Lower Triangular Systems Research Articles

Quantized-output-feedback practical prescribed-time design strategy for decentralized tracking of a class of interconnected nonlinear systems with unknown interaction delays

This paper proposes a decentralized practical prescribed-time (PT) tracking design using quantized output feedback (QOF) for uncertain interconnected lower-triangular systems with unknown time-delay interconnections. The local output signals are assumed to be only measured and quantized for the PT tracker design under a band-limited network. By employing a PT-dependent scaling function, a decentralized memoryless PT observer based on quantized local outputs is developed to estimate local unmeasurable state variables. Owing to output quantization, the available output feedback signals become discontinuous. As a result, the tracking error between the actual (i.e., unquantized) local output and the local desired signal cannot be utilized in the local virtual controller. To address this issue, a novel adaptive compensation mechanism is derived to design the local PT neural network tracking laws using only quantized local outputs and estimated states. The proposed PT tracking controller does not require information on the interconnected nonlinear functions and interaction delays. During the Lyapunov stability analysis, the boundary layer error decomposition approach is employed to address the issue of non-differentiability in the local virtual control laws. The proposed QOF control system achieves practical PT stability. It is shown that the settling time of local tracking errors can be predetermined, regardless of the design parameters and initial conditions. Finally, the proposed QOF decentralization strategy is supported with illustrative examples and a comparison to demonstrate its benefits.

Switching Event-Triggered Adaptive Resilient Dynamic Surface Control for Stochastic Nonlinear CPSs With Unknown Deception Attacks.

This work concentrates on the adaptive resilient dynamic surface controller design problem for uncertain nonlinear lower triangular stochastic cyber-physical systems (CPSs) subject to unknown deception attacks based on a switching threshold event-triggered mechanism. The adverse effect of deception attacks on the stochastic CPSs is that the exact system state variables become unavailable. Furthermore, it should be emphasized that the coexistence of unknown nonlinearities, stochastic perturbations, and unknown sensor and actuator attacks makes it a very difficult and challenging event to implement the control design. To get the desired controller, radial basis function (RBF) neural networks (NNs) are introduced so that the design obstacle caused from the unknown nonlinearities can be easily solved. On this basis, in order to save resources and effectively transmit, the event-triggered control scheme based on a switching threshold strategy is further considered. In the backstepping design process, the dynamic surface control (DSC) technique is presented to deal with the issue of "explosion of complexity." By skillfully designing a new coordinate transformation and the attack compensators, the problem of unknown deception attacks is successfully handled. Under our proposed control scheme, all the closed-loop signals are bounded in probability and the stabilization errors converge to an adjustable neighborhood of the origin in probability. Finally, the simulation results on the double chemical reactor show the validity of the proposed design scheme.

Prescribed Performance Tracking Control of Lower-Triangular Systems with Unknown Fractional Powers

This paper is concerned with the tracking control problem for the lower-triangular systems with unknown fractional powers and nonparametric uncertainties. A prescribed performance control approach is put forward as a means of resolving this problem. The proposed control law incorporates a set of barrier functions to guarantee error constraints. Unlike the previous works, our approach works for the cases where the fractional powers, the nonlinearities, and their bounding functions or bounds are totally unknown; no restrictive conditions on the powers, such as power order restriction, specific size limitation or homogeneous condition, are made. Moreover, neither the powers and system nonlinearities nor their bounding functions or bounds are needed. It achieves reference tracking with the preassigned tracking accuracy and convergence speed. In addition, our controller is simple, as it does not necessitate parameter identification, function approximation, derivative calculation, or adding a power integrator technique. At the end, a comparative simulation demonstrates the effectiveness and advantage of the proposed approach.

Output regulation problem with prescribed performance for nonlinear lower-triangular system

ABSTRACT This paper addresses the output regulation problem with prescribed performance for nonlinear lower-triangular systems. To achieve the transient prescribed performance, the output transformation technique and barrier Lyapunov function technique are introduced to solve the output regulation problem. The prescribed performance control scheme based on normal backstepping is constructed not only to restrict the output error of the closed-loop system keeping in a prescribed performance bound, but also to achieve that the output tracks a designed trajectory asymptotically. The effectiveness of the proposed method is illustrated by two simulation examples.

