Nonlinear Control Lecture # 14 Tracking & Regulation Nonlinear - - PowerPoint PPT Presentation

Nonlinear Control Lecture # 14 Tracking & Regulation

Nonlinear Control Lecture # 14 Tracking & Regulation

Normal form: ˙ η = f0(η, ξ) ˙ ξi = ξi+1, for 1 ≤ i ≤ ρ − 1 ˙ ξρ = a(η, ξ) + b(η, ξ)u y = ξ1 η ∈ Dη ⊂ Rn−ρ, ξ = col(ξ1, . . . , ξρ) ∈ Dξ ⊂ Rρ Tracking Problem: Design a feedback controller such that lim

t→∞[y(t) − r(t)] = 0

while ensuring boundedness of all state variables Regulation Problem: r is constant

Nonlinear Control Lecture # 14 Tracking & Regulation

Assumption 13.1 b(η, ξ) ≥ b0 > 0, ∀ η ∈ Dη, ξ ∈ Dξ Assumption 13.2 ˙ η = f0(η, ξ) is bounded-input–bounded-state stable over Dη × Dξ Assumption 13.2 holds locally if the system is minimum phase and globally if ˙ η = f0(η, ξ) is ISS Assumption 13.3 r(t) and its derivatives up to r(ρ)(t) are bounded for all t ≥ 0 and the ρth derivative r(ρ)(t) is a piecewise continuous function of t. Moreover, R = col(r, ˙ r, . . . , r(ρ−1)) ∈ Dξ for all t ≥ 0

Nonlinear Control Lecture # 14 Tracking & Regulation

The reference signal r(t) could be specified as given functions

• f time, or it could be the output of a reference model

Example: For ρ = 2 ω2

n

s2 + 2ζωns + ω2

n

, ζ > 0, ωn > 0 ˙ y1 = y2, ˙ y2 = −ω2

ny1 − 2ζωny2 + ω2 nuc,

r = y1 ˙ r = y2, ¨ r = ˙ y2 Assumption 13.3 is satisfied when uc(t) is piecewise continuous and bounded

Nonlinear Control Lecture # 14 Tracking & Regulation

Change of variables: e1 = ξ1 − r, e2 = ξ2 − r(1), . . . , eρ = ξρ − r(ρ−1) ˙ η = f0(η, ξ) ˙ ei = ei+1, for 1 ≤ i ≤ ρ − 1 ˙ eρ = a(η, ξ) + b(η, ξ)u − r(ρ) Goal: Ensure e = col(e1, . . . , eρ) = ξ − R is bounded for all t ≥ 0 and converges to zero as t tends to infinity Assumption 13.4 r, r(1), . . . , r(ρ) are available to the controller (needed in state feedback control)

Nonlinear Control Lecture # 14 Tracking & Regulation

Feedback controllers for tracking and regulation are classified as in stabilization State versus output feedback Static versus dynamic controllers Region of validity

local tracking regional tracking semiglobal tracking global tracking

Local tracking is achieved for sufficiently small initial states and sufficiently small R, while global tracking is achieved for any initial state and any bounded R.

Nonlinear Control Lecture # 14 Tracking & Regulation

Practical tracking: The tracking error is ultimately bounded and the ultimate bound can be made arbitrarily small by choice of design parameters local practical tracking regional practical tracking semiglobal practical tracking global practical tracking

Nonlinear Control Lecture # 14 Tracking & Regulation

Tracking

˙ η = f0(η, ξ), ˙ e = Ace + Bc

• a(η, ξ) + b(η, ξ)u − r(ρ)

Feedback linearization: u =

• −a(η, ξ) + r(ρ) + v
• /b(η, ξ)

˙ η = f0(η, ξ), ˙ e = Ace + Bcv v = −Ke, Ac − BcK is Hurwitz ˙ η = f0(η, ξ), ˙ e = (Ac − BcK)e Ac − BcK Hurwitz ⇒ e(t) is bounded and limt→∞ e(t) = 0 ⇒ ξ = e + R is bounded ⇒ η is bounded

