Input Delay Compensation X ( t ) = AX ( t )+ BU ( t D ) Assume: ( - - PDF document

Input Delay Compensation

˙ X(t) = AX(t)+BU(t −D)

Assume: (A,B) controllable and matrix K found such that A+BK is Hurwitz. Predictor-based control law:

U(t) = K

• eADX(t)+

Z t

t−DeA(t−θ)BU(θ)dθ

• Compensates the delay but what about:

stability? disturbance attenuation? parametric robustness? robustness to dynamic uncertainties? Need a Lyapunov functional. Construct one with a backstepping transformation of the actuator state:

W(θ) = U(θ)−K Z θ

t−DeA(θ−σ)BU(σ)dσ+eA(θ+D−t)X(t)

• V(t) = X(t)TPX(t)+2 |PB|2

λmin(Q)

Z t

t−D(1+θ+D−t)W(θ)2dθ

Is there any benefit to having a Lyapunov function besides proving stability?

Inverse optimality and robustness to actuator lag. Theorem 1 There exists c∗ such that the feedback system with the controller

U(t) = c s+c

• K
• eADX(t)+

Z t

t−DeA(t−θ)BU(θ)dθ

• ,

is exponentially stable in the sense of the norm

N(t) =

• |X(t)|2 +

Z t

t−DU(θ)2dθ+U(t)2

1/2

for all c > c∗. Furthermore, there exists c∗∗ > c∗ such that, for any c ≥ c∗∗, the feedback minimizes the cost functional

J =

Z ∞

• Q(t)+ ˙

U(t)2 dt ,

where Q(t) ≥ µN(t)2 for some µ(c) > 0, which is such that µ(c) → ∞ as c → ∞. With a Lyapunov function, one can even quantify disturbance attenuation

˙ X(t) = AX(t)+BU(t −D)+Gd(t)

Theorem 2 ∃c∗ s.t. ∀c > c∗, the feedback system is L∞-stable, i.e., ∃ positive constants

β1,β2,γ1 s.t. N(t) ≤ β1e−β2tN(0)+γ1 sup

τ∈[0,t]

|d(τ)|.

Furthermore, ∃c∗∗ > c∗ s.t. ∀c ≥ c∗∗ the feedback minimizes the cost functional

J = sup

d∈D

lim

t→∞

• 2cV(t)+

Z t

• Q(τ)+ ˙

U(t)2 −cγ2d(τ)2 dτ

• for any

γ2 ≥ γ∗∗

2 = 8λmax(PBBTP)

λmin(Q) ,

where Q(t) ≥ µN(t)2 for some µ(c,γ2) > 0, which is such that µ(c,γ2) → ∞ as c → ∞, and D is the set of linear scalar-valued functions of X.

Robustness to Delay Mismatch

The biggest open question in robustness of predictor feedbacks.

˙ X = AX +BU(t −D0 −ΔD) U(t) = K

• eAD0X(t)+

Z t

t−D0

eA(t−θ)BU(θ)dθ

• ΔD either positive or negative

Theorem 3 ∃δ > 0 s.t. ∀ΔD ∈ (−δ,δ) the closed-loop system is exp. stable in the sense

• f the state norm

N2(t) =

• |X(t)|2 +

Z t

t− DU(θ)2dθ

1/2 ,

where

D = D0 +max{0,ΔD}.

Corollary 1 ∃δ > 0 s.t. ∀D0 ∈ [0,δ) the system

˙ X = AX +BU(t), U(t) = K

• eAD0X(t)+

Z t

t−D0

eA(t−θ)BU(θ)dθ

• is exp. stable in the sense of the norm
• |X(t)|2 +

R t

t−D0U(θ)2dθ

1/2

.

