Quadratic Stability Research Articles

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.

Periodic compensation of continuous‐time linear systems with norm‐bounded parametric uncertainties

AbstractThis article employs a recently proposed high frequency, continuous‐time periodic controller to compensate single‐input‐single‐output linear plants with norm‐bounded time‐varying uncertainties. To this end, the design of periodic controller is converted into designing a linear time‐invariant (LTI) controller for a single‐input‐two‐output equivalent uncertain system, following averaging technique. To represent this equivalent uncertain system into a standard form of linear uncertain systems, where uncertainties appear in system matrix A and input matrix B, the observer canonical form description of the original uncertain plant is exploited. Using the relationship between quadratic stability and ‐norm minimization result available in literature, it is proved that the presence of virtual output with the periodic feedback introduces a feedforward action on the disturbance input of the generalized plant, unlike an LTI one. By virtue of this, the proposed periodic controller can tolerate more uncertainty bound in the input matrix B than its LTI counterpart, while keeping the bound on system matrix A the same. Finally, a linear matrix inequality‐based method is used to design the periodic controller to achieve quadratic stability along with robust linear quadratic performance. Suitable examples are considered to show the superiority of the periodic control as compared to LTI ones.

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.

Advanced controller synthesis for fuzzy parameter varying systems

A novel nonlinear time-varying model termed as the fuzzy parameter varying (FPV) system is proposed in this research, which inherits both advantages of the conventional T-S fuzzy system in dealing with nonlinear plants and strengths of the linear parameter varying (LPV) system in handling time-varying features. It is, therefore, an attractive mathematical model to efficiently approximate a nonlinear time-varying plant or to serve as a type of time-varying controller. Using the full block S-procedure, sufficient stability conditions have been derived in the form of linear matrix inequalities (LMIs) to test quadratic stability of the open-loop FPV system. Moreover, sufficient conditions have been derived on synthesizing both state feedback and dynamical output feedback fuzzy gain-scheduling controllers that can stabilize the FPV system. An inverted pendulum with a variable length pole is utilized to demonstrate advantages of the FPV system compared to the conventional T-S fuzzy system in representing a practical time-varying nonlinear plant and to validate the controller synthesis conditions.

Quantized Feedback Control Based on Spherical Polar Coordinate Quantizer

Different from uniform quantizer and logarithmic quantizer, this article proposes a novel quantizer for the design problems of linear systems with quantized feedback. This quantizer is based on spherical polar coordinates and has a desired relation between the quantized vector and the corresponding quantization error. Utilizing the proposed quantizer with infinite data rate, the quadratic stabilization problems of linear systems via state feedback under quantization can be converted, respectively, to the same problems of the corresponding robust control systems with sector bound uncertainties. From a practical point of view, we are more concerned with quantized feedback control with finite data rate, so a quantizer with finite data rate is proposed for the abovementioned problems. Different from the quantizer with infinite data rate, a time-varying bounded quantization region needs to be determined to contain the quantized vector as the system evolves. Under this circumstance, the vector needed to be quantized cannot be quantized directly, so we give a quantization method to solve this problem.

Robust multi-agent differential games with application to cooperative guidance

Robust multi-agent differential games and its application to cooperative guidance are investigated in this paper. The multi-agent system is modeled by general linear dynamics with norm-bound parameter uncertainties. The players in the game are divided into two teams. One team consists of a leader, and the other team consists of a fixed number of followers agents that aim to achieve leader-following consensus. The two teams constitute the adversaries of a two-player zero-sum differential game. Based on quadratic stabilization techniques, a set of saddle point strategies of the game are designed to stabilize the closed-loop multi-agent system. By modifying the optimal solution of the multi-agent differential game of the nominal model, the sufficient conditions are presented such that the stabilization of the uncertain system is guaranteed. It is proved that the modified solution achieves quasi-optimality. It is also shown that L2-bounded consensus error can be achieved, if the control of the leader deviates from its quasi-optimal strategy. The model of cooperative guidance problem is established, based on which the robust multi-agent differential games approach is applied to design the cooperative guidance laws. Numerical examples are given to verify the effectiveness of the theoretical results and the performance of the proposed cooperative guidance law.

