ROBUST H2/LQG DESIGN FOR A CROSS COUPLED PROCESS

K. Cooney1, G. Caffrey2, E Coyle3

Dublin Institute of Technology

Ireland E-mail: keith.Cooney@dit.ie 1, gerard.caffrey@dit.ie 2, eugene.coyle@dit.ie 3

Keywords: LQG, H2 Control, Modern Paradigm. Abstract This paper examines the application of modern control techniques on a cross-coupled multivariable process model perturbed by a process disturbance. The designs use time and frequency domain weights for performance and robustness criteria while minimising the generalised energy. 1. Introduction Lewis [6] stated that “naturally occurring systems exhibit optimality in their motion and so it makes sense to design man-made control systems in an optimal fashion”. Indeed the notion of optimal control has been around for many years but it is only relatively recently that practical solutions to important control problems have been available. Kalman introduced important techniques in linear quadratic design for both control and estimation problems using Lyapunov equations and state variable descriptions. Control theory entered the modern era and has grown considerably since. Kwakernaak and Siven [5], Saberi et al [9], Lublin and Athans [7] outline the solution to a range of deterministic optimal control problems in the time domain. Most notable are the l inear quadratic regulator LQR (state feedback control) and l inear quadratic gaussian regulator LQG (observer based compensation) problems, which achieve optimality (and a certain degree of robustness) by minimising a quadratic cost function thus keeping the generalised energy required to regulate the system small. Further algorithms such as loop transfer recovery LTR were developed by Doyle and Stein [3] to improve the robustness of the more practical LQG when compared to the more robust but less practical LQR. Unfortunately, these methods do not tackle robustness issues such as modell ing errors and sensitivity to process parameter variation directly as they are time domain approaches. It is possible to adjust the transient response in an iterative search but loop shaping the important singular values of the closed loop system is not the primary focus. A frequency domain approach is obviously more applicable to a robustness problem. How then to achieve both optimal H2 control and robust stabili ty to parameter changes in the process model. Solutions have been proposed by Doyle et al [2], Francis [4] and Lublin et al [8] to achieve H2 and H∞ norm specifications. Other design methods, which can be used to achieve this type of control, are based on coprime factors and yoala parameterisation, but this paper will discuss the design of state space, central multivariable controllers from both the LQ

approach and the more modern robust design of H2 and is structured to show the close mathematical relationship between them. The modern paradigm outlined by Boyd and Barrett [1] is well suited to visualise both types of problem and perform the synthesis. THE PROCESS MODEL

)(1 sU

)(2 sU

)(1 sY

)(2 sY

)(1 sd4

10

+s

1

1

+s

2

1

+s

6

8

+s

+

+ ++

++

+

+

5.0

7.0

+s

5.0

7.0

+s)(2 sd

Fig. 1: Plant and disturbance model.

The diagram in figure 1 shows the 2 input 2 output cross-coupled plant perturbed by disturbances. The cross coupling paths have slower dynamics than the feed forward paths which allows design without decoupling. Integral control is required for observer based compensation, so in these cases the plant will be augmented with integrators at the plant input i.e. on signals )(1 sU and )(2 sU . The disturbed system is realisable in

state space by the conventional equation given by (1.1). The subscript to denote time will be omitted henceforth.

)()()()( tLwtButAxtx ++=•

(1.1)

)()()( tDutCxty +=

nnRA ×∈ ; lnRB ×∈ ; nrRC ×∈ ; lrRD ×∈ ; pnRL ×∈ . The modern framework relies on the solution of the controllabil ity/observability problem. For frequency (H2) and time domain LQR/LQG designs, the condition that the system is detectable and stabilisable is sufficient for the design of linear time-invariant (LTI) controllers. The conditions of the relevant grammians were investigated using the Lyaponov equations given below.

