Block approach to analysis and design of the invariant nonlinear tracking systems

Consideration was given to the nonlinear multiple-input multiple-output systems under the action of unmatched exogenous perturbations. For the general systems where the conditions for bounded problem of autonomous control are not met, formalized were the organization principles and existence conditions for the affine block input–output form underlying the block design of the basic law of open-plus-closed-loop control providing invariance of the output variables with respect to the exogenous perturbations. The constructions are augmented by the derivatives of the exogenous and control actions, but in distinction from the existing approaches their generating dynamic models are not included. To realize the basic law of control, used is an observer of the mixed phase and exogenous variables based on a virtual system with respect to the tracking errors with closed local links. An original decomposition procedure to design correcting observer actions was presented as a linear saturation functions estimating the mixed variables with the desired precision.     

Adaptive suboptimal robust control of the first-order plant

For a linear first-order plant, consideration was given to minimization of the deviation of its output from the specified value in an uncertain environment. Both the parameters of the plant itself and the upper bounds of the external disturbances and operator perturbations in output and control were assumed to be unknown. The suboptimal adaptive control was constructed on the basis of an approximate solution of the problem of optimal identification from the measurement data where the performance of the problem of control was used as an identification criterion.     

Hierarchical Design of Sigmoidal Generalized Moments of Manipulator under Uncertainty

For the orientation control system of the manipulator’s end effector with electrical actuators, we develop a decomposition pr ocedure of feedback law design to track given trajectories in the end effector’s coordinate system. Owing to the S-shaped smooth sigma-functions used as the local feedback laws and corrections of the state observer, the tracking system is invariant with a given accuracy with respect to existing uncertainties under constraints imposed on the variables of the mechanical subsystem. The suggested approach does not involve the solution of the inverse kinematics and dynamics problems and also relaxes the requirements to the volume of a priori information about the plant and external perturbations.     

Cascaded Design of the State Observer for Nonlinear Systems in the Presence of External Perturbations

Applicability of the block approach to the problem of observation of nonlinear systems in the presence of uncontrollable external perturbations was demonstrated. A block-observable form of the general nonlinear systems was obtained with regard for perturbations. It underlies two types of cascaded procedures for designing the state observers in the class of systems with separable motions: (i) with discontinuous controls and (ii) with high-gain feedbacks. The first procedure relies on the method of equivalent control to restore in a finite time the nonmeasurable components of the state vector and also perturbations. The second procedure estimates the nonmeasurable components of the state vector in the prelimit situation with a given accuracy and retention of the decomposition of the design procedure.     

Robust output‐based controller design for enlarging the region of attraction of input saturated linear systems

The aim of this study was to design a constrained robust output feedback control strategy for stabilizing linear systems affected by uncertainties and output perturbations. The input is constrained by a saturation function. A classical Luenberger observer reconstructed the states from output observations. The state feedback control employed the estimated states given by the observer. This work proposed a Barrier Lyapunov Function (BLF) to manage the saturation problem in the control input. Moreover, an extended version of the attractive ellipsoid method (AEM) characterized the zone of convergence due the presence of perturbations in the output. A convex optimization procedure, formulated as a set of matrix inequalities, yielded the control parameters and the region of attraction for the close‐loop system as well as a minimal ultimately bounded set for the system trajectories. Numerical simulations supported the theoretical results formulated in this study.     

Sigma function in observer design for states and perturbations

For nonlinear systems operating under uncertainty, this paper involves the principle of motion separation to design a state observer with nonlinear corrections in the form of sigma functions. For the systems representable in the regular form with respect to the external perturbations, the above approach yields the current estimates of the unmeasurable state variables and external perturbations without extending the dynamic order of the observer by a model that simulates the action of the external perturbations. The developed algorithms are applied in the control system of an asynchronous drive with an incomplete set of measuring devices.     

Algorithm to control linear plants with measurable quantized output

Consideration was given to the control of linear plants under external perturbations and measurement of the quantized plant output. The “consecutive compensator” method was used to design the controller. The obtained algorithm tracks the quantized plant output with respect to the reference signal with precision depending on the quantization step. The simulations illustrate the efficiency of the proposed scheme.     

Robust control of multidimensional nonstationary linear plants

Solved was the problem was of constructing a robust control system with linear nonstationary multidimensional control plant compensating the parametric and external bounded perturbations to within δ if the derivatives of the output vector are not measured and fully if the derivatives are measured.     

Design of adaptive block backstepping controllers for uncertain nonlinear dynamic systems with n blocks

Based on the Lyapunov stability theorem, a design methodology of adaptive block backstepping control is proposed in this article for a class of multi-input systems with matched and mismatched perturbations to solve regulation problems. The systems to be controlled contain n blocks' dynamic equations, hence n???1 virtual input controllers are firstly designed so that the state variables of first n???1 blocks are asymptotically stable if each virtual control input is equal to the state variable of next block. Then the control input is designed in the last nth block to ensure asymptotic stability for the whole state variables even if the perturbations exist. In addition, adaptive mechanisms are embedded in each virtual input function and control input, so that the upper bound of perturbations is not required to be known beforehand. Finally, a numerical example is given for demonstrating the feasibility of the proposed control scheme.     

A physics-based approach to control systems design using compensation of controlled plant dynamics and perturbations

This paper introduces a new physical considerations-based approach to the development of control algorithms for dynamic plants, which involves a compensation principle as follows. Being external with respect to a controlled plant, control actions (compensating signals) are applied with the opposite sign to the corresponding variables of the inverse mathematical model of the plant. Regularization of the resulting system is performed by incorporating etalon filters in compensation loops. The described method is used to solve a control problem for the linear stable multidimensional plants with additive external influences and compensation of the perturbations affecting the output variables.     

