High-Gain Observers in Nonlinear Feedback Control. Lecture # 3 Regulation

Transcription

1 High-Gain Observers in Nonlinear Feedback Control Lecture # 3 Regulation High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 1/5

2 Internal Model Principle d r Servo- Stabilizing u y + Compensator Plant Controller Measured Signals Advantages: Zero steady-state error for exogenous signals generated by a known model (constants and sinusoids with known frequencies) Robustness for parameter uncertainty that does not destroy stability Drawback: Degradation of transient response High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 2/5

3 Problem Formulation ẋ = f(x, θ) + g(x, θ)u + β(x, d, θ) y = h(x, θ) + γ(d, θ) where x R n is the state, u R is the control input, y R is the measured output, d R p is a time-varying disturbance input. The functions f, g, β, h and γ are smooth in x and continuous in θ, a vector of unknown constant parameters, which belongs to a compact set Θ R l. β(x,, θ) = and γ(, θ) = The goal is to design an output feedback controller such that the output y(t) is asymptotically regulated to a reference signal y r (t); that is lim [y(t) y r(t)] = t High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 3/5

4 Assumption 1: With d =, the system has relative degree r n, uniformly in θ, and there is a diffeomorphism [ ] η = T(x), (possibly dependent on θ) ξ that transforms the system into the normal form a(η, ξ, θ) k > η = φ(η, ξ, θ) ξ i = ξ i+1, 1 i r 1 ξ r = b(η, ξ, θ) + a(η, ξ, θ)u y = ξ 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 4/5

5 Assumption 2: With d, the foregoing change of variables transforms the system into the form η = φ a (η, ξ 1,..., ξ m, d, θ) ξ i = ξ i+1 + Ψ i (ξ 1,..., ξ i, d, θ), 1 i m 1 ξ i = ξ i+1 + Ψ i (η, ξ 1,..., ξ i, d, θ), m i r 1 ξ r = b(η, ξ, θ) + a(η, ξ, θ)u + Ψ r (η, ξ, d, θ), y = ξ 1 + γ(d, θ) where 1 m r 1. The functions Ψ i vanish at d = When m = 1, the first ξ i -equations are dropped See Marino and Tomei (1995) for geometric conditions under which a system is transformable into this form High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 5/5

6 Assumption 3: Let r be the disturbance relative degree and r = r r D T (t) = [d(t) d ( r) (t)] is bounded and piecewise continuous Y T (t) = [y r (t) y r (r) (t)] is bounded and piecewise continuous lim t [D(t) D(t)] = ; lim t [Y(t) Ȳ(t)] = D(t) and Ȳ(t) are generated by the known exosystem ẇ = S w, [ D Ȳ ] = Γ w S has distinct eigenvalues on the imaginary axis High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 6/5

7 Let d(w) = D 1, ȳ r (w) = Ȳ 1, π 1 = ȳ r γ( d, θ) π i+1 = π i w S w Ψ i (π 1,, π i, d, θ), 1 i m 1 Assumption 4: There exists a unique λ(w, θ) that solves the PDE where λ w S w = Φ a (λ, π 1,, π m, π m+1, d, θ) π m+1 = π m w S w Ψ m (λ, π 1,, π m, d, θ) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 7/5

8 Let π i+1 = π i w S w Ψ i (λ, π 1,, π i, d, θ), m+1 i r 1 π T = [π 1 π r ] The steady-state value of the control u on the zero-error manifold {η = λ(w, θ), ξ = π(w, θ)} is given by [ ] 1 πr χ(w, θ) = a(λ, π, θ) w S w b(λ, π, θ) Ψ r (λ, π, d, θ) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 8/5

