Single-input Control Research Articles

To the Linearization Problem for Single-Input Control Affine Systems

State feedback linearization is well-known as an effective tool to solve various problems for nonlinear control systems. An affine system is called to be state feedback linearizable if there exist smooth invertible changes of the state and the inputs which transform the system into a linear controllable system. If an affine system is not state feedback linearizable one can try to use orbital feedback linearization. An affine system is called to be orbital feedback linearizable if there exists time scaling which transforms the system into a state feedback linearizable system. As usual for affine systems, time scaling is considered to be depending only on the state. Recently, it has been shown that if time scaling depends both on the state and on the inputs, then it becomes possible to linearize affine systems which are not orbital feedback linearizable. In this paper, while considering such transformations, we suggest the new sufficient condition for linearizability of single-input control affine systems. We derive a system of partial differential equations which has to be solved in order to find an appropriate time scaling. We provide en example how the proposed approach can be applied to solve a terminal problem.

Modeling and Control of a Diesel Engine With Regenerative Hydraulic-Assisted Turbocharger

Diesel engines are of great challenges due to stringent emission and fuel economy requirements. Compared with the conventional turbocharger system, regenerative assisted system provides additional degrees-of-freedom for turbocharger speed control. Hence, it significantly improves control capability for exhaust-gas-recirculation (EGR) and boost pressure. This paper focuses on modeling and control of a diesel engine air-path system equipped with an EGR subsystem and a variable geometry turbocharger (VGT) coupled with a regenerative hydraulic-assisted turbocharger (RHAT). The challenges lie in the inherent coupling among EGR, turbocharger performance, and high nonlinearity of the engine air-path system. A control-oriented nonlinear RHAT system model is developed; and a linear quadratic (LQ) control design approach is proposed in this paper to regulate the EGR mass flow rate and boost pressure simultaneously and the resulting closed-loop system performance can be tuned by properly selecting the LQ control weighting matrices. Multiple LQ controllers with integral action are designed based on the linearized system models over a gridded engine operational map and the final gain-scheduling controller for a given engine operational condition is obtained by interpreting the neighboring LQ controllers. The gain-scheduling LQ controllers for both traditional VGT-EGR and VGT-EGR-RHAT systems are compared with the in-house baseline controller, consisting of two single-input and single-output (SISO) controllers, against the nonlinear plant. The simulation results show that the designed multi-input and multi-output LQ gain-scheduling controller is able to manage the performance trade-offs between EGR mass flow and boost pressure tracking. With the additional assisted and regenerative power available on the turbocharger shaft for the RHAT system, engine transient boost pressure performance can be significantly improved without compromising the EGR tracking performance, compared with the baseline control.

Tetrahedron formation of nanosatellites with single-input control

The present paper studies the formation flight of four nanosatellites forming a tetrahedron. The main goal of this research is to find the relative orbits of these satellites that, at least in the linear Hill–Clohessy–Wiltshire model, ensure finite relative motion and keep the volume and shape of the tetrahedron configuration. Since real motions of these satellites will differ from the linear ones, especially under the influence of the $J_{2}$ perturbation, active control is necessary. In addition, the limited size of the satellites does not allow us to use a complex 3-axis attitude control system. In the present paper we consider the passive magnetic attitude control system and suppose that the thrust direction is always aligned with the local geomagnetic field. In order to increase mission lifetime the control algorithm that minimizes the propellant consumption and keeps the tetrahedron volume and shape is investigated.

Funnel Control Via Funnel Precompensator for Minimum Phase Systems With Relative Degree Two

We consider tracking control for linear minimum phase single-input, single-output systems with relative degree two. For a class of sufficiently smooth reference signals, we introduce a dynamic controller which achieves that the tracking error evolves within a prespecified performance funnel. This controller is based on the recently developed funnel precompensator combined with a proportional-derivative funnel controller. Altogether, this yields a dynamic controller which satisfies the control objective and uses only the output of the system and NOT the derivative of the output. The system parameters do not have to be known for the controller design.

Mathematical model of a serine integrase-controlled toggle switch with a single input.

Dual-state genetic switches that can change their state in response to input signals can be used in synthetic biology to encode memory and control gene expression. A transcriptional toggle switch (TTS), with two mutually repressing transcription regulators, was previously used for switching between two expression states. In other studies, serine integrases have been used to control DNA inversion switches that can alternate between two different states. Both of these switches use two different inputs to switch ON or OFF. Here, we use mathematical modelling to design a robust one-input binary switch, which combines a TTS with a DNA inversion switch. This combined circuit switches between the two states every time it receives a pulse of a single-input signal. The robustness of the switch is based on the bistability of its TTS, while integrase recombination allows single-input control. Unidirectional integrase-RDF-mediated recombination is provided by a recently developed integrase-RDF fusion protein. We show that the switch is stable against parameter variations and molecular noise, making it a promising candidate for further use as a basic element of binary counting devices.

