LMI Conditions Research Articles

Novel reachable set estimation results for delayed linear systems with bounded disturbance

Reachable set estimation is a prominent research topic in control theory and systems science. Reachable set estimation for time-delay systems has drawn a lot of interest. Until now, most approaches to the reachable set estimation problem of time-delay systems have involved building intricate Lyapunov functionals and utilizing LMI techniques. This has resulted in complicated LMI conditions and a proof procedure. The reachable set estimation issue for delayed linear systems with bounded disturbance will be reviewed in this paper. A novel reachable set estimation lemma with a time-delay term is proposed, and based on the acquired lemma, a sufficient condition is given to bound the reachable set estimation of the systems under discussion. In addition, an event-based sufficient criterion for reachable set estimation is also provided. The theoretical analysis in this paper is succinct when compared to previous findings, and the obtained conditions are simple to validate. Lastly, two examples are presented to confirm that the theoretical results hold true.

Toward enhancing nonlinear observers for Lipschitz system: Exploiting the matrix multipliers‐based LMIs

AbstractThis article is focused on the design of an LMI‐based observer for the class of disturbance‐affected nonlinear systems. Two novel LMI conditions are derived by deploying a more general form of the matrix multiplier compared to the one used in the literature. The first method is based on the use of the criterion, while the second one utilizes an ISS notion. Both LMIs are developed by employing the reformulated Lipschitz property, a well‐known LPV approach and the new variant of Young inequality. The key element of the proposed LMI conditions is the incorporation of the novel matrix multipliers which allow us to include some additional decision variables as compared to the methods proposed in the literature. These additional variables add extra degrees of freedom, thus enhancing the LMI feasibility. Furthermore, the effectiveness of the proposed methodologies is showcased through a numerical example.

Simplified LMI Conditions for Takagi-Sugeno Fuzzy Observer Design with Unmeasured Premise Variables

Simplified LMI Conditions for Takagi-Sugeno Fuzzy Observer Design with Unmeasured Premise Variables

A T-S Fuzzy Approach with Extended LMI Conditions for Inverted Pendulum on a Cart

The Inverted Pendulum On a Cart (IPOC) system poses a challenge in control engineering due to its inherent instability, nonlinearity, and underactuation. This addresses the fundamental issues arising from its underactuated nature and introduces an approach that combines Takagi-Sugeno (T-S) fuzzy control with an awareness of real-world constraints to create a control system ensuring both stability and practicality. By aligning theoretical insights with extended considerations, the Linear Matrix Inequality (LMI)-based control design is demonstrated in a comprehensive framework. Theorems are introduced and validated, leading to the derivation of LMI conditions. The simulation results are assessed with accompanying comments to demonstrate the effectiveness of the theorems. Through this integration of T-S fuzzy control with additional considerations, the paper aims to bridge the gap between theory and practical applications, advancing the field of control engineering.

Global stability of multi-agent systems with heterogeneous transmission and perception functions

In this brief, we consider an extended opinion dynamic framework by including both transmitting and perceiving behaviors in the agents’ interactions: the first represents how the agent transmits his own opinion to the neighbors, while the latter models how personal features of the agent affect the final perception of external opinions. The agents’ interactions are modeled by general piecewise linear functions that can be heterogeneous and not necessarily monotone, thus generalizing the analytical framework usually considered in the literature, in particular, the well-known interval consensus (Fontanet al., 2020). In the considered novel multi-agent scenario, we formulate sufficient operative LMI conditions to assess global network asymptotic stability for either a consensus or a cluster equilibrium, without necessarily requiring strong network connectivity. The proposed approach is validated through illustrative examples.

Control Design for Linear Systems With Asymmetric Input Backlash and Dead-Zone Through LMI Conditions

Control Design for Linear Systems With Asymmetric Input Backlash and Dead-Zone Through LMI Conditions

Discrete-time Negative Imaginary Systems from ZOH Sampling

A new definition of discrete-time negative imaginary (NI) systems is provided. This definition characterizes the dissipative property of a zero-order hold sampled continuous-time NI system. Under some assumptions, asymptotic stability can be guaranteed for the closed-loop interconnection of an NI system and an output strictly negative imaginary system, with one of them having a one step advance. In the case of linear systems, we also provide necessary and sufficient frequency-domain and LMI conditions under which the definition is satisfied. Also provided is a simple DC gain condition for the stability results in the linear case.

On state feedback control of a class of NonLinear PDE systems in finite dimension

In this note, we propose some new results on state feedback stabilization of a class of non linear PDE systems that are described by Vlasov-Poisson equations. For doing so, thanks to the Galerkin discontinuous discretization approach, we establish first an explicit state space model in finite dimension. A state feedback controller is then proposed. One of the main futures is that the asymptotic stabilization may be guaranteed under an easy tractable LMI condition. Extension to the observer based control case, in the presence of disturbances or not, was also established. We provide an LMI condition to assure convergence in the H∞ sense.

