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Multi-model direct adaptive decoupling control with application to the wind tunnel system
ISA TRANSACTIONS® ISA Transactions 44 共2005兲 131–143
Multi-model direct adaptive decoupling control with application to the wind tunnel system Xin Wang,a,* Shaoyuan Li,a Wenjian Cai,b Heng Yue,c Xiaojie Zhou,c Tianyou Chaic a
Institute of Automation, Shanghai Jiao Tong University, Shanghai, 200030, People’s Republic of China b School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore c Research Center of Automation Northeastern University, Shenyang, 110004, People’s Republic of China
共Received 25 March 2003; accepted 24 June 2004兲
Abstract In this paper, a new multi-model direct adaptive decoupling controller is presented for multivariable processes, which includes multiple fixed optimal controllers, one free-running adaptive controller, and one re-initialized adaptive controller. The fixed controllers provide initial control to the process if its model lies in the corresponding region. For each controller selected, the re-initialized adaptive controller uses the values of this particular controller to improve the adaptation speed. This controller may replace the fixed controller at a later stage according to the switching criterion which is to select the best one among all controllers. A free-running adaptive controller is also added to guarantee the overall system stability. Different from the multiple models adaptive control structure proposed in Narendra, Balakrishnan, and Ciliz 关Adaptation and learning using multiple models, switching, and tuning. IEEE Control Syst. Mag. 15, 37–51 共1995兲兴, the method not only is applicable to the multi-input multi-output processes but also identifies the decoupling controller parameters directly, which reduces both the computational burden and the chances of a singular matrix during the process of determining controller parameters. Several examples for a wind tunnel process are given to demonstrate the effectiveness and practicality of the proposed method. © 2005 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Multi-model; Direct adaptive control; Decoupling control; Pole placement; Global convergence; Wind tunnel process
1. Introduction When a system boundary condition changes, a subsystem fails, or an external disturbance occurs, these cause the parameter variation of the system that will generally result in slower reaction, slower convergence, poorer transient response, and possibly an unstable system 关1兴. To deal with these kinds of processes, an adaptive control scheme, by adding an on-line parameter estimation algorithm to construct control systems, can be employed 关2兴. *Corresponding author. [email protected]
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However, the conventional adaptive controller based on a single model can only be effective for systems with unknown time invariant or slowly time varying parameters. For systems with large uncertainties or abruptly changed system parameters, the multi-model adaptive controller 共MMAC兲 has been demonstrated to be more effective 关3兴. The earliest MMAC appeared in the 1970s where multiple Kalman filter-based models were proposed to improve the accuracy of state estimation 关4,5兴. However, no switching and stability results were presented. Two MMAC schemes with switching conditions were later developed during the 1980s, namely, direct switching MMAC and
0019-0578/2005/$ - see front matter © 2005 ISA—The Instrumentation, Systems, and Automation Society.
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indirect switching MMAC. For the first scheme, multiple state feedback controllers were used and switched based on the predetermined switching sequence to guarantee the stability of the closedloop system 关6兴. For the second scheme, multiple convex region parameter estimators were adopted to overcome the problem of stablizability of the estimated model 关7兴. The choice of when and which controller to switch is determined by a switching criterion. Another important method in this field was the work of Morse 关8兴. His method focused on the superior control of the families of linear set-point controllers using hysteresis switching logic to deal with noise, disturbances, and unmodeled dynamics 关9兴. In order to improve the transient response, Narendra et al. 关10兴 utilized multiple adaptive models to simultaneously identify the parameters of unknown systems. According to a switching criterion, the best-fit model is selected and the corresponding controller is fired at the time. However, after a certain period of operation, multiple adaptive models might converge to a particular neighborhood which defeats the purpose of the multiple models control principle. To improve the performance of the multiple adaptive models control system, a control method with multiple fixed and two adaptive models were developed for both continuous and discrete-time systems 关11,12兴. Unfortunately, all of these methods are intended for single-input single-output 共SISO兲 systems. In addition, they adopt the indirect adaptive algorithm, which may make the control system less robust to estimated parameter errors. The repeated inverse matrices in the algorithm may also cause the singularity problem for those not so well-conditioned processes 关13兴. In this paper, a new multi-model direct adaptive decoupling controller 共MMDADC兲 is presented to extend Narendra’s indirect adaptive control structure for SISO processes 关12兴 into a direct adaptive control structure for multi-input multi-output 共MIMO兲 processes. The control structure includes a set of multiple fixed optimal controllers, a freerunning adaptive controller and a re-initialized adaptive controller. The effect of the fixed controllers is to provide initial control actions to the process if its model lies in the corresponding region. The re-initialized adaptive controller utilizes the parameters of the selected controller as its initial value for the adaptation algorithm to improve the
transient response. As the value of the reinitialized controller is very close to that of the true optimal one, it has faster adaptation speed compared with the conventional adaptive controller with other initial values. Therefore the reinitialized adaptive controller can be used to replace the selected ones at a later stage according to a switching criterion in which the value of a controller nearest to the true optimal one is selected. A free-running adaptive controller using single model structure is also added to guarantee the overall system stability. The main contributions of this paper are 共i兲 the proposed method is applicable to MIMO processes and it is an optimal control with dynamic decoupling, 共ii兲 it adopts the direct adaptive algorithm to reduce both the computational burden and the chances of a singular matrix during the process of determining controller parameters, 共iii兲 the global stability of the overall system and the convergence of the tracking error to zero using this control method are theoretically proved. Several simulation examples in an injector driven transonic wind tunnel process show that the MMDADC can improve the overall system performances compared with the conventional adaptive control method and can meet the strict requirements. 2. Fixed decoupling control Consider a discrete-time linear time-varying 共LTV兲 MIMO system described in the determinant autoregressive moving average 共DARMA兲 form as
A 共 t,z ⫺1 兲 y 共 t 兲 ⫽B 共 t,z ⫺1 兲 u 共 t⫺k 兲 ⫹d 共 t 兲 , 共1兲 with
A 共 t,z ⫺1 兲 ⫽I⫹A 1 共 t 兲 z ⫺1 ⫹¯⫹A n a 共 t 兲 z ⫺n a ,
共2兲
B 共 t,z ⫺1 兲 ⫽B 0 共 t 兲 ⫹B 1 共 t 兲 z ⫺1 ⫹¯⫹B n b 共 t 兲 z ⫺n b , 共3兲 where u ( t ) , y ( t ) , d ( t ) , k, and z ⫺1 are the n⫻1 input vector, n⫻1 output vector, n⫻1 steadystate offset vector, process time delay, and unit delay operator, respectively, A ( t,z ⫺1 ) with order of n a and B ( t,z ⫺1 ) with order of n b are polynomial matrices with comparable dimensions, A i ( t ) and B i ( t ) are time-varying parameter matrices with infrequent large jumps, B 0 ( t ) is a nonsingular known matrix for any t, and I is a unit matrix.
Wang, Li, Cai, Yue, Zhou, Chai / ISA Transactions 44 (2005) 131–143
It is assumed that the system satisfies the following conditions: 共i兲 The period between two adjacent jumps is sufficiently large and the parameters in between can be viewed as time constant; 共ii兲 The parameters of the system model, ⌽ ( t ) ⫽ 关 ⫺A 1 ( t ) ,...;B 0 ( t ) ,...;d ( t ) 兴 , change in a compact set ⌺, i.e., there exists a constant value 0 ⬍K⬍⬁ such that 储 ⌽ ( t ) 储 ⬍K for all t; 共iii兲 n a , n b , and k are known a priori; 共iv兲 The system is minimum phase. According to the system operating conditions, the overall working range can be partitioned into multiple regions. In each region, a fixed system model is elected and all these models constitute the multiple fixed system models. Mathematically, it can be described as follows: Partition ⌺ into m subsets ⌺ s ( s⫽1,...,m ) and each ⌺ s has the following properties: m 共i兲 艛 s⫽1 ⌺ s 傶⌺, ⌺ s is not empty; s⫽1,...,m; 共ii兲 For any ⌽苸⌺ s , s⫽1,...,m, there exists a ⌽ s 苸⌺ s and 0⭐ra ⬍⬁, satisfying 储 ⌽⫺⌽ s 储 ⭐ra . s
133
In order to find unique polynomial matrices F s ( z ⫺1 ) and G s ( z ⫺1 ) , the orders of F s ( z ⫺1 ) and G s ( z ⫺1 ) are chosen, respectively, as
n f ⫽k⫺1, s
n g ⫽n a ⫺1. s
共7兲
Multiply Eq. 共4兲 by F s ( z ⫺1 ) from the left and substitute Eq. 共6兲 into the resultant equation, and we obtain
P s 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽G s 共 z ⫺1 兲 y 共 t 兲 ⫹F s 共 z ⫺1 兲 B s 共 z ⫺1 兲 u 共 t 兲 ⫹F s 共 z ⫺1 兲 d s .
共8兲
Substitute Eq. 共8兲 into Eq. 共5兲, and the optimal control law is derived by minimizing the cost function which results in
G s 共 z ⫺1 兲 y 共 t 兲 ⫹H s 共 z ⫺1 兲 u 共 t 兲 ⫹r¯ s ⫽R s 共 z ⫺1 兲 w 共 t 兲 , 共9兲
s
⌽ s is the center of the subset ⌺ s and r a is the s
corresponding radius. The existence of r a is guars
anteed by assumption 共ii兲. Let the centers of all subsets be the fixed system models which cover the entire set ⌺, then each model of ⌽ s can be described as
A s 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽B s 共 z ⫺1 兲 u 共 t 兲 ⫹d s ,
共4兲
H s 共 z ⫺1 兲 ⫽F s 共 z ⫺1 兲 B s 共 z ⫺1 兲 ⫹Q s 共 z ⫺1 兲 , ¯r s ⫽F s d s ⫹r s .
共11兲
Combine Eq. 共9兲 with Eq. 共4兲, and the closed-loop system is formulated by
⫽R s 共 z ⫺1 兲 w 共 t 兲 ⫹Q s B s⫺1 共 1 兲 d s ⫺r s .
