Design of Decentralized Repetitive Control of Linear MIMO System Edi Kurniawan, Zhenwei Cao and Zhihong Man Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Australia
[email protected],
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Abstract— This paper proposes decentralized repetitive control (RC) design for multiple inputs multiple outputs (MIMO) system. The design comprises in both continuous and discrete-time domain. A decentralized stabilizing controller is firstly developed in continuous-time to achieve a desired complimentary sensitivity function. Relative Gain Array (RGA) analysis is used to obtain the best pairings of inputs and outputs. Then, robustness analysis is conducted to determine the impact of neglected couplings. Secondly, the RC is designed in discrete-time based on the information of complimentary sensitivity function. A set of low order, stable and causal repetitive control compensators are developed using a new frequency domain method. The proposed RC compensator works for both minimum and non-minimum phase system. Simulation results for linear MIMO model of pick and place robot arm are presented to validate the effectiveness of the design. Keywords— repetitive control; decentralized control; MIMO system
I.
INTRODUCTION
Tracking periodic commands are common tasks found in many control systems such as disk drive, optical disc player, pick-and-place robot. Repetitive Control (RC) gives superior performance for tracking purpose compared to a nonpredictive control scheme [1]. Repetitive controller requires two main designs; an internal model and a RC compensator design. The internal model refers to the internal model principle originated from [2], which states that the reference model needs to be attached to the controller in order to achieve zero tracking error. The RC compensator is part of RC used to stabilize closed-loop system. Some RC designs to track/reject repetitive signal have been proposed in [3-14]. The RC designs in [3-10] aim to track/reject repetitive signal with fixed period, while the designs in [11-14] are intended to handle time-varying repetitive signal. The designs in [3-10] basically formulated a compensator that is used to compensate the dynamic of single input single output (SISO) model, in which the designs end up with either non-causal or unstable compensator. The designs of RC compensator for MIMO system are still few to be found, and some of them were proposed in [15,16]. Jeong and Fabien [15] proposed the design of Phase Cancellation Inverse Matrix (PCI) which operates by compensating the phase lag in the diagonal elements of the plant model. The idea is initiated from Zero Phase Tracking Error Controller (ZPETC) by [17] targeting to cancel the phase response of the plant. Xu [16]
c 978-1-4673-6322-8/13/$31.00 2013 IEEE
proposed an optimization based compensator to mimics the inverse of the MIMO model. The approaches [15,16] are based on the full MIMO design which results of a controller with the same dimension to the plant. This implies that if we have a ݉ ݉ݔMIMO system ( ݉ଶ transfer functions), then we need to design ݉ଶ RC compensators. Moreover, the designs also end up with non-causal compensator which needs to be merged with the internal model to make it realizable. This paper proposes the design of RC compensator for MIMO system based on decentralized control, which means that the MIMO system is considered as a set of SISO systems. The design is motivated by the fact that most of MIMO control problems can be treated on decentralized basis [18]. In addition, the decentralized control is easier as it can apply simpler SISO theories [18]. Another advantage of decentralized RC here is only ݉ RC compensators are required. The design of decentralized RC involves both continuous and discrete-time techniques. Firstly, a decentralized stabilizing controller is designed in continuous time to obtain the desired complementary sensitivity function. Before that, Relative Gain Array (RGA) and Robust analysis are performed to determine whether decentralized control is applicable or not. Secondly, a set of RC compensators are designed in discrete time based on the information of the complimentary sensitivity function. The RC design uses an optimization to obtain a low order, stable and causal compensator. This paper is organized as follows. Section 2 presents the design of decentralized stabilizing controller in continuous time, which also covers the RGA and Robustness analysis. Section 3 covers RC compensator design using optimization. Simulation results are given in Section 4. Section 5 draws the conclusion II.
