Here, u is the control input with m components, y is the measured output with k componenets, and the channels w j z j from disturbance inputs to contr

Transcription

1 From Mixed to Multi-Objective ontrol arsten W. Scherer Mechanical Engineering Systems and ontrol Group Delft University of Technology Mekelweg, 8 D Delft, The Netherlands Paper ID: Reg Abstract. We revisit a technique for solving multi-objective control problems through anely parameterizing the closed-loop system with the Youla parameterization and con- ning the search of the Youla parameter to nite-dimensional subspaces. It is pretty well-known how to solve such problems if the closed-loop specication are formulated in terms of the solvability of linear matrix inequalities. However, all approaches proposed so far suer from a substantial ination of size of the resulting optimization problems if improving the approximation accuracy. On the basis of a novel state-space approach to solve static output feedback control problem by convex optimization for a specic class of plants, we reveal how the ination of the size of the optimization problems can be avoided to arrive at more ecient algorithms. As an additional ingredient we discuss how to use a so-called mixed controller as a starting point for a genuine multi-objective design in order to improve the algorithms. Notation. All plants are described in discrete-time, and a (transfer) matrix is said to be stable if all its (poles) eigenvalues are contained in the open unit disk. For A nm, kak F denotes the Frobenius norm kak the spectral norm of A. RH denotes the set of all proper and R stable transfer matrices. This algebra is equipped either with the H -norm kt k = kt (e it )k F dt or the H -norm kt k = max t[;] kt (e it )k: Introduction Suppose a linear nite dimensional time-invariant generalized plant (including all weightings) is described (with a minimal realization of McMillan degree n) as z. z q y A = A q D E q D q E q F F q w. w q u A : ()

2 Here, u is the control input with m components, y is the measured output with k componenets, and the channels w j z j from disturbance inputs to controlled outputs serve to specify robustness and/or performance objectives. Sometimes we collect the signals as z = (z T z T q ) T and w = (w T w T q ) T. A controller is any nite dimensional linear time-invariant system " # A c c u = y () c D c specied through the parameter matrices A c, c, c, D c. The closed loop system is denoted as z = T w = where the channel w j z j has the realization " # A j z j = T j w j = w j = j D j " A D # w A + D c c j + D c F j c A c c F j j + E j D c E j c D j + E j D c F j w j : A controller is said to be stabilizing if A is stable. Stabilizing controllers exists i (A; ) is stabilizable and (A; ) is detectable (with respect to the open unit disk); this is assumed from now on. A typical multi-objective control problem imposes dierent specication on dierent channels of the closed-loop system [,,,,,,, 8, 9]. In this paper we concentrate on those requirements that can be formulated in terms of the solvability of a linear matrix inequality (LMI). For a pretty comprehensive list of possible choices we refer to [,, ]. Only to simplify the exposition we conne ourselves to the discussion of a paradigm example that has received considerable attention in the literature [,,, ], the so-called multi-objective H =H control problem. In this problem the goal is to keep bounds on the H -norm of, say, the rst channel w z and on the H -norm of, say, the scond channel of the controlled system: kt k < and kt k < : () The H -constraint is imposed to specify desired performance requirements on the controller such as reducing the asymptotic output-variance against white noise input or the output energy against impulse inputs. The H -constraint on z = T w guarantees that the controller is robustly stabilizing against uncertainties w = z with RH fullling the norm bound k kkt k <. oth specications can as well be interpreted as loop-shaping requirements. Note that we assume all required frequency dependent weightings to be incorporated into the system description (). The goal might be to optimize performance under the constraint of keeping the controller robustly stabilizing against uncertainties of a xed size; this amounts to minimizing for a xed such that the stabilizing controller satises the constraints (). In order to

