AUSTRALIAN JOURNAL OF BASIC AND APPLIED …ajbasweb.com/old/ajbas/2017/March/107-112.pdf108 Farzaneh Shahidinoghabi and Rajab Asghariyan, 2017 Australian Journal of Basic and Applied

Australian Journal of Basic and Applied Sciences, 11(4) March 2017, Pages: 107-112

AUSTRALIAN JOURNAL OF BASIC AND

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ToCite ThisArticle:Farzaneh Shahidinoghabi and Rajab Asghariyan.,Zeros Removal In Robust H∞ Control Using Analysis and Synthesis

of μ. Aust. J. Basic & Appl. Sci., 11(4): 107-112, 2017

Zeros Removal In Robust H∞Control Using Analysis and Synthesis of μ

1Farzaneh Shahidinoghabi and2Rajab Asghariyan

1Departmentof Electronics, Islamic Azad University of Science and Research Branch, Borujerd, Iran. 2Departmentof Electronics, Ferdowsi University, Mashhad, Iran.

Address For Correspondence:

Farzaneh Shahidinoghabi, Department of Electronics, Islamic Azad University of Science and Research Branch, Borujerd, Iran.

A R T I C L E I N F O A B S T R A C T

Article history: Received 18 January 2017

Accepted 22 March 2017

Available online 28 March 2017

Keywords: Robust controller design H∞, remove

zeros and poles method, analysis and

synthesis of μ, D-K iteration method.

BACKGROUND: In order to avoid phenomena have been proposed to remove zeros and poles to design of robust H∞ controller, most of these methods are in appropriate

and correct adjustment of mass matrix. OBJECTIVE: μ Analysis and Synthesis has

been proposed as a solution of that leads to controller design of H∞ in which, removing zeros and poles phenomenon does not happen. RESULTS: This transformation

matrices sensitivity and frequency response curves or supplements - along with the

response sensitivity of the control outputs when the show on scope page design in the software as well as trial and error process makes it extremely easy to access.

CONCLUSION: The first proposed solution on a single input - single output linear

and non-linear model of the complete solution on two inputs - two outputs are implemented as a turbine-generators.

INTRODUCTION

Through the setup process the controller parameters have physical limitations such as saturation operator

and other nonlinear factors are considered which is performed based on settings on the nonlinear model system.

The function generator of the model for 11 degree, a turbine system model is shown. This controller also has the

first swing and the subsequent swing well damped. In fact, LQG controller has the best performance among

different types of controllers, of course, on when the final version of the model name, the system gives to the

designers. But in this paper we show that such a high performance controller when there is uncertainty

(Dynamic deleted) unstable performance of their shows. In (Asgharian, 1994) the first successful application of

robust control H∞ design algorithms on a nonlinear model of the turbine generator was introduced where the

comparative performance of LQG control that have shown good performance in which, in similar circumstances

have the ability to deal with uncertainty and instability that has led to (Asgharian, 1988) that certain conditions

can cause undesirable volatility in output closed loop control system. In (Ahmed,S.S, 1996, Doyle, J.C, 1982,

Parniani, M, 1998 and Tsai,M.C, 1992) to remove the zeros and poles phenomena studied the phenomenon

adversely affect the performance and system controller controls.

Analysis and Synthesis μ:

In 1986, with the development of an efficient method for calculating the properties and concepts μ were

presented (Fan, M. K. H.,, 1986). Multivariate accurate calculation of margins for the stability of the system

described in 1988, and at the same time μ calculation method was presented in a manner that duplicates a scalar

units (Packard,A. K, 1988). Then provide a powerful way to calculate μ and effective steps were taken to obtain

accurate values (Packard,A. K, 1988). Uncertainty may be real or complex blocks consist of a mixture of the

two. In 1990, the method for calculating μ in the presence of uncertainty was real and had a complex

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108 Farzaneh Shahidinoghabi and Rajab Asghariyan, 2017


expression(Young, P. M, 1990). The first software analysis and design topics μ was used for the system, in 1991

by some of the most famous researchers in this field, were presented. μ -Tools is the software that runs under

