Faculty of Electrical and Computer Engineering Institute of Control Theory STRUCTURE MATTERS: Some Notes on High Gain Observer Design for Nonlinear Systems Klaus Röbenack Int. Conf. on Systems, Analysis and Automatic Control 2012
Structure of the Talk 1. Introduction to High-Gain Observer Design 2. Structured Systems 3. Design in Observability Canonical Form 4. Conclusions TU Dresden High Gain Observer Design for Nonlinear Systems slide 2 of 15
1. Introduction to High-Gain Observer Design Nonlinear System (decomposed into linear and nonlinear part) ẋ = Ax + Φ(x, u) y = Cx Lipschitz condition with a Lipschitz constant γ > 0: x, ˆx R n : Φ(x, u) Φ(ˆx, u) γ x ˆx High-Gain observer with constant observer gain L: ˆx = Aˆx + Φ(ˆx, u) + L(y C ˆx) Observation error x = x ˆx is governed by the error dynamics x = (A LC) x + Φ(x, u) Φ(ˆx, u) Earlier results, conservative bounds: Thau 1973, Raghaven/Hedrick 1994 TU Dresden High Gain Observer Design for Nonlinear Systems slide 3 of 15
Full characterization of the convergence Theorem (Rajamani 1998) Let (A, C) observable and γ > 0 be the Lipschitz constant of the nonlinearity. The error dynamics of the high-gain observer is asymptotically stable if L can be chosen such that A LC is stable and min ω 0 σ min(a LC jωi) > γ, (1) where σ min denotes the smallest singular value. Interpretation of condition (1) from H -Theory min σ min(a LC jωi) = ω 0 [ ] 1 sup σ max(a LC jωi) 1 ω 0 = (si (A LC)) 1 1 Remark: This condisiton depends explicitly on the observer gain L! > γ TU Dresden High Gain Observer Design for Nonlinear Systems slide 4 of 15
Motivation of this work Upper bound for the Lipschitz constant γ γ max = sup (si (A LC)) 1 1 L Observation: For general systems we often have (A, C) = γ max < High-gain observer design in normal forms allows arbitrary large Lipschitz constants - Observability canonical form: γ max = (Gauthier, Hammouri 1992,... ) - Observer canonical form: γ max = (Röbenack, Lynch 2007) TU Dresden High Gain Observer Design for Nonlinear Systems slide 5 of 15
2. Structured Systems Structured nonlinearity with appropriate (possibly rectangular) matrices B and D Φ(x, u) = BF (Dx, u) Structured system ẋ = Ax + BF (Dx, u) y = Cx High-gain Observer ˆx = Aˆx + BF (Dˆx, u) + L(y C ˆx) Observation error x = x ˆx is governed by the error dynamics x = (A LC) x + B[F (Dx, u) F (Dˆx, u)] TU Dresden High Gain Observer Design for Nonlinear Systems slide 6 of 15
Convergence of the structured system Lyapunov candidate function with positive de nite P > 0 V ( x) = x T P x Time derivative V = x T [(A LC) T P + P (A LC)] x + 2 x T P B[F (Dx, u) F (Dˆx, u)] The Lipschitz bound and standard inequalities yields V x T [(A LC) T P + P (A LC) + γ 2 P BB T P + D T D] x < 0 Result: We require a positive de nite solution P > 0 of the Riccati inequality (A LC) T P + P (A LC) + γ 2 P BB T P + D T D < 0 TU Dresden High Gain Observer Design for Nonlinear Systems slide 7 of 15
Computation of the observer gain The Riccati inequality has to be solved simultaneously w.r.t. P > 0 and L: (A LC) T P + P (A LC) + γ 2 P BB T P + D T D < 0 Remove bilinear terms P LC, C T L T P with substitution Y = P L: A T P + P A C T Y T Y C + γ 2 P BB T P + D T D < 0 Schur's complement: Solve linear matrix inequality (LMI) in P and Y : P >0 [ A T P P A+C T Y T +Y C D T ] D γp B γb T >0 P I The upper bound γ max can be calculated with a bisection algorithm (if γ max < ). TU Dresden High Gain Observer Design for Nonlinear Systems slide 8 of 15
