Multi-input Multi-output Nonlinear Systems Research Articles

An Adaptive Learning Control for MIMO Nonlinear System with Nonuniform Trial Lengths and Invertible Control Gain Matrix

In the traditional iterative learning control (ILC) method, the operational time interval is conventionally fixed to facilitate a seamless learning process along the iteration axis. However, this condition may frequently be contravened in real-time applications owing to unknown uncertainties and unpredictable factors. In essence, replicating a control system at a consistent time interval proves challenging in practical scenarios. This paper proposes an adaptive iterative learning control (AILC) method for the multi-input–multi-output (MIMO) nonlinear system with nonuniform trial lengths and an invertible control gain matrix. Compared to the existing AILC research that features nonuniform trial lengths, the control gain matrix of the system in this paper is assumed to be invertible. Hence, the general requirement in the conventional AILC method that the control gain matrix of the system is positive-definite (or negative-definite) or even known is relaxed. Moreover, the tracking reference allows it to be iteration-varying. Finally, to prove the convergence of the system, the composite energy function is introduced and to verify the validity of the AILC method, a robot movement imitation with an uncalibrated camera system is used. The simulation results show that the actual output can track the desired reference trajectory well, and the tracking error converges to zero after 30 iterations.

Adaptive fault-tolerant control for ascent HSV with wing damage and function constraints on states

In this article, a fault-tolerant tracking control technique is developed for the ascent of hypersonic vehicles (HSV) experiencing wing damage and time-varying state constraints. The difficulty of control problem lies in handling state constraints associated with both state and time, while determining the unknown control directions of the multi-input multi-output (MIMO) nonlinear system. The presence of constraints in both state and time complicates the design of the control algorithm. This paper proposes the construction of a novel state transition function to establish a new unconstrained system, thereby eliminating the conservative feasibility conditions of Barrier Lyapunov Function (BLF) and integral BLF methods when dealing with time-varying constraints. The fusion of time-varying scaling functions and Nussbaum-type functions not only resolves the issue of unknown control directions in the MIMO nonlinear system but also enhances the transient and steady-state performance of the closed-loop system. Neural Networks are employed to approximate the unknown nonlinear functions. Using Lyapunov stability theory, it is proven that state constraints are not violated, and all signals in the closed-loop system are bounded. Simulation results demonstrate the merits of the designed fault-tolerant control (FTC) algorithm.

Saturation-Tolerant Prescribed Control for Nonlinear Systems With Unknown Control Directions and External Disturbances.

In this article, saturation-tolerant prescribed control (SPC) is investigated for a class of multiinput-multioutput (MIMO) nonlinear systems. The key challenge lies in how to guarantee both input and performance constraints simultaneously for nonlinear systems especially under external disturbance and unknown control directions. We propose concise finite-time tunnel prescribed performance (FTPP) for better tracking performance, which features tight allowable set and user-specified settling time. To comprehensively tackle the conflict between the above two constraints, an auxiliary system is designed to explore their interconnections instead of neglecting their contradictions. By introducing its generated signals into FTPP, the obtained saturation-tolerant prescribed performance (SPP) has the ability to degrade or recover the performance boundaries in the light of different saturation conditions. Consequently, the developed SPC, together with nonlinear disturbance observer (NDO), can effectively improve the robustness and reduce the conservatism against external disturbances, input, and performance constraints. Finally, comparative simulations are presented to showcase these theoretical findings.

Higher Order Sliding Mode Control of MIMO Induction Motors: A New Adaptive Approach

In this paper the objective is to force the outputs of nonlinear nonaffine multi-input multi-output (MIMO) systems to track those of a linear system with the desired properties. The approach is based on designing higher order sliding mode controller (HOSMC) with the definition of a new proportional-integral (PI) sliding surface. To this end, a linear state feedback with an adaptive switching gain (ASG) is applied to the nonlinear MIMO systems. Therefore, the switching gain can increase or decrease based on the system conditions. Then, the chattering is completely removed using a combination of HOSMC and ASG. Moreover, the proposed procedure is independent from the upper bound of the matched uncertainty, which is in the direction of system inputs. The finite time convergence to the sliding surface is also proved, which provides an invariance property in finite time. Note that invariance is the most important property of SMC. Finally, the general model of MIMO induction motors (IM) is used to address and to verify the proposed controller.

