Semi-Decentralized Approximation of Optimal Control of Distributed Systems Based on a Functional Calculus
This paper discusses a new approximation method for operators which are solution to an operational Riccati equation (ORE). The latter is derived from the theory of optimal control of linear problems posed in Hilbert spaces. The approximation is based on the functional calculus of self-adjoint operators and the Cauchy formula. Under a number of assumptions the approximation is suitable for implementation on a semi-decentralized computing architecture in view of real-time control. Our method is particularly applicable to problems in optimal control of systems governed by partial differential equations with distributed observation and control. Some relatively academic applications are presented for illustration. More realistic examples relating to microsystem arrays have already been published.
This work is a contribution to the area of semi-decentralized optimal control of large linear distributed systems for real-time applications. It applies to systems modeled by linear partial differential equations with observation and control distributed over the whole domain. This is a strong assumption, but it does not mean that actuators and sensors are actually continuously distributed. The models satisfying such assumption may be derived by homogenization of systems with periodic distribution of actuators and sensors.
In this paper we consider two classes of systems, those with bounded control and bounded observation operators as in R. Curtain and H. Zwart , and those with unbounded control but bounded observation operators as in H.T. Banks and K. Ito . In an example, we show how the method may also be applied to a particular boundary control problem. We view possible applications in the field of systems including a network of actuators and sensors, see for instance  dedicated to arrays of Atomic Force Microscopes.
We consider four linear operators , , and the Linear Quadratic Regulator (LQR) problem stated classically as a minimization problem,
constrained by a state equation,
Under usual assumptions there exists a unique solution , where is a solution of the operational Riccati equation (ORE),
In the framework of , , and consequently for some linear spaces and . To derive our semi-decentralized realization of , we further assume that there exists a linear self-adjoint operator , three one-to-one mappings
with appropriate integers and and four continuous matrix-valued functions , and such that
We notice that the functions of the self-adjoint operator used in the above formulae are defined using spectral theory of self-adjoint operators (having a real spectrum) with compact or not compact resolvent so that to encompass bounded and unbounded domains. From (6), it follows that the Riccati operator is factorized as
where is a continuous function, solution of the algebraic Riccati equation (ARE)
Our goal is reached once separate efficient semi-decentralized approximations of , and are provided for the realization of through (7). This is generally not an issue for and for then the point is the semi-decentralized approximation of . It might be build by a polynomial approximation,
or a rational approximation,
Then, for practical implementations, the operator could be replaced by a discretizations with parameter We emphasize that the formulae (9) or (10) yield large approximation errors, with respect to due to the high powers of . To overcome this defect, we use an approximation based on the Cauchy integral which requires to know the poles of . In practice, we first approximate the function by a polynomial approximation or a rational approximation with degrees or sufficiently high to insure a very small error. When is known its poles also, so we can state the Cauchy formula for . This yields to introduce the equations of the complex function for each input
where is the contour of the Cauchy formula. Denoting by the solution corresponding to a quadrature point of the contour and some quadrature weights, the final approximation of is
Remark that the number of quadrature points is the only important parameter governing the approximation error. For real-time computation, the expression of is pre-computed, so the approximation cost is also governed by only. With this method, we do not observe a lack of precision when is replaced by its discretizations and is large. In the sequel, we show that the same derivation can be done directly for provided that the isomorphisms and are also some functions of
This approach based on functional calculus is relatively simple, but in each case it requires to determine the isomorphisms (5). The theory has already been applied in  to a LQG control problem with a bounded operator that is not a function of . It has been shown how the control approximation can be implemented through a distributed electronic circuit. In  and  it has also been applied to a one-dimensional array of cantilevers with regularly spaced actuators and sensors for which the operator is not a function of . The underlying model was derived with a multiscale method, an implementation of the semi-decentralized control was provided in the form of a periodic network of resistors, and the numerical validations of the complete strategy was carried out. In the present paper, we illustrate the theory with four simpler examples ranging from a simple heat equation with internal bounded control and observation operators, a heat equation with an unbounded control operator, a vibrating Euler-Bernoulli beam, and a heat equation with a boundary controls.
