Flatness based analysis and control of distributed parameter systems Elgersburg Workshop PDF Free Download

Flatness based analysis and control of distributed parameter systems Elgersburg Workshop 2018 Frank Woittennek Institute of Automation and Control Engineering Private University for Health Sciences, Medical Informatics and Technology (UMIT) Hall in Tirol, Austria 01.03.2018 F. Woittennek Flatness of d.p.s Elgersburg 1 / 40

Outline Introduction: Flatness of finite dimensional linear Systems Flatness as parametrizability Wave equation: Boundary controlled string with boundary load Flatness State controllability Control design Numerical implementation by late lumping Some examples More general hyperbolic p.d.e. Coupled strings with one control Nonlinear systems Summary F. Woittennek Flatness of d.p.s Elgersburg 2 / 40

Flatness of finite dimensional systems Several abstract definitions: Nonlinear systems: differential algebraic [2] and differential geometric definitions [1] Linear systems: simple algebraic and geometric definitions (state space, polynomial matrices, modules over polynomial rings) Essence of the definitions: for the engineer system described by a set of (non-)linear differential equations involving several variables Flat output is a function of the system variables and derivatives. System variables can be expressed by flat output and derivatives No nontrivial differential equation in the components of the flat output. F. Woittennek Flatness of d.p.s Elgersburg 4 / 40

Flatness of a linear gantry crane model Linear model: for small deflections (angles) momentum balance for the cart: M D(t) = u(t) + mgθ(t) momentum balance for the load: mÿ(t) = mgθ(t) geometrical constraint: y(t) = D(t) + lθ(t) Flat output of the linear model: load position y θ(t) = 1 g ÿ(t), D(t) = y(t) + l Ml ÿ(t), u(t) = g g y(4) (t) + (m + M)ÿ(t) F. Woittennek Flatness of d.p.s Elgersburg 5 / 40

Flatness based open loop control design Control design: with cart position as input Prescribed: load position t y d (t) Apply computed input trajectory Computed: cart position t D d (t) = y d (t) + l g ÿd(t) F. Woittennek Flatness of d.p.s Elgersburg 6 / 40

Flatness based open loop control design Control design: with cart position as input 1.0 Flacher Ausgang y 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Zeit t Prescribed: load position t y d (t) Computed: cart position Apply computed input trajectory t D d (t) = y d (t) + l g ÿd(t) F. Woittennek Flatness of d.p.s Elgersburg 6 / 40

Flatness based open loop control design Control design: with cart position as input 1.0 1.0 Flacher Ausgang y 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Zeit t Prescribed: load position t y d (t) Wagenposition D 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Zeit t Computed: cart position Apply computed input trajectory t D d (t) = y d (t) + l g ÿd(t) F. Woittennek Flatness of d.p.s Elgersburg 6 / 40

Feedback design Assumption: Reference trajectory t y d (t) for flat output already planned Introduce: Tracking error e(t) = y(t) y d (t) Prescribe: Desired dynamics for tracking error: ë(t) + k 1 ė(t) + k 0 e(t) = 0, k 0, k 1 0 Control law: Solve for ÿ substitute into input-parametrization: D(t) = y(t) + l g ÿ(t) = y(t) + l g (ÿ d(t) k 1 ė(t) k 0 e(t)) F. Woittennek Flatness of d.p.s Elgersburg 7 / 40

Extensions to more general systems Flatness: structural property of ordinary differential equations Extensions: several extensions for models of different structure available (e.g., delay differential systems, more general convolutional systems) Problems: Different model structures require different notions some phenomena can be described by models of different structure (definition of flatness should no depend on it) no rigorous definition for boundary value problems available Alternative: definition on the basis of the trajectories F. Woittennek Flatness of d.p.s Elgersburg 9 / 40

Trajectory definition Willems behavioral approach: associate dynamic system with the set B of its trajectories the behavior Time invariant linear systems: B is a shift invariant (w.r.t. t) vector-space Equivalence of dynamic systems: behaviors are isomorphic Concatenability of trajectories: trajectories τ 1, τ 2 B are concatenable in time T if there exists τ B such that { τ 1 (t), for t < 0 τ(t) = τ 2 (t), for t > T Concatenability of behaviors: trajectories are concatenable behavior is concatenable if any pair of Flatness: B is isomorphic to behavior B concatenable in any time ɛ > 0 (corresponds to controllability of B in time ɛ > 0 à la Willems) F. Woittennek Flatness of d.p.s Elgersburg 10 / 40

String with boundary load Normalized wave equation on [0, 1]: 2 x z 2 (z, t) 2 x (z, t) = 0, z [0, 1] (1a) t2 Left boundary: momentum balance for point mass m x(0, t) = y(t), Right boundary: force control: Behavior: x z (0, t) = m 2 t y(t) x (1, t) = u(t) z (1c) (1b) B = { (x, u, y) H 1 loc([0, l] R) (L 2 loc(r)) 2 is a limit of classical solutions of (1) } F. Woittennek Flatness of d.p.s Elgersburg 13 / 40

