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).     

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.     

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.     

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.     

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.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.     

On improving control-loop robustness of model-matching controllers

This note first characterizes the class of all stabilizing controllers for a two-degree-of-Freedom control system which achieve a prescribed achievable transfer function. The characterization is in terms of an arbitrary proper stable transfer function. With this characterization, robust model matching is formulated as a standard H^∞-optimization problem. This means that standard controller designs for a nominal plant, such as linear-quadratic Gaussian ones, can be enhanced to give improved robustness properties using H^∞-design techniques.     

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.     

On the weighted sensitivity minimization problem for delay systems

We consider the H^∞-optimal sensitivity problem for delay systems. In particular, we consider computation of μ:= inf {|W-φq|_∞ : q ε H^∞(j )} where W(s) is any function in RH^∞(j ), and φ in H^∞(j ) is any inner function. We derive a new explicit solution in the pure delay case where φ = e^−sh, h > 0.     

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∞ control design in bilinear systems: a tensor formal series approach

In this paper, the H_∞ disturbance attenuation problem of bilinear system is discussed. Dynamic game theory is used to solve this bilinear minimax problem. The solvability of H_∞ disturbance attenuation in bilinear system is also discussed. The techniques of tensor products and formal power series are employed to solve the nonlinear Bellman-Isaac differential equation. Furthermore, the convergence for the tensor formal series approach of this H_∞ control problem is discussed, and the radius of convergence for this H_∞ control to be well defined is also obtained.     

H∞ filtering for a class of uncertain nonlinear systems

This paper investigates the problem of H_∞ filtering for a class of uncertain continuous-time nonlinear systems with real time-varying parameter uncertainty and unknown initial state. We develop an infinite horizon H_∞ filtering methodology which provides both robust stability and a guaranteed H_∞ performance for the filtering error irrespective of the parameter uncertainty.     

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.     

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.     

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.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

On decoupling the H-optimal sensitivity problem for products of plants

In this note we show how to solve the H^∞-optimal sensitivity problem for a SISO plant P(s) = P₁(s)P₂(s), given the solutions for P₁(s), P₂(s). This allows us to solve the problem for systems of the form e^−hsP₀(s), where P₀(s) is the transfer function of a stable, LTI, finite dimensional system.     

Nonlinear robust H∞ control via dynamic output feedback

The problem on robust H_∞ control for a class of nonlinear systems with parameter uncertainty is studied. Sufficient conditions for the existence of the dynamic output feedback controller are obtained. Under these conditions, the closed-loop systems have robust H_∞-performance. A numerical example is given to illustrate the design of a robust controller using the proposed approach.     

Robust H∞ observer design of linear time-delay systems with parametric uncertainty

This paper deals with the problem of H_∞ observer design for a class of uncertain linear systems with delayed state and parameter uncertainties. This problem aims at designing the linear state observers such that, for all admissible parameter uncertainties, the observation process remains robustly stable and the transfer function from exogenous disturbances to error state outputs meets the prespecified H_∞ norm upper bound constraint, independently of the time delay. The time delay is assumed to be unknown, and the parameter uncertainties are allowed to be norm-bounded and appear in all the matrices of the state-space model. An effective matrix inequality methodology is developed to solve the proposed problem. We derive the conditions for the existence of the desired robust H_∞ observers, and then characterize the analytical expression of these observers in terms of some free parameters. A numerical example demonstrates the validity and applicability of the present approach.     

Sample complexity of worst-case H-identification

We show that the sample complexity of qorst-case H^∞-identification is of order n², by proving that the minimal length of a fractional H^∞-cover for C_n, regarded as the linear space of complex-valued sequences of length n, is of order n². A unit vector u in ^∞ is a fractional H^∞-cover for C_n if for some

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for all rh ε C_n, where is the z-transform of h. We also give similar results for real-valued sequences.