Advanced Model Predictive Control Part 13 pot

Therefore, solving Problem 2 produces control input command deviation Δv k|k which is robust against the uncertain delays at the control input satisfying 3 and all possible turbulence 3.

measured turbulence data, is defined below.

Considering that the turbulence are measured for N steps ahead, the horizon step number in

MPC, which denotes the step number during which control performance is to be optimized, is

where matrices Q and S are appropriately defined positive semidefinite matrices, matrix R is

at step k.

There usually exist preferable or prohibitive regions for the state, the performance output, andthe control input command deviation For the consideration of these regions, constraints for

u

have no more worse performance than the worst Considering this, the design objective is to

obtainΔv k +j|k which minimizes the maximum of J i(ˆx i,Δv, z i) Thus, the addressed problem

paper is to obtain the optimal control input command by solving an optimization problemonline using a family of plant models That is, the proposed control strategy is MPC

It is easily confirmed that solving Problem 1 is equivalent to solving the following problem

Problem 2. Find Δv k +j|k(j=0,· · · , N −1)which minimize the following performance index.

max

˜

w∈Ω i∈ {maxd,··· ,d } J i(ˆx

i,Δv, z i)subject to(17),(18),(19),(10)with(12)and(13)

Remark 6. Note that Problem 2 seeks the common control input command deviation for all i and for all possible ˜ w ∈ Ω Therefore, solving Problem 2 produces control input command deviation Δv k|k which

is robust against the uncertain delays at the control input satisfying (3) and all possible turbulence

3.1 No measurement error case

Then, the state equation and the performance output equation of P u i are respectively given asfollows:

˜x i= I N ⊗ Aˆi0(n+n

u d )N,n+n u d

i k|k

γmin:=1N ⊗ γmin, ˜γmax:=1N ⊗ γmax,

˜δmin:=1N ⊗ δmin, ˜δmax:=1N ⊗ δmax,

Using these definitions, the following proposition, which is equivalent to Problem 2, is directlyobtained

Proposition 1. Find ˜v which minimizes q subject to (22), (23), and (24).

Remark 7. If Proposition 1 is solved, then the state is bounded by γmin and γmax; that is, the boundedness of the state is assured.

3.2 Measurement error case

˜

Q=Q ˆˆQ T, ˜R=R ˆˆR T, ˜S=S ˆˆS T

Then, inequality (22) is equivalently transformed to the following inequality by applying theSchur complement (Boyd & Vandenberghe, 2004).

⎡

⎢

⎣

q (˜x i)T Q ˜vˆ T Rˆ (˜z i)T Sˆˆ

˜

w is affine with respect to each element ofΔw Similarly, ˆx i

k +m|k (m = 1,· · · , N −1) and

z i

k +m|k(m = 0,· · · , N −2)are also affine with respect to each element ofΔw Considering

Φ=p= [p1 · · · p n w]T ∈ R n w : p i = ± 1, i=1,· · · , n w



Under these preliminaries, the following proposition, which is equivalent to solvingProblem 2, is directly obtained

Proposition 2. Find ˜v which minimizes q subject to (22), (23) and (24) for allΔw ∈ Φ.

Similarly to Proposition 1, as Proposition 2 is also an SOCP problem, its global optimum iseasily obtained with the aid of some software, e.g (Sturm, 1999)

(13) for the real turbulence, then the addressed problem, i.e Problem 1, is solved by virtue ofProposition 2 without introducing any conservatism (see Remarks 2 and 5)

Remark 8. The increases of the numbers N, n w and i lead to a huge numerical complexity for solving Proposition 2 Thus, obtaining the delay time bounds precisely is very important to reduce i On the other hand, in general, n w cannot be reduced, because this number represents the number of channels of turbulence input The remaining number N has a great impact on controller performance, which will

be shown in the next section with numerical simulation results.

4 Numerical example

Several numerical examples are shown to demonstrate that the proposed method works wellfor GA problem under the condition that there exist bounded uncertain delays at the control

input and the measurement errors in a priori measured turbulence data.