[ ] 0=������++=

L

BLBANANN T

ccc (2)

0=++= CCANNAN Too

To

(3)

Control 2004, University of Bath, UK, September 2004 ID-092

The matrices cN and oN have all positive eigenvalues

describing a system that is both controllable and observable. The synthesis of H2 and LQ optimal controllers is therefore straightforward. 2. Time Domain Approach The control strategy focuses on the LTI infinite horizon optimal control were the steady state solution is appropriate for the design of the control law. Steady state regulation in the time domain is based on the design of an LQ full state feedback gain usually combined with an optimal statistical observer or Kalman filter. Initially, access to all the states was assumed in order to arrive at the conventional LQR solution. 2.1 Time Domain LQR The purpose of the regulator is to drive the states from initial conditions to zero while minimising the generalised energy. For this, it will be assumed that the plant is not affected by disturbances at the output, so 0)( =tLw and the state matrix ( 44×∈ RA ) does not comprise any disturbance states and is not augmented with integrators. The design criteria for an infinite horizon problem can be represented by the desire to minimise the time domain cost function (4).

[ ]� ∞++=

0)()()()(2)()( dttuRtutuRtxtxRtxJ uu

Txuxx

TS

(4)

The plant is augmented to the generalised state space form given by the following equations.

uDwDxCy

uDwDxCz

uBwBAxx

22212

12111

21

++=++=

++=•

The real system is given by the following.

BuwAxx ++=•

0

uR

wxQ

z

z ������++��

����=������ 0

002

1

uwxy 00 ++= The regulated outputs ( z vector) are the signals to be minimised. Large entries on symmetric, positive semi definite Q penalise deviation of the state from the origin of

state space. Similarly, large entries of positive R penalise the control action. Therefore the choice of CCQ Tα= and

IR β= can be made to affect the transient characteristics.

The cost function is related by QCCR T

xx == 11,

0121 == DCR Txu

, RDDR Tuu == 1212

. The term 0=xuR

implies a standard asymptotically stable LQR solution with guaranteed robustness margins i.e. gain margin is infinite and phase margin ±60°.

Conditions For Controller Realisation. The standard conditions require the plant to be at least stabil isable and detectable. For a well-posed problem 011 =D [2] is required as no exogenous signals can be allowed to affect the regulated variables directly. The weightings must satisfy the following condition.

0)(

≥����uu

Txu

xuxx

RR

RR

The optimal FSFB gain is given by the following [8].

)( 221

2 hTT

xuuuh XBRRF += − (5)

221

22220 hT

uuhrhT

rrh XBRBXGXPPX −−++= (6)

)( 12

Txuuur RRBAP −−=

)( 1 Txuuuxuxxr RRRRG −−=

Weight Selection for LQR

Initially the control weight IR β= should be considered, as the control signal cannot exceed 100%. In this respect

1=β would be the necessary choice. Holding this term constant, the speed of regulation of initial conditions on the

states to zero is set by CCQ Tα= were α is a scalar to be

varied. The larger the value of α , the more the states are penalised from deviating from the origin of state space. Figure 2 shows the state trajectories for different values of

]511.001.0[=α . An initial condition vector of

[ ]Tx 1001= is applied i.e. the cross coupling states are assumed at zero.

Fig 2: Regulation of states x1(s) and x4(s) achieved by LQR’s. The quadratic cost’s for these LQR designs are as follows: State Weight: 01.0=α 1.0=α 1=α 5=α

)( 2hLQR XTrJ = 0.0387 0.3307 1.883 4.653

Table 1: Choice of state weighting and associated LQ cost. The larger the state weighting, the larger the FSFB gain and hence the greater the LQ cost.

Control 2004, University of Bath, UK, September 2004 ID-092

2.2 Observer Design for LQG The augmented plant, which now contains integrators at the inputs and disturbances on both outputs, is given by the following with 1010×∈ RA for a 10th order system.

