The Geometric Approach to Control - unibo.itThe Geometric Approach to Control a Light Presentation of Theory and Applications Dipartimento di Elettronica, Informatica e Sistemistica

Giovanni Marro

The Geometric Approach to Control

a Light Presentation of Theory and Applications

Dipartimento di Elettronica, Informatica e Sistemistica

Universita di Bologna, Viale Risorgimento 2, I-40136 Bologna, Italy

Phone: +39 -51-2093046, Fax: +39 -51-2093073, E-Mail: [email protected]

Preface

The geometric approach is a collection of tools, properties and algorithms thatenable us to analyze and state in precise terms the main properties of dynamicsystems and suggest how to modify them with control. It has gradually beendeveloped for about forty years, as new linear and nonlinear problems aroseand were investigated.

Statements in terms of subspaces (or, more generally, sets) instead ofmatrices are compact and clean, and understanding of their meaning is highlyfacilitated.

Unfortunately, although the basic geometric methods and tools shouldbe profitably taught in most system and control theory courses, at least inthe very simplified form herein presented that naturally links them to theconcepts of controllability, observability and stability, by now they have beennearly abandoned even in the research literature.

The motive of this scarce interest is that, although the geometric tools arevery simple and supported by exhaustive computational machinery, it is ratherdifficult to get a complete panorama of them, since their presentation in theliterature by several authors over a period of about forty years is not uniformin style. In many cases it is misconceived in choosing the most convenientway to the solution (often far from being quick and simple) and frequentlypresented with unnecessarily heavy mathematics, that covers their inherentsimplicity.

This contribution aims at giving an extremely simplified presentation ofthe basic insights into the system structure on which the geometric approachrelies and a short review of the use of the geometric approach tools to classify,analyze and solve a certain number of basic control problems. In fact, someproblems already considered and solved in the literature from a strict and rig-orous mathematical standpoint can profitably be revisited in the geometricapproach context to form a different insight, more suited to realize connec-tions between apparently different problems, and an efficient computationalenvironment. Evident examples of problems where this unifying treatment is

VI Preface

particularly profitable are the synthesis of unknown-input observers and ofunits for fault detection and identification.

This presentation is organized as follows: Chapter 1 is a “notes and refer-ences” section, i.e., it reports in chronological order the main contributions tothe geometric approach, from the very beginning that, in the author’s opinion,goes back to the dawn of the state space analysis of dynamical systems. InChapter 2 the cornerstones of the geometric approach are defined and somecomputational tools that allow their use in practice are listed. In Chapter 3some new properties of linear systems, such as left and right invertibilityand relative degree, are defined by using the geometric tools introduced inthe previous chapter; “new” simply means that these definitions are specificto the geometric approach environment and rather far from those currentlyadopted in the literature, that refer to the system representation throughtransfer matrices (in the frequency domain). The importance of using dualityto avoid repetition of the same approach for dual problems is also pointedout in Chapter 3. In Chapter 4 the most important regulation and observa-tion problems that can be approached and solved exactly in geometric termsare considered and conditions for their solvability are stated; the word “ex-actly” means that tracking or observation must be done without error. Thisrequirement is relaxed in Chapter 5, where for some of the previously statedproblems the non-exact but optimal solution (with respect to the H2 norm)is also considered in geometric terms, thus achieving a unified setting of thegeometric approach with very basic well known regulation and observationproblems, such as the Kalman regulation and filtering. Finally, in Chapter 6the possible insertion of some of the previously presented particular problemsin a more general regulation layout is briefly described and some possiblechallenges for future research are conjectured.

Bologna, April 2007 Giovanni Marro

Notation

∈ belonging to⊆ contained in∀ for allR the set of all the real numbersR

n the set of all the n-tuples of real numbersAT the transpose of matrix A

(the conjugate transpose if A is complex)σ(A) the spectrum of matrix A

〈x, y〉 the scalar product of vectors x and y

Cg the subset of all the complex numbers related to stable modes(with real part negative or absolute value less than one)

H a generic subspaceH⊥ the orthogonal complement of HAH the image of H in the linear transformation A

A−1 H the inverse map of H in the linear transformation A

Σ a generic linear time invariant (LTI) systemJ a generic invariantR the reachability subspace of ΣU the unobservability subspace of ΣV a generic controlled invariantS a generic conditioned invariantV∗ the maximum output-nulling controlled invariant of ΣR∗ the maximum output-nulling reachable subspace of ΣS∗ the minimum input-containing conditioned invariant of ΣZ(V) the set of the internal unassignable eigenvalues of VZ(Σ) the set of the invariant zeros of Σ

Contents

1 A sketch of history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The three books on the geometric approach . . . . . . . . . . . . . . . . . 4

2 The basic geometric tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Some preliminary definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Controlled invariants and V∗, the first main tool . . . . . . . . . . . . . 122.4 Conditioned invariants and S∗, the second main tool . . . . . . . . . . 132.5 R∗, the reachable subspace on V∗, and the invariant zeros of Σ . 142.6 Extension to quadruples and computational tools . . . . . . . . . . . . . 15

3 Geometrically-stated system properties . . . . . . . . . . . . . . . . . . . . 193.1 Left invertibility, right invertibility, and relative degree . . . . . . . . 193.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Geometrically-solved regulation problems . . . . . . . . . . . . . . . . . . 234.1 Disturbance decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Disturbance decoupling by dynamic output feedback . . . . . . . . . . 294.3 Model following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 The multivariable regulator with internal model . . . . . . . . . . . . . . 334.5 Noninteraction and fault detection and identification . . . . . . . . . . 35

5 Geometric approach to H2-optimal regulation and filtering . 415.1 Disturbance decoupling in minimal H2-norm . . . . . . . . . . . . . . . . . 41

6 Conclusions and challenges for the future . . . . . . . . . . . . . . . . . . 49

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

1

A sketch of history

1.1 The papers

The enhancement of the geometric approach has been almost uniform in time,with new concepts and tools gradually added as the need arose. We will tryto present the contributions by decades, even though this custom has beendropped.

• The sixties (1960-69)In the early sixties the research on systems and control was directed to-

wards using state space techniques, probably because of the growing interestfor dynamic optimization techniques, based on calculus of variations, dynamicprogramming, Pontryagin maximum principle. The concept of controllabilityand observability were introduced by Kalman [1], and a great interest for re-search in optimization techniques related to control problems arose from thepublication in English of the book by Pontryagin et al. [2]. An early geomet-ric interpretation of the Pontryagin maximum principle was given by Basileand Marro [3]. At those times a significant research work concerned systeminvertibility (Sain and Massey [4], Silverman [5]) and decoupling (Morgan[6], Gilbert [7]). These problems were destined to be further investigated andclarified later by using geometric techniques. The basic tools of the geometricapproach were first introduced by Basile and Marro in 1969 [8], in a paperwhere controlled invariants and conditioned invariants were defined, theirmain properties were investigated, duality was pointed out and some efficientcomputational algorithms, that are still profitably used to solve the analysisand synthesis problems described in Chapters 4 and 5, were proposed. Nev-ertheless this is not the only contribution to the geometric approach in thesixties. There are other papers by Basile, Laschi and Marro published in anoutstanding Italian scientific journal [9] and presented in the most importantItalian scientific convention (AEI-ANIPLA) [10, 11]. In [10] the disturbancerejection problem without the internal stability requirement was approachedand solved with geometric tools, and in [11] the dual problem, unknown-input

2 1 A sketch of history

observation of a linear function of the state, was considered and also solvedin the geometric approach context, thus completing a previous result by thesame authors [12].

• The seventies (1970-79)At the beginning of the seventies Wonham and Morse independently in-

troduced a similar coordinate-free approach to control problems [13]. Theexpression geometric approach was first used in the title of their initial pa-per in this context, and an algorithm similar to that introduced in [8] forthe computation of the maximal controlled invariant was presented, aimingat deriving the controllability subspace, i.e., the state set reachable from theorigin with trajectories belonging to a given subspace. It should be pointedout that in [13, 14] there is no trace of geometric tools similar to controlledand conditioned invariants. The controllability subspace was finalized to thesolution of the noninteracting control problem. In [13] this problem was solvedwith state feedback and algebraic feedforward units, while subsequently thesame problem was solved by Morse and Wonham with state extension and al-gebraic feedforward units [15], and by Basile and Marro with pre-stabilizationof the controlled system and dynamic feedforward units [16]. The concepts ofsystem invertibility were introduced by Morse and Wonham in [14] and re-visited in the light of duality by Basile and Marro [17], while a deep insightinto the structure of LTI systems and their connection with their geometricproperties was given by Morse [18]. In 1974 Wonham published the first bookon the geometric approach [19], where he first adopted the new term (A,B)-invariant in place of controlled invariant. Due to the circulation of this book,this term was largely used in the literature and only recently there has been acertain tendency to restore the original name. Other significant contributionsin the seventies were made by Bhattacharyya [20, 21] and Silverman, who firsttraced out a connection between the geometric approach and optimal controlin the discrete-time case [22]. Extensions from triples to quadruples of themain definitions and properties, thus defining output-nulling subspaces andstrong controllability and observability concepts were introduced by Anderson[23] and Molinari [24, 25].

• The eighties (1980-89)Duality and conditioned invariance, that were completely ignored in the

Wonham’s book and papers, began to awaken a certain interest. At thosetimes conditioned invariants were called (A,C)-invariants or (C,A)-invariants,so that the symbols of the system matrices were required to be (A,B,C). Anearly investigation on the structure of the set of all conditioned invariantswith the same dimension was come out by Hinrichsen, Munzer and Pratzel-Woltzer [26]. Of course, rediscovering conditioned invariance brought someimportant new contributions to the geometric approach theory, namely anew understanding of unknown-input observers as dual of disturbance decou-

1.1 The papers 3

pling controllers and their use in solving the disturbance decoupling problemby output dynamic feedback, a problem which was solved by Schumacherwithout internal stability [27] and by Willems and Commault with internalstability [28]. Schumacher and Aling also gave a more complete interpreta-tion of the linear system structure in geometric terms [29]. Basile and Marro[30] and Schumacher [31] provided a new tool to deal with stabilizability insynthesis problem, the self-bounded controlled invariant , that was used byBasile, Marro and Piazzi in a new solution of the disturbance decouplingproblem by output feedback with stability [32], aiming at designing minimal-dimension controllers. An extended mathematical interpretation of controlledand conditioned invariants when the admissible signals include distributionswas proposed by Willems who introduced the almost controlled and the al-most conditioned invariants. [33, 34]. A new, very broad field of research inthe geometric approach context was started at the beginning of the eighties,i.e., the extension to the nonlinear control theory, with significant contribu-tions by Brockett [35], Hirschorn [36], Isidori et al. [37], Nijemeijer and Vander Shaft [38], Krener [39], where controlled invariant distributions, and con-ditioned invariant distributions were defined and used. It is remarkable thatalso in the nonlinear case the concepts of controlled and conditioned invariantsdistributions descended from the extensions of the concepts of controllabilityand observability (see the early contribution by Krener and Hermann [40]).In this context the seminal book by Isidori [41] is very remarkable, since itcleared the way for numerous new appplications of these new geometric con-cepts. Other approaches to take into account nonlinearities or dependence ofsystem matrices on a finite set of parameters by means of local linear approxi-mation were proposed by Bhattacharyya [42] and Basile and Marro [43]. Soonafter Massoumnia, Verghese and Willsky applied the geometric approach tofault detection and identification problems [44, 45], thus giving a new, inter-esting interpretation to the duality of the noninteracting control problem. Aformer, interesting interpretation of the singular optimal control using almostcontrolled and conditioned invariants was proposed by Willems, Kıtapci, andSilverman [46].

