A very brief history of feedback: before and after Maxwell Jean-Michel Coron Laboratoire J.-L. Lions, Université Pierre et Marie Curie http://www.ann.jussieu.fr/coron/ BCAM OPTPDE summer school, Bilbao, July 4-8 2011 The cart-inverted pendulum C F The cart-inverted pendulum: An equilibrium C Instability of this equilibrium C Instability of this equilibrium C Stabilization of the unstable equilibrium F : depending on the state C F is a feedback Doubleinverted pendulum (CAS, ENSMP/La Villette) The Clepsydra (water clock) The Clepsydra (water clock) Ctésibius (3th century BC) towards tank 1 tank 2 tank 3 James Watt 1736. Birth, Greenock, Scotland. 1769. Patent of the separate condensation chamber. 1782. Double action. 1788. Adaptation to the steam engine of a regulator used in windmills. 1819. Death, Heathfield, England. Thomas Newcomen 1663-1729 *** Thomas Savery 1650-1715 History of the fly ball (centrifugal governor) In fact, this regulator is directly inspired from regulators used in windmills. There exists a letter, dated May 28 1788, where Boulton, Watt’s partner in the firm Boulton & Watt, describes this regulator: [the regulation of the top mill stone] is produced by the centrifugal force of 2 lead weights which rise up horizontal when in motion and fall down when the motion is decreased, by which means they act on a lever that is divided as 30 to 1, but to explain it requires a drawing. In the windmills, these regulators where used to regulate • the distance between the two mill stones, • the speed of rotation of the top mill stone. 29 30 30 30 30 30 30 46 46 31 32 31 45 45 B 28 44 44 27 26 38 37 33 35 A 45 34 40 35 41 38 37 39 42 38 C 43 39 42 38 39 42 Maxwell’s article “On governors” On governors, Proceedings of the Royal Society, vol. 16, 1868, pages 270–283, James Clerk Maxwell (1831-1879) James Clerk Maxwell 1831. Birth, Edinburgh, Scotland. 1857. Adams Prize for his article “On the Stability of the Motion of Saturn’s Rings”. Alexis Bouvard 1821 : Uranus does not follow the expected trajectory (that one can compute using the Laplace perturbations theory and the informations on the known planets)! John Couch Adams and Urbain Le Verrier Le Verrier (1811-1877) Adams (1819-1892) 1845 : beginning of the 1843: Beginning of the computations. computations. 1846 (June) : First results. 1845 (October): Results given to G. B. Airy and J. Challis. 1846 (August) : New results. 1846 (August) : J. Challis is 1846 (September) : J. G. Galle looking for the planet. sees the planet at the place indicated by Le Verrier. John Couch Adams and Urbain Le Verrier Le Verrier (1811-1877) Adams (1819-1892) 1845 : beginning of the 1843: Beginning of the computations. computations. 1846 (June) : First results. 1845 (October):Results given to G. B. Airy and J. Challis. 1846 (August) : New results. 1846 (August): J. Challis is 1846 (September) : J. G. Galle looking for the planet. sees the planet at the place indicated by Le Verrier. A picture of “la belle inconnue” Sonde Voyager 2; 1989 Distance to sun : 4,5 milliards de kms. Radius : 25 000 kms. Temperature : -226 degrés C. Period of revolution 164 ans 323 jours. James Clerk Maxwell 1831. Birth, Edinburgh, Scotland. 1857. Adams Prize for his article “On the Stability of the Motion of Saturn’s Rings”. James Clerk Maxwell 1831. Birth, Edinburgh, Scotland. 1857. Adams Prize for his article “On the Stability of the Motion of Saturn’s Rings”. 1861. First color photograph. Works on the perception of color 1855 to 1872. James Clerk Maxwell 1831. Birth, Edinburgh, Scotland. 1857. Adams Prize for his article “On the Stability of the Motion of Saturn’s Rings”. 1861. First color photograph. Works on the perception of color 1855 to 1872. 1866. Kinetic theory of gases. 1868. Article: “On governors”. 1873. Book: “A treatise on electricity and magnetism” (équations de Maxwell). 1879. Death, Cambridge, England. Aleksandr Lyapunov (1857-1918) Thesis: The general problem of the stability of motion (1892 in Russian, 1907 in French -see Numdam, English 1992) • Definition of the stability and of the asymptotic stability. Remark In the case of Mechanical systems, there were important previous works, in particular by J.-L. Lagrange, P.