H∞ robust control based on event‐triggered sampling for hybrid systems with singular Markovian jump

The problem of H_∞ robust control based on event‐triggered sampling for a class of singular hybrid systems with Markovian jump is considered in this paper. The primary object of this paper here is to design the event‐triggered sampling controller for a class of uncertain singular Markovian systems, and two fundamental issues on mean square exponential admissibility and H_∞ robust performance are fully addressed. By making use of a suitable Lyapunov functional, in combination with both infinitesimal operator and linear matrices inequalities(LMIs), the sufficient criteria are derived to guarantee the controlled singular hybrid system with Markovian jump is robustly exponentially mean‐square admissible and has a prescribed H_∞ performance γ. Finally, a typical RLC circuit system is given to show the effectiveness of the proposed control method.     

The stabilization of program motions of controlled nonlinear mechanical systems

We consider a controlled nonlinear mechanical system described by the Lagrange equations. We determine the control forcesQ ₁ and the restrictions for the perturbationsQ ₂ acting on the mechanical system which allow to guarantee the asymptotic stability of the program motion of the system. We solve the problem of stabilization by the direct Lyapunov's method and the method of limiting functions and systems. In this case we can use the Lyapunov's functions having nonpositive derivatives. The following examples are considered: stabilization of program motions of mathematical pendulum with moving point of suspension and stabilization of program motions of rigid body with fixed point.     

A sequential method for preventive maintenance and replacement of a repairable single-unit system

This paper is concerned with when to implement preventive maintenance (PM) and replacement for a repairable ‘single-unit’ system in use. Under the main assumption that a ‘single-unit’ system gradually deteriorates with time, a sequential method is proposed to determine an optimal PM and replacement strategy for the system based on minimising expected loss rate. According to this method, PM epochs are determined one after the other, and consequently we can make use of all previous information on the operation process of the system. Also the replacement epoch depends on the effective age of the system. A numerical example shows that the sequential method can be used to solve the PM and replacement problem of a ‘single-unit’ system efficiently. Some properties of the loss functions W(L? _n ,b? _n ) and W? _r (N) with respect to PM and replacement respectively are discussed in the appendix.     

Time-delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott-rotor system

Time-delay is an unavoidable phenomenon in active control systems. Measuring of the system states, processing of the measured signals, executing the control laws, conditioning and enforcing the control actions are the main reasons of time-delayed systems. This paper studies the vibration control of a horizontally suspended Jeffcott-rotor system having cubic and quadratic nonlinearities via time-delayed position-velocity controller. The intervals of the time-delays (τ₁ and τ₂) at which the system response is stable has been studied. The τ₁ − τ₂ plane is constructed to illustrate the area at which the system solutions are stable. The influences of the controller gains on the stable-solutions area in τ₁ − τ₂ plane are explored. The analysis revealed that the time-delay increases the vibration amplitudes and can destabilize the system solution in the case of negative position feedback control, while at positive position feedback control it improves the vibration suppression performance. The time-delays mechanism in stabilizing and destabilizing the dynamical systems is explained. Then, we proposed a simple and concrete method to determine the optimal value for time-delays that can improve the vibrations suppression efficiency. The acquired analytical results are confirmed numerically and the optimal working conditions of the system are concluded. Finally, a comparison with the papers that published previously is included.     

Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ₀ close enough to an eigenstate and ψ_f close enough to another eigenstate, the wave function can be moved exactly from ψ₀ to ψ_f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash-Moser implicit function theorem, the return method and expansion to the second order.     

Optimal feedback control for constrained mechanical systems

An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we want to use it to control constrained dynamical systems (differential algebraic equations of Index 3). To describe their discrete dynamics, a constrained variational integrators [1] is used. Using a discrete version of the Lagrange-d’Alembert principle yields a forced constrained discrete Euler-Lagrange equation in a position-momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (q_opt, p_opt) and according control input u_opt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input u_R. Adding u_opt and u_R to the discrete Euler-Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of a nonlinear time-optimal control problem

Nonlinear systems with a stationary (i.e., explicitly time independent) right-hand side are considered. For time-optimal control problems with such systems, an iterative method is proposed that is a generalization of one used to solve nonlinear time-optimal control problems for systems divided by phase states and controls. The method is based on constructing finite sequences of simplices with their vertices lying on the boundaries of attainability domains. Assuming that the system is controllable, it is proved that the minimizing sequence converges to an ɛ-optimal solution after a finite number of iterations. A pair {T, u(·)} is called an ɛ-optimal solution if |T − T _opt| − ɛ, where T _opt is the optimal time required for moving the system from the initial state to the origin and u is an admissible control that moves the system to an ɛ-neighborhood of the origin over the time T.     

Robust L ∞ reliable control for impulsive switched nonlinear systems with state delay

This paper investigates the problem of robust L _∞ reliable control for a class of uncertain impulsive switched nonlinear systems with time-delay in the presence of actuator failure. Based on the dwell time approach, we firstly obtain a sufficient condition of exponential stability for the impulsive switched nonlinear system with time-delay, and L _∞ performance for the considered system is also analyzed. Then, based on above results, a state feedback controller, which guarantees the exponential stability with L _∞ performance of the corresponding closed-loop system, is constructed. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method.     

