Page 731 - Mechanical Engineers' Handbook (Volume 2)

722 State-Space Methods for Dynamic Systems Analysis

y(t) e (t) x (t) (25)
out
2
The system and coupling matrices for the electrical circuit are
R 1 0
1
L
A 1 L B L (26)
0
C 1
C [0 1] D 0
For sampled-data control applications involving digital control of continuous-time sys-
tems, state-variable selection is often based on continuous-time system equations and needs
to be retained in the discrete-time formulation. Let the time-invariant, continuous-time system
equations be of the form given by Eqs. (6) and (7). The solution for the vector equation of
state x(t) is given by Eqs. (32) and (33) later in this chapter. Applying this solution form
over the time interval from kT to (k 1)T, we get
x[(k 1)T] e x(kT) (k 1)T A[(k 1)T t]
AT
kT e Bu(t) dt (27)
In sampled-data control applications, the input vector u(t) is the control input computed by
the digital computer and applied to the continuous-time system by a digital–analog converter.
Usually, the digital–analog converter has a latch that maintains the output constant between
the time instants kT and (k 1)T:
u(t) u(kT) kT t (k 1)T (28)
Equation (27) then simplifies to
x(k 1) (e )x(k) e T A(T t) B dt u(k)
AT
0
Fx(k) Gu(k) (29)
Equation (29) establishes the relationship between the system matrices in the discrete-time
and continuous-time formulations of the system equations.

3.1 Canonical Forms for Continuous-Time Systems

For high-order, single-input–single-output (SISO) systems or multiple-input–multiple-output
(MIMO) systems, the number of elements in the system matrices is large. Selection of state
variables that simplify the state-space representation is thus desirable. Such representations
of the state-space equations also exhibit significant properties of the system more clearly
and are referred to as canonical forms. However, the names and corresponding structures of
the canonical forms are not completely standardized.
The controllable canonical form is useful in control system design applications. The
controllable canonical form for an nth-order, SISO, LTI, continuous-time system is described
in Table 1. The selection of state variables x , x , . .., x that results in the controllable
1
n
2
canonical form of the state equations is indicated on the simulation diagram in the table.
For the case where all the , i 1, n, are zero and 0, the transfer function Y(s)/U(s)
0
i
has no numerator dynamics. The n state variables are then simply the output and n 1
successive derivatives of the output. These are referred to as the phase variables. The phase-
variable canonical form is thus a special case of the controllable canonical form. The system