automatic control and mixed sensitivity hinf control

Contents Automatic control Background Mixed Sensitivity Benchmark of a Mechanical System Conclusions

Automatic control and mixed sensitivityH∞

control

René Galindo Orozco

FIME - UANL

CINVESTAV-Monterrey, 20 de Febrero de 2008


Contents

1 Automatic control1 Basic notions2 Degree of mechanization3 History4 Class of systems

2 Mixed sensitivityH∞ control1 Motivation2 Basic notions3 Background4 Loop-shaping5 Mixed sensitivity6 Standard solutions7 Parity interlacing property

1 Mixed sensitivityH∞ control in anon conventional scheme

1 A mixed sensitivity problem2 Direct solutions3 Controllers4 Tuning procedure5 Benchmark of a mechanical

system6 Conclusions


Basic notions

Automatic control [Wikipedia]

A research area and theoretical base for mechanization andautomation , employing methods from mathematics andengineering

Theory that deals with influencing the behavior of dynamicsystems

MechanizationMachinery to assist human operators with the physicalrequirements

Automation (ancient Greek: = self dictated) [Salvatencyclopedia]Control systems for industrial machinery andprocesses, replacing human operatorsReduces the need for human sensory and mentalrequirements


Basic notions

Automatic control [Wikipedia]

A research area and theoretical base for mechanization andautomation , employing methods from mathematics andengineeringTheory that deals with influencing the behavior of dynamicsystems

MechanizationMachinery to assist human operators with the physicalrequirements

Automation (ancient Greek: = self dictated) [Salvatencyclopedia]Control systems for industrial machinery andprocesses, replacing human operatorsReduces the need for human sensory and mentalrequirements


Basic notions

Components [Wikipedia]

u (t) -

f (�)

O& O �O%#&O"

O"

-y (t)

?

d (t)

System , set of interactingentities, real or abstract,forming an integrated whole

Sensor , measure some physical stateController , manipulates u (t) to obtain the desired y (t)Actuator effect a response under the command of the controllerReference or set point , a desired y (t)


Controller improved by J. Watt


Basic notions

Control objectives1 Regulation. Ex. controller improved by Watt

limt!∞

x (t)! 0, or limt!∞

[x (t)� xd]! 0, xd 2 <

2 Tracking or servo. Ex. radar

limt!∞

x (t)! xd (t)

3 Model matching4 Input / output decoupling5 Disturbance rejection or attenuation, etc.


Basic notions

Type of controllerOpen-loop controller

-r K(s) -Ldi?u- P∆ (s) -Ldo

? -y

Can not compensate d (t), ∆ and noiseClosed-loop controller

r-L -e K(s) -Ldi?u- P∆ (s) -Ldo

? -� y

?L dm�6�16

Feedback on the performance allows the controller to dynamicallycompensate for d (t), ∆ and noise


Basic notions

Feedback [Wikipedia]Basic mechanism by which systems maintain their equilibrium orhomeostasis

Types of feedback

Negative, tends to reduce output,Positive, tends to increase output, orBipolar.


por flotante

4.pdf


Kalman decomposition


Degree of mechanization

[Salvat encyclopedia]


History

[Wikipedia]Ktesibios, -270 Float regulator for a water clockPhilon, -250 Keep a constant level of oil in a lampIn China, 12th cen-tury

South-pointing chariot used for navigationalpurposes

14th century Mechanical clock1588 Mill-hopper, a device which regulated the flow

of grain in a millC. Drebbel, 1624 An automatic temperature control system for a

furnaceP. de Fermat, 1600’s Minimum-time principle in opticsD. Papin, 1681 A safety valve for a pressure cookerBernoulli, 1696

Principle of Optimality in connection with theBrachistochrone Problem

T. Newcomen, 1712 Steam engineE. Lee, 1745 Fantail for a windmill


curve

7.jpg

t12 =R P2

P1

s1+ (y0)2

2gydx

Brachistochrone curve


J. Brindley, 1758 Float valve regulator in a steam boilerI.I. Polzunov, 1765 A float regulator for a steam enginePontryagin, Boltyan-sky, Gamkrelidze, andMishchenko 1962

