Linear Control Theory 2001

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References

Wonham 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 Linear System Theory, Journal of Optimization The- ∗DEIS, University of Bologna, Italy ory and Applications, vol. 3, n. 5, 1969. ‡DII, 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. d 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 Σ and a feedforward part Σ . c f Σr rp e d1 d2

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. • y controlled output 1 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 cr An example

r + va ω e PI M PI controller _ K ampliﬁer motor T + ω cr + 1 + va r _ e T z vc Fig. 1.4. The simpliﬁed block diagram.

Σ w e tachometer e 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. error. 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 intheresponsetow. 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

reduction controller gear Let us consider the velocity control of a motor shown rp in Fig. 1.3 and its reduced block diagram (Fig. 1.5): + delay r e Σ w e motor e u Σ y

v0 gage transducer d Σr Fig. 1.6. Rolling mill control. Mathematical model of Σ:

dia This example fits the general control scheme given in va(t)=Ra ia(t)+La (t)+vc(t) (1.1) Fig. 1.1. dt dω The gage control has an inherent transportation de- cm(t)=Bω(t)+J (t)+cr(t) (1.2) dt 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,andω 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 dz 1 feedforward (preview control) possible. (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)=Ke(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 ofthesametype: x˙(t)=Ax(t)+B1 u(t)+B2 d(t) (1.5) y(t)=Cx(t)+D1 u(t)+D2 d(t) xˆ˙(t)=Aˆxˆ(t)+Bˆ1 u(t)+Bˆ2 d(t) (1.7) T yˆ(t)=Cˆ xˆ(t)+Dˆ u(t)+Dˆ d(t) where for Σ, x := [ia ω] , u := va, d := cr, y := ω and 1 2 −R /L −k /L T A = a a 1 a where for x := [ia ωz] , u := r, d := cr y := ω and k2/J −B/J   − − 1/La 0 Ra/La (k1 + K)/La 1/La B1 = B2 =   0 −1/J Aˆ = k2/J −B/J 0 0 −1/T 0 C = 01 D =0 D =0 1 2     K/La 0 while for Σr, xr := z, ur := e, yr := va and     Bˆ1 = 0 Bˆ2 = −1/J 1 0

Ar =0 Br =1/T Cr =1 Dr = K Cˆ = 010 Dˆ1 =0 Dˆ2 =0

Mathematical model of Σe: The regulator design problem is: determine T and K dr dc such that the system (1.7) is internally stable,i.e.the =0 r =0 (1.6) dt dt eigenvalues of Aˆ have stricly negative real parts and this property is maintained in presence of admissible This corresponds to an autonomous system (without parameter variations. T input) having xe = y := [rcr] and 00 10 A = C = e 00 e 01

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)=Ax(t)+Bu(t) (1.9) xˆ˙(t)=Aˆxˆ(t) y(t)=Cx(t)+Du(t) (1.8) yˆ(t)=Cˆ xˆ(t) with the state x ∈X = Rn,theinput u ∈U= Rp,the output y ∈Y= Rq and A, BC, D real matrices of suit- T able dimensions. The system will be referred to as the where for x := [ia ωzrcr] , y := ω and quadruple (A, B, C, D)orthetriple (A, B, C)ifD =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 001/J  Discrete-time systems: ˆ =  0 −1 01 0 A  /T  00000 x(k+1) = A x(k)+B u(k) 00000 d d (1.10) y(k)=Cd x(k)+Dd u(k) ˆ C = 01000 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 such that the autonomous system (A,ˆ Cˆ)isexternally 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 N − y(k)= l=0 W (l) u(k l) (1.11) where W (k)(k =0,...,N)isaq × p real matrix, referred 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 multivariable linear systems based on: Y (s)=G(s) U(s)with (1.12) G(s)=C (sI − A)−1 B + D • linear transformations and • subspaces ( )= ( ) ( )with Y z Gd z U z (The alternative approach is the transfer function ap- −1 (1.13) Gd(z)=Cd (zI − Ad) Bd + Dd proach) respectively. The geometric approach consists of The H2 norm in the continuous-time case is • an algebraic part (theoretical) ∞ 1/2 1 ∗ • G2 = tr G(jω) G (jω) dω (1.14) an algorithmic part (computational) 2π −∞ ∞ 1/2 Most of the mathematical support is developed in = tr g(t) gT(t) dt (1.15) coordinate-free form, to take advantage of simpler 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 1 jω ∗ jω Gd2 = tr Gd(e ) G (e ) dω (1.16) system is defined. 2π d −π ∞ 1/2 T = tr gd(k) gd (k) dt (1.17) k=0 jω where Gd(e ) 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 maximum 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 X∩Y ⊥ X ⊥ Y⊥ Gauss–Jordan or Gram–Schmidt type. ( ) = + A (X∩Y) ⊆ A X∩A Y Basic Operations A (X + Y)=A X + A Y −1 X∩Y −1 X∩ −1 Y • sum: Z = X + Y A ( )=A A −1 X Y ⊇ −1 X −1 Y • linear transformation: Y = A X A ( + ) A + A • orthogonal complementation: Y = X ⊥ Remarks: • 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 contained 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. A X⊆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- Algorithm 2.1 Computation of minJ ( B) space J⊆X is an A-invariant if A, J⊆J Z1 = B A Z B Z i = + A i−1 (i =2, 3,...) (2.1) Property 2.1 Given the subspaces D, E contained in minJ ( B)=B + minJ ( B) X and such that D⊆E, and a linear map A : X→X, A, A A, 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 -invariants Z C∩ −1 Z A i = A i−1 (i =2, 3,...) (2.2) contained in E)andwithminJ (A, D) the mini- −1 mal A-invariant containing D (the intersection of maxJ (A, C)=C∩A maxJ (A, C) all the A-invariants containing D): the above lat- D⊆ J E tice is non-empty if and only if max (A, )or Property 2.2 Dualities minJ (A, D) ⊆E. maxJ (A, C)=minJ (AT, C⊥)⊥ E minJ (A, B)=maxJ (AT, B⊥)⊥

