Robust stability analysis of DCFDOB - Robust Stability Analysis

Chapter 4 Uncertainty and Robustness …

4.2 Robust Stability Analysis

4.2.1 Robust stability analysis of DCFDOB

According to small gain theorem and Fig. 4.2, one modified Fig. 4.1 as M_Δ −Δ loop type and shown in Fig. 4.3.

⎥⎦

⎢ ⎤

⎣

⎡ Δ

= Δ Δ

M N

] )) (

(

[ M_n¹QM_nY_r M_n¹ I Q I M_nX_r M_Δ = − ⁻ − ⁻ − −

⎥⎦

⎢ ⎤

⎣

⎡

1 2

Z Z

Fig. 4.3 M_Δ −Δ loop stability analysis of Fig. 4.1

The interconnected system is internally stable if the following inequalities are satisfied：

⎥ ≤

⎦

⎢ ⎤

⎣

⎡ Δ Δ

∞ M

N if and only if [ ¹ ¹( ( ))] ⁻¹

∞

−

− − − − <

−M_n QM_nY_r M_n I Q I M_nX_r ε

(4.3a)

If Eq. (4.3b) is pre-multiplied by normalized coprime factorization _⎥

⎦

N , the robustness bound of DCFDOB can be obtained from the following

form: large uncertainties, however,

)) 1

the maximum allowable bound of

be rewritten as follows.

According to Eqs. (3.6) and (3.7), although one can enlarge the bandwidth of N~_nQY_l to increase disturbance rejection capability, however, according to Eq. (4.9), ε⁻¹ increases with increasing the bandwidth of N~_nQY_l

. For simply explanation, one assumes that the system is SISO, minimum phase and then substitute Eq. (2.9) into Eq.

(4.9) to yield Eq. (4.10).

Obviously, the stability margin ε⁻¹ can be obtained by designing the low-pass filter )

obtained. A tradeoff must be done between rejection capability and robust stability.

In addition, the output y(t) of plant with uncertainties is represented as follows.

Since one design P_n(I−Q(I −M_nX_r))≈0 and I−P_nQM_nY_r ≈0 in low frequency ranges to reject input and output disturbances, Eq. (4.11) can be approximated as

follows.

⋅

−

⋅

≈ P_n r P_nQM_nY_r

y (4.12) That is, the perturbation will be suppressed as well. Thus, the system will behave like nominal plant in low frequency ranges, and an outer loop controller for stabilizing and better performance can be designed easily. The complete feasible form of DCFDOB is shown in Fig. 4.4.

−1

Mn N_n

Xr Y_r

Zˆ

r e^r

−

edi

edo

−

ΔM Δ_N

Plant Actual

Observer e

Disturbanc

Controller Loop

Outer−

Fig. 4.4 The complete feasible form of DCFDOB with an outer-loop controller

4.2.2 Robust stability analysis of DCFDOB-VS

Recalling the DCFDOB-VS structure shown in Fig. 2.17, we modified it as

Δ −

M loop as shown in the following figure.

⎥⎦

⎢ ⎤

⎣

⎡ Δ

= Δ Δ

M N

~ ] )

~ ( ) (

[ Y_r H ¹ I Q Y_lM_n X_r H ¹ I Q Y_lN_n M_Δ = − + ⁻ − − − ⁻ −

⎥⎦

⎢ ⎤

⎣

⎡

1 2

Z Z

Fig. 4.5 M_Δ −Δ loop of DCFDOB-VS

According to small gain theorem, the DCFDOB-VS is guaranteed internally stable for

all Δ _∞ <1 if and only if：

[

⁻ ⁺ ¹⁽ ⁻ ⁾ ^~ ⁻ ⁻ ¹⁽ ⁻ ⁾ ^~

]

^≤¹

⎥ ⋅

⎦

⎢ ⎤

⎣

⎡ Δ Δ

∞

−

∞

n l r

N Y H I Q YM X H I QY N (4.13)

If the relation Q(s)=I −HQ_YY_l⁻¹ is substituted into M_Δ(s) of Eq. (4.13), we can obtain the following equation.

[ ]

_∞ ^∞

−

− Δ ∞

−

− +

−

− +

−

n Y r n

Y r

n l r

N Q X M

Q Y

N Y Q I H X M

Y Q I H Y M

) ~

~ ( )

( ¹

(4.14)

We found that the parameter H(s) does not appear in Eq. (4.14) and the value of

Δ(s) ∞

M is only influenced by the independent parameter Q_Y(s). That is, the

advantage is that it will simplify the robustness tuning procedure and disturbances rejection by using only one independent instead of two parameters H(s) and Q(s). Furthermore, we can modify Fig. 2.17 as Fig. 4.6, which is further equaled to Figs. 4.7(a)

and 4.7(b) through I/O equivalence.

