LMI characterization of reduced-order H∞ controllers via invariant zero structure

In this paper, we clarify a new relationship between invariant zeros of a generalized plant and the order reduction of H_∞ controllers by using linear matrix inequalities in both continuous-time and discrete-time cases. In contrast with our recent paper, where a relationship between an unstable transmission-zero structure and the H_∞ controller order reduction is initiated in a fundamental manner, results obtained in this paper are more flexible in two senses: assumptions that are made for the generalized plant are relaxed, and stable as well as unstable invariant zeros are characterized to obtain a reduced-order H_∞ controller.     

On Steiner minimal trees withLp distance

LetL _p be the plane with the distanced _p (A ₁ ,A ₂ ) = (¦x ₁ –x ₂¦ ^p + ¦y₁ –y ₂¦^p)^/1p wherex _i andy _i are the cartesian coordinates of the pointA _i . LetP be a finite set of points inL _p . We consider Steiner minimal trees onP. It is proved that, for 1 <p < , each Steiner point is of degree exactly three. Define the Steiner ratio _p to be inf{L _s (P)/L _m (P)¦PL _p } whereL _s (P) andL _m (P) are lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. Hwang showed ₁ = 2/3. Chung and Graham proved ₂ > 0.842. We prove in this paper that {} = 2/3 and (2/2)_{12 p} 3/2 for anyp.This work was supported in part by the National Science Foundation of China and the President Foundation of Academia Sinica.     

Analyzing the statistical features of CIXL2 crossover offspring

We cannot deny the effort that the scientific community is devoting to the explanation of the features of the crossover operator of real-coded genetic algorithms and its effect over the evolutive process. This paper is another step in that direction, we analyze the behavior of the Confidence Interval Based Crossover using L₂ Norm (CIXL2). This crossover is based on the learning of the statistical features of localization and dispersion of the best individuals of the population. The crossover obtains, by means of a L₂ norm, the estimators of the parameters of localization and dispersion of the distributions of the fittest individuals. From this estimation three virtual parents are created using the localization parameter and the lower and upper bounds of the bilateral confidence intervals of the gene values of the best individuals of the population. This paper studies the statistical features of the offspring generated by this crossover and corroborates this study showing the behavior of the crossover in a set of test functions.     

Optimal low-order controller design via ‘LQG-like’ parametrization

A new LQG-like parametrization scheme is introduced for low-order controller design. Descent-based design methodologies are developed for optimizing an H₂ cost with respect to the controller design parameters, in this case, the reduced-order regulator and estimator gain matrices. In a modified version, the reduced-order model of the plant is also updated. Numerical experiments using large-space structure models show the efficacy of this approach.     

H∞ model reduction for negative imaginary systems

This paper is concerned with the H_∞ model reduction for negative imaginary (NI) systems. For a given linear time-invariant system that is stable and NI, our goal is to find a stable reduced-order NI system satisfying a pre-specified H_∞ approximation error bound. Sufficient conditions in terms of matrix inequalities are derived for the existence and construction of an H_∞ reduced-order NI system. Iterative algorithms are provided to solve the matrix inequalities and to minimise the H_∞ approximation error. Finally, a numerical example is used to demonstrate the effectiveness of the proposed model reduction method.     

A linear-time algorithm for linearL1 approximation of points

In this paper we present a linear-time algorithm for approximating a set ofn points by a linear function, or a line, that minimizes theL ₁ norm. The algorithmic complexity of this problem appears not to have been investigated, although anO(n ³) naive algorithm can be easily obtained based on some simple characteristics of an optimumL ₁ solution. Our linear-time algorithm is optimal within a constant factor and enables us to use linearL ₁ approximation of many points in practice. The complexity ofL ₁ linear approximation of a piecewise linear function is also touched upon.     

Simultaneous Lp-stabilization and internal stabilization of linear systems subject to input saturation — state feedback case

We consider the problem of simultaneous finite gain L_p-stabilization and internal stabilization of linear systems subject to input saturation via linear static state feedback. We show that bounded input finite-gain L_p-stabilization and local asymptotic stabilization can always be achieved simultaneously no matter where the poles of the open-loop system are, and the locations of these poles play a role only when bounded input finite gain L_p-stabilization and global or semi-global stabilization are required simultaneously.     

