2 Scherer The main goal of this chapter is to generalize these ideas to robust performance specications if the matrices describing the system can be r

Transcription

1 Robust Mixed ontrol and LPV ontrol with Full lock Scalings arsten W. Scherer bstract For systems aected by time-varying parametric uncertainties, we devise in this chapter a technique that allows to equivalently translate robust performance analysis specications characterized through a single Lyapunov function into the corresponding analysis test with multipliers. Out of the multitude of possible applications of this so-called full block S-procedure, we will concentrate on a discussion of robust mixed control and of designing linear parametrically-varying (LPV) controllers. ntroduction n the past years mixed control problems have attracted considerable interest, and the LM approach to controller design has shown benecial to tackle design specications that have deemed untractable. For the design of output feedback controllers, it has been revealed quite recently how a suitable non-linear change of controller parameters allows to proceed in a straightforward fashion from an analysis specication for a controlled system formulated in terms of matrix inequalities to the corresponding synthesis inequalities for controller design [6, 2, 2]. One purpose of this chapter is to extend this general paradigm to robust performance specications if the underlying system is aected by time-varying parametric uncertainties [3, 22]. We introduce a general technique that allows to equivalently translate robust performance objectives formulated in terms of a common Lyapunov function into the corresponding analysis test with multipliers. Similar approaches, as those in [3, 22], are usually based on the so-called S-procedure that introduces conservatism since the resulting multipliers have, in general, a block-diagonal structure. nstead, we propose to use full block multipliers that are only indirectly described by linear matrix inequalities such that the reformulation will not introduce conservatism. Hence we call the underlying abstract result for quadratic forms a full block S-procedure. For robust stability specications with ane dependence on the uncertainties, such anequivalent reformulation has been provided before in [5]. Mechanical Engineering Systems and ontrol Group, Delft University of Technology, Mekelweg 2, 2628 D Delft, The Netherlands

2 2 Scherer The main goal of this chapter is to generalize these ideas to robust performance specications if the matrices describing the system can be rational functions of the parameters, and to base the transformation on one abstract result that can be applied with ease to several performance specications. Moreover, we will provide a discussion of the (straightforward) consequence for robust controller design. Related results for the robust stabilization problem have appeared in the inspiring paper []. Finally, we will turn to the design of linear parametrically varying controllers that has been fully tackled for blockdiagonal multipliers [4, 9, 2]. Our purpose is to show how to overcome this structural restriction to arrive at a solution even if the multipliers are full block matrices that are only indirectly described by LMs. ontrary to previous approaches one has to modify the structure of the scheduling function: one cannot just employ a copy of the parameters to schedule the controller but one has to use a quadratic function thereof. 2 System Description We concentrate on systems that are aected by time-varying parametric uncertainties. Since we deal with linear fractional system representations, we assume that all these parameters are collected in the matrix. The admissible setofvalues that can be taken by the uncertainties is denoted as R kl with 2 and assumed to be compact. Note that captures both the size of the uncertainties as well as their structure. Typically, for a rational dependence of the system description on the parameters, the matrix can be taken blockdiagonal, and the blocks take a real repeated structure for the results in this chapter, such a structure is not required. Given the value set, the actual time-varying parametric uncertainties consist of all continuous curves () _x. z m y Let us now lookat z z 2 :[ ) : 2 m D D 2 D m E 2 D 2 D 22 D 2m E m D m D m2 D mm E m F F 2 F m x. w m u w w 2 w (t)z a linear time-invariant system in which the uncertainties enter in a linear fractional fashion to dene the time-varying uncertain system under investigation. Obviously, w z constitutes the uncertainty channel, whereas w j z j, j 2 ::: m, are the performance channels used to describe the desired performance specications, and u y is the control channel.

3 Robust Mixed and LPV ontrol with Full lock Scalings 3 n fact, u is the control input and y is the measured output, and any linear time-invariant system that closes the loop as _x c c c x (2) c u c D c y is said to be a controller. description The resulting controlled system admits the (3) _ z z 2. 2 m D D 2 D m 2 D 2 D 22 D 2m w w 2. w (t)z z m m D m D m2 D mm w m with the realization (4) i j D ij + D c c j + D c F j c c c F j i + E i D c E i c D ij + E i D c F j : The multi-objective robust controller design problem can be sketched as follows: Find a controller that exponentially stabilizes the closed-loop system, and that achieves a mixture of various performance specications on the dierent performance channels of the controlled system for all possible uncertainties aecting the system. n the next section we extend well-known analysis tests for nominal performance as described in [2] to the corresponding robust performance tests against time-varying uncertainties in terms of a constant quadratic Lyapunov function. The essential new ingredient will be the ease how one can derive the corresponding multiplier test by just referring to the abstract full block S-procedure that is presented in Lemma. in the appendix. 3 Robust Performance nalysis with onstant Lyapunov Matrices n this section, we provide analysis tests that are based on nding a constant quadratic Lyapunov function in order to guarantee the following properties: Well-posedness of the linear fractional transformation used to describe the uncertain system. Uniform (in the uncertainty) exponential stability. Robust performance, specied as quadratic performance (such as bounding the L 2 -gain or dissipativity) or as bounding the H 2 norm, the generalized H 2 norm, or the peak-to-peak gain.

