Multi-output Linear Systems Research Articles

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.

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.

A Cyclic Small Phase Theorem for MIMO LTI Systems*

This paper introduces a brand-new phase definition called the segmental phase for multi-input multi-output linear time-invariant systems. The underpinning of the definition lies in the matrix segmental phase which, as its name implies, is graphically based on the smallest circular segment covering the matrix normalized numerical range in the unit disk. The matrix segmental phase has the crucial product eigen-phase bound, which makes itself stand out from several existing phase notions in the literature. The proposed bound paves the way for stability analysis of a cyclic feedback system consisting of multiple subsystems. A cyclic small phase theorem is then established as our main result, which requires the loop system phase to lie between -π and π. The proposed theorem complements the celebrated small gain theorem.

Fault detection in uncertain systems via proportional integral observer

This article focuses on the design of a proportional integral observer (PIO) for fault detection in uncertain systems. For multi-output linear systems with actuator fault and disturbances, an approach using two PIOs is designed to estimate both the system states and disturbances simultaneously when the actuator fault is free. Then a residual generator based on the output function is used to identify whether the fault is occurring. For a class of nonlinear systems with constant disturbances, a full-order PIO is designed to estimate the disturbances. Similarly, the residual generator is used to identify whether the fault is occurring. Numerical examples are given to show the efficiency of the proposed approach.

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.

Continuous-Discrete Time Observer Design for Multi-Output Linear Continuous Systems With Aperiodic Asynchronous Measurements

In this letter, a new observer is proposed for the state estimation of time-invariant systems with multiple outputs providing non periodic and asynchronous measurements. The state of the proposed hybrid observer is made up of two parts, the first one corresponding to the estimate which is continuous and the second one corresponding to a kind of output error predictor which is updated when new measurements are received. The convergence of the observer is analyzed and ensured with a condition on the maximum value of the outputs sample rates, with the help of a Lyapunov approach and is further recast using a Sum of Squares decomposition approach. Results of various simulations illustrate the efficiency of the observer.

DrStabilization: A Matlab-based Software to Design Structured Time-delay Controllers

A new Matlab-based software called DrStabilization is introduced. The main purpose of the software is to stabilize multi-input multi-output linear time-invariant systems which may have an arbitrary number of discrete time-delays. The stabilization problem is formulated as the minimization of spectral abscissa function over the controller parameters. By modifying this problem, it is also possible to consider strong stability, i.e., stability achieved by a stable controller. The software allows designing structured dynamic output feedback controllers. By structuring the controller matrices, one can design decentralized controllers and/or well-known servocompansators such as PI/PID/PIR. Furthermore, the software also allows designing time-delay controllers in which case an arbitrary number of artificial time-delays can be introduced both to the I/O channels and the dynamics of the controller itself. The utilization of the software is illustrated through an example.

Generalized algorithms for the approximate matrix polynomial GCD of reducing data uncertainties with application to MIMO system and control

Computation of (approximate) polynomials common factors is an important problem in several fields of science, like control theory and signal processing. While the problem has been widely studied for scalar polynomials, the scientific literature in the framework of matrix polynomials seems to be limited to the problem of exact greatest common divisor computation. In this paper, we generalize two algorithms from scalar to matrix polynomials. The first one is fast and simple. The second one is more accurate but computationally more expensive. We test the performances of the two algorithms and observe similar behavior to the one in the scalar case. Finally we describe an application to multi-input multi-output linear time-invariant dynamical systems.

On mode reconstructability and reconstructability sets of piecewise linear systems

This paper focuses on the problem of mode reconstructability of piecewise linear multi-output systems for the case when a partition of the state space characterises the switching modes. The partition is assumed to be generated by an arbitrary number of parallel hyperplanes. The main results obtained are the necessary and sufficient conditions that the system must satisfy to be mode reconstructible on the whole state space. Furthermore, some other relevant results are also given, such as the characterisation of subsets of the state space where the system might be mode reconstructible whereas being mode unreconstructible anywhere else; and the conditions for mode discernibility/indiscernibility in some particular cases. The results can be summarised as a simple and easy criterion to verify the mode reconstructability property for the considered class of systems. For the sake of clarity, examples are included to illustrate the obtained results.

Non-sequential QFT Design Methodology for Disturbance Rejection Problem in Uncertain Multivariable Systems

A new design approach for quantitative feedback synthesis (QFT) to the disturbance rejection problem in an uncertain multi-input multi-output (MIMO) linear time-invariant system is proposed. The proposed method is centered on converting the problem on uncertain system into a nominal system and the uncertainties are transferred as a output disturbances whose range is known. In particular, we derive the controller bounds for each of the equivalent nominal single-input single-output (SISO) problems. The proposed non-sequential approach is attractive as it brings simplicity and less conservative than an existing independent SISO method. The effectiveness of the proposed approach is illustrated via challenging MIMO systems with both parametric and non-parametric uncertainty. The proposed solution is less conservative than the existing design which leads to reduced feedback cost.