Exhaustive search methods for CNS polynomials

In this paper, we present a method for finding all expansive polynomials with a prescribed degree n and constant term c ₀. Our research is motivated by the fact that expansivity is a necessary condition for number system constructions. We use the algorithm for an exhaustive search of CNS polynomials for small values of n and c ₀. We also define semi-CNS polynomials and show that producing them the same search method can be used.     

The approximation of reachable sets of control systems with integral constraint on controls

In this paper an approximation method for the construction of reachable sets of control systems with integral constraints on the control is considered. It is assumed that the control system is non-linear with respect to the phase state vector and is linear with respect to the control vector. The admissible control functions are chosen from the ball centered at the origin with radius μ₀ in L_p, p > 1. The reachable set is replaced by the set which consists of finite number of points. The estimated accuracy of the Hausdorff distance between the reachable set and the set which is approximately constructed is obtained.     

A Relation between formal and Asymptotic Equivalence for linear Systems Containing A Parameter

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ε^Pdy/dx = A(x,ε)y with A holomorphic at (x₀,0), suppose that x₀ is formally not a turning point in the sense that no singularities appear at x₀ during the standard formal solution procedure with formal fractional power series in ε. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x₀. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.     

Solution of a Certain Class of Problems of Optimal Control Theory in Domains of an Arbitrary Form

To solve optimal control problems for elliptic systems in domains that have an arbitrary form, we use a combination of the method of dummy domains and the grid method. We show that the proposed difference scheme has precision of order O(h ^1/2) with respect to the grid norm of W₂ ².     

The linear quadratic minimum-time problem for a class of discrete systems

We consider a linear discrete-time systems controlled by inputs on L ²([0, t _N ], U), where (t _i )_{1?≤?i?≤?N} is a given sequence of times. The final time t _N (or N) is considered to be free. Given an initial state x ₀ and a final one x _d , we investigate the optimal control which steers the system from x ₀ to x _d with a minimal cost J(N, u) that includes the final time and energy terms. We treat this problem for both infinte and finite dimensional state space. We use a method similar to the Hilbert Uniqueness Method. A numerical simulation is given.     

Finite-time L1 control for positive switched linear systems with time-varying delay

This paper is concerned with the problem of finite-time L₁ control for a class of positive switched linear systems with time-varying delay. Firstly, by using the average dwell time approach, sufficient conditions which can guarantee the L₁ finite-time boundedness of the underlying system are given. Then, in virtue of the results obtained, a state feedback controller is designed to ensure that the resulting closed-loop system is finite-time bounded with L₁-gain performance. All the obtained results are formulated in terms of linear matrix inequalities (LMIs), which can be solved conveniently. Finally, an example is given to illustrate the efficiency of the proposed method.     

Robust H∞ control for a class of switched nonlinear cascade systems via multiple Lyapunov functions approach

This paper addresses the problem of robust H_∞ control for a class of switched nonlinear cascade systems with parameter uncertainty using the multiple Lyapunov functions (MLFs) approach. Each subsystem under consideration is composed of two cascade-connected parts. The uncertain parameters are assumed to be in a known compact set and are allowed to enter the system nonlinearly. Based on the explicit construction of Lyapunov functions, which avoids solving the Hamilton-Jacobi equations, sufficient conditions for the solvability of the robust H_∞ control problem are presented. As an application, the hybrid robust H_∞ control problem for a class of uncertain non-switched nonlinear cascade systems is solved when no single continuous controller is effective. Finally, a numerical example is provided to demonstrate the feasibility of the proposed method.     

On an approach to solving problems of optimal control of elliptic systems in domains of an arbitrary shape

We use the method of dummy domains to solve the problem of optimal control for elliptic systems in domains that have an arbitrary shape. An estimate of the convergence rate in the norms W₂ ¹ and L₂ is obtained. Bibliography: 4 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 72–78.     

Finite-time H∞ control for a class of discrete-time switched time-delay systems with quantized feedback

This paper is concerned with the finite-time quantized H_∞ control problem for a class of discrete-time switched time-delay systems with time-varying exogenous disturbances. By using the sector bound approach and the average dwell time method, sufficient conditions are derived for the switched system to be finite-time bounded and ensure a prescribed H_∞ disturbance attenuation level, and a mode-dependent quantized state feedback controller is designed by solving an optimization problem. Two illustrative examples are provided to demonstrate the effectiveness of the proposed theoretical results.     

On the computation of the infimum in H∞‐optimization

In this paper, a new method for the computation of the infimum for a large class of continuous‐time H_∞ optimal control problem by state feedback is presented. The main ingredients of the new method include three generalized eigenvalue problems whose coefficient matrices are from a condensed form of the given system. This condensed form is computed using only orthogonal transformations which can be implemented via a numerically stable way. The superiority of the new method over the existing one given in Chen (H_∞ Control and its Applications, Chapter 5. Springer: Berlin, 1997) is verified by some numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.     

Group Foliation Approach to the Complex Monge–Ampère Equation

We apply the group foliation method to find noninvariant solutions of the complex Monge–Ampère equation (CMA₂). We use the infinite symmetry subgroup of the CMA₂ to foliate the solution space into orbits of solutions with respect to this group and correspondingly split the CMA₂ into an automorphic system and a resolvent system. We propose a new approach to group foliation based on the commutator algebra of operators of invariant differentiation. This algebra together with Jacobi identities provides the commutator representation of the resolvent system. For solving the resolvent system, we propose symmetry reduction, which allows deriving reduced resolving equations.