The Behavioral Approach to Systems Theory

The Behavioral Approach to Systems Theory... and DAEs Patrick Kürschner July 21, 2010 1/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Outline 1 Introduction 2 Mathematical Models 3 Linear Time Invariant Differential Systems 4 Properties of LTIDSs 5 Outlook 2/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Introduction LTI Control Systems Linear Time Invariant System E ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t) Coefficient matrices: system matrices A, E R n n, input matrix B R n m, output matrix C R p n, direct transmission term D R p m. Involved functions: state / descriptor vector x(t) R n with initial condition x(t 0 ) = x 0, input / control u(t) R m, output y(t) R p. 3/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Introduction LTI Control Systems Linear Time Invariant System E ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t) Coefficient matrices: system matrices A, E R n n, input matrix B R n m, output matrix C R p n, direct transmission term D R p m. Involved functions: state / descriptor vector x(t) R n with initial condition x(t 0 ) = x 0, input / control u(t) R m, output y(t) R p. Is this an adequate start from a modeling point of view? 3/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Introduction LTI Control Systems Linear Time Invariant System E ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t) Coefficient matrices: system matrices A, E R n n, input matrix B R n m, output matrix C R p n, direct transmission term D R p m. Involved functions: state / descriptor vector x(t) R n with initial condition x(t 0 ) = x 0, input / control u(t) R m, output y(t) R p. Or: Can we find another representation which does not require an immediate distinction between input / state / output? 3/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Introduction LTI Control Systems Linear Time Invariant System E ẋ(t) = A x(t) + B u(t) Define y(t) = C x(t) + D u(t) ω := x u R n+m+p y and rewrite. 3/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Introduction LTI Control Systems Linear Time Invariant System E ẋ(t) = A x(t) + B u(t) Define y(t) = C x(t) + D u(t) ω := x u R n+m+p y and rewrite. [ A d E B 0 ] ω = 0 C D I p }{{} =:R( d ) with R(ξ) R[ξ] n+p n+m+p. 3/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Introduction LTI Control Systems Linear Time Invariant System E ẋ(t) = A x(t) + B u(t) Define y(t) = C x(t) + D u(t) Kernel Representation [ A d E B 0 ] ω = 0 C D I p }{{} =:R( d ) with R(ξ) R[ξ] n+p n+m+p. ω := and rewrite. x u R n+m+p y In a behavioral setting, every control system is a DAE!! Even if E = I n. 3/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Introduction Miscellaneous People Jan C. Willems, (Katholieke Universiteit Leuven, B) Jan Willem Polderman, (University of Twente, NL) Paolo Rapisarda, (University Southampton, UK)... THE Book Useful link A very nice introductory lecture was held by Jan C. Willems and Paolo Rapisarda at the Elgersburg School 2010. The slides are available at http://www.tu-ilmenau.de/fakmn/home.9078.0.html 4/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Introduction Notation R[ξ] g q - g q matrices whose entries are real univariate polynomials, e.g., [ ] ξ R(ξ) = 2 ξ ξ + 1 3 ξ 4 + ξ 2 ξ 1 ξ 3. For reasons of convenience, the complex analog C[ξ] g q of the above is often used instead. deg(p) N - the degree of scalar polynomials p R[ξ] (or C[ξ]). R[ξ 1,..., ξ n ] g q - g q matrices whose entries are real polynomials in n variables, e.g., [ ] ξ1 ξ R(ξ) = 2 ξ3 2 ξ 1 + 1 3 ξ 1 ξ 2 ξ 3 ξ 4 + ξ4 2 ξ 3 1 ξ 2 ξ3 2. W T - maps from T to W. Often: T R and W R w, w N. 5/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models 1 Introduction 2 Mathematical Models 3 Linear Time Invariant Differential Systems 4 Properties of LTIDSs 5 Outlook 6/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models A Behavorial Approach Goal: We want to find a mathematical model that describes a real-world phenomenon which produces events. 