STATE-SPACE REPRESENTATION OF SINGLE-INPUT-SINGLEOUTPUT SYSTEMS
In the following a short introduction into the representation of systems using statevariable techniques is given. For this purpose, the example of the
network from
Figure 4.16 is used. The dynamical behavior of this network is completely defined for
, if the

initial conditions
,
and the

input variable
for
are known. For these specifications the variables
. The variables
and
and
can be determined for all
characterise the 'state' of the network and are
therefore called state variables of the network.
One can also use the two original differential equations and can write them in vector
notation so that the 1st-order vector differential equation
(12.1)
with the initial condition
is obtained. This linear 1st-order vector differential equation describes the connection
between the input variable and the state variables. To complete a state-space system, one
needs an additional equation that describes the dependence of the output variable on the
state variables. In this example, it is the direct relationship
Introducing the state vector
into Eq. (12.1), with the vectors
and
with the matrix
and with the scalar variables
and
one obtains the general state-space representation of a linear time-invariant single-inputsingle-output system:
(12.2)
initial condition
(12.3)
The Eq. (12.2) is the state equation, and in the general case it is a linear system of 1storder differential equations of
state variables
the state vector
, which are combined in
. Eq. (12.3) is the output equation, which maps the
states and inputs linearly to the output. This is an algebraic equation, whereas the state
equation is a differential equation.
CONTROLLABILITY
Controllability
A system with internal state vector x is called controllable if and only if the
system states can be changed by changing the system input.
A state x1 is called reachable at time t1 if for some finite initial time t0 there exists an
input u(t) that transfers the state x(t) from the origin at t0 to x1.
Similarly, we can more precisely define the concept of controllability:
A state x0 is controllable at time t0 if for some finite time t1 there exists an input u(t) that
transfers the state x(t) from x0 to the origin at time t1.
A system is called controllable at time t0 if every state x0 in the state-space is
controllable.
Controllability Matrix
For LTI systems, a system is reachable if and only if its controllability matrix, ζ, has a
full row rank of p, where p is the dimension of the matrix A, and p × q is the dimension
of matrix B.
[Controllability Matrix]
A system is controllable or "Controllable to the origin" when any state x1 can be driven to
the zero state x = 0 in a finite number of steps.
A system is controllable when the rank of the system matrix A is p, and the rank of the
controllability matrix is equal to:
Rank(ζ) = Rank(A − 1ζ) = p
If the second equation is not satisfied, the system is not .
If
Rank(A) < p
Then controllability does not imply reachability.
Gramians
Gramians are complicated mathematical functions that can be used to determine specific
things about a system. For instance, we can use gramians to determine whether a system
is controllable or reachable. Gramians, because they are more complicated then other
methods, are typically only used when other methods of analyzing a system fail (or are
too difficult).
All the gramians presented on this page are all matrices with dimension p × p (the same
size as the system matrix A).
All the gramians presented here will be described using the general case of Linear timevariant systems. To change these into LTI (time-invariant equations), the following
substitutions can be used:
Where we are using the notation X' to denote the transpose of a matrix X (as opposed to
the traditional notation XT)..
Where we are using the notation X' to denote the transpose of a matrix X (as opposed to
the traditional notation XT).
Observability
The state-variables of a system might not be able to be measured for any of the following
reasons:
1. The location of the particular state variable might not be physically accessible (a
capacitor or a spring, for instance).
2. There are no appropriate instruments to measure the state variable, or the statevariable might be measured in units for which there does not exist any
measurement device.
3. The state-variable is a derived "dummy" variable that has no physical meaning.
If things cannot be directly observed, for any of the reasons above, it can be necessary to
calculate or estimate the values of the internal state variables, using only the input/output
relation of the system, and the output history of the system from the starting time. In
other words, we must ask whether or not it is possible to determine what the inside of the
system (the internal system states) is like, by only observing the outside performance of
the system (input and output)? We can provide the following formal definition of
mathematical observability:
Observability
A system with an initial state, x(t0) is observable if and only if the value of the
initial state can be determined from the system output y(t) that has been observed
through the time interval t0 < t < tf. If the initial state cannot be so determined, the
system is unobservable.
Complete Observability
A system is said to be completely observable if all the possible initial states of
the system can be observed. Systems that fail this criteria are said to be
unobservable.