Multivariable non-singular terminal composite sliding mode control of gas-water-coal mixture lifting system

In order to ensure the stability of flow rate and valve pressure difference during gas-water-coal mixture lifting, a multivariable non-singular terminal composite sliding mode (MNTCSM) controller based on accurate feedback linearization is proposed. The multi-input multi-output nonlinear system with time delay and coupling is transformed into a multi-input multi-output uncoupled linear system by using an improved Smith predictor and accurate feedback linearization. At the same time, the MNTCSM controller is designed to make the flow and pressure tracking errors of the decoupling system converge to zero in a finite time. Finally, simulations and experiments are designed for different control methods to verify the feasibility and effectiveness of the proposed scheme.

Improved Model-Free Adaptive Control for MIMO Nonlinear Systems With Event-Triggered Transmission Scheme and Quantization.

In this article, an improved model-free adaptive control (iMFAC) is proposed for discrete-time multi-input multioutput (MIMO) nonlinear systems with an event-triggered transmission scheme and quantization (ETQ). First, an event-triggered scheme is designed, and the structure of the uniform quantizer with an encoding-decoding mechanism is given. With the concept of partial form dynamic linearization based on event-triggered and quantization (PFDL-ETQ), a linearized data model of the MIMO nonlinear system is constructed. Then, an improved model-free adaptive controller with the ETQ process is designed. By this design, the update of the pseudo partitioned Jacobean matrix (PPJM) estimates and control inputs occurs only when the trigger conditions are met, which reduces the network transmission burden and saves the computing resources. Theoretical analysis shows that the proposed iMFAC with the ETQ process can achieve a bounded convergence of tracking error. Finally, a numerical simulation and a biaxial gantry motor contour tracking control system simulation are given to illustrate the feasibility of the proposed iMFAC method with the ETQ process.

Iterative Learning Control of Constrained Systems With Varying Trial Lengths Under Alignment Condition.

This brief is concerned with iterative learning control (ILC) of constrained multi-input multi-output (MIMO) nonlinear systems under the state alignment condition with varying trial lengths. A modified reference trajectory is constructed to meet the alignment condition by adjusting the reference trajectory to be spatially closed. Resorting to the barrier composite energy function (BCEF) approach, an adaptive ILC scheme is built to guarantee the bounded convergence of the resultant closed-loop system. Illustrative examples are presented to verify the validity of the proposed iteration scheme.

Adaptive Containment Control for Heterogeneous MIMO Nonlinear Multiagent Systems With Unknown Direction Actuator Faults

In this article, a novel output-feedback adaptive containment control scheme is proposed for a class of heterogeneous multi-agent systems, where the agents are nonlinear multi-input multi-output (MIMO) systems whose relative degrees are allowed to be different. Unlike existing results, we only require the leading principal minors of agents' control gain matrices (CGMs) to be nonzero and take into account unknown direction actuator faults, which relaxes the restrictions on CGMs and enhances system reliability. The difficulties jointly caused by the unknown CGMs, unknown parameters, and unknown jumps introduced by the actuator faults are successfully overcome by a novel recursive contradiction argument based on some Nussbaum functions and a matrix similarity transformation. Moreover, an event-triggering mechanism is introduced to avoid continuous communication among agents and reduce the communication burden. It is shown that all closed-loop signals are globally uniformly bounded and the containment errors converge to a residual set that can be made arbitrarily small. Simulation results illustrate the effectiveness of the proposed scheme.

