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Design of time delay controller using variable reference model
Control Engineering Practice 8 (2000) 581}588
Design of time delay controller using variable reference model Jae-Bok Song*, Kyung-Seok Byun Department of Mechanical Engineering, Korea University, 5-Ka Anam-Dong Sungbuk-Ku, Seoul, 136-701, South Korea Received 10 August 1998; accepted 27 August 1999
Abstract Reference models are used in many control algorithms to improve transient response characteristics. Most control systems have bounded control inputs to avoid saturation of the plant. If the reference models do not account for the limits of the control inputs, the control performance of the system may deteriorate. In this paper a new approach to avoid saturation is proposed by varying the reference model for time delay control-based systems subject to step changes in the reference input. In this scheme, the variable reference model is determined based on the information on control inputs and the size of the step changes in the reference inputs. This scheme was veri"ed by application to the BLDC motor position control system in simulations and experiments. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Time delay control; Variable reference model; Saturation avoidance; Tracking control; Nominal control input
1. Introduction Time delay control (TDC) schemes have been suggested as e!ective control techniques for nonlinear time-varying systems with uncertain dynamics and/or unpredictable disturbances. Basically, it cancels out the undesired dynamics and disturbances, and substitutes them with the desired dynamics given in terms of the reference model. Thus, the state vector of a plant is made to follow the reference model accurately. Reference models are used in model reference adaptive control (Slotine & Li, 1991) or model reference learning control (Cheah & Wang, 1997) as well as in TDC. The selection of the reference model has a great impact on control performance because it provides the desired trajectory for the plant state vector. In most control systems the control signal to the plant has some bounds. Beyond these bounds, the control signal may cause a saturation phenomenon, where the change in input cannot bring about changes in the plant output. Large control e!orts are usually desired for quick responses, but they can produce large overshoots or saturation problems. On the other hand, small control
* Corresponding author. Tel.: #82-2-3290-3363; fax: #82-2-9289769. E-mail address: [email protected] (J.-B. Song).
e!orts avoiding overshoots or saturation may cause slow responses. Thus it is preferable to make use of large control signals within the range of no saturation. In this paper, a way of satisfying this ideal condition is proposed in relation to the TDC scheme. This is achieved by taking the bounds of the saturation region into account when determining the reference model for the TDC scheme. Some research has been done on control systems with bounded control e!orts. Lu and Chen (1995) suggested global sliding mode control (GSMC), where the sliding line of the GSMC is designed so that the initial state is located on it. The maximum and minimum values of control e!orts generated according to this GSMC can be estimated and the range of allowable reference inputs is then obtained under the control bound and the sliding line. Spong, Thorp and Kleinwaks (1986) studied optimal decision strategy (ODS), where the decision strategy is to minimize a Euclidean norm of the di!erence between the actual vector of the instantaneous joint accelerations and the desired joint acceleration vector which may be generated by a reference model. However, they did not suggest the criterion for the selection of the reference model. In this paper, a new method for avoiding saturation is proposed by varying the reference model for the TDCbased systems subject to the step changes in reference inputs. The variable reference model in this scheme is determined based on the information about the control e!orts and the size of the step changes in the reference
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inputs. This reference model is not "xed, but varies depending on the size of the change in the reference input. In Section 2 the time delay control law is brie#y introduced. Then, a detailed discussion of the idea of the variable reference model is given in Section 3. Section 4 will show the experimental results of the servo system based on the TDC scheme with a variable reference model.
In this section, time delay control (TDC) law (YoucefToumi & Ito, 1990) will be brie#y explained. Consider a nonlinear time-varying plant in the state-space model (1)
where x(t) represents the state vector (n;1), u(t) the control vector (r;1), d(t) the unpredicted disturbance vector. Dynamics of the plant is divided into the known dynamics vector f(x, t) and the unknown dynamics vector h(x, t). Also, control matrix B(x, t) is assumed temporarily to be known. Consider the reference model represented by the following nonlinear time-varying system: xR (t)"'(x (t), r(t)), (2) K K where x (t) and r(t) denote the state vector of the referK ence model and the reference input vector, respectively. This reference model yields the ideal state vector that the plant state vector should follow. De"ning the error vector e(t)"x (t)!x(t), the error K dynamics can be written by eR (t)"A e(t), (3) C where A is the error system matrix. When A is selected C C so that all eigenvalues are placed in the left-half of the s-plane, the error dynamics becomes asymptotically stable. After substitution of the relation xR (t)"xR (t)!A e(t) K C from (3) into (1), the following equation is obtained:
xR 0 Tx 0 0 } } } u"! } } ! } } ! } } # }KO B f h d xR P P P P KP
Hence,
2. Time delay control law
xR (t)"f(x(t), t)#h(x(t), t)#B(x(t), t)u(t)#d(t),
where x , f , h and d are r;1 vectors, x a (n!r);1 P P P O vector, and B a r;r nonsingular matrix. Substitution of P (5) into (4) yields
T ! } } e. A CP
B (x, t)u(t)"!f (x, t)!h (x, t)!d (t) P P P P #xR (t)!A e(t). (7) KP CP Since h (x, t)#d (t) is an unknown function, the control P P e!ort u(t) cannot be obtained from (7), and thus the estimate of this function h) (x, t)#d) (t) must be found. P P Rearranging (7) gives h (x, t)#d (t)"xR (t)!f (x(t), t)!B (x(t), t)u(t). (8) P P P P P Assuming that the time delay ¸ is su$ciently small and h #d is a continuous function, h (x, t)#d (t) at time P P P P t is close to h (x, t!¸)#d (t!¸) at time t!¸. Hence, P P h #d estimates to P P h) (x(t), t)#d) (t)+h (x(t!¸), t!¸)#d (t!¸) P P P P "xR (t!¸)!f (x(t!¸), t!¸) P P !B (x(t!¸), t!¸)u(t!¸). (9) P If B (x, t) is unknown or uncertain, then its estimate B) (t) P P is used. After substituting (9) into (7), the following TDC control law is obtained: u(t)"B) \(t)+!f (x(t), t)!xR (t!¸) P P P #f (x(t!¸), t!¸)#B) (t!¸)u(t!¸) P P #xR (t)!A e(t),. (10) KP CP This control law can be easily implemented in digital control system by taking time delay ¸ as an integer multiple of the sampling period ¹ . Q
B(x(t), t)u(t)"!f(x(t), t)!h(x(t), t)!d(t)
3. Variable reference model
#xR (t)!A e(t). (4) K C On the other hand, a nonlinear MIMO system satisfying the conditions of controllability and structural matching, without loss of generality, can be represented by the following partitioned matrices and vectors:
3.1. Why variable reference model?
x 0 0 x" } O} , f" }Tx} , h" } } , B" } } , x f h B P P P P
x 0 T d" } } , x " }KO } , A " }} , K C x A d KP CP P
(5)
(6)
Most plants have some bounds within which they are capable of accepting the control inputs without saturation. For example, all actuators saturate at some level. If they do not, their output would increase to in"nity, which is physically impossible. However, the characteristics of the bounded inputs usually have adverse e!ects on control system performance. The features of the bounded inputs can be modeled by the saturation element shown in Fig. 1, which is described by a linear region and a saturation region. That is, if the control
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Fig. 1. The saturation of control signals.
Fig. 3. Time responses of the BLDC motor position control system using the TDC with fast reference model (experimental results).
Fig. 2. Reference model subject to stepwise reference input.
e!ort u is smaller than the upper bound ; and larger
than the lower bound ; , the input u to the plant is the
same as the controller output u; otherwise, the input u would be at either saturation level. Reference models are frequently used in many control laws including TDC. The role of the reference model is to provide the reference state the plant should follow. Consider a "rst-order linear reference model for the sake of discussion x (t)"a x (t)#b r(t), (11) K K K K where x (t) denotes the state of the reference model and K r(t) the reference input, respectively. Fig. 2 illustrates a special case where the reference inputs are given in the form of step functions as in most servo problems. In Fig. 2 the di!erence between the current and the past reference inputs is referred to as a step change size in the reference input *r(t). If *r(t) is large, the control signal will also have a sudden large change, which may cause overshoots or saturation e!ects. Such a problem can be alleviated by smoothing the shape of the reference input. The reference model can act as a low-pass "lter and produce the smooth reference trajectory x (t), which can K be followed by the plant more easily than the stepwise reference input r(t) itself. The degree of low-pass "ltering is determined by the parameters a and b in case of the K K "rst-order system, thus the shape of x (t) is strongly K dependent upon these parameters. Suppose that a reference model is selected so that the reference state x (t) follows the reference input r(t) very K
quickly. Since it is very likely to directly track r(t), large overshoots may occur and the control signal computed using this large error also becomes large. To illustrate this, consider the response characteristics of the TDCbased position control system with the fast reference model shown in Fig. 3. The details of the experimental setup will be given in Section 4.1. Quick responses are observed for small step changes, while large overshoots occur and the control signal goes into saturation for large step changes. In this case, the time delay controller considers the controller output (u in Fig. 2) as the input to the plant and uses its time delay term (i.e., u(t!¸)) to compute the control input (refer to Eq. (10)). The control e!ort becomes large, due to the di!erence between the computed control input u and the actual control input u to the plant, thus resulting in large overshoot or limit cycles in some cases. Fig. 4 shows the responses of the TDC-based position control system when the reference model is selected so that the reference state x (t) follows the reference input K r(t) rather slowly. In this case no overshoot is observed even for large step changes, but responses are sluggish for small step changes when overshoots are unlikely to occur. From these observations, it follows that it is preferable to make the reference model slow (or fast) when the size of the step change (i.e., *r in Fig. 2) in the reference input is large (or small). However, there are two di$culties. One is that the size of the step change is constantly changing during tracking. Therefore, the reference model should be updated whenever *r changes. The other di$culty is to determine the criterion for `speeda of the
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where u represents the nominal control input when L e"0. Arranging (13) gives B) (t)u (t)!B) (t!¸)u (t!¸) P L P L "!f (x(t), t)#f (x(t!¸), t!¸) P P #xR (t)!xR (t!¸). (14) KP KP By rearranging (14) so that the same time index terms are separated on each side, we obtain
Fig. 4. Time responses of the BLDC motor position control system using the TDC with slow reference model (experimental results).
