Control Design for Autonomous Vehicles: A Dynamic ... - Science Direct

European Journal of Control (2001)7:178±202 # 2001 EUCA

Control Design for Autonomous Vehicles: A Dynamic Optimization Perspective Fernando Lobo Pereira DEEC, FEUP and ISR-Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

Control design for autonomous vehicles involves a number of issues that are not satisfactorily addressed in classical control systems theory. There is typically the need for prescribing and commanding a collection of interacting dynamic control systems in order to meet the desired requirements for overall behavior, whereas conventional control design has only one system to govern. This context requires a whole new set of concepts, methods, and tools that are adapted to capture the integration of logical commands, continuous time evolution, and discrete event dynamics. A rapidly growing and extensive body of literature testifies to the valuable heritage of control systems that will address the challenges ahead. Several new issues arise in attempting a principled design of a specific system, and it is essential that they be resolved in order to ground the effort for a suitable control design framework. Furthermore, control design cannot be dissociated from the implementation phase, which introduces a wide range of challenges in communication and computation. This paper provides an overview of three control projects designed at the Underwater Systems and Technology Laboratory, and discusses the new challenges from the dynamic optimization viewpoint. Keywords: Autonomous vehicles; Heterogenous systems; Interacting control; Optimization; Underwater systems

Correspondence and offprint requests to: F.L. Pereira, DEEC, FEUP and ISR-Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. Email: [email protected].

1. Introduction The 20th century witnessed dramatic developments and successes of control systems which is, now, contemplating even more remarkable challenges that will foster further progress. These have been propelled by the need to deal with a number of novel, and somewhat interrelated, aspects of the plant to be controlled: high complexity, networked and/or decentralized operation, real-time constraints, and life-cycle integration. Control architectures, formal models spanning design and implementation, design principles and new control design frameworks (hybrid and networked control) have been considered to address the design issues for these systems. Their interdiscipinary character calls for a, stronger than ever, conjugation of efforts of systems, control, information and computation sciences. Furthermore, the design of concrete practical systems constitutes a natural and privileged environment to bring in new perspectives needed to shape future control design developments. With particular emphasis for the role played by dynamic optimization, we report on perspectives towards system's design supported by lessons extracted in three R&D projects involving autonomous vehicles, namely, autonomous mobile systems for industrial applications, inspection of underwater structures, and networked autonomous vehicles for ocean observation, Received 28 May 2001; Accepted 28 May 2001. Recommended by Martins de Carvalho, J.M.G. SaÂ da Costa and AntoÂnio Dourado.

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and, outline the approach to technological and tools development underlying our R&D strategy. Innovative applications and systems are the main source of motivation for technological and tools developments. This strategic orientation was reinforced by our participation in R&D projects in mobile robotics in the early nineties, [9,80,95,96]. Given our predominant background in control systems, this effort required an extensive literature survey which contributed not only to identify areas of potential technological development, but also to acquire a hands-on experience in the Systems Engineering Process [43], thus reinforcing the understanding of the importance of a principled approach to systems design ± in this case, in the context of autonomous and semi-autonomous vehicles. Our first designs soon gave way to more ambitious concepts for the operation of autonomous and semiautonomous multi-vehicle systems that require further technological developments which are starting to receive the attention of the control systems and computer science communities. Credibility comes with successful designs and implementations of innovative systems. In 1995, the Underwater Systems and Technologies Laboratory (LSTS) was created with the goal of providing the required experimental set-up for an intensive and low cost operation. LSTS started to operate the Autonomous Underwater Vehicle (AUV) Isurus as a test-bed for advanced control systems design. This AUV and a Remotely Operated Vehicle have been in operation for a while, and, now, a new generation of AUVs to test the networked control of multi-vehicle systems is being developed. This is a hot topic in robotics and one that strongly motivates cooperative research. In parallel, the last decade has witnessed unprecedented interactions between technological developments in computing, communications and control, and the design and implementation of interacting dynamical systems. These developments enable engineers to design new classes of systems, whose implementation, in turn, lead to: a better understanding of the underlying technological issues, the consideration of new conceptual approaches and the development of new tools (see [84]). Hybrid systems represent an exciting new field where advances are propelled by theoretical control systems and computer science. Several topics in control theory are of value for the research on hybrid systems.

produced a body of concepts and results whose interpretation in the context of hybrid systems has been fundamental for research: early ground-breaking work in this direction produced by Witsenhausen [107]; characterization of the reachable set [20,36, 98,103]; interpretation of reachable sets and solvability sets in terms of the level sets of value functions of some speci®c optimization problems (see [48,104]), and the associated implications for state feedback control synthesis; impulsive control, where the control space is enlarged to encompass measures, thus allowing for discontinuous trajectories [8,106]; relations between optimality and several forms of invariance in terms of Hamiltonian equations (see [24,46]); control of ordinary differential equation subject to positive switching costs is addressed in [17], where it was proved that the corresponding value functions are viscosity solutions to the dynamic programming quasi-variational inequalities.

Optimal control. Centered in the principle of optimality underlying dynamic programming, and in the necessary conditions of optimality, particularly, the Pontryagin maximum principle, optimal control

The control of distributed hybrid systems has presented a new challenge to control theory. The challenge comes from the distributed nature of the problem. For example, in networked multi-vehicle

Viability theory. Concepts and techniques from viability theory using set-valued analysis techniques (see [4]) have a natural extension to hybrid systems. Differential games. The setting here is that of a dynamic optimization problem where the control inputs are partitioned into at least two classes: (1) those available for controlling the system, (2) those used by the adversary or the disturbance. This setting encompasses systems where uncertainty prevails and a stochastic characterization of disturbances is not available. Automata and supervisory control. These bodies of concepts and results are used to model and control the behavior of discrete event dynamical systems (see [18,41]), an important component of the behavior of hybrid systems. Switched controls. The problem consists in selecting a strategy to switch between several control laws to achieve some goal [52]. If, on the one hand, switching controls arise naturally in systems with discrete actuator settings, then on the other, switching is sometimes essential, e.g., some systems cannot be stabilized by a continuous feedback law [22]. However, the introduction of discontinuous feedback laws is not trivial, since those are, in general, quite sensitive to measurement errors [89]. Krasovskii proposed what amounts to be a precursor to hybrid controllers to address this problem [46].

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systems, information and commands are exchanged among vehicles, and their roles, relative positions, and dependencies change in the course of operation. In fact, this challenge entails a shift in the focus of control theory: from prescribing and commanding the behavior of isolated systems to prescribing and commanding the behavior of interacting systems. The control and computer science communities addressed this challenge in the context of distributed hybrid systems, and contributed complementary views and techniques. The inter-disciplinary nature of hybrid systems requires a new description language. Meanwhile, control engineers have developed a collection of idioms, patterns and styles of organization that serves as a shared, semantically rich, vocabulary among them [79]. However, this shared vocabulary is still deeply rooted in the underlying mathematical framework and lacks some key concepts invoked by distributed computing. The cause may be that experience and functionality in computing are acquired at a rate unmatched by that of evolution of concepts in control systems. For example, it was only recently that the expressiveness of the language of differential equations and dynamic optimization was enlarged with concepts from mathematical logic, under the denomination of hybrid control (see, for example [12,34,53,67]). This article is organized as follows. In Section 2, each one of the three selected systems are presented in some detail and remarks concerning modeling and control design are provided in the last subsection. Then, the main issues on system's design are addressed and, in particular, those concerning the control architecture which, being the backbone of the overall systems control, provides the framework to articulate control systems design techniques. In Section 4, the main issues in control design problems are discussed. Finally, a couple of examples extracted from the considered R&D projects are presented in the last section.

2. Selected Systems A selection of systems that operate on land and sea is discussed. This selection illustrates a common approach for the integrated design and implementation of control systems for autonomous vehicles, and for the integration of the underlying technologies and tools. The issues of complexity, high level control, subsystems coordination and low level control design are addressed in the three considered systems, AMSIA, IES, and PISCIS, so that the emergent behaviors of

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the overall system fulfill the specified requirements. Some of these issues become particularly acute in the PISCIS system, where questions concerning vehicle heterogeneity, functional coordination, and resource allocation are best illustrated with the coordinated operation of AUVs. Networked systems control is a key paradigm in the PISCIS system, where the overall behavior control within the control architecture, the decentralization within a hierarchical architecture, and the stability of coordinated motions play a key role. Communication is a critical aspect in control design as only very limited data can be exchanged to ensure the emergence of the required collective behaviors. The role of human intervention is tackled within the AMSIA and IES systems, where tele-operation and tele-programming capabilities are present. 2.1. Autonomous Mobile Systems for Industrial Applications We will focus on the main issues of the AMSIA (Autonomous Mobile Systems for Industrial Applications) project which has its roots on the results of the PO-Robot project funded by the NATO Sciencefor-Stability Programme [80,96]. Consider a typical factory floor where parts are to be transported between workstations by one or more autonomous wheeled platforms. Typically, the autonomous vehicle will move in a 2D, easy to model, geometrically well structured environment, being the dock station locations and the paths connecting them well defined a priori. The variability to be considered might be due either to the reconfiguration of the manufacturing process layout or to random unexpected changes in the environment such as: obstacles generated by unintentionally dropped objects or occasional moving objects or persons, changes in the ground properties or noise disturbance that might affect communications and navigation sensors. Both low-level and topological navigation, the later involving the identification of environmental features, are required. Precise localization at docking stations is extremely important. Wide-bandwidth communications between each platform and system's external devices is available to support both tele-operation and vehicle coordination. To pick an object from one working station and to deliver it at another one is a basic typical example of transportation mission. Given a set of mission specific data, a detailed plan of activities is constructed by taking into account the expected world

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map. Relevant environmental features such as walls, doors, specific landmarks, and eventual obstacles are sensed during motion. This data is interpreted at various levels of abstraction and fed into the decision-making and control structures. The topological nature of the environment representation at a high abstraction level constitutes a natural framework to organize motion into simpler segments amenable to conventional control design techniques and endows the overall motion control feedback with robustness. The autonomous transportation system was designed to be integrated in a factory environment. It consists of wheeled mobile platforms, a communication infra-structure and a coordination center that schedules transportation tasks for each platform. The coordination center also uploads the on-board computer of each platform with the world map, that is, all features assumed to be invariant during the course of the mission, which can be sensed and properly interpreted as requirements for the execution of the mission activities, e.g., navigation, docking, object uploading or downloading, etc. However, we will not dwell on this problem for now. The wheeled mobile platform that we used as a prototype is described in [1] and a summary appears in Table 1. A complete set of functionalities was identified using the Systems Engineering Process, [43], out of which the following are examples: Elementary functionalities Trim motion regulation ± straight ahead or backwards, turn left or right while going ahead, turn left or right while going backwards; Data processing: ®ltering and fusing sensed data, resources optimization, trajectory planning, generating signals for the actuators, etc.;

Commands processing: tele-operation or teleprogramming. Complex functionalities

Tele-operation and tele-programming; Docking, parking and maneuvering in tight curves; Topological localization; Sonar management and sensor data fusion; Localization and piloting.

The tele-operation functionality allows for human intervention in exceptional situations, which is typically required in order to recover the proper operating conditions. Notice that most of these requirements are common to other classes of industrial applications, such as, cleaning, painting, inspection, monitoring, and surveillance. 2.2. Underwater Inspection System The project IES concerns the design and development of an advanced, versatile, low cost, highly operational, open system for the inspection of underwater structures based on a Remotely Operated Vehicle (ROV) shown in Fig. 1 [55,82,83,95]. Although tailored to the requirements of the Port Authority of Douro and LeixoÄes applications, the large spectrum of its inspection activities which range from the detection and evaluation of damaged underwater structures (e.g., corrosion in metallic structures, fissures in concrete walls), to the inspection of ship hulls and including the follow-up of a wide range of underwater civil works, make it a relatively general purpose system. The approach followed in this project consists in enhancing a conventional small size inspection-class ROV by incorporating onboard processing and

Table 1. Main mobile platform features. Mechanical features Sensing capabilities

Computation/ communications

Car-like platform with two front omnidirectional free wheels, two differentially actuated rear wheels. Odometric sensors in the rear wheels, belt with 24 sonars, rotating sonar, infra-red and laser sensors, inclinometers, accelerometers and compass. VME bus with a 68060 CPU, axis controller, 2 Mbites/s sensor network, and ethernet radio link

Fig. 1. The IES remotely operated vehicle.

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advanced sensing capabilities, making it possible to implement control schemes more elaborate than the ones usually available in commercial vehicles of this type. More specifically, this enhancement will allow advanced tele-operation and tele-programming which involves the ability to accept high level commands and perform simple ``missions'' autonomously. In particular, this means that some navigation and piloting tasks and tedious motion control, such as stability, and other positional control activities, can be automated, thus freeing the operator attention from the specialized training demanding activities of piloting the platform to the more specific ones of inspecting a given underwater structure. The system (Fig. 2) consists of: a platform possessing an onboard computational system, payload and navigation sensors, and actuators providing four degrees of freedom, a console with an interface through which the operator receives payload and navigation data from the platform and sends it high and low level commands, and an umbilical cord to supply power and transmit data and commands. Navigation sensors include inertial motion unit, acoustic doppler profiler, pressure depth cell, magnetic compass, inclinometers and an optical encoder in each thruster. An acoustic positioning and homing systems are also available to support navigation. A video camera supported on a pan-and-tilt unit, whose operation is coupled with that of an illumination system, constitute the only payload sensor suite for the simplest version. Systematic survey of a given wall while enabling the intervention of the operator is an example of a

Console Umbilical Cable

ROV Computational System

Interface Devices

Power Management

Inspection Sensors

Navigation Sensors

Camera Sonar

Compass Depth Acoustics Thrusters Lights Pan & Tilt Inclination IMU Doppler

...

Fig. 2. The IES system.

Actuators

typical mission. This mission may be decomposed in a number of activities: Moving from the platform launching location to the point at which the inspection starts with the camera pointing perpendicularly to the wall to be scanned. This segment can be performed by sending a high level command specifying the motion terminal con®guration. The con®guration of the video camera is stabilized by a two-stage stabilization scheme: a ®rst signi®cant attenuation of motion perturbations by controlling the platform, and a second stability ®ne-tuning by controlling the pan-and-tilt unit. A scanning pattern de®ned by the operator: The wall can be scanned with either vertical, or horizontal segments, or yo-yo type of motion with a speed which can be set by the operator within a range of values that depends on the environmental conditions. The scanning operation can be interrupted by high level commands issued by the operator forcing the platform to hover. Then, in order to better observe a certain feature, several other commands might be issued via either teleprogramming or tele-operation, such as: changing the orientation of the camera and its distance to the wall, or tracking a path generated by the operator. This can be done by controlling either or both the platform and the pan-and-tilt unit. During the execution of these commands, safety monitoring and stabilization of the platform are performed autonomously. Whenever con¯icts arise between autonomous and operator issued commands, action is undertaken to preserve safety and alternative viable commands are suggested to the operator. At some point, the operator might command the platform to continue its autonomous scanning mission from the location where it was interrupted. Then, the onboard system will automatically generate a set of maneuvers to recover the corresponding motion con®guration and the scanning operation continues from there. Once the scanning operation is concluded, a message is sent to the operator, the platform hovers for a while at the ®nal position and returns to the mission departure point. Clearly, this typical mission illustrates the main operating modes: free tele-operation, tele-programming in the horizontal or in the vertical planes, free motion, tracking a trajectory specified by the operator, hover, positional control.

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2.3. The PISCIS System The PISCIS project [30,55±57,65,92] addresses the design and implementation of an underwater data gathering system based on the coordinated control of multiple heterogeneous autonomous underwater vehicles that survey a given region of the ocean where communication and navigation devices as well as logistic stations are deployed. The data to be gathered may include bathymetry, salinity, oxygen, temperature, acoustic landscape, etc., which, once interpreted, may play a crucial role in detecting episodic events in the underwater milieu, improving the understanding of oceanographic phenomena, supporting navigation surveillance, managing ocean resources and monitoring their exploitation. The relevance of these concepts is attested by the worldwide increasing effort in the design and development of AUVs which will make it possible to envisage their operation in complex application scenarii. Given the specific features of the sub-aquatic phenomena, most of these applications impose highly strict requirements, such as: (1) collection of spatially distributed, temporally correlated measurements, (2) response in a timely fashion to episodic events, (3) gathering of time series of spatially distributed phenomena, and (4) interaction with measurement platforms in the course of observations. which are insufficiently addressed by the currently available data sampling techniques involving satellite remote sensing, moored instrumentation or towed bodies from ships. These are the so-called real-time oceanography requirements, to be fulfilled by the system being designed within the PISCIS project and are at the root of the Autonomous Ocean Sampling Network (AOSN) concept presented in [32]. At the current development stage, while the construction of a second vehicle is taking place, a first one has already been developed and field tested in several missions involving CTD (ConductivityTemperature-Depth) and bathimetric data measurements in very diverse environments such as the mouth of river Minho, a damn reservoir of River Douro and an ocean coastal area just south of Porto. Isurus (Fig. 3) is an underactuated vehicle whose main features are listed in Table 2. The simplest implemented navigation system involves the fusion of dead reckoning data with Long Base Line (LBL)1 information. While the first has the advantage of being available at a high sampling rate by appropriately processing the measured

Fig. 3. The Isurus vehicle.

Table 2. The Isurus vehicle. Shape Dimensions Weight Actuators Navigation sensors Payload sensors

Cylinder 1.5 m long and 20 cm diameter 30 kg Propeller and two independent orthogonal surfaces in the back Altimeter, depth cell, gyro-compass, and omnidirectional transducer. CTD, video camera, side scan sonar, acoustic doppler current profiler, and optical backscatter

rotation speed of the propeller, it has the disadvantage of a time-increasing variance. To improve the position estimate and lower its variance, an external position measurement has to be performed from time to time. That is where the LBL navigation system enters. Depth cell information is used to further enhance localization. A typical mission for one of such vehicles consists in scanning a given volume of water according to a predefined pattern while sampling at a given rate the required data, say CTD and bathimetric data and, then, returning to the point of departure. Before launching the vehicle, the mission is downloaded on the onboard computer and all systems are checked to go. The scanning pattern may consist, for example, in a set of horizontal planes, ordered by altitude, to be covered by the vehicle moving along a set of parallel straight line segments. A combination of dead reckoning, gyrocompass, depth cell sensory data and LBL

1 The LBL navigation system is, loosely speaking, the underwater equivalent to the GPS system: The vehicle interrogates each one of a number of acoustic transponders whose position is known. Each acoustic transponder replies with a specific frequency, being the distance to the vehicle obtained by measuring the time sound takes to travel. The position of the vehicle in absolute coordinates is determined by triangulation.