Define the cost function Jc s
共12兲
Since it is minimum phase, the poles of the system can be placed arbitrarily through
Jc ⫽ 储 P s 共 z ⫺1 兲 y 共 t⫹k 兲 ⫺R s 共 z ⫺1 兲 w 共 t 兲 s
共5兲
Q s 共 z ⫺1 兲 ⫽R1 B s 共 z ⫺1 兲 ,
共13兲
P s 共 z ⫺1 兲 ⫹R 1 A s 共 z ⫺1 兲 ⫽T 共 z ⫺1 兲 ,
共14兲
s
where w ( t ) is the known reference signal, P s ( z ⫺1 ) , R s ( z ⫺1 ) , Q s ( z ⫺1 ) , and r s are the weighting polynomial matrices, respectively. For the optimal controller design, the objective is to minimize the cost function 共5兲, which is determined by the following: Introduce the identity
P s 共 z ⫺1 兲 ⫽F s 共 z ⫺1 兲 A s 共 z ⫺1 兲 ⫹z ⫺k G s 共 z ⫺1 兲 .
共10兲
关 P s 共 z ⫺1 兲 ⫹Q s 共 z ⫺1 兲 B s⫺1 共 z ⫺1 兲 A s 共 z ⫺1 兲兴 y 共 t⫹k 兲
s⫽1,...,m.
⫹Q s 共 z ⫺1 兲 u 共 t 兲 ⫹r s 储 2 ,
with
共6兲
s
where
R1 s
is a constant matrix, the polynomial matrix T ( z ⫺1 ) is assumed to be stable with the form of
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T 共 z ⫺1 兲 ⫽I⫹T 1 z ⫺1 ⫹¯⫹T n t z ⫺n t ,
共15兲
where T i , i⫽1,...,n t are designed matrices with comparable dimensions. Substituting Eqs. 共13兲 and 共14兲 into Eq. 共12兲 gives
identify the controller parameters and the formulation procedures are given as follows. Multiplying Eq. 共14兲 by y ( t⫹k ) from the right, we have
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽ P 共 z ⫺1 兲 y 共 t⫹k 兲 ⫹R 1 A 共 z ⫺1 兲 y 共 t⫹k 兲 . 共20兲
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽R s 共 z ⫺1 兲 w 共 t 兲 ⫹R 1 d s ⫺r s . s
共16兲
Then, the steady-state error and the effect of d s can be eliminated by selecting
R s 共 z ⫺1 兲 ⫽T 共 z ⫺1 兲 ,
共17兲
r s ⫽R 1 d s .
共18兲
s
Substituting P ( z ⫺1 ) and A ( z ⫺1 ) y ( t⫹k ) by Eqs. 共6兲 and 共4兲, respectively, the system estimation equation can be derived as
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽F 共 z ⫺1 兲 A 共 z ⫺1 兲 y 共 t⫹k 兲 ⫹G 共 z ⫺1 兲 y 共 t 兲 ⫹R 1 B 共 z ⫺1 兲 u 共 t 兲 ⫹R 1 d
By such a selection, Eq. 共16兲 is simplified to
T共 z
⫺1
兲 y 共 t⫹k 兲 ⫽T 共 z
⫺1
兲w共 t 兲.
⫽G 共 z ⫺1 兲 y 共 t 兲
共19兲
⫹F 共 z ⫺1 兲 B 共 z ⫺1 兲 u 共 t 兲 ⫹F 共 1 兲 d
⫺1
Remark 1. Since the polynomial matrix T ( z ) is to be designed, we can assign its element matrices T i , i⫽1,...,n t as diagonal matrices with arbitrary diagonal elements. Consequently, the resulting system can be not only dynamically decoupled but also its poles can be placed arbitrarily. Remark 2. Regardless of the difference of the models in each region, the overall system has to have the same pole locations to guarantee the smoothness of the response. As the elements of the designed polynomial matrix T ( z ⫺1 ) can be arbitrarily assigned, we can have the same T ( z ⫺1 ) for all controllers.
⫹R 1 B 共 z ⫺1 兲 u 共 t 兲 ⫹R 1 d ⫽G 共 z ⫺1 兲 y 共 t 兲 ⫹ 关 F 共 z ⫺1 兲 B 共 z ⫺1 兲 ⫹R 1 B 共 z ⫺1 兲兴 u 共 t 兲 ⫹F 共 1 兲 d ⫹R 1 d.
共21兲
Considering the definitions of Q and r in Eqs. 共13兲 and 共18兲, Eq. 共21兲 can be simplified to
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽G 共 z ⫺1 兲 y 共 t 兲 ⫹ 关 F 共 z ⫺1 兲 B 共 z ⫺1 兲 ⫹Q 共 z ⫺1 兲兴 u 共 t 兲 ⫹F 共 1 兲 d⫹r ⫽G 共 z ⫺1 兲 y 共 t 兲 ⫹H 共 z ⫺1 兲 u 共 t 兲 ⫹r¯ , 共22兲
3. Direct adaptive control Even though the fixed controllers are designed for the local regions, the free-running adaptive controller must be designed based on overall system structure. By assumption 共i兲, however, the difference between the overall system and local systems is only their parameters. Therefore the results derived in Section 2 for controller design are still valid for real-time calculations of controller parameters. For both free-running and re-initialized adaptive controllers, either model parameters or controller parameters have to be identified. In the direct adaptive control scheme, the standard recursive estimation algorithm is employed to direct
where
H 共 z ⫺1 兲 ⫽F 共 z ⫺1 兲 B 共 z ⫺1 兲 ⫹Q 共 z ⫺1 兲 ,
共23兲
¯r ⫽Fd⫹r.
共24兲
Then, the recursive estimation algorithm and optimal control law are described as 关14兴
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽G 共 z ⫺1 兲 y 共 t 兲 ⫹H 共 z ⫺1 兲 u 共 t 兲 ⫹r¯ , 共25兲
ˆ i 共 t 兲 ⫽ ˆ i 共 t⫺1 兲 ⫹a 共 t 兲
X 共 t⫺k 兲 1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲
⫻ 关 y f i 共 t 兲 T ⫺X 共 t⫺k 兲 T ˆ i 共 t⫺1 兲兴 ,
共26兲
Wang, Li, Cai, Yue, Zhou, Chai / ISA Transactions 44 (2005) 131–143
135
Fig. 1. The structure of the MMDADC.
and
ˆ 共 z ⫺1 兲 y 共 t 兲 ⫹H ˆ 共 z ⫺1 兲 u 共 t 兲 ⫹rC ⫽R 共 z ⫺1 兲 w 共 t 兲 , G 共27兲 ⫺1 respectively, where y f i ⫽T ii ( z ) y i ( t ) is the ith element of the transformation of the system output vector, X 共 t 兲 ⫽ 关 y 共 t 兲 T ,...;u 共 t 兲 T ,...,1兴 T
共28兲
is the data vector, ⌰⫽ 关 1 ,..., n 兴 is the parameter 0 0 1 1 matrix with i ⫽ 关 g i1 ,...,g in ;g i1 ,...,g in ,...; 0 0 T h i1 ,...,h in ;... 兴 , i⫽1,2,...,n, the scalar a ( t ) is used to avoid the singularity problem in estimating ˆ ( 0 ) , i.e., if H ˆ ( 0 ) is singular such that u ( t ) canH not be calculated, change the a ( t ) value in the interval ⬍a ( t ) ⬍2⫺ , 0⬍⬍1 to estimate ˆ ( 0 ) again 关14兴. H Remark 3. The same as the conventional direct adaptive control concept, the adaptive controller can direct estimate the controller parameter matrices G ( z ⫺1 ) , H ( z ⫺1 ) , ¯r through Eqs. 共25兲 and 共26兲 and presume the estimated controller matrices ˆ ( z ⫺1 ) , H ˆ ( z ⫺1 ) , rC as true ones. In this way, the G
procedure of calculating controller parameters is eliminated, which reduces both the computational burden and the chances of a singular matrix.
4. Multi-model direct adaptive decoupling control The overall multiple controllers are composed of m fixed controllers ⌰ 1 ,...,⌰ m , one freerunning adaptive controller ⌰ m⫹1 , and one reinitialized adaptive controller ⌰ m⫹2 共see Fig. 1兲. The selection of which controller to be used in the time instant is determined as follows:
J s⫽
储 ef 共 t 兲 储 2 s
1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲
,
s⫽1,...,m,m⫹1,m⫹2
共29兲
and
ef 共 t 兲 ,y f 共 t 兲 ⫺y f 共 t 兲 ⫽T 共 z ⫺1 兲 e s 共 t 兲 , s
s
共30兲
136
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where y f ( t ) ⫽T ( z ⫺1 ) y ( t ) is the transformation of the real system output and
Then, there must exist an instant t s , when t⬎t s , it holds
y f 共 t 兲 ⫽T 共 z ⫺1 兲 y s 共 t 兲
ef
s
m⫹1 i
is the transformation of the sth model output,
共 t 兲2
1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲
⭐
ef 共 t 兲 is the transformation of the output error e s ( t ) between the real system and the sth model. The switching criterion based on Eq. 共29兲 is given as
J j ⫽min共 J s 兲 ,
共31兲
which means that the jth controller is chosen as the controller to be fired if its transformation output error is minimum. The adaptive controllers include a free-running adaptive controller and a reinitialized adaptive controller. The difference between them is that the free-running adaptive controller uses a global model to determine the controller parameters, while the re-initialized adaptive controller utilizes the parameters of the controller selected by Eq. 共31兲 as its initial conditions to speed up the convergence speed. As the switching criterion is to select the one closest to the optimal value and if the free-running adaptive controller has global convergence property, the global convergence of the overall system is guaranteed. Therefore it is necessary to prove the global convergence of the multi-model adaptive controller which is given by the following theorem. Theorem 1. Subject to assumptions 共i兲–共iv兲, if the controller determined by Eq. 共27兲 is applied to the system 共4兲, 兵 y ( t ) 其 , 兵 u ( t ) 其 are bounded and limt→⬁ 储 e ( t ) 储 ⫽0. Proof: 共i兲 If
si
For the ( m⫹1 ) th free-running adaptive controller, the recursive estimation algorithm 共26兲 has the property 关14兴
lim
1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲
m⫹1 i
共 t 兲2
t→⬁ 1⫹X 共 t⫺k 兲
T
X 共 t⫺k 兲
⭐
m⫹1 i
共 t 兲2
. 1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲 共34兲
Consequently,
e f i共 t 兲 2
lim
t→⬁ 1⫹X 共 t⫺k 兲
T
X 共 t⫺k 兲
⫽0,
i⫽1,...,n. 共35兲
Since T ( z ⫺1 ) is designed to be stable, the system is minimum phase, w i ( t ) and d i are both bounded, it follows that 关14兴
兩 y i 共 t 兲 兩 ⭐K 1 ⫹K 2 max 兩 e f j 共 兲 兩 ,
0⬍K 1 ⬍⬁,
1⭐ ⭐t 1⭐ j⭐n
0⬍K 2 ⬍⬁,
1⭐t⭐N,
i⫽1,...,n,
兩 u i 共 t⫺k 兲 兩 ⭐K 3 ⫹K 4 max 兩 e f j 共 兲 兩 , 1⭐ ⭐t 1⭐ j⭐n
1⭐t⭐N,
共36兲
0⬍K 3 ⬍⬁,
i⫽1,...,n.