Low-complexity observer–based output-feedback tracking control for a class of nonlinear lower-triangular systems

The tracking control problem for a group of nonlinear lower-triangular systems with multiple uncertainties is investigated in this work. Wherein, a novel performance constraint is first constructed to guarantee the fixed-time convergence of the output tracking error. Subsequently, a linear extended high-gain observer is employed to estimate the system uncertainties including the unmeasured states and the external disturbances. Based on the observer estimations, a novel output-feedback tracking control approach is formulated via using the backstepping technique, to ensure the boundedness of the closed-loop system. Compared with the existing works, the primary advantages of the proposed design are that (1) the problem of "explosion of terms" is avoided by eliminating the need for the derivative of the virtual control signals and (2) without the use of extra auxiliary techniques, the fixed-time convergence can be guaranteed, where the convergence time is independent of the system states and initial conditions. Then, the results about system stability are proved by the theoretical analysis. Moreover, an extension of the proposed approach to multi-input multi-output systems is deduced in this paper to further present its versatility. Finally, two groups of numerical examples are performed to validate the effectiveness of the proposed controller.

Finite-Time Stabilization for Continuous Triangular Systems via Vector Lyapunov Function Approach

This article provides finite-time stabilizing control laws for the continuous triangular systems, in which the nominal dynamics is the chain of power integrators with the power of a positive odd rational number. For a class of continuous lower-triangular systems, the linear controllers are designed. By invoking a vector Lyapunov function theory, there is no need to use the usual back-stepping technique and a quite number of inequality computations are avoided. For a class of continuous upper-triangular systems, the nested-saturation controllers are constructed. In this case, the use of a vector Lyapunov function theory also helps to reduce computational burden, and allows us to summarize a same group of parameter conditions guaranteeing both the reduction of saturated terms and the finite-time stability of the reduced system.

On active disturbance rejection control for lower-triangular systems with mismatched nonlinear uncertainties and unknown time-varying control coefficients

On active disturbance rejection control for lower-triangular systems with mismatched nonlinear uncertainties and unknown time-varying control coefficients

Adaptive NN Cross Backstepping Control for Nonlinear Systems With Partial Time-Varying State Constraints and Its Applications to Hyper-Chaotic Systems

This paper proposes a novel cross backstepping technique-based controller design for a class of nonlinear constrained systems with its applications to hyper-chaotic systems. Considering the strict feedback systems with partial time-varying state constraints, we divide the special systems into constrained subsystems and unconstrained subsystems. However, the normal backstepping method is only an effective method structured to control lower-triangular systems. Since the method is strictly limited to the system structure, the previous works cannot solve the problem considered. Thus, we employ the cross backstepping method to solve the problem of partial and alternate time-varying state constraints. For the constrained subsystems, the time-varying barrier Lyapunov function (TVBLF) is employed to ensure that the violation of any time-varying constraint does not occur. Besides, the radial basis function neural network (RBFNN) is used to approximate the uncertainties. Then, based on the stability analysis, it is concluded that the output follows the desired trajectory as closely as possible, all the signals in closed-loop systems are bounded, and the alternate time-varying state constraints are never violated. Finally, the simulation results demonstrate the effectiveness of the proposed strategy.

Adaptive State-Quantized Control of Uncertain Lower-Triangular Nonlinear Systems with Input Delay

In this paper, we investigate the adaptive state-quantized control problem of uncertain lower-triangular systems with input delay. It is assumed that all state variables are quantized for the feedback control design. The error transformation method using an auxiliary time-varying signal is presented to deal with the compensation problem of input delay. Based on the error surfaces with the auxiliary variable, a neural-network-based adaptive state-quantized control scheme is constructed with the design of the input delay compensator. Different from existing results in the literature, the proposed method exhibits the following features: (i) compensating for the input delay effect by using quantized states; and (ii) establishing the stability of the adaptive quantized feedback control system in the presence of input delay. Furthermore, the boundedness of all the signals in the closed-loop and the convergence of the tracking error are analyzed. The effectiveness of the developed control strategy is demonstrated through the simulation on a hydraulic servo system.

Active disturbance rejection control for lower triangular uncertain stochastic nonlinear systems driven by coloured noises

In this paper, the active disturbance rejection control approach is addressed for a class of lower triangular uncertain stochastic nonlinear systems driven by coloured noises. An extended state observer is first constructed to estimate the stochastic total disturbance including the total effects of internal uncertainties and external disturbances affecting system performance, and then an active disturbance rejection controller based on estimation and compensation is designed to obtain desired closed-loop performance. It is shown with a rigorous analysis that the closed-loop state tracks a reference trajectory generated from the target system in practically mean square and almost sure sense. Some numerical simulations are performed to validate the practicality of the proposed control strategy.

