Multivariable Controller Design with Integrity - Semantic Scholar
2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
Multivariable Controller Design with Integrity∗ P. Kallakuri1 , L.H. Keel1 and S.P. Bhattacharyya2
Abstract— In this paper, a new approach to multivariable controller design with integrity is introduced. This method is based on the reinterpretation of individual transmission channel representation of MIMO systems defined in the individual channel design (ICD) method. It provides a procedure for finding stabilizing PID controllers for two input - two output (TITO) systems which maintain stability of the multivariable system even when one of the controllers fails to operate. The PID controllers are directly designed for the unknown systems from frequency response data, without developing mathematical models and without system identification. A key ingredient for the success of the method is that complete sets of stabilizing controllers can be found since such sets have to be intersected. It is shown that strategic processing of a few measurements can achieve this goal.
I. I NTRODUCTION Arbitrary failures of one or more controllers in the MIMO systems can have considerable effects on normal operation, most importantly, stability of control systems. Designing controllers such that the system remains stable for a class of failures addresses this problem. Such systems are said to possess integrity. Research on controller integrity to increase the fault tolerance has been an important area for the past several decades. In [1], one plant and two controller configuration was considered to obtain a reliable design. In [2], a stable factorization approach was used to design controllers with fault tolerance. Subsequently, this work led to design a software package implemented in Mathematica [3]. In [4] a state feedback control law which retains stability against arbitrary actuator failures and parameter perturbations is derived using a positive definite symmetric solution of a new Riccati-type matrix equation. In [5], a method to design a decentralized H∞ output feedback controller to maintain stability of the system and a certain performance under failure of any one of the local controllers. Readers should refer to the references therein for more details. In this paper, we propose a new control design method for TITO systems against arbitrary controller failures. Our approach is motivated by the result that effectively and completely determines the set of stabilizing PID controllers for a given system described by its frequency response data without mathematical models [8]. Our framework utilizes the classical Individual Channel Design (ICD) method that ∗ This work was supported in part by DOD Grant W911NF-08-0514 and NSF Grants CMMI-0927664 and CMMI-0927652 1 P. Kallakuri and L.H. Keel is with Department of Electrical & Computer Engineering, Tennessee State University, Nashville, TN 37209, USA.
[email protected]
allows the analysis and synthesis of multivariable control systems using the Multivariable Structure Function (MSF) by applying classical techniques based on the Bode and Nyquist plots [6]. It is based on the definition of individual transmission channels. Once the channels are defined it is possible to form a feedback loop with the compensator specially designed to meet design specifications. In this manner the multivariable control design problem can be reduced to the design of a SISO control for each channel [6], [7]. The rest of the paper is organized as follows. Section II gives an overview of the Individual Channel Design (ICD) method for TITO controller design. Following the problem formulation in Section III, Section IV details the measurement based PID controller design approach given in [8]. In Section V, our proposed multivariable controller design process is explained. An illustrative example is given in Section VI. II. I NDIVIDUAL C HANNEL D ESIGN Consider a stable, proper TITO plant G and TITO diagonal controller given by the transfer functions (the argument s is suppressed): K given by, g11 g12 K1 0 G= , K= (1) g21 g22 0 K2 The unity feedback configuration of plant G and controller K is as shown in Fig. 1. K r1 + r2
Fig. 1.
G
k1 k2
# "
g11 g21
g12 g22
#
y1 y2
A TITO unity feedback System
The feedback arrangement in Fig. 1 can be redrawn as Fig. 2. The individual channel representation which is equivalent to TITO system with diagonal controller in Fig. 2 is shown in Fig. 3. The channels are given by, Ci = Ki gii (1 − γhj ), where
2 S.P.
Bhattacharyya is with Department of Electrical & Computer Engineering, Texas A&M University, College Station, TX 77843, USA.
[email protected] 978-1-4799-0176-0/$31.00 ©2013 AACC
− + −
"
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γ=
g12 g21 g11 g22
i, j = 1, 2.
(2)
ki gii . 1 + ki gii
(3)
and hi =
r1 − +
G k1
g11
+
Integrity of TITO systems: Find a controller set K which satisfies the following: (1) (K1 , K2 ) stabilizes the unity feedback system around g22 (2) K1 stabilizes the unity feedback system R1 (s) and or equivalently, (1) K1 stabilizes the unity feedback system around g11 and (2) (K1 , K2 ) stabilizes the unity feedback system R2 (s). For the case of 3 input 3 output systems, we note the following.
y1
g12 g21 r2
k2
+ −
−
+
+
y2
A 2 × 2 feedback system (component-wise view)
Fig. 2.
r1
g22
Channel C1 k1
g11 (1 − γh2 )
+
Integrity of 3 × 3 systems: Find a controller set K satisfies the following: Let G be a 3 × 3 transfer function. (1) K stabilizes the unity feedback system with G. (2) For failure of any one controller, i.e., Ki = 0 for i, j, k = 1, 2, 3 and i 6= j 6= k, Kjk stabilizes the TITO unity feedback system with the plant Gjk where gjj gjk Kj Gjk = , Kjk = . (5) gkj gkk Kk
y1
g12 h2 g22 g21 h1 g11
r2
k2
+ − Fig. 3.
g22 (1 − γh1 )
+
Channel C2
y2
(3) For failure of any two controllers, i.e., Ki = Kj = 0 for i, j, k = 1, 2, 3 and i 6= j 6= k, Kk stabilizes the SISO unity feedback with the plant gkk for k = 1, 2, 3.
