This article presents a solution to the problem of achieving optimal prescribed-time stability and stabilization for nonlinear dynamical systems. In contrast to existing prescribed-time control methods, this article initiates by establishing sufficient conditions for prescribed-time stability through the use of continuous Lyapunov candidate functions. Building upon these conditions, we introduce an optimal prescribed-time stabilization method that incorporates specific differential inequalities. This method complies with the Hamilton-Jacobi-Bellman steady-state equation, ensuring both optimality and prescribed-time stability. Furthermore, we derive a set of optimal prescribed-time stabilizing control laws for a class of affine nonlinear dynamical systems. Finally, we demonstrate the effectiveness of the proposed approach through simulations and experiments involving the reference level tracking of a coupled tank system, thus ensuring that the tracking performance aligns with practical user specificationsNote to Practitioners—This article was instigated by the challenge of devising optimal feedback control strategies for a specific class of nonlinear dynamical systems at predetermined time instances. In recent years, there has been a growing interest in prescribed time stability and stabilization approaches, driven by their potential applications across diverse fields, including control engineering, robotics, and aerospace engineering. These methods facilitate the regulation of nonlinear dynamical systems to reach a desired steady state within a predefined finite time, offering a valuable solution for situations demanding rapid stabilization. In this article, we introduce a novel optimal prescribed-time stabilization method that relies on specific differential inequalities. This method not only adheres to the Hamilton-Jacobi-Bellman steady-state equation but also guarantees both optimality and prescribed-time stability. Furthermore, we derive a family of optimal prescribed-time stabilizing control laws tailored to a particular class of affine nonlinear dynamical systems. To validate the effectiveness of our proposed stabilization approach, we conduct experiments focusing on tracking the desired water level within a coupled tank system. Ultimately, the presented prescribed-time optimal feedback control strategy marks a significant stride forward in the advancement of optimal and efficient control methods for nonlinear dynamical systems, offering solutions that hold immense promise in practical applications.

This paper presents super-twisting sliding mode control (ST-SMC) integrated with a Hopf oscillator for islanded microgrids. The ST-SMC is realized to reduce the effect of uncertainties due to parameter variation in coordination with the Hopf oscillator for synchronization among inverters. This approach provides robustness to the system and contributes overall system performance and enhances stability. The mathematical analysis and simulated results of this control technique are presented.

Over the past few years, rated convergence has gained importance and has acquired tremendous interest from the research community. In this regard, some nomenclatures have emerged, broadly falling under the spectrum of finite time stability. However, these notions have certain aspects which differentiate them. Therefore it becomes imperative to bring out a comparative study among these notions. Moreover, such recent concepts are being applied to several application problems. Thus it is essential to understand even the small details associated with them. The purpose of the present paper is to address these objectives. In addition, the convergence of states and boundedness of the control in case of arbitrary time convergence has also been discussed.

A novel unknown input observer (UIO) using sliding modes is proposed for a general class of uncertain, nonlinear systems. The system generalization over-comes requirement of observability matching condition for the uncertainties and also does not rely on any transformation techniques, typically used for nonlinear systems. The UIO philosophy of disturbance decoupling is used for compensating the matched disturbances and overall robustness is engineered using sliding modes. Fault detection is proposed using the estimations. The efficacy of the method is illustrated through simulations on a quadruple tank process (QTP) plant.

This paper describes the implementation of an integral sliding mode (ISM) based robust composite non-linear feedback (CNF) controller design for a single input single output (SISO), continuous linear time invariant (LTI) descriptor system with input saturation. This scheme combines the salient features of the CNF controller and the sliding mode controller i.e good transient behavior while tracking and matched disturbance rejection respectively. Also, this scheme guarantees a general reference tracking for the descriptor systems even in presence of input saturation.

