H₂ Design of Decoupled Control Systems Based on Directional Interpolations

Abstract

$H_2$ design of decoupled control systems is treated in the generalized plant model. The existence condition of a decoupling controller is stated and a parameterized form of all achievable decoupled closed loop transfer matrices is presented by using the directional interpolation approaches under the assumption of simple transmission zeros. The class of all decoupling controllers that yield finite cost function is obtained as a parameterized form and an illustrative example to find the optimal controller is provided.

1. Introduction

One important characteristic of the multivariable systems is coupling interactions between input and output variables. Efforts to eliminate these interactions lead to finding controllers that make the transfer matrix from the inputs to the outputs diagonal. Once a closed loop system is decoupled, engineers can exploit the well-established design methods of single-input-single-output control system for each channel. The existence condition of a decoupling controller is now well known. For two-degreeof- freedom (2DOF) configuration, Doseor and Gündes [1] and Lee and Bongiorno [2, 3] show that a decoupling controller always exists if the plant is internally stabilizable. On the other hand, a decoupling controller of the 1DOF control system does not always exist. For 1DOF configuration, necessary and sufficient conditions for the existence of decoupling controllers are presented by [4, 5, 6]. While the existence condition of the decoupling controllers has been sufficiently studied, not many papers treat performance issues of decoupled systems. The robust stability problem of decoupling controllers is first addressed by Safonov and Chen [7]. They obtain the stabilizing controllers maximizing stability margin in the H∞ norm context under decoupling and output regulation constraints. Brinsmead and Goodwin [8] investigate the inherent limits of a decoupled system via H2 cost of tracking error. Optimal H2 design considering both of the tracking error and the plant saturation in decoupling problems is treated by Lee and Bongiorno [2, 3] for 2DOF and 3DOF systems and by Youla and Bongiorno [5] and Bongiorno and Youla [9] for 1DOF configuration with non-unity feedback.

Decoupling design in the generalized plant model is very compact and effective in that the derived formulas are applied to the most various models including 1DOF, 2DOF, and 3DOF configurations with non-unity feedback cases. The existence condition of a decoupling controller for the generalized plant is treated by [10, 11]. As for H2 decoupling design, Park [12] extends the work of Youla and Bongiorno [5] to the generalized plant model. However, the freedom of controller configuration in [12] is limited to one and hence the potential of the generalized plant model for including all possible feedback configurations is not fully appreciated. In [13], optimal H2 block decoupling problem is treated with the general setting of 2DOF controller configuration. The class of all decoupling controllers is parameterized by a free parameter and this parameter is used to obtain the optimal controller which minimizes the H2 norm of the transfer matrix from the reference input to the error. In formulating the cost functional, however, the control variable is not considered.

In this paper, H2 design of decoupled system for the generalized plant model is treated based on directional interpolation approaches [14]. The assumption of onedegree- of-freedom configuration in [12] is eliminated and the only assumption needed on the plant is the condition of simple transmission zeros. The approach using the vector operation in [10, 11] is not taken here and hence dimension inflation problem is also avoided to describe the existence condition of a decoupling controller. In this paper, the class of all decoupled transfer matrices is parameterized and the optimal H2 controller is obtained together with the ones that yield finite H2 cost. It is shown that the optimal controller is strictly proper under the reasonable order assumptions on the generalized plant.

Notations; Throughout the paper, we consider only real rational matrices whose elements are from the set of all real rational functions, which are not necessarily proper. Since this set is the quotient field associated with the ring of real polynomials, we will adopt fractional representtations of a real rational matrix using real polynomial matrices. For any real rational matrix G(s) , the notation G∗(s) stands for G'(−s) (G'(s) denotes the transpose of G(s) ). In the partial fraction expression of G(s) , the contribution made by all its finite poles in Re s ≤ 0 and Re s > 0 are denoted by {G}+ and {G}− , respectively. The notation G(s) ≤ 0(sν ) means that no entry in G(s) grows faster than sν as s →∞ . A rational matrix G(s) is said to be stable if it is analytic in Re s ≥ 0 . The Kronecker product of two matrices is denoted as G⊗R . The Schur product G ◦ R of two equi-size matrices G= [gij] and R= [rij] is the matrix whose i-row, j-column entry is gij rij . The Khatri-Rao product of two matrices is denoted as G⊙ R and is the matrix whose i - column is given by gi ⊗ ri where gi and ri the i- column of G and i -column of R , respectively [15]. The notation vecG implies the vector formed by stacking all the columns of the matrix G . For a diagonal matrix, vecd G denotes the vector formed by stacking all the diagonal elements of the matrix G . Tba(s) represents the transfer matrix from a to b . The notations C , C+ and denotes the complex number plane, the open right half plane of C and the closure of C+ , respectively. The notation ξ ∗ denotes the conjugate transpose of a vector ξ .

