Systems-within-systems: a unifying perspective Wayne J. Davis Professor Emeritus Industrial and Enterprise Systems Engineering University of

Past Future (a) Past Future (b) Past Future (c) (a)Physical process evolves in real time. Blue dot indicates current state, tail its observed trajectory (b)Process controller resides in the future. It cannot access the present: it observes immediate past as it attempts to influence the imminent future Its planning seeks the trajectory from last observed state toward a future goal state Often goal state corresponds to end of an assigned task

Past Future (a) Past Future (b) Past Future (c) (c) Multiple physical processes evolve concurrently in real time Each process exists independently in real world Each process has a dedicated controller that projects its future state at a common future time These projections evolve in real time as additional observations are made

(d) Process controllers’ projected states at a common future time are aggregated into an initial planning state for a composite planner Planner seeks trajectory from aggregated future state toward an assigned future goal state Both boundary states are time-variant The planned trajectory must be dynamic also Planning is a process, not a task Past Future

(d) Continued Planners’ response initiates from and thus contains the processes’ response Planner relies upon processes to implement its prior plans==>planner cannot execute its plan Processes are not subordinate to planner Planner does not exist with its processes Past Future

(e) Continued Additional processes are included and aggregated to specify initial planning states for two planners In this example, two planners share middle process This is not a hierarchy Past Future (e)

(f) The two planners behave as aggregate process encapsulating the responses of their contained processes. Another planner initializes its planning from their projected states at a common time Aggregated processes execute this planner’s prior plans Past Future (f)

(f) Continued Recursive structure is revealed All planners behave the same Past Future (f)

Critical observation One: All systems have a primal and dual configurations Primal system configuration: fix state definition and describe state evolution as function of time Dual system configuration: Fix time and describe transitions among coupled state definitions Past Future (f)

Critical observation Two: Super-symmetry Both primal and dual formulations have their dedicated primal and dual formulations Traditional planning only considers the primal formulation of the primal configuration between two time-variant future states Past Future (f)

Critical observation Three: Planning is a process, not a task Equilibration replaces optimization Equilibration underlies energy analyses within classical mechanics Past Future (f)

Major conclusion: Trajectory planning includes three components Identifying trajectory between two future states Collaborate with its executors to plan the initial state from which it will initialize its planning Collaborate with planners with which its goal state is coupled For time-variant systems there are other modes of concurrent “optimizations” to be addressed collaboratively in real time.

Additional accomplishment: Unify classical mechanics, controls and optimization Temporally unify past and future with the present instantiates imminent future into immediate past Mathematical formulations for linear, nonlinear and discrete-event systems exist