Algorithms for H-Infinity Optimization

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Algorithms for H-Infinity Optimization1 This lecture covers the algorithmic side of H-Infinity optimization. It presents classical necessary and sufficient conditions of existence of a β-suboptimal controller, stated in terms of stabilizing solutions of Riccati equations, and provides explicit formulae for the so-called “central” β-suboptimal controller. 7.1 Problem Formulation and Algorithm Objectives This section examines the specifics of the way in which the H-Infinity optimal LTI feedback design problem is formulated (and, eventually, solved). 7.1.1 Suboptimal Feedback Design H-infinity optimization problem appears to be formulated as the task of designing a sta­ bilizing controller K(s), which minimizes the H-infinity norm of the closed-loop transfer matrix Tzw from w to z for a given open loop plant P (s) (see Figure 7.1), defined by state space equations ẋ = Ax + B1 w + B2 u z = C1 x + D11 w + D12 u y = C2 x + D21 w + D22 u A set of standard well posedness constraints is imposed on the setup: 1 Version of March 29, 2004 2 w � � � P (s) z y u K(s) Figure 7.1: General LTI design setup • for output feedback stabilizability, the pairs (A, B2 ) and (C2 , A) must be respectively stabilizable and detectable • for non-singularity, D21 must be right invertible (full measurement noise), D12 must be left invertible (full control penalty), and matrices ⎦ ⎡ ⎦ ⎡ A − sI B1 A − sI B2 , C2 D21 C1 D12 must be respectively left and right invertible for all s ∀ jR However, in contrast with the case of H2 optimization, basic H-Infinity algorithms solve a suboptimal controller design problem, formulated as that of finding whether, for a given β > 0, a controller achieving the closed loop L2 gain ≡Tzw ≡� < β exists, and, in case the answer is affirmative, calculating one such controller. 7.1.2 Why Suboptimal Controllers? There are several reasons to prefer suboptimal controllers over the optimal one in HInfinity optimization. One of the most compelling reasons is that the optimal closed loop transfer matrix Tzw can be shown to have a 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 off at high frequencies. Example 7.1 Consider the standard LTI feedback optimization setup with ⎦ 1 ⎡ −1 s+1 P (s) = 1−s , 0 1+s 3 (a case of state estimation). Since there is no feedback loop to close here, the controller transfer function K(s) must be a stable, and the cost is the H-Infinity norm ≡Tzw ≡� � min, Tzw = 1−s 1 K(s) − . 1+s 1+s Since Tzw (1) = 0.5 for all K, and ≡T ≡� � |T (1)| for all T , the norm ≡Tzw ≡� cannot be less than 0.5. Also, there is the only way to make it identically equal to 0.5, by using K0 (s) → 0.5. Hence K0 (s) = 0.5 is the (only) H-Infinity optimal controller, and the optimal Tzw has a flat (at 0.5) amplitude of frequency response. Note that the H2 optimal K(s) equals zero in this case. Another good reason is that, in a setup with more than one control variable or more than one sensor variable, the optimal H-Infinity controller is frequently not unique (indeed, there could be a continuum of H-Infinity 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 2 sensors, 2 controls, and with � ⎤ 1 0 −1 0 s+1 � 0 1 1 0 −1 ⎢ 2 s+1 ⎢ P (s) = � 1−s � 0 0 ⎣ 0 1+s 1−s 0 0 0 1+s (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-Infinity norm from w to z cannot be made smaller than 0.5. However, there are plenty of “controllers” K(s) which will make the closed loop H-Infinity norm exactly equal to 0.5: for example, one can take ⎦ ⎡ 0.5 0 K(s) = , 0 k2 which will be an optimal controller for every k2 ∀ [0, 0.5]. A somewhat less compelling reason for not using H-Infinity optimal controllers is that, while it it known how to find them using ideal algebraic calculations, implementing these formulae in an efficient numerical algorithm is a tough problem. 