7.3 Difficulties of MIMO Control Design

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7.3 Difficulties of MIMO Control Design

The fundamental reason MIMO controller design is more difficult than SISO controller design is the presence of cross-coupling between plant inputs and outputs. If there were no cross-coupling (i.e., if each plant input influenced only one plant output), it would be perfectly correct to assume the MIMO system consists of a group of independent SISO systems and deal with them separately.

MIMO systems in the real world often are more complicated than this. In many everyday systems, each plant input primarily affects one output but also has a smaller influence on other outputs. In the most complex case, each plant input significantly affects all of the outputs.

Example 7.1: Helicopter control cross-coupling.

One example of control cross-coupling occurs in a helicopter. A helicopter has three primary flight controls.

Collective This control changes the simultaneous tilt of all the rotor blades. Pulling the collective up causes the blade leading edges to tilt up, causing the helicopter to move upward.

Cyclic This controls the varying tilt of the rotor blades as they rotate. Moving the cyclic to the left alters the blade tilt so that the helicopter responds by tilting left. Right-ward cyclic motion tilts the helicopter to the right. Similar motion controls the front-back tilt of the helicopter.

Tail Rotor The foot pedals control the tilt of the tail rotor blades. The functions of the tail rotor are to cancel out the torque on the helicopter body produced by the main rotor and to allow the nose of the helicopter to be pointed in the desired direction.

Each of the three controls has one primary effect on helicopter flight behavior. However, changes to any of the control inputs cause additional effects that the control system (usually a pilot) must counteract.

For example, the primary effect of pulling up on the collective is increased upward force. However, this control input also causes the torque from the main rotor to increase, which causes the helicopter body to begin turning about the main rotor axis. This body rotation is usually undesirable and can be prevented by adjusting the tail rotor tilt to balance out the additional main rotor torque. The result of this cross-coupling is that a desired change in one plant output (helicopter altitude) requires manipulation of (at least) two of the control inputs: the collective and the tail rotor.

The need to simultaneously adjust multiple control inputs to produce a desired system response is a typical attribute of MIMO systems. Obviously, this makes the control system development procedure more difficult. Although it is often possible to apply techniques such as PID controller design (Chapter 2) and root locus design (Chapter 4) in these situations, the results are seldom satisfactory. Because the SISO design techniques do not account for cross-coupling within the plant, the resulting controllers might be unable to satisfy the design requirements.

The pole placement (Chapter 5) and optimal (Chapter 6) state-space design techniques are inherently capable of designing MIMO controllers. These approaches create controllers that are tuned to deal effectively with the cross-coupling within the plant.

The remainder of this chapter consists of two MIMO controller design examples. Because these systems are more complex than the plant models used in previous chapters, the linear model development process will be discussed in some detail before the controller design work begins. I will then determine performance specifications and proceed with development of the control systems. Both pole placement and optimal design techniques will be demonstrated.

This controller functions as a regulator: All the states, inputs, and outputs are defined as deviations from their equilibrium values. Because of this, it is not necessary to develop a feedforward gain matrix N as was described in Chapter 5.

Because the controller is based on a linearized model about nonzero equilibrium states and inputs, it is important to remember to adjust the inputs and outputs accordingly. All inputs to the observer-controller must first have the equilibrium values subtracted from them, and all outputs must have the equilibrium values added back.