H∞ Closed–Loop Control for Uncertain DiscreteInput–Shaped Systems

John StergiopoulosPh.D. Researcher

Department of Electrical & Computer EngineeringUniversity of PatrasRio, Achaia 26500

GreeceEmail: stergiopoulos@ece.upatras.gr

Anthony Tzes ∗

ProfessorDepartment of Electrical & Computer Engineering

University of PatrasRio, Achaia 26500

GreeceEmail: tzes@ece.upatras.gr

ABSTRACT

The article addresses the problem of stabilization for uncertain discrete input–shaped systems. The uncertainty

affects the autoregressive portion of the transfer function of the system. A discrete input shaper compensator is

designed in order to reduce the oscillations of the plant’s response. The input–shaped system’s dynamics are

appropriately reformulated for robust controller synthesis, and a robust H∞–controller is used in an outer–loop,

in order to guarantee stability of the uncertain input–shaped plant. Simulation results confirm the efficacy of the

proposed combined scheme in comparison with open–loop input shaping and closed–loop LQ–control.

1 Introduction

Input shaping is a technique primarily used for the suppression of undesired residual vibration in oscillatory under-

damped systems [1,2]. The main advantage is that only estimates of the natural frequencies and damping ratios of the plant

under control are needed [3]. The input shaper can easily be implemented, but suffers from the fact that it is an open–loop

precompensator.

∗Corresponding Author

Robustness issues of the classical scheme have been analyzed thoroughly, while modifications have been made in the

original concept in order to decrease even more its sensitivity to uncertainties in the plant model [4]. In cases where the plant

is completely unknown, adaptive identification techniqueshave been proposed in order to tune online the shaper [3, 5, 6],

based on the current estimates of the modes of the plant. Modifications have also been made to the classical scheme when

the input to the system is constrained [7,8] or in cases of multi–mode systems [9,10]. Although these methods target linear

systems, its application to nonlinear or time–varying onesneeds special treatment [11,12].

A robust control scheme for continuous–time plants has beenproposed by [13], where the loop is closed around the

shaper and the plant, in order to deal with uncertainties in the system model. In this paper, a robust control scheme for linear

discrete–time systems is proposed in order to face the system’s uncertainties. Due to the discrete nature of the plant, “dis-

crete” classical shapers must be applied [14,15]. AnH∞–controller is designed to close the loop around both the shaper and

the uncertain plant, while guaranteeing stability of the closed–loop system for any uncertainty laying inside predetermined

interval sets [16].

Most of the effort made in previous works was in designing closed–loop controllers responsible to stabilize the plant

itself, while in the sequel an input–shaper compensator wasdesigned to amend for suppressing residual vibration of the

controlled plant [17,18]. The main difference of the proposed scheme compared to others in the existing literature is that the

controller is implemented around the input–shaped plant. The insertion of the shaping filter in the main loop of the complete

control scheme can destabilize the control system in the existence of uncertainties, and a proper controller is required for

robust stabilization.

In Section 2 the relationship between discrete and continuous–time LTI systems is discussed. In Section 3 the discrete

version of the classical input shaping technique is presented, while in Section 4 the design of anH∞–controller for the

stabilization of input–shaped discrete systems is analyzed. Finally, in Section 5 simulation results are provided in order to

prove the efficacy of the proposed controller in comparison to those obtained by standard input shaping open–loop control

and closed–loop LQ–control. Concluding remarks are provided in the last section.

2 System Description

Consider the discrete SISO LTI plant described as:

G(

z−1) =B

(

z−1)

A(z−1)=

n

∏i=1

Gi(

z−1) =n

∏i=1

Bi(

z−1)

Ai (z−1), (1)

where

Bi(

z−1) =nb

∑j=1

b j,iz− j , Ai

(

z−1) = 1+a1,iz−1 +a2,iz

−2. (2)

The number of modesn, and theBi polynomials are assumed to be known a–priori.

If the coefficientsa1,i ,a2,i are known exactly, then the plant is a nominal one, consistedof G i (z−1), where superscript

“” denotes the nominal case. In this paper theAi–polynomials’ coefficients are considered to be constant but unknown,

residing in a subset ofR2n. The solutionsρz,i of the characteristic equationsAi(

z−1)

= 0 are complex (not real) and lay

inside the unit circle in thez–plane.