Delay-Robustness of Predictor Feedback

uncertain delay LTI-ODE plant

U(t) Delay-Adaptive Control

uncertain delay LTI-ODE plant

U(t)

unknown delay LTI-ODE plant

U(t)

Motivation: control of thermoacoustic instabilities in gas turbine combustors

transport PDE with unknown propagation speed 1/D LTI-ODE plant

U(t) X(t)

estimator of D

u(x,t)

certainty equivalence version of predictor feedback

Update law

d dt ˆ D(t) = −γ

Z 1

0 (1+x)

reg. error

w(x,t)

regressor

• KeA ˆ

D(t)xdx (AX(t)+Bu(0,t))

1+X(t)TPX(t)+b

Z 1

0 (1+x)w(x,t)2dx

• normalization

w(x,t) = u(x,t)− ˆ D(t)

Z x

0 KeA ˆ D(t)(x−y)Bu(y,t)dy−KeA ˆ D(t)xX(t).

Update law

d dt ˆ D(t) = −γ

Z 1

0 (1+x)

reg. error

w(x,t)

regressor

• KeA ˆ

D(t)xdx (AX(t)+Bu(0,t))

1+X(t)TPX(t)+b

Z 1

0 (1+x)w(x,t)2dx

• normalization

w(x,t) = u(x,t)− ˆ D(t)

Z x

0 KeA ˆ D(t)(x−y)Bu(y,t)dy−KeA ˆ D(t)xX(t).

Theorem 4

∃R,ρ > 0 s.t. ϒ(t) ≤ R

• exp
• ρϒ(0)
• −1
• (exp. growing class K ∞ glob. stab. bound)

where

ϒ(t) = |X(t)|2 +

Z 1

0 u(x,t)2dx+

• D− ˆ

D(t) 2 .

Furthermore,

X(t),U(t) → 0.

2 4 6 8 10 0.5 1 1.5 2 t D(t) D(0) = 2 D(0) = 0

^ ^ ^

2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 1.4 t X(t) D(0) = 2 D(0) = 0

^ ^

2 4 6 8 10 −2.5 −2 −1.5 −1 −0.5 0.5 t U(t) D(0) = 2 D(0) = 0

^ ^

X(s) = e−s s−0.75U(s)

Simulations by Delphine Bresch-Pietri

0–1 sec The delay precludes any influence of the control on the plant, so X(t) shows an exponential open-loop growth. 1–3 sec The plant starts responding to the control and its evolution changes qualitatively, resulting also in a qualitative change of the control signal. 3–4 sec When the estimation of ˆ

D(t) ends at about 3 seconds, the controller structure

becomes linear. However, due to the delay, the plant state X(t) continues to evolve based on the inputs from 1 second earlier, so, a non-monotonic transient continues until about 4 seconds. 4 sec and onwards The (X,U) system is linear and the delay is sufficiently well compen- sated, so the response of X(t) and U(t) shows a monotonically decaying exponential trend of a first order system.

Time-Varying Input Delay

Basic idea introduced by Artstein (TAC, 1982) , but only conceptually (nor explicitly), for LTV systems with TV delays. Explicit design for LTI plants presented by Nihtila (CDC, 1991) , but no analysis of stability

• r of feasibility of the controller.

Time-Varying Input Delay

˙ X(t) = AX(t)+BU(φ(t)) φ(t) = t −D(t) := “delayed time”

Predictor feedback

U(t) = K

• eA
• φ−1(t)−t
• X(t)+

Z t

φ(t)eA

• φ−1(t)−φ−1(θ)
• B

U(θ) φ′ φ−1(θ) dθ

• Time-Varying Delay

˙ X(t) = AX(t)+BU(φ(t))

Predictor feedback

U(t) = K

• eA
• φ−1(t)−t
• X(t)+

Z t

φ(t)eA

• φ−1(t)−φ−1(θ)
• B

U(θ) φ′ φ−1(θ) dθ

• Transport PDE representation

u(x,t) = U

• φ
• t +x
• φ−1(t)−t
• Time-varying backstepping transformation

w(x,t) = u(x,t)−KeAx

• φ−1(t)−t
• X(t)−K

Z x

0 eA(x−y)

• φ−1(t)−t
• Bu(y,t)
• φ−1(t)−t
• dy

Need a Lyapunov functional. Construct one with a backstepping transformation of the actuator state:

W(θ) = U(θ)−K

X(φ−1(θ)P(θ)