Equivalent Channel Representation of TITO system
The loop interactions are preserved in the individual channels.The channels C1 and C2 together are structurally equivalent the TITO system.The closed loop system is given by
Remark 2 As seen above, the TITO result given here may be an effective tool to solve the general case beyond TITO systems. However, developing a complete design method for the general case requires much further research.
Our solution of the TITO integrity problem will rely on a recent result in which the complete set of PID controllers yi (s) = Ri (s)ri (s) + Pi (s)Qi (s)rj (s), i 6= j (4) stabilizing a single loop has been presented. Moreover, the result is based on frequency response measurement. The fact where that the complete set is obtained is the key feature which 1 gij hj allows us to obtain integrity by intersecting over stabilizing Ci (s) . , Pi (s) = , Qi (s) = Ri (s) = 1 + Ci (s) 1 + Ci (s) gjj sets.
Theorem 1 Write Ri (s), Pi (s), Qi (s) as ratios of two polynomials in s. Then the denominator polynomials of Ri (s), Pi (s)Qi (s), i = 1, 2, and the numerator polynomial of det(I + GK) are all same. Consequently, the given TITO system is stable if and only if K ensures stability of the transfer functions Ri (s), i = 1 or 2. Proof is simple calculation and omitted here. Remark 1 Note that the denominators of R1 and R2 are identical and also denominators of each of the closed-loop transfer functions in Fig. 3 are the same. Hence it follows that if we design a controller K for stabilizing either channel C1 or channel C2 in the closed loop, it will also stabilize the closed loop TITO system. III. P ROBLEM F ORMULATION The problem of determining stabilizing controllers with integrity for a given TITO system in Fig. 1 can now be stated as follows.
IV. M EASUREMENT BASED PID D ESIGN This section briefly explains the data based PID controller design method demonstrated in [8]. The key result is that the complete stabilizing set can be found only from frequency response data. This facilitates the design of controllers for integrity. Consider a continuous-time LTI plant, with underlying transfer function P (s) with n(m) poles (zeros). We assume that the only information available to the designer is: • Knowledge of the frequency response magnitude and phase, equivalently, P (jω), ω ∈ [0, ∞) if the plant is stable. Knowledge of a known stabilizing controller and the corresponding closed loop frequency response if the plant is unstable. + • Number of unstable poles p of the system. Write
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P (jω) = |P (jω)|ejφ(ω) = Pr (ω) + jPi (ω)
(6)
Let the PID controller be of the form K i + Kp s + K d s 2 C(s) = , for T > 0 s(1 + sT )
For strictly proper plants, P (∞) = 0 and this constraint vanishes. (7) V. D ESIGN OF C ONTROLLER
where T is assumed to be fixed and small. The complete set of stabilizing PID gains can be computed by the following procedure.For stable systems, the available data is the frequency response of the plant P (jω). 1. Determine the relative degree of the plant rP = n − m from the high frequency slope of the Bode magnitude plot of P (jω). 2. Let ∆∞ 0 [φ(ω)] denote the net change of phase of P (jω) for ω ∈ [0, ∞). Determine the number of unstable zeros z + from + π ∆∞ 0 [φ(ω)] = − (n − m) + 2z 2 + with p = 0. 3. Fix Kp = Kp∗ , solve Kp∗ = −
Pr (ω) + ωT Pi (ω) =: g(ω) |P (jω)|2
F¯i (ω, Kp∗ ) = 0,
(9)
4. Set ω0 = 0, ωl = ∞ and j = sgnF¯i (−∞− , Kp∗ ). Determine all strings of integers it ∈ {+1, −1} such that: For n − m even, i0 − 2i1 + · · · + (−1)i−1 2il−1 + (−1)l il ·(−1)l−1 j = n − m + 2z + + 2. (10) For n − m odd, i0 − 2i1 + 2i2 + · · · + (−1)i−1 2il−1 (−1)l−1 j = n − m + 2z + + 2. (11) 5. For the fixed Kp = Kp∗ chosen in Step 1, solve for the stabilizing (Ki , Kd ) values from ωt Pi (ωt ) − ωt2 T Pr (ωt ) it > 0, Ki − Kd ωt2 + |P (jωt )|2 or ωt sin φ(ωt ) − ωt2 T cos φ(ωt ) 2 Ki − Kd ω t + it > 0, |P (jωt )|
for t = 0, 1, · · · , l. 6. Repeat the previous three steps by updating Kp over prescribed ranges. The ranges over which Kp must be swept is determined from the requirements that (10) or (11) is satisfied. We emphasize that all computations are based on the data P (jω) and knowledge of the transfer function P (s) is not required. For well-posed ness of the loop, it is necessary that Kd 6= −
T . P (∞)
C1 = K1 g11 (1 − γh2 ) where γ=
g12 g21 g11 g22
r1
and
h2 =
+
(12)
k2 g22 1 + k2 g22
(13)
y1
g11 (1 − γh2 )
k1 − Fig. 4.
(8)
and let ω1 < ω2 < · · · < ωl−1 denote the distinct frequencies of odd multiplicities which are solutions of
WITH I NTEGRITY
Considering the Individual Channel Design method we can say that, stabilization of the TITO system in Fig. 1 is equivalent to that of stabilization of a closed loop SISO system of channel C1 shown in Fig. 4. C1 is given by
SISO channel C1
The controllers K1 and K2 designed for stabilizing SISO system in Fig. 4 will in turn stabilize the TITO system in closed loop. The unity feedback loop shown in Fig. 4 is redrawn considering the definition of h2 . We consider h2 as the unity feedback loop of g22 with controller K2 . The block diagram shown in Fig. 5 is equivalent to Fig. 4.
r1 + −
k1
+
k2 −
Fig. 5.
