Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science
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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science

245 MULTIVARIABLE CONTROL SYSTEMS by A Megretski Algorithms for HIn57356nity Optimization This lecture covers the algorithmic side of HIn57356nity optimization It presents classical necessary and su57358cient conditions of existence of suboptimal con

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science




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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Algorithms for H-Innity Optimization This lecture covers the algorithmic side of H-Innity optimization. It presents classical necessary and sucient conditions of existence of -suboptimal controller, stated in terms of stabilizing solutions of Riccati equations, and provides explicit formulae for the so-called \central" -suboptimal controller. 7.1 Problem Formulation and Algorithm Objectives This section

examines the specics of the way in which the H-Innity optimal LTI feedback design problem is formulated (and, eventually, solved). 7.1.1 Suboptimal Feedback Design H-innity optimization problem appears to be formulated as the task of designing sta/ bilizing controller ), which minimizes the H-innity norm of the closed-loop transfer matrix zw from to for given open loop plant (see Figure 7.1), dened by state space equations Ax 11 12 21 22 set of standard well posedness constraints is imposed on the setup: Version of March 29, 2004
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Figure

7.1: General LTI design setup for output feedback stabilizability, the pairs A;B and ;A must be respectively stabilizable and detectable for non-singularity, 21 must be right invertible (full measurement noise), 12 must be left invertible (full control penalty), and matrices sI sI 12 21 must be respectively left and right invertible for all However, in contrast with the case of H2 optimization, basic H-Innity algorithms solve suboptimal controller design problem, formulated as that of nding whether, for given 0, controller achieving the closed loop L2 gain zw exists, and, in

case the answer is armative, calculating one such controller. 7.1.2 Why Suboptimal Controllers? There are several reasons to prefer suboptimal controllers over the optimal one in H- Innity optimization. One of the most compelling reasons is that the optimal closed loop transfer matrix zw can be shown to have constant largest singular number over the complete frequency range. In particular, this means that the optimal controller is not strictly proper, and the optimal frequency response to the cost output does not roll o at high frequencies. Example 7.1 Consider the standard LTI

feedback optimization setup with +1 1+
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(a case of state estimation). Since there is no feedback loop to close here, the controller transfer function must be stable, and the cost is the H-Innity norm zw min ; T zw Since zw (1) for all and j (1) for all the norm zw cannot be less than 0.5. Also, there is the only way to make it identically equal to 0.5, by using 5. Hence ) = 0 is the (only) H-Innity optimal controller, and the optimal zw has at (at 0.5) amplitude of frequency response. Note that the H2 optimal equals zero in this case. Another

good reason is that, in setup with more than one control variable or more than one sensor variable, the optimal H-Innity controller is frequently not unique (indeed, there could be continuum of H-Innity optimal controllers). While some of those optimal controllers could be arguably better than the other, the \most optimal" ones tend to have an order much larger than the necessary minimum (which equals the order of the plant). Example 7.2 Consider the standard LTI feedback optimization setup with sensors, controls, and with +1 1+ +1 1+ (this is essentially the optimization task

from the previois example, repeated twice with some scaling). Due to the same arguments as before, the closed loop H-Innity norm from to cannot be made smaller than 5. However, there are plenty of \controllers" which will make the closed loop H-Innity norm exactly equal to 0.5: for example, one can take 5 0 which will be an optimal controller for every [0 5]. somewhat less compelling reason for not using H-Innity optimal controllers is that, while it it known how to nd them using ideal algebraic calculations, implementing these formulae in an ecient

numerical algorithm is tough problem. 7.1.3 What is done by the software The -Analysis and Synthesis Toolbox provides function hinfsyn.m for calculating suboptimal output feedback:
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function [k,g,gfin]=hinfsyn(p,nmeas,ncon,gmin,gmax,tol) (some additional optional inputs and outputs are possible here). Here \p" is the packed model of ), \ncon" and \nmeas" are the numbers of actuators and sensors, \gmin" and \gmax" are lower and an upper a-priori bounds of the achievable closed-loop H-innity norm. The program performs binary search for the \level of optimality" parameter