Distributed Stochastic MPC for Networked Linear Systems With a Multirate Sampling Mechanism

In this article, a distributed stochastic model predictive control (MPC) algorithm with a multirate sampling mechanism is proposed for a networked linear system with multiple dynamic subsystems. A delta operator approach is used for the multiple dynamic subsystems with different sampling periods in the multirate sampling mechanism. Both stochastic disturbances and probabilistic constraints of the multiple dynamic subsystems are satisfied by the distributed stochastic MPC algorithm. Packets dropout are considered by the stochastic MPC algorithm and state predicted errors are compensated. Recursive feasibility and quadratic stability are obtained for the networked linear system under the distributed stochastic MPC algorithm. A numerical example is given to illustrate effectiveness of the proposed algorithm.

Maximum time step for the BDF3 scheme applied to gradient flows

For backward differentiation formulae (BDF) applied to gradient flows of semiconvex functions, quadratic stability implies the existence of a Lyapunov functional. We compute the maximum time step which can be derived from quadratic stability for the 3-step BDF method (BDF3). Applications to the asymptotic behaviour of sequences generated by the BDF3 scheme are given.

Observer Design for Positive Uncertain Discrete-time Lipschitz Systems

Abstract For positive uncertain discrete-time Lipschitz systems this paper proposes a way to reflect matched uncertainties, structural system parameter constraints, positiveness and Lipschitz continuity in solving the problem of the state observer quadratic stability. The design conditions are proposed in the set of linear matrix inequalities to guarantee the observer strict positiveness, system parameter constraint representation and estimation error bounding in terms of achieved quadratic stability and nonnegative feedback gain matrix. It follows from the results obtained that the impact of nonnegative system matrix structures can be reflected in uncertainty matching problems. A numerical example is included to assess the feasibility of the technique and its applicability.

Control Design for Linear Uncertain Positive Discrete-time Systems

Abstract For linear uncertain positive discrete-time systems this paper proposes an extended way to reflect structural system parameter constraints and positiveness in solving the problem of the system quadratic stability. The new design conditions are proposed, exploiting a set of system parameter constraint representation in the form of linear matrix inequalities to guaranty the closed-loop strictly positiveness, while a Lyapunov principle is focused to guaranty the system quadratic stability. Moreover, only time-invariant strictly positive control law gain is used in the feedback loop. Closely connected to the obtained results is the fact that the impact of nonnegative system matrix structures can be successfully implemented. A numerical example is included to assess the feasibility of the technique and its applicability.

Robust Distributed Filters Design for Discrete-Time Spatially Interconnected Systems with LFT Uncertainties

In this paper, we investigate discrete-time uncertain spatially interconnected systems (USISs), where uncertainties are modeled by linear fractional transformation (LFT). First, the well-posedness, quadratic stability and contractiveness of discrete-time USISs are introduced. Second, a sufficient condition is proposed to guarantee that discrete-time USISs are well-posed, quadratically stable and contractive. Then, a more tractable condition is derived to check the well-posedness, quadratic stability and contractiveness of discrete-time USISs via a modified bilinear transformation. Besides, the robust distributed filters which inherit the structure of the plants are designed. A sufficient and necessary condition is presented to guarantee the existence of the robust distributed filters. Finally, a vehicle platoon model demonstrates the effectiveness of the proposed scheme.

Vehicle-trailer lateral velocity estimation using constrained unscented transformation

Using a 3-degree-of-freedom lateral dynamic vehicle-trailer system model with the LuGre tyre model, this paper proposes a constrained unscented transformation based scheme for estimating the lateral velocity of a vehicle trailer. It is shown that the proposed lateral velocity estimation scheme is robust to the road condition. An affine quadratic stability approach is used to analyze the stability of the proposed estimation scheme. Simulations studies in CarSim/Simulink along with experimental tests are used to validate the proposed scheme. The test results confirm the accuracy of the developed estimation scheme and convergence of the vehicle-trailer lateral velocity estimate to the actual value.