7/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models A Behavorial Approach Goal: We want to find a mathematical model that describes a real-world phenomenon which produces events. Phenomenon Event 7/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models A Behavorial Approach Goal: We want to find a mathematical model that describes a real-world phenomenon which produces events. Phenomenon A deterministic model should prescribe which events of the phenomenon can and which events cannot occur. Event 7/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models A Behavorial Approach Goal: We want to find a mathematical model that describes a real-world phenomenon which produces events. Phenomenon A deterministic model should prescribe which events of the phenomenon can and which events cannot occur. Event We think of a mathematical model as a subset of all possible unmodeled events. 7/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models A Behavorial Approach Universum The set of events that are - in principle - possible is called the universum and is denoted by U. 8/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models A Behavorial Approach Universum The set of events that are - in principle - possible is called the universum and is denoted by U. Behavior The subset B U containing all allowed events is referred to as the behavior. 8/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models A Behavorial Approach Universum The set of events that are - in principle - possible is called the universum and is denoted by U. Behavior The subset B U containing all allowed events is referred to as the behavior. Mathematical Model A mathematical model is a pair (U, B) with B U. 8/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Examples Discrete event phenomena If U is a finite set, we speak about discrete event systems (DESs). Example: U = {all finite strings consisting of characters in {a,..., z, A,..., Z}}, B = all legal words (e.g., words recognized by a spell checker). homogeneity B, but hohmochenejedi / B Continuous phenomena If U is a subset of a finite vector space, e.g., C w, R w with w N, we speak about continuous models. Example: The Gas Law describes the relation between the events temperature T, volume V, pressure P and quantity N. U = [0, ) 4, B = {(T, V, P, N) U PV = NT }. 9/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Examples Dynamical phenomena If U is a set of functions of time, we speak about dynamical systems. More precisely, a dynamical system is a triple (T, W, B) with T R the time set, W the signal space, and B W T the behavior. In other words: B contains all trajectories ω : T W which are allowed by the model. Usually T (R, R + ) (continuous time systems) or T (Z, N) (discrete time systems). 10/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Examples Dynamical phenomena If U is a set of functions of time, we speak about dynamical systems. More precisely, a dynamical system is a triple (T, W, B) with T R the time set, W the signal space, and B W T the behavior. In other words: B contains all trajectories ω : T W which are allowed by the model. Usually T (R, R + ) (continuous time systems) or T (Z, N) (discrete time systems). Main topic of this talk: Behavioral approach for dynamical systems described by ODEs with T R, W R w, that is, w B are univariable functions taking their values in a finite dimensional real vector space. 10/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Examples for Dynamical Systems Capacitors and inductors Denote by I, V the events current and voltage as scalar functions of time and let C, L be capacitance and inductance of a capacitor and a inductor, respectively. Universum: T = R, W = R R, U = (R R) R. Behavior for Capacitor: B = { ω = (I, V ) U C d V = I }. Behavior for Inductor: B = { ω = (I, V ) U L d I = V }. Newton s second law Consider a point mass m with position q (R 3 ) R and let F (R 3 ) R be a force acting on m. Universum: 11/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Examples for Dynamical Systems Capacitors and inductors Denote by I, V the events current and voltage as scalar functions of time and let C, L be capacitance and inductance of a capacitor and a inductor, respectively. Universum: T = R, W = R