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A New Approach to Multi-Objective Control Design from the Viewpoint of the Inverse Optimal Control Problem
Copyright © lFAC System, Structure and Control Oaxaca. Mexico. USA, 8-10 December 2004
ELSEVIER
IFAC PUBLICATIONS www.elsevier.comllocalelifac
A NEW APPROACH TO MULTI-OBJECTIVE CONTROL DESIGN FROM THE VIEWPOINT OF THE INVERSE OPTIMAL CONTROL PROBLEM Daniel Alazard· Olivier Voinot·· Pierre Apkarian ••
• SUPAERO, 10 av. Edouard Belin, 31055 Toulouse, FRANCE Email: alazard(Osupaero . fr •• ONERA/DCSD, 2 av. Edouard Belin, 31055 Toulouse, FRANCE
Abstract: This paper introduces the Cross Standard Form (CSF) as a solution to the generalized inverse optimal control problem. The CSF is a canonical augmented standard plant whose Hoo or H2 optimal controller is a given controller. The definition of the CSF relies on the possibility to construct minimal observer-based realization of a given controller for a given model. From the multi-objective or robust control point of view , the general idea is to apply the CSF to a first controller satisfying nominal performances in order to initialize an Hoc design procedure to handle frequencydomain constraints or robustness specifications. Copyright © 2004 IFAC Keywords: Hoo design, mixed design, observer-based, inverse control problem
1. INTRODUCTION
(Kreindler and Jameson 1972). Nevertheless, such a trivial solution could be interesting when one aims at mixing a given state feedback with an quadratic index. Note also that the infinite gain margin property is restricted to state feedback and is no more valid if we seek the inverse optimal control problem associated with a static or dynamic output feedback.
This paper introduces the Cross Standard Form (CSF) for a given nth-order system G(s) and a given stabilizing nKth-order controller K(s) (n ~ nK). The CSF can be seen as a generalization of the well-known inverse LQ (Linear Quadratic) problem posed in (Kalman 1964) . The inverse LQ control problem consists in finding a quadratic performance index whose minimization restores a given state feedback . In (Kalman 1964) and (Molinari 1973), cross product terms between state and control are not considered in the quadratic performance index for two reasons . Firstly to find a meaningful performance index and secondly to take advantage of the stability margins of LQ state feedback control laws (for instance: infinite gain margin in single input case). In (Fujii 1987), an LQ design procedure based on asymptotic modal properties of the regulator is proposed. When cross product terms between state and control are introduced in the performance index, such properties are lost but the solution of the inverse control problem becomes trivial
More recently, solutions to the inverse optimal problem has motivated design procedures to mix various approaches (Sugimoto and Yamamoto 1987), (Sugimoto 1998), (Shimomura and Fujii 1997) : in (Sugimoto 1998). the author gives a necessary and sufficient condition under which a state feedback achieves partial pole placement. being optimal for an LQ cost with specific weightings. Such an approach has not been generalized to (static or dynamic) output feed backs and the number of assigned closed-loop poles is restricted (and must be equal) to n - m where n and m are the dimensions of the state and input vectors, respectively. In (Shimomura and Fujii 1997), authors address the inverse optimal control problem 439
for observer based controllers (that is: dynamic output feedbacks) and propose a sufficient condition for such a controller to be positive real and simultaneously H2 optimal.
observer-based realization of an arbitrary controller (Alazard and Apkarian 1999). The CSF can be used to mix various synthesis techniques. From a practical point of view, this approach is particularly interesting as many control problems involve various constraints and specifications. The general idea is to use a first synthesis technique to satisfy some specifications, mainly performance specifications; for example: eigen-structure assignment for modal shaping specifications. LQG technique for quadratic specifications. In a second step, the CSF is applied to this first solution to initialize a standard plant which will be subsequently enriched to handle frequency-domain and/or robustness specifications in an H00 synthesis procedure.
More generally, one can also define inverse Hoc (resp. H 2 ) optimal control problem, that is. to find an augmented plant whose Hoc (resp. H 2 ) optimal central controller is an arbitrary given stabilizing controller. The Cross Standard Form proposed in this paper solved both Hoc and H2 optimal control problems for full- or augmented-order controllers. The Hao inverse control problem has been firstly addressed in (K. E. Lenz and Doyle 1988) and (Fujii and Khargonekar 1988) and presented as the H00 counterpart of the LQ inverse optimal cont.rol problem posed by KALMAN in (Kalman 1964). The results presented in (Fujii and Khargonekar 1988) are restricted to state feedback while in (K. E. Lenz and Doyle 1988). the problem was to find conditions under which a given (single input single output) controller is Hao optimal for a specific mixed sensitivity problem where the frequency weighting functions are specified in terms of the given controller. A more general H00 inverse problem (multi input multioutput. dynamic output feedback and arbitrary given interconnection structure) is addressed in (Sebe 2001): for a given weight system W(s) and a given controller K(s), find all the plants G(s) such tha.t 1 11F1(FI(W, G), K)lIoo < -y (see Figure 1). Note that the problem addressed in this paper is different since we impose that the plant G(s) (that is, the lower right-hand transfer matrix of the standard augmented plant P = Fl (W, G)) must be equal to the model of the system between the control input and the measured output . From a practical point of view, it seems more interesting to find the weight system W(s) for a given controller K(s) and a given system G(s) even if we obtain a less meaningful weight system W(s).