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information is used in order to determine the position of the vehicle in global coordinates. An auto-pilot scheme of the line of sight type yields the motion controller reference in vehicle coordinates from the global position data determined by the localization system. During the mission, key parameters concerning safety (e.g., leak threshold and power consumption rate) and navigation precision are continuously monitored. A decision to abort the mission or to pursue any of its downgraded versions (e.g., sparser sampling than planned) has to be made in case safety is at stake. Data is stored onboard and will be processed after the vehicle recovery. The reasons for multiple vehicle missions are diverse: from simple economics, i.e., relative return of the overall cost of used resources in achieving a certain goal is better than the one corresponding to single vehicle systems, to the unique capabilities that may be achieved when more than one vehicle are used. These unique capabilities might be due to the nature of sampling (e.g., spatial-temporal constraints on data sampling), multiplicity of required payload sensors which can not be transported by a single vehicle (note that, like energy, space is at a premium), diversity of functional roles that the different vehicles may have to play (scouting, navigation support, data gathering, communications relay, etc.), and strict reliability requirements which can only be achieved with signi®cant redundancy. It should be noted that communication among the different vehicles is crucial to enable the coordinated mission execution. However, the fact that only low data rate of acoustic communication among vehicles are possible, their interaction will have to involve the exchange of commands and of small volumes of data. 2.4. Some Remarks on Control Systems Modeling and Design Let us take a close look at what is involved in the control design for these systems and relate it to traditional control design. The literature is abundant in control techniques which are mainly tailored to solve low level control problems2 formulated in the framework of continuous time and differential equations. However, vehicle automation entails 2

The level of control that directly interfaces with actuators such as, auto-pilots, etc . . .

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the realm of logic, and of discrete event models interacting with differential equations that model vehicle dynamics. This is why a new control framework is required. Some of the elements of this framework are: Formal methods. The requirements of vehicle automation encompass some control logic. Examples of these include fairness and liveness. Formal methods from computer science (see [51]) are needed in order to capture this logic, to express it mathematically, and to check it for consistency and correctness. Models. The low level control context involves continuous time dynamics, typically expressed by differential equations. A richer framework encompassing logic and allowing to model systems driven by the interaction between discrete event and continuous time dynamics is needed. The example of ROV maneuvers illustrates these interactions: a certain motion of the ROV might be triggered by the reception of some message that, in turn, is generated by the fact that the ROV crosses some safety boundary. Hierarchic control. In all three systems, a high level mission speci®cation provided by the system's end user will have to be translated into a set of simpler activities whose execution requires the timely recruitment of vehicle resources. A hierarchic control structure is instrumental in order to deal with complexity of the overall system as well as the speci®city of each one of the composing subsystems: control of resources and activities is, on the one hand, performed on a subsystem basis and, on the other hand, articulated globally so that mission goals and requirements are met. Communications and control. Control actions depend on interactions with other systems. The communications requirement for these projects are diverse and a variety of communication protocols and technologies are needed. Traditionally, communication and control have been addressed separately. These issues are solved within protocol design that can be formally veri®ed for correctness (see [84] for further details). Logic based control. The actions of each vehicle follow some control logic. This control logic is complicated because it involves not only discrete event behaviors, but also complex continuous dynamics and interactions with other vehicles. The traditional practice of if-then-else programming is no longer adequate. It is not required an expert programmer to realize that the amalgamation of ifthen-else statements is very dif®cult to verify and often leads to unpredictable behavior.

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Safety and predictability. Two important requirements for automated operation are predictability and safety. The controller has to perform according to what it is expected to do while respecting safety constraints. Disturbances play against predictability and safety. The design of predictable and safe controllers involves not only the synthesis of control laws but also the determination of regions for safe operation. Furthermore, error detection and fault recovery are essential considerations which have to be embedded in the control structure and closely linked to reliability and resources redundancy management. Flexibility. There is no need to over-constrain vehicle motions by imposing constraints which might be appealing from the formal point of view but are too strict in practice. This suggests the consideration of least restrictive controllers. Informally, a least restrictive controller is one that outputs sets of control settings, from which the commanded control setting can be selected.3 Models for the coordination of multiple vehicles. These are required to express the joint operation of multiple vehicles [94]. Models of coordination are addressed in Section 3. Finally, a note on the specific design challenges posed by each one of the considered systems. First, it should be noted that the AMSIA, IES and PISCIS systems occupy quite distinct positions in the spectrum of issues arising from the conjugation of environment structure, available sensing capabilities and the corresponding motion requirements. Human intervention in the course of the mission execution differs substantially from system to system. While in the PISCIS system, it occurs only at the highest level of the hierarchy, in both IES and AMSIA systems, it spans all the control layers. Resources scarcity in the IES system is less critical than that on the other two for which computational power and energy are at a premium. Furthermore, communications constraints in the PISCIS system are a truly critical design issue in what concerns coordinated control of multiple vehicles.

3. Systems Design 3.1. Introduction One of the most salient features common to the systems presented in the previous section concerns the 3 Steering of a car is a good example of a least restrictive controller: the driver selects one of several possible steering angles at each moment.

design methodology and, in particular, the experimental validation of systems and concepts. This similarity entails early prototyping, simulation and testing, and multi-disciplinary teams. This, in turn, involves the development of a conceptual control architecture in the early phases of each project which is further refined and developed in the ensuing software implementation. The fact of the matter is that the control of every large-scale system is organized in a distributed hierarchy. It is widely reckoned that control design for autonomous systems is a very difficult area for which well established methodologies based on proper formal frameworks are not yet available. 3.2. Design Principles Here, we enumerate a set principles that we extracted from design practice over the years. These are inspired by the Systems Engineering Process4 which enables the proper consideration of the relevant intervening issues in the design of a system in spite of the often strong interdependence of multiple disciplines, and of the inherent complexity, uncertainty and variability. In this way the definition of a trade-off between tractability and expressiveness is supported. 1. Decompose motion coordination and control problems into elementary sub-problems. These yield the elemental maneuvers that can be veri®ed, or at least analyzed, and from which solutions to all control problems can be derived. The consideration of this principle enables us to partition a complex problem into a number of sub-problems that can be addressed independently. This organization also introduces 4

The Systems Engineering Process is a systematic approach in the form of a standard described in [43]. It consists of a set of tasks to transform end-user objectives, requirements, and constraints into a system solution. In order to better understand how this methodology ensures the integration of control design in the overall systems design, a brief description is included in this note: (a) Requirements analysis and validation. By considering operational scenarii, modes of operation, performance specifications, human factors, external constraints, interfaces, and measures of effectiveness, a requirements baseline composed of operational, functional and physical views is extracted and validated. (b) Functional analysis and verification. It translates the validated requirements baseline into a functional architecture. While the analysis involves the functional decomposition, performance requirements, function state and modes, data control and flows, interfaces, failure modes and monitoring functions, the verification requires the definition of verification procedures, and verification evaluation which includes the architecture completeness, functional and performance measures, and constraints satisfaction.

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2.

3.

4.

5.

structure to the problem thus providing the structuring principle for the control architecture. The organization of motion as the composition of elemental maneuvers within a control architecture has been adopted independently in the works of several authors [76,92,95,102]. The work of Varaiya [100] was quite in¯uential in terms of architecture design. For related work on the coordinated motion of cars, helicopters and submarines see [84,91,99]. Design the control architecture using, as basic building blocks, the elemental maneuvers. The elemental maneuvers should ful®ll the high level requirements and accommodate changes in those requirements, while maintaining the structural integrity. Structural invariance is probably a good measure of the quality of an architectural design. Maintain a uniform representation of the control architecture in all phases of design, simulation, evaluation and deployment. Note that conceptual control architectures tend not to be implemented but subsumed by the corresponding software architecture. Capability-driven design. A requirements-driven design is bound to fail if it does not take into account previously available capabilities. A capability-driven design speeds up the design process by eliminating tedious, and time consuming tasks which can be reused from previous designs. Results-driven design. The control architecture provides the template that can be used to guide the overall implementation process since it de®nes functional dependencies and capabilities. This knowledge can be used to organize and articulate parallel developments so that basic functional components ± hence results ± exist at the early stages of the project. Having a basic structure

(c) Synthesis. It generates a design from the verified functional architecture. A number of steps, such as, functional grouping and allocation, safety issues and failure modes assessment, quality factors and technological requirements for effectiveness, to name just a few, are carried out at this stage. (d) Physical verification. It ensures that the requirements of the lowest level of the physical architecture are traceable to the verified functional architecture, and that the physical architecture requirements satisfies the validated requirements baseline. (e) System analysis has to be performed to support the various design steps, in order to resolve requirements conflicts, examine functional and solution alternatives, identify risks and assess system and cost effectiveness. (f) Control of the design activity with the purpose of managing and documenting all the involved activities.

