6.2. 1 Comparison of LQG and Hoo designs . . . ... . . . ... . . ... . . . ... . . 1 06 6.2.2 Proposed robust LQG design approach . . ... . . ... . . ... . . ... . .
107
6.3. Design of Robust LQG Controllers .. . . . ... . . ... . . ... . ... . . 1 086.3. 1 Systems with norm bounded uncertainty . . ... . . ... . . .. . . .. . . 1 08
6.3.2 Systems with matched norm bounded uncertainty . .. . . .. .. . . 1 1 5 6.4 Examples . . .
1 1 7
6.5 Discussion . . . ... ... . . ... . . ... .. ... .. ... . . .. . .. . . ... . . . .. . 1 226. Robust Controller 1 04
From chapter
3,
the standard LQG control design theory can not guarantee good stability robustness and its stability margin can be arbitrarily small, the existence of system uncertainty may degrade the system performance or even destabilise the closed loop system. So in recent years, increasing attention has been paid to the design of dynamic output feedback controllers that can stabilise the uncertain system and guarantee the cost performance, the Hoo performance or both to lie within a specified bound. For the system with modelled uncertainties and residual unknown uncertainty, a design technique of the robust state feedback controller (RLQR) has been presented in chapter 4. It is proved that this robust LQR can provide optimal robustness of stability and performance. However, since it is often not possible to measure all state variables, the output feedback controller is more practicable.In this chapter, a new design methodology has been developed for robust dynamic output feedback controllers that is applicable for systems with parametric uncertainty. The method guarantees robust stability of the system for all uncertainties within a given admissible domain, the magnitude of this domain is treated as a design parameter.
As
discussed in §2.2.4, for simplicity, the residual unknown uncertainty is considered to be nonparametric, robustness subject to it may be measured by the Hoo norm bound of the system. Thus the desired Hoo norm bound is also treated as a design parameter for the method. Cost performance with respect to a quadratic cost function is explicitly considered and the problem is initially posed in the LQG format. A relationship established between LQG solutions and Hoo solutions allows the robust LQG design problem to be translated to an Hoo problem with explicit reference to the cost function. Thus the controller is designed by employing the established solution technique to the Hoo problem for the specified level of non parametric robustness. System performance is also inherently considered in the design with respect to a quadratic cost function. Though no optimality is proved, informal analysis and example applications have produced good performance subject to the robustness constraints specified. In essence the method enables the designer to trade off robustness to parametric uncertainty, robustness to nonparametric uncertainty and cost performance.After the problem statement, a robust LQG design method will be developed for the system with only nominal model in §6.2, then the system with parametric uncertainties will be considered in §6.3, the robust LQG controller will be found for the system with norm bounded or matched norm bounded uncertainties, it will also be demonstrated how the
RLQG controller design method can be posed as an Hoo control design problem for a scaled certain system hence allowing the existing Hoo numerical techniques to be used. In
6. Robust Controller 1 05
6. 1 PROBLEM STATEMENT
As
mentioned in
§2.2.3,both parametric and nonparametric uncertainties effect the
stability and perfonnance robustness of dynamic output feedback controlled systems, to
avoid conservative design, for a system with modelled (parametric) uncertainty, robust
controller design should refer to the Wlcertainty model. The system with modelled
uncertainty to be studied may
bedescribed by the following state-space representation:
x(t) = (A + M(t»x(t) + (B2 + Ml(t»u(t) + Ed
yet) = (C2 + AC(t» x(t) + Fv
(6. 1 . 1 )Where x
E9\0 is the state vector, u
E9\m is the control vector,
y E9\r is the observation
vector,
dis the vector of disturbance inputs and
Vis the vector of measurement noises.
The system is linear and
alldisturbance and noise processes are assumed to be
uncorrelated Gaussian white noise processes, and have covariances W and V respectively.
Related to
§2.2. 1 ,the plant uncertainties M ( t), Ml(t), AC(t) are assumed to lie within an
admissible domain and may
bedescribed by:
{
M (t) = N.<1>l (t)M.
n = AB( t) =Nb<1>2 (t)Mb
AC(t) = Nc<1>3(t)Mc
(6. 1 .2)
Where
N. , M. , Nb , Mb , Nc , Mc
are constant matrices describing the structure of the
admissible domain and the magnitude is constrained by a(<1>j (t» $ E for i= l , 2 and
3,where, a denotes maximum singular value.
Asa special case of this,
bounded uncertainties may be described by
n = {[M ( t), AB (t)] = N.b<1>(t)[M. , M b ]
AC( t) = Nc<1>{t)Mc
with a(<l>(t» $ E .
matched nonn
(6. 1 .3)
The controller is to be designed with reference to a quadratic cost criterion of the fonn:
1 T T
J = lim
(x Qx + u Ru)dt
(6. 1 .4)To -400
Where R
> 0and Q
�O.
