Robust Synthesis and Robustness Analysis Techniques and Tools

Abstract

This entry provides a brief summary of the synthesis and analysis tools that have been developed by the robust control community. Many software tools have been developed to implement the major theoretical techniques in robust control. These software tools have enabled robust synthesis and analysis techniques to be successfully applied to numerous industrial applications.

Introduction

¹

Robust control is a methodology to address the effect of uncertainty on feedback systems. This approach includes techniques and tools to model system uncertainty, assess stability and/or performance characteristics of the uncertain system, and synthesize controllers for uncertain systems. The theory was developed over a number of years. The foundational results can be found in classical papers Packard and Doyle (1993a), Desoer et al. (1980), Doyle (1978, 1982), Doyle et al. (1989), Doyle and Stein (1981), Megretski and Rantzer (1997), Safonov (1982), Willems (1971), and Zames (1981) and more recent textbooks Boyd et al. (1994), Desoer and Vidyasagar (2008), Dullerud and Paganini (2000), Francis (1987), Skogestad and Postlethwaite (2005), Vidyasagar (1985), and Zhou et al. (1996). It should be emphasized that this entry is not meant to be a survey and more complete references to the literature can be found in the cited textbooks. The remainder of this entry discusses the main theoretical and computational tools for robust synthesis and robustness analysis.

Notation

R and C denote the set of real and complex numbers, respectively. $\mathbf{R}^{m\times n}$ and $\mathbf{C}^{m\times n}$ denote the sets of m × n matrices whose elements are in $\mathbf{R}$ and $\mathbf{C}$, respectively. A single superscript index is used for vectors, e.g., $\mathbf{R}^{n}$ denotes the set of n × 1 vectors whose elements are in $\mathbf{R}$. For a matrix $M \in \mathbf{C}^{m\times n}$, M^T denotes the transpose and M^∗ denotes the complex conjugate transpose. A matrix M is Hermitian (Skew-Hermitian) if M = M^∗ ($M = -M^{{\ast}}$). The maximum singular value of a matrix M is denoted by $\bar{\sigma }(M)$. The trace of a matrix M, denoted tr[M], is the sum of the diagonal elements. M = M^∗ is a positive semidefinite matrix, denoted M ⪰0, if all eigenvalues are nonnegative. M = M^∗ is negative semidefinite, denoted M ⪯0, if − M ⪰0. $\mathcal{L}_{2}^{n}[0,\infty )$ is the space of functions $u : [0,\infty ) \rightarrow \mathbf{R}^{n}$ satisfying $\|u\| <\infty$ where $\|u\| := \left [\int _{0}^{\infty }u(t)^{T}u(t)\,dt\right ]^{0.5}$. For $u \in \mathcal{L}_{2}^{n}[0,\infty )$, u _T denotes the truncated function u _T (t) = u(t) for t ≤ T and u(t) = 0 otherwise. The extended space, denoted $\mathcal{L}_{2e}$, is the set of functions u such that $u_{T} \in \mathcal{L}_{2}$ for all T ≥ 0. The Fourier transform $\hat{v} := \mathcal{F}(v)$ maps the time domain signal $v \in \mathcal{L}_{2}^{n}[0,\infty )$ to the frequency domain by

$$\displaystyle\begin{array}{rcl} \hat{v}(j\omega ) :=\int _{ 0}^{\infty }e^{-j\omega t}v(t)dt& &{}\end{array}$$

(1)

Capital letters are used to represent dynamical systems. For linear systems, the same letter is used to represent the system, its convolution kernel, as well as its frequency-response function. Lowercase letters denote time-signals, and ω represents the continuous-time frequency variable. For an m × n system G, define the H _∞ and H₂ norms as $\|G\|_{\infty } =\sup _{\omega }\bar{\sigma }\left (G(j\omega )\right )$ and $\|G\|_{2} = \sqrt{\frac{1} {2\pi }\int _{-\infty }^{\infty }tr[G(j\omega )^{{\ast}}G(j\omega )]d\omega }$. The $\mathcal{L}_{1}$ norm of G is defined as $\|G\|_{1} =\max _{1\leq i\leq m}\sum _{j=1}^{n}\int _{0}^{\infty }\vert g_{ij}(t)\vert dt$ where g _ij (t) is the response of the ith output due to a unit impulse in the jth input. The entry describes continuous-time systems. Most results carry over, in a similar form, to discrete-time systems.

