Lmi Method

8/13/2019 Lmi Method

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DECLARATION

I hereby certify that this report which is being presented in the seminar entitled Linear

Matrix Inequality in Control System in partial fulfilment of the requirement of award of

Degree of Master of Technology in Electrical Engineering with specialization in System &

Control, submitted to the Department of Electrical Engineering, Indian Institute of Technology,

Roorkee, India is an authentic record of the work carried out during a period from May 2013 to

September 2013 under the supervision of Dr. G. N. Pillai, Associate Professor, Department of

Electrical Engineering, Indian Institute of Technology, Roorkee. The matter presented in this

seminar has not been submitted by me for the award of any other degree of this institute or any

other institute.

Date:

Place: Roorkee (Lokesh kumar Dewangan)

CERTIFICATE

This is to certify that the above statement made by the candidate is correct to best of my

knowledge.

(Dr. G. N. Pillai)Associate Professor

Department of Electrical Engineering

Indian Institute of Technology

Roorkee-247667, India


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ACKNOWLEDGEMENT

I wish to express my deep regards and sincere gratitude to my respected supervisor Dr. G. N.

Pillai, Associate Professor, Department of Electrical Engineering, Indian Institute of Technology

Roorkee for being helpful and a great source of inspiration. His keen interest and constant

encouragement gave me the confidence to complete my work. I wish to extend my sincere thanks

for his excellent guidance and suggestion for the successful completion of my work.

LOKESH KUMAR DEWANGAN


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ABSTRACT

This report gives a brief overview of Linear Matrix Inequality (LMI) and various types of

problem related to this. Synthesis of linear output feedback controller using LMI approach done

to get robust performance. For better response, LMI regions (half plane, circular and conical

sector) are defined for closed loop poles. The Hconstraint is formulated with the help of LMI toimprove robustness of the controller. LMI region and Hconstraints are combined to design statefeedback controller and output feedback controller of order k. Also the concept of solvability of

controller which gives the feasibility of LMI formulation is discussed.


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TABLE OF CONTENTS

DECLARATION ...................................................................................................................... iACKNOWLEDGEMENT ....................................................................................................... iiABSTRACT ........................................................................................................................... iii1 INTRODUCTION ............................................................................................................ 12 LINEAR MATRIX INEQUALITY ................................................................................. 3

2.1 INTRODUCTION .................................................................................................... 32.2 LINEAR MATRIX INEQUALITY FORMULATION............................................ 32.3 SOME STANDARD PROBLEMS .......................................................................... 52.4 SOLVING THESE PROBLEM ................................................................................ 6

3 POLE PLACEMENT IN LMI REGIONS ....................................................................... 83.1 LYAPUNOV CONDITIONS FOR POLE CLUSTERING ..................................... 83.2 KRONECKER PRODUCTS .................................................................................... 83.3 LMI REGION ........................................................................................................... 93.4 DIFFERENT LMI REGIONS DEFINED IN LMI ................................................. 103.5 INTERSECTION OF LMI REGIONS ................................................................... 13

4 HDESIGN VIA LMI OPTIMIZATION ..................................................................... 144.1 Schur Complements ................................................................................................ 144.2 LMI FORMULATION FOR HPERFORMANCE ............................................. 15

5 FEEDBACK CONTROLLER DESIGN ........................................................................ 185.1 STATE FEEDBACK HDESIGN WITH POLE PLACEMENT ........................ 185.2 OUTPUT FEEDBACK HDESIGN WITH POLE PLACEMENT ..................... 205.3 LINEARIZING CHANGE OF VARIABLE .......................................................... 25

6 CONCLUSION .............................................................................................................. 28


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7 REFERENCES ............................................................................................................... 298 APPENDIX .................................................................................................................... 30


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LIST OF FIGURE

Figure 3-1 Intersection of different LMI regions [3] ............................................................. 12Figure 5-1 Augmented block diagram plant [10] .................................................................. 20


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1 INTRODUCTIONStability is a minimum requirement for control systems. In most practical situations,

however, a good controller should also deliver sufficiently fast and well-damped time responses.

