F.J. Kraus

Iterative controller optimization for nonlinear systems

Recently, a data-driven model-free control design method has been proposed in Hjalmarsson et al. (Proceedings of the Conference on Decision and Control, Orlando, FL, 1994, pp. 1735–1740; IEEE Control Systems Mag. 18 (1998) 26) for linear systems. It is based on the minimization of a control criterion with respect to the controller parameters using an iterative gradient technique. In this paper, we extend this method to the case where both the plant and the controller can be nonlinear. It is shown that an estimate of the gradient of the control criterion can be constructed using only signal-based information obtained from closed-loop experiments. The obtained estimate contains a bias which depends on the local nonlinearity of the noise description of the closed-loop system which can be expected to be small in many practical situations. As a side effect the linear model-free control design method is reobtained in a new way.

Robust stability of polynomials with multilinear parameter dependence

Abstract The problem is studied of testing for stability a class of real polynomials in which the coefficients depend on a number of variable parameters in a multilinear way. We show that the testing for real unstable roots can be achieved by examining the stability of a finite number of corner polynomials (obtained by setting parameters at their extreme values), while checking for unstable complex roots normally involves examining the real solutions of up to m + 1 simultaneous polynomial equations, where m is the number of parameters. When m = 2, this is an especially simple task.

Robust stability of control systems with polytopical uncertainty: a Nyquist approach

For technical plants, a bounded continuum of parameter values describes admissible process models. A robust controller guarantees the observance of minimal requirements for all of the allowed process models. In this paper a frequency domain approach is proposed that permits the instability of a family of polynomials to be checked. This method can be extended in an easy manner to investigate the Instability of a closed loop. The plant is thereby described by a family of models with a polytopical region of uncertainty in the parameter space. As an important special case of the general D-stability, the Hurwitz stability of the closed loop is investigated and a minimal set of necessary and sufficient conditions for robust Hurwitz stability is given. A generalization for other stability domains is possible. Besides an algebraic criterion based on a zero inclusion check for a minimal set of exposed edges of the polytopical region, a graphic criterion of Nyquist type can also be used.

Stabilized Least-Squares Estimators for Time-Variant Processes

Abstract The paper presents the analysis of a class of modified recursive least-squares algorithms (RLS-SF) devoted to track parameters of time-varying plants in the presence of poor excitation. The algorithms are stabilized in the sense of bounding the covariance matrix from above and below and thus avoiding blow-up. They combine linear-forgetting recursive least-squares algorithm with exponential covariance matrix stabilization. The numerical burden imposed by the modification is of O(n) and thus can be neglected. Under the assumption that the data are generated by a deterministic linear time-invariant system the RLS-SI algorithm is exponentially convergent for persistently exciting signals. For not persistently exciting signals the normalized prediction errors and estimate changes are square summable and the estimates are bounded. Asymptotic behavior of the covariance matrix is considered for not persistently exciting signals: the eigenvectors of the matrix can be asymptotically partitioned into two groups, where the first group spans entirely the excited subspace and the second one is orthogonal to the excited subspace. The good perfonnance of the algorithm is verifyed via simulation studies.

FIFO Stable Control Systems

Abstract Motivated by the dead-beat and shortest-correlation control strategies, the paper introduces the notion of finite-input finite-output (FIFO) stability for linear discrete-time control systems. A simple parametrization of all controllers that FIFO-stabilize a given plant is obtained. The best controller is then chosen for various applications, including H 2 optimization, disturbance rejection and transfer function shaping

Identification and Control of a Servo System

Abstract Identification for control is demonstrated on a small servo system. It is shown how to go through the entire cycle of modeling and identification with the aim of obtaining models suitable for controller design. Controllers are designed and implemented in the real time from the identified models.

Computation of Value Sets of Uncertain Transfer Functions

The determination of value sets of uncertain polynomials or transfer functions plays a major role for analysis and design of robust control systems. A method for computation of such value sets is presented for a special case of decomposable systems where each uncertain part is described by a simple value set — an axis parallel box.

Use of Hypersurfaces for Fault Detection, Isolation and Reconstruction

Abstract Hypersurface models are useful means for process monitoring for they handle data redundancy in a natural way. Within a unified framework the paper discusses application of such (linear and nonlinear static) models for data smoothing, fault detection, isolation, and reconstruction. In the linear case the approach is known as principal component analysis (PCA), in the nonlinear case neural networks can be utilized to construct the models.

On Duality of Exponential and Linear Forgetting

Abstract Regularized (stabilized) versions of exponential and linear forgetting in parameter tracking are shown to be dual to each other. Both are derived by solving essentially the same Bayesian decision-making problem where Kullback-Leibler divergence is used to measure (quasi)distance between posterior probability distributions of estimated parameters. The type of forgetting depends solely on the order of arguments in Kullback-Leibler divergence. This general view indicates under which conditions one technique is superior to the other. Applied to the case of ARX models, the approach results in a class of regularized (stabilized) forgetting strategies that are naturally robust with respect to poor system excitation.

Robust stability of control systems: extreme point results for the stability of edges

For the investigation of the robust stability of control systems with structured uncertainties many results have been presented recently that lead to stability tests of the edges of a polytope. In this paper some results are discussed where the stability of an edge is guaranteed by the stability of the vertices of the edge. Given the characteristic polynomial of a closed loop system with structured uncertainties in the parameters; for special types of uncertainties the family of polynomials is defined by a polytope in the space of parameters. In the frequency domain the corresponding value set of the family, for s = jw fixed, is a polygon whose edges are formed by the so-called exposed edges of the polytope. For the investigation of the stability of the polynomial family these exposed edges must be tested (Kraus and Truol 1991 a). This test can be simplified if an edge has the vertex property, i.e. if the stability of the edge is guaranteed by the stability of the two vertices.