4 4 Scherer 3. Well-Posedness and Robust Stability We call the description (3) well-posed if (5) ; D is non-singular for every 2 : Then the channel w j z i of the uncertain closed-loop system admits the alternative representation _ ((t)) j((t)) (6) i ((t)) D ij ((t)) z i w j with the function () j () i () D ij () + ( ; D ) ; i + D i ( ; D ) ; j + ( ; D ) ; D j D ij + D i ( ; D ) ; D j that is, due to (5), continuous (and even smooth) on. f the interconnection is well-posed, (3) or (6) are uniformly exponentially stable if there exist constants K and > such that, for every uncertainty (:) and for every unforced (w 2 ::: w m ) system trajectory (:), k(t)k Ke ;(t;t ) k(t )k for all t t : Under the hypothesis (5), it is well-known that uniform exponential stability is guaranteed by the existence of an X > that satises the Lyapunov inequality () T X + X () < for all in the admissible value set. n the following result the Lyapunov inequality is rewritten into a form that facilitates the application of the full block S-procedure. Theorem 3.. Suppose the interconnection (3) is well-posed, and suppose there exists an X > satisfying (7) () T X X () < for all 2 : Then the uncertain system (3) is uniformly exponentially stable. Proof. The very standard proof is only included for the convenience of the reader. y compactness of, there exists an > such that () T X X () + X < for all 2. Now suppose (:) isany admissible uncertainty curve, and let (:) be an arbitrary unforced system trajectory. With v(t) :(t) T X (t), we obtain _v(t)+v(t) for all t

5 Robust Mixed and LPV ontrol with Full lock Scalings 5 by left-multiplying (t) T and right-multiplying (t). With the integrating factor e t, this implies v(t) v(t )e ;(t;t ) for all t t : Due to min (X )k(t)k 2 v(t) max (X )k(t)k 2 we nally obtain k(t)k s max (X ) min (X ) e; 2 (t;t ) k(t )k for all t t what nishes the proof with explicit formulas for the constants and K. The condition (7) is formulated in terms of the rational function () on the whole set such that it is pretty hard to verify directly. The main purpose of the full block S-procedure proposed in this chapter is to equivalently reformulate this test into a more explicit condition that makes use of multipliers. Here, the relevant set of multipliers is dened as (8) 8 < P : : P 2 R(k+l)(k+l) : P P T T P 9 > for all 2 : Whenever required we will tacitly assume that any such multiplier is partitioned as Q S P S T conformable to : R For robust stability, we will reveal in detail how to apply Lemma. such that we can be very short in extending the technique to robust performance tests. Theorem 3.2. The interconnection (3) is well-posed and X satises (7) if and only if there exists a multiplier P 2P with (9) D T X X Q S S T R Proof. To apply Lemma., introduce N X X S im D D < : S im D and U ; T :

6 6 Scherer Hence S U ker(ut) \S is described as 8 >< >: D 2 9 > : ; +( ; D ) 2 : > Therefore, we conclude that S U is complementary to S i ; D is nonsingular. Moreover, if ; D is non-singular, we arrive at the alternative description of S U as im () ( ; D ) ; + D ( ; D ) ; Then it is obvious that the following conditions are equivalent: The inequality (7) and N being negative denite on S U, the inequality (9) and N + T T PT being negative on S, and, nally, the condition P 2 P and P being positive on ker(u). The latter just follows from the observation that the kernel of U is nothing but the image of. ll this shows that Theorem 9 is a reformulation of Lemma. what nishes the proof. 3.2 Robust Quadratic Performance Suppose P pi is a symmetric matrix that denes a performance index. Then the robust quadratic performance (QP) specication on the channel i is formulated as follows: There exists an > such that () Z w i (t) z i (t) T P pi w i (t) z i (t) dt ; Z : w i (t) T w i (t) dt holds for any trajectory of the uncertain system (3) with (). The following technical hypothesis on the performance index is both indispensable to apply the full block S-procedure, and to arrive at nominal controller synthesis procedures that can be based on solving LMs: () P pi Q pi S pi Spi T R pi satises R pi : mong others, this hypothesis holds true for the following very important special cases: Robust quadratic performance with P pi ; guarantees that the L 2 -gain of the channel w i z i is robustly smaller than.

7 Robust Mixed and LPV ontrol with Full lock Scalings 7 ; P pi guarantees strict robust dissipativity ofthechannel ; w i z i, generalizing positive realness. ased on Lyapunov arguments, one can easily obtain a sucient LM condition for robust quadratic performance. Theorem 3.3. Suppose the interconnection (3) is well-posed, and suppose there exists an X satisfying (2) X > T X X Q pi S pi Spi T R pi () i () i () D ii () < for all 2. Then the uncertain system (3) is uniformly exponentially stable and satises the robust quadratic performance specication for the channel w i z i. Proof. The left-upper block of (2) reads as () T X X () + i () T R pi i () < : t this point we exploit () to infer (7) such that we can conclude uniform robust exponential stability. The proof of robust performance is, again, straightforward. y continuity and compactness of, we can add on the left-hand side of (2) for some small > without violating the inequality. For any w i 2 L 2, let (:) and z i (:) be the corresponding state- and output trajectory of the uncertain (t) system (3) with (). y right- and left-multiplication with w i (t) and its transpose, we infer d dt (t)t X (t)+ w i(t) z i (t) T P pi w i (t) z i (t) + w i (t) T w i (t) : ntegrating over [ T] and letting T go to innity implies the desired inequality () if we recall() and lim T (T ). On the basis of Lemma., we can obtain with ease the equivalent multiplier test. Theorem 3.4. The interconnection (3) is well-posed and X satises (2)