Phase model-based neuron stabilization into arbitrary clusters

Deep brain stimulation (DBS) is a common method of combating pathological conditions associated with Parkinson's disease, Tourette syndrome, essential tremor, and other disorders, but whose mechanisms are not fully understood. One hypothesis, supported experimentally, is that some symptoms of these disorders are associated with pathological synchronization of neurons in the basal ganglia and thalamus. For this reason, there has been interest in recent years in finding efficient ways to desynchronize neurons that are both fast-acting and low-power. Recent results on coordinated reset and periodically forced oscillators suggest that forming distinct clusters of neurons may prove to be more effective than achieving complete desynchronization, in particular by promoting plasticity effects that might persist after stimulation is turned off. Current proposed methods for achieving clustering frequently require either multiple input sources or precomputing the control signal. We propose here a control strategy for clustering, based on an analysis of the reduced phase model for a set of identical neurons, that allows for real-time, single-input control of a population of neurons with low-amplitude, low total energy signals. After demonstrating its effectiveness on phase models, we apply it to full state models to demonstrate its validity. We also discuss the effects of coupling on the efficacy of the strategy proposed and demonstrate that the clustering can still be accomplished in the presence of weak to moderate electrotonic coupling.

A Parametrization of Controllers Quadratically Stabilizing Single-Input Discrete-Time Systems under Bounded Disturbances

Abstract This paper presents a parametrization of static state-feedback controllers that stabilize a discrete-time linear system with single control input corrupted by disturbances with bounded amplitude. All stabilizing controllers that suppress a quadratic form of the state are parametrized by a positive definite matrix corresponding to the weight on the state in cheap control. Asymptotic stability is also guaranteed by such controllers except for some extreme cases. This parametrization enables to evaluate the decrement of a quadratic Lyapunov function and the magnitude of control values.

Modeling and Control of Stochastic Systems With Poorly Known Dynamics

This paper is concerned with controlling poorly known systems, for which only a simplified and rough model is available for control design. There are many systems that cannot be reasonably probed for the sake of identification, yet they are important for areas such as economy, populations, or medicine. The ideas are developed around an alternative way to account for the bare modeling in a stochastic-based setting, and to heighten the control features for such a modified model. The mathematical framework for the optimal control reveals important features such as the raising of a precautionary feedback policy of “keep the action unchanged” (inaction for short), on a certain state-space region. This feature is not seen in the robust approach, but has been pointed out and permeates part of the economics literature. The control problem relies on the viscosity solution for the Hamilton–Jacobi–Bellman equation, and the value of the problem is shown to be convex. When specialized to the quadratic problem with discounted cost, the exact solution inside the inaction region is given by a Lyapunov type of equation, and asymptotically, for large state values, by a Riccati-like equation. This scenario bridges to the stochastic stability analysis for the controlled model. The single control input is developed in full, part analytically, part numerically, for the scalar case, and an approximation is tested for the multidimensional case. The advantage of the precautionary policy is substantial in some situations.

Robust stabilizing control of nonholonomic systems with uncertainties via adaptive integral sliding mode

This article presents a robust stabilizing control for nonholonomic underwater systems that are affected by uncertainties. The methodology is based on adaptive integral sliding mode control. Firstly, the original underwater system is transformed in a way that the new system has uncertainties in matched form. A change of coordinates is carried out for this purpose, and the nonholonomic system is transformed into chained form system with matched uncertainties. Secondly, the chained form system with uncertainties is transformed into a special structure containing nominal part and some unknown terms through input transformation. The unknown terms are computed adaptively. Afterward, the transformed system is stabilized using integral sliding mode control. The stabilizing controller for the transformed system is constructed which consists of the nominal control plus some compensator control. The compensator controller and the adaptive laws are derived in a way that the derivative of a suitable Lyapunov function becomes strictly negative. Two different cases of perturbation are considered including the bounded uncertainty present in any single control input and the uncertainties present in the overall system model of the underwater vehicle. Finally, simulation results show the validity and correctness of the proposed controllers for both cases of nonholonomic underwater system affected by uncertainties.

Design and Implementation of a Feedback Linearization Controlled IM Drive via Simplified Neuro-Fuzzy Approach

ABSTRACTThis research paper introduces an outline of a simplified version of single-input neuro-fuzzy controller (NFC) method via a decoupling-controlled intuitive feedback linearization (FBL) approach of induction motor (IM) model. The proposed NFC with FBL extensively reduces the torque and speed ripple along with better performance enhancement. A simplified state-space feedback linearization technique-based nonlinear IM model is designed and simulated in the stationary reference frame. Again, the parameter and plant uncertainties in the feedback-linearized model of the IM are continuous phenomenon which led to the design of a robust, simplified NFC to overcome these issues in the real-time industrial application. The proposed reduced membership function (MF) based simplified NFC is the integration of fuzzy logic and neural network concept with single input as an error (speed and torque) unlike two inputs error and change in error of conventional NFC. This has the advantage of improving the computational efficiency due to its simple structure and robustness over conventional NFC. This makes the system easy and simple to implement in a realistic situation. The performance and effectiveness of the proposed method using FBL concept of IM drive is analyzed using MATLAB software in different modes of operation and is distinguished with the conventional PI-controller and conventional NFC to show its superiority. The proposed system with the different control strategies is also validated by extensive experimental results using DSP 2812.

