is studied to incorporate technological advances in many fields, such as a close-coupled canard configuration, aero elastic tailing with composite structures, relaxed static stability, and advanced transonic aerodynamics[34−35] . To derive the linear system model from nonlinear dynamics of the full-envelope vehicle[14−16] , Jacobian linearization is performed on the 20 characteristic operating points within the full envelope presented in [34], which are chosen based on the altitude H and Mach number M a, as depicted in Fig. 1. As each linear model of the chosen operating point can describe the dynamics in its vicinity, the 20 models can cover the nonlinear dynamic behaviors of the HiMAT vehicle within the full envelope[34] .

Fig. 1.

Flight envelope and the operating points.

Considering that the longitudinal motion stability and maneuverability primarily depend on the short period motion, with the aid of 20 continuous-time longitudinal linear models in [34], the longitudinal short period model of the i-th oper-

ating point can be modeled as the discrete time model (1) by setting the system0 s sampling period to be T , which is ½ x (k + 1) = Aix (k) + Biu (k) + Bd,id (k), (1) y (k) = Cix (k) + Dd,id (k). Here, x = (α, q)T is the state vector, and α, q denote angle of attack, pitch rate, respectively. u = (ξe , ξv , ξc )T is the control input, ξe , ξv , ξc denote elevator, elevon and canard deflection, respectively. y is the output, and d ∈ L2 [0, ∞) is the unknown external disturbances. The real system matrices Ai , Bi , Ci , Bd,i , Dd,i for the i-th characteristic operating point, i ∈ Ω = {1, 2, · · · , 20}, are of appropriate dimensions with the index set Ω denoting the collection of all characteristic operating points within the envelope, and the trim condition for every operating point is illustrated in Table I. TABLE I THE 20 OPERATING POINTS OF HIMAT VEHICLE Operating point

Mach

Altitude (m)

Angle of attack ( ˚ )

1

0.29

762.5

3.18

2

0.4

762.5

1.49

3

0.6

762.5

0.69

4

0.5

1525

1.02

5

0.4

3050

2.17

6

0.7

3050

0.73

7

0.4

6100

3.60

8

0.6

6100

1.48

9

0.7

6100

1.08

10

1.4

6100

2.06

11

0.7

7625

1.38

12

0.9

7625

1.19

13

1.2

7625

2.15

14

0.9

9150

1.36

15

0.8

10675

1.77

16

0.7

12200

2.98

17

0.9

12200

1.96

18

1.2

12200

2.23

19

1.4

12200

2.03

20

0.68

13725

4.11

To reduce designing conservatisms and solving complexities, the large-scale flight envelope shown in Fig. 1 is divided into N regions referring to the switched LPV approach. The index set Ω is accordingly divided into N index subsets which are defined as Ω1 , Ω2 , · · · , ΩN to represent the collection of the characteristic operating points within each region. In our work, a polytopic system is adopted to represent the system dynamics in each region, and its vertices are those characteristic operating points within this region. Inspired by the gain-scheduled method, the system dynamics of current operating point within this region is obtained by interpolating among the system dynamics on vertices. Considering the flight dynamics can only switch from the current region to another adjacent region in the full-envelope flight, a locally overlapped switched polytopic system is furthermore established as follows: ½ ak )x x(k)+Bσ(k) (a ak )u u(k)+Bd,σ(k) (a ak )dd(k), x (k + 1) = Aσ(k) (a ak )x x(k)+Dd,σ(k) (a ak )dd(k), y (k) = Cσ(k) (a (2)

WANG et al.: ASYNCHRONOUS H∞ CONTROL FOR UNMANNED AERIAL VEHICLES: SWITCHED POLYTOPIC SYSTEM APPROACH

where k ∈ N is time instant, and the switching laws σ(k) ∈ Γ stands for variation of the flight region according to k, where the set Γ = {1, 2, · · · , N } is the collection of the flight region indices, with N being a finite integer denoting the number of flight regions. The system matrices of (2) satisfy that ¸ · ak ) Bm (a ak ) Bd,m (a ak ) Am (a = ak ) ak ) Cm (a 0 Dd,m (a X i∈Ωm

