The matrix equation E X F − AX = C is investigated in this study, and three approaches are provided to solve this equation. The first approach is to transform it into a real matrix equation with the help of real representation of complex matrices. In the second approach, the solution is given in terms of characteristic polynomial of a constructed matrix pair. In the third approach, the solution can be neatly expressed in terms of controllability matrices and observability matrices. By specialising the obtained solutions of E X F − AX = C, some new expressions of the generalised Sylvester matrix equations are also provided.

The authors consider the problem of robust output estimation in the discrete-time setting. The uncertainty in the system is characterised by integral quadratic constraints with dynamic multipliers. The main technical contribution of this study is to derive two different synthesis conditions in terms of linear matrix inequalities (LMI) by two different approaches. The LMI condition derived by the first approach allows one to find the robust estimator and the performance certificate simultaneously, whereas the condition by the second approach has a smaller size but requires an additional step for recovering the estimator. The synthesis conditions are validated by numerical examples. The results verify the equivalence of the two approaches and show that the two-step approach is in fact computationally favourable when the system has large number of state variables.

This study investigates the observability problem and the observer design of partially observed finite automata via a matrix approach. Using semi-tensor product of matrices, finite automata are modelled in the form of discrete-time bilinear systems. Matrix-form necessary and sufficient conditions for both the initial and current state observability, either with or without input information, are first proposed. Based on that, a constructive method for the observer design is provided.

An output-feedback sliding-mode controller is proposed for multivariable uncertain systems with unknown time-varying state delays and unmatched non-linear terms. The non-linearities are allowed to depend not only on the system output, but also on its unmeasurable state, and can be polynomial. This scheme is based on model-reference unit vector control under a mild Hurwitz condition for the high-frequency gain matrix. To achieve global stability properties, the amplitude of the control signal is given by a norm observer for the unmeasured state. For the first time, the design of norm observers is applied to a class of time-delay systems. This approach guarantees finite-time convergence of the output-tracking error to zero. Simulation results illustrate the application of the developed control scheme.

In this study, a fractional-order adaptive-sliding mode control (SMC) scheme is proposed to stabilise commensurate fractional-order polytopic non-linear differential inclusion systems containing sector and dead-zone nonlinearities in the control inputs and unknown bounded disturbances. The suggested control method is composed of fractional-order sliding surfaces, adaptive-SMC inputs and adaptation laws for unknown bounds of disturbances. The Lyapunov stability theorem is used to prove the stability of the closed-loop system. A practical system and two numerical examples are simulated to show the effectiveness and performance of the proposed control technique.

This study investigates the optimal tracking performance of multiple-input multiple-output linear time-invariant systems over a noisy channel in the feedback path. The communication channel is also subject to data dropouts, which was modelled as a binary stochastic process. The problem under consideration amounts to determining the minimal error in tracking a Brownian motion random process, which emulates a step reference signal in the deterministic setting. With the unity feedback and two-parameter control structure, a lower bound and an exact expression of optimal tracking performance are obtained respectively. It is shown how the noise may degenerate the tracking performance and how the data dropouts effect may intertwine with unstable poles and non-minimum phase zeros, including the location and direction, which are intrinsic characteristics of the plant. Finally, computer simulations are provided to illustrate the analytical results obtained.

The authors are concerned in this study by bounded control of single input non-linear under-actuated mechanical systems. The authors focus the exposition on a feedback-based stabilisation-bounded control action shaped by saturation functions, and the proposed approach was illustrated via the design and the experimental evaluation of a simple stabilising controller for the cart–pendulum system, a well-known control benchmark. The proposed simple control strategy is built around a lumped linear continuous time-invariant description of the concerned under-actuated non-linear system. Namely, a model consisted of a cascade non-linear dynamical system constituted by a chain of four integrators affected by a high-order smooth non-linear perturbation. Assuming initialisation of the under-actuated system to the upper-half plane, the proposed feedback-based regulation design procedure involves the simultaneous combination of two control actions: one bounded linear and one bounded quasi-linear. Control boundedness is provided in both involved control actions by specifically designed saturation functions. The first bounded control action brings the non-actuated coordinate near to the upright position and keep it inside of a well characterised small vicinity, whereas the second bounded control action asymptotically brings the whole state of the dynamical system to the origin. The necessary closed-loop stability analysis uses standard linear stability arguments as well as the traditional well-known Lyapunov method and the LaSalle's invariance principle. The proposed control law ensures global stability of the closed-loop system in the upper-half plane.

This study is concerned with the problem of exponential stability for a class of switched positive linear systems consisting of both stable and unstable subsystems. The sufficient conditions of exponential stability are established in continuous-time and discrete-time domains. Based on the average dwell-time approach, new stability results for such kind of systems are first derived, which allows the ascent of the multiple linear copositive Lyapunov functions caused by unstable subsystems. Furthermore, when all subsystems are stable, the exponential stability condition for switched positive systems is presented. Finally, numerical examples are given to illustrate the effectiveness of the results.

This article studies the problem of distributed filtering for jump Markov linear systems in a not fully connected sensor network. A distributed consensus filter is developed by applying an improved interacting multiple model approach in which the mode-conditioned estimates are derived by the Kalman consensus filter and the mode probabilities are obtained in the sense of linear minimum variance. A numerical example is provided to demonstrate the effectiveness of the proposed algorithm for tracking a manoeuvring target in a sensor work with eight nodes.

This study is concerned with the problem of ℋ₂ state-feedback control design for discrete-time Markov jump linear systems (MJLS), assuming that the transition probability matrix is not precisely known, but belongs to a polytopic domain, or contains unknown or bounded elements. As a first contribution, the uncertainties of the transition probability matrix are modelled in terms of the Cartesian product of simplexes, called multi-simplex. Thanks to this representation, the problem of robust mean square stability analysis with an ℋ₂ norm bound can be solved through convergent linear matrix inequality (LMI) relaxations constructed in terms of polynomial solutions. The proposed conditions yield a better trade-off between precision and computational effort when compared with other methods. As a second contribution, new conditions in terms of LMIs with a scalar parameter lying in the interval (− 1, 1) are proposed for ℋ₂ state-feedback control with complete, partial or no observation of the Markov chain. Owing to the presence of the scalar parameter, less conservative results when compared with other conditions available in the literature can be obtained, at the price of increasing the associated computational effort. Numerical examples illustrate the advantages of the proposed methodology.