Keywords

Introduction

Can mathematics help us deal with the inevitable theory-practice gap? Should we be optimistic and assume that discrepancies between models and nature are random and neutral towards our actions or be pessimistic and design for the worst such discrepancies? Feedback control theory has struggled with these questions, perhaps more so than other fields.

During the surge of optimal control in the 1960s, optimism carried the day. A prominent example is the LQG (\(\mathcal{H}_{2}\)) regulator, which minimizes the effect of random disturbances and has an elegant state-space solution; in comparison, the frequency-domain designs of classical control appeared primitive and conservative. But the pessimists struck back in the late 1970s, showing things could go very wrong (unstable) with LQG, when a parameter variation was introduced in the plant model. This ushered in the robust control era of the 1980s, with its worst-case analysis of stability over deterministic sets of plants, leading to other design metrics such as \(\mathcal{H}_{\infty }\) control. In this mentality, exogenous disturbances were also treated as an adversary to be protected against in the worst case, perhaps an excess of pessimism.

The robust \(\mathcal{H}_{2}\) problem incarnates the search for a middle ground, where stability is treated with the conservatism it deserves, but performance is optimized for a more neutral noise. This entry summarizes efforts made around the 1990s to seek this compromise.

\(\mathcal{H}_{2}\) Optimal Control

In the feedback diagram of Fig. 1, signals are vector valued, and we focus on continuous time. G is a linear system with a given state-space representation. Initially omit the upper loop (set \(\Delta = 0\)). The LQG regulator is the controller K that internally stabilizes the feedback loop and minimizes the variance of the error variable z, assuming the input v is white Gaussian noise.

Robust H2 Performance in Feedback Control, Fig. 1
figure 1730figure 1730

Feedback control and model uncertainty

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For an alternative description, denote by \(\hat{T}_{zv}(s)\) the closed-loop transfer function from v to z; we wish to design K such that \(\hat{T}_{zv}(s)\) is analytic in Re(s) ≥ 0 and has minimum \(\mathcal{H}_{2}\) norm, defined by

$$\displaystyle{ \|\hat{T}_{zv}\|_{\mathcal{H}_{2}} = \left (\int _{-\infty }^{\infty }\mathbf{Tr}(\hat{T}_{ zv}(j\omega )^{{\ast}}\hat{T}_{ zv}(j\omega ))\frac{d\omega } {2\pi }\right )^{\frac{1} {2} }; }$$
(1)

here Tr denotes matrix trace and∗ denotes conjugate transpose. The equivalence between this \(\mathcal{H}_{2}\)-optimal control and LQG follows from classical filtering, modeling v as uncorrelated components of unit power spectral density over all frequency. By adding a filter in the input of G, noise of known, colored spectrum can be accommodated as well.

A different motivation, in the case of scalar v, is to observe that \(\|\hat{T}_{zv}\|_{\mathcal{H}_{2}}^{2}\) is the energy (\(\mathcal{L}_{2}\)-norm square) of the system impulse response. Thus it measures the transient error in response to known inputs or initial conditions which may be generated by an impulse.

The \(\mathcal{H}_{2}\) (LQG) optimal feedback has an elegant solution, computable in state-space through two algebraic Riccati equations (AREs). Its quick popularity was, however, hampered due to its lack of stability margins: a small error in model parameters can make the closed-loop unstable (Doyle 1978). This motivated methods to explicitly address such modeling errors.

Model Uncertainty and Robustness

Suppose some parameter in the model of G is uncertain, \(\alpha =\alpha _{0}+\kappa \delta\), δ ∈ [−1, 1]; often, the normalized variation δ can be “pulled out” into the uncertainty block\(\Delta\) of Fig. 1. The same technique can account for unmodeled linear time-invariant (LTI) dynamics, e.g., high frequency effects: they can be “covered” by a normalized transfer function \(\hat{\Delta }(j\omega )\) and frequency weights that connect it to G. Even further, a nonlinear or time-varying (NL,TV) modeling error can be represented through an operator\(\Delta\) in signal space. The references contain details on this modeling technique.

To analyze the effect of such errors, suppose K has been chosen to stabilize G and M is the resulting closed-loop system, with state-space representation

$$\displaystyle\begin{array}{rcl} \dot{x}& =& Ax + B_{p}p + B_{v}v, \\ q& =& C_{q}x, \\ z& =& C_{z}x. {}\end{array}$$
(2)

A is an n × n stable (Hurwitz) matrix, and for simplicity there are no feed-through terms. Figure 2, represents the interconnection of M with the uncertainty.

