Controller reduction for linear systems based on subspace balanced ...
Controller reduction for linear systems based on subspace balanced realization Kenji Fujimoto, Sayaka Ono and Yoshikazu Hayakawa Abstract— Model order reduction is an important tool in control systems theory. In particular, it is useful for controller design since the dimension of the controller becomes very high when we use advanced control theory. Balanced truncation is one of the most useful model order reduction methods. But in general the stability of the feedback system is not maintained when the order of the controller is reduced by balanced truncation. This paper proposes a novel method of controller reduction by which we can preserve the stability of the feedback system. A numerical example demonstrates the effectiveness of the proposed method.
I. I NTRODUCTION Recently, the development of CAD softwares enable one to execute controller design and stability analysis for large scale systems. Accordingly, the importance of the model order reduction increases. So far, many techniques for model order reduction were proposed. See e.g. [7], [1]. Among them, balanced realization and its application to model order reduction is known as one of the most effective model order reduction methods [6]. Balanced realization is a statespace realization whose states are balanced in a sense that the relationship between the input and the output, and the relationship among the coordinate axes are balanced. On these coordinates, one can easily eliminate less important states and consequently obtain a reduced order model. This model order reduction method is called balanced truncation. There are many extensions of this result. In this paper, we consider a controller reduction problem. Since conventional balanced truncation is only applicable to an open-loop system, it was difficult to reduce the dimension of the controller and maintain the stability of the whole closed-loop system. This problem is called controller reduction and investigated within the framework of weighted balanced realization [2], [10]. Section 7.6 of [1] and Section 4 of [7] discuss the basic ideas and the references on this research topic. For instance, Liu et al. provides a framework for controller reduction using coprime factorizations [5], and Wortelboer et al. proposes a method to choose appropriate frequency weights for controller reduction [11]. Although the existing results are enough effective, it is difficult to ensure the stability of the resulting feedback system with the original plant and the reduced order controller. Recently, Li [4] and Sandberg et al. [8], [9] proposed a variation of balanced realization applicable to a feedback system. Although these method can solve a class of controller reduction problems, K. Fujimoto is with Department of Mechanical Science and Engineeirng, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
[email protected]
there are some restrictions on the system to be applied. A similar idea was proposed by the authors [3] which derives a controller reduction method in the state feedback case. This paper is devoted to its extension to the output feedback case using state observers. It also ensures the stability of the resulting closed loop system with the original plant and the reduced order controller. The authors believe that the present paper gives the first result on controller reduction based on balanced truncation which maintains the stability of the original feedback system. Furthermore, a numerical example will demonstrate its effectiveness compared to the conventional one. II. M ODEL ORDER REDUCTION FOR INTERCONNECTED SYSTEMS VIA STATE FEEDBACKS
This section refers to conventional model order reduction method based on balanced truncation [7] and the authors former results on partial balanced realization for state-feedback systems [3]. A. Balanced truncation Consider the following linear system. { x˙ = Ax + Bu, x(0) = 0 y = Cx Here x(t) ∈ Rn , u(t) ∈ Rm , and y(t) ∈ Rr . The system matrix A is Hurwitz. The controllability and observability Gramians P and Q are the symmetric solutions to the following Lyapunov equations. AP + P AT + BB T = 0, AT Q + QA + C T C = 0 (1) The positive definiteness of the Gramians P and Q is equivalent to the controllability and observability of the system. In addition, they are quantitative indices of the controllability and observability of the system. A state-space realization on which the following equation holds is called a balanced realization. P = Q = diag(σ1 , · · · , σn ), σ1 ≥ · · · ≥ σn
(2)
Here the relation between the input and the output, and the relation among the coordinate axes are balanced. The scalars σi ’s indicating the importance of the states are called Hankel singular values. Remark 1: Instead of using the Lyapunov equations (1), we can use the following Lyapunov inequalities (LMIs). ˇ + QA ˇ + C T C ≺ 0 (3) APˇ + Pˇ AT + BB T ≺ 0, AT Q
(Non-unique) symmetric solutions to these inequalities are called pseudo controllability Gramian and pseudo observability Gramian, respectively. Furthermore, the Gramians P and Q define the controllability and observability functions Lc and Lo as Lc (x) = (1/2)xT P −1 x, and Lo (x) = (1/2)xT Q x. The procedure of model order reduction based on balanced realization (balanced truncation) is as follows. Suppose that the Hankel singular values defined in Equation (2) satisfy σk σk+1 , 1 ≤ k ≤ n − 1. This means that the set of the states x1 , · · · , xk have a bigger effect to the input-output map than the rest xk+1 , · · · , xn . Therefore we can obtain a k-dimensional reduced order model by substituting 0 for the unimportant states xk+1 , · · · , xn . B. Problem setting This section focuses on balanced realization for interconnected systems via state feedbacks and its application to model oder reduction. The biggest advantage of the proposed method is that the reduced order interconnected system preserves the stability of the original one. Fig. 1 shows the configuration of the proposed model order reduction problem. Here P and K denote state-space realizations with the state x and that ξ, respectively. C denotes a matrix determining the output signal y from x and ξ. •
u
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ξ
P
x
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C
y
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•
u ξ
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P /
•
Ko /
(i) Original system Fig. 1.
x
•
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C
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Kred o
(ii) Reduced system Outline
Suppose that the feedback system with P and K as in Fig.1 (i) is stable. The problem considered here is to obtain a reduced order controller Kred in such a way that the feedback system with P and Kred as in Fig.1 (ii) maintains its stability and furthermore the input output map u-y is also preserved. C. Subspace balanced realization Let us consider the following linear system depicted as in Fig.1 (i). x˙ = A11 x + A12 ξ + B1 u ξ˙ = A21 x + A22 ξ + B2 u y = C1 x + C2 ξ
(4)
Here x(t) ∈ Rl is the state of the plant (the state to be preserved), ξ(t) ∈ Rn is the state of the controller (the state to be reduced), and u(t) ∈ Rm and y(t) ∈ Rr are the input and the output, respectively. In order to reduce the order of the controller K, let us replace its state ξ by a lower order vector ξˆr ∈ Rk , k < n
and obtain a reduced order controller Kred in Equation (5). x˙ = A11 x + A12 ξ + B1 u ˆ˙ ˆ2 u ξr = Aˆ21r x + Aˆ22r ξˆr + B r (5) ξ = E1 x + E2 ξˆr y =C x+C ξ 1
2
Here we employ the following assumption for this system. Assumption 1: The system (4) is asymptotically stable, controllable and observable. The controllability and observability functions Lc and Lo of the system (4) are defined as follows. 1 1 Lc (x, ξ) = xT R11 x + xT R12 ξ + ξ T R22 ξ (6) 2 2 1 1 (7) Lo (x, ξ) = xT Q11 x + xT Q12 ξ + ξ T Q22 ξ 2 2 Here the matrices Pij ’s and Qij ’s are defined by the controllability and observability Gramians P ∈ R(l+n)×(l+n) and Q ∈ R(l+n)×(l+n) in the following way where the division of the matrices corresponds to the division of the states x and ξ. ] [ ] [ R11 R12 Q11 Q12 −1 (8) , Q= P = T R22 QT12 Q22 R12 Since we are interested in the effect of the partial state ξ to the input-output behavior of the whole system, it is natural to define Gramians describing the properties of the partial state ξ. To this end, let us define the partial controllability Gramian Psub and the partial observability Gramian Qsub respectively as follows. Definition 1: Partial controllability and observability Gramians Psub and Qsub are defined as follows. −1 Psub := R22 , Qsub := Q22 (9) Note that if P, Q 0 then Psub , Qsub 0. We are interested in the Hankel singular values with respect to the partial Gramians Psub and Qsub . That is, we should obtain the Hankel singular value σ ∈ R and the corresponding coordinate axis ζ ∈ Rn by solving the following equation for singular value analysis.
