International Journal of Electrical and Computer Engineering (IJECE) Vol.
No.
April 2013, pp.
ISSN: 2088-8708
Robust Multivariable Controller Design with the simultaneous H2/HO/AA for a Single Person Aircraft Javad Mashayekhi Fard*.
Mohammad Ali Nekoui**.
Ali Khaki Sedigh**.
Roya Amjadifard*** * Department of Electrical Engineering.
Science and Research Branch.
Islamic Azad University.
Tehran.
Iran ** Department of Electrical Engineering.
Toosi University of Technology.
Tehran.
Iran ***Department of Computer engineering.
Kharazmi University.
Tehran.
Iran
Article Info
ABSTRACT
Article history:
In a physical system several targets are normally being considered in which each one of nominal and robust performance has their own strengths and In nominal performance case, system operation without uncertainty has decisive effect on the operation of system, whereas in robust performance one, operation with uncertainty will be considered.
The purpose of this paper is a balance between nominal and robust performance of the state feedback.
The new approach of present paper is the combination of two controllers of and H2/HO.
The controller for robust stability status, nominal performance, robust performance and noise rejection are designed The controller will be achieved by solving constraint optimization problem.
The paper uses a simultaneous H2/HO/AA robust multivariable controller design over an X-29 Single Person aircraft.
This model has three inputs and three outputs, where the state space equations of the system correspond to an unstable one.
Simulation results show the effectiveness and benefits of the method.
Received Feb 6, 2013 Revised Mar 19, 2013 Accepted Mar 30, 2013 Keyword:
Constraint optimization H2 /HO/AA robust controller Multi-objective control Single person aircraft Copyright A 2013 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Javad Mashayekhi Fard.
Department of Electrical Engineering.
Science and Research Branch.
Islamic Azad University Hesarak.
Tehran.
Iran.
Box 14515/775.
Email: mashayekhijavad@yahoo.
INTRODUCTION
Unmodeled dynamics, non linearity of systems and the availability of disturbances usually cause that linear control systems theory never reaches the ideal solution.
For this, several targets are attended in a control system.
Robust Stability: meaning that system will be stable with uncertainty.
Nominal performance: without considering uncertainty, the fault of system will be minimized.
Robust performance:
with considering uncertainty, the fault of system will be minimized.
To consider robust performance, we use AA analysis.
Usually, the availability of robust performance causes extreme limitations on the controller and prevents it from reaching its feasible condition, and in the case of achieved feasible condition, it increases the grade of the controller, and the resulted controlling signal increases and causes to saturate actuator.
But in some systems it will be used, compulsorily.
For example, in power systems which transient response is decisive, robust performance will be considered.
Operating limitation on controlling signal: increase of controlling signal causes saturation of actuators.
H2 norm essence can be responsible for such target.
Minimizing disturbance effect: distortion can cause undesirable effect of transient response, so reducing the effect of disturbance, is one of the controlling targets.
Mixed norm of H2 and HO can be a useful strategy, to reach noted controlling targets.
To date, number of studies has been performed on the mixed norm and multiobjective control.
This paper attempts to present an implementation approach for controlling systems.
One of the useful techniques used on controlling system is the robust state feedback.
Prevalently, it is possible to Journal homepage: http://iaesjournal.
com/online/index.
php/IJECE A ISSN:2088-8708 design a controller which regularly includes 5 noted targets above.
In a more explicit description, controller includes two parts.
the first one uses mixed H2 and HO norm and the other uses AA synthesis.
These two parts, include some weights which have important roles in systems control, because each one of 2 and 3 targets, has its own definiteness and their combinations can create a new solution.
Practically, we look for minimizing If the available error function is not desirable, usage of a suitable weight function can lead to the So, designing weight function is extremely important.
At first, a controlling problem will be changed to LFT standard form, considering uncertainty, then status equations will be written and constraint's weight function will be determined to reach the robust controlling targets.
Then, using robust state feedback method, it will be is the time to select them again, using state feedback and weight functions repetition methods to supply the robust performance of system in an acceptable way.
The first formulation of HO control problem performed in 1981 by Zames.
Next to Zames.
Doyle.
Zhou, and Glover were premiers of robust control.
date, a vast number of researches have performed studies in robust control.