B) (t)u (t)#+f (x(t), t)!xR (t), P L P KP "B) (t!¸)u (t!¸)#+f (x(t!¸), t!¸) P L P !xR (t!¸),. (15) KP Note that both sides of Eq. (15) are of the same form, but with di!erent time indices. In order for this condition to be satis"ed, they must be either a constant or a periodic function with period ¸. The possibility of a periodic function can be excluded, since the time delay ¸ is usually chosen as a small value. Thus there must exist certain constant vector C satisfying B) (t)u (t)#+f (x(t), t)!xR (t),"C. P L P KP Then the nominal control input can be given by
reference model. In order to solve these di$culties, a new approach is proposed in this paper. The key idea is that the reference model is made as fast as possible, provided the control e!ort does not enter the saturation region of the plant. This notion of bounding the magnitude of the control e!ort is di!erent in that the proposed method places no limitation on the control e!ort. In other words, the control e!ort is prevented from being saturated by the appropriate selection of reference model. 3.2. Derivation of the variable reference model In this section, the relation between the control e!ort and the reference model in the TDC law will be found. Then, a way of selecting the reference model is suggested, which provides the reference trajectory for a good transient response. To begin with, the condition in which the control signal stays in the allowable linear range without saturation is found. An appropriate reference model is selected assuming that the control signal given by Eq. (10) is in the linear range, and the plant state vector tracks the reference state vector perfectly. Then, x(t) x (t) or x (t) x (t). (12) K P KP Since the error vector e(t)"0, substitution of (12) into (10) yields u(t) u (t)"B) \(t)+B) (t!¸)u (t!¸)!f (x(t), t) L P P L P #f (x(t!¸), t!¸)#xR (t)!xR (t!¸),, P KP KP
(13)
(16)
u (t)"B) \(t)+xR (t)!f (x(t), t)#C,. (17) L P KP P Note that Eq. (17) is the condition for perfect tracking since the nominal control input u is obtained with the L perfect tracking assumption. Now the condition for no saturation can be easily found by (18) U )u (t))U ,
L
where U and U represent the lower and upper
bounds of the linear region. In most control systems, perfect tracking can be obtained only when the control e!orts are not bounded. Thus the best performance in practical situations can be achieved when two conditions (17) and (18) are met. The criterion for the reference model is obtained by investigating the term xR after the substitution of Eq. (17) into KP condition (18) because the other terms in (17) and (18) are already known or readily computed. Note that no restrictions have been imposed in this procedure. If the reference model is described by parameters, the selection of the reference model corresponds to the selection of the parameters. For example, the time constant for a "rstorder system and the natural frequency and damping ratio for a second-order system can be determined to satisfy conditions (17) and (18). This point will be detailed in Section 3.3. Remember that in the beginning of the derivation of the reference model the tracking error was zero, since the plant state was assumed to track the reference state perfectly. This assumption is only for obtaining the condition of the reference model that enables perfect
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Fig. 6. Block diagram of the BLDC motor system. Fig. 5. Block diagram of the TDC with variable reference model.
tracking, but it does not mean that the actual plant state can or should follow the reference state perfectly. In summary, whenever step changes in the reference input occur, the reference model is updated so as to satisfy conditions (17) and (18). This will be referred to as a variable reference model. With this variable reference model, the controller can utilize the maximum allowable range within the plant's capability without going into the saturation region. Fig. 5 represents the block diagram of the TDC scheme with a variable reference model. 3.3. Variable reference model for the second-order system In order to investigate the e!ectiveness of the proposed method, a second-order system will be considered in this section. A brushless DC (BLDC) motor was chosen as a second-order system. Since the motor can rotate in either direction, control inputs have negative lower bounds and positive upper bounds. Modeling of the BLDC motor including the driver is quite similar to that of the DC motor as shown in Fig. 6. (Ong, 1998). If the armature inductance L is neglected, the model can be ? represented in state space by
x
0
1
" 1 K K x 0 ! f# 2 # J R ?
x x
or xK (t)#2fu x (t)#ux (t)"ur(t), K L K L K L
x (t)"r #+1!e\SL R(1#u t),*r. K L By di!erentiating (21) successively, x (t)"x (t)"+u e\SL R(1#u t)!u e\SL R,*r K K L L L "u t e\SL R*r L and xK (t)"x (t)"+ue\SL R!ute\SL R,*r K K L L is obtained. Substituting (23) into (17) yields
0 # K K u. (19) ? 2 JR ? where the state variables x and x denote the angular position and velocity of the motor, J, f, K , K , R , 2 # ? K are moment of inertia, viscous friction coe$cient, ? torque constant, back emf constant, armature resistance, and ampli"er gain, respectively. The second-order reference model can be characterized by the natural frequency u and damping ratio f L x (t) 0 1 x (t) 0 K K " # r(t) x (t) !u !2fu x (t) u K L L K L
where r(t) represents the current reference input. In this case, a variable reference model means that two parameters are updated whenever the reference input changes stepwise to satisfy conditions (17) and (18). Although possible, varying two parameters at the same time is complicated, so the damping ratio is "xed to f"1, which means critical damping. This simpli"cation can also be justi"ed because the critically damped system provides the fastest response without overshoot. Now updating the reference model corresponds to determining a new value for the natural frequency. Suppose that a new reference input is given at t"0 and the plant has successfully followed the previous reference input r , then the initial state vector becomes +x (0) x (0),2"+r 0,2. Also, it is assumed that all plant dynamics are unknown (i.e., f "0). Then the P response to the step change with magnitude *r"r!r can be easily obtained by
(20)
1 u (t)" +ue\SL R!ute\SL R,*r. L L b L
(21)
(22)
(23)
(24)
This equation relates the control input to the natural frequency of the reference model. When *r'0, the maximum and minimum control e!orts occur at t"0 and t"2/u , respectively, and the values are as follows: L u*r u*r u (t)" L , u (t)"!e\ L . (25)
b b On the other hand, when *r(0, the maximum and minimum control e!orts occur at t"2/u and t"0, L respectively, and the values are as follows: u*r u*r u (t)"!e\ L , u (t)" L .
b b
(26)
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By substituting (25) and (26) into (18), the natural frequency range in which the control e!orts do not go into the saturation region is obtained by
b;
, *r
!7.41
b;
*r
for *r'0, (27)
b;
, *r
!7.41
b;
*r
for *r(0. (28)
u "min L
u "min L
Note that the values inside the square roots are always positive since ; '0 and ; (0.
Whenever a new reference input is given, the natural frequency is newly computed according to Eqs. (27) or (28). Note that the natural frequency varies depending on the size of the change in the reference input; it becomes smaller for a larger change in the reference input, and vice versa. This means that the overshoot that usually occurs as a result of the sudden large change in the reference input can be prevented with this variable reference model.
