young acc 06 control

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f 22(θ, φ) = 12m1 + 36m2

l22m2[4m1 + 12m2 − 9m2 cos2 (φ − θ)]

+ 18l2 cos (φ − θ)

l1l22[4m1 + 12m2 − 9m2 cos2 (φ − θ)],

f 13(θ, θ,φ, φ) = 9gm2 sin (2φ − θ)

l1[15m2 + 8m1 − 9m2 cos (2φ − 2θ)] −

9l1m2 θ2 sin(2φ − 2θ)

l1[15m2 + 8m1 − 9m2 cos (2φ − 2θ)] +

12l2m2 φ2 sin(φ − θ)

l1[15m2 + 8m1 − 9m2 cos (2φ − 2θ)] −

(15gm2 + 12gm1)sin(θ)

l1[15m2 + 8m1 − 9m2 cos (2φ − 2θ)],

f 23(θ, θ,φ, φ) = 12l1 θ2 sin(φ − θ)(m1 + 3m2)

l2[15m2 + 8m1 − 9m2 cos (2φ − 2θ)] −

9l2m2

˙φ

2

sin(2φ − 2θ)l2[15m2 + 8m1 − 9m2 cos (2φ − 2θ)] −

9g sin(φ − 2θ)(m1 + 2m2)

l2[15m2 + 8m1 − 9m2 cos (2φ − 2θ)] +

3g sin(φ)(m1 + 6m2)

l2[15m2 + 8m1 − 9m2 cos (2φ − 2θ)].

The equations in (1) can be solved with respect to θ(t) and

φ(t):

θ(t) = f 11(θ, φ) M 1(θ(t), φ(t), u1(t))+f 12(θ, φ) M 2(θ(t), φ(t), u2(t))+

f 13(θ, θ,φ, φ)

(2)

φ(t) = f 21(θ, φ) M 1(θ(t), φ(t), u1(t))+f 22(θ, φ) M 2(θ(t), φ(t), u2(t))+

f 23(θ, θ,φ, φ),

θ(0) = θ0, θ(0) = θ0, φ(0) = φ0, φ(0) = φ0,

where θ0, θ0, φ0, φ0 are initial conditions. Letting

x1(t) = θ(t), x2(t) = θ(t), x3(t) = φ(t), x4(t) = φ(t),

the double pendulum system in (1) can be written in state-

space form:

x1(t)

x2(t)x3(t)x4(t)

=

0 1 0 0

0 0 0 00 0 0 10 0 0 0

x1(t)

x2(t)x3(t)x4(t)

+

0 01 00 00 1

f 1(x(t), u(t))f 2(x(t), u(t))

,

y(t) =

1 0 0 00 0 1 0

x(t), x(0) = x0 ,

where x(t) =

x1(t) x2(t) x3(t) x4(t)

is the

vector of measurable states, y(t) =

y1(t) y2(t)

is

the vector of regulated outputs, u(t) =

u1(t) u2(t)

is the vector of control inputs, and

f 1(x, u) = f 11(x)M 1(x, u1) + f 12(x)M 2(x, u2)+f 13(x)

f 2(x, u) = f 21(x)M 1(x, u1) + f 22(x)M 2(x, u2)+f 23(x).

(3)

It is straightforward to verify that f 11, f 12, f 13, f 21, f 22,

f 23 are well-defined for all x ∈ R4. We notice that the

system has full relative degree with respect to the regulated

outputs so that there are no internal dynamics present in

the system. To ensure controllability, we assume that the

nonlinear functions of control input satisfy:

∂M i(x, ui)

∂ui

≥ β i > 0 (4)

for some β i > 0 and i = 1, 2. The control objective is todesign control input u(t) so that the output y(t) tracks a

desired bounded reference trajectory rd(t) asymptotically.

