Discrete‐time polynomial systems approach to combined fault monitoring and control

The combined fault monitoring and control problem is discussed, using an H₂ optimization framework. The main aim is to investigate the frequency‐domain and pole‐zero properties of the combined control and fault estimation compensator. The solution of the LQG stochastic optimal multivariable control and fault estimation problem is obtained using a polynomial matrix systems approach. The combined control and fault estimation problem involves a performance index including the usual regulating error and control weighting terms and also fault estimation error terms. The system model includes input disturbances, reference, coloured noise signals and actuator/sensor fault signals. The proposed numerical algorithm has common polynomial terms, so that the computation of the fault estimates and control law, is not significantly more difficult than the calculation of the control law alone. Copyright © 2003 John Wiley & Sons, Ltd.     

Unified approach for open‐loop optimal control

We develop a unified approach for the necessary conditions for optimization of open‐loop control systems, starting from the basic principles of calculus of variations. The unified approach results are simultaneously applicable to both shift (q)‐operator‐based, discrete‐time systems and the derivative (d/dt)‐operator‐based, continuous‐time systems. It is shown that the optimal condition results that are now obtained separately for continuous‐time and discrete‐time systems can be easily obtained from the unified approach. An illustrative example is given. Copyright © 2006 John Wiley & Sons, Ltd.     

Minimal‐power control of hydrogen evolution reactions

An integral approach to solve finite‐horizon optimal control problems posed by set‐point changes in electrochemical hydrogen reactions is developed. The methodology extends to nonlinear problems with regular, convex Hamiltonians that cannot be explicitly minimized, i.e. where the functional dependence of the H‐minimal control on the state and costate variables is not known. The Lagrangian functions determining trajectory costs will not have special restrictions other than positiveness, but for simplicity the final penalty will be assumed quadratic. The answer to the problem is constructed through the solution to a coupled system of three first‐order quasi‐linear partial differential equations (PDEs) for the missing boundary conditions x(T), λ(0) of the Hamiltonian equations, and for the final value of the control variable u(T). The independent variables of these PDEs are the time‐duration T of the process and the characteristic parameter S of the final penalty. The solution provides information on the whole (T, S)‐family of control problems, which can be used not only to construct the individual optimal control strategies online, but also for global design purposes. Copyright © 2009 John Wiley & Sons, Ltd.     

Hierarchical optimal control of a binary distillation column

This paper presents a two‐level optimal control for the binary distillation column. From the control point of view, the binary distillation column is a high‐order system consisting of several interconnected subsystems (trays) that aim to maximize output purity. Controlling the system in a hierarchical manner not only decreases complexity and solution time but also can improve the control structure's reliability. In this study, the distillation column is decomposed into N_T second‐order subsystems, each representing one stage (one tray). Accordingly, the cost function is decomposed, and thus, the overall problem is converted into N_T lower‐order subproblems where each tray has its own controller. A two‐level interaction prediction approach provides optimal control for the overall system. The task of the first level is to solve the subproblems using the predicted values of the coordination parameters, whereas the second level acts as a coordinator to update the coordination parameters. Simulation results show the capability and efficacy of the proposed two‐level control method in finding the optimal solution with less complexity and lower solution time than those of the centralized method.     

Robust stabilization of switched positive discrete‐time systems with asynchronous switching and input saturation

This paper investigates the stabilization problems of switched positive discrete‐time systems with asynchronous switching and input saturation jointly. The asynchronous switching means that state feedback controllers lag behind the subsystems for a period. Next, the input saturation comes from the saturation effect of state feedback controllers. In this paper, the exponential stability condition is firstly obtained based on mode‐dependent average dwell time method. Then, under zero initial condition, we analyze the boundedness and l₁ ‐gain performance of the system with disturbance. Finally, considering that state feedback controllers for some subsystems cannot be designed, we study the exponential stability condition of the system with unstable subsystems. Moreover, optimization methods are given to get the optimal solution to the nonlinear inequality constraints.     

Robust H∞ control of linear time‐varying systems with mixed delays in the Hilbert space

In this paper, a class of linear infinite‐dimensional systems with discrete and distributed time‐varying delays is studied. New sufficient conditions for the robust H_∞ control of linear time‐varying control systems with mixed time‐varying delays in Hilbert spaces have been established. The conditions are given in terms of the solution of a Riccati differential equation. Furthermore, it is proved for the first time that the solution to the robust H_∞ control problem can be solved by the global controllability of an appropriate linear control delay‐like system. Copyright © 2010 John Wiley & Sons, Ltd.     

