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					       Definition of Systems
• aggregation of things so combined to form
  an integral or complex whole [Encyclopedia
• interdependent group of items forming a
  unified whole [Webster]
• combination of components that act
  together to perform a function not possible
  with any of the individual parts [IEEE:
  Electronic Terms]
        Systems Engineering
• General systems can be : physical, human
  made objects as well as ``population
  dynamics’’ ``economic mechanisms’’
• SE is the art of designing and optimising
  systems, starting from the desired or
  identified need (for a system) to the
  specifications for all the system elements
       SE applied to Product
• SE provides a structured and logical
  framework to develop a customised method
  of product development
• Life cycle of product Development:
  – Need Analysis, Product Planning and Analysis,
    Evolution from System Research to System
    Design, Production, Evaluation, Customer
           Product Planning
• Extracting the system specifications from
  the objectives of the project definition
• "design/develop/manufacture" something
  (=system) that will produce a
  product/service (=outputs) which satisfies
  certain needs (=requirements)
• the system needs information, knowledge,
  and intelligence, human and/or other
  resources (=inputs) to produce outputs
• the system transfers (=functionality) inputs
  to outputs
    Product (System) Analysis
• Analysis begins with partitioning the
  functionality into components (interfacing
  different functions)
• components must be complete and
  unambiguous and implementable (as
  software, mechanical/electronic hardware)
• interaction between components should be
     Product Analysis (Cont..)
• main outcome is the system specification
  that may also contain High level decisions:
  – choice of technology for each component
  – basic (e.g. control) structure:
  – assignment of responsibilities
• examples of Mechatronic products that
  passed this stage: automated highways and
  smart cars.
           Other Steps in SE
• Evolution from Research to Design
  – Design and Research (e.g. Smart Cars)
• conceptual design considering financial and
  other restrictions
• Production or implementation
  – Rapid-prototypes (Smart Bolts)
• Evaluation and verification
• Customer Support
        Customisation of SE
• Mechatronic Systems Engineering considers
  the specific elements of integration or
  fusion of component disciplines in the
  systems Engineering context. Examples:
  – concurrency known from Manufacturing
  – inherent parallelism to be exploited in
    implementing an embedded system with an
    intelligent neural network algorithm.
        Mechatronic Systems
• traditional Electronics+Mechanical systems
  (sequential design method)
• Mechatronic systems (concurrent design
  and synergistic integration)
• General System Theory applicable to
  Mechatronic Systems
• Goals of GST :) Modeling and Analysis,
  Design and Synthesis, Control, Performance
  Evaluation, Optimization
             System Behavior

• can be defined by state equations
• The state of a system at t0 is the information
  required at t0 such that
  – output y(t), for all t greater or equal to t0, is
    uniquely determined from this information and
    from input u(t) for t greater or equal to t0.
• State Space X of a system is the set of all
  possible values state can take
       General State Equations
• State equations are the set of equations
  required to specify the state x(t) for all t
  greater or equal to t0 given x(t0) and u(t) for
  t greater or equal to t0
   – x(t) = f(x(t),u(t),t) x(t0)=x0
   – y(t) = g(x(t),u(t),t)
    Behavioral Classification of
• static (independent of past values) or
• dynamic systems are dependent of past
  values and therefore differential/difference
  equations are often used to describe the
  behavior. Dynamic systems can be:
  – time varying (behavior does change with time)
    or time invariant
 Discrete Event Systems (Intro)
• time-invariant systems can be linear or
  nonlinear. Nonlinear systems can be:
  – continuous (state variable can take any value)
    or discrete state (state variable є discrete set),
• discrete state systems can be time driven
  (state changes with the time)or event
  driven (state changes or transitions are
  forced by asynchronously generated
  discrete events)
• event driven systems can be deterministic
  or stochastic (at least one output is random)
         DES (Intro, Cont...)
• DES are discrete state and event driven
  processes (obviously they are nonlinear,
  dynamic and time-invariant processes):
• State space X is a discrete set, state
  transitions driven by the events are of the
  form: IF action AND the current state is S1
  THEN next state become S2.
