A Direct Algorithm for Pole Placement by State-derivative Feedback for Single-input Linear Systems

This paper deals with the direct solution of the pole placement problem for single-input linear systems using state-derivative feedback. This pole placement problem is always solvable for any controllable systems if all eigenvalues of the original system are nonzero. Then any arbitrary closed-loop poles can be placed in order to achieve the desired system performance. The solving procedure results in a formula similar to the Ackermann formula. Its derivation is based on the transformation of a linear single-input system into Frobenius canonical form by a special coordinate transformation, then solving the pole placement problem by state derivative feedback. Finally the solution is extended also for single-input time-varying control systems. The simulation results are included to show the effectiveness of the proposed approach

LTV controller flatness-based design for MIMO systems

In this paper, a flatness-based control strategy for multi-input multi-output linear time-varying systems is proposed in order to track desired trajectories. The control design, based on the use of an exact observer, leads to a polynomial two-degree-of-freedom controller without resolving Bézout’s equation in a time-varying framework. The proposed approach is illustrated with the control of a nonlinear model of the satellite SPOT-5

Nonlinear Tracking Control Using a Robust Differential-Algebraic Approach.

This dissertation presents the development and application of an inherently robust nonlinear trajectory tracking control design methodology which is based on linearization along a nominal trajectory. The problem of trajectory tracking is reduced to two separate control problems. The first is to compute the nominal control signal that is needed to place a nonlinear system on a desired trajectory. The second problem is one of stabilizing the nominal trajectory. The primary development of this work is the development of practical methods for designing error regulators for Linear Time Varying systems, which allows for the application of trajectory linearization to time varying trajectories for nonlinear systems. This development is based on a new Differential Algebraic Spectral Theory. The problem of robust tracking for nonlinear systems with parametric uncertainty is studied in relation to the Linear Time Varying spectrum. The control method presented herein constitutes a rather general control strategy for nonlinear dynamic systems. Design and simulation case studies for some challenging nonlinear tracking problems are considered. These control problems include: two academic problems, a pitch autopilot design for a skid-to-turn missile, a two link robot controller, a four degree of freedom roll-yaw autopilot, and a complete six degree of freedom Bank-to-turn planar missile autopilot. The simulation results for these designs show significant improvements in performance and robustness compared to other current control strategies

An iterative approach to eigenvalue assignment for nonlinear systems

In this paper, the authors present a method for controlling a nonlinear system by using the ideas of eigenvalues assignment. A time-varying approach to nonlinear exponential stability via eigenvalue placement is studied based on an iteration technique that approaches a nonlinear system by a sequence of linear time varying equations. The convergent behaviour of this method is shown and applied to a practical nonlinear example in order to illustrate these ideas

Algebrogeometric and topological methods in control theory

The aim of this thesis is to provide a unifying framework and tools for the study of a number of Control Theory problems of the determinantal type. These problems are known as Frequency Assignment Problems (FRT) and they include the constant, dynamic, pole, zero assignment by centralised as well as decentralised output feedback and the zero assignment problems via squaring down. It has been shown [Kar.1],[Gia.2] that all such problems may be formulated under the unifying framework of the Determinantal Assignment Problem (DAP), and it can be studied using tools from exterior algebra and algebraic geometry. The main objective of this thesis is to develop further the DAP framework, unify it with other algebrogeometric approaches and develop issues related to computation and parametrisation of solutions when such solutions exist. The natural setup for the study of solutions of the DAP framework has been the intersection theory of projective varieties. This has been extended by developing the topological properties of the pole, zero placement maps and introducing an equivalent formulation for real intersection based on cohomology theory. The properties of this map, with respect to standard system invariants are also established. This approach allows the derivation of new conditions for constant pole, zero assignment with centralised and decentralised controllers, using conditions based on the height of an appropriate cohomology class. Affine algebraic geometry methods are also used for the derivation of partial results for the dynamic case corresponding to PI and OBD controllers. An entirely new approach for the study of solvability of DAP, as well as computation of solutions is introduced in terms of the notion of global linearisation of the corresponding pole, zero assignment map around a degenerate point. This is based on the special “blow up” property of the pole placement map at degenerate feedbacks and permits the reduction of the overall DAP to a globally linear problem, the solvability of which is defined by the properties of a new local invariant, the “blow up” matrix