Parametrization by boundary trajectories Characteristics: lines in (z, t)-plane with slope +1 resp. 1 Riemann-Invariants: sum and difference of space and time-derivatives r 1 (z, t) = x (z, t) + ẋ(z, t), r 2 (z, t) = x (z, t) ẋ(z, t) are constant along the characteristics Consequences: Riemann-Invariants determined by its boundary values r 1 (z, t z) = x (z, t z) + ẋ(z, t z) = x (0, t) + ẋ(0, t) r 2 (z, t + z) = x (z, t + z) ẋ(z, t + z) = x (0, t) ẋ(0, t) Displacement profile: integration of x = r 1 + r 2 w.r.t. space x(z, t) = 1 2 ( x(0, t + z) + x(0, t z) ) + 1 2 z z x (0, t + σ)dσ Conclusion: Solution of the wave equation determined by boundary values (x(0, ), x (0, )) H 1 loc (R) L2 loc (R) F. Woittennek Flatness of d.p.s Elgersburg 14 / 40

Boundary conditions and flat output Starting point: Parametrization by x(, 0), x (, 0) H 1 loc (R) L2 loc (R) x(z, t) = 1 2 ( ) 1 z x(0, t + z) + x(0, t z) + x (0, t + σ)dσ 2 z Left boundary: boundary gradient and displacement from load position x(0, t) = y(t), x (0, t) = mÿ(t) Displacement profile: parametrization by load position y H 2 loc (R) x(z, t) = 1 2 ( ) m y(t + z) + y(t z) + (ẏ(t + z) ẏ(t z)) 2 Control input: evaluate solution at right boundary u(t) = 1 2 (ẏ(t+1) ẏ(t 1))+ m 2 (ÿ(t+1)+ÿ(t 1)) Conclusion: load position y Hloc 2 (R) is flat output F. Woittennek Flatness of d.p.s Elgersburg 15 / 40

Boundary conditions and flat output Starting point: Parametrization by x(, 0), x (, 0) H 1 loc (R) L2 loc (R) x(z, t) = 1 2 ( ) 1 z y(t + z) + y(t z) + x (0, t + σ)dσ 2 z Left boundary: boundary gradient and displacement from load position x(0, t) = y(t), x (0, t) = mÿ(t) Displacement profile: parametrization by load position y H 2 loc (R) x(z, t) = 1 2 ( ) m y(t + z) + y(t z) + (ẏ(t + z) ẏ(t z)) 2 Control input: evaluate solution at right boundary u(t) = 1 2 (ẏ(t+1) ẏ(t 1))+ m 2 (ÿ(t+1)+ÿ(t 1)) Conclusion: load position y Hloc 2 (R) is flat output F. Woittennek Flatness of d.p.s Elgersburg 15 / 40

State associated with flat parametrization Starting point: equivalent System y Hloc 2 (R) with input u = Uy given by u(t) = Uy = 1 2 (ẏ(t+1) ẏ(t 1))+ m (ÿ(t+1)+ÿ(t 1)). (*) 2 State space (w.r.t. u): portion of system behavior independent of u Ȳ = ker U (does not depended on the representative of equivalence class) State: projection of system behavior to the state space Observation: initial value problem associated with (*) when prescribing y on[ 1, 1]: y(θ) = ȳ 0 (θ), θ [ 1, 1], ȳ 0 H 2 [ 1, 1] Consequence: ȳ = H 2 [ 1, 1] is a state space and the restriction of y to [ 1, 1] is a state F. Woittennek Flatness of d.p.s Elgersburg 17 / 40

Controllability as interpolation problem Starting point: input-state system with input u and state ȳ Controllability problem: given T > 0 and initial and final states ȳ 0, ȳ T y find a control u L 2 loc ([0, T ]) that drives the system from ȳ 0 to ȳ T in time T Solution: compute y H 2 ([ 1, T + 1]) that coincides with ȳ 0 on [ 1, 1] and with ȳ T on [T 1, T + 1] Initial state Transition Final state 1 0 1 T 1 T T + 1 Consequence: exact controllability of the system with input u for T > 2 F. Woittennek Flatness of d.p.s Elgersburg 18 / 40

