My Research Activities in Control Theory and Engineering: An OverviewQing-Chang Zhong

Department of Electrical Engineering and ElectronicsThe University of Liverpool

AbstractMy research activities have been focusing on control theory and con-

trol engineering. On the theoretical side, siginificant contributions

have been made towards time-delay systems, algebraic Ricatti equa-

tions, J -spectral facotrisation and infinite-dimensional systems. On

the application side, the focus is on power electronics and renew-

able energy, automotive electronics, rapid control prototyping and

hardware-in-the-loop simulation, embedded systems etc.

1. Research on Control Theory

Algebraic Riccati equation and similarity transformation

Let A, R and E be real n×n matrices with R and E symmetric. The

following equation is called an algebraic Riccati equation (ARE):

[

− X I]

H

[

I

X

]

= 0 (1)

with H =[

A R

−E −A∗

]

called a Hamiltonian matrix.

Lemma 1 Suppose that H has no imaginary eigenvalues. Then

a stabilising solution X = Ric(H) exists if and only if the (1, 1)-

block X1 of T in the following similarity transformation is non-

singular:

T−1HT =

[

A− ?

0 ?

]

,

Furthermore, X = X2X−11 .

The formula (1), together with

AX =[

I 0]

H

[

I

X

]

, (2)

can be represented by the block diagram shown below. This can be

used to simplify the relationship between algebraic Riccati equations.

H XX

+ -

U

V

W

Y

U1

V1=0

W1

Y1 (=0)

Lemma 2 Let H =[

A R

−E −A∗

]

be a Hamiltonian matrix with X =

Ric(H) and P be the solution of the Lyapunov equation

AXP + PA∗X + R = 0,

where AX = A + RX is stable. Then the nonsingular matrix

T =

[

I 0

X I

] [

I P

0 I

]

=

[

I P

X I + XP

]

similarly transforms H into the block diagonal form given below:

T−1HT =

[

AX 0

0 −A∗X

]

.

A stabilising solution X does not stabilise A, but group all the stable

eigenvalues into AX .

J-spectral factorisation

J -spectral factorisation is defined as

Λ(s) = W∼(s)JW (s),

where the J -spectral factor W (s) is bistable and Λ(s) is a para-

Hermitian matrix: Λ(s) = Λ∼(s).= ΛT (−s). Assume that Λ, having

no poles or zeros on the jω-axis including ∞, is realised as

Λ =

[

Hp BΛ

CΛ D

]

= D + CΛ(sI − Hp)−1BΛ (3)

and denote the A-matrix of Λ−1as Hz, i.e.,

Hz = Hp − BΛD−1CΛ.

There always exist nonsingular matrices ∆p and ∆z (e.g. via Schur

decomposition) such that

∆−1p Hp∆p =

[

? 0

? A+

]

and

∆−1z Hz∆z =

[

A− ?

0 ?

]

,

where A+ is antistable and A− is stable (A+ and A− have the same

dimension).

Lemma 3 Λ admits a J-spectral factorisation if and only if there

exists a nonsingular matrix ∆ such that

∆−1Hp∆ =

[

Ap− 0

? Ap+

]

, ∆−1Hz∆ =

[

Az− ?

0 Az+

]

where Az− and A

p− are stable, and Az

+ and Ap+ are antistable.

When this condition is satisfied, a J-spectral factor W is given

as

W =

[

I 0]

∆−1Hp∆

[

I

0

]

[

I 0]

∆−1BΛ

Jp,qD−∗W CΛ∆

[

I

0

]

DW

, (4)

where DW is a nonsingular solution of D∗WJp,qDW = D.

One possible ∆ is

∆ =[

∆z

[

I

0

]

∆p

[

0I

] ]

.

The Delay-type Nehari problem

Given a minimal state-space realisation Gβ =[

A B

−C 0

]

, which is not

necessarily stable, and h ≥ 0, characterise the optimal value

γopt = inf{∥

∥Gβ(s) + e−shK(s)∥

∥

L∞ : K(s) ∈ H∞}

and for a given γ > γopt, parametrise the suboptimal set of proper

K ∈ H∞ such that∥

∥Gβ(s) + e−shK(s)∥

∥

L∞ < γ.