MatLab toolbox (Balas, G. J., 1991). In the same year robust analysis in the presence of non-Qatari uncertainties

(Chen, J., 1991) and parameter (] Fan, M. K. H., 1986) stated. A study in which μ reduced through a number of

variables acquires optimization (] Latchman, 1992). In 1993 an article was published long training where a

review of all topics singular values with its mathematical structure, with emphasis on the prospects for

categories were presentedin (Packard,A. K, 1993). The limitations of computing μ topics of robust analysis

methods for linear systems, linear feedback systems have robust properties, stability and performance tests

robust, methods for analysis and synthesis robust system modeling, designing μ, the use of LFT and to compare

the methods and frequency domains discussed. In 1994 it was proved that the exact μ calculation is not always

possible, and in some places we do not find an explanation may never access the original value of μ (Braatz, S.,

et, al., 1994), so efforts focused on the calculation of the lower and upper bounds. In 1995, approximate μ

calculated (Young, P. M. et, al., 1995) and its properties are described in the following year (Young, P. M.

1996).

Being robust insensitive to the dynamic changes that are not considered in the analysis and design stages,

turbulence, noise measurement and dynamics of these are not modeled. The system must be able to withstand

the effects of neglected and control objectives for which it was designed, to be brought. Robust control H∞

design project aims to grade 11 systems for the turbine generators in the article (Asgharian, R., 1994) is a GUI

(Graphic Utility Interface) application using MatLab to achieve these demands, then this shows that the real

model system for models 21 degree is tested. Finally, it is observed that on both occasions by putting the

controller model comes out better than the situation without controller. As previously mentioned robust control

H∞ due to uncertainty and changes in circumstances and ensure the satisfactory performance of closed-loop

control systems, an appropriate method is applied on many issues.

Infinity Controller Design of Soft Robust:

According to the agreement aims to gain the stability of the closed-loop controller to tolerate changes in the

dynamics of the high frequency closed-loop transfer function which are not modeled. The controller recognizes

that a wide range of operating conditions of the system have good performance and have robust to changes in

transmission line and other parameters. Using PSS (Power System Stabilizer) controller is a supplementary

controller, i.e., by applying two additional control signal to the AVR and governor loop synchronization,

damping moment can be safely torsional excitation modes, respectively, with improved magnetic mode (the first

swing transient stability of the system is faster in the presence of large disturbances) and Mechanical modes

(dynamic stability to the presence of small disturbance and the subsequent swing damping or attenuation is

concerned) increased. To design robust the best controller for the first time 11 turbo generator mass matrices,

W1 و W2 W3using GUI tools in MATLAB determined.

P(S) =

[ W1 − W1G 0 W2

0 W3GI − G

]

(1)

Fig. 1: Issue on Control Standard of Robust H∞

S = (I + GK)−1 (2)

T = I – S = GK (I + GK)−1 (3)

R = K(I + GK)−1 (4)



Fig. 2: Robust Generator Load Angle With and Without Controller H∞As can be seen high system efficiency is

not satisfactory, while system robust is very high.

Fig. 3: Robust Generator Load Angle With and without controller H∞ to change the work As Much As 20% +

Response with controller without controller does not show a better performance of behavior is acceptable,

but robust system.

Case study:

The paper presents a case study of a power system consists of three generators and three times have been

connected to each other. The method is provided from occurring phenomenon will prevent the removal of zeros

and poles in the linear system and finally, we found a more optimal performance. In this method, the size of the

imaginary axis on the left, we shift the poles of the plant to be transferred to the right axis. A matrix system is

enough to shift the system with a constant diagonal matrix that is the identity matrix we get. Then we designed

H∞ controller for the virtual system. Virtual Plant in accordance with unstable poles then zeros controller (mirror

image of the poles of the imaginary axis) are placed. The imaginary axis is then shifted again to first place and

the virtual plant is stable and improved attenuation poles of the main plant. The only wrong in this paper is that

the linear system is discussed in this article but our system is nonlinear.

In this paper, the standard H∞ control limits can be expressed as:

1. Inability to deal effectively with the uncertainty when a batching system, stable name by adding

uncertainty becomes unstable.