7.1.3 What is done by the software The µ-Analysis and Synthesis Toolbox provides function hinfsyn.m for calculating a suboptimal output feedback: 4 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 P (s), “ncon” and “nmeas” are the numbers of actuators and sensors, “gmin” and “gmax” are a lower and an upper a-priori bounds of the achievable closed-loop H-infinity norm. The program performs a binary search for the “level of optimality” parameter β. At any point, it operates with a “current guess” β of the minimal achievable closed loop H-infinity norm, a lower bound β− and an upper bound β+ of β. The initial values of β− and β+ are supplied by “gmin” and “gmax”, and, initially, β = β+ . At each iteration of the algorithm, it is checked whether it is possible to design a controller with the resulting closed loop H-infinity gain less than β. If the answer is positive, new values of β and β + are assigned, according to β new = 0.5(β−old + β old ), β+new = β old . If the answer is negative, new values of β and β− are assigned, according to β new = 0.5(β+old + β old ), β−new = β old . This process is continued until the relative difference between β+ and β− becomes smaller than the tolerance parameter “tol”. Then, actual suboptimal feedback design is per­ formed, with β+ being the target H-infinity performance. For a given β, in order to check that a β-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-infinity control problem, stabilizing solutions X and Y of some Riccati equations must exist and be non-negative definite. In addition, a coupling condition �max (Y X) < β 2 must be satisfied. When checking solvability of the Riccati equations, “hinfsyn” uses the “Hamiltonian matrices approach”, which associates stabilizing (i.e. such that λ − αæ is a Hurwitz matrix) solutions æ = æ� of an algebraic Riccati equation � + æλ + λ � æ = æαæ with the stable invariant subspace of the auxiliary Hamiltonian matrix ⎦ ⎡ λ α H= . � −λ � Unlike in the Riccati equations and Hamiltonian matrices associated with the usual KYP Lemma, matrix α will not be positive semidefinite. This will be somewhat compensated by validity of inequality � � 0. 5 7.2 Background Results This section contains general mathematical results which play an important role in under­ standing H-Infinity optimization. In addition to providing a specific version of the KYP Lemma and a statement on computing stabilizing solutions of general algebraic Riccati equation, it presents the Parrot’s theorem and its generalizations. 7.2.1 A Specific Version of the KYP Lemma In general, KYP Lemma does not provide information about sign definiteness of matrix P = P � of coefficients of the associated quadratic storage function. However, there exists a number of situations when positivity of P is implied by the setup. To derive a solution of H-Infinity optimization problem, we need one such result. Theorem 7.1 The following conditions are equivalent: (a) a is a Hurwitz matrix, and G(s) = d + c(sI − a)−1 b has H-Infinity norm less than β; (b) there exists p = p� > 0 such that the quadratic form ψp (x, w) = β 2 |w|2 − |cx + dw|2 − 2x� p(ax + bw) is strictly positive definite. Proof Taking the KYP lemma into account, the only thing to prove here is that, subject to positive definiteness of ψp , p > 0 whenever a is a Hurwitz matrix. Indeed, ψp ≥ 0 implies the Lyapunov inequality pa + a� p + c� c < 0, hence p > 0 if and only if a is a Hurwitz matrix. 