Output Regulation Problem for Differentiable Families of Linear Systems

Abstract

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.

1. 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 𝑣₁,…,𝑣_𝑠 are vectors of ℝ^𝑛, then [𝑣₁,…,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.

2. 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 by𝑥̇₁=𝐴₁𝑥₁+𝐴₃𝑥₂+𝐵₁𝑥̇𝑢,₂=𝐴₂𝑥₂,𝑦=𝐶₁𝑥₁+𝐶₂𝑥₂,𝑧=𝐷₁𝑥₁+𝐷₂𝑥₂, (2.1) where 𝑥₁ is the plant state vector, 𝑢 the control input, 𝑥₂ 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 𝒳₁, 𝒰, 𝒳₂, 𝒴, and 𝒵, of dimensions 𝑛₁, 𝑚, 𝑛₂, 𝑛_𝑦, and 𝑛_𝑧, respectively; the linear maps or real matrices 𝐴₁,𝐴₂, 𝐴₃, 𝐵₁, 𝐶₁, 𝐶₂, 𝐷₁, and 𝐷₂ are time-invariant.

The objective is to construct a controller modelled by the equations𝑥̇_𝑐=𝐴_𝑐𝑥_𝑐+𝐵_𝑐𝑦,𝑢=𝐹_𝑐𝑥_𝑐+𝐺_𝑐𝑦, (2.2) 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 as𝑥̇_𝐿=𝐴_𝐿𝑥_𝐿+𝐵_𝐿𝑥₂,𝑧=𝐷_𝐿𝑥_𝐿+𝐷₂𝑥₂, (2.3) where𝑥_𝐿=𝑥₁𝑥_𝑐,𝒳_𝐿=𝒳₁⊕𝒳_𝑐,𝐴_𝐿=𝐴₁+𝐵₁𝐺_𝑐𝐶₁𝐵₁𝐹_𝑐𝐵_𝑐𝐶₁𝐴_𝑐,𝐵_𝐿=𝐴₃+𝐵₁𝐺_𝑐𝐶₂𝐵_𝑐𝐶₂,𝐷_𝐿=𝐷₁0. (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 𝑥₂(0).

We next introduce the mathematical setting used in [4]. For any linear space ℛ, bring in the linear space ℛ=Hom(𝒳₂,ℛ). Given maps 𝐴∶𝒳→𝒳 and 𝐶∶𝒳→𝒴, define, respectively, the linear maps 𝐴∶𝒳→𝒳 and 𝐶∶𝒳→𝒴 by 𝐴𝑋=𝐴𝑋−𝑋𝐴₂,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(F1)𝜎(𝐴₂)⊂ℂ⁺,(F2)Im𝐶₁+Im𝐶₂=𝒴,(F3)Im𝐷₁=𝒵,(F4)(𝐴₁,𝐵₁)isstabilizable,(F5)(𝐶₁,𝐴₁)isdetectable,(F6)(𝐶,𝐴) is detectable, where 𝐴=(^𝐴1𝐴3_0𝐴2) and 𝐶=(^𝐶₁^𝐶₂).Then, a controller exists if and only if (F7)(^𝐴3_𝐷2)∈Im(^𝐴1𝐵1_𝐷10).

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_𝑐𝐶 is stable.

Step 2. Select 𝐹₁ so that 𝐴₁+𝐵₁𝐹₁ is stable.

Step 3. Select 𝐹₂ so that 𝐴₃+𝐵₁𝐹₂𝐷₂𝐴∈Im¹+𝐵₁𝐹₁𝐷₁. (2.6)

Step 4. Set 𝐹_𝑐=(𝐹₁𝐹₂), 𝐴_𝑐=𝐴−𝐵_𝑐𝐶+𝐵𝐹_𝑐, and 𝐺_𝑐=0.

3. 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 equations𝑥̇₁=𝐴₁(𝜏)𝑥₁+𝐴₃(𝜏)𝑥₂+𝐵₁𝑥̇(𝜏)𝑢,₂=𝐴₂(𝜏)𝑥₂,𝑦=𝐶₁(𝜏)𝑥₁+𝐶₂(𝜏)𝑥₂,𝑧=𝐷₁(𝜏)𝑥₁+𝐷₂(𝜏)𝑥₂, (3.1) or, equivalently, it will be represented by a differentiable family of matrices (𝐴₁(𝜏),𝐴₂(𝜏),𝐴₃(𝜏),𝐵₁(𝜏),𝐶₁(𝜏),𝐶₂(𝜏),𝐷₁(𝜏),𝐷₂(𝜏)).

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 𝐴(𝜏)𝑢(𝜏)=𝑣(𝜏) (3.2) for any 𝜏∈𝑀.

