## Department of Electrical and Computer Engineering Technical Reports

Thesis

#### Abstract

The contribution of this thesis is the introduction and development of the variable gradient method of generating Liapunov functions. A Liapunov function, V, is considered to be generated if the form of V is not known before the generating procedure is applied. Two previous attempts at the generation of Liapunov functions to prove global asymptotic stability for nonlinear autonomous systems have been made. These attempts are summarized and evaluated in some detail, as they form the basis for the variable gradient approach proposed in this thesis. It is assumed that the system whose stability is being investigated is represented by n first order, ordinary, nonlinear differential equations in state variable form The particular state variables used throughout the thesis are the phase variables. This was done for convenience. The problem of finding a scalar V(x) to satisfy a particular Liapunov theorem is recast into the problem of finding a vector function, \nabla V, having suitable properties. As the name implies, \nabla V is assumed to be a vector of n elements, \nabla Vi, each of which has n arbitrary coefficients. These coefficients, designated as α ij may be constants or functions of the state variables, In its most general form, the variable gradient is assumed to be V may be determined as a line integral of \nabla V if the following (n-l)n/2 partial differential equations are satisfied. Here \nabla V^ are the elements of the vector \nabla V. The equations (3) are referred to as generalized curl equations. dv/dt may also be determined from \nabla V. An outline of the procedure by which a suitable V and dY/dt may be determined for a particular problem, starting from the variable gradient of (2) is as follows, 1. Assume a gradient of the form (2), 2. From the variable gradient, determine dV/dt by equation (4). 3. In conjunction with and subject to the requirements of the generalized curl equations (3), constrain dV/dt to be at least negative semi- definite, 4. From the now known \nabla V, determine V, 5. Invoke the necessary theorem to establish stability, Numerous examples are worked to illustrate the procedure outlined above, V functions are generated that involve higher order terms in x, integrals, and terms involving three state variables as factors. The problem of determining Hurwitz like criteria for nonlinear systems is considered in some detail. The last chapter attempts to extend .the variable gradient approach to nonautonosnous systems. The results of this chapter, though somewhat marginal, are of interest from the point of view of further research

4-1-1962

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