The design criteria are to produce a dynamic output feedback controller that
willguarantee stability for
alladmissible values of parametric uncertainties M (t) , AB(t) and
AC(t) , to guarantee an Hoc nonn bound such that a prescribed level of non parametric
plant Wlcertainty AG(s) may
betolerated. Furthennore, the reference cost function
is1 06
implicitly considered in the controller synthesis and good cost perfonnance is sought subject to the above robustness constraints.
6.2 DESIGN OF ROBUST LQG CONTROLLERS BASED ON NOMINAL
MODELS
In §3.2. 1 and §3.2.3, the LQG and Hoo controller design techniques are presented, and from the robustness analysis in §2.4. 1 , it follows that the robustness of the closed loop system against nonparametric uncertainty can be measured by the Hoo nonn bound of the transfer function
Tuoyo
(s) , the smallerI/TuoYo (S)II..
the greater the robustness. In this section, a design approach is proposed in this section which offers a compromise between cost perfonnance and robustness against nonparametric uncertainty, and it is also found that the LQG and Hoo controller design techniques are equivalent under certain conditions. This leads to the development of a robust LQG controller design approach that enables a compromising controller to be found.6.2. 1 Comparison of LQG and
Hoodesigns
The certain part of the system (6. 1 . 1 ) will be considered here
x
= Ax + B2u +Ed
y = C2x+ Fv
(6.2. 1 )A stabilising controller that minimises the cost function (6. 1 .4) can be found by the LQG design method in §3.2. 1 . By the use of the separation principle, a dynamic LQG output feedback controller is combined by a Kalman filter and a full state feedback controller as:
where
Ac = A + B2Kc - K rC2 , Kc = _R-1B�Pc'
Pc: A Tpc + PeA - PcB2R-1B;Pc + Q =
0K - P CT (FVFT)-l r - ( 2
Pr: APr + PrAT - PrC; (FVFTrl C2Pr + EWET =
0(6.2.2)
For comparison the Hoo in (6.2. 1 ), this system may be described in the fonnat used for the standard Hoo design method (3.2.26) by choosing (0
= [dT
'llf
and the noise/disturbanceinput matrices with respect to the LQG fonnat as:
(6.2.3) So
6. Robust Controller 1 07
Consider also the selection of a particular perfonnance vector z with reference to the cost
function such that:
(6.2.4)
So (6.2.5)
It may be noted that the constraints (A I ) and (A2) in §3.2.3 are satisfied. The H'"' design method provides a controller that will stabilise the system (3.2.26) and satisfy the following H'"' nonn bound:
(6.2.6)
where 'Yo may be treated as a design parameter. Then according to Lemma 3.3.2, Yo =
00
implies Pc =
X_,
p( = Y_,
soAc = A_, Kc
=F_, K(
=L_
Thus the LQG controllersolution (6.2.2) is a special case of the H'"' controller solution for this particular perfonnance vector and disturbance/noise vector, giving an equivalent solution when Yo
= 00.
Reference to Corollary 2.4.3 illustrates that the LQG solution offers no stability robustness guarantee against nonparametric uncertainty, this is consistent with the analysis in chapter 3.
6.2.2 Proposed robust LQG design approach
It is proposed that a compromise between cost perfonnance and robusmess to nonparametric uncertainty may be found by employing the H'"' design technique to the system with particular perfonnance vector (6.2.4) for a fInite H'"' nonn bound Yo . The robusmess analysis results in Corollary 2.4.3 and (6.2.5) have shown that Yo gives a measure of nonparametric robustness, with a lower bound relating to greater robusmess. It has also been shown that if the bound is chosen to be infinite then the solution is
equivalent to that of the LQG approach that is known to be cost optimal. It has been found from example applications that the cost monotonically increases as 'Yo is reduced. Thus, it is proposed that suitable choice of Yo can yield a suitable compromise controller solution. This is consistent with (Mustafa and
Glover, 1 990) where this result is the subject of an unproved conjecture. To complete the analysis, a result is given to calculate the cost perfonnance for any given dynamic output controller:
6. Robust Controller 1 08
Lemma 6.2.1 For the system (6.2. 1 ) and a given stabilising controller K (s)
= Cc (sI - AJ-1 Bc'
the cost perfonnance (6. 1 .4)J(Ac,Bc,CJ
is givenby tr
{
PR)
where P:
AoP + PA� + V = O
_
r
Q 01
_r EWET
0
1
R
=l
0C�
RCJ
'V =l
0BcFVFTB� J
r
A
and
Ao
is the system matrix of the closed loop system,Ao =lB C c
2
Proof: lbis result can be directly found from Lemma 2.3.3 by using the above
Ao
and-
-
-
T
substituting R for Qo ' and