Large Scale Systems

Model Order Reduction and Controller DesignTechniques

Dr. S. Janardhanan

Contents

1 Introduction to Large Scale Systems 11.1 What are Large Scale Systems ? . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Hierarchial Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Decentralized Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Large Scale System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Large Scale System Model Order Reduction and Control - Modal AnalysisApproach 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Davison Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Reduced Order Model Using Davison Technique . . . . . . . . . . . . 82.2.2 Alternative Method to Obtain Reduced Order Model through Davison

Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Improved Davison Technique . . . . . . . . . . . . . . . . . . . . . . . 132.2.4 Suboptimal Control Using Davison Model . . . . . . . . . . . . . . . 142.2.5 Control Law Reduction Approach Using Davison Model . . . . . . . . 14

2.3 Chidambara Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Reduced Order Model Using Chidambara Technique . . . . . . . . . . 152.3.2 Suboptimal Control Using Chidambara Model . . . . . . . . . . . . . 162.3.3 Control Law Reduction Approach Using Chidambara Model . . . . . 17

2.4 Marshall Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 Reduced Order model by Marshall Technique . . . . . . . . . . . . . 17

2.5 Choice of Reduced Model Order . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.1 Model Order Selection Criterion by Mahapatra . . . . . . . . . . . . 182.5.2 Another Criterion for Order Selection and Mode Selection . . . . . . 20

3 Model Order Reduction and Control - Aggregation Methods 233.1 Aggregation of Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Properties of Aggregated System Matrix . . . . . . . . . . . . . . . . 243.1.2 Error in Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Determination of Aggregation Matrix . . . . . . . . . . . . . . . . . . . . . . 263.3 Modal Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Reduced Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.2 Stability of Feedback System . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Aggregation by Continued Fraction . . . . . . . . . . . . . . . . . . . . . . . 32

4 Large Scale System Model Order Reduction - Frequency Domain BasedMethods 374.1 Moment Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Pade Approximation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 Pade Approximation Method for SISO Systems . . . . . . . . . . . . 404.2.2 Modal-Pade Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.3 Pade Approximation for Multivariable Systems in Frequency Domain 434.2.4 Stable Pade for Multivariable Systems . . . . . . . . . . . . . . . . . 454.2.5 Reduction of non-asymptotically stable systems . . . . . . . . . . . . 464.2.6 Time-Domain Pade Approximation for Multivariable Systems . . . . 474.2.7 Time-Domain Modal-Pade Method . . . . . . . . . . . . . . . . . . . 51

4.3 Routh Approximation Techniques . . . . . . . . . . . . . . . . . . . . . . . . 534.3.1 Routh Approximation Method Using Parameters . . . . . . . . 534.3.2 Routh Approximation Technique Using Parameters . . . . . . . 544.3.3 Aggregated Model of Routh Approximants . . . . . . . . . . . . . . . 564.3.4 Optimal Order of Routh Approximant . . . . . . . . . . . . . . . . . 59

4.4 Continued Fraction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.1 The Three Cauer Forms . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.2 A Generalized Routh Algorithm . . . . . . . . . . . . . . . . . . . . . 624.4.3 Simplified Models Using Continued Fraction Expansion Forms . . . . 64

5 Large Scale System Model and Controller Order Reduction - Norm BasedMethods 675.1 Introductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.1.1 Norms of Vectors and Matrices . . . . . . . . . . . . . . . . . . . . . 675.1.2 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 685.1.3 Grammian Matrices and Hankel Singular Values . . . . . . . . . . . . 705.1.4 Matrix Inversion Formulae . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Model Reduction by Balanced Truncation . . . . . . . . . . . . . . . . . . . 735.2.1 Balanced Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2.2 Balanced Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2.3 Steady State Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.4 Reduction of Unstable Systems by Balanced Truncation . . . . . . . . 765.2.5 Properties of Truncated Systems . . . . . . . . . . . . . . . . . . . . 765.2.6 Frequency-Weighted Balanced Model Reduction . . . . . . . . . . . . 81

