Showing papers on "Riccati equation published in 2018"

Guaranteed‐cost consensus for multiagent networks with Lipschitz nonlinear dynamics and switching topologies

Abstract: Guaranteed-cost consensus for high-order nonlinear multi-agent networks with switching topologies is investigated. By constructing a time-varying nonsingular matrix with a specific structure, the whole dynamics of multi-agent networks is decomposed into the consensus and disagreement parts with nonlinear terms, which is the key challenge to be dealt with. An explicit expression of the consensus dynamics, which contains the nonlinear term, is given and its initial state is determined. Furthermore, by the structure property of the time-varying nonsingular transformation matrix and the Lipschitz condition, the impacts of the nonlinear term on the disagreement dynamics are linearized and the gain matrix of the consensus protocol is determined on the basis of the Riccati equation. Moreover, an approach to minimize the guaranteed cost is given in terms of linear matrix inequalities. Finally, the numerical simulation is shown to demonstrate the effectiveness of theoretical results.

Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method

Abstract: The generalized projective Riccati equation method is proposed to establish exact solutions for generalized form of the reaction Duffing model in fractional sense namely, Khalil’s derivative. The compatible traveling wave transform converts the governing equation to a non linear ODE. The predicted solution is a series of two new variables that solve a particular ODE system. Coefficients of terms in the series are calculated by solving an algebraic system that comes into existence by substitution of the predicted solution into the ODE which is the result of the wave transformation of the governing equation. Returning original variables give exact solutions to the governing equation in various forms.

Guaranteed-cost consensus for multiagent networks with Lipschitz nonlinear dynamics and switching topologies

Abstract: Guaranteed-cost consensus for high-order nonlinear multi-agent networks with switching topologies is investigated. By constructing a time-varying nonsingular matrix with a specific structure, the whole dynamics of multi-agent networks is decomposed into the consensus and disagreement parts with nonlinear terms, which is the key challenge to be dealt with. An explicit expression of the consensus dynamics, which contains the nonlinear term, is given and its initial state is determined. Furthermore, by the structure property of the time-varying nonsingular transformation matrix and the Lipschitz condition, the impacts of the nonlinear term on the disagreement dynamics are linearized and the gain matrix of the consensus protocol is determined on the basis of the Riccati equation. Moreover, an approach to minimize the guaranteed cost is given in terms of linear matrix inequalities. Finally, the numerical simulation is shown to demonstrate the effectiveness of theoretical results.

Fault Detection for Linear Discrete Time-Varying Systems Subject to Random Sensor Delay: A Riccati Equation Approach

Abstract: This paper mainly studies the fault detection (FD) problem for linear discrete time-varying systems subject to random sensor delay. By assuming that the measurement channel is with Transmission Control Protocol (TCP), an observer-based FD filter (FDF) is provided as a residual generator via embedding the packet indicator into the filter. To construct this FDF, its design issue is formulated into two sub-problems. One is to maximize the $\mathcal {H}_{-}/\mathcal {H}_{\infty }$ or $\mathcal {H}_{\infty }/\mathcal {H}_{\infty }$ FD performance index, which aims to enhance the ratio of fault sensitivity/disturbance attenuation. The other one is to find the filter parameter matrices such that the error between the residual and the fault is minimized in the $\mathcal {H}_{\infty }$ sense. By employing stochastic analysis and introducing some adjoint operator based optimization approaches, analytical solutions to the aforementioned FDF design problem are derived via solving recursive Riccati equations. An illustrative example is given to show the effectiveness of the proposed methodologies.

Rogue waves, bright–dark solitons and traveling wave solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

Abstract: In this paper, a ( 3 + 1 ) -dimensional generalized Kadomtsev–Petviashvili (gKP) equation is investigated, which describes the dynamics of nonlinear waves in plasma physics and fluid dynamics. By employing the extended homoclinic test method, we construct a new family of two wave solutions, rational breather wave and rogue wave solutions of the equation. Moreover, by virtue of some ansatz functions and the Riccati equation method, its analytical bright soliton, dark soliton and traveling wave solutions are derived. Finally, we obtain its exact power series solution with the convergence analysis. In order to further understand the dynamics, we provide some graphical analysis of these solutions.

