Benginner's Guide to State Space Design

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Intro to State Space Design

A state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form. The state-space representation provides a simple and small way to represent and analyze MIMO systems. We would otherwise have to express Laplace transforms to encode all the information about a system. Unlike the classical control methods, the use of the state-space equations is not limited to systems with only linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The states of the mechanism can be modeled as in vectorized form within that space.

State Variables

The internal state representation are the smallest possible subset of system equations that can represent the entire state of the model at any moment in time. State variables must be linearly independent. The least number of state equations required to model a given model, n, is typically equal to the size of the system's defining differential equations. If the system is modeled in transfer function form, the least number of state equations is equal to the size of the transfer function's denominator following its reduction to a proper fraction. It is important to appreciate that changing a state-space realization to an s-domain form will sometimes lose internal information about the system, and will sometimes model a description of a mechanism which is stable, when the state space model is unstable at certain points.

State space Controller Analysis

In many Single-Input Single Output mechanisms a compensator can be designed through a methodology named ''Loop Shaping'' using Classical Control techniques; the Bode plot is the primary tool used for ''Loop Shaping''. For Multi-Input Multi-Output mechanisms the singular value decomposition (SVD) provides a tool similar to the Bode plot. ''Loop Shaping'' is more difficult for Multi-Input Multi-Output mechanisms than SISO.

Since ''Loop Shaping'' is diffcult for MIMO mechanisms other techniques were created for designing state-space controllers. The two most popular state-space compensators are the Linear Quadratic Regulator (LQR controller) and the Linear Quadratic Gaussian (LQG).

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