h2
g22
γ
−
+ g11
y1
g11 (1 − γh2 )
SISO channel equivalent to Fig. 4
Consider that the frequency response data of the plant G ¯ 2 of the unity feedback is available. The stabilizing PID set K around g22 is know. Pick a controller K2∗ from this set. With this controller in place, the plant seen by controller K1 is g11 (1 − γh2 ). The frequency response of this could be calculated or measured. Then the controller K1∗ can be picked from the stabilizing set K1 (K2∗ ) from the new plant. The design strategy is as follows, Design Procedure: ¯ 1 stabilizing the closed-loop system 1. Determine the set K with g11 . ¯ 2 stabilizing the closed-loop system 2. Determine the set K with g22 . ¯ 2 . Using 3. Select a fixed controller K2∗ from the set K selected K2∗ , determine the set of K1 s to stabilize the closed-loop system R1 . This gain set is referred as K1 (K2∗ ). 4. If the set is empty, repeat step 3 with another choice of K2∗ ∈ K2 .
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¯ 1. 5. Find the intersection K1 (K2∗ ) ∩ K ∗ ∗ ¯ 1. 6. Select K1 from the set K1 (K2 ) ∩ K This choice of controllers (K1∗ , K2∗ ) will ensure integrity of the given TITO system.
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¯ 1 and K ¯ 2 be the sets stabilizing the feedTheorem 2 Let K back systems around g11 and g22 , respectively. Let K1 (K2∗ ) be the set stabilizing the feedback system around the channel ¯ 2 . Then the TITO closed-loop C1 with a choice of K2∗ ∈ K ∗ ¯ 1 and K ∗ is system with a controller K1 ∈ K1 (K2∗ ) ∩ K 2 stable with integrity.
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(a) Bode plot of g11
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(b) Bode plot of g12 20
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Proof: (1) Case 1: Both K1 and K2 are ON The TITO feedback system is stable if the numerator of det[I + GK] is Hurwitz stable. Here, we have
−80 −1 10 0
(14)
¯ 2 which stabilizes the feedback Since K2∗ is chosen from K system around g22 , the TITO system remains stable. (3) Case 3: K2 fails - With K2 = 0, (15)
¯ 1 which stabilizes the feedback Since K1∗ is chosen from K system around g11 , the TITO system remains stable. Hence, we conclude that the controller ensures integrity of the TITO feedback system. Remark 3 Similarly, the procedure can be repeated by ¯ 1 first, determine K2 (K ∗ ), and then select selecting K1∗ ∈ K 1 ∗ ¯ 2. K2 from the set K2 (K1∗ ) ∩ K
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Fig. 6.
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(d) Bode plot of g22
Frequency response data for G
¯ 2 by using data based design method in [8]. The following K is the procedure for the controller design. The set of frequency response points collected is considered as g22 (jω). g22 (jω) = g22r (ω) + jg22i (ω)
(16)
From the Bode plot in Fig.6 the estimated high frequency slope is approximately -20db/decade and thus rp = n − m = 1. From the phase plot in Fig.6 it is shown that, the total change of phase is − π2 . Also p+ = 0. Thus, we have −rP + p+ − π2 ∆∞ 0 ∠g22 (jω) = 0. 2 The stability of the closed-loop system consisting of g22 with K2 requires the signature condition z+ =
σ(F¯ (s)) = (n − m) + 2z + + 2 = 3. Since n-m is odd, i0 − 2i1 + 2i2 + · · · + (−1)l−1 2il−1 (−1)l−1 j = 3. (17) where h i j := sgn F¯i (∞− , Kp∗2 ) = −sgn lim g(ω) = −1.
VI. A N E XAMPLE Consider a TITO plant G defined as, g11 (s) g12 (s) G(s) = g21 (s) g22 (s) and the controller K with T > 0 K1 0 K= 0 K2 K i 1 + Kp 1 s + Kd 1 s 2 0 s(1 + sT ) = Ki 2 + K p 2 s + K d 2 s 2 0 s(1 + sT )
1
10
(c) Bode plot of g21
Note that K1∗ and K2∗ are chosen from the set K1 (K2∗ ) that stabilizes the feedback system R1 . From Theorem 1, the stability of R1 is equivalent to that of the TITO system. (2) Case 2: K1 fails - With K1 = 0,
det[I + GK] = 1 + g11 K1 .
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det[I + GK] = (1 + g11 K1 ) (1 + g22 K2 ) − g12 g21 K1 K2 .
det[I + GK] = 1 + g22 K2 .
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ω→∞
It is clear from the signature value that at least one frequency term is required to satisfy (17). Therefore,
i0 − 2i1 = 3
.
(18)
The only feasible string satisfying (18) is {i0 , i1 } = {1, −1}
The open-loop frequency data (magnitude and phase), g11 (jω), g12 (jω), g21 (jω), g22 (jω), of plant G is shown in Fig. 6.
(19)
The next step is to calculate the feasible range of Kp2 . The feasible Kp2 values are the g(ω) values at which the equation (20) has at least one positive frequency as solution.
¯ 2: Characterization of stabilizing controller set K The Bode data of g22 is used to determine the PID gains of 5169
Kp2 = −
g22r (ω) + ωT g22i (ω) =: g2 (ω) |g22 |2
(20)
the PID design for g11 to obtain the stabilizing controller set ¯ 1 shown in Fig. 9(b). K 3
3
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d
k
d
1
1
1
1 k
The feasible range of Kp2 is such that it intersects the graph of g(ω) at least one time. Thus, the feasible range is Kp2 ≥ −2.980. We now fix the value of Kp2 to any feasible proportional gain and compute the set of ω’s that satisfies (20). Then we solve for the stabilizing (Ki2 , Kd2 ) values using the set of inequalities, ωt g22i (ωt ) − ωt2 T g22r (ωt ) 2 Ki2 − Kd2 ωt + it > 0, |g22 (jωt )|2 ¯ 2 is shown in The calculated stabilizing controller set K Fig. 7.
