IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 7, NO. 4, JULY 2020 919
Parallel Control for Continuous-Time Linear Systems: A Case Study Qinglai Wei, Member, IEEE, Hongyang Li, and Fei-Yue Wang, Fellow, IEEE
Abstract—In this paper, a new parallel controller is developed of industrial control systems, the intelligent control theory, for continuous-time linear systems. The main contribution of the such as fuzzy control [6], neural network control [7], adaptive method is to establish a new parallel control law, where both dynamic programming [8], [9], is attracted by researchers. state and control are considered as the input. The structure of the parallel control is provided, and the relationship between the Among these previous stages, most system control problems parallel control and traditional feedback controls is presented. are analyzed by state feedback control methods in present Considering the situations that the systems are controllable and study: we generally design state feedback controllers to form incompletely controllable, the properties of the parallel control closed-loop systems, that is, the control laws are functions of law are analyzed. The parallel controller design algorithms are the system states. However, the state feedback controllers have given under the conditions that the systems are controllable and incompletely controllable. Finally, numerical simulations are some disadvantages: carried out to demonstrate the effectiveness and applicability of 1) The traditional state feedback controllers are only related the present method. to the system states rather than the properties of the controllers Index Terms—Continuous-time linear systems, digital twin, and it causes that the control signals may change greatly with parallel controller, parallel intelligence, parallel systems. the system states, which brings great difficulty to the execution of the controllers. 2) The control signals are generated passively, and it is I.INTRODUCTION difficult to generate control signals under the condition that VER the past decades, with the rapid development of the system states have no changes or the system states cannot O science and technology, control theory and technology be obtained. are playing increasingly important roles. The development 3) The structure of the state feedback controllers is onefold, of control theory has generally gone through three stages: which forces the system into a closed-loop one. It causes classical control theory, modern control theory, and intelligent difficulties in performance improvements of the systems. control theory [1]. Based on frequency domain analysis, the Therefore, it is necessary to build a new type of controller classical control theory mainly solves the control problems to overcome the above problems. of single input single output linear time-invariant systems. Parallel control theory, proposed by Wang [1], [10], [11], is Based on state space description, the modern control theory an effective method to obtain the control laws of the control mainly solves the control problems of multi-input and multi- systems [12]−[16]. The basic structure of parallel systems is output systems. Comparing with classical control theory, the shown in Fig. 1. The basic idea of parallel control is expanding modern control theory is more suitable for the analysis of the practical problems into virtual space, then the control tasks time-varying nonlinear systems. The typical modern control can be realized by means of virtual-reality interaction. theory includes optimal control [2], adaptive control [3] and so To be specific, parallel control is the application of ACP (Ar- on [4], [5]. With the increase of complexity and nonlinearity tificial systems, computational experiments, parallel execution) theory [12] in control theory, where artificial systems (A) are Manuscript received May 8, 2020; accepted June 9, 2020. This work was supported in part by the National Key Research and Development Program used for modeling the physical systems, computational exper- of China (2018AAA0101502, 2018YFB1702300) and the National Natural iments (C) are used for analysis, evaluation and learning, and Science Foundation of China (61722312, 61533019, U1811463, 615330 parallel executions (P) are utilized for control, management, 17). Recommended by Associate Editor Jun Zhang. (Corresponding author: Qinglai Wei.) and optimization. Comparing with parallel systems, a similar Citation: Q. L. Wei, H. Y. Li, and F.-Y. Wang, “Parallel control for concept is digital twins. The parallel systems and digital twins continuous-time linear systems: A case study,” IEEE/CAA J. Autom. Sinica, manage and control systems which are difficult to analyze with vol. 7, no. 4, pp. 919−928, Jul. 2020. Q. L. Wei and H. Y. Li are with the State Key Laboratory of Manage- mathematical models by establishing the virtual systems cor- ment and Control for Complex Systems, Institute of Automation, Chinese responding to physical systems [17]. However, there are some Academy of Sciences, Beijing 100190, and with the University of Chinese differences between parallel systems and digital twins. The Academy of Sciences, Beijing 100049, and also with Qingdao Academy of Intelligent Industries, Qingdao 266109, China (e-mail: [email protected]; research objects of digital twins are cyber-physical systems [email protected]). (CPS) which are composed of information space and physical F.-Y. Wang is with the State Key Laboratory of Management and Control space. And parallel systems mainly focus on cyber-physical- for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, and with the Institute of Systems Engineering, Macau social systems (CPSS) which refer to the deep integration of University of Science and Technology, and also with Qingdao Academy of social networks, information resources, and physical space. In Intelligent Industries, Qingdao 266109, China (e-mail: [email protected]). addition to the research objects, there are certain differences in Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. core ideas, frameworks, mathematical descriptions, implemen- Digital Object Identifier 10.1109/JAS.2020.1003216 920 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 7, NO. 4, JULY 2020
Fig. 1. The basic structure of parallel systems [12].