A framework of robust fault estimation observer design for continuous‐time/discrete‐time systems

A design framework of the observer‐based robust fault estimation for continuous‐time/discrete‐time systems is presented in this paper. Firstly, a multiconstrained fault estimation observer under the H_∞ performance specification with the regional pole constraint is proposed to achieve robust fault estimation. Then, the existence conditions for both continuous‐time/discrete‐time systems are derived explicitly. Furthermore, by introducing slack variables, improved results on the multiconstrained fault estimation observer design are obtained such that different Lyapunov functions can be separately designed for each constraint. Finally, simulation results of a vertical takeoff and landing aircraft are presented to illustrate our contributions. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal trajectories for angular systems on the projective line

We analyze two optimal problems for a class of nonlinear system on the real projective line Popf¹ induced by a class of bilinear control system: the angular system. Two functional costs are considered: time‐optimal and quadratic. According to the Pontryagin Maximum Principle, in the time‐optimal case we show that if the angle system ?Σ satisfies the controllability property, then there exists a minimal time bang‐bang trajectory connecting any two points on ?¹, the noncontrollable case was discussed in closed form in (SIAM J. Control Optim. 2009; 48 (4):2636–2650). On the other hand, in the quadratic cost, the optimal control is a continuous function (Proyecciones J. Math. 2010; 29 (2):145–164). A comparison is also established between the structure of the solutions for the two optimal problems: time‐optimal and quadratic in the controllable and noncontrollable cases. The extremals are obtained from the adjoint system given by the Pontryagin Maximum Principle onto ?¹ via radial projection. An example is given. Copyright © 2011 John Wiley & Sons, Ltd.     

A new result for closed‐loop poles region of LQ‐optimal discrete‐time systems

This paper deals with a new result for pole location of linear quadratic (LQ) optimal discrete‐time systems. In this paper a new region in which the closed‐loop eigenvalues of an LQ‐optimal discrete time system are located is determined. It has been shown that this region is bounded by four lines parallel to the real and imaginary axes, creating a square region. This region is symmetric with respect to the real axis of z‐plane, but not with respect to the imaginary axis. Inequality constraints, which bound the closed‐loop eigenvalues in a discrete‐time LQ‐optimal system, are also presented via two theorems. These theorems cover all systems except the ones where A is singular and H is positive semidefinite at the same time. The result is also extended to the problem of the prescribed degree of stability for LQ optimal discrete‐time systems. Copyright © 2007 John Wiley & Sons, Ltd.     

An extended model for pairwise conflict resolution in air traffic management

A conflict resolution model for aircraft traversing planar intersecting trajectories is developed and solved. The model incorporates the flight dynamics of each aircraft. Three types of optimal conflict avoidance manoeuvres are examined. The first is the minimum‐time safe deviation to a designated objective and the second and third are the minimum‐time safe deviation to and from a designated ground track. These objectives lead to singular optimal control problems where the control variable constraint is defined in terms of the maximum allowable turn rate ∣u(t)∣⩽u_M,0⩽t⩽t_f, for the aircraft executing the avoidance manoeuvre and the non‐autonomous state variable constraint is defined in terms of the radius r_p of the protected zone about the potentially conflicting aircraft. A fast, accurate procedure is derived for computing the optimal steering program which requires the solution of a coupled differential‐algebraic system of equations to determine the control law on any boundary arc of an extremal trajectory. An example pairwise trajectory conflict is analysed in detail. The incorporation of variable airspeed v(t) and an acceleration/deceleration control u₂(t) is also discussed. Copyright © 1999 John Wiley & Sons Ltd.     

Observer‐based H∞ control for uncertain singular time‐delay systems with actuator saturation

In this paper, the observer‐based H_∞ control problem for uncertain singular time‐delay systems with actuator saturation is concerned. First, a delay‐dependent linear matrix inequality (LMI) condition is obtained which, guarantees that the uncertain singular time‐delay systems with actuator saturation are regular, impulse free, and asymptotically stable with H_∞ performance condition. Then, with this condition, the estimation of stability region and the design method of observer‐based H_∞ controller are given by solving LMIs and convex optimization problem. Finally, some numerical examples are provided to demonstrate the merit of the obtained results. Copyright © 2015 John Wiley & Sons, Ltd.     

A new method for suboptimal control of a class of non‐linear systems

In this paper, a new non‐linear control synthesis technique (θ–D approximation) is discussed. This approach achieves suboptimal solutions to a class of non‐linear optimal control problems characterized by a quadratic cost function and a plant model that is affine in control. An approximate solution to the Hamilton–Jacobi–Bellman (HJB) equation is sought by adding perturbations to the cost function. By manipulating the perturbation terms both semi‐global asymptotic stability and suboptimality properties are obtained. The new technique overcomes the large‐control‐for‐large‐initial‐states problem that occurs in some other Taylor series expansion based methods. Also this method does not require excessive online computations like the recently popular state dependent Riccati equation (SDRE) technique. Furthermore, it provides a closed‐form non‐linear feedback controller if finite number of terms are taken in the series expansion. A scalar problem and a 2‐D benchmark problem are investigated to demonstrate the effectiveness of this new technique. Both stability and convergence proofs are given. Copyright © 2004 John Wiley & Sons, Ltd.     

