Chapter 3 The H 2 -optimal control problem In this chapter we present the solution of the H 2 -optimal control problem. We consider the control system in Figure 1.2. Introducing the partition of G according to z y G11 G 12 G 21 G 22 v u è3.1è the closed-loop system z F èg; Kèv has the transfer function F èg; Kè given by F èg; Kè G 11 + G 12 èi, KG 22 è,1 KG 21 è3.2è è3.3è The H 2 -optimal control problem consists of ænding a causal controller K which stabilizes the plant G and which minimizes the cost function J 2 èkè kfèg; Kèk 2 2 è3.4è where kf èg; Kèk 2 is the H 2 norm, cf. Section 2.2.1. The control problem is most conveniently solved in the time domain. We will assume that the plant G has the state-space representation _xètè Axètè+B 1 vètè+b 2 uètè zètè C 1 xètè+d 12 uètè yètè C 2 xètè+d 21 vètè è3.5è Remark 3.1. In the optimal and robust control literature, the state-space representation Gèsè CèsI, Aè,1 B + D is commonly written using the compact notation A B Gèsè : è3.è C D 24
Hence the systems in è3.5è can also be written as where D 11,D 22. Gèsè : 2 4 A C 1 B 1 D 11 B 2 D 12 C 2 D 21 D 22 3 5 è3.è In è3.5è, the direct feedthrough from v to z has been assumed zero in order to obtain a ænite H 2 norm for the closed-loop system. The direct feedthrough from u to y has been assumed zero because physical systems always have a zero gain at inænite frequency. It is convenient to make the following assumptions on the system matrices. Assumptions: èa1è èa2è The pair èa; B 2 è is stabilizable. D T 12 D 12 is invertible. èa3è D T 12 C 1. èa4è èb1è èb2è The pair èc 1 ;Aè has no unobservable modes on the imaginary axis. The èc 2 ;Aè is detectable. D 21 D T 21 is invertible. èb3è D 21 B T 1. èb4è The pair èa; B 1 è has no uncontrallable modes on the imaginary axis. The ærst set of assumptions èa1èíèa4è is related to the state feedback control problem, while the second set of assumptions èb1èíèb4è is related to the state estimation problem. Assumptions èa1è and èb1è are necessary for a stabilizing controller u Ky to exist. Assumptions èa2è, èa3è and èb2è, èb3è can be relaxed, but they are not very restrictive, and as we will see they imply convenient simpliæcations in the solution. Assumptions èa4è and èb4è are required for the Riccati equations which characterize the optimal controller to have stabilizing solutions. These assumptions can be relaxed, but the solution of the optimal control problem must then be characterized in an alternative way,for example in terms of linear matrix inequalities èlmisè. It is appropriate to introduce the time-domain characterization in è2.45è of the cost è3.4è, giving J 2 èkè mx Z 1 ë zètè T zètèdt : v e k æètèë è3.8è Notice, however, that the characterization è3.8è is introduced solely because the solution of the H 2 -optimal control problem can be derived in a convenient way using è3.8è. On the other hand, the motivation of the H 2 -optimal is often more naturally stated in terms of the average frequency-domain characterization in è2.15è or the stochastic characterization in è2.4è. 25
Remark 3.2. In the linear quadratic optimal control literature, the cost function is usually deæned as J LQ èkè mx Z 1 ëxètè T Qxètè+uètè T Ruètèëdt : v e k æètè è3.9è where Q is a symmetric positive semideænite matrix and R is a symmetric positive deænite matrix. It is easy to see that the cost functions è3.8è and è3.9è are equivalent, because the positive èsemièdeænite matrices Q and R can always be factorized as Q èq 12 è T Q 12 ; RèR 12 è T R 12 è3.1è The factorizations in è3.1è are called Cholesky factorizations, cf. the MATLAB routine chol. Deæning the matrices Q 12 C 1 ; D 12 è3.11è R 12 it follows that z T z èc 1 x+d 12 uè T èc 1 x + D 12 uèx T Qx + u T Ru è3.12è and the costs è3.8è and è3.9è are thus equivalent. Notice also that with C 1 and D 12 deæned by è3.11è, assumption èa3è holds. The solution of the H 2 -optimal control problem can be done in two stages. In the ærst stage, an optimal state-feedback law is constructed. The second stage consists of ænding an optimal state estimator. 