- Email: [email protected]
Hybrid Control of Structures
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y(t+
At) = E J,BAl/ (t+ j-1)+EJ,DA,f (1+ j-l)+
(13)
(7)
F .. / (t) j'
Substituting for the vector y yields.
Redetining variables yields, y(t+
At) = GJ,AI/ F, /
.!'
(t
+ j-l)+ E ,DAJ u+ j-l)+ .I
(8)
(r)
The extreme value if J based OIl control effort can be found by completing the partial derivative.
a.J/a.ii=O where Gj=EjB. Note that u(t+i-I )=0 for i>Nu and v(t+i1)=0 for i>Np. The term GjArI(t+j-l) may be split into two parts. The tirst part consists of the portion known before time t and the second portion is determined by invoking the Diophantine equation.
(15)
Thus, the optimal control effort can be defined.
It can be seen that u is the minimizer of the cost function because the second partial of the cost function with respect to the control variable is greater than zero.
The optimal predictor can then be detined, Thus, Equation 17 is evaluated at each time step and the tirst element Au(t) is used as the control input.
,vCr + jltl = G'AI/ (t + j -1) + rjAI/ (1-1) + (10)
F,/ (I)
E .DAJ (1+ j -1)+ J .I
The formation of the prediction equations for j E [NI, ... N2] requires the solution of two Diophantine equations for the polynomials G'j, r j , Ej and F j • Recursive solutions are offered in the literature by Clarke (1987a and I987b) and Astrom (1995). The set of prediction equations can be written in matrix/vector form.
y:::: Gii +f
The fixed controller design scheme is based on frequency domain techniques using loop shaping methods on the sensitivity Nichols chart. This design methodology utilizes only experimental frequency domain data that is obtained from the plant in different operating conditions. This process will help to quantify plant variations/uncertainties and guarantees a stable closed-loop system at each operating condition. Coherence measurements are also recorded versus frequency and low values of coherence are included in the controller design as additional sources of plant uncertainty,
(11 )
The matrix G is N,-N 2xNu , and the following vectors are defined.
y = [y(t + NI ), ... , y(t + N)21 T ii = [Llu(t) .... ,Llu(t + Nil -l)f
2.2. Fixed Controller Design Scheme
(12)
Using this matrix notation, the performance index can be written in the following vectorized form,
The design process begins with data collection on the real system, Transfer function measurements and coherence measurements are recorded versus frequency and the uncertainty bounds are determined. The power spectrum of the disturbance is also recorded versus frequency to determine what frequencies ;tre critical in the design process. Design goals are then set based on a minimum gain and phase margin that can be directly related to a maximum value of closed loop sensitivity. A controller is then designed using loop shaping methods on the Nichols chart and expected values of closed loop sensitivity versus frequency arc determined. Dettils of this design technique
5555
can be found in open literature (0' Azzo and Houpis, 1988 and Horowitz, 1993).
schemes. Preliminary results show great promise. These results will be the topic of future study.
The tinal step in the controllcr design process is to implement the controller on the real system. This process involves either digital implementation where the NO and D/A conversion is of utmost importance or analog implementation where inductors, capacitors, resistors and OP amps are used within electrical circuitry to replicate the controller transfer function.
2.3. Proposed Hybrid Controller The proposed hybrid control scheme consists of a iixed control loop and a secondary adaptive control loop. The tixed controller is designed using loop shaping methods and is robust with respect to plant variations and measurement uncertainties. This loop is required to handle disturbance rejection and to meet the design specifications when the adaptive controller is not operative. The role of the adaptive controller is to supervise the performance of the tixed controller and to adaptively tilter error signals to increase the performance of the fixed controller. The adaptive controller will also be used to reduce closed loop sensitivity in specified frequency ranges. The standard block diagram for feedback control is shown below as Figure 1 for the regulator problem.
