Hannu T. Toivonen

Sampled-data control of continuous-time systems with an H ∞ optimality criterion

Abstract The synthesis of sampled-data controllers for continuous-time systems is studied in the context of H∞ control theory. The sampled-data H∞ optimal control problem consists of finding a sampled-data controller which minimizes the L2-induced norm from the continuous-time disturbance to the continuous-time output. The problem is here solved via a discrete system representation. This is achieved by describing the closed-loop system as a discrete system with finite state dimension, and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite-dimensional spaces. Via finite-rank approximations of a compact integral operator associated with the system, the sampled-data H∞ control problem can be solved to any degree of accuracy. From the structure of the problem it follows that irrespective of the order of the approximation, a feasible sampled-data controller can be found which has the same order as the system.

Paper: Constrained linear quadratic gaussian control with process applications

Constrained linear quadratic Gaussian control is considered. Important practical design constraints including restrictions in control signal variations and in regulator structure are introduced quantitatively into the control problem formulation. Various topics in the resulting extension of the standard LQG design procedure are discussed, for instance optimality conditions, design of optimal low-order controllers and variance-constrained self-tuning control. Numerical algorithms for solving the constrained LQG control problems are given facilitating the application of the design procedure. Three industrial applications of linear quadratic Gaussian design are described. The examples are taken from the cement industry and from a process for the production of plastic film.

Computer-aided design procedure for multiobjective LQG control problems

Abstract A procedure for computing a satisfactory non-inferior solution to a multiobjective LQG control problem is given. An efficient computational algorithm is developed by exploiting the structure of the LQG problem. The proposed multiobjective optimization procedure is based on a sequence of constrained min-max control problems, which are efficiently solved by applying Newtons method to an associated dual maximization problem.

Approximation of delay systems—a case study

Abstract The approximation of delay systems is studied. A numerically well-behaved method for computing Hankel optima) rational approximations for delay systems is presented based on certain properties of Hankel matrices and the theory of Laguerre polynomials. The CF method of Trefethen and certain Pade approximations of delay systems are also considered. The importance of a certain Wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the L∞ norm and the Hankel norm is shown, completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions. A case study and numerical comparison is presented for the approximation of the parametric family of first-order stable delay systems G(s) = exp ( —τs)/(Ts + 1). Numerical experience indicates that L°° optimized CF approximations based on short truncated Maclaurin series give, in general, somewhat smaller L∞ approximation errors than Hankel optim...

NEURAL NETWORK APPROXIMATION OF A NONLINEAR MODEL PREDICTIVE CONTROLLER APPLIED TO A PH NEUTRALIZATION PROCESS

Abstract Model predictive control of nonlinear sampled-data systems is studied, with a particular focus on computational efficiency. In order to reduce the computational requirements associated with the solution of the continuous-time nonlinear system equations, the process is modelled by a set of linear models constructed by velocity-based linearization. The resulting quasi-linear models also simplify the estimation of the system state from the measured outputs. The on-line computational burden associated with the controller calculation is reduced by using a neural network function approximator to approximate the optimal model predictive control strategy. The accuracy of the neural network controller approximation which is required to ensure stability and performance is shown to be related to the fragility of the model predictive controller, which can be characterized in terms of an l 2 -induced norm defined for the closed-loop system. The proposed approach is applied to a simulated nonlinear pH neutralization process.

Technical communique: A descent anderson-moore algorithm for optimal decentralized control

A convergent version of the Anderson-Moore algorithm for the optimal output feedback problem is applied to a class of optimal decentralized control problems. The algorithm is based on a positive definite approximation of the second-order Taylor-series expansion of the loss function. Numerical results show that the algorithm performs favourably when compared with the method of Geromel and Bernussou (1979) (Automatica, 15, 489) and with variable metric function minimization techniques.

Brief paper: Variance constrained self-tuning control

A self-tuning regulator for a variance constrained optimal control problem is given. The criterion for control is to minimize the stationary variance of the output. In the cases when the regulator which gives minimum variance requires too large control signals an inequality constraint on the input variance is introduced. In practice it is easier to select a constraint on the variance of the input than to choose the relative weights in a quadratic loss function. The self-tuning regulator applies the Robbins-Monro scheme to adjust the Lagrange multiplier of the variance constrained control problem. The behaviour of the algorithm is illustrated by a simulated example. The asymptotic behaviour of the regulator is studied using a set of associated ordinary differential equations.

State-space solution to the periodic multirate H/sub /spl infin// control problem: a lifting approach

A two-Riccati equation solution to the H/sub /spl infin// control problem for periodic multirate systems is derived via the lifting method. The solution is expressed in terms of two algebraic Riccati equations. The causality constraints are represented by a set of positive definiteness conditions and coupling criteria. As a by-product, the study shows that there is a close connection between the solution of the causally constrained lifted problem and the solution obtained by solving the problem directly as a periodic H/sub /spl infin// problem defined for the multirate system.

H ∞ and LGQ control of asynchronous sampled-data systems

A class of dual-rate sampled-data systems is considered, where the sampling rate is asynchronously related to the rate at which the hold element is operating. Such asynchronous sampled-data systems are aperiodic, and hence the lifting technique, which is used to represent synchronous sampled-data systems as single-rate discrete systems, cannot be applied to the asynchronous case. In this study, optimal H∞ and LQG controllers for asynchronous sampled-data systems are derived using a two-Riccati equation approach, which does not rely on the lifting method. A distribution theorem is used to evaluate the optimal stationary performance levels in the two cases.

H ∞ control of multirate sampled-data systems: a state-space approach

A state-space solution of the H∞ control problem for periodic multirate sampled-data systems is presented. The solution is characterized in terms of a pair of discrete algebraic Riccati equations with a set of associated matrix positive-definiteness conditions and coupling criteria. The solution is derived using two different approaches. In the first approach, the solution is obtained by discretizing two coupled periodic Riccati equations with jumps which characterize the H∞-optimal controller. The second approach is based on the lifting technique. The H∞-optimal controller for a lifted representation of the multirate system is characterized by a two-Riccati equation solution, in which the so-called causality constraint is represented by a set of positive-definiteness conditions and coupling criteria.