 

2. Decoupling Problem and its Solution

The model under consideration is shown in Fig. 1. The vectors r(s) and w(s) are the exogenous inputs. The vector v(s) is the output variable in regard of decoupling design. The vector z(s) is the regulated variable. The vectors u(s) and y(s) are the control input and the measured variable, respectively. The variables r(s) and v(s) are the ones such that the transfer matrix Tvr(s) is to be decoupled. In most cases, r(s) is the reference input and v(s) is the plant output.

Fig. 1.The generalized plant model.

The transfer matrix of the generalized plant is given by

The variables v and r have the same dimension n×1 . The variables w and u have the dimensions q1 ×1 and q2 ×1, respectively. The variables z and y have the dimensions l1 ×1 and l2 ×1 , respectively. In this paper, we consider the multi-objective design of decoupling and H2 cost minimization. The following assumption is necessary and sufficient for the existence of a stabilizing controller [16]. In the below, the notation ΨP denotes the characteristic denominator [17] of the rational matrix P(s) and ΨP+ is the polynomial which absorbs all the zeros of ΨP in

Assumption 1: The general plant block P(s) is free of hidden modes in and ΨP+=ΨP22+.

Let

denote polynomial coprime fractional expressions. There always exist polynomial matrices X(s), Y(s), X1(s) and Y1(s) such that

with det X (s) ⋅det X1 (s) ≢ 0 . It is well known [18] that the condition in Assumption 1 is equivalent to the one that the three matrices

are stable. As explained, the transfer matrix Tvr(s) is the one to be decoupled (i.e., to be diagonalized) and it is given by

In the generalized plant model in Fig. 1, decoupling design is to find stabilizing controllers C(s) that make the transfer matrix Tvr(s) diagonal and invertible. The approach taken here for solving the decoupling problem is to characterize the diagonal matrices Tvr(s) that admit stabilizing controllers C(s) in (5) and hence we define the realizability of a diagonal matrix T(s) as follows:

Definition 1: A diagonal stable rational matrix T(s) is said to be realizable for the given plant P(s) if there exists a stabilizing controller C(s) that realizes the transfer matrix Tvr(s) of the system as the matrix T(s) .

In decoupling design, we ask Tvr(s) to be diagonal and invertible so that the normal rank of Tvr(s) should be n . In almost all cases, the matrix P00 is a null matrix and in this case Tvr(s)=P02(I– CP22)−1CP20. In view of this, it can be concluded that we need assumptions on the rank of the matrices P02 and P20. In this paper, we assume the following.

Assumption 2: n=q2=l2 and P02 and P20 are invertible.

Now we seek to find the realizability condition for diagonal matrices T(s) . Consider the class of all stabilizing controllers characterized by the formula

where K(s) arbitrary real rational stable matrices such that det(X1 − KB) ≢0 and det(X − B1K) ≢0 . In this case we have

where

Notice that T00 , T02 and T20 , and hence Tvr(s), are stable by the properties in (4). Since C(s) in (6) characterizes the class of all stabilizing controllers, the formula for Tvr(s) in (8) describes the structure of realizable T(s) and this is the basis to start for characterizing realizable T(s) . Before we proceeding further, we assume the following.

Assumption 3: The matrices T02 and T20 have the distinct simple transmission zeros zi ∈ , i = 1➔m1 and ∈ , i = 1➔m2 , respectively , and zi ≠ for any i and j .

Since T02 and T20 are invertible and zi and are zeros of T02 and T20 , respectively, we can find nonzero vectors ξi and μi [19] such that

Hence, for a diagonal stable matrix T(s) to be realizable, it is necessary from (8) and (11) that

Since a realizable T(s) is a diagonal matrix, say T(s) = diag{tk (s)}, k = 1→ n , the above directional interpolation conditions reduce to the interpolation constraints to the scalar function tk(s),k = 1→ n. Let’s denote the k − th element of a (row or column) vector xi as xik. The conditions in (12) are changed to

for k = 1→n . Obviously, a rational function tk(s) to satisfy the interpolation conditions in (13) does not exist if

for some i. Since the interpolation condition for tk(s) in (13) is a necessary condition for the existence of a decoupling controller, if the interpolation problem in (13) does not have a solution tk(s) for some k , a decoupling controller does not exist. On the other hand, if the data sets {ξik,εik} and {μik, δik} are free from the non-existence condition in (14), a solution tk(s) exists. That is, a decoupling controller exists in the following cases:

or,

for 1 ≤ k ≤ n. This existence condition is conveniently described by the following rank description. That is, a decoupling controller exists if and only if

for any i, j and k . From now on we will assume that the data sets satisfy the conditions in (15.d).