At any point, it operates with \current guess" of the minimal achievable closed loop H-innity norm, lower bound and an upper bound of The initial values of and are supplied by \gmin" and \gmax", and, initially, At each iteration of the algorithm, it is checked whether it is possible to design controller with the resulting closed loop H-innity gain less than If the answer is positive, new values of and are assigned, according to new 5( old old ; new old If the answer is negative, new values of and are assigned, according to new 5( old old ; new old This process is continued

until the relative dierence between and becomes smaller than the tolerance parameter \tol". Then, actual suboptimal feedback design is per/ formed, with being the target H-innity performance. For given in order to check that -suboptimal solution exists, the algorithm calls for forming two auxiliary abstract H2 optimization problems. Roughly speaking, one of these equations corresponds to the limitations in the \full information feedback" part of the stabilization task, and the other corresponds to the \zero sensor input" part of the stabilization task. For feasibility of the

suboptimal H-innity control problem, stabilizing solutions and of some Riccati equations must exist and be non-negative denite. In addition, coupling condition max YX < must be satised. When checking solvability of the Riccati equations, \hinfsyn" uses the \Hamiltonian matrices approach", which associates stabilizing (i.e. such that is Hurwitz matrix) solutions of an algebraic Riccati equation with the stable invariant subspace of the auxiliary Hamiltonian matrix Unlike in the Riccati equations and Hamiltonian matrices associated with the usual KYP Lemma, matrix will

not be positive semidenite. This will be somewhat compensated by validity of inequality 0.
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7.2 Background Results This section contains general mathematical results which play an important role in under/ standing H-Innity optimization. In addition to providing specic version of the KYP Lemma and statement on computing stabilizing solutions of general algebraic Riccati equation, it presents the Parrots theorem and its generalizations. 7.2.1 Specic Version of the KYP Lemma In general, KYP Lemma does not provide information about sign

deniteness of matrix of coecients of the associated quadratic storage function. However, there exists number of situations when positivity of is implied by the setup. To derive solution of H-Innity optimization problem, we need one such result. Theorem 7.1 The following conditions are equivalent: (a) is Hurwitz matrix, and ) = sI has H-Innity norm less than (b) there exists such that the quadratic form x; jj cx dw xp ax bw is strictly positive denite. Proof Taking the KYP lemma into account, the only thing to prove here is that, subject to

positive deniteness of whenever is Hurwitz matrix. Indeed, implies the Lyapunov inequality pa cc hence if and only if is Hurwitz matrix. 7.2.2 Stabilizing Solutions of General Riccati Equations To present solution to the H-Innity optimization problem, we need an important remark on uniqueness of stabilizing solutions (i.e. such that is Hurwitz matrix) of the general Riccati equation (7.1) where are given -by- matrices with real coecients is not required to be sign denite), and is an -by- real matrix to be found. Formally speaking, is not required to be

symmetric, but it will be shown that stabilizing solution always is.
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Theorem 7.2 Riccati equation (7.1) has stabilizing -by- solution if and only if the maximal stable invariant subspace of the associated Hamiltonian matrix (7.2) has dimension and contains no non-zero vectors in which the rst components are equal to zero. stabilizing solution (if exists) is unique, symmetric, and dened by ::: ::: (7.3) where is an arbitrary basis in Thus, given algebraic Riccati equation cannot have more than one stabilizing solu/ tion. Moreover, checking existence and

nding the solution, when it exists, is relatively straightforward linear algebra operation. Proof First, let us note that columns of matrix span an invariant subspace of square matrix if and only if MV VS for some square matrix Moreover, this subspace is stable if and only if is Hurwitz matrix, and anti-stable if and only if is Hurwitz matrix. The proof follows easily from the observation that (7.1) is equivalent to   I = ( ) as well as to = (  I Hence, if is stabilizing solution of (7.1) then the columns of ] form basis in stable invariant

subspace of and the columns of [; form basis in an anti- stable invariant subspace of Since the dimension of the maximal stable or anti-stable invariant subspace does not change with transposition of the matrix, the -by-2 matrix has an -dimensional stable, and an -dimensional anti-stable subspaces, which means that is actually the dimension of the maximal stable invariant subspace of and the columns of ] form basis in this subspace.
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Conversely, if is the -dimensional stable invariant subspace of the columns of = [ form basis in and det( then   F

