Stochastic Approximation and Its Application to Problems of Prediction and Control Synthesis

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.

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.

In this thesis, a number of search techniques are proposed as a solution for program and discrete controller synthesis (DCS). Classic synthesis techniques facilitate exhaus- tive search, while genetic programming has recently proven the potential of generic search techniques. But is genetic programming the right search technique for the synthesis prob- lem? In this thesis we challenge this belief and argue in favor of simulated annealing, a different class of general search techniques. We show that, in hindsight, the success of genetic programming has drawn from what is arguably a hybrid between simulated annealing and genetic programming, and compare the fitness of classic genetic program- ming, the hybrid form, and pure simulated annealing. Our experimental evaluation suggests that pure simulated annealing offers better results for automated programming than techniques based on genetic programming. Discrete Controller Synthesis (DCS) and Program Synthesis have similar goals: they are automated techniques to infer a control strategy and an implementation, respectively, that is correct by construction. We also investigate the application of the search tech- niques that we have been used for program synthesis for the computation of deterministic strategies solving symbolic Discrete Controller Synthesis (DCS) problems, where a model of the system under control is given along with desired objective behaviours. We experi- mentally confirm that relative performance results are similar to program synthesis, and give a complexity analysis of our simulated annealing algorithm for symbolic DCS. From the performance results we obtain, we draw the conclusion that simulated annealing, when combined with efficient model-checking techniques, is worth further investigating to solve symbolic DCS problems. A tool is designed to explore the parameter space of different synthesis techniques. Besides using it to synthesise a discrete control strategies for reactive systems (controller synthesis) and for protocol adapters for the coordination of different threads (software synthesis), we can also use it to study the influence of turning various screws in the syn- thesis process. For simulated annealing, PranCS allows the user to define the behaviour of the cooling schedule. For genetic programming, the user can select the population size.

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.

We propose a general class of randomized gradient estimates to be employed in a recursive search for the minimum of an unknown multivariate regression function. Here only two observations per iteration step are used. Special cases include random direction stochastic approximation (Kushner and Clark), simultaneous perturbation stochastic approximation (Spall) and a special kernel based stochastic approximation method (Polyak and Tsybakov). If the unknown regression is p-smooth ($p\\ge 2$) at the point of minimum, these methods achieve the optimal rate of convergence $O(n^{-(p-1)/(2p)})$. For both the classical stochastic approximation scheme (Kiefer and Wolfowitz) and the averaging scheme (Ruppert and Polyak) the related asymptotic distributions are computed.

This study considers the problem of control synthesis via static output feedback for time‐delay systems with restricted frequency domain specifications. A static output‐feedback (SOF) controller is designed by a two‐stage method. The first stage is to design an initial full‐information (FI) controller, and the second stage is to explore an SOF controller heuristically from the initial FI controller. An iterative algorithm to optimise the initial FI controller is constructed. Moreover, a more effective algorithm to explore an SOF controller is also formulated. Finally, the effectiveness of the proposed method is illustrated via the classical ACC benchmark problem of cart‐spring system.

Two approaches to control system design for constrained systems are studied. The first involves theoretical investigations of constrained model predictive control algorithms. The second involves extensions of robust linear control theory to handle the nonlinear control schemes commonly used in practice for constrained systems. A novel model predictive control algorithm, with attractive functional and numerical characteristics is developed. This algorithm minimizes peak excursions in the controlled outputs and is particularly suited to regulatory control problems common in continuous process systems. Model predictive control concepts are extended to uncertain linear systems. An on-line optimizing control scheme (RMPC) is developed which has as its objective the minimization of worst-case tracking error for an entire family of linear plants. For model uncertainty descriptions which provide plant impulse response coefficients as affine functions of uncertain parameters, it is shown that the required minimax optimization problem can be recast as a single linear program. The discrete time optimal averaging level control problem is formulated and solved. A finite horizon approximation to the problem is introduced and analytical solutions are obtained in important special cases. A model predictive control formulation is introduced which provides optimal flow filtering and integral action. Analysis tools are provided to characterize the trade-off between flow filtering and rapid integral action. A complete theory is developed for the multivariable anti-windup, bumpless transfer (AWBT) problem. The theoretical framework allows the consideration of any linear time invariant (LTI) control system subject to plant input limitations and substitutions. A general AWBT compensation scheme, applicable to multivariable controllers of arbitrary structure and order, is developed. Conditions are derived under which this general AWBT method reduces to any one of several well-known heuristics for AWBT (e.g., PI anti-reset windup and IMC). The design issues which affect AWBT performance are identified and quantitative analysis methods are developed. Sufficient conditions for nonlinear stability of the AWBT compensated system are provided. These results are a generalization of, and are less conservative than, those available in the AWBT literature. The definition of AWBT performance objectives which are independent of controller structure allows the formulation of a general AWBT synthesis problem. This formal synthesiproblem addresses each of the identified performance objectives in a quantitative manner. The synthesis problem is shown to be a special case of a constrained structure controller synthesis (CSCS) problem. A solution method via reduction to static output feedback is presented and the engineering trade-offs available in the AWBT design are discussed.