· ai,k

Ai Ci

Bi 0

Bd,i Dd,i

¸ ,

(3)

P where i∈Ωm ai,k = 1, ai,k ≥ 0, ∀σ(k) = m ∈ Γ. Considering the locally overlapped property of σ(k) within the full-envelope flight, the flight regions must be divided under the following S conditions: Condition 1. m∈Γ Ωm = Ω. Condition 2. Ωσ(k) ∩ Ωσ(k+1) 6= ∅, ∀k ∈ N. Taking the flight trajectory depicted in Fig. 1 for example, the flight envelope can be partitioned into three flight regions, and one has   Γ = {1, 2, 3} , Ω1 = {1,2,3,4,5,6,7} , Ω2 = {6,7,8,9,11,12,16} , (4)  Ω = {10,12,13,14,15,16,17,18,19,20} . 3 Remark 1. Condition 1 guarantees the index subsets must involve all the characteristic operating points within the flight envelope. Condition 2 guarantees the switching occurs between two adjacent regions that share common characteristic operating points. Thus, the switching on the boundary of two adjacent regions will not induce non-smooth change of the weighted coefficient a k , provided a k is only calculated by using these common vertices of the adjacent polytopic subsystems. For example, the controller gains on the boundary of the Region 1 and Region 2 are only interpolated by the gains on characteristic operating Points 6 and 7. III. P ROBLEM FORMULATION In this section, a switched parameter-dependent state feedback H∞ controller is designed to guarantee that the system is stable and satisfies a prescribed performance level. Firstly, the following assumptions are made. Assumption 1. System state x is measurable, and the missing measurements of x are described by a Bernoulli distributed stochastic variable θ(k) ∈ {0, 1}, where Pr{θ(k) = 1} = ρ, Pr{θ(k) = 0} = 1−ρ. θ(k) = 1 stands for the normal case, while θ(k) = 0 stands for the missing measurement case, accordingly[27] . Thus, the state feedback controller0 s input z (k) is given by x(k) + (1 − θ(k))zz (k − 1). z (k) = θ(k)x

(5)

With the locally overlapped switching laws ∀σ(k) ∈ Γ, the discrete state feedback switched parameter-dependent controller (6) is designed as ak )zz (k), u (k) = Kσ(k) (a

(6)

ak ), ∀σ(k) = m ∈ Γ of polytopic where subcontroller Km (a subsystem m is synthesized as follows: X X ak ) = ai,k = 1, ai,k ≥ 0, (7) ai,k Ki , Km (a i∈Ωm

i∈Ωm

209

where Ki is the controller gain to be determined for the i-th characteristic operating point. Remark 2. Because the partition information of the flight regions is stored in controller unit previously, σ(k) can be determined online by judging which region the current Mach number and altitude belong to, and a k could also be calculated referring to the finite element theory[23] . Remark 3. Considering the measured Mach number and altitude data will be missing at some instants, there will be updating lags for σ(k) and a k , which will further induce asynchronous switching phenomena between the system modes and controllers. Moreover, the asynchronous switching here will be more complex than that in [32−33], because neither σ(k) nor a k can be obtained in time. Due to the asynchronous switching, the Lyapunov function value may moderately increase even during the active time of certain subsystems except at the switching instants, but the increase rate must be bounded[32] . For concise notation, let kv and kv+1 , v ∈ N denote the starting time and ending time of some active subsystem, while T↑ (kv , kv+1 ) and T↓ (kv , kv+1 ) represent the unions value of the dispersed intervals during which Lyapunov function is increasing and decreasing within the interval, where T↓ (kv+1 − kv ) and T↑ (kv+1 − kv ) denote the length of T↑ (kv , kv+1 ) and T↓ (kv , kv+1 ), respectively[33] . As a result of the updating lags of σ(k), the Lyapunov function value of the polytopic subsystem may moderately increase, and the corresponding T↑ (kv , kv+1 ) will be only the interval close to the switching instants. As T↑ (kv+1 − kv ) may vary at different instants, without loss of generality, the maximal lag of the Mach number and altitude data is assumed to be δm T , where δm is known a priori here[33] . Thus, the controller in (6) will be transformed to (8) under the asynchronous switching, i.e., ak−δm )zz (k). u (k) = Kσ(k−δm ) (a