3 determine the trade-o between the H - and H -norm constraint, one rather minimizes the functional + () for various real parameters, over the constraints () in order to (approximately) determine Pareto-optimal controllers [, 9]. For xed,, let us consider from now on the following Multi-objective (MO) control problem. Minimize () over all stabilizing controller and all real numbers, that satisfy (). The essence of the approach to be proposed in this paper is the possibility to equivalently reformulate the constraints () in terms of the solvability of LMIs. In fact, A is stable and kt k < holds if and only if there exist symmetric X and Z that satisfy the following inequalities: X A X I X X A X X A > ; X I D A > ; tr(z) < : () D Z Similarly, A is stable and kt k < is equivalent to the existence of a symmetric X with X A X I X D X A X X D I A > : () Recall that A, j, j, D j depend on the controller parameters. Hence, searching for both the controller parameter A c, c, c, D c and the matrices X j, Z to satisfy these inequalities turns out to be a non-convex (bilinear matrix inequality) problem and is, therefore, in general very hard to solve. Only pretty recently it has been claried in this generality [, ] how this non-convex problem can be transformed into a convex feasbility problem by imposing the extra (articial) constraint The resulting problem is called the X = X : () Mixed control problem. Minimize () over all stabilizing controller for which ()-() are solvable. The optimal value of the mixed control problem can be computed by solving an SDP. Since the mixed problem results from the genuine multi-objective control problem by adjoining an (articial) extra constraint, the optimal value of the former is an upper bound of that of the latter. This leads us to the topics of this paper. In Section we provide a novel state-space solution of the genuine multi-objective control problem by static output feedback if the

4 transfer matrix from u to y vanishes. It is claried in Section how this strong structural property is enforced by the Youla parameterization to solve multi-objective control problems by dynamic output-feedback for general plants. We also discuss the benet of our approach by a direct comparison with that in [9]. In Section we show how a mixed controller can be transformed such that in can serve as a starting point for the Youla parameterization to potentially improve the algorithms. The possibility for improvement is illustrated by means of an academic example in Section. Remark. All the results in this paper fully generalize to multiple specications on more than two channels of the controlled system that admit an LMI formulation. MO ontrol by Static Output-Feedback Let us consider in this section controllers of the form u = Ny where N is a static gain. Even if neglecting one of the constraints in () and considering the single objective H - or H -control problems, it not known how to compute the optimal value by solving a convex optimization problem, or whether this is possible at all. Under the hypothesis that the controller enters the generalized plant () in an ane fashion, not only single-objective control problems but even multi-objective control problems by static-output feedback can be solved via convex optimization techniques. As the main technical contribution of this article, we will reveal how a suitable parameter transformation allows to arrive at an ecient solution of these problems where eciency is related to both the size and the number of variables of the resulting LMI problems. Algebraically, the relevant property amounts to (si? A)? = : (8) Under this hypothesis, we can choose specic state-space coordinates in the realization of () to arrive at A j j D j E j F j A = A A b j b A j j j D j E j b Fj This implies that the closed-loop system matrix after static output-feedback reads as A A b + N b b j + NFj b A j = A j j D j j A : j + E j N b Dj + E j NF j A : (9) Remark. learly, since (A; ) and (A; ) are stabilizable, the matrix A itself must be already stable. Moreover, this representation claries why the closed-loop transfer matrices T j depend anely on the controller gain N.

5 Let us now partition X j = X j Z j Z according to A. We observe that j Y j X j A = X j Z j Z j Y j A b + N b b A depends non-linearly on the unknowns X j, Y j, Z j and N. We introduce the new variables P j := Q j S j S = j R j The transformation X Z Z Y A X? j?x? j Z j?z j X? j Y j? Z j X? Q S S R j Z j maps the set of all positive denite matrices into the set of all matrices whose diagonal blocks are positive denite. It is easily shown that this map is bijective, and that its inverse is given by X j Z j Q? j?q? j S j Z = j Y j?s j Q? j R j + S : j Q? j S j The main reason for introducing these new variables is found in the factorization Q j X j Z j I Z = I?S j : j Y j R j If we introduce we conclude S j Q j := Q j S ; j I Q j X j AQ j = A Q j A S j? S j A + A b + N b b =: A(N; P j ): R j A Therefore, the block X j A depending non-linearly on the parameters X j, Y j, Z j, N can be transformed by congruence into Q j X j AQ j which indeed depends anely on the new variables Q j, S j, R j and on the original controller parameter N. The same structural property holds for the blocks Q j X j Q j = Q j =: X(P j ); R j Q j X j j = j Q j = j + NFj b? S j j =: j (N; P j ); R j j j Q j j + E j N b + j S j = j (N; P j ); :

