Uncertain Continuous-time Systems Research Articles

Quadratic stability of uncertain large-scale networked systems

In this paper, the quadratic stability of continuous-time uncertain large-scale networked systems is considered. The systems are composed of many subsystems distributed in different locations in space and interconnected according to a certain network topology. A computationally efficient sufficient condition for the quadratic stability of large-scale networked systems is established, utilising effectively the block diagonal structure of the system parameter matrix and the sparseness of the subsystem connection matrix (SCM). Furthermore, a stability condition is presented, which is entirely contingent on the parameters of each subsystem. The simulation results show that the obtained conditions have obvious advantages in analysing the quadratic stability of large-scale networked systems.

Robust H∞ Performance Analysis of Uncertain Large-Scale Networked Systems

This paper considers the robust H∞ performance problem of continuous-time uncertain large-scale networked systems (LSNSs). The systems consist of numerous arbitrarily connected subsystems, each of which has different dynamics. Currently it is computationally difficult to manage systems with the existing lumped analysis method; therefore, exploiting the structural properties of the systems, sufficient conditions are derived for robust H∞ performance. Based on these results, an analysis condition that depends only on the parameters of each single subsystem is further obtained. Some numerical simulations are proposed to verify the validity and superiority of the developed conditions.

Non-fragile robust model predictive controller design for uncertain time-delay systems with input constraints

ABSTRACT This paper addresses a non-fragile robust model predictive control design for a class of continuous-time uncertain systems with multiple state-delay and constrained control signals. The parameters of the system and the control gain are assumed to have perturbation in the additive form. The Lyapunov–Krasovskii functional approach is employed to derive sufficient conditions for determining a non-fragile robust state-feedback controller for all admissible uncertainties by minimising the upper bound of the defined quadratic cost function with respect to some linear matrix inequalities (LMIs). An additional inequality condition is imposed to address the constraint associated with the control signals. Numerical simulations are provided to assess the performance of the proposed controller.

Design of a preview repetitive control with equivalent-input-disturbance system based on a continuous-discrete 2D model

This paper deals with the problem of two-dimensional (2D) system-based preview repetitive control (PRC) with equivalent-input-disturbance (EID) for uncertain continuous-time systems. First, to use the available values of the reference signal, we construct an equality constraint which includes the output of a basic repetitive controller, preview compensation and tracking error. Next, to compensate the unknown external disturbances, we incorporate an EID estimator into the PRC controller. Then, by employing the 2D system theory together with the linear matrix inequality (LMI) approach, we derive a sufficient condition to ensure the robust stability of the closed-loop system. By solving an LMI, the gains of the controller and state observer can be obtained. The results obtained in this paper generalize and include some results in the existings. literature. Finally, a numerical simulation demonstrates the effectiveness and superiority of the proposed method.

Reliable induced [formula omitted] control for polytopic uncertain continuous-time singular systems with dynamic quantization

This paper considers the problem of state feedback control for a class of polytopic uncertain continuous-time singular systems with input quantization and actuator faults. The aim of this paper is to design the reliable controller and dynamic quantizer that ensure the considered closed-loop system is asymptotically stable and satisfies the prescribed induced L∞ performance constraint. The introduced proportional plus derivative state feedback controller can eliminate the structural restrictions of the system matrices, thus a more general sufficient condition is obtained for the problem. Finally, the four-mesh circuit is provided to demonstrate the effectiveness of the proposed induced L∞ control approach.

