# Sampled-Data \(\mathcal{H}_{\infty }\) Optimization

**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_157-1

## Abstract

\(\mathcal{H}_{\infty }\) optimization is central in robust control. When controllers are implemented by computers, sampled-data control systems arise. Designing \(\mathcal{H}_{\infty }\)-optimal controllers in purely continuous time or in purely discrete time is standard in robust control; in this entry, we discuss the process of sampled-data optimization, namely, designing digital controllers based on a continuous-time \(\mathcal{H}_{\infty }\) performance measure.

## Keywords

Robust control Sampled-data systems Computer control \(\mathcal{H}_{\infty }\) discretization## Introduction

Robust control deals mainly with controller design against uncertainties in system modeling and disturbances. The central tool used is \(\mathcal{H}_{\infty }\) optimization.

*G*is the generalized plant and

*K*is the controller;

*G*has two inputs (

*w*, the exogenous input, and

*u*, the control input) and two outputs (

*z*, the output to be controlled, and

*y*, the measured output);

*K*processes

*y*to generate

*u*. The \(\mathcal{H}_{\infty }\)-optimal control problem is to design

*K*to stabilize

*G*and minimize the \(\mathcal{H}_{\infty }\) norm of the closed-loop system in Fig. 1 from

*w*to

*z*, denoted

*T*

_{ zw }. When both

*G*and

*K*are continuous-time, linear time-invariant (LTI), the \(\mathcal{H}_{\infty }\) norm, \(\|T_{zw}\|\), relates to the frequency response matrix \(\widehat{T}_{zw}(j\omega )\) as follows:

## Sampled-Data Control

*K*in Fig. 1 by sampler

*S*(with period

*h*), discrete-time controller

*K*

_{ d }, and zero-order hold

*H*(synchronized with

*S*), we obtain a sampled-data control system shown in Fig. 2; here,

*S*converts

*y*into a discrete-time sequence

*ψ*;

*K*

_{ d }, a real-time algorithm in the computer, inputs

*ψ*and computes another sequence

*υ*, which is converted by

*H*into

*u*. There are in general three approaches to design a digital controller

*K*

_{ d }: design a continuous-time controller

*K*and then implement digitally via approximation, discretize the plant and then design

*K*

_{ d }in discrete time, and finally, design

*K*

_{ d }directly based on continuous-time performance specifications (Chen and Francis 1995). The last approach is followed in the \(\mathcal{H}_{\infty }\) optimization framework.

## Sampled-Data \(\mathcal{H}_{\infty }\) Discretization

*K*

_{ d }

*directly*to stabilize

*G*in Fig. 2 and minimize \(\|T_{zw}\|\). Notice that even if

*G*is LTI in continuous time and

*K*

_{ d }is LTI in discrete time, the closed-loop system

*T*

_{ zw }is no longer LTI, due to the presence of

*S*and

*H*in the control loop; in this case, the \(\mathcal{H}_{\infty }\) norm is interpreted as the \(\mathcal{L}_{2}\)-induced norm:

*γ*> 0, construct an LTI discrete-time system

*G*

_{ eq, d }connected to

*K*

_{ d }as in Fig. 3; the two systems,

*T*

_{ zw }in Fig. 2 and

*T*

_{ ζω }:

*ω*↦

*ζ*in Fig. 3, are

*equivalent*in that \(\|T_{zw}\| <\gamma\) if \(\|T_{\zeta \omega }\| <\gamma\), where the latter norm is

*ℓ*

_{2}-induced, and since

*T*

_{ ζω }is LTI in discrete time, it equals the \(\mathcal{H}_{\infty }\) norm of the corresponding transfer function \(\widehat{T}_{\zeta \omega }(z)\). Thus, pure discrete-time techniques are immediately applicable.

There are several ways to present this discretization. However, the computation is quite involved and hence is not given here; interested readers can find details in the papers by Kabamba and Hara (1993), Bamieh and Pearson (1992), and Toivonen (1992), or the book by Chen and Francis (1995). Note that the \(\mathcal{H}_{\infty }\) discretization process is not quite *exact* in the sense that *G* _{ eq, d } depends on *γ* (Chen and Francis 1995).

## Summary and Future Directions

In sampled-data \(\mathcal{H}_{\infty }\) optimization, the key idea is to address the hybrid nature of the problem, considering intersample behavior in formulation; the main tool is the so-called continuous lifting (Yamamoto 1994; Bamieh and Pearson 1992), making use of periodicity of sampled-data systems.

The ideas and tools developed in sampled-data control theory are still being used in emerging areas such as hybrid systems and networked control systems. For example, in event-triggered control systems, information exchange and control updating are not time driven but are done by certain event-triggering schemes, resulting in necessarily nonlinear and time-varying closed-loop dynamics; the analysis and synthesis issues in such systems are still challenging.

## Cross-References

\(\mathcal{H}_{\infty }\) Control

LMI Approach to Robust Control

Optimization Based Robust Control

Sampled-Data Optimal Control

## Recommended Reading

The continuous-time \(\mathcal{H}_{\infty }\) control problem and its solutions are discussed extensively in several textbooks, e.g., Zhou et al. (1996) and Dullerud and Paganini (2000). The discrete-time \(\mathcal{H}_{\infty }\) control problem was solved via the approach of Riccati equations in Iglesias and Glover (1991). The sampled-data \(\mathcal{H}_{\infty }\) control problem was solved simultaneously with different methods in Kabamba and Hara (1993), Bamieh and Pearson (1992), and Toivonen (1992); details of the solution discussed here can be found in the book by Chen and Francis (1995).

## Bibliography

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