Saturated Linear Feedback Research Articles

Fuzzy control of singular fractional order multi-agent systems with actuator saturation

In this paper, a T-S fuzzy model is established to describe nonlinear fuzzy singular fractional order multi-agent systems (SFOMASs) exhibiting actuator saturation when the order is between 0 and 2. We focus on the case where the system state is difficult to measure or cannot be directly measured, so it needs to be estimated based on other measurable signals. To this end, a fuzzy observer is designed, which can process fuzzy or imprecise input and output signals through fuzzy processing. Then the observer-based leader-following consensus problem of fuzzy SFOMASs with actuator saturation is addressed. To address input saturation, we use some mathematical tools to represent the saturated linear feedback on the convex hull, which is then transformed into seeking the optimal solution of linear matrix inequalities (LMIs). Finally, the effectiveness of the proposed approach is verified through the adoption of a numerical example.

Stabilization of the complex double integrator by means of a saturated linear feedback

Stabilization of the complex double integrator by means of a saturated linear feedback

Analysis of Positive Systems With Input Saturation: Invariant Hyperpyramids and Hyperrectangles

This article addresses the problem of estimating the domain of attraction of positive systems under a saturated linear feedback. Compared with the analysis of general saturated linear systems, one remarkable difference is that only the domain of attraction inside the first orthant is of interest. Hence, we tackle this problem by exploiting different shapes of invariant sets confined in the first orthant, including hyperpyramids and hyperrectangles, which are, respectively, the level sets of sum-separable and max-separable Lyapunov functions. For an open-loop positive system, a sufficient condition is given to ensure that a pyramid (or a hyperrectangle) is contractively invariant for the closed loop. This condition is nonconservative for the single-input case. Moreover, for a system without open-loop positivity, we establish conditions such that the closed loop has a hyperrectangular invariant set. Examples show the superiority of the results, in comparison with those existing ones using invariant ellipsoids.

A comparison of tracking step inputs with a piezo stage using model predictive control and saturated linear quadratic Gaussian control

Compressed Sensing for Atomic Force Microscopy is a newer imaging mode that requires the piezo stage be driven rapidly between measurement locations. In contrast to raster scanning applications, this translates to a setpoint tracking problem. This paper considers the setpoint tracking performance of a piezo nano-positioning stage subject to rate-of-change limitations on the control signal, which is derived from the current limit of the power amplifier. To compensate the vibrational dynamics of the stage, a model predictive control scheme (MPC) and a linear quadratic Gaussian (LQG) controller which saturates the control increment are considered. In both cases, hysteresis and drift are compensated via dynamic inversion. To design the weighting matrices required by the MPC and linear feedback designs, an extension to classic reciprocal root locus ideas is proposed. The robustness of both schemes using classical methods like gain margin, phase margin, and gain of the sensitivity function at low frequencies is analyzed. The overall settle times achieved by both controllers (in both simulation and experiment) across a range of control weights where the reference input is a sequence of step inputs of varying amplitudes are compared. The results show that the best simulation settle time is achieved by MPC using the smallest control weight. However under experimental conditions, the best settle time is achieved by a much larger control weight and the performance of MPC becomes comparable with that of saturated linear feedback. This result is explained by showing that robustness increases with larger control weights.

Composite anti-disturbance control for stochastic systems with multiple heterogeneous disturbances and input saturation

This paper develop a novel composite anti-disturbance control (CADC) method for a class of stochastic nonlinear systems in which input saturation and multiple heterogeneous disturbances occur. The disturbances include additive white noise, multiplicative white noise and modeled disturbances with coupling of energy-limited white noise and state signals. The input saturation item of system is approximated by the polytopic bounding method. In order to estimate the modeled disturbances, a novel stochastic saturating disturbance observer (SSDO) is designed. The CADC scheme is presented by integrating SSDO with convex hull expression of saturated linear feedback law, such that the superior anti-disturbance performance to be achieved while overcoming the effects of input saturation. Finally, the validity and usability of the developed scheme are testified by the numerical simulation and practical application example.