5.3 Model Reduction by Impulse/Step Error Minimization . . . . . . . . . . . . 825.3.1 Impulse Error Minimization . . . . . . . . . . . . . . . . . . . . . . . 835.3.2 Step Error Minimization . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Optimal Model Order Reduction Using Wilsons Technique . . . . . . . . . . 905.4.1 Impulse Error / White Noise Error Minimization . . . . . . . . . . . 90

6 Pole Placement Techniques 956.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.2 What Poles to Choose ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.2 Interpretation of Responses from Pole-Zero Locations . . . . . . . . . 966.3 Pole Assignment in Single Input Systems . . . . . . . . . . . . . . . . . . . . 97

6.3.1 State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.3.2 Optimal State Feedback ( Brief Introduction to LQR ) . . . . . . . . 996.3.3 Static Output Feedback in Single Input Systems . . . . . . . . . . . . 1026.3.4 Dynamic Output Feedback ( SISO Case ) . . . . . . . . . . . . . . . . 103

6.4 Pole Assignment and Placement in Multi-Input Systems . . . . . . . . . . . 1046.4.1 Concepts of Multivariable Systems . . . . . . . . . . . . . . . . . . . 1046.4.2 State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.4.3 Design of Optimal Control Systems with Prescribed Eigenvalues . . . 1086.4.4 Static Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 1166.4.5 Two Time-Scale Decomposition and State Feedback Design . . . . . . 123

7 Fast Output Sampling (FOS) 1277.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.2 Controller Deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.3 Closed Loop Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.4 Techniques for Determining Fast Output Sampling Controller Gain . . . . . 130

7.4.1 Two Time-Scale Approach for Conditioning of State Feedback Gain F 1307.4.2 Singular Value Decomposition of Measurement Matrix C . . . . . . . 1327.4.3 Approach for Multi-Plant Systems . . . . . . . . . . . . . . . . . . . 132

7.5 An LMI Formulation of the design problem . . . . . . . . . . . . . . . . . . . 1337.6 A Modified Approach for Fast Output Sampling Feedback . . . . . . . . . . 133

8 Periodic Output Feedback (POF) 1358.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358.2 Periodic Output Feedback Controller Deduction . . . . . . . . . . . . . . . . 1358.3 Multimodel Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9 Robust Control of Systems with Parametric Uncertainty 1399.1 Concepts Related to Uncertain Systems . . . . . . . . . . . . . . . . . . . . . 139

9.1.1 Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399.1.2 Hermite-Bieler Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1409.1.3 Kharitonov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1429.1.4 Gerschgorin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1459.1.5 Simultaneous Stabilization of Interval Plant Family Based on Kharitonov

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469.2 Bhattacharyyas Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1479.3 Jayakumars Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9.3.1 Routh Table Based Kharitonov Algorithm in Controller Design . . . 1549.3.2 An Alternative Proof for Existence of a Simultaneously Stabilizing

State Feedback for Interval Systems . . . . . . . . . . . . . . . . . . . 1569.4 Smagina & Brewers Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9.4.1 The Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . 1579.4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

9.4.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.5 State Feedback for Uncertain Systems Based on Gerschgorin Theorem . . . . 163

A Numerical Problems in Large Scale Systems 171A.1 Davison, Chidambara and Marshall Techniques . . . . . . . . . . . . . . . . 171A.2 Routh and Pade Approximations . . . . . . . . . . . . . . . . . . . . . . . . 172A.3 State and Output Feedback Design . . . . . . . . . . . . . . . . . . . . . . . 173A.4 Periodic Output Feedback and Fast Output Sampling . . . . . . . . . . . . . 174A.5 Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Chapter 1

Introduction to Large Scale Systems

1.1 What are Large Scale Systems ?

A great number of problems are brought about by the present day technology and societal andenvironmental processes which are highly complex and large in dimension and stochastic bynature. The notion of large scale is highly subjective one in that one may ask : How largeis large?. There has been no accepted definition for what constitutes a large scale system.Many viewpoints have been presented on this issue. One viewpoint has been that a systemis considered large scale if it can be decoupled or partitioned into a number of interconnectedsubsystems or small scale systems for either computational or practical reasons. Anotherviewpoint is that a system is large scale when its dimensions are so large that conventionaltechniques of modeling, analysis, control, design and computation fail to give reasonablesolutions with reasonable computational efforts. In other words a system is large when itrequires more than one controller.