maxJ (A, E) Computational support with Matlab Φ0 { minJ (A, D) Q = mininv(A,B) Minimal A-invariant containing imB 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)=Ax(t) x(0) = x0 (2.5) J is denoted by |J ; J is said to be internally stable if A or 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) J J | space 2/ 1 is denoted by A J2/J1 .Inparticular,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 x(0) and A|X /J up to an isomorphism, are derived as follows. Let us consider the similarity transformation T := [JT2],withimJ = 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 x(0) −1 A11 A12 A = T AT = (2.3) OA22 The requested matrices are deﬁned as P := A11, Q := A22. J Complementability of an Invariant An A-invariant J⊆Xis said to be complementable if an A-invariant Jc exixts such that J⊕Jc = X ;ifso, 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 complementable if and only if the Sylvester equation Computational support with Matlab A X − XA = −A (2.4) 11 22 12 [P,Q] = stabi(A,X) Matrices for the internal J admits a solution. If so, a basis matrix of c is given and external stability of by Jc := JX+ T2. the A-invariant imX 17 18 Controllability and Observability If R!= X , but R is externally stabilizable, (A, B)issaid 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)=Ax(t)+Bu(t) (2.7) y(t)=Cx(t) Pole Assignment

v Let B := imB. The reachability subspace of (A, B), + u y u y i.e., the set of all the states that can be reached + Σ Σ 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 F G 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). State feedback If Q = {0},(A, C)issaidtobecompletely observable. x˙(t)=(A + BF) x(t)+Bv(t) (2.8) y(t)=Cx(t) Output injection x˙(t)=(A + GC) x(t)+Bu(t) (2.9) R y(t)=Cx(t)

The eigenvalues of A + BF are arbitrarily assignable by a suitable choiche of F iﬀ the system is completely 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 + a subspace B⊆X a subspace V⊆X is an (A, B)- + Σ Σ controlled invariant if

F −G A V⊆V+ B (2.10)

x˜ Let B and V be basis matrices of B and V respectively: Σ x˜ Σ c o the following statements are equivalent to (2.10):

-amatrix exists such that ( + ) V⊆V Fig. 2.5. Dynamic pre-compensator and observer F A BF - matrices X and U exist such that AV = VX+ BU - V is a locus of trajectories of the pair (A, B) y v + u

+ Σ

−G

x˜ V Σo

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 completely 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 S∩C ⊆S ⊆, +, hence admits a supremum, the maximal (A, B)- A ( ) (2.11) controlled invariant contained in E, that is denoted by V B E V∗ V∗ Let C be a matrix such that C =kerC. The following max (A, , )(orsimply (B,E)). We use the symbol for maxV(A, imB,kerC), which is the most important statement is equivalent to (2.11): controlled invariant concerning the triple (A, B, C). -amatrixG exists such that (A + GC) S⊆S

Referring to the pair ( ), we denote with RV the A, B 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 ( B)-controlled invari- A, 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)- ( + )-invariant, it also is an ( B)-controlled in- A BF A, conditioned invariant containing D, that is denoted variant. S C D S∗ by min (A, , )(orsimply (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 | other. Controlled invariants are used in control prob- of (A + BF) V/RV is independent of F ; it is called the 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 internal 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 E D⊆V∗ to if, besides (2.10), the following relations hold Property 2.3 Let (B,E). The infimum of the B V⊆V∗ (2.12) lattice Φ of all the (A, )-controlled invariants self- (B,E) bounded with respect to E and containing D is ex- V∗ ∩B⊆V (B,E) (2.13) pressed by V V∗ ∩S∗ m = (B,E) (E,B+D) (2.14) The set of all the (A, B)-controlled invariants self- bounded with respect to E is a nondistributive lattice ∗ ∗ R ∗ V ∩S ∗ Note, in particular, that V = B E E B .The with respect to ⊆, +, ∩, whose supremum is V and (B,E) ( , ) ( , ) (B,E) dual of Property 2.3 is whose infimum is RV ∗ . (B,E) S∗ ⊆E Property 2.4 Let C D . The infimum of the lat- Given subspaces D, E contained in X and such ( , ) tice Ψ of all the (A, C)-conditioned invariants self- that D⊆V∗, the infimum of the lattice of all hidden with respect to D and contained in E is ex- the ( B)-controlled invariants self-bounded with A, pressed by respect to E and containing D is the reach- ∗ S S∗ V∗ able set on V with forcing action B + D, i.e., M = (C,D) + (D,C∩E) (2.15) V J V∗ ∩ B D m := min (A + BF, (B,E) ( + )), with F such V∗ ⊆V∗ that (A + BF) (B,E) (B,E).

25 26 E E The Algorithms

∗ V∗ Algorithm 2.5 Computation of S =minS(A, C, B) (B,E) SM S1 = B Φ Ψ S B S ∩C i = + A ( i−1 )(i =2, 3,...) (2.16) V S∗ m (C,D) S∗ = B + A (S∗ ∩C)

∗ D D Algorithm 2.6 Computation of V =maxV(A, B, C)

V1 = C V C∩ −1 V B Fig. 2.8. The lattices Φ and Ψ. i = A ( i−1 + )(i =2, 3,...) (2.17) V∗ = C∩A−1(V∗ + B) The main theorem and its dual D⊆V∗ Property 2.5 Dualities Theorem 2.1 Let (B,E). There exists at least ⊥ ⊥ ⊥ one internally stabilizable (A, B)-controlled invariant maxV(A, B, C)=minS(AT, B , C ) V such that D⊆V⊆E if and only if V is internally m T ⊥ ⊥ ⊥ stabilizable. minS(A, C, B)=maxV(A , C , B ) S∗ ⊆E Theorem 2.2 Let C D . There exists at least ( , ) Remark: Refer to the discrete-time triple (A ,B ,C ), one externally stabilizable (A, C)-conditioned invariant d d d i.e., to equations (1.10) with D = 0. Algorithm 2.5 S such that D⊆S⊆E if and only if S is internally d M with A = A , B =imB and C =kerC at the generic stabilizable. d d d i-th step provides the set of all states reachable from the origin with trajectories having all the states but the last one belonging to kerCd, hence invisible at the output. Thus S∗ has a control meaning in the discrete-time dynamics: it is the maximum subspace of the state space reachable from the origin with this type of trajectories in ρ steps, being ρ the number of iterations required for (2.16) to converge to S∗.