• The nineties (1990-99)In 1992 Basile and Marro published the second book on the geometric

approach [47]. Stoorvogel [48] first applied the geometric approach techniquesto H2 optimal control problems. His approach was significantly different fromthat reported here in Chapter 5, since it uses linear matrix inequalities (LMI)to solve a reduced-order Riccati equation, thus departing from the applica-tion of the classical geometric approach algorithms to the linear time invari-ant (LTI) system obtained by differentiating the Hamiltonian function corre-sponding to the optimization problem. Further, relevant publications in thiscontext are by Saberi, Sannuti and Chen [49, 50].

4 1 A sketch of history

• The new century (2000-2006)In 2002 Trentelman, Stoorvogel and Hautus published the third book on

the geometric approach [51]. Almost exact decoupling with preaction in thenonminimal system case was interpreted as a geometric approach problemand solved by Marro, Prattichizzo and Zattoni [52]. The singular and cheapLQR problem in the discrete-time domain was solved by the same authorswith geometric techniques applied to the LTI system derived from the Hamil-tonian function [53]. The complete geometric solution to the model followingproblem both feedforward and feedback was investigated by Marro and Zat-toni [54]. Some new results on the solution of finite-horizon LQ problems bothin the continuous and discrete-time, directed to the solution of the H2 opti-mal regulation problem with preview or H2 optimal filtering with delay werederived by Ferrante, Marro and Ntogramatzidis [55, 56].

1.2 The three books on the geometric approach

Fig. 1.1. The three book on the geometric approach.

Fig. 1.1 shows the covers of the books on the geometric approach publishedup to now. Wonham has the great merit of having widely publicized thegeometric approach from the very beginning. However the three editions ofthe book (1974, 1979, 1985) have been only slightly modified compared to eachother, so that new developments of the theory and new applications reportedmeanwhile in the literature were neglected and the initial heavy mathematicalformalism was preserved.

Basile and Marro (1992) emphasize duality and dual problems, that wereneglected in the Wonham book, but deeply investigated in the literature,

1.2 The three books on the geometric approach 5

particularly at the beginning of the eighties. The need for considering togetherboth the controlled and the conditioned invariants for a clean solution ofnumerous control problem through both regulation and observation is alsopointed out in the title.

Trentelman, Stoorvogel and Hautus (2002) also give prominence to duality.In their book connections between the geometric approach and transfer func-tion techniques are often analyzed within a simple but rigorous mathematicalbackground and the need to extend signals from functions to distributionsfor the most complete solution of many control and observation problems isconsidered. Furthermore, linear quadratic optimal control is presented from amodern standpoint, and an interesting bridge towards achieving minimal H2

and H∞ norm solutions is set.

2

The basic geometric tools

The geometric approach deals with subspaces. Properties of systems are ex-pressed in terms of subspaces. The most important analysis and synthesis al-gorithms are based on operations on subspaces, like sum, intersection, directand inverse linear transformation, calculation of the orthogonal complement.

2.1 Some preliminary definitions

The concepts of linear algebra that are required to build the geometric controltheory belong to a very restricted set. For the sake of completeness they willbe briefly recalled.

Given a vector space X over a field F (usually the field of real or complexnumbers) a subset V of X is a subspace of X if

αx + βy ∈ V ∀α, β ∈ F , ∀x, y ∈ V (2.1)

Any subspace of X is a vector space over F . The origin is a subspace ofX that is contained in all other subspaces of X ; it will be denoted by {0}.

A function A : X →Y , where X and Y are vector spaces over the samefield F , is a linear function or linear map or linear transformation if

A(αx + βy) = α A(x) + β A(y) ∀α, β ∈ F , ∀x, y ∈ X (2.2)

In other words, a linear function is one that preserves all linear combina-tions. The particular linear function I : X →X such that I(x) = x ∀x∈X iscalled the identity function.

Let A : X →Y be a linear function. The set

imA :={

y : y = A(x), x∈X}

(2.3)

is called the image of A. The dimension of imA is called the rank of A anddenoted by ρ(A).

8 2 The basic geometric tools

Let A : X →Y be a linear function. The set

kerA :={

x : x∈X , A(x) = 0}

(2.4)

is called the kernel or null space of A. The dimension of kerA is called thenullity of A and denoted by ν(A).

Both imA and kerA are subspaces (of Y and X respectively).Let {x1, . . . , xh} be any set of vectors belonging to a vector space X over a

field F . The span of {x1, . . . , xh} (denoted by sp{x1, . . . , xh}) is the subspaceof X

sp{x1, . . . , xh} :={

x : x =h∑

i=1

αi xi , αi ∈ F (i = 1, . . . , h)}

(2.5)

A set of vectors {x1, . . . , xh} belonging to a vector space V over a field Fis a basis of V if it is linearly independent and sp{x1, . . . , xh} = V .

V

v1

v2w

X

Fig. 2.1. The representation of a subspace.

A specific feature of the geometric approach is to be coordinate-free. Inrelated computations, quite detached from theory, subspaces and linear trans-formations are represented by matrices. Referring to Fig. 2.1, the subspace Vof a vector space X = R

n is represented by a basis matrix V whose columnsv1 and v2, linearly independent and possibly orthonormal, span V , or by itsorthogonal complement W with basis matrix W (in the figure, the single vec-tor w). It follows that our subspace can be indifferently expressed as V = imV

or V = ker W T. The matrices V or W are numerical expressions of V , clearlynon-unique, whose availability is generally assumed when dealing with sub-spaces.

To mechanize computational procedures on basis matrices for operationson subspaces, the Gram-Schmidt orthonormalization process or a column se-lection algorithm based on the SVD (singular value decomposition) - bothprovided with a suitable linear dependence test - can be used. The basicoperations on subspaces and the corresponding processing of basis matricesare:

2.1 Some preliminary definitions 9

1. Sum:Z = X + Y =

{z : z = x + y , x∈X , y ∈Y

}(2.6)

Let X,Y be basis matrices of subspaces X ,Y ⊆Fn, with F = R or F = C.A basis matrix Z of Z =X +Y is obtained by orthonormalizing thecolumns of matrix [X Y ].

2. Linear transformation:

Y = AX ={y : y = Ax , x∈X

}(2.7)

Let X be a basis matrix of subspace X ⊆Fn. A basis matrix Y of Y = AXis obtained by orthonormalizing the columns of matrix AX.

3. Orthogonal complementation:

Y = X⊥ ={y : 〈y, x〉 = 0 , x∈X

}(2.8)

Let X, n×h, be a basis matrix of subspace X ⊆Fn. A basis matrix Y

of Y =X⊥ is obtained by orthonormalizing the columns of matrix [X In]and selecting the last n−h of the n vectors obtained.

4. Intersection:Z = X ∩ Y =

{z : z ∈X , z ∈Y

}(2.9)

It is reduced to sum and orthogonal complementation owing to (2.13),(2.15).

5. Inverse linear transformation:

X = A−1 Y ={x : y = Ax , y ∈Y

}(2.10)

It is reduced to direct transform and orthogonal complementation owingto (2.13), (2.21).

The following relations are often considered in algebraic manipulationsregarding subspaces.

X ∩ (Y + Z) ⊇ (X ∩ Y) + (X ∩ Z) (2.11)

X + (Y ∩ Z) ⊆ (X + Y) ∩ (X + Z) (2.12)

(X⊥)⊥ = X (2.13)

(X + Y)⊥ = X⊥ ∩ Y⊥ (2.14)

(X ∩ Y)⊥ = X⊥ + Y⊥ (2.15)

A (X ∩ Y) ⊆ AX ∩ AY (2.16)

A (X + Y) = AX + AY (2.17)

A−1 (X ∩ Y) = A−1 X ∩ A−1 Y (2.18)

A−1 (X + Y) ⊇ A−1 X + A−1 Y (2.19)

The following well-known properties are also useful when dealing withsubspaces.


1. Relations (2.11), (2.12) hold with the equality sign if any one of the in-volved subspaces X ,Y ,Z is contained in any of the others.

2. Consider a linear map A :Fn →Fm (F = R or F = C) and any two sub-spaces X ⊆Fn and Y ⊆Fm. In the real and complex field (where AT refersto the conjugate transpose) the following relation holds:

AX ⊆ Y ⇔ AT Y⊥ ⊆ X⊥ (2.20)

3. Consider a linear map A :Fn →Fm (F = R or F = C) and a subspaceY ⊆Fm. In the real and complex field (where AT refers to the conjugatetranspose) the following relation holds:

(A−1 Y)⊥ = AT Y⊥ (2.21)

2.2 Invariant subspaces

Let X be a vector space and A : X →X a linear map. A subspace J ⊆X issaid to be an invariant under A or an A-invariant , if

AJ ⊆ J (2.22)

Consider the change of basis defined by the transformation T = [T1, T2]with im T1 = J . The new matrix A′ = T−1AT has the structure

A′ =

[A′

11 A′12

0 A′22

](2.23)

The A-invariant J is said to be internally stable if σ(A′11)∈Cg and exter-

nally stable if σ(A′22)∈Cg.

It is easily proven that the sum and the intersection of two or more A-invariants are A-invariants; hence any subset of the set of all the A-invariantscontained in a given subspace C is closed with respect to the sum and the setof all the A-invariants containing a given subspace B is closed with respectto the intersection. Hence these sets are lattices with respect to the partialordering expressed by ⊆ and the binary operations + and ∩, and admit asupremum J ∗

↑ (the sum of all the A-invariant contained in C) and an infimumJ ∗

↓ (the intersection of all the A-invariant containing B), respectively.If A :Fn →Fn (F = R or F = C) the subspace J ∗

↑ can be computed withthe sequence

J1 = CJi = C ∩ A−1 Ji−1 (i = 2, 3, . . .)

