Matlab Robust Control Toolbox

1
Matlab Robust Control Toolbox
2
Purpose

• Increase Enthusiasm for Robust Controls
• Learn how to simulate control algorithms with
  uncertainty
• Increase your knowledge in Robust Controls

3
Outline

• What is the Robust Control Toolbox
• Uncertainty
• Uncertain Elements
• Uncertain Matricies and Systems
• Manipulation of Uncertain Models
• Interconnection of Uncertain Models
• Model Order Reduction
• Robustness
• Robustness and Worst Case Analysis
• Parameter-Dependent Systems
• MIMO Control
• Controller Synthesis
• µ-synthesis
• Sampled Data Systems
• Gain Scheduling
• Supporting Utilities
• LMI
• Specification of Systems of LMIs
• LMI Characteristics

4
What is the Robust Control Toolbox

• A collection of functions and tools that help you
  to analyze and design MIMO control systems with
  uncertain elements
• Ability to build uncertain LTI system models
  containing uncertain parameters and uncertain
  dynamics
• You get tools to analyze MIMO system stability
  margins and worst case performance
• Ability to simplify and reduce the order of
  complex models with model reduction tools that
  minimize additive and multiplicative error bounds
  
• It provides tools for implementing advanced
  robust control methods like H , H2, linear matrix
  inequalities (LMI), and µ-synthesis robust
  control
• You can shape MIMO system frequency responses and
  design uncertainty tolerant controllers

5
Modeling Uncertainty

• Arises when system gains or other parameters are
  not precisely known, or can vary over a given
  range
• Examples of real parameter uncertainties include
  uncertain pole and zero locations and uncertain
  gains
• With the Robust Control Toolbox you can create
  uncertain LTI models as objects specifically
  designed for robust control applications

6
Sources of Uncertainty
Uncertainty in an system can occur in various
forms and from various sources.
Additive Uncertainty
Multiplicative Uncertainty
Feedback Uncertainty
7
Uncertain Elements

1. UComplex() is a function to define complex
   uncertain parameters
2. Ucomplexm() is a function for the creation of
   complex valued uncertain matricies.
3. Udyn() creates an unstructured uncertain dynamic
   system class, with input/output dimension
   specified by iosize.
4. Ultidyn() is a function to create an uncertain
   linear time inveriant object where only bounds on
   the frequency response are known.
5. Ureal() is a function to define real uncertain
   parameters used in various analysis and design
   functions in the robust control toolbox.

8
UComplex(Name,nominal value)
UComplex() is a function to define complex
uncertain parameters

• Name Variable name for the uncertain parameter
• Nominal Value center nominal value for the
  parameter
• radius paramter name to specify a radius of
  uncertainty around the nominal. Radius value
  should follow the parameter specification.
• percentage parameter specifying an
  uncertainty percentage of the nominal for the
  radius. Value should follow the parameter
  specificaiton

EX A ucomplex('A',43j) Uncertain Complex
Parameter Name A, Nominal Value 43i, Radius 1
9
Ucomplexm(Name, nominal value)
Ucomplexm() is a function for the creation of
complex valued uncertain matricies.

• Name Variable name for the uncertain parameter
• Nominal Value center nominal value for the
  parameter
• WL,WR WL and WR are square, invertible, and
  weighting matrices that quantify the size and
  shape of the ball of matrices represented by this
  object. Each parameter name is to be followed by
  the respective matrix. (Assumed to be the
  identity if left out.)
• AutoSimplify parameter to set the type of
  simplification to be done on outputs from the
  uncertain matrix. Setting values are off,
  basic, and full.

EX F ucomplexm('F',1 2 34 5
6,'WL',diag(.1 .3),... 'WR',diag(.4 .8
1.2)) Uncertain Complex Matrix Name F, 2x3
10
n udyn('name',iosize)
Udyn() creates an unstructured uncertain dynamic
system class, with input/output dimension
specified by iosize.

• Input
• Name Variable name for output
• Iosize - input/output dimension specification.
  Input,Output
• Output
• N - an unstructured uncertain dynamic system class

EX N udyn('N',2 3) Uncertain Dynamic
System Name N, size 2x3 size(N) ans 2
3 get(N) Name 'N' NominalValue
2x3 double AutoSimplify 'basic'
11
ultidyn('Name',iosize)
Ultidyn() is a function to create an uncertain
linear time inveriant object where only bounds on
the frequency response are known.

• Parameters Include
• Name variable name
• Iosize the size of input, output
• GainBounded upper bound on the magnitude of
  the response.
• PositiveReal lower bound on the real part
• Bound Specifies the actual bound for the
  frequency response.
• SampleStateDim Specifies the state dimension
  to be used by usample().
• AutoSimplify specifies the amount of result
  simplification.

EX B ultidyn('B',1 1,'Type','PositiveReal','B
ound',2.5) B.SampleStateDim 3
nyquist(usample(B,30)) Uncertain PositiveReal
LTI Dynamics Name B, 1x1, MM' gt 2(2.5)
12
Ureal(Name,Nominal Value)
Ureal() is a function to define real uncertain
parameters used in various analysis and design
functions in the robust control toolbox.

• Parameters Include
• Name variable name
• Nominal Value
• Mode Choose uncertainty type amongst
• Percentage percentage area around the
  nominal.
• Range a given interval with the nominal
  inside.
• PlusMinus add and subtract from nominal
• Autosimplify choose amount of simplification
  for result.

EX c ureal('c',4,'Mode','Range','Percentage',25
) Name 'c' NominalValue 4 Mode
'Range' Range 3 5 PlusMinus -1 1
Percentage -25 25 AutoSimplify
'basic'
13
Uncertain Matricies and Systems

1. Umat() Uncertain matrices
2. Uss() creates uncertain state space objects given
   the uncertain state space system matricies
   a,b,c,d.
3. Ufrd() is a function to create an uncertain
   frequency response model which often arises when
   converting uncertain state space objects to
   frequency response objects.
4. Randatom() generates a 1-by-1 type uncertain
   object.
5. Randumat() Generate random uncertain umat objects
6. Randuss() Generate stable, random uss objects

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h umat(m)

• Uncertain matrices are usually created by
  manipulation of uncertain atoms (ureal, ucomplex,
  ultidyn, etc.), double matrices, and other
  uncertain matrices
• The command umat is rarely used
• There are two situations where it may be useful
• If M is a double, then H umat(M) recasts M as
  an uncertain matrix (umat object) without any
  uncertainties
• In both cases, simplify(H,'class') is the same as
  M

15
uss(a,b,c,d)
Uss() creates uncertain state space objects given
the uncertain state space system matricies
a,b,c,d.

• Parameters Include
• A,B,C,D umat or other uncertain state space
  parameters to describe the system.
• Ts sample time for a discrete time state space
  object.
• RefSys if a refrence uss is given, the
  parameters of that object are included.
• Note if only one matrix is include, the function
  assumes that it represents D and a static gain
  matrix.

