Chapter 7 - State-Space Models

In this chapter we introduce state-space models for describing system dynamics,
an alternative to the transfer function models presented in Chapter 5 and 6. The
idea of state space is to characterize how the system operates in terms of one or
more variables. Such state variables need not be measured outputs. Indeed, they
may not even be directly measurable. However, the state variables must be able
to express the dynamics of the system. State-space models provide a scalable
approach to modeling systems with a large number of inputs and outputs. It
turns out that many of the techniques and results for transfer functions, such as
dominant pole analysis, apply to state space as well.

7.1 STATE VARIABLES

It is sometimes convenient to describe system dynamics in terms of variables
other than the control input and the measured output. These auxiliary variables
are referred to as state variables. We motivate their use with an example.

Example 7.1: Modeling a tandem queue This example is motivated by complex
systems such as multitiered e-commerce environments in which there are
multiple interconnected components. A common way to model such systems is
as a network of queueing systems.

Fig. 7.1 Architecture diagram of a tandem queue.

One of the simplest queueing networks is a tandem queue. As depicted in
Figure 7.1, the tandem queue consists of two queueing systems in series. Incoming
requests arrive at system 1. If the server is idle, the request begins service
immediately. Otherwise, the request waits in buffer 1 until the server becomes
available. If buffer 1 is full, the request is discarded (e.g., as in packet-switching
networks).¹ Departures from the first system become arrivals at the second system.
The second queueing system handles incoming requests in the same manner
as the first system. Departures from the second system are outgoing requests.

For the purposes of this example, we assume that buffer 2 is sufficiently large
so that departures from queue 1 are never discarded. We seek to control the
end-to-end response time of the tandem queue, which is the sum of the response
times of the two queues. Thus, the size of buffer 1 K(k) is the control input, and
end-to-end response time R(k) is the measured output.

Although this is a SISO system, it is natural to model it differently. Let R₁(k)
be the average response time during the kth interval of requests entering the
first queueing system, and let R₂(k) be the same metric for the second queueing
system. It is natural to construct separate models for the dynamics of R₁(k)
and R₂(k) since each queueing system can be modeled using a first-order ARX
model. However, neither of these variables is being controlled. We only want
to control their sum, R(k).

< Previous Excerpt Next Excerpt >

Purchase This Book

TABLE OF CONTENTS

Chapter 7 - State-Space Models

Contact Preferences

This is embarrasing...

Customize Your GlobalSpec Experience

Select Your Free Newsletters

Industry Newsletters

Select Your Free Product Alerts

This is embarrasing...