Quadratic Stability Condition Research Articles

Parameter-Estimation-Based Gain-Scheduling Control of Weight on Bit in Drilling Process With Uncertain Penetration Resistance Coefficient.

It is inevitable to encounter through different formations in the drilling process for deep exploration, and the penetration resistance coefficient (PRC) is an uncertain parameter related to lithology. In this article, a parameter-estimation-based gain-scheduling controller is developed to eliminate undesired system performance deterioration due to the uncertain parameter. First, a drill-string axial finite element model with the uncertain PRC is established, and a control-oriented low-order model is derived via mode selection. A gain-scheduling controller is computed based on the quadratic stability condition of the closed-loop system, which can cope with the system's parameter uncertainty using variable low-frequency gain. An adaptive observer is designed to estimate the unmeasurable scheduling variable. Field data from a geothermal drilling well is obtained to validate our model. According to this drilling well scenario, both numerical and experimental results illustrate the effectiveness of our method. The explicit relationship between the controller gain and the uncertain parameter is presented. It is found that the performance of the closed-loop system is more sensitive to the controller gain when drilling in soft formations, requiring more attention in such scenarios.

DIGITAL STABILIZATION OF A SWITCHABLE LINEAR SYSTEM WITH COMMENSURATE DELAYS

An approach to the construction of a digital controller that stabilizes a non-continuous switchable linear system with commensurate delays in control is proposed. The approach to stabilization sequentially includes the construction of a switchable continuously discrete closed system with a digital controller, the transition to its discrete model, represented as a switchable system with modes of various orders and the construction of a discrete dynamic controller based on the quadratic stability condition of a closed switchable discrete system.

Iterative Design Algorithm for Robust Disturbance-Rejection Control

An iterative design algorithm is developed for robust disturbance–rejection control of uncertain systems with time-varying parameter perturbations in this paper. For more design degrees of freedom, a generalized equivalent-input-disturbance estimator is adopted to approximate the effect of both disturbances and uncertainties. By the bound real lemma, the H∞ norm is used to evaluate the robust disturbance–rejection performance of the closed-loop uncertain system. To avoid the constraints introduced by the widely used commutative condition, the control gains are divided into two groups and calculated by steps. Further, two robust quadratic stability conditions are derived, and an iterative design algorithm is developed to optimize the robust H∞ disturbance–rejection performance. Finally, the effectiveness and advantages of the developed method are demonstrated by a case study of a suspension system of modern vehicles.

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.

Quantized Control Design for Linear Systems Using Reinforcement Learning

In this paper, a quadratic stabilizing controller minimizing a cost function is designed through model-free and online reinforcement learning for systems with logarithmic quantized input. By introducing a new gain dependent on quantization density, the input and related weighting matrix in the cost function are deviated from their original ones. Then, using these deviated parameters, the controller is trained through reinforcement learning such that the closed-loop system satisfies the quadratic stability condition with the cost function minimized. An inverted pendulum example is used to show the effectiveness and merits of the proposed method.

Stochastic MPC With Dynamic Feedback Gain Selection and Discounted Probabilistic Constraints

This paper considers linear discrete-time systems with additive disturbances, and designs a Model Predictive Control (MPC) law incorporating a dynamic feedback gain to minimise a quadratic cost function subject to a single chance constraint. The feedback gain is selected online and we provide two selection methods based on minimising upper bounds on predicted costs. The chance constraint is defined as a discounted sum of violation probabilities on an infinite horizon. By penalising violation probabilities close to the initial time and assigning violation probabilities in the far future with vanishingly small weights, this form of constraints allows for an MPC law with guarantees of recursive feasibility without a boundedness assumption on the disturbance. A computationally convenient MPC optimisation problem is formulated using Chebyshev's inequality and we introduce an online constraint-tightening technique to ensure recursive feasibility. The closed loop system is guaranteed to satisfy the chance constraint and a quadratic stability condition. With dynamic feedback gain selection, the closed loop cost is reduced and conservativeness of Chebyshev's inequality is mitigated. Also, a larger feasible set of initial conditions can be obtained. Numerical simulations are given to show these results.