8 8 Scherer if and only if there exists a P 2P such that (3) X > T X X Q S S T R Q pi S pi Spi T R pi i D D i i D i D ii < : Proof. s in the proof of Theorem 3.2, apply Lemma. to N X X Q pi S pi Spi T R pi S im D D i (4) S im i D D i i D i D ii and U ; T : 3.3 Robust H 2 Performance f the disturbance w i that aects the system is stochastic in nature, we can also consider the H 2 criterion as a performance measure for the perturbed system. Let us assume that (), that w i is white noise, and that we are interested in bounding the variance of the output z i byagiven number. We need to assume that the uncertainty model and the controller are such that (5) 8 2 : D ii () : Then it is easy to arrive at the following analysis result.

9 Robust Mixed and LPV ontrol with Full lock Scalings 9 Theorem 3.5. Suppose the interconnection (3) is well-posed, and suppose there exist X >, Z> satisfying, for all 2, (6) () i () i () D ii () T X X ; () i () i () D ii () < (7) i () T ;X Z ; i () < Tr(Z) < : Then the uncertain system (3) is uniformly exponentially stable, and the variance of the output z i on the time-interval [ ) is smaller than. Due to the zero row and column, the inequality in (6) could be simplied. We kept it in this form to display the similarity to the robust quadratic performance test. Proof. Uniform exponential stability follows as in the proof of Theorem 3.3. For verifying robust performance, one can easily rewrite (6) with Y X ; to ()Y + Y() T + i() i () T < : Using Lemma.2, (7) is seen to be equivalent to i ()Y i () T <Z Tr(Z) < : Let us now recall that the state covariance matrix of the uncertain system is given as the solution of the initial value problem _K ((t))k + K((t)) T + i ((t)) i ((t)) T K() : Standard comparison results for linear matrix dierential equations imply K(t) <Y for t and hence we infer h i h i Tr i ((t))k(t) i ((t)) T Tr i ((t))y i ((t)) T < for all t what leads to the desired bound on the output variance. We end up with two inequalities in the parameter. Therefore, we have to apply Lemma. to each of these inequalities individually. This leads to two independentmultipliers to equivalently reformulate the robust H 2 analysis condition to the multiplier version.

10 Scherer Theorem 3.6. The interconnection (3) is well-posed and X, Z satisfy (6)-(7) if and only if there exist P P 2 2P such that (8) i D D i i D i D ii T X X Q S S T R ; i D D i i D i D ii < (9) D i D i T ;X Q 2 S 2 S2 T R 2 Z ; D i D i < Tr(Z) < : Remark. nstead of the stochastic interpretation, the criterion derived here admits as well a deterministic interpretation it guarantees a bound on the sum of the energy of the output responses to initial conditions taken as the columns of i (()). 3.4 Robust Generalized H 2 Performance The generalized H 2 norm is the gain of the system mapping w i 2 L 2 into z i 2 L, i.e., the energy to peak gain [6]. The corresponding performance specication is to robustly guarantee the bound (2) kz i k sup <kw i k 2 < kw i k 2 <: Note that the gain can only be nite if (5) holds what is assumed throughout this section. Theorem 3.7. Suppose the interconnection (3) is well-posed, and that there exists an X > such that, for all 2, the inequality (6) and (2) i () T ;X i () hold true. Then (3) is robustly exponentially stable, and the gain of L 2 3 w i z i 2 L is smaller than. Proof. s for quadratic performance, the proof of stability is obvious, and one can infer from (6) that there exists an > such that d dt (t)t X (t) ( ; )w i (t) T w i (t) <

11 Robust Mixed and LPV ontrol with Full lock Scalings for any w i 2 L 2. ntegrating over [ T] leads to (T ) T X (T ) ( ; ) R T w i(t) T w i (t) dt and hence (T ) T X (T ) ( ; )kw i k 2 2 for all T : The second inequality (2) implies i () T i () <X and therefore, due to D ii (), z i (T ) T z i (T ) (T ) T X (T ) for all T : oth relations taken together lead to (2). Similarly as for the H 2 performance specication, we have to introduce two independent multipliers to equivalently reformulate these conditions to the corresponding multiplier versions. Theorem 3.8. The interconnection (3) is well-posed and X satises (6) and (2) if and only if there exist multipliers P P 2 2P with (8) and (22) D i D i T ;X Q 2 S 2 S2 T R 2 D i D i < : 3.5 Robust ound on Peak-to-Peak Gain Let us nally include in our list the important time-domain specication to keep the peak-to-peak gain L 3 w i z i 2 L smaller than. To be precise, we intend to robustly guarantee (23) kz i k sup <: <kw i k< kw i k gain, simple Lyapunov arguments provide sucient conditions for this inequality tohold. Theorem 3.9. Suppose the interconnection (3) is well-posed. f there exist X, >, such that, for all 2, (24) () i () i () D ii () T X X X ; () i () i () D ii () < (25) i () D ii () T ;X ( ; ) i () D ii () < then (3) is robustly exponentially stable, and the gain of L 3 w i z i 2 L is robustly smaller than.

