A review of methods for input/output selection <s> One DOF control system set-up <s> J.C. Doyle et al. (1982) have shown that a necessary and sufficient condition for robust stability or robust performance in the H/sup infinity /-frame work may be formulated as a bound on the structured singular value ( mu ) of a specific matrix M which includes information on the system model, the controller, the model uncertainty, and the performance specifications. Often it is desirable to express the robust stability and performance conditions as norm bounds on transfer matrices (T) which are of direct interest to the engineer, e.g. sensitivity or complementary sensitivity. The present paper shows how to derive bounds on sigma (T) from bounds on mu (M). > <s> BIB001 </s> A review of methods for input/output selection <s> One DOF control system set-up <s> The proper selection and pairing of manipulated and controlled variables is crucial for successful control system design. A test for such a purpose is developed in this paper. The criteria for which a set of variables is deemed adequate for control purposes are 1) robustness to model uncertainty, 2) stability despite decentralized control, and 3) constraints (on manipulated and associated variables) satisfied at steady-state. The basis for this test is the Structured Singular Value, which provides a generalized framework for simultaneously addressing the performance and robustness aspects of control systems. This test, which is independent of a specific controller design methodology, is a definitive test - alternatives can be evaluated precisely on whether they satisfy requirements, without requiring the user to specify a scaling or some arbitrary "cut-off" point for acceptable configurations. The effectiveness of the test is illustrated with a case study involving the control of a Heavy Oil Fractionator, which forms part of the Shell Standard Control Problem. <s> BIB002 </s> A review of methods for input/output selection <s> One DOF control system set-up <s> Two major tasks that are required to obtain a control system utilizing secondary measurements are measurement selection and inferential control system design. The first involves choosing an appropriate subset of the available measurements and the second involves designing a feedback controller based on the chosen measurements. The important issues to be addressed are not only the theoretical performance of the closed-loop system, but also the effects arising from the factors prevalent in practical environments such as model/plant mismatch, constraints, and failures of actuators and sensors. General measurement selection methodology is developed accounting for all the factors that can affect the measurement selection in signifcant ways. These factors include model uncertainty, signal-to-noise ratios, and measurement dynamics. The underlying philosophy is to reduce the number of candidates to a sufficiently low level before going onto detailed analysis by eliminating those candidates for which there does not exist a linear time-invariant controller meeting the required level of robust performance. Based on this philosophy and using the Structured Singular Value theory as a vehicle, a number of numerically efficient screening tools are developed. Conditions are derived under which some of the new criteria reduce to previously published measurement selection criteria. The proposed tools are applied to the measurement selection problems in a multi-component distillation column and a high-purity distillation column. Two different approaches are considered for inferential control system design: an output estimation based design approach and a state estimation based design approach. The former approach involves independent design of an output estimator and a feedback controller while the latter involves direct one step design although the design can be actually separated into those of a state estimator and of a feedback regulator using the separation principle argument. For the former approach, design of the output estimator was examined for two different cases: the case where a full dynamic model is available and the case where only the time records of the primary and secondary measurements are available either from simulations or from process measurements. For the former case, multi-rate Kalman filter design and μ-Synthesis design are discussed. For the latter case, the estimator design problem is formulated as a regression problem and various regression techniques are evaluated in terms of their suitability to the output estimator design problem. For design of the feedback controller, traditional techniques such as LQG, IMC, and MPC were combined into a control technique that has nice algorithmic properties as well as many operational merits such as straightforward constraint handling and simple, intuitive on-line tuning. A heavy-oil fractionator was used as an example application. For the latter approach, general state estimation techniques (e.g., multi-rate Kalman filtering) used in LQG and finite receding horizon control used in traditional MPC were integrated into a control technique that can incorporate general disturbances and multi-rate sampled measurements and has desirable operational characteristics. The concept of classical IMC was extended to equip the control system with on-line tuning parameters that have direct connections with the speed of the closed-loop responses. Application to a high purity distillation column demonstrates the effectiveness of the control technique in terms of closed-loop performance and operational flexibility. <s> BIB003 </s> A review of methods for input/output selection <s> One DOF control system set-up <s> Strong trends in chemical engineering and plant operation have made the control of processes increasingly difficult and have driven the process