A system state xi is unobservable at a given time ti if the zero-input response of the
system is zero for all time t. If a system is observable, then the only state that produces a
zero output for all time is the zero state. We can use this concept to define the term stateobservability.
State-Observability
A system is completely state-observable at time t0 or the pair (A, C) is
observable at t0 if the only state that is unobservable at t0 is the zero state x = 0.
Observability Matrix
The observability of the system is dependant only on the system states and the system
output, so we can simplify our state equations to remove the input terms:
Matrix Dimensions:
A: p × p
B: p × q
C: r × p
D: r × q
x'(t) = Ax(t)
y(t) = Cx(t)
Therefore, we can show that the observability of the system is dependant only on the
coefficient matrices A and C. We can show precisely how to determine whether a system
is observable, using only these two matrices. If we have the observability matrix Q:
[Observability Matrix]
we can show that the system is observable if and only if the Q matrix has a rank of p.
Notice that the Q matrix has the dimensions pr × p.
POLE PLACEMENT DESIGN
orthogonal (and thus linearly independent) eigenvectors (one for each eigenvalue).
THE RELATIONSHIP BETWEEN TRANSFER FUNCTIONS AND
THE STATE-SPACE REPRESENTATION
or rearranged
12.5
The solution of the state equation in the
domain is then given by
(12.6)
with
(12.7)
Similarly, for Eq. (12.5) yields
Substituting
from Eq. (12.6), the system output in the
domain is
To obtain the relationship with transfer functions, the initial condition
has to be set
to zero. For a single-input-single-output system the system output is
(12.8)
The matrix
from Eq. (12.7) is a matrix of rational functions of
, which can always
be represented by
(12.9)
where
is a matrix with polynomial elements in
.
(12.10)
and
(12.11)
which is the characteristic polynomial of the system. The zeros of this polynomial are the
poles of the transfer function and at the same time eigenvalues of the system matrix
. If
the system in the state-space representation is fully controllable and observable, then the
number of poles are equal to the number of eigenvalues.
BLOCK DIAGRAM REPRESENTATION OF LINEAR SYSTEMS
DESCRIBED BY STATE EQUATIONS
The matrix-based state equations express the derivatives of the state-variables explicitly
in terms of the states themselves and the inputs. In this form, the state vector is expressed
as the direct result of a vector integration. The block diagram representation is shown in
Fig. 2. This general block diagram shows the matrix operations from input to output in
terms of the A, B, C, D matrices, but does not show the path of individual variables. In
state-determined systems, the state variables may always be taken as the outputs of
integrator blocks. A system of order n has n integrators in its block diagram. The
derivatives of the state variables are the inputs to the integrator blocks, and each state
equation expresses a derivative as a sum of weighted state variables and inputs. A
detailed block diagram representing a system of order n may be constructed directly from
the state and output equations as follows:
Step 1: Draw n integrator (S−1) blocks, and assign a state variable to the output of each
block.
Step 2: At the input to each block (which represents the derivative of its state variable)
draw a summing element.
Step 3: Use the state equations to connect the state variables and inputs to the summing
elements through scaling operator blocks.
Step 4: Expand the output equations and sum the state variables and inputs through a set
of scaling operators to form the components of the output.
STATE-TRANSITION MATRIX
In control theory, the state-transition matrix is a matrix whose product with the state
vector x at an initial time t0 gives x at a later time t. The state-transition matrix can be
used to obtain the general solution of linear dynamical systems.
Overview
Consider the general linear state space model
The general solution is given by
The state-transition matrix
, given by
where
is the fundamental solution matrix that satisfies
is a
matrix that is a linear mapping onto itself, i.e., with
state
, given the
at any time τ, the state at any other time t is given by the mapping
While the state transtion matrix φ is not completely unknown, it must always satisfy the
following relationships:
φ(τ,τ) = I
And φ also must have the following properties:
1. φ(t2,t1)φ(t1,t0) = φ(t2,t0)
2. φ − 1(t,τ) = φ(τ,t)
3. φ − 1(t,τ)φ(t,τ) = I
4.
If the system is time-invariant, we can define φ as:
In the time-variant case, there are many different functions that may satisfy these
requirements, and the solution is dependent on the structure of the system. The statetransition matrix must be determined before analysis on the time-varying solution can
continue.