Controller Design For Uncertain Systems Research Articles

Robust non-monotonic Lyapunov based stability and stabilization methods for continuous-time systems: Applied on bilateral teleoperation system

This study introduces a Non-monotonic Lyapunov (NML) framework aimed at stability evaluation and controller design for continuous-time systems, particularly under conditions of uncertainty. Conventional Lyapunov techniques often exhibit a conservative nature, particularly in the context of uncertain systems, which necessitates the development of less conservative alternatives like NML. The NML methodology distinguishes itself by not imposing strict monotonicity requirements for demonstrating the decrease of a Lyapunov functional. Consequently, this paper derives new stability and stabilization criteria framed as matrix inequalities applicable to a specific class of uncertain systems. The practical applicability of the introduced approach is illustrated through controller design for uncertain systems, exemplified by a nonlinear bilateral teleoperation model. Assorted demonstrative examples and simulation outcomes support the findings, underscoring the NML approach’s efficaciousness.

Zero/low overshoot conditions based on maximally‐flatness for PID‐type controller design for uncertain systems with time‐delay or zeros

AbstractThis paper extends the characteristic ratio approach using novel inequalities to ensure zero/low overshoot for linear‐time‐invariant systems with zeros. The extension provided by this paper is based on the maximally‐flatness property of a transfer function, where the square‐magnitude of the transfer function is ensured to be a low‐pass filter. In order to be able to design low‐order/fixed structure controllers, a partial pole‐assignment approach is used instead of the full pole‐assignment used in the Characteristic Ratio Assignment (CRA) method. The developed inequalities and additional stability conditions are combined into an optimization problem using time domain restrictions when necessary. Although the method given in the paper is general, particular inequalities are developed for PI and PI‐PD controller cases, due to their frequent use in industrial applications. Similarly, First‐Order‐Plus‐Delay‐Time (FOPDT) and Second‐Order‐Plus‐Delay‐Time (SOPDT) systems are considered specifically, since most of the practical systems can be approximated by one of these types. The study is extended to plants with uncertainties where a theorem is developed to decrease computation time dramatically. The benefits of the proposed methods are demonstrated by several examples.

Optimal interval controller design for uncertain systems

AbstractThe interval controller design is a hot issue for uncertain systems, whereas how to design an optimal interval controller under the premise of ensuring system stability is a difficult problem that needs further study. This paper mainly aims at the single input single output uncertain system to propose an optimal interval controller based on the Kharitonov theorem and an interval optimization algorithm, which can guarantee the stability and optimization of a closed‐loop interval system. According to the Kharitonov theorem, the optimal interval controller design can be transformed into an optimal controller synthesis issue of multiple vertex objects. An interval particle swarm optimization (IPSO) algorithm is then used to optimize the quadratic performance index with interval variables for each vertex object to obtain the solution domains of the controller parameters, and the vertex method is utilized to prevent interval width expansion or divergence in the iteration. Finally, the intersections of the solution domains for all vertex objects are obtained as the optimal interval solution of interval controller parameters. In addition, the stability verification approach of the closed‐loop system and the empirical rule to select the interval particle width are given. Simulation results for typical examples demonstrate that the designed interval controller not only performs optimally but also can robustly stabilize the interval system.

Robust Fixed-Order Controller Design for Uncertain Systems With Generalized Common Lyapunov Strictly Positive Realness Characterization

This article investigates the design of a robust fixed-order controller for single-input–single-output (SISO) polytopic systems with interval uncertainties, with the aim that the closed-loop stability is appropriately ensured and the performance specifications on sensitivity shaping are conformed in a specific finite frequency range. Utilizing the notion of generalized common Lyapunov strictly positive realness (CL-SPRness), the equivalence between strictly positive realness (SPRness) and strictly bounded realness (SBRness) is established; and then, the specifications on robust stability and performance are transformed into the SPRness of newly constructed systems and further characterized in the framework of linear matrix inequality (LMI) conditions. The proposed methodology avoids the tedious yet mandatory evaluations of the specifications on all vertices of the uncertain polytopic system in an explicit form. Instead, solving five LMIs exclusively suffices for ensuring the robust stability and performance regardless of the number of vertices, and thus, the typically heavy computational burden is considerably alleviated. It is also noteworthy that the proposed methodology additionally provides the necessary and sufficient conditions for this robust controller design with the consideration of a prescribed finite frequency range, and therefore, significantly less conservatism is attained in the system performance.