−1

~ ) ) (

(−HY_r+ I−QY_lM_n

r ^r

e −

ΔM Δ_N d^o

−

) ~

[(I−QY_lN_n+HX_r ⁻

Fig. 4.6 The modification of DCFDOB-VS

−1

~ ) ) (

(−Y_r+H⁻¹ I−QY_lM_n

ξ r

er −

ΔM Δ_N d^o

−

1 ~ ]

) (

[X_r+H⁻ I−QY_lN_n ⁻

−1

(a)

−1

~ ) (Y_r−Q_YM_n

ξ r

er −

ΔM ΔN d^o

(X_r+Q_YN~_n ⁻

−1

(b)

Fig. 4.7 (a) The modification of DCFDOB-VS

(b) the equivalent block diagram of Fig. 4.7 (a) with two independent parameters, )

H and Q_Y(s)

According to Fig. 4.7(b), we knew that the DCFDOB-VS can be modified as the well-known Youla-Kucera controller structure with a pre-filter H⁻¹(s) when

Y s H I Q Y

Q ( )= ⁻¹( − ) . Moreover, Fig. 4.7(b) can explain more clearly why the loop properties, e.g. M_Δ(s) _∞, is only influenced by the independent parameter Q_Y(s). These modifications from DCFDOB-VS to Fig. 4.7(b) can further validate the following properties that we stated and provided in the forgoing sections.

1. Vidyasagar’s structure has the subset stabilizing solutions of the Youla-Kucera parameterization and provides the tracking properties [33].

2. The last two columns of Eq. (2.46), i.e. the loop properties of Youla-Kucera parameterization structure are the same as those of Eq. (2.48) when

) ( )

(s Q s Q_YK = _Y .

Recalling the numerical example in part c of paragraph 2.4.2.2, we gave three different bandwidth low-pass filters and three different α(s)s, i.e. three different H(s) parameters to observer M_Δ _∞ behaviors. Table 4.1 shows the results for nine cases, which indicate the robust stability is only influenced by the low-pass filter J(s), i.e.

the independent parameter Q_Y(s).

Table 4.1 Plots of M_Δ(jω) with different bandwidth of J(s), )α(s and corresponding M_Δ _∞

10000 0 4

4.3 Robust DCFDOB

One discussed the robust stability of DCFDOB and DCFDOB-VS in sections 4.1 and 4.2. The maximum allowable bound of plant uncertainties ε can be obtained by designing an adequate Q(s) and system robustness can be guaranteed when the small gain theorem is satisfied. In this section, one will discuss the design method of robust DCFDOB in H_∞ frameworks. Consider again Eq. (4.3a), one rearranges the inequality as follows.

[

^Mn⁻¹⁽^I ⁻^Q⁽^I ⁻^Mn^Xr^))[⁽^I⁻^Q⁽^I ⁻^Mn^Xr⁾⁾⁻¹^QMn^Yr ^I^]

]

_∞ ^<^ε⁻¹ (4.15)

According Eqs. (4.17) and (4.18), one rewrote Eq. (4.16) as Eq. (4.19) and rearranged DCFDOB in form of an output feedback type in Fig. 4.8.

1 1

1( ) [ ] ⁻

∞

−

− I +K P K I <ε

M_n _DOB _n _DOB (4.19)

) 1

(I−QY_lN~_n ⁻

−

di d_o

r nY QM r

−1

Mn N_n

ΔM Δ_N

−

Fig. 4.8 Right coprime factor perturbed system with reduced DCFDOB

Theorem 4.1[36, chapter 8]: Consider a right coprime factor perturbed plant described

in Fig. 4.8 and P=(N_n +Δ_N)⋅(M_n +Δ_M)⁻¹ with N , _n M , _n Δ and _N Δ_M ∈RH_∞. Assume the output feedback controller K_DOB internally stabilizes the nominal system

P , then the closed-loop system is well-posed and internally stable for all n Δ _∞ ≤ε if and only if

1 1

1( ) [ ] ⁻

∞

−

− I +K P K I <ε

M_n _DOB _n _DOB (4.20)□

A design objective is to find a reduced DCFDOB K_DOB(s) which satisfied Eq.

