Output Regulation Problem for Differentiable Families of Linear Systems

Given a family of linear systems depending on a parameter varying in a differentiable manifold, we obtain sufficient conditions for the existence of a (global or local) differentiable family of controllers solving the output regulation problem for the given family. Moreover, we construct it when these conditions hold.


Introduction
The output regulation problem arose as one of the main research topics in linear control theory in the 1970s.This problem considers controlling a given plant such that its output tracks a reference signal or rejects a disturbance.The reference and disturbance signals are typically generated by an external system called exosystem.The output regulation problem for linear time-invariant systems has been well studied by many authors like, for example, 1-8 .Usually, these LTI systems are obtained by linearizing around an operating point or by using system identification techniques.However, in some cases this approach is restrictive, and it would be better to allow the system to depend differentiably on a specific parameter.We think that this situation is interesting to consider and, in this case, the controller must be able to maintain the properties of closed-loop stability and output regulation when it is modelled as a global or a local differentiable family.We notice that recently a regulator design approach based on the parameterization of a set of controllers that can achieve regulation for switched bimodal linear systems has been presented in 9 .
Thus, we lead to the quite general question of whether pointwise solvability implies the existence of a nicely parameterized solution 10 .There is an abundance of literature concerning global parameterized families of linear systems, and in particular the key point of global pole assignment, mainly when pointwise controllability holds.
For a general introduction to families of linear systems, see, for example, 11 .Some problems that justify the study of families of systems are presented there, and one tackles the classification of families fine moduli spaces , the existence of global canonical forms, and some others.In fact, 11 deals with a more general class of families of linear systems in terms of bundles over the space of parameters, which includes the parameterized ones.An alternative generalization is the consideration of systems over rings, that is to say, pairs of matrices with entries on a commutative ring.The particular case of parameterized families arise, when rings of functions defined in the space of parameters are considered.See 12 for a general introduction and 10 for a survey and many references mainly centered on control and stabilization problems.
In particular, the global pole assignment of parameterized families has been widely studied see, e.g., 10 , always under the hypothesis of pointwise controllability.The problem is solved in 13, 14 by means of algebrogeometric and algebraic techniques, respectively, if provided constant controllability indices.Other conditions are considered, for example, in 15 constant rank of B, ring controllability or in 16 one-dimensional manifold of parameters .
The general case of nonnecessarily pointwise controllable families is solved in 17 .In fact, it provides a general machinery for extending the local case to the global parametrical one if one has a geometrical description of the construction in the constant case.Notice that this method does not need further references to algebrogeometric or algebraic techniques, so that it seems more accessible to engineers.Here, we show that this machinery is also applicable to more sophisticated problems, such as the construction of global families of suitable controllers.
Secondly, we tackle the same problem for local differentiable perturbations of the systems, that is to say, for parameterized families defined not globally on a contractible manifold but locally in some neighborhood of the original point.The pointwise construction is easily extended for the controllable case, but up to our knowledge no technique exists for the general case beyond the trivial approach as the restriction of a global family .In this sense, a significative contribution is 18 , where one proves that local differentiable pole assignment exists even for the noncontrollable case without any pointwise hypothesis on the local family of perturbations, in contrast with the global case above.The proof is based in the use of Arnold's techniques, reducing the problem to a suitable versal deformation of the central system.Here we show that this quite surprising result can be used in more complex processes, such as the output regulation.
In summary, this work deals with the existence of differentiable families of controllers following the pattern in 4 , both for global and local families of systems.In particular, we assume that the hypotheses there hold.We will denote them by F1 -F7 see Theorem 2.1 , and we add ' for the natural generalization to parameterized families see Theorems 3.3 and 4.2 .We remark see 4 that F1 -F6 either involve no loss of generality or are necessary for the existence of the controller.
As we have pointed out, the assumptions concerning controllability or detectability of a system are easily transferred from the pointwise case to parameterized families.Obstructions arise when subsystems are considered and in general for geometrical conditions.The key tools are the techniques in 17 for the global case and the results in 18 for the local one.
The paper is organized as follows.First we present, as a reminder, the main result in 4 , emphasizing the pointwise hypotheses that will be considered later.In the following sections we deal with the global and the local cases.
Notations R and C denote the fields of real and complex numbers, respectively.C is the closed right-half complex plane and C − is the open left-half one.If v 1 , . . ., v s are vectors of R n , then v 1 , . . ., v s denotes the subspace spanned by them.We write M n,m R for the vector space of matrices with n rows and m columns with entries in R. We identify M n,n m R with the pairs of matrices M n,n R × M n,m R .If A is a matrix, A t is its transpose matrix.I denotes the identity matrix and I k the identity k-matrix.σ • denotes the spectrum of the corresponding matrix or pairs of matrices, each eigenvalue being repeated as many times as its multiplicity.For linear spaces R and S, R ∼ S means that R and S are isomorphic and Hom R, S is the linear space of all linear maps R → S.

