J-spectral factorization and equalizing vectors

For the Wiener class of matrix-valued functions we provide necessary and sufficient conditions for the existence of a J-spectral factorization. One of these conditions is in terms of equalizing vectors. The second one states that the existence of a J-spectral factorization is equivalent to the invertibility of the Toeplitz operator associated to the matrix to be factorized. Our proofs are simple and only use standard results of general factorization theory (Clancey and Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Operator Theory: Advances and Applications, Vol. 3, Birkhäuser, Basel, 1981). Note that we do not use a state space representation of the system. However, we make the connection with the known results for the Pritchard–Salamon class of systems where an equivalent condition with the solvability of an algebraic Riccati equation is given.     

Synthesis of stable H controller via the chain scattering framework

A sufficient condition for the existence of suboptimal stable stabilizing H^∞ controllers is given. By exploiting the free parameter in the parameterization of stabilizing controllers and using the chain scattering framework, we reformulate the H^∞ strong stabilization problem as an equivalent H^∞ optimization problem which can be solved via only one algebraic Riccati equation. A parameterization of all suboptimal stable stabilizing H^∞ controllers is also given.     

Algebraic Riccati equation and J-spectral factorization for estimation

In this paper we investigate on the existence of the stabilizing solution of the algebraic Riccati equation (ARE) related to the filtering problem with a prescribed attenuation level γ. It is well known that such a solution exists and is positive definite for γ larger than a certain γ_F and it does not exist for γ smaller than a certain γ₀. We consider the intermediate case γ(γ₀,γ_F] and show that in this interval the stabilizing solution does exist, except for a finite number of values of γ. We show how the solution of the ARE may be employed to obtain a minimum-phase J-spectral factor of the J-spectrum associated with the filtering problem.     

Discrete J-spectral factorization

This paper deals with the J-spectral factorization for general discrete rational matrices. A simple approach based on the Kalman filtering in Krein space is proposed. The main idea is to construct a stochastic state space filtering model in Krein space such that the spectral matrix of the output is equal to the rational matrix to be factorized. The spectral factor is then easily derived by using the generalized Kalman filtering in Krein space, which is similar to the H₂ spectral factorization. Our approach unifies the treatment of the H₂ spectral factorization and the J-spectral factorization. The applications of the derived results in H_∞ and risk-sensitive estimation for both nonsingular and singular systems are demonstrated.     

On inner-outer factorization

The calculation of an inner-outer factorization of a proper, rational matrix by state-space methods is considered. Particular attention is given to the extraction of the inner factor, so that a minimal realization is obtained for this factor.     

On pure H reduced-order dynamic controllers: an erratum

There are at least two approaches advocated to obtain a pure H^∞ reduced-order dynamic controller for a given augmented plant. One approach is to eliminate completely the H² aspect from a standard H²/H^∞ setting. A second approach is to equate the H² aspect with the H^∞ aspect in that same setting. This paper invalidates the first approach but affirms the second approach and produces the correct equations resulting therefrom.     

H optimization problem with sign-indefinite quadratic form

This paper presents the solution to min-max control problem arising when the matrix C₁^TC₁ of the cost function in the standard H^∞ control problem (Doyle et al., 1989) is replaced by an arbitrary matrix Q 0. This difference is proved to be sufficient for results obtained in (Doyle et al., 1989) not to cover such the case. Their derivations essentially base on the cost function being H^∞ norm and can not be adjusted to deal with sign-indefinite quadratic form. With some sort of strict frequency condition assumed, state space technique is fruitful to obtain the necessary and sufficient conditions of the solvability of the problem. The solution is given by two Riccati equations and has some difference when compared to that of (Doyle et al., 1989).     

An H-minimax approach to the design of robust control systems, part II: All solutions, all-pass form solutions and the ‘best’ solution

We characterize all solutions to a robustness optimization problem as the solutions of a two-parameter interpolation problem. From this characterization it is easy to show that an all-pass form solution always exists as long as a solution exists. We also study the possibility of using non-all-pass form solutions and by introducing other optimization objectives (motivated by improvements in disturbance rejection and robust stability) we search for the 'best' solution.     

Algorithm of J-spectral factorization of polynomial matrices

Numerical algorithms for the canonical J-factorization of polynomial matrices with respect to the imaginary axis are given. The factorization problems for the non-regular polynomial matrices and for polynomials, whose determinants have zero roots, are considered. The constructed algorithms are not connected to the finding of roots of the polynomial. The basic calculation procedure is the construction of the stabilizing solution of the matrix algebraic Riccati equation.     

Stabilization and H control of symmetric systems: an explicit solution

In this paper we address the H^∞ control analysis, the output feedback stabilization, and the output feedback H^∞ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system we obtain an explicit expression to compute the H^∞ norm of the system. For the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the H^∞ control synthesis problem we derive an explicit expression for the optimally achievable closed-loop H^∞ norm and the optimal control gains. Extension to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H^∞ control synthesis problems using simple matrix algebraic tools.     

H disturbance attenuation for state delayed systems

In this note, the differential game and dissipation inequality are applied to the disturbance attenuation or H^∞-control for linear systems with delayed state. Firstly, a simple sufficient condition on the existence of a γ-suboptimal H^∞ state feedback controller is given, which is independent of delay, and an observer-based dynamic output feedback solution is presented in terms of Riccati inequalities (or Riccati equations). Secondly, a sufficient condition on the existence of a delay-dependent state feedback is presented and the criterion is presented by a matrix inequality which can be solved by numerical methods.     

Some remarks on Hamiltonians and the infinite-dimensional one block H problem

In this note, a simple proof is given for the Hamiltonian solution to the H^∞ optimal sensitivity for plants with arbitrary inner transfer function. The approach combines skew Toeplitz theory and state-space representations, and gives rise to a straightforward and basis-free methodology.     

H robust stabilizability

In this paper, we improve on the results of robust stabilizability obtained by Kimura. We do this in a constructive way by using an H^∞-approach, and exhibit an upper bound for the order of robust controllers.     

Achievable H performance in sampled-data smoothing: beyond the -barrier

The lifting technique is a powerful tool for handling the periodically time-varying nature of sampled-data systems. Yet all known solutions of sampled-data H^∞ problems are limited to the case when the feedthrough part of the lifted system, , satisfies , where γ is the required H^∞ performance level. While this condition is always necessary in feedback control, it might be restrictive in signal processing applications, where some amount of delay or latency between measurement and estimation can be tolerated. In this paper, the sampled-data H^∞ fixed-lag smoothing problem with a smoothing lag of one sampling period is studied. The problem corresponds to the a-posteriori filtering problem in the lifted domain and is probably the simplest problem for which a smaller than performance level is achievable. The necessary and sufficient solvability conditions derived in the paper are compatible with those for the sampled-data filtering problem. This result extends the scope of applicability of the lifting technique and paves the way to the application of sampled-data methods in digital signal processing.     

State-space formulae for all stabilizing controllers that satisfy an H∞-norm bound and relations to relations to risk sensitivity

Given a linear system, all stabilizing controllers such that a specified closed-loop transfer function has H_∞ norm less than a given scalar, are parametrized. This characterization involves the solution to two algebraic Riccati equations, each with the same order as the system, and further gives gives feasible controllers also with this order. The relationship to the risk-sensitive LQG stochastic control problem is established, giving an equivalence between robust and stochastic control.     

Electrochemical DNA biosensor for the detection of interaction between di[azino-di(5,6-azafluorene)-κ-NN′]dichlormanganous and DNA

A new Mn(II) complex of MnL₂Cl₂ (L = azino-di(5,6-azafluorene)-κ²-NN′) was synthesized and utilized as an electrochemical indicator for the determination of hepatitis B virus (HBV) based on its interaction with MnL₂Cl₂. The electrochemical behavior of interaction of MnL₂Cl₂ with salmon sperm DNA was investigated on glassy carbon electrode (GCE). In the presence of salmon sperm DNA, the peak current of [MnL₂]²⁺ was decreased and the peak potential was shifted positively without appearance of new peaks. The binding ratio between [MnL₂]²⁺ and salmon sperm DNA was calculated to be 2:1 and the binding constant was 3.72 × 10⁸ mol² L⁻². The extent of hybridization was evaluated on the basis of the difference between signals of [MnL₂]²⁺ with probe DNA before and after hybridization with complementary sequence. Control experiments performed with non-complementary and mismatch sequence demonstrated the good selectivity of the biosensor. With this approach, a sequence of the HBV could be quantified over the range from 1.76 × 10⁻⁸ to 1.07 × 10⁻⁶ mol L⁻¹, with a linear correlation of r = 0.9904 and a detection limit of 6.80 × 10⁻⁹ mol L⁻¹. Additionally, the binding mechanism was preliminarily discussed. The mode of interaction between MnL₂Cl₂ and DNA was found to be primary intercalation binding.     

A Hamiltonian-based solution to the two-block H problem for general plants in H and rational weights

A complete skew-Toeplitz-type solution to the two-block H^∞ problem for infinite-dimensional stable plants with rational weights is derived with a basis-free proof. The solution consists of one Riccati equation with a rank criterion for a transcendental function of a certain Hamiltonian. This gives a natural extension of the well-known formula for the one-block case. An example is given to illustrate the result.     

An inequality governing nonlinear H∞ control

This note gives necessary and sufficient conditions for solving a reasonable version of the nonlinear H^∞ control problem. The most objectionable hypothesis is elegant and holds in the linear case, but every possibly may not be forced for nonlinear systems. What we discover in distinction to Isidori and Astolfi (1992) and Ball et al. (1993) is that the key formula is not a (nonlinear) Riccati partial differential inequality, but a much more complicated inequality mixing partial derivatives and an approximation theoretic construction called the best approximation operator. This Chebeshev-Riccati inequality when specialized to the linear case gives the famous solution to the H^∞ control problem found in Doyle et al. (1989). While complicated the Chebeshev-Riccati inequality is (modulo a considerable number of hypotheses behind it) a solution to the nonlinear H^∞ control problem. It should serve as a rational basis for discovering new formulas and compromises. We follow the conventions of Ball et al. (1993) and this note adds directly to that paper.     

Achieving nonvanishing stability regions with high-gain cheap control using H techniques: the second-order case

This paper demonstrates how to use an asymptotically H^∞-optimal controller to stabilize a second-order system subject to unknown disturbances such that the stability region does not vanish as the feedback gains increase. The high-gain feedback arises when one attempts to achieve the lowest achievable limit of the disturbance attenuation under the H^∞ design. This type of gain increase can cause the stability region to vanish if the disturbance contains nonlinear terms. The analysis using Lyapunov techniques derives a sufficient condition on the design parameters to prevent the stability region from vanishing. In addition to describing exact solutions for six different cases, the paper provides simulations to illustrate the results.     

An H-minimax approach to the design of robust control systems

A realistic feedback design problem is posed based on the minimization of a weighted combination of the sensitivity and complementary sensitivity matrices. A solution is obtained which makes use of the recently proposed methods for minimizing the sensitivity function alone.