(4.20) for a given ε . Suppose the stable nominal plant P has the minimal _n realization )(A,B,C,D . A state-space construction for the normalized right coprime factorization can be obtained in terms of solution to the generalized control (respectively,

filter) algebraic Riccati equation as follows, i.e. Generalized Control Algebraic Riccati Equation (GCARE):

~ 0

) (

)

(A−BR_∞⁻¹D^*C ^*X_∞+ X_∞ A−BR_∞⁻¹D^*C −X_∞BR_∞⁻¹B^*X_∞+C^*R_∞⁻¹C = (4.21) and Generalized Filter Algebraic Riccati Equation (GFARE):

~ 0 )

( )

(A−BR_∞⁻¹D^*C Y_∞+Y_∞ A−BR_∞⁻¹D^*C ^*−Y_∞C^*R_∞⁻¹CY_∞+BR_∞⁻¹B^* = (4.22)

where ~ ^*

DD I

R_∞ = + and R_∞ =I+D^*D. Then, the normalized RCF is given as

∞

−

∞

∞−

∈

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡ + +

⎥ =

⎦

⎢ ⎤

⎣

⎡ RH

DR DF C

R F

BR BF A N

n n

2 1 2 1

2 1

: (4.23)

where )F =−R_∞⁻¹(B^*X_∞ +D^*C . The normalized RCF of P means _n

M_n^T(−jω)M_n(jω)+N_n^T(−jω)N_n(jω)= I for all ω. (4.24) Moreover, [41] showed that the solution satisfying Eq. (4.20) is obtained as follows:

⎥⎦

⎢ ⎤

⎣

⎡ + + + −

∞

−

∞

−

∞

* 1

* 2

* 1

2( ) ( ) ( )

D X

C Y W DF

C C Y W BF

K_DOB A γ γ

(4.25)

where )W_∞ =I +(X_∞Y_∞ −γ²I , )F =−R_∞⁻¹(D^*C+B^*X_∞ and γ =ε⁻¹ . Also, a maximum value of ε can be obtained by a non-iterative method, and is given by

¹² _min¹

max (1 ⎥ ) = ⁻

⎦

⎢ ⎤

⎣

− ⎡

= γ

n H n

N (4.26)

where • _Hdenotes the Hankel norm, and ε_max is called the maximum stability margin.

That is, the stability of the closed-loop can be guaranteed for all

εmax

⎥ <

⎦

⎢ ⎤

⎣

⎡ Δ Δ

M ∞

N (4.27)

The parameter Q(s) of robust DCFDOB can be obtained.

r n n

DOB s I QYN QM Y

K = − ~ )⁻¹⋅ (

)

( (4.28) )) 1

~ ( ~

( )

(s =K_DOB⋅ Y_l M_n +N_nK_DOB ⁻

Q (4.29) In the above section, one discussed the design method of robust DCFDOB that satisfied (4.15) in H_∞ frameworks. In the following section, one will introduce the loop - shaping methods to obtain performance / robust stability tradeoffs, and a particular H_∞ optimization problem to guarantee closed - loop stability and a level of robust stability at all frequencies.

4.4 Robust DCFDOB using H_∞- loop shaping design

This section considers the H_∞- loop shaping design which is developed by McFarlane and Glover [42] to obtain the robust DCFDOB. The objective of the H_∞

- loop shaping is to incorporate simple performance / robustness tradeoff obtained in the loop - shaping with the guaranteed stability properties of H_∞ design methods. The

H∞ - loop shaping is an open - loop shaping approach, which follows the elementary open - loop shaping principles specifying the closed - loop objectives in terms of requirements on the open - loop singular values, denoted σ(•). σ(•) and σ(•) denote the maximum and minimum singular values, respectively. To complete a robust DCFDOB, we have to consider the following objectives:

1) Input sensitivity: Recall from Eq. (4.18) that minimizing σ((I +K_DOB⋅P_n)⁻¹) minimizes the effect of input disturbance on the plant input. The following inequality relates this objective to an open - loop singular value condition:

1 2) Similar, the inequality of output sensitivity:

1 3) Robustness of coprime uncertainty on the nominal plant can be obtained by minimizing both σ(K_DOB(I+P_nK_DOB)⁻¹P_n) and σ(P_n(I+K_DOBP_n)⁻¹K_DOB) . One

In each of cases 1) - 3), one has approximated a closed - loop objective by a condition on the singular values of P and _n K_DOB over a particular frequency range.

1) and 2) are rejection performance objectives, while 3) is robust stability objective.

Good rejection performance requires that

1 ) ( , 1 ) (

, 1 )

(P_nK_DOB >> σ K_DOBP_n >> σ K_DOB >>

σ (4.33)

in a low frequency range [0,ω_l].