Cascading With Inner Systems: Hankel Singular Values and Characteristic Values

In this paper, new properties of the cascade between a multi-input multi-output linear time-invariant system, referred to in the following as the original system, and an inner system are dealt with. In particular, attention is devoted to the relationship between the Hankel singular values and characteristic values of the cascade system and those of the original one, proving that, if the inner system has gain greater than or equal to one, then the first $n$ Hankel singular values (characteristic values) of the cascade system are greater than or equal to those of the original system. The results are then applied to derive a new property of minimum phase systems.

Employing phase trajectory length concept as performance index in linear power oscillation damping controllers

In this paper, phase trajectory length concept is employed to introduce a performance index in order to investigate the oscillation of linear power systems. At first, Phase Trajectory Length (PTL) concept in the state space of a Multi Input Multi Output (MIMO) linear system is defined as the traversed distance from a certain point in the state-space to the equilibrium point of the system. Moreover, lower and upper bounds of the PTL are computed. In order to evaluate and compare the oscillatory nature of the power system, oscillation number is defined as the ratio of the PTL to the radial distance of a certain point to the equilibrium. Based on this criterion, it is demonstrated that shortening the PTL leads to effective oscillation damping for all variables which are linear combinations of the main states of the intended system. It is proven that considering the aforementioned index in controller design leads to satisfy a certain finite boundary for Integral Absolute Error (IAE) of system states. Predicated on expressed features of the PTL, corresponding Hamilton Jacobi Bellman (HJB) equations with Minimum Length Controller (MLC) is represented to design damping controller. In order to reduce the settling time in oscillation damping, the desired time weight is augmented in the calculation of the considered objective function. The proposed index can be employed to tune the controller parameters. In this regard, a numerical algorithm is suggested to design a full state feedback controller as MLC. At the end, the given linear power examples, both in simulation and experimental results, show the benefits of this approach for analyzing and designing the oscillation damping controller in linear power systems.

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.

An optimal stochastic multivariable PID controller: a direct output tracking approach

ABSTRACTThis paper deals with the design of an optimal stochastic controller possessing tracking capability of any reference output trajectory in the presence of measurement noise. We consider multi-input multi-output linear time-invariant systems and a proportional-integral-derivative (PID) controller. The system under consideration needs not be stable. A recursive algorithm providing optimal time-varying PID gains is proposed for the case where the number of inputs is larger than or equal to the number of outputs. The development of the proposed algorithm aims for per-time-sample minimisation of the mean-square output error in the presence of erroneous initial conditions, measurement noise, and process noise. Necessary and sufficient conditions are provided for the convergence of the output error covariance. In addition, convergence results are presented for discretised continuous-time plants. Simulation results are included to illustrate the performance capabilities of the proposed algorithm. Performance comparison with an optimal stochastic iterative learning control scheme, an optimal PID controller, an adaptive PID controller, and a recent optimal stochastic PID controller are also included.

Data-driven sensor fault diagnosis systems for linear feedback control loops

This paper develops a sensor fault diagnosis (SFD) scheme for a multi-input and multi-output linear dynamic system under feedback control to identify different types of sensor faults (bias, drift and precision degradation), particularly for the incipient sensor faults. Feedback control, leading to fault propagation and disguised fault rectification, imposes the challenge on the data-driven SFD. With only available output data in closed loop, the proposed scheme comprises two stages of residual generation and residual evaluation. In the residual generation, a data-driven identification of the residual generator for the feedback control system is proposed. One class of parameters in the residual generator are estimated using process delays while another class of parameters describing the output dynamic are derived by the Bayes’ formula. The means and variances control charts of online calculated residuals are made to judge the root cause. Two case studies are performed to illustrate the effectiveness of the proposed method.

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.

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.

A stochastic PID controller for a class of MIMO systems

ABSTRACTThis paper deals with the design of a controller possessing tracking capability of any realisable reference trajectory while rejecting measurement noise. We consider discrete-time-varying multi-input multi-output stable linear systems and a proportional-integral-derivative (PID) controller. A novel recursive algorithm estimating the time-varying PID gains is proposed. The development of the proposed algorithm is based on minimising a stochastic performance index. The implementation of the proposed algorithm is described and boundedness of trajectories and convergence characteristics are presented for a discretised continuous-time model. Simulation results are included to illustrate the performance capabilities of the proposed algorithm.