Systems-within-Systems: A Unifying Paradigm

This paper introduces a new paradigm for the design and operation of complex systems. Traditional approaches, including the system-of-systems perspective, attempt to reduce the overall into its fundamental components. Such approaches inherently seek to differentiate, and often isolate, the components. The proposed paradigm seeks a shared mission for all components to exploit recursive design practices. An underlying feature of the proposed recursive designs is containment, where identified components are characterized as a system within another system. While developing the proposed paradigm, the core system technologies— mechanics, controls and planning—were also unified. This unification was accompanied with an unanticipated unification of time—the past, present and future. In fact, a second temporal axis has been introduced to facilitate the on-line concurrent implementation of planning and control responsibilities. The paradigm discusses the inherent deficiencies of planning in general, and the specific limitations of applying optimization in a real-world planning situations. It establishes an insurmountable need for another agent to implement an entity’s plan while refuting the subordinate stature that traditional hierarchical architectures would assign to an implementing agent. Rather, this interdependency establishes the need for expanded interaction among the planner and implementing agent, including the necessity of collaborative planning among the interacting systems. Introduction This paper explores a new paradigm for design and operations of systems, one that I suspect many will find disturbing. This proposed paradigm rejects many fundamental principles underlying traditional system engineering. It seeks to unify technologies. It seeks to unify time while placing severe constraints upon one’s ability to observe the present. It views planning and control as collaborative activities while severely diminishing the potential contributions that established approaches as optimization or feedback control might provide. Given the expansive nature of the proposed paradigm and its associated consequences, it cannot be adequately portrayed within a single paper. Thus, this paper itself is only an introduction. Much of what is presented has been substantiated by over a hundred pages of comprehensive mathematical derivations and extensions to existing theories and principles. Only the most basic mathematical arguments are presented here. As an introduction, the paper focuses upon listing concerns and issues. Systems-within-Systems—W.J. Davis 2 Two Introductory Examples Example 1: You are about to enter a limited access highway. Before accelerating into traffic, however, you carefully study the situation before you, including the flow of traffic and the highway’s geometry. After formulating your plan, you close your eyes and begin implementing the plan as you merge into traffic. With closed eyes, you continue implementing the plan until it is no longer possible to do so, (i.e. until you encounter a barrier, run off the highway or collide with another car.) At that point, if you are capable of doing so, you open your eyes and initiate the next planning cycle. Obviously, the above scenario is not quite realistic because planners seldom execute their plan. Let us further assume that you are a blindfolded passenger who must rely upon the driver’s description in your assessment of the situation. After planning your response, you relate it to the driver. The driver then interprets and implements your plan until it fails. Meanwhile, you wait for the next planning cycle. The above example depicts planning as a task. The planner first formulates the problem, often without immediate knowledge of the considered situation. After solving the problem, the planner typically relies upon another agent to manage the implementation. The discovery of the optimal solution terminates the current planning task. The planner then initiates the next planning task, which often revisits a prior problem, when its solution can no longer be implemented. The implications associated with seeking an optimal solution are seldom discussed. First, the planner needs to consider a perfect statement of the situation to be faced. Should an alternative situation be encountered during the solution’s implementation, the current solution will probably not be optimum, and may cease to be feasible. On the other hand, seeking an optimal solution becomes an irrelevant pursuit if one considers the inherent uncertainties. One might minimize the expected cost of addressing the situation, but any solution that is generated will still be optimal to a particular scenario, which has little probability of occurring. When uncertainties are considered, there are several criteria for seeking an optimal solution. A pessimist might seek to minimize the maximum cost while the optimist might maximize the maximum payoff. There are also additional facets of planning that are seldom addressed. In order to specify its current problem, the planner must quantify any future interactions with another entity during the adopted planning horizon. These quantifications isolate the planner from the other agents with which he routinely interacts. The same situation