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¯1 (b) K
(a) K1 (K2∗ ) Fig. 9.
¯1 K1 (K2∗ ) vs. K
Implementable PID region of K2
Then the intersection of these two sets are found as shown in Fig. 10. ∗ ¯ 1 is the stabilizing PID gains of • Region 1 - K1 (K2 ) ∩ K K1 designed for both g11 (1 − γh2 ) and g11 . ∗ • Region 2 - K1 (K2 ) is the stabilizing PID gains of K1 designed for g11 (1 − γh2 ). ¯ 1 is the stabilizing PID gains of K1 • Region 3 - K designed for g11 .
2 1.5 1
Kp
0.5 0 −0.5 −1 −1.5 50 0 −50
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Kd
4
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7
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Ki
Fig. 7.
¯2 PID gain region K 3
¯ 1: Characterization of stabilizing region K1 (K2∗ ) ∩ K ∗ ¯ We first select K2 ∈ K2 as follows. Kp2 = 1.5,
Ki2 = 2,
2.5 Region 1 2
Kd2 = −0.05
1.5 1 kd1
Using K2∗ , we construct the frequency response of P1 as shown in Fig. 8 where
0.5 Region 3 0
P1 = g11 (1 − γh2 ).
−0.5 −1 Region 2
P1 frequency data
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Fig. 8.
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¯ 1 with Kp1 = −0.5 Integrity set K1 (K2∗ ) ∩ K
The step responses of the closed loop system when controller K1 is selected from three regions are shown in Figs. 11, 12, and 13. From the step responses depicted, we can see that, when ¯ 1 , the stability K1∗ is selected from region 1, i.e., K1 (K2∗ )∩ K of the system is retained even when any one of the controllers is turned off (Fig. 11). When K1∗ is selected from region 2, ¯ 1 , the stability is not i.e., K1 (K2∗ ) but not in K1 (K2∗ ) ∩ K retained when K2∗ is off (Fig. 12). When K1∗ is selected ¯ 1 , the stability ¯ 1 but not in K1 (K ∗ ) ∩ K from region 3, i.e., K 2 is not retained when K1∗ is off (Fig. 13).
w
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Fig. 10.
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VII. C ONCLUSIONS
P1 Frequency response data
The Bode data of P1 is used to determine the stabilizing PID gain set K1 (K2∗ ) by using measurement based PID design method. The set is shown in Fig. 9(a). We repeat
Hence we have introduced a new approach to design controllers for TITO systems with integrity. This method is based on the reinterpretation of individual transmission
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[2] A.N. G¨undes, “Stability of feedback systems with sensor or actuator failures: analysis,” International Journal of Control, vol. 56, no. 4, pp. 735 - 753, 1992. [3] A. Bhaya and J.B.L. Vieira, “CONFAL: a stable-factorization based fault-tolerant controller design software package,” Proceedings of the 33rd IEEE Conference on Decision and Control, vol. 2, pp. 1556 1557, Lake Buena Vista, FL, Dec. 14 - 16, 1994. [4] Qing-Long Han, Jin-Shou Yu and Zheng-Zhi Tang, “Design of Controller Possessing Integrity for Uncertain Continuous-Time Systems,” Proceedings of the IEEE International Conference on Industrial Technology, pp. 545-549,2-6 Dec, 1996. [5] N. Chen and M. Ikeda, “Fault-tolerant design of decentralized H∞ control systems using homotopy method,” Proceedings of the 2004 SICE Annual Conference, vol. 3, pp. 1927 - 1931, Sapporo, Japan, Aug. 2-4, 2004. [6] J. O’Reilly and W.E. Leithead, “Multivariable control by Individual Channel Design,” International Journal of Control, vol. 54, pp. 1 - 46, 1991. [7] Carlos E. Ugalde-Loo and Eduardo Licaga-Castro “2x2 Individual Channel Design MATLAB Toolbox,” Proceedings of the 44th IEEE Conference on Decision and Control and the European Control Conference, 2005. [8] L.H. Keel and S.P. Bhattacharyya, “Controller synthesis free of analytical models: Three term controllers,” IEEE Transactions on Automatic Control, vol. 53, no. 6, pp. 1353 - 1369, July 2008.
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∗ ¯ 1 but K ∗ ∈ K1∗ ∈ K 1 / K1 (K2 )
channel representation of MIMO systems defined in individual channel design method. The PID controllers are designed using the measurement based approach. The designed PID gain regions are tested for stability with a step input. It is illustrated that the designed controllers are fault tolerant. An open problem not addressed here is the construction of the controller set (K1 , K2 ) such that robust integrity is maintained over arbitrary choices of the controller (K1 , K2 ) from these sets. Research is ongoing for extending this theory to the systems beyond TITO. R EFERENCES ˘ [1] X.-L. Tan, D.D. Siljak, and M. Ikeda, “Reliable stabilization via factorization methods,” IEEE Transactions on Automatic Control, vol. 37, no. 11, pp. 1786 - 1791, 1992.
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Multivariable Controller Design with Integrity∗ P. Kallakuri1 , L.H. Keel1 and S.P. Bhattacharyya2
Abstract— In this paper, a new approach to multivariable controller design with integrity is introduced. This method is based on the reinterpretation of individual transmission channel representation of MIMO systems defined in the individual channel design (ICD) method. It provides a procedure for finding stabilizing PID controllers for two input - two output (TITO) systems which maintain stability of the multivariable system even when one of the controllers fails to operate. The PID controllers are directly designed for the unknown systems from frequency response data, without developing mathematical models and without system identification. A key ingredient for the success of the method is that complete sets of stabilizing controllers can be found since such sets have to be intersected. It is shown that strategic processing of a few measurements can achieve this goal.