Event-triggered controllers based on the supremum norm of sampling-induced error

The paper proposes a novel event-triggered control scheme for nonlinear systems. Specifically, the closed-loop system is associated with a pair of auxiliary input and output. The auxiliary output is defined as the derivative of the continuous-time input function, while the auxiliary input is defined as the input disturbance caused by the sampling or equivalently the integral of the auxiliary output over the sampling period. As a result, it forms a cyclic mapping from the input to the output via the system dynamics and back from the output to the input via the integral. The event-triggered law is constructed to make the mapping contractive such that the stabilization is achieved and an easy-to-check Zeno-free condition is provided. Within this framework, we develop a theorem for the event-triggered control of interconnected nonlinear systems which is employed to solve the event-triggered control for lower-triangular systems with dynamic uncertainties.

Distributed Quantized Feedback Design Strategy for Adaptive Consensus Tracking of Uncertain Strict-Feedback Nonlinear Multiagent Systems With State Quantizers.

This study investigates a quantized feedback design problem for distributed adaptive leader-following consensus of uncertain strict-feedback nonlinear multiagent systems with state quantizers. It is assumed that all system nonlinearities of followers are unknown and heterogeneous, all state variables of each follower are quantized by a uniform state quantizer, and quantized states of followers are only communicated under a directed network. Compared with previous approximation-based distributed consensus tracking methods for uncertain lower triangular multiagent systems, the main contribution of this article is addressing the distributed quantized state communication problem in the adaptive leader-following consensus tracking field of uncertain lower triangular multiagent systems. A quantized-states-based local adaptive control law for each follower is derived by designing quantized-signals-based weight tuning laws for neural-network-based function approximators. By analyzing the boundedness of the local quantization errors, it is shown that the total closed-loop signals are uniformly ultimately bounded and the consensus tracking errors converge to a sufficiently small domain around the origin. Finally, simulation examples, including multiple ship steering systems, are considered to verify the effectiveness of the proposed theoretical approach.

Actuator Failure Compensation Control Scheme of the Nonlinear Triangular Systems by Static Gain Technique

In this paper, the technique of static gain as a new research approach is applied in the nonlinear triangular systems to investigate the issue of actuator fault. By employing the static gain technique, the nonlinear lower-triangular systems are converted into the form which is easy to find its Lyapunov function. The fault parameters of the actuator are subsequently processed by the efficient adaptive estimation tactic, after that, the goal of guaranteeing the global boundness of all closed-loop signals can be achieved by utilizing a state controller with the hyperbolic functions. Moreover, by adopting the same strategy, the actuator failure compensation problem is also solved for the nonlinear upper-triangular systems. Last but not least, the effectiveness of the design scheme is verified by two numerical simulation examples.

Barrier Lyapunov Function Based Adaptive Cross Backstepping Control for Nonlinear Systems with Time-varying Partial State Constraints

This paper focuses on an adaptive cross-backstepping control for a class of nonstrict-feedback nonlinear systems affected by time-varying partial state constraints. The nonstrict-feedback nonlinear systems considered in this paper are divided into two strict-feedback nonlinear subsystems. One constrained subsystems and another unconstrained subsystems. In view of the normal backstepping method is only an effective method to control lower-triangular systems, cross-backstepping control is introduced which successfully solves the problem of alternate time-varying state constraints. For the constrained subsystems, novel time-varying tan-type barrier Lyapunov function (TBLF) is employed in each step of backstepping design to guarantee the boundedness of the fictitious or actual state tracking errors. Besides, proper adaptive laws are designed for the upper bound of uncertain parameters, which successfully neutralize the influence of parametric uncertainties. Based on the stability analysis, it is concluded that the states of the closed-loop system maintain in the predefined time-varying compact sets and the output can be guaranteed to be as close to the desired trajectory as possible. The effectiveness of the proposed technique is illustrated by a constrained hyperchaotic system in two cases.

