sliding mode manual

A Sliding Mode Control Matlab Toolbox

Christopher Edwards � Sarah K� Spurgeon

Control Systems ResearchDepartment of EngineeringUniversity of LeicesterLeicester LE� �RH

U�K�

Revised ��

Report No � ��

Abstract

This document serves as a manual for a suite of Matlab functions for designingsliding mode controllers for linear systems with bounded nonlinear uncertainty� Thealgorithms are described in detail elsewhere and in this document the emphasis is ondescribing the software which implements these ideas� A number of worked examplesare included to demonstrate di�erent facets of the design procedures together withdiaries of relevant Matlab sessions�

� An Introduction to Variable Structure Control

The focus of much of the research in the area of control systems theory duringthe seventies and eighties has addressed the issue of robustness � i�e� designingcontrollers with the ability to maintain stability and performance in the presenceof discrepancies between the plant and model� One nonlinear approach to robustcontroller design which emerged during this period is the Variable structure controlsystems methodology� Variable structure control systems evolved from the pioneer�ing work in Russia of Emel�yanov and Barbashin in the early ��s� The ideasdid not appear outside the Soviet Union until the mid ��s when a book by Itkis�� and a survey paper by Utkin � were published in English� Variable structuresystems concepts have subsequently been utilised in the design of robust regulators�model�reference systems� adaptive schemes� tracking systems and state observers�The ideas have successfully been applied to problems as diverse as automatic �ightcontrol� control of electrical motors� chemical processes� helicopter stability aug�mentation� space systems and robotics�

Variable structure control systems comprise a collection of di�erent� usually quitesimple� feedback control laws and a decision rule� Depending on the status of thesystem� a decision rule� often termed the switching function� determines which ofthe control laws is �on�line� at any one time� Unlike� for example� a gain�schedulingmethodology� the decision rule is designed to force the system states to reach� andsubsequently remain on� a pre�de�ned surface within the state�space� The dynam�ical behaviour of the system when con�ned to the surface is described as the idealsliding motion� The advantages of obtaining such a motion are twofold� �rstlythere is a reduction in order and secondly the sliding motion is insensitive to pa�rameter variations implicit in the input channels� The latter property of invariancetowards so�called matched uncertainty makes the methodology an attractive one fordesigning robust controllers for uncertain systems�

The following notation �which is explained in detail in �� will be used throughout�IR and C will represent the �eld of real and complex numbers respectively and C�will represent the open left half of the complex plane� N �� and R�� will denote thenull and range spaces of a matrix respectively and k � k will represent the Euclideannorm for vectors and the induced spectral norm for matrices�

� State�feedback Sliding Mode Control

�� Introduction

Consider the nth order linear time invariant system with m inputs given by

�x�t� � Ax�t� �Bu�t� ��

where A � IRn�n and B � IRn�m with � � m � n� Without loss of generality it canbe assumed that the input distribution matrix B has full rank� De�ne a switchingfunction s � IR� IRm to be

s�t� � Sx�t� ��

State�feedback Sliding Mode Control �

where S � IRm�n is of full rank and let S be the hyperplane de�ned by

S � fx � IRn � Sx � g ��

Suppose u�s�t�� x�t�� represents a variable structure control law where the changesin control strategy depend on the value of the switching function� It is natural toexplore the possibility of choosing the control action and selecting the switchingstrategy so that an ideal sliding motion takes place on the hyperplane� i�e� thereexists a time ts such that

s�t� � Sx�t� � for all t � ts � �

Suppose at time t � ts the systems states lie on the surface S and an ideal slidingmotion takes place� This can be expressed mathematically as Sx�t� � and �s�t� �S �x�t� � for all t � ts� Substituting for �x�t� from �� gives

S �x�t� � SAx�t� � SBu�t� � for all t � ts ��

Suppose the matrix S is designed so that the square matrix SB is nonsingular �inpractise this is easily accomplished since B is full rank and S is a free parameter��The equivalent control� written as ueq� is de�ned to be the unique solution to thealgebraic equation �� namely

ueq�t� � ��SB��SAx�t� ��

This represents the control action which is required to maintain the states on theswitching surface� The ideal sliding motion is then given by substituting the ex�pression for the equivalent control into equation �� which results in a free motion

�x�t� ��In � B�SB��S

�Ax�t� for all t � ts and Sx�ts� � ��

It can be seen from equation �� that the sliding motion is a control independentfree motion which depends on the choice of sliding surface� although the precisee�ect is not readily apparent� A convenient way to shed light on the problem is to�rst transform the system into a suitable canonical form� In this form the systemis decomposed into two connected subsystems� one acting in R�B� and the otherin N �S�� Since by assumption rank�B� � m there exists an orthogonal matrixTr � IR

n�n such that

TrB �

�B�

��

where B� � IRm�m and is nonsingular� Let z � Tx and partition the new coordi�

nates so that

z �

�z�z�

��

where z� � IRn�m and z� � IR

m� The nominal linear system �� can then be writtenas

�z��t� � A��z��t� � A��z��t� ��

�z��t� � A��z��t� � A��z��t� �B�u�t� ��


which is referred to as regular form� Equation �� is referred to as describingthe null�space dynamics and equation �� as describing the range�space dynamics�Suppose the matrix de�ning the switching function �in the new coordinate system�is compatibly partitioned as

STTr �

�S� S�

��

where S� � IRm��n�m� and S� � IR

m�m� Since SB � S�B� it follows that a neces�sary and su�cient condition for the matrixSB to be nonsingular is that det�S�� By design assume this to be the case� During an ideal sliding motion

S�z��t� � S�z��t� � for all t � ts ��

and therefore formally expressing z��t� in terms of z��t� yields

z��t� � �Mz��t� ��

where M � S�� S�� Substituting in equation �� gives

�z��t� � �A�� A��M � z��t� ��

and thus the problem of hyperplane design may be considered to be a state feedbackproblem for the system �� where z��t� is considered to play the role of the controlaction� In the context of designing a regulator� the matrix governing the slidingmotion �A�� A��M � must have stable eigenvalues� The switching surface designproblem can therefore be considered to be one of choosing a state feedback matrixM to stabilise the reduced order system �A�� A�� Because of the special structureof the regular form� it follows that the pair �A�� A�� is controllable if and onlyif �A�B� is controllable� It can be seen from equation �� that S� has no directe�ect on the dynamics of the sliding motion and acts only as a scaling factor for theswitching function� The choice of S� is therefore somewhat arbitrary� A commonchoice however� which stems from the so�called hierarchical design procedure� is tolet S� � �B�� for some diagonal design matrix � � IRm�m which implies SB �� By selecting M and S� the switching function in equation �� it completelydetermined�

Several approaches have been proposed for the design of the feedback matrix Mincluding quadratic minimisation� eigenvalue placement and eigenstructure assign�ment methods �see Chapter in �� These approaches will be discussed in thefollowing sections�

Example

A linearisation of the nonlinear equations of motion of an inverted pendulum aboutthe equilibrium point at the origin �x�� in �� is given by

A �

��

� B �

��

��

� ��

This system is not in regular form because the input enters the last two state�spaceequation� The following Matlab session performs the necessary transformationand identi�es the important sub�blocks�

State�feedback Sliding Mode Control

Session �� regfor

�� A��A��B��Tr��regfor�A�B�

A��

��

� � ��

��

A��

��

��

��

B� �

��

Tr �

�

��

��

��

��

Notice the transformation Tr is symmetric and in fact orthogonal� This is desirablefrom the numerical viewpoint�

�� Robust Eigenstructure Assignment

For the case of a scalar controlled problem� speci�cation of the �n � �� eigenvaluesassociated with the sliding mode will uniquely determine the matrixM of equation�� For multi�input systems this is not the case� In such a situation the avail�able degrees of freedom may be used to modally shape the system response by ajudicious choice of eigenvector form and�or ensure that the resulting closed�loopsystem is maximally robust to system parameter variations� However the eigenvec�tor corresponding to a given eigenvalue must lie in an allowable sub�space which isdetermined by the system matrix� the input matrix and the eigenvalue itself� Toevaluate robustness� a bound upon the individual eigenvalue sensitivities ci is givenby

maxfcig � ��V � � kV kkV ��k ��

Here ��V � denotes the condition number of the matrix V of right eigenvectors andis a measure of the orthogonality of the eigenvectors vi� The closer the eigenvectorsof a matrix are to being orthogonal� the smaller is the associated condition numberand the greater the robustness of the eigenvalue locations to changes in the elementsof the matrix� In robust eigenstructure assignment the feedback matrix is obtainedby assigning a set of linearly independent right eigenvectors corresponding to the


required eigenvalues such that the matrix of eigenvectors is as well�conditioned aspossible �x �� in �� This can be achieved via the Matlab command place�

Example

Here the use of robust pole placement will be demonstrated on a dc motor examplefrom x�� in�� The state and input distribution matrices are given by

A �

��

� B �

��

�

� ��

Notice the system is already in regular form and the sliding motion will be a secondorder system� In the following session the reduced order sliding motion is assigneda pair of complex poles with damping ratio � � �� and natural frequency wn � ��

Session �� rpp

zeta� ��

wn��

�� sp�roots�� zeta�wn wn�wn��

sp �

�� i

�� i

�� S�rpp�A�B�sp�

place� ndigits� ��

S �

� ��

��

This compares with the �speci�c� analytic approach described in x�� Noticethe hyperplane matrix S is returned in the original coordinate system� The trans�formation to regular form and the subsequent computations take place internallywithin the function�

�� Direct Eigenstructure Assignment

In the previous subsection� the additional degrees of freedom in the pole placementproblem for multi�input systems were used to minimise the condition number ofthe associated eigenvalues� If information is known about a desirable weightingof the system states for each mode� it is possible to choose a desired eigenvectorspeci�cation� Again this will not necessarily be achievable because it may not liewithin the prescribed allowable sub�space� details are given in x �� in ��

A numerical example using the approach appears later in x��


�� Quadratic Minimisation

Consider the problem of minimising the quadratic performance index

J ��

�

Z �

ts

x�t�TQx�t� dt ��

where Q is both symmetric and positive de�nite and ts is the time at which slidingmotion commences� The aim is to minimise equation �� subject to the systemequation �� under the assumption that sliding takes place� It is assumed thatthe state of the system at time ts� x�ts�� is a known initial condition and is suchthat x�t� � as t � �� The matrix Q from equation �� is transformed andpartitioned compatibly with z so that

TrQTTr �

�Q�� Q��

QT�� Q��

��

and subsequently de�ne�Q � Q�� Q��Q

�� Q��

andv � z� �Q�� Q��z� ��

After some algebraic manipulation equation �� may then be written in the newcoordinate system as

J ��

�

Z �

ts

zT��Qz� � vTQ��v dt ��

Recall the constraint equation may be written as

�z��t� � A��z��t� � A��z��t� ��

Eliminating the z� contribution from equation �� using equation �� the modi�edconstraint equation becomes

�z��t� � �Az��t� �A��v�t� ��

where�A � A�� A��Q

�� Q��

The positive de�niteness of Q ensures that Q�� so that Q�� exists� and also

that �Q � � Furthermore� the controllability of the original �A�B� pair ensures thatthe pair � �A�A�� is controllable� The problem thus becomes that of minimising thefunctional �� subject to the system �� and thus can be interpreted as a standardlinear�quadratic optimal state�regulator problem� A more detailed description ofthis approach is given in x �� in ��

Example

Consider once again the linearised equations of motion of an inverted pendulumgiven in �� The following session designs a hyperplane for this system by min�imising a quadratic cost function�


Session �� lqcf

�� Q�diag��

Q �

�

�

�

�

�� S�E��lqcf�A�B�Q�

S �

��

E �

�� i

�� i

��

Notice the hyperplane matrix S is returned in the original coordinate system and allthe transformations take place internally� The remainder of this section is devotedto the synthesis of a control law which will theoretically bring about ideal sliding�

�� State�feedback Control Laws

Of the many di�erent multivariable sliding mode control structures which exist theone that will be considered here is essentially that of Ryan Corless �� and maybe described as a unit vector approach� Consider an uncertain system of the form

�x�t� � Ax�t� � Bu�t� � f�t� x� u� ��

where the function f � IR� � IRn� IRm � IRm which represents the uncertaintiesor nonlinearities satis�es the so�called matching condition� i�e�

f�t� x� u� � B��t� x� u� ��

where � � IR� � IRn� IRm � IRm and is unknown but satis�es

k��t� x� u�k � k�kuk� ��t� x� ��

where � � k� � is a known constants and �� is a known function� The proposedcontrol law comprises two components� a linear component to stabilise the nominallinear system� and a discontinuous component� Speci�cally

u�t� � ul�t� � un�t� ��

where the linear component is given by

ul�t� � �� SA �!S� x�t� ��


where ! � IRm�m is any stable design matrix and � � SB� The nonlinear compo�nent is de�ned to be

un�t� � ��t� x�� P�s�t�

kP�s�t�kfor s�t� ��

where P� � IRm�m is a symmetric positive de�nite matrix satisfying the Lyapunov

equationP�! �!