The coefficientsa1,i ,a2,i of polynomialsAi can be associated with the natural frequenciesωi and damping ratiosζi ,

0 < ζi < 1, of their equivalent s–plane eigenvalues as:

ρz,i = −a1,i

2± j

√

a2,i −a2

1,i

4, (3)

ωi = ‖ln(ρz,i)‖ , (4)

ζi = −cos

(

arctan

(

ℑln(ρz,i)ℜln(ρz,i)

))

, (5)

whereℑ andℜ denote the imaginary and real part of a complex variable, respectively. The inverse mapping(ωi ,ζi) →

(a1,i ,a2,i) can similarly be computed via the following formulae:

ρz,i = e−ζiωi± jωi

√1−ζ 2

i , (6)

a1,i = −2ℜρz,i , (7)

a2,i = ‖ρz,i‖2 . (8)

The plant (1) is underdamped and stable at the same time when 0< ζi < 1 and 0< ωi < ωN, ∀i = 1, . . .n, where

ωN is the underlying Nyquist frequency. Furthermore, according to Jury’s criterion, the condition for (1) to be stable is

(a1,i ,a2,i) ∈ Ω =

(a1,i ,a2,i) ∈ R2 : (−1 < a2,i < 1)

⋂

(a2,i > a1,i −1)⋂

(a2,i > −a1,i −1)

, ∀i = 1, . . .n. Figure 1 shows

graphically the admissible workspace for each pair of the parametersa1,i ,a2,i for stable underdamped systems. The triangle

defines the stability regionΩ. The variations of the corresponding natural frequenciesωi and damping ratiosζi , shown in

Fig. 1, are computed by (3)–(5). It can be seen that the valuesof ωi are large for positive values ofa1,i , while negative values

of a1,i correspond to large values ofζi .

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

1.5

a1,i

a2,i

ωi

0.5

1

1.5

2

2.5

3

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

1.5

a1,i

a2,i

ζi

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 1. Admissible workspace of the parameters a1,i ,a2,i and variations of ωi ,ζi for stable underdamped plants.

When dealing with uncertain plants, usually some bounds are set on the amount of uncertainty in the system model. In

this paper the uncertainty lays in the polynomial coefficients of the denominator. Thus, given the bounds ona1,i ,a2,i , one can

a–priori know the uncertainty in the natural frequency and damping ratio of each mode of vibration via Fig. 1. Alternatively,

if bounds are set onωi andζi [19], the corresponding ones fora1,i ,a2,i can be obtained by (6)–(8).

3 Input shaping technique for discrete systems

3.1 Standard input shaping technique for continuous–time systems

The classical input shaping technique is used in most of the cases where the system to be controlled has an oscillatory

response due to lightly–damped eigenvalues. A shaper is a precompensator consisting of a number of impulses with which

the reference signal is convolved, so that the system reaches the desired state with no (or minimum) oscillation.

In cases where the system under control has multiple modes—like the one defined in (1)—then for each modei a shaper

is designed, based on the estimates ofωi andζi in order to cancel the oscillating effect of the current mode. Then shapers

are then convolved with each other to result in a train of∏ni=1Ni impulses, whereNi are the number of impulses of each

shaper, as shown in Fig. 2. It should be noted that the number of impulses in each shaper design can differ between each

other, depending on the amount of uncertainty in the currentmode parametersωi , ζi ; the aforementioned method may result

in a shaper that does not behave well when the original shapers have negative impulses [7,8,20]. An alternative method for

dealing with multi–mode systems is to solve all vibration constraints at the same time, leading in shapers with less number

of impulses [9,10,21,22].

mode 1 mode n

A1,1

t1,1t1,n t2,n t3,n

tN n,t2,1tN ,1

A1,n

A2,n

A3,n

AN n,

A2,1AN ,1

1

n

n1

Fig. 2. Implementation of an n–modes shaper via convolution of independent ones.