Nonlinear Control Lecture # 14 Tracking & Regulation

Example 13.1 (Pendulum equation) ˙ x1 = x2, ˙ x2 = − sin x1 − bx2 + cu, y = x1 We want the output y to track a reference signal r(t) e1 = x1 − r, e2 = x2 − ˙ r ˙ e1 = e2, ˙ e2 = − sin x1 − bx2 + cu − ¨ r u = 1 c[sin x1 + bx2 + ¨ r − k1e1 − k2e2] K = [k1, k2] assigns the eigenvalues of Ac − BcK at desired locations in the open left-half complex plane

Nonlinear Control Lecture # 14 Tracking & Regulation

Simulation r = sin(t/3), x(0) = col(π/2, 0) Nominal: b = 0.03, c = 1 Figures (a) and (b) Perturbed: b = 0.015, c = 0.5 Figure (c) Reference (dashed) Low gain: K =

• 1

1

• , λ = −0.5 ± j0.5

√ 3, (solid) High gain: K = 9 3 , λ = −1.5 ± j1.5 √ 3, (dash-dot)

Nonlinear Control Lecture # 14 Tracking & Regulation

2 4 6 8 10 −0.5 0.5 1 1.5 2

Time Output (a)

2 4 6 8 10 −0.5 0.5 1 1.5 2

Time Output (b)

2 4 6 8 10 −0.5 0.5 1 1.5 2

Time Output (c)

2 4 6 8 10 −10 −5 5

Time Control (d) Nonlinear Control Lecture # 14 Tracking & Regulation

Robust Tracking

˙ η = f0(η, ξ) ˙ ei = ei+1, 1 ≤ i ≤ ρ − 1 ˙ eρ = a(η, ξ) + b(η, ξ)u + δ(t, η, ξ, u) − r(ρ)(t) Sliding mode control: Design the sliding surface ˙ ei = ei+1, 1 ≤ i ≤ ρ − 1 View eρ as the control input and design it to stabilize the

• rigin

eρ = −(k1e1 + · · · + kρ−1eρ−1) λρ−1 + kρ−1λρ−2 + · · · + k1 is Hurwitz

Nonlinear Control Lecture # 14 Tracking & Regulation

s = (k1e1 + · · · + kρ−1eρ−1) + eρ = 0 ˙ s =

ρ−1

• i=1

kiei+1 + a(η, ξ) + b(η, ξ)u + δ(t, η, ξ, u) − r(ρ)(t) u = v or u = − 1 ˆ b(η, ξ) ρ−1

• i=1

kiei+1 + ˆ a(η, ξ) − r(ρ)(t)

• + v

˙ s = b(η, ξ)v + ∆(t, η, ξ, v) Suppose

• ∆(t, η, ξ, v)

b(η, ξ)

• ≤ ̺(η, ξ) + κ0|v|,

0 ≤ κ0 < 1 v = −β(η, ξ) sat s µ

• ,

β(η, ξ) ≥ ̺(η, ξ) (1 − κ0) + β0, β0 > 0

Nonlinear Control Lecture # 14 Tracking & Regulation

s ˙ s ≤ −β0b0(1 − κ0)|s|, |s| ≥ µ ζ = col(e1, . . . , eρ−1), ˙ ζ = (Ac − BcK)

• Hurwitz

ζ + Bcs V0 = ζTPζ, P(Ac − BcK) + (Ac − BcK)TP = −I ˙ V0 = −ζTζ+2ζTPBcs ≤ −(1−θ)ζ2, ∀ ζ ≥ 2PBc |s|/θ 0 < θ < 1. For σ ≥ µ {ζ ≤ 2PBc σ/θ} ⊂ {ζTPζ ≤ λmax(P)(2PBc/θ)2σ2} ρ1 = λmax(P)(2PBc/θ)2, c > µ Ω = {ζTPζ ≤ ρ1c2} × {|s| ≤ c} is positively invariant

Nonlinear Control Lecture # 14 Tracking & Regulation

For all e(0) ∈ Ω, e(t) enters the positively invariant set Ωµ = {ζTPζ ≤ ρ1µ2} × {|s| ≤ µ} Inside Ωµ, |e1| ≤ kµ k = LP −1/2√ρ1, L = 1 . . .

