Polynomial State Feedback Research Articles

Controlled invariant varieties for second order polynomial control systems

We investigate controlled invariant varieties for second order input-affine control systems with polynomial nonlinearity. For autonomous, first order systems, a variety V is said to be invariant if any trajectory starting in V remains in V for all times. We discuss how to adapt this notion in the second order case and give conditions to decide whether a variety is invariant for a second order system using symbolic computation. This results serve as a foundation to characterise controlled invariant varieties, i.e., varieties that can be rendered invariant by a polynomial state feedback.

Gaussian inference for data-driven state-feedback design of nonlinear systems

Data-driven control of nonlinear systems with rigorous guarantees is a challenging problem as it usually calls for nonconvex optimization and often requires the knowledge of the true basis functions of the unknown system dynamics. To tackle these drawbacks, this work is based on a data-driven polynomial representation of general nonlinear systems exploiting Taylor polynomials. Thereby, we design state-feedback laws that render a known equilibrium globally asymptotically stable while operating with respect to a desired quadratic performance criterion. The calculation of the polynomial state feedback boils down to a sum-of-squares optimization problem, and hence to computationally tractable linear matrix inequalities. Moreover, we examine state-input data in presence of Gaussian noise by Bayesian inference to overcome the conservatism of deterministic noise characterizations from recent data-driven control approaches for Gaussian noise.

Analysis and Design of Robust Controller for Polynomial Fractional Differential Systems Using Sum of Squares

This paper discusses the robust stability and stabilization of polynomial fractional differential (PFD) systems with a Caputo derivative using the sum of squares. In addition, it presents a novel method of stability and stabilization for PFD systems. It demonstrates the feasibility of designing problems that cannot be represented in LMIs (linear matrix inequalities). First, sufficient conditions of stability are expressed for the PFD equation system. Based on the results, the fractional differential system is Mittag–Leffler stable when there is a polynomial function to satisfy the inequality conditions. These functions are obtained from the sum of the square (SOS) approach. The result presents a valuable method to select the Lyapunov function for the stability of PFD systems. Then, robust Mittag–Leffler stability conditions were able to demonstrate better convergence performance compared to asymptotic stabilization and a robust controller design for a PFD equation system with unknown system parameters, and design performance based on a polynomial state feedback controller for PFD-controlled systems. Finally, simulation results indicate the effectiveness of the proposed theorems.

Disturbance Decoupling for Polynomial Systems

We study nonlinear control systems x˙(t) = f(x(t)) + g(x(t))u(t) + d(x(t))w(t), y(t) = h(x(t)), where f,g,d,h are polynomial functions. The output y is called decouplable from disturbances if there exists a polynomial state feedback u(t) = α(x(t)) + β(x(t))v(t) with β as an invertible matrix, which renders the output y invariant under disturbances. The question whether this is possible leads to the concept of controlled invariant modules. We give algebraic characterisations to decide whether an output is invariant or decouplable and present algorithms to constructively verify these criteria by symbolic computational methods.

Nonlinear state-feedback design for vehicle lateral control using sum-of-squares programming

ABSTRACT This work addresses the lateral stabilisation problem of four-wheels ground vehicles. The objective is to estimate the largest state-space region such that the closed-loop vehicle lateral stability can be guaranteed. Sum-of-squares (SOS) programming technique is applied to find these maximum invariant sets while accounting for steering and yaw moment input saturations. The algorithm allows the region of attraction (RoA) to be approximated by a level set of a Lyapunov function (LF) and the computation of polynomial state feedback control laws. The method is applied for both straight-line motion and cornering manoeuver. Finally, a Monte-Carlo analysis is presented to show that the proposed SOS-based methodology can be used as a valid analysis and design tool considering a real vehicle application.

