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Cyclic Control: The Case of Static Output Feedback
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Cyclic Control: The Case of Static Output Feedback Y. Eun ∗ E.M. Gross ∗ P.T. Kabamba ∗∗ S.M. Meerkov ∗∗∗ A.A. Menezes ∗∗ H.R. Ossareh ∗∗∗ ∗
Xerox Corporation, Webster, NY 14580 USA (e-mail: Eric.Gross, [email protected]) ∗∗ Aerospace Engineering Department, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: kabamba, [email protected]) ∗∗∗ Electrical Engineering Department, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: smm, [email protected]) Abstract: This paper develops a control theory for systems comprised of multiple plants controlled by a single actuator and sensor. Such systems arise in rotating machinery with actuators and sensors fixed in inertial space. Representing the rotating device as a set of discrete wedges leads to a model mentioned above. In such systems, each plant can be controlled only intermittently, e.g., once per revolution. We refer to this situation as cyclic control (CC). The paper investigates the problems of stabilizability, reference tracking, and disturbance rejection in CC systems using static output feedback. Keywords: Cyclic control, Periodic control, Stabilizability, Reference tracking, Disturbance rejection, Lifting, Gershgorin circles. C1 (s)
u1
P1 (s) S
. . .
r
S
S Actuator
. . .
ua
Sensor
uN
PN (s)
Fig. 2. Rotating machinery with N discrete wedges.
Fig. 1. Cyclic control (CC) system. 1. INTRODUCTION This paper considers a system comprised of a finite number of plants P1 (s), . . . , PN (s) that may or may not interact. If each plant is equipped with an actuator and a sensor, a controller, Ci (s), i = 1, . . . , N , can be designed for each Pi (s) using a plethora of techniques (Anderson and Moore (2007), Kwakernaak and Sivan (1972)) to ensure satisfactory system performance. We refer to such an architecture as permanently acting control (PAC). There are, however, situations where only one actuator and one sensor are available to control all N plants (see the examples below). In these situations, the actuation and sensing can be allocated to each Pi (s) only intermittently, for instance, in a round-robin or cyclic manner. This implies that within a period, T , each plant is controlled T . We refer to exactly once during the time slot τ = N such an architecture as cyclic control (CC, see Fig. 1). The purpose of this paper is twofold: 1) to develop a control-theoretic methodology for CC systems using static output feedback of individual plants, Pi (s), i = 1, . . . , N ; and 978-3-902661-93-7/11/$20.00 © 2011 IFAC
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CN (s)
ys
2) to quantify the performance losses due to CC in comparison with PAC. Systems where only intermittent or stroboscopic control is possible, arise in rotating machinery with actuators and sensors fixed in inertial space. In these systems, the rotating device can be represented as N discrete wedges (or plants), which leads to the CC architecture (see Fig. 2). Examples of such systems are the charging and fusing stages of the xerographic process (Hamby and Gross (2004)). During charging, the rotating photoreceptor, which is not uniform due to material and surface irregularities, must be charged uniformly by an inertially fixed charger to ensure high-quality print. To accomplish this, each wedge of the photoreceptor must be charged using a different charging voltage to compensate for the irregularities. Similarly, during fusing, each wedge of the fuser must be heated uniformly by an inertially fixed heater; however, different actuation is required to compensate for material non-uniformity and fuser eccentricity (see Eun et al. (2011) for modeling and CC of this system). Similar situations arise in drilling and milling machines (Drozda (1983)) and turbo machinery (Childs (1993)).
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Systems with CC architecture are periodic in the sense that their actuation and sensing are periodic. Obviously, the general theory of linear periodic systems (e.g., Floquet/Lyapunov theory) can be applied to their analysis and design. However, since the periodicity of CC is a specific one — time invariant within each time slot — more powerful, constructive control-theoretic techniques can be developed. At present, no results in this direction are available with the exception of Ching et al. (2010), where the stability problem for the case of one-dimensional noninteracting plants has been investigated. In the current paper, we address all standard problems of control — stabilization, reference tracking, and disturbance rejection — for all possible cases, including the general case of multidimensional plants with interactions. Accordingly, the outline of this paper is as follows. Section 2 is devoted to modeling CC systems and problem formulation. Section 3 presents the main results on stabilizability. Section 4 addresses the problem of reference tracking and disturbance rejection. Section 5 provides the conclusions and topics for future research. Due to space limitations, an application to a xerographic process is not presented here, but can be found in Eun et al. (2011). 2. MODELING AND PROBLEM FORMULATION 2.1 Models Assume that each plant Pi (s), i = 1, . . . , N , is a SISO system described by x˙ i = Aii xi + bi ui , (1) yi = c i x i , xi ∈ RMi , ui , yi ∈ R, Aii ∈ RMi ×Mi , bi , cTi ∈ RMi . T Then, with x = xT1 , . . . , xTN , the overall system of N interacting plants under PAC can be formalized as x˙ = Ax + Bu, y = Cx, y = [y1 , . . . , yN ]T , u = [u1 , . . . , uN ]T , A11 A12 . . . A1N A21 A22 . . . A2N A= . , (2) .. . . .. . . AN 1 AN 2 . . . AN N B = [b1 , . . . , bN ], C = [cT1 , . . . , cTN ]T , T bi = 0, . . . , 0, bTi , 0, . . . 0 , i = 1, . . . , N, ci = [0, . . . , 0, ci , 0, . . . 0] , i = 1, . . . , N,
where xi , yi , ui , Aii , bi , and ci are as in (1) and Aij , i 6= j, are matrices of appropriate dimensions that describe interactions between Pi (s) and Pj (s). To model plants (1) under CC, consider a vector-function v(t) ∈ RV , V ≥ 1, and introduce the following stroboscope operator: v(t), t ∈ (nT + (i − 1)τ, nT + iτ ], stri (v(t)) = (3) 0, otherwise, i = 1, . . . , N ; n ∈ {0, 1, . . .}, where T and τ are, as before, the period of cyclic control and the time slot, respectively. Clearly, for any v1 (t) ∈ RV1 , V1 ≥ 1, and v2 (t) ∈ R,
v1 (t)stri (v2 (t)), i = j, 0, i= 6 j, ∀t ≥ 0.
stri (v1 (t))strj (v2 (t)) =
(4)
Using the stroboscope operator, each plant Pi (s), i = 1, . . . , N , under stroboscopic actuation and sensing is described by x˙ i = Aii xi + bi stri (ui ), (5) yi = strmodN (i+r−1)+1 (ci xi ), where r ∈ {0, . . . , N −1} and all other variables are defined in (1). If r = 0, the actuator and sensor are collocated; otherwise, they are not. We assume throughout this paper that r = 0; the case of non-collocated actuator and sensor is a topic of future work. (Note that Fig. 1 illustrates a collocated case, while Fig. 2 illustrates a non-collocated one.) The overall system of N plants under CC can be formalized as x˙ = Ax + BuCC , uCC = [ str1 (u1 (t)) , . . . , strN (uN (t)) ]T ,
(6)
T
yCC = [ str1 (y1 (t)) , . . . , strN (yN (t)) ] , where ui and yi are as in (1), and x, A and B are defined in (2). While (6) highlights the differences between CC and PAC, it is not convenient for analytical investigation since it does not present the periodicity of CC systems explicitly. An alternative representation can be given as follows: x˙ = Ax + b(t)ua , PN ua = i=1 stri (ui ), ys = c(t)x, (7) PN PN b(t) = i=1 stri (bi ), c(t) = i=1 stri (ci ),
where bi and ci are defined in (2), and ua ∈ R and ys ∈ R are the actuator and sensor outputs, respectively. Note PN that ys can also be represented as ys = i=1 stri (yi ), where yi is as in (1). To show that (6) and (7) are indeed equivalent, observe that, using (4), P P N N b(t)ua = i=1 stri (ui ) i=1 stri (bi ) = b1 str1 (u1 ) + . . . + bN strN (uN ) ≡ BuCC .
