Polynomial Control Systems 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.

On the small-time local controllability

A class of polynomial control systems is considered. The local properties of the corresponding reachable sets are studied by using a general differential–geometrical approach based on the Campbell–Baker–Hausdorff (C–B–H) formula. The main result is a new sufficient condition of small-time local controllability. Two four-dimensional examples illustrate this result.

Invariant sets for a class of nonlinear control systems tractable by symbolic computation

A set S is said to be controlled invariant with respect to a control system if a state feedback law exists such that the closed loop system has S as an invariant set. In the present paper we generalise results on input-affine polynomial control systems and algebraic varieties (i.e. sets described by the zeros of polynomial equations) considered in Zerz and Walcher (2012) to an extended class of vector fields. More precisely, we consider vector fields of the form f = F ° h, where F is a polynomial vector and h is a continuously differentiable function with certain (algebraic) properties, as well as sets Vh as the preimages of varieties under h. We will see that for example polynomial expressions in sine and cosine satisfy the mentioned properties. The main advantage of the considered function class is that it is accessible to symbolic computation. We give computational methods (based on the theory of Gröbner bases) to decide the controlled invariance of Vh.

SOS-based policy iteration for H∞ control of polynomial systems with uncertain parameters

In this article, the H∞ control problem is investigated for polynomial systems with uncertain parameters by utilising an adaptive dynamic programming (ADP) approach. The goal of this problem is to design a controller that stabilises the system and renders the system L2 -gain less than some disturbance attenuation level for all possible values of the uncertain parameters. A policy iteration (PI) algorithm is proposed to approximately solve the Hamilton-Jacobi-Isaacs (HJI) equation based on the sum-of-squares (SOS) optimisation. The convergence of the iterative algorithm and robustness of the closed-loop system with respect to both external disturbances and parametric uncertainties are guaranteed. Moreover, a two-loop iterative algorithm is presented to design a parametric robust H∞ controller with a smaller L2 -gain. The efficiency and advantage of the proposed SOS-based algorithm are demonstrated by simulation examples.

The Investigation of Nonlinear Polynomial Control Systems

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

Iterative stability analysis for general polynomial control systems

The main challenge of the stability analysis for general polynomial control systems is that non-convex terms exist in the stability conditions, which hinders solving the stability conditions numerically. Most approaches in the literature impose constraints on the Lyapunov function candidates or the non-convex related terms to circumvent this problem. Motivated by this difficulty, in this paper, we confront the non-convex problem directly and present an iterative stability analysis to address the long-standing problem in general polynomial control systems. Different from the existing methods, no constraints are imposed on the polynomial Lyapunov function candidates. Therefore, the limitations on the Lyapunov function candidate and non-convex terms are eliminated from the proposed analysis, which makes the proposed method more general than the state-of-the-art. In the proposed approach, the stability for the general polynomial model is analyzed and the original non-convex stability conditions are developed. To solve the non-convex stability conditions through the sum-of-squares programming, the iterative stability analysis is presented. The feasible solutions are verified by the original non-convex stability conditions to guarantee the asymptotic stability of the general polynomial system. The detailed simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed approach is more capable to find feasible solutions for the general polynomial control systems when compared with the existing ones.

Invariance of a class of semi-algebraic sets for polynomial systems with dynamic compensators

We derive a constructive sufficient condition for the existence of a compensator such that a prescribed semi-algebraic set is invariant for the resulting closed-loop system. The construction procedure consists of symbolic computations. Although the target system is restricted to polynomial control systems, non-polynomial smooth compensators can be designed. We also derive the procedure for extracting static controllers from the set of dynamic compensators. Moreover, we explain the similarities and differences between our method and the conventional slack-variable method of Jacobson and Lele (1969).

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.

Attracting and Natural Invariant Varieties for Polynomial Vector Fields and Control Systems

We discuss real and complex polynomial vector fields and polynomially nonlinear, input-affine control systems, with a focus on invariant algebraic varieties. For a given real variety we consider the construction of polynomial ordinary differential equations dot{x}=f(x) such that the variety is invariant and locally attracting, and show that such a construction is possible for any compact connected component of a smooth variety satisfying a weak additional condition. Moreover we introduce and study natural controlled invariant varieties (NCIV) with respect to a given input matrix g, i.e. varieties which are controlled invariant sets of {dot{x}}=f(x)+g(x)u for any choice of the drift vector f. We use basic tools from commutative algebra and algebraic geometry in order to characterize NCIV’s, and we present a constructive method to decide whether a variety is a NCIV with respect to an input matrix. The results and the algorithmic approach are illustrated by examples.

Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

We present a sum of squares (SOS) method for the synthesis of nonlinear polynomial control systems. As an emerging numerical solution method in recent years, SOS targets polynomials as the research object. It guarantees that the polynomial we solve for is always nonnegative. In this paper, we give a generalized S-procedure to solve the SOS problem. As an illustration of how the SOS method can be used, the region of attraction (ROA) in a nonlinear polynomial system is analyzed in detail. The method of determining decision variables is given in the SOS problem. We discuss the determination and solution of set-containment constraints and the conservatism problem in solving the SOS problem. SOS provides a convenient numerical method to solve nonlinear problems that are not easy to solve analytically.

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.