A leader-follower stochastic linear quadratic differential game with time delay

Abstract: In this paper, we are concerned with the leader-follower stochastic differentialgame of Itotype with time delay appearing in the leaders control.The open-loop solution is explicitly given in the form of the conditionalexpectation with respect to several symmetric Riccati equations.The key technique is to establish the nonhomogeneousrelationship between the forward variables and the backward onesobtained in the optimization problems of both the follower and the leader.

Riccati Observers for the Nonstationary PnP Problem

Abstract: This paper revisits the problem of estimating the pose (position and orientation) of a body in 3-D space with respect to (w.r.t.) an inertial frame by using 1) the knowledge of source points positions in the inertial frame, 2) the measurements of the body angular velocity expressed in the body's frame, 3) the measurements of the body translational velocity, either in the body frame or in the inertial frame, and 4) source points bearing measurements performed in the body frame. An important difference with the much studied static Perspective-n-Point problem addressed with iterative algorithms is that body motion is not only allowed but also used as a source of information that improves the estimation possibilities. With respect to the probabilistic framework commonly used in other studies that develop extended Kalman filter solutions, the deterministic approach here adopted is better suited to point out the observability conditions, that involve the number and disposition of the source points in combination with body motion characteristics, under which the proposed observers ensure robust estimation of the body pose. These observers are here named Riccati observers because of the instrumental role played by the continuous Riccati equation in the design of the observers and in the Lyapunov stability and convergence analysis that we develop independently of the well-known complementary (either deterministic or probabilistic) optimality properties associated with Kalman filtering. The set of these observers also encompasses extended Kalman filter solutions. Another contribution of this study is to show the importance of using body motion to improve the observers performance and, when this is possible, of measuring the body translational velocity in the inertial frame rather than in the body frame to allow for the body pose estimation from a single source point taken as the origin of the inertial frame. This latter possibility finds a natural extension in the Simultaneous Localisation And Mapping (SLAM) problem in Robotics.

Exact lower tail large deviations of the KPZ equation

Abstract: Consider the Hopf--Cole solution $ h(t,x) $ of the KPZ equation with narrow wedge initial condition. Regarding $ t\to\infty $ as a scaling parameter, we provide the first rigorous proof of the Large Deviation Principle (LDP) for the lower tail of $ h(2t,0)+\frac{t}{12} $, with speed $ t^2 $ and an explicit rate function $ \Phi_-(z) $. This result confirms existing physic predictions [Sasorov, Meerson, Prolhac 17], [Corwin, Ghosal, Krajenbrink, Le Doussal, Tsai 18], and [Krajenbrink, Le Doussal, Prolhac 18]. Our analysis utilizes the formula from [Borodin, Gorin 16] to convert LDP of the KPZ equation to calculating an exponential moment of the Airy point process. To estimate this exponential moment, we invoke the stochastic Airy operator, and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.

The Hawking–Penrose Singularity Theorem for C 1,1 -Lorentzian Metrics

Abstract: We show that the Hawking–Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of C 1,1-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for C 1,1-metrics, and of C 0-trapped submanifolds. By regularisation, we show that, under these weak conditions, causal geodesics necessarily become non-maximising. This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest.

Analytic solution for the space-time fractional Klein-Gordon and coupled conformable Boussinesq equations

Abstract: In this paper, we constructed a travelling wave solution for space-time fractional nonlinear partial differential equations by using the modified extended Tanh method with Riccati equation. The method is used to obtain analytic solutions for the space-time fractional Klein-Gordon and coupled conformable space-time fractional Boussinesq equations. The fractional complex transforms and the properties of modified Riemann-Liouville derivative have been used to convert these equations into nonlinear ordinary differential equations.

Adaptive Model-Free Control Based on an Ultra-Local Model With Model-Free Parameter Estimations for a Generic SISO System

Abstract: In this paper, a new structure to design model-free control (MFC) based on the ultra-local model is presented for an unknown nonlinear single-input single-output dynamic system. The proposed structure includes two adaptive laws corresponding to the unknown linear and nonlinear terms. Utilizing the adaptive law for linear term, the controller gain is going to be updated online using a differential Riccati equation. Subsequently, the control policy which includes an optimal term as well as a term for compensating the system unknown dynamics is generated. Here, the two proposed adaptive laws are model-free estimation algorithms, in which the need for any regressor parameter and also the persistent excitation condition is eliminated. Finally, two simulation studies are presented to show that the proposed adaptive MFC (AMFC) policy outperforms the two well-known controllers. Moreover, the AMFC is applied on a Duffing-Holmes chaotic oscillator plant and the convincing performance of the algorithm is observed through the simulation results.