Design of suboptimal robust controllers under unknown weights of perturbations

In the classical formulations of the problems of design of the robust optimal controllers, the equations of the nominal plant and the weights of the permissible perturbations are assumed to be known. In the present paper, consideration was given to a nonclassical formulation of the design problem where the weights of perturbations are assumed to be unknown and subject to estimation from the measurement data. The multivariable discrete-time controlled plant was described by a given transfer matrix with perturbations in the irreducible factors and a bounded external perturbation. The problem of determination with a predefined accuracy of the perturbation weights that are best coordinated with the measurements and of their corresponding suboptimal controller was solved.     

Design of block backstepping controllers for a class of perturbed multiple inputs and state-delayed systems in semi-strict-feedback form

Based on the Lyapunov stability theorem, an adaptive block backstepping control scheme is proposed in this paper for a class of multi-input systems with mismatched multiple state-delayed perturbations to solve regulation problems. The traditional backstepping control method is modified so that it can be directly applied to systems in block semi-strict-feedback form. The terms in the dynamic equations which do not satisfy the block strict-feedback form are accumulated in the last design step and are suppressed effectively by the adaptive gains, so that the property of asymptotic stability is achieved. Adaptive mechanisms are employed in each of the virtual input controllers as well as the robust controller, hence the least upper bounds of perturbations are not required to be known in advance. A numerical example and a practical application are also given for demonstrating the feasibility of the proposed control scheme.     

Block Decoupling by Precompensation Revisited

The block decoupling problem by admissible dynamic precompensation for LTI systems is considered. Admissibility refers to the preservation of the class of controlled output trajectories, i.e. functional output controllability is concerned, which is more demanding than just pointwise output controllability. This problem has been solved by Hautus and Heyman, within a transfer function matrix approach. Different new equivalent solvability conditions in terms of controllability subspaces, transfer function matrices or matrix pencils are given. One of these conditions (expressed in the input space) is at the origin of new necessary and sufficient conditions for block decoupling by general precompensation (possibly non admissible and nonsquare), in the wider sense of Basile and Marro     

Robust Sliding Control of Robotic Manipulators Based on a Heuristic Modification of the Sliding Gain

A task space robust trajectory tracking control is developed for robotic manipulators. A second order linear model, which defines the desired impedance for the robot, is used to generate the reference position, velocity and acceleration trajectories under the influence of an external force. The control objective is to make the robotic manipulator’s end effector track the reference trajectories in the task space. A sliding mode based robust control is used to deal with system uncertainties and external perturbations. Thus, a sliding manifold is defined by a linear combination of the tracking errors of the system in the task space built from the difference between the real and the desired position, velocity and acceleration trajectories in comparison with previous works where the sliding manifold was defined by the desired impedance and the external force. Moreover, the ideal relay has been substituted by a relay with a dead-zone in order to fit in with the actual way in which a real computational device implements the typical sign function in sliding mode control. Furthermore, a higher level supervision algorithm is proposed in order to reduce the amplitude of the high frequency components of the output associated to an overestimation of the system uncertainty bounds. Then, the robust control law is applied to the case of a robot with parametric uncertainty and unmodeled dynamics. The closed-loop system is proved to be robustly stable with all signals bounded for all time while the control objective is fulfilled in practice. Finally, a simulation example which shows the usefulness of the proposed scheme is presented.     

Global approximate output tracking for nonlinear systems

Addresses the global output tracking problem for nonlinear systems with singular points. For nonlinear systems which satisfy a suitable observability condition, the authors identify a class of smooth output trajectories which the system can track using continuous open-loop controls. This class includes all output trajectories generated by smooth state feedback. They then study the problem of approximate output tracking using discontinuous time-varying feedback controllers. Given a smooth output trajectory for which exact tracking is possible, the authors construct a discontinuous feedback controller which achieves robust tracking of the desired output trajectory in the face of perturbations. Finally, it is shown that their results can be applied to the control of a chain system, and some numerical results are presented to illustrate the performance of their controller     

Output tracking control of MIMO fuzzy nonlinear systems using variable structure control approach

The output tracking control problem for nonlinear systems in the presence of both parameter perturbations and external disturbances is studied. Our approach is based on the Takagi-Sugeno (T-S) fuzzy modeling method and a variable structure control (VSC) technique. Therefore, the systems considered are not and need not be in the triangular and parametric strict-feedback form, which are prevalent among adaptive model following control for nonlinear systems, or in the normal form, which pervades almost all existing results in neuro-fuzzy model following control approach. We first study the problem of stabilization of T-S fuzzy systems by using a VSC technique. A method for the design of a switching surface based on linear matrix inequalities is developed and a stabilizing controller based on a reaching law concept in the presence of both parameter perturbations and external disturbances is proposed. Then, the method is extended to design controllers for output tracking of T-S fuzzy nonlinear systems in two cases, i.e. systems which possess the so-called strong passive subsystems and strong stable zero dynamics, respectively. Finally, illustrative examples are presented to demonstrate the whole design procedure from the original nonlinear systems to their fuzzification and finally to the realization of the desired controllers. Simulation results show that the goal of output tracking can be achieved by the proposed controllers.     

A new algorithm for the on-line multi-level control of large interconnected systems with fast dynamics

A now two-level algorithm is developed for the on-line optimal control of largo interconnected systems with fast dynamics. On the first level, independent optimization problems are solved for given ’ desired ’ output trajectories which are supplied by the second level. On the second level these ’ desired ’ output trajectories are iteratively improved. For ‘ Linear Quadratic ’ problems, the algorithm has a particularly simple form and for such systems a general purpose computer programme has been written for its implementation. A simple simulation study illustrates the approach.