Computation of the observer gain Error dynamics x = (A LC) x + B[F (Dx, u) F (Dˆx, u) ] }{{} =: F (D x, t) TU Dresden High Gain Observer Design for Nonlinear Systems slide 9 of 15
Computation of the observer gain Error dynamics x = (A LC) x + B[F (Dx, u) F (Dˆx, u) ] }{{} =: F (D x, t) Transfer function of the full linear part (system & observer gain) G(s) = D (si (A LC)) 1 B Observer gain L stabilizes the error dynamics if γ < D(sI (A LC)) 1 B 1 Maximum admissible Lipschitz constant (upper bound) γ max = sup D(sI (A LC)) 1 B 1 L TU Dresden High Gain Observer Design for Nonlinear Systems slide 10 of 15
3. Design in Observability Canonical Form Observability canonical from ẋ 1 = x 2. ẋ n 1 = x n ẋ n = F (x) y = x 1 ẋ = Ax + BF (Dx) y = Cx Linear part (A, C) is in Brunovsky form A = 0 1 0.............. 1 0 0 C = ( 1 0 0 ), Nonlinearities: B = e n and D = I (acts on last equation, depends on all variables) TU Dresden High Gain Observer Design for Nonlinear Systems slide 11 of 15
Nominal design for the linear part Observer gain yields characteristic polynomial L = p n 1. p 0 ρ(s) = det[si (A LC)] = s n + p n 1 s n 1 + + p 1 s + p 0 = (s s 1 ) (s s n) Linear design: Choose coef cients p 0,..., p n 1 such that ρ(s) ist a Hurwitz polynomial, i.e., all roots s 1,..., s n have negative real parts TU Dresden High Gain Observer Design for Nonlinear Systems slide 12 of 15
High-gain design via scaling Observer gain scaled with small ɛ > 0 p n 1 /ɛ L ɛ =. p 0 /ɛ n yields characteristic polynomial (pole scaling to the left) ρ ɛ (s) = s n + p n 1 s n 1 + + p 1 ɛ ɛ n 1 s + p ( 0 ɛ n = s s ) ( 1 s sn ) ɛ ɛ Transfer function of the linear part G ɛ (s) = D (si (A L ɛ C)) 1 B = 1 ρ ɛ (s) 1 s + p n 1 ɛ 1 s 2 + p n 1 ɛ 1 s + p n 2 ɛ 2 Result: Scaling of the poles implies downscaling of the amplitude G ɛ 0 for ɛ 0 = γ max =. TU Dresden High Gain Observer Design for Nonlinear Systems slide 13 of 15
4. Conclusions High-Gain design can be improved using the structure of the nonlinear part Strict bound on the Lipschitz constant can be formulated Staightforward calculation of the observer gain using LMIs Alternative proof for the stabilization in canonical forms TU Dresden High Gain Observer Design for Nonlinear Systems slide 14 of 15
References J. P. Gauthier, H. Hammouri, and S. Othman. A simple observer for nonlinear systems application to bioreactors. IEEE Trans. on Automatic Control, 37(6):875 880, 1992. S. Raghavan and J. K. Hedrick. Observer design for a class of nonlinear systems. Int. J. Control, 59(2):515 528, 1994. R. Rajamani. Observers for Lipschitz nonlinear systems. IEEE Trans. on Automatic Control, 43(3):397 401, 1998. K. Röbenack and A. F. Lynch. High-gain nonlinear observer design using the observer canonical form. IET Control Theory & Applications, 1(6):1574 1579, 2007. F. E. Thau. Observing the state of nonlinear dynamical systems. Int. J. Control, 17(3):471 479, 1973. TU Dresden High Gain Observer Design for Nonlinear Systems slide 15 of 15