Different problems in control system can be formulated into Linear Matrix Inequality problem

and these LMI problems can be solved by using higher mathematics method like interior point

method [1]. A customary way to guarantee satisfactory transients is to place the closed-loop

poles in a suitable region of the complex plane. We refer to this technique as regionalpole

placement, where the poles are assigned to specific locations in the complex plane. In case of

direct pole placement technique, the poles are kept in particular location and it gives better

performance because the location of poles decide performance of system (Rise time, Peak time,

Damping ratio, Settling time etc) . But it is not sure that the controller gain calculated by thismethod is robust when changes the system parameter or disturbance on the system [2].

For robust Performance of the controller Hperformance must be added into controller.But both simultaneously not possible in direct pole placement method [2].

In Linear Matrix Inequality both better time response and frequency response can be

obtained. Linear Matrix Inequality first time used in control system by Lyapunov to check the

stability of the system. Different convex problem and convex optimization problem can be

formulated in Linear Matrix Inequality and it can be solved by using different method like

interior point method, ellipsoid method and newton method. Since development of fast interior

point method for solving LMI, controller design by this method become very easy and different

tools available in Matlab to solve LMI [2].

In pole Eigenvalue problem poles are restricted to lies in a region. In this report we designed

half plane, circular region and conic convex region. By putting constraints on the closed loop

poles the performance of the system can be improved, because it depends on location of the

poles. For example, fast decay, good damping, and reasonable controller dynamics can be

imposed by confining the poles in the intersection of a shifted half-plane, a sector, and a disk [2].

Since in real time systems always involve some amount of uncertainty, so it is natural to

worry about the robustness of pole clustering, i.e. whether the pole remain in the prescribed

region when the nominal model is perturbed. Robustness is the characteristic of the system or


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controller that maintain the system stable and performance of the system remain same after

system subjected to some disturbance and changes in parameter. Different type of uncertainty

may be present in the system and their H value is calculated in this report. HPerformance isformulated into LMI problem and solvability condition for the general suboptimal

Hproblem

are derived. Then the controller is designed with putting both pole placement constraints and HPerformance constraint.


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2 LINEAR MATRIX INEQUALITY2.1 INTRODUCTION

The different convex problem can be formulated into Linear Matrix Inequality Problem.

When the Linear Matrix Inequality is in the of diagonal matrix, then it is called linear

optimization problem. The problems arising in system and control theory can be reduced to a few

standard convex or quasi-convex optimization problems involving linear matrix inequalities

(LMIs). Since these resulting optimization problems can be solved numerically very efficiently

using recently developed interior-point methods, our reduction constitutes a solution to the

original problem. In comparison, the more conventional approach is to seek an analytic or

frequency-domain solution to the matrix inequalities. The LMI technique is first time used by

Lyapunov in control system to check the stability of the system. There are many advantage of

LMI which attract the researcher to do work in this field. One of the most important reason is

that many LMIs can be converted into single LMI i.e. a number of constraint on variable can be

put simultaneously [1].

2.2 LINEAR MATRIX INEQUALITY FORMULATIONA linear matrix inequality (LMI) has the form

F x F xF > 0= (2.1)where x R is the variable and the symmetric matrices F= F R, i = 0,1,2..,m.are given. The inequality symbol in (2.1) means that F(x) is positive-definite, i.e., uF(x) u > 0for all nonzero u R. The LMI (2.1) is equivalent to a set of n polynomial inequalities in x, i.e.,the leading principal minors of F(x) must be positive. LMI can be written in non-strict form like

[1],

Fx 0 (2.2)The LMI (2.1) is a convex constraint on x, i.e., the set {x| F(x)> 0} is convex. The LMI

(2.1) is a standard form, it can represent a wide variety of convex constraints on x. In particular,

linear inequalities, (convex) quadratic inequalities, matrix norm inequalities, and constraints that


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arise in control theory, such as Lyapunov and convex quadratic matrix inequalities, can all be

cast in the form of an LMI. When the matrices Fi are diagonal matrix, the LMI F(x) > 0 is a set

of linear inequalities, then this convex optimization problem convert in linear optimization

problem. Nonlinear (convex) inequalities are converted to LMI form using Schur complements.