EX p1 ureal('p1',5,'Range',2 6) p2
ureal('p2',3,'Plusminus',0.4) A -p1 0p2
-p1 B 0p2 C 1 1 usys
uss(A,B,C,0)
16
ufrd(usys,frequency)
Ufrd() is a function to create an uncertain
frequency response model which often arises when
converting uncertain state space objects to
frequency response objects.

• Parameters Include
• Name uncertain system
• Frequency Range of frequencies for ufrd model.
• Units specify units
• rad/s default
• hz

EX p1 ureal('p1',5,'Range',2 6) p2
ureal('p2',3,'Plusminus',0.4) p3
ultidyn('p3',1 1) Wt makeweight(.15,30,10)
A -p1 0p2 -p1 B 0p2 C 1 1
usys uss(A,B,C,0)(1Wtp3) usysfrd
ufrd(usys,logspace(-2,2,60)) bode(usysfrd,'r',us
ysfrd.NominalValue,'b')
17
A randatom(Type)

• Generate random uncertain atom objects
• generates a 1-by-1 type uncertain object
• Valid values for Type include 'ureal', 'ultidyn',
  'ucomplex', and 'ucomplexm'

Example xr randatom('ureal') Uncertain Real
Parameter Name BMSJA, NominalValue -6.75, Range
-7.70893 -1.89278
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um randumat(ny,nu)

• Generate random uncertain umat objects
• generates an uncertain matrix of size ny-by-nu
• andumat randomly selects from uncertain atoms of
  type 'ureal', 'ultidyn', and 'ucomplex

Example The following statement creates the umat
uncertain object x1 of size 2-by-3. Note that
your result can differ because a random seed is
used x1 randumat(2,3) UMAT 2 Rows, 3 Columns
ROQAW complex, nominal 9.924.84i, radius
0.568, 1 occurrence UEPDY real,
nominal -5.81, variability -1.98681
0.133993, 3 occurrences VVNHL complex,
nominal 5.64-6.13i, radius 1.99, 2
occurrences
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usys randuss(n,p,m)

• Generate stable, random uss objects
• generates an nth order uncertain continuous-time
  system with p outputs and m inputs

Example The statement creates a fifth order,
continuous-time uncertain system s1 of size
2-by-3. Note your display can differ because a
random seed is used. s1 randuss(5,2,3) USS
5 States, 2 Outputs, 3 Inputs, Continuous System
CTPQV 1x1 LTI, max. gain 2.2, 1 occurrence
IGDHN real,
nominal -4.03, variability -3.74667
22.7816, 1 occurrence MLGCD complex, nominal
8.363.09i, /- 7.07, 1 occurrence
OEDJK complex, nominal -0.346-0.296i,
radius 0.895, 1 occurrence
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Manipulation of Uncertain Models

1. isuncertain() Checks whether argument is an
   uncertain class type
2. simplify() performs model-reduction-like
   techniques to detect and eliminate redundant
   copies of uncertain elements
3. usample() substitutes N random samples of the
   uncertain objects in A, returning a certain array
   of size size(A) N
4. usubs() used to substitute a specific value for
   an uncertain element of an uncertain object.
5. gridureal() Grid ureal parameters uniformly over
   their range
6. lftdata() Decompose uncertain objects into fixed
   normalized and fixed uncertain parts
7. Ssbal() yields a system whose input/output and
   uncertain properties are the same as usys, a uss
   object.

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B isuncertain(A)

• Checks whether argument is an uncertain class
  type (checks the class)
• does not actually verify that the input argument
  is truly uncertain
• Returns true if input argument is uncertain,
  false otherwise
• Uncertain classes are umat, ufrd, uss, ureal,
  ultidyn, ucomplex, ucomplexm, and udyn

Example isuncertain(rand(3,4)) ans 0
isuncertain(ureal('p',4)) ans 1
isuncertain(rss(4,3,2)) ans 0
isuncertain(rss(4,3,2)ureal('p1',4) 60 1)
ans 1
22
B simplify(A)

• Simplify representations of uncertain objects
• performs model-reduction-like techniques to
  detect and eliminate redundant copies of
  uncertain elements
• After reduction, any uncertain element which does
  not actually affect the result is deleted from
  the representation

Example Create a simple umat with a single
uncertain real parameter. Select specific
elements, note that result remains in class umat.
Simplify those same elements, and note that class
changes. p1 ureal('p1',3,'Range',2 5) L
2 p1 L(1) UMAT 1 Rows, 1 Columns L(2)
UMAT 1 Rows, 1 Columns p1 real, nominal
3, range 2 5, 1 occurrence simplify(L(1))
ans 2 simplify(L(2)) Uncertain Real
Parameter Name p1, NominalValue 3, Range 2 5
23
B usample(A,N)

• substitutes N random samples of the uncertain
  objects in A, returning a certain array of size
  size(A) N

Example A ureal('A',5) Asample
usample(A,500) size(A) ans 1 1
size(Asample) ans 1 1 500
class(Asample) ans double hist(Asample())
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B usubs(M,StrucArray)

• Substitute given values for uncertain elements of
  uncertain objects

Example p ureal('p',5) m 1 pp2 4
size(m) ans 2 2 m1
usubs(m,'p',5) m1 1 5 25 4
NamesValues.p 5 m2 usubs(m,NamesValues)
m2 1 5 25 4 m1 - m2 ans
0 0 0 0
25
B gridreal(A,N)

• Grid ureal parameters uniformly over their range
• substitutes N uniformly spaced samples of the
  uncertain real parameters in A
• The N samples are generated by uniformly
  gridding each ureal in A across its range

Example gamma ureal('gamma',4) tau
ureal('tau',.5,'Percentage',30) p
tf(gamma,tau 1) KI 1/(2tau.Nominalgamma.No
minal) c tf(KI,1 0) clp
feedback(pc,1) subplot(2,1,1)
step(gridureal(p,20),6) title('Open-loop plant
step responses') subplot(2,1,2)
step(gridureal(clp,20),6)
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M,Delta lftdata(A)

• Decompose uncertain objects into fixed normalized
  and fixed uncertain parts
• partially decompose an uncertain object into an
  uncertain part and a normalized uncertain part
• Uncertain objects (umat, ufrd, uss) are
  represented as certain objects in feedback with
  block-diagonal concatenations of uncertain
  elements
• separates the uncertain object A into a certain
  object M and a normalized uncertain matrix Delta
  such that A is equal to lft(Delta,M), as shown
  below

Example p1 ureal('p1',-3,'perc',40) p2
ucomplex('p2',2) A p1 p1p21 p2
M,Delta lftdata(A) simplify(A-lft(Delta,M))
ans 0 0 0 0 M M
0 0 1.0954 1.0954 0
0 0 1.0000 1.0954 1.0000
-3.0000 -1.0000 0 1.0000
1.0000 2.0000
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usysout ssbal(usys,Wc)
Ssbal() yields a system whose input/output and
uncertain properties are the same as usys, a uss
object.