共37兲
According to the technical lemma in Ref. 关15兴, it results that 兵 y ( t ) 其 , 兵 u ( t ) 其 is bounded and lim 储 e f ( t ) 储 ⫽0.
⑀ t ⫽ min 兩 ef 共 t 兲 兩 .
ef
ef
e f i共 t 兲 2
0⬍K 4 ⬍⬁,
there must exist a constant value 1⭐s⭐m
共33兲
which means that for t⬎t s , no fixed controllers can be fired. The fired controller must be selected between the free-running adaptive controller ⌰ m⫹1 and the re-initialized adaptive controller ⌰ m⫹2 according to the switching criterion 共31兲. However, as the error of the free-running adaptive controller is equal to or larger than that of the re-initialized adaptive controller, it follows that
s苸 兵 1,...,m 其 ,
s
1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲 i⫽1,...,n,
s
ef 共 t 兲 ⫽0,
⑀ i共 t 兲 2
t→⬁
共ii兲 For any time instant, the system has only one true parameter set ⌰ 0 . If
e f 共 t 兲 ⫽0, s
⫽0.
共32兲
s苸 兵 1,...,m 其 ,
which means that there exists at least one fixed controller whose controller parameter matrix ⌰ s is
Wang, Li, Cai, Yue, Zhou, Chai / ISA Transactions 44 (2005) 131–143
equal to the real value of the system ⌰ 0 or ⌰ s ⫺⌰ 0 is orthogonal to the data vector X ( t⫺k ) . Then, J s ⫽0 and ⌰ s is chosen to be the controller parameter matrix until it jumps again. Combining cases 共i兲 and 共ii兲, for the transformation of the output error e f ( t ) ⫽T ( z ⫺1 ) e ( t ) and T ( z ⫺1 ) is stable, it concludes that 关15兴
lim 储 e 共 t 兲 储 ⫽0.
共38兲
t→⬁
The overall control system including multiple fixed controllers, one free-running adaptive controller, and one re-initialized adaptive controller can be summarized in the following algorithm. Algorithm 1. 共i兲 Partition the system model into m fixed models as in Eq. 共4兲; 共ii兲 Design the corresponding fixed controllers by Eq. 共9兲; 共iii兲 Form the data vector X ( t ) as in Eq. 共28兲; ˆ and ⌰ ˆ ( t ) by Eq. 共26兲; 共iv兲 Estimate ⌰ m⫹1
m⫹2
共v兲 Choose the best controller j based on the switching criterion 共31兲; ˆ ˆ 共vi兲 Initialize ⌰ m⫹2 as ⌰m⫹2 ( t ) ⫽⌰ j if j⫽m ⫹2; 共vii兲 Set the fired controller as ⌰ ( t ) ⫽⌰ j and obtain u ( t ) as in Eq. 共27兲 accordingly; 共viii兲 For t⫽t⫹1, go to step 共iii兲. Remark 4. The re-initialized adaptive controller is used to speed up the convergence speed, which causes the transient response to be greatly improved. At every instant, if it is not the best model chosen according to the switching criterion 共31兲, its parameters must be reinitialized as those of the fired model whose parameters are nearest to the real ones, then it can estimate the system parameter values from these nearest values to increase the approaching speed. Remark 5. The fixed controllers have dual functions: 共i兲 they are utilized to localize the region where the system parameters lie in and present the initial value to the re-initialized adaptive controller; 共ii兲 they provide the initial control until the re-initialed adaptive controller replaces them later. Remark 6. For a conventional adaptive controller, the controller is fired all the time and it may prove that the input and the output sequence of the closed-loop system has the stability property. However, for the multi-model controllers, according to the switching criterion 共31兲, each controller
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has the possibility to control the system and each one is not fired all the time, so it is difficult to conclude the stability property. The free running adaptive controller must be added to guarantee the overall system stability. 5. Applications to the wind tunnel system A wind tunnel is a facility used for aerodynamic tests of airplane and missile models, which is very important for national defense and civil aviation. After the stable air flow in the wind tunnel has been established, aerodynamic data of scale models are measured at a given Mach number and stagnation pressure. The experiment requires step changes of Mach number and constant air pressure throughout a series of experiments and measurements regardless of big changes in the angle of attack of the scale model. The control problem is to regulate the wind speed around a Mach number set point with a constant air pressure. The difficulty of the control system design is that the scale model has a time-varying load, which produces different wind tunnel dynamics at each testing stage. Even though there exist many control methods for wind tunnels 关16 –19兴, they are designed for certain wind tunnel structures. Therefore a particular control structure has to be designed for a particular wind tunnel process. A 2.4⫻2.4-m2 injector driven transonic wind tunnel located at China Aerodynamics Research and Development Center 共CARDC兲 is the biggest wind tunnel in Asia 关20兴. It is constructed for the aeronautical research and testing airplane performances 关21兴. The air in the storage tank flows into the wind tunnel through the main control hydraulic servo valve, part of it is ejected out through the main exhaust hydraulic servo valve, and the remaining part is injected back into the wind tunnel through the injector, as shown in Fig. 2 关20,21兴. The experiment requires the Mach number in the test section vary from 0.3 to 0.7 with step changes through adjusting the choke finger, while the stagnation total pressure is set to be a constant 1.5 by regulating the main exhaust hydraulic servo valve. Because the experiment time of the wind tunnel is limited, the system must satisfy that 关20兴 共i兲 In the initial stage, the time to establish the stable flow field with the Mach number 0.3 in the test section is less than 7.0 sec and the tracking error is within 0.2%.
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Fig. 2. The structure of the transonic wind tunnel.
共ii兲 In the experiment stage, when the Mach number in the test section varies with step changes, the settling time of the system is required to be within 0.8 sec with the steady-state error less than 0.2%. Since there exist strong interactions between the inputs and the outputs and the experiment requires very strict conditions, the controller must be so designed such that it is adaptive and decoupled with good transient performance and disturbance rejection capabilities 关20兴. For this particular wind tunnel, the linear reduced-order model is established from the input and output data as 关22兴
冋 册 y 1共 s 兲 y 2共 s 兲
⫽
冋
⫺ ⫺
1 e ⫺0.3s ␣ 1 s⫹1  3 s⫹1
共 ␣ 3 s⫹1 兲
冋 册
u 1共 s 兲 ⫻ u s兲 , 2共
e ⫺0.3s 2
⫺
2 共 ␣ 2 s⫹1 兲 2
e ⫺0.3s
4 e ⫺0.3s ␣ 4 s⫹1
册
共39兲
where y 1 ( s ) , y 2 ( s ) , u 1 ( s ) , u 2 ( s ) are the Mach number in the test section, the stagnation total
pressure, the plenum exhaust valve, and the main exhaust hydraulic servo valve, respectively, ␣ i ,  i are parameters which satisfy ␣ i 苸 关 ␣ i min ,␣i max兴,  i 苸 关  i min ,i max兴. When the Mach number is smaller than 0.8, the wind tunnel system can be viewed as a minimum phase system. By selecting the sampling period as 0.1 sec, the linear discretetime multivariable minimum phase system can be described as 共 I⫹A 1 z ⫺1 ⫹A 2 z ⫺2 兲 y 共 t 兲
⫽ 共 B 0 ⫹B 1 z ⫺1 兲 u 共 t⫺3 兲 ⫹d,
共40兲
where the system is of second order and the time delay equals 3. For every 60 steps, the Mach number in the test section varies with step change 0.1 from 0.3 to 0.7, which causes a simultaneous jump of the system parameters. In order to compare the performance of the proposed control scheme, the following three methods are used in simulation. Method 1 共CADC兲: Conventional adaptive decoupling controller. In this scheme, the initial value is chosen close to the true controller parameter. The responses of the system outputs are shown in Figs. 3共a兲 and 共b兲, respectively. In the initial stage, after 7 sec operation, the overshoots of the system are all less than 0.2%, which satisfies the requirement 关see Table 1 part 共a兲兴. But
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Fig. 3. The simulation response of the wind tunnel using CADC. 共a兲 The output test-section-Mach-number y 1 . 共b兲 The output stagnation-total-pressure y 2 .
