Design of Optimally Convex Controller for Pitch Control of an Aircraft

Design of Optimally Convex Controller for Pitch Control ofan Aircraft

P. S. Khuntia, A. S. Chaudhuri

Abstract

The convex optimization problems for development of linear time-

invariant controllers are more prevalent in practice than was previously

thought. Since 1990 many applications have been discovered in areas of

automatic control systems of aircraft. The solution methods are reliable

enough to be embedded in a computer-aided design or analysis tool, or

even a real-time automatic control system. There are also theoretical or

conceptual advantages of formulating a problem as a convex optimization

problem. Our main goal is to develop a working knowledge of convex

optimization, i.e., to develop the skills and background needed to

recognize, formulate, and solve convex optimization problems pertaining

to aircraft control. In this paper a design method is proposed to solve

control system design problems in which a set of multiple closed loop

performance specification must be simultaneously satisfied. To utilize

this approach all close loop performance specification considered must

have the property that they are convex with respect to closed loop

system transfer matrix. For close loop performance specification a close

loop controller chosen from a set of all linear controllers determined

by trial and error such that the specification is satisfied. The

transfer matrix of the final system is determined through the convex

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combination of the transfer matrices of the plant and the controllers.

With the close loop transfer matrices given the closed loop controller

structures and its gains are solved algebraically. In this paper we

established conditions for the existence of a designed parameter

inherent in convexity. The experimental verification deals with a

problem of pitch control of flight dynamics of a rigid body aircraft.

Key words: convex controller, parameter optimization, transfermatrices, integral square error(ISE), pitch control,

Introduction

Convex controller design is an approach which is increasingly used to

solve close loop system design problems of robotics, mechatronics, high

performance aircraft and flexible space structures[1-2] .such problems

typically required that a set of designed parameters and control gains

be adjusted simultaneously so that a prescribed close loop system

performance is achieved. This system design is termed as convex

controller design in the relevant literature [3-5]. Optimization methods

are extensively used to solve these design problems. To apply the convex

controller methods all performance specifications must be convex with

respect to the close loop transfer matrix. The close loop transfer

matrices of the systems are combined in a convex combination to form a

single transfer matrix, which satisfies that close loop performance

specification. Boyd et al [ 6 ] first pointed out that many commonly used

performance specifications, such as percentage overshot, control

efforts, robust stability are convex with respect to the close loop

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transfer matrix. Controller design can be greatly simplified by taking

advantages of this convex property. Based on

this convex property the present paper proposes linear controller

methods for design pitch control of an aircraft.

This paper is meant for the researcher, scientist, or engineer who uses

mathematical optimization, or more generally, computational mathematics.

This includes, naturally, those working directly in optimization and

operations research, and also many others who use optimization, in

fields like computer science, economics, finance, statistics, data

mining, and many fields of science and engineering. The primary focus is

on the latter group, the potential users of convex optimization.

The goal of control engineering is to improve, or in some cases enable,

the performance of a system by the addition of sensors, which measure

various signals in the system and external command signals, control

processors, which process the measured signals to drive actuators, which

affect the behavior of the system. A schematic diagram of a general

control system is shown in Fig. 1.

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Fig. 1. A schematic diagram of ageneral control system.

The use of the sensed response of the system (and not just the

command signals) in the computation of the actuator signals is called

the feedback control, an old idea which has been developed and applied

with great success in this century .

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Other signals which affect system (disturbances)

System to be controlled

sensors

Sensed signals

Command signals(operator inputs)

Actuator signals

Operator display warning indicators

ControlProcessor(s)

actuators

The purpose of this paper is to describe how the fundamental problem of

controller design can be solved for a restricted set of systems and a

restricted set of design specifications, by combining a recent theo-

retical result with numerical convex optimization techniques [6-8]. The

restriction of the systems is that they must be linear and time-

invariant (LTI). If the specifications are achievable, it is possible

to find a controller which meets the specifications, although the

controller found may be complex and higher order. It may be possible

to find a simpler controller which meets the specifications, for

example by some model reduction technique [9].

Figure 2:classical feedback control setup

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K

P+_

er ye

Classical Synthetic Open-Loop DesignClassical synthetic open-loop design methods have their origin in the

work of Bode . These methods are extremely widely studied and applied,

and are described in many current introductory control texts, such as

[10-13]. Examples of the numerous works developing and extending syn-

thetic open-loop techniques are [14-17]. In this section we will briefly

describe and comment on this kind of approach to control design.

Consider the classical feedback control setup shown in Fig. 2. Given

the plant, P, the designer must find a controller K such that results

should give satisfactory closed-loop performance. Classical open-loop

methods concentrate on designing the loop gain, L = PK, the closed-loop

transfer function from r to y, PK / (1 + PK).

