state space analysis of control system.pdf

State Space Analysis of State Space Analysis of C lC lControl systemControl system

TopicsTopicsTopicsTopics

A practical control system.A practical control system.p yp yState space representationState space representationDefinitions.Definitions.Critical considerations while selecting state variables.Critical considerations while selecting state variables.State variable selection.State variable selection.Advantages of state variable representation.Advantages of state variable representation.Generic state space representation.Generic state space representation.Block diagram representation of linear systems.Block diagram representation of linear systems.Writing differential equations in First Companion formWriting differential equations in First Companion form

29-Apr-13 3State Space Analysis of Control System

Mugdha Salvi, VCET

A practical control systemA practical control systemA practical control systemA practical control system

(bi i )

Wi d i / l i

(bias noise)

Window opening/closing(random noise)

Temperature control system in a car29-Apr-13 4

State Space Analysis of Control System Mugdha Salvi, VCET

Another Practical Control SystemAnother Practical Control SystemAnother Practical Control SystemAnother Practical Control System

noise

Water level control in an overhead tank29-Apr-13 5

State Space Analysis of Control System Mugdha Salvi, VCET

State Space RepresentationState Space RepresentationState Space RepresentationState Space Representation

Input variable: Systemcontrolnoise

Y →Y*Input variable:

Manipulative (control)Non-manipulative (noise)

System

Controller

Z

p ( )Output variable:Variables of interest that can be either be measured or

Controller

Variables of interest that can be either be measured or calculated

State variable:Minimum set of parameters which completely summarize the system’s status.y


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DefinitionsDefinitionsDefinitionsDefinitions

State:The state of a dynamic system is the smallest number ofvariables (called state variables) such that the

d bknowledge of these variables at t = t0, together with theknowledge of the input for t ≥ t0, completely determinethe behavior of the system for any time t ≥ t0.the behavior of the system for any time t ≥ t0.

Note:State variables need not be physically measurable orobservable quantities. This gives extra flexibility.


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DefinitionsDefinitionsDefinitionsDefinitions

State vector:State vector:A n - dimensional vector whose components are n state variables that describe the system completely.

St t SState Space:the n - dimensional space whose co-ordinate axes consist of the x1 axis, x2 axis, …., xn axis is called a state 1 , 2 , , nspace.

N tNote: For any dynamical system, the state space remains unique, but the state variables are not unique. q , q


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Critical Considerations while Critical Considerations while selecting State Variables.selecting State Variables.

Minimum number of variablesMinimum number of first-order differential equations needed to describe the system dynamics completelyLesser number of variables: won’t be possible to describe theLesser number of variables: won t be possible to describe the system dynamicsLarger number of variables:

Computational complexityCo p o co p e yLoss of either controllability, or observability or both.

Linear independence. If not, it may result in:Bad: May not be possible to solve for all other systemBad: May not be possible to solve for all other system variablesWorst: May not be possible to write the complete state equationsequations


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Sate Variable SelectionSate Variable SelectionSate Variable SelectionSate Variable Selection

Typically, the number of state variables (i.e. theTypically, the number of state variables (i.e. the order of the system) is equal to the number of independent energy storage elements. However, p gy gthere are exceptions!

Is there a restriction on the selection of the state variables ?YES! All state variables should be linearly independent and they must collectively describe the system completely.y y y p y


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Advantages of State Space Advantages of State Space R iR iRepresentationRepresentation

Systematic analysis and synthesis of higher order y y y gsystems without truncation of system dynamicsConvenient tool for MIMO systemsU if l tf f ti ti i i tUniform platform for representing time-invariant systems, time-varying systems, linear systems as well as nonlinear systemsCan describe the dynamics in almost all systems (mechanical systems, electrical systems, biological systems, economic systems, social systems etc.)y , y , y )

Note: Transfer function representations are valid for l f li i i i (LTI)only for linear time invariant (LTI) systems


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State variable representation of a State variable representation of a systemsystem


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Generic State SpaceRepresentation


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Generic State SpaceRepresentation

