## A D Truss

The truss structure in Fig. 1.2 is a more complex example of a structure, which can still easily be simulated by the reader, if necessary. For this structure, 1 15 cm, 2 20 cm are dimensions of truss components. Each truss has a cross-sectional area of 1 cm2, elastic modulus of 2.0x107 N cm2, and mass density of 0.00786 kg cm3. This structure has 32 states (or 16 degrees of freedom). Its stiffness and mass matrices are given in Appendix C.1.

## A H Norm of a Mode

function This function finds an approximate Hm norm for each mode of a structure with displacement, rate, and om - vector of natural frequencies z - vector of modal damping bm - modal matrix of actuator location cmq - modal matrix of displacement sensor location cmr - modal matrix of rate sensor location cma - modal matrix of accelerometer location function norm norm_Hinf (om, z, bm, cmq, cmr, cma)

## A HBalanced Representation

function This function finds the H_inf-balanced representation (Ab,Bbl,Bb2,Cbl,Cb2) so that HCARE (Sc) and HFARE (Se) solutions are equal and diagonal Input parameters (A,B1,B2,C1,C2) Output parameters (Ab,Bbl,Bb2,Cbl,Cb2) function Ab, Bbl, Bb2, Cbl, Cb2, Mu_inf, R bal_H_inf (A, Bl, B2, CI, C2, ro) nl,n2 size(A) Qc Cl'*Cl gi l (ro*ro) Rc B2*B2'-gi*Bl*Bl' HCARE solution Qe Bl*Bl' HFARE solution if(norm(imag(Se))> le-6 norm(imag(Sc))> le-6) Uc,Ssc,Vc svd(Sc) Pc sqrt(Ssc)*Vc' Pc Ue,Sse,Ve...

## Low Authority Controller

In the following we distinguish between the low- and high-authority controllers. This distinction allows us to design controllers that significantly suppress the flexible vibrations of structures (which is done by the low-authority controller), and to follow a command precisely (which is done by the high-authority controller). The control forces that act on a structure can be divided into tracking forces and damping forces. The tracking forces move the structure to follow a target and the...

## Open Loop Balanced Representation

function Ab,Bb,Cb,Gamma,R balan2(A,B,C) This function finds the open-loop balanced representation (Ab,Bb,Cb) so that controllability (Wc) and observability (Wo) grammians are equal and diagonal (A,B,C) - system state-space representation (Ab,Bb,Cb) - balanced representation R - transformation to the balanced representation Gamma - Hankel singular values function Ab, Bb, Cb, Gamma, R balan2(A, B, C) controllability grammian observability grammian SVD of the controllability grammian SVD of the...

## Transformation from an Arbitrary State Space Representation to the Modal State Space Representation

The modall state-space representation is obtained by the initial transformation of an arbitrary representation (A,B,C) to the modal representation (An, Bn, Cn) the system matrix An is complex and diagonal. Its diagonal entries are the eigenvalues of A, This transformation, denoted V, is obtained using the Matlab command eig(A). In the next step the representation (An, Bn, Cn) is turned into the modal form 1 (Am, Bm, Cm) by applying the following transformation The system matrix Am in the...

## Transformation from Modal Parameters to the Modal State Space Representation

If coord 1 this function determines the modal state-space representation in form 2, as in (2.53), or if coord 0 this determines the state-space representation in modal coordinates in form 1, as in (2.52). The input data include natural frequencies, modal damping, modal mass, a modal matrix, an input matrix, and displacement and rate output matrices. function am, bm, cm modallm(om, z, mm, phi, b, cq, cv, coord) the determination of modal form 1 (am,bm,cm) om - vector of natural frequencies (nxl)...

## Transformation from Nodal Parameters to the Modal State Space Representation

If coord 1 this function determines the modal state-space representation in form 2, as in (2.53), or if coord 0 this determines the state-space representation in modal coordinates in form 1, as in (2.52). The input data include mass, stiffness, and damping matrices, an input matrix, and displacement and rate output matrices. function am, bm, cm modalln(m, damp, k, b, cq, cv, n, coord) the determination of modal form 1 (am,bm,cm) from nodal data n - number of modes nd - number of degrees of...

## Actuator and Sensor Placement

how to set up a test procedure and controCstrategy Experimentalists think that it is a mathematical theorem while the mathematicians believe it to be an experimental fact. A typical actuator and sensor location problem for structural dynamics testing can be described as a structural test plan. The plan is based on the available information on the structure itself, on disturbances acting on the structure, and on the required structural performance. The preliminary information on structural...

## Actuator Placement Strategy

Place sensors at all accessible degrees of freedom. 2. Based on engineering experience, technical requirements, and physical constraints select possible actuator locations. In this way, S candidate actuator locations are selected. 3. For each mode (k) and each selected actuator location (i), determine the actuator placement index < rk (i). 4. For each mode select the s1 most important actuator locations (those with the largest < rk (i)). The resulting number of actuators s2 for all the...

## Approximate Solutions of CARE and FARE

The design of the LQG controller seems to be a straightforward task since it goes as follows for given weights Q and V we obtain the gains Kc and Ke from (11.7) and (11.9), and the controller representation from (11.4). However, from the implementation point of view, this approach is not appropriately defined, since the design process typically starts from the definition of the required closed-loop system performance, such as the norm of the tracking error, or the location of the closed-loop...

## Approximate Solutions of HCARE and HFARE

We obtain the approximate solutions of HCARE and HFARE in closed-form, in order to understand the relationship between structural parameters and the closed-loop system performance. For flexible structures in modal coordinates, we use Properties 12.3 and 11.1 to obtain the Riccati equations (12.5), (12.6) in the following form Keiseoii + ser4 wc1i - 0, i 1, ,n, The solutions of the ith equation are Pa 41 + 4Wo1i ci 41 + 4 21i - VVu, Pei 41 + 4Wc1i ei 41 + 4 122i 4P hi , and YjU is the ith Hankel...

## Approximately Balanced Structure in Modal Coordinates

Second-order modal models are not unique, since they are obtained using natural modes that are arbitrarily scaled. Hence we have a freedom to choose the scaling factor. By a proper choice of the scaling factors we introduce a model that is almost balanced, i.e., its controllability and observability grammians are approximately equal and diagonally dominant. The second-order almost-balanced model is obtained by scaling the modal displacement (qm) as follows qm Rqab, (4.61) and qab is the...

## State Space Representation of the Deep Space Network Antenna

The state-space representation (a, b, c) of the Deep Space Network antenna in azimuth axis motion was obtained from the field test data. The following are the state matrices after reduction to 18 states. The state matrix a is in the block-diagonal form -0.645922 12.561336 -12.561336 -0.645922 -0.934874 18.937362 -18.937362 -0.934874 -0.842839 36.140547 -36.140547 -0.842839 -3.569534 48.508185 -48.508185 -3.569534 1.004771 -0.206772 -0.093144 0.048098 0.051888 1.292428 -0.024689 0.245969...

## Collocated Controller Design Examples

Two examples of modal collocated controller design are presented the controller design for the simple flexible system, and for the 2D truss structure. The Matlab code for this example is in Appendix B. The system is shown in Fig. 1.1, with masses m1 m2 m3 1, stiffness k1 10, k2 k4 3, k3 4, and the damping matrix D as a linear combination of the mass and stiffness matrices, D 0.004 + 0.001M. The input force is applied to mass m3 and the output is the rate of the same mass. The poles of the...

## Contents

Series Preface vii Preface ix List of Symbols xix 1 Introduction to Structures 1 1.1 Examples 1 1.1.1 A Simple Structure 1 1.1.2 A 2D Truss 2 1.1.3 A 3D Truss 2 1.1.4 A Beam 3 1.1.5 The Deep Space Network Antenna 3 1.1.6 The International Space Station Structure 6 1.2 Definition 6 1.3 Properties 7 2 Standard Models 13 2.1 Models of a Linear System 14 2.1.1 State-Space Representation 14 2.1.2 Transfer Function 15 2.2 Second-Order Structural Models 16 2.2.1 Nodal Models 16 2.2.2 Modal Models 17...

## Continuous Time Systems

Controllability, as a measure of interaction between the input and the states, involves the system matrix A and the input matrix B. A linear system, or the pair (A, B), is controllable at to if it is possible to find a piecewise continuous input u(t), t e to, t1 , that will transfer the system from the initial state, x(to), to the origin x(t1) 0, at finite time t1 > to. If this is true for all initial moments to and all initial states x(to) the system is completely controllable. Otherwise,...

## Controllability and Observability of a Second Order Modal Model

In this section we present the controllability and observability properties of a structure given by the second-order model. The grammians and the balanced models are defined exclusively in the state-space representation, and they do not exist in the second-order form. This is a certain disadvantage since the second-order structural equations are popular forms of structural modeling. We will show, however, that for flexible structures one can find a second-order model which is almost balanced,...

## Controller Design Examples

Here we present examples of the design of a modal LQG controller for a simple structure, for the 3D truss structure, and for the Deep Space Network antenna. The Matlab code for this example is in Appendix B. Design the LQG controller for the system shown in Fig. 1.1. The system masses are m1 m2 m3 1, stiffness k1 10, k2 3, k3 4, and k4 3, and a damping matrix D 0.004K + 0.001M, where K, M are the stiffness and mass matrices, respectively. The input force is applied to mass m3 , the output is...

