Embedding Norm-Bounded Model Predictive Control

Embedding Norm-Bounded Model Predictive Control Allocation strategy for the High Altitude Performance Demonstrator (HAPD) Aircraft Giuseppe Franz`e, Massimiliano Mattei and Valerio Scordamaglia Abstract— In this paper the implementation of a robust Model Predictive Control (MPC) algorithm for constrained uncertain discrete-time linear systems subject to norm-bounded Linear Fractional Representation (LFR) model uncertainties is used to deal with the control allocation problem of an High Altitude Performance Demonstrator (HAPD) unmanned aircraft with redundancy control surfaces. Specifically, the HAPD nonlinear model is described by means of LFR differential inclusions achieved via embedding arguments while the surface deflection limitations (amplitudes and rates of variations) are formulated in terms of convex constraints. As a consequence, the overall control problem can be recast in terms of Linear Matrix Inequalities (LMIs) that are affordable from a computational point of view. Numerical simulations show the effectiveness of the proposed strategy.

I. I NTRODUCTION Modern advanced aircraft are characterized by the presence of more control effectors than controlled variables. The implication of this characteristic is that the control system possesses a certain degree of redundancy and can in principle achieve multiple control objectives and maintain certain performance when control authority is limited by failures. Control algorithm design for systems with effectors redundancy is challenging since multiple combinations of the available control effectors can generate the same desired control. In addition to this, actuator constraints (e.g. limited deflection) can be more efficiently managed. In order to systematically deal with such control system challenges, one may decompose the control problem in two parts: a controller that commands a desired virtual control and a control allocation module that maps the virtual control input into the redundant actuators. Since there are more degree of freedom available than virtual control variables, the available degrees of freedom can be used to satisfy actuator constraints and to meet secondary objectives such as fault tolerance, power consumption minimization and actuator wear minimization. In general, the control allocation problem can be formulated as an optimization problem with a main objective to minimize the difference between the resulting control effect and the desired virtual control command. Due to overactuation and coupling of control surface effects, it is difficult Giuseppe Franz`e is with DIMES, Universit´a degli Studi della Calabria, Via Pietro Bucci, Cubo 42-C, Rende (CS), 87036, ITALY

[email protected] Massimiliano Mattei is with DIII - Seconda Universit`a degli Studi di Napoli, Real Casa dell’Annunziata Via Roma, 29, Aversa (CE), 81031, ITALY, [email protected] Valerio Scordamaglia is with DIIES, Universit`a degli Studi di Reggio Calabria, Reggio Calabria, Via Graziella, loc. Feo di Vito, 89060, ITALY

[email protected]

to determine an appropriate method of how to translate a flight control command into a control surface command. In addition, rate and position limits of the control surfaces must be considered in order to achieve a realistic solution. Some of the simplest control allocation techniques are explicit ganging, pseudo control, pseudo inverse, pseudo-inverseredistribution [11] and daisy chaining. Unfortunately, each suffers from difficulty in guaranteeing that rate and position limits will not be violated and some can be difficult to apply due to the need to derive a control mixing law a priori. Another control allocation method, called direct allocation (see [1], [17],[18]) finds the control vector that results in the best approximation of the command vector in a given direction. Unconstrained least squares control allocation methods, that account for rate and position limits, through the use of penalty functions, have also been developed [2], while one of the first instances of linear programming based control allocators was proposed in [3]. Moving from these considerations, in this paper we propose to explore the chance to use the Model Predictive Control algorithm, developed in [14] for norm-bounded uncertain systems, for solving the control allocation problem of over-actuated systems. In particular the HAPD aircraft, whose the nonlinear model can be embedded into a linear norm-bounded differential inclusion, will be analyzed in detail when the constrained tracking problem of time-varying reference trajectories is considered. Therefore, the focus is to show that this robust MPC scheme can be efficiently adapted to take care of tight control allocation requirements. To this end, we have proposed a control architecture consisting of two modules: •

•

a reference trajectory module that allows to determine a command input sequence compatible with the prescribed state reference trajectory; a tracking MPC controller that makes use of finite input and state sequences and is obtained from the basic scheme of [14] by resorting to standard system theory arguments.

The paper is organized as follows: the mathematical model of the High Altitude Performance Demonstrator is detailed in Section II. In Section III the basic MPC control scheme for norm-bounded uncertain linear systems is summarized. The problem is stated in Section IV and there the proposed scheme is developed and discussed. Finally, a set of simulation are reported in Section V and some conclusions end the paper.