The paper is organized as follow . In the next section, we define the CSF as a solution of the H2 and Hoo inverse optimal control problems, for an nth-order Linear Time Invariant (LTI) syst.em and a full-order LTI controller (nK = n). In the third section , the CSF is generalized to LT! controllers of order nK :::: n . Finally, a multichannel objective design procedure is proposed ans illustrated on an numerical example. 2. THE CROSS STANDARD FORM (CSF)
Nomenclature : A transposed : n x n identity matrix : time derivation (i: = dx/dt) : LAPLACE variable S FI (P, K) : lower Linear Fractional Transformation of P and K IIG(s)lIoo: Hoc norm of the stable system G(s)
G(s) :=
:
shorthand for G(s) = C(sI-
A)-lB+D. The general standard plant between exogenous input W, control input u, controlled output z and measurement output y is denoted:
:-----------------------------------P: w :
[~I~]
z
u with corresponding state space realization:
L-----------~K ~----------~
Fig. 1. Block diagram of standard plant P, weight function W. model G and controller K .
2.1 General inverse optimal problem statement
The demonstration of the properties of the CSF is based on the possibility to determine a minimal
Consider the stabilizable and detectable nth-order system G(s) with minimal state-space realization:
I
and definitions
(1)
see latter for the notations
440
Consider also the stabilizing nKth order controller Ko(s) (with nK 2: n) with minimal state-space realization:
[J
(2)
Using the change of variables:
Definition 2. 1. [Inverse H2 optimal problem] Find a nKth order standard plant P(s) such that :
• Pyu(s) = G(s), • Ko stabilizes P(s) , • Ko(s) = argminK(s) IIFI(P(s), K(s))112 '
the new representation of F1(PCSF , Kob) becomes:
(namely: Ko(s) minimizes IIF1(P(s) , K(s))1I2)·
[ ~] = f:
Definition 2.2. [Inverse Hoc optimal problem] Find a uKt.h order standard plant P(s) such that :
• Pyu(s) = G(s), • Ko stabilizes P(s), • Ko(s) = arg minK(s) IIFi(P(s) , K(s))lIoo .
z
[A-BKc 0 A -BKc KfG Kf] 0 0 Kc 0
[X] e
.
w
We stress that the states e are uncontrollable by X are unobservable by z . The transfer matrix between wand z:
w whereas the states
Fi(PCSF(s), Kob(S)) = 0
Definition 2.3. (Cross Standard Form). If the standard plant P(s) is such that:
(4)
o Remark: Note the uniqueness of arg minK(s) IIFI(P(S), K(s))112 and argminK(s) IIFI(P(s),K(s))ll oo , in definitions 2.1 and 2.2, has not been considered since we can show that the solution of the Cross Standard Problem is unique: Pzu and Pyw are left and right inversible respectively in (3) (see (Chen and Saberi 1992) for more details) .
• Pyu(s) = G(s) , • Ko stabilizes P(s) , • FI(P(.s), Ko(s)) = 0, then P( s ) is called the Cross Standard Form (CSF) associated with the system G(s) and the controller Ko(s) and will be denoted PCSF(S) in the sequel.
The CSF used as such is not of interest since it is necessary to know the gains K c and Kf to set-up the standard plant Pcsds) and to finally find the initial observer-based controller. On the other hand, from an initial controller Ko or a state feedback Kc satisfying some time domain specifications or modal shaping constraints, the CSF can be very useful to initialize a standard setup which will be enriched by dynamic weightings to take into account some complementary frequency domain specifications (see (Alazard 2002) and (Voinot et al. 2003)) .
By construction, the CSF solves the inverse H2 optimal problem and the inverse Hoc optimal problem.
2.2 Full-order observer-based controller case Consider a full order observer-based stabilizing controller Kob(.s) with minimal state-space realization:
2.3 Generalization
where K c and Kf are the state-feedback and the state estimation gains, respectively. It is well known that KOb(S) stabilises G(s) provided Kc and Kf are chosen such that A - BKe and A KfG are stable, which is assumed throughout .
In a more general way, it could be interesting t.o extend the CSF notion to arbitrary controllers and not particularly to full-order observer-based controllers. Then, a straightforward way to solve this problem proceeds in two steps. The first step consists in applying the procedure proposed in (Alazard and Apkarian 1999) to compute a minimal observe- based realization for the initial controller. This realization is depicted in Figure 2 and involves a state feedback gain Kc, a state estimation gain K, and a dynamic Youla parameter Q(s). The second step consists in generalizing the notion of CSF to the observer-based structure
Lemma 2.{ The CSF associated with Kob(S) and the system G(s) reads: PCSF
Proof: The state-space representation of F1(PCSF , K ob ) reads: 441
fitted wit.h a dynamic Youla's parameter Q(s) . So we consider a general plant (equation (1)) and a controller of order nK ~ n.
'. w
~------j
,
-------
u
'---- ----IDI--- - - --
l~.~~~ ................... .
Fig. 3. Block diagram of Cross Standard Pcs F .
Fig. 2. Observer-based structure and YOULA parameterization (direct feed-through D is ignored for reason of clearness) .