Theoretical Tools

Uncertainty Modeling

In order to analyze and/or design for the degrading effects of uncertainty, it is imperative that explicit models of uncertainty be characterized. Two distinct forms of uncertainty are considered: signal uncertainty and model uncertainty. Signal uncertainty represents external signals (plant disturbances, sensor noise, reference signals) as sets of time functions, with explicit descriptions. For example, a particular reference input might be characterized as belonging to the set $\left \{ \frac{4} {2s+1}d : d \in \mathcal{L}_{2},\left \|d\right \|_{2} \leq 1\right \}$. This set is often referred to as a weighted ball in$\mathcal{L}_{2}$. The transfer function $\frac{4} {2s+1}$ is called a weighting function and it shapes the normalized signals d, in a manner that its output represents the actual traits of the reference inputs that occur in practice.

Model uncertainty represents unknown or partially specified gains (more generally, operators) that relate pairs of signals in the model. For example, z and w are signals within a model and are related by an operator $\mathcal{N}$ as $w = \mathcal{N}(z)$. Typical partial specifications either constrain $\mathcal{N}$ to be drawn from a specified set or describe the set of signals (w, z) that $\mathcal{N}$ allows. An uncertain parameter δ is modeled as time-invariant (i.e., constant), belonging to the interval [a, b] and relating z and w as w(t) = δ z(t). An uncertain linear dynamic element, $\Delta$ is modeled as linear, time-invariant, causal system, described by a convolution kernel δ whose frequency-response function (i.e., Fourier transform) satisfies $\max _{\omega }\left \vert \hat{\delta }(j\omega )\right \vert \leq 1$, and relating z and w as w = δ ⋆ z. More generally, consider an $\mathcal{L}_{2}$ bounded, causal operator, mapping $\mathcal{L}_{2,e} \rightarrow \mathcal{L}_{2,e}$ relating the signals as $w = \Delta (z)$. The behavior of $\Delta$ is unknown but constrained by a family of multipliers, $\left \{\Pi _{\alpha }\right \}_{\alpha \in \mathcal{A}}$. Specifically, each $\Pi _{\alpha }$ is a Hermitian, matrix-valued function of frequency, and for any $z \in \mathcal{L}_{2}$, the mapping $\Delta$ is known to satisfy

$$\displaystyle{\int _{-\infty }^{\infty }\left [\begin{array}{c} \hat{z}(j\omega ) \\ \hat{w}(j\omega ) \end{array} \right ]^{{\ast}}\Pi _{\alpha }(\omega )\left [\begin{array}{c} \hat{z}(j\omega ) \\ \hat{w}(j\omega ) \end{array} \right ]d\omega \geq 0}$$

This is called an integral quadratic constraint (IQC) description of $\Delta$, as the input/output pairs of $\Delta$ satisfy a family of quadratic, integral constraints. These different descriptions of model uncertainty are related. For example, if w(t) = δ z(t), with w(t) ∈ Rⁿ and z(t) ∈ Rⁿ, and δ ∈ R, | δ | ≤ 1, then for any Hermitian-valued $X : \mathbf{R} \rightarrow \mathbf{C}^{n\times n}$ with X(ω)⪰0 for all ω ∈ R and Skew-Hermitian $Y : \mathbf{R} \rightarrow \mathbf{C}^{n\times n}$,

$$\displaystyle\begin{array}{rcl} & & \int _{-\infty }^{\infty }\left [\begin{array}{c} \hat{z}(j\omega ) \\ \hat{w}(j\omega ) \end{array} \right ]^{{\ast}}\left [\begin{array}{cc} X(\omega ) & Y (\omega ) \\ Y ^{{\ast}}(\omega )& - X(\omega ) \end{array} \right ]\left [\begin{array}{c} \hat{z}(j\omega ) \\ \hat{w}(j\omega ) \end{array} \right ]d\omega {}\\ & & \quad =\int _{ -\infty }^{\infty }\hat{z}^{{\ast}}(j\omega )\left [(1 -\delta ^{2})X(\omega )\right ]\hat{z}(j\omega )d\omega {}\\ \end{array}$$

which is always ⪰0. Hence, the uncertain parameter can be recast as an operator satisfying an infinite family of IQCs. Nonlinear operators may also satisfy IQCs and it is common to “model” known nonlinear elements (e.g., saturation) by enumerating IQCs that they satisfy (Megretski and Rantzer 1997). An uncertain dynamic model is made up of an interconnection of these uncertain elements with a known (usually) linear system G.