Simultaneous state and unknown input set‐valued observers for quadratically constrained nonlinear dynamical systems

AbstractIn this article, we propose fixed‐order set‐valued (in the form of ‐norm hyperballs) observers for several classes of quadratically constrained nonlinear dynamical systems with unknown input signals that simultaneously/jointly find bounded hyperballs of states and unknown inputs that include the true states and inputs. Necessary and sufficient conditions in the form of linear matrix inequalities (LMIs) for the stability (in the sense of quadratic stability) of the proposed observers are derived for ()‐quadratically constrained (()‐QC) systems, which includes several classes of nonlinear systems: (I) Lipschitz continuous, (II) ()‐QC* and (III) linear parameter‐varying (LPV) systems. This new quadratic constraint property is at least as general as the incremental quadratic constraint property for nonlinear systems and is proven in the paper to embody a broad range of nonlinearities. In addition, we design the optimal observer among those that satisfy the quadratic stability conditions and show that the design results in uniformly bounded‐input bounded‐state (UBIBS) estimate radii/error dynamics and uniformly bounded sequences of the estimate radii. Furthermore, we provide closed‐form upper bound sequences for the estimate radii and sufficient conditions for their convergence to steady state. Finally, the effectiveness of the proposed set‐valued observers is demonstrated through illustrative examples, where we compare the performance of our observers with some existing observers.

An MPC-LQR-LPV Controller with Quadratic Stability Conditions for a Nonlinear Half-Car Active Suspension System with Electro-Hydraulic Actuators

The active suspension system of a vehicle manipulated using electro-hydraulic actuators is a challenging nonlinear control problem. In this research work, a novel Linear Parameter Varying (LPV) State-Space (SS) model with a fictional input is proposed to represent a nonlinear half-car active suspension system. Four different scheduling parameters are used to embed the nonlinearities of both the suspension and the electro hydraulic actuators to represent its nonlinear behavior. A recursive least squares (RLS) algorithm is used to predict the future behavior of the scheduling parameters along the prediction horizon. A Model Predictive Control-Linear Quadratic Regulator (MPC-LQR) is implemented as the control strategy and, to ensure stability, Quadratic Stability conditions are imposed as Linear Matrix Inequalities (LMI) constraints. Furthermore, the inclusion of attraction sets to overcome the conservative performance imposed by the Quadratic Stability conditions is included, as well as a terminal set were the switching between the MPC and the LQR controller is made. Simulations results for the half-car active suspension model over a typical road disturbance are tested to show the effectiveness of the proposed MPC-LQR-LPV controller with quadratic stability conditions in terms of comfort and road-holding.

Discrete-Time Takagi-Sugeno Stabilization Approach Applied in Autonomous Vehicles

This paper deals with a new robust control design for autonomous vehicles. The goal is to perform lane-keeping under various constraints, mainly unknown curvature and lateral wind force. To reach this goal, a new formulation of Parallel Distributed Compensation (PDC) law is given. The quadratic Lyapunov stability and stabilization conditions of the discrete-time Takagi–Sugeno (T-S) model representing the autonomous vehicles are discussed. Sufficient design conditions expressed in terms of strict Linear Matrix Inequalities (LMIs) extracted from the linearization of the Bilinear Matrix Inequalities (BMIs) are proposed. An illustrative example is provided to show the effectiveness of the proposed approach.

Asymptotically Necessary and Sufficient Quadratic Stability Conditions of T-S Fuzzy Systems Using Staircase Membership Function and Basic Inequality

Asymptotically necessary and sufficient quadratic stability conditions of Takagi-Sugeno (T-S) fuzzy systems are obtained by utilizing staircase membership functions and a basic inequality. The information of the membership functions is incorporated in the stability analysis by approximating the original continuous membership functions with staircase membership functions. The stability of the T-S fuzzy systems was investigated based on a quadratic Lyapunov function. The asymptotically necessary and sufficient stability conditions in terms of linear matrix inequalities were derived using a basic inequality. A fuzzy controller was also designed based on the stability results. The derivation process of the stability results is straightforward and easy to understand. Case studies confirmed the validity of the obtained stability results.

