## 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...

## 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...

## 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 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...

## 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...

## State Space Representation

We assume a continuous-time model in the form of the state-space representation (A,B,C,D). The discrete-time sequences of this model are sampled continuous-time signals, i.e., for k 1, 2, 3 The corresponding discrete-time representation for the sampling time At is (Ad, Bd, Cd, Dd), where and the corresponding state-space equations are The discretization can be carried out numerically using the c2d command of Matlab. Similarly to the continuous-time models the discrete-time models can also be...

## 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 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 H Norm

For a structure, the approximate Hm norm is proportional to its largest Hankel singular value max. The modal Hm norms can be calculated using the Matlab function norm_Hinf.m given in Appendix A.10. Property 5.5. H Norm of a Structure. Due to the almost independence of the modes, the system Hm norm is the largest of the mode norms, i.e., GIL max Gi I, i 1, n. (5.26) This property is illustrated in Fig. 5.6(b), and it says that for a single-input-single-output system the largest modal peak...

## 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...

## 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,...