Multi-output Linear Systems Research Articles

Robust adaptive attenuation of unknown periodic disturbances in uncertain multi-input multi-output systems

In high-performance high-accuracy systems, the attenuation of vibrational disturbances is essential. In this paper, we design and analyze a robust output-feedback adaptive control scheme to attenuate noise-corrupted vibrational disturbances with unknown characteristics for multi-input multi-output linear time-invariant systems in the presence of plant unstructured modeling uncertainties. The conditions and the design parameters that contribute to the performance of the closed-loop system are investigated. The main ingredients of the proposed scheme include the use of a compensator to properly shape the singular values of the plant model, a robust adaptive law to handle unmodeled dynamics and output random noise, as well as an over-parametrization in the estimated parameters in order to provide flexibility for performance improvement. Analysis and simulations together with practical design considerations are presented to demonstrate the efficacy of the proposed scheme.

Design of switching hyperplanes for multi‐inputs multi‐outputs discrete‐time linear systems

The design of sliding switching surfaces consist in finding a manifold where the controlled system dynamics are asymptotically stable. This problem has been studied for a long time, during which a wide range of designing approaches has been proposed, which can be classified into two research methodologies; the first one corresponding to eigenvalues designing approaches and the second one is based on optimal approaches. In addition, some research approaches only can be applied to specific type of systems, e.g. single-input single-output systems or continuous time systems. Finally, the performance or the level of the achieved accuracy among the different approaches varies significantly. The proposed approach facilitates the task of designing sliding surfaces applied to multi-input multi-output (MIMO) linear systems. The presented methodology introduces a switching surface for MIMO linear systems through a simple equation that allows both eigenvalues placement design or optimal methods without any coordinate system representation transformation. In addition, a design parameter is also included for the adjustment of the non-ideal dynamics. Moreover, this approach guarantees high accuracy of the system dynamics, either for arbitrary eigenvalues assignment or for optimal switching surface definitions. In addition, these benefits are attained using a single switching surface equation. An illustrative example is included to show the advantages of the proposed design methodology.

An Adaptive Rational Block Lanczos-Type Algorithm for Model Reduction of Large Scale Dynamical Systems

Multipoint moment matching based methods are considered as powerful methods for model-order reduction problems. They are related to rational Krylov subspaces (classical or block ones) and are based on the selection of some interpolation points which is the major problem for these methods. In this work, an adaptive rational block Lanczos-type algorithm is proposed and applied for model order reduction of dynamical multi-input and multi-output linear time independent dynamical systems. We give some algebraic properties of the proposed algorithm and derive an explicit formulation of the error between the original and the reduced transfer functions. An adaptive method for choosing the interpolation points is also introduced. Finally, some numerical experiments are reported to show the effectiveness of the proposed adaptive rational block Lanczos-type process.

Robust model reference adaptive output feedback tracking for uncertain linear systems with actuator fault based on reinforced dead-zone modification

In this paper, robust model reference adaptive tracking controllers are considered for Single-Input Single-Output (SISO) and Multi-Input Multi-Output (MIMO) linear systems containing modeling uncertainties, unknown additive disturbances and actuator fault. Two new lemmas are proposed for both SISO and MIMO, under which dead-zone modification rule is improved such that the tracking error for any reference signal tends to zero in such systems. In the conventional approach, adaption of the controller parameters is ceased inside the dead-zone region which results tracking error, while preserving the system stability. In the proposed scheme, control signal is reinforced with an additive term based on tracking error inside the dead-zone which results in full reference tracking. In addition, no Fault Detection and Diagnosis (FDD) unit is needed in the proposed approach. Closed loop system stability and zero tracking error are proved by considering a suitable Lyapunov functions candidate. It is shown that the proposed control approach can assure that all the signals of the close loop system are bounded in faulty conditions. Finally, validity and performance of the new schemes have been illustrated through numerical simulations of SISO and MIMO systems in the presence of actuator faults, modeling uncertainty and output disturbance.

