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    • Abstract: ReferencesWonhamLinear Multivariable Control – A Geometric Approach,Summer School 3rd edition, Springer Verlag, 1985.on Time Delay Equations and Control TheoryDobbiaco, June 25–29 2001 Basile and Marro

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Linear Multivariable Control – A Geometric Approach,
Summer School 3rd edition, Springer Verlag, 1985.
on Time Delay Equations and Control Theory
Dobbiaco, June 25–29 2001 Basile and Marro
Controlled and Conditioned Invariants in Linear Sys-
tem Theory, Prentice Hall, 1992
Trentelman, Stoorvogel and Hautus
Control Theory for Linear Systems, Springer Verlag,
Linear Control Theory 2001
Giovanni MARRO∗, Domenico PRATTICHIZZO‡
Early References
Basile and Marro
Controlled and Conditioned Invariant Subspaces in
∗ DEIS, Linear System Theory, Journal of Optimization The-
University of Bologna, Italy
‡ DII, ory and Applications, vol. 3, n. 5, 1969.
University of Siena, Italy
Wonham and Morse
Decoupling and Pole Assignment in Linear Multivari-
able Systems: a Geometric Approach, SIAM Journal
on Control, vol. 8, n. 1, 1970.
Introduction to Control Problems
u Σ y
Consider the following figure that includes a controlled
system (plant) Σ and a controller Σr , with a feedback
part Σc and a feedforward part Σf .
d1 d2 rp e
Fig. 1.2. A reduced block diagram.
rp r e u y1
Σf + Σc + Σ
y2 In the above figure d := {d1 , d2 }, y := {y1 , y2 , d1 }.
Σr All the symbols in the figure denote signals, repre-
sentable by real vectors varying in time.
The plant Σ is given and the controller Σr is to be
Fig. 1.1. A general block diagram for regulation. designed to (possibly) maintain e(·) = 0.
Both the plant and the controller are assumed to be
linear (zero state and superposition property).
• rp previewed reference
The blocks represent oriented systems (inputs, out-
• r reference
puts), that are assumed to be causal.
• y1 controlled output
In the classical control theory both continuous-time
• y2 informative output systems and discrete-time systems are considered.
• e error variable
• u manipulated input
• d1 non-measurable disturbance
• d2 measurable disturbance
0 t 0 k
1 2
An example
r + e va ω
PI controller _
K amplifier motor
+ ω cr
+ 1 +
r e T z va
vc Fig. 1.4. The simplified block diagram.
w e
tachometer Σe u Σ y
Fig. 1.3. The velocity control of a dc motor. Σr
The PI controlled yields steady-state control with no Fig. 1.5. The reduced block diagram.
This property is robust against parameter variations, In Fig. 1.5 w accounts for both the reference and
provided asymptotic stability of the loop is achieved. the disturbance. The control purpose is to achieve a
This is due to the presence of an internal model of “minimal” error e in the response to w.
the exosystem that reproduces a constant input sig- If w is assumed to be generated by an exosystem Σe
nal (an integrator). like in the previous example, the internal model en-
Thus, a step signal r of any value is reproduced with sures zero stedy-state error.
no steady-state error and the disturbance cr is steady- This approach can easily be extended to the multi-
state rejected. This is called a type 1 controller. variable case with geometric techniques.
Similarly, a double integrator reproduces with no Modern approaches consider, besides the internal
steady-state error any linear combination of a step model, the minimization of a norm (H2 or H∞) of
and a ramp and rejects disturbances of the same type the transfer function from w to e to guarantee a sat-
This is a type 2 controller. isfactory transient.
3 4
A more complex example Mathematical Models
controller reduction Let us consider the velocity control of a motor shown
rp gear
in Fig. 1.3 and its reduced block diagram (Fig. 1.5):
delay re w e
motor Σe u Σ y
v0 gage
transducer d
Fig. 1.6. Rolling mill control.