4. Experiments and results 4.1. Experimental setup Various simulations and experiments on position control with TDC with a variable reference model have been carried out in order to verify the proposed variable reference model. Fig. 7 shows the experimental setup, where the BLDC motor is coupled to the load whose moment of inertia is comparable to that of the rotor. Table 1 represents parameters of this system. As shown in Table 1, the electric time constant (¸ /R "0.0033 s) is very ? ? small, compared with the time constant (J/f"2.99 s) of the mechanical part. Thus, ¸ is neglected. ? At every sampling time, the control signal computed from the controller is converted to PWM signal, which is used to switch each transistor of the inverter on and o!. The angular position of the motor is measured by counting the pulse signals from the encoder, and the angular velocity and acceleration are estimated by numerical di!erentiation of the angular position. 4.2. Experimental results Applying the TDC control law described by Eq. (10) to the second-order reference model of Eq. (20) gives 1 u(t)" +!x (t!¸)#bK u(t!¸)#x (t)#u (x (t) K LC K bK !x (t))#2f u (x (t)!x (t)),, (29) C LC K where u and f denote the natural frequency and LC C damping ratio of the error dynamics, and are selected so
Fig. 7. Experimental setup of the BLDC motor control system.
Table 1 Speci"cations of a BLDC motor control system Torque constant (K ) 2 Moment of inertia (J) Armature resistance (R ) ? Armature inductance (¸ ) ? Damping constant ( f ) Ampli"er gain (K ) ? Bounds of control input (; }; )
Current limits Sampling time (¹ ) Q Encoder
0.176 N m/A 7.87;10\ kg m 3.02 ) 10 mH 2.63;10\ N m sec 8 !5 to #5V 3A 5 ms 2048 pulses/rev
that the error dynamics are stable. The value of the control parameter bK can be estimated from Eq. (19) K K bK " ? 2 . JR ?
(30)
The TDC control law represented by Eq. (29) was used for both simulations and experiments. Figs. 8 and 9 illustrate the simulation and experimental results showing the position responses of the BLDC servo system using the TDC scheme with the variable reference model. The reference inputs were randomly selected so that they included both large and small step changes, ranging from a few degrees to 1403. Note that this reference input is the same as in Figs. 3 and 4. The two responses in Figs. 8 and 9 are very similar. No overshoots were observed even for large changes in the reference input, while very quick responses were obtained for small changes in the reference input. In addition, the control e!orts always stayed within $5 V, which coincides with the allowable linear range since the motor is driven by !40 to #40 V and the ampli"er gain is 8. It was noted that the values of the natural frequency vary depending on the size of the change in the reference input. In other words, the natural frequency was selected as a small value (e.g., 40 rad/s) for a large change in reference input, while a large value was
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Fig. 8. Responses of the BLDC motor position control system using the TDC with variable reference model (simulation results).
587
The control signal in Fig. 9 shows a somewhat oscillatory variation, and this can be explained as follows. The TDC scheme uses position, velocity and acceleration information. The position is obtained by counting the pulses from the optical encoder, but the velocity and acceleration are obtained by numerical di!erentiation. This numerical di!erentiation (especially, the acceleration computation) introduces some ripples in the computed values, thus leading to oscillatory behavior in the control signal computed using the acceleration value. However, the position output of the motor does not oscillate much since the motor itself has substantial inertia and thus it cannot respond to such a high-frequency noise in the control voltage. Therefore, it has little in#uence on the control performance. Fig. 10 shows the experimental results when a PID controller is used for the same system. It is observed that the integral term causes windup and thus overshoot for a large step change. If a PD control is adopted, such a windup does not occur, but a steady state error is likely to be introduced. The well-tuned PID control system for some operating point yields satisfactory performance around that operating range, but di!erent sets of PID gains should be used for di!erent operating ranges to achieve good performance. Therefore, there were di$culties in "nding many sets of PID gains to cover the entire operating conditions. However, the TDC with a variable reference model updates the reference models every time the new reference inputs are commanded, and thus this amounts to updating the controller parameters on-line. This is why the TDC scheme with a variable reference model shows a better performance than the PID controller.
Fig. 9. Responses of the BLDC motor position control system using the TDC with variable reference model (experimental results).
chosen (e.g., 180 rad/s) for a small change. As a result, the transient response characteristics using TDC with the variable reference model are better than those of the TDC with "xed reference model shown in Figs. 3 and 4.
Fig. 10. Responses of the BLDC motor position control system with a PID controller (experimental results).
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5. Conclusions In this research a time delay control (TDC) scheme with a variable reference model was proposed. The key idea was to improve the transient response characteristics by updating the reference model subject to stepwise reference inputs. That is, the reference input chosen depended on the size of the changes in the reference input. In order to investigate e!ectiveness of the proposed method, the selection procedure and experimental results for a typical second-order system was shown. As a consequence of this approach, accurate responses without overshoots can be obtained for large changes, as well as fast responses for small changes in the reference input. Thus the TDC with a variable reference model yields better transient response behavior than the original TDC scheme and the PID controller. The idea of improving the transient response by means of the smart selection of the reference model can be extended to other control schemes, and research to
extend the concept of a variable reference model is currently under way.
References Cheah, C. C., & Wang, D. (1997). A model reference learning scheme for a class of nonlinear systems. International Journal of Control, 66(2), 271}287. Lu, Y. S., & Chen, J. S. (1995). Design of a global sliding-mode controller for a motor drive with bounded control. International Journal of Control, 62(5), 1001}1019. Ong, C. M. (1998). Dynamic simulation of electric machinery: Using Matlab and Simulink. Englewood Cli!s, NJ: Prentice-Hall. Slotine, J. J., & Li, W. (1991). Applied nonlinear control. Englewood Cli!s, NJ: Prentice-Hall. Spong, M. W., Thorp, J. S., & Kleinwaks, J. M. (1986). The control of robot manipulators with bounded input. IEEE Transaction on Automatic Control, AC-31(6), 483}490. Youcef-Toumi, K., & Ito, O. (1990). A time delay controller for systems with unknown dynamics. ASME Journal of Dynamic Systems, Measurement, and Control, 112(1), 133}142.