F.L. Pereira

in place, it is then easier to improve each of its components. 3.3. Architectural Issues At the design level, the control architecture addresses the overall structure and properties of the controlled system. Hence, it provides a focus for certain aspects of design and development that are not to be found within the constituent modules. Yet, this level of design lacks a formal expression. Surprisingly, in the field of software design, Architecture Description Languages provide formal representations of software architectures [39]. This is not the case with control architectures. Moreover, there is no formal basis to study properties at this level. 3.3.1. Design Methodology Conceptual architectural design. This is a first step whereby the identification of elemental maneuvers which, in spite of sharing basic feedback control laws, will be implemented and tested independently. Then, middleware services that will coordinate the execution and control of elemental maneuvers are assembled. Hence, models of coordination are defined that will result in constraints for communications and maneuver design. Mapping the conceptual architectural design onto the software architecture. This entails selecting models of computation that implement the models of coordination from the conceptual architecture. The implementation of the basic software architecture and communication mechanisms will provide the structure to incremental modeling and design of control systems and simulation blocks. At the level of software implementation, controllers exist first and foremost in a computational environment that mediates their interactions with the plant to be controlled. However, in general, control design only takes into account models of the physical plant. Thus, the behavior of a situated controller ± a controller in a computational environment ± may be quite different from the expected one.5 This is the reason we suggest the same computational environment for simulation and real-time implementation. 5

This situation is properly accounted for in digital control design where a specific model of computation is used. However, this is no longer the case with other models of computation, as the ones required by networked multi-vehicle systems. It is not a coincidence that control methods for networked systems are receiving significant attention from the control community [84].

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Incremental development within the architecture. By incrementally replacing empty place-holders with appropriate controllers and models, which can be further refined (multi-models with increasing complexity of the same entity), early designs can be tested under simplified world and device models. Informally, we can say that the frame to check correctness of the behavior is determined first by the architecture design, and, then, by the specific controllers that implement the architecture.

process, a reduced abstract representation of each sub-system is used in order to deal with tractability without losing essential expressiveness. This level encompasses the following two layers mentioned in the literature [100]: Supervisory layer. This layer ensures the total automation of the system operation according to the mission plan: It supervises the execution of the motion plan by commanding and monitoring the execution of elemental and complex maneuvers so that safety is ensured and ef®ciency is maximized. Its implementation involves mechanisms to check pre-, post-, and co- conditions underlying the execution of the activities composing the mission as well as the eventual change of con®guration.6

3.3.2. Control Architecture Clearly, the control architecture involves two distinct and complementary roles:

Maneuver layer. This is the second level of automation responsible for the safe execution of maneuvers. Interactions with the regulation layer are mediated by the elemental maneuvers. Each elemental maneuver sends low-level commands to the regulation layer and receives events signaling their completion or failure. Elemental maneuver control is given in terms of hybrid automata. Complex maneuver control is formed of a control law and a protocol which is used to coordinate the elemental maneuvers involved in its execution. The current design uses protocols in the form of ®nite state machines.

(a) Mission oriented. In the sense that it provides ``feed-forward control'' and speci®c requirements for ``feed-back control'' at the various levels of abstraction. (b) Adaptation. In the sense that it uses gathered data in order to evaluate the properness of semantics and adjust models of subsystems accordingly. This allows for embedded fault detection as well as for system recon®guration. The ``feed-forward control'' consists in the generation of reference inputs for control problems by solving a decision making problem at a higher abstraction layer. Examples arise, in mission replanning or in on-line path generation. Reconfiguration of resources being used in a given activity and selection of environment-adjusted feedback controllers are examples of adaptability features of the control architecture. The control architecture is organized in terms of the following categories of layers and the respective interfaces [95]: Organization layer. Also designated by strategic level, it consists of top-down problem solving providing for feed-forward control. It uses global information in order to a priori de®ne an open-loop control strategy (control reference) or decision-making for the various sub-systems at appropriate time horizons. Examples of such kind of activities are: mission organization, i.e., transformation of high level mission speci®cation into a set of viable activities, path planning, and management of the human-system interaction. Coordination layer. It consists in the generation of joint feedback policies for the various subsystems which have to interact in order to carry out a given activity. Typically, given the complexity of this

Regulation layer. This ®rst level of automation, also denoted by lowest level functional layer, deals with continuous signals, and interfaces directly with the vehicle hardware whose dynamical models are given in terms of differential equations or inclusions. Conventional control laws designed for, for example, stabilization, regulation, and path tracking, are given as vehicle state or output feedback policies. Control laws at this level correspond to low level commands such as course keeping, turning, hovering, etc . . . . 3.4. Modeling, Simulation and Implementation The experimental nature of these projects places important roles first on modeling and simulation, and then on implementation and evaluation. Usually, these involve different tools and models. Our experience suggests that, at least structurally, uniformity should be preserved. Otherwise, at some point, some 6

By configuration of a system it is meant a specific set of active links between its subsystems. Two different configurations are obtained when, for example, two different suites of sensing devices might be alternatively used to produce data to close a certain motion control loop.

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of the models and relations lose adherence to the underlying physical reality and are no longer useful, thus compromising reuse, and, as a consequence, architectural uniformity. The description of requirements is complex since the underlying problem domains involve both continuous time and discrete event driven dynamics (commonly designated by hybrid systems) evolving in several time scales. The dynamic nature of the problem stems from the existence of multiple devices whose roles and dependencies change during operations. To meet these complex system description requirements, all these projects share the same modeling framework: dynamic networks of hybrid automata. Any model of operation of multiple interacting dynamic systems should be able to capture two essential features of these systems: (1) switched mode operation, and, (2) dynamic interactions. Hybrid automata, [2], are quite convenient to model switched mode control systems. Modeling dynamic interactions poses significant difficulties. First, we want to be able to express links and exchange of information among different dynamic systems. Second, there are several modalities for interactions: information exchange can take time, and depend on the communications environment. Dynamic networks of hybrid automata (DNHA) have been used to model dynamic interactions [33]. Informally, DNHS allow for interacting automata to create and destroy links among themselves, and for the creation and destruction of automata. Formally, for each hybrid automaton, there are two types of interactions: (1) Differential equations or inclusions, guards, jump and reset relations are also functions of variables from other automata, and (2) Exchange of events among automata. The range of models for hybrid control systems arise from the different ways in which continuous time (specified by, for example, ordinary differential equations) and discrete event driven (represented by a digital automaton) dynamics can be combined. For general representative references see, for example, [12,15,41,54]. An attempt to produce a model which subsumes all others appear in [12]. These structures yield time-driven state variables which may either present discontinuities or correspond to switching velocity vector fields. Along those lines, we consider

F.L. Pereira

the following model which incorporates the formal requirements extracted from the described projects: Q, a countable set of discrete states , a collection of controlled dynamic systems, : fq : q 2 Qg, being the conventional controlled dynamic system q, de®ned by q : Xq , fq , Uq , Dq , Pq , Cq , where ± Xq state space of system, ± fq dynamics: x_ q fq t, xq , uq , ± Uq control space, ± Dq disturbance space, ± Pq performance measure, and ± CqS soft and hard constraints, S : q2Q fqg Xq the hybrid state space, V: fVq : q 2 Qg the discrete transition control set, J : fJaq , Jcq : q 2 Qg with Jaq , Jcq Xq the autonomous and the controlled jump sets, G : fGq : q 2 Qg, Gq : Jaq Vq ! S, the autonomous jump transition map, and F : fFq : q 2 Qg, Fq : Jaq ! S, the controlled jump transition map. Obviously, interactions are mediated by means of communication. Hence, a model for dynamic interaction has to include a description of the mechanisms by which automata interact. At the software implementation level, mechanisms by which modules interact are called models of computation, or semantic frameworks. Hence, models of computation provide the formal basis for dynamic interactions. The choice of the model (or mix of models) of computation for a specific implementation is quite dependent on the properties of the underlying problem domain [50]. 3.5. Coordination and Control A control architecture spans several levels of abstraction, from concepts to software implementation. At the level of software implementation the choice of the model (or mix of models) of computation for a specific implementation is quite dependent on the properties of the underlying problem domain [50]. At the conceptual level, we can draw an analogy, in terms of the mechanisms by which control modules interact. Let us call those mechanisms models of interaction or coordination. Several models of computation have been formalized. This is not the case with models of coordination. Yet, models of computation implement informal models of coordination, thus providing a different view of the same

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architecture. Since the underlying problem domain is the same, it is reasonable to expect that the two types of models are closely related.7 Hybrid systems are used to incorporate communication protocols and control theory in a natural manner. These protocols and properties are formulated within a formal framework such as finite state machines or timed automata and message exchanges are modeled as events. Hybrid systems admit a compact representation of these protocols. Furthermore, the DNHA computation model admits a compact representation for dynamic scheduling and network reconfiguration. The 1-to-1 synchronization is sufficient to model a static communications network. However, when the network is dynamic (nodes can be created or destroyed), then more sophisticated data structures are needed. DNHA provides such constructs based on first-order predicate logic. 3.6. Maneuver Design All motions are composed from elemental maneuvers. By elemental maneuver, it is understood a prototypical solution to an equivalent class of motion problems.8 The set of all equivalent classes of motion problems is complete in the sense that total coverage of the range of motions is ensured. The first design step is to find one such a set of elemental maneuvers. There are several ways to do this. We consider both systems engineering principles and control design techniques, possibly combined, to list and classify all the required elemental maneuvers. From the control perspective, each elemental maneuver defines a category of problems that share the same structure and the same solution method. This structure involves the specification of: (1) (2) (3) (4)

7

objective, dynamic models, hard constraints (those that cannot be violated), soft constraints (those that can be negotiated under special circumstances),

For example, in the underwater domain, the reduced communication bandwidth limits interactions to asynchronous high level message passing, thus restricting the choice of computational models at this level. Obviously, this precludes the consideration of models of interaction involving closed low-level control loops (hence fast ones) among distinct vehicles. 8 We adopt a meaning to this term more general than the one that appears in some literature. In [60], the word maneuver means: `à controlled change of course of a vehicle or a vessel''. In some motion control design literature, e.g. [35], this term is reserved to motions associated with transitions between trim trajectories. Trim trajectories correspond to constant control inputs.

