Stability and boundary stabilization of physical networks represented by 1-D hyperbolic balance laws

Stability and boundary stabilization of physical networks represented by 1-D hyperbolic balance laws G. Bastin http://perso.uclouvain.be/georges.bastin/

Georges Bastin Jean Michel Coron Progress in Nonlinear Differential Equations and Their Applications Subseries in Control 88 Stability and Boundary Stabilization of 1-D Hyperbolic Systems Outline Hyperbolic 2x2 systems of balance laws Steady-state and characteristic form Networks of balance laws Boundary conditions Exponential stability Lyapunov stability analysis Real-life application : hydraulic networks Non uniform systems

Open channels, St Venant equations (1871) th + x (hv) = tv + x (gh + 1 2 v2 )=gs Cv 2 /h h = water level, v = water velocity, g= gravity, S = canal slope, C = friction coe cient Road traffic, Aw-Rascle equations (2) t + x ( v) = t(v + p( )) + v x v + p( )) = (V ( ) v)/ = tra c density, v = tra c velocity, p( ) = tra c pressure, V ( ) = preferential velocity, = constant time constant

Telegrapher equations, Heaviside (1892) @ t (Li)+@ x v = @ t (Cv)+@ x i = Ri, Gv, i =currentintensity,v = voltage, L = line inductance, C = capacitance, R = line resistance, G = dielectric admittance (per unit length). Optical fibers : Raman amplifiers (1927) @ t S + s @ x S = s @ t P p@ x P = p s S + p P s SP, psp, S(t, x) = transmitted signal power, P (t, x) = pump laser beam power, s and p = propagation velocities, s and p = attenuation coe cients, s and p = amplification gains

Euler isentropic equations (1757) @ t + @ x ( v) = @ t ( v)+@ x ( v 2 + P ( )) = C v v = gas density, v = gas velocity, P =pressure,c = friction coe cient Fluid flow in elastic tubes (e.g. blood flow) @ t A + @ x (AV )=, @ t (AV )+@ x AV 2 + applea 2 = CV A = tube cross-section, V = fluid velocity,, apple,c = constant coe cients

Hyperbolic 2x2 systems of balance laws Space x [,L] Time t [, + ) State Y (t, x) y 1 (t, x) y 2 (t, x) ty + x f(y )=g(y ) ty + A(Y ) x Y = g(y ) A(Y ) has 2 distinct real eigenvalues

Uniform steady state ty + A(Y ) x Y = g(y ) A steady-state is a constant solution Y (t, x) Ȳ which satisfies the equation g(ȳ )= and (obviously) the state equation tȳ + A(Ȳ ) xȳ = g(ȳ ) Road traffic = density v = velocity Open channels h = water depth v = velocity Steady state v = V ( ) Steady state r gs v = C h (Toricelli formula)

Characteristic form Hyperbolic system : ty + A(Y ) x Y = g(y ) Change of coordinates : (Y )= 1 2 ( Riemann coordinates ) @ t + c1 ( ) c 2 ( ) with c 1 ( ) = c 2 ( ) eigenvalues of A(Y ) @ x = h( ) ( Characteristic velocities ) The change of coordinates (Y ) is defined up to a constant. It can therefore be selected such that (Ȳ )= h() =.

Physical netwoks of 2x2 hyperbolic systems (e.g. hydraulic networks (irrigation, waterways) or road tra c networks) 1 7 6 2 1 8 3 11 4 9 5 directed graph n edges one 2x2 hyperbolic system of balance laws attached to each arc L x arrow = orientation of the x axis @ t i n+i + ci ( ) i @ c n+i ( ) x n+i = h i n+i (i =1,...,n)

Characteristic form @ t + C( )@ x = h( ) (characteristic velocities) C( ) = diag(c 1 ( ),...,c 2n ( )) c i ( ) > c i ( ) < i =1,...,n i = n +1,...,2n Notations + =( 1, 2,..., n ) Boundary conditions + (t, ) (t, L) =( n+1,..., 2n ) = G + (t, L) (t, ) x 2 [,L] L

Boundary conditions = Physical constraints + (t, ) (t, L) = G + (t, L) (t, ) F (Y (t, ),Y(t, L)) = Road traffic 1 = density v = velocity 3 Open channels h = water depth v = velocity 2 1 2 Flow conservation at a junction 3(t, )v 3 (t, ) = 1(t, L)v 1 (t, L)+ 2 (t, L)v 2 (t, L) Modelling of hydraulic gates h 2 (t, )v 2 (t, ) = h 1 (t, L) u 3/2

Boundary conditions = boundary feedback control Road traffic = density v = velocity q = v = flux Open channels h = water depth v = velocity 1 3 h 2 Feedback implementation of ramp metering Feedback control of water depth in navigable rivers G function of the control tuning parameters : How to design the control laws to make the boundary conditions stabilizing?

t 2 [, +1) x 2 [,L] @ t + C( )@ x = h( ) + (t, ) + (t, L) = G (t, L) (t, ) (,x)= (x) Conditions on C, h and G such that =is exponentially stable?