• Input
• Usys - a continuous-time uncertain system
• Wc - the critical frequency for the bilinear
  prewarp transformation from continuous time to
  discrete time. (Default is 1)
• Output
• Usysout - a system whose input/output and
  uncertain properties are the same as usys, a uss
  object. (The numerical conditioning of usysout is
  usually better than that of usys.)

EX p2ureal('p2',-17,'Range',-19 -11)
p1ureal('p1',3.2,'Percentage',0.43) A -12
p1.001 p2 B 120 -809503 24 C .034
.0076 .00019 2 usys ss(A,B,C,zeros(2,2)) US
S 2 States, 2 Outputs, 2 Inputs, Continuous
System p1 real, nominal 3.2, variability
-0.43 0.43, 1 occurrence p2 real, nominal
-17, range -19 -11, 1 occurrence usysout
ssbal(usys) USS 2 States, 2 Outputs, 2
Inputs, Continuous System p1 real, nominal
3.2, variability -0.43 0.43, 1 occurrence
p2 real, nominal -17, range -19 -11, 1
occurrence
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ndist actual2normalized(A,V)

• the normalized distance between the nominal
  value of the uncertain atom A and the given value
  V
• If V is an array of values, then ndist is an
  array of normalized distances

29
avalue normalizedactual2(A,NV)

• Convert the value for an atom in normalized
  coordinates to the corresponding actual value
• converts the value for atom in normalized
  coordinates, NV, to the corresponding actual value

Example a ureal('a',3,'range',1 5)
actual2normalized(a,1 3 5) ans -1.0000
-0.0000 1.0000 normalized2actual(a,-1 1)
ans 1.0000 5.0000 normalized2actual(a,
-1.5 1.5) ans 0.0000 6.0000
30
umatout stack(arraydim,umat1,umat2,...)

• Construct an array by stacking uncertain
  matrices, models, or arrays along array
  dimensions of an uncertain array
• All models must have the same number of columns
  and rows

Example Consider usys1 and usys2, two
single-input/single-output uss models zeta
ureal('zeta',1,'Range',0.4 4) wn
ureal('wn',0.5,'Range',0.3 0.7) P1 tf(1,1
2zetawn wn2) P2 tf(zeta,1 10) You
can stack along the first dimension to produce a
2-by-1 uss array. stack(1,P1,P1) USS 2
States, 1 Output, 1 Input, Continuous System
array, 2 x 1 wn real, nominal 0.5,
range 0.3 0.7, 3 occurrences zeta real,
nominal 1, range 0.4 4, 1 occurrence
31
B squeeze(A)

• Remove singleton dimensions for umat objects
• returns an array B with the same elements as A
  but with all the singleton dimensions removed
• A singleton is a dimension such that
  size(A,dim)1
• 2-D arrays are unaffected by squeeze so that row
  vectors remain rows

32
Interconnection of Uncertain Models

1. Imp2exp() transforms a linear constraint between
   variables into an explicit input/output
   relationship
2. Sysic Builds interconnections of certain and
   uncertain matrices and systems
3. Iconnect Builds complex interconnections of
   uncertain matrices and systems
4. Icsignal Specifies signal constraints described
   by the interconnection of components

33
B imp2exp(A,yidx,uidx)

• transforms a linear constraint between variables
  Y and U of the form A(,yidxuidx)YU 0
  into an explicit input/output relationship Y
  BU
• The vectors yidx and uidx refer to the columns
  (inputs) of A as referenced by the explicit
  relationship for B
• The constraint matrix A can be a double, ss, tf,
  zpk and frd object as well as an uncertain
  object, including umat, uss and ufrd
• The result B will be of the same class

34
sysout sysic

• Build interconnections of certain and uncertain
  matrices and systems
• requires that 3 variables with fixed names be
  present in the calling workspace systemnames,
  inputvar and outputvar
• systemnames is a char containing the names of the
  subsystems that make up the interconnection
• nputvar is a char, defining the names of the
  external inputs to the interconnection
• outputvar is a char, describing the outputs of
  the interconnection
• Also requires that for every subsystem name
  listed in systemnames, a corresponding variable,
  input_to_ListedSubSystemName must exist in the
  calling workspace
• This variable is similar to outputvar - it
  defines the input signals to this particular
  subsystem as linear combinations of individual
  subsystem's outputs and external inputs

35
H iconnect

• Create empty iconnect (interconnection) objects
• Interconnection objects (class iconnect) are an
  alternative to sysic, and are used to build
  complex interconnections of uncertain matrices
  and systems
• An iconnect object has 3 fields to be set by the
  user, Input, Output and Equation
• Input and Output are icsignal objects, while
  Equation is a cell-array of equality constraints
  (using equate) on icsignal objects
• the System property is the input/output model,
  implied by the constraints in Equation relating
  the variables defined in Input and Output

36
v icsignal(n,'name')

• Create an icsignal object
• creates an icsignal object of length n, which is
  a symbolic column vector
• used with iconnect objects to specify signal
  constraints described by the interconnection of
  components
• internal name identifiers given by the character
  string argument name

37
Model Order Reduction and Application

• Complex models are not always required for good
  control
• Optimization methods generally tend to produce
  controllers with at least as many states as the
  plant model
• The Robust Control Toolbox offers you an
  assortment of model-order reduction commands
• Less complex low-order approximations to plant
  and controller models

38
Model Order Reduction Tools
The Robust Control Toolbox has various controller
synthesis tools outlined below

1. reduce() returns a reduced order model
2. balancmr() Balanced model truncation via square
   root method
3. bstmr() Balanced stochastic model truncation
   (BST) via Schur method
4. hankelmr() Hankel Minimum Degree Approximation
   (MDA) without balancing
5. hankelsv() Computes Hankel singular values for
   stable/unstable or continuous/discrete system

39
Model Order Reduction Tools cont

1. modreal() returns a set of state-space LTI
   objects in modal form
2. ncfmr() Balanced model truncation for normalized
   coprime factors.
3. shurmr() Balanced model truncation via Schur
   method
4. slowfast() computes the slow and fast modes
   decompositions of a system
5. stabproj() computes the stable and antistable
   projections of a minimal realization
6. imp2ss() produces an approximate state-space
   realization of a given impulse response

40
GRED reduce(G,order)

• Returns a reduced order model of G
• Groups all the Hankel SV based model reduction
  routines
• Hankel singular values of a stable system
  indicate the respective state energy of the
  system.
• In many cases, the additive error method
  GREDreduce(G,ORDER) is adequate to provide a
  good reduced order model.
• for systems with lightly damped poles and/or
  zeros, a multiplicative error method (namely,
  GREDreduce(G,ORDER,'ErrorType','mult')) that
  minimizes the relative error between G and GRED
  tends to produce a better fit.

41
GRED balancmr(G,order)

• balanced truncation model reduction for
  continuous/discrete stable/unstable plant.
• With only one input argument G, the function will
  show a Hankel singular value plot of the original
  model and prompt for model order number to reduce.

