7. THE CLASSICAL APPROACH TO TIME-VARIANT (NON- AUTONOMOUS), LINEAR & NONLINEAR SYSTEMS

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1 7. THE CLASSICAL APPROACH TO TIME-VARIANT (NON- AUTONOMOUS), LINEAR & NONLINEAR SYSTEMS that complicates the analysis of TV systems. Thus, we will also learn how to overcome this problem by adding additional constraints on the system. In section 7.4, we discuss the extension of the Lyapunov direct method for stability analysis of non-autonomous systems. We will learn again how the initial time is excluded from the various definitions. Finally in section 7.5, we look at the extension of some autonomous systems' analysis tools to non-autonomous systems. From these limited discussions, it should become clear that the topdown approach is not an efficient way for the design and application of nonautonomous systems. 7.2 Non-autonomous System Models and Examples 7.1 Introduction to the chapter In this chapter, we briefly discuss time-varying or (TV) non-autonomous systems and present some basic concepts of stability for time-varying systems. Again, the emphasis is to present the basic results (that are derived from a mathematical/analytical perspective) in as simply as possible. In the next chapter, we will look at non-autonomous systems from a physical realization perspective and derive a number of new results and techniques for designing stable non-autonomous systems. The reasons for considering non-autonomous systems is that in a number of physical systems, the parameters associated with physical systems (that may or may not appear in the dynamical equations used to represent the systems) may vary with time. Some example parameters that play a major role in many aeronautical systems are temperature and pressure. The key here are the two statements: 1) "parameters associated with physical systems," and 2) "vary with time," shown highlighted here. Usually, greater emphasis is placed on the latter statement in the analytical approach to dealing with non-autonomous systems. Thus, time-varying is used to imply the appearance of terms such as "t, exp[t], sin [t]" etc. in the dynamical equations and results based on such modeling techniques appear in the literature. In this chapter, results based on such an approach are presented. In the next chapter, we will discuss a new approach that makes use of the above two key statements in defining physically meaningful devices and their use in forming complex systems and or dynamics. The chapter is organized as follows. In section 7.2, we provide the general mathematical model commonly used and a number of first- or second-order examples. It should be obvious from these examples and definitions that we do not have any physical guiding principles other than our mastery of differential equations in arriving at the dynamics. In section 7.3, we introduce the concept of equilibrium points of non-autonomous systems and the definitions for stability. We will find, by necessity, we need to include the initial time in their definitions We can simply extend the state-space representation to model non-autonomous systems as: x = f[ x,t] (7.1) Here, the independent variable 't ' appears along with the state vector x in the nonlinear vector function f [ ] indicating that it depends on the time variable (or it is an explicit function of the time variable). That is, f [ ] is a vector function that varies with respect to time. Examples of non-autonomous systems appearing in circuits and control theory literature are: where x (t) = a(t)x(t) (7.2) a(t) x (t) = x(t) (7.3) 1 x 2 (t) x (t) = tx(t) (7.4) x (t) = x(t) (7.5) (1 t) 2 x (t) = x(t) (1 t) x(t) x (t) = 1 sin 2 [x(t)] g x (t) = (t)x(t) g(t) (7.6) (7.7) (7.8a)

2 with g 2 (τ)dτ < e τ (τ)dτ 2 n = 2 (7.8b) 0 0 n =1 x (t) y (t) = 0 1 x(t) x 2 (t)1 ε y(t) 0 rcos[ωt] ; ε, r > 0 (7.9) x 2t 1 (t) 1 e x 2 (t) = x 1(t) 1 1 x 2 (t) (7.10) x = x (t) c[t] x (t) x(t) = 0 c[t]= 2 e t > 0 for all t. (7.11a) (7.11b) x (t) (2 8t) x (t) 5x(t) = 0 (7.12) We have given these examples to illustrate a number of key points. The examples that are used to describe non-autonomous systems are mostly limited to first- and second-order dynamic equations since we can only obtain closed form solutions for first-order dynamic equations and some second-order dynamic equations. For example, the solution of a first-order dynamic equation expressed in the form given in equation (7.2) is: x(t) = x(t 0 )exp t t 0 a(τ)dτ (7.13) which can be used to obtain the closed form solutions for equations (7.3) to (7.7). When higher order dynamic equations are used, they are carefully handcrafted based on these equations to drive home some point. An example belonging to this category is: x 1 (t) (5 x 2 3 (t)) x 4 2 (t)x 1 (t) 0 x 1 (t) x 2 (t) = 0 1 4x 3 (t) x 2 (t) x 3 (t) 0 0 (2 sin[t]) x 3 (t) (7.14) In this model, the derivative of the third state variable depends on this (and only