Fig. 2. The architecture of parallel control and management for CPSS [20], [21]. tation methods, and so on [17], [18]. Fig. 2 demonstrates the [22]−[25]. However, it is worth pointing out that the present architecture of parallel control and management for CPSS. The parallel control methods focus on the artificial systems on the detailed description can be found in [19]−[21]. reconstruction of the system dynamics and the computational It is pointed out that parallel execution is an important and experiments focus on the performance evaluation with state distinctive step to guarantee the performance of the control feedback controllers. Furthermore, the properties analysis of systems. The basic block diagram of parallel execution [10] the parallel control methods are scarce, which are necessary to is shown in Fig. 3. guarantee the performance of the control laws. These motivate It is shown in Fig. 3 that the parallel execution is established our research. based on the parallel system theory. Based on the parallel In this paper, a new parallel control structure is developed execution between the artificial systems and physical systems, for continuous-time linear systems. The main contribution of we can convert passive computer simulations to the active the method is to establish a new parallel control law, where artificial systems, and give full play to the role of artificial the state and control input are both considered to construct systems in management and control of physical systems. Many the variation of the control, such that the system states are tasks, such as learning and training, experiment and evaluation, forced to converge to the equilibrium point and simultaneously management and control, and so on, can be executed based on analyze the performance of the parallel control laws. First, parallel execution. The parallel control theory is a hot research the basic structure of the parallel control is provided. The spot in resent study, and it has sparked a great deal of attention relationship between the parallel control and traditional feed- WEI et al.: PARALLEL CONTROL FOR CONTINUOUS-TIME LINEAR SYSTEMS: A CASE STUDY 921
Fig. 3. The basic structure of parallel execution [10]. back controls is presented and the advantages of the parallel where the variation of the control is explicitly depended with control are explained. Second, considering the continuous-time the state and the current control. linear systems, the expression of parallel controller is shown. Third, considering two situations including system control- lable and incompletely controllable (uncontrollable in brief), respectively, the properties of the parallel control method are analyzed. The detailed controller design algorithms are also given under the conditions that the systems are controllable and uncontrollable. Next, two simulation examples are pro- vided which verify the effectiveness of the developed method and the conclusion is finally drawn. The rest of this paper is organized as follows. In Section II, the structure of parallel controller is introduced and the controller design problem is formulated. In Section III, the existence of parallel controller is analyzed and the parallel controller design algorithms are presented. Simulation results Fig. 4. Structure of parallel controller. are provided and discussed in Section IV. Some concluding remarks are given in Section V. In parallel control method, system (1) and parallel controller (2) are executed in parallel with information interaction. It is shown that the parallel control is not a traditional feedback II.PROBLEM FORMULATIONS control, where traditional control laws are function of the In this section, the design ideas of the parallel control are states, i.e., x˙ = f(x, K(x)) under u = K(x). However, it presented. First, the basic structure of the parallel control is is worth pointing out that the parallel control