4.1 Small aircraft example

Let us first consider the linearized longitudinal aircraft motions of JAXA’s research aircraft

This aircraft is based on Dornier Do-228, which is a twin turbo-prop commuter aircraft

4.1.1 Simulation setting

It is supposed that only the elevator is used for aircraft motion control The transfer function

representing the linearized longitudinal motions with the modeled actuator dynamics is given

axes, inertial vertical velocity in body axes, pitch rate, pitch angle, elevator deflection, verticalturbulence in inertial axes, elevator command, and vertical acceleration deviation in inertialaxes

are calculated (The state-space matrices are omitted for space problem.) The augmented state

ˆx i

u i w i q θ δ e δ e c (−4) δ e c (−3) δ e c (−2) δ e c (−1) T, whereδ e c (−l) denotes the elevator

command created at l step before The objective is to obtain the elevator input command,

δ e c(0), which minimizes the effect of vertical turbulence to vertical acceleration for all possibledelays.

as follows:

γmax= 10 10 10180π 10180π 1805π ×15 T, γmin= − γmax,

δmax= π

180, δmin = − δmax

output has no constraints

the frequency of the turbulence

4.1.2 Simulation results without measurement errors in turbulence data

Let us first show the results of simulations in which turbulence is supposed to be exactlymeasured

longitudinal motions and the first-order elevator actuator model, and the proposed MPC in

various constant weighting matrices R, and various constant receding horizon step numbers

N are used from the following sets:

For comparison, the following scenarios are simultaneously carried out

Scenario A: MPC in which Proposition 1 is solved online is applied,

Scenario B: no control is applied,

Scenario C: MPC in which Proposition 1 is solved online but with the measured turbulencedata being set as zeros, i.e MPC without prior turbulence data, is applied

denote the following performance indices for the corresponding scenarios, which are obtainedfrom the simulations:

max

ˆt d ∈{1, 2, 3, 4}

 20

For comparison, mesh planes at J A /J B =1 and J A /J C=1 are drawn J A /J B <1 means that

the a priori measured turbulence data are useful for the improvement of GA performance.

The following are concluded from Fig 3

8[rad/s]

Proposition 1 is solved online improves GA performance for low and middle frequency

The first item is reasonable because aircraft motion model has a direct term from the verticalturbulence to the vertical acceleration and it is supposed that there exists uncertain delay atits control input The second item is interesting, because there is a limit for the improvement

of GA performance even when a priori measured turbulence data are available.

cases These figures illustrate the usefulness of the a priori measured turbulence data.

4.1.3 Simulation results with measurement errors in turbulence data

Let us next show the results of simulations in which measured turbulence data havemeasurement errors

longitudinal motions and the first-order elevator actuator model, and the proposed MPC in

various constant weighting matrices R, and various constant receding horizon step numbers

N are used from the following sets:

Three possibilities are considered in the simulations; that is, (i) the real turbulence is the same

For comparison, the following scenarios are simultaneously carried out

100

100

0 0.5 1 1.5

20 30

30 40

40 50

50

100

100

0 0.5 1 1.5

20 30

30 40

40 50

50

100

100

0 0.5 1 1.5

20 30

30 40

40 50

0 0.5 1

-0.1 0 0.1

-1 0 1

-1 0 1

-0.5 0 0.5

-2 0 2

-2 0 2

-2 0 2

-5 0 5

-5 0 5

Scenario A: MPC in which Proposition 2 is solved online is applied,

Scenario B: no control is applied,

Scenario C: MPC in which Proposition 2 is solved online but with the measured turbulencedata being set as zeros, i.e MPC without prior turbulence data, is applied

denote the following performance indices for the corresponding scenarios, which are obtainedfrom the simulations:

max

w k +j={ w k +j|k , w k +j|k ±X j } ˆt d ∈{1, 2, 3, 4}max

20

The following are concluded from Fig 5

proposed method is larger than the uncontrolled case

reduced even if prior turbulence data are obtained

Proposition 2 is solved online improves GA performance for middle frequency turbulence,

The first item does not hold true for no measurement error case (see also Fig 3) Thus, GAperformance deterioration for low frequency turbulence is caused by the measurement errors

in the measured turbulence data The second item is reasonable for considering that it isdifficult to suppress turbulence effect on aircraft motions caused by high frequency turbulenceeven when the turbulence is exactly measured (see also Fig 3) The fourth item illustrates

that the a priori measured turbulence data improve GA performance even when there exist

measurement errors in the measured turbulence data

usefulness of the a priori measured turbulence data for middle frequency turbulence (e.g.

turbulence data deteriorate GA performance; that is, if the real turbulence is smaller than the

elevator deflections and this causes extra downward accelerations The converse, i.e the case

performance to measure turbulence exactly

To evaluate the impact of the rate limit for elevator command on GA performance, the same

are carried out The results for (35) are shown in Fig 7

Comparison between Figs 5 and 7 concludes the following

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0 0.5 1

-2 0 2

-5 0 5

-5 0 5

-2 0 2

-1 0 1

-2 0 2

-2 0 2

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

0.5 1 1.5

command under measurement errors in turbulence data

• The rate limit for elevator command does not have so large impact on GA performance

except for the cases using small R.

