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system. An effort of abstraction is therefore required, which leads to postulating a new quantity, called the state, which
summarizes information about the past and the present of the system.t) taken by the state at
Specifically, the value x(
timetmust be sufficient to determine the output at the same point in time.t1 ) and u[t t2 ),
Also, knowledge of both x(
1
thatt1t1t
is, of the state at time and the input over the interval
thet2. For the mass attached to a spring, for instance, the state could be the position and velocity of the
output) at time
mass. In fact, the laws of classical mechanics allow computing the new position andt2
velocity of the mass at time
givent1 and thet1 t2). Furthermore, in this example,
its position and velocity at time forces applied over the interval [
the output y of the system happens to coincide with one of the two state variables, and is therefore always deducible
from the latter.
Thus, in a dynamic system the input affects the state, and the output is a function of the state. For a discrete
system, the way that the input changes thekintok+1 can
state at time instant number the new state at time instant
be represented by a simple equation:
xk+1f(xk uk k)
=
wherefis some function that represents the change,k. Similarly, the relation between state
and uk is the input at time
and output can be expressed by another function:
ykh(xk k):
=
A discrete dynamic system is completely described by these two equations and an initial state x0. In general, all
quantities are vectors.
For continuous systems, time does not come in quanta, so one cannot compute xk+1 as a function of xk, uk, and
k,t2) oft1 ) and thet1 t2):
but rather compute x( as a functional x( entire input u over the interval [
x(t2) (x(t1 u( ) t1 t2)
= )
where u( ) represents the entire function u, not just one of its values. A description of the system in terms of
functions, rather than functionals, can be given in the case of a regular system, is continuous,
for which the functional
differentiable, and with continuous first derivative. In that case, one can showfsuch that
that there exists a function
thet) of the system satisfies the differential equation
state x(
_ =
x(t)f(x(t) u(t) t)
where the dot denotes differentiation with respect to time. The relation from state to output, on the other hand, is
essentially the same as for the discrete case:
y(t)h(x(t) t):
=
Specifying the initial state x0 completes the definition of a continuous dynamic system.
7.1.2 Uncertainty
The systems defined in the previous section are called deterministic, since the evolution is exactly determined once
the initial state x at time 0 is known. Determinism implies thatfand the output function
both the evolution function
hare known exactly. This is, however, an unrealistic state of affairs. In practice, the laws that govern a given physical
7.2. AN EXAMPLE: THE MORTAR SHELL 85
system are known up to some uncertainty. In fact, the equations themselves are simple abstractions of a complex
reality. The coefficients that appear in the equations are known only approximately, and can change over time as a
result of temperature changes, component wear, and so forth. A more realistic model then allows for some inherent,
unresolvablefandh. This uncertainty can be represented as noise that perturbs the equations we
uncertainty in both
have presented so far. A discrete system then takes on the following form:
xk+1f(xk uk k) k
= +
ykh(xk k) k
= +
and for a continuous system
_ = +
x(t)f(x(t) u(t) t) (t)
y(t)h(x(t) t) (t):
= +
Without loss of generality, the noise distributions can be assumed to have zero mean, for otherwise the mean can be
incorporated into the deterministicforh. The mean may not be known, but this is a different
part, that is, in either
story: in general the parameters thatfandhmust be estimated by some method, and the
enter into the definitions of
mean perturbations are no different.
A common assumption, which is sometimes valid and always simplifies and are
the mathematics, is that
zero-mean Gaussian random variables withQandR, respectively.
known covariance matrices
7.1.3 Linearity
The mathematics becomes particularly simple whenfandhare linear.
both the evolution function the output function
Then, the system equations become
xk+1FkxkGkuk k
= + +
= +
ykHkxk k
for the discrete case, and
_ = + +
x(t)F(t)x(t)G(t)u(t) (t)
y(t)H(t)x(t) (t)
= +
for the continuous one. It is useful to specify the sizes of the matrices involved. We assume that the input u is a vector
in Rp, the state x is in Rn, and the output y is in Rm. Then,Fisn n, the input matrix
the state propagation matrix
Gisn p,Hism n.Qof isn n, and the
and the output matrix The covariance matrix the system noise
covariance ism m.
matrix of the output noise
7.2 An Example: the Mortar Shell
In this section, the example of the mortar shell will be discussed in order to see some of the technical issues involved [ Pobierz całość w formacie PDF ]