Discrete-Time Robust Hierarchical Linear-Quadratic Dynamic Games

Abstract: In this paper, the theory of robust min–max control is extended to hierarchical and multiplayer dynamic games for linear quadratic discrete time systems. The proposed game model consists of one leader and many followers, while the performance of all players is affected by disturbance. The Stackelberg–Nash-saddle equilibrium point of the game is derived and a necessary and sufficient condition for the existence and uniqueness of such a solution is obtained. In the infinite time horizon, it is shown that the solution of the Riccati equation is upper bounded under a condition that is called individual controllability. By assuming such a condition and using a time-varying Lyapunov function the input-to-state stability of the hierarchical dynamic game is achieved, considering the optimal feedback strategies of the players and an arbitrary disturbance as the input.

Sensor Choice for Minimum Error Variance Estimation

Abstract: A Kalman filter is optimal in that the variance of the error is minimized by the estimator. It is shown here, in an infinite-dimensional context, that the solution to an operator Riccati equation minimizes the steady-state error variance. This extends a result previously known for lumped parameter systems to distributed parameter systems. It is shown then that minimizing the trace of the Riccati operator is a reasonable criterion for choosing sensor locations. It is then shown that multiple inaccurate sensors, that is, those with large noise variance, can provide as good an estimate as a single highly accurate (but probably more expensive) sensor. Optimal sensor location is then combined with estimator design. A framework for calculation of the best sensor locations using approximations is established and sensor location as well as choice is investigated with three examples. Simulations indicate that the sensor locations do affect the quality of the estimation and that multiple low-quality sensors can lead to better estimation than a single high-quality sensor.

Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations

Abstract: Abstract The numerical analysis of linear quadratic regulator design problems for parabolic partial differential equations requires solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. The coefficient matrices of the resulting RDE often have a given structure, e.g., sparse, or low-rank. The associated RDE usually is quite stiff, so that implicit schemes should be used in this situation. In this paper, we derive efficient numerical methods for solving RDEs capable of exploiting this structure, which are based on a matrix-valued implementation of the BDF and Rosenbrock methods. We show that these methods are suitable for large-scale problems by working only on approximate low-rank factors of the solutions. We also incorporate step size and order control in our numerical algorithms for solving RDEs. In addition, we show that within a Galerkin projection framework the solutions of the finite-dimensional RDEs converge in the strong operator topology to the solutions of the infinite-dimensional RDEs. Numerical experiments show the performance of the proposed methods.

Control of active suspension system considering nonlinear actuator dynamics

Abstract: Application of the state-dependent Riccati equation and approximating sequence of Riccati equation techniques for the control of active suspension system considering nonlinear actuator dynamics will be investigated. First, equation of motion of the vehicle model is written in terms of the nonlinear state equations. Then, a performance index is formed for the minimization of the acceleration of the sprung mass, suspension deflection, velocity of the sprung mass, tire deflection, velocity of the unsprung mass, pressure decrease through the piston, and rate of pressure decrease through the piston. A sinusoidal bump and road roughness are considered as the road disturbances. After that, control input is expressed in terms of the Riccati differential equation variables and the state variables. Finally, quarter vehicle suspension system model of a Ford Fiesta Mk2 is taken into consideration in numerical simulations to test the performances of the both techniques. The results are compared to those of equivalent passive suspension system.

Linear Quadratic Stochastic Control Problems with Stochastic Terminal Constraint

Abstract: We study a linear quadratic optimal control problem with stochastic coefficients and a terminal state constraint, which may be in force merely on a set with positive, but not necessarily full, probability. Under such a partial terminal constraint, the usual approach via a coupled system of a backward stochastic Riccati equation and a linear backward equation breaks down. As a remedy, we introduce a family of auxiliary problems parametrized by the supersolutions to this Riccati equation alone. The target functional of these problems dominates the original constrained one and allows for an explicit description of both the optimal control policy and the auxiliary problem's value in terms of a suitably constructed optimal signal process. This suggests that, for the minimal supersolution of the Riccati equation, the minimizers of the auxiliary problem coincide with those of the original problem, a conjecture that we see confirmed in all examples understood so far.

