5th International Conference on “Technical and Physical Problems of Power Engineering”
TPE-2009
3-5 September 2009
www.iotpe.com
University of the Basque Country Bilbao, Spain
ROBUST CONTROLLER DESIGN FOR THREE PHASE INDUCTION MOTOR USING QUANTITATIVE FEEDBACK THEORY M.A. Ataei
R.Pouraghababa
M.R.Nouhi
S.Najafi
Esfahan Regional Electric Company- Esfahan- IRAN
[email protected]
Keywords: Quantitative Feedback theory, Robust controller, There phase induction motor, none linear system. Abstract- There is uncertainty in modeling of systems and also, input distortion to systems is unknown. So, according to Feedback base it is necessary to design controllers that for all available system model uncertainties and unknown distortion, closed loop system provide desired performance such as input routing, distortion elimination, elimination of noise and etc. In other hand, closed loop system must be robust against variation and & unreliability. In this paper, robust controller design by Quantitative feedback theory method (QFT) which is an efficient approach for none linear systems are described. Then, its application for robust controller design of three phase induction motor speed, which is one linear system with uncertainty in parameters is studied. Simulation results by using "MATLAB" software have been presented. I. INTRODUCTION Most physical systems are determined by none linear equations, which have uncertainty in parameters and system obtained model do not specify exactly the behavior of system. According to feedback rule, if a complete and accurate model of a system was available, there isn't need to using feedback in the process of system controlling. Because of having uncertainty in modeling of systems and unknown input distortions, controllers must be designed so that the closed loop system provide suitable specifications such as input routing, elimination of distortion and noise elimination. One way for providing above condition is using QFT for robust controller design which presented by Horowitz. Benefits of this method to other methods like H∞ ،µSynthesis are simple controller design, specified closed loop controller parameter and ability of distortion elimination during the designing process. If robust controller design for a system by QFT method has not any response other methods absolutely hasn't either. This theory first is proposed for single input single output systems(SISO) and then for multi input multi output
systems(MIMO), time variable systems and none linear system is extended. In this paper, robust controller design for none linear systems by QFT is explained and then applied for there phase induction motor speed controller design, which is none linear and uncertain system. Simulation results show that this system is stable and has a desired output. II. QUANTITATIVE FEEDBACK THEORY QFT is a complete theory that present using feedback to obtain desired performance of system in presence of uncertainty in model of system and distortion. These factors are modeled in QFT theory as follow:
τ ={T}
Collection R R from suitable relations of input – output routing and
τD ={TD}
from
suitable
relations
of
distortion
elimination.
p={P}
Collection from changes of system operating point (system uncertainty). QFT theory is generally using the structure of system with two degree of freedom, figure (1). G controller and the pre filter "F" designed in order to set control ratio of TR in the collection of τ R and control ratio of TD in collection of τ D .
Fig(1): System structure with two degree freedom
This theory consists of five main steps: 1- Calculatin system performance variation in frequency domain base on time response characteristics. 2- Determination magnitude & phase of system for various frequencies in Nicholz diagram.
5th International Conference on “Technical and Physical Problems of Power Engineering”
3- Determination borders on loop gain function for providing suitable & high performance tolerance and system uncertainty in Nicholz diagram. 4- Designing nominal loop gain function with regards to borders on the loop gain function that is leading to design compensator (shaping nominal loop gain function) 5-disigning pre-filter. III. DESIGNING ROBUST CONTROLLER FOR NONE LINEAR SYSTEMS BY QFT Linearization for designing Controller in nonlinear systems is custom. Local linearization method can replace none linear & uncertain system with a collection of uncertain & linear with time systems that are designed with robust controller methods such as QFT method. Main idea of robust controller designing with QFT method for none linear systems is changing & converting none linear system to equivalent linear system addition to noise in output as regards to character of suitable output. Therefore, difference between none linear system & equivalent linear system is considered as a noise in out put of linear system. Hence, controller is designed by method of QFT for equivalent linear system and modeled noise. The obtained results is valid for none linear system. This idea is proposed by Horvitz and is according to the theory of "SHADER". This method is formulated in below cases: Case 1: figure.2 shows the nonelinear system of SISO, that N, as nonlinear system of "SISO" is inside {N} collection and {y} is a collection of admissible output responses. The object is designing G and Fr in a way that system output for all N ∈ {N } be inside {y} collection.