Thus, system (6) and (7) describe the same dynamics. Regarding the output, note that during control of Pi (s), yCC = [0, . . . , 0, yi , 0, . . . , 0]T and ys = yi . Therefore, yCC and ys carry the same information at any time. Since vectors b(t) and c(t) are T -periodic, (7) is a linear periodic system. This system is the main object of study in this paper. 2.2 Problems Both PAC and CC systems can be controlled by static output feedback: ui = −ki yi , ki ∈ R, i = 1, . . . , N. (8) For CC systems, (8) can be restated in terms of ua and ys : PN (9) ua = −k(t)ys , k(t) = i=1 stri (ki ), and this expression is used throughout this paper. For the case of PAC, all basic control problems — stabilization, reference tracking and disturbance rejection —
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are well understood and various methods for their solution have been developed. For the case of CC, none of these problems has been solved at any level of generality. Therefore, the problems addressed in this paper are: • Derive conditions for stabilizability of system (7) under control (9) and characterize reference tracking and disturbance rejection in the framework of the CC architecture. • Characterize the limitations of CC as compared with PAC. 3. APPROACHES 3.1 Taylor series approach Consider the linear periodic system PN x˙ = A(t)x, A(t) = i=1 stri (Ai ), (10) PN Ai ∈ RM ×M , i = 1, . . . , N, M = i=1 Mi . Here, A(t) could be viewed as the closed loop system matrix for a stroboscopically controlled system. Stability of (10) is determined by the eigenvalues of the monodromy matrix, which can be written as Φ = e AN τ · · · e A1 τ . (11) The eigenvalues of Φ are all but impossible to derive analytically. Therefore, a simplification is necessary. We use here the Taylor series approach. Specifically, expanding each matrix exponential in (11), we obtain: Φ = (I + AN τ + O((AN τ )2 )) · · · (I + A1 τ + O((A1 τ )2 )) P =I +τ N + H. O. T. i=1 Ai Neglecting the higher order terms, we obtain a simplified monodromy matrix: ˆ = I + τ PN Ai . (12) Φ i=1
Due to continuity of eigenvalues with respect to matrix elements, ∀ǫ > 0, ∃ τ ∗ > 0 such that τ < τ ∗ implies ˆ < ǫ, i = 1, . . . , N . Thus, for sufficiently |λi (Φ) − λi (Φ)| small time slots, the stability of (10) can be analyzed using the simplified monodromy matrix (12).
To simplify this expression, assume that ∀n ≥ 0, v(t) satisfies v(t) = v((n + 1)τ ), ∀t ∈ (nτ, (n + 1)τ ]. (16) This assumption implies that changes in v occur only at the beginning of each time slot. Then, the integrals in (15) simplify to Z nT +iτ eAi (nT +iτ −s) bi v(s)ds = Γi bi v(nT + iτ ). nT +(i−1)τ
Finally, define the lifted signals ve[n] ∈ RN and ye[n] ∈ RN as follows: ve[n] = [v(nT + τ ), v(nT + 2τ ), . . . , v((n + 1)T )]T ,
ye[n] = [ys (nT + τ ), ys (nT + 2τ ), . . . , ys ((n + 1)T )]T . Clearly, the lifted signals ve and ye are the sampled versions of v and ys , respectively. Therefore, introducing the notation ξ[n] := x(nT ), system (13) can be rewritten as e v [n], ξ[n + 1] = Φξ[n] + Be
e e v [n], ye[n] = Cξ[n] + De e = [ΦN · · · Φ2 Γ1 b1 , . . . , ΦN ΓN −1 bN −1 , ΓN bN ], B e C = [(c1 Φ1 )T , (c2 Φ2 Φ1 )T , . . . , (cN ΦN · · · Φ1 )T ]T , c1 Γ1 b1 0 ... 0 c 2 Φ 2 Γ1 b 1 c2 Γ2 b2 ... 0 e , D= .. .. .. . . . c N Φ N · · · Φ 2 Γ1 b 1 c N Φ N · · · Φ 3 Γ2 b 2 . . . c N ΓN b N (17) e ∈ RM ×N , C e ∈ RN ×M , D e ∈ RN ×N , M is as in where B (10), and Φ is the monodromy matrix (11). This N -input N -output discrete time LTI system is referred to as the lifted version of (13). This system is used in Section 5 for reference tracking and disturbance rejection. Note that a similar lifting approach has been used by Chen and Francis (1996). As a final remark in this section, observe that the system matrix of the lifted system (17) is the monodromy matrix of the original system (13); therefore, the lifted system is asymptotically stable iff the original system is asymptotically stable.
3.2 Lifting approach
4. STABILIZABILITY
Consider the linear periodic system x˙ = A(t)x + b(t)v, (13) ys = c(t)x, where A(t) is defined in (10), ys , b(t) and c(t) are as in (7), and v ∈ R. Here also, A(t) could be viewed as the closed loop system matrix of a stroboscopically controlled system, while v is an external reference signal. To describe the lifting approach, introduce the notation Z τ eAi s ds, (14) Φi = eAi τ , Γi = 0
and express x ((n + 1)T ) in terms of x(nT ) as follows: x((n + 1)T ) = ΦN ΦN −1 · · · Φ1 x(nT ) Z nT +τ eA1 (nT +τ −s) b1 v(s)ds + ΦN ΦN −1 · · · Φ2 (15) nT Z (n+1)T eAN ((n+1)T −s) bN v(s)ds. + ... + nT +(N −1)τ
The closed-loop system (7), (9) is given by x˙ = (A − b(t)k(t)c(t)) x. (18) For the case of non-interacting, one-dimensional plants, P1 (s), . . . , PN (s), the stability criterion of (18) was found by Ching et al. (2010) to be ki > NbiAciii . In this section, we consider the general case of interacting, multidimensional plants and specialize the obtained result to several particular cases. 4.1 Main result Let diag(Ji ) denote the block-diagonal matrix with the diagonal blocks Ji , i = 1, . . . , N . Let Dr (p) and Dr (p) be the open and closed disks, respectively, of radius r centered at p, i.e., Dr (p) = {s ∈ C : |s − p| < r}, (19) Dr (p) = {s ∈ C : |s − p| ≤ r}.
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Theorem 1. There exists τ ∗ such that for all τ < τ ∗ system (18) is asymptotically stable iff ki 1 λ A − diag(bi ci ) ∈ D T1 (− ). (20) N T
P 1 (s) 4 2
i=1
i=1
Taking into account the structure of bi and ci , the eigenvalues of this matrix are in the open unit disk D1 (0) iff condition (20) holds. Thus, for sufficiently small τ , the monodromy matrix Φ also has all eigenvalues in D1 (0), implying asymptotic stability of (18). Q.E.D.
Note that PAC system (2) under static output feedback (8) is described by x˙ = (A − diag(bi ki ci ))x, (23) implying that it is asymptotically stable iff Re{λ (A − diag(bi ki ci ))} < 0. (24) Thus, comparing (20) and (24), we conclude: • While the domain of asymptotic stability under PAC is the open left half plane, the domain of asymptotic stability under CC is the open disk D T1 (− T1 ). • To place the eigenvalues of the closed loop system at desired locations, the gains under CC must be N times larger than those under PAC.
P 2 (s)
1.5
D 1 (− T1 )
λ(Φ) ˆ λ( Φ)
T
1
0 −2
Proof. The monodromy matrix of system (18) can be written as Φ = e(A−bN kN cN )τ · · · e(A−b1 k1 c1 )τ , (21) where bi and ci are defined in (2). For system (18), the ˆ defined by (12), becomes simplified monodromy matrix Φ, P P N ˆ = I+τ Φ (A−bi ki ci ) = I+T A−τ N bi ki ci . (22)
2
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(a) Root loci of the plants.
0 0
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(b) An eigenvalue of Φ vs. the corˆ responding eigenvalue of Φ.
Fig. 3. Example 1 eigenvalues of the monodromy matrix Φ of the system at hand for various T ’s and k1 = k2 = 2. The behavior of one of these eigenvalues as a function of T , along with the corresponding eigenvalue of the simplified monodromy ˆ are shown in Fig. 3(b). As one can see, for matrix Φ, T < 0.3, i.e., τ < 0.15, the approximation is accurate. 4.3 Interacting one-dimensional plants Corollary 2. For interacting one-dimensional plants, condition (20) is satisfied if for each i = 1, . . . , N , either 1 when bi ci 6= 0, (26) Ri < T or 1 DRi (Aii ) ⊂ D T1 (− ) when bi ci = 0, (27) T PN where Ri = j=1,j6=i |Aij |.