9 Assumption 4: There exist real numbers c,, c q 1, independent of θ, such that χ(w, θ) satisfies ( ) χ L q s χ = c χ+c 1 L s χ+ +c q 1 L q 1 s χ, L s χ = w S w and the polynomial p q c q 1 p q 1 c has distinct roots on the imaginary axis Remark: Assumption 4 allows χ(w, θ) to be generated by a linear internal model independent of θ. It reflects the fact that for nonlinear systems the controller must be able to reproduce not only the sinusoidal signals generated by the exosystem, but also higher-order harmonics induced by nonlinearities High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 9/5

10 Linear Internal Model: Let 1 1 S =.. 1 c c q 1, τ = χ L s χ. Ls q 2 χ χ L q 1 s Γ = [1 ] 1 q χ(w, θ) is generated by the internal model τ(w, θ) w S w = Sτ(w, θ), χ(w, θ) = Γτ(w, θ) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 1/5

11 Example: ẋ 1 = x 2, ẋ 2 = 2θ(x 1 x 3 1 ) + u, y = x 1 y r (t) = α sin βt α and θ are unknown but β is known. Regulate (y y r ) to zero [ ] [ ] β ẇ = w, w() = β α y r = w 1 χ(w) = (2θ β 2 )w 1 2θw 3 1 The internal model should generate the first and third harmonics High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 11/5

12 χ 4 + 1β 2 χ 2 + 9β 4 χ = Roots are ±jβ, ±3jβ S = β 4 1β 2 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 12/5

13 Change of Variables z = η λ(w, θ), e i = y (i 1) y r (i 1), 1 i r ż = φ (z, e, ν, w, θ) ė i = e i+1, 1 i r 1 ė r = b (z, e, ν, w, θ) + a (z, e, ν, w, θ)u, y m = e 1 where ν T (t) = [D T (t) D T (t), Y T (t) Ȳ T (t)] φ (,,, w, θ) = a (,,, w, θ) = a(λ(w, θ), π(w, θ), θ) b (,,, w, θ) = χ(w, θ) a(λ(w, θ), π(w, θ), θ) lim t ν(t) =. The zero-error manifold is {z =, e = } High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 13/5

14 Minimum Phase Assumptions Assumption 5 [ISS]: There exist a proper function V z (z, w, θ) and class K functions α i (i = 1, 2, 3) and δ, independent of (w, θ), such that α 1 ( z ) V z (z, w, θ) α 2 ( z ), V z z φ (z, e, ν, w, θ) + V z w S w α 3 ( z ) for all z δ( (e T, ν T ) ) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 14/5

15 Assumption 6 [Exponential Stability]: There exists a Lyapunov function V zz (z, w, θ), defined in some neighborhood of z =, and positive constants λ 1 to λ 4, independent of (w, θ), such that λ 1 z 2 V zz (z, w, θ) λ 2 z 2 V zz z φ (z,,, w, θ) + V zz w S w λ 3 z 2 V zz z λ 4 z High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 15/5

16 Controller Design Partial State Feedback: Augment the plant with the servocompensator [ ] σ = Sσ + Je 1, J T =... 1 (S, J) is controllable s = K 1 σ + r 1 i=1 k i e i + e r def = K 1 σ + K 2 e 1 e 2. r r 1 + e r High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 16/5

17 Choose K 1 and K 2 such that the matrix [ S JC B K 1 (A B K 2 ) ] is Hurwitz; the triple (A, B, C ) is a canonical form representation of a chain of r integrators. This is possible because the pair is controllable ([ S JC A ], [ B ]) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 17/5

18 ( ) s u = β sat µ ṡ = ( ) βa ( )sat(s/µ) β ( ) a ( ) + β, β > The continuously-implemented sliding mode control ensures that, in finite time, the trajectory will be in a neighborhood Ω µ of {σ =, z =, e = }, where the size of Ω µ shrinks as µ decreases High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 18/5

19 Inside Ω µ it can be shown that, when ν =, there is an exponentially stable invariant manifold M µ = {σ = (µ/β)mτ(w, θ), z =, e = } where SM = MS and K 1 M = Γ. Moreover, the trajectory approaches M µ as t tends to infinity; hence lim e(t) = t The combination of continuously-implemented sliding mode control with servocompensator enables us to achieve Nonlocal (even semiglobal) results Regulation (e(t) ) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 19/5

