Structure assignment problems in linear systems: Algebraic and geometric methods

The Determinantal Assignment Problem (DAP) is a family of synthesis methods that has emerged as the abstract formulation of pole, zero assignment of linear systems. This unifies the study of frequency assignment problems of multivariable systems under constant, dynamic centralized, or decentralized control structure. The DAP approach is relying on exterior algebra and introduces new system invariants of rational vector spaces, the Grassmann vectors and Plücker matrices. The approach can handle both generic and non-generic cases, provides solvability conditions, enables the structuring of decentralisation schemes using structural indicators and leads to a novel computational framework based on the technique of Global Linearisation. DAP introduces a new approach for the computation of exact solutions, as well as approximate solutions, when exact solutions do not exist using new results for the solution of exterior equations. The paper provides a review of the tools, concepts and results of the DAP framework and a research agenda based on open problems

Selection of Decentralised Schemes and Parametrisation of the Decentralised Degenerate Compensators

The design of decentralised control schemes has two major aspects. The selection of the decentralised structure and then the design of the decentralised controller that has a given structure and addresses certain design requirements. This paper deals with the parametrisation and selection of the decentralized structure such that problems such as the decentralised pole assignment may have solutions. We use the approach of global linearisation for the asymptotic linearisation of the pole assignment map around a degenerate compensator. Thus, we examine in depth the case of degenerate compensators and investigate the conditions under which certain degenerate structures exist. This leads to a parametrisation of decentralised structures based on the structural properties of the system

Decentralized pole assignment for interconnected systems

Given a general proper interconnected system, this paper aims to design a LTI decentralized controller to place the modes of the closed-loop system at pre-determined locations. To this end, it is first assumed that the structural graph of the system is strongly connected. Then, it is shown applying generic static local controllers to any number of subsystems will not introduce new decentralized fixed modes (DFM) in the resultant system, although it has fewer inputoutput stations compared to the original system. This means that if there are some subsystems whose control costs are highly dependent on the complexity of the control law, then generic static controllers can be applied to such subsystems, without changing the characteristics of the system in terms of the fixed modes. As a direct application of this result, in the case when the system has no DFMs, one can apply generic static controllers to all but one subsystem, and the resultant system will be controllable and observable through that subsystem. Now, a simple observer-based local controller corresponding to this subsystem can be designed to displace the modes of the entire system arbitrarily. Similar results can also be attained for a system whose structural graph is not strongly connected. It is worth mentioning that similar concepts are deployed in the literature for the special case of strictly proper systems, but as noted in the relevant papers, extension of the results to general proper systems is not trivial. This demonstrates the significance of the present work

The Optimal Projection Equations for Fixed-Order Dynamic Compensation

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57875/1/OptimalProjectionRedOrdDynCompTAC1984.pd

Modelling and Control of an Annular Momentum Control Device

The results of a modelling and control study for an advanced momentum storage device supported on magnetic bearings are documented. The control challenge posed by this device lies in its dynamics being such a strong function of flywheel rotational speed. At high rotational speed, this can lead to open loop instabilities, resulting in requirements for minimum and maximum control bandwidths and gains for the stabilizing controllers. Using recently developed analysis tools for systems described by complex coefficient differential equations, the closed properties of the controllers were analyzed and stability properties established. Various feedback controllers are investigated and discussed. Both translational and angular dynamics compensators are developed, and measures of system stability and robustness to plant and operational speed variations are presented

Comparative studies of global and targeted control of walkway bridge resonant frequencies

In this paper, three controllers are investigated for active vibration control (AVC) of a pedestrian walkway structure. They comprise of direct velocity feedback (DVF), observer-based and independent modal space (IMSC) controllers that are implemented in single-input single-output (SISO), multi-SISO and multiple-input multiple-output (MIMO) configurations. The objective of the SISO controller schemes is to compare vibration mitigation performances arising from global control versus selective control of structural resonant frequencies in a given frequency bandwidth. The objectives set out for the multi-SISO and MIMO controllers are to realise global control within the same frequency bandwidth considered in the SISO studies. A novel aspect of these latter studies is the independent control of selected resonant frequencies at different locations on the structure with the aim of imposing global control. Vibration mitigation performances are evaluated using frequency response function measurements and uncontrolled and controlled responses to a synthesized walking excitation force. In the SISO studies, selective control of specific resonant frequencies has a slight degradation in the global vibration mitigation performance although it reflects better performance around the target frequencies. For the multi-SISO and MIMO controller studies, the selective control of the two lowest and dominant frequencies of the structure at two different locations still offers comparative vibration mitigation performances with the controllers considered as global in the sense that they target both structural frequencies at both locations. Attenuations of between 10-35 dB are achieved.The authors would like to acknowledge the financial assistance provided by the UK Engineering and Physical Sciences Research Council (EPSRC) through Platform Grant (Ref. EP/G061130/2) and Leadership Fellowship Grant (Ref. EP/J004081/2)