• eA
• φ−1(θ)−t
• X(t)+

Z θ

φ(t)eA

• φ−1(θ)−φ−1(σ)
• B

U(σ) φ′ φ−1(σ) dσ

• φ(t) ≤ θ ≤ t

Need a Lyapunov functional. Construct one with a backstepping transformation of the actuator state:

W(θ) = U(θ)−K

X(φ−1(θ)P(θ)

• eA
• φ−1(θ)−t
• X(t)+

Z θ

φ(t)eA

• φ−1(θ)−φ−1(σ)
• B

U(σ) φ′ φ−1(σ) dσ

• φ(t) ≤ θ ≤ t

V(t) = X(t)TPX(t)+a

Z t

φ(t)

e

bφ−1(θ)−t

φ−1(t)−t

• φ−1(t)−t
• φ′

φ−1(θ) W(θ)2dθ

Theorem 1 ∃G,g > 0 s.t.

|X(t)|2 +

Z t

t−D(t)U2(θ)dθ ≤ Ge−gt

• |X0|2 +

Z 0

−D(0)U2(θ)dθ

• ,

∀t ≥ 0,

where G (but not g) depends on the function D(·). Target system

˙ X(t) = (A+BK)X(t)+Bw(0,t), wt(x,t) = π(x,t)wx(x,t), w(1,t) = 0,

where the variable speed of propagation is

π(x,t) = 1+x    d

• φ−1(t)
• dt

−1    φ−1(t)−t

Theorem 5 Let the delay function δ(t) =t −φ(t) be strictly positive and uniformly bounded from above. Let the delay rate function δ′(t) be strictly smaller than 1 and uniformly bounded from below. There exist positive constants G and g (the latter one being in- dependent of φ) such that

|X(t)|2 +

Z t

φ(t)U2(θ)dθ ≤ Ge−gt

• |X0|2 +

Z 0

φ(0)U2(θ)dθ

• ,

for all t ≥ 0.

Nonlinear systems with constant delay

Nonlinear Systems with Constant Input Delay

˙ X(t) = f(X(t),U(t −D))

Assumptions:

˙ X = f(X,κ(X)) is g.a.s. ˙ X = f(X,U) is forward complete

Predictor-based controller (predictor given implicitly in general):

U(t) = κ(P(t)) P(t) = X(t)+

Z t

t−D f(P(θ),U(θ))dθ

Nonlinear stabilization when delay is a function of state

Miroslav Krstic

Sontagfest, May 2011 Conditions on the delay function D(t) = t −φ(t):

• D(t) ≥ 0 (causality);
• D(t) is uniformly bounded from above (all inputs applied to the plant eventually reach

the plant);

• D′(t) < 1 (plant never feels input values that are older than the ones it has already

felt— input signal direction never reversed );

• D′(t) is uniformly bounded from below (delay cannot disappear instantaneously, but
• nly gradually).

Achilles heel:

φ−1(t) > t > φ(t)

t

D(t) needs to be known sufficiently far in advance ⇒ method appears not to be usable for state-dependent delays

Nonlinear Systems with State-Dependent Input Delay

˙ X(t) = f

• X(t),U
• t − D(X(t))
• Challenge:

P(t) =

value of the state at the time when the control applied at t reaches the system

= X

• t +D(P(t))
• P(θ)

= X(t)+

Z θ

t−D(X(t))

f (P(s),U(s)) 1−∇D(P(s)) f (P(s),U(s))ds, t −D(X(t)) ≤ θ ≤ t

Nonlinear Systems with State-Dependent Input Delay

˙ X(t) = f

• X(t),U
• t − D(X(t))
• Challenge:

P(t) =

value of the state at the time when the control applied at t reaches the system

= X

• t +D(P(t))
• P(θ)

= X(t)+

Z θ

t−D(X(t))

f (P(s),U(s)) 1−∇D(P(s)) f (P(s),U(s))ds, t −D(X(t)) ≤ θ ≤ t

Controller (possibly time-varying)

U(t) = κ((t +D(P(t)),P(t))

Example 1 (stabilizing, but not global even for linear systems)

˙ X(t) = X(t)+U

• t −X(t)2

Simulations with input initial conditions

U(θ) = 0,−X(0)2 ≤ θ ≤ 0.