R, U = (R R) R. Behavior for Capacitor: B = { ω = (I, V ) U C d V = I }. Behavior for Inductor: B = { ω = (I, V ) U L d I = V }. Newton s second law Consider a point mass m with position q (R 3 ) R and let F (R 3 ) R be a force acting on m. Universum: T = R, W = R 3 R 3, U = (R 3 R 3 ) R, 11/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Examples for Dynamical Systems Capacitors and inductors Denote by I, V the events current and voltage as scalar functions of time and let C, L be capacitance and inductance of a capacitor and a inductor, respectively. Universum: T = R, W = R R, U = (R R) R. Behavior for Capacitor: B = { ω = (I, V ) U C d V = I }. Behavior for Inductor: B = { ω = (I, V ) U L d I = V }. Newton s second law Consider a point mass m with position q (R 3 ) R and let F (R 3 ) R be a force acting on m. Universum: T = R, W = R 3 R 3, U = (R 3 R 3 ) R, B = { ω = (q, F ) U m d2 2 q = F }. 11/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Examples Distributed phenomena If U is a set of functions of space and time, we speak about distributed parameter systems. Example: 3-dimensional heat-diffusion. Denote temperature and heat flow as multivariate functions T, Q of space x R 3 and time t R. Universum: U = (R + R) R4, T = R 4, W = R + R, Behavior: B = { ω = (T, Q) U t T = T + Q}. 12/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Examples Distributed phenomena If U is a set of functions of space and time, we speak about distributed parameter systems. Example: 3-dimensional heat-diffusion. Denote temperature and heat flow as multivariate functions T, Q of space x R 3 and time t R. Universum: U = (R + R) R4, T = R 4, W = R + R, Behavior: B = { ω = (T, Q) U t T = T + Q}. Exercise for you Write down the time set T, signal space W, universum U and the behavior B of the distributed parameter system governed by the Maxwell s equations. 12/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Representations of Behaviors Kernel representation Specify ω B as solution of equations: f : U, B = {ω U f (ω) = 0} Image representation Specify ω B as image of a map: f : U, B = {ω U ψ such that f (ψ) = ω} Latent variable representation Specify ω B via a projection: B Ext. U L B = {ω U ψ L such that (ω, ψ) B Ext. } 13/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Mathematical Models Representations of Behaviors Kernel representation Specify ω B as solution of equations: f : U, B = {ω U f (ω) = 0} Examples Gas law: U = R 4 +, B = {ω = (P, V, N, T ) U f (ω) = 0}, f (ω) = PV NT. Capacitor: U = (R 2 ) R, B = {ω U f (ω) = 0}, f (ω) = C d V I. Newton s second law: U = (R 3 R 3 ) R, B = {ω = (q, F ) U f (ω) = 0}, f (ω) = m d2 2 q F. 13/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems 1 Introduction 2 Mathematical Models 3 Linear Time Invariant Differential Systems 4 Properties of LTIDSs 5 Outlook 14/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Dynamical Systems Governed by ODEs In the sequel we consider dynamical systems (R, W R w, B), with B of the form { B = ω W R sufficiently smooth ( dω f ω,, d 2 ) } ω 2,..., dn ω n = 0, t R, f : W R w R w R. Meaning of sufficiently smooth in this talk: ω C (R, W). Other solution concepts (strong, weak, distributional) also possible and may be even more reasonable from an application oriented point of view (see THE book). 15/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Important Properties Linearity A dynamical system (T, W, B) is linear iff W is a vector space over a field F {C, R} and ω 1, ω 2 B, γ F ω 1 + γω 2 B, or, in other words, iff the superposition principle holds. Time-invariance A dynamical system (T, W, B) is time-invariant iff T {R, R +, Z, N} and ω B, t T σ t ω B, with the backward-t-shift σ t ω(t ) := ω(t + t). 