A standard shaper ofNi impulses filters its input according to the following equation:

rshi (t) = r i (0)+

Ni

∑j=1

A j,i [r i (t − t j,i)− r i (0)], (9)

wherer i(t) andrshi (t) are the signals entering and exiting thei–th shaper, respectively,Ni is the length of the impulse train,

A j,i andt j,i are the amplitudes and time–instances of each impulse. The latter can be computed by [1]:

A j,i =

(Ni−1j−1

)

K j−1i

∑Ni−1ℓ=0

(Ni−1ℓ

)

Kℓi

, (10)

t j,i = ( j −1)π

ωi

√

1−ζ 2i

, (11)

Ki = e−ζiπ/√

1−ζ 2i , (12)

whereωi andζi are the natural frequency and damping ratio of the mode to be suppressed.

Considering (10)–(12), it is profound that the design of a shaper depends only on the natural frequency and damping

ratio of the modes of the nominal plant. Although the zeros ofthe system under control affect its response, they are not

needed in the shaper design procedure.

3.2 Discrete input shaping technique

In cases, though, where the system to be controlled is discrete by nature or its input is given in standard sample–instances,

it is clear that the impulses of a standard shaper cannot be applied exactly at the desired time–instances.

This problem is remedied [14,23] by applying two impulses instead of one at two consecutive sample–instances,k and

k+1, such thatkT < t j,i < (k+1)T, whereT is the sampling period. Other approaches to handle this issue have appeared

in [15]. Consequently, ifNi is the number of impulses of a standard shaper, then the discrete version consists ofN∗i = 2Ni −1

impulses. After determining the “new” time–instancest∗j,i at which the impulses are to be applied, then their amplitudes A∗j,i

are computed by:

A∗1,i

A∗2,i

...

A∗N∗

i −1,i

A∗N∗

i ,i

=

m1,1,i . . . m1,N∗i ,i

m2,1,i . . . m2,N∗i ,i

..... .

...

mN∗i −1,1,i . . . mN∗

i −1,N∗i ,i

1 . . . 1

−1

0

0

...

0

1

, (13)

wheremk, j,i are defined as:

mk, j,i =

t∗⌊ k+1

2 ⌋−1

j,i e−ζiωi

(

t∗N∗

i ,i−t∗j,i

)

sin

(

t∗j,iωi

√

1−ζ 2i

)

, k odd

t∗⌊ k+1

2 ⌋−1

j,i e−ζiωi

(

t∗N∗

i ,i−t∗j,i

)

cos

(

t∗j,iωi

√

1−ζ 2i

)

, k even

. (14)

Once the appropriate amplitudesA∗j,i and the time–instancest∗j,i of the impulses are computed for each modei, the input

shaper transfer functionScan be computed as the convolution of then separate shapers:

S=n

∏i=1

N∗i

∑j=1

A∗j,iz

−t∗j,iT . (15)

It should be noted that the sampling period of a sampled–datacontinuous–time system affects the response of an input–

shaped plant only when the sampling period is large enough. Since, the plants under consideration in this article are purely

discrete ones, the discrete version of the input shaper compensator is not affected by the selected sampling period, since the

delays that the latter induces in the system dynamics are sample–delays rather than time–delays, considering (15).

4 H∞ robust controller synthesis

4.1 Uncertainty model

A discrete input shaper of Section 3 is designed for each of the nominal plantsG i . However from a practical point of

view, there is always some uncertainty in the model of the plant. In this paper the parametric uncertainty is assumed to be

embedded within the coefficientsa1,i ,a2,i of theAi–polynomials of denominator. Let each of the coefficientsa1,i ,a2,i lay in

a rectangular regionAi =

(a1,i , a2,i) ∈ R2 : amin

j,i ≤ a j,i ≤ amaxj,i , j = 1,2

. Thus, the latter can be written as:

a j,i = aj,i + r j,iδ j,i , j = 1,2 , i = 1, . . .n, (16)

whereaj,i represents the nominal values andδ j,i are unknown constants for which∥

∥δ j,i∥

∥

∞ < 1 holds; by selectingaj,i =

12

(

amaxj,i +amin

j,i

)

, r j,i can then be expressed in terms of the bounds for each parameter as:

r j,i ,amax

j,i −aminj,i

2= amax

j,i −aj,i = aj,i −aminj,i . (17)

Let us denote the serial interconnection of the discrete input shaper with the nominal plant as:

L (

z−1) = S(

z−1)G (

z−1) . (18)

It should be noted thatL is an FIR filter, thus its impulse response reaches equilibrium in finite time (more specifically in

