Output Feedback Control

4.4 Adaptive NN Control Design

4.4.2 Output Feedback Control

74 4 Altitude Control of Helicopters with Unknown Dynamics forj D1; 2; :::; , where

AD 2 66 66 64

0 1 0 0 0 0 1 0

0 0 0 1 1 1 2 1

3 77 77 75

; bD 2 66 66 64 0 0 ::: 0 1

3 77 77 75

(4.56)

Since belongs to a compact set, and u is bounded, we know that jy^{.j /}j Yj. Therefore, there exists a constantt > 0such that for t > t,j^{.j /}.t /j Dj, whereDjis a positive constant independent of. This leads toj^{.j /}j hj WDBDj,

whereBis the normofŒ1 12::: 1^T. ut

Remark 4.16. Note that_k1^k converges to a neighborhood ofk, provided thatyand its derivatives up to the-th order are bounded. Hence, ^kC1

^k is a suitable observer to estimate thekth order output derivative.

To prevent peaking [52], saturation functions are employed on the observer signals whenever they are outside the domain of interest:

_i^s D Ni sat _i

N _i

; Ni max

.z;W ;Q V /2˝Q .i/; sat.a/D 8<

:

1fora <1 a forjaj 1 1 fora > 1

(4.57) foriD1; 2; :::; , whereQDŒQ1; :::;Q^T, and the compact set˝WD˝z˝W˝V, where˝z,˝W, and˝V are defined in (4.47), (4.48), and (4.49) respectively, denotes the domain of interest.

Now, we revisit the control law (4.24)–(4.26) and adaptation laws (4.30)–(4.31) for the full-state feedback case. Via the certainty equivalence approach, we modify them by replacing the unavailable quantities zi andZ with their estimates,zOi WD

_i^s

ⁱ¹ O˛i1 andZO WD Œ1;^2s; :::;

s

¹; ;Oz;˛PO₁^T respectively, fori D 2; :::; . Therefore, the control laws are given by

O

˛₁D k₁z1C Py_d; O

˛_i D k_iOzi Oz_i1C PO˛_i1; (4.58)

unnD OW^TS.VO^TZ/;O (4.59)

ub D kzO Oz1kb

ZOWO^TSO_o⁰²

F CSOoCSO_o⁰VO^TZO ²

O z

(4.60)

Due to the fact that the actual NN is in terms ofZO while the ideal NN is in terms of Z, the following Lemma is needed.

Lemma 4.17. [35] The error between the actual and ideal NN output can be written as

WO^TS.VO^TZ/O W^TS.V^TZ/D QW^T.SOo OS_o⁰VO^TZ/O C OW^TSO_o⁰VQ^TZO Cdu

(4.61) whereSOo WDS.VO^TZ/;O SO_o⁰ WDdiagfOs⁰_o1; :::;sO⁰_olgwith

O

s⁰_oi Ds⁰.vO^T_iZ/O D ds.za/ dza

ˇˇˇz_aDOv^T_iZO; iD1; 2; :::; l; (4.62)

and the residual termduis bounded by

du kWkSO_o⁰VO^TZOCSOo C kVkFZOWO^TSO_o⁰

F (4.63)

Accordingly, the adaptation laws are designed as WPO D W

h

.SOo OS_o⁰VO^TZ/O OzCWWO i

(4.64) VPO D V

hZOWO^TSO_o⁰zOCVVO i

(4.65) Using the backstepping procedure similar to Sect.4.4.1, and substituting (4.59), (4.64), and (4.65) into the derivative ofValong the closed loop trajectories, it can be shown that

VP

1X

jD1

k_jz²_j

g₀C gP

2g

z² g

X1 jD2

k_jzjQzjC X1 jD2

zj.˛PO_j1 P˛_j1/

C

1

X

jD3

zjQzj1z" QW^T h

.SOo OS_o⁰VO^TZ/QO zCWWO i

trn VQ^T

hZOWO^TSO_o⁰QzC_VVO

ioC jzjh

kWkSO_o⁰VO^TZOCSO_o CkVkFZOWO^TSO_o⁰

F

iCzub (4.66)

76 4 Altitude Control of Helicopters with Unknown Dynamics From the inequalities in (4.33) and (4.34), we know that