this) state variable and a time-varying term. That is, we have a third-order system in which there is no coupling from the first two state variable to the dynamics of the third state variable. Of course, all these example are used to prove a number of properties of non-autonomous systems, as we will demonstrate later. They all use the maxim that, though a large number of examples cannot prove conclusively a proposition, a single counter example can disapprove a proposition. Thus, a standard set of mathematical techniques are used to highlight key aspects of non-autonomous systems Another conclusion that can be derived from the above examples is that there is no clear delineation between time-varying linear systems (equation 7.2 for example), nonlinear autonomous systems driven by an external source (as we can interpret equation 7.9, also known as Duffing's equation), and nonautonomous systems. For example, Chua used non-autonomous systems with a term that changes periodically and shows that such systems can be converted to a autonomous system of one-order higher (than the order of the non-autonomous system) by appending an extra state given by: θ = ωt = 2πt T (7.15) where T is the period of the time-periodic term. Thus, equation (7.9) can be converted to a third-order autonomous system given by: x (t) y(t) y (t) = x(t)(1 x 2 (t)) εy(t) rcos[θ(t)] θ (t) 2π T (7.16) Finally, the examples given here use time functions such as 't', sin[ ω 0 t], 1/(1t), functions that are motivated from an analytical perspective, and often, no connection is made between such terms and physical devices. 7.3 Equilibrium points and Stability Concepts Similar to the definitions for autonomous systems, we can define equilibrium points and consider their stability etc. However, when we define equilibrium point(s) and their stability, their dependence on time has to be shown explicitly. Thus, equilibrium points of non-autonomous systems x equ can be defined as those for which f[ x equ,t] = 0 for all t t 0 (7.17) That is, the vector function f [ ] becomes zero for x = x equ and for t t 0. Similarly, we can consider the stability of an equilibrium point by considering an initial time, norm of the initial state, which may explicitly depend on this initial time, and norm of the state at any time after the initial time. Thus:

3 The equilibrium point, assumed to be the origin, is stable at t 0 if for any R f > 0, there exists a positive scalar r i R f,t 0 [ ] such that x(t 0 ) < r i [ R f,t 0 ] x(t) < R f for all t > t 0 (7.18) Otherwise, the equilibrium point is unstable. Similarly, we can define asymptotic stability and exponential stability for non-autonomous systems as: The equilibrium point 0 is asymptotically stable at time t 0 if: 1) It is stable and 2) For all r i [ t 0 ] > 0 such that x(t 0 ) < r i t 0 t (7.19) [ ] x(t) 0 as The equilibrium point is exponentially stable if there exist two positive scalar values such that x(t) α x(t 0 ) exp[ λ(t t 0 )] for all t > t 0 (7.20) A major limitation of these definitions as far as application to practical systems is the dependence of the behavior on the initial time. It is highly desirable to have definitions that describe the behavior of non-autonomous systems without involving the starting time of operation as follows: The equilibrium point is uniformly stable if the scalar r i [ R f,t 0 ] used in the definition for stability of equilibrium can be chosen independent of the value t 0. That is, if r i [ R f,t 0 ] = r i [ R f ]. The equilibrium point is uniformly asymptotically stable 1) If it is uniformly stable 2) If the state converges to 0, the origin, uniformly. That is, the convergence rate and time is independent of the initial time t 0. That is, finally, we arrived at definitions which require a) independence from the initial time, and b) the state vector to become zero (assuming 0 to be the equilibrium point). We will use these statements which perhaps look useless or redundant at this point to define nonlinear and time-varying circuit elements in next chapter. 7.4 Lyapunov's Direct Method for Stability Analysis of Non-autonomous Systems The success in using energy-like functions and a study of their derivatives along the system trajectory in the Lyapunov's direct method for the stability analysis of autonomous systems, makes a similar approach a natural candidate for the stability analysis of non-autonomous systems. First, we start with a scalar energy-like function V[ x,t ] which of course has to show explicitly the timedependence of non-autonomous systems. Thus, we define a time-varying positive definite function as one which is characterized by: and 1) V[ 0,t]= 0 (7.21a) 2) lower bounded by (or dominates) a time-invariant positive definite function for all time t t 0. That is: V[ x,t ] V L [ x] for all t t 0 (7.21b) Modifiers such as local or global can be added, as necessary, to this definition. Note the requirement