27 28 Algorithm 2.7 Computation of matrix F such that Computational support with Matlab (A + BF) V⊆V.LetV be a basis matrix of the (A, B)- controlled invariant V. First, compute Q = mainco(A,B,X) Maximal (A, imB)-controlled invariant contained in imX X =[VB]+ AV Q = miinco(A,C,X) Minimal (A, imC)-conditioned U invariant containing imX where the symbol + denotes the pseudoinverse. Then, F = eﬀe(A,B,X) State feedback matrix such that compute (A + BF)imX ⊆ imX − F := −U (V T V ) 1 V T [P,Q] = stabv(A,B,X) Matrices for the internal and external stability of the (A, imB)-controlled Algorithm 2.8 Computation of the internal unas- invariant imX signable eigenstructure of an (A, B)-controlled invari- | F = eﬀest(A,B,X,ei,ee) Stabilizing feedback matrix ant. A matrix P representing the map (A + BF) V/RV up to an isomorphism, is derived as follows. Let us setting the assignable eigenvalues as ei and the assignable external eigenvalues as ee consider the similarity transformation T := [T1 T2 T3], with imT1 = RV, imT2 = V and T3 such that T is nonsingular. In the new basis matrix A + BF is expressed by   A11 A12 A13 −1   (A + BF) = T (A + BF) T = OA22 A23 OOA33 The requested matrix is P := A22.

29 30 The Geometric Characterization Left and Right Invertibility of Some Properties of Linear Systems Definition 3.1 Assume that B has maximal rank. System (3.1) is said to be invertible (left-invertible) if, given any output function y(t), t ∈ [0,t ] t > 0 be- Consider the standard continuous-time system – triple 1 1 longing to imT , there exists a unique input function (A, B, C) f u(t), t ∈ [0,t1), such that (3.3) holds. x˙(t)=Ax(t)+Bu(t) (3.1) y(t)=Cx(t) Definition 3.2 Assume that Bd has maximal rank. System (3.2) is said to be invertible (left-invertible) or the standard discrete-time system – triple if, given any output function y(k), k ∈ [0,k1], k1 ≥ n (Ad,Bd,Cd) belonging to imTf there exists a unique input function x(k+1) = A x(k)+B u(k) u(k), k ∈ [0,k1 − 1] such that (3.4) holds. d d (3.2) y(k)=Cd x(k) Definition 3.3 Assume that C has maximal rank. (we consider triples since they provide a better insight System (3.1) is said to be functionally controllable and extension to quadruples is straightforward – ob- (right-invertible) if there exists an integer ρ ≥ 1 such tainable with a suitable state extension) that, given any output function y(t), t ∈ [0,t1], t1 > 0 with ρ-th derivative piecewise continuous and such Systems (3.1) and (3.2) with x(0) = 0 define linear that y(0) = 0, ... y(ρ)(0) = 0, there exists at least one T U →Y U maps f : f f from the space f of the admissi- input function u(t), t ∈ [0,t1) such that (3.3) holds. ble input functions to the functional space Yf of the The minimum value of ρ satisfying the above state- zero-state responses. These maps are defined by the ment is called the relative degree of the system. convolution integral and the convolution summation t Definition 3.4 Assume that Cd has maximal rank. y(t)=C eA (t−τ) Bu(τ) dτ (3.3) System (3.2) is said to be functionally controllable 0 (right-invertible) if there exists an integer ρ ≥ 1 such k−1 that, given an output function y(k), k ∈ [0,k1], k1 ≥ ρ (k−h−1) y(k)=Cd Ad Bd u(h) (3.4) such that y(k)=0, k ∈ [0,ρ−1], there exists at least h=0 one input function u(k), k ∈ [0,k1 − 1] such that (3.4) The admissible input functions are: holds. The minimum value of ρ satisfying the above - piecewise continuous and bounded functions of time statement is called the relative degree of the system. t for (3.3); - bounded functions of the discrete time k for (3.4). 31 32 ∗ ∗ Σi Σ Let V :=maxV(A, imB,kerC)andS :=minV(A, kerC, imB) + e _ Theorem 3.1 System (3.1) or (3.2) is invertible if Σ and only if f V∗ ∩S∗ = {0} (3.5) + Σ Σi Theorem 3.2 System (3.1) or (3.2) is functionally _ e controllable if and only if V∗ + S∗ = X (3.6) Σf

Note the duality: if system (A, B, C)or(Ad,Bd,Cd)is Fig. 3.1. Connections for right and left inversion invertible (functionally controllable), the adjont sys- T T T T T T tem (A ,C ,B )or(Ad ,Cd ,Bd ) is functionally controllable (invertible). In Fig. 3.1 Σf denotes a suitable relative-degree ﬁlter in the continuous-time case or a relative degree delay Relative Degree in the discrete-time case. The inverse system Σi is to be designed to null the error e. Property 3.1 Assume that (3.6) holds and consider If the system is nonminimum phase, i.e, has some un- the conditioned invariant computational sequence Si (i =1, 2,...). The relative degree is the least integer stable zeros, the inverse system is internally unstable, ρ such that so that the time interval considered for the system ∗ inversion must be ﬁnite. V + Sρ = X

Computational support with Matlab

r = reldeg(A,B,C,[D]) Relative degree of (A, B, C) or (A, B, C, D)

33 34 Invariant Zeros Property 3.2 Let W be a real m × m matrix having the invariant zero structure of (A, B, C) as eigenstruc- Roughly speaking, an invariant zero corresponds to ture. A real p × m matrix L and a real n × m matrix X a mode that, if suitably injected at the input of a exist, with (W, X) observable, such that by applying dynamic system, can be nulled at the output by a to (A, B, C) the input function suitable choice of the initial state. Wt u(t)=Le v0 (3.7) m Deﬁnition 3.5 The invariant zeros of (A, B, C) are where v0 ∈ R denotes an arbitrary column vector, and V∗ the internal unassignable eigenvalues of .The starting from the initial state x0 = Xv0, the output y(·) invariant zero structure of (A, B, C) is the internal is identically zero, while the state evolution (on ker ) ∗ C unassignable eigenstructure of V . is described by ∗ ∗ Wt Recall that RV ∗ =V ∩S . The invariant zeros are the x(t)=Xe v0 (3.8) | ∗ eigenvalues of the map (A + BF) V /RV∗ ,whereF denotes any matrix such that (A + BF)V∗ ⊆V∗. v0 x0 = Xv0 v u y unstable L zero Σe Σ

RV ∗ Fig. 3.3. The meaning of Property 3.2

Remark. In the discrete-time case equations k ∗ (3.7) and (3.8) are replaced by u(k)=LW v0 and V stable k zero x(k)=XW v0, respectively.