CL-G_C~-~~I Gp Fig. 1. Standard Feedback Control Block Diagram The proposed adaptive control scheme would supervise the input and output of the plant and adaptively changes a predeiined set of controller parameters. A simple case would be one where a fixed controller has been designed and the gain of the controller is set adaptively to ensure high performance under different operating conditions. There are obviously much more complex scenarios where more dynamics could be induded in the adaptive block and would prove more useful when controlling complex systems. A block diagram of the hybrid, gain controlled system is shown as Figure 2. It should be noted that the adaptive portion of the loop could be implemented digitally or using digitally programmahle analog circuitry. Our current research project investigates extensions of the proposed technique extensions to comhined feed forward and feedhack control
Fig. 2. Proposed Hybrid Controller Structure
3. EXPERIMENTAL RESULTS Experimental results are necessary in control applications to validate proposed control schemes. The experimental testing procedure often exposes weakness in the original design philosophy that require~ modification and redesign. Additionally, real constraints sHch as processor speed (e.g. the tradeoff between controller complexity and sample rate), the physical realizability of the controller, integrator windup, aliasing, NO and 01 A conversions, and system modeling hecome a major factor in the design. The controller must he ahle to handle these sources of uncertainty and disturbance. Two independent experimental tests are presented that validate the individual controller design methodologies. One case utilizes a fixed controller designed, using robust loop shaping methods, in an automotive application. The second test is performed on .1 cable-stayed bridge using adaptive control techniques. A brief summary of each experimental setup is given and the controller results are provided.
3.1. Cable-Stayed Bridge Experiment A 1/150 dynamically scaled model of a cable-stayed bridg,~ has been constructed in a lahoratory environment. Th,~ experimental bridge structure has a total length of 3.2 meters with tower height of 0.5 meters. It has a multiple fa:1 cable arrangement with A-li"arne towers and box sectiol girders. The structure was conmucted using machined steel and aluminum parts and high tt'nsile strength piano wire tl) represent the cable stays. An electromagnetic shaker is used to produce disturbance vibrations and a smaller electromagnetic shaker is used as the control actuator. A three dimensional representation of the structure is shown as Figure 3. Details of the experimental structure can be found in related papers hy Bell (1995) and Shoureshi (1994).
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y(t+
At) = E J,BAl/ (t+ j-1)+EJ,DA,f (1+ j-l)+
(13)
(7)
F .. / (t) j'
Substituting for the vector y yields.
Redetining variables yields, y(t+
At) = GJ,AI/ F, /
.!'
(t
+ j-l)+ E ,DAJ u+ j-l)+ .I
(8)
(r)
The extreme value if J based OIl control effort can be found by completing the partial derivative.
a.J/a.ii=O where Gj=EjB. Note that u(t+i-I )=0 for i>Nu and v(t+i1)=0 for i>Np. The term GjArI(t+j-l) may be split into two parts. The tirst part consists of the portion known before time t and the second portion is determined by invoking the Diophantine equation.
(15)
Thus, the optimal control effort can be defined.
It can be seen that u is the minimizer of the cost function because the second partial of the cost function with respect to the control variable is greater than zero.
The optimal predictor can then be detined, Thus, Equation 17 is evaluated at each time step and the tirst element Au(t) is used as the control input.
,vCr + jltl = G'AI/ (t + j -1) + rjAI/ (1-1) + (10)
F,/ (I)
E .DAJ (1+ j -1)+ J .I
The formation of the prediction equations for j E [NI, ... N2] requires the solution of two Diophantine equations for the polynomials G'j, r j , Ej and F j • Recursive solutions are offered in the literature by Clarke (1987a and I987b) and Astrom (1995). The set of prediction equations can be written in matrix/vector form.
y:::: Gii +f
The fixed controller design scheme is based on frequency domain techniques using loop shaping methods on the sensitivity Nichols chart. This design methodology utilizes only experimental frequency domain data that is obtained from the plant in different operating conditions. This process will help to quantify plant variations/uncertainties and guarantees a stable closed-loop system at each operating condition. Coherence measurements are also recorded versus frequency and low values of coherence are included in the controller design as additional sources of plant uncertainty,
(11 )
The matrix G is N,-N 2xNu , and the following vectors are defined.