[ ] Buw

wLWAxx +

������+=

•

2

10

uRw

wx

Qz

z ����+��������+

����=

���� 0

00

00

0 2

1

2

1

[ ] [ ] uw

wIVCxy 00

2

1 +� ����+=

The separation structure inherent in the LQG problem means the observer and optimal FSFB gain can be designed separately. In this respect, an observer is sought to minimise the effects of the disturbances entering the system through the channel [ ]TDB 211 with joint correlation function given by

[ ] )()(

)()()(

)(

2121211

21111 τδττ −��

����=� �

��� � ��

����t

DDDB

DBBBvd

tv

tdE

TTT

TTTT

A necessary condition for the existence of an optimal solution to this dual optimisation problem is that

( ) ( ) 02121211

21111 ≥���

!!"#=��

!"#TTT

TT

yyT

xy

xyxx

DDDB

DBBB

VV

VV

Synthesis

1222 )( −+−= yyxy

Thh VVCYL (7)

221

22220 hyyT

hoT

ohho YCVCYGPYYP −−++= (8)

)( 21CVVAP yyxyo

−−=

)( 1 Txyyyxyxxo VVVVG −−=

The state space form of the dynamic compensator combining the FSFB and observer designs is given by $%

&'() −+++=

02

222222222

h

hhhhhC F

LFDLCLFBAK (9)

We seek to design desirable set point tracking and disturbance rejection characteristics based on the specifications via iterative adjustment of the constant matrices Q , R , W and V .

LQG Weight Selection Weights are chosen to meet the following specifications.

1. Zero steady state error to a step input. 2. ± 2% settling time for step input < 0.25s 3. Disturbance magnitude < 5% of step.

4. Control Signal no more than 100% Initially all weights were set to unity and set point tracking and disturbance rejection was observed with the disturbances affecting the outputs at td = 20s for Y1(s) and 45s for Y2(s). Refer to fig. 3 part (a). The first important consideration is the sensor noise affecting the outputs. Large values of V imply a lot of noise, so the optimisation will make the control action

small. A value of IV 05.0= was chosen to capture a realistic sensor inaccuracy. The result of this change on the transients can be seen in fig. 3 part (b) i.e. a large increase in ± 5% settling time (1s) for both channels and the effects of the disturbances have been reduced. The second important consideration is the control signal, which should be no more than 100%. In this respect the control weight R is kept at unity. Increasing the state weighting Q via 100=α will

speed up the response further and give a ± 2% settling time of 0.35s in both channels as well as reducing the disturbances to about 6% of the magnitude of the step. This can be referred to in part (c). W determines the amount of impact the disturbance will have on the system. By choosing a larger

value of W , the response will speed up and cause the disturbances to be attenuated further. A choice of

[ ]5.22.2diagW = achieved the time domain specification with the disturbance around 4% of the magnitude of the step and a ± 2% settling time 0.14s. Fig. 3 part (e) and (f) show enlarged versions of set-point tracking and disturbance rejection of fig. 3 (d).

Fig. 3: Tracking and disturbance rejection achieved by LQG. . The optimal LQG cost can be calculated from the equation below.

)2( 2222cos hhhht AYXQYWXTrJ ++=

3. Frequency Domain Approach 3.1 H2 (FSFB) The states are assumed to be directly measurable. The augmentation of the plant is carried out using stable, proper

Control 2004, University of Bath, UK, September 2004 ID-092

minimum phase 1st order transfer functions. These being a combination of phase lead and phase lag filters. It is assumed that the weights affect each regulated output i.e. IswW ss )(=

with sssss BAsICDsw 1)()( −−+= and IswW cc )(= with

ccccc BAsICDsw 1)()( −−+= . The augmented plant can be

described by the following equations [7].

