Frequency response algorithms for H∞optimization with time domain constraints

A very broad framework for control system design is considered that encompasses frequency-response methodologies for H_∞ optimization that solve various aspects of the control design problem and that are less well known that state-space methods. The focus is on linear programming, Lawson's algorithm, and Trefethen's algorithm. A modified Lawson's algorithm is proposed and related to Trefethan's method. The modified algorithm is shown to be significantly faster than linear programming and Lawson's algorithm. It is also shown how to extend the modified Lawson's algorithm so as to handle time-domain constraints in addition to frequency-domain specifications, which distinguishes it from other H_∞ optimization methods. Some steps are taken toward dealing with time domain constraints within an H_∞ optimization framework     

Robust stabilization of normalized coprime factor plantdescriptions with H∞-bounded uncertainty

The problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by H _∞ bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system. This problem can be reduced to a Nehari extension problem directly and gives an optimal stability margin. All controllers satisfying a suboptimal stability margin are characterized, and explicit state-space formulas are given     

Direct state-space solution to the discrete-timeH∞ one-block problem

A general state-space representation is used to allow a complete formulation of the H_∞ optimization problem without any invertibility condition on the system matrix, unlike existing solutions. A straightforward approach is used to solve the one-block H_∞ optimization problem. The parameterization of all solutions to the discrete-time H_∞ suboptimal one-block problem is first given in transfer function form in terms of a set of functions in H _∞ that satisfy a norm bound. The parameterization of all solutions is also given as a linear fractional representation     

L2-gain analysis of nonlinear systems andnonlinear state-feedback H∞ control

Previously obtained results on L2-gain analysis of smooth nonlinear systems are unified and extended using an approach based on Hamilton-Jacobi equations and inequalities, and their relation to invariant manifolds of an associated Hamiltonian vector field. On the basis of these results a nonlinear analog is obtained of the simplest part of a state-space approach to linear H_∞ control, namely the state feedback H_∞ optimal control problem. Furthermore, the relation with H_∞ control of the linearized system is dealt with     

One-step extension approach to optimal Hankel-norm approximationand H∞-optimization problems

A methodology is presented for Hankel approximation and H ^∞-optimization problems that is based on a new formulation of a one-step extension problem which is solved by the Sarason interpolation theorem. The parameterization of all optimal Hankel approximants for multivariable systems is given in terms of the eigenvalue decomposition of an Hermitian matrix composed directly from the coefficients of a given transfer function matrix φ. Rather than starting with the state-space realization of φ, the authors use polynomial coefficients of φ as input data. In terms of these data, a natural basis is given for the finite dimensional Sarason model space and all computations involve only manipulations with finite matrices     

The graph topology on nth-order systems is quotientEuclidean

It is shown that on the set of m-input p-output minimal nth-order state-space systems the graph topology and the induced Euclidean quotient topology are identified. The author considers the set L_n^p×m of m -input p-output nth-order minimal state-space systems. The author presents three lemmas and a corollary from which a theorem is proved stating that the graph topology and the quotient Euclidean topology are identical on a quotient space L_n ^p×m/~. Since the graph topology is constructed to be weak, and the quotient Euclidean topology is intuitively strong, this result is unexpected     

Robust H/sub /spl infin// control for uncertain discrete-time-delay fuzzy systems via output feedback controllers

This work investigates the problem of robust output feedback H/sub /spl infin// control for a class of uncertain discrete-time fuzzy systems with time delays. The state-space Takagi-Sugeno fuzzy model with time delays and norm-bounded parameter uncertainties is adopted. The purpose is the design of a full-order fuzzy dynamic output feedback controller which ensures the robust asymptotic stability of the closed-loop system and guarantees an H/sub /spl infin// norm bound constraint on disturbance attenuation for all admissible uncertainties. In terms of linear matrix inequalities (LMIs), a sufficient condition for the solvability of this problem is presented. Explicit expressions of a desired output feedback controller are proposed when the given LMIs are feasible. The effectiveness and the applicability of the proposed design approach are demonstrated by applying this to the problem of robust H/sub /spl infin// control for a class of uncertain nonlinear discrete delay systems.     