Parameter optimization methods

Decomposition of the plant inputs and outputs are shown in figure-3.

The inputs to the model are divided into two vector signals. The

actuator or control signal vector, denoted by u, will consist of those

inputs to the model that can be manipulated by the controller. The

actuator signal u is the signal generated by the controller, other

input signals to the model will be lumped into vector signal w, called

the exogenous input. The sensor or measured signal vector, denoted by

y, will consist of those output signals that are accessible to

controller. The sensor signal y will be the input signal to the

controller. The output signals from the model will be lumped into a

vector signal z, called the regulated variables.

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Figure-3 :Decomposition of the

plants inputs and outputs

Algebraic formulation of the decomposed plant

The plant as shown in the figure-4 can be described by the set of

transfer functions from each of its inputs (the components of the

vectors w and u) to each of its outputs (the components of z and y),

organized into a matrix, the transfer matrix. P and K denote the

transfer matrices of the plant and controller, respectively. Plant

transfer matrix P is partitioned as

z regulated outputs

y sensed output

exogenous inputs w

actuator inputs u

P

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where Pzw is the transfer matrix from w to z, Pzu is the transfer matrix

from u to z, Pyw is the transfer matrix from w to y, and Pyu is the

transfer matrix from u to y. This decomposition is shown in Fig. 4.Now

suppose the controller is operating, as shown in Fig. 5. We can solve

for the closed-loop transfer matrix from w to z, which we denote

…(1)

w

u

Pzw

Pyw

Pzu

Pyu

K

∑

∑ z

y

+

+

+

+

Figure 4. The decomposed plant

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The entries of the transfer matrix H are the closed-loop transfer

functions from each exogenous input to each regulated variable.

Various entries might represent, for example, closed-loop transfer

functions from some disturbance to some actuator, some sensor to some

internal variable, and some command signal to some actuator signal. The

equation (1) above shows exactly how each of these closed-loop transfer

functions depends on the controller K. The central theme of this

paper is that H should contain every closed-loop transfer function of

interest to us. The close loop matrix from w to z is

Design of Closed-Loop Convex Controllers

If two controllers K and each stabilize P and yield closed-loop

transfer matrices H and, respectively, then for each , there is

P

K

+ ++

+

w

v1

u

z

v2

y

Figure 5: The closed loop regulator

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some controller that stabilizes P and yields closed-loop transfer

matrix .

Thus we can find an entire family of controllers that stabilize P, and

the corresponding closed-loop transfer matrices will lie on a line in

..A very important point is that the controller that yields

closed-loop transfer matrix is generally not .

Straightforward but tedious algebra yields

Where

may be defined as the coefficient of affine criterion of convexity .

An infinite no of such stabilizing controllers may be obtained by

varying a single parameter between 0 and 1. Out of these infinite

combination ,to select the optimal setting of convex controller we

impose certain design specifications. In the present system, the

specification chosen are internal stability, time domain and frequency

domain performance specifications and optimization of a time domain

performance index (PI) for the close loop system.

Time domain and frequency domain constrains

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Performance specifications can be considered as convex constrains.

Primary purpose of feed back system is to ensure that one for the

regulated outputs be nearly tracking the command input. This is called

tracking performances. The following specifications are required

Steady state error 0.001 ,phase margin 450, gain margin 3 dB

Figure 6: bode plot of close loop

transfer functions

gain margin and phase margin satisfy our desired specifications of

the example discussed below[18].

Optimization value of

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The convex optimization can be obtained for all the set of closed loop

transfer matrices given by the above equations. The set of transfer

matrices all such specifications are thus given by

H spec= H stab H t1 Ht2 H f1 H f2

As the design specification becomes stronger and stronger .the range of

parameter narrows down within limit 0 and 1. Still an infinite number

of controllers are obtained which satisfy all the specifications. as the

time domain specification are closed loop convex and H H , the optimal

value of will also produce a convex controller. The parameter is

optimized by optimizing time domain index (integral squared error, ISE

for continuous time system or sum of the squared errors SSE for discrete

time system) thus producing the optimal set of transfer matrices

H opt= H spec H SSE

Example: Modelling a Pitch Controller

Physical setup and system equations

The equations[18] governing the motion of an aircraft are a very

complicated set of six non-linear coupled differential equations.

However, under certain assumptions, they can be decoupled and

liberalized into the longitudinal and lateral equations. Pitch control

is a longitudinal problem, and in this example, we will design a convex

controller that controls the pitch of an aircraft.

