Uncertain Discrete-time Systems Research Articles

On-line data-driven control for uncertain systems based on greedy algorithm

Considering a result that persistently exciting data can be used to replace the linear system model, this paper is devoted to applying this result in the field of data-driven control of nonlinear systems. An on-line iteration based on greedy algorithm to stabilize uncertain discrete-time systems is proposed. The method tends to obtain approximate optimal control through solving a series of programming problems. Every programming problem is linear for the convenience of solving. Besides, in particular, the method requires few prior conditions, as long as the system is controllable and observable and the equilibrium state of the system is known. First, we prove that under certain circumstances, the solution to our linear matrix inequality can stabilize the system. Next, a multi-objective programming problem is proposed to deal with situations where the required conditions are unknown. Finally, an on-line iteration is used to enhance robustness as well as real-time evaluation. The method is illustrated to be effective through a simulation under repeated experiments.

Lyapunov-based robust model predictive tracking controller design for discrete-time linear systems with ellipsoidal terminal constraint

In this article, a robust model predictive control (MPC) method is introduced based on Lyapunov-theory to control the discrete-time uncertain linear systems due to external disturbances. The structure of the proposed controller consists of two parts designed based on the offline/online schemes. In the offline-scheme, an H ∞ -based robust tracking controller is provided to satisfy the robust and tracking performance. Based on the H ∞ performance, a new linear matrix inequality problem is developed to calculate the offline-controller gains. By considering the Lyapunov-function of the offline-controller as the terminal cost of the MPC and the designed offline-controller, the online optimisation problem (OOP) is structured. This means, the MPC is designed based on the stabilised system to satisfy the physical limitations of the overall controller and the system states. Moreover, to improve the feasibility problem of the OOP, an ellipsoidal terminal constraint is added to the proposed MPC problem. Therefore, compared with the existing min–max MPC and tube MPC, the advantages of the overall controller are feasibility improvement and reducing the computational complexity with less conservatism while the robustness against parametric uncertainties and disturbances is achieved. Two simulation examples are employed to show the superiorities of the proposed robust MPC.

Energy-to-peak filtering for uncertain discrete-time singular bilinear systems with multipath data packet dropouts

In this paper, the energy-to-peak filtering problem for the uncertain discrete-time singular bilinear systems is studied. This paper considers the uncertainties that consist of real systems and the possible data packet dropouts when performance output signals and measurement output signals are transmitted over digital channels. In the course of the study, the phenomenon is elucidated utilising the Bernoulli random binomial distribution. The goal is to design the desired filter, so that the filtering error system is admissible, and meets the specified energy-to-peak performance index. The design conditions of energy-to-peak filters for uncertain discrete-time singular bilinear systems are given by introducing the Lyapunov function and using linear matrix inequalities (LMIs). The impact of uncertainty on singular bilinear systems is addressed using a two-step approach. At last, a numerical illustration is provided to demonstrate the validity of the design.

Pareto-based guaranteed cost control of discrete-time uncertain stochastic systems

The Pareto-based guaranteed cost control (GCC) of discrete-time uncertain stochastic systems in the infinite horizon is studied. Firstly, the convexity of the weighted sum objective function is proved. Based on this convexity, we demonstrate the relationship between Pareto optimality and minimization of the weighted sum objective function. This relationship confirms that the Pareto-based GCC strategy can be obtained by optimizing the weighted sum objective function. Secondly, to address the Pareto-based GCC issue for discrete-time uncertain stochastic systems, we derive the generalized algebraic Riccati inequality (GARI) specific to these systems. Thirdly, the necessary condition for a Pareto-based guaranteed cost controller is derived using the Karush–Kuhn–Tucker (KKT) condition. Lastly, we introduce a method based on linear matrix inequality (LMI) to determine the Pareto-based GCC strategy. This method reduces computational complexity and establishes the sufficient conditions for a Pareto-based guaranteed cost controller. To validate our findings, we provide a example implemented in MATLAB, confirming the accuracy of the proposed approach.

Set-valued state estimators for a class of uncertain discrete-time nonlinear systems

Set-membership state estimation problem for a certain class of discrete-time nonlinear systems is tackled in this work. First, this problem is formulated as a reachability analysis problem. Then, to avoid the wrapping effect, explicit expression of the general solution of this class of systems is used to design new set-valued state estimation schemes. Moreover, for practical considerations, two periodically re-initialized algorithms based on both interval analysis and zonotope computation are introduced. The merit of the proposed approach is illustrated on a numerical example and its performance is compared to that of other methods borrowed from the literature.

