Definition and Properties - Advanced Structural Dynamics and Active Control of Structures

Structures

4.1 Definition and Properties

4 Controllability and Observability

ª how to excite and monitor a structure

It is the theory which decides what we can observe

—Albert Einstein

Controllability and observability are structural properties that carry information useful for structural testing and control, yet they are not fully utilized by structural engineers. The usefulness can be found by reviewing the definitions of the controllability and observability of a structure. A structure is controllable if the installed actuators excite all its structural modes. It is observable if the installed sensors detect the motions of all the modes. This information, although essential in many applications (e.g., in the placement of sensors and actuators), is too limited. It answers the question of mode excitation or detection in terms of yes or no. The more quantitative answer is supplied by the controllability and observability grammians, which represent a degree of controllability and observability of each mode.

In this chapter we discuss the controllability and observability properties of flexible structures. The fundamental property of a flexible structure in modal coordinates consists of a set of uncoupled modes, as shown in Property 2.1. It allows us to treat the properties of each individual mode separately and to combine them into a property of the entire structure. This also refers to the controllability and observability properties of the whole system, which are combined out of the properties of individual modes. These controllability and observability properties are used later in this book in the evaluation of structural testing and in control analysis and design.

variable (x) is excited by the input (u) and measured by the output (y). However, the input may not be able to excite all states (or, equivalently, to move them in an arbitrary direction). In this case we cannot fully control the system. Also, not all states may be represented at the output (or, equivalently, the system states cannot be recovered from a record of the output measurements). In this case we cannot fully observe the system. However, if the input excites all states, the system is controllable, and if all the states are represented in the output, the system is observable. More precise definitions follow.

4.1.1 Continuous-Time Systems

Controllability, as a measure of interaction between the input and the states, involves the system matrixA and the input matrixB. A linear system, or the pair (A,B), is controllable at if it is possible to find a piecewise continuous input u(t),

, that will transfer the system from the initial state, to

[ , ]_o 1

t t t x t( )_o , to the origin

at finite time If this is true for all initial moments and all initial states

( ) 0,1

x t t₁!t_o. t_o

( )_o

x t the system is completely controllable. Otherwise, the system, or the pair (A,B), is uncontrollable.

Observability, as a measure of interaction between the states and the output, involves the system matrixA and the output matrix C. A linear system, or the pair (A,C), is observable at if the state t_o x t( )_o can be determined from the outputy(t),

, where is some finite time. If this is true for all initial moments and all initial states

[ , ]_o 1

t t t t₁!t_o t_o

( )_o

x t the system is completely observable. Otherwise, the system, or the pair (A,C), is unobservable. Note that neither the controllability nor observability definition involves the feed-through term D.

There are many criteria to determine system controllability and observability; see [88], [143]. We consider two of them. First, a linear time-invariant system (A,B,C), withs inputs is completely controllable if and only if theNu sNmatrix

2 N 1

B AB A B A B

ª º

¬ ! ¼

(4.1)

has rank N. A linear time-invariant system (A, B,C) with r outputs is completely observable if and only if therNu N matrix of

N 1

C CA CA CA

ª º

« »

¬ ¼

' (4.2)

has rank N.

The above criteria, although simple, have two serious drawbacks. First, they answer the controllability and observability question in yes and no terms. Second, they are useful only for a system of small dimensions. The latter can be visible if we assume, for example, that the system is of dimensionN = 100. In order to answer the controllability and observability question we have to find powers of A up to 99.

Finding for a 100u100 matrix is a numerical task that easily results in numerical overflow.

A99

The alternative approach uses grammians to determine the system properties.

Grammians are nonnegative matrices that express the controllability and observability properties qualitatively, and are free of the numerical difficulties mentioned above. The controllability and observability grammians are defined as follows, see, for example, [88]:

( ) exp( ) exp( ) ,

( ) exp( ) exp( ) .

t T T

W t = A BB A d

W t = A C C A d

W W W

³

(4.3)

We can determine them alternatively and more conveniently from the following differential equations:

(4.4)

T T

c c c

T T

o o o

W = AW +W A + BB W = A W +W A+ C C

The solutions and are time-varying matrices. At the moment we are interested in the stationary, or time-invariant, solutions (the time-varying case is discussed later). For a stable system, we obtain the stationary solutions of the above equations by assuming In this case, the differential equations (4.4) are replaced with the algebraic equations, called Lyapunov equations,

c( )

W t W t_o( )

c o 0.

W W

(4.5) 0,

T T

c c

T T

o o

AW +W A + BB = A W +W A+ C C =

For stable A, the obtained grammiansW_c andW_o are positive definite.