H2 control and HO control.
Doyle et al in .
analyzed the state space with HO and H2 standard form and its solving.
The conditions of solving problem and its solution using Hamiltonian matrix introduction are of importance of this paper.
This paper is a comprehensive reference that has been beneficial in many other research works.
Doyle et al in .
present a tutorial overview of linear fractional transformations (LFT.
and the role of the structured singular value, , and linear matrix inequalities (LMI.
in solving LFT problems.
Rotea et al in .
combined H2/HO.
Two important approaches are presented.
Minimal H2-norm subject to an HO-norm constraint.
Simultaneous H2/HO optimal control.
In each step, problem formulation and controller were performed.
Doyle et al in .
are shown that different of a mixed H2 and HO infinity norm arise from different assumptions on the input Lanzon in his PHD thesis chooses the weight functions in and HO design .
Akbar et al in .
, study a mixed H2/HO control law is derived using auxiliary cost minimization approach for continuous time linear time invariant singularly Perturbed System (SPS).
The time responses of closed-loop LQG, mixed H2/HO and HO control system for a unit step input and their robustness measures such as Gain and Phase margin for the open-loop systems are analyzed.
See for instance the references .
Ae.
for the mixed norm Tan et al in .
Robust load frequency control for power systems is discussed.
Keel et al in .
, show by examples that optimum and robust controllers, designed by using the L1.
HO.
H2, and formulations.
The rest of this paper is as follows.
Section 2 the discription of aircraft model.
Section 3 establishes the problem to be addressed, the H2/HO.
AA and H2/HO/ combination control will be demonstrated.
Section 4, illustrates the approach and the results of simulations will be discussed.
Section 5 presents the conclusions.
AIRCRAFT MODEL
In an airplane, five main sections could be listed as: motor, body section, landing system and wheels, wing, and tail.
The pitch angle of an airplane is controlled by adjusting the angle .
nd therefore the lift forc.
of the rear elevator.
The aerodynamic forces .
ift and dra.
as well as the airplane's inertia are taken into account.
The X-29 aircraft is a recent example of a control configured vehicle that was designed with a high degree of longitudinal static instability .
p to 35 percent at low subsonic speed.
The vehicle is stabilized by a full-authority, flyAebyAewire flight control system.
Linear models were used extensively prior to flight to determine the close loop stability, controllability, and handling qualities with the various control system modes through the flight envelope.
This section describes the commercials aircraft models which is In .
which is a comprehensive report of NASA, it has been researched over X-29 state equation and model.
In .
has been designed only the HO controller over X-29.
The X-29 airplane is a relatively small, single seat, high-performance aircraft powered by a single F404-GE-400 engine (General Electric.
Lynn.
Massachusett.
Empty weight is 6350 kg.
The vehicle incorporates a forward-swept wing with close-coupled canards to provide a low-drag configuration.
The airplane physical characteristics are presented in table 1.
The aircraft model is obtained by linearizing the nonlinear equations of motion about a 280 ft/sec .
Km/h.
landing configuration .
The three input three output model which describes the longitudinal dynamics is given as follows .
Table 1.
X-29 physical characteristics Maximum trust force Angel attack Strake flap position Pitch Euler angel 8130 Nm stf Weight Canard area Symmetric flap position Pitch rate IJECE Vol.
No.
April 2013: 279Ae286
6350 kg 437 m2
Wing span
Wing area
Canard position Horizontal speed
185 m2
IJECE
ISSN: 2088-8708
xA = Ax Bu , y = Cx , x = E - .
t / se.
- .
- .
ad / se.
E E
- .
E c u = Esf E stf E - .
- .
- .
E , y = E v - .
t / se.
E E
- .
E E - .
E0.
A 8.
4451 A 32.
E .
B A E0.
EE 0
E0 0 0 57.
C A E1 0 0 0 E
E0 57.
3 0 0 E
E0.
A A EE
EE 0
PROBLEM STATEMENT
H2/HO Controller Existence of uncertainty created due to an uncertain and erratic input .
oise and disturbanc.
and Un-modeled dynamics cannot be described completely and precisely as a true system by a mathematical On the other hand, a true system should contain the following important objects: robust stability, robust and nominal performance, settling time, maximum over shoot and etc, which try to gain these objectives about the controlling problem .
