Extension of a Purely Pulse Control under Boundary Conditions

The control of a linear system under boundary conditions for the choice of a purely pulse program of limited potentialities, the asymptote of the set of admissible controls under different constraint perturbations, and the influence of perturbations on the reachability sets in the space of trajectories generated by admissible controls are studied. The asymptotic insensitivity of the problem under perturbations of boundary conditions is determined.     

External ellipsoidal estimates of the attainability sets of differential impulse systems

The problems of estimating a tube of solutions of dynamic systems described by differential equations containing generalized (impulse) control functions are considered. It is supposed that a special constraint of an ellipsoidal type is imposed on the control, and, in particular, that the vectors of jumps of generalized equations are to belong to a specified ellipsoid of the appropriate finite-dimensional space. To obtain external estimates of the attainability sets of the system, the results of ellipsoidal calculus, which is well developed nowadays, are applied. A new method for constructing estimates based on the external ellipsoidal approximation of the convex set of a union of ellipsoids is also proposed. Particular examples of constructing estimates of the attainability set are considered based on the proposed methods.     

Control of multiplicative noise stochastic gene regulation systems by the attractive ellipsoid technique

This article describes the application of the so-called attractive ellipsoidal method to solve the trajectory stabilization problem for a class of genetic network systems modelled by a stochastic model. The genetic network model is described by a stochastic quasi-linear system affected by additive and multiplicative noises simultaneously. The solution of the control design provided in this study is based on a linear feedback structure. In this paper the algorithm to construct a suboptimal gain for adjusting the control design is introduced. The attractive ellipsoidal method is the key stone for designing the so-called suboptimal gain. Moreover, the practical stability of the genetic network trajectories is demonstrated on the mean and in almost sure senses. Some numerical simulations show how a set of stochastic trajectories are stabilized by the controller suggested in this study and how the predicted ellipsoid region is achieved by these trajectories.     

Regularity properties of solutions to linear quadratic optimal control problems with state constraints

We establish regularity properties of solutions of linear quadratic optimal control problems involving state inequality constraints. Under simply stated and directly verifiable hypotheses on the data, it is shown that if the state constraint has index k> 0 then the optimal control is k times differentiable; the kth derivative may be discontinuous but it is a function of bounded variation (and consequently it has left and right limits at each point in its domain). If on the other hand the state constraint has index k = 0 then the optimal control is continuous. The latter property is perhaps surprising because it implies that for the class of problems considered optimal state trajectories cannot abruptly change direction when they strike the boundary of the state constraint region. These findings are significant because they justify assumptions which underlie analysis of junction conditions (i.e. properties of state trajectories at contact points with the boundary of the state constraint set) provided elsewhere in the literature.     

Ellipsoidal techniques for reachability analysis: internal approximation

For the reach tube of a linear time-varying system with ellipsoidal bounds on the control variable consider the following approximation problem. Find a tight ellipsoid-valued tube inside the reach tube that touches it along any prespecified smooth curve on the boundary. We construct this ellipsoidal tube, and show that if the given curve is itself a system trajectory, the tube can be calculated recursively, with minimum computational burden.     

Ellipsoidal approximations of the attraction domain in the path following problem for a wheeled robot with constrained resource

The path following problem for a wheeled robot with constrained resource moving along a given curvilinear path is studied. With the help of an earlier introduced change of variables, the path following problem is reduced to that of stability of the zero solution, and a control law linearizing the system in the case of the unconstrained control resource is synthesized. For the closed-loop system, the problem of finding the best ellipsoidal approximation of the attraction domain of the target path is set. To take into account the control constraint, an approach based on absolute stability theory is used. In the framework of this approach, construction of an approximating ellipse reduces to solving a parameterized system of linear matrix inequalities. The LMI system in the considered case can be solved analytically. Owing to this, construction of the best ellipsoidal approximation is reduced to solving a standard constrained optimization problem for a function of two variables. The proposed method is further extended to finding the best ellipsoidal approximation with an additional constraint on the maximum deviation from the target path. The discussion is illustrated by numerical examples.     

Algorithms to estimate the reachability sets of the pulse controlled systems with ellipsoidal phase constraints

Methods to construct the ellipsoidal estimates of the reachability sets of a nonlinear dynamic system with scalar pulse control and uncertainty in the initial data were proposed. The considered pulse system was rearranged in the ordinary differential inclusion already without any pulse components by means of a special discontinuous time substitution. The results of the theory of ellipsoidal estimation and the theory of evolutionary equations of the multivalued states of dynamic systems under uncertainty and ellipsoidal phase constraints were used to estimate the reachability sets of the resulting nonlinear differential inclusion.     

