Krisztián Kósi

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.

Chaos patterns in a 3 Degree of Freedom control with Robust Fixed Point Transformation

The controllers designed by Lypunovs 2nd method normally have global stability but do not concentrate on the details of the primary intent of the designer: the details of the tracking error relaxation. They have a huge number of arbitrary adaptive control parameters that -from the engineering point of view- are hard to design for prescribed detailed behavior of the controlled system. In the past few years a new method that concentrates on the design intent, easy to produce and has only a few adaptive control parameters was invented: the Robust Fixed Point Transformation (RFPT). According to ample simulations it is seems to be a good choice, but it has only local stability yet. Though its present form can be satisfactory for solving most of the cases, sometimes ancillary methods are needed for maintaining or restoring its convergence. It was recently discovered for one and two degree of freedom systems that when the controller quits the region of stability it still guarantees good tracking at the price of huge chattering that also was reduced and stopped. In this paper it will be shown that in the adaptive control of the 3Degree Of Freedom (DOF) system similar phenomena happen and the controller can be stabilized by similar methods.

Adaptive controllability of the brusselator model with input coupling

For replacing Lyapunovs ingenious but complicated “2nd method” in designing adaptive controllers for nonlinear systems the use of “Robust Fixed Point Transformations (RFPT)” was extensively studied in the past few years mainly for “Classical Mechanical Systems (CMS)”. In spite of the strongly nonlinear coupling that is typical in the Euler-Lagrange equations of motion CMS are simple in the sense that their state variables (i.e. the generalized coordinates and their time-derivatives), driving force or torque components, as well as the tracking error signals are physically well interpreted both in the positive and the negative domains. Furthermore, the time-derivatives of the control forces do not occur in the equations of motion. Therefore simple PIDtype controllers with great feedback gains as well as RFPT-based adaptive ones of smaller feedback gains but of the aptitude for introducing strong nonlinear transient fluctuations after their switching on can successfully deal with such systems. In contrast to CMS “Chemical Systems (CS)”, besides their multiplication and power-type terms in the reaction equations also have further strong nonlinearities due to phenomenological restrictions. Neither negative concentrations, nor negative ingress rates of pure reagents can occur in the case of “Continuous Stirring Tank Reactors (CSTR)”. Such effects were recently investigated by assuming the ingress of very concentrated reagents in the control that do not considerably dilute the other reagents in the CTRS. In this paper the far reaching consequences of this mutual diluting effect are studied. It is shown that the RFPT-based adaptive controller still can be useful but the control strategy has to take far more complicated form.

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.

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.

Increased cycle time achieved by fractional derivatives in the adaptive control of the Brusselator model

In the control of chemical reactions the goal is to stabilize the control and take care of the design that has to integrate engineering aspects, as restrictions concerning the control signals, and the phenomenological limits that are not necessarily expressed by the reaction kinetic equations (e.g. no negative concentrations can be physically interpreted, and the reactants at the side of ingress cannot be purely extracted from the stirring tank reactor). In this paper the Robust Fixed Point Transformation (RFPT)-based adaptive approach was chosen for the control of an approximately modeled Brusselator reaction. The main reason of that was the fact that this methodology concentrates on the primary design goals as precise realization of the prescribed concentrations while the more conventional design methods that apply some Lyapunov function mainly concentrate on guaranteeing global stability without providing quasi-optimal solutions for the primary goals. Though the RFPT-based design has only local stability, its region of stability may be quite satisfactory for several practical applications. For controlling chemical reactions Continues Stirring Tank Reactors (CSTR) are widely used engines. The Busselator model will be represented in a CSTR in the simulations. In the present example two different reactants can be injected into the tank and the mixture is taken out in a single outlet. The necessary sampling frequency is a practically important design factor. It is shown that by the use of fractional order derivatives in the prescribed error relaxation considerably increases the necessary sampling time so it decreases the sampling frequency. This statement is substantiated by simulation results.

Iterative adaptive control of a three degrees-of-freedom aeroelastic wing model

The aeroelastic wing model is an underactuated 3 Degrees-of-Freedom (DoF) system in which only a single control signal can be applied. Its model-based control attracted much attention recently. In the present contribution the effects of 20% modeling error in two significant system parameters are investigated in PI and P-type controllers that can be made adaptive by the application of Robust Fixed Point Transformations (RFPT). It is shown that by the use of this simple adaptive technique operating with altogether 4 control parameters precise control of the trailing-edge surface deflection is possible. This statement is substantiated by simulation results.

Nonlinear order-reduced adaptive controller for a DC motor driven electric cart

The precise control of a voltage-controlled, DC motor driven electric cart would be a 3rd order task since Classical Mechanics relates the acceleration of the cart to driving torques, while these torques are proportional to the current that cannot be modified abruptly. Abrupt jumps in the control voltage cause abrupt change only in the time-derivative of the motor current. Therefore a 2nd order controller can only approximate the operation of a 3rd order one. Modeling imprecisions further complicate the control task. Furthermore, if the cart has two independently driven main wheels and a supporting caster, no exact trajectory tracking can be prescribed simultaneously for the location of the cart in the (x, y) plane and its rotational orientation (θ) since for three prescribed quantities only two control signals are available. For the tracking error of these components, some kinematically formulated compromise must be applied. The aim of the present paper is to show that the simultaneous problems caused by the order reduction, the modeling errors and the necessary kinematic compromises can be elegantly solved by the adaptive controllers designed on the basis of Robust Fixed Point Transformations (RFPT). This statement is substantiated by simulation results.

Novel method for quadcopter controlling using nonlinear adaptive control based on robust fixed point transformation phenomena

In the past few years quadcopters have become an integral part of life. There are many applications for these objects including entertaining purpose, agricultural monitoring, or gathering information from places what humans can not reach. An interesting problem is controlling these vehicles from remote locations in order to follow a desired trajectory. While many solutions exist to this task the vast majority of them heavily rely on linear control methods which are susceptible to parameter uncertainties or outer disturbances, which affect the tracking capability of the quadcopter adversely. In this paper a novel nonlinear approach, called Robust Fixed Point Transformation based adaptive control (RFPT) is presented which can provide remedy to the above mentioned disturbances.