3.3 State Equations
113
Starting from this observation, the next section focuses on the description of
dynamic nonlinear networks via the state equations perspective.
The tableau equations (3.30), (3.31), and (3.32) are frequently referred to as
sparse tableau equations because there always exist a large number of zeros in the
matrices involved in these equations. For the class of linear dynamic networks, a
conventional method for solving linear integro-differential equations (3.30), (3.31),
and (3.32) exploits the Laplace transform and efficient algorithms for sparse
systems.
The next sections discuss the state equation approach for RLC networks. Readers
interested in a deep understanding of circuit analysis methodologies are also invited
to refer to the specialized books [1] and [2].
3.3 State Equations
The state equations (SEs) of any dynamic lumped network are written in normal
form
˙
x = f(x),
t ≥ t 0
(3.43)
where x = (x 1 , x 2 , . . . , x n ) T ∈ R n is the vector of state variables, t 0 is a given finite
initial instant,
x(t 0 ) ∈ R
n
(3.44)
is the initial state (initial condition), and f : R n → R n is the vector field defining the
SEs. The order of the SEs (3.43) is the dimension n of the vector x of state variables.
Equations (3.43), together with the initial condition (3.44), define an initial value
problem (IVP) (or Cauchy problem) associated with the SEs. If x(t), t ∈ I , where
I ⊆ R is an interval such that t 0 ∈ I , is a solution of the IVP, then the locus
{x(t) ∈ R n , t ∈ I } is a curve in R n passing through x 0 that is named trajectory or
orbit through x 0 of (3.43) and R n is named state space or phase space.
An equilibrium point (EP) is a constant (i.e., a stationary) solution of (3.43). Note
that ¯
x is an EP if and only if we have f(¯ x) = 0. A (nontrivial) periodic solution is a
nonequilibrium solution of (3.43) such that x(t +T ) = x(t) for some T > 0 and any
t. While the image in the state space of an EP is a singleton, the image of a periodic
solution is a closed trajectory usually called a periodic orbit or a closed orbit. An
isolated periodic orbit is called a limit cycle. The reader is referred to [3] for the
concept of local stability, asymptotic stability, and instability of an EP or a periodic
solution.
Equation (3.43) is usually called an autonomous SE in the literature and includes
dynamic circuits containing only linear time-invariant elements and dc sources. In
order to include non-autonomous circuits with time-varying elements or sources
113
Starting from this observation, the next section focuses on the description of
dynamic nonlinear networks via the state equations perspective.
The tableau equations (3.30), (3.31), and (3.32) are frequently referred to as
sparse tableau equations because there always exist a large number of zeros in the
matrices involved in these equations. For the class of linear dynamic networks, a
conventional method for solving linear integro-differential equations (3.30), (3.31),
and (3.32) exploits the Laplace transform and efficient algorithms for sparse
systems.
The next sections discuss the state equation approach for RLC networks. Readers
interested in a deep understanding of circuit analysis methodologies are also invited
to refer to the specialized books [1] and [2].
3.3 State Equations
The state equations (SEs) of any dynamic lumped network are written in normal
form
˙
x = f(x),
t ≥ t 0
(3.43)
where x = (x 1 , x 2 , . . . , x n ) T ∈ R n is the vector of state variables, t 0 is a given finite
initial instant,
x(t 0 ) ∈ R
n
(3.44)
is the initial state (initial condition), and f : R n → R n is the vector field defining the
SEs. The order of the SEs (3.43) is the dimension n of the vector x of state variables.
Equations (3.43), together with the initial condition (3.44), define an initial value
problem (IVP) (or Cauchy problem) associated with the SEs. If x(t), t ∈ I , where
I ⊆ R is an interval such that t 0 ∈ I , is a solution of the IVP, then the locus
{x(t) ∈ R n , t ∈ I } is a curve in R n passing through x 0 that is named trajectory or
orbit through x 0 of (3.43) and R n is named state space or phase space.
An equilibrium point (EP) is a constant (i.e., a stationary) solution of (3.43). Note
that ¯
x is an EP if and only if we have f(¯ x) = 0. A (nontrivial) periodic solution is a
nonequilibrium solution of (3.43) such that x(t +T ) = x(t) for some T > 0 and any
t. While the image in the state space of an EP is a singleton, the image of a periodic
solution is a closed trajectory usually called a periodic orbit or a closed orbit. An
isolated periodic orbit is called a limit cycle. The reader is referred to [3] for the
concept of local stability, asymptotic stability, and instability of an EP or a periodic
solution.
Equation (3.43) is usually called an autonomous SE in the literature and includes
dynamic circuits containing only linear time-invariant elements and dc sources. In
order to include non-autonomous circuits with time-varying elements or sources