Decentralized adaptive control of uncertain interconnected systems with triggering state signals

Backstepping is a powerful technique of control design for uncertain systems and has been well received. As it involves differentiating virtual control signals recursively, the involved signals must be differentiable. However, when the measured states are only transmitted based on event-triggered conditions, such requirement will not be met, making the recursive backstepping design procedure inapplicable. Thus it is still an open problem to study how an event-triggered scheme is employed to transmit state signals in decentralized control of uncertain interconnected systems with dynamic interactions. In this paper, we present a solution by constructing a new decentralized adaptive backstepping-based control algorithm capable of coping with discontinuity caused by state-triggering. With aid of appropriately chosen Lyapunov functions, all signals in the closed-loop system are ensured globally uniformly bounded. Simulation results are presented to demonstrate the effectiveness of the proposed methodology.

Non-fragile exponential [formula omitted] observer-based event-triggered control of uncertain systems with disturbance input: Theoretical studies and application investigations

This paper is concerned with the problem of exponential H∞ event-triggered controller design for uncertain systems with disturbance input. The proposed controller is based on a non-fragile observer which is resilient to gain variations. A new event-triggering condition is introduced deciding on the transmission of the current signal. An event-driven state-feedback controller updates the control signals according to the transmitted data. Utilizing the Lyapunov stability theorem, the sufficient conditions for exponential stability of the estimated states and the error system are proposed. It is proven that the state estimates and the error vector converge to a limited region with a pre-specified exponential rate. The stability conditions are given in the form of linear matrix inequalities (LMIs), by solving which, the controller and the observer gains are attained. It is also shown that a positive scalar limits the inter-execution intervals, by analyzing the Zeno behavior. The design procedure has been stated in a step-by-step manner which makes the design very convenient and straightforward. Eventually, the proposed approach is applied to the level control of a four-tank system. The simulation results illustrate the applicability and effectiveness of the method.

Dual Adaptive Model Predictive Control with Disturbances

Abstract Model-based control requires good accuracy of model parameters to achieve high performance. Controller design under parametric uncertainties is therefore a challenging topic in control system engineering. One well known control design for uncertain systems is adaptive control. An adaptive controller has two tasks: process regulation and parameter learning. Dual control explores the trade-off between the two seemingly conflicting tasks. The control structure in an adaptive system consists of a model-based controller and a recursive update rule for parameter estimation. In this paper, the adaptive control framework consists of model predictive control for system regulation and recursive least squares for parameter estimation. The requirement for persistent excitation is shown to be necessary for systems with high-dimensional parameter space. Then, a dual formulation is proposed in an attempt to generate excitation signals while maintaining control performance in the adaptive control scheme. The algorithm is implemented and tested on a simulated SISO system.

On Robust Stability and Performance With a Fixed-Order Controller Design for Uncertain Systems

Typically, it is desirable to design a control system that is not only robustly stable in the presence of parametric uncertainties but also guarantees an adequate level of system performance. However, most of the existing methods need to take all extreme models over an uncertain domain into consideration, which then results in costly computation. Also, since these approaches attempt rather unrealistically to guarantee the system performance over a full frequency range, a conservative design is always admitted. Here, taking a specific viewpoint of robust stability and performance under a stated restricted frequency range (which is applicable in rather many real-world situations), this article provides an essential basis for the design of a fixed-order controller for a system with bounded parametric uncertainties, which avoids the tedious but necessary evaluations of the specifications on all the extreme models in an explicit manner. A Hurwitz polynomial is used in the design and the robust stability is characterized by the notion of positive realness, such that the required robust stability condition is then successfully constructed. Also, the robust performance criteria in terms of sensitivity shaping under different frequency ranges are constructed based on an approach of bounded realness analysis. Furthermore, the conditions for robust stability and performance are expressed in the framework of linear matrix inequality (LMI) constraints, and thus can be efficiently solved. Comparative simulations are provided to demonstrate the effectiveness and efficiency of the proposed approach.