3.1 The optimal state-feedback problem The following result gives the optimal state-feedback law ^uèsè K x èsè^xèsè which minimizes the quadratic cost è3.8è or è3.9è. The result is classical in optimal linear quadratic èlqè control theory. Theorem 3.1 H 2 -optimal state feedback control. Consider the system è3.5è. Suppose that the assumptions èa1èíèa4è hold. Assume that the control signal uètè has access to the present and past values of the state, xèè; t. Then the cost è3.8è is minimized by the static state state-feedback controller uètè K opt xètè è3.13è where K opt,èd T 12 D 12è,1 B T 2 S è3.14è and where S is the unique symmetric positive èsemièdeænite solution to the algebraic Riccati equation èareè A T S + SA, SB 2 èd T 12 D 12è,1 B T 2 S + CT 1 C 1 è3.15è 2
such that the matrix A + B 2 K opt è3.1è is stable, i.e. all its eigenvalues have negative real parts. Moreover, the minimum value of the cost è3.8è achieved by the control law è3.13è is given by min J 2 èk x è trèb T 1 SB 1è è3.1è Kx Proof: By the theory of Riccati equations it follows from assumptions èa1è and èa4è that the algebraic Riccati equation è3.15è has a positive èsemièdeænite solution S such that the matrix in è3.1è is stable. Consider the integral in è3.8è. It can be decomposed as Z 1 Z + Z 1 zètè T zètèdt zètè T zètèdt + + zètèt zètèdt è3.18è where + denotes the time immediately after the impulse input at time t. With xèè and vètè e k æètè, we have formally Z + xè + è e Aè+, è ëb 1 e k æèè+b 2 uèèëd B 1 e k è3.19è It follows that the ærst integral in è3.18è is zero. Assuming that the algebraic Riccati equation è3.15è has a positive èsemièdeænite solution, the second integral in è3.18è can be expanded as where Z 1 + zètèt zètèdt Z 1 + ëc 1xètè+D 12 uètèë T ëc 1 xètè+d 12 uètèëdt hëc 1 xètè+d 12 uètèë T ëc 1 xètè+d 12 uètèë + dèxètèt Sxètèè Z 1 +,xè1è T Sxè1è+xè + è T Sxè + è Z 1 dt i dt + ëuètè, u ètèë T D T 12 D 12ëuètè, u ètèëdt + xè + è T Sxè + è è3.2è u ètè K opt xètè; è3.21è and xè + è denotes the state immediately after the impulse input at time t. Wehave also assumed that xè1è due to stability, and used the fact that S satisæes è3.15è. Introducing è3.2è and è3.19è, the cost è3.8è can be expressed as J 2 èkè mx Z 1 mx Z 1 mx Z 1 zètè T zètèdt : v e k æètè ëuètè, u ètèë T D T 12 D 12ëuètè, u ètèëdt + xè + è T Sxè + è:ve k æètè ëuètè, u ètèë T D T 12 D 12ëuètè, u ètèëdt + e T k BT 1 SB 1e k : v e k æètè è3.22è 2
Here mx e T k BT 1 SB 1e k mx trèb T 1 SB 1e k e T k è trèbt 1 SB 1 mx e k e T k è trèbt 1 SB 1è è3.23è Hence, J 2 èkè mx Z 1 ëuètè, u ètèë T D T 12 D 12ëuètè, u ètèëdt : v e k æètè + trèb T 1 SB 1è è3.24è Since the integrals in è3.24è are non-negative, and are equal to zero when uètè u ètè, it follows that the cost J 2 èk x è is minimized by the static state feedback è3.13è, or è3.21è, and the minimum cost is given by è3.1è. 2 Problem 3.1. Verify the last step in è3.2è. Notice that the optimal state-feedback controller consists of a static state feedback, although no restrictions on the controller structure was imposed. Another point to notice about the optimal controller is that it does not depend on the disturbance v, since the controller is independent of the matrix B 1. This matrix enters only in the expression è3.1è for the minimum cost. Remark 3.3. By equation è3.2è, the control law è3.21è is optimal for all initial states xè + è. This implies that the solution of the H 2 control problem can be interpreted in a worst-case setting as the controller which minimizes the