## Controller Design Procedure

The following steps help to design an LQG controller 1. Put the structural model into modal coordinates 1 or 2. 2. Define the performance criteria, such as bandwidth, settling time, overshoot, etc. 3. Assign initial values of weighting matrices Q and V (remember these matrices are diagonal). 4. Solve the Riccati equations (11.8) and (11.10), find controller gains from (11.7) and (11.9), and simulate the closed-loop performance. Check if the performance satisfies the performance criteria. If...

## Definition and Gains

A block diagram of an LQG control system is shown in Fig. 11.1. It consists of a stable plant or structure (G) and controller (K). The plant output y is measured and supplied to the controller. Using the output y the controller determines the control signal u that drives the plant. The inside structure of the plant and controller is shown in Fig. 11.2. The plant is described by the following state-space equations as shown in Fig. 11.2. In the above description the plant state vector is denoted...

## Definition and Properties

The controllability and observability properties of a linear time-invariant system can be heuristically described as follows. The system dynamics described by the state variable (x) is excited by the input (u) and measured by the output (y). However, the input may not be able to excite all states (or, equivalently, to move them in an arbitrary direction). In this case we cannot fully control the system. Also, not all states may be represented at the output (or, equivalently, the system states...

## Determining Markov Parameters

From measurements one obtains the input and output time histories, rather than the Markov parameters themselves (the exceptions are impulse response measurements). However, the above presented algorithm identifies the state-space representation from the Hankel matrices, which are composed of Markov parameters. Therefore, in this section we describe how to obtain the Markov parameters from the input and output measurements. In order to do this, denote the Markov matrix H that contains p + 1...

## Discrete Time Grammians in Limited Time and Frequency Range

The above time- and frequency-limited grammians were determined for the continuous time and frequencies. If the time or frequency range is discrete, the grammians are determined differently. Let the discrete-time state-space representation be (A, B, C), and let the sampling time be At. We obtain from (4.11) the discrete-time controllability grammian Wc (k) over the time interval 0, kAt , where Ck is the controllability matrix Similarly we find the discrete-time observability grammian Wo (k) for...

## Discrete Time Systems

Consider now a discrete-time system as given by (3.46). For the sampling time At the controllability matrix Ck is defined similarly to the continuous-time systems, as follows Ck B AB Ak B . (4.9) The controllability grammian Wc (k) over the time interval 0, kAt is defined as Wc (k) AiBBT (Ai )T. (4.10) Unlike the continuous-time systems we can use the controllability matrix of the discrete-time system to obtain the discrete-time controllability grammian Wc (k). Namely, The stationary grammian...

## Examples

In this section we perform the identification of the models of a simple structure (in order to illustrate the method in a straightforward manner) the 2D truss a more complicated structure, and the Deep Space Network antenna where the model is identified from the available field data. The Matlab code for this example is in Appendix B. Analyze a simple system with k1 10, k2 50, 3 50, k4 10, m1 m2 m3 1, and with proportional damping matrix, D 0.005K + 0.1M. The input is applied to the third mass...

## FfliGrt i n

Wherea> i is the ith natural frequency and . denotes either H2, Hm , or Hankel norms. The above equations show that the norm of the ith mode with an accelerometer sensor is obtained as a product of the norm of the ith mode with a rate sensor and the ith natural frequency. Example 6.11. Consider the truss from the previous example. The longitudinal input force is applied to node 21 and the longitudinal acceleration is measured at node 14. Determine the Hm norms of the modes for the structure...

## Frequency Weighting

The LQG controller can be designed to meet tracking requirements and, at the same time, maintain the disturbance rejection properties. In order to achieve this, the problem should be appropriately defined in quantitative terms. For this purpose we use the frequency shaping filters to define tracking requirements, or disturbance rejection performance of the closed-loop system. Although these filters are used only in the controller design stage, they add to the complexity of the problem. This is...

## Frequency Limited Grammians

In this section we interpret the controllability and observability grammians in frequency domain. In order to do this, note that from the Parseval theorem the time integrals (4.3), for the time span (0, < ), can be substituted with the following integrals in the frequency domain is the Fourier transform of eAt, and H* is a complex-conjugate transpose of H. The above grammians are defined over the entire frequency range. The frequency band (-< , < ) can be narrowed to (- , a> ) where <...

