Returning to the antenna position control system as considered in Chapter 3 (Section 3.6) and Chapter 4 (Section 4.2), the system is described using a transfer-function model and then using a state model. Although either model could be uti-lized in a simulation using MATLAB, the state-model format allows a simultaneous consideration of both inputs, and use of this format generates output data that include all of the state variables and output variables.
The reference input to the antenna is assumed to be the function as described in Figure 5.8. The reference angle is composed of a constant rate of change of angle in the period from 0 to 2 s, followed by a constant angle of 0.5 rad. The wind distur-bance torque on the antenna is assumed to be a burst of 20 N-m that exists only
Figure 5.8 The reference input angle and the wind disturbance torque.
effect of the disturbance torque is apparent. The developed torque of the motor is shown in Figure 5.10. Since the feedback operation acts to reduce the error, the antenna angle follows changes in the reference angle. The system tends to maintain the antenna angle at the position described by the input reference function. It is apparent, however, that a significant angular error occurs when the antenna is following the ramp motion as described by the input and again when the wind disturbance occurs. Various tech-niques that will improve the control are considered in the following chapters.
The use ofFOR loops to generate the input vectors (the reference input and the wind disturbance) versus time is one of several possible alternatives. Since matri-ces can be created by adding parts, the inputs can be generated by adding portions that occur in different time periods. For example, the angular input reference can be created as follows:
tl=[0:.05:2]'; thetal=.25*tl; % The reference input t2=[2.05:.05:8]'; theta2=.5+0*t2;
t=[tl;t2]; theta=[thetal;theta2]; % Time; Reference input completed Because the program creates and manipulates a number of matrices, care must be taken to develop matrices that are not transposed or otherwise inconsistent with dimensions as required for mathematical operations and plotting. A request for
Sec. 5.7 Simulation Using SIMULINK 133 5.7 SIMULATION USING SIMULINK
SIMULINK is a supplement to MATLAB(see Section 3.9) that is used primarily as a tool to simulate dynamic systems. The representation of a model is developed in an interactive environment by using graphical representations of all of the simulation elements. The block diagram format is offered with a large selection of operational blocks, including transfer functions, state models, nonlinear relationships, and user-defined functions. When using SIMULINK, the major programming tasks are all performed internally. The user must construct the diagram, enter the parameters as required, select a simulation algorithm, and then request a simulation.
The following discussion introduces some additional SIMULINK blocks, and the simulation procedure is demonstrated with two examples.
Some SIMULINK Blocks
The Sources Library contains a number of blocks for generating various excitation signals. The blocks labeled Constant, Step Input, Sine Wave, Pulse Generator, and Signal Generator will produce signals that are commonly used to test dynamic sys-tems. The characteristics of each block can be determined by double clicking on the box and examining the corresponding dialog boxes. If the waveforms are piecewise linear, additional periodic signals can be generated with the Repeating Sequence block. The signals are constructed by specifying the vertices of the signal over one period. For example, specifying [0 2 4] for Time values and [0 10] for Output val-ues produces a triangle wave that oscillates between 0 and 1with a period of 4.A signal that is not directly available can be produced by creating the time and signal vector in MATLABand importing the data into SIMULINK using the From Work-space block.
The connecting lines can represent a single variable or a vector. Blocks labeled Mux and Demux from the Connections Library are used to combine variables (or vectors) into a vector of increased dimension or to split a vector into individual vari-ables. This is a very useful capability when working with multivariable systems or when combining signals for1~e display of multiple output signals.
Before a simulation is requested, a numerical integration algorithm is selected in the Parameters dialog box. Linear system solutions do not normally introduce any special computational difficulties, and an efficient solution technique (using a matrix exponential algorithm) is Linsim. With nonlinear models, a Runge-Kutta 3 or Runge-Kutta 5 algorithm is usually sufficient. However, certain nonlinear sys-tems can exhibit special computational problems. For example, if the system is non-linear and smooth but stiff (i.e., some variables change much faster than others), then Gear's algorithm may be required to obtain a satisfactory result.
Examples
Consider again the multiloop feedback system of Section 4.8 and Figure 4.13. Sup-pose that both state variables, Xl (t) and x2(t), are to be observed, and they are to be plotted together with the excitation signal, r(t), in the same figure. Then the block
The ability to perform system simulation studies in the time domain is a valuable resource for a system designer. The use of digital simulation is a prevalent technique, but analog computation displays some desirable characteristics that can be utilized in an academic environment.
Sec. 5.9 Connectionsto Further Study 137
The state model affords an excellent format for the development of simulation models (of any order) using either analog or digital simulation. The conversion of a state model to an analog simulation diagram is obtained without any uncertainty with regard to the identificatjon of signals that appear at the outputs of integrators.
Thus, a nonzero initial statdis freely implemented.