A state-space approach to super-optimalH∞ control problems

A numerically stable algorithm is presented for solving the strengthened (or super-optimal) model-matching problem. The steps of the algorithm follows closely those given by N.J. Young (1986) except that for each step a reliable implementation using state-space models is provided     

H∞ optimal control as a weightedWiener-Hopf problem

It is shown that H_∞ optimization is equivalent to weighted H₂ optimization in the sense that the solution of the latter problem also solves the former. The weighting rational matrix that achieves this equivalence is explicitly computed in terms of a state-space realization. The authors do not suggest transforming H_∞ optimization problems to H₂ optimization problems as a computational approach. Rather, their results reveal an interesting connection between H_∞ and H₂ optimization problems which is expected to offer additional insight. For example, H₂ optimal controllers are known to have an optimal observer-full state feedback structure. The result obtained shows that the minimum entropy solution of H_∞ optimal control problems can be obtained as an H₂ optimal solution. Therefore, it can be expected that the corresponding H_∞ optimal controller has an optimal observer-full state feedback structure     

Robust H/sub /spl infin// and mixed H/sub 2//H/sub /spl infin// filters for equalization designs of nonlinear communication systems: fuzzy interpolation approach

Generally, it is difficult to design equalizers for signal reconstruction of nonlinear communication channels with uncertain noises. In this paper, we propose the H/sub /spl infin// and mixed H/sub 2//H/sub /spl infin// filters for equalization/detection of nonlinear channels using fuzzy interpolation and linear matrix inequality (LMI) techniques. First, the signal transmission system is described as a state-space model and the input signal is embedded in the state vector such that the signal reconstruction (estimation) design becomes a nonlinear state estimation problem. Then, the Takagi-Sugeno fuzzy linear model is applied to interpolate the nonlinear channel at different operation points through membership functions. Since the statistics of noises are unknown, the fuzzy H/sub /spl infin// equalizer is proposed to treat the state estimation problem from the worst case (robust) point of view. When the statistics of noises are uncertain but with some nominal (or average) information available, the mixed H/sub 2//H/sub /spl infin// equalizer is employed to take the advantage of both H/sub 2/ optimal performance with nominal statistics of noises and the H/sub /spl infin// robustness performance against the statistical uncertainty of noises. Using the LMI approach, the fuzzy H/sub 2//H/sub /spl infin// equalizer/detector design problem is characterized as an eigenvalue problem (EVP). The EVP can be solved efficiently with convex optimization techniques.     

Model reduction for state-space symmetric systems

In this paper, the model reduction problem for state-space symmetric systems is investigated. First, it is shown that several model reduction methods, such as balanced truncation, balanced truncation which preserves the DC gain, optimal and suboptimal Hankel norm approximations, inherit the state-space symmetric property. Furthermore, for single input and single output (SISO) state-space symmetric systems, we prove that the H_∞ norm of its transfer functions can be calculated via two simple formulas. Moreover, the SISO state-space symmetric systems are equivalent to systems with zeros interlacing the poles (ZIP) under mild conditions.     

A stable state-space realization in the formulation of H∞norm computation

In the two block H^inftyoptimization problem, usually we are given the state-space realizations of the proper rational matricesR_{1}(s)andR_{2}(s)whose poles are all the open right-half plane. Two problems are studied in the note. The first is the evaluation ofphi(s)R_{1}(s)ats = s_{k}, k = 1, 2, ..., n, wherephi(s)is an inner function whose zeros{s_{k}, k = 1, 2, ..., n }are the poles ofR_{1}(s). This evaluation is essential if Chang and Pearson's method is used for computing the optimal H^inftynorm. The problem is solved in state space via the solutions of Lyapunov equations. Neither polynomial matrix manipulations nor numerical pole-zero cancellations are involved in the evaluation. The second problem is to find a stable state-space realization ofS(s) = U(s)R_{2}(s)whereU(s)is an inner matrix. This problem arises in the spectral factorization ofgamma^{2} - R_{2}^{ast}R_{2}. Doyle and Chu had a method for constructing stableS(s)based on a minimal realization ofR_{2}(s). An alternate method is proposed. The alternate method does not require a minimal realization ofR_{2}(s)and only a Lyapunov equation is involved.     

State-space solutions to the ℋ︁∞/LTR design problem

The LTR design problem using an ??∞ optimality criterion is presented for two types of recovery errors, the sensitivity recovery error and the input-output recovery error. For both errors two different approaches are presented. First, following the classical LTR design philosophy, a Luenberger observer based approach is proposed, where the ??∞ part of the controller is appended to a standard full-order observer. Second, allowing for general controllers, an ??∞ state-space problem is formulated directly from the recovery errors. Both approaches lead to controller orders of at most 2n. In the minimum phase case, though, the order of the controllers can be reduced to n in all cases. The control problems corresponding to the various controller types are given as four different singular ??∞ state-space problems, and the solutions are given in terms of the relevant equations and inequalities.     

