Renormalized measure of regular languages

The signed real measure of regular languages has been introduced and validated in recent literature for quantitative analysis and synthesis of discrete-event supervisory (DES) control systems, where all events are assumed to be observable. This paper presents a modification of the language measure for supervisory control under partial observation and shows how to generalize the analysis when some of the events may not be observable at the supervisory level. Thus, for synthesis of DES control systems, the language measure of partially observable discrete-event processes is expressed in a closed form, which is structurally similar to that of completely observable discrete-event processes.

Supervisory control theory for discrete event systems, introduced by Ramadge and Wonham, is based on a non-probabilistic formal language framework. Building on the concept of signed real measure of regular languages, this paper formulates a comprehensive theory for optimal control of finite-state probabilistic processes. It is shown that the resulting discrete-event supervisor is optimal in the sense of elementwise maximizing the renormalized language measure vector for the controlled plant behavior and is efficiently computable.

In this chapter, we will further study the concept of dissipative systems which is a very useful tool in the analysis and synthesis of control laws, for linear and nonlinear dynamical systems. One of the key properties of a dissipative dynamical system is that the total energy stored in the system decreases with time. Dissipativeness can be considered as an extension of PR systems to the nonlinear case. Some relationships between positive real and passive systems have been established in Chap. 2. There exist several important subclasses of dissipative nonlinear systems with slightly different properties which are important in the analysis. Dissipativity is useful in stabilizing mechanical systems like fully actuated robots manipulators [], robots with flexible joints [, , , , ], underactuated robot manipulators, electric motors, robotic manipulation [], learning control of manipulators [, ], fully actuated and underactuated satellites [], combustion engines [], power converters [, , , , , ], neural networks [, , , ], smart actuators [], piezoelectric structures [], haptic environments and interfaces [, , , , , , , , , ], particulate processes [], process and chemical systems [, , , , ], missile guidance [], model helicopters [], magnetically levitated shafts [, ], biological and physiological systems [, ], flat glass manufacture [], and visual feedback control [] (see Sect. 9.4 for more references). Some of these examples will be presented in the following chapters.

A cyclic unary regular language is a regular language over a unary alphabet that is represented by a cyclic automaton. We propose a similarity measure for cyclic unary regular languages by modifying the Jaccard similarity coefficient and the Sørensen coefficient to measure the level of overlap between such languages. This measure computes the proportion of strings that are shared by two or more cyclic unary regular languages and is an upper bound of the Jaccard coefficient and the Sørensen coefficient. By using such similarity measure, we define a dissimilarity measure for cyclic unary regular languages that is a semimetric distance. Moreover, it can be used for the non-cyclic case.

The objective of this paper is to illustrate that the standard form or Linear Fractional Transformation LFT framework is a unique and powerful framework to treat with nonlinear systems during control law synthesis and analysis. It is also shown that the expression linear in the acronym LFT is too restrictive as a nonlinear behaviour can be simulated by using the LFT framework. Within this paper, a precise and as small as possible LFT for a nonlinear 2-DOF aircraft with time varying parameters is derived. This LFT is validated both in a linear and nonlinear framework.

We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous-time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract super exponentially to a constant-dimensional subsystem. Under a non-degeneracy condition on the rank deficiency of the associated Poincaré map, the contraction occurs in finite time regardless of the stability properties of the orbit. Hybrid transitions may be removed from the resulting subsystem via a topological quotient that admits a smooth structure to yield an equivalent smooth dynamical system. We demonstrate reduction of a high-dimensional underactuated mechanical model for terrestrial locomotion, assess structural stability of deadbeat controllers for rhythmic locomotion and manipulation, and derive a normal form for the stability basin of a hybrid oscillator. These applications illustrate the utility of our theoretical results for synthesis and analysis of feedback control laws for rhythmic hybrid behavior.

This paper addresses the synthesis of control laws for the reorientation of a small nanosatellite spacecraft of the Cube- Sat standard in the case of severe degradation of actuator efficiency. Severe degradation (or reduction of efficiency) of actuators refers to a condition where the control torque generated by the actuator is reduced to 5 % of its maximum value under nominal operating conditions. The angular motion model accounts for gravitational, aerodynamic, and disturbance torques. The spacecraft reorientation problem is solved for the case of degradation in two control channels using sliding mode control. An original nonlinear sliding surface is used for the synthesis of the control laws. During the problem-solving process, quasi-adaptive and adaptive control laws were obtained, and their performance was compared using mathematical modelling. А quasi-adaptive control law is understood as a law in which the coefficients of the sliding surface are constant but depend on the initial angular velocity. An adaptive control law, on the other hand, is one in which the coefficients of the sliding surface change according to predefined differential equations. For adjusting the two coefficients of the sliding surface, an approach based on experimentally derived relationships between these two coefficients and the initial angular velocity was proposed and implemented. Based on the results of the numerical simulations, it can be concluded that the adaptive control law provides a shorter solution time compared to the quasi-adaptive control law. The quasi-adaptive law, however, is simpler to implement, as it does not require the adaptation of the coefficients.

We consider the information constrained optimization problem for stochastic dynamical systems governed by quasi-linear Ito equations. Let us describe the information constraints. We suppose that each control vector component depends on a prespecified set of precisely measured state vector components. In this article we present an algorithm for synthesis of the suboptimal control law. This control law is the linear feedback regulator. The linear parameter and the constant term of the regulator are polynomial functions of time. The algorithm is successfully applied to the problem of two-link robotic arm optimal control. These manipulators may be effectively used at space stations, e.g. for moving cargo in outer space.