Psub Qsub ζ = σζ
(10)
Then the partial Gramians are balanced (diagonalized) on the new coordinates ζ’s. Consequently we can obtain a reduced order controller Kred of K by balanced truncation. A question arises now: Is the feedback system (5) with the reduced order controller Kred stable? The answer is generally no. Here we use a trick to ensure the stability of the feedback system (5) by employing a freedom in choosing the partial state ξ itself. For this purpose, the following property is useful. Theorem 1: [3] Consider the system (4). For any linear coordinate transformation in the following form, the Hankel singular values based on the partial Gramian with respect to ξˆ are equivalent to that with respect to ξ. ˆ = θ(x, ξ) := (x, M x + N ξ), (x, ξ) M ∈ Rn×l , N ∈ Rl×l , det N 6= 0
(11)
Proof: See [3]. This theorem suggests that the freedom in choosing the coordinate transformation θ(·) can be used for the partial balanced realization and truncation. D. Partial balanced truncation with stability Let us consider a candidate Lyapunov function V V (x, ξ) := (1 − α)Lc (x, ξ) + αLo (x, ξ), 0 ≤ α ≤ 1 (12) Here Lc and Lo are the controllability and observability functions given in Equations (6) and (7). Assumption 1 implies V 0. The procedure of model order reduction is as follows. First of all, apply a (pre-)coordinate transformation (11) satisfying −1
M = N ((1 − α)R22 + αQ22 )
T
((1 − α)R12 + αQ12 ) . (13)
Then apply the partial balanced realization. Let us describe the partially balanced realization by the following equation. x˙ = A11 x + A¯12 ζa + A¯13 ζb + B1 u ζ˙ = A¯ x + A¯ ζ + A¯ ζ + B ¯2 u a 21 22 a 23 b (14) ¯3 u ζ˙b = A¯31 x + A¯32 ζa + A¯33 ζb + B y = C1 x + C¯2 ζa + C¯3 ζb Here the division of the state is ζ = (ζaT , ζbT )T , ζa = (ζ1 , · · · , ζk )T , ζb = (ζk+1 , · · · , ζn )T , k < n. Suppose that the partial states ζa and ζb have bigger and smaller effect to the input-output behavior, respectively. The system matrices ¯ B, ¯ C), ¯ and the Gramians P¯ and Q ¯ are divided according (A, to the division of the state. ζ ζ ζ ζ Q11 Qζ12 Qζ13 R11 R12 R13 ζT ¯ = QζT Σa P¯ −1 = R12 0 0 ,Q Σ−1 a 12 ζT ζT −1 0 Σb Q13 0 Σb R13 (15) ζ Here Rij ’s and Qζij ’s are matrices with the appropriate dimensions. Σa = diag(σ1 , · · · , σk ), and Σb = diag(σk+1 , · · · , σn ) hold. The reduced order model can be obtained by substituting ζb ≡ 0 for Equation (14). Stability of the corresponding feedback system is proven in the following theorem. Theorem 2: [3] Suppose that Assumption 1 holds. The reduced order systems obtained by substituting ζb ≡ 0 for Equation (14) is stable. Furthermore, if the pseudo Gramians satisfying (3) are used instead of the conventional Gramians, then the resultant reduced order system is asymptotically stable. Proof: See [3]. Finally, let us summarize the proposed model order reduction procedure as follows.
Procedure of the partial model order reduction 1) Apply the pre-coordinate transformation in Eq. (11) with the matrix M given in Eq. (13). 2) Compute subspace balanced realization using the parˆ tial Gramians Psub and Qsub with respect to ξ.
3) Truncating the less important state ζ b , we can obtain the reduced order system. III. C ONTROLLER REDUCTION BASED ON SUBSPACE BALANCED REALIZATION
This section gives the main result of the present paper, that is, controller reduction based on subspace balanced realization. The basic idea is to apply the model order reduction method in the state-feedback case explained in the previous section to output feedback systems. This method allows us to obtain a reduced order output feedback controller which stabilizes the plant. Let us consider the feedback system with P and K as in the previous section. suppose that P and K denote the plant and the controller, respectively. Then the proposed method in Section II directly gives a controller reduction method (in the state-feedback case). However, in order to apply the feedback M x, we need to know the state x of the plant P . The information of the state x is not available in the output feedback case. In this section, we consider a controller reduction problem as shown in Fig. 2. Here G and K denote a generalized plant and a stabilizing controller. The signals w(t) ∈ Rm , z(t) ∈ Rr , u(t) ∈ Rp and y(t) ∈ Rq denote the external signal, the controlled signal, the input signal and the output signal, respectively. Let us denote the states of G w u
/ /G
z y
w
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K o (i) Original system Fig. 2.
u
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z
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y
(ii) Reduced system
Partial model reduction for output feedback systems
and K by x(t) ∈ Rl and ξ(t) ∈ Rn . Then the dynamics in Fig. 2 (i) can be described by x˙ = A1 x + B11 w + B12 u G: (16) z = C11 x + D11 u + D13 w y = C12 x + D12 u + D14 w { ξ˙ = A2 ξ + B2 y K: . (17) u = C2 ξ + D2 y Here we suppose that the feedback system is well-posed. The problem considered here is to obtain a reduced order controller Kred in such a way that the feedback system in Fig.2 (ii) is stable. Let ξˆr ∈ Rk , k < n the state of the reduced controller. ˙ ˆ2 y ξˆr = Aˆ2r ξˆr + B r ˆ Kred (18) ξ = E1 y + E2 ξr u = C2 ξ + D2 y A. State estimation via a state-observer As stated in the previous section, we need to use the information of the state of the plant (more precisely, the feedback M x in Fig.3 (ii) is needed) when we apply the model order reduction procedure in Section II to the output feedback system.