Active Suspension Control Using an MPC-LQR-LPV Controller with Attraction Sets and Quadratic Stability Conditions

The control of an automotive suspension system by means of a hydraulic actuator is a complex nonlinear control problem. In this work, a linear parameter varying (LPV) model is proposed to reduce the complexity of the system while preserving the nonlinear behavior. In terms of control, a dual controller consisting of a model predictive control (MPC) and a Linear Quadratic Regulator (LQR) is implemented. To ensure stability, quadratic stability conditions are imposed in terms of Linear Matrix Inequalities (LMI). Simulation results for quarter-car model over several disturbances are tested in both frequency and time domain to show the effectiveness of the proposed algorithm.

Quadratic Stabilization of Linear Uncertain Positive Discrete-Time Systems

The paper provides extended methods for control linear positive discrete-time systems that are subject to parameter uncertainties, reflecting structural system parameter constraints and positive system properties when solving the problem of system quadratic stability. By using an extension of the Lyapunov approach, system quadratic stability is presented to become apparent in pre-existing positivity constraints in the design of feedback control. The approach prefers constraints representation in the form of linear matrix inequalities, reflects the diagonal stabilization principle in order to apply to positive systems the idea of matrix parameter positivity, applies observer-based linear state control to assert closed-loop system quadratic stability and projects design conditions, allowing minimization of an undesirable impact on matching parameter uncertainties. The method is utilised in numerical examples to illustrate the technique when applying the above strategy.

Fast Simulation of Analog Circuit Blocks Under Nonstationary Operating Conditions

This article proposes a black-box behavioral modeling framework for analog circuit blocks (CBs) operating under small-signal conditions around nonstationary operating points. Such variations may be induced either by changes in the loading conditions or by event-driven updates of the operating point for system performance optimization, e.g., to reduce power consumption. An extension of existing data-driven parameterized reduced-order modeling techniques is proposed, which considers the time-varying bias components of the port signals as nonstationary parameters. These components are extracted at runtime by a low-pass filter and used to instantaneously update the matrices of the reduced-order state-space model realized as a SPICE netlist. Our main result is a formal proof of quadratic stability of such linear parameter varying (LPV) models, enabled by imposing a specific model structure and representing the transfer function in a basis of positive functions whose elements constitute a partition of unity. The proposed quadratic stability conditions are easily enforced through a finite set of small-size linear matrix inequalities (LMIs), used as constraints during model construction. Numerical results on various CBs, including voltage regulators, confirm that our approach not only ensures the model stability but also provides speedup in runtime up to two orders of magnitude with respect to full transistor-level circuits.

Robust mixed-sensitivity [formula omitted] control of weight on bit in geological drilling process with parameter uncertainty

In this paper, a robust mixed-sensitivity H∞ controller is designed to control surface weight on bit using surface measurement feedback. First, we construct a drill-string longitudinal model with uncertain penetration resistance coefficient based on finite element method, and a reduced-order model is derived for controller synthesis. In order to solve the fixed point tracking problem of weight on bit control with parameter uncertainty, the mixed-sensitivity H∞ control strategy is developed to design on the model with uncertain parameter. The boundary of the uncertain parameter is obtained using field data. A quadratic stability condition of the closed-loop system is derived by using a single Lyapunov function, based on which a suitable dynamic output feedback H∞ controller is obtained. The uncertain penetration resistance coefficient and system’s high-order mode are considered in modeling and control of weight on bit for the first time. Comparison with field data is provided to validate our modeling method. Numerical results are given on a set of models with different penetration resistance coefficient. The integral of time multiplied by the squared error criterion of the closed-loop system with parameters obtained through equidistant sampling are provided. The simulation results demonstrate that the proposed controller can improve the robustness of the system’s tracking performance to parameter uncertainty and suppress the tracking oscillation results from the high-order mode.