(8)

¡ ¢T Denote ξ (k) = x T (k), z T (k − 1) , then, by combining (2), (5)-(8), the augmented locally overlapped switched polytopic stochastic system (9) is given as ½

¯ Aˆ1,n (a ˆn (a ak¯ )ξξ (k)+θ(k) ak¯ )ξξ (k)+B ak¯ )dd(k), ξ (k + 1) = Aˆn (a ˆ ˆ ak¯ )ξξ (k)+ Dn (a ak¯ )dd(k), ∀k ∈ [kv , kv + δm ) , y (k) = Cn (a ½ ¯ A¯1,m (a ¯m (a ak¯ )dd(k), ak¯ )ξξ (k)+θ(k) ak¯ )ξξ (k)+B ξ (k + 1) = A¯m (a ¯ ¯ ak¯ )dd(k), ∀k ∈ [kv + δm , kv+1 ) , ak¯ )ξξ (k)+ Dm (a y (k) = Cm (a (9) ¯ ¯ = where k¯ = k − δm , θ(k) = θ(k) − ρ, ∀{σ(kv ) = m, σ(k) n} ∈ Γ × Γ, m 6= n, Ωm ∩ Ωn 6= ∅, and one has ª © ª © ¯ ¯ θ(k) ¯ E θ(k) = 0, E θ(k) = ρ(1 − ρ), · ¸ ˆn (a ak¯ ) Aˆ1,n (a ak¯ ) B ak¯ ) Aˆn (a = ˆ n (a ak¯ ) ak¯ ) Cˆn (a 0 D · ¸ X X Aˆil Aˆ1,il Bd,i ai,k al,k¯ , Ci 0 Dd,i i∈Ωm l∈Ωn · ¸ ¯m (a ak¯ ) A¯1,m (a ak¯ ) B ak¯ ) A¯m (a = ¯ m (a ak¯ ) ak¯ ) C¯m (a 0 D

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IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 2, NO. 2, APRIL 2015

X

al,k¯

X i∈Ωm

l∈Ωm

·

· ai,k

Aˆil Ci

Aˆ1,il 0

Ai + ρBi Kl (1 − ρ)Bi Kl ρ (1 − ρ) · ¸ Bi Kl −Bi Kl = . I −I

Aˆil = Aˆ1,il

¸

Bd,i Dd,i

¸ ,

,

Then, the H∞ controller design for the HiMAT vehicle with missing measurements is summarized as follows. Considering the locally overlapped switched polytopic system (2) with missing measurements described in (5), for a positive constant γ, determine the matrices Ki (i ∈ Ω) of asynchronous controller in (8), such that system (9) is stable and has a weighted l2 -gain performance (10) under zero initial conditions, that is, (∞ ) ∞ X X s T y y E (1−λ) (s)y (s) ≤ γ 2dT (s)dd(s), (10) s=0

s=0

IV. M AIN RESULTS In this section, the sufficient existing conditions of the desired controller will be derived in the form of LMIs. Firstly, we present the following definition and lemmas on the stability and l2 -gain analysis here for later use. Definition 1[32] . For switching signal σ(·) and any k ≥ 1, let Nσ [0, k) be the switching numbers of σ(·) over the interval [0, k). If for any given N0 ≥ 0 and τa > 0, we have Nσ [0, k) ≤ N0 + k/τa , then τa and N0 are called average dwell time and the chatter bound, respectively. Without loss of generality, we choose N0 = 0 for convenience as commonly used in the literature. Lemma 1. Consider system (9), and let 0 < λ < 1, β ≥ 0 and µ ≥ 1 be given constants. Suppose that there exist C 1 functions Vσ(k) , σ(k) = m ∈ Γ, and two class κ∞ functions κ1 , κ2 such that

