Multivariable MFA Control System

In control applications, most processes have multiple inputs and multiple outputs with interactions in between. The level and density loops of an evaporator, and the temperature loops of a multi-zoned furnace are good examples of multivariable processes. Lacking general-purpose multivariable controllers, a large percentage of multivariable processes are treated as single variable processes resulting in poor control, wasted energy and materials, inconsistent quality, and plant upsets.

This graph illustrates a multivariable Model-Free Adaptive (MFA) control system, which consists of a multi-input multi-output (MIMO) process and an MIMO MFA controller.

Multivariable MFA control system

Similar to a SISO system, the MIMO system has controller setpoints r(t), error signals e(t), controller outputs u(t), process variables y(t), and disturbance signals d(t). Since it is a multivariable system, all the signals here are vectors represented in bold case.

2-Input-2-Output MFA Control System

Without losing generality, we will show how a MIMO MFA controller works with a 2-input-2-output (2x2) system as illustrated in following graph. In the 2x2 MFA system, the 2x2 MFA controller consists of two main controllers C11, C22, and two compensators C21, and C12. The process has four sub-processes G11, G21, G12, and G22.

2-input-2-output MFA control system

The measured process variables y1 and y2 are used as the feedback signals for the main control loops. They are compared with the setpoints r1 and r2 to produce errors e1 and e2. The output of each controller associated with one of the inputs e1 or e2 is combined with the output of the compensator associated with the other input to produce control signals u1 and u2. The output of each sub-process is cross-added to produce measured process variables y1 and y2. Notice that in real applications the outputs from the sub-processes are not measurable and only their combined signals y1 and y2 can be measured. Thus, by the nature of the 2x2 process, the inputs u1 and u2 to the process are interconnected with outputs y1 and y2. The change in one input will cause both outputs to change.

The control objective for this 2x2 MFA control system is to produce control outputs u1(t) and u2(t) to force the process variables y1(t) and y2(t) to track their setpoints r1(t) and r2(t), respectively. The minimization of e1(t) and e2(t) is achieved by (i) the regulatory control capability of the MFA controllers, (ii) the decoupling capability of the MFA compensators, and (iii) the adjustment of the MFA weighting factors that allow the controllers to deal with the dynamic changes, large disturbances, and other uncertainties.

2x2 MFA Controller Configuration

A 2x2 MFA controller can be considered to have 2 main controllers C11 and C22. For each main controller, the parameters to configure are: (1) Sample Interval, Ts - the interval between two samples or calculations in seconds. A high speed MFA controller can run at a 1 millisecond rate; (2) Controller Gain, Kc1 - use of a default value is recommended, (3) Time Constant, Tc - a rough estimate of the process Time Constant in seconds; (4) Acting Type - direct or reverse acting of the process; and (5) Compensator Gain, Kc2 - to deal with the interaction from the other loop.

MIMO MFA Controller Application Guide

A MIMO system can be much more complex than a SISO system, therefore precautious must be taken when applying a MIMO MFA controller. When designing a multivariable control system, the first step is to decide which process variable is paired with a manipulated variable. A MIMO MFA control system should follow these pairing rules: (1) Each main process (G11, G22) has to be controllable, open-loop stable, and either reverse or direct acting; (2) A process with a large static gain should be included in the main loop as the main process (G11, G22), and a process with a small static gain should be treated as a sub-process. (G21, G12); (3) A faster process should be paired as the main process, and a slower process or processes with time delays should be treated as sub-processes; and (4) If Pairing Rules 2 and 3 are in conflict, a tradeoff is the only option.

As a general guide, an MFA control system should be designed based on the degree of interactions between the loops. This table lists the control system design strategy based on the degree of interaction of a MIMO process.