## Gsi i n

Where .11 denotes either H2, Hw, or Hankel norms, where The variable ksi is the ith modal stiffness of the structure. Proof. The force fo acting on the structure is related to the actuator force f as in (6.23). Hence, replacing fo with f in the structural model gives (6.26). S In addition to conditions (6.25), consider the following ones a)o fflj and k minksi, (6.30) where m1 is the fundamental (lowest) frequency of the structure. These conditions say that the actuator natural frequency should...

## Il ik lit o II i I It

2 i 11 2 Introducing (5.47) to the above equation. we obtain (5.50). S Equations (5.49) and (5.50) show that the H norm of a mode with a set of actuators (sensors) is the rms sum of the Hm norms of this mode with a single actuator (sensor). This is illustrated in Fig. 5.9(a),(b). Note, however, that unlike the H2 norm, this property does not hold for the whole structure. Instead, the maximum norm rule is applied see (5.26) and Fig. 5.10. Since the Hankel norm is approximately one-half of the Hm...

## List of Symbols

Each equation in the book would halve the sales. Stephen Hawking A complex-conjugate transpose of matrix A A-1 inverse of square nonsingular matrix A tr(A) trace of a matrix A, tr(A) . aii A Euclidean (Frobenius) norm of a real-valued matrix A diag(a -) diagonal matrix with elements ai along the diagonal eig(A) eigenvalue of a square matrix A Ai (A) ith eigenvalue of a square matrix A Anax (A) maximal eigenvalue of a square matrix A ai (A) ith singular value of a matrix A max (A) maximal...

## Modal Actuators and Sensors Through Grammian Adjustment

In the method presented above the modal actuator does not depend on the sensors' location. But, the sensors can be located such that the actuated mode can be unobservable. We can notice a similar situation in the modal sensor procedure it is independent on the actuator location. But a sensed mode can be uncontrollable. The method proposed below allows us to avoid that undesirable situation. It was shown in Chapter 4 that in modal coordinates the controllability and observability grammians are...

## Modal Actuators and Sensors Through Modal Transformations

In the above equation R+ is a pseudoinverse of R, R+ VI, 1UT, where U, 2, and V are obtained from the singular value decomposition of R, i.e., from R U I.VT. Note that a structure with a modal actuator excites nm modes only (other modes are uncontrollable) therefore, the implementing modal actuator is equivalent to model reduction, where the structure has been reduced to nm modes, or to 2nm states. The input matrix Bo in (8.2) that defines the modal actuator can be determined alternatively from...

## Modal Models

The second-order models are defined in modal coordinates. These coordinates are often used in the dynamics analysis of complex structures modeled by the finite elements to reduce the order of a system. It is also used in the system identification procedures, where modal representation is a natural outcome of the test. Modal models of structures are the models expressed in modal coordinates. Since these coordinates are independent, it leads to a series of useful properties that simplify the...

## Model with Inertial Actuators

In the inertial actuator, force is proportional to the square of the excitation frequency. It consists of mass m and a spring with stiffness k, and they are attached to a structure at node, say, na. The force acts on mass m exclusively (Fig. 3.5 at position nb). It is assumed that the stiffness of the actuator is much smaller than the dynamic stiffness of the structure. This configuration is shown in Fig. 3.5, position nb. The force acting on mass m is proportional to the squared frequency...

## Model with Proof Mass Actuators

Proof-mass actuators are widely used in structural dynamics testing. In many cases, however, the actuator dynamics are not included in the model. The proof-mass actuator consists of mass m and a spring with stiffness k, and they are attached to a structure at node na. This is a reaction-type force actuator, see 144 , 57 . It generates a force by reacting against the mass m, thus force f acts on the structure, and -f acts on the mass m (Fig. 3.5 at position na). Typically, the stiffness of the...

## Models in Modal Coordinates

Frequently the order of the nodal representation is unacceptably high. For example, it is not uncommon that the number of degrees of freedom of the finite-element model exceeds 1000. Therefore, the nodal state representation is rarely used in structural dynamics. An alternative approach is to obtain the state-space representation using the modal coordinates and the second-order modal form (2.19), where the number of equations is significantly lower, while the accuracy of the analysis has not...

## Models with Rigid Body Modes

Many structures are free or unrestrained they are not attached to a base. An example is the Deep Space Network antenna structure shown in Fig. 1.5 if uncontrolled, it can rotate freely with respect to the azimuth (vertical) axis and its dish can freely rotate with respect to the elevation (horizontal) axis. Modal analysis for such structures shows that they have zero natural frequency, and that the corresponding natural mode shows structural displacements without flexible deformations. A mode...

