Model Reference Adaptive Control (MRAC)

MRAS

The Model-Reference Adaptive system (MRAS) was originally proposed to solve a problem in which the performance specifications are given in terms of a reference model. This model tells how the process output ideally should respond to the command signal. The Adaptive Controller has two loops. The inner loop consists of the process and an ordinary feedback controller. The outer loop adjusts the controller parameters in such a way that the error, which is the difference between the process output y and model output ym is small. The MRAS was originally introduced for flight control.In this case, the reference model describes the desired response of the aircraft to joystick motions. Model reference adaptive systems were originally derived for deterministic continuous systems.

The mechanism for adjusting the parameters in a model reference adaptive system can be obtained in two ways: Using a gradient method and Appling a stability theory

MIT Rule

MIT Rule for adaptation of feed forward gain

Let the process be described by the transfer function kG(s),

where G(s) is known and k is an unknown parameter.

The design problem is to find a feedforward controller that gives a system

with transfer function Gm(s) = k0 G(s)

where k0 is a given constant.

With the feed forward controller u = uc ,

where u is the control signal and uc the command signal,

the transfer function from the command signal to output becomes

kG(s)

MIT Rule based MRAS for first order system

Determination of adaptation gain

Problem:

Design of MRAS using Lyapunov theory

The drawback of MIT rule based MRAS design is that there is no guarantee that the resulting closed loop system will be stable. To overcome this difficulty, the Lyapunov theory based MRAS can be designed, which ensures that the resulting closed loop system is stable.

Lyapunov theory based MRAS for first order system

Relation between MRAS and STR

MRAS and STR were regarded as two quite different approaches to adaptive control. But later it was proved that they are actually very closely related. In particular the direct self tuning regulator with cancellation of process zeros can be interpreted as a MRAS.

Gain scheduling

Gain scheduling is an adaptive control strategy, where the gain of the system is determined and based on its value the controller parameters are changed.In many cases, it is possible to find measurable variables that correlate well with changes in process dynamics. These variables can then be used to change the controller parameters. This approach is called gain scheduling because the scheme was originally used to measure the gain and then change, that is, schedule the controller to compensate for changes in the process gain.

The system can be viewed as having two loops. There is an inner loop composed of the process and the controller Outer loop contains components that adjust the controller parameters on the basis of the operating conditions. Gain scheduling can be regarded as mapping from process parameters to controller parameters.It can be implemented as a function or a table lookup.The concept of gain scheduling originated in connection with the development of flight control systems.In this application, the Mach number and the altitude are measured by air data sensors and used as scheduling variables. This was used, for instance, in the X-15.In process control the production rate can be often chosen as a scheduling variable, since time constants and time delays are often inversely proportional to production rate. Gain scheduling is thus a very useful technique for reducing the effects of parameter variations.

Advantages:

Parameters can be changed quickly in response to changes in plant dynamicsThis strategy is very easy to apply if the plant dynamics depends in a well knownfashion on a relatively few easily measurable variables

Drawbacks:

It is an open-loop adaptation scheme, with no real learning or intelligenceThe design required for its implementation is enormous.

Design of gain scheduling controllers

The key issue in the design of gain scheduling controllers is the determination of variables that can be used as scheduling variables. One criterion for selection of the scheduling variable is that these auxiliary variables must reflect the operating conditions of the plant. Ideally there should be simple expressions for how the controller parameters relate to the scheduling variables.It is thus necessary to have a good insight into the dynamics of the process if gain scheduling is to be used.

Design of gain scheduling controllers can be carried out by one of the following techniques:

Design of gain scheduling controllers can be carried out by one of the following techniques: Gain scheduling based on measurement of auxiliary variables Time scaling based on the production rate and Nonlinear transformationsIt should be noted here that by linearizing of nonlinear actuators, we get a very improved performance, but this should not be regarded as gain scheduling because, gain scheduling should consist of a measurement of variable related to the operating condition of the process.

Gain scheduling based on measurement of auxiliary variables

From the above expression, it is clear that it is sufficient to make the gain proportional to the cross section of the tank.

So it can be seen that we have established a relation between the gain (auxiliary variable) and the area of the tank (variable relating operating condition of process).

The above example illustrates that it is sufficient to measure one or two variables in the process and use them as scheduling variables.

But often it is not easy to determine the controller parameters as a function of measured variables.

The design of controller must then be redone for different working points of the process. Some care must also be taken if the measured signals are

noisy. They should be filtered properly before they are used as scheduling variables.

Time scaling based on the production rate

Consider concentration control for a fluid that flows through a pipe, with no mixing, and through a tank, with perfect mixing. A schematic diagram of the process is shown

Notice that the sampled data model has only one parameter, a, that does not depend on q.

A constant gain controller can easily be designed for the sampled data system.

The gain scheduling is realized simply by having a controller with constant parameters, in which the sampling rate is inversely proportional to the flow rate.

This will give the same response independent of the flow, in looking at the sampling instants, but the transients will be scaled in time.

To implement the gain scheduling controller for the case where flow is varying, it is necessary to measure not only the concentration but also the flow.

Errors in flow measurement will result in jitter in the sampling period. To avoid this, it is necessary to filter the flow measurement.

Nonlinear Transformations

It is of great interest to find transformations such that the transformed system is linear and independent of the operating conditions.

In case of concentration control problem, time scaling was used to make the model independent of flow.

i.e ts = (Vd/q)t was used to make the model independent of flow.

All processes associated with material flows like rolling mills, band transporters, flows in pipes, etc. have this property.

The design procedure making use of nonlinear transformation is as described below:

The system is first transformed into a fixed linear system. The transformation is usually nonlinear and depends on the states of the process.

A controller is then designed for the transformed model, and the control signals of the model are retransformed into the original control signals.

The result is a special type of nonlinear controller, which can be interpreted as a gain scheduling controller.

The nonlinear transformation u = g1(x,v) and z = g2(x) makes the relation between v and z linear. A state feedback controller from z is then computed that gives v. The control signal v is then transformed into the original signal u. Feedback linearization requires good knowledge about the nonlinearities of the process.The method of nonlinear transformations is described with the following examples.

Example: Nonlinear transformation of a Pendulum

Example: Nonlinear transformation of a second order system

Auto-Tuning of PID regulators

The commonly used techniques for auto tuning of PID regulators are:Open loop response method (transient response method)Closed loop response methodTuning by use of external equipment Use of expert systems

Transient response methods

Methods based on Relay Feedback: (Ultimate cycle method)

The main drawback of the transient response method is that it is sensitive to disturbance because it relies on open loop experiments. The relay based methods avoid this difficulty because the required experiments are performed in closed loop.The key idea here is the fact that many processes have limit cycle oscillations under relay feedback.

Here the idea is to determine the critical gain and critical period first and then determine the controller parameters.

The critical period (Tu) is the period of oscillations when relay feedback is applied.

The gain at which the oscillations just begin is termed as critical period (Ku).

Ziegler and Nichols have devised a simple method for determining the parameters of the controller based on the values of critical gain and critical period

The controller parameters are given by following

Systems with better
damping can be obtained by relay auto tuner

When tuning is demanded, the switch is set to T, which means that relay feedback is activatedand the PID regulator is disconnected. When a stable limit cycle is established, the PIDparameters are computed, and the PID controller is then connected to the process. Naturally themethod will not work for all systems. First, there will not be unique limit cycle oscillations for anarbitrary transfer function. Second, PID control is not appropriate for all processes. This type oftuning works well for a large class of systems encountered in process control.