Good robustness requires that

δ σ

σ(P_nK_DOB)<<1, (K_DOBP_n)<<1, (K_DOB)≤ (4.34)

in a high frequency range [ω_u,∞] where δ is not too large. Figure 4.9 indicates graphically how the requirements on these closed - loop objective constrain the shape of the open - loop singular values in design.

Fig. 4.9 Open - loop singular value shaping

4.4.1 The H_∞ - loop shaping design procedure of the robust DCFDOB

The H_∞- loop shaping design procedure which we utilize to develop the robust DCFDOB is developed by McFarlane and Glover [41] and is stated next.

(1) The H_∞ - loop shaping uses a pre-weighting matrix W₁ and/or a post - weighting matrix W₂ to shape the singular values of the nominal plant P as a desired _n open - loop shape P_S =W₂P_nW₁ and its normalized right coprime factor P_S = N_SM_S⁻¹.

W1 and W₂ are selected such that P contains no hidden modes. _S (2) Robust Stability:

a) Calculate ε_S_,_max, where

[ ]

1 ) 1

(

) (

inf

12 2

1 1

1 max

⎥ <

⎦

⎢ ⎤

⎣

− ⎡

⎟⎠

⎜ ⎞

⎝⎛ +

= ⁻

∞ ∞

∞ −

−

∈ _∞

∞

S H S

S RH S

S K

M N

I K P K I ε M

(4.35)

If 1ε_S_,_max << return to step (1) and adjust W₁ and W₂.

b) Select ε_S ≤ε_S_,_max; then synthesize the solution K_∞ that satisfies

S S S

S I K P K I

M + ≤ε⁻ =γ

∞ ∞

−

∞

−1 1 1

] [

)

( (4.36) (3) The final reduced DCFDOB K_DOB is the constructed by combining the solution K_∞ with the shaping functions W₁ and W₂.

2 1K W W

K_DOB = _∞ (4.37) (4) The final parameter Q(s) can be obtained according to Eqs. (4.28), (4.29) and (4.37).

1 and forms the robust DCFDOB, we will explain how all the closed-loop objectives of Eqs. (4.33) and (4.34) are incorporated. Note that Eq. (4.36) is not only the criterion for robustness but implicitly considers minimizing the H_∞ norm of the transfer functions from d~_o d~_i]^T

where the inner function _⎥

⎦

The corollary 16.7 of [36] shows the Eq. (4.39) also equals Eq. (4.40) by interchanging

K∞ and P . _S

Equation (4.40) presents the transfer functions from d~_i d~_o]^T

[ to [y₁ y₂]^T in Figure 4.10(b)

) 1

(I−QY_lN~_n ⁻

−

d~i

d~o

r nY QM

W1 1

1 1

W− W₂ W₂⁻¹

di y1 y₂

(a)

) 1

(I −QY_lN~_n ⁻

−

d~i

d~o

r nY QM

1 2

W−

W2 1

W−

di d_o

ˆy1

ˆy2

(b)

Fig. 4.10 Two cases of the transfer functions from ~ ~ ) (d_i d_o to

~ )

(~y₁ y₂ and (y₁ y₂)

Equations (4.39) and (4.40) show how all the closed-loop objectives of Eqs. (4.33) and (4.34) are incorporated.

4.4.2 On the achieved loop shape

As described above, the desired loop shaped was specified as W₂P_nW₁, but the finally achieving loop shape is in fact given by W₁K_∞W₂P_n at plant input and

2 1K W W

P_n _∞ at plant output. Figure 4.11 illustrates the discrepancies that may occur

between specified and achieved loop shapes. It can be seen in Figure 4.11 that, at low frequency (in particular ω∈(0,ω_l)), the deterioration in loop shape at plant output can be obtained by comparing σ(P_nW₁K_∞W₂) with σ(W₂P_nW₁).

Fig. 4.11 Specified and achieved loop shapes Note that: deterioration of the loop shapes at low / high frequencies. Note that the condition

numbers κ(W₁) and κ(W₂) are selected by the designer. Moreover, theorems 4.2 and 4.3 show that σ(K_∞) / σ(K_∞) is bounded by function of γ_S and σ(P_S) /

) (P_S

σ . Hence by Eqs. (4.41) and (4.42), K_∞ will only have a limited effect on the

specified loop shape at low-frequency.