VP

1X

jD1

k_jz²_j

g₀C gP

2g

z² g

X1 jD2

k_jzjQzjC X1 jD2

zj.˛PO_j1 P˛_j1/

C

1

X

jD3

zjQzj1z" QW^T.SOo OS_o⁰VO^TZ/QO z OW^TSO_o⁰VQ^TZOzQ

C_W

2 .k QWk²C kWk²/C_V

2 .k QVk²_FC kVk²_F/C1 2kWk² C1

2kVk²F C1 2z²

SO_o⁰VO^TZOCSO_o ²CZOWO^TSO_o⁰²

F

Czub

(4.67) Substituting the bounding control (4.60) into (4.66) yields

VP

X1 jD1

k_jz²_j

k1 2

z²

g₀C gP

2g

z² g W

2 k QWk²V

2 k QVk²F

QW^T.SO_o OS_o⁰VO^TZ/O zQ OW^TSO_o⁰VQ^TZOzQ X jD2

k_jzjzQj C X jD2

zj.˛PO_j1

P˛j1/C X jD3

zjzQj1C _W C1

2 kWk²C _V C1

2 kVk²_F C1 2"N²

kb1

2 SO_o⁰VO^TZOCSOo ²CZOWO^TSO_o⁰²

F

.z²CzQz/ (4.68) The following lemma is useful for handling the terms containing the estimation errorszQj, forj D1; 2; :::; .

Lemma 4.18. There exist positive constantsF_iandG_i, which are independent of, such that, fort > t,i D1; 2; :::; 1, the estimates˛PO_i andzOisatisfy the following inequalities:

j PO˛_i P˛_ij F_i; jQzij WD jOzizij G_i: (4.69)

Proof. Starting fromiD1, we know that PO

˛₁ P˛₁ D k₁.POz₁ Pz₁/D k₁Q₂: (4.70) Subsequently, it is easy to obtain that

PO

˛2 P˛2 D k2.Q3.˛PO1 P˛1// Q2k1Q3

D .1Ck1k2/Q2.k1Ck2/Q3DWa₂^T2 (4.71) wherea2WDŒ.1Ck1k2/ .k1Ck2/^Tand2WDŒQ2Q3^T. Suppose that

PO

˛_i2 P˛_i2Da^T_i2i2; PO

˛_i1 P˛_i1Da^T_i1i1; (4.72)

wherej DŒQ2:::QjC1^Tandaj is a vector of constants, forj Di2; i1. Then, by induction, it can be shown that

PO

˛i P˛i D ki

QiC1.˛POi1 P˛i1/

Qi .˛POi2 P˛i2/ C RO˛i1 R˛i1; D k_i

Q_iC1a_i^T_1i1

Q_ia^T_i2i2 Ca_i^T₁P_i1Da_i^T_i (4.73) fori D 3; :::; 1. From Lemma4.15, we know thatk_ik H_i, whereH_i WD Œh₂; :::; h_iC1^T, which leads to the fact thatj PO˛_i P˛_ij ka_ikH_i DWF_i, and thus (4.69) is proven.

To prove (4.69), note that Q

zi D Q_i.˛O_i1˛_i1/

D Q_i.k_i1zQ_i1 Qz_i2C PO˛_i1 P˛_i1/ (4.74) By following a similar inductive procedure, starting fromQz1 D Q₁ andQz2 D Q₂ .˛O₁˛₁/D Q₂, it can be shown thatzQi Da^T_zii, wherea_ziis a constant vector. Using the property in (4.69), it is straightforward to see thatjQzij ka_zikH_i DWG_i. The

proof is now complete. ut

Using Lemma4.18, it is clear that, fort > t, the following inequalities hold:

QW^T.SOo OS_o⁰VO^TZ/O zQ

2k QWk²C 2

k OSok C k OS_o⁰VO^TZOk ²G²; (4.75)

OW^TSO_o⁰VQ^TZOzQ

2k QVk²_FC 2

ZOWO^TSO_o⁰²

FG²; (4.76)

78 4 Altitude Control of Helicopters with Unknown Dynamics

X jD2

kjzjzQj X jD2

k_j 2

z²_jC²G_j² ; (4.77)

1

X

jD2

zj.˛PO_j1 P˛_j1/ X jD2

1 2

z²_j C²F_j² ; (4.78)

