H∞-optimal control with state-feedback

A H_∞-optimal control problem in which the measured outputs are the states of the plant is considered. The main result shows that the infimum of the norm of the closed-loop transfer function using linear static state-feedback equals the infimum of the norm of the closed-loop transfer function over all stabilizing dynamic (even, nonlinear time-varying) state-feedback controllers     

Mixed H2/H∞ control:a convex optimization approach

The problem of finding an internally stabilizing controller that minimizes a mixed H₂/H_∞ performance measure subject to an inequality constraint on the H_∞ norm of another closed-loop transfer function is considered. This problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. Both the state-feedback and output-feedback problems are considered. It is shown that in the state-feedback case one can come arbitrarily close to the optimal (even over full information controllers) mixed H₂/H_∞ performance measure using constant gain state feedback. Moreover, the state-feedback problem can be converted into a convex optimization problem over a bounded subset of (n×n and n ×q, where n and q are, respectively, the state and input dimensions) real matrices. Using the central H_∞ estimator, it is shown that the output feedback problem can be reduced to a state-feedback problem. In this case, the dimension of the resulting controller does not exceed the dimension of the generalized plant     

Necessary and sufficient conditions for mixedH2 and H∞ optimal control

It is shown that D.S. Bernstein and W.M. Hadad's (ibid., vol.34, no.3, p.293, 1989) necessary condition for full-order mixed H ₂ and H_∞ optimal control is also sufficient, and that J.C. Doyle et al.'s (Proc. Amer. Control Conf., p.2065, 1989) sufficient condition for full-order mixed H₂ and H_∞ optimal control is also necessary. They are duals of one another     

H∞-control by state-feedback and fastalgorithms for the computation of optimalH∞-norms

The suboptimality of some parameter for H_∞ -optimization by dynamic state-feedback is characterized in terms of the solvability of Riccati inequalities. This is done without restricting the finite zero structure of the plant. If there are no system zeros on the imaginary axis, the H_∞-problem can be treated in a complete and satisfactory way. Explicit characterizations optimum to be achieved are provided, and a closed formula for the optimal value is derived in terms of the H_∞-norm of some fixed transfer matrix. If the optimum is not attained, any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high-gain. A globally and quadratically convergent algorithm to compute the optimal value is proposed. This algorithm is generalized to the H_∞-optimization problem by measurement feedback     

H2-optimal sampled-data control

The H₂-optimal control of continuous-time linear time-invariant systems by sampled-data controllers is discussed. Two different solutions, state space and operator theoretic, are given. In both cases, the H₂ sampled-data problem is shown to be equivalent to a certain discrete-time H₂ problem. Other topics discussed include input-output stability of sampled-data systems, performance recovery in digital implementation of analog controllers, and sampled-data control of systems with the possibility of multiple-time delays     

Correction of the FI result in H∞ controland parameterization of H∞ state feedbackcontrollers

The authors correct the parameterization of the H_∞ controller of the full-information (FI) problem derived by J.C. Doyle et al. (1989). Then they parameterize the H_m0 state feedback controller and explain how dynamical free parameters implied in it are related to constant feedback gains different from the central solution F_∞     

L1 analysis and design of sampled-data systems

The focus of this work is L₁-optimal control of sampled-data systems. A converging approximation procedure is derived to compute the L_∞-induced norm of closed-loop finite-dimensional linear time-invariant (LTI) sampled-data systems. An approximation method is developed to synthesize L₁-optimal sampled-data regulators. Finally, an example is provided that illustrates the L₁ analysis and design techniques presented     

H∞ control and filtering for sampled-datasystems

H_∞ control and filtering problems for sampled-data systems are studied. Necessary and sufficient conditions are obtained for the existence of controllers and filters that satisfy a specified H_∞ performance bound. When these conditions hold, explicit formulas for a controller and a filter satisfying the H_∞ performance bound are also given     

Robust H∞ control for linear systems withnorm-bounded time-varying uncertainty

Robust H_∞ control design for linear systems with uncertainty in both the state and input matrices is treated. A state feedback control design which stabilizes the plant and guarantees an H_∞-norm bound constraint on disturbance attenuation for all admissible uncertainties is presented. The robust H_∞ control problem is solved via the notion of quadratic stabilization with an H_∞ -norm bound. Necessary and sufficient conditions for quadratic stabilization with an H_∞-norm bound are derived. The results can be regarded as extensions of existing results on H_∞ control and robust stabilization of uncertain linear systems     

