1 - Control law design techniques

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Introduction

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Introduction

The main function of the flight control system (FCS) of an aircraft is to ensure that the desired flight missions can be accomplished in safety. A fly-by-wire FCS consists of these basic components: sensors providing air speed and air flow data, inertial motion data and cockpit/pilot command inputs; a flight control computer (FCC) in which flight control laws are implemented to determine the commands for the actuations systems of the aircraft control surfaces and throttles for engines demands; an actuation system.

Modern "Fly by wire" systems allow the pilot to control the aircraft states, as an alternative to the conventional direct control of the engines and control surfaces.

Within the flight control law, feedback control is used to provide tight pilot command tracking, to attenuate external disturbances, such as gusts and turbulence and provide robust performance against parameters (for example aircraft mass) or flight condition variations.

For a FBW aircraft, within the limit of performance and constraints given by the airframe, the primary control laws and its design have a fundamental role in defining the handling characteristics. At the same time, control laws design is a rather complex task.

The aim of this thesis is to design the lateral-directional primary control laws for the M346 "Master" Advanced Trainer Aircraft, using modern design techniques.

Classical design is based on the successive loop closure, guided by the experience and intuition of the designer. Classical design approach involves the use of tools as root locus, Bode and Nyquist plots, etc, that help the designer to visualize how the system dynamics are being modified. The design procedure becomes increasingly difficult as more loops

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Introduction

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are added and doesn't guarantee success when the dynamics are multivariable: multiple inputs, multiple outputs, or multiple feedback loops

Modern control techniques, in contrast, allows all the control gains to be computed simultaneously, so that all loops are closed at the same times.

The first part of the thesis presents the design techniques used in this work: a brief comparison between classical and modern approach is presented, then Optimal Control and Linear Quadratic (LQ) theory are analyzed more in detail. At the end of this first chapter the problem of gain scheduling is discussed.

The second part describes the linear stability and handling criteria that will be used to analyze effects and characteristics of the proposed control laws.

The third part presents the control laws Synthesis.

At the beginning the reader can find an introduction at the problem: flight envelope, Mass properties variations and Aerodynamic Tolerance used for the synthesis are illustrated. In the first part of this chapter, linear design is described in detail. After a brief introduction on the Lateral AC model, the problem of the simplification of the model is presented, and the FCS model used for the synthesis is illustrated. The core of this part is based on the construction of the synthesis model (using Matlab and Simulink) which will allow us to compute the control laws gains: the choice of the control laws structure is explained, with the description of the frequency shaping method, then feed-forward gains are computed. The second part of this chapter illustrates the non linear implementation of the control laws: signal preprocessing, authority, gain scheduling are analyzed and described.

In the fourth part the reader will find the design validation for the controller: in the first part design validation for the linear controller (Linear Analysis) is described: results obtained by the synthesis model are compared against the complete model, without simplification, and with a more complete and accurate FCS model. The second part presents the design validation of the non linear controller: time responses obtained by non linear simulation of manoeuvres with small amplitude commands are compared with results obtained in the linear analysis.

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Introduction

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The fifth part is dedicated to the results presentation. Here aircraft response obtained with the new control laws constructed with modern design techniques will be compared against the aircraft controlled with the FCS primary control laws developed by AleniaAermacchi.

Finally, the reader can find the conclusions of this work.

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1 - Control law design techniques

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1 - Control law design techniques

In this chapter different aspects of control law design are discussed. The first part deals with linear design techniques used to define control laws gains. Two design approaches are presented: the first based on classical control theory, then a second approach on

"modern" control theory. The second part considers some non linear aspects of the control laws design, like gain scheduling and its implementation methods.

1.1 Classical control

1.1.1 Introduction

In classical control theory, the main results emerged in the years 1930–1950, the initial period of development of the field of feedback and control engineering.

Classical control theory development started with H. Nyquist's stability criterion, H. W.

Bode’s frequency domain analysis, and W. R. Evans’ root locus method.

Today these methods continue to be of great importance for the practical design of control systems, especially for the case of single-input, single-output systems.

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1.1.2 Compensator design

In classical control the design of feedback compensation relies on trial-and-error procedures.

As explained in ref [2], the classical procedure involves the following steps:

• Determine the plant transfer function P based on a linearized model of the plant.

• Investigate the shape of the frequency response, to understand the plant properties.

• Consider the desired steady-state error and transient properties of the system. Choose a compensator structure (for example by introducing integrating action or lag compensation) that provides the required characteristics..

• Plot the Bode, Nyquist or Nichols diagram of the loop frequency response of the compensated system. Adjust the gain to obtain a desired amount of stability of the system. The gain and phase margins are typical criteria to quantify stability.

• If the specifications are not met adjust the loop gain frequency response: use lag, lead, lag-lead or other compensation to realize the necessary modification of the loop frequency response function.

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1.2 Robust design

Robustness deals with the discrepancy between models and reality. It is basically concerned with whether a controlled systems will work satisfactorily under the circumstance it will meet in practice.

At various stages of the flight control laws design model uncertainty can be introduced, for example when linearized versions of complex models are derived. Similarly, if known variations in the position of the centre of gravity, or a time delay in the system, are neglected in the design model.

Model uncertainty can also be introduced unintentionally due to modeling errors, unknown characteristics of the aircraft in relation to the environment, or inaccurate information about the signal flowing through the systems (for example, the precise value of aerodynamic stability derivatives).

In classical control, the concept of robustness can be involved in the design procedure from the beginning, providing sufficient gain margin and sufficient phase margin, in order to take into account the effects of inaccurate modeling or disturbances. That means that in the Bode magnitude plot, loop gain should be high at low frequencies for performance robustness, and low at high frequencies, where unmodeled dynamics may be present, for stability robustness and sensor noise rejection.

Classical controls design techniques are typically in the frequency domain, so they imply convenient approach to robust design for single input/single output systems. However, the compliance of the individual gain margins, phase margins of all the SISO transfer functions in a multi input/multi output (MIMO) system doesn't guarantee its overall robustness.

Instead modern control techniques give to the designer a direct way to design multiloop controllers for MIMO systems by closing all the loops simultaneously. Performance is guaranteed in term of minimizing a quadratic performance index (as it will be explain in the next section) which implies closed loop stability.

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The classical frequency-domain robustness measures can be extended to MIMO systems, by using the notion of the singular values.

In this work the design technique used for the control law synthesis are robust and multivariable techniques, while the analysis has been conducted with standard classical techniques in frequency-domain.

1.3 Modern control theory

1.3.1 Introduction

Modern control theory had a considerable impact on the aircraft industry in recent years.

Modern control system design is based on two central concepts. The first is that the design is based directly on the state variable model; the second concept is the expression of performance specification in terms of a mathematically precise performance criteria which then gives matrix equations for the control gains. These matrix equations are solved using readily available computer software.

As introduced in the previous section, with the classical successive loop closure approach control gains are selected individually, instead the solution of matrix equations allows us to compute all the control gains simultaneously. That means that using the modern approach all loops are closed at the same time.

"The modern control formulation means that the trial and error of one loop at time design disappears. Instead, the fundamental engineering decision is the selection of a suitable performance criterion" (ref [1])

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1.3.2 Optimal control

Optimal Control Theory (OCT) is an extension of Calculus of Variation that deals with the problem of finding the control profile that optimizes a given performance index.

"The system resulting from an optimal design is not supposed merely to be stable, to have a certain bandwidth, or to satisfy any one of the constraints associated with classical control, but it is supposed to be the best possible system of a particular type, hence, the word optimal" (ref [3]).

The application of optimal control theory to the practical design of MIMO systems attracted much attention in the years '60-'80. This theory deals with linear finitedimensional systems represented in state space form, and with quadratic performance criteria.

The system may be affected by disturbances and measurement noise represented as stochastic processes, in particular, by Gaussian white noise. This class of design methods is called LQG theory. The deterministic part is called LQ theory. In the period since 1980 the theory has been further extended to address robustness aspects within theory framework.

The technique used in this work is the LQ theory, selected for its computational simplicity, and the associated robustness properties (as discussed below).

1.3.3 LQ theory

This paragraph of the thesis is intended to give only a short introduction to LQ theory; the reader can find in ref [3] and ref [4] an exhaustive explanation of the theory.

1.3.3.1 Performance index

Consider a linear time-invariant system represented in state space form as

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9

= +

=

For each t > 0 the state x(t) is an n-dimensional vector, the input u(t) a k-dimensional vector, and the output z(t) an m-dimensional vector.

We would like to control the system from any initial state x(0) such that the output z is reduced to a very small value as quickly as possible without making the input u unduly large. To this task, we introduce the performance index

= +

where the cost weight matrices are symmetric positive definite.

The first term in the integral criterion measures the accumulated deviation of the output from zero. The second term measures the accumulated amplitude of the control input.

The numerical choices of the matrices Q and R are very important in achieving performance and robustness in the closed-loop system, and it will be discussed in next paragraph.

The problem of controlling the system such that the performance index is minimal along all possible trajectories of the system is the optimal linear regulator problem.

An optimal trajectory is generated by choosing the input for t > 0 as

= −

The LQR Hamiltonian is, with ^∗ being the optimal cost

= + + ^∗ +

= 2 + ∇_! ^∗ , = 0

where the optimal control is

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∗ =1

2 ^%& ∇_! ^∗ ,

substituting ^∗we find the Hamiltonian-Jacobi equation

− ^∗ = +1

4

∗ %& ∇_! ^∗+ ^∗ −1

2

∗ %& ∇_! ^∗

It's possible to demonstrate that the optimal cost ^∗is a quadratic time-varying function of the system state

∗ = ^∗ , = (

Substituting we get

∗ , = (

)−( − ( − ( − + ( ^%& ( * = 0

Since this must be satisfied for any state , we find the Riccati equation:

−( − ( − ( − + ( ^%& ( = 0

and so

∗ , = −1 2 ^%&

∗ , = − ^%& ( = −

In this way, we find the expression of the optimal control ^∗ , .

With the infinite-time form of the linear quadratic regulator problem ( ₊ = 0 , = +∞)

= +

+

solving the algebric Riccati equation (ARE)

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( + ( + − ( ^%& ( = 0

we get the optimal feedback solution

= − ^%& ( = −

where K is a constant matrix.

Software that solve the Linear Quadratic problem are available and efficient.

It's possible to modify the performance index J, in order to weight the output z, instead of the state x.

= +

+

The engineering judgment in modern LQ design appears in the selection of Q and R.

1.3.3.2 Weight selection

The choice of the weighting matrices Q and R is a trade-off between control performance (Q large) and low input energy (R large).

Increasing both Q and R by the same factor leaves the optimal solution invariant

The Q and R parameters generally need to be tuned until satisfactory behavior is obtained, or until the designer is satisfied with the result.

An initial guess is to choose both Q and R diagonal

= . ^& 0 0 0 ⋱ 0

0 0 ₀1 ; = . ^& 0 0 0 ⋱ 0

0 0 ₃1

where Q and R have positive diagonal entries such that

4 5 = 1

506! ; 4 5 = 1

506!