New iterative linear matrix inequality based procedure for H2 and H∞ state feedback control of continuous‐time polytopic systems

SummaryThis article provides new linear matrix inequality (LMI) sufficient conditions for a generalized robust state feedback control synthesis problem for linear continuous‐time polytopic systems. This generalized problem includes the robust stability, H2‐norm, and H∞‐norm problems as special cases. Using a novel general separation result, which separates the state feedback gain from the Lyapunov matrix but with the state feedback gain synthesized from the slack variable, then allows the formulation of LMI sufficient conditions for the generalized problem. Compared to existing parameterized LMI based conditions, where auxiliary scalar parameters are introduced in order to include the quadratic stability conditions (ie, assuming a constant Lyapunov matrix) as a special case, the proposed new conditions are true LMIs and contain as a particular case the optimal quadratic stability solution. Utilizing any initial solution derived by the quadratic or some existing methods as a starting solution, we propose an algorithm based on an iterative procedure, which is recursively feasible in each update, to compute a sequence of nonincreasing upper bounds for the H2‐norm and H∞‐norm. In addition, if no feasible initial solution can be found for some uncertain systems using any existing methods, another algorithm is presented that offers the possibility of obtaining a robust stabilizing gain. Numerical examples from the literature demonstrate that our algorithms can provide less conservative results than existing methods, and they can also find feasible solutions where all other methods fail.

SPR based design conditions for quadratic stability of multi-mode switched linear systems

In this contribution we show that under some circumstances the Kalman–Yakubovich–Popov (KYP) lemma can be applied for testing quadratic stability of multi-mode switched linear systems. Based on this observation, we introduce several constructive algorithms which enable us to associate a transfer function matrix to a switched linear system given in general form. The strict positive realness test of the resulting transfer function matrix can be easily achieved by available known algebraic conditions. By means of some examples we illustrate the step-by-step implementation of these algorithms.

On Gain-Scheduled State-Feedback Controller Synthesis With Quadratic Stability Condition

This letter shows that, as long as continuous-time linear parameter-varying (LPV) systems are concerned, quadratic-stability-based gain-scheduled state-feedback controller synthesis offers no advantage over quadratic-stability-based fixed (parameter-independent) state-feedback controller synthesis in typical control performance specifications. We derive this counterintuitive result by properly extending the previous results on the robust versions of Finsler's lemma and the elimination lemma. We also show that this counterintuitive result is continuous-time LPV system specific, and in the discrete-time LPV system case quadratic-stability-based gain-scheduled state-feedback controller synthesis does bring improvement. These results give a proper warning about the effectiveness of the quadratic-stability-based gain-scheduled state-feedback controller synthesis.

Dual switched positive systems – a less conservative condition for diagonal quadratic stability

ABSTRACTThe key result of the paper exploits the duality of arbitrary switching positive linear systems, in order to derive a sufficient condition for the existence and construction of diagonal quadratic copositive Lyapunov functions. This condition is less conservative than a result also relying on duality, recently reported in the literature; our condition operates with milder inequalities than those required by the existing result. We prove that the algebraic relaxation of these inequalities is related to a relaxation of the primal and dual systems’ properties, referring to the trajectory growth rates, or, equivalently, to the spectra of the associated column representatives. The existing result is incorporated into the new one as a particular case, and the advantages offered by the latter are illustrated by a relevant numerical example. Our developments focus on the continuous-time case, so as to permit direct comparisons with the existing result; some brief discussion suggests how the proposed approach can be applied, mutatis mutandis, to discrete-time dynamics.

Robust model reference control of linear parameter‐varying systems with disturbances

This paper deals with the design and analysis of robust model reference control systems for a class of linear parameter-varying (LPV) systems with disturbances. Two types of robust model reference control systems are proposed to achieve a prescribed tracking performance. The first type is the LPV H ∞ state-feedback model reference control system and the second type is the LPV H ∞ output-feedback model reference control system. Based on the notion of quadratic stability, necessary and sufficient conditions for the existence of LPV reference models are given. Also, a parameterisation of LPV reference models is derived in this study. A practical example of a single-link flexible-joint robot is given to illustrate the design steps and to show the effectiveness of the proposed LPV H ∞ model reference controllers.

Hysteresis Switching Control of the Ćuk Converter Operating in Discontinuous Conduction Modes

In this brief, the behaviors of the Cuk converter operating in discontinuous conduction modes (DCMs) under a hysteresis controlled switching are studied. The hysteresis switching is defined in terms of values of a common quadratic Lyapunov function and Lyapunov stability conditions are established for all possible switchings, which represent a DCM of operation. A set of simulation results are provided to illustrate the converter’s behaviors when the hysteresis switching control results in discontinuous modes of operation.