State observer

In control theory, a state observer is a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. It is typically computer-implemented, and provides the basis of many practical applications.

Knowing the system state is necessary to solve many control theory problems; for example, stabilizing a system using state feedback. In most practical cases, the physical state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by way of the system outputs. A simple example is that of vehicles in a tunnel: the rates and velocities at which vehicles enter and leave the tunnel can be observed directly, but the exact state inside the tunnel can only be estimated. If a system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer.

The state of a linear, time-invariant physical discrete-time system is assumed to satisfy

where, at time ${\displaystyle k}$, ${\displaystyle x(k)}$ is the plant's state; ${\displaystyle u(k)}$ is its inputs; and ${\displaystyle y(k)}$ is its outputs. These equations simply say that the plant's current outputs and its future state are both determined solely by its current states and the current inputs. (Although these equations are expressed in terms of discrete time steps, very similar equations hold for continuous systems). If this system is observable then the output of the plant, ${\displaystyle y(k)}$, can be used to steer the state of the state observer.

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