worst-case cost J 2;worst èkè : max xèè max xèè n kzk 2 2 : xt èèxèè 1 Z 1 o z T ètèzètèdt : x T èèxèè 1 è3.25è The optimal state-feedback control result suggests how an optimal output feedback controller ^uèsè Kèsè^yèsè can be obtained. In particular, the integral in the expansion è3.22è cannot be made equal to zero if the whole state vector xètè is not available to the controller. In the output feedback problem, one should instead make this integral as small as possible by using an optimal state estimate ^xètè instead. Therefore, we give next the solution to an H 2 -optimal state estimation problem. 3.2 The optimal state-estimation problem In this section we consider the system _xètè Axètè+B 1 vètè zètèc 1 xètè yètè C 2 xètè+d 21 vètè è3.2è 28
Notice that in the estimation problem, we need not consider the input u. As u is a known input, its eæect on the state is exactly known. We consider stable causal state estimators F such that the state estimate ^xètè is determined from the measured output y according to èin the Laplace-domainè ^xèsè Fèsèyèsè. In the H 2 -optimal estimation problem, the problem is to construct an estimator such that the quadratic H 2 -type cost J e èf è mx Z 1 ë ëxètè, ^xètèë T C T 1 C 1ëxètè, ^xètèëdt : v e k æètèë è3.2è is minimized. The solution of the optimal state estimation problem is given as follows. Theorem 3.2 H 2 -optimal state estimation. Consider the system è3.2è. Suppose that the assumptions èb1èíèb4è hold. The stable causal state estimator which minimizes the cost è3.2è is given by _^xètè A^xètè+Lëyètè,C 2^xètèë è3.28è where L PC T 2èD 21 D T 21è,1 è3.29è and where P is the unique symmetric positive èsemièdeænite solution to the algebraic Riccati equation such that the matrix AP + PA T,PC T 2èD 21 D T 21è,1 C 2 P + B 1 B T 1 è3.3è A, LC 2 è3.31è is stable, i.e. all its eigenvalues have negative real parts. Moreover, the minimum value of the cost è3.2è achieved bythe estimator è3.28è is given by min J eèf è trèc 1 PC1è T è3.32è F Remark 3.4. Notice that the optimal state estimator is independent ofthe matrix C 1 in the cost è3.2è. The matrix enters only in the expression è3.32è for the minimum cost. Remark 3.5. The solution deæned by è3.29è and è3.3è is the dual to the solution è3.14è and è3.15è of the optimal state-feedback problem in the sense that è3.14è and è3.15è reduce to è3.29è, è3.3è by making the substitutions A! A T C 1! B T 1 è3.33è B 2! C T 2 D 12! D T 21 29
and the costs è3.1è and è3.32è are dual under the substitution B 1! C T 1. The estimator è3.28è is the celebrated Kalman ælter due to Rudolf Kalman and R. S. Bucy è19, 191è. Originally, ithas been introduced in a stochastic framework for recursive state estimation in stochastic systems. Notice that the cost è3.2è equals the H 2 norm of the transfer function from the disturbance v to the estimation error C 1 èx, ^xè. From what was mentioned previously in Section 2.2.1 it follows that if v is stochastic white noise, the cost è3.2è can be characterized as J e èf è lim t f!1 Eë 1 Z tf ëxètè, ^xètèë T C 1 C 1 ëxètè, ^xètèëdtë t f è3.34è Remark 3.. In the standard formulation of the stochastic estimation problem the system is deæned as _xètè Axètè +v 1 ètè zètè C 1 xètè yètè C 2 xètè +v 2 ètè è3.35è where v 1 ètè and v 2 ètè are mutually independent white noise processes with covariance matrices R 1 and R 2, respectively. This formulation reduces to the one above by the identiæcations B1 ë B T 1 D T 21 D ë è3.3è 21 and R1 R 2 v1 v 2 B1 v D 21 è3.3è The optimal estimator result is more diæcult to prove than the optimal statefeedback controller. We shall only provide a proof based on duality between the optimal state-feedback and estimation problems. Proof of Theorem 3.2: Consider a state estimator ^xèsè F èsèy with transfer function F èsè. Then we have in terms of operators, C 1 èx, ^xè ëc 1 G 1,C 1 FG 2 ëv è3.38è where ècf. è3.2èè G 1 èsè èsi, Aè,1 B 1 ; G 2 èsè C 2 èsi, Aè,1 B 1 + D 21 è3.39è The cost è3.2è can then be characterized as J e èf èkc 1 G 1,C 1 FG 2 k 2 2 è3.4è But by the deænition è2.15è, the H 2 norm of a transfer function matrix is equal to the H 2 norm of its transpose. Hence J e èf èkg T 1 CT 1,G T 2 FT C T 1k 2 2 è3.41è 3
