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On Centralized Optimal Control
the optimal baseline and the universal algorithm seen in Fig. 2 demonstrates that more dynamic distortion control is required to maximize system performance. VI. CONCLUDING COMMENT
Yu-Chi Ho Index Terms—Decentralization, feedvack control, optimality.
In this note, we focus on signals that share a common distortion measure. As mentioned in Section II, the universal policy is directly applicable to situations where the distortion measure is signal specific. This is relevant for systems that handle different classes of data, e.g., audio versus image or video. Different distortion measures could also be used to give different qualities-of-service to different data classes, or data sources, and to implement a fair allocation of resources between them. Distortion measures can be designed to give priority to delay-sensitive data (such as voice), to favored data (as in the digital camera example), and even to be time-dependent so that the marginal utility of each piece of information decreases the longer it remains enqueued. As long as all distortion measures are convex, the priority storage and transmission protocols presented herein will remain sample-path optimal. REFERENCES [1] M. Alasti, K. Sayrafian-Pour, A. Ephremides, and N. Farvardin, “Multiple description coding in networks with congestion problem,” IEEE Trans. Inf. Theory, vol. 47, no. 3, pp. 891–902, Mar. 2001. [2] W. H. R. Equitz and T. M. Cover, “Successive refinement of information,” IEEE Trans. Inf. Theory, vol. 37, no. 3, pp. 269–275, Mar. 1991. [3] N. Farvardin and J. W. Modestino, “Adaptive buffer-instrumented entropy-coded quantizer performance for memoryless sources,” IEEE Trans. Inf. Theory, vol. IT-32, no. 1, pp. 9–22, Jan. 1986. [4] D. D. Harrison and J. W. Modestino, “Analysis and further results on adaptive entropy-coded quantization,” IEEE Trans. Inf. Theory, vol. 36, no. 9, pp. 1069–1088, Sep. 1990. [5] K. Ramchandran and M. Vetterli, “Multiresolution joint source-channel coding,” in Wireless Communications: Signal Processing Perspectives, H. V. Poor and G. W. Wornell, Eds. Upper Saddle River, NJ: PrenticeHall, 1998, ch. 7, pp. 282–329. [6] B. E. Rimoldi, “Successive refinement of information: Characterization of the achievable rates,” IEEE Trans. Inf. Theory, vol. 40, no. 1, pp. 253–259, Jan. 1994. [7] J. M. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3445–3462, Dec. 1993. [8] D. Tse, R. Gallager, and J. Tsitsiklis, “Optimal buffer control for variable-rate lossy compression,” in Proc. 31st Allerton Conf. Communication, Control, and Computing, Monticello, IL, Sep. 1993, pp. 727–736.
It can be argued that the Holy Grail of control theory is the determination of the optimal feedback control law or simply the feedback control law.1 This is understandable given the huge success of the linear quadratic Gaussian (LQG) theory and applications for the past half-century. It is not an exaggeration to say that the entire aerospace industry, from the Apollo moon landing to the latest global positioning system (GPS), owe a debt to this control-theoretic development in the late 1950s and early 1960s. As a result, the curse of dimensionality notwithstanding, finding the optimal control law for more general dynamic systems remains an idealized goal for all problem solvers. We continue to hope that with each advance in computer hardware and mathematical theory, we will move one step closer to this ultimate goal. Efforts such as feedback linearization and multimode adaptive control [3] and [4] can be viewed as such successful attempts. It is the thesis of this note to argue that this idealized goal of control theory is somewhat misplaced. We have been seduced by our early successes with the LQG theory and its extensions. The simple but often not emphