Stability, Controllability and Observability

Introduction

The chapter contains a discussion on some fundamental system properties. Stability from a geometric point of view, is related to the properties of system trajectories around an equilibrium point. Elementary Lyapunov techniques are employed to analyze and quantify the stability of a linear system . controllability is another grometric property of a system, describing the ability to �drive� the system states to arbitrary values through the control input. Its dual notion of absorbability describes the ability to infer the system states given output measurements in an interval. An elegant analysis of these structural properties is presented using sector space methods

Stability

This notion of stability is different from the input-output (operator) stability where a system is L- stable if any input in L produces an output in L. here L is a sector space, eg bounded functions, energy functions etc .the input in describing performance specifications. On the other hand Lyapunov stability is suitable to describe convergence properties and provides a more appealing computational framework. While in the case of linear time invariant systems the two stability notions are closely related, their differences become more pronounced (and technically involved) in the general nonlinear case.

The basic Lyapnov analysis begins with a positive definite function of the states, interpreted as the energy stored in the system, e.g. V(x) = xT px where P= PT>0. a sufficient condition for the asymptotic stability (stability ) of the zero equilibrium is that the derivative of thus function along the trajectories of the system (dV/dt =(oV0x)x) is negative definite (semi-definite). This can be viewed as a condition on the energy dissipation within the system. It is also a necessary condition in the sense that if an equilibrium is asymptotically stable, then there exists a Lyapunov function with the above properties . in general it is difficult to construct such a function. Nevertheless, in the case of linear systems the lyapunov functions are quadratic making their computation a straightforward exercise in matrix algebra.

To demonstrate the application of Lyapunov analysis, let us consider the system x= Ax and the function V = xT px the derivative of V along the trajectories of the system is computed as follows:

Notice that, for a Hurwitz matrix A not every positive definite P produces a positive definite Q : only the converse holds . the equation is referred to as Lyapunov . it is linear in P and can be solved as a system of linear equation has a unique solution (positive definite or not ) iff any two eigenvalues of P satisfy .from a system theoretic , a more interesting property of Lyapunov equations is that for a Hurwitz A their solution has the form

The last expression is extremely important for its analytical value. Among other applications, it allows an easy computation of controllability and observability Gramians as solutions of linear Lyapunov equations. These are an integral component of general model order reduction algorithms.

Lyapunov equations play an important role in several recent results on the design of control system via numerical optimization. For example consider the intermediate) problem where given a matrix A we would like to estimate the exponential rate of decay of the states to zero. This can be found as the real part of the eigenvalue of A closest to the jw-axis. However eigenvalues are nonlinear functions of the entries of A and are not suitable objectives for any (additional) optimization. Alternatively, we can ask to find the matrix Q that maximizes the ratio as previously shown this ratio provides an estimate of the rate of decay of the states can be shown that the optimal Q for this purpose is the identity. This problem can be cast as the optimization of a convex objective subject to convex constraints (linear matrix inequalities), and its solution can be obtained with numerically efficient and reliable algorithms. The value estimate the jw-axis. However Of this approach lies in its ability to handle case where the matrix A is itself a convex function of other parameters. A simple example of that is to find a single Lyapunov function if it exists, that has a negative definite derivative for two systems , i.e,

The existence of such a P would imply the stability of a system whose matrix A undergoes arbitrarily fast transitions between the values A1 and A2. this type of problems arises in the analysis and design of gain-scheduled control systems.

For linear systems it is straightforward to show that exponential stability of the zero equilibrium (A being Hurwitz) also implies the input-output stability (in a BIBO or energy sense) of the system [A,B,X,D], for any B,C,D. the converse is not always true unless some additional conditions are imposed, e.g., controllability and observability. Furthermore, a somewhat similar statement is valid in a general nonlinear setting but with significantly more involved technical conditions.