The basic idea is as follows [1]:

Qx SxSx Rx > 0 (2.3)where Q(x) = Q(x, R(x) = R(x, and S(x) depend affinity on x, is equivalent to

Rx > 0, QxSxRx

Sx

> 0 (2.4)

In other words, the set of nonlinear inequalities (2.4) can be represented as the LMI (2.3).

Multiple LMIs , . , > 0can be expressed as the single LMI by keepingall LMIs into diagonal of matrix and can be expressed as, . , > 0[1].

2.2.1 MATRICES AS VARIABLESIn some problems where the variables are in the form of matrix, e.g., the Lyapunov

inequality

P A AP < 0 (2.5)where A Ris given and P= P Ris the variable. In this case it will not write out theLMI explicitly in the form F(x) > 0. As another related example, consider the quadratic matrix

inequality

AP P A P B RBP Q < 0 (2.6)

where A, B, Q = QT, R = RT> 0 are given matrices of appropriate sizes, and P = PT is the

variable. Note that this is a quadratic matrix inequality in the variable P. It can be expressed as

the linear matrix inequality [1].


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AP PA Q BPBP R (2.7)

2.3 SOME STANDARD PROBLEMS2.3.1 LMI PROBLEMS

Suppose we have an LMI F(x) > 0, the corresponding LMI Problem (LMIP) means to find

xfeassuch that it satisfy above condition F(xfeas) > 0 or determine that the LMI is infeasible. This

is a convex feasibility problem where the above LMI is checked whether it is feasible or not. It

means solving the LMI F(x) > 0 means solving the corresponding LMIP. For example we

consider the simultaneous Lyapunov stability problem, here given that A

Rand we

have to find the optimal value of P such that it satisfy the constraints [1],

P > 0 (2.8)AP P A < 0 (2.9)

2.3.2 EIGENVALUE PROBLEMSIn eigenvalue problem (EVP) the maximum eigenvalue of a matrix is minimize and the

matrix depends on a variable, subject to an LMI constraint (or determine that the constraint is

infeasible),

Minimize Subject to I Ax > 0; Bx > 0

where A and B are symmetric matrices that depend on the optimization variable x. This is a

convex optimization problem. EVPs can be written in other form like minimizing a linear

function subject to an LMI, i.e.

Minimize cTx

Subject to F(x) > 0


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where F a function of x. In the special case when the matrices Fi are all diagonal, this problem

reduces to the general linear programming problem: minimizing the linear function cT x subject

to a set of linear inequalities on x [1].

2.3.3

GENERALISED EIGENVALUE PROBLEM

The generalized eigenvalue problem (GEVP) is to minimize the maximum generalized

eigenvalue of a pair of matrices that depend on a variable, subject to an LMI constraint. The

general form of a GEVP is [2]:

Minimize Subject to Bx Ax > 0; Bx > 0; Cx > 0

where A, B and C are symmetric matrices that are functions of x. We can express this as

Minimize maxAx,BxSubject to > 0; > 0

2.3.4 CONVEX OPTIMIZATION PROBLEMMostly we prefer LMIP, EVP and GEVP. There is another form of problem in LMI that is

called convex optimization problem. It can be written as

Minimize

logdetAx

Subject to A(x) > 0; B(x) > 0

where A and B are symmetric matrices that depend on x [1].

2.4 SOLVING THESE PROBLEM2.4.1 INTERIOR POINT METHODConsider the LMI

F x F xF > 0= (2.10 )where F F R, i = 0, 1, 2... m. This LMI can rewritten as


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x logdetFx Fx > 0 otherwise (2.11)This function is finite if and only if F(x) > 0, and as x approaches the boundary of the feasible

setx|Fx > 0 it became infeasible, i.e., it is a barrier function for the feasible set, It can beshown that x is strictly convex on the feasible set, so it has a unique minimizer, which wedenote x.