also arises with respect to the other planners. These planners are also trying to anticipate their interactions with the other agents. In short, every planner is trying to optimize their portion of the overall problem from their own perspective. Unfortunately, one cannot demonstrate that the ensemble of optimal solutions to the individual problems can be assembled into an optimal solution to the overall problem. In competitive situations, the notion of a global optimal solution becomes confused. In non-competitive cases, however, the potential benefits of collaborative Systems-within-Systems—W.J. Davis 3 planning are obvious. Moreover, it is probable that each planner interacts with a unique ensemble of other planners. In such situations, a network of concurrent collaborations evolves among the planners. Example 2. A person in Champaign discovers that she must drive to a particular address in Chicago by 3 p.m. the next day. Several questions emerge immediately. When should she leave? What route should she take? In this case, the obvious solution is to take I-57 from Champaign to Chicago. After this decision has been made, a trio of tasks emerges. Task 1 is to drive from the starting point in Champaign to a selected entrance onto I-57. Task 2 involves driving along I-57 to the selected exit. Task 3 is to drive from the selected exit to the specified downtown destination. Task 1 starts with her departure. A route to the desired entrance onto I-57 must have been selected. This first task then evolves into a sequence of subtasks of driving along specified streets between appropriate intersections. Now consider the driving along a given street between two intersections. She enters the street by turning into a particular lane and later exits the street from a particular lane. During the intervening drive, she can transfer between lanes within reason to expedite her movement in traffic. While driving in any lane, she must observe appropriate constraints upon her speed as follow other cars. Because these tasks are usually executed with little or no conscious thought, her thoughts might be anticipating what lies ahead. The lead car in her lane of traffic suddenly brakes. Without making a conscious choice, she instinctively brakes while her conscious effort turns to the car which is braking. Sometime later, she encounters a detour and quickly reconfigures the remaining route to her selected entrance. She might even decide to travel to another entrance. When she finally enters the interstate highway, the second task begins. While driving along I-57, she manages her flow with the traffic. She also continues to compare her prior success against her original driving plan. With this comparison, she can update the anticipated arrival time. In this example, the rationale for optimal planning is less obvious. There are numerous tradeoffs to be considered. The inherent uncertainties cannot be ignored. The continued utility and feasibility of any plan must be reassessed. Implementing any plan involves several concurrent activities, each addressing different elements of the response at different resolutions. Clearly, the latter example involves the most sophisticated planning. It also involves an intelligent planning capability which current planning methodologies cannot replicate. As humans, we plan automatically and incessantly throughout our existence. We continually reassess our situations, choose among our available options and respond. Our responses usually can be partitioned into tasks. While implementing any particular task, a sequence of subtasks often emerges. Our dynamic response crosses numerous Systems-within-Systems—W.J. Davis 4 time scales, ranging from milliseconds to hours, hours, days, and beyond. Almost everyone has formulated fiveand ten-year goals at one point or another. Now let’s address a new paradigm for coordinating several interacting systems. The new paradigm arose from my professional goal to unify three principal technologies: mechanics, planning and control. This goal emerged during my graduate studies and was later reinforced while engineering several complex real-world studies that ranged from designing Illinois’ emergency response plans to assessing the efficacy of a firstgeneration, computer-integrated-manufacturing hierarchy for an integrated steel mill to assisting the National Institute Standards and Technology in developing new on-line planning capabilities for their experimental manufacturing execution systems. Before introducing the new paradigm, however, I will first list several assertions upon which the new paradigm is based. Given their scope, it will be difficult to rigorously demonstrate their validity. No Time like the Present The Heisenberg Uncertainty Principle places insurmountable constraints upon one’s ability to simultaneously measure a physical entity’s position and momentum. It also implicitly limits one’s ability to observe the present. In fact, the new paradigm assumes that it is impossible to observe the present. As a mental exercise, let us ponder what we might see if we could instantaneously freeze the moment, i.e. stop time. In our universe, physical objects are comprised of vibrating molecules. The molecules, in turn, are comprised of vibrating atoms. The atoms are comprised of