Synthesis of a stabilizing control problem for a multimachine power system with phase shifter

A systematic procedure is considered for the synthesis of a stabilizing control method for a multimachine power system with phase shifter, taking into account the velocity governor. A new approach is presented, which uses a coordinate-transformation technique and an optimization technique. The application of this method to a stabilizing control problem for a power system is illustrated by considering a 3-machine power system with phase-shifter control, taking into account an additional control vector for the governing system with one time constant. The synthesized controls are then used to improve the power-system transient stability to a remarkable degree and to restore the power-system transients rapidly to the stable-equilibrium point. Numerical results arc given.     

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

Optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria (that permit state- and control-dependent time-varying weighting parameters), is solved classically using a sequence of linear- quadratic and time-varying problems. The proposed method introduces an “approximating sequence of Riccati equations” (ASRE) to explicitly construct nonlinear time-varying optimal state-feedback controllers for such nonlinear systems. Under very mild conditions of local Lipschitz continuity, the sequences converge (globally) to nonlinear optimal stabilizing feedback controls. The computational simplicity and effectiveness of the ASRE algorithm is an appealing alternative to the tedious and laborious task of solving the Hamilton–Jacobi–Bellman partial differential equation. So the optimality of the ASRE control is studied by considering the original nonlinear-nonquadratic optimization problem and the corresponding necessary conditions for optimality, derived from Pontryagin's maximum principle. Global optimal stabilizing state-feedback control laws are then constructed. This is compared with the optimality of the ASRE control by considering a nonlinear fighter aircraft control system, which is nonaffine in the control. Numerical simulations are used to illustrate the application of the ASRE methodology, which demonstrate its superior performance and optimality.     

MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

Stabilization of linear autonomous systems of differential equations with distributed delay

Consideration is given to the problem of optimal stabilization of differential equation systems with distributed delay. The optimal stabilizing control is formed according to the principle of feedback. The formulation of the problem in the functional space of states is used. It was shown that coefficients of the optimal stabilizing control are defined by algebraic and functional-differential Riccati equations. To find solutions to Riccati equations, the method of successive approximations is used. The problem for this control law and performance criterion is to find coefficients of a differential equation system with distributed delay, for which the chosen control is a control of optimal stabilization. A class of control laws for which the posed problem admits an analytic solution is described.     

Transient Stability Enhancement of Multi–Machine Power Systems: Synchronization via Immersion of a Pendular System

In this paper the problem of designing excitation controllers to improve the transient stability of multi‐machine power systems is addressed adopting two new perspectives. First, instead of the standard formulation of stabilization of an equilibrium point, we aim here at the more realistic objective of keeping the difference between the generators rotor angles bounded and their speeds equal—which is called synchronization in the power literature—and translates into a problem of stabilization of a set. Second, we adopt the classical viewpoint of power systems as a set of coupled nonlinear pendula, and express our control objective as ensuring that some suitable defined pendula dynamics are (asymptotically) immersed into the power system dynamics. Our main contribution is the explicit computation of a control law for the two–machine system that achieves global synchronization. The same procedure is applicable to the n–machine case, for which the existence of a locally stabilizing solution is established.     

Stochastic nonlinear stabilization — II: Inverse optimality

After considering the stabilization of a specific class of stochastic nonlinear systems in a companion paper, in this second part, we address the classical question of when is a stabilizing (in probability) controller optimal and show that for every system with a stochastic control Lyapunov function it is possible to construct a controller which is optimal with respect to a meaningful cost functional. Then we return to the problem from Part I and design an optimal backstepping controller whose cost functional includes penalty on control effort and which has an infinite gain margin.     

Stochastic minimax control for stabilizing uncertain quasi- integrable Hamiltonian systems

A procedure for designing feedback control to asymptotically stabilize, with probability one, quasi-integrable Hamiltonian systems with bounded uncertain parametric disturbances is proposed. First, the partially averaged Itô stochastic differential equations are derived from given system by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Second, the Hamilton-Jacobi-Issacs (HJI) equation for the ergodic control problem of the averaged system and a performance index with undetermined cost function is established based on the principle of optimality. This equation is then solved to yield the worst disturbances and the associated optimal controls. Third, the asymptotic Lyapunov stability with probability one of the optimally controlled system with worst disturbances is analyzed by evaluating the maximal Lyapunov exponent of the fully averaged Itô equations. Finally, the cost function and feedback control are determined by the requirement of stabilizing the worst-disturbed system. A simple example is worked out to illustrate the application of the proposed procedure and the effects of optimal control on stabilizing the uncertain system.     