## Models with Small Nonproportional Damping

The damping properties of structures are often assumed in the modal form, i.e., they are introduced as damping coefficients in the modal equations (2.19) or (2.26). This is done not only for the sake of analytical simplicity, but also because it is the most convenient way to measure or estimate it. This is the way, for example, to estimate the material damping in the finite-element analysis of large flexible structures, where the modal analysis is executed, the low-frequency modes retained, and...

## Nodal Models

In order to obtain a state representation from the nodal model as in (2.7) we rewrite the latter equation as follows (assuming that the mass matrix is nonsingular) q + M xDq + M Kq M lBou, j C q + Cvq. A state is a vector that contains the minimal number of physical variables that enable us to calculate uniquely the output using the applied input. For a structure, nodal displacements and velocities allow for such determination of the outputs. Therefore, we define the state vector x as a...

## Norm Comparison

Comparing (5.21), (5.22), and (5.23) we obtain the approximate relationships between H2, Hm, and Hankel norms The above relationship is illustrated in Fig. 5.4, using (5.21), (5.22), and (5.23), assuming the same actuator and sensor locations. Figure 5.4. Modal norms versus a . Figure 5.4. Modal norms versus a . Example 5.3. The Matlab code for this example is in Appendix B. Consider a truss presented in Fig. 1.2. Vertical control forces are applied at nodes 9 and 10, and the output rates are...

## Norms of a Generalized Structure

Consider a structure as in Fig. 3.10, with inputs w and u and outputs z andy. Let Gwz be the transfer matrix from w to z, let Gwy be the transfer matrix from w to y, let Guz be the transfer matrix from u to z, and let Guy be the transfer matrix from u to y. Let Gwzi, Guyi, Gwyi, and Guzi be the transfer functions of the ith mode. The following multiplicative properties of modal norms hold Property 5.18. Modal Norms of a General Plant. The following norm relationships hold Gwzi Guyi Gwy Guz ,...

## Norms of a Single Mode

From the definition of the H2 norm and (4.45) we obtain We determine the norm of the second-order modal representation (mt,bt,ct) by replacing Bmi, Cmi with bt, ct, respectively. Note also that G( 2 is the modal cost of Skelton 124 , Skelton and Hughes 126 . The Matlab function norm_H2.m in Appendix A.9 can be used to compute modal H2 norms. Figure 5.1. The determination of the half-power frequency, H2 norm and H norm for the second mode. Example 5.1. In this example we illustrate the...

## Norms of the Continuous Time Systems

Three system norms, H2, Hm, and Hankel are analyzed in this book. Their properties are derived and specified for structural applications. Let (A, b, c) be a system state-space representation of a linear system, and let g( ) c(jml - a)'1 b be its transfer function. The H2 norm of the system is defined as Gil 2 y tr(G* ( )G( )) d . (5.1) Note that tr(G* (a)G(a)) is the sum of the squared magnitudes of all of the elements of G( ), i.e., tr(G*( )G( )) gkl(j ) 2. Thus, it can be interpreted as an...

## Norms of the Discrete Time Structures

The norms of discrete-time structures are obtained in a similar way to the norms of the continuous-time structures. First of all, the system matrix A in discrete-time modal coordinates is block-diagonal, similar to the continuous-time case. For a diagonal A the structural norms are determined from the norms of structural modes, as described previously in this chapter. However, the norms of modes in discrete time are not exactly the same as the norms of modes for the continuous-time case. Later...

## Placement for Large Structures

In the case of the placement of a very large number of sensors, the maximization of the performance index alone may be either a sufficient or satisfactory criterion. Suppose that a specific sensor location gives a high-performance index. Inevitably, locations close to it will have a high-performance index as well. But the locations in the neighborhood of the original sensor are not necessarily the best choice, since the sensors at these locations can be replaced by the appropriate gain...

## Placement Indices and Matrices

Properties 7.1(a),(b) are the basis of the actuator and sensor search procedure of a general plant. The actuator index that evaluates the actuator usefulness in test is defined as follows where Gu 112 puy II + Guz 112, while the sensor index is The indices are the building blocks of the actuator placement matrix 2, The placement index of the kth actuator (sensor) is determined from the kth column of X In the case of the H2 norm it is the rms sum of the kth actuator indexes over all modes, k 4 ,...

## Placement of a Large Number of Sensors

For the placement of a large number of sensors the maximization of the performance index alone is not a satisfactory criterion. These locations can be selected using the correlation of each sensor modal norm. Define the kth sensor norm vector, which is composed of the squares of the modal norms where Guy- denotes the transfer function of the ith mode at the kth sensor. The norm denotes the H2, H , or Hankel norm. We select the sensor locations using the correlation coefficient rik, defined as...

