Embedded model control (EMC) should not be confused with ‘embedded electronic systems’. Embedded is a more cogent attribute than internal, as it suggests that modern control systems should be designed and built around a dynamic model (the embedded model, EM) of the real system (the plant) to be controlled.

Discrepancies and their rejection

The first EM function is to reveal real-time discrepancies between real system and model. It becomes viable if plant and model are simultaneously driven by the same input signal (the command) and produce comparable measurements (output signals) which can be each other subtracted. Plants are equipped with suitable actuators and sensors and input-output models are implemented on numerical computers in the form of state equations. The difference between plant measurement and model output, the model error, may be either off-line or on-line elaborated for model updating until error fluctuations become bounded and smaller than a predefined threshold.

In Figure 1, the plant is drawn as a yellow 3D box, the computer as a pink 3D box and EM as a cyan 2D box. The command is generated inside the computer by an unnamed white box.

Figure 1 – The model error

Focusing on real-time elaboration, the model is endowed with a secondary input signal (the disturbance) – the primary being the command – which is synthesized by model error through a feedback algorithm. The algorithm is expected of being capable of saving the past elaboration in the form of a state vector (the disturbance state) and, at any new measurement acquisition, of refreshing the state with model error components that are sufficient for keeping the error within the target threshold. A closed-loop system of this kind is known as a state observer, but we should point out that, now, two kinds of information become available: the state of the command-driven model (the controllable state) and the disturbance state saved by the feedback algorithm. The latter state actualizes the past discrepancies, and, being a time signal, it can be included, upon a sign change, into the plant command. The aim is to compensate plant discrepancies and make plant and model closer (the action is known as disturbance rejection/cancellation). As a result, the model error will reveal only the residual plant-to-model discrepancies, due to imperfect compensation and unpredictable components.

Figure 2 shows the stabilizing feedback (2D green box) from model error to the disturbance signal entering the embedded model. In turn, the feedback dispatches the disturbance state to a white box, where after elaboration is subtracted from the current command and dispatched to model and plant, so as to implement the disturbance rejection. The controllable state is drawn in grey color since its use is not yet clear.

Figure 2 – The feedback from model error to disturbance.
Plant stabilization and the unrejectable discrepancy

From this standpoint, the model error becomes a key error signal of control design, implementation and maintenance, necessary but not sufficient to guarantee overall stability and performance. The fact is that model errors can be made bounded also in presence of unbounded plant measurements! In other terms, disturbance rejection is not sufficient for bounding plant measurements, since unrejected residuals, though small, may be progressively accumulated by plant internal energy into long-term drifts which push plant output to deviate from the target time profile (the output reference).

The remedy stems from a second EM function: to provide the command with the necessary and sufficient information for stabilizing the plant output close to the target profile, having assumed that disturbance rejection is in operation. The difference between a model variable and the reference signal to be copied will be referred as tracking error, which becomes the new error to make bounded and within a predefined thereshold. Since now measurements do not play any role, being replaced by  model state and output which, as already pointed out, can track unbounded measurements with a bounded error (model-based control law).

At first sight, the consequence of a model-based control law looks of benefit, since one may expect that model variables like model and reference output (also the reference is coming from a model) may track each other with a very small error, which becomes true under certain conditions. On the other hand, the effort of tuning the feedback design to the peculiarities of the plant-to-model discrepancies cannot be retained complete. In fact, not all the components of the model error should be saved into the disturbance state in view of their cancellation. The most dangerous are the effects of unmodelled dynamics such as delays, body deformations and interconnections, which though small, must be discriminated and blocked inside the model error itself not to spill into model variables. The relevant design effort, though delicate, will guarantee that the model output, once stabilized around the reference, will entrain the plant measurements to do the same, but with a larger error because of the unrejectable discrepancies.

Figure 3 shows the decomposition of the plant into model and two kinds of discrepancies: 1) the neglected dynamics, not to be rejected, command driven and adding to measurements, 2) output-to-input interconnections producing a disturbance signal to be rejected. Both discrepancies are usually affected by uncertainties here symbolized by clouds.

Figure 3 – Plant decomposition into model and discrepancies.

In summary, discrepancies, though partly unknown, must be carefully studied for decomposing them into rejectable and not. The plant command synthesized from EM variables (the white box in the above Figures) will cancel the rejectable discrepancies and stabilize the plant, though excited by small unrejected residuals. The output of the unmodelled plant dynamics, if excited by the command, will remain bounded and confined within the model error.

Coming back to the second EM function, the necessary information for the command synthesis cannot be restricted to output signals, but must include the whole controllable state, in other words, the minimal set of variables which the command has to change for forcing the model output to track arbitrary profiles. The relevant stabilization algorithm is a second feedback, driven by the tracking error, i.e., by the difference between controllable and reference state. The latter state is the result of a third feedback system (the reference generator) which, fed by the reference output, provides reference command and state. Injection of the stabilizing feedback and of the reference command into an EM which is only affected by the bounded residuals of the disturbance rejection, guarantees a bounded and close to zero tracking error, whichever be the output reference. As aready discussed, the command components of a model-based stabilization will be effective on the plant, too, if the effects of the unmodelled plant dynamics remain confined in the model error. Of course, model and plant tracking errors will be completely different. Let us consider the classical control error which is defined as the difference between output measurement and reference. Let us rewrite the error by adding and subtracting the model output, which means to add model error and model tracking error. If the latter is brought close to zero, the classical control error becomes fairly equal to the model error.

In summary, driven by  the embedded model,  we have identified three functions, which provide the command with the necessary and sufficient information for bounding plant measurements close to their reference signals. 1) The EM feedback bounds the model error, where the unrejectable discrepancies must be confined, and saves the rejectable components in the disturbance state. 2) The reference generator, briefly mentioned, is driven by control requirements, which are converted into reference command and state. 3) The control law is the algorithm which combines all the model-based information into a command to be dispatched to plant and model. In turn, the command is the combination (usually the addition) of three components: reference command, stabilizing feedback and disturbance rejection. The ensemble of embedded model and of the three functions is referred to as control unit.

Figure 4 shows the block-diagram of the control unit built arround the embedded model. Fault detection isolation and recovery functions as well as man-machine interfaces are not shown.

Figure 4 – The three main functions of a control unit around the EM.
Control unit design

The design of the control unit may be roughly split as follows. 1) Definition of the plant, together with commands (actuators), measurements (sensors) and control requirements. 2) Construction of a faithful model, enlightening environment interactions and uncertainty, to be run as a plant surrogate. 3) Design of the embedded model and definition of rejectable and unrejectable discrepancies. 4) Design of the control unit functions and test on the plant surrogate, before of the field test. The sequence may be iterated.

The embedded model as the cornerstone of the control unit guarantees that any control functions stems as the simplest as possible. Pluralitas non est ponenda sine necessitate.