in the experiment stage, when the parameters of the system jump, i.e., t⫽1, 130, 190, 250, 310, after 0.8 sec operation, the maximum overshoot of the system is 68.74%, which is much larger than the specification 共0.2%兲 关see Table 2 part 共a兲兴. In fact, during the entire experiment period, i.e., after 6.0 sec operation, the overshoots of the system are all larger than 0.2% 关see Table 2 part 共b兲兴. Method 2 共MFDC兲: Multiple fixed decoupling controllers. In this scheme, 625 decoupling controllers which do not include the true system models are designed. The responses of the system outputs are shown in Figs. 4共a兲 and 4共b兲, respectively. The fired model is shown in Fig. 4共c兲. During the experiment stage, the steady-state error exists and cannot be eliminated, which does not satisfy the requirement of the wind tunnel 关see Fig. 4共d兲兴. Method 3 共MMDADC兲: Proposed in this paper. In this scheme, 625 fixed and two adaptive controllers are designed. The 625 fixed controllers are the same as those in the MFDC scheme and the two adaptive controllers’ initial values are same as those of CADC. The responses of the system outputs are shown in Figs. 5共a兲 and 5共b兲, respectively. The fired model is shown in Fig. 5共c兲. When the parameters of the system jump, the fixed models are first chosen to control the system. After several steps, however, the adaptive controller will be fired to further improve the transient response. In both the initial stage and the experiment stage, the overshoots of the system are all less than 0.2%, which satisfies the requirement 关see Table 1 part
共a兲 and Table 2 part 共a兲兴. Especially, in the initial stage, the overshoot of the system is less than 0.2%, and the settling time is about 1 sec 关see Table 1 part 共b兲兴. Comparing these three schemes, the following can be easily seen. 共i兲 Settling time: In the initial stage, all three schemes satisfy this requirement. But scheme 共i兲 is longer than that of schemes 共ii兲 and 共iii兲. In the experiment stage, scheme 共i兲 does not meet the requirement while both schemes 共ii兲 and 共iii兲 are satisfactory. 共ii兲 Steady stage error: In both initial and experiment stages, only schemes 共i兲 and 共iii兲 can meet this specification. Table 1 The simulation results in the initial stage. 共a兲 The overshoot after 7 sec operation
CADC MMDADC
Mach number in the test section y1
Stagnation total pressure y 2
0.15% 2⫻10⫺9
0.06% 8⫻10⫺10
共b兲 The settling time when the overshoot is less than 0.2% 共sec兲
CADC MMDADC
Mach number in the test section y1
Stagnation total pressure y 2
6.5 1.0
2.8 1.1
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Table 2 The simulation results in the experiment stage. 共a兲 The overshoot after 0.8 sec operation Mach⫽0.4 y1 CADC MMDADC
Mach⫽0.5 y2
2.05% 0.19%
1.86% 0.0027%
y1 2.25% 0.12%
Mach⫽0.6 y2
0.91% 0.025%
y1
Mach⫽0.7 y2
0.25% 0.18%
0.43% 0.0823%
y1
y2
18.23% 68.74% 0.13% 0.0016%
共b兲 The settling time when the overshoot is less than 0.2% 共sec兲 Mach⫽0.4 y1 CADC MMDADC
5.9 0.8
Mach⫽0.5
y2 5.9 0.4
y1 5.9 0.7
y2 5.9 0.3
Mach⫽0.6
Mach⫽0.7
y1
y1
5.9 0.8
y2 5.9 0.3
5.9 0.8
y2 5.9 0.4
Fig. 4. The simulation response of the wind tunnel using MFDC. 共a兲 The output test-section-Mach-number y 1 . 共b兲 The output stagnation-total-pressure y 2 . 共c兲 The fired model. 共d兲 The output test-section-Mach-number y 1 when Mach number is 0.6.
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共iii兲 System interactions: Although all three schemes adopt the same decoupling design strategy, scheme 共iii兲 has many fewer interactions than those of schemes 共i兲 and 共ii兲. In summary, the proposed MMDADC has much better performance and can meet the experiment requirement under the big parameter jumping condition.
6. Conclusion In this paper, a MMDADC system was developed on the discrete-time LTV systems with infrequent parameters jump. The multiple fixed optimal controllers were directly derived from their corresponding system models. By properly selecting the weighting polynomial matrices, it not only decouples the system dynamically, but also places the poles of the closed-loop system arbitrarily. Both the re-initialized and free-running adaptive controllers employed conventional estimation algorithms and were designed using the same methodology as that of the fixed controllers. While the re-initialized adaptive controller used the selected controller value as its initial value for adaptation to improve the adaptation speed, the free-running adaptive controller was to guarantee the overall system stability. The applications of this method to the 2.4⫻2.4-m injector driven transonic wind tunnel process had demonstrated its effectiveness and practicality. It is noted, however, that both schemes 共ii兲 and 共iii兲 had 625 fixed controllers to guarantee the system performances that may be a big obstacle for the method to be applied to wide industry processes. The work to reduce the number of the fixed controllers plus the extension of the method to the nonlinear process is currently under investigation and the results will be reported later.
Acknowledgments
Fig. 5. The simulation response of the wind tunnel using MMDADC. 共a兲 The output test-section-Mach-number y 1 . 共b兲 The output stagnation-total-pressure y 2 . 共c兲 The fired model.
Part of this paper was presented at Asian Control Conference 2002 in Singapore. This research was supported in part by the National Science Foundation of China 共Grant No. 60074004兲 and in part by the High Technology Research and Development Program of China 共Grant No. 2002AA412130兲.
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References 关1兴 Goodwin, G. C., Hill, D. J., and Palaniswami, M., A perspective on convergence of adaptive control algorithms. Automatica 20, 519–531 共1984兲. 关2兴 Astrom, K. J. and Wittenmark, B., On self-tuning regulators. Automatica 9, 195–199 共1973兲. 关3兴 Johanson, T. A. and Murray-Smith, R., Multiple Models Approaches to Modeling and Control. Taylor & Francis Inc., London, 1997. 关4兴 Lainiotis, D. G., Partitioning: A unifying framework for adaptive systems—I: Estimation. Proc. IEEE 64, 1126 –1143 共1976兲. 关5兴 Athans, M., et al., The stochastic control of the F-8c aircraft using multiple model adaptive control method—Part 1: Equilibrium flight. IEEE Trans. Autom. Control 22, 768 –780 共1977兲. 关6兴 Fu, M. and Barmish, B. R., Adaptive stabilization of linear systems via switching control. IEEE Trans. Autom. Control 31, 1097–1103 共1986兲. 关7兴 Middleton, R. H., Goodwin, G. C., Hill, D. J., and Mayne, D. Q., Design issues in adaptive control. IEEE Trans. Autom. Control 33, 50–58 共1988兲. 关8兴 Morse, A. S., Supervisory control of families of linear set-point controller—Part I. Exact matching. IEEE Trans. Autom. Control 41, 1413–1431 共1996兲. 关9兴 Hespanha, J. P., Liberzon, D., and Morse, A. S., Hysteresis-based switching for surpervisory control of uncertain systems. Automatica 39, 263–272 共2003兲. 关10兴 Narendra, K. S. and Balakrishnan, J., Improving transient response of adaptive control systems using multiple models and switching. IEEE Trans. Autom. Control 39, 1861–1866 共1994兲. 关11兴 Narendra, K. S. and Balakrishnan, J., Adaptive control using multiple models. IEEE Trans. Autom. Control 42, 171–187 共1997兲. 关12兴 Narendra, K. S. and Xiang, C., Adaptive control of discrete-time systems using multiple models. IEEE Trans. Autom. Control 45, 1669–1686 共2000兲. 关13兴 Wittenmark, B. and Astrom, K. J., Practical issues in the implementation of self-tuning control. Automatica 20, 595– 605 共1984兲. 关14兴 Goodwin, G. C., Ramadge, P. J., and Caines, P. E., Discrete-time multivariable adaptive control. IEEE Trans. Autom. Control 25, 449– 456 共1980兲. 关15兴 Landau, I. D. and Lozano, R., Unification of discrete time explicit model reference adaptive control designs. Automatica 17, 593– 611 共1981兲. 关16兴 Nelson, D. M., Wind Tunnel Computer Control System and Instrumentation. Proceedings of the ISAAerospace Instrumentation Symposium, Orlando, FL, Vol. 35, pp. 87–101, 1989. 关17兴 Pels, A. F., Closed-loop Mach number control in a transonic wind tunnel. Journal A 30, 25–32 共1989兲. 关18兴 Soeterboek, R. A. M., Pels, A. F., Verbruggen, H. B., and van Langen, G. C. A., A predictive controller for the Mach number in a transonic wind tunnel. IEEE Control Syst. Mag. 11, 63–72 共1991兲. 关19兴 Motter, M. A. and Principe, J. C., Neural control of the NASA Langley 16-foot transonic tunnel. Proceedings of the 1997 American Control Conference, pp. 662– 663, 1997.
关20兴 Zhang, G. J., Chai, T. Y., and Shao, C., A synthetic approach for control of intermittent wind tunnel. Proceedings of the American Control Conference, pp. 203–207, 1997. 关21兴 Yu, W. and Zhang, G. J., Modelling and controller design for 2.4 m injector powered transonic wind tunnel. Proceedings of the 1997 American Control Conference, pp. 1544 –1545, 1997. 关22兴 CARDC, Measurement and Control System Design in High and Low Speed Wind Tunnel, National Defense Industry Press, Beijing, 2002 共in Chinese兲.
Xin Wang was born in 1972, Shenyang, P. R. China. He received his B.S. degree in Electrical Engineering from Shanghai Jiao Tong University in 1993, Master and Ph.D. degrees in Control Theory & Engineering from Northeastern University, P. R. China in 1998 and 2002, respectively. Now he is a Postdoctoral Research Fellow in the Institute of Automation, Shanghai Jiao Tong University. His research interests include multiple models adaptive control, multivariable intelligent decoupling control, modeling, control and optimization of complex industrial processes, and so on.
Shaoyuan Li was born in 1965. He received his B.S. and M.S. degrees from Hebei University of Technology in 1987 and 1992, respectively. He received his Ph.D. degree from the Department of Computer and System Science of Nankai University, China in 1997. Now he is a professor of the Institute of Automation, Shanghai Jiao Tong University. His research interest includes fuzzy systems, nonlinear system control, and so on.
Wenjian Cai was born in 1957. He received his B.S. and M.S. degrees from Harbin Institute of Technology in 1980 and 1983, respectively. He received his Ph.D. degree in Systems Engineering, Oakland University, USA in 1992. Now he is a associate professor of School of Electrical & Electrical Engineering, Nanyang Technological University, Singapore. His research interest includes advanced process control, fuzzy logic control, robust control and estimation techniques, and so on.
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Heng Yue received the Ph.D. degree in Control Theory & Engineering from Northeastern University, China, in 1999. He is associate professor in the Research Center of Automation, Northeastern University, China. His research interests include: neural network, multivariable intelligent decoupling control, optimizing control, and modeling of complex industrial processes.
Xiaojie Zhou received the M. Eng. degree in Industrial Automation from Northeastern University, China, in 1996. Now she is a Ph.D. candidate in the Research Center of Automation, Northeastern University, China. Her research interests include: intelligent control, multivariable decoupling control, optimizing control, and modeling of complex industrial processes.
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Tianyou Chai received the Ph.D. degree in Industrial Automation from Northeastern University of Technology, China, in 1985. He is professor and head of the Research Center of Automation, Northeastern University, China. He was the Member of IFAC’s Technical Committee and the Chairman of IFAC’s Coordinating Committee on Manufacturing and Instrumentation during 1996 –1999. He was elected as the Member of Chinese Academy of Engineering in 2003. His research interests include: adaptive control, multivariable intelligent decoupling control, optimizing control, and integrated automation of process industry.