7.2.2 Stabilizing Solutions of General Riccati Equations To present a solution to the H-Infinity optimization problem, we need an important remark on uniqueness of stabilizing solutions æ (i.e. such that λ − αæ is a Hurwitz matrix) of the general Riccati equation � + æλ + λ � æ = æαæ, (7.1) where � = �� , λ, α = α� are given n-by-n matrices with real coefficients (α is not required to be sign definite), and æ is an n-by-n real matrix to be found. Formally speaking, æ is not required to be symmetric, but it will be shown that a stabilizing solution always is. 6 Theorem 7.2 Riccati equation (7.1) has a stabilizing n-by-n solution æ if and only if the maximal stable invariant subspace V+ of the associated Hamiltonian matrix ⎦ ⎡ λ α H= (7.2) � −λ � has dimension n and contains no non-zero vectors in which the first n components are equal to zero. A stabilizing solution æ (if exists) is unique, symmetric, and defined by æ = −[�1 �2 . . . �n ] · [x1 x2 . . . xn ]−1 , where ⎦ x1 �1 (7.3) ⎡ ⎦ ⎡ ⎦ ⎡ xn x2 , , �2 �n is an arbitrary basis in V+ . Thus, a given algebraic Riccati equation cannot have more than one stabilizing solu­ tion. Moreover, checking existence and finding the solution, when it exists, is a relatively straightforward linear algebra operation. Proof First, let us note that columns of a matrix V span an invariant subspace of a square matrix M if and only if M V = V S for some square matrix S. Moreover, this subspace is stable if and only if S is a Hurwitz matrix, and anti-stable if and only if −S is a Hurwitz matrix. The proof follows easily from the observation that (7.1) is equivalent to ⎦ ⎡⎦ ⎡ ⎦ ⎡ λ α I I = (λ − αæ), −æ −æ � −λ � as well as to ⎦ λ� � α −λ ⎡⎦ æ I ⎡ = ⎦ æ I ⎡ (αæ − λ). Hence, if æ is a stabilizing solution of (7.1) then the columns of [I; −æ] form a basis in a stable invariant subspace of H, and the columns of [æ; I] form a basis in an antistable invariant subspace of H� . Since the dimension of the maximal stable or anti-stable invariant subspace does not change with transposition of the matrix, the 2n-by-2n matrix H has an n-dimensional stable, and an n-dimensional anti-stable subspaces, which means that n is actually the dimension of the maximal stable invariant subspace of H, and the columns of [I; −æ] form a basis in this subspace. 7 Conversely, if V+ is the n-dimensional stable invariant subspace of H, the columns of V = [F ; G] form a basis in V+ , and det(F ) = ∈ 0 then ⎦ ⎡⎦ ⎡ ⎦ ⎡ λ α F F = S � −λ � G G for some n-by-n Hurwitz matrix S. Hence (7.1) holds for æ = −GF −1 , and S = λ − αæ. 7.2.3 Riccati Equalities and Inequalities This section presents two frequently used relations between solutions of Riccati equations and Riccati inequalities. The first relation is a straihtforward implication of the KYP lemma. Theorem 7.3 Let α = α� � 0, � = �� and λ be n-by-n matrices with real coefficients, such that the pair (λ, α) is stabilizable. Then the following conditions are equivalent: (a) the algebraic Riccati equation (7.1) has a stabilizing solution æ = æ � ; (b) the strict Riccati inequality � + æ0 λ + λ � æ0 > æ0 αæ0 has a solution æ0 = æ�0 . Moreover, æ > æ0 whenever (a) and (b) are true. Proof Let α = f f � . A stabilizing solution æ from (a) defines a completiion of squares relation x� �x + |u|2 + 2x� æ(λx + f u) = |u + f � æx|2 , where λ − f f � æ is a Hurwitz matrix. A solution æ0 from (b) means strict positive defi­ niteness of the quadratic form ψ0 (x, u) = x� �x + |u|2 + 2x� æ0 (λx + f u). According to the KYP lemma, the two conditions are equivalent. When both (a) and (b) hold, substituting u = −f � æx and comparing the coefficient matrices of the resulting quadratic forms with respect to x yields (æ0 − æ)(λ − f f � æ) + (λ − f f � æ)� (æ0 − æ) > 0, which implies æ0 − æ < 0, since λ − f f � æ is a Hurwitz matrix. 