The Riccati Equation

Towards the turn of the seventeenth century, when the baroque was giving way to the enlightenment, there lived in the Republic of Venice a gentleman, the father of nine children, by the name of Jacopo Franceso Riccati. On the cold New Year’s Eve of 1720, he wrote a letter to his friend Giovanni Rizzetti, where he proposed two new differential equations. In modern symbols, these equations can be written as follows: where m is a constant. This is probably the first document witnessing the early days of the Riccati Equation, an equation which was to become of paramount importance in the centuries to come.

This review is concerned with two algebraic Riccati equations. The first is a quadratic matrix equation for an unknown n × n matrix X of the form where A, D, C are n × n complex matrices with C and D hermitian. Further hypotheses are imposed as required, although Section 2.3 contains some discussion of more general non-symmetric quadratic equations. The second equation has the fractional form where R and Q are hermitian m × m and n × n matrices, respectively, and A, B, C are complex matrices with respective sizes n × n, n × m, and m × n. The two equations are frequently referred to as the “continuous” and “discrete” Riccati equations, respectively, because they arise in physical optimal control problems in which the time is treated as a continuous variable, or a discrete variable.

Let A, B, and C be constant n × n matrices with entries in C, the field of complex numbers. Let B and C be hermitian, i.e., B = B* and C = C*, where an asterisk is used to denote the conjugate transpose of a matrix. The quadratic equation for the n × n complex matrix X is called the algebraic Riccati equation.

This is a tutorial paper which describes a “geometric approach” to the description of the phase portrait of the Riccati differential equation. As such, no new results are presented. Our intention is to show how the geometric viewpoint gives insight into many of the properties of the Riccati differential equation. It is not our purpose to present a comprehensive exposition of all that is presently known on the subject Instead, we will willingly make (mostly) generic assumptions and focus on the properties of the differential equation under these simplifying assumptions. For more details, some generalizations and additional references, the reader is referred to the papers [4.14] (time-invariant coefficients) and [4.13] (periodic coefficients).

The mathematics used to study the matrix Riccati equation is as widely varied as its applications and occurrences. In this chapter we consider some of the geometric aspects of the equation and by necessity must consider some basic differential and algebraic geometry. This mathematics is at least as simple as the classical analysis used to study ordinary differential equations and is probably more intuitive. However a certain level of understanding is necessary. In the book of Doolin and Martin, [5] there is an introduction to the subject that is designed to appeal to ones intuition rather than to ones mathematical skills. For the considerations of length we refer the reader to that book and hope that this will encourage the reader to pursue more advanced monographs for the future study of the geometry of the matrix Riccati equation.

The history of the time-varying Riccati equation can be traced back to Riccati’s original manuscripts of 1715–1725. Indeed, the major concern of Count Riccati was to study the problem of the separation of variables in quadratic and time-varying scalar differential equations [1]. The equation has been the subject of several contributions in the subsequent centuries. In recent times, the importance of the Riccati equation in Control, Systems, and Signals has led to the development of a considerable research activity on the subject, see e.g., [2], [3], [4] for the time-varying matrix Riccati equation.

In this tutorial paper, an overview is given of progress over the past ten to fifteen years towards reliable and efficient numerical solution of various types of Riccati equations. Our attention will be directed primarily to matrix-valued algebraic Riccati equations and numerical methods for their solution based on computing bases for invariant subspaces of certain associated matrices. Riccati equations arise in modeling both continuous-time and discrete-time systems in a wide variety of applications in science and engineering. One can study both algebraic equations and differential or difference equations. Both algebraic and differential or difference equations can be further classified according to whether their coefficient matrices give rise to so-called symmetric or nonsymmetric equations. Symmetric Riccati equations can be further classified according to whether or not they are definite or indefinite.

Undoubtedly one of the most important concepts in linear systems and control, both from a theoretical as well as from a practical point of view, is the algebraic Riccati equation. Since its introduction in control theory by Kaiman [16] the beginning of the sixties, the algebraic Riccati equation has known an impressive range of applications, such as linear quadratic optimal control, stability theory, stochastic filtering and stochastic control, stochastic realization theory, synthesis of linear passive networks, differential games and, most recently, H _∞ optimal control and robust stabilization. The purpose of the present paper is to give an expository survey of the main concepts, results and applications related to the algebraic Riccati equation.

One of the main reasons why the Riccati equation and its generalizations has become very important in the theory of control, systems, and signals, is that it shows up in a very straightforward way in the analysis of two benchmark problems in control system design and signal filtering.

The main theme of this Chapter will be the connections between various Riccati equations and the closed loop stability of control schemes based on Linear Quadratic (LQ) optimal methods for control and estimation. Our presentation will encompass methods applicable both for discrete time and continuous time, and so we discuss concurrently the difference equations (discrete time) and the differential equations (continuous time) — the intellectual machinery necessary for the one suffices for the other and so it makes sense to dispense with both cases in one fell swoop.

The discrete- and continuous-time Riccati equations, which play a prominent role in linear-quadratic control and filtering theory (as discussed extensively in other chapters of this book), appear also in discrete- and continuous-time dynamic games, albeit in more general forms. Both the existence and the characterization of nonco-operative equilibria in zero-sum and nonzero-sum linear-quadratic dynamic games, under saddle-point, Nash and Stackelberg equilibrium concepts, involve the solutions of these generalized matrix Riccati (differential or algebraic) equations.