Infinite Zero Structure Research Articles

Modeling and Model Reference Adaptive Control of Aircraft with Asymmetric Damage

This paper addresses some fundamental issues in adaptive control of aircraft with struc- tural damage. It presents a thorough study of linearized aircraft models with damage to obtain new details of system descriptions, such as coupling and partial derivatives of lateral and longitudinal dynamics. A detailed study of system invariance under damage conditions is performed for generic aircraft models to obtain key system characterizations for model reference adaptive control (MRAC), such as infinite zero structure and signs of high frequency gain matrices. A comprehensive study of multivariable MRAC systems in the presence of damage is performed to obtain critical design specifications for adaptive flight control, such as system and controller parametrizations and adaptive parameter up- date laws. Both analytical and simulation results are given to illustrate the design and performance of adaptive control systems for aircraft flight control. Adaptive control of aircraft in the presence of damage has been an important topic in the research of flight control design for aircraft safety. Damage can cause uncertain parametric and structural variations, which requires new aircraft modeling and control approaches. In Reference (1), a study of aircraft dynamics with damage is presented, and a neural network based adaptive control algorithm is introduced for control of aircraft in the presence of structure uncertainties. In (2), equations of motion are introduced in detail for aircraft with asymmetric mass loss. In (3), we introduced a nonlinear aircraft model with partial wing damage, and illustrated linearization of such a model. In (4), real time identification of a damaged aircraft model is studied. A two-step identification process is introduced, which consists of an aircraft state estimation phase and an aerodynamic model identification step. With such a two-step process, the nonlinear part of the model identification is isolated in the first phase, and the aerodynamic parameter identification procedure is simplified to a linear one. A hybrid adaptive control method is proposed in (5) for control of aircraft with damage. The control design is based on a neural network parameter estimation blended with a direct adaptive law. A stability and convergence analysis is presented for this adaptive control methodology. For accommodating unknown changes in the structure and parameters, multivariable MRAC designs offer many advantages. In (6), we introduced an MRAC design based on the LDS decomposition of the high frequency gain matrix for the control of aircraft with multiple wing damage. The key design conditions are that, both the nominal and post-damage systems should have a uniform known modified interactor matrix, and the leading principal minors of their high frequency gain matrices should be nonzero with their signs unchanged. In (7), we studied linearization of nonlinear aircraft models under damage conditions and designed a multivariable MRAC scheme which does not require the knowledge of the signs of the high frequency gain matrix. Potential extension to aircraft flight control systems with changing signs of the high frequency gain matrix remains a topic of future research.

Squaring-Down Descriptor Systems: Constructive Solutions and Numerical Algorithms

We consider the problem of squaring-down a general descriptor linear time-invariant system. Squaring-down consists in finding a pre- and a post-compensator such that the system is turned into a square invertible one. We consider three classes of solutions: static, dynamic, and norm-preserving. All characterization are made by using generalized state-space realizations while the associated computations are performed by employing orthogonal transformations and standard reliable procedures for eigenvalue assignment. Usual benefits of classical squaring-down schemes like the stability of the designed compensators or preservation of minimum phase, stabilizability, detectability, and the infinite zero structure of the original system are recovered as well.

Structured transfer function matrices and integer matrices: the computation of the generic McMillan degree and infinite zero structure

The computation of the McMillan degree and structure at infinity of a transfer function model is considered for the family of early design models, referred to as Structured Transfer Function (STF) matrices. Such transfer functions have certain elements fixed to zero, some elements being constant and other elements expressing some identified dominant dynamics of the system. For the family of large dimension STF matrices the computation of the generic McMillan degree and structure at infinity are considered using genericity arguments which lead to optimization problems of integer matrices. A novel approach is introduced here that uses the notion of “irreducibility” of integer matrices, which is developed as the equivalent of irreducibility (properness) of polynomial matrices. This new notion provides the means for exploiting the structure of integer matrices and enables the termination of searching processes in a reduced number of steps, thus leading to an efficient new algorithm for the computation of the generic value of the McMillan degree and the structure at infinity of STFs. Links are made to standard optimization problems and to graph theory. The formulation of the optimization algorithm in terms of bipartite graphs offers better results and reduces the computational effort.

Squaring down LTI systems: A geometric approach

In this paper, the problem of reducing a given LTI system into a left or right invertible one is addressed and solved with the standard tools of the geometric control theory. First, it will be shown how an LTI system can be turned into a left invertible system, thus preserving key system properties like stabilizability, phase minimality, right invertibility, relative degree and infinite zero structure. Moreover, the additional invariant zeros introduced in the left invertible system thus obtained can be arbitrarily assigned in the complex plane. By duality, the scheme of a right inverter will be derived straightforwardly. Moreover, the squaring down problem will be addressed. In fact, when the left and right reduction procedures are applied together, a system with an unequal number of inputs and outputs is turned into a square and invertible system. Furthermore, as an example it will be shown how these techniques may be employed to weaken the standard assumption of left invertibility of the plant in many optimization problems.