42
GRED bstmr(G,order)

• Balanced stochastic truncation (BST) model
  reduction for continuous/discrete
  stable/unstable plant.
• bstmr returns a reduced order model GRED of G
• With only one input argument G, the function will
  show a Hankel singular value plot of the phase
  matrix of G and prompt for model order number to
  reduce
• This method guarantees an error bound on the
  infinity norm of the multiplicative
  GRED-1(G-GRED) or relative error
  G-1(G-GRED) for well-conditioned model
  reduction problems

43
GRED hankelmr(G,order)

• Optimal Hankel norm approximation for
  continuous/discrete stable/unstable plant
• hankelmr returns a reduced order model GRED of G
• With only one input argument G, the function will
  show a Hankel singular value plot of the original
  model and prompt for model order number to reduce
• This method guarantees an error bound on the
  infinity norm of the additive error G-GRED
  for well-conditioned model reduced problems

44
hankelsv(G)

• Compute Hankel singular values for
  stable/unstable or continuous/discrete system
• draws a bar graph of the Hankel singular values
  such as the following

45
G1,G2 modreal(G,cut)

• State-space modal truncation/realization
• returns a set of state-space LTI objects G1 and
  G2 in modal form given a state-space G and the
  model size of G1, cut
• The modal form realization has its A matrix in
  block diagonal form with either 1x1 or 2x2
  blocks. The real eigenvalues will be put in 1x1
  blocks and complex eigenvalues will be put in 2x2
  blocks. These diagonal blocks are ordered in
  ascending order based on eigenvalue magnitudes
• G can be stable or unstable

46
GRED ncfmr(G,order)

• balanced truncation model reduction for
  normalized coprime factors of continuous/discrete
  stable/unstable G
• ncfmr returns a reduced order model GRED formed
  by a set of balanced normalized coprime factors
• With only one input argument G, the function will
  show a Hankel singular value plot of the original
  model and prompt for model order number to reduce
• The left and right normalized coprime factors are
  defined as

47
GRED schurmr(G,order)

• Schur balanced truncation model reduction for
  continuous/discrete stable/unstable plant
• schurmr returns a reduced order model GRED of G
• With only one input argument G, the function will
  show a Hankel singular value plot of the original
  model and prompt for model order number to reduce
• This method guarantees an error bound on the
  infinity norm of the additive error G-GRED
  for well-conditioned model reduced problems

48
G1,G2 slowfast(G,cut)

• State-space slow-fast decomposition
• computes the slow and fast modes decompositions
  of a system G(s) such that

49
G1,G2,m stabproj(G)

• State-space stable/anti-stable decomposition
• stabproj computes the stable and antistable
  projections of a minimal realization G(s) such
  that

50
a,b,c,d,totbnd,hsv imp2ss(y,ts,nu,ny,tol)

• System identification via impulse response
• The function imp2ss produces an approximate
  state-space realization of a given impulse
  response impmksys(y,t,nu,ny,'imp')
• A continuous-time realization is computed via the
  inverse Tustin transform if t is positive
  otherwise a discrete-time realization is returned
• In the SISO case the variable y is the impulse
  response vector in the MIMO case y is an
  N1-column matrix containing N 1 time samples
  of the matrix-valued impulse response H0, ..., HN
  of an nu-input, ny-output system stored row-wise.
• The variable tol bounds the H norm of the error
  between the approximate realization (a, b, c, d)
  and an exact realization of y
• The inputs ts, nu, ny, tol are optional if not
  present they default to the values ts 0, nu
  1, ny (number of rows of y)/nu, tol
  1.01sigmabar1

51
Robustness and Worst Case Analysis

• To be robust, your control system should meet
  your stability and performance requirements for
  all possible values of uncertain parameters
• Monte Carlo parameter sampling via usample can be
  used for this purpose, but Monte Carlo methods
  are inherently hit or miss
• The Robust Control Toolbox gives you a powerful
  assortment of robustness analysis commands that
  let you directly calculate upper and lower bounds
  on worst-case performance without random
  sampling

52
Performance Analysis
The hyperbola is used to define the performance
margin. Systems whose performance degradation
curve intersects high on the hyperbola curve
represent "non-robustly-performing systems" in
that very small deviations of the uncertain
elements from their nominal values can result in
very large system gains. Conversely, an
intersection low on the hyperbola represent
"robustly-performing systems."
53
Small Gain Theorem
When the plant modeling uncertainty is not too
big, you can design high-gain, high-performance
feedback controllers. High loop gains
significantly larger than 1 in magnitude can
attenuate the effects of plant model uncertainty
and reduce the overall sensitivity of the system
to plant noise. But if your plant model
uncertainty is so large that you do not even know
the sign of your plant gain, then you cannot use
large feedback gains without the risk that the
system will become unstable. Thus, plant model
uncertainty can be a fundamental limiting factor
in determining what can be achieved with
feedback. The Small Gain Theorem asserts that the
overall system gain must therefore be kept below
1 to assure stability.
54
Robustness and Worst Case Performance Tools
The Robust Control Toolbox has various controller
synthesis tools outlined below

1. cpmargin() calculates the normalized coprime
   factor/gap metric robust stability of the
   multivariable feedback loop consisting of C in
   negative feedback with P.
2. Gapmetric() calculates upper bounds on the gap
   and nugap metric between systems
3. loopmargin() analyzes the multivariable feedback
   loop consisting of the loop transfer matrix.
4. loopsens() Creates a Struct, loops, whose fields
   contain the multivariable sensitivity,
   complementary and open-loop transfer functions.
5. mussv() calculates upper and lower bounds on the
   structured singular value, or µ, for a given
   block structure.
6. Mussvextract() is used to extract the compressed
   information within muinfo into a readable form.

55
Robustness and Worst Case Performance Tools cont

• ncfmargin() Calculates the normalized coprime
  stability margin of the plant-controller feedback
  loop
• popov() uses the Popov criterion to test the
  robust stability of dynamical systems with
  possibly nonlinear and/or time-varying
  uncertainty.
• robustperf() Calculates the robust performance
  margin of an uncertain multivariable system.
• robuststab() Calculate robust stability margins
  of an uncertain multivariable system.
• robopt() Creates an options object for use with
  robuststab and robustperf
• wcnorm() Calculate the worst-case norm of an
  uncertain matrix

56
Robustness and Worst Case Performance Tools cont

1. wcgain() Calculates bounds on the worst-case gain
   of an uncertain system
2. wcgopt() creates a wcgain, wcsens and wcmargin
   options object called options in which specified
   properties have specific values.
3. wcmargin() calculates the combined worst-case
   input and output loop-at-a-time gain/phase
   margins of the feedback loop
4. wcsens() Calculate the worst-case sensitivity and
   complementary sensitivity functions of a
   plant-controller feedback loop

57
MARG,FREQ cpmargin(P,C,TOL)
Cpmargin() calculates the normalized coprime
factor/gap metric robust stability of the
multivariable feedback loop consisting of C in
negative feedback with P.