12 2 Scherer Proof. s before we observe that (24) implies d dt (t)t X (t)+(t) T X (t) ; w i (t) T w i (t) : We infer (t) T X (t) R t e;(t; ) kw i ()k 2 d and hence (t) T X (t) kw ik 2 for all t : The inequality (25) leads (due to ; >) to ; kz i(t)k 2 (t) T X (t)+( ; )kw i (t)k 2 (t) T X (t)+( ; )kw i k 2 for some small >. f we combine we inferkz i k 2 ( ;)kw i k 2 and hence (23). The reformulation into the corresponding multiplier version is, again, straightforward. Theorem 3.. The interconnection (3) is well-posed and (24)-(25) hold if and only if there exist multipliers P P 2 2P that satisfy (26) T X X X Q S S T R ; i D D i i D i D ii < (27) T ;X Q 2 S 2 S2 T R 2 ( ; ) D D i i D i D ii < : Note that X and enter these inequalities non-linearly. n an analysis problem, it is advisable to search for the best upper bound on the peak-topeak gain by xing >, and minimizing the bound over the LMs (26)-(27) (what is a convex optimization problem) to get the optimal value (). Then one can perform an additional line-search over to further minimize () and to get to the smallest achievable bound. 4 Dualization On the basis of Lemma.2, one can dualize all robust performance tests we have provided in the previous section. Let us concentrate on quadratic

13 Robust Mixed and LPV ontrol with Full lock Scalings 3 performance only with an index matrix Q pi Spi T S pi R pi that is non-singular and whose inverse is denoted as ~Q pi ~ Spi ~S pi T ~R pi Q pi S pi Spi T R pi ; : Note that this property can be enforced by a slight perturbation that neither changes the problem formulation nor its solution. For an arbitrary matrix M, we observe that (28) im M? im ;M T : Hence the set of dual scalings has to be dened as 8 < ~P : P ~ 2 R (k+l)(k+l) : P ~ P ~ T 8 2 : : ; T T ~ P ; T 9 < and each P ~ 2 P ~ is partitioned in the same fashion as P. orollary 4.. There exists an X and a multiplier P 2P with (2) if and only if there is a dual Lyapunov matrix Y > and a dual multiplier P ~ 2 P ~ with T Y Y Q ~ S ~ S ~ T R ~ Qpi ~ Spi ~ S ~ T pi Rpi ~ The Lyapunov matrices and multipliers are related as X Y ; and P ~ P ; : ; T T T i ; T ;D T ;D T j ; T i ;D T j ;D T i > Proof. Suppose X and P satisfy (2). Since is in the value set, we conclude that R>. Hence we infer T X X Q S S T R Q pi S pi Spi T R pi {z } N :

14 4 Scherer This implies that S as dened in (4) is, in fact, a negative subspace of N of T maximal dimension. Moreover, since (3) implies P < P is non-singular. Hence the same is true of N. Due to (28) (after a row permutation), the proof is nished by applying Lemma.2. 5 How to Verify the Robust Performance Tests The set of multipliers P is obviously convex. However, since this set is described by innitely many inequalities, and since it is not obvious, in general, how to reduce to nitely many conditions, it is hard to even decide whether a matrix P is indeed contained in P or not. This precludes the verication of the robust performance tests in Section 3 by standard algorithms. This is the motivation to conne, at the expense of conservatism, the search to a smaller set of multipliers that admits a simple description, preferably in terms of nitely many LMs. For that purpose we assume that the value set is the convex hull of nitely many prespecied matrices: D D (29) of ::: N g: Let us then introduce the multiplier set P : fp 2P: Q<g which satises P P: Note that, in general, this is a strict inclusion such that the restriction of the search to P introduces conservatism. However, a straightforward convexity argument (based on Q<) reveals that P 2P if and only if Q< T P > for all ::: N: Hence, the set P can indeed be fully described by nitely many linear matrix inequalities, and the search for P 2 P renders our robust performance test veriable by solving standard LM problems. Remark. f D such that the parameters enter (3) anely,we observe that all the robust performance inequalities already imply that any multiplier in P satises Q< and is, hence, contained in P. Hence, in this case, P and P conincide and no conservatism has been introduced. t is possible to reduce the conservatism for block-diagonal uncertainties. For that purpose let us assume that 8 >< >:... m m 9 > : i 2 [; ] > :

15 Robust Mixed and LPV ontrol with Full lock Scalings 5 Then we obtain (29) where the N 2 m generators are dened by letting each i vary in f; g. Let us now partition the left-upper block Q of the multiplier P as what denes m diagonal blocks Q ::: Q m. f we introduce P 2 : fp 2P: Q < ::: mg satisfying P P 2 P we infer P 2P 2 i Q < ::: m T P > ::: N on the basis of a simple multi-convexity argument. gain, the set P 2 is described in terms of nitely many LMs. s expected, one can demonstrate by simple examples that the set P 2 is in general larger than P, and that P 2 can lead to less conservative robust performance tests. ll these techniques apply in the same fashion to the corresponding dual inequalities as provided in Section.2 for the corresponding sets of dual multipliers ~ P, ~ P and ~ P2. 6 Mixed Robust ontroller Design We have listed several important analysis tests for robust performance. n a typical multi-objective robust controller design problem one tries to nd a controller that meets a selection of all these specications on various channels of the controlled system. n order to render the underlying analysis problems algorithmically tractable, we constrain ourselves to the multiplier set P or, for block-diagonal uncertainties, to P 2 respectively. Even for the nominal performance specications, the corresponding multiobjective control problems are still hard and mostly open. t is wellknown that the main diculty arises due to the fact that each of the performance specication requires, in general, a dierent Lyapunov matrix to render it satised. The presently known design techniques do not allow to easily overcome this obstacle. This has been the motivation to consider, instead, the so-called mixed control problems in which the goal is to render the specications satised with a common Lyapunov function X for all specications under investigation. To be specic, we consider the robust mixed QP/H 2 problem: Try to robustly achieve quadratic performance with index P pi on channel w i z i and an H 2 bound on channel w j z j. Note that the generalization to any other combination of (possibly repeated) performance specications on other channels is obvious. The robust mixed QP/H 2 control problem hence aims at nding a controller, multipliers P P P 2 2 P and a Lyapunov matrix X as well as an auxiliary variable Z > that render the LMs (3) and (8)-(9) with i replaced by j satised.

