For a system described by the triple (A9B9C) where the matrix A has a set of distinct eigenvalues and a wellconditioned modal matrix, we propose measures of modal controllability and observability. The angles between the left eigenvectors of A and the columns of the matrix B are used to propose modal controllability measures and the angles between the rows of the matrix C, and the right eigenvectors of A are used to propose modal observability measures. Gross measures of controllabili ty of a mode from all inputs and its observability in all outputs are also proposed. These measures are related to other measures suggested in the literature. A closed-form relation between the norm of the residue and the proposed measures is given, thus linking the residue to the unobservability or uncontrollability of the mode. We finally show that the proposed measures can be applied directly to second-order models.

Measures of Modal Controllability and Observability for First- and Second-Order Linear Systems

A method of finding the optimal sensor and actuator locations for the control of flexible structures is presented. The method is based on the orthogonal projection of structural modes into the intersection subspace of the controllable and observable subspaces corresponding to an actuator/sensor pair. The controllability and observability grammians are then used to weight the projections to reflect the degrees of controllabili ty and observability. This method produces a three-dimensio nal design space wherein sets of optimal actuators and sensors may be selected. A novel parameter is introduced that is potentially useful for studying the problem of the number of actuators and sensors, in addition to their optimal locations. UPPOSE a specific number of actuators and sensors is given and they are placed at specific locations on a flexible structure such that the effectiveness of the chosen actuator and sensor locations could be analyzed. If it turns out that the a priori chosen number and locations for the actuators and sensors are not sufficiently effective, the question naturally arises as to how the locations could be changed to improve the system. Furthermore, it is possible that the a priori number of actuators and/or sensors used is insufficient or redundant. Thus, there is clearly a need for a computationall y feasible technique that is capable of determining an optimal set of locations and the minimal number of actuators and sensors. In general, there will be many more candidate locations (perhaps an order of magnitude more) than the number of actuators and sensors actually available. If the number of actuators and sensors is known a priori, all possible combina- tions could be evaluated, and in principle, the global optimum could then be found. Unfortunately, the number of possible combinations increases factorially, and therefore an exhaus- tive search for a global optimal is usually computationally infeasible, while nonlinear programming based techniques typically produce local minimums. In the past, various definitions of the degree of controllabil- ity and observability have been used in guiding the search for optimal actuator and sensor locations. Among these, the de- gree of controllabili ty defined by scalar measures of recovery regions appears useful for the purpose of actuator and sensor placement.1'3 A second approach4 uses the projection magni- tudes of eigenvectors into the input and output matrices to define gross measures of modal controllability and observabil- ity. However, only little attention is given to the development of a systematic search strategy for actually solving for an optimal set of actuators and sensors, and most attention has been directed toward defining what constitutes most suitable actuator and sensor locations. In this paper, the problem of defining and obtaining the optimal actuator and sensor locations is addressed. A method that is based on the controllability and observability of an actuator/sensor pair is introduced. An outline of the present paper follows. First, the model of actuator and sensor loca- tions for a linear, second-order dynamical system is presented. The basic assumption is that we are given a set of significant modes whose control is desired via feedback. In the next section, controllable, observable, and their intersection sub- spaces are presented, which forms the basis of the method presented in the sequel. The following section presents the main results of this paper. A cost function that is based on the weighted projection of structural modes into the intersection subspace of the controllable and observable subspaces is intro- duced, and a simple interpretation in terms of balanced coor- dinates is given. A novel method for selecting optimal sensor and actuator locations based on the preceding cost function is outlined. The weighted projections of the structural modes can be viewed as a scalar field in three-dimensi onal design space wherein a designer can easily select a set of actuators and sensors based on his or her own criteria without resorting to elaborate nonlinear programming strategies. The method also allows for the comparison of many actuator and sensor candi- date locations since the computational effort depends only on the product of the number of actuator and sensor location candidates rather than combinatorially based search strategies whpse computational effort is in the order of factorials. In the next section, the method of finding optimal locations is ap- plied to an existing laboratory structure to demonstrate the algorithm. Finally, a few concluding remarks are given.

Method for Optimal Actuator and Sensor Placement for Large Flexible Structures

Active Control Technology for Large Space Structures

Starting from basic physical considerations, this paper develops a concept of the degree of controllability of a control system, and then develops numerical methods to generate approximate values of the degree of controllability for any linear time-invariant system. In many problems, such as the control of future, very large, flexible spacecraft and certain chemical process control problems, the question of how to choose the number and locations of the control system actuators is an important one. The results obtained here offer the control system designer a tool which allows him to rank the effectiveness of alternative actuator distributions, and hence to choose the actuator locations on a rational basis. The degree of controllability is shown to take a particularly simple form when the dynamic equations of a satellite are in second-order modal form. The degree of controllability concept has still other fundamental uses - it allows one to study the system structural relations between the various inputs and outputs of a linear system, which has applications to decoupling and model reduction.

A degree of controllability definition - Fundamental concepts and application to modal systems

A method is presented which allows an integrated determination of actuator/sens or positions and feedback gains for control of flexible structures. This method is based on the maximization of dissipation energy due to control action. The optimality criterion is determined via a Liapunov equation, and it is maximized with a recursive quadratic programming algorithm. The application of this method to a simple flexible structure yields several in-general dislocated actuator and sensor locations, which are locally optimal. An extension of the method to treat spillover effects is implemented as an additional constraint to the optimization criterion.

Dislocated actuator/sensor positioning and feedback design for flexible structures

The unsolved problem of how to control the attitude and shape of future very large flexible satellite structures represents a challenging problem for modern control theory. One aspect of this problem is the question of how to choose the number and locations throughout the spacecraft of the control system actuators. Starting from basic physical considerations, this paper develops a concept of the degree of controllability of a control system, and then develops numerical methods to generate approximate values of the degree of controllability for any spacecraft. These results offer the control system designer a tool which allows him to rank the effectiveness of alternative actuator distributions, and hence to choose the actuator locations on a rational basis. The degree of controllability is shown to take a particularly simple form when the satellite dynamics equations are in modal form. Examples are provided to illustrate the use of the concept on a simple flexible spacecraft.

C. N. Viswanathan

Papers

A degree of controllability definition - Fundamental concepts and application to modal systems

A definition of the degree of controllability - A criterion for actuator placement