Figure-2: none linear system of SISO (i. c+ Initial conditions)
The process of designing by Horvitz method is represented as follow: Rule (1): Imagine below maps on collection {y}
ϕ ( y) =
PN , y GFr + y N , y + d N , y 1 + PN , y G
y = Nu = p N , y u + y N , y + d N , y
(1) (2)
Where "u" is calculated for each γ by equation γ =Nu. Equation (2) is used for selecting an equivalent fixed linear system with time ( PN .γ ) and noise signal (d N .γ ) and then
ϕ (γ
) in equation (1) is calculated. If resulted
map has a fix point for each N ∈ {N } in collection { γ } then G and Fr can be solution of this problem. According to above mentioned rule, process of designing is: 1- For each N ∈ {N } and γ ∈ {γ } , value of PN .γ and
DN .γ based on equation(2) is chosen.
TPE-2009
2- G and Fr are designed in a way that ϕ (γ ) has fix point in collection of { γ }. In this process, two basically questions are proposed: A: the way of choosing the ( PN .γ ) and (d N .γ ) (which is explained in reference [1]) B. The way of ensuring that the resulted value has fixed point in { γ }. For reaching to answer of this question, Horvitz used the theory of SHADER fix point and it is according to below rule: Rule (2): Imagine { γ } be a close & complete collection in the space of "Banakh" and ϕ is map of {y} towards itself. In this manner, there is a fix point in {y}. Threfore, If the complete and close collection of {y} be in space of "Banakh" and G, Fr be in a way that ϕ (y) as flat map for N ∈ {N } be from {y} towards itself G and Fr can be a solution for problem (1). If the PN.y and dN.y have been selected in correct way according to mentioned conditions, the case (1) is changed into below case: Case (2): Imaging the shown system in figure (3) that (P) and (d) represent a LTI system and noise respectively and they are into the collection {P,d}. y0(s) is a nominal output and e(w) is a characteristic function. Design controller G & Fr in a way that for all of {P,d}: - System be stable - Output of system is limited as below:
y ( jw) − y ( w) < e( w)
(3)
Figure 3: structure of feedback system of "LTI" as a substitution of main system.
The main step for solving the problem is to calculate margins of G(jw) by using the equation (3) , which is solving equation 4 (The value of Fr is assumed already).
PGFr + d − y0 ( jω ) < e( jω ), ∀p, d ∈ { p, d } (4) 1 + PG One method for solving above equation is explained bellow: - Regarding p0 , d0 choose an average state and obtain Fr in a way that left side of this equation be zero. Therefore we have: y − d0 (5) Fr = y 0 + 0 P0 G And Equation (4) can be written as follow:
( y0 − d0 )P / P0 + d − y0 < e(ω), ∀P, d ∈{P, d} (6) 1 + PG Accordingly, designing steps can be explained as below:
5th International Conference on “Technical and Physical Problems of Power Engineering”
I. Equation (6) should be used for calculating Bands on G(jw) (MATLAB QFT toolbox can be used). II.G(s) controller should be designed in a way that resulted bands and calculated Fr be confirmed in equation (5). IV. DESIGNING SPEED CONTROLLER OF THREE PHASE ASYNCHRONY MOTOR Asynchrony motors have plenty advantages such as suitable efficiency, speed controlling and etc. Due to this fact that three phase asynchrony motor equations is not completely accurate and motor parameter can be changed with time and during usage, therefore designing robust controller for reaching expected results in all conditions of utilization is necessary. Quantitative feedback theory has been used for designing robust speed controller in these kinds of motors. A: dynamic model of three phase Asynchrony motor
γ = ( X 2′ + X th ) 2 + Rth2
If electric Circuit, as shown in Figure – 4 is used to describe three phase Asynchrony motor, these equations can be written as blow:
T=
3p (λds′ iqs − λqs ids ) 4
R 1 = 0 . 435 ( Ohm ), L 1 = 0 . 002 ( H ).
R 2 = 0 . 816 ( Ohm ), L 2 = 0 . 002 ( H ). Lm = 0 . 0693 ( H ). J m = [0 . 25
B m = [0 . 006
T L = [0 . 5
0 . 1 ]( kg . m 2 ).
0 . 06 ]( kg .m / rad / sec).
20 ]( N . m ).