Proof. The proof is based on the concept of Gershgorin disks (G-disks). Let A ∈ RM ×M be a matrix with elements Aij ∈ R. Then, (see Lancaster and Tismenetsky (1985)) the eigenvalues of A lie in the union of the G-disks DRi (Aii ), i = 1, . . . , M , where Ri is defined in Corollary 4.2 Non-interacting multidimensional plants 2. Based on this result, condition (20) is satisfied if every ki 1 Corollary 1. For non-interacting multidimensional plants, G-disk of A − diag(bi N ci ) lies within D T1 (− T ). Since for one-dimensional plants bi ∈ R and ci ∈ R, the Gcondition (20) is satisfied iff disks of A − diag(bi kNi ci ) are given by DRi (Aii − bi kNi ci ), 1 ki (25) i = 1, . . . , N . Clearly, the radii of DR (Aii − bi ki ci ) are λ Aii − bi ci ∈ D T1 (− ), i = 1, . . . , N. i N N T fixed. However, their centers can be placed arbitrarily Proof. Follows from (20), taking into account that for on the real axis by the choice of ki , if bi ci 6= 0. Thus, non-interacting plants, matrix A is block diagonal. Q.E.D. conditions (26) and (27) ensure that there exist ki ’s such that DRi (Aii − bi kNi ci ) ⊂ D T1 (− T1 ), implying that (20) is In terms of the root locus of each Pi (s), i = 1, . . . , N , satisfied. Q.E.D. Corollary 1 implies that, for sufficiently small τ , the CC system is stabilizable iff there is a segment of the root locus Example 2. Consider the plants in Example 1 and reduce corresponding to each Pi (s), which belongs to D T1 (− T1 ). them to one-dimensional models by keeping only the dominant poles: The following example illustrates this situation. 1.5 2 P1 (s) = , P2 (s) = . Example 1. Consider a CC system with two non-interacting s−1 s + 0.5 plants defined by A state space description of these plants along with inters+3 s+4 acting terms, can be represented as follows: P1 (s) = 2 , P2 (s) = 2 . s +s−2 s + 2.5s + 1 1 0.4 A= , b1 = b2 = 1, c1 = 1.5, c2 = 2. The root loci of these plants along with the disk D5 (−5) 0.4 −0.5 (i.e., T = 0.2) are shown in Fig. 3(a). As one can determine The G-disks D0.4 (1 − 0.75k1 ) and D0.4 (−0.5 − k2 ) along from the root loci, condition (25) is satisfied with this T for k1 ∈ [1.33, 25.14] and k2 ∈ [0, 25.33]. As it follows from with D1 (−1) (i.e., T = 1) are plotted in Fig. 4(a) for Theorem 1, these ki ’s ensure asymptotic stability of the k1 = k2 = 0 and in Fig. 4(b) for k1 = 3.2, k2 = 0. Since in CC system if τ = T2 is sufficiently small. To illustrate the latter case, the G-disks are within D1 (−1), condition how small τ should be, we numerically computed the (20) is satisfied ∀ T ≤ 1, because for any T1 < T2 , Below, we apply Theorem 1 to several particular cases, where condition (20) becomes more specific.
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(a) Gershgorin disks for the (b) Gershgorin disks open loop system. closed loop system.
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Fig. 4. Example 2 1 1 D T1 (− ) ⊂ D T1 (− ). 2 1 T2 T1
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4.4 Interacting multidimensional plants
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Corollary 3. For interacting multidimensional plants with M1 = . . . = MN , condition (20) is satisfied if there exists ki , i = 1, . . . , N , such that 1 Gi (ki ) ⊂ D T1 (− ), T where ki Gi (ki ) = {s ∈ C : ||(Aii − bi ci − sI)−1 ||−1 ≤ Ri }, (28) N α 1−α P P N N , (29) Ri = j=1,j6=i ||Aji || j=1,j6=i ||Aij ||
0 ≤ α ≤ 1, and ||.|| is any induced matrix norm with the convention that ||B −1 ||−1 = 0 iff B is singular. Proof. The proof is based on the concept of Generalized Gershgorin sets (GG sets). Let A be a matrix partitioned into blocks Aij ∈ RM ×M , M > 1. Then, (see Feingold and Varga (1962)) every eigenvalue of A is contained in the union of the GG sets given by Gi = {s ∈ C : ||(Aii − sI)−1 ||−1 ≤ Ri }, i = 1, . . . , N, where Ri is defined in (29). Now, for the matrix A − diag(bi kNi ci ), the GG sets are given by (28), (29). Thus, if there exist gains ki , i = 1, . . . , N , such that Gi (ki ) ⊂ D T1 (− T1 ), condition (20) is satisfied. Q.E.D. It turns out that, as ki is varied over the positive reals, the set Gi (ki ) includes segments of the root locus of Pi (s). This is because for any ki , λ Aii − bi kNi ci ∈ Gi (ki ). The relationship between the root locus and the GG sets is illustrated in the example given below. Example 3. Assume that the plants of Example 1 are interacting according to the following overall system matrix: −1 2 0.4 0 1 0 0 0.4 A= 0.4 0 −2.5 −1 .
0 0.4 1 0 Using the 2-norm for matrices and α = 0, we obtain R1 = R2 = 0.4. The sets G1 (k1 ) and G2 (k2 ) for the open and closed loop systems, along with the root loci for P1 (s) and P2 (s), are shown in Fig. 5, which also includes the disk D5 (−5). Specifically, Fig. 5(a) and 5(b) show G1 (0) and G2 (0) for the open loop system; none of these sets is in D5 (−5). Fig. 5(c) and 5(d) demonstrate G1 (3.2) and G2 (1) for the closed loop system, respectively, both of
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Fig. 5. Example 3 which are in D5 (−5). Thus, with these gains, condition (20) is satisfied for all T < 0.2. Lastly, as can be seen from the figure, Gi (ki ) moves along the root locus of Pi (s), i = 1, 2. 5. REFERENCE TRACKING AND DISTURBANCE REJECTION 5.1 General equations With a disturbance d(t), plant Pi (s), i = 1, . . . , N , given in (1), is described by x˙ i = Aii xi + bi ui + βi d(t), (30) yi = c i x i , where βi ∈ RMi . Let r(t) denote the reference to be tracked. Then, with the controller ui (t) = ki (r(t) − yi ), (31) the closed loop PAC system becomes x˙ = (A − B diag(ki )C)x + B[k1 , . . . , kN ]T r(t) + Bd [1, . . . , 1]T d(t), y = Cx,
(32)
Bd = [β 1 , . . . , β N ], β i = [0, . . . , 0, βiT , 0, . . . , 0]T , where A, B, and C are defined in (2). Guided by the application described in Eun et al. (2011), we assume that r(t) = r0 1(t), d(t) = d0 1(t), (33) where r0 , d0 ∈ R, and 1(t) denotes the unit step function. The methods for selecting ki ’s so that (32) has the desired tracking and disturbance rejection properties are wellknown (Kwakernaak and Sivan (1972)). For instance, if each Pi (s), i = 1, . . . , N , is minimum-phase and of relative degree not higher than 2, the steady state error can be made arbitrarily small. To investigate the steady state error under CC, we first formulate the CC version of the tracking and disturbance rejection problem. As far as the disturbance is concerned, assume that d(t) acts strobosopically. Such a disturbance arises, for example, in the fusing stage of the xerographic process, where paper, viewed as a disturbance, affects
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each wedge once per revolution. Under the stroboscopic disturbance, CC system (7) is modified to x˙ = Ax + b(t)ua + β(t)d(t), ys = c(t)x, (34) PN str (β β(t) = ), i modN (i+m−1)+1 i=1
where β i is as in (32) and m ∈ {0, . . . , N − 1}. If m = 0, the disturbance and the actuator are collocated; otherwise, they are not. As far as tracking is concerned, introduce the stroboscopic performance output z(t): PN z(t) = h(t)x, h(t) = stri (cmodN (i+l−1)+1 ), (35) i=1
where ci is defined in (2) and l ∈ {0, . . . , N − 1}. If l = 0, the performance output and the actuator are collocated. Note that in this case, z(t) = ys (t) for all t, where ys is the sensor output defined in (7). An example of this situation is the charging stage of the xerographic process, where the voltage in each wedge of the photoreceptor must track the reference voltage when it leaves the actuator (charger). If l ≥ 1, the performance output and the actuator are not collocated, and z(t) 6= ys (t). This case arises in the fusing stage of the xerographic process, where z is the temperature of the wedge at the paper nip (i.e., the point where paper comes in contact with the fuser), while ys is the temperature of the wedge at the actuator (external heating roll). If the actuator, the disturbance, and the performance output z(t) are all collocated, we use the controller ua = k(t) (r(t) − ys ) , (36) where k(t) is as in (9). With (36) and the performance output (35), the closed loop CC system becomes x˙ = (A − b(t)k(t)c(t))x + b(t)k(t)r(t) + β(t)d(t), (37) z = h(t)x. Unlike PAC, tracking in CC systems for all time moments is not generally possible because of stroboscopic actuation. As an example, consider a plant Pi (s) in a system composed of N non-interacting plants. When the actuator is not acting on this plant, Pi (s) undergoes its natural dynamics, which implies that the performance output z(t) cannot constantly track the reference over an entire period. Under similar reasoning, the output z(t) cannot constantly reject disturbances over an entire period. As a result, the problem of tracking and disturbance rejection in CC systems considers z(t) only at the end of each time slot and investigates the error e(nτ ) = r0 − z(nτ ). (38) Below, we first consider the case of collocated actuator, disturbance, and performance output and then investigate the non-collocated case. 5.2 Collocated case For system (37), the lifted performance output ze[n] and the lifted tracking error ee[n] are given by: ze[n] = [z(nT + τ ), z(nT + 2τ ), . . . , z((n + 1)T )]T , (39) ee[n] = [1, . . . , 1]T r0 − ze[n], where, due to collocation, z(nT + iτ ) = ci x(nT + iτ ), i = 1, . . . , N . Applying the lifting approach to (37), as shown in Section 3.2, yields the lifted system
e [k1 r0 , . . . , kN r0 ]T + B ed [d0 , . . . , d0 ]T , ξ[n + 1] = Φξ[n] + B e e 1 r 0 , . . . , k N r0 ] T + D e d [d0 , . . . , d0 ]T , ze[n] = Hξ[n] + D[k (40) where ξ[n] = x(nτ ); Φ = ΦN · · · Φ1 , Φi = e(A−bi ki ci )τ ; Rτ e and D e are as Γi = 0 e(A−bi ki ci )s ds, i = 1, . . . , N ; B e is the same as C e in (17); and B ed and defined in (17); H e d are the same as B e and D e defined in (17) except that D all bi ’s are replaced by β i .