-S. Laplace, G. Dirichlet. Stability Stability Stability Stability Stability Attractor Attractor Attractor Attractor Attractor does not imply stable Attractor does not imply stable Attractor does not imply stable Attractor does not imply stable Attractor does not imply stable Attractor does not imply stable Attractor does not imply stable Asymptotically stable Définition (Asymptotically stable) n n n Let xe be an equilibrium point of X : R → R , i.e. a point in R such that X(xe) = 0. One says that xe is asymptotically stable for x˙ = X(x) if xe is stable and attractor. One says that xe is unstable for x˙ = X(x) if it is not stable. First Lyapunov’s theorem Theorem 1 Let us assume that X is of class C and that xe is an equilibrium point of ′ x˙ = X(x). If all the eigenvalues of X (xe) have a strictly negative real ′ part, then xe is locally asymptotically stable for x˙ = X(x). If X (xe) has an eigenvalue with a strictly positive real part, then the equilibrium xe is unstable for x˙ = X(x). Remark This theorem was assumed to be true before Lyapunov proved it. See, for example, the fundamental paper “On governors” by J.C. Maxwell (1868). Second Lyapunov’s theorem Theorem The point xe is asymptotically stable for x˙ = X(x) if and only if there n ∞ exists η> 0 and V : B(xe, η) : {x ∈ R ; |x − xe| < η}→ R of class C such that (1) V (x) > V (xe), ∀x ∈ B(xe, η) \ {xe} n ∂V (2) ∇V (x) · X(x)= X (x) < 0, ∀x ∈ B(x , η) \ {x }. ∂x i e e i=1 i Remark • In the cases of mechanical systems; there are important previous results by J.-L Lagrange (1788) and G. Dirichlet (1847). • For the only if part: J. Massera and J. Kurzweil. • This theorem explains why “asymptotic stability” is the good notion. Robustness and Lyapunov function Robustness and Lyapunov function Key innovations of Maxwell’s paper 1. Regulation is a dynamical problem. Before Maxwell’s paper, the engineers and scientists were looking only to equilibria. 2. Linearization of the dynamical system. 3. The stability holds if all the roots of the characteristic polynomial have a strictly negative real part. Remark In fact, part of the above innovations already appeared in two papers by G.B. Airy (1840 and 1850). G.B. Airy was “Astronomer Royal”. For a telescope, having a very good regulation of speed is an essential point: One needs to compensate exactly the earth rotation in order to make astronomical observations. The two Airy papers are on the dynamical properties of regulators used for telescopes. Watt’s regulator: Notations S l θ m ϕ Dynamic equations Let x1 = θ, x2 = θ˙ and x3 =ϕ ˙. The dynamic equations are (1) x˙ 1 = x2, 2 g C (2) x˙ 2 = sin(x1) cos(x1)x3 − sin(x1) − x2, l 2ml2 Γr Γ0 k (3) x˙ 3 = − + − (1 − cos(x1)), J J J where g is the gravitational constant, Cx2 = Cθ˙, with C > 0 is the friction term (it includes the friction at the pivot point S), Γr > 0 is the load torque applied to the steam engine, Γ0 − k(1 − cos(x1)) is the engine torque produced by the steam J is the moment of inertia. The feedback law is Γ0 −k(1 − cos(x1)). The constants Γ0 and k are part of the design of the feedback law. They can be essentially be chosen freely by modifying the throttle valve and beam linkage. Watt’s regulator: The equilibrium point x1 = x1, x2 = x2 and x3 = x3 is an equilibrium of the regulator if and only if 2 g (1) x2 = 0, sin(x1) cos(x1)x3 − sin(x1) = 0, l Γr Γ0 k (2) − + − (1 − cos(x1)) = 0. J J J For physical reasons sin(x1) = 0. Hence, using (1), g (3) cos(x1)= 2 . lx3 ′ The desired angular velocity is x3 = ϕ = ω. Then, using (2), one imposes on Γ0 and k to satisfy g (4) Γ =Γ0 − k 1 − . r lω2 Maxwell’s key idea: The linearization One writes x1 = x1 + y1, x2 = x2 + y2 = y2, x3 = x3 + y3 = ω + y3. We assume that the yi’s are small. Up to the order 1 in the yi’s, we have y˙ = Ay, with 0 1 0 2 2 C (1) A = −ω sin (x1) − ω sin(2x1) . 2ml2 k − sin(x1) 0 0 J The eigenvalues of A are the root of the following polynomial (called characteristic polynomial): 3 C 2 2 2 2kg 2 (2) z + z + ω sin (x1)z + sin (x1) = 0. 2ml2 Jlω In his 1868 paper, J. C. Maxwell presents other regulators for which the characteristic polynomial is of larger degree (n ∈ {3, 4, 5}). This leads naturally to the following problem. Let n be a positive integer. Let a1, a2, ..., an be n real numbers. Under which condition on the ai’s, all the roots of the polynomial n n−1 n−2 (1) P (z)= z + a1z + a2z + . + an−1z + an have a strictly negative real part. Remark J.C. Maxwell says “the possible roots and the possible parts of the impossible roots” for the real roots and the real parts of the nonreal roots. A necessary condition on the ai’s One easily checks that the following theorem holds.