Event‐triggered constrained control of positive systems with input saturation

SummaryThis paper addresses the problem of event‐triggered stabilization for positive systems subject to input saturation, where the state variables are in the nonnegative orthant. An event‐triggered linear state feedback law is constructed. By expressing the saturated linear state feedback law on a convex hull of a group of auxiliary linear feedback laws, we establish conditions under which the closed‐loop system is asymptotically stable with a given set contained in the domain of attraction. On the basis of these conditions, the problem of designing the feedback gain and the event‐triggering strategy for attaining the largest domain of attraction is formulated and solved as an optimization problem with linear matrix inequality constraints. The problem of designing the feedback gain and the event‐triggering strategy for achieving fast transience response with a guaranteed size of the domain of attraction is also formulated and solved as an linear matrix inequality problem. The effectiveness of these results is then illustrated by numerical simulation.

Global stabilization of the linearized three-axis axisymmetric spacecraft attitude control system by bounded linear feedback

In this paper, the three-axis attitude stabilization of the axisymmetric spacecraft with bounded inputs is studied. By constructing some novel state transformations, saturated linear state feedback controllers are constructed for the considered attitude control system. By constructing suitable quadratic plus integral Lyapunov functions, globally asymptotic stability of the closed-loop systems is proved if the feedback gain parameters satisfy some explicit conditions. By solving some min–max optimization problems, a global optimal feedback gain for the underactuated attitude stabilization system is proposed such that the convergence rate of the linearized closed-loop system is maximized. Numerical simulations show the effectiveness of the proposed approaches.

Global Stabilization of Linearized Spacecraft Rendezvous System by Saturated Linear Feedback

The problem of global stabilization of the circular orbit spacecraft rendezvous system subject to actuator saturation is studied in this brief. Based on the linearized Clohessy–Wiltshire equations, a class of saturated linear state feedback controllers is designed by utilizing Lyapunov functions that are frequently used in the absolute stability theory. The global stability of the closed-loop system with the proposed saturated linear feedback is proved by providing explicit conditions on the parameters in the feedback gains. Optimal linear feedback gains are also discussed. The most significant advantages of the proposed controllers are that the controllers are linear and the global stability of the resulting closed-loop system can be guaranteed. Numerical simulations show the effectiveness of the proposed approaches.

Magnetic attitude control of bias momentum spacecraft by bounded linear feedback

This paper addresses the three-axis magnetic attitude control of small spacecraft with momentum bias configuration by bounded linear feedback. Necessary and sufficient conditions are derived to guarantee that the linearized dynamics and kinematics are stable in the Lyapunov sense. Explicit solutions to some Lyapunov equations associated with the considered open-loop system are obtained. Based on the explicit solutions to the Lyapunov equation, both explicit saturated linear state feedback controller and function observer based output feedback controller are designed. Globally asymptotic stability of the closed-loop systems is proved by constructing explicit Lyapunov functions. Simulations show the effectiveness of the proposed approach.

Event‐triggered probabilistic robust control of linear systems with input constrains: By scenario optimization approach

SummaryThis paper addresses the problem of probabilistic robust stabilization for uncertain systems subject to input saturation. A new probabilistic solution framework for robust control analysis and synthesis problems is addressed by a scenario optimization approach, in which the uncertainties are not assumed to be norm bounded. Furthermore, by expressing the saturated linear feedback law on a convex hull of a group of auxiliary linear feedback laws, we establish conditions under which the closed‐loop system is probabilistic stable. Based on these conditions, the problem of designing the state feedback gains for achieving the largest size of the domain of attraction is formulated and solved as a constrained optimization problem with linear matrix inequality constraints. The results are then illustrated by a numerical example.