The grammians depend on the system coordinates, and for a linear transformation of a state x into a new state x_n, such that x_n Rx, they are transformed to new grammiansW_cn and W_on as follows:

1 , .

cn c T

on T o

W = R W R W = R W R

(4.6)

The eigenvalues of the grammians change during the coordinate transformation.

However, the eigenvalues of the grammian product are invariant. It can be shown as follows:

1 1

( ) ( ^{T T} ) ( ) (

i W Wcn on = i R W R R W R =c o i R W W R =c o i W Wc o

O O O O ). (4.7)

These invariants are denoted ,J_i

( ), 1,...,

i i W Wc o i N

J O , (4.8)

and are called the Hankel singular values of the system.

4.1.2 Discrete-Time Systems

Consider now a discrete-time system as given by (3.46). For the sampling time the controllability matrix is defined similarly to the continuous-time systems, as follows:

t '

1 .

k ªB AB AkBº

¬ " ¼

(4.9)

The controllability grammianW k_c( ) over the time interval [0,k t'] is defined as

(4.10)

( ) ( ) .

k i T i T

c i

W k

¦

A BB A

Unlike the continuous-time systems we can use the controllability matrix of the discrete-time system to obtain the discrete-time controllability grammian . Namely,

c( ) W k

(4.11) ( ) ^T.

c k

W k _k

The stationary grammian (for ko f) satisfies the discrete-time Lyapunov equation

, W_cAW A_c ^T BB (4.12)

but can still be obtained from (4.11) using large enough k, since for Similarly the observability matrix is defined as follows:

k 0 A o

ko f. '_k

1 k ,

C CA CA

ª º

« »

¬ ¼

' # (4.13)

and the discrete-time observability grammian for the time interval [ is defined as

o( )

W k 0,k t']

(4.14)

( ) ( ) ,

k i T T i

o i

W k

¦

A C CA

which is obtained from the observability matrix

(4.15)

( ) ^T .

o k

W k ' '_k

For (stationary solution) the observability grammian satisfies the following Lyapunov equation:

ko f

. W_oA W A C C^T _o (4.16)

Similarly to the continuous-time grammians, the eigenvalues of the discrete-time grammian product are invariant under linear transformation. These invariants are denotedJ_i,

( ), 1,..., ,

i i W Wc o i N

J O (4.17)

and are called the Hankel singular values of the discrete-time system.

4.1.3 Relationship Between Continuous- and Discrete-Time Grammians

Let be the state-space representation of a discrete-time system. From the definitions (4.10) and (4.14) of the discrete-time controllability and observability grammians we obtain

( , , )A B C

(4.18)

2 2

( ) ,

( ) .

T T T T T

T T T T T

W BB ABB A A BB A

W C C A C CA A C CA

We show that the discrete-time controllability and observability grammians do not converge to the continuous-time grammians when the sampling time approaches zero, see [109]. Indeed, consider the continuous-time observability grammian

T .

At T A t ocont

W e BB e

³

^dt

This can be approximated in discrete time, at time moments t 0,'t, 2 , ...,'t as

0 0

( )

iA t TT iA t i T T i

ocont d d

i i

W e C Ce t A C CA t

f f

' '' '

¦ ¦

Introducing the second equation of (4.18) one obtains

(4.19)

lim0 .

ocont odiscr

W t t W

' o '

Obtaining the controllability grammians is similar. First note that for a small sampling time one has

d .

B # 't B (4.20)

Indeed, from the definition of B_d, one obtains

1 2 2 2

0 0

2 2 3

1 1

2 2

( )

t t

d A c c c

c c c c c

B e B d I A A B d

B t A B t A B t t B

W W W W

' '

' ' ' # '

³ ³

t '

Now, from the definition of the continuous-time controllability grammian, the following holds:

0 0 0

lim .

T T

c c c c

A T A iA t T iA t

ccont c c c c

t i

W e ^WB B e ^WdW e B B e

f f

' '

' o

¦

³

Using (4.20) and A_d e^{A t}^c^' we obtain

0 0

1 1

lim ⁱ ^T( ^{i T}) lim ;

ccont d d d d cdiscr

t i t

W A B B A

t t

' o '

¦

' o ' ^W hence,

lim 1 .

ccont cdiscr

W t W

' o ' (4.21)

Note, however, from (4.19) and (4.21), that the product of the discrete-time controllability and observability grammians converge to the continuous-time grammians,

(4.22)

lim (0 );

ccont ocont cdiscr odiscr

W W t W W

' o

therefore, the discrete-time Hankel singular values converge to the continuous-time values, as the sampling time approaches zero:

(4.23) lim0 .

icont idiscr

J t J

' o

Dalam dokumen Advanced Structural Dynamics and Active Control of Structures (Halaman 88-94)