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The number ₅^06! denotes the maximally acceptable deviation value for the i th component of the output z. The other quantity ₅^06! has a similar meaning for the i th component of the input u. Starting with this initial guess the values of the diagonal entries of Q and R may be adjusted by systematic trial and error.

The choice of weighting matrices has been widely discussed in literature (ref [2]-[3]-[4]).

1.3.3.3 Guaranteed Stability Margins

The LQR solution has excellent stability robustness properties at the input to the plant.

This can be shown by examining the return difference matrix in the frequency domain.

The loop transfer function L(s), with the loop break point at the plant input (fig 1.1), is defined as

7 8 = 89 − ^%&

Figure 1. 1 Open loop at the plant input

Re-arranging the Riccati equation we find (all the analytical steps are explained in ref [5])

+ 7 8 + 7 −8 + 7 −8 7 8 = :^∗ : +

where : is the open loop dynamics : = 89 − ^%& .

The term :^∗ : is a Hermitian positive semidefinite matrix.

Removing this term on the right side and assuming that = ;9 with ρ>0 we obtain the inequality

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<9 + 7 8 =^∗<9 + 7 8 = > 9

Figure 1. 2 Frequency domain Analysis of LQR transfer functions

This implies a minimum gain margin of ?−6 ∶ +∞B dB and a phase margin of at least 60°.

This property is guaranteed for any choice of Q matrix.

1.3.3.4 Frequency shaping

Frequency shaping design technique blends classical control design with linear quadratic method.

The key point of the frequency shaping is to augment the plant with a standard filter (generally PI or PID filter), and then apply linear quadratic regulator to the resultant plant.

Standard LQ design for this augmented plant can be interpreted as a frequency-shaped design for the original plant (here the reader can find a resume of the procedure, for an exhaustive explanation see ref [3]).

The steps of frequency shaping procedure are:

1. Define output variables (termed regulated variables) as linear combinations of the states

& = C

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The elements of _&, will usually consist of actual plant measurements or combinations of such. The intention is to regulate the variables _& to zero, or have them track a constant reference r.

2. Augment the plant: append proportional plus integral plus derivative (PID) filters to each element of _&. The PID gains are selected to achieve minimum phase target zeros intended to attract the closed-loop poles in a closed-loop design. The zero assignments can be considered as an approximate closed-loop pole assignments, so should be suitably damped and in appropriate frequency ranges.

We have now constructed the augmented plant shown in fig. 1.3

Figure 1. 3 Augmented plant

with transfer function matrix

( 8 = D< _EC + _FC = + _GC1

8H 89 − ^%&

The state-space form representation of the augmented plant is

= +

& = C

with state name and _& = I &

3. Apply linear quadratic state feedback control laws for this augmented plant (which penalize the augmented plant output _J).

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15 The control law obtained is:

= ^K + _&^K _& = ^K + _&^K C

So, instead of proportional state feedback K‘, we have dynamic state feedback

K 8 = ^K+ _&^KC1 8

Fig 1.4 shows the resultant plant, controlled with dynamic state feedback

Figure 1. 4 Resultant plant, controlled with dynamic state feedback

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1.4 Gain scheduling

1.4.1 Introduction

Gain-scheduling is one of the most popular approaches to nonlinear control design.

With the term “gain-scheduling” we refer to a wide class of control methods, where the nonlinear control design task is decomposed into a number of linear sub-problems (see ref [6]).

Considering the nonlinear plant

= L , M = N ,

The classical gain-scheduling design approach is based on the family of equilibrium linearization of the plant.

The design procedure typically involves the following steps (as explained in ref [20]):

1. Linearize the plant about a finite number of representative operating points.

2. Design linear controllers for the plant linearizations at each operating point.

3. Interpolate the parameters of the linear controllers at all points where the plant is expected to operate.

4. Implement the gain-scheduled controller on the nonlinear plant.

The resulting gain-scheduled controller is nonlinear, its parameters evolving as functions of the plant states, inputs, outputs, or any combination thereof.

In this work, the method adopted to implement gain-scheduled controllers, is the D- Method.

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1.4.2 D-Method

As discussed before, a common approach to the design of a gain-scheduled controller consists in designing a family of linear controllers for a finite number of plants, and then interpolating between controllers, in order to achieve adequate performance for all the linearized plants.

During real-time operation, the controller parameters are updated as functions of the gain-scheduling variable O. A gain scheduling unintended consequence is the introduction of feedback implicit loop, due to the variation of the scheduling variable O.

The D-method is based on the observation that linear controllers are designed to operate on the perturbations of the plant’s inputs and outputs about the equilibrium points. So the different controllers should work with such perturbations. This is achieved by differentiating some of the measured outputs before they are fed back to the gain- scheduled controller. Then integral action is provided at the input to the plant, in order to preserve the input-output behavior of the feedback system.

Furthermore, the integrator presence, mitigates the possible disturbance introduced by the implicit loop in O (as will be demonstrated in cap 3).

This is the method selected to be used in this work: the resulting nonlinear gain- scheduled controller is easy to obtain, and its structure is similar to the original linear controller structure.

Figure 1. 5 D-method

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Resuming, the D-method consists in providing integral action at the inputs to the plant and differentiating some of the outputs before they are fed back to the scheduled controller.

The reader can find an accurate analytical demonstration of D-method in ref [7].

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2 - Stability and Handling criteria

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2 - Stability and Handling Criteria

2.1 Introduction

Chapter 2 describes the criteria used to valuate stability and handling performances of the proposed controller.

Ref [8] defines the following flight phases in line with current military standards (ref [15]- [16]).