Proof. In [19, (IV-1-6)], it is proved that Im𝐴(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 𝑢₁(𝜏),…,𝑢_𝑠(𝜏) of linearly independent vectors such that ℝ^𝑝=Ker𝐴(𝜏)⊕[𝑢₁(𝜏),…,𝑢_𝑠(𝜏)] (3.3) for every 𝜏∈𝑀. Clearly 𝐴(𝜏)𝑢₁(𝜏),…,𝐴(𝜏)𝑢_𝑠(𝜏) is a differentiable basis of Im𝐴(𝜏). So, the coordinates of 𝑣(𝜏) in this basis vary differentiably, that is, there exist differentiable families 𝑎₁(𝜏),…,𝑎_𝑠(𝜏) of real numbers such that 𝑣(𝜏)=𝑎₁(𝜏)𝐴(𝜏)𝑢₁(𝜏)+⋯+𝑎_𝑠(𝜏)𝐴(𝜏)𝑢_𝑠(𝜏). (3.4) Finally, it suffices to take 𝑢(𝜏)=𝑎₁(𝜏)𝑢₁(𝜏)+⋯+𝑎_𝑠(𝜏)𝑢_𝑠(𝜏). (3.5)

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.

Theorem 3.2 (see [17]). Let M be a contractible manifold, (𝐴(𝜏),𝐵(𝜏)), 𝜏∈𝑀, a differentiable family of pairs of matrices of 𝑀_{𝑛,𝑛+𝑚}(ℝ) having constant Brunovsky type, 𝜆₁(𝜏),…,𝜆_𝑞(𝜏)∈ℂ giving the distinct eigenvalues of (𝐴(𝜏),𝐵(𝜏)), and 𝑚₁,…,𝑚_𝑞 their respective algebraic multiplicities. If 𝜇_𝑖(𝜏)∈ℂ, 1≤𝑖≤𝑠, is a set of maps closed under conjugation, then there exists a differentiable family of matrices 𝐹(𝜏)∈𝑀_𝑚,𝑛(ℝ) such that the eigenvalues of 𝐴(𝜏)+𝐵(𝜏)𝐹(𝜏) are 𝜇₁(𝜏),…,𝜇_𝑠(𝜏), 𝜆₁(𝜏),…,𝜆_𝑞(𝜏), with the latter having multiplicities 𝑚₁,…,𝑚_𝑞.

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 (𝐴₁(𝜏), 𝐴₂(𝜏), 𝐴₃(𝜏), 𝐵₁(𝜏), 𝐶₁(𝜏), 𝐶₂(𝜏), 𝐷₁(𝜏), 𝐷₂(𝜏)), 𝜏∈𝑀, a differentiable family of linear systems verifying that for any 𝜏∈𝑀(F1’)𝜎(𝐴₂(𝜏))⊂ℂ⁺,(F2’)Im𝐶₁(𝜏)+Im𝐶₂(𝜏)=𝒴,(F3’)Im𝐷₁(𝜏)=𝒵,(F4’)(𝐴₁(𝜏),𝐵₁(𝜏))isstabilizable,(F5’)(𝐶₁(𝜏),𝐴₁(𝜏))isdetectable,(F6’)(𝐶(𝜏),𝐴(𝜏))isdetectable,(F7’)(^𝐴3_𝐷^(𝜏)_2(𝜏))∈Im(^{𝐴1(𝜏)𝐵1}_𝐷^(𝜏)_1(𝜏)0).
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: (i)(𝐶(𝜏),𝐴(𝜏))hasconstantBrunovskytype,(ii)(𝐴₁(𝜏),𝐵₁(𝜏))hasconstantBrunovskytype,(iii)rank(^{𝐼𝑛2⊗𝐴1(𝜏)−𝐴2(𝜏)𝑡⊗𝐼𝑛1𝐼𝑛2⊗𝐵1}_𝐼^(𝜏)_{𝑛2⊗𝐷1(𝜏)0}) is constant.