G G for some -by- Hurwitz matrix Hence (7.1) holds for GF and . 7.2.3 Riccati Equalities and Inequalities This section presents two frequently used relations between solutions of Riccati equations and Riccati inequalities. The rst relation is straihtforward implication of the KYP lemma. Theorem 7.3 Let and be -by- matrices with real coecients, such that the pair ; is stabilizable. Then the following conditions are equivalent: (a) the algebraic Riccati equation (7.1) has stabilizing solution (b) the strict Riccati inequality has solution Moreover, whenever (a)

and (b) are true. Proof Let ff stabilizing solution from (a) denes completiion of squares relation x ( x fu where ff is Hurwitz matrix. solution from (b) means strict positive de/ niteness of the quadratic form x;u x x fu According to the KYP lemma, the two conditions are equivalent. When both (a) and (b) hold, substituting and comparing the coecient matrices of the resulting quadratic forms with respect to yields ( )( ff ) ff ) ( ) which implies 0, since ff is Hurwitz matrix.
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The

derivation of H-Innity optimal control results relies on less straightforward relation between Riccati equalities and inequalities. Theorem 7.4 Let and be -by- matrices with real coecients, such that the pair ; does not have unobservable modes on the imaginary axis (i.e. matrix j!I is left invertible for all ). Then the following conditions are equivalent: (a) the algebraic Riccati equation (7.1) has stabilizing solution (b) the strict Riccati inequality has solution Moreover, whenever (a) and (b) are true, and, for given in (a), the dierence can be made

arbitrarily small by selecting an appropriate solution in (b). Proof To show that (a) implies (b), note that, since is Hurwitz matrix, equation ( ) ) has (unique) solution 0. For we have I which is negative denite matrix for all suciently small 0. To prove that (b) implies (a), note that multiplying the inequality by = on both side yields 0 0 p (7.4) When the pair is stabilizable, this inequality, according to Theorem 7.3, yields existence of stabilizing solution of pp: Multiplying this equality by shows that is solution of (7.1).

Since we also have p and p is Hurwitz matrix, is also Hurwitz matrix, i.e. is stabilizing solution of (7.1).
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In the case when the pair ; is not stabilizable, consider the block decomposition 11 12 0 0 22 ; 22 where the pair 22 ; 22 is stabilizable, and 11 has no eigenvalues with positive real part. Since it was assumed that the pair ; does not have unobservable modes on the imaginary axis, 11 has no eigenvalues on the imaginary axis. Considering the compatible block decompositions 0 0 11 12 ; 11 12 0 0 21 22 21 22 comparing the (2,2) blocks in (7.4)

yields 0 0 0 22 22 22 22 22 22 22 22 Since the pair 22 ; 22 is stabilizable, Theorem 7.3 yields existence of stabilizing solution 22 22 of the Riccati equation 22 22 22 22 22 22 22 22 22 Hence 22 is stabilizing solution of Riccati equation 22 22 22 22 22 22 22 22 22 and 0 0 0 22 is stabilizing solution of (7.1). 7.2.4 The Parrots Lemma and Its Generalizations The classical Parrots lemma establishes an explicit solution formulae for the following matrix optimization question: given real matrices a;b;c of dimensions -by- -by- and -by- respectively, minimize a b c L over the set of all -by-

real matrices
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10 Theorem 7.5 The minimum of equals max a b and is achieved, in particular, with dened (uniquely) by Ly arg max inf j aw by jj cw Theorem 7.5 is about dening \control variable" as linear function Lg of the \sensor variable" with an objective of satisfying given quadratic inequality f; v; 0, where f; v; j af bg jj cf Since would imply Lg 0, the inequality f; 0) must be satised, which yields (7.5) In addition, for every given pair f; ), the maximum of f; v; must be non-negative, which yields the inequality a

b (7.6) The non-trivial part of the Parrots theorem is concerned with establishing that inf sup inf f; v; g: Since (7.6) means that sup f; v; the Parrots lemma follows form the minimax statement inf sup f; v; sup inf f; v; g; which in turn follows from the standard minimax theorem since f; v; is strictly con/ cave with respect to and (7.5) means that f; v; is convex with respect to In future derivations, the following generalization of the Parrots lemma will be used. Theorem 7.6 Let f; v; be quadratic form which is concave with respect to its second argument (i.e. (0 v; 0) for all ). Then

