Polynomial Control Research Articles

Polynomial Controller Synthesis for Uncertain Large-Scale Polynomial T-S Fuzzy Systems.

This paper proposes a novel method to synthesize a controller for stabilizing the nonlinear large-scale system which is represented by a large-scale polynomial Takagi-Sugeno (T-S) fuzzy system. The large-scale system consists of a set of the uncertain polynomial T-S fuzzy system with interconnection terms. Modeling the large-scale nonlinear system under the framework of the polynomial form will decrease both the modeling errors and the number of fuzzy rules with respect to the conventional large-scale T-S fuzzy system. In addition, because of the existence of uncertainties, the synthesizing controller for the large-scale polynomial fuzzy system becomes much more challenging and has not been investigated in the previous studies. In this paper, a controller is synthesized to simultaneously eliminate the impact of the uncertainties and stabilize the system. With the aid of Lyapunov theory, sum-of-square technique, and S-procedure, the conditions for controller synthesis are derived in the main theorems. Finally, two examples are illustrated to show the effectiveness and merit of the proposed method.

Robot Manipulator Control System with Dynamic Moment Compensation

The paper offers the variants of manipulator control system design with PID and polynomial controllers. A manipulator dynamics model is used for compensating the moments affecting its sections during the motion.

Feedback polynomial filtering and control of non-Gaussian linear time-varying systems

This paper deals with the optimal filtering and optimal output-feedback control of discrete-time, linear time-varying non-Gaussian systems. In the hypothesis that the time-varying and non-Gaussian distributions of the state and measurement noises have bounded and known moments up to a given order, this work extends previous results about polynomial filtering and optimal control to the time-varying case. The properties of the resulting filtering and control algorithms are discussed in the light of a stable recursive representation of the Kronecker powers of the system obtained through a suitable rewriting of the system with an output injection term. The resulting sub-optimal algorithm inherits the structure and the properties of the classical LQG approach but with enhanced performance.

Optimal control of linear PDEs using occupation measures and SDP relaxations

This paper addresses the problem of solving a class of optimal control problems (OCPs) with infinite-dimensional linear state constraints involving Riesz-spectral operators. Each instance within this class has time/control-dependent polynomial Lagrangian cost and control constraints described by polynomials. We first perform a state-mode discretization of the Riesz-spectral operator. Then we approximate the resulting finite-dimensional OCPs by using a previously known hierarchy of semidefinite relaxations. Under certain compactness assumptions, we provide a converging hierarchy of semidefinite programming relaxations whose optimal values yield lower bounds for the initial OCP. We illustrate our method by two numerical examples, involving a diffusion partial differential equation and a wave equation. We also report on the related experiments.

Control strategies for Hopf bifurcation in a chaotic associative memory

Abstract This paper presents a set of control approaches that can deal with unwanted dynamics such as the Hopf Bifurcation in a Chaotic Associative Memory (CAM) for two distinct goals: the bifurcation critical point shift to anticipate or to postpone the start of a Hopf bifurcation; or to extinguish it. A Washout Filter Control, a Polynomial Control, and an adaptive control for data-driven called the Model-Free Adaptive Control were used in this study. We present the results of numerical simulations in a CAM with three neurons (a system with six dimensions), which were trained classically to retrieve a set of memories. Our analysis includes the dynamics of the bifurcated system, the ability of the network to recover the memories learned and how it is affected by the control strategies. The generality of the proposed control approaches was tested to control a different system and for a different behavior (chaotic dynamics).

Polynomial control on stability, inversion and powers of matrices on simple graphs

Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B(ℓp),1≤p≤∞, the space of all bounded operators on the space ℓp of all p-summable sequences. The ℓp-stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓp-stabilities of matrices in Beurling algebras for different exponents 1≤p≤∞, with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B(ℓ2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network.

Polynomial control design for polynomial systems: A non-iterative sum of squares approach

This paper proposes a non-iterative state feedback design approach for polynomial systems using polynomial Lyapunov function based on the sum of squares (SOS) decomposition. The polynomial Lyapunov matrix consists of states of the system leading to the non-convex problem. A lower bound on the time derivative of the Lyapunov matrix is considered to turn the non-convex problem into a convex one; and hence, the solutions are computed through semi-definite programming methods in a non-iterative fashion. Furthermore, we show that the proposed approach can be applied to a wide range of practical and industrial systems that their controller design is challenging, such as different chaotic systems, chemical continuous stirred tank reactor, and power permanent magnet synchronous machine. Finally, software-in-the-loop (SiL) real-time simulations are presented to prove the practical application of the proposed approach.

Development of a decentralized nonlinear controller for a class of uncertain polynomial interconnected systems: Application for a large scale power system

This paper presents a new robust decentralized control of nonlinear interconnected systems, which is applied and validated on a large scale power system. Our work is performed in three steps. Firstly, we have developed the polynomial description of the nonlinear uncertain and interconnected system using odd Kronecker power of state vectors, which is an easy-manipulation model for such complex systems. Then we applied Lyapunov’s direct method of stability analysis, associated with a quadratic function, in order to determine a sufficient condition for global asymptotic stability by applying a nonlinear, decentralized and optimal polynomial control. Finally, we carried out a simulation study on a nonlinear uncertain power system with three interconnected machines. We considered different cases of perturbations on its state variables as well as different cases of fault locations. We prove via advanced simulations, the effectiveness of the proposed control technique which is able to mitigate the successive amplitudes of the oscillations, to limit the control actions and to enhance the power system transient stability.

Learning control lyapunov functions from counterexamples and demonstrations

We present a technique for learning control Lyapunov-like functions, which are used in turn to synthesize controllers for nonlinear dynamical systems that can stabilize the system, or satisfy specifications such as remaining inside a safe set, or eventually reaching a target set while remaining inside a safe set. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function, and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. We show how the verifier can be constructed efficiently using convex relaxations of the verification problem for polynomial systems to semi-definite programming (SDP) problem instances. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.

Stability and stabilization of periodic piecewise linear systems: A matrix polynomial approach

In this paper, new conditions of stability and stabilization are proposed for periodic piecewise linear systems. A continuous Lyapunov function is constructed with a time-dependent homogeneous Lyapunov matrix polynomial. The exponential stability problem is studied first using square matricial representation and sum of squares form of homogeneous matrix polynomial. Constraints on the exponential order of each subsystem used in previous work are relaxed. State-feedback controllers with time-varying polynomial controller gain are designed to stabilize an unstable periodic piecewise system. The proposed stabilizing controller can be solved directly and effectively, which is applicable to more general situations than those previously covered. Numerical examples are given to illustrate the effectiveness of the proposed method.

Polynomial Chaos-Based Controller Design for Uncertain Linear Systems With State and Control Constraints

For linear dynamic systems with uncertain parameters, design of controllers which drive a system from an initial condition to a desired final state, limited by state constraints during the transition is a nontrivial problem. This paper presents a methodology to design a state constrained controller, which is robust to time invariant uncertain variables. Polynomial chaos (PC) expansion, a spectral expansion, is used to parameterize the uncertain variables permitting the evolution of the uncertain states to be written as a polynomial function of the uncertain variables. The coefficients of the truncated PC expansion are determined using the Galerkin projection resulting in a set of deterministic equations. A transformation of PC polynomial space to the Bernstein polynomial space permits determination of bounds on the evolving states of interest. Linear programming (LP) is then used on the deterministic set of equations with constraints on the bounds of the states to determine the controller. Numerical examples are used to illustrate the benefit of the proposed technique for the design of a rest-to-rest controller subject to deformation constraints and which are robust to uncertainties in the stiffness coefficient for the benchmark spring-mass-damper system.

Algorithm for Bernstein Polynomial Control Design

This paper considers control synthesis for polynomial control systems. The developed method leans upon Lyapunov stability and Bernstein certificates of positivity. We strive to develop an algorithm that computes a polynomial control and a polynomial Lyaponov function in the simplicial Bernstein form. Subsequently, we reduce the control synthesis problem to a finite number of evaluations of a polynomial within Bernstein coefficient bounds representing controls and Lyapunov functions. As a consequence, the equilibrium is asymptotically stable with this control.

The Polynomial Approach to the LQ non-Gaussian Regulator Problem through Output Injection

In this paper, an improved approach for the solution of the regulator problem for linear discrete-time dynamical systems with non-Gaussian disturbances and quadratic cost functional is proposed. It is known that a suboptimal recursive control can be derived from the classical linear quadratic Gaussian (LQG) solution by substituting the linear filtering part with a quadratic, or in general polynomial, filter. However, we show that when the system is not asymptotically stable the polynomial control does not improve over the classical LQG solution, due to the lack of the internal stability of the polynomial filter. In order to enlarge the class of systems that can be controlled, we propose a new method based on a suitable rewriting of the system by means of an output injection term. We show that this allows us to overcome the problem and to design a polynomial optimal controller also for non asymptotically stable systems. Numerical results show the effectiveness of the method.

About the $L^2$ analyticity of Markov operators on graphs

Let $\Gamma$ be a graph and $P$ be a reversible random walk on $\Gamma$. From the $L^2$ analyticity of the Markov operator $P$, we deduce that an iterate of odd exponent of $P$ is `lazy', that is there exists an integer $k$ such that the transition probability (for the random walk $P^{2k+1}$) from a vertex $x$ to itself is uniformly bounded from below. The proof does not require the doubling property on $\Gamma$ but only a polynomial control of the volume.

High performances of Polynomial and Nonlinear Backstepping Control Strategies of an Induction Motor fed by Matrix Converter

High performances of Polynomial and Nonlinear Backstepping Control Strategies of an Induction Motor fed by Matrix Converter

Polynomial Control Systems: Invariant Sets given by Algebraic Equations/Inequations

Consider a nonlinear input-affine control system x(t) = f(x(t)) + g(x(t))u(t), y(t) = h(x(t)), where f, g, h are polynomial functions. Let S be a set given by algebraic equations and inequations (in the sense of =). Such sets appear, for instance, in the theory of the Thomas decomposition, which is used to write a variety as a disjoint union of simpler subsets. The set S is called controlled invariant if there exists a polynomial state feedback law u(t) = α(x(t)) such that S is an invariant set of the closed loop system x = (f + gα)(x). If it is possible to achieve this goal with a polynomial output feedback law u(t) = β(y(t)), then S is called controlled and conditioned invariant. These properties are discussed and algebraically characterized, and algorithms are provided for checking them with symbolic computation methods.

Interactive modeling of smooth manifold meshes with arbitrary topology: G1 stitched bi-cubic Bézier patches

In this paper, we present a new piecewise parametric approach to obtain smooth surfaces from any given orientable 2-manifold polynomial control mesh. Our approach provides C2 continuous Bi-Cubic Bézier patches that are guaranteed to be stitched with G1 continuity regardless of the underlying mesh topology. A high level summary of the approach can be given as a two-step process: (1) Starting from an orientable 2-manifold mesh we apply vertex-insertion, which is remeshing algorithm of Catmull–Clark subdivision, to break that surface into a set of all quadrilateral patches. (2) for each such quadrilateral patch, we construct a Bi-Cubic Bézier patch, where the positions of the 16 control points are obtained from the limit surface of the Doo–Sabin subdivision scheme.This approach, when combined with topological modeling, provides a framework for real-time interactive modeling of smooth manifold meshes with arbitrary topology. Based on this approach, we have developed an interactive modeling system that allows users to work with smooth surfaces in real-time by opening or closing holes, creating handles, and combining and disconnecting surfaces. The paper has three main contributions: (1) We guarantee that our method will produce only smooth manifold meshes with C2 continuous Bi-Cubic Bézier patches. (2) We guarantee visual smoothness for any arbitrary topological structure by providing geometric G1 continuity in boundaries of patches. (3) We demonstrate that smooth surfaces with arbitrary topology can be created interactively in real-time using parametric patches.

Semi-Active Control of Structures Equipped With MR Dampers Based on Uniform Deformation Theory

Optimization based on uniform deformation theory (UDT) was proposed as a new technique in the field of optimum design of structures. In this paper, the concept of UDT has been utilized for semi-active control of buildings equipped with magneto-rheological (MR) dampers. Based on UDT, a method for determining the optimum control voltage using a polynomial controller has been presented. The coefficients of the polynomial controller have been determined by minimizing the standard deviation of the maximum inter-story drifts of a structure and by utilizing the Particle Swarm Optimization (PSO) algorithm. The proposed controller based on UDT has been applied to control three- and six-story nonlinear shear buildings subjected to various earthquakes. The performance of this controller was evaluated regarding the uniform distribution of the maximum inter-story drifts and the reduction of the structural responses. Using this method, the maximum inter-story drifts of the structures were remarkably uniform in comparison to the uncontrolled structures. In addition, the maximum structural responses, especially the maximum inter-story drifts of the structures, significantly decreased.

Synthesis of polynomial control laws for continuous dynamic objects

Synthesis of polynomial control laws for continuous dynamic objects

Control design for discrete‐time bilinear systems using the scalarized Schur complement

SummaryIn this paper, controller design for discrete‐time bilinear systems is investigated by using sum of squares programming methods and quadratic Lyapunov functions. The class of rational polynomial controllers is considered, and necessary conditions on the degree of controller polynomials for quadratic stability are derived. Next, a scalarized version of the Schur complement is proposed. For controller design, the Lyapunov difference inequality is converted to a sum of squares problem, and an optimization problem is proposed to design a controller, which maximizes the region of quadratic stability of the bilinear system. Input constraints can also be accounted for. Copyright © 2017 John Wiley & Sons, Ltd.