Temporal logic guided safe model-based reinforcement learning: A hybrid systems approach

This paper studies the problem of synthesizing control policies for uncertain continuous-time nonlinear systems from linear temporal logic (LTL) specifications using model-based reinforcement learning (MBRL). Rather than taking an abstraction-based approach, we view the interaction between the LTL formula’s corresponding Büchi automaton and the nonlinear system as a hybrid automaton whose discrete dynamics match exactly those of the Büchi automaton. To find satisfying control policies, we pose a sequence of optimal control problems associated with states in the accepting run of the automaton and leverage control barrier functions (CBFs) to prevent specification violation. Since solving many optimal control problems for a nonlinear system is computationally intractable, we take a learning-based approach in which the value function of each problem is learned online in real-time. Specifically, we propose a novel off-policy MBRL algorithm that allows one to simultaneously learn the uncertain dynamics of the system and the value function of each optimal control problem online while adhering to CBF-based safety constraints. Unlike related approaches, the MBRL method presented herein decouples convergence, stability, and safety, allowing each aspect to be studied independently, leading to stronger safety guarantees than those developed in related works. Numerical results are presented to validate the efficacy of the proposed method.

Robust approximate symbolic models for a class of continuous-time uncertain nonlinear systems via a control interface

Discrete abstractions have become a standard approach to assist control synthesis under complex specifications. Most techniques for the construction of a discrete abstraction for a continuous-time system require time–space discretization of the concrete system, which constitutes property satisfaction for the continuous-time system non-trivial. In this work, we aim at relaxing this requirement by introducing a control interface. Firstly, we connect the continuous-time uncertain concrete system with its discrete deterministic state-space abstraction with a control interface. Then, a novel stability notion called η-approximately controlled globally practically stable, and a new simulation relation called robust approximate simulation relation are proposed. It is shown that the uncertain concrete system, under the condition that there exists an admissible control interface such that the augmented system (composed of the concrete system and its abstraction) can be made η-approximately controlled globally practically stable, robustly approximately simulates its discrete abstraction. The effectiveness of the proposed results is illustrated by two simulation examples.

Higher-Order Sliding Mode design with Bounded Integral Control generation

In this paper uncertain continuous-time nonlinear systems affine in the control variable and with saturated actuators are considered. The finite-time regulation problem of the system output to zero is then solved by proposing a generic Higher-Order Sliding Mode (HOSM) controller equipped with a novel mechanism to encounter the saturation limits, thus extending previous results on saturated control inputs valid only in case of specific r-order sliding mode algorithms. The so-called Bounded Integral Control (BIC) method is reformulated into the framework of continuous HOSM, so as to replace the traditional integrator used to generate the continuous signal directly fed into the plant. Stability conditions for tuning the proposed algorithm are provided, and a numerical example finally assesses the effectiveness of the proposed technique.

Robust Tracking Control for Non-Zero-Sum Games of Continuous-Time Uncertain Nonlinear Systems

In this paper, a new adaptive critic design is proposed to approximate the online Nash equilibrium solution for the robust trajectory tracking control of non-zero-sum (NZS) games for continuous-time uncertain nonlinear systems. First, the augmented system was constructed by combining the tracking error and the reference trajectory. By modifying the cost function, the robust tracking control problem was transformed into an optimal tracking control problem. Based on adaptive dynamic programming (ADP), a single critic neural network (NN) was applied for each player to solve the coupled Hamilton–Jacobi–Bellman (HJB) equations approximately, and the obtained control laws were regarded as the feedback Nash equilibrium. Two additional terms were introduced in the weight update law of each critic NN, which strengthened the weight update process and eliminated the strict requirements for the initial stability control policy. More importantly, in theory, through the Lyapunov theory, the stability of the closed-loop system was guaranteed, and the robust tracking performance was analyzed. Finally, the effectiveness of the proposed scheme was verified by two examples.

Robust boundary of stability and design of robust controller in uncertain polytopic linear systems

Abstract This paper is devoted to design of a robust controller for an uncertain continuous-time linear polytopic system. The design procedure will take place in two steps. In the first step, we calculate the robust stability boundary of a closed loop system. Then on the basis of the result obtained in this way, we select a robust controller design method that accepts the calculated stability boundary. This procedure allows better implementation of the second step of the designed robust controller. Two examples of application of the proposed method follow finally, which illustrate its effectiveness.

