Finite Horizon Minimax Optimal Control of Nonlinear Continuous Time Systems with Stochastic Uncertainty

Robust controller synthesis attracts reviving research interest, driven by\nthe rise of learning-based systems where uncertainty and perturbation are\nubiquitous. Facing an uncertain situation, a robustly stabilizing controller\nshould maintain stability while operating under a perturbed system deviating\nfrom its nominal specification. There have been numerous results for robust\ncontroller synthesis in multiple forms and with various goals, including\nmu-synthesis, robust primal-dual Youla, robust input-output, and robust system\nlevel parameterizations. However, their connections with one another are not\nclear, and we lack a general approach to robust controller analysis and\nsynthesis.\n To serve this purpose, we derive robust stability conditions for general\nsystems and formulate the general robust controller synthesis problem. The\nconditions hinge on the realization-stability lemma, a recent analysis tool\nthat unifies existing controller synthesis methods. Not only can the conditions\ninfer a wide range of existing robust results, but they also lead to easier\nderivations of new ones. Together, we demonstrate the effectiveness of the\nconditions and provide a unified approach to robust controller analysis and\nsynthesis.\n

Automated Synthesis of Fixed Structure QFT Controller using Interval Constraint Satisfaction Techniques

The problem of optimal control synthesis of complex nonlinear and multidimensional systems oriented on the building of integrated control systems with the adaptation and self-organization properties is of much current interest. The development of optimization methods employed in control problems demonstrated their effectiveness and has led to the construction of algorithms of approximate solution of optimization problems that significantly extends the class of control problems that can be set. In this work we consider a synthesis of optimal control for affine nonlinear system, in which a special emphasis is placed on the degenerate problems of optimal control. Our approach is based on the extension of the method of minimization of generalized work function by properly transforming the initial nondegenerate problem of optimal control into degenerate problem of optimal control synthesis. The search for the solution is based on the suboptimal strategy that includes the construction of minimizing sequences that converge interval-wise to the optimal solution of the initial problem. We obtain some optimality conditions as well as the necessary and sufficient conditions of existence of optimal solution.

The paper introduces a new method for finding optimal control of algebraic dynamic systems. The structure of algebraic dynamical systems is nonlinear with quadratic and bilinear terms. A new hybrid extended Fourier series is introduced, and state and control variables of the system are expanded by this series. Moreover, properties of new series are presented, and integration and product operational matrices are obtained. Using operational matrices, optimal control of the systems is converted to a set of simultaneous nonlinear algebraic relations. An illustrative example is included to compare our results with those in the literature.

Stochastic optimal control based on reliability plays a crucial role in mitigating structural failure and ensuring safe operation. For general stochastic vibration systems, a data-driven method for reliability-based optimal control is proposed based on random state data. Firstly, the feedback control is split into conservative and dissipative components in coherence with physical intuition, and each component is expanded upon using pre-selected basic functions, resulting in a modified system with undetermined coefficients. Then, from discrete random samples, the expressions for the probability densities of the first-passage time and the stationary response can be identified, explicitly including system and excitation parameters, as well as initial and safety boundary conditions. Finally, a performance index, combining system reliability and control cost, is constructed and can be explicitly expressed in terms of the undetermined coefficients of the control forces. The optimal control problem for determining the optimal control forces is reformulated as an optimization problem for determining the coefficients that minimize the performance index. This unconstrained optimization problem can be easily solved. Three typical nonlinear stochastic systems, a controlled Duffing oscillator, a controlled nonlinear hysteretic system, and a controlled 2-DOF nonlinear dynamical system, are given to illustrate the validity and accuracy of the proposed data-driven method. Additionally, the control performance of the optimal control strategy in improving the system reliability is also discussed.

This paper addresses the problems of robust stability analysis and L1-gain controller synthesis for uncertain impulsive positive systems. First, by employing the idea of impulse interval partitioning, an impulse-time-dependent discretized copositive Lyapunov function is proposed to analyze the robust stability and L1-gain performance of the considered system without control inputs, and several stability conditions and L1-gain criteria are respectively derived. Subsequently, a sufficient condition is formulated for the existence of state-feedback controllers with which not only the positivity and robust uniform asymptotic stability of the resulting closed-loop system are guaranteed, but also a prescribed L1-gain performance is satisfied simultaneously. Furthermore, to make the controller synthesis problem numerically tractable, we propose an iterative convex optimization algorithm to compute the desired controller parameters. Finally, three numerical examples and two realistic examples are presented to show the effectiveness and applicability of the proposed methodology.

In this paper, we consider the problem of robust controller synthesis with noise spectral uncertainties and nonlinear time‐varying unmodeled dynamics in the LQG (linear‐quadratic‐Gaussian) optimal systems. We derive a robust stabilization criterion for multivariable stochastic systems with nonlinear time‐varying unmodeled dynamics. Meanwhile, we shall frame up an algorithm for synthesizing the robust controller not only to minimize the least favorable cost functional J but also to satisfy the robust stabilization criterion via specifying an appropriate weighting scalar in the cost functional. The Wiener‐Hopf's technique in LQG optimal system, the saddle point theory, the Nevanlinna‐Pick interpolation theory, and a property of the contractive operator are employed to treat this problem. Furthermore, there are no stable and minimum phase constraints on the controlled plant. Key words: LQG optimal controlnoise spectral uncertaintiesnonlinear time‐varying perturbations Notes To whom all correspondence should be addressed.