DECENTRALIZED CONTROL
Let a ݉ ݉ݔMIMO system is represented by a following transfer function:
ܩሺݏሻ ൌ
݃ଵଵ ሺݏሻ ڭ ݃ଵ ሺݏሻ
ڮ ڰ ǥ
݃ଵ ሺݏሻ ڭ൩ ݃ ሺݏሻ
(1)
The plant model ܩሺݏሻ has ݉ inputs and ݉ outputs, where
427
the relation between inputs and outputs can be formulated as follows: ݕଵ ሺݐሻ ݃ଵଵ ሺݏሻݑଵ ሺݐሻ ڮ ݃ଵ ሺݏሻݑ ሺݐሻ ݕଶ ሺݐሻ ݃ଶଵ ሺݏሻݑଵ ሺݐሻ ݃ଶଶ ሺݏሻݑଶ ሺݐሻǤ Ǥ ݃ଶ ሺݏሻݑ ሺݐሻ ൦ ൪ൌ൦ ൪ (2) ڭ ڭ ݕ ሺݐሻ ݃ଵ ሺݏሻݑଵ ሺݐሻǤ Ǥ ݃ ሺݏሻݑ ሺݐሻ ሺݐሻǡ ǥ ǡ ݕ ሺݐሻ and ݑଵ ሺݐሻǡ ǥ ǡ ݑ ሺݐሻ are plant outputs where ݕଵ and inputs respectively. Decentralized control aims to approximate the MIMO system into a set of independent SISO systems. This is different with decoupling control which tries to convert the full MIMO model into a perfect set of independent SISO models. The assumption in the design of decentralized control is ignoring dynamics that result in weak interactions. Each of the system outputs is approximated from the input response that gives dominant contribution. Therefore, the degree of interaction is necessary to be quantified in the decentralized control. RGA introduced by [19] is one of the techniques termed as dominant interaction control method that can be used to determine the best input output pairings for multivariable control [20]. RGA is defined as matrix Ȧ which is formulated as follows: (3) Ȧ ൌ ܩሺͲሻǤ כሾି ܩଵ ሺͲሻሿ் ିଵ ሺͲሻ
are system dc gain matrix and its where ܩሺͲሻ and ܩ inverse, notation Ǥ כand ܶ operate as element wise multiplication and transpose of matrix respectively. For simplicity, suppose the RGA suggestsሾݑ ǡ ݕ ሿpairings, where ൌ ͳǡ ǥ ǡ ݉ . This means that we only need to consider transfer function ݃ ሺݏሻ in the design. Now, we need to choose complimentary sensitivity function ሺሻ as the desired stabilized closed-loop model of ݃ ሺݏሻǤ (4) ሺሻ ൌ ሾ ሺሻǡ ǥ ǡ ሺሻሿ ଵ
(5)
ܩሺݏሻ =ܩ ሺݏሻሾ ܫ ܩሺݏሻሿ where Ͳ ڰ Ͳ
Ͳ Ͳ
൩
(6)
݃ ሺݏሻ
ܩሺݏሻ ൌ ሾܩሺݏሻ െ ܩ ሺݏሻሿൣܩ ିଵ ሺݏሻ൧
(7)
and ܫis a ݉ ݉ݔidentity matrix. Then, robustness check is given by stability condition as follows [18]: (8) ߪതሾ ܩሺ݆߱ሻሺɘሻሿ ൏ ͳ א ߱Թ
Eq. (8) states that the maximum singular values of ሾ ܩሺ݆߱ሻ ሺɘሻሿ has to be less than one.
428
ଵ
୫
where
୧ ሺሻǡ ݅ ൌ ͳǡ ǥ ǡ ݉ are obtained to satisfy the following equality
୧ ሺሻ݃ ሺݏሻ (10) ൌ ୧ ሺሻ ͳ
୧ ሺሻ݃ ሺݏሻ Transfer function
୧ ሺሻ, ݃ ሺݏሻ, and ୧ ሺሻ can be expressed in the form of numerator and denumerator as follow: ݊ܿ ሺݏሻ ݊݃ ሺݏሻ ݊ݐ ሺݏሻ (11) ǡ ୧୧ ሺሻ ൌ ǡ ୧ ሺሻ ൌ ݀ܿ ሺݏሻ ݀݃ ሺݏሻ ݀ݐ ሺݏሻ The polynomial ݊ܿ ሺݏሻ and ݀ܿ ሺݏሻ can be obtained by solving the following Diophantine Equation
୧ ሺሻ ൌ
݊ܿ ሺݏሻሾ݊݃ ሺݏሻ݊ݐ ሺݏሻ െ ݊݃ ሺݏሻ݀ݐ ሺݏሿ
(12)
݀ܿ ሺݏሻሾ݀݃ ሺݏሻ݊ݐ ሺݏሻሿ ൌ Ͳ The relative degree of complimentary sensitivity function ݐ ሺݏሻneeds to be chosen carefully so it gives proper stabilizing controller
୧ ሺሻ. III.