## Properties of Collocated Controllers

As a corollary, consider a system with the state-space representation (A, B, C), which has collocated sensors and actuators, that is, C BT. In this case, a closed-loop system with the proportional feedback gain is stable, for K diag(k), i 1, , r and ki > 0. This particularly useful configuration can be used only if there is the freedom to choose the collocated sensors and actuators, and if the number of available sensors and actuators is large enough to satisfy the performance requirements....

## Reduction Errors

We use H2, Hm, and Hankel norms to evaluate the reduction errors. The first approach, based on the H2 norm, is connected to the Skelton reduction method, see 125 . The second method, based on the Hm and Hankel norms, is connected with the Moore reduction method see 109 . The H2 reduction error is defined as where G is the transfer function of the full model and Gr is the transfer function of the reduced model. Note that in modal coordinates the transfer function is a sum of its modes (see...

## Reduction in the Finite Time and Frequency Intervals

We introduced the time- and frequency-limited grammians in Chapter 4. They are used in model reduction such that the response of the reduced system fits the response of the full system in the prescribed time and or frequency intervals. This approach is useful, for example, in the model reduction of unstable plants (using time-limited grammians) or in filter design (using band-limited grammians). Figure 6.2. (a) Magnitude of the transfer function and (b) impulse responses of the full (solid...

## Reduction Through Truncation

In this chapter we consider a structural model in modal coordinates, namely, modal models 1, 2, and 3, as in (2.52), (2.53), and (2.54)). The states of the model are ordered as follows where xt is the state corresponding to the ith mode. It consists of two states see (2.55), (2.56), and (2.57) Let Gi denote either H2, Hm, or Hankel norms of the ith mode, and order the states in the state vector (6.1) in the descending norm order. Now, the norm of the first mode is the largest one, and the norm...

## References

1 Aidarous, S.E., Gevers, M.R., and Installe, M.J., Optimal Sensors' Allocation Strategies for a Class of Stochastic Distributed Systems, International Journal of Control, vol. 22, 1975, pp. 197-213. 2 Anderson, B.D.O., A System Theory Criterion for Positive Real Matrices, SIAM Journal of Control, vol. 5, 1967, pp. 171-182. 3 Anderson, B.D.O., and Moore, J.B., Optimal Control, Prentice Hall, Englewood Cliffs, NJ, 1990. 4 Athans, M., On the Design of PID Controllers Using Optimal Linear...

## Relationship Between Continuous and Discrete Time Grammians

Let (A, B, C) be the state-space representation of a discrete-time system. From the definitions (4.10) and (4.14) of the discrete-time controllability and observability grammians we obtain Wc BBt + ABBtAt + A2 BBt ( A2)t + , We show that the discrete-time controllability and observability grammians do not converge to the continuous-time grammians when the sampling time approaches zero, see 109 . Indeed, consider the continuous-time observability grammian This can be approximated in discrete...

## Root Locus

Using the diagonally dominant solutions of CARE and FARE we determine the relationship between the weights and the pole location and noise suppression, which is a useful tool in controller design. Property 11.3(a). LQG Root-Locus. Let the weight Q be Q diag(0,0, ,qiI2 , ,0,0) (11.36) then for the low-authority controller (qi < qoi) the closed-loop pair of flexible poles (Acri jAcii) relates to the open-loop poles (Aori jAoii) as follows (An j ai) iPcikri j onX i 1, ,n, OUT) or, the real part...

## Root Locus of Collocated Controllers

Here we present the relationship between the controller gains and the closed-loop pole locations. In order to determine the properties of the collocated controllers in modal coordinates, consider further the dissipativity conditions (10.1) for a structure in the modal coordinates 2. Consider also a feedback as in (10.5). In this case the closed-loop equations are as follows where uo is a control command (uo 0 in the case of vibration suppression). Since the matrix A is in the modal form 2 and K...

## Second Order Representation

For a structure with accelerometers we obtain the second-order equations similar to (2.7). Note that from this equation we obtain the acceleration as q -M lDq-M lKq + M lBou. (3.15) Let the accelerometer locations be defined by the output matrix Ca, that is, thus, introducing (3.15) to the above equation yields Similar equations can be obtained in modal coordinates. Namely, using (2.19), we arrive at the following acceleration output equation y Cmqm Cmvqm + Cmqqm + Dmau, (3.19) Cmq Cma , Cmv...

## Sensor and Actuator Properties

Consider a plant as in Fig. 3.10, with inputs w and u and outputs z andy. Let Gwz be the transfer matrix from w to z, let Gwy be the transfer matrix from w to y, let Guz be the transfer matrix from u to z, and let Guy be the transfer matrix from u to y. Let Gwzi, Guyi, Gwyi, and Guzi be the transfer functions of the ith mode. The following multiplicative property of modal norms holds, see (5.52), Gwzi Guyi Gwyi Guzi , for i n, (7.25) where II .11 denotes either H2, H , or Hankel norms. We show...