III. CONTROL D ESIGN FOR THE P ENDULUM S YSTEM

A. Reference Model

Consider a reference system given by:

xr(t) = Arxr(t) + Brr(t), t ≥ 0, xr(0) = xr,0

yr(t) = C rxr(t), (5)

where the matrices Ar, Br, and C r are defined as

Ar =

0 1 0 0

−ar

21 −ar

22 0 00 0 0 10 0 −ar43 −ar44

,

Br =

0 0br21 00 00 br42

, C r =

1 0 0 00 0 1 0

,

xr(t) =

xr1(t) xr2(t) xr3(t) xr4(t)

denotes the

state vector of the reference system, ar21, ar22, ar43, ar44 are

such that Ar is Hurwitz. Let rd(t) =

rd1(t) rd2(t)

be the vector of desired smooth bounded reference trajec-

tories. Due to the time-varying nature of rd(t) and the

definition of the reference model, if one sets r(t) = rd(t),then yr(t) will track rd(t) with bounded errors. To force

yr(t) track the desired signal rd(t) asymptotically and

guarantee zero steady state tracking error, we introduce the

following reference input:

r(t) =

ar

21

br21

0

0 ar

43

br42

rd(t) +

ar

22

br21

0

0 ar

44

br42

rd(t) +

1br21

0

0 1br42

rd(t). (6)

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It is straightforward to verify that limt→∞

(yr(t) − rd(t)) = 0,

where yr(t) has been defined in (5).

B. Control Law

Let e(t) = x(t) − xr(t) be the tracking error. Then the

open-loop tracking error dynamics are given by:

e(t) = F (e(t) + xr(t), u(t)) − Arxr(t) − Brr(t), (7)

with e(0) = e0 and

F (x, u) =

x2 f 1(x, u) x4 f 2(x, u)

,

where f i(x, u) have been defined in (3). Dynamic inversion

based control is to be determined from the solution of the

following system of equations f 1(x, u)f 2(x, u)

=

−ar21x1 − ar22x2 + br21r1−ar43x3 − ar44x4 + br42r2

, (8)

yielding the following asymptotically stable error dynamics:

e(t) = Are(t), e(0) = e0.However, due to the nonlinear expressions in (3), an explicit

expression for u from (8) cannot be obtained, in general.

Following [18], introduce the following fast dynamics to

approximate its solution:

u(t) = −Λ(t, e(t), u(t))Q−1(t, e(t))f (t, e(t), u(t)), (9)

u(0) = u0, 0 < 1,

where

f (t,e,u) =

f 1(e + xr(t), u) + ar21(xr1(t) + e1)+

ar22(xr2(t) + e2) − br21r1(t)

f 2(e + xr(t), u) + ar43(x

r3(t) + e3)+

ar44(xr4(t) + e4) − br42r2(t)

, (10)

Q(t, e) =

f 11(e + xr(t)) f 12(e + xr(t))f 21(e + xr(t)) f 22(e + xr(t))

, (11)

and

Λ(t,e,u) =

∂M 1(e+xr(t),u1)

∂u10

0 ∂M 2(e+xr(t),u2)∂u2

. (12)

We note that Q(t, e)Λ(t,e,u) is the 2 x 2 Jacobian matrix

∂f 1∂u1

(e + xr(t), u1) ∂f 1∂u2

(e + xr(t), u2)∂f 2

∂u1(e + xr(t), u1) ∂f 2

∂u2(e + xr(t), u2) ,

where

∂f 1∂u1

(e + xr(t), u1) = f 11(e + xr(t))∂M 1(e + xr(t), u1)

∂u1

∂f 1∂u2


∂u2

∂f 2∂u1


∂u1

∂f 2∂u2


∂u2.

We also notice that the matrix Q(x) can be row reduced us-

ing the explicit forms of f 11(x), f 12(x), f 21(x) and f 22(x)and rewritten as

1 q 1(x)0 q 2(x)

, (13)

in which

q 2(x) = 4l1m1 + 12l1m2 − 9l1m2 cos2 (x3 − x1)

2l2m2= 0

for any x ∈ R4, which implies that Q(x) is full rank, so

that its inverse, used in (9), is well-defined. We argue that

the solution of the singularly perturbed system (7), (9) will

solve the tracking problem. Towards that end, we recall

Tikhonov’s theorem from singular perturbations theory.