(2.24)

and J ∗↓ with the sequence

2.2 Invariant subspaces 11

u

Σy

x

Fig. 2.2. A system with input, state and output.

J1 = BJi = B + A Ji−1 (i = 2, 3, . . .)

(2.25)

Both the above sequences converge in a finite number of steps (at most n).Consider the system Σ shown in Fig. 2.2, with input u∈R

p, state x∈Rn,

output y ∈Rq. For the sake of generality and to achieve a better insight into

the meaning of some geometric features of the state trajectories and a betterunderstanding of the related algorithms, we will consider both continuous-time and discrete-time linear time-invariant (LTI) systems, whose equationsand state trajectories are represented in Fig. 2.3 where X denotes the statespace. In this figure, and as a rule in all the figures, the state space is assumedto be three-dimensional for easy graphics. These systems are often referredto as the triples (A,B,C). Matrices B and C are assumed to be maximumrank. Let B= imB. The control action consists in the possibility of choosingthe state velocity in the subspace B in the continuous-time case or the nextstate in the linear variety B+ x(k) in the discrete-time case. This choice,represented by arrows in Fig. 2.3, leads to the possibility of obtaining differenttrajectories x(·) in the state space by taking different control functions u(·).Systems (2.26) and (2.27) are said to be internally stable if σ(A)∈Cg.

Continuous-time:

x(t) = A x(t) + B u(t)y(t) = C x(t)

(2.26)

X

x(t)

x(t)

0

Discrete-time:

x(k+1) = A x(k) + B u(k)y(k) = C x(k)

(2.27)

. . .

. . .

X

x(k)

x(k + 1)

0

Fig. 2.3. System equations and state trajectories.


The geometric approach is not a unified discipline. Different authors preferto use different tools and layouts of the theory in explanation. In this note theoverall theory will be based on three very intuitive concepts: the controlledinvariant , the conditioned invariant and the invariant zero.

2.3 Controlled invariants and V∗, the first main tool

Given a linear map A : X →X and a subspace B⊆X , a subspace V ⊆X isan (A,B)-controlled invariant if the following inclusion holds

AV ⊆ V + B (2.28)

Controlled invariants have the following properties:

1. The sum of any two (A,B)-controlled invariants is an (A,B)-controlledinvariant.

2. Let B= imB, C = kerC. A state trajectory x(·) of the continuous ordiscrete-time LTI system Σ can be made invisible at the output by meansof a suitable control action if and only if the initial state x(0) belongs toan (A,B)-controlled invariant contained in C.

3. For any (A,B)-controlled invariant V there is at least one matrix F (calleda friend of V) such that (A + BF )V ⊆ V. If, furthermore, a matrix F existssuch that V is an internally stable and/or an externally stable (A + BF )-invariant, V is said to be internally stabilizable and/or externally stabiliz-able, respectively.

Property 1 implies that the set of all the controlled invariants contained ina given subspace C ⊆X , that is a semilattice with respect to ⊆ and +, admitsa supremum, that coincides with their sum. It is computed with the sequence

V1 = CVi = C ∩ A−1 (Vi−1 + B) (i = 2, 3, . . .)

(2.29)

that converges in a finite number of steps. In (2.29) the inverse map A−1(Vi−1 +B)denotes the set of all the points whose image according to A belongs to(Vi−1 +B). Referring to the system Σ (continuous or discrete-time), the se-quence (2.29) with B= imB, C = kerC yields V∗, the maximal (A,B) con-trolled invariant contained in C, that is the locus of all the possible state tra-jectories of Σ invisible at the output. V∗ is called the maximal output-nullingcontrolled invariant subspace of Σ.

The meaning of Properties 2 and 3 above is pointed out in Fig. 2.4: a statetrajectory that crosses a controlled invariant V can be maintained on it with asuitable control action (Fig. 2.4.a) and this control action can be obtained bymeans of state-to-input algebraic feedback, called state feedback (Fig. 2.4.b).Since V ⊆V∗ ⊆C, the segment of trajectory on V is invisible at the output.

2.4 Conditioned invariants and S∗, the second main tool 13

V

x(0) X

a)

u y

x

Σ

Fb)

Fig. 2.4. The meaning of Properties 2 and 3 of controlled invariants.

2.4 Conditioned invariants and S∗, the second main

tool

The conditioned invariants are dual to the controlled invariants. Given a lin-ear map A : X →X and a subspace C ⊆X , a subspace S ⊆X is an (A, C)-conditioned invariant if

A (S ∩ C) ⊆ S (2.30)

Conditioned invariants have the following properties:

1. The intersection of any two (A, C)-conditioned invariants is an (A, C)-conditioned invariant.

2. Let B= imB, C = kerC. The state trajectories of the discrete-time systemΣ that originate at x(0) = 0 can be made invisible at the output for acertain number of steps ρ while belonging to a given subspace S if andonly if S is a (A, C)-conditioned invariant containing B.

3. For any (A, C)-conditioned invariant S there is at least one matrix G

(called a friend of S) such that (A + GC)S ⊆ S. If, furthermore, a matrixG exists such that S is an externally stable and/or an internally stable(A + GC)-invariant, S is said to be externally stabilizable and/or internallystabilizable, respectively.

Property 1 implies that the set of all the conditioned invariants containinga given subspace B⊆X , that is a semilattice with respect to ⊆, ∩, admitsan infimum, that coincides with their intersection. It is computed with thesequence

S1 = BSi = B + A (Si−1 ∩ C) (i = 2, 3, . . .)

(2.31)

that converges to it in a finite number of steps. Referring to the system Σ, thesequence (2.31) with B= imB, C = kerC yields S∗, the minimal (A, C) condi-tioned invariant containing B, that for discrete-time systems is the maximalsubspace of the state space reachable from the origin in at most ρ steps withtrajectories that have all the states except the last one invisible at the output.The value of ρ is the number of steps required for convergence of sequence


(2.31). S∗ is also called the minimal input-containing conditioned invariantsubspace of Σ.

Property 2 has no counterpart in the continuous-time case unless controlfunctions, that are usually considered to be piecewise continuous, are extendedto include distributions. This aspect is beyond the scope of this monographand is therefore omitted.

S 0

X

a)

u y

f

Σ

Gb)

Fig. 2.5. The meaning of Properties 2 and 3 of conditioned invariants.

The meaning of Properties 2 and 3 above is pointed out in Fig. 2.5: a statetrajectory of a discrete-time system starting from the origin can be maintainedon a subspace S for a certain number of steps with a suitable control actionif and only if S is a conditioned invariant containing B (Fig. 2.5.a) and thiscontrol action can be obtained by means of output-to-state algebraic feedback(Fig. 2.5.b), where f denotes an input directly acting on the state. This typeof feedback is called output injection.

2.5 R∗, the reachable subspace on V∗, and the

invariant zeros of Σ

Still referring to the system Σ (continuous or discrete-time), consider the pre-viously defined maximal output-nulling subspace V∗ and define the subspaceR∗ ⊆V∗ as the maximum subspace reachable from the origin with state tra-jectories completely belonging to V∗. It has been shown by Morse [18] that

R∗ = V∗ ∩ S∗ (2.32)

Hence R∗ can be computed by using again the sequences (2.29) and (2.31).Note that R∗ is an (A,B)-controlled invariant. In fact, it is easily proven thatthe intersection of an (A,B)-controlled invariant contained in C and an (A, C)-conditioned invariant containing B is an (A,B)-controlled invariant. ConsiderFig. 2.6, that shows both V∗ and R∗. While any state in R∗ is completelyreachable from any other state in R∗, i.e., with state trajectories invisible atthe output, the trajectories on V∗ corresponding to initial states not belonging

2.6 Extension to quadruples and computational tools 15

V∗ R∗

unstablezero

stablezero

X

Fig. 2.6. R∗ and some trajectories corresponding to invariant zeros.

to R∗ are strictly constrained to be linear combinations of modes of the typetmk−1 eζit, called fixed modes . The values of ζi, in general complex, are calledthe invariant zeros of Σ and the maximum value of the integer mk is said themultiplicity of ζi.

Let us get some insight into the structure of Σ. Let F be such that(A + BF )V∗ ⊆V∗. Consider the change of basis defined by the transformationT = [T1, T2, T3] with im T1 = R∗, im[T1 T2] = V∗, T3 such that T is nonsin-gular. The new matrices A′ = T−1(A + BF )T , B′ = T−1B, C ′ = CT have thestructures

A′ =

A′

11 A′12 A′

13

0 A′22 A′

23

0 0 A′33

, B′ =

B′

1

0B′

3

, C ′ =[0 0 C ′

3

](2.33)

Note that the structural decomposition in (2.33) is an extension of thatin (2.23). The invariant zeros of Σ are the eigenvalues of matrix A′

22. Boththe pairs (A′

11, B′1) and (A′

33, B′3) are stabilizable if Σ is stabilizable. The self-

bounded controlled invariants of Σ are defined, in the new basis, as the sumof R∗ with the invariants of matrix A′

22. The system is said to be minimumphase if all its invariant zeros are stable, i.e. with the real parts negative in thecontinuous-time case or with absolute value less than one in the discrete-timecase. Hence the invariant zeros of Σ are the internal unassignable eigenvaluesof V∗ and will be referred to with the symbol Z(V∗) or Z(Σ). By extension,we will denote with Z(V) the internal unassignable eigenvalues of any con-trolled invariant V . If V is a self-bounded controlled invariant, i.e., satisfiesR∗ ⊆V ⊆V∗, Z(V) is a subset of the invariant zeros of Σ.

2.6 Extension to quadruples and computational tools

For an easier understanding of their meaning, the above geometric tools havebeen defined for purely LTI systems represented by a triple (A,B,C), withoutany feedthrough matrix D, like (2.26) and (2.27). However they can easily beextended to to systems represented by a quadruple (A,B,C,D), like


x(t) = Ax(t) + B u(t)y(t) = C x(t) + D u(t)

(2.34)x(k + 1) = Ax(k) + B u(k)

y(k) = C x(k) + D u(k)(2.35)

in the continuous and discrete-time case, respectively.In these cases an output-nulling subspace and its friend are defined as

a pair (V ,F) such that (A + BF )V ⊆V and V ⊆ ker(C + DF ). Likewise, aninput-containing subspace and its friend are defined as a pair (S,G) such that(A + GC)S ⊆S and S ⊇ im(B + GD).