16 6 Scherer 6. Synthesis nequalities for Output Feedback ontrol Quite recently it has been observed [2, 6, 2, 8] how to step in a formal manner from the analysis to the corresponding synthesis inequalities. This procedure is based on transforming the Lyapunov matrix X and the controller parameters as (3) X c c c D c X Y K L M N : v into the new symmetric blocks X, Y and the transformed controller parameters K, L, M, N that are collected in the variable v. Let us now introduce the functions X(v) Y X and (v) j (v) i (v) D ij (v) Y j X X j i Y i D ij + E i K L M N F j that are ane in v. n order to transform the synthesis inequalities, one needs to nd a formal congruence transformation involving Z such that the blocks in the analysis inequalities transform as X Z T XZ X i X j D ij ZT [X]Z Z T [X j ] i Z D ij : Then it suces to perform the substitution Z T XZ X(v) Z T [X]Z Z T [X j ] i Z D ij (v) j(v) i (v) D ij (v) to arrive at the synthesis inequalities in the new variables v and all other variables that appear in the analysis test (such as multipliers and auxiliary parameters). Once having solved the synthesis inequalities, the inversion of (3) leads back to the Lyapunov matrix and to the desired controller parameters. This inverse is easily calculated by nding non-singular matrices U, V with ;XY UV T and solving the equations K M L N Y V XY X + U X X U c c c D c V T Y

17 Robust Mixed and LPV ontrol with Full lock Scalings 7 for X and c, c, c, D c. f performing these two steps for the robust mixed QP/H 2 problem, one arrives at the synthesis inequalities D j D j T T Y > X Q S S T R Q pi S pi Spi T R pi Q S S T R ; T ;X Q 2 S 2 S2 T R 2 Z ; i D D i i D i D ii j D D j j D j D jj D j D j < < < Tr(Z) < where we suppress the variable v for reasons of space. Unfortunately, testing the feasibility of these inequalities does, in general, not amount to solving a convex optimization or LM problem. Hence one has to resort to controller scalings iteration as they are known from -theory [3]. One might suggest the following procedure: onsider the scaled uncertainty set r and try to maximize r such that the synthesis inequalities are satised for the value set r. Start with a nominal design for r. Then the iteration proceeds as follows: n the rst step, x K, L, M, N and nd X, Y and multipliers P, P, P 2 which correspond to r such that the synthesis inequalities hold and r is maximal. f parametrizing the multipliers by nitely many LMs, testing the feasibility for xed r corresponds to an analysis problem and hence reduces to a standard LM feasibility test the maximization of r can be performed by bisection. n the second step, one xes P, P, P 2, and maximizes r by varying the whole variable v. For xed r, this corresponds to a nominal performance design problem (such that it reduces to an LM feasibility test), and the maximization of r is done by bisection. One can immediately advise many variations of this principal procedure. n particular, one might choose other combinations of xed and varying parameters in the optimization steps to increase r [8].

18 8 Scherer 6.2 Synthesis nequalities for State-Feedback ontrol The procedure described in the previous section applies literally to statefeedback design. One just needs to employ the functions (v) X(v) j (v) Y + M Y i (v) D ij (v) i Y + E i M and the equations to reconstruct the Lyapunov matrix and the (static) controller read as X Y ; D c MY ; : s such, the synthesis inequalities are not ane in all variables. However, as demonstrated for the analysis inequalities, one can straightforwardly dualize the synthesis inequalities that we have derived above. Due to the mere fact that j (v) andd ij (v) do actually not depend on v, the dual inequalities dene convex constraints even if letting both the multipliers, the auxiliary parameter Z, and the whole parameter v vary in fact, they can be easily rearranged to become ane in all unknowns. Hence, what is known from single-objective control problems or for block-diagonal multipliers completely generalizes to robust mixed control problems with full block scalings. 6.3 Elimination of ontroller Parameters t is well-known how to eliminate the controller parameters in single-objective nominal design problems. This is also possible in single-objective robust control problems. Let us consider, as an example, the robust quadratic performance problem with index P p on the channel w 2 z 2. We will provide a variation of the standard procedure based on the projection lemma that leads, even for a general quadratic performance index, to particularly simple formulas. We just apply Lemma.3 and observe we can work with basis matrices K and K 2 of the kernels of ker T E T E2 T and ker F F 2 respectively. Then we arrive at the following equivalent synthesis test: Find X, Y and multipliers P 2P, P ~ 2 P ~ that satisfy Y (3) > X j D ij (32) K T 2 T Q S S T R Q p S p Sp T R p X X X 2 D D 2 2 D 2 D 22 K 2 <