For X(0) ≥ X∗ =

1 √ 2e = 0.43, the controller never “kicks in” (dashed)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t x(t) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 φ(t) t

Result not global because of feasibility condition “delay rate < 1”

To keep the prediction horizon finite and control bounded, the initial conditions and solu- tions must satisfy

Fc :

∇D(P(θ)) f (P(θ),U(θ)) < c,

for all θ ≥ −D(X(0)), for some c ∈ (0,1]. We refer to F1 as the feasibility condition of the controller.

state: X, U(•+s), -D(X) ≤ s ≤ 0

. Theorem 2 (local u.a.s. in sup-norm of U)

∃ψRoA ∈ K , ρ ∈ K C , and β ∈ K L s.t. ∀ initial cond. that satisfy B0(c) : |X(0)|+ sup

−D(X(0))≤θ≤0

|U(θ)| < ψRoA(c)

for some 0 < c < 1,

|X(t)|+ sup

t−D(X(t))≤θ≤t

|U(θ)| ≤ β

• ρ
• |X(0)|+

sup

−D(X(0))≤θ≤0

|U(θ)|,c

• ,t
• ,

∀t ≥ 0.

If U is locally Lipschitz on the interval [−D(X(0)),0), there exists a unique solution to the closed-loop system with X Lipschitz on [0,∞), U Lipschitz on (0,∞)

Assumption 1 D ∈ C1 (Rn;R+) Assumption 2

˙ X = f (X,ω) is forward complete

Assumption 3

˙ X = f (X,κ(t,X)) is g.u.a.s.

Lemma 1 (infinite-dimensional backstepping transformation of the actuator state)

W(θ) = U(θ)−κ(σ(θ),P(θ)), t −D(X(t)) ≤ θ ≤ t,

transforms the closed-loop system into the “target system”

˙ X(t) = f (X(t),κ(t,X(t))+W (t −D(X(t)))) W(t) = 0, ∀t ≥ 0.

Lemma 2 (u.a.s. of target system)

∃ρ∗ ∈ K C , β2 ∈ K L s.t., for all solutions satisfying Fc for 0 < c < 1, |X(t)|+ sup

t−D(X(t))≤θ≤ t

|W(θ)| ≤ β2  ρ∗  |X(0)|+ sup

−D(X(0))≤θ≤ 0

|W(θ)|,c  ,t  ,

U(· + τ) τ ∈ [-D(X),0] W(· + τ) τ ∈ [-D(X),0]

.

level set of Lyapunov functional

state: X, U(•+s), -D(X) ≤ s ≤ 0

.

Lemma 3 (norm equivalence between the original system and target system)

∃ρ2 ∈ K C ∞, α9 ∈ K ∞ s.t., for all solutions satisfying Fc for 0 < c < 1, |X(t)|+ sup

t−D(X(t))≤θ≤t

|U(θ)| ≤ α−1

9

• |X(t)|+

sup

t−D(X(t))≤θ≤t

|W(θ)|

• |X(t)|+

sup

t−D(X(t))≤θ≤t

|W(θ)| ≤ ρ2

• |X(t)|+

sup

t−D(X(t))≤θ≤t

|U(θ)|,c

• Lemma 4

(finding a ball

B around the origin and within the feasibility region) ∃ ρc ∈ K C ∞ s.t. Fc (0 < c < 1) is satisfied by all solutions that satisfy B(c) : |X(t)|+ sup

t−D(X(t))≤θ≤t

|U(θ)| < ρc(c,c) ∀t ≥ 0.

Lemma 5 (finding a ball B0 of initial conditions s.t. all solutions are confined in

B ⊂ Fc) ∃ψRoA ∈ K s.t. for all initial conditions in B0(c), the solutions remain in B(c) ⊂ Fc for

some 0 < c < 1.

