Partial State Feedback Research Articles

H ∞ control for non-linear stochastic systems: the output-feedback case

The H ∞ output-feedback control problem for non-linear stochastic systems is considered. A solution for a large class of non-linear stochastic systems is introduced (including non-linear diffusion systems as a subclass). This solution is based on a bounded real lemma for non-linear stochastic systems that was previously established via a stochastic dissipativity concept. The theory yields sufficient conditions for the closed-loop system to possess a prescribed L 2-gain bound in terms of two Hamilton Jacobi inequalities: one that is associated with the state feedback part of the problem is n-dimensional (where n is the underlying system's state dimension) and the other inequality that stems from the estimation part is 2n-dimensional. Both stationary and non–stationary systems are considered. Stability of the closed-loop system is established, both in the mean-square and the in-probability senses. As the solution to the Hamilton Jacobi inequalities may, in general, lead to a non–realisable state estimator, a modification of the associated 2n-dimensional Hamilton Jacobi inequality is made in order to circumvent this realisation problem, while preserving the system's L 2-gain bound. For time-invariant systems, the problem of robust output-feedback is considered in the case of norm-bounded uncertainties. A solution is then derived in terms of linear state-dependent matrix inequalities.

Reduced Order Robust Gain-Scheduling Control Of The Diesel Apu For Series Hybrid Vehicles

This paper presents a reduced order robust gain-scheduling approach for the control of the diesel auxiliary power unit (APU) for series hybrid vehicles. The nonlinear plant dynamics are converted into a linear parameter-varying (LPV) form with parametric uncertainties, in which only partial information of the plant states is available. For this type of LPV system, a new reduced order robust gain-scheduling synthesis method is proposed based on partial state feedback. The parametric uncertainties are considered using multipliers to reduce the conservatism. The reduced order synthesis problem is solved offline by means of linear matrix inequalities (LMIs), and the synthesis result requires much simpler online computation than the explicit controller formulas do. The synthesis method is applied to the diesel APU controller design, and simulation results are given to demonstrate the controller performance.

Partial-state feedback control of high-order cascade systems with unknown control direction

Without a priori knowledge of the signs of control directions, this paper investigates the robust asymptotic regulating problem for a class of high-order non-linear cascade systems with both dynamic and static uncertainties. Under some weaker assumptions, a partial-state feedback controller is proposed which ensures that all the signals of closed-loop system are globally uniformly bounded and all states can be regulated to the origin. The design procedure is illustrated through an example.

H∞ Static Control Law for a Three-phase Shunt Active Power Filter

The aim of this work is to design a static feedback gain used to force a three-phase shunt active power filter to act as a current source capable of delivering the appropriate harmonics demanded by non-linear loads. This filter is connected in parallel to a three-phase power distribution system in order to keep the current supplied by the latter almost sinusoidal. The control design is performed using the H ∞ suboptimal control theory based on linear matrix inequalities (LMIs), with both partial and full state feedback. Numerical simulations will be presented in order to investigate the advantages of the full state feedback structure over the partial state feedback one, and to compare the performances of the corresponding H ∞ static control laws.

Asymptotic ε-Nash Equilibrium for 2nd Order Two-Player Nonzero-sum Singular LQ Games with Decentralized Control

We propose a new equilibrium concept: asymptotic ε-Nash equilibrium for 2***nd order two-player nonzero-sum games where each player has a control-free cost functional quadratic in the system states over an infinite horizon and each player's control strategy is constrained to be continuous linear state feedback. Based on each player's singular control problem, the asymptotic ε-Nash equilibrium implemented by partial state feedback is constructed and the feedback gains can be found by solving a group of algebraic equations which involves the system coefficients and weighting matrices in the cost functionals. As an illustration of the theories discussed in this paper, a numerical example is given where the partial state feedback gains can be found explicitly in terms of the system coefficients and weighting matrices in the cost functionals.

Adaptive backstepping control of an induction motor under time-varying load torque and rotor resistance uncertainty

A new global adaptive controller is designed for induction motor speed control based on measurements of speed and stator current. The designed partial state feedback controller is singularity free, and guarantees asymptotic tracking of smooth reference trajectories for the speed of the motor under time-varying load torque and rotor resistance uncertainty, for any initial condition. The new control algorithm generates estimates for unknown time-varying load torque, rotor resistance and unmeasured state variables (rotor fluxes) which asymptotically tracks and converges to their true values. The rotor flux modulus asymptotically tracks a desired reference signal which allows the motor to operate within its specifications. The control strategy yields decoupled rotor speed and rotor flux amplitude tracking control goals which allows the selection of an appropriate flux modulus for the rotor to maximise efficiency.