Structured squaring down and zero assignment

The problem of zero assignment by squaring down is considered for a system of p-inputs, n-outputs and n-states (m > p), where not all outputs are free variables for design. We consider the case where a k-subset of outputs is preserved in the new output set, and the rest are recombined to produce a total new set of p-outputs. New invariants for the problem are introduced which include a new class of fixed zeros and the methodology of the global linearization, developed originally for the output feedback pole assignment problem, is applied to this restricted form of the squaring down problem. It is shown that the problem can be solved generically if (p − k)(m − p) > δ*, where k (k < p) is the number of fixed outputs and δ* is a system and compensation scheme invariant, which is defined as the restricted Forney degree

A Near-Optimal Decentralized Servomechanism Controller for Hierarchical Interconnected Systems

This paper is concerned with decentralized output regulation of hierarchical systems subject to input and output disturbances. It is assumed that the disturbance can be represented as the output of an autonomous LTI system with unknown initial state. The primary objective is to design a decentralized controller with the property that not only does it reject the degrading effect of the disturbance on the output (for a satisfactory steady-state performance), it also results in a small LQ cost function (implying a good transient behavior). To this end, the underlying problem is treated in two phases. In the first step, a number of modified systems are defined in terms of the original system. The problem of designing a LQ centralized controller which stabilizes all the modified systems and rejects the disturbance in the original system is considered, and it is shown that this centralized controller can be efficiently found by solving a LMI problem. In the second step, a method recently presented in the literature is exploited to decentralize the designed centralized controller. It is proved that the obtained controller satisfies the pre-determined design specifications including disturbance rejection. Simulation results elucidate the efficacy of the proposed control law

Topological Conditions for Wireless Control Networks

We study the problem of stabilizing a linear system over a wireless control network. We propose a scheme where each wireless node maintains a scalar state, and periodically updates it as a linear combination of neighboring plant outputs and node states. We make connections to decentralized fixed modes and structured system theory to provide conditions on the network topology that allow the system to be stabilized. Our analysis provides the minimal number of feedback edges that have to be introduced to stabilize the system over a network, and shows that as long as the network connectivity is larger than the geometric multiplicity of any unstable eigenvalue, stabilizing controllers can be constructed at each actuator. A byproduct of our analysis is that by co-designing the network dynamics with the controllers, delays in the network are not a factor in stabilizing the system

A decentralized linear quadratic control design method for flexible structures

A decentralized suboptimal linear quadratic control design procedure which combines substructural synthesis, model reduction, decentralized control design, subcontroller synthesis, and controller reduction is proposed for the design of reduced-order controllers for flexible structures. The procedure starts with a definition of the continuum structure to be controlled. An evaluation model of finite dimension is obtained by the finite element method. Then, the finite element model is decomposed into several substructures by using a natural decomposition called substructuring decomposition. Each substructure, at this point, still has too large a dimension and must be reduced to a size that is Riccati-solvable. Model reduction of each substructure can be performed by using any existing model reduction method, e.g., modal truncation, balanced reduction, Krylov model reduction, or mixed-mode method. Then, based on the reduced substructure model, a subcontroller is designed by an LQ optimal control method for each substructure independently. After all subcontrollers are designed, a controller synthesis method called substructural controller synthesis is employed to synthesize all subcontrollers into a global controller. The assembling scheme used is the same as that employed for the structure matrices. Finally, a controller reduction scheme, called the equivalent impulse response energy controller (EIREC) reduction algorithm, is used to reduce the global controller to a reasonable size for implementation. The EIREC reduced controller preserves the impulse response energy of the full-order controller and has the property of matching low-frequency moments and low-frequency power moments. An advantage of the substructural controller synthesis method is that it relieves the computational burden associated with dimensionality. Besides that, the SCS design scheme is also a highly adaptable controller synthesis method for structures with varying configuration, or varying mass and stiffness properties