∑ni=1

t∗N∗

i ,i

T samples, whereN∗i is the number of impulses of the discrete shaper designed formodei). Substitution of (16) into

(1) results in:

Gi(

z−1) =G

i

(

z−1)

1+G i (z−1)(W1,i (z−1)δ1,i +W2,i (z−1)δ2,i)

, (19)

where the weighting filtersWj,i are defined as:

Wj,i(

z−1) ,r j,iz− j

Bi (z−1), j = 1,2. (20)

Considering (19), the uncertain input–shaped plant can be transformed as shown in Fig. 3. Note thatr is an exogenous

signal,y is the input–shaped plant’s output,e , y− r is the error to be kept small andu is the output of an outer–loop

controller that is to be designed.

r + + ++

++

++

-

- --

u

ud

ud ud

ud ydyd

ydyd ud

ud

e

y

S Gn

oG1

o

W1,nW1,1

W2,nW2,1

d1,nd1,1

d2,nd2,1

2,1 2,1

1,1 1,1

2,n 2,n

1,n

1, -1n 2, -1n

1,n

=

Fig. 3. Interconnection of blocks considering parametric uncertainty for the underdamped plant under consideration.

4.2 H∞–controller design

In order to increase the insensitivity of the open–loop input–shaped system with respect to the uncertainty in the denom-

inators’ coefficients, anH∞ closed–loop controller is to be designed. According to the design procedure [16], a generalized

block–form of the system representation must be firstly derived, as shown in Fig. 4;∆ is the uncertainty matrix of the system

in block–diagonal form,P is the interconnection matrix, andK is the output–feedback controller to be designed. It should

udyd

u

r e

y

D

P

K

Fig. 4. Generalized block–form of an uncertain system under control.

be noted that:

uδ =(

uδ1,1,uδ2,1

, . . .uδ1,n,uδ2,n

)T,

yδ =(

yδ1,1,yδ2,1

, . . .yδ1,n,yδ2,n

)T,

∆ = diagδ1,1,δ2,1, . . .δ1,n,δ2,n .

In order to find an analytic expression for the transfer function matrixP of Fig. 4, the signalsuδ ,y,e must be expressed

in terms ofyδ ,u, r. For the system under consideration, this can be obtained byobserving the corresponding signals’ flow in

Fig. 3 as:

uδ1,i= uδ2,i

=

G 1

(

−W1,1yδ1,1−W2,1yδ2,1

+S(r −u))

, i = 1

G i

(

−W1,iyδ1,i−W2,iyδ2,i

+uδ1,i−1

)

, i = 2, . . .n

, (21)

y = uδ1,n= uδ2,n

, (22)

e = y− r = uδ1,n− r. (23)

The non–recursive expression of (21) is:

uδ1,i= uδ2,i

= −i

∑k=1

W1,k

i

∏m=k

G myδ1,k

−i

∑k=1

W2,k

i

∏m=k

G myδ2,k

+Si

∏k=1

G k r −S

i

∏k=1

G k u. (24)

Consequently, the full interconnection matrixP of the system (considering Fig. 4) can be derived by (22)–(24) in the form

shown in Fig. 5.