DISCRETE RC DESIGN FOR SISO
In this section, we introduce a new RC design using optimization in the frequency domain which is also discussed in [11]. The design is based on the information of complimentary sensitivity function ሺሻ. The general structure of SISO RC system is shown in Fig 1
z −N
୫
In the decentralized control, robustness analysis is needed to check the impact of the neglected dynamics [18]. To perform robustness analysis, the ܩሺݏሻ firstly is represented as the nominal model ܩ ሺݏሻ with additive uncertainty ܩሺݏሻ
݃ଵଵ ሺݏሻ ܩ ሺݏሻ ൌ Ͳ Ͳ
Once robustness check has been performed, then the decentralized stabilizing controller ܥሺݏሻ can be designed based on the chosen complimentary sensitivity function ሺሻ (9) ሺሻ ൌ ሾ
ሺሻǡ ǥ ǡ
ሺሻሿ
Fig. 1. Block diagram of discrete-time RC system
where ܩ ሺݖሻ is the digital repetitive controller, t ሺݖሻ is the stabilized plant model, ݎሺ݇ሻ is the periodic reference signal, ݁ሺ݇ሻ is tracking error, and ݕሺ݇ሻ is tracking output. The digital repetitive controller has the following transfer function. ܳሺݖሻ (13) ܨሺݖሻ ܩ ሺݖሻ ൌ ே ݖെ ܳሺݖሻ where ܰ ൌ
்ೝ ்ೞ
אԳ , ܶ is reference signal period, ܶ௦ is the
sampling period, ܨሺݖሻ is the RC compensator, ܳሺݖሻ is zero phase low pass filter. The RC system is stable if the following conditions are fulfilled [4,21] : 1) ݐሺݖሻ is stable. గ (14) 2) ห൫ͳ െ ݐሺݖሻܨሺݖሻ൯ܳሺݖሻห ൏ ͳ Ͳ൏ ߱ ൏
2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA)
்ೞ
The RC compensator designs mostly aim to mimics the inverse of the SISO plant model, where end up with either non-causal or unstable compensator. Here, we develop a design scheme to obtain a low order, stable and causal compensator. ݍ ݖ ݍଵ ݖିଵ ڮ ݍ (15) ǡ ݉ Ͳ ܨሺݖሻ ൌ ݖ ݎଵ ݖିଵ ڮ ݎ The ݐሺݖሻ is a known stable transfer function, while ܳሺݖሻ is a chosen filter to give specific tracking bandwidth. Here, the RC compensator ܨሺݖሻ is simply designed to satisfy stability condition (14) as both ݐሺݖሻ and ܳሺݖሻ are already known. Let ܯ௧ and ߠ௧ , ܯொ andߠொ , ܯி and ߠி are magnitude and phase response of ݐሺݖሻ , ܳሺݖሻ and ܨሺݖሻ at frequency ߱ respectively. Denote the magnitude response of ൣ൫ͳ െ ݐሺݖሻܳݖܨሺݖሻ at frequency ߱݅ as ݄݅ (16) ݄ ൌ หͳ െ ݁ ܯఏಷ ݁ ܯఏ หห ݁ ܯఏೂ ห ൏ ͳ
The first constraint guarantees that all poles of ܨሺݖሻ are inside the unit circle with a safe minimum distance ߜ to the unit circle. The second constraint guarantees that the closed loop system is stable with a positive margin of ɒ . The optimization problem (20) is a nonlinear optimization to findሺʹ݉ ͳሻ parameters of ܨሺݖሻ that satisfy both bound and nonlinear constraints.
