Lena S. Valavani

Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics

This paper examines the robustness properties of existing adaptive control algorithms to unmodeled plant high-frequency dynamics and unmeasurable output disturbances. It is demonstrated thai there exist two infinite-gain operators in the nonlinear dynamic system which determines the time-evolution of output and parameter errors. The pragmatic implication of the existence of such infinite-gain operators is that 1) sinusoidal reference inputs at specific frequencies and/or 2) sinusoidal output disturbances at any frequency (including dc), can cause the loop gain to increase without bound, thereby exciting the unmodeled high-frequency dynamics, and yielding an unstable control system. Hence, it is concluded that existing adaptive control algorithms as they are presented in the literature referenced in this paper, cannot be used with confidence in practical designs where the plant contains unmodeled dynamics because instability is likely to result. Further understanding is required to ascertain how the currently implemented adaptive systems differ from the theoretical systems studied here and how further theoretical development can improve the robustness of adaptive controllers.

Stable adaptive controller design, part II: Proof of stability

The paper presents a proof of stability of model reference adaptive control systems using direct control. The structure of the adaptive system is similar to that considered by Monopoli [1] and Narendra and Valavani [2] but contains an additional feedback term which ensures that the time derivative of the parameter error vector belongs to the L2space. The output of the plant is shown to be bounded by expressing the plant feedback loop as an exponentially stable system with a time-varying gain \dot{\phi}(\cdot)\in L^{2} in the feedback path.

Stable adaptive controller design--Direct control

This paper deals with the design of stable adaptive controllers using a model reference approach. A systematic procedure is developed using the concept of positive realness and is generalized to arbitrarily located coutrol parameters. For plants with n poles and (\geq)(n-2) zeros, the uniform asymptotic stability in the large of the adaptive loop is demonstrated. For plants with (\leq)(n-3) zeros the stability problem is clawed and is stated as a conjecture. Simulation studies of the adaptive control of both stable and unstable plants are included towards the end of the paper.

Paper: Direct and indirect model reference adaptive control

The paper considers the control of an unknown linear time-invariant plant using Direct and Indirect Model Reference Adaptive Control. Employing a specific controller structure and the concept of positive realness, adaptive laws are derived using Indirect Control which are identical to those obtained in the case of Direct Control. The stability questions that arise are also shown to be the same. Simulation results using the new scheme are presented for the control of both stable and unstable plants.

Modeling for control of rotating stall

Abstract An analytical model for control of rotating stall has been obtained from the basic fluid equations describing the process at inception. The model describes rotating stall as a traveling wave packet, sensed—in spatial components—via the Fourier decomposition of measurements obtained from a circumferential array of evenly distributed sensors (hot wires) upstream of the compressor. A set of “wiggly” inlet guide vanes (IGVs) equally spaced around the compressor annulus constitute the “forced” part of the model. Control is effected by launching waves at appropriate magnitude and phase, synthesized by spatial Fourier synthesis from individual IGV deflections. The effect of the IGV motion on the unsteady fluid process was quantified via identification experiments carried out on a low speed, single-stage axial research compressor. These experiments served to validate the theoretical model and refine key parameters in it. Further validation of the model was provided by the successful implementation of a complex-valued proportional control law, using a combination of first and second harmonic feedback; this resulted in an 18% reduction of stalling mass flow, at essentially the same pressure rise.

H ∞ and H 2 -optimal controllers for periodic and multirate systems

In this paper we present the solutions to the optimal l2 to l2 disturbance rejection problem (H∞) as well as to the LQG (H2) problem in periodic systems using the lifting technique. Both problems involve a causality condition on the optimal LTI compensator when viewed in the lifted domain. The H∞ problem is solved using Neharis theorem whereas in the H2 problem the solution is obtained using the Projection theorem. Exactly the same methods of solution can be applied in the case of multirate sampled data systems. Finally, stability robustness issues in periodic and multirate systems are analyzed.

Direct and Indirect Adaptive Control

Abstract The paper considers the control of an unknown linear time-invariant plant using Direct and Indirect Control. Using a specific controller structure and the concept of positive realness, adaptive laws which are very similar are derived for the two cases. The stability questions that arise are also shown to be analogous and are discussed in some detail. Simulation results are presented towards the end of the paper to demonstrate the effectiveness of the two schemes.

Stable adaptive controller design part I: Direct control

The adaptive control problem for the case of an unknown plant with a single input and a single output is considered. The objective is to design a differentiator-free controller so that the output of the plant evolves asymptotically to the output of a given model. A method of choosing the controller-structure is presented in the paper and methods for adjusting the parameters to achieve asymptotic stability in the whole are discussed. Simulation results are included to indicate the generality of the approach and its potential in other control situations.

Stability and Convergence of Adaptive Control Algorithms: A Survey and Some New Results.

Abstract : Although adaptive controllers have been designed for a number of years, the central question of global asymptotic stability of the overall feedback adaptive loop and its associated error equations remained open until recently. The two main approaches used to design controllers from a stability viewpoint - Lyapunovs Direct Method and Popovs Hyperstability Theory - only assure boundedness of the (augmented) state and parameter errors. The difficulties encountered in both approaches are analyzed and shown to be exactly analogous. Subsequently, a generic model for an adaptive controller is proposed from which already existing adaptive algorithms can be derived with minor modifications. The model unifies various algorithms, under the essential requirement for positive reality of the associated error transfer function, and its proof of stability contains others suggested to date as special cases. Finally some general comments are made regarding rates of convergence, performance of the algorithms in the presence of disturbances and unmodeled dynamics, ease of implementation, etc. (Author)

A comparsion of lyapunov and hyperstability approaches to adaptive control

Abstract : The aim of this brief paper is to examine the conditions which have to be satisfied for the two approaches to be successfully applied to adaptive observers and controllers and also to demonstrate that they are entirely equivalent. Using a typical error model it is shown that hyperstability and asymptotic hyperstability are achieved under exactly the same conditions as stability and asymptotic stability in the sense of Lyapunov. In those cases where the adaptive control problem remains unresolved, the difficulties encountered using Lyapunovs method are shown to have their counterparts in hyperstability theory as well.