Applied Mathematics > Control Theory > Modern Control
Modern Control is a sophisticated subfield of Control Theory within Applied Mathematics that focuses on the design and analysis of systems to ensure their stability, performance, and robustness in real-world applications. Unlike classical control methods, which are typically limited to single-input, single-output (SISO) systems and rely on simpler mathematical techniques such as transfer functions and frequency response, Modern Control employs advanced mathematical tools to handle more complex, multi-variable systems.
In Modern Control, the state-space representation is a fundamental concept. A system’s state is represented by a vector of variables, capturing all necessary information to describe the system’s behavior over time. The state-space form of a linear time-invariant (LTI) system is given by:
\[
\begin{aligned}
\dot{\mathbf{x}}(t) &= \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t), \\
\mathbf{y}(t) &= \mathbf{C}\mathbf{x}(t) + \mathbf{D}\mathbf{u}(t),
\end{aligned}
\]
where:
- \(\mathbf{x}(t)\) is the state vector,
- \(\mathbf{u}(t)\) is the input vector,
- \(\mathbf{y}(t)\) is the output vector,
- \(\mathbf{A}\) is the state matrix,
- \(\mathbf{B}\) is the input matrix,
- \(\mathbf{C}\) is the output matrix,
- \(\mathbf{D}\) is the feedforward (or direct transmission) matrix.
Key techniques in Modern Control include optimal control, robust control, and adaptive control. Optimal control aims to determine the control inputs that will minimize or maximize a certain performance criterion, often formulated as an integral cost function. The Linear Quadratic Regulator (LQR) is a quintessential example of an optimal control problem, where the goal is to minimize a quadratic cost function of the state and control input:
\[
J = \int_{0}^{\infty} (\mathbf{x}^T(t) \mathbf{Q} \mathbf{x}(t) + \mathbf{u}^T(t) \mathbf{R} \mathbf{u}(t)) \, dt,
\]
with \(\mathbf{Q}\) and \(\mathbf{R}\) being weight matrices that balance state and control penalties.
Robust control deals with systems under uncertainty, ensuring the system performs satisfactorily despite variations in system parameters or the presence of disturbances. Techniques like \(H_{\infty}\) control and \(\mu\)-synthesis fall into this category, providing guarantees on system performance and stability within specified bounds of uncertainty.
Adaptive control is designed for systems with parameters that change over time or are initially unknown. This approach modifies the control law in real-time based on the observed behavior of the system, using techniques such as Model Reference Adaptive Control (MRAC) and Self-Tuning Regulators (STR).
Modern Control theory is not only theoretical but also widely applied in various engineering fields, including aerospace, automotive, robotics, and economics. Its development has been pivotal in advancing technology, allowing for precise and reliable control of complex systems.