L2-induced Norm Research Articles

Robust induced [formula omitted] optimal control of discrete-time systems having magnitude and rate-bounded actuators

The design of robust state- and output-feedback control for uncertain discrete-time systems with physical magnitude and rate constraints on their actuator dynamics was addressed. Unlike the traditional methods such as anti-windup (AW) methods, nested ellipsoids, model predictive controllers (MPCs) and integral quadratic constraints(IQCs) formulated by sector bounded inequalities, this paper uses a transformation of the system dynamics to a form which considers control signal and its rate as controlled outputs and using discrete-time ℓ∞ induced (peak-to-peak) norm from disturbance inputs to these outputs. To cope with the magnitude and rate bound non-linearities together, the induced ℓ∞ norm from disturbance input to the outputs involving control signal and its rate is utilised. On the other hand, discrete-time(DT) induced ℓ2 norm from disturbance input to the main controlled output is used to mitigate the effects of disturbances. We can tackle this ambitious non-linear control problem in the domain of linear convex multi-objective optimal control problem, which can be solved by effective semi-definite optimisation methods by using the proposed transformation and handling the control constraints in terms of worst case peak-to-peak gain of the system. Extended Linear Matrix Inequalities (LMIs) and full block S-procedure based design conditions developed over Linear Fractional Representation(LFR) framework allow the user to obtain robust state- and output-feedback control solutions with reduced conservatism. For the first time, this paper introduces an extended LMI based robust output-feedback control design for magnitude and rate bounded (MRB) systems, using full block S-procedure. We demonstrate the performance of the proposed controller through several simulations over benchmark examples covering systems having multi-variable structures and uncertainties. Our study also involves comparison results with a recently introduced technique based on multi-stage AW technique. The simulation results show that the proposed method of this paper is much effective and less conservative compared to the recent AW method provided in the literature.

An Improved Integral Inequality for Delay-Dependent Gain-Scheduled LPV Control

The present work examines the delay-dependent gain-scheduling feedback control with guaranteed closed-loop stability and induced L2 norm performance for continuous-time linear parameter-varying (LPV) systems with arbitrary time-varying delay. An extension of Lyapunov stability utilizing Krasovskii functionals is considered to derive stability analysis and synthesis conditions for delay-dependent dynamic output feedback LPV control design. The main challenges associated with this approach are selecting appropriate Lyapunov-Krasovskii functionals (LKFs) and finding efficient integral inequalities to bound the derivative of the LKF. Accordingly, a novel modified parameter-dependent LKF candidate along with an affine version of Jensen’s inequality bounding technique are employed leading to the derivation of less conservative sufficient conditions expressed in terms of convex linear matrix inequalities (LMIs). The proposed methodology is compared with past work in the literature in terms of conservatism reduction and performance improvement through a numerical example. Finally, the application of the proposed output-feedback LPV control design is evaluated on the automated mean arterial blood pressure (MAP) regulation in critical patient resuscitation via vasoactive drug infusion. Closed-loop simulation results are presented to illustrate the potential of the introduced LPV gain-scheduling design to provide MAP set-point tracking in the presence of disturbances and varying input delays.

LPV control and analysis under uncertain initial conditions

This paper is focused on the control of discrete-time linear parameter-varying (LPV) systems with uncertain initial conditions. The synthesis objective is to design a feedback LPV controller that ensures that the closed-loop system is asymptotically stable and some performance inequality is satisfied for all permissible parameter trajectories. The performance measure used is similar to the ℓ2-induced norm except that the initial state and the disturbance input are considered to reside in two separate norm balls; that is, the supremum of the ℓ2-norm of the performance output is minimized subject to the initial state taking values in a unit Euclidean ball and the disturbance input taking values in a unit ℓ2-norm ball. The paper provides convex analysis and synthesis conditions for solving this problem and demonstrates that using nonstationary LPV controllers instead of standard LPV ones could result in a significantly improved performance. The paper also shows how to determine bounding envelopes for the performance output using the analysis result. An illustrative example concludes the paper.