6 and D j = D j + E j NF j = D j (N): It just remains to perform congruence transformations with diag(q ; I; Q ) and diag(q ; I; I) on () to get Q X Q I Q X AQ Q X Q X Q and diag(q ; I; Q ; I) for () leads to A > ; Q X Q I Q X Q I A > ; tr(z) < ; Q D Z Q X AQ X Q X Q A > : Q D I We arrive in a straightforward fashion at the inequalities X(P ) I A(N; P ) (N; P ) X(P ) A > ; X(P ) I (N; P ) D (N) Z X(P ) I A > ; tr(z) < ; () A(N; P ) (N; P ) X(P ) A > () (N; P ) D (N) I that depend anely on the variables Q j, R j, S j, Z, N. We conclude that there exists a (stabilizing) static output-feedback controller N that renders () satised i there exist P, P, Z and N that solve the LMIs ()-(). Theorem Suppose that (si? A)? =. Then the optimal value of the multiobjective control problem by static output feedback u = N y equals the minimal value of () if varying P, P, Z, N,, over the LMI constraints ()-(). Remarks. The cost functional could also be chosen to depend anely on any of the other variables involved in the LMIs. The controller gain N appears directly in the synthesis LMIs. Hence one could without any trouble incorporate structural requirements on this controller gain. This reveals that one could even design decentralized controller gains (in which N is blockdiagonal) by convex optimization techniques.

7 The synthesis inequalities ()-() are only coupled via the controller gain N. Let us nally stress the benet of the present approach over those that appear in the literature [, 9]. The set of all static gains N R mk can be parameterized as N P = l = N with l = km basis matrices N and with free parameters R. If we dene T j; = A A b j A j j j D j and introduce the realization T j; := controlled system can be described as T j = T j; + lx = T j; = " A ; T j; = " # A j; j; j; D j; # b j E j " # A j N b F j, we infer that the j-th channel of the A j; j; A j; j;.... A j;l j;l j; j; l j;l D j; + P l D j; : () Due to this description the inequalities () and () are convex in all the variables X j, Z, without any parameter transformation. Note, however, that A j; has, generically, the size n such that the size of the LMIs as well as the number of variables is, in particular for MIMO systems, considerably larger than those in our novel approach. Let us be more specic and quantify the dierence for the H constraint in () if T j has dimension k j m j : Size of LMI Size of Matrix Variable Standard approach n (l + ) + k j + m j n (l + ) Novel approach n + k j + m j n Dierence n l n l Since the number of variables grows quadratically in the size of the matrix variables, the parameter transformation suggested in this section can save a considerable amount of computation time. MO ontrol by Dynamic Output-Feedback In a general interconnection structure the property (8) does typically not hold. It is, however, well-known that the Youla parameterization [] is the tool to enforce this condition. Indeed, let us choose K, L such that A + K, A + L are stable. Then the set of all stabilizing controllers for () can be parameterized as u by = A + K + L?L K I? I y bu ; bu = Qby; Q RH :

8 The corresponding closed-loop system then admits the description z jby = A + K?K j A + L j + LF j j + E j K?E j K D j E j F j with the Youla parameter Q RH entering as bu = Qby. The system () indeed has the structure that is required to apply our previous results. We can conclude that the search for a constant Youla parameter to solve the MO control problem is amenable to the technique in Section and can be reduced to an ecient LMI problem. This particular case can be easily generalized to the search over an arbitrary nitedimensional subspace of the innite-dimensional space RH. To be specic, let us represent the Youla parameter as Q(z) = Q (zi? A Q )? Q + D Q = w j bu Q D Q A Q Q I I If we x A Q (stable) and Q, the Youla parameter depends anely on Q and D Q. Moreover, connecting Q to () is the same as post-compensating () with y = by y to obtain z j y A = y A Q Q I I A Q Q Q F j A + K?K j A + L j + LF j j + E j K?E j K D j E j I F j and then applying the static output feedback control bu = Q D Q y y : w j bu : () ; () Again, () obeys the crucial structural property to apply the results in Section in order to eciently reduce the search for Q, D Q in the MO control problem to an LMI problem. Let us nally recall that the specic choice Q(z) = Q + Q z + + Q p z = p I I I I Q p Q p? Q Q Q () 8