Multi-model direct adaptive decoupling control with application to the wind tunnel system Xin Wang,a,* Shaoyuan Li,a Wenjian Cai,b Heng Yue,c Xiaojie Zhou,c Tianyou Chaic a
Institute of Automation, Shanghai Jiao Tong University, Shanghai, 200030, People’s Republic of China b School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore c Research Center of Automation Northeastern University, Shenyang, 110004, People’s Republic of China
共Received 25 March 2003; accepted 24 June 2004兲
Abstract In this paper, a new multi-model direct adaptive decoupling controller is presented for multivariable processes, which includes multiple fixed optimal controllers, one free-running adaptive controller, and one re-initialized adaptive controller. The fixed controllers provide initial control to the process if its model lies in the corresponding region. For each controller selected, the re-initialized adaptive controller uses the values of this particular controller to improve the adaptation speed. This controller may replace the fixed controller at a later stage according to the switching criterion which is to select the best one among all controllers. A free-running adaptive controller is also added to guarantee the overall system stability. Different from the multiple models adaptive control structure proposed in Narendra, Balakrishnan, and Ciliz 关Adaptation and learning using multiple models, switching, and tuning. IEEE Control Syst. Mag. 15, 37–51 共1995兲兴, the method not only is applicable to the multi-input multi-output processes but also identifies the decoupling controller parameters directly, which reduces both the computational burden and the chances of a singular matrix during the process of determining controller parameters. Several examples for a wind tunnel process are given to demonstrate the effectiveness and practicality of the proposed method. © 2005 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Multi-model; Direct adaptive control; Decoupling control; Pole placement; Global convergence; Wind tunnel process
1. Introduction When a system boundary condition changes, a subsystem fails, or an external disturbance occurs, these cause the parameter variation of the system that will generally result in slower reaction, slower convergence, poorer transient response, and possibly an unstable system 关1兴. To deal with these kinds of processes, an adaptive control scheme, by adding an on-line parameter estimation algorithm to construct control systems, can be employed 关2兴. *Corresponding author. [email protected]
address:
However, the conventional adaptive controller based on a single model can only be effective for systems with unknown time invariant or slowly time varying parameters. For systems with large uncertainties or abruptly changed system parameters, the multi-model adaptive controller 共MMAC兲 has been demonstrated to be more effective 关3兴. The earliest MMAC appeared in the 1970s where multiple Kalman filter-based models were proposed to improve the accuracy of state estimation 关4,5兴. However, no switching and stability results were presented. Two MMAC schemes with switching conditions were later developed during the 1980s, namely, direct switching MMAC and
0019-0578/2005/$ - see front matter © 2005 ISA—The Instrumentation, Systems, and Automation Society.
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indirect switching MMAC. For the first scheme, multiple state feedback controllers were used and switched based on the predetermined switching sequence to guarantee the stability of the closedloop system 关6兴. For the second scheme, multiple convex region parameter estimators were adopted to overcome the problem of stablizability of the estimated model 关7兴. The choice of when and which controller to switch is determined by a switching criterion. Another important method in this field was the work of Morse 关8兴. His method focused on the superior control of the families of linear set-point controllers using hysteresis switching logic to deal with noise, disturbances, and unmodeled dynamics 关9兴. In order to improve the transient response, Narendra et al. 关10兴 utilized multiple adaptive models to simultaneously identify the parameters of unknown systems. According to a switching criterion, the best-fit model is selected and the corresponding controller is fired at the time. However, after a certain period of operation, multiple adaptive models might converge to a particular neighborhood which defeats the purpose of the multiple models control principle. To improve the performance of the multiple adaptive models control system, a control method with multiple fixed and two adaptive models were developed for both continuous and discrete-time systems 关11,12兴. Unfortunately, all of these methods are intended for single-input single-output 共SISO兲 systems. In addition, they adopt the indirect adaptive algorithm, which may make the control system less robust to estimated parameter errors. The repeated inverse matrices in the algorithm may also cause the singularity problem for those not so well-conditioned processes 关13兴. In this paper, a new multi-model direct adaptive decoupling controller 共MMDADC兲 is presented to extend Narendra’s indirect adaptive control structure for SISO processes 关12兴 into a direct adaptive control structure for multi-input multi-output 共MIMO兲 processes. The control structure includes a set of multiple fixed optimal controllers, a freerunning adaptive controller and a re-initialized adaptive controller. The effect of the fixed controllers is to provide initial control actions to the process if its model lies in the corresponding region. The re-initialized adaptive controller utilizes the parameters of the selected controller as its initial value for the adaptation algorithm to improve the
transient response. As the value of the reinitialized controller is very close to that of the true optimal one, it has faster adaptation speed compared with the conventional adaptive controller with other initial values. Therefore the reinitialized adaptive controller can be used to replace the selected ones at a later stage according to a switching criterion in which the value of a controller nearest to the true optimal one is selected. A free-running adaptive controller using single model structure is also added to guarantee the overall system stability. The main contributions of this paper are 共i兲 the proposed method is applicable to MIMO processes and it is an optimal control with dynamic decoupling, 共ii兲 it adopts the direct adaptive algorithm to reduce both the computational burden and the chances of a singular matrix during the process of determining controller parameters, 共iii兲 the global stability of the overall system and the convergence of the tracking error to zero using this control method are theoretically proved. Several simulation examples in an injector driven transonic wind tunnel process show that the MMDADC can improve the overall system performances compared with the conventional adaptive control method and can meet the strict requirements. 2. Fixed decoupling control Consider a discrete-time linear time-varying 共LTV兲 MIMO system described in the determinant autoregressive moving average 共DARMA兲 form as
A 共 t,z ⫺1 兲 y 共 t 兲 ⫽B 共 t,z ⫺1 兲 u 共 t⫺k 兲 ⫹d 共 t 兲 , 共1兲 with
A 共 t,z ⫺1 兲 ⫽I⫹A 1 共 t 兲 z ⫺1 ⫹¯⫹A n a 共 t 兲 z ⫺n a ,
共2兲
B 共 t,z ⫺1 兲 ⫽B 0 共 t 兲 ⫹B 1 共 t 兲 z ⫺1 ⫹¯⫹B n b 共 t 兲 z ⫺n b , 共3兲 where u ( t ) , y ( t ) , d ( t ) , k, and z ⫺1 are the n⫻1 input vector, n⫻1 output vector, n⫻1 steadystate offset vector, process time delay, and unit delay operator, respectively, A ( t,z ⫺1 ) with order of n a and B ( t,z ⫺1 ) with order of n b are polynomial matrices with comparable dimensions, A i ( t ) and B i ( t ) are time-varying parameter matrices with infrequent large jumps, B 0 ( t ) is a nonsingular known matrix for any t, and I is a unit matrix.
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It is assumed that the system satisfies the following conditions: 共i兲 The period between two adjacent jumps is sufficiently large and the parameters in between can be viewed as time constant; 共ii兲 The parameters of the system model, ⌽ ( t ) ⫽ 关 ⫺A 1 ( t ) ,...;B 0 ( t ) ,...;d ( t ) 兴 , change in a compact set ⌺, i.e., there exists a constant value 0 ⬍K⬍⬁ such that 储 ⌽ ( t ) 储 ⬍K for all t; 共iii兲 n a , n b , and k are known a priori; 共iv兲 The system is minimum phase. According to the system operating conditions, the overall working range can be partitioned into multiple regions. In each region, a fixed system model is elected and all these models constitute the multiple fixed system models. Mathematically, it can be described as follows: Partition ⌺ into m subsets ⌺ s ( s⫽1,...,m ) and each ⌺ s has the following properties: m 共i兲 艛 s⫽1 ⌺ s 傶⌺, ⌺ s is not empty; s⫽1,...,m; 共ii兲 For any ⌽苸⌺ s , s⫽1,...,m, there exists a ⌽ s 苸⌺ s and 0⭐ra ⬍⬁, satisfying 储 ⌽⫺⌽ s 储 ⭐ra . s
133
In order to find unique polynomial matrices F s ( z ⫺1 ) and G s ( z ⫺1 ) , the orders of F s ( z ⫺1 ) and G s ( z ⫺1 ) are chosen, respectively, as
n f ⫽k⫺1, s
n g ⫽n a ⫺1. s
共7兲
Multiply Eq. 共4兲 by F s ( z ⫺1 ) from the left and substitute Eq. 共6兲 into the resultant equation, and we obtain
P s 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽G s 共 z ⫺1 兲 y 共 t 兲 ⫹F s 共 z ⫺1 兲 B s 共 z ⫺1 兲 u 共 t 兲 ⫹F s 共 z ⫺1 兲 d s .
共8兲
Substitute Eq. 共8兲 into Eq. 共5兲, and the optimal control law is derived by minimizing the cost function which results in
G s 共 z ⫺1 兲 y 共 t 兲 ⫹H s 共 z ⫺1 兲 u 共 t 兲 ⫹r¯ s ⫽R s 共 z ⫺1 兲 w 共 t 兲 , 共9兲
s
⌽ s is the center of the subset ⌺ s and r a is the s
corresponding radius. The existence of r a is guars
anteed by assumption 共ii兲. Let the centers of all subsets be the fixed system models which cover the entire set ⌺, then each model of ⌽ s can be described as
A s 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽B s 共 z ⫺1 兲 u 共 t 兲 ⫹d s ,
共4兲
H s 共 z ⫺1 兲 ⫽F s 共 z ⫺1 兲 B s 共 z ⫺1 兲 ⫹Q s 共 z ⫺1 兲 , ¯r s ⫽F s d s ⫹r s .
共11兲
Combine Eq. 共9兲 with Eq. 共4兲, and the closed-loop system is formulated by
⫽R s 共 z ⫺1 兲 w 共 t 兲 ⫹Q s B s⫺1 共 1 兲 d s ⫺r s .
Define the cost function Jc s
共12兲
Since it is minimum phase, the poles of the system can be placed arbitrarily through
Jc ⫽ 储 P s 共 z ⫺1 兲 y 共 t⫹k 兲 ⫺R s 共 z ⫺1 兲 w 共 t 兲 s
共5兲
Q s 共 z ⫺1 兲 ⫽R1 B s 共 z ⫺1 兲 ,
共13兲
P s 共 z ⫺1 兲 ⫹R 1 A s 共 z ⫺1 兲 ⫽T 共 z ⫺1 兲 ,
共14兲
s
where w ( t ) is the known reference signal, P s ( z ⫺1 ) , R s ( z ⫺1 ) , Q s ( z ⫺1 ) , and r s are the weighting polynomial matrices, respectively. For the optimal controller design, the objective is to minimize the cost function 共5兲, which is determined by the following: Introduce the identity
P s 共 z ⫺1 兲 ⫽F s 共 z ⫺1 兲 A s 共 z ⫺1 兲 ⫹z ⫺k G s 共 z ⫺1 兲 .