If the eigenvalues of the monodromy Φ are within D1 (0), system (40) is asymptotically stable, and in the steady state, the lifted error ee[∞] is given by e − Φ)−1 B e + D)[k e 1 , . . . , k N ] T r0 ee[∞] = [1, . . . , 1]T − (H(I e − Φ)−1 B ed + D e d [1, . . . , 1]T d0 . + H(I
Observe that if τ is sufficiently small, the results of Section 4 can be applied to analyze stability of (40). Specifically, the gains ki , i = 1, . . . , N , can be chosen to place the eigenvalues of A − diag(bi kNi ci ) within D T1 (− T1 ), or equivalently, the eigenvalues of Φ within D1 (0). Note that the range of gains ki that achieve this pole placement is always finite, because for any T , there exist segments of the root loci, corresponding to large ki ’s, that are outside of D T1 (− T1 ). Therefore, unlike the PAC case: Proposition 1. For any τ > 0, if condition (20) is satisfied, the norm of the lifted steady state error ke e[∞]k in the collocated case is bounded from below by a positive constant. As far as controller design is concerned, assuming that τ is sufficiently small, the gains ki must be chosen such that condition (20) is satisfied and the desired dynamic and steady state tracking and disturbance rejection specifications are met. If no ki ’s achieve the specifications, the period T must be decreased, so that the domain of stability D T1 (− T1 ) is enlarged, and the process of selecting ki ’s is repeated anew. Example 4. Consider a stroboscopic disturbance d(t) acting on the plants of Example 1. Assume that the disturbance and performance output z(t) are collocated with the actuator, d(t) = −0.51(t), β1 = b1 , β2 = b2 and the reference signal is r(t) = 1(t). To illustrate Proposition 1, we use controller (36) and, for various T ’s, plot the minimum ke e[∞]k attainable such that the eigenvalues of A − bi kNi ci are within D T1 (− T1 ) (see Fig. 6(a)). Clearly, the minimum error is non-zero and is monotonically increasing in T . (Note that the cusp in the curve of Fig. 6(a) is due to the break-in points of the root loci of the plants.) Fig. 6(b) shows the transient response of the sampled performance output z(nτ ), for k1 = k2 = 3 and for k1 = k2 = 20. Obviously, with higher gains, z(nτ ) tracks r(t) and rejects d(t) faster and with a smaller error. 5.3 Non-collocated case For the case in which either the performance output or the disturbance is not collocated with the actuator, we modify the controller (36) as follows: ua = k(t) (g(t)r(t) − ys ) + q(t)d(t), PN PN (41) g(t) = i=1 stri (gi ), q(t) = i=1 stri (qi ),
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Note that to employ (45) and (46), the matrices A, bi , ci , i = 1, . . . , N , of (2) and the disturbance d0 must be known precisely. Generally, this is not the case, and therefore we have: Proposition 2. For any τ > 0, if condition (20) is satisfied and the system parameters are not known precisely, then the norm of the lifted steady state error ke e[∞]k in the non-collocated case is bounded from below by a positive constant.
p erformance output, z
minimum ke e[∞]k
0.8 0.6 0.4 0.2 0 0
k1 = k2 = 3 1.5
k 1 = k 2 = 20
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0
0.2
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(a) Minimum error vs. T
1
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1
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(b) Transient response.
Fig. 6. Example 4 where gi ∈ R and qi ∈ R, i = 1, . . . , N , are to be selected to compensate for the non-collocation. System (34), (35) under feedback (41) is given by x˙ = (A − b(t)k(t)c(t))x + (β(t) + b(t)q(t))d(t) + b(t)k(t)g(t)r(t), z = h(t)x.
(42)
To simplify notation, let col(ai ) denote the column vector [a1 , . . . , aN ]T . Then, applying the lifting approach to (42) yields the system e col(ki gi )r0 ξ[n + 1] = Φξ[n] + B e col(qi ) + B ed col(1))d0 , + (B e e col(ki gi )r0 ze[n] = Hξ[n] +D e col(qi ) + D e d col(1))d0 , + (D
Note that (41) is a feedback/feedforward controller where gi ’s and qi ’s account for the feedforward part, while ki ’s account for the feedback. As a final remark, observe that in the case of nonstroboscopic disturbances, β 1 = . . . = β N , and β i has no special structure. An example of such disturbance arises in the fusing stage of the xerographic process, where the power delivered to the internal lamp, viewed as a disturbance, enters the system non-stroboscopically. The procedure presented in this section remains valid for such disturbances, as is demonstrated in Eun et al. (2011). 6. CONCLUSIONS AND FUTURE WORK
(43)
where ξ[n] = x(nτ ); col(1) = [1, . . . , 1]T , Φ = ΦN · · · Φ1 , Rτ Φi = e(A−bi ki ci )τ ; Γi = 0 e(A−bi ki ci )s ds, i = 1, . . . , N ; e is defined in (17); H e and D e are the same as C e B e and D in (17), respectively, except that all ci ’s are reed and D e d are the same placed by cmodN (i+l−1)+1 ; and B e e as B and D defined in (17) except that all ci ’s are replaced by cmodN (i+l−1)+1 and all bi ’s are replaced by β modN (i+m−1)+1 . Assuming that system (43) is asymptotically stable, the lifted tracking error (39) satisfies e − Φ)−1 B e+D e col(gi ki ) r0 ee[∞] = col(1) − H(I e − Φ)−1 (B e col(qi ) + B ed col(1)) − H(I e col(qi ) + D e d col(1) d0 . +D (44) Thus, to ensure perfect tracking, i.e., ee[∞] = 0, gi ’s and qi ’s must satisfy 1 col(gi ) = diag( )Φ−1 col(1), (45) ki CL e − Φ)−1 B ed + D e d col(1), col(qi ) = −Φ−1 H(I (46) CL
e −Φ)−1 B e + D. e Observe that gi ’s and qi ’s where ΦCL = H(I can be calculated from (45) and (46) if ΦCL is invertible. It can be shown that this matrix is indeed invertible iff # " e Φ−I B = M + N. (47) rank e D e H
As far as controller design is concerned, assuming that τ is sufficiently small, the gains ki must be chosen such that condition (20) is satisfied and that the desired transient specifications are met. The gi ’s and qi ’s can then be calculated from (45) and (46), respectively, to achieve ee[∞] = 0.
This paper developed CC theory for the case of static output feedback and compared it with PAC. Future work includes: cyclic control using state space feedback; cyclic control using dynamic output feedback; extension of the theory developed to the case of non-collocated actuators and sensors; and disturbance rejection when the disturbances are unknown. Solutions to these problems will provide a relatively complete theory of cyclic control systems. REFERENCES Anderson, B. and Moore, J. (2007). Optimal Control: Linear Quadratic Methods. Dover Publications. Chen, T. and Francis, B. (1996). Optimal Sampled-Data Control Systems. Springer. Childs, D. (1993). Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis. Wiley. Ching, S., Eun, Y., Gross, E., Hamby, E., Kabamba, P., Meerkov, S., and Menezes, A. (2010). Modeling and control of cyclic systems in xerography. Proceedings of the American Control Conference. Drozda, T. (ed.) (1983). Tool and Manufacturing Engineers Handbook. Society of Manufacturing Engineers, fourth edition. Eun, Y., Gross, E.M., Kabamba, P., Meerkov, S., Menezes, A., and Ossareh, H. (2011). Cyclic control: The case of static output feedback. http://www-personal.umich.edu/~hamido/IFAC11/ifacconf.pdf .