20 In output feedback, e 2 to e r are substituted by their estimates ê 2 to ê r, provided by the high-gain observer ê i = ê i+1 + (α i /ε i )(e 1 ê 1 ), 1 i r 1 ê r = (α r /ε r )(e 1 ê 1 ) In this case, the zero-error manifold is given by {σ = (µ/β)mτ(w, θ), z =, e =, η = } where η is the scaled estimation error Remark: This controller achieves asymptotic regulation, but usually at the expense of the transient response which is degraded due to the inclusion of the servocompensator High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 2/5

21 Conditional Servocompensator ( ) s σ = (S JK 1 )σ + µj sat µ Choose K 1 such that (S JK 1 ) is Hurwitz s = K 1 σ + Choose k 1 to k r 1 such that r 1 i=1 k i e i + e r is Hurwitz λ r 1 + k r 1 λ r k 2 λ + k 1 ( ) s u = β sat µ High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 21/5

22 Within finite time, the trajectory enters the boundary layer { s µ}; then µ sat(s/µ) = s σ = (S JK 1 )σ + J σ = Sσ + J ( ( r 1 K 1 σ + i=1 r 1 i=1 k i e i + e r ) At steady state, e = k i e i + e r ) σ() = O(µ) σ(t) = O(µ), t High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 22/5

23 High-Gain Observers ê i = ê i+1 + (α i /ε i )(e 1 ê 1 ), 1 i r 1 ê r = (α r /ε r )(e 1 ê 1 ) ε > (small) (λ r + α 1 λ r α r 1 λ + α r ) is Hurwitz s = K 1 σ + k 1 e 1 + r 1 i=2 k i ê i + ê r High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 23/5

24 Performance Recovery for sufficiently small µ and ε, all variables are bounded and lim t e(t) = Let (z, e ) be the state of the closed-loop system under (ideal) sliding mode state feedback control and (z, e) be the corresponding state under output feedback control, with z() = z () and e() = e (). Then, for every δ >, z(t) z (t) δ and e(t) e (t) δ, t for sufficiently small µ and ε High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 24/5

25 Example: Magnetic Levitation Controller m Light source High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 25/5

26 Normalized Model: ẋ 1 = x 2 ẋ 2 = c x g(x 1 ) + g(x 1 )u g(x 1 ) =.2 m ( x 1 ) 2.1 m u 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 26/5

27 Sliding Mode Control: s = k 1 (x 1 x 1d ) + x 2, k 1 = 1 sṡ = s[(k 1 c)x g(x 1 )] + sg(x 1 )u u = { if s > 1 if s < For sṡ < we need (k 1 c)x g(x 1 ) g(x 1 ) < 4.265, for s > (k 1 c)x 2 < 1, for s < High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 27/5

28 x Ideal Sliding Mode Control t s t 2 u 2 s = (x 1.5)+x t High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 28/5

29 Continuous Implementation: φ(y) = Boundary Layer: u = φ ( ) µ s for y > y for 1 y for y < 1.35µ s 1.65µ High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 29/5

30 x Continuous Sliding Mode Control t s t 2 u 2 µ =.5 4 inin t High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 3/5

31 Integral action: ẋ = x 1 x 1d ẋ 1 = x 2 ẋ 2 = c x g(x 1 ) + g(x 1 )u φ(y) = s = k x + k 1 (x 1 x 1d ) + x 2 ( ) u = φ 2µ s for y > y for 1 y for y < 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 31/5

32 .7.6 Continuous SMC with Integrator.5.4 Ideal SMC x t High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 32/5