16/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Important Properties Restriction in the sequel: Linear Time-Invariant Differential Systems (LTIDSs) 17/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Important Properties Restriction in the sequel: Linear Time-Invariant Differential Systems (LTIDSs) LTIDSs A dynamical system (R, R w, B) is a LTIDS iff { B = ω (R w ) R dω R 0 ω + R 1 + R d 2 ω 2 2 +... + R d n } ω n n = 0. with R j R g w. This admits an extremely elegant kernel representation via polynomial matrices: { ( ) } ( ) d d B = ω (R w ) R R ω = 0 = kernel R, R(ξ) = R 0 + R 1 ξ + R 2 ξ 2 +... + R n ξ n R[ξ] g w. 17/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Examples of LTIDSs Examples Capacitor: U = (R 2 ) R, B = { ω = R(ξ) = [Cξ, 1] R[ξ] 1 2. Newton s second law: U = (R 3 R 3 ) R, B = R(ξ) = [mi 3 ξ 2, I 3 ] R[ξ] 3 6. [ ] V U I { [ ] q ω = U F ( d R ( d R ) } ω = 0, ) } ω = 0, Scalar homogeneous differential equation: { ( ) } U = R R, B = ω U d p ω = 0, p(ξ) R[ξ]. 18/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Examples of LTIDSs Distributed Parameter Systems By introducing a parameter set T R q and polynomial matrices R[ξ 1,..., ξ q ] R[ξ 1,..., ξ q ] g w, systems governed by linear constant coefficient PDEs can be represented by similar means. 19/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Examples of LTIDSs Distributed Parameter Systems By introducing a parameter set T R q and polynomial matrices R[ξ 1,..., ξ q ] R[ξ 1,..., ξ q ] g w, systems governed by linear constant coefficient PDEs can be represented by similar means. Example for you Give a kernel representation for the system governed by the Maxwell equations. 19/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Polynomial Matrices Rings and units Recall that R[ξ] and R[ξ] g w are rings, but only the first one is a commutative ring. An element r of a (commutative) ring R is called unit if it has a multiplicative inverse r R such that r r = rr = 1 (r =: r 1 ). Scalar polynomials Among the scalar polynomial p R[ξ] only the non-zero ones of degree zero are units, i.e. they have a multiplicative inverse p 1. Polynomials matrices A square polynomial matrix U(ξ) R[ξ] w w has a multiplicative inverse U 1 (ξ) := V (ξ) R[ξ] w w such that V (ξ)u(ξ) = I w iff det U(ξ) is a non-zero polynomial of degree zero. Polynomial matrices with this property are called unimodular. 20/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Polynomial Matrices Some Examples R(ξ) = [ ] ξ 2 1 1 ξ 2 R[ξ] 2 2 ξ R 1 (ξ) =, 20/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Polynomial Matrices Some Examples R(ξ) = R 1 (ξ) = [ ] ξ 2 1 1 ξ 2 ξ 1 ξ 3 + ξ 2 1 R[ξ] 2 2 [ ] ξ 1 ξ 2 1 ξ 2 / R[ξ] 2 2, 20/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Polynomial Matrices Some Examples R(ξ) = R 1 (ξ) = U(ξ) = [ ] ξ 2 1 1 ξ 2 ξ 1 ξ 3 + ξ 2 1 R[ξ] 2 2 [ ] ξ 1 ξ 2 1 ξ 2 / R[ξ] 2 2, [ ] ξ 2 + 2ξ 1 2 + ξ 2 + ξ 3 1 ξ 2 R[ξ] 2 2 U 1 (ξ) =. 20/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Polynomial Matrices Some Examples R(ξ) = R 1 (ξ) = [ ] ξ 2 1 1 ξ 2 ξ 1 ξ 3 + ξ 2 1 R[ξ] 2 2 [ ] ξ 1 ξ 2 1 ξ 2 / R[ξ] 2 2, [ ] ξ U(ξ) = 2 + 2ξ 1 2 + ξ 2 + ξ 3 1 ξ 2 R[ξ] 2 2 U 1 (ξ) = 1 [ ] 1 ξ 2 2 ξ 3 2 ξ 3 1 ξ 2 R[ξ] 2 2. Hence, U(ξ) is unimodular, but R(ξ) is not! 20/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Polynomial Matrices The Smith Form For every M(ξ) R[ξ] g w there exist unimodular matrices U(ξ) R[ξ] g g and V (ξ) R[ξ] w w, such that [ ] diag (d1,..., d UMV = r ) O r w r O r g r O g r w r with d 1,..., d r R[ξ] and r = rank (M). Furthermore, d j+1 (ξ) = q j (ξ)d j (ξ), q j R[ξ], j = 1,..., r 1. The Smith form provides a useful tool for proofs involving polynomial matrices, expecially with respect to kernel representations of LTIDSs. It is also used for linearizations of polynomial eigenvalue problems. 21/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Differential Operators Scalar ODEs Consider the scalar, linear, homogenous ODE dω p 0 ω + p 1 + p d 2 ω 2 2 +... + p d n ω n n = 0 with p 0,..., p n C (for convenience), or in polynomial representation Which functions ω : R C solve the ODE? p( d )ω = 0, p C[ξ]. (1) 22/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Differential Operators Scalar ODEs p( d )ω = 0, p C[ξ]. (1) Let λ 1,..., λ r be the r n distinct roots of p(ξ) with multiplicities m 1,..., m r (m 1 +... + m r = deg(p)). A function y : R C is a solution of (1) iff it is of the form 22/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Differential Operators Scalar ODEs p( d )ω = 0, p C[ξ]. (1) Let λ 1,..., λ r be the r n distinct roots of p(ξ) with multiplicities m 1,..., m r (m 1 +... + m r = deg(p)). A function y : R C is a solution of (1) iff it is of the form y(t) = q 1 (t)e λ1t +... + q r (t)e λr t, where q j (t) C[ξ], deg(q j ) < m j, j = 1,..., r. 22/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Differential Operators Multivariable ODEs Consider the system of ODEs represented by ( ) d P ω = 0, P(ξ) = P 0 + P 1 ξ +... + P n ξ n C[ξ] w w. (2) Let λ 1,..., λ r be the r n distinct roots of det (P(ξ)) with multiplicities m 1,..., m r (m 1 +... + m r = deg(det P)). The solutions y : R C w of (2) are given in the form y(t) = Q 1 (t)e λ1t +... + Q r (t)e λr t, where the polynomial vectors Q j C[ξ] w with deg(q j ) < m j vary over an m j -dimensional subspace V j C[ξ] w for j = 1,..., r. 23/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Structure of the Kernel Representations of LTIDSs Equivalent representations Consider the behavior B = kernel { ( )} d R, R R[ξ] g w. The polynomial matrix R determines the behavior B. Does the converse hold, too? No! Stated differently, when do R 1 ( d define the same system? ) ω = 0 and R 2 ( d ) ω = 0 24/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Structure of the Kernel Representations of LTIDSs Equivalent representations Consider the systems of ODEs ω 1 + d2 d ω 2 1 = 0 ω 2 + d2 ω 2 2 = 0 and ω 1 + ω 2 1 = 0 d 2 ω 2 1 + d4 ω 4 1 ω 2 + d2 ω 2 2 = 0 which define the same system. ω 1 + d2 ω 2 1 = 0 implies d2 ω 2 1 + d4 ω 4 1 = 0 which can then be added to the second row of the first system. 2 24/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Structure of the Kernel Representations of LTIDSs Equivalent representations Consider the systems of ODEs ω 1 + d2 d ω 2 1 = 0 ω 2 + d2 ω 2 2 = 0 and ω 1 + ω 2 1 = 0 d 2 ω 2 1 + d4 ω 4 1 ω 2 + d2 ω 2 2 = 0 which define the same system. In polynomial form R 1 ( d ) ω = 0, R2 ( d ) ω = 0 with R 1 (ξ) = [ 1 + ξ 2 0 ] 0 1 + ξ 2 and R 2 (ξ) = 2 [ 1 + ξ 2 0 ] ξ 2 + ξ 4 1 + ξ 2 we can transform R 1 (ξ) into R 2 (ξ) using U(ξ) R[ξ] 2 2 unimodular [ ] [ ] [ ] 1 0 1 + ξ 2 0 1 + ξ 2 0 ξ 2 1 0 1 + ξ 2 = ξ 2 + ξ 4 1 + ξ 2. }{{}}{{}}{{} =:U(ξ) =R 1(ξ) =R 2(ξ) 24/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Structure of the Kernel Representations of LTIDSs Equivalent representations Consider the behavior B = kernel { ( )} d R, R R[ξ] g w. The polynomial matrix R determines the behavior B. Does the converse hold, too? No! Stated differently, when do R 1 ( d ) ω = 0 and R 2 ( d ) ω = 0 define the same system? Answer: If there exists an unimodular U(ξ) R[ξ] g g such that UR 1 = R 2. 24/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Structure of the Kernel Representations of LTIDSs Minimum kernel representation Let R ( d ) ω = 0 be a kernel representation of the behavior B. If it has, among all other kernel representations, the smallest number of rows, it is called minimal kernel representation. Minimal kernel theorem Let B be the behavior of a LTIDSs. Then the following statements are equivalent: 1 R ( d ) ω = 0 is a minimal kernel representation of B. 2 R has linearly independent rows. 3 R has full row rank. Note that all minimal kernel representations of B are generated from one single polynomial matrix R(ξ) via transformations R UR with U unimodular. 25/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Input/Output Systems and Free Variables Free variables Consider a behavior B with signal space R w which we partition as R w = R m R p with w = m + p. The trajectories ω C (R, R w ) are also partitioned as ω = (ω 1, ω 2 ) with ω 1 C (R, R m ), ω 2 C (R, R p ). This partition is an input/output partition if 1 ω 1 is free: ω 1 C (R, R m ), ω 2 C (R, R p ) s.t. (ω 1, ω 2 ) B. 2 ω 1 is also maximally free, that is, for a given ω 1 no component of ω 2 is free. Naturally, ω 1 is called input variable and ω 2 output variable. 