Feingold, D. and Varga, R. (1962). Block diagonally dominant matrices and generalizations of the gerschgorin circle theorem. Pacific J. Math. Hamby, E. and Gross, E. (2004). A control-oriented survey of xerographic systems: Basic concepts to new frontiers. Proceedings of the American Control Conference, 2615– 2629. Kwakernaak, H. and Sivan, R. (1972). Linear Optimal Control Systems. Wiley. Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices. Academic Press, second edition.
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Cyclic Control: The Case of Static Output Feedback Y. Eun ∗ E.M. Gross ∗ P.T. Kabamba ∗∗ S.M. Meerkov ∗∗∗ A.A. Menezes ∗∗ H.R. Ossareh ∗∗∗ ∗
Xerox Corporation, Webster, NY 14580 USA (e-mail: Eric.Gross, [email protected]) ∗∗ Aerospace Engineering Department, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: kabamba, [email protected]) ∗∗∗ Electrical Engineering Department, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: smm, [email protected]) Abstract: This paper develops a control theory for systems comprised of multiple plants controlled by a single actuator and sensor. Such systems arise in rotating machinery with actuators and sensors fixed in inertial space. Representing the rotating device as a set of discrete wedges leads to a model mentioned above. In such systems, each plant can be controlled only intermittently, e.g., once per revolution. We refer to this situation as cyclic control (CC). The paper investigates the problems of stabilizability, reference tracking, and disturbance rejection in CC systems using static output feedback. Keywords: Cyclic control, Periodic control, Stabilizability, Reference tracking, Disturbance rejection, Lifting, Gershgorin circles. C1 (s)
u1
P1 (s) S
. . .
r
S
S Actuator
. . .
ua
Sensor
uN
PN (s)
Fig. 2. Rotating machinery with N discrete wedges.
Fig. 1. Cyclic control (CC) system. 1. INTRODUCTION This paper considers a system comprised of a finite number of plants P1 (s), . . . , PN (s) that may or may not interact. If each plant is equipped with an actuator and a sensor, a controller, Ci (s), i = 1, . . . , N , can be designed for each Pi (s) using a plethora of techniques (Anderson and Moore (2007), Kwakernaak and Sivan (1972)) to ensure satisfactory system performance. We refer to such an architecture as permanently acting control (PAC). There are, however, situations where only one actuator and one sensor are available to control all N plants (see the examples below). In these situations, the actuation and sensing can be allocated to each Pi (s) only intermittently, for instance, in a round-robin or cyclic manner. This implies that within a period, T , each plant is controlled T . We refer to exactly once during the time slot τ = N such an architecture as cyclic control (CC, see Fig. 1). The purpose of this paper is twofold: 1) to develop a control-theoretic methodology for CC systems using static output feedback of individual plants, Pi (s), i = 1, . . . , N ; and 978-3-902661-93-7/11/$20.00 © 2011 IFAC
–
CN (s)
ys
2) to quantify the performance losses due to CC in comparison with PAC. Systems where only intermittent or stroboscopic control is possible, arise in rotating machinery with actuators and sensors fixed in inertial space. In these systems, the rotating device can be represented as N discrete wedges (or plants), which leads to the CC architecture (see Fig. 2). Examples of such systems are the charging and fusing stages of the xerographic process (Hamby and Gross (2004)). During charging, the rotating photoreceptor, which is not uniform due to material and surface irregularities, must be charged uniformly by an inertially fixed charger to ensure high-quality print. To accomplish this, each wedge of the photoreceptor must be charged using a different charging voltage to compensate for the irregularities. Similarly, during fusing, each wedge of the fuser must be heated uniformly by an inertially fixed heater; however, different actuation is required to compensate for material non-uniformity and fuser eccentricity (see Eun et al. (2011) for modeling and CC of this system). Similar situations arise in drilling and milling machines (Drozda (1983)) and turbo machinery (Childs (1993)).
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Systems with CC architecture are periodic in the sense that their actuation and sensing are periodic. Obviously, the general theory of linear periodic systems (e.g., Floquet/Lyapunov theory) can be applied to their analysis and design. However, since the periodicity of CC is a specific one — time invariant within each time slot — more powerful, constructive control-theoretic techniques can be developed. At present, no results in this direction are available with the exception of Ching et al. (2010), where the stability problem for the case of one-dimensional noninteracting plants has been investigated. In the current paper, we address all standard problems of control — stabilization, reference tracking, and disturbance rejection — for all possible cases, including the general case of multidimensional plants with interactions. Accordingly, the outline of this paper is as follows. Section 2 is devoted to modeling CC systems and problem formulation. Section 3 presents the main results on stabilizability. Section 4 addresses the problem of reference tracking and disturbance rejection. Section 5 provides the conclusions and topics for future research. Due to space limitations, an application to a xerographic process is not presented here, but can be found in Eun et al. (2011). 2. MODELING AND PROBLEM FORMULATION 2.1 Models Assume that each plant Pi (s), i = 1, . . . , N , is a SISO system described by x˙ i = Aii xi + bi ui , (1) yi = c i x i , xi ∈ RMi , ui , yi ∈ R, Aii ∈ RMi ×Mi , bi , cTi ∈ RMi . T Then, with x = xT1 , . . . , xTN , the overall system of N interacting plants under PAC can be formalized as x˙ = Ax + Bu, y = Cx, y = [y1 , . . . , yN ]T , u = [u1 , . . . , uN ]T , A11 A12 . . . A1N A21 A22 . . . A2N A= . , (2) .. . . .. . . AN 1 AN 2 . . . AN N B = [b1 , . . . , bN ], C = [cT1 , . . . , cTN ]T , T bi = 0, . . . , 0, bTi , 0, . . . 0 , i = 1, . . . , N, ci = [0, . . . , 0, ci , 0, . . . 0] , i = 1, . . . , N,
where xi , yi , ui , Aii , bi , and ci are as in (1) and Aij , i 6= j, are matrices of appropriate dimensions that describe interactions between Pi (s) and Pj (s). To model plants (1) under CC, consider a vector-function v(t) ∈ RV , V ≥ 1, and introduce the following stroboscope operator: v(t), t ∈ (nT + (i − 1)τ, nT + iτ ], stri (v(t)) = (3) 0, otherwise, i = 1, . . . , N ; n ∈ {0, 1, . . .}, where T and τ are, as before, the period of cyclic control and the time slot, respectively. Clearly, for any v1 (t) ∈ RV1 , V1 ≥ 1, and v2 (t) ∈ R,
v1 (t)stri (v2 (t)), i = j, 0, i= 6 j, ∀t ≥ 0.
stri (v1 (t))strj (v2 (t)) =
(4)
Using the stroboscope operator, each plant Pi (s), i = 1, . . . , N , under stroboscopic actuation and sensing is described by x˙ i = Aii xi + bi stri (ui ), (5) yi = strmodN (i+r−1)+1 (ci xi ), where r ∈ {0, . . . , N −1} and all other variables are defined in (1). If r = 0, the actuator and sensor are collocated; otherwise, they are not. We assume throughout this paper that r = 0; the case of non-collocated actuator and sensor is a topic of future work. (Note that Fig. 1 illustrates a collocated case, while Fig. 2 illustrates a non-collocated one.) The overall system of N plants under CC can be formalized as x˙ = Ax + BuCC , uCC = [ str1 (u1 (t)) , . . . , strN (uN (t)) ]T ,
(6)
T
yCC = [ str1 (y1 (t)) , . . . , strN (yN (t)) ] , where ui and yi are as in (1), and x, A and B are defined in (2). While (6) highlights the differences between CC and PAC, it is not convenient for analytical investigation since it does not present the periodicity of CC systems explicitly. An alternative representation can be given as follows: x˙ = Ax + b(t)ua , PN ua = i=1 stri (ui ), ys = c(t)x, (7) PN PN b(t) = i=1 stri (bi ), c(t) = i=1 stri (ci ),
where bi and ci are defined in (2), and ua ∈ R and ys ∈ R are the actuator and sensor outputs, respectively. Note PN that ys can also be represented as ys = i=1 stri (yi ), where yi is as in (1). To show that (6) and (7) are indeed equivalent, observe that, using (4), P P N N b(t)ua = i=1 stri (ui ) i=1 stri (bi ) = b1 str1 (u1 ) + . . . + bN strN (uN ) ≡ BuCC .