IV. CONVERGENCE A NALYSIS OF THE N ONAFFINE

CONTROL L AW

A. Tikhonov’s Theorem

Consider the problem of solving the following singularly

perturbed system

Σ0 :

x(t) = f (t, x(t), u(t), ), x(0) = ζ ()u(t) = g(t, x(t), u(t), ), u(0) = η()

, (14)

where ζ : → ζ () and η : → η() are smooth, f and g are continuously differentiable in their arguments for

(t,x,u,) ∈ [0, ∞] × Dx × Du × [0, 0], where Dx ⊂ Rn

and Du ⊂ Rm are domains, and 0 > 0. Let Σ0 be in

standard form, that is,

0 = g(t,x,u, 0) (15)

has k ≥ 1 isolated real roots u = hi(t, x), i ∈ 1,...,k

for any (t, x) ∈ [0, ∞] × Dx. Choose one particular i anddrop the subscript henceforth. Let ν (t, x) = u − h(t, x).

Then the system given by

Σ00 : x(t) = f (t, x(t), h(t, x(t), 0), x(0) = ζ (0), (16)

is called the reduced system, and the system given by

Σb : dν

dτ = g(t,x,ν + h(t, x), 0), ν (0) = η0 − h(t, ζ 0) (17)

is called the boundary layer system, where η0 = η(0) and

ζ 0 = ζ (0) are fixed initial parameters, and (t, x) ∈ [0, ∞)×Dx . The new time scale τ is related to the original one via

τ = t

. The next theorem is due to Tikhonov.

Theorem 4.1: Consider the singular perturbation systemΣ0 given in (14) and let u = h(t, x) be an isolated root

of (15). Assume that the following conditions are satisfied

for all [t,x,u − h(t, x), ] ∈ [0, ∞) × Dx × Dv × [0, 0]for some domains Dx ⊂ R

n and Dv ⊂ Rm, which contain

their respective origins:

A1. On any compact subset of Dx × Dv, the functions f ,g, their first partial derivatives with respect to (x,u,),

and the first partial derivative of g with respect to t are

continuous and bounded, h(t, x) and∂g∂u

(t,x,u, 0)

have bounded first derivatives with respect to their

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arguments,∂f ∂x

(t,x,h(t, x))

is Lipschitz in x, uni-

formly in t, and the initial data given by ξ and η are

smooth functions of .

A2. The origin is an exponentially stable equilibrium point

of the reduced system Σ00 given by equation (16).

There exists a Lyapunov function V : [0, ∞) × Dx →[0, ∞) that satisfies

W 1(x) ≤ V (t, x) ≤ W 2(x)∂V ∂t

(t, x) + ∂V ∂x

(t, x)f (t,x,h(t, x), 0) ≤ −W 3(x)

for all (t, x) ∈ [0, ∞) × Dx, where W 1, W 2, W 3are continuous positive definite functions on Dx,

and let c be a nonnegative number such that x ∈Dx | W 1(x) ≤ c is a compact subset of Dx.

A3. The origin is an equilibrium point of the boundary

layer system Σb given by equation (17), which is

exponentially stable uniformly in (t, x).

Let Rv ⊂ Dv denote the region of attraction of the

autonomous system dvdτ

= g(0, ξ 0, v + h(0, ξ 0), 0), and let

Ωv be a compact subset of Rv. Then for each compact setΩx ⊂ x ∈ Dx | W 2(x) ≤ ρc, 0 < ρ < 1, there exists

a positive constant ∗ such that for all t ≥ 0, ξ 0 ∈ Ωx,

η0 − h(0, ξ 0) ∈ Ωv and 0 < < ∗, Σ0 has a unique

solution x on [0, ∞) and

x(t) − x00(t) = O()

holds uniformly for t ∈ [0, ∞), where x00(t) denotes the

solution of the reduced system Σ00 in (16).

We will refer to the following Remark which will be used

to prove exponential stability of the boundary layer system.

Remark 1: Verification of Assumption A3 can be done

via a Lyapunov argument: if there is a Lyapunov function

V : [0, ∞) × Dx × Dv that satisfies

c1v2 ≤ V (t,x,v) ≤ c2v2 (18)

∂V

∂v g(t,x,v + h(t, x)) ≤ −c3v2, (19)

for all (t,x,v) ∈ [0, ∞) × Dx × Dv, then Assumption A3

is satisfied.

B. Convergence Result

The condition in (4) implies existence of an isolated root

for f (t,e,u) = 0. Let u = h(t, e) denote this isolated root.