We notice that our method together improves and generalizes a previous paper . It was related to a specific application, namely vibration control problem for a plate with a periodic distribution of piezoelectric actuators and sensors. There, the general isomorphisms (5) and the general factorization (6) were not introduced, and was approximated by a polynomial as in (9) which were severely limiting the accuracy of the approximation. In both papers, the control method is a LQR, but the theory is applicable to Riccati equations that may arise in a number of other control problem, for instance for or dynamic compensators. Other extensions are also possible, for instance, we may want to deal with functions of a non self-adjoint operator . In such a case, another functional calculus, like these in  or in , could be used instead of the spectral theory. Other frameworks for control problems of infinite dimensional systems could also be used, for instance this of  for optimal control with unbounded observations and unbounded controls.
Other techniques have already been established, see , , , ,  and the references therein. But they are mostly focused on the infinite length systems, see , ,  and  for systems governed by partial differential equations, and  for discrete systems. Finally, in  we developed another theoretical framework based on the diffusive realization applicable to a broad range of linear operators on bounded or unbounded domains. In principle this approach allows to cover general distributed control problems with internal or boundary control. However, in this first paper in the subject, only one-dimensional domains and linear operational equations (e.g. Lyapunov equations) are covered.
The paper is organized as follows. Notations and basic definitions are recalled in Section 2. In Section 3 the abstract approximation method is stated in the framework of bounded control and observation operators. The framework of unbounded control operators is treated in Section 4. Some extensions are outlined in Section 5. Most proofs are concentrated in Section 6. The illustrative examples are detailed in Section 7 and finally the paper is concluded by Section 8.
2 Preliminary Results and Notations
The norm and the inner product of an Hilbert space are denoted by and For a second Hilbert space denotes the space of continuous linear operators from to In addition, is denoted by One says that is an isomorphism from to if is one-to-one and if its inverse is continuous.
Since the approximation method of is based on the concept of matrices of functions of a self-adjoint operator, this section is devoted to their definition. Let be a self-adjoint operator on a separable Hilbert space with domain , we denote by its spectrum and by an open interval that includes . We recall that if is compact then is bounded and is only constituted of eigenvalues They are the solutions to the eigenvalue problem where is an eigenvector associated to chosen normed in , i.e. such that . For a given real valued function , continuous on , is the linear self-adjoint operator on defined by
with domain Then, if is a matrix of real valued functions continuous on , is a matrix of linear operators with domain
In the general case, where is not compact and where is still a continuous function, the self-adjoint operator is defined on by the Stieltjes integral
and its domain is where is the spectral family associated to , see . When is a matrix, is a matrix of linear operators with entries defined by the above formula and with domain
3 Bounded Control Operators
In this section, we state the approximation result in the framework of bounded input operators. We follow the mathematical setting  of the LQR problem (1-3). So, is the infinitesimal generator of a continuous semigroup on a separable Hilbert space with dense domain , , and where and are two Hilbert spaces. We assume that is stabilizable and that is detectable, in the sense that there exist and such that and are the infinitesimal generators of two uniformly exponentially stable continuous semigroups. For each the LQR problem (1-3) admits a unique solution where is the unique self-adjoint nonnegative solution of the ORE
for all The adjoint of the unbounded operator is defined from to by the equality for all and . The adjoint of the bounded operator is defined by , the adjoint being defined similarly.
Now, we state specific assumptions for the approximation method. Here, is a given self-adjoint operator on a separable Hilbert space which is chosen to be easily approximable on a semi-decentralized architecture. Generally, is chosen with regard to then and can be chosen so that to have also a natural semi-decentralized approximation.