Thus, system (6) and (7) describe the same dynamics. Regarding the output, note that during control of Pi (s), yCC = [0, . . . , 0, yi , 0, . . . , 0]T and ys = yi . Therefore, yCC and ys carry the same information at any time. Since vectors b(t) and c(t) are T -periodic, (7) is a linear periodic system. This system is the main object of study in this paper. 2.2 Problems Both PAC and CC systems can be controlled by static output feedback: ui = −ki yi , ki ∈ R, i = 1, . . . , N. (8) For CC systems, (8) can be restated in terms of ua and ys : PN (9) ua = −k(t)ys , k(t) = i=1 stri (ki ), and this expression is used throughout this paper. For the case of PAC, all basic control problems — stabilization, reference tracking and disturbance rejection —
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are well understood and various methods for their solution have been developed. For the case of CC, none of these problems has been solved at any level of generality. Therefore, the problems addressed in this paper are: • Derive conditions for stabilizability of system (7) under control (9) and characterize reference tracking and disturbance rejection in the framework of the CC architecture. • Characterize the limitations of CC as compared with PAC. 3. APPROACHES 3.1 Taylor series approach Consider the linear periodic system PN x˙ = A(t)x, A(t) = i=1 stri (Ai ), (10) PN Ai ∈ RM ×M , i = 1, . . . , N, M = i=1 Mi . Here, A(t) could be viewed as the closed loop system matrix for a stroboscopically controlled system. Stability of (10) is determined by the eigenvalues of the monodromy matrix, which can be written as Φ = e AN τ · · · e A1 τ . (11) The eigenvalues of Φ are all but impossible to derive analytically. Therefore, a simplification is necessary. We use here the Taylor series approach. Specifically, expanding each matrix exponential in (11), we obtain: Φ = (I + AN τ + O((AN τ )2 )) · · · (I + A1 τ + O((A1 τ )2 )) P =I +τ N + H. O. T. i=1 Ai Neglecting the higher order terms, we obtain a simplified monodromy matrix: ˆ = I + τ PN Ai . (12) Φ i=1
Due to continuity of eigenvalues with respect to matrix elements, ∀ǫ > 0, ∃ τ ∗ > 0 such that τ < τ ∗ implies ˆ < ǫ, i = 1, . . . , N . Thus, for sufficiently |λi (Φ) − λi (Φ)| small time slots, the stability of (10) can be analyzed using the simplified monodromy matrix (12).
To simplify this expression, assume that ∀n ≥ 0, v(t) satisfies v(t) = v((n + 1)τ ), ∀t ∈ (nτ, (n + 1)τ ]. (16) This assumption implies that changes in v occur only at the beginning of each time slot. Then, the integrals in (15) simplify to Z nT +iτ eAi (nT +iτ −s) bi v(s)ds = Γi bi v(nT + iτ ). nT +(i−1)τ
Finally, define the lifted signals ve[n] ∈ RN and ye[n] ∈ RN as follows: ve[n] = [v(nT + τ ), v(nT + 2τ ), . . . , v((n + 1)T )]T ,
ye[n] = [ys (nT + τ ), ys (nT + 2τ ), . . . , ys ((n + 1)T )]T . Clearly, the lifted signals ve and ye are the sampled versions of v and ys , respectively. Therefore, introducing the notation ξ[n] := x(nT ), system (13) can be rewritten as e v [n], ξ[n + 1] = Φξ[n] + Be
e e v [n], ye[n] = Cξ[n] + De e = [ΦN · · · Φ2 Γ1 b1 , . . . , ΦN ΓN −1 bN −1 , ΓN bN ], B e C = [(c1 Φ1 )T , (c2 Φ2 Φ1 )T , . . . , (cN ΦN · · · Φ1 )T ]T , c1 Γ1 b1 0 ... 0 c 2 Φ 2 Γ1 b 1 c2 Γ2 b2 ... 0 e , D= .. .. .. . . . c N Φ N · · · Φ 2 Γ1 b 1 c N Φ N · · · Φ 3 Γ2 b 2 . . . c N ΓN b N (17) e ∈ RM ×N , C e ∈ RN ×M , D e ∈ RN ×N , M is as in where B (10), and Φ is the monodromy matrix (11). This N -input N -output discrete time LTI system is referred to as the lifted version of (13). This system is used in Section 5 for reference tracking and disturbance rejection. Note that a similar lifting approach has been used by Chen and Francis (1996). As a final remark in this section, observe that the system matrix of the lifted system (17) is the monodromy matrix of the original system (13); therefore, the lifted system is asymptotically stable iff the original system is asymptotically stable.
3.2 Lifting approach
4. STABILIZABILITY
Consider the linear periodic system x˙ = A(t)x + b(t)v, (13) ys = c(t)x, where A(t) is defined in (10), ys , b(t) and c(t) are as in (7), and v ∈ R. Here also, A(t) could be viewed as the closed loop system matrix of a stroboscopically controlled system, while v is an external reference signal. To describe the lifting approach, introduce the notation Z τ eAi s ds, (14) Φi = eAi τ , Γi = 0
and express x ((n + 1)T ) in terms of x(nT ) as follows: x((n + 1)T ) = ΦN ΦN −1 · · · Φ1 x(nT ) Z nT +τ eA1 (nT +τ −s) b1 v(s)ds + ΦN ΦN −1 · · · Φ2 (15) nT Z (n+1)T eAN ((n+1)T −s) bN v(s)ds. + ... + nT +(N −1)τ
The closed-loop system (7), (9) is given by x˙ = (A − b(t)k(t)c(t)) x. (18) For the case of non-interacting, one-dimensional plants, P1 (s), . . . , PN (s), the stability criterion of (18) was found by Ching et al. (2010) to be ki > NbiAciii . In this section, we consider the general case of interacting, multidimensional plants and specialize the obtained result to several particular cases. 4.1 Main result Let diag(Ji ) denote the block-diagonal matrix with the diagonal blocks Ji , i = 1, . . . , N . Let Dr (p) and Dr (p) be the open and closed disks, respectively, of radius r centered at p, i.e., Dr (p) = {s ∈ C : |s − p| < r}, (19) Dr (p) = {s ∈ C : |s − p| ≤ r}.
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Theorem 1. There exists τ ∗ such that for all τ < τ ∗ system (18) is asymptotically stable iff ki 1 λ A − diag(bi ci ) ∈ D T1 (− ). (20) N T
P 1 (s) 4 2
i=1
i=1
Taking into account the structure of bi and ci , the eigenvalues of this matrix are in the open unit disk D1 (0) iff condition (20) holds. Thus, for sufficiently small τ , the monodromy matrix Φ also has all eigenvalues in D1 (0), implying asymptotic stability of (18). Q.E.D.
Note that PAC system (2) under static output feedback (8) is described by x˙ = (A − diag(bi ki ci ))x, (23) implying that it is asymptotically stable iff Re{λ (A − diag(bi ki ci ))} < 0. (24) Thus, comparing (20) and (24), we conclude: • While the domain of asymptotic stability under PAC is the open left half plane, the domain of asymptotic stability under CC is the open disk D T1 (− T1 ). • To place the eigenvalues of the closed loop system at desired locations, the gains under CC must be N times larger than those under PAC.
P 2 (s)
1.5
D 1 (− T1 )
λ(Φ) ˆ λ( Φ)
T
1
0 −2
Proof. The monodromy matrix of system (18) can be written as Φ = e(A−bN kN cN )τ · · · e(A−b1 k1 c1 )τ , (21) where bi and ci are defined in (2). For system (18), the ˆ defined by (12), becomes simplified monodromy matrix Φ, P P N ˆ = I+τ Φ (A−bi ki ci ) = I+T A−τ N bi ki ci . (22)
2
6
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−8
−6
−4
−2
0
2
(a) Root loci of the plants.
0 0
0.2
0.4
T
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(b) An eigenvalue of Φ vs. the corˆ responding eigenvalue of Φ.
Fig. 3. Example 1 eigenvalues of the monodromy matrix Φ of the system at hand for various T ’s and k1 = k2 = 2. The behavior of one of these eigenvalues as a function of T , along with the corresponding eigenvalue of the simplified monodromy ˆ are shown in Fig. 3(b). As one can see, for matrix Φ, T < 0.3, i.e., τ < 0.15, the approximation is accurate. 4.3 Interacting one-dimensional plants Corollary 2. For interacting one-dimensional plants, condition (20) is satisfied if for each i = 1, . . . , N , either 1 when bi ci 6= 0, (26) Ri < T or 1 DRi (Aii ) ⊂ D T1 (− ) when bi ci = 0, (27) T PN where Ri = j=1,j6=i |Aij |.