On/off relay control as optimal control

L. Euler (1707-1783)Calculus of variations . System moves in sucha way as to minimize the time integral of thedifference between the kinetic and potentialenergies

W. Henry, 1771 Sentinel registerBonnemain, 1777 A temperature regulator suitable for indus-

trial useJ. Watt, 1788,#Industrial revolu-tion

Centrifugal flyball governor

A.-L. Breguet, 1793 A closed-loop feedback system to synchro-nize pocket watches


R. Delap, M. Murray, 1799 Pressure regulatorBoulton, Watt, 1803 Combined a pressure regulator with a

float regulatorI. Newton (1642-1727),G.W. Leibniz (1646-1716),brothers Bernoulli (late1600’s, early 1700’s), J.F.Riccati (1676-1754)

Infinitesimal calculus

G.B. Airy, 1840 A feedback device for pointing a tele-scope. Discuss the instability of closed-loopsystems . Analysis using differential equa-tions

J.L. Lagrange (1736-1813), W.R. Hamilton(1805-1865)

Motion of dynamical systems using differ-ential equations

C. Babbage, 1830 Computer principlesJ.C. Maxwell, 1868, I.I.Vishnegradsky, 1877,"Prehistory

Analyzed the stability of Watt’s flyballgovernor, Re froots (G (s))g < 0


E.J. Routh, 1877, A.Hurwitz, 1895

Determining when a characteristic equationhas stable roots. Generalized the results ofMaxwell for linear systems

A.B. Stodola, 1893 Included the delay of the actuating mecha-nism. System time constant

Lyapunov, 1892 Stability of nonlinear differential equationsusing a generalized notion of energy

O. Heaviside, 1892-1898

Operational calculus. The transfer function

P.-S. de Laplace(1749-1827), J. Fourier(1768-1830), A.L.Cauchy (1789-1857)

Frequency domain approach

Wright Brothers, 1903,"Primitive period

Successful test flights

C.R. Darwin Feedback over long time periods is responsi-ble for the evolution of species


E.A. Sperry, 1910Gyroscope

J. GroszkowskiDescribing function approach

H. S. Black, 1927 Apply negative feedback to electrical amplifiersH. Nyquist, 1930s Regeneration theory for the design of stable am-

plifiers. Nyquist stability criterion for feedbacksystems

A. Einstein The motion of systems occurs in such a way asto maximize the time, in 4-D space-time

H.W. Bode, 1938Magnitude and phase frequency response plots.Closed-loop stability using gain and phase margin

N. Minorsky, 1922 Proportional-integral-derivative (PID ) con-troller. Nonlinear effects in the closed-loopsystem

A. Rosenbluethand N. Wiener,1943

Set the basis for cybernetics


plot

9.jpgNyquist plot


V. Volterra, 1931 Explained the balance between two populationsof fish using feedback

H.L. Házen, 1934 Theory of ServomechanismsA.N. Kolmogorov,1941

Theory for discrete-time stationary stochasticprocesses

N. Wiener, 1942 Statistically optimal filter for stationarycontinuous-time signals

A.C. Hall, 1946 Confront noise effects in frequency-domainN.B. Nichols, 1947

Nichols ChartIvachenko, 1948

Relay controlW.R. Evans, 1948

Root locus techniqueJ. R. Ragazzini,1950s Digital control and the z-transform

Tsypkin, 1955 Phase plane for nonlinear controls


J. von Neumann, 1948 Construction of the IAS stored-programSperry Rand, 1950 Commercial data processing machine, UNI-

VAC IR. Bellman, 1957 Dynamic programming to the optimal con-

trol of discrete-time systemsL.S. Pontryagin, 1958 Maximum principleC.S. Draper, 1960,"Classical period#Modern period

Inertial navigation system

V.M. Popov, 1961Circle criterion for nonlinear stability analysis

Kalman, 1960’s Linear quadratic regulator (LQR ). Discreteand continuous Kalman filter . Linear algebraand matrices. Internal system state

