Riccati Equation Research Articles

A nonlinear optimal control approach for 3-DOF four-cable driven parallel robots

In this article, a nonlinear optimal control approach is proposed for the dynamic model of 3-DOF four-cable driven parallel robots (CDPR). To solve the associated nonlinear optimal control problem, the dynamic model of the 3-DOF cable-driven parallel robot undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the 3-DOF cable-driven parallel robot a stabilizing optimal (H-infinity) feedback controller is designed. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs and a minimum dispersion of energy.

Inverse design of ultrawideband one-dimensional metamaterial photonic filters using Riccati equation constrained gradient descent

One-dimensional photonic wave devices exhibit a pivotal role in wave engineering. Despite their relative simplicity, designing 1D wave devices that implement complex functionalities over a broad frequency range is challenging and requires careful sculpting and multiple optimizations. This paper theoretically and experimentally demonstrates a new inverse design paradigm to achieve a desired broadband frequency response efficiently. Specifically, we calculate the required dielectric profile along the device using constrained gradient descent optimization to minimize the L2 norm between the desired and actual responses. In each optimization step, we avoid the need to solve the complete set of Maxwell equations by using Riccati’s equation or its discrete ancestor as the optimization constraint for calculating the local reflection coefficient. Using this approach, we design several unorthodox filters, such as dual-band narrowband bandpass filters located within a wideband bandstop and ultrawideband first and second-order differentiators. The technique produces excellent results for ultrawideband frequency ranges, with very low computational complexity and, remarkably, with a single trivial guess for the optimization starting point. We experimentally implemented the two differentiator designs in radio frequencies using electronic circuit elements that comprise a metamaterial transmission line structure.

Robust distributed Nash equilibrium solution for multi‐agent differential graphical games

AbstractThis paper studies the differential graphical games for linear multi‐agent systems with modelling uncertainties. A robust optimal control policy that seeks the distributed Nash equilibrium solution and guarantees the leader‐following consensus is designed. The weighting matrices rely on modelling uncertainties, leading to the Nash equilibrium solution, and the solution can be obtained by solving a decoupled algebraic Riccati equation. Simulation studies are finally reported to illustrate the effectiveness of proposed policy.

Wavelet LQR based gain scheduling for power regulation and vibration control of floating horizontal axis wind turbine with double-pitched rotor

This paper introduces a wavelet-based linear quadratic regulator (LQR) control strategy for double-pitched rotor systems of an offshore horizontal axis wind turbines. A comprehensive aero-servo-elastic model incorporating bending-torsion coupling is developed to assess the performance of the proposed controller under combined turbulent wind and wave loading. The weight matrices of the classical LQR algorithm are optimized using a nonlinear optimization scheme. Additionally, frequency-dependent gain scheduling is proposed using wavelet transform where the analytic Morse wavelet serves as the basis function. The resulting LQR algorithm operates in the time-frequency domain and solves scale-dependent Algebraic Riccati equations to determine optimal gains. Numerical simulations demonstrate the superior performance of the proposed gain scheduling compared to the classical LQR approach. The effectiveness of the algorithm is evaluated across various flow conditions and wind-wave misalignment, with benchmark results used for performance comparison.

An operator Riccati equation and reflectionless Schrodinger operators

In this paper, we study a connection between the operator Riccati equation $\displaystyle S'(x)=KS(x)+S(x)K-2S(x)KS(x), \quad x\in\mathbb{R},$ and the set of reflectionless Schr\"odinger operators with operator-valued potentials.Here $K\in \mathcal{B}(H)$, $K>0$ and $S:\mathbb{R}\to \mathcal{B}(H)$, where $\mathcal{B}(H)$ is the Banach algebra of all linear continuous operators acting in a separable Hilbert space $H$. Let $\mathscr{S}^+(K)$ be the set of all solutions $S$ of the Riccati equation satisfying the conditions $0< S(0)< I $ and $S'(0)\ge 0$, with $I$ being the identity operator in $H$. We show that every solution $S\in \mathscr{S}^+(K)$ generates a reflectionless Schr\"odinger operator with some potential $q$ that is an analytic function in the strip $\displaystyle \Pi_K:=\left\{z=x+iy \mid x,y\in\mathbb{R}, \,\, |y|<\tfrac{\pi}{2\|K\|} \right\};$ moreover, $\displaystyle \|q(x+iy)\|\le2\|K\|^2\cos^{-2}(y\|K\|), \quad (x+iy)\in\Pi_K .$