t 2 [, +1) x 2 [,L] @ t + C( )@ x = h( ) + (t, ) + (t, L) = G (t, L) (t, ) (,x)= (x) Conditions on C, h and G such that =is exponentially stable? In a linear space of functions from [,L] into R 2n Definition of exponential stability for the norm k k 9 ",, such that solutions are defined 8 t 2 [, +1) and k k 6 " )k (t,.)k 6 e t k k (+ usual compatibility conditions adapted to )

Lyapunov stability : we start with the linear case System Boundary condition t + x = B (t, ) = K (t, L) = diag{ i > } Lyapunov function V = Z L T (t, x)p (t, x)e µx dx µ> P = diag {p i > } V = Z L (t, x) µp [B T P + PB] (t, x)e µx dx T (t, L)[P e µl K T P K] (t, L)

System t + x = B (t, ) = K (t, L) (,x)= (x) V = V = Z L Z L Lyapunov function T (t, x)p (t, x)e µx dx (t, x) µp [B T P + PB] (t, x)e µx dx T (t, L)[P e µl K T P K] (t, L) If 9 P (diag. pos.) and µ> such that: 1) µp [B T P + PB] 2) P e µl K T P K ( : positive definite) Then 9 > s.t. V 6 V ) exponential stability for L 2 -norm (B-C chapter 5)

Let us now consider the nonlinear case System Boundary condition @ t + C( )@ x = h( ) c i ( ) > (t, ) = G( (t, L)) Notations: = C(), B = h (), K = G () (linearization) If 9 P (diag. pos.) and µ> such that: 1) µp [B T P + PB] 2) P e µl K T P K then the steady-state is exponentially stable for the H 2 -norm (B-C chapter 6)

Let us now consider the nonlinear case System Boundary condition @ t + C( )@ x = h( ) c i ( ) > (t, ) = G( (t, L)) Notations: = C(), B = h (), K = G () (linearization) If 9 P (diag. pos.) and µ> such that: 1) µp [B T P + PB] 2) P e µl K T P K then the steady-state is exponentially stable for the H 2 -norm Lyapunov function V = Z L ( T P + t P t + tt P tt )e µx dx (B-C chapter 6)

Special case: a more explicit stability condition @ t + C( )@ x = h( ) (t, ) = G( (t, L)) = C(), B = h (), K = G () Theorem If B =,orifkbksu ciently small, then is exponentially stable for H 2 -norm if 2 (K) < 1 2 (K) =min(k K 1 k, positive diagonal) (boundary damping) (t, ) = K (t, L) (k k : 2-norm) (B-C chapters 4 and 6)

Special case: Li Ta Tsien Condition @ t + C( )@ x = h( ) (t, ) = G( (t, L)) = C(), B = h (), K = G () Theorem If B =,orifkbk su ciently small, then is exponentially stable for C 1 -norm if ( K ) < 1 ( K ) = spectral radius of the matrix h i K ij (boundary damping) (t, ) = K (t, L) (Li Ta Tsien 1994 ) (B-C chapter 6)

H 2 /C 1 exponential stability @ t + C( )@ x = h( ) (t, ) = G( (t, L)) = C(), B = h (), K = G () For every K 2 M m,m 2 (K) 6 ( K ) Example: for K = 1 1, 1 1 2 (K) = p 2 < 2= ( K ) (B-C chapter 4 + Appendix) For the C 1 -norms, the condition ( K ) < 1 is su cient for the stability the condition 2 (K) < 1isnot su cient (Coron and Nguyen 215)

H 2 /C 1 exponential stability @ t + C( )@ x = h( ) (t, ) = G( (t, L)) = C(), B = h (), K = G () For every K 2 M m,m Example: for K = 2 (K) 6 ( K ) 1 1, 1 1 2 (K) = p 2 < 2= ( K ) (B-C chapter 4 + Appendix) For the C 1 -norms, the condition ( K ) < 1 is su cient for the stability the condition 2 (K) < 1isnot su cient There are boundary conditions that are su ciently damping for the H 2 -norm but not for the C 1 -norm

Another special case @ t + C( )@ x = h( ) (t, ) = G( (t, L)) = C(), B = h (), K = G () If 9 P (diag. pos.) such that: 1) B T P + PB 4 (e.g. dissipative friction term) 2) 2 (K) =(k K 1 k < 1 with 2 = P (boundary damping)

The stability conditions hold also, with appropriate modifications, for systems with positive and negative characteristic velocities c i ( ), as we shall see in the example.

Example: Boundary control for a channel with multiple pools Pool i H i V i u i H i+1 V i+1 x x Pool i +1 u i+1 Meuse river (Belgium)

Real-life application : level control in navigable rivers. Meuse river (Belgium).

