D. Grant Fisher

Continuous sliding mode control

Classical sliding mode control (SMC) uses a discontinuous control action to drive the state from an arbitrary initial state to the origin along a user-specified path and exhibits excellent robustness to disturbances and parameter uncertainty. However, the control chattering due to the discontinuity in the control law is undesirable in most processes applications. The continuous sliding mode control (CSMC) approach developed in this paper satisfies the sliding condition using a continuous control law. It therefore retains the positive properties of SMC but without the disadvantage of control chattering. The concept of boundary layer equivalence is used to show that in the presence of unknown disturbances and/or parameter uncertainty, CSMC keeps the state trajectories within a boundary layer of user-specified width. It is also shown that CSMC is equivalent to a cubic feedback control law and can be reduced to a linear form (LSMC) which provides a useful link between sliding mode control and traditional line...

Solution algorithms for fuzzy relational equations with max-product composition

The conditions for the existence of an inverse solution to the max-min composition of fuzzy relational equations have been well documented since the original work by Sanchez [30, 31]. These same existence theorems have been extended to the t-norm composition of relational equations, in which the max-product composition is a member [5,13,26]. Several studies [8,15, 24, 33, 34, 38] have shown that the max-min operator may not always be the most desirable fuzzy relational composition and in fact the max-product operator was superior in these instances. This paper reviews the algorithms necessary to determine the complete solution of the inverse for fuzzy relational equations with max-product composition.

Improved least squares identification

An improved recursive least squares algorithm for parameter estimation is presented which includes: on/off criteria to prevent parameter drift during periods of low excitation; a variable forgetting factor which maintains the trace of the covariance matrix at a user-specified value; data preprocessing and normalization to improve numerical accuracy; scaling of the regressor vector to minimize the condition number of the covariance matrix; plus independent estimation of the mean values of the input/output data which can be used to eliminate errors due to d.c. bias or slowly drifting elements in the regressor vector. The algorithm can also include parameter projection to constrain the estimates to a priori specified regions and retains the formal properties, such as convergence, of a true weighted least squares algorithm. The proposed algorithm is compared with other modifications suggested in the literature, and its advantages are demonstrated by a simulated example.

A stable adaptive predictive control system

An adaptive, predictive control system (APCS) is described and a formal proof of global stability is presented for the case of multi variable, stable – in verse, time – invariant processes in the presence of bounded, unmeasured disturbances plus process and measurement noise. Global stability is proven in the sense that the process input / output vector remains bounded and the control error is minimized. The proof of stability is based on certain properties of the aposteriori estimation error and convergence properties of the estimated parameters. Reference is also made to successful experimental and simulation applications of APCS.

An augmented UD identification algorithm

An augmented UD identification (AUDI) algorithm for system identification is developed by rearranging the data vectors and augmenting the covariance matrix of Biermans UD factorization algorithm. The structure of the augmented information (covariance) matrix is particularly easy to interpret and it is shown that the AUDI algorithm is a direct extension of the familiar recursive least squares (RLS) algorithm. The proposed algorithm permits simultaneous identification of the model parameters plus loss functions for all orders from 1 to n at each time step with approximately the same calculation effort as «th order RLS. This provides a basis for simultaneous model order and parameter identification so that problems due to over- and under-estimation of model can be avoided. Based on its least-squares properties, numerical robustness, theoretical basis and the fact that it simultaneously estimates multiple models, the proposed AUDI algorithm is recommended for use in place of RLS and Biermans UD factorizatio...

Convergence, eigen fuzzy sets and stability analysis of relational matrices

Abstract If stable control applications are formulated with a max-product composition, then instability in the relational matrix under any conditions is undesirable. This paper reviews the stability analysis of relational matrices combined with the max-min composition [19] and then presents an analysis of the stability of relational matrices combined with the max-product composition. This analysis includes results defining the convergence properties of the relational matrix, determination of the eigen fuzzy sets of the stable matrices and algorithmic solutions for several important cases. A method is also presented to stabilize unstable relational matrices for some control applications.

Multirate constrained adaptive control

An adaptive servo control law with input constraints is derived for application to linear time-invariant systems with unknown parameters and two sampling rates: a slower one for the output, and a faster one for the input. The error between the actual and a reference performance index is shown to be bounded by the product of a finite gain and the parameter estimation error in the limit sense. Sufficient conditions for parameter convergence are proven under which the performance of the multirate, adaptive constrained control system asymptotically approaches that of the analogous single (fast) rate, constrained control system with known, constant parameters. The advantages of the multirate, adaptive constrained control algorithm are demonstrated by numerical simulations.

Identification algorithms for fuzzy relational matrices, part 2: optimizing algorithms

Abstract This paper is the first of a two part series that reviews and critiques several identification algorithms for fuzzy relational matrices. Part 1 reviews and evaluates algorithms that do not optimize or minimize a specified performance criteria [3,9,20,24]. It compliments and extends a recent comparative identification analysis by Postlethwaite [17]. Part 2 [1] evaluates algorithms that optimize or minimize a specified performance criteria [6,8,23,26]. The relational matrix, learned by each algorithm from the Box–Jenkins gas furnace data [2], is compared for effectiveness of the prediction based on a minimum distance from actual. A new, non-optimized identification algorithm with an on-line formulation that guarantees the completeness of the relational matrix, if sufficient learning has taken place, is also presented. Results show that the proposed new algorithm ranks as the best among the non-optimized algorithms with prediction results very close to the optimization methods of Part 2.

Multiple Prediction Models for Long Range Predictive Control

Abstract A new multi-step prediction formulation is developed and used to generate the long range predictions of the future process outputs required for predictive control. The basic idea behind this new approach is to simultaneously and directly construct a separate j-step prediction model for each future output y ( k + j ) where j = 1, 2,…., N , and N is the prediction horizon. This is different from the conventional approach, which constructs only the one-step prediction model and calculates the N multi-step output predictions either by repeated use of the one-step prediction model or by use of the Diophantine equation. A simulated example is given which shows that the extra computation inherent in the proposed approach is justified by much better predictions than the conventional approach. The proposed multiple model prediction approach is then combined with a multiple model control technique to create a long range predictive controller. Results from an experimental application of this control strategy to a 2 × 2 pilot-scale level process demonstrate the excellent control that can be obtained on a real process.

An experimental evaluation of state estimation in multivariable control systems

This paper describes the application of two state estimation techniques, Kalman filters and Luenberger observers, to a computer-controlled pilot plant evaporator. The estimation techniques are used to provide state estimates for an optimal feedback control system and are evaluated using both simulated and experimental data.