Proof. The proof is based on the concept of Gershgorin disks (G-disks). Let A ∈ RM ×M be a matrix with elements Aij ∈ R. Then, (see Lancaster and Tismenetsky (1985)) the eigenvalues of A lie in the union of the G-disks DRi (Aii ), i = 1, . . . , M , where Ri is defined in Corollary 4.2 Non-interacting multidimensional plants 2. Based on this result, condition (20) is satisfied if every ki 1 Corollary 1. For non-interacting multidimensional plants, G-disk of A − diag(bi N ci ) lies within D T1 (− T ). Since for one-dimensional plants bi ∈ R and ci ∈ R, the Gcondition (20) is satisfied iff disks of A − diag(bi kNi ci ) are given by DRi (Aii − bi kNi ci ), 1 ki (25) i = 1, . . . , N . Clearly, the radii of DR (Aii − bi ki ci ) are λ Aii − bi ci ∈ D T1 (− ), i = 1, . . . , N. i N N T fixed. However, their centers can be placed arbitrarily Proof. Follows from (20), taking into account that for on the real axis by the choice of ki , if bi ci 6= 0. Thus, non-interacting plants, matrix A is block diagonal. Q.E.D. conditions (26) and (27) ensure that there exist ki ’s such that DRi (Aii − bi kNi ci ) ⊂ D T1 (− T1 ), implying that (20) is In terms of the root locus of each Pi (s), i = 1, . . . , N , satisfied. Q.E.D. Corollary 1 implies that, for sufficiently small τ , the CC system is stabilizable iff there is a segment of the root locus Example 2. Consider the plants in Example 1 and reduce corresponding to each Pi (s), which belongs to D T1 (− T1 ). them to one-dimensional models by keeping only the dominant poles: The following example illustrates this situation. 1.5 2 P1 (s) = , P2 (s) = . Example 1. Consider a CC system with two non-interacting s−1 s + 0.5 plants defined by A state space description of these plants along with inters+3 s+4 acting terms, can be represented as follows: P1 (s) = 2 , P2 (s) = 2 . s +s−2 s + 2.5s + 1 1 0.4 A= , b1 = b2 = 1, c1 = 1.5, c2 = 2. The root loci of these plants along with the disk D5 (−5) 0.4 −0.5 (i.e., T = 0.2) are shown in Fig. 3(a). As one can determine The G-disks D0.4 (1 − 0.75k1 ) and D0.4 (−0.5 − k2 ) along from the root loci, condition (25) is satisfied with this T for k1 ∈ [1.33, 25.14] and k2 ∈ [0, 25.33]. As it follows from with D1 (−1) (i.e., T = 1) are plotted in Fig. 4(a) for Theorem 1, these ki ’s ensure asymptotic stability of the k1 = k2 = 0 and in Fig. 4(b) for k1 = 3.2, k2 = 0. Since in CC system if τ = T2 is sufficiently small. To illustrate the latter case, the G-disks are within D1 (−1), condition how small τ should be, we numerically computed the (20) is satisfied ∀ T ≤ 1, because for any T1 < T2 , Below, we apply Theorem 1 to several particular cases, where condition (20) becomes more specific.
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Corollary 3. For interacting multidimensional plants with M1 = . . . = MN , condition (20) is satisfied if there exists ki , i = 1, . . . , N , such that 1 Gi (ki ) ⊂ D T1 (− ), T where ki Gi (ki ) = {s ∈ C : ||(Aii − bi ci − sI)−1 ||−1 ≤ Ri }, (28) N α 1−α P P N N , (29) Ri = j=1,j6=i ||Aji || j=1,j6=i ||Aij ||
0 ≤ α ≤ 1, and ||.|| is any induced matrix norm with the convention that ||B −1 ||−1 = 0 iff B is singular. Proof. The proof is based on the concept of Generalized Gershgorin sets (GG sets). Let A be a matrix partitioned into blocks Aij ∈ RM ×M , M > 1. Then, (see Feingold and Varga (1962)) every eigenvalue of A is contained in the union of the GG sets given by Gi = {s ∈ C : ||(Aii − sI)−1 ||−1 ≤ Ri }, i = 1, . . . , N, where Ri is defined in (29). Now, for the matrix A − diag(bi kNi ci ), the GG sets are given by (28), (29). Thus, if there exist gains ki , i = 1, . . . , N , such that Gi (ki ) ⊂ D T1 (− T1 ), condition (20) is satisfied. Q.E.D. It turns out that, as ki is varied over the positive reals, the set Gi (ki ) includes segments of the root locus of Pi (s). This is because for any ki , λ Aii − bi kNi ci ∈ Gi (ki ). The relationship between the root locus and the GG sets is illustrated in the example given below. Example 3. Assume that the plants of Example 1 are interacting according to the following overall system matrix: −1 2 0.4 0 1 0 0 0.4 A= 0.4 0 −2.5 −1 .
0 0.4 1 0 Using the 2-norm for matrices and α = 0, we obtain R1 = R2 = 0.4. The sets G1 (k1 ) and G2 (k2 ) for the open and closed loop systems, along with the root loci for P1 (s) and P2 (s), are shown in Fig. 5, which also includes the disk D5 (−5). Specifically, Fig. 5(a) and 5(b) show G1 (0) and G2 (0) for the open loop system; none of these sets is in D5 (−5). Fig. 5(c) and 5(d) demonstrate G1 (3.2) and G2 (1) for the closed loop system, respectively, both of
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Fig. 5. Example 3 which are in D5 (−5). Thus, with these gains, condition (20) is satisfied for all T < 0.2. Lastly, as can be seen from the figure, Gi (ki ) moves along the root locus of Pi (s), i = 1, 2. 5. REFERENCE TRACKING AND DISTURBANCE REJECTION 5.1 General equations With a disturbance d(t), plant Pi (s), i = 1, . . . , N , given in (1), is described by x˙ i = Aii xi + bi ui + βi d(t), (30) yi = c i x i , where βi ∈ RMi . Let r(t) denote the reference to be tracked. Then, with the controller ui (t) = ki (r(t) − yi ), (31) the closed loop PAC system becomes x˙ = (A − B diag(ki )C)x + B[k1 , . . . , kN ]T r(t) + Bd [1, . . . , 1]T d(t), y = Cx,
(32)
Bd = [β 1 , . . . , β N ], β i = [0, . . . , 0, βiT , 0, . . . , 0]T , where A, B, and C are defined in (2). Guided by the application described in Eun et al. (2011), we assume that r(t) = r0 1(t), d(t) = d0 1(t), (33) where r0 , d0 ∈ R, and 1(t) denotes the unit step function. The methods for selecting ki ’s so that (32) has the desired tracking and disturbance rejection properties are wellknown (Kwakernaak and Sivan (1972)). For instance, if each Pi (s), i = 1, . . . , N , is minimum-phase and of relative degree not higher than 2, the steady state error can be made arbitrarily small. To investigate the steady state error under CC, we first formulate the CC version of the tracking and disturbance rejection problem. As far as the disturbance is concerned, assume that d(t) acts strobosopically. Such a disturbance arises, for example, in the fusing stage of the xerographic process, where paper, viewed as a disturbance, affects
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each wedge once per revolution. Under the stroboscopic disturbance, CC system (7) is modified to x˙ = Ax + b(t)ua + β(t)d(t), ys = c(t)x, (34) PN str (β β(t) = ), i modN (i+m−1)+1 i=1
where β i is as in (32) and m ∈ {0, . . . , N − 1}. If m = 0, the disturbance and the actuator are collocated; otherwise, they are not. As far as tracking is concerned, introduce the stroboscopic performance output z(t): PN z(t) = h(t)x, h(t) = stri (cmodN (i+l−1)+1 ), (35) i=1
where ci is defined in (2) and l ∈ {0, . . . , N − 1}. If l = 0, the performance output and the actuator are collocated. Note that in this case, z(t) = ys (t) for all t, where ys is the sensor output defined in (7). An example of this situation is the charging stage of the xerographic process, where the voltage in each wedge of the photoreceptor must track the reference voltage when it leaves the actuator (charger). If l ≥ 1, the performance output and the actuator are not collocated, and z(t) 6= ys (t). This case arises in the fusing stage of the xerographic process, where z is the temperature of the wedge at the paper nip (i.e., the point where paper comes in contact with the fuser), while ys is the temperature of the wedge at the actuator (external heating roll). If the actuator, the disturbance, and the performance output z(t) are all collocated, we use the controller ua = k(t) (r(t) − ys ) , (36) where k(t) is as in (9). With (36) and the performance output (35), the closed loop CC system becomes x˙ = (A − b(t)k(t)c(t))x + b(t)k(t)r(t) + β(t)d(t), (37) z = h(t)x. Unlike PAC, tracking in CC systems for all time moments is not generally possible because of stroboscopic actuation. As an example, consider a plant Pi (s) in a system composed of N non-interacting plants. When the actuator is not acting on this plant, Pi (s) undergoes its natural dynamics, which implies that the performance output z(t) cannot constantly track the reference over an entire period. Under similar reasoning, the output z(t) cannot constantly reject disturbances over an entire period. As a result, the problem of tracking and disturbance rejection in CC systems considers z(t) only at the end of each time slot and investigates the error e(nτ ) = r0 − z(nτ ). (38) Below, we first consider the case of collocated actuator, disturbance, and performance output and then investigate the non-collocated case. 5.2 Collocated case For system (37), the lifted performance output ze[n] and the lifted tracking error ee[n] are given by: ze[n] = [z(nT + τ ), z(nT + 2τ ), . . . , z((n + 1)T )]T , (39) ee[n] = [1, . . . , 1]T r0 − ze[n], where, due to collocation, z(nT + iτ ) = ci x(nT + iτ ), i = 1, . . . , N . Applying the lifting approach to (37), as shown in Section 3.2, yields the lifted system
e [k1 r0 , . . . , kN r0 ]T + B ed [d0 , . . . , d0 ]T , ξ[n + 1] = Φξ[n] + B e e 1 r 0 , . . . , k N r0 ] T + D e d [d0 , . . . , d0 ]T , ze[n] = Hξ[n] + D[k (40) where ξ[n] = x(nτ ); Φ = ΦN · · · Φ1 , Φi = e(A−bi ki ci )τ ; Rτ e and D e are as Γi = 0 e(A−bi ki ci )s ds, i = 1, . . . , N ; B e is the same as C e in (17); and B ed and defined in (17); H e d are the same as B e and D e defined in (17) except that D all bi ’s are replaced by β i .