Since the early 1950s, when classical control theory was being established, engineers havedevised several procedures, both within the classical and modern control contexts, whichanalyze or design a given system. These procedures can be summarized as follows.

1. Modeling procedures which consist of differential equations, input-output transfer func-tions and state space formulations.

2. Behavioral procedures of systems such as controllability, observability and stabilitytests and application of such criteria as Routh - Hurwitz, Nyquist, Lyapunovs secondmethod etc.,.

3. Control procedures such as series compensation, pole placement, optimal control etc.,.

The underlying assumption for all such control and system procedures has been cen-trality i.e., all the calculations based upon system information and the informationitself are localized at a given center, very often a geographical position.

A notable characteristic of most large scale systems is that centrality fails to hold updue to either the lack of centralized computing capability or centralized information. Need-less to say, many real problems considered are large scale by nature and not by choice.

1

2 Large Scale Systems

The important points regarding large scale systems are that their hierarchial (multilevel)and decentralized structures depict systems dealing with society, business, management, theeconomy, the environment, energy, data networks, power networks, space structures, trans-portation, aerospace, water resources, ecology and flexible manufacturing networks to namea few. These systems are often separated geographically, and their treatment requires con-sideration of not only economic costs as is common in centralized systems but also suchimportant issues as reliability of communication links, value of information etc.,. It is forthe decentralized and hierarchial control properties and potential applications that manyresearchers throughout the world have devoted a great deal of effort to large scale systemsin recent years.

1.2 Hierarchial Structures

One of the earlier attempts in dealing with large scale systems was to decompose a givensystem into a number of subsystems for computational efficiency and design simplification.The idea of decomposition was first treated theoretically in mathematical programming byDantzig and Wolfe [1] by treating large linear programming problems possessing specialstructures. The coefficient matrices of such large linear programs often have relatively fewnonzero elements, i.e., they are sparse matrices. There are two basic approaches for dealingwith such problems, coupled and decoupled. The coupled approach keeps the problemsstructure intact and takes advantage of the structure to perform efficient computations. Thecompact basis triangularization and generalized upper bounding are two such efficientmethods. The decoupled approach divides the original system into a number of subsystemsinvolving certain values of parameters. Each subsystem is solved independently for a fixedvalue of so-called decoupling parameter, whose value is subsequently adjusted by a coordina-tor in an appropriate fashion so that the subsystems resolve their problems and the solutionto the original system is obtained. Recently the decoupled approach has been termed asmultilevel or hierarchial approach. Consider a two level system shown in Fig .(1.1). Atthe first level, N subsystems of the original large scale system are shown. At the second levela coordinator receives the local solutions of the N subsystems, Si, i = 1, 2, , N and thenproduces a new set of interaction parameters ai, i = 1, 2, , N . The goal of the coordinatoris to arrange the activities of the subsystems to provide a feasible solution to the overallsystem.

1.3 Decentralized Control

Most large scale systems are characterized by a great multiplicity of measured outputs andinputs. For example, an electric power system has several control substations, each beingresponsible for the operation of a part of the overall system. This situation arising in acontrol system design is often referred to as decentralization. The designer for such systemsdetermines a structure for control which assigns system inputs to a given set of local con-trollers(stations), which observe only local system outputs. In other words, this approach,called decentralized control, attempts to avoid difficulties in data gathering storage require-ments, computer program debuggings and geographic separation of system components.

1.3 Decentralized Control 3

Figure 1.1: Hierarchial Control

4 Large Scale Systems

Figure 1.2: Two Controller Decentralized Large Scale System

Fig. (??) shows a two controller decentralized system. The basic characteristic of anydecentralized system is that the transfer of information from one group of sensors or actuatorsto others is quite restricted. For example, in the system of Fig. (??), only the output y1 andthe external input u1 are used to find the control v1 and likewise the control v2 is obtainedthrough only the output y2 and external input u2. The determination of control signals v1and v2 based on the output signals y1 and y2 respectively is nothing but two independentoutput feedback problems which can be used for stabilization or pole placement purposes. Itis therefore clear that the decentralized control scheme is of feedback form, indicating thatthis method is very useful for large scale linear systems. This is a clear distinction from thehierarchial control scheme, which was mainly intended to be an open loop structure.