industry's demand for improved control techniques. Improved control leads to savings in resources, smaller downtimes, improved safety, and reduced pollution. Though the need for improved process control is clear, advanced control methodologies have had only limited acceptance and application in industrial practice. The reason for this gap between control theory and practice is that existing control methodologies do not adequately address all of the following control system requirements and problems associated with control design: ::: ::: * The controller must be insensitive to plant/model mismatch, and perform well under unmeasured or poorly modeled disturbances. ::: ::: * The controlled system must perform well under state or actuator constraints. ::: ::: * The controlled system must be safe, reliable, and easy to maintain. ::: ::: * Controllers are commonly required to be decentralized. ::: ::: * Actuators and sensors must be selected before the controller can be designed. ::: ::: * Inputs and outputs must be paired before the design of a decentralized controller. ::: ::: A framework is presented to address these control requirements/problems in a general, unified manner. The approach will be demonstrated on adhesive coating processes and distillation columns. <s> BIB004 </s> A review of methods for input/output selection <s> One DOF control system set-up <s> This paper studies deviations of open-loop properties in the presence of modeling uncertainties. Our aim is to gain insights into how open-loop properties and thus potentially closed-loop properties may vary in the face of a diagonally structured uncertainty. We give several estimates for the worst case deviations of the open-loop transfer function in terms of certain structured singular values and their bounds, and also in terms of certain scaled plant condition numbers, the relative gain array, and the block relative gains. Our analysis shows that the estimates in terms of the structured singular values and bounds are tight in general, so are those in terms of the condition numbers for certain cases studied previously in the literature. We show that the worst case deviation will be large when the estimates stated in terms of the structured singular values, or under certain circumstances in terms of the condition numbers, are large. On the other hand, an example is constructed to show that the relative gain array and block relative gains may be optimistic measures in assessing these deviations. The developments here support and reinforce previous conjec- tures and results which assert that plants with large condition numbers and/or relative gains are potentially difficult to control. <s> BIB005 </s> A review of methods for input/output selection <s> One DOF control system set-up <s> Abstract When a plant-wide control structure is being designed, there are often more measurements and manipulations than are required. Therefore the designer must judiciously select a subset of them that will be used for feedback control. If the control structure is to be decentralized, a set of pairings between the selected measurements and manipulations must also be established. This subset of process variables and their feedback interconnections has been termed the control configuration, and must be decided prior to the design of the controller. However the number of possible control configurations is usually very high, and the complexity of the controller depends upon the control configuration chosen. This paper presents a systematic way to design the control configuration so as to reduce controller complexity, and yet meet control objectives in the presence of uncertainty This method, called control configuration design, is then applied to a plant-wide control problem posed by Tennessee Eastman <s> BIB006 </s> A review of methods for input/output selection <s> One DOF control system set-up <s> Abstract Screening tools are developed which provide nonconservative estimates of the achievable performance in the presence of general structured model uncertainty. These tools allow the rational selection among plant designs, or can be applied to provide recommedations on how to modify a plant design to improve the closed loop performance. Many of these tools are applicable to the pairing or partitioning of inputs and outputs of decentralized controllers. It is shown through examples that ignoring or improperly characterizing plant/model mismatch while selecting among plant designs can lead to erroneous results. <s> BIB007 </s> A review of methods for input/output selection <s> One DOF control system set-up <s> The Robust Performance Number (RPN) and the Robust Performance Number of a Plant Set (RPPN) are introduced as measures to determine the controllability of a system. These numbers indicate how potentially difficult it is for a given system to achieve the desired performance robustly. The RPN and the RPPN reflect both the attainable performance of a system and its degree of directionality. The RPN is used for the analysis of the nominal system, whereas the RPPN can be used to include the effects of the nonlinearities and uncertainties of the model. It is shown how the attainable performance can be determined. As appropriate scaling is crucial for the analysis and for the design of multivariable systems, a strategy for system scaling based on the RPN is also presented. <s> BIB008
