Disturbance Decoupling of Multi-input Multi-output Discrete-time Nonlinear Systems by Static Measurement Feedback

This paper addresses the disturbance decoupling problem (DDP) for nonlinear systems, extending the results for continuous-time systems into the discrete-time case. Sufficient conditions are given for the solvability of the problem. The notion of the rank of a one-form is used to find the static measurement feedback that solves the DDP whenever possible. Moreover, necessary and sufficient conditions are given for single-input single-output systems, as well as for multi-input multi-output systems under the additional assumption.


INTRODUCTION
The disturbance decoupling problem (DDP) for a discrete-time nonlinear control system by state feedback has been addressed in many papers (see [2,3,7,8,14,16]).Most papers extend the results known for continuous-time systems into the discrete-time domain (e.g.[6,10,17]), describing the control system by smooth or analytic difference equations.Few studies address the DDP for discrete-time nonlinear control systems using output feedback (e.g.[15,18,20], see also [13]), whereas only [18] treats explicitly the case of static measurement feedback, which is the topic of our paper.However, in [18] necessary and sufficient conditions are given only for single-input single-output (SISO) systems.Papers [15] and [20] focus on dynamic measurement feedback.In [20] the controlled output is a vector function of the measured output, having possibly less components than the measured output itself.Therefore, the above solution may be considered only as a partial solution.Paper [15] provides a full algorithmic solution for the problem using dynamic feedback.In both papers the novel algebraic approach, called the algebra of functions (see [22]), is applied.
Only a few papers address the problem of continuous-time nonlinear control systems ( [1,11,19,21]).Paper [19] studies the problem using static measurement feedback, in [11] the feedback considered is restricted to pure dynamic measurement feedback, and papers [1,21] focus on dynamic measurement feedback.
Our goal is to extend the results of [19] for discrete-time nonlinear control systems.Moreover, the results of [19] were given for multi-input single-output (MISO) systems, whereas the present paper addresses the multi-input multi-output (MIMO) case.The preliminary results of this paper (for MISO systems only) were presented at the 18th International Conference on Process Control ( [12]).

PRELIMINARIES
Consider a discrete-time nonlinear control system where the state x(t) ∈ R n , the control input u(t) ∈ R m , the disturbance input w(t) ∈ R ν , the output to be controlled y(t) ∈ R p , and the measured output z(t) ∈ R µ .Assume that f , h, and k are meromorphic functions of their arguments.Throughout the paper we also assume that system (1) is generically submersive, i.e.
everywhere except on the set of zero measure.
Under submersivity assumption we can construct the inversive difference field1 of meromorphic functions in variables x(t), u(t), w(t) and a finite number of their (independent) forward and backward shifts associated with system (1), which we denote as K * .Note that not all the variables are independent because of the relationships defined by (1) and in the computations the dependent variables have to be expressed via the independent ones.For example, x(t + 1) has to be replaced by f (x(t), u(t), w(t)).See [9] for the details how to construct K * .
Define the vector spaces 4]).The relative degree r of the output y(t) is defined by If such an integer does not exist, define r := ∞.
The static measurement feedback of the form u(t) = F(z(t), v(t)) is called regular if F is invertible with respect to v(t), i.e. if there exists an inverse function α Problem Statement.Given a nonlinear system of the form (1), the goal is to find, if possible, a regular static measurement feedback of the form such that every controlled output y i (t), i = 1, . . ., p, of the closed-loop system satisfies the following conditions: , where r i is the relative degree of y i (t) with respect to u(t).Condition (i) represents the independence of the output of the closed-loop system from the disturbance, whereas condition (ii) represents the output controllability of the closed-loop system.
Analogously to the continuous-time case (see [19]), define the subspaces Ω i ⊂ X for every output y i (t) (i = 1, . . ., p) by The subspaces Ω i will be important in solving the DDP, because the forward shifts of a one-form ω(t) ∈ Ω i do not depend explicitly on inputs u(t) and w(t).
To simplify the presentation of the proof of Lemma 1 below, we omit the index i.That is, instead of y i and Ω i , i = 1, . . ., p, we just write y and Ω, respectively.Lemma 1.The subspace Ω may be computed as the limit of the following algorithm: Proof.We show below that sequence Ω k converges and in the limit we get Ω.Consider a subspace Ω k .By (2), . Thus sequence (2) converges and the limit is Ω k * .We show now that Ω = Ω k * .Suppose ω(t) ∈ Ω k * .Then, by ( 2) and so ω(t + 1) = ω(t) + ξ dy(t + r) for some ω(t) and so the forward shift of ω(t + 1) is Continuing in the same way, we get ω(t + k * ) ∈ Ω 0 + span K * {dy(t + r), . . . ,dy(t + r + k * − 1)}, which means that ω(t) ∈ Ω.We showed that if ω(t As where ω(t) ∈ Ω 0 and ξ 1 , . . ., Continuing in the same way, we get Thus ω(t) ∈ Ω k * and we have shown that Next we will show how Ω changes under the regular static measurement feedback u(t) = F(z(t), v(t)).Denote by K * the field of meromorphic functions in variables x(t), v(t), w(t) and a finite number of their independent forward and backward shifts and define the vector spaces [21] we can prove that there exists an isomorphism Φ : E → E such that if Ω cl is the subspace for the closed-loop system, then Ω cl = Φ(Ω).
Let ω(t) ∈ E .In general, ω(t) is a linear combination over K * of a certain number of standard basis elements of E .However, it is often possible to find a linearly independent set of exact one-forms with less elements than those basis elements of E in terms of which ω(t) can be expressed.For example, a oneform ω(t) = (x 2 (t)u 1 (t) + u 2 (t))dx 1 (t) + x 2 (t)x 1 (t)du 1 (t) + x 1 (t)du 2 (t), as a linear combination of dx 1 (t), du 1 (t), and du 2 (t), can be expressed as a linear combination of two exact one-forms, d(x 1 (t)u 1 (t)) and d(x 1 (t)u 2 (t)).