ܳሺݖሻ is a low pass filter with zero phase for all frequencies. Thus, (16) can be rewritten as: (17) ݄ ൌ หͳ െ ݁ ܯఏಷ ݁ ܯఏ หܯ
where ݃ଵଵ ሺݏሻǡ ݃ଵଶ ሺݏሻǡ ݃ଶଵ ሺݏሻ , and ݃ଶଶ ሺݏሻ are given in the bottom of this page. This is a stable and minimum phase MIMO plant as all zeros and poles of ݀݁ݐሾܩሺݏሻሿ are on the left half plane. The RGA of ܩሺݏሻ is calculated as follows:
ி
௧
ி
SIMULATION RESULTS
Simulation is now performed to validate the effectiveness of the design. A two-input two -output MIMO model of pick and place robot arm [22] is used in the simulation. ܩሺݏሻ ൌ
݃ଵଵ ሺݏሻ ݃ଶଵ ሺݏሻ
݃ଵଶ ሺݏሻ ൨ ݃ଶଶ ሺݏሻ
(21)
ொ
௧
ொ
Eq (17) can be further derived to భ
݄ ൌ ൣͳ െ ʹ்ܯ ܿݏ൫ߠ் ൯ ்ܯ ଶ ൧మ ܯொ
(18)
where ்ܯ ൌ ܯி ܯ௧ , and ߠ் ൌ ߠி ߠ௧ To be stable, the stability condition (14) states that ݄ has to be less than one for all frequencies. Here we only consider harmonic frequencies and define the objective function as follows: ேȀଶ
݄ ்௧ ൌ ݄ ߱ ൌ ʹߨ ୀଵ
IV.
݅ ݅ ൌ ͳǡʹǤ Ǥ ǡ ܰȀʹ ܰܶ௦
(19)
Then, the following optimization problem is introduced: ݄ ்௧
ݐሺݏሻ ܶሺݏሻ ൌ ଵ Ͳ
ሺభǡ ǡǥǡǡబ ǡǥǡ ሻ
ǣ ଵ െͳ ߜ ͳെߜ Ǥ 1. ൦ Ǥ ൪ ൏ ൏ Ǥ Ǥ Ǥ Ǥ െͳ ߜ ͳെߜ ݅ ൌ ͳǡʹǤ Ǥ ǡ ܰȀʹ 2. ݄ ൏ ͳ െ ߬ǡ ߱ ൌ ʹߨ
்
(22) ͳ Ͳ ͳ Ͳ ͳ Ͳ ିଵ ቃ ቃ Ǥ כ൬ቂ ቃ ൰ ൌቂ Ͳ ͳ Ͳ ͳ Ͳ ͳ The RGA value Ȧ shows that the dynamic ݃ଵଵ ሺݏሻ and ݃ଶଶ ሺݏሻ give dominant interaction to output ݕଵ ሺݐሻ and ݕଶ ሺݐሻ respectively. The RGA suggests that the best pairings are ሾݑଵ ǡ ݕଵ ሿ and ሾݑଶ ǡ ݕଶ ሿ. Robust analysis needs to be performed to check the impact of the neglected dynamics ݃ଵଶ ሺݏሻ and ݃ଶଶ ሺݏሻ . Let the desired complimentary sensitivity function ܶሺݏሻ is Ȧൌቂ
(20)
ே்ೞ
Ͳ ൨ ݐଶ ሺݏሻ
ͶͲͲ ۍଶ ሺ ݏ ݏ ͳሻሺ ݏ ʹͷሻ ൌێ ێ Ͳ ۏ
(23) ې ۑ ͶͲͲ ۑ ሺ ݏଶ ݏ ͳሻሺ ݏ ʹͷሻے Ͳ
where ଵǡ ଶ ǡ ǥ ǡ are ݉ real poles of ܨሺݖሻ , ߜ and ߬ are
small positive constants. ݃ଵଵ ሺݏሻ ൌ ͷݏଵଶ
ͷǤ͵݁ െ ݃ଵଶ ሺݏሻ ൌ
ͷݏଵ
ͷǤ͵݁ െ ݃ଶଵ ሺݏሻ ൌ
ͲǤͳ ݏଽ ͳͶǤͷͳ ଼ ݏ ͷͺǤʹ ݏ ͳǤ͵ͻʹ݁Ͷ ݏ ʹǤʹ݁ͷ ݏହ ʹǤͷͺ݁ ݏସ ʹǤͲͻ݁ ݏଷ ͳǤͳ݁ͺ ݏଶ ͶǤʹͳ݁ͺ ݏ Ǥ݁ͺ ͲǤͲʹݏଵଵ ͲǤͻͳݏଵ ͵ͳǤʹ ݏଽ ͳͶǤͳ ଼ ݏ ͳǤʹ݁Ͷ ݏ ͳǤͶͷ݁ͷ ݏ ͳǤͶ݁ ݏହ ͳǤͲͳ݁ ݏସ ͷǤ݁ ݏଷ ʹǤ͵݁ͺ ݏଶ ͷǤͻ݁ͺ ݏ Ǥ݁ͺ