Notice that Hence and the transposed system C1 G 1 èsè C1 èsi, Aè,1 B 1 + G 2 èsè C 2 D 21 è3.42è ë G 1 èsè T C T 1 G 2 èsè T ëb T 1 èsi, AT è,1 ë C T 1 C T 2 ë+ë DT 21 ë è3.43è thus has the state-space representation r ëg T 1 CT 1 G T 2 ë è3.44è _pètè A T pètè+c T 1 ètè+ct 2 ètè rètè B T 1 pètè+dt 21 ètè Using the control law,f T C T 1 the closed-loop system è3.44è, è3.4è is described by è3.45è è3.4è r èg T 1 CT 1,GT 2 FT C T 1è è3.4è Hence the H 2 -optimal estimation problem is equal to the dual control problem which consists of ænding a controller è3.4è such that the H 2 norm of the closed-loop system is minimized. The dual control problem is, however, equivalent to a state-feedback control problem, because the fact that C T 1 is available to the controller means that the state pètè in è3.45è is also available. More speciæcally, the problem of ænding a control law è3.4è which minimizes the H 2 norm of the closed-loop transfer function, equation è3.41è, is equivalent to the problem of ænding a state-feedback law ètè,fp T pètè è3.48è where pètè is available through equation è3.45è and knowledge of C 1 ètè and ètè. But the problem of ænding an optimal state-feedback controller for the dual system è3.45è is known, and from our previous result it is given by è3.14è and è3.15è under to the substitutions è3.33è. This gives the optimal stationary state-feedback with ètè,f T opt pètè; F T opt èd 21D T 21 è,1 C 2 P è3.49è where P is given by the algebraic Riccati equation è3.3è. In order to derive the optimal state estimator, notice that combining è3.49è and è3.45è, the dual state-feedback controller è3.49è corresponds to a controller,f T C T 1, è3.4è, with state-space representation _pètè A T pètè +C T 1 ètè,ct 2 FT opt pètè èa, F opt C 2 è T pètè +C T 1ètè è3.5è ètè,f T opt pètè Taking the negative of the transpose of è3.5è gives the state-space representation of èf T C T 1 èt C 1 F, and thus the optimal estimator ^x Fy, _^xètè èa,f opt C 2 è^xètè+f opt yètè ^zètè C 1^xètè è3.51è 31
which is equal to è3.28è. 2 Remark 3.. As noted above, the H 2 -optimal estimator is dual to an optimal state feedback problem. It follows that for all the properties of the latter there is a corresponding dual property of the former. In particular, it follows from Remark 3.3 that the optimal estimator can be interpreted as a worst-case estimator as follows. By Remark 3.3 the dual state-feedback problem gives the optimal controller è3.49è which minimizes the worst-case performance from pèè 2 R n to r 2 L 2 ë; 1è. By duality of the optimal estimation and the optimal state feedback problem, it follows that the optimal estimator can be interpreted in a worst-case setting as the estimator which minimizes the worst-case estimation error J e;worst èkè : max v max v n kxètè, ^xètèk 2 : kvk 2 1 ëxètè, ^xètèë T ëxètè, ^xètèë : o,1 vt èèvèèd 1 Z t è3.52è This observation gives a nice deterministic interpretation of the Kalman ælter as the ælter which minimizes the pointwise worst-case estimation error. This property can be compared with the H1 estimator, which will be studied in section 4.2. 