in (2) can be introduced and the comparisons between the parallel control transformed into a open-loop or a closed-loop control law. and traditional control are illustrated, where the advantages First, according to (2), there is a function G , such that u = of the parallel control are emphasized. Second, the problem G (x, t), which indicates the closed-loop control law. On the formulations of the parallel control for continuous-time linear other hand, according to (1) and (2), letting augmented state £ ¤T systems are presented. variable z be z = xT , uT , the system function can be written as A. Basic Structure of Parallel Control · ¸ f(z) z˙ = = F(z) (3) In this subsection, the basic structure of parallel control is g(z) introduced. Consider the following systems x˙ = f(x, u) (1) which establishes a closed-loop system by designing the control function g. This is an obvious merit of the parallel where x ∈ Rn is the n-dimensional state vector, u ∈ Rm is control method. Second, if parallel control (2) is reduced to the m-dimensional control vector, and f(x, u) is the system u˙ = g(u), then there exists a function Gˆ, such that u = Gˆ. function. A new parallel control method is established. The In this situation, the parallel control law is reduced to a open- structure of the parallel control is shown in Fig. 4, and the loop control law. Thus, the flexible structure is another merit parallel control can be expressed by of the parallel control. In the following, we focus on the parallel control method for continuous-time linear systems and u˙ = g(x, u) (2) properties of the parallel control method will be analyzed. 922 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 7, NO. 4, JULY 2020
B. Parallel Control of Continuous-Time Linear Systems The matrices A˜ij, i = 1, 2, . . . , r, j = 1, 2, . . . , r can be In this subsection, the parallel control of continuous- expressed as time linear systems is displayed. Consider the following continuous-time linear system 0 0 ··· • . 1 0 ··· . x˙ = Ax + Bu (4) A˜ = , i = 1, 2, . . . , r (9) ii . . .. .. • where x ∈ Rn, u ∈ Rm are system state and control 1 • input, respectively, and A ∈ Rn×n, B ∈ Rn×m are the (li×li) ˜ system matrices. According to (2), we can design the parallel Aij = 0, i > j (10) controller as and u˙ = Cx + Du (5) 0 ··· 0 • where C ∈ Rm×n and D ∈ Rm×m. 0 ··· 0 • We would like to find the appropriate matrices C, D, such ˜ Aij = . . . . , i < j (11) that system (4) is stable. Defining augmented state variable z . .. . . £ ¤T as z = xT , uT , we can obtain a closed-loop type system, 0 ··· 0 • (li×lj ) such that · ¸ AB respectively, where l1, l2, . . . , lr are the Kronecker constants z˙ = z = Gz. (6) ˜ CD of system (4) and l1 + l2 + ··· + lr = n. The matrix B is expressed by Assume that the desired characteristic polynomial of matrix G is 1 0 ··· 0 0 0 ··· 0 • m+n m+n−1 |λI − G| = λ + β1λ + ··· + βn+m−1λ + βn+m. . . . . . .. . (7) . . . . 0 0 ··· 0 In the following, we will analyze the parallel controller 0 1 ··· 0 design from the perspective of system controllability. • 0 0 ··· 0 . . . . ˜ −1 . . . . B = T B = . . . . . (12) III.MAIN RESULTS 0 0 ··· 0 In this section, considering system (4) is controllable and . . . . . ...... uncontrollable, respectively, the design of the parallel control . . . . is presented and the properties of the parallel control will be 0 0 ··· 1 • analyzed. 0 0 ··· 0 . . . . . . .. . 