ͲǤͲͳͶ ݏଽ
െͲǤͲʹʹ ݏെ ͵ǤʹͶ ݏെ ͺͺǤ͵ ݏହ െͳ͵Ͷ ݏସ െ ͳǤͲ݁Ͷ ݏଷ െ ͶǤͷʹ݁Ͷ ݏଶ ͲǤʹ ଼ ݏ ʹͲ ݏ ͵͵ ݏ ͶͶͷ ݏହ ͶǤ͵݁Ͷ ݏସ ʹǤͻ݁ͷ ݏଷ ͳǤͶ݁ ݏଶ ͶǤͳͺ݁ ݏ Ǥ͵ʹ݁
െͲǤͳ ݏെ ͺǤ ݏെ ͳͻͶ ݏହ െʹͶͻͺ ݏସ െ ͳǤͺ݁Ͷ ݏଷ െ ǤͶ݁Ͷ ݏଶ ͷǤʹ݁ െ ͷݏଵ ͲǤͲͳͶ ݏଽ ͲǤʹ ଼ ݏ ʹͲ ݏ ͵͵ ݏ ͶͶͷ ݏହ ͶǤ͵݁Ͷ ݏସ ʹǤͻ݁ͷ ݏଷ ͳǤͶ݁ ݏଶ ͶǤͳͺ݁ ݏ Ǥ͵ʹ݁ ݃ଶଶ ሺݏሻ ൌ ͲǤͲʹ ݏଽ ͶǤͻͷ ଼ ݏ ʹͶ ݏ ͵ͻͶ ݏ ͳǤ͵݁ͷ ݏହ ͳǤͻ݁ ݏସ ͳǤͷ݁ ݏଷ ͻǤͶ݁ ݏଶ ͵Ǥͺ݁ͺ ݏ Ǥ݁ͺ ͷǤ͵݁ െ ͷݏଵଶ ͲǤͲͳͶݏଵଵ ͲǤͻ ݏଵ ͵ͳ ݏଽ ͳͶǤͳ ଼ ݏ ͳǤͳͻ݁Ͷ ݏ ͳǤͶͺ݁ͷ ݏ ͳǤͶ݁ ݏହ ͳǤͲͶ݁ ݏସ ͷǤ݁ ݏଷ ʹǤ͵݁ͺ ݏଶ ͷǤͻ݁ͺ ݏ Ǥ݁ͺ
2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA)
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Both ݐଵ ሺݏሻ and ݐଶ ሺݏሻ are chosen to have relative degree 3, and to give tracking bandwidth 3.7 rad/s. The uncertainty model ܩሺݏሻ is given as follows: Ͳ ܩሺݏሻ ൌ ݃ଶଵ ሺݏሻȀ݃ଵଵ ሺݏሻ
݃ଵଶ ሺݏሻȀ݃ଶଶ ሺݏሻ ൨ Ͳ
(24)
Singular values of ܩሺ݆߱ሻሺɘሻ is shown in Fig.2
ͺǤ͵݁ െ Ͷሺ ݖ ͵ǤͲͻሻሺ ݖ ͲǤʹʹሻ (28) ሺ ݖെ ͲǤͷ͵ሻሺ ݖଶ െ ͳǤͺͷ ݖ ͲǤͺሻ Both ݐଵ ሺݖሻ and ݐଶ ሺݖሻare non-minimum phase model because they have a zero outside the unit circle. The Q-filterܳሺݖሻ, the order of compensator݊, and positive constants ߜ and ߬ are chosen respectively as follow: ܳଵ ൌ ܳଶ ሺݖሻ ൌ ͲǤʹͷ ݖ ͲǤͷ ͲǤʹͷି ݖଵ , ݊ ൌ ͵, (29) ߜ ൌ ͲǤͲͷ, and ߬ ൌ ͲǤͲͷ The RC compensator ܨሺݖሻ is obtained by solving optimization problem (20) using an Optimization Toolbox from Matlab. ݐଵ ሺݖሻ ൌ ݐଶ ሺݖሻ ൌ
ܨሺݖሻ ܨሺݖሻ ൌ ଵ Ͳ
Ͳ ൨ ܨଶ ሺݖሻ
(30)
where ͵ͻͶǤ ݖଷ െ ͻʹǤʹ ݖଶ ͲͳǤ ݖെ ͳǤ (31) ݖଷ ʹǤͲͺ ݖଶ ͳǤʹͺ ݖ ͲǤͳͻ In this simulation, two tracking schemes are presented. The first scheme is the first channel ݕଵ is required to track triangle reference signal, while the second channel ݕଶ needs to stay idle. The second scheme is both channels are required to track triangle reference signals with different amplitudes. The tracking output and tracking error for the first scheme is shown in Fig 3. ܨଵ ൌ ܨଶ ሺݖሻ ൌ
Fig. 2. Singular values of ۵ઢ ሺܒሻ܂ሺܒሻ
