Non-square Linear Systems Research Articles

Non-Convex Optimization: Using Preconditioning Matrices for Optimally Improving Variable Bounds in Linear Relaxations

The performance of branch-and-bound algorithms for solving non-convex optimization problems greatly depends on convex relaxation techniques. They generate convex regions which are used for improving the bounds of variable domains. In particular, convex polyhedral regions can be represented by a linear system A.x=b. Then, bounds of variable domains can be improved by minimizing and maximizing variables in the linear system. Reducing or contracting optimally variable domains in linear systems, however, is an expensive task. It requires solving up to two linear programs for each variable (one for each variable bound). Suboptimal strategies, such as preconditioning, may offer satisfactory approximations of the optimal reduction at a lower cost. In non-square linear systems, a preconditioner P can be chosen such that P.A is close to a diagonal matrix. Thus, the projection of the equivalent system P.A.x=P.b over x, by using an iterative method such as Gauss–Seidel, can significantly improve the contraction. In this paper, we show how to generate an optimal preconditioner, i.e., a preconditioner that helps the Gauss–Seidel method to optimally reduce the variable domains. Despite the cost of generating the preconditioner, it can be re-used in sub-regions of the search space without losing too much effectiveness. Experimental results show that, when used for reducing domains in non-square linear systems, the approach is significantly more effective than Gauss-based elimination techniques. Finally, the approach also shows promising results when used as a component of a solver for non-convex optimization problems.

Fast and Practical Algorithms for Solving Non-square Linear Systems

Fast and Practical Algorithms for Solving Non-square Linear Systems

Sensor blending and Control allocation for non-square linear systems to achieve negative imaginary dynamics

This paper deals with the design of static pre- and post- compensators to transform stable, non-square LTI systems into the class of strongly strict negative imaginary systems. The pre-compensator plays the role of a control allocator while the post-compensator does sensor blending in order to make a non-square system square along with satisfying the strongly strict negative imaginary property. A specific structure of the post-compensator is also given that guarantees a feasible solution of the LMI conditions when applied to systems with number of outputs greater than or equal to number of inputs. The proposed pre- and post- compensators can also stabilize a non-square plant in closed-loop upon satisfying a particular DC-gain condition and furthermore, they can be utilized to develop a simple constant input tracking framework for non-square systems. The LMI-based design methodology offers a numerically tractable solution framework and hence the easy implementation of the proposed scheme in practical applications. Illustrative examples are provided throughout the paper to demonstrate the usefulness of the proposed results in widening the scope of the negative imaginary theory to non-square LTI systems (e.g. safety-critical systems having redundant sensors and actuators).

On a geometric characterisation of zeros for non-square linear systems with time-delay in state

The concept of invariant zeros in a linear time-invariant system with point delay in state vector is discussed in the state space framework. These zeros are treated as the triples: complex number, non-zero state-zero direction and input-zero direction. Such treatment is strictly related to the output-zeroing problem and in that spirit the zeros can be easily interpreted. As is shown, for systems with matrix CB of full row-rank, general formulas for output-zeroing inputs can be obtained as well as a characterisation of invariant zeros as the roots of a certain quasi-polynomial can be given. The question of degeneracy/non-degeneracy of the system is also addressed. Moreover, it is shown that diagonal decoupling can be achieved by constant state feedbacks and a pre-compensator. The transfer matrix of the decoupled system is square and does not contain delay. The mathematical tools used in the analysis are the Moore–Penrose pseudo-inverse and singular value decomposition of a matrix.

Transformation of high order linear differential-algebraic systems to first order

We study general nonsquare linear systems of differential-algebraic systems of arbitrary order. We analyze the classical procedure of turning the system into a first order system and demonstrate that this approach may lead to different solvability results and smoothness requirements. We present several examples that demonstrate this phenomenon and then derive existence and uniqueness results for differential-algebraic systems of arbitrary order and index. We use these results to identify exactly those variables for which the order reduction to first order does not lead to extra smoothness requirements and demonstrate the effects of this new formulation with a numerical example.

선형 시스템 수동화를 위한 병렬 앞먹임 보상기 설계방법 연구

A passivity-based dynamic output feedback controller design is considered for a finite collection of non-square linear systems. Design of a single controller for a set of plants i.e. simultaneous stabilization is an important issue in the area of robust control design. We first determine a squaring gain matrix and an additional dynamics that is connected to the systems in a feedforward way, then a static passivating control law is designed. Consequently, the actual feedback controller will be the static control law combined with the feedforward dynamics. A necessary and sufficient condition for the existence of the parallel feedforward compensator is given by the static output feedback formulation. In contrast to the previous result [1], a technical condition for constructing the parallel feedforward compensator is removed by proposing a new type of the parallel compensator.

Discussion on: “Passification of Non-square Linear Systems and Feedback Yakubovich-Kalman-Popov Lemma”

Discussion on: “Passification of Non-square Linear Systems and Feedback Yakubovich-Kalman-Popov Lemma”

Passification of Non-square Linear Systems and Feedback Yakubovich-Kalman-Popov Lemma

The concepts of G-passivity and G-passifiability (feedback G-passivity) are introduced extending the concepts of passivity and passifiability to nonsquare systems (systems with different numbers of inputs and outputs). Necessary and sufficient conditions for strict G-passifiability of nonsquare linear systems by output feedback are given. Simple description of a broad subclass of passifying feedbacks is proposed. The proofs are based on a version of the celebrated Yakubovich-Kalman-Popov lemma.

수동화 기법에 의한 비정방 선형 시스템의 강인 제어기 설계

We present a state-space approach to design a passivity-based dynamic output feedback control of a finite collection of non-square linear systems. We first determine a squaring gain matrix and an additional dynamics that is connected to the systems in a feedforward way, then a static passivating (i.e. rendering passive) control law is designed. Consequently, the actual feedback controller will be the static control law combined with the feedforward dynamics. A necessary and sufficient condition for the existence of the parallel feedfornward compensator (PFC) is given by the static output feedback fomulation, which enables to utilize linear matrix inequality (LMI). The effectiveness of the proposed method is illustrated by some examples including the systems which can be stabilized by the proprotional-derivative (PD) control law.

Bond-Graph Modeling and Geometric Approach: Analysis of Non Square Systems

In this paper, the geometric approach and the bond-graph methodology are combined to characterize the structure of non-square linear systems modeled by bondgraph. A new graphical methodology is proposed to emphasize, directly on the bond graph representation, the symbolic expressions of non regular static state feedbacks ensuring decouplability with stability.