Design of time delay controller using variable reference model Jae-Bok Song*, Kyung-Seok Byun Department of Mechanical Engineering, Korea University, 5-Ka Anam-Dong Sungbuk-Ku, Seoul, 136-701, South Korea Received 10 August 1998; accepted 27 August 1999
Abstract Reference models are used in many control algorithms to improve transient response characteristics. Most control systems have bounded control inputs to avoid saturation of the plant. If the reference models do not account for the limits of the control inputs, the control performance of the system may deteriorate. In this paper a new approach to avoid saturation is proposed by varying the reference model for time delay control-based systems subject to step changes in the reference input. In this scheme, the variable reference model is determined based on the information on control inputs and the size of the step changes in the reference inputs. This scheme was veri"ed by application to the BLDC motor position control system in simulations and experiments. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Time delay control; Variable reference model; Saturation avoidance; Tracking control; Nominal control input
1. Introduction Time delay control (TDC) schemes have been suggested as e!ective control techniques for nonlinear time-varying systems with uncertain dynamics and/or unpredictable disturbances. Basically, it cancels out the undesired dynamics and disturbances, and substitutes them with the desired dynamics given in terms of the reference model. Thus, the state vector of a plant is made to follow the reference model accurately. Reference models are used in model reference adaptive control (Slotine & Li, 1991) or model reference learning control (Cheah & Wang, 1997) as well as in TDC. The selection of the reference model has a great impact on control performance because it provides the desired trajectory for the plant state vector. In most control systems the control signal to the plant has some bounds. Beyond these bounds, the control signal may cause a saturation phenomenon, where the change in input cannot bring about changes in the plant output. Large control e!orts are usually desired for quick responses, but they can produce large overshoots or saturation problems. On the other hand, small control
* Corresponding author. Tel.: #82-2-3290-3363; fax: #82-2-9289769. E-mail address: [email protected] (J.-B. Song).
e!orts avoiding overshoots or saturation may cause slow responses. Thus it is preferable to make use of large control signals within the range of no saturation. In this paper, a way of satisfying this ideal condition is proposed in relation to the TDC scheme. This is achieved by taking the bounds of the saturation region into account when determining the reference model for the TDC scheme. Some research has been done on control systems with bounded control e!orts. Lu and Chen (1995) suggested global sliding mode control (GSMC), where the sliding line of the GSMC is designed so that the initial state is located on it. The maximum and minimum values of control e!orts generated according to this GSMC can be estimated and the range of allowable reference inputs is then obtained under the control bound and the sliding line. Spong, Thorp and Kleinwaks (1986) studied optimal decision strategy (ODS), where the decision strategy is to minimize a Euclidean norm of the di!erence between the actual vector of the instantaneous joint accelerations and the desired joint acceleration vector which may be generated by a reference model. However, they did not suggest the criterion for the selection of the reference model. In this paper, a new method for avoiding saturation is proposed by varying the reference model for the TDCbased systems subject to the step changes in reference inputs. The variable reference model in this scheme is determined based on the information about the control e!orts and the size of the step changes in the reference
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inputs. This reference model is not "xed, but varies depending on the size of the change in the reference input. In Section 2 the time delay control law is brie#y introduced. Then, a detailed discussion of the idea of the variable reference model is given in Section 3. Section 4 will show the experimental results of the servo system based on the TDC scheme with a variable reference model.
In this section, time delay control (TDC) law (YoucefToumi & Ito, 1990) will be brie#y explained. Consider a nonlinear time-varying plant in the state-space model (1)
where x(t) represents the state vector (n;1), u(t) the control vector (r;1), d(t) the unpredicted disturbance vector. Dynamics of the plant is divided into the known dynamics vector f(x, t) and the unknown dynamics vector h(x, t). Also, control matrix B(x, t) is assumed temporarily to be known. Consider the reference model represented by the following nonlinear time-varying system: xR (t)"'(x (t), r(t)), (2) K K where x (t) and r(t) denote the state vector of the referK ence model and the reference input vector, respectively. This reference model yields the ideal state vector that the plant state vector should follow. De"ning the error vector e(t)"x (t)!x(t), the error K dynamics can be written by eR (t)"A e(t), (3) C where A is the error system matrix. When A is selected C C so that all eigenvalues are placed in the left-half of the s-plane, the error dynamics becomes asymptotically stable. After substitution of the relation xR (t)"xR (t)!A e(t) K C from (3) into (1), the following equation is obtained:
xR 0 Tx 0 0 } } } u"! } } ! } } ! } } # }KO B f h d xR P P P P KP
Hence,
2. Time delay control law
xR (t)"f(x(t), t)#h(x(t), t)#B(x(t), t)u(t)#d(t),
where x , f , h and d are r;1 vectors, x a (n!r);1 P P P O vector, and B a r;r nonsingular matrix. Substitution of P (5) into (4) yields
T ! } } e. A CP
B (x, t)u(t)"!f (x, t)!h (x, t)!d (t) P P P P #xR (t)!A e(t). (7) KP CP Since h (x, t)#d (t) is an unknown function, the control P P e!ort u(t) cannot be obtained from (7), and thus the estimate of this function h) (x, t)#d) (t) must be found. P P Rearranging (7) gives h (x, t)#d (t)"xR (t)!f (x(t), t)!B (x(t), t)u(t). (8) P P P P P Assuming that the time delay ¸ is su$ciently small and h #d is a continuous function, h (x, t)#d (t) at time P P P P t is close to h (x, t!¸)#d (t!¸) at time t!¸. Hence, P P h #d estimates to P P h) (x(t), t)#d) (t)+h (x(t!¸), t!¸)#d (t!¸) P P P P "xR (t!¸)!f (x(t!¸), t!¸) P P !B (x(t!¸), t!¸)u(t!¸). (9) P If B (x, t) is unknown or uncertain, then its estimate B) (t) P P is used. After substituting (9) into (7), the following TDC control law is obtained: u(t)"B) \(t)+!f (x(t), t)!xR (t!¸) P P P #f (x(t!¸), t!¸)#B) (t!¸)u(t!¸) P P #xR (t)!A e(t),. (10) KP CP This control law can be easily implemented in digital control system by taking time delay ¸ as an integer multiple of the sampling period ¹ . Q
B(x(t), t)u(t)"!f(x(t), t)!h(x(t), t)!d(t)
3. Variable reference model
#xR (t)!A e(t). (4) K C On the other hand, a nonlinear MIMO system satisfying the conditions of controllability and structural matching, without loss of generality, can be represented by the following partitioned matrices and vectors:
3.1. Why variable reference model?
x 0 0 x" } O} , f" }Tx} , h" } } , B" } } , x f h B P P P P
x 0 T d" } } , x " }KO } , A " }} , K C x A d KP CP P
(5)
(6)
Most plants have some bounds within which they are capable of accepting the control inputs without saturation. For example, all actuators saturate at some level. If they do not, their output would increase to in"nity, which is physically impossible. However, the characteristics of the bounded inputs usually have adverse e!ects on control system performance. The features of the bounded inputs can be modeled by the saturation element shown in Fig. 1, which is described by a linear region and a saturation region. That is, if the control
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Fig. 1. The saturation of control signals.
Fig. 3. Time responses of the BLDC motor position control system using the TDC with fast reference model (experimental results).
Fig. 2. Reference model subject to stepwise reference input.
e!ort u is smaller than the upper bound ; and larger
than the lower bound ; , the input u to the plant is the
same as the controller output u; otherwise, the input u would be at either saturation level. Reference models are frequently used in many control laws including TDC. The role of the reference model is to provide the reference state the plant should follow. Consider a "rst-order linear reference model for the sake of discussion x (t)"a x (t)#b r(t), (11) K K K K where x (t) denotes the state of the reference model and K r(t) the reference input, respectively. Fig. 2 illustrates a special case where the reference inputs are given in the form of step functions as in most servo problems. In Fig. 2 the di!erence between the current and the past reference inputs is referred to as a step change size in the reference input *r(t). If *r(t) is large, the control signal will also have a sudden large change, which may cause overshoots or saturation e!ects. Such a problem can be alleviated by smoothing the shape of the reference input. The reference model can act as a low-pass "lter and produce the smooth reference trajectory x (t), which can K be followed by the plant more easily than the stepwise reference input r(t) itself. The degree of low-pass "ltering is determined by the parameters a and b in case of the K K "rst-order system, thus the shape of x (t) is strongly K dependent upon these parameters. Suppose that a reference model is selected so that the reference state x (t) follows the reference input r(t) very K
quickly. Since it is very likely to directly track r(t), large overshoots may occur and the control signal computed using this large error also becomes large. To illustrate this, consider the response characteristics of the TDCbased position control system with the fast reference model shown in Fig. 3. The details of the experimental setup will be given in Section 4.1. Quick responses are observed for small step changes, while large overshoots occur and the control signal goes into saturation for large step changes. In this case, the time delay controller considers the controller output (u in Fig. 2) as the input to the plant and uses its time delay term (i.e., u(t!¸)) to compute the control input (refer to Eq. (10)). The control e!ort becomes large, due to the di!erence between the computed control input u and the actual control input u to the plant, thus resulting in large overshoot or limit cycles in some cases. Fig. 4 shows the responses of the TDC-based position control system when the reference model is selected so that the reference state x (t) follows the reference input K r(t) rather slowly. In this case no overshoot is observed even for large step changes, but responses are sluggish for small step changes when overshoots are unlikely to occur. From these observations, it follows that it is preferable to make the reference model slow (or fast) when the size of the step change (i.e., *r in Fig. 2) in the reference input is large (or small). However, there are two di$culties. One is that the size of the step change is constantly changing during tracking. Therefore, the reference model should be updated whenever *r changes. The other di$culty is to determine the criterion for `speeda of the
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where u represents the nominal control input when L e"0. Arranging (13) gives B) (t)u (t)!B) (t!¸)u (t!¸) P L P L "!f (x(t), t)#f (x(t!¸), t!¸) P P #xR (t)!xR (t!¸). (14) KP KP By rearranging (14) so that the same time index terms are separated on each side, we obtain
Fig. 4. Time responses of the BLDC motor position control system using the TDC with slow reference model (experimental results).
