Abstract:
Systems for controlling flow effector control surfaces. The system comprises a flow effector control coupled to an elongated bar at one end of the bar. The elongated bar is coupled at the other end to the middle of a T-shaped member. A compliant link is coupled to the bar between the control surface and the T-shaped member. At each end of the T-shaped member is coupled a shape memory alloy wire which acts as an actuating means. When one of the shape memory alloy wire contracts, the elongated bar pivots about the compliant link and activates or retracts the flow effector control surface.

Description:
This application claims benefit of priority under 35 USC 119(e) from U.S. provisional patent application 60/924,859 filed 1, Jun. 2007. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to control systems for control surfaces on aircraft or airborne munitions. Specifically, the present invention relates to methods and devices for controlling control surfaces which affect velocity and pressure fields. 
     BACKGROUND TO THE INVENTION 
     Efficient guidance systems for airborne munitions and efficient control systems for aircraft have always been in high demand. The main objectives for new technologies in aerospace have always been increased speed and range in aerovehicles and reduced volume and weight of vehicle components. In recent years, smart structures have been introduced to replace traditional control actuation systems while interest in active flow control technologies have developed. For smart structures to be effect in active flow control depends on a micro-flow effector&#39;s ability to influence the macroscopic flow around the aircraft body. One approach which contributes to this control is the use of flow control surfaces deployed at various points on the aircraft superstructure. For airborne munitions, such control surfaces can be deployed in the forebody of the munition as missiles with slender forebodies face significant yawing moments under asymmetric vortices. Controlling forebody vortex asymmetry is dependent on the sensitivity of the asymmetric flow to the distance between the micro-flow effector and the nose top. The closer the micro-flow effector is to the nose tip, the lower the power required to trigger flow changes. 
     As such, flow control using micro-surfaces can be applied where manipulation of the velocity and pressure field is desired. Micro-flow effectors can be used in place of traditional control surfaces to reduce weight and volume while maintaining control authority. Boundary layer separation on aircraft wings needs to be controlled because it can result in reduction of lift and micro-flow effectors can be used for separation control. 
     Current state-of-the-art actuators are based on electromechanical devices and are often used in conventional missile control actuation systems. Examples of such actuators may be found in U.S. Pat. No. 6,685,143 issued to Prince et al, and U.S. Pat. No. 7,070,144 issued to DiCocco et al., the contents of both are being incorporated herein by reference. However due to the nature of flow effectors, the control surface must be close to the nose tip where volume is highly constrained. Since electromechanical systems, such as those disclosed by Prince and DiCocco, require electric motors to be connected to control surfaces through gear trains, the volume used by the components exceed the volume available in the nose envelope. There is therefore a need for a compact actuation system that mitigates if not overcomes the shortcomings of the prior art. 
     SUMMARY OF THE INVENTION 
     The present invention relates to systems for controlling flow effector control surfaces. The system comprises a flow effector control surface coupled to an elongated member at one end of the member. The elongated member is coupled at the other end to the middle of a T-shaped bar. A compliant link is coupled to the member between the control surface and the T-shaped bar. At each aned of the T-shaped bar is coupled a shape memory alloy wire which acts as an actuating means. When one of the shape memory alloy wire contracts, the elongated member pivots about the compliant link and activates or retracts the flow effector control surface. 
     In one aspect of the invention, there is provided a system for controlling a control surface on a device, the system comprising:
         a control surface   an elongated member coupled to said control surface, said elongated member having a first member end and a second member end   a T-shaped bar coupled to said elongated member, said bar having a first bar end and a second bar end, said   a pivotable link coupled to said elongated member   first actuator means coupled to said T-shaped bar at said first bar end   second actuator means coupled to said T-shaped bar at said second bar end   wherein   said control surface is coupled to said elongated member at said first member end of said elongated member   said T-shaped bar is coupled to said elongated member at said second member end of said elongated member, said second member end being coupled to said T-shaped bar at a point between said first bar end and said second bar end   when either said first or second actuator means is actuated, said elongated member pivots about said pivotable link to deploy or retract said control surface.       

     In another aspect, there is provided a flow effector position control system, the system comprising:
         an elongated member coupled to said flow effector   a T-shaped bar coupled to said elongated member   a compliant link coupled to said elongated member between said bar and said flow effector   actuator means coupled to said bar   wherein   when said actuator means are actuated, said elongated member pivots about said compliant link.       