The computation of the maximum output-nulling subspace V∗ and theminimum input-containing subspace S∗ is still possible with algorithms (2.29)and (2.31), applied to a suitably extended system. Refer to the overall system

+ +

D

u(A, B, C)

y zΣe

Σ

Fig. 2.7. An artifice to reduce a quadruple to a triple.

Σ shown in Fig. 2.7, where Σe is a set of integrators in the continuous-timecase or a set of unit delays in the discrete-time case. It can be described bythe extended state x and extended triple A, B, C) defined as

x =

[x

u

], A =

[A 0C 0

], B =

[B

D

], C =

[0 Iq

](2.36)

Let us compute with algorithm (2.29) the output nulling V∗ with basis

matrix V for system Σ and a corresponding state feedback matrix F anddenote with

V =

[V1

V2

], F =

[F1 F2

]

their partitions according to (2.36). Owing to the structure of C, it turns outthat V2 = 0 and F2 = 0 and the maximum output nulling controlled invariantof the quadruple (A,B,C,D) is V∗ = imV1 and that F1 is a correspondingstate feedback matrix adapted to its basis matrix V1.

A similar procedure can be used for S∗. The definitions of R∗ and invariantzeros considered in Section 2.5 are still valid, provided they are referred tothese V∗ and S∗.

2.6 Extension to quadruples and computational tools 17

Computational tools for LTI system analysis

Computational routines in Matlab performing the previously described oper-ations on subspaces (represented by basis matrices), are

Q = sums(A,B) sum of subspaces

Q = ints(A,B) intersection of subspaces

Q = invt(A,X) inverse transform of a subspace

Q = ortco(X) orthogonal complement of a subspace

[V,F] = vstar(A,B,C,[D]) computation of V∗ and friend F

[S,G] = sstar(A,B,C,[D]) computation of S∗ and friend G

Z = gazero(A,B,C,[D]) invariant zeros of an LTI system

Z = gazero(A,B,ortco(V)’) internal unassignable eigenvalues of V

r = reldeg(A,B,C,[D]) relative degree (see the next section)

These are downloadable from the author’s web page

http://www3.deis.unibo.it/Staff/FullProf/GiovanniMarro/geometric.htm

3

Geometrically-stated system properties

The concepts of controllability and observability indexobservability were intro-duced in the early 60s to state mathematical conditions connected with theexistence and uniqueness of Kalman regulators and filters. Kalman himselfpointed out that the reachable subspace R of a LTI system Σ is the minimumA-invariant containing B and the unobservable subspace U the maximum A-invariant contained in C [1]. These were the first system properties stated ingeometric terms, and the related subspaces can be computed with recursivealgorithms similar to those for V∗ and S∗. In fact, algorithm (2.31) with C =X(the whole state space) is equivalent to algorithm (2.25) and provides R andalgorithm (2.29) with B={0} (the origin of the state space) is equivalent toalgorithm (2.24) and provides U .

Recall that if Σ is completely reachable (i.e., with R=X ), the eigen-values of the system with state feedback shown in Fig. 2.4.b are completelyassignable by a suitable choice of matrix F . If Σ is completely observable (i.e.,with U = {0}), the eigenvalues of the system with output injection shown inFig. 2.5.b are completely assignable by a suitable choice of matrix G. Thisis the pole assignability property of algebraic state feedback and algebraicoutput injection.

There are other system properties that are suitably expressed in geometricterms: left invertibility , right invertibility , relative degree.

3.1 Left invertibility, right invertibility, and relative

degree

Consider the continuous-time LTI system (2.26) or (2.34) with x(0) = 0 andassume that its input function u(t) is bounded and piecewise continuous. Weset the following definitions.

1. The system is said to be left invertible if, given any admissible output func-tion y(t), t∈ [0, t1], t1 > 0, there is a unique corresponding input functionu(t), t∈ [0, t1) producing that output function.

20 3 Basic Geometric Tools and System Properties

2. The system is said to be right invertible if there is an integer ρ≥ 1 suchthat, given any output function y(t), t∈ [0, t1], t1 > 0 with ρ-th deriva-tive piecewise continuous and such that y(0) = 0, . . . , y(ρ− 1)(0) = 0, thereis at least one input function u(t), t∈ [0, t1) producing that output func-tion. The minimum value of ρ satisfying the above statement is called therelative degree of the system.

Consider the discrete-time LTI system (2.27) or (2.35) with x(0) = 0 andassume that its input function u(t) is bounded.

1. The system is said to be left invertible if, given any admissible outputfunction y(k), k ∈ [0, k1], k1 ≥n, there is a unique corresponding inputfunction u(k), k ∈ [0, k1) producing that output function.

2. The system is said to be right invertible if there is an integer ρ≥ 1 suchthat, given an output function y(k), k ∈ [0, k1], k1 ≥ ρ such that y(k) = 0,k ∈ [0, ρ−1], there is at least one input function u(k), k ∈ [0, k1 − 1] pro-ducing that output function. The minimum value of ρ satisfying the abovestatement is called the relative degree of the system.

The geometric necessary and sufficient conditions for left and right invert-ibility of both continuous and discrete-time systems are very simple, whenexpressed in terms of V∗ and S∗.

• Left invertibility

V∗ ∩S∗ = {0} (3.1)

• Right invertibilityV∗ +S∗ =X (3.2)

• Relative degreeFor a right invertible system without feedthrough the relative degree isthe minimal value of ρ such that V∗ +Sρ =X where Si (i = 1, 2, . . .) isprovided by sequence (2.31). For systems with feedthrough this still holdsbut referred to a suitably extended system, like that shown in Fig. 2.7(note that in this case the computed relative degree has to be reduced byone).

3.2 Duality

Duality is an important strategy in the geometric approach context. It willbe shown in the next Chapter that numerous system theory problems caneasily be solved by duality, even if generally this feature has not been pointedout in the literature, thus leading to unnecessary repetition of arguments andprocedures.

3.2 Duality 21

Given an LTI system Σ : (A,B,C,D), we define its dual system asΣT : (AT, CT, BT, DT). Duality entails some basic connections between char-acteristic subspaces of LTI systems. Referring to the dual system with thesubscript d, we can easily prove the following identities.

• Controllability and observability

R = U⊥d , U = R⊥

d (3.3)

Hence Σ is completely controllable if and only if ΣT is completely observ-able and vice versa.

• Left and right invertibility

V∗ = (S∗d)⊥ , S∗ = (V∗

d)⊥ (3.4)

Hence Σ is left invertible if and only if ΣT is right invertible and vice versa.The relative degree of a left invertible system is assumed to be equal tothat of its dual system.

• Impulse responseg(t) = gd(t)

T (3.5)

Σ1

Σ2

Σ3

u y

a)

+

+

ΣT3

ΣT1

ΣT2

u y

b)

Fig. 3.1. An interconnection of systems and its duality.

Fig. 3.1.a shows the connection of the three systems Σi : (Ai, Bi, Ci)(i = 1, 2, 3). Denote the overall system with Σ : (A,B,C). It is easily checkedthat

A =

A1 0 00 A2 0

B3,1C1 B3,2C2 A3

, B =

B1

B2

0

, C =[0 0 C3

]

The overall dual system ΣT : (AT, CT, BT) is represented by

AT =

AT

1 0 CT1 BT

3,1

0 AT2 CT

2 BT3,2

0 0 AT3

, BT =

00

CT3

, CT =[BT

1 BT2 0

]

The corresponding connection is shown in Fig. 3.1.b. Hence, the overalldual system is obtained by reversing the order of serially connected systemsand interchanging branching points with summing junctions and vice versa.

4

Geometrically-solved regulation problems

In this section we will refer to some basic control problems and present thenecessary and sufficient conditions for their solvability. The statements willstrictly be in terms of the geometric tools presented in Chapter 2. For everyproblem, only the necessary and sufficient conditions for solvability are hereinpresented without proof, but, as a rule, efficient computational tools, directlyrelated to the proofs of the sufficiency conditions, are also available withinthe geometric approach software facilities.

4.1 Disturbance decoupling

When the output of a system must be decoupled from an input signal wemust distinguish three cases

1. (inaccessible) disturbance decoupling

2. measurable signal decoupling

3. previewed signal decoupling

whose difference is basic for a correct statement and solution of the corre-sponding control problem. Feedback is strictly required only in the first case,since the second and third case are solvable with feedforward, possibly appliedto a system pre-strengthened with feedback.

Inaccessible disturbance decoupling by state feedback

The disturbance decoupling problem by state feedback is the basic problem ofthe geometric approach. It was studied by Basile and Marro [10] and Wonhamand Morse [13] as one of the earliest applications of the new concepts.

Consider the continuous-time LTI system Σ with two inputs shown inFig. 4.1, described by

x(t) = Ax(t) + B u(t) + H h(t)y(t) = C x(t)

(4.1)

24 4 Geometrically-solved regulation problems

where u denotes the manipulable input, h the disturbance input. Let B= imB,H= imH, C = kerC. The (inaccessible) disturbance decoupling problem is setas follows: determine, if possible, a state feedback matrix F such that distur-bance h has no influence on output y.

u

hy

x

Σ

F

Fig. 4.1. The inaccessible disturbance decoupling problem.

In spite of its apparent simplicity, the disturbance decoupling problemwas not completely solved at the outset. The system with state feedback isdescribed by

x(t) = (A + BF ) x(t) + H h(t)

y(t) = C x(t)

i.e., by the triple (A + BF,H,C). It behaves as requested if and only if itsreachable set by h, i.e., the minimum (A + BF )-invariant containing H, is con-tained in C. Denote by V∗

(B,C) the maximum output-nulling (A,B)-controlled

invariant contained in C. Since any (A + BF )-invariant is an (A,B)-controlledinvariant, the inaccessible disturbance decoupling problem has a solution ifand only if

H ⊆ V∗(B,C) (4.2)

This is a necessary and sufficient structural condition and does not ensureinternal stability. If stability is requested, we have the inaccessible disturbancedecoupling problem with stability. Stability is conveniently handled by usingself-bounded controlled invariants, introduced by Basile and Marro [30] andSchumacher [31]. Recall that any (A,B)-controlled invariant V self-boundedwith respect to V∗

(B,C) satisfies the relation

R∗(B,C) ⊆ V ⊆ V∗

(B,C) (4.3)

and the set of all the self-bounded controlled invariant is closed with respectto both sum and intersection, so it is possible to define the minimum self-bounded controlled invariant containing H, provided that (4.2) holds. This isthe reachable subspace with both inputs u and h, that clearly contains H. Itis defined as

4.1 Disturbance decoupling 25

Vm = V∗(B,C) ∩ S∗

(C,B+H) (4.4)

The change of basis shown in (2.33) also holds for Vm instead of V∗ with thestabilizability property of (A′

33, B′3) still valid. Hence there is a state feedback

Fm achieving disturbance decoupling and making the overall system stable ifand only if Vm is internally stabilizable. Thus the necessary and sufficient con-ditions for solvability of the disturbance decoupling problem with stability areboth the structural condition (4.2) and the following stabilizability condition.

