Followed by an unrelated technical article on
The Discrete Time System & Differential Equation Analysis
within an Automotive Context
These two volumes contain sub-sections devoted to production, capital and the value of labor. Baranzini includes a brief discussion (Chapter Heading 2.2, pages 48 and 49) of early macro-economic theory, couched historically in the Harrod-Domar equilibrium model, a paradigm which states that aggregate savings are equal to growth multiplied times the capital/output ratio. He states that there are dilemmas and concerns related, firstly, to definition of terms and, secondly, to ways of generating growth that is demonstrably "in equilibrium" with the other factors. He suggests that the whole "production-function" model needs to be reexamined.
Pen, in his Fourth Chapter (pp. 76 ff) also refers to production at some length. He addresses the dynamics of labor utilizing the theoretical bases mentioned by Baranzini. The capital/output ratio is considered all-critical and the relationship between the two elements of this ratio are reviewed with care. Definitions of a "unit of labor" are offered and productivity is concretely defined in terms of the equations presented. Supply, demand, elasticity -- elementary concepts -- are covered and emphasized in ways which could be applied to a study on "International Trade and Distribution of Income". Indeed productivity, value and capital are fundamental concepts of use in realigning the distribution of income efficiently and equitably, whether within the context of world trade or internal commerce.
The reviewer traces the development of Mansfield's thoughts in an in-depth and organized manner, pointing out that considerable effort had been expended by the author to document his positions through inclusion of data and empirical, rather than anecdotal, evidence.
In the estimation of Cohen, Power, Trade and War which focuses on the pivotal issues of concentration of authority, and of inequality in the distribution of financial resources, examines the crucial factors tipping the balance between trade and war among nation-states. It is an eloquent discussion of the complex elements which determine the success or failure of a nation to prosper or to perish.
Although Mansfield's conclusion, that there is a positive correlation between trade and income, may be obvious to even the casual statistician, the content of this book is revealing and applicable to a paper on "Distribution of Income" by virtue of its assertions dealing with the oligopoly model and the concentration of wealth.
The authors demonstrate that market share of a given country in its international trade relations correlates well with a power quotient, and, in turn, impacts distribution of wealth and the country's trade policies. The focus of this analysis deals, as the source would imply, with commodities, wheat among others, and examines in statistical detail the dynamics of import/export activity, deriving graphic models for a wide range of scenarios.
The models produced reflect governmental preferences with respect to free trade, domestic objectives, internal economic goals, and other variables. Outcomes, expressed as a function of financial success or profitability, are examined in light of global cooperation, non-cooperation or "cheating" strategies. It is found that free trade, under honest conditions, benefits most nations, most of the time, although conclusions are mixed.
The merit and content that this study could bring to bear on a study of the "Distribution of Income" within the context of World Trade are clear. Indeed, it is crucial to examine statistical evidence in order to best assess how distribution of income could be enhanced, providing more nations the opportunity to expand, to the ultimate advantage of all.
This General Accounting Office document was compiled for submission to the Committee on Foreign Affairs, and was specifically addressed to Representative Sam Gejdenson, Chairman of the Economic Policy Subcommittee. It outlines the nature and extent of "tied-aid", which is basically comprised of offers of assistance contingent upon willingness by the host country to enlist the services of the donor country. The nations most active in this regard are, after the United States, predictably Canada, France, Germany, Italy, Japan and the U.K. Major recipient nations include China and Algeria, among others, as displayed cartographically in the GAO Report.
This document claims that distortions in the international trade picture occur when aid is tied to grants and assistance (page 2). The losses for US firms occur at the hands of competing donor-nations who undercut existing agreements, or compete directly with U.S. firms attempting to gain access to recipient-nation markets. Figures are provided breaking down percentages of such loans among competitors.
A brief, but revealing analysis of capital projects as opposed to human needs is presented, as well.
For a study on the "Distribution of Income", in relation to world trading patterns, this government analysis will prove indispensable in terms of demonstrating (a) flow of capital in addition to (b) the nature of the distortions that occur, deviating from the pure laissez-faire, free-trade approach recommended, on balance, by the GAO.
The policies of former President George Bush were externally oriented, specifically toward a policy favoring expansion of world trade. Bush believed, according to Ingwerson, that strengthening the U.S. foreign trade position in relation to both the developed world and the less developed countries was tantamount to improving America's overall security. In fact, former President Bush left a lasting impression in the sector of trade and foreign relations.
This Monitor article could be credibly utilized in bolstering the thesis of a paper on the "Effects of International Trade on the Distribution of Income" because it centers on the U.S./Japan trading relationship, where a dichotomous balance sheet between these two primary trade-partners influences distribution patterns measurably. Ingwerson's insightful analysis is largely of value because it points out the merits of developing strong commercial ties, enabling the U.S. to take advantage of other nations' efficiency and productivity, thus building its own economic base.