B) (t)u (t)#+f (x(t), t)!xR (t), P L P KP "B) (t!¸)u (t!¸)#+f (x(t!¸), t!¸) P L P !xR (t!¸),. (15) KP Note that both sides of Eq. (15) are of the same form, but with di!erent time indices. In order for this condition to be satis"ed, they must be either a constant or a periodic function with period ¸. The possibility of a periodic function can be excluded, since the time delay ¸ is usually chosen as a small value. Thus there must exist certain constant vector C satisfying B) (t)u (t)#+f (x(t), t)!xR (t),"C. P L P KP Then the nominal control input can be given by
reference model. In order to solve these di$culties, a new approach is proposed in this paper. The key idea is that the reference model is made as fast as possible, provided the control e!ort does not enter the saturation region of the plant. This notion of bounding the magnitude of the control e!ort is di!erent in that the proposed method places no limitation on the control e!ort. In other words, the control e!ort is prevented from being saturated by the appropriate selection of reference model. 3.2. Derivation of the variable reference model In this section, the relation between the control e!ort and the reference model in the TDC law will be found. Then, a way of selecting the reference model is suggested, which provides the reference trajectory for a good transient response. To begin with, the condition in which the control signal stays in the allowable linear range without saturation is found. An appropriate reference model is selected assuming that the control signal given by Eq. (10) is in the linear range, and the plant state vector tracks the reference state vector perfectly. Then, x(t) x (t) or x (t) x (t). (12) K P KP Since the error vector e(t)"0, substitution of (12) into (10) yields u(t) u (t)"B) \(t)+B) (t!¸)u (t!¸)!f (x(t), t) L P P L P #f (x(t!¸), t!¸)#xR (t)!xR (t!¸),, P KP KP
(13)
(16)
u (t)"B) \(t)+xR (t)!f (x(t), t)#C,. (17) L P KP P Note that Eq. (17) is the condition for perfect tracking since the nominal control input u is obtained with the L perfect tracking assumption. Now the condition for no saturation can be easily found by (18) U )u (t))U ,
L
where U and U represent the lower and upper
bounds of the linear region. In most control systems, perfect tracking can be obtained only when the control e!orts are not bounded. Thus the best performance in practical situations can be achieved when two conditions (17) and (18) are met. The criterion for the reference model is obtained by investigating the term xR after the substitution of Eq. (17) into KP condition (18) because the other terms in (17) and (18) are already known or readily computed. Note that no restrictions have been imposed in this procedure. If the reference model is described by parameters, the selection of the reference model corresponds to the selection of the parameters. For example, the time constant for a "rstorder system and the natural frequency and damping ratio for a second-order system can be determined to satisfy conditions (17) and (18). This point will be detailed in Section 3.3. Remember that in the beginning of the derivation of the reference model the tracking error was zero, since the plant state was assumed to track the reference state perfectly. This assumption is only for obtaining the condition of the reference model that enables perfect
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Fig. 6. Block diagram of the BLDC motor system. Fig. 5. Block diagram of the TDC with variable reference model.
tracking, but it does not mean that the actual plant state can or should follow the reference state perfectly. In summary, whenever step changes in the reference input occur, the reference model is updated so as to satisfy conditions (17) and (18). This will be referred to as a variable reference model. With this variable reference model, the controller can utilize the maximum allowable range within the plant's capability without going into the saturation region. Fig. 5 represents the block diagram of the TDC scheme with a variable reference model. 3.3. Variable reference model for the second-order system In order to investigate the e!ectiveness of the proposed method, a second-order system will be considered in this section. A brushless DC (BLDC) motor was chosen as a second-order system. Since the motor can rotate in either direction, control inputs have negative lower bounds and positive upper bounds. Modeling of the BLDC motor including the driver is quite similar to that of the DC motor as shown in Fig. 6. (Ong, 1998). If the armature inductance L is neglected, the model can be ? represented in state space by
x
0
1
" 1 K K x 0 ! f# 2 # J R ?
x x
or xK (t)#2fu x (t)#ux (t)"ur(t), K L K L K L
x (t)"r #+1!e\SL R(1#u t),*r. K L By di!erentiating (21) successively, x (t)"x (t)"+u e\SL R(1#u t)!u e\SL R,*r K K L L L "u t e\SL R*r L and xK (t)"x (t)"+ue\SL R!ute\SL R,*r K K L L is obtained. Substituting (23) into (17) yields
0 # K K u. (19) ? 2 JR ? where the state variables x and x denote the angular position and velocity of the motor, J, f, K , K , R , 2 # ? K are moment of inertia, viscous friction coe$cient, ? torque constant, back emf constant, armature resistance, and ampli"er gain, respectively. The second-order reference model can be characterized by the natural frequency u and damping ratio f L x (t) 0 1 x (t) 0 K K " # r(t) x (t) !u !2fu x (t) u K L L K L
where r(t) represents the current reference input. In this case, a variable reference model means that two parameters are updated whenever the reference input changes stepwise to satisfy conditions (17) and (18). Although possible, varying two parameters at the same time is complicated, so the damping ratio is "xed to f"1, which means critical damping. This simpli"cation can also be justi"ed because the critically damped system provides the fastest response without overshoot. Now updating the reference model corresponds to determining a new value for the natural frequency. Suppose that a new reference input is given at t"0 and the plant has successfully followed the previous reference input r , then the initial state vector becomes +x (0) x (0),2"+r 0,2. Also, it is assumed that all plant dynamics are unknown (i.e., f "0). Then the P response to the step change with magnitude *r"r!r can be easily obtained by
(20)
1 u (t)" +ue\SL R!ute\SL R,*r. L L b L
(21)
(22)
(23)
(24)
This equation relates the control input to the natural frequency of the reference model. When *r'0, the maximum and minimum control e!orts occur at t"0 and t"2/u , respectively, and the values are as follows: L u*r u*r u (t)" L , u (t)"!e\ L . (25)
b b On the other hand, when *r(0, the maximum and minimum control e!orts occur at t"2/u and t"0, L respectively, and the values are as follows: u*r u*r u (t)"!e\ L , u (t)" L .
b b
(26)
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By substituting (25) and (26) into (18), the natural frequency range in which the control e!orts do not go into the saturation region is obtained by
b;
, *r
!7.41
b;
*r
for *r'0, (27)
b;
, *r
!7.41
b;
*r
for *r(0. (28)
u "min L
u "min L
Note that the values inside the square roots are always positive since ; '0 and ; (0.
Whenever a new reference input is given, the natural frequency is newly computed according to Eqs. (27) or (28). Note that the natural frequency varies depending on the size of the change in the reference input; it becomes smaller for a larger change in the reference input, and vice versa. This means that the overshoot that usually occurs as a result of the sudden large change in the reference input can be prevented with this variable reference model.