On Contraction Analysis of Switched Systems With Mixed Contracting-Noncontracting Modes via Mode-Dependent Average Dwell Time

This paper studies contraction analysis of switched systems that are composed of a mixture of contracting and noncontracting modes. The first result pertains to the equivalence of the contraction of a switched system and the uniform global exponential stability of its variational system. Based on this equivalence property, sufficient conditions for a mode-dependent average dwell/leave-time based switching law to be contractive are established. Correspondingly, LMI conditions are derived that allow for numerical validation of contraction property of nonlinear switched systems, which include those with all non-contracting modes.

Extending the TPTool MATLAB toolbox with LMI based observer and disturbance rejection design

Present paper demonstrates the extensions of MATLAB TPtool toolbox. The toolbox does not include the LMI based observer and disturbance rejection design. Thus, the suggested extensions include the observer and H∞ based disturbance rejection LMI conditions for TP models. Furthermore this toolbox contains obsolete algorithms. Therefore, in this paper, the recommended algorithms are presented. Showing the proposed design method is based on a nonlinear dynamic system; modified TP model of the TORA example is used. Thus, the detailed observer and controller design, the disturbance rejection design are presented with LMI stability conditions through the HOSVD based method in the PDC framework.

Dynamic Output Feedback Fault Tolerant Control for Switched Fuzzy Systems With Fast Time Varying and Unbounded Faults

This paper focuses on the fault reconstruction and fault-tolerant control problems of switched fuzzy systems with actuator and sensor faults, where the faults may be fast time-varying and unbounded. The multiple coordinate transformations are introduced, then the reduced-order observer based fault reconstruction method is considered, which can reconstruct the system state, actuator and sensor faults, simultaneously. For the case that there is no disturbance, the LMI conditions are achieved to insure that for any switching signal, the error dynamics can be exponentially stable. Utilizing the estimated fault information, the fuzzy dynamic output feedback fault tolerant controllers are designed to compensate the fault effect and stabilize the systems. The root-mean-square gain is considered to reduce the disturbance influence on fault estimation and fault-tolerant control, in which the zero initial condition is not needed. Finally, simulations are conducted to demonstrate the effectiveness of the considered techniques.

Hidden Markov Model-Based Control for Cooperative Output Regulation of Heterogeneous Multi-Agent Systems under Switching Network Topology

This paper investigates the problem of stochastically cooperative output regulation of heterogeneous multi-agent systems (MASs) subject to hidden Markov jumps using observer-based distributed control. In order to address a more realistic situation than prior studies, this paper focuses on the following issues: (1) asynchronous phenomena in the system mode’s transmission to the controller; (2) the impact of system mode switching on network topology; and (3) the emergence of coupled terms between the mode-dependent Lyapunov matrix and the control gain in control design conditions. Specifically, to reduce the complexity arising from the asynchronous controller-side mode, the leader–state observer is developed so that the solution pair of regulator equations can be integrated into the observer. Furthermore, a linear decoupling method is proposed to handle the emergence of the aforementioned coupled terms; this provides sufficient LMI conditions to achieve stochastically cooperative output regulation for heterogeneous MASs. Finally, the validity of the proposed method is shown through two illustrative examples.

Output regulation of stochastic sampled-data systems with post-processing internal model

This paper deals with the output regulation problem (ORP) of a linear time-invariant (LTI) system in the presence of sporadically sampled measurement streams with the inter-sampling intervals following a stochastic process. Under such sporadically available measurement streams, a regulator consisting of a hybrid observer, continuous-time post-processing internal model, and stabilizer are proposed, which resets with the arrival of new measurements. The resulting system exhibits a deterministic behavior except for the jumps that occur at random sampling times and therefore the overall closed-loop system can be categorized as a piecewise deterministic Markov process (PDMP). In existing works on ORPs with aperiodic sampling, the requirement of boundedness on inter-sampling intervals precludes extending the solution to the random sampling intervals with possibly unbounded support. Using the Lyapunov-like theorem for the stability analysis of stochastic systems, we offer sufficient conditions to ensure that the overall closed-loop system is mean exponentially stable (MES) and the objectives of the ORP are achieved under stochastic sampling of measurement streams. The resulting LMI conditions lead to a numerically tractable design of the hybrid regulator. Finally, with the help of an illustrative example, the effectiveness of the theoretical results are verified.

Classical Control System Analysis Over Lossy Networks: An Easy-to-Use Nyquist Plot Approach

This article presents an approach based on the Nyquist plot to analyze the mean-square stability of a networked control system in which data transmission is subject to the occurrence of packet losses. This result is obtained by translating well known LMI conditions for mean-square stability to conditions that take the form of upper bounds on the <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> norms of the sensitivity and complementary sensitivity transfer matrices. Though the results obtained provide only sufficient conditions, the proposed approach allows to inspect if the stochastic feedback system is mean-square stable by conveniently using classical Nyquist plots. A numerical example is given to illustrate the ease of use of the proposed tool.