In this paper, an equivalent output injection based sliding mode observer is proposed for estimating the states of a class of perturbed n-dimensional nonholonomic systems. In the proposed design, we introduce a new transformation based on state scaling which renders the perturbed system in a triangular input form, amenable for the design of the observer. Thereafter, the equivalent control method is adopted in a sequential manner to estimate each of the system states one after another, in finite time such that ultimately all the states are estimated. Thus the entire state estimates are available for feedback after a finite time, in the presence of structured perturbations affecting the system. Simulations are performed on a unicycle mobile robot, to demonstrate the effectiveness of the proposed observer design.

This paper introduces a discrete time sliding mode by using a periodic intermittent control with multirate fast output feedback technique for the robust stabilization of discretized linear time invariant systems. In discrete time periodic intermittent control technique, the control input is applied to the system for the first few sampling time interval and then the control is not applied for next few sampling time interval, this control cycle is repeated. The proposed periodic intermittent control drives the system to discrete time sliding mode to achieve the robust stabilization of discretized linear time invariant system. The proposed theory is demonstrated by a simulation example on industrial plant emulator.

This paper presents a multivariable state feedback tracking control with integral sliding mode which can achieve the output tracking of a reference step signal without any overshoot by rejecting the disturbances. The control designed is able to completely reject the matched-bounded disturbances by using the integral sliding mode technique. The multivariable feedback controller is capable of achieving arbitrarily small rise time without any overshoot for square multivariable system under some mild assumptions. The proposed control is verified using a numerical example of quadruple-tank setup.

The paper presents an event-triggering based sliding mode control law for delta operator systems with disturbance. The discrete domain representation of a system using shift operator becomes numerically ill-conditioned at very high sampling rates. To circumvent this problem, the system is represented using delta operator. The delta operator creates a rapprochement between continuous-time and discrete-time system model at very high sampling rates. A discrete-time sliding mode (DTSM) control law is developed to achieve robustness against external disturbances. Further to reduce the resource utilization, an event-triggered DTSM control law is proposed such that the closed loop system is stable. Due to the inherent discrete nature of the control there is no accumulation of triggering instants at any instant of time. Hence the zeno free execution of the control updating instants is always guaranteed. Simulation results are also shown to illustrate the effectiveness of the proposed method.

This paper discusses the robust stabilization of a linear time-invariant system based on event triggering strategy. In most practical system, it is not possible to update control in continuous manner, so it is given to plant at some periodic discrete instants only. Recently, the event based control technique for control law update has become popular as it uses the resources efficiently while guaranteeing closed loop system stability. We address here the event based sliding mode control for a linear system in the presence of disturbances and it is shown that the closed loop system achieves stability with respect to measurement errors. The expression for minimum control execution time, where the control law gets updated only after this time interval is also derived. Simulation results are given to verify the theoretical analysis.

The paper presents some new results on digital implementation of sliding mode control when designed from a continuous time system. When a continuous sliding mode control is digitally implemented, it cannot bring the sliding variable to the surface in finite time and stay there, but the sliding variable remains within an ultimate band around the surface. Analysis is done which relates such an ultimate band with the continuous controller parameters. If the controller parameters are chosen accordingly, one can be sure that the sliding motion will remain inside the specified ultimate band, and hence stability can be assured for such a digital implementation of the continuous sliding mode control.

In this paper, a suboptimal flow controller design for connection-oriented communication network via reduced model is proposed. Network is modeled as an nthorder discrete system whose available bandwidth variations at the bottlenecked link act as an exogenous unmatched disturbance. The proposed scheme is characterized by a simple first order design which is applied to nthorder systems through aggregation. Effect of bandwidth variation at the congested node which is modeled as a disturbance, is well accounted in controller design with widely adopted delay-based disturbance estimation. Performance of the proposed scheme is validated through Matlab and ns simulations. Simulation results shows that all the proposed reduced order control scheme tracks the desired queue length effectively under the adverse influence of unmatched disturbances.

