14.6: Comments Regarding Control Theory

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In the words of Brogan, 1974, page 1, “Control theory can be divided into two major categories, classical and modern.” Chapters 14, 15, 16, and 17 of this book constitute an introduction to the former, namely, classical control theory for linear, time-invariant (LTI) systems. Whereas classical control theory focuses on relatively simple single-input-single-output (SISO) systems, modern control theory, which is based upon the state-space mathematical representation of systems, is very well suited for design and analysis of control for both simple systems and much more complicated multiple-input-multiple-output (MIMO) systems.¹

The state-space representation of systems is a generalization of the standard 1^st order ODE Equation 1.2.1, \(\dot{x}-a x=b u(t)\), which has only the single input \(u(t)\) and the single dependent variable \(x(t)\). In the generalization to an LTI-MIMO system of, say, \(r\) inputs and \(n\) dependent variables, scalar \(x(t)\) becomes the \(n \times 1\) state vector (column matrix) \(\mathbf{x}(t)\) of dependent variables, scalar \(u(t)\) becomes the \(r \times 1\) vector of inputs \(\mathbf{u}(t)\), single constant \(a\) becomes the \(n \times n\) matrix \(\mathbf{A}\) of system constants, single constant \(b\) becomes the \(n \times r\) matrix \(\mathbf{B}\) of input constants, and single ODE \(\dot{x}-a x=b u(t)\) becomes the system of \(n\) coupled 1^st order ODEs written in matrix form, \(\dot{\mathbf{x}}-\mathbf{A} \mathbf{x}=\mathbf{B u}(t)\). All of the 2^nd order systems analyzed in this book can be expressed in state-space form with \(\mathbf{x}(t)\) being \(2 \times 1) state vectors; for example, the two scalar Equations 1.9.3 and 1.9.4 can clearly be written as the following single matrix ODE:

\[\left[\begin{array}{c}
\dot{x} \\
\dot{v}
\end{array}\right]-\left[\begin{array}{cc}
0 & 1 \\
-k / m & -c / m
\end{array}\right]\left[\begin{array}{c}
x \\
v
\end{array}\right]=\left[\begin{array}{c}
0 \\
1 / m
\end{array}\right] f_{x}(t) \nonumber \]

Further, the 2-DOF systems of Chapters 11 and 12 can be expressed in state-space form with \(\mathbf{x}(t)\) being \(4 \times 1\) state vectors.

Chapters 14, 15, 16, and 17 of this book should provide adequate preparation for study of the fundamental aspects of modern control theory. Three appropriate textbooks, among the many available, are Brogan, 1974, Franklin, et al., 1991, and Ogata, 2001.

¹Franklin, et al., 1991, page 361, wrote that the adjective “modern” is misleading “… since the state-space method of description for differential equations is over 100 years old and was introduced to control design in the late 1950s, … .” Those authors prefer the designations “transform methods” rather than “classical control” and “state-space methods” rather than “modern control”