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The basic coordinate axes and forces acting on an aircraft are shown in

the figure below:

Assume that the aircraft is in steady-cruise at constant altitude and

velocity; thus, the thrust and drag cancel out and the lift and weight

balance out each other. Also, it may be assumed that change in pitch

angle does not change the speed of an aircraft under any circumstance

(unrealistic but simplifies the problem a bit). Under these assumptions,

the longitudinal equations of motion of an aircraft can be written as:

…(2)For this system, the input will be the elevator deflection angle, andthe output will be the pitch angle.

Design requirements

The next step is to set some design criteria. It is desired to design a

feedback controller so that the output has an overshoot of less than

10%, rise time of less than 2 seconds, settling time of less than 10

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seconds, and steady-state error of less than 2%. For example, if the

input is 0.2 rad (11 degrees), then the pitch angle will not exceed 0.22

rad, reaches 0.2 rad within 2 seconds, settles to 2% of the steady-state

within 10 seconds, and stays within 0.196 to 0.204 rad at the steady-

state.

Overshoot: Less than 10%

Rise time: Less than 2 seconds

Settling time: Less than 10 seconds

Steady-state error: Less than 2%

Transfer function and the state-space

Before finding transfer function and the state-space model, let's plug

in some numerical values to simplify the modeling equations (2) shown

above.

… (3) These values are taken from the data from one of Boeing's commercialaircraft [8].

Transfer functionTo find the transfer function of the above system, we need to take the

Laplace transform of the above modeling equations (3). The Laplace

transform of the above equations are shown below.

After few steps of algebra, the following transfer function is obtained.

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State-spaceKnowing the fact that the modeling equations (3) are already in the

state-variable form, we can rewrite them into the state-space model.

Since our output is the pitch angle, the output equation is:

MATLAB representation and open-loop response

MATLAB is used to observe the system characteristics. First an open-loop

system to a step input is obtained and if it is necessary system

characteristics need to be improved. Let the input (delta e) be 0.2 rad

(11 degrees).

.

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From the plot, we see that the open-loop response does not satisfy the

design criteria at all. In fact the open-loop response is unstable.

Closed-loop transfer functionTo solve this problem, a feedback controller will be added to improve

the system performance. The figure shown below is the block diagram of a

typical unity feedback system.

Figure 7: Closed-loop transfer function

A convex controller needs to be designed so that the step responsesatisfies all design requirements.

Model transfer function of a pitch controller

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The general equation of a PID controller. K(s+a)2/s [13].The

proportional–integral-derivative (PID) controller is the widely used

controller structure in control system application. Its structural

simplicity and sufficient ability many practical control problems to its

wide acceptance[190].

Possible values ok K, a, m(maximum overshot) are respectively:

3.0000 0.5000 1.0262

2.8000 0.5000 1.0273

2.6000 0.5000 1.0283

2.4000 0.5000 1.0294

2.2000 0.5000 1.0304

2.0000 0.5000 1.0313

2.4000 0.7000 1.0324

2.2000 0.7000 1.0326

2.0000 0.7000 1.0353

2.6000 0.7000 1.0353

2.8000 0.7000 1.0376

3.0000 0.7000 1.0393

3.0000 0.9000 1.0952

2.0000 0.9000 1.0960

2.8000 0.9000 1.0961

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2.6000 0.9000 1.0967

2.2000 0.9000 1.0968

2.4000 0.9000 1.0970

Two best values are taken with respect to the minimum overshoot, K =2.4000,a = 0.9000 and K = 2.8000,a = 0.7000

Transfer function of controllers:

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Figure 8 :plot fortwo best values

Close loop transfer function (H ):

Close loop transfer function ( )

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Figure 9: plotfor various values of lambda

In the present paper it has been shown the closed loop system which

satisfies all the performance specification is the convex combination

of the plant and the controller. Using an algebraic procedure design

parameter, controller structures and controller gains are

simultaneously determined which successfully solves the problems of

system design with multiple simultaneous closed loop performance

specifications.

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ConclusionThis makes a sensible formulation of the controller design problem

by considering simultaneously all the closed-loop transfer functions

of interest. The paper stresses that the closed-loop transfer matrix

H should include every closed loop transfer function necessary to

evaluate a candidate controller In the convex controller design each

design specification has been associated with a set of transfer

matrices. Most of the design specifications for pitch control of an

aircraft are closed loop convex. There might be infinite no of

stable controllers for any real value of the λ in the range 0 to

1.The overshoot and the settling time of a pitch maneuvering

dynamics would be brought down to an appreciable limit by a properly

designing a convex controller. In the paper the integral squared

error (ISE) method of optimizing the parameter λ to produce optimal

performance specification in all possible combination of pitch

dynamics has been suggested. The combine system with plant and

convex controller is stable and it satisfies the design

specification.

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