广义双线性系统的最优控制

最优控制是自动控制理论的重要研究分支,本文首次对广义双线性系统的最优控制问题进行研究.利用李雅普诺夫稳定性理论和广义李雅普诺夫方程的解来设计最优控制器,使得闭环系统全局渐近稳定且使广义二次性能指标最小.此外,还给出最优化控制器的设计方法,整个设计过程简单,具有较少的保守性,例子表明设计方法的有效性和合理性.     

Output‐based H2 optimal controllers for a class of discrete‐time stochastic linear systems with periodic coefficients

The aim of the paper is to present a design procedure of the optimal controller minimizing the H₂‐type norm of discrete‐time stochastic linear systems with periodic coefficients simultaneously affected by a nonhomogeneous but periodic Markov chain and state and control multiplicative white noise perturbations. Firstly, two H₂‐type norms for the linear stochastic systems under consideration were introduced. These H₂‐type norms may be viewed as measures of the effect of the additive white noise perturbations on the regulated output of the considered system. Before deriving of the state space representation of the optimal controller, some useful formulae of the two H₂‐type norms were obtained. These formulae are expressed in terms of periodic solutions of some suitable linear equations and are derived in the absence of some additional assumptions regarding the Markov chain other than the periodicity of the sequence of the transition probability matrices. Further, it is shown that the optimal H₂ controller depends on the stabilizing solutions of some specific systems of coupled Riccati equations, which generalize the well‐known control and filtering equations from linear time invariant case. For the readers convenience, the paper presents iterative numerical algorithms for the computations of the stabilizing solutions of these Riccati type systems. The theoretical developments are illustrated by numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal control using an algebraic method for control-affine non-linear systems

A deterministic optimal control problem is solved for a control-affine non-linear system with a non-quadratic cost function. We algebraically solve the Hamilton–Jacobi equation for the gradient of the value function. This eliminates the need to explicitly solve the solution of a Hamilton–Jacobi partial differential equation. We interpret the value function in terms of the control Lyapunov function. Then we provide the stabilizing controller and the stability margins. Furthermore, we derive an optimal controller for a control-affine non-linear system using the state dependent Riccati equation (SDRE) method; this method gives a similar optimal controller as the controller from the algebraic method. We also find the optimal controller when the cost function is the exponential-of-integral case, which is known as risk-sensitive (RS) control. Finally, we show that SDRE and RS methods give equivalent optimal controllers for non-linear deterministic systems. Examples demonstrate the proposed methods.     

Stabilization of a synchronous generator with a controllable series capacitor via immersion and invariance

The immersion and invariance (I&I) technique (it IEEE Trans. Automat. Control 2003; 48 (4):590–606) is a recently proposed control methodology for stabilizing nonlinear systems. Here we apply this philosophy to stabilize a single machine infinite bus (SMIB) system using a controllable series capacitor (CSC). The synchronous generator is modeled using the second‐order swing equation and a first‐order model is used for the CSC. The control objective here is to immerse a desired second‐order dynamic model into the higher order system manifold and design an asymptotically stabilizing control law. Copyright © 2011 John Wiley & Sons, Ltd.     

On Kalman filtering over fading wireless channels with controlled transmission powers

We study stochastic stability of centralized Kalman filtering for linear time-varying systems equipped with wireless sensors. Transmission is over fading channels where variable channel gains are counteracted by power control to alleviate the effects of packet drops. We establish sufficient conditions for the expected value of the Kalman filter covariance matrix to be exponentially bounded in norm. The conditions obtained are then used to formulate stabilizing power control policies which minimize the total sensor power budget. In deriving the optimal power control laws, both statistical channel information and full channel information are considered. The effect of system instability on the power budget is also investigated for both these cases.     