Following Theorem 4.1, the reduced system for (7), (9) is

given by

e(t) = Are(t), e(0) = e0, (20)

and the boundary layer system is given by

dν

dτ = Λ(t,e,ν + h(t, e))Q−1(t, e)f (t,e,ν + h(t, e)), (21)

with ν (0) = ν 0.

The main theorem of this paper is stated as follows:

Theorem 4.2: Subject to the condition in (4), the origin

of (21) is exponentially stable. Moreover, there exists an

∗ > 0 and a T > 0, such that for all t ≥ T and 0 < < ∗,

the system given by (3), (9) has a unique solution on [0, ∞),

x(t) = xr(t) + O() holds uniformly for t ∈ [T, ∞).

Proof. We will verify that the assumptions in Tikhonov’s

theorem are satisfied. Verification of A1 is straightforward,

since the condition in (4) implies that for any compact

subset, the function f and its first partial derivatives with

respect to t are continuous and bounded, h(t, e) and∂ f ∂u

(t,e,ν + h(t, e)) have bounded first derivatives with

respect to their arguments, and ∂ f ∂e

(t,e,ν + h(t, e)) is

Lipschitz in e, uniformly in t. Since Ar is Hurwitz by

definition, the origin is an exponentially stable equilibrium

point for the reduced system given in (20), and, hence, A2

is satisfied.

For verification of A3, consider the following Lyapunov

function candidate for the boundary layer system (21):

V (t,e,ν (τ )) =Q−1(t, e)f (t,e,ν (τ ) + h(t, e))

Q−1(t, e)f (t,e,ν (τ ) + h(t, e))

. (22)

Since f (t,e,h(t, e)) = 0, then Q−1(t, e)f (t,e,h(t, e)) = 0,

and hence V (t,e, 0) = 0. Therefore

V (t,e,ν ) =

L

∇V (t,e,s)ds, (23)

where L

assumes integration along the

pass L from zero to ν , and ∇V (t,e,ν ) =Q−1(t, e)f (t,e,ν + h(t, e))

Λ(t,e,ν + h(t, e)).

Since

∂ f (t,e,ν + h(t, e))

∂ν = Q(t, e)Λ(t,e,ν + h(t, e)), (24)

then f (t,e,ν + h(t, e)) = LQ(t, e)Λ(t,e,s + h(t, e))ds.

Hence, it follows from (23) that

V (t,e,ν ) = L

L1

Q(t, e)Λ(t,e,s1 + h(t, e))ds1

(Q−1(t, e))Λ(t,e,s + h(t, e))ds,

and consequently,

V (t,e,ν ) =

L

L1

(ds1)Λ(t,e,s1 + h(t, e))

Λ(t,e,s + h(t, e))ds, (25)

where L

, L1

assume integration along the pass L and L1

from zero to ν , s, respectively. From (4), (12), one can de-

rive that λmin

Λ(t,e,s1 + h(t, e))Λ(t,e,s + h(t, e))

≥

min(β 21 , β 22), where λmin(·) denotes the minimum eigen-

value. Consequently,

V (t,e,ν ) ≥

L

L1

λmin

Λ(t,e,s1 + h(t, e))

Λ(t,e,s + h(t, e)))(ds1)ds

=

||ν ||0

χ0

λmin

Λ(t,e,s1 + h(t, e))

Λ(t,e,s + h(t, e))) dξdχ. (26)

Therefore,

V (t,e,ν ) ≥ c1||ν ||2. (27)

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Condition A1 implies that there exists c2 such that

V (t,e,ν ) ≤ c2||ν ||2, which verifies (18). Using the rela-

tionships in (21) and (22), one obtains

V (t,e,ν ) = (Q−1(t, e)f (t,e,ν + h(t, e)))

Λ(t,e,ν + h(t, e))Λ(t,e,ν + h(t, e))

Q−1(t, e)f (t,e,ν + h(t, e))

(28)

= f (t,e,ν + h(t, e))(Q−1(t, e))

Λ(t,e,ν + h(t, e))Λ(t,e,ν + h(t, e))

Q−1(t, e)f (t,e,ν + h(t, e)),

which implies that V (t,e,ν ) ≤ −2c1V (t,e,ν ). Con-

sequently, the inequality in (27) leads to V (t,e,ν ) ≤−2c21||ν ||2, and this proves that (19) holds with c3 =2c21. Hence, A3 holds, and the boundary layer (21) has

exponentially stable origin. Following Tikhonov’s theorem,

there exists an ∗ > 0 and a T > 0, such that for all t ≥ T and 0 < < ∗, the system given by (3), (9) has a unique

solution, x(t) on [0, ∞), and x(t) = xr(t) + O() holdsuniformly for t ∈ [T, ∞).