Quadratically saturated regulator for constrained linear systems

The problem of stabilization and null-controllability of open-loop unstable discrete-time multi-input systems with constraints on the inputs and the controls is addressed in this paper. First necessary and sufficient conditions for solvability of the problem are derived. They guarantee the existence of a linear controller leaving the state constraint set for the closed-loop system positively invariant. An optimal control law is computed, and the admissible set of initial conditions is given such that along trajectories of the closed-loop system the state and input constraints are satisfied. Then the domain of feasible initial conditions is enlarged using a saturating control if such is feasible     

Fast suboptimal predictive control with guaranteed stability

Efficient algorithms for constrained minimization of infinite horizon predictive control costs are presented. Control laws with guaranteed stability and asymptotic tracking are derived on the basis of appropriately chosen one-dimensional and ellipsoidal constraint set approximations. These achieve comparable performance with existing QP-based control laws with significant reductions in computational burden.     

Coordination control of multiple ellipsoidal agents with collision avoidance and limited sensing ranges

This paper contributes a design of cooperative controllers that force N mobile agents with an ellipsoidal shape and a limited sensing range to track desired trajectories and to avoid collision between them. A separation condition for ellipsoidal agents is first derived. Smooth step functions are then introduced. These functions and the separation condition between the ellipsoidal agents are embedded in novel pairwise collision avoidance functions to design coordination controllers. The proposed control design guarantees (1) smooth coordination controllers despite the agents’ limited sensing ranges, (2) no collision between any agents, (3) asymptotical stability of desired equilibrium set, and (4) instability of all other undesired critical sets of the closed loop system.     

An ellipsoidal off-line MPC scheme for uncertain polytopic discrete-time systems

An off-line Model Predictive Control (MPC) method based on ellipsoidal calculus and viability theory is described in order to address control problems in the presence of state and input constraints for uncertain polytopic linear plants subject to persistent disturbances. In order to reduce the computational burdens and conservativeness of traditional polytopic MPC schemes, the present approach carries out off-line most of the computations and it makes use of closed-loop predictions to improve the control performance. This is done by recursively pre-computing suitable ellipsoidal inner approximations of the exact controllable sets and solving on-line a simple and numerically low-demanding optimization problem subject to a set-membership constraint. Comparisons with three other recent off-line MPC approaches are also provided in the final example.     

输入受限系统的滚动时域预测控制

针对输入受限系统,提出滚动时域预测控制方法,用线性矩阵不等式和不变椭圆集处理输入受限问题,给出稳定性和可行性条件,并用半正定规划求解控制律.仿真例子表明了本文方法的优点.     

Control improvement method for impulsive systems

An optimal impulsive control problem is considered under constraint on the total impulse of control. In the problem there are conventional bounded controls and vector-valued measures (impulsive control). Based on a needle-shaped variation procedure an algorithm for optimal control of impulsive dynamic systems is elaborated. A discontinuous time reparameterization technique is applied. Control improvement property is established. Convergence analysis is provided.     

Pulse control of flexible multibody systems

An active pulse control method is developed to reduce the vibrations of multibody systems resulting from impact loadings. The pulse, which is a function of system generalized coordinates and velocities, is determined analytically using energy and momentum balance equations of the impacting bodies. Elastic components in the multibody system are discretized using the finite element method. The system equations of motions and nonlinear algebraic constraint equations describing mechanical joints between different components are written in the Lagrangian formulation using a finite set of coupled reference position and local elastic generalized coordinates. A set of independent differential equations are identified by the generalized coordinate partitioning of the constraint Jacobian matrix. These equations are written in the state space formulation and integrated forward in time using a direct numerical integration method. Dependent coordinates are then determined using the constraint kinematic relations. Points in time at which impact occurs are monitored by an impact predictor function, which controls the integration algorithms and forces for the solution of the momentum relation, to define the jump discontinuities in the composite velocity vector as well as the system reaction forces. The effectiveness of the active pulse control in reducing the vibration of flexible multibody aircraft during the touchdown impact is investigated and numerical results are presented.     

