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60 3 State Variable Feedback Control Theory

It can be seen that there is a large steady state error. To overcome this problem, the
roots must be moved away from the imaginary axis as far as possible or to use an
integrator. When an integrator is added to the transfer function, the order of charac-
teristic equation raises by one. This means that there will be four state variables and
the system becomes more complicated. The reader is encouraged to study the above
system with an integrator in the transfer function.

3.5 Dynamic Observer

In state variables control theory, it is assumed that all state variables are measur-
able for direct feedback. If all state variables are not measurable or impractical,
then state observer must be used. A dynamic observer is a computer program that
by knowing the mathematical model of the system and input variables, calculates
the estimate of the state variables. In principle, if the mathematical model and input
variable is known, then it should be possible to solve the state equations to obtain
the state variables. Therefore, for a state equation with system matrix A and input
matrix B and input variable u, the observer equation becomes

d X := AX +
o
o
dt Bu
In the above equation, the vector X represents the estimate of state variables. Other
0
parameters have the usual meanings. The above equations can be solved numeri-
cally by the method explained in the previous section assuming that the initial val-
ues of X are known. In practice, the initial values may not be known and, therefore,
a forcing function must be added to ensure that the estimates converge to the state
variables of the system. First, it is useful to study the convergence of the error. The
error is defined as the difference between the state variables and the observed state
variables,

X := X X− o
e
where X are the errors.
e
Subtracting the observer state equation from the state equations and with some
algebraic manipulation gives
d X := AX (3.24)
dt e e
Equation (3.24) shows that for stable A, the error becomes zero when the initial
conditions are

X () :0 = X( )0 − X ( )0
o
e
The only problem with the above equation is that the speed of response of the sys-
tem matrix A is the same for the system and observer. Therefore, the observed vari-