F017n
3. A tvIULTI CHANNEL DESIGN PROCEDURE 3.1 Methodology general description
The state-space representation of such a controller, denoted KobQ , reads:
5:= Ax+Bu+Kt(y-Cx-Du) + BQ(Y - Cx - Du) { == AQxQ - Kcx + CQxQ + DQ(Y - Cx xQ u
We consider here the two channel control problem described by the following state space realization:
Du)
where AQ, BQ, CQ and DQ are the 4 matrices of a realization of Q(s) (order nK - n) with the state vector xQ . Note that if the initial controller is a nth order controller with a direct feed-through term then the YOULA parameter is static and coincides with this direct feed-through .
The performance problem corresponds to the min-· imization of the Hoo (resp . H 2 ) norm of the transfer from exogenous input Wl to controlled output Zl while the second channel (between W2 and Z2) corresponds to a robustness problem against frequency domain uncertainties. It is well known that ignoring the structure of this problem, that is seeking to minimize T(s) = F/(P(s),K(s)) without any distinction between Zl, Z2 and Wl, W2 leads to a too much conservative solution. On the other hand, the /l-synthesis approach (Balas et al. 1991) , which can handle such structured problems is very cumbersome to use and often yield to high order controller. The actual problem is to minimize IITwlzlll oo (resp. IITwl zlll2) and IITw,z,lIoo simultaneously and independently.
Lemma 2.5. The CSF PCSF(s), associated with the controller defined by (2.3), such that :
is given as:
.
(6)
We will assume that D12 = 0 and D21 = 0 (we will also assume Du = 0 in the case of an H2 performance channel).
Proof: analogous to the previous ones and omitted for brevity.
The procedure we propose is based on the CSF previously presented and requires 3 steps.
o The general block diagram associated with PCSF(S) is depicted in Figure 3. Thus, from the extended parameterization (Kc, Kt and Q(s)) of an arbitrary controller, one can build a minimal standard plant PCSF(S) on which both H2 and Hoo syntheses provide the same initial controller. That could be interesting to mix both approaches in both senses or to initialize an H2 design from an Hoo controller (and vice-versa).
442
Step 1: pure performance design
In a first time, an H (resp. H 2) synthesis IS performed on the standard plant defined by: OX)
(8)
This design provides a nth-order controller K 1 (s) where n is the size of A .
Step 2: using the CSF The previous controller KI (s) is turned into an observer-based controller using the procedure proposed in (Alazard and Apkarian 1999) to find the general parameterization Kc, Kf and DQ (the Youla parameter is static). Note that if the first synthesis is an H2 synthesis then the computation of Kc, Kf is obvious and DQ = o. Then we built a new plant as follows:
Fig. 4. A 2 channel control problem. • from W2 to Z2, that is 12 = 11F1(P, K)W2- Z2I1oo : this channel corresponds to a robustness specification against unstructured uncertainties (rejection of plant input disturbances) with a weighting function W 2 (s) . A state space representation of this third order standard problem P( s) is:
~l ~
We recognize the CSF between Wc and Zc completed by the channel (W2 -+ Z2) dedicated to the robustness problem.
r
0 1 o 00 0 -1 -0.01 0 011 -9.9 0 -0.1 9.900 -1 0 1 1 00 10 0 0 0 00 -1 0 0 1 o0
~;l
r
Numerical values of all the matrices (A, ..., D) of equation (7) can thus be obtained.
Step 3: final design Since we have shown that the Hoo (resp. H 2 ) synthesis on the CSF yields the controller K 1 (s) which satisfies the performance problem (now expressed by the transfer between Wc and zc) , one can expect that the Hoo synthesis on the new plant (9) will be inflected toward the first solution K 1 (s) while minimizing the H cc norm of the transfer between W2 and Z2, which leads to better robustness guarantees in face of uncertainties. Note that the inflection toward K 1 (s) is ensured since the new performance channel vanishes with K 1 (s). Although there is no guarantee that this last synthesis will not degrade too much the performance channel, this approach allows the multi-channel feature to be handled very easily and the controller order is restricted to the order of the initial problem (7) . Note also that the very same procedure can be iterated when more than 2 channels are present.
All Hoo syntheses presented here have been performed using LMI solver (macro-function hinflmi from the MATLABTM LMI control toolbox) 2. For comparison , the result obtained with a classical Hoo synthesis is given in the first column of table 1. This synthesis provides a third-order controller Ko(s) such that :
Ko(s) = arg
~V~ Ill~~~ I OX) .
Let us apply the multi-channel design procedure: Step 1: the first Hoo synthesis considers only the first channel from Wl to ZI and is thus performed on the following standard problem :
Pds):=
0 -0.01 1 -1 00 00 01 1 -9.9 0 -0.1 9.9 0 0 1 1 0 [ -1 -1 0 0 1 0
3.2 Numerical example The resulting controller KI (s) reads: This numerical example is given to illustrate this methodology and to show the way to link the various steps of this procedure. Let us consider the control problem depicted in Figure 4 where: G(s) "+o.bls+I' W 1 (s) = :t~~, W 2 (s) = 10 .
K!(s) =
9.56910 8 s2 + 9.5710 6 s + 9.56910 8 s3 + 6 .419104 s 2 + 9.7371O's + 9.73710 6
'
the corresponding result, presented in the second column of table 1, highlights that the first channel is efficiently minimized while the second channel is completely relaxed.
