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### 关于一般控制问题的所有控制器：LMI存在条件与状态空间公式 #### 摘要本文通过纯粹的代数方法解决了广义$ \mathcal{H}_\infty $控制问题。给出了$ \mathcal{H}_\infty $控制器存在的线性矩阵不等式（LMI）条件，并且所有$ \mathcal{H}_\infty $控制器在状态空间中被显式地参数化。 #### 引言在20世纪80年代末，$ \mathcal{H}_\infty $控制理论达到了一个相对成熟的状态，其中包含了状态空间表述（Glover和Doyle, 1988）以及与广泛熟知的$ \mathcal{H}_2 $（或LQG）控制问题的全面比较（Doyle等人, 1989a）。$ \mathcal{H}_\infty $控制器的存在条件通常由两个Riccati方程及一个谱半径条件描述，而所有$ \mathcal{H}_\infty $控制器集则使用唯一稳定解来参数化。 #### 1. 一般控制问题的背景一般控制问题涉及到设计控制器，使得系统的性能指标满足特定的要求，特别是在不确定性和干扰下保持系统的稳定性。$ \mathcal{H}_\infty $控制是现代控制系统理论的一个分支，它关注于设计控制器以最大化系统的鲁棒性，即在不确定环境下确保系统性能的最大化。该理论特别适用于处理具有不确定性的系统，这些不确定性可能来自模型误差、环境变化或其他未知因素。 #### 2. LMI存在条件线性矩阵不等式（LMI）是一种数学工具，在控制理论中用于解决复杂的优化问题，特别是那些涉及稳定性分析的问题。本文给出了一般控制问题中$ \mathcal{H}_\infty $控制器存在的三个LMI条件。这三个LMI条件可以看作是Scherer的结果的另一种形式，但具有更基础的推导过程。具体来说，这三条LMI条件为： 1. **LMI条件一**：描述了控制器的稳定性。 2. **LMI条件二**：确保控制器能够使系统的$ \mathcal{H}_\infty $性能达到指定水平。 3. **LMI条件三**：确保闭环系统满足所需的性能指标。 #### 3. 状态空间公式本文还提供了一种新的方法来参数化所有的$ \mathcal{H}_\infty $控制器，这种方法是在状态空间中显式地表示这些控制器。传统的参数化方法通常涉及到一个称为Q的自由变量，它是一个实有理稳定的传递函数，其$ \mathcal{H}_\infty $范数被限定在一个特定的数值范围内。相比之下，本文提出的方法将这个自由度替换为一个固定维度的矩阵L，它的范数受到约束，同时还有LMI条件下的正定解(X,Y)。这种方法的一个优点在于，它将自由矩阵L和正定解对(X,Y)定义在一个有限维的设计空间内，而不是像传统Q参数化方法那样在无限维的空间中进行。 #### 4. 凸可行性问题 LMI的存在条件将问题转化为一个凸可行问题。通过这种方法，控制器的设计问题被转换成了一个标准的凸优化问题，这对于实际工程应用而言是非常有益的。这种转化不仅简化了问题的求解过程，而且还提供了更广泛的优化工具的应用可能性。 #### 结论通过纯粹的代数方法，本文解决了广义$ \mathcal{H}_\infty $控制问题。通过给出$ \mathcal{H}_\infty $控制器存在的LMI条件以及所有控制器的状态空间参数化表达，本文不仅为控制器的设计提供了一个新的视角，而且也为解决复杂控制问题提供了一种有效的途径。这种方法相比于传统的Q参数化方法具有明显的优势，因为它能够在有限维设计空间中进行优化，从而提高了设计的效率和实用性。

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Pergamon

O00S-IO~Oa)EOO~-X

Automatica,

Vol. 30, No. 8, pp. 1307-1317, 1994

0005-1098194 $7.00 + 0.00

All Controllers for the General Control

Problem: LMI Existence Conditions and State

Space Formulas*

T. IWASAKIt and R. E. SKELTONt

This paper solves the general ~(~ control problem by a purely algebraic

approach. Existence conditions for an ~(® controller are given in terms of

linear matrix inequalities, and all ~® controllers are parametrized explicitly

in state space.

Key

Words--Linear systems; robust control; optimal control; matrix algebra; convex programming.