Transf. to physical coordinates State variables: x t = ( x 1 t, ȳ 1 t, x 2 t, ȳ 2 t ) = (x(, t), y(t), ẋ(, t), ẏ(t)) X State space X = X 1 X 2 X 1 = { ( x 1, ȳ 1 ) H 1 ([0, 1]) R : x 1 (0) = ȳ 1}, Map ϕ : Ȳ X from flat parametrization: x 1 (z) = 1 2 (ȳ(z)+y( z))+ m 2 ( ȳ(z) ȳ( z)) x 2 (z) = 1 2 ( ȳ(z)+ẏ( z))+ m 2 ( ȳ(z) ȳ( z)) ȳ 1 (z) = ȳ(0) ȳ 2 (z) = ȳ(0) Inverse map as solution of inhomogeneous linear o.d.e.: { ȳ + (θ), t > 0 ȳ(θ) = ȳ (θ), t < 0 with X2 = L p ([0, 1]) R ȳ+ (θ) = ȳ 1 + m(1 e θ/m )ȳ 2 + θ 0 (1 e (θ τ)/m )( x 1 (τ) + x 2 (τ))dτ ȳ (θ) = ȳ 1 + m(e θ/m 1)ȳ 2 + θ 0 (e(θ τ)/m 1)( x 1 ( τ) x 2 ( τ))dτ. Result: transformation to physical state x F. Woittennek Flatness of d.p.s Elgersburg 19 / 40

State transformation (illustration) displacement profiles (bottom) correspond with restriction of load trajectory (above) on moving interval (blue) F. Woittennek Flatness of d.p.s Elgersburg 20 / 40

Flatness based control design Starting point: parametrization of input u(t) = 1 2 (ẏ(t+1) ẏ(t 1))+ m 2 (ÿ(t+1)+ÿ(t 1)) Desired dynamics: (mit k 0, k 1 > 0 und α < 1) 0 = ÿ α (t) + k 1 ẏ α (t) + k 0 y α (t), y α (t) = y(t + 1) + αy(t 1) Feedback: solve desired dynamics for ÿ(t + 1) and use in input parametrization u(t) = 1 2 ((1 mk 1)ẏ(t+1) (1+mαk 1 )ẏ(t 1))+ m 2 ((1 α)ÿ(t 1) k 0y α (t)) As state feedback: with ȳ(θ, t) = y(t + θ) Unbounded part in physical coordinates: mȳ ( 1, t) = x (1, t) ẋ(1, t) + ȳ ( 1, t) = u(t) ẋ(1, t) + ȳ ( 1, t) State feedback: unbounded collocated feedback extended by bounded terms u(t) = 1 1 + α ((1 mk 1)ȳ (1, t) α(1+mk 1 )ȳ ( 1, t) mk 0 y α (t)) 1 α t) 1 + αẋ(1, F. Woittennek Flatness of d.p.s Elgersburg 22 / 40

Numerical implementation Starting point: Control law u(t) = 1 1 + α ((1 mk 1)ȳ (1, t) α(1+mk 1 )ȳ ( 1, t) mk 0 y α (t)) 1 α t) 1 + αẋ(1, Approximation of the state: sum of N ansatz functions ϕ N,k x N (t) = N x k (t)ϕ N,k, k=1 x(t) = lim N x N(t) State in flat coordinates: transformation of ansatz functions N ȳ N (t) = x k (t)ψ N,k, ȳ(t) = lim N ȳn (t), ψ N,k = Φ 1 (ϕ N,k ) k=1 Approximated feedback: u(t) = N k=1 x k (t)r k 1 α t) 1 + αẋ(1, with real gains R k = 1 1 + α ((1 mk 1)ψ k(1) α(1 + mk 1 )ψ k( 1) mk 0 (ψ k (1) + αψ k ( 1)) Implementation: simple if pairs ψ N,k = Φ(ϕ N,k ) are known F. Woittennek Flatness of d.p.s Elgersburg 24 / 40

Simulation: Controller Controller parameters: α = 0.0, k 0 = 4, k 1 = 4 Simulation model: modal approximation using 20 modes Feedback law: approximation with 5 eigenfunctions (late lumping) F. Woittennek Flatness of d.p.s Elgersburg 25 / 40

Spatially dependent coefficients Point of depart: hyperbolic p.d.e. a i,j (z) j i w t j z i (z, t) = 0, a 0,2(z) < 0, a 0,2 (z) > 0 i+j 2 coupled with differentially flat system n w(0, t) = l i,1 y (i) (t), w (0, t) = i=0 n l i,2 y (i) (t) i=0 ( ) w Control input: u(t) = r T (L, t) ẇ(l, t) Flat output: flat output y Hloc n (R) of boundary system F. Woittennek Flatness of d.p.s Elgersburg 28 / 40

Gantry: Distributed parameter model Inner force: weight of the remaining rope σ(s) = g(m + sρ) Horizontal part: F (s, t) = σ(s) x s (s, t) Momentum balance: ϱ 2 x F (s, t) = (s, t) t2 s Eliminate force: second order p.d.e. Boundary conditions: ϱ 2 x t 2 (s, t) = s (σ(s) x (s, t)) s Free end: momentum balance for load m 2 x (0, t) = F (0, t) = σ(0) x(0, t) t2 s Top: control via horizontal force u(t) = σ(l) x (l, t) s F. Woittennek Flatness of d.p.s Elgersburg 29 / 40