u

B

B

w

x

x

x

A

ACB

A

x

x

x

dt

d

cc

s

c

ss

c

s ���

��

�����

++���

��

�����

���

��

�����

=���

��

�����

00

00

0

00

uD

w

x

x

x

C

CCD

z

z

cc

sc

ss ����

++���

��

���

��=��

�� 00

00

0

2

1

[ ]Tcs xxxy =

We seek to minimise the same cost function given by equation (4) and based on the state space model without the effects of the disturbances ( 01 =B ) or integral action as discussed for the LQR. The frequency-weighted version does not possess the same guaranteed robustness margins due to Rxu ≠ 0. The unique solution to this optimisation problem is achieved using equation (6) and the optimal FSFB gain is given by equation (5). The necessary system performance is achieved by the choice of weights. The controller itself uses the states of the weights as well as the process states for FSFB control. Weight Selection The state weight ws(s) is chosen to have a large magnitude were the process output is desired to be small while the control weight wc(s) is chosen to have a large magnitude were the control signal is required to be small. This provides a constraint on the bandwidth and desensitises the closed loop system to the effects of high frequency un-modelled dynamics. 3.2 Observer Design for H2 When the states are not assumed to be measurable, an observer-based dynamic compensator can be designed. The process outputs are affected by the disturbances )0( 1 ≠B as in

fig. 1 and the process inputs are augmented with integrators on

1U and 2U . The state matrix is dimensioned as 1010×∈ RA i .e.

10th order.

u

B

B

w

wL

x

x

x

A

ACB

A

x

x

x

dt

d

cc

s

c

ss

c

s ���

�

�����

+�

����

���

�

�����

+���

�

�����

���

�

�����

=���

�

�����

0

00

00

0

00

0

00

2

1

u

Dw

w

x

x

x

C

CCD

z

z

cc

sc

ss ������

+��������

����+��

���

�������

����=��

���� 0

00

00

00

0

2

1

2

1

[ ] [ ] uw

wIV

x

x

x

Cy

c

s 00002

1 +������+��

���

�����

=

The optimal estimator gain matrix for the Kalman fil ter is achieved by observing the realisation conditions set for the LQG and by calculation of a unique solution to the algebraic Ricatti equation (ARE) given by equation (8). The H2 dynamic compensator is given by equation (9). The robustness specifications depend on the weighted closed loop transfer matrix with IswW ss )(= and IswW cc )(= .

min)()()()()(

)()()(

2

2==

sSsKVWsDsSsKW

sTVWsDsSWT

CcCc

sszw

Where LAsICsD 1)()( −−= is the open loop transfer function

matrix from disturbances 1d , 2d to Y1, Y2.

BAsICsGp1)()( −−= is the open loop transfer function matrix

of the process, [ ] 1)()()( −−= sKsGIsS Cp the sensitivity

function, [ ] 1)()()()()( −−= sKsGIsKsGsT CpCpis the complementary

sensitivity function. The state and control weights shape the singular of the closed loop transfer matrix and frequency domain bounds are obtained from the following inequalities.

)(/1))()((max jwwjwDjwS s<σ (10)

))(/(1))((max jwwVjwT s<σ (11)

)(/1))()()((max jwwjwDjwSjwK cC <σ (12)

))(/(1))()((max jwwVjwSjwK cC <σ (13)

Inequality (13) is important as it guarantees the robustness of the system to an unstructured additive error defined by

)(swV c. This is as a result of the standard additive robust

stabil ity test given by the small gain theorem for a unitary negative feedback system.

1)]()()][([max <jwSjwKjwwV Ccσ (14)

Weight Selection for H2 The choice of weights was carried out in the same manner as the LQG but with the added design consideration of the weights bandwidth. We also seek transient performance to be comparable to the LQG. Initially all the weights were assumed to be unity gains as in fig. 3(a). A realistic choice of sensor noise was assigned at 05.0=V which increases the rise time (and also bandwidth) considerably as with the LQG. With the dc-gain of )(sws

at 10 and Iswc =)( , the bandwidth was

adjusted. Smaller bandwidths ofsw have the effect of

increasing rise time and settl ing time of the system and allow the disturbances to have more of an impact. Similarly, if the magnitude of

cw is large over a wide range of frequencies

especially around the bandwidth of the process the performance wil l be reduced. For a frequency domain design, it was required that the system meet the following specifications.

