Multi-input Multi-output Linear Time-invariant Systems Research Articles

Theoretical and Algorithmic Study of Inverses of Arbitrary High-Dimensional Multi-Input Multi-Output Linear-Time-Invariant Systems

Nowadays, systems need to be built and/or characterized to handle exceptional circumstances or adapt to a world itself more complex. A typical phenomenon can often be found that systems are required to accommodate high-dimensional inputs and outputs. Although the theories and methods for inverting a single-input single-output (SISO) linear-time-invariant (LTI) system have been well established, the generalized framework (consisting of theories and algorithms) for extending to arbitrary high-dimensional multi-input multi-output (MIMO) scenarios is still unsubstantial in the existing literature as this extension is far from trivial. In this work, we would like to develop such a new framework for governing the inversion of arbitrary high-dimensional discrete-time MIMO LTI systems, where any individual transfer function from a certain input to a certain output may have the infinite-impulse-response (IIR) characteristics. We propose two new inversion algorithms to invert the transfer-function tensors (TFTs) of arbitrary MIMO LTI systems. The pertinent computational complexities are also investigated for our proposed two TFT-inversion algorithms. The approximation of the inverse of an arbitrary TFT by a finite-impulse-response (FIR) TFT is studied and the corresponding approximation-error analysis is derived as well. Finally, numerical evaluations are presented to study the computational complexities with respect to different TFT dimensions and ranks.

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.

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.

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.

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.

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.

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.

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.

Minimal unknown-input functional observers for multi-input multi-output LTI systems

Designing minimum possible order (minimal) disturbance-decoupled proper functional observers for multi-input multi-output (MIMO) linear time-invariant (LTI) systems is studied. It is not necessary that a minimum-order unknown-input functional observer (UIFO) exists in our proposed design procedure. If the minimum-order observer cannot be attained, the observer's order is increased sequentially through a recursive algorithm, so that the minimal order UIFO can be obtained. To the best of our knowledge, this is the first time that this specific problem is addressed. It is assumed that the system is unknown-input functional detectable, which is the least requirement for the existence of a stable UIFO. This condition also is a certificate for the convergence of our observer's order-increase algorithm. Two methodologies are demonstrated to solve the observer design equations. The second presented scheme, is a new design method that based on our observations has a better numerical performance than the first conventional one. Numerical examples and simulation results in the MATLAB/Simulink environment describe the overall observer design procedure, and highlight the efficacy of our new methodology to solve the observer equations in comparison to the conventional one.

Controller Design for Multivariable Linear Time-Invariant Unknown Systems

This paper deals with the design of multivariable controllers for stable linear time-invariant multi-input multi-output systems, with an unknown mathematical model, subject to constant reference/disturbance signals. We propose a new controller parameter optimization approach, which can be carried out experimentally without knowledge of the plant model or the order of the system. The approach has the advantages that controllers can be tuned by perturbing only the initial conditions of the servocompensator, and that the order of the resulting controller can be specified by the designer. Implementation of the proposed controller design approach is described, and an experimental application study of the proposed method applied to a multivariable system with industrial sensor/actuator components is presented to illustrate the feasibility of the design method in an industrial environment.

Closed-loop Optimal Input Design: the Partial Correlation Approach

We consider optimal experiment design for parametric prediction error system identification of linear time-invariant multi-input multi-output systems in closed-loop. The optimization is performed jointly over the controller and the external input. We use a partial correlation approach, i.e. parametrize the set of admissible controller - external input pairs by a finite set of matrix-valued trigonometric moments. Our main contribution is to derive a description of the set of admissible finite-dimensional moment vectors by a linear matrix inequality. Optimal input design problems with constraints and criteria which are linear in these moments can then be cast as semi-definite programs and solved by standard semi-definite programming packages. Our results can be applied to most of the usual model structures, but we assume that the true system is in the model set.

Robust adaptive [formula omitted]-gain control of polytopic MIMO LTI systems — LMI results

Passification-based direct adaptive control is considered for polytopic uncertain linear time-invariant multi-input multi-output systems. Linear Matrix Inequality based results are provided to guarantee that the adaptive algorithm passifies the system whatever the uncertain parameters in some given set. Contributions are based on the introduction of a parallel feed-forward shunt that liberates the strong equality constraint P B = C T G T often used for strict passification. The shunt, combined with the introduction of slack variables, allows a conclusion with easy to test conditions without restrictions on which data is uncertain in the process model. The resulting adaptive control is such that control gain is bounded in a chosen set whatever the bounded disturbance. Moreover, it proves to be not worse than computable parameter-dependent static output-feedback controls with respect to L 2 gain attenuation. A simple academic example illustrates the results.

LMI conditions for robust adaptive control of MIMO LTI systems1

Passification-based direct adaptive control is considered for polytopic uncertain linear time-invariant multi-input multi-output systems. Linear Matrix Inequality based results are provided to guarantee that the adaptive algorithm passifies the system whatever the uncertain parameters in some given set. The contributions are based on the introduction of a parallel feed-forward shunt that liberates from the strong equality constraint PB = CT GT often used for strict passification. The shunt, combined with the introduction of slack variables, allows to produce results without assumptions on which data is uncertain in the process model. Formulas proving robust passification are formulated as Linear Matrix Inequality and can therefore be efficiently tested. A simple academic example illustrates the results.

A unified approach to scrambling filter design

Most speech scrambling systems are either linear periodic filters or can be modeled as such. It is well-known that from an input-output point of view, a periodic filter is equivalent to a multi-input multi-output linear time-invariant system and thus a rational transfer matrix. In this paper, we propose a framework, based on transfer matrices and frequency domain descriptions, for analysis and design of speech scrambling filters. We derive necessary and sufficient conditions for a scrambling system to be insensitive to frame synchronization error. We propose a procedure for the design of scrambling filters that are insensitive to synchronization error and have zero bandwidth expansion. An illustrative design example is given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stabilization of Multi-Input Multi-Output Systems with Variable Operating Conditions

A stabilization problem is considered for linear time-invariant multi-input multi-output systems with variable operating conditions. The plant is assumed to be modeled as an interpolation of two nominal plants described by proper stable factorizations of transfer function matrices. To stabilize the closed-loop system, two kinds of controllers are proposed. One is a fixed controller, and the other is an interpolation of stabilizing controllers for the nominal plants. For each case, sufficient conditions for stabilization are reduced to solvability of certain H∞ control problems. Design algorithms for the proposed controllers are presented in the state space. The relation to the simultaneuos stabilization problem for two nominal plants is also considered.

Algebraic theory of linear time-invariant feedback systems with two-input two-output plant and compensator

This paper present an algebraic theory for linear time-invariant multi-input multi-output systems with two-input two-output plant and compensator, using the factorization approach. This system configuration considered by Doyle and Nett is most general in that any interconnection of two systems can be represented in terms of this scheme. Other system configurations encountered in compensator design problems are special cases of this system configuration. The analysis and synthesis applies to continuous-time as well as discrete-time systems, and to lumped as well as distributed systems, since the algebraic setting is completely general. The compensator parametrization has four degrees of freedom. The paper is self-contained and is tutorial in the sense that it develops the main results of Nett without introducing the concept of containment.

Decoupling linear multiinput multioutput plants by dynamic output feedback: An algebraic theory

This paper presents an algebraic theory for the design of a decoupling compensator for linear time-invariant multiinput multioutput systems. The design method uses a two-input one-output compensator, which gives a convenient parametrization of all diagonal input-output (I/ O) maps and all disturbance-to-output (D/O) maps achievable by a stabilizing compensator for a given plant. It is shown that this method has two degrees of freedom: any achievable diagonal I/O map and any achievable D/O map can be realized simultaneously by a choice of an appropriate compensator. The difference between all achievable diagonal and nondiagonal I/O maps and the cost of decoupling is discussed for some particular algebraic settings.

Pole placement using dynamic compensators

The problem of designing a compensator to obtain arbitrary pole placement in the system consisting of the plant and compensator in cascade is considered. The design uses only those state variables which can be measured. It is shown that for a controllable observable plant a compensator of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta = \min(\nu_{c} - 1, \nu_{o} - 1)</tex> is sufficient to achieve this result. Here <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu_{c}(\nu_{o})</tex> is the controllability (observability) index of the plant. This result is obtained by first showing that any multi-input multi-output linear time-invariant system may be made controllable (observable) from a single input (output) using only output feedback. The main result is then proved in a constructive manner which explicitly relates the compensator parameters to the coefficients of the desired characteristic polynomial.