8 The derivation of H-Infinity optimal control results relies on a less straightforward relation between Riccati equalities and inequalities. Theorem 7.4 Let � = �� � 0, α = α� , and λ be n-by-n matrices with real coefficients, such that the pair (�, λ) does not have unobservable modes on the imaginary axis (i.e. matrix [�; λ − jσI] is left invertible for all σ ∀ R). Then the following conditions are equivalent: (a) the algebraic Riccati equation (7.1) has a stabilizing solution æ = æ � � 0; (b) the strict Riccati inequality � + æ0 λ + λ � æ0 < æ0 αæ0 has a solution æ0 = æ�0 > 0. Moreover, æ < æ0 whenever (a) and (b) are true, and, for a given æ in (a), the difference æ0 − æ can be made arbitrarily small by selecting an appropriate solution æ 0 in (b). Proof To show that (a) implies (b), note that, since λ − αæ is a Hurwitz matrix, equation �(λ − αæ) + (λ − αæ)� � = −I has a (unique) solution � = �� > 0. For æ0 = æ + ρ � we have � + æ0 λ + λ � æ0 − æ0 αæ0 = −ρ I − ρ 2 �α�, which is a negative definite matrix for all sufficiently small ρ > 0. To prove that (b) implies (a), note that multiplying the inequality by p 0 = æ−1 0 on both side yields α + p0 (−λ � ) + (−λ � )� p0 > p0 �p0 . (7.4) When the pair (−λ � , �) is stabilizable, this inequality, according to Theorem 7.3, yields existence of a stabilizing solution p = p� > p0 of α + p(−λ � ) + (−λ � )� p = p�p. Multiplying this equality by æ = p−1 shows that æ is a solution of (7.1). Since we also have p(−λ � − �p)p−1 = λ − αæ, and −λ � − �p is a Hurwitz matrix, λ − αæ is also a Hurwitz matrix, i.e. æ is a stabilizing solution of (7.1). 9 In the case when the pair (−λ � , �) is not stabilizable, consider the block decomposition ⎦ ⎡ ⎦ ⎡ λ11 λ12 0 0 , λ= , �= 0�22 0 λ22 � , �22 ) is stabilizable, and λ11 has no eigenvalues with positive real where the pair (−λ22 part. Since it was assumed that the pair (�, λ) does not have unobservable modes on the imaginary axis, λ11 has no eigenvalues on the imaginary axis. Considering the compatible block decompositions ⎡ ⎦ ⎡ ⎦ 0 α11 α12 p11 p012 0 , α= , p = α21 α22 p021 p022 comparing the (2,2) blocks in (7.4) yields � � � 0 0 α22 + p022 (−λ22 ) + (−λ22 ) p22 > p022 �22 p22 . � Since the pair (−λ22 , �22 ) is stabilizable, Theorem 7.3 yields existence of a stabilizing � solution p22 = p22 > p022 of the Riccati equation � � � α22 + p22 (−λ22 ) + (−λ22 ) p22 = p22 �22 p22 . Hence æ22 = p−1 22 is a stabilizing solution of Riccati equation � �22 + æ22 λ22 + λ22 æ22 = æ22 α22 æ22 , and æ= ⎦ 0 0 0 æ22 ⎡ is a stabilizing solution of (7.1). 7.2.4 The Parrot’s Lemma and Its Generalizations The classical Parrot’s lemma establishes an explicit solution formulae for the following matrix optimization question: given real matrices a, b, c of dimensions q-by-n, n-by-k, and m-by-n respectively, minimize ⎥⎦ ⎡⎥ ⎥ a b ⎥ ⎥ J(L) = ⎥ ⎥ c L ⎥ over the set of all m-by-k real matrices L. 10 Theorem 7.5 The minimum of J(L) equals ⎥⎦ ⎡⎥� � ⎥� �⎥ ⎥ a ⎥ ⎥ β = max ⎥ a b ⎥ , ⎥ ⎥ c ⎥ , and is achieved, in particular, with L defined (uniquely) by Ly = arg max inf {β 2 (|w|2 + |y|2 ) − |aw + by|2 − |cw + v|2 }. v w Theorem 7.5 is about defining a “control variable” v as a linear function v = Lg of the “sensor variable” g, with an objective of satisfying a given quadratic