There exist three integers and , three isomorphisms and and four matrices of functions , and continuous on such that
One of the consequences of this assumption, for a system governed by a partial differential equation posed in a domain is that both the control and the observation must be distributed throughout the domain, in conformity with what has been stated from the beginning
In case where all operators are function of , then the isomorphisms are or not useful or can be chosen as function of . In both cases is also a function of .
Introducing the isomorphisms , and allows to deal with problems where operators , and are not functions of .
For boundary control or observation problems, it is impossible to find such isomorphisms. Nevertheless, in Subsection 7.4 we show how to proceed to address some boundary control problems.
We introduce the ARE
For all , the ARE (14) admits a unique nonnegative symmetric solution denoted by .
This assumption is stronger than the typical sufficient condition for the mere existence of a solution to the Riccati equation [give ref].
We make the following choices for the inner products of , and :
Thus , and , are related as follows.
If (H1) and (H2) are fulfilled then
where the controller admits the factorization with
Now, we focus on a semi-decentralized approximation of which reduces to provide such an approximation for . We restrict the presentation to the case of bounded operators since they have a bounded spectra. This is sufficient for applications to systems governed by partial differential equations in bounded domains.
The operator is bounded and its spectrum is bounded, so there exists with
This assumption can be relaxed, see Section 5.
The operators , and admit semi-decentralized approximations for all with .
Now, we introduce two successive approximations and of that play a key role in our method.
The rational approximation : Since the interval is bounded, each entries of the matrix admits a rational approximation on . This defines a matrix of rational approximations of ,
to be understood componentwise, so each , is a matrix and is a pair of matrices of polynomial degrees. The particular case corresponds to a classical polynomial approximation. For any the degrees of approximations can be chosen so that the uniform estimate
Approximation by quadrature of the Cauchy integral: For any complex valued function continuous on we introduce a quadrature rule for the integral , denoting the nodes of a regular subdivision of and the associated quadrature weights. The quadrature rule is assumed to satisfy an error estimate as
For and a sufficiently regular complex contour enlacing and not surrounding any pole of We parameterize it by a parameter varying in . We further introduce the solution of the system
and the second approximation of through its realizations
We notice that two approximations and of the function can be constructed by following the same steps. The next theorem states the approximations of the operators and
Under the assumptions (H1-H4), and can be approximated by one of the two semi-decentralized approximations
Moreover, for any there exist and such that
, and being independent of and .
In the case of a polynomial approximation, i.e. , we can set a circle as contour . For actual rational approximations, the contour must leave the poles outside, so we choose an ellipse centered at parameterized by where and are for the major and minor radii and is small enough.
The approximation of used in  is based on Taylor series, so it is applicable only if the interval is sufficiently small. The approximation proposed in our paper does not suffer from this drawback.
In case where the solution of a Riccati equation is a kernel operator (see  for optimal control of systems governed by partial differential equations) i.e. and if is a compact operator then the kernel may be decomposed on a basis of eigenvectors of ,
The truncation technique used in  can be applied to build a semi-decentralized approximation of . However, when the decay of is not very fast, this technique is not efficient, see for example the case that may yield from a LQR problem.
For concrete real-time computations one can use either of the two formulae (15) or (19) given that both are semi-decentralized, but we prefer the second since it does not make use of powers of The reason will become clearer when discretizing. In a real-time computation, the realization requires solving systems (18) corresponding to complex values . So the parameter is essential to evaluate the cost of our algorithm. The matrix is pre-computed off-line once and for all and we choose sufficiently large that is a very good approximation of . Consequently, is the only parameter that influences the accuracy of the method, except the parameter space discretization that is discussed now.
The end of the section is devoted to spatial discretization. For the sake of simplicity, the interval is meshed with regularly spaced nodes separated by a distance .
Spatial discretization with polynomial approximation: First, we introduce the finite differences discretizations of , with . For , the discretization of in (15) can be written as
where is the vector of nodal values of . Their discretization yields very high errors because the powers of . This can be avoided by using the Cauchy formula, i.e. the equation (18).