Theorem 4.2 [36]:

Any K_∞ satisfying Eq. (4.36), where P is assumed square, also satisfies _S

1 )) ( ( 1

1 ))

( )) (

(

( 2

−

≥ −

∞ γ σ ω

γ ω ω σ

j P j j P

S S

S (4.43)

for all ω such that σ(P_S(jω))> γ_S²−1. Furthermore, if σ(P_S)>> γ_S²−1, then

1 )) 1

(

( _∞ ≈> 2 −

K ω γ

σ , where ≈> denotes asymptotically greater than or equal to as

∞

→ ) (P_S

σ . □

Theorem 4.3 [36]:

Any K_∞ satisfying Eq. (4.36), where P is assumed square, also satisfies _S

)) ( ( 1 1

)) ( ( )) 1

(

( 2

ω σ

ω γ σ

j P

j j P

S S

−

− +

≤ −

∞ (4.44)

for all ω such that

1 )) 1

(

( < 2−

S S j

P ω γ

σ . Furthermore, if

1 ) 1

( << 2 −

σ γ , then

1 ))

(

(K_∞ jω <≈ γ_S²−

σ , where <≈ denotes asymptotically less than or equal to as

0 ) (P_S →

σ . □

4.4.3 Bounds of the robust DCFDOB

In this paragraph, we discuss each bounded magnitudes of the robust DCFDOB via

H∞ - loop shaping design procedure. Let P be the nominal plant and let _n

2 1K W W

K_DOB = _∞ be the associated reduced DCFDOB obtained from the loop shaping design procedure. Then if M_S I+K P_S K I ≤γ_S

∞ ∞

−

∞

−¹( ) ¹[ ] one has

{

⁽ ^), ⁽ ⁾ ⁽ ⁾

}

min )

(S_i γ_Sκ W₁ γ_Sσ M_S κ W₁

σ ≤ (4.45)

where σ(S_i) notes the gain from input disturbance d to plant input _i y₁ of Figure 4.10(a), κ(•)=σ(•) σ(•) denotes the condition number and

1 2

2( )

1 ) 1 ( ~ )

( ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

= +

= M W PW

n S

S σ σ

σ . One also has

⎭⎬

⎫

⎩⎨

≤ ⎧

) ( ) (

) , (

) ( ) min (

) (

2 1 2

1 W W

N W

S W

P_n _i ^S ^S ^S

σ σ

σ γ σ

σ γ (4.46)

where σ(P_nS_i) denotes the gain from input disturbance d to plant output _i y₂ of

Figure 4.10(a) and

1 2 2

) (

) ) (

(~ )

( ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

= +

= W PW

W P N W

n n S

S σ

σ σ

σ . Furthermore, one has

{

⁽ ⁾ ⁽ ^), ⁽ ⁾ ⁽ ⁾ ⁽ ⁾

}

min )

(S_iK_DOB γ_Sσ W₁ σ W₂ γ_Sσ M_S σ W₁ σ W₂

σ ≤ (4.47)

where σ(S_iK_DOB) denotes the gain from output disturbance d to plant input _o y₁ of

Figure 4.10(a).

σ⁽P_nS_iK_DOB⁾≤^min

{

γ_Sκ⁽W2^), γ_Sσ⁽N_S⁾κ⁽W2⁾

}

(4.48) where σ(P_nS_iK_DOB) denotes the gain from output disturbance d to plant output _o

y2 of Figure 4.10(a).

Moreover, we also can obtain Eqs. (4.49)-(4.52).

{

⁽ ^), ⁽^~ ⁾ ⁽ ⁾

}

min )

(K_DOBS_oP_n γ_Sκ W₁ γ_Sσ N_S κ W₁

σ ≤ (4.49)

{

⁽ ⁾ ⁽ ^), ⁽ ^~ ⁾ ⁽ ⁾ ⁽ ⁾

}

min )

(K_DOBS_o γ_Sσ W₁ σ W₂ γ_Sσ M_S σ W₁ σ W₂

σ ≤ (4.50)

⎭⎬

⎫

⎩⎨

≤ ⎧

) ( ) (

~ ) , (

) ( ) min (

) (

2 1 2

1 W W

N W

P W

S_o _n ^S ^S ^S

σ σ

σ γ σ

σ γ (4.51)

{

⁽ ^), ⁽ ^~ ⁾ ⁽ ⁾

}

min )

(S_o γ_Sκ W₂ γ_Sσ M_S κ W₂

σ ≤ (4.52)

where σ(K_DOBS_oP_n) , σ(K_DOBS_o) ,σ(S_oP_n) and σ(S_o) denotes the gain from input disturbance d to DCFDOB output _i ˆy₁ , output disturbance d to DCFDOB _o output ˆy₁, input disturbance d to system output _i ˆy₂ and output disturbance d to _o system output ˆy₂of Figure 4.10(b), respectively.