Control 2004, University of Bath, UK, September 2004 ID-092

4.0))()((max <sDsSσ for 1001.0 ≤≤ w

This is the performance bound for disturbance rejection and requires the choice of a state weight )(sws

. For robustness, the

standard stabil ity test (14) must be achieved [8]. This requires the choice of control weight

cw . The following weights were

found to achieve the inequalities (10) – (13) and the specifications above. The bounds on the singular values can be seen in fig. 4.

10

5.33033.0)(

++=

s

ssw s

7.39

9.19)(

++=

s

sswc

Fig. 4: Singular Values of Tzw and associated bounds. Part (a) and (d) achieve the frequency domain specifications i.e. 4.0))()((max <sDsSσ for 1001.0 ≤≤ w and the transfer

function matrix )()( sSsKC is robustly stable with respect to

the unstructured additive uncertainty given by (15) i.e. the bound in part (d).

)(

1995.005.07.39

)(swVs

ss

c

a =+

+=∆ (15)

Fig. 5: Set Point tracking (a) and Disturbance Rejection (b). Conclusion The performance has been chosen to be somewhat similar to the LQG design but the robustness bound given by fig. 4 (d) is

not tight. We can increase the robustness of the system to additive uncertainty dramatically by choice of wc(s). Choosing a weight that tightly bounds the singular values of fig. 4 part (d) will guarantee the robustness to a much larger uncertainty than specified but this will cause the performance to reduce dramatically causing settl ing times to increase and allowing disturbances to have a greater impact on regulation. This is of no surprise due to the inherent performance/robustness trade off associated with all controller design. A limitation that should be considered is the full order closed loop system realisation. The H2 design is 24th order with the LQG 20th order. Designs with such a high order may be undesirable. Therefore reduced order models should be sought if possible and any redundancy in the plant augmentation should be eliminated. These full order realisations generate 2×2 cross coupled controllers whose elements contain polynomials of very high order (n = 11 or 12) and so it has not been possible to include the actual transfer function matrices of either the LQG or H2 controllers in this paper due to space restrictions. Further work will focus on the design of state space H∞ controllers for MIMO systems. H∞ control is best associated with a robustness problem and has been of celebrated success over recent years. References

1. Boyd, S., Barrett, C., 1991, “Linear Controller

Design: Limits of Performance”, Prentice Hall. 2. Doyle, J., Glover, K., Khargonekar, P., Francis, B.,

1989, “State Space Solutions to the Standard H2 and H∞ Control Problems”, IEEE Trans. Automatic. Control., AC-34, 8, 831-847.

3. Doyle, J., Stein, G., 1981 “Multivariable Feedback

Design: Concepts for a Classical Modern Synthesis” , IEEE Trans. Automatic. Control. AC-26, 4-16.

4. Francis, B., 1987, “A Course in H∞ Theory”, Springer

Verlag.

5. Kwakernaak, H., Siven, R., 1972 “Linear Optimal Systems”, John Wiley and Sons, New York.

6. Lewis F., 1992, “Applied Optimal Control and

Estimation”, Prentice Hall and Digital Signal Processing series (Texas Instruments).

7. Lublin, L., Athans, M., 1996, “Linear Quadratic

Regulator Control” , “The Controls Handbook”, IEEE and CRC Press.

8. Lublin, L., Grocott, S., Athans, M., 1996, “H2 (LQG)

And H∞ Control” , “The Controls Handbook”, IEEE and CRC Press.

9. Saberi, A., Sannuti, P., Chen, B., 1995, “H2 Optimal

Control” , Prentice Hall.

Control 2004, University of Bath, UK, September 2004 ID-092