Flatness based control F. Woittennek Flatness of d.p.s Elgersburg 30 / 40

Simulation F. Woittennek Flatness of d.p.s Elgersburg 31 / 40

Simple network: Coupled strings m 2 x x 2 (z 2, t) m 1 x 1 (z 1, t) 0 Normalized wave equations: z 1 z 2 1 u z zx 2 i (z, t) t 2 x i (z, t) = 0, i = 1, 2, z [0, 1] Left boundaries: Momentum balances for the loads m 1 t 2 x 1 (0, t) = z x 1 (0, t), m 2 t 2 x 2 (0, t) = z x 2 (0, t) Right boundary: strings joined to each other x 1 (1, t) x 2 (1, t) = 0 force control u z x 1 (1, t) + z x 2 (1, t) = u(t) F. Woittennek Flatness of d.p.s Elgersburg 33 / 40

Simple network: Coupled strings m 2 x x 2 (z 2, t) m 1 x 1 (z 1, t) 0 Parametrization by the load positions: z 1 z 2 1 u z x i (z, t) = 1 2 (y i(t+z)+y i (t z))+ m i 2 (ẏ i(t+z) ẏ i (t z)) System of convolution equations for (y 1, y 2, u) H 1 loc (R)2 L 2 loc (R) u(t) = 0 = 2 i=1 2 i=1 1 2 (ẏ i(t+1) ẏ i (t 1))+ m i 2 (ÿ i(t+1)+ÿ i (t 1)) ( 1) i ( 1 2 (y i(t+1)+y i (t 1))+ m i 2 (ẏ i(t+1) ẏ i (t 1)) ) F. Woittennek Flatness of d.p.s Elgersburg 33 / 40

Simple network: Coupled strings m 2 x x 2 (z 2, t) m 1 x 1 (z 1, t) 0 z 2 Flat output (by solving a Bezout equation): 1 y(t) = 1 2 (α(t + 1) + α(t 1)) + 1 2 β(t τ)dτ 1 z 1 α(t) = m 1y 1 (t) m 2 y 2 (t) m 1 m 2, β(t) = y 1(t) y 2 (t) m 1 m 2 belongs to Y = {y H 2 loc (R) : (y( + 1) y( 1)) H3 loc (R)} Observations and consequences: space Y is not exactly concatenable only approximate controllability for the system 1 u z F. Woittennek Flatness of d.p.s Elgersburg 33 / 40

Water-waves in a tube Plant: tube with moving wall (control input) Model: quasi-linear system of hyperbolic p.d.e. for (s, t) (h(s, t), v(s, t)) (Saint-Venant-equations) Mass balance: Momentum balance: t A(h) + ( ) A(h)v = 0 s t (A(h)v) + s (A(h)v2 ) + ga(h) h s = 0 Boundary conditions: v(0, t) = 0, v(u(t), t) = u(t) Flat output: water level h(0, t) at uncontrolled boundary F. Woittennek Flatness of d.p.s Elgersburg 35 / 40

Water-waves: Setup F. Woittennek Flatness of d.p.s Elgersburg 36 / 40

Water-waves: no motion planning Transition without motion planning F. Woittennek Flatness of d.p.s Elgersburg 37 / 40

Water-waves: flatness based Flatness based rest-to-rest transition F. Woittennek Flatness of d.p.s Elgersburg 38 / 40

Summary Flatness: fundamental property of many systems with distributed parameters Useful: flatness based parametrization simplifies controllability analysis, open loop control design, and feedback design Related results and outlook: Observer design Generalization of controller resp. observer canonical forms to hyperbolic b.v.p systematic search for flat outputs parabolic equations F. Woittennek Flatness of d.p.s Elgersburg 40 / 40

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F. Woittennek. On flatness and controllability of simple hyperbolic distributed parameter systems. In Proc. 18th IFAC World Congress, pages 14452 14457, Milano, Italy, 2011. F. Woittennek and H. Mounier. Controllability of networks of spatially one-dimensional second order p.d.e. an algebraic approach. SIAM J. Control Optim., 48(6):3882 3902, 2010. F. Woittennek, M. Riesmeier, and S. Ecklebe. On approximation and implementation of transformation based feedback laws for distributed parameter systems. In Proc. 20. IFAC World Congress 2017, 2017. F. Woittennek, S. Wang, and T. Knüppel. Backstepping design for parabolic systems with in-domain actuation and robin boundary conditions. In Proc. 19. IFAC World Congress 2014, 2014. 5175-5180. F. Woittennek Flatness of d.p.s Elgersburg 40 / 40