János F. Bitó

Replacement of Lyapunov's direct method in Model Reference Adaptive Control with Robust Fixed Point Transformations

The “Model Reference Adaptive Control (MRAC)” is a popular and efficient controller that normally is designed by the use of Lyapunovs 2nd (“direct”) method guaranteeing global asymptotic stability. However, the use of Lyapunov function can entail a relatively complicated tuning that may have disadvantages whenever very fast applications are needed. In this paper an alternative background, the application of “Robust Fixed Point Transformations (RFPT)” working with local basin of attraction of convergence is recommended to simplify this task. As a potential application example the adaptive control of an “Electrostatic Micro-actuator (EμA)” is considered that contains considerable non-linearities. The conclusions of the paper are substantiated by simulation results.

Possible adaptive control by tangent hyperbolic fixed point transformations used for controlling the - 6 -type van der pol oscillator

In this paper a further step towards a novel approach to adaptive nonlinear control developed at Budapest Tech in the past few years is reported. Its main advantage in comparison with the complicated Lyapunov function based techniques is that its fundament is some simple geometric consideration allowing to formulate the control task as a Fixed Point Problem for the solution of which various Contractive Mappings can be created that generate Iterative Cauchy Sequences for Single Input - Single Output (SISO) systems. These sequences can converge to the fixed points that are the solutions of the control tasks. Recently alternative potential solutions were proposed and sketched by the use of special functions built up of the response function of the excited system under control. These functions have almost constant values apart from a finite region in which they have a wrinkle in the vicinity of the desired solution that is the proper fixed point of these functions. It was shown that at one of their sides these fixed points were repulsive, while at the opposite side they were attractive. It was shown, too, that at the repulsive side another, so called false fixed points were present that were globally attractive, with the exception of the basins of attraction of the proper ones. This structure seemed to be advantageous because no divergences could occur in the iterations, the convergence to the false values could easily be detected, and by using some ancillary tricks in the most of the cases the solutions could be kicked from the wrong fixed points into the basins of attraction of the proper ones. It was expected that via adding simple rules to the application of these transformations good adaptive control can be developed. However, due to certain specialties of these functions practical problems arose. In the present paper novel transformations are presented that seem to evade these difficulties. Their applicability is illustrated via simulations in the ad

Chaos formation and reduction in robust fixed point transformations based adaptive control

In the design of adaptive controllers for roughly modeled nonlinear dynamic plants the most popular prevalent fundamental mathematical tool is Lypunovs “direct” method. Though normally it guarantees global stability several controller performance parameters of practical engineering significance cannot directly be addressed in this manner. In general simulation investigations or GA-based parameter optimization is needed for refining the controller. A possible alternative of the Lyapunov function technique is the application of Robust Fixed Point Transformation (RFPT) that has only local region of convergence but directly addresses practical needs as error relaxation. In this paper the details of quitting the region of convergence and its consequences are investigated. In the control of a 2 Degree Of Freedom (DOF) paradigm it will be shown that though this process has chaotic features it does not has drastic consequences in the control quality. Furthermore, it also is shown that by a simple smoothing trick this chaos can be refined and reduced to a limited amplitude of chattering that much probably is tolerable in many practical applications.

A novel approach to the Model Reference Adaptive Control of MIMO systems

The “Model Reference Adaptive Control (MRAC)” is a popular approach from the early nineties to our days. Its basic idea is the application of proper feedback that makes the behavior of the controlled system identical to that of the “reference model” that normally is simple enough to control. The idea has many particular variants with the common feature that they are designed by the use of Lyapunovs 2nd (“direct”) method that normally applies a quadratic Lyapunov function constructed of the tracking error and further additional terms. Though this approach normally guarantees global asymptotic stability, its use can entail complicated tuning that may have disadvantages whenever very fast applications are needed. In this paper an alternative problem tackling, the application of “Robust Fixed Point Transformations (RFPT)” in the MRAC technique is recommended. This approach applies strongly saturated, multiplicative nonlinear terms causing a kind of “deformation” of the input of the available imprecise system model. Instead parameter tuning that is typical in the traditional MRAC it operates with a simple convergence guaranteed only within a local basin of attraction. This technique can well compensate the simultaneous consequences of modeling errors and external disturbances that normally can “fob” the more traditional, tuning based approaches. As a potential application paradigm the novel MRAC control of a “cart - beam - hamper” system is considered. The conclusions of the paper are illustrated by simulation results.

Comparative analysis of a traditional and a novel approach to Model Reference Adaptive Control

In this paper the operation of a recently introduced novel version of the popular “Model Reference Adaptive Controller (MRAC)” is compared with that of a simple version of its possible traditional implementations. The “traditional implementations” normally use Lyapunovs 2nd (“direct”) method for adaptive tuning of the controllers parameters. This method yields global asymptotic stability but its application technically is difficult. In the current control papers pages are occupied with sophisticated estimations to guarantee the non-positive time-derivative of the Lyapunov function. The here proposed novel approach avoids the use of any Lyapunov function, it works with simple iterations that yield Cauchy sequences that converge to the solution of the control problem. Its advantages are its simplicity and precision, its weak side is the local basin of attraction of the convergence applied. However, simulations testify that for practical applications satisfactory “width” can be obtained for the region of convergence. In this paper a 3 Degree Of Freedom (DOF) paradigm, the cart+beam+hamper system is used in numerical simulations to demonstrate the advantages of the novel approach in comparison with a traditional MRAC controller. It is worth noting that the novel method can be completed with additional tuning for guaranteeing its convergence.