Model : Saint-Venant equations Pool i H i V i u i H i+1 V i+1 x x Pool i +1 u i+1 t Hi V i + x 1 2 V i 2 H i V i + gh i Boundary conditions = g[s i C i V 2 i H 1 slope i ] friction, i =1...,n. (width = W ) 1) Conservation of flows H i (t, L)V i (t, L) =H i+1 (t, )V i+1 (t, ) i =1,...,n 1 H i (t, L)V i (t, L) =k G H i (t, L) u i (t) 3 2) Gate models i =1,...,n 3) Input flow WH 1 (t, )V 1 (t, ) = Q (t) = disturbance input in navigable rivers or control input in irrigation networks

Riemann coordinates i =(V i V i )+(H i H i ) Boundary control design g H i n+i =(V i V i ) (H i H i ) (H i,v i ) = steady-state g H i i =1,..., n. Control objective n+i(t, L) = k i i (t, L) i =1,...,n k i = control tuning parameters Gate model H i (t, L)V i (t, L) =k G H i (t, L) u i (t) 3 i =1,...,n u i (t) =H i (t, L) Control law H i H i (t, L) 1 k i g Si H (H i (t, L) H k G 1+k i ) + i i H i C = level set points 2/3 i =1,...,n

Linearized Saint-Venant equations in Riemann coordinates i =1,...,n @ t i n+i + @ x i n+i @ x i n+i = i i i i i n+i n+i = V i gh i < < i = V i + gh i < n+i < i < i = gs i 1 V i 1 2 gh i < i = gs i 1 V i + 1 2 gh i

Lyapunov stability @ t i n+i + @ x i n+i Subcritical flow condition @ x i n+i = < n+i < i < i < i i i i i i n+i V = n L i=1 p i 2 i e µx + p n+i 2 n+ie µx dx, p i,p n+i,µ> If parameters p i selected such that p i i = p n+i i p i+1 = p i and µ su ciently small (dissipative friction) control tuning parameters k i such that k 2 i i i n+i i < 1 (boundary damping) steady-state exponentially stable (B-C chapter 5)

Experimental result (Dinant) meters above sea level change of set point water level low flow rate high flow rate gate levels = control action 16 to 23 october 212

The nonuniform case : example Now we consider a pool of a prismatic horizontal open channel with a rectangular cross section and a unit width. H(x, t) = water depth V (x, t) = water velocity C = friction coe cient th + x (HV )= V 2 tv + x 2 + gh + gc V 2 H = Steady-state H (x),v (x) for a constant flow rate Q = H (x)v (x) dv dx = gc Q (V (x)) 5 gq (V (x)) 3 Nonuniform steady state H (x) V (x) L x

th + x (HV )= V 2 tv + x 2 + gh + gc V 2 H = Linearization about the steady-state h(t, x) :=H(t, x) H (x), v(t, x) :=V (t, x) V (x) Linearized model in physical coordinates h t v t + V H g V h x v x + V x gc V 2 H H x V x +2gC V H h = v

h t v t + V H g V h x v x + V x gc V 2 H H x V x +2gC V H h v = Lyapunov function V = Z L gh 2 + H v 2 dx = Z L h v @ g 1 A H h v dx dv dt = Z L Y T N(x)Y dx Y T M(x)Y L Y = h v g 2 CV 3 H (gh V 2 ) N(x) = B @ gcv 2 H 1 gcv 2 H C 2gCV 3 (gh V 2 ) +4gCV A M(x) = gv gh gh H V

h t v t + V H g V h x v x + V x gc V 2 H H x V x +2gC V H h v = Lyapunov function V = Subcritical flow (i.e. fluvial): gh V 2 > Z L gh 2 + H v 2 dx = Z L h v @ g 1 A H h v dx dv dt = Z L Y T N(x)Y dx Y T M(x)Y L Y = h v g 2 CV 3 H (gh V 2 ) N(x) = B @ gcv 2 H 1 gcv 2 H C 2gCV 3 (gh V 2 ) +4gCV A M(x) = gv gh gh H V positive definite

h t v t + V H g V h x v x + V x gc V 2 H H x V x +2gC V H h v = Lyapunov function V = Subcritical flow (i.e. fluvial): gh V 2 > Z L gh 2 + H v 2 dx = Z L h v @ g 1 A H h v dx dv dt = Z L Y T N(x)Y dx Y T M(x)Y L boundary damping conditions? Y = h v g 2 CV 3 H (gh V 2 ) N(x) = B @ gcv 2 H 1 gcv 2 H C 2gCV 3 (gh V 2 ) +4gCV A M(x) = gv gh gh H V positive definite

disturbance Q (t) H(t, x) V (t, x) control Q L (t) L x control Q L (t) =Q (t)+k P (H(t, L) H sp ) feedforward feedback

disturbance Q (t) H(t, x) V (t, x) control Q L (t) L x measured measured set point control Q L (t) =Q (t)+k P (H(t, L) H sp ) control tuning

disturbance Q (t) H(t, x) V (t, x) control Q L (t) L x control Q L (t) =Q (t)+k P (H(t, L) H sp ) Y T M(x)Y L > =) > boundary damping V apple () gh H () (gh () V 2 ())h 2 (L) V 2 (L) (t, )+ H (2k P V (L)) + V (L) (L) H (L) k2 P h 2 (t, L) =) exponential stability if k P is su ciently large...

back : closed loop front : open loop Thank you Meuse river