The digital simulation techniques as presented are all developed using the state-model format. A set of linear differential equations can be converted to a set of linear difference equations using several different techniques. The forward-Euler algorithm is easily implemented as a simple program of repeated matrix computa-tions, but the calculation interval must be relatively small. The trapezoidal algorithm is somewhat more complex, but the error is considerably smaller if considering a fixed step size. A linear system simulation can also be realized using the matrix exponential function. The input can be described using a piecewise-constant func-tion or a piecewise-linear funcfunc-tion.
The simulation of a nonlinear state model is readily achieved using the forward-Euler algorithm, but a satisfactory implementation may require the application of a higher-order algorithm.
The MATLABsoftware package with the Control System Toolbox provides a set of computer-generated computational procedures that afford efficient simulation procedures for linear or nonlinear systems. MATLABfunctions that are applicable to the study of time-domain behavior with linear continuous system models include impulse, step, andinitial. The functions assume an impulse input, a step input, and a nonzero initial state, respectively. Considering impulse and step, the system descrip-tion can be entered either as a state model or a transfer-funcdescrip-tion model, but the solutions are realized using a state-model description. The use ofIsimallows the con-sideration of a user-defined input. If the system model is entered as a state model, Isim allows the consideration of multiple inputs and a nonzero initial state. Non-linear system simulation can be pursued using ode23 or ode45 to introduce an applicable integration algorithm.
The application of SIMULINK provides a graphical approach with the devel-opment of a block diagram. The blocks offer a wide selection of operational func-tions that include transfer funcfunc-tions, state models, nonlinear transfer relafunc-tionships, and user-defined functions.
The circuits as utilized for analog simulation are similar to circuits that are consid-ered in subsequent chapters for the design of continuous controllers. The applica-tions include the design of analog PID (proportional-integral-derivative) controllers and various compensation functions. The development of discrete-time relationships for digital simulation also includes some concepts that appear in other areas of study. The functions are similar in many respects to the mathematical models that are considered with the study of sampled-data systems.
138 Simulation Chap. 5 The simulation programs are, of course, applicable to linear and nonlinear continuous system models as considered throughout this text.MATLAB functions are utilized as an aid to other areas of study, including the development of root loci, frequency-response plots, discrete-time system simulation, and pole placement techniques. The ability to combine time-domain simulation with the application of s-plane, z-plane, or frequency-response techniques is a valued skill.
REFERENCES
1. L. o.Chua and P. Lin, Computer-Aided Analysis of Electronic Circuits: Algorithms and Computational Techniques. Englewood Cliffs, N.J.: Prentice Hall, 1975.
2. H. Saadat, Computational Aids in Control Systems Using Matlab. New York: McGraw-Hill, 1993.
3. B. C. Kuo and D. C. Hanselman, Matlab Tools for Control System Analysis and Design.
Englewood Cliffs, N.J.: Prentice Hall, 1994.
4. B. Shahian and M. Hassul, Control System Design Using Matlab. Englewood Cliffs, N.J.:
Prentice Hall, 1993.
5. K. Ogata, Solving Control Engineering Problems with Matlab. Englewood Cliffs, N.J.:
Prentice Hall, 1994.
Problems
6.1 INTRODUCTION
The dynamic stability of a control system is a critical property that is reflected in the character of the transient response. Considering the control of a linear system, an important component of a successful design is to obtain a natural response that exhibits a rapid and well-behaved asymptotic decay. If the natural response does not decay, the effect is observed as a loss of the desired control action. Although a lin-ear system can theoretically display a transient response that neither increases nor decays, the usual manifestation of instability is a natural response that exhibits an exponentially increasing magnitude or an oscillation with an exponentially increas-ing magnitude. The runaway action of the natural response tends to overshadow the forced response quickly; thus, the normal control action is defeated. The growth of the transient is eventually bounded by some form of nonlinear behavior, but the limitation may not occur gracefully. Depending on the application, the unstable behavior may produce a hazardous situation or catastrophic failure.
The intricate relationship between stability and the manner in which feedback is employed is one of the very interesting aspects of control system design. A poorly conceived feedback configuration can create an unstable situation when applied to the control of a stable plant. On the other hand, a well-conceived feedback configu-ration can create a stable performance when applied to an unstable plant. Thus, sta-bility is a fascinating and important topic. A system designer must also be concerned with the robustness of a control system with regard to stability-it is important that the ability to display a stable response is not significantly altered by changes that occur in the value of system parameters.
An investigation of stability, however, is not necessarily difficult. If a system model is linear and time invariant, the evaluation of system stability is relatively straightforward and uncomplicated. It is an intrinsic property of linear systems that stability is not dependent on signal level-linear system stability depends only on characteristics of the system model.
143
144 Stability Chap. 6
6.2 STABILITY CRITERIA AS APPLIED TO