1. Category A

Non terminal phases requiring rapid manoeuvre, precision tracking or precise flight path control

• Air-to-air combat

• Ground attack

• Weapon delivery or launch

• Reconnaissance

• In-flight refueling

• Formation flying

2. Category B

Non terminal phase normally accomplished with gradual manoeuvre and without precision tracking

• Climb

• Cruise

• Loiter

• Descent

• Emergency descent

• Emergency deceleration

3. Category C

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20 Terminal phases

• Takeoff

• Approach

• Wave-off/go-around

• Landing

For each flight phase category, three level of flying qualities are identified :

Level Description

Level 1 Flying qualities clearly adequate for the mission Flight Phase

Level 2 Flying qualities adequate to accomplish the mission Flight Phase, but some increase in pilot workload or degradation in mission effectiveness, or both, exists

Level 3 Flying qualities such that the airplane can be controlled safely, but pilot workload is excessive or mission effectiveness is inadequate, or both.

Category A flight phase can be terminated safely, and category B and C Flight Phases can be completed

Tab 2.1 Flying qualities levels

Lateral stability and handling quality discussed in this chapter are discussed in detail in ref [8].

2.2 Lateral Stability

2.2.1 Stability margins

For the Nominal condition, encompassing the full range of aircraft CG, mass and inertia, the lateral and directional open-loop frequency responses with loops broken at aileron

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(or Rolling-Moment demand if both aileron and diff-HST are commanded) and rudder actuators shall comply with the Nominal boundaries of ref [8].

Single-loop lateral stability margins shall be measured via linear analysis of the separate Aileron and Rudder actuator-actuator open-loop frequency responses.

All other feedback loops shall be closed, including any roll/yaw cross feeds and including any implicit feedbacks induced by scheduling, unless these are separately analyzed and shown to have no effect for the system in its trimmed equilibrium state.

For conditions where both Aileron and Differential-HST are active and commanded from a single output rolling-moment demand, they shall be treated as effectively a single control surface and the loop opened accordingly at the Rolling-Moment demand point.

2.2.2 Controlled modes

Lateral/directional controlled-aircraft modes shall be determined via linear analysis of the closed-loop model.

All applicable feedback loops shall be closed, including any implicit feedbacks induced by scheduling, unless these are separately analyzed and shown to have no effect for the system in its trimmed equilibrium state.

No use is made of equivalent-system analysis, and the criteria here all apply to the directly computed closed loop modes.

A conventional design will generally result in distinct and recognizable Dutch-Roll (DR) and "FCS" modes together with a roll-mode time-constant.

These modes are assessed here for damping and frequencies: low mode-damping is a direct indicator of potentially unsatisfactory handling or turbulence sensitivity, and DR frequency gives an indication of basic bandwidth that relates to aircraft response and disturbance rejection.

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For the Nominal condition, encompassing the full range of aircraft CG, mass and inertia the lateral/directional Spiral Mode, Roll Mode, Dutch-Roll and FCS modes shall comply with the Nominal boundaries of ref [8]

2.3 Lateral Handling

2.3.1 Long period response

If the spiral pole is stable, there aren't particular requirements to meet.

Instead any regions of unstable spiral mode at Nominal and Tolerance conditions shall be documented for the purpose of manned simulation assessment to confirm that no difficulty or hazardous flight path deviation is induced during short periods of pilot head- down attention.

2.3.2 Short period response

2.3.2.1 Linear Roll time response

The linear roll time-response at all AoA shall comply with the numerical values of ref [8]

under Nominal and Tolerance conditions for each handling Level.

This identifies a deadbeat roll-rate response with a target range for the roll time-constant resulting from command path design, and a target limit on roll-acceleration lag and magnitude.

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Figure 2. 3 Roll linear time response

The parameter _Q has a clear physical interpretation. Lateral stick command is thought to be a step, applied by the pilot for a finite temporal interval.

0 < < _S

The target is to obtain a desired φ.

When the command is removed p returns to zero with the approximated expression:

T = T_UU 1 − V^{% WX}

The roll steady-state value (before the stick is moved back to zero) is defined as:

T_UU =T< _S= − T _S− 2 2

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YZ is defined as the difference between the bank angle reached before pilot removes the command, and the final bank angle obtained.

Q = YZ

T_UU = I T₊

T_UU =

So _Q gives an important indication of the Z overshoot.

2.3.2.3 Linear pedal time-response

The linear pedal time-response shall comply with the numerical values of Figure under Nominal and Tolerance conditions for each handling Level.

This identifies a settling time of the sideslip response to pedal together with the related roll response characteristic.

Figure 2. 5 Sideslip time response

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25 2.2.2.4 Dihedral effect

Appropriate command crossfeed shall be used at all AoA to produce a stable (positive) effective dihedral characteristic, whose magnitude may be selected by pilot simulation assessment to lie between near-neutral and a conventional moderate roll-to-sideslip characteristic.

For Level 1 and 2 handling however, the selected dihedral characteristic shall lie between a minimum level that produces neutral rolling in response to pedal for a worst-case Tolerance condition, and a maximum level that can be trimmed by the smaller of 75% or 10lb roll inceptor input to hold wings level at full pedal, also under Tolerance conditions.

For Level 3 the rolling response to full pedal shall be balanced by a roll inceptor input of less than 20lb.

2.3.3 Gust response

The requirement reported in ref [8] gives an upper limit on the yaw- and bank-to-gust response transfer-functions to allow identification of any likely problematic characteristic.

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3 – Control law Synthesis

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3 – Control laws Synthesis

3.1 Problem definition

3.1.1 Introduction

The M-346 Master is an advanced military trainer, showed in fig. 3.1. Its Flight Control System is a full authority digital Fly-By-Wire, multiple-redundant, reconfigurable, full time system, providing both pilot manual and automatic capability.

