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4.4  Coupling in MIMO systems

As stated in examples in Section 2.9, coupling is typically negligible in well designed control systems where the number of the actuators is kept small. However, at some frequencies the coupling can be large, uncertain, and create stability problems.

In mechanical structures with multi-dimensional control, the actuators are typically applied in mutually orthogonal directions so that the coupling between the corresponding feedback loops is relatively small. However, the plant might include some flexible attachment, like an antenna, a solar panel, or a magnetometer boom on a spacecraft, as shown in Fig. 4.39. The attachment's flexible mode can be excited by any of the actuators (reaction wheels, thrusters), and will provide signals to all the sensors. This coupling may occur at any frequency within a certain frequency range defined by uncertainty in the mass and stiffness of the appendage.

Fig. 4.39  Mechanical plant with a flexible appendage	Fig. 4.40  Block diagram for the coupled attitude control loops

Because of the coupling, the block diagram for the coupled loops looks like that shown in Fig. 4.40. Here, K(s) is the coupling transfer function. The return ratios for the controllers in x and y calculated without taking the coupling into account are T_x = C_x A_x P_x B_x and T_y = C_y A_y P_y B_y .

The system can be designed with the Bode-Nyquist multiloop system stability criterion as follows. First, the y-actuator is disabled and T_x is shaped so that the x-loop is stable and robust. Then, A_y is switched on (while the x-actuator is kept on). The transfer function in the y-channel between A_y and P_y is 1 + (T_x/F_x)K². The compensator C_y is then shaped properly to make the system stable (sometimes, this is not possible). The gain-stabilization in the range of the flexible mode is the best choice since phase-stabilization is difficult because the transfer function K² = a²M/ (s² + 2 + _v²)² contains double complex poles with large associated phase uncertainty; here, a is some coefficient. If gain-stabilization cannot be used and the system needs to be phase-stabilized, C_x should be modified to make the response in the x-loop shallower over the frequency range of coupling. The associated reduction of available feedback is unavoidable. Bode diagrams for stability analysis with the successive loop closure criterion, for the x- and y-loops, may look like those shown in Fig. 4.41.

(a)	(b)

Coupling between x- and y-controllers can be also caused by the effects of rotation about the z-axis (even in a spacecraft without a flexible appendage). In this case, x-actuators produce rotation about the y-axis, and y-actuators produce rotation about the x-axis. The effect is only profound at frequencies close to the frequency of rotation about the z-axis. The system analysis and design are similar to those in the case of flexible mode coupling.

Fig. 4.42  x-y positioner

An x-y positioning table is shown in Fig. 4.42. The translational motions may become coupled via rotational motion due to the load asymmetry and due to structural flexibility, especially at the frequency of the structural mode of the rotation.

When the number of the actuators is large (there are many thousand separate muscles in the trunk of an elephant), each control loop should use position (or velocity) and force sensors. This compound feedback makes the loop transfer function less sensitive to plant parameter variations, and makes the output mobility of the actuator dissipative and damping the plant. This also reduces the variations of the loop coupling that are caused by variations of the load and the plant parameters. Loop decoupling algorithms can then be used effectively. Design of a loop with prescribed actuator mobility will be discussed in Chapter 7.