Adaptive low-power high-gain observers for lower-triangular systems with input-dependent Lipschitz constant

We combine the low-power high-gain observer recently proposed in Astolfi and Marconi (2015) with the updated-gain technique used in Andrieu et al. (2009). The resulting adaptive low-power high-gain observer inherits the advantages of both techniques and can be used to address the state-estimation problem for Lipschitz systems in lower triangular form with nonlinearities having a Lipschitz constant that depends on a known external input.

Implicit method for the solution of supersonic and hypersonic 3D flow problems with Lower-Upper Symmetric-Gauss-Seidel preconditioner on multiple graphics processing units

The paper describes a numerical method for the solution of stationary gas dynamics 3D spatial equations on unstructured grids that is designed for multiple graphics processing unit (GPU) computational architecture. Discretization of governing equations is done using first and second order TVD schemes. The Newton's method with simple pseudo time-step homotopy is used to solve the problem. Each iteration step involves solution of the linear system originated from the linearization of gas dynamics equations. Krylov subspace iterative methods are used to solve the linear system. The main aim of the paper is to describe a preconditioning Lower-Upper Symmetric-Gauss-Seidel (LU-SGS) method and its adaptation on multiple GPU computational systems. It is shown that deliberately reordered matrices with rearranged solution process of arising lower and upper triangular linear systems allow one to obtain close algebraic properties to the original single threaded LU-SGS. The method is benchmarked against published results. The analysis of computational efficiency and acceleration is presented for different flows with Mach number ranging from 1.2 up to 25.

Output feedback regulation of a class of triangular structural nonlinear systems with unknown measurement sensitivity

ABSTRACTThis paper investigates the problem of global regulation via output feedback for a class of triangular structural nonlinear systems with unknown measurement sensitivity. Two kinds of triangular structure nonlinear systems, namely upper triangular systems and lower triangular systems, are considered here, and the key features of our considered systems are that there are uncertain linear growth condition in the nonlinear terms. Firstly, for a class of upper triangular nonlinear systems with unknown measurement sensitivity, an output feedback controller is designed such that global regulation of the system is achieved. Then, for a class of lower triangular nonlinear systems with unknown measurement sensitivity, global regulation is realised in a unifying framework. Finally, two simulation examples are respectively given to demonstrate the effectiveness of the theoretical results.

Back‐stepping stabilization of fractional‐order triangular system with applications to chaotic systems

AbstractThis paper addresses the stabilization of fractional‐order nonlinear lower triangular systems and applies the obtained results to the synchronization of fractional‐order chaotic systems (FOCSs). Three cases are considered as follows: (i) FOCSs have a lower triangular structure themselves. (ii) FOCSs can be converted into the form in full by appropriate nonsingular coordinate transformations. (iii) FOCSs can be transformed into the structure in part by the conversions. It is shown that the above three cases combining with back‐stepping approach are useful for the control of FOCSs. Different from the previous literature, the system structure studied is more concise with more applications. A numerical example is presented to show the validity of the proposed results.

A time-varying high-gain approach to feedback regulation of uncertain time-varying nonholonomic systems

In this paper, novel state feedback and output feedback control strategies are proposed to regulate a class of uncertain time-varying nonholonomic systems with lower-triangular nonlinearities. The nonlinearities admit an incremental rate depending on an unknown parameter and an essential time-varying function. By means of the input-state-scaling technique, the involved systems are firstly transformed into a class of lower-triangular systems. The time-varying high-gain method is then introduced to tackle the system uncertainties and essential time-varying features simultaneously, and based on this, the time-varying state feedback controllers are constructed to achieve the state feedback regulation of the involved systems. Finally a time-varying high-gain observer is proposed to effectively compensate for system uncertainties and essential time-varying features, and on the basis of this, the time-varying output feedback controllers are constructed to achieve the output feedback regulation of the involved systems. Two illustrative examples are provided to demonstrate the effectiveness of the proposed control strategies.

Globally exponential stabilization for a class of nonlinear systems with time delays both in nonlinearities and input

This paper stabilizes a class of nonlinear systems with lower-triangular structure and different time delays at exponential convergence rate via output feedback domination method. An explicit construction for the output feedback stabilizer guaranteeing the globally exponential stabilization of the system is proposed, in which a scaling gain is introduced and can be designed to dominate the nonlinearities. We then analyze the exponential stability of the system via constructing a Lyapunov functional, and result in some constraints subject to the scaling gain and time-delays. Note that it is the first time to analyze the effects from time delays on the stability of lower-triangular systems. Two simulation results are provided to support the theoretical developments.