Mathematical model of Σ:
d ia
This example fits the general control scheme given in va (t) = Ra ia(t) + La (t) + vc (t) (1.1)
Fig. 1.1. dt

The gage control has an inherent transportation de- cm (t) = B ω(t) + J (t) + cr (t) (1.2)
lay. If the aim of the control is to have given amounts
of material (in meters) at a specified thickness, it is In (1.1) va is the applied voltage, Ra and La the arma-
necessary to have a preview of these amounts, that is ture resistance and inductance, ia and vc the armature
taken into account with the delay. current and counter emf, while in (1.2) cm is the mo-
Of course, this preview can be used with negligible tor torque, B, J, and ω the viscous friction coefficient,
error if the cilinder rotation is feedback controlled by the moment of inertia, and the angular velocity of the
measuring the amount of material with a type 2 con- shaft, and cr the externally applied load torque.
troller. Mathematical model of Σr :
Thus, robustness is achieved with feedback and makes
feedforward (preview control) possible. dz 1
(t) = e(t) (1.3)
There are cases in which preaction (action in advance) dt T
on the controlled system significantly improves track- va (t) = K e(t) + z(t) (1.4)
ing of a reference signal. The block diagram shown where z denotes the output of the integrator in the
in Fig. 1.1 also accounts for these cases. PI controller.
5 6
Their state space representation is The overall system (controlled system and controller)
can be represented with a unique mathematical model
of the same type:
x(t) = A x(t) + B1 u(t) + B2 d(t)
y(t) = C x(t) + D1 u(t) + D2 d(t)
˙ ˆˆ ˆ ˆ
x(t) = A x(t) + B1 u(t) + B2 d(t)
ˆˆ ˆ ˆ (1.7)
where for Σ, x := [ia ω]T, u := va , d := cr , y := ω and y (t) = C x(t) + D1 u(t) + D2 d(t)
−Ra/La −k1 /La where for x := [ia ω z]T, u := r, d := cr y := ω and
k2 /J −B/J
 
1/La 0 −Ra/La −(k1 + K)/La 1/La
B1 =
B2 =
−1/J A =  k2 /J
ˆ −B/J 0 
0 −1/T 0
C= 0 1 D1 = 0 D2 = 0
   
K/La 0
while for Σr , xr := z, ur := e, yr := va and B1 =  0 
ˆ B2 =  −1/J 
1 0
Ar = 0 Br = 1/T Cr = 1 Dr = K
C= 0 1 0 ˆ
D1 = 0 ˆ
D2 = 0
Mathematical model of Σe :
The regulator design problem is: determine T and K
dr d cr such that the system (1.7) is internally stable, i.e. the
=0 =0 (1.6) ˆ
eigenvalues of A have stricly negative real parts and
dt dt
this property is maintained in presence of admissible
This corresponds to an autonomous system (without parameter variations.
input) having xe = y := [r cr ]T and
0 0 1 0
Ae = Ce =
0 0 0 1
7 8
If only its behavior with respect to step inputs must State Space Models
be considered, the overall system in Fig. 1.3 can be
represented as the autonomous system Continuous-time systems:
x(t) = A x(t) + B u(t)
˙ ˆˆ
x(t) = A x(t)
ˆ y(t) = C x(t) + D u(t)
y (t) = C x(t)
ˆ with the state x ∈ X = Rn, the input u ∈ U = Rp, the
output y ∈ Y = Rq and A, B C, D real matrices of suit-
able dimensions. The system will be referred to as the
where for x := [ia ω z r cr ]T, y := ω and quadruple (A, B, C, D) or the triple (A, B, C) if D = 0.
Most of the theory will be derived referring to triples
  since extension to quadruples is straightforward.
−Ra/La −(k1 + K)/La 1/La K/La 0
 k2 /J −B/J 0 0 −1/J 
  Discrete-time systems:
ˆ 0 −1/T 0 1 0 
 0 0 0 0 0 
x(k+1) = Ad x(k) + Bd u(k)
0 0 0 0 0 (1.10)
y(k) = Cd x(k) + Dd u(k)
C= 0 1 0 0 0 Recall that a continuous-time system is internally
asymptotically stable iff all the eigenvalues of A be-
The regulator design problem is: determine T and K long to C− (the open left half plane of the complex
ˆ ˆ
such that the autonomous system (A, C) is externally plane) and a discrete-time system is internally asymp-
stable, i.e., limt→∞ y(t) = 0 for any initial state and totically stable iff all the eigenvalues of Ad belong to
this property is maintained in presence of admissible C (the open unit disk of the complex plane).
parameter variations.
In the discrete-time case a significant linear model
is also the FIR (Finite Impulse Response) system,
defined by the finite convolution sum
y(k) = l=0 W (l) u(k − l) (1.11)
where W (k) (k = 0, . . . , N ) is a q × p real matrix, re-
ferred to as the gain of the FIR system, while N is
called the window of the FIR system.