    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A better understanding of the invention will be obtained by considering the detailed description below, with reference to the following drawings in which: 
         FIG. 1  illustrates a system for actuating a control flow effector according to one embodiment of the invention. 
         FIG. 1A  illustrates an airborne munition with micro-flow effectors deployed in the nose cone of the munition. 
         FIG. 2  is a critical stress-temperature transition diagram for a 0.1 mm diameter shape memory alloy (SMA) wire 
         FIG. 3  illustrate open-loop results for an antagonistic SMA actuator subjected to a 2V square wave excitation at 0.1 Hz 
         FIG. 3A  is a schematic diagram of a micro-flow effector according to one embodiment of the invention. 
         FIG. 4  shows a circuit for variable structure control of the system illustrated in  FIG. 1   
         FIG. 5  is a feedback control diagram of the SMA discrete-time control in the vicinity of the steady state 
         FIGS. 6   a  and  6   b  illustrates the experimental results of the SMA micro-flow effector set-point regulation.  FIG. 6   a  shows the results with a piecewise constant command of 0.25 Hz while  FIG. 6   b  shows the results with a square wave command of 0.5 Hz. 
     
    
    
     DETAILED DESCRIPTION 
     Referring to  FIG. 1 , a system  10  for actuating a control surface, in this case a control flow effector, is illustrated. The system  10  has a control surface  20  attached to an elongated member  30  at one member end  30 A of the member  30 . At the other end  30 B of the member  30 , is attached a T-shaped bar  40  with a first bar end  50  and a second bar end  60 . Between the control surface  20  and the T-shaped bar  40  is coupled a compliant link  70  to the elongated member  30 . Two actuating means,  80 A and  80 B are attached to the ends of the T-shaped bar with actuating means  80 A being attached to the first bar end  50  and actuating means  80 B being attached to the second bar end  60 . 
     When either of the actuating means  80 A,  80 B is actuated, it pulls in the direction of arrow  90 . This activating causes the elongated member  30  to pivot about the pivot point  95  and thereby about the compliant link  70 . Depending on which actuating means is activated, the control surface  20  thus moves in either the direction of arrow  100 A or  100 B. If the actuating means  80 A is activated, the control surface  20  moves in the direction of arrow  100 A. If the actuating means  80 B is activated, then the control surface  20  moves in the direction of arrow  100 B. In one embodiment, activating actuating means  80 A deploys or actuates the control surface  20  while activating actuating means  80 B retracts the control surface  20 . 
     In one embodiment, the actuating means  80 A,  80 B are shape memory alloy (SMA) wires which contract when actuated. Actuation of the SMA wires involves running a current through the SMA wire. While regular wires, whose actuation may involve pulling the wires, may be used, it has been found that SMA wires provide better results as lesser mechanisms are required. 
     The pivot point  95  provides a coupling between the link  70  and the elongated member  30 . Ideally, the link  70  is a compliant link but other embodiments may use non-compliant links as long as the elongated member  30  is able to pivot about pivot point  95 . 
     It should be noted that in the description of the embodiment that follows, the control surface is a micro-flow effector for use in constrained space applications such as in the nose cone of a missile.  FIG. 1A  illustrates a missile with the micro-flow effectors deployed at the nose cone. However, the flow-effector may also be used in other parts of airborne munitions or, indeed, in other parts of aircraft. Furthermore, the control surface may be other than micro-flow effectors—any control surface whose actuation involves a range of motion at right angles to a rectilinear activating motion may be used. As can be seen from  FIG. 1 , the rectilinear motion illustrated be arrow  90  is at right angles to the motion illustrated by arrows  100 A,  100 B. 
     The shape memory alloy actuator performance is described using a hybrid micro-macroscopic constitutive law and a one-dimensional heat transfer equation. At the macroscopic level, the global strain is governed by a Reuss-type rule of mixtures law
 
ε ij =(1−Φ)ε ij   A +Φε ij   M   (1)
 
     where ε is the global strain, Φ is the total martensite fraction, ε A  is the strain in the austenite phase and ε M  is the strain in the martensite phase. 
     Strains are assumed to remain within the linear elastic limits of the SMA. The elastic strain in the austenite is given by 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     ij 
                     A 
                   
                   = 
                   
                     
                       
                         1 
                         
                           E 
                           A 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   υ 
                                   A 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               σ 
                               ij 
                             