Let us consider a feedback system in Fig.2 (i) where the state x of the system G is not measurable. Design a minimum-dimensional state observer Ob with a state x ˇ ∈ Rl−q . Here x ¯ is the estimation of the state x of G produced by x ˇ. Note that the dimension of the controller becomes higher by adding the state observer Ob. Therefore we need to reduce the order of the controller by more than that of the state observer. In the figure, CK denotes a matrix producing the control input u from ξ and y. Here the dynamics of the observer Ob is described as follows. { x ˇ˙ = A3 x ˇ + B31 y + B32 u Ob : (19) x ¯ = C3 x ˇ + D31 y Fig.3 denotes the feedback system with the state observer. Let us employ the following assumption for this system accordingly to Assumption 1. Assumption 2: The feedback system (16) and (17) with the observer (19) is asymptotically stable, controllable and observable. Here let us apply the model order reduction method proposed in Section II to the feedback system in Fig.3 to reduce the order of the controller Kξ . The procedure converting (to reduce its order) the system in Fig.3 (i) into that in Fig.3 (ii) corresponds to converting the system in Fig.3 into that in Fig.3 (i). Since Fig.3 (i) uses a state feedback of x which is not available, we use the formulation in Fig.3 (ii) where the estimated value x ¯ is used instead of the true state x. Here w •
CK O ξ •O o
+
−
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M o ˆξ o K
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(i) State feedback Fig. 3.
/ •
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(ii) Output feedback
State and output feedback systems with observers
B. Model order reduction of output feedback controllers This section focuses on model order reduction of the output feedback controller in Fig.3 (ii). Let χ denote χ = (xT , x ˇT )T ∈ R2l−q . Then the whole dynamics with an additional controlled output z is described as follows. χ˙ = A11 χ + A12 ξ + B1 w (20) ξ˙ = A21 χ + A22 ξ + B2 w z = C1 χ + C2 ξ + Dw Equation (20) can be obtained by substituting x for χ in Equation (4). Therefore, stability of the reduced order system of Fig.3 (i) can be proven in a similar way to the state feedback case. The detailed procedure is given below. Let us denote the controllability and the observability Gramians by P ∈ R(2l−q+n)×(2l−q+n) and Q ∈ R(2l−q+n)×(2l−q+n) , respectively. Then the controllability and observability functions Lc and Lo are given below. 1 T 1 χ R11 χ + χT R12 ξ + ξ T R22 ξ (21) 2 2 1 1 Lo (χ, ξ) = χT Q11 χ + χT Q12 ξ + ξ T Q22 ξ (22) 2 ] [ 2 ] [ Q11 Q12 R11 R12 0 0, Q = P −1 = T R22 QT12 Q22 R12 Lc (χ, ξ) =
Here the matrices P −1 and Q are divided accordingly to the states χ and ξ. Next, we introduce a coordinate transformation before applying the balanced truncation by which the stability of the feedback system is preserved. Let us consider the following Lyapunov function candidate as in Equation (12). V (χ, ξ) := (1 − α)Lc (χ, ξ) + αLo (χ, ξ), 0≤α≤1
Positive definiteness of the controllability and observability functions Lc , Lo 0 in Assumption 2 implies that of the function V 0. Here let us employ the following coordinate transformation θ(·) in such a way that ˆ = χT Vχ χ + ξˆT Vξ ξˆ V (θ−1 (χ, ξ))
the matrix M is that in Equation (13). How to compute it will be explained later. We can prove the following lemma concerning the stability of the systems in Fig.3. Lemma 1: The system in Fig.3 (i) is (asymptotically) stable if and only if so is the system in Fig.3 (ii). Proof: Lemma can be proven by directly comparing the eigenvalues of the closed-loop system. The proof is omitted for the reason of space. Assumption 2 implies that the system in Fig.3 (i) is asymptotically stable. Then Lemma 1 proves asymptotic stability of the system in Fig.3 (ii). By applying the model order reduction procedure in Section II to the system in Fig.3 (ii) and reduce the order of the controller, then we can obtain a reduced order controller which stabilizes the original plant system. The detail of this procedure will be explained in the following section.
(23)
(24)
holds. This equation can be achieved by any coordinate transformation in the following form. ˆ = θ(χ, ξ) = (χ, M χ + N ξ), (χ, ξ) −1
M = N ((1−α)R22 + αQ22 )
T
((1−α)R12 + αQ12 )
The partial Gramian with respect to the state of the controller ξ reduces to −1 Psub = R22 0, Qsub = Q22 0.
(25)
By solving the balancing equation in (10) we obtain the singular values σ ∈ R with respect to the state ξ and the corresponding coordinate transformation achieving the (partial) balanced realization. Converting the state-space realization with respect to ξˆ into a partial balanced realization with respect to ζ and truncating subspace of ζ, one can obtain a reduced order output feedback controller. We call this
realization as subspace balanced realization. This procedure can be summarized as follows. Theorem 3: Suppose that Assumption 2 holds. Then the closed-loop system with the plant G and the corresponding reduced order output feedback controller Kred is stable. Furthermore, if the pseudo Gramians satisfying the LMIs (3) is used instead of the conventional Gramians in the balanced truncation procedure, then the closed-loop system is asymptotically stable. Proof: Lemma 2 with Assumption 2 proves the (asymptotic) stability of the closed-loop system in Fig.3 (i) substiˆ ξ by its reduced order version (when the generalized tuted K Gramians are used, respectively). Furthermore, the stability of the above system is equivalent to that of the closedˆ ξ by its reduced order loop system in Fig.3 (ii) substituted K version. This proves the theorem. Remark 2: Increasing the convergence speed of the estimation error x − x ¯ of the observer Ob, the behavior of the closed-loop system converges to that of the corresponding state-feedback system.
This section gives a numerical example. Let us consider a two mass spring damper system depicted in Fig.4. The
Two carts system
dynamics of this system can be described by the following equation. m1 q¨1 = −c1 (q˙1 − q˙2 ) − k1 (q1 − q2 ) + u m2 q¨2 = −c2 q˙2 − k2 q2
(26) (27)
Here the physical parameters and variables are summarized in Table I.