Performance Metric

The main goal of robustness analysis is to assess the degrading effects of uncertainty. For this, a concrete notion of performance is needed, resulting in a mathematical/computational exercise to quantify the average or worst-case effects of the two types of uncertainty, signal and model, described earlier. In the robust control framework, adequate performance is characterized in terms of the variability of possible behavior of particular signals. For instance, in the presence of reference inputs and disturbance inputs, as well as parameter uncertainty, it is required that tracking errors (e) and control inputs (u) remain small. A common measure of smallness is the $\mathcal{L}_{2}$ norm of signals. Typically, frequency-dependent weighting functions are used to preferentially weight one frequency range over another and/or to weight one signal relative to another. In this way, adequate performance be defined as $\left \|W\left [\begin{matrix}\scriptstyle e \\ \scriptstyle u\end{matrix}\right ]\right \|_{2} \leq 1$, where W is a stable, linear system, called the “output” weighting function. Weighting functions are often used to transform a collection of performance objectives into a single norm bound objective in the robust control framework.

Robustness Analysis

Robustness analysis refers to the task of ascertaining the stability and/or performance characteristics of the uncertain system, given the limited knowledge about the uncertain information. The main result from Megretski and Rantzer (1997) concerns the stability of the interconnection shown in Fig. 1, where G is a known, stable, linear system and $\Delta$ is an operator that satisfies the IQC defined by $\Pi$. Under some important technical conditions, the theorem states “if there exists an ε > 0 such that

$$\displaystyle\begin{array}{rcl} \left [\begin{array}{c} G(j\omega )\\ I \end{array} \right ]^{{\ast}}\Pi (\omega )\left [\begin{array}{c} G(j\omega ) \\ I \end{array} \right ]\preceq -\epsilon I& &{}\end{array}$$

(2)

for all ω ∈ R, then the interconnection is stable.” Stability here refers to finite $\mathcal{L}_{2}$ gain from inputs (r₁, r₂) to loop signals (u₁, u₂).

Robust Synthesis and Robustness Analysis Techniques and Tools, Fig. 1

Feedback interconnection for IQC stability test

Multiple IQCs satisfied by $\Delta$ can be incorporated into the analysis. In particular, assume that $\Delta$ satisfies the IQCs defined by the multipliers $\{\Pi _{k}\}_{k=1}^{N}$. Then $\Delta$ satisfies the IQC defined by any mutliplier of the form $\Pi ^{\alpha } :=\sum _{ k=1}^{N}\alpha _{k}\Pi _{k}$ where α _k ≥ 0. The stability test amounts to a semi-infinite, semidefinite feasibility problem: find nonnegative scalars $\{\alpha _{k}\}_{k=1}^{N}$ such that for some ε > 0,

$$\displaystyle\begin{array}{rcl} \left [\begin{array}{c} G(j\omega )\\ I \end{array} \right ]^{{\ast}}\Pi ^{\alpha }(\omega )\left [\begin{array}{c} G(j\omega ) \\ I \end{array} \right ]\preceq -\epsilon I& &{}\end{array}$$

(3)

for all ω ∈ R. This infinite family of matrix inequalities (one for each frequency) can be equivalently expressed as a finite-dimensional linear matrix inequality (LMI) under some additional restrictions.

The structured singular value (μ) approach provides an alternative robust stability test in the case of only linear, time-invariant uncertainty (parametric or dynamic). Suppose $\Delta$ is drawn from a set of matrices, $\boldsymbol{\Delta } \subseteq \mathbf{C}^{m\times n}$ of the form

$$\displaystyle{\begin{array}{ll} {\boldsymbol{\Delta }} = \left\{\mathrm{diag} \left[\delta_{1}^{r}{I_{t_{1}}},\ldots,\delta_{V }^{r}{I_{t_{V }}},\right.\right. \\ \qquad \qquad \qquad \left.\delta_{1}^{c}{I_{r_{1}}},\ldots,\delta_{S}^{c}{I_{r_{S}}},\Delta_{1},\ldots,\Delta_{F}\right ] : \\ \left. \qquad \qquad \qquad \delta_{k}^{r} \in \mathbf{R},\delta_{i}^{c} \in \mathbf{C},\Delta_{j} \in \mathbf{C}^{m_{j}\times n_{j}}\right\}\end{array}}$$