0 0 0 0

1,1 1 2,1 1 1 1

0 0 0 0

1,1 1 2,1 1 1 1

0 0 0 0 0 0 0 0 0 0

1,1 1 2 2,1 1 2 1,2 2 2,2 2 1 2 1 2

0 0 0 0 0 0 0 0 0 0

1,1 1 2 2,1 1 2 1,2 2 2,2 2 1 2 1 2

10

1,1

1

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0

n

i

i

W G W G SG SG

W G W G SG SG

W G G W G G W G W G SG G SG G

W G G W G G W G W G SG G SG G

W G W-

=

- - -

- - -

- - - - -

- - - - -

- -Õ

L

L

M M O M M

1 1 1 1 10 0 0 0 0 0 0

2,1 1,2 2,2 1, 1 1 2, 1 1

1 2 2 1 1

1 1 1 1 10 0 0 0 0 0 0 0

1,1 2,1 1,2 2,2 1, 1 1 2, 1 1

1 1 2 2 1 1

0 0

0 0

n n n n n

i i i n n n n i i

i i i i i

n n n n n n

i i i i n n n n i i

i i i i i i

G W G W G W G W G S G S G

W G W G W G W G W G W G S G S G

- - - - -

- - - -

= = = = =

- - - - - -

- - - -

= = = = = =

- - - - -

- - - - - - -

Õ Õ Õ Õ Õ

Õ Õ Õ Õ ÕL

1

0 0 0 0 0 0 0 0 0 0

1,1 2,1 1,2 2,2 1, 1 2, 1 1, 2,

1 1 2 2 1 1 1 1

0 0 0 0 0

1,1 2,1 1,2 2,2 1, 1 2, 1

1 1 2 2 1

n n n n n n n n

i i i i n i n i n n n n i i

i i i i i n i n i i

n n n n n

i i i i n i n

i i i i i n

W G W G W G W G W G W G W G W G S G S G

W G W G W G W G W G W

- -

= = = = = - = - = =

- -

= = = = = -

- - - - - - - - -

- - - - - -

Õ

Õ Õ Õ Õ Õ Õ Õ Õ

Õ Õ Õ Õ Õ

L

0 0 0 0 0

1, 2,

1 1 1

0 0 0 0 0 0 0 0 0 0

1,1 2,1 1,2 2,2 1, 1 2, 1 1, 2,

1 1 2 2 1 1 1 1

0 0

1,1 2,1 1,2

1 1

1

n n n

i n n n n i i

i n i i

n n n n n n n n

i i i i n i n i n n n n i i

i i i i i n i n i i

n n

i i

i i

G W G W G S G S G

W G W G W G W G W G W G W G W G S G S G

W G W G W

= - = =

- -

= = = = = - = - = =

= =

- - -

- - - - - - - - - -

- - -

Õ Õ Õ

Õ Õ Õ Õ Õ Õ Õ Õ

Õ Õ

L

0 0 0 0 0 0 0 0

2,2 1, 1 2, 1 1, 2,

2 2 1 1 1 1

n n n n n n

i i n i n i n n n n i i

i i i n i n i i

G W G W G W G W G W G S G S G- -

= = = - = - = =

- - - - - -Õ Õ Õ Õ Õ ÕL

yd ud

r e

u y

M M

Fig. 5. Analytic form of interconnection matrix P for the uncertain system under consideration.

The objective is to find an output–feedback controllerK(

z−1)

that minimizes the following cost function:

‖Tw→z‖∞ = supω

σmax(Tw→z ( jω)) , (25)

wherew =(

yTδ , r

)Tis a vector containing the exogenous signals,z =

(

uTδ ,e

)Tis a concatenation of the signals to be kept

small, andTw→z is the transfer function matrix fromw to z. Practically, the cost (25) represents minimization of theeffect

of uncertainty on the signals that need to be kept small.

The state–space realization of the interconnection transfer function matrixP [24] (shown in Fig. 6) can be written as:

P :

x = Ax+B1w+B2u

z = C1x+D11w+D12u

y = C2x+D21w+D22u

, (26)

wherex is the corresponding state vector, whileu andy are the output and input to the controllerK(

z−1)

, respectively. The

aforementioned minimization problem is transformed into the existence of symmetric solutionsR,Sof the following system

A B1

C1 D11

D21

D12

D22C2

B2w z

u y

Fig. 6. Packed state–space realization of interconnection matrix P.

of LMIs [25]:

N12 0

0 I

T

ARAT −R ARCT1 B1

C1RAT −γI +C1RCT1 D11

BT1 DT

11 −γI

N12 0

0 I

< 0, (27)

N21 0

0 I

T

ATSA−S ATSB1 CT1

BT1SA −γI +BT

1SB1 DT11

C1 D11 −γI

N21 0

0 I

< 0, (28)

R I

I S

≥ 0, (29)

whereN12 andN21 denote bases of the null spaces of(

BT2 ,DT

12

)

and(

CT2 ,DT

21

)

, respectively, whileγ is a positive real constant.

The aforementioned optimization problem is a convex one [25]; thus, the solutionK(

z−1)

can be obtained numerically via

“γ–iteration algorithm” implemented in specialized software toolboxes [26], for the minimum possible positive value ofγ

satisfying (27)–(29). Essentially,γ represent the minimum value of the cost defined in (25),‖Tw→z‖∞ ≤ γ, for which the

existence of stabilizing controller is guaranteed.

