Control Systems

This page is going to be used to maintain a list of all the different versions available for the [Control Systems]( book. Each different version will contain different amounts of material presented in different orders. Each version will target a different audience. Instructors can pick and choose wh

All images added to this book should be properly documented in the [Licensing Page]( ## Advertising This book has a good book template located at [Template:Goodbook/Control Systems]( As a featured book, this template may be rotated onto the main page. That template should be formatted in a similar m

This page is going to serve as a local manual of style for the [Control Systems]( book. Guidelines listed in this page should be followed when adding new content to this book. Sections of the book not conforming to this manual should be changed. To discuss changes to the guidelines on this page, mak

The early sections of this book will focus primarily on **linear time-invariant** (LTI) systems. LTI systems are the easiest class of system to work with, and have a number of properties that make them ideal to study. This chapter discusses some properties of systems. Later chapters in this book wil

Physically realizable control systems must have a number of poles greater than the number of zeros. Systems that satisfy this relationship are called **Proper**. We will elaborate on this below. ## Time-Domain Relationships Let's say that we have a transfer function with 3 poles: $H ( s ) = \frac{a}

The Laplace transform moves a system out of the time-domain into the complex frequency domain so we can study and manipulate our systems as algebraic polynomials instead of linear ODEs. Given the complexity of differential equations, why would we ever want to work in the time domain? It turns out th

Once the state equation has been solved for x, that solution can be plugged into the output equation. The resulting equation will show the direct relationship between the system input and the system output, without the need to account explicitly for the internal state of the system. The sections in

However, the state-transition matrix does follow some basic properties that we can use to determine the state-transition matrix. In a time-variant system, the general solution is obtained when the state-transition matrix is determined. For that reason, the first thing (and the most important thing)

Notice that we set our start time t0 = kT, because we are only interested in the behavior of our system during a single sample period: $u ( t ) = u ( k T ) , k T ≤ t ≤ ( k + 1 ) T$ $x ( t ) = Φ ( t , k T ) x ( k T ) + {∫}_{k T}^{t} Φ ( t , τ ) B d τ u ( k T )$ We were able to remove u(kT) from the i

Non-square matrices cannot be analyzed using the methods below. The word "eigen" comes from German and means "own" as in "characteristic", so this chapter could also be called "Characteristic values and characteristic vectors". The terms "Eigenvalues" and "Eigenvectors" are most commonly used. Eigen

Notice that we know beforehand Aw and Bw, since we know both the form of the matrices and the coefficients of the equation (e.g. a linear ODE with constant coefficients or a transfer function). We can form ζw if we know these two matrices. We can then use this matrix to create our transformation mat

To represent multiple inputs we expand the input u(t) into a vector U(t) with the desired number of inputs. Likewise, to represent a system with multiple outputs, we expand y(t) into Y(t), which is a vector of all the outputs. For this method to work, the outputs must be linearly dependent on the in

The state-space representation, for instance, is preferable when it comes time to move the system design from the drawing board to a constructed physical device. For that reason, we call the process of converting a system from the Laplace representation to the state-space representation "realization

Consider the given second-order system: $T ( s ) = \frac{1}{{s}^{2} + 2 s + 1}$ We can include an arbitrary gain term, K in this system that will represent an amplification, or a power increase: $T ( s ) = K \frac{1}{{s}^{2} + 2 s + 1}$ In a state-space system, the gain term k can be inserted as fol

This is performed by adding controllers, compensators, and feedback structures to the system. ## Basic Feedback Structure  This is a basic feedback structure. Here, we are using the output value of the system to help us prepare the ne

The Magnitude plot is typically on the top, and the Phase plot is typically on the bottom of the set. (Image: User:Netnet, Public domain) The frequency of the bode plots are plotted against a logarithmic frequency axis. Every tickmark on the frequency axis represents a power of 10 times the previous