9 with an FIR structure of the Youla parameter has the following favorable property. If letting p go to innity, the optimal value of the problem with () converges to the optimal value of the original MO control problem. Remark. Dually to what has been discussed, we could as well let A Q, Q in Q(z) = Q (zi? A Q )? Q + D Q be xed and search over the parameters Q, D Q. Since the size of A Q inuences the size of the resulting optimization problems, this latter remark is particularly important for the FIR structure (). If Q of size m k is tall (m k), one should work with () and A Q of size kp, and if Q is fat (m < k), the dual parameterization leads to the size mp for A Q. In general, the smallest size of A Q is hence given by p minfk; mg. Recall again that McMillan degree of () determines the sizes of the LMIs and the number of variables in the resulting LMI solution of the MO control problem. This leads us to the fundamental benet of our approach if compared to that in [9]. Indeed, the realization () with () in our approach has the dimension n + p minfk; mg; whereas that in [9] (based on Kronecker calculus similarly to what has been discussed at the end of Section ) has the dimension n + n k m + p k m k j : We observe that there can be a substantial gap between these dimensions, in particular for MIMO control problems. We reiterate that both the number of constraints and the number of variables depend quadratically on the sizes of these realization what makes it even more important for practical applications to keep them as small as possible Let us illustrate the advantage for a pretty moderately sized plant of order with control inputs, measured outputs, and component in the performance output signal. In Figure we have plotted the number variables required for one Lyapunov matrix versus the length l of the FIR representation. Youla Parameterization based on Mixed ontroller A central step in the design of dynamic MO controllers is the Youla parameterization. Instead of starting with an arbitrary stabilizing controller, it is expected - and will be demonstrated by means of an example - that one should employ a mixed controller to dene the Youla parameterization. However, controllers designed with the techniques in [, ] do in general not admit an observer structure. Of course, it is well-known that this causes no principal problem since the Youla parameterization can be based on an arbitrary stabilizing controller without any hypothesis on its structure []. If following the usual arguments in the frequency domain to derive the corresponding state-space formulas, one arrives at the closed-loop parameterization z jby = ~A ~ ~ ~ j D j E j ~ F j w j bu ; bu = Qby with ~ A of size n. () 9

10 Number of variables: Previous (dotted) compared to presented approach 9 Size of Lyapunov matrix for one specification Length l of FIR filter Figure : Number of variables versus length of basis for previous (dotted) and novel (solid) approach As shown in Section, observer based controllers lead to such a representation of degree n. We have already stressed that it is of considerable importance to keep this McMillan degree as small as possible. It was a big surprise to us that there seem no results available in the literature that allow to base the Youla parameterization on a mixed controller and, still, reduce the size of the realization in () to n. On the other hand, is has been demonstrated in [] how to compute a state-coordinate change that transforms a general controller realization into one with an observer structure. Unfortunately, this paper gives no conditions for such a transformation to exist, and it seems not to exists for an arbitrary stabilizing controller. We are in the specic situation that the set of all controllers that meet the requirements () and () (for some bounds j ) is open. Hence one could try to slightly perturb a given mixed controller and then show that one can indeed obtain an observer-based representation of this perturbed controller. To the best of our knowledge it seems a new result that this is always possible by an arbitrarily small perturbation of A c. Let us assume that the controller () stabilizes (). learly, the closed-loop system can be obtained by controlling the system z j y = A + D c j + D c F j j + E j D c D j + E j D c F j E j F j w j u ()