II. HAPD

MATHEMATICAL MODEL

The High Altitude Performance Demonstrator (HAPD) is an over-actuated aircraft, see Fig. 1. In particular, it has three pairs of elevators (inboard, middle, and outboard), two pairs of ailerons (inboard and outboard), two rudders (upper and lower rudders) and 8 independent propellers. Note that the HAPD is an original project developed by C.I.R.A. [12] which is at an early stage of the controllability studies.

Fig. 1.

HAPD model: twelve control surfaces and eight available engines

A mathematical model, which takes into account flexibility phenomena, was developed under the following hypotheses: 1) the inertia matrix I is independent from the aircraft elastic deformations; 2) the linear elastic theory can be used to model the aeroelastic dynamics; 3) aero-elastic modes are quasi-stationary; 4) inertial effects due to actuator forces are negligible. Under such assumptions, the classical 6DoF mathematical model and the Eulers angles dynamics can be written as follows:   = FB + Fg (1) M V˙ + ωB × VB − VW I ω˙ B + ωB × IωB = MB   θ˙ sin ψ − φ˙ sin θ cos ψ = p θ˙ cos ψ + φ˙ sin θ cos ψ = q  φ˙ cos θ + ψ˙ = r

(2)

(3) T

where M is the aircraft mass, VB = [u, v, w] and ωB = [p, q, r]T the linear and rotational velocity vectors, VW = [uw , vw , ww ]T the atmospheric wind velocity vector, FB = [FX , FY , FZ ]T and MB = [MX , MY , MZ ]T the external forces and moments vectors acting on the aircraft and Fg the gravity force vector. Moreover by resorting to the generalized state variables ηi and η˙ i , aero-elastic modes are modelled by means of a second order linear state space description: Mηi η¨ + ζηi η˙ + Mηi ωηi ηi = Qηi , i = 1, . . . , na

(4)

where Mηi is the generalized mass of the i−th mode, ζηi the generalized damping coefficient, ωηi the generalized natural frequency and Qηi the generalized force. Note that the evolution of the external forces FB and moments MB acting on (1)-(2) depends on both the rigid body and the generalized flexibility state variables. An example of the latter is shown in (5). There, the aero¯ , L, ¯ N ¯ ]T ) of forces (moments) dynamic quote [L, D, Y ]T ([M ¯ ηi of vector FB (MB ) as well as the aerodynamic quote Q the generalized force Qηi are below reported:  # " n X  2  ρV S 0  L= CLηi ηi CL (α, β, p, q, r, δsup)+   2   i=1    n  X  ρV02 Sc   + CLη˙ i η˙ i  2    i=1 # "   n  X  ρV02 S   CDηi ηi CD (α, β, p, q, r, δsup)+ D=  2    i=1    n  X  ρV02 Sc   + CDη˙ i η˙ i  2    i=1 # "   n  X 2  ρV S  0  CYηi ηi CY (α, β, p, q, r, δsup)+  Y = 2    i=1    n  X  ρV02 Sc   + CYη˙ i η˙ i  2    i=1 # "   n  X  ρV02 Sc  ¯  Cmηi ηi Cm (α, β, p, q, r, δsup)+  M= 2 i=1

                                                                  

ρV02 Sc2

+

ρV02 Sb 2

¯= L

2

"

ρV02 Sb2 2

ρV02 Sb 2

"

¯ ηi Q

n X

n X

#

Clηi ηi

i=1

Clη˙ i η˙ i

i=1

Cn (α, β, p, q, r, δsup)+

ρV02 Sb2 2

+

Cmη˙ i η˙ i

i=1

Cl (α, β, p, q, r, δsup)+

+ ¯ = N

n X

n X i=1

n X

#

Cnηi ηi

Cnη˙i η˙ i i=1 h ρV 2 S = 02 c C0i + Cαi α + Cβi β + Cpi p + Cqi q + Cri r  n n X X ρV02 Sc2 ij i  Cηij Cηi ηi + 2 +Cδsup δsup + ˙ i η˙ i j=1

j=1

(5) with δsup the control surfaces deflection,  ρ the air density  w and V0 = kVB − VW k, α = arctan w−w , β = u−uw   v−vw . arcsin V0 Notice that the aerodynamic coefficients depend on the aircraft geometry and can be derived by resorting to computational fluid dynamics features or by using experimental tests. Similar arguments apply to the propulsive actions. Moreover, the numerical values of the HAPD geometrical parameters and of the constraints on maximum slew rate and maximum

TABLE I

Then, the aim is to find a state feedback regulation u(t) = g(x(t)), which asymptotically stabilizes (7) subject to (9)(10).