4. Experiments and results 4.1. Experimental setup Various simulations and experiments on position control with TDC with a variable reference model have been carried out in order to verify the proposed variable reference model. Fig. 7 shows the experimental setup, where the BLDC motor is coupled to the load whose moment of inertia is comparable to that of the rotor. Table 1 represents parameters of this system. As shown in Table 1, the electric time constant (¸ /R "0.0033 s) is very ? ? small, compared with the time constant (J/f"2.99 s) of the mechanical part. Thus, ¸ is neglected. ? At every sampling time, the control signal computed from the controller is converted to PWM signal, which is used to switch each transistor of the inverter on and o!. The angular position of the motor is measured by counting the pulse signals from the encoder, and the angular velocity and acceleration are estimated by numerical di!erentiation of the angular position. 4.2. Experimental results Applying the TDC control law described by Eq. (10) to the second-order reference model of Eq. (20) gives 1 u(t)" +!x (t!¸)#bK u(t!¸)#x (t)#u (x (t) K LC K bK !x (t))#2f u (x (t)!x (t)),, (29) C LC K where u and f denote the natural frequency and LC C damping ratio of the error dynamics, and are selected so
Fig. 7. Experimental setup of the BLDC motor control system.
Table 1 Speci"cations of a BLDC motor control system Torque constant (K ) 2 Moment of inertia (J) Armature resistance (R ) ? Armature inductance (¸ ) ? Damping constant ( f ) Ampli"er gain (K ) ? Bounds of control input (; }; )
Current limits Sampling time (¹ ) Q Encoder
0.176 N m/A 7.87;10\ kg m 3.02 ) 10 mH 2.63;10\ N m sec 8 !5 to #5V 3A 5 ms 2048 pulses/rev
that the error dynamics are stable. The value of the control parameter bK can be estimated from Eq. (19) K K bK " ? 2 . JR ?
(30)
The TDC control law represented by Eq. (29) was used for both simulations and experiments. Figs. 8 and 9 illustrate the simulation and experimental results showing the position responses of the BLDC servo system using the TDC scheme with the variable reference model. The reference inputs were randomly selected so that they included both large and small step changes, ranging from a few degrees to 1403. Note that this reference input is the same as in Figs. 3 and 4. The two responses in Figs. 8 and 9 are very similar. No overshoots were observed even for large changes in the reference input, while very quick responses were obtained for small changes in the reference input. In addition, the control e!orts always stayed within $5 V, which coincides with the allowable linear range since the motor is driven by !40 to #40 V and the ampli"er gain is 8. It was noted that the values of the natural frequency vary depending on the size of the change in the reference input. In other words, the natural frequency was selected as a small value (e.g., 40 rad/s) for a large change in reference input, while a large value was
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Fig. 8. Responses of the BLDC motor position control system using the TDC with variable reference model (simulation results).
587
The control signal in Fig. 9 shows a somewhat oscillatory variation, and this can be explained as follows. The TDC scheme uses position, velocity and acceleration information. The position is obtained by counting the pulses from the optical encoder, but the velocity and acceleration are obtained by numerical di!erentiation. This numerical di!erentiation (especially, the acceleration computation) introduces some ripples in the computed values, thus leading to oscillatory behavior in the control signal computed using the acceleration value. However, the position output of the motor does not oscillate much since the motor itself has substantial inertia and thus it cannot respond to such a high-frequency noise in the control voltage. Therefore, it has little in#uence on the control performance. Fig. 10 shows the experimental results when a PID controller is used for the same system. It is observed that the integral term causes windup and thus overshoot for a large step change. If a PD control is adopted, such a windup does not occur, but a steady state error is likely to be introduced. The well-tuned PID control system for some operating point yields satisfactory performance around that operating range, but di!erent sets of PID gains should be used for di!erent operating ranges to achieve good performance. Therefore, there were di$culties in "nding many sets of PID gains to cover the entire operating conditions. However, the TDC with a variable reference model updates the reference models every time the new reference inputs are commanded, and thus this amounts to updating the controller parameters on-line. This is why the TDC scheme with a variable reference model shows a better performance than the PID controller.
Fig. 9. Responses of the BLDC motor position control system using the TDC with variable reference model (experimental results).
chosen (e.g., 180 rad/s) for a small change. As a result, the transient response characteristics using TDC with the variable reference model are better than those of the TDC with "xed reference model shown in Figs. 3 and 4.
Fig. 10. Responses of the BLDC motor position control system with a PID controller (experimental results).
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5. Conclusions In this research a time delay control (TDC) scheme with a variable reference model was proposed. The key idea was to improve the transient response characteristics by updating the reference model subject to stepwise reference inputs. That is, the reference input chosen depended on the size of the changes in the reference input. In order to investigate e!ectiveness of the proposed method, the selection procedure and experimental results for a typical second-order system was shown. As a consequence of this approach, accurate responses without overshoots can be obtained for large changes, as well as fast responses for small changes in the reference input. Thus the TDC with a variable reference model yields better transient response behavior than the original TDC scheme and the PID controller. The idea of improving the transient response by means of the smart selection of the reference model can be extended to other control schemes, and research to
extend the concept of a variable reference model is currently under way.
References Cheah, C. C., & Wang, D. (1997). A model reference learning scheme for a class of nonlinear systems. International Journal of Control, 66(2), 271}287. Lu, Y. S., & Chen, J. S. (1995). Design of a global sliding-mode controller for a motor drive with bounded control. International Journal of Control, 62(5), 1001}1019. Ong, C. M. (1998). Dynamic simulation of electric machinery: Using Matlab and Simulink. Englewood Cli!s, NJ: Prentice-Hall. Slotine, J. J., & Li, W. (1991). Applied nonlinear control. Englewood Cli!s, NJ: Prentice-Hall. Spong, M. W., Thorp, J. S., & Kleinwaks, J. M. (1986). The control of robot manipulators with bounded input. IEEE Transaction on Automatic Control, AC-31(6), 483}490. Youcef-Toumi, K., & Ito, O. (1990). A time delay controller for systems with unknown dynamics. ASME Journal of Dynamic Systems, Measurement, and Control, 112(1), 133}142.