Galerkin approximation for [formula omitted]-control of the stable parabolic system under Dirichlet boundary control

In this paper, we explore state feedback control for the H∞ disturbance-attenuation problem in stable parabolic systems with in-domain distributed disturbances under Dirichlet boundary control. Calculating the state feedback control involves solving an operator algebraic Riccati equation, which poses challenges in finding an analytic solution. A practical approach is to seek an approximate solution via finite-dimensional approximation. Specifically, we employ the Galerkin approximation, which generates a sequence of finite-dimensional systems that approximate the original infinite-dimensional system. All corresponding finite-dimensional disturbance-attenuation problems are solvable, and it is demonstrated that the sequence of solutions to the associated finite-dimensional algebraic Riccati equations converges in norm to the solution of the infinite-dimensional operator algebraic Riccati equation. Furthermore, the state feedback controls derived from the finite-dimensional algebraic Riccati equations are proven to be γ-admissible controls for the original system.

A class of ADI method for the nonsymmetric coupled algebraic Riccati equations

A class of ADI method for the nonsymmetric coupled algebraic Riccati equations

[formula omitted] tracking control for de-oiling hydrocyclone systems via event-triggered reinforcement learning

De-oiling hydrocyclone system is a crucial device in offshore oil and gas production. To reduce action frequency of actuators as well as energy costs, it is desired to apply event-triggered control (ETC) technique for coordination control of de-oiling hydrocyclone systems. However, the triggering errors and slugging disturbances will greatly affect oil-water separation efficiency and need to be well treated. Towards this end, this work studies the coordination control problem of de-oiling hydrocyclone systems subject to slugging disturbances and unknown system dynamics. Robust H∞ control approach is applied to reduce disturbances sensitivity and guaranteeing satisfactory oil-water separation performance. The coordination control problem is transformed into a 2-player zero-sum game problem. A model-based solution by solving the game algebraic Riccati equation (GARE) is obtained and an event-triggered (ET) off-policy reinforcement learning (RL) algorithm is developed to implement the model-based solution. The salient feature of the proposed algorithm is that the proposed model-free H∞ control method alleviates the affects of both triggering errors and slugging disturbances, which are handled together as a generalized disturbance in the algorithm. Stability and Zeno behavior avoidance of the proposed algorithm are analyzed. Simulation studies demonstrate the effectiveness of the proposed algorithm.

Dynamical behavior of the fractional nonlinear Kadoma equation in plasma physics and optics

The nonlinear Kadoma equation with M-truncated derivatives (NLKE-MTD) is taken into consideration here. By using generalized Riccati equation method (GRE method) and Jacobi elliptic function method, new hyperbolic, rational, trigonometric and elliptic solutions are discovered. Because the NLKE is widely employed in optics, fluid dynamics and plasma physics, the resulting solutions may be used to analyze a wide variety of important physical phenomena. The dynamic behaviors of the different derived solutions are interpreted using 3D and 2D graphs to explain the effects of M-truncated derivatives. We may conclude that the surface moves to the right as the order of M-truncated derivatives increases.

Tracking optimal feedback control under uncertain parameters

Optimal control problems of tracking type for a class of linear systems with uncertain parameters in the dynamics are investigated. An affine tracking feedback control input is obtained by considering the minimization of an energy-like functional depending on a finite ensemble of training/sample parameters. It is computed from the solution of an associated differential Riccati equation. Simulations are presented showing the tracking performance of the computed input for trained as well as untrained parameters.