  – g(a1U1+ a2U2 ) NOT = a1 g( U1 ) + a2 g( U2 )
  – Output depends on past values of input
  – behavior when a specific input is applied does
    not change with time
          Examples of DES
• Queueing Systems containing entities,
  resources, and the queue (e.g. ATM)
• Computer Systems with jobs/tasks
  competing for resources (CPU, memory, ..)
• Manufacturing Systems with production
  parts and pieces competing for machines
  and robots...
• Interactive multimedia introduction at
 Mathematical Models for DES
• untimed models: sample paths represent a
  sequence of states; S1->e1->S2->e2->S3
• timed models: sample paths also have time
  instants at which state transitions take place;
• language is the ``universal set’’ of all
  possible orderings of events that could
  happen in a system (for untimed models).
 Mathematical Models (Cont..)
• Languages, timed-languages, stochastic
  timed languages represent three levels of
  abstractions at which DES are modeled.
  Automata and Petrinets are the modeling
  formalisms discussed in the lecture.
• E= Set of all possible events => alphabet of
  the language. Event sequences => words
• Can we build a system that represent a given
  language ? Can we develop the language
  represented by an existing system ? Do two
  systems ``speak’’ identical languages ?
         Language of a DES
• language over an event set E is a set of
  finite-length strings (or words) formed from
  events in E
   – eg. E ={start, print-status-report, done} =
   – L= {ε, s, sp, spd, sppd, spppd, sd} and ε =
     empty string
  – Concatenation(s,p) = > sp, Concatenation(sp,d)
    = > spd
  – Concatenation(s, ε) = Concatenation(ε ,s) = s
   Terminology and Operations
• E* is the set of all finite strings of elements
  of E including ε
• E* = {ε,s,ss,sss….,sp,spp,….} is an infinite
• all languages L possible with the event set E
  are subsets of E*
• * is called the Kleene closure operator:
   Terminology and Operations
• Set operations (intersection,union…) can be
  applied to languages L1,L2... over the event
  set E (all L1,L2… are subsets of E*)
• Concatenation:
  LaLb :={s ε E*: (S=sa,sb) and (sa ε La) and
  (sb ε Lb)}
• Prefix-closure of L: (L is a subset of E*)
  L = {s ε E*: for all t ε E* (st ε L)}
• L is usually subset or equals to L, if L = L
  then L is prefix-closed
• A language which specifies all possible sequences
  of events is a formal way of describing the
  behavior of a DES.
• An automata represents a language according to
  defined set of rules.
   –    nodes of automata represents the states of DES
   –   labeled arcs represent transitions between the states
   –   the set of transition labels is the event set
   –   initial state and marked states need to be defined
        Deterministic Automata
• G = (set of states, set of Events associated with
  transitions in G, transition_function,
  active_event_set, initial_state,marked_states)
• Fig 2.1 :
   – f(x,ε) = x
   – f(x,se) = f(f(x,s),e) for s ε E* and e ε Ε
   – f(x,gba) = y,f(x,aagb)=z, f(z,b^) = z (n greater/equals 0)
• Language generated by G: L(G)= {s ε E*: f(x_initial,s)
• Language marked by G : Lm(G) = {s ε L(G): f(x_initial,s)
  ε Xm} (fig. 2.2)
• Automata G1 and G2 are equivalent if they generate and
  mark the same language (fig 2.3)
• Lm(G) subset/equals Lm(G) subset/equals L(G)
• blocking if G reach x : Active event set at x = 0 and x
  NOT ε Xm
• Automation G is blocking if deadlock or livelock happens,
  when Lm(G) subset (ONLY) of L(G). Nonblocking is the
  case if Lm(G) = of L(G).
  Petri Nets (instead of automata)
• PN graph is weighted bipartite graph (P,T,A,w)
   – P is the finite set of places (one type of nodes)
   – T is the finite set of transitions (another type of nodes)
   – A is a set of arcs from P --> T and T -->P
   – w: A ---> {1,2,3,...} is the weight function on the arcs
• Marked Petri Net is a five-tuple (P,T,A,w,x): where x is a
  marking of the set of places P; x=[x(p1), x(p2),...]
• row vector x is also considered as the state of the petri net
• a transition is enabled if x(pi) greater-equals w(pi,tj)