HPDA: MAIN PARAMETERS Parameters Wing Area (S) Wing Span (Sb ) Mean Chord (Sc ) Mass (M) Inboard, Middle board Elevators Slew Rates Outboard Elevators Slew Rate Ailerons Slew Rate Rudders Slew Rate Ailerons deflections Elevators deflections Rudders deflections

Value 13.5 16.55 0.557 184.4 ±200

Units m2 m m kg deg/s

±200 ±200 ±200 ±25 ±25 ±25

deg/s deg/s deg/s deg deg deg

First, recall that in [13] it has been shown that a linear state-feedback control law u = Kx is able to quadratically stabilise the uncertain system (7) if there exist a matrix P = P T > 0 and a scalar λ > 0 such that the following matrix inequality is satisfied  T ΦK P ΦK − P + K T Ru K+ T  Rx + λCK CK   BpT P ΦK

deflection pertaining to all the actuators are reported in Table I. For more technical details on the proposed mathematical model, the interested reader can refer to [12]. III. N ORM -B OUNDED E MBEDDING MPC

SCHEME

where Ω(∆) is a convex set of matrices of appropriate dimensions. Then, any property ensured for the uncertain linear system = = = =

Φx(t) + Gu(t) + Bp p(t) Cx(t) Cq x(t) + Dq u(t) ∆(t)q(t)

(7)

with p, q ∈ Rnp denoting additional variables which account for the uncertainty, holds true locally also for the nonlinear system (6), see [13]. In (7) the set n o ˜ C˜ | k∆k ≤ 1 Ω(∆):= A˜ + B∆ (8) with

A˜ =



Φ C

G 0



˜= ,B



Bp 0



  ≤ 0 (11) 

BpT P Bp − λInx

with ΦK := Φ + GK, CK := Cq + Dq K and Rx > 0, Ru > 0 shaping symmetric matrices. Then, the following set  S(t) := p ∈ Rnp | kpk22 ≤ kCK x(t)k22 (12)

(6)

with x ∈ Rnx denoting the state, u ∈ Rnu the control input, y ∈ Rny the output. Suppose that for each x, u, and each t there exists a matrix function Γ(·, ·) : Rnx × Rnu → Ω(∆), such that   x f (x, u) = Γ(x, u) u

x(t + 1) y(t) q(t) p(t)



represents the plant uncertainty regions at each time instant t for the state evolutions of (7) under the action of u = Kx. Moreover, a bound to the quadratic performance index

Consider the system x(t + 1) = f (x(t), u(t)) y(t) = h(x(t), u(t))

ΦTK P Bp

, C˜ = [Cq Dq ]

is the image of the matrix norm unit ball under a matrix linear-fractional mapping. In the sequel we shall assume that the plant is subject to the following ellipsoidal input and state evolution constraints  ¯ ∈ R+ (9) u(t) ∈ Ωu , Ωu , u ∈ Rnu : uT u ≤ u¯2 , u  ¯2 , x¯ ∈ R+ Cx(t) ∈ Ωx , Ωx , x ∈ Rnx : (Cx)T (Cx) ≤ x (10)

J(x(0), Kx(·)) :=

max

p(t)∈S(t)

∞ X  kx(t)k2Rx + kKx(t)k2Ru t=0

(13)

is given by J(x(0), Kx(·) ≤ x(0)T P x(0) and the ellipsoidal set  C(P, ρ) := x ∈ Rnx | xT P x ≤ ρ, ρ > 0

(14)

(15)

is a robust positively invariant region. Then, a feasible couple (K, Q) which solves (11) under the input and state constraints (9)-(10) for a given initial state x ∈ Rnx can be achieved by simultaneously satisfying the set of linear matrix inequalities given in [15]. If solvable, the latter pair ensures the existence of a robustly stabilizable state-feedback control law K that is capable to drive the state from x to zero without constraint violations. In order to add predictive capability to the above receding horizon controllers in [14] was proposed to introduce N additional free control moves, as usual in standard predictive control algorithms, over which the optimization can take place. Then to exploit this argument, the following family of virtual commands has been adopted  Kx ˆk (t) + ck (t), k = 0, 1, . . . , N − 1 u(·|t) := (16) Kx ˆk (t), k≥N where the vectors ck (t) provide N free perturbations to the action of a stabilizing and admissible controller K and x ˆk (t) := ΦK x(t) +

k−1 X i=0

k−1−i ΦK (Gci (t) + Bp pi (t)) (17)

represent convex set-valued state predictions, computed under the conditions pi (t) ∈ Si (t)   2 2 Si (t) := p | kpk2 ≤ max kCK xˆi (t) + Dq ci (t)k2 , x ˆi (t)

i = 0, 1, . . . , k − 1.