$H_\infty$ Event-Triggered Control With Performance Guarantees Vis-à-Vis the Optimal Periodic Solution

The paper puts forward an event-triggered controller with continuous-time H-inf (L2-induced norm) performance guarantees. The sampling rate of the proposed controller never exceeds that of the optimal periodic sampled-data controller for the same performance level. The proposed event-triggered controller yields actually slower sampling than that in the optimal periodic case, unless certain internal signal belongs to a class of worst-case analogue signals. The class of such signals, termed spoilers, is exhaustively characterized and shown to be atypical.

On unknown input observers designs for discrete-time LPV systems with bounded rates of parameter variation

This paper concerns the design of a novel unknown input observer (UIO) for discrete-time linear parameter-varying (LPV) systems with bounded rates of parameter variation. The synthesis conditions have been formulated using a less conservative approach to obtain two different structures: a proportional UIO and a proportional-integral UIO. The main highlights of the present design conditions are the ability to deal with discrete-time LPV systems subject to both states and outputs parameter-varying matrices in the state-space representation and also the possibility of bounded rates of parameter variation. Such conditions may be useful to achieve better performance than considering only arbitrarily fast time-varying parameters for the discrete-time LPV representations. In order to obtain the UIO designs, stability and induced L2 norm performance conditions using a poly-quadratic approach in terms of Linear Matrix Inequalities are addressed. Numerical examples are presented to demonstrate the effectiveness of the proposed design methods.

Induced Norm-Based Analysis for Computed Torque Control of Robot Systems

This paper is concerned with providing a new quantitative analysis method of computed torque control of robot systems by using three types of induced norm. This paper first considers design of a computed torque controller to achieve a trajectory tracking performance of a robot manipulator. Taking into account of the effects of unknown elements on the tracking performance, this paper next divides the unknown elements into model uncertainty and disturbance, and introduces various closed-loop representations of robot systems consisting of robot manipulators together with computed torque controllers and unknown elements. This paper further derives a readily applicable robust stability condition for the model uncertainty by using two types of induced norm from L 2 to L 2 and from L ∞ to L ∞ . Regarding a performance analysis for the disturbance, this paper also proposes to take the induced norm from L 2 to L ∞ , by which the relation between the maximum tracking errors caused by the disturbance and the corresponding parameters of the computed torque controllers are dealt with. Finally, this paper gives some experiments to validate the effectiveness of the performance analysis methods based on the L ∞ /L 2 -induced norm.

LPV Control Design for Autonomous Underwater Vehicles Using Robustness Analysis Tools

This paper deals with the design and analysis of a linear parameter-varying (LPV) path-following controller for an autonomous underwater vehicle (AUV). The LPV controller is designed to provide guaranteed performance for any planar path whose inverse radius of curvature is bounded. Such a flexibility provided by the LPV controller offers significant advantages in AUV missions such as seabed mapping and mine countermeasures. The control design approach uses a lumped system model that is based on a virtual vehicle formulation and combines the six degrees-of-freedom AUV dynamics with the path-following dynamics. An LPV path-following controller is designed using the ℓ2-induced norm as the performance measure and the inverse radius of curvature of the path as the scheduling parameter. The penalty weights of the controller are tuned using a tool that utilizes integral quadratic constraint (IQC) theory to perform robustness analysis. The robustness and performance of the LPV controller are studied using IQC analysis and non-deterministic MATLAB simulations where hydrodynamic model uncertainties, unmodeled servo dynamics, AUV nonlinear dynamics, and sensor noise are considered.

Periodically Time-varying State Observer Design for Multi-rate Systems Using <i>l</i><sub>2</sub>-induced Norm

A multi-rate state observer with a periodically time-varying gain is proposed in this paper. The proposed observer can be regarded as a periodically time-varying system. By using the cycling technique, the periodically time-varying system can be dealt with as a time-invariant system. This paper presents a method to design a multi-rate state observer using the cycling technique. Then, analysis about the l2-induced norm of the estimation error by the proposed observer is reduced to an LMI optimization problem. Then, the design problem of the time-varying gain is also described as an LMI optimization problem. The effectiveness of the multi-rate state observer is shown by numerical examples.