Definition 2. ([5]
).Let γ be the minimal number of linearly independent exact one-forms necessary to express a one-form ω(t).Then ω(t) is said to be of rank γ.
Note that 1 ≤ γ ≤ n.For example, if the rank γ of a one-form ω(t) is 1, then ω(t) = ξ dα and thus ω(t) ∧ dω(t) = 0.In the general case, if the rank γ is k, then ω(t) ∧ (dω(t)) (k) = 0, where We prove the following lemma for MIMO systems, providing an alternative formulation of the system to be disturbance decoupled.It allows us to check whether the system is disturbance decoupled or not.The lemma will be used later in the proof of the main result of the paper (i.e.Theorem 1).Lemma 2. Under the assumption that the relative degrees r i of the outputs y i (t) are finite, system (1) is disturbance decoupled iff
We are going to use the subspaces Ω i (i = 1, . . ., p) and the concept of the rank of a one-form to give a sufficient condition for the DDP.

MAIN RESULTS
The following theorem gives sufficient conditions for solvability of the DDP by static measurement feedback.
Remark.We point to the fact that condition (ii) is not a direct extension of the respective condition in the single-output case.Even if we can find individual one-forms ω i (t) such that dy i (t + r i ) − ω i (t) ∈ Ω i , for all i = 1, . . ., p, this does not necessarily mean that we can find a single one-form ω(t) as given in (ii) of Theorem 1.
In case of SISO systems when m = 1, ( 7) and (ii) of Theorem 1 yield Thus, condition (iii) of Theorem 1 is satisfied if and only if γ = 1.For SISO systems we can conclude from Theorem 1 a necessary and sufficient condition.
In general there is no necessary and sufficient condition for MIMO systems, but under additional assumptions Ω i ∩ Z = Ø and dy i (t + r i ) ∈ Ω i ⊕ Z + U we can find a necessary and sufficient condition for MISO systems.

