### Linear State Space Models

As introduced in Book II [449, Appendix G], in the linear,
time-invariant case, a discrete-time *state-space model* looks
like a vector first-order finite-difference model:

where is the length

*state vector*at discrete time , is in general a vector of inputs, and the output vector. is the

*state transition matrix*, and it determines the

*dynamics*of the system (its

*poles*, or

*modal resonant frequencies and damping*).

The state-space representation is especially powerful for
*multi-input, multi-output* (MIMO) linear systems, and also for
*time-varying* linear systems (in which case any or all of the
matrices in Eq.(1.8) may have time subscripts ) [220].

To cast the previous force-driven mass example in state-space form, we
may first observe that the state of the mass is specified by its
velocity and position
, or
.^{2.9}Thus, to Eq.(1.5) we may add the explicit difference equation

with being a typical initial state.

General features of this example are that the entire physical state of the system is collected together into a single vector, and the elements of the matrices include physical parameters (and the sampling interval, in the discrete-time case). The parameters may also vary with time (time-varying systems), or be functions of the state (nonlinear systems).

The general procedure for building a state-space model is to label all
the state variables and collect them into a vector
, and then
work out the state-transition matrix , input gains , output
gains , and any direct coefficient . A state variable
is needed for each lumped energy-storage element (mass,
spring, capacitor, inductor), and one for each sample of delay in
sampled distributed systems. After that, various equivalent (but
numerically preferable) forms can be generated by means of
*similarity transformations* [449, pp. 360-374]. We
will make sparing use of state-space models in this book, because
they can be linear-algebra intensive, and therefore rarely used in
practical real-time signal processing systems for music and audio
effects. However, the state-space framework is an important
general-purpose tool that should be kept in mind [220], and
there is extensive support for state-space models in the matlab
(``matrix laboratory'') language and its libraries. We will use it
mainly as an analytical tool from time to time.

As noted earlier, a point mass only requires a first-order model:

#### Impulse Response of State Space Models

As derived in Book II [449, Appendix G], the impulse response of the state-space model can be summarized as

Thus, the th ``sample'' of the impulse response is given by for . Each such ``sample'' is a matrix, in general.

In our force-driven-mass example, we have , , and . For a position output we have while for a velocity output we would set . Choosing simply feeds the whole state vector to the output, which allows us to look at both simultaneously:

Thus, when the input force is a *unit pulse*, which corresponds
physically to imparting momentum at time 0 (because the
time-integral of force is momentum and the physical area under a unit
sample is the sampling interval ), we see that the velocity after
time 0 is a constant , or , as expected from
conservation of momentum. If the velocity is constant, then the
position must grow linearly, as we see that it does:
. The finite difference approximation to the time-derivative
of now gives
, for , which
is consistent.

#### Zero-Input Response of State Space Models

The response of a state-space model Eq.(1.8) to *initial
conditions*, *i.e.*, its *initial state*
, is given by

*complete response*of a linear system is given by the sum of its

*forced response*(such as the impulse response) and its

*initial-condition response*.

In our force-driven mass example, with the external force set to zero, we have, from Eq.(1.9) or Eq.(1.11),

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Transfer Functions

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State Space Models