½ E (∆Vm (ξξ (k))) ≤

−λVm (ξξ (k)), βVm (ξξ (k)),

If average dwell time τa satisfies (14), define κ := . ˜ − ln λ (δm ln ~ + ln µ), then −δ

˜ ln λ

˜

λ ˜ δm /τa µ1/τa < λ~ ˜ δm lnm~+ln µ µ δm −lnln~+ln µ = λ~ . ¡ δm ¢ κ ¡ δm ln ~+ln µ ¢κ ˜ ~ µ =λ ˜ e ˜ λ ˜ = 1. λ =λ

(16)

¡

where 0 < λ < 1, and γ > 0. Remark 4. In the presence of missing measurements and the locally overlapped switching laws, a Bernoulli distributed stochastic variable is introduced to describe the missing measurements, and an augmented locally overlapped switching polytopic stochastic system (9) is further established. Obviously, the asynchronous H∞ control problem for (9) in our work is more complicated than the existent H∞ control problem for the switched polytopic system.

κ1 (kξξ (k)k) ≤ Vm (ξξ (k)) ≤ κ2 (kξξ (k)k) ,

Proof. Referring to the method in [33], ∀k ∈ [kv , kv+1 ), it holds from (12) that ¡ ¢ ¡ ¢ ˜ T↓ (k−kv ) β˜T↑ (k−kv ) E Vσ(k ) (ξξ (kv )) ≤ E Vσ(k) (ξξ (k)) ≤ λ v ¡ ¢ ˜ (k−kv ) ~δm E Vσ(k ) (ξξ (kv )) ≤ λ v ¡ ¢ ˜ (k−kv ) ~δm µE Vσ(k −1) (ξξ (kv )) ≤ · · · ≤ λ v ¢ ¡ ˜ (k−0) ~δm µ Nσ [0,k) Vσ(0) (ξξ (0)) ≤ λ ³ ´k ˜ δm /τa µ1/τa Vσ(0) (ξξ (0)). µN0 ~N0 δm λ~ (15)

(11)

∀k ∈ [kv + δm , kv+1 ) , ∀k ∈ [kv , kv + δm ) , (12)

Vm (ξξ (k)) ≤ µVn (ξξ (k)), ∀(m, n) ∈ Γ × Γ, m 6= n. (13) Thus, the system is globally uniformly asymptotically stable in the mean (GUAS-M) for any switching signal with average dwell time τa satisfying . ˜ + ln µ} ln λ, ˜ τa > τa∗ = −{δm [ln β˜ − ln λ] (14) . ˜ = 1 − λ, β˜ = 1 + β, and ~ = β˜ λ. ˜ where λ

¢ Therefore, we can get that E Vσ(k) (ξξ (k)) → 0 as k → ∞, and system (9) is GUAS-M with the aid of (11) by using the same method as in [36]. ¤ Lemma 2. Consider system (9), and let 0 < λ < 1, β ≥ 0, µ ≥ 1 and γm > 0 (∀m ∈ Γ) be given constants. Suppose that there exist C 1 functions Vσ(k) , σ(k) = m ∈ Γ, such that (13) is satisfied and E (∆Vm (ξξ (k))) ≤ ½ −λVm (ξξ (k)) − φ(k), ∀k ∈ [kv + δm , kv+1 ) , βVm (ξξ (k)) − φ(k), ∀k ∈ [kv , kv + δm ) .