Abstrad--This paper presents all controllers for the general ~'®

control problem (with no assumptions on the plant

matrices). Necessary and sufficient conditions for the

existence of an ~® controller of any order are given in terms

of three Linear Matrix Inequalities (LMIs). Our existence

conditions are equivalent to Scherer's results, but with a

more elementary derivation. Furthermore, we provide the

set of all ~(= controllers explicitly parametrized in the state

space using the positive definite solutions to the LMIs.

Even under standard assumptions (full rank, etc.), our

controller parametrization has an advantage over the

Q-parametrization. The freedom Q (a real-rational stable

transfer matrix with the ~® norm bounded above by a

specified number) is replaced by a constant matrix L of fixed

dimension with a norm bound, and the solutions (X, Y) to

the LMIs. The inequality formulation converts the existence

conditions to a convex feasibility problem, and also the free

matrix L and the pair (X, Y) define a finite dimensional

design space, as opposed to the infinite dimensional space

associated with the Q-parametrization.

1. INTRODUCTION

IN THE LATE 1980S, ~ control theory reached a

fairly mature state, complete with state space

formulations (Glover and Doyle, 1988) and

comprehensive comparisons with the widely

known ~2 (or LQG) control problem (Doyle et

al., 1989a). The existence conditions for an ~®

controller were described by two Riccati

equations and a spectral radius condition, and

the set of all ~ controllers was parametrized

using the (unique) stabilizing solutions to the

* Received 2 October 1992; revised 10 November 1992;

revised 1 December 1992; revised 7 June 1993; received in

final form 28 September 1993. This paper was not presented

at any IFAC meeting. This paper was recommended for

publication in revised form by Associate Editor Ruth Curtain

under the direction of Editor Huibert Kwakernaak.

Corresponding author T. Iwasaki. Tel. +1 317 494 4209; Fax

+ 1 317 494 2351; E-mail iwasaki@gn.ecn.pudue.edu.

t Space Systems Control Laboratory, Purdue University,

West Lafayette, IN 47907, U.S.A.

1307

Riccati equations with free parameter Q, a

real-rational stable transfer matrix with an ~

norm less than a specified number. Since there

are many ~f~ controllers (beyond the central

controller), the remaining freedom should be

utilized to meet other specifications. However, a

major deficiency of some of the existing ~®

control theory as a practical tool for controller

design is the fact that the free parameter Q is

hard to choose [see the computational ap-

proaches in Boyd and Barratt (1991)]. The

mixed ~2/~f= control problem has been solved

directly (without optimizing over the free

parameter Q) in Bernstein and Haddad (1989)

and Doyle et al. (1989b) where necessary and

sufficient conditions for optimality are obtained

in terms of highly nonlinear coupled matrix

equations, which are not easily solved. Khar-

gonekar and Rotea (1991) overcame this

difficulty via finite dimensional convex program-

ming and obtained a feasible controller of order

equal to the plant. Our paper is about ~= and

not mixed problems. But, our result can be

applied to solve the mixed problem directly, in

exactly the same way as in Khargonekar and

Rotea (1991), but we do not require any

assumptions on the plant and the set of all

feasible controllers (of any order) is para-

metrized explicitly.

th order linear time-invariant

Consider the np

generalized plant

! =Ax + BlW + BEU

(Zp) = C~x + Dl~w + D~2u (1)

= C2X + D21w +

022,

where x e ~np is the plant state, w e .~l n" is any

exogenous input, including plant disturbances,

1308 T. IWASAKI and R. E. SKELTON

measurement noise, etc., u ~ ~"° is the control

input, z e YI~ "~ is the regulated output and y e .*}t",

is the measured output. Recall that most of the

existing ~= control theory imposes the following

"standard" assumptions:

D(2D~2

> 0 and

D2~D r >0. There are a few exceptions.

Petersen (1987) and Khargonekar

et al.

(1988)

solved the singular state feedback case where

Dlz=0

and D2~ = 0 via the "algebraic Riccati

equation approach". Zhou and Khargonekar

(1988) considered the state feedback case

(C2=1, D21=0 and /:)22--0) without any

assumption on the rank of D~z. Their approach

for dealing with the singularity of

D(2D~2

basically to split the (constant) state feedback

gain into two components; one belonging to the

nullspace of D12 and the other orthogonal to it.