Nonlinear Control Lecture # 14 Tracking & Regulation

Example 13.2 (Reconsider Example 13.1) ˙ e1 = e2, ˙ e2 = − sin x1 − bx2 + cu − ¨ r r(t) = sin(t/3), 0 ≤ b ≤ 0.1, 0.5 ≤ c ≤ 2 s = e1 + e2 ˙ s = e2 − sin x1 − bx2 + cu − ¨ r = (1 − b)e2 − sin x1 − b˙ r − ¨ r

• (1 − b)e2 − sin x1 − b˙

r − ¨ r c

• ≤ |e2| + 1 + 0.1/3 + 1/9

0.5 u = −(2|e2| + 3) sat e1 + e2 µ

• Nonlinear Control Lecture # 14 Tracking & Regulation

Simulation: µ = 0.1, x(0) = col(π/2, 0) b = 0.03, c = 1 (solid) b = 0.015, c = 0.5 (dash-dot) Reference (dashed)

Nonlinear Control Lecture # 14 Tracking & Regulation

2 4 6 8 10 −0.5 0.5 1 1.5 2

Time Output (a)

0.2 0.4 0.6 0.8 1 −0.2 0.2 0.4 0.6 0.8 1 1.2

Time s (b)

Nonlinear Control Lecture # 14 Tracking & Regulation

Robust Regulation via Integral Action

˙ η = f0(η, ξ, w) ˙ ξi = ξi+1, 1 ≤ i ≤ ρ − 1 ˙ ξρ = a(η, ξ, w) + b(η, ξ, w)u y = ξ1 Disturbance w and reference r are constant Equilibrium point: = f0(¯ η, ¯ ξ, w) = ¯ ξi+1, 1 ≤ i ≤ ρ − 1 = a(¯ η, ¯ ξ, w) + b(¯ η, ¯ ξ, w)¯ u r = ¯ ξ1

Nonlinear Control Lecture # 14 Tracking & Regulation

Assumption 13.5 0 = f0(¯ η, ¯ ξ, w) has a unique solution ¯ η = φη(r, w) ¯ u = − a(¯ η, ¯ ξ, w) b(¯ η, ¯ ξ, w)

def

= φu(r, w) Augment the integrator ˙ e0 = y − r z = η − ¯ η, e =      e1 e2 . . . eρ      =      ξ1 − r ξ2 . . . ξρ     

Nonlinear Control Lecture # 14 Tracking & Regulation

˙ z = f0(z + ¯ η, ξ, w)

def

= ˜ f0(z, e, r, w) ˙ ei = ei+1, for 0 ≤ i ≤ ρ − 1 ˙ eρ = a(η, ξ, w) + b(η, ξ, w)u Sliding mode control: s = k0e0 + k1e1 + · · · + kρ−1eρ−1 + eρ λρ + kρ−1λρ−1 + · · · + k1λ + k0 is Hurwitz ˙ s =

ρ−1

• i=0

kiei+1 + a(η, ξ, w) + b(η, ξ, w)u u = v

• r

u = − 1 ˆ b(η, ξ) ρ−1

• i=0

kiei+1 + ˆ a(η, ξ)

• + v

Nonlinear Control Lecture # 14 Tracking & Regulation

˙ s = b(η, ξ, w)v + ∆(η, ξ, r, w)

• ∆(η, ξ, r, w)

b(η, ξ, w)

• ≤ ̺(η, ξ)

v = −β(η, ξ) sat s µ

• ,

β(η, ξ) ≥ ̺(η, ξ) + β0, β0 > 0 Assumption 13.6 α1(z) ≤ V1(z, r, w) ≤ α2(z) ∂V1 ∂z ˜ f0(z, e, r, w) ≤ −α3(z), ∀ z ≥ α4(e) Assumption 13.7 z = 0 is an exponentially stable equilibrium point of ˙ z = ˜ f0(z, 0, r, w)

Nonlinear Control Lecture # 14 Tracking & Regulation

Theorem 13.1 Under the stated assumptions, there are positive constants c, ρ1 and ρ2 and a positive definite matrix P such that the set Ω = {V1(z) ≤ α2(α4(cρ2)} × {ζTPζ ≤ ρ1c2} × {|s| ≤ c} where ζ = col(e0, e1, . . . , eρ−1), is compact and positively invariant, and for all initial states in Ω lim

t→∞ |y(t) − r| = 0

Special case: β = k (a constant) and u = v u = −k sat k0e0 + k1e1 + · · · + kρ−1eρ−1 + eρ µ