The problem is to find a stabilizing controller K (s) that minimizes H 00 norms of both channnels: • from WI to ZI, that is'y\ = IIFl(P, K)wl_zllloo: this channel corresponds to a performance specification on the sensitivity function S = (I +GK) - 1 with a weighting function W 1 (s) ,
443
The reader will find the MATLAD™ source file required to run this example at the Web address http://vvv.8upaero . fr/page-perso/autom/alazard/ demos/demoSSSC04.html 2
Step 2: Following the procedure described in (Alazard and Apkarian 1999) to compute the observer based realizat ion of K I (s), the 6 closedloop eigenvalues of Fi(P, K.) must be distributed between the state-feedback dynamics (that is: spec(A-BKc)) and the state estimation dynamics (that is: spec(A - KfC)). The choice: spec(A - BKc)
=
{-9.9927, -0.005
+ i,
-0.005 - i},
spec(A-K,C) = {-62635, -1544.4, -0.1} ,
leads to the following parameters: Kc = [-0.8903
KJ = [-6.418010
4
9.8927
- 1.0083] ,
-9.673110 7
9.144510 1 ],
Q=O.
Then, the following standard problem P 2 (s) is formed:
~1
P2(8) :=
-99 10
1 _0 01 0 0
~ ~=~':~~~~~~~l -0.1 0
0 9144510 1 0 0 0 0
r
-0.8903 9 8927 -1.00830 0 -1 0 0 0
0 1
1 0
Step 3: the Hoo synthesis on problem P2 (s) provides a third-order controller K2(S): K2(S) = 83
8.8610 9 5 2 + 6 .997 1010 8 + 2.02 109 + 6 .437104s2 + 1.0271088 + 1.027107
whose results are displayed on the third column of table 1. One can notice that '1'1 is a little bit degraded by comparison with the previous controller KI(s) but '1'2 is now quite minimized. 1'1 1'2
KO(8) 4.0306 2.32 29
Kl(8) 1.0073 100.72
K2(S) 1.0429 0.0506
Table 1. Comparative results. 4. CONCLUSIONS
The Cross Standard Form (CSF) we have presented in this paper and the observer-based realization of arbitrary controller previously published (Alazard and Apkarian 1999) are complementary tools to build a standard plant whose H2 or H'X) (optimal or sub-optimal) controller is equal to a given controller, that is to solve the generalized inverse optimal control problem. Indirectly, these techniques allow to mix various approaches and to solve practical multi-channel control problems. The reader will find in (Alazard 2002) and (Voinot et al. 2003) quite realistic applications in the context of flight control and launcher control which demonstrate the flexibility and practical value of the CSF methodology. 5. REFERENCES Alazard, D. (2002). Robust H.2 design for lateral flight control of a highly flexible aircraft. Journal of Guidance, Control and Dynamics. 444
Alazard, D. and P. Apkarian (1999) . Exact oberserver-based structures for arbitrary compensators. Internationnal Journal of Robust and Non-Linear Control 9 , 101-118. Balas. G. J ., J . C. Doyle and K. Glover (1991). J.LAnalysis and Synthesis toolbox: User 's Guide. The Mathworks, Inc. Chen, B. ivl. and A. Saberi (1992) . Necessary and sufficient conditions under which an H. 2 optimal control problem has a unique solution. In: Proceedings of the 31th Conference on Decision and Cont1·ol. IEEE. Tucson (Arizona). pp. 1105- 1110. Fujii, T . (1987). A new approach to the Iq design from the viewpoint of the inverse regulator problem. IEEE. Transactions on Automat'ic Control vol. AC32., No. 11 .. 995-1004. Fujii, T . and P.P. Khargonekar (1988). Inverse problems in 'Hoc control thery ans linearquadratic differential games. In: Proceedings of the 27th Confe1'ence on Decision and Control. IEEE. Austin (Texas) . pp. 26- 31. K. E . Lenz, P. P. Khargonekar and J. C. Doyle (1988). When is a controller 'Hoc optimal? Mathematics of Control, Signals and Systems 1, 107-122. Kalman, R.E. (1964). When is a linear control optimal? Transactions of the ASME, JO'urnal of Basic Engineering pp. 51-60. Kreindler, E. and A. Jameson (1972) . Optimality of linear control systems. IEEE Trans . on A utomatic Control AC-17, 349-351. Molinari, B. P. (1973). The stable regulator problem and its inverse. IEEE Trans. on Automatic Controlvol. AC-18., No. 5.,454-459. Sebe, N. (2001). A characterization of solut.ions to the inverse 'Hoo optimal control problem. In: Proceedings of the 40th Conference on Decision and Control. IEEE. Orlando (Florida). pp. 273-278. Shimomura, T . and T . Fujii (1997). Strictly positive real 'H 2 controller synthesis from the viewpoint of the inverse problem. In: Proceedings of the 36th Conference on Decision and Control. IEEE. San Diego (California). pp. 1014-1019. Sugimoto, K. (1998). Partial pole-placement by LQ regulator: An inverse problem approach . IEEE. Transactions on Automatic Control vol. 43., No. 5., 706-708. Sugimoto, K. and Y. Yamamoto (1987) . New solution to the inverse regulator problem by the polynomial matrix method. International Journal of Control 45-5, 1627- 1640. Voinot, 0 ., D.Alazard, P. Apkarian, S. Mauffrey and B. Clement (2003). Launcher attitude control: Discrete-time robust design and gainscheduling. IFAC Control Engineering Practice 11, 1243- 1252.