Event-triggered H∞ control for uncertain discrete-time singular systems via sliding mode approach

This article analyzes the event-triggered (ET) scheme for uncertain discrete-time singular systems via [Formula: see text] sliding mode control (SMC). The issue of network delay is taken into account, and a Bernoulli stochastic variable is used to explain its distribution feature. A sliding mode surface (SMS) function is built by using ET scheme, and the corresponding sliding mode dynamics (SMD) is combined. Then, based on Lyapunov-Kravoskii functional (LKF) stability theory and linear matrix inequality (LMI) technology, [Formula: see text]-admissibility criterion for closed-loop SMD is deduced. Additionally, an improved SMC rule based on ET strategy is created so that the system trajectories can be driven onto the prescribed SMS in finite time. Finally, the effectiveness of the approach is demonstrated by three simulated examples.

Reliable stabilization of discrete-time nonlinear singular systems with time-varying delays

The problem of delay-dependent reliable control for uncertain discrete-time nonlinear singular system with time-varying delays is investigated. A new set of delay-dependent sufficient conditions are established for the existence of a reliable controller which ensures the singular system to be regular, causal (absence of impulses) and stable for all admissible parameter uncertainties. First, the controller for singular system in the presence of known actuator fault matrix is designed. Further, the result is extended to study the robust stabilization of singular systems with unknown actuator fault matrix. The sensor failures of actuator are considered by a variable, which is varying in a given interval. The conditions for the existence of proposed controller design are established in the form of linear matrix inequality (LMI), which can be easily solved via standard numerical software. Finally, numerical examples with simulation results are provided to illustrate the usefulness of control design and results are compared with existing results to show the less conservatism of the proposed theory.

Output feedback pole-placement in damping region for discrete-time uncertain polytopic systems

ABSTRACT Pole placement with specified damping and decay rate is one of the primary criteria for ensuring good transient response. This control design problem has known solutions for continuous-time systems in terms of the linear matrix inequality (LMI) region. However, a systematic solution for a discrete-time system is an open problem due to the non-convexity of the constant damping locus. This paper addresses the problem of placing the closed-loop poles of linear discrete-time uncertain polytopic systems in a constant damping region. The non-convex damping region is approximated to an elliptical segment that represents an LMI region. Criteria for designing state and output feedback (static as well as dynamic) controllers are then derived. The effectiveness of the proposed approach is demonstrated with examples including a boost converter to improve its transient behaviour in the presence of input voltage and load variations.

Reducing the wrapping effect of set computation via Delaunay triangulation for guaranteed state estimation of nonlinear discrete-time systems

Set computation methods have been widely used to compute reachable sets, design invariant sets and estimate system state for dynamic systems. The wrapping effect of such set computation methods plays an essential role in the accuracy of their solutions. This paper studies the wrapping effect of existing interval, zonotopic and polytopic set computation methods and proposes novel approaches to reduce the wrapping effect for these set computation methods based on the task of computing the dynamic evolution of a nonlinear uncertain discrete-time system with a set as the initial state. The proposed novel approaches include the partition of a polytopic set via Delaunay triangulation and also the representation of a polytopic set by the union of small zonotopes for the following set propagation. The proposed novel approaches with the reduced wrapping effect has been further applied to state estimation of a nonlinear uncertain discrete-time system with improved accuracy. Similar to bisection for interval and zonotopic sets, Delaunay triangulation has been introduced as a set partition tool for polytopic sets, which has opened new research directions in terms of novel set partition, set representation and set propagation for reducing the wrapping effect of set computation.

Robust Observer-Based Proportional Derivative Fuzzy Control Approach for Discrete-Time Nonlinear Descriptor Systems with Transient Response Requirements

This paper proposes an observer-based proportional Derivative (O-BPD) fuzzy controller for uncertain discrete-time nonlinear descriptor systems (NDSs). Representing NDSs with the Takagi–Sugeno fuzzy model (T-SFM), the proportional derivative (PD) feedback method can be utilized in the fuzzy controller design via the Parallel Distributed Compensation (PDC) concept, such that the noncausal problem and impulse behavior are avoided. A fuzzy observer is proposed to obtain unmeasured states to fulfill the PD fuzzy controller. Moreover, uncertainties and transient response performances are taken into account for the NDSs. Then, a stability analysis process and corresponding stability conditions are derived from the Lyapunov theory with the robust control method and the pole constraint. Different from existing research, the Singular Value Decomposition (SVD) and the projection lemma are utilized to transfer the stability conditions into the Linear Matrix Inequation (LMI) form. Because of this reason, the conservatism of the analysis process can be reduced by eliminating the restriction on the positive definite matrix in the Lyapunov function. By giving the proper center and radius parameters of the pole constraint in the O-BPD fuzzy controller design process, the expected transient responses can be obtained for different designers and different practical applications. Finally, the effectiveness and applicability of the proposed O-BPD fuzzy controller are demonstrated by two examples of the simulation.