V. SIMULATIONS

For simplicity, let m1 = m2 = 1 kg, l1 = l2 = 1 m,

and let g = 9.81 m/s2. Since actuator models are often

represented by hyperbolic tangent functions of control input

u, we assume that M 1 and M 2 are given as

M 1(u1) = tanh(u1) + 0.35u1

M 2(u2) = tanh(u2) + 0.25u2 ,

so that

∂M 1(u1)

∂u1≥ 0.35 > 0,

∂M 2(u2)

∂u2≥ 0.25 > 0.

From (13) it follows that the matrices Q(t, x) and Λ(u) are

given by

Q(x) =

f 11(x) f 12(x)f 21(x) f 22(x)

,

Λ(u) =

1.35 − tanh2 (u1) 0

0 1.25 − tanh2 (u2)

.

We consider the following reference input:

rd(t) =

sin(t) + e−t

sin(2t) + e−t. (29)

This parametrization ensures that as the exponential term

decays, x1 and x3 follow a figure ∞ pattern as described

in [20]. Let

Ar =

0 1 0 0−20 −9 0 0

0 0 0 10 0 −6 −5

, Br =

0 020 00 00 6

.

In order to ensure asymptotic convergence of the states

x1(t) and x3(t) to the reference input, we define r(t) as

r(t) =

1 00 1

rd(t) +

920 00 5

6

rd(t) +

120 00 1

6 rd(t).

The fast dynamics are designed as:

0.003u(t) = −Λ(u(t))(Q−1(t, e))f (t,e,u(t)),

u0 =

0 0

,

where

f (t,e,u) =

f 1(e + xr(t), u) + 20(e1 + xr1(t))+9(e2 + xr2(t)) − 20

rd1(t) + 9

20 rd1(t)

−rd1(t)

f 2(e + xr(t), u) + 6(e3 + xr

3

(t))+5(e4 + xr4(t)) − 6

rd2(t) + 5

6 rd2(t)

−rd2(t)

.

The initial values used for this numerical simulation are:

x0 = [0 0 0 0] and xr0 = [0 0 0 0]. The plots in Figures

2, 3 show the simulation results. Figure 2 shows the closed-

loop tracking performance of the reference state xr(t) to

the desired reference trajectory rd(t) and the closed-loop

tracking performance of the reference state xr(t) by system

state x(t), while Figure 3 shows the phase plot of the states

x1(t) and x2(t) and the control effort u(t). Note that all

signals are bounded and x(t) tracks rd(t).

0 5 10 15 20−60

−40

−20

0

20

40

60Reference State Tracking of Reference Input

time, t in sec

d e g r e e s

0 5 10 15 20−60

−40

−20

0

20

40

60System State Tracking of Reference State

time,t in sec

d e g r e e s

Fig. 2. (top) Output y(t) (dashed lines) and reference input rd(t) (solidlines), (bottom) system output y(t) (dashed lines) and reference outputyr(t) (solid lines).

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−60 −40 −20 0 20 40 60−100

−50

0

50

100Phase Plot

state x1

s t a t e x 3

0 5 10 15 20−40

−20

0

20

40Control Effort, u

time,t in sec

m a g n i t u d e

Fig. 3. (top) Phase plot of system states x1 versus x3, and (bottom)control input u(t) (u1 solid line, u2 dashed line).

VI . CONCLUSIONS

In this paper, using tools from singular perturbations the-

ory, we have solved the tracking problem for the nonaffine

pendulum dynamics. An approximate dynamic inversion

control method was developed that achieved time-scale

separation between the system dynamics and the controller

dynamics. This was verified using Tikhonov’s theorem from

singular perturbations theory.

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