                           
                           - 
                           
                             
                               υ 
                               A 
                             
                             ⁢ 
                             
                               δ 
                               ij 
                             
                             ⁢ 
                             
                               σ 
                               kk 
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         δ 
                         ij 
                       
                       ⁢ 
                       
                         α 
                         ij 
                         A 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       T 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where E A  is the austenite elastic modulus (Pa), υ A  is the austenite Poisson ratio, σ is the applied stress (Pa), α A  is the austenite thermal expansion coefficient (K −1 ). 
     The martensite strain consists of an elastic component (eq. 2, except martensite materials properties are used), a stress-induced phase transformation component and a temperature-induced phase transformation component. The martensite strain is 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     ij 
                     M 
                   
                   = 
                   
                     
                       ɛ 
                       ij 
                       el 
                     
                     + 
                     
                       
                         
                           Φ 
                           σ 
                         
                         Φ 
                       
                       ⁢ 
                       
                         ɛ 
                         ij 
                         σ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where ε el  is the elastic strain, ε σ  is the stress-induced strain and Φ σ  is the stress-induced martensite fraction. The total martensite fraction is
 
Φ=Φ T +Φ σ   (4)
 
     where Φ T  is the temperature-induced martensite fraction. 
     The stress-induced strain, ε σ , is a preferential deformation of the martensite variants in response to an external stress. The strain is given by 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     ij 
                     σ 
                   
                   = 
                   
                     
                       1 
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             f 
                             n 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 θ 
                                 1 
                               
                               , 
                               
                                 θ 
                                 2 
                               
                               , 
                               
                                 θ 
                                 3 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           R 
                           ik 
                           n 
                         
                         ⁢ 
                         
                           R 
                           jl 
                           n 
                         
                         ⁢ 
                         
                           ɛ 
                           kl 
                           
                             σ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             n 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where N is the number of grains, f n (θ 1 ,θ 2 ,θ 3 ) is a frequency distribution function, R ik   n R jl   n  is the coordinate rotation matrices that rotate the local grain coordinate system to the global coordinate system and ε σn  is the average stress-induced variant strain of grain n in the local coordinate system. 
     The martensite transformation kinetics is defined on a global basis. The model parameters are quantified by a critical stress-temperature diagram (see  FIG. 2 ) derived from constant temperature tensile tests carried out over a temperature range T&lt;T Mf  to T&gt;T Af  where T Mf  is the martensite finish temperature (K) and T Af  is the austenite finish temperature (K). The stress-induced martensite fraction is calculated as a function of critical stress at a specific temperature. The martensite fraction versus critical stress relationship is described using a linear function in the form of a Heaviside model. 
     The temperature-induced martensite fraction, Φ T , is assumed to be linearly dependent on temperature between the martensite start and finish temperatures. 
     
       
         
           
             
               
                 
                   
                     Φ 
                     T 
                   
                   = 
                   
                     
                       
                         T 
                         Ms 
                       
                       - 
                       T 
                     
                     
                       
                         T 
                         Ms 
                       
                       - 
                       
                         T 
                         Mf 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     where T Ms  is the martensite start temperature and T Mf  is the martensite finish temperature. The fraction of temperature-induced martensite is subject to the inequality
 
Φ T ≦1−Φ σ   (7)
 
     The shape memory alloy actuator is a wire with a large length to diameter ratio and is uniformly heated from resistive heating. The one-dimensional heat transfer equation for heat flow in the radial direction including exo- and endothermic behaviour is 
     
       
         
           
             
               
                 
                   
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       V 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               C 
                               p 
                             
                             ⁢ 
                             
                               
                                 ⅆ 
                                 T 
                               
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                           - 
                           
                             H 
                             ⁢ 
                             
                               
                                 ⅆ 
                                 Φ 
                               
                               
                                 ⅆ 
                                 t 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         E 
                         2 
                       
                       
                         R 
                         el 
                       
                     
                     - 
                     
                       
                         h 
                         th 
                       
                       ⁡ 
                       
                         ( 
                         
                           T 
                           - 
                           
                             T 
                             a 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     where ρ is density (kg/m 3 ), V is the wire volume (m 3 ), C p  is specific heat (J/kg K) and H is latent heat of formation (J/kg), E is the applied voltage (V), R el  is the wire resistance (ohms), T is the wire temperature (K), T a  is the ambient temperature (K), h th  is the convective thermal conductance (W/K). 
     Equations 1 through 8 are used to optimize the open-loop actuator force-displacement-frequency characteristics based on the wire geometry and electrical power input. For a SMA actuator with the material properties given in Table 1,  FIG. 3  shows a typical result where a ±2 mm displacement at 0.1 Hz performance is sought. ( FIG. 3  shows the open-loop results for an antagonistic SMA actuator subjected to a 2V square wave excitation at 0.1 Hz.) 
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Constants for SMA actuator 
               