• Input Parameters
• P LTI plant
• C only feedback controller
• Tol - specifies a relative accuracy for
  calculating the normalized coprime factor/gap
  metric robust stability margin

• Output Parameters
• Marg - contains upperand lower bound for the
  normalized coprime factor/gap metric robust
  stability margin
• Freq - the frequency associated with the upper
  bound

58
gap,nugap gapmetric(p0,p1,tol)
Gapmetric() calculates upper bounds on the gap
and nugap metric between systems p0 and p1.
EX p1 tf(1,1 -0.001) p2 tf(1,1 0.001)
g,ng gapmetric(p1,p2) K 1 H1
loopsens(p1,K) H2 loopsens(p2,K)
subplot(2,2,1) bode(H1.Si,'-',H2.Si,'--')
subplot(2,2,2) bode(H1.Ti,'-',H2.Ti,'--')
subplot(2,2,3) bode(H1.PSi,'-',H2.PSi,'--')
subplot(2,2,4) bode(H1.CSo,'-',H2.CSo,'--') g
0.0029 ng 0.0020
Output Gap gap metric value Nugap nugap
metric value (For gap or nugap, a value close to
0 means the plants are far apart and a value
close to 1 means they are close together.)

• Input
• P0,P1 plants of the same size
• Tol tolerance for the gap metric (default is
  .001)

59
loopmargin(L)

• SM, DM, MM Loopmargin() is a tool to
  determine the closed loop margins for an
  uncertain system.
• Outputs include
• SM A structure containing the nominal system
  margins.
• DM also called disk margin. Describes the
  disk of deviations from the nominal system.
• MM establishes guaranteed bounds on the system
  with variations from all input channels.

60
loopsens(P,C)
Loopsens() creates a structure which contains the
multivariable sensitivity, complementary and
open-loop transfer functions.

• Structure Output Fields
• Poles - Closed-loop poles. NaN for frd/ufrd
  objects.
• Stable - 1 if nominal closed loop is stable, 0
  otherwise. NaN for frd/ufrd objects.
• Si - Input-to-plant sensitivity function
• Ti - Input-to-plant complementary sensitivity
  function
• Li - Input-to-plant loop transfer function
• So - Output-to-plant sensitivity function
• To - Output-to-plant complementary sensitivity
  function
• Lo - Output-to-plant loop transfer function
• PSi - Plant times input-to-plant sensitivity
  function
• CSo - Compensator times output-to-plant
  sensitivity function

EX int tf(1,1 0) addunc
ultidyn('addunc',1 1,'Bound',0.2) uncint
int addunc int3 blkdiag(int,uncint,int) p
1 .2 .5-.5 0 11 0 00 1 0int3 gain
ureal('gain',0.9, 'Range',-3 1.5) c .02 .3
0 01 -.3 .32 gain 0 1 -.2 .1 loops
loopsens(p,c)
61
bounds,muinfo mussv(M,BlockStructure)
Mussv() calculates upper and lower bounds on the
structured singular value, or µ, for a given
block structure.
EX M randn(5,5) sqrt(-1)randn(5,5) F
randn(2,5) sqrt(-1)randn(2,5) BlockStructure
-1 0-1 01 12 0 ubound,Q
mussv(M,F,BlockStructure) points completed (of
1) ... 1 bounds mussv(M,BlockStructure)
optbounds mussv(MQF,BlockStructure,'C5') o
ptbounds(1) ubound 1.5917 1.5925
bounds(1) bounds(2) 3.8184 3.7135

• Input
• M - is a double or a frd object
• BlockStructure - a matrix specifying the
  perturbation block structure consisting of 2 rows
  and as many columns as uncertainty blocks.
• Output
• Bounds upper and lower bounds on the mu value
  of the system.
• Muinfo a structure containing additional
  information

62
VDelta,VSigma,VLmi mussvextract(muinfo)
Mussvextract() is used to extract the compressed
information within muinfo into a readable form.

• Input
• Muinfo a variable returned by mussv() with
  information
• Ouput
• VDelta used to verify the lower bound
• VSigma a structure of information used to
  verify the Newlin/Young upper bound
• VLmi a structure of information used to verify
  the LMI upper bound

EX M randn(4,4) sqrt(-1)randn(4,4)
BlockStructure 1 11 12 2 bounds,muinfo
mussv(M,BlockStructure) VDelta,VSigma,VLmi
mussvextract(muinfo)
63
marg,freq ncfmargin(P,C,tol)
Ncfmargin() calculates the normalized coprime
factor/gap metric robust stability margin b(P,
C), marg, of the multivariable feedback loop
consisting of C in negative feedback with P.

• Input
• P LTI plant model
• C Compensator in the feedback path
• Tol tolerance for the ncfmargin must be
  between .00001 and .01
• Output
• Marg - the normalized coprime factor/gap metric
  robust stability margin b(P, C)
• Freq destabilizing frequency

EX x tf(4,1 0.001) clp1
feedback(x,1) clp2 feedback(x,10) marg1,freq1
ncfmargin(x,1) marg2,freq2
ncfmargin(x,10) marg1 0.7071 freq1
Inf marg2 0.0995 freq2 Inf
64
t,P,S,N popov(sys,delta,flag)

• Perform the Popov robust stability test
• uses the Popov criterion to test the robust
  stability of dynamical systems with possibly
  nonlinear and/or time-varying uncertainty
• The uncertain system must be described as the
  interconnection of a nominal LTI system sys and
  some uncertainty delta

65
perfmarg,perfmargunc,Report robustperf(sys)
Robustperf() is a tool to determine the robust
performance margin which sets bounds on the
robustness of a nominally stable system to given
uncertainty.
EX P tf(1,1 0) ultidyn('delta',1
1,'bound',0.4) BW 0.8 K tf(BW,1/(25BW)
1) S feedback(1,PK) perfmargin,punc
robustperf(S) perfmargin
UpperBound 7.4305e-001 LowerBound
7.4305e-001 CriticalFrequency 5.3096e000

• Structure Output Fields
• Perfmarg performance margin lower bound, upper
  bound, and critical frequency.
• Perfmargunc structure of values of critical
  uncertain elements of the system.
• Report A text description of the robustness
  analysis.

Note Output may not contain true critical
frequency because of the assigned frequency range
over which the function operates. For
limitations, see the function definition in the
matlab help.
66
stabmarg,destabunc,Report robuststab(sys)
Robuststab() is used to determine the robust
stability margin for a nominally stable uncertain
system up to the closest instability from the
stable nominal.
EX P tf(4,1 .8 4) delta
ultidyn('delta',1 1,'SampleStateDim',5) Pu
P 0.25tf(1,.15 1)delta C tf(1 1,.1
1) tf(2,1 0) S feedback(1,PuC)
stabmarg,destabunc,report,info
robuststab(S) report Uncertain System is NOT
robustly stable to modeled uncertainty.
-- It can tolerate up to 81.8 of
modeled uncertainty.
-- A destabilizing combination of 81.8 the
modeled uncertainty exists, causing an
instability at 9.13 rad/s.
-- Sensitivity with respect
to uncertain element ...
'delta' is 100. Increasing 'delta' by
25 leads to a 25 decrease in the margin.