3.3 The optimal output feedback problem We are now ready to present the solution to the H 2 -optimal control problem for the system shown in Fig. 1.2. The solution will be based on the cost function expansion in equation è3.24è. Recall that è3.24è can be written as J 2 èkè mx Z 1 ëuètè, K opt xètèë T D T 12 D 12ëuètè, K opt xètèëdt : v e k æètè +trèb T 1 SB 1è è3.53è In contrast to the state-feedback case, when only the output y is available to the controller the integrals cannot be made equal to zero. Instead, the cost è3.53è is equivalent to an H 2 -optimal state estimation problem, because the best one can do is to compute an estimate ^x of the state, and to determine the control signal according to uètè K opt^xètè è3.54è giving the cost J 2 èkè mx Z 1 +trèb T 1 SB 1è ë^xètè, xètèë T K T opt DT 12 D 12K opt ë^xètè, xètèëdt : v e k æètè è3.55è Hence, it follows that the minimum of the quadratic cost è3.8è is achieved by the controller è3.54è, where ^x is an H 2 -optimal state estimate for the system è3.5è. We summarize the result as follows. 32
Theorem 3.3 H 2 -optimal control. Consider the system è3.5è. Suppose that the assumptions èa1èíèa4è and èb1èíèb4è hold. Then the controller u Ky which minimizes the H 2 cost è3.8è is given by the equations _^xètè èa + B 2 K opt è^xètè+lëyètè,c 2^xètèë uètè K opt^xètè è3.5è where and K opt,èd T 12 D 12è,1 B T 2 S L PC T 2èD 21 D T 21è,1 è3.5è è3.58è and S and P are the symmetric positive èsemièdeænite solutions of the algebraic Riccati equations è3.15è and è3.3è, respectively. Moreover, the minimum value of the cost achieved by the controller è3.5è is min K J 2èKè trèd 12 K opt PK T opt DT 12è + trèb T 1 SB 1è è3.59è Proof: First we show that the optimal controller è3.5è gives a stable closed loop. Introduce the estimation error ~xètè : xètè,^xètè. Then the closed loop consisting of the system è3.5è and the controller è3.5è is given by _x A + B2 K opt B 1 _~x + v è3.è B 1,LD 21,B 2 K opt A, LC 2 x ~x Stability ofthe closed-loop follows from stability of the matrices A + B 2 K opt and A, LC 2. The fact that the controller è3.5è minimizes the H 2 cost è3.8è follows from the expansion è3.53è of Theorem 3.1 and Theorem 3.2. 2 The optimal controller è3.5è consists of an H 2 -optimal state estimator, and an H 2 -optimal state feedback of the estimated state. A particular feature of the solution is that the optimal estimator and state feedback can be calculated independently of each other. This feature of H 2 -optimal controllers is called the separation principle. Remark 3.8. The optimal controller è3.5è is a classical result in linear optimal control, where one usually assumes that the disturbances are stochastic white noise processes with a Gaussian distribution ècf. Remark 3.è, and the cost is deæned as " 1 Z tf J LQG èkè lim t f!1 E ëxètè T Qxètè +uètè T Ruètèëdt t f è è3.1è cf. Remark 3.1. problem. This problem is known as the Linear Quadratic Gaussian èlqgè control 33
3.4 Some comments on the application of H 2 optimal control The H 2 -optimal èlqgè control problem solves awell-deæned optimal control problem deæned by the quadratic cost in è3.4è, è3.8è, or è3.1è. Practical applications of the method require, however, that the cost be deæned in a manner which corresponds to the control objectives. This is not always very straightforward to achieve, and some observations are therefore in order. Here, we will only discuss two particular problems in H 2 controller design. The ærst one concerns the problem of applying the approach to the case with general deterministic inputs, not necessarily impulses. The second problem is concerned with the selection of weights in the quadratic cost è3.1è. H 2 -optimal control against general deterministic inputs In the deterministic interpretation of the H 2 cost, equation è3.8è, it is in many cases not very realistic to assume an impulse disturbance. For example, a step disturbance would often be a more realistic disturbance form. It is therefore important to generalize the control problem to other types of deterministic