Further results on H ∞ control for discrete‐time uncertain singular systems with interval time‐varying delays in state and input

This paper investigates the state feedback robust H _∞ control problem of a class of discrete‐time singular systems with norm‐bounded uncertainties and interval time‐varying delays in state and input. A new bounded real lemma for discrete‐time singular systems with a pair of time‐varying interval state delays is first investigated. Mathematical comparisons of the new bounded real lemma and two existing ones are presented. Then, on the basis of the bounded real lemma proposed here, a sufficient condition in the form of nonlinear matrix inequality, such that the considered state feedback robust H _∞ control problem is solvable, is given. In order to solve the nonlinear matrix inequality, a cone complementarity linearization algorithm is offered. Several numerical examples are presented to show the applicability of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.     

A differential dynamic programming algorithm for differential games

We develop and prove the convergence of a first‐order differential dynamic programming algorithm for the solution of a zero‐sum two‐person differential game with perfect information. The algorithm extends a first‐order strong variation algorithm for optimal control given by Mayne and Polak. Assuming separability of the Hamiltonian, we decompose the differential game problem into two control subproblems, C₁ and C₂. The objective is to determine a point (u^*, v^*) in U×V, where U and V are the control spaces for C₁ and C₂, respectively, that satisfies an integral form of Pontryagin's maximum principle for differential games. Copyright © 2001 John Wiley & Sons, Ltd.     

A minimal energy control problem for second‐order linear hyperbolic systems with two independent variables

In this paper, we investigate controllability and minimal energy optimal control for Goursat–Darboux problem for the second‐order linear hyperbolic systems with two independent variables. The equation describes a relation between functions u: D→?^m and z: D→?ⁿ. We get an integral representation of the Goursat–Darboux problem by means of Riemann's matrix. The first half of the paper considers conditions under which there exists a control u for which the solution z of dynamics satisfies z(x₁, y₁) = p for any given p. The studied problem is reduced to the moments problem. The optimal control was found in a closed analytic form. Further, degeneracy of the matrix constructed by means of Riemann's matrix is shown to be a necessary and sufficient condition of controllability. Copyright © 2011 John Wiley & Sons, Ltd.     

Controller design for optimal tracking response in discrete‐time systems

The problem of designing a controller, which results in a closed‐loop system response with optimal time‐domain characteristics, is considered. In the approach presented in this paper, the controller order is fixed (higher than pole‐placement order) and we seek a controller that results in closed‐loop poles at certain desired and pre‐specified locations; while at the same time the output tracks the reference input in an optimal way. The optimality is measured by requiring certain norms on the error sequence—between the reference and output signals—to be minimum. Several norms are used. First, l₂‐norm is used and the optimal solution is computed in one step of calculations. Second, l_∞‐norm (i.e. minimal overshot) is considered and the solution is obtained by solving a constrained affine minimax optimization problem. Third, the l₁‐norm (which corresponds to the integral absolute error‐(IAE)‐criterion) is used and linear programming techniques are utilized to solve the problem. The important case of finite settling time (i.e. deadbeat response) is studied as a special case. Examples that illustrate the different design algorithms and demonstrate their feasibility are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal control problems with Atangana‐Baleanu fractional derivative

In this paper, we study fractional‐order optimal control problems (FOCPs) involving the Atangana‐Baleanu fractional derivative. A computational method based on B‐spline polynomials and their operational matrix of Atangana‐Baleanu fractional integration is proposed for the numerical solution of this class of problems. With this numerical technique, the FOCPs are reduced to a system of equations which are solved for the unknown parameters with the help of Mathematica® software. Our results show the applicability and usefulness of the numerical technique.     

A novel approach for the numerical investigation of optimal control problems containing multiple delays

This paper deals with the numerical solution of optimal control problems with multiple delays in both state and control variables. A direct approach based on a hybrid of block‐pulse functions and Lagrange interpolating polynomials is used to convert the original problem into a mathematical programming one. The resulting optimization problem is then solved numerically by the Lagrange multipliers method. The operational matrix of delay for the presented framework is derived. This matrix plays an imperative role to transfer information between 2 consecutive switching points. Furthermore, 2 upper bounds on the error with respect to the L²‐norm and infinity norm are established. Several optimal control problems containing multiple delays are carried out to illustrate the various aspects of the proposed approach. The simulation results are compared with either analytical or numerical solutions available in the literature.     

A priori and a posteriori error estimates of finite‐element approximations for elliptic optimal control problem with measure data

We analyze both a priori and a posteriori error analysis of finite‐element method for elliptic optimal control problems with measure data in a bounded convex domain in (d = 2or3). The solution of the state equation of such type of problems exhibits low regularity due to the presence of measure data, which introduces some difficulties for both theory and numerics of the finite‐element method. We first prove the existence, uniqueness, and regularity of the solution to the optimal control problem. To discretize the control problem, we use continuous piecewise linear elements for the approximations of the state and co‐state variables, whereas piecewise constant functions are used for the control variable. We derive a priori error estimates of order for the state, co‐state, and control variables in the L²‐norm. Further, global a posteriori upper bounds for the state, co‐state, and control variables in the L²‐norm are established. Moreover, local lower bounds for the errors in the state and co‐state variables and a global lower bound for the error in the control variable are obtained in the case of two space dimensions (d = 2). Numerical experiments are provided, which support our theoretical results.