Trustable autonomous systems using adaptive control

Abstract

A long standing problem in adaptive control is the derivation of robustness properties in the presence of unmodeled dynamics, a necessary and highly desirable property for designing adaptive flight control for systems with trustable autonomy. We provide a solution to this problem in this paper for linear time-invariant plants whose states are accessible for measurement. This is accomplished by using a Lipschitz continuous projection algorithm that allows the utilization of properties of a linear system when the adaptive parameter lies on the projection boundary. This in turn helps remove the restriction on plant initial conditions, as opposed to the currently existing proofs of semi-global stability. A direct implication of this result is the robustness of adaptive control systems to time-delays, and the guarantee that the underlying adaptive system will have a delay margin.

Guaranteed delay margins for adaptive systems with state variables accessible

“…Several attempts have been made to extend the robustness properties of adaptive systems to time-delays and unmodeled dynamics (see for example [3]- [6] and more recently [7]- [9]) by introducing modifications to the underlying adaptive law. Either these results are semiglobal or global where the delay margin can be shown to exist but is not otherwise computable, or they are restricted to a small class of plants [7]. In contrast to these results, it was recently proven in [11] that global boundedness can be achieved for a first-order plant with a guaranteed delay margin using an adaptive law which includes a modification based on projections.…”

“…The adaptive law used in this paper was originally proposed in [12], rigorously analyzed in [5], [6], and revised and refined in [7], [13]. Unlike the standard practice of Lyapunov function based arguments which suffice when states are measurable, extensive arguments based on first principles are employed in order to prove the boundedness.…”

Adaptive architectures for control of uncertain dynamical systems with actuator and unmodeled dynamics

“…In addition, and are matrices associated with the modeled dynamics with the pair ( A , B ) being controllable, is an unknown weight matrix, and , , and are matrices associated with the unmodeled dynamics. Since the state and output vectors of the unmodeled dynamics are unmeasurable, we consider (as in, for example, the works of Matsutani et al and Dogan et al) that F is Hurwitz for the solvability of the problem, and therefore, there is a unique satisfying the Lyapunov equation 0= F T S + SF + I p . Furthermore, in is the output of the actuator dynamics given by with being the control input and being the actuator bandwidth of all control channels.…”

“…While some authors [1][2][3][4][5][6][7][8] study actuator dynamics problem and some others [10][11][12][13][14][15] study unmodeled dynamics problem for model reference adaptive control architectures, the effects of both dynamics are strictly present in feedback loops for a wide array of applications. To elucidate this point, for example, consider the flight control problem of flexible aerospace vehicles (eg, see other works [16][17][18].…”

“…In addition, A ∈ ℝ n×n and B ∈ ℝ n×m are matrices associated with the modeled dynamics with the pair (A, B) being controllable, W ∈ R n×m is an unknown weight matrix, and F ∈ ℝ p×p , G ∈ ℝ p×n , and H ∈ ℝ m×p are matrices associated with the unmodeled dynamics. Since the state and output vectors of the unmodeled dynamics are unmeasurable, we consider (as in, for example, the works of Matsutani et al 13 and Dogan et al 15,19 ) that F is Hurwitz for the solvability of the problem, and therefore, there is a unique S ∈ ℝ p×p + satisfying the Lyapunov equation 0 = F T S + SF + I p . Furthermore, v(t) ∈ ℝ m in (1) is the output of the actuator dynamics given by…”