disturbance signals. Assume that the system è3.5è is subject to an input v with a rational Laplacetransform. Such an input can be characterized in the time domain by a state-space equation _x v ètè A v x v ètè; vètè C v x v ètè xèè b è3.2è or _x v ètè A v x v ètè+bwètè; vètèc v x v ètè wètè æètè è3.3è where the input w is an impulse function. By augmenting the system è3.5è with the disturbance model è3.3è, the problem has thus been taken to the standard form with an impulse input. One problem with the above approach is that for some important disturbance types, such as step, ramp and sinusoidal disturbances, the disturbance model è3.3è is unstable. As the disturbance dynamics is not aæected by the controller, the augmented plant consisting of è3.5è and è3.3è will not be stabilizable. It follows that the Riccati equation è3.15è does not have a stabilizable solution in these cases. This problem can be handled in the following way, which we will here demonstrate for the case with a step disturbance. Assume that the disturbance v is a step disturbance, described by vètè ; té v step ; t where v step is a constant vector. The disturbance can be characterized by the model è3.4è _vètè v step æètè è3.5è 34
Notice that this model is not stable. One approach would be to make the model stable, by using _vètè,avètè+v step æètè for some aéé. A more direct approach is, however, to eliminate the unstabilizable mode. This is achieved by diæerentiating the system equations è3.5è, giving íxètè A _xètè+b 1 _vètè+b 2 _uètè _yètè C 2 _xètè+d 21 _vètè è3.è Notice that if the input u is weighted in the cost, a steady-state oæset after the step disturbance will result. Therefore, it is more realistic to weight the input variation _uètè instead. The controlled output z is therefore redeæned according to zètè C 1 xètè+d 12 _uètè è3.è Introducing z x : C 1 x, the system can be written as 2 3 2 4 íxètè _z x ètè 5 4 A C 1 _yètè C 2 3 2 5 2 3 4 _xètè z x ètè 5 + yètè 3 2 4 B 1 D 21 3 2 5 _vètè+ 4 B 2 3 5_uètè zètèë I ë 4 _xètè z x ètè 5+D 12 _uètè è3.8è yètè 2 yètè ë Ië 4 _xètè z x ètè yètè 3 5 This characterization is in standard form with an impulse disturbance input _v given by è3.5è. Notice, however, that assumption èb2è associated with the measurement noise does not hold. In order to satisfy this assumption as well, we can add a noise term v meas on the measured output, 2 3 yètè ë Ië 4 _xètè z x ètè 5+v meas ètè è3.9è yètè As there is often a measurement noise present in practical situations, this modiæcation seems to be realistic one. Selection of weighting matrices in H 2 -optimal control The second design topic which we will discuss is the speciæcation of the controlled output z, or equivalently, the choice of weighting matrices Q and R in the quadratic cost è3.9è or è3.1è. For convenience, we will focus the discussion in this section on the stochastic problem formulation, and the associated cost è3.1è. It has been stated that the LQG problem is rather artiæcial, because the quadratic cost è3.1è seldom, if ever, corresponds to a physically relevant cost in practical applications. This may hold for the quadratic cost è3.1è. However, it should be noted that 35
it is often meaningful to consider the variances of the individual variables. Therefore, we introduce the individual costs " 1 Z tf J i èkè lim t f!1 E x i ètè 2 dt t f " 1 Z tf J n+i èkè lim t f!1 E u i ètè 2 dt t f è è ; i 1;:::;n ; i 1;:::;m è3.è Thus, the costs J i èkè;i 1;:::;n denote the variances of the state variables, and J n+i èkè denote the variances of the inputs. In contrast to the quadratic cost è3.1è, the individual costs J i èkè are often physically relevant. If the states x i represent quality variables èsuch as concentrations, basis weight etcè, then the variance is a measure of the quality variations. The input variances, again, show how much control eæort is required. It is therefore well motivated to control the system in such awaythat all the individual costs è3.