1 Introduction to Multiple Input-Multiple Output
Linear control theory, as taught to undergraduate students, is primarily concerned with Single Input-Single Output (SISO) systems. Many real world systems are linear and while technically Multiple Input-Multiple Output (MIMO) the coupled axes are so weakly coupled that the coupling can be neglected. As a result the system can be approximated as SISO. An example of this is the hovering helicopter where at hover the pitch attitude and horizontal speed can be decoupled because they should both be nearly zero minimizing any coupling between them.
MIMO systems are typically more complex than SISO systems. The interactions of the dynamics make simple linear predictions of system performance difficult. Singular value decomposition (SVD) is the equivalent of a Bode diagram for an SISO system. From an SVD frequency domain techniques can be applied. FOR MIMO systems the SISO frequency domain techniques require some adjustment.
SISO techniques exist for transfer functions and state equations. The same is true for MIMO. However, most of the effort and focus is on state equations for MIMO and transfer functions for SISO. Everyone has their own favorite of doing things, however, and there are as many different ways to do things as there are people to do them.
However, the focus in MIMO control system theory appears to be on Optimal and Robust control which involve designing a controller that minimizes a cost function. An example cost function would energy required for control (I believe this is cost function minimized for H∞ control).
Often these Optimal and Robust techniques require full Observability of all states and so a state estimator is required. The most common estimator is the Kalman Filter. Estimators are a large topic area and deserve their own subcategory.
What follows are some design goals to keep in mind.
2 Big Picture
Design the pre-filter (W) and controller (K) on the basis of a nominal model for the plant such that the feedback system exhibits the following properties:
- Stability: if the system is perturbed then the system will return to equilibrium
- Small tracking error
- Good low frequency command following
- Good low frequency disturbance attenuation
- Good high frequency noise attenuation
The stated goals must be achieved in the presence of the following sources of uncertainty
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