Constrained control of uncertain nonhomogeneous Markovian jump systems

SummaryThis paper addresses the problem of robust stabilization for uncertain systems subject to input saturation and nonhomogeneous Markovian jumps, where the uncertainties are assumed to be norm bounded and the transition probabilities are time‐varying and unknown. By expressing the saturated linear feedback law on a convex hull of a group of auxiliary linear feedback laws and the time‐varying transition probabilities inside a polytope, we establish conditions under which the closed‐loop system is asymptotically stable. On the basis of these conditions, the problem of designing the state feedback gains for achieving fast transience response with a guaranteed size of the domain of attraction is formulated and solved as a constrained optimization problem with linear matrix inequality constraints. The results are then illustrated by numerical examples including the application to a DC motor speed control example. Copyright © 2017 John Wiley & Sons, Ltd.

The maximal contractively invariant ellipsoids for discrete-time linear systems under saturated linear feedback

In this paper, we consider the problem of determining the maximal contractively invariant ellipsoids for discrete-time linear systems with multiple inputs under saturated linear feedback. We propose an algebraic computational approach to determining such maximal contractively invariant ellipsoids. We divide the state space into several regions according to the saturation status of each input, and compute the possible maximal contractively invariant ellipsoids on each region except the region where none of inputs saturate and on their intersections. The minimal one among these possible maximal contractively invariant ellipsoids is the maximal contractively invariant ellipsoids of the system. Simulation results demonstrate the effectiveness of our methods.

Robust MPC for Linear Systems with Structured Time‐Varying Uncertainties and Saturating Actuator

AbstractIn this paper, a synthesis of model predictive control (MPC) algorithm is presented for uncertain systems subject to structured time‐varying uncertainties and actuator saturation. The system matrices are not exactly known, but are affine functions of a time varying parameter vector. To deal with the nonlinear actuator saturation, a saturated linear feedback control law is expressed into a convex hull of a group of auxiliary linear feedback laws. At each time instant, a state feedback law is designed to ensure the robust stability of the closed‐loop system. The robust MPC controller design problem is formulated into solving a minimization problem of a worst‐case performance index with respect to model uncertainties. The design of controller is then cast into solving a feasibility of linear matrix inequality (LMI) optimization problem. Then, the result is further extended to saturation dependent robust MPC approach by introducing additional variables. A saturation dependent quadratic function is used to reduce the conservatism of controller design. To show the effectiveness, the proposed robust MPC algorithms are applied to a continuous‐time stirred tank reactor (CSTR) process.

Robust distributed model predictive control under actuator saturations and packet dropouts with time‐varying probabilities

In this study, the authors investigate distributed model predictive control for polytopic uncertain systems subject to actuator saturations and packet dropouts with time-varying probabilities in controller-to-actuator channels. For the sake of less computational efforts and a flexible system structure, the global system is decomposed into several lower-dimensional subsystems and the subsystems optimise a global performance function in parallel. A novel augmented distributed controller model is proposed to describe both the actuator saturations and packet dropouts in one unified expression by adopting a stochastic variable taking the value 1 or 0 with time-varying probabilities and accordingly a new result is obtained. By placing the saturated linear feedback law into the convex hull of a group of linear feedback laws, the distributed controller for each subsystem is obtained from the solution of the convex optimisation problem involving linear matrix inequalities. A numerical example is supplied to demonstrate the effectiveness of the developed techniques.

Stabilization on null controllable region of delta operator systems subject to actuator saturation

Summary This paper focuses on stabilization on null controllable region of delta operator systems subject to actuator saturation via constructing suitable controllers. Under a saturated linear feedback law, a set of stable equilibrium points is obtained for the delta operator system with actuator saturation. A comprehensive controller is given to show global stabilization of the delta operator system. Thereby, semi-global stabilization of the delta operator system is also achieved. For higher-order systems with no more than two unstable exponentially poles, a new feedback law is constructed to achieve semi-global stabilization on the null controllable region. Finally, three examples are given to illustrate the effectiveness of the proposed techniques. Copyright © 2016 John Wiley & Sons, Ltd.