             
          
           
               
                 Constant 
                 Symbol 
                 Value 
               
               
                   
               
             
          
           
               
                 austenite modulus (GPa) 
                 E A   
                 45 
               
               
                 austenite Poisson ratio (-) 
                 ν A   
                 0.33 
               
               
                 martensite modulus (GPa) 
                 E M   
                 20 
               
               
                 martensite Poisson ratio (-) 
                 ν m   
                 .33 
               
               
                 austenite finish temperature (K) 
                 T Af   
                 367 
               
               
                 austenite start temperature (K) 
                 T As   
                 358 
               
               
                 martensite start temperature (K) 
                 T Ms   
                 314 
               
               
                 martensite finish temperature (K) 
                 T Mf   
                 300 
               
               
                 critical martensite start stress at T Ms  (MPa) 
                 σ* Ms   
                 42 
               
               
                 critical martensite finish stress at T Ms  (MPa) 
                 σ* Mf   
                 102 
               
               
                 austenite slope (MPa/K) 
                 C A   
                 6.5 
               
               
                 martensite slope above T Ms  (MPa/K) 
                 C M   
                 5.75 
               
               
                 martensite slope below T Ms  (MPa/K) 
                 C* M   
                 — 
               
               
                 thermal conductance (W/K) 
                 h th   
                 850 
               
               
                 electrical resistance (ohm) 
                 R el   
                 15 
               
               
                 density (kg/m 3 ) 
                 ρ 
                 6500 
               
               
                 specific heat (J/kg · K) 
                 C p   
                 350 
               
               
                 latent heat of formation (kJ/kg) 
                 H 
                 20 
               
               
                   
               
             
          
         
       
     
     The volume restrictions in a missile nose preclude the use of complex linkages to transform the horizontal motion of the SMA actuator to the vertical motion required by the flow effector. A compliant link was coupled to the micro-flow effector to transform the SMA force and displacement into an output force and displacement required by the flow effector while maximizing mechanical efficiency. 
     A schematic of the micro-flow effector with compliant link is shown in  FIG. 3A . The flow effector is actuated by applying one of the SMA forces at the T-section. The geometry was optimized to minimize the bending stresses in the compliant link and SMA attachment points while maximizing the tip displacement. The lever arm ratio ‘L2LEN’ to ‘L1LEN’ controlled the amplification of the upward displacement at ‘FEWID’ for a horizontal displacement to the right at ‘THGT’. 
     The micro-flow effector is fabricated from a 787 micron thick Ti-6A14V sheet stock. The tip of the micro-flow effector is required to displace 1 mm while subjected to a maximum aerodynamic load of 0.172 N. No yielding of the material is permitted. The SMA wire described in Table 1 can generate between 150 g and 185 g force at voltages starting at 0.5 VDC. Table 2 lists the dimensions of the optimized micro-flow effector and the expected performance. 
     Closed loop control of the micro-flow effector requires a feedback sensor to monitor the tip displacement magnitude. Two semi-conductor strain gages are mounted at the base of the compliant link in a half-bridge arrangement to measure the bending strains which are proportional to the tip displacement. 
     
       
         
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Optimized micro-flow effector mechanism dimensions 
               
             
          
           
               
                 Parameter 
                 Value 
                 Performance 
                 Value 
               
               
                   
               
             
          
           
               
                 L1LEN (micron) 
                 3607 
                 F SMA  low top only SMA disp 
                 +370 
               
               
                 L2LEN (micron) 
                 21132 
                 (micron) 
               
               
                 L3HGT (micron) 
                 3429 
                 Effector tip disp (micron) 
                 +1820 
               
               
                 L4HGT (micron) 
                 2540 
                 Effector post stress (MPa) 
                 800 
               
               
                 L5HGT (micron) 
                 2286 
                 F SMA  high top plus F Aero   
                 +220 
               
               
                 TOPWID (micron) 
                 254 
                 SMA disp (micron) 
               