In the next, we seek to characterize all rational tk(s) satisfying the interpolation conditions in (13). When both ξik and εik are zeros as in (15.b), this imposes no interpolation constraint at zi in (13.a) on tk(s) and the same is true for μjk and δjk in (15.c). So let’s define the polynomials dkl(s) and dkr(s) as follows;

We can now describe all solutions of tk(s),k = 1➔ n, satisfying the interpolation conditions of

and

as

where tk0(s) is any stable rational function satisfying the interpolation conditions in (17) (when there are no interpolation conditions in (17), we set tk0(s)=0) and tka(s) any arbitrary stable rational function. Since we don’t exclude improper Tvr(s) from our consideration, tk0(s) can be chosen as a polynomial and an easy choice in this case is the one obtained from Lagrange interpolation formula [20]. Hence we can express T(s) as

where

and Δ(s) is an arbitrary n× n diagonal stable rational matrix. In (19), when T02(s) (T20 (s)) does not have a zero in , Δl = I (Δr = I ).

The matrix T(s) formulated by the parameterized form in (19) is sufficient to be realizable as Tvr(s). In fact, consider the matrix K(s) obtained from (8) with Tvr(s) in (19). It can be shown in the below that the matrix K(s) formed by

is stable where

Since zi and are simple zeros of T02 and T20 , respectively, they are simple poles of T02−1 and T20−1 , respectively. Consider their partial fractional expressions

where F1 and F2 are stable. Using the results in Lemma 1 (Appendix), We can easily show that Ka and Kb are stable since MiΔl(zi)=0 (notice that ikdkl(zi) = 0 for any i and k ) and Δr()Nj. To show that K0 is stable, we insert the equalities in (27) into (25) so that

It now follows from (64) of Lemma 1 (Appendix) that Mi(T00(zi)– Δ0(zi))=kiξ∗i(T00(zi)-Δ0(zi))=ki(εi∗–εi∗)=0 and hence the matrix Mi(T00(s)– Δ0(s)) has the factor (s −zi). Similarly, we can show that the matrix (T00(s)– Δ0(s))Nj has the factor (s-zi)(s-). Therefore, the first, the second, and the third terms of (28) are stable. The fourth term is obviously stable and this completes the proof and now we can state the following theorem without further proof.

Theorem 1: Under Assumptions 1~3, a decoupling controller for the plant (1) exists if and only if the data sets {ξik, εik} and {μjk, δjk} satisfy the conditions in (15.d). When decoupling controllers exist, the class of all decoupled transfer matrices Tvr(s) is characterized by the formula

as in (19).

From the previous developments, we see that not all non-minimum phase zeros of T02(s) and T20(s) appear as the zeros of realizable Tvr(s). In view of (18), the nonminimum phase zero zi () appears as a zero at the k -th channel of Tvr(s), tk(s), only when

which is the condition that both tk0(s) and dkl(s)(dkr(s)) ahve the factor (s−zi)((s−.))

 

3. H2 Design of Decoupled Control Systems

In this section, we formulate an H2 design problem for the decoupled systems and present its solution. The H2 design problem for the system in Fig. 1 is to find the decoupling controller which minimizes a given quadratic cost associated with the regulated variable z when the system is stimulated by the exogenous input [r '(s) w'(s) ]' . To allow the exogenous input to include shapedeterministic components, we assume that q(t) = [r '(t) w'(t)]' is the output of the square block Pq(s) driven by a stochastic vector qo(t)so that

Although allowing the block Pq(s)to possess the poles in makes the H2 problem more general [21], we assume here for simplicity that Pq(s) is stable and qo(t)is a white noise vector with spectral density of unity. It now follows that

where

Let’s denote the transfer matrix from qo to z as Tzo(s). With the setup in (31) and the assumption on qo(t), a meaningful quadratic cost that considers the transient performance and the steady state performance is given by

where

The H2 problem here is to find the decoupled loop transfer matrix T(s) that minimizes the cost index in (34). Since P02 and P20 are invertible, it follows from (5) that

and inserting (36) with the parameterized formula for Tvr(s)in (29) to (35) yields

where

And Δ(s) an arbitrary diagonal stable rational matrix. Knowing that K0(s) , Ka(s) and Kb(s)in (25) and (26) are stable, we can easily show that T11 , Ta and Tb are stable (see Lemma 2 in Appendix). Notice that the expression for Tzo(s)in (37) is the standard form to develop the optimal H2 solution and it can be obtained under the following standard assumptions [12].