H∞ model reduction for discrete-time singular systems

This paper investigates the problem of H_∞ model reduction for linear discrete-time singular systems. Without decomposing the original system matrices, necessary and sufficient conditions for the solvability of this problem are obtained in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. When these conditions are feasible, an explicit parametrization of the desired reduced-order models is given. Particularly, a simple LMI condition without rank constraint is derived for the zeroth-order H_∞ approximation problem. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.     

Internal positivity preserved model reduction

This article studies model reduction of continuous-time stable positive linear systems under the Hankel norm, H _∞ norm and H ₂ norm performance. The reduced-order systems preserve the stability as well as the positivity of the original systems. This is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the Lyapunov matrices are decoupled with the system matrices. In this way, the positivity constraints in the reduced-order model can be imposed in a natural way. As the model reduction performances are expressed in linear matrix inequalities with equality constraints, the desired reduced-order positive models can be obtained by using the cone complementarity linearisation iterative algorithm. A numerical example is presented to illustrate the effectiveness of the given methods.     

H ∞ structured model reduction algorithms for linear discrete systems via LMI-based optimisation

In this article, H _∞ structured model reduction is addressed for linear discrete systems. Two important classes of systems are considered for structured model reduction, i.e. Markov jump systems and uncertain systems. The problem we deal with is the development of algorithms with the flexibility to allow any structure in the reduced-order system design, such as the structure of an original system, decentralisation of a networked system, pole assignment of the reduced system, etc. The algorithms are derived such that an associated model reduction error guarantees to satisfy a prescribed H _∞ norm-bound constraint. A new condition for the existence of desired reduced-order models preserving a certain structure is presented in a set of linear matrix inequalities (LMI) and non-convex equality constraints. Effective computational algorithms involving LMI are suggested to solve the matrix inequalities characterising a solution of the structured model reduction problem. Numerical examples demonstrate the advantages of the proposed model reduction method.     

On stability, -gain and control for switched systems

This paper addresses the issues of stability, L₂-gain analysis and H_∞ control for switched systems via multiple Lyapunov function methods. A concept of general Lyapunov-like functions is presented. A necessary and sufficient condition for stability of switched systems is given in terms of multiple generalized Lyapunov-like functions, which enables derivation of improved stability tests, an L₂-gain characterization and a design method for stabilizing switching laws. A solution to the H_∞ control problem for switched systems is also provided.     

On the gap between H2 and entropy performance measures in H∞ control design

Recent papers have considered the problem of minimizing an entropy functional subject to an H_∞ performance constraint. Since the entropy is an upper bound for the H₂ cost, there remains a gap between entropy minimization and H₂ minimization. In this paper we consider a generalized cost functional involving both H₂ and entropy aspects. This approach thus provides a means for optimizing H₂ performance within H_∞ control design.     

Stability and robustness analysis of MIMO MRAC using Kp = L 2 D 2 S 2 factorization

For a class of MIMO systems with input and output unmodeled dynamics, bounded disturbances and any relative degree, using the idea of K_p = L ₂ D ₂ S ₂ factorization, the design and analysis of robust direct model reference adaptive control are further investigated in this article. By establishing the L_p and L _2δ relationship properties between the input and output, multivariable swapping lemmas and relating all the signals in the closed-loop system with the normalizing signal, stability and robustness of adaptive system are analyzed rigorously. Compared with the existing results, the proof procedure is more compact and simple. A simulation example verifies the effectiveness of the control scheme.     

Nonlinear almost disturbance decoupling

The L₂-gain almost disturbance decoupling problem for SISO nonlinear systems is formulated. Sufficient conditions are identified for the existence of a parametrized state feedback controller such that the L₂-gain from disturbances to output can be made arbitrarily small by increasing its gain. The controller is explicity constructed using a Lyapunov-based recursive scheme. Sufficient conditions for the solvability of the almost disturbance decoupling problem and the explicit construction of teh controller are given for a more restrictive class of nonlinear systems.     

H∞ control for a class of structured time-delay systems

In this paper, we design an H_∞ controller for a class of lower-triangular time-delay systems. Backstepping is applied to construct an explicit feedback controller, and the closed-loop system maintains internal stability and an L₂-gain from the disturbance input to the output. The design is delay-dependent. Simulations on an example system demonstrate the good performance of the proposed design.     

L1 impulse response error bound for balanced truncation

This paper presents an upper bound in L₁ for the impulse response error between a system and its balanced truncation. It is an a priori bound and can be computed easily. Numerical examples are used to illustrate its applications and to compare with other available error bounds.     

Reduced-order filtering for linear systems with Markovian jump parameters

This paper addresses the reduced-order H_∞ filtering problem for continuous-time Makovian jump linear systems, where the jump parameters are modelled by a discrete-time Markov process. Sufficient conditions for the existence of the reduced-order H_∞ filter are proposed in terms of linear matrix inequalities (LMIs) and a coupling non-convex matrix rank constraint. In particular, the sufficient conditions for the existence of the zero-order H_∞ filter can be expressed in terms of a set of strict LMIs. The explicit parameterization of the desired filter is also given. Finally, a numerical example is given to illustrate the proposed approach.     

Nonlinear H∞ control around periodic orbits

In this paper, we study the nonlinear H_∞ control of systems with periodic orbits. We develop the notion of an induced L₂ gain (so-called nonlinear H_∞ norm) for systems where the no-disturbance behavior of the system is a periodic orbit and provide conditions under which the induced L₂ gain of the system (around the orbit) can be made less than a specified value by state feedback. This work is a natural extension of results on nonlinear H_∞ control of nonlinear systems in a neighborhood of a stable equilibrium point to the periodic orbit case. Synthesis of a nonlinear H_∞ state feedback controller is facilitated by the use of transverse coordinates and, in particular, the transverse linearization of the system.     

Low-rank matrix decomposition in L1-norm by dynamic systems

Low-rank matrix approximation is used in many applications of computer vision, and is frequently implemented by singular value decomposition under L₂-norm sense. To resist outliers and handle matrix with missing entries, a few methods have been proposed for low-rank matrix approximation in L₁ norm. However, the methods suffer from computational efficiency or optimization capability. Thus, in this paper we propose a solution using dynamic system to perform low-rank approximation under L₁-norm sense. From the state vector of the system, two low-rank matrices are distilled, and the product of the two low-rank matrices approximates to the given measurement matrix with missing entries, in L₁ norm. With the evolution of the system, the approximation accuracy improves step by step. The system involves a parameter, whose influences on the computational time and the final optimized two low-rank matrices are theoretically studied and experimentally valuated. The efficiency and approximation accuracy of the proposed algorithm are demonstrated by a large number of numerical tests on synthetic data and by two real datasets. Compared with state-of-the-art algorithms, the newly proposed one is competitive.     

Extensive theoretical/numerical comparative studies on H2 and generalised H2 norms in sampled-data systems

This paper is concerned with linear time-invariant (LTI) sampled-data systems (by which we mean sampled-data systems with LTI generalised plants and LTI controllers) and studies their H₂ norms from the viewpoint of impulse responses and generalised H₂ norms from the viewpoint of the induced norms from L₂ to L_∞. A new definition of the H₂ norm of LTI sampled-data systems is first introduced through a sort of intermediate standpoint of those for the existing two definitions. We then establish unified treatment of the three definitions of the H₂ norm through a matrix function G(τ) defined on the sampling interval [0, h). This paper next considers the generalised H₂ norms, in which two types of the L_∞ norm of the output are considered as the temporal supremum magnitude under the spatial 2-norm and ∞-norm of a vector-valued function. We further give unified treatment of the generalised H₂ norms through another matrix function F(θ) which is also defined on [0, h). Through a close connection between G(τ) and F(θ), some theoretical relationships between the H₂ and generalised H₂ norms are provided. Furthermore, appropriate extensions associated with the treatment of G(τ) and F(θ) to the closed interval [0, h] are discussed to facilitate numerical computations and comparisons of the H₂ and generalised H₂ norms. Through theoretical and numerical studies, it is shown that the two generalised H₂ norms coincide with neither of the three H₂ norms of LTI sampled-data systems even though all the five definitions coincide with each other when single-output continuous-time LTI systems are considered as a special class of LTI sampled-data systems. To summarise, this paper clarifies that the five control performance measures are mutually related with each other but they are also intrinsically different from each other.