William Paul Heath

Orthogonal functions for cross-directional control of web forming processes

The cross-directional behaviour of web forming processes can be represented in terms of orthogonal basis functions. By choosing a suitable basis set, it is possible to separate controllable and uncontrollable components of the profile into low- and high-order spectral components and hence obtain a parsimonious representation. In most cases if there are sufficient actuators to control all low-order spectral components then the interaction matrix is likely to be ill-conditioned, which necessitates the incorporation of input constraints into the control design. The representation in terms of orthogonal functions allows the design of finite-horizon predictive controllers with hard inequality constraints on the input that are computationally feasible for commercial machines. It is also possible to exploit such a representation to construct efficient estimators for the process.

Barrier function based model predictive control

A new formulation of nonlinear model predictive control (MPC) is developed by including a weighted barrier function in the control objective. While the barrier ensures that inequality constraints are strictly satisfied it also provides a smooth transition between points in the interior and those on the boundary of the constraint set. In addition, the resulting optimisation problem, to be solved at each control step, is effectively unconstrained and thus amenable to elegant optimisation techniques. The barrier must satisfy certain conditions in order that the state converges to the origin and we show how to construct such a barrier. Conventional MPC may be seen as a limiting case of the new class as the barrier weighting itself approaches zero. We pay particular attention to the novel approach of fixing the weighting parameter to some positive value-possibly large-and observe that this provides a degree of controller caution near constraint boundaries. We construct an ellipsoidal invariant set by exploiting the geometry of self-concordant functions and show nominal closed-loop stability for this class of controllers under full state feedback.

Zames-Falb Multipliers for Quadratic Programming

In constrained linear model predictive control, a quadratic program must be solved on-line at each control step, and this constitutes a nonlinearity. If zero is a feasible point for this quadratic program then the resultant nonlinearity is sector bounded. We show that if the nonlinearity is static then it is also monotone and slope restricted; hence, we show the existence of Zames-Falb multipliers for such a nonlinearity. The multipliers may be used in a general and versatile analysis of the robust stability of input constrained model predictive control.

Second-order counterexamples to the discrete-time Kalman conjecture

The Kalman conjecture is known to be true for third-order continuous-time systems. We show that it is false in general for second-order discrete-time systems by construction of counterexamples with stable periodic solutions. We discuss a class of second-order discrete-time systems for which it is true provided the nonlinearity is odd, but false in general. This has strong implications for the analysis of saturated systems.

Identification of cross-directional behaviour in web production: Techniques and experience

Abstract The paper concerns the problem of parameter estimation in web manufacturing processes such as paper making, plastics extrusion and sheet coating processes. The key feature of such processes is that the manufactured web has width and length. In order to control or monitor the process, it is necessary to be able to identify the behaviour of the web width (cross-direction, in short CD) as well as the web length (machine-direction, in short MD). The aim of the paper is: (1) to review the identification techniques, (2) present a new method based upon basis function sets, (3) to illustrate typical results using industrial data.

Comments on “On the Existence of Stable, Causal Multipliers for Systems With Slope-Restricted Nonlinearities”

The above technical note presents a novel convex search within the Zames-Falb multiplier class. The aim of this correspondence is to correct the misuse of a relaxation on one condition in the search. This relaxation leads to nontrivial numerical errors in all six examples discussed in the technical note. Correct application of the conditions still gives an improvement over absolute stability criteria in the literature for at least one example, but some of the claims for lack of conservativeness in the above technical note should be moderated.

Bias of indirect non-parametric transfer function estimates for plants in closed loop

Expressions for the bias of indirect non-parametric estimates in closed loop are derived in terms of the experimental conditions. Application of the results to finite data experiments using both spectral estimates and empirical transfer function estimates are considered and illustrated with simulations.

Zames-Falb multipliers for quadratic programming

In constrained linear model predictive control a quadratic program must be solved on-line at each control step. If zero is feasible the resultant static nonlinearity is sector bound. We show that the nonlinearity is also monotone nondecreasing and slope restricted; furthermore it may be expressed as the gradient of a convex potential function. Hence we show the existence of Zames-Falb multipliers for such a nonlinearity. For completeness, we construct such multipliers both for the general case of multi-input multi-output static nonlinearities and for the particular case where the nonlinearity arises from a quadratic program. We also express the results in terms of integral quadratic constraints. These multipliers may be used in a general and versatile analysis of the robust stability of constrained model predictive control.

Self-tuning prediction and control for two-dimensional processes Part 1: Fixed parameter algorithms

Least-squares optimal prediction, minimum variance control and generalized minimum variance control algorithms for a two-dimensional CARMA process are developed. Each algorithm involves the algebraic solution of a two-dimensional diophantine equation, and may be embedded within ‘classical’ two-dimensional systems theory. We show how the algorithms must be modified for any practical implementation to take into account the edges of the data field. In this case we show how we may analyse the process using multivariable theory, and explore the linkages between multivariable representations and two-dimensional systems.

Self-tuning prediction and control for two-dimensional processes. Part 2: parameter estimation, set-point tracking and offset handling

We consider some of the issues of implementing the prediction and control algorithms of the companion paper (Heath and Weltstead 1994). The algorithms are suitable for setting in self-tuning mode by combining them with recursive parameter estimation algorithms. The problems of set-point tracking and offset handling are also considered. It is shown that if the two-dimensional process can be modelled as incremental, then the solution is particularly staightforward. Throughout, the results are illustrated by simulations.