The Behavioral Approach to Systems Theory Paolo Rapisarda, Un. of Southampton, U.K. & Jan C. Willems, K.U. Leuven, Belgium MTNS 2006 Kyoto, Japan, July 24–28, 2006
Lecture 1: General Introduction Lecturer: Jan C. Willems
Questions • What is a mathematical model , really?
Questions • What is a mathematical model , really? • How is this specialized to dynamics ?
Questions • What is a mathematical model , really? • How is this specialized to dynamics ? • How are models arrived at? • From basic laws: ‘first principles’ modeling • Combined with interconnection: tearing, zooming & linking • From measured data: SYSID (system identification)
Questions • What is a mathematical model , really? • How is this specialized to dynamics ? • How are models arrived at? • From basic laws: ‘first principles’ modeling • Combined with interconnection: tearing, zooming & linking • From measured data: SYSID (system identification) • What is the role of (differential) equations ?
Questions • What is a mathematical model , really? • How is this specialized to dynamics ? • How are models arrived at? • From basic laws: ‘first principles’ modeling • Combined with interconnection: tearing, zooming & linking • From measured data: SYSID (system identification) • What is the role of (differential) equations ? • Importance of latent variables
Static models
The seminal idea Consider a ‘phenomenon’; produces ‘outcomes’, ‘events’ . Mathematization: events belong to a set, U .
The seminal idea Consider a ‘phenomenon’; produces ‘outcomes’, ‘events’ . Mathematization: events belong to a set, U . Modeling question: Which events can really occur ? The model specifies: Only those in the subset B ⊆ U ! ⇒⇒ a mathematical model , with behavior B ⇐⇐
The seminal idea Consider a ‘phenomenon’; produces ‘outcomes’, ‘events’ . Mathematization: events belong to a set, U . Modeling question: Which events can really occur ? The model specifies: Only those in the subset B ⊆ U ! ⇒⇒ a mathematical model , with behavior B ⇐⇐ Before modeling: events in U are possible After modeling: only events in B are possible Sharper model smaller B . ❀
Examples Gas law Gas Phenomenon: A balloon filled with a gas ¡¡ Model the relation between volume, quantity, pressure, & temperature !!
Examples Gas law Gas Phenomenon: A balloon filled with a gas ¡¡ Model the relation between volume, quantity, pressure, & temperature !! Event: ( vol. V , quant. N , press. P , temp. T ) ❀ U = R 4 +
Examples Gas law Gas Phenomenon: A balloon filled with a gas ¡¡ Model the relation between volume, quantity, pressure, & temperature !! Event: ( vol. V , quant. N , press. P , temp. T ) ❀ U = R 4 + Charles Boyle and Avogadro PV NT = a universal constant =: R ❀ model PV � � ( T , P , V , N ) ∈ R 4 ⇒⇒ B = | NT = R ⇐⇐ +
Examples An economy Phenomenon: trading of a product ¡¡ Model the relation between price, sales & production !!
Examples An economy Phenomenon: trading of a product ¡¡ Model the relation between price, sales & production !! Event: ( price P , demand D ) ❀ U = R 2 + Typical model: B = graph of a curve Demand Price
Examples An economy Phenomenon: trading of a product ¡¡ Model the relation between price, sales & production !! ❀ U = R 2 Event: ( price P , supply S ) + Typical model: B = graph of a curve Supply Price
Examples An economy Phenomenon: trading of a product ¡¡ Model the relation between price, sales & production !! ❀ U = R 3 Event: ( price P , demand D , supply S ) + B = intersection of two graphs : ❀ usually point(s) Supply Demand Price
Examples An economy Phenomenon: trading of a product ¡¡ Model the relation between sales & production !! Price only to explain mechanism ❀ U = R 2 Event: ( demand D , supply S ) + B = intersection of two graphs : ❀ usually point(s) Supply Demand The price P becomes a ‘hidden’ variable. Modeling using ‘hidden’, ‘auxiliary’, ‘latent’ intermediate variables is very common. How shall we deal with such variables?
Examples Newton’s 2-nd law MASS FORCE Phenomenon: A moving mass ¡¡ Model the relation between force, mass, & acceleration !!
Examples Newton’s 2-nd law MASS FORCE Phenomenon: A moving mass ¡¡ Model the relation between force, mass, & acceleration !! Event: ( force F , mass m , acceleration a ) ❀ U = R 3 × R + × R 3
Examples Newton’s 2-nd law MASS FORCE Phenomenon: A moving mass ¡¡ Model the relation between force, mass, & acceleration !! Event: ( force F , mass m , acceleration a ) ❀ U = R 3 × R + × R 3 Model due to Newton: F = ma B = { ( F , m , a ) ∈ R 3 × R + × R 3 | F = ma } ⇒⇒ ⇐⇐
Examples Newton’s 2-nd law MASS FORCE Phenomenon: A moving mass But, the aim of Newton’s law is really: ¡¡ Model the relation between force, mass, & position !!
