A review of methods for input/output selection <s> General control system set-up <s> Abstract Given a linear system, all stabilizing controllers such that a specified closed-loop transfer function has H ∞ norm less than a given scalar, are parametrized. This characterization involves the solution to two algebraic Riccati equations, each with the same order as the system, and further gives gives feasible controllers also with this order. The relationship to the risk-sensitive LQG stochastic control problem is established, giving an equivalence between robust and stochastic control. <s> BIB001 </s> A review of methods for input/output selection <s> General control system set-up <s> Abstract Screening tools for control structure selection in the presence of model/plant mismatch are developed in the context of the Structured Singular Value (μ) Theory. The developed screening tools are designed to aid engineers in the elimination of undesirable control structure candidates for which a robustly performing controller does not exist. The screening tools are examined on a multi-component distillation column control problem and compared with previously published methods such as the Condition Number Criterion. <s> BIB002 </s> A review of methods for input/output selection <s> General control system set-up <s> A new method to select sensors and actuators for linear control systems is presented. The key idea is to eliminate candidate sensor/actuator combinations for which a controller achieving a desired level of nominal performance and/or robust stability against unstructured uncertainties cannot be designed. All combinations are subjected to six viability tests, which are necessary conditions for the existence of stabilizing controllers meeting the required H/sub /spl infin// norm bound. The new selection method is used for the active suspension of a truck. This application illustrates the need for a method dealing with structured uncertainties. <s> BIB003 </s> A review of methods for input/output selection <s> General control system set-up <s> This paper deals with the treatment of model uncertainty in a Q-Parametrization framework for dynamic operability assessment. Structured nonlinear and/or time-varying model uncertainty is considered using the 1 robust control theory of Khammash and Pearson (1991). In the case of unstructured model uncertainty, dynamic operability assessment is posed as a convex quadratic programming problem and solved efficiently using sparse matrix techniques. If the uncertainty is structured, the resultant problem is nonconvex and is solved at present using a hybrid approach, with more sophisticated global optimization methods being investigated. The approach is applied to a multivariable distillation column containing all four performance limiting factors and the results are discussed. <s> BIB004 </s> A review of methods for input/output selection <s> General control system set-up <s> A recently presented method for actuator and sensor selection for linear control systems is applied and evaluated for an active vehicle suspension control problem. The aim is to eliminate the actuator/sensor combinations for which no controller exists that achieves a specified level of robust performance. Complete controller synthesis is avoided by using necessary conditions for robust performance. Due to this, it cannot be guaranteed that a stabilizing and robustly performing controller exists for the combinations satisfying the conditions. This is a major shortcoming of the method. Depending on the application and implementation, the computation time consumed by the selection procedure can be large. <s> BIB005
The IO selection methods in this section use the set-up in Fig. 6 . The uncertainties are isolated from the nominal plant model G and w and z are split into parts related to uncertainty (w , z ) and performance (w , z ). Many control system set-ups (including those in Figs. 2, 4, and 5) can be cast into the set-up of Fig. 6 . may act upon the plant in a multiplicative, additive, or many other ways and it may be unstructured or structured. BIB004 consider combined RS and NP. In Fig. 6 , w and z are required to have bounded amplitudes and the control problem is formulated in the l system norm setting. The uncertainties are treated as nonlinear and/or time varying, which may be conservative if they are linear and time invariant. The performance is in terms of minimizing tracking errors to setpoint changes. A controller-independent condition is derived that could be used to check if, for a given IO set, there exists any linear time-invariant controller meeting NP and RS. For unstructured (as for NP only), this test involves solving a convex, constrained optimization problem. For structured , the optimization is not convex and an approximation is invoked. The global optimum may not be found, so the IO selection may not be e!ective. In BIB003 and Van de Wal (1998, Section 4.1) , an IO selection method is proposed for separate or combined RS and NP. is assumed to be unstructured, linear, and time invariant. An H control problem is formulated, with the H norm of a TFM H(s) de"ned as Weighting "lters imposing speci"cations on the uncertainties and the exogenous and controlled variables are incorporated into G. For a given IO set and corresponding G, it can be checked whether there exists a stabilizing controller achieving ""M"" ( . Here, is a speci"ed value and M : "F J (G, K) is the nominal closed-loop system, denoted as a lower linear fractional transformation (Zhou et al., 1996, Chapter 10) . This could be read as`G closed by Ka. In case of RS (NP), w and z (w and z ) in Fig. 6 are absent and the H norm requirement is imposed on the relevant nominal closed-loop system denoted by M 01 (M ,. ). In case of combined RS and NP, the H norm requirement becomes ""[M 01 M ,. ]"" ( . The IO selection method amounts to checking the set of controller existence conditions proposed by BIB001 . These involve requirements on the solutions of two algebraic Riccati equations. For a given IO set, the conditions are checked in succession, up to failure. For IO selection aimed at RP or RS against structured uncertainties, G can be modi"ed such that the same method can be used, see Section 3.7. In and Van de Wal (1998, Section 4 .1), the IO selection method sketched above is used for nonlinear systems via linearizations in a grid of stationary operating points. For the nonlinear H problem to be locally solvable, it is su$cient that the corresponding linear H problem is solvable (Van der Schaft, 1996, Chapter 7). Due to this, IO sets may be incorrectly