Examples

Example 2 Non-holonomic unicycle with D(x,y) = x2 +y2 A predictor-based version of Pomet’s (1992) time-varying controller:

ω = −5P2cos(3σ(t))− pq

• 1+25cos(3σ(t))2

−Θ v = −P+5Q(sin(3σ(t))−cos(3σ(t)))+Qω,

where

P = X cos(Θ)+Y sin(Θ) Q = X sin(Θ)−Y cos(Θ),

and the predictor is given by

X(t) = x(t)+

Z t

t−D(x(t),y(t))

˙ σ(s)v(s)cos(Θ(s))ds Y(t) = y(t)+

Z t

t−D(x(t),y(t))

˙ σ(s)v(s)sin(Θ(s))ds Θ(t) = θ(t)+

Z t

t−D(x(t),y(t))

˙ σ(s)ω(s)ds σ(t) = t +D(X(t),Y(t)) ˙ σ(s) = 1 1−2(X(s)v(s)cos(Θ(s))+Y(s)v(s)sin(Θ(s)))

−0.4 −0.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 x(t) y(t) Trajectory of the robot for t ∈ [0, 15]

−20 −10 10 20 −15 −10 −5 5 10 15 Trajectory of the robot for t ∈ [0, 500] x(t) y(t)

5 10 15 1 2 3 4 5 6 7 8 9 10 t D(t)

Solid: with delay compensation; dashed: without.

Example 3 (global stabilization—not with D unif. bdd but with ˙

D = ∇Df < 1) ˙ X(t) = X(t)+U (t −D(X(t))) 1+U (t −D(X(t)))2 , D(X) = 1 4 log

• 1+X2

.

In the delay-free case, the controller U = −2X yields the closed-loop system ˙

X = −

X 1+4X2.

5 10 15 20 0.5 1 1.5 2 2.5 X (t) t

• 1. X(t) grows exponentially,
• 2. X(t) decays as “backwards” square root,
• 3. X(t) decays exponentially.

Example 4 (forward completeness not needed for local stabilization)

˙ X(t) = X4(t)+2X5(t)+X2(t)(1+X(t))U(t −X2(t)).

Origin not reachable for X0 < −1, hence not glob. stabilizable. Origin not loc. exp. stabilizable. Delay-free controller U = −X yields ˙

X = −X3 +2X5, with RoA = 1

√ 2 ≈ 0.7.

1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t X (t)

Solid: controller with delay compensation. Dotted: delay-free case. Dashed: U = −X applied to the plant with delay. The initial condition X0 = 0.54 is large. The state

X(σ∗) is almost at R =

1 √ 2 when control kicks in.

Theorem 3 (loc. asymp. stabilization of ODE ⇒ loc. asymp. stabilization ∀ delay fcn) If in the absence of delay there exist R > 0 and β1 ∈ K L s.t. ∀t ≥ 0,

|X(0)| < R ⇒ |X(t)| ≤ β1(|X(0)|,t),

then there exist δ > 0 and β2 ∈ K L s.t. ∀t ≥ 0,

|X(0)|+ sup

−D(X(0))≤θ≤0

|U(θ)| < δ ⇓ |X(t)|+ sup

t−D(X(t))≤θ≤t

|U(θ)| ≤ β2

• |X(0)|+

sup

−D(X(0))≤θ≤0

|U(θ)|,t

• .

Extra challenge: Make δ so small that, when control kicks in, |X| < R. (Estimate the time the control kicks in from a fixed pt. problem on a delay bound, which is a contraction for sufficiently small initial condition.) Example 5 (state-dependent delay on state)

˙ X1(t) = X2

• t −asin2X1(t)
• ,

a ≥ 0 ˙ X2(t) = U(t) U(t) = −c2(X2(t)+c1P1(t))−c1 X2(t) 1−asin(2P1(t))X2(t) P1(θ) = X1(t)+

Z θ

t−asin2X1(t)

X2(s)ds 1−asin(2P1(s))X2(s)

2 4 6 8 10 −5 5 10 15 20 t X2(t) X1(t) 2 4 6 8 10 −6 −4 −2 2 4 6 8 t U (t)

2 4 6 8 10 2 4 6 8 10 φ(t) = t − 0.3 sin

2(X1(t))

σ(t) t