H∞ analysis and synthesis ofdiscrete-time systems with time-varying uncertainty

The problems of H_∞ analysis and synthesis of discrete-time systems with block-diagonal real time-varying uncertainty are considered. It is shown that these problems can be converted into scaled H_∞ analysis and synthesis problems. The problems of quadratic stability analysis and quadratic stabilization of these types of systems are dealt with as a special case. The results on synthesis are established for general linear dynamic output feedback control     

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     

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     

The robust H2 control problem: a worst-casedesign

The worst-case effect of a disturbance system on the H ₂ norm of the system is analyzed. An explicit expression is given for the worst-case H₂ norm when the disturbance system is allowed to vary over all nonlinear, time-varying and possibly noncausal systems with bounded L₂-induced operator norm. An upper bound for this measure, which is equal to the worst-case H₂ norm if the exogeneous input is scalar, is defined. Some further analysis of this upper bound is done, and a method to design controllers which minimize this upper bound over all robustly stabilizing controllers is given. The latter is done by relating this upper bound to a parameterized version of the auxiliary cost function studied in the literature     

On the parameterization H∞ controllers

The author clarifies some of the results of J.C. Doyle et al. (ibid., vol.34, no.8, p.831-47, Aug. 1989) and gives some new interpretations. In particular, the author parameterizes all suboptimal H_∞ controllers for the full information (FI) and state feedback control problems and indicates when this FI H_∞ control problem can or cannot be given a differential game saddle point interpretation     

Optimal L∞ bounds for disturbancerobustness

Robust disturbance rejection bounds obtained recently by Zhu et al. are tight with respect to the disturbance set that has a specified L₂ norm bound. It is shown that the bounds are also tight with respect to a weighted L₂ disturbance set that has a specified bound on its outer product by choosing a special weight matrix in the L₂ norm. The matrix can computed by numerical iteration     

Disturbance attenuation and H∞-controlvia measurement feedback in nonlinear systems

A solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system. It is shown that the concept of disturbance attenuation, in the sense of truncated L₂ norms, can be given an interpretation in terms of the response to periodic inputs in the sense of RMS amplitude, even in the nonlinear setup. In the case of state feedback, a family of controllers is also provided. The proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem     

Rational approximation of L1 optimalcontrollers for SISO systems

The L₁ optimal control problem with rational controllers for continuous-time systems is considered in which it is shown that the optimal L₁ performance index with rational controllers is equal to that of irrational controllers. A sequence of rational controllers that approximates the optimal index is constructed. Convergence properties of such a sequence are studied. That the corresponding sequence of objective transfer functions is shown to converge in weak-* topology in BV(R₊) in the time domain and uniformly in a wider sense in the frequency domain     

Controller reduction by H∞-balancedtruncation

H_∞-balanced truncation may be used to obtain reduced-order plants or controllers. The plant (possibly unstable) is compensated using a particular robustly stabilizing controller. The two Riccati equations involved are then used to define a set of closed-loop input-output invariants called the H_∞-characteristic values. That part of the plant or controller corresponding to small H_∞-characteristic values is discarded to give a reduced-order plant or controller. By exploiting an intimate connection with coprime factorization, a simple a priori test is derived for the ability of such a reduced-order controller to stabilize the full-order plant. The performance of the resulting closed-loop may also be bounded a priori, i.e. in terms of the prespecified level of robustness and the discarded H_∞-characteristic values     

The L2-polynomial spline pyramid

The authors are concerned with the derivation of general methods for the L₂ approximation of signals by polynomial splines. The main result is that the expansion coefficients of the approximation are obtained by linear filtering and sampling. The authors apply those results to construct a L₂ polynomial spline pyramid that is a parametric multiresolution representation of a signal. This hierarchical data structure is generated by repeated application of a REDUCE function (prefilter and down-sampler). A complementary EXPAND function (up-sampler and post-filter) allows a finer resolution mapping of any coarser level of the pyramid. Four equivalent representations of this pyramid are considered, and the corresponding REDUCE and EXPAND filters are determined explicitly for polynomial splines of any order n (odd). Some image processing examples are presented. It is demonstrated that the performance of the Laplacian pyramid can be improved significantly by using a modified EXPAND function associated with the dual representation of a cubic spline pyramid     

Linear and nonlinear algorithms for identification in H∞ with error bounds

A linear algorithm and a nonlinear algorithm for the problem of system identification in H_∞ posed by Helmicki et al. (1990) for discrete-time systems are presented. The authors derive some error bounds for the linear algorithm which indicate that it is not robustly convergent. However, the worst-case identification error is shown to grow as log(n), where n is the model order. A robustly convergent nonlinear algorithm is derived, and bounds on the worst-case identification error (in the H_∞ norm) are obtained