33 Conditional Integrator: σ = σ + 2µ φ Inside the Boundary Layer: ( ) 5.265s 2µ s = σ + k 1 (x 1 x 1d ) + x 2 ( ) 5.265s u = φ 2µ σ = σ + σ + k 1 (x 1 x 1d ) + x 2 = k 1 (x 1 x 1d ) + x 2 u = 5.265s 2µ High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 33/5

34 x Ideal Continuous with Integrator Continuous with Conditional Integrator t High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 34/5

35 State Feedback.5.4 x Ideal SMC µ =.5 µ = t High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 35/5

36 .6 Output Feedback.5.4 x State Feedback µ =.5 Output Feedback ε =.1 Output Feedback ε = t Deviation from State Feedback.5 ε = ε = t High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 36/5

37 Connection Between the Conditional Integrator and Antiwindup ( ) γσ + s u = β sat µ ( ) γσ + s σ = γσ + µ sat µ + Ref KH(s, ε) Output K = β/µ + γ s + + β β + Control High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 37/5

38 Ref + KH(s, ε) Output K = β/µ H = 1 for r = 1 H = k 1 + H = k γ s + + β β + s (εs) 2 /α 2 + (εs)α 1 /α 2 + 1, for r = 2 Control k 2 s + (1 + εk 2 α 2 /α 3 )s 2 (εs) 3 /α 3 + (εs) 2 α 2 /α 3 + (εs)α 1 /α 3 + 1, r = 3 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 38/5

39 + Ref KH(s, ε) Output K = β/µ + γ s + + β β + Control Ref + KH(s, ε) s+γ s Output Control High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 39/5

40 Internal Model Perturbation χ(w, θ) is the steady-state control needed to achieve zero steady-state error χ(w, θ) is the function generated by the internal model Sources of Error: χ(w, θ) χ(w, θ) Uncertainty in the frequencies of the internal model Truncation of the frequencies represented by the internal model χ(w, θ) = χ(w, θ) χ(w, θ) χ(w, θ)a (,,, w, θ) δ (w, θ) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 4/5

41 Approximation Result Theorem: There exists µ > and for each µ (, µ ), there exist ε = ε (µ) >, δ = δ (µ) > and c > such that for each < δ δ, < µ µ and < ε ε, the regulation error e 1 is ultimately bounded by cµδ High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 41/5

42 Proof ẋ 1 = f 1 ( ) + MNx 3 µẋ 2 = a( )(x 2 Nx 3 ) + µf 2 ( ) + µφ(y) εẋ 3 = Ax 3 + B 2 [εf 3 ( ) ε ] µ a( )(x 2 Nx 3 ) + εφ(y) ẏ = By ( ) = (x 1, x 2, y), φ δ, A Hurwitz, B neutrally stable With φ(y) = {x 1 = h 1 (y), x 2 = h 2 (y), x 3 = } is an invariant manifold where e 1 = z 1 = x 1 h 1 (y), z 2 = x 2 h 2 (y), z 3 = x 3 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 42/5

43 ż 1 = F 1 ( ) + MNz 3 µż 2 = α( )(z 2 Nz 3 ) + µf 2 ( ) + µφ(y) εż 3 = Az 3 + B 2 [εf 3 ( ) ε ] µ α( )B 2(z 2 Nz 3 ) + εφ(y) c 1 z 1 V 1 (z 1, y) c 2 z 1 V 1 z 1 F 1 (z 1,, y) + V 1 y B 1y c 3 V 1 V 3 = z T 3 Pz 3, V 2 = z 2 2 PA + A T P = I High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 43/5

44 V = V 1 V 2 V 3 V c 3 k 1 k 2 k 4 c µ + k 5 k 3 µ k 6 k 7 + k 8 µ c 4 ε + k 9 µ V + δ k 1 δ V ΨV + Υ Comparison Lemma: V U U = ΨU + Υ, U() = V () Singular Perturbation Analysis: U = O(µδ) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 44/5