共10兲
关 P s 共 z ⫺1 兲 ⫹Q s 共 z ⫺1 兲 B s⫺1 共 z ⫺1 兲 A s 共 z ⫺1 兲兴 y 共 t⫹k 兲
s⫽1,...,m.
⫹Q s 共 z ⫺1 兲 u 共 t 兲 ⫹r s 储 2 ,
with
共6兲
s
where
R1 s
is a constant matrix, the polynomial matrix T ( z ⫺1 ) is assumed to be stable with the form of
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T 共 z ⫺1 兲 ⫽I⫹T 1 z ⫺1 ⫹¯⫹T n t z ⫺n t ,
共15兲
where T i , i⫽1,...,n t are designed matrices with comparable dimensions. Substituting Eqs. 共13兲 and 共14兲 into Eq. 共12兲 gives
identify the controller parameters and the formulation procedures are given as follows. Multiplying Eq. 共14兲 by y ( t⫹k ) from the right, we have
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽ P 共 z ⫺1 兲 y 共 t⫹k 兲 ⫹R 1 A 共 z ⫺1 兲 y 共 t⫹k 兲 . 共20兲
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽R s 共 z ⫺1 兲 w 共 t 兲 ⫹R 1 d s ⫺r s . s
共16兲
Then, the steady-state error and the effect of d s can be eliminated by selecting
R s 共 z ⫺1 兲 ⫽T 共 z ⫺1 兲 ,
共17兲
r s ⫽R 1 d s .
共18兲
s
Substituting P ( z ⫺1 ) and A ( z ⫺1 ) y ( t⫹k ) by Eqs. 共6兲 and 共4兲, respectively, the system estimation equation can be derived as
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽F 共 z ⫺1 兲 A 共 z ⫺1 兲 y 共 t⫹k 兲 ⫹G 共 z ⫺1 兲 y 共 t 兲 ⫹R 1 B 共 z ⫺1 兲 u 共 t 兲 ⫹R 1 d
By such a selection, Eq. 共16兲 is simplified to
T共 z
⫺1
兲 y 共 t⫹k 兲 ⫽T 共 z
⫺1
兲w共 t 兲.
⫽G 共 z ⫺1 兲 y 共 t 兲
共19兲
⫹F 共 z ⫺1 兲 B 共 z ⫺1 兲 u 共 t 兲 ⫹F 共 1 兲 d
⫺1
Remark 1. Since the polynomial matrix T ( z ) is to be designed, we can assign its element matrices T i , i⫽1,...,n t as diagonal matrices with arbitrary diagonal elements. Consequently, the resulting system can be not only dynamically decoupled but also its poles can be placed arbitrarily. Remark 2. Regardless of the difference of the models in each region, the overall system has to have the same pole locations to guarantee the smoothness of the response. As the elements of the designed polynomial matrix T ( z ⫺1 ) can be arbitrarily assigned, we can have the same T ( z ⫺1 ) for all controllers.
⫹R 1 B 共 z ⫺1 兲 u 共 t 兲 ⫹R 1 d ⫽G 共 z ⫺1 兲 y 共 t 兲 ⫹ 关 F 共 z ⫺1 兲 B 共 z ⫺1 兲 ⫹R 1 B 共 z ⫺1 兲兴 u 共 t 兲 ⫹F 共 1 兲 d ⫹R 1 d.
共21兲
Considering the definitions of Q and r in Eqs. 共13兲 and 共18兲, Eq. 共21兲 can be simplified to
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽G 共 z ⫺1 兲 y 共 t 兲 ⫹ 关 F 共 z ⫺1 兲 B 共 z ⫺1 兲 ⫹Q 共 z ⫺1 兲兴 u 共 t 兲 ⫹F 共 1 兲 d⫹r ⫽G 共 z ⫺1 兲 y 共 t 兲 ⫹H 共 z ⫺1 兲 u 共 t 兲 ⫹r¯ , 共22兲
3. Direct adaptive control Even though the fixed controllers are designed for the local regions, the free-running adaptive controller must be designed based on overall system structure. By assumption 共i兲, however, the difference between the overall system and local systems is only their parameters. Therefore the results derived in Section 2 for controller design are still valid for real-time calculations of controller parameters. For both free-running and re-initialized adaptive controllers, either model parameters or controller parameters have to be identified. In the direct adaptive control scheme, the standard recursive estimation algorithm is employed to direct
where
H 共 z ⫺1 兲 ⫽F 共 z ⫺1 兲 B 共 z ⫺1 兲 ⫹Q 共 z ⫺1 兲 ,
共23兲
¯r ⫽Fd⫹r.
共24兲
Then, the recursive estimation algorithm and optimal control law are described as 关14兴
T 共 z ⫺1 兲 y 共 t⫹k 兲 ⫽G 共 z ⫺1 兲 y 共 t 兲 ⫹H 共 z ⫺1 兲 u 共 t 兲 ⫹r¯ , 共25兲
ˆ i 共 t 兲 ⫽ ˆ i 共 t⫺1 兲 ⫹a 共 t 兲
X 共 t⫺k 兲 1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲
⫻ 关 y f i 共 t 兲 T ⫺X 共 t⫺k 兲 T ˆ i 共 t⫺1 兲兴 ,
共26兲
Wang, Li, Cai, Yue, Zhou, Chai / ISA Transactions 44 (2005) 131–143
135
Fig. 1. The structure of the MMDADC.
and
ˆ 共 z ⫺1 兲 y 共 t 兲 ⫹H ˆ 共 z ⫺1 兲 u 共 t 兲 ⫹rC ⫽R 共 z ⫺1 兲 w 共 t 兲 , G 共27兲 ⫺1 respectively, where y f i ⫽T ii ( z ) y i ( t ) is the ith element of the transformation of the system output vector, X 共 t 兲 ⫽ 关 y 共 t 兲 T ,...;u 共 t 兲 T ,...,1兴 T
共28兲
is the data vector, ⌰⫽ 关 1 ,..., n 兴 is the parameter 0 0 1 1 matrix with i ⫽ 关 g i1 ,...,g in ;g i1 ,...,g in ,...; 0 0 T h i1 ,...,h in ;... 兴 , i⫽1,2,...,n, the scalar a ( t ) is used to avoid the singularity problem in estimating ˆ ( 0 ) , i.e., if H ˆ ( 0 ) is singular such that u ( t ) canH not be calculated, change the a ( t ) value in the interval ⬍a ( t ) ⬍2⫺ , 0⬍⬍1 to estimate ˆ ( 0 ) again 关14兴. H Remark 3. The same as the conventional direct adaptive control concept, the adaptive controller can direct estimate the controller parameter matrices G ( z ⫺1 ) , H ( z ⫺1 ) , ¯r through Eqs. 共25兲 and 共26兲 and presume the estimated controller matrices ˆ ( z ⫺1 ) , H ˆ ( z ⫺1 ) , rC as true ones. In this way, the G
procedure of calculating controller parameters is eliminated, which reduces both the computational burden and the chances of a singular matrix.
4. Multi-model direct adaptive decoupling control The overall multiple controllers are composed of m fixed controllers ⌰ 1 ,...,⌰ m , one freerunning adaptive controller ⌰ m⫹1 , and one reinitialized adaptive controller ⌰ m⫹2 共see Fig. 1兲. The selection of which controller to be used in the time instant is determined as follows:
J s⫽
储 ef 共 t 兲 储 2 s
1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲
,
s⫽1,...,m,m⫹1,m⫹2
共29兲
and
ef 共 t 兲 ,y f 共 t 兲 ⫺y f 共 t 兲 ⫽T 共 z ⫺1 兲 e s 共 t 兲 , s
s
共30兲
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where y f ( t ) ⫽T ( z ⫺1 ) y ( t ) is the transformation of the real system output and
Then, there must exist an instant t s , when t⬎t s , it holds
y f 共 t 兲 ⫽T 共 z ⫺1 兲 y s 共 t 兲
ef
s
m⫹1 i
is the transformation of the sth model output,
共 t 兲2
1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲
⭐
ef 共 t 兲 is the transformation of the output error e s ( t ) between the real system and the sth model. The switching criterion based on Eq. 共29兲 is given as
J j ⫽min共 J s 兲 ,
共31兲
which means that the jth controller is chosen as the controller to be fired if its transformation output error is minimum. The adaptive controllers include a free-running adaptive controller and a reinitialized adaptive controller. The difference between them is that the free-running adaptive controller uses a global model to determine the controller parameters, while the re-initialized adaptive controller utilizes the parameters of the controller selected by Eq. 共31兲 as its initial conditions to speed up the convergence speed. As the switching criterion is to select the one closest to the optimal value and if the free-running adaptive controller has global convergence property, the global convergence of the overall system is guaranteed. Therefore it is necessary to prove the global convergence of the multi-model adaptive controller which is given by the following theorem. Theorem 1. Subject to assumptions 共i兲–共iv兲, if the controller determined by Eq. 共27兲 is applied to the system 共4兲, 兵 y ( t ) 其 , 兵 u ( t ) 其 are bounded and limt→⬁ 储 e ( t ) 储 ⫽0. Proof: 共i兲 If
si
For the ( m⫹1 ) th free-running adaptive controller, the recursive estimation algorithm 共26兲 has the property 关14兴
lim
1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲
m⫹1 i
共 t 兲2
t→⬁ 1⫹X 共 t⫺k 兲
T
X 共 t⫺k 兲
⭐
m⫹1 i
共 t 兲2
. 1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲 共34兲
Consequently,
e f i共 t 兲 2
lim
t→⬁ 1⫹X 共 t⫺k 兲
T
X 共 t⫺k 兲
⫽0,
i⫽1,...,n. 共35兲
Since T ( z ⫺1 ) is designed to be stable, the system is minimum phase, w i ( t ) and d i are both bounded, it follows that 关14兴
兩 y i 共 t 兲 兩 ⭐K 1 ⫹K 2 max 兩 e f j 共 兲 兩 ,
0⬍K 1 ⬍⬁,
1⭐ ⭐t 1⭐ j⭐n
0⬍K 2 ⬍⬁,
1⭐t⭐N,
i⫽1,...,n,
兩 u i 共 t⫺k 兲 兩 ⭐K 3 ⫹K 4 max 兩 e f j 共 兲 兩 , 1⭐ ⭐t 1⭐ j⭐n
1⭐t⭐N,
共36兲
0⬍K 3 ⬍⬁,
i⫽1,...,n.