• Nonlinear Control Lecture # 14 Tracking & Regulation

Example 13.4 (Pendulum with horizontal acceleration) ˙ x1 = x2, ˙ x2 = − sin x1 − bx2 + cu + d cos x1, y = x1 d is constant. Regulate y to a constant reference r 0 ≤ b ≤ 0.1, 0.5 ≤ c ≤ 2, 0 ≤ d ≤ 0.5 e1 = x1 − r, e2 = x2 ˙ e0 = e1, ˙ e1 = e2, ˙ e2 = − sin x1 − bx2 + cu + d cos x1 s = e0 + 2e1 + e2 ˙ s = e1 + (2 − b)e2 − sin x1 + cu + d cos x1

Nonlinear Control Lecture # 14 Tracking & Regulation

• e1 + (2 − b)e2 − sin x1 + d cos x1

c

• ≤ |e1| + 2|e2| + 1 + 0.5

0.5 u = −(2|e1| + 4|e2| + 4) sat e0 + 2e1 + e2 µ

• For comparison, SMC without integrator

s = e1 + e2, ˙ s = (1 − b)e2 − sin x1 + cu + d cos x1 u = −(2|e2| + 4) sat e1 + e2 µ

• Simulation: With integrator (dashed), without (solid)

µ = 0.1, x(0) = 0, r = π/2, b = 0.03, c = 1, d = 0.3

Nonlinear Control Lecture # 14 Tracking & Regulation

2 4 6 8 10 0.5 1 1.5 2

Time Output

9 9.2 9.4 9.6 9.8 10 1.5 1.52 1.54 1.56 1.58 1.6

Time Output

Nonlinear Control Lecture # 14 Tracking & Regulation

Output Feedback

Tracking: ˙ η = f0(η, ξ) ˙ ei = ei+1, 1 ≤ i ≤ ρ − 1 ˙ eρ = a(η, ξ) + b(η, ξ)u + δ(t, η, ξ, u) − r(ρ)(t) Regulation: ˙ η = f0(η, ξ, w) ˙ ξi = ξi+1, 1 ≤ i ≤ ρ − 1 ˙ ξρ = a(η, ξ, w) + b(η, ξ, w)u y = ξ1 Design partial state feedback control that uses ξ Use a high-gain observer

Nonlinear Control Lecture # 14 Tracking & Regulation

Tracking sliding mode controller: u = −β(ξ) sat k1e1 + · · · + kρ−1eρ−1 + eρ µ

• Regulation sliding mode controller:

u = −β(ξ) sat k0e0 + k1e1 + · · · + kρ−1eρ−1 + eρ µ

• ˙

e0 = e1 = y − r β is allowed to depend only on ξ rather than the full state

• vector. On compact sets, the η-dependent part of ̺(η, ξ) can

be bounded by a constant

Nonlinear Control Lecture # 14 Tracking & Regulation

High-gain observer: ˙ ˆ ei = ˆ ei+1 + αi εi (y − r − ˆ e1), 1 ≤ i ≤ ρ − 1 ˙ ˆ eρ = αρ ερ (y − r − ˆ e1) λρ + α1λρ−1 + · · · + αρ−1λ + αρ Hurwitz e → ˆ e ξ → ˆ ξ = ˆ e + R β(ˆ ξ) → βs(ˆ ξ) (saturated)

Nonlinear Control Lecture # 14 Tracking & Regulation

Tracking: u = −βs(ˆ ξ) sat k1ˆ e1 + · · · + kρ−1ˆ eρ−1 + ˆ eρ µ

• Regulation:

u = −βs(ˆ ξ) sat k0e0 + k1ˆ e1 + · · · + kρ−1ˆ eρ−1 + ˆ eρ µ

• We can replace ˆ

e1 by e1 Special case: When βs is constant or function of ˆ e rather than ˆ ξ, we do not need the derivatives of r, as required by Assumption 13.4

Nonlinear Control Lecture # 14 Tracking & Regulation

The output feedback controllers recover the performance of the partial state feedback controllers for sufficiently small ε. In the regulation case, the regulation error converges to zero Relative degree one systems: No observer u = −β(y) sat y − r µ

• ,

u = −β(y) sat k0e0 + y − r µ

• Nonlinear Control Lecture # 14 Tracking & Regulation