Optimal disturbance rejection for discrete-time nonlinear systems with time-delays via high-gain observer-based control

A new feedforward and feedback optimal control law for a class of nonlinear systems with persistent disturbances is presented in this paper. By using a successive approximation approach (SAA), the original nonlinear optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The optimal control law obtained consists of analytical linear feedforward and feedback terms and a nonlinear compensation term which is the limit of the solution sequence for the adjoint vector differential equations. By using the finite-step iteration of nonlinear compensation sequence, we can obtain a feedforward and feedback suboptimal control law. A disturbance observer is designed to get a physically realizable controller. Simulation examples show the effectiveness of the approach.

ABSTRACTThis paper focuses on composite nonlinear feedback (CNF) controller design for tracking control problem of strict-feedback nonlinear systems with input saturation to address the improvement of transient performance. First, without considering the input saturation, a stabilisation control law is designed by using standard backstepping technique for the nonlinear system, then a feedforward control law is added to the backstepping-based stabilisation control law to construct a tracking control law. The tracking control law is tuned to drive the output of the closed-loop system to track a command input with quick response. Then, an additional nonlinear feedback law is constructed and combined with the tracking control law to obtain a CNF control law. The role of this additional nonlinear feedback law is to smoothly change the damping ratio of the closed-loop system while the system output approaches the command input, and to reduce overshoot caused by the tracking control law. It is shown that the extra-adding nonlinear feedback part does not cause the loss of stability of the closed-loop system in its attractive basin.

Conditions for the solvability of the discrete Lyapunov and the discrete Riccati equations subject to linear equality constraints are derived. These problems arise naturally in the context of output min-max robust control. It is shown that the following problems are equivalent to one another: (a) the solvability of the constrained discrete Riccati equation; and (b) the existence of a feedback gain that guarantees the solvability of the constrained discrete Lyapunov equation of the resulting closed loop. A simple criterion for the existence of a solution to both problems is presented. These problems are shown to be related to the discrete positive real property.

This paper shows that the analysis and the design of discrete-time models of continuous-time systems can be carried out within a single framework. Briefly, all classes of discrete-time models, namely the invariant models, the mapping models and the matched pole-zero models, are shown to incorporate the common structure of a hold followed by a continuous-time system and the ideal sampler. This interpretation of discrete-time models of continuous-time systems facilitates the understanding of the differences and similarities among the various discrete-time models. A numerical example illustrates the concepts involved.

In this paper, a model predictive control (MPC) strategy for nonlinear homogeneous reaction systems is proposed. The nonlinear system is first transformed to a linear parameter varying (LPV) system by means of a linear transformation known as extents decomposition. The resulting LPV system is further converted to a linear time invariant (LTI) system by means of a parametric state feedback and feedforward control laws. The use of this description results in linear MPC with a quadratic performance index and nonlinear state parameter constraints. However, based on the polytopic nature of the parameter of the LPV systems, the constraints are transformed to a set of intersected polyhedrons. The final result is a linear MPC problem with linear constraints that can be easily converted and solved as a quadratic programming problem. Finally, the performance of the control strategy is illustrated in simulation and compared with a controller based on a constant-parameter LTI model.