ELSEVIER
IFAC PUBLICATIONS www.elsevier.comllocalelifac
A NEW APPROACH TO MULTI-OBJECTIVE CONTROL DESIGN FROM THE VIEWPOINT OF THE INVERSE OPTIMAL CONTROL PROBLEM Daniel Alazard· Olivier Voinot·· Pierre Apkarian ••
• SUPAERO, 10 av. Edouard Belin, 31055 Toulouse, FRANCE Email: alazard(Osupaero . fr •• ONERA/DCSD, 2 av. Edouard Belin, 31055 Toulouse, FRANCE
Abstract: This paper introduces the Cross Standard Form (CSF) as a solution to the generalized inverse optimal control problem. The CSF is a canonical augmented standard plant whose Hoo or H2 optimal controller is a given controller. The definition of the CSF relies on the possibility to construct minimal observer-based realization of a given controller for a given model. From the multi-objective or robust control point of view , the general idea is to apply the CSF to a first controller satisfying nominal performances in order to initialize an Hoc design procedure to handle frequencydomain constraints or robustness specifications. Copyright © 2004 IFAC Keywords: Hoo design, mixed design, observer-based, inverse control problem
1. INTRODUCTION
(Kreindler and Jameson 1972). Nevertheless, such a trivial solution could be interesting when one aims at mixing a given state feedback with an quadratic index. Note also that the infinite gain margin property is restricted to state feedback and is no more valid if we seek the inverse optimal control problem associated with a static or dynamic output feedback.
This paper introduces the Cross Standard Form (CSF) for a given nth-order system G(s) and a given stabilizing nKth-order controller K(s) (n ~ nK). The CSF can be seen as a generalization of the well-known inverse LQ (Linear Quadratic) problem posed in (Kalman 1964) . The inverse LQ control problem consists in finding a quadratic performance index whose minimization restores a given state feedback . In (Kalman 1964) and (Molinari 1973), cross product terms between state and control are not considered in the quadratic performance index for two reasons . Firstly to find a meaningful performance index and secondly to take advantage of the stability margins of LQ state feedback control laws (for instance: infinite gain margin in single input case). In (Fujii 1987), an LQ design procedure based on asymptotic modal properties of the regulator is proposed. When cross product terms between state and control are introduced in the performance index, such properties are lost but the solution of the inverse control problem becomes trivial
More recently, solutions to the inverse optimal problem has motivated design procedures to mix various approaches (Sugimoto and Yamamoto 1987), (Sugimoto 1998), (Shimomura and Fujii 1997) : in (Sugimoto 1998). the author gives a necessary and sufficient condition under which a state feedback achieves partial pole placement. being optimal for an LQ cost with specific weightings. Such an approach has not been generalized to (static or dynamic) output feed backs and the number of assigned closed-loop poles is restricted (and must be equal) to n - m where n and m are the dimensions of the state and input vectors, respectively. In (Shimomura and Fujii 1997), authors address the inverse optimal control problem 439
for observer based controllers (that is: dynamic output feedbacks) and propose a sufficient condition for such a controller to be positive real and simultaneously H2 optimal.
observer-based realization of an arbitrary controller (Alazard and Apkarian 1999). The CSF can be used to mix various synthesis techniques. From a practical point of view, this approach is particularly interesting as many control problems involve various constraints and specifications. The general idea is to use a first synthesis technique to satisfy some specifications, mainly performance specifications; for example: eigen-structure assignment for modal shaping specifications. LQG technique for quadratic specifications. In a second step, the CSF is applied to this first solution to initialize a standard plant which will be subsequently enriched to handle frequency-domain and/or robustness specifications in an H00 synthesis procedure.
More generally, one can also define inverse Hoc (resp. H 2 ) optimal control problem, that is. to find an augmented plant whose Hoc (resp. H 2 ) optimal central controller is an arbitrary given stabilizing controller. The Cross Standard Form proposed in this paper solved both Hoc and H2 optimal control problems for full- or augmented-order controllers. The Hao inverse control problem has been firstly addressed in (K. E. Lenz and Doyle 1988) and (Fujii and Khargonekar 1988) and presented as the H00 counterpart of the LQ inverse optimal cont.rol problem posed by KALMAN in (Kalman 1964). The results presented in (Fujii and Khargonekar 1988) are restricted to state feedback while in (K. E. Lenz and Doyle 1988). the problem was to find conditions under which a given (single input single output) controller is Hao optimal for a specific mixed sensitivity problem where the frequency weighting functions are specified in terms of the given controller. A more general H00 inverse problem (multi input multioutput. dynamic output feedback and arbitrary given interconnection structure) is addressed in (Sebe 2001): for a given weight system W(s) and a given controller K(s), find all the plants G(s) such tha.t 1 11F1(FI(W, G), K)lIoo < -y (see Figure 1). Note that the problem addressed in this paper is different since we impose that the plant G(s) (that is, the lower right-hand transfer matrix of the standard augmented plant P = Fl (W, G)) must be equal to the model of the system between the control input and the measured output . From a practical point of view, it seems more interesting to find the weight system W(s) for a given controller K(s) and a given system G(s) even if we obtain a less meaningful weight system W(s).