Decentralized robust preview control for uncertain continuous-time interconnected systems

This paper investigates the problem of robust preview tracking control for uncertain continuous-time interconnected systems. We adopt the technology of decentralized control. The tracking controller for each uncertain isolated subsystem is first designed. With the help of an uncertain error system, the considered tracking problem is converted into a regulation problem. The preview tracking controller for the nominal system is obtained by utilizing existing results. A novel method is proposed to tackle the uncertainty in the systems. In this method, the uncertainty bound is incorporated into the cost function. As a result, the preview tracking controller of the nominal system can be treated as a robust preview controller of the uncertain isolated subsystem. Then, the controllers of the uncertain isolated subsystems are assembled as a robust preview controller of uncertain interconnected systems. Finally, an academic numerical example is presented to show the effectiveness and superiority of the proposed controller.

Designing of a non‐fragile robust predictive controller for uncertain systems with time‐varying delay: A delay‐range‐dependent approach

This study addresses a new delay-dependent non-fragile robust model predictive controller for a class of uncertain continuous-time systems with time-varying delay subjected to constrained input and perturbed controller gain. The uncertainties in system parameters and controller gain are considered as general additive forms. The proposed controller can provide a suitable linear feedback control law depending on delay such that robust performance is achieved to the possible existence of uncertainties. To this end, an optimisation problem is formulated to minimise the upper-bound of the cost function involving some conditions in terms of linear matrix inequalities, utilising the information of lower-and upper-bounds of time-varying delay. By including information regarding the delay range in the quadratic function, the ability to have less conservative results compared to the related works in literature is provided. The effectiveness of the proposed technique is illustrated through a comparison with recent works.

LMI conditions for evaluating and increasing stability margins of uncertain continuous-time linear systems

This article investigates the robust stability of uncertain continuous-time linear systems with parameters belonging to intervals. Differently from the techniques available in the literature, the proposed method, formulated in terms of an iterative algorithm based on LMIs, treats the interval bounds as optimization variables and can be used, for instance, to search for the maximum variation around nominal values. Experiments involving the fragility analysis of controllers and volume maximization of the polyhedral region centered on nominal parameters are presented to illustrate possible applications of the proposed technique.

H∞Model Match Output-Feedback Control by Means of an LMI-Based Algorithm

This letter addresses the problem of model match control for uncertain continuous-time linear systems by means of fixed order dynamic output-feedback. The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm is used as performance metric for the approximation error and the synthesis conditions are formulated in terms of LMIs, which are solved iteratively. The novelty of the approach with respect to previous results is that the matrices of the controller, as well as the matrices of the reference model, appear affinely in the conditions, allowing the immediate treatment of structured controllers and providing extra degrees of freedom for the reference model, in the sense that some of the entries of the matrices do not need to be fixed a priori and, therefore, can be determined by the synthesis conditions. This latter feature allows an extension of the method to address the problem of order reduction of uncertain systems. The results are illustrated by an example based on a model borrowed from the literature.

Optimal control for uncertain random continuous-time systems

Chance theory is a useful tool to deal with the analysis of indeterminacy, including both uncertainty and randomness. Based on chance theory, an optimal control for uncertain random continuous-time systems is introduced to design dynamic optimization problems. Applying the method of dynamic programming, the principle of optimality is presented and then the equation of optimality is provided to solve the proposed problem. Meanwhile, three special cases of optimal control problems are discussed by using the equation obtained. Finally, a numerical example and an optimal cash balance problem are given to show the effectiveness of the results achieved.

Optimal control and zero-sum differential game for Hurwicz model considering singular systems with multifactor and uncertainty

Uncertainty theory is a new branch of mathematics concerned with the analysis of subjective indeterminacy. Based on the concept of uncertain theory, this paper considers optimal control and zero-sum differential game for multi-factor uncertain continuous-time singular systems with Hurwicz criterion, where dynamic systems are modelled by a type of uncertain differential equation driven by canonical Liu process. Using the approach of dynamic programming, the principle of optimality is proposed and then the equation of optimality is derived to resolve an uncertain control model. A two-player zero-sum uncertain differential game is further investigated based on the optimality equation above, and an equilibrium equation is presented to locate a saddle-point in the game. Finally, an example is discussed to illuminate the effectiveness of the results.