5 Simulation results

In this section theH∞–controller described in Section 4 is designed for two input–shaped discrete stable underdamped

plants. In the first example the system has one mode of vibration, while in the second one a dual–mode plant is examined.

In both cases, results are to be compared to LQ state–feedback control of the input–shaped systems examined. According

to the LQR design procedure [27], the goal is to find the optimal static–gains matrixK, such that the state–feedback control

law u [k] = −Kx [k] minimizes the quadratic cost function:

J(u) =∞

∑k=0

(

x [k]T Qx [k]+u [k]T Ru [k]+2x [k]T Nu [k])

(30)

for a discrete plant described byx [k+1] = Ax [k]+Bu [k]. The LQ–controller is designed for the nominal input–shaped plant

L; thus, it does not take into account the uncertainty in the plant model, contrary to theH∞–controller. The controllers are

compared via their performance at regulation problems.

5.1 Single–mode uncertain system

Let the nominal plant under control be:

G =0.05z−1 +0.05z−2

1+0.65z−2

The corresponding natural frequency and damping factor areω1 = 1.5855 andζ1 = 0.1359, respectively. The impulse

response of the nominal plant is shown in Fig. 7. Let the system parameters be bounded as−0.9 < a1,1 < 0.9 and 0.4 <

0 5 10 15 20 25 30−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

samples

y

Fig. 7. Impulse response of the nominal single–mode plant.

a2,1 < 0.9. It should be noted thatall systems of the form (1) are stable for the given bounds ofa1,1,a2,1.

In order to eliminate the oscillations in the response of thenominal plant, a discrete input shaper is designed. Two kinds

of shapers are examined: a Zero–Vibration shaper (ZV) consisted of two impulses and an Extra–Insensitive (EI) consisted of

three impulses. The choice of an EI shaper was based on the fact that the amount of uncertainty in the system model is large

enough (considering the bounds on the denominator’s polynomial–coefficients). The expressions that provide the impulses’

amplitudes and time–instances of the EI shaper, given the mode’s natural frequency and damping ratio, can be found in [4],

while the corresponding ones for the ZV are (10)–(12). The nominal system’s responses when controlled in open–loop with

each of the shapers are shown in the left (ZV) and right (EI) part of Fig. 8 (solid line), respectively. As expected, after afew

samples the oscillation has been eliminated completely in both cases.

0 5 10 15 20 25 30−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

samples

y

open-loop (ZV )

LQR + ZV

H∞ + ZV

0 5 10 15 20 25 30−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

samples

y

open-loop (EI)

LQR + EI

H∞ + EI

Fig. 8. Impulse response of the nominal ZV (left) and EI (right) input–shaped single–mode plant a) in open–loop (solid), b) with LQ closed–

loop control (dotted), c) with H∞ closed–loop control (dashed–dotted).

In the sequel, two LQ state–feedback controllers are designed for the nominal input–shaped plantsL. The matrices

in the cost (30) are chosen asQ = 0.1I, R= 10, N = 51, whereI (1) is the identity matrix (column vector with ones) of

appropriate dimensions. These parameters were selected inan ad–hoc manner so that the control effort for the LQR and the

H∞ has approximately similar maximum values. The system’s responses are shown in the left (ZV) and right (EI) part of Fig.

8 (dotted line). The resulting LQ–controllers are:

K = ( 0.4556, 0.5141) : ZV case,

K = ( 0.3720, 0.4244, 0.4755, 0.5243) : EI case,

for the ZV and EI cases, respectively. Finally, twoH∞–controllers (one for each case) are designed to close the loop around

the shaper and the plant, such that all closed–loop systems for anya1,1,a2,1 residing inside the uncertainty box are stable,

which are derived as:

KZV (

z−1) =−0.4−21.85z−1−0.54z−2−14.52z−3−0.18z−4−0.21z−5

1+1z−1 +1.3z−2 +1.3z−3 +0.42z−4 +0.42z−5 ,

KEI (z−1) =−0.41−36.1z−1−36.52z−2−24.62z−3−24.35z−4−1.15z−5−0.72z−6−0.26z−7−0.13z−8

1+2z−1 +2.95z−2 +3.9z−3 +3.21z−4 +2.53z−5 +1.54z−6 +0.55z−7 +0.27z−8 .

