STATE SPACE ANALYSIS in .NET

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STATE SPACE ANALYSIS
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[CHAP. 7
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ments, the outputs of these memory elements can be chosen to be the state variables of the system (Probs. 7.4 and 7.5). If the system is described by the difference or differential equation, the state variables can be chosen as shown in the following sections. Note that the choice of state variables of a system is not unique. There are infinitely many choices for any given system.
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7 3 STATE SPACE REPRESENTATION OF DISCRETE-TIME LTI SYSTEMS .
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A. Systems Described by Difference Equations:
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Suppose that a single-input single-output discrete-time LTI system is described by an Nth-order difference equation
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y [ n ] + a , y [ n - 11
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+ a N y [ n- N ] = x [ n ]
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(7.1)
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We know from previous discussion that if x [ n ] is given for n 2 0, Eq. (7.1) requires N initial conditions y [ - I ] , y [ - 2 1 , . . ., y [ - N ] to uniquely determine the complete solution for n > 0. That is, N values are required to specify the state of the system at any time. Let us define N state variables q , [ n ] ,q 2 [ n ] ,. . . ,q N [ n ]as
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y [ n ] = - a N q , [ n ] - a N - , q 2 [ n ]-
- .. -
a , q ~ [ + x l[ n l ~
In matrix form Eqs. ( 7 . 3 ~ and (7.36) can be expressed as )
Now we define an N x 1 matrix (or N-dimensional vector) q [ n ] which we call the state
CHAP. 71
STATE SPACE ANALYSIS
vector :
Then Eqs. ( 7 . 4 ~and (7.46) can be rewritten compactly as )
where
Equations ( 7 . 6 ~and (7.66) are called an N-dimensional state space representation (or ) state equations) of the system, and the N x N matrix A is termed the system matriu. The solution of Eqs. ( 7 . 6 ~and (7.66) for a given initial state is discussed in Sec. 7.5. )
Similarity Transformation: As mentioned before, the choice of state variables is not unique and there are infinitely many choices of the state variables for any given system. Let T be any N X N nonsingular matrix (App. A) and define a new state vector v [ n ]= m [ n ] (7.7) where q [ n ] is the old state vector which satisfies Eqs. ( 7 . 6 ~ ) and (7.66). Since T is nonsingular, that is, T-I exists, and we have q [ n ]= T - ' v [ n ] Now v [ n + 1 = T q [ n + 11 1
= TAq[n] =T
(7.8)
( ~ q [ n ]b x [ n ] ) + (7.9~)
+ T b x [ n ]= T A T - ' v [ n ]+ T b x [ n ]
Thus, if we let
b=Tb
;=(q-'
then Eqs. ( 7 . 9 ~and (7.9b) become ) v [ n + 11 = R [ n ] +b x ~ n l
STATE SPACE ANALYSIS
[CHAP. 7
Equations (7.11a) and (7.11b) yield the same output y[n] for a given input x[n] with different state equations. In matrix algebra Eq. (7.10a) is known as the similarity transformation and matrices A and are called similar matrices (App. A).
C. Multiple-Input Multiple-Output Systems:
If a discrete-time LTI system has m inputs and p outputs and N state variables, then a state space representation of the system can be expressed as
where
+ 11 = Aq[n] + Bx[n]
y[nI
= Cq[n]
+ Dx[n]
-NXrn
STATE SPACE REPRESENTATION OF CONTINUOUS-TIME LTI SYSTEMS Systems Described by Differential Equations:
Suppose that a single-input single-output continuous-time LTI system is described by an Nth-order differential equation
where y(k)(O= One possible set of initial conditions is y(O), y(l)(O),. . . , Y(~-')(O), dky(t)/dt '. Thus, let us define N state variables ql(0,q,( 0,.. . ,qN(O as
q1W = y ( t ) qz(1) = Y ( ' ) ( ~ )
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