(5) rules, and (6) information sets. The design of controllers for an elemental maneuver in the framework of hybrid systems involves the following phases: (a) expression of maneuver speci®cations as a target set and/or constraint sets in the controlled dynamic system state space, (b) formulation of a control problem or a differential game of the appropriate type, (c) speci®cation of synthesis conditions (e.g., Hamilton±Jacobi equations, whose solutions are related to the boundary of the reachable set ± the set of states that can be reached at a given time from a given set ± and of the solvability set ± the set of states from which a given target set can be reached), and (d) synthesis of the hybrid controller from the conditions in (c). It happens that the set of elemental maneuvers are not enough to describe all the actions involved in the execution of a practical maneuver ± the case of maneuvers involving the coordination of multiple devices or vehicles. This is where the composition of maneuvers enters the picture, and where the notion of complex maneuver becomes relevant. The joint execution of these maneuvers is coordinated through the exchange of specific patterns of messages that implement a cooperation protocol. Therefore, a complex maneuver is a prototypical solution to an equivalent class of motion control problems characterized by: (1) objective; (2) set of elemental maneuvers; (3) hard and soft constraints; (4) information sets; and (5) controller that coordinates the execution of the elemental maneuvers. The hybrid controller consists of: (1) a specification of the solvability set; (2) a least restrictive controller that assumes the form of a set-valued feedback control law for the continuous and discrete variables which guarantees that the hybrid system remains in a ``solvability subset''; (3) a controller that selects the final control setting from the set of control options given by the least restrictive controller. Prior to execution, the controller performs a feasibility verification for a maneuver specification, thus ensuring its correctness. This is done by checking that the set of feasible controls is not empty. The second controller selects the commanded control setting from this set. This two-level control design allows for the consideration of qualitative design techniques and may account for the integration of human decision-making in the control loop. Note that issues addressing

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complexity and partial information can naturally be incorporated in this hybrid control design framework.9

4. Control Design: A Dynamic Optimization Perspective 4.1. Introduction In this section, several control design issues arising in the context of the systems described in Section 2 are discussed. As it was pointed out in the previous section, the addressed control problems are organized into a distributed hierarchy enabling collections of interacting dynamic systems to perform a number of activities underlying the given mission. Now, we discuss the use of dynamic optimization to address the control problems formulated in the context of the above mentioned systems. Typically, a system can be regarded as composed by several subsystems among which links are established, thus defining a configuration. The set of viable configurations as well as the mappings describing the feasible transitions between any pair of configurations are specified by some of the constraints associated with the model of the physical system. Now, let us examine the different types of control design problems that may arise in this modeling framework. Control synthesis for elemental maneuvers are addressed by conventional feedback control design methods for which a wide range of techniques for linear and nonlinear, stochastic and deterministic systems are documented in a vast literature [24,44,45, 48,78,86]. The generic problem of elemental maneuver control synthesis can be described as follows: given a dynamic system, or a collection of interacting dynamic systems, synthesize a controller so that the system(s) satisfies(y) the given maneuver specifications. The type of maneuver specification dictates the type of control problem formulation. Paradigmatic classes include: Invariance, i.e., staying inside some region; attaining a given target set while the trajectories of the system remain inside some other set; optimizing some criteria; and stabilizing a system. Inherent to most of these control formulations is the problem of the 9 A thorough discussion of complexity for decentralized control problems can be found in [61]. Complexity results for centralized and decentralized supervisory control problems appear in [74]. Extensions of complexity results to hybrid systems are presented in [11]. In some cases, control synthesis can be reduced to that of parameter synthesis that can be resolved with some verification and control synthesis tools [42].

F.L. Pereira

computation or approximation of the reachable set, i.e., the set of points in the state space that the dynamic system can reach from a given initial set in a given time interval. From the mission plan and the specific execution instance, control references for each elemental maneuver will have to be generated by solving organization problems. These are open loop control or optimization problems solved `òff-line'' which should ensure that global requirements are met. This set of references should correspond to an instance of the overall mission such that the resulting feedback control strategies are feasible and `àdapted'' to the respective specific execution contexts. Optimal open loop control synthesis, and reachable set approximation are well known areas [24,48,49,97,101,103] addressing this class of problems. Mechanisms to articulate the execution of maneuvers (elemental or not) give rise to coordination problems. These consist in finding joint feedback control strategies for a set of dynamic control systems that interact in order to perform a given activity. This class of problems, also designated by hybrid control problems, underlies the mechanisms of coordination and control referred to in the previous section. In the rest of this section we will address the following deeply intertwined main categories of issues which play a fundamental role in the analysis and control of dynamic system: Reachability, invariance, control feedback, and optimization. 4.2. Reachability As we have seen, a significant class of practical design and verification problems can be formulated as a reachability problem which consists in checking whether or not a given point belongs to the reachable set of the considered dynamic system. The reachable (or attainable) Rt;t0 , x0 is the set of points of Rn that can be reached from t0 , x0 at time t. A general verification problem that arises in the designed systems can be formulated as follows. Let C : t0 , 1,!PRn be a given closed-valued setvalued map representing a certain safety property underlying a given maneuver. Then a practical verification problem amounts to show the existence of an open-valued set-valued map A with a connected graph and satisfying 8 t t0 , At Ct \ Rt;t0 , x0 : The vast body of literature in Control Theory addressing the characterization of reachable sets of dynamic control systems attests their important role.

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This characterization has been provided by: Integral forms which have been given by time- sections of set-valued attainability tubes de®ned forward in time, by funnel equations [48], or, backward in time, by solvability set evolution equations [47,48]. The semigroup property of set-valued maps generating solvability tubes or reachable sets makes them convenient to synthesize feedback control strategies for problems of terminal target control [48]. These conditions as well as the associated control synthesis can be readily extended to dynamic systems with uncertainty or under closed and open loop controls. An alternative characterization may also be given in terms of, respectively, forward and backward value functions. In fact, the reachable set of a dynamic system can be interpreted in terms of level sets of some value functions [48]. Differential forms given by conditions involving equations or inequalities of the Hamilton±Jacobi± Bellman type have been the subject of extensive research (see [7,16,20,24,36,37,105]) in both control and game theoretic contexts. A particularly, simple derivation appears in [20] where the dynamics are given by _ 2 Ft, xt, xt

xt0 2 C:

Here, it is derived, under mild assumptions, the following proximal Hamilton±Jacobi characterization of the reachable set Ht, x, 0 8 , 2 @p Rt, x, 8 t, x 2 t0 , 1 Rn , with the boundary condition limt#t0 Rt; t0 , C C, being R : ft, x : x 2 Rt; t0 , Cg, Ht, x, p, the Hamiltonian, given by Maxfp v : v 2 Ft, xg and @p Ss the set of proximal normals10 to the set S at s. A constructive approach is provided by the socalled exponential formula derived in [108], RT; 0, lim RN T; 0, , N!1

where RN T; 0, I T=NFN , I is the identity operator, the power of I T=NF is in the sense of composition of multifunctions and the limit is considered in the sense of Kuratowski. 10

is a proximal normal to S at s 2 S, if x s for some x not in S that it has s as closest point. For additional definitions and properties see [24].

From the complexity point of view, the reachability problem is undecidable for general hybrid systems [34]. The main techniques for reachability analysis can be organized in two categories [41,42]: (1) Purely symbolic methods based on the existence of analytic solutions to the differential equations and on the representation of the state space in a decidable theory of the real numbers. (2) Methods that combine numeric integration of differential equations and symbolic representations of approximations of reachable and state space sets by polyhedra or ellipsoids. The well established geometric characterization of reachable sets of linear dynamic control systems [98] has been giving rise to a number of approaches addressing constructive techniques to compute polyhedral approximations [40,64,101]. The basic idea is the one behind proximal aiming strategies [46], i.e., to generate reachable set boundary points by picking a feasible velocity that, at each point, drives the state variable at the final time the fastest along a given direction. This same idea underlies a recursive algorithm presented in [64] that combines dynamic programming and the exponential formula in order to generate inner and outer polyhedral approximations to the reachable sets of nonlinear systems. In [49], ellipsoidal set-valued evolution equations are used and an appropriate calculus adopted. From the computational point of view, this approach is advantageous relative to the one based on polyhedral approximations. However, the approximations can be considerably more conservative. 4.3. Invariance Let us consider the following problem: a vehicle is to be controlled in such a way that its state is to be maintained in a specified region. In the IES system, this might correspond to keep the ROV at a safe distance from a wall to be inspected but still sufficiently close so that clear video images of the wall can be obtained for inspection purposes. This is the problem of invariance. Invariance is a classical notion of the theory of dynamic systems, which has proved extremely useful for control design. Furthermore, the interpretation of level sets of value functions in terms of reachable sets [24,48] provides, in the hybrid systems context, a convenient geometric framework to accommodate logical propositions that naturally arise from specifications.