Fractional-Order Model-Free Adaptive Control with High Order Estimation

This paper concerns an improved model-free adaptive fractional-order control with a high-order pseudo-partial derivative for uncertain discrete-time nonlinear systems. Firstly, a new equivalent model is obtained by employing the Grünwald–Letnikov (G-L) fractional-order difference of the input in a compact-form dynamic linearization. Then, the pseudo-partial derivative (PPD) is derived using a high-order estimation algorithm, which provides more PPD information than the previous time. A discrete-time model-free adaptive fractional-order controller is proposed, which utilizes more past input–output data information. The ultimate uniform boundedness of the tracking errors are demonstrated through formal analysis. Finally, the simulation results demonstrate the effectiveness of the proposed method.

Robust H∞ observer-based model predictive controller for uncertain linear discrete-time systems due to external disturbances

This article investigates an observer-based robust model predictive control (RMPC) design to control the uncertain discrete-time linear systems with disturbances. To make a more practical scheme, it is supposed that the uncertain system has been faced with unknown disturbance and input constraints. The proposed RMPC approach is based on a state feedback control design that ensures the [Formula: see text] performance criterion to attenuate disturbance affections. Furthermore, in view of practical application, the control law is constructed based on the estimated states obtained from the Luenberger observer. Based on Lyapunov’s theory, the input to state practically stability (ISPS) of the closed-loop system is ensured. Appropriate conditions for the ISPS of the closed-loop system and the estimation error are obtained in terms of online linear matrix inequalities (LMIs) which lead to obtaining the time-varying gain matrices of both controller and observer. Finally, to validate the obtained results, the proposed approach is applied to a numerical example and it is compared with the existing control scheme and the superiority is proved.

A New Neural Dynamic Learning Framework for Discrete-Time Strict-Feedback Systems: Internal Interaction-Based Weight Adaptive Laws.

This article investigates internal interaction-based dynamic learning control (LC) for uncertain discrete-time strict-feedback systems. On the basis of predict technology, the original system is converted into a common n -step-ahead input-output predict model. The predict model causes every estimated neural weight to converge to n different constants using the existing control framework. To solve such a problem, the predict model is further decomposed into n one-step-ahead subsystems, which can be viewed as n independent agents. Subsequently, the distributed cooperative weight adaptive laws are designed by introducing an undirected and connected interconnection topology among subsystems. By constructing the variable relationship between the subsystems and the n -step-ahead predict model, a new internal weight interaction-based neural dynamic LC framework is proposed for the whole closed-loop system, in which estimated weights at different times share their weight knowledge. The proposed framework ensures the ultimately uniform boundedness of the closed-loop system and achieves the excellent control performance. By combining the consensus theory and a cooperative persistent excitation condition, every estimated weight along the neural input orbit is verified to exponentially converge to a close vicinity of a unique ideal constant, rather than n different constants. Consequently, the developed LC framework facilitates constant weights storage, saves the knowledge storage space, and improves the robustness of knowledge utilization. These characteristics are verified by simulation results.

Reference Governor for Linear Uncertain Systems with Time-Varying Constraints

This paper considers the control of uncertain discrete-time linear systems subject to time-varying polytopic constraints. The proposed methodology recasts the problem into one with time-invariant constraints and an unmeasured disturbance input. Through the use of an extended observer the disturbance input is estimated and canceled. Then, a time dependent sequence of safe sets is employed to bound the estimation error. A robust reference governor based on such a sequence is designed to robustly ensure constraint enforcement. Simulation results for several dynamical systems are reported.

