A review of methods for input/output selection <s> Controllability and observability in a quantitative sense <s> One of the first questions facing the designer of the control system for a large space structure is how many components actuators and sensors - to specify and where to place them on the structure. This paper presents a methodology which is intended to assist the designer in making these choices. A measure of controllability is defined which is a quantitative indication of how well the system can be controlled with a given set of actuators. Similarly, a measure of observability is defined which is a quantitative indication of how well the system can be observed with a given set of sensors. Then the effect of component unreliability is introduced by computing the average expected degree of controllability (observability) over the operating lifetime of the system accounting for the likelihood of various combinations of component failures. The problem of component location is resolved by optimizing this performance measure over the admissible set of locations. The variation of this optimized performance measure with number of actuators (sensors) is helpful in deciding how many components to use. <s> BIB001 </s> A review of methods for input/output selection <s> Controllability and observability in a quantitative sense <s> The optimal location problem of distributed sensor/actuator for observation and control of a flexible structure is investigated. Using a property of controllability and observability grammian matrices, this approach employs a nonlinear optimization technique to determine the optimal placement of a distributed sensor/actuator. The effect of unimportant modes that do not strongly affect the structural behavior of a system is minimized and the effect of important modes is maximized. The final objective function is expressed as the combinational form of two different objective functions. This technique is applied to several types of beam support conditions and the corresponding optimal locations are determined. <s> BIB002 </s> A review of methods for input/output selection <s> Controllability and observability in a quantitative sense <s> This paper explores the application of recent techniques to the problem of structure selection and robust multivariable controller design for a high performance gas turbine engine. The engine under consideration is a Rolls Royce: 2-spool reheated turbofan, used to power modern military aircraft. Efficient and effective tools for control structure design are described and applied to the problem at hand. It is stressed that a good structure selection is very important, it results in simpler controllers, and in general, a simpler design exercise. A 2-DOF H/sub infinity / controller design is presented. Work is currently in progress on gain scheduling of such controllers. <s> BIB003 </s> A review of methods for input/output selection <s> Controllability and observability in a quantitative sense <s> In this paper, we discuss the optimal location of sensors and actuators for both linear and nonlinear dynamical systems, in both the continuous-time and discrete-time case, on the basis of observability and controllability functions. The optimal location of sensors can be viewed as the problem of maximizing the output energy generated by a given state. On the other hand, the optimal location of actuators can be viewed as the problem of minimizing the input energy required to reach a given state. Such design problems occur in many applications, such as the control of distributed parameter systems, arising in mechanical, hydraulic or chemical processes. In this paper, some new results on observability and controllability functions for nonlinear systems are also provided. Furthermore, we propose a general procedure for computing the optimal design parameters, based on both integer programming and a branch and bound method, suitable for large-scale systems. The effectiveness of this approach is demonstrated for a practical example. <s> BIB004 </s> A review of methods for input/output selection <s> Controllability and observability in a quantitative sense <s> Considers the placement of distributed actuators in motion control of flexible structures. By representing the system dynamics in terms of natural modes of vibration, simple conditions are developed from which degrees of controllability of individual modes for a given actuator distribution are determined. Systematic procedures for optimal placement of distributed actuators are proposed, which are extensions of the methods developed earlier for point actuators. The proposed optimality criteria for actuator distribution seek to minimise the actuator input energy under a variety of operating conditions. Minimal levels of controllability and observability of each controlled mode are. Achieved with this approach. The method is illustrated for a simply supported beam and a rectangular plate controlled by distributed actuators of various shapes. <s> BIB005
Due to the binary nature of (structural) state controllability and state observability, they are unlikely to provide IO selection conditions which are selective enough. The accepted number of IO sets may be too large for controller design and closed-loop evaluation. It should be possible to draw conclusions on the strength of the coupling between inputs, states, and outputs. Various quantitative measures for state controllability and observability have been proposed for this. In the upcoming IO selection methods, the controllability Gramian = (t) and the observability Gramian = (t) play a role, see, e.g., Zhou et al. (1996, Chapter 3) . If A in (2) is stable,¸ : "lim R = (t) and¸ : " lim R = (t) are the unique, positive semi-de"nite solutions to the following Lyapunov equations: A2¸#¸A#C2C"0. Georges (1995) selects optimal actuator and sensor locations (n S and n W are "xed) based on maximizing the minimum eigenvalues of = (¹) and = (¹), respectively, with ¹ a given, possibly in"nite, time. The idea is to minimize the input energy to reach a given state and to maximize the output energy generated by a given state. BIB004 extends this idea to nonlinear systems x "f (x)#Bu, y"Cx. IO selection then requires the solution of nonlinear, partial di!erential equations. MuK ller and embed the minimum eigenvalues of = (t) and = (t) in an inxnite set of quantitative controllability and observability measures. Among those are the determinants of = (t) and = (t) and the reciprocals of the traces of =\ (t) and =\ (t). IO selection could aim at maximizing any of these measures. HacH and Liu (1993) and BIB002 de"ne performance indices that involve the eigenvalues of¸ and¸. The IO selection criteria proposed by  are based on the energy of the input and the output in case of transient responses or persistent disturbances. The criteria are especially useful for #exible structures. They provide a balance between the importance of the lower-and the higher-order modes. For systems with small damping ratios and well-separated natural frequencies, the locations of force actuators and velocity sensors tend to be collocated (also for many other criteria). The selection criterion for point actuators in  is extended to distributed actuators in BIB005 . Assume that the plant is controllable, observable, and stable. A balanced realization of (2)}(3) can then be obtained, with¸ and¸ equal and diagonal  . The diagonal entries of these Gramians are called the Hankel singular values (HSVs), re#ecting the joint controllability and observability of the balanced states. Candidate IO sets with large HSVs are preferred in BIB003 . HSVs also form the basis for IO selection in  and  , but they employ special properties of #exible structures. As in Fig. 1 ,  explicitly de"nes exogenous and controlled variables and performance weightings to improve the IO selection criterion of  . Another quantitative controllability measure could be the size of a region of initial states from which the state can return to the origin in a "xed time t C with an admissible input signal.  terms u admissible if it obeys a magnitude bound, while Vander BIB001 call u admissible if it obeys an energy bound. Both measures may be helpful for actuator selection. Vander BIB001 also discuss a dual approach to sensor selection and they address the e!ect of possible actuator and sensor failures from a statistic viewpoint.