Adaptive control strategies for cooperative dual-arm manipulators

The article describes three strategies for adaptive control of cooperative dual-arm robots. In the position-position control strategy, the adaptive controllers ensure that the end-effector positions of both arms track desired trajectories in Cartesian space despite unknown time-varying interaction forces exerted through the load. In the position-hybrid control strategy, the adaptive controller of one arm controls end-effector motions in the free directions and applied forces in the constraint directions; while the adaptive controller of the other arm ensures that the end-effector tracks desired position trajectories. In the hybrid-hybrid control strategy, the adaptive controllers ensure that both end-effectors track reference position trajectories while simultaneously applying desired forces on the load. In all three control strategies, the coupling effects between the arms through the load are treated as “disturbances” which are rejected by the adaptive controllers while following desired commands in a common frame of reference. The adaptive controllers do not require the complex mathematical model of the arm dynamics or any knowledge of the arm dynamic parameters or the load parameters such as mass and stiffness. The controllers have simple structures and are computationally fast for on-line implementation with high sampling rates. Simulation results are given to illustrate the proposed adaptive control strategies.     

Convexity and convex approximations of discrete-time stochastic control problems with constraints

We investigate constrained optimal control problems for linear stochastic dynamical systems evolving in discrete time. We consider minimization of an expected value cost subject to probabilistic constraints. We study the convexity of a finite-horizon optimization problem in the case where the control policies are affine functions of the disturbance input. We propose an expectation-based method for the convex approximation of probabilistic constraints with polytopic constraint function, and a Linear Matrix Inequality (LMI) method for the convex approximation of probabilistic constraints with ellipsoidal constraint function. Finally, we introduce a class of convex expectation-type constraints that provide tractable approximations of the so-called integrated chance constraints. Performance of these methods and of existing convex approximation methods for probabilistic constraints is compared on a numerical example.     

Robustness analysis for systems with ellipsoidal uncertainty

This note derives an explicit expression for computing the robustness margin for affine systems whose real and complex coefficients are related by an ellipsoidal constraint. The expression, which is an application of a result by Chen, Fan, and Nett for rank-one generalized structured singular-value problems, extends and unifies previous results on robustness margin computation for systems with ellipsoidal uncertainty. © 1998 John Wiley & Sons, Ltd.     

Probabilistic tubes in linear stochastic model predictive control

This paper considers constrained control of linear systems with additive and multiplicative stochastic uncertainty and linear input/state constraints. Both hard and soft constraints are considered, and bounds are imposed on the probability of soft constraint violation. Assuming the plant parameters to be finitely supported, a method of constraint handling is proposed in which a sequence of tubes, corresponding to a sequence of confidence levels on the predicted future plant state, is constructed online around nominal state trajectories. A set of linear constraints is derived by imposing bounds on the probability of constraint violation at each point on an infinite prediction horizon through constraints on one-step-ahead predictions. A guarantee of the recursive feasibility of the online optimization ensures that the closed loop system trajectories satisfy both the hard and probabilistic soft constraints. The approach is illustrated by a numerical example.     

Ellipsoidal Techniques for Reachability Analysis of Discrete-Time Linear Systems

This paper describes the computation of reach sets for discrete-time linear control systems with time-varying coefficients and ellipsoidal bounds on the controls and initial conditions. The algorithms construct external and internal ellipsoidal approximations that touch the reach set boundary from outside and from inside. Recurrence relations describe the time evolution of these approximations. An essential part of the paper deals with singular discrete-time linear systems     

Sliding mode force,motion control,and stabilization of elastic manipulator in the presence of uncertainties

This article considers the question of position and force control of three-link elastic robotic systems on a constraint surface in the presence of robot parameter and environmental constraint geometry uncertainties. The approach of this article is applicable to any multi-link elastic robot. A sliding mode control law is derived for the position and force trajectory control of manipulator. Unlike the rigid robots, sliding mode control of an end point gives rise to unstable zero dynamics. Instability of the zero dynamics is avoided by Controlling a point that lies in the neighborhood of the actual end point position. The sliding mode controller accomplishes tracking of the end-effector and force trajectories on the constrained surface; however, the maneuver of the arm causes elastic mode excitation. For point-to-point control on the constraint surface, a stabilizer is designed for the final capture of the terminal state and vibration suppression. Numerical results are presented to show that in the closed-loop system position and force control is accomplished in spite of payload and constraint surface geometry uncertainty. © 1995 John Wiley & Sons, Inc.