The impulse responses of the closed–loop system are shown inthe left (ZV) and right (EI) part of Fig. 8 (dashed–dotted

line). Although the system does not reach equilibrium as soon as in the open–loop case, it still has a very fast response, since

almost after the tenth sample the oscillations are eliminated, in both cases.

It should be noted that ifN = 0 in (30), the resulting LQ–controller results inK = 0, since theL system (input shaper

in cascade with the nominal plant) corresponds to a stable Finite–Impulse–Response (FIR) filter. Any nonzero feedback will

convert this system to an Infinite–Impulse–Response (IIR) system, resulting in a larger costJ(u) in (30).

Let now the true values for the uncertain parameters bea1,1 = 0.72, a2,1 = 0.875, which correspond toδ1,1 = 0.8, δ2,1 =

0.9. The system’s responses when controlled in open–loop withthe shapers, in closed–loop with the LQ–controllers, and

in closed–loop with theH∞–controllers are shown in the left (ZV) and right (EI) part ofFig. 9. The oscillations in the

0 5 10 15 20 25 30−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

samples

y

open-loop (ZV )

LQR + ZV

H∞ + ZV

0 5 10 15 20 25 30−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

samples

y

open-loop (EI)

LQR + EI

H∞ + EI

Fig. 9. Impulse response of the perturbed ZV (left) and EI (right) input–shaped single–mode plant a) in open–loop (solid), b) with LQ

closed–loop control (dotted), c) with H∞ closed–loop control (dashed–dotted).

H∞ closed–loop case have almost been eliminated after the twelfth sample in contrast with the open–loop and LQR case,

where there is a quite large amount of residual vibration. Interesting though is the fact that the LQ–controller is incapable of

stabilizing the input–shaped plant for anya1,1,a2,1 residing inside the uncertainty box. In fact, the perturbedLQ–controlled

closed–loop ZV input–shaped system becomes unstable, since two unstable poles (−0.4169± j 0.9109) arise for the afore-

mentioned uncertainty.

5.2 Dual–mode uncertain system

In this example, the examined system has two modes of vibration, while it still is stable and underdamped. Let the

nominal plant be:

G =

(

0.005z−1 +0.004z−2

1−0.5z−1 +0.4z−2

)(

0.08z−1 +0.12z−2

1−0.1z−1 +0.5z−2

)

The corresponding natural frequency and damping factor are: ω1 = 1.2513, ω2 = 1.5395 andζ1 = 0.3661, ζ2 = 0.2251,

respectively. The impulse response of the nominal plant is shown in Fig. 10. The system’s parametersa1,1,a2,1 anda1,2,a2,2

0 5 10 15 20 25 30−1

−0.5

0

0.5

1

1.5x 10

−3

samples

y

Fig. 10. Impulse response of the nominal dual–mode plant.

are uncertain, while their bounds are given by−0.6 < a1,1 < −0.4, 0.3 < a2,1 < 0.5 and−0.2 < a1,2 < 0, 0.4 < a2,2 < 0.6.

Since the bounds on theAi polynomial coefficients are not as large as in the single–mode example, two discrete ZV input

shapers of lengthsN1 = N2 = 2 are chosen to be designed for the dual–mode system. The nominal system’s response when

controlled in open–loop with the convolution of the shapersis shown in Fig. 11 (solid line), in which after seven samplesthe

oscillation has been eliminated.

0 5 10 15 20 25 30−2

−1

0

1

2

3

4

5

6x 10

−4

samples

y

open-loop (ZV )

LQR + ZV

H∞ + ZV

Fig. 11. Impulse response of the nominal ZV input–shaped dual–mode plant a) in open–loop (solid), b) with LQ closed–loop control (dotted),

c) with H∞ closed–loop control (dashed–dotted).

In the sequel, an LQ state–feedback controller was designed, where the design parameters were chosen asQ= 0.1I, R=

3, N = 51; its static–gains matrix is given by:

K = ( 0.8093, 1.1011, 1.5799, 1.6654, 1.7126, 1.7235) .