0 0 ··· 0 A. System (4) Is Controllable (n×m) In this subsection, considering the situation that system (4) ˜ is controllable, the existence and the solution of the parallel Then we can define matrix F as control are discussed. The following lemma, which is inspired h ˜ by [26], is necessary to facilitate the analysis. F = 0 ··· 0 e2 ··· 0 ··· 0 er | {z } | {z } Lemma 1: Let B denote the range of B. If the system (A, B) l1 lr−1 is controllable, then there exists a nonzero vector b ∈ B and i 0 ··· 0 (13) an m × n matrix F , such that the system (A + BF, b) is | {z } controllable. £ ¤ lr Proof: Denote matrix B as B = b1 b2 ··· bm m×1 and r = rankB. For the situation that the system (A, B) where ei ∈ R , i = 1, 2, . . . , r are unit vectors whose ith is controllable, according to [27], there exists a nonsingular element is 1. Then, let matrix F = FT˜ −1, we can obtain that matrix T , which can transform the system (A, B) into first (A + BF, b ) is controllable [28]. ¥ ˜ ˜ 1 controllable canonical form (A, B). Let “• ” denote an element Then, according to the above result, the existence of parallel ˜ that is unrelated to the analysis. The matrix A can be expressed controller (5) can be shown. as Theorem 1: If system (4) is controllable, then there exists A˜11 A˜12 ··· A˜1r a parallel controller (5), such that system (6) has the desired A˜ ······ characteristic polynomial in the form of (7). ˜ −1 22 A = T AT = . . (8) Proof: If system (4) is controllable, then according to .. ··· Lemma 1, there exist matrix F1 and vector v1, such that A˜rr system (A + BF1, Bv1) is controllable. Then there exists a WEI et al.: PARALLEL CONTROL FOR CONTINUOUS-TIME LINEAR SYSTEMS: A CASE STUDY 923
∗ ∗ ∗ ∗ nonsingular matrix P which can transform system (A + BF1, and matrices Aˆ , Bˆ , Cˆ and Dˆ have dimension (n + 1) × Bv1) into controllable canonical form as (n+1), (n + 1)×(m − 1), (m − 1)×(n + 1) and (m − 1)× (m − 1), respectively. From (20), we can see that (Aˆ∗, Bˆ∗) is 0 1 0 0 ··· 0 0 controllable. Then repeating steps (14)−(20) m − 2 times, we 0 0 1 0 ··· 0 0 can obtain the matrix Gˆ as −1 ...... m−2 P (A + BF1) P = ...... · ¸ Aˆ∗ Bˆ∗ Gˆ = m−2 m−2 (21) 0 0 0 0 ··· 0 1 m−2 ˆ∗ ˆ ∗ Cm−2 Dm−2 −αn · · · · · · −α1 where (14) and 0 1 ··· 0 • £ ¤ . . .. . . −1 T ˆ∗ . . . . ˆ∗ . P Bv1 = 0 ··· 0 1 (15) Am−2 = , Bm−2 = 0 0 ··· 1 • where α1, α2, . . . , αn are coefficients of characteristic poly- 0 0 ··· 0 1 nominal of matrix A + BF , such that £ ¤ 1 ˆ∗ ˆ Cm−2 = cˆm−2,1 ··· cˆm−2,n+m−2 dm−2,1 |λI − A − BF | = λn + α λn−1 + ··· + α . (16) 1 1 n ˆ ∗ ˆ £ ¤ Dm−2 = dm−2,2. Define the vector ζ as ζ = α α ··· α , and n n−1 1 For (21), execute steps (14)−(17). Then we can get the define£ the matrix Γ as Γ = F¤1P +v1ζ. Define the matrix V as matrix Gˆm−1 as V = v1 v2 ··· vm , where v2, . . . , vm are arbitrary · ¸ vectors, such that the matrix V is nonsingular. Then, for Aˆm−1 Bˆm−1 Gˆm−1 = (22) system (6), we can take a nonsingular transformation as Cˆm−1 Dˆ m−1 · ¸ −1 −1 −1 Aˆ Bˆ where Gˆ = (T3) (T2) (T1) GT1T2T3 = (17) Cˆ Dˆ 0 1 ··· 0 0 . . . . . where ˆ . . .. . ˆ . · ¸ · ¸ · ¸ Am−1 = , Bm−1 = . P I I 0 0 ··· 1 0 T = ,T = ,T = . 1 I 2 Γ I 3 V −1 0 0 ··· 0 1 ˆ ˆ Then, the matrices Aˆ and Bˆ are expressed as For (22), let matrices Cm−1 and Dm−1 be £ ¤ ˆ 0 1 ··· 0 0 • ··· • Cm−1 = −βn+m, −βn+m−1 · · · −β2 . . . . . . . . ˆ . . . . . . . . Dm−1 = −β1. (23) Aˆ = . . . . , Bˆ = . . . . . (18) 0 0 ··· 1 0 • ··· • Then, from the above analysis, we can obtain the matrix 0 0 ··· 0 1 • ··· • Gˆm−1 with characteristic polynominal (7), and matrix Gˆm−1 is similar to matrix G. Therefore, there exist matrices C and The matrices Cˆ and Dˆ are expressed as D, such that matrix G has desired characteristic polynominal 0 0 ··· 0 0 1 ··· 0 (7). ¥ . . . . . . . . According to Theorem 1, Algorithm 1 can be used to find . . . . . . .. . Cˆ = . . . . , Dˆ = . . . . parallel controller when system (4) is controllable. 0 0 ··· 0 0 0 ··· 1 cˆ cˆ ··· cˆ dˆ dˆ ··· dˆ 1 2 n 1 2 m Algorithm 1 Parallel controller design algorithm when system (4) is (19) controllable ˆ According to (17), we can repartition the matrix G as Initialization: · ¸ Aˆ∗ Bˆ∗ Give the system matrices A, B. Gˆ = (20) Give the desired characteristic polynominal (7). Cˆ∗ Dˆ ∗ Execution: where 1: If m = 1, then 0 1 ··· 0 0 1) Find matrix P , such that system (A, B) can be transformed into • • ··· • . . . . . controllable canonical form. h i ...... . . . . . . . . . . . . 2) Define the matrix Γ as Γ = αn αn−1 ··· α1 . Aˆ∗ = , Bˆ∗ = . . . . 0 0 ··· 1 0 3) Take a nonsingular transformation according to (17). Let matrices • • ··· • 0 0 ··· 0 1 ˆ ˆ 1 0 ··· 0 C, D be 0 0 ··· 0 0 h i ˆ ˆ C = −βn+m −βn+m−1 · · · −β2 , D = −β1 0 ··· 0 0 0 1 ··· 0 . . . . . . . . then matrices C, D can be obtained from ∗ . .. . . ∗ . . .. . Cˆ = , Dˆ = ˆ −1 −1 −1 0 ··· 0 0 0 0 ··· 1 G = T1T2T3G(T3) (T2) (T1) (24) ˆ ˆ ˆ ˆ cˆ1 ··· cˆn d1 d2 d3 ··· dm where T3 = I. 924 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 7, NO. 4, JULY 2020