This paper presents a direct digital control law synthesis procedure for a large order, sampled data, linear feedback system using constrained optimization techniques to meet multiple design requirements. A linear quadratic Gaussian type cost function is minimized while satisfying a set of constraints on the design loads and responses. General expressions for gradients of the cost function and constraints, with respect to the digital control law design variables are derived analytically and computed by solving a set of discrete Liapunov equations. The designer can choose the structure of the control law and the design variables, hence a stable classical control law as well as an estimator-based full or reduced order control law can be used as an initial starting point. Selected design responses can be treated as constraints instead of lumping them into the cost function. This feature can be used to modify a control law, to meet individual root mean square response limitations as well as minimum single value restrictions. Low order, robust digital control laws were synthesized for gust load alleviation of a flexible remotely piloted drone aircraft.

The synthesis of a discrete adaptive control law with simultaneous generation of a non-periodic sampling law for linear systems by means of an approximate bilinear model and standard quadratic optimization techniques is present. The usual adaptivity concept is extended to the optimally concept over a predefined planning horizon. In this way, an approximate discrete DARMA bilinear model from a state-space model is obtained by giving sufficient uniform observability requirements. The application of these bilinear models on simulated examples shows their validity in the adaptive sampling design of imperfectly known discrete linear plants under adaptive control. Global asymptotic stability of the overall (closed-loop) adaptive scheme is guaranteed under direct extensions of standard assumptions from the case of periodic sampling to the adaptive sampling case. In this way, adaptive sampling is designed in order to improve the adaptation transient performances.

An analytical procedure is described for synthesis and stability assessment of control systems for adaptive flying-qualities adherence with linear and nonlinear aircraft dynamics. This procedure directly incorporates desired low-order equivalent system (LOES) frequency and damping parameter values into the synthesis of control laws that provide optimal adherence to MIL-STD-1797A guidelines. The procedure is used to derive a longitudinal control law for a contemporary fighter aircraft. Simulation results are presented. It is believed that this approach offers improved flying qualities and may be suitable for mitigation of pilot induced oscillation problems because its closed-loop phase lag is essentially that of the LOES specification.

A method is proposed for synthesizing adaptive systems with a combined control law for the state vector for the nonlinear and non-stationary astatic objects with functional uncertainty on the state in the direct circuit and in the chain of their own feedbacks, as well as nonzero initial values of state variables. Moreover, the control objects of the synthesized systems can have an unstable equilibrium state, and their exit matrix can have right eigenvalues. The algorithm for the structural synthesis of the proposed method is based on defining the required equation of motion of the system and solving the inverse problem of dynamics. In this case, an observer of state variables is used in the system, a proportional controller in its direct circuit as a multiplication unit, and a self-tuning loop with a controller gain controller, the value of which changes inversely proportional to the current value of the control object’s gain. The parametric synthesis algorithm is reduced to solving the problem on the conditional minimum of the integral control error when the characteristic frequency polynomial is limited to an algorithmically linearized system.

Theory and Software Package for Simulation and Smart Control Design for Complex Flexible Aerospace Vehicles

The paper considers the problem of constructing a driver assistance system for a car with a trailer when reversing, functioning under apriori uncertainty of parameters of the controlled object car-trailer and uncontrollable disturbances. The mathematical model of the controlled object car-trailer and the basic relations, necessary for the synthesis of the control law are substantiated. Stability conditions of motion taking into account the notions of short and long trailer and the influence of disturbances are evaluated. The relative angular speed of the trailer, i.e., the angular speed of the trailer divided by the speed of the vehicle, is used as the controlled variable. An adaptive control law on the deviation of the the vehicle steered wheels to achieve the value of the controlled variable set by the driver using an implicit reference model is formed. Apriori uncertainty is eliminated by approximate estimates of trailer length and current parametric identification of the control object mathematical model. The current estimates delivered by the identifier are immediately used to synthesize the control law. The rationale for the operation of the adaptive control system is based on simplified adaptability conditions. The synthesized control law is aimed at preventing from reaching the folding angle with a short trailer and limiting the coupling angle with a long trailer. Automatic formation of warning signals, in particular about trailer folding, is proposed, and model examples showing efficiency of the proposed control system are given

: New handling qualities specifications are currently being developed for attack helicopters. Most unaugmented helicopters will not meet these specifications and feedback control is necessary to improve handling qualities so that safe operation close to the earth in poor weather conditions and/or at night is possible. In this paper a methodology for the direct design of helicopter flight control systems which meet handling qualities specifications is presented. This methodology uses full state feedback to place closed loop eigenvalues to achieve bandwidth specifications and to shape closed loop eigenvectors to decouple lateral and longitudinal responses to control inputs. Full state feedback requires that all state variables be known; however, only angular rates and normal acceleration are measured by sensors. Thus, a state estimator is required in the feedback loop in order to convert sensor outputs to control inputs. This estimator is designed using eigenstructure assignment so as to achieve loop transfer recovery. Design of a feedback system for use in precise hovering control for a modern attack helicopter is used to illustrate the method. Control law synthesis is accomplished using an eighth order model which includes only rigid body modes. Control law performance is evaluated using a 37th order model which includes rigid body, actuator, rotor, sensor, and flexure dynamics. It is found that a notch filter must be added to the design in order to eliminate a high frequency instability. Once this is accomplished, both the time and frequency response characteristics of the augmented helicopter are much improved compared with the unaugmented helicopter.