Multi-input Multi-output Linear Systems Research Articles

Model reference adaptive controller with interpretable fuzzy rules for linear MIMO systems

This paper presents a method for fuzzy model reference adaptive control system design for a partially known multi-input multi-output (MIMO) linear dynamic plant. Differential or convolutional equations can describe the plant to be controlled, and the fuzzy controller is assumed to be from the MIMO sector-bounded nonstationary nonlinearities class. The task is to obtain robust adaptive stabilization. Some absolute stability theorems and integral evaluations of the state and control vectors are applied in the MIMO fuzzy adaptive controller design procedure for the first time. A method for automatically obtaining a near-optimal nonlinear dynamic reference model is described. A practical step-by-step procedure of the MIMO adaptive fuzzy controller design is proposed, resulting in highly interpretable MIMO fuzzy control rules based on triangular fuzzy sets and strong triangular fuzzy partition. An example of the fuzzy controller design according to the proposed procedure is given.

Almost sure and mean square convergence of iterative learning control system with state feedback under random packet dropouts

This paper investigates the almost sure and mean square convergence of a state feedback-based iterative learning control scheme for a class of discrete-time multi-input–multi-output linear systems with a direct feedback term in the presence of random data dropouts. Firstly, the realisability of the control system is considered. It is shown that the realisability of the control system is only determined by the direct feedback matrix, which has nothing to do with the input–output coupling matrix, and the direct feedback matrix to be full-row rank is enough to guarantee the system realisable. Secondly, the convergence of output tracking errors is analysed in both the almost sure and mean square senses, respectively. By the contraction mapping method combined with matrix decomposition and transformation techniques, it is strictly proved that the output sequence can converge to any given desired trajectory in the sense of almost sure and mean square, respectively, if and only if the general spectral radius condition is satisfied. Finally, an example is given to show the correctness of the theoretical analysis.

Performance enhancement of large-scale linear dynamic MIMO systems using GWO-PID controller

The multi-input multi-output (MIMO) technique is becoming grown and integrated into wireless wideband communication. MIMO techniques suffer from a large-scale linear dynamic problem, it will be easy to adjust the proportional-integral-derivative (PID) of a continuous system, unlike the nonlinear model. This work displays the tuning of the PID controller for MIMO systems utilizing a statistical grey wolf optimization (GWO) and evaluated by objective function as integral time absolute error (ITAE). The instantaneous adjusting characteristic GWO approach is the criterion that distinguishes such a combination-proposed strategy from that existing in the traditional PID approach. The GWO algorithm searching-based methodology is used to determine the adequate gain factors of the PID controller. The suggested approach guarantees stability as the initial scheme for a steady state condition. A combination of ITAE combined with the GWO reduction method is adopted to reduce the steady-state transient time responses between the higher-order initial scheme and the unit amplitude response. Simulation outcomes are illustrated using MATLAB software to show the capability of adopting the GWO scheme for PID controlling.

A convex optimization approach to multi-objective design of two-stage compensators for MIMO linear systems

The previous design of two-stage compensators with multi-objectives concerns the closed-loop stability, the low-sensitivity, and the satisfactory time-domain performance. However, the design only considers the single-input single-output system. This paper aims to present the design of two-stage compensators for multi-input multi-output (MIMO) linear systems which contain dynamical interaction between inputs and outputs. When Youla parameterization is applied to the MIMO linear system, the design criteria can be cast as a nonlinear optimization. In addition, the infinite dimension optimization problem is approximated to a finite dimension problem with Ritz approximation. The suitable form of design parameters is proposed to ensure the low sensitivity of the MIMO system. The two-stage compensator design is formulated as convex optimization which is efficiently solved using available solvers. Finally, the two-stage compensators are designed for Wood and Berry’s model of binary distillation column. The numerical results indicate that all design objectives of frequency domain and time domain can be achieved by the two-stage compensators.

Multivariable Robust Proportional-Integral-Derivative Control for Linear Quadratic Compensation of a Class of Norm-Bounded Uncertain Systems

AbstractThis paper is concerned with designing multi-input multi-output (MIMO) proportional-integral-derivative (PID) control having filtered derivative terms to achieve optimal linear quadratic performance for a class of linear uncertain MIMO systems with norm-bounded time-varying uncertainties. A necessary and sufficient condition for existence of the PID controller is obtained in terms of rank constrained linear matrix inequalities (LMIs). Using an existing penalty-function-based approximation for matrix rank minimization, an algorithm is proposed to solve these rank constrained LMIs and find PID gains. A suitable numerical example is considered to show the efficacy of the proposed approach in ensuring robust performance as compared to the existing ones. The proposed algorithm is also tested on several benchmark examples and found to be computationally much more efficient than some well-known methods.

