Petter Lundström

Performance weight selection for H-infinity and μ-control methods:

The paper discusses, from a process control perspective, different approaches to performance weight selection when using H-infinity objectives. Approach A considers direct bounds on important transfer functions such as sensitivity and complementary sensitivity. Approach B considers the output response to sinusoidal disturbances, setpoints and noise. We also give some insight into the practical use of H-infinity and μ methods. μ is the structured singular value (SSV) introduced by Doyle (1982). μ-synthesis is generally not a convex optimisation problem and is presently not straightforward. We will discuss some of the problems we have encountered.

Representation of uncertain time delays in the H∞ framework‡

Abstract How to represent an uncertain time delay in a form suitable for robust control in terms of the H∞-norm and the structured singular value (µ) is discussed. To use µ-synthesis the uncertain delay has to be approximated to yield a proper rational interconnection matrix M(s) and a purely complex perturbation matrix? (s). The parametric average of the delay is usually included in the nominal model, while the uncertain perturbation covers variation around the average delay. It is proposed to model the nominal time delay as uncertainty, i.e. using a larger uncertainty set. This yields a delay-free nominal model, which simplifies the controller synthesis. For the cases studied the delay free nominal model does not yield a more conservative design than the average delay model, despite its larger uncertainty set.

Mu-optimal LV-control of distillation columns

Abstract Two-point composition control using the LV -configuration is examined. In spite of high steady-state RGA-values it is shown for seven example columns that the LV -configuration may yield acceptable robust performance using single-loop controllers provided measurement delays are not too large. In order to investigate the level of modelling detail that is required for effective control, five process models of different complexity are studied. It is demonstrated that flow dynamics should be included in a column model which is to be used for controller design. The structured singular value (Mu or μ) is used as a robust performance index. To use this index one must define performance using the H ∞ framework. The issue of selection of performance and uncertainty weights is critical in this procedure, and is discussed in detail in the paper.

Opportunities and difficulties with 5 × 5 distillation control

Abstract Multivariable 5 × 5 distillation control, i.e.. control of levels, pressure and compositions by one multivariable controller, provides opportunities to improve the control performance as compared to decentralized control. Multivariable interactions can be counteracted with a 5 × 5 controller. However, the main advantage is automatic constraint handling which cannot be realized by a fixed linear 5 × 5 controller, but requires a solution based on on-line optimization, for example, using a model predictive control (MPC) strategy. A multivariable control scheme also presents some difficulties. Unconsidered model uncertainty may be a severe problem. It may also be difficult to tune the multivariable controller. In this paper the MPC approach is combined with the H x μ framework in order to obtain a robust design.

Uncertainty weight selection for H-infinity and mu-control methods

The authors give some insight into H/sub infinity /-weight selection from a process control perspective, with emphasis on uncertainty modeling. They study how to transform parametric gain-delay uncertainties into frequency-dependent weights specifying norm-bounded uncertainties. They postulate a parametric gain delay uncertainty set as the true uncertainty. They then try to minimize an uncertainty set generated by a complex additive perturbation which covers the true set. In addition, mu -optimal controllers are designed for the example problem considered. Simulation results are presented.<<ETX>>

Selecting the Best Distillation Control Structure

Abstract The selection of an appropriate control structure (configuration) is the most important decision when designing distillation control systems. The steady state RGA is commonly used in industry as a tool for selecting the best structure. In this paper it is stressed that decisions regarding controller design should be based on the initial response (high-frequency behavior) rather than the steady state. However, for most distillation configurations the steady state RGA turns out to be a good indicator of expected control quality. One counterexample is the DB-configuration which has infinite steady-state RGA-values. However, the RGA-values at higher frequency are close to one, and good control performance is possible. Based on a frequency-dependent RGA-analysis and optimal PI controller designs, the (L/D)(V/B)-configuration is found to be the best choice for two-point composition control. The traditional LV-configuration performs much poorer, but is preferable if one-point control is used.

RIGOROUS DYNAMIC SIMULATION OF DISTILLATION COLUMNS BASED ON UV-FLASH

Abstract The paper demonstrates that a rigorous dynamic simulation for distillation columns, including varying pressure and vapor holdup is feasible using a UV -flash approach. A strategy is presented for calculating this flash efficiently when used in dynamic simulation. The paper also considers use of local thermodynamic models combined with look-up tables, where all variables are continuous in time. The results are illustrated on a propane-butane fractionator where the vapor holdup is considerable and should not be neglected.

Modelling Requirements for Robust Control of Distillation Columns

Abstract In order to investigate the level of modelling detail that is required for effective control, five process models of different complexity are studied. It is demonstrated that the high-frequency behavior (initial time response) of the model is much more important for controller design than it’s steady-state. characteristics. Flow dynamics should be included in a column model which is to be used for controller design.

Modelling of Gain and Delay Uncertainty in the Structured Singular Value Framework

Abstract Gain and delay uncertainty is commonly used to quantify plant-model mismatch in the process control community. This type of uncertainty description cannot be directly used for robustness analysis and design in the structured singular value (μ) framework. This paper provides analytical expressions for some tight approximations of gain-delay uncertainty based on complex perturbations in a form suitable for analysis within the μ-framework. Simple bounds suitable for synthesis are also presented. A model that covers the gain uncertainty exactly and closely approximates the delay uncertainty, is derived using real perturbations. Finally these uncertainty models are applied to a distillation example

Non-uniqueness of robust H ∞ decentralized PI-control

In the paper we use as an example the control of identical processes operating in parallel. Such systems are quite common in industry, for example in distribution networks or when parallel units (reactors, heat exchangers, etc.) are used. When performance is measured in terms of the H∞ norm then the optimal single-loop PI or PID-tunings are not necessarily equal for the individual loops. The same applies when model uncertainty is included and the structured singluar value μis used as a performance measure. This is contrary to what one intuitively would expect, and also implies that the optimal solution is non-unique.