On the algebraic structure of quadratic and bilinear dynamical systems

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Abstract

A number of important control and estimation problems in the field of aerospace and avionics involve dynamical models that are quadratic and bilinear functions of system states and inputs. In this paper, formal definitions of free and forced quadratic/bilinear dynamical systems are given and the algebraic structure of this class of systems is explored with a view to setting up a systematic approach for deriving state and measurement models. Properties of quadratic and bilinear vectors are investigated, and relationships between these and linear vectors established. Systematic procedure for constructing quadratic state and bilinear state–input vectors is derived. The quadratic/bilinear vector modeling technique is applied to the formulation of a state–space model for a second-order approximation of a general non-linear system.

Introduction

A number of important control and estimation problems in the field of aerospace and avionics involve dynamical models that are quadratic and bilinear functions of system states and inputs. Examples of some of these problems are: target tracking, vehicle navigation and guidance and autopilot design [1], [2], [3], [4], [5], [6], [7], [8]. While there are a number of different ways of posing (modeling) the above problems, it appears that by appropriate choice of state and input variables, systems involving translational and rotational kinematics can be modelled as a set of differential equations that contain linear plus quadratic and/or bilinear expressions. The well-known Riccati differential equation [9], [10] that occurs in the synthesis of optimal control and estimation problems, is also of this class of differential equations.

Other applications of bilinear systems are in the field of industrial process control, economics, biological systems, and chemical process modelling and control [11], [12], [13]. In addition to the applications, various researchers have also reported on important developments in bilinear systems theory and related stabilization, observability and controllability issues, including the Lie-algebra approach [14], [15], [16]. The control synthesis problem, involving bilinear systems, has been addressed by a number of authors utilising the optimum control theory [17], [18], [19], [20]. While the above research lays down a fundamental and, in many respects, rigorous basis, for system analysis and synthesis for this class of non-linear systems, there still remains a need to apply the results of these investigations to high dimension systems (involving large number of state and control variables) that arise in many practical engineering problems. Motivated by this need, the current report addresses issues relating to the algebraic structure of quadratic and bilinear dynamical systems and proposes an approach for state–space formulation of the dynamical models is presented that should allow for systematic and generalized procedures in dealing with practical control analysis and synthesis problems.

In this report, formal definitions of free and forced quadratic/bilinear dynamical systems are given and the algebraic structure of this class of systems is explored with a view to setting up a systematic approach for deriving state dynamics and measurement models. This class of dynamical systems is characterized by a set of first-order differential equations with the RHS that contains linear and quadratic terms in system states as well as bilinear terms involving state and input (or control) variables. Properties of quadratic and bilinear vectors are investigated and relationships between these and linear vectors are established. Systematic procedure for constructing quadratic state and bilinear state–input vectors is derived. The quadratic algebraic structure, developed in this report, is applied in the formulation of a state–space model for a second-order approximation to a general (analytic) non-linear system.

The concept of quadratic and bilinear generator matrices is introduced that allows linear (state and state–input) vectors to be mapped onto quadratic and bilinear vectors. Properties of these generator matrices are explored and it is shown that corresponding inverse generator matrices may be defined that allow quadratic and bilinear vectors to be mapped onto linear state and linear input vectors. State–space representation of the system output or measurement model is also derived for the case where this contains linear and quadratic terms in states and bilinear terms in states and disturbance (noise) variables.

The state–space structure of the quadratic/bilinear dynamical systems considered in this report should facilitate analysis of this class of non-linear systems and possibly lead to general synthesis techniques or extension of linearised techniques for applications to these problems.

The algebraic structure of free quadratic dynamical systems is considered in Section 2 of this report and includes consideration of various properties of quadratic state vectors and their relationships with linear state vectors. Bilinear vectors are considered in Section 3 along with the properties of these vectors and their relationships with linear vectors. 4 Forced quadratic-bilinear dynamical system, 5 Quadratic-bilinear system output model consider forced dynamical systems and measurement models where the latter includes linear as well as quadratic terms in state variables and bilinear terms involving state and disturbance variables. In Section 6, general non-linear dynamical systems with analytic non-linearity is considered and a second-order perturbation state–space model of the system is derived in a quadratic/bilinear vector form.

Section snippets

Free dynamical system

In this section, we formalise the definition of a class of non-linear dynamical systems whose time evolution of states may be expressed via a set of first-order differential equations, the right-hand side (RHS) of which contain both linear and quadratic state terms. A formal definition will lead to a state vector characterisation (or a state–space representation) of these systems. The class of dynamical systems being considered in this section will be referred to as free (or uncontrolled)

Bilinear state–input vector

Before we consider forced-QBDS, we shall formalize the structure and definition of a bilinear state–input vector (bsiv) generated from two linear vectors, namely, the n-dimensional state vector x[1]∈Rn and the m-dimensional input vector u[1]∈Rm. In order to develop this concept we shall utilize the familiar scalar bilinear forms and its various properties. Accordingly, we consider the following scalar bilinear form:Πi(x[1],u[1])=∑k=1nj=1msk,j[i]xkuj=∑k=1mj=1nsj,k[i]xjuk.In a more familiar

Forced quadratic-bilinear dynamical system

We shall now consider the algebraic structure of a forced-quadratic/bilinear system where the RHS of the differential equations consists of linear and quadratic terms identical to those in the free-QDS as well as additional linear input terms and bilinear terms involving inputs (forcing functions or control inputs) and system states.

Quadratic-bilinear system output model

Utilising the definitions given in previous sections, dynamical system output or measurement models may be derived. Following on from the results of previous sections, the system output model may be written as:z[1]=[H[1]]x[1]+[J[2]]x[2]+Π[2](x[1],v[1])+[L[1]]v[1]z[1]=[H[1]]x[1]+[J[2]]x[2]+[M[2]]π[2](x[1],v[1])+[M[2]]π[2](x[1],v[1])+[L[1]]v[1].Or using qgm and bgms asz[1]=[H[1]]+[J[2]]X[1][2]x[1]+[L[1]]+[M[2]]X[1][2]+[M[2]]X[1][2]v[1],z[1]=[H[1]]+[J[2]]X[1][2]+[M[2]]V[1][2]+[M[2]]V[1][2]x

Extension to a second-order approximation of analytical functions

One obvious application of the quadratic/bilinear algebraic structure discussed in the previous sections is in the second-order approximation of analytical functions. Non-linear dynamical systems with RHS that consists of analytical functions can be conveniently set up as a set of quadratic/bilinear differential equations. In order to demonstrate this we consider the following non-linear dynamical system of the form:ddtxi=fi(x,u);i,1,2,…,nwith the output model given byzj=hj(x,v);j=1,2,…,r,where

Conclusion

In this report, formal definitions of free and forced quadratic/bilinear dynamical systems were given and the algebraic structure of this class of systems was explored with a view to setting up a systematic approach for deriving state dynamics and measurement models. This class of dynamical systems is characterized by a set of first-order differential equations with the RHS that contains linear and quadratic terms in system states as well as bilinear terms involving state and input (or control)

Acknowledgements

The author gratefully acknowledges the constant encouragement and support, over the years, by Mr. R.V. Lawrence (DSTL, UK) for the work on quadratic/bilinear dynamical systems; he was instrumental in sponsoring initial work on this topic some twenty years ago. The author also acknowledges contributions of his former Ph.D. students Dr. Ken Turner and Dr. Subhash Challa, both for furthering the state-of the-art in this field and for lively and imaginative discussions on the topic. Finally, the

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