Multimode power processing interface for fuel cell range extender in battery powered vehicle

Uninhabited electric vehicle typically utilize batteries as an exclusive power source. In order to remedy the limited energy density and thus operation time of the battery-powered unmanned vehicles, fuel based range extenders are often utilized. The operation strategy for the range extender is to maximize the mission endurance therefore it should operate at minimum specific fuel consumption point. Furthermore, the source and load voltage level may differ, hence non-inverting Buck-Boost converter (NBC) is conjoint them while keeping the desire operation point. This paper proposes a fuel cell (FC) based range extender, interfaced to the main power bus by cascade output power regulation strategy for NBC. The NBC is based on unique modulator that produces switching sequence for both converter legs. The modulator receive single control input from the dual loop regulator, inner inductor current loop and outer output power loop, which is common for energy management strategies of hybrid energy sources. System design commences with regulator design based on small signal analysis of NBC, and then it is shown that the suggested control architecture allows tight output power control throughout the entire operating range despite various plants. Finally a case study is presented, in which the converter operates as power processing interface of a fuel cell, operating as range extender in an all-electrical aircraft. The revealed findings are well-supported by simulation and experimental results.

Iterative Learning Control and System Identification of the Antagonistic Knee Muscle Complex During Gait Using Functional Electrical Stimulation

Functional Electrical Stimulation (FES) can be used to support the gait of stroke patients. By measuring joint angles and adjusting the stimulation intensities automatically to the current need of the patient, setup times can be reduced and time-variant effects like muscle fatigue can be compensated. This was achieved in recent publications by using Iterative Learning Control (ILC) on the ankle complex. In this paper we consider FES of the antagonistic knee muscle complex (quadriceps and hamstring muscles) that controls knee flexion/extension. We used a coactivation strategy in order to map the two stimulation channels to a single control input. A large class of dynamic models was obtained by system identification based on data from two experiments: one with standing subjects and one with subjects walking on a treadmill while being stimulated during different time segments of the gait cycle. Time delays, system poles, and in particular the system gains were found to vary largely. Furthermore, large differences were observed between muscle dynamics in standing pose and during walking. We designed an iterative learning controller that is stable for almost all models. In experiments with eight healthy subjects walking on a treadmill, the ILC was found to reduce deviations from a reference trajectory to about five degrees within two strides.

Strong local optimality for bang-bang-singular extremals in single input control problems

In this paper we consider a Mayer problem for a single input control affine dynamics and we give sufficient conditions for strong local optimality of a reference trajectory, consisting of two bang arcs followed by a singular one. We use a Hamiltonian approach and its connection with the second order conditions. A case study is proposed.

Dual-Control of Autothermal Thermophilic Aerobic Digestion Using Aeration and Solid Retention Time

Autothermal thermophilic aerobic digestion (ATAD) is an advanced sewage sludge treatment which allows compliance with increasingly demanding regulations. Concerning sludge pasteurization, a certain average temperature must be assured in the digester during batch treatment. Aeration flow is the variable most manipulated to regulate the digester temperature. Additionally, the manipulation of the batch sludge flow—which is related to the solid-retention-time—is considered to improve temperature regulation despite variations in air and sludge temperatures and the variability of raw sludge organic content. Thus, a dual-input control structure was provided where the aeration and solid-retention-time contributed as faster and slower inputs, respectively. Two controllers intervened, and the set-point for the batch average temperature was chosen to meet the minimum effluent quality established by the US regulations or European recommendations, considering that lower set point temperatures save aeration costs. A set-point for the aeration allowed us to achieve an extra goal, which aimed at either reducing operation costs or increasing production rates. The two feedback controllers were designed following the robust control methodology known as quantitative feedback theory (QFT). Improvements were compared with single-input (aeration-flow) control strategy and open-loop control strategy. Simulations were performed on a benchmark non-linear simulation model for ATAD.

Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology

This paper is devoted to demonstrate how a class of fractional-order chaotic systems can be controlled in a given finite time using just a single control input. First a novel fractional switching sliding surface is proposed with desired properties such as fast convergence to zero equilibrium and no steady state errors. At the second phase, a smooth reaching control law is derived to guarantee the occurrence of the sliding motion with a finite settling time. Owing to the integration of the control signal discontinuity, chattering oscillations are hindered from the controller. Rigorous stability analysis is performed to validate the design claims. The effects of high frequency external noises as well as modeling errors and dynamic variations are also taken into account, and the robustness of the closed-loop system is ensured. The proposed robust controller is realized for a class of chaotic fractional-order systems with one control input. In accordance, some remarks regarding the inclusion of mismatched uncertainties in the system dynamics are given. The robust functionality and quick convergence property as well as chatter-free attribute of the introduced non-smooth sliding mode technology are demonstrated using oscillation suppression of fractional-order chaotic Lorenz and financial systems.

Optimal Control of a Constrained Bilinear Dynamic System

In this paper, an optimal feedback, for a free vibrating semi-active controlled plant, is derived. The problem is represented as a constrained optimal control problem of a single input, free vibrating bilinear system, and a quadratic performance index. It is solved by using Krotov’s method and to this end, a novel sequence of Krotov functions that suits the addressed problem, is derived. The solution is arranged as an algorithm, which requires solving the states equation and a differential Lyapunov equation in each iteration. An outline of the proof for the algorithm convergence is provided. Emphasis is given on semi-active control design for stable free vibrating plants with a single control input. It is shown that a control force, derived by the proposed technique, obeys the physical constraint related with semi-active actuator force without the need of any arbitrary signal clipping. The control efficiency is demonstrated with a numerical example.

Convergence of iterative learning control for SISO nonrepetitive systems subject to iteration-dependent uncertainties

This paper studies the robust convergence properties of iterative learning control (ILC) for single-input, single-output (SISO), nonrepetitive systems subject to iteration-dependent uncertainties that arise in not only initial states and external disturbances but also plant models. Given an extended relative degree condition, it is possible to propose necessary and sufficient (NAS) conditions for robust ILC convergence. The tracking error bound is shown to depend continuously on the bounds of the iteration-dependent uncertainties. When the iteration-dependent uncertainties are bounded, NAS conditions exist to guarantee bounded system trajectories and output tracking error. If the iteration-dependent uncertainties converge, then NAS conditions ensure bounded system trajectories and zero output tracking error. The results are also extended to a class of affine nonlinear systems satisfying a Lipschitz condition. Simulation tests on a representative batch process demonstrate the validity of the obtained robust ILC convergence results.

VAM: Hypocycloid Mechanism for Efficient Bioinspired Robotic Gaits

We present variable amplitude module orthogonal slider (VAMOS), a novel two-motor hexapedal robot that is able to walk and turn with arbitrary curvature. The VAMOS walking mechanism is based around a variable amplitude module (VAM), a compact and modular gearbox that produces reciprocating sinusoidal output of continuously variable amplitude. The inputs to the VAM are a drive motor, rotating at a constant speed, and a single control input for changing the output amplitude. Unlike other methods for amplitude modulation, the VAM does not require modulation of the main drive input and we can, therefore, operate it at peak efficiency thoughout the gait cycle. As demonstrated by VAMOS, the VAM provides a versatile building block for biomimetic robotic gaits.

Predictor-based control for an inverted pendulum subject to networked time delay

The inverted pendulum is considered as a special class of underactuated mechanical systems with two degrees of freedom and a single control input. This mechanical configuration allows to transform the underactuated system into a nonlinear system that is referred to as the normal form, whose control design techniques for stabilization are well known. In the presence of time delays, these control techniques may result in inadequate behavior and may even cause finite escape time in the controlled system. In this paper, a constructive method is presented to design a controller for an inverted pendulum characterized by a time-delayed balance control. First, the partial feedback linearization control for the inverted pendulum is modified and coupled with a state predictor to compensate for the delay. Several coordinate transformations are processed to transform the estimated partial linearized system into an upper-triangular form. Second, nested saturation and backstepping techniques are combined to derive the control law of the transformed system that would complete the design of the whole control input. The effectiveness of the proposed technique is illustrated by numerical simulations.

Backstepping stabilization of the linearized Saint-Venant–Exner model

Using backstepping design, exponential stabilization of the linearized Saint-Venant–Exner (SVE) model of water dynamics in a sediment-filled canal with arbitrary values of canal bottom slope, friction, porosity, and water–sediment interaction, is achieved. The linearized SVE model consists of two rightward convecting transport Partial Differential Equations (PDEs) and one leftward convecting transport PDE. A single boundary input control strategy with actuation located only at the downstream gate is employed. A full state feedback controller is designed which guarantees exponential stability of the desired setpoint of the resulting closed-loop system. Using the reconstruction of the distributed state through a backstepping observer, an output feedback controller is established, resulting in the exponential stability of the closed-loop system at the desired setpoint. The proposed state and output feedback controllers can deal with both subcritical and supercritical flow regimes without any restrictive conditions.