A High Order P-type Iterative Learning Control Scheme for Unknown Multi Input Multi Output Nonlinear Systems with Unknown Input Saturation

In this paper, a new high order P-type iterative learning control scheme is presented to solve the trajectory tracking problem of Multi Input Multi Output (MIMO) nonlinear systems with unknown input saturation. It is well known that most systems can be affected by input uncertainties such as saturation. This undesirable input has the potential to destabilize the system. Thus, it is important to mitigate the effect of saturation. In this paper, we take this problem into account. In addition, the controller scheme is very simple, in which the control input in each trial is adjusted by using the tracking error signals obtained from previous and current trials. As the iterations continue, the control system eventually learns the task and follows the desired trajectory with little or no errors. The asymptotic stability of the closed loop system under unknown input saturation is guaranteed over the whole finite time by using the λ-norm method. Finally, to illustrate the effectiveness of the proposed method, simulation results are presented.

General fuzzy adaptive fault-tolerant control based on Nussbaum-type function with additive and multiplicative sensor and state-dependent actuator faults

The present paper devotes to propose a general active fault-tolerant control (GAFTC) for a class of unknown multi-input multi-output (MIMO) nonlinear systems that are subject to simultaneous additive and multiplicative sensor and nonaffine nonlinear state-dependent actuator faults, exogenous disturbances and multiple unknown control directions. By combining fuzzy logic systems and back-stepping technique, an adaptive control unit along with a robust control unit are developed. In the proposed approach, the unknown dynamics along with sensor and non-affine nonlinear state-dependent actuator faults are approximated to avoid developing fault detection and isolation unit. Furthermore, the difficulty caused by the unknown control gain directions is solved by using the Nussbaum-type function. The adaptation mechanism is proposed to overcome fuzzy approximation errors and exogenous disturbances, and the Butterworth low-pass filter is adopted to get rid of the algebraic loop appearing in the actuator faults approximation. The stability of the closed-loop system is proved by Lyapunov method where all the signals involved in the resulting closed-loop system are shown to be bounded, and the tracking errors converge asymptotically to the origin. Quadrotor attitude dynamics stabilization is proposed in the simulation study to test the accuracy and the effectiveness of the proposed control scheme.

Prescribed-Time Tracking Control of MIMO Nonlinear Systems With Nonvanishing Uncertainties

It is highly desirable yet challenging to achieve zero-error tracking within a given short time period for uncertain multi-input multi-output (MIMO) nonlinear systems. Previous results are primarily state-regulation oriented and are valid only for single-input single-output (SISO) linear systems or nonlinear systems in normal-form. This note presents a tracking control solution for high-order MIMO nonlinear systems in strict-feedback-like form. The proposed control is able to achieve precise tracking within the prescribed time irrespective of initial conditions in the presence of mismatched uncertainties and non-vanishing disturbances.

Model-Free Adaptive Containment Control for Unknown Multi-Input Multi-Output Nonlinear MASs With Output Saturation

In this work, a model-free adaptive containment control scheme is investigated for a class of multi-input multi-output nonlinear multiagent systems, where the agents’ dynamics are unknown with output saturation. Firstly, the dynamics of followers are transformed into a equivalently data model by using full form dynamic linearization technology. Secondly, the distributed containment control algorithm is proposed only using the input data and the saturated output data of followers and neighbors. Further, the boundedness of the containment error is proved by using the contraction mapping principle and mathematical induction method. It is shown that the developed scheme can ensure that the followers move into the convex hull composed of the leaders. Last, the effectiveness of the developed scheme can be verified through numerical simulations.