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For each maneuver, Lyapunov functions can be constructed11 which have, as level sets, S0 , S1 , S2 , and S3 , in the t x-space (where x 2 Rn denotes the state variable), which are compact subsets of, respectively, optimal set (in a minimax sense, is also designated by stable bridge,12 see [46]), termination set, the set from where all trajectories corresponding to the feasible maneuver terminate, initial set, the set of points satisfying conditions required for the initiation of the maneuver, and maneuver stability set, the set from where the state can be driven to successful maneuver completion by a low-level feedback control. Note that, if, by construction, the relations S0 S1 S2 S3 are satisfied, a relation of relative ``restrictivity'' of the associated synthesized controllers is provided. A subset of the state-space is said invariant with respect to a given dynamic system if the fact that contains the value of the trajectory at a given time implies that it contains all its values. The qualification positive or negative is added if a particular direction of time flow is considered. Furthermore, for a dynamic control system, an invariant set can also be qualified as strong and weak, depending, respectively, if all trajectories or at least one trajectory remain in the set. The last concept coincides with the notion of viability [4]. In [4,6,27], the tangential condition Fx \ TBS x 6 ;, with TBS x being the Bouligand tangent cone13 to S at x, is proved, under reasonable assumptions, to be necessary and sufficient for weak invariance or viability of the set S with respect to the dynamic system x_ 2 Fx. This and additional characterizations of weak and strong invariance are given in [24]. In particular, it is shown that the weakly invariance of the pair (F,S) is equivalent to any one of the conditions: (a) hx, 0, 8 x 2 S, 8 2 NpS x. (b) 8 x0 2 S, 8 " > 0, S \ Rt0 ; t0 , x0 6 ; for some 2 0, ".

11 E.g., as the value function associated with a parametrized family of auxiliary optimal control problems [21]. 12 In case there are no uncertainties or disturbances, this set is reduced to an optimal path or point. 13 For a definition, see [6].

Here, NpS x and hx, are, respectively, the proximal normal cone of S at x and the lower Hamiltonian of F. Definitions, properties as well as necessary and sufficient conditions for strong invariance are given in [24]. The fact that the proof of these conditions involve the construction of a proximal aiming feedback fp x 2 Fx, i.e., fp x minimizes v ! v x

s

for some s 2 S such that x s 2 NpS s, being NpS s the cone generated by the proximal normals to S at s, clearly reveals the role of invariance in control synthesis. The notion of invariance is addressed in the context of hybrid systems in [5] by formulating an optimal impulsive control problem. The main idea is to re-establish the viability of a given arbitrary set K by re-initializing the state trajectory by a reset map whenever the stopping set is attained. Therefore, a typical trajectory exhibits discontinuities which can be associated to discrete transitions of a hybrid automaton. In this reference, Lyapunov stability results, a viability theorem and a characterization of the value function are derived as extensions of the corresponding results in the context of viability theory. A thorough review of the role of set invariance, and, in particular, of positive invariance, in control is presented in [10]. Besides surveying its role in constrained control, robustness analysis, control synthesis, disturbance rejection and optimization, methods for the construction of ellipsoidal and polyhedral invariant sets for control synthesis are presented. A discussion on the trade-off between complexity and control performance for ellipsoidal and polyhedral sets is also included. 4.4. Feedback Control Let us consider now the problem of stabilizing an ROV platform in the context of the IES system described in Section 2. Stabilization is a critical issue in the visual inspection of underwater infra-structures. The relation between the power of the ROV actuators and that of disturbances (e.g., waves generated by a passing ship) and the stability requirements underlying the continuous visualization impose the need of a two stage stabilization system: disturbances are first attenuated by controlling the vehicle propellers and further reduced by controlling a fast pan-and-tilt unit on which the video camera is installed. This and other problems, such as regulation, tracking, and disturbance attenuation, are typically

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addressed by feedback control synthesis techniques14 which enable the enforcement of a behavior to the dynamic system so that, either a given performance function is optimized ± optimal feedback control ± or the state variable is maintained or driven into a given set ± positional feedback control. Furthermore, in either case, uncertainty can be incorporated by considering a game theoretic framework [48]. Lyapunov methods, exact-feedback linearization, linear techniques applied to a linearization of the dynamics in a neighborhood of the operating point and gain-scheduling of linear controllers to encompass several operating points, are the main approaches of the positional kind of feedback control. We will focus only on the main issues arising in the former. Let us consider the dynamic control system x_ fx, u, x0 x0 , assumed to have an equilibrium point15 by appropriately choosing ut 2 as a function of the state variable, i.e., u kx. A smooth function V : Rn ! R is a Control Lyapunov Function (CLF), if it is endowed with the following properties [24]: (a) positive de®nite: Vx > 0 for x 6 0 and V0 0, (b) proper: limkxk!1 Vx 1, and (c) in®nitesimal decrease: for some continuous function W positive on Rn f0g, minu2 frVx fx, ug Wx, 8 x 6 0. The Lyapunov design method involves two stages: (a) Find a Control Lyapunov Function and (b) Select a function u kx satisfying rVx fx, kx Wx, 8x 2 Rn . Clearly, a number of questions are of utmost importance and have been fueling very active research, (see [14,21,28,68,87±89], to name just a few references): (a) Existence of regular Lyapunov functions, (b) existence of continuous feedback functions, (c) solution concept for ordinary differential equations with discontinuous r.h.s. (due to discontinuous feedback law) and (d) robustness to errors, being measurement errors of particular importance for discontinuous feedback. Brockett condition [75] states that a dynamic system x_ fx, u is stabilizable by continuous feedback only if, for every neighbourhood N of 0, fN , contains the origin. A famous counterexample is the nonholonomic integrator which even is globally asymptotically controllable (GAC). In [22] it is shown that a GAC system possesses a, possibly discontinuous, 14

Treated in a wide range of excellent publications [24,38,44, 45,48,86]. 15 To which the state variable can be driven from a certain set in the state space by a control function.

stabilizable feedback controller being the adopted solution concept, x : limN!1 xN where xN satisfies x_ N t fxN t, kxN i ,

8t 2 ti , ti1 ,

N with xN i x ti , and N ft0 , t1 , . . . , tN g is a partition of the time interval defined so that maxi jti ti 1 j ! 0 as N ! 1 [24]. It is shown in [85] that a GAC system always has a continuous Lyapunov function satisfying

inf DVx; fx, u

u2

Wx < 0,

8x 6 0,

where DVx; f represents the Dini derivative of V at x in the direction f (for definitions [24]). In [21], it is shown that a locally Lipschitz Lyapunov function always exists that ``practically stabilizes the system'', i.e., provides a feedback driving the system to an arbitrarily small neighborhood of the equilibrium point in finite time and a converse Lyapunov theorem is shown. In [68], it is shown that a global locally Lipschitz Lyapunov function always exists for a GAC system. Robustness to measurement errors is ensured for sufficiently smooth Lyapunov functions. In the nonsmooth case, robustness can only be guaranteed if dynamic feedback is considered [87]. The construction of a Lyapunov function remains an important issue in control design which has been addressed when either there is a clear physical interpretation or the specific structure of the problem can be exploited. In this later category, backstepping [38,45] and sliding mode [112] are two of the most popular classes of methods.

4.5. Dynamic Optimization 4.5.1. Problem Formulation Let us consider the following optimal control problem: P Min subject to

gxT, _ ft, xt, ut L xt

a:e:, 1

m

ut 2 R , xt 2 X Rn , x0, x1 2 C:

2

This is by no means the most general optimal control problem which may include other ingredients such as, for example, joint state and control constraints, integral cost functionals, free initial and terminal times,

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intermediate constraints, and integral constraints. A wide range of issues for this and other dynamic optimization paradigms such as calculus of variations, differential inclusions, and differential games, has been addressed in a vast literature that includes a number of excellent treatises. Let us associate some specific semantics to the data of (P). The differential equation represents the dynamic constraints of the vehicle. For an underwater vehicle, and by letting x col, , the dynamic constraint (1) is given by the general model, see [35], _ J, M_ C D g ,

(P): one based on the necessary conditions of optimality, among which the maximum principle is well known, and the dynamic programming approach. For simplicity, let us consider, in what follows, problem (P) with smooth data, X : Rn and C : fx0 g Rn . Dynamic programming is a widely documented approach for optimal control feedback synthesis, (see [7,19,24,37,70,71,104,105,109], to name just a few references). The first step of this approach consists in solving the Hamilton±Jacobi equation in order to compute a value function. For problem (P), these conditions can loosely17 be stated as follows

3

where : colx, y, z, , , , and : colu, v, w, p, q, r, are in R6 , being the position and orientation in the external navigation frame and the linear and angular velocities in the body frame. M is the inertial matrix which includes the effects of added mass (fluid displacement inertia) of the vehicle, C is the centripetal and Coriolis matrix, D is the matrix corresponding to lift and drag forces/moments, J is a transformation matrix, g is the restoring forces and moments caused by gravity and buoyancy and the external forces and moments. Notice that, typically, the value of most of the parameters in this equation are difficult to estimate. However, for several specific classes of trajectories, a simplified model with relatively well structured uncertainties suffices. On the other hand, some ingredients such as models of actuators and sensors, are of interest for specific control problems. The state constraint (2) clearly indicates that some safety condition has to be satisfied, e.g., the distance of the vehicle to a underwater wall subject of a visual inspection must be above a certain value. Another constraint could be the orientation of the vehicle which has to be in a cone centered in a direction orthogonal to the wall. While, for an autonomous vehicle, a natural performance criterion is a weighted function of total fuel consumption16 and a measure of the quality of the retrieved data, for the ROV system, a reasonable performance criterion would include only the second part. Typically, this performance criterion is closely linked to a measure of how far the actual motion is from the desired one. 4.5.2. Main Techniques There are two main dynamic optimization approaches towards the definition of the solution to problem