19 Robust Mixed and LPV ontrol with Full lock Scalings 9 (33) K T T Q ~ S ~ S ~ T R ~ Qp ~ Sp ~ S ~ T p Rp ~ ;Y T ;Y T ;Y T 2 ; T ;D T ;D T 2 ; T 2 ;D T 2 ;D T 22 K > and the duality coupling condition (34) ~Q S ~ ~S T R ~ Q S T S R ; : Non-convexity enters via the generally non-convex constraint (34). lthough this result has not appeared in the literature, it is a straightforward (and notationally simple) extension of those in [5, ] to robust quadratic performance. The main purpose for being so detailed is to point out the relation to LPV control in the next section. 7 LPV Design with Full Scalings n LPV control, it is assumed that the parameters (t) that enter the system are not unknown but that they can be measured on-line. This allows, among other applications, to approach (robust) gain-scheduling synthesis problems for non-linear control systems. So far, the LPV problem has been solved for blockdiagonal parameter matrices with the corresponding block-diagonal scalings [, 9, 4, 2]. n [9] we have sketched how to extend these techniques to full block scalings, and in this chapter we give for the rst time the full problem solution. djusted to the structure of (), we assume that the measured parameter curve enters the controller also in a linear fractional fashion. Hence an LPV controller is dened by scheduling the LT system (35) _x c c x c + c y w c with the actual parameter curve as u z c w c c ((t))z c : c x c + D c y w c The controller is hence parameterized through the matrices c, c, c, D c, and through a possibly non-linear scheduling function c (). Note that previous approaches were based on c () such that the controller is scheduled with an identical copy of the parameters. However, full block scalings require the extension to more general scheduling function that will - a posteriori - turn out to be quadratic functions.

20 2 Scherer The goal is to construct an LPV controller that renders the quadratic performance specication with index P p for the channel w 2 z 2 for all possible parameter curves satised. s standard, the solution of this problem is obtained with a simple trick. n fact, the controlled system can, alternatively, be obtained by scheduling the LT system (36) _x z z c z 2 y w c with the parameters as (37) w w c 2 D D 2 E 2 D 2 D 22 E 2 F F 2 (t) c ((t)) and then controlling this parameter dependent system with the LT controller (35). Hence, with the chosen controller structure, the LPV problem we started out with is equivalently reformulated to a robust performance design problem as discussed previously: Find an LT controller (35) that renders the system (36)- (37) uniformly exponentially stable such that the robust quadratic performance specication for w 2 z 2 with non-singular index P p is satised. Due to the fact that the parameters are measured on-line, the uncertainties enter the system in a very specic form what is reected in the particular structure of the describing matrices in (36) and of the parameters in (37). This particular structure implies that the synthesis inequalities related to this robust performance problem are standard LM problems. Hence they can be solved (without conservatism) using existing algorithms. For guaranteeing robust stability and performance of the closed-loop system, we employ extended multipliers adjusted to the extended uncertainty structure that are given as (38) P e Q e S e Se T R e and that satisfy c () (39) P e Q Q 2 S S 2 Q 2 Q 22 S 2 S 22 R R 2 R 2 R 22 c () z z c x w w c w 2 u z c with Q e < R e > > forall2:

21 Robust Mixed and LPV ontrol with Full lock Scalings 2 The corresponding dual multipliers ~ Pe P ; e (4) ~P e ~Q e Se ~ ~R e ~S T e ~Q ~ Q2 ~ S ~ S2 ~Q 2 ~ Q22 ~ S2 ~ S22 R ~ R2 ~ R2 ~ R2 ~ are partitioned similarly as with Qe ~ < Re ~ > : s indicated by this notation, it will turn out that the LPV synthesis inequalities will be only inuenced by the multipliers blocks in P e and Pe ~ without indices, and they will be actually identical to those of the robust control problem apart from the coupling condition (34). Therefore, testing these synthesis conditions indeed amounts to solving a standard LM problem. Theorem 7.. There exists a controller (35) and a scheduling function such that the system (36)-(37) controlled with (35) satises the analysis conditions for robust quadratic performance with multipliers (38)-(39) if and only if there exist X, Y and scalings P 2 P, P ~ 2 P ~ that satisfy the linear matrix inequalities (3)-(33). Proof. The proof of `only if' is straightforward: Eliminate the controller parameters in the analysis inequalities. Due to the specic structure of the describing matrices in (36), the resulting synthesis inequalities simplify to (3)- (33) such that only the multiplier parts Q S P S T and P ~ Q ~ S ~ R ~S T R ~ appear. The constructive proof of `if'is more involved. Let us assume that we have found a solution to (3)-(33). First step: Extension of Scalings. Let us dene the matrices Z and ~ Z with the row partition of P. Note that im( ~ Z) is the orthogonal complement of im(z), and recall (4) Z T PZ < ~ Z T P ~ Z>aswell as Z T ~ PZ < ~ Z T ~ P ~ Z>: For the given P and P ~,we try to nd an extension Pe with (38) such that the dual multiplier Pe ~ P e ; is related to the given P ~ as in (4). fter a suitable permutation, this amounts to nding an extension ; P T ~P P T (42) T T T T with NT T T T T NT where the specic parametrization of the new blocks in terms of a non-singular matrix T and some symmetric N willturn out convenient. Such an extension is