Application of Elliptic Jerk Motion Profile to Cartesian Space Position Control of a Serial Robot

The paper discusses the application of a motion profile with an elliptic jerk to Cartesian space position control of serial robots. This motion profile is obtained by means of a kinematic approach, starting from the jerk profile and then calculating acceleration, velocity and position by successive integrations. Until now, this profile has been compared to other motion laws (trapezoidal velocity, trapezoidal acceleration, cycloidal, sinusoidal jerk, modified sinusoidal jerk) considering single-input single-output systems. In this work, the comparison is extended to nonlinear multi-input multi-output systems, investigating the application to Cartesian space position control of serial robots. As case study, a 4-DOF SCARA-like architecture with elastic balancing is considered; both an integer-order and a fractional-order controller are applied. Multibody simulation results show that, independently of the controller, the behavior of the robot using the elliptic jerk profile is similar to the case of adopting the sinusoidal jerk and modified sinusoidal jerk laws, but with a slight reduction in the position error (−3.8% with respect to the sinusoidal jerk law and −0.8% with respect to the modified sinusoidal jerk law in terms of Integral Square Error) and of the control effort (−8.2% with respect to the sinusoidal jerk law and −1.3% with respect to the modified sinusoidal jerk law in terms of Integral Control Effort).

Fast finite-time observer-based sliding mode controller design for a class of uncertain nonlinear systems with input saturation

In this paper, a novel fast finite-time velocity observer-based terminal sliding mode controller is proposed for a class of multi-input multi-output nonlinear systems in the presence of unknown time-varying matched uncertainties and input saturation constraint. The considered nonlinear system is a chain of interdependent second-order nonlinear subsystems that can describe a multitude of real practical applications. The notion of an auxiliary system is introduced and utilized to deal with the input saturation. Due to the hardship of measuring and accessibility to all the system states, a robust output feedback sliding mode controller is designed based on nonsingular fast terminal sliding surfaces exploiting the estimated states and the auxiliary system’s states which is capable of alleviating the control signal gains and enhancing the convergence rate. By noticing that the separation principle does not hold in nonlinear systems, analyzing the finite-time stability of the closed-loop system is a challenging problem and will be assured via an innovative Lyapunov function candidate. Moreover, an explicit adjustable finite convergence time is derived. Simulation results on two practical systems consisting of a two-degree-of-freedom rigid robotic manipulator and Van der Pol oscillator illustrate the effectiveness and superiority of the suggested control scheme.

An improved model-free adaptive control for nonlinear systems: An LMI approach

This paper proposes two model-free adaptive control (MFAC) schemes by using the linear matrix inequality (LMI) approach for the single-input single-output (SISO) and multi-input multi-output nonlinear (MIMO) systems, respectively. For each control scheme, with the aid of the dynamic linearization technique, the nonlinear system is transformed into an equivalent linear data model. In such a transformation, the nonlinear characteristic of systems is compressed into a time-varying parameter. Then, with the help of the introduction of the observer method, the tracking control problem can be converted into an optimization problem and the controller parameters can be obtained by using the LMI technique. This conversion cannot only reduce the complexity of the stability analysis but also find the appropriate controller parameters, especially for the MIMO case. Finally, three examples with comparisons are provided to illustrate the validity of the devised MFAC schemes.

Virtual Hardware-in-the-Loop FMU Co-Simulation Based Digital Twins for Heating, Ventilation, and Air-Conditioning (HVAC) Systems

In this paper, a novel self-adaptive control method based on a digital twin is developed and investigated for a multi-input multi-output (MIMO) nonlinear system, which is a heating, ventilation, and air-conditioning system. For this purpose, hardware-in-loop (HIL) and software-in-loop (SIL) are integrated to develop the digital twin control concept in a straightforward manner. A nonlinear integral backstepping (NIB) model-free control technique is integrated with the HIL (implemented as a physical controller) and SIL (implemented as a virtual controller) controllers to control the HVAC system without the need for dynamic feature identification. The main goal is to design the virtual controller to minimize the distinction between system outputs in the SIL and HIL setups. For this purpose, Deep Reinforcement Learning (DRL) is applied to update the NIB controller coefficients of the virtual controller based on the measured data of the physical controller. Since the temperature and humidity of HVAC systems should be regulated, the NIB controllers in the HIL and SIL are designed by the DRL algorithm in a multi-objective scheme (MO). In particular, the simulations of the HIL and SIL environments are coupled by a new advanced tool: function mockup interface (FMI) standard. The Functional Mock-up Unit (FMU) is adopted into the FMI interface for data exchange. The extensive research of HIL and SIL controllers shows that the system outputs of the virtual controller are controlled exactly according to the physical controller.