Preliminaries: The Nonparameterized Case
In this section, we summarize the problem and the results stated in 4 , which have been the basis of our work.This reference deals with the regulation of the linear system represented by where x 1 is the plant state vector, u the control input, x 2 the vector of exogenous signals, y the vector of measurements available for control, and z the output to be regulated.These vectors belong to fixed finite-dimensional real linear spaces X 1 , U, X 2 , Y, and Z, of dimensions n 1 , m, n 2 , n y , and n z , respectively; the linear maps or real matrices The objective is to construct a controller modelled by the equations where x c is the compensator state vector, which belongs to a finite-dimensional real linear space X c , and the linear maps A c , B c , F c , and G c are time-invariant.This system must achieve two properties: closed-loop stability and output regulation of the associated closed loop defined as where

2.4
Closed-loop stability means that A L is stable, that is, σ A L ⊂ C − , and output regulation means that z t → 0 as t → ∞ for all x L 0 and x 2 0 .We next introduce the mathematical setting used in 4 .For any linear space R, bring in the linear space R Hom X 2 , R .Given maps A : X → X and C : X → Y, define, respectively, the linear maps A : X → X and C : X → Y by CX CX, X ∈ X.

2.5
Analogous notation will be used for the subsystems involved in the construction.Now we state the main result obtained in 4 .

Theorem 2.1 see 4 . Assume that
Then, a controller exists if and only if Following 4 again, an algorithm to compute a controller can be organized into the following four steps.
Step 1. Select B c so that A − B c C is stable.
Step 2. Select F 1 so that A 1 B 1 F 1 is stable.
Step 3. Select F 2 so that Step 4.
, and G c 0.

The Globally Parameterized Case
Our aim is to generalize the above results when the involved matrices depend on a parameter varying in a differentiable manifold.That is to say, we consider a family of linear systems depending differentiably on a parameter, and we study sufficient conditions to ensure the existence of a differentiable solution of the output regulation problem for the given family.Thus, we consider a differentiable family of linear systems modelled by the equations or, equivalently, it will be represented by a differentiable family of matrices To this end, basic tools are introduced in 19 .More specifically, we will need the following result.Lemma 3.1.Let M be a contractible manifold.If A τ , τ ∈ M, is a differentiable family of q × p real matrices having constant rank and v τ , τ ∈ M, a differentiable family of vectors in ImA τ ⊂ R q , then there exists a differentiable family of vectors u τ ∈ R p such that for any τ ∈ M.
Proof.In 19, IV-1-6 , it is proved that Im A τ and KerA τ are differentiable families of subspaces of R q and R p , respectively.Then, from 19, IV-2-3 , we can choose differentiable families u 1 τ , . . ., u s τ of linearly independent vectors such that for every τ ∈ M. Clearly A τ u 1 τ , . . ., A τ u s τ is a differentiable basis of Im A τ .So, the coordinates of v τ in this basis vary differentiably, that is, there exist differentiable families a 1 τ , . . ., a s τ of real numbers such that Finally, it suffices to take In addition, we will use the following result from 17 , where these tools are applied to prove the existence of a global parameterized pole assignment.Recall that if M is a contractible manifold and A τ , B τ , τ ∈ M, is a differentiable family of pairs of matrices of M n,n m R , the family A τ , B τ has constant Brunovsky type if i the controllability indices are constant, ii the Jordan invariants have constant type, that is to say, the number of distinct eigenvalues and the list of sizes of the Jordan blocks corresponding to different eigenvalues are independent of τ.
These results allow us to study the existence of a global differentiable family of controllers.In the next section we address the local case.