An Evolutionary Optimization Technique for Time Domain Modelling

The authors proposed an evolutionary-optimization technique known as the Genetic Algorithm (GA) for the optimization and order reduction of high order systems (HOSs) in the time domain. As we know a lot of optimization techniques are available in the frequency domain but limited in the time domain. The proposed technique is applicable for time-domain systems. The reduced model (RM) obtained by the proposed technique can be replaced with the original HOS as it retains all the important time and frequency response specifications of the original HOS. The efficacy of the proposed method has been tested on few numerical examples from the literature. The important time and frequency response specifications of the proposed RM are compared with RM obtained by recent techniques using MATLAB/Simulink. The proposed algorithm is applicable for Single-input single-output (SISO) and multi-input multi-output (MIMO) linear, nonlinear, time-invariant, and time-variant systems.

Robust output-feedback SOSM control subject to unmatched disturbances and its application: A fixed-time observer-based method

This paper investigates the output-feedback control for a general class of multi-input multi-output (MIMO) linear systems in the presence of unmatched disturbances. Firstly, a new observer composed of a Luenberger observer and a novel hierarchical high-order sliding mode (HOSM) observer is proposed to identify the system states and disturbances, simultaneously. As one of the most remarkable properties, the convergence time of the proposed observer is bounded by a positive constant which is free of the system initial error conditions. Secondly, based on the proposed observer, a new second-order sliding mode (SOSM) controller is constructed by using a novel sliding surface with unmatched disturbances compensation. The proposed control law is a simply continuous function of time and thus can certainly reduce numerical chattering. Finally, to show the effectiveness of the theoretical results, an application to inverted pendulum system is used to make simulation comparison.

Blending based multiple-model adaptive control of multivariable systems with application to lateral vehicle motion control

This paper develops multiple fixed model blending based adaptive parameter identification schemes for multi-input multi-output (MIMO) systems with polytopic parameter uncertainty. The developed identification schemes are proven to be asymptotically stable for uncertain linear time-invariant (LTI) MIMO systems, and is shown to provide fast adaptation for even uncertain linear time-varying (LTV) systems. Furthermore, utilizing the proposed parameter identification schemes, a linear-quadratic (LQ) optimal multiple-model adaptive control (MMAC) scheme is developed for linear MIMO systems with polytopic uncertainties. The proposed MMAC scheme is proven to be asymptotically stable for LTI MIMO systems and applied to tracking control of uncertain lateral vehicle dynamics. A set of simulation test results are presented to verify the stability, effectiveness, and comparison of the proposed MMAC scheme.

Multivariable Nonadaptive Controller Design

Although proportional-integral-derivative control remains one of the most common control schemes used in industry, its tuning still remains inadequately understood in many applications. This task becomes much more challenging when applied to multi-input multi-output (MIMO) systems. This article presents the design of a discrete-time robust multivariable (nonadaptive) tracking controller that comes with a simple structure, requires very limited information on the plant model, and is relatively easy to tune. In addition to being easy to tune and implement, an objective of this controller is to deal with a class of large-scale systems with complex dynamics. We analytically demonstrate the robustness and convergence of the closed-loop system for a class of MIMO linear time-varying systems. The overall superiority of the proposed controller is experimentally validated on a Barrett robot arm in a laboratory environment. The article also provides a stochastic framework of the general setting of the controller. Within this framework, two minimum mean square error optimal solutions of the controller are provided; one is designed for the case where the number of inputs is not greater than the number of outputs, and the other is for the antithesis.

Robust Feedback Stabilization of Linear MIMO Systems Using Generalized Homogenization

A robust nonlinear control is designed for stabilizing linear multi-input–multi-output systems. The presented control law homogenizes a linear system (without its transformation to a canonical form) with a specified degree and stabilizes it in a finite time (or with a fixed-time attraction to any compact set containing the origin) if the degree of homogeneity is negative (positive). The tuning procedure is formalized in an linear matrix inequalities (LMI) form. Performance of the approach is illustrated by numerical and experimental examples.

Robust model predictive control for a small reverse osmosis desalination unit subject to uncertainty and actuator fault

Abstract Actuator faults are inevitable in small reverse osmosis desalination plants. They may cause energy losses and reduce the quality of the freshwater, which may endanger human life. Model predictive control (MPC) is a model-based approach widely used to control process systems such as reverse osmosis, while considering a set of constraints. In this paper, three methods of predictive model controllers are considered for the control of a multi-input multi-output (MIMO) reverse osmosis desalination system in the presence of noise, model mismatch, and actuator fault. Formulation of enhanced constrained receding horizon predictive control via bounded data uncertainties (CRHPC-BDU) are extended for linear time-invariant MIMO systems. Permeate flow rate and conductivity of the water produced are controlled by a retentate valve and a bypass valve, respectively. The simulation results show the robustness of the suggested approach in the presence of both noise and uncertainties. CRHPC-BDU has a better performance subject to systems with model uncertainty and actuator fault up to a reasonable limit. By increasing the actuator fault up to 34%, the robustness of CRHPC-BDU is further highlighted in permeate conductivity, where the fluctuations of permeate conductivity dampen sooner than in the other two controllers.