vibrating atomic particles. The atomic particles are comprised of vibrating subatomic particles. This recursion continues until one eventually encounters what some believe to be vibrating strings of energy. Given this conjectured universe, an instantaneous image of the universe would likely be empty (void of any physical entity). Obviously, this is purely conjecture because it is impossible to take an instantaneous snapshot of any physical entity. Our observations of any physical entity arise from the intrinsic blurring that arises from observing the trajectories of constituent elements over a finite time interval. Moreover, the perceived image is comprised of photons emitted across a spectrum of times. To better appreciate the consequences that arise from this, consider one of the many pictures captured by Hubble telescope as it peers into distant space. Often, such images are accompanied with a caption that asserts one is viewing the universe so many billions of years ago. In reality, the recorded image never occurred. Because the recorded objects are different distances from the earth, we are seeing each object at different time. Pondering such issues is beyond my knowledge of cosmology. However, one would do well to accept the reality that the present cannot be observed. Observations are simply approximations or virtual images of what has occurred. Moreover, any physical response we seek to manage must reside in the object’s virtual future. Our image of the present might be best described as the difference between two open sets: our current conceptualizations of the past and the future. Should you question this assertion, you might recall a time when you were totally engrossed in a particular activity, say reading a book. Whenever your activity is temporarily interrupted, you will likely not be able to estimate the current time without first consulting a clock. On the other hand, had your attention been less focused, you likely could estimate the current time. The human instinct continuously assesses its recent observations as it anticipates the immediate Systems-within-Systems—W.J. Davis 5 future. This instinct caused the example’s driver to react subconsciously to the car’s braking in front of her even before her conscious attention returned to the emerging situation. What’s Next? The notion of a present pervades the traditional system technologies. In the field of controls, one often assumes that the current state is known. When planning, the temporal horizon typically originates from the present. Both assertions are physically impossible. With regard to controls, one observes the recent past while seeking to manage the immediate future. The size of the deadband about the present depends upon several physical factors. Consider the factors that arose while managing the Mar’s explorer, Rover. The delay between recording an image on Mars and receiving that image on earth prohibits the employment of earth-based feedback control measures to control Rover’s real-time activity on Mars. The earth-based control is restricted to assigning future tasks or goals which the Rover’s on-board controllers subsequently executed using its on-board feedback control systems. It is also impossible to plan for the present because time continues to pass while the plan is generated. Planners may only consider the future. After the earth-based planners specified future assignments for Rover, they relied upon its on-board controllers to implement their assigned tasks. Rover consequently served as an agent for implementing the planner’s assignments. This situation is not unique for every planner must rely upon a local agent to execute the response that it had previously planned for the current moment. Planners must plan between two future times. Unfortunately, the planner does not have total control upon the state from which it initiates its plan or the goal state it should seek. Because the planner must rely upon a local agent to act upon its behalf, the outcome of the local agent’s execution determines the future state from which the planner initiates its planning. Moreover, the planner must anticipate that outcome even while the local agent’s executes upon its behalf. If the planner postpones its planning until its agent completes its current assignment, then the agent must wait while the planners decide which goal should be pursued next. The planner is also an agent that attempts to achieve its assigned goals. Consider Rover’s handler. The manager must decide which tasks should be pursued next. Several alternatives often exist. The quality of each alternative must be evaluated. Usually, such evaluations assess the given alternative’s contribution toward achieving a more comprehensive goal or assignment. Realistically, the necessary goals must be assigned by another entity. One should not conclude, however, that the agent’s role is subordinate to the planner. This is not a hierarchical relationship. Rather, it is a symbiotic relationship. The planner accomplishes nothing without the implementing agent. The agent lacks direction or mission without a goal. Interestingly, however, a given entity cannot be characterized exclusively as a planner or an agent. Each entity serves both roles. As an agent, it assigned goals direct its planning. As a planner, each