Fig. 3.2. Decomposition of the map (A + BF)|V ∗ Computational support with Matlab

z = gazero(A,B,C,[D]) Invariant zeros of (A, B, C) or (A, B, C, D)

35 36 Extension to Quadruples The addition of integrators at inputs or outputs does not affect the system right and left invertibility, while Extension to quadruples of the above definitions and the relative degree of (A,ˆ B,ˆ Cˆ) must be simply reduced properties can be obtained through a simple con- by1tobereferredto(A, B, C, D) trivance. In the discrete-time case Σd is described by v u y u(k +1)=v(k) Σd Σ and the overall system by integrators or delays xˆ(k +1) = Aˆd xˆ(k)+Bˆb v(k) y(k)=Cˆd yˆ(t) y u z ˆ ˆ ˆ Σ Σd with the extended matrices Ad, Bd, Cd defined like in the continuous-time case in terms of Ad, Bd, Cd, Dd. integrators or delays This contrivance can also be used in most of the synthesis problems considered in the sequel. Fig. 3.4. Artifices to reduce a quadruple to a triple

Refer to the ﬁrst ﬁgure: system Σd is modeled by u˙(t)=v(t) and the overall system by xˆ˙(t)=Aˆxˆ(t)+Bvˆ (t) y(t)=Cˆ xˆ(t) with x AB xˆ := Aˆ := u 00 0 Bˆ := Cˆ := CD Ip

37 38 V∗ V B E Disturbance Decoupling Let (B,E) := max (A, , ). Since any (A + BF)- invariant is an (A, B)-controlled invariant, the inacces- The disturbance decoupling problem is one of the ear- sible disturbance decoupling problem has a solution if liest (1969) applications of the geometric approach. and only if D⊆V∗ (3.11) d (B,E) e Equation (3.11) is a structural condition and does not u Σ ensure internal stability. If stability is requested, we have the disturbance decoupling problem with stabil- x F ity. Stability is easily handled by using self-bounded controlled invariants. Assume that (A, B) is stabilizable (i.e., that R =minJ (A, B) is externally stable) Fig. 3.5. Disturbance decoupling and let with state feedback V V∗ ∩S∗ m := (B,E) (E,B+D) (3.12) Let us consider the system This subspace has already been deﬁned in Property 2.3. The following result, providing both the struc- x˙(t)=Ax(t)+Bu(t)+Dd(t) (3.9) tural and the stability condition, is a direct conse- e(t)=Ex(t) quence of Theorem 2.1. where u denotes the manipulable input, d the distur- Corollary 3.1 The disturbance decoupling problem bance input. Let B := imB, D := imD, E := kerE. with stability admits a solution if and only if The disturbance decoupling problem is: determine, if D⊆V∗ (B,E) (3.13) possible, a state feedback matrix F such that distur- V is internally stabilizable bance d has no inﬂuence on output e. m

The system with state feedback is described by If conditions (3.13) are satisfied, a solution is provided by a state feedback matrix such that (A + BF) Vm ⊆ x˙(t)=(A + BF) x(t)+Dd(t) (3.10) Vm and σ(A + BF)isstable. e(t)=Ex(t) If the state is not accessible, disturbance decoupling It behaves as requested if and only if its reachable set may be achieved through a dynamic unit similar to a by d, i.e., the minimum (A + BF)-invariant containing state observer. This is called disturbance decoupling D, is contained in E. problem with dynamic measurement feedback,and will be considered later. 39 40 Feedforward The solvability conditions once again are consequence of Theorem 2.1. Decoupling of Measurable Signals Corollary 3.2 The measurable signal decoupling Consider now the system problem with stability admits a solution if and only if x˙(t)=Ax(t)+Bu(t)+Hh(t) ∗ (3.14) H⊆V + B e(t)=Ex(t) (B,E) (3.16) Vm is internally stabilizable The triple (A, B, E) is assumed to be stable. This is similar to (3.9), but with a different symbol for the The feedforward unit Σ has state dimension equal non-manipulable input, to denote that it is accessible c to the dimension of Vm and includes a state feedback for measurement. Let H := imH. Signals d1 and rp in matrix F such that (A + BF)|V is stable. It is not the general block diagram in Fig. 1.1 are of this type. m necessary to reproduce ( + )| in Σ since it A BF X /Vm c h is not influenced by input h. The assumption that e Σ is stable is not restrictive. It can be relaxed to Σ u Σ being stabilizable and detectable, so that the stabilizing feedback connection shown in Fig. 2.5 can be Σc used. This does not influence conditions (3.16) since input v in Fig 2.5 clearly overrides the feedback signal Fig. 3.6. Measurable signal decoupling through F . V The measurable signal decoupling problem is: deter- Note that internal stabilizability of m is ensured if the plant is minimum phase (with all the invariant zeros mine, if possible, a feedforward compensator Σc such that the input h has no inflence on the output e. Con- stable), since the internal unassignable eigenvalues of V V∗ ditions for this problem to be solvable with stability m are a part of those of (B,E), that are invariant zeros are similar to those of disturbance decoupling prob- of the plant. lem, but state feedback is not required (a feedforward solution with a pre-compensator of the type shown in It is possible to include feedthrough terms in (3.14) Fig. 2.5 is possible). Define by using the extensions to quadruples previously described. In this case addition of a dynamic unit with V V∗ ∩S∗ m := (B,E) (E,B+H) (3.15) relative degree one at the output achieves our aim.

41 42 The Dual Problem: Unknown-Input Observation Decoupling of Previewed Signals (Discrete-Time)