45 Simulation ẋ 1 = x 2, ẋ 2 = sin(x 1 ) + 3u, y = x 1 y y r = α sin(ωt) ẇ 1 = Ωw 2, ẇ 2 = Ωw 1 [ ] w() =, y r (t) = w 1 α χ = ( Ω 2 w 1 + sin(w 1 ))/3 sin(w 1 ) p q (w 1 ) = q i=1 ( 1) i 1 w 2i 1 1 (2i 1)! χ = ( Ω 2 w 1 + p q (w 1 ))/3 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 45/5

46 Tracking error (transient) Tracking error (steady state) Time(sec) 1.5 x q=4,mu= q=4,mu=.5 q=4,mu= q=4,mu=.1,.5, Time(sec) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 46/5

47 Function value 1 1 sin(w 1 ) p 2 (w 1 ) p 4 (w 1 ) p 6 (w 1 ) Tracking error (steady state) Tracking error (steady state) 2 x q=2 vs. q=4 1 q=4,mu=.1 q=2,mu= x Time(sec) q=4 vs. q=6 q=4,mu=.1 q=6,mu= Time(sec) w 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 47/5

48 Tracking error (transient).5 1 q=4,mu=.1,.5,.25 Tracking error (steady state) Time(sec) 5 x 1 5 q=4,mu=.5 q=4,mu=.1 q=4,mu= Time(sec) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 48/5

49 Adaptive Internal Models ẋ 1 = x 2, ẋ 2 = 2(x 1 x 3 1 ) + u, y = x 1 r(t) = α sin βt α and β are unknown. Regulate (y y r ) to zero [ ] [ ] β ẇ = w, w() = β α y r = w 1 χ = (2 β 2 )w 1 2w 3 1 The internal model should generate the first and third harmonics High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 49/5

50 S = β 4 1β 2 S = β 4 1β 2, J = 1 Choose λ to assign the eigenvalues of (S Jλ T ) at.5, 1, 1.5, 2 λ 1 = 1.5 9β 4, λ 2 = 6.25, λ 3 = β 2, λ 4 = 5 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 5/5

51 If β was known s = 4 i=1 λ i σ i + x 1 y r }{{} e 1 σ = (S Jλ T )σ + µj sat u = k sat ( s µ + x 2 ẏ r }{{} e 2 ) ( s µ ) High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 51/5

52 When β is unknown s = ˆλ 1 σ 1 + λ 2 σ 2 + ˆλ 3 σ 3 + λ 4 σ 4 + e 1 + e 2 Adaptive law: ˆλ i = Proj {γ(e 1 + e 2 )σ i }, i = 1, 3, γ > High-Gain Observer: e 2 is replaced by ê 2 (saturated) Simulation: α = 1, µ =.1, ε = 1 3, γ = 1 4 x 1 () = 1, x 2 () =, ˆλ 1 () = ˆλ 3 () =, σ() = β = 1, 1 2, 1 2, 1 ˆλ i 1 High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 52/5

53 1.8.6 β = 1 3 x x 1 r.4 x 1 r Time Time 4 15 Estimate of λ Time Estimate of λ Time High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 53/5

54 1.8.6 β =.5 1 x x 1 r.4 x 1 r Time Time 2 1 Estimate of λ Estimate of λ Time Time High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 54/5

55 For β =.5 and β = 1, a PE condition is satisfied For β = 1/ 2 the PE condition is not satisfied χ = (2 β 2 )α sin βt 2α 3 sin 3 βt χ = [ (2 β 2 )α 1.5α 3] sin βt +.5α 3 sin 3βt α = 1 & β = 1/ 2 χ =.5 sin 3βt High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 55/5

56 1.8.6 β = x 1 8 x 1 r.4 x 1 r Time Time 4 8 Estimate of λ Time Estimate of λ Time High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 56/5

57 .8 β = 1 β =.77 β =.5.6 x 1 r Time High-Gain ObserversinNonlinear Feedback ControlLecture # 3Regulation p. 57/5