the following conditions are equivalent:
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11 (a) there exists matrix and such that f; Lg; f; (b) there exists such that f; 0) sup f; v; Moreover, when (b) is satised, is dened by the condition that the quadratic form min f; Ly; is strictly positive denite. Proof Following the arguments explaining the classical Parrots theorem, it is easy to see that (a) implies (b). The fact that (b) implies (a) follows from the standard minimax theorem. 7.3 H-Innity Optimization for Simplied Setup In this section, we present derivation of necessary and

sucient conditions of existence of -suboptimal feedback law, as well as formula for one such feedback (the so-called central controller ), for slightly simplied version of the standard feedback design setup. 7.3.1 The Simplied Setup Solution To keep the formulae simple, commonly used simplied setup is considered here: noises do not enter the cost, no cross-term penalties between control and state variables are allowed, and sensor noises and plant disturbances are decoupled: _ = Ax u; (7.7) (7.8) (7.9) i.e. ; C ; D 12 ; D 21
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12 The

simplied setup is non-singular if and only if matrices j!I j!I (7.10) are right invertible for all real In addition, for stabilizing controllers to exist, the pair A; must be stabilizable, and the pair must be detectable. Consider the following Riccati equations dened by the coecients of (7.7)-(7.9) and suboptimality level 0: XA X; (7.11) AY Y; (7.12) with respect to symmetric matrices Theorem 7.7 Assume that the pair A;B is stabilizable, the pair is detectable, and matrices in (7.10) are right invertible for all Then an LTI controller such that the feedback Ky

stabilizes system (7.7)-(7.9) and yields closed loop L2 gain strictly less than exists if and only if Riccati equations (7.11),(7.12) have stabilizing solutions and (7.13) While the proof of Theorem 7.7 is constructive, and shows simple way of calculat/ ing suboptimal controller whenever it exists, the following more explicit statement is frequently used. Theorem 7.8 Under the conditions for the existence of suboptimal H-Innity controller from Theorem 7.7 are satised, an LTI controller is suboptimal if and only if it has realization (7.14) )) (7.15) ( )( (7.16) (7.17)

where and ( is stable LTI system with H-Innity norm strictly less than
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13 When 0, Theorem 7.8 yields the so-called central controller The proof of Theorem 7.7, given in the rest of this section, can be extended easily to the general non-simplied setup. In addition, the theorem can be strengthened by noting that using nonlinear controllers does not reduce the minimal achievable L2 gain. Also, nice parameterization of all stabilizin LTI controllers which yield zw can be given in terms of and the coecients of (7.7)-(7.9). 7.3.2 Proof of Theorem

7.7 An LTI controller y; denes closed loop system bw; dw; dt with input output and state = [ ]. According to Theorem 7.1, the controller is -suboptimal if and only if there exists matrix such that the quadratic form f; Lg; is strictly positive denite for some matrix where f; v; j jj Ax h x ; v ; g According to the generalized Parrots lemma, this is equivalent to positive deniteness of f; 0) j xp 11 Ax and sup f; v; for some 0, where 11 is the upper left corner of Consider rst the condition of positivity of f; 0). It is equivalent to

Riccati inequality 11 11 11 11
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14 or, equivalently, AY YA Y; where p > 0. According to Theorem 7.4, existence of such is equivalent to 11 existence and positive semideniteness of the stabilizing solution of (7.12). Moreover, and the dierence can be made arbitrarily small whenever exists. Now consider the condition of positivity of sup f;v;g ). Note that the supremum equals plus innity unless in which case f;v;g 11 j 11 j Aq 11 where 11 is the upper left corner of and the supremum equals 11 j 11 Aq 11 Hence, positive

deniteness of the supremum is equivalent to the matrix inequality 11 11 Aq 11 11 which can be written in an equivalent form XA AX X; where 11 According to Theorem 7.4, existence of such is equivalent to existence and positive semideniteness of the stabilizing solution of (7.11). Moreover, and the dierence can be made arbitrarily small whenever exists. direct calculation of matrix inverse shows that 11 11 12 22 12 where 11 12 12 22 Hence 11 11 Equivalently, YX
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15 which implies (7.13). This proves necessity of the conditions for existence of -suboptimal

controller. Conversely, when (7.13) is satised, inequality >X will hold when the solutions and of the corresponding Riccati inequalities are suciently close to and Then  X 1 2 is positive denite and the corresponding satises all conditions of the generalized Parrots lemma, which implies existence of -suboptimal controller.