Now turn to the problem of computing the analytic center of an LMI. (This is a special

form of our problem CP.) Newton's method, with appropriate step length selection, can be used

to efficiently compute x, starting from a feasible initial point. We consider the algorithm:

x+ x aHxgx (2.12)

where a is the damping factor of the kth iteration, and g(x) and H(x) denote the gradient andHessian of x, respectively, at x.

Their damping factor is:

a 1 if x 1 / 411 x otherwise

(2.13)

where x gxHxgx is called the Newton decrement of at x. Nesterov andNemirovskii show that this step length always results in x+feasible, i.e., F(x+)) > 0, andconvergence of xKto x[1].


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3 POLE PLACEMENT IN LMI REGIONS3.1 LYAPUNOV CONDITIONS FOR POLE CLUSTERING

Let us consider D be a sub region of the complex left-half plane. A dynamical system

x=

Ax is called D stable only if all its poles lie in D (that is, all eigenvalues of the matrix A lie in D).

When D is the entire left-half plane, which is characterized in LMI terms by the Lyapunov

theorem. Specifically, A is stable if and only if there exists a symmetric matrix X satisfying [3],

P A AP < 0, P > 0 (3.1)This Lyapunov characterization of stability has been extended to a variety of regions by

Gutman [4]. The regions considered there are polynomial regions of the form

D z C : ckzkz k, < 0 (3.2)Where the coefficients ckare real and satisfy ck=ck.The polynomial regions are not fully

general since, e.g., the region S,r, cannot be represented in this form. For polynomialregions, Gutman's fundamental result states that a matrix A is D-stable if and only if there exists

a symmetric matrix X such that [3]

D { z C : ckAkPAk, 0 (3.3)By replacing zkz with AkXAin equation (3.2) we get equation (3.3). From above result

it is clear that the Lyapunov theorem is a particular case of this result [3].

3.2 KRONECKER PRODUCTSThe Kronecker product is an important tool for the subsequent analysis. Recall the

Kronecker product of two matrices A and B is a block matrix C with generic block entry

C=

AB, that is, B AB (3.4)


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The following properties of the Kronecker product are easily established

1A A

C AC BC

BCD AC BDB A BB A B

The eigenvalues of (AB) are the pairwise products (A)(B) of the eigenvalues of A andB . As a result, the Kronecker product of two positive-definite matrices is a positive-definite

matrix. Finally, the singular values of (A

B) consist of all pairwise products

(A)

(B) of

singular values of A and B [2].

3.3 LMI REGIONDefinition 3.1 (LMI Regions): Asubset Dof the complex plane is called an LMI region if

there exist a symmetric matrix L = L =Lk Rand matrix M = Mk Rsuch thatD z C Fz < 0 (3.5)

Fz L z M z M (3.6)D z C: L z M z M < 0 (3.7)In other words, an LMI region define above is a subset of the complex plane that can be

represented in terms of LMI, or equivalently, an LMI in x = Re(z) and y = Im(z). As a result,

LMI regions are convex. Moreover, LMI regions are symmetric with respect to the real axis for

any value of z D,Interestingly, there is a complete counterpart of Gutman's theorem for LMIregions. Specifically, pole location in a given LMI region can be characterized in terms of the

mm block matrix [3], MA, P LP M PA MAP (3.8)


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Theorem 3.2: The matrix A is D-stable if and only if there exists a symmetric matrix X such

that

MA, P < 0 and P > 0 (3.9)

3.4 DIFFERENT LMI REGIONS DEFINED IN LMI3.4.1 HALF-PLANE:

Suppose we have to design half-plane in left hand side, means all poles lies in left hand side

R (z) < - : (3.10)

Now this expression can be written in terms of LMI region like that,

Fx z z 2 < 0 (3.11)where z = + j and z= j, From Gutmans expression

Fx L z M z M < 0 (3.12)Comparing equation (3.11) and (3.12), we get

L 2 , M 1 , M 1Put these value in given equation (3.8), we get

MA, P LP M PA MAP (3.13)MA, P PA AP 2P (3.14)

Above expression represent the LMI formulation of the half plane in complex plane and by

varying the value of half plane can be shiftedeither left side or right side. If is zero then this

result is equivalent to lyapunov stability condition [3].