2: If m > 1, then If (7) cannot be divided by the characteristic polynominal ∗ 1) Find matrix F1 and vector v1, such that system (A + BF1, Bv1) |λI − A1|, from the above analysis, we can obtain that there is controllable. does not exist parallel controller (5) such that system (6) 2) Find matrix P , such that system (A + BF1, Bv1) can be has the desired characteristic polynomial. Otherwise, for the transformed into controllable canonicalh form. i characteristic polynominal |λI − G|, define new system and 3) Define the vector ζ as ζ = αn αn−1 ··· α1 , parallel controller as the matrix Γ as Γ = F P + v ζ, and the matrix V as V = h i1 1 X˙ = AX + BU (29) v1 v2 ··· vm . 4) Take a nonsingular transformation according to (17). Let matrices U˙ = CX + DU. (30) Cˆ and Dˆ be as (19). Then repartition the matrix Gˆ as (20). ˆ Then we can obtain a new closed-loop system as 5) Repeat Steps 1)−4) m − 2 times, and obtain the matrix Gm−2 · ¸ as (21). AB ˆ Z˙ = Z = GZ (31) 6) For matrix Gm−2, execute Steps 1)−3), take a nonsingular CD transformation as (17) and obtain the matrix Gˆm−1 as (22). ˆ ˆ £ ¤T 7) Let matrices Cm−1 and Dm−1 be as (23), then matrices C, D where Z = X T , U T . Let M be the column number and can be obtained from N be the row number of matrix B. Then we can obtain the −1 −1 G = Tm−1,1Tm−1,2Tm−1,3 ··· T1T2T3Gˆ(T3) (T2) desired characteristic polynomial of matrix G according to (7) −1 −1 −1 −1 ×(T1) ··· (Tm−1,3) (Tm−1,2) (XTm−1,1) . and (28) as (25) ∗ |λI − G| = |λI − G|/|λI − A1| 3: return C, D. M+N ˜ M+N −1 ˜ = λ + β1λ + ··· + βM+N −1λ ˜ + βM+N . (32) B. System (4) Is Uncontrollable As system (29) is controllable, for (29), there exist matrix In this subsection, considering the situation that system F1 and vector V1, such that system (A + BF1, BV1) is con- (4) is incompletely controllable (uncontrollable in brief), the trollable. Then there exists a nonsingular matrix P which can existence and the solution of the parallel control are discussed. transform system (A + BF1, BV1) into controllable canonical If system (4) is uncontrollable, the uncontrollable modes of the form as system cannot be changed, and there exist nonsingular matrix 0 1 0 0 ··· 0 0 Q which can transform system (A, B) as · ¸ · ¸ 0 0 1 0 ··· 0 0 A∗ 0 0 −1 ...... Q−1AQ = 1 ,Q−1B = (26) P (A + BF1)P = ...... A∗ A B ...... 21 0 0 0 0 ··· 0 1 ∗ where (A1, 0) is uncontrollable subsystem and (A, B) is −α˜N · · · · · · −α˜1 controllable subsystem. Then for system (6), we can obtain (33) that and · ¸ · ¸ A∗ 0 0 Q−1 Q 1 £ ¤T G∗ = G = A∗ AB . P−1BV = 0 ··· 0 1 . (34) I I 21 1 C∗ CD 1 ˜ ˜ (27) Define the vector ζ as ζ = [ α˜N α˜N −1 ··· α˜1 ], and ˜ define the matrix Γ˜ as Γ˜ = F1P + V1ζ, and the matrix Then for (27), the existence of parallel controller (5) can be V as V = [ V1 V2 ···VM ], where V2,..., VM are shown. arbitrary vectors, such that the matrix V is nonsingular. Take Theorem 2: If system (4) is uncontrollable, there exists a a nonsingular transformation as parallel controller (5) such that system (6) has the desired · ¸ ˆ ˆ characteristic polynominal (7), if and only if (7) can be divided ˆ −1 −1 −1 A B ∗ G = (T3) (T2) (T1) GT1T2T3 = ˆ ˆ (35) by the characteristic polynominal |λI − A1|. C D Proof: From (26) and (27), the matrix G∗ is similar to G. where For (27), we can obtain that · ¸ · ¸ · ¸ ¯ · ¸¯ P I I ¯ AB ¯ T = , T = , T = . |λI − G∗| = |λI − A∗| ¯λI − ¯ 1 I 2 ˜ 3 V−1 1 ¯ CD ¯ Γ I = |λI − A∗| |λI − G| . (28) Then we can obtain that 1 From (28), we can obtain that the characteristic polynominal 0 1 ··· 0 0 • ··· • ∗ . . . . . . . . |λI − G | is composed of two parts: one is the characteristic ˆ . . .. . ˆ . . .. . ∗ A = , B = . (36) polynominal of uncontrollable subsystem |λI − A1| which 0 0 ··· 1 0 • ··· • cannot be changed by parallel control; the other is related to 0 0 ··· 0 1 • ··· • the controllable subsystem, and this part can be changed by the design of parallel control. Define matrices Cˆ and Dˆ as WEI et al.: PARALLEL CONTROL FOR CONTINUOUS-TIME LINEAR SYSTEMS: A CASE STUDY 925
0 0 ··· 0 0 1 ··· 0 similar to matrix G. Therefore, there exist matrices C and D, . . . . . . . . ˆ . . .. . ˆ . . .. . such that matrix G has desired characteristic polynominal (32). C = , D = . ∗ 0 0 ··· 0 0 0 ··· 1 Let matrix C1 in (27) be any matrix with suitable dimension, ˆ ˆ ˆ ˆ ˆ ˆ we can obtain that there exist matrices C and D such that the C1 C2 ··· CN D1 D2 ··· DM ∗ (37) matrix G has desired characteristic polynomial (7). ¥ Remark 1: The uncontrollable modes of system (4) cannot According to (35), we can repartition the matrix Gˆ as be changed by parallel controller (5). Therefore, when system · ¸ (4) is controllable, we can find parallel controller such that Aˆ∗ Bˆ∗ Gˆ = (38) the system has desired steady-state characteristics as Theorem ˆ∗ ˆ∗ C D 1. If system (4) is uncontrollable, the existence of parallel where controller is connected with the steady-state characteristics of 0 1 ··· 0 0 uncontrollable subsystem of system (4) which has been proven • • ··· • . . . . . in Theorem 2. ...... . . . . Then, according to Theorem 2, parallel controller design ˆ∗ ˆ∗ . . .. . A = 0 0 ··· 1 0 , B = when system (4) is uncontrollable can be summarized in • • ··· • 0 0 ··· 0 1 Algorithm 2. 1 0 ··· 0 0 0 ··· 0 0 Algorithm 2 Parallel controller design algorithm when system (4) is 0 ··· 0 0 0 1 ··· 0 uncontrollable . . . . . . . .. . . . . .. . Initialization: Cˆ∗ = . . . . , Dˆ∗ = . . . . 0 ··· 0 0 0 0 ··· 1 Give the system matrices A, B. Give the desired characteristic polynominal (7). Cˆ1 ··· CˆN Dˆ1 Dˆ2 Dˆ3 ··· DˆM Execution: and matrices Aˆ∗, Bˆ∗, Cˆ∗ and Dˆ∗ have dimension (N + 1) × 1: Take nonsingular transformation as (27). ¯ ¯ ¯ ∗¯ (N + 1), (N + 1) × (M − 1), (M − 1) × (N + 1) and (M − 2: If (7) cannot be divided by λI − A1 , 1) × (M − 1), respectively. From (38), we can see that (Aˆ∗, then there does not exist parallel controller, and go to Step 8; Bˆ∗) is controllable. Then for (38), repeat steps (33)−(38) M else go to next step. − 2 times, we can obtain the matrix GˆM−2 as 3: Define new system and parallel controller as (29), (30), define new · ¸ closed-loop system as (31) and characteristic polynominal as (32). Aˆ∗ Bˆ∗ Gˆ = M−2 M−2 (39) 4: For system (29), if M = 1, then M−2 ˆ∗ ˆ∗ CM−2 DM−2 1) Find matrix P, such that system (A, B) can be transformed into where controllable canonical form.h i ˜ 0 1 ··· 0 • 2) Define the matrix Γ = α˜N α˜N −1 ··· α˜1 . . . . . . 3) Take a nonsingular transformation according to (35). Let matrices ˆ∗ . . .. . ˆ∗ . AM−2 = , BM−2 = Cˆ, Dˆ be • 0 0 ··· 1 h i ˆ ˜ ˜ ˜ ˆ ˜ 0 0 ··· 0 1 C = −βN +M −βN +M−1 · · · −β2 , D = −β1 £ ¤ ˆ∗ ˆ ˆ ˆ CM−2 = CM−2,1 ··· CM−2,N +M−2 DM−2,1 then matrices C, D can be obtained from ∗ Dˆ = DˆM−2,2. −1 −1 −1 M−2 G = T1T2T3Gˆ(T3) (T2) (T1) (42) For (39), execute steps (33)−(35). Then we can obtain the where T3 = I. matrix GˆM−1 as · ¸ 5: If M > 1, then Aˆ Bˆ 1) Find matrix F1 and vector V1, such that system (A + BF1, BV1) Gˆ = M−1 M−1 (40) M−1 is controllable. CˆM−1 DˆM−1 2) Find matrix P, such that system (A + BF1, BV1) can be where transformed into controllable canonicalh form. i 0 1 ··· 0 0 ˜ ˜ 3) Define the vector ζ as ζ = α˜N α˜N −1 ··· α˜1 , . . . . . ˜ ˜ ˜ . . .. . . defineh the matrix Γ as Γ = Fi 1P + V1ζ, and the matrix V as AˆM−1 = , BˆM−1 = . 