Vm internally stabilizable (4.5)

that is equivalent toZ(Vm) ⊆ Cg (4.6)

Since Z(Vm) is a part of Z(V∗(B,C)), minimality of phase ensures stabilizability.

Measurable signal decoupling and unknown-input state

observation

indexunknown-input state observation The measurable signal decouplingproblem is stated as follows: determine, if possible, an algebraic or dynamiccompensator such that the measurable signal h has no influence on output y.The system equations are still (4.1). This problem appears as a slight exten-sion of the previous inaccessible disturbance decoupling problem, but indeedit is very different and opens out to many types of solutions.

The first solution considered in the literature, based on state feedback andalgebraic feedforward, is represented in Fig. 4.2. The structural necessary andsufficient condition for its solvability (Bhattacharyya [20]) is

H ⊆ V∗(B,C) + B (4.7)

In fact if (4.7) holds, by assuming as matrix G in Fig. 4.2 a projection ofHh on V∗

(B,C) along B, the action of input h is driven on V∗(B,C), hence it is

invisible at the output.

+ +

u

hy

x

Σ

F

G

Fig. 4.2. Measurable signal decoupling with state feedback.

It can be proven that the stabilizability condition for this problem is againexpressed by (4.5) or (4.6), provided (4.7) is satisfied (Basile and Marro [47]).


However, the algebraic layout shown in Fig. 4.2 is not the most convenientfor achieving decoupling of a measurable signal since it requires accessibilityof state. Let us consider instead the layout shown in Fig. 4.3 where Σc isa replica of the system with feedback shown in Fig. 4.2. It is worth notingthat the dynamics of Σc can be restricted to Vm, that is an internally stable(A + BFm)-invariant containing the projection of Hh on Vm along B.

h

u Σ

Σc

y

Fig. 4.3. Measurable signal decoupling with a feedforward compensator.

However in this case Σ must be stable. But, if the system is unstable, howcan we avoid feedback? This can be done, very easily, as shown in Fig. 4.4.

+

+

h

u Σ

Σc

Σs

y

ym

−ym

xs = 0

Fig. 4.4. Using both a compensator and a stabilizer.

If Σ is stabilizable from u and detectable from a suitable measurable out-put ym (possibly coinciding with y), it can be stabilized with a feedback unitΣs which can be maintained at zero by the pre-compensator since this repro-duces the state evolution (hence the output) of Σ, restricted to Vm. If ym = y

this connection is not necessary. The stabilizer is standard, based on statefeedback Fs such that A+BFs is strictly stable and output injection Gs suchthat A+GsC1 is strictly stable, i.e., Σs : (A + BFs + GsC1 , −Gs , Fs). Inputmatrix −Gs is referred to both inputs. Note that the output of the stabilizeris identically zero, since its inputs due to the action of h on Σ and Σc canceleach other.

Consider the continuous-time LTI system Σ with two outputs, shown inFig. 4.5, described by

4.1 Disturbance decoupling 27

+ +

ue

y

−e

Σ

Σo

η

Fig. 4.5. Unknown-input observer of a linear function of the state.

x(t) = Ax(t) + B u(t)e(t) = E x(t)y(t) = C x(t)

(4.8)

The unknown-input observation problem of a linear function of the state(possibly the whole state) is stated as follows: design a stable feedforward unitthat, connected to the output y, provides an exact estimation of the output e.

This problem was the object of very early investigation (1969) by Basile,Marro and Laschi [9, 12]. More recently, owing to its connection with thefault detection problem, it has been the object of hundreds of non-geometricscientific papers based on appalling manipulations of matrices, but dualitywith the measurable signal decoupling problem was never recognized andpointed out.

The overall system in Fig. 4.5 is clearly the dual of measurable signal de-coupling, considered in Fig. 4.3 (the summing junction on the right is onlyexplanatory), so the design of an unknown-input observer can be considereda very standard problem in the geometric approach context. Of course thenecessary and sufficient conditions to build a stable unknown-input observerΣo are still (4.2), (4.5) for the dual of system (4.8), obtained with the sub-stitutions AT → A, BT → C, CT → B, ET → H. However, if an equivalentset of geometric conditions directly referred to the system matrices in (4.8) issought-after, these are

S∗(C,B) ∩ C ⊆ E (4.9)

the structural condition, with C = kerC, E = kerE,

SM externally stabilizable (4.10)

the stabilizability condition, where SM is defined as

SM = S∗(C,B) + V∗

(B,C ∩E) (4.11)

Also in this case, a stabilizer can be used if the system is unstable, as shownin Fig. 4.6. By duality, the output η is independent of both u and d (anydisturbance acting on the stabilizer).


+

+

u e

Σ

Σo

Σs

η

d

y

−e

u1

Fig. 4.6. Unknown-input observer with a stabilizer.

Previewed signal decoupling and delayed state estimation

It has previously been pointed out that a sufficient condition for stability inthe above disturbance and measurable signal decoupling problems is mini-mality of phase of Σ. If Σ is not minimum-phase, however, it is still possibleto obtain decoupling if signal h is known in advance by a certain amount oftime (several times the maximum time constant of the unstable zeros). In thiscase the necessary structural condition is still (4.7), while the stabilizabilitycondition is

Z(Vm) ∩ C0 = ∅ (4.12)

where C0 denotes the boundary between the set of stritcly stable and thatof strictly unstable zeros (the imaginary axis in the continuous-time case andthe unit circle in the discrete-time case), while ∅ denotes the empty set.

hpdelay h

u Σ

Σc

y

a)

+ +

ue

y

ed

−ed

delay

Σo

Σ η

b)

Fig. 4.7. Previewed decoupling and delayed unknown-input observation.

In Fig. 4.7.a h denotes the previewed signal, hp its value t0 seconds inadvance, so that hp(t) = h(t + t0). The feedforward unit Σc includes a convo-lutor, also called a finite impulse response (FIR) system. Refer to Fig. 4.8 andsuppose that a single impulse is applied at input h, i.e., assume h(t) = hδ(t),causing an initial state Hh at time zero. Since H⊆Vm +B owing to (4.7),this initial state can be projected on Vm along B, and decomposed into threecomponents: a component on R∗

(B,C), a component on the subspace of Vm cor-responding to strictly stable zeros, and a component on the subspace of Vm

corresponding to strictly unstable zeros. While the former two (state α) canbe driven to the origin along stable trajectories on Vm, the latter, that corre-sponds to unstable motions on Vm, can be nulled by a preaction on u prior to

4.2 Disturbance decoupling by dynamic output feedback 29

unstable zeroVm

BuHh

stable zero

α

−α

Fig. 4.8. Preaction along the unstable zeros.

its occurrence (in the time interval [−t0, 0]), thus cancelling it at t = 0 (state−α). This is obtained by means of the above-mentioned FIR system, wherethe convolution profile corresponding to the control action along the unstablezeros, computed backward in time for good and all, is suitably stored.

The overall system in Fig. 4.7.a is dualized as shown in Fig. 4.7.b, thusobtaining an unknown-input observer with delay. The FIR system includedin Σc to steer the system along the unstable zeros is simply dualized bytransposing its convolution profile for all t.

4.2 Disturbance decoupling by dynamic output

feedback

This is an extension of the inaccessible disturbance decoupling problem withstate feedback, considered in Section 4.1.

h e

yu Σ

Σc

Fig. 4.9. Disturbance decoupling by dynamic output feedback.

Consider the continuous-time LTI system Σ with two inputs and two out-puts, shown in Fig. 4.9, described by


x(t) = Ax(t) + B u(t) + H h(t)y(t) = C x(t)e(t) = E x(t)

(4.13)

where u denotes the manipulable input, h the disturbance input. Let B= imB,H= imH, C = kerC, E = kerE. Σc denotes a feedback dynamic compensator,described by

z(t) = N z(t) + M y(t)u(t) = Lz(t) + K y(t)

(4.14)

The disturbance decoupling problem by dynamic output feedback is setas follows: design, if possible, a dynamic compensator (N,M,L,K) such thatthe disturbance h has no influence on the regulated output e and the overallsystem is stable.

This problem is a basic one in control. The structural conditions for itssolution were investigated by Basile, Laschi and Marro [9, 11] and stated inprecise terms by Schumacher [27], while conditions including stability werefirst stated by Willems and Commault [28] and restated in terms of self-bounded controlled invariants and their duals by Basile, Marro an Piazzi[32, 47]. The structural condition is

S∗(C,H) ⊆ V∗

(B,E) (4.15)

Let us recall definition (4.4) of Vm, the minimum self-bounded controlledinvariant of Σ containing H, and definition (4.11) of SM , its duality. Thestabilizability conditions are

SM externally stabilizableVM = Vm +SM internally stabilizable

(4.16)

Conditions (4.15), (4.16) are constructive, since they trace the way tobuild a full-order unknown-input observer providing information to make VM

a locus of state trajectories due to h while stabilizing the overall system.

4.3 Model following

The model following problem has been deeply investigated by numerous au-thors. In the geometric approach context, it was first approached by Morse[57] and refined by Descusse, Malabre and Kucera [58, 59, 60]. Model follow-ing is a particular case of measurable signal decoupling, and its solution inthis context is straightforward and appealing. Using self-bounded controlledinvariants and the geometric interpretation of invariant zeros make the most

4.3 Model following 31

+

_ h

u y

ym

yΣΣc

Σm

Σ

Fig. 4.10. Model following.

recent contributions very complete. In fact, they consider both feedforwardand feedback and the nonminimum-phase case. The theory herein presentedis a summary of a contribution by Marro and Zattoni [54].

Refer to Fig. 4.10. The model following problem is: determine, if possible,a dynamic feedforward compensator Σc such that the output of the system Σstrictly follows (is equal to) the output of a given model Σm. The block diagramshown in Fig. 4.10 clearly repeates that in Fig. 4.3, provided the model isconsidered a part of the controlled system.