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3.4.2 DISK CENTERED AT (-Q, 0) WITH RADIUS R:The equation of circle of radius r with center at (-q, 0) in real axis is given as,

(3.15)

This equation can be written in matrix form like as

Fx r q zq z r (3.16)Fx L z M z M < 0 (3.17)

Comparing above equation (4.7) and (4.9), we get

L r qq r , M 0 10 0 , M 0 01 0Put these value in given below equation (4.5), we get

MA, P rP qP PAqP AP rP (3.18)Above expression represent the LMI formulation of the circular plane of radius r and center

at (-q, 0) in complex plane and by varying the value of q circular plane can be shifted either left

side or right side and by r radius can be changed [3] [5].

3.4.3 CONIC SECTOR WITH APEX AT THE - AND INNER ANGLE 2:Its characteristic function is

Fx 2 cos ze z e < 0, (3.19)This equation can be written in matrix form like as

Fx 2coscosz z sinz z sinz z 2coscosz z > 0 (3.20)Comparing above equation (3.17) and (3.20), we get


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L 2cos 00 2cos ; M cos sinsin cosPut these value in given below equation (3.13), we get

MA, P 2cossinAPPA cosAP PAcosPA AP 2cos sinAP PA > 0 (3.21)Equation (3.21) shows a conical sector having angle and the origin of the conic is at [2].After combining above all regions we get a intersectional region which is shown in fig.3.1

FIGURE 3-1INTERSECTION OF DIFFERENT LMIREGIONS [3]


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3.5 INTERSECTION OF LMI REGIONSIn LMI one of the important advantage is that the many LMIs can be combined into single

LMI. LMI regions are often specified as the intersection of elementary regions, such as conic

sectors, disks, or vertical half-planes. Given LMI regions D1, D2,..DN, the intersection of

these region is D

D D D . . D (3.22)has characteristic function

Fz diagFz= (3.23)Corollary 3.1: Given two LMI regions D1and D2, a matrix A is both D1-stable and D2-stable if

and only if there exists a positive definite matrix X such that MA,X< 0 and MA,X< 0[3].


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4 DESIGN VIA LMI OPTIMIZATION4.1 Schur Complements

The equivalence between the Riccati inequality and the LMI can be seen by the following

well-known fact:

Lemma 1 (Schur Complement). Suppose R and S are Hermitian, i.e. R Rand S SThen, the following conditions are equivalent:

R < 0 , S GRG < 0AndS GG R < 0

Proof. Post-multiply (29) by the nonsingular I 0RG I and pre-multiply by its transpose: I 0RG I S GG R I 0RG I S GRG 00 R (4.1)Using Schur complements we can infer that if a matrix is positive definite then an arbitrary

diagonal square sub-block is also positive definite. For instance, if any diagonal element P of amatrix P is negative or zero the matrix P is not positive definite [1] [5].

Lemma 2. Let Q, U and V be given matrices. Then

Q VK U UKV < 0 (4.2)has a solution K if and only if

WQW < 0WQW < 0Where Wand Ware the null spaces of U and V respectively. Detail of Q, V and U matricesare given below [5].


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4.2 LMI FORMULATION FOR PERFORMANCEAssume we have plant transfer function H(s) with impulse response h(t), input to plant is u(t) and

output is y(t),

yt hut ht ut ht ud (4.3)Norm-2 of y(t) can be written as from appendix [6]

yt |yt|dt (4.4)

y 12 |yjw|dw (4.5)y 12 |Hjw||ujw|dw (4.6)

According to the Cauchy-Schwartz inequality

y 12 |Hjw||ujw|dw

H 12 |ujw|dw

H u (4.7)

Now we now the Hcondition for robust stability isH < (4.8)

Now comparing equation (5.3) and (5.4), we get

y

u < (4.9)

y u < 0 (4.10) y ud t > 0 (4.11)


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Lyapunov functions is define as [7]

v(xt) xtPxt y ud t > 0 (4.12)Now differentiating above energy function with respect to time and after differentiating it

must be less zero, which show that the system releasing the energy and state comes into

equilibrium state.