2. Plant low attenuation poles with zeros removed by controller

The first problem solving (problem-solving using A.N.D Uncertainty):

In this paper, a mathematical discussion and G (s) splits as follows:

G(s) = D−1(s)N(s) (5)

D(s) = D0(s) + Q(s)∆d(s)W1(s) (6)

N(s) = N0(s) + Q(s) ∆N(s)W2(s) (7)

Since we have no zeros and poles in nonlinear system. Hence, all of these behaviors are disrupted and

transient behavior changes. The method presented in paper [3] carried out in a manner beyond the proposed

articles and the weight is based on non-linear system. In this method, can be seen that near the poles despite the

imaginary axis with nonlinear simulation-based regulation is no concern of damage caused by the removal of

zeros and poles will have the response. . So that the output of the system is achieved by selecting the appropriate

mass matrix will be no swinging. This means that the reference method [3] has been able to solve this problem

as well.

Modeling Uncertainty:

D matrix may be fixed and no variable dynamic or variable frequency which is not the state variables of the

state vector Plant added and because the controller is in the design phase of the plant P will increase as a result

of the controller K. Hence, the design consists of two phases reproducible repeat D-K is called:

1. Find a diagonal matrix D (S) to optimize the structure according to ∆ = diag (∆n,⋯,∆1) so

σ(DMD−1)at different frequencies minimize as much as possible. To this end, we can help optimize D computer



program and then using its existing fixed matrix with at least some degree or approximated. In the first iteration

we do not have K and assuming D = 1.

2. With the help H∞ and design controller K (s) that softens extremely DF𝑙(P, K)D−1 is minimized as

much as possible to be found.

3. In stage 1, and this is repeated until all frequencies μ(F𝑙(P, K)) < 1or greater reduction in earning the

minimum possible of μ.

According to the rules of closed-loop feedback matrix M = F𝑙(p , k)can be obtained as follows:

M = F𝑙(p , k) = [W1k(I − GK)−1 W1k(I − KG)−1

W2(I − GK)−1 W2G(I − KG)−1]

∆ = diag( ∆1 , ∆2)

(8)

If K is produced by a synthesis μ to be designed: For all frequencies:

μ(M) < 1

(9)

Fig. 4: Uncertainty Model

Fig. 5: Balanced Model



Fig. 6: Stop responses

Fig.7: Stop responses

Conclusion:

To overcome the negative effects of this phenomenon proposed four solutions that were examined

individually. These solutions mainly depend on selection criteria of the appropriate mass matrices in controller

design robust H∞ which makes this phenomenon does not happen or the system or the imaginary axis movement

or by converting two inverse linear phenomenon and return to the original system to remove zeros and poles

will be taken. These solutions are based on linear model system that acts as the controller will remain on non-

linear dynamics. In this thesis two solutions to avoid any fluctuations in response closed-loop system were

presented. In the first solution using analysis and synthesis μ controller design H∞ that is the inverse function of

the batching function and thus remove the zeros and poles. These solutions are based on linear model plant takes

place and applied problems on the nonlinear model is capable. In the second solution in the second quarter of

this thesis was presented, using systematic regulation on non-linear matrix weighted plant were regulated so that

the nonlinear system response is not swinging. This was done using MatLab graphics capabilities which does

not only eliminates fluctuations relating to the phenomenon of the nonlinear dynamics of the system zeros and

poles but also destroys. Nonlinear Simulation-based design methods, systems needed a software that can be

changed simultaneously parameter controller tuning, control outputs evaluated. The innovative method of "pot-

scope" is called, the application presents the parameters of the matrix of mass as potentiometers and frequency

response matrices R, S and T together with the response when variable control (angle bar) generators as the

scope were introduced the basic result of this thesis controller design H∞ matrices using systemic regulator is

weighting on nonlinear simulation plant.

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AUSTRALIAN JOURNAL OF BASIC AND APPLIED …ajbasweb.com/old/ajbas/2017/March/107-112.pdf108 Farzaneh Shahidinoghabi and Rajab Asghariyan, 2017 Australian Journal of Basic and Applied