This fact is reasonable because it is difficult to suppress high frequency turbulence effect even

when there are no measurement errors in the turbulence data (see Fig 3), and if R is set

if R is set small then the proposed GA flight controller allows high rate elevator commands,

which lead to severe oscillatory accelerations Thus, GA performance deteriorates

Finally, CPU time to solve Proposition 2 is shown in Table 1 The simulation setting is thesame as for obtaining the results in Fig 5 The simulations are conducted with Matlab® usingSeDuMi (Sturm, 1999) along with a parser YALMIP (Löfberg, 2004) with a PC (Dell PrecisionT7400, Xeon®3.4 GHz, 32 GB RAM; PT7400) and a PC (Dell Precision 650, Xeon®3.2 GHz,

2 GB RAM; P650) Although CPU time with PT7400 is just about 30 % of P650, at the present

moment, solving Proposition 2 online is impossible with these PCs even when N is set as 10.

Thus, the reduction of numerical complexity for solving Proposition 2 is to be investigated

4.2 Large aircraft example

Let us next consider the linearized longitudinal aircraft motions of large aircraft Boeing 747

4.2.1 Simulation setting

Then, the continuous-time system representing the linearized longitudinal motions with themodeled actuator dynamics is given as (1), where the state, the turbulence, the control input,

After the discretization of (1) with sampling period T s[s]being set as 0.1 using a zero-orderhold, the discrete-time system (5) is given as (36).

command deviation and the performance output, matrices Q and S in (16), the turbulence,

4.2.2 Simulation results

The same numerical simulations in section 4.1.3 but the aircraft motion model being replaced

by the B747 model are carried out for the following parameter setting

The following are concluded from Fig 8

proposed method is larger than the uncontrolled case

reduced even if prior turbulence data are obtained

• It is sufficient for B747 to measure turbulence for 20 steps ahead

Proposition 2 is solved online improves GA performance for middle frequency turbulence,

be reduced by the proposed GA controller; however, middle frequency turbulence effect,

Fig 8 GA performance comparison for B747 under measurement errors in turbulence data

-1 0 1

-2 0 2

-2 0 2

-0.5 0 0.5

scenario A scenario B scenario C

-1 0 1

-5 0 5

-5 0 5

-1 0 1

scenario A scenario B scenario C

for turbulence measurement does not depend on aircraft models However, it is not sure thatthis fact indeed holds for other aircraft, which is to be investigated.

this consequently leads to severely oscillatory vertical accelerations However, the proposed

aircraft motions

5 Conclusions

This paper tackles the design problem of Gust Alleviation (GA) flight controllers exploiting

a priori measured turbulence data for suppressing aircraft motions driven by turbulence For

this problem, a robust Model Predictive Control (MPC) considering the plant uncertaintiesand the measurement errors in the turbulence data is proposed In the usual setting, MPC foruncertain plant requires to solve an optimization problem with infinitely many conditions ifconservatism is avoided However, it is shown that if the plant uncertainties are represented asthe bounded time-invariant uncertain delays at the control input, then the associated problemfor the robust MPC is equivalently transformed to an optimization problem for finitely manyplant models, which consequently means that the optimization problem has finitely manyconditions

In our problem setting, the measurement errors in the a priori measured turbulence data

are represented as affine with respect to a constant uncertain vector, whose elements areall bounded Using this property, it is shown that it is necessary and sufficient to evaluatethe performance index in MPC at the maxima and minima of the uncertain vector Thisconsequently means that the robust MPC has finitely many conditions even when themeasurement errors are considered

Several numerical examples illustrate that the proposed GA flight controller withappropriately chosen controller parameters effectively suppresses turbulence effect on aircraftmotions, and reveal that it is very difficult to suppress high frequency turbulence effect even

when the a priori measured turbulence data are exploited.

To guarantee the feasibility of the proposed MPC at every step is an important issue forthe implementation of the proposed method to real systems Thus, this topic is now underinvestigation

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