Monotone Control System. Brad C. Yu SEACS, National ICT Australia And RSISE, The Australian National University June, 2005

Transcription

1 Brad C. Yu SEACS, National ICT Australia And RSISE, The Australian National University June, 005

2 Foreword The aim of this presentation is to give a (primitive) overview of monotone systems and monotone control systems. The slides are prepared based mainly on by Angeli & Sontag (IEEE Trans Auto Ctrl 003), with several other reference papers. The main contribution of the paper is to extend the notion of monotone systems to systems with i/o The technical results of the main paper are summarized, wherever possible, institutive proofs/explanations are given. Audience are advised to refer the paper or literature for most accurate statements of theorems and/or mathematically rigorous proofs.

3 Outline Introduction & Motivation History s Monotone Systems Characterizations Static i/s & i/o characteristics Feedback interconnections Applications Conclusion 3

4 Motivation Monotone System (M.S.): One of the most important dynamic systems in theoretical (mathematical) biology and chemistry Many models in bio & chem typically treat quantities intrinsically positive (e.g. population density, concentration of chemicals) Apart from preserving positivity of solutions, models often preserve also mononicity or state orders 4

5 Introduction M.S. : The trajectory preserve a partial ordering on states. Subclasses: Cooperative systems (Positive feedback: diff. state variables reinforce each other) More general negative feedback (Competitive systems) mixed feedback 5

6 History 1900s: Essential ideas behind M.S. (Lyapnov, Poincare) (stability theory, dynamic sys. approach to D.E.) 191: Mono. iteration scheme for solving nonlinear elliptic equations (Courant, Hilbert) : Foundation works Work on dynamic systems (Birkhoff) Use DEs to describing interactive bio populations (Lotka,Volterra) Strong maximum principle Monotonicity results of ODEs : More foundation works (Hopf) (Muller & Kamke) Func. Analysis, positive operators, semigroup, linear partial DEs s: Three major trends > Nonlinear partial differential equations > Ext. of dynamic systems theory to infinite-dim settings > Math. Model in biology, chem., econ., other than engr./phy. 6

7 History (cont d) 1980s 90s: a synthesis of 3 trends Hirsch: unifying theories & notations (widely used since then) Matano: independent contribution Smith et al: Continuous time theory Application to delay equations Application to biological models Hess. : Discrete time theory 000s: ( A century since the idea emerged) Sontag et al.: Extend monotone control with non-constant input functions => (M.C.S.) A beginning of a new type of nonlinear systems 7

8 Monotone Systems An ordered Banach space is a real Banach space B together with a distinguished nonempty closed subset K of B, its positive cone. ( Remark: B in this paper will all be Euclidean spaces, however the basic definitions allow more to be more general and thus useful for more applications) Set K s properties : A cone αk K forα R + Convex K + K K Transitive Pointed K I{ K} = {0} Reflexive Ordering is defined as x1 x iff x1 x K Strict ordering x > 1 x means that x1 x and x1 x Stricter ordering x1 >> x x1 x int( K) where Int(K) denotes the nonempty interior of positive cone K. 8

9 Monotone Systems A typical example: (The positive orthant) So n n B = R and K = R 0 i i x1 x iff x1 xfor i = 1 n, where i represents the ith coordinate of the variable 9

10 s In the following (14 slides with tedious definitions and theories) we seek to summarize some of the main technical results of the sontag s paper 10

11 Monotone System (autonomous: no external input ) A dynamic system φ : R 0 X X is monotone if x1 x φ ( t, x1 ) φ( t, x ) for all t 0 Similarly, the system is Strongly Monotone if x > x φ t, x ) >> φ( t, x ) for all t 0 ( with external input ) 1 ( 1 First, define partially ordered input value space U as for state variable x, So u u if u1 u K u, where K is the positive cone in ordered Banach space 1 u Bu Definition II.1: A controlled dynamical system φ : R 0 X U X is monotone if following holds for all t 0: u1 u, x1 x φ( t, x1, u1 ) φ( t, x, u ) Further Extension to Notion ( with measurable output ) We define a monotone mapping from state-space X to output-space Y, where Y is subset of ordered Banach Space B y

12 The system to be analyzed in this presentation (and the paper) is defined by D.E. with inputs x & = f ( x, u ) Assumptions: f is defined on X U where X is some open subset of B which contains X. f(x, u) is continuous in (x,u), moreover, f is locally Lipschitz continuous in x any solution with initial state in X is well defined for all t>=0 set X is forward (strongly) invariant (Recall that the subset S (nonempty closed subset relative to V) is said to be strongly invariant under the differential inclusion if the following property holds: for every solution x:[ 0, T] int(x) which has the property that x( 0) S, x( t) S for all t [ 0, T ] )

13 III. Infinitesimal characterization of Monotonicity We note the interior of X, int(x), having the approximability property: i i For all ξ 1,ξ X such that ξ1 ξ, there exist sequences { ξ },{ ξ } int( X ) 1 i i such that ξ for all i and ξ and ξ i as i 1 ξ i ξ 1 1 ξ We also use the standard notion of tangent cone in nonsmooth analysis: Let S be a subset of a Euclidean space, and choose anyξ S. The tangent cone to S at ξ is the set T ξ S consisting all limits of the type lim (1/ )( ξ ξ ) such that i t i (1) ξ i ξ. as i and that ξ i t 0. () i as i and that i t 1 s t, i

14 Theorem 1: The system is monotone if and only if, for all ξ ξ int( ) ξ 1, X 1 ξ, ζ u1 u f ( ξ1, u1) f ( ξ, u ) Tξ K 1 Remarks: (1) Theorem 1 is valid even if the ordering is defined with respect to an arbitrary closed set K as oppose to a closed convex cone. (Because the proof of theorem 1 does not require a closed convex cone) So we may generalize as follows, Given an arbitrary close subset Γ For X ξ ξ,ξ, 1 1 ξ ( ξ1, ξ ) Γ X X with the relation, Then define monotonicity just as Definition II.1.