Robust H2 Performance in Feedback Control, Fig. 2
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Robustness analysis setup

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To quantify the size of uncertainty, it is convenient to use an induced norm (gain) in signal space and constrain \(\Delta\) to the normalized ball \(\{\|\Delta \| \leq 1\}\). If the subsystem M qp in feedback with \(\Delta\) satisfies itself the induced norm constraint \(\|M_{qp}\| <1\), the small gain theorem implies robust stability over the entire ball. Focusing for the rest of this article on the \(\mathcal{L}_{2}\) signal space (square-integrable functions), the latter induced norm is equivalent to the \(\mathcal{H}_{\infty }\) norm of the transfer function:

$$\displaystyle{\|\hat{M}_{qp}(s)\|_{\mathcal{H}_{\infty }} := \text{ess}\sup \limits _{\omega \in \mathbb{R}}\bar{\sigma }(\hat{M}_{qp}(j\omega )),}$$

where \(\bar{\sigma }(\cdot )\) denotes matrix maximum singular value.

This motivates \(\mathcal{H}_{\infty }\)-optimal control: design K to minimize the above quantity with internal stability. This problem also admits state-space solutions based on AREs and thus is a valid competing paradigm to \(\mathcal{H}_{2}\).

To accommodate multiple sources of uncertainty within Fig. 1, we can use a block diagonal structure:

$$\displaystyle{ \Delta = \mbox{ diag}\left [\Delta _{1},\ldots,\Delta _{d}\right ]. }$$
(3)

Here, different uncertainty blocks (parametric, LTI, LTV, or NL) enter in separate “channels”; \(\mathbf{B}_{\boldsymbol{\Delta }}\) denotes the unit ball of operators with the prescribed structure. For stability studies, causality of the operator is required.

Robust stability under structured uncertainty is a rich topic: we refer to the article on the structured singular value (μ) in this encyclopedia. We invoke here robustness conditions based on the set \(\mathbf{\Lambda }\) of positive definite matrix scalings or multipliers of the form:

$$\displaystyle{ \Lambda = \mbox{ diag}\left [\lambda _{1}I,\ldots,\lambda _{d}I\right ], }$$
(4)

with submatrices of the same dimensions as the blocks in (3), thus commuting with a matrix \(\Delta\) of that structure.

Consider the frequency family of matrix inequalities

$$\displaystyle\begin{array}{rcl} & & \hat{M}_{qp}^{{\ast}}(j\omega )\Lambda (\omega )\hat{M}_{ qp}(j\omega ) - \Lambda (\omega ) <0\quad \forall \omega ; \\ & & \quad \Lambda (\omega ) \in \mathbf{\Lambda }. {}\end{array}$$
(5)

At each ω, this is a linear matrix inequality (LMI); testing its feasibility is a convex, tractable problem. A solution implies the scaled-small gain condition

$$\displaystyle{\bar{\sigma }\left (\Lambda (\omega )^{\frac{1} {2} }\hat{M}_{qp}(j\omega )\Lambda ^{-\frac{1} {2} }(\omega )\right ) <1;}$$

this “μ upper bound” implies robust stability when uncertainty is LTI, through commuting \(\Lambda (\omega )\) with \(\hat{\Delta }(j\omega )\).

If uncertainty is NLTV, (5) must be strengthened to enforce \(\Lambda (\omega ) \equiv \Lambda\), constant in frequency. This condition turns out to be both necessary and sufficient for robust stability. Here the LMIs would be coupled in frequency; however, the Kalman-Yakubovich-Popov lemma reduces them to an equivalent LMI in terms of the state-space matrices in (2), with variables \(\Lambda \in \boldsymbol{ \Lambda }\) and an n × n matrix P > 0:

$$\displaystyle{ \left [\begin{array}{cc} A^{{\ast}}P + PA + C_{q}^{{\ast}}\Lambda C_{q}&\;\;PB_{p} \\ B_{p}^{{\ast}}P & \;\;-\Lambda \end{array} \right ] <0. }$$
(6)

What about performance? The mapping \(T_{zv}(\Delta )\) between the disturbance v and the error z now depends on the uncertainty. The default procedure in robust control has been to measure performance with the same induced norm, evaluating \(\|T_{zv}(\Delta )\|_{\mathcal{L}_{2}\rightarrow \mathcal{L}_{2}}\) in the worst-case over \(\Delta \in \mathbf{B}_{\boldsymbol{\Delta }}.\) This can be computed with similar complexity to establishing robust stability. It amounts, however, to treating noise with the same worst-case mentality as stability, a questionable choice. For instance, in LTI systems the worst-case signals are sinusoids at the worst frequency and spatial direction; while one should protect against such signals arising in the \(\Delta\)-loop due to instability, it is not natural to expect them as external disturbances, which are usually of broad spectrum.