As the existence condition for a controller, this

approach yields an algebraic Riccati equation

with a positive scalar parameter e>0. To

eliminate the extra work of searching for a

suitable value of the parameter •>0,

Stoorvogel and Trentelman (1990) and Stoorv-

ogel (1991) dealt with the singular ~= control

problem more directly for the state feedback

case and extended the result to include the

dynamic output feedback case. In Stoorvogel

and Trentelman (1990), and Stoorvogel (1991),

assuming that the plant has no invariant zeros on

the ]w-axis, the conditions for the existence of a

controller were given in terms of Quadratic

Matrix Inequalities (QMIs) and a set of rank

conditions on the positive semidefinite solutions

to the QMIs. It was shown that such solutions

could be obtained as solutions to certain

reduced-order Riccati equations. However, the

author only gives a "conceptual algorithm" for

constructing a controller of order equal to the

generalized plant.

Further generalization is available from

Scherer (1992a, b) for the state feedback case

and for the dynamic output feedback case. The

rank conditions and Sroorvogel's assumption

"no invariant zeros on the jw-axis" are removed

in Scherer's work. The only restriction on the

plant in Scherer (1992b) is that DH--0 and

Dzz

= 0. In the specialized case DH = 0, one of

our results (the existence conditions in Theorem

4.2) is shown to be equivalent to the result of

Scherer (1992b, Theorem 1). Scherer's main

focus was the existence of a controller (of any

order) and the computation of the optimal Y(=

norm bound. For this purpose, he provided an

algebraically verifyable method to check the

existence of a solution to a Riccati inequality

including a controller parameter, and gave only

one (strictly proper) full order controller. Our

focus is an explicit formula for parametrizing all

9~= controllers. Our result (Theorem 4.2)

removes the dependence of Scherer's Riccati

inequality on the controller parameters, and

shows the convexity of the set associated with

the Riccati inequality solutions. In our formula-

tion, the convex set is used to capture all Y(~

controllers, and is shown to be useful to

incorporate additional performance specifica-

tions. Moreover, our new results provide the

existence conditions for a fixed order ~

controller (e.g. static output feedback).

This paper considers a general ~ control

problem without any assumptions on the plant

matrices

(A,

B1,

Be,

CI, C2,

D11, D12, Dzl).

also make no assumption on 022 except for the

well-posedness of the feedback connection.

,h order linear time-invariant

Consider the

dynamic (no > O) and static (no = O) controllers

,[-/c =

A<x,. + B,.y

(Z'~d){u = Cox< + O<y '

(Zcs){u =

Gy,

(2)

where xc e ~,c is the controller state. We shall

denote the whole class of controllers (no >- 0) by

(Zc). Suppose one defines a fictitious measured

output

S,:=C2x+D21w

of (1) (which has no

term in u), and designs a controller (Z~d)

assuming )? is available for feedback, and then

replaces 2 by

y - D22u.

Then from (2),

u = (I + DcOzz)-'(Ccxc + Dcy)

(?,.xc + DcY

(3)

which is of the same form as (2). Of course,

well-posedness assumes that

(1 + D~Dz2)-

exists. Hence, under a well-posedness assump-

tion, we need only a theory for treating (1) with

Dzz

= 0. We consider the design of a stabilizing

controller (y) which yields the closed-loop

transfer matrix with ~(= norm bounded above by

a specified number. Such a controller is said to

be an ~= controller. Necessary and sufficient

conditions for the existence of an ~ controller

of some (unspecified) order are given in terms of

three Linear Matrix Inequalities (LMIs). Posi-

tive definite solutions to the LMIs form a convex

set. The controller order can be fixed by

imposing an additional rank condition (at the

expense of convexity) on the solutions to the

LMIs. Moreover, the set of all ~ controllers is

characterized explicitly in the state space. Of

course, the results can be specialized to the

standard case. In this event, there is a strong

similarity to the Q-parametrization (Doyle

et al.,

1989a; Glover and Doyle, 1988) except that our

existence conditions involve two Riccati in-

equalities instead of two Riccati equations, and

the set of ~= controllers associated with a given

solution pair to the Riccati inequalities is

parametrized by a constant matrix of fixed

dimension with a norm bound (this is a finite

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