inequality ψ(f, v, g) � 0, where ψ(f, v, g) = β 2 (|f |2 + |g|2 ) − |af + bg|2 − |cf + v|2 . Since g = 0 would imply v = Lg = 0, the inequality ψ(f, 0, 0) � 0 must be satisfied, which yields ⎥⎦ ⎡⎥ ⎥ a ⎥ ⎥ β�⎥ (7.5) ⎥ c ⎥. In addition, for every given pair (f, g), the maximum of ψ(f, v, g) must be non-negative, which yields the inequality ⎥� �⎥ β � ⎥ a b ⎥. (7.6) The non-trivial part of the Parrot’s theorem is concerned with establishing that sup inf ψ(f, v, g) � 0 � g. v f Since (7.6) means that inf sup ψ(f, v, g) � 0 � , f v the Parrot’s lemma follows form the minimax statement inf sup ψ(f, v, g) = sup inf ψ(f, v, g) � g, f v v f which in turn follows from the standard minimax theorem since ψ(f, v, g) is strictly con­ cave with respect to v, and (7.5) means that ψ(f, v, g) is convex with respect to f . In future derivations, the following generalization of the Parrot’s lemma will be used. Theorem 7.6 Let ψ = ψ(f, v, g) be a quadratic form which is concave with respect to its second argument (i.e. ψ(0, v, 0) � 0 for all v). Then the following conditions are equivalent: 11 (a) there exists matrix L and γ > 0 such that ψ(f, Lg, g) � γ(|f |2 + |g|2 ) � f, g; (b) there exists γ > 0 such that ψ(f, 0, 0) � γ|f |2 , sup ψ(f, v, g) � γ(|f |2 + |g|2 ). v Moreover, when (b) is satisfied, L is defined by the condition that the quadratic form ψL (y) = min ψ(f, Ly, y) f is strictly positive definite. Proof Following the arguments explaining the classical Parrot’s 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-Infinity Optimization for a Simplified Setup In this section, we present a derivation of necessary and sufficient conditions of existence of a β-suboptimal feedback law, as well as a formula for one such feedback (the so-called central controller), for a slightly simplified version of the standard feedback design setup. 7.3.1 The Simplified Setup Solution To keep the formulae simple, a commonly used simplified 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: ¯1 w1 + B2 u, ẋ = Ax + B ⎦ ⎡ C¯1 x z = , u y = C 2 x + w2 , i.e. B1 = � ¯1 0 B � , C1 = ⎦ C¯1 0 ⎡ , D12 = ⎦ 0 I ⎡ (7.7) (7.8) (7.9) , D21 = � 0 I � . 12 The simplified setup is non-singular if and only if matrices � � � � � A − jσI C1� A − jσI B1 , (7.10) are right invertible for all real σ. In addition, for stabilizing controllers to exist, the pair (A, B2 ) must be stabilizable, and the pair (C2 , A) must be detectable. Consider the following Riccati equations defined by the coefficients of (7.7)-(7.9) and a suboptimality level β > 0: ¯1 B ¯1� )X, C¯1� C¯1 + XA + A� X = X(B2 B2� − β −2 B (7.11) ¯1� B ¯1 + Y A� + AY = Y (C2 C2� − β −2 C¯1 C¯1� )Y, B (7.12) with respect to symmetric matrices X = X � , Y = Y � . Theorem 7.7 Assume that the pair (A, B2 ) is stabilizable, the pair (C2 , A) is detectable, and matrices in (7.10) are right invertible for all σ ∀ R. Then an LTI controller K = K(s) such that the feedback u = Ky stabilizes system (7.7)-(7.9) and yields a closed loop L2 gain strictly less than β > 0 exists if and only if Riccati equations (7.11),(7.12) have stabilizing solutions X = X� � 0, Y = Y� � 0, and 1/2 1/2 β 2 I > X� Y� X� . (7.13) While the proof of Theorem 7.7 is constructive, and shows a simple way of calculat­ ing a suboptimal controller whenever it exists, the following more explicit statement is frequently used. Theorem 7.8 Under the conditions for the existence of a suboptimal H-Infinity controller from Theorem 7.7 are satisfied, an LTI controller u = K(s)y is suboptimal if and only if it has a realization u(t) ˙ x̂(t) v(t) ŵ1 (t) = = = = −B2� X� x̂(t) + v(t) Ax̂(t) + B̄1 