9 10
Transfer Matrix Models Geometric Approach (GA)
By taking the Laplace transform of (1.9) or the Z
transform of (1.10) we obtain the transfer matrix rep-
resentations Geometric Approach: is a control theory for multivari-
Y (s) = G(s) U (s) with able linear systems based on:
G(s) = C (sI − A)−1 B + D • linear transformations
and • subspaces
Y (z) = Gd(z) U (z) with
(1.13) (The alternative approach is the transfer function ap-
Gd (z) = Cd (zI − Ad)−1 Bd + Dd proach)
The geometric approach consists of
The H2 norm in the continuous-time case is
• an algebraic part (theoretical)
∞ 1/2
1 ∗ • an algorithmic part (computational)
G 2 = tr G(jω) G (jω) dω (1.14)
2π −∞
∞ 1/2 Most of the mathematical support is developed in
T coordinate-free form, to take advantage of simpler
= tr g(t) g (t) dt (1.15)
0 and more elegant results, which facilitate insight into
where g(t) denotes the impulse response of the system the actual meaning of statements and procedures; the
(the inverse Laplace transform of G(s)), and in the computational aspects are considered independently
discrete-time case it is of the theory and handled by means of the standard
π 1/2 methods of matrix algebra, once a suitable coordinate
Gd 2 = tr Gd (e jω
) G∗ (ejω ) dω
d (1.16) system is defined.
2π −π
∞ 1/2
= tr gd (k) gd (k) dt (1.17)
where Gd (ejω )
denotes the frequency response of the
discrete-time system for unit sampling time and gd(k)
the impulse response of the system (the inverse Z
transform of Gd(z)).
11 12
A Few Words on the Algorithmic Part Basic relations
A subspace X is given through a basis matrix of max-
imum rank X such that X = imX.
X ∩ (Y + Z) ⊇ (X ∩ Y) + (X ∩ Z)
The operations on subspaces are all performed X + (Y ∩ Z) ⊆ (X + Y) ∩ (X + Z)
through an orthonormalization process (subroutine (X ⊥ )⊥ = X
ima.m in Matlab) that computes an orthonormal ba-
(X + Y)⊥ = X ⊥ ∩ Y⊥
sis of a set of vectors in Rn by using methods of the
Gauss–Jordan or Gram–Schmidt type. (X ∩ Y)⊥ = X ⊥ + Y⊥
A (X ∩ Y) ⊆ AX ∩ AY
Basic Operations A (X + Y) = AX + AY
• sum: Z = X + Y A (X ∩ Y) = A−1 X ∩ A−1 Y
• linear transformation: Y = A X A−1 (X + Y) ⊇ A−1 X + A−1 Y
• orthogonal complementation: Y = X ⊥
• intersection: Z = X ∩ Y
• inverse linear transformation: X = A−1 Y
1. The first two relations hold with the equality sign
if one of the involved subspaces X , Y, Z is con-
tained in any of the others.
Computational support with Matlab
2. The following relations are useful for computa-
Q = ima(A,p) Orthonormalization. tional purposes:
Q = ortco(A) Complementary orthogonalization. AX ⊆ Y ⇔ AT Y ⊥ ⊆ X ⊥
Q = sums(A,B) Sum of subspaces.
(A−1 Y)⊥ = AT Y ⊥
Q = ints(A,B) Intersection of subspaces.
where AT denotes the transpose of matrix A.
Q = invt(A,X) Inverse transform of a subspace.
Q = ker(A) Kernel of a matrix.
In program ima the flag p allows for permutations of
the input column vectors.
13 14
Invariant Subspaces The Algorithms
Definition 2.1 Given a linear map A : X → X , a sub-
space J ⊆ X is an A-invariant if Algorithm 2.1 Computation of minJ (A, B)
AJ ⊆ J Z1 = B
Zi = B + A Zi−1 (i = 2, 3, . . .) (2.1)
Property 2.1 Given the subspaces D, E contained in
X and such that D ⊆ E, and a linear map A : X → X , minJ (A, B) = B + A minJ (A, B)
the set of all the A-invariants J satisfying D ⊆ J ⊆ E
is a nondistributive lattice Φ0 with respect to ⊆, +, ∩. Algorithm 2.2 Computation of maxJ (A, C)
We denote with maxJ (A, E) the maximal A-invariant Z1 = C
contained in E (the sum of all the A-invariants Zi = C ∩ A−1 Zi−1 (i = 2, 3, . . .) (2.2)
contained in E) and with minJ (A, D) the mini-
mal A-invariant containing D (the intersection of maxJ (A, C) = C ∩ A−1 maxJ (A, C)
all the A-invariants containing D): the above lat-
tice is non-empty if and only if D ⊆ maxJ (A, E) or Property 2.2 Dualities
minJ (A, D) ⊆ E.