Assume that Σ is described by the triple (A,B,C) and Σm by the triple

(Am, Bm, Cm). The overall sistem Σ is described by

A =

[A 00 Am

], B =

[B

0

], H =

[0

Bm

], C =

[C −Cm

](4.17)

Both the system and the model are assumed to be stable, square, leftand right invertible. The structural condition expressed by inclusion (4.7) issatisfied if

ρ(Σ) ≤ γ(Σm) (4.18)

i.e., if the relative degree ρ of Σ is less or equal to the minimum delay γ

of Σm. The minimum delay of a triple (A,B,C) is defined as the minimumvalue of i such that C Ai B is nonzero. Hence the structural condition issatisfied if a model is chosen with a sufficiently high minimum delay. Considerthe stabilizability condition (4.6). Assuming that Σ and Σm have no equal

invariant zeros, it can be shown that the internal eigenvalues of Vm are theunion of the invariant zeros of Σ and the eigenvalues of Am, so that in generalmodel following with stability is not achievable if Σ is nonminimum-phase.Hence the stabilizability condition to be considered together with (4.18) is

Z(Σ) ⊆ Cg (4.19)

Condition (4.19) can be evaded by properly replicating in Σm the unstablezeros of Σ. This can be achieved, for instance, by assuming a model Σm con-sisting of q independent single-input single-output systems all having as zeros


the unstable invariant zeros of Σ, so that these are cancelled as internal eigen-values of Vm. This makes it possible to achieve both input-output decouplingand internal stability, but restricts the model choice. If Σ is nonminimumphase, perfect or almost perfect following of a minimum phase model mayalso be achieved if h is previewed by a significant time interval t0, as pointedout in Section 4.1.

+

+ _

_

Σ

Σm

Σc

r h u

y

y

ym

Fig. 4.11. Model following with feedback.

In the geometric approach context the model following problem with feed-back, corresponding to the block diagram shown in Fig. 4.11, is also easilysolvable. Like in the feedforward case, both Σ and Σm are assumed to bestable and Σm to have at least the same relative degree as Σ.

+

+ _

_

Σ

Σm

Σc

r h u

y

y

ym

Fig. 4.12. A structurally equivalent connection.

Replacing the feedback connection with that shown in Fig. 4.12 does notaffect the structural properties of the system. However, it may affect stability.The new block diagram represents a feedforward model following problem. Infact, note that h is obtained as the difference of r (applied to the input of themodel) and ym (the output of the model). This corresponds to the parallelconnection of Σm and a diagonal algebraic system with gain −1, that is invert-ible, having zero relative degree. Its inverse is Σm with a feedback connectionthrough the identity matrix, as shown in Fig. 4.13. Let the model consist of q

independent single-input single-output systems all having as zeros the unsta-ble invariant zeros of Σ. Since the invariant zeros of a system are preserved

4.4 The multivariable regulator with internal model 33

+

+

_ +

Σ

Σm

Σc

h

r

u

y

Σ′m

Fig. 4.13. Another structurally equivalent connection.

under any feedback connection, a feedforward model following compensatordesigned with reference to the block diagram in Fig. 4.13 does not includethem as poles. It is also possible to include multiple internal models in thefeedback connection shown in the figure (this is well known in the single in-put/output case), that are repeated in the compensator, so that both Σ′

m andthe compensator may be unstable systems. In fact, zero output in the modi-fied system may be obtained as the difference of diverging signals. However,stability is recovered when going back to the original feedback connectionrepresented in Fig. 4.11.

4.4 The multivariable regulator with internal model

The multivariable regulator problem was first approached and solved by Fran-cis in 1977 [61]. A reformulation using the geometric approach tools is reportedin the Basile and Marro book [47]. A convenient reference block diagram for

+ _ ΣΣe Σr

r u

d

e y

Fig. 4.14. The multivariable regulator with internal model.

the multivariable regulator with internal model is shown in Fig. 4.14. It ex-tends to the multivariable case the design philosophy usually adopted in thestandard approach to the single input-single output control system design.The exosystem Σe generates the exogenous signals, r and d, applied to theregulation loop including the plant Σ and a regulator Σr to be determined. For


instance, the exogenous signals may be steps, ramps, sinusoids. The eigenval-ues of Σe are assumed to belong to the imaginary axis of the complex plane.The overall system considered, included the exosystem, is described by a lin-ear homogeneous set of differential equations, whose initial state is the onlyvariable affecting evolution in time. The plant and the exosystem are mod-elled as a unique regulated system which is not completely controllable orstabilizable (the exosystem is not controllable). The corresponding equationsare

x(t) = Ax(t) + B u(t)

e(t) = E x(t)(4.20)

with

x =

[x1

x2

], A =

[A1 A3

0 A2

], B =

[B1

0

], E =

[E1 E2

]

In (4.20) the plant corresponds to the triple (A1, B1, E1). Note that theexosystem state x2 influences both the plant through matrix A3 and the errore through matrix E2. The pair (A1, B1) is assumed to be stabilizable and(A,E) detectable. The regulator is modelled by equations (4.14), like in thedisturbance decoupling problem by dynamic output feedback.

The autonomous regulator problem is stated as follows: derive, if possi-ble, a regulator Σr : (N,M,L,K) such that the closed-loop system with theexosystem disconnected is stable and limt→∞ e(t) = 0 for all the initial statesof the autonomous extended system.

The overall system is referred to as the autonomous extended system

˙x(t) = A x(t)

e(t) = E x(t)(4.21)

with

x =

x1

x2

z

A =

A1 + B1KE1 A3 + B1KE2 B1L

0 A2 0ME1 ME2 N

, E =[E1 E2 0

]

Let x1 ∈Rn1 , x2 ∈R

n2 , z ∈Rm. If the internal model principle is used to

design the regulator, the autonomous extended system is characterized byan unobservability subspace containing these modes, that are all not strictlystable by assumption. In geometric terms, an A-invariant W ⊆ kerE havingdimension n2 exists, which is internally not strictly stable, but externally

4.5 Noninteraction and fault detection and identification 35

strictly stable. The geometric conditions for the multivariable regulator prob-lem to admit a solution are

V∗ + P = X

Z(A1, B1, E1) ∩ σ(A2) = ∅(4.22)

where V∗ is the maximum output-nulling controlled invariant of the triple(A,B,E), while P is plant , defined as

P = {x : x2 = 0} = im

[In1

0

]

clearly an A-invariant. It is worth noting that conditions (4.22) are againa structural condition and a stability condition in terms of invariant zeros.However, the stability condition is very mild in this case since it only requiresthat the plant has no invariant zeros equal to eigenvalues of the exosystem.Hence the autonomous regulator problem may also be solvable if the plant isnonminimum phase. Minimality of phase is only required for perfect tracking ,not for asymptotic tracking .

4.5 Noninteraction and fault detection and

identification

Another problem that in the geometric approach context was approached withstate feedback at the beginning, even though involving measurable exogenoussignals, is noninteracting control.

The noninteracting control problem is stated as follows: given an LTIsystem Σ whose output is partitioned by blocks (y1, y2, . . .), derive a controllerwith again as many inputs (α1, α2, . . .) such that input αi allows completereachability of output yi while maintaining at zero all the other outputs .

Only two output blocks y1 and y2 will herein be considered for the sake ofsimplicity. Therefore Σ is described by

x(t) = Ax(t) + B u(t)y1(t) = C1 x(t)y2(t) = C2 x(t)

(4.23)

Fig. 4.15.a shows the solutions proposed by Wonham and Morse in theirfirst paper on geometric approach [13], based on state feedback and algebraicfeedforward units. This solution was probably suggested by the measurablesignal decoupling layout with algebraic feedforward and feedback shown inFig. 4.2. The completely algebraic solution shown in Fig. 4.15.a includes two


+ + +

+

G1

G2

α1

α2

uΣ

y1

y2

x

Fa)

+ + +

+

G1

G2

α1

α2

uΣ

y1

y2

stateextensionF

b)

Fig. 4.15. Noninteracting control with state feedback.

feedforward units G1, G2 and state feedback F . Achieving the most com-plete noininteraction with this technique is more restrictive than with othermethods since, in general, the same state feedback cannot transform anytwo controlled invariants into simple (A+BF )-invariants. This drawback canbe overcome by extending the state with a suitable bank of integrators asproposed by Wonham and Morse in their second paper [15] and shown inFig. 4.15.b.

+

+

α1

α2

Σ1

Σ2

Σu

y1

y2

Fig. 4.16. Noninteracting control with dynamic feedforward units.

An alternative solution, inspired by the measurable signal decoupling bymeans of a dynamic feedforward unit whose layout is shown in Fig. 4.3, wasproposed by Basile and Marro [16]. Refer to Fig. 4.16 and suppose for a whilethat the controlled system Σ is stable. Let yi ∈R

qi , Ci = kerCi (i = 1, 2). Themaximum subspace that can be reached from the origin while being invisibleat output y2 is R∗

(B,C2) and the maximum subspace that can be reached fromthe origin while being invisible at output y1 is R∗

(B,C1). These subspaces are

controlled invariants internally stabilizable (or, more exactly, pole assignable)whose dynamics can be reproduced in the feedforward units Σ1 and Σ2. Hencenoninteraction is possible if and only if

C1 R∗(B,C2) = R

q1

C2 R∗(B,C1) = R

q2(4.24)

There is a certain degree of freedom in the choice of inputs α1 and α2.They can be assumed of full dimension, i.e., corresponding to input matri-ces spanning the whole subspaces R∗

(B,C2) and R∗(B,C1). On the other hand,

since only dynamic reachability is required, owing to a well known Heymann


lemma [62, 63] they can also be assumed to be scalar without affecting prob-lem solvability. It is not necessary that the controlled system Σ is stable, but,similarly to the measurable disturbance decoupling problem, only stabiliz-ability and detectability are required. The overall block diagram for this case,similar to that in Fig. 4.4, is shown in Fig. 4.17.

+

+

+

+

+

+

Σ1

Σ2

α1

α2

uΣ

y1

y2

ym

−ym Σs xs =0

Fig. 4.17. Using a stabilizer in feedforward noninteracting control.

+

+

+ +

+

+

α1

α2

Σ1

Σ2

Σu

y1

y2

ym 0

−ym

Fig. 4.18. Nulling a measurable output.

Note that the stabilizer shown in Fig. 4.17 is not influenced by the inputsα1 and α2 since the measured output ym (from which Σ is detectable) can benulled by a signal −ym generated in the feedforward units, where a replica ofthe state evolution produced by α1 and α2 is available. This detail is pointedout in Fig. 4.18.

u1

u2 Σy Σ1

Σ2

α1

α2

um

Fig. 4.19. A block diagram for fault detection and identification.