Theorem 3.3. Let M be a contractible manifold and
Then, there exists a global differentiable family of controllers A c τ , B c τ , F c τ , G c τ if, when τ varies in M, one has: iii rank Proof.Let us see that conditions i , ii , and iii allow us to transfer the pointwise algorithm in 4 summarized in the above section to the parameterized case.
Step 2. Analogously, if condition ii holds, we can select a differentiable family Mathematical Problems in Engineering 7 Step 3. By hypothesis F7' , and with rank being constant because of condition iii it is guaranteed that the dimension of this subspace is constant in M. Using the Kronecker product and the vecfunction, this condition can be reformulated as condition iii in terms of the matrices defining the system.
Right multiplying by matrix I 0 F 1 τ I , this dimension does not change and hypothesis F7' turns to Then, from Lemma 3.1 there exist differentiable families X τ , Y τ such that or, equivalently, Therefore, taking F 2 τ −Y τ , we have for any τ ∈ M. Step Clearly, they are differentiable families in M, and, from Theorem 2.1, they define a controller for any τ ∈ M.

The Locally Parameterized Case
In this section, we tackle the local case, that is, we deal with local differentiable families.This means that the parameter varies only in an open neighborhood of the origin of R k .
Similarly to the global case, we will use the existence of a local differentiable family of pole assignments for a local differentiable family of stabilizable pairs.As above, a pointwise construction does not guarantee the differentiability of the of feedbacks.It is so by means of the following result, based on the Arnold's techniques about versal deformations.
Then, there exists a local differentiable family of controllers A c τ , B c τ , F c τ , G c τ in some open neighborhood of the origin W ⊂ V if, when τ varies in V , one has: iii rank Proof.It is easy to adapt the proof of Theorem 3.3 bearing in mind the following remarks.Notice that hypotheses F2 and F3 mean, respectively, that matrices C 1 C 2 and D 1 have maximal rank.Hence, matrices C 1 τ C 2 τ and D 1 τ have maximal rank for any τ small enough or, equivalently, In a similar way, since the set of detectable pairs is open, hypothesis F5 guarantees that pairs C 1 τ , A 1 τ are detectable for τ small enough.
Moreover, hypotheses F4 and F6 allow us to apply Theorem 4.1 in the first two steps of the synthesis algorithm.Notice that, thanks to Theorem 4.1, hypotheses i and ii can be avoided.Finally, for Step 3 we use F7' and iii as in the proof of Theorem 3.3.

Theorem 4 . 1 Theorem 4 . 2 .
see 18 .Let A, B ∈ M n,n m R be a stabilizable pair and A τ , B τ , τ ∈ V , a local differentiable family of pairs with A 0 , B 0 A, B .Then there is a local differentiable family of feedbacks F τ ∈ M m,n R , defined in some open neighborhood of the origin W ⊂ V , such that σ A τ B τ F τ ⊂ C − .Finally, we apply this result to our output regulation problem.Let A 1 τ , A 2 τ , A 3 τ , B 1 τ , C 1 τ , C 2 τ , D 1 τ , D 2 τbe a differentiable family of linear systems defined in some open neighborhood of the origin V ⊂ R k .Assume that, for any τ ∈ V ,