Switched Second‐Order Sliding Mode Control with Partial Information: Theory and Application

This paper proposes a novel switched second order sliding mode (S‐SOSM) control strategy in a partial information setting, i.e., when only the sliding variable is accessible for measurements. Such a control approach allows one to deal with systems characterised by different levels of uncertainties associated with different regions of the state space and to accommodate different control objectives in the different regions by switching among appropriate SOSM controllers. The proposed approach is shown to ensure global finite‐time convergence to the origin of the closed‐loop system trajectory. The braking control of two‐wheeled vehicles is considered as a case‐study to test the controller performance.     

STABILIZATION AND H∞ CONTROL FOR UNCERTAIN STOCHASTIC TIME‐DELAY SYSTEMS VIA NON‐FRAGILE CONTROLLERS

This paper considers the problems of robust non‐fragile stochastic stabilization and H_∞ control for uncertain time‐delay stochastic systems with time‐varying norm‐bounded parameter uncertainties in both the state and input matrices. Attention is focused on the design of memoryless state feedback controllers which are subject to norm‐bounded uncertainties. For both the cases of additive and multiplicative controller uncertainties, delay‐independent sufficient conditions for the solvability of the above problems are obtained. The desired state feedback controller can be constructed by solving a certain linear matrix inequality.     

IQC‐synthesis with general dynamic multipliers

Analogously to the existing μ‐synthesis tools, we propose an alternative algorithm for the systematic design of robust controllers based on an iteration of standard nominal controller synthesis and integral quadratic constraint (IQC) analysis with general dynamic multipliers. The suggested algorithm enables us to perform robust controller synthesis for a significantly larger class of uncertainties if compared with the existing methods. Indeed, while the classical approaches are restricted to the use of real/complex time invariant or arbitrarily fast time‐varying parametric uncertainties, the IQC framework also offers, for example, the possibility to efficiently handle sector‐bounded and slope restricted nonlinearities or time‐varying parametric uncertainties and uncertain time‐varying time‐delays, both with bounds on the rate‐of‐variation. Secondly, in contrast to the classical approaches, the proposed techniques completely avoid gridding and curve‐fitting. We present new insights that allow us to reformulate the robust IQC analysis LMIs into a standard quadratic performance problem. This enables us to generate suitable initial conditions for each subsequent iteration step. Depending on the size of the problem, this can significantly speed up the synthesis process. The results are illustrated by means of two numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.     

On the boundedness of the set of stabilizing controllers

This paper shows that the set of rational, strictly proper, robustly stabilizing controllers for single‐input single‐output linear‐time invariant plants will form a bounded (can even be empty) set in the controller parameter space if and only if the order of the stabilizing controller cannot be reduced any further; if the set of proper stabilizing controllers of order r is not empty and the set of strictly proper controllers of order r is bounded, then r is the minimal order of stabilization. The paper also extends this result to characterize the set of controllers that guarantee some pre‐specified performance specifications. In particular, it is shown here that the minimal order of a controller that guarantees specified performance is l iff (1) there is a controller of order l guaranteeing the specified performance and (2) the set of strictly proper, robustly stabilizing controllers of order l and guaranteeing the performance is bounded. Moreover, if the order of the controller is increased, the set of higher‐order controllers which satisfies the specified performance will necessarily be unbounded. This characterization is provided for performance specifications, such as gain margin and robust stability, which require a one‐parameter family of real polynomials to be Hurwitz, where the parameter is in a closed interval. Other performance specifications, such as phase margin and ℋ︁_∞ norm, can be reduced to the problem of determining a set of stabilizing controllers that renders a family of complex polynomials Hurwitz. The characterization of the set of controllers for the stabilization of complex polynomials is provided and is used to show the boundedness properties for the set of controllers that guarantee a given phase margin or an upper bound on the ℋ︁_∞ norm. Copyright © 2007 John Wiley & Sons, Ltd.     