that the function become zero when the value of the state vector equals zero (or equals the equilibrium point) regardless of the value of the time. If our interest is in defining a time-varying positive definite function only, the above definition which requires a lower bound in the form of a time-variant positive function for the time-variant function will suffice. However, we are interested in defining a function that will serve as an energy-like function. Thus, we also need to bound this function on the upper or the maximum side. Thus, we require: V[ x,t ] V u [ x] for all t t 0 (7.22) That is, V[ x,t ] is dominated by another time-invariant positive definite function V u [ x]. Such a time-variant function V[ x,t ] is called a decrescent function. Given a time-varying positive definite function, V[ x,t ], of the state variables x of a non-autonomous system that fits the description of energy left within that system, we can calculate its derivative along the system trajectory as:

4 V [ x,t ] x =fx,t = [ ] dv [ x,t] dt x =fx,t [ ] = V t = V t V x x t x =f [ x,t] V x f x,t [ ] (7.23) Using the scalar function V[ x,t ] of the state and its derivative along the system trajectory, we can state the main Lyapunov stability results for nonautonomous systems as follows: A non-autonomous system is stable if 1) V[ x,t ] is positive definite, 2) V [ x,t ] is negative semi-definite, 3) V [ x,t ] is uniformly continuous in time [A differentiable function f(t) which has a finite limit as t will be uniformly continuos if its derivative is bounded]. By this condition, we imply that V [ x,t ] alongsystrajec. 0 as t. Thus, V[ x,t ] will approach a finite limiting value V where V is less than or equal to V[ x(0),0]. It is uniformly stable if, in addition to 1), 2) and 3) above: 4) V[ x,t ] is decrescent. It is uniformly asymptotically stable if V [ x,t ] is not just negative semi-definite but is negative definite. It is globally uniformly asymptotically stable if: 5) V[ x,t ] is radially unbounded in x. Note that the radial unboundedness applies to the state variables only and not to the time-variable t. Otherwise, the requirement that V[ x,t ] is dominated by a time-invariant function V u [ x] will be violated. Note in the above definitions, the decrescent condition follows immediately after the negative semi-definiteness and uniformly continuos conditions, and then only the negative definiteness condition. Thus, just changing the negative semi-definiteness requirement to negative definiteness with out the decrescent condition does not guarantee asymptotic stability as is the case with nonlinear autonomous systems. We omit presenting here analytical proofs of these theorems as they are not really needed for this books objectives. Further, we have modified the theorems slightly so that they will conform to results that can be obtained from a network or systems' perspective. Because of the complexities involved in nonautonomous systems, these subtle changes may enable one to come up with counter-examples to prove or disprove the necessity of some of these conditions. However, these conditions make sense from a practical perspective. In the next chapter, we will discuss these requirements and the various definitions from a network or systems' perspective. 7.5 Analysis of Non-Autonomous Systems Having seen the conditions for the stability of non-autonomous systems, we can ask how we can analyze non-autonomous systems to characterize their behavior. We will discuss some of the possibilities. The aim here is to provide just enough information to highlight the problems in dealing with non-autonomous systems using the classical analytical approach. We can use computer simulation and draw phase-plane portraits (in the case of lower order systems) to get some feel for the response of non-autonomous systems. However, the dependence on the initial condition and the presence of time-varying terms makes the situation complicated. One can study the timevarying component (s) in particular and use intuition to come up with useful conclusions. However, even here many problems can be found. For example, equation (7.4) reproduced below: x (t) = tx(t) (7.4) may give the impression that x (t) will become large as t becomes large. However, it can be shown that this system is exponentially stable. On the other hand, the systems (7.11) and (7.12) reproduced below: with and x (t) c[t] x (t) x(t) = 0 c[t]= 2 e t > 0 for all t. (7.11a) (7.11b) x (t) (2 8t) x (t) 5x(t) = 0 (7.12) correspond to second-order systems with coefficients that are always positive. Thus, one may (and indeed can) associate the time-varying terms (that multiplies the first derivative) to power dissipative elements and hence conclude that both systems are asymptotically stable. However, it can be shown that the system with the dynamics in equation (7.11) is not asymptotically stable where as the system (7.12) is.