(18) with Si (t) characterizing all admissible perturbations along the system trajectories corresponding to the virtual command sequences (16). Then in virtue of (17), the closed-loop dynamics is given by x ˆk+1 (t) yˆk (t)

= ΦK x ˆk (t) + Gck (t) + Bp pk (t) = Cx ˆk (t), ∀pk (t) ∈ Sk (t)

(19)

and a convenient upper-bound to cost (13) is given by the following quadratic index V (x(t), P, ck (t)) :=kx(t)k2Rx  N −1  X 2 2 + max kˆ xk (t)kRx + kck−1 (t)kRu k=1

x ˆk (t)

(20)

+ max kˆ xN (t)k2P + kcN −1 (t)k2Ru x ˆN (t)

to be minimized w.r.t. ck (t), k = 0, 1, . . . , N − 1. Then, at each time instant t, the solution will consist of computing the sequence of increments c∗k (t) := arg min V (x(t), P, ck (t)) ck (t)

(21)

subject to Kx ˆk (t) + ck (t) ∈ Ωu , k = 0, 1, . . . , N − 1, Cx ˆk (t) ∈ Ωx , k = 0, 1, . . . , N − 1,

(22) (23)

xˆN (t) ⊂ C(P, ρ), Kz ∈ Ωu , Cz ∈ Ωx , ∀z ∈ C(P, ρ) (24) In order to derive a computable scheme, we need to find a tight upper-bound to the cost (20), which can be simply determined by introducing a sequence of non-negative slack variables Ji (t) (see [14] for details), i = 0, 1, . . . , N − 1, such that for arbitrary P, K and ck , k = 0, . . . , N − 1, the following inequalities max

max

p0 ∈S0 x ˆTk+1

pi ∈Si i=0,....,k k=1,...,N −2

max

pi ∈Si i=0,....,N −1

x ˆT1

Rx x ˆ1 +

cT0

Ru c0

≤ J0

(25)

Rx x ˆk+1 +

cTk

Ru ck

≤ Jk ,

(26)

x ˆTN P x ˆN + cTN −1 Ru cN −1

≤ JN −1 (27)

y

.

ref( )

Reference Trajectory Module

x(t)

MPC Controller

u(t)

y(t) Process

u(t)

Fig. 2.

Control allocation architecture

Then, the control allocation problem can be stated as follows: Control Allocation (CA) problem - Given an initial time instant t0 and a reference trajectory yref (·), determine at each time instant t ≥ t0 a command input u(t) such that the output y(t) of the plant model (7)-(8) subject to (9)-(10) tracks yref (t), ∀t ≥ t0 , as closely as possible in a 2-norm sense. ✷ In the sequel, the reference to be tracked will be denoted as r(·) = yref (·), yref (·) ∈ Y, where Y accounts for limitations in the angular acceleration provided by the actuators. A. Reference Trajectory module In what follows we shall consider time-varying reference trajectories and therefore the solution of the CA problem is subject to the computation over a horizon of a finite length N ∈ N of a feasible input sequence such that the corresponding state trajectory {xi (t)}N i=0 is such that the sequence output {yi (t) = Cxi (t)}N i=0 corresponds to the given reference sequence {ri (t)}N . i=0 To deal with the latter, it is clearly mandatory that a nominal plant model is available: within the proposed normbounded framework this translates to consider the so-called central dynamics, i.e. x(t + 1) = Φx(t) + Gu(t) y(t) = Cx(t)

ALLOCATION M ODEL SCHEME

P REDICTIVE

In this section, an MPC-based algorithm will be proposed for control allocation purposes. By resorting to the ideas developed in [16], we propose a two-stage optimization procedure. Specifically, in the first phase, hereafter denoted as Reference Trajectory module, a feasible reference trajectory is made available to the MPC

(28)

Then, the following problem must be solved: Reference Trajectory (RF) problem - Given the reference sequence {rk (t)}N k=0 ⊂ Y and an initial state condition x(t) such that Cx(t) = r(t), determine an input sequence −1 {uk (t)}N k=0 ⊂ Ωu such that the corresponding solution of xk (t + 1) = Φxk (t) + Guk (t), yk (t) = Cxk (t)

hold true. IV. C ONTROL

controller. Then, the control allocation problem is formulated as a receding horizon tracking problem with the reference trajectory achieved at the previous step and attacked by means of the MPC framework discussed in Section 3. A sketch of the proposed architecture is depicted in Fig. 2.

yields yk (t) = rk (t), i {xk (t)}N k=1 ⊂ Ωx .