Non‐iterative SOS‐based approach for guaranteed cost control design of polynomial systems with input saturation

This study proposes a novel sum of squares (SOS) decomposition-based constrained guaranteed cost function controller for polynomial systems. The sufficient non-linear controller design conditions are derived such that an upper bound of a quadratic cost function subjected to the input amplitude constraint is minimised and asymptotic stability of the polynomial system is guaranteed. First, the constrained optimisation problem is formulated by an alternative unconstrained optimisation problem with the vanishing disturbance input. Then, by considering the S-procedure, the sufficient conditions are derived in terms of SOS decomposition. The main novelty of this study is that the stability issue of input saturated polynomial systems together with the non-linear control law is considered. In addition, the other advantage of this approach over the recently guaranteed cost controller design methods is that the proposed conditions can be solved without any iterative techniques. Finally, to show the effectiveness of the proposed approach, it is applied to a piezoelectrically actuated clamped–clamped micro-beam system and the results are compared with those of existing ones.

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.

Input‐delay approach to sampled‐data control of polynomial systems based on a sum‐of‐square analysis

In this study, the authors develop an H ∞ stabilisation condition for polynomial sampled-data control systems with respect to an external disturbance. Generally, continuous-time and sampled state variables are mixed in polynomial sampled-data control systems, which is the main drawback to numerically solving the stabilisation conditions of these control systems. To overcome this drawback, this study proposes novel stabilisation conditions that address the mixed-states problem by casting the mixed states as a time-varying uncertainty. The stabilisation conditions are derived from a newly proposed polynomial time-dependent Lyapunov–Krasovskii functional and are represented as a sum-of-squares, which can be solved using existing numerical solvers. Some additional slack variables are further introduced to relax the conservativeness of the authors' proposed approach. Finally, some simulation examples are provided to demonstrate the effectiveness of their 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.

A general U-block model-based design procedure for nonlinear polynomial control systems

ABSTRACTThe proposition of U-model concept (in terms of ‘providing concise and applicable solutions for complex problems’) and a corresponding basic U-control design algorithm was originated in the first author's PhD thesis. The term of U-model appeared (not rigorously defined) for the first time in the first author's other journal paper, which established a framework for using linear polynomial control system design approaches to design nonlinear polynomial control systems (in brief, linear polynomial approaches → nonlinear polynomial plants). This paper represents the next milestone work – using linear state-space approaches to design nonlinear polynomial control systems (in brief, linear state-space approaches → nonlinear polynomial plants). The overall aim of the study is to establish a framework, defined as the U-block model, which provides a generic prototype for using linear state-space-based approaches to design the control systems with smooth nonlinear plants/processes described by polynomial models. For analysing the feasibility and effectiveness, sliding mode control design approach is selected as an exemplary case study. Numerical simulation studies provide a user-friendly step-by-step procedure for the readers/users with interest in their ad hoc applications. In formality, this is the first paper to present the U-model-oriented control system design in a formal way and to study the associated properties and theorems. The previous publications, in the main, have been algorithm-based studies and simulation demonstrations. In some sense, this paper can be treated as a landmark for the U-model-based research from intuitive/heuristic stage to rigour/formal/comprehensive studies.

A practical criterion for positivity of transition densities

We establish a simple criterion for locating points where the transition density of a degenerate diffusion is strictly positive. Throughout, we assume that the diffusion satisfies a stochastic differential equation (SDE) on Rd with additive noise and polynomial drift. In this setting, we will see that it is often the case that local information of the flow, e.g. the Lie algebra generated by the vector fields defining the SDE at a point x ∈ Rd, determines where the transition density is strictly positive. This is surprising in that positivity is a more global property of the diffusion. This work primarily builds on and combines the ideas of Arous and Léandre (1991 Décroissance exponentielle du noyau de la chaleur sur la diagonale. II Probab. Theory Relat. Fields 90 377–402) and Jurdjevic and Kupka (1985 Polynomial control systems Math. Ann. 272 361–8).

Convex Computation of the Region of Attraction of Polynomial Control Systems

We address the long-standing problem of computing the region of attraction (ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving an infinite-dimensional convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description. The approach is demonstrated on several numerical examples.

Output mini-max control for polynomial systems: analysis and applications

This paper presents a solution to a robust optimal regulation problem for a nonlinear polynomial system affected by parametric and matched uncertainties, which is based only on partial state information. The parameters describing the dynamics of the nonlinear polynomial plant depend on a vector of unknown parameters, which belongs to a finite parametric set, and the application of a certain control input is associated with the worst or least favourable value of the unknown parameter. A high-order sliding mode state reconstructor is designed for the nonlinear plant in such a way that the previously designed control can be applied for a system with incomplete information. Additionally, the matched uncertainty is also compensated by means of the same output-based regulator. The obtained algorithm is applied to control an uncertain nonlinear inductor circuit of the third order and a mechanical pendulum of the third order, successfully verifying the effectiveness of the developed approach.

Controlled Invariant Hypersurfaces of Polynomial Control Systems

We study input-affine control systems with polynomial nonlinearity. A variety V is said to be controlled invariant if there exists a feedback law of polynomial type that causes the closed loop system to have V as an invariant variety. Using the theory of Grobner bases, we show how to constructively decide whether a given variety is controlled invariant for a given system, and if so, how to determine all feedback laws achieving the task. We also describe a set of “trivial” vector fields for which V is invariant. If V is a smooth hypersurface, then V is only invariant for its trivial vector fields. We discuss conditions under which the converse is also true.