Summary:


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Block diagram representation oflinear systems

tionstate equatut(t) : )(B)(Axx +=&

ationoutput eqututty : )(D)(Cx)( +=


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State Model for Linear systemState Model for Linear systemState Model for Linear systemState Model for Linear system

SISO systemu yInitial state : x(0)Input Output

State variables: )(b)(Axx tionstate equatut(t) +=&

)( : )()(cx)(

)()(

1112111 baaatxationoutput equtdutty

q( )

n⎥⎤

⎢⎡

⎥⎤

⎢⎡

⎥⎤

⎢⎡

+=

Λ

[ ]c;b;A;

)(

)(x 21

2

21

222212 ccc

b

b

aaa

aaa

tx

tx(t) n

nnnnn

n

n

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=Δ

ΛΜ

ΛΜΜΜ

ΛΜ


Mugdha Salvi, VCET

)()()( 21

txtybaaatx nnnnnn

=⎦⎣⎦⎣⎦⎣

State Model for Linear systemState Model for Linear system

: )(D)(Cx)(: )(B)(Axx

+=+=

ationoutput eqututtytionstate equatut(t)&

State Model for Linear systemState Model for Linear system

MIMO systemu1 - upInput

y1 - yqOutput

;A;)()(

x

)()()(

22221

11211

2

1

⎥⎥⎥⎤

⎢⎢⎢⎡

=⎥⎥⎥⎤

⎢⎢⎢⎡

=Δ

n

n

aaaaaa

txtx

(t)

p qy

ΛΛ

Initial state : x(0)Input Output

;;

)( 21⎥⎥

⎦⎢⎢

⎣⎥⎥

⎦⎢⎢

⎣ nnnnn aaatx

( )

ΛΜΜΜΜ

State variables

112111121111211 dddcccbbb ⎤⎡⎤⎡⎤⎡ ΛΛΛ

D;C;B 22221

11211

22221

11211

22221

11211

dddddd

cccccc

bbbbbb

p

p

n

n

p

p

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=

⎥⎥⎥⎥⎤

⎢⎢⎢⎢⎡

=ΜΜΜ

ΛΛ

ΜΜΜΛΛ

ΜΜΜΛΛ

)()(212121

txty

dddcccbbb qpqqqnqqnpnn

=⎥⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣⎥

⎥⎦⎢

⎢⎣ ΛΛΛ


Mugdha Salvi, VCET

Writing Differential Equations in First Companion Writing Differential Equations in First Companion FormForm

(Phase variable form/Controllable canonical form)(Phase variable form/Controllable canonical form)


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First Companion FormFirst Companion Form(Controllable Canonical Form)(Controllable Canonical Form)


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Example Example –– 11First Companion FormFirst Companion FormFirst Companion FormFirst Companion Form

(Controllable Canonical Form)(Controllable Canonical Form)


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Example Example –– 22(spring(spring--massmass--damper system)damper system)


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Example Example –– 33(R(R LL C circuit)C circuit)(R (R –– L L –– C circuit)C circuit)

First Companion Form (Controllable Canonical Form)First Companion Form (Controllable Canonical Form)


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Realization of First Companion Realization of First Companion Form (Controllable Canonical Form)Form (Controllable Canonical Form)

Consider only the following transfer function

Corresponding differential equation is:where,where,

Solving for highest derivative of z(t) we obtain


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Mugdha Salvi, VCET

First Companion FormFirst Companion Form(Controllable Canonical Form)(Controllable Canonical Form)


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Example Example –– 4 4 (Constant as numerator)(Constant as numerator)

By cross multiplication:

The corresponding differential equation is found by taking the inverse Laplace transform, assuming zero initial conditions:

Choose state variables as:


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ExampleExample –– 44Example Example 44

Vector – matrix form it can be written as:


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Example Example –– 55TT(T.F. with polynomial in numerator)(T.F. with polynomial in numerator)


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ExampleExample –– 55Example Example 55Introduce the effect of the block with the numerator., where b1 =1; b2= 7, and b3=2, states that