5 x 2 (t) = x 2 (0) 1 s a 2 e bt 1 s a 1 x 1 (t) We can also use the linearization approach to study the stability of nonlinear non-autonomous systems. We can write a Taylor expansion of f [ ] near the equilibrium point 0 as: x (t) x 0 = f[ x,t] x 0 = f[ 0,t] f[ x,t]. x higher order terms x x 0 (7.25) Figure 7-1. Flowgraph of a second-order time-varying dynamics to show possible problems in the use of eigen analysis for finding the stability of non-autonomous systems. where it can be noted that the higher order terms we need to ignore are not just functions of the state x, but also of time t. Thus, we need to ensure that they are small enough for x close to 0, the equilibrium point, and for all t. A commonly used criteria based on the L 2 -norm is: sup f hot [x,t] x x 0 0 for all t (7.26) Due to our experience with eigen analysis, we may think of obtaining the time-varying eigenvalues of a linear, non-autonomous system x = A(t)x (7.23) and see if stability can be established if all the eigen values of A(t) have negative real parts for all time t t 0. For example, the second order system: x 1 (t) x 2 (t) = a 1 e bt 0 a x 1(t) 2 x 2 (t) ; a 1,a 2,b > 0 (7.24) has two eigenvalues which are negative and constant (-a1, -a2). However, we can show that the state variable x 1 (t) as t indicating that the system is unstable. By looking carefully at the dynamics and the flow graph of the dynamics in Fig. 7.1, we can note that we have two stable linear systems that are not fully coupled. The state variable x 2 (t) (which tends to zero as t ) modulated by exp[ bt] becomes the input (which tends to infinity as t ) to another stable linear system. The lack of coupling between the state variables is indicated by the A(t) matrix being an upper diagonal or lower diagonal matrix, and it prevents the use of eigen analysis for testing the asymptotic stability. We can overcome this problem by studying the eigen values of A(t) A t (t). We can show that a sufficient condition for asymptotic stability is that the eigenvalues of this new matrix all have negative real parts. We can see that it is not easy to prove this when we deal with fairly high order systems. Incorporating this criteria in any design will be much more difficult if not altogether impossible. Assuming that the above equation is satisfied, we get the approximate state equation around the equilibrium point as: x (t) = A(t)x(t) (7.27) That is, the linearized model can in general be non-autonomous. We can use methods developed for linear time-varying systems such as the eigen analysis approach to ascertain the stability of this approximate model and hence the local stability of the original non-autonomous system. Needless to say, caution needs to be exercised before reaching any conclusion, due to the approximation, as we noted for the case of nonlinear autonomous systems. From the above discussions, we can appreciate the problems involved. Some of the problems are due to the complexity inherent in the kind of system (nonlinear non-autonomous) that we are dealing with. However, the top down approach, that is starting from the abstract mathematical relationship from a global level and trying to figure out the stability etc. aggravate the conditions further. Given a non-autonomous I/O relationship, we need to first find a function that qualifies as a Lyapunov function. There is no systematic procedure to arrive at a Lyapunov function of a system known to be stable. To assure us that Lyapunov functions do exist for such systems (and, note, they still do not tell us how to arrive at the Lyapunov function; they merely serve as a first step of assuring us), a number of theorems called converse Lyapunov theorems have been developed. For example:

6 If the vector function f[ x,t] of a non-autonomous system of (7.1) has continuos and bounded partial derivatives with respect to x and t for all x in a ball B r and for all t 0, then the equilibrium point x e = 0 is exponentially stable iff there exists a function V[ x,t ] and strictly positive constants α 1, α 2 and α 3 such that for all x in B r and for all t 0 : α 1 x 2 V[ x,t ] α 2 x 2 (7.28a) V [ x,t ] α 3 x (7.28b) V x α 4 x (7.28c) Results such as these can be used to construct a Lyapunov function for the subsystem of nonlinear system which may be known to possess some stability properties. This, in turn may enable us to find a Lyapunov function for the whole system. In summary, we find that designing a stable system is our main goal. In the next chapter, we will use the building block approach to achieve the same. 7.6 Summary In this chapter, we briefly considered time-varying or non-autonomous systems and basic concepts of stability for time-varying systems. Again, the emphasis was on presentation of basic results (that are derived from a mathematical/analytical perspective) in simple terms as possible. The problems in dealing with time-varying systems from an analytical perspective without intuition and guidance from the physical world are obvious from the results presented. In the next chapter, we will look at non-autonomous systems from a physical realization perspective and derive a number of new results and techniques for designing stable non-autonomous systems. 7.7 Notes and References Comprehensive information on non-autonomous systems including the proofs for various results and theorems presented here can be found in a number of books; see for example, [Vidyasagar, 1978]. The materials presented here are based on the book by [Slotine and Li, 1991].