Proof.
Necessity.Assume that system (1) is disturbance decoupled by the regular static measurement feedback v(t) = α(z(t), u(t)).By Lemma 2, dy cl is the output of the closed-loop system.Because of the isomorphism Φ : E → E described above and feedback α(z(t), u(t)), we can write Thus, there exist a one-form ω(t) ∈ Ω i and ξ ∈ K * such that dy i (t + r i ) = ω(t) + ξ dα(z(t), u(t)).
The assumption dy i (t + r i ) for some ω0 (t) ∈ Ω i and ωz (t) ∈ Z .As in the proof of Lemma 2, we can show that ωz (t) ∈ Ω i .Due to the assumption Ω i ∩ Z = 0, we have ωz (t) = 0. Then define ω(t) = ξ dα(z(t), u(t)) and the necessity of condition (i) is fulfilled.
As the rank of a one-form ω(t) is γ, where β i ∈ K * , i = 1, . . ., γ. Suppose, contrarily to the claim of Theorem 2, that (ii) is not fulfilled.Then there exists a one-form Assume without loss of generality that ξ 1 = 0. Then ω(t) can be decomposed into for i = 2, . . ., γ.As shown before, if ωz (t) ∈ Z , then necessarily ωz (t) ∈ Ω i and since Ω i ∩ Z = 0, this yields a contradiction.Thus condition (ii) has to be fulfilled.Sufficiency.As all conditions of Theorem 1 are satisfied, the sufficiency is fulfilled.

EXAMPLES
In this section we give some examples to illustrate the theory.
Example 1.In the first example we consider the SISO system x 4 (t + 1) = w(t) + x 4 (t), Note that the relative degree of the output y(t) is 2, because Next we find the vector space Ω, using the algorithm defined by (2).First, From we can conclude that In the next step we get ).Thus the rank of ω(t) is 1 and condition (ii) of Corollary 1 is satisfied.The disturbance decoupling feedback may be found as the solution of the equation v(t) = z(t) u(t) with respect to u(t).Example 2. Consider the MISO system x 1 (t + 1) = x 2 (t) + x 3 (t) + x 5 (t), x 2 (t + 1) = ln(u 1 (t)x 4 (t)), x 5 (t + 1) = x 5 (t), The relative degree of output y(t) is 2 and Still, the following feedback solves the DDP: .
Example 4. Consider the MIMO system Note that the relative degree of the outputs y 1 (t) and y 2 (t) is 1 and the relative degree of y 3 (t) is 3.As Thus the first condition of Theorem 1 is satisfied.Now, we must choose ω(t) such that dy i (t + r i ) − ω(t) ∈ Ω i for every output y i (t).If we choose ω(t) to be the previous conditions are satisfied and the rank of ω(t) is 2. Thus the second condition is also satisfied.Finally, the third condition of Theorem 1 is satisfied because The feedback that solves the DDP is Recall that all results of this paper hold under submersivity assumption.The following example demonstrates that wrong conclusions can be reached if the system is not submersive, but nevertheless Theorem 1 is blindly used to check the solvability of the DDP by regular static measurement feedback.
Example 5. Consider the system x 5 (t + 1) = x 4 (t), System ( 16) is not submersive.If we forget about this fact and check the conditions of Theorem 1, we will get that the DDP is not solvable for system (16) by static measurement feedback.Really, the relative degree of output y(t) is 2 and the subspace Ω = span K * {dx 1 (t), dx 2 (t)}.Since dy(t + 2) = dx 3 (t) − d(x 1 (t)x 5 (t)) + d(z(t)u 1 (t)u 2 (t)), condition (i) of Theorem 1 is not satisfied.However, since the system is not submersive, the forward shift of x 3 (t) − x 1 (t)x 5 (t) is 0 and because of that, the following static measurement feedback solves the DDP: , u 2 (t) = v 2 (t).

CONCLUSION
In this paper the notion of the rank of a one-form and the subspace Ω of differential one-forms was used to solve the DDP for nonlinear discrete-time control systems by static measurement feedback.Sufficient conditions for solvability of the DDP were found for MIMO systems.Necessary and sufficient conditions were derived from the above conditions for SISO systems and for MIMO systems under the additional assumption.These sufficient conditions also provided a procedure to find the static measurement feedback to solve the DDP.As these conditions are very restrictive, further research is necessary.The next step is to extend the results of [21] by addressing dynamic measurement feedback in the framework of differential forms for discrete-time systems.The results can then be compared with those by [15], which were obtained using the tools of the algebra of functions.In addition to the above theoretical problems, the functions in the Mathematica program have been developed for solving the DDP and integrated into the symbolic software package NLControl, developed in the Institute of Cybernetics at Tallinn University of Technology.