Taking Inverse Laplace Transform:

)()()( 1322

1 sXsssC βββ ++=

But

HenceHence,

123132231 27)()( xxxxxxtcty ++=++== βββ


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[ ]321 βββ



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Second Companion Form / Second Companion Form / Observable Canonical FormObservable Canonical Form


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Second Companion Form Second Companion Form (Observable Canonical Form)(Observable Canonical Form)


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Second Companion Form Second Companion Form (Observable Canonical Form)(Observable Canonical Form)

We observe that A, b and c matrices of one companion form corresponds to the transpose of the A, c and b matrices, respectively of the other.


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Diagonal Canonical FormDiagonal Canonical FormDiagonal Canonical FormDiagonal Canonical FormThe poles of the transfer function appear in the main diagonalThis form follows directly from the partial fraction expansion of the transfer functionThis form follows directly from the partial fraction expansion of the transfer function


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Case 1: Poles are Real and DistinctCase 1: Poles are Real and DistinctCase 1: Poles are Real and DistinctCase 1: Poles are Real and Distinct

The Л matrix is a diagonal matrix with the poles of G(s) as its diagonal l


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elements

Case 2: Real and Complex polesCase 2: Real and Complex polesCase 2: Real and Complex polesCase 2: Real and Complex poles


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Case 3: Real and Repeated polesCase 3: Real and Repeated polesCase 3: Real and Repeated polesCase 3: Real and Repeated poles

Partial fraction expansion is given by:

where


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Mugdha Salvi, VCET

Case 3: Real and Repeated polesCase 3: Real and Repeated polesCase 3: Real and Repeated polesCase 3: Real and Repeated poles


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Jordan Canonical FormJordan Canonical FormJordan Canonical FormJordan Canonical Form


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Where each of the submatrices Лi is in the Jordan form .The b and c matrices of the overall system are the concatenations of the bi and cimatrices respectively of each of the subsystems:


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The state variable model derived for the case of distinct poles, is a special case of Jordan canonical form where each Jordan block is of 1 x 1 dimension



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Example Example –– 66Second Companion Form (Observable Canonical Form)Second Companion Form (Observable Canonical Form)

Obtain state-space representations in the controllable canonical form, observable canonical form,and diagonal canonical formand diagonal canonical form.

controllable canonical form


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ExampleExample –– 66Example Example 66observable canonical form

diagonal canonical form


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Converting from State Space to aConverting from State Space to aTTTransfer FunctionTransfer Function

Given the state and output equations

take the Laplace transform assuming zero initial conditions:

Solving for X(s)or where I is the identity matrix.

Substituting X(s)into Y(s)yields

if U(s) = U(s) and Y(s) = Y(s) are scalars, we can find the transfer function,if U(s) U(s) and Y(s) Y(s) are scalars, we can find the transfer function,


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ExampleExample -- 77Example Example 77Given the system defined below, find the transfer function, T(s) / Y(s)=U(s), where U(s) is the input and Y(s) is the output.


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ExampleExample -- 77Example Example 77

Calculate (sI - A)-1:

Therefore, the transfer function is:


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EigenvaluesEigenvaluesEigenvaluesEigenvalues

The roots of the characteristic equation areThe roots of the characteristic equation areThe roots of the characteristic equation are The roots of the characteristic equation are referred to as eigenvalues of the matrix A.referred to as eigenvalues of the matrix A.

0)AI(0)AI(λwhich is called the characteristic equation of the which is called the characteristic equation of the

0)AI(0)AI( =−⇔=− siλ

system.system.


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Properties of eigenvaluesProperties of eigenvaluesProperties of eigenvaluesProperties of eigenvalues

1.1. If coefficients of A are all real, its eigenvalues either If coefficients of A are all real, its eigenvalues either , g, greal or in complex conjugate pairs.real or in complex conjugate pairs.