I. I NTRODUCTION Arbitrary failures of one or more controllers in the MIMO systems can have considerable effects on normal operation, most importantly, stability of control systems. Designing controllers such that the system remains stable for a class of failures addresses this problem. Such systems are said to possess integrity. Research on controller integrity to increase the fault tolerance has been an important area for the past several decades. In [1], one plant and two controller configuration was considered to obtain a reliable design. In [2], a stable factorization approach was used to design controllers with fault tolerance. Subsequently, this work led to design a software package implemented in Mathematica [3]. In [4] a state feedback control law which retains stability against arbitrary actuator failures and parameter perturbations is derived using a positive definite symmetric solution of a new Riccati-type matrix equation. In [5], a method to design a decentralized H∞ output feedback controller to maintain stability of the system and a certain performance under failure of any one of the local controllers. Readers should refer to the references therein for more details. In this paper, we propose a new control design method for TITO systems against arbitrary controller failures. Our approach is motivated by the result that effectively and completely determines the set of stabilizing PID controllers for a given system described by its frequency response data without mathematical models [8]. Our framework utilizes the classical Individual Channel Design (ICD) method that ∗ This work was supported in part by DOD Grant W911NF-08-0514 and NSF Grants CMMI-0927664 and CMMI-0927652 1 P. Kallakuri and L.H. Keel is with Department of Electrical & Computer Engineering, Tennessee State University, Nashville, TN 37209, USA.
[email protected]
allows the analysis and synthesis of multivariable control systems using the Multivariable Structure Function (MSF) by applying classical techniques based on the Bode and Nyquist plots [6]. It is based on the definition of individual transmission channels. Once the channels are defined it is possible to form a feedback loop with the compensator specially designed to meet design specifications. In this manner the multivariable control design problem can be reduced to the design of a SISO control for each channel [6], [7]. The rest of the paper is organized as follows. Section II gives an overview of the Individual Channel Design (ICD) method for TITO controller design. Following the problem formulation in Section III, Section IV details the measurement based PID controller design approach given in [8]. In Section V, our proposed multivariable controller design process is explained. An illustrative example is given in Section VI. II. I NDIVIDUAL C HANNEL D ESIGN Consider a stable, proper TITO plant G and TITO diagonal controller given by the transfer functions (the argument s is suppressed): K given by, g11 g12 K1 0 G= , K= (1) g21 g22 0 K2 The unity feedback configuration of plant G and controller K is as shown in Fig. 1. K r1 + r2
Fig. 1.
G
k1 k2
# "
g11 g21
g12 g22
#
y1 y2
A TITO unity feedback System
The feedback arrangement in Fig. 1 can be redrawn as Fig. 2. The individual channel representation which is equivalent to TITO system with diagonal controller in Fig. 2 is shown in Fig. 3. The channels are given by, Ci = Ki gii (1 − γhj ), where
2 S.P.
Bhattacharyya is with Department of Electrical & Computer Engineering, Texas A&M University, College Station, TX 77843, USA.
[email protected] 978-1-4799-0176-0/$31.00 ©2013 AACC
− + −
"
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γ=
g12 g21 g11 g22
i, j = 1, 2.
(2)
ki gii . 1 + ki gii
(3)
and hi =
r1 − +
G k1
g11
+
Integrity of TITO systems: Find a controller set K which satisfies the following: (1) (K1 , K2 ) stabilizes the unity feedback system around g22 (2) K1 stabilizes the unity feedback system R1 (s) and or equivalently, (1) K1 stabilizes the unity feedback system around g11 and (2) (K1 , K2 ) stabilizes the unity feedback system R2 (s). For the case of 3 input 3 output systems, we note the following.
y1
g12 g21 r2
k2
+ −
−
+
+
y2
A 2 × 2 feedback system (component-wise view)
Fig. 2.
r1
g22
Channel C1 k1
g11 (1 − γh2 )
+
Integrity of 3 × 3 systems: Find a controller set K satisfies the following: Let G be a 3 × 3 transfer function. (1) K stabilizes the unity feedback system with G. (2) For failure of any one controller, i.e., Ki = 0 for i, j, k = 1, 2, 3 and i 6= j 6= k, Kjk stabilizes the TITO unity feedback system with the plant Gjk where gjj gjk Kj Gjk = , Kjk = . (5) gkj gkk Kk
y1
g12 h2 g22 g21 h1 g11
r2
k2
+ − Fig. 3.
g22 (1 − γh1 )
+
Channel C2
y2
(3) For failure of any two controllers, i.e., Ki = Kj = 0 for i, j, k = 1, 2, 3 and i 6= j 6= k, Kk stabilizes the SISO unity feedback with the plant gkk for k = 1, 2, 3.
Equivalent Channel Representation of TITO system
The loop interactions are preserved in the individual channels.The channels C1 and C2 together are structurally equivalent the TITO system.The closed loop system is given by
Remark 2 As seen above, the TITO result given here may be an effective tool to solve the general case beyond TITO systems. However, developing a complete design method for the general case requires much further research.
Our solution of the TITO integrity problem will rely on a recent result in which the complete set of PID controllers yi (s) = Ri (s)ri (s) + Pi (s)Qi (s)rj (s), i 6= j (4) stabilizing a single loop has been presented. Moreover, the result is based on frequency response measurement. The fact where that the complete set is obtained is the key feature which 1 gij hj allows us to obtain integrity by intersecting over stabilizing Ci (s) . , Pi (s) = , Qi (s) = Ri (s) = 1 + Ci (s) 1 + Ci (s) gjj sets.