Robust \(\mathcal{H}_{2}\) Performance Analysis

In the absence of uncertainty, the \(\mathcal{H}_{2}\) norm of the nominal mapping T zv (0) = M zv provides a natural performance criterion, measuring the response to flat-spectrum disturbances or the transient response. When uncertainty is present, it motivates a worst-case analysis of stability; a natural combination is to impose robust\(\mathcal{H}_{2}\)performance: evaluating the worst-case \(\mathcal{H}_{2}\) norm of \(T_{zv}(\Delta )\) over the uncertainty class \(\mathbf{B}_{\boldsymbol{\Delta }}.\) We will highlight some methods based on semidefinite programming to perform such evaluations; for further details and comparisons, we refer to :̧def :̧def Paganini and Feron (1999).

A Frequency Domain Robust Performance Criterion

Consider the following optimization:

$$\displaystyle\begin{array}{rcl} & & \mathbf{J}_{\mathbf{f}} :=\inf \int _{ -\infty }^{\infty }\mathbf{Tr}(Y (\omega ))\frac{d\omega } {2\pi },\mbox{ subject to } \\ & & \hat{M}(j\omega )^{{\ast}}\left [\begin{array}{*{10}c} \Lambda (\omega )&0 \\ 0 &I \end{array} \right ]\hat{M}(j\omega ) -\left [\begin{array}{*{10}c} \Lambda (\omega )& 0\\ 0 &Y (\omega ) \end{array} \right ] < 0{}\end{array}$$
(7)

for each \(\omega\), and \(\Lambda (\omega ) \in \mathbf{\Lambda }\).

Here \(\hat{M}\) is the transfer function in Fig. 2; a submatrix of the above includes (5), implying robust stability under structured LTI uncertainty. Furthermore, we have the robust \(\mathcal{H}_{2}\) performance bound (Paganini 1999):

$$\displaystyle{ \sup _{\Delta \in \mathbf{B}_{\boldsymbol{\Delta }}^{\mathbf{LTI}}}\|T_{zv}(\Delta )\|_{\mathcal{H}_{2}}^{2} \leq \mathbf{J}_{\mathbf{ f}}. }$$
(8)

We sketch the argument based on the Fourier transforms \(\hat{p}(j\omega ),\) etc., for signals in Fig. 2. Applying the quadratic form in (7) to the joint vector of \(\hat{p}\) and \(\hat{v}\) gives

$$\displaystyle{\sum _{i=1}^{d}\lambda _{ i}(\omega )\vert \hat{q}_{i}\vert ^{2} + \vert \hat{z}\vert ^{2} \leq \sum _{ i=1}^{d}\lambda _{ i}(\omega )\vert \hat{p}_{i}\vert ^{2} +\hat{ v}^{{\ast}}Y (\omega )\hat{v}.}$$

The subvectors \(\hat{p}_{i}\), \(\hat{q}_{i}\) correspond to uncertainty blocks, \(\hat{p}_{i} =\hat{ \Delta }_{i}(j\omega )\hat{q}_{i}\); since \(\bar{\sigma }(\hat{\Delta }_{i}(j\omega )) \leq 1\), \(\vert \hat{q}_{i}\vert \geq \vert \hat{p}_{i}\vert\). Also λ i (ω) > 0, so these terms can be simplified, leading to

$$\displaystyle{\vert \hat{T}_{zv}(j\omega )\hat{v}\vert ^{2} = \vert \hat{z}\vert ^{2} \leq \hat{ v}^{{\ast}}Y (\omega )\hat{v}.}$$

This means \(\hat{T}_{zv}(j\omega )^{{\ast}}\hat{T}_{zv}(j\omega ) \leq Y (\omega )\) for every \(\Delta\), and therefore the \(\mathcal{H}_{2}\) norm bound

$$\displaystyle{\|T_{zv}(\Delta )\|_{\mathcal{H}_{2}}^{2} \leq \int _{ -\infty }^{\infty }\mathbf{Tr}(Y (\omega ))\frac{d\omega } {2\pi }}$$

holds, from which (8) follows.