rejected. On the other hand, for the nonlinear H problem to be globally solvable in the operating region, it is necessary that the linear H problems are solvable for all equilibria. This is caused by the fact that the sizes of the controllers' regions of attraction around the equilibria are unknown during IO selection and may not be large enough to overlap. Due to this, IO sets may be incorrectly accepted. Nevertheless, the IO selection method is useful for initial screening of many IO sets. The IO sets accepted for the "rst operating point are checked for the second operating point and so on, until all grid points have been checked. If the number of accepted IO sets is small enough, (nonlinear) controller design and closed-loop evaluation could be invoked to rigorously address IO set viability. Lee (1991, Chapter 3) suggests necessary conditions for RP-based IO selection which are tied to LQG control and model predictive control (MPC) or to integral controllers. The conditions are in terms of a bound on weighted plant data. Their application is restricted to square IO sets and steady state. This is circumvented for the methods in the rest of this section, but these are computationally more involved. Consider Fig. 6 and denote the nominal closed-loop system relating w to z by M. The upcoming IO selection methods are posed in the H system norm setting and they aim at selecting those IO sets for which there exists a stabilizing controller achieving RP or RS against structured uncertainties. The RP requirement is expressed as a requirement on the value of a TFM H(s), de"ned as Strictly speaking, ""H"" I is not a norm. In contrast to the H norm requirement ""M"" ( , there is no set of conditions that can be used to verify whether there exists a stabilizing controller achieving ""M"" I ( . Controller design aimed at minimizing ""M"" I or achieving a particular value of ""M"" I could be invoked instead. This -synthesis problem is still unresolved, but an approximate design, called D}K iteration (Zhou et al., 1996, Chapter 11) , is possible. The aim is to make the following upper bound small enough: with (H( j )) : The D-scales D( j ) belong to a set determined by the structure of (Zhou et al., 1996, Chapter 11) . For notational simplicity, it is assumed in (36) that each individual block in is square and that w and z have the same dimension. Each iterative step in D}K iteration alternates between computing (M) and the D-scales for a frequency grid, "tting a TFM through D( j ), and performing an H optimization of G augmented with the D-scale approximations. The iteration could be stopped if ""M"" I is small enough or does not further converge. This is the basic idea of IO selection in Van de Wal (1998, Section 6.6 ). The IO selection method in De Jager, Van de Wal, and Kamidi (1998) combines D}K iteration with H controller existence checks (Section 3.6.2) for a scaled plant. Due to the fact that achieving ""M"" I ( is su$cient for achieving ""M"" I ( , IO sets may be incorrectly rejected. The methods in the remainder of this section circumvent the time-consuming D}K iterations for distinct candidate IO sets, at the price of reduced e!ectiveness. The starting point for the IO selection method proposed by BIB002 is to check whether there exists a stabilizing controller achieving ""M"" I ( . To arrive at controller-independent IO selection conditions, the requirement of the controller being stabilizing is dropped, which is equivalent to dropping the controller's causality. According to BIB002 , the consequences are only signi"cant in the crossover region, where joint stability and performance requirements are especially di$cult to be met. Note that dropping the stabilizing property amounts to dropping the interdependence of gain and phase as imposed by Bode's gain-phase relationships: the performance speci-"cations, expressed as magnitude requirements, are more easily met if the phase requirements can be neglected. The IO selection conditions are in terms of two LMIs that must be jointly feasible across frequency. Neglecting crucial frequencies in the grid is another source of necessity. Only for a few special cases, like RP with unstructured , checking the joint feasibility of the LMIs is convex. For more general cases, only the feasibility of the individual LMIs can be checked. This implies that the input (output) set is checked under the assumption of a perfect output (input) set, but that the combination of the actual input and output sets cannot be checked. The pros and cons of this IO selection method are discussed in more detail by BIB005 . The last two methods to be reviewed reduce the structured RP problem to an unstructured one, such that H controller existence conditions can be used for IO selection, as in Section 3.6.2. Both methods proceed along four steps that are mainly the same: (1) D}K iteration for the full IO set until ""M"" I does not further decrease, (2) -analysis for the full IO set's optimal closed-loop system and generation of particular TFMs to be used in the next step, (3) modi"cation of the candidate IO sets' plants G with these TFMs, and (4) subjecting the modi"ed plants to the H controller existence conditions. The two IO selection methods use di!erent TFMs in the second step. For both methods, using the TFMs related to the full IO set also for the other IO sets is based on the assumption that these TFMs are representative for IO sets which are almost as good as the full one. The "rst method is described in Van de Wal (1998, Section 4.2.1) and . As in D}K iteration, G is augmented with approximations of the D-scale data resulting from the upper bound computation. These TFMs, called`D-scale estimatesa, are constructed for the full IO set, but they are also used for the other IO sets. The D-scale estimates may not be optimal for other IO sets than the full one, which is the main reason for the su$ciency of this method. So, IO sets may be incorrectly rejected. The second method is described in Van de Wal (1998, Section 4.2.2) and . A particular TFM ) (s), representing an uncertainty, is absorbed into G for the distinct candidate IO sets. The magnitude of ) is such that the RP level of the full IO set's optimal closed-loop system is just violated, at least for one frequency, and therefore it is called a`worst case uncertaintya. This method has a necessary character and IO sets may be incorrectly accepted. This is mainly due to two sources. First, ) is not necessarily worst case for all frequencies. Second, even if it is, it may not be worst case for other IO sets than the full one for which it was constructed.