(17)

Then the systemp has a weighted l2 -gain no greater than N0 δm N0 γ } like (10), where φ(k) = γ ¡= maxm∈Γ m ¢{ µ ~ E y T (k)yy (k) − γmd T (k)dd(k). x(k)) ¢and ∆Vm (x x(k)) with ¡Proof. By ¢replacing ¡ Vm (x E Vσ(k) (ξξ (k)) and E ∆Vσ(k) (ξξ (k)) in [33], Lemma 2 for the switched polytopic system with missing measurement can be directly obtained with the aid of the theoretical analysis for the stochastic switched system in [37], and the proofs are omitted here. ¤ A. Stability and l2 -Gain Performance Analysis pTheorem 1. Given scalars 0 < λ < 1, β ≥ 0, ϑ = ρ(1 − ρ) and γm > 0, ∀m ∈ Γ, if there exist matrices Pi > 0, i ∈ Ωm , ∀m ∈ Γ, such that the following LMIs (18) and (19) hold for ∀ (m, n) ∈ Γ × Γ:   Pj 0 0 Pj Aˆil Pj Bd,i ∗ P 0 ϑPj Aˆ1,il 0  j    Υi,j,l= ∗ ∗ −I Ci Dd,i   < 0, ∗ ∗ ∗ −(1 − λ)Pi 0  2 ∗ ∗ ∗ ∗ −γm I ∀i, j, l ∈ Ωm , (18)   Pj 0 0 Pj Aˆil Pj Bd,i ∗ P 0 ϑPj Aˆ1,il 0  j   ¯  Υi,j,l =  ∗ ∗ −I Ci Dd,i  < 0, ∗ ∗ ∗ −(1 + β)Pi 0  2 ∗ ∗ ∗ ∗ −γm I ∀i, j ∈ Ωm , l ∈ Ωn , Ωm ∩ Ωn 6= ∅, m 6= n. (19)

WANG et al.: ASYNCHRONOUS H∞ CONTROL FOR UNMANNED AERIAL VEHICLES: SWITCHED POLYTOPIC SYSTEM APPROACH

Then system (9) is GUAS-M when d (k) = 0 for any switching signal with average dwell time τa satisfying . ˜ ˜ τa > τa∗ = −{δm [ln β˜ − ln λ]} ln λ, (20) and has the weighted l2 -gain performance index (10) under ˜= zero initial conditions, where γ = max{γm }, ∀m ∈ Γ, λ 1 − λ, and β˜ = 1 + β. Proof. Firstly, we choose the following Lyapunov-like function (21) based on the switched parameter-dependent Lyapunov function method[38] : ak , ξ (k)) = ξ T (k)Pm (a ak )ξξ (k), Vm (ξξ (k)) = Vm (a

(21)

where¡ the weighting matrices for the polytopic subsys¢ ¯m (a ¯1,m (a ¯m (a ¯ m (a a a a a ak ) = tem A ), A ), C ), D ) are P (a k k k k m P i∈Ωm ai,k Pi , ∀σ(k) = m ∈ Γ. From Remark 1, it is clear that switchings on the boundary of two adjacent regions will not induce non-smooth change of the weighted coefficient a k , because a k is calculated by using only these common vertices of the adjacent polytopic subsystems. So, when the switching occurs, we have T ak , ξ (k ak Vm (a )ξξ (kv ) = v v )) = ξ (kv )Pm (a X ξ T (kv )  as,kv Ps  ξ (kv ), s∈{Ωm ∩Ωn }

T ak , ξ (k a kv  Vn (a )ξξ (kv ) = v )) = ξ (kv )Pn (a X ξ T (kv )  as,kv Ps  ξ (kv ).