22 22 Scherer very simple to obtain. However, we also need to obey the positivity/negativity constraints in (38) that amount to (43) Z Z T P T T T T T NT Z Z < and (44) ~Z ~ Z T P T T T T T NT ~Z ~ Z > : We can assume w.l.o.g. (perturb, if necessary) that P ; ~ P ; is non-singular. Then we set N (P ; ~ P ; ) ; and observe that (42) holds for any non-singular T. The main goal is to adjust T to render (43)-(44) satised. We will in fact construct the subblocks T TZ and T 2 T ~ Z of T (T T 2 ). Due to (4), the conditions (43)-(44) read in terms of these blocks as (45) T T h N ; Z(Z T PZ) ; Z T i T < and T T 2 h N ; ~ Z( ~ Z T P ~ Z) ; ~ Z T i T 2 > : f we denote by n + (S), n ; (S) the number of positive, negative eigenvalues of the symmetric matrix S, we hence have to calculate n ; (N ; Z(Z T PZ) ; Z T ) and n + (N ; ~ Z( ~ Z T P ~ Z) ; ~ Z T ). Simple Schur complement arguments reveal that equals Z T PZ Z T n ; Z N n ; (N ; Z(Z T PZ) ; Z T )+n ; (Z T PZ) and n ; (Z T ~ P ; Z)+n ; (N): y Lemma.2 and ~ Z T P ~ Z >, we infer Z T ~ P ; Z <, and hence n ; (Z T PZ)n ; (Z T ~ P ; Z). This leads to n ; (N ; Z(Z T PZ) ; Z T )n ; (N) and n + (N ; ~ Z( ~ Z T P ~ Z) ; ~ Z T )n + (N): This implies that there exist T, T 2 with n ; (N), n + (N) columns that satisfy (45). Since n + (N)+n ; (N) equals the number of rows of T, these two blocks indeed dene a square T that can be assumed (after perturbation, if necessary) to be non-singular. This nishes the construction of the extended multiplier (38) where we observe that the dimensions of Q 22 /R 22 equal the number of columns of T /T 2 which are, in turn, identical to n ; (N)/n + (N).

23 Robust Mixed and LPV ontrol with Full lock Scalings 23 Second Step: onstruction of the scheduling function. Let us recall that T T P > and ~ P ; T ; T < on. y Lemma.3, we can hence infer that, for each 2, there indeed exists a c () that satises (39). Due to the structural simplicity,wecaneven provide an explicit formula which shows that c () can be selected to depend smoothly on. ndeed, by a straightforward Schur-complement argument, (39) is equivalent to U U 2 (W +) T W2 T U 2 U 22 W2 T (W 22 + c ()) T W + W 2 V V 2 W 2 W 22 + c () V 2 V 22 for U R e ; Se T Q; e S e >, V ;Q ; e >, W Q ; e a solution, the inequality can be rearranged to U 22 ; W 22 + c () V 22 ; U 2 W T 2 W 2 V 2 U W + V ; > S e. Since there exists U 2 W2 T W 2 V 2 > and it is clear that one solution is obtained by rendering the (2 )-block to vanish: ; U c () ;W 22 + U 2 W 2 V 2 : W + V W 2 Note that c () has dimension n ; (N) n + (N). Third Step: LT controller construction. fter having reconstructed the scalings, the design of the LT part of the controller amounts to solving a nominal quadratic performance problem what can be done along standard lines, as e.g. shown in [2]. Remark. The proof reveals that the scheduling function c () has a many rows/colums as there are negative/positive eigenvalues of P ; P ~ ; (if assuming w.l.o.g. that the latter is non-singular.) f it happens that P ; P ~ ; is posititve ornegative denite, there is no need to schedule the controller at all then we obtain a controller that solves the robust quadratic performance problem. The search for X and Y and for the multipliers P 2 P and P ~ 2 P ~ to satisfy (3)-(33) amounts to testing the feasibility of a standard LM. Moreover, the controller construction in the proof of Theorem 7. is constructive. Hence we conclude that we have found a full solution to the quadratic performance LPV control problem (including L 2 -gain and dissipativity specications) for

24 24 Scherer full block scalings P e that satisfy Q e <. The more interesting general case without this still restrictive negativity hypotheses is dealt with in a forthcoming paper. 8 onclusions n this chapter we have discussed how to handle multi-objective robust performance analysis and synthesis problems for systems that are aected by time-varying parametric uncertainties. We introduced a general technique, the so-called full block S-procedure, that allows to formally introduce full block multipliers in order to reduce the conservatism that could result from the restriction to standard block-diagonal multipliers. Finally, we provided a solution to the single-objective LPV control problem if employing a subclass of full block scalings the fully general case is left to a forthcoming paper. uxiliary Results on Quadratic Forms. Full lock S-Procedure Suppose S is a subspace of R n, T 2 R ln is a full row rank matrix, and U R kl is a compact set of matrices of full row rank. Dene the family of subspaces S U : S\ker(UT)fx 2S: UTx g fx 2S: Tx 2 ker(u)g indexed by U 2 U. Suppose N 2 R nn is a xed symmetric matrix. The goal is to render the implicit negativity condition 8U 2 U : N < ons U explicit. We want to relate this property, under certain technical hypotheses, to the existence of a symmetric multiplier P that satises N + T T PT < ons and 8U 2 U : P > onker(u): s a technical hypothesis, we require that all subspaces S U are complementary to a xed subspace S S that has the two properties dim(s ) k and N ons : n the intended applications, S is an unperturbed system, T picks the interconnection variables that are constrained by the uncertainties, the elements of U 2 U dene kernel representations of the uncertainties, and S U is the uncertain system. The complementarity condition amounts to a wellposedness property of the uncertain system description, and the non-negativity of N is a condition on the performance index of interest. Lemma.. The two conditions (46) 8U 2 U : S U \S fg and N < on S U