The type of uncertainty is another important factor in the system analysis.
Theorem 1.
mall gain Theore.
supposes M Ea RH Cu and let A 0 , then the interconnected system shown in Figure 1 is well-posed and internally stable for all AE ( s ) Ea RH Cu with AE Cu A A A1 if and only if M Cu A A .
Consider additive uncertainty shown in Figure 2.
In this case, robust stability task is:
q A ( I A KG ) A1
KP E
( I A KG )
HCu
A A
Figure 1.
M A i Model Figure 2.
Additive uncertainty The objective for the inner loop control is to design a state feedback law such that the closed loop system satisfies the following performance specifications.
Objective 1: if AE A 0 then FS Cu A 1 .
ominal performanc.
S A ( I A GK )
Objective 2: if AE C 0 then the system is robust stability.
M A ( I A KG )
if A ( AE ( jA )) C A ( jA ) E A ( S ) M (S is sensitivity functio.
Objective 3: n is white noise with one PSD .
ower spectral densit.
H2 norm, caused due to a decrease in the controlling signal.
RTnU A 1 (To minimize U1 variance with noise inpu.
and A .
) weighting functio.
from parseval equation and objective 3:
Robust Multivariable Controller Design with the simultaneous H2/HO/a (Javad Mashayekhi Far.
A ISSN:2088-8708 Cu Y 2 A E0 Y T .
)Y .
)dt A 1 EACu Y * ( jA )Y ( jA )dA .
Y ( jA ) A G( jA )U ( jA ) E Y 2 A 1 EACuU * ( jA )G* ( jA )G( jA )U ( jA )dA 2A C 1 supA (G* ( jA )G ( jA ))EACuU * ( jA )U ( jA )dA A G .
Cu U ( .
2 E sup 22 A G ( .
Cu C Figure 3.
LFT Model
2C T
1Cu Figure 4.
Graphical model of H2/HO, combination problem with additive uncertainty Then we have three tasks for controller design ( FS Cu A 1 .
A ( S ) M Cu A 1 TnU 1 Cu A 1 ), such that.
E FS (K ,G ) E
EA M (K ,G ) E A 1 .
A large class of systems with uncertainty can be treated as LFT (Linear Factorial E E RT ((K ,G )E Cu Transfor.
LFT model has been shown in Figure 3 .
in which.
W: the disturbance signals to the system which wonAot be a function of states of system.
Z: the variable that will be controlled.
P: the nominal open loop system.
Y: the system measurable output.
AA Controller Here we try to assess robust performance of this closed-loop system using AA -analysis.
Robust performance condition is equivalent to the following structured singular value AA test .
Twz ( M .
AE) Cu
A AE
A A Eu AAEP ( M ) A A AW The complex structured singular value A AE ( M ) is defined as A AE ( M ) A A A min A ( AE ) det( I A M AE ) A 0 Lower and upper bounnd of AA can be shown to be P(UM ) C AAE (M ) A min A ( DMD ) .
New approach: The simultaneous H2/HO/ problem
Theorem 2: consider a AECuA
EAE 0 E
E0 AE f E system with AE p A E M ,AEp , internally stable, and Twz (M .
AE ) Cu let A A 0 for all AE( s ) Ea M (AE) with A A if and only if sup AAE ( M ) A A .
This can be shown as follows:
AAE p C M 11 ( j A ) A M 22 ( j A ) A 2 M 12 ( j A ) M 21 ( j A ) .
EAM M E
At additive uncertainty M A E
such that S Cu A M
A 1 and at multiplication uncertainty Cu
S EE
EAS
E AT T E
M AE
E such that S Cu A T Cu A 1 .
E AS S E
IJECE Vol.
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April 2013: 279Ae286
IJECE
ISSN: 2088-8708
Lemma 1: consider a system with additive uncertainty.
Then H2/HO, controller will be designed in a way E FS ( K .
G ) E
that EEA M ( K .
G ) EE A 1 and simultaneously S Cu A M A 1 .
We have shown the noted mathematical Cu EE RT (( K .
G ) EE
Cu problem, in Figure.
State space equation for Figure 4 will be .
Lemma 2: consider a system with multiplication uncertainty.