G. Zames, 1966, I.W.Sandberg, K.S. Naren-dra, Goldwyn, 1964,C.A. Desoer, 1965

Nonlinear stability


W. Hoff, 1969 MicroprocessorJ.R. Ragazzini, G.Franklin, L.A. Zadeh,C.E. Shannon, 1950’s, E.I.Jury, 1960, B.C. Kuo, 1963

Theory of sampled data systems

Åström, Wittenmark,1984

Industrial process control

Gelb, 1974 Digital filtering theoryH.H. Rosenbrock, 1974,A.G.J. MacFarlane, I.Postlethwaite, 1977

Extend frequency-domain techniques tomultivariable systems. Characteristic lo-cus, diagonal dominance and the inverseNyquist array

I. Horowitz, 1970’s Quantitative feedback theoryJ. Doyle, G. Stein, M.G.Safonov, A.J. Laub, G.L.Hartmann, 1981

Singular value plots in robust multivariabledesign


Class of systems

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

Non-causal oranticipative or predictive f

Causal ornon-anticipative

8>>>>>>>>>><>>>>>>>>>>:

Stochasticwith noise f

Deterministic

8>>>>>><>>>>>>:

Static orwithout memory f

Dynamic

8>><>>:DistributedParameters fLumpedParameters f8>><>>:

Non-linear f

Linear

8<: Discrete fContinuous

�Time varyingTime invariant


Motivation

Robust : some property is preservedUncertainty ∆

x (t0)u (t)

�; y (t)

∆ always exists due to frequency dependent elements,unmodeled dynamics and failures

Zames in 1981 describes ∆ (s) in the frequency domain asclassical control

Infinity norm kargk∞

kargk∞ := supw

σ (arg)

kargk∞ is “good” for specifying the ∆ level and the effect ofkd (t)k2 < ∞ over ky (t)k2


Motivation

Robust : some property is preservedUncertainty ∆

x (t0)u (t)

�; y (t)

∆ always exists due to frequency dependent elements,unmodeled dynamics and failuresZames in 1981 describes ∆ (s) in the frequency domain asclassical control

Infinity norm kargk∞

kargk∞ := supw

σ (arg)

kargk∞ is “good” for specifying the ∆ level and the effect ofkd (t)k2 < ∞ over ky (t)k2


Basic notions

RobustH∞ control objective

r�L�eK(s)

-

w- G (s)z-

- e∆ (s)�

?�1?

Uncertainty and kw (t)k2 attenuationin a bandwidth, over kz (t)k2,guaranteeing stability

Minimize,

J := kz (t)k2 :=�Z ∞

�∞z2 (t) dt

�1/2

For linear time invariant systems, minimizeJ := kTzew (s)k∞ := supew(t):kew(t)k2�1 σ (Tzew), or J := σ (Tzew (s))


Basic notions

Uncertainty models8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

Structured or parametric

Finite number of uncertainty parameters

Ex. diag�k∆1 (s)k∞ , . . . ,

∆q (s)

∞

�diag

�m1, . . . , mq

Non structured

The frequency response is in a set 8w

Ex.: a) phase and gain marginsb) k∆ (s)k∞ � m

u - P(s) -L -yd?�

- ∆a

Figure:


D.C. Youla, H. A.Jabr, J.J. Bongiorno;1976

Explicit formula for the optimal controller basedon a least-square Wiener-Hopf minimization of acost functional

C.A. Desoer, R. Liu,J. Murray, R. Saeks;1980

Controllers placing the feedback system in a ringof operators with the prescribed properties. Theplant is modeled as a ratio of two operators in thatring

M. Vidyasagar, H.Schneider, B.A.Francis, 1982

Necessary and sufficient conditions for a giventransfer function matrix to have a coprime factor-ization . Characterization of all stabilizing compen-sators

C.N. Nett, C.A. Ja-cobson, M. J. Balas;1984

Give explicit formulas for a doubly coprime frac-tional representation of the transfer function instate-space

K. Glover, D. Mc-Farlane, 1989

An optimal stability margin. Characterizationof all controllers satisfying a suboptimal stabil-ity margin, in state-space


Loop-shaping

d (t), usually in w < wl

σ (P∆ (s)) " and any phase of P∆ (s), in w > wh =)worst ∆ (s) inw > wh

σ (arg) gives a measure of the “gain” of ∆ (s) or d (t)