Let us now consider the block diagram in Fig. 4.19, which clearly is thedual of that in Fig. 4.18. The mathematical model of Σ is


u1

u2

uh

v1 v2 vk

Σy

unit 1

unit 2

unit h+k

α1

α2

umαh+k

d

Fig. 4.20. A more general layout for the FDI problem.

x(t) = Ax(t) + Bm um(t) + B1 u1(t) + B2 u2(t)

y(t) = C x(t)(4.25)

where um refers to a measurable input, while both u1 and u2 are assumedto be inaccessible. The fault detection and identification problem is statedas follows: given an LTI system Σ having an inaccessible input partitionedby blocks (u1, u2, . . .), derive an observer with again as many scalar outputs(α1, α2, . . .) such that output αi is different from zero if any component of ui

is different from zero while all the other outputs are maintained at zero.A geometric approach to this problem was first proposed by Massoum-

nia [44] and restated in improved terms by Massoumnia, Verghese and Will-sky [45].

A more general layout for the fault detection and identification problemherein considered is shown in Fig. 4.20. System Σ is described by

x(t) = Ax(t) + B um(t) + H d(t) +h∑

i=1

Bi ui (4.26)

y(t) = C x(t) [ +k∑

j=1

Dj vj(t)] (4.27)

where (A,B,C) is the nominal system (without any fault or disturbance in-put) with input um (accessible for measurement), and

• ui (i = 1, . . . , h) the actuator fault inputs,

• Bi = (∆B)i the corresponding signatures,

• vj (j = 1, . . . , k) the sensor fault inputs,

• Dj = (∆C)j the corresponding signatures,

• d the disturbance input.


The fault detecting units provide the outputs

• αℓ (ℓ = 1, . . . , h + k), called the residuals. The residuals are herein assumedto be scalar.

The fault inputs are assumed to be

• white noises in on-line processes,

• impulses in single-throw processes or batch processes.

They are usually applied through filters or exosystems whose dynamicsis included in matrix A, so that the actual faults are modelled as colorednoise or a specific transient, respectively. Hence the sensor faults often aretaken into account with further terms of the type Bi ui. This is the reason forthe square brackets in equation (4.27). Thus the overall system whose layoutshown in Fig. 4.20 is led back to that in Fig. 4.19, provided that disturbanced is considered as fault to be rejected by all the detecting units.

5

Geometric approach to H2-optimal regulation

and filtering

The study of the Kalman linear-quadratic regulator (LQR) is the central topicof most courses and treatises on advanced control systems. See, for instance,the books by Kwakernaak and Sivan (1973), Anderson and Moore (1989),Syrmos and Lewis (1995).

More recently, in the nineties, a certain attention was given to the H2

optimal control, which is substantially a restatement of the LQR with somestandard and well settled problems of the geometric approach (for instance,disturbance decoupling with output feedback). Feedthrough is not present ingeneral, so that the standard Riccati equation-based solutions are not imple-mentable and the existence of optimal solution is not ensured. Books on thissubject are by Stoorvogel (1992), Saberi, Sannuti and Chen (1995). The com-putational method they use to solve the H2-optimal problem are linear matrixinequalities (LMI), supported by a “special coordinate basis” that points outthe geometric features of the systems dealt with.

An alternative approach, which will be briefly presented in the following,is to treat the singular and cheap problems, where feedthrough is not present,by directly referring to the LTI system obtained by differentiating the Hamil-tonian function, which can be considered as a generic dynamic system, withall the previously described features.

5.1 Disturbance decoupling in minimal H2-norm

Let us recall that the H2-norm of a continuous-time LTI system Σ representedby the differential equations (2.26) is defined as

‖Σ‖2 =

√

tr(∫ ∞

0

g(t) gT(t) dt)

(5.1)

where “tr” denotes the trace of a matrix, namely the sum of the elementson the main diagonal and g(t) denotes the impulse response of the system,

42 5 Geometric approach to H2-optimal regulation and filtering

namely the matrix of time functions whose columns are the solutions of theautonomous system x(t) = Ax(t), y(t) = C x(t) with initial conditions corre-sponding to the columns of B.

Likewise, for a discrete-time LTI system Σ represented by the differenceequations (2.27), the H2-norm is

‖Σ‖2 =

√√√√ tr( ∞∑

k=0

g(k) gT(k))

(5.2)

where the impulse response g(k) is defined like in the continuous-time case,but referring to the autonomous system x(k + 1) = Ax(k), y(k) = C k(t).

The continuous-time case will be primarily considered herein. Only themain distinguishing differences will be added on for the discrete-time case.

The standard linear quadratic regulator (LQR) problem is stated as fol-lows: given a stabilizable LTI system whose state evolution is described by

x(t) = Ax(t) + B u(t) , x(0) = x0 (5.3)

determine a control function u(·) such that the corresponding state trajectoryminimizes the performance index

J =

∫ ∞

0

(x(t)TQx(t) + u(t)TR u(t) + 2 x(t)TS u(t)

)dt (5.4)

where Q, R and S are symmetric positive semidefinite matrices of suitablesizes. If R > 0 (positive definite), the problem is said to be regular and itssolution is standard, if R≥ 0 (positive semidefinite) the problem is said to besingular , while if R =0 the problem is said to be cheap.

u

x0δ(t)

x

Σ

F

Fig. 5.1. The standard LQR problem.

It is well known that the regular LQR problem is solved by a state feedbackF , independent of the initial state x0. The layout of the solution is shown inFig. 5.1.

The equivalence of the minimum H2-norm disturbance rejection problemand the classical Kalman regulator problem is a simple consequence of the

5.1 Disturbance decoupling in minimal H2-norm 43

expression of the H2-norm in terms of the impulse response of the triple(A,B,C). In fact, since Q, R and S are symmetric and nonnegative semidef-inite, there exist matrices C and D such that

[C D

]T [C D

]=

[Q S

ST R

](5.5)

Consider the two-input system (4.1) with also a possible feedthrough, i.e.,

x(t) = Ax(t) + B u(t) + H h(t)

y(t) = C x(t) + D u(t)(5.6)

where C and D are defined in (5.5), and refer to Fig. 4.1. The system (5.6) isassumed to be left-invertible, not necessarily stabilizable, and with no zeroson the imaginary axis. It is easily seen that in our case the H2-norm is thesquare root of (5.4) for the system (A + BF,H,C) described by

x(t) = (A + BF ) x(t) , x(0) = H

y(t) = (C + DF ) x(t)(5.7)

where state and output are now matrices instead of vectors, so that it isminimized for any H by the Kalman feedback matrix F .

The LQR problem is solvable with the standard geometric tools. Accordingto the classical optimal control approach, consider the Hamiltonian function

M(t) = x(t)TQx(t) + u(t)TR u(t) + 2 x(t)TS u(t) + p(t)T(Ax(t) + B u(t)

)

and set the state, costate equations and stationary condition as

x(t) =∂M(t)

∂p(t), p(t) = −

∂M(t)

∂x(t), 0 =

∂M(t)

∂u(t)

We derive the following Hamiltonian system

x(t) = Ax(t) + B u(t) , x(0) = hi

p(t) = −2 Qx(t) − ATp(t) − 2 S u(t)

0 = 2 STx(t) + BTp(t) + 2 R u(t)

(5.8)

where hi denotes a generic column of H, or, in more compact form,

˙x(t) = A x(t) + B u(t) + H h(t)

0 = C x(t) + D u(t)(5.9)

with


x =

[x

p

], H =

[H

0

]

A =

[A 0

−2 Q −AT

], B =

[B

−2 S

], C =

[2 ST BT

], D = 2 R

The set of equations (5.9) can be considered as referring to an LTI dynamicsystem whose output is constrained to be zero. It follows that minimizing theH2 norm of system (5.7) is equivalent to the perfect decoupling problem for

the quadruple (A, B, C, D), that admits a solution if and only if there is an

internally stable (A, B)-controlled invariant V∗, output-nulling for the overallextended system whose projection on the state space of the original system,defined as

P V∗ =

{x :

[x

p

]∈ V∗

}(5.10)

contains the image of the matrix initial state H. It can be proven that theinternal unassignable eigenvalues of V∗ having nonzero real parts are stable-unstable by pairs. Hence a solution of the LQR problem is obtained as follows:

1. compute V∗;

2. compute a matrix F such that (A + BF )V∗ ⊆ V∗;

3. compute Vs, the maximum internally stable (A + BF )-invariant contained

in V∗ (this is a standard eigenvalue-eigenvector problem) and define V∗H2

as P Vs;

4. if H∈V∗H2

, the problem admits a solution F , that is easily computabledirectly from V∗

H2, which is an (A,B)-controlled invariant.

The above procedure provides a state feedback matrix F correspondingto the minimum H2 norm of the LTI system with input h and output y. Thisimmediately follows from the previously recalled expression of the H2 normin terms of the impulse response. In fact, the impulse response correspondsto the set of initial states defined by the column vectors of matrix H.

Let us briefly consider the extension to the discrete-time case, correspond-ing to the two-input system

x(k + 1) = Ax(k) + B u(k) + H h(k)

y(k) = C x(k) + D u(k)(5.11)

In this case the Hamiltonian function is

M(k) = x(k)TQx(k) + u(k)TR u(k) + 2 x(k)TS u(k)

+ p(k + 1)T(Ax(k) + B u(k)

)

and the state, costate equations and stationary condition are


x(k + 1) =∂M(k)

∂p(k + 1), p(k) =

∂M(k)

∂x(k), 0 =

∂M(k)

∂u(k)

i.e.x(k + 1) = Ax(k) + B u(k) , x(0) = hi

p(k) = 2 Qx(k) + AT p(k + 1) + 2S u(k)

0 = 2 ST x(k) + BT p(k + 1) + 2R u(k)

Like in the continuous-time case, it is convenient to express this systemin the compact form

x(k + 1) = A x(k) + B u(k) + H h(k)

0 = C x(k) + D u(k)(5.12)

with

x =

[x

p

], H =

[H

0

]

A =

[A 0

−2 A−T Q −A−T

], B =

[B

−2 A−T S

]

C =[−2 BTA−TQ + 2 ST BTA−T

], D = 2 R − 2 BTA−TS

The drawback due to the term A−T when A is singular can be overcomeby using a stabilizing state feedback to be subtracted to the final state feed-back solving the problem. The solution is obtained again with a geometricprocedure, but, unlike the continuous-time case, in this case a dead-beat likemotion is also feasible and V∗

H2covers the whole state space of system (5.11)

if (A,B) is stabilizable. Hence the problem of minimizing the H2 norm fromh to y is always solvable in the discrete-time case. An interesting geomet-ric interpretation of the dead-beat part of the solution in terms of S∗, thatinfluences the state space trajectory as shown in Fig. 2.5.a, is given in [53].

The Kalman regulator

The Kalman regulator is the minimum H2 norm extension of the exact dis-turbance decoupling problem with stability considered in Section 4.1.

Referring to Fig. 5.2, recall that the H2 norm of a system Σ : (A,B,C) isthe mean power of the output signal when the input is white noise with zeromean and unitary variance. The problem of minimizing the H2 norm from h

to y has a solution if and only if

H⊆V∗H2

(5.13)


u

h

x

y

Σ

F

Fig. 5.2. Minimal H2 norm decoupling (Kalman regulator).