Consider the system The role of controlled and conditioned invariants is very clearly pointed out by the previewed signal de- ˙( )= ( )+ ( ) x t Ax t Dd t coupling problem in the discrete-time case. Consider y(t)=Cx(t) (3.17) again signal decoupling, but suppose that there is e(t)=Ex(t) some preview (knowledge in advance) of the signal Triple (A, D, C) is assumed to be stable. Output e h to be decoupled. To take into account preview, denotes a linear function of the state to be estimated replace the block diagram in Fig. 3.6 with that in (possible the whole state). Fig. 3.8. e delay hp h d + e Σ y + < u Σ Σo − ˜e Σc Fig. 3.7. Unknown-input observation Fig. 3.8. Previewed signal decoupling The unknown-input observation problem is: deter- a) relative-degree preview mine, if possible, an observer Σo such that the input u has no inflence on the output <. Conditions for this If a relative-degree preview is available, the structural problem to be solvable with stability are dual to those condition in Corollary 3.2 is relaxed as follows. of the measurable signal decoupling problem. The problem can be solved by duality. Define Corollary 3.4 The relative-degree previewed signal decoupling problem with stability admits a solution S S∗ V∗ M = (C,D) + (D,C∩E) (3.18) if and only if like in (2.15). The solvability conditions are conse- H⊆V∗ + S∗ quence of Theorem 2.2. (B,E) (E,B) (3.20) Vm is internally stabilizable Corollary 3.3 The unknown-input observation problem with stability admits a solution if and only if where Vm is defined again by (3.15). S∗ ∩C⊆E Note that the first condition in (3.20) is satisfied if (C,D) (3.19) SM is externally stabilizable Σ is right invertible and the second is satisfied if it is minimum-phase. 43 44 a) large preview Two different strategies are outlined according to whether condition 2 in Corollary 3.4 is satisfied or A large preview time enables to overcome the stability not. The basic idea is synthesized as follows. condition, thus making it possible to obtain signal de- H⊆V∗ S coupling also in the nonminimum-phase case. “Large” Denote by ρ the least integer such that (B,E) + ρ. means significantly greater than the time constant of Let us recall that Vm is a locus of initial states in E the unstable zero closest to the unit circle. corresponding to trajectories controllable indefinitely in E, while (Sρ) is the maximum set of states that can Property 3.3 The “largely” previewed signal decou- be reached from the origin in ρ steps with all the states pling problem with stability admits a solution if and in E except the last one. Suppose that an impulse is only if applied at input h atthetimeinstantρ, producing an H⊆V∗ S∗ initial state x ∈H, decomposable as x = x + x , (B,E) + (E,B) (3.21) h h h,s h,v with xh,s ∈Sρ and xh,v ∈Vm. Let us apply the control sequence that drives the state from the origin to −xh,s Suppose that an impulse is scheduled at input h at along a trajectory in Sρ, thus nulling the first compo- time ρ. It can be decoupled with an input signal u of nent. The second component can be maintained on the type shown in the following figure with preaction Vm by a suitable control action in the time interval concerning unstable zeros and postaction stable zeros. ρ ≤ k<∞ while avoiding divergence of the state if all the internal unassignale modes of Vm arestableor dead-beat stabilizable. If not, it can be further decomposed as xh,v = xh,v + xh,v,withxh,v belonging to the subspace of the stable or stabilizable internal modes of Vm and preaction postaction xh,v to that of the unstable modes. The former component can be maintained on Vm as before, while the latter can be nulled by reaching −x with a control − ρ h,v ka 0 action in the time interval −∞

Localization of a previewed generic signal h(·)is achievable through a FIR system having such type of functions as gain.

45 46 Unknown-input Delayed Observation Feedforward Model Following

The feedforward model following problem reduces to delay e e d decoupling of measured signals, as the following ﬁgure d + Σ y + < shows. Σ o −˜ed u y Σc Σ Fig. 3.10. Unknown-input delayed observation + e

The dual problem is the unknown-input observation _ of a linear function of the state with relative degree h ym delay if Σ is minimum phase or “large” delay if not. Σm The duals of Corollary 3.4 and Property 3.3 are stated as follows. Σˆ Corollary 3.5 The unknown-input observation prob- Fig. 3.11. Feedforward model following lem of a linear function of the state with relative degree delay and stability admits a solution if and only Assume that system Σ is described by the triple if (A, B, C) and model Σm by the triple (Am,Bm,Cm). V∗ ∩S∗ ⊆E The overall sistem Σˆ is described by (D,C) (C,D) (3.22) S is externally stabilizable M A 0 B Aˆ := Bˆ := S 0 A 0 where M is deﬁned again by (3.18). m (3.24) Note that the unknown-input observation of any linear 0 Hˆ := Eˆ := C −Cm function of the state (possibly the whole state) with Bm relative degree delay is achievable if Σ is left-invertible and minimum phase. Both system and model are assumed to be stable, Property 3.4 The unknown-input observation prob- square, left and right invertible. The structural con- lem of a linear function of the state with “large” delay dition expressed by the former of (3.16) is satisﬁed if and stability admits a solution if and only if and only if the relative degree of Σm is at least equal V∗ ∩S∗ ⊆E to that of Σ. (D,C) (C,D) (3.23)

47 48 It can be shown that the internal eigenvalues of Vˆm Feedback are the union of the invariant zeros of Σ and the Disturbance Decoupling by Dynamic Output Feedback eigenvalues of Am, so that in general model following with stability is not achievable if Σ is nonminimum- d e phase. If, on the other hand, the model Σm consists of q independent single-input single-output systems all u Σ y having as zeros some invariant zeros of Σ, these are canceled as internal eigenvalues of Vˆm.Thismakesit possible to achieve both input-output decoupling and internal stability, but restricts the model choice. Σc Note that the right inversion layout shown in Fig. 3.1 is achievable with a model consisting of q independent Fig. 4.1. Disturbance decoupling relative-degree ﬁlters in the continuous-time case or q by dynamic output feedback independent relative-degree delays in the discrete-time case. Model of Σ: The dual problem of model following is model follow- x˙(t)=Ax(t)+Bu(t)+Dd(t) ing by output feedforward correction, that reduces to y(t)=Cx(t) (4.1) the left inversion layout shown in Fig. 3.1 if a model e(t)=Ex(t) consisting of p independent relative-degree ﬁlters in the continuous-time case or p independent relative- The inputs u and d are the manipulable input and the degree delays in the discrete-time case is adopted. disturbance input, respectively, while outputs y and e are the measured output and the controlled output, respectively.

Model of Σc: z˙(t)=Nz(t)+My(t) (4.2) u(t)=Lz(t)+Ky(t) The disturbance decoupling problem by dynamic output feedback is stated as follows: determine, if possible, a dynamic compensator (N,M,L,K) such that the disturbance d has no influence on the regulated output e and the overall system is internally stable. 49 50 It has been shown that output dynamic feedback of Stated in very simple terms, disturbance decoupling the type shown in Fig. 4.1 enables stabilization of is achieved if and only if the overall system (A,ˆ D,ˆ Eˆ) the overall system provided that (A, B) is stabilizable exibits at least one Aˆ-invariant Wˆ such that and (A, C) detectable. Since overall system stability D⊆ˆ W⊆ˆ Eˆ is required, these conditions on (A, B)and(A, C)are (4.5) still necessary. Wˆ is internally and externally stable The overall system is described by Necessary and sufficient conditions for solvability of our problem are stated in the following theorem. xˆ˙(t)=Aˆxˆ(t)+Ddˆ (t) (4.3) e = Exˆ (t) Theorem 4.1 The dynamic measurement feedback disturbance decoupling problem with stability admits with at least one solution if and only if there exist an x A + BKC BL (A, B)-controlled invariant V and an (A, C)-conditioned xˆ := Aˆ := z MC N invariant S such that: (4.4) D D⊆S⊆V⊆E Dˆ := Eˆ := E 0 0 S is externally stabilizable (4.6) V is internally stabilizable i.e., it can de described by a unique triple (A,ˆ B,ˆ Cˆ). A short outline of the “only if” part of the proof. d e Define the following operations on subspaces of the Σˆ extended state spacex ˆ:

projection: Fig. 4.2. The overall system x P (Wˆ )= x : ∈ Wˆ (4.7) Output e is decoupled from input d if and only if z J ˆ ˆ min (A, imD) (the reachable subpace of the pair intersection: (A,ˆ Dˆ)) is contained in kerEˆ or, equivalently, imDˆ is contained in maxJ ( ˆ ker ˆ). Furthermore, in order x A, E I(Wˆ )= x : ∈ Wˆ (4.8) the stability requirement to be satisfied, Aˆ must be 0 astablematrixorminJ (A,ˆ imDˆ)andmaxJ (A,ˆ kerEˆ) must be both internally and externally stable. 51 52 Clearly, I(Wˆ ) ⊆ P (Wˆ ), D = I(Dˆ)=P (Dˆ), E = P (Eˆ)= A more constructive set of necessary and sufficient I(Eˆ). The “only if” part of the proof of Theorem 4.1 conditions, based on the dual lattice structures af self- follows from (4.5) and the following lemmas. bounded controlled invariants and their duals, providing a convenient set of resolvent pair, is stated in the Lemma 4.1 Subspace Wˆ is an internally and/or ex- following theorem. ternally stable Aˆ-invariant only if P (Wˆ ) is an internally and/or externally stabilizable (A, B)-controlled invari- Theorem 4.2 Consider the subspaces Vm and SM de- ant. fined in (2.14) and (2.15). The dynamic measurement feedback disturbance decoupling problem with stabil- Lemma 4.2 Subspace Wˆ is an internally and/or ex- ity admits at least one solution if and only if ternally stable Aˆ-invariant only if I(Wˆ ) is an internally and/or externally stabilizable (A, C)-conditioned S∗ ⊆V∗ (C,D) (B,E) invariant. SM is externally stabilizable (4.9) V V S The “if” part of the proof is constructive, i.e., if a M := m + M is internally stabilizable resolvent pair (S, V) is given, directly provides a compensator ( ) satisfying all the requirements N,M,L,K If Theorem 4.2 holds, (SM , VM ) is a convenient resol- in the statement of the problem. This consists of a vent pair. Similarly, define Sm := Vm ∩SM . It can easily special type of state observer fed by the measured be proven that (Sm, Vm) is also a convenient resolvent output y plus a special feedback connection from the pair. observer state to the manipulable input u. Note that conditions (4.9) consist of a structural condition ensuring feasibility of disturbance decoupling without internal stability and two stabilizability conditions ensuring internal stability of the overall system.

53 54 The layout of the possible resolvent pairs in the dual The Autonomous Regulator Problem lattice structure is shown in the following figure, that also points out the correspondences between any self- Consider the block diagram shown in the following bounded controlled invariant belonging to the first figure. lattice and an element of the second and viceversa. This enables to derive other resolvent pairs satisfying Σ

Theorem 4.1. e2 d V∗ (B,E) Σ r e Σ u Σ y e1 + r p _

VM SM ∩SM Fig. 4.4. The closed-loop control scheme. + V V m m Sm The regulator Σr achieves: (i) closed-loop asymptotic stability or, more generally, pole assignability; S∗ (C,D) (ii) asymptotic (robust) tracking of reference r and asymptotic (robust) rejection of disturbance d. Fig. 4.3. The resolvents with minimum ﬁxed poles Both the reference and disturbance inputs are steps, ramps, sinusoids, that can be generated by the exosystems Σe1 and Σe2. The eigenvalues of the exosystems are assumed to belong to the closed rigth half-place of the complex plane.

The overall system considered, included the exosystems, is described by a linear homogeneous set of differential equations, whose initial state is the only variable affecting evolution in time. 55 56 regulated system The plant and the exosystems are modelled as a Σ unique regulated system which is not completely con- Σ trollable or stabilizable (the exosystem is not control- e u Σ e lable). The corresponding equations are exosystem x2 x˙(t)=Ax(t)+Bu(t) regulated system (4.10) u e e(t)=Ex(t) Σ p plant Σ with r x1 A1 A3 x := A := regulator x2 0 A2 Σ a) b) B regulatorr B := 1 E := E E 0 1 2 In (4.10) the plant corresponds to the triple (A1,B1,E1). Note that the exosystem state x2 in- Fig. 4.5. Regulated system and regulator connection fluences both the plant through matrix A3 and the error e through matrix E2.(A1,B1)isassumedtobe The overall system is referred to as the autonomous stabilizable and (A, E) detectable. extended system The regulator is modelled like in the disturbance de- xˆ˙(t)=Aˆxˆ(t) (4.12) couplig problem by measurement feedback, i.e. e(t)=Eˆ xˆ(t) z˙(t)=Nz(t)+Me(t) with (4.11)   u(t)=Lz(t)+Ke(t) x1   xˆ := x2 z   A1 + B1KE1 A3 + B1KE2 B1L   Aˆ := OA2 O ME1 ME2 N Eˆ := E1 E2 O

57 58 n n m Let x1 ∈ R 1 , x2 ∈ R 2 , z ∈ R . If the internal model In the extended state space Xˆ with dimension principle is used to design the regulator, the au- n1 + n2 + m,definetheAˆ-invariant extended plant Pˆ tonomous extended system is characterized by an unas observability subspace containing these modes, that   In O are all not strictly stable by assumption. In geometric  1  Pˆ := { xˆ : x2 =0} =im OO (4.14) terms, an Aˆ-invariant W⊆ˆ kerEˆ having dimension n2 exists, that is internally not strictly stable. OIm By a dimensionality argument, the Aˆ invariant Wˆ , Since the eigenvalues of Aˆ are clearly those of A plus 2 besides (4.13), must satisfy those of the regulation loop, that are strictly stable, Wˆ is externally strictly stable. Hence Aˆ|Wˆ has the W⊕ˆ Pˆ = Xˆ (4.15) eigenstructure of A2 (n2 eigenvalues) and Aˆ that Xˆ/Wˆ The main theorem on asymptotic regulation simply of the control loop (n + n eigenvalues). 1 2 translates the extended state space conditions (4.13) The existence of this Aˆ-invariant W⊆ˆ kerEˆ is pre- and (4.15) into the plant plus exosystem state space served under parameter changes. where matrices A, B and E are defined. Define the A-invariant plant P through The autonumous regulator problem is stated as fol- In1 lows: derive, if possible, a regulator (N,M,L,K)such P := { x : x2 =0} =im (4.16) that the closed-loop system with the exosystem dis- O connected is stable and lim →∞ e(t) = 0 for all the ini- t Theorem 4.3 Let E := kerE. The autonomous regu- tial states of the autonomous extended system. lator problem admits a solution if and only if an (A, B)- V In geometric terms it is stated as follows: refer to the controlled invariant exists such that extended system (A,ˆ Eˆ)andletEˆ:= kerEˆ. Given the V⊆E mathematical model of the plant and the exosystem, (4.17) V⊕P X determine, if possible, a regulator (N,M,L,K)such = that an Aˆ-invariant Wˆ exists satisfying W⊆ˆ Eˆ The “only if” part of the proof derives from (4.13) (4.13) and (4.15), while the “if” part provides a quadruple ( ˆ| ) ⊆ C− σ A Xˆ/Lˆ (N,M,L,K) that solves the problem.