Stabilize the system under ∆ (s) ,

σ (To (s)) # ()_σ (Lo (s)) # , in w > wh

Regulation of y (t) under d (t) ,

_σ(So (s)) # () σ (Lo (s)) " , in w < wl

9>>>>>>>>=>>>>>>>>;Loop-shaping


Loop-shaping

W2 (s) and W1 (s) high and low pass weightings

wl and wh depends onthe specific applicationthe knowledge of d (t) and ∆ (s)the Bode Phase-Gain Relation


Mixed sensitivity

r-L -e K(s) -Ldi?u- P∆ (s) -Ldo

? -� y

?L dm�6�16

Robust stabilityBy the small gain theorem if

e∆ (s) ∞< 1, stability is guaranteed if, W2 (s)Tu∆y∆ (s)

∞ < 1

Robust performancekW1 (s) So (s)k∞ < 1, minK(s) ke (t)k2

So (s) = Tdoy (s) = Ter (s) = (I+ P (s)K (s))�1 : output sensitivity

P∆ (s) : uncertain plant


Mixed sensitivity

Minimize

kW1 (s) So (s)k∞ and W2 (s)Tu∆y∆ (s)

∞

in the frequency range in which kd (t)k2 and k∆ (s)k∞ aresignificative by a stable K (s) designed for P (s), guaranteeing robustperformance and stability, i.e. minimize,

J1 := � W1 (s) So (s)

W2 (s)Tu∆y∆ (s)

� ∞

Uncertainty model Tu∆y∆ (s)Additive K (s) So (s)Multiplicative at the output To (s) := So (s)P (s)K (s)Feedback at the input So (s)P (s)


Standard solutions

General StandardH∞ Optimal Problem[Doyle, 1981], [Glover, 1984], [Francis, Doyle, 1987], [Chiang, Safonov,1997]K (s) stabilizing P (s), and minimizing, J := sup

w:kwk22�k

kz (t)k2,

J = kTzw (s)k∞

[Nett, Jacobson, Balas, 1984]Formula for the YJBK-parametrization,using static state feedback to stabilize P (s)

=) Recursiveprocedures


Parity interlacing property

P (s) 2 Lpxm∞ is strongly stabilizable()

the unstable poles of P (s) between every

even real and unstable zeros of P (s), is even

9=;) 9K (s) 2 RH∞

Strong stability)

8<: For loop breaking

For closed-loop bandwidth "


Problem

J1 is transformed into [Galindo, Malabre, Kucera, 2004]:

J2 := � Sol

Tu∆y∆h

� ∞

R (s) is fixed solving a MSP without an augmented systemSol and Tu∆y∆h becomes real matrices

J2 involves the simultaneous minimization of kSolk∞ and Tu∆y∆h

∞,

minK(s)kSolk∞

subject to kSolk∞ = Tu∆y∆h

∞

that is equivalent to minimize the Lagrange function [Galindo,Herrera, Martínez, 2000],

f := kSolk∞ � η�kSolk∞ �

Tu∆y∆h

∞

�η Lagrange multiplier


Direct solutions

[Galindo, Sanchez, Herrera, 2002]Suppose that det f(s+ a) In � R (s)g is a Hurwitz polynomial, R (s) 2 <H∞Define

X (s) = eX (s) = aIn +A 2 <H∞Y (s) = eY (s) = In 2 <H∞eNp (s) = Np (s) =

1s+ a

In 2 <H∞eDp (s) = Dp (s) =1

s+ a(sIn �A) 2 <H∞

NpD�1p =

�1

s+ a

� �1

s+ a(sIn �A)

��1

XNp + YDp = (aIn +A)�

1s+ a

�+

1s+ a

(sIn �A) = In

eNp (s), Np (s), eDp (s) and Dp (s) are of low order) lesscomputational effort


Direct solutions[Galindo, Sanchez, Herrera, 2002]Then, a proper stabilizing K (s) 2RH∞ is:K (s) = A+ [(s+ a) In � R (s)]�1 [(s+ a) aIn + R (s) s]

and,kSolk∞ = 1

a2 k(aIn � Rl)Ak∞kKhSohk∞ = kA+ aIn + Rhk∞kTohk∞ = 1

whkA+ aIn + Rhk∞

kSohPhk∞ = 1wh

kSolk∞ # by a ", and kTohk∞ # by wh "For P (s) strictly proper Toh

�= LohSelect rIn for Rh and Rl, and r < a,

kSolk∞ =a� r

a2 kAk∞

A solution of kTohk∞ = kSolk∞ for Tu∆y∆h = SohPh is,

re = a�

1� awh kAk∞

�(1)

implying kSolk∞ = 1/wh, and J2 is satisfied increasing wh


Direct Solutions

9 [�a, a] in which kTohk∞ # (") and kSolk∞ " (#) as linear functions ofr [Galindo, Malabre, Kucera, 2004],