Condition (5.13) replaces (4.2) and (4.6) for a minimal norm solution. Italways holds in the regular case, since V∗

H2= R

n in this case, but also appliesto the singular and cheap cases, where the dimension of V∗

H2is always re-

duced. V∗H2

is an internally stabilizable controlled invariant, but, for synthesispurposes it may be replaced with the minimum controlled invariant containedin it and containing H by using (4.4) with V∗

H2instead of C, to reduce the

number of fixed modes.

The Kalman dual filter and the Kalman filter

Refer now to the problems considered in Section 4.1, measurable signal de-coupling and unknown-input observation of a linear function of the state andconsider their minimum H2 norm extensions, that are represented by the blockdiagrams shown in Fig. 5.3.

h

u Σ

Σc

y

a)

Measurable signalH2-optimal decoupling(Kalman dual filter)

+ +

ue

y

−e

ΣΣo

η

b)

Unknown-input H2-optimalstate observation(Kalman filter)

Fig. 5.3. Minimal H2 norm decoupling problems, primal and dual.

The necessary and sufficient condition for the solution of the minimal H2

norm problem shown in Fig. 5.3.a is

H⊆V∗H2

+ B (5.14)

similar to condition (4.7) for the exact decoupling. It directly ensures stabilityand also holds in the singular and cheap cases. The block diagram in Fig. 5.3.brefer to the Kalman filter, which is here deduced by duality. The transpose ofthe matrix on the right of (5.5) represents the covariance matrix of a global


white noise injected on the state and the output of Σ. The singular and cheapcases correspond to incomplete or absent measurement noise (noise injectedat the output).

Other H2-optimal control and filtering problems

The problems considered in Section 4.1 can also be revisited in the minimalH2 norm context. The corresponding layouts are shown in Fig. 5.4.

hpdelay h

u Σ

Σc

y

a)

Previewed signalH2-optimal decoupling

(Kalman dual smoother)

+ +

ue

y

ed

−ed

delay

Σo

Ση

b)

Unknown-input H2-optimalstate observation with delay

(Kalman smoother)

Fig. 5.4. Other problems solved in minimal H2 norm.

Solution of the minimum H2 norm decoupling of a previewed signal witha feedforward unit as shown in Fig. 5.4.a is related to the solution of the finitehorizon LQR problem, that has been the object of numerous contributionin the literature. It has been profitably revisited introducing a significantcomputational shortcut for the stable and antistable invariant subspaces ofthe Hamiltonian system by Ferrante, Marro and Ntogramatzidis [55, 56], thatmakes it possible to include also this problem in the geometric approachcontext. The solution is restricted to the regular case for continuous-timesystems, but is rather general in discrete-time case. Duality yields the optimalsmoother shown in Fig. 5.4.b.

u

h

y

e

Σ

Σc

Fig. 5.5. H2 optimal decoupling with dynamic output feedback.

Fig. 5.5 refers to H2 optimal decoupling with dynamic output feedback.This is another problem already deeply investigated in the literature but forwhich the geometric approach through the Hamiltonian system may give a


significant insight, in particular into the features of the singular and cheapcases, and provide new solutions. It has been investigated with geometric toolsin the regular continuous-time case by Saberi, Sannuti, Stoorvogel and Chen[49, 50] and in the book by Trentelman, Hautus and Stoorvogel [51].

6

Conclusions and challenges for the future

At the beginning of most books on multivariable control systems the blockdiagram in Fig. 6.1 is generally shown as representing the most general con-trol problem. The input signal h is specified as including all the exogenousinfluences on the system behavior, like disturbances, measurable disturbancesand references to be tracked.

h

Σu y

Σr

e

Fig. 6.1. A standard block diagram for control.

As pointed out at the beginning of Chapter 4, this is quite misleading,since, while inaccessible disturbances must be faced with feedback, the otherinputs could profitably be processed through feedforward units, and, if knownin advance, tracking and rejection could also be improved by using preaction.This situation is frequent in robotics, where the signal to be tracked is oftenknown in advance and the controlled system is often nonminimum phasebecause of the flexibility of the controlled structures.

Hence, it is necessary to resort to the more complex block diagram shownin Fig. 6.2, including both a feedback section and a feedforward section. Feed-back through Σc of the controlled output y1 and, possibly, of some otherinformative output y2 is mandatory for total or partial rejection of the inac-cessible disturbances d2, directly acting on the plant Σ and d3, pertaining tosensors. Σc can be designed according to Section 4.2 in the exact case or 5.1(Fig. 5.5), in the minimal H2 or H∞ norm case. The accessible disturbance d1

is globally or partially rejected by means of the feedforward unit Σf according

50 6 Conclusions and challenges for the future

_

+

+

+

rp

Σf

r eΣc

d1 d2

uΣ

y2

ΣFIR

feedforward

feedback

delayy1

d3

Fig. 6.2. A complete block diagram for tracking and regulation

to Section 4.1, by acting both on Σ through u and Σc, considered as a stabi-lizer. Lastly, the reference r, that may be previewed by a significant intervalof time (signal rp in the figure), can be processed through a suitable FIRsystem ΣFIR for a reduction of the tracking error in the presence of delaysor nonminimality of phase, as pointed out in Section 4.1 for the exact case orSection 5.1 (Fig. 5.4.a) for the minimal norm case.

As far as challenges for the future are considered, it is worth pointingout that the geometric approach techniques, although not very widespread,are still valid and adopted by a significant number of research groups in theworld, with different aims.

For instance, an interesting application of the geometric approach to con-trol in aeronautics was proposed by Crouch [64]. Another object of investiga-tion with geometric techniques is fault detection and identification in nonlin-ear systems, based on the extension to nonlinear systems of the theory hereinpresented in Section 4.5, by De Persis and Isidori [65]. In this context remark-able contributions are those by Mattone and De Luca [66] and Castaldi et al.[67], that trace the way to set up a collection of solved problems each regard-ing a particular type of system, thus improving knowledge and experience indealing with the nonlinear case. It is likely and desirable that future researchon the nonlinear geometric approach will follow this trend.

Another possible context where the use of some geometric control theoryconcepts could improve insight and computational speed is model predictivecontrol, that may be considered a major framework where also the controlwith preview (implemented by the part on the left in the general diagram inFig. 6.2) may be included. For a general review on model predictive control seethe survey paper by Garcia, Prett and Morari [68] and the paper by Mayneat al. [69], while features related to nonlinearities were recently investigatedby Magni et al. [70]. The bridge between model predictive control and thegeometric control techniques is provided by the numerous investigations on

6 Conclusions and challenges for the future 51

the problem of controlling a system within a bounded set (see the survey byBlanchini [71]).

The possible improvement provided by preview in dealing with optimalcontrol problems was first pointed out by Tomizuka [72], investigated refer-ring to perfect tracking in the nonlinear case by Devasia, Chen and Paden[73], but completely solved in a rigorous mathematical framework for the lin-ear case only in recent times by Tadmor and Mirkin [74, 75]. For the sameproblem Ferrante, Marro and Ntogramatzidis [55, 56] provided an efficientcomputational setting, where the finite-horizon H2 optimal control problemis expressed in finite terms as a by-product of the well known solution forthe infinite-horizon case. A possible and likely expansion of the geometricapproach towards joining with some aspects of control on bounded sets andmodel predictive control is investigation on properties of trajectories lying onlinear varieties, that may constitute the boundary of a finite set, and workingout efficient algorithms for control and optimal welding of subarcs subject todifferent constraints.

To conclude, if the geometric approach is identified with the branchof teaching and research that uses the concepts of controlled invariant or(A,B)-invariant, conditioned invariant or (A,C)-invariant and invariant zeroas herein recalled, it certain to continue for a lomg time, since new applica-tions of these concepts will certainly be proposed in the future, in connectionwith investigations of new control problems, both linear and nonlinear, in thestate space.

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Index

autonomous regulator problem, 34

basic operations on subspaces, 8basis, 8basis matrix, 8batch processes, 39

computation of J ∗↓ , 10

computation of J ∗↑ , 10

computation of S∗, 13computation of V∗, 12computational tools for analysis, 17conditioned invariant, 13conditioned invariant externally

stabilizable, 13conditioned invariant internally

stabilizable, 13controllability, 19controlled invariant, 12controlled invariant externally stabiliz-

able, 12controlled invariant internally stabiliz-

able, 12

delayed state estimation, 28disturbance decoupling by dynamic

output feedback, 29dual system, 21

extension to quadruples, 15

fault detection and identification, 35fixed modes, 15friend of a conditioned invariant, 13

friend of a controlled invariant, 12

Gram-Schmidt orthonormalizationprocess, 8

H2 optimal decoupling with dynamicoutput feedback, 47

H2-norm, 41Hamiltonian function, 43Hamiltonian system, 43

identity function, 7image, 7inaccessible disturbance decoupling, 23infimum of a lattice, 10internal model, 33intersection of subspaces, 9invariant externally stable, 10invariant internally stable, 10invariant subspace, 10invariant zeros, 14inverse linear transformation of a

subspace, 9

Kalman dual filter, 46Kalman dual smoother, 47Kalman filter, 46Kalman regulator, 45Kalman smoother, 47kernel, 8

lattice, 10linear function, 7linear map, 7

60 Index

linear transformation, 7linear transformation of a subspace, 9LQR problem, 42

maximal ouput-nulling reachablesubspace R∗, 14

maximal output-nulling controlledinvariant V∗, 12

measurable signal decoupling, 25minimal input-containing conditioned

invariant S∗, 14model following, 30model following with feedback, 32multiplicity of invariant zeros, 15multivariable regulator problem, 33

next state, 11noninteraction, 35null space, 8nullity, 8

on-line processes, 39orthogonal complementation of a

subspace, 9output injection, 14

performance index, 42pole assignment by output injection, 19pole assignment by state feedback, 19previewed signal decoupling, 28

rank, 7

relative degree, 20

routine gazero, 17

routine ints, 17

routine invt, 17

routine ortco, 17

routine reldeg, 17

routine sstar, 17

routine sums, 17

routine vstar, 17

self-bounded controlled invariant, 15

semilattice of conditioned invariants, 13

semilattice of controlled invariants, 12

signature of a fault, 38

single-throw processes, 39

singular value decomposition, 8

span, 8

state feedback, 12

state velocity, 11

subspace, 7

sum of subspaces, 9

supremum of a lattice, 10

system internally stable, 11

system left invertible, 19

system minimum phase, 15

system right invertible, 20

vector space, 7

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