Where nominal values of system is assumed as below:
J mo = 0.18(kg.m2 ), Bmo = 0.01(kg.m / rad / sec),, TL = 15(N.m) Therefore, equivalent linear system for none linear system of three phases Asynchrony motor is obtained as below: R′ ( 2 ) αω s (15) P ( s) = J m S 2 + B m S + TL If time response range (expected output) is considered for system then:
120 S + 15S 2 + 120 S 3
(7)
n denotes Asynchrony speed 60
And also in constant state it can be written:
Vth =
Xm R + ( X 1 + X th ) 2 2 1
Z th = Rth + jX th =
.V1
jZX m ( R1 + jX 1 ) R1 + j ( X 1 + X m )
(8)
(9) Figure 5: time response for system
And dynamic three phase Asynchrony motor equation is:
Jm
dω m + Bmω m = T − TL dt
Where results:
(10)
TL is Load torque. Substituting in equation (9)
dω (αωs2 − 2βωsωm + γωm2 ) = J m m + Bmω + TL (11) dt Where:
α = ( X 2′ + X th ) 2 + Rth2 + 2 Rth R2 + R2′ β = ( X 2′ + X th ) 2 + Rth2 + 2 Rth R2′
(16)
120 (17) S 3 + 34 S 2 + 120 S Figures 5,6 show time response domain and frequency response domain: ydown =
Where : T denotes torque
ω S = 2π
(14)
B: Controller design None linear system of three phase Asynchrony motor is according to relation (10). For example, the values of system parameters are assumed as below: Voltage: 220 V
yup = Figure 4. Circuit diagram of three phase Asynchrony motor.
TPE-2009
(12) (13) Figure 6: frequency response for system
5th International Conference on “Technical and Physical Problems of Power Engineering”
Where due to expected output and calculation of difference between linear system & main none linear system for all system changes and by considering it as a noise in output of equivalent linear system, by using QFT and by using the MATLAB software, we can design the robust controller, which The figure (7) shows Bands and the way of shaping of nominal loop gain function in Nicholz diagram. Accordingly, G controller and (F) prefilter are obtained as below: 1.977 ×106 (S 2 + 20.6S + 1.5) (18) G( s) = 3 S + 59.65S 2 + 32.19S + 1.023 − 0.00276 S + 94 .18 (19) F (s) = S 2 + 20 .19 S + 94.18
According to obtained controller & pre-filter, Simulation is done for all changes of system parameters (Jm , Bm , TL) and results are shown in figure (7). It is shown that output of closed loop none linear system has been placed in expected range for all system changes. It means that system is robust for all changes in system parameters.
Figure (7): Bands & the way of shaping nominal loop gain function.
Figure (8): output of closed loop none linear system for all changes in parameters of system.
Figure9: Expected controlling signal for all changes in parameters.
TPE-2009
V. CONCLUSIONS figures (8) and (9) shows that output of none linear system has been placed in expected range for all variation in system parameters and good controlling signal is used for controlling system. It means that designed controllers are robust against all variation of system parameters and the output of closed loop has been placed in suitable and expected range. In addition usability for all systems, considering all expected controlling characters during designing and giving controller with fix structure are other advantages of QFT method. One disadvantage of QFT is more controlling energy consuming because of designing fixed structure controller. If energy consumption is reduced by using a special method, we can have a better & suitable controller. One suggested methods for this purpose is to combine this method with other methods like phase control. REFERENCES [1] H.Khalil,"Nonlinear System",translated by G.Montazer,Tarbeat Modares Publication ,2000. [2] M.A.Ataei," Quantitative Feedback theory in Nonlinear System Robust controller design",Tabriz University,2001. [3] Bimbahara, "Electric Machin",Ghaem Publication ,translated by J.Soltani,1996. [4] P.C.Sen , "Electric Machin",Baser Publication, ,translated by M.Abede,1996. [5] M.A. Ataei, R. Esmaeilzadeh, Gh. Alizadeh., "Quantitative Feedback Theory for Nonlinear Syatems", WSEAS Transaction on CirCuits and System, Issue 11, Volume 4, November 2005. [6] Rasoul Esmaeilzadeh, Mohammad Ali Ataei, Majid Amjadi, Ghasem Alizadeh, "Generalized quantitative feedback theory for MIMO nonlinear systems", WSEAS Transaction on Systems, Issue 4, Volume 5, April 2006. [7] Yaniv O., “Quantitative Feedback Design of Linear & Nonlinear Control Systems”, Kluwer Academic Publishers, 1999. [8] Jacques. J, Slotine. E, Li. W., “Applied nonlinear control”, Prentice-Hall International Editions, 1991. [9] D,Azzo J., Hocepisc., “Linear control system Analysis and design conventional and modern”.,3rded, McGraw-Hill, new York, 1988. [10] Horowitz I., “Quantitative Feedback Theory”, IEEE Proc., Vol. 129, Pt. D, No. 6, NOV. 1982. [11] Horowitz I., “Application of quantitative feedback theory to highly uncertain nonlinear time-varying plants”, European Control Conference Grenoble, France, July 25, 1991.