ŵ1 (t) + B2 u(t) + L1 (C2 x̂(t) − y(t)) + L2 v(t), �(s)(y(t) − C2 x̂(t), β −2 B̄1� X� x̂(t), where L1 = −(I − β −2 Y� X� )−1 Y� C2� , L2 = β −2 Y� X� (I − β −2 Y� X� )−1 B2 , and � = �(s) is a stable LTI system with H-Infinity norm strictly less than β. (7.14) (7.15) (7.16) (7.17) 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-simplified setup. In addition, the theorem can be strengthened by noting that using nonlinear controllers does not reduce the minimal achievable L2 gain. Also, a nice parameterization of all stabilizin LTI controllers which yield ≡T zw ≡� < β can be given in terms of X� , Y� and the coefficients of (7.7)-(7.9). 7.3.2 Proof of Theorem 7.7 An LTI controller ẋf = Af xf + Bf y, u = Cf xf + Df y defines a closed loop system dx̄ = a¯ x + bw, z = c¯ x + dw, dt with input w, output z, and state x̄ = [x; xf ]. According to Theorem 7.1, the controller is β-suboptimal if and only if there exists a matrix p = p� > 0 such that the quadratic form ψp = ψp (f, Lg, g) is strictly positive definite for some matrix L, where ψp (f, v, g) = β (|w1 | + |y − C2 x| ) − |C¯1 x| − |u| 2 2 2 f= ⎦ w1 x 2 ⎡ , v= L= ⎦ ⎦ h u ⎡ 2 , Af Bf C f Df ⎡� ⎦ ⎡ ¯ 1 w1 + B 2 u x Ax + B −2 p , xf h ⎦ ⎡ xf g= , y ⎡ . ⎦ According to the generalized Parrot’s lemma, this is equivalent to positive definiteness of ¯ 1 w1 ) ψp (f, 0, 0) = β 2 (|w1 |2 + |C2 x|2 ) − |C̄1 x|2 − 2x� p11 (Ax + B and sup ψp (f, v, g) v for some p = p > 0, where p11 is the upper left corner of p. Consider first the condition of positivity of ψp (f, 0, 0). It is equivalent to Riccati inequality ¯1 B ¯ � p11 , β 2 C2� C2 − C¯1� C¯1 − p11 A − A� p11 > β −2 p11 B 1 � 14 or, equivalently, ¯1 B ¯1� + AY + Y A� < Y (C2� C2 − β −2 C¯1� C̄1 )Y, B where Y = β 2 p−1 11 > 0. According to Theorem 7.4, existence of such Y is equivalent to existence and positive semidefiniteness of the stabilizing solution Y� of (7.12). Moreover, Y > Y� , and the difference Y − Y� can be made arbitrarily small whenever Y� exists. Now consider the condition of positivity of supv ψp (f, v, g). Note that the supremum equals plus infinity unless ⎦ ⎡ ⎦ ⎡ x � p , = xf 0 in which case ¯1 w1 + B2 u), ψp (f, v, g) = β 2 (|w1 |2 + |y − C2 q11 �|2 ) − |C¯1 q11 �|2 − |u|2 − 2� � (Aq11 � + B where q11 is the upper left corner of q = p−1 , and the supremum equals ¯1 w1 ) + |B2� �|2 . β 2 (|w1 |2 + |y − C2 q11 �|2 ) − |C̄1 q11 �|2 − 2� � (Aq11 � + B Hence, positive definiteness of the supremum is equivalent to the matrix inequality ¯1 B ¯1� , q11 C̄1� C¯1 q11 + Aq11 + q11 A� < B2 B2� − β −2 B which can be written in an equivalent form ¯1 B ¯1� )X, C¯1� C¯1 + XA + A� X < X(B2 B2� − β −2 B −1 where X = q11 . According to Theorem 7.4, existence of such X is equivalent to existence and positive semidefiniteness of the stabilizing solution X� of (7.11). Moreover, X > X� , and the difference X − X� can be made arbitrarily small whenever X� exists. A direct calculation of a matrix inverse shows that −1 � q11 = p11 − p12 p−1 22 p12 , where p= ⎦ p11 p12 � p12 p22 ⎡ > 0. Hence −1 β 2 Y −1 = p11 > q11 = X � X� . Equivalently, 1/2 1/2 β 2 I > X� Y X� , 15 which implies (7.13). This proves necessity of the conditions for existence of a β-suboptimal controller. Conversely, when (7.13) is satisfied, inequality β 2 Y −1 > X will hold when the solutions X and Y of the corresponding Riccati inequalities are sufficiently close to X� and Y� . Then ⎦ ⎡ β 2 Y −1 X − β 2 Y −1 p= X − β 2 Y −1 β 2 Y −1 − X is positive definite and the corresponding ψp satisfies all conditions of the generalized Parrot’s lemma, which implies existence of a β-suboptimal controller.