Primary flight control laws are the core of the FCS, and they are of the type “manoeuvre demand”.

The goal of this synthesis is to evaluate an alternative design the lateral directional controller of the primary flight control laws, for a limited flight envelope (in term of AoA and speed range). The resulting closed loop system shall tracks roll rate and angle of sideslip (AoS) as commanded by the pilot by means of lateral stick and pedal deflections.

Figure 3. 1 M346 Master

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The synthesis process of the control laws follows these steps:

• Design conditions definition: Mass/CG/Inertia, Altitude, AoA, EAS, and configuration).

• Linear Design of the controller parameters related with the stability augmentation (feedback)

• Linear Design of the controller parameters related with handling quality (feedforward, command path)

• Non linear implementation of the linear controller (gain scheduling, control allocation,etc)

• Non linear functions implementation (authority scheduling, carefree protections, etc.)

Definition of the controller structure is an implicit and transversal step to all the other steps listed before.

3.1.2 Flight envelope

Having only a demonstration purpose, this study was conducted on a limited flight envelope to reduce the effort. For each selected flight condition, the task of the synthesis process is to return a gains set which assure the request performances and stability margins for each case obtained (combining different Mass cases with Aerodynamic tolerance cases

The design points chosen for the synthesys are resumed in tab 3.1

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28 Flight envelope

Mach [0.2 0.4 0.6 0.8]

Altitude [ft] 15000

AoA [deg] [-5 0 5 10 15]

Aircraft configuration

External Store Configuration Clean

Flap Cruise

Landing gear Up

Airbrake Retracted

CG/mass par 3.1.3

Aero tolerance par 3.1.4

Tab 3.1

3.1.3 Mass Properties Variations

As is well known by flight mechanic study, the CG position influences the aircraft stability.

This is the reason why, in order to design a robust controller, several mass/CG configurations have to be analyzed in the same flight point.

Fig 3.3 shows the target mass/x_cg envelope of the clean aircraft (similarly in fig. 3.4 shows the inertia envelope). To cover all possible condition this study considers six combinations of mass properties and CG position, chosen at the corners of the envelope (which are the most critical conditions).

For brevity, each mass case is indicated with two letters: first represent the CG position (F-forward, A-aft, M-medium), the second represent the inertial class (L-light, M-medium, H-heavy).

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3.1.3 Aerodynamic Tolerance

Aerodynamic Simulation Model and Dataset are the best possible representation of M346 Stability & Control information.

Eventual deviations from flight data may be due to wind tunnel constraints, Reynolds’

number effect, uncertainties in structural flexibility calculations, data discretization error.

A consequence of neglecting such deviations could be a reduction of FCS stability margins leading to deterioration in handling qualities and possible loss of control.

Tolerances have been determined on the basis of M346 models wind tunnel tests data repeatability, deviation between data acquired in different wind tunnels and on comparisons between wind tunnel M346 models data and M-346 flight test data.

DEF. TOLL

Nominal -

C

_[_\ ^pClmp ^+Y C^[^\

nClmp −Y C_[_\

C

_]_{^_`} ^pCnmdr ^+Y C^]^{^_`}

nCnmdr −Y C_]_{^_`}

C

[abc

pClmda +Y C_[_abc nClmda −Y C_[_abc

C

_[_d ^pClmbe ^+Y C^[^d

nClmbe −Y C_[_d

C

_]_d ^pCnmbe ^+Y C^]^d

nCnmbe −Y C_]_d

Tab 3.3

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3.2 Linear design

In this paragraph, all the steps made to obtain the control laws structure and respective gains are explained.

3.2.1 Lateral-Directional AC Model

The linearized equations used to describe the lateral aircraft dynamic have been derived by the non linear model of rigid body dynamic.

e fg = Lg

ffg − fg˄ifg = jffg

Under the assumption of pole coincident with CG, flat and non rotating heart, and reference system fixed with the aircraft CG the equations become

kl mifg + nfg˄ifgo = Lg + pfffg 9nfg + nfg˄ 9nfg = jffg

where assuming x-z as a symmetry plane

9 = q9_! 0 9_!r

0 9_s 0

9_!r 0 9_rt

For the small perturbation theory each variable of a dynamic problem can be expressed as the sum of a steady state value (trim condition) and a perturbation share.

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31 e ifg = ifffg + ug₊

nfg = nffffg + vffg₊ Lg = Lfffg + YLg₊

under the assumption of small perturbation ^wfg

xffffg≪ 1 etc.

where

ug = zu

{| , vffg = zT

}~| , nffffg = •₊ T₊ }₊

~₊€ , ifffg = •₊ u⁺₊ {₊€

Substituting these relations in the aircraft dynamic equations, removing the infinitesimal of higher order and the steady state parts, the mathematical linearized model is obtained.

The complete set of linear equations for the decoupled lateral small perturbation dynamics, expressed in body axes, is the follower.