These following derivations present how these boundaries in Eqs. (4.45)-(4.52) be obtained. Firstly, note that

S S

S I K P K I

M + ≤γ

∞ ∞

−

∞

−¹( ) ¹[ ] (4.53)

If Eq. (4.53) is pre-multiplied by normalized coprime factorization _⎥

⎦

⎢ ⎤

⎣

⎡

S S

N . Since

⎥⎦

⎢ ⎤

⎣

⎡

S ∞ S

N , one can obtained Eq. (4.54).

[ ]

Then by Eq. (4.57), one can obtain Eqs. (4.49)-(4.52). Furthermore, recalling Eq.

(4.57), we have

and we can immediately show (4.59) and (4.60).

Similarly, one can obtain

)

Similar, one has

i seen that, by (4.45)-(4.52), all of the closed – loop objectives are guaranteed to have bounded magnitude and the bounds depend only on γ_S, W₁, W₂ and P . _n

In this paragraph, one has incorporated the normalized DCFDOB into a loop shaping based systematic design technique. This enables both performance and robust stability objective to be traded off, and preserves the exact solution associated with this

particular H_∞ problem. The following chapter, we will give some numerical examples and one experimental result to demonstrate the design steps and verify the correctness of our derivations of the DCFDOB structure.

C

HAPTER

5 N

UMERICAL

E

XAMPLES AND EXPERIMENTAL

R

ESULTS

In this chapter, we demonstrate four parts numerical examples and an experimental result to illustrate the design steps for each plant case. In part 1, two examples, SISO and MIMO, of minimum phase systems are given. In part 2, we show the design steps for a SISO, non-minimum phase plant in part 2-example 1, and for MIMO, non-minimum phase plant in part 2-example 2. In part 3, two types of non-square plant, thin and wide systems, are given to express the inherent restriction on input / output disturbances rejections. In part 4, a robust DCFDOB developed by

H∞-loop shaping design procedures are presented. Finally, an experimental result of a positioning control for an AC brushless servomotor system and cogging force suppressing is illustrated.

5.1 Numerical examplepart 1:minimum phase plants

In the first example of part 1, a stable, minimum phase, SISO plant with uncertainty is used and the second example of part 1, a stable, minimum phase, MIMO

plant is considered.

Example1:Assume the actual plant P(s) and the nominal plant P_n(s) are given as

follows.

L , the corresponding coprime factors are obtained as:

120

40rad bandwidth, are given and the corresponding parameters Q₁(s) and )

Figure 5.1 illustrates frequency response of input sensitivity. From Fig. 5.1, obviously, the rejection capability and rejection bandwidth are directly related to the bandwidth of low-pass filter. However, Fig. 5.2 displays that the bandwidth from measurement noise ξ to system output y also increase with an increase in bandwidth of J(s). That is, bandwidth of the low-pass filter is a basic criterion to be considered and tradeoff when designs a DOB.

10^-1 10⁰ 10¹ 10² 10³ 10⁴ -70

-60 -50 -40 -30 -20 -10 0 10

Magnitude (dB)

Frequency (rad/sec) Q₁

Q₂

18.23 rad/sec

199.52 rad/sec

Fig. 5.1 Frequency responses of input sensitivity of part 1 - example 1

10⁰ 10¹ 10² 10³ 10⁴

-120 -100 -80 -60 -40 -20 0 20

Magnitude (dB)

Frequency (rad/sec) Q₂

Q₁

Fig. 5.2 Frequency responses from noise to system output of part 1 – example 1

In time domain simulation, a unit-step input disturbance d_i(t)= tu( −5) is given, where )u(t−τ denotes a unit-step and u(t)= t1, ≥τsec and u(t)= t0, <τsec.

0 1 2 3 4 5 6 7 8 9 10

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Time(sec)

y_Q

Fig. 5.3 Simulation results of part 1 – example 1 In Fig. 5.3,

y and

y denote system outputs of DCFDOB with parameters Q₁

and Q₂, respectively and this figure demonstrates that when a wide bandwidth low-pass filter is adopted, the disturbance eliminate rate is faster than a narrow one. In the following, we consider both input and output disturbances and a unit-step u(t−7) is given as an output disturbance, where u(t)= t1, ≥7sec and u(t)= t0, <7sec. The simulation results are shown in Fig. 5.4(a)-(c).