The paper is organized as follow . In the next section, we define the CSF as a solution of the H2 and Hoo inverse optimal control problems, for an nth-order Linear Time Invariant (LTI) syst.em and a full-order LTI controller (nK = n). In the third section , the CSF is generalized to LT! controllers of order nK :::: n . Finally, a multichannel objective design procedure is proposed ans illustrated on an numerical example. 2. THE CROSS STANDARD FORM (CSF)
Nomenclature : A transposed : n x n identity matrix : time derivation (i: = dx/dt) : LAPLACE variable S FI (P, K) : lower Linear Fractional Transformation of P and K IIG(s)lIoo: Hoc norm of the stable system G(s)
G(s) :=
:
shorthand for G(s) = C(sI-
A)-lB+D. The general standard plant between exogenous input W, control input u, controlled output z and measurement output y is denoted:
:-----------------------------------P: w :
[~I~]
z
u with corresponding state space realization:
L-----------~K ~----------~
Fig. 1. Block diagram of standard plant P, weight function W. model G and controller K .
2.1 General inverse optimal problem statement
The demonstration of the properties of the CSF is based on the possibility to determine a minimal
Consider the stabilizable and detectable nth-order system G(s) with minimal state-space realization:
I
and definitions
(1)
see latter for the notations
440
Consider also the stabilizing nKth order controller Ko(s) (with nK 2: n) with minimal state-space realization:
[J
(2)
Using the change of variables:
Definition 2. 1. [Inverse H2 optimal problem] Find a nKth order standard plant P(s) such that :
• Pyu(s) = G(s), • Ko stabilizes P(s) , • Ko(s) = argminK(s) IIFI(P(s), K(s))112 '
the new representation of F1(PCSF , Kob) becomes:
(namely: Ko(s) minimizes IIF1(P(s) , K(s))1I2)·
[ ~] = f:
Definition 2.2. [Inverse Hoc optimal problem] Find a uKt.h order standard plant P(s) such that :
• Pyu(s) = G(s), • Ko stabilizes P(s), • Ko(s) = arg minK(s) IIFi(P(s) , K(s))lIoo .
z
[A-BKc 0 A -BKc KfG Kf] 0 0 Kc 0
[X] e
.
w
We stress that the states e are uncontrollable by X are unobservable by z . The transfer matrix between wand z:
w whereas the states
Fi(PCSF(s), Kob(S)) = 0
Definition 2.3. (Cross Standard Form). If the standard plant P(s) is such that:
(4)
o Remark: Note the uniqueness of arg minK(s) IIFI(P(S), K(s))112 and argminK(s) IIFI(P(s),K(s))ll oo , in definitions 2.1 and 2.2, has not been considered since we can show that the solution of the Cross Standard Problem is unique: Pzu and Pyw are left and right inversible respectively in (3) (see (Chen and Saberi 1992) for more details) .
• Pyu(s) = G(s) , • Ko stabilizes P(s) , • FI(P(.s), Ko(s)) = 0, then P( s ) is called the Cross Standard Form (CSF) associated with the system G(s) and the controller Ko(s) and will be denoted PCSF(S) in the sequel.
The CSF used as such is not of interest since it is necessary to know the gains K c and Kf to set-up the standard plant Pcsds) and to finally find the initial observer-based controller. On the other hand, from an initial controller Ko or a state feedback Kc satisfying some time domain specifications or modal shaping constraints, the CSF can be very useful to initialize a standard setup which will be enriched by dynamic weightings to take into account some complementary frequency domain specifications (see (Alazard 2002) and (Voinot et al. 2003)) .
By construction, the CSF solves the inverse H2 optimal problem and the inverse Hoc optimal problem.
2.2 Full-order observer-based controller case Consider a full order observer-based stabilizing controller Kob(.s) with minimal state-space realization:
2.3 Generalization
where K c and Kf are the state-feedback and the state estimation gains, respectively. It is well known that KOb(S) stabilises G(s) provided Kc and Kf are chosen such that A - BKe and A KfG are stable, which is assumed throughout .
In a more general way, it could be interesting t.o extend the CSF notion to arbitrary controllers and not particularly to full-order observer-based controllers. Then, a straightforward way to solve this problem proceeds in two steps. The first step consists in applying the procedure proposed in (Alazard and Apkarian 1999) to compute a minimal observe- based realization for the initial controller. This realization is depicted in Figure 2 and involves a state feedback gain Kc, a state estimation gain K, and a dynamic Youla parameter Q(s). The second step consists in generalizing the notion of CSF to the observer-based structure
Lemma 2.{ The CSF associated with Kob(S) and the system G(s) reads: PCSF
Proof: The state-space representation of F1(PCSF , K ob ) reads: 441
fitted wit.h a dynamic Youla's parameter Q(s) . So we consider a general plant (equation (1)) and a controller of order nK ~ n.
'. w
~------j
,
-------
u
'---- ----IDI--- - - --
l~.~~~ ................... .
Fig. 3. Block diagram of Cross Standard Pcs F .
Fig. 2. Observer-based structure and YOULA parameterization (direct feed-through D is ignored for reason of clearness) .