-�a

��

��

��

��

6

1whkA+ aInk∞ �

1whkA+ 2aInk∞ �

1a kAk∞�

@@

@@

@@

@@

@@

@@

�a r

kSolk∞

kTohk∞


Direct solutions[Galindo, Malabre, Kucera, 2004]Let rIn be for R (s) 2 <H∞

Then, a value for r is:

r �= a�

1� γmina(wh + 1) kAk∞

�where

γmin = [1+ λmax (YX)]1/2

being Y and X the solutions of the Riccati equationsATX+XA�X2 + In = 0AY+ YAT � Y2 + In = 0

The optimal value for r lies in,r 2 [rb, a]

and a lower bound rb for r is:

rb =a�(wh � a) kAk∞ � a2�

wh kAk∞ + a2

limwh�!0

rb = � (a+ kAk∞), limwh�!∞

rb = a


Let rIn be for R (s) 2 <H∞

Then, an optimal value for r is:

re =b2 � b1

m1 �m2where

b1 :=1akAk∞ , m1 :=

�1a2 kAk∞

b2 :=1

whkA+ aInk∞

m2 :=1

awh(kA+ 2aInk∞ � kA+ aInk∞)

Moreover

kSolk∞ =kA+ 2aInk∞ kAk∞

wh kAk∞ + a (kA+ 2aInk∞ � kA+ aInk∞)


Controllers

x (t) = Ax (t) + Bu (t)Define v1 (t) := Bu (t) ) x (t) = Ax (t) + v1 (t)

+u (t) = BLv1 (t) ) x (t) = Ax (t) + BBLv1 (t)

) E1 := BBL � In

xd-L - K1(s)v1- BL -u

(sIn �A)�1 B -Ld1? -� x

?L d2�6�16

A 2 <n�n, B 2 <n�m, C 2 <p�n

v1 (t) the output of the precompensator K1 (s)BL a left inverse of Bxd (t) the input reference


Controllers

Let�A, BBL, C

�,

u (s) = BLK1 (s) (xd (s)� x (s))

Dual system�

AT, CT,�BBL�T

�,

u (s) =�

CT�L

KT2 (s) (y (s)� by (s))

In original coordinates,

u (s) = K2 (s)CR (y (s)� by (s))x-L ξ- C - CR - K2(s) - BL -v2

(sIn �A)�1 B -�bx6�16

CR a right inverse of Cv2 (t) the output of K2 (s)bx (t) and by (t), the estimated state and output


Controllers

( �bx (t) = Abx (t) + v2 (t)by (t) = Cbx (t) Definebφ (t) := bx (t) )( �bx (t) = Abx (t) + v2 (t)bφ (t) = bx (t)

+bx (t) = CRby (t) )( �bx (t) = Abx (t) + v2 (t)bφ (t) = CRCbx (t)

) E2 := CRC� In


Controllers

Nominal plant P (s)

(A, B, C) a stabilizable and detectable realization of P (s) satisfyingthe parity interlacing property

A =�

A11 A12A21 A22

�B =

�0

B1

�C =

�C eC �

B1 2 <m�m non-singular

For P (s) proper, transform quadruples into and extended triples[Basile, Marro, 1992]


Controllers [Galindo, 2006]

exd-L e- K1(s)v1- BL -�

?

uP(s) -Ld1

? -� y

?L?

d2�6�16

L - (sIn �A)�1 Bbx� - C - �1 -L

�CR�K2(s)v2 �BL

6

bx (t) : estimated statee1 (t) : deviation from the desired state trajectory xd (t)exd (s) := Wr (s) xd (s) : filtered state reference

Satisfies the separation principleAllows to get a stableH∞ compensatorbxss = xss, lim

ri!aixss �! xdss, Toh �! 0

The class of systems depends of the observability in closed loopSome of the poles are fixed. For eC = 0, det (sIn�m �A11) must beHurwitz


Controllers

[Galindo, 2007]

exd-L e1- K1(s)v1-�

?