• Structure Output Fields
• Stabmarg robust stability margin lower bound,
  upper bound, destabilizing frequency.
• Destabunc structure of uncertain values closest
  to the nominal that cause instability.
• Report text description of the stability
  analysis

Note Output may not contain all unstable
frequencies because of the assigned frequency
range over which the function operates. For
limitations, see the function definition in the
matlab help.
67
opts robopt('name1',value1,'name2',value2,...)
Robopt() creates an options object for use with
robuststab and robustperf.
EX opt robopt Property Object Values
Display 'off' Sensitivity 'on'
VaryUncertainty 25 Mussv 'sm9'
Default 1x1 struct
Meaning 1x1 struct
68
maxnorm,wcu wcnorm(m)
Wcnorm() determines the maximum norm over all
allowable values of the uncertain elements and is
referred to as a worst-case norm analysis.
EX aureal('a',5,'Range',4 6)
bureal('b',2,'Range',1 3) bureal('b',3,'Ran
ge',2 10) cureal('c',9,'Range',8 11)
dureal('d',1,'Range',0 2) M a bc d
Mi inv(M) maxnormM wcnorm(M) maxnormM
LowerBound 14.7199 UpperBound
14.7327 maxnormMi wcnorm(Mi) maxnormMi
LowerBound 2.5963 UpperBound 2.5979

• Input
• M - a umat or a uss
• Output
• Maxnorm a structure with fields
• Wcu - a structure that includes values of
  uncertain elements and maximizes the matrix norm

69
maxgain,maxgainunc wcgain(sys)
Wcgain() calculates the worst case gain frequency
response of a given system due to uncertain
elements. Pointwise-in-frequency analysis
creates a response curve for the system.
EX P tf(1,1 0) ultidyn('delta',1
1,'bound',0.4) BW1 0.8 K1
tf(BW1,1/(25BW1) 1) S1 feedback(1,PK1)
BW2 2.0 K2 tf(BW2,1/(25BW2) 1) S2
feedback(1,PK2) maxgain1,wcunc1
wcgain(S1) maxgain1 LowerBound
1.5070e000 UpperBound 1.5080e000
CriticalFrequency 5.3096e000

• Structure Output Fields
• Maxgain maximum gain lower bound, upper bound,
  critical frequency.
• Maxgainunc structure containing uncertain
  elements that maximize the gain.
• Options set in wcopt()
• Pointwise-in-frequency- creates a response
  curve for the system.
• Peak-over-frequency computes the largest gain
  over all freqencies.

Note Output may not contain all unstable
frequencies because of the assigned frequency
range over which the function operates. For
limitations, see the function definition in the
matlab help.
70
options wcgopt('name1',value1,'name2',value2,...
)
Wcgopt() creates a wcgain, wcsens and wcmargin
options object called options in which specified
properties have specific values.
EX opt wcgopt('MaxTime',10000,'Sensitivity','of
f') Property Object Values Sensitivity
'off' LowerBoundOnly 'off'
FreqPtWise 0 ArrayDimPtWise
VaryUncertainty 25 Default 1x1
struct Meaning 1x1 struct
AbsTol 0.0200 RelTol 0.0500
MGoodThreshold 1.0400 AGoodThreshold
0.0500 MBadThreshold 20
ABadThreshold 5 NTimes 2
MaxCnt 3 MaxTime 10000
71
wcmargi,wcmargo wcmargin(L)
Wcmargin() describes a disk within with the
nominal gain margin and phase margin are stable
for an uncertain system.
EX a 0 10-10 0 b eye(2) c 1 8-10
1 d zeros(2,2) G ss(a,b,c,d) K 1
-20 1 ingain1 ureal('ingain1',1,'Range',0.97
1.06) b ingain1 00 1 Gunc
ss(a,b,c,d) unmod ultidyn('unmod',2
2,'Bound',0.08) Gmod (eye(2)unmod)Gunc
Gmodg ufrd(Gmod,logspace(-1,3,60)) wcmi,wcmo
wcmargin(Gmodg,K) wcmi(1)
GainMargin 0.3613 2.7681 PhaseMargin
-50.2745 50.2745 Frequency 0.1000
Sensitivity 1x1 struct

• Output Structure Fields
• Wcmargi contains gain margin bound, phase
  margin in degrees, frequency associate with worst
  case, and a structure containing the sensitivity
  to all uncertain elements.
• Wcmargo same structure as wcmargi but gives
  single loop values for each.

72
wcst wcsens(L)
Wcsens() determines the worst case sensitivity of
a system to various input and disturbance
parameters in an uncertain system L.

• Output Structure Fields
• Si - Worst-case input-to-plant sensitivity
  function
• Ti - Worst-case input-to-plant complementary
  sensitivity function
• So - Worst-case output-to-plant sensitivity
  function
• To - Worst-case output-to-plant complementary
  sensitivity function
• PSi - Worst-case plant times input-to-plant
  sensitivity function
• CSo - Worst-case compensator times
  output-to-plant sensitivity function
• Stable - 1 if nominal closed loop is stable, 0
  otherwise. NaN for frd/ufrd objects.

EX delta ultidyn('delta',1 1) tau
ureal('tau',5,'range',4 6) P tf(1,tau
1)(10.25delta) Ctf(4 4,1 0)
looptransfer loopsens(P,C) Snom
looptransfer.Si.NominalValue wcst wcsens(P,C)
Swc wcst.Si.BadSystem omega
logspace(-1,1,50) bodemag(Snom,'-',Swc,'-.',omeg
a) legend('Nominal Sensitivity','Worst-Case
Sensitivity',... 'Location','SouthEast')
73
Robust Analysis for Parameter Dependent Systems

1. psys() specifies state-space models where the
   state-space matrices can be uncertain,
   time-varying, or parameter-dependent.
2. psinfo() a multi-usage function for queries about
   a polytopic or parameter-dependent system
3. ispsys() returns 1 if sys is a polytopic or
   parameter-dependent system
4. pvec() used in conjunction with psys to specify
   parameter-dependent systems.
5. pvinfo() retrieves information about a vector of
   real parameters declared with pvec and stored in
   pv
6. polydec() takes an uncertain parameter vector PV
   taking values ranging in a box, and returns the
   corners or vertices of the box as columns of the
   matrix vertx

74
Robust Analysis for Parameter Dependent Systems
cont.

1. aff2pol() Derives a polytopic representation
   polsys of the affine parameter- dependent system.
   