Ball Balancing on 3-RRS Parallel Manipulator Using PD and LQR Control

Ball and plate system (BPS) is known as a nonlinear multi-input multi-output (MIMO) system that has been usually used to investigate and develop control algorithms. In this paper the study deals with a 3-RRS parallel manipulator which has a metal ball, three DC motors with linkage mechanism to a fixed base that can be a resistive touch screen . The moving base has two rotational movements, it can be rotated around the x-axis and y-axis and translated along the z-axis. The main purpose of this study is obtaining dynamics equations of motions and designing PD and LQR controllers. Through simulation and experiment, both control algorithms are proved to work well on this model.

An Output Feedback Stabilizer for Uncertain MIMO Nonlinear Systems: Nonlinear Nominal Input Gain Function

This article presents a robust output feedback stabilizer for multi-input–multi-output nonlinear systems subject to external disturbances and system uncertainties. We assume that the system has a well-defined vector relative degree and the zero dynamics is input-to-state stable. Based on the assumption that there exists a state feedback controller that makes the origin of the nominal closed-loop system asymptotically stable, we present an output feedback stabilizer that recovers the stability of the nominal closed-loop system in the practical sense. We allow that the nominal system can have a nonlinear input gain matrix that is a function of system state, and this is done by modifying the structure of the disturbance observer-based robust outputfeedback controller. Simulation results on a practical system are included to validate the proposed controller.

Sufficient Conditions for Global Integral Action via Incremental Forwarding for Input-Affine Nonlinear Systems

In this article, we study the problem of constant output regulation for a class of input-affine multi-input multi-output nonlinear systems, which do not necessarily admit a normal form. We allow the references and the disturbances to be arbitrarily large and the initial conditions of the system to range in the full-state space. We cast the problem in the contraction framework, and we rely on the common approach of extending the system with an integral action processing the regulation error. We then present sufficient conditions for the design of a state-feedback control law able to make the resulting closed-loop system incrementally stable, uniformly with respect to the references and the disturbances. Such a property ensures uniqueness and attractiveness of an equilibrium on which output regulation is obtained. To this end, we develop an incremental version of the forwarding (mod <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lbrace L_gV\rbrace$</tex-math></inline-formula> ) approach. Finally, we provide a set of sufficient conditions for the design of a pure (small-gain) integral-feedback control. The proposed approach is also specialized for two classes of systems that are linear systems having a Lipschitz nonlinearity and a class of minimum-phase systems whose zero dynamics are incrementally stable.

Observer-Based Event-Triggered Adaptive Fuzzy Control for Fractional-Order Time-Varying Delayed MIMO Systems Against Actuator Faults

This article presents the observer-based event-triggered adaptive hybrid fuzzy dynamic surface control strategy for a category of uncertain nonstrict-feedback fractional-order nonlinear multi-input multi-output systems, including unknown time-varying delays and actuator faults. First, an adaptive hybrid fuzzy state observer and a serial-parallel estimation system are constructed to estimate the unmeasured system states and incorporate them into the control design scheme, respectively, where some appropriate fuzzy logic systems are introduced to approximate the unknown nonlinear functions. According to the dynamic surface control technique, the designed adaptive fuzzy control approach can surmount the deficiency of “complexity explosion.” Then, an observer-based adaptive event-triggered control algorithm is developed by constructing the Lyapunov–Krasovskii functionals and estimating the compounded disturbances. Furthermore, it is proved that under the drive of the reference signals, all the signals in the closed-loop system are semiglobally uniformly ultimately bounded and Zeno behavior can be successfully excluded. Finally, an example with numerical simulations is utilized to exhibit the applicability of the obtained observer-based event-triggered adaptive fuzzy control approach.