Complexiton and solitary wave solutions of the coupled nonlinear Maccari’s system using two integration schemes

Abstract: In this paper, we consider a coupled nonlinear Maccari’s system (CNMS) which describes the motion of isolated waves localized in a small part of space. There are some integration tools that are adopted to retrieve the solitary wave solutions. They are the modified F-Expansion and the generalized projective Riccati equation methods. Topological, non-topological, complexiton, singular and trigonometric function solutions are derived. A comparison between the results in this paper and the well-known results in the literature is also given. The derived structures of the obtained solutions offer a rich platform to study the nonlinear CNMS. Numerical simulation of the obtained solutions are presented with interesting figures showing the physical meaning of the solutions.

Stability in mean for multi-dimensional uncertain differential equation

Abstract: Liu process is an uncertain process with stationary and independent increments. Multi-dimensional uncertain differential equation is a type of differential equation driven by multi-dimensional Liu process to model a multi-dimensional dynamic system. This paper aims at proposing a definition of stability in mean for multi-dimensional uncertain differential equations. Then a stability theorem for a multi-dimensional uncertain differential equation being stable in mean is proved. Furthermore, some examples are given to show what is stable in mean.

Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

Abstract: The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension However, a reduced rank representation of the estimated covariance leaves a large dimensional complementary subspace unfiltered Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, providing a novel theoretical interpretation for the role of covariance inflation in ensemble-based Kalman filters Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically

Direct Speed Control of a PMSM Drive Using SDRE and Convex Constrained Optimization

Abstract: The challenge for control of permanent magnet synchronous motor (PMSM) drives is to achieve high dynamics, accurate steady-state performance, and to respect all constraints on input voltage and stator currents. Many partial results on each of these aspects are available. Recently, it has been shown that the existing techniques can be combined with ideas from predictive control to achieve satisfaction of state constraints such as maximum current amplitude. In this article, we propose to complement the direct-speed-control-based state-dependent Riccati equation (SDRE) approach by explicit constraints on the current amplitude and the field-weakening curve. Since the cost-to-go function for the SDRE is available, the problem is formulated as quadratic programming with a quadratic constraint. The resulting controller achieves an excellent steady-state solution due to SDRE and satisfies constraints on the maximum current amplitude and field-weakening operation. Experimental tests of the proposed cascade-free speed control are performed on a laboratory prototype of a 10.7-kW PMSM drive. The proposed optimization routine can be used to enforce state constraints in other unconstrained control methods.

Static Output-Feedback Stabilization for MIMO LTI Positive Systems using LMI-based Iterative Algorithms

Abstract: This letter proposes two iterative algorithms for solving static output feedback stabilization problem for LTI multi-input multi-output systems. Contrary to the existing literature, in this letter, no restrictive assumption has been made on the state-space description of the open-loop plant for which a static output feedback controller is to be designed to synthesize a stable internally positive system in closed-loop. Algorithm 1 involves the cone complementarity linearization technique to overcome the underlying difficulties of BMI problem, whereas the success of Algorithm 2 depends on a scalar search associated with a positive definite matrix obtained as the stabilizing solution of a Riccati equation. Numerical examples show that, in some cases where the existing algorithms fail, the proposed algorithms can find a static output feedback stabilizing controller ensuring the positive system properties.

Direct Adaptive Optimal Control for Uncertain Continuous-Time LTI Systems Without Persistence of Excitation

Abstract: This brief presents a novel direct adaptive optimal controller design for uncertain continuous-time linear time-invariant systems. The optimal gain parameter, obtained from the Riccati equation, is continuously estimated without using knowledge of the system dynamics, rather information rich past, and current data along the system trajectory is used for parameter estimation. This approach guarantees parameter convergence to the close neighborhood of the optimal controller by relaxing the restrictive persistence of excitation condition, typically required for achieving parameter convergence in approximate/adaptive optimal control methods. A Lyapunov-based analysis establishes the uniformly ultimately bounded stability of the designed controller. Further a simulation example demonstrates the efficacy of the proposed result.