An input‐to‐state stable model predictive control framework for Lipschitz nonlinear parameter varying systems

SummaryThis article develops a state‐feedback model predictive control (MPC) framework for nonlinear parameter varying (NLPV) systems with explicit Lipschitz nonlinearities along the input trajectory. Considering a guess for the evolution of the scheduling parameters along the prediction horizon, the proposed optimization procedure for the MPC design includes a terminal stage cost, a contracting terminal region constraint and nominal predictions scheduled with respect to this guess. The terminal region is taken so that it is monotonically decreasing to a pre‐determined set, which guarantees recursive feasibility of the algorithm with respect to a bound on the admissible uncertainties (introduced from the scheduling parameter guess). The terminal set is a scheduled robust control invariant set for Lipschitz nonlinear systems, computed through some proposed LMIs. The inclusion of the terminal ingredient also serves to demonstrate input‐to‐state quadratic stability. This article ends with a successful numerical simulation example of the technique applied to the control of a semiactive automotive suspension system equipped with electro‐rheological dampers. Comparisons are given with respect to open‐loop behavior and to a robust LQR controller.

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.

Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions

In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system.

From Noisy Data to Feedback Controllers: Nonconservative Design via a Matrix S-Lemma

In this article, we propose a new method to obtain feedback controllers of an unknown dynamical system directly from noisy input/state data. The key ingredient of our design is a new matrix S-lemma that will be proven in this article. We provide both strict and nonstrict versions of this S-lemma, which are of interest in their own right. Thereafter, we will apply these results to data-driven control. In particular, we will derive nonconservative design methods for quadratic stabilization, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> control, all in terms of data-based linear matrix inequalities. In contrast to previous work, the dimensions of our decision variables are independent of the time horizon of the experiment. Our approach, thus, enables control design from large datasets.

A robust least squares based approach to min‐max model predictive control

SummaryThis article deals with the model predictive control (MPC) of linear, time‐invariant discrete‐time polytopic (LTIDP) systems. The 2‐fold aim is to simplify the treatment of complex issues like stability and feasibility analysis of MPC in the presence of parametric uncertainty as well as to reduce the complexity of the relative optimization procedure. The new approach is based on a two degrees of freedom (2DOF) control scheme, where the output r(k) of the feedforward input estimator (IE) is used as input forcing the closed‐loop system ∑f. ∑f is the feedback connection of an LTIDP plant ∑p with an LTI feedback controller ∑g. Both cases of plants with measurable and unmeasurable state are considered. The task of ∑g is to guarantee the quadratic stability of ∑f, as well as the fulfillment of hard constraints on some physical variables for any input r(k) satisfying an “a priori” determined admissibility condition. The input r(k) is computed by the feedforward IE through the on‐line minimization of a worst‐case finite‐horizon quadratic cost functional and is applied to ∑f according to the usual receding horizon strategy. The on‐line constrained optimization problem is here simplified, reducing the number of the involved constraints and decision variables. This is obtained modeling r(k) as a B‐spline function, which is known to admit a parsimonious parametric representation. This allows us to reformulate the minimization of the worst‐case cost functional as a box‐constrained robust least squares estimation problem, which can be efficiently solved using second‐order cone programming.

Systematic development and application of a fuzzy logic equipped generic energy storage system for dynamic stability reinforcement

Even though fuzzy logic has been tremendously utilized in power systems, it has often been termed as far from complete due to the in-existence of a systematic procedure. Here, we report a systematic development and application of a fuzzy logic equipped generic energy storage system (GESS) for dynamic stability reinforcement in a conglomerate power system. While ensuring a stabilizing performance by minimizing a quadratic stability index, the reactive and real-power trajectories of GESS are determined. The proposed control strategy is optimized offline using a genetic algorithm (GA) optimization and cast with mixed integer programming mathematics to dispatch an optimal template of series weights, grades of membership and decisive fuzzy rules. The dynamic behavior of the dc link is also assimilated in the detailed GESS modeling using cascaded proportional-integral controllers. A perspective of a combination of flywheel energy storage system and superconducting magnetic energy storage is traversed by approximating the inherent loop dynamics of the storages by a GA-tuned first-order system. The role of the said storage devices in manifestation of exogenous wind power intermittency as well as concentrated short circuits is investigated. A wind farm integrated, reoriented form of an exemplar multibus network is adopted as the model system to authenticate the potency of the presented control strategy.

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.