Stability Analysis of an UAV Controller using Singular Perturbation Theory

This paper presents the stability analysis of a hierarchical controller for an Unmanned Aerial Vehicle, using singular perturbation theory. Position and attitude control laws are successively designed by considering a time-scale separation between the translational dynamics and the orientation dynamics of a six degrees of freedom Vertical Take Off and Landing UAV model. In addition, for the design of the position controller, we consider the case where the linear velocity of the vehicle is not measured. A partial state feedback control law is proposed, based on the introduction of virtual states in the translational dynamics of the system.

Tracking and robustness analysis for controlled microelectromechanical relays

AbstractThe feedback control problem for microelectromechanical (MEM) relays is complicated by a quadratic nonlinearity in the dynamic model. We show that this nonlinearity imposes constraints on the reference trajectories that can be tracked and on the global convergence rate of the tracking error. Using a dynamic model that is applicable to both electrostatic and electromagnetic MEM relays, we introduce a new class of nonlinear tracking controls. In particular, we use Lyapunov theory to construct a state feedback that yields uniform global asymptotic stability and arbitrarily fast local exponential convergence of the tracking error. We then show how our control can be redesigned with partial‐state feedback under the assumption that only the movable electrode position and the electrical state (i.e. charge or flux) are fed back. Finally, we utilize input‐to‐state stability theory to quantify the robustness of our state feedback controller to parametric uncertainties. Our simulation results illustrate the good stability and tracking performance of the proposed control. They also illustrate how to craft a reference trajectory that satisfies the aforementioned constraints while being compatible with a typical relay operation. Copyright © 2007 John Wiley & Sons, Ltd.

Partial state feedback control of chaotic neural network and its application

The chaos control in the chaotic neural network is studied using the partial state feedback with a control signal from a few control neurons. The controlled CNN converges to one of the stored patterns with a period which depends on the initial conditions, i.e., the set of control neurons and other control parameters. We show that the controlled CNN can distinguish between two initial patterns even if they have a small difference. This implies that such a controlled CNN can be feasibly applied to information processing such as pattern recognition.

Semiglobal ISpS Disturbance Attenuation With Output Tracking via Direct Adaptive Design

Direct adaptive partial state feedback control is presented to achieve semiglobally input-to-state practically stable (ISpS) disturbance attenuation with output tracking for a class of uncertain time-varying nonlinear systems in which the unmeasured dynamics do not possess a constant disturbance attenuation level (CDAL). Identifying a necessary condition for the existence of a CDAL, direct adaptive neural networks (NNs) control is developed, where the universal approximation property of NNs and the domination design are employed together to overcome the difficulties due to the lack of state information, unknown system nonlinearities, and unknown state-dependent disturbance attenuation gain. The proposed method is coherent in the sense that it is applicable to the case in which a CDAL exists.

Robust Control System Design Using Proportional Plus Partial Derivative State Feedback

Based on a general parametric eigenstructure assignment result proposed for descriptor linear systems via proportional plus partial derivative state feedback and a result for generalized eigenvalue sensitivity problem of matrix pairs, parametric representation of the closed-loop eigenvalue sensitivities to the perturbed elements in the open-loop system matrices is obtained. An effective algorithm for eigenvalue assignment with minimum sensitivity in descriptor linear systems via proportional plus partial derivative state feedback is then proposed. The algorithm does not contain ‘going back’ procedures, and allows the closed-loop eigenvalues to be conveniently optimized within desired regions. An example demonstrates its effectiveness and simplicity.

Design of Critical Control Systems for Non-Minimum Phase Plants via LTR Technique

An application of the loop transfer recovery (LTR) technique to critical control systems design is proposed for non-minimum phase plants. The controller structure is chosen as the Davison type integral controller with the Kalman filter. First, a critical control system is designed on the assumption that the state of the minimum phase part of the plant can be used for the feedback. A quadratic performance index with tuning parameters is used for determining the partial state feedback gain matrix. Second, the Kalman filter gain matrix is determined such that the output feedback controller performs as in the partial state feedback controller. The formal partial loop recovery procedure using the Riccati equation is adopted for this purpose. The proposed design method requires much simpler numerical search than the conventional one-step approach. An illustrative design example is presented.

A Small-Gain Theorem for a Wide Class of Feedback Systems with Control Applications

A small-gain theorem, which can be applied to a wide class of systems that includes systems satisfying the weak semigroup property, is presented in the present work. The result generalizes all existing results in the literature and exploits notions of weighted, uniform, and nonuniform input-to-output stability properties. Applications to partial state feedback stabilization problems with sampled-data feedback applied with zero order hold and positive sampling rate are also presented.