Stabilizing feedback design for time delayed polynomial systems using kinetic realizations

A new stabilizing feedback design method is proposed in this paper for time delayed polynomial systems with a linear input structure. The task is to transform the open loop system to a time delayed complex balanced kinetic system by using a polynomial state feedback structure which guarantees stability with arbitrary time delays.It is shown that the required computations can be performed by simple linear programming, when the only goal is the semistability of the chosen equilibrium point. If one wants to achieve additionally the uniqueness of the closed loop equilibrium point to ensure local asymptotic stability, then the extended optimization problem requires the application of semidefinite programming. The existence of the solution and computability of the feedback do not depend on the magnitude of the delays.It is shown that involving additional monomials into the feedback beyond the ones contained in the open-loop model does not improve the solvability of the semistabilization problem, but it may ensure the uniqueness of the prescribed complex balanced equilibrium point. Thus, two variants of a systematic method are proposed to find appropriate extra monomials for the feedback. One of these requires to solve a linear programming optimization problem even in this extended case.

Regional Stabilization of Input-Delayed Uncertain Nonlinear Polynomial Systems

This paper addresses the problem of local stabilization of nonlinear polynomial control systems subject to time-varying input delay and polytopic parameter uncertainty. A linear matrix inequality approach based on the Lyapunov–Krasovskii theory is proposed for designing a nonlinear polynomial state feedback controller ensuring the robust local uniform asymptotic stability of the system origin along with an estimate of its region of attraction. Two convex optimization procedures are presented to compute a stabilizing controller ensuring either a maximized set of admissible initial states for given upper bounds on the delay and its variation rate or a maximized lower bound on the maximum admissible input delay considering a given set of admissible initial states. Numerical examples demonstrate the potentials of the proposed stabilization approach.

Sum of squares approach for ground vehicle lateral control under tire saturation forces

This work presents the stability analysis and design of a lateral controller for a nonlinear ground vehicle applying the concept of polynomial sum of squares relaxations. The system is approximated by a polynomial vector field that describes the lateral vehicle dynamics. The resulting polynomial system falls on a class of non-affine in input system, which makes the control synthesis more involved. This issue is circumvented by an input-affine approximation, simplifying the stability analysis and the design procedure of a polynomial state-feedback controller able to enlarge the region of attraction (RoA). We also compare the estimated region of attraction with the standard LQR optimal controller.

Polynomial static output feedback H∞ control via homogeneous Lyapunov functions

SummaryIn this paper, we study a polynomial static output feedback (SOF) stabilization problem with H∞ performance via a homogeneous polynomial Lyapunov function (HPLF). It is shown that the quadratic stability ascertaining the existence of a single constant Lyapunov function becomes a special case. With the HPLF, the proposal is based on a relaxed two‐step sum of square (SOS) construction where a stabilizing polynomial state feedback gain K(x) is returned at the first stage and then the obtained K(x) gain is fed back to the second stage, achieving the SOF closed‐loop stabilization of the underlying polynomial fuzzy control systems. The SOS equations obtained thus effectively serve as a sufficient condition for synthesizing the SOF controllers that guarantee polynomial fuzzy systems stabilization. To demonstrate the effectiveness of the proposed polynomial fuzzy SOF H∞ control, benchmark examples are provided for the new approach.

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.

Polynomial-Approximation-Based Control for Nonlinear Systems

This paper is concerned with the stabilization problem for nonlinear systems. A new polynomial-approximation-based approach for modeling nonlinear systems is first proposed. The nonlinearity is approximated by polynomials, and the approximation errors are treated as modeling uncertainties. The original nonlinear systems are converted into polynomial systems with modeling uncertainties. In order to highlight the approximation accuracy, the piecewise polynomial approximation functions are utilized. A novel polynomial state-feedback controller is designed to solve the stabilization problem. Furthermore, switched polynomial state-feedback controllers are designed to improve the performance. The stabilization conditions are presented in terms of sum of squares, which can be numerically solved via SOSTOOLS. Finally, simulation examples are provided to demonstrate the feasibility of the proposed method and show its advantage over the polynomial-fuzzy-model-based approach.

Design of networked polynomial control systems with random delays: sum of squares approach

This paper studies the problem of networked polynomial state feedback control using the sum of square (SOS) approach considering network-induced varying delay. The stabilisation conditions are achieved based on the Lyapunov-Krasovskii stability theory. The proposed control approach is based on the SOS decomposition numerical method which applies a control input to stabilise the networked control system states. Finally, numerical simulation is done to demonstrate the effectiveness of the proposed method.