In this paper, the problem about optimal iterative learning control of general nonlinear discrete-time systems has been studied. Based on contraction mapping principle, if some mild assumptions are given, we can obtain some sufficient conditions for the existence of optimal iterative learning control of general nonlinear discrete-time systems. These conditions are generalizations of the existence conditions of optimal iterative learning control of linear discrete-time systems.

This paper considers the control synthesis problem and its solution using symbolic regression. Symbolic regression methods, which were previously called genetic programming methods, allow one to use a computer to find not only the parameters of a given regression function but also its structure. Unlike other works on solving the control synthesis problem using symbolic regression, the novelty of this paper is that for the first time this work employs a training dataset to address the problem of general control synthesis. Initially, the optimal control problem is solved from each point in a given set of initial states, resulting in a collection of control functions expressed as functions of time. A reference model is then integrated into the control object model, which generates optimal motion trajectories using the derived optimal control functions. The control synthesis problem is framed as an approximation task for all optimal trajectories, where the control function is sought as a function of the deviation of the object from the specified terminal state. The optimization criterion for solving the synthesis problem is the accuracy of the object’s movement along the optimal trajectory. The paper includes an example of solving the control synthesis problem for a mobile robot using a supervised machine learning method. A relatively new method of symbolic regression, the method of variational complete binary genetic programming, is studied and proposed for the solution of the control synthesis problem.

Synthesis of smoother robust control for nonlinear uncertain dynamical systems is considered. The existing saturation-type robust controller guarantees asymptotic stability or stability of uniform ultimate boundedness. It is shown that, mainly due to the part of uncertainties that is bounded by a constant, the controller may potentially have a very large rate of change in order to achieve a small ultimate bound for the state. This study, on its gain or the property of its partial derivative, reveals the need for designing smoother nonlinear robust controller along two directions. First, in a recursive design of robust control, fictitious robust control must be designed to be as smooth as possible so that the actual robust control will not have too large of a magnitude for implementation purposes. Second, if the uncertainties are known to have bounded rate of change, it is both practically and theoretically important to design a robust controller which rate of change is limited by that of the uncertainties. The objective of this paper is to design smoother robust controls by providing results on those two topics. >

The proposed event-triggered control design uses the adaptive dynamic programming (ADP) technique to solve the infinite-horizon optimal control of nonlinear continuous time system in affine form with complete unknown system dynamics in a forward time and online manner. The approximation property of the neural network (NN) is used to estimate the system dynamics and the value function with event-based sampling of state vector. Subsequently the estimated values are used to design the near optimal control policy. In addition, the NN weights are updated as a jump at every trigger instant, hence aperiodic in nature, to save computation when compared to the traditional NN-based approaches. Further, the closed-loop dynamics are formulated as a nonlinear impulsive dynamical system and the extension of the Lyapunov technique is utilized to prove the locally ultimate boundedness of all the closed-loop signals by deriving an adaptive event-trigger condition. Nonetheless, a positive lower bound on the inter-event time is guaranteed to avoid accumulation point. Finally, the analytical design is evaluated by using an example.

This paper is focused on usability of multiestimation scheme approach in the area of identification and control of nonlinear systems. A multiestimation scheme is introduced and subsequently used for the adaptive control of real-time nonlinear system. The multiestimation scheme integrates on-line identification of suitable models of a controlled system and a controller synthesis on base of the identified model. The real-time testing has been carried out by the control of nonlinear laboratory model of interconnected tanks (DTS200 by Amira company).

Robust control synthesis for seat suspension systems with actuator saturation and time-varying input delay

In this two-part paper we study stabilization and optimal control of linear time-invariant systems with stochastic multiplicative uncertainties. We consider structured multiplicative perturbations, which, unlike in robust control theory, consist of static, zero-mean stochastic processes, and we assess the stability and performance of such systems using mean-square measures. While Part 2 of this paper tackles and solves optimal control problems under the mean-square criterion, Part 1 is devoted to the stabilizability problem. We develop fundamental conditions of mean-square stabilizability which ensure that an open-loop unstable system can be stabilized by output feedback in the mean-square sense. For single-input single-output systems, a general, explicit stabilizability condition is obtained. This condition, both necessary and sufficient, provides a fundamental limit imposed by the system's unstable poles, nonminimum phase zeros and time delay. For multi-input multi-output systems, we provide a complete, computationally efficient solution for minimum phase systems possibly containing time delays, in the form of a generalized eigenvalue problem readily solvable by means of linear matrix inequality optimization. Limiting cases and nonminimum phase plants are analyzed in depth for conceptual insights, revealing, among other things, how the directions of unstable poles and nonminimum phase zeros may affect mean-square stabilizability in MIMO systems. Other than their independent interest, stochastic multiplicative uncertainties have found utilities in modeling networked control systems pertaining to, e.g., packet drops, network delays, and fading. Our results herein lend solutions applicable to networked control problems addressing these issues.

An identification control is a control of an object which mathematical model is unknown. For such objects initially the problem of identification is solved, and then for the resulting object model the problem of control synthesis is solved. As a result of control synthesis we obtain a multi-dimensional function that describes the dependence of control on the object state. After the implementation of this function in the control unit of the real object, it achieves the goal of control with optimal value of the quality criteria. To solve the identification and synthesis problems we use the method of network operator, which with the help of evolutionary algorithms finds the structure and parameters of the functions describing the model and functional dependence of control from the object state. In this paper, an example of using a network operator to solve the problem of identification control synthesis for a multilink robot is given.