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State Transition Matrix Properties
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The homogeneous part of eqn (343) is x(t) = F(t)x(t) (353)
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De nition 32 A continuous and di erentiable matrix function (t) : R1 Rn n is the fundamental solution of eqn (353) on t [0, T ] if and only if (0) = I and (t) = F(t) (t) for all t [0, T ] The fundamental solution is important since it will serve as the basis for nding the solution to both eqns (343) and (353) If x(t) = (t)x(0) then x(t) satis es the initial value problem corresponding to eqn (353) with initial condition x(0): x(t) d (t)x(0) dt = (t)x(0) = F(t) (t)x(0) = F(t)x(t) =
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If (t) is nonsingular, then 1 (t)x(t) = x(0) and x( ) = ( )x(0) = ( ) 1 (t)x(t) = ( , t)x(t)
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(354)
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CHAPTER 3 DETERMINISTIC SYSTEMS
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where ( , t) = ( ) 1 (t) is called the state transition matrix from time t to time The state transition matrix transforms the solution of the initial value problem corresponding to eqn (353) at time t to the solution at time The state transition matrix has the following properties: (t, t) = I (t, ) ( , t) F( ) ( , t) ( , t)F(t)
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( , t) = ( , ) ( , t) = d ( , t) = d d ( , t) = dt
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(355) (356) (357) (358) (359)
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The general solution to eqn (343) is
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x(t)
(t, t0 )x(t0 ) +
(t, )G( )u( )d
(360)
which can be veri ed as follows: d d d x(t) = (t, t0 )x(t0 ) + dt dt dt x(t) = F(t) (t, t0 )x(t0 )
(t, )G( )u( )d
F(t) (t, )G( )u( )d + (t, t)G(t)u(t)
x(t) x(t)
= F(t) (t, t0 )x(t0 ) +
(t, )G( )u( )d + G(t)u(t)
= F(t)x(t) + G(t)u(t),
where Leibnitz rule d dt
b(t) b(t)
f (t, )d =
a(t) a(t)
db da f (t, )d + f (b(t), ) f (a(t), ) t dt dt
has been used to move derive the second equation from the rst
Linear Time-Invariant Systems
In the case that F is a constant matrix, it can be shown by direct di erentiation that (t) = ( , t) = eFt , and eF( t) (361)
35 STATE SPACE ANALYSIS
The matrix exponential, its properties, and its computation are discussed in Section B12 Given this special case of ( , t) we have that eqn (360) becomes
x(t) = eF(t t0 ) x(t0 ) +
eF(t ) G( )u( )d
(362)
Assuming that G is time-invariant, the output is determined by multiplying eqn (362) on the left by the measurement matrix H
y(t) = HeF(t t0 ) x(t0 ) +
HeF(t ) Gu( )d
(363)
De ning m(t) = HeFt G, which is the impulse response of the linear system, we see that eqn (363) is the linear combination of the response due to initial conditions and the response due to the input The response to the input u(t) is determined as the convolution of the impulse response m(t) with u(t) The Laplace transform of the impulse response is the transfer 1 function M(s) = H (sI F) G derived in eqn (351) An approximate method to compute (t + , t) for small when F(t) is slowly time-varying is discussed in Section 7252
Discrete-Time Equivalent Models
It is often the case that a system of interest is naturally described by continuous-time di erential equations, but that the system implementation is more convenient in discrete-time In these circumstances, it is of interest to determine a discrete-time model that is equivalent to the continuous-time model at the discrete-time instants tk = kT for some xed value T > 0 and k = 0, 1, 2, Equivalence meaning that the discrete and continuous-time models predict the same system state at the speci ed discrete-time instants If F is a constant matrix, then from eqn (362) xk+1 xk+1 = = eF
(k+1)T kT (k+1)T
xk +
(k+1)T
(k+1)T
G( )u( )d (364)
(k+1)T
xk +
G( )u( )d
where xk = x(kT ) and = eFT Simpli cation of the second term on the right hand side is possible under various assumptions The most common assumption is that G(t) is a constant vector and that u(t) has the constant value uk for t (kT, (k + 1)T ] With this assumption, eqn (364) reduces to (365) xk+1 = xk + uk where =
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