15 () One may interest in system such that the state dynamics are not necessarily monotone, but the output is monotone. Γ ( 1, ξ) ξ such that h ξ ) h( ) in the output- Let now be the set of all pairs of states value order. ( 1 ξ We introduce a state-space [] x & = f ( x, u) in block form [] X X and input-value set U whose dynamics is (Essentially, these are two copies of the same system driven by different inputs u) Theorem : The system is monotone iff, for all ξ 1, ξ int( X ) [] ξ ξ, u u f ( ξ, u) T Γ 1 1 ξ

16 IV. Cascade of Monotone Systems An example: cascade structures with triangular form x& 1 = f1( x1, x, L, xn, u) x& = f ( x, L, xn, u) x & N = f N ( x, u) N The proof is to use induction, first applying definition II.1 from N=. The system may be visualized as Xn Xn-1 X1 u

17 V. Static Input-State and Input-Output Characteristics For MCS satisfying an additional property, it is possible to obtain tight estimates of Cauchy Gain (a biological term to quantify amplification of signals) Definition V.1: A controlled dynamic system x & = f ( x, u ) is endowed with the static input-output characteristic k x ( ) : U X If for each constant input u( t) u there exists unique globally asympototicallly stable equilibrium k x (u ) Consequentially, Static Input-Output Characteristics, k y ( ) exists for same system if k x ( ) exists and h(.), state-output mapping, is continuous. Remarks: (1) k x ( ) is nondecreasing () kx ( ) is continuous

18 Implication (with weak assumptions): Existence of k x ( ) means system behaves well (i.e. stable) w.r.t. arbitrary bounded inputs w.r.t. inputs that converge to some limit For convenience, we introduce following terminology: The order on X is bounded if following holds: (1) Bounded subset S of X are contained by the set of all elements of X lies in between an ordered pair [a, b] () For each a, b B, the set [a, b] is bounded. Proposition V.4: (Stability) Considering monotone system x & = f ( x, u ) with k ( x ), and Order on state space X is bounded, and Order on input space U is bounded, Then, bounded input u implies trajectory x(t) to u is bounded subset of X.

19 Proposition V.5: (Robustness of Stability) Considering monotone system x & = f ( x, u ) with k x ( ), and For each constant input u U and corresponding equilibrium x = kx (u ), 1) There are associated (bounded) neighbourhoods in initial-state space, state-space, and input-space. ) If, in addition, the order in state-space X is bounded, Then for each bounded input (as time goes to infinity) converging to u, and all initial states in X, necessarily state converges to x as time goes to infinity Corollary V.6: Static input-state characteristic can be propagated through Cascade of system, in which the previous block (of system) must be monotone, input-state static and input-output static.

20 VI. Feedback Interconnections Theorem 3: Considering the following interconnection of two SISO dynamic systems: x& = z& = f ( x, w) x With x z y = h (x) f z ( z, y) w = hz (z) U = Y and x U z = Y x Suppose that 1) the first system is monotone when its input w and output y are ordered according to the standard order induced by the positive real semi-axis; ) the second system is monotone when its input y is ordered according to the standard order induced by the positive real semi-axis and its output w is ordered by the opposite order, i.e.. the one induced by the negative real semi-axis;

21 3) the respective static input-state characteristics k x ( ) and kz ( ) exists (thus the static input-output characteristics exists too and k y ( ) is monotonically increasing and k w ( ) is monotonically decreasing; 4) Every solution of the closed-loop system is bounded. Then, the system above has a globally attractive equilibrium provided that the following scalar discrete time dynamic system, evolving in U x : uk + 1 = ( kw o k y )( uk ) has a unique globally attractive equilibrium u. Remark: 1) traditional small-gain theorems also provide sufficient conditions for global existence and boundedness of solutions. Thus, for M.S., boundedness of trajectories follows at once provided that at least one of the interconnected systems has a uniformly bounded output map (equivalent to require, for instance, the state space of the corresponding system is compact) ===

22 What if both output maps are unbounded? (or such boundedness of trajectories is not a priori known condition) The input-state characteristic k x ( ) is unbounded (relative to X) if ξ X u, u U s. t. k ( u ) ξ k ( ), 1 x 1 y u Proposition IV.3 establishes that for the same system as described for Theorem 3, but endowed with unbounded input-state static characteristics. If the conditions of theorem 3 are still satisfied then the solutions still exist for all positive time t, and are bounded.

23 Applications In the paper, the application example is on how to guarantee the nonexistence of oscillations in certain biological inhibitory feedback loops, and specially in a model named MAPK (mitogen-activated protein kinase) cascade It is a cascade connection of 3 SISO systems, each of which is either 1- or - dim system. The paper shows that Every MAPK cascade is monotone with static input-state characteristics 11

24 Conclusions Monotone systems have been studied since 1900s While Monotone control systems has only recently been proposed. It finds various applications in mathematical modeling of Biology, Chemistry, Economics and more, where the quantities of interest are inherently/intrinsically monotone Monotone (control) systems and some of its subclasses, such as cooperative systems, can be related to other field of theories of dynamic systems, such as positive systems, etc. The main paper s extends the notion of monotone systems with i/o, pointing out a new direction/type of nonlinear systems. 1

25 Thank You 13