Decentralized adaptive stabilization of nonlinear systems

Decentralized adaptive control schemes using the principle of dominant subsystems are presented for time-varying nonlinear dynamic large-scale interconnected systems. Sufficient conditions for the existence of local decentralized adaptive control laws stabilizing a given large-scale system (LSS) are derived in terms of controller parameters for incompletely known composite systems. The approach proposed in this paper is applied to nonlinear stabilizing adaptive decentralized control (ADC) of multimachine power systems. The stability of the multimachine power systems with the ADC is illustrated by the simulation results for a two machine system.     

Output‐feedback stabilization for planar output‐constrained switched nonlinear systems

In this paper, globally asymptotical stabilization problem for a class of planar switched nonlinear systems with an output constraint via smooth output feedback is investigated. To prevent output constraint violation, a common tangent‐type barrier Lyapunov function (tan‐BLF) is developed. Adding a power integrator approach (APIA) is revamped to systematically design state‐feedback stabilizing control laws incorporating the common tan‐BLF. Then, based on the designed state‐feedback controllers and a constructed common nonlinear observer, smooth output‐feedback controllers, which can make the system output meet the predefined constraint during operation, are proposed to deal with the globally asymptotical stabilization problem of planar switched nonlinear systems under arbitrary switchings. A numerical example is employed to verify the proposed method.     

Robust decentralized fast-output sampling technique based power system stabilizer for a multi-machine power system

Power-system stabilizers (PSSs) are added to excitation systems to enhance the damping during low-frequency oscillations. In this paper, the design of robust decentralized PSS for four machines with a 10-bus system using fast-output sampling feedback is proposed. The nonlinear model of a multimachine system is linearized at different operating points, and 16 linear state space models are obtained. For all of these plants, a common stabilizing state feedback gain, F, is obtained. A robust decentralized fast-output sampling feedback gain which realizes this state feedback gain is obtained using LMI approach. This method does not require all the states of the system for feedback and is easily implementable. This robust decentralized fast-output sampling control is applied to a nonlinear plant model of several machines at different operating (equilibrium) points. This method yields encouraging results for the design of power-system stabilizers.     

Uniting local and global controllers for uncertain nonlinear systems: beyond global inverse optimality

This paper provides a solution to a new problem of global robust control for uncertain nonlinear systems. A new recursive design of stabilizing feedback control is proposed in which inverse optimality is achieved globally through the selection of generalized state-dependent scaling. The inverse optimal control law can always be designed such that its linearization is identical to linear optimal control, i.e. optimal control, for the linearized system with respect to a prescribed quadratic cost functional. Like other backstepping methods, this design is always successful for systems in strict-feedback form. The significance of the result stems from the fact that our controllers achieve desired level of ‘global’ robustness which is prescribed a priori. By uniting locally optimal robust control and global robust control with global inverse optimality, one can obtain global control laws with reasonable robustness without solving Hamilton–Jacobi equations directly.     

Stabilizing a class of nonlinear parameter-varying systems using interval observer based on a piecewise framework

This article addresses an interval observer-based control for stabilizing a class of nonlinear parameter-varying systems with noisy output by designing a switching surface. An input-dependent interval observer is firstly developed to estimate the lower and upper bounds of the states. Next, a switching-based controller is designed to stabilize the interval observer which implies the stability of the main parameter-varying system. The developed stabilizing switching surfaces are designed based on the outputs of the main system and the bounds of the states of the observer. By choosing an appropriate piecewise Lyapunov function, the closed-loop stability analysis of the interval observer system leads to a set of linear matrix inequalities including stability and Metzler constraints, simultaneously. The effectiveness of the proposed method is verified using the simulation results.     

Data‐driven–based attitude control of combined spacecraft with noncooperative target

In this paper, the attitude control of combined spacecraft with noncooperative target is studied. For the linearized system of the attitude control system, which possesses uncertainties on both its control and system matrices caused by the noncooperative target, this paper proposes an approximated iteration method to obtain the optimal controller such that the closed‐loop system possesses a prescribed convergence rate. An explicit stabilizing gain is established to initiate the iteration method. Based on the proposed iteration method and the initial stabilizing gain, a data‐driven algorithm is proposed to obtain the optimal controller for the attitude control system without using the system parameter information. A simulation is given to verify the effectiveness of the proposed method.     

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

In this paper, two methods for approximating the stabilizing solution of the Hamilton–Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton–Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.