H2/HO, controller will be designed in a way E FS ( K .
G ) E
that EEA M ( K .
G ) EE A 1 and simultaneously S Cu A T Cu A 1 .
We draw the noted mathematical problem, in EE RT (( K .
G ) EE Cu Figure 6.
State space equation for Figure 6 will be .
Figure 5.
Multiplication uncertainty E
xA EE
xA f EE
E A
EAB C A
E f
AxA EE E 0
xAR e EE 0
z1 EE EAD C
z2EE E 0
z3e E 0
z4e EE A C
E xA E E A E xA E EAB C A E f EAE f
E xA E E 0
E E E
EE xAR EE EE 0
E z1 E EADf C Cf
E E E
E z2 E A E DA C 0
E z3 E E 0
E E E
EE z4 EE EE A C
0 E EE x EE E0
Ex E
0 E E f E E Bf A Bf D A Bf D EE EE
BA E EEa
0 E EE xA EE E0
EE E E
EE W
AR EE EE x EE EE0
BR EE
E RE
0 E EE x EE E Df
E Ex E E
0 E EE f EE E0
CR E EE xA EE E0
EE E E
0 EE EE x EE EE I E RE Figure 6.
Graphical model of H2/HO, combination problem With multiplication uncertainty E
A Df D AD
A Df D E
DA E E
EEr
DR E
I A DEE
n ueT
0 E E x E E0
B E
0 EE EE xf EE EEBf A Bf D A Bf D EE E
r n u
AA 0 E E xA E E0
BA D
BA DE EA E
EE E E
E EE W
0 AR EE EE xR EE EE0
BR EE
0 E Ex E EDf A Df D A Df DE
EE E E
DA D
DA D E
CA 0 E Exf E E0
Au r n uAy E
0 CR xA
EE E E
I AD
I A DEE
0 EE EExR EE EEI
SIMULATION RESULTS
These simulations were performed using robust multi-objective control toolbox .
and AA-Analysis and Synthesis Toolbox .
of MATLAB Ee .
In this section, simulation result of X-29 aircraft model was At first the design weight functions were drawn in Figure 7-a-b.
By considering the practical experiments, the weighting functions are selected.
Then, step response of S.
M and T function in Figure 8-bc-d are shown.
Note that, as regards, the system is multi input-output, the weight and sensitivity functions are Robust Multivariable Controller Design with the simultaneous H2/HO/a (Javad Mashayekhi Far.
A ISSN:2088-8708 in matrix form.
The maximum controlling signal contains 12 units.
The sign of success is the combination of nominal and robust performance, together.
Reaching the targets with the minimum controlling signal is of the gains of noted controller.
The results shown in the figures indicate that the unstable system, but the closedloop system has appropriate results whereas it has been considered three objects .
for itself.
comparison with previous control methodologies, we see that the proposed H2/HO/AA combination is although simpler and perhaps of lower implementation cost, gives acceptable and proper robust performance.
Figure 7.
Singular value for weighting functions Figure 8.
Singular value for weighting functions.
Step response for S.
T IJECE Vol.
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April 2013: 279Ae286
IJECE
ISSN: 2088-8708
CONCLUSION
Each of nominal and robust performance has their own strengths and weaknesses.
Nominal performance means considering system operation without uncertainty, and has decisive effect on the operation of system.
Robust performance means considering operation with uncertainty.
It is obvious that whenever the singular values of controller are higher, systems performance is more desirable, but, also it provides higher chances of saturation occurances.
By using H2 norm, in conjunction with the design of appropriate filters or selection of other weighting functions, one can avoid over-increasing of the controlling signal which has an important role in determining the response of the close-loop system.
This paper tends to reduce controlling signal, robust performance, and robust stability and to provide a new design for weight The new approach presented in this paper utilizes a combination of two controllers of and H2/HO.
The controller for robust stability case, nominal performance, robust performance and noise rejection are designed simultaneously.
Controller will be achieved from solving constraint optimization problem.
Using two low pass filters and one high pass filter, we tended to optimize the solutions.
This approach has been applied to X-29 air craft state space equations.
The results shown in the figures indicate that the unstable system becomes stable in the presence of uncertainty using the proposed controller, and also show an appropriate desired performance.
REFERENCES .