v(xt) xAP P Ax uBPx xPBu yy uu < 0 (4.13)v(xt) xAP P Ax uBPx xPBu xCD u uDCx

xCC x uDDu


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BX, P MA, P MPB MXCMBP XI XDXMC PD PI < 0 (4.21)P > 0, X > 0

where, MA, P LP M PA MAP (4.22)


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5 FEEDBACK CONTROLLER DESIGN5.1 STATE FEEDBACK HDESIGN WITH POLE PLACEMENT

In this section, the gain of the controller is designed via Linear Matrix Inequality with

H

performance and restricting the closed loop poles in LMI regions for better performance.

Suppose K is controller gain, then according to control law

(5.1)The state space model of system is

x A x B

w B

u

z Cx Dw Duy Cx Dw Du(5.2)

Where A is system matrix, B is input matrix, C is output matrix and D is transfer matrix.Now, put u = -Kx in above equation, closed loop state space will be

x A x B Kx A BKx Ax Bwy Cx DKx C DKx Cx Dw

(5.3)

Now put the value of closed loop state space matrices in equation given below

v(xt) A P P A PB CBP I DC D I < 0 (5.4)

v(xt) A BKP PA BK PB C DK

BP I DCC DK D I < 0 (5.5)

Above equation gives robust performance and for better performance add pole placement

constraints on this LMI. Different pole placement region is derived in last section are given as


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Half plane

MA,P = PA +

AP+ 2P < 0 (5.6)

Disk at centered (-q, 0) and radius r

MA, P rP qP PAqP AP rP < 0 (5.7)Conical sector starting atand angle

MA,P 2cossinAPPA

cosAP PA

cosPA AP 2cos sinAP PA < 0 (5.8)

Now combining above two constraints Hand half plane constraints, combine expression willbe

A B KP PA B K 2P PB CBP I DC D I < 0 (5.9)And to obtain better robustness of the controller has to be minimize. So optimization problemis formulated as

Objective function = minA B KP PA B K 2P PB CBP I DC D I < 0 (5.10)

By solving interior point method or any other optimization method the value of K can be obtain


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5.2 OUTPUT FEEDBACK HDESIGN WITH POLE PLACEMENTThe LTI plant can be represented in state-space form as

x A x Bw Bu

z Cx Dw Duy Cx Dw Du (5.11)where u is control input, w is a vector of exogenous inputs (such as reference signals, disturbance

signals, sensor noise), y is the measured output, and z is a vector of output signals related to the

performance of the control system. Fig 5.1 shows augmented block diagram plant with controller

is in feedback path.

FIGURE 5-1AUGMENTED BLOCK DIAGRAM PLANT [10]Let T denote the closed-loop transfer functions from w to z. the output-feedback control law

u = -Ky such that, closed loop poles lies in LMI region and T < . The controller can berepresented in state space form by

xk Ax Byu Cx Dy (5.12)Then the closed-loop transfer functionTSfrom w to z is given by [3] [6] [5]TS D CSI AB (5.13)where


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A A BDC BCBkC A B B BDD

BD

C C DDC DCkD= D DDDFrom equation (4.20), the closed loop Hdesign is derived as

PA A P PB CB P I DC D I < 0 (5.14)The conical region derived for closed loop pole in LMI with cone at origin and angle asMA, P sinAP PA cosAP PA cosPA AP sinAP PA < 0 (5.15)These two LMIs can be solved using optimization method, but before solving check whether

above given condition is feasible nor not, equation (4.20) is feasible only when P is positive

definite and satisfy following solvability condition as derived below

PA A P PB CB P I DC D I < 0 (5.16)Let P and P-1be partitioned as [2] [3] [7]:

P R MM U, P S NN V (5.17)Where R R and S R. Put value of P and closed loop state space matrices inequation (5.16)


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[ R MM U A BDC BCBkC A A BDC BCBkC A R MM U R MM U B BDDBD C DDC DCkB BDDBD R MM U 1 00 1 D DDDC DDC DCk D DDD I ]

< 0

(5.18)