Rapid advances in information processing, communication and sensing technologies have enabled more and more devices to be provided with embedded processors, networking capabilities and sensors. For the field of estimation and control, it is now possible to consider an architecture in which many simple components communicate and cooperate to achieve a joint team goal. This distributed (or networked) architecture promises much in terms of performance, reliability and simplicity of design; however, at the same time, it requires extending the traditional theories of control, communication and computation and, in fact, looking at a unified picture of the three fields. A systematic theory of how to design distributed systems is currently lacking. This dissertation takes the first steps towards understanding the effects of imperfect information flow in distributed systems from an estimation and control perspective and coming up with new design principles to counter these effects. Designing networked systems is difficult because such systems challenge two basic assumptions of traditional control theory - presence of a central node with access to all the information about the system and perfect transmission of information among components. We formulate and solve many problems that deal with the removal of one, or both, of these assumptions. The chief idea explored in this dissertation is the joint design of information flow and the control law. While traditional control design has concentrated on calculating the optimal control input by assuming a particular information flow between the components, our approach seeks to synthesize the optimal information flow along with the optimal control law that satisfies the constraints of the information flow. Thus besides the question of 'What should an agent do?', the questions of 'Whom should an agent talk to?', 'What should an agent communicate?', 'When should an agent communicate?' and so on also have to be answered. The design of the information flow represents an important degree of freedom available to the system designer that has hitherto largely been ignored. As we demonstrate in the dissertation, the joint design of information flow and the optimal control input satisfying the constraints of that information flow yields large improvements in performance over simply trying to fit traditional design theories on distributed systems. We begin by formulating a distributed control problem in which many agents in a formation need to cooperate to minimize a joint cost function. We provide numerical algorithms to synthesize the optimal constrained control law that involve solving linear equations only and hence are free from numerical issues plaguing the other approaches proposed in the literature. We then provide and analyze a model to understand the issue of designing the topology according to which the agents interact. The results are very surprising since there are cases when allowing communication to happen between two agents may, in fact, be detrimental to the performance. We then move on to consider the effects of communication channels on control performance. To counter such effects, we propose the idea of encoding information for the purpose of estimation and control prior to transmission. Although information theoretic techniques are not possible in our problem, we are able to solve for a recursive yet optimal encoder / decoder structure in many cases. This information flow design oriented approach has unique advantages such as being optimal for any packet drop pattern, being able to include the effect of known but random delays easily, letting us escape the limits set by reliability for transmission of data across a network by using intermediate nodes as 'repeaters' similar to a digital communication network and so on. We finally take a look at combining the effects of multiple sources of information and communication channels on estimation and control. We look at a distributed estimation problem in which, at every time step, only a subset out of many sensors can transmit information to the estimator. This is also a representative resource allocation problem. We propose the idea of stochastic communication patterns that allows us to include the effects of communication channels explicitly during system design. Thus, instead of tree-search based algorithms proposed in the literature, we provide stochastic scheduling algorithms that can take into account the random packet drop effect of the channels. We also consider a distributed control problem with switching topologies and solve for the optimal controller. The tools that we develop are applicable to many other scenarios and we demonstrate some of them in the dissertation. Along the way, we look at many other related problems in the dissertation. As an example, we provide initial results about the issue of robustness of a distributed system design to a malfunctioning agent. This notion is currently lacking in the control and estimation community, but has to be a part of any effective theory for designing networked or distributed systems.

This paper develops an approximate design procedure of optimal tracking controllers for a class of nonlinear systems with persistent disturbances in the infinite time domain. By using the successive approximation approach (SAA), the two-point boundary value (TPBV) problem, which is derived from the optimal control theory, is transformed into solving a sequence of linear TPBV problems. The solution sequence of the linear TPBV problems uniformly converges to the solution of the original optimal tracking control (OTC) problem. A feedforward and feedback optimal tracking control (FFOTC) law is derived from a Riccati equation, matrix equations and a sequence of adjoint vectors. The existence and uniqueness conditions of the FFOTC are given. At last, a design algorithm of solving the approximate FFOTC law is proposed, and simulation results demonstrate the validity of the algorithm.

MTN optimal control of MIMO non-affine nonlinear time-varying discrete systems for tracking only by output feedback

We consider an optimal control problem of MIMO systems affected by external disturbances with known dynamic characteristics but unknown initial conditions based on stability degree constraint. The objective is to find an optimal control, by which the cost function minimum and the state with the optimal control having a higher mean-square convergence rate can be obtained. A feedforward and feedback optimal control law is derived from a Riccati equation and a Matrix equation. We give the existence and uniqueness conditions of the feedforward and feedback optimal control law, and present an actualizing algorithm of solving the optimal control law. Finally, a practical example is given to illustrate the effectiveness of the theory.