(22)

s∈{Ωm ∩Ωn }

Thus, we get µ = 1 at the switching instant from (13), and it can be derived from (14) that the dwell time τa of system (9) must satisfy (20). Secondly, we prove the stability of system (9). To this end, assume that d (k) = 0. For any non-zero ξ (k), ∀k ∈ [kv + δm , kv+1 ), and any switching signal σ(k) = m ∈ Γ, one has (23) holds. Similarly, for any non-zero ξ (k), ∀k ∈ [kªv , kv + δm ), and © ¯ = n ∈ Γ × Γ, Ωm ∩ any switching signal σ(k) = m, σ(k) Ωn 6= ∅, m 6= n, one has (24) holds. As Pi ≥ 0, i ∈ Ωm , ∀m ∈ Γ, (25) and (26) will be obtained by performing a congruence © ª transformation to (18) and (19) by diag Pj−T , Pj−T , I, I, I and the Schur complement. ¯ i,j,l can imply Noticing the (1,1) block of Θi,j,l and Θ 2 ˆT ˆ ˆ Λi,j,l = AˆT il Pj Ail + ϑ A1,il Pj A1,il − (1 − λ)Pi < 0, T 2 ˆT ˆ ˆ ˆ ˆ Λi,j,l = Ail Pj Ail + ϑ A1,il Pj A1,il − (1 + β)Pi < 0, (27)

211

we can have[38] ak , a k+1 )= Λ1,m (a

P

l∈Ωm

ak , a k+1 )= Λ2,m (a

P

l∈Ωn

al,k¯ al,k¯

P

ai,k

i∈Ωm

P

i∈Ωm

P

aj,k+1 (Λi,j,l ) 0 and Ui , i ∈ Ωm , ∀m ∈ Γ, such that 0 S¯i the LMIs hold for ∀ (m, n) ∈ Γ × Γ, i.e.,  −Sj  ∗   ∗  ∗ ∗

0 −Sj ∗ ∗ ∗

0 0 −I ∗ ∗

 φ¯14 Bd,i ¯ φ24 0   Ci Sl Dd,i  < 0, −(1 − λ)(Si − Sl − SlT ) 0  2 ∗ −γm I ∀i, j, l ∈ Ωm , (34)

 −Sj  ∗   ∗  ∗ ∗

 0 0 φ¯14 Bd,i ¯ −Sj 0 φ24 0   ∗ −I Ci Sl Dd,i  < 0, T ∗ ∗ −(1 + β)(Si − Sl − Sl ) 0  2 ∗ ∗ ∗ −γm I ∀i, j ∈ Ωm , l ∈ Ωn , Ωm ∩ Ωn 6= ∅, m 6= n. (35)

Then under the asynchronous delay δm , system (9) is GUASM for any switching signals with τa satisfying (20), and has a weighted l2 -gain performance index γ = max{γm }, ∀m ∈ Γ. Moreover, if a feasible solution exists, the desired controller gains in (8) are given by

Ki = Ui S¯i−1 , i ∈ Ωm , ∀m, ∈ Γ,

(36)