Matrix inversion-free quasi-differential approach in solving the inverse kinematic task

The traditionally viable way for the solution of the inverse kinematic task for general open kinematic structure is the differential approach in which the Jacobian of the robot arm usually is inverted by the use of some generalized matrix inverse. These approaches suffer from the kinematic singularities nearby which these inverses are ill-conditioned and behave in a very inconvenient way. For dealing with the singularities complementary tricks used to be introduced that so deform the original task that the obtained solution behaves conveniently though it cannot solve the original task that does not have exact solution. In this paper an alternative approach is suggested that requires only the computation of the Jacobian but does not need the calculation of its generalized inverse. Instead of that it applies an iterative sequence that has nice convergence properties. The method automatically handles the problem of the singularities, ambiguity, redundancies, and non-existing exact solutions without the application of any complementary trick or artificial parameter. Its operation is demonstrated for a simple 2 Degree of Freedom (DoF) arm, and for an 8 DoF arm, that is an irregular extension of a 6 DoF PUMA-type robot.

On the effects of strong asymmetries on the adaptive controllers based on Robust Fixed Point Transformations

For replacing Lyapunovs sophisticated “2nd Method” in the design of adaptive controllers a novel approach based on Robust Fixed Point Transformations (RFPT) was proposed that directly concentrates on the designers intent instead of forcing global stability. It guarantees convergence only in a bounded basin while iteratively generating the sequence of the appropriate control signals. In the initial phase of this iterative learning considerable fluctuation may occur in the control signal that otherwise may be limited due to phenomenological reasons. While in mechanical systems positive or negative force or torque components can be allowed, in controlling chemical reactions negative ingress rates of pure reactants into a stirring tank reactor phenomenologically cannot be realized. While velocity components may have well interpreted positive or negative values, negative concentrations physically cannot make sense. On this reason the mathematical models of chemical reactions normally containing the products of various powers of the concentrations must be completed with truncation-type nonlinearities that introduce strong asymmetric nonlinearities. In this paper the effects of these phenomena are investigated via computer simulations in the adaptive control of a Classical Mechanical and a chemical system. It was found that in spite of these limitations the adaptive controller can still work at least in certain segments of the whole control section.

Adaptive Tackling of the Swinging Problem for a 2 DOF Crane – Payload System

The control of a crane carrying its payload by an elastic string corresponds to a task in which precise, indirect control of a subsystem dynamically coupled to a directly controllable subsystem is needed. This task is interesting since the coupled degree of freedom has little damping and it is apt to keep swinging accordingly. The traditional approaches apply the input shaping technology to assist the human operator responsible for the manipulation task. In the present paper a novel adaptive approach applying fixed point transformations based iterations having local basin of attraction is proposed to simultaneously tackle the problems originating from the imprecise dynamic model available for the system to be controlled and the swinging problem, too. The most important phenomenological properties of this approach are also discussed. The control considers the 4th time-derivative of the trajectory of the payload. The operation of the proposed control is illustrated via simulation results.

On the simulation of RFPT-based adaptive control of systems of 4 th order response

As an alternative of Lyapunov functions based design methods the “Robust Fixed Point Transformations (RFPT)”-based adaptive control design was developed in the past years. The traditional approaches emphasize the global stability of the controlled phenomena while leaving the details of the trajectory tracking develop as a not very clear consequence of the control settings the novel design directly concentrates on the observable response of the controlled system therefore it can concentrate on the tracking details as a primary design intent. Whenever a Classical Mechanical system that normally produces 2nd order response (i.e. acceleration) is forced through an elastic component its immediate response becomes 4th order one. Practical observation of the 4th order derivatives of a variable may suffer from measurement noises. Furthermore, when in simulation studies the higher order derivatives are numerically integrated and later numerically differentiated to provide the appropriate feedback signals the non-smooth jumps in the numerical integrator can destroy the simulation results. By the use of a simple 4th order model in this paper it is shown that the chained use of the built-in differentiators of the simulation package SCILAB is inappropriate for simulation purposes. It is also shown that by the use of a simple 4th order polynomial differentiator this problem can be solved. This statement is substantiated by simulation results.

Robust Fixed Point Transformations in the Model Reference Adaptive Control of a Three DoF Aeroelastic Wing

The Model Reference Adaptive Controllers (MRAC) of dynamic systems have the purpose of simulating the dynamics of a reference system for an external control loop while guaranteeing precise tracking of a prescribed nominal trajectory. Such controllers traditionally are designed by the use of some Lyapunov function that can guarantee global and sometimes asymptotic stability but pays only little attention to the primary design intent, has a great number of arbitrary control parameters, and also is a complicated technique. The Robust Fixed Point Transformations (RFPT) were recently introduced as substitutes of Lyapunov’s technique in the design of adaptive controllers including MRACs, too. Though this technique guarantees only stability (neither global nor asymptotic), it works with a very limited number of control parameters, directly concentrates on the details of tracking error relaxation, and it is very easily can be designed. In the present paper this novel technique is applied for the MRAC control of a 3 Degrees-of-Freedom (DoF) aeroelastic wing model that is an underactuated system the model-based control of which attracted much attention in the past decades. To exemplify the efficiency of the method via simulations it is applied for PI and PID-type prescribed error relaxation for a reference model the parameters of which considerably differ from that of the actual system.