Economic sensor/actuator selection and its application to flexible structure control

Transcription

1 To appear in Proc. SPIE Int. Soc. Opt. Eng Economic sensor/actuator selection and its application to flexible structure control Robert E. Skelton a and Faming Li a a Department of Mechanical and Aerospace Engineering, University of California, San Diego 9500 Gilman Drive, La Jolla, California, USA 1. ABSTRACT A systematic design method is proposed for the selecting of actuators and sensors in the structural control in order to minimize the instrumental cost. With actuators and sensors placed at all the admissible locations initially, an iterative minimization algorithm is carried out to identify the sensor/actuator that requires the least precision. By deleting the roughest sensor/actuator each time till loss of feasibility, one can conclude simultaneously the necessary number and type of sensor/actuator, and the location and precision for each sensor/actuator. A tensegrity structure example has been solved as an application of the proposed algorithm. Keywords: Sensor/actuator selection(sas), maximal accuracy, economic design, tensegrity structure 2. THE VECTOR SECOND-ORDER SYSTEM MODEL We consider a flexible structure with n degrees of freedoms which is defined as M q + D q + Kq = Bu (1) In the above equation, q = [q 1, q 2,... q n ] T is the displacement vector, u = [u 1,... u nu ] T is the input vector. The matrices M and K are n n mass and stiffness matrices. The state space representation of the control system is: ẋ = Ax + B(u + w) + De y = Cx z = Mx + v (2a) (2b) (2c) where y is the output that we are interested; z is measurement. A IR 2n 2n. Matrices B and M are 2n n u and n y 2n matrices. e is the external perturbation such as wind. It is assumed e is white noise with covariance E. w and v are actuator noise and sensor noise. Conventionally w and v are modelled as white noise with covariance W, V respectively. [ ] w E = 0, E v [ w(t) v(t) ] [ w(τ) v(τ) ] T = [ W 0 0 V ] δ(t τ) Assume all these white noises are independent, then W and V are diagonal matrices, with the ith diagonal entry equivalent to the inequality constrained problem stated as corresponding to the ith component precision. 3. MOTIVATION OF THE SAS ALGORITHM The technique that we shall employ to determine the location, type and precision of each sensor/actuator is called economic design 1. The economic design problem minimizes the total required precision, while satisfying the system performance constraints. It is reasonable to assume that the price of a component is proportional to its precision 2. Hence the instrumental cost can be expressed in term of the noise covariance matrix W and V. bobskelton@ucsd.edu, faming@ucsd.edu.

2 [ W 1 0 Define Γ = 0 V 1 vector indicating the relative cost. ]. In the sequel we shall use p T Γp as the instrumental cost, where p is a coefficient Given the total cost and output variance requirement, it is desired to find the most economic strategy to select sensor/actuator 1. In other words, seek the selection of sensor/actuator such that the control energy is minimized. This problem can be stated as min U = W,V,G EuT u (3) subject to [Y] i,i σ i, $ $. i = 1, 2... n y where U denotes the control energy. Y is the closed-loop output covariance. $ = p T Γp stands for the instrument cost. $ is a given upper bound of the cost. G is an output feedback controller. (3) is a typical optimization problem in the form of given performance and cost A, minimize cost B. Its permuted counterparts given performance and cost B, minimize cost A or given cost A B, seek the best performance can be stated in a similar fashion. In the sequel, we shall call (3) the economic OVC problem. The output variance constrained (OVC) problem 3 is to design a dynamic output feedback controller such that the control energy is minimized while each output variance constrain is satisfied. (3) reduces to an OVC problem if sensor/actuator are fixed, which can be solved iteratively by using the weight selection algorithm. But when W and V are involved to be optimized, this presents a difficult problem(3). This paper intends to shed some light on this problem. In the output feedback control design, the actuator and measurement noises will deteriorate the output performance no matter how large the control effort is. For instance, in a regulator problem, there exists steady state output error due to the actuator and measurement noises. Maximal accuracy characterizes the best performance one can attain with the existence of actuator and measurement noises 3. The rougher the sensor/actuator, the worse the performance. Hence the maximal accuracy of an output feedback system can be related to the instrument cost. Algorithm 1 Given the total instrument cost $ and the desired output variance upper bound σ i, solve the twostep optimization problem: (i) Denote Y as the maximal accuracy of the system. Solve the maximal accuracy minimization problem with fixed instrument cost $. min tr[y ] (4) P,W,V subject to p T Γp $ (5) CP C T = Y (6) AP + P A T P M T V 1 MP + BW B T + DED T = 0 (7) (ii) Fix W and V generated from (i), solve the OVC problem subject to min G U = EuT u (8) [Y] i,i σ i, i = 1, 2... n y Algorithm 1 is a sequential optimization problem. Either (i) or (ii) is tractable with existing methods. Unlike minimizing the maximal accuracy, sometimes an economic sensor/actuator selection is desirable, which can be stated as follows:

3 Algorithm 2 Given the total maximal accuracy requirement Y and the output variance upper bound σ i, (i) Minimize the instrument cost $. min P,W,V $ = pt Γp (9) subject to (6) and (7). (ii) Fix W, V and $ resulted from (i), solve the OVC problem (8). Algorithm 2 minimizes the cost given performance requirement. The economic OVC problem (3) is proposed to be approximated by algorithm THE SAS ALGORITHM In sensor/actuator selection, one has to decide where to place sensor/actuator, how many sensor/actuator are required, what is the precision for each sensor/actuator. In this section, a systematic design method is proposed for the selection of actuators and sensors in the structural control aiming to minimize the instrument cost. With actuators and sensors located at all the possible locations initially, the minimization algorithm (9) is carried out to identify the sensor/actuator that requires the least precision. By deleting the roughest sensor/actuator each time till loss of feasibility, one can conclude simultaneously how many sensor/actuator are necessary, and the location and precision for each sensor/actuator. The equation constrained optimization (9) is equivalent to the inequality constrained problem stated as follows: min $ = P,W,V pt Γp (10) where Y is the specified upper bound of the maximal accuracy. We shall let Y = αi for simplicity. That is, we assume the price ratio between each actuator and sensor is µ. (10) can be transferred to an LMI optimization by using Schur complements and change of variables 4. Denote W = W 1, V = V 1, Ē = E 1, P = P 1. Then (10) can be rewritten as min P, W, V $ = p T Γp (11) subject to [ ] Y C C T 0, P > 0, V > 0 P A T P + P A M T V M P B P D B T P W 0 < 0 D T P 0 Ē Now all the constraints in (11) are linear with respect to W, V and P. Hence (11) can be solve by any linear matrix inequalities (LMI) solvers. List the diagonal of W and V given by the minimization in descent sequence, one can see that the last entry requires the least precision. We now know where to spend money, making the components corresponding to large entries reliable, because performance is critical to these components. The components corresponding to the least entries might be taken off the shelf, or even deleted. In our algorithm, we shall delete the least demanding sensor/actuator each time and repeat the minimization till loss of feasibility. And for each selection of sensor/actuator, an output variance constrained (OVC) control design is carried out. Compare the sensor/actuator cost and control energy for each selection of sensor/actuator, we choice the selection which requires the least total cost. This selection gives the economic solution while satisfies the performance requirements. In conclusion, we state the SAS algorithm as follows: Algorithm 3: The SAS algorithm Step 1. Place sensor/actuator at all admissible locations. Set i = Na + Ns, where Na and Ns stand for the number of actuators and sensors respectively. Fix maximal accuracy Y and the output variance upper bound Y.

4 Step 2. Run the cost minimization algorithm (11) with initial point P j, j = 1, 2, 2n + 1. Yield [W j i, V j i, $ j i]. If (11) is infeasible, go to step 4. Otherwise, run the OVC control design algorithm (8) for each set [W j i, V j i, $ j i]. Yield U j i, j = 1, 2, 2n + 1. Choose the minimal U j i, denoted as U i, and the corresponding [W j i, V j i, $ j i], denoted as [W i, V i, $ i ]. Step 3. If there is no solution to the OVC problem, either loose the performance constraint or go to step 4. Otherwise, delete the actuator that requires the least precision. If U i < U i 1, set i = i 1, go to step 2, Otherwise, go to step 4. Step 4. Delete sensor with smallest precision. If Y < Ȳ, go to step 2. Otherwise, stop. Put back the last deleted sensor or actuator, then go to step 5. Step 5. Consider both the instrument cost and control energy cost, choose the most economic scheme as the final selection of sensor/actuator. 5. THE TENSEGRITY STRUCTURE EXAMPLE The tensegrity structure is under development at the structural systems and control laboratory at UCSD. 5 Figure 1 shows a deployable tensegrity boom. The bottom nodes P 1, P 3 and P 5 are fixed. There are 6 free Figure 1. A tensegrity deployable boom nodes (middle nodes: P2,P4,P6; top nodes: P7,P8,P9), 6 bars and 21 strings (including the knuckle strings). The structure has 18 degrees of freedom (DOF). The control objective is to keep the top and bottom surface parallel. The potential locations for actuator/sensor are all the strings, i.e., we shall collocate 21 sensors and 21 actuators initially. A finite element linearized model of this structure is given in the form of (2). In the state