WANG et al.: ASYNCHRONOUS H∞ CONTROL FOR UNMANNED AERIAL VEHICLES: SWITCHED POLYTOPIC SYSTEM APPROACH

where

·

φ¯14 φ¯24

Ai S¯l + ρBi Ul (1 − ρ)Bi Ul = ¯ ρS¯l ¸(1 − ρ)Sl · Bi Ul −Bi Ul . =ϑ S¯l −S¯l

¸ ,

Proof. Suppose that there exist matrices Si and Ui , i ∈ Ωm , ∀m ∈ Γ satisfying (34) and (35). Now suppose · matrices ¸ P¯i 0 Pi in Theorem 1 have the form of Pi := , 0 P¯i · ¸ S¯i 0 let Si = , S¯i := P¯i−1 , and define Ui := 0 S¯i Ki S¯i , i ∈ Ωm , ∀m ∈ Γ. Considering the fact that T (Si − Sl ) Sl−1 (Si − Sl ) ≥ 0, we have Si − Sl − SlT ≥ T −1 −Sl Si Sl . Then replace the (4,4) block of (34) and (35) T −1 by −S Si Sl , and do ª a congruence transformation via © l−T diag Sj , Sj−T , I, Sl−T , I , which will lead to (18) and (19). Meanwhile, the switched parameter-dependent controller gains are given by (36) according to the definition that Ui := Ki S¯i , i ∈ Ωm , ∀m ∈ Γ. ¤ Remark 5. Referring to Theorem 2, a similar conclusion can be obtained for the input-to-state stability of the homologous systems which have reference inputs. This corollary can be directly obtained from Lemma 1 in [18] which will be validated later in the numerical example section. V. N UMERICAL EXAMPLE In this section, a numerical example is provided to illustrate the effectiveness of the proposed method. Suppose that the sampling period is T = 0.025 s, then corresponding system matrices Ai , Bi in (1) can be obtained with the aid of the continuous-time system matrix parameters in [34]. Moreover, other system matrices are given below, where i ∈ Ωm , ∀m ∈ Γ: £ ¤ £ ¤T Ci = 1 0 , Bd,i = 0.01 0.01 , Dd,i = 0.05. The external disturbance d is supposed to be a random sequence uniformly distributed over [−0.1, 0.1]. Moreover, it is assumed that E{θ(k)} = ρ = 0.95, which means that the probability of the measured system state x being successfully transmitted to the controller is 0.95, and Fig. 2 shows the actual response of θ(k).

Fig. 2.

Missing measurement mode θ(k).

The performance of the proposed switched parameterdependent controller for the full-envelope HiMAT vehicle is

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validated by a large-scale flight trajectory, i.e., 2-4-5-7-8-9-1214-15-17, which is depicted in Fig. 1. The variation of altitude and Mach number are depicted in Fig. 3.

Fig. 3.

Variation of the altitude and Mach number.

The flight envelope can be partitioned into three regions which are given in (4), and the flight trajectory includes two switchings between the three polytopic subsystems. First switching occurs when the HiMAT vehicle enters characteristic operating Point 7, and the second switching occurs when the HiMAT vehicle enters characteristic operating Point 12 which can give the fact that the average dwell time τa = 1 600 (40 s) from Definition 1. Supposed that the maximal lag δm = 5, it can be calculated that τa∗ = 5 (0.1250 s) which implies the constraint condition in (20) can be easily satisfied. To analyze the input-to-state stability[18] , the controller in (6) is modified into the following form: ak ) (zz (k) − r (k)) , u (k) = Kσ(k) (a

(37)

where r (k) = [αc qc ]T is the command input, including angle of attack command and pitch rate command. By setting γm as the optimization variable at the same time, the YALMIP toolbox in Matlab is used to solve the optimization problem in Theorem 2, where λ = 0.01 and β = 0.01 are given. Then, we can get γ ∗ = 1.9237, and the controller gains can be obtained, which are shown as follows:     3.0284 0.3215 3.4061 0.3054 K2 =  2.0639 0.2191  , K4 =  2.2868 0.2050  ,  -0.3264 -0.0347   -0.0136 -0.0012  3.8160 0.4595 4.7516 0.6742 K5 =  2.6204 0.3156  , K7 =  3.2841 0.4660  ,   -0.4047 -0.0574  -0.3823 -0.0460 1.6363 0.1859 2.4448 0.2869 K8 =  1.5827 0.1857  , K9 =  1.1127 0.1264  , 0.0504 0.0057     0.2334 0.0274 0.9897 0.1341 0.8494 0.1017 K12 =  0.5943 0.0712  , K14 =  0.7026 0.0952  ,  -0.0271 -0.0037   -0.0202 -0.0024  1.5329 0.2341 1.9732 0.2757 K15 =  1.4165 0.1979  , K17 =  1.1137 0.1701  . -0.0482 -0.0074 -0.1031 -0.0144 The controller gains for the characteristic operating points are given above, so the gain-scheduled subcontroller for each polytopic subsystem is interpolated by (7). In this work, the gain-scheduled subcontrollers are interpolated in triangular regions, namely, three characteristic operating points, which are the closest to the current one, are chosen.