y = [y(t + NI ), ... , y(t + N)21 T ii = [Llu(t) .... ,Llu(t + Nil -l)f
2.2. Fixed Controller Design Scheme
(12)
Using this matrix notation, the performance index can be written in the following vectorized form,
The design process begins with data collection on the real system, Transfer function measurements and coherence measurements are recorded versus frequency and the uncertainty bounds are determined. The power spectrum of the disturbance is also recorded versus frequency to determine what frequencies ;tre critical in the design process. Design goals are then set based on a minimum gain and phase margin that can be directly related to a maximum value of closed loop sensitivity. A controller is then designed using loop shaping methods on the Nichols chart and expected values of closed loop sensitivity versus frequency arc determined. Dettils of this design technique
5555
can be found in open literature (0' Azzo and Houpis, 1988 and Horowitz, 1993).
schemes. Preliminary results show great promise. These results will be the topic of future study.
The tinal step in the controllcr design process is to implement the controller on the real system. This process involves either digital implementation where the NO and D/A conversion is of utmost importance or analog implementation where inductors, capacitors, resistors and OP amps are used within electrical circuitry to replicate the controller transfer function.
2.3. Proposed Hybrid Controller The proposed hybrid control scheme consists of a iixed control loop and a secondary adaptive control loop. The tixed controller is designed using loop shaping methods and is robust with respect to plant variations and measurement uncertainties. This loop is required to handle disturbance rejection and to meet the design specifications when the adaptive controller is not operative. The role of the adaptive controller is to supervise the performance of the tixed controller and to adaptively tilter error signals to increase the performance of the fixed controller. The adaptive controller will also be used to reduce closed loop sensitivity in specified frequency ranges. The standard block diagram for feedback control is shown below as Figure 1 for the regulator problem.
CL-G_C~-~~I Gp Fig. 1. Standard Feedback Control Block Diagram The proposed adaptive control scheme would supervise the input and output of the plant and adaptively changes a predeiined set of controller parameters. A simple case would be one where a fixed controller has been designed and the gain of the controller is set adaptively to ensure high performance under different operating conditions. There are obviously much more complex scenarios where more dynamics could be induded in the adaptive block and would prove more useful when controlling complex systems. A block diagram of the hybrid, gain controlled system is shown as Figure 2. It should be noted that the adaptive portion of the loop could be implemented digitally or using digitally programmahle analog circuitry. Our current research project investigates extensions of the proposed technique extensions to comhined feed forward and feedhack control
Fig. 2. Proposed Hybrid Controller Structure
3. EXPERIMENTAL RESULTS Experimental results are necessary in control applications to validate proposed control schemes. The experimental testing procedure often exposes weakness in the original design philosophy that require~ modification and redesign. Additionally, real constraints sHch as processor speed (e.g. the tradeoff between controller complexity and sample rate), the physical realizability of the controller, integrator windup, aliasing, NO and 01 A conversions, and system modeling hecome a major factor in the design. The controller must he ahle to handle these sources of uncertainty and disturbance. Two independent experimental tests are presented that validate the individual controller design methodologies. One case utilizes a fixed controller designed, using robust loop shaping methods, in an automotive application. The second test is performed on .1 cable-stayed bridge using adaptive control techniques. A brief summary of each experimental setup is given and the controller results are provided.
3.1. Cable-Stayed Bridge Experiment A 1/150 dynamically scaled model of a cable-stayed bridg,~ has been constructed in a lahoratory environment. Th,~ experimental bridge structure has a total length of 3.2 meters with tower height of 0.5 meters. It has a multiple fa:1 cable arrangement with A-li"arne towers and box sectiol girders. The structure was conmucted using machined steel and aluminum parts and high tt'nsile strength piano wire tl) represent the cable stays. An electromagnetic shaker is used to produce disturbance vibrations and a smaller electromagnetic shaker is used as the control actuator. A three dimensional representation of the structure is shown as Figure 3. Details of the experimental structure can be found in related papers hy Bell (1995) and Shoureshi (1994).
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