(Edited on behalf of a Foreign Graduate Student at New Mexico State University)
ABSTRACT
In the field of digital control systems, it is desirable to have a system that is reliable, stable and controllable. Accomplishing this objective requires more than just obtaining the data, analyzing the data and, if possible, coming up with an equation, and building upon it. Basically, most systems can be represented by a "hybrid system." Such a system contains 'discrete time controller as well as continuous time plant'. Between the controller and the plant, the necessary A/D and D/A conversion must be implemented.
There are many mathematical forms that can be used to represent the same discrete or same continuous system. Some of them are: state-space form, differential equations describing the discrete-time system, differential equation for continuous and transfer functions. One can easily convert from one form to the other form. These transformations can be done by hand or can use MATLAB command. It would be very helpful to test the design first before actually implementing it "real world", so that he, she or they know what to expect from the real system.
PROJECT 2 / EVALUATION OF AUTOMOBILE CRUISE CONTROL
INTRODUCTION
This report is intended for the reader who is interested in analyzing the Hybrid Control System. This project can be divided into two parts. The first part is to simulate the whole system using predefined MATLAB function "Hybsim". This function requires that both systems be in "state-space" form. It also requires the "initial conditions" of the system. If these requirements are not met, similar transformation must be done before we can use this function. The MATLAB script used in "hybsim" is attached in Appendix C. The second part is to implement the controller using an Analog Computer together with C program. The program is supposed to be user friendly, do the A/D and D/A conversion, produce both graphic and digital display of the output (i.e. speedometer). The code listing of C program is given in APPENDIX B. The simulation output and the hardware output are obtained and compared. The comparison is based on controller constraints such as overshoot, settling time and steady-state error. Both outputs are expected to be the same.
THEORETICAL DESCRIPTION
The automobile cruise control is a hybrid control system. The configuration shown in Figure 1 illustrates the systems. The overall system contains both "continuous-time signal" and "discrete-time signal". The combination of different types of signals creates difficulties. It is difficult to describe the behavior of the whole system especially between sampling instants.
(Non-Reproducible Schematic Insert)
Figure 1: Cruise Control Configuration
The output from the plant, y(t) is a continuous time signal. The analog signal is converted into a digital signal by the A/D converter at the sampling instants, kT. The output of the A/D converter is a discrete sequence of numbers , y (kT) and can be compared to the "references input" to form an error signal, e(kT).
The error signal is processed using an algorithm, C(z), and a new sequence, u(kT) is generated which is then converted into another algorithm, P(s). C(z) and P(s) represents the transfer function of controller and plant, respectively. It is also expressible in state-space form or continuous-state form. In order to go from one form to another, a similar transformation can be done by hand or MATLAB can be used.
In this configuration, the speed of the car is sensed with a velocity sensor. The information from the sensor is then sent to the controller which is actually an algorithm that produces voltage. The voltage is an input to the throttle actuator. The output actuator is then sent to the engine throttle which controls the speed of the engine, which in turn controls the speed of the car.
PROJECT DESCRIPTION
The purpose of this project is to evaluate an automobile cruise control. It is desired that the user determine the cruising speed of the car, but the cruising speed can range from 45 mph to 75 mph. This implies the controller must be user friendly. The controller must take in the user's desired cruising speed and changes the current speed of the car (i.e. if desired speed is not equal to the current speed). If the desired speed is greater than the current (reference speed), then the controller would cause the car to accelerate and settle at the desired speed. However, if the desired speed is less than the current speed, the car must slow down and settle at the desired level. If desired speed is equal to current speed, or out of the range specified above, then the controller does nothing.
There are two assumptions made in this project. The first assumption is that the car is in high gear when the model shown in Figure 1 was derived. The second assumption is that the car initially settled at 45 mph. The second assumption implies that the car must be running at 45 mph when the driver wants to settle to cruising speed.
By using finite time/settling time controller design, the controller is designed to drive the automobile from the initial speed to final speed in some interval of time which could be a design criterium. The time it takes to be plus/minus 5% of its final value is the settling time of the system. In this project, the settling time was determined to be 4 seconds. During this time interval, on going from initial speed to final speed, the speed of the car changes. These changes could result in overshoot. The overshoot of this system was set to be less than 5 mph. The overshoot is another design criterium. For example, if the driver's desired speed is 60 mph and he is currently running at 50 mph, the maximum speed that the controller can have is 65 mph. Another design criterium that must be satisfied is the steady-state error. Steady-state error was defined to be the difference between the input signal and the output signal. For this project, steady-state error is the difference between actual speed and final speed at a time greater than settling time and its value must be less than 2 mph. Another design criterium what must be met is that the amplitude of [V(t)] <10V. This design criterium will be further discussed when we describe implementation of the controller.
IMPLEMENTION OF THE PROJECT
Situation of the Control System
We are given the plant transfer functions which may be written as
Y(s) 10
H(s) = U(s) = (s+10)(s+1)(s+0.1) and the controller is given in discrete-time form.
We assume the controller for the systems operates at T=1.5 seconds.