Robust optimal feedback control design for uncertain systems based on artificial neural network approximation of the Bellman’s value function

In this study, a local approximated solution for the Hamilton–Jacobi–Bellman equation based on differential neural networks is proposed. The approximated Value function is used to obtain a feedback control law for a class of uncertain dynamical systems. The approach to deal with the uncertainties of the dynamical system is a min–max method that yields obtaining a robust-like solution for an optimal control problem. The proposed cost functional is presented in the Bolza form and the necessary and sufficient conditions for getting the min–max optimal solution with the designed robust version of the dynamic programming approach are provided. Moreover, the effect of the neural network approximation is studied. All the analysis considers a smoothness assumption regarding the Value function. The practical stability of the system with the neural network feedback optimal control is formally demonstrated by means of the Lyapunov method. Finally, the performance of the control law is compared in simulation with a classical optimal controller. The proposed controller overcomes the results produced by the classical controller, which is confirmed by the functional evaluation.

Robust H∞-Control for Uncertain Stochastic Systems with Impulsive Effects

Robust stabilization and H ∞ controller design for uncertain systems with impulsive and stochastic effects have been deeply discussed. Some sufficient conditions for the considered system to be robustly stable are derived in terms of linear matrix inequalities (LMIs). In addition, an example with simulations is given to better demonstrate the usefulness of the proposed H ∞ controller design method.

Self-adjusting leakage type adaptive robust control design for uncertain systems with unknown bound

Abstract The control problem for an uncertain mechanical system is addressed. The mechanical system under consideration, which is to follow a class of prescribed constraints, contains (possibly fast) time-varying uncertainty. The uncertainty is bounded, but the bound is not necessarily known. A new class of robust control schemes is proposed, which is based on on-line adaptation of a design parameter. Two categories of adaptation laws, both are based on self-adjusting leakage, are proposed. The control guarantees uniform boundedness and uniform ultimate boundedness of a performance measure. Comparing with the previous control scheme, the importance of this new control is that it can compensate the uncertainty in a very effective way. It also avoids over compensation and renders modest control effort. For demonstration purpose, the Toda Lattice, an ideal model of mechanical system with asymmetrical behaviour is chosen. The energy control of the Toda Lattice demonstrates the validity and reliability of control schemes and adaptation laws.

Dynamic Compensator-Based Output Feedback Controller Design for Uncertain Systems with Adjustable Robustness

A low-order dynamics compensator-based output feedback stabilization method for a class of uncertain linear MIMO systems with mismatched disturbances is presented. Since all the system states are not measurable, the proposed controller inherently has low-pass filter property in which it can successfully replace the derivative terms of the system output and hence effectively estimate the input disturbance. The control scheme proposed here can simultaneously consider the input saturation problem and obtain the desired performance. Although the system has unknown uncertainties and disturbances, the uniformly ultimate boundedness of system states in the closed-loop system is analytically shown using the Lyapunov method. Finally, two numerical examples are presented to demonstrate the applicability of the proposed scheme.