               
                 BASWID (micron) 
                 279 
                 Effector tip disp (micron) 
                 +1010 
               
               
                 ARMWID (micron) 
                 762 
                 Effector post stress (MPa) 
                 640 
               
               
                 LEVHGT (micron) 
                 762 
                 F SMA  low bottom only 
                 +21 
               
               
                 FEWID (micron) 
                 2667 
                 SMA disp (micron) 
               
               
                 LEVLEN (micron) 
                 24739 
                 Effector tip disp (micron) 
                 +56 
               
               
                 THGT (micron) 
                 1207 
                 Effector post stress (MPa) 
                 250 
               
               
                 Yield Stress (MPa) 
                 882 
               
               
                 Modulus (MPa) 
                 115 
               
               
                 F SMA  low (g) 
                 150 
               
               
                 F SMA  high (g) 
                 185 
               
               
                 F Aero  (N) 
                 0.172 
               
               
                   
               
             
          
         
       
     
     To control the positioning of the control surface, a digital controller was designed to control the effector position of an SMA actuation mechanism over a range of 1 mm and within a bandwidth of [0 Hz, 1 Hz]. To meet these requirements, a two-step variable structure control law is proposed (see circuit/control diagram in  FIG. 4  and the feedback control diagram in  FIG. 5 ). The control law consists of: 
     A bang-bang control, υ=Vsign(e), which is triggered whenever |e| is greater than a threshold ε. 
     A discrete-time control law which ensures set-point regulation near the equilibrium and prevention of high frequency chattering. 
     The choice of the threshold ε results from a trade-off between a large value to avoid fast switching from one controller to another and a small value to warrant reliable computation of the controller&#39;s state-space variables at the switching time. A small ε indicates that the closed-loop system is close to its steady state. 
     The direct digital design ensures that pole placement of the closed-loop system leads to the computation of coefficients l 0 , l 1 , l 2 , p 0 , p 1 , p 2  in polynomials L(q −1 ) and P(q −1 ). The discrete-time control law, as shown in  FIG. 5 , is comprised of an integrator 1/(1−q −1 ) in series with P(q −1 )/L(q −1 ). The model of the plant in series with the integrator yields q −1 B(q −1 )/Ā(q −1 ) where Ā(q −1 )=(1−q −1 )A(q −1 ) and B(q −1 )=b 1 +b 2 q −1 . Polynomial A(q −1 ) is of degree two. Henceforth, 
                             A   _     ⁡     (     q     -   1       )       =       ⁢         (     1   -     q     -   1         )     ⁢     (     1   +       a   1     ⁢     q     -   1         +       a   2     ⁢     q     -   2           )     ⁢   1     +                     ⁢         (       a   1     -   1     )     ⁢     q     -   1         +       (       a   2     -     a   1       )     ⁢     q     -   2         -       a   2     ⁢     q     -   3                       =       ⁢         a   _     0     +         a   _     1     ⁢     q     -   1         +         a   _     2     ⁢     q     -   2         +         a   _     3     ⁢       q     -   3       .                       (   9   )               
The model of the closed-loop system is written as,
 
     
       
         
           
             
               
                 
                   
                     
                       d 
                       m 
                     
                     
                       d 
                       * 
                     
                   
                   = 
                   
                     
                       
                         
                           q 
                           
                             - 
                             d 
                           
                         
                         ⁢ 
                         PB 
                       
                       
                         
                           L 
                           ⁢ 
                           
                             A 
                             _ 
                           
                         
                         + 
                         
                           
                             q 
                             
                               - 
                               d 
                             
                           
                           ⁢ 
                           BP 
                         
                       
                     
                     = 
                     
                       
                         
                           q 
                           
                             - 
                             d 
                           
                         
                         ⁢ 
                         PB 
                       
                       
                         A 
                         * 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where A*(q −1 ) is a polynomial of degree 5. Let z i =e s     i     T     s   , for i=1, . . . 5, the zeros of A*. Therefore, poles of the closed-loop transfer function d m /d* are equal to z i , i=1 . . . 5, if the coefficients of L(q −1 ) and P(q −1 ) satisfy the following system when d=1 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 a 
                                 _ 
                               
                               0 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 a 
                                 _ 
                               
                               1 
                             
                           
                           
                             
                               
                                 a 
                                 _ 
                               
                               0 
                             
                           
                           
                             0 
                           
                           
                             
                               b 
                               1 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 a 
                                 _ 
                               