Before we simulate the controller, it is very useful to find whether the system is controllable or not. To show the controller is controllable, we must test its controllability. The controller is controllable if we can find a control sequence, u(k) that can drive the system from any initial condition, X to any final condition X in a finite number of steps. To test controllability such that controllability matrix Q has rank 3, where Q = [G FG FG], where F and G are given above. This rank is solved by using MATLAB command 'rank'. The controllability was found to be:
0.2668 0.2668 0.2668
Q = 0.7476 -0.3182 0.1355 since it is a 3 by 3 square matrix, 0.0452 -0.0082 0.0000
The rank is 3. So we can say that the system is controllable.
Note that the system describing the controller is in discrete-time state-space form. And, we also know the transfer function of the plant. In order to simulate the whole system using the MATLAB command "HYBSIM", we must obtain continuous-time state-space matrices of this plant. Obtaining these matrices is outlined in APPENDIX A.
Before doing the simulation, there is one more thing to consider. That is, in order to use the HYBSIM command, the plant must be strictly proper which means that the relative degree of the model of the system, M(s)= Y(s) must be greater than the relative R(s) - degree of the controller. Let's find out...
First, we have to convert C(z) to C(s). The conversion is skipped
here. If we let C(s) = N (s) such that N (s) and D (s) is a
D (s) - non-zero number and is raised to some power of s, then M(s) can be
written as M(s) = C(s)P(s). Note that M(s) is basically a product
of C(s) and P(s) which implies that M(s) has a relative degree
greater than either C(s) or P(s). So, we could say that the plant
is strictly proper. Now we are ready to do the simulation.
The MATLAB commands used are as follows:
ac=[010;001;-1-11.1-11.1];
bc=[0;0;1];
cc=[10 0 0];
xc0=[0.45;0;0];
ad=[1 0 0;0 -0.42659 0;0 0 -.004699];
bd=[0.26681;0.7476;0.045249];
cd=[0.22681 -0.7476 0.045249];
dd=0.94288;
xd0=[1.7208;0;0];
k=[0.00001;1;10];
T=1.5;
r=(6.5*k./k)';
[tc,y,td,u]=hybsim(ac,bc,cc,ad,bd,cd,dd,xc0,xd0,r,T);
subplot(211);
stairs(td,u);
axis([0 8 0 3]);
xlabel('Time in second');
ylabel('u(t)');
title ('FIGURE :Control signal vs time');
grid
subplot(212);
plot(tc,y);
axis([0 8 4 6.7]);
xlabel ('Time in second');
ylabel('y(t)');
title ('FIGURE:Analog output vs time')
grid
e=y(10)-r(10)
The figure for this MATLAB command can be found in Appendix C.
IMPLEMENTATION OF THE CONTROLLER
Using the PC interfaced with the analog computer simulation for the automobile cruise controller, a program in C is written which inputs the value of y(t), calculates the error signal into the difference equation of the controller, and outputs the control signal to the analog computer. It prompts the user to enter the desired speed and displays the speed in real time once the controller is triggered. The user is also given the freedom to change the speed while the controller is still running. The user can do this by just pressing any keyboard-key; thus, the user will be asked to enter "new speed". However, despite the user having freedom to change the speed, the software has the ability to adjust the user's input to the preset limit which is /10/V. The C program, in which the controller is implemented, makes use of the A/D and D/A converters with the help of "b2v and v2b" functions available at the <dac.h.> library. Actual sampled data are stored in an array and then loaded into an output file compatible with MATLAB for plotting. A complete source-code listing of the software programs is provided in APPENDIX B.
Below, we discuss results using the hybsim command in the software tool MATLAB to the response of the system designed.
Summary of Results
In the graphs shown in APPENDIX E, the first page shows the simulated result of the hybrid closed loop control system. From the graph we have, we can see that the maximum overshoot of this ideal (theoretical) hybrid system is 67 mph, the settling time is around 1.5 seconds, the control amplitude is around 2.3 Volts, and the steady-state error is negligible.
Conclusion
Thus, digital control of a continuous-time automobile was achieved by the discrete-time, finite settling controller design.
The controller worked reasonably well by settling to the final desired speed within 4 seconds. The overshoot was negligible in most of the test drives. The maximum control amplitude was well within the range and the steady-state error was less than 2 mph. In addition to noise generated due to the analog components of the plant, the "plots" of the test drives were not smooth. In the first part of the project, simulation of the throttle control system was successfully performed on the GP-6 panel of the analog computer and in the second part, the cruise controller was designed and ultimately implemented in the final stage of the project. The design used for implementing an automobile cruise control can also be applied in various control system applications such as robotics, aircraft control, medical operations, process control, biomedical systems, and satellite operations.
References
Houpis, C.H., Digital Control Systems Theory: Hardware/Software, McGraw-Hill, 1992.
Lafore, R., The Waite Group's Turbo C Programming for the PC, H.W. Samms Publisher, 1989.
Paz, R.A., Computer Control Systems, 1993.