A lot of dynamic plants to be controlled have constant or slowly-varying uncertain pa- rameters. For those plants there are various examples like fire-fighting aircraft, power systems and underwater vehicles. To control such systems, often, the conventional con- trollers can not achieve the desired performance and can not stabilize suchlike systems. Therefore, adaptive controllers are applied to control systems with uncertainties which provide techniques for automatic adjustment of the controller (estimate uncertain plant
parameters) in real-time, in order to fulfill the aimed requirements when the parameters of the dynamic system are unknown and/or changing in time [62].
The design of the adaptive controller relies mainly on the plant dynamics to be con- trolled. For linear or linearized nonlinear plant models, many techniques are developed to control those plants under uncertainties in their parameters, among them, are:
• Gain Scheduling Control,
• Self-Tuning Regulator (STR) and
• Model Reference Adaptive Control (MRAC).
Gain Scheduling was developed, originally, for trajectory control of aircrafts. The non- linear plant is linearized at certain operating points which cover the whole desired oper- ation range. At each point, a linear feedback controller with constant gains is designed to achieve the control requirements at the considered point. The global controller of the nonlinear plant over the regarded range is then an interpolation or a scheduling of the linear controllers at the chosen operating points. The main disadvantage of this technique is that a rapid changing in the controller gain, may lead to instability in the closed-loop system. In addition to its simplicity, for many applications in which the gains are changing slowly, this technique is a convenient control approach. The Gain Scheduling Control is not an actual adaptive control, but it is a kind of open-loop adap- tive control, where the controller gain is adapted depending on auxiliary measurements and off-line look-up tables [62, 63, 64, 65].
An on-line adaptation approach (Self-Tuning Regulator (STR)) is introduced in [62, 63]. This Regulator consists of a controller, designed based on pole placement, PID, LQR (Linear Quadratic Regulator), . . . , and an estimator, which could be designed with many techniques, the most common one is the least squares method.
In STRs, the parameters of the controller are designed based on the estimation of the plant parameters, by replacing the real values of the plant parameters with the estimated parameters, which is known as the certainty equivalent principle. This controller is able to tune its own parameters, therefore is called self-tuning. Beside the flexibility with designing the controller and the estimator, the applicability of this controller to control of minimum and non-minimum phase systems is actually an obvious advantage. On the downside, the analyzing of the (STR) is not simple.
MRAC consists of an adaptation law, a controller and a reference model. The idea of the MRAC, indeed, is based on the canceling of the zeros of the plant transfer function and replacing them with those of the reference model by using a feedback controller. This implies that the plant must be minimum phase (stable zeros) because the cancellation of unstable zeros leads to unbounded signals. The MRAC system can be constructed in two different ways, the MRAC-series high-gain scheme and the MRAC-parallel scheme. The most used one is the parallel structure because it has more benefits in comparison with the MRAC-series high-gain which despite its simplicity has some problems such as oscillation and saturation due to high-gain. In the MRAC-parallel scheme one can distinguish two loops: A regulator loop, which involves the unknown plant (but the structure is known) and the ordinary controller, and an adaptation loop that adjusts the parameters of the controller using a certain adaptation mechanism. The goal of the adaptation loop is to estimate the controller parameters such that the error between the output of the plant and the output of the reference model is zero.
A combination of robust control techniques, which deal with the unmodeled uncertain- ties and/or disturbances, and adaptive control techniques, which handle the structural uncertainties, gives a new field of work: robust adaptive control [65]. In this kind of combination, the robust controller would be enhanced by using an adaptive controller which increases the operation range of the closed loop system. On the other side, the robust controller may enhance the performance of the adaptive one as well [62]. An example for this combination is the Adaptive Sliding Mode Control (ASMC) [66] which can deal with a wide range of perturbed linear or nonlinear plants with uncertainties. The above mentioned approach of adaptive control, can be extended to cover many classes of nonlinear systems.
Remark 5.1.1 In the approach of Self-Tuning Regulation (STR) and the Model Ref- erence Control (MRAC) the plant parameters can be estimated and then the controller parameters are computed. Such a scheme is called usually indirect (explicit) adaptive control, because, one must translate the estimated parameters of the plant into controller tuning parameters. In other approaches it is possible to eliminate this intermediate step of the computation by reparameterizing the plant dynamics using the controller param- eters which are also unknown and to be adjusted. Therefore, this kind of adaptation is called a direct (implicit) adaptive control.
Remark 5.1.2 There are related techniques which deal with linear and nonlinear plants such as the Extremum Seeking method, which is presented in [67].
Remark 5.1.3 It is worth noticing, that the important difference between the STR and MRAC techniques lies in regarding the parameter estimation. The parameter estimation of the plant in STR can be understood, actually, as the procedure of finding (estimation) a set of parameters that matches the available input-output data from a plant. But on the other hand, this is unlike the parameter adaptation in MRAC systems, where the parameters in MRAC are adjusted so that the tracking errors converge to zero.