•‚

ƒ u + ₊~ − {₊T − „ cos ˆ₊sin ˆ₊: = ‹

9_!T − 9_!r~ − m<9_s− 9_r=}₊− 29_sr~₊− 9_!sT₊o ~ + <9_!s~₊− 9_!r}₊=T = 7 9_r~ − 9_!rT − m<9_!− 9_s=}₊+ 29_!sT₊− 9_sr~₊o T + <9_!r}₊− 9_srT₊=~ = Œ

When lateral force Y, roll and yaw moments (L,N) are expressed as follows:

e

‹ = ‹w<u − u•= + ‹65[Ž•• + ‹‘U Fℎ8 + ‹Q“F~ + ‹ET + ‹”<~ − u•⁄ = + ‹i+ w <u − u•= + ‹ET /j 7 = 7w<u − u•= + 765[Ž•• + 7‘U Fℎ8 + 7Q“F~ + 7ET + 7”<~ − u•⁄ = + 7i+ w <u − u•= + 7ET Œ = Œw<u − u•= + Œ65[Ž•• + Œ‘U Fℎ8 + ŒQ“F~ + ŒET + Œ”<~ − u•⁄ = + Œi+ w <u − u•= + ŒET

and

‹_w ‹_65[ ‹_{‘U F} 7_w 7_65[ 7_{‘U F} Œ_w Œ_65[ Œ_{‘U F}

‹_Q“F ‹_” ‹_w 7_Q“F 7_” 7_w Œ_Q“F Œ_” Œ_w

‹_E 7_E Œ_E

are the aerodynamic stability derivatives .

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32 In the standard state space form representation:

= +

M = C +

Inputs:

















g d

v rud hst

ail

, States:

























Ψ r p v

φ , Outputs:





















Ψ

cg

nY

r r p p

,

&

φ β

.

3.2.2 Model reduction

The previous model has been simplified, in order to be applied to the selected design technique. Since the Linear Quadratic Regulator requires a full-state it was necessary to reduce as much as possible the state variables used to describe the dynamic system, in order to avoid the use of more complex reduction techniques for the controller size.

The synthesis model is expressed in wind axis, through the transformation

—T_˜

~_˜™ = šcos O sin O sin O −cos O› —T_œ

~_œ™

The AC Lateral model reduced is described by the following representation:

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The state variables chosen to represent the Aircraft model are the roll rate sideslip β and •_Fž . The latter is obtained through the approximated relationship:

Assuming this approximation,

Removing the state φ we a

explanation: first, we wouldn’t like to have a state feedback in φ (control laws we are looking for are aimed to control fast dynamics, i.e. roll and Dutch roll), so we have to delete it from the states; second this is generally a good approximation (especially for high speed condition).

Anyway, the validity of this assumption, and of all the other simplifications, will be discussed in cap 4.

•‚

ƒ u

9_!T − 9_!r~ − m<9 9_r~ − 9_!rT − m<9_!

Resuming, the state space form representation of the reduced model is:

33

Figura 3. 2 Lateral-directional AC model

variables chosen to represent the Aircraft model are the roll rate . The latter is obtained through the approximated relationship:

• = T sin O ~ cos O

Assuming this approximation, • ~_˜.

Removing the state φ we are neglecting spiral mode. The reason of this choice has dual explanation: first, we wouldn’t like to have a state feedback in φ (control laws we are looking for are aimed to control fast dynamics, i.e. roll and Dutch roll), so we have to e states; second this is generally a good approximation (especially for

Anyway, the validity of this assumption, and of all the other simplifications, will be

u ₊~ {₊T „ cos ˆ₊sin ˆ₊: ‹ m<9_s 9_r=}₊ 29_sr~₊ 9_!sT₊o ~ <9_!s~₊ 9_!r m< _! 9_s=}₊ 29_!sT₊ 9_sr~₊o T <9_!r}₊ 9_sr

Control law Synthesis

variables chosen to represent the Aircraft model are the roll rate T, the angle of . The latter is obtained through the approximated relationship:

re neglecting spiral mode. The reason of this choice has dual explanation: first, we wouldn’t like to have a state feedback in φ (control laws we are looking for are aimed to control fast dynamics, i.e. roll and Dutch roll), so we have to e states; second this is generally a good approximation (especially for

Anyway, the validity of this assumption, and of all the other simplifications, will be

!r}₊=T 7

srT₊=~ Œ

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• State name:

• Input name:

• Output name

:

3.2.3 FCS model

This paragraph shows the simplified model used to describe the FCS hardware and software components of the system. The task is to obtain a model with the lowest possible number of states, but yet representative of the physical problem we are analyzing.

After a preliminary study of the complete model characteristics, the choices made to represent all FCS contributions are the following:

• 1st order lag filter;

• A lumped delay represented by a

The first order Lag could be thought as a fictitious actuators (even if it doesn't correspond to the real actuator model, because it incorp

34 •, T_˜, •_Fž

Ail, dHst, Rud, Vg

•, T_˜, •_Fž , Ÿ_s , ~_˜ , T_Fž

This paragraph shows the simplified model used to describe the FCS hardware and components of the system. The task is to obtain a model with the lowest possible number of states, but yet representative of the physical problem we are

After a preliminary study of the complete model characteristics, the choices made to ent all FCS contributions are the following:

A lumped delay represented by a 1st order Padè

The first order Lag could be thought as a fictitious actuators (even if it doesn't correspond to the real actuator model, because it incorporates also part of the HW delay).

Figura 3. 3 FCS representation

This paragraph shows the simplified model used to describe the FCS hardware and components of the system. The task is to obtain a model with the lowest possible number of states, but yet representative of the physical problem we are

After a preliminary study of the complete model characteristics, the choices made to

The first order Lag could be thought as a fictitious actuators (even if it doesn't correspond orates also part of the HW delay).

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N

₆

In the state space form, Matlab transfer function) the following results:

Input name: ~¡•

Output name:

State name: ~¡••

In this way we have introduc

will be used for the feedback by the LQR technique The FCS system has been concatenated at

the simplified model (aircraft+FCS), to be used for the design, has been completed.

Figure 3.