In the previous part the concept of a large scale system and two basic hierarchial anddecentralized control structures were briefly introduced. Although there is no universaldefinition of a large scale system, it is commonly accepted that such systems possess thefollowing characteristics

1. Large scale systems are often controlled by more than one controller or decision makerinvolving decentralized computations.

2. The controllers have different but correlated information available to them, possiblyat different times.

3. Large scale systems can also be controlled by local controllers at one level whose controlactions are being coordinated at another level in a hierarchial(multilevel) structure.

4. Large scale systems are usually represented by imprecise aggregate models.

1.4 Large Scale System Modeling 5

5. Controllers may operate in a group as a team or in a conflicting manner with single-or multiple-objective or even conflicting-objective functions.

6. Large scale systems may be satisfactorily optimized by means of suboptimal or nearoptimum controls, sometimes termed as a satisfactory strategy.

An attempt is made here to consider primarily modeling and control of large scale systems.Most of the discussions are focussed on large scale linear, continuous time, stationary anddeterministic systems.

1.4 Large Scale System Modeling

Scientists and engineers are often confronted with the analysis, design and synthesis of real-life problems. The first step in such studies is the development of a mathematical modelwhich can be a substitute for the real problem.

In any modeling task, two often conflicting factors prevail - simplicity and accuracy. Onone hand, if a system model is oversimplified, presumably for computational effectiveness,incorrect conclusions may be drawn from it in representing an actual system. On the otherhand, a highly detailed model would lend to a great deal of unnecessary complicationsand should a feasible solution be attainable, the extent of resulting details may becomeso vast that further investigations on the system behavior would become impossible withquestionable practical values. Clearly a mechanism by which a compromise can be madebetween a complex, more accurate model and a simple, less accurate model is needed. Sucha mechanism is not a simple undertaking. The key to a valid modeling philosophy is to setforth the following outline.

1. The purpose of the model must be clearly defined, no single model can be appropriatefor all purposes.

2. The systems boundary separating the system and the outside world must be defined.

3. A structural relationship among different system components which would best repre-sent desired or observed effects must be defined.

4. Based on the physical structure of the model, a set of system variables of interest mustbe defined. If a quantity of important significance cannot be labelled, step (3) mustbe modified accordingly.

5. Mathematical descriptions of each system component, sometimes called elementalequations, should be written down.

6. After the mathematical description of each system component is complete, they arerelated through a set of physical laws of conservation (or continuity) and compatibility,such as Newtons, Kirchoffs or D Alemberts.

7. Elemental, continuity and compatibility equations should be manipulated and themathematical format of the model should be finalized.

6 Large Scale Systems

8. The last step to a successful modeling is the analysis of the model and its comparisonwith real situations.

Chapter 2

Large Scale System Model OrderReduction and Control - ModalAnalysis Approach

2.1 Introduction

It is usually possible to describe the dynamics of physical systems by a number of simulta-neous linear differential equations with constant coefficients.

x = Ax+ bu

But for many processes ( like chemical plants and nuclear reactors) the order of the matrixA may be quite large. It would be difficult to work with these complex systems in theiroriginal form. In such cases, it is common to study the process by approximating it to asimpler model. For instance, the response of an airplane is quite commonly approximated bya second order transfer function. These mathematical models correspond to approximatinga system by its dominant pole-zeros in the complex plane. They generally require empiricaldetermination of the system parameters. Many different methods have been developed toaccomplish the purpose by estimating the dominant part of the large system and finding asimpler ( or reduced order) system representation that has its behavior akin to the originalsystem.

2.2 Davison Technique

A structured approach to the model reduction problem was given by E. J. Davison in [2].The method suggests that a large (n n) system can be reduced to a simpler (l l) model(l n) by considering the effects of the l most dominant ( dominant in the sense of beingclosest to instability) eigenvalues alone. The principle of the method is to neglect eigenvaluesof the original system that are farthest from the origin and retain only dominant eigenvaluesand hence dominant time constants of the original system in the reduced order model.