Continuous Time Algebraic Riccati Equation Solution for Second Order Systems

Abstract The continuous time algebraic Riccati equation (ARE) is often utilized in control, estimation, and optimization. For a linear system with a second order structure, the ARE required to be solved to get the control values in standard control problems results in complex sub-equations in terms of the second order system matrices. The computational costs of solving the algebraic Riccati equation through standard methods such as the Hamiltonian matrix pencil approach increase substantially as matrix sizes increase for a second order system, due to the eigendecomposition of the 2n x 2n system matrices involved. This work introduces a new solution that does not require the eigendecomposition of the 2n x 2n system matrices, while satisfying all of the requirements of the solution to the Riccati equation (e.g., detectability, stabilizability, positive semi-definite solution matrix).

Quiescent optical solitons for Fokas–Lenells equation with nonlinear chromatic dispersion and a couple of self-phase modulation structures

The focus of the current paper is on the retrieval of quiescent optical solitons from Fokas–Lenells equation with nonlinear chromatic dispersion and having quadratic–cubic as well as quadratic–cubic–quartic forms of self-phase modulation structures. Two integration algorithms are implemented to carry out to seek such soliton solutions. They are the enhanced Kudryashov’s approach and the projective Riccati equation approach. In this context, both linear temporal evolution and generalized temporal evolution effects are addressed. A full spectrum of quiescent optical solitons is thus recovered.

Observer-Based Suboptimal Controller Design for Permanent Magnet Synchronous Motors: State-Dependent Riccati Equation Controller and Impulsive Observer Approaches

Permanent Magnet Synchronous Motors (PMSMs) with high energy efficiency, reliable performance, and a relatively simple structure are widely utilised in various applications. In this paper, a suboptimal controller is proposed for PMSMs without sensors based on the state-dependent Riccati equation (SDRE) technique combined with customised impulsive observers (IOs). Here, the SDRE technique facilitates a pseudo-linearised display of the motor with state-dependent coefficients (SDCs) while preserving all its nonlinear features. Considering the risk of non-available/non-measurable states in the motor due to sensor and instrumentation costs, the SDRE is combined with IOs to estimate the PMSM speed and position states. Customised IOs are proven to be capable of obtaining quality, continuous estimates of the motor states despite the discrete format of the output signals. The simulation results in this work illustrate an accurate state estimation and control mechanism for the speed of the PMSM in the presence of load torque disturbances and reference speed changes. It is clearly shown that the SDRE-IO design is superior compared to the most popular existing regulators in the literature for sensorless speed control.

Zero order meromorphic solutions of q-difference equations of Malmquist type

We consider the first order q-difference equation(†)f(qz)n=R(z,f), where q≠0,1 is a constant and R(z,f) is rational in both arguments. When |q|≠1, we show that, if (†) has a zero order transcendental meromorphic solution, then (†) reduces to a q-difference linear or Riccati equation, or to an equation that can be transformed to a q-difference Riccati equation. In the autonomous case, explicit meromorphic solutions of (†) are presented. Given that (†) can be transformed into a difference equation, we proceed to discuss the growth of the composite function f(ω(z)), where ω(z) is an entire function satisfying ω(z+1)=qω(z), and demonstrate how the proposed difference Painlevé property, as discussed in the literature, applies for q-difference equations.

Distributed Minimum-Energy Containment Control of Continuous-Time Multi-Agent Systems by Inverse Optimal Control

Dear Editor, This letter focuses on the distributed optimal containment control of continuous-time multi-agent systems (CTMASs) with respect to the minimum-energy performance index over fixed topology. To achieve this, we firstly investigate the optimal containment control problem using the inverse optimal control method, where all states of followers asymptotically converge to the convex hull spanned by the leaders while some quadratic performance indexes get minimized. A sufficient condition for existence of the distributed optimal containment control protocol is derived. By introducing the parametric algebraic Riccati equation (PARE), it is strictly proved that the global performance index can be used to approximate the standard minimum-energy performance index as the parameters tends to infinity. In consequence, the standard minimum-energy cooperative containment control can be solved by local steady state feedback protocols. Finally, numerical examples are given to demonstrate the effectiveness of theoretical results.

