Metrology
Measurement Science and Technology
ETH Zurich, Switzerland
Institute of Machine Tools and Manufacturing
Contact the Author
Cooperation
© Copyright
d0000881; rev00
Module
Dynamic System in Steady State Condition – State Space
Karl H. Ruhm
Content
Introduction
1
Static Transfer Response Behaviour of the Dynamic System
2
Special Cases
1
1
2
Key Words
dynamic system, static transfer response behaviour, steady state condition, state space
Abstract
There is a special condition of stable dynamic systems, the steady state condition. The appropriate static
transfer response behaviour is a special case of the general dynamic behaviour and is derivable directly from
it. This is the topic of the following sections, treated in the state space domain.
Introduction
Besides the usual transient condition, where at least one quantity changes with time, stable dynamic systems may be in a steady state condition (equilibrium state, system at rest). This special condition is possible
after some time, when all input and output quantities are and remain constant. We call a relation between
input and output quantities for a system in a steady state condition
Static Transfer Response of the Dynamic System
Many processes are designed and analysed concerning this condition only. Often, the dynamic behaviour is
not considered. Whether such a restriction in a system description is allowed concerning a certain task, is
questionable and has to be investigated as the case arises.
Remark
This seemingly contradictory statement, using the terms static and dynamic at the same time, will indicate
that the system is always a dynamic system, even if it remains momentarily in a steady state condition. The
opposite of dynamic is not static but non-dynamic!
systems
B1221
may be
dynamic
static
1
non-dynamic
the transfer response behaviour is
transient
proportional
Static Transfer Response Behaviour of the Dynamic System
In order to get the steady state description of an asymptotic stable dynamic system, the temporal derivations
of all variables in the differential equations have to be zero. The set of differential equations degrades to a
set of algebraic equations. This means for the state space description (Supplement → Module "Basic Structures of the State Space Description"):
(t) = A x(t) + Bu(t)
x
N
=0
y(t) = C x(t) + Du(t)
Since the variables in the steady state condition do not depend on time t anymore, this has to be indicated
unconditionally by omitting the time arguments t in all equations (relation of the steady state values). It may
be reasonable to subsume all algebraic equations and to eliminate the steady state values of the state variables x by insertion into the output equation y. We mark the common static transfer response value matrix of
the dynamic system by G.
2
Definition: Set of Static Transfer Response Equations of the Dynamic System
x = − A −1 Bu
y = C x + Du
summarised:
y = Gu
with the transfer response value matrix
G = D − C A −1 B
In fact, we must not change the signal effect diagram and the appropriate simulation programs of a stable
dynamic system at all costs. Because, if we provide constant input quantities u, the output quantities y(t) will
be constant after some time anyway: y.
However, the signal effect diagram is considerably simpler and more clearly arranged in case of steady state
condition. The structure changes insofar as the integrators as dynamic sub-systems are inactive in a steady
state condition. We map the remaining algebraic set of equations in the signal effect diagram.
D
u
B
(N)
B0092
(M)
–A
–1
y
x
Σ
C
(N)
(P)
dynamic system in steady state condition
u
y
–1
D – CA B
(P)
(M)
u
G
(M)
y
(P)
From the transformation of the set of differential equations we recognise that matrix A, which was located
until now in the feedback path, now appears in form of its inverse A–1 in the feed-forward path. The invertibility of the quadratic (N)x(N)-matrix A is guarantied with stable systems.
2
Special Cases
a. monovariable system of Nth order (SISO)
b. monovariable system of first order (SISO)
y = g u [{y}]
with the transfer response value
y = g u [{y}]
with the transfer response value
b
g = d−c
[{y u−1}]
a
g = d − cA −1b
[{y u−1}]
Remark
In an experiment we are able to determine the steady state values of the state space quantities x from the
measured constant values of the input quantities u (first equation: x = A–1B u). In the contrary, it is not possible to draw conclusions from the measured values of the output quantities y on to the steady state values of
the state space quantities x normally, since the output matrix C is invertible (quadratic) not to often (second
equation: x = C–1(y – D u)) and since we need the constant values of the input vector u too.
(Supplement → Exercise "Dynamic System in the Steady State Condition")
Reference to this document
Please refer to this document by the following mode:
Ruhm, K. H.; Dynamic System in the Steady State Condition – State Space
Internet Portal "Measurement Science and Technology"; Document: www.mmm.ethz.ch/dok01/d0000881.pdf
Alternate Versions
There is a German version of this document: d0000321
Modifications
Rev. Date
Modification
00
First Edition
11.12.2008