Stability and Performance Analysis of Networked Control Systems: A Lifted Sample-Time Approach with [formula omitted] Induced Norm

The lifted sample-time approach has been extensively applied to analyze the behavior of a Networked Control System (NCS) under instances of packet-drop and/or variable communication latency (jitter) of varying durations. This paper combines the simplicity of the lifted sample-time approach in modeling an NCS under arbitrary but bounded instances of packet-drop and/or jitter with the proven L2induced norm based approach of controller design to realize an NCS which is capable of preserving the designed value of the norm under assumed conditions of packet-drop and/or jitter and hence a satisfactory time response under these conditions. The design methodology presented in this paper assumes a state feedback controller and uses a simple criterion based on a Linear Matrix Inequality (LMI). The methodology is validated using credible offline and real-time simulations.

Model reduction of distributed nonstationary LPV systems

This paper is on the structure-preserving model reduction of distributed systems formed by heterogeneous, discrete-time, nonstationary linear parameter-varying subsystems interconnected over arbitrary directed graphs. The subsystems are formulated in a linear fractional transformation (LFT) framework, and a communication latency of one sampling period is considered. The balanced truncation method is extended to the class of systems of interest, and upper bounds on the ℓ2-induced norm of the resulting error system are derived. Balanced truncation suffers from conservatism since it only applies to stable systems which possess structured solutions to the generalized Lyapunov inequalities. The coprime factors reduction method is then provided as a partial remedy to this conservatism. An illustrative example is given to demonstrate the efficacy of the proposed approaches.

Balanced truncation model reduction of nonstationary systems interconnected over arbitrary graphs

This paper deals with the balanced truncation of discrete-time, linear time-varying, heterogeneous subsystems interconnected over finite arbitrary directed graphs. The information transfer between the subsystems is subject to a communication latency of one time-step. The method guarantees the preservation of the interconnection structure and further allows for its simplification. In addition to truncating temporal states associated with the subsystems, the method allows for the order reduction of spatial states associated with the interconnections between the subsystems and even the removal of whole interconnections. Upper bounds on the ℓ2-induced norm of the resulting error system are derived. The method is illustrated through an example.

Distributed control of nonstationary LPV systems over arbitrary graphs

This paper deals with the ℓ2-induced norm control of discrete-time, nonstationary linear parameter-varying (NSLPV) subsystems, represented in a linear fractional transformation (LFT) framework and interconnected over arbitrary directed graphs. Communication between the subsystems is subjected to a one-step time-delay. NSLPV models have state-space matrix-valued functions with explicit dependence on time-varying terms that are known a priori, as well as parameters that are not known a priori but are available for measurement at each discrete time-step. The sought controller has the same interconnection and LFT structures as the plant. Convex analysis and synthesis results are derived using a parameter-independent Lyapunov function. These conditions are infinite dimensional in general, but become finite dimensional in the case of eventually time-periodic subsystems interconnected over finite graphs. The method is applied to an illustrative example.

A Parameter Dependent Controller Design Approach for Delayed LPV System

AbstractThis paper investigates the memory state feedback controller synthesis problem for linear parameter‐varying (LPV) systems with time‐varying input delay. A novel state transformation is proposed to reduce the influence of time delay, and two types of Lyapunov‐Krasovskii functional are employed to guarantee the stability of the closed‐loop system and induced L2 norm performance. The controller gain is formulated to parameter dependent, and design method is then led to a convex optimization problem and solved by parameter gridding technique. A numerical example is provided to validate the effectiveness of the proposed approach.

Optimal control of a small fixed-wing UAV about concatenated trajectories

The paper focuses on the control of a small fixed-wing unmanned aerial vehicle (UAV) along trajectories. The trajectories are generated in real-time from a library of pre-specified motion primitives. Two concatenated primitive trajectories are considered: one formed from seven primitives and exhibiting a figure-8 geometric path and another composed of a Split-S maneuver that settles into a level-turn trim trajectory. Switched control systems stemming from ℓ2-induced norm synthesis approaches are designed for discrete-time linearized models of the nonlinear UAV system. These controllers are analyzed based on simulations in a realistic operational environment and are further implemented on the physical UAV.