F017n
3. A tvIULTI CHANNEL DESIGN PROCEDURE 3.1 Methodology general description
The state-space representation of such a controller, denoted KobQ , reads:
5:= Ax+Bu+Kt(y-Cx-Du) + BQ(Y - Cx - Du) { == AQxQ - Kcx + CQxQ + DQ(Y - Cx xQ u
We consider here the two channel control problem described by the following state space realization:
Du)
where AQ, BQ, CQ and DQ are the 4 matrices of a realization of Q(s) (order nK - n) with the state vector xQ . Note that if the initial controller is a nth order controller with a direct feed-through term then the YOULA parameter is static and coincides with this direct feed-through .
The performance problem corresponds to the min-· imization of the Hoo (resp . H 2 ) norm of the transfer from exogenous input Wl to controlled output Zl while the second channel (between W2 and Z2) corresponds to a robustness problem against frequency domain uncertainties. It is well known that ignoring the structure of this problem, that is seeking to minimize T(s) = F/(P(s),K(s)) without any distinction between Zl, Z2 and Wl, W2 leads to a too much conservative solution. On the other hand, the /l-synthesis approach (Balas et al. 1991) , which can handle such structured problems is very cumbersome to use and often yield to high order controller. The actual problem is to minimize IITwlzlll oo (resp. IITwl zlll2) and IITw,z,lIoo simultaneously and independently.
Lemma 2.5. The CSF PCSF(s), associated with the controller defined by (2.3), such that :
is given as:
.
(6)
We will assume that D12 = 0 and D21 = 0 (we will also assume Du = 0 in the case of an H2 performance channel).
Proof: analogous to the previous ones and omitted for brevity.
The procedure we propose is based on the CSF previously presented and requires 3 steps.
o The general block diagram associated with PCSF(S) is depicted in Figure 3. Thus, from the extended parameterization (Kc, Kt and Q(s)) of an arbitrary controller, one can build a minimal standard plant PCSF(S) on which both H2 and Hoo syntheses provide the same initial controller. That could be interesting to mix both approaches in both senses or to initialize an H2 design from an Hoo controller (and vice-versa).
442
Step 1: pure performance design
In a first time, an H (resp. H 2) synthesis IS performed on the standard plant defined by: OX)
(8)
This design provides a nth-order controller K 1 (s) where n is the size of A .
Step 2: using the CSF The previous controller KI (s) is turned into an observer-based controller using the procedure proposed in (Alazard and Apkarian 1999) to find the general parameterization Kc, Kf and DQ (the Youla parameter is static). Note that if the first synthesis is an H2 synthesis then the computation of Kc, Kf is obvious and DQ = o. Then we built a new plant as follows:
Fig. 4. A 2 channel control problem. • from W2 to Z2, that is 12 = 11F1(P, K)W2- Z2I1oo : this channel corresponds to a robustness specification against unstructured uncertainties (rejection of plant input disturbances) with a weighting function W 2 (s) . A state space representation of this third order standard problem P( s) is:
~l ~
We recognize the CSF between Wc and Zc completed by the channel (W2 -+ Z2) dedicated to the robustness problem.
r
0 1 o 00 0 -1 -0.01 0 011 -9.9 0 -0.1 9.900 -1 0 1 1 00 10 0 0 0 00 -1 0 0 1 o0
~;l
r
Numerical values of all the matrices (A, ..., D) of equation (7) can thus be obtained.
Step 3: final design Since we have shown that the Hoo (resp. H 2 ) synthesis on the CSF yields the controller K 1 (s) which satisfies the performance problem (now expressed by the transfer between Wc and zc) , one can expect that the Hoo synthesis on the new plant (9) will be inflected toward the first solution K 1 (s) while minimizing the H cc norm of the transfer between W2 and Z2, which leads to better robustness guarantees in face of uncertainties. Note that the inflection toward K 1 (s) is ensured since the new performance channel vanishes with K 1 (s). Although there is no guarantee that this last synthesis will not degrade too much the performance channel, this approach allows the multi-channel feature to be handled very easily and the controller order is restricted to the order of the initial problem (7) . Note also that the very same procedure can be iterated when more than 2 channels are present.
All Hoo syntheses presented here have been performed using LMI solver (macro-function hinflmi from the MATLABTM LMI control toolbox) 2. For comparison , the result obtained with a classical Hoo synthesis is given in the first column of table 1. This synthesis provides a third-order controller Ko(s) such that :
Ko(s) = arg
~V~ Ill~~~ I OX) .
Let us apply the multi-channel design procedure: Step 1: the first Hoo synthesis considers only the first channel from Wl to ZI and is thus performed on the following standard problem :
Pds):=
0 -0.01 1 -1 00 00 01 1 -9.9 0 -0.1 9.9 0 0 1 1 0 [ -1 -1 0 0 1 0
3.2 Numerical example The resulting controller KI (s) reads: This numerical example is given to illustrate this methodology and to show the way to link the various steps of this procedure. Let us consider the control problem depicted in Figure 4 where: G(s) "+o.bls+I' W 1 (s) = :t~~, W 2 (s) = 10 .
K!(s) =
9.56910 8 s2 + 9.5710 6 s + 9.56910 8 s3 + 6 .419104 s 2 + 9.7371O's + 9.73710 6
'
the corresponding result, presented in the second column of table 1, highlights that the first channel is efficiently minimized while the second channel is completely relaxed.
