# (Generalized) KYP lemma and applications

**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_160-1

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## Abstract

Various properties of dynamical systems can be characterized in terms of inequality conditions on their frequency responses. The Kalman-Yakubovich-Popov (KYP) lemma shows equivalence of such frequency domain inequality (FDI) and a linear matrix inequality (LMI). The fundamental result has been a basis for robust and optimal control theories in the past several decades. The KYP lemma has recently been generalized to the case where an FDI on a possibly improper transfer function is required to hold in a (semi)finite frequency range. The generalized KYP lemma allows us to directly deal with practical situations where design parameters are sought to satisfy FDIs in multiple (semi)finite frequency ranges. Various design problems, including FIR filter and PID controller, reduce to LMI problems which can be solved via semidefinite programming.

## **Keywords**

Bounded real Positive real Frequency domain inequality Linear matrix inequality Robust control Optimal control Multi-objective design ## Introduction

In linear systems analysis and control design, dynamical properties are often characterized by frequency responses. The shape of a frequency response, as visualized by the Bode or Nyquist plot, is closely related to various performance measures including the steady state error, fast and smooth transient, and robustness against unmodeled dynamics. Hence, desired system properties can be formalized in terms of a set of frequency domain inequalities (FDIs) on selected transfer functions. The analysis and design problems then reduce to verification and satisfaction of the FDIs.

The Kalman-Yakubovich-Popov (KYP) lemma (Kalman 1963; Anderson 1967; Willems 1971; Rantzer 1996) establishes the equivalence between an FDI and a linear matrix inequality (LMI). The LMI is defined by state space matrices of the transfer function in the FDI so that the FDI holds true if and only if the LMI admits a solution. The LMI characterization of an FDI is useful since it replaces the process of checking the FDI at infinitely many frequency points by the search for a symmetric matrix satisfying a finite dimensional convex constraint defined by the LMI. In addition to exact and tractable computations, benefits of the LMI conditions include analytical understanding of robust and optimal controls through spectral factorizations and storage/Lyapunov functions. The KYP lemma is a fundamental result in the systems and control field that has provided, in the past half century, a theoretical basis for developments of various tools for system analysis and design.

A drawback of the KYP lemma is its inability to characterize an FDI in a finite frequency range. Feedback control designs typically involve a set of specifications given in terms of multiple FDIs in various frequency ranges. However, the KYP lemma is not capable of treating such FDIs directly since it has to consider the entire frequency range. To address this deficiency, the KYP lemma has recently been generalized to characterize an FDI in a finite frequency range exactly (Iwasaki et al. 2000). Further generalizations (Iwasaki and Hara 2005) are available for FDIs within various frequency ranges for both continuous- and discrete-time, possibly improper, rational transfer functions. The generalized KYP lemma allows for direct multi-objective design of filters, controllers, and dynamical systems.

## KYP Lemma

*G*(

*s*) is the transfer function from

*u*to

*x*. If

*u*is a disturbance to the system and

*x*represents the error from a desired operating point, we may be interested in how large the state variables can become for a given magnitude of the disturbance. The gain ∥

*G*(

*jω*) ∥ captures this property for the case of a sinusoidal disturbance at frequency

*ω*, where ∥ ⋅ ∥ denotes the spectral norm ( = absolute value for a scalar). If ∥

*G*(

*jω*) ∥ <

*γ*holds for all frequency

*ω*with a small

*γ*, then the system has a good disturbance attenuation property.

*G*(

*jω*) ∥ <

*γ*with

*γ*= 1 holds for all frequency

*ω*if and only if there exists a symmetric matrix

*P*satisfying the LMI:

*P*satisfying the LMI is enough to conclude that the gain is less than one for all, infinitely many, frequencies. This result is known as the bounded real lemma and has played a fundamental role in the robust and

*H*

_{ ∞ }control theories.

*G*(

*jω*) ∥ < 1 and the LMI condition can equivalently be written as

*A*,

*B*, and \(\Theta = {\Theta }^{\mathsf{T}}\), the FDI (1) holds for all frequency

*ω*if and only if there exists a matrix \(P = {P}^{\mathsf{T}}\) satisfying the LMI (2), provided

*A*has no eigenvalues on the imaginary axis.

*L*(

*s*). For instance, the FDI (3) with

*L*(

*jω*) +

*L*(

*jω*)

^{∗}> 0. This is called the positive real property, with which the phase angle remains between ± 90

^{∘}when

*L*(

*jω*) is a scalar.

## Generalization

The standard KYP lemma deals with FDIs that are required to hold for all frequencies. To allow for more flexibility in practical system designs, the KYP lemma has been generalized to deal with FDIs in (semi)finite frequency ranges.

*ω*| ≤

*ϖ*

_{ ℓ }if and only if there exist matrices \(P = {P}^{\mathsf{T}}\) and \(Q = {Q}^{\mathsf{T}} > 0\) satisfying

*A*has no imaginary eigenvalues in the frequency range. In the limiting case where

*ϖ*

_{ ℓ }approaches infinity and the FDI is required to hold for the entire frequency range, the solution

*Q*to (5) approaches zero, and we recover (2).

*Q*is to enforce the FDI only in the low frequency range. To see this, consider the case where the system is stable and a sinusoidal input \(u = \mathfrak{R}[\hat{u}{e}^{j\omega t}]\), with (complex) phasor vector \(\hat{u}\), is applied. The state converges to the sinusoid \(x = \mathfrak{R}[\hat{x}{e}^{j\omega t}]\) in the steady state where \(\hat{x} := G(j\omega )\hat{u}\). Multiplying (5) by the column vector obtained by stacking \(\hat{x}\) and \(\hat{u}\) in a column from the right, and by its complex conjugate transpose from the left, we obtain

*ω*is outside of the range, however, the first term is negative, and the FDI is not required to hold.