Fig. 2 shows that maximum singular values of ܩሺ݆߱ሻሺɘሻ is still less thanͲ݀ܤ. Thus, the impact of the neglected dynamics is acceptable. The decentralized stabilizing controller ܥሺݏሻ is obtained by solving the following Diophantine Equations ݊ܿଵ ሺݏሻሾ݊݃ଵଵ ሺݏሻ݊ݐଵ ሺݏሻ െ ݊݃ଵଵ ሺݏሻ݀ݐଵ ሺݏሿ
(25)
݀ܿଵ ሺݏሻሾ݀݃ଵଵ ሺݏሻ݊ݐଵ ሺݏሻሿ ൌ Ͳ ݊ܿଶ ሺݏሻሾ݊݃ଶଶ ሺݏሻ݊ݐଶ ሺݏሻ െ ݊݃ଶଶ ሺݏሻ݀ݐଶ ሺݏሿ
(a)
(26)
݀ܿଶ ሺݏሻሾ݀݃ଶଶ ሺݏሻ݊ݐଶ ሺݏሻሿ ൌ Ͳ The controller ܥሺݏሻ is ܿଵ ሺݏሻ Ͳ (27) ൨ Ͳ ܿଶ ሺݏሻ whereܿଵ ሺݏሻ and ܿଶ ሺݏሻ are shown in the bottom of this page Let the frequency of reference signal and sampling rate be 0.2 Hz and 40 Hz respectively. The discretized model of ݐଵ ሺݏሻ and ݐଶ ሺݏሻ are: ܥሺݏሻ ൌ
ܿଵ ሺݏሻ ൌ
(b)
͵Ǥͻ݁ െ ͺ ݏଵଶ ͳǤͳ݁ െ ͷ ݏଵଵ Ǥͺ݁ െ Ͷ ݏଵ ͲǤͲʹ ݏଽ ͲǤͷ͵ ଼ ݏ ͺǤͻ ݏ ڮ ʹǤͻ݁ െ ݏଵଶ ͵Ǥ݁ െ ͷ ݏଵଵ ͳǤͺ݁ െ ͵ ݏଵ ͲǤͲ ݏଽ ͳǤͶ ଼ ݏ ʹʹǤʹ ݏ ʹͷͺǤ ݏ ڮ ͳͲͺǤͶ݁Ͷ ݏ ͳͲͶ ݏହ ͷͷʹ ݏସ ͶǤ͵݁Ͷ ݏଷ ͳǤ݁ͷ ݏଶ ͶǤͶ݁ͷ ݏ ͷǤ݁ͷ ʹʹ͵Ͳ ݏହ ͳǤͶ݁Ͷ ݏସ Ǥʹ݁Ͷ ݏଷ ͳǤͷ݁ͷ ݏଶ ʹǤ͵ݏ
ܿଶ ሺݏሻ ൌ
ʹǤͷ݁ െ ͳʹݏଵଶ Ǥͺ݁ െ ͳͲݏଵଵ ͶǤͶ݁ െ ͺݏଵ ͳǤͷ݁ െ ݏଽ ͵Ǥͷ݁ െ ͷ ଼ ݏ ͷǤ݁ െ Ͷ ݏ ڮ ͵Ǥʹ݁ െ ͳʹݏଵଶ ݁ െ ͳͲݏଵଵ ͷǤͳ݁ െ ͺݏଵ ͳǤͻͺ݁ െ ݏଽ ͶǤͺͺ݁ െ ͷ ଼ ݏ ͺǤͶ݁ െ Ͷ ݏ ڮ Ǥʹ݁ െ ͵ ݏ Ǥͺ݁ െ ʹ ݏହ ͲǤͷ ݏସ ʹǤ ݏଷ ͳͳǤͳ ݏଶ ʹͺǤͷ ݏ ͵Ǥͺ ͳǤͲ݁ െ ʹ ݏ ͲǤͳ ݏହ ͲǤ ݏସ ͵ǤͶ ݏଷ ͳͲǤͶͺ ݏଶ ͳͷǤʹݏ
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V.
(c) Fig. 3. The first scheme a) Tracking output ࢟ ,b) Tracking output ࢟ , c).Tracking error ࢋ and ࢋ
The first scheme aims to emphasize the effect of neglected dynamics in this decentralized control. On Channel 1, tracking output ݕଵ is able to follow periodic reference ݎଵ after 4 repetitions. On channel 2, a small excitation due to coupling effect appears. The tracking output and tracking error for the second scheme is shown in Fig 4.
This paper proposes decentralized RC design for MIMO system. The decentralized stabilizing controller is firstly developed in continuous-time to achieve the desired complimentary sensitivity function. RGA analysis is used to obtain the best pairing of inputs and outputs. Then, robustness analysis is conducted to determine the impact of neglected