This work eonsiden the H, norm control of structural systems with collocated sensors and actuators using an analytical upper bound approach. A parametrization of the output feedback control gains that achieve a desired H, norm bound is obtained using an analytical solution of the Bounded Real Lemma. This approach is verified experimentally on a cantilevered aluminum beam with two collocated pairs of piezoceramic patches that serve as sensors and actuators. The finite element model of this beam is developed and the simulations show a drop of -6 dB in the H, norm of the closed loop system. Experimental results verify the simulated results and demonstrate the effectiveness of this method. The proposed control methodology has obvious computational advantages compared to traditional H, control design methods, especially for lnrge scale systems. The control of structural systems with collocated sensors and actuators has been shown to provide great advantages from a stability, passivity, robustness and an implementation viewpoint. For example, collocated control can easily be achieved in a space structure when an attitude rate sensor is placed at the same point as a torque actuator.(l)(2) The actuator placement and sizing optimization for vibration sup pression of a collocated beam structure is discussed in Ref. (3).Collocation of sensors and actuators leads to symmetric transfer functions. Several other classes of engineering sys- tems, such as circuit systems, chemical reactors and power networks, can be modelled as systems with symmetric trans- fer functions. Stabilization, robustness, model reduction and control of such systems has been examined recently.(4)(5) The H, norm control of structural systems has attracted a lot of attention recently due to the interpretation of the H, norm in the frequency domain. In early work, D-K iteration method was used to design H, controllers for a non-collocated flexible shucture. Now state space H, control based on the standard Riccati equation approach or the linear matrix inequality (LMI) formulation is a well developed control synthesis tool. The optimal static state feedback and full-order dynamic output feedback H, con- trol synthesis problems can be solved using iterations on the corresponding Riccati solutions or via the computational solution of a convex LMI optimization problem.(7)(8) On the other hand, the static output feedback and the fixed-order dynamic output feedback H, control synthesis problems are difficult computational problems since they require the

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

In this work, we consider a simplified version of the distributed control synthesis problem where the system is modeled as a Boolean network and specifications are given as propositional formulae on environment and system variables. In this relatively static Boolean setting, (centralized) reactive control synthesis problem reduces to a quantified Boolean satisfiability problem which can be solved by off-the-shelf SAT solvers. For the distributed control synthesis, we propose a recursive algorithm that starts from a global assume-guarantee specification and maps it to local assume-guarantee specifications for individual subsystems. Finding these local assume- guarantee specifications leads to a modular design framework. For instance, local controllers for subsystems can be synthesized independently via SAT or a local subsystem can be replaced with another subsystem that satisfies the same local assume-guarantee specifications. We also investigate how the interconnection structure of the network affects the performance of this algorithm. In particular, we show that if the interconnection structure is a multi-tree and the global assumptions and guarantees satisfy certain independence conditions, the algorithm is complete.

SUMMARY In this paper, several concepts of switching frequency are introduced to analyze the properties and performance of switched systems in infinite as well as finite-time intervals. The observation is very motivating that different system properties and performances depend on different switching frequencies. Sufficient conditions ensuring asymptotic stability, l2 gain performance, and state boundness are derived on the basis of the notions of switching frequency, respectively. Then, on the basis of the analysis results, the control synthesis problems are addressed. LMI-based design algorithms are proposed to meet different control synthesis requirements. Numerical design examples are provided to demonstrate our results. Copyright © 2011 John Wiley & Sons, Ltd.

Chapter 7 - Advanced approaches to multiagent coordination

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.

The paper is concerned with a problem of adaptive tracking system control synthesis. It is assumed that a nonlinear object dynamics (model structure) is (partially) unknown and some of its nonlinear characteristics can be approximated by a sort of functional approximators. It has been proven that a nonlinear state feedback plus parameter adaptation are able to ensure its asymptotic stability. The proposed controller design is based on modified adaptive backstepping method. In this approach, as opposite to the standard backstepping, the design process has been reduced to a single iteration, which enables avoiding complicated recursive procedure. An extra task here is to find a control synthesis of linear-like, chained system. The solution of the last problem in general framework is also presented. The obtained thereby controller has several attractive features of both the feedback linearization and backstepping schemes.