Polynomial Matrix Inequalities Research Articles

Dissipativity-Based Integrated Fault Estimation and Fault Tolerant Control for IT2 Polynomial Fuzzy Systems With Sensor and Actuator Faults

The integrated fault estimation and fault-tolerant control scheme is developed in this paper for a series of interval type-2 polynomial fuzzy systems with both sensor faults and actuator faults, where the bi-directional influence between fault estimation unit and fault-tolerant control unit is investigated. Considering the existence of sensor faults, unmeasurable premise variables are investigated for more general situations and Class III state/fault estimation observers are established for the final fault estimation and fault-tolerant control purposes. To increase design flexibility and reduce physical implementation complexity, the proposed method allows the observer and original system to share asynchronous membership functions and a different number of fuzzy rules. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\mathcal{Q,S,R}) \,-\, \alpha$</tex-math></inline-formula> dissipative performance index is also introduced to fulfill a wider vary of perfor-mance requirements. Membership-function-dependent stability constraints are given in the format of bi-linear polynomial matrix inequalities to obtain less conservative results, which are computed by a two-step path-following method. Superiority and validity are demonstrated by an inverted pendulum example in terms of estimation errors, fault-tolerant control performance and control inputs.

H2model order reduction of bilinear systems via linear matrix inequality approach

AbstractThis paper proposes an H2‐optimal model order reduction (MOR) method for bilinear systems based on the linear matrix inequality (LMI) approach. In this method, to reduce the computational complexity, at first, a reduced middle‐order approximation of the system is derived based on common bilinear MOR methods. Next, the H2norm of the error system is minimized to obtain the reduced‐order bilinear model. Generalized Lyapunov equations are added to the optimization problem as LMI constraints to guarantee the specification of type II Gramians of the bilinear system to improve accuracy. Besides, two stability conditions are included to the optimization problem as its constraints to preserve stability of reduced‐order bilinear model. One of advantages of the proposed method is the need for only one of the Gramians of controllability or observability. Since the proposed H2‐optimal MOR problem is a polynomial matrix inequality (PMI) problem, an iterative method is used to convert the PMI to the LMI problems and solve the optimization problem. Three bilinear test systems are considered to show the proposed method's efficiency, while its performance is compared with some classical methods. Results show that the proposed methods lead to a more accurate reduced‐order model than other MOR methods.

Stability Verification for Heterogeneous Complex Networks via Iterative SOS Programming

This letter is to investigate the stability verification for heterogeneous polynomial complex networks through iterative sum-of-squares programming approach. With polynomial Lyapunov functions, a global asymptotic stability criterion is established for the heterogeneous complex networks under the directed topology. Based on the proposed criterion, the stability verification problem is then reduced to a sum-of-squares optimization problem for solving polynomial matrix inequalities. To process non-convex terms of the polynomial matrix inequalities, we present an iterative sum-of-squares programming approach to turn them into the convex polynomial matrix inequalities, yielding polynomial Lyapunov functions effectively to realize the stability verification. Finally, a numerical example is given to show the fully automatic verification of global asymptotic stability for the heterogeneous polynomial complex networks, which demonstrates the theoretical and algorithmic advantages of the proposed method.

Robust fuzzy observer‐based fault‐tolerant control: A homogeneous polynomial Lyapunov function approach

In this paper, a homogeneous polynomial Lyapunov function (HPLF) is employed in robust observer-based Integrated Fault-Tolerant Control (IFTC) of nonlinear systems modelled by the polynomial fuzzy model (PFM). This makes the design problem benefit from the conservative reduction property of polynomial Lyapunov functions (PLFs) and simultaneously avoid the emergence of non-convex terms due to differentiation of the polynomial Lyapunov matrix. As a result, the less conservative sum of square (SOS) conditions are obtained in the form of polynomial matrix inequalities (PMI), which are solved via SOSTOOLS. For the first time, a non-quadratic Lyapunov function is used to design a polynomial fuzzy unknown-input observer that estimates the system's states and actuator faults in the presence of model uncertainties and external disturbances. The effectiveness of the proposed approach in providing more relaxed and less conservative results along with the better fault tolerance is illustrated through three simulating examples and compared with the quadratic Lyapunov function (QLF) approach.

Sum-of-squares chordal decomposition of polynomial matrix inequalities

We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert–Artin, Reznick, Putinar, and Putinar–Vasilescu Positivstellensätze. First, we establish that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all xin mathbb {R}^n if and only if there exists a sum-of-squares (SOS) polynomial sigma (x) such that sigma P is a sum of sparse SOS matrices. Second, we show that setting sigma (x)=(x_1^2 + cdots + x_n^2)^nu for some integer nu suffices if P is homogeneous and positive definite globally. Third, we prove that if P is positive definite on a compact semialgebraic set mathcal {K}={x:g_1(x)ge 0,ldots ,g_m(x)ge 0} satisfying the Archimedean condition, then P(x) = S_0(x) + g_1(x)S_1(x) + cdots + g_m(x)S_m(x) for matrices S_i(x) that are sums of sparse SOS matrices. Finally, if mathcal {K} is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for (x_1^2 + cdots + x_n^2)^nu P(x) with some integer nu ge 0 when P and g_1,ldots ,g_m are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones.

A Matrix Positivstellensatz with Lifting Polynomials

Given the projections of two semialgebraic sets defined by polynomial matrix inequalities, it is in general difficult to determine whether one is contained in the other. To address this issue we propose a new matrix Positivstellensatz that uses lifting polynomials. Under some general assumptions (e.g., the archimedeanness, nonempty interior, convexity), we prove that such a containment holds if and only if the proposed matrix Positivstellensatz is satisfied. The corresponding certificate can be searched for by solving a semidefinite program. An important application is to certify when a spectrahedrop (i.e., the projection of a spectrahedron) is contained in another one.

A mixed PI/VI design method for nonlinear H∞ Control

SummaryThis paper proposes a new mixed policy iteration and value iteration (PI/VI) design method for nonlinear H∞ control based on the theories of polynomial optimization and Lasserre's hierarchy. The design of a mixed PI/VI controller can be carried out in four steps: firstly, initialize design parameters and expand nonlinear system matrices; secondly, obtain a polynomial matrix inequality for policy improvement; thirdly, obtain the Lasserre's hierarchy of a global polynomial optimization problem for value improvement; fourthly, perform the mixed PI/VI algorithm to approximate the optimal nonlinear H∞ control law. The novelty of this work lies in that the problem of designing a nonlinear H∞ controller is translated into a polynomial global optimization problem, which can be solved by Lasserre's hierarchy directly, and then, the mixed PI/VI algorithm is presented to approximate the optimal nonlinear H∞ control law by updating global optimizers iteratively. The main results of this paper consist of the mixed PI/VI algorithm and the related three theorems, which guarantee robust stability and performance of the closed‐loop nonlinear system. Numerical simulations show that the mixed PI/VI algorithm converges very fast and achieves good robust stability and performance in transient behavior, disturbance rejection, and enlarging the domain of attraction of the close‐loop system.

Exponential Stability Analysis for Delayed Semi-Markovian Recurrent Neural Networks: A Homogeneous Polynomial Approach.

This paper investigates the exponential stability analysis issue for a class of delayed recurrent neural networks (RNNs) with semi-Markovian parameters. By constructing a stochastic Lyapunov functional and using some zoom techniques to estimate its weak infinitesimal operator, the exponential mean square stability criteria have been proposed for the Markovian neural networks with certain transition probabilities. We then generalize the homogeneous polynomial approach for the delayed Markovian RNNs with uncertain transition probabilities during the stability analysis. Theoretical results have obtained by introducing an appropriate technique for dealing with a large number of complex homogeneous polynomial matrix inequalities. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed technique.

Stabilisation of discrete-time polynomial fuzzy systems via a polynomial lyapunov approach

ABSTRACTThis paper deals with the problem of designing a controller for a class of discrete-time nonlinear systems which is represented by discrete-time polynomial fuzzy model. Most of the existing control design methods for discrete-time fuzzy polynomial systems cannot guarantee their Lyapunov function to be a radially unbounded polynomial function, hence the global stability cannot be assured. The proposed control design in this paper guarantees a radially unbounded polynomial Lyapunov functions which ensures global stability. In the proposed design, state feedback structure is considered and non-convexity problem is solved by incorporating an integrator into the controller. Sufficient conditions of stability are derived in terms of polynomial matrix inequalities which are solved via SOSTOOLS in MATLAB. A numerical example is presented to illustrate the effectiveness of the proposed controller.

Guaranteed cost fuzzy tracking control for nonlinear polynomial systems with input saturation

This paper is concerned with the output tracking problem for a class of nonlinear systems represented by polynomial T–S fuzzy model. The input saturation and external disturbances are considered simultaneously. Static output-feedback fuzzy controllers are designed such that the resulting closed-loop system is locally stable and satisfies a regional H∞ performance from the disturbance to tracking error. The guarantee of cost function is also considered during the design procedure. Sufficient conditions ensuring the existence of the desired controllers are obtained in terms of polynomial matrix inequalities, which could be solved by using the sum-of-squares (SOS) tools in Matlab. A numerical example is provided finally to illustrate the effectiveness of the proposed design method.

Dual form reduction in the atomic optimization method

One-dimensional optimization problems with a polynomial objective function and polynomial matrix inequality constraints are considered. For problems dual to their linear relaxations, a transformation is presented that makes them compatible with the atomic optimization method, both in its basic and in the generalized form with a reduced number of atoms.

2D system analysis via dual problems and polynomial matrix inequalities

Application of the Lyapunov method to 2D system stability and performanceanalysis yields algebraic systems that can be interpreted as eithersum-of-squares problems for nontrivial matrix polynomials, or parameterizedlinear matrix inequalities that need to be satisfied for certain ranges ofparameter values. In this paper we show that dualizing core inequalities inthe latter forms allows converting these systems to conventionaloptimization problems on sets described by polynomial matrix inequalities.Methods for solving these problems include moment-based methods or the“atomic optimization” method proposed earlier by the author. As aresult, we obtain necessary conditions for 2D system stability and lowerbounds on system performance. In particular, we demonstrate respectiveresults for discrete-discrete system stability and mixedcontinuous-discrete system $\mathcal{H}_\infty$ performance. A numerical example isprovided.

Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates

The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be viewed, in particular, as solution sets to problems of generalized semi-infinite programming with polynomial data. Exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry, we establish a far-going extension of the \L ojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (H\"older type) local error bounds results with explicitly calculated exponents. The obtained results are applied to higher-order quantitative stability analysis for various classes of optimization problems including generalized semi-infinite programming with polynomial data, optimization of real polynomials under polynomial matrix inequality constraints, and polynomial second-order cone programming. Other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semialgebraic settings.

Atomic optimization. II. Multidimensional problems and polynomial matrix inequalities

The paper studies multidimensional optimization problems with a polynomial objective function and constraints in the form of polynomial matrix inequalities. The author suggests a transformation of the solution method based on the theory of moments. This transformation allows to reduce appreciably the computational complexity of the method, still preserving its applicability to optimization problems of the above class.

Robust Nonlinear H ∞ State Feedback Control of Polynomial Discrete-Time Systems: An Integrator Approach

This paper investigates the problem of designing a nonlinear H ∞ state feedback controller for polynomial discrete-time systems with norm-bounded uncertainties. In general, the problem of designing a controller for polynomial discrete-time systems is difficult, because it is a nonconvex problem. More precisely, in general, its Lyapunov function and control input are not jointly convex. Hence, it cannot be solved by semidefinite programming. In this paper, a novel approach is proposed, where an integrator is incorporated into the controller structure. In doing so, a convex formulation of the controller design problem can be rendered in a less conservative way than the available approaches. Furthermore, we establish the interconnection between robust H ∞ control of polynomial discrete-time systems with norm-bounded uncertainties and H ∞ control of scaled polynomial discrete-time systems. This establishment allows us to convert the robust H ∞ control problems to H ∞ control problems. Then, based on the sum of squares (SOS) approach, sufficient conditions for the existence of a nonlinear H ∞ state feedback controller are given in terms of solvability of polynomial matrix inequalities (PMIs), which can be solved by the recently developed SOS solvers. A tunnel diode circuit is used to demonstrate the validity of this integrator approach.

Nonlinear state feedback control for a class of polynomial nonlinear 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 nonlinear discrete-time systems with norm-bounded uncertainties. In general, the problem of stabilising nonlinear discrete-time systems cannot be formulated as a convex problem. This is due to the fact that the Lyapunov function and the control input are 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 this nonconvex controller design problem. Furthermore, based on the sum of squares approach, sufficient conditions for the existence of a polynomial nonlinear state feedback controller for polynomial nonlinear discrete-time systems are given in terms of solvability of polynomial matrix inequalities. These inequalities are then solved by the sum of squares (SOS) solvers. Finally, numerical examples are provided to demonstrate the validity of this integrator approach.

Robust H∞ static output feedback controller design for parameter dependent polynomial systems: An iterative sums of squares approach

This paper considers the problem of designing a robust H∞ static output feedback controller for polynomial systems with parametric uncertainties. Sufficient conditions for the existence of a nonlinear H∞ static output feedback controller are given in terms of solvability conditions of polynomial matrix inequalities. An iterative sum of squares decomposition is proposed to solve these polynomial matrix inequalities. The proposed controller guarantees that the closed-loop system is stable and the L2-gain of the mapping from exogenous input noise to the controlled output is less than or equal to a prescribed value. Numerical examples are provided to demonstrate the validity of applied methods.

Input-output LPV model identification with guaranteed quadratic stability

The problem of identifying linear parameter-varying (LPV) systems, a priori known to be quadratically stable, is considered in the paper using an input-output model structure. To solve this problem, a novel constrained optimization-based algorithm is proposed which guarantees quadratic stability of the identified model. It is shown that this estimation objective corresponds to a nonconvex optimization problem, defined by a set of polynomial matrix inequalities (PMI), whose optimal solution can be approximated by means of suitable convex semidefinite relaxations. Applicability of such relaxation-based estimation approach in the presence of either stochastic or deterministic bounded noise is discussed. A simulation example is also given to demonstrate the effectiveness of the resulting identification method.

Inner Approximations for Polynomial Matrix Inequalities and Robust Stability Regions

Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modeled by parametrized polynomial matrix inequalities (PMIs). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modeled by a single polynomial superlevel set. Those inner approximations converge in a well-defined analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case, they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximations.

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.