35

6 &

W_a¢£U¤&

₆ _{1 v}X ₆

N_F¥[6s 2 8 1

2 8 1

In the state space form, Matlab gives (as one of the possible representation given a transfer function) the following results:

~¡••_0F , ~ _0F Output name: Ail, dHst, Rud

~¡••¦[s , ~¡••₆ , ~ ¦[s , ~ ₆

introduced four new states, ~¡••_{¦[s ,}~¡••₆ , ~ e feedback by the LQR technique.

The FCS system has been concatenated at the bare reduced aircraft model. In this way, the simplified model (aircraft+FCS), to be used for the design, has been completed.

Figure 3. 4 Series of FCS simplified model+AC model

X

gives (as one of the possible representation given a

~ _{¦[s ,}~ ₆, which

the bare reduced aircraft model. In this way, the simplified model (aircraft+FCS), to be used for the design, has been completed.

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36 The complete state space rappresentation is:

Input name: ~¡••_0F , ~ _0F , u_•“U Output name:

•, T, •_Fž , Ÿ_s , ~ , T_Fž

State name: •, T, •_Fž , ~¡••¦[s , ~¡••₆ , ~ ¦[s , ~ ₆

3.2.4 Linear Quadratic Regulator

3.2.4.1 Frequency shaping

This step of the synthesis process consists in augmenting the plants with suitable filter, in order to apply the LQ regulator to the augmented plant (this procedure can be interpreted in terms of a frequency-shaped design for the original plant).

The filter chosen to augment the plant is the PID filter.

Figura 3. 5 Augmented plant

M_J = _EM_&+ _G M_& + _FM_&

PID filter is applied to the state variable T and β

TE_U = 8 + ^GE ^EE+ _¦E8

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37

§

•

G§

8 _E§+ _¦§8

In closed loop poles are attracted by zeros of the transfer function, so the assignation of the zeros filter in open loop, corresponds approximately to the desired position of poles in closed loop.

• =5 8^J+ 2 ¨ ₅v ₅8 + v₅^J 8

Damping of the complex zeros is chosen always very high, in order to obtain the require damping for the Dutch roll mode.

¨_§ = 0.9

¨_E = 0.85

Zero frequency is chosen near Dutch roll frequency

3.2.4.2 Construction of the “Performance Criteria” system

The "Performance criteria" system is the plant which will be connected in series to the Aircraft model, in order to obtain the augmented plant used for the synthesis. In the states space representation it has the following form:

=

_E

+

_E

= C

_E

+

_E

E = š0 00 0› ^E= š1 0 0 0−1 0

0 1 0 0

0 −1›

CE ¬ ^9T 0

0 _9•- E D ^TT _T

0 0

0 _T• 0 0

_• 0H

with

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_G5 v₅^J, _E5 = 2

Inputname: T_5] , T_Fž Outputname:

_E _§

Statename: T_5] •_5]

The performance criteria system, connected with aircraft+

synthesis model shown in fig 3.16

38 2 ¨ ₅v ₅ , ¦5 1

Figure 3. 6 Performance criteria system

Fž , T_0F •_5] , •_Fž , • 0F

5]

The performance criteria system, connected with aircraft+FCS model, gives the complet shown in fig 3.16

model, gives the complete

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3.2.4.3 LQR with output weighting

Rearranging input and output, we find the plant which will be controlled with LQR

Note that T_0F and •_0F

regulator problem; the tracking problem will be dealt in the next paragraph (with the introduction of feed forward gains).

The controller, in the form of

39

Figure 3. 7 Complete synthesis model

3.2.4.3 LQR with output weighting

input and output, we find the plant which will be controlled with LQR

Figure 3. 8 Complete augmented plant

0F are been neglected: the problem dealt in the first step is the

; the tracking problem will be dealt in the next paragraph (with the introduction of feed forward gains).

The controller, in the form of

= −

input and output, we find the plant which will be controlled with LQR

are been neglected: the problem dealt in the first step is the

; the tracking problem will be dealt in the next paragraph (with the

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40

is chosen in order to minimize the performance index J (see chapter 1)

1

2 I +

Where

= m

^Qž[[_Q“F^¢®`_¢®`

o = m

_r^r^\_d

o

3.2.4.4 Choice of matrix weight

As explained in chapter 1, the choice of matrix weight in the performance index is an iterative procedure, based on a trial and error process.

There are some guidelines which have been used in this study: as suggested in literature (ref [1]) Q and R have been chosen diagonal.

E§= D T 0 0 _§ H

E§ = D ^65[ 0 0 _Q“F H

The strategy used in the choice of Q is the following:

E§ = ¬ ~¡••V¯¯V° ^%J∗ }_E 0

0 MŽ{V¯¯V° ^%J∗ }_§ -

Where ~¡••V¯¯V° e MŽ{V¯¯V° change according to the selected design condition.

In this way, _E e _§ are normalized with respect to the roll effect and yaw effect of each flight point analyzed: if roll effect increase, the weight of p in the performance index decrease. Similarly, the choice of _65[ and _Q“F is based on consideration about aileron and rudder control power.

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41

Anyway, even if these assumptions have been useful for the construction of the matrix structure, the precise choice of each weight was completed with an iterative procedure.

3.2.4.4 Results

After the assignation of the system and the matrix weight, LQR gains feedback are computed directly with Matlab routine.