2.2. If If λλ11, , λλ2, ……, 2, ……, λλnn are the eigenvalues of are the eigenvalues of AA, then, then

i e the trace ofi e the trace of AA is sum of all the eigenvalues ofis sum of all the eigenvalues of AA

∑=

=n

ii

1tr(A) λ

i.e. the trace of i.e. the trace of AA is sum of all the eigenvalues of is sum of all the eigenvalues of AA..3.3. If If λλii, , ii = 1, 2, ……, = 1, 2, ……, nn is an eigenvalue of is an eigenvalue of AA, it is an , it is an

eigenvalue of eigenvalue of AA''..4.4. If A is nonsingular, with eigenvalues of If A is nonsingular, with eigenvalues of λλii, , ii = 1, 2, = 1, 2,

……, ……, nn , then 1/, then 1/ λλii, , ii = 1, 2, ……, = 1, 2, ……, nn , are the , are the eigenvalues of Aeigenvalues of A--11eigenvalues of Aeigenvalues of A ..


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EigenvectorsEigenvectorsEigenvectorsEigenvectors

Any nonzero vector xAny nonzero vector xii that satisfies the matrixthat satisfies the matrixAny nonzero vector xAny nonzero vector xii that satisfies the matrix that satisfies the matrix equationequation

0x)AI(xAx iii =−⇔= ii λλwhere where λλii, , ii = 1, 2, ……, = 1, 2, ……, nn , denotes the , denotes the iiththeigenvalue ofeigenvalue of AA, is called the eigenvector of, is called the eigenvector of AA

0)( iii ⇔ ii λλ

eigenvalue of eigenvalue of AA, is called the eigenvector of , is called the eigenvector of AAassociated with the eigenvalue associated with the eigenvalue λλii..IfIf AA has distinct real roots, thenhas distinct real roots, then AA can becan beIf If AA has distinct real roots, then has distinct real roots, then AA can be can be diagonalizeddiagonalized by the transformation matrix by the transformation matrix PPwhose basis vectors are the eigenvectors of whose basis vectors are the eigenvectors of AA..gg


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Generalized EigenvectorsGeneralized EigenvectorsGeneralized EigenvectorsGeneralized Eigenvectors

If A has multiple order eigenvalues and isIf A has multiple order eigenvalues and isIf A has multiple order eigenvalues and is If A has multiple order eigenvalues and is nonsymmetricnonsymmetric..Assume that A hasAssume that A has qq (<(<nn) distinct eigenvalues) distinct eigenvaluesAssume that A has Assume that A has qq (<(<nn) distinct eigenvalues, ) distinct eigenvalues, then then qq eigenvectors are given byeigenvectors are given by

ii 0 1i lthih0)AI( λλ

Among remaining highAmong remaining high--order eigenvalues, let order eigenvalues, let λλjj

qiiii 0,1,...,,eigenvalueth is where,0x)AI( i ==− λλ

jjbe of the be of the mmthth order (order (mm ≤ ≤ nn –– qq); corresponding ); corresponding generalized eigenvectors are given by:generalized eigenvectors are given by:


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Generalized EigenvectorsGeneralized EigenvectorsGeneralized EigenvectorsGeneralized Eigenvectors

1

xx)AI(

0x)AI( +

=

=− n-qj

λ

λ

23

12

xx)AI(

xx)AI(

++

++

−=−

−=−

n-qn-qj

n-qn-qj

λ

λ

xx)AI(λΜ

1xx)AI( −++ −=− mn-qmn-qjλ


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Example Example –– 88(Eigenvectors)(Eigenvectors)

Find the eigenvectors of the matrix

the eigenvalues are λ= -2, and -4


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ThThus,

∴Using the other eigenvalue, -4, we have

Thus, eigenvectors is


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DiagonalizationDiagonalization –– IIDiagonalizationDiagonalization IINote that if an n x n matrix A with distinct eigenvalues is given by:

the transformation x = Pz, where

λ1, λ2, ……, λn = n distinct eigenvalues of A

will transform P-1 AP into the diagonal matrix, or


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Example Example –– 99((DiagonalizationDiagonalization –– I)I)