0 0 ··· 1 0 V = V1 V2 ···Vm . 0 0 ··· 0 1 4) Take a nonsingular transformation according to (35). Let matrices Cˆ, Dˆ be as (37). Then repartition the matrix Gˆ as (38). ˆ ˆ For (40), let matrices CM−1 and DM−1 be 5) Repeat Steps 1)−4) M − 2 times, and obtain the matrix Gˆ £ ¤ M−2 ˆ ˜ ˜ ˜ as (39). CM−1 = −βN +M −βN +M−1 · · · −β2 ˆ ˜ 6) For matrix GM−2, execute Steps 1)−3), take a nonsingular DˆM−1 = −β1. (41) transformation as (35) and obtain the matrix GˆM−1 as (40). 7) Let matrices Cˆ and Dˆ be as (41), then matrices C, D From the above analysis, we can obtain the matrix GˆM−1 M−1 M−1 can be obtained from with characteristic polynominal (32), and matrix GˆM−1 is 926 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 7, NO. 4, JULY 2020
−1 −1 G = TM−1,1TM−1,2TM−1,3 ···T1T2T3Gˆ(T3) (T2) −1 ¡ ¢−1¡ ¢−1¡ ¢−1 ×(T1) ··· TM−1,3 TM−1,2 TM−1,1 . (43)
∗ 6: Let C1 in (27) be any matrix with suitable dimension, then we can obtain matrices C, D from (27) as " # " # Q Q−1 G = G∗ (44) I I
7: return C, D. 8: return “There does not exist parallel controller.”
IV. NUMERICAL ANALYSIS In this section, two simulations are employed to evaluate the effectiveness of the proposed method. Fig. 5. System states in Example 1. Example 1: In the first example, we consider the following linearized model of the power system [29], [30] −0.0665 8 0 0 0 −3.663 3.663 0 x˙ = x −6.86 0 −13.736 −13.736 0.6 0 0 0 0 0 + u. (45) 13.736 0 It is easy to derive that system (45) is controllable. The de- sired poles are −1, −1, −2, −2, −3, which have characteristic polynominal f (λ) = (λ + 1)2(λ + 2)2 (λ + 3) 5 4 3 2 = λ + 9λ + 31λ + 51λ + 40λ + 12. (46) Fig. 6. Control law in Example 1. According to Algorithm 1, we can obtain that The poles of controllable subsystem are 1, 2, and the pole £ ¤ of uncontrollable subsystem is −1. The desired poles are −1, C = −3.7523 3.8300 −9.2736 −8.5152 −1, −2 and −2. According to Algorithm 2, we can obtain D = 8.4655 matrices C and D as and the parallel controller can be expressed as £ ¤ C = c −30 −c − 48 ,D = −8 £ ¤ u˙ = −3.7523 3.8300 −9.2736 −8.5152 x + 8.4655u. (47) where c is an arbitrary constant. And the parallel controller can be obtained as follows. The initial state and control law are x = £ ¤ 0 £ ¤ 0 0.1 0 0 and u0 = 0.1, respectively. Then we u˙ = c −30 −c − 48 x − 8u. (49) can obtain the simulation results for the trajectories of the system states and control, which are shown in Figs. 5 and 6, £ ¤ The initial state and control law are x = 0.5 1 1 respectively. The trajectories of the system states are shown in 0 and u = 0.1, respectively. Letting c = −3, we can obtain the Fig. 5, and the trajectory of the control law is shown in Fig. 6. 0 simulation results as Figs. 7 and 8. From the figures, we can see that the system is stable after control law is applied to system. Therefore, the correctness of The trajectories of the system states are shown in Fig. 7, and the proposed method can be verified. the trajectory of the control law is shown in Fig. 8. The cor- Example 2: In the second example, we consider the follow- rectness of the proposed control method can be demonstrated. ing uncontrollable system 0 1 2 0 V. CONCLUSION x˙ = 0 1 0 x + 1 u. (48) 1 1 1 0 This paper has concerned a new control, that is the parallel WEI et al.: PARALLEL CONTROL FOR CONTINUOUS-TIME LINEAR SYSTEMS: A CASE STUDY 927