The computation involved in (7) at each frequency is a semidefinite program (SDP): minimizing the linear cost Tr(Y (ω)) subject to an LMI constraint, a tractable problem. Adding a frequency sweep, we have a practical method to bound the desired robust performance.

The inequality (8) is in general strict. Beyond the usual conservatism of convex bounds for μ, when noise is of dimension m, a conservatism of up to this order may appear; an improvement to address this issue with augmented SDPs is given in Sznaier et al. (2002). Finally, causality of the uncertainty is not imposed in the frequency-domain criterion.

As in the study of robust stability, we wish to extend the analysis to NLTV uncertainty blocks. Now the mapping \(T_{zv}(\Delta )\) can no longer be represented by a transfer function, so what is the “\(\mathcal{H}_{2}\)” cost? We return to our motivation for this performance notion: to measure the effect of disturbances of flat spectrum.

In Paganini (1999), the flat-spectrum property is imposed as a deterministic constraint on the input disturbances. For the scalar v case, define \(W_{\eta,B} \subset \mathcal{L}_{2}\) by the family of integral quadratic constraints:

$$\displaystyle\begin{array}{rcl} \int _{-\beta }^{\beta }\vert v(j\omega )\vert ^{2}\frac{d\omega } {2\pi }\left \{\begin{array}{@{}l@{\quad }l@{}} \leq \frac{\beta }{\pi }+\eta \quad &\forall \beta ; \\ \geq \frac{\beta }{\pi }-\eta,\quad &\beta \in [0,B]. \end{array} \right.& &{}\end{array}$$
(9)

This imposes that the cumulative spectrum is approximately linear (to a tolerance η > 0), up to bandwidth B, and has sublinear growth beyond that. Extensions to vector-valued signals are also given. For a stable LTI system T zv , it is not difficult to verify that

$$\displaystyle{\|T_{zv}\|_{\mathcal{H}_{2}}^{2} =\lim _{\stackrel{\eta \rightarrow 0}{B\rightarrow \infty }}\sup _{v\in W_{\eta,B}}\|T_{zv}v\|_{2}^{2},}$$

but the right-hand side applies to NLTV systems as well. The following result can be established in the latter case:

$$\displaystyle{\lim _{\stackrel{\eta \rightarrow 0}{B\rightarrow \infty }}\sup _{v\in W_{\eta,B},\Delta \in \mathbf{B}_{\boldsymbol{\Delta }}^{\mathbf{NLTV}}}\|T_{zv}(\Delta )v\|_{2}^{2} = \mathbf{J^{\prime}}_{\mathbf{ f}},}$$

where the right-hand side is the variant of (7) with the restriction that \(\Lambda (\omega ) \equiv \Lambda\), constant in frequency. In this case the characterization is exact, with equality above. This follows from a duality argument in function space, where Y (ω) appears as the multiplier for the constraint in (9). While coupled in frequency, J′ f is again equivalent to a finite-dimensional SDP in state space.

Let us review, instead, a different state-space method, motivated by alternate definitions of the \(\mathcal{H}_{2}\) cost.

A State-Space Criterion Invoking Causality

Consider the semidefinite program

$$\displaystyle\begin{array}{rcl} & & \mathbf{J}_{\mathbf{s}} :=\inf \; \mathbf{Tr}(B_{v}^{{\ast}}PB_{ v})\mbox{ subject to $P> 0$, $\Lambda \in \mathbf{\Lambda }$,} \\ & & \left [\begin{array}{cc} A^{{\ast}}P + PA + C_{q}^{{\ast}}\Lambda C_{q} + C_{z}^{{\ast}}C_{z}&\;\;PB_{p} \\ B_{p}^{{\ast}}P & \;\;-\Lambda \end{array} \right ] <0.{}\end{array}$$
(10)

The LMI above is very similar to (6); indeed it provides a robust stability certificate and in addition a bound on a generalized \(\mathcal{H}_{2}\) cost, for arbitrary (NLTV) causal uncertainty blocks. Again, we sketch the argument.