=

(29)

1, . . . , N and satisfies ✷

In order to ensure the resolvability of the RF problem, we shall further assume that: Assumption 1: The nominal system (28) is controllable and observable. ✷ Assumption 2: At each time instant t the reference trajectory r(t) is known over the horizon It = [t, t + N ], i.e.

the sequence {rk (t)}N k=0 is available to the controller for all k ∈ N. ✷ It is important to underline that, even if the RF problem is well-posed (the requirement r(t) ∈ Y, ∀t ≥ 0, holds true), it may happen that a admissible solution (x(t), u(t)) could give rise to a state trajectory that does not fulfil the prescribed constraints at each time instant t. Then, to avoid such a drawback a reference trajectory algorithm must be capable to determine a steady-state solution that provides a trade-off between the reference tracking properties and the constraints fulfilment. Such an idea can be pursued by relaxing the exact tracking requirement yi (t) = ri (t), i = 0, 1, . . . , N and formulating the RF problem as a constrained optimization problem. To this end, a set of adequate slack variables is introduced and at each time interval It the state/input reference sequences are computed via the following quadratic optimization: min J RF (t) (30) s s r¯ (t),¯ x (t),¯ u(t)

such that            

−¯ rs (t) −¯ xs (t) u ¯(t) −¯ u(t) Du¯(t) − r¯s (t) −Du¯(t) − r¯s (t) G¯ u(t) + F x ¯(t) − x ¯s (t) −G¯ u(t) + F x¯(t) − x ¯s (t)





          ≤          

0 0 u ¯·1 −¯ u·1 r¯(t) −¯ r (t) x ¯·1 −¯ x·1

           

(31)

Moreover, r¯s and x¯s are vectors of nonnegative slack variables, Ψr , Ψx and Ψu are symmetric and positive definite matrices and wr and wx are vectors of positive weights. Finally, u¯p (t) is a known vector chosen to take care of the virtual control allocation purposes and it is obtained as the solution of the unconstrained weighted minimum energy problem u¯p (t) = W −1 DT (DW −1 DT )−1 r¯(t) with W > 0 a weighting matrix. B. MPC tracking scheme This section is devoted to adapt the basic MPC scheme of Section III to deal with the proposed CA problem. To this end, the tracking problem will be formulated as a regulation problem by using the reference sequences {xk (t)}N k=1 and −1 {uk (t)}N k=0 and by means of standard coordinate transformations. Specifically, by defining x ˜k (t) = xk (t) − x ¯k (t), u ˜k (t) = uk (t) − u ¯k (t)

T

V (˜ x(t), P, c˜k (t)) := k˜ x(t)k2Rx + ! N −1 X 2 2 ˆ ck−1 (t)kRu max kx ˜k (t)kRx + k˜ + max

(32)

Note that by defining the vectors x ¯(t), u¯(t) and y¯(t) as follows     uN −1 (t) xN (t)  uN −2 (t)   xN −1 (t)      ¯(t) =  x ¯(t) =  , , u .. ..     . . u(t) x1 (t)   yN (t)  yN −1 (t)    y¯(t) =  , ..   . y1 (t)

and under the assumption that the initial state x(t) is known, the solution of (29) satisfies an equation of the form x ¯(t) = F x(t) + G¯ u(t), y¯(t) = Hx(t) + Du¯(t) As a consequence r¯(t) is given by   rN (t)  rN −1 (t)    r¯(t) =   − Hx(t) ..   . r(t)

ˆ x ˜k (t)

k=1

T

r¯s (t) Ψr r¯s (t) + x ¯s (t) Ψx x¯s (t)+ p T (¯ u(t) − u ¯ (t)) Ψu (¯ u(t) − u ¯p (t))+ T s T s wr r¯ + wx x¯

(33)

