Controller synthesis with guaranteed closed-loop phase constraints

In this paper, we present an analysis and synthesis approach for guaranteeing that the phase of a single-input, single-output closed-loop transfer function is contained in the interval [−α,α] for a given α>0 at all frequencies. Specifically, we first derive a sufficient condition involving a frequency domain inequality for guaranteeing a given phase constraint. Next, we use the Kalman–Yakubovich–Popov theorem to derive an equivalent time domain condition. In the case where , we show that frequency and time domain sufficient conditions specialize to the positivity theorem. Furthermore, using linear matrix inequalities, we develop a controller synthesis approach for guaranteeing a phase constraint on the closed-loop transfer function. Finally, we extend this synthesis approach to address mixed gain and phase constraints on the closed-loop transfer function.     

Stability margins of nonlinear optimal regulators with nonquadratic performance criteria involving cross-weighting terms

In this paper we derive guaranteed gain, sector, and disk margins for nonlinear optimal and inverse optimal regulators that minimize a nonlinear-nonquadratic performance criterion involving cross-weighting terms. Specifically, sufficient conditions that guarantee gain, sector, and disk margins are given in terms of the state, control, and cross-weighting nonlinear-nonquadratic weighting functions. The proposed results provide a generalization of the “meaningful” inverse optimal nonlinear regulator stability margins as well as the classical linear-quadratic optimal regulator gain and phase margins.     

Adaptive Input Estimation Methods for Improving the Bandwidth of Microgyroscopes

Unlike an accelerometer, the input to a vibratory microgyroscope is a product of the periodic excitation and the angular velocity to be detected. Consequently, the detection of the angular velocity requires a demodulation process involving a signal whose frequency is close to the resonance. Due to this special circumstance the dynamic performance characteristics such as the bandwidth of the microgyroscope sensors is severely limited. In this paper, we present adaptive input estimation methods to achieve wide bandwidth dynamic characteristics. Specifically, we present an adaptive estimator-based technique to estimate the angular motion by providing the Coriolis force as the input to the adaptive estimator. We present two methods, namely, the open-loop method and the closed-loop method. Both methods provide identical dynamic characteristics. But for the open-loop method, we require complete knowledge of all the parameters of the microgyroscopes whereas the closed-loop method can also be used where precise knowledge of the dynamic characteristics of the microgyroscopes is not available     

Non-linear impulsive dynamical systems. Part II: Stability of feedback interconnections and optimality

In a companion paper (Nonlinear Impulsive Dynamical Systems. Part I: Stability and Dissipativity) Lyapunov and invariant set stability theorems and dissipativity theory were developed for non-linear impulsive dynamical systems. In this paper we build on these results to develop general stability criteria for feedback interconnections of non-linear impulsive systems. In addition, a unified framework for hybrid feedback optimal and inverse optimal control involving a hybrid non-linear-non-quadratic performance functional is developed. It is shown that the hybrid cost functional can be evaluated in closed-form as long as the cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the non-linear closed-loop impulsive system. Furthermore, the Lyapunov function is shown to be a solution of a steady-state, hybrid Hamilton‐Jacobi‐Bellman equation.     

Stability margins of discrete-time nonlinear-non-quadratic optimal regulators

The paper derives guaranteed gain, sector and disk margins for discrete-time nonlinear optimal and inverse optimal regulators that minimize a nonlinear-non-quadratic performance criterion. The proposed results provide a generalization of the classical discrete-time, linear-quadratic optimal regulator gain and phase margins.     

Robust nonlinear feedback control for uncertain linear systems with nonquadratic performance criteria

In this paper we develop a unified framework to address the problem of optimal nonlinear robust control for linear uncertain systems. Specifically, we transform a given robust control problem into an optimal control problem by properly modifying the cost functional to account for the system uncertainty. As a consequence, the resulting solution to the modified optimal control problem guarantees robust stability and performance for a class of nonlinear uncertain systems. The overall framework generalizes the classical Hamilton–Jacobi–Bellman conditions to address the design of robust nonlinear optimal controllers for uncertain linear systems. © 1998 Elsevier Science B.V.     

Limit cycle stability analysis and adaptive control of a multi-compartment model for a pressure-limited respirator and lung mechanics system

Acute respiratory failure due to infection, trauma or major surgery is one of the most common problems encountered in intensive care units, and mechanical ventilation is the mainstay of supportive therapy for such patients. In this article, we develop a general mathematical model for the dynamic behaviour of a multi-compartment respiratory system in response to an arbitrary applied inspiratory pressure. Specifically, we use compartmental dynamical system theory and Poincaré maps to model and analyse the dynamics of a pressure-limited respirator and lung mechanics system, and show that the periodic orbit generated by this system is globally asymptotically stable. Furthermore, we show that the individual compartmental volumes, and hence the total lung volume, converge to steady-state end-inspiratory and end-expiratory values. Finally, we develop a model reference direct adaptive controller framework for the multi-compartmental model of a pressure-limited respirator and lung mechanics system where the plant and reference model involve switching and time-varying dynamics. We then apply the proposed adaptive feedback controller framework to stabilise a given limit cycle corresponding to a clinically plausible respiratory pattern.     

Lyapunov function proof of Poincaré's theorem

One of the most fundamental results in analysing the stability properties of periodic orbits and limit cycles of dynamical systems is Poincaré's theorem. The proof of this result involves system analytic arguments along with the Hartman–Grobman theorem. Using the notions of stability of sets, lower semicontinuous Lyapunov functions are constructed to provide a Lyapunov function proof of Poincaré's theorem.     

Neural network adaptive control for nonlinear nonnegative dynamical systems

Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in engineering and life sciences and typically involve the exchange of nonnegative quantities between subsystems or compartments wherein each compartment is assumed to be kinetically homogeneous. In this paper, we develop a full-state feedback neural adaptive control framework for adaptive set-point regulation of nonlinear uncertain nonnegative and compartmental systems. The proposed framework is Lyapunov-based and guarantees ultimate boundedness of the error signals corresponding to the physical system states and the neural network weighting gains. In addition, the neural adaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state-space for nonnegative initial conditions.     

A unification between nonlinear-nonquadratic optimal control and integrator backstepping

In this paper we develop an optimality-based framework for backstepping controllers. Specifically, using a nonlinear-nonquadratic optimal control framework we develop a family of globally stabilizing backstepping controllers parametrized by the cost functional that is minimized. Furthermore, it is shown that the control Lyapunov function guaranteeing closed-loop stability is a solution to the steady-state Hamilton–Jacobi–Bellman equation for the controlled system and thus guarantees both optimality and stability. The results are specialized to the cases of integrator backstepping and block backstepping for cascade systems with linear and nonlinear input subsystems. © 1998 John Wiley & Sons, Ltd.