maxJ (A, C) = minJ (AT, C ⊥ )⊥
minJ (A, B) = maxJ (AT, B ⊥)⊥
maxJ (A, E)
Φ0 Computational support with Matlab
minJ (A, D) Q = mininv(A,B) Minimal A-invariant containing
Q = maxinv(A,C) Maximal A-invariant contained
D in imC
Fig. 2.1. The lattice Φ0.
15 16
Internal and External Stability of an Invariant Refer to the autonomous system
The restriction of map A to the A-invariant subspace x(t) = A x(t)
˙ x(0) = x0 (2.5)
J is denoted by A|J ; J is said to be internally stable if
A|J is stable. Given two A-invariants J1 and J2 such
that J1 ⊆ J2 , the map induced by A on the quotient x(k + 1) = Ad x(k) x(0) = x0 (2.6)
space J2/J1 is denoted by A|J2/J1 . In particular, an The behavior of the trajectories in the state space with
A-invariant J is said to be externally stable if A|X /J respect to an invariant can be represented as follows.
is stable.
Algorithm 2.3 Matrices P and Q representing A|J
and A|X /J up to an isomorphism, are derived as fol- x(0)
lows. Let us consider the similarity transformation
T := [J T2 ], with imJ = J (J is a basis matrix of J )
and T2 such that T is nonsingular. In the new basis
the linear transformation A is expressed by
A11 A12
A = T −1 A T = (2.3)
O A22
The requested matrices are defined as P := A11 ,
Q := A22.
Complementability of an Invariant
An A-invariant J ⊆ X is said to be complementable
if an A-invariant Jc exixts such that J ⊕ Jc = X ; if so, Fig. 2.2. External and internal stability of an
Jc is called a complement of J . invariant.
Algorithm 2.4 Let us consider again the change of
basis introduced in Algorithm 2.3. J is comple-
mentable if and only if the Sylvester equation
Computational support with Matlab
A11 X − X A22 = −A12 (2.4)
[P,Q] = stabi(A,X) Matrices for the internal
admits a solution. If so, a basis matrix of Jc is given and external stability of
by Jc := J X + T2. the A-invariant imX
17 18
Controllability and Observability If R = X , but R is externally stabilizable, (A, B) is said
to be stabilizable.
Consider a triple (A, B, C), i.e., refer to
If Q = {0}, but Q is internally stabilizable, (A, C) is
said to be detectable.
x(t) = A x(t) + B u(t)
y(t) = C x(t) Pole Assignment
Let B := imB. The reachability subspace of (A, B), v + u y u y
i.e., the set of all the states that can be reached +
from the origin in any finite time by means of control
actions, is R = minJ (A, B). If R = X , the pair (A, B) x
is said to be completely controllable.
Let C := kerC. The unobservability subspace of (A, C), Fig. 2.4. State feedback and output injection
i.e., the set of all the initial states that cannot be rec-
ognized from the output function, is Q = maxJ (A, C).
If Q = {0}, (A, C) is said to be completely observable. State feedback
x(t) = (A + BF ) x(t) + B v(t)
y(t) = C x(t)
Output injection
x(t) = (A + GC) x(t) + B u(t)
y(t) = C x(t)
The eigenvalues of A + BF are arbitrarily assignable
by a suitable choiche of F iff the system is com-
pletely controllable and those of A + GC are arbitrarily
assignable by a suitable choice of G iff the system is
completely observable.
Fig. 2.3. The reachability subspace.