The optimal value γopt is

γopt = max{γ : det Σ22 = 0},

with

Σ22 =[

−Lc I]

Σ

[

Lo

I

]

,

where Lo and Lc are stabilising solutions, respectively, to

[

−Lc I]

[

A γ−2BB∗

0 −A∗

] [

I

Lc

]

= 0,

[

I −Lo

]

[

A 0

−C∗C −A∗

] [

Lo

I

]

= 0.

Σ =

[

Σ11 Σ12

Σ21 Σ22

]

= exp(

[

A γ−2BB∗

−C∗C −A∗

]

h)

The block diagram of K is shown below.

����

Gβ Z

e−shI

W−1

����

Q

K@

@@

�

-

-

u

y

z

w

�

-

��

6

6

6

?

?

Example: Gβ(s) = − 1s−a

(a > 0)

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ah

aγ

aγopt

Since I − LcLo = 1 − 4a2γ2, there is∥

∥

∥ΓGβ

∥

∥

∥= 1

2a. As a result, the

optimal value γopt satisfies 0.5 ≤ aγopt ≤ 1.

The standard H∞ problem of single-delay systems

Given a γ > 0, find a proper controller K such that the closed-loop

system is internally stable and∥

∥Fl(P, Ke−sh)∥

∥

∞< γ.

P

e−shI

K

�

� �

�

y

z

u

w

u1

-

The problem can be simplified as shown below.

Cr(P )

@@@ e−shI

K

-

� � �

w

z u

y

u1

6

Cr(P )

@@@

Gα

@@@

Cr(Gβ)

@@@ e−shI

K

Delay-free problem 1-block delay problem

-

�

-

�

-

� � �

w

z u

y

u1

6

w1

z1

y

u1

Gα is the controller generator of the delay-free problem. Gβ is defined

such that Cr(Gβ).= G−1

α . Gα and Cr(Gβ) are all bistable.

Solvability ⇐⇒ :

• H0 ∈ dom(Ric) and X = Ric(H0) ≥ 0;

• J0 ∈ dom(Ric) and Y = Ric(J0) ≥ 0;

• ρ(XY ) < γ2;

• γ > γh, where γh = max{γ : det Σ22 = 0}.

Z V −1

����

Q

@@@

--

u

y-

��

6

?

?

V −1 =

A + B2C1 B2 − Σ12Σ−122 C∗

1 Σ−∗22 B1

C1 I 0−γ−2B∗

1Σ∗21 − C2Σ

∗22 0 I

Implementation of the controller

As seen above, the control laws associated with delay systems nor-

mally include a distributed delay like

v(t) =

∫ h

0

eAζBu(t − ζ)dζ,

or Z(s) = (I − e−(sI−A)h) · (sI − A)−1inthes−domain.

The implementation of Z is not trivial because A may be unstable.

This problem had confused the delay community for several years

and was proposed as an open problem in Automatica in 2003. It was

reported that the quadrature implementation might cause instability

however accurate the implementation is.

My investigation shows that: The quadrature approximation error

converges to 0 in the sense of H∞-norm.

10−2

10−1

100

101

102

103

10−4

10−3

10−2

10−1

100

101

Frequency (rad/sec)

N=1

N=5

N=20

App

roxi

mat

ion

erro

r

Rational implementation

1x2xΠ

Nx 1−Nx

B1−Φbu

u

rv

…

ΦΦ+−=Π −1)( AsI

Π Π

…

Π = (sI − A + Φ)−1Φ, Φ = (∫ h

N0 e−Aζdζ)−1.

Feedback stabilisation of delay systems

The feedback stabilizability of the state–input delay system

x(t) = A0x(t) + A1x(t − r) + Pu(t) + P1u(t − r)

is equivalent to the condition

Rank[

(P + e−rλiP1)∗ · ϕi

]

= di, i = 1, 2, · · · , l.

where λi ∈ {λ1, λ2, · · · , λl} = {λ ∈ C : det∆(λ) = 0 and Reλ ≥ 0}

with ∆(λ) := λI − A0 − A1e−rλ. The dimension of Ker∆(λi)

∗ is di

and the basis of Ker∆(λi)∗ is ϕi

1, ϕi2, · · · , ϕ

idi

for i = 1, 2, · · · , l .