If the eigenvalues of the monodromy Φ are within D1 (0), system (40) is asymptotically stable, and in the steady state, the lifted error ee[∞] is given by e − Φ)−1 B e + D)[k e 1 , . . . , k N ] T r0 ee[∞] = [1, . . . , 1]T − (H(I e − Φ)−1 B ed + D e d [1, . . . , 1]T d0 . + H(I
Observe that if τ is sufficiently small, the results of Section 4 can be applied to analyze stability of (40). Specifically, the gains ki , i = 1, . . . , N , can be chosen to place the eigenvalues of A − diag(bi kNi ci ) within D T1 (− T1 ), or equivalently, the eigenvalues of Φ within D1 (0). Note that the range of gains ki that achieve this pole placement is always finite, because for any T , there exist segments of the root loci, corresponding to large ki ’s, that are outside of D T1 (− T1 ). Therefore, unlike the PAC case: Proposition 1. For any τ > 0, if condition (20) is satisfied, the norm of the lifted steady state error ke e[∞]k in the collocated case is bounded from below by a positive constant. As far as controller design is concerned, assuming that τ is sufficiently small, the gains ki must be chosen such that condition (20) is satisfied and the desired dynamic and steady state tracking and disturbance rejection specifications are met. If no ki ’s achieve the specifications, the period T must be decreased, so that the domain of stability D T1 (− T1 ) is enlarged, and the process of selecting ki ’s is repeated anew. Example 4. Consider a stroboscopic disturbance d(t) acting on the plants of Example 1. Assume that the disturbance and performance output z(t) are collocated with the actuator, d(t) = −0.51(t), β1 = b1 , β2 = b2 and the reference signal is r(t) = 1(t). To illustrate Proposition 1, we use controller (36) and, for various T ’s, plot the minimum ke e[∞]k attainable such that the eigenvalues of A − bi kNi ci are within D T1 (− T1 ) (see Fig. 6(a)). Clearly, the minimum error is non-zero and is monotonically increasing in T . (Note that the cusp in the curve of Fig. 6(a) is due to the break-in points of the root loci of the plants.) Fig. 6(b) shows the transient response of the sampled performance output z(nτ ), for k1 = k2 = 3 and for k1 = k2 = 20. Obviously, with higher gains, z(nτ ) tracks r(t) and rejects d(t) faster and with a smaller error. 5.3 Non-collocated case For the case in which either the performance output or the disturbance is not collocated with the actuator, we modify the controller (36) as follows: ua = k(t) (g(t)r(t) − ys ) + q(t)d(t), PN PN (41) g(t) = i=1 stri (gi ), q(t) = i=1 stri (qi ),
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Note that to employ (45) and (46), the matrices A, bi , ci , i = 1, . . . , N , of (2) and the disturbance d0 must be known precisely. Generally, this is not the case, and therefore we have: Proposition 2. For any τ > 0, if condition (20) is satisfied and the system parameters are not known precisely, then the norm of the lifted steady state error ke e[∞]k in the non-collocated case is bounded from below by a positive constant.
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Fig. 6. Example 4 where gi ∈ R and qi ∈ R, i = 1, . . . , N , are to be selected to compensate for the non-collocation. System (34), (35) under feedback (41) is given by x˙ = (A − b(t)k(t)c(t))x + (β(t) + b(t)q(t))d(t) + b(t)k(t)g(t)r(t), z = h(t)x.
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To simplify notation, let col(ai ) denote the column vector [a1 , . . . , aN ]T . Then, applying the lifting approach to (42) yields the system e col(ki gi )r0 ξ[n + 1] = Φξ[n] + B e col(qi ) + B ed col(1))d0 , + (B e e col(ki gi )r0 ze[n] = Hξ[n] +D e col(qi ) + D e d col(1))d0 , + (D
Note that (41) is a feedback/feedforward controller where gi ’s and qi ’s account for the feedforward part, while ki ’s account for the feedback. As a final remark, observe that in the case of nonstroboscopic disturbances, β 1 = . . . = β N , and β i has no special structure. An example of such disturbance arises in the fusing stage of the xerographic process, where the power delivered to the internal lamp, viewed as a disturbance, enters the system non-stroboscopically. The procedure presented in this section remains valid for such disturbances, as is demonstrated in Eun et al. (2011). 6. CONCLUSIONS AND FUTURE WORK
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where ξ[n] = x(nτ ); col(1) = [1, . . . , 1]T , Φ = ΦN · · · Φ1 , Rτ Φi = e(A−bi ki ci )τ ; Γi = 0 e(A−bi ki ci )s ds, i = 1, . . . , N ; e is defined in (17); H e and D e are the same as C e B e and D in (17), respectively, except that all ci ’s are reed and D e d are the same placed by cmodN (i+l−1)+1 ; and B e e as B and D defined in (17) except that all ci ’s are replaced by cmodN (i+l−1)+1 and all bi ’s are replaced by β modN (i+m−1)+1 . Assuming that system (43) is asymptotically stable, the lifted tracking error (39) satisfies e − Φ)−1 B e+D e col(gi ki ) r0 ee[∞] = col(1) − H(I e − Φ)−1 (B e col(qi ) + B ed col(1)) − H(I e col(qi ) + D e d col(1) d0 . +D (44) Thus, to ensure perfect tracking, i.e., ee[∞] = 0, gi ’s and qi ’s must satisfy 1 col(gi ) = diag( )Φ−1 col(1), (45) ki CL e − Φ)−1 B ed + D e d col(1), col(qi ) = −Φ−1 H(I (46) CL
e −Φ)−1 B e + D. e Observe that gi ’s and qi ’s where ΦCL = H(I can be calculated from (45) and (46) if ΦCL is invertible. It can be shown that this matrix is indeed invertible iff # " e Φ−I B = M + N. (47) rank e D e H
As far as controller design is concerned, assuming that τ is sufficiently small, the gains ki must be chosen such that condition (20) is satisfied and that the desired transient specifications are met. The gi ’s and qi ’s can then be calculated from (45) and (46), respectively, to achieve ee[∞] = 0.
This paper developed CC theory for the case of static output feedback and compared it with PAC. Future work includes: cyclic control using state space feedback; cyclic control using dynamic output feedback; extension of the theory developed to the case of non-collocated actuators and sensors; and disturbance rejection when the disturbances are unknown. Solutions to these problems will provide a relatively complete theory of cyclic control systems. REFERENCES Anderson, B. and Moore, J. (2007). Optimal Control: Linear Quadratic Methods. Dover Publications. Chen, T. and Francis, B. (1996). Optimal Sampled-Data Control Systems. Springer. Childs, D. (1993). Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis. Wiley. Ching, S., Eun, Y., Gross, E., Hamby, E., Kabamba, P., Meerkov, S., and Menezes, A. (2010). Modeling and control of cyclic systems in xerography. Proceedings of the American Control Conference. Drozda, T. (ed.) (1983). Tool and Manufacturing Engineers Handbook. Society of Manufacturing Engineers, fourth edition. Eun, Y., Gross, E.M., Kabamba, P., Meerkov, S., Menezes, A., and Ossareh, H. (2011). Cyclic control: The case of static output feedback. http://www-personal.umich.edu/~hamido/IFAC11/ifacconf.pdf .
Feingold, D. and Varga, R. (1962). Block diagonally dominant matrices and generalizations of the gerschgorin circle theorem. Pacific J. Math. Hamby, E. and Gross, E. (2004). A control-oriented survey of xerographic systems: Basic concepts to new frontiers. Proceedings of the American Control Conference, 2615– 2629. Kwakernaak, H. and Sivan, R. (1972). Linear Optimal Control Systems. Wiley. Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices. Academic Press, second edition.
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