A numerically reliable solution for the squaring-down problem in system design

In this paper, matrix pencil theory is used to study the squaring-down problem where a linear time-invariant system with an unequal number of inputs and outputs is turned into an invertible square system with an equal number of inputs and outputs. Both static and dynamic compensators are considered for squaring down. In the case of static compensation, the infinite-zero structure of the original system is preserved after squaring down. In the case of dynamic compensation, key system properties including stabilizability, detectability and the infinite-zero structure of the original system are also preserved after squaring down. Furthermore, one can additionally assign the invariant zeros induced by squaring down to the open left half plane, provided that the original system is stabilizable and detectable. This means that squaring down by dynamic compensation preserves minimum phaseness as well. The preservation of these system properties is highly desirable for subsequent feedback design of the squared-down system. Unlike existing squaring-down methods which do not address the issue of numerical properties, our solution is based on a condensed form derived using only orthogonal transformations which are numerically stable. Explicit formulas which can be implemented in a numerically reliable manner are given for determining the squaring-down compensators. Examples are presented to illustrate the numerical superiority of the proposed method.

Generic properties and control of linear structured systems: a survey

In this survey paper, we consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A, B, C and D, is either a fixed zero or a free parameter. The location of the fixed zeros in these matrices constitutes the structure of the system. Indeed a lot of man-made physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly, a lot of classical properties of linear systems can be studied in terms of genericity. Moreover, these generic properties can, in general, be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system-specific properties like controllability, the finite and infinite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input–output decoupling, and so on. In this paper, we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool. We emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only. Moreover, the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well-known polynomially bounded combinatorial techniques for larger systems.

On the problem of general structural assignments of linear systems through sensor/actuator selection

A systematic method is developed for determining an output matrix C for a given matrix pair ( A, B) such that the resulting linear system characterized by the matrix triple ( A, B, C) has the pre-specified system structural properties, such as the finite and infinite zero structure and the invertibility structures. Since the matrix C describes the locations of the sensors, the procedure of choosing C is often referred to as sensor selection. The method developed in this paper for sensor selection can be applied to the dual problem of actuator selection, where, for a given matrix pair ( A, C), a matrix B is to be determined such that the resulting matrix triple ( A, B, C) has the pre-specified structural properties.

On the numerical computation of a structural decomposition in systems and control

In this paper, we develop a new numerical method for a special coordinate basis of a linear time invariant system. Such a special coordinate basis is essentially a structural decomposition which explicitly displays the finite and infinite zero structures, as well as the invertibility structures of the given system. The technique is playing important roles in numerous topics in system and control theory, such as robust control, H/sub /spl infin// and H/sub 2/ optimal control almost disturbance decoupling, and zero placement of linear systems, just to name a few. Our method consists of three steps: reduction by orthogonal transformations, reduction by generalized Sylvester equations, and extraction of infinite zero structure. The performance of our method is illustrated by some numerical examples.

Generic Properties and Control of Linear Structured Systems

In this paper we deal with generic properties and control of linear systems taking into account prior structural knowledge available from physical considerations. We consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A, B, C and D, is either a fixed zero or a free parameter. The location of the fixed zeros in these matrices constitutes the structure of the system. Indeed, most physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly a lot of classical properties of linear systems can be studied in terms of genericity . Moreover, these generic properties can in general be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system specific properties like controllability, the finite and infinite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input-output decoupling, and so on. In this paper we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider a wonderful modeling and analysis tool. We emphasize the fact that this modeling technique allows us to get a number of important results based on poor information on the system only. Moreover the graph theoretic conditions are intuitive and are easy to check by hand for small systems, and by means of well-known combinatorial techniques for larger systems.

A direct and numerically stable state-space design of an interactor for a non-square system

Abstract The construction of an interactor canceling the infinite zeros of a non-square proper transfer matrix is discussed in this paper. Some new properties of the infinite eigenstrncture of the system matrix pencil of the transfer function are presented by using a numerically stable algorithm. Then, a direct and simple state-space design of the interactor is proposed. Unlike the previous result, this paper does not use the relation between the infinite zero structure of the transfer function in the frequency domain and the infinite eigenstrncture of the system matrix pencil of the transfer function for the derivation of the interactor.

Limiting zeros of decouplable MIMO systems

For sufficiently rapid zero-order hold (ZOH) sampling, it is known that the zeros of single-input/single-output discrete-time systems expressed in the forward shift operator converge to values determined only by the degrees of the finite and infinite zeros of the underlying continuous-time system. In this paper, it is shown how this result can be generalized to multi-input/multi-output (MIMO) systems decouplable by static-state feedback (equivalently, having a diagonal interactor matrix). In the fast sampling limit, the authors show how invariant and infinite zero structure is mapped under ZOH sampling for discrete-time systems expressed in either the delta (forward difference) or forward shift operators.