Reduced-Complexity LMI Conditions for Admissibility Analysis and Control Design of Singular Nonlinear Systems

We present a reduced-complexity control approach for a class of descriptor nonlinear systems with a nonlinear derivative matrix, possibly singular. To this end, a systematic approach is proposed to obtain an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">equivalent</i> polytopic representation of a given nonlinear system within a compact set of the state-space. This modeling approach has two particular features compared to the related Takagi–Sugeno (T–S) fuzzy model-based framework. First, the model complexity only grows <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">proportionally</i> , rather than exponentially, with the number of premise variables. Second, the vertices of the proposed polytopic models can admit an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infinite</i> number of representations for the same predefined set of premise variables. This <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonuniqueness</i> feature allows introducing some specific slack variables at the modeling step to reduce the control design conservatism. Based on the proposed polytopic representation and Lyapunov stability theory, we derive reduced-complexity admissibility analysis and design conditions, expressed in terms of linear matrix inequalities, for the considered class of descriptor systems. In particular, a new nonlinear control law is proposed for regular descriptor systems to avoid using the extended redundancy form, which may yield numerically complex and conservative results due to the imposed special control structure. Both numerical and physically motivated examples are given to demonstrate the interests of the new control approach with respect to existing T–S fuzzy model-based control results.

Stability analysis and synthesis of discrete-time semi-Markov jump singular systems

This paper investigates the σ-error pth 0) $ ]]> ( p > 0 ) -moment stability and synthesis problems for a class of discrete-time semi-Markov jump singular systems (DTSMJSSs). In view of the Lyapunov functions approach, we first derive the σ-error pth 0) $ ]]> ( p > 0 ) -moment stable behaviour of interested systems. Based on semi-Markov kernel approach and the techniques of eliminating power of matrices (EPMs), we then obtain several LMI conditions for σ-error mean square stability. Subsequently, due to the time-varying singular E-matrix in DTSMJSSs, two different forms of controller gain are exploited in order to design stabilising state-feedback strategies. In particular, computational analysis and comparison of the proposed control schemes are provided, which shows the numerical complexity of the proposed control schemes is vastly lower than that of the existing literature. Finally, an example has been performed to verify the obtained analytical results.

A Data-Driven Approach to the L₂ Stabilization of Linear Systems Subject to Input Saturations

This paper revisits the L2 stabilization problem for linear systems subject to saturating input and energy-bounded additive disturbance from a data-driven point of view. The backbone of the results resides in writing the stability conditions in a compact dedicated structure allowing to exhibit LMI conditions. Based on the assumption that the data collection is informative, the finite L2-gain stabilization of the closed-loop system is addressed by considering a static state-feedback control law. Then, the L2-gain of the closed loop and an inner-approximation of the basin of attraction of the origin for the disturbance-free closed-loop system are characterized. The potential of the method is discussed along the treatment of an academic example.

Stability by averaging via time-varying Lyapunov functions

We study linear continuous-time systems with fast-varying almost periodic coefficients that are piecewise-continuous in time. Recently, a constructive time-delay approach to periodic averaging of systems with a single fast time-scale was introduced and employed to averaging of systems with small time-varying delays (of the order of the small parameter). In this paper we present a novel transformation of the fast varying coefficients. This transformation is suitable for averaging over multiple time-scales, and is applicable to averaging of systems with constant delays, where the value of delay is not small (i.e. essentially larger than the small parameter). We carry out stability analysis by employing time-varying Lyapunov functions (or functionals for the delayed case). The analysis leads to LMI conditions that are always feasible for small enough parameters. Numerical examples demonstrate the efficiency of the proposed approach and its conservatism.

Internal stabilization of three interconnected semilinear reaction-diffusion PDEs with one actuated state

This work deals with the exponential stabilization of a system of three semilinear parabolic partial differential equations (PDEs), written in a strict feedforward form. The diffusion coefficients are considered distinct and the PDEs are interconnected via both a reaction matrix and a nonlinearity. Only one of the PDEs is assumed to be controlled internally, thereby leading to an underactuated system. Constructive and efficient control of such underactuated systems is a nontrivial open problem, which has been solved recently for the linear case. In this work, these results are extended to the semilinear case, which is highly challenging due the coupling introduced by the semilinearity. Modal decomposition is employed, where due to the semilinearity, the finite-dimensional part of the solution is coupled with the infinite-dimensional tail. A transformation is then employed to map the finite-dimensional part into a target system, which allows for an efficient design of a static linear proportional state-feedback controller. Furthermore, a high-gain approach is employed in order to compensate for the semilinear terms. Lyapunov stability analysis is performed, leading to LMI conditions guaranteeing exponential stability with an arbitrary decay rate. The LMIs are shown to always be feasible, provided the number of actuators and the value of the high gain parameter are large enough. Numerical examples demonstrate the proposed approach.

LMI-Based H∞ Observer Design for Nonlinear Lipschitz System

The problem of circle criterion-based H∞ observer design for nonlinear systems is addressed in this article. An LMI-based observer is developed for the nonlinear Lipschitz systems with linear outputs under the presence of noise/disturbances. The proposed LMI condition is obtained by using reformulated Lipschitz property, standard Young's relation and a newly defined positive definite matrix multiplier. The inclusion of the matrix multiplier adds additional numbers of decision variables in the LMI, which provides extra degrees of freedom from a feasibility point of view. Further, the performance of the proposed observer is validated using numerical examples.