(36)

the following modifications of the basic MPC algorithm result: 1) Family of virtual commands:  Kx ˆk (t)+ck (t)− u ¯k (t), k = 0, . . . , N −1 u˜(·|t) := Kx ˆk (t), k≥N (37) 2) Quadratic index:

where 0 = [0 0 · · · 0]T , 1 = [1 1 · · · 1]T are vectors of appropriate dimensions and the optimization cost index is J RF (t) :=

(35)

ˆ x ˜N (t)

ˆ˜N (t)k2P kx

+

(38)

k˜ cN −1 (t)k2Ru

where ˆ˜k (t) = x

ΦK x(t) +

k−1 X

k−1−i ΦK (G(ci (t) − u¯i (t))+

i=0

Bp pi (t)) − x ¯k (t) (39) and c˜k (t) = ck (t) − u¯k (t), k = 0, . . . , N − 1.

(40)

3) Upper bound to (38): ˆ˜1 + c˜T0 Ru c˜0 ˜ˆT1 Rx x max x

≤ J0 (41)

ˆ˜Tk+1 Rx x ˆ˜k+1 + c˜Tk Ru c˜k x

≤ Jk (42)

ˆ˜N + c˜TN −1 Ru c˜N −1 x ˜ˆTN P x

≤JN−1(43)

˜0 p0 ∈S

max

˜i pi ∈S i=0,....,k k=1,...,N −2

max

˜i pi ∈S i=0,....,N −1

where S˜i (t) := (34)

(

ˆ˜i (t) + Dq c˜i (t)k22 p | kpk22 ≤ max kCK x ˆ x ˜i (t)

)

,

i = 0, 1, . . . , k − 1. (44)

Moreover, the following component-wise constraints on the maximum control rates have to be imposed:  T ˆ˜k (t) + c˜k (t) − u(t − 1) Kx ei ·   (45) T ˆ ei K x ˜k (t) + c˜k (t) − u(t − 1) ≤ δ¯ u2i , δ¯ ui > 0, i = 1, . . . , nu , where u(t − 1) is the computed command at the previous time instant t − 1 and ei is the i − th vector of the canonical basis. Finally, also the input constraints are recast into a component-wise form as follows:    T ˆ ˆ˜k (t) + c˜k (t) ˜k (t) + c˜k (t) ≤ u ¯2i , Kx ei eTi K x u ¯i > 0, i = 1, . . . , nu . (46) Then, the tracking MPC scheme is as follows: NB-MPCOff-line 1 Given the initial state x ˜(0), compute the triplet (K, Q, ρ) by solving the optimization problem subject: min

Q,Y,X,ρ,λ,t

ρ

(47)





Q  1/2  Ru Y  1/2  R x Q   Cq Q + Dq Y ΦQ + GY

where

1 x˜(t)T x ˜(t) Q 1/2

Y T Ru ρ Inu 0 0 0



≥ 0,

(48)

1/2

Q Rx QCqT +Y T DqT 0 0 0 ρ Inx 0 λ Inx 0 0  G   ≥0 (49)  

Q ΦT + Y T 0 0 0 Q − λ Bp BpT λ > 0,

(50)



X Y ≥ 0, Xjj ≤ δ¯ u2j , j = 1, . . . , nu , YT Q P = ρQ−1 , K = Y Q−1



˜i pi ∈S i=0,...,k−1

2

k = 0, 1, . . . , N − 1, i = 1, . . . , nu max

˜i pi ∈S i=0,...,k−1

  2

T ˆ˜k (t) + c˜k − u ˜(t − 1) ≤ δ¯ u2i ,

ei K x 2

k = 0, 1, . . . , N − 1, i = 1, . . . , nu max

˜i pi ∈S i=1,...,k−1

2

ˆ

˜k (t) ≤ x ¯2 , k = 1, 2, . . . , N

C x 2

2 feed the plant with u ˜(t) = K x ˜(t) + c˜∗0 (t); 3 t = t + 1, and go to step 1.1 Proposition 1: Let the NB-MPC scheme have solution at time t = 0. Then, it has solution at each future time instant t, satisfies input and output constraints and yields an asymptotically (quadratically) stable closed-loop system. Proof - Proof follows by resorting to the same arguments of [14]. ✷ V. S IMULATIONS

subject to



subject to (41)-(43) and

  2

ˆ˜k (t) + c˜k ¯2i , max eTi K x

≤u

x ¯2 Q Cq Q + Dqu Y C(ΦQ + GY )

(Cq Q + Dqu Y )T t−1 Inx 0

(51)