The system’s response when controlled in closed–loop with the LQ–controller is shown in Fig. 11 (dotted line). AnH∞

controller is also designed to close the loop around the shaper and the plant, such that all closed–loop systems for any

a1,1,a2,1 anda1,2,a2,2 residing inside the uncertainty boxes are stable. The resulted controller from the optimization scheme

has 42 states and its zero–pole map is shown in Fig. 12 (crosses (×) stand for the poles, while circles () are for the zeros).

The impulse response of the closed–loop system when theH∞–controller is implemented is shown in Fig. 11 (dashed–dotted

line). As it is seen, both the open–loop and theH∞ closed–loop schemes handle the oscillations for the nominal system.

Let now the real values for the uncertain parameters bea1,1 = −0.59, a2,1 = 0.49 anda1,2 = −0.18, a2,2 = 0.58, which

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real axis

Imagin

ary

axis

Fig. 12. Zero–pole locations of the designed H∞–controller for the dual–mode plant.

correspond toδ1,1 = −0.9, δ2,1 = 0.9 andδ1,2 = −0.8, δ2,2 = 0.8. The system responses when controlled in open–loop

with the shaper, in closed–loop with the LQ–controller, andin closed–loop with theH∞–controller are shown in Fig. 13.

Comparing Fig. 11 and Fig. 13, it is seen that the combined input–shaping/H∞ control scheme is more robust to uncertainties

in the system modeling.

It should be noted that theH∞–controller guarantees stability of the closed–loop input–shaped system in contrast with

LQR scheme that can possible destabilize the system in presence of uncertainties (as shown in the first example). Although

robustness of the closed–loop system is increased via utilization of theH∞ controller, the latter is more complex (from an

implementation point of view) compared to open–loop/LQ schemes. However, approximation techniques exist in order to

reduce the order of the resulting controller [16], though these are beyond the scope of this article.

6 Conclusions

In this paper a combinedH∞ and input shaping control scheme has been designed for uncertain discrete LTI systems.

When the nominal plant is controlled with the discrete version of the input shaping technique, the latter reaches equilibrium

in finite time. However, the system response deteriorates incases where there is parametric uncertainty in the system model.

The control scheme proposed is the design of a robustH∞–controller to close the loop around both the shaper and the plant.

0 5 10 15 20 25 30

−2

−1

0

1

2

3

4

5

6x 10

−4

samples

y

open-loop (ZV )

LQR + ZV

H∞

Fig. 13. Impulse response of the perturbed ZV input–shaped dual–mode plant a) in open–loop (solid), b) with LQ closed–loop control

(dotted), c) with H∞ closed–loop control (dashed–dotted).

Stability of the closed–loop system is guaranteed for the range of uncertainty for which theH∞–controller has been designed.

Simulation results show the efficacy of this scheme in comparison with the standard open–loop input shaping and LQ control.

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List of figures’ captions:

1. Admissible workspace of the parametersa1,i ,a2,i and variations ofωi ,ζi for stable underdamped plants.

2. Implementation of ann–modes shaper via convolution of independent ones.

3. Interconnection of blocks considering parametric uncertainty for the underdamped plant under consideration.

4. Generalized block–form of an uncertain system under control.

5. Analytic form of interconnection matrixP for the uncertain system under consideration.

6. Packed state–space realization of interconnection matrix P.

7. Impulse response of the nominal single–mode plant.

8. Impulse response of the nominal ZV (left) and EI (right) input–shaped single–mode plant a) in open–loop (solid), b)

with LQ closed–loop control (dotted), c) withH∞ closed–loop control (dashed–dotted).

9. Impulse response of the perturbed ZV (left) and EI (right)input–shaped single–mode plant a) in open–loop (solid), b)

with LQ closed–loop control (dotted), c) withH∞ closed–loop control (dashed–dotted).

10. Impulse response of the nominal dual–mode plant.

11. Impulse response of the nominal ZV input–shaped dual–mode plant a) in open–loop (solid), b) with LQ closed–loop

control (dotted), c) withH∞ closed–loop control (dashed–dotted).

12. Zero–pole locations of the designedH∞–controller for the dual–mode plant.

13. Impulse response of the perturbed ZV input–shaped dual–mode plant a) in open–loop (solid), b) with LQ closed–loop

control (dotted), c) withH∞ closed–loop control (dashed–dotted).