Design scheme of discrete time variable structure controllers applied to multi‐input‐multi‐output linear systems with chattering elimination

Designing discrete time variable structure control with sliding mode (SM) schemes implies finding an asymptotically stable sliding surface with a control policy which assures the existence of a SM. Based on a recently innovative approach, where the authors developed a flexible design of sliding surfaces applied to discrete time linear multi-input multi-output (MIMO) systems, a novelty controller design strategy is proposed, where a switching surface parameter is used for the adjustment of the control signal limitations and for the improvement of the controlled system dynamics against model uncertainties. It is the first time that these drawbacks are solved modifying the switching surface values because it has been always resolved via control law adjustment. Moreover, a new chattering elimination method using adjustable sigmoid functions is also implemented in the controller. The main advantages of the proposed methodology are: firstly, it can be easily applied to high-order linear MIMO systems; secondly, system dynamics coordinates representation transformations are not needed and thirdly, the non-ideal SM dynamics is considered in the design procedure. Both the sliding surface and the control law specification are included in the proposed methodology. An illustrative example is included.

Optimal input design for system identification using spectral decomposition

ABSTRACT The aim of this paper is to design a band-limited optimal input with power constraints for identifying a linear multi-input multi-output system, with nominal parameter values specified. Using spectral decomposition theorem, the power spectrum is written as . The matrix is expressed in terms of a truncated basis for , where is the cut-off frequency. The elements of the Fisher Information Matrix and the power constraints become homogeneous quadratics in basis coefficients. The optimality criterion used are -optimality, -optimality, -optimality and -optimality. This optimization problem is not known to be convex. A bi-linear formulation gives a lower bound on the optimum, while an upper bound is obtained through a convex relaxation. These bounds can be computed efficiently. The lower bound is used as a suboptimal solution, its sub-optimality determined by the difference between the bounds. Simulations reveal that the bounds match in many instances, implying global optimality.

Improved disturbance rejection for predictor-based control of MIMO linear systems with input delay

ABSTRACTIn this paper, we are concerned with the predictor-based control of multi-input multi-output (MIMO) linear systems with input delay and disturbances. By taking the future values of disturbances into consideration, a new improved predictive scheme is proposed. Compared with the existing predictive schemes, our proposed predictive scheme can achieve a finite-time exact state prediction for some smooth disturbances including the constant disturbances, and a better disturbance attenuation can also be achieved for a large class of other time-varying disturbances. The attenuation of mismatched disturbances for second-order linear systems with input delay is also investigated by using our proposed predictor-based controller.

A global rational Arnoldi method for model reduction

In this paper, we propose two new approaches for model order reduction of large-scale multi-input multi-output (MIMO) linear time invariant dynamical systems (LTI). These methods are based on a generalization of the global Arnoldi algorithm which is used to generate projection subspaces. An adaptive procedure for the selection of shift parameters is proposed and some simple Arnoldi relations are established in order to compute error bounds on the transfer function. Numerical experiments showing the effectiveness of these approaches are provided.

Extended State Observer-Based Sliding Mode Control for Multi-input Multi-output System with Multiple Disturbances

This paper proposes a generalized design procedure for the extended state observer-based sliding mode control for multi-input multi-output linear systems with multiple disturbances (matched and/or mismatched) using an extended state observer. First, the system is transferred to a regular form to decouple the states having matched and mismatched uncertainties; then, a novel sliding surface is designed based on the state and disturbance estimation obtained through extended state observer (ESO). The proposed generalized ESO-based sliding mode controller (SMC) is designed to counteract the effect of matched and/or mismatched disturbances which are bounded, while the classical SMC fails to provide the desired performance in the presence of mismatched disturbance. This paper develops a simple method for the design of sliding surface coefficients and the selection of switching gain. The proposed control strategy is validated through numerical simulations, and the comparison with ESO-based state feedback control shows a reduced-order closed-loop dynamics and better transient performance.