The inclusion of complex-valued, uncertain matrices within $\boldsymbol{\Delta }$ may seem unusual and hard to motivate. However, in terms of their effect on stability, these are equivalent to the uncertain linear dynamic element introduced earlier in the Uncertainty Modeling section. This is discussed in more detail in the entry Structured Singular Value and Applications: Analyzing the Effect of Linear Time-Invariant Uncertainty in Linear Systems.

Using the Nyquist stability criterion, the $(G,\Delta )$ interconnection is stable for all $\Delta \in \boldsymbol{ \Delta }$, with $\bar{\sigma }\left (\Delta \right ) <\beta$ if and only if G is stable, and

$$\displaystyle{\det (I - G(j\omega )\Delta )\not =0}$$

for all $\Delta \in \boldsymbol{ \Delta }$ with $\bar{\sigma }\left (\Delta \right ) <\beta$ and all ω ∈ R including ω = ∞. The importance of the nonvanishing determinant condition warrants a definition of its own, the structured singular value. For a matrix $M \in \mathbf{C}^{n\times m}$, and $\boldsymbol{\Delta }$ as given, define

$$\displaystyle{\mu _{\boldsymbol{\Delta }}\left (M\right ) := \frac{1} {\min \left \{\bar{\sigma }\left (\Delta \right ) : \Delta \in \boldsymbol{ \Delta },\det \left (I - M\Delta \right ) = 0\right \}}}$$

unless no $\Delta \in \boldsymbol{ \Delta }$ makes $\left (I - M\Delta \right )$ singular, then $\mu _{\boldsymbol{\Delta }}\left (M\right ) := 0$. In this parlance, the $(G,\Delta )$ interconnection is stable for all $\Delta \in \boldsymbol{ \Delta }$, with $\bar{\sigma }\left (\Delta \right ) <\beta$ if and only if

$$\displaystyle{\mu _{\boldsymbol{\Delta }}\left (G(j\omega )\right ) \leq \frac{1} {\beta } }$$

for all ω ∈ R including ω = ∞.

In summary, the structured singular value approach employs a Nyquist-based argument, resulting in a nonvanishing determinant condition, which must hold over all frequency and all possible frequency-response values of the uncertain elements. However, checking the nonvanishing determinant is difficult, and sufficient conditions, in the form of semidefinite programs (Doyle 1982; Fan et al. 1991) to ensure this are derived. This results in semidefinite feasibility problems which must hold at all frequencies. It is common to verify these only on a finite grid of frequencies, which is equivalent to ensuring that the closed-loop poles cannot migrate across the stability boundary at these frequencies. Semidefinite programs can be defined which carve out intervals around these fixed frequencies to completely guarantee stability.

Robust Synthesis

Synthesis refers to the mathematical design of the control law. The nominal synthesis problem (with no uncertainty) is formulated using the generic feedback structure shown in Fig. 2. The various signals in the diagram are the control inputs u, measurements y, exogenous disturbances d, and regulated variables e. P is a generalized plant that contains all information required to specify the synthesis problem. This includes the dynamics of the actual plant being controlled as well as any frequency domain weights that are used to specify the performance objective. The objective of an optimal control problem is to synthesize a controller K that minimizes the closed-loop (e.g., H₂, H _∞ , $\mathcal{L}_{1}$) norm from disturbances (d) to regulated variables (e), i.e., solve

$$\displaystyle{\min _{\mathrm{allowable}\ K}\left \|F_{L}(P,K)\right \|}$$

where F _L (P, K) denotes the system obtained by closing the controller K around the lower loop of P. The H₂, H _∞ , and $\mathcal{L}_{1}$ optimal control problems refer to the choice of the specific norm $\left \|F_{L}(P,K)\right \|$ used to specify the performance. A generalization of the H _∞ performance objective is simply to require that the closed-loop map from $d \rightarrow e$ satisfy an IQC defined by a given multiplier $\Pi$, called the performance multiplier, Apkarian and Noll (2006). The H₂, H _∞ and $\mathcal{L}_{1}$ optimal control problems formulated as in Fig. 2 only involve signal uncertainty. In other words, these design problems do not explicitly account for the effects of model uncertainty.