Rearranging above matrix and expresses in the form of lemma 2 [5],

Q YKZ ZK Y < 0 (5.19)

Q [

R A AR AMMA 0 RB CMB 0BR BMC 0 I DD I]

(5.20)

Y BM BUR M 0 00 D (5.21)Z 0 IC 0 0 0D 0 (5.22)

K A BC D

(5.23)

According to lemma 2above expression will solvable if and only if

WQW < 0WQW < 0Where Wand Ware null space of Y and Z,

W W 00 00 IW 0 (5.24)One difficulty in LMI in output feedback case is that it include product of controller matrix

and P which make LMI non-linear. In case of state feedback new variable L = KP is defined that


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makes LMI linear. But in output feedback it is not possible. To make LMI linear proper change

of variable is derived in below

Theorem 6.1 Consider a proper plant P(s) of minimal realization and assume ( A, B, C isstabilizable and detectable and D 0withY 0 BI 0 0 00 D (5.25)

Z 0 IC 0 0 0D 0 (5.26)Let Wand Wbe two matrices whose column span the null spaces of Y and Z respectively,

Then the set of

-suboptimal controller of order k is non empty if and only if there exists some

(n+k)*(n+k) positive definite matrix Psuch that [5]:WQW < 0WQ1W < 0where

Q PA AP PB C

B

P I D

C D I

Q 1 AP PA B PCB I DCP D I A A 00 0

B B0

C C 0

P matrix and inverse of P are replaced by its partisan value then new Q and Q1 will be as


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Q

[

R A AR AMMA 0 RB CMB 0BR BMC 0

I DD I]

(5.27)

Q1 A S S A ANNA 0 B SC0 NCB 0CS CN I DD I (5.28)

Null space of the matrix Y and Z are calculated using eigenvalue of Y and Z matrix or other

method.

W W 00 00 IW 0 (5.29)

W 0 W30 0I 00 W (5.30)In null space second row is zero hence second row and second column can be removed from

equations (5.27) and (5.28)

Q R A AR RB CBR I DC D I (5.31)

Q 1 A S S A B SCB I DCS D I

(5.32)From equations (5.31) and (5.32) it is clear if R and S are symmetric and positive definite

then the controller will be non-empty. Now we have to transform P matrix into terms of R and S

[5].


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5.3 LINEARIZING CHANGE OF VARIABLEThe controller variables are implicitly defined in terms of the (unknown) matrix P. Let P and

P-1be partitioned as [2] [6] [7]:

P R MM U, P S NN V (5.33)Where R Rand S R

P I R S M N R N M VMS U N MN U V (5.34)

P

P R IM 0 , I S0 NPre and post-multiplying the variable matrix P by and, respectively; and carrying out

appropriate change of variables, the following LMI is obtained.

R II S > 0

(5.35)

Similarly, pre- and post-multiplying the inequality (5.15) by diag (, I , I and diag (, I, I),respectively; and carrying out appropriate change of variables, the following LMI is obtained.

AP A (5.36) A R BCM DCR A BDCNAM NBCR S BCM SA BDCR SA NB BDC (5.37)

PA PA A (5.38) RA CM DCRB NAM NBCR S BCM SA BDCRA CDB AS CNB BD (5.39)


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AP A A R BChat A BDhatCAhat SA BhatC (5.40)

PA PA A R

A

chat

B

AhatA CDhatB AS CBhat (5.41)

Where,

AhatNAM NBCR S BCM SA BDCRBhatNB SBDCh a t CM DCR

D h a t D (5.42)

MA, P diag , I , IsinAP PA cosAP PA cosPA AP sinAP PA diag , I, I < 0 (5.43)

MA, P =A R BChatRA ChatB A BDhatC Ahat

AhatA

C

Dhat

B

SABhatC

A

S

C

Bhat RA chatB AR Chat Ahat A DhatC

A

C

Dhat

B

Ahat A

S

C

Bhat

SABhatC

RA chatB A R BChat Ahat A BDhatCA CDhatB Ahat AS CBhat SABhatC A R BChatRA ChatB A BDhatC AhatAhatA CDhatB SABhatC AS CBhat< 0(5.44)

< 0 (5.45)Similarly, pre- and post-multiplying the inequality (5.14) by diag (, I , I and diag (, I,

I), respectively; and carrying out appropriate change of variables, the following LMI is obtained.