Implicit Volterra series interpolation for model reduction of bilinear systems

We propose a new interpolatory framework for model reduction of large-scale bilinear systems. The input–output representation of a bilinear system in frequency domain involves a series of multivariate transfer functions, each representing a subsystem of the bilinear system. If a weighted sum of these multivariate transfer functions associated with a reduced bilinear system interpolates a weighted sum of the original multivariate transfer functions, we say that the reduced system satisfies a Volterra series interpolation (Flagg and Gugercin, 2015). These interpolatory conditions can also ensure the necessary conditions for H2-optimal model reduction (Flagg and Gugercin, 2015; Benner and Breiten, 2012). We observe that, by carefully selecting the weights of the series, the Volterra series interpolatory conditions are transformed to the problem of interpolating a linear system with an affine parameter dependence. Such linear parametric systems can then be reduced by some method for parametric model order reduction.Linear systems where the affine parameter dependence is given as low-rank variation in the state matrix can be mapped into a non-parameterized multi-input multi-output linear system. This allows us to utilize the standard (non-parametric) linear IRKA (Gugercin et al., 2008) for the problem of parameterized/bilinear interpolation. Numerical results show that the approximations are of comparable accuracy to those obtained from the bilinear iterative rational Krylov algorithm (Benner and Breiten, 2012). The proposed approach, however, has the advantage that it reduces the computational costs as it involves computations associated with solving linear systems only.

Solvability conditions of problem of synthesizing proportional-integral control of a linear MIMO system

The solvability conditions and just the solution of the problem of the regular and irregular proportional-integral (PI) control are found in accordance with the properties of invariant zeros of a multi-input multioutput (MIMO) system. It is proved that the problem of synthesizing the control of the MIMO system is solvable if and only if the pair of matrices (A, B) that describes a control plant is controllable and the matrix BL?ACR? (where BL? is the left zero divisor of the matrix B and CR? is the right zero divisor of the output matrix C) has a complete row rank.

Predictive Control of Suspension Systems Through Combining Dynamic Matrix and Constrained Variable Structure Controllers

In this paper, a controller called dynamic matrix constrained variable structure controller (DM-CVSC) is proposed. The controller takes advantages of both dynamic matrix (DM) and constrained variable structure controllers. As a result, DM-CVSC is a robust trajectory tracking controller dealing with the constraints on control inputs and also makes decision based on the future behavior of the vehicle. The controller is applied to a linearized model of half-car suspension systems which are subject to different types of road disturbances and measurement noises. In this paper, it is shown that there is a simple formulation for calculating the range of sliding gains for single-input single-output (SISO) linear control systems. As for the multi-input multi-output (MIMO) linear control systems, the calculation of upper sliding gain profile for controller leads to a search problem. To show the efficiency of the proposed controller, it is applied to four different cases involving specific road disturbances and measurement noises. The performance of the proposed controller is compared to various control techniques.

Predictor-Based Adaptive Output Feedback Control: Application to Functional Electrical Stimulation of a Human Arm Model

A simulation study to control the motion of a human arm using muscle excitations as inputs is presented to validate a recently developed adaptive output feedback controller for a class of unknown multi-input multi-output (MIMO) systems. The main contribution of this paper is to extend the results of Nguyen and Leonessa (2014, “Adaptive Predictor-Based Output Feedback Control for a Class of Unknown MIMO Linear Systems,” ASME Paper No. DSCC2014-6214; 2014, “Adaptive Predictor-Based Output Feedback Control for a Class of Unknown MIMO Linear Systems: Experimental Results,” ASME Paper No. DSCC2014-6217; and 2015, “Adaptive Predictor-Based Output Feedback Control for a Class of Unknown MIMO Systems: Experimental Results,” American Control Conference, pp. 3515–3521) by combining a recently developed fast adaptation technique and a new controller structure to derive a simple approach for a class of high relative degree uncertain systems. Specifically, the presented control approach relies on three components: a predictor, a reference model, and a controller. The predictor is designed to predict the systems output for any admissible control input. A full state feedback control law is then derived to control the predictor output to approach the reference system. The control law avoids the recursive step-by-step design of backstepping and remains simple regardless of the system relative degree. Ultimately, the control objective of driving the actual system output to track the desired trajectory is achieved by showing that the system output, the predictor output, and the reference system trajectories all converge to each other. Thelen and Millard musculotendon models (Thelen, D. G., 2003, “Adjustment of Muscle Mechanics Model Parameters to Simulate Dynamic Contractions in Older Adults,” ASME J. Biomech. Eng., 125(1), pp. 70–77; Millard, M, Uchida, T, Seth, A, and Delp, Scott L., 2013, “Flexing Computational Muscle: Modeling and Simulation of Musculotendon Dynamics,” ASME J. Biomech. Eng., 135(2), p. 021005) are used to validate the proposed controller fast tracking performance and robustness.