This paper examines the fundamental limitations imposed by unstable (Right Half Plane; RHP) zeros and poles in multivariable feedback systems. We generalize previously known controllerindependent lower bounds on the À ½ -norm of closed-loop transfer functions Ï Î , Ï Î×µ ½ , where is input or output sensitivity or complementary sensitivity, to include multivariable unstable and non-minimum phase weights Ï and Î . The bounds are tight for cases with only one RHP-zero or pole. For plants with RHP-zeros we obtain bounds on the output performance for reference tracking and disturbance rejection. For plants with RHP-poles we obtain new bounds on the input performance. This quanitifies the minimum input usage needed to stabilize an unstable plant in the presence of disturbances or noise. For a one degree-of-freedom controller the combined effect of RHP-zeros and poles further deteriorate the output performance, whereas there is no such additional penalty with a two degrees-of-freedom controller where also the disturbance and/or reference signal is used by the controller.

Summary Severe slugging in multiphase pipelines can cause serious and troublesome operational problems for downstream receiving production facilities. Recent results demonstrating the feasibility and the potential of applying dynamic feedback control to unstable multiphase flow like severe slugging and casing heading have been published.1–5 This paper summarizes our findings on terrain-induced slug flow.5 Results from field tests as well as those from dynamic multiphase flow simulations are presented. The simulations were performed with the pipeline code OLGA2000.* The controllers applied to all of these cases aim to stabilize the flow conditions by applying feedback control rather than coping with slug flow in the downstream processing unit. The results from simulations with feedback control show stable process conditions at both the pipeline inlet and outlet in all cases, whereas without control, severe slug flow is experienced. Pipeline profile plots of the liquid volume fraction through a typical slug flow cycle are compared against corresponding plots with feedback control applied. The comparison is used to justify the internal stability of the pipeline. In many cases, feedback control enables a reduced pipeline inlet pressure, which, again, means an increased production rate. This paper summarizes the experience gained with active feedback control applied to severe slugging. The focus is on extracting similarities and differences between the cases. The main contribution is demonstrating that dynamic feedback control can be a solution to the severe slugging problem. Introduction Multiphase pipelines connecting remote wellhead platforms and subsea wells are already common in offshore oil production, and there will be even more of them in the years to come. In addition, the proven feasibility of using long-distance tieback pipelines to connect subsea processing units directly to onshore processing plants makes it likely that these will also appear in the future. Such developments are turning the spotlight on one of the biggest challenges for control and operation of offshore processing facilities and subsea separation units - controlling the feed disturbance to the separation process (that is, smoothing or avoiding flow variations at the outlet of multiphase pipelines connecting wells and remote installations to the processing unit). Common forms of flow variations are slug flow in multiphase pipelines and casing heading in gas-lifted oil wells. In both cases, the liquid flows intermittently along the pipe in a concentrated mass called a slug. The unstable behavior of slug flow and casing heading has a negative impact on the operation of offshore production facilities. Severe slugging can even cause platform trips and plant shutdown. More frequently, the large and rapid flow variation causes unwanted flaring and limits the operating capacity in separation and compression units. This reduction is caused by the need for larger operating margins for both separation (to meet the product specifications) and compression (to ensure safe operation with minimum flaring). Backing off the plant's optimal operating in this way reduces its throughput. A lot of effort and money have been spent trying to avoid the operational problems with severe slugging and reduce the effects of the slugs. Roughly speaking, there are three main categories of principles for avoiding or reducing the effects of slugs.Design changes.Operational changes and procedures.Control methods, including feed-forward control, slug choking, and active feedback control. An example of a typical slug-handling technique involving design changes is to install slug catchers (onshore) or increase the size of the first-stage separator(s) to provide the necessary buffer capacity. A different, compact, process-design change is reported in Ref. 6, in which the authors introduce an additional small, pressurized, closed vessel upstream of the first-stage separator to cope with slug flow. An example of operational change is to increase the flowline pressure so that operation of the pipeline/well is outside the slug flow regime.7,8 This is not a viable option for older wells with reduced lifting capacity. For gas-lifted wells, an option would be to increase the gas injection rate (see Ref. 2). These design and operational changes may not be appropriate for already existing installations with slug flow problems or for compact separation units. Control methods for slug handling are characterized by the use of process and/or pipeline information to adjust available degrees of freedom (pipeline chokes, pressure, and levels), reducing or eliminating the effect of slugs in the downstream separation and compression units. The idea of feed-forward control is to detect the buildup of slugs and prepare the separators accordingly to receive them (e.g., via feed-forward control to the separator level and pressure control loops). The aim of slug choking is to avoid overloading the process facilities with liquid or gas. These methods make use of a topside pipeline choke by reducing its opening in the presence of a slug, thereby protecting the downstream equipment. The slug choking may use measurements in the separation unit and/or the output from a slug-detection device/algorithm. For a more complete assessment of the current technology for slug handling, refer to Ref. 9. In this assessment, however, active control methods are not properly addressed. Recently, results have been published that demonstrate the feasibility and potential of applying dynamic feedback control to unstable multiphase flow like severe slugging and casing heading. 1–5 Like slug choking, active feedback control makes use of a topside choke. With dynamic feedback control, however, the approach is to solve the slug problem by stabilizing multiphase flow. Despite the promising results first reported in 1990,1 the use of active slug control on multiphase flow has been limited. To our knowledge, only two installations in operation have stabilizing controllers. These are the Dunbar-Alwyn5,2 and the Hod-Valhall pipelines.5 One reason for this might be that control engineering and fluid flow dynamics usually are separated technical fields (i.e., the control engineers have limited knowledge about multiphase flow and the experts in fluid flow dynamics have limited insights into what can be achieved with feedback control). Indeed, when presenting the results on the Hod-Valhall pipeline,5 we had a hard time convincing several of the fluid-flow-dynamics engineers that one can avoid slug formation in severe slugging by active control. Hence, one objective of this paper is to provide insight and understanding into how feedback control can be used to avoid severe slugging, thereby contributing to bridging the gap between control and petroleum engineering.

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This paper was prepared for presentation at the 1999 SPE Annual Technical Conference and Exhibition held in Houston, Texas, 3–6 October 1999.

For a linear multivariable plant, it is known from earlier work that the easy computable pole vectors provide useful information about in which input channel (actuator) a given mode is controllable and in which output channel (sensor) it is observable. In this paper we provide a rigorous theoretical basis for the use of pole vectors, by providing a link to previous results on performance limitations for unstable plants.