Examples Newton’s 2-nd law MASS FORCE Phenomenon: A moving mass But, the aim of Newton’s law is really: ¡¡ Model the relation between force, mass, & position !! Event: ( force F , mass m , position q ) a = d 2 F = ma , dt 2 q not ‘instantaneous’ relation between F , m , q ❀ dynamics How shall we deal with this?
Dynamic models
Dynamical systems Phenomenon produces ‘events’ that are functions of time Mathematization: It is convenient to distinguish domain (‘independent’ variables) T ⊆ R ‘time-axis’ co-domain (‘dependent’ variables) ‘signal space’ W
Dynamical systems Phenomenon produces ‘events’ that are functions of time Mathematization: It is convenient to distinguish domain (‘independent’ variables) T ⊆ R ‘time-axis’ co-domain (‘dependent’ variables) ‘signal space’ W A dynamical system := B ⊆ ( W ) T Σ = ( T , W , B ) the behavior
Dynamical systems Phenomenon produces ‘events’ that are functions of time Mathematization: It is convenient to distinguish domain (‘independent’ variables) T ⊆ R ‘time-axis’ co-domain (‘dependent’ variables) ‘signal space’ W A dynamical system := B ⊆ ( W ) T Σ = ( T , W , B ) the behavior T = R , R + , or interval in R : continuous-time systems T = Z , N , etc.: discrete-time systems Later: set of independent variables = R n , n > 1, PDE’s.
Dynamical systems Phenomenon produces ‘events’ that are functions of time Mathematization: It is convenient to distinguish domain (‘independent’ variables) T ⊆ R ‘time-axis’ co-domain (‘dependent’ variables) ‘signal space’ W A dynamical system := B ⊆ ( W ) T Σ = ( T , W , B ) the behavior W = R w , etc. lumped systems W = finite: finitary systems T = Z or N , W finite: DES (discrete event systems) W = function space: DPS (distributed parameter systems)
Dynamical systems Phenomenon produces ‘events’ that are functions of time Mathematization: It is convenient to distinguish domain (‘independent’ variables) T ⊆ R ‘time-axis’ co-domain (‘dependent’ variables) ‘signal space’ W A dynamical system := B ⊆ ( W ) T Σ = ( T , W , B ) the behavior W vector space, B ⊂ ( W ) T linear subspace: linear systems controllability, observability, stabilizability, dissipativity, stability, symmetry, reversibility, (equivalent) representa- tions, etc.: to be defined in terms of the behavior B The behavior is all there is
Examples Newton’s 2-nd law MASS FORCE ¡¡ Model the relation between force & position of a pointmass !!
Examples Newton’s 2-nd law MASS FORCE ¡¡ Model the relation between force & position of a pointmass !! Event: ( force F (a f’n of time) , position q (a f’n of time) ) ❀ T = R , W = R 3 × R 3
Examples Newton’s 2-nd law MASS FORCE ¡¡ Model the relation between force & position of a pointmass !! Event: ( force F (a f’n of time) , position q (a f’n of time) ) ❀ T = R , W = R 3 × R 3 Model: a = d 2 F = ma , dt 2 q Σ = ( R , R 3 × R 3 , B ) ❀ with ( F , q ) : R → R 3 × R 3 | F = m d 2 � � ⇒⇒ B = dt 2 q ⇐⇐
Examples RLC circuit Phenomenon: the port voltage and current, f’ns of time I I R R L L C C + + V V environment system C C − − R L R L Model voltage/current histories as a f’n of time !
Examples RLC circuit Σ = ( R , R 2 , B ) ❀ behavior B specified by: CR C � = L Case 1: R L � R C d 2 � 1 + R C � CR C d dt + CR C L � � 1 + CR C d � � 1 + L d � R L + V = R C I R L R L dt 2 dt R L dt CR C = L Case 2: R L � R C � R L + CR C d V = ( 1 + CR C ) d dt R C I dt behavior all solutions ( V , I ) : R → R 2 of this DE ❀
Examples input/output models u 1 y 1 u 2 y 2 SYSTEM output input u u m p � u � y ( t ) = f ( y ( t − 1 ) , · · · , y ( t − n ) , u ( t ) , u ( t − 1 ) , u ( t − n )) , w = y Differential equation analogue P ( d dt ) y = P ( d � u � dt ) u , w = , P , Q : polynomial matrices y or matrices of rational functions as in y = G ( s ) u How shall we define the behavior with the rat. f’ns?
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