The procedure to obtain the reduced order model can be described thus

7

8 Large Scale Systems

2.2.1 Reduced Order Model Using Davison Technique

Suppose the original system is represented as

X = AX +Bu, where A = n n matrix (2.1)and the new mathematical model is given by

Y = AY +Bu, where A = l l matrix (2.2)where, l < n, and Bu and Bu are respectively their forcing functions.It can be shown that if x1, x2, , xn are the normalized eigenvectors corresponding to

the eigenvalues 1, 2, , n of the matrix A with Re(1) Re(2) Re(n), thenthe transformation

Z = P1X (2.3)

P =[v1 v2 vn

]vi =

[x1,i x2,i xn,i

]Twould transform the system into

Z = AzZ +Bzu

where,

Az = P1AP,

Bz = P1B

and Az would be either in the diagonal or the Jordan canonical form. Truncation ofnon-dominant eigenvalues is simpler in this case.

In this case, the state response of the system for an input Bu = Bu, can be shown to be

X =

t0

PeAz(t)P1Bud (2.4)

P1 = [ij], i = 1..n, j = 1..n

which would give

X =ni=1

1 + eiti

x1,ix2,i...xn,i

(i1b1 + i2b2 + + inbn) (2.5)If l states that need to be considered for the reduced order model are represented as

xc1, xc2, , xcl then the lth order reduced model can be derived as

2.2 Davison Technique 9

A = A0 + A1P1P

10 (2.6)

B = P0[P1B] (2.7)

Y =

xc1xc2

xcl

(2.8)where,

A0 =

ac1,c1 ac1,c2 ac1,clac2,c1 ac2,c2 ac2,cl...

......

acl,c1 acl,c2 acl,cl

(2.9)

A1 =

ac1,1 acl,1ac1,2 acl,2...

...ac1,c11 acl,c11ac1,c1+1 acl,c1+1...

...ac1,c21 acl,c21ac1,c2+1 acl,c2+1...

...ac1,cl1 acl,cl1ac1,cl+1 acl.cl+1...

...ac1,n acl,n

T

(2.10)

P0 =

xc1,1 xc1,2 xc1,lxc2,1 xc2,2 xc2,l...

......

xcl,1 xcl,2 xcl,l

(2.11)

P1 =

x1,1 x1,2 x1,lx2,1 x2,2 x2,l

xc11,1 xc11,2 xc11,lxc1+1,1 xc1+1,2 xc1+1,l

xcl1,1 xcl1,2 xcl1,lxcl+1,1 xcl+1,2 xcl+1,l

xn,1 xn,2 xn,l

(2.12)

10 Large Scale Systems

[P1B] = first l rows of P1B (2.13)

Derivation :

The principle involved in reducing the matrix is to neglect higher-order time constants ofthe system. The following equation is obtained if the first alone are retained from Eqn..(2.4),.i.e., considering only the first l terms of Eqn. (2.5), and taking into account only thel states considered (i.e., equating the other states to zero), the following is obtained.

Y =l

i=1i

xc1,ixc2,i...xcl,i

(2.14)i =

1 + eii

(i1b1 + i2b2 + + inbn)

and therefore

12...l

= P10 Y (2.15)

then

x1x2...xc11xc1+1...xcl1xcl+1...xn

= P1

12...l

(2.16)

Substituting Eqn. (2.15) in Eqn. (2.16)

2.2 Davison Technique 11

x1x2...xc11xc1+1...xcl1xcl+1...xn

= P1P

10 Y (2.17)

Considering the equations for xc1, xc2, , xcl alone from Eqn. (2.1), the following equa-tion can be obtained

Y = A0Y + A1

x1x2...xc11xc1+1...xcl1xcl+1...xn

+ P0PlBzu (2.18)

Pl =[Il 0

]Substituting Eqn. (2.17) in Eqn. (2.18),

Y = A0Y + A1P1P

1o + P0[P

1B]u (2.19)