Global stabilization of periodic linear systems by bounded controls with applications to spacecraft magnetic attitude control

We study in this paper the three-axis magnetic attitude control of small spacecraft by considering the actuator saturation. By noticing that the linearized dynamics is a neutrally stable linear periodic system, a general problem of global stabilization of linear periodic systems with bounded feedback is formulated and solved by both state feedback and observer based output feedback. Explicit saturated linear feedback control laws are established by using solutions to some Lyapunov differential equations associated with the open-loop systems and global stability of the closed-loop system is proved by constructing an explicit Lyapunov function. To apply the theoretical results on the three-axis magnetic attitude control systems, necessary and sufficient conditions are established to guarantee that the linearized open-loop system is neutrally stable and explicit solutions to the associated Lyapunov equations are obtained consequently. The controllers for the three-axis magnetic attitude control systems are then given in closed-form. Simulations show the effectiveness of the proposed approach.

A Complete Characterization of the Maximal Contractively Invariant Ellipsoids of Linear Systems Under Saturated Linear Feedback

As level sets of quadratic Lyapunov functions, ellipsoids have been extensively used as estimates of the domain of attraction of a linear system under saturated linear feedback. For a linear system with a single input subject to actuator saturation, based on a convex hull representation of saturation functions, a necessary and sufficient condition for an ellipsoid to be contractively invariant was previously established, which, through the solution of an LMI problem, leads to the maximal ellipsoidal invariant set. In this technical note, for a linear system with multiple inputs subject to actuator saturation, we develop a complete characterization of the maximal ellipsoidal invariant set by algebraic computation. Simulation results demonstrate the effectiveness of our results.

Saturation-based switching anti-windup design for linear systems with nested input saturation

This paper proposes a saturation-based switching anti-windup design for the enlargement of the domain of attraction of a linear system subject to nested saturation. A nestedly saturated linear feedback is expressed as a linear combination of a set of auxiliary linear feedbacks, which form a convex hull where the nestedly saturated linear feedback resides. This set of auxiliary linear feedbacks is then partitioned into several subsets. The auxiliary linear feedbacks in each of these subsets form a convex sub-hull of the original convex hull. When the value of the nestedly saturated linear feedback falls into a convex sub-hull, it can be expressed as a linear combination of the subset of all the auxiliary feedbacks that form the convex sub-hull. A separate anti-windup gain is designed for each convex sub-hull by using a common quadratic Lyapunov function and is implemented when the value of the nestedly saturated linear feedback falls into this convex sub-hull. Simulation results indicate that such a saturation-based switching anti-windup design has the ability to significantly enlarge the domain of attraction of the closed-loop system.

Analysis and design for delta operator systems with actuator saturation

This paper studies the problem of estimating the domain of attraction for delta operator systems under a pre-designed saturated linear feedback. Ellipsoids are used as the estimates of domain of attraction for delta operator systems. A condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractive invariant. Moreover, some necessary and sufficient conditions are given for single input delta operator systems. The results are also further extended to determine the invariant sets for delta operator systems with persistent disturbances. Linear matrix inequalities based on the methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements. Some numerical examples are given to illustrate the effectiveness of the developed techniques.

Design of Saturation-Based Switching Anti-Windup Gains for the Enlargement of the Domain of Attraction

This technical note proposes a saturation-based switching anti-windup design for the enlargement of the domain of attraction of the closed-loop system. A widely adopted method of dealing with an m-dimensional saturated linear feedback is to express it as a linear combination of a set of 2m auxiliary linear feedbacks. For each value of the state, these auxiliary linear feedbacks form a convex polyhedron of 2m vertices in the input space. We propose to divide this convex polyhedron into several convex sub-polyhedrons, each of which is defined as the convex hull of a subset of the auxiliary feedbacks. Whenever the value of the saturated linear feedback falls into a sub-polyhedron, it can be expressed as a linear combination of a subset of the 2m auxiliary linear feedbacks that define the sub-polyhedron and thus a less conservative result can be expected. A separate anti-windup gain is designed for each sub-polyhedron by using a common quadratic Lyapunov function and implemented when the value of the saturated linear feedback falls into this sub-polyhedron. Simulation results indicate that such a saturation-based switching anti-windup design has the ability to significantly enlarge the domain of attraction of the closed-loop system.