*Q*in (5) can be expressed as the Kronecker product \(\Psi \otimes Q\) with \(\Psi \) being a diagonal matrix with entries \((-1,\varpi _{\ell}^{2})\). The matrix

*Ψ*arises from characterization of the low frequency range:

Table 1

Low | Middle | High | ||
---|---|---|---|---|

\(\boldsymbol{\Omega }\) | \(\vert \omega \vert \leq \varpi _{\ell}\) | \(\varpi _{1} \leq \omega \leq \varpi _{2}\) | \(\vert \omega \vert \geq \varpi _{h}\) | |

| \(\left [\begin{array}{@{}c@{\quad }c@{}} - 1\quad &0\\ 0 \quad & \varpi _{\ell}^{2 } \end{array} \right ]\) | \(\left [\begin{array}{@{}c@{\quad }c@{}} - 1 \quad & j\varpi _{c} \\ - j\varpi _{c}\quad &-\varpi _{1}\varpi _{2}\end{array} \right ]\) | \(\left [\begin{array}{@{}c@{\quad }c@{}} 1\quad & 0 \\ 0\quad &-\varpi _{ h}^{2} \end{array} \right ]\) |

*P*and

*Q*> 0 satisfying

*A*has no eigenvalues in \(\boldsymbol{\Omega }\), where

*F*. More specifically, the choices

*L*(

*z*) = (

*zE*−

*A*)

^{− 1}(

*B*−

*zO*).

## Applications

The generalized KYP lemma is useful for a variety of dynamical system designs. As an example, let us consider a classical feedback control design via shaping of a scalar open-loop transfer function in the frequency domain. The objective is to design a controller *K*(*s*) for a given plant *P*(*s*) such that the closed-loop system is stable and possesses a good performance dictated by reference tracking, disturbance attenuation, noise sensitivity, and robustness against uncertainties.

*L*(

*s*) : =

*P*(

*s*)

*K*(

*s*) in various frequency ranges as shown in Fig. 1. The controller

*K*(

*s*) should be designed so that the frequency response

*L*(

*jω*) avoids the shaded regions. For instance, the gain should satisfy |

*L*(

*jω*) | ≥ 1 for |

*ω*| <

*ω*

_{2}and |

*L*(

*jω*) | ≤ 1 for |

*ω*| >

*ω*

_{3}to ensure the gain crossover occurs in the range

*ω*

_{2}≤

*ω*≤

*ω*

_{3}, and the phase bound

*∠L*(

*jω*) ≥

*θ*in this range ensures robust stability by the phase margin.

*γ*=

*γ*

_{1},

*γ*

_{4}, or 1, and the + ∕ − signs for upper/lower gain bounds. These FDIs in the corresponding frequency ranges can be converted to inequalities of the form (6) with \(\Theta \) given by (4).

The control problem is now reduced to the search for design parameters satisfying the set of inequality conditions (6). In general, both coefficient matrices *F* and \(\Theta \) may depend on the design parameters, but if the poles of the controller are fixed (as in the PID control), then the design parameters will appear only in \(\Theta \). If in addition an FDI specifies a convex region for *L*(*jω*) on the complex plane, then the corresponding inequality (6) gives a convex constraint on *P*, *Q*, and the design parameters. This is the case for specifications of gain upper bound (disk: | *L* | < *γ*) and phase bound (half plane: *θ* ≤ *∠L* ≤ *θ* + *π*). A gain lower bound | *L* | > *γ* is not convex but can often be approximated by a half plane. The design parameters satisfying the specifications can then be computed via convex programming.

Various design problems other than the open-loop shaping can also be solved in a similar manner, including finite impulse response (FIR) digital filter design with gain and phase constraints in a passband and stop-band and sensor or actuator placement for mechanical control systems (Hara et al. 2006; Iwasaki et al. 2003). Control design with the Youla parametrization also falls within the framework if a basis expansion is used for the Youla parameter and the coefficients are sought to satisfy convex constraints on closed-loop transfer functions.

## Summary and Further Directions

The KYP lemma has played a fundamental role in systems and control theories, equivalently converting an FDI to an LMI. Dynamical systems properties characterized in the frequency domain are expressed in terms of state space matrices without involving the frequency variable. The resulting LMI condition has been found useful for developing robust and optimal control theories.

A recent generalization of the KYP lemma characterizes an FDI for a possibly improper rational function in a (semi)finite frequency range. The result allows for direct solutions of practical design problems to satisfy multiple specifications in various frequency ranges. A design problem is essentially solvable when transfer functions are affine in the design parameters and are required to satisfy convex FDI constraints. An important problem, which falls outside of this framework and remains open, is the design of feedback controllers to satisfy multiple FDIs on closed-loop transfer functions in various frequency ranges. There have been some attempts to address this problem, but none of them has so far succeeded to give an exact solution.

The KYP lemma has been extended in other directions as well, including FDIs with frequency-dependent weights (Graham and de Oliveira 2010), internally positive systems (Tanaka and Langbort 2011), full rank polynomials (Ebihara et al. 2008), real multipliers (Pipeleers and Vandenberghe 2011), a more general class of FDIs (Gusev 2009), multidimensional systems (Bachelier et al. 2008), negative imaginary systems (Xiong et al. 2012), symmetric formulations for robust stability analysis (Tanaka and Langbort 2013), and multiple frequency intervals (Pipeleers et al. 2013). Extensions of the KYP lemma and related S-procedures are thoroughly reviewed in Gusev and Likhtarnikov (2006). A comprehensive tutorial of robust LMI relaxations is provided in Scherer (2006) where variations of the KYP lemma, including the generalized KYP lemma as a special case, are discussed in detail.

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