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He joined the University of Arizona in 1990 and became a Professor and the [30] D. Vrabie, O. Pastravanu, M. Abu-Khalaf, and F. L. Lewis, “Adaptive Director of the Robotics and Automation Laboratory optimal control for continuous-time linear systems based on policy and the Program in Advanced Research for Complex iteration,” Automatica, vol. 45, no. 2, pp. 477−484, Feb. 2009. Systems. In 1999, he founded the Intelligent Control and Systems Engineering Center at the Institute of Automation, Chinese Academy of Sciences (CAS), Beijing, China, under the support of the Outstanding Chinese Talents Program from the State Planning Council, and in 2002, was appointed as the Director of the Key Laboratory of Complex Systems and Qinglai Wei (M’11) received the B.S. degree in au- Intelligence Science, CAS. In 2011, he became the State Specially Appointed tomation, and the Ph.D. degree in control theory and Expert and the Director of The State Key Laboratory for Management and control engineering, from Northeastern University, Control of Complex Systems. Shenyang, China, in 2002 and 2009, respectively. His current research focuses on methods and applications for parallel From 2009−2011, he was a Postdoctoral Fellow intelligence, social computing, and knowledge automation. He is a Fellow with the State Key Laboratory of Management and of INCOSE, IFAC, ASME, and AAAS. In 2007, he received the National Control for Complex Systems, Institute of Automa- Prize in Natural Sciences of China and became an Outstanding Scientist of tion, Chinese Academy of Sciences, Beijing, China. ACM for his work in intelligent control and social computing. He received He is currently a Professor of the institute and the the IEEE ITS Outstanding Application and Research Awards in 2009 and Associate Director of the laboratory. He has authored 2011, respectively. In 2014, he received the IEEE SMC Society Norbert four books, and published over 80 international Wiener Award. Since 1997, he has been serving as the General or Program journal papers. His research interests include adaptive dynamic programming, Chair of over 30 IEEE, INFORMS, IFAC, ACM, and ASME conferences. He neural-networks-based control, optimal control, nonlinear systems and their was the President of the IEEE ITS Society from 2005 to 2007, the Chinese industrial applications. Association for Science and Technology, USA, in 2005, the American Zhu He is the Secretary of IEEE Computational Intelligence Society (CIS) Kezhen Education Foundation from 2007 to 2008, the Vice President of the Beijing Chapter since 2015. He was guest editors for several international ACM China Council from 2010 to 2011, the Vice President and the Secretary journals. He was a recipient of IEEE/CAA Journal of Automatica Sinica Best General of the Chinese Association of Automation from 2008 to 2018. He was Paper Award, IEEE System, Man, and Cybernetics Society Andrew P. Sage the Founding Editor-in-Chief (EiC) of the International Journal of Intelligent Best Transactions Paper Award, IEEE Transactions on Neural Networks and Control and Systems from 1995 to 2000, the IEEE ITS Magazine from 2006 Learning Systems Outstanding Paper Award, the Outstanding Paper Award of to 2007, the IEEE/CAA Journal of Automatica Sinica from 2014 to 2017, Acta Automatica Sinica, IEEE 6th Data Driven Control and Learning Systems and the China’s Journal of Command and Control from 2015 to 2020. He Conference (DDCLS2017) Best Paper Award, and Zhang Siying Outstanding was the EiC of the IEEE Intelligent Systems from 2009 to 2012, the IEEE Paper Award of Chinese Control and Decision Conference (CCDC). He was Transactions on Intelligent Transportation Systems from 2009 to 2016, and a recipient of Shuang-Chuang Talents in Jiangsu Province, China, Young is the EiC of the IEEE Transactions on Computational Social Systems since Researcher Award of Asia Pacific Neural Network Society (APNNS), Young 2017, and the Founding EiC of China’s Journal of Intelligent Science and Scientist Award and Yang Jiachi Tech Award of Chinese Association of Technology since 2019. Currently, he is the President of CAA’s Supervision Automation (CAA). He is a Board of Governors (BOG) member of the Council, IEEE Council on RFID, and Vice President of IEEE Systems, Man, International Neural Network Society (INNS) and a council member of CAA. and Cybernetics Society.