Output‐feedback stabilization for planar output‐constrained switched nonlinear systems

In this paper, globally asymptotical stabilization problem for a class of planar switched nonlinear systems with an output constraint via smooth output feedback is investigated. To prevent output constraint violation, a common tangent‐type barrier Lyapunov function (tan‐BLF) is developed. Adding a power integrator approach (APIA) is revamped to systematically design state‐feedback stabilizing control laws incorporating the common tan‐BLF. Then, based on the designed state‐feedback controllers and a constructed common nonlinear observer, smooth output‐feedback controllers, which can make the system output meet the predefined constraint during operation, are proposed to deal with the globally asymptotical stabilization problem of planar switched nonlinear systems under arbitrary switchings. A numerical example is employed to verify the proposed method.     

Output‐Feedback Control for Continuous‐time Interval Positive Systems under L1 Performance

This paper is concerned with the design of an L₁‐induced output‐feedback controller for continuous‐time positive systems with interval uncertainties. A necessary and sufficient condition for stability and an L₁‐induced performance of interval positive linear systems is proposed in terms of linear inequalities. Based on this, conditions for the existence of robust static output‐feedback controllers are established and an iterative convex optimization approach is developed to solve the conditions. For special single‐input‐multiple‐output (SIMO) positive systems, the problem of controller synthesis is completely solved with the help of an analytical formula for the L₁‐induced norm. An illustrative example is provided to show the effectiveness and applicability of the theoretical results.     

Decay rate constrained stabilization of positive systems using static output feedback

This paper investigates the stabilization problem by means of a static output feedback control for positive systems with/without interval uncertainties. The designed closed‐loop system is ensured to be positive and stable with a prescribed decay rate. Necessary and sufficient conditions for the existence of such controllers are established in terms of linear matrix inequalities together with a matrix equality constraint. Furthermore, the desired static output feedback controllers can be derived directly via the cone complementarity linearization techniques. Finally, an illustrative example is presented to show the validity of results obtained in this paper. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust H∞ control for uncertain discrete‐time stochastic bilinear systems with Markovian switching

This paper is concerned with the problems of robust stochastic stabilization and robust H_∞ control for uncertain discrete‐time stochastic bilinear systems with Markovian switching. The parameter uncertainties are time‐varying norm‐bounded. For the robust stochastic stabilization problem, the purpose is the design of a state feedback controller which ensures the robust stochastic stability of the closed‐loop system irrespective of all admissible parameter uncertainties; while for the robust H_∞ control problem, in addition to the robust stochastic stability requirement, a prescribed level of disturbance attenuation is required to be achieved. Sufficient conditions for the solvability of these problems are obtained in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, explicit expressions of the desired state feedback controllers are also given. An illustrative example is provided to show the effectiveness of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Delay‐dependent robust H∞ control for uncertain stochastic time‐delay systems

This paper investigates the robust H_∞ control problem for stochastic systems with a delay in the state. Sufficient delay‐dependent conditions for the existence of state‐feedback controllers are proposed to guarantee mean‐square asymptotic stability as well as the prescribed H_∞ performance for the closed‐loop systems. Moreover, the results are further extended to the stochastic time‐delay systems with parameter uncertainties, which are assumed to be time‐varying norm‐bounded appearing in both the state and the input matrices. The appealing idea is to partition the delay, which differs greatly from the most existing results and reduces conservatism by thinning the delay partitioning. Numerical examples are provided to show the advantages of the proposed techniques. Copyright © 2009 John Wiley & Sons, Ltd.     

ROBUST H∞ CONTROL AND QUADRATIC STABILIZATION OF DISCRETE‐TIME SWITCHED SYSTEMS WITH NORM‐BOUNDED TIME‐VARYING UNCERTAINTIES

We focus on robust H_∞ control analysis and synthesis for discretetime switched systems with norm‐bounded time‐varying uncertainties. Sufficient conditions are derived to guarantee quadratic stability along with a prescribed H_∞‐norm bound. Each of them can be dealt with as a linear matrix inequality (LMI) which can be tested with efficient algorithms. All the switching rules are constructively designed, and do not rely on any uncertainties.     