Dynamics analysis and chatter control of a polishing and milling nonideal flexible manipulator

Some robots are designed to be lightweight and flexible, enabling them to access small and challenging paths in various applications. These features enable robots to collaborate with humans in performing specific production tasks. However,the movement of the flexible manipulator can become self-excited when handling a cutting tool on a workpiece, which can lead to a control problem. This article presents a control solution for lightweight robotic manipulators with rotating tools, such as polishing and milling, using smart actuators. The control discretization method is also introduced to facilitate integration into digital controllers. The paper starts by describing the governing equations of the non-ideal flexible manipulator for polishing and milling and analysing its dynamic behavior. Subsequently, models for the controllers of the DC motor-only actuators and the hybrid (shape memory alloy and DC motors) actuators were formulated using shape memory alloy and a suboptimal control scheme known as the discrete state-dependent Riccati equation. The coupled system was analysed dynamically, and it was observed that it exhibits chaotic behavior. The results suggest that incorporating smart actuators into the motor group of the system could decrease the positioning error of the manipulator and significantly reduce the oscillation of the robot’s end-effector.

ℒ2 gain tracking control of linear completely unknown discrete‐time networked control systems with dropout

AbstractWe introduce an online, model‐free algorithm to address the gain optimal tracking problems in discrete linear networked control systems. The algorithm is specifically proposed to handle stochastic information dropout in the feedback loop. Our goal is to design a control law that achieves the system output reference tracking while attenuating the effect of the disturbance input. We first construct an augmented system consisting of the original system and the command generator system. Then, the performance index is expressed in a quadratic form taking into account packet loss. Next, we obtain the optimal solution by solving the dropout generalized algebraic riccati equation (GARE). Finally, a ‐learning algorithm is utilized to estimate the control and disturbance feedback gains of the system, using only measurement data with unknown system dynamics in the presence of dropout. Two algorithms are tested on a numerical example to demonstrate the validity and effectiveness of the proposed methodology.

Dynamics of switching optical soliton in fiber with sixth order dispersion and inter modal dispersion

This work pertains to the study of switching soliton in fiber for nonlinear Schrödinger (NLS) equation with the presence of higher order dispersion and inter modal dispersion (IMD). The nonlinear wave in the optical fiber are described by the NLS equation which having the second order, fourth order, fifth order, sixth order dispersion, higher order nonlinearity and IMD. The main goal of the paper is to examine the sixth order dispersion on nonlinear wave in the fiber with the conditions of IMD. Hence, we employ the Bäcklund transformation of the Riccati equation (BTRE) approach to NLS equation and obtain the soliton solution. By the use of soliton solution along with graphical snapshots, we provide the conditions for forming switching solition in optical fiber and also analyze the effect of sixth order dispersion in fiber. The stability of the solution of NLS equation is also addressed.

A note on a q-difference equation related to Riccati equations

In this paper, we consider the existence of analytic solutions of associated with q-Airy equation y ( q 2 t ) + qty ( qt ) − y ( t ) = 0. By the position of q, we study the existence of analytic solutions of the form of power functions, the convergence radius of the formal solutions for some cases is also given.

A Lie symmetry approach to traveling wave solutions, bifurcation, chaos and sensitivity analysis of the geophysical Korteweg–de Vries equation

Strong waves known as tsunamis are caused by earthquakes, landslides, or volcanic eruptions that traverse oceans. This article examines the geophysical Korteweg–de Vries (GPKdV) equation, which controls the propagation of tsunami waves in seas. The study involves exploring symmetry diminution exerted Lie group analysis, examining the properties of the dynamical structure with the help of bifurcation phase pictures, and researching the dynamic demeanor of the perturbed dynamical system utilizing chaos theory. Techniques such as 3D and 2D phase portraits, time series analysis, poincaré maps, looking into the existence of multistability in the autonomous structure under diverse beginning circumstances, lyapunov exponent, and bifurcation diagram are applied to identify chaotic demeanor. Furthermore, the study establishes general forms of solitary wave results, containing periodic, trigonometric, and singular soliton results, by using the unified Riccati equation expansion approach to address the examined problem analytically. These results are visually represented as 2D and 3D graphs with cautiously chosen parameters, along with their accompanying constraint conditions. Moreover, the sensitivity evaluation of the investigated equation is discussed and demonstrated pictorially. The discoveries revealed are intriguing, novel, and potentially helpful in understanding a wide range of physical events in engineering and science.