description displacement of cart 1 displacement of cart 2 input force to cart 1 mass of cart 1 mass of cart 2 elastic constant for cart 1 elastic constant for cart 2 viscous resistance constant for cart 1 viscous resistance constant for cart 2 reference input
Aξ =
0.0 0.0 −96.0 4.0 0.0
4.52 × 103 −78.3 −4.61 × 104 −2.23 × 103 0.0
1.0 0.0 −25.0 1.0 0.0
0.0 1.0 −109.0 −1.0 0.0
0.0 0.0 176.0 0.0 −0.01
Bξ1 = [−4.52 × 103 , 78.3, 4.59 × 104 , 2.22 × 103 , −1.0]T
Next, let us apply the proposed model order reduction procedure to this system. First of all, we need to design a minimum dimensional state observer Ob as in Fig.3. The observer gain is selected as gm = (−1.6 × 103 , 9.05 × 103 , 89.7)T and the corresponding poles of the estimation error are {−31.5, −31.25, −31.0}. The state of this observer is denoted by x ˇ ∈ R3 . Then its dynamics is described as follows. { ˇx + B ˇ1 y + B ˇ2 u x ˇ˙ = Aˇ (29) Ob : ˇ x ¯ = Cˇ x ˇ + Dy Here x ¯ ∈ R4 denotes the estimation of the state x of the plant G. The parameters in Equation (29) is given as follows. [ ] 3 ˇ= A
0.0 −8.0 4.0
value variable variable variable 1.0 1.0 4.0 4.0 2.0 1.0 variable
Here the state and the output are defined by x = (x1 , x2 , x3 , x4 )T = (q1 , q2 , q˙1 , q˙2 )T and y = q2 . Next, let us design an output feedback stabilizing controller as depicted in Fig.2 (i) for this system. In the figure, G denotes the
1.0 −3.0 1.0
1.6 × 10 −9.05 × 103 −90.7
ˇ1 = [1.53 × 105 , −8.26 × 105 , −5.5 × 103 ]T B ˇ2 = [ 0.0, 1.0, 0.0 ]T B
TABLE I P HYSICAL PARAMETERS parameter q1 q2 u m1 m2 k1 k2 c1 c2 r
The parameters in Equation (28) are obtained as follows.
Bξ2 = [ 0.0, 0.0, 0.0, 0.0, 1.0 ]T
IV. N UMERICAL EXAMPLE
Fig. 4.
plant, K denotes the output feedback controller designed later, d denotes the external noise. The external signals are defined as w = (r, d)T , z = q2 , and y = (q2 , r)T . The controller K is made of a state observer, a pseudo integrator KI /(s + ε) and a state feedback KP ξ. The observer gain gs and the feedback gains KP and KI are defined as gs = (−4.52×103 , 78.3, 4.59×104 , 2.22×103 )T , ε = 10−2 , KP = (88, 176, 22, 110), KI = 176. Then the dynamics of the controller is described as follows. { ξ˙ = Aξ ξ + Bξ1 y + Bξ2 r K: (28) u = [−KP KI ]ξ + r
1.0 0.0 0.0 ˇ = 0.0 0.0 0.0 C 0.0 1.0 0.0 0.0 0.0 1.0 ˇ = [−1.6 × 103 , 1.0, 9.05 × 103 , 89.7]T D
Comparison of three different feedback systems are given. One is the feedback system with the original controller, another is that with the reduced order controller obtained by the proposed method, and the other is that with another reduced order controller obtained by simply applying conventional balanced truncation to the original controller directly. Fig.5 shows the bode diagrams of the controllers (from the input q2 to the output u). Fig.6 shows that of the feedback systems (from the input r to the output z). Fig.7 shows their time responses. In the figures, the (blue) plus marks + denote the results with the original controller,
Bode Diagram
Magnitude (dB)
80 60 40 20
Phase (deg)
0 360 270
0.7 0.6 0.5 0.4 0.3 Original model Proposed model Traditional model
0.2 0.1 0 0
2
4
6
8
10
t[sec]
Fig. 7.
The time response (4 dim.)
method derives a static (0 dimensional) stabilizing controller with the state observer whereas the conventional method does not give any stabilizing controller. These results show that the proposed method gives a reduced order controller which preserves the stability of the closed-loop system and the behavior of a certain input-output map of the closed-loop system. V. C ONCLUSION This paper proposes a novel balanced realization method for a controller reduction problem. It characterizes a partial balanced realization for a feedback system with respect to a prescribed input-output map. It can be directly applied to a controller reduction problem and the resulting reduced order controller stabilizes the original plant. Furthermore, numerical examples demonstrate the effectiveness of the proposed method. The authors believe that the present paper gives the first result on controller reduction based on balanced truncation which maintains the stability of the original feedback system. R EFERENCES
0 −1 10
0
10
1
10
Frequency (rad/sec)
Fig. 5.
Bode plots of the reduced controllers (4 dim.)
Bode Diagram
0 −10 Magnitude (dB)
0.8
180 90
−20 −30 −40 −50 0
Phase (deg)
1 0.9
displacement[m]
the (green) times marks × denote those with the proposed reduced controller, and the (red) circle marks ◦ denote those with the conventional reduced controller. The dimensions of the original, the proposed and the conventional controllers are 5, 4 and 4. The Hankel singular values for the above two reduction methods (the proposed and the conventional) are computed as follows: the Hankel singular values for the subspace balanced realization are σ1 = 7.17 × 10−2 , σ2 = 3.72 × 10−4 , σ3 = 7.00 × 10−5 , σ4 = 9.48 × 10−6 , and 2.83 × 10−6 ; the conventional Hankel singular values are σ1 = 7.10 × 103 , σ2 = 5.86 × 103 , σ3 = 5.42 × 103 , σ4 = 1.30 × 103 , and σ5 = 9.52. The original controller is 4 dimensional and we want to obtain a reduced order controller with 3 dimensional state-space. Since the designed minimum dimensional state observer is 3 dimensional, the proposed method needs to obtain 1 dimensional reduced order system in order to obtain a 4 dimensional reduce order controller. The double lines in the table denote the truncated dimensions, that is, the proposed method derives a 1 dimensional controller (with a 3 dimensional state observer) from the 5 dimensional original controller and the conventional method derives a 4 dimensional controller from a 5 dimensional original one. Please note that both methods yield 4 dimensional controllers.
−90 −180 −270 −1 10
0
10
1
10
Frequency (rad/sec)
Fig. 6.
Bode plots of the feedback systems (4 dim.)
