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基于Matlab实现H-infinity鲁棒控制仿真（源码+说明文档）.rar （3个子文件）

基于Matlab实现H-infinity鲁棒控制仿真（源码+说明文档）

robustcontrol.m 7KB

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Institute for Systems Theory and Automatic Control

Prof. Dr.-Ing. F. Allgwer

Robust Control

http://www.ist.uni-stuttgart.de/education/courses/robust/

Uni Stuttgart

02.07.2009

jj, ssc

∞

Control in Matlab

1 Introduction

Typically the concept of H

∞

controller design is fairly easy to grasp. However, as

controller synthesis is done numerically, a major problem for people new to the subject

is how to write the Matlab code. I w ill here try to give a short overview of some useful

Matlab functions. Hopefu lly this will help you when trying to design your ﬁrst H

∞

controller.

There are many H

∞

related functions available in Matlab and its toolboxes. The im-

portant toolboxes are, in addition to the Contr ol System Toolbox, the mu-Analysis and

Synthesis Toolbox (mu-tools), the Robust Control Toolbox (RCT) and the LMI Control

Toolbox. LMI and mu-tools are both included in RCT v.3.0.1 which comes with Matlab

7, in earlier versions they are separate.

I have also prepared an m-ﬁle where I have tried to use as many of the fun ctions discussed

here as possible. The m-ﬁle is included in the appendix and can also be downloaded

from the r ob ust control webpage.

A mixed S/KS synthesis problem will be used to illustrate the use of a handful of useful

functions. Let’s take a look at the this problem ﬁrst.

2 Shaping closed loop transfer functions

The mixed S/KS problem can be illustrated with the block diagram shown in Figure 1.

The closed loop transfer fun ction T = F

(P, K) from w to z can be found by visual

inspection as









r. (1)

The generalized plant P (s) (see Figure 2) is













−W

0 W

I −G









. (2)

If we have the state space realizations



A B

C D



, W





, W





it can be s hown that a possible state space r ealization for P (s) is given by







0 −B

−B

0 A

0 0 B

0 0 A 0 B

0 −D

C D

−D

0 C

0 D

0 0 −C

I −D







. (3)

- - -

K G

−

r y

e u

Figure 1: Mixed S/KS problem.

- -



w z

Figure 2: General control problem.

−4

−2

−45

−40

−35

−30

−25

−20

−15

−10

−5

Frequency [rad/sec]

Amplitude [dB]

−1

Figure 3: Inverse of weighting functions W

and W

(I leave this as an exercise for you.)

The weights W

and W

are your tuning parameters, and it typically requires some

iterations to obtain weights which will yield a good controller. That being said, a good

starting point is to choose

s/M + ω

s + ω

; W

= const. (4)

where A < 1 is the maximim allowed steady state oﬀset, w

is the desired bandwidth and

M is the sensitivity peak (typically A = 0.01 and M = 2). For the controller synthesis,

the inverse of W

is an upper bound on the desired sensitivity loop shape, and W

−1

will

eﬀectively limit the controller output u.

In some cases, you would also like to shape the complementary sensitivity function

T = GK(I + GK)

−1

(done by adding an extra output z

= W

y in Figure 1). A

starting point is to choose

s + ω

As + ω

, (5)

which is symmetric to W

around the line ω = ω

. The two weighting fu nctions are

shown in Figure f or the parameter values A = 0.01(= −40dB), M = 2(= 6dB) and

= 1 r ad /sec.

3 Obtaining the subsystems

There are several ways to obtain the dy namical systems G, W

and W

in Mat-

lab. Methods you probably already have heard about are ss, tf and zpk in Con-

trol System Toolbox. Mu-tools oﬀer a variety of similar possibilities like pck , nd2sys

and zp2sys. Other methods are mksys and tree. You should be aware however,

that mu-tools uses a diﬀerent representation than the Control Sys tem Toolbox, called

a system matrix. Thus you cannot just pass a system generated with e.g. Gcst =

ss(A,B,C,D) in Contr ol S ystem Toolbox to a function fou nd in mu-tools (with RCT

v.3.0.1 th is is no longer so, most functions have been rewritten to accept both system

representations). Which one to choose is a matter of convenience, you can transfer

back an d forth between the diﬀerent representations quite easily. One possibility is

to write [A,B,C,D]=ssdata(Gcst); Gmu=pck(A ,B,C,D). The opposite way would be

[A,B,C,D]=unpck( Gmu); Gcst = ss(A ,B,C,D). Take a look at the docu mentation to

see other options.

4 Obtaining the generalized pla nt P

Also in creating P you have many options. I list ﬁve:

a) Write down th e transfer f unction matrix in (2) directly. I prefer to use mu-tools for

this option. I f you afterwards convert to state-space, you shou ld use e.g. minreal

to obtain a minimal realization. Useful commands: sbs (side-by-side), abv (above),

mmult (multiply), minv (inverse).

b) Write down the state space matrices A,B,C,D in (3) and use P = pck(A,B,C,D ).

c) Use sysic (system interconnect), an m-ﬁle in mu-tools where you specify your

subsystems and the interconnection between them.

d) Use sconnect , a fun ction in L MI-tools where subsystems, inputs and outputs are

passed as parameters, and scon ncet returns the connected system.

e) Use iconnect in RCT v3.0.1, functionally similar to sysic.

Of these methods I personally prefer sysic and iconnect because they are ﬂexible and

easy to use also for more complex systems where method 1 and 2 are no longer feasible.

Generally it is a good idea to u s e a balanced realization to avoid num erical problems. A

balanced realization can be obtained e.g. with balreal in Control S ystem Toolbox.

5 Synthesizing controller

The H

∞

S/KS synthesis problem is to ﬁnd a controller K which stabilizes G and mini-

mizes the H

∞

cost function

(P, K)k

∞



∞

I guess by now you are not surprised to hear that there are several methods available to

synthesize H

∞

controllers. Typically you would use hinfsyn, hinf ric or hinflmi which

all have P in the System (mu-tools) representation as an input. In RCT v3.0.1, there

is the f unction mixsyn with G, W

, W

(and W

, a weight for the complementary

sensitivity function) as inputs, that is, you do not need the generalized plant P at all.

The main diﬀerence between th e methods is whether they use Riccati equations and

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