## Documentation |

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Modeling Controllers in Simulink and Plants in SimMechanics Software Control Transfer Function Forms and Units |

You should have some experience with these tools before proceeding. To understand trimming better, work through Trimming and Linearizing Through Inverse Dynamics earlier in this chapter. |

A classic engineering problem is the design of *controllers* for
a physical system, the *plant* [2].
A SimMechanics model can represent a complex mechanical system
and helps you design and implement a control system for the plant,
in conjunction with Simulink and related control design products.

In the next two case studies, you use SimMechanics software to model the plant and Simulink to analyze and synthesize controllers. You explore a basic challenge of control design, the tradeoff between responsiveness and stability, by implementing first a simple controller, then a more complex and robust one [4]. This section is preliminary to those studies.

The motion of an uncontrolled physical system is represented
by its position and velocity variables arranged into a *state
vector* ** X**. The dynamics
of the system is described by a force law:

$$\text{d}X/\text{dt}=f(X)$$

Introducing control means introducing sensors and actuators
that modify the system's otherwise natural motion. The actuators impose
artificial forces — collectively, the *inputs* ** U** — on the system, while the sensors
detect motions and report

$$\begin{array}{l}\text{d}X/\text{dt}=f\left(X,U\right)\\ Y=g\left(X,U\right)\end{array}$$

The ** U** and

By selecting the proper set of ** U** and

Selecting ** c** is the fundamental
problem of control design. The desired trajectory of

The controller and plant transfer functions are often called *C* and *G*,
respectively. The combined controller-plant transfer function forms
are the *open-loop* *CG* and the *closed-loop* *CG/(1+CG)*.

Controller and plant response magnitudes are measured in decibels (dB).

The next two case studies use these files, in addition to the initialization script and function.

File | Purpose |
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mech_stewart_controlmech_stewart_control | Main model |

mech_stewart_control_derivmech_stewart_control_deriv | Configurable subsystem: Derivative block or transfer function (filtered) |

mech_stewart_controllersmech_stewart_controllers | Configurable subsystem: Null, PID, or H-infinity controller |

mech_stewartplatform_legmech_stewartplatform_leg | Library model of Stewart platform leg; used six times in the Plant subsystem of the main model |

The problems and techniques of the next two case studies only touch the basics of control design. In practice, you need to consider additional issues and goals. Also consult References.

To fully understand the plant, you need to find plant operating points other than the simple ones used here and optimize the controller in other representative states.

See the preceding case study, Trimming and Linearizing Through Inverse Dynamics.

To design more robust controllers, you should consider the effect of parameter uncertainty and signal noise. This step involves comparing typical plant motion frequencies, noise frequencies, and filtered derivative cutoffs.

The following toolboxes can help with such tasks:

To move toward hardware implementation, you must consider discretizing the controller [8]. Among other requirements, this necessitates using a fixed-step solver, optimizing the solver step size and sample rate, and adjusting the filtered derivative cutoff.

See the final two case studies:

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