couplings. Secondly, the RC is designed in discrete-time based on the information of complimentary sensitivity. The RC design uses an optimization to obtain a low order, stable and causal compensator. Simulation results have been presented to validate the effectiveness of the design. REFERENCES [1]
[2]
[3]
[4]
[5] (a) [6]
[7]
[8]
(b)
[9]
[10]
[11]
[12] (c) Fig. 4. The second scheme a) Tracking output࢟ ,b) Tracking output ࢟ , c).Tracking error ࢋ and ࢋ [13]
Fig. 4 shows good tracking performance for both channel 1 and 2. The simulation results verify the effectiveness of decentralized RC.
CONCLUSION
[14]
[15]
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[16] K. Xu, "Multi-Input Multi-Output Repetitive Control Theory And Taylor Series Based Repetitive Control Design," Phd thesis, Graduate School of Arts and Sciences –Columbia University, 2012. [17] M. Tomizuka, "Zero phase error tracking algorithm for digital control," Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, vol. 190, pp. 65-68, 1987. [18] G. C. Goodwin, S. F. Graebe, and M. E. Salgado, Control System Design , Prentice Hall, 2000. [19] E. Bristol, "On a new measure of interaction for multivariable process control," IEEE Transactions on Automatic Control, vol. 11, pp. 133134, 1966. [20] M. T. Tham, "Multivariable Control: An Introduction to Decoupling Control " Dept.Of Chemical and Process Engineering, University of Newcastle upon Tyne, 1999. [21] K. K. Chew and M. Tomizuka, "Steady-state and stochastic performance of a modified discrete-time prototype repetitive controller” Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, vol. 112, pp. 35-41, 1990. [22] L. Wang, S. Chai, E. Rogers, and C. T. Freeman, "Multivariable Repetitive-Predictive Controllers Using Frequency Decomposition," IEEE Transactions on Control Systems Technology, vol. 20, pp. 15971604, 2012.
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