2.2.2 Alternative Method to Obtain Reduced Order Model throughDavison Technique

The Davison Model can also be computed thusInitially, the system states are rearranged in such a manner that the eigenvectors corre-

sponding to the states to be retained from Eqn. (2.1) are placed first. If P is representedas

P =

[P11 P12P21 P22

](2.20)

then, if the state vector X is partitioned into dominant and non-dominant parts as X1(consisting of the states to be retained) and X2 (consisting of the states to be discarded),

12 Large Scale Systems

and the state and input matrices are also partitioned in appropriate fashion, then X can berepresented as

X =

[X1X2

]=

[A11 A12A21 A22

] [X1X2

]+

[B1B2

]u (2.21)

X =

[X1X2

]=

[P11 P12P21 P22

] [Z1Z2

], (2.22)

where, Z1 and Z2 are the states of the decoupled system representation. Thus,

X1 = P11Z1 + P12Z2, (2.23)

X2 = P21Z1 + P22Z2. (2.24)

If the decoupled representation of the system is

Z =

[Z1Z2

]=

[1 00 2

] [Z1Z2

]+

[12

]u (2.25)

where,

[1 00 2

]= P1AP, (2.26)[

12

]= P1B. (2.27)

The modes in Z2 are non-dominant and therefore can be ignored (according to Davison[2]). Thus, setting Z2 to zero, and substituting in Eqns. (2.23 and 2.24) the following areobtained.

X1 = P11Z1 (2.28)

X2 = P21Z1 (2.29)

X2 = P21P111 X1 (2.30)

and from Eqn. (2.21)

X1 = A11X1 + A12X2 +B1u (2.31)

Substituting Eqn. ( 2.29) in Eqn. (2.31),

X1 = A11X1 + A12P21Z1 +B1u

X1 =(A11 + A12P21P

111

)X1 +B1u (2.32)

Eqn. (2.32) gives the reduced order model for the system computed through the alternatemethod. Both models in Eqn. (2.2) and Eqn. (2.32) represent the same system dynamics.

2.2 Davison Technique 13

2.2.3 Improved Davison Technique

It was noted that the above said methods were able to reproduce the transient behaviorof the large scale system well. But, as pointed out in [35] and rectified in [6], there is anerror in the steady state value of the system response. An improvement in the method wasproposed by Davison in [6]. In the improved Davison Technique, the state equation in Eqn.(2.2) is replaced by

Y = DAD1Y +DBu (2.33)

where,

D =

d1

d2. . .

dl

(2.34)dj =

[A1B]j

[A1B]j, if [A1B]j 6= 0

= 1 if [A1B]j = 0

j = 1, 2, , l (2.35)where [A1B]j is the j

th element of the l vector A1B and [A1B]j is the element ofthe n vector A1B which corresponds to the jth state retained in the simplified system. Thenew simplified system is equivalent to the following system.

X = AX +Bu (2.36)

Y = DX (2.37)

and so it can be seen that the response of the new system will have correct steady-state values for a step-function input (provided that [A1B]j 6= 0), and will still maintainsatisfactory dynamic behavior.. It should be noted that if [A1B]j = 0 for some j, thenthe steady-state values of the variable yj may be in error. Variables to be retained in thereduced order model should, therefore, always be chosen so that [A1B]j 6= 0.

For the case of multi-input systems, the corresponding model would be

Xi = AXi +B

i ui, i = 1, 2, , r

Y =r

i=1DiX

i (2.38)

where,

B =[B1 B

2 Br

](2.39)

and Di, i = 1, 2, , r is determined from Eqns. (2.34 - 2.35), using Bi in place of Bwhere

B =[B1 B2 Br

](2.40)

14 Large Scale Systems

2.2.4 Suboptimal Control Using Davison Model

For the large scale system represented in Eqn. .(2.1), an optimal controller may be deducedby minimizing the cost function

J =

0

(XTQX + uTRu

)dt (2.41)

where, Q and R are the weights of the states and inputs of the system respectively.Partitioning Q and using Eqn. (2.30),

XTQX = XT1 Q11X1 + 2XT1 Q12X2 +X

T2 Q22X2 (2.42)