59 60 Unfortunately the necessary and sufficient conditions Proof of Theorem 4.4: stated in Theorem 4.3 are nonconstructive. The fol- ∗ ∗ lowing theorem provides constructive sufficient and al- Let F be a matrix such that (A + BF)V ⊆V.In- ∗ most necessary conditions in terms of the invariant troduce the similarity transformation T := [T1 T2 T3], ∗ ∗ zeros of the plant. with imT1 = V ∩P,im[T1 T2]=V and T3 such that im[T1 T3]=P. Theorem 4.4 Let us define V∗ := maxV(A, B, E). The autonomous regulator problem admits a solution In the new basis the linear transformation A + BF has the structure if   ∗ V + P = X A11 A12 A13 (4.18) −1   Z ∩ ∅ A = T (A + BF) T = OA22 O (4.19) (A1,B1,E1) σ(A2)= OOA33 Remark: P We have again a structural condition and a stability Recall that is an A-invariant and note that, owing condition in terms of invariant zeros. However, the to the particular structure of B,itisalsoan(A + BF)- stability condition is very mild in this case since it is invariant for any F . only required that the plant has no invariant zeros By a dimensionality argument the eigenvalues of the equal to eigenvalues of the exosystem. Hence the au- exosystem are those of A22, while the invariant zeros tonomous regulator problem may be also solvable if R ∗ of (A1,B1,E1) are a subset of σ(A11)since V is the plant is nonminimum phase. In other words, min- ∗ contained in V ∩P. All the other elements of σ(A ) imality of phase is only required for perfect tracking, 11 are arbitrarily assignable with F . Hence, owing to non for asymptotic tracking. (4.18), the Sylvester equation Corollary 4.1 (Uniqueness of the resolvent) If the A X − XA = −A (4.20) plant is invertible and conditions (4.18) are satisfied, 11 22 12 a unique (A, B)-controlled invariant V satisfying con- admits a unique solution. ditions (4.17) exists. The matrix V := T X + T ∗The conditions become necessary if the boundedness 1 2 of the control variable u is required. This is possible is a basis matrix of an (A, B)-controlled invariant V also when the output y is unbounded if a part of the satisfying the solvability conditions (4.17). internal model is contained in the plant.

61 62 Remarks: Feedback Model Following

• The proof of Theorem 4.4 provides the computa- The reference block diagram for feedback model fol- tional framework to derive a resolvent when the lowing is shown in Fig. 4.6. Like in the feedforward suﬃcient conditions stated (that are also neces- case,bothΣandΣm are assumed to be stable and sary if the boundedness of the plant input is re- Σ to have at least the same relative degree as Σ. quired) are satisﬁed. m

r + h u y • Relations (4.18) are respectively a structural con- Σ dition and a spectral condition; they are easily − c Σ checkable by means of the algorithms previously + e described. − ym • When a resolvent has been determined by means Σm of the computational procedure described in the proof of Theorem 4.4, it can be used to derive a Fig. 4.6. Feedback model following regulator with the procedure outlined in the “if” part of the proof of Theorem 4.3. Replacing the feedback connection with that shown in Fig. 4.7 does not affect the structural properties • The order of the obtained regulator is n (that of the plant plus that of the exosystem) with of the system. However, it may affect stability. The new block diagram represents a feedforward model the corresponding 2n1 + n2 closed-loop eigenvalues completely assignable under the assumption following problem. that (A1,B1) is controllable and (E,A) observ- r + u able. h Σ − c Σ • The internal model principle is satisfied since the + e from the proof of the “if” part of Theorem 4.3 − it follows that the eigenstructure of the regulator system matrix N contains that of A2. Σm • It is necessary to repeat an exosystem for every Fig. 4.7. A structurally equivalent connection regulated output to achieve independent steady- state regulation (different internal models are obtained in the regulator).

63 64 In fact, note that h is obtained as the diﬀerence of Geometric Approach to LQR Problems r (applied to the input of the model) and ym (the output of the model). This corresponds to the parallel Consider again the disturbance decoupling problem by connection of Σm and the opposite of the identity state feedback, corresponding to the state equations matrix, that is invertible, having zero relative degree. x˙(t)=Ax(t)+Bu(t)+Dd(t) Its inverse is Σm with a feedback connection through (5.1) the identity matrix, as shown in Fig. 4.8. e(t)=Ex(t) in the continuous-time case and to the equations h u Σ x(k +1) = A x(k)+B u(k)+D d(k) c Σ d d d (5.2) + e e(k)=Ed x(k) − + r in the discrete-time case. The corresponding block Σm diagram is represented in Fig. 5.1. + Σm d e Fig. 4.8. A structurally equivalent block diagram u Σ