FRITを用いた多入出力むだ時間系に対するスミス補償器のチューニング―出力にむだ時間を持つ場合

In this paper, we develop a controller tuning method of the Smith compensator for multi input multi output (MIMO) linear time-invariant systems with output time delays. Particularly, we apply fictitious reference iterative tuning (FRIT), which is one of the controller tuning methods with only one-shot experimental data, to such systems. Since the Smith compensator contains a model of a plant, it is expected that the model is also obtained together with control parameters by FRIT. To realize this expectation, we introduce a special structure of the Smith compensator with tunable parameters. Then, we also show that the application of FRIT to the parameterized Smith compensator yields not only a desired control parameter but also an appropriate mathematical model of a plant. To show the validity of the proposed method, we give an illustrative numerical example.

Design of CNF-based nonlinear integral sliding surface for matched uncertain linear systems with multiple state-delays

This paper presents an approach of achieving high performance and robustness for matched uncertain multi-input multi-output linear systems with external disturbances and multiple state-delays, which are often encountered in practice and are frequently the sources of instability. This scheme is based on composite nonlinear feedback and integral sliding mode control methods. The selection of nonlinear function and the existence of sliding mode are two important issues, which have been addressed. The control law is designed to guarantee the existence of the sliding mode around the nonlinear surface, and the damping ratio of the closed-loop system is increased as the output approaches the set-point. Simulation results are presented to show the effectiveness of the proposed method as a promising way for controlling similar nonlinear systems.

Adaptive observers design for a class of linear descriptor systems

In this paper, we propose a new approach to an adaptive observer design for a class of multi-input–multi-output linear descriptor systems. Global exponential convergence for joint state-parameter estimation is established for noise-free systems. For noise corrupted descriptor systems, it is proved that the state and parameter estimation errors are bounded and converge in the mean to zero, when the noises are bounded and have zero means. Potential applications of the presented adaptive observer are online system identification, fault detection and adaptive control of descriptor systems. Numerical example is presented to illustrate the performance of the proposed adaptive observer.

LTV controller flatness-based design for MIMO systems

In this paper, a flatness-based control strategy for multi-input multi-output linear time-varying systems is proposed in order to track desired trajectories. The control design, based on the use of an exact observer, leads to a polynomial two-degree-of-freedom controller without resolving Bézout’s equation in a time-varying framework. The proposed approach is illustrated with the control of a nonlinear model of the satellite SPOT-5.

Discrete time second order sliding mode observer for uncertain linear multi-output system

This paper presents a second order sliding mode observer (SOSMO) design for discrete time uncertain linear multi-output system. The design procedure is effective for both matched and unmatched bounded uncertainties and/or disturbances. A second order sliding function and corresponding sliding manifold for discrete time system are defined similar to the lines of continuous time counterpart. A boundary layer concept is employed to avoid switching across the defined sliding manifold and the sliding trajectory is confined to a boundary layer once it converges to it. The condition for existence of convergent quasi-sliding mode (QSM) is derived. The observer estimation errors satisfying given stability conditions converge to an ultimate finite bound (within the specified boundary layer) with thickness O(T2) where T is the sampling period. A relation between sliding mode gain and boundary layer is established for the existence of second order discrete sliding motion. The design strategy is very simple to apply and is demonstrated for three examples with different class of disturbances (matched and unmatched) to show the effectiveness of the design. Simulation results to show the robustness with respect to the measurement noise are given for SOSMO and the performance is compared with pseudo-linear Kalman filter (PLKF).

Filtered-dynamic-inversion control for unknown minimum-phase systems with unknown-and-unmeasured disturbances

This article presents a filtered-dynamic-inversion controller for multi-input multi-output linear minimum-phase systems. The controller requires limited model information, specifically, knowledge of the relative degree and the first nonzero Markov parameter. Furthermore, the controller is effective for stabilisation, command following and rejection of unknown-and-unmeasured disturbances. This article analyses the stability and closed-loop performance (i.e. command following error) of filtered dynamic inversion, which is a single-parameter high-parameter-stabilising controller. We show that for sufficiently large parameter the closed-loop is asymptotically stable, and the average power of the performance signal is arbitrarily small.