共37兲
According to the technical lemma in Ref. 关15兴, it results that 兵 y ( t ) 其 , 兵 u ( t ) 其 is bounded and lim 储 e f ( t ) 储 ⫽0.
⑀ t ⫽ min 兩 ef 共 t 兲 兩 .
ef
ef
e f i共 t 兲 2
0⬍K 4 ⬍⬁,
there must exist a constant value 1⭐s⭐m
共33兲
which means that for t⬎t s , no fixed controllers can be fired. The fired controller must be selected between the free-running adaptive controller ⌰ m⫹1 and the re-initialized adaptive controller ⌰ m⫹2 according to the switching criterion 共31兲. However, as the error of the free-running adaptive controller is equal to or larger than that of the re-initialized adaptive controller, it follows that
s苸 兵 1,...,m 其 ,
s
1⫹X 共 t⫺k 兲 T X 共 t⫺k 兲 i⫽1,...,n,
s
ef 共 t 兲 ⫽0,
⑀ i共 t 兲 2
t→⬁
共ii兲 For any time instant, the system has only one true parameter set ⌰ 0 . If
e f 共 t 兲 ⫽0, s
⫽0.
共32兲
s苸 兵 1,...,m 其 ,
which means that there exists at least one fixed controller whose controller parameter matrix ⌰ s is
Wang, Li, Cai, Yue, Zhou, Chai / ISA Transactions 44 (2005) 131–143
equal to the real value of the system ⌰ 0 or ⌰ s ⫺⌰ 0 is orthogonal to the data vector X ( t⫺k ) . Then, J s ⫽0 and ⌰ s is chosen to be the controller parameter matrix until it jumps again. Combining cases 共i兲 and 共ii兲, for the transformation of the output error e f ( t ) ⫽T ( z ⫺1 ) e ( t ) and T ( z ⫺1 ) is stable, it concludes that 关15兴
lim 储 e 共 t 兲 储 ⫽0.
共38兲
t→⬁
The overall control system including multiple fixed controllers, one free-running adaptive controller, and one re-initialized adaptive controller can be summarized in the following algorithm. Algorithm 1. 共i兲 Partition the system model into m fixed models as in Eq. 共4兲; 共ii兲 Design the corresponding fixed controllers by Eq. 共9兲; 共iii兲 Form the data vector X ( t ) as in Eq. 共28兲; ˆ and ⌰ ˆ ( t ) by Eq. 共26兲; 共iv兲 Estimate ⌰ m⫹1
m⫹2
共v兲 Choose the best controller j based on the switching criterion 共31兲; ˆ ˆ 共vi兲 Initialize ⌰ m⫹2 as ⌰m⫹2 ( t ) ⫽⌰ j if j⫽m ⫹2; 共vii兲 Set the fired controller as ⌰ ( t ) ⫽⌰ j and obtain u ( t ) as in Eq. 共27兲 accordingly; 共viii兲 For t⫽t⫹1, go to step 共iii兲. Remark 4. The re-initialized adaptive controller is used to speed up the convergence speed, which causes the transient response to be greatly improved. At every instant, if it is not the best model chosen according to the switching criterion 共31兲, its parameters must be reinitialized as those of the fired model whose parameters are nearest to the real ones, then it can estimate the system parameter values from these nearest values to increase the approaching speed. Remark 5. The fixed controllers have dual functions: 共i兲 they are utilized to localize the region where the system parameters lie in and present the initial value to the re-initialized adaptive controller; 共ii兲 they provide the initial control until the re-initialed adaptive controller replaces them later. Remark 6. For a conventional adaptive controller, the controller is fired all the time and it may prove that the input and the output sequence of the closed-loop system has the stability property. However, for the multi-model controllers, according to the switching criterion 共31兲, each controller
137
has the possibility to control the system and each one is not fired all the time, so it is difficult to conclude the stability property. The free running adaptive controller must be added to guarantee the overall system stability. 5. Applications to the wind tunnel system A wind tunnel is a facility used for aerodynamic tests of airplane and missile models, which is very important for national defense and civil aviation. After the stable air flow in the wind tunnel has been established, aerodynamic data of scale models are measured at a given Mach number and stagnation pressure. The experiment requires step changes of Mach number and constant air pressure throughout a series of experiments and measurements regardless of big changes in the angle of attack of the scale model. The control problem is to regulate the wind speed around a Mach number set point with a constant air pressure. The difficulty of the control system design is that the scale model has a time-varying load, which produces different wind tunnel dynamics at each testing stage. Even though there exist many control methods for wind tunnels 关16 –19兴, they are designed for certain wind tunnel structures. Therefore a particular control structure has to be designed for a particular wind tunnel process. A 2.4⫻2.4-m2 injector driven transonic wind tunnel located at China Aerodynamics Research and Development Center 共CARDC兲 is the biggest wind tunnel in Asia 关20兴. It is constructed for the aeronautical research and testing airplane performances 关21兴. The air in the storage tank flows into the wind tunnel through the main control hydraulic servo valve, part of it is ejected out through the main exhaust hydraulic servo valve, and the remaining part is injected back into the wind tunnel through the injector, as shown in Fig. 2 关20,21兴. The experiment requires the Mach number in the test section vary from 0.3 to 0.7 with step changes through adjusting the choke finger, while the stagnation total pressure is set to be a constant 1.5 by regulating the main exhaust hydraulic servo valve. Because the experiment time of the wind tunnel is limited, the system must satisfy that 关20兴 共i兲 In the initial stage, the time to establish the stable flow field with the Mach number 0.3 in the test section is less than 7.0 sec and the tracking error is within 0.2%.
138
Wang, Li, Cai, Yue, Zhou, Chai / ISA Transactions 44 (2005) 131–143
Fig. 2. The structure of the transonic wind tunnel.
共ii兲 In the experiment stage, when the Mach number in the test section varies with step changes, the settling time of the system is required to be within 0.8 sec with the steady-state error less than 0.2%. Since there exist strong interactions between the inputs and the outputs and the experiment requires very strict conditions, the controller must be so designed such that it is adaptive and decoupled with good transient performance and disturbance rejection capabilities 关20兴. For this particular wind tunnel, the linear reduced-order model is established from the input and output data as 关22兴
冋 册 y 1共 s 兲 y 2共 s 兲
⫽
冋
⫺ ⫺
1 e ⫺0.3s ␣ 1 s⫹1  3 s⫹1
共 ␣ 3 s⫹1 兲
冋 册
u 1共 s 兲 ⫻ u s兲 , 2共
e ⫺0.3s 2
⫺
2 共 ␣ 2 s⫹1 兲 2
e ⫺0.3s
4 e ⫺0.3s ␣ 4 s⫹1
册
共39兲
where y 1 ( s ) , y 2 ( s ) , u 1 ( s ) , u 2 ( s ) are the Mach number in the test section, the stagnation total
pressure, the plenum exhaust valve, and the main exhaust hydraulic servo valve, respectively, ␣ i ,  i are parameters which satisfy ␣ i 苸 关 ␣ i min ,␣i max兴,  i 苸 关  i min ,i max兴. When the Mach number is smaller than 0.8, the wind tunnel system can be viewed as a minimum phase system. By selecting the sampling period as 0.1 sec, the linear discretetime multivariable minimum phase system can be described as 共 I⫹A 1 z ⫺1 ⫹A 2 z ⫺2 兲 y 共 t 兲
⫽ 共 B 0 ⫹B 1 z ⫺1 兲 u 共 t⫺3 兲 ⫹d,
共40兲
where the system is of second order and the time delay equals 3. For every 60 steps, the Mach number in the test section varies with step change 0.1 from 0.3 to 0.7, which causes a simultaneous jump of the system parameters. In order to compare the performance of the proposed control scheme, the following three methods are used in simulation. Method 1 共CADC兲: Conventional adaptive decoupling controller. In this scheme, the initial value is chosen close to the true controller parameter. The responses of the system outputs are shown in Figs. 3共a兲 and 共b兲, respectively. In the initial stage, after 7 sec operation, the overshoots of the system are all less than 0.2%, which satisfies the requirement 关see Table 1 part 共a兲兴. But
Wang, Li, Cai, Yue, Zhou, Chai / ISA Transactions 44 (2005) 131–143
139
Fig. 3. The simulation response of the wind tunnel using CADC. 共a兲 The output test-section-Mach-number y 1 . 共b兲 The output stagnation-total-pressure y 2 .