25 Robust Mixed and LPV ontrol with Full lock Scalings 25 hold i there exists a matrix P that satises (47) 8U 2 U : N + T T PT < on S and P > on ker(u):.2 Dualization The dualization of robust performance tests is most easily achieved with the following well-known auxiliary result that is the abstract version of a dualization argument in [] which is based on manipulating block matrices. Lemma.2. Suppose that N is symmetric and non-singular, and that S is a negative subspace of N of maximal dimension. (N is negative denite on S, and the number of negative eigenvalues of N coincides with the dimension of S.) Then N ; is positive denite on S? :.3 Solvabiltiy Test for a Quadratic nequality The following explicit solvability characterization for a quadratic matrix inequality serves to conveniently eliminate controller parameters in synthesis tests. onsider the quadratic inequality (48) T X + T Q S T S R T X + in the unstructured unknown X. We assume that Q S R and that S T is non-singular. R y Lemma.2, we can dualize to equivalently reformulate the inequality as ; T X T ; T T Q S T S R ; ; T X T ; T < > : The solvability test makes use of matrices? and? whose columns form a basis of the kernels of and respectively. Lemma.3. The quadratic inequality (48) has a solution X if and only if T?? Q S T S R? <? and ; T?? T Q S T S R ; ; T?? > :

26 26 Scherer The proof of `only if' is obvious. The proof of the converse can be based on the projection lemma and is constructive if a solution is known to exist, one can explicitly calculate it. References [] P. pkarian and P. Gahinet, convex characterization of gain-scheduled H controllers, EEE Trans. utomat. ontrol, 4 (995), pp [2] P. pkarian, P. Gahinet, and G. ecker, Self-scheduled H control of linear parameter-varying systems, Proc. mer. ontr. onf., (994), pp [3] G. ecker,. Packard, D. Philbrick, and G. alas, ontrol of parametricallydependent linear systems: single quadratic Lyapunov approach, Proc. mer. ontr. onf., San Francisco,, (993), pp [4] G. ecker,. Packard, Robust performance of linear parametrically varying systems using parametrically dependent linear feedback, Systems ontrol Lett., 23 (994), pp [5] S.P. oyd, L. El Ghaoui, E. Feron, and V. alakrishan, Linear Matrix nequalities in Systems and ontrol Theory. SM Studies in pplied Mathematics 5, SM, Philadelphia, 994. [6] M. hilali, P. Gahinet, H Design with Pole Placement onstraints: an LM pproach, EEE Trans. utomat. ontrol, 4 (996), pp [7] P. Gahinet,. Nemirovskii,.J. Laub, and M. hilali, The LM control toolbox, Proc. 33rd EEE onf. Decision ontr., (994), pp [8] L. El Ghaoui and J.P. Folcher, Multiobjective robust control of LT systems subject to unstructured perturbations, Systems ontrol Lett., 28 (996), pp [9]. Helmersson, Methods for Robust Gain-Scheduling, Ph.D. Thesis, Linkoping University, Sweden, 995. [] T. wasaki and S. Hara, Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations, EEE Trans. ut. ontrol, 43 (998), pp [].E. Kose, F. Jabbari, W.E. Schmittendorf, direct characterization of L 2 -gain controllers for LPV systems, Proc. EEE onf. Decision ontr., (996), pp [2]. Masubuchi,. Ohara, N. Suda, LM-based controller synthesis: a unied formulation and solution, Proc. merican ontr. onf., (995), pp [3]. Masubuchi,. Ohara, N. Suda, Robust multi-objective controller design via convex optimization, Proc. EEE onf. Decision ontr., (996), pp [4]. Packard, Gain-scheduling via linear fractional transformations, Systems ontrol Lett., 22 (994), pp [5]. Rantzer and. Megretski, System analysis via integral quadratic constraints, Proc. EEE onf. Decision ontr., (994), pp [6] M.. Rotea, Generalized H 2 control, utomatica, 29 (993), pp [7].W. Scherer, Mixed H 2 H ontrol, in:. sidori, Ed., Trends in ontrol, European Perspective, Springer-Verlag, erlin 995, pp [8].W. Scherer, From LM nalysis to Multichannel Mixed LM Synthesis: General Procedure, Selected Topics in dentication, Modelling and ontrol, 8 (995), pp. -8. [9].W. Scherer, Robust generalized H 2 control for uncertain and LPV systems with general scalings, Proc. EEE onf. Decision ontr., (996), pp [2].W. Scherer, P. Gahinet, M. hilali Multi-Objective Output-Feedback ontrol via LM Optimization, EEE Trans. utomat. ontrol 42 (997), pp

27 Robust Mixed and LPV ontrol with Full lock Scalings 27 [2] G. Scorletti, L. El Ghaoui, mproved linear matrix inequality conditions for gain scheduling, Proc. EEE onf. Decision ontr., (995), pp [22] H. Tokunaga, T. wasaki, S. Hara, Multi-objective robust control with transient specications, Proc. EEE onf. Decision ontr., (996), pp