Reeves (1991) discusses a method to reject those IO sets for which there does not exist a controller achieving joint RS and NP. The original idea is due to Nett (1989) . The method is also summarized by BIB006 and implemented in a MATLAB toolbox  . The starting point is RS for the control system in Fig. 5 , where P and K are square. The unstructured, additive uncertainty obeys the relative, additive uncertainty bound: Reeves (1991) uses P? to specify the amount of uncertainty to be tolerated. A necessary and su$cient condition for RS is: there exists a controller that stabilizes all P "P# with the same number of unstable poles as P if and only if (P) (P\(I!S))( 1 The necessary conditions for IO selection in BIB004 and BIB007 and the necessary conditions for secondary measurement selection in BIB003 and  are derived in the context of a particular controller design method. In Braatz (1993, Chapter 6) this is robust loopshaping. This technique is based on magnitude bounds for TFMs which are of special interest and which are used to parameterize the controller BIB001 . For instance, for a tracking or disturbance rejection problem in the set-up of Fig. 5 , K could be parameterized in terms of the sensitivity S in (6). If the RP control goal is appropriately quanti"ed, the following is a necessary condition for existence of a robustly performing controller designed via loopshaping of S: with the structured singular value of a matrix. A formal de"nition of can be found in many textbooks (Zhou et al., 1996, Chapter 11) . Here it su$ces to see as an extension of the maximum singular value that permits (structured) uncertainty characterizations and performance speci"cations to be captured simultaneously and nonconservatively. In (31), G , G , and G correspond to the partitioning of G in Fig. 6 , while G equals !P in, e.g., Fig. 5 . Condition (31) is thus applicable to control problems which are "rst formulated in a one DOF control system set-up and which, after de"ning w and z, are transformed into the set-up of Fig. 6 . BIB004 and BIB007 show that (31) is also necessary for existence of an RP controller with integral action, i.e. K(s)"(1/s)K K (s), with K K (0) nonsingular and S(0)"0. BIB007 propose additional necessary conditions in terms of which can be used for joint IO and CC selection. RP is actually a dynamic system property and hence the steady-state condition (31) may not be very useful for IO selection.  discuss conditions which are in essence equivalent to (31), but which are derived for loopshaping in estimation-based inferential control systems where estimates of the controlled variables are used. A TFM like S may not be relevant for performance and other TFMs are used instead. Application of all the above-mentioned IO selection conditions is restricted to square IO sets and steady state. BIB002 speci"es NP in terms of magnitude constraints on the inputs u and on variables in z. The IO selection method involves three steps: the candidates are successively tested for (1) NP in terms of satisfaction of the magnitude constraints (using ), (2) RS for uncertainties bounded by the H norm (using ), and (3) satisfaction of the constraints in the face of the uncertainties, i.e., RP (using ). The constraints cannot be incorporated directly into the H norm setting that is common for RP. The L system norm setting would be more appropriate for handling constraints. However, by employing an inequality relating the H and L norm, Rivera (1989) accounts for magnitude constraints, though possibly quite conservatively. The IO selection conditions are restricted to steady state and square IO sets and they assume integral control also in the presence of uncertainty, i.e., S (0)"0, with S : "(I#(P# )K)\ for the system in Fig. 5 with unstructured . In BIB008 and  , the`robust performance numbera (P, ¹ Q ) is proposed as a potential IO selection tool which is not restricted to steady state. For a control system in the set-up of Fig. 2 , This measure is based on the measure for RP in Zhou et al. (1996, Section 11.3.3) . The latter is valid in case of a plant-inverting controller and the conservative, unstructured, multiplicative input uncertainty in Fig. 4 . The measure also involves the plant condition number (P). To reduce conservatism, BIB008 replace (P) by the minimum plant condition number (P) in (21). In case of the less conservative, structured, multiplicative input uncertainty, (P) is more appropriate, as illustrated by BIB005 . Furthermore, the "rst term below the root in (32) replaces an expression in the original measure that represents the RS and NP requirements. Here, ¹ Q is a specixcation for the complementary sensitivity function that is de"ned as The choice of ¹ Q is determined by (1) the performance speci"cations and (2) the occurrence of RHP zeros, RHP poles, and time delays in P. Due to (2), the choice of ¹ Q is not completely free, since closed-loop stability must be guaranteed. BIB008 state that ((I!¹)¹) is largest in the crossover region and so (P, ¹ Q ) automatically emphasizes the frequency region where suppressing the e!ect of uncertainties is usually the most important. The IO sets yielding a small (P, ¹ Q ) are preferred, since these will give good RP if an inversebased controller`K"P\a is used. BIB008 also propose an extension to (P, ¹ Q ) that applies to polytopic models, i.e., sets of linear system models which together represent the original model. This extension makes it possible to consider various types of uncertainties explicitly and to examine nonlinear systems with varying operating conditions. However, IO selection may be conservative if the nonlinear system is not accurately described by the polytopic model. Three drawbacks of IO selection with the RP number (P, ¹ Q ) are mentioned. First, the derivation of (P, ¹ Q ) is mainly heuristic and lacks a sound theoretical foundation. It seems an arbitrary combination of di!erent numbers, which makes (P, ¹ Q ) di$cult to assess. Second, due to the use of the condition number larger IO sets are more likely to be eliminated (see Section 3.6) and in case of the optimally scaled condition number the physical meaning of the inputs and outputs of P may be lost. Third, it may be necessary to re-compute ¹ Q for distinct IO sets, e.g., if the RHP zeros are not the same for all IO sets.