TP� � �I ��

and the scalar function ��t� x�� which depends only on the magnitude of the uncer�tainty� is any function satisfying

��t� x� � �k�kulk� ��t� x� � �� k��

where � � is a design parameter� In this equation it is assumed that the scalingparameter has been chosen so that

k��

It can easily be established that any function satisfying equation �� also satis�es

��t� x� � k��t� x� u�k� � ��

and therefore ��t� x� is greater in magnitude than the matched uncertainty occurringin equation �� It is straightforward to verify that V �s� � sTP�s guaranteesquadratic stability for the switching states s and in particular

�V � �sTs � ��kP�sk ��

This control law guarantees that the switching surface is reached in �nite timedespite the disturbance or uncertainty and once the sliding motion is attained it iscompletely independent of the uncertainty� The relationship between the controllaw in �� and the original description of Ryan Corless is described in x��in ��

The following session considers the inverted pendulum from Session � and constructsthe components of the control law described in �� Session �� contl

�� Phi��

�� L�P�Lam��contl�A�B�S�Phi�

L �

��

P �

��

Lam �

� ��

��


Thus far only regulation problems have been considered� In practise tracking prob�lems are often encountered whereby �usually� the output of the system is requiredto follow a pre�de�ned reference signal� One approach is to use a model�followingsliding mode control law� In this situation a state error system is formed betweenthe plant states and the ideal model states and a sliding mode �regulation�like� con�trol law is designed to force the error system to zero �x � �� in �� In the followingsection an integral action based method is considered for output tracking�

�� Output Tracking with Integral Action

Consider the development of a tracking control law for the nominal linear system

�x�t� � Ax�t� �Bu�t� ��

y�t� � Cx�t� ��

which is assumed to be square� In addition� for convenience� assume the matrix pair�A�B� is in regular form� The control law described here utilises an integral action

methodology� Consider the introduction of additional states xr � IRp satisfying

�xr�t� � r�t�� y�t� ��

where the di�erentiable signal r�t� satis�es

�r�t� � " �r�t��R� � �

with " � IRp�p a stable design matrix and R a constant demand vector� Augmentthe states with the integral action states and de�ne

#x �

�xrx

��

The associated system and input distribution matrices for the augmented system

are

#A �

� �C A

�and #B �

�B

��

and assuming the pair �A�B� is in regular form� the pair � #A� #B� is in regular form�The proposed controller seeks to induce a sliding motion on the surface

S � f#x � IRn�p � S#x � Srrg � ��

where S � IRm��n�p� and Sr � IRp�p are design parameters which govern thereduced order motion� Partition the hyperplane system matrix as

n

m

S ��S� S�

� � �

and the system matrix

#A �

n

m�

#A��#A��

#A��#A��

�ln

lm

� ��


and assume � � S #B is nonsingular� If a controller exists which induces an idealsliding motion on S and the augmented states are partitioned as

#x �

�x�x�

��

where x� � IRn and x� � IR

m� then the ideal sliding motion is given by

�x��t� � � #A�� #A��M �x��t� ��#A��S

�� Sr � Br

r�t� � ��

where M � S�� S� and Br ��Ip n�p

�T� In order for the hyperplane design

methods described earlier to be valid� it is necessary for the matrix pair � #A�� #A��to be completely controllable� Necessary conditions on the original system are that�A�B�C� is completely controllable and has no invariant zeros at the origin �x � ��in ��

The development that follows mirrors the approach in x�� where ! is any stabledesign matrix� The overall control law is then given by

u � ul�#x� r� � un�#x� r� � ��

where the discontinuous vector

un�s� r� �

��c�uL� y��

P��S�x�Srr�kP��S�x�Srr�k

if S#x �� Srr

otherwise� ��

where P� is a symmetric positive de�nite matrix satisfying

P�! �!TP� � �I ��

The positive scalar function which multiplies the unit vector component can beobtained from arguments similar to those in in x�� It follows that� in terms of theoriginal coordinates

un�#x� r� � L#x� Lrr � L �r �r ��

with gains de�ned as

L � ��S #A � !S� ��

Lr � �� !Sr � S�Br� ��

L �r � ��Sr ��

The parameter Sr can take any value and does not a�ect the stability of the closedloop system� One common choice is to let Sr � for simplicity� Another option�which has been found to give good results with practical applications� is to chooseSr so that at steady state the integral action states are zero in the absence of anyuncertainty �x�� in ��

This session repeats the design aircraft design study in x �� in �� which uses integralaction and a hyperplane calculated using direct eigenstructure assignment� Thesystem triple given below represents the longitudinal motion of of an aircraft at

State�feedback Sliding Mode Control ��

�� Mach and the outputs correspond to �ight path angle and pitch angle

A �

��

� B �

��

� �

�

C �

��

��

First the augmented system represented by �Aa�Ba� is constructed� The commandeigbuil is then used to form the matrix which contains the information identifyingthe eigenvector components which are to be speci�ed and the associated desiredvalues� For more details of the design and closed�loop simulations see x �� in ��

Session �� intac eigbiul and dea

�� Aa�Ba��intac�A�B�C�

Aa �

��

��

��

��

� ��

��

��

Ba �

�

�

�� nocomp��

�� specent�eigbuil��nocomp�

Eigenvector � �Real Part�

Enter position of specified entry � to terminate�

Followed by the value of specified entry

Enter position �

Enter associated value

Enter position �

Enter associated value �

Enter position

��


Eigenvector � �Imaginary part�



Enter position �


Enter position �


Enter position

��

Eigenvector �



Enter position �


Enter position �


Enter position �


Enter position

��

Eigenvector �



Enter position �


Enter position �


Enter position

��

Eigenvector �



Enter position �


Enter position �



Enter position �


Enter position

��

specent �

NaN NaN NaN NaN NaN

NaN NaN NaN NaN NaN

� �

� NaN �

NaN � NaN

NaN NaN NaN NaN NaN

NaN NaN NaN NaN NaN

�� lambda��

lambda �

��

�� S�V��dea�Aa�Ba�specent�lambda��

S �

��

��

V �

��

��

� ��

� � � ��

� ��

��

� ��

��

The vector V represents the achieved eigenstructure� Notice when only two elementshave been speci�ed in an eigenvector the requirements have been achieved exactly�This is not the case for the two eigenvectors in which three elements were speci�ed�This is to be expected as a maximum of m entries can be speci�ed exactly�

The associated control law which includes feed�forward terms for both the referenceand its derivative can be obtained as shown below� The hyperplane is �rst scaledso that � � I� and the two range�space poles are selected to be at ��

Session �� contlia

�� S�inv�S�Ba��S

S �


��

��

�� L�Lr�Lrdot�Sr�Lam�P��contlia�A�B�C�S��eye��

L �

��

��

Lr �

��

� � � �

Lrdot �

��

� � ��

Sr �

��

� � ��

Lam �

�

�

P �

��

��

��

The following session reconstructs the case study from Chapter � in �� and designsa state feedback output tracking controller utilising integral action for the furnacesystem linearisation

A �

��

��

� B �

��

��

� ��

�

C �

��

��

��

The robust pole placement method has been chosen for the design of the switchingfunction� Once the matrix S has been calculated the �feedback� gain matrix M iscomputed �this is easily accomplished in this case since the system is in regular form�and the system matrix governing the reduced order sliding motion �A�� A��M �is formed� The condition number of the associated eigenvectors is also computedpurely for pedagogical purposes to verify that a robust design has been achieved�


Session � intac and rpp

�� sp�� j ��j��

�� Aa�Ba��intac�A�B�C��

�� S�rpp�Aa�Ba�sp�


S �

��

��

�� M�inv�S��S��

M �

��

��

�� A��Aa��

A��

��

��

��

��

�� A��Aa��

A��

��

��

� ��

��

�� eig�A��A��M�

ans �

��

��

�� i

�� i

�� V�D��eig�A��A��M��

�� cond�V�

ans �

��

��

The condition number of the eigenvectors is reasonably small indicating that thedesign is relatively robust with respect to unmatched uncertainty�

Output Based Hyperplane Design ��

Up to this point it has been assumed that all the states are available for use in thecontrol law� This hypothesis will be dropped in the remaining sections�

� Output Based Hyperplane Design

Consider the linear system in �� and suppose that only the measured outputs

y�t� � Cx�t� ��

where C � IRp�n are available� One strategy is to restrict the class of slidingsurfaces and control laws to those which require only output information� Recentwork has isolated the class of systems for which this approach is applicable and adesign framework has been proposed for synthesising appropriate sliding surfacesand control laws �x�� in �� Here the case when there are more outputs thaninputs is considered since� in the square case� no design freedom exists in terms ofselecting the dynamics of the sliding motion� Two assumptions will be made�

A�� the Markov parameter CB is full rank

A�� any invariant zeros of �A�B�C� are in the C�

These assumptions will be central to the output feedback based sliding mode controland observer methods used here� The following lemma provides a canonical formfor the system triple �A�B�C� which will be used in the subsequent analysis�

Lemma � Let �A�B�C� be a linear system with p � m and rank �CB� � m� Then

a change of coordinates exists so that the system triple with respect to the new

coordinates has the following structure�

a� the system matrix can be written as

A �

�A�� A��

A�� A��

�where A�� IR

�n�m��n�m� ��

and the sub�block A�� when partitioned has the structure

A��

�� Ao

�� Ao��

Ao��

Am��

Ao�� Am

��

� ��

where Ao�� IR

r�r� Ao�� IR

�n�p�r��n�p�r� and Ao�� IR

�p�m��n�p�r�

for some r � and the pair �Ao�� A

o�� is completely observable� Further�

more the invariant zeros of the system are the r eigenvalues of Ao��

b� the input distribution matrix has the form

B �

�B�

�where B� � IR

m�m and is nonsingular ��


c� the output distribution matrix has the form

C �� T

�where T � IRp�p and is orthogonal ��

Example

The following session considers the inverted pendulum example from �� and as�sumes that only the �rst three states are measured� This session considers a changeof coordinates which puts the system triple into the output�feedback canonical form�

Session �� outfor

�� Ac�Bc�Cc�Tc�r��outfor�A�B�C�

Ac �

��

�

� ��

� � � ��

Bc �

�

��

Cc �

��

�

��

Tc �

��

��

�

��

r �

��

Notice r � which indicates that the system �A�B�C� does not have any invariantzeros�

Under the premise that only output information is available� the switching functionmust be of the form

s�t� � FCx�t� ��


where F � IRm�p� Suppose a controller exists which induces a stable sliding motionon the surface

S � fx � IRn � FCx � g ��

For a unique equivalent control� to exist the matrix FCB � IRm�m must have fullrank� this implies that rank�CB� � m since rank�F � � m� Therefore in all theanalysis which follows it can be assumed without loss of generality that the systemis already in the canonical form of Lemma �� Let

p�m

m�

F� F�� FT

��

then the matrix which de�nes the switching function can be written as

FC ��F�C� F�

�where

C� ��p�m��n�p� I�p�m�

��

In this way FCB � F�B� and in particular the square matrix F� is nonsingular�The canonical form of Lemma � can be viewed as a special case of the �regular form�normally used in sliding mode controller design and therefore arguing as in x�� thereduced order sliding motion is governed by a free motion with system matrix

As�� A�� A��F

�� F�C� ��

De�ne K � F�� F� then the problem of designing a suitable hyperplane is equivalentto an output feedback problem for the system �A�� A�� C�� It is well establishedthat invariant zeros play an important part in the sliding motion and� in particular�any invariant zeros of �A�B�C� must appear in the sliding mode dynamics whenonly output information is used� Intuitively� if the original system has invariantzeros then by Lemma � the eigenvalues of Ao

�� must appear in the spectrum of As��

and so the poles of A�� A��KC� cannot be assigned arbitrarily by choice of K�To this end partition the matrices A�� and Am

�� as

A��

�A��

A��

�and Am

��

�Am��

Am��

��

where A�� IR�n�m�r��m and Am

�� IR�n�p�r��p�m� and form a new sub�system

represented by the triple � #A�� A�� #C�� where

#A��

�Ao�� Am

��

Ao�� Am

��

�and #C� �

��p�m��n�p�r� I�p�m�

��

As a result it can be shown that the spectrum of As�� contains the invariant zeros

of �A�B�C� and in particular

��A�� A��KC�� Ao�� #A�� A��K #C��

�For comprehensive details see Chapter � in Utkin ��


It follows directly that for a stable sliding motion the invariant zeros of the system�A�B�C� must lie in the open left half plane and the triple � #A�� A�� #C�� mustbe stabilisable with respect to output feedback�� It can be shown that if the pair�A�B� is completely controllable then the pair � #A�� A�� is completely control�lable� the pair � #A�� #C�� is completely observable by construction and thus if thetriple � #A�� A�� #C�� satis�es the Kimura�Davison conditions� output feedback poleplacement methods can be used to place the poles appropriately�

It is well established in the static output feedback literature that if the Kimura�Davison conditions are not satis�ed by the nominal triple� the system can be aug�mented by an appropriately dimensioned compensator� Speci�cally let

�xc�t� � Hxc�t� �Dy�t� ��

where the matrices H � IRq�q and D � IRq�p are to be determined� De�ne a newhyperplane in the augmented state space� formed from the plant and compensatorstate spaces� as

Sc � f�x� xc� � IRn�q � Fcxc � FCx � g ��

where Fc � IRm�q and F � IRm�p� De�ne D� � IR

q��p�m� and D� � IRq�m as�

D� D�

�� DT ��

If the the input distribution matrix is partitioned conformably as

A��

�A��

A��

�ln�p�r

lp�m��

then it can be shown �x�� in �� that a parametrisation of the compensator is

H � Ao�� LoAo

��

D� � Am�� LoAm

�� Ao�� LoAo

��Lo ��

D� � A�� LoA��

where Lo � IR�n�p�r��p�m� is any gain matrix so that Ao�� LoAo

�� is stable�Let K be any state feedback matrix for the controllable pair � #A�� A�� so that#A�� A��K is stable and partition the state feedback matrix so that

n�r�p

p�m�

K� K�

�� K

��

and de�ne

K � K� � K�Lo ��

Kc � K� ��

If r � de�ne a new dynamical system by

�zr�t� � Ao��zr�t� � Ao

��xc�t� � �Am�� Ao

��Lo�x��t� �A��x��t� ��

�The phrase �stabilisable with respect to output feedback� will indicate that for the linear system�A�B�C� there exists a �xed gain K such that the matrix A� BKC is stable�

Output Based Hyperplane Design �

and augment �� with �� to form a new compensator

��xc�t� � �H�xc�t� � �Dy�t� ��

where

�H �

�Ao�� Ao

��

H

�and �D �

��Am

�� Ao��L

o� A��

D� D�

�TT

If

�x �

��xcTTy

��

then the sliding surface Sc can be written as f�x � IRn � S�x � g where

S � F��m�r Kc K Im

��

A control law to induce a sliding motion on the sliding surface Sc is given by

u�t� � L�x�t� � un�t� ��

where L is de�ned in x�� of �� and un�t� is the scaled unit�vector law in ��

Example

The following session calculates a compensator based output dependant slidingmode controller� The canonical form from Session � underpins the synthesis andthe speci�ed poles obtained from Session � are used to specify part of the slidingmode dynamics�

Session � comrobs

�� Hhat�Dhat�S�L�P��comrobs�A�B�C�

Enter � pole for the error dynamics ��

Enter � pole for sliding motion ��j ��j ��

Enter � pole for the range space dynamics ��



Hhat �

��

Dhat �

��

S �

��

L �

� ��

Sliding Mode Observers ��

P �

��

��

This example is consider in greater detail in x�� in ��

� Sliding Mode Observers

Despite fruitful research and development activity in the area of variable structurecontrol theory� relatively few authors have considered the application of the un�derlying principles to the problem of observer design� Consider the nominal linearsystem subject to a class of uncertainty described by

�x�t� � Ax�t� �Bu�t� �D��t� y� u� ��

y�t� � Cx�t� ��

where A � IRn�n� B � IRn�m� C � IRp�n� D � IRn�q with q � p n and thematrices B�C and D are of full rank� The function ��t� y� u� is assumed to beunknown but bounded by a known function� It is further assumed that the statesof the system are unknown and only the signals u�t� and y�t� are available�

The objective is to synthesise an observer to generate a state estimate �x�t� suchthat the error

e�t� � �x�t� � x�t� ��

tends to zero despite the presence of the uncertainty� Furthermore the intention isto induce a sliding motion on the surface in the error space

So � fe � IRn � Ce � g ��

The particular observer structure that will be considered can be written in the form

��x�t� � A�x�t� �Bu�t� � GlCe�t� � Gn� ��

where Gl� Gn � IRn�p are appropriate gain matrices and � represents a discontinu�

ous switched component to induce a sliding motion on So�

Consider the dynamical system given in �� and �� and assume that

� rank�CD� � q

� the invariant zeros of �A�D�C� lie in the C��

It can be shown �x�� in �� that under these assumptions there exists a linearchange of coordinates x �� Tx such that in the new coordinate system

�x��t� � A��x��t� � A��x��t� �B�u�t� ��

�x��t� � A��x��t� � A��x��t� �B�u�t� �D��t� y� u� ��

y�t� � x��t� ��


where x� � IR�n�p�� x� � IRp and the matrix A�� has stable eigenvalues� Thefreedom of choice associated with A�� depends on the number of invariant zeros of�A�B�C�� The coordinate system above will be used as a platform for the designof a sliding mode observer� Consider a dynamical system of the form

��x��t� � A��x��t� �A��x��t� �B�u�t�� A��ey�t� ��

��x��t� � A��x��t� �A��x��t� �B�u�t�� A�� As��ey�t� � kD�k� ��

�y�t� � �x��t� ��

where As�� is a stable design matrix and ey � �y � y� The discontinuous vector � is

de�ned by

� �

��t� y� u� P�ey

kP�eykif ey ��

otherwise��

where P� � IRp�p is a Lyapunov matrix for As

�� satisfying

P�As�� A

s��

TP� � �I ��

and the scalar function ��t� y� u� is chosen so that

k��t� y� u�k ��t� y� u� ��

If the state estimation errors are de�ned as e� � �x� � x� and e� � �x� � x� then itis straightforward to show

�e��t� � A��e��t� ��

�ey�t� � A��e��t� �As��ey�t� � kD�k� �D��t� y� u� ��

Furthermore the nonlinear error system in �� is quadratically stable and asliding motion takes place on the surface So de�ned in �� The dynamical systemin �� may thus be regarded as an observer for the system in �� Itfollows that if the linear gain

Gl � T��

A��

A�� As��

��

and the nonlinear gain

Gn � kD�kT��

�Ip

��

then the observer given in �� can be written in terms of the original coor�dinates in the form of ��

Example

The following session places the aircraft system

A �

��

��

��

�


B �

��

��

� C �

��

�

from x�� in �� in the canonical form for observer design under the assumptionthat the matrix which distributes the uncertainty is D � B�

Session �� obsfor

�� Ac�Bc�Cc�Tc��obsfor�A�B�C�

Enter � desired stable sliding mode pole�s� ��


Ac �

��

��

��

��

�

Bc �

�

��

��

� �

Cc �

�

�

� �

�

Tc �

��

� ��

�

� �

�

��

Notice in this example because no invariant zeros are present the dynamics associ�ated with the upper left element �which govern the sliding motion� can be placedarbitrarily�

The next session uses the same example and computes the observer gains fromequations �� and ��

Sliding Mode Observers �

Session �� disobs

�� Gl�Gn�P��disobs�A�B�C�



Enter � output estimation error pole�s� ��

Gl �

� �

��

��

� ��

��

Gn �

��

� ��

��

� ��

� � ��

P �

��

��

��

�

�� eig�A�Gl�C�

ans �

��

��

��

��

��

��

Another observer� introduced by Walcott �Zak �� considers the special case whenthe uncertainty is matched i�e� when D � B� They propose the observer structuregiven by

�z�t� � Az�t� � Bu�t� �GCe�t� � B�o ��

where z represents an estimate of the true states x� and e � z � x is the stateestimation error� The output error feedback gain matrix G is chosen so that theclosed loop matrix A� � A�GC is stable and has a Lyapunov matrix P satisfying


bothPA� � AT

� P � �Q ��

for some positive de�nite design matrix Q and the structural constraint

PB � CTFT ��

for some nonsingular matrix F � IRm�p� The discontinuous vector �o is given by

�o �

��o�t� y� u�

FCekFCek if Ce ��

otherwise��

where �o�t� y� u� is a scalar function which bounds the uncertainty� The observergiven in �� may be viewed as a Luenberger observer with an additional nonlinearterm� It can be shown �x�� in �� that assumptions A� and A� relating to thetriple �A�B�C� are necessary and su�cient conditions for the existence of such anobserver which is insensitive to matched uncertainty and induces a sliding motionon

Sw � f e � IRn � FCe � g ��

The original formulation of Walcott �Zak required the use of symbolic manipu�lation to synthesise the matrices G and P which completely de�ne the observer�More recently an analytic solution has been proposed based on the canonical formdescribed in �� An appropriate choice of G is that given in �� and anappropriate choice of

F � �P�B��T ��

This session considers the same example as Session �� but this time computes aWalcott� �Zak observer�Session �� wzobs

�� G�F��wzobs�A�B�C�



Enter � desired output estimation error pole�s� ��

G �

� �

��

��

� ��

��

F �

� � ��

��

��


In the special case of a square system an even more explicit solution can be obtained�This does not require the attainment of the canonical form �� explicitly andin certain circumstances produces an observer with better numerical properties�

This �nal session consider the furnace system from �� and computes a Walcott��Zak observer as in x�� in �� The observer is computed in two ways� �rst usingthe square system approach and secondly using the generic method� The numericalproperties of the two resulting error systems are compared�

Session �� sqobs

�� G�F��sqobs�A�B�C�

Enter � real stable poles for the output error system ��

G �

��

��

��

��

F �

��

��

�� V�D��eig�A�G�C��

�� cond�V�

ans �

��

�� G�F��wzobs�A�B�C�

The system has � stable invariant zero�s�

Enter � desired output estimation error pole�s� ��

G �

��

��

��

��

F �

��

��

�� V�D��eig�A�G�C��

�� cond�V�

Summary ��

ans �

��

��

It can be seen that the square system approach has in this case furnished eigenvec�tors with better conditioning�

� Summary

This document has described a collection ofMatlab m�les which have been devel�oped to design sliding mode controllers for linear systems with bounded nonlinearuncertainty� The algorithms are described in detail elsewhere and indeed this doc�ument is best viewed as an addendum to �� Here the emphasis is on describingthe software and providing examples of its use�

References

�� C� Edwards and S�K� Spurgeon� Sliding Mode Control� Theory and Applications�Taylor Francis� ��

�� U� Itkis� Control Systems of Variable Structure� Wiley� New York� ��

�� E�P� Ryan and M� Corless� Ultimate boundedness and asymptotic stability of aclass of uncertain dynamical systems via continuous and discontinuous control�IMA Journal of Mathematical Control and Information� ��

� V�I� Utkin� Variable structure systems with sliding modes� IEEE Transactions

on Automatic Control� ��

�� V�I� Utkin� Sliding Modes in Control Optimization� Springer�Verlag� Berlin��

�� B�L� Walcott and S�H� �Zak� Combined observer�controller synthesis for uncer�tain dynamical systems with applications� IEEE Transactions on Systems� Man

and Cybernetics� ��

� Appendix Toolbox Commands

comrobs

Purpose

To generate a compensator based output sliding mode controller

Syntax

Hhat�Dhat� S� L� P� Lam� T ��comrobs�A�B�C�

Description

Generates a compensator and the associated components of the slidingmode control law for the triple �A�B�C�� In order for this to be possibleconditions A� and A� need to be satis�ed� These are checked during theexecution� If n� m� p and r are the number of states� inputs� outputs andinvariant zeros respectively� then� provided the approach is applicable� theuser is prompted to supply three vectors�

� an n � p � r dimensional vector which represents the poles of theestimation error system� �In the notation of x�� the speci�ed polesare assigned to the matrix H from �� via pole placement��

� an n �m � r dimensional vector which constitute the poles of thesliding motion� �In the notation of x�� the gain matrixK is computedso that the eigenvalues of � #A�� A��K� are the speci�ed values��

� anm dimensional vector which will be used to assign the poles whichgovern the range space dynamics� �In the notation of x�� the matrix! will be built from the speci�ed poles��

The matrices Hhat� Dhat which are returned are the compensators� stateand input distribution matrices respectively� S is the matrix which de�nesthe switching function� L is the linear feedback component in the controllaw� P is a Lyapunov matrix which scales the switching function andLam represents the gain which pre�multiplies the unit vector componentin the nonlinear control� The orthogonal matrix T is obtained from thecanonical form for output feedback design from x� and is used to scale theoutputs�

Algorithm

First the canonical form for output feedback design is obtained and thecomponents that constitute the triple � #A�� A�� #C�� are identi�ed� Areduced order linear observer structure is then obtained in conjunctionwith a state feedback gain for the pair � #A�� A�� This e�ectively pa�rameterises both the compensator matrices and the switching function�The approach is described in detail in x�� in ��

Appendix� Toolbox Commands ��

contl

Purpose

This assembles the components of the state feedback unit vector controllaw

Syntax

L�P� Lam��contl�A�B� S� Phi�

Description

The pair �A�B� constitutes the state and input distribution matrices of thesystem and S represents a previously designed matrix which de�nes theswitching function� If the system has m inputs� then Phi is a stable m�mmatrix which will govern the linear part of the range space dynamics� Thefunction returns a matrix L which constitutes the linear state feedbackcomponent of the control law� the matrix Lyapunov matrix P which scalesthe hyperplane and Lam which is the gain which pre�multiplies the unitvector component in the nonlinear control law�

Algorithm

The control law is the unit vector control structure of Ryan Corless ��Details of the components are given in �� The justi�cation of theformula appears in x�� in ��

Appendix� Toolbox Commands �

contlia

Purpose

This assembles the components of the state feedback unit vector controllaw in the case when integral action is employed for tracking purposes

Syntax

L�Lr� Lrdot� Sr� Lam�P ��contlia�A�B�C�S� Phi� Sr�L�Lr� Lrdot� Sr� Lam�P ��contlia�A�B�C�S� Phi�

Description

The triple �A�B�C� constitutes the state� input and output distributionmatrices of the system� The matrix S represents a previously designedmatrix which de�nes the augmented state component of the switchingfunction and the parameter Sr represents the reference dependant com�ponent of the switching function� If the system has m inputs then Phi is astable m�m matrix which will govern the linear part of the range spacedynamics� The function returns the controller parameters in the casewhere integral action is employed to provide tracking and all the statesare available for use in the control law� L represents the linear augmentedstate feedback gain and Lr and Ldot represent feed�forward gains for thereference and derivative of the reference respectively� The symmetric posi�tive de�nite matrix P is the Lyapunov matrix which scales the hyperplaneand Lam is the gain associated with the unit vector component�If only �ve right hand arguments are supplied the reference dependantcomponent of the switching function Sr is calculated so that� in the nom�inal case� the steady state value of the integral action states tend to zero�

Algorithm

The control law is the unit vector control structure described in � �� Details of the construction� particularly with regard to the choice of Sr �is given in x�� in ��

See also

dea� lqcf� rpp


dea

Purpose

Hyperplane design using so�called direct eigenstructure assignment � i�e�specifying both the sliding mode poles and elements of the associatedeigenvectors

Syntax

S�dea�A�B� specent� lambda�S� V ��dea�A�B� specent� lambda�S�dea�A�B� specent� lambda� nocomp�S� V ��dea�A�B� specent� lambda� nocomp�

Description

The pair �A�B� constitutes the state and input distribution matrices of thesystem� When four right hand arguments are supplied� the vector lambdarepresents the n�m speci�ed sliding mode eigenvalues which must be real�n and m represent the number of states and inputs respectively�� Then� �n�m� matrix specent indicates the position and value of the desiredeigenvector entries� Elements of the eigenvector which can take arbitraryvalues are indicated by NaN�s� Thus the �rst column of specent is thedesired eigenvector associated with the �rst element in lambda etc� Thefunction returns the matrix S which de�nes the switching function� If twoleft hand arguments are provided an n��n�m� matrixV is returned whichrepresents the attained eigenvectors� If complex eigenvalues are desired��ve right hand arguments must be supplied and the scalar nocomp de�nesthe number of pairs of complex conjugate poles� Again the vector lambdade�nes the desired sliding mode poles� the �rst �i � � �i � �� nocomp�entries contain the real parts of the complex conjugate poles and the�rst �i �i � �� nocomp� contain the imaginary parts� Likewise the �rst�i � � �i � �� nocomp� columns of specent are associated with the realparts of the complex eigenvectors and the �rst �i �i � �� nocomp� areassociated with the imaginary parts�

Algorithm

The algorithm is that described in Appendix B�� in �� For implemen�tation purposes� the variable specpos in Appendix B�� is obtained fromspecent by interrogating the user about the speci�ed elements using theMatlab command isnan�

See also

eigbuil


disobs

Purpose

Calculates the gains of a sliding mode observer which is insensitive to aparticular class of uncertainty

Syntax

Gl�Gn� P��disobs�A�D�C�

Description

The pair �A�C� represents the state and output distribution matrix ofthe system and D represents a distribution matrix which describes thechannels in which the uncertainty occurs� It is assumed that if n and prepresent the number of states and outputs respectively� then D is n� qwhere q � p� The function returns the linear gain Gl and non�linear gainGn along with the Lyapunov matrix P� which constitutes a discontinuousobserver� For the theory to be valid both C and D should be full rank�the Markov parameter CD must be full rank and any invariant zeros oftriple �A�D�C� must lie in the C�� During the computations a prompt isgiven for a vector of dimension n� p� r where r the number of invariantzeros of �A�D�C�� This vector speci�es the desired stable poles of thesliding motion� In addition the user is prompted for p stable real poleswhich form part of the state estimation error dynamics�

Algorithm

The function puts the given system in the canonical form for observerdesign described in �� The gains are constructed from equations�� and �� and from solving the Lyapunov equation �� For detailssee x�� in ��

See also

sqobs� wzobs


eigbuil

Purpose

Forms the matrix which determines the speci�ed components within eacheigenvector for use in the design of a hyperplane by direct eigenstructureassignment

Syntax

specent�eigbuil�n�m�specent�eigbuil�n�m� nocomp�

Description

The scalar n is the size of the state space vector and m is the number ofinputs� If three right hand arguments are presented then nocomp is thenumber of complex pairs of eigenvalues that make up the desired polesfor the sliding motion� The function returns a matrix of size n� �n�m�whose columns indicate the position and value of the speci�ed elementsin the associated eigenvector� The user is prompted for the position andvalues of the speci�ed entries eigenvector by eigenvector� The remainingentries in the appropriate columns of specent are �lled automatically byNaN�s�

See also

dea


intac

Purpose

Calculates the augmented state and input distribution matrices for de�signing a control law incorporating integral action

Syntax

Aa�Ba��intac�A�B�C�

Description

For the system represented by the triple �A�B�C� the augmented stateand input distribution matrices Aa and Ba as in � �� If this approach isnot applicable because of the presence of any invariant zeros of �A�B�C� atthe origin� empty matrices are returned and an appropriate error messageis displayed�

See also

contlia


lqcf

Purpose

Hyperplane design using a quadratic cost function

Syntax

S�lqcf�A�B�Q�S�E��lqcf�A�B�Q�

Description

For a given system represented by the state and input distribution ma�trices �A�B� the function returns a matrix S which de�nes the switchingfunction by minimising the cost function

J �

Z �

ts

xTQxdt

where Q is a symmetric positive de�nite matrix of the same order asthe state space and ts is the time at which the sliding motion begins�In the coordinates of the regular form� and thinking of the hyperplanedesign problem as a state feedback problem for a certain subsystem� thequadratic minimisation problem becomes a conventional well�posed LQRproblem which can be solved using the standard control system toolboxroutines�

Algorithm

Regular form is �rst established and a further series of transformationsestablishes the cost function �� and the reduced order system matrix�� In this form the problem becomes a standard Linear QuadraticRegulator problem which is solved using the routines from the ControlSystem Toolbox� Details are given in x � �� in ��


obsfor

Purpose

Performs the necessary state transformations to put a system triple intothe canonical form for observer design

Syntax

Ac�Dc�Cc� T c��obsfor�A�D�C�

Description

Here the matrices A and C represent the state and output distributionmatrices respectively and D is a matrix which describes the channelsin which the uncertainty acts� The function returns the nominal triple�A�D�C� in the canonical form used in the construction of the observergains given in �� In order for this to be possible CD must befull rank and any invariant zeros of the triple �A�D�C� must be in theC�� Checks are made during the execution of the function to verify theseconditions� If the number of columns of D is less than the number ofrows of C then some �exibility exists in choosing the eigenvalues of thesliding motion� In this case the user is prompted to supply an appropriatenumber of stable poles� The state transformation matrix to induce thisform is given in Tc�

Algorithm

The canonical form for output feedback sliding mode control is �rst estab�lished� A further transformation using all the available degrees of freedomis made to achieve the canonical form in �� For further details seex�� in ��


outfor

Purpose

Performs the necessary state transformation to put a system triple intothe canonical form for output feedback design

Syntax

Af�Bf�Cf� T can� r��outfor�A�B�C�

Description

For the system represented by the triple �A�B�C� the function computes�where possible� the canonical form for output feedback design� This re�quires that assumptions A� and A� are valid� If these are found not tohold then the function returns empty matrices� The variable r representsthe number of invariant zeros of the system �A�B�C�� The matrix Tcanrepresents the coordinate transformation necessary to bring about thecanonical form�

Algorithm

Details of the state�space transformations are given in x�� in ��

See also

comrob


regfor

Purpose

Transforms a given state and input distribution matrix into regular form

Syntax

A�� A�� B�� T r��regfor�A�B�

Description

For the system with state and input distribution matrices A and B thefunction returns the sub�matrices and the orthogonal transformation ma�trix that brings about the regular form described in ��

Algorithm

A QR factorisation is �rst obtained for the input distribution matrix �the required transformation is then obtained by reordering the rows� Theapproach is described in more detail in x �� in ��


rpp

Purpose

Hyperplane design using pole placement with robust eigenstructure as�signment

Syntax

S � rpp�A�B� sp�

Description

For the system represented by the state and input distribution matricesA and B the function calculates the matrix S which de�nes the switchingfunction using robust eigenstructure assignment� If n and m representthe number of states and inputs respectively then sp is a vector of ordern�m containing the desired eigenvalues� Any complex eigenvalues in thevector sp must appear in consecutive complex conjugate pairs and theeigenvalues cannot be placed with multiplicity greater than the numberof inputs�

Algorithm

Regular form is �rst achieved then the modi�ed pole placement commandvplace is utilised �which relies on the Control System Toolbox routineplace�� For details see Appendix B�� in ��

See also

dea

Appendix� Toolbox Commands

sqobs

Purpose

Calculates the gain matrices for a Walcott� �Zak observer in the specialcase of a square system

Syntax

G�F ��sqobs�A�B�C�

Description

For the system represented by the triple �A�B�C� this function produces aWalcott� �Zak observer for the special case of square systems i�e� when thenumber of inputs equals the number of outputs� Necessary and su�cientconditions for the observer to exist are that assumptions A� and A� aresatis�ed� The arguments returned are the linear gain matrixG along withthe matrix F which scales the switching function� During the executionthe user is prompted for a vector of order p containing stable real poles�where p is the number of outputs�� These are assigned to form a subsetof the poles of the state estimation error system�

Algorithm

The design is made from regular form rather than from the observer canon�ical form� The explicit solution is described in x�� in ��

See also

wzobs� disobs


vplace

Purpose

Performs pole placement with robust eigenstructure assignment

Syntax

K�vplace�A�B� sp�

Description

For the system represented by the state and input distribution matrix Aand B this function returns a state feedback matrix K which assigns tothe closed loop the poles contained in the vector sp� This command isa modi�cation of the control system tool box command pole placementroutine place and performs an initial transformation to remove any re�dundant columns of the input distribution matrix�

Algorithm

An initial QR factorisation of the input distribution matrix is made andany redundant columns are removed before invoking the place commandfrom the Control System Toolbox �see Appendix B�� in �� for details��


wzobs

Purpose

Computes the gains for a Walcott� �Zak observer

Syntax

G�F ��wzobs�A�B�C�

Description

For the system represented by the triple �A�B�C�� this function calculatesthe gains for a Walcott� �Zak observer� Necessary and su�cient conditionsfor the observer to exist are that assumptions A� and A� are satis�ed� Thearguments returned are the linear gain matrix G along with the matrix Fwhich scales the switching function� During the computations a promptis given for a vector of dimension n�p�r where n the number of states� pis the number of outputs and r the number of invariant zeros of �A�B�C��This vector allows the user to specify the assignable poles in the top leftsub�block of the observer canonical form� The user is prompted for avector of order p containing stable real poles� These poles are assigned toform a subset of the poles of the state estimation error system�

Algorithm

The function puts the given system in the canonical form for observerdesign described in �� The gains are constructed from equations�� and �� For details see x�� in ��

See also

sqobs


comrobs generates a compensator based output sliding mode controller

contl assembles the components of the state feedback unit vector controllaw

contlia assembles the components of the state feedback unit vector con�trol law in the case when integral action is employed for trackingpurposes

dea hyperplane design using direct eigenstructure assignment

disobs calculates the gains of a sliding mode observer which is insensitiveto a particular class of uncertainty

eigbuil form the matrix which determines the speci�ed components withineach eigenvector for use in the design of a hyperplane by directeigenstructure assignment

intac calculates the augmented state and input distribution matrices fordesigning a control law incorporating integral action

lqcf hyperplane design using a quadratic cost function

output performs the necessary state transformations to put a system tripleinto the canonical form for observer design

obsfor performs the necessary state transformation to put a system tripleinto the canonical form for output feedback design

regfor transforms a given state and input distribution matrix into regularform

rpp hyperplane design using pole placement with robust eigenstructureassignment

sqobs calculate the gain matrices for a Walcott� �Zak observer in the specialcase of a square system

vplace pole placement with robust eigenstructure assignment

wzobs computes the gains for a Walcott� �Zak observer

Table �� Summary of Matlab commands

sliding mode manual

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