19 20
Complete Pole Assignment through an Observer Controlled and Conditioned Invariants
v + u y u y Definition 2.2 Given a linear map A : X → X and
Σ Σ a subspace B ⊆ X a subspace V ⊆ X is an (A, B)-
controlled invariant if
F −G AV ⊆ V + B (2.10)
Σc x
˜ Let B and V be basis matrices of B and V respectively:
˜ Σo the following statements are equivalent to (2.10):
- a matrix F exists such that (A + BF ) V ⊆ V
Fig. 2.5. Dynamic pre-compensator and observer
- matrices X and U exist such that A V = V X + B U
- V is a locus of trajectories of the pair (A, B)
v + u y
˜ V
Fig. 2.6. Pole assignment through an observer
Fig. 2.7. The controlled invariant as a locus of
The eigenvalues of the overall system are the union trajectories.
of those of A + BF and those of A +GC, hence com-
pletely assignable if the triple (A, B, C) is completely
controllable and observable.
21 22
The sum of any two controlled invariants is a con- Definition 2.3 Given a linear map A : X → X and
trolled invariant, while the intersection is not; thus a subspace C ⊆ X a subspace S ⊆ X is an (A, C)-
the set of all the controlled invariants contained in a conditioned invariant if
given subspace E ⊆ X is a semilattice with respect to
⊆, +, hence admits a supremum, the maximal (A, B)- A (S ∩ C) ⊆ S (2.11)
controlled invariant contained in E, that is denoted by
maxV(A, B, E) (or simply V(B,E) ). We use the symbol V ∗
∗ Let C be a matrix such that C = kerC. The following
for maxV(A, imB, kerC), which is the most important statement is equivalent to (2.11):
controlled invariant concerning the triple (A, B, C).
- a matrix G exists such that (A + GC) S ⊆ S
Referring to the pair (A, B), we denote with RV the
The intersection of any two conditioned invariants is
reachable subspace from the origin by trajectories con-
a conditioned invariant while the sum is not; thus
strained to belong to a generic (A, B)-controlled invari-
the set of all the conditioned invariants containing
ant V. Owing to the first property above, it is derived
a given subspace D ⊆ X is a semilattice with respect
as RV = minJ (A + BF, V ∩ B) and, clearly being an
to ⊆, ∩, hence admits an infimum, the minimal (A, C)-
(A + BF )-invariant, it also is an (A, B)-controlled in-
conditioned invariant containing D, that is denoted
variant. ∗
by minS(A, C, D) (or simply S(C,D) ). We use the simple
A generic (A, B)-controlled invariant V is said to be in- symbol S ∗ for minS(A, kerC, imB), which is the most
ternally stabilizable or externally stabilizable if at least important conditioned invariant concerning the triple
one matrix F exists such that (A + BF )|V is stable or (A, B, C).
at least one matrix F exists such that (A + BF )|X /V
is stable. It is easily proven that the eigenstructure Controlled and conditioned invariants are dual to each
of (A + BF )|V/RV is independent of F ; it is called the other. Controlled invariants are used in control prob-
internal unassignable eigenstructure of V. V is both lems, while conditioned invariants are used in obser-
internally and externally stabilizable with the same F vation problems.
if and only if its internal unassignable eigenstructure
The orthogonal complement of an (A, C)-conditioned
is stable and the A-invariant V + R = V + minJ (A, B)
invariant is an (AT, C ⊥ )-controlled invariant, hence the
is externally stable. This latter is ensured by the sta-
orthogonal complement of an (A, C)-conditioned in-
bilizability property of the pair(A, B).
variant containing a given subspace D is an (AT, C ⊥ )-
controlled invariant contained in D⊥ . External and in-
ternal stabilizability of conditioned invariants are easily
defined by duality.
23 24
Self-bounded Controlled Invariants The infimum of the lattice of all the (A, B)-controlled
invariants self-bounded with respect to a given sub-
Definition 2.4 Given a linear map A : X → X and two space E can be expressed in terms of conditioned in-
subspaces B ⊆ X , E ⊆ X , a subspace V ⊆ X is an variants as follows.
(A, B)-controlled invariant self-bounded with respect
to E if, besides (2.10), the following relations hold ∗
Property 2.3 Let D ⊆ V(B,E) . The infimum of the

V ⊆ V(B,E) (2.12) lattice Φ of all the (A, B)-controlled invariants self-

bounded with respect to E and containing D is ex-
V(B,E) ∩B ⊆V (2.13) pressed by
∗ ∗
Vm = V(B,E) ∩ S(E,B+D) (2.14)
The set of all the (A, B)-controlled invariants self-
bounded with respect to E is a nondistributive lattice ∗ ∗
Note, in particular, that RV(B,E) = V(B,E) ∩ S(E,B). The

Use: 0.1313