è are as small as possible. As all the costs cannot be made arbitrarily small simultaneously, the problem deæned in this way is a multiobjective optimization problem, and it can be solved eæciently using techniques developed in multiobjective optimization. The set of optimal points in multiobjective optimization is characterized by the set Pareto-optimal, or noninferior, points. Deænition 3.1. èpareto-optimalityè Consider the multiobjective control problem associated with the individual costs J i èkè, i 1;:::;n+m in equation è3.è. A stabilizing controller K is a Pareto-optimal, or noninferior, controller if there exists no other stabilizing controller K such that J i èkè J i èk è for all i 1;:::;n+m and J j èkè éj j èk è for some j 2f1;:::;n+mg Thus, a Pareto-optimal controller is simply any controller such that no individual cost can be improved without making at least one other cost worse. It is clear that the search for a suitable controller should be made in the set of Pareto-optimal controllers. The multiobjective optimal control problem is connected to the H 2 or LQG optimal control problem as follows. It turns out that the set of Pareto-optimal controllers consists of precisely those controllers which minimize linear combinations of the individual costs, i.e., where nx J LQG èkè q i J i èkè+ r i J n+i èkè i1 i1 " 1 Z tf E ëxètè T Qxètè+uètè T Ruètèëdt t f mx è è3.1è Q diag èq 1 ;:::;q n è; R diag èr 1 ;:::;r m è è3.2è 3
and q i ;r i are positive. The multiobjective control problem is thus reduced to a selection of the weights in the LQ cost in such awaythat all the individual costs J i èkè havesatisfactory values. Thus, although the LQ cost may not be physically well motivated as such, it is still relevant via the individual costs J i èkè, which often can be motivated physically. There are systematic methods to ænd suitable weights è3.2è in such away that the individual costs have satisfactory values. A particularly simple case is when one cost, J j èkè say, is the primary cost to be minimized, while the other costs should be restricted. In this case the problem can be formulated as a constrained minimization problem, subject to Minimize J j èkè è3.3è J i èkè c 2 ; i 1;:::;n+m; i j è3.4è 3.5 Literature In addition to the books referenced in Chapter 1, there are some excellent classic texts on the optimal LQ and LQG problems. Standard texts on the subject are for example Anderson and Moore è191, 199è, Astríom è19è and Kwakernaak and Sivan è19è. A newer text with a very extensive treatment of the theory is Grimble and Johnson è1988è. The minimax property of the Kalman ælter, which was mentioned in Remark 3., has been observed by many people, see for example Krener è198è. The multiobjective approach to LQG design described in Section 3.4 has been discussed in Toivonen and Míakilía è1989è. References Anderson, B. D. O. and J. B. Moore è191è. Linear Optimal Control. Prentice-Hall, Englewood Cliæs, N.J. Anderson, B. D. O., and J. B. Moore è191è. Optimal Filtering. Prentice-Hall, Englewood Cliæs, N.J. Astríom, K. J. è19è. Introduction to Linear Stochastic Control Theory. Academic Press, New York. Grimble, M. J. and M. A. Johnson è1988è. Optimal Control and Stochastic Estimation. Volumes 1 and 2. Wiley, Chichester. Krener, A. J. è198è. Kalman-Bucy and minimax æltering. IEEE Transactions on Automatic Control Theory AC-25, 291í292. Kwakernaak, H. and R. Sivan è192è. Linear Optimal Control Systems. Wiley, New York. Toivonen, H. T. and P. M. Míakilía è1989è. Computer-aided design procedure for multiobjective LQG control problems. Int. J. Control 49, 55í. 3