An Adaptive Distributed Observer for a Class of Discrete-Time Uncertain Linear Leader Systems Over Jointly Connected Switching Networks

An Adaptive Distributed Observer for a Class of Discrete-Time Uncertain Linear Leader Systems Over Jointly Connected Switching Networks

Recursive Estimator-Based Fuzzy Adaptive Control for Discrete-Time Uncertain Systems with State Saturations and Missing Measurements

Recursive Estimator-Based Fuzzy Adaptive Control for Discrete-Time Uncertain Systems with State Saturations and Missing Measurements

Online control synthesis for uncertain systems under signal temporal logic specifications

Signal temporal logic (STL) formulas have been widely used as a formal language to express complex robotic specifications, thanks to their rich expressiveness and explicit time semantics. Existing approaches for STL control synthesis suffer from limited scalability with respect to the task complexity and lack of robustness against the uncertainty, for example, external disturbances. In this paper, we study the online control synthesis problem for uncertain discrete-time systems subject to STL specifications. Different from existing techniques, we propose an approach based on STL, reachability analysis, and temporal logic trees. First, based on a real-time version of STL semantics, we develop the notion of tube-based temporal logic tree (tTLT) and its recursive (offline) construction algorithm. We show that the tTLT is an under-approximation of the STL formula, in the sense that a trajectory satisfying a tTLT also satisfies the corresponding STL formula. Then, an online control synthesis algorithm is designed using the constructed tTLT. It is shown that when the STL formula is robustly satisfiable and the initial state of the system belongs to the initial root node of the tTLT, it is guaranteed that the trajectory generated by the control synthesis algorithm satisfies the STL formula. We validate the effectiveness of the proposed approach by several simulation examples and further demonstrate its practical usability on a hardware experiment. These results show that our approach is able to handle complex STL formulas with long horizons and ensure the robustness against the disturbances, which is beyond the scope of the state-of-the-art STL control synthesis approaches.

Decentralized Controller Design for Large-Scale Uncertain Discrete-Time Systems with Non-Block-Diagonal Output Matrix

This research paper addresses the challenge of designing a decentralized controller for a discrete-time uncertain polytopic system with a linear large-scale (LSS) structure. Specifically, we investigate this problem in cases where the subsystem’s output matrix lacks a decentralized structure. Firstly, the proposed novel procedure of a decentralized controller design transforms the LSS model to have a fully decentralized structure (both input and output matrices are block-diagonal). Then, the robust stability boundary parameter is calculated for the open-loop system. This stability boundary parameter is considered in robust decentralized controller design where an appropriate controller design method is used. The entire process of designing a robust decentralized controller takes place at the subsystem level, and the influence of interaction is considered through the robust stability boundary parameter. Lastly, we present an example of a five-order system comprising two subsystems to show the effectiveness of the new method.

Reference regulation of constrained uncertain discrete-time systems via robust predictive integral control

Reference regulation of uncertain systems is investigated in the discrete-time case. Hence, including the uncertainties and constraints, through a prediction-based controller, the set-point following will be robustly fulfilled in a class of uncertain systems. A novel idea is addressed to achieve this goal optimally. Thus the control synthesis is deliberately transformed into an optimisation issue subjected to some matrix inequality. In the real-time implementation, the controller’s parameters would be obtained when a minimisation problem is solved. The applicability of the method is evaluated in some examples. The superiority and robustness of the suggested procedure are remarkably validated by comparing it with a similar one.

Convex Temporal Convolutional Network-Based Distributed Cooperative Learning Control for Multiagent Systems.

Due to its great efficiency, scalability, and inclusivity, distributed cooperative learning control has gotten a lot of attention. For complex uncertain multiagent systems, it is challenging to model the uncertainties and exploit the cooperative learning ability of the systems. To address these issues, we proposed a novel convex temporal convolutional network-based distributed cooperative learning control for uncertain discrete-time nonlinear multiagent systems. A new concept of using a convex temporal convolutional network (CTCNet) is proposed for estimating the uncertain agent dynamics in a cooperative way. Unlike previous methods that require adjustment of network weights for different control tasks, the proposed CTCNet can map the high-dimensional input-output space into a deep space spanned by basis features that represent the inherent properties of the system, so it has good robustness for different tasks. Consequently, to improve the control performance, a CTCNet-based distributed cooperative learning control method that shares learned knowledge through the communication topology among adaptive laws of CTCNet is proposed. Furthermore, the asymptotic convergence of system tracking errors to an arbitrarily small neighborhood of the origin is strictly proved. Finally, the simulation results are given to illustrate that our suggested method has higher control accuracy, stronger robustness, and anti-interference ability than the existing methods.