0 1 2 3 4 5 6 7 8 9 10 0

0.5 1

(a)

4 4.5 5 5.5 6

0 0.02 0.04 0.06 0.08

(b)

6 6.5 7 7.5 8

0 0.5 1

(c)

Time(sec)

y_Q

2 y_Q

y_Q

Fig. 5.4 (a) Simulation results when considered both input and output disturbances of part 1 – example 1

(b) Detailed response from 4 sec to 6 sec of Fig. 5.4(a) (c) Detailed response from 6 sec to 8 sec of Fig. 5.4(a)

In Fig. 5.4(a)-(c), we observe that suppressing the input disturbance will also reduce the influence of the output disturbance. That is because the output sensitivity function is the same as the input sensitivity function in SISO system. Consequently, large bandwidth low-pass filter, i.e. Q₁(s) indicates better disturbance rejection. In Fig.

5.5(a)-(b), we plot the frequency responses and phases of the nominal plant and the ones from I/O points of DCFDOB with Q₁(s) and Q₂(s).

-80 -60 -40 -20 0

Magnitude (dB)

10^-1 10⁰ 10¹ 10² 10³ 10⁴

-135 -90 -45 0

Phase (deg)

Frequency (rad/sec) (a)

(b)

Nominal Plant : P_n

Frequency response from I/O : Q₁ Frequency response from I/O : Q₂

Nominal Plant : P_n Phase : Q₁

Phase : Q₂

Fig. 5.5 (a) Frequency responses (b) phase plots of nominal plant and the ones from I/O points of DCFDOB with Q₁(s) and Q₂(s)

Fig. 5.5(a)-(b) show that the DCFDOB can push the actual plant to nominal plant within the rejection bandwidth. In Fig. 5.5(a), the frequency response from I/O points of DCFDOB with Q₁(s) and overlaps the one of nominal plant before 100 rad/sec, but the one with Q₂(s) overlaps only before 50 rad/sec. In Fig. 5.5(b), the phase of DCFDOB with Q₁(s) overlaps the one of nominal plant before 10 rad/sec while the one with Q₂(s) overlaps only before 2 rad/sec. In general, the wider bandwidth the filter has, the closer the approximation will be.

Next, we will apply the DCFDOB to a MIMO, stable, minimum phase system in

numerical part 1-example 2.

Example 2: Suppose a simple MIMO system with 2× dimension is given as 2

follows.

⎥⎥

The transmission zeros of the system lie in -8.2527 and -4.7373 and the control gain matrix F and the observer gain matrix L are given as follows.

The corresponding coprime factorization factors can be obtained by Eqs. (1.10)-(1.11).

To eliminate both input and output disturbances, we adopt the solution, Q=N^~_n⁻¹⋅J⋅Y_l⁻¹, which stated in section 3.1 to achieve this aim and is shown in Eq. (5.10).

⎥⎥

The frequency responses from _⎥

⎦

Fig. 5.6(a)-(d) and the simulation results are shown in Fig. 5.6(e) where the input

disturbances are _⎥

⎦

, , output disturbances are given as

⎥⎦

10^-2 10⁰ 10² 10⁴ -100

-80 -60 -40 -20 0

(a)

Magnitude(dB)

10^-2 10⁰ 10² 10⁴

-100 -80 -60 -40 -20 0

(b)

10^-2 10⁰ 10² 10⁴

-100 -80 -60 -40 -20 0

Frequency(rad/sec)

Magnitude(dB)

10^-2 10⁰ 10² 10⁴

-100 -80 -60 -40 -20 0

Frequency(rad/sec)

0 1 2 3 4 5 6 7 8 9 10

-10 -5 0 5 10

Regulation Results Output Magnitude

sec d_o1→y₁

d_i1→y₁

Uncompensated Output y₁

Uncompensated Output y₂ Compensated Outputs

(e)

d_o2→y₁

d_o1→y₂ d_o2→y₂

d_i2→y₁

d_i1→y₂ d_i2→y₂

Fig. 5.6 Frequency responses from (a) ₁

1 , 1

, y

d d

i ⎥→

⎦

⎢ ⎤

⎣

⎡ (b) ₁

2 ,

, y

d d

i ⎥→

⎦

⎢ ⎤

⎣

⎡

1 , 1

, y

d d

i ⎥→

⎦

⎢ ⎤

⎣

⎡ (d) ₂

2 ,

, y

d d

i ⎥→

⎦

⎢ ⎤

⎣

⎡ and (e) simulation results of part 1 - example 2

From Figs. 5.6(a)-(d) it is clear that in low frequency ranges, the DCFOB attenuates both input and output disturbances to system outputs very well. In Figs. 5.6 (a) and (d), we observer that the magnitudes of frequency responses from d_o_,₁ to y₁ and d_o_,₂ to

y2 close to 0dB in high frequency ranges, i.e. high frequency output disturbances will directly influence the outputs beyond rejection bandwidth. From Fig. 5.6(e), the uncompensated outputs i.e. the open loop, are much larger than the ones compensated

by DCFDOB, it reveals that the DCFDOB structure can not only apply to MIMO system but also eliminate various kinds of input and output disturbances at the same time.