Fig.5 shows that the conventional method gives the better approximation for the controller reduction itself. Figs.5-7 show that the proposed method gives the better approximation for the whole closed-loop system. Furthermore, if we try to obtain 3 dimensional controller, then the proposed
[1] A. C. Antoulas. Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia, 2005. [2] D. F. Enns. Model reduction with balanced realizations: an error bound and frequency weighted generalization. In Proc. 23rd IEEE Conf. on Decision and Control, pages 127–132, 1984. [3] K. Fujimoto, S. Ono, and Y. Hayakawa. Balanced truncation for linear interconnected systems: the state feedback case. To appear in Proc. Mathematical Theory and Network Systems, 2010. [4] L. Li and F. Paganini. Structured coprime factor model reduction based on LMIs. Automatica, 41:145–151, 2005. [5] Y. Liu, B. D. O. Anderson, and U.-L. Ly. Coprime factorization controller reduction procedure with bezout identity induced frequency weighting. Automatica, 26:233–249, 1990. [6] B. C. Moore. Principal component analysis in linear systems: controllability, observability and model reduction. IEEE Trans. Autom. Contr., AC-26:17–32, 1981. [7] G. Obinata and B. D. O. Anderson. Model Reduction for Control System Design. Springer-Verlag, London, 2001. [8] H. Sandberg and R. M. Murray. Model reduction of interconnected linear systems using structured gramians. In Proc. 47th IEEE Conf. on Decision and Control, pages 8725–8730, 2008. [9] H. Sandberg and R. M. Murray. Model reduction of interconnected linear systems. To appear in Optim. Control Appl. Meth., 2009. [10] V. Sreeram, B. D. O. Anderson, and A. G. Madievski. New results on frequency weighted balanced reduction technique. In Proc. American Control Conference, pages 4004–4009, 1995. [11] P. M. Wortelboer, M. Steinbuch, and O. Bosgra. Iterative model and controller reduction using closed-loop balancing, with application to a compact disc mechanism. Int. J. Robust and Nonlinear Control, 9:123–142, 1999.
I. I NTRODUCTION Recently, the development of CAD softwares enable one to execute controller design and stability analysis for large scale systems. Accordingly, the importance of the model order reduction increases. So far, many techniques for model order reduction were proposed. See e.g. [7], [1]. Among them, balanced realization and its application to model order reduction is known as one of the most effective model order reduction methods [6]. Balanced realization is a statespace realization whose states are balanced in a sense that the relationship between the input and the output, and the relationship among the coordinate axes are balanced. On these coordinates, one can easily eliminate less important states and consequently obtain a reduced order model. This model order reduction method is called balanced truncation. There are many extensions of this result. In this paper, we consider a controller reduction problem. Since conventional balanced truncation is only applicable to an open-loop system, it was difficult to reduce the dimension of the controller and maintain the stability of the whole closed-loop system. This problem is called controller reduction and investigated within the framework of weighted balanced realization [2], [10]. Section 7.6 of [1] and Section 4 of [7] discuss the basic ideas and the references on this research topic. For instance, Liu et al. provides a framework for controller reduction using coprime factorizations [5], and Wortelboer et al. proposes a method to choose appropriate frequency weights for controller reduction [11]. Although the existing results are enough effective, it is difficult to ensure the stability of the resulting feedback system with the original plant and the reduced order controller. Recently, Li [4] and Sandberg et al. [8], [9] proposed a variation of balanced realization applicable to a feedback system. Although these method can solve a class of controller reduction problems, K. Fujimoto is with Department of Mechanical Science and Engineeirng, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
[email protected]
there are some restrictions on the system to be applied. A similar idea was proposed by the authors [3] which derives a controller reduction method in the state feedback case. This paper is devoted to its extension to the output feedback case using state observers. It also ensures the stability of the resulting closed loop system with the original plant and the reduced order controller. The authors believe that the present paper gives the first result on controller reduction based on balanced truncation which maintains the stability of the original feedback system. Furthermore, a numerical example will demonstrate its effectiveness compared to the conventional one. II. M ODEL ORDER REDUCTION FOR INTERCONNECTED SYSTEMS VIA STATE FEEDBACKS
This section refers to conventional model order reduction method based on balanced truncation [7] and the authors former results on partial balanced realization for state-feedback systems [3]. A. Balanced truncation Consider the following linear system. { x˙ = Ax + Bu, x(0) = 0 y = Cx Here x(t) ∈ Rn , u(t) ∈ Rm , and y(t) ∈ Rr . The system matrix A is Hurwitz. The controllability and observability Gramians P and Q are the symmetric solutions to the following Lyapunov equations. AP + P AT + BB T = 0, AT Q + QA + C T C = 0 (1) The positive definiteness of the Gramians P and Q is equivalent to the controllability and observability of the system. In addition, they are quantitative indices of the controllability and observability of the system. A state-space realization on which the following equation holds is called a balanced realization. P = Q = diag(σ1 , · · · , σn ), σ1 ≥ · · · ≥ σn
(2)
Here the relation between the input and the output, and the relation among the coordinate axes are balanced. The scalars σi ’s indicating the importance of the states are called Hankel singular values. Remark 1: Instead of using the Lyapunov equations (1), we can use the following Lyapunov inequalities (LMIs). ˇ + QA ˇ + C T C ≺ 0 (3) APˇ + Pˇ AT + BB T ≺ 0, AT Q
(Non-unique) symmetric solutions to these inequalities are called pseudo controllability Gramian and pseudo observability Gramian, respectively. Furthermore, the Gramians P and Q define the controllability and observability functions Lc and Lo as Lc (x) = (1/2)xT P −1 x, and Lo (x) = (1/2)xT Q x. The procedure of model order reduction based on balanced realization (balanced truncation) is as follows. Suppose that the Hankel singular values defined in Equation (2) satisfy σk σk+1 , 1 ≤ k ≤ n − 1. This means that the set of the states x1 , · · · , xk have a bigger effect to the input-output map than the rest xk+1 , · · · , xn . Therefore we can obtain a k-dimensional reduced order model by substituting 0 for the unimportant states xk+1 , · · · , xn . B. Problem setting This section focuses on balanced realization for interconnected systems via state feedbacks and its application to model oder reduction. The biggest advantage of the proposed method is that the reduced order interconnected system preserves the stability of the original one. Fig. 1 shows the configuration of the proposed model order reduction problem. Here P and K denote state-space realizations with the state x and that ξ, respectively. C denotes a matrix determining the output signal y from x and ξ. •
u
/ /
ξ
P
x
•
/ /
C
y
/
•
u ξ
/
/
P /
•
Ko /
(i) Original system Fig. 1.