Theorem 1 Write Ri (s), Pi (s), Qi (s) as ratios of two polynomials in s. Then the denominator polynomials of Ri (s), Pi (s)Qi (s), i = 1, 2, and the numerator polynomial of det(I + GK) are all same. Consequently, the given TITO system is stable if and only if K ensures stability of the transfer functions Ri (s), i = 1 or 2. Proof is simple calculation and omitted here. Remark 1 Note that the denominators of R1 and R2 are identical and also denominators of each of the closed-loop transfer functions in Fig. 3 are the same. Hence it follows that if we design a controller K for stabilizing either channel C1 or channel C2 in the closed loop, it will also stabilize the closed loop TITO system. III. P ROBLEM F ORMULATION The problem of determining stabilizing controllers with integrity for a given TITO system in Fig. 1 can now be stated as follows.
IV. M EASUREMENT BASED PID D ESIGN This section briefly explains the data based PID controller design method demonstrated in [8]. The key result is that the complete stabilizing set can be found only from frequency response data. This facilitates the design of controllers for integrity. Consider a continuous-time LTI plant, with underlying transfer function P (s) with n(m) poles (zeros). We assume that the only information available to the designer is: • Knowledge of the frequency response magnitude and phase, equivalently, P (jω), ω ∈ [0, ∞) if the plant is stable. Knowledge of a known stabilizing controller and the corresponding closed loop frequency response if the plant is unstable. + • Number of unstable poles p of the system. Write
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P (jω) = |P (jω)|ejφ(ω) = Pr (ω) + jPi (ω)
(6)
Let the PID controller be of the form K i + Kp s + K d s 2 C(s) = , for T > 0 s(1 + sT )
For strictly proper plants, P (∞) = 0 and this constraint vanishes. (7) V. D ESIGN OF C ONTROLLER
where T is assumed to be fixed and small. The complete set of stabilizing PID gains can be computed by the following procedure.For stable systems, the available data is the frequency response of the plant P (jω). 1. Determine the relative degree of the plant rP = n − m from the high frequency slope of the Bode magnitude plot of P (jω). 2. Let ∆∞ 0 [φ(ω)] denote the net change of phase of P (jω) for ω ∈ [0, ∞). Determine the number of unstable zeros z + from + π ∆∞ 0 [φ(ω)] = − (n − m) + 2z 2 + with p = 0. 3. Fix Kp = Kp∗ , solve Kp∗ = −
Pr (ω) + ωT Pi (ω) =: g(ω) |P (jω)|2
F¯i (ω, Kp∗ ) = 0,
(9)
4. Set ω0 = 0, ωl = ∞ and j = sgnF¯i (−∞− , Kp∗ ). Determine all strings of integers it ∈ {+1, −1} such that: For n − m even, i0 − 2i1 + · · · + (−1)i−1 2il−1 + (−1)l il ·(−1)l−1 j = n − m + 2z + + 2. (10) For n − m odd, i0 − 2i1 + 2i2 + · · · + (−1)i−1 2il−1 (−1)l−1 j = n − m + 2z + + 2. (11) 5. For the fixed Kp = Kp∗ chosen in Step 1, solve for the stabilizing (Ki , Kd ) values from ωt Pi (ωt ) − ωt2 T Pr (ωt ) it > 0, Ki − Kd ωt2 + |P (jωt )|2 or ωt sin φ(ωt ) − ωt2 T cos φ(ωt ) 2 Ki − Kd ω t + it > 0, |P (jωt )|
for t = 0, 1, · · · , l. 6. Repeat the previous three steps by updating Kp over prescribed ranges. The ranges over which Kp must be swept is determined from the requirements that (10) or (11) is satisfied. We emphasize that all computations are based on the data P (jω) and knowledge of the transfer function P (s) is not required. For well-posed ness of the loop, it is necessary that Kd 6= −
T . P (∞)
C1 = K1 g11 (1 − γh2 ) where γ=
g12 g21 g11 g22
r1
and
h2 =
+
(12)
k2 g22 1 + k2 g22
(13)
y1
g11 (1 − γh2 )
k1 − Fig. 4.
(8)
and let ω1 < ω2 < · · · < ωl−1 denote the distinct frequencies of odd multiplicities which are solutions of
WITH I NTEGRITY
Considering the Individual Channel Design method we can say that, stabilization of the TITO system in Fig. 1 is equivalent to that of stabilization of a closed loop SISO system of channel C1 shown in Fig. 4. C1 is given by
SISO channel C1
The controllers K1 and K2 designed for stabilizing SISO system in Fig. 4 will in turn stabilize the TITO system in closed loop. The unity feedback loop shown in Fig. 4 is redrawn considering the definition of h2 . We consider h2 as the unity feedback loop of g22 with controller K2 . The block diagram shown in Fig. 5 is equivalent to Fig. 4.
r1 + −
k1
+
k2 −
Fig. 5.
h2
g22
γ
−
+ g11
y1
g11 (1 − γh2 )
SISO channel equivalent to Fig. 4
Consider that the frequency response data of the plant G ¯ 2 of the unity feedback is available. The stabilizing PID set K around g22 is know. Pick a controller K2∗ from this set. With this controller in place, the plant seen by controller K1 is g11 (1 − γh2 ). The frequency response of this could be calculated or measured. Then the controller K1∗ can be picked from the stabilizing set K1 (K2∗ ) from the new plant. The design strategy is as follows, Design Procedure: ¯ 1 stabilizing the closed-loop system 1. Determine the set K with g11 . ¯ 2 stabilizing the closed-loop system 2. Determine the set K with g22 . ¯ 2 . Using 3. Select a fixed controller K2∗ from the set K selected K2∗ , determine the set of K1 s to stabilize the closed-loop system R1 . This gain set is referred as K1 (K2∗ ). 4. If the set is empty, repeat step 3 with another choice of K2∗ ∈ K2 .