Overview of System Identification for Power Systems from Measured Responses

Large interconnected power systems are arguably some of the most complicated man-made systems to understand and to characterize. The scale of the problem is immense, involving large numbers of generators, controllers, and transmission lines covering millions of square kilometers. Measurement technology has reached a point where Phasor Measurement Units (PMUs) are being widely installed in power systems all over the world. These devices provide time synchronized (via GPS) phasor measurements from throughout the power grid to Phasor Data Concentrators (PDCs) at power system control centers. These time series can be used to better characterize the system and hopefully, in the long term, to better control the system. This paper presents a tutorial on estimating power system characteristics from measured responses. About a given operating point, power system low-frequency dynamics are well modeled as a high-order, multi-input, multi-output linear system. Of primary interest is the estimation of the inter-area electromechanical modes of the system. These inter-area modes involve generators from one area of the system oscillating against generators in another area of the system. The modes are characterized by their frequency, damping, and shape. In August 1996, the western United States experienced a massive wide spread black out caused by an unstable inter-area mode, involving generators in the north swinging against generators in the south. This paper overviews the problem and examines several methods of estimating the electromechanical modes under different signal conditions. Several real-world examples are given for estimating the electromechanical modes from ambient, transient, or probing situations. When the system is probed, more general state-space and transfer-function models are estimated. Probing a power system with known inputs is challenging and is discussed in this paper. Estimation performance issues are also discussed.

Output feedback adaptive stabilization and command following for minimum phase dynamical systems with unmatched uncertainties and disturbances

In this article, we develop an output feedback adaptive control framework for continuous-time minimum phase multivariable dynamical systems for output stabilisation and command following. The approach is based on a nonminimal state-space realisation that generates an expanded set of states using the filtered inputs and filtered outputs and their derivatives of the original system. Specifically, a direct adaptive controller for the nonminimal state-space model is constructed using the expanded states of the nonminimal realisation and is shown to be effective for multi-input, multi-output linear dynamical systems with unmatched disturbances, unmatched uncertainties and unstable dynamics. The proposed adaptive control architecture requires only knowledge of the open-loop system's relative degree as well as a bound on the system's order. Several illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.

Bounded-error uncertainty domain description for continuous-time state-space model

A method is introduced to determine the uncertainty domains of the estimated parameters for multi-input multi-output linear time-invariant continuous-time systems. This method uses a continuous-time subspace-based algorithm in order to identify the parameters of the system. This identification method is specific because the estimated state-space realisation is described with the help of a state-space canonical form. Then, a bounded-error approach is considered in order to characterise the uncertainty domains of the coefficients of the state-space matrices. A simulation example is included to demonstrate the efficiency of the developed method.

Adaptive control of MI-MO systems with input saturations

In all actual systems, there exist certainly input saturations. However, in conventional adaptive control schemes, the input saturations were ignored. The adaptive controller designed with ignoring the input saturations may give poor performance or lead instability of the closed loop systems. To overcome the problem, in this paper, we propose an adaptive control scheme for multi-input multi-output linear systems with input saturations. In the proposed scheme, it is assumed that there exist all leading principal minors of the high frequency gain matrix of the controlled object and the signs of leading principal minors are known. Under the assumptions, even for unstable controlled objects, we can design a stable adaptive controller for systems with input saturations. Carrying out numerical simulations, it is shown that the control performance of the closed loop system using the proposed adaptive controller can be easily improved by setting only one design parameter.

Robust state controller via reflection coefficient assignment

A solution to the robust pole assignment problem via reflection coefficients of polynomials is provided for discrete-time single-input single-output (SISO) and multi-input multi-output (MIMO) linear systems. For SISO systems a robust state controller and the polytopic uncertain plant which is stabilized by this controller have been found. For MIMO systems the problem is solved for an uncertain interval plant.

Multimodel-based techniques for the identification and adaptive control of delayed multi-input multi-output systems

This study introduces a novel method to identify each individual delay in multi-input multi-output linear systems with unknown external delays, and then to apply an adaptive control strategy. The solution is based on a multi-model scheme consisting of a set of tentative delay models running in parallel along with a switching mechanism aimed at progressively updating them. Moreover, the time-varying tentative delays are proved to converge to the actual delays under certain persistent excitation-type conditions on the input signals. The presented approach can be regarded as an online implementation of an heuristic optimisation method known as pattern search, very uncommon in control systems applications. Finally, numerical examples showing the effectiveness of the proposed method are included, as well as the convergence proofs and stability properties of the closed loop.

Orthogonal Eigenstructure Control for Vibration Suppression

Orthogonal eigenstructure control is a novel active control method for vibration suppression in multi-input multi-output linear systems. This method is based on finding an output feedback control gain matrix in such a way that the closed-loop eigenvectors are almost orthogonal to the open-loop ones. Singular value decomposition is used to find the matrix, which spans the null space of the closed-loop eigenvectors. This matrix has a unique property that has been used in this new method. This unique property, which has been proved here, can be used to regenerate the open-loop system by finding a coefficient vector, which leads to a zero gain matrix. Also several vectors, which are orthogonal to the open-loop eigenvectors, can be found simultaneously. The proposed method does not need any trial and error procedure and eliminates not only the need to specify any location or area for the closed-loop eigenvalues but also the requirements of defining the desired eigenvectors. This method determines a set of limited number of closed-loop systems. Also, the elimination of the extra constraints on the locations of the closed-loop poles prevents the excessive force in actuators.

Computation of positive realization of MIMO hybrid linear systems in the form of second Fornasini-Marchesini model

Computation of positive realization of MIMO hybrid linear systems in the form of second Fornasini-Marchesini modelThe realization problem for positive multi-input and multi-output (MIMO) linear hybrid systems with the form of second Fornasini-Marchesini model is formulated and a method based on the state variable diagram for finding a positive realization of a given proper transfer matrix is proposed. Sufficient conditions for the existence of the positive realization of a given proper transfer matrix are established. A procedure for computation of a positive realization is proposed and illustrated by a numerical example.

Orthogonal Eigenstructure Control with Non-collocated Actuators and Sensors

An orthogonal eigenstructure control method with collocated actuators and sensors was recently developed by the authors. In this paper the application of the method is extended beyond the collocation of the actuators and sensors, including the cases that different numbers of actuators and sensors are used. Orthogonal eigenstructure control is an output feedback control for multi-input multi-output linear systems. This method uses singular value decomposition to find the matrix that spans the null space of the closed-loop eigenvectors. This method regenerates the open-loop system while simultaneously determines a set of eigenvectors that are orthogonal to the open-loop eigenvectors. This method does not attempt to place the eigenvalues of the closed-loop system, nor does it require defining a set of closed-loop eigenvectors. The closed-loop eigenvectors will be within the achievable eigenvector set and the closed-loop poles will be consistent with them. As a result, no extra constraints have been imposed to the system trying to place the closed-loop poles at certain locations such that excessive force in actuators is prevented. Since there are usually some limitations on the location of the actuators and sensors, collocation of actuators and sensors is not always possible. Also, some systems might not have equal number of actuators and sensors. The proposed method in this paper is based on adding virtual actuators and sensors to the closed-loop system in order to extend the application of the orthogonal eigenstructure control to the systems with non-collocated actuators and sensors as well as the systems with different numbers of actuators and sensors.

Design Method of Multi-Input Multi-Output Minimum-Phase State Control System with Decoupling and Feedforward Compensation

In this paper a design of the decoupling control of multi-input multi-output (MIMO) linear system is discussed. A new configuration of the prepositional tandem matrix is shown as a decoupling compensator, and the minimum-phase state control is applied to the resulting decoupled system. In general non minimum-phase characteristics is often accompanied to the decoupled systems. The feedforward compensation makes the non minimum-phase effect of each decoupled scalar system change to delay time. A numerical example is given for the MIMO linear system which conventionally results in non minimum-phase systems.