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Model-Following Control Systems

Description

Although the classic PID control features some robustness and insensitivity to process parameter variations, it appears to be often ineffective if coping with strongly nonlinear or time-variant processes. Despite its limitations PID control have found a wide utility in industry due to its easiness of parameterization and low costs of implementation. High requirements set on control quality of highly perturbed process parameters often force one to employ more advanced control structures or robust control design methods. An interesting compromise between the low-robust (however, easy-to-use) classic PID structure and the high-robust (however, usually complicated to synthesize) systems are constant-parameter systems belonging to the Model-Following Control (MFC) group.

mfc1.jpg
Fig. 1: General block diagram for four control structures: 2DOF IMC (sw1 = 0, sw2 = 0), MFC-m (sw1 = 1, sw2 = 0), MFC-p (sw1 = 0, sw2 = 1) and MFC-mp (sw1 = 1, sw2 = 1).

Systems Overview

Four control systems are depicted in a single Figure (Fig. 1) by employing switches sw1 and sw2. So, for different combinations of both switches we get:

1. 2DOF IMC (sw1 = 0, sw2 = 0) This is a well-known control system, which finds a frequent utility in, amongst others, chemical processes. The synthesis of such a system is not trivial. The design of IMC structures involve casting the design problems in therms of H2 or Hinf norms of sensitivity functions in order to obtain the parameters of a robust controller. In frequency domain, these require the use of weighting functions. Since IMC systems are widely covered in the literature, we shall not go into details of their properties in the subsequent text.
2. MFC-m (sw1 = 1, sw2 = 0) This is the classic MFC system (here is termed MFC- m (model feedback)), which appeared in the literature in the early 1990s. In this concept the compensator C is replaced by the controller Rm driven by the model error. Hence, two loops may be distinguished in this structure: the model loop RmM and the process loop RkP. This system offers a much higher robustness to process parameter variations than the classic PID structure. However, MFC-m is not free from some limitations. The most substantial of them is the necessity to simplify the model M to such a form that the control performance in the model loop be satisfying. This is essential, because the model output ym represents a reference signal for the process loop. So, for example, an overshoot obtained in the model loop will result in the same overshoot (or even worse) obtained at the process output.
3. MFC-p (sw1 = 0, sw2 = 1) This was the first attempt to modify MFC-m. Here the model- dependent feedback is replaced by the process-dependent one. This slight change in the structure imparts higher robustness than that achievable in the original MFC-m systems. Moreover, the both controllers Rm and Rk are involved actively in suppression of disturbances, which makes the MFC-p (plant feedback) system more attractive. The shortcoming of this structure is less favorable stability conditions.
4. MFC-mp (sw1 = 1, sw2 = 1) To combine advantages offered my MFC-m and MFC-p the next modification was made by introducing an additional feedback. In doing so, a control structure with three loops: RmM, RmP and RkP has been obtained. Indeed, the system provides still higher robustness and ability to suppress disturbances than MFC-p with its stability conditions being comparable to those of MFC-m.

In MFC systems the task performed by the compensator C is taken over by the controller Rm, which generates manipulated variables from the error. Hence, it may be spoken of an indirect linearization of the process. It is essential that MFC systems do not require an exact process model for them to operate properly. To employ MFC the knowledge of only basic dynamic/static properties of the process is needed. The model inexactness is compensated here by the corrective controller Rk, the parameterization of which is much easier than that of a controller in the classic single-loop PID system, because Rk is not subject to stepwise varying signals. This allows high Rk gain values to be set, which results in improved robustness or suppression of system disturbances z.

Simulation

The aim of simulation tests described below is to provide an illustrative example of robustness and ability to suppress disturbances by the MFC systems. Since the design of MFC systems is only slightly more complicated than that of the most frequently employed classic single-loop PID structure, therefore the control performance offered by MFC has been always related to that offered by PID. It should be stressed that the results presented here have been obtained for controller settings once found and remained unchanged for all the structures, i.e. PID, MFC-m, MFC-p and MFC-mp.
The tests have been carried out on a SISO plant that is strongly nonlinear (statically, for the major part), and also may be time-variant. This is a single-link robotic manipulator characterized by its mass m, arm length lx and friction coeffcient b.

sim1.jpg
Fig 2: Controllers for each structure have been tuned up for the operation point
q = 90deg.

The simulation result is displayed in Fig. 2. The control performance offered by PID, MFC- m, MFC-p i MFC-mp is similar. However, the situation will be different if the operation point is changed to q = 270deg.

sim2.jpg
Fig 3: Control performance at the change in the operation point to q = 270deg.

In Fig. 7 one may notice distinct differences between the robustness provided by the PID system and that by MFC systems. The change in the loop gain gives rise to significant oscillations experienced by the PID system. The same change performed in MFC structures produces an unnoticeable effect. The manipulator dynamics seen from the level of the first link drive may fluctuate due to changes occurring in robot configuration and/or mass transport even by an order of magnitude. Figure 4 illustrate the obtained control performance provided by the systems under comparison in the case that the link mass is increased 10-fold. Here the MFC systems also provide a significant improvement in reference tracking.

sim3.jpg
Fig 4: Robustness of the control systems at the 10-fold change in the link mass.

Suppression of disturbances plays a vital part in selecting a control system. Particular emphasis is laid on suppression of system disturbances z and output disturbances v. Figure 5 illustrate to what extent the systems under study are effective in suppression of low and medium frequency disturbances. The result is not surprising, since the MFC systems are those with two degrees of freedom. Therefore, the tasks of reference tracking and disturbance suppression can be shared here between the individual controllers.

sim4.jpg
Fig 5: Suppression performance of medium-frequency disturbances (7 rad/sec).

MFC systems based on a simple process model! This is one of more important advantages offered by MFC systems because the basic component of the manipulated variable is worked out in the loop, which makes the use of compensators (usually being dificult to synthesize) needless. That the parameters of the MFC structures are constant is the next advantage. This entails easiness of parametrization and implementation, which is comparable with the well-known single-loop PID system. The above-mentioned features make the design of a robust control system quite simple. As for the obtained theoretical findings, it should be remembered that the MFC systems not always could yield expected results. This is especially true where the model M displays too strong nonlinearities that may impair the operation of the control system as a whole. The multi-loop n-MFC control overcomes the impediment, however on condition that the model can be presented as gradually complex. If you are interested on this project, please contact Rafal Osypiuk.

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