Quadratic stabilizability and H∞ control of linear discrete‐time stochastic uncertain systems

This paper considers quadratic stabilizability and H_∞ feedback control for stochastic discrete‐time uncertain systems with state‐ and control‐dependent noise. Specifically, the uncertain parameters considered are norm‐bounded and external disturbance is an l²‐square summable stochastic process. Firstly, both quadratic stability and quadratic stabilization criteria are presented in the form of linear matrix inequalities (LMIs). Then we design the robust H_∞ state and output feedback H_∞ controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust H_∞ controllable. Sufficient conditions for the existence of the desired robust H_∞ controllers are obtained via LMIs. Finally, some examples are supplied to illustrate the effectiveness of our results.     

Second‐order sliding mode controller design subject to mismatched unbounded perturbations

The dynamics of the second‐order sliding mode (SOSM) can be obtained by directly taking the second derivative on the sliding variable when it has a relative degree of 2 with respect to the control input. However, there will always appear some state‐dependent certain or uncertain terms in the first derivative of the sliding variable, and the derivative directly imposed on these terms could enlarge the uncertainties in the control channel. One method to reduce the uncertainties in the control channel is to hold this information in the dynamics of the first derivative of the sliding variable, while the original SOSM dynamics could be transformed to be a SOSM system with a mismatched unbounded perturbation. This paper focuses on the controller design problem for SOSM dynamics subject to mismatched unbounded perturbation. By using Lyapunov analysis, a novel backstepping‐like design methodology will be proposed. The rigorous mathematical proof will show that under the derived SOSM controller, the closed‐loop sliding mode dynamics is globally finite‐time stable. Meanwhile, the frequently used constant upper bound assumptions for the standard SOSM system can also be extended to the state‐dependent hypotheses in this paper. An academic example is illustrated to verify the effectiveness of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

Robust fault estimation of uncertain systems using an LMI‐based approach

General recent techniques in fault detection and isolation (FDI) are based on H_∞ optimization methods to address the issue of robustness in the presence of disturbances, uncertainties and modeling errors. Recently developed linear matrix inequality (LMI) optimization methods are currently used to design controllers and filters, which present several advantages over the Riccati equation‐based design methods. This article presents an LMI formulation to design full‐order and reduced‐order robust H_∞ FDI filters to estimate the faulty input signals in the presence of uncertainty and model errors. Several cases are examined for nominal and uncertain plants, which consider a weight function for the disturbance and a reference model for the faults. The FDI LMI synthesis conditions are obtained based on the bounded real lemma for the nominal case and on a sufficient extension for the uncertain case. The conditions for the existence of a feasible solution form a convex problem for the full‐order filter, which may be solved via recently developed LMI optimization techniques. For the reduced‐order FDI filter, the inequalities include a non‐convex constraint, and an alternating projections method is presented to address this case. The examples presented in this paper compare the simulated results of a structural model for the nominal and uncertain cases and show that a degree of conservatism exists in the robust fault estimation; however, more reliable solutions are achieved than the nominal design. Copyright © 2008 John Wiley & Sons, Ltd.     

Design of sliding mode controllers with bounded L 2 gain performance: an LMI approach

This paper is devoted to the design of robust controllers for sliding mode control systems. A linear fractional transformation (LFT) framework is adopted to describe the systems subject to model uncertainties and external disturbances. A linear matrix inequality (LMI) technique is then developed into the design of both sliding mode and reaching phase control laws. It is shown that by solving a set of LMIs, the switching surface can be designed such that the dynamics in the sliding mode achieve robust stability and bounded L ₂ gain performance with respect to matched and unmatched uncertainties in the presence of disturbances. Furthermore, a reaching phase control law is presented to eliminate the undesirable chattering effect and maintain the robustness property all the time whether the system evolves into its sliding mode in finite time or not. Finally, an observer based control law is also developed to achieve the robust performance during the reaching phase. The results are illustrated by an example.     