Vt t, x

VT, x gx,

Given the formulation of (P), this consideration requires an extension of the dynamics just specified.

rx Vt, x 0,

4

where Ht, x, p is the Hamiltonian defined by Maxf p ft, x, u: u 2 g. Then, the optimal trajectory corresponds to a control function that yields the supremum in the definition of the Hamiltonian evaluated at x, rx Vt, x, thus defining a feedback form u kt, x. A particularly clear use of this approach for the minimum time problem is given in [109]. This simplistic description implicits a quite important number of difficulties to which an impressive amount of research work has been providing increasingly interesting answers: (a) existence, uniqueness and regularity of solution to (3) [7], (b) the regularity of the synthesized feedback control law [105], (c) solution concepts [29], and trajectories to dynamic systems discontinuous in the state variable [24], and (d) constructive procedures [72]. Although in a different context, most of these issues are addressed in the previous subsection. It is of interest to remark that a recent promising approach which takes advantage of the pivotal role played by convexity in optimization to determine value functions without having to solve an HamiltonJacobi equation for each particular values of the boundary data is presented in [70,71]. Counterparts of some of these results have been developed in the context of differential games [23, 46,53,111], thus catering for problems with uncertainty. Assumptions under which the value function exists and satisfies a certain partial differential equation (Hamilton±Jacobi±Bellman±Isaacs) have been erived thus providing a method for state feedback control synthesis. Necessary conditions of optimality have been the subject of a huge research effort in the past forty years 17

16

Ht, x,

Note that, typically, there is no smooth V satisfying (3) even for problems with smooth data and, therefore, this equation have to be understood in some appropriate sense.

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[3,19,24,66,98,105]. These conditions, of which the maximum principle of Pontryagin [66], is probably the best known result, serve to identify the class of potential candidates for optimality. Let Ht, x, u, p : p ft, x, u be the pseudoHamiltonian. Then, if x , u is an optimal control process, then u t maximizes almost everywhere in

the map v ! Ht, x t, v, pt, where p 2 AC0, t; Rn is the adjoint variable satisfy_ pt rx ft, x t, u t with pT ing pt rgx T . There is an extensive literature examining a number of issues pertinent for this result (and encompassing also counterparts to other classes of problems, e.g., free-time, nonsmooth data, with state constraints, intermediate constraints, or mixed state and control constraints) such as sensitivity, regularity, robustness, and conditions of degeneracy [3,19,24,66,73,105]. A minimum cost sensitivity interpretation of the associated multipliers has been provided by establishing their relations with the generalized gradient of the value function [105]. Optimality conditions for impulsive control problems has been addressed by a number of authors [5,8,13,62,69,81,106]. For problems with vector valued control measures and non commutative singular vector fields, new solution concepts have been defined. Existence of solution to minimum fuel type of problems by enlarging the control space to include measures was the initial motivation. This motivation was further reinforced by the engineering interpretation associated with problems where state-dependent jumps are considered to model the controlled reconfiguration of a hybrid dynamic system [63]. An important class of hybrid systems control problems can be formulated as the dynamic optimization of switching systems in both control and game frameworks [12,17,67,111]. The problem consists in finding a strategy to switch between several vector fields in order to optimize some performance measure while satisfying the problem constraints, for example, reaching some target set. Typically, the cost functional involves a strictly positive cost to pay for the decision to switch. Generalized quasi-variational inequalities providing a ``differential'' characterization of the value function have been derived as an application of the optimality principle to problems in which the state variable is driven by switching vector fields [12,17, 111]. These conditions indicate, at any given time, whether to jump, i.e., select another vector field, and pay the corresponding price, or to pursue with the

currently selected vector field. In [93], an extension for graph constrained switching is addressed in order to extend the switching games framework to problems treating games on graphs [67]. Quasi-variational inequalities have also been developed for optimal control problems with impulses [8], i.e., problems that admit discontinuous trajectories, thus providing a synthesis method for a class of hybrid systems (see discussion in [12]). 4.5.3. On the Optimization of Coordinated Dynamic Systems Consider the following mission involving the coordinated control of two vehicles which arises in the context of the PISCIS system. The mission consists in collecting salinity and temperature data uniformly in a specified surface at a given constant depth and transmit it with a specified regularity during the course of the mission to an external processing unit. Data should be sampled at a rate proportional to an averaged data variance. One important requirement is the rapid deployability of this system which precludes the a priori installation of a long base line navigation system based on acoustic transponders. A simplified mission evaluation criterion is the weighted sum of two components: a measure of the selected area coverage and the overall cost of retrieved data for the predefined quality specifications. The set up to carry out such a mission consists of two small size AUVs that should travel aligned in the vertical reference axis and use the vertical acoustic channel to communicate navigation and or payload data. At any given time, while one vehicle travels submerged but sufficiently close to the surface so that an antenna is used to transmit mission data and to obtain GPS information, the other one follows a reference path at the desired depth. Note that, given the performance measure of the overall mission, the control synthesis for each vehicle involves the specification of different cost functionals: while the surface vehicle attempts to track the best coverage path while minimizing time and fuel consumption, the vehicle at the desired depth will try to track the surface vehicle while minimizing its fuel consumption. Given the different disturbances to which they are subject to, these objective functionals may yield conflicting control strategies. In this set-up, the path tracking error can be decomposed in three components: (1) The error in the horizontal plane of the vehicle at the surface that uses GPS for navigation, (2) depth error and (3) the error associated with the vertical alignment of both

196

vehicles. In order to get all the required information to support motion control, a GPS, a short baseline mounted in each vehicle, and an acoustic modem are needed besides the usual inertial system, compass and depth sensor. Clearly, a better solution might be obtained if vehicles can switch their roles in the course of mission execution. This will depend on the conjugation of a number of factors such as the rate at which data is gathered (notice that this is not known a priori), the vertical channel data rate transmission, and onboard data storage limitations. Under this context, the optimization of the mission performance criterion might dictate the need of the system's reconfiguration which would ideally amount to a controlled ``discontinuity'' of the system's trajectory triggered by a discrete event such as onboard data storage memory full or unsatisfactory vertical channel data transmission. This example illustrates the interest of considering a hybrid systems framework enabling the coordinated control of both vehicles. The coordinated control of dynamic systems can be performed either via joint constraints or via objective functions. Let us address two paradigms proposed for this purpose. Necessary conditions of optimality have been derived for several classes of optimal control problems which are of interest for the control of hybrid systems: (a) Multiprocesses. This class of dynamic optimization problems is introduced in [26] and consists in considering a ®nite collection of dynamic systems for which control strategies have to be found so that a global cost function is minimized, and some joint state and time endpoint constraints have to be satis®ed. In fact, this problem deals with the coordinated control of multiple dynamic systems being the articulation among them provided by the endpoint constraints. In [25], a maximum principle is provided in both control and differential inclusion contexts. (b) Bilevel optimization. Problems of hierarchic optimization have been considered by several authors [77,100]. In [110], a maximum principle is considered for a two-level problem where a higher level control is determined by the leader in order to minimize a global cost functional by taking into account the fact that, based on this, the follower determines its own control in order to minimize its own cost functional. In this paradigm the coordination is effected through cost functionals. A problem suggested by control requirements extracted from the above described systems combines the paradigms discussed in (a) and (b), i.e., it involves the bilevel optimization of multiprocesses. This

F.L. Pereira

problem can be formulated as follows PH Minimize JH fei g, u, fvi g subject to u 2 U and solutions ei , vi to PiL u, fe j gj6i where the problem PiL u, fe j gj6i , i 1, . . . , N, is defined by Minimize J iL ei , u, vi subject to x_ i t f i t, xi t, ut, vi t ti0 , ti1 i

v 2V

a:e:

i

and fei g 2 , where fei g denotes the Cartesian product JH fei g, u, fvi g : f0 fei g, U : e 1 eN , m 2 U is a closed bounded set in R , R L t0 , t1 ; U, PN (with i1 2ni 2), and, for i 1, . . . , N, J iL ei , u, vi : ei colti0 , xi ti0 , ti1 , xi ti1 R2ni 2 , i i i g e , V is the set of measurable functions on some interval ti0 , ti1 taking values on a bounded closed set Vi Rmi , and ti0 , ti1 t0 , t1 . Note that coordination takes place both via the cost functional and the joint endpoint constraints. If cost functionals for PiL u, fe j gj6i , i 1, . . . , N are dropped and the goal is just to find a feasible control process for the collection of dynamic systems that minimizes the global cost, then the overall problem reduces to the one in the multiprocesses context [25]. On the other hand, if : fi g, then this framework becomes a direct extension of the bilevel dynamic optimization problem [110], to multiple ``follower'' control processes. This formulation addresses only the centralized case, i.e., all the needed information is available for synthesizing the coordination control. When that is not the case, a game framework should be considered in order to find the coordinating control u and the boundary points fei g that minimize the worst case for the global cost minimization which might arise due to the selection of the vi s which attempt to minimize the corresponding cost functionals. 4.6. Examples Now, two specific examples where dynamic optimization and control techniques play a crucial role in the design of the systems in Section 2 are described. Space limitations in this article, made it difficult to present examples providing full coverage of the various classes of problems described in previous sections. Therefore, I left out the more conventional type of low-level control design arising in the regulation layer of the control architecture. Also, instead of a more complete