• T •Fž ~ ¦[s ~ 6 ~¡••¦[s ~¡••6 T5] •5]

[±

= z ²

&,&

²

&,J

²

&,³

²

&,´

²

&,µ

²

&,¶

²

&,·

²

&,¸

²

&,¹

²

_J,&

²

_J,J

²

_J,³

²

_J,´

²

_J,µ

²

_J,¶

²

_J,·

²

_J,¸

²

_J,¹

|

3.2.5 Feed-Forward gains

The gains matrix obtained solving the linear quadratic regulator problem is augmented adding feed forward gains, using the following relations (see ref [9]):

= − _º» + ^K

K = ^%& 0 ~

where 8 = C 89 − + _º» ^%&

~ = m^§_E^¢®`_¢®`o

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42

¼½ _¼½ C_¼½ are the matrix which describe the lateral AC model.

Solving the previous equations matrix feed forward gains is computed

ºº = − C _º»− ^%& ^%& ?2¾2B

3.2.6 Closed loop design model

The complete gains matrix is the following:

• T •_Fž ~ _¦[s ~ ₆~¡••_¦[s ~¡••₆ T_5] •_5] •_0F T_0F

ž = z²&,& ²_&,J ²_&,³ 0 ²_&,µ 0 ²_&,· ²_&,¸ ²_&,¹ ²&,&+ ²&,&&

²_J,& ²_J,J ²_J,³ 0 ²_J,µ 0 ²_J,· ²_J,¸ ²_J,¹ ²_J,&+ ²J,&&|

Gains feedback correspondent to the state name ~¡••_¦[sand ~ _¦[sare set to zero.

This choice has two reasons: first, looking at the numerical results, it's possible to see that this feedback gives a smaller contribute than the other; second, these states don't correspond to a real physic state of the system, but derive from the simplified FCS model.

Then, other simplifications were made, in order to further reduce the number of feedback: after the observation of the numerical value, feedbacks of p_rud, p_cmd_rud, p_int_rud, ail_rud are neglected. In this way we have considerably reduced the number of cross-feed path.

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Figura 3.

43

Figura 3. 9 Complete synthesis model + control laws

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3.2.6 Choice of gains set

The described procedure

aerodynamic tolerance case, this providing (Altitude, Mach, AoA fixed)

Each gains set, represents the optimal controller for the particular conditions.

For each flight condition, we have implemented in the scheduler

condition, each gains set is applied at all possible combination of mass variation and aero tolerance (for the fixed flight point). Among all gains sets that meet the requirements (Gain margin, phase margin ecc), the lowest bandwidth solution is

3.3 Surfaces use strategy

M346 has five primary surfaces: two ailerons, two fully movea rudder.

44

3.2.6 Choice of gains set

described procedure has been applied for each combination of mass/CG case, this providing 66 gains set for each flight condition (Altitude, Mach, AoA fixed)

flight condition, we have then to select only one gains set, which will be in the scheduler. The logic used to choose the gains set is: fixed a flight set is applied at all possible combination of mass variation and aero olerance (for the fixed flight point). Among all gains sets that meet the requirements (Gain margin, phase margin ecc), the lowest bandwidth solution is then

Surfaces use strategy

M346 has five primary surfaces: two ailerons, two fully moveable horizontal tail, and

Figure 3. 10 M346 primary control surfaces

has been applied for each combination of mass/CG - 66 gains set for each flight condition

only one gains set, which will be . The logic used to choose the gains set is: fixed a flight set is applied at all possible combination of mass variation and aero olerance (for the fixed flight point). Among all gains sets that meet the requirements

then chosen.

ble horizontal tail, and

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45

The aerodynamic rolling moment (produced by surfaces deflection) is generally modeled as a function of wing area (S), wing span (b), dynamic pressure (q) as:

7 }¿ÀC_[

The parameter C_[ is a function of aircraft configuration, sideslip angle, AoA angle, rudder deflection and aileron deflection. In a symmetric aircraft with no sideslip and low AoA, this coefficient is linearly modeled as:

C_[ C_[_ÁaÂ₆

The parameter C_[_Áa is referred to as the aircraft rolling moment-coefficient-due-to- aileron-deflection derivative and is also called the aileron roll control power.

By the analysis of aerodynamic dataset, the dependence of aileron control power with aileron deflection is evidenced

The aileron control power quickly decreases with the increasing of the surface deflection:

when the aileron is deflected of 10° the control power is reduced of 25% respect at the equilibrium position, while at a 20° deflection correspond a reduction of 50%.

Differently the horizontal tail control power remain almost constant up to 20° deflection.

So, on the base of the above aerodynamic considerations, in order to have an almost- constant control power (important for the final closed-loop robustness) and to provide an efficient allocation of control demand to the available control surfaces, a split ratio between aileron and Hdst has been defined.

The selected split ratio is a function of aileron deflection, AoA and Mach

Â

_{ÃU F}

Â

_65[_UE[5

jŽ°ℎ, ¡ , Â

_65[

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46

3.4 Non Linear Design

3.4.1 Non linear controller

Control law has been implemented in the simulation model using the Fortran language.

Fig 3.24 shows a block diagram simplified, which doesn't correspond to the real complete implementation, but it allows to understand the different paths which form the control law.

Figure 3. 11 Control law block diagram

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47 Signal preprocessing Command path Feed-forward path Feed-back path Exe-dmd path

In feedback path and feed-forward path, we can find gains computed in the linear design, implemented with D-method, and scheduled as function of Mach and AoA.

Command path and exe-dmd paths contain non linear functions and care free protections.

The following paragraphs explain in detail the different paths and the non linear implementation logic.

3.4.2 Signal pre-processing

In the signal pre-processing path we have the transformation from body axis to wind axis, and the complete computation of •_Fž .

T_˜ cos O ∗ T_œ sin O ∗ ~_œ

•_Fž sin O ∗ T_œ− cos O ∗ ~_œ+ „:°¡8 ˆ i₊ + Ÿ_s

i₊„