Two-degree-of-freedom H∞-optimization ofmultivariable feedback systems

A solution to the two-degree-of-freedom H_∞ -minimization problem that arises in the design of multivariable optimal continuous-time stochastic control systems is derived. A decoupling approach that enables a partially independent design of the prefilter and the feedback controller and yields a simple solution to the optimization problem is applied. This solution is obtained by transforming the optimization problem into two standard form (four-block) problems     

A pole-placement design approach for systems with multipleoperating conditions

The author proposes design procedures based on state-space pole-placement techniques for systems with multiple operating conditions. This is the so-called simultaneous pole-placement problem. First, the full state feedback problem is studied, in which a nonlinear local pole-placement solution is proposed. The design condition is formulated in terms of the rank condition of a multimode controllability matrix. Then, the output feedback problem is approached using a multimodel controller design, which is an extension of the observer design to multimode systems. The design is decomposed into separated global pole-placement subproblems and a local pole-placement subproblem. For a system with some operating conditions having modes on the j ω-axis, but no modes at the origin in the open right-half of the complex plane, stabilizability and detectability conditions for the design of an asymptotically stabilizing control are established, without any restriction on the number of inputs or outputs. Relations of this approach to other simultaneous control design approaches are pointed out     

Self-tuning filters and predictors for two-dimensional systems Part 1: Algorithms

The filtering of two-dimensional (2-D) signals is treated using a self-tuning technique based on a truncated innovations model of the data. The resultant algorithms offer two key advantages over their fixed-coefficient counterparts. First, the self-tuning filters quickly and automatically set their own coefficients, thus avoiding the normal off-line design cycle. Secondly, self-tuning filters can function in an adaptive manner, such that the filter retunes to track time variations in the two-dimensional data. The self-tuning algorithms are formulated in terms of input/output models and thus complement the more usual state-space approach to the 2-D filtering problem.     

Rational matrix GCDs and the design of squaring-down compensators-astate-space theory

A state-space construction for rational matrix greatest common divisors (GCDs) of rational transfer matrices is given. It is shown how the GCD results can be used to solve the problem of designing stable minimum-phase squaring-down compensators for multi-variable plants. One application is a direct state-space construction for such compensators and a state-space solution to `fat-plant' H-infinity control problems. The results make use of the concepts of strongly observable systems and maximally observable systems and build upon the concepts introduced in the state-space GCD extraction results for polynomial matrices of L.M. Silverman and P. Van Dooren (1979)     

Linear state-space systems in infinite-dimensional space: the roleand characterization of joint stabilizability/detectability

Several fundamental results from the theory of linear state-space systems in finite-dimensional space are extended to encompass a class of linear state-space systems in infinite-dimensional space. The results treated are those pertaining to the relationship between input-output and internal stability, the problem of dynamic output feedback stabilization, and the concept of joint stabilizability/detectability. A complete structural characterization of jointly stabilizable/detectable systems is obtained. The generalized theory applies to a large class of linear state-space systems, assuming only that: (i) the evolution of the state is governed by a strongly continuous semigroup of bounded linear operators; (ii) the state space is Hilbert space; (iii) the input and output spaces are finite-dimensional; and (iv) the sensing and control operators are bounded. General conclusions regarding the fundamental structure of control-theoretic problems in infinite-dimensional space can be drawn from these results     

A practical two-dimensional Thiele-type matrix Pade/spl acute/ approximation

In view of several potential applications in multivariable two-dimensional (2-D) systems theory, a practical 2-D matrix Pade/spl acute/ approximation is introduced by using a generalized inverse of the matrices. The approximants are expressed in the form of the 2-D Thiele-type continued fractions and are computed by an efficient recursive algorithm. As it's an application, the state-space realization problem of the 2-D filters is discussed.     

LQG control with an H∞ performance bound:a Riccati equation approach

An LQG (linear quadratic Gaussian) control-design problem involving a constraint on H^∞ disturbance attenuation is considered. The H^∞ performance constraint is embedded within the optimization process by replacing the covariance Lyapunov equation by a Riccati equation whose solution leads to an upper bound on L₂ performance. In contrast to the pair of separated Riccati equations of standard LQG theory, the H^∞-constrained gains are given by a coupled system of three modified Riccati equations. The coupling illustrates the breakdown of the separation principle for the H^∞ -constrained problem. Both full- and reduced-order design problems are considered with an H^∞ attenuation constraint involving both state and control variables. An algorithm is developed for the full-order design problem and illustrative numerical results are given