Approximate controllability and observability measures in control systems design

The selection of systems of inputs and outputs (input and output structure) forms part of early system design, which is important since it preconditions the potential for control design. Existing methodologies for input, output structure selection rely on criteria expressing distance from uncontrollability (unobservability). The thesis introduces novel measures for evaluating and estimating the distance to uncontrollability and relatively unobservability. At first, the modal measuring approach is studied in detail, providing a framework for the ”best” structure selection. Although controllability (observability) is invariant under state feedback (output injection), the corresponding degrees expressing distance from uncontrollability (unobservability) are not. Hence, the thesis introduces new criteria for the distance problem from uncontrollability (unobservability) which is invariant under feedback transformations. The approach uses the restricted input-state (state-output) matrix pencil and then deploys exterior algebra that reduces the overall problem to the standard problem of distance of a set of polynomials from non-coprimeness. Results on the distance of the Sylvester Resultants from singularity provide the new measures. Since distance to singularity of the corresponding Sylvester matrix is the key in evaluating the distance to uncontrolability it is of the particular interest in the present work. In order to find the solution two novel methods are introduced in the thesis, namely the alternating projection algorithm and a structured singular value approach. A least-squares alternating projection algorithm, motivated by a factorisation result involving the Sylvester resultant matrix, is proposed for calculating the ”best” approximate GCD of a coprime polynomial set. The properties of the proposed algorithm are investigated and the method is compared with alternative optimisation techniques which can be employed to solve the problem. It is also shown that the problem of an approximate GCD calculation is equivalent tothe solution of a structured singular value (µ) problem arising in robust control for which numerous techniques are available. Motivated by the powerful concept of the structured singular values, the proposed method is extended to the special case of an implicit system that has a wide application in the behavioural analysis of complex systems. Moreover, µ-value approach has a potential application for the general distance problem to uncontrollability that is numerically hard to obtain. Overall, the proposed framework significantly simplifies and generalises the input-output structure selection procedure and evaluates alternative solutions for a variety of distance problems that appear in Control Theory

一般化出力誤差の最小化に基づくデータ指向型PID制御器の設計

広島大学(Hiroshima University)博士(工学)Engineeringdoctora

A unified approach to decentralised control based on the exterior algebra and algebraic geometry methods

The aim of the thesis is to provide a unifying framework and tools for the study of a number of control theory problems arising in Decentralised control. These problems are known as frequency assignment problems and they include the decentralised problem of pole assignment by state, output feedback and zero assignment by decentralised squaring-down. It is further shown that decentralised dynamic problems where the dynamic complexity of the controller is fixed , may also be reduced to the same formulation given for the constant Determinantal Assignment Problem(DAP). The unifying mathematical problem studied here is the Decentralised - Determinantal Assignment Problem (D- DAP) which is a special form of the general DAP defined for multivariable systems. The study of D-DAP involves the use of tools from exterior algebra in an essential way and also tools from classical algebraic geometry, specifically the theory of Grassmann varieties. In this thesis the mathematical framework of D-DAP is fully developed and used for the study of pole, zero assignment problems by decentralised constant controllers and new solvability conditions are given. The mathematical framework of D-DAP also allows the study of structural properties of Decentralised controllers. In fact new invariants for decentralised control are introduced in the form of Pliicker matrices and Decentralised Indices and a new characterisation of fixed modes and fixed zeros is given based on tools from exterior algebra. The classical notion of fixed modes, fixed zeros is extended to almost fixed modes, almost fixed zeros for certain families of systems. Finally it is shown that the general framework of D-DAP is suitable for the study of dynamic problems such as pole assignment by decentralised classical controllers of the P-D, P-I, P-I-D type

Linear systems and control structure selection

This thesis is concerned with the development of concepts and results to facilitate study in two areas of control methodology. The two notions investigated are measures of controllability and observability and eigenstructure assignment. The link between these two areas is exposed, and it is demonstrated how the eigenvectors of a system play an important role in determining the degree of controllability and observability. The main concerns are issues dealing with the complexity of the instrumentation, and in particular the development of techniques that may assist in the development of methodology for sensor and actuator placement. The research involves the development of notions that help to structure a system on which control design is based. There are two areas of investigation. The first is the development of concepts and tools that aid in the selection and placement of sensors and actuators based on properties related to degrees of controllability and observability. The second is the investigation of the eigenstructure of a system and its properties, which enable the development of design procedures based on eigenstructure properties. A study of existing measures of controllability and observability leads to new techniques which take into consideration the problem of coordinate transformations, which is often overlooked. It is shown that the degree of controllability is influenced by changes in the structure of the state feedback matrix, as well as how controllability properties can be determined from Pliicker matrices of transfer function matrices. It is also shown that the energy required to move a system from one state to another is linked to the singular values of the output controllability grammian. A review of the problem of eigenstructure assignment paves the way for the development of a new technique of assigning the closed loop eigenstructure. This is based on matrix fraction description algorithms, and stems from an algebraic description of the total system behaviour, leading to a systematic study of closed loop eigenvectors by using a parametric approach. A new algebraic characterisation of the family of closed loop eigenvectors and related input and output directions is shown. Closed loop system robustness to parameter variations is also considered, where it is shown that there is a link with the orthogonality of the matrix of eigenvectors. As a result, the notion of strong stability is introduced, where it is shown that the shape of the eigenframe plays a role in the system response by way of overshoots. The work develops concepts and results which are important steps in the development of an integrated methodology for input, output structure selection