Spatial discretization with Cauchy formula approximation: For each quadrature point , the discrete approximation of is the solution of the discrete set of equations
Thus we deduce the discretization of the approximation in (19),
Under the Assumption (H4), we introduce and the semi-decentralized approximations of and . So, the approximations of and by a spatial discretization are
This constitutes two different final semi-decentralized approximations of .
The approximations and are given in the general case where the isomorphisms and are not function of only. Therefore, we use our approximation technique to represent . In some cases and are function of and then is also and the approximation is developed directly on it that we denote by ,
In the case where and are functions of then the approximation is developed on , we will also denote it by without risk of confusion,
4 Unbounded Control Operators
When the input operator is unbounded from to and the observation operator is bounded from to , we use the framework of  where is another Hilbert space, is its dual space with respect to the pivot space and . A number of other technical assumptions are not detailed here. The state equations are written in the sense of with The optimal control is where is the unique nonnegative solution of the Riccati operatorial equation
for all The adjoint is defined by when is defined as the adjoint of a bounded operator. We keep the same inner products for and , and those of and are
Moreover, we choose as the canonical isomorphism from to and the duality product between and is
Same statement as (H1) excepted that
where and are two additional isomorphisms. Moreover,
are some functions of .
Here, the ARE is
For all , the ARE (25) admits a unique nonnegative solution denoted by .
If (H1’,H2’) are fulfilled, then
where admits the factorization with
The following assumptions are necessary for the semi-decentralized approximation of .
Same statement that (H4) completed by , and admit a semi-decentralized approximation.
Under the Assumptions (H1’,H2’,H3,H4’), and can be approximated by one of the two semi-decentralized approximations
Moreover, for any there exist and such that
, , and being independent of and .
An example of unbounded control operators is given in the Subsection 7.2.
The approximations of and are constructed using the same method as in the case of bounded control operators.
In this section, we mention possible extensions of the theoretical framework presented above.
The same strategy applies directly to dynamic estimators and compensators derived by the to the theories. For instance, the condition on the spectral radius of the product of the solution of the two Riccati equation can be expressed under the form of a condition on the spectral radius of the product of two parameterized matrices for all , see Lemma 6.2 (6).
The spectral theory of self-adjoint operators has been chosen for its relative simplicity. We are aware of its limitation, so we mention possible extensions based on more general functional calculi like these developped in  or  to cite only two.
Other frameworks for the well-posedness of the LQR problem can be used. In particular, this of  for optimal control with unbounded observation and control may be incorporated in this approach.
First, we remark that for and two Hilbert spaces and an isomorphism from to if is equipped with the inner product then . In the next lemma, we state few functional calculus properties.
For a self-adjoint operator on a separable Hilbert space , and for , two functions continuous on
for , on ;
when the range of is included in ;
if in then exists and is equal to ;
if for all then ;
for all .
The proofs of the first five statements can be found in . We prove (6) i.e. that . First, assume that is bounded. We recall that for a function continuous on and for the integral is defined as the strong limit in of the Riemann sums, see , when vanishes, where and . When is not bounded, we use a subdivision of a bounded interval and the integral is defined by passing to the limit in the integral bounds. Let us establish that the Riemann sum is nonnegative, so the result will follow by passing to the limit. Since where then the Riemann sum is the sum over of the nonnegative terms which in turn is nonnegative.
Now we prove (7):
For two integers , , a matrix of functions continuous on and two Hilbert spaces , isomorphic with and by and respectively, we introduce the so-called generalized matrix of functions of : with domain . For the sake of shortness, the spaces and do not appear explicitly in the notation , so they will be associated to each matrix at the beginning of their use. Then, no confusion will be possible. In the next lemma, we state some properties of generalized matrices of functions.
For any generalized matrices of functions of and , and any real number ,
for another Hilbert space and ,