## Simultaneous Placement of Actuators and Sensors

In this section we present a simultaneous selection of sensor and actuator locations this is an extension of the actuator and sensor placement algorithm presented above. The latter algorithm describes either actuator placement for given sensor locations, or sensor placement for given actuator locations. The simultaneous placement is an issue of some importance, since fixing the locations of sensors while placing actuators (or vice versa) limits the improvement of system performance. The...

## State Space Representation

A linear time-invariant system of finite dimensions is described by the following linear constant coefficient differential equations with the initial state x(0) xo . In the above equations the N-dimensional vector x is called the state vector, xo is the initial condition of the state, the s-dimensional vector u is the system input, and the r-dimensional vector y is the system output. The A, B, and C matrices are real constant matrices of appropriate dimensions (A is NxN, B is Nxs, and C is...

## Structural Testing and Control

The formulation of structural testing is based on a block diagram as in Fig. 3.10. In this diagram the structure input is composed of two inputs not necessarily collocated the vector of disturbances (w) and the vector of actuator inputs (u). Similarly, the plant output is divided into two sets the vector of the performance (z) and the vector of the sensor output (y). The actuator inputs include forces and torque applied during a test. The disturbance inputs include disturbances, noises, and...

## Structures with Rigid Body Modes

Structures with rigid-body modes have poles at zero, therefore they behave like systems with integrators. The corresponding H2, Hm, and Hankel norms for systems with poles at zero do not exist as their values tend to infinity. However, the infinite values of the norms of some modes should not be an obstacle in the reduction process. These values indicate that the corresponding states should be retained in the reduced model, regardless of the norms of other modes. The reduction problem can be...

## The Balanced LQG Controller

The solutions of the CARE and FARE depend on the states we choose. Among the multiple choices there exists a state-space representation such that the CARE and FARE solutions are equal and diagonal, see 82 , 113 , and 49 , assuming that the system is controllable and observable. In this case we obtain Se M diagC , , , ), (11.15) where u1 > n2 > > nN > 0 and M is a diagonal positive definite M diag( ), i 1, ,N, nt > 0. A state-space representation with condition (11.15) satisfied is...

## The Deep Space Network Antenna

The NASA Deep Space Network antenna structure illustrates a real-world flexible structure. The Deep Space Network antennas, operated by the Jet Propulsion Laboratory, consist of several antenna types and are located at Goldstone (California), Madrid (Spain), and Canberra (Australia). The Deep Space Network serves as a communication tool for space exploration. A new generation of Deep Space Network antenna with a 34-m dish is shown in Fig. 1.5. This antenna is an articulated large flexible...

## The H and Hankel Norms

A similar relationship can be obtained for the H norm. From (5.22) one obtains and from Properties 5.14 and 5.15 the additive property of the H norm has the following form are the H norms of the th mode with the jth actuator only, or the th mode with the kth sensor only. Equation (7.6) shows that the H norm of a mode with a set of actuators (sensors) is the rms sum of the H norms of this mode with a single actuator (sensor). Hankel norm properties are similar to the H norm properties and follow...

## The H Norm

The discrete-time H2 norm is defined as an rms sum of integrals of the magnitudes of its transfer function, or as an rms sum of the impulse response ( 1 2 1 2 i to IG4 (e> d)ddJ l g2d(i& ) In the above equation 6 coAt, and gd (iAt) is the impulse response of the discrete-time system at t iAt. Similarly to the continuous-time case we calculate the H2 norm using the discrete-time grammians Wdc and Wdo, Odf2 tr(CTCWdc) tr(BdBTdWdo). (5.14) A relationship between the discrete- and...

## The Hankel Norm

The Hankel norm of a discrete-time system is its largest Hankel singular value where subscript d denotes a discrete-time system. In Chapter 4 we showed that the discrete-time Hankel singular values converge to the continuous-time Hankel singular values, see (4.23) therefore, the discrete-time Hankel norms converge to the continuous-time Hankel norms when the sampling time approaches zero, For structures in the modal representation, each mode is independent, thus the norms of a single mode are...

## The International Space Station Structure

The Z1 module of the International Space Station structure is a large structure of a cubical shape with a basic truss frame, and with numerous appendages and attachments such as control moment gyros and a cable tray. Its finite-element model is shown in Fig. 1.9. The total mass of the structure is around 14,000 kg. The finite-element model of the structure consists of 11,804 degrees of freedom with 56 modes, of natural frequencies below 70 Hz. This structure was analyzed for the preparation of...