, I , IPA A P B PCB I DCP D I diag , I, I < 0 (5.46)


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< 0 (5.47)

A R R A

BChat Chat

B

B BDhatDCB BDhatD rI

(5.48)

Ahat A BDhatC SB BhatDCR DChat D DDhatD (5.49) ASSABhatC CBhat C DDhatCC DDhatC rI (5.50)

After changing the variable matrices into new matrix, three LMIs are obtained

R II S > 0 (5.51) < 0 (5.52)

< 0 (5.53)Now solving above constraints with objective function minimize gamma for increasing the

stability margin of the system, the values of R, S, Ahat, Bhat, Chat and Dhat are obtained. From

these value controller State space matrix can be calculated as [2] [6],

D DhatB NBhatSBDC ChatDCRMA NAhatNBCR S BCM SA BDCRM

(5.54)


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6 CONCLUSION In Linear Matrix Inequality different regions can be derived and closed loop poles

can be placed in anywhere in complex plane. By varying the ranges of LMI region

the better time response can be obtained.

In LMI one or more number of constraints can be put simultaneously and hence bothrobust and better time response can obtained.

Since LMI is solved by optimization method the optimum objective function can beobtained in case of H performance. The value of gamma can be minimized toobtain increased stability margin.


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7 REFERENCES

[1] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems

and Control Theory, Philadelphia: SIAM, 1994.

[2] Mahmoud chilali, Pascal Gahinet and Pierre Apkarian, "Robust pole placement in LMI

regions,"IEEE Trans. Automat. Contr., vol. 44, p. 12, DECEMBER 1999.

[3] M. Chilali and P. Gahinet, "H_ design with pole placement constraints: An LMIapproach,"IEEE Trans. Automat. Contr., vol. 44, p. 358367, 1996.

[4] S. Gutman and E. I. Jury, "A general theory for matrix root clustering in sub-regions of the

complex plan,"IEEE Trans. Automat. Contr., Vols. AC-26, pp. 853-863, 1981.

[5] G. Garcia and J. Bemussou, "Pole assignment for uncertain systems in a specified disk by

state-feedback,"IEEE Trans. Automat. Contr., vol. 40, pp. 184-190, 1995.

[6] Bikash Pal and Balarko Chaudhuri, Robust Control in Power Systems, New York, NY

10013. USA: Springer Science+Business Media, Inc, 2005.

[7] P. Gahinet and P. Apkarian, "A linear matrix inequality approach to H_ control,"Int. J.

Robust Nonlinear Contr., vol. 40, pp. 184-190, 1994.

[8] Kemin Zhou and John C. Doyle, Essentials of Robust Control, Prentice-Hall, 1998.

[9] C. Scherer, P. Gahinet and M. Chilali, "Multiobjective output-feedback control via LMI

optimization,"IEEE Trans. Automat. Contr., vol. 42, p. 896911, 1997.

[10] John Doyle, Bruce Francis and Allen Tannenbaum, Feedback Control Theory, Macmillan

Publishing Co., 1990.


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8 APPENDIX

Norms for Signals, Matrices and Systems

Norms of a continuous time varying signal is define as

ut |ut|dt

By using above definition we can calculate different norms of the signal like,

1-Norm ut utdt 2-Norm ut ( |ut|dt )-Norm ut sup|ut|

Suppose we have vectorX x xx3, then Norms isX |x|

1-Norm

X |x|

2-Norm X |x| -Norm X maxxMatrix A =x xx3 x

Maximum absolute column sum norm A ma x x Maximum absolute row sum norm A ma x x A maximum eigenvalue ofAA

Let we have system G(s)


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2-Norm Gs GjwGjWdt -Norm Gs sup||Gjw|| supGjwNow according parsvel theorem |ut|dt 12 |ujw|dt This theorem gives relationship between time domain and frequency domain [6] [9].

Lmi Method

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