The problem is to find a stabilizing controller K (s) that minimizes H 00 norms of both channnels: • from WI to ZI, that is'y\ = IIFl(P, K)wl_zllloo: this channel corresponds to a performance specification on the sensitivity function S = (I +GK) - 1 with a weighting function W 1 (s) ,
443
The reader will find the MATLAD™ source file required to run this example at the Web address http://vvv.8upaero . fr/page-perso/autom/alazard/ demos/demoSSSC04.html 2
Step 2: Following the procedure described in (Alazard and Apkarian 1999) to compute the observer based realizat ion of K I (s), the 6 closedloop eigenvalues of Fi(P, K.) must be distributed between the state-feedback dynamics (that is: spec(A-BKc)) and the state estimation dynamics (that is: spec(A - KfC)). The choice: spec(A - BKc)
=
{-9.9927, -0.005
+ i,
-0.005 - i},
spec(A-K,C) = {-62635, -1544.4, -0.1} ,
leads to the following parameters: Kc = [-0.8903
KJ = [-6.418010
4
9.8927
- 1.0083] ,
-9.673110 7
9.144510 1 ],
Q=O.
Then, the following standard problem P 2 (s) is formed:
~1
P2(8) :=
-99 10
1 _0 01 0 0
~ ~=~':~~~~~~~l -0.1 0
0 9144510 1 0 0 0 0
r
-0.8903 9 8927 -1.00830 0 -1 0 0 0
0 1
1 0
Step 3: the Hoo synthesis on problem P2 (s) provides a third-order controller K2(S): K2(S) = 83
8.8610 9 5 2 + 6 .997 1010 8 + 2.02 109 + 6 .437104s2 + 1.0271088 + 1.027107
whose results are displayed on the third column of table 1. One can notice that '1'1 is a little bit degraded by comparison with the previous controller KI(s) but '1'2 is now quite minimized. 1'1 1'2
KO(8) 4.0306 2.32 29
Kl(8) 1.0073 100.72
K2(S) 1.0429 0.0506
Table 1. Comparative results. 4. CONCLUSIONS
The Cross Standard Form (CSF) we have presented in this paper and the observer-based realization of arbitrary controller previously published (Alazard and Apkarian 1999) are complementary tools to build a standard plant whose H2 or H'X) (optimal or sub-optimal) controller is equal to a given controller, that is to solve the generalized inverse optimal control problem. Indirectly, these techniques allow to mix various approaches and to solve practical multi-channel control problems. The reader will find in (Alazard 2002) and (Voinot et al. 2003) quite realistic applications in the context of flight control and launcher control which demonstrate the flexibility and practical value of the CSF methodology. 5. REFERENCES Alazard, D. (2002). Robust H.2 design for lateral flight control of a highly flexible aircraft. Journal of Guidance, Control and Dynamics. 444
Alazard, D. and P. Apkarian (1999) . Exact oberserver-based structures for arbitrary compensators. Internationnal Journal of Robust and Non-Linear Control 9 , 101-118. Balas. G. J ., J . C. Doyle and K. Glover (1991). J.LAnalysis and Synthesis toolbox: User 's Guide. The Mathworks, Inc. Chen, B. ivl. and A. Saberi (1992) . Necessary and sufficient conditions under which an H. 2 optimal control problem has a unique solution. In: Proceedings of the 31th Conference on Decision and Cont1·ol. IEEE. Tucson (Arizona). pp. 1105- 1110. Fujii, T . (1987). A new approach to the Iq design from the viewpoint of the inverse regulator problem. IEEE. Transactions on Automat'ic Control vol. AC32., No. 11 .. 995-1004. Fujii, T . and P.P. Khargonekar (1988). Inverse problems in 'Hoc control thery ans linearquadratic differential games. In: Proceedings of the 27th Confe1'ence on Decision and Control. IEEE. Austin (Texas) . pp. 26- 31. K. E . Lenz, P. P. Khargonekar and J. C. Doyle (1988). When is a controller 'Hoc optimal? Mathematics of Control, Signals and Systems 1, 107-122. Kalman, R.E. (1964). When is a linear control optimal? Transactions of the ASME, JO'urnal of Basic Engineering pp. 51-60. Kreindler, E. and A. Jameson (1972) . Optimality of linear control systems. IEEE Trans . on A utomatic Control AC-17, 349-351. Molinari, B. P. (1973). The stable regulator problem and its inverse. IEEE Trans. on Automatic Controlvol. AC-18., No. 5.,454-459. Sebe, N. (2001). A characterization of solut.ions to the inverse 'Hoo optimal control problem. In: Proceedings of the 40th Conference on Decision and Control. IEEE. Orlando (Florida). pp. 273-278. Shimomura, T . and T . Fujii (1997). Strictly positive real 'H 2 controller synthesis from the viewpoint of the inverse problem. In: Proceedings of the 36th Conference on Decision and Control. IEEE. San Diego (California). pp. 1014-1019. Sugimoto, K. (1998). Partial pole-placement by LQ regulator: An inverse problem approach . IEEE. Transactions on Automatic Control vol. 43., No. 5., 706-708. Sugimoto, K. and Y. Yamamoto (1987) . New solution to the inverse regulator problem by the polynomial matrix method. International Journal of Control 45-5, 1627- 1640. Voinot, 0 ., D.Alazard, P. Apkarian, S. Mauffrey and B. Clement (2003). Launcher attitude control: Discrete-time robust design and gainscheduling. IFAC Control Engineering Practice 11, 1243- 1252.