x
•
/ /
C
y
/
•
Kred o
(ii) Reduced system Outline
Suppose that the feedback system with P and K as in Fig.1 (i) is stable. The problem considered here is to obtain a reduced order controller Kred in such a way that the feedback system with P and Kred as in Fig.1 (ii) maintains its stability and furthermore the input output map u-y is also preserved. C. Subspace balanced realization Let us consider the following linear system depicted as in Fig.1 (i). x˙ = A11 x + A12 ξ + B1 u ξ˙ = A21 x + A22 ξ + B2 u y = C1 x + C2 ξ
(4)
Here x(t) ∈ Rl is the state of the plant (the state to be preserved), ξ(t) ∈ Rn is the state of the controller (the state to be reduced), and u(t) ∈ Rm and y(t) ∈ Rr are the input and the output, respectively. In order to reduce the order of the controller K, let us replace its state ξ by a lower order vector ξˆr ∈ Rk , k < n
and obtain a reduced order controller Kred in Equation (5). x˙ = A11 x + A12 ξ + B1 u ˆ˙ ˆ2 u ξr = Aˆ21r x + Aˆ22r ξˆr + B r (5) ξ = E1 x + E2 ξˆr y =C x+C ξ 1
2
Here we employ the following assumption for this system. Assumption 1: The system (4) is asymptotically stable, controllable and observable. The controllability and observability functions Lc and Lo of the system (4) are defined as follows. 1 1 Lc (x, ξ) = xT R11 x + xT R12 ξ + ξ T R22 ξ (6) 2 2 1 1 (7) Lo (x, ξ) = xT Q11 x + xT Q12 ξ + ξ T Q22 ξ 2 2 Here the matrices Pij ’s and Qij ’s are defined by the controllability and observability Gramians P ∈ R(l+n)×(l+n) and Q ∈ R(l+n)×(l+n) in the following way where the division of the matrices corresponds to the division of the states x and ξ. ] [ ] [ R11 R12 Q11 Q12 −1 (8) , Q= P = T R22 QT12 Q22 R12 Since we are interested in the effect of the partial state ξ to the input-output behavior of the whole system, it is natural to define Gramians describing the properties of the partial state ξ. To this end, let us define the partial controllability Gramian Psub and the partial observability Gramian Qsub respectively as follows. Definition 1: Partial controllability and observability Gramians Psub and Qsub are defined as follows. −1 Psub := R22 , Qsub := Q22 (9) Note that if P, Q 0 then Psub , Qsub 0. We are interested in the Hankel singular values with respect to the partial Gramians Psub and Qsub . That is, we should obtain the Hankel singular value σ ∈ R and the corresponding coordinate axis ζ ∈ Rn by solving the following equation for singular value analysis.
Psub Qsub ζ = σζ
(10)
Then the partial Gramians are balanced (diagonalized) on the new coordinates ζ’s. Consequently we can obtain a reduced order controller Kred of K by balanced truncation. A question arises now: Is the feedback system (5) with the reduced order controller Kred stable? The answer is generally no. Here we use a trick to ensure the stability of the feedback system (5) by employing a freedom in choosing the partial state ξ itself. For this purpose, the following property is useful. Theorem 1: [3] Consider the system (4). For any linear coordinate transformation in the following form, the Hankel singular values based on the partial Gramian with respect to ξˆ are equivalent to that with respect to ξ. ˆ = θ(x, ξ) := (x, M x + N ξ), (x, ξ) M ∈ Rn×l , N ∈ Rl×l , det N 6= 0
(11)
Proof: See [3]. This theorem suggests that the freedom in choosing the coordinate transformation θ(·) can be used for the partial balanced realization and truncation. D. Partial balanced truncation with stability Let us consider a candidate Lyapunov function V V (x, ξ) := (1 − α)Lc (x, ξ) + αLo (x, ξ), 0 ≤ α ≤ 1 (12) Here Lc and Lo are the controllability and observability functions given in Equations (6) and (7). Assumption 1 implies V 0. The procedure of model order reduction is as follows. First of all, apply a (pre-)coordinate transformation (11) satisfying −1
M = N ((1 − α)R22 + αQ22 )
T
((1 − α)R12 + αQ12 ) . (13)
Then apply the partial balanced realization. Let us describe the partially balanced realization by the following equation. x˙ = A11 x + A¯12 ζa + A¯13 ζb + B1 u ζ˙ = A¯ x + A¯ ζ + A¯ ζ + B ¯2 u a 21 22 a 23 b (14) ¯3 u ζ˙b = A¯31 x + A¯32 ζa + A¯33 ζb + B y = C1 x + C¯2 ζa + C¯3 ζb Here the division of the state is ζ = (ζaT , ζbT )T , ζa = (ζ1 , · · · , ζk )T , ζb = (ζk+1 , · · · , ζn )T , k < n. Suppose that the partial states ζa and ζb have bigger and smaller effect to the input-output behavior, respectively. The system matrices ¯ B, ¯ C), ¯ and the Gramians P¯ and Q ¯ are divided according (A, to the division of the state. ζ ζ ζ ζ Q11 Qζ12 Qζ13 R11 R12 R13 ζT ¯ = QζT Σa P¯ −1 = R12 0 0 ,Q Σ−1 a 12 ζT ζT −1 0 Σb Q13 0 Σb R13 (15) ζ Here Rij ’s and Qζij ’s are matrices with the appropriate dimensions. Σa = diag(σ1 , · · · , σk ), and Σb = diag(σk+1 , · · · , σn ) hold. The reduced order model can be obtained by substituting ζb ≡ 0 for Equation (14). Stability of the corresponding feedback system is proven in the following theorem. Theorem 2: [3] Suppose that Assumption 1 holds. The reduced order systems obtained by substituting ζb ≡ 0 for Equation (14) is stable. Furthermore, if the pseudo Gramians satisfying (3) are used instead of the conventional Gramians, then the resultant reduced order system is asymptotically stable. Proof: See [3]. Finally, let us summarize the proposed model order reduction procedure as follows.
Procedure of the partial model order reduction 1) Apply the pre-coordinate transformation in Eq. (11) with the matrix M given in Eq. (13). 2) Compute subspace balanced realization using the parˆ tial Gramians Psub and Qsub with respect to ξ.
3) Truncating the less important state ζ b , we can obtain the reduced order system. III. C ONTROLLER REDUCTION BASED ON SUBSPACE BALANCED REALIZATION
This section gives the main result of the present paper, that is, controller reduction based on subspace balanced realization. The basic idea is to apply the model order reduction method in the state-feedback case explained in the previous section to output feedback systems. This method allows us to obtain a reduced order output feedback controller which stabilizes the plant. Let us consider the feedback system with P and K as in the previous section. suppose that P and K denote the plant and the controller, respectively. Then the proposed method in Section II directly gives a controller reduction method (in the state-feedback case). However, in order to apply the feedback M x, we need to know the state x of the plant P . The information of the state x is not available in the output feedback case. In this section, we consider a controller reduction problem as shown in Fig. 2. Here G and K denote a generalized plant and a stabilizing controller. The signals w(t) ∈ Rm , z(t) ∈ Rr , u(t) ∈ Rp and y(t) ∈ Rq denote the external signal, the controlled signal, the input signal and the output signal, respectively. Let us denote the states of G w u
/ /G
z y
w
/
K o (i) Original system Fig. 2.
u
/ /G Kred o
z
/
y
(ii) Reduced system
Partial model reduction for output feedback systems
and K by x(t) ∈ Rl and ξ(t) ∈ Rn . Then the dynamics in Fig. 2 (i) can be described by x˙ = A1 x + B11 w + B12 u G: (16) z = C11 x + D11 u + D13 w y = C12 x + D12 u + D14 w { ξ˙ = A2 ξ + B2 y K: . (17) u = C2 ξ + D2 y Here we suppose that the feedback system is well-posed. The problem considered here is to obtain a reduced order controller Kred in such a way that the feedback system in Fig.2 (ii) is stable. Let ξˆr ∈ Rk , k < n the state of the reduced controller. ˙ ˆ2 y ξˆr = Aˆ2r ξˆr + B r ˆ Kred (18) ξ = E1 y + E2 ξr u = C2 ξ + D2 y A. State estimation via a state-observer As stated in the previous section, we need to use the information of the state of the plant (more precisely, the feedback M x in Fig.3 (ii) is needed) when we apply the model order reduction procedure in Section II to the output feedback system.