2. quadstab() seeks a fixed Lyapunov function that
   establishes quadratic stability
3. quadperf() Compute the quadratic H-inf
   performance of a polytopic or parameter-dependent
   system
4. pdlstab() uses parameter-dependent Lyapunov
   functions to establish the stability of uncertain
   state-space models over some parameter range or
   polytope of systems.
5. decay() returns the quadratic decay rate drate
6. pdsimul() simulates the time response of an
   affine parameter-dependent system

75
pols psys(syslist) affs psys(pv,syslist)

• Specify a parameter-dependent system
• psys specifies state-space models where the
  state-space matrices can be uncertain,
  time-varying, or parameter-dependent.

• a polytopic model with vertex systems S1, . . .,
  S4 is created by pols psys(s1,s2,s3,s4)
  while an affine parameter-dependent model with 4
  real parameters is defined by
  affs psys(pv,s0,s1,s2,s3,s4)
• The output is a structured matrix storing all the
  relevant information

76
type,k,ns,ni,no psinfo(ps)

• Query characteristics of a P-system
• displays the type of system (affine or polytopic)
• the number k of SYSTEM matrices involved in its
  definition
• the numbers of ns, ni, no of states, inputs, and
  outputs of the system

77
bool ispsys(sys)

• True for parameter-dependent systems
• returns 1 if sys is a polytopic or
  parameter-dependent system

78
pv pvec('box',range,rates) pv
pvec('pol',vertices)

• Specify a vector of uncertain or time-varying
  parameters used in conjunction with psys to
  specify parameter-dependent systems
• The type 'box' corresponds to independent
  parameters ranging in intervals
  
  (extremal values)
• The parameter vector p then takes values in a
  hyperrectangle of Rn called the parameter box
• range is an n-by-2 matrix that stacks up the
  extremal values
• type 'pol' corresponds to parameter vectors p
  ranging in a polytope of the parameter space Rn
• efined by a set of vertices V1, . . ., Vn
  corresponding to "extremal" values of the vector
  p
• second argument is the concatenation of the
  vectors v1,...,vn

79
typ,k,nv pvinfo(pv)

• pvinfo(pv) displays the type of parameter vector
  ('box' or 'pol'), the number n of scalar
  parameters, and for the type 'pol', the number of
  vertices used to specify the parameter range

80
vertx polydec(PV)

• Compute polytopic coordinates wrt. box corners
• takes an uncertain parameter vector PV taking
  values ranging in a box, and returns the corners
  or vertices of the box as columns of the matrix
  vertx

81
polsys aff2pol(affsys)

• Convert affine P-systems to polytopic
  representation
• derives a polytopic representation polsys of the
  affine parameter- dependent system

• Where p (p1, . . ., pn) is a vector of
  uncertain or time- varying real parameters
  taking values in a box or a polytope. The
  description affsys of this system should be
  specified with psys

82
tau,P quadstab(ps,options)

• Assess quadratic stability of parameter-dependent
  system
• Quadratic stability of polytopic or affine
  parameter-dependent systems

83
perf,P quadperf(ps,g,options)

• Assess quadratic Hinf performance of P-systems

84
tau,Q0,Q1,... pdlstab(pds,options)

• Assess the robust stability of a polytopic or
  parameter-dependent system
• uses parameter-dependent Lyapunov functions to
  establish the stability of uncertain state-space
  models over some parameter range or polytope of
  systems

85
drate,P decay(ps,options)

• Compute quadratic decay rate

86
pdsimul(pds,'traj',tf,'ut',xi,options)

• Time response of a parameter-dependent system
  along a given parameter trajectory along a
  parameter trajectory p(t) and for an input signal
  u(t)
• The parameter trajectory and input signals are
  specified by two time functions ptraj(t) and
  uut(t)
• The affine system pds is specified with psys but
  can also use the polytopic representation of such
  systems as returned by aff2pol(pds) or hinfgs
• The final time and initial state vector can be
  reset through tf and xi
• options gives access to the parameters
  controlling the ODE integration

87
Robust Controller Synthesis Tools
The Robust Control Toolbox has various controller
synthesis tools outlined below

1. Augw() computes a state-space model of an
   augmented LTI plant with weighting functions
2. Hinfmix() creates a suboptimal controller K(s)
   based on the optimization of a mixed H2/ H8
   criterion.
3. H2syn() computes a stabilizing LTI/SS controller
   K for a partitioned plant P(s).
4. Hinfsyn() creates an optimal H-inifinity
   controller for a partitioned plant matrix P.
5. Sdhinfsyn() synthesizes a sampled data H-infinity
   controller.
6. Loopsyn() is an H infinity optimal way for
   loopshaping controller synthisis.

88
Robust Controller Synthesis Tools cont

1. Ltrsyn() computes a reconstructed-state
   output-feedback controller K for LTI plant G
   using Kalman filtering to asymptotically recover
   the lost states.
2. Mixsyn() is H-infinity mixed-sensitivity
   synthesis method for robust control loopshaping
   design.
3. Ncfsyn() is a normalized coprime factorization
   based parameterized controller synthesis using
   the Glover-McFarlane method for loopshaping.
4. Mkfilter() returns a single-input, single-output
   analog low pass filter sys as an ss object.

89
P AUGW(G,W1,W2,W3)
Augw() computes a state-space model of an
augmented LTI plant P(s) with weighting functions
W1(s), W2(s), and W3(s) penalizing the error
signal, control signal and output signal
respectively.

• Input
• G Nominal Plant Model
• W1 weight for the error signal
• W2 weight for the control signal
• W3 weight for the output signal
• Output
• P augmented plant model

EX szpk('s') G(s-1)/(s1) W10.1(s100)/(100
s1) W20.1 W3 Paugw(G,W1,W2,W3) K,CL,GA
Mhinfsyn(P) K2,CL2,GAM2h2syn(P) LGK
Sinv(1L) T1-S sigma(S,'k',GAM/W1,'k-.',T,'r'
,GAMG/W2,'r-.')
90
gopt,h2opt,K,R,S hinfmix(P,r,obj,region,dkbnd,
tol)
Hinfmix() creates a suboptimal controller K(s)
based on the optimization of a mixed H2/ H8
criterion that optimizes the following function
subject to T lt
gamma, T22 lt v. The closed-loop poles lie in
some prescribed LMI region D.

• Input Parameters
• P lmi object such as ss or tf.
• R z2, y, u
• Obj gamma, v, alpha, beta
• Region identifies the pole placement region.
  Default is the open LHP.
• Dkbnd set bound on norm of controller feed
  through matrix DK.
• Tol tolerance for the trade-off parameter

• Output Parameters
• Gopt - guaranteed Hinf performance
• H2opt guaranteed H2 performance
• K Controller K(s) coefficients
• R,S optimal values of the LMI objects.

91
K,CL,GAM,INFOH2SYN(P,NMEAS,NCON)
H2syn() computes a stabilizing LTI/SS controller
K for a partitioned plant P(s).

• Input Parameters
• P LTI/SS plant model
• Nmeas B2 column size
• Ncon C2 row size
• Output Parameters
• K vector of Feedback Controller Coefficients
• CL Closed loop system matrix
• Gam the H2 norm of the closed loop system
• Info structure containing additional
  information.