Dynamical adaptive integral backstepping variable structure controller design for uncertain systems and experimental application

SummaryIn this study, a dynamical adaptive integral backstepping variable structure control (DAIBVSC) system based on the Lyapunov stability theorem is proposed for the trajectory tracking control of a nonlinear uncertain mechatronic system with disturbances. In this control scheme, no prior knowledge is required on the uncertain parameters and disturbances because it is estimated by two types of dynamical adaptive laws. These adaptive laws are integrated into the dynamical adaptive integral backstepping control and variable structure control (VSC) parts of the DAIBVSC. The dynamical adaptive law in the dynamical adaptive integral backstepping control part updates parametric uncertainties, while the other in the VSC part adapts upper bounds of non‐parametric uncertainties and disturbances. In order to achieve a more robust output tracking and better parameter adaptation, the control system is extended by one integrator and sliding surface is augmented by an integral action. Experimental evaluation of the DAIBVSC is conducted with respect to performance and robustness to parametric uncertainties. Experimental results of the DAIBVSC are compared with those of a traditional VSC. The proposed DAIBVSC exhibits satisfactory output tracking performance, good estimation of the uncertain parameters and can reject disturbances with a chattering free control law. Copyright © 2017 John Wiley & Sons, Ltd.

On the sample size of random convex programs with structured dependence on the uncertainty

The “scenario approach” provides an intuitive method to address chance constrained problems arising in control design for uncertain systems. It addresses these problems by replacing the chance constraint with a finite number of sampled constraints (scenarios). The sample size critically depends on Helly’s dimension, a quantity always upper bounded by the number of decision variables. However, this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability. We derive improved bounds of Helly’s dimension for problems where the chance constraint has certain structural properties. The improved bounds lower the number of scenarios required for these problems, leading both to improved objective value and reduced computational complexity. Our results are generally applicable to Randomized Model Predictive Control of chance constrained linear systems with additive uncertainty and affine disturbance feedback.

Dissipative reliable controller design for uncertain systems and its application

This paper address the reliable robust strictly (Q, R, S) dissipative control problem for a class of uncertain continuous time systems with input delay and linear fractional uncertainties despite possible actuator failures in system model. In particular, by employing a novel Lyapunov functional together with delay fractioning approach, a reliable control law is designed in terms of the solution of certain linear matrix inequalities which makes the considered system strictly (Q, R, S) dissipative which can be easily solved by using the available software. At the same time, as special cases the H∞, passivity, mixed H∞ and passivity control problems can obtained from the proposed dissipative control formulation. This explains the fact that the dissipative control unifies H∞ control, passive control, and mixed H∞ and passivity control in a single framework. Finally, a numerical example based on a mechanical system is given to demonstrate the effectiveness and applicability of the proposed design approach. More precisely, the proposed results have been compared through numerical simulation which reveals that the obtained criteria are considerably less conservative than some existing results.

Saturated robust control with regional pole placement and application to car active suspension

In this paper a new saturated control design for uncertain systems is proposed. The developed saturated control scheme is based on linear matrix inequality (LMI) optimization to achieve prescribed dynamic performance measures e.g., settling time and damping ratio. In this design, the closed-loop poles are forced to lie within a desired region. The proposed design provides robustness against system uncertainties. The existence of the saturated robust control with regional pole placement is shown using LMI and Lyapunov function analysis. Simulation results of two illustrative examples are given to validate the effectiveness of the proposed controllers. Application to car active suspension to achieve comfortable dynamic performance by pole placement and avoiding actuator saturation is also considered.

An efficient reliability method for probabilistic H-infinity robust control of uncertain linear dynamic systems

Achieving balance between robustness and performance is always a fundamental challenge we are faced with in control design of systems in the presence of uncertainties. H∞ robust control theory has been extensively used to deal with the problems of robust stabilization and disturbance attenuation of uncertain systems. However, in traditional H∞ robust control, the most frequently used method for dealing with uncertainties is adopting the assumption of norm-bounded perturbations of arbitrary structure about a nominal plant and the design techniques are based on the deterministic worst-case scenarios that may never occur in a particular control system and thus often led to over-conservative results. In this paper we focus on developing a reliability method for probabilistic H∞ robust control of linear uncertain systems by describing the uncertain parameters as random variables. A new efficient reliability method for robust control of dynamic systems with probabilistic parametric uncertainties is presented systematically. Robust control design is carried out by solving a reliability-based optimization problem where the disturbance attenuation and control cost are minimized under the condition that reliability requirement is satisfied. One of the advantages of the presented reliability method is that it can be used directly for robust control design of parametric uncertain systems. Another advantage of the presented method is that it makes it possible to consider the factors such as system performance, control cost, and reliability simultaneously in an integrated framework and provides an essential basis for the coordination and tradeoffs between these important factors in control design of uncertain systems. Compared to traditional H∞ robust control, the presented method is less-conservative and more reasonable for dealing with probabilistic uncertainties. The presented formulations are within the framework of linear matrix inequality and thus can be carried out conveniently. Two numerical examples are investigated to demonstrate the effectiveness and feasibility of the presented method.