= XT1

(Q11 + 2Q12P21P

111 +

(P111

)TP T21Q22P21P

111

)X1 (2.43)

Thus,

J = JM =

0

(XT1 QMX1 + u

TRu)dt (2.44)

QM =(Q11 + 2Q12P21P

111 +

(P111

)TP T21Q22P21P

111

)(2.45)

If the reduced order system is represented as

X1 = FX1 +Gu (2.46)

then, the suboptimal controller is

u = R1GT (2.47)where, is the solution of the Riccati equation,

F + F T GR1GT +QM = 0 (2.48)

2.2.5 Control Law Reduction Approach Using Davison Model

A large scale system (as in Eqn. (2.1)) with a performance criterion described by Eqn. (2.41)can have its optimal control law described as

u = Kx (2.49)

=[K1 K2

] [ X1X2

](2.50)

Using the Davison model in Eqn. (2.30),

u =(K1 +K2P21P

111

)X1 (2.51)

2.3 Chidambara Technique 15

2.3 Chidambara Technique

Noting that the basic model of Davison in [2] does not give accurate steady-state response,Chidambara, in his correspondence with Davison ( [35]) had suggested an approach formodel order reduction. In this model only the transient response of the left out states areignored (i.e. X2), the steady-state contribution of these states are taken into account inorder to nullify the steady-state error seen in the basic Davison technique.

2.3.1 Reduced Order Model Using Chidambara Technique

For the Chidambara model, the system is represented as in Eqn. (2.21). The transformationP is computed as earlier and represented as in Eqn. (2.20).

Consider the differential equation for Z2 as in Eqn. (2.25).

Z2 = 2Z2 + 2u (2.52)

Performing Laplace transform on Eqn. (2.52),

(sI 2)Z2(s) = 2U(s) (2.53)

Since only the steady-state contribution of Z2 is to be considered, setting s = 0,

2Z2(s) = 2U(s)Z2 = 12 2u (2.54)

Now substituting the value of Z2 in the relation in Eqn. (2.24),

X2 = P21Z1 P2212 2u (2.55)

Solving for Z1 in Eqn. (2.23) and substituting it in Eqn. (2.55),

X2 = P21P111 X1 +

(P21P

111 P12 P22

)12 2u (2.56)

= LX1 +Hu (2.57)

Substituting Eqn. (2.57) in Eqn. (2.31), the reduced order model of the large scalesystem is obtained using Chidambara technique as

X1 =(A11 + A12P21P

111

)X1 +

(A12

(P21P

111 P12 P22

)12 2 +B1

)u (2.58)

= FX1 +Gu (2.59)

16 Large Scale Systems

2.3.2 Suboptimal Control Using Chidambara Model

The method of deducing a suboptimal control law for the reduced order Chidambara modelof a system was proposed by Rao and Lamba in [7]. The modified performance criterion ofthe reduced order system reflects the performance criterion desired from the original system.

If the cost function is defined as in Eqn. (2.41) and the original system as in Eqn. (2.21),then as in the case of the Davison model, the value of XTQX can be deduced using Eqn.(2.57) as,

XTQX = XT1 Q11X1 + 2XT1 Q12(LX1 +Hu) + (X

T1 L

T + uTHT )Q22(LX1 +Hu)

The cost function can therefore be represented as

JM =

0

(XT1 (Q11 + 2Q12L+ L

TQ22L)X1 + 2XT1 (Q12H +H

TQ22L)u+ uT (R +HTQ22H)u

)dt

(2.60)Defining

Q1 = Q11 + 2Q12L+ LTQ22L, (2.61)

R1 = R +HTQ22H, (2.62)

S1 = Q12H +HTQ22L, (2.63)

and

u = u+R11 STX1, (2.64)

the simplified model represented in Eqn. (2.59) is equivalent to

X1 =(F GR11 ST1

)X1 +Gu (2.65)

and the performance criterion in Eqn. (2.60) is equivalent to

JM =

0

(XT1 (Q1 S1R11 ST1 )X1 + uTR1u

)dt (2.66)

If the matrices R1 and QM =(Q1 S1R11 ST1

)are positive definite and positive semi-

definite, respectively, then an optimal solution of the problem represented be Eqns. (2.65and 2.66) is given as

u = R11 GTX1 (2.67)where is the solution of the Riccati equation

(F GR11 ST1

)+(F GR11 ST1

)T GR11 GT +Q1 S1R11 ST1 = 0 (2.68)

Thus

2.4 Marshall Technique 17

u = u R11 STX1= R1 (GT + ST )X1 (2.69)

This optimal controller to the simplified system could also serve as a suboptimal controllerto the original system.