                               2 
                             
                           
                           
                             
                               
                                 a 
                                 _ 
                               
                               1 
                             
                           
                           
                             
                               
                                 a 
                                 _ 
                               
                               0 
                             
                           
                           
                             
                               b 
                               2 
                             
                           
                           
                             
                               b 
                               1 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 a 
                                 _ 
                               
                               3 
                             
                           
                           
                             
                               
                                 a 
                                 _ 
                               
                               2 
                             
                           
                           
                             
                               
                                 a 
                                 _ 
                               
                               1 
                             
                           
                           
                             0 
                           
                           
                             
                               b 
                               2 
                             
                           
                           
                             
                               b 
                               1 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 a 
                                 _ 
                               
                               3 
                             
                           
                           
                             
                               
                                 a 
                                 _ 
                               
                               2 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               b 
                               2 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               
                                 a 
                                 _ 
                               
                               3 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               l 
                               0 
                             
                           
                         
                         
                           
                             
                               l 
                               1 
                             
                           
                         
                         
                           
                             
                               l 
                               2 
                             
                           
                         
                         
                           
                             
                               p 
                               0 
                             
                           
                         
                         
                           
                             
                               p 
                               1 
                             
                           
                         
                         
                           
                             
                               p 
                               2 
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             a 
                             0 
                             * 
                           
                         
                       
                       
                         
                           
                             a 
                             1 
                             * 
                           
                         
                       
                       
                         
                           
                             a 
                             2 
                             * 
                           
                         
                       
                       
                         
                           
                             a 
                             3 
                             * 
                           
                         
                       
                       
                         
                           
                             a 
                             4 
                             * 
                           
                         
                       
                       
                         
                           
                             a 
                             5 
                             * 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     where a i *, for i=1, . . . 5, are such that 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             A 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             
                               q 
                               
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             a 
                             0 
                             * 
                           
                           + 
                           
                             
                               a 
                               1 
                               * 
                             
                             ⁢ 
                             
                               q 
                               
                                 - 
                                 1 
                               
                             
                           
                           + 
                           
                             
                               a 
                               2 
                               * 
                             
                             ⁢ 
                             
                               q 
                               
                                 - 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               a 
                               3 
                               * 
                             
                             ⁢ 
                             
                               q 
                               
                                 - 
                                 3 
                               
                             
                           
                           + 
                           
                             
                               a 
                               4 
                               * 
                             
                             ⁢ 
                             
                               q 
                               
                                 - 
                                 4 
                               
                             
                           
                           + 
                           
                             
                               a 
                               5 
                               * 
                             
                             ⁢ 
                             
                               q 
                               
                                 - 
                                 5 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 q 
                                 
                                   - 
                                   1 
                                 
                               
                               - 
                               
                                 z 
                                 1 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 q 
                                 
                                   - 
                                   1 
                                 
                               
                               - 
                               
                                 z 
                                 2 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 q 
                                 
                                   - 
                                   1 
                                 
                               
                               - 
                               
                                 z 
                                 3 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 q 
                                 
                                   - 
                                   1 
                                 
                               
                               - 
                               
                                 z 
                                 4 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   q 
                                   
                                     - 
                                     1 
                                   
                                 
                                 - 
                                 
                                   z 
                                   5 
                                 
                               
                               ) 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     The identified model gives rise to undershoot that is caused by unstable zeros. Locating dominating poles of d m /d* at s 1 =s 2 =−3, while s 3 =s 4 =s 5 =−30, gives satisfactory responses in terms of the rise time and the transients. 
     The two-step variable structure controller was implemented on a LabView platform with a sample period of 10 ms. The voltage applied to the SMA wires was generated by a pulse width modulator (PWM), with a switching frequency of 400 Hz. The output voltage of the PWM for the bang-bang control is approximately 3 V while the discrete-time control voltage is approximately 0.8 V.  FIGS. 6A and 6B  show typical set-point-regulated flow effector position schedules. The actuator under feedback control shows fast responses during the rising part of the motion and zero steady error. Furthermore, there is no chattering because the digital controller replaces the bang-bang law for small tracking errors. The response time is less than 0.3 sec with an overshoot of about 5% of the steady state value. When the heat transfer rate is increased by forced convection, a frequency of 1 Hz can be attained. 
     A person understanding this invention may now conceive of alternative structures and embodiments or variations of the above all of which are intended to fall within the scope of the invention as defined in the claims that follow.