• Plant Partitioning (if done by mktito(), nmeas
  and ncon can be omitted)
• B1 Inputs are disturbances
• B2 Inputs are control signals
• C1 Outputs are errors to be minimized
• C2 Outputs are inputs to control K

92
K,CL,GAM,INFO hinfsyn(P,NMEAS,NCON)
Hinfsyn() creates an optimal h-inifinity
controller for a partitioned plant matrix P.

• Input Parameters
• P Partitioned plant matrix
• Nmeas row size of the C2 matrix
• Ncon column size of the B2 matrix
• Output Parameters
• K controller coefficient vector
• CL closed loop system matrix
• Gam H-inifinity norm for the closed loop system
• Info - structure containing additional information

Plant Partitioning (if done by mktito(), nmeas
and ncon can be omitted) B1 Inputs are
disturbances B2 Inputs are control signals C1
Outputs are errors to be minimized C2 Outputs
are inputs to control K
93
K,GAMsdhinfsyn(P,NMEAS,NCON)
Sdhinfsyn() synthesizes a sampled data H-infinity
controller.

• Input Parameters
• P partitioned system matrix
• Nmeas row size of C2
• NCon column size of B2
• Output parameters
• K H infinity controller coefficients
• Gam H infinity norm cost for the system

Where

• Plant Partitioning
• B1 Inputs are disturbances
• B2 Inputs are control signals
• C1 Outputs are errors to be minimized
• C2 Outputs are inputs to control K

94
K,CL,GAM,INFOloopsyn(G,Gd,RANGE)
Loopsyn() is an H infinity optimal way for
loopshaping controller synthisis.

• Input Parameters
• G LTI plant
• Gd desired loop shape
• Range Wmin,Wmax desired loop shaping
  frequency range. (10Wmin lt Wmax)
• Output Parameters
• K Controller Coefficient vector
• CL Closed Loop System Matrix
• Gam H infinity norm for the closed loop
  system
• Info structure containing additional
  information.

EX rand('seed',0)randn('seed',0) stf('s')
w05 Gd5/s G((s-10)/(s100))rss(3,4,5)
K,CL,GAM,INFOloopsyn(G,Gd)
sigma(GK,'r',GdGAM,'k-.',Gd/GAM,'k-.',.1,100)
95
K,SVL,W1 ltrsyn(G,F,XI,THETA,RHO)
Ltrsyn() computes a reconstructed-state
output-feedback controller K for LTI plant G
using Kalman filtering to asymptotically recover
the lost states.

• Input Parameters
• G LTI System matrix
• F - LQ full-state-feedback gain matrix
• XI plant noise intensity
• Theta sensor noise intensity
• Rho vector containing a set of recovery gains
• Output Parameters
• K controller coefficient vector
• Svl sigma plot data for recovered system
• W1 - frequencies for SVL plots

EX stf('s')Gss(1e4/((s1)(s10)(s100)))A,
B,C,Dssdata(G) Flqr(A,B,C'C,eye(size(B,2)))
Lss(A,B,F,0FB) XI100C'C
THETAeye(size(C,1)) RHO1e3,1e6,1e9,1e12Wlo
gspace(-2,2) nyquist(L,'k-.')hold K,SVL,W1lt
rsyn(G,F,XI,THETA,RHO,W)
96
K,CL,GAM,INFOmixsyn(G,W1,W2,W3)
Mixsyn() is H-infinity mixed-sensitivity
synthesis method for robust control loopshaping
design.

• Input Parameters
• G LTI system
• W1,W2,W3 Weighting matricies associated with
  S(s), R(s) and T(s)
• Output Parameters
• K Controller Coefficient vector
• CL Closed Loop System Matrix
• Gam H infinity norm for the closed loop
  system
• Info structure containing additional
  information.

Mixsyn() creates the h-infinity controller based
on the sensitivity function S(s) and the
complimentatry sensitivitiy T(s) function
according to the weighting matricies W1, W2, W3.
97
K,CL,GAM,INFOncfsyn(G,W1,W2)
Ncfsyn() is a normalized coprime factorization
based parameterized controller synthisis using
the Glover-McFarlane method for loopshaping.

• Input Parameters
• G LTI plant model
• W1,W2 Pre and Post compensator to give Gd
  W1GW2 a good high frequency and low frequency
  response
• Output Parameters
• K controller coefficient vector
• CL closed loop system matrix
• Gam H infinity optimal cost
• Info structure containing additional information

EX szpk('s') G(s-1)/(s1)2 W10.5/s
K,CL,GAMncfsyn(G,W1) sigma(GK,'r',GW1,'r-.'
,GW1GAM,'k-.',GW1/GAM,'k-.')
98
sys mkfilter(fc,ord,type,psbndr)
Mkfilter() returns a single-input, single-output
analog low pass filter sys as an ss object.

• Input
• Fc cutoff frequency of the filter
• Ord order of the filter
• Type type of the filter.
• 'butterw- Butterworth filter
• cheby - Chebyshev filter
• 'bessel - Bessel filter
• 'rc - Series of resistor/ capacitor filters
• Psbndr - the magnitude of the passband
• Output
• Sys Designed filter

EX butw mkfilter(2,4,'butterw') cheb
mkfilter(4,4,'cheby',0.5) rc
mkfilter(1,4,'rc') bode(butw,'-',cheb,'--',rc,'-
.')
99
Mu-synthesis
Mu-synthesis functions enable the synthesis of
robust optimal controllers. Approximation tools
for this type of design are shown below

1. Cmsclsyn() approximately solves the
   constant-matrix, upper bound µ-synthesis problem.
2. Dksyn() synthesizes a robust controller via D-K
   iteration which is an approximation to
   µ-synthesis control design.
3. Dkitopt() creates a dkitopt object called options
   with specific values assigned to certain
   properties.
4. Drawmag() interactively uses the mouse in the
   plot window to create pts and sysout, which
   approximately fits the frequency response
   (magnitude) in pts.
5. Fitfrd() fits D-scaling frequency response data
   with state-space model.
6. Fitmagfrd() fits frequency response magnitude
   data with a stable, minimum-phase state-space
   model.

100
qopt,bnd cmsclsyn(R,U,V,BlockStructure)
Cmsclsyn() approximately solves the
constant-matrix, upper bound µ-synthesis problem
by minimization, for given matrices
and
. This applies to constant matrix
data in R, U, and V.

• Input
• R,U,V Constant matrices of appropriate
  structure
• BlockStructure a matrix specifying the
  perturbation blockstructure as defined for mussv.

• Output
• Qopt - the optimum value of Q
• Bnd - the upper bound of mussv(RUQV,BLK)

101
k,clp,bnd dksyn(p,nmeas,ncont)
Dksyn() synthesizes a robust controller via D-K
iteration which is an approximation to
µ-synthesis control design.

• Input
• P Partitioned plant matrix