A Stewart platform manipulator is 6 degrees of freedom parallel robot with a superior performance over serial robots. The forward kinematic problem of a Stewart platform is to determine the actual position and orientation of the movable platform with respect to the base for a given set of leg lengths. The existing analytical methods to solve this problem give multiple solutions, out of which an exact and practical solution has to be selected. A novel method based on non linear autoregressive model with exogenous input (NLARX) using wavelet network is proposed in this paper for the identification of forward kinematic model of a Stewart platform. For a given position of the platform, the leg lengths are estimated using inverse kinematics. This information is used to identify and validate the NLARX model. The simulation results show the ability of the estimated NLARX model to represent the dynamic behavior of the Stewart platform in the forward kinematic mode.

The robust stabilization of roll angle is considered in this paper. The roll dynamics is stable but it has large parametric variations and slow response behaviour. Here, a new method for sliding mode control is presented to robustly stabilize the roll angle within prescribed time. The dynamics of servo actuator and gyroscope sensors are also considered which makes the problem more realistic.

In this paper, a novel discrete-time integral sliding-mode control (DISMC) scheme for connection-oriented communication network is proposed. Network is modeled as an nthorder discrete system whose available bandwidth variations at the bottlenecked link act as an exogenous unmatched disturbance. The proposed scheme is characterized by discrete-time integral sliding manifold which inherits the desired properties such as elimination of reaching phase, full order sliding manifold with pole assignment etc. In particular, comparing with the existing discrete flow control scheme, the new scheme ensures more precise tracking with O(T2) steady-state error for state regulation with widely adopted delay-based disturbance estimation. Surface parameters are so chosen to minimise the effect of unmatched uncertainty. Simulation results shows that discrete-time ISMC achieves more effective control performance under the adverse influence of unmatched disturbances, meanwhile eliminating the reaching phase.

In this paper the concept of sliding mode is introduced in the design of functional observers. Conditions for the existence of the observer are given. How to design the observer parameters are also shown. Under special circumstances the sliding mode functional observer proposed in this paper reduces to the Utkin state observer.

The paper presents a method to design a spatial control system based on sliding mode observer. The non-linear model of pressurized heavy water reactor (PHWR) including xenon and iodine dynamics is characterized by 70 state variables and 14 inputs and outputs each. Linear nodal model is obtained by linearizing the non-linear dynamic equations of the reactor about the full power operating point. A sliding mode spatial controller is designed for the linearized model of the reactor and the state computation is successfully done by the sliding mode observer. From simulation of the non-linear model of the reactor in representative transients, the proposed control and observer scheme is found to be satisfactory.

Power System Stabilizer (PSS) are added to excitation system to enhance the damping of electric power system during low frequency oscillations. Design of decentralized PSS for 4 machines with 10 buses using periodic output feedback is proposed. The nonlinear model of multi-machine system is linearized and linear state space model is obtained. An output injection gain is obtained using LQR technique. A decentralized periodic output feedback gain which realizes this output injection gain is obtained using LMI approach. This method doesn't require state of the system for feedback. It uses only the output for feedback. Thus it is easily implementable.

The paper presents the design of periodic output feedback control for a multivariable smart structure system. The method is illustrated by an example.

This note presents a method of designing the state feedback gain which places the closed-loop poles of a given discrete interval system inside some region. The Levy-Hadamard and Bendixson theorems have been used to derive algebraic relations which set bounds on the real and imaginary parts of the eigenvalues of the closed-loop system matrix. This helps in placing the closed-loop poles in a specified region, either inside a vertical strip, or inside a horizontal strip, or inside a rectangular region. It turns out that the relations are easily computable and the feedback gain can be determined in a very simple way. A numerical example illustrates the proposed procedure.

This paper presents model reduction of linear interval system using Pade approximation method. In the first part a full Pade approximation method for interval system is presented, where as in the second part stable Pade approximation is discussed. A numerical example illustrates the procedure.