Adaptive Motion/Force Tracking Control for a Class of Mobile Manipulators

In this paper, modeling and adaptive motion/force tracking control is considered for a class of mobile manipulators under the holonomic and affine constraints with the presence of uncertainties and disturbances. Based on a suitable reduced dynamic model, adaptive controllers are proposed to ensure that the states of a closed‐loop system asymptotically track desired trajectories while the constraint force remains bounded by tuning design parameters. Detailed simulation results confirm the effectiveness of the control strategy.     

ROBUST H∞ CONTROL FOR UNCERTAIN STOCHASTIC SYSTEMS WITH MARKOVIAN SWITCHING AND TIME‐VARYING DELAYS

This paper is concerned with the problem of robust H_∞ control for uncertain stochastic systems with Markovian jump parameters and time‐varying state delays. A linear matrix inequality approach is developed and state feedback controllers are designed, which guarantee mean square asymptotic stability of the closed‐loop system and a prescribed H_∞ performance level for all modes and admissible uncertainties. A numerical example is provided to demonstrate the application of the proposed method.     

Robust stabilization of mimo nonlinear time-varying mismatched uncertain systems

This paper considers the problem of robust stabilization of multi-input-multi-output (MIMO) nonlinear systems in the presence of mismatched time-varying uncertainties. Combining the differential geometric feedback linearization with the deterministic approach, it derives two robust stabilizing controllers i.e. the high-gain feedback controller and the variable structure controller. These two controllers only require the knowledge of the nominal system and the bounds of the uncertainties. Therefore, they are more feasible to be implemented. Both controllers reduce the effects of arbitrarily large bounded mismatched uncertainties successfully to achieve uniform ultimate boundedness stabilization of the MIMO nonlinear timevarying uncertain systems.     

H∞ output feedback control for stochastic systems with randomly occurring convex‐bounded uncertainties and channel fadings

In this paper, the H_∞ output feedback control problem for a class of stochastic discrete‐time systems with randomly occurring convex‐bounded uncertainties and channel fadings is investigated. A sequence of mutually independent random variables with known probabilistic distributions are utilized to describe the randomness that convex‐bounded uncertainties appear in practical systems. The measurements with channel fadings are given by a stochastic Rice fading model which is regulated by a set of random variables with certain probability density functions. The purpose of this paper is to design an output feedback controller such that the closed‐loop control system is asymptotically stable with a prescribed H_∞ performance level. The less conservative results are obtained by employing the stochastic Lyapunov technique. Numerical examples are presented to illustrate effectiveness of the proposed approach.     

Adaptive output feedback tracking control of stochastic nonlinear systems with dynamic uncertainties

In this paper, adaptive output feedback tracking control is developed for a class of stochastic nonlinear systems with dynamic uncertainties and unmeasured states. Neural networks are used to approximate the unknown nonlinear functions. K‐filters are designed to estimate the unmeasured states. An available dynamic signal is introduced to dominate the unmodeled dynamics. By combining dynamic surface control technique with backstepping, the condition in which the approximation error is assumed to be bounded is avoided. Using It ô formula and Chebyshev's inequality, it is shown that all signals in the closed‐loop system are bounded in probability, and the error signals are semi‐globally uniformly ultimately bounded in mean square or the sense of four‐moment. Simulation results are provided to illustrate the effectiveness of the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust ℋ︁∞ PID control for multivariable networked control systems with disturbance/noise attenuation

In this paper, we study the design problem of PID controllers for networked control systems (NCSs) with polyhedral uncertainties. The load disturbance and measurement noise are both taken into account in the modeling to better reflect the practical scenario. By using a novel technique, the design problem of PID controllers is converted into a design problem of output feedback controllers. Our goal of this paper is two‐fold: (1) To design the robust PID tracking controllers for practical models; (2) To develop the robust ??_∞ PID control such that load and reference disturbances can be attenuated with a prescribed level. Sufficient conditions are derived by employing advanced techniques for achieving delay dependence. The proposed controller can be readily designed based on iterative suboptimal algorithms. Finally, four examples are presented to show the effectiveness of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.