197


problem of the coordination type (as I would like to have included), an automated maneuvering example is given. In spite of its delimited scope, the distinct flavour of the approach to coordinated control is provided. Finally, the organization type of dynamic optimization problems is illustrated by a trajectory optimization example. 4.6.1. Trajectory Optimization This is a classic problem in Robotics for which there is a vast body of literature documenting a wide variety of approaches. A general and versatile scheme for AUV trajectory optimization in an environment with obstacles [31], is presented. Versatility arises from the fact that specific criteria can be easily incorporated in order to generate the desired optimal trajectory, and the level of suboptimality can be selected. A three layer structure provides successive refinements of the trajectory. In the highest layer, path information, performance criteria, environment information and temporal constraints are used in order to organize the free space into cells. A first approximation to the desired trajectory is computed by searching the cells graph. In a second stage, dynamic programming in a discretized time-space is then used to further refine the first approximation as a solution to the optimization problem. Finally, a more precise optimal trajectory is computed by using either the maximum principle or the Hamilton-Jacobi conditions. An on-line version of this scheme consists in embedding either the last two layers or only the last one (depending on the disturbances magnitude) in a receding horizon scheme [59]. This version allows local replanning which is particularly useful when mobile or unanticipated obstacles are to be taken into account. Note that this stage could be enhanced by adapting the algorithms proposed in [95], where a serial and a parallel algorithm for trajectory optimization based on the discretization of the Hamilton± Jacobi equation and exploiting the structure of the problem are proposed. Results of [31] in the stochastic context for an autonomous underwater vehicle with a simplified version of the model (3) are illustrated in Fig. 4. The cost function is defined on the boundary of the free space in such a way that the optimal trajectory drives the vehicle to the target through a path such that the approximation to obstacles is penalized. A different approach is adopted in [58] to address the shortest path problem. First, a generalization for differential inclusion optimization problems with state constraints of the maximum principle for multiprocesses in [25] is

Fig. 4. Optimal AUV trajectory.

derived. These conditions are then used to find the shortest trajectory for a car-like vehicle in an environment with obstacles (only the kinematic component of the dynamics is considered [1]). A multiprocesses optimal control problem is formulated in order to optimize the set of intermediate points connecting segments of a feasible extremal trajectory, each one joining a pair of adjacent points, so that the total length is minimized. 4.6.2. Automated Maneuvering Next, we address an example for which dynamic optimization methods are combined in order to synthesize an hybrid controller that allows a car-like vehicle to perform a tight curve. By a tight curve, it is meant one for which the vehicle's minimum radius of curvature is too large to go from the entry to the exit points without changing the sign of the linear velocity (component tangential to the trajectory). Several discretization schemes and dedicated algorithms have been proposed [97] to address this class of problems. Our specific example involves an additional ingredient that increases complexity substantially: state constraints. However, a scheme taking advantage of the problem structure was defined in order to deal with it in a satisfactory way. See [1] for details. The basic idea consists in considering a collection of control problems without state constraints and interpret the activation of a constraint as an event. The way this event affects the behavior of the system is reflected in the adjoint variable which together with the state variable will have to satisfy the Hamiltonian inclusion

198

F.L. Pereira

as well as all the constraints. A dynamic programming search is performed over this restricted set of Hamiltonian trajectories satisfying a set of boundary conditions which, under certain conditions, are shown to depend on a finite number of finite dimensional parameters. These reflect the penalization of attempting to violate state constraint. The criterion to optimize is the total Rvariation of the front wheels orientation, given by 0, tf dt where tf is the final time and d=dt is the front wheels turning rate which might exhibit impulses. Let x, y, , be the state variable as indicated in Fig. 5, v, play the role of control variables, being v 2 V, V for some V > 0, the linear velocity. Note that this model admits discontinuities in the orientation of the front wheels, , even for v 0. The kinematic model is x_ v cos y_ v sin 1 v sin2 _ 2L d d

5

where L is the distance between the midpoints of both wheel axis. Let z colx, y, , and p col px , py , p , p . The initial state is given z0 z0 , and terminal curve exit constraints, ztf 2 Sf , as well as state constraints, zt 2 S, 8 t 2 0, tf , are to be satisfied. In order to characterize the Hamiltonian trajectories, let the pseudo-Hamiltonian, Hz, p, v, be given by v, , px , py , p where is px cos py sin

1 p sin2: 2L

R1

y

W2 = R1y – Resqy P1

Exit

xrobot P2 yrobot R2

P

P4

Resq Lf

Pd W

P3 Lb

Entrance x W1 = Resqx – R2x Fig. 5. Car-like vehicle.

Extremal trajectories correspond to controls v , d where v maximizes H and is supported on the set ft 2 0, tf : p t 1g where the adjoint variable p satisfies dpx x d, dpy y d, dp v px sin py cosdt d 1 v cos2p d dp L

6

for some positive scalar measure 2 C 0, tf ; R and for some colx , y , , 2 NS z, p tf 0 and ptf 2 NSf ztf . Furthermore, p satisfies p t 1 for all t 2 0, tf . Let us consider an assumption that, from the practical point of view, does not introduce a significant loss of generality: The initial vehicle configuration is such that state constraints are active either at isolated points of the time interval or continuously on certain time sub-intervals. In the first case, px , py , and are piecewise constant, more precisely, constant in the subintervals, say tk , tk1 , in whose endpoints state constraints become active. Then, it follows that grows piecewise linearly in time and, from the constancy of the Hamiltonian, it can be concluded that p satisfies p K p

h h

if > 0, otherwise,

7

where K V=2L sin2. Then, expressions for p and can be derived directly. In the second case, it is easily understood the control strategy t ! 0, t is a continuous function and a closed form for the control can be obtained as a function of certain parameters. This means that, for each segment tk , tk1 , extremal trajectories are fully defined by (5), (7), z(0), ptf , ptk , for all discontinuity points tk as well as pt k ti for all ti for which p ti 1. from t i Therefore, extremal control processes can be expressed as a function of a finite dimensional parameter, q, which lies in a certain set Q. Note that, from the above, one concludes that Ht, zq, pq 0, for all q 2 Q and all t 2 t0 , tf . Each segment of the extremal trajectory corresponds to an elemental maneuver which is articulated with all the others in order to produce the overall maneuver. Observe that there is a finite number of classes of elemental maneuvers, each one parametrized by values of q. On the other hand, a finite number of discrete events can also be identified by considering the several ways in which constraints

199

Control Design for Autonomous Vehicles eend2

xtf 2 Sf as a function of q . Then, apply the control for some relatively small time interval t, measure the state at t t and repeat the procedure. The sampling rate is limited by the computational resources available and depends on both, the level disturbances and the required motion sub-optimality. Note that, the proposed receding horizon scheme is amenable to real-time computation, since the main effort can be performed off-line and, by assuming sufficiently small disturbances, the optimal parameters at tk1 will not differ very much from qtk . An approach somewhat closer to this effort, without state constraints, is described in [90]. A partition of the configuration space where classes of optimal paths to reach the origin are defined by combining a local characterization provided by the maximum principle with global information from geometric reasoning. In this way, a shortest path synthesis is provided to steer a car-like robot to a given point.

eend1 v = vnom

7 c = climR

e6 e6

vnom 6 cv == –c max e5

nom 1 cv == c–vmax e1

e2

e5

e4

e1 e1

e7 e3

v = vnom 5 c = climR1

1

e3

v = –vnom 3 c = –cmax

e2

2 v = –vnom c=0

e3

v = vnom

4 c = cmax

ebegin

Fig. 6. Automaton for turning maneuver.

1.5

1

Conclusions 0.5

0

–0.5

–1

–1.5 –2

–1.5

–1

–0.5

0

0.5

1

Fig. 7. Optimal turning maneuver.

become active or the optimality conditions force a control switch. For the considered example, the transition map associating the occurrence of given event when a certain elemental maneuver is being executed with the resulting elemental maneuver defines the hybrid automaton in Fig. 6 which, for a certain initial vehicle configuration, generates a trajectory of the midpoint between rear wheels represented in Fig. 7. A ``realtime'' feedback control strategy can be easily obtained by applying a receding horizon scheme. That is, for a given time t, consider zt zm , where zm is the measured state of the system, find q 2 Q maximizing the pseudo-Hamiltonian, and compute u in the interval t, tf where tf is the first time such that

In this article, ingredients for autonomous vehicles control design framework based on the lessons extracted from the effort carried out by LSTS in the AMSIA, IES, and PISCIS projects are outlined. After presenting the main formal requirements underlying the coordinated control of dynamic systems, essential for the control of autonomous vehicles, which lead to the consideration of a hybrid systems framework, the main design principles are described and architectural issues are discussed. By addressing the overall structure and properties of the controlled system, the control architecture dictates the organization of the overall system's control design into specific subproblems that can be dealt with by adequate control design and dynamic optimization techniques. In this context, the relevance of important concepts and results of nonlinear control and, in particular, dynamic optimization is stressed.

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