5.2 Numerical examplepart2:non-minimum phase plants

In the first example of part 2, an unstable, non-minimum phase, SISO plant with output feedback controller is used and the second example of part 2, a stable, non-minimum phase, MIMO plant is considered.

Example 1: An unstable and non-minimum phase SISO plant is given as

20 20 2 . ) 0

( −

= − s s s

P_n (5.11) Its unstable pole locates at 20 and a single real RHP-zero locates at 100 . The control gain matrix and observer gain matrix are given as F =[−80], ]L=[38.75 and then the corresponding coprime factorization factors are obtained as:

∞

+ ∈

= − + ∈

= +

+ ∈

= − + ∈

= −

s RH Y

s RH X s

s RH N s

s RH M s

r r

n n

600 , 3100

600 1300

60 20 2 . , 0

60 20

(5.12)

In this example, one gives two different weighting functions as in Eqs. (5.13) and (5.14) to show the flexibility in designing Q(s).

10 10 5

10 1

6 8

1 × +

= ×

W s (5.13) )

100 1 . 0 )(

10 2 (

) 120 6 . 1 ( 78 . 16

2 6 2

2 + × + +

= ₋ +

s s

W s (5.14)

W1 is a low-pass filter with high DC-gain and W₂ contains large magnitude peak at sec

10 rad . According to paragraph 2.2.2, one obtains the corresponding optimum parameters )(

1 s

Moreover, model reduction method proposed by [15] is also applied to this example and the approximated minimum phase plant G_approx is

6356

To stabilize this unstable system, an output feedback controller

given. Figure 5.7 shows these three corresponding input sensitivity functions. From Fig. 5.7, it reveals that the sensitivity functions which added W₁(s),W₂(s) are almost the same in the whole frequency ranges except a notch response located at 10 rad sec, and the rejection capability of our method is better than the one proposed by [15]. The waterbed effect caused by the RHP-zero is obviously and the limitation of crossover frequency ω_c is 47 rad sec. In time domain simulation, the reference r=0 and a complex input disturbance d_i =0.2×sin(10⋅t)+u(t−5) is given and output responses are shown in Fig. 5.8(a)-(c).

10^-2 10⁰ 10² 10⁴ -80

-70 -60 -50 -40 -30 -20 -10 0 10 20

Magnitude (dB)

Frequency (rad/sec)

47 rad/sec

Solid Line : W₂

Dash-dot Line : Reduction Method [15]

Dash Line : W₁

Fig. 5.7 Frequency responses of part 2 – example 1

0 1 2 3 4 5 6 7 8 9 10

-10 -5 0 5 10

Time(sec)

0 1 2 3 4 5 6 7 8 9 10

-1.5 -1 -0.5 0 0.5

Time(sec)

0 1 2 3 4 5 6 7 8 9 10

-1.5 -1 -0.5 0 0.5

Time(sec)

(a)

(b)

(c)

Reduction Method [15]

W₁ W₂

y_W

Fig. 5.8 (a) System outputs of simulation results of part 2 – example 1 (b)System output

y of DCFDOB with parameter ( )

1 s

Q_W (c) System output

y of DCFDOB with parameter ( )

2 s

Q_W

In Fig. 5.8(a), a large disturbance influence also remains in the output by using reduction method [15] but almost be reduced in the ones by our structure. In Figs. 5.8 (b) and (c), the output

y compensated by DCFDOB with parameter ( )

1 s

Q_W remains slight oscillation cased by the specific sinusoid disturbance and

y decays toward zero when ( )

2 s

Q_W is adopted in DCFDOB. This is beneficial and flexible in

designing when a system encounters a specific frequency of sinusoid wave type disturbance with unknown magnitude such like cogging force of constant speed motor, unbalance force of magnetic levitation rotor system and cutting force of milling system.

Example2: In the second example of part 2, one gives a simple 2× MIMO, stable 2

and non-minimum phase plant as a numerical example and the nominal plant and its state-space realization are given as follows.

⎥⎥ locates at -4.0494 and a RHP-zero locates at 5.8070. The corresponding coprime factors are obtained and given in appendix A and then the parameter Q(s) can be obtained according to section 3.2. Furthermore, assume the input disturbances is

⎥⎥

Output disturbances are given as

⎥⎥

frequency disturbances and keeping small DC-gain of S and _i S , a weighting _o

function matrix

frequency and 35.6dB DC-gain is given. The frequency responses of sensitivity

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