Let us consider a feedback system in Fig.2 (i) where the state x of the system G is not measurable. Design a minimum-dimensional state observer Ob with a state x ˇ ∈ Rl−q . Here x ¯ is the estimation of the state x of G produced by x ˇ. Note that the dimension of the controller becomes higher by adding the state observer Ob. Therefore we need to reduce the order of the controller by more than that of the state observer. In the figure, CK denotes a matrix producing the control input u from ξ and y. Here the dynamics of the observer Ob is described as follows. { x ˇ˙ = A3 x ˇ + B31 y + B32 u Ob : (19) x ¯ = C3 x ˇ + D31 y Fig.3 denotes the feedback system with the state observer. Let us employ the following assumption for this system accordingly to Assumption 1. Assumption 2: The feedback system (16) and (17) with the observer (19) is asymptotically stable, controllable and observable. Here let us apply the model order reduction method proposed in Section II to the feedback system in Fig.3 to reduce the order of the controller Kξ . The procedure converting (to reduce its order) the system in Fig.3 (i) into that in Fig.3 (ii) corresponds to converting the system in Fig.3 into that in Fig.3 (i). Since Fig.3 (i) uses a state feedback of x which is not available, we use the formulation in Fig.3 (ii) where the estimated value x ¯ is used instead of the true state x. Here w •
CK O ξ •O o
+
−
o
/ u
z
/G x
ξˆ
M o ˆξ o K
• y
/ Ob o x ¯ x ˇ
(i) State feedback Fig. 3.
/ •
CK O +
•O o
o
w • u
x
ξ −
/ /G
ξˆ
M o ˆξ o K
z
/
• y x ¯
/ Ob o
•
x ˇ
(ii) Output feedback
State and output feedback systems with observers
B. Model order reduction of output feedback controllers This section focuses on model order reduction of the output feedback controller in Fig.3 (ii). Let χ denote χ = (xT , x ˇT )T ∈ R2l−q . Then the whole dynamics with an additional controlled output z is described as follows. χ˙ = A11 χ + A12 ξ + B1 w (20) ξ˙ = A21 χ + A22 ξ + B2 w z = C1 χ + C2 ξ + Dw Equation (20) can be obtained by substituting x for χ in Equation (4). Therefore, stability of the reduced order system of Fig.3 (i) can be proven in a similar way to the state feedback case. The detailed procedure is given below. Let us denote the controllability and the observability Gramians by P ∈ R(2l−q+n)×(2l−q+n) and Q ∈ R(2l−q+n)×(2l−q+n) , respectively. Then the controllability and observability functions Lc and Lo are given below. 1 T 1 χ R11 χ + χT R12 ξ + ξ T R22 ξ (21) 2 2 1 1 Lo (χ, ξ) = χT Q11 χ + χT Q12 ξ + ξ T Q22 ξ (22) 2 ] [ 2 ] [ Q11 Q12 R11 R12 0 0, Q = P −1 = T R22 QT12 Q22 R12 Lc (χ, ξ) =
Here the matrices P −1 and Q are divided accordingly to the states χ and ξ. Next, we introduce a coordinate transformation before applying the balanced truncation by which the stability of the feedback system is preserved. Let us consider the following Lyapunov function candidate as in Equation (12). V (χ, ξ) := (1 − α)Lc (χ, ξ) + αLo (χ, ξ), 0≤α≤1
Positive definiteness of the controllability and observability functions Lc , Lo 0 in Assumption 2 implies that of the function V 0. Here let us employ the following coordinate transformation θ(·) in such a way that ˆ = χT Vχ χ + ξˆT Vξ ξˆ V (θ−1 (χ, ξ))
the matrix M is that in Equation (13). How to compute it will be explained later. We can prove the following lemma concerning the stability of the systems in Fig.3. Lemma 1: The system in Fig.3 (i) is (asymptotically) stable if and only if so is the system in Fig.3 (ii). Proof: Lemma can be proven by directly comparing the eigenvalues of the closed-loop system. The proof is omitted for the reason of space. Assumption 2 implies that the system in Fig.3 (i) is asymptotically stable. Then Lemma 1 proves asymptotic stability of the system in Fig.3 (ii). By applying the model order reduction procedure in Section II to the system in Fig.3 (ii) and reduce the order of the controller, then we can obtain a reduced order controller which stabilizes the original plant system. The detail of this procedure will be explained in the following section.
(23)
(24)
holds. This equation can be achieved by any coordinate transformation in the following form. ˆ = θ(χ, ξ) = (χ, M χ + N ξ), (χ, ξ) −1
M = N ((1−α)R22 + αQ22 )
T
((1−α)R12 + αQ12 )
The partial Gramian with respect to the state of the controller ξ reduces to −1 Psub = R22 0, Qsub = Q22 0.
(25)
By solving the balancing equation in (10) we obtain the singular values σ ∈ R with respect to the state ξ and the corresponding coordinate transformation achieving the (partial) balanced realization. Converting the state-space realization with respect to ξˆ into a partial balanced realization with respect to ζ and truncating subspace of ζ, one can obtain a reduced order output feedback controller. We call this
realization as subspace balanced realization. This procedure can be summarized as follows. Theorem 3: Suppose that Assumption 2 holds. Then the closed-loop system with the plant G and the corresponding reduced order output feedback controller Kred is stable. Furthermore, if the pseudo Gramians satisfying the LMIs (3) is used instead of the conventional Gramians in the balanced truncation procedure, then the closed-loop system is asymptotically stable. Proof: Lemma 2 with Assumption 2 proves the (asymptotic) stability of the closed-loop system in Fig.3 (i) substiˆ ξ by its reduced order version (when the generalized tuted K Gramians are used, respectively). Furthermore, the stability of the above system is equivalent to that of the closedˆ ξ by its reduced order loop system in Fig.3 (ii) substituted K version. This proves the theorem. Remark 2: Increasing the convergence speed of the estimation error x − x ¯ of the observer Ob, the behavior of the closed-loop system converges to that of the corresponding state-feedback system.
This section gives a numerical example. Let us consider a two mass spring damper system depicted in Fig.4. The
Two carts system
dynamics of this system can be described by the following equation. m1 q¨1 = −c1 (q˙1 − q˙2 ) − k1 (q1 − q2 ) + u m2 q¨2 = −c2 q˙2 − k2 q2
(26) (27)
Here the physical parameters and variables are summarized in Table I.
description displacement of cart 1 displacement of cart 2 input force to cart 1 mass of cart 1 mass of cart 2 elastic constant for cart 1 elastic constant for cart 2 viscous resistance constant for cart 1 viscous resistance constant for cart 2 reference input
Aξ =
0.0 0.0 −96.0 4.0 0.0
4.52 × 103 −78.3 −4.61 × 104 −2.23 × 103 0.0
1.0 0.0 −25.0 1.0 0.0
0.0 1.0 −109.0 −1.0 0.0
0.0 0.0 176.0 0.0 −0.01
Bξ1 = [−4.52 × 103 , 78.3, 4.59 × 104 , 2.22 × 103 , −1.0]T
Next, let us apply the proposed model order reduction procedure to this system. First of all, we need to design a minimum dimensional state observer Ob as in Fig.3. The observer gain is selected as gm = (−1.6 × 103 , 9.05 × 103 , 89.7)T and the corresponding poles of the estimation error are {−31.5, −31.25, −31.0}. The state of this observer is denoted by x ˇ ∈ R3 . Then its dynamics is described as follows. { ˇx + B ˇ1 y + B ˇ2 u x ˇ˙ = Aˇ (29) Ob : ˇ x ¯ = Cˇ x ˇ + Dy Here x ¯ ∈ R4 denotes the estimation of the state x of the plant G. The parameters in Equation (29) is given as follows. [ ] 3 ˇ= A
0.0 −8.0 4.0
value variable variable variable 1.0 1.0 4.0 4.0 2.0 1.0 variable
Here the state and the output are defined by x = (x1 , x2 , x3 , x4 )T = (q1 , q2 , q˙1 , q˙2 )T and y = q2 . Next, let us design an output feedback stabilizing controller as depicted in Fig.2 (i) for this system. In the figure, G denotes the
1.0 −3.0 1.0
1.6 × 10 −9.05 × 103 −90.7
ˇ1 = [1.53 × 105 , −8.26 × 105 , −5.5 × 103 ]T B ˇ2 = [ 0.0, 1.0, 0.0 ]T B
TABLE I P HYSICAL PARAMETERS parameter q1 q2 u m1 m2 k1 k2 c1 c2 r
The parameters in Equation (28) are obtained as follows.
Bξ2 = [ 0.0, 0.0, 0.0, 0.0, 1.0 ]T
IV. N UMERICAL EXAMPLE
Fig. 4.
plant, K denotes the output feedback controller designed later, d denotes the external noise. The external signals are defined as w = (r, d)T , z = q2 , and y = (q2 , r)T . The controller K is made of a state observer, a pseudo integrator KI /(s + ε) and a state feedback KP ξ. The observer gain gs and the feedback gains KP and KI are defined as gs = (−4.52×103 , 78.3, 4.59×104 , 2.22×103 )T , ε = 10−2 , KP = (88, 176, 22, 110), KI = 176. Then the dynamics of the controller is described as follows. { ξ˙ = Aξ ξ + Bξ1 y + Bξ2 r K: (28) u = [−KP KI ]ξ + r
1.0 0.0 0.0 ˇ = 0.0 0.0 0.0 C 0.0 1.0 0.0 0.0 0.0 1.0 ˇ = [−1.6 × 103 , 1.0, 9.05 × 103 , 89.7]T D
Comparison of three different feedback systems are given. One is the feedback system with the original controller, another is that with the reduced order controller obtained by the proposed method, and the other is that with another reduced order controller obtained by simply applying conventional balanced truncation to the original controller directly. Fig.5 shows the bode diagrams of the controllers (from the input q2 to the output u). Fig.6 shows that of the feedback systems (from the input r to the output z). Fig.7 shows their time responses. In the figures, the (blue) plus marks + denote the results with the original controller,
Bode Diagram
Magnitude (dB)
80 60 40 20
Phase (deg)
0 360 270
0.7 0.6 0.5 0.4 0.3 Original model Proposed model Traditional model
0.2 0.1 0 0
2
4
6
8
10
t[sec]
Fig. 7.
The time response (4 dim.)
method derives a static (0 dimensional) stabilizing controller with the state observer whereas the conventional method does not give any stabilizing controller. These results show that the proposed method gives a reduced order controller which preserves the stability of the closed-loop system and the behavior of a certain input-output map of the closed-loop system. V. C ONCLUSION This paper proposes a novel balanced realization method for a controller reduction problem. It characterizes a partial balanced realization for a feedback system with respect to a prescribed input-output map. It can be directly applied to a controller reduction problem and the resulting reduced order controller stabilizes the original plant. Furthermore, numerical examples demonstrate the effectiveness of the proposed method. The authors believe that the present paper gives the first result on controller reduction based on balanced truncation which maintains the stability of the original feedback system. R EFERENCES
0 −1 10
0
10
1
10
Frequency (rad/sec)
Fig. 5.
Bode plots of the reduced controllers (4 dim.)
Bode Diagram
0 −10 Magnitude (dB)
0.8
180 90
−20 −30 −40 −50 0
Phase (deg)
1 0.9
displacement[m]
the (green) times marks × denote those with the proposed reduced controller, and the (red) circle marks ◦ denote those with the conventional reduced controller. The dimensions of the original, the proposed and the conventional controllers are 5, 4 and 4. The Hankel singular values for the above two reduction methods (the proposed and the conventional) are computed as follows: the Hankel singular values for the subspace balanced realization are σ1 = 7.17 × 10−2 , σ2 = 3.72 × 10−4 , σ3 = 7.00 × 10−5 , σ4 = 9.48 × 10−6 , and 2.83 × 10−6 ; the conventional Hankel singular values are σ1 = 7.10 × 103 , σ2 = 5.86 × 103 , σ3 = 5.42 × 103 , σ4 = 1.30 × 103 , and σ5 = 9.52. The original controller is 4 dimensional and we want to obtain a reduced order controller with 3 dimensional state-space. Since the designed minimum dimensional state observer is 3 dimensional, the proposed method needs to obtain 1 dimensional reduced order system in order to obtain a 4 dimensional reduce order controller. The double lines in the table denote the truncated dimensions, that is, the proposed method derives a 1 dimensional controller (with a 3 dimensional state observer) from the 5 dimensional original controller and the conventional method derives a 4 dimensional controller from a 5 dimensional original one. Please note that both methods yield 4 dimensional controllers.
−90 −180 −270 −1 10
0
10
1
10
Frequency (rad/sec)
Fig. 6.
Bode plots of the feedback systems (4 dim.)
Fig.5 shows that the conventional method gives the better approximation for the controller reduction itself. Figs.5-7 show that the proposed method gives the better approximation for the whole closed-loop system. Furthermore, if we try to obtain 3 dimensional controller, then the proposed
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