Robust Synthesis and Robustness Analysis Techniques and Tools, Fig. 2

Feedback interconnection for H₂, H _∞ , and $\mathcal{L}_{1}$ optimal control

Robust synthesis refers to control design that explicitly accounts for model uncertainty. It is usually formulated as a worst-case optimization, where the controller is chosen to minimize the worst-case effect of the signal and model uncertainty, loosely

$$\displaystyle{\min _{\mathrm{allowable}\ K}\ \max _{\mathrm{allowable}\ d,\Delta }\left \|T(d,\Delta,K)\right \|}$$

where d is a set of exogenous disturbances and $\Delta$ corresponds to the model uncertainty set. T represents the closed-loop relationship between d, $\Delta$ and the controller K. μ-synthesis is a specific technique developed to synthesize control algorithms which achieve robust performance, i.e., performance in the presence of signal and model uncertainty. The objective of μ-synthesis is to minimize over all stabilizing controllers K, the peak value of $\mu _{\Delta }\left (F_{L}(P,K)\right )$ of the closed-loop transfer function defined by the interconnection in Fig. 3. P is the generalized plant model. The $\Delta$ block is the uncertain element from the set $\boldsymbol{\Delta }$, which parameterizes all of the assumed model uncertainty in the problem. The μ-synthesis optimization has high computational complexity (so-called NP-hard problem), though practical algorithms and software have been developed to design controllers using this control technique (Balas et al. 2013). Alternative robust synthesis approaches exist and often involve nonlinear optimization algorithms (Apkarian and Noll 2006). Drastic simplification regarding the models and uncertainty can be made resulting in problems that can be solved using LMI and semidefinite programming techniques (Boyd and Barrat 1991; Boyd et al. 1994).

Robust Synthesis and Robustness Analysis Techniques and Tools, Fig. 3

Feedback interconnection for μ synthesis

Computational Tools

The MATLAB Robust Control Toolbox is a commercially available software product that is part of the Mathworks control product line. It is tightly integrated with Control System Toolbox and Simulink products (Balas et al. 2013). The Robust Control Toolbox includes tools to analyze and design multi-input, multi-output control systems with uncertain elements. The primary building blocks, called uncertain elements or atoms, are uncertain real parameters and uncertain linear, time-invariant objects. These can be used to create coarse and simple or detailed and complex descriptions of model uncertainty. The uncertain object data structure eliminates the need to generate models of uncertainty and control analysis and design problem formulations, thereby allowing the practicing engineer to apply advanced robust control theory to their applications. Functions are available to analyze the robust stability, robust performance, and worst-case performance of uncertain multivariable system models using the structured singular value, μ. The Robust Control Toolbox also includes multivariable control synthesis tools to compute controllers that optimize worst-case performance and identify worst-case parameter values.

The IQC-Beta Toolbox is a publicly available robust analysis toolbox based on the IQC framework (Jönsson et al. 2004). A wide range of robust stability and performance analysis tests are available for uncertain, nonlinear, and time-varying systems. IQC-Beta is written in MATLAB and works seamlessly with the Control System Toolbox objects and basic interconnection functions. The Users manual nicely complements the literature on IQCs. The Computer Aided Control System Design package in Scilab, an open source numerical computation software, includes functionality for robustness analysis and the synthesis of robust control algorithms for multivariable systems (http://www.scilab.org/).

Conclusions

Robust control analysis and synthesis software tools are widely available and have been extensively used by industry since the late 1980s. The availability of software tools for robustness analysis and synthesis played a major role in their wide and ubiquitous adoption in industry. They have been successfully applied to a variety of applications including aircraft flight control, launch vehicles, satellites, compact disk players, disk drives, backhoe excavators, nuclear power plants, helicopters, thin film extrusion, gas- and diesel-powered engines, missile autopilots, heating and ventilation systems, process control, and active suspension systems.

Cross-References

• LMI Approach to Robust Control

• Optimization Based Robust Control

• Structured Singular Value and Applications: Analyzing the Effect of Linear Time-Invariant Uncertainty in Linear Systems