EE363 Winter 2008-09Lecture 11Invariant sets, conservation, and dissipation• invariant sets• conserved quantities• dissipated quantities• derivative along trajectory• discrete-time case11–1Invariant setswe consider autonomous, time-invariant nonlinear system ˙x = f(x)a set C ⊆ Rnis invariant (w.r.t. system, or f) if for every trajectory x,x(t) ∈ C =⇒ x(τ) ∈ C for all τ ≥ t• if trajectory enters C, or starts in C, it stays in C• trajectories can cross into boundary of C, but never out of CCInvariant sets, conservation, and dissipation 11–2Examples of invariant setsgeneral examples:• {x0}, where f(x0) = 0 (i.e., x0is an equilibrium point)• any trajectory or union of trajectories, e.g.,{x(t) | x(0) ∈ D, t ≥ 0, ˙x = f(x)}more specific examples:• ˙x = Ax, C = span{v1, . . . , vk}, where Avi= λivi• ˙x = Ax, C = {z | 0 ≤ wTz ≤ a}, where wTA = λwT, λ ≤ 0Invariant sets, conservation, and dissipation 11–3Invariance of nonnegative orthantwhen is nonnegative orthant Rn+invariant for ˙x = Ax?(i.e., when do nonnegative trajectories always stay nonnegative?)answer: if and only if Aij≥ 0 for i 6= jfirst assume Aij≥ 0 f or i 6= j, and x(0) ∈ Rn+; we’ll show that x(t) ∈ Rn+for t ≥ 0x(t) = etAx(0) = limk→∞(I + (t/k)A)kx(0)for k large enough the matrix I + (t/k)A has all nonnegative entries, so(I + (t/k)A)kx(0) has all nonnegative entrieshence the limit above, which is x(t), has nonnegative entriesInvariant sets, conservation, and dissipation 11–4now let’s assume that Aij< 0 for some i 6= j; we’ll find trajectory withx(0) ∈ Rn+but x(t) 6∈ Rn+for some t > 0let’s take x(0) = ej, so for small h > 0, we have x(h) ≈ ej+ hAejin particular, x(h)i≈ hAij< 0 f or small positive h, i.e., x(h) 6∈ Rn+this shows that if Aij< 0 f or some i 6= j, Rn+isn’t invariantInvariant sets, conservation, and dissipation 11–5Conserved quantitiesscalar valued function φ : Rn→ R is called integral of the motion, aconserved quantity, or invariant for ˙x = f(x) if for every trajectory x,φ(x(t)) is constantclassical examples:• total energy of a lossless mechanical system• total angular momentum about an axis of an isolated system• total fluid in a closed systemlevel set or level surface of φ, {z ∈ Rn| φ(z) = a}, are invariant setse.g., trajectories of lossless mechanical system stay in surfaces of constantenergyInvariant sets, conservation, and dissipation 11–6Example: nonlinear lossless mechanical systemmq(t)Fqm¨q = −F = −φ(q), where m > 0 is mass, q(t) is displacement, F isrestoring force, φ is nonlinear spring characteristic with φ(0) = 0with x = (q, ˙q), we have˙x =˙q¨q=x2−(1/m)φ(x1)Invariant sets, conservation, and dissipation 11–7potential energy stored in spring isψ(q) =Zq0φ(u) dutotal energy is kinetic plus potential: E(x) = (m/2) ˙q2+ ψ(q)E is a conserved quantity: if x is a trajectory, thenddtE(x(t)) = (m/2)ddt˙q2+ddtψ(q)= m ˙q¨q + φ(q) ˙q= m ˙q(−(1/m)φ(q)) + φ(q) ˙q= 0i.e., E(x(t)) is constantInvariant sets, conservation, and dissipation 11–8Derivative of function along trajectorywe have function φ : Rn→ R and ˙x = f(x)if x is trajectory of system, thenddtφ(x(t)) = Dφ(x(t))dxdt= ∇φ(x(t))Tf(x)we define˙φ : Rn→ R as˙φ(z) = ∇φ(z)Tf(z)intepretation:˙φ(z) givesddtφ(x(t)), if x(t) = ze.g., if˙φ(z) > 0, then φ(x(t)) is increasing when x(t) passes through zInvariant sets, conservation, and dissipation 11–9if φ is conserved, then φ(x(t)) is constant along any trajectory, so˙φ(z) = ∇φ(z)Tf(x) = 0for all zthis means the vector field f(z) is everywhere orthogonal to ∇φ, which isnormal to the level surfaceInvariant sets, conservation, and dissipation 11–10Dissipated quantitieswe say that φ : Rn→ R is a dissipated quantity for system ˙x = f(x) if forall trajectories, φ(x(t)) is (weakly) decreasing, i.e., φ(x(τ)) ≤ φ(x(t)) forall τ ≥ tclassical examples:• total energy of a mechanical system with damping• total fluid in a system that leakscondition:˙φ(z) ≤ 0 for all z, i.e., ∇φ(z)Tf(z) ≤ 0−˙φ is sometimes called the dissipation functionif φ is dissipated quantity, sublevel sets {z | φ(z) ≤ a} are invariantInvariant sets, conservation, and dissipation 11–11Geometric interpretationφ = const.∇φ(z)f(z)x(t)• vector field points into sublevel sets• ∇φ(z)Tf(z) ≤ 0, i.e., ∇φ and f always make an obtuse angle• trajectories can only “slip down” to lower values of φInvariant sets, conservation, and dissipation 11–12Examplelinear m echanical system with damping: M ¨q + D ˙q + Kq = 0• q(t) ∈ Rnis displacement or configuration• M = MT> 0 is mass or inertia matrix• K = KT> 0 is stiffness matrix• D = DT≥ 0 is damping or loss matrixwe’ll use state x = (q, ˙q), so˙x =˙q¨q=0 I−M−1K −M−1DxInvariant sets, conservation, and dissipation 11–13consider total (potential plus kinetic) energyE =12qTKq +12˙qTM ˙q =12xTK 00 Mxwe have˙E(z) = ∇E(z)Tf(z)= zTK 00 M0 I−M−1K −M−1Dz= zT0 K−K −Dz= − ˙qTD ˙q ≤ 0makes sense:ddt(total stored energy) = − (power dissipated)Invariant sets, conservation, and dissipation 11–14Trajectory limit with dissipated quantitysuppose φ : Rn→ R is dissipated quantity for ˙x = f(x)• φ(x(t)) → φ∗as t → ∞, where φ∗∈ R ∪ {−∞}• if trajectory x is bounded and˙φ is continuous, x(t) converges to thezero-dissipation set:x(t) → D0= {z |˙φ(z) = 0}i.e., dist (x(t), D0) → 0, as t → ∞ (more on this later)Invariant sets, conservation, and dissipation 11–15Linear functions and linear dynamical systemswe consider linear system ˙x = Axwhen is a linear function φ(z) = cTz conserved or dissipated?˙φ = ∇φ(z)Tf(z) = cTAz˙φ(z) ≤ 0 for all z ⇐⇒˙φ(z) = 0 for all z ⇐⇒ ATc = 0i.e., φ is dissipated if only if it is conserved, if and only if if ATc = 0(c is left eigenvector of A with eigenvalue 0)Invariant sets, conservation, and dissipation 11–16Quadratic functions and linear dynamical systemswe consider linear system ˙x = Axwhen is a quadratic form φ(z) = zTP z conserved or dissipated?˙φ(z) = ∇φ(z)Tf(z) = 2zTP Az = zT(ATP + P A)zi.e.,˙φ is also a quadratic form• φ is conserved if and only if ATP + P A = 0(which means A and −A share at least Rank(P ) eigenvalues)• φ is dissipated if and only if ATP + P A ≤ 0Invariant sets, conservation,