Robust reliability method for non-fragile guaranteed cost control of parametric uncertain systems

The problem of non-fragile guaranteed cost control of uncertain systems is studied from a new point of view of reliability against uncertainties. An efficient robust reliability method for the analysis and design of non-fragile guaranteed cost controller of parametric uncertain systems is presented systematically. By the method, a robust reliability measure of an uncertain control system satisfying required robust performance can be obtained, and the robustness bounds of uncertain parameters such that the control cost of a system is guaranteed can be provided. The optimal non-fragile guaranteed cost controller obtained in the paper may possess optimal guaranteed cost performance satisfying the precondition that the system is robustly reliable with respect to uncertainties occurring in both the controlled plant and controller gain. The presented formulations are in the framework of linear matrix inequality and thus can be carried out conveniently. The presented method provides an essential basis for the tradeoff between reliability and control cost in controller design of uncertain systems. Two numerical examples are provided to demonstrate the efficiency and feasibility of the presented method. It is shown that the coordination and simultaneous realization of the system performance, control cost, and robust reliability in control design of uncertain systems are significant.

Robust Reliability Based Optimal Design of H∞ Control of Parametric Uncertain Systems

An efficient nonprobabilistic robust reliability method for H∞ robust controller design of parametric uncertain systems is presented by describing the uncertain parameters as interval variables. Design optimization of H∞ robust controller is carried out by solving a robust reliability based optimization problem, by which the disturbance attenuation, control cost, and robust reliability can be taken into account simultaneously. By the method, a robust reliability measure of an uncertain control system satisfying required H∞ robust performance can be obtained, and the robustness bounds of uncertain parameters such that the control cost of the system is guaranteed can be provided. The presented formulations are in the framework of linear matrix inequality and thus can be carried out conveniently. The presented method provides an essential basis for reasonable tradeoff between reliability and control cost in controller design of uncertain systems. Active control design of vehicle suspension is employed for illustrating the effectiveness and feasibility of the presented method.

Non-probabilistic reliability method and reliability-based optimal LQR design for vibration control of structures with uncertain-but-bounded parameters

Uncertainty is inherent and unavoidable in almost all engineering systems. It is of essential significance to deal with uncertainties by means of reliability approach and to achieve a reasonable balance between reliability against uncertainties and system performance in the control design of uncertain systems. Nevertheless, reliability methods which can be used directly for analysis and synthesis of active control of structures in the presence of uncertainties remain to be developed, especially in non-probabilistic uncertainty situations. In the present paper, the issue of vibration control of uncertain structures using linear quadratic regulator (LQR) approach is studied from the viewpoint of reliability. An efficient non-probabilistic robust reliability method for LQR-based static output feedback robust control of uncertain structures is presented by treating bounded uncertain parameters as interval variables. The optimal vibration controller design for uncertain structures is carried out by solving a robust reliability-based optimization problem with the objective to minimize the quadratic performance index. The controller obtained may possess optimum performance under the condition that the controlled structure is robustly reliable with respect to admissible uncertainties. The proposed method provides an essential basis for achieving a balance between robustness and performance in controller design of uncertain structures. The presented formulations are in the framework of linear matrix inequality and can be carried out conveniently. Two numerical examples are provided to illustrate the effectiveness and feasibility of the present method.