For stability, consider the system of Fig. 2 with \(v \equiv 0\), initial condition x(0) = x0. Define the storage function V (x) = x∗Px; differentiating it under (2) and applying the LMI (10) to the joint vector of x(t), p(t) yield

$$\displaystyle{\dot{V } + \vert z\vert ^{2} \leq -q^{{\ast}}\Lambda q + p^{{\ast}}\Lambda p =\sum _{ i=1}^{d}\lambda _{ i}(\vert p_{i}\vert ^{2} -\vert q_{ i}\vert ^{2}).}$$

Integrating the above over (0, t), the sum on the right becomes nonpositive because λ i > 0 and the operator \(\Delta _{i} : q_{i} \rightarrow p_{i}\) is causal and contractive. This leads to

$$\displaystyle{ V (x(t)) +\int _{ 0}^{t}\vert z(\tau )\vert ^{2}d\tau \leq V (x_{ 0}), }$$
(11)

which implies Lyapunov stability; the bound can be sharpened to prove asymptotic stability. Also, letting t → ∞ yields the energy bound \(\|z\|_{2}^{2} \leq V (x_{0})\).

Suppose now that x0 is generated by applying to the (causal) system at rest, an impulse v(t) = δ(t), assumed scalar. The result is \(x(0+) = B_{v}\), so \(V (x_{0}) = B_{v}^{{\ast}}PB_{v}\); the impulse response energy of \(T_{zv}(\Delta )\) is thus bounded. Minimizing over P, \(\Lambda\) leads to the robust \(\mathcal{H}_{2}\) performance bound

$$\displaystyle{\sup _{\Delta \in \mathbf{B}_{\boldsymbol{\Delta }}^{\mathbf{NLTV}}}\|T_{zv}(\Delta )\delta (t)\|_{2}^{2} \leq \mathbf{J}_{\mathbf{ s}},}$$

where the \(\mathcal{H}_{2}\) cost is generalized as the impulse response energy. An extension to multiple impulse channels is available. This kind of result was first obtained by Stoorvogel (1993) for unstructured uncertainty.

An alternate notion of \(\mathcal{H}_{2}\) cost for NLTV systems, also considered in Stoorvogel (1993), is the average output variance when the input is random white noise. This is formalized by replacing (2) with a stochastic differential equation (e.g., :̧def :̧def Oksendal 1985) and extending the bound (11) using Ito calculus; for details see :̧def :̧def Paganini and Feron (1999). The following robust \(\mathcal{H}_{2}\) performance bound is obtained:

$$\displaystyle{\limsup _{\tau \rightarrow \infty }\frac{1} {\tau } \int _{0}^{\tau }E\vert z(t)\vert ^{2}dt. \leq \mathbf{J}_{\mathbf{ s}}\quad \forall \Delta \in \mathbf{B}_{\boldsymbol{\Delta }}^{\mathbf{NLTV}}.}$$

What if the uncertainty is time invariant? Incorporating frequency-dependent scalings, with causality, into the state-space approach must be done approximately, generating \(\hat{\Lambda }(j\omega )\) through the span of a predefined finite basis of causal, rational transfer functions. Searching over this basis for a bound on the impulse response energy can be pursued with state-space SDPs, now of a size increasing with the basis dimensionality. We refer to Feron (1997) for details.

Robust \(\mathcal{H}_{2}\) Synthesis

Prior sections have focused on the robustness analysis of a closed-loop system M, obtained from G after designing a nominally stabilizing controller. Can we synthesize K with robust \(\mathcal{H}_{2}\) performance as an objective? We overview some contributions to this question.

Multiobjective \(\mathcal{H}_{2}\)/\(\mathcal{H}_{\infty }\) Control

Let us discuss first the more modest objective of optimizing nominal\(\mathcal{H}_{2}\) performance while guaranteeing robust stability. If the uncertainty block \(\Delta\) in Fig. 2 is unstructured, the problem is equivalent to

$$\displaystyle{\mbox{ Minimize }\|\hat{M}_{zv}\|_{\mathcal{H}_{2}},\mbox{ subject to }\|\hat{M}_{qp}\|_{\mathcal{H}_{\infty }} <1.}$$

Using a Youla parameterization of stabilizing controllers, \(\hat{M}(s)\) depends affinely on a stable parameter \(\hat{Q}(s)\); this makes the optimization over \(\hat{Q}\) convex. However it has been shown to give infinite-dimensional solutions that must be approximated by suitable truncations; see Sznaier et al. (2000) and references therein.