BL -uP(s) -Ld1

? -� y

?L?

d2�6�16

L- (sIn �A)�1 bx� - C - �1 -L�CRe2�K2(s)

v2

6

A simplified version of the one of [Galindo, 2006]The separation principle is not satisfiedKi (s), become PI as ri ! ai, low complexity controllersThe closed loop poles depends on the selection of the freeparameters of BL and CR, and the rest s = �ai, are stable polesSome of the poles are fixed. For eC = 0, det (sIm �A22) and forC = 0, det (sIn�m �A11) must be Hurwitz


Tuning procedure

Tuning Procedure [Galindo, Malabre, Kucera, 2004]For a desired time response, attenuation of kd1 (t)k2, i.e., for a given a1

1 Select a2 = a1 > 02 Find the largest free parameters of BL and CR, and the lowest wh,

satisfyinga) The stationary state error specifications,b) ri � ai (1� ai), i = 1, 2,c) and minimizing kE1 (ρ1A+ In)k∞ and k(ρ2A+ In)E2k∞

ρi :=�(ai � ri) /a2

i if ri 6= aiwl/a2

i if ri = aiwl a fixed frequency in the low frequency bandwidth of Ki (s)

3 If possible select xd 2 Im B to assure that limri!ai

bxss�! xss

4 If needed, use a pre-filter Wr (s) for the reference


Controllers

[Galindo, 2007]

exd�

- φ(s)?- C -L e- K2(s)CR -L- V(s) -u P(s) -

Ld1? -� y

?L d2�6�16

V (s) := σ (s)BLK1 (s) Γ�1 (s), Γ (s) := In + σ (s)K2 (s)CRC,σ (s) := s+a1�r1

(s+a1)2 , Φ (s) = sIn �A

For P∆ (s), we must satisfy also the Small Gain TheoremTu∆y∆ (s) does not depend on exd (t), indeed Tu∆y∆ (s) becomesK (s) So (s), To (s) := P (s)K (s) So (s), and So (s)P (s), whereK (s) = V (s)K2 (s)CR


Controllers

Assignment of part of the poles by a change of basisLet a change of basis, [Galindo, 2007]

T =�

Iq 0T21 Iq

�, T�1 =

�Iq 0�T21 Iq

�preserving the structure of B, where q � m, and

A =� eA11 eA12eA21 eA22

�be partitioned accordingly with the block partition of T. So,A = TAT�1 =" eA11 � eA12T21 eA12

T21eA11 + eA21 ��

T21eA12 + eA22

�T21 T21eA12 + eA22

#Then,

T21 = eAR12

�eA11 �Λ11

�assigns a desired dynamics Λ11 to A11, and,

T21 =�

Λ22 � eA22

� eAL12

assigns a desired dynamics Λ22 to A22


Mixed sensitivity

[Galindo, 2007] The norm-∞ of Tu∆y∆h is,

kKhSohk∞ = 1wh

BLD1D2CR

∞kTohk∞ = 1

w2h

CBBLD1D2CR

∞

kSohPhk∞ = 1whkCBk∞

Di := A+ (ai + ri) In, i = 1, 2.


Mixed sensitivity

[Galindo, Malabre, Kucera, 2004] lim

s!0limρi!0

So (s)

∞

�= wl

a2ikAk∞

Robust stability is achievedai # , but the performance is amelioratedwh " , but the high frequency bandwidth is decreased

Tuning Procedure1 Look for the highest values of ai, i = 1, 2, satisfying stability

conditions, minimizing Tu∆y∆h

∞ and satisfying plant input

specifications2 Fix the value of the free parameters of BL and CR

3 Select wh, satisfying stability conditions, stationary state errorspecifications and minimizing

Tu∆y∆h

∞


Benchmark of a Mechanical System

-u

m1

7�! x1(t)

kb

m2

7�! x2(t)