entity relies upon an agent to execute its plans. The entities differ not with their roles, but rather in their mission and in the time horizons they address. Systems-within-Systems—W.J. Davis 6 One Size Fits All Reductionism currently provides the primary modus operandi for system analysis. The recent characterization of large systems as systems-of-systems further substantiates the fundamental contribution that reductionism provides as it attempts to identify the fundamental components comprising the overall system. Once the components are known, each can be singularly analyzed. One needs only to design a central interface with which each component interacts after the solution for each component has been engineered. Reductionism has at least three critical flaws: it emphasizes the difference among the components rather than their similarities; it isolates an individual component from direct interactions with the other components; and it does not distinguish between the core systems that physically exist and the virtual systems that manage their existence. On the other hand, by emphasizing the similarities among the system’s components, one can employ one of the most useful concepts in algorithmic design— recursion. Because most entities function both as planner and executing agent, a common design for their concept of operation and their interaction with other entities emerges. More importantly, the entities interact directly with each other rather than through a central communication device. The three principal system technologies—mechanics, controls and planning—can also be unified under the singular phenomenon of equilibration. This technological unification was accompanied with a temporal unification of the past, present and future. Although the temporal unification was unexpected, both unifications were essential. Mechanics provides the principles for modeling past performance. Planning chooses a desirable future response from the potential responses. Control manages the instantiation of the immediate future into the immediate by bridging the temporal threshold, which we designate as the present. From the combined technological and temporal unifications, a second temporal axis naturally emerged, which the paradigm characterizes as virtual time. The origin of the virtual time axis is fixed to the current real time. Future plans continuously evolve for a designated future time interval along the positive virtual time axis while system identification concurrently models the system’s observed response along the negative virtual axis. Control straddles the origin of virtual time axis where it manages the instantiation of the future into the past. A New Paradigm The existence of an integrated planning and executing agent creates the potential for a recursive system architecture. The new paradigm assumes that the considered system manages the real-time responses of an ensemble of physical processes or entities. Their physical behavior can be modeled using the principles of mechanics because these entities are physical. We will also assume that the physical entities are time variant, implying that on-line system identification will be essential. We are guaranteed that we can delineate each physical process because no two objects can occupy the same region of space and time. On the other hand, we will assume that individual entities interact with each other for different purposes and upon different time scales. Processes physically exist in the real-world, and their responses explicitly evolve with real time. That is, each exists in the present. Newton’s First Law asserts that every Systems-within-Systems—W.J. Davis 7 physical object exhibits an inertial response in real time unless an external force accelerates its motion. As an entity traverses its inertial trajectory, no external force performs work upon the system and its energy remains constant. Newton’s Second Law states that an external force can accelerate the entity. Moreover, the entity’s acceleration is directly proportional to the applied force. The ratio of force to acceleration is defined to be the entity’s inertial mass. Finally, Newton’s Third Law roughly states that any external force will be counteracted by an internal force of the same magnitude but opposite direction. Let us consider an object whose state trajectory between an initial and final time will be denoted as ( ) { } f i t t t t t t t S f i < ≤ = < ≤ s Let us further assume that an energy metric exists for this state trajectory such that energy at state will be evaluated as ( ) t s ( ) ( ) t E s . We then assert that the state trajectory is an inertial trajectory, if the energy at every state f i t t t S < ≤ ( ) t s along the trajectory initiating from is constant or ( ) i t s ( ) ( ) ( ) ( ) ( ) i S t t E t E f t t i t s s s = < ≤ ∈ In Figure 1, the lower state trajectory prior to the time corresponds to an inertial response, denoted by . The energy at any point along the curve prior to (the bold black segment on the upper figure) can be evaluated and is equal to because the energy (upper curve) is constant over this inertial trajectory. On the subsequent time interval between and , a dynamic force function is applied causing the considered entity to accelerate from its inertial trajectory. i t