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 1. If condition (i) holds, we can select a differentiable family 𝐵_𝑐(𝜏) such that 𝐴(𝜏)−𝐵_𝑐(𝜏)𝐶(𝜏) is stable, for any 𝜏∈𝑀, by means of Theorem 3.2: 𝜆₁(𝜏),…,𝜆_𝑞(𝜏)∈ℂ⁻ because of (F6'), so that it suffices to take 𝜇₁(𝜏),…,𝜇_𝑠(𝜏)∈ℂ⁻.Step 2. Analogously, if condition (ii) holds, we can select a differentiable family 𝐹₁(𝜏) such that 𝐴₁(𝜏)+𝐵₁(𝜏)𝐹₁(𝜏) is stable, for any 𝜏∈𝑀.Step 3. By hypothesis (F7'), 𝐴₃𝐷(𝜏)₂𝐴(𝜏)∈Im¹(𝜏)𝐵₁𝐷(𝜏)₁(𝜏)0, (3.6)and with rank(^{𝐴1(𝜏)𝐵1}_𝐷^(𝜏)_1(𝜏)0) being constant (because of condition (iii)) it is guaranteedthat the dimension of this subspace is constant in 𝑀. Using the Kronecker product and the vec-function, this condition can be reformulated as condition (iii) in terms of the matrices defining the system.
Right multiplying by matrix (_𝐹^𝐼0_1(𝜏)𝐼), this dimension does not change and hypothesis (F7') turns to 𝐴₃𝐷(𝜏)₂𝐴(𝜏)∈Im¹(𝜏)+𝐵₁(𝜏)𝐹₁(𝜏)𝐵₁𝐷(𝜏)₁(𝜏)0. (3.7)
Then, from Lemma 3.1 there exist differentiable families 𝑋(𝜏),𝑌(𝜏) such that 𝐴₃𝐷(𝜏)₂=𝐴(𝜏)¹(𝜏)+𝐵₁(𝜏)𝐹₁(𝜏)𝐵₁𝐷(𝜏)₁(𝜏)0𝑋(𝜏)𝑌(𝜏) (3.8) or, equivalently, 𝐴₃(𝜏)−𝐵₁(𝜏)𝑌(𝜏)=(𝐴₁(𝜏)+𝐵₁(𝜏)𝐹₁𝐷(𝜏))𝑋(𝜏),₂(𝜏)=𝐷₁(𝜏)𝑋(𝜏). (3.9)
Therefore, taking 𝐹₂(𝜏)=−𝑌(𝜏), we have 𝐴₃(𝜏)+𝐵₁(𝜏)𝐹₂𝐷(𝜏)₂𝐴(𝜏)∈Im¹(𝜏)+𝐵₁(𝜏)𝐹₁𝐷(𝜏)₁(𝜏) (3.10) for any 𝜏∈𝑀.Step 4. Set 𝐹_𝑐(𝜏)=(𝐹₁(𝜏)𝐹₂(𝜏)), 𝐴_𝑐(𝜏)=𝐴(𝜏)−𝐵_𝑐(𝜏)𝐶(𝜏)+𝐵(𝜏)𝐹_𝑐(𝜏), 𝐺_𝑐(𝜏)=0.
Clearly, they are differentiable families in 𝑀, and, from Theorem 2.1, they define a controller for any 𝜏∈𝑀.

4. 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 family of feedbacks. It is so by means of the following result, based on the Arnold's techniques about versal deformations.

Theorem 4.1 (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.

Theorem 4.2. Let (𝐴₁(𝜏),𝐴₂(𝜏),𝐴₃(𝜏),𝐵₁(𝜏),𝐶₁(𝜏),𝐶₂(𝜏),𝐷₁(𝜏),𝐷₂(𝜏)) be a differentiable family of linear systems defined in some open neighborhood of the origin 𝑉⊂ℝ^𝑘. Assume that, for any 𝜏∈𝑉, (F1’)𝜎(𝐴₂(𝜏))⊂ℂ⁺,(F2)Im𝐶₁+Im𝐶₂=𝒴,(F3)Im𝐷₁=𝒵,(F4)(𝐴₁,𝐵₁)isstabilizable,(F5)(𝐶₁,𝐴₁)isdetectable,(F6)(𝐶,𝐴)isdetectable,(F7’)(^𝐴3_𝐷^(𝜏)_2(𝜏))∈Im(^{𝐴1(𝜏)𝐵1}_𝐷^(𝜏)_1(𝜏)0).
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(^{𝐼𝑛2⊗𝐴1(𝜏)−𝐴2(𝜏)𝑡⊗I𝑛1𝐼𝑛2⊗𝐵1}_𝐼^(𝜏)_{𝑛2⊗𝐷1(𝜏)0}) is constant.

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 (𝐶₁𝐶₂) and 𝐷₁ have maximal rank. Hence, matrices (𝐶₁(𝜏)𝐶₂(𝜏)) and 𝐷₁(𝜏) have maximal rank for any 𝜏 small enough or, equivalently, Im𝐶₁(𝜏)+Im𝐶₂(𝜏)=𝒴,Im𝐷₁(𝜏)=𝒵. (4.1)
In a similar way, since the set of detectable pairs is open, hypothesis (F5) guarantees that pairs (𝐶₁(𝜏),𝐴₁(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.

Acknowledgment

This work was supported by the Ministerio de Educación y Ciencia under Project MTM2007-67812-C02-02.