International audience

Nonlinear acoustic systems are often described by means of nonlinear maps which act as instantaneous constraints on the solutions of a system of linear differential equations. This description leads to discrete-time models exhibiting non-computable loops. This paper presents a solution to this computability problem by means of geometrical transformation of the nonlinearities and algebraic transformation of the time-dependent equations. The proposed leads to stable and accurate simulations even at relatively low sampling rates.

This paper considers the global optimal control for nonlinear interconnected large-scale systems with respect to linear quadratic functional. A successive approximation procedure is proposed to design the feedforward feedback nearly optimal control law. By introducing the Successive Approximation Approach of the nonlinear differential equation theory, a nonhomogeneous linear two-point boundary value problem sequence is constructed to approximate the nonlinear large-scale two-point boundary value problem, which is the necessary condition of the nonlinear control problem. By iteratively solving this nonhomogeneous linear approximating sequence, a global nearly optimal control law that composed of a linear closed-loop feedback compensatory term and an open-loop feedforward compensatory term is designed. An algorithm is presented to design certain qualified nearly optimal control law by truncating an infinite iteration result. An illustrative example shows the validity of the algorithm.

In this paper, the nonlinear optimal control problem is investigated for spacecraft formation flying with collision avoidance. Based on a nonlinear model of formation flying, two optimal tracking control laws are proposed to ensure the formation pattern converges to the predetermined configuration. The first optimal control law which combines Lyapunov optimizing control with a trajectory-following optimization technique is developed to solve the finite-time nonlinear optimal control problem. For the second controller, an extended [Formula: see text]-[Formula: see text] method is applied to design a closed-form feedback control scheme for nonlinear control problem of spacecraft formation flying with non-standard cost functions. By taking into account the flying safety, a repulsive control scheme is incorporated in the optimal controller to ensure collision avoidance and minimize the performance index. Finally, a numerical example is performed to demonstrate the effectiveness of the proposed approaches.

In the design of digital control systems, the time delay due to computation time of a processor cannot be often neglected. In this paper, we consider the problem of synthesizing a type-1 optimal digital servosystem taking account of controller delay of one sampling period.It is first shown that the optimal digital servo problem in which a quadratic criterion function is defined for a given continuous-time plant can be reasonably reformulated to a discrete optimal regulator problem. To assure the solvability of this problem, some conditions for the preservation of stabilizability and detectability of the plant under sampling are given. In the discrete optimal regulator problem considered here, not only control input but also the initial state of the servocompensator are referred as free design parameters. The obtained optimal digital control law contains a feedforward term of reference signal in addition to a feedback term. It is also shown that the optimal control law has a function of estimating the state variables at the following sampling period in order to recover the controller delay.

Feedback linearization control is a simple and effective strategy whenever a faithful model of the system and its states are available. Feedback linearization may suffer from the mismatches between the model used in the design and the actual system due to e.g. uncertain parameter values, parasitic dynamics, or because of the impossibility to measure some states of the system. To aleviate such an issue, we suggest a novel robust adaptive control approach using the evolving participatory learning algorithm together with a high-gain observer. The robust evolving granular high-gain observers (RegHGO) controller is suitable to control nonlinear systems that can be input-output linearized by feedback. The approach is robust to modeling mismatches and does not require full state availability because, once the system is in a suitable canonical form, a high gain observer can be constructed to supply the state information required for control. The usefulness and efficacy of the approach is shown using a fan and plate system, and an DC motor driven angular arm-position control. The fan and plate evaluates the controller in a regulation process, and the angular arm position control evaluates reference tracking perfromance. In both cases, time-varying parameter uncertainties disturb the closed-loop control system. Both, qualitative and quantitative performance evaluation of the RegHGO controller are done. Additionally, we compare the performance of the RegHGO controller with well-established methods such as exact feedback linearization with high-gain state observer and extensions. The results show that robust evolving granular control with high-gain observers achieves better performance than its counterparts.