To better exploit the state-space structure common to \(\mathcal{H}_{2}\) and \(\mathcal{H}_{\infty }\) synthesis, Bernstein and Haddad (1989) proposed a simplification: minimize an auxiliary cost that upper bounds the \(\mathcal{H}_{2}\) norm while imposing the \(\mathcal{H}_{\infty }\) constraint, through a common storage function. This cost is optimized by controllers of the order of the plant, characterized in terms of coupled AREs; later on Khargonekar and Rotea (1991) recast this problem using convex optimization. Also Zhou et al. (1994) and Doyle et al. (1994) studied the dual (transpose) structure.

The latter version is in fact directly related to the analysis condition (10), with a fixed \(\Lambda =\lambda I\). A matrix P satisfying this condition imposes the \(\mathcal{H}_{\infty }\) norm restriction and upper bounds the nominal \(\mathcal{H}_{2}\) cost. This idea of imposing multiple objectives through a common storage function has more general applicability: Scherer et al. (1997) showed that all such problems admit tractable synthesis based on LMIs, with solutions of the same order as the plant.

Synthesis for Robust Performance

We have seen that rather than just an upper bound on nominal performance, (10) ensures the more stringent robust \(\mathcal{H}_{2}\) performance requirement; therefore it becomes the basis of a robust \(\mathcal{H}_{2}\) synthesis technique. In Stoorvogel (1993) this method is laid out for unstructured uncertainty: search linearly over the scalar λ and solve the auxiliary cost synthesis problem for each λ.

What about structured uncertainty? We run here into a general difficulty of such synthesis questions, even for robust stability alone. In that case, seeking simultaneously a controller \(K\) and a scaling \(\Lambda\) so that conditions (5) or (6) are satisfied by the resulting M is not a computationally friendly problem. In the absence of a general solution method, iterating between an \(\mathcal{H}_{\infty }\) design of K for fixed \(\Lambda\) and the analysis conditions to find \(\Lambda\) is commonly used for design.

Things can be no easier for robust \(\mathcal{H}_{2}\) performance, but the iterative procedure does generalize to the conditions in (10): for fixed K, the SDP will return structured \(\Lambda\)’s, which can then be fixed for a multiobjective synthesis step based on the “auxiliary cost” in (10) as discussed above. If constant \(\Lambda\) are used (designing for NLTV uncertainty), all controllers obtained are of the order of the plant.

If uncertainty is LTI, an alternative is to carry out the analysis step in the frequency domain, finding a \(\Lambda (\omega )\), Y (ω) through (7). In the corresponding situation for μ-synthesis, where only \(\Lambda (\omega )\) is found, a step of fitting and spectral factorization is needed to approximate such scalings through a rational weights, which are then incorporated into \(\mathcal{H}_{\infty }\) synthesis. A similar frequency weight in the performance channel can approximate the effect of Y (ω), thus relying on weighted \(\mathcal{H}_{\infty }\) synthesis to pursue the \(\mathcal{H}_{2}\) performance objective. Of course, the order of the resulting controllers is increased.

Summary and Future Directions

The tradeoff between performance and robustness is essential to feedback control. In the case of linear multivariable design, it motivated a compromise between \(\mathcal{H}_{2}\) performance and \(\mathcal{H}_{\infty }\)-type robustness, pursued with the state-space and frequency-domain tools common to these metrics. We have highlighted robust \(\mathcal{H}_{2}\) analysis conditions obtained in the 1990s based on semidefinite programming, which provided the greatest flexibility to integrate the aforementioned tools and different points of view (worst-case, average case) present in this problem. As in other situations, the robust synthesis question has proven more difficult: design cannot be “automated” to the degree that was once envisioned.

The passage of time makes issues that once attracted strong attention look narrow in scope, so it is not natural to indicate directions that directly follow on this work. Perhaps the best legacy that the robust \(\mathcal{H}_{2}\) generation can take to other problems is the willingness to integrate various disciplines (dynamics, operator theory, stochastics, optimization) to face the demands of applied mathematical research.

Cross-References

Recommended Reading

LQG control is covered in many textbooks, e.g., Anderson and Moore (1990). A standard text for robust control with an \(\mathcal{H}_{\infty }\) perspective, including structured singular values, the Youla parameterization, and the Riccati equation solution for \(\mathcal{H}_{\infty }\) synthesis, is Zhou et al. (1996); see also Sánchez-Peña and Sznaier (1998) with application examples. The textbook of Dullerud and Paganini (2000) incorporates the more recent developments based on LMIs; see Boyd and Vandenberghe (2004) for background on semidefinite programming.