in the experiment stage, when the parameters of the system jump, i.e., t⫽1, 130, 190, 250, 310, after 0.8 sec operation, the maximum overshoot of the system is 68.74%, which is much larger than the specification 共0.2%兲 关see Table 2 part 共a兲兴. In fact, during the entire experiment period, i.e., after 6.0 sec operation, the overshoots of the system are all larger than 0.2% 关see Table 2 part 共b兲兴. Method 2 共MFDC兲: Multiple fixed decoupling controllers. In this scheme, 625 decoupling controllers which do not include the true system models are designed. The responses of the system outputs are shown in Figs. 4共a兲 and 4共b兲, respectively. The fired model is shown in Fig. 4共c兲. During the experiment stage, the steady-state error exists and cannot be eliminated, which does not satisfy the requirement of the wind tunnel 关see Fig. 4共d兲兴. Method 3 共MMDADC兲: Proposed in this paper. In this scheme, 625 fixed and two adaptive controllers are designed. The 625 fixed controllers are the same as those in the MFDC scheme and the two adaptive controllers’ initial values are same as those of CADC. The responses of the system outputs are shown in Figs. 5共a兲 and 5共b兲, respectively. The fired model is shown in Fig. 5共c兲. When the parameters of the system jump, the fixed models are first chosen to control the system. After several steps, however, the adaptive controller will be fired to further improve the transient response. In both the initial stage and the experiment stage, the overshoots of the system are all less than 0.2%, which satisfies the requirement 关see Table 1 part
共a兲 and Table 2 part 共a兲兴. Especially, in the initial stage, the overshoot of the system is less than 0.2%, and the settling time is about 1 sec 关see Table 1 part 共b兲兴. Comparing these three schemes, the following can be easily seen. 共i兲 Settling time: In the initial stage, all three schemes satisfy this requirement. But scheme 共i兲 is longer than that of schemes 共ii兲 and 共iii兲. In the experiment stage, scheme 共i兲 does not meet the requirement while both schemes 共ii兲 and 共iii兲 are satisfactory. 共ii兲 Steady stage error: In both initial and experiment stages, only schemes 共i兲 and 共iii兲 can meet this specification. Table 1 The simulation results in the initial stage. 共a兲 The overshoot after 7 sec operation
CADC MMDADC
Mach number in the test section y1
Stagnation total pressure y 2
0.15% 2⫻10⫺9
0.06% 8⫻10⫺10
共b兲 The settling time when the overshoot is less than 0.2% 共sec兲
CADC MMDADC
Mach number in the test section y1
Stagnation total pressure y 2
6.5 1.0
2.8 1.1
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Table 2 The simulation results in the experiment stage. 共a兲 The overshoot after 0.8 sec operation Mach⫽0.4 y1 CADC MMDADC
Mach⫽0.5 y2
2.05% 0.19%
1.86% 0.0027%
y1 2.25% 0.12%
Mach⫽0.6 y2
0.91% 0.025%
y1
Mach⫽0.7 y2
0.25% 0.18%
0.43% 0.0823%
y1
y2
18.23% 68.74% 0.13% 0.0016%
共b兲 The settling time when the overshoot is less than 0.2% 共sec兲 Mach⫽0.4 y1 CADC MMDADC
5.9 0.8
Mach⫽0.5
y2 5.9 0.4
y1 5.9 0.7
y2 5.9 0.3
Mach⫽0.6
Mach⫽0.7
y1
y1
5.9 0.8
y2 5.9 0.3
5.9 0.8
y2 5.9 0.4
Fig. 4. The simulation response of the wind tunnel using MFDC. 共a兲 The output test-section-Mach-number y 1 . 共b兲 The output stagnation-total-pressure y 2 . 共c兲 The fired model. 共d兲 The output test-section-Mach-number y 1 when Mach number is 0.6.
Wang, Li, Cai, Yue, Zhou, Chai / ISA Transactions 44 (2005) 131–143
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共iii兲 System interactions: Although all three schemes adopt the same decoupling design strategy, scheme 共iii兲 has many fewer interactions than those of schemes 共i兲 and 共ii兲. In summary, the proposed MMDADC has much better performance and can meet the experiment requirement under the big parameter jumping condition.
6. Conclusion In this paper, a MMDADC system was developed on the discrete-time LTV systems with infrequent parameters jump. The multiple fixed optimal controllers were directly derived from their corresponding system models. By properly selecting the weighting polynomial matrices, it not only decouples the system dynamically, but also places the poles of the closed-loop system arbitrarily. Both the re-initialized and free-running adaptive controllers employed conventional estimation algorithms and were designed using the same methodology as that of the fixed controllers. While the re-initialized adaptive controller used the selected controller value as its initial value for adaptation to improve the adaptation speed, the free-running adaptive controller was to guarantee the overall system stability. The applications of this method to the 2.4⫻2.4-m injector driven transonic wind tunnel process had demonstrated its effectiveness and practicality. It is noted, however, that both schemes 共ii兲 and 共iii兲 had 625 fixed controllers to guarantee the system performances that may be a big obstacle for the method to be applied to wide industry processes. The work to reduce the number of the fixed controllers plus the extension of the method to the nonlinear process is currently under investigation and the results will be reported later.
Acknowledgments
Fig. 5. The simulation response of the wind tunnel using MMDADC. 共a兲 The output test-section-Mach-number y 1 . 共b兲 The output stagnation-total-pressure y 2 . 共c兲 The fired model.
Part of this paper was presented at Asian Control Conference 2002 in Singapore. This research was supported in part by the National Science Foundation of China 共Grant No. 60074004兲 and in part by the High Technology Research and Development Program of China 共Grant No. 2002AA412130兲.
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References 关1兴 Goodwin, G. C., Hill, D. J., and Palaniswami, M., A perspective on convergence of adaptive control algorithms. Automatica 20, 519–531 共1984兲. 关2兴 Astrom, K. J. and Wittenmark, B., On self-tuning regulators. Automatica 9, 195–199 共1973兲. 关3兴 Johanson, T. A. and Murray-Smith, R., Multiple Models Approaches to Modeling and Control. Taylor & Francis Inc., London, 1997. 关4兴 Lainiotis, D. G., Partitioning: A unifying framework for adaptive systems—I: Estimation. Proc. IEEE 64, 1126 –1143 共1976兲. 关5兴 Athans, M., et al., The stochastic control of the F-8c aircraft using multiple model adaptive control method—Part 1: Equilibrium flight. IEEE Trans. Autom. Control 22, 768 –780 共1977兲. 关6兴 Fu, M. and Barmish, B. R., Adaptive stabilization of linear systems via switching control. IEEE Trans. Autom. Control 31, 1097–1103 共1986兲. 关7兴 Middleton, R. H., Goodwin, G. C., Hill, D. J., and Mayne, D. Q., Design issues in adaptive control. IEEE Trans. Autom. Control 33, 50–58 共1988兲. 关8兴 Morse, A. S., Supervisory control of families of linear set-point controller—Part I. Exact matching. IEEE Trans. Autom. Control 41, 1413–1431 共1996兲. 关9兴 Hespanha, J. P., Liberzon, D., and Morse, A. S., Hysteresis-based switching for surpervisory control of uncertain systems. Automatica 39, 263–272 共2003兲. 关10兴 Narendra, K. S. and Balakrishnan, J., Improving transient response of adaptive control systems using multiple models and switching. IEEE Trans. Autom. Control 39, 1861–1866 共1994兲. 关11兴 Narendra, K. S. and Balakrishnan, J., Adaptive control using multiple models. IEEE Trans. Autom. Control 42, 171–187 共1997兲. 关12兴 Narendra, K. S. and Xiang, C., Adaptive control of discrete-time systems using multiple models. IEEE Trans. Autom. Control 45, 1669–1686 共2000兲. 关13兴 Wittenmark, B. and Astrom, K. J., Practical issues in the implementation of self-tuning control. Automatica 20, 595– 605 共1984兲. 关14兴 Goodwin, G. C., Ramadge, P. J., and Caines, P. E., Discrete-time multivariable adaptive control. IEEE Trans. Autom. Control 25, 449– 456 共1980兲. 关15兴 Landau, I. D. and Lozano, R., Unification of discrete time explicit model reference adaptive control designs. Automatica 17, 593– 611 共1981兲. 关16兴 Nelson, D. M., Wind Tunnel Computer Control System and Instrumentation. Proceedings of the ISAAerospace Instrumentation Symposium, Orlando, FL, Vol. 35, pp. 87–101, 1989. 关17兴 Pels, A. F., Closed-loop Mach number control in a transonic wind tunnel. Journal A 30, 25–32 共1989兲. 关18兴 Soeterboek, R. A. M., Pels, A. F., Verbruggen, H. B., and van Langen, G. C. A., A predictive controller for the Mach number in a transonic wind tunnel. IEEE Control Syst. Mag. 11, 63–72 共1991兲. 关19兴 Motter, M. A. and Principe, J. C., Neural control of the NASA Langley 16-foot transonic tunnel. Proceedings of the 1997 American Control Conference, pp. 662– 663, 1997.
关20兴 Zhang, G. J., Chai, T. Y., and Shao, C., A synthetic approach for control of intermittent wind tunnel. Proceedings of the American Control Conference, pp. 203–207, 1997. 关21兴 Yu, W. and Zhang, G. J., Modelling and controller design for 2.4 m injector powered transonic wind tunnel. Proceedings of the 1997 American Control Conference, pp. 1544 –1545, 1997. 关22兴 CARDC, Measurement and Control System Design in High and Low Speed Wind Tunnel, National Defense Industry Press, Beijing, 2002 共in Chinese兲.
Xin Wang was born in 1972, Shenyang, P. R. China. He received his B.S. degree in Electrical Engineering from Shanghai Jiao Tong University in 1993, Master and Ph.D. degrees in Control Theory & Engineering from Northeastern University, P. R. China in 1998 and 2002, respectively. Now he is a Postdoctoral Research Fellow in the Institute of Automation, Shanghai Jiao Tong University. His research interests include multiple models adaptive control, multivariable intelligent decoupling control, modeling, control and optimization of complex industrial processes, and so on.
Shaoyuan Li was born in 1965. He received his B.S. and M.S. degrees from Hebei University of Technology in 1987 and 1992, respectively. He received his Ph.D. degree from the Department of Computer and System Science of Nankai University, China in 1997. Now he is a professor of the Institute of Automation, Shanghai Jiao Tong University. His research interest includes fuzzy systems, nonlinear system control, and so on.
Wenjian Cai was born in 1957. He received his B.S. and M.S. degrees from Harbin Institute of Technology in 1980 and 1983, respectively. He received his Ph.D. degree in Systems Engineering, Oakland University, USA in 1992. Now he is a associate professor of School of Electrical & Electrical Engineering, Nanyang Technological University, Singapore. His research interest includes advanced process control, fuzzy logic control, robust control and estimation techniques, and so on.
Wang, Li, Cai, Yue, Zhou, Chai / ISA Transactions 44 (2005) 131–143
Heng Yue received the Ph.D. degree in Control Theory & Engineering from Northeastern University, China, in 1999. He is associate professor in the Research Center of Automation, Northeastern University, China. His research interests include: neural network, multivariable intelligent decoupling control, optimizing control, and modeling of complex industrial processes.
Xiaojie Zhou received the M. Eng. degree in Industrial Automation from Northeastern University, China, in 1996. Now she is a Ph.D. candidate in the Research Center of Automation, Northeastern University, China. Her research interests include: intelligent control, multivariable decoupling control, optimizing control, and modeling of complex industrial processes.
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Tianyou Chai received the Ph.D. degree in Industrial Automation from Northeastern University of Technology, China, in 1985. He is professor and head of the Research Center of Automation, Northeastern University, China. He was the Member of IFAC’s Technical Committee and the Chairman of IFAC’s Coordinating Committee on Manufacturing and Instrumentation during 1996 –1999. He was elected as the Member of Chinese Academy of Engineering in 2003. His research interests include: adaptive control, multivariable intelligent decoupling control, optimizing control, and integrated automation of process industry.