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and the output
x(t) = Q(t)x(0) + [,' <P(t - r)Bu(r)dr
(5.189)
y(t) = C [q>(,)x(0) + [r q>(t _ r)B.(,) d,] +Du (5.190)
The integral in Eq. (5.189) is the convolution integral, which was introduced
earlier in Eq. (5.39). The first term on the right-hand side of Eq. (5.189) represents
the solution to the homogeneous part of the state equation and gives the free (tran-
sien0 response. The second term represents Lhe forced response and is independent
of the initial conditions x(0).
Properties of state transition matrix. The state transition matrix cP(t) has
the following properties. The proof of these identities is left as an exercise to the
reader.
q>(0) -- 1
<p-l(t) = q>(-t)
<P(t2 - tI)Q(tl - t0) : <p(t2 - t0)
[<P(t)lk = Q(kt)
(5.191)
(5.192)
(5.193)
(5.194)
Characteristic equation. Given a square matrix A, the equation
A(1) = lAl - Al - 0 (5.195)
is called the characteristic equation of matrix A. Here, I-I denotes the deternunant
of the argument (square) matrix.
Eigenva/ues and eigenvectors. The roots ofcharacteristic Eq. (5.195) are
called the eigenvalues of the matrix A and are usually denoted by Ar, / : 1, . . . , n,
where n is the number of rows or columns of the matrix A. As an example,let
so that
A=[2 ;]
AI-A = [A_41 A,-ll]
(5.196)
(5.197)
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 511
A(A) : IAI - Al = A2 - 2A - 3 = 0
(5.198)
Solving, we get Ai - 3 and A2 - -1.
An important property of the eigenvalues is that they remain invariant under any
linear tra~sformation. A direct consequence of this property is that the closed-loop
characteristic equation also remains invariant under any linear transformation.
To make this p9int clear, suppose we are given a linear, time-invariant system
x - Ax + Bu and we transform this system using a linear transformation x - Pz
so that the transformed system is z = P-lAPz + P-IBu. Then, the eigenvalues
of matrix A and those of P-lAP are identical. Interested readers may verify this
statement by working out the details.
If A, is an eigenvalue of the square matrix A, then any vector x that satisfies the
equation
Ax - A/x
(5.199)
is called the eigenvector corresponding to the eigenvalue A.,, In other words, every
eigenvalue will have an associated eigenvector. Returning to the above example,
[A 1] [f:] = c-i) [f:]
or
2xi +X2 = O
4xi +2x2 -. O
(5.200)
(5.201)
(5.202)
Note that the two equations are identical, stating that the eigenvector is not unique
and depends on our choice ofone ofthe two variables. Let X2 -. 1 SO that xi = -1l2.
The eigenvector corresponding to A = -1 is [-1l2 1]T. Here, the superscript "T"
denotes the matrix transpose. It can be shown that the eigenvector obtained by
choosing any other value for X2 would be a scalar multiple of this eigenvector.
Similarly, the eigenvector corresponding to A2 = 3 is [1 2) r .
To understand the physical meaning of eigenvalues and eigenvectors, consider
a system with two first-order, coupled linear differential equations
x] - XI +x2
X2 -. 4xi +X2
(5.203)
(5.204)
Assume that the solution is given by xi(t) = uieAt and X2(t) = u2eA: Substitut.ing,
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