## The Low Authority H Controller

We extend the properties of flexible structures to Hm control design. These properties are valid for a low-authority controller of moderate action. In this case flexible structure properties are reflected in the properties of the Hm controller. Let (A, B1, B2, Cj, C2) be the open-loop modal representation of a flexible structure (in the modal form 1 or 2), and let Ac1 A - B2B S , Ac2 A - SoSwC2C2 be the closed-loop matrices, where Sxc and S e are the solutions of the HCARE and HFARE equations,...

## The Low Authority LQG Controller

For LQG controllers we modify the definition of the low-authority controller of a structure as known from Chapter 10. Let (A, B, C) be the open-loop modal representation of a flexible structure (in the modal form 1 or 2), and let Ac1 A - BBTSc, Ac2 A - SeCTC be the closed-loop matrices where Sc and Se are the solutions of the CARE and FARE equations, respectively. The LQG controller is of low authority if its closed-loop matrices have the following property eig(Ad ) eig(A - BBTSC ) - eig(A -...

## The Reduced Order H Controller

The order of the state-space representation of the Hm controller is equal to the order of the plant, which is often too large for implementation. Order reduction is therefore a design issue worth consideration. The reduction of a generic Hm controller is not a straightforward task however, an Hm controller for flexible structures inherits special properties that are used for controller reduction purposes. We introduce the following reduction index for the Hm controller In this index y22i is the...

## The Tracking LQG Controller

Previously considered LQG controllers were designed for vibration suppression purposes, where the commanding signal was zero. A more complex task includes a tracking controller, where a structure must follow a command. It requires tracking performance in addition to vibration suppression properties. This is the case of controllers for radar and microwave antennas, such as the NASA Deep Space Network antennas. This kind of controller should assure zero steady-state tracking error, which is...

## Time and Frequency Limited Grammians

The time- and frequency-limited grammians characterize a system in a limited-time interval and in a limited-frequency window. They are obtained from the full time grammians using time and frequency transformation or vice versa. The results are identical in both cases, since the time and frequency transformations commute, as will be shown below. Consider the controllability grammian in the finite-time interval, defined in (4.80). From the Parseval theorem, the grammian (4.97) in the...

## Transfer Function

Besides the state-space representation a linear system can be alternatively represented by its transfer function. The transfer function G(s) is defined as a complex gain between y(s) and u(s), where y(s) and u(s) are the Laplace transforms of the output y(t) and input u(t), respectively. Using the Laplace transformation of (2.1) for the zero initial condition, x(0) 0, we express the transfer function in terms of the state parameters (A,B,C), The transfer function is invariant under the...

## Controllability and Observability of the Discrete Time Structural Model

Consider now a structure in modal coordinates. Similar to the continuous-time grammians the discrete-time grammians in modal coordinates are diagonally dominant, where Wci and W0i are 2 x 2 blocks, such that Wci wCiI2 and W0i w0iI2, see 98 , where IK II2 2 1 - cos fflj At 2 1 - cos fflj At wci TT--Y- wci cont-Yl-- 4.8 Q In the above equations Bmi is the ith block of Bm in modal coordinates, and Cmi is the ith block of Cm in modal coordinates, where Cm CmqQ_1 Cmv , see 2.42 for Zs0. In the...

## Time Limited Grammians

The steady-state grammians, defined over unlimited time integrals, are determined from the Lyapunov equations 4.5 . The grammians over a finite-time interval T t1, t2 where 0 lt t1 lt t2 lt lt are defined by 4.3 , and can be obtained from the matrix differential equations 4.4 . In many cases these equations cannot be conveniently solved, and the properties of their solutions are not readily visible. However, using the definitions from 4.3 we will derive the closed-form grammians over the finite...

## Diagonally Dominant Grammians

Assuming small damping, such that 1, where max , i 1, , n, the balanced and modal representations of flexible structures are closely related. One indication of this relationship is expressed in the grammian form. The balanced grammians are equal and diagonal similarly, the grammians in modal coordinates are diagonally dominant, and by using appropriate scaling, they are approximately equal. This is expressed in the following property Property 4.1. Diagonally Dominant Grammians in Modal...

## Sensor Placement Strategy

Actuator locations are already determined. 2. Select the areas where the sensors can be placed, obtaining the R candidate sensor locations. 3. Determine the sensor placement indices lt rk i for all the candidate sensor locations i 1, , R , and for all the modes of interest k 1, , n . 4. For each mode, select r1 for the most important sensor locations. The resulting number of sensors r2 for all the modes considered i.e., r2 lt n x r1 is much smaller than the number of candidate locations,...