The Behavioral Approach to Systems Theory

Transcription

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

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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 The slides are available at 4/31 Patrick Kürschner The Behavioral Approach to Systems Theory

10 Introduction Notation R[ξ] g q - g q matrices whose entries are real univariate polynomials, e.g., [ ] ξ R(ξ) = 2 ξ ξ 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 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

11 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

12 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

13 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

14 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

15 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

16 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

17 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

18 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

19 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 9/31 Patrick Kürschner The Behavioral Approach to Systems Theory

20 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

21 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

22 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

23 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 }. 11/31 Patrick Kürschner The Behavioral Approach to Systems Theory

24 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

25 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

26 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

27 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

28 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

29 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

30 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

31 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

32 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

33 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. 16/31 Patrick Kürschner The Behavioral Approach to Systems Theory

34 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

35 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

36 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 R d n } ω n n = 0. with R j R g w. 17/31 Patrick Kürschner The Behavioral Approach to Systems Theory

37 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 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 R n ξ n R[ξ] g w. 17/31 Patrick Kürschner The Behavioral Approach to Systems Theory

38 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

39 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

40 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

41 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 ). 20/31 Patrick Kürschner The Behavioral Approach to Systems Theory

42 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. 20/31 Patrick Kürschner The Behavioral Approach to Systems Theory

43 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

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

45 Linear Time Invariant Differential Systems Polynomial Matrices Some Examples R(ξ) = R 1 (ξ) = [ ] ξ 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

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

47 Linear Time Invariant Differential Systems Polynomial Matrices Some Examples R(ξ) = R 1 (ξ) = [ ] ξ 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 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

48 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. 21/31 Patrick Kürschner The Behavioral Approach to Systems Theory

49 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

50 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 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

51 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 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

52 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 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

53 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 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

54 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

55 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 d4 ω 4 1 ω 2 + d2 ω 2 2 = 0 which define the same system. 2 24/31 Patrick Kürschner The Behavioral Approach to Systems Theory

56 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 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 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

57 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 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 ] ξ 2 and R 2 (ξ) = 2 [ 1 + ξ 2 0 ] ξ 2 + ξ ξ 2 we can transform R 1 (ξ) into R 2 (ξ) using U(ξ) R[ξ] 2 2 unimodular [ ] [ ] [ ] ξ ξ 2 0 ξ ξ 2 = ξ 2 + ξ ξ 2. }{{}}{{}}{{} =:U(ξ) =R 1(ξ) =R 2(ξ) 24/31 Patrick Kürschner The Behavioral Approach to Systems Theory

58 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

59 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. 25/31 Patrick Kürschner The Behavioral Approach to Systems Theory

60 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

61 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

62 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

63 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

64 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

65 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

66 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

67 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

68 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

69 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

70 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

71 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