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0
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¯ 1. 5. Find the intersection K1 (K2∗ ) ∩ K ∗ ∗ ¯ 1. 6. Select K1 from the set K1 (K2 ) ∩ K This choice of controllers (K1∗ , K2∗ ) will ensure integrity of the given TITO system.
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¯ 1 and K ¯ 2 be the sets stabilizing the feedTheorem 2 Let K back systems around g11 and g22 , respectively. Let K1 (K2∗ ) be the set stabilizing the feedback system around the channel ¯ 2 . Then the TITO closed-loop C1 with a choice of K2∗ ∈ K ∗ ¯ 1 and K ∗ is system with a controller K1 ∈ K1 (K2∗ ) ∩ K 2 stable with integrity.
−40
−100 −120
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−140 −160
−80
−180 −200 −1 10
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−100 −1 10
4
10
(a) Bode plot of g11
0
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10
(b) Bode plot of g12 20
0
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0
21
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−30 −40
−50 −50 −60
Proof: (1) Case 1: Both K1 and K2 are ON The TITO feedback system is stable if the numerator of det[I + GK] is Hurwitz stable. Here, we have
−80 −1 10 0
(14)
¯ 2 which stabilizes the feedback Since K2∗ is chosen from K system around g22 , the TITO system remains stable. (3) Case 3: K2 fails - With K2 = 0, (15)
¯ 1 which stabilizes the feedback Since K1∗ is chosen from K system around g11 , the TITO system remains stable. Hence, we conclude that the controller ensures integrity of the TITO feedback system. Remark 3 Similarly, the procedure can be repeated by ¯ 1 first, determine K2 (K ∗ ), and then select selecting K1∗ ∈ K 1 ∗ ¯ 2. K2 from the set K2 (K1∗ ) ∩ K
2
3
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−120 −1 10
Fig. 6.
0
10
1
10
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10
3
10
4
10
(d) Bode plot of g22
Frequency response data for G
¯ 2 by using data based design method in [8]. The following K is the procedure for the controller design. The set of frequency response points collected is considered as g22 (jω). g22 (jω) = g22r (ω) + jg22i (ω)
(16)
From the Bode plot in Fig.6 the estimated high frequency slope is approximately -20db/decade and thus rp = n − m = 1. From the phase plot in Fig.6 it is shown that, the total change of phase is − π2 . Also p+ = 0. Thus, we have −rP + p+ − π2 ∆∞ 0 ∠g22 (jω) = 0. 2 The stability of the closed-loop system consisting of g22 with K2 requires the signature condition z+ =
σ(F¯ (s)) = (n − m) + 2z + + 2 = 3. Since n-m is odd, i0 − 2i1 + 2i2 + · · · + (−1)l−1 2il−1 (−1)l−1 j = 3. (17) where h i j := sgn F¯i (∞− , Kp∗2 ) = −sgn lim g(ω) = −1.
VI. A N E XAMPLE Consider a TITO plant G defined as, g11 (s) g12 (s) G(s) = g21 (s) g22 (s) and the controller K with T > 0 K1 0 K= 0 K2 K i 1 + Kp 1 s + Kd 1 s 2 0 s(1 + sT ) = Ki 2 + K p 2 s + K d 2 s 2 0 s(1 + sT )
1
10
(c) Bode plot of g21
Note that K1∗ and K2∗ are chosen from the set K1 (K2∗ ) that stabilizes the feedback system R1 . From Theorem 1, the stability of R1 is equivalent to that of the TITO system. (2) Case 2: K1 fails - With K1 = 0,
det[I + GK] = 1 + g11 K1 .
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−10
det[I + GK] = (1 + g11 K1 ) (1 + g22 K2 ) − g12 g21 K1 K2 .
det[I + GK] = 1 + g22 K2 .
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ω→∞
It is clear from the signature value that at least one frequency term is required to satisfy (17). Therefore,
i0 − 2i1 = 3
.
(18)
The only feasible string satisfying (18) is {i0 , i1 } = {1, −1}
The open-loop frequency data (magnitude and phase), g11 (jω), g12 (jω), g21 (jω), g22 (jω), of plant G is shown in Fig. 6.
(19)
The next step is to calculate the feasible range of Kp2 . The feasible Kp2 values are the g(ω) values at which the equation (20) has at least one positive frequency as solution.
¯ 2: Characterization of stabilizing controller set K The Bode data of g22 is used to determine the PID gains of 5169
Kp2 = −
g22r (ω) + ωT g22i (ω) =: g2 (ω) |g22 |2
(20)
the PID design for g11 to obtain the stabilizing controller set ¯ 1 shown in Fig. 9(b). K 3
3
2.5
2.5
2
2
1.5
1.5
d
k
d
1
1
1
1 k
The feasible range of Kp2 is such that it intersects the graph of g(ω) at least one time. Thus, the feasible range is Kp2 ≥ −2.980. We now fix the value of Kp2 to any feasible proportional gain and compute the set of ω’s that satisfies (20). Then we solve for the stabilizing (Ki2 , Kd2 ) values using the set of inequalities, ωt g22i (ωt ) − ωt2 T g22r (ωt ) 2 Ki2 − Kd2 ωt + it > 0, |g22 (jωt )|2 ¯ 2 is shown in The calculated stabilizing controller set K Fig. 7.