Introduction to a special coordinate basis for multivariable linear systems

The features of a special coordinate basis for proper linear and time-invariant multivariable systems are illustrated in the paper. Through a particular choice of state, input and output coordinates, this special coordinate basis demonstrates the physical meaning and importance of finite and infinite zero structure, the conditions for the existence of right and left inverse systems, and how to construct right and left inverse systems when they exist.

Numerically Reliable Computation of Optimal Performance in Singular $H_{\inf}$ Control

A numerically stable algorithm is described for the computation of the optimal $H_{\inf}$ performance $\gamma_{{\rm opt}}$ when the feedthrough matrices D12 or D21 are rank-deficient. Using only orthogonal transformations, the singular $H_{\inf}$ problem is reduced to a regular subproblem and a standard Riccati-based $\gamma$-iteration is applied to this subproblem to compute $\gamma_{{\rm opt}}$. Various interpretations of this scheme are given in terms of the infinite zero structure of the plant and the deflation of Hamiltonian pencils. The implementation of this algorithm is straightforward and its performance and reliability are confirmed by extensive numerical testing.

Structure Assignment of State-Output Systems by Choice of the Output Equation

Necessary and sufficient conditions to assign a given finite and infinite zero structure to the state-output system E ẋ= Ax + Bu, y = Cx + Dv by choice of C and D are given.

The state-feedback H∞-problem at optimality

We discuss the infinite horizon H ∞-control problem by state-feedback for a linear time invariant plant which is not restricted with respect to its finite or infinite zero structure. We provide an algebraic characterization for the optimal value to be attained in terms of the solvability of a nonstrict Riccati inequality and propose possibilities for verifying these criteria. If the optimum is not achieved we give a complete test for the need of high-gain feedback in order to approach the optimal value.

Zero cancelling compensators for singular control problems and their application to the inner–outer factorization problem

AbstractMany control problems fall into the category of what we call singular control problems, i.e., problems for which known solutions fail owing to the fact that the relevant transfer functions have zeros on the extended imaginary axis, 𝒞0e. An example of this is the inner–outer factorization (IOF) problem for such transfer functions.In this paper, we show how to design compensators which would allow for a direct solution to these problems through cancellation of the offending 𝒞0e zeros. This is demonstrated for the IOF problem mentioned above. Our interest lies primarily with infinite zeros but we develop formulae for finite 𝒞0e compensators as well. The paper also presents some discussions on the infinite zero structure of linear time‐invariant systems.

Explicit expressions for cascade factorization of general nonminimum phase systems

Explicit expressions for two different cascade factorizations of any detectable left invertible nonminimum phase systems are given. The first one is a well known minimum phase/all-pass factorization by which all nonminimum phase zeros of a transfer function G(s) are collected into an all-pass factor V(s), and G(s) is written G/sub m(s)V/$ where G/sub m/s is considered as a minimum phase image of G(s). The second one is a new cascade factorization by which G(s) is rewritten as G/sub M/(s)U(s) where U(s) collects all 'awkward' zeros including all nonminimum phase zeros of G(s). Both G/sub m/(s) and G/sub M/(s) retain the given infinite zero structure of G(s). Further properties of G/sub m/(s), G/sub M/(s), and U(s) are discussed. These factorizations are useful in several applications including loop transfer recovery.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Reduced form of the interactor matrix

Some structural results about the interactor of a given right invertible system are presented. An original form for this operator, which directly exhibits structural information such as infinite-zero structure or essential orders, is provided. Moreover, this form allows the linkage of the infinite and essential structures of a system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Special coordinate basis for order reduction of linear multivariable systems

A method of nested feedback loop decomposition of a given system is developed based on a special coordinate basis which reveals the finite and infinite zero structure of the system. This decomposition leads to a model order reduction procedure by which any arbitrary subset of the system invariant zeros can be retained in the model while the infinite zero structure of the model and the system are one and the same. Thus this method of model order reduction is a method of reducing the dynamic order via the zero structure of the given system. Several examples illustrate the method

The block essential orders and the dynamic block decoupling problem

In recent papers [8], it has been pointed out that two invariant lists of integers, the ‘infinite zero orders’ and the ‘essential orders’ play a key role in the solution of the so called ‘Morgan's problem’, i.e. the row by row decoupling problem. The purpose of the present paper is to define the ‘block essential orders’ of a linear system with respect to the block decoupling problem. This is done through a new dynamic solution to the latter, based upon a procedure which extends to the block case, that previously given in [15] for Morgan's problem. This procedure is minimal in the sense that it leads to the least infinite zero structure for any decoupled system reachable from the original one.