 (ΦQ+GY )T C T ≥ 0 0 I−t−1GBpBpTC T (52)

where t>0

(53)

On-line 1 At each time instant t, given x ˜(t), solve the following optimization problem [Jk∗ (t), c˜∗k (t)] , arg min

Jk ,ck

N −1 X k=0

Jk

By exploiting the ideas developed in the seminal papers [19] and [20], the HAPD nonlinear model has been recast into a Polytopic Linear Differential Inclusion (PLDI) obtained by using nine linearized models characterizing different flight conditions within the following operating envelope: • the true air speed belongs to [17, 23] m/s; • the altitude varies between 300 m and 700 m. Then, the PLDI has been outer-approximated as the Normbound Linear Differential Inclusion (NDLI) (7) by applying the optimization procedure described in [13]. Moreover, the following assumptions have been made: • actuator and sensor dynamics have been considered negligible; • by resorting to well-known residual stiffness techniques [12], aeroelastic dynamics have been assumed to be instantaneous. The validity of such assumptions and the effectiveness of the proposed norm-bounded model predictive control allocation strategy have been verified by using the full nonlinear HAPD aircraft model whose structure includes symmetrical and asymmetrical flexible modes, sensors and actuators dynamics. Specifically, we have considered the following simulation scenario mainly exciting the roll rate dynamics: Doublet step manoeuvre on roll-rate demand (pref )- At t = 1 s pref is set to 8 deg/s for a duration of 2 s. Then, at t = 5 s the reference on p is assumed to be equal to −8 deg/s for the same time interval. ✷ All the computations have been carried out on a PC Intel Quad Core with the Matlab LMI and Optimization

5

10

15

r [deg/s]

10

15

[deg]

[deg]

−25 0

15

5 10 Elevator MID−DX

15

5 10 Elevator OB−DX

15

5 10 Time [s]

15

25

5 10 Elevator OB−SX

0 −25 0

15

25

5 10 Time [s]

15

0 −25 0

Rudder SUP

[deg]

25

0

−25 0

5

Rudder INF

10

15

10

15

25

[deg]

A tracking model predictive control allocation scheme for time-varying references of an High Altitude Performance Demonstrator aircraft subject to input and control rate of variation constraints has been proposed. A norm-bounded differential inclusion of the nonlinear aircraft model has been derived for MPC controller design purposes. As one of its main merits, the proposed architecture is capable under critical flight conditions both to satisfy tight allocation requirements and to satisfactorily track time-varying references on the relevant aircraft variables. Numerical experiments have been performed by considering the full nonlinear model and by including the flexibility dynamics of the aircraft. The simulation studies have shown the effectiveness of the proposed approach both in terms of constraints satisfaction and control allocation capabilities.

0

Fig. 5. Elevators control effort. The dashed lines represent the prescribed constraints.

Fig. 3. Response to the reference signal on p. The dashed lines are the reference signals.

VI. C ONCLUSIONS

[deg]

[deg] [deg]

0

Time [s]

5 10 Elevator MID−SX

0 −25 0

5

Elevator IB−DX 25

0

25

2

−2 0

15

−25 0

15

0

−25 0

15

4

5 10 Time [s]

0

Elevator IB−SX

25

10

5 10 Time [s]

25

0.2

5

15

Fig. 4. Ailerons control effort. The dashed lines represent the prescribed constraints.

−25 0

−0.2 0

5 10 Aileron OB−DX

25

0

−25 0

[deg]

−10 0 6

−25 0

15

25

0.4

0

5 10 Aileron OB−SX

0

[deg]

−5

0

−25 0

[deg]

q [deg/s]

p [deg/s]

0

Aileron IB−DX 25

[deg]

0.6

10

Aileron IB−SX 25

[deg]

Toolboxes by choosing a prediction horizon N = 1 and the results for this numerical experiment are reported in Figs. 3-9. In particular, Fig. 3 shows the capability of the proposed NB-MPC strategy to track the required reference signal with an acceptable residual coupling on pitch and yaw dynamics, whereas Figs. 4-6 show control surfaces dynamical behaviours and Figs. 7-9 rates of variation on the control surface deflections. It is interesting to underline how the strategy is capable to take advantage from the aircraft redundant actuation capability: in fact the actuation deficit on the ailerons (see Fig. 4) is compensated by some of the available elevators, Fig. 5. As a consequence, this allow to have sufficient authority to achieve an almost exact tracking on the p reference signal (the dashed line in Fig. 3) without any constraint violations. Although not reported here for the sake of space, an exhaustive simulation campaign has been conducted to demonstrate that proposed algorithm provides stability and performance of the closed-loop system by using different flight conditions on the aircraft operating envelope.