Hamiltonian Feedback Design for Mass Action Law Chemical Reaction Networks

A special polynomial state feedback structure is proposed for open chemical reaction networks obeying the mass action law (MAL-CRNs) that stabilizes them for any of their admissible positive set of parameters. The proposed feedback makes the closed-loop system a reversible CRN that enables a generalized Hamiltonian description assuming that their number of reversible reactions is less or equal that the number of species. The design is based on solving a mixed integer linear optimization problem (MILP). A simple example is used to illustrate the basic concepts and the design method.

The Tensor Product Based Analytic Solutions of Nonlinear Optimal Control Problem

In this paper, the attention is focused on the optimization of a particular class of nonlinear systems. The optimum linear solution is not the best one so the problem of determining a nonlinear state feedback optimal control law with quadratic performance index over infinite time horizon is considered. It isn't an easy task and the most discouraging obstacle is the resolution of the Hamilton-Jacobi equation. Thus our contribution, based on the use of the tensor product and its algebraic laws, provide analytic solutions of the studied optimal control problem. The polynomial state feedback solution is computed through a numerical procedure. A numerical example is treated to illustrate the proposed solutions and some conclusions are drawn.

Lie Derivative Inclusion for a Class of Polynomial State Feedback Controls

Lie Derivative Inclusion for a Class of Polynomial State Feedback Controls

Output Regulation Control Experiment for Inverted Pendulum (Nonlinear Control via Center Manifold Approximation Theory)

This paper reports experimental verification of nonlinear output regulation for an inverted pendulum system. The output regulation problem, which is alternatively referred to as servo system design problem, is posed so that the cart of the pendulum tracks an arbitrarily given sinusoidal signal while holding the pendulum up above. To solve the problem, it is necessary to solve the so-called nonlinear regulator equation and the computation for its solution is based on a newly proposed theory to approximate center manifolds in a system of ordinary differential equations. This computation theory, which was proposed by the authors previously, provides an iterative algorithm that enables one to obtain higher order approximations. The algorithm is suitable for computer implementation and a 7-th order polynomial state feedback law is obtained. The experiments show that the cart of the pendulum tracks sinusoidal signals with different frequencies. The paper also reviews the general theory of nonlinear output regulation as well as the theories of center manifold and its approximate computation.

Nonlinear State Feedback Control for A Class of Polynomial Discrete-time Systems with Norm-Bounded Uncertainties: An Integrator Approach

This paper investigates the problem of designing a nonlinear state feedback controller for a class of uncertain polynomial discrete-time systems using rational Lyapunov functions. The uncertainty that under consideration is modelled as a norm-bounded uncertainty. In general, the problem of designing a controller for polynomial discrete-time systems cannot be formulated as a convex problem. This is due to the fact that the Lyapunov function and the control input is not jointly convex, hence it cannot be solved by a semidefinite programming (SDP). In this paper, we propose a novel approach where an integrator is introduced to convexify the nonconvex controller design problem, so that it can be solved easily by SDP. Furthermore, based on the sum of squares approach, sufficient conditions for the existence of a rational polynomial state feedback controller for a polynomial discrete-time systems are given in terms of solvability of polynomial matrix inequalities. These inequalities are then solved by the recently developed sum of squares (SOS) solvers. Finally, numerical example is provided to demonstrate the validity of this integrator approach.

Stabilizing polynomial approximation of explicit MPC

A given explicit piecewise affine representation of an MPC feedback law is approximated by a single polynomial, computed using linear programming. This polynomial state feedback control law guarantees closed-loop stability and constraint satisfaction. The polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage.

Feedback passivation of an electrostatic microactuator: a semidefinite programming approach

The purpose of this paper is to study an application of the numerical controller design procedure for the class of polynomial control systems proposed by the authors in an earlier paper. This design procedure is applied to design a polynomial state feedback controller for an electrostatic micro actuator. The proposed controller design procedure consists of two concepts: Feedback passivation and semidefinite programming. The combination of these two concepts leads to a physically motivated controller design procedure which can be efficiently solved on a computer with the help of the sum of squares decomposition.

On spectrum placement for linear time invariant delay systems

A ring of delay operators is used to obtain a ring model description of a linear time invariant delay system. With this aid an algorithm is developed for the computation of the polynomial state-feedback matrix which assigns an arbitrary set of poles of the closed-loop system (spectrum placement). An example is included to illustrate the method.