2. Research on control engineering

Power electronics & renewable energy

Control problems involved in distributed generation

• voltage control: e = Vref − Vc as small as possible

• neutral point control: to provide a non-drifting neutral point

• power control: to regulate the active/reactive power

• phase-locked loop (PLL): to synchronise the converter with the

grid

Voltage control of DC-AC converters

The single-phase circuit:

The objective is to make sure that the output voltage Vout or Vc is

a clean sinusoidal signal even when the load is nonlinear and/or the

public grid is polluted with harmonics.

Structure of voltage controller

Techniques used:

• H∞ control

• Repetitive control, where a delay is introduced into the controller

Nyquist plot of the system

−2 −1 0 1 2 3 4 5 6−8

−6

−4

−2

0

2

4

6

8−L(jω)

Re

Im

Simulation results

0 0.05 0.1 0.15 0.2−400

−300

−200

−100

0

100

200

300

400

Time (sec)

Vol

tage

(v)

Vc e

0.36 0.37 0.38 0.39 0.4−400

−300

−200

−100

0

100

200

300

400

Vol

tage

(V

) micro−grid

(external) grid

Time (sec)

(a) Transient response (b) Steady-state response

Neutral-point control

Control objective is to force ic ≈ 0 so that the point N will be the

mid-point of DC supply.

• No need to re-design the converter;

• The controller is decoupled.

H∞ control design

Experimental results

Vave

0.2V/div

iN

50A/div

iL

50A/div

ic

20A/div

0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27

Time (sec)

Regulation of induction generators for wind power

Q

P

Control of wind turbines

Patented by Nheolis, France

Experiments show that the new wind turbine is very efficient. The

maximum mechanical power of a prototype with a 2m (diameter)

rotor reached 12kW at a wind speed of 20m/s. The nominal power

is 4.1kW at 14 m/s. A 1-meter 3-bladed prototype recorded 2.8kW

mechanical power at 14 m/s. This is much more efficient than any

commercial wind turbines available.

Damping control of inter-area oscillations in large-scale

power systems

AC-DC converters: DC drives

AC-DC-AC converters: AC drives

Philips Semiconductors

VVVF speed control by:

• using the PWM circuit HEF4752V shown above

• using Intel 8051 microcomputer to generate space vector PWM

signal

Rapid control prototyping & hardware-in-the-loop sim-

ulation

dSPACE+Matlab/Simulink/SimPower

• Single-board PCI hardware for use in PCs

• powerful development system for RCP

• Real-Time Interface provides Simulink r© blocks for graphical con-

figuration of A/D, D/A, digital I/O lines, incremental encoder

interface and PWM generation

Embedded systems

Different development kits for embedded control:

• MPC5567

• EasyPIC4

• dsPICPro2

• LabJack U12

Chemical process control

3. Practical experiences

• Software design

– Intel 8086 assembly language: ¿ 100kB binary code

– C language: ¿ 10,000 lines

– Database/Javascript

• Hardware design

– Micro-computers: Intel 8051, Zilog Z80, Motorola ...

– DC, AC drives etc

– Lift control systems

– System design experience

4. EPSRC-funded New-ACE

Leading a nation-wide collaborative network: New-ACE.

• Partners: Imperial, Sheffield, Loughborough and Queen’s

Belfast.

• Advisory members: D.J.N. Limebeer (Imperial),

D.H. Owens (Sheffield), R.M. Goodall (Loughborough),

G. Irwin (Queen’s Belfast), Q.H. Wu (Liverpool).

• Main activities and outcomes:

– to organise six workshops in subject areas including renewable

energy and control in power electronics

– to submit 6˜12 joint proposals in the coming three years.

5. Current funded projects

• EPSRC: EP/C005953/1, 126k

• EPSRC: EP/E055877/1, 88k

• EPSRC: one DTA studentship

• EPSRC and Add2: Dorothy Hodgkin Postgraduate Award, 90k

• ESPRC and Nheolis: Dorothy Hodgkin Postgraduate Award, 90k

6. Future research topics

Control Theory

• Industrial collaboration to consolidate research: CHP and auto-

motive electronics

• Theoretical research to deepen the depth of research

Acknowledgement

The support from EPSRC, Add2 Ltd and Nheolis, France is greatly

appreciated.