ADAPTIVE SLIDING MODE CONTROL OF FED-BATCH PROCESSES USING SPECIFIC GROWTH RATE ESTIMATION FEEDBACK

Regulation of the biomass specific growth rate is an important goal in many fermentation fed-batch processes. Inspired in an invariant control law, we propose in this paper a controller with the structure of a partial state feedback with gain dependent on the output error. Further, to make the desired state trajectory effectively invariant despite modeling errors and parameter variations, the feedback gain is continuously adapted by means of a sliding mode algorithm. The stability properties of the closed–loop process are investigated in detail.

Globally stabilizing control of fed-batch processes with Haldane kinetics using growth rate estimation feedback

The regulation of the biomass specific growth rate is an important goal in many biotechnological applications. To achieve this goal in fed-batch processes, several control strategies have been developed employing a closed loop version of the exponential feeding law, an estimation of the controlled variable and some error feedback term. In the case of non-monotonic kinetics, the specified growth rate can be achieved at two different substrate concentration values. Because of the inherent unstable properties of the system in the decreasing portion of the kinetics function, stabilization becomes a crucial problem in this high-substrate operating region. In this context, the dynamic behavior of fed-batch processes with Haldane kinetics is further investigated. In particular, some conditions for global stability and performance improvement are derived. Then, a stabilizing control law based on a partial state feedback with gain dependent on the output error feedback and gain saturation is proposed. Although particular emphasis is put on the critical case of high-substrate operation, low-substrate regulation is also treated.

Robust pole assignment in matrix descriptor second-order linear systems

The present paper considers eigenvalue assignment with minimum sensitivity in matrix descriptor second-order linear (MDSOL) systems via proportional-derivative plus partial second-derivative state feedback. Based on a result in the perturbation theory of the eigenvalue problem of MDSOL systems, a closed-loop eigenvalue sensitivity measure in terms of the closed-loop normalized right and left eigenvectors is established directly in the matrix second-order framework. Also a general parametric eigenstructure assignment for MDSOL systems via proportional-derivative plus partial second-derivative state feedback is proposed. Based on this, the robust pole assignment problem is converted into an independent minimization problem. The closed-loop eigenvalues may be easily taken as part of the design parameters and optimized within certain desired fields on the complex plane to improve robustness. Using a simple sequential-order algorithm, the optimality of the obtained solution to the robust pole assignment problem is totally dependent on the solution to the independent minimization problem. An example of mass-spring system demonstrates the effect of the proposed approach.

Authors' reply [to comments on 'A remark on partial-state feedback stabilization of cascade systems using small gain theorem'

Zhong-Ping Jiang (see ibid. vol.50, no.6, p.927-80, 2005) made some comments on the authors' short remark (see ibid. vol.48, no.3, p.497-500, 2003). Two major points were made in the comment. The main purpose of the reply is to point out that the two points are not really an issue. Another purpose is to correct several inaccurate statements in the comment.

Comments on "A remark on partial-state feedback stabilization of cascade systems using small gain theorem"

This note points out that the results of the paper by Lin and Gong (see ibid., vol.48, no.3, p.497-500, 2003) are easy consequences of known results centered around nonlinear ISS small-gain theorems. In particular, the restrictive assumptions are relaxed.

Global partial-state feedback and output-feedback tracking controllers for underactuated ships

New global partial-state feedback and output-feedback control schemes for tracking control of an underactuated surface ship without sway force available are presented. For the case of partial-state feedback, we do not require measurements of the ship sway and surge velocities, while, for the case of output-feedback, none of the ship velocities are required for feedback. The reference trajectory to be tracked can be a curve including a straight-line. Global nonlinear coordinate changes are introduced to transform the ship dynamics to a system affine in the ship velocities to design observers to globally exponentially estimate unmeasured velocities. The controllers’ development is based on Lyapunov's direct method and backstepping technique, and utilizes the passive property of ship dynamics and their interconnected structure. Numerical simulations are provided to illustrate the effectiveness of the proposed controllers.

Continuous Suboptimal Control with Partial State Feedback

It is well known that the classical optimal control method requires availability of all state variables of the controlled system for control feedback. This requires a measurable system, namely that all state variables may be obtained by measurement of sensors or by estimation of observers. However, some state variables in control feedback actually have a small effect on control performance. Neglecting these state variables does not greatly affect the performance of control. In this paper, we investigate a continuous suboptimal control method with partial-state feedback. First, the state variables with significant effect on the performance of control are determined using second-order sensitivity (i.e. second-order derivative of performance index with respect to control gain). Then, the suboptimal controller is designed utilizing only the determined state variables. Numerical examples are presented to demonstrate the application of the proposed control algorithm. It is shown that the relative importance of each state variable can be indicated clearly by the second-order sensitivity. The suboptimal control method presented is effective in reducing maximum responses of the structure. Furthermore, the continuous suboptimal control method may achieve almost the same level of control effectiveness as the discrete suboptimal control method.