A hierarchical approach for primitive-based motion planning and control of autonomous vehicles

A hierarchical approach for motion planning and control of nonlinear systems operating in obstacle environments is presented. To reduce the computation time during the motion planning process, dynamically feasible trajectories are generated in real-time through concatenation of pre-specified motion primitives. The motion planning task is posed as a search over a directed graph, and the applicability of informed graph search techniques is investigated. Specifically, we develop a locally greedy algorithm with effective backtracking ability and compare this algorithm to weighted A* search. The greedy algorithm shows an advantage with respect to solution cost and computation time when larger motion primitive libraries that do not operate on a regular state lattice are utilized. Linearization of the nonlinear system equations about the motion primitive library results in a hybrid linear time-varying model, and an optimal control algorithm using the ℓ2-induced norm as the performance measure is provided to ensure that the system tracks the desired trajectory. The ability of the resulting controller to closely track the trajectory obtained from the motion planner, despite various disturbances and uncertainties, is demonstrated through simulation.

On finite-horizon [formula omitted]-induced norms of discrete-time switched linear systems

The finite-horizon ℓ2-induced norm evaluation problem of discrete-time switched linear systems is considered in this paper. The set of worst-case switching laws is characterized, which is a subset of the whole set of switching laws under consideration. Based on the properties of the worst-case switching law set, a low-complexity complete solution to the ℓ2-induced norm evaluation problem is provided for general switched linear systems. It is shown that under the assumption of exponential stability, the finite-horizon ℓ2-induced norm monotonically converges to the infinite-horizon norm, and the volume of the worst-case switching law set gradually becomes constant as the time horizon approaches infinity. Some numerical properties of the worst-case switching law set are also exploited. A numerical example is presented to illustrate the proposed results.

Induced [formula omitted] norm control for LPV system with specified class of disturbance inputs

This paper presents a novel controller design approach for linear parameter-varying (LPV) systems to guarantee the stability with an induced L2 norm performance. Motivated by the idea of the generalized Kalman–Yakubovich–Popov (KYP) lemma, the spectral information of the disturbance inputs can be captured by a matrix-valued integral quadratic constraint. To a certain extent, this formulation not only weakens the function of weighting functions in the control synthesis but also reduces the order of the designed controller. The obtained controller design conditions for LPV systems are presented in terms of linear matrix inequalities (LMIs), which can be solved efficiently by the standard Matlab toolbox. The effectiveness of the proposed method is verified by solving a missile benchmark problem.

LPV Control of Nonstationary Systems: A Parameter-Dependent Lyapunov Approach

The paper presents a parameter-dependent Lyapunov approach for the control of nonstationary linear-parameter varying systems. The work is motivated by the challenges encountered in controlling nonlinear systems about pre-specified aggressive trajectories, specifically ones which eventually settle into periodic orbits. The control design method uses the l2-induced norm performance measure, and the analysis and synthesis results are given in terms of parameterized linear matrix inequalities. Two fast and easy-to-implement algorithms for online controller construction are also provided.

A Hybrid Architecture for Maneuver-Based Motion Planning and Control of Agile Vehicles

AbstractThe paper presents an integrated approach for motion planning and control of nonlinear systems in obstacle environments. To reduce online computation time, dynamically-feasible trajectories will be generated in real-time from a library of pre-specified maneuvers, with the sequence of maneuvers to be traversed determined by a heuristic search algorithm which incorporates some features of greedy search along with effective backtracking. Linearizing the nonlinear system equations about the library maneuvers results in a hybrid linear time-varying model. The paper provides an optimal control algorithm, with the ℓ2-induced norm as the performance measure, for such a hybrid model, which ensures that the system tracks the desired trajectory as accurately as possible despite various disturbances and uncertainties. The paper also compares the performance of the developed motion planning algorithm to that of dynamically weighted A*.

Induced L2 Norm Improvement by Interpolating Controllers for Discrete-time LPV Systems

The paper shows an interpolation-based control solution asapossible technique to formulate the constrained H∞ control problem for discrete-time linear parameter varying (LPV) systems. The control policy is constructed by interpolating amongapriori designed, unconstrained, constant, state feedback controllers. Invariant set theory is used to introduce the measure of the domain of applicability. It is shown that the `trade-off' between the performance and the size of the domain of applicability can be significantly reduced by controller interpolation. Hence, the interpolation-based controller becomes applicable overamuch larger region than any other single state feedback. The proposed method gives stabilizing solution not only under hard constraints, but also allows the online modification of the induced L² norm from the generalized disturbance input to the predefined performance output. Moreover, the suggested method can be applied in real-time environment.