Reduced-order interval observer based consensus for MASs with time-varying interval uncertainties

In this paper, the consensus problem of a continuous-time uncertain multi-agent system (MAS) with time-varying interval uncertainties in the system matrix and unknown initial states is investigated via output feedback. Relating only to the interval bounds on the uncertainties, a reduced-order framer, which includes two Luenberger-like observers, is first derived. Then, a distributed control algorithm relying only on the information of the framers associated with each agent and its neighbors is proposed. It is shown that the control algorithm can render the reduced-order framer to be an exponential reduced-order interval observer, and simultaneously drive the uncertain MAS to reach robust consensus exponentially. Finally, numerical simulations for a large-scale MAS are used to verify the theoretical results.

Robust gain-scheduling [formula omitted] control of uncertain continuous-time systems having magnitude- and rate-bounded actuators: An application of full block S-procedure

This paper addresses a novel robust gain-scheduling(GS) state-feedback (SF) H∞ (induced L2) control method for uncertain continuous-time(CT) systems having actuators with hard rate and magnitude bounds. The proposed method relies on acceleration form representation of the uncertain CT system. The technique enables the user to represent the control signal and its slew rate as auxiliary outputs along with the main controlled output. Two different norms, namely the induced L∞ and L2 are used to control the system effectively. While the L∞ gain from the exogenous disturbance inputs to the control signal related outputs is used to deal with actuator saturation problem, L2 gain from the disturbance inputs to the performance outputs is utilised in attenuation of the effects of the disturbances. To achieve these goals with minimal conservatism, we develop new forms of dilated Matrix Inequality (MI) conditions for peak-to-peak gain and L2 gain with no additional conservatism. Then, we propose a novel robust GS control methodology to deal with the problem via dilated MIs and a modified full block S-procedure method (MFBSPM). The utilisation of MFBSPM ensures that the robust control design method of this note applies to any uncertain system having rational parameter dependence. Finally, the efficiency of the presented method is portrayed through extensive simulations of the responses of a marine vessel to wave excitations in which the vessel model is assumed to have magnitude and rate limited active fin stabiliser.

Output-Feedback Robust Control of Uncertain Systems via Online Data-Driven Learning.

Although robust control has been studied for decades, the output-feedback robust control design is still challenging in the control field. This article proposes a new approach to address the output-feedback robust control for continuous-time uncertain systems. First, we transform the robust control problem into an optimal control problem of the nominal linear system with a constructive cost function, which allows simplifying the control design. Then, a modified algebraic Riccati equation (MARE) is constructed by further investigating the corresponding relationship with the state-feedback optimal control. To solve the derived MARE online, the vectorization operation and Kronecker's product are applied to reformulate the output Lyapunov function, and then, a new online data-driven learning method is suggested to learn its solution. Consequently, only the measurable system input and output are used to derive the solution of the MARE. In this case, the output-feedback robust control gain can be obtained without using the unknown system states. The control system stability and convergence of the derived solution are rigorously proved. Two simulation examples are provided to demonstrate the efficacy of the suggested methods.

Detection of actuator faults for continuous-time systems with intermittent state feedback

This paper considers the problem of detecting actuator faults on uncertain continuous-time systems, where the state can be measured only intermittently. Initially, we describe the class of faults which we want to identify, as those that create a state evolution that is incompatible with the model of the system. We then derive a necessary and sufficient condition, the verification of which allows us to detect the aforementioned inconsistencies. Relaxations are employed to make this verification computationally easier and an equivalent condition, in the ϵ − closeness sense, is obtained, through the solution of a boundary-value problem. We finally present simulation results that showcase the effectiveness of our proposed algorithm, in terms of speed and validity.