-d

Consider a model of a mechanical system,�x (t) = Ax (t) + Bu (t) +Ψd (t)y (t) = Cx (t)

where x (t)T :=�

x1 (t) x2 (t) x2 (t) x1 (t)�,

A =

266640 0 0 10 0 1 0

km2

�km2

�bm2

bm2

�km1

km1

bm1

�bm1

37775 , B =

2664000

1m1

3775 , Ψ =

2664001

m20

3775C =

�0 1 0 0

�



Non-collocated case,The control input acts only on one uncertainty mass 0.1 � m1 � 3and the output is the position of m2The nominal value of m1 = 1

d (t) unknown disturbancek and b the elasticity and friction coefficientsm1 and m2 the massm2 = k = b = 1



Let a := a1 = a2So, r := r1 = r2 which implies K1 (s) = K2 (s)

(A, B, C) is a minimal realization and B has the desired structureP (s) satisfies the parity interlacing propertyeC = 0, det (sIm �A22) = s+ 1 is Hurwitz

A desired dynamics Λ22 =diagf�2, � 2g, and T with q = 2 isrealized, getting,

A =

26641 1 0 11 1 1 0�1 �3 �2 0�3 �1 0 �2

3775 , B = B, C = C

det�sIm �A22

�= s+ 2 is Hurwitz



Select a = 2 and,

BL =�

g1 0 0 1�

CR =�

g2 1 0 g3�T

g1 = 1.6, g2 = 0.3, g3 = �0.55

CB = 0 =) kTohk∞ = 0 and kSohPhk∞ = 0,

wh kKhSohk∞ kKhSohk∞ kKhSohk∞with r with rb with re

1 1.9 stable 0.05 unstable 0.909 unstable3 0.675 stable 0.487 unstable 0.57 unstable5 0.412 unstable 0.35 unstable 0.377 stable10 0.209 unstable 0.195 stable 0.201 stable12 0.175 unstable 0.165 stable 0.169 stable15 0.14 unstable 0.134 stable 0.136 unstable17 0.124 unstable 0.119 stable 0.121 unstable18 0.117 unstable 0.113 stable 0.114 unstable20 0.105 unstable 0.102 unstable 0.103 unstable



Select wh = 3 rad/sec., wh = 17 rad/sec. and wh = 12 rad/sec.=) r = 1.61, rb = 1.622, and re = 1.631

with r with rb with rekE1 (ρ1A+ In)k∞ 2.067 2.052 2.042k(ρ2A+ In)E2k∞ 1.627 1.625 1.623

The characteristic polynomials det (sI�AK) of the overallcompensators are stables

Tol = �CclA�1cl Bcl =) exd (t) = (1/To2l) xd (t)

xd (t) =�

0 yd 0 0�T /2 Im B



yd = 5, under d (t) = 0.1 (sin (10t) + sin (100t))

Plant output y (t)



yd = 5, under d (t) = 0.1 (sin (10t) + sin (100t))

Plant input u (t)



x2 (t) tracks the reference signal with r, rb and re, under the d (t)and the variation of the parameter m1

d (t) remains as very small oscillations at y (t)Sinusoidal functions of frequencies over wh = 3 rad/sec.,wh = 17 rad/sec. and wh = 12 rad/sec. for r, rb and re, are wellattenuated at y (t)Bigger time response with re and less control energy, the contrarywith r, and rb in the middleSmooth control energyAs m1 ", more energy is required, and the peaks and frequencyof the oscillations decrease


Conclusions

1 A methodology to design a mixed sensitivityH∞ compensatorfor a LTI MIMO plant is proposed

2 A nominal compensator is designed for the nominal plantsolving a mixed sensitivityH∞ problem, in a non-conventionalobserver-compensator scheme

3 A mixed sensitivityH∞ control law and necessary and sufficientstability conditions are given

4 Good performance guaranteeing stability, in spite of theuncertainties and of the external disturbances that are attenuated

5 The controllers with r, rb and re have good performance and theirselection depends on the desired time response and the plantinput specifications

6 An analytic or a numerical method replacing the tuningprocedure is still an open problem


Thank you

automatic control and mixed sensitivity hinf control

Engineering