2.3.3 Control Law Reduction Approach Using Chidambara Model

If a control law u has been designed to control the original system then it can be applied onthe reduced order model thus.

Assuming the derived control law is of the form

u = KX

=[K1 K2

] [ X1X2

]= K1X1 +K2X2

Using the relation between X2 and X1 described in Eqn. (2.57), the control law can berepresented as

u = (K1 +K2L)X1 +K2Hu (2.70)

(I K2H)u = (K1 +K2L)X1If (I K2H) is invertible then the control law can be represented for the reduced order

system as

u = (I K2H)1(K1 +K2L)X1 (2.71)

2.4 Marshall Technique

S. A. Marshall had proposed an alternate way to compute the reduced order model in [8].This technique is quite similar to the Chidambara technique since it too takes into accountthe steady-state values of the X2 states. The difference exists in the manner in which thereduced order state equation is obtained.

2.4.1 Reduced Order model by Marshall Technique

If a large scale system is represented as in Eqn. (2.21) and the transformation that producesa decoupled system be as in Eqn. (2.22), then its reduced order model may be deduced thus.

If the transformation relating the X states to the Z states be given as Q then,[Z1Z2

]= Z = P1X = QX =

[Q11 Q12Q21 Q22

] [X1X2

](2.72)

18 Large Scale Systems

Observing Eqn. (2.25), it can be seen that the submatrix 1 is associated with largertime constants of the system, whereas the response of any element in Z2 settles very fast.Thus, it may be approximated as an instantaneous step change. This is the essence of thetechnique.

Mathematically, this approximation is equivalent to putting

Z2 = 0 (2.73)

Substituting Eqn. (2.73) and P1 = Q in Eqn. (2.25),

Z1 = 1Z1 + 1u (2.74)

0 = 2Z2 + 2u

and using Eqn. (2.72) in Eqn. (2.74),

Z2 = Q21X1 +Q22X2 = 12 2uX2 = Q122 Q21X1 Q122 12 2u (2.75)

Substituting Eqn. (2.75) in Eqn. (2.31) and using the relationships between Pij and Qij,one obtains the reduced order model be Marshall technique as

X1 = P111P111 X1 +

(B1 A12Q122 12 2

)u (2.76)

2.5 Choice of Reduced Model Order

The methods presented above give a reduced order model that is an approximation of theoriginal system. However, it is not yet clear how small the approximate model can be andyet accurately represent the process. It is seen that the smallness of the approximate modelcan be decided in terms of the largest eigenvalue neglected, the size of the original plant,and the reduced plant. This criterion is judged by comparing the time responses of variouslow order systems.

2.5.1 Model Order Selection Criterion by Mahapatra

A criterion for selecting the model order for a reduced order Davison model was proposed byMahapatra [9]. For a large scale system model in Eqn. (2.1), if the transformation matrixP is partitioned as

P =

[P11 P12P21 P22

]=[P1 P2

](2.77)

Mahapatra had made the assumption that the eigenvalues are real, negative and distinct.The initial conditions and the inputs are considered as zero and unit step respectively. Thenthe solution of the system would be

2.5 Choice of Reduced Model Order 19

X(t) = P1[e1t I]11 1 + P2 [e2t I]12 2 (2.78)

The approximate solution provided by the reduced order model is

X(t) = P1[e1t I]11 1

Hence, the error involved in ignoring the higher order modes l+1, , n in state equationEqn. (2.78) is given by

E(t) = P2[e2t I]12 2 (2.79)

E(t) P2 .e2t I . 12 . 2 (2.80)

exp (2t) I