Let the model consist of q independent single-input x single-output systems all having as zeros the unstable F invariant zeros of Σ. Since the invariant zeros of a system are preserved under any feedback connection, Fig. 5.1. Disturbance decoupling a feedforward model following compensator designed by state feedback with reference to the block diagram in Fig. 4.8 does not include them as poles. Assume that the necessary and sufficient conditions for its solvability with internal stability It is also possible to include multiple internal models in the feedback connection shown in the figure (this is D⊆V∗ (B,E) (5.3) well known in the single input/output case), that are V m is internally stabilizable repeated in the compensator, so that both Σm and the compensator may be unstable systems. In fact, zero are not satisfied. In this case a convenient resort is to output in the modified system may be obtained as minimize the H2 norm of the matrix transfer function the difference of diverging signals. However, stability from input d to output e, defined by equation (1.14) is recovered when going back to the original feedback or (1.15) in the continuous-time case and equation connection represented in Fig. 4.6. (1.16) or (1.17) in the discrete-time case. 65 66 The continuous-time case Problem 5.1 is solvable with the geometric tools. Ac- cording to the classical optimal control approach, con- Consider the following problem: sider the Hamiltonian function Problem 5.1 Referring to system (5.1), determine a H(t):=x(t)TETEx(t)+p(t)T (Ax(t)+Bu(t)) state feedback matrix F such that A + BF is stable and derive the state, costate equations and stationary and the corresponding state trajectory for any initial condition as state x(0) minimizes the performance index T ∞ ∞ ∂H(t) T T T x˙(t)= = Ax(t)+Bu(t) J = e(t) e(t) dt = x(t) E Ex(t) dt (5.4) ∂p(t) 0 0 ∂H(t) T p˙(t)= = −2 ETEx(t) − ATp(t) This problem is the so-called “cheap version” of ∂x(t) the classical Kalman regulator problem or Linear Quadratic Regulator (LQR) problem. In the Kalman ∂H(t) T 0= = BTp(t) problem the performance index is ∂u(t) ∞ This overall Hamiltonian system can also be written J = x(t)TQx(t)+u(t)TRu(t) dt as 0 ∞ xˆ˙(t)=Aˆxˆ(t)+Buˆ (t) = x(t)TCTCx(t)+u(t)TDTDu(t) dt (5.5) 0 0=Eˆ xˆ(t) where matrices Q and R are symmetric positive with semidefinite and positive definite respectively, hence 0 factorizable as shown. It can be proven that the cheap x ˆ A xˆ = A = − T − T version is the more general, since the input to output p 2E E A T T (5.6) feedthrough term u(t) D Du(t) can be accounted for B Bˆ = Eˆ = 0 BT with a suitable state extension. 0 Problem 5.1 admits a solution if and only if there exixts an internally stable (A,ˆ Bˆ)-controlled invariant of the overall Hamiltonian system contained in Eˆ whose projection on the state space of system (5.1), defined as in (4.7), contains the initial state x(0).

67 68 It can be proven that the internal unassignable eigen- The discrete-time case values of Vˆ∗ := maxV(A,ˆ Bˆ, Eˆ) are stable-unstable by pairs. Hence a solution of Problem 5.1 is obtained as The discrete-time cheap LQR problem is stated as follows: follows. Problem 5.2 Referring to system (5.2), determine a 1. compute Vˆ∗; state feedback matrix Fd such that Ad + BdFd is stable 2. compute a matrix Fˆ such that (Aˆ+BˆFˆ)Vˆ∗ ⊆ Vˆ∗ and the corresponding state trajectory for any initial and the assignable eigenvalues (those internal to state x(0) minimizes the performance index R Vˆ∗ )arestable; ∞ ∞ Vˆ T T T 3. compute s, the maximum internally stable J = e(k) e(k)= x(k) Ed Ed x(k) (5.7) ∗ (Aˆ+BˆFˆ)-invariant contained in Vˆ ; k=0 k=0

4. if x(0) ∈ P (Vˆs) the problem admits a solution F , that is easily computable as a function of Vˆs and In this case the Hamiltonian function is Fˆ; if not, the problem has no solution. T T T H(k):=x(k) Ed Ed x(k)+p(k) (Ad x(k)+Bd u(k)) Refer to Fig. 5.1. The above procedure also provides and the state, costate equations and stationary con- a state feedback matrix F corresponding to the mini- dition are mum H2 norm from d to e. This immediately follows ∂H(k) T from expression (1.15) of the H2 norm in terms of x(k +1) = = Ad x(k)+Bd u(k) ∂p(k +1) the impulse response. In fact, the impulse response corresponds to the set of initial states deﬁned by the T ∂H(k) T T column vectors of matrix D. Thus, the problem of p(k)= =2E Ed x(k)+A p(k +1) ∂x(k) d d minimizing the H2 norm from d to e has a solution if and only if ∂H(k) T 0= = BTp(k +1) D⊆P (Vˆs) ∂u(k)

Thus, the minimum H2 norm disturbance almost decoupling problem has no solution if the above condition is not satisﬁed. The discrete-time case is partic- ularly interesting since a solution always exist. The reason for this will be pointed out below.

69 70 dead-beat Like in the continuous-time case, it is convenient to state the overall Hamiltonian system in compact form:

xˆ(k +1) = Aˆd xˆ(k)+Bˆd u(k) postaction (5.8) 0=Eˆd xˆ(k) with 0 ρ A 0 x ˆ d Fig. 5.2. Cheap H2 optimal control. xˆ = Ad = −T T −T p −2 A E Ed −A d d d A typical control sequence is shown in Fig. 5.2: as B − − Bˆ = d Eˆ = −2 BTA TETE BTA T the sampling time approaches zero, the dead beat d 0 d d d d d d d segment tend to a distribution, which is not obtainable (5.9) with state feedback. For this reason solvability of the A solution of Problem 5.2 is obtained again with a H2 optimal decoupling problem is more restricted in geometric procedure, but, unlike the continuous-time the continuous-time case. case, in this case a dead-beat like motion is also feasi- If the signal to be optimally decoupled is measurable Vˆ ble and P ( s) covers the whole state space of system and the system considered is stable, state feedback (5.2). Hence both Problem 5.2 and the problem of can be used in an auxiliary feedforward unit of the minimizing the H2 norm from d to e are always solvable type shown in Fig. 3.6, while the dual layout shown in in the discrete-time case. Fig 3.7 realizes the H2-optimal observation of a linear function of the state or possibly of the whole state (Kalman ﬁlter).

However, if the signal is not measurable and state is not accessible, the problem of H2-optimal decoupling with dynamic output feedback can be stated and solvability conditions derived by using geometric techniques again.

71 72 Conclusions

The three types of input signals:

• disturbance (eliminable only with feedback) • measurable • previewed

The seven characterizing properties of systems:

• (internal) stability • controllability • observability • invertibility • functional controllability • relative degree • minimality of phase

In general, the necessary and suﬃcient conditions for solvability of control problems consist of

• a structural condition • s stability condition

When a tracking or disturbance rejection problem is not perfectly solvable with internal stability, it is possible to resort to H2 optimal solutions that can also be obtained through the standard geometric tools and algorithms. http://www.deis.unibo.it/Staﬀ/FullProf/GiovanniMarro/geometric.htm