26/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Input/Output Systems and Free Variables Systems in i/o form For a given kernel representation with R(ξ) R[ξ] p w this leads to a similar partition of R: R(ξ) = [ Q(ξ), [ ] P(ξ)], [ P(ξ) R[ξ] p p, Q(ξ) R[ξ] p m. Consequently, ω1 u with ω = =: we find ω 2 y] ( ) ( ) ( ) d d d R ω = 0 P y = Q u. This is referred to as system in input/output form if 1 det P(ξ) 0, 2 the entries of the transfer matrix G(ξ) := P 1 (ξ)q(ξ) are proper rational functions: deg(numerator(g ij )) deg(denominator(g ij )), i, j. 26/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Input/Output Systems and Free Variables I/o partition theorem 1 Let R ( d ) ω = 0, ω (R w ) R be a minimal kernel representation of B. Then there exist a partition of the index set {1,..., w} into such that the partition of ω as {k 1,..., k m(b) } and {ˆk 1,..., ˆk p(b) } u = ( ω k1,..., ω km(b) ) and y = {ωˆk1,..., ωˆkp(b) } is an input/output partition of B. Note that, although this partition is not unique, the numbers m(b) and p(b) are! 26/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Linear Time Invariant Differential Systems Input/Output Systems and Free Variables I/o partition theorem 2 Let R ( d ) ω = 0, ω (R w ) R be a minimal kernel representation of B. Then there exist a partition of the index set {1,..., w} into such that the partition of ω as {k 1,..., k m(b) } and {ˆk 1,..., ˆk p(b) } u = ( ω k1,..., ω km(b) ) and y = {ωˆk1,..., ωˆkp(b) } is an input/output partition of B with proper transfer matrix. 26/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Properties of LTIDSs 1 Introduction 2 Mathematical Models 3 Linear Time Invariant Differential Systems 4 Properties of LTIDSs 5 Outlook 27/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Properties of LTIDSs Stability Recall that a dynamical system is stable iff all its trajectories go to zero. Stability of LTIDSs behaviors Let B be a behavior defining a LTIDSs. Then the following statements are equivalent: 1 B is stable. 2 Every complexified trajectory t ae λt B, a C w has Re (λ) < 0. 3 B admits a kernel representation R ( d ) ω = 0 with rank (R) = w and it holds rank (R(λ)) < w, λ C Re (λ) < 0. 4 B has a minimal kernel representation R ( d ) ω = 0 with R(ξ) Hurwitz. 28/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Properties of LTIDSs Controllability A time - invariant dynamical system is controllable if we can always concatenate two trajectories: ω 1, ω 2 B there exists t T, T 0, ω B s.t. { ω 1 (t) : t 0 ω(t) = ω 2 (t T ) : t T. Controllability of LTIDSs behaviors Let B be a behavior defining a LTIDSs. Then the following statements are equivalent: 1 B is controllable. 2 B admits a kernel representation R ( d ) ω = 0 with rank (R) = w λ C. 3 It exists a direct summand B of B such that B B = C (R, R w ). 29/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Properties of LTIDSs Controllability A time - invariant dynamical system is controllable if we can always concatenate two trajectories: ω 1, ω 2 B there exists t T, T 0, ω B s.t. { ω 1 (t) : t 0 ω(t) = ω 2 (t T ) : t T. Controllability of SISO LTIDSs behaviors A single-input-single-output system given in input/output form ( ) ( ) [ ] d d u p y = q u, ω = y is controllable iff the polynomials p, q R[ξ] are coprime, i.e. they have no common factors. 29/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Properties of LTIDSs Stabilizability A dynamical system is stabilizable if its trajectories can be steered towards zero. Stabilizability of LTIDSs behaviors Let B be a behavior defining a LTIDSs. Then the following statements are equivalent: 1 B is stabilizable. 2 B has a kernel representation R ( d ) ω = 0 with constant rank (R) λ C +. 30/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Outlook Stabilizability Further important topics Observability, detectability. Latent variables and states Input/state/output systems = State stability, controllability,.... System interconnection. Feedback. Observers. 31/31 Patrick Kürschner The Behavioral Approach to Systems Theory

Outlook Stabilizability Further important topics Observability, detectability. Latent variables and states Input/state/output systems = State stability, controllability,.... System interconnection. Feedback. Observers. Thank you for your attention! 31/31 Patrick Kürschner The Behavioral Approach to Systems Theory