Assumption 4: is strictly proper.

Assumption 5: Ta(jω) has full column rank for finite ω .

Assumption 6: Tb(jω) has full row rank for finite ω .

Assumption 7: P12 (s) and (s) behave as P12 (s) → M12sk and(s)→M21sl as s →∞ with the condition that M12 and M21 have full column rank and full row rank, respectively, and k + l ≥ 0 . The transfer matrices P00(s), P02(s), P20(s) and P22(s) are proper.

Let Ω(s) be the Wiener-Hopf spectral factor of the equation

and define

It should be noticed that U∗U = I and hence U is inner. Now we present the H2 solution formula whose proof is omitted since it proceeds as similarly as that of [12].

Theorem 2: Suppose that Assumptions 1~7 are satisfied and the plant P(s) admits the existence condition in (15.d). The class of all decoupling loop transfer matrices that yield finite cost is given by

where

and f (s) an arbitrary strictly proper stable vector. The optimal loop transfer matrix is the one with f = 0 and the cost E for the transfer matrix in (42) is given by E=+||f||2≥, where denotes the cost for the optimal one. The loop transfer matrix T(s) in (42) is proper. The corresponding controller C(s) is obtained form the equation

and it is strictly proper.

We remark that when P00 is diagonal, γ0 in (45) simplifies to γ0 = Ωvecd (Δl−1 Δr−1 P00). In most cases, P00 = 0 and in this case T(s) in (42) is strictly proper.

 

4. An Illustrative Example

As a brief example to show the procedures to obtain the optimal transfer matrix T(s) in (42), we consider the case that the generalized plant is given by

This case corresponds to the one-degree-of-freedom controller system in which a measurement noise exists and the regulated variables are set as the tracking error and the plant input. Decoupling and minimization problems are as explained in the previous sections. Suppose that Pq(s)=diag{Pr(s),I} and and Pr are given as

Since ΨP+=ΨP22+=1in (47), Assumption 1 is satisfied and Assumption 2 is obviously satisfied. We can find coprime fractions for – (s) as

We see that T02=B1 has a simple zero at z1 = 2 and T20=A has no non-minimum phase zero (Assumption 3 is satisfied). The output zero vector and the value are obtained as =[1 0.25] and

The interpolation conditions for t1(s) and t2(s) are given by t1(2) = 0 and t2(2) = 0 . Therefore t1(s) = t2(s) = 0 and the decoupled transfer matrix in (29) is obtained as T(s) = Δ0 + Δl ΔΔr with Δ0 = 0 , Δl = (s − 2)I2 and Δr = I2 . It can be confirmed that this result is identical with that of [5] in which all realizable T(s) is described by the formula T = ΔθΔα + ΔθΔΔ ψ. In fact, we can easily obtain that Δα= 0 , Δθ = (s − 2)I2 , and Δψ = I2 by following the definitions of them in [5].

It now follows from (39) that

and

It is easy to verify that Assumptions 4~7 are satisfied and the Wiener-Hopf factor Ω and other values in Theorem 2 are obtained as

With and . The optimal decoupled transfer matrix in (42) is given by

and the corresponding controller matrix C(s) can be calculated from (46).

 

5. Conclusion and Discussion

Decoupling design of lineal multivariable control systems is treated for the generalized plant model within the H2 framework. A necessary and sufficient condition for the existence of decoupling controllers is obtained based on interpolation approaches. It is shown that directional interpolation problems associated with the decoupling design are changed to simple interpolation constrains of scalar functions whose solutions can be easily obtained. The class of all decoupled closed loop transfer matrices is parameterized and the optimal transfer matrix is obtained using this parameterized formula. The existence condition of a decoupling controller for the generalized plant is treated also in [10, 11] by using vector operations. A major disadvantage of those works is that the vector operation causes dimension inflation and hence the method suffers a difficulty when applied to large-size plants. The existence condition formula developed in this paper is free of dimension inflation.

In this paper, the matrices T02 and T20 are assumed to be square but extension to the rectangular case can be readily done if the transformation of T02 and T20 using unimodular matrices in [3, 9, 11] is adopted. The constraint that zi ≠ for any i, j in Assumption 3 can be loosened but it requires more complex descriptions in Theorem 1 and it will be presented in future publications. In Assumption 3, non-minimum phase zeros of T02 and T20 are assumed to be simple and one of future research topics would be generalization of the results in this paper to the multiple zero case. It is expected that the methods of the multiple directional interpolation approach [22] and the generalized characteristic vectors in [23, 24] will play a role for the generalization.