0

−25 0

5 Time [s]

Fig. 6. Rudders control effort. The dashed lines represent the prescribed constraints.

SR−Aileron IB−SX

SR−Aileron IB−DX

5 10 SR−Aileron OB−SX

5 10 SR−Aileron OB−DX

0

−200 0

15

5 10 Time [s]

5 10 Time [s]

15

[deg/sec]

[deg/s]

200

0

5 10 SR−Elevator MID−SX

[deg/sec]

[deg/s]

0 5 10 SR−Elevator OB−SX

[deg/sec]

[deg/s]

0 −200 0

5 10 Time [s]

15

5 10 SR−Elevator MID−DX

15

5 10 SR−Elevator OB−DX

15

5 10 Time [s]

15

200 0 −200 0

15

200

0 −200 0

15

200

−200 0

10

15

10

15

0

−200 0

5

Fig. 9. Rates of variation on rudder deflections. The dashed lines represent the prescribed constraints.

SR−Elevator IB−DX

200

−200 0

SR−Rudder INF

Time [s]

Fig. 7. Rates of variation on aileron deflections. The dashed lines represent the prescribed constraints. SR−Elevator IB−SX

5

200

[deg/s]

0

0

−200 0

15

200 [deg/sec]

[deg/s]

200

−200 0

0

−200 0

15

200

[deg/s]

0

−200 0

SR−Rudder SUP

200 [deg/sec]

[deg/s]

200

200 0 −200 0

Fig. 8. Rates of variation on elevator deflections. The dashed lines represent the prescribed constraints.

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[8] J. A. M. Petersen and M. Bodson, “ Interior-point algorithms for control allocation. Journal of Guidance, Control and Dynamics, Vol. 28, No. 3, pp. 471-480, 2005. [9] T. A. Johansen, T. I. Fossen and P. Tondel, “ Efficient optimal constrained control allocation via multiparameteric programming. Journal of Guidance, Control and Dynamics,Vol. 28, No. 3, pp. 506-514, 2005. [10] M. Bodson, “ Evaluation of optimization methods for control allocation”, Journal of Guidance, Control and Dynamics, Vol. 25, No. 4, pp. 703-711, 2002. [11] J. Jin, “ Modified pseudoinverse redistribution methods for redundant controls allocation”, Journal of Guidance, Control and Dynamics, Vol. 28, No. 5, pp. 1076-1079, 2005. [12] M. Cicala and A. Sollazzo, “HAPD Modello di Velivolo Elastico orientato al controllo”, Technical Report, CIRA-CF-08-0346, 2008. [13] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, “Linear Matrix Inequalities in System and Control Theory”, SIAM Studies in Applied Mathematics, 15, SIAM, London. [14] A. Casavola, D. Famularo and G. Franz`e, “Robust constrained predictive control of uncertain norm-bounded linear systems”, Automatica, Vol. 32, pp. 1865–1876, 2004. [15] M. V. Kothare,V. Balakrishnan and M. Morari, “Robust constrained model predictive control using linear matrix inequalities”, Automatica, Vol. 32, pp. 1361–1379, 1996. [16] Y. Luo, A. Serrani, S. Yurkovich, M. W. Oppenheimer and D. B. Doman “Model-Predictive Dynamic Control Allocation Scheme for Reentry Vehicles”, Journal of Guidance, Control, and Dynamics, Vol. 30, No. 1, pp. 100–112, 2007. [17] V. Scordamaglia, M. Mattei, C. Calabro, A. Sollazzo and F. Corraro “Comparison of control allocation methods for the High Altitude Performance Demonstrator”, 8th European Workshop on Advanced Control and Diagnosis, Ferrara (IT), 2010. [18] V. Scordamaglia, A. Sollazzo and M. Mattei “Fixed Structure Flight Control Design of an Over-actuated Aircraft in the Presence of Actuators with Different Dynamic Performance”, 7th IFAC Symposium on Robust Control Design, Aalborg (DK), 2012. [19] R. W. Liu, “Convergent systems”, IEEE Trans. Aut. Control, Vol. 13, No. 4, pp. 384-391, 1968. [20] R. Liu, R. Saeks, and R. J. Leake, “On global linearization”, SIAMAMS, pp. 93-102, 1969.