5 representation, A IR ; When actuators/sensors are located on each string, B IR 36 21, M = B T. To keep the top and bottom surfaces parallel is equivalent to keep the following output zero. [ ] P 7z P 8 y = z (12) P 8 z P 9 z where P n z, n = 7, 8, 9 stands for the z coordinate of node P n with respect to the origin of the universal coordinate system, which is located at the center of the bottom surface. The nodes P 7, P 8 and P 9 determine the top surface. The top and bottom surfaces are parallel if and only if y = 0. The strings are numbered from 1 to 21 as shown in figure 1. 1, 2, 3 are the top surface strings; 4, 5, 6 are the middle surface strings. 7, 8, 9 are diagonal strings of the 1st stage; 10, 11, 12 are the diagonal strings of the 2nd stage. 13, 14, 15 are the knuckle strings of the 1st stage. 16, 17, 18 are the knuckle strings of the 2nd stage. 19, 20, 21 are the reach strings. Similarly, the numbering of the sensor/actuator coincides with the numbering of the strings they are attached. Choose α = 0.001, that is, specify the upper bound of the maximal accuracy Y = 0.001I; Let µ = 1, namely, the price ratio between actuators and sensors is 1. Apply algorithm 3, keep deleting the least important sensor/actuator till 4 sensor/actuator left, the total sensor/actuator cost vs. the number of sensor/actuator is shown as in Figure Number of Actuatos/Sensors vs. Cost,alpha = 0.001; mu= cost ($) Number of actuators and sensors Figure 2. When α = 0.001, µ = 1 The deleting sequence is shown in table 1. It can be seen that when µ = 1, the sensors are more prone to be deleted. And among the 21 sensors, those located at the top surface are of the least importance, followed by those located on the knuckle strings in the 1st stage, then those on the strings in the middle surface. Among the actuators, those located at top surface are least relevant to the performance, then those at the middle surface. The sensors/actuators on the reach strings require significantly greater precision than the rest sensors/actuators, thus are most sensitive to the output performance. Table 1. When α = 0.001, µ = 1 Deleting sequence (1-21) Sen. # Act. # Deleting sequence (22-42) Sen. # Act. #

6 24 Number of Actuatos/Sensors vs. Cost; alpha=0.001, mu= cost ($) Number of actuators and sensors Figure 3. When α = 0.001, µ = 100 Choose α = 0.001; µ = 100, namely, the price ratio between actuators and sensors is 100. Apply algorithm 1, the total sensor/actuator cost vs. the number of sensor/actuator is shown as in Figure 3. In table 2, as µ = 100, the actuators located at the top and middle surfaces turn out to be costly but marginally relevant to the performance. And it can be seen that the existence of actuators have effects on the sensitivity of the sensors. In this case, the sensors on the knuckle strings and the middle surface, rather than the top surface, are among those that require the least precision. Again the sensor/actuator on the reach strings are most critical to the output performance. Table 2. When α = 0.001, µ = 100 Deleting sequence (1-21) Sen. # Act. # Deleting sequence (22-42) Sen. # Act. # Choose α = 0.001, µ = Apply algorithm 1, the total sensor/actuator cost vs. the number of sensor/actuator is shown as in Figure 4. When µ = 1000, the unit price of the actuator is assumed to be much higher than that of the sensor. To minimize the cost, most of the actuators are thrown away before sensors to be deleted. However, at least 2 actuators that on the reach strings stand firm till the end. See table 3. Table 3. When α = 0.001, µ = 1000 Deleting sequence (1-21) Sen. # Act. # Deleting sequence (22-42) Sen. # Act. # 21 Next we illustrate the effect of the performance constraint on the instrumental cost. It is observed from the previous simulations that the economical selection of sensor/actuator is with sensors and actuators only on the reach strings, that is, there are 3 sensors and 3 actuators collocated on the reach strings. We shall study the

7 80 Number of Actuatos/Sensors vs. Cost ; alpha = 0.001; mu= cost ($) Number of actuators and sensors Figure 4. When α = 0.001, µ = 1000 relationship between the cost and performance based on this configuration. Fix mu = 50, when α changes from 10 5 to 100, the cost changes as in Figure 5. It can be seen that when the performance constraint is strict, the financial cost on sensor/actuator increases accordingly Cost vs. Performance Cost: tr(v i nv)+ mu tr(w i nv) Performance: alpha is from 1e 5 to 100 Figure 5. When α changes from 10 5 to CONCLUSION This paper presents a novel method on sensor/actuator selection. The objective is to minimize the instrument cost while preserve control performance. As maximal accuracy depends on the sensor/actuator precisions, and is relevant to the control performance, it is employed in this sensor/actuator selection for control design. A tensegrity structure example is studied to show the effectiveness of this method. REFERENCES 1. R. E. Skelton, System design: The grand challenge of system theory, in plenary lecture of the American Control Conference, 1999.

8 2. J.. Lu and R. E. Skelton, Integrating instrumentation and control design, International journal of control 72(9), pp , R. E. Skelton, Dynamic systems control: linear systems analysis and synthesis, John Wiley & Sons, New York, S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakerishnan, Linear matrix inequalities in system and control theory, SIAM, PA, D. M. R. E. Skelton, JP Pinaud, Dynamics of the shell class of tensegrity structures, Journal of the franklin institute-engineering and applied mathematics (338), pp , 2001.