Consider the following state-space representation of a system:

Eigen values are:

If we define a set of new state variables zl, z2, and z3 by the transformation


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Transformation to Jordon Canonical Transformation to Jordon Canonical FormForm

If the matrix A involves multiple eigenvalues, then diagonalization is impossible. But it is then transformed into the Jordan Canonical Form.then transformed into the Jordan Canonical Form.

has the eigenvalues λ1, λ1, λ3 then the transformation x = Sz, where

will yield


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Transformation of State variablesTransformation of State variablesTransformation of State variablesTransformation of State variables

The original system dynamics are given by,The original system dynamics are given by,g y y g yg y y g y

Th f i h d i d ib dTh f i h d i d ib d )()(cx)(

xx; )(b)(Axx 00

tdutty)(ttut(t)

+==+=Δ

&

Then on transformation the system dynamics are described Then on transformation the system dynamics are described as: as:

tttut(t) +=•

)(xP)(x; )(b)(xAx 01-

0

dd

tudtty +=

cPcbPbAPPA

where, )()(xc)(

1-1-

(Note: Both systems have identical output response to the (Note: Both systems have identical output response to the same input)same input)

dd ==== cP,cb,PbAP,PA

p )p )


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DiagonalizationDiagonalization –– IIIIDiagonalizationDiagonalization IIIIIf A has distinct real roots, then A can be diagonalized by the transformation matrix P whose basis vectors are the eigenvectors of Atransformation matrix P whose basis vectors are the eigenvectors of A.


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Example Example –– 1010((DiagonalizationDiagonalization –– II)II)

System is given by:


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Example Example –– 1010((DiagonalizationDiagonalization –– II)II)


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Solution of Homogeneous state Solution of Homogeneous state TTequation in Time domainequation in Time domain

We can write the solution of the homogeneousWe can write the solution of the homogeneousWe can write the solution of the homogeneous We can write the solution of the homogeneous state equationstate equationasas ororasas ororwhere where ΦΦ(t) is an (t) is an nn x x nn matrix and is the unique matrix and is the unique

l i fl i fsolution ofsolution ofAlso,Also,


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Solution of Homogeneous state Solution of Homogeneous state equation in Laplace domainequation in Laplace domain

We can write the solution of the homogeneousWe can write the solution of the homogeneousWe can write the solution of the homogeneous We can write the solution of the homogeneous state equationstate equationasas ororasas oror

The matrix exponential is computed as,The matrix exponential is computed as,


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Solution of Non Solution of Non –– Homogeneous Homogeneous state equation in Time domainstate equation in Time domain

d h hConsider the non – homogeneous state equation:

The solution in the time domain is given by:

Where by definition, and which is called the state-transition matrix


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transition matrix.

The The first term on the rightfirst term on the right--hand side of the equation is hand side of the equation is gg qqthe the response due response due to the initial state vector, x(0). to the initial state vector, x(0). Notice Notice also that it is the only term dependent also that it is the only term dependent on the on the initial state vector and not the input.initial state vector and not the input.initial state vector and not the input. initial state vector and not the input. We We call this part of the response the call this part of the response the zerozero--input input responseresponse, since it is the total response if the input is zero. , since it is the total response if the input is zero. ThTh dd ll dll d hh l i i ll i i l iiThe The second termsecond term, called , called the the convolution integralconvolution integral, is , is dependent only on the input, u, and the dependent only on the input, u, and the input matrixinput matrix, B, , B, not the initial state vector. not the initial state vector. We We call this part of the response the call this part of the response the zerozero--state state responseresponse, since it is the total response if the initial state , since it is the total response if the initial state vector is zero.vector is zero.