11 with the controller u = " A c c c # y: (8) The intention is to show that (8) admits, after a state-coordinate change, the structure of an observer for (). Hence we are required to show that there exists a non-singular T and K, L with T? A c T = [A + D c ] + K + L; T? c =?L; c T = K: (9) We can eliminate K, L and observe that any such T satises the quadratic equation A c T? T [A + D c ]? T c T + c = : () onversely, if () has a non-singular solution T, we arrive at (9) with K := c T and L :=?T? c. We conclude that (8) admits an observer structure if and only if () has a non-singular solution. The main goal is to show that a small perturbation of A c guarantees that () has a nonsingular solution. For that purpose we choose A c () := [A + D c ] + c? c such that () with A c replaced by A c () has the solution T = I. This motivates to introduce the analytic family A c (z) := za c + (? z)([a + D c ] + c? c ); z and consider the parameter dependent quadratic equation A c (z)t (z)? T (z)[a + D c ]? T (z) c T (z) + c = : () Recall that we are interested in solving this equation in a neighborhood of z =, and that we know the solution I for z = by construction. We can even prove that the exceptional set z for which () does not have a non-singular solution is at most discrete (and hence nite in a compact neighborhood of z = ). The crucial technical result is found in Theorem.. of []. This result implies that there exists a function T (z) that is analytic on n S for some discrete set S, and that satises the quadratic equation () as well as the interpolation condition T () = I. (The solution T () = I admits an extension that is analytic o a discrete set.) This implies that the exceptional set for which () has no solution is at most discrete. The same holds if we consider non-singular solutions since det(t (z)) is analytic and not identically zero (det(t ()) = ) on the connected set n S such that its set of zeros is at most discrete. Therefore, in any neighborhood of z = and on the line [; ] there exist (innitely many) parameters for which () indeed has a non-singular solution. This proves that any controller () for () admits, after possibly a slight perturbation of A c, a representation with an observer structure. Theorem Given A c, c, c, D c and >, there exists a Ac c with kc Ac? A c k < such # " # that " cac c c for some matrices K and L. = A + D c + K + L?L K

12 Remark. It is easily seen that () has a non-singular solution i there exists a subspace S of dimension n that is A + D c c -invariant and complementary to im I ; im : c A c I If S = im S, both blocks S and S are non-singular due to the complementarity S condition, and the non-singular T = S S? satises () due to the invariance property. Similarly as for symmetric Riccati equations, T can be computed in a numerically stable fashion on the basis of a Schur decomposition of the matrix A []. Example Let us consider the simple unstable system z z A = y : :? :??:?? with two performance channels. Let us rst solve the problem of minimizing kt k + kt k over all internally stabilizing controllers. A mixed design leads to the upper bound : (straight line in Figure ) with controll K mix. Performing the Youla parametrization with K mix and searching the Youla parameters in the spaces spanf; z ; : : : ; g for z = ; : : : ; leads to the controllers K with the solid optimal value curve in Figure. If we base the Youla parametrization on an arbitrary stabilizing controller (design by poleplacement), we arrive at the dashed curve in Figure - the algorithm performs far worse. However, by playing with the closed-loop poles, we could nd a pole-placing controller that leads to the dotted curve in Figure which is, actually, better suited than the mixed controller. However, mixed controllers form the only systematic choice of a starting point for the algorithm suggested in this article. It is interesting to observe the eect of performing a multi-objective design by comparing the closed-loop magnitude plots of both channels for the controllers K mix (Figure ) and K (Figure ). For reasons of comparison we show the same results in Figures - if performing the same steps for the cost kt k + kt k : One can nicely observe the change of the character of the closed-loop magnitude plots due to replacing the hard frequency constraint kt k by the average constraint kt k. w w u A

13 Optimal values based on mixed and pole placing controllers 8 Optimal Value Number of poles of FIR expansion Figure : Optimal value versus length of basis Singular Values Singular Values (d) Frequency (rad/sec) Figure : Magnitude plots of two closed-loop channels for K mix

14 Singular Values Singular Values (d) Frequency (rad/sec) Figure : Magnitude plots of two closed-loop channels for K Optimal values based on mixed and pole placing controllers 8 Optimal Value Number of poles of FIR expansion Figure : Optimal value versus length of basis

15 Singular Values Singular Values (d) Frequency (rad/sec) Figure : Magnitude plots of two closed-loop channels Singular Values Singular Values (d) Frequency (rad/sec) Figure : Magnitude plots of two closed-loop channels for

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