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IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 2, NO. 2, APRIL 2015

For example, for the triangular regions which are constructed by the characteristic operating Points 2, 4 and 5 in Region 1, the gain-scheduled subcontroller is interpolated by three gains K2 , K4 and K5 . Given the flight condition in this flight region with altitude and Mach number (ha , Ma ) at instant k, then the interpolated gain-scheduled subcontroller gain can be calculated as ak ) = a2,k K2 + a4,k K4 + a5,k K5 , K1 (a and the weighted coefficient a k satisfies

Fig. 6.

Elevator deflection.

Fig. 7.

Elevon deflection.

Fig. 8.

Canard deflection.

  a2,k + a4,k + a5,k = 1, a2,k h2 + a4,k h4 + a5,k h5 = ha ,  a M +a M +a M =M , 2,k a,2 4,k a,4 5,k a,5 a where hi , Ma,i , i ∈ Ω are altitude and Mach number for the i-th characteristic operating point. Fig. 4 demonstrates the response of angle of attack, and pitch rate is shown in Fig. 5 where the pitch rate command is set to be zero. We can conclude from Figs. 4 and 5 that the tracking performance of angle of attack is acceptable in the full flight envelope, and the pitch rate response is satisfying.

Fig. 4.

Angle of attack response.

VI. C ONCLUSION

Fig. 5.

Pitch rate response.

The control surface deflections are shown in Figs. 6 ∼ 8, which are achievable and practical. From the simulation results above, it can be concluded that the HiMAT vehicle is stable in the full-envelope flight with robustness against the external disturbances, and can also restrain the effect of asynchronous switching to guarantee a desired performance.

This study proposed an improved gain-scheduled controller design approach for the full-envelope UAVs with missing measurements and external disturbances. A locally overlapped switched polytopic system is established to describe flight dynamics within full flight envelope, and an asynchronous robust H∞ synthesis approach is utilized to obtain a prescribed disturbance attenuation level. The proposed approach could also be expanded to other applications where the locally overlapped switched polytopic system model could be established, and the missing measurements and asynchronous switching phenomena exist. Future works will analyze the impact of system states that cannot be measured, and the output feedback control problem will be concerned. R EFERENCES [1] Dierks T, Jagannathan S. Output feedback control of a quadrotor UAV using neural networks. IEEE Transactions on Neural Networks, 2010, 21(1): 50−66

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Zhaolei Wang received his bachelor degree from Beijing Institute of Technology, Beijing, China in 2009. He received his Ph. D. degree in navigation guidance and control from School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing, China, in 2015. He is an engineer of Beijing Aerospace Automatic Control Institute, Beijing, China. His research interests include networked control system, fault diagnosis, switched system theory and its application on flight vehicle. Corresponding author of

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this paper.

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Qing Wang received her master and Ph. D. degrees in flight vehicle control guidance and simulation from Northwestern Polytechnical University, in 1993 and 1996, respectively. She worked as a post-doctoral researcher in the Department of Automatic Control, Beijing University of Aeronautics and Astronautics from 1996 to 1998. She worked in China Academy of Space Technology in 1999, and visited Moscow College of Aeronautics as a visitor during this period. She is now a professor in the Department of Guidance, Navigation and Control at the School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics. Her research interests include guidance and control of aerospace vehicles, intelligent flight control system.

IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 2, NO. 2, APRIL 2015

Chaoyang Dong received his master degree from Northwestern Polytechnical University in 1992, and Ph. D. degree in guidance, navigation and control from Beijing University of Aeronautics and Astronautics. From 2002 to 2008, he was an associate professor in the Department of Automatic Control, Beijing University of Aeronautics and Astronautics. He is now a professor of flight mechanics at the School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics. His research interests include kinetic analysis of flexible vehicle and control of the complex dynamics system.