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−1
−1
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i
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i
1
1
¯1 (b) K
(a) K1 (K2∗ ) Fig. 9.
¯1 K1 (K2∗ ) vs. K
Implementable PID region of K2
Then the intersection of these two sets are found as shown in Fig. 10. ∗ ¯ 1 is the stabilizing PID gains of • Region 1 - K1 (K2 ) ∩ K K1 designed for both g11 (1 − γh2 ) and g11 . ∗ • Region 2 - K1 (K2 ) is the stabilizing PID gains of K1 designed for g11 (1 − γh2 ). ¯ 1 is the stabilizing PID gains of K1 • Region 3 - K designed for g11 .
2 1.5 1
Kp
0.5 0 −0.5 −1 −1.5 50 0 −50
3
2
1
0
Kd
4
6
5
7
8
Ki
Fig. 7.
¯2 PID gain region K 3
¯ 1: Characterization of stabilizing region K1 (K2∗ ) ∩ K ∗ ¯ We first select K2 ∈ K2 as follows. Kp2 = 1.5,
Ki2 = 2,
2.5 Region 1 2
Kd2 = −0.05
1.5 1 kd1
Using K2∗ , we construct the frequency response of P1 as shown in Fig. 8 where
0.5 Region 3 0
P1 = g11 (1 − γh2 ).
−0.5 −1 Region 2
P1 frequency data
−1.5
P1 magnitude
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0 −2 −4 −6 0
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w
Fig. 8.
1
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3
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¯ 1 with Kp1 = −0.5 Integrity set K1 (K2∗ ) ∩ K
The step responses of the closed loop system when controller K1 is selected from three regions are shown in Figs. 11, 12, and 13. From the step responses depicted, we can see that, when ¯ 1 , the stability K1∗ is selected from region 1, i.e., K1 (K2∗ )∩ K of the system is retained even when any one of the controllers is turned off (Fig. 11). When K1∗ is selected from region 2, ¯ 1 , the stability is not i.e., K1 (K2∗ ) but not in K1 (K2∗ ) ∩ K retained when K2∗ is off (Fig. 12). When K1∗ is selected ¯ 1 , the stability ¯ 1 but not in K1 (K ∗ ) ∩ K from region 3, i.e., K 2 is not retained when K1∗ is off (Fig. 13).
w
−8 −1 10
0.5
Fig. 10.
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0
VII. C ONCLUSIONS
P1 Frequency response data
The Bode data of P1 is used to determine the stabilizing PID gain set K1 (K2∗ ) by using measurement based PID design method. The set is shown in Fig. 9(a). We repeat
Hence we have introduced a new approach to design controllers for TITO systems with integrity. This method is based on the reinterpretation of individual transmission
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[2] A.N. G¨undes, “Stability of feedback systems with sensor or actuator failures: analysis,” International Journal of Control, vol. 56, no. 4, pp. 735 - 753, 1992. [3] A. Bhaya and J.B.L. Vieira, “CONFAL: a stable-factorization based fault-tolerant controller design software package,” Proceedings of the 33rd IEEE Conference on Decision and Control, vol. 2, pp. 1556 1557, Lake Buena Vista, FL, Dec. 14 - 16, 1994. [4] Qing-Long Han, Jin-Shou Yu and Zheng-Zhi Tang, “Design of Controller Possessing Integrity for Uncertain Continuous-Time Systems,” Proceedings of the IEEE International Conference on Industrial Technology, pp. 545-549,2-6 Dec, 1996. [5] N. Chen and M. Ikeda, “Fault-tolerant design of decentralized H∞ control systems using homotopy method,” Proceedings of the 2004 SICE Annual Conference, vol. 3, pp. 1927 - 1931, Sapporo, Japan, Aug. 2-4, 2004. [6] J. O’Reilly and W.E. Leithead, “Multivariable control by Individual Channel Design,” International Journal of Control, vol. 54, pp. 1 - 46, 1991. [7] Carlos E. Ugalde-Loo and Eduardo Licaga-Castro “2x2 Individual Channel Design MATLAB Toolbox,” Proceedings of the 44th IEEE Conference on Decision and Control and the European Control Conference, 2005. [8] L.H. Keel and S.P. Bhattacharyya, “Controller synthesis free of analytical models: Three term controllers,” IEEE Transactions on Automatic Control, vol. 53, no. 6, pp. 1353 - 1369, July 2008.
y1
1 No failure
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Fig. 11.
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∈
2
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1 0 −1 −2
K1 fail
No failure K2 fail 0
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250 t
2
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1 0 −1 No failure −2
K fail 1
K2 fail 0
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Fig. 12.
200
250 t
∈
K1 (K2∗ )
300
but
K1∗
¯1 6∈ K
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1
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2
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500 t
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600
∗ ¯ 1 but K ∗ ∈ K1∗ ∈ K 1 / K1 (K2 )
channel representation of MIMO systems defined in individual channel design method. The PID controllers are designed using the measurement based approach. The designed PID gain regions are tested for stability with a step input. It is illustrated that the designed controllers are fault tolerant. An open problem not addressed here is the construction of the controller set (K1 , K2 ) such that robust integrity is maintained over arbitrary choices of the controller (K1 , K2 ) from these sets. Research is ongoing for extending this theory to the systems beyond TITO. R EFERENCES ˘ [1] X.-L. Tan, D.D. Siljak, and M. Ikeda, “Reliable stabilization via factorization methods,” IEEE Transactions on Automatic Control, vol. 37, no. 11, pp. 1786 - 1791, 1992.
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