CONCEPT OF MODEL-FREE ADAPTIVE CONTROL
The formal definition of Model-Free Adaptive Control (MFAC) is given in the following. The issues related to control methodology and practice are also discussed. It is recommended that the user take the time to read this section so that they can see the creative yet logical thinking processes underneath the MFAC technology.
Definition of MFAC
A Model-Free Adaptive Control (MFAC) system shall be defined to have at least the following properties or features:
(1) No precise quantitative knowledge of the plant is available;
(2) No plant identification mechanism or identifier is included in the system;
(3) No controller design for a specific plant is needed;
(4) No complicated manual tuning of controller parameters is required; and
(5) Closed-loop system stability analysis and criteria should be available to guarantee the system stability.
Plant Knowledge Issue
Most advanced control techniques for designing control systems are based on a good understanding of the plant and its environment. Laplace transfer functions or dynamic differential equations are usually used to represent the plant dynamics. In many process control applications, however, the plant may be too complex and basic physical processes in it are not well understood. The quantitative knowledge of the plant is then not available. This is what is usually called a "black box" problem that we will deal with.
It is necessary to clarify the following points:
(1) There is neither intention nor claim in the theory of Model-Free Adaptive Control to discard the usefulness of well established control theories and methods. In fact, the development of MFAC is based on the solid foundation of control theory;
(2) The most suitable situation to use a model-free adaptive controller is when we face a black box - the unknown plant. In later chapters, we will show our approach to deal with this fundamental problem;
(3) In many cases, we may have some uncertain knowledge of the plant but are not sure if the knowledge will benefit our system design or mislead us. In such cases, we would rather take a more conservative approach by assuming the plant is unknown. Then, we will still use the model-free adaptive control approach.
(4) If quantitative knowledge of the plant becomes available, there is no reason not to take advantage of it. However, designing a controller based on the understanding of the plant is what many traditional control methods do, but that is not the topic for this research;
(5) A model-free adaptive controller shall be useful in the cases when the plant is known, semi-unknown, or unknown. In other words, the box could be white, gray, or black.
Plant Identification Issue
In most traditional adaptive control methods, if the plant under study is too complex or the basic physical processes in it are not fully understood, an identification mechanism is usually required in the adaptive system to obtain the plant dynamics on-line or off-line.
As discussed in Chapter 1, this contributes to a number of fundamental problems such as the headache of off-line training that might be required, the tradeoff between the persistent excitation of signals for correct identification and the steady system response for control performance, the assumption of the plant structure, the model convergence and system stability issues in real applications, etc. One of the main ideas in MFAC is to stay away from these fundamental problems by not using any identification mechanism. We will see in later chapters what kind of "freedom" an MFAC has acquired after the burden of identification has become a non-issue.
Controller Design Issue
One of the main reasons why PID is still so popular in process control is that people do not need any complicated controller design procedures in order to use it. Designing an advanced controller is not a task that an average control engineer or technician know how to do.
Therefore, to make a model-free adaptive controller practically useful and acceptable, we have to greatly simplify or even eliminate the controller design procedures that it may require. This is one of the major differences between a model-free adaptive controller and a traditional adaptive controller. In later chapters, we will see how to startup a new model-free adaptive control loop in practical applications.
Controller Parameter Tuning Issue
Requiring basically no manual tuning for controller parameters is a natural feature for an adaptive controller to have. It is of course a feature that a model-free adaptive controller should have.
However, there are two points we need to make clear:
(1) A traditional adaptive controller is usually tailored to fit the given or obtained plant dynamics and control tasks. Then, controller parameter tuning should not be required since the controller should perform in the range that it is designed to work. For a model-free adaptive controller, there is no controller design required. The tradeoff is then that a few parameters may need to be tuned based on the qualitative plant knowledge understood. We will discuss this in a great detail in later chapters.
(2) From an engineering point of view, it is usually a good idea to leave some kind of flexibility to the controller user (an operator) because the controller may need to be adjusted to perform as desired by the user. In Chapter 5, 6, and 7, we will show that the parameters of a model-free adaptive controller need to be properly tuned.
Controller Parameter Tuning Issue
The closed-loop control system stability analysis is always a very important issue for a controller to be practically useful. Once a set of system stability criteria is available, it can guide the user to study the system stability and even decide if the controller can or cannot be used in the system.
In traditional adaptive control, the stability of the overall closed system is related to the plant, the controller, and the identifier in the following matter: (1) the stability of the plant is usually assumed or we say the plant is open-loop stable; (2) the stability of the control loop must be guaranteed by the convergence of the identifier; but (3) the convergence of the identifier is dependent on the stability and persistent excitation of signals originating from the control loop. This is a circular argument that it is difficult to resolve.
Therefore, many practically useful adaptive controllers are running in a semi-on-line and semi-off-line fashion. That is, although the learning algorithm could be a recursive one that can be implemented in an on-line fashion, it is not turned on all the time to avoid the poor identification results caused by the smooth control situation which does not generate sufficient excitation signals that good identification requires.
Since the model-free adaptive controller does not have an identification mechanism, the stability for the overall closed-loop system is much relaxed compared to the traditional adaptive control. It is easy to assume the stability proofs for an MFAC should become simpler as well. However, in this study, an artificial neural network (ANN) is used as the adaptive controller.
In contrast to the large number of ANN or ANN based control system architectures, few research results on the problem of stability of dynamic systems with ANN controller have been reported.
The difficulties in stability analysis of ANN based control systems arise mainly from the high degree of nonlinearity in ANN architecture that leads to nonlinear characteristics of the overall control system, even though the dynamics of a plant may be linear. Therefore, the development of system stability criteria for ANN based model free adaptive control is one of the main challenges in the study of MFAC.
In the advanced control field, various definitions and concepts of stability have been proposed. A classical definition is the stability in the sense of Lyapunov.
For practical reasons, however, a Lyapunov-type definition of stability is not sufficient for adaptive systems. As we know, this definition is a local property. It says that when the state starts sufficiently close to an equilibrium point, the Lyapunov stability will guarantee that the trajectories remain arbitrarily close to the equilibrium point. In adaptive systems, we do not have any control on how close the initial conditions are to the equilibrium values. In addition, in most circumstances, there are no fixed equilibrium points due to the changes in system parameters, etc.
For the reasons mentioned above, we will use the bounded-input bound-output (BIBO) stability concept. That is, for any bounded setpoint, or bounded disturbance, the output remains bounded. More generally, the Lp stability concept will be used.