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Solution of Non Solution of Non –– Homogeneous Homogeneous state equation in Laplace domainstate equation in Laplace domainConsider the state and output equation:

Taking the Laplace transform of both sides of the state equation yields OR

Taking the Laplace transform of the output equation yields


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Taking the Laplace Inverse of the state equation yieldsg p q y

WhereWhere,


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Properties of State Transition Matrix Properties of State Transition Matrix TT(STM)(STM)

For the time-invariant system:For which

We have the following:


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if the matrix A is diagonal, then

h i l λ λ λ f h i A di ithe eigenvalues, λ1, λ2, ……, λn of the matrix A are distinct,


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If there is a multiplicity in the eigenvalues forIf there is a multiplicity in the eigenvalues, for example, if the eigenvalues of A are:λ1 λ1 λ1 λ4 λ5 λλ1, λ1, λ1, λ4, λ5, ……, λn

then Φ(t) will contain in addition to thethen Φ(t) will contain, in addition to the exponentials eλ1t, eλ4t, eλ5t, ……, eλnt , terms like teλ1t t2eλ1tte , t e .


Mugdha Salvi, VCET

CayleyCayley –– Hamilton TheoremHamilton TheoremCayleyCayley Hamilton TheoremHamilton TheoremThe Cayley-Hamilton theorem is very useful in proving theorems involving matrix equations or solving problems involving matrixinvolving matrix equations or solving problems involving matrixequations.

C id i A d i h i i iConsider an n x n matrix A and its characteristic equation:

The Cayley-Hamilton theorem states that the matrix A satisfies its own characteristic equation, or that


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Proof for the Proof for the CayleyCayley –– Hamilton Hamilton TTTheoremTheorem

To prove this theorem, note that adj(λ I - A) is a polynomial in A of degree n - 1. That is,

where B1 = I. Since

we obtain

From this equation, we see that A and Bi (i = 1,2,. . . , n) commute. Hence, the productof (λ I - A) and adj(λ I - A) becomes zero if either of these is zero. If A is substitutedfor λ in this last equation, then clearly λ I - A becomes zero. Hence, we obtain

Thi th C l H ilt


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This proves the Cayley-Hamilton theorem

Methods for computingMethods for computing eeAtAtMethods for computing Methods for computing eeMethod 1:If matrix A can be transformed into a diagonal form, then eAt can be given by:

where P is a diagonalizing matrix for A.

If matrix A can be transformed into a Jordan canonical form, then eAt can be given by


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Methods for computingMethods for computing eeAtAtMethods for computing Methods for computing ee

Method 2:


Mugdha Salvi, VCET

Methods for computingMethods for computing eeAtAtMethods for computing Methods for computing eeMethod 3:The third method is based on Sylvester's interpolation method. We shall first consider the case where the roots of the minimal polynomial φ ( λ ) of A are distinct..

Solve the determinant to find value of eAt about the last column


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Sol ing the determinant is similar to sol ing the eq ation beloSolving the determinant is similar to solving the equation below:

Where, αi can be found by solving the following equations simultaneously,


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the case where the roots of the minimal polynomial φ ( λ ) of A are multiple..Solve the determinant to find value of eAt about the last column


Mugdha Salvi, VCET

the case where the roots of the minimal polynomial φ ( λ ) of A are multiple..


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Example Example –– 1111(STM and x(t))(STM and x(t))

Consider the following matrix A:Compute eAt the 3 methods

Method 1:Method 1:

The eigenvalues of A are 0 and -2 (λ1, = 0, λ2 = -2). A necessary transformation matrix P may be obtained as


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ExampleExample –– 1111Example Example 1111Method 2:


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b f dSubstituting 0 for λ1, and -2 for λ2 in this equation, we obtain

Expanding the determinant, we obtain


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Since λ1, = 0 and λ2 = -2, the above two equations become1, 2 , q

Solving for α0(t) and α1 (t) gives


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Example Example –– 1212TT(STM)(STM)

For the state equation and initial state vector shown find the state-transition matrix and then solve for x(t) where u(t) is a unit stepmatrix and then solve for x(t), where u(t) is a unit step

and

To find STM which is given by,

Calculate (sI – A)-1( )


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taking the partial fractionsg p

taking the inverse Laplace transform of each term, we obtain


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Mugdha Salvi, VCET

state space analysis of control system.pdf

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