X jD3

zjQzj1 X jD3

1 2

z²_j C²G_j²₁ : (4.79) By substitution of the inequalities (4.75)–(4.79) into (4.68), it is straightforward to obtain the following expression:

VP

1

X

jD1

kjz²_j

k1 2

z²

g0C gP

2g

z²

g .W / 2 k QWk² ._V /

2 k QVk²_FC

k OSok C k OS_o⁰VO^TZkO ²CZOWO^TSO_o⁰²

F

2G² C

X jD2

kj

2 .z²_j C²G_j²/C X jD2

1

2.z²_j C²F_j²/C X jD3

1

2.z²_j C²G_j²₁/ CW C1

2 kWk²C V C1

2 kVk²_F C 1 2"N² 1

2

kb1

2 SO_o⁰VO^TZOCSOo ²CZOWO^TSO_o⁰²

F z²²G² (4.80) The RHS terms can be rearranged into a more convenient form for analysis:

VP k1z²₁1

2.k21/z²₂

1

X

jD3

1

2.kj 2/z²_j1

2.k3/z²

g0C gP

2g z²

g .W /

2 k QWk².V /

2 k QVk²FC X jD2

k_j

2 ²G_j²C X jD2

1 2²F_j² C

X jD3

1

2²G_j1² C W C1

2 kWk²C V C1

2 kVk²_F C 1 2"N² 1

2

k_b1

2 SO_o⁰VO^TZOCSO_o ²CZOWO^TSO_o⁰²

F z²

²C 2 2kb1

G²

: (4.81)

Finally, by appropriately choosing the control parameterskandk_bas follows:

k2 > 1; k3; :::; k1> 2; k> 3; kb> 1

2; W; V > ; (4.82) it can be shown that

VP c1VCc2K

z²c3 ; (4.83)

where c1WDmin

2k1; .k21/; .k32/; :::; .k₁2/; g.k3/; .W / max._W¹/; .V /

_max._W¹/

; (4.84)

c2WD 1

2"N²C _W C1

2 kWk²C_V C1 2 kVk²_F C1

2² 0

@X

jD2

F_j²C

1

X

jD2

.kjC1/G_j²CG² 1

A; (4.85)

c3WD

C 2

2k_b1

G²; (4.86)

KWD 1 2

kb1

2 SO_o⁰VO^TZOCSOo ²CZOWO^TSO_o⁰²

F

: (4.87)

It can be shown that

VP.t / c_o1V.t /Cc_o2; t t c_o1WD min

(

2_min.K₁/; _min

K2³₂I

_max.M / ; miniD1;2;3f₄ⁱk Qik²g _max.¹/

)

(4.88) c_o2WD

X3 iD1

1C i

2 k_ik²C 1

2k"k²C2_max.KN₂KN₂^TC/₂ (4.89) To ensure that > 0, the control gainsK1 and K2 are chosen to satisfy the following conditions:

_min.K1/ > 0; _min

K23 2I

> 0: (4.90)

80 4 Altitude Control of Helicopters with Unknown Dynamics We are ready to summarize our results for the output feedback case under the following theorem.

Theorem 4.19. Consider the helicopter dynamics (4.1) under Assumptions 4.1–

4.5, with output feedback control laws (4.59)–(4.60), adaptation laws (4.64)–(4.65), and high gain observer (4.53) which is turned on at timetin advance. For initial conditions .0/, .0/, W .0/,Q V .0/Q starting in any compact set ˝₀, all closed loop signals are SGUUB, and the tracking error z₁ converges to the steady state compact set:

˝z₁ WD (

z12R ˇˇˇˇ ˇjz1j

s 2cN2

c1

)

; (4.91)

wherecN₂WDc₂Cc₃K, andN c₁is as defined in (4.84).

Proof. We consider the following two cases for the stability analysis:

Case 1:jzj>p c₃

For this case, the last term of (4.83) is negative, thus yielding

VP c1VCc2; (4.92)

which straightforwardly implies that all closed loop signals are SGUUB, according to Lemma4.9. However, whenjzj p

c3, the last term of (4.83) may not be negative, leading to a more complicated analysis, as shown in the subsequent case.

Case 2:jzj p c3

For this case, we want to show that, as a result of z being bounded, the function Kin (4.87) is also bounded, for which (4.83) can be expressed in the form of (4.9), convenient for establishing SGUUB property. To this end, note that the derivative of V₁is given by

VP1

1

X

iD1

kiz²_i

1

X

iD2

kizizQiC

1

X

iD2

zi.˛POi1 O˛i1/C

1

X

iD3

ziQzi1Cz1z

(4.93) According to Lemma4.18, we can show that

VP1

1

X

iD1

kiz²_i C

1

X

iD2

kijzijGi C

1

X

iD2

jzijFiC

1

X

iD3

jzijGi1C jz1jp c3

k₁z²₁1

2.k₂1/z²₂ X2 iD3

1

2.k_i 2/z²₃1

2.k3/z²₁

C1 2c₃C1

2 X1 iD3

²G_i²₁C 1 2

X1 iD2

²F_i²C1 2

X1 iD2

k_i²G_i²

c₄V₁Cc₅; (4.94)

where the positive constantsc4andc5are defined by

c4WDminf2k1; k21; k32; :::; k12; k3g; (4.95) c₅WD

2

"

c₃C

1

X

iD2

F_i²C

2

X

iD2

.1Ck_i/G_i²Ck₁G₁²

!#

: (4.96)

This implies that z.t /satisfies the inequality kz.t /k

s 2

V₁.0/C c5

c4

Cc₃ DW Nz: (4.97)

According to Lemma4.11, it follows from the boundedness of z.t /that the internal states.t /are also bounded, i.e.,

k.t /k a₁.zNC N_d/Ca₂DW N; (4.98) wherek_d.t /k N_d for constantN_d > 0, based on Assumption4.1. Thus, the vector of NN inputsZO is also bounded as follows

ZO

_1d C Nz;N2

; :::; N

¹;;N p

c₃CG;˛NP₁CF₁ T

DW NOZ (4.99) where the constant ˛NP₁ > 0 is an upper bound for ˛P₁

z; _1d; _1d^.1/; :::; _1d^./ . Exploiting the properties of sigmoidal NNs [30], it can be shown that

SO_o⁰VO^TZOCSOo

F 1:224p

l: (4.100)

As a result, from the adaptation law (4.64), the dynamics of the neural weights WO DŒWO1; :::;WOi; :::;WOl^Tcan be shown to satisfy the inequality

WPO _min.W/WWO C1:224p

l_min.W/.p

c3CG/; (4.101) which results in

k OW .t /k k OW .0/k C 1:224p l.p

c3CG/

_W DW NOW; (4.102)

82 4 Altitude Control of Helicopters with Unknown Dynamics

whereWNO is a positive constant. Accordingly, from (4.87), and the fact thatkS_o⁰kF 0:25 p

l, we can show thatKis bounded as follows:

K l 2

kb1

2 1:498C0:0625

ZNOW NO ²

DW NK; (4.103)

whereKN is a positive constant. From (4.83) and (4.103), we obtain that

VP c₁VC Nc₂; (4.104)

wherecN₂WDc₂Cc₃KNis a positive constant.

Having obtained (4.104) for Case 2, we can compare it with (4.92) of Case 1 to see that (4.104) describes a larger compact set in which the closed loop signals remain, by virtue of the fact that cN₂ c₂. Hence, the performance bounds can be analyzed from (4.104), as a conservative approach. A nice property is that as diminishes to zero, we havecN₂ ! c₂, and the performance can be analyzed from (4.92) instead, albeit conservatively.

Based on (4.104), we can directly invoke Lemma4.9to conclude SGUUB for all closed loop signals. Since it is straightforward to prove that the tracking error z1 D_11d converges to the compact set˝z₁, by following the steps outlined in the proof of Theorem4.12, we have omitted the proof. ut Remark 4.20. It follows from Theorem4.19that the size of the steady state compact set˝_z1, to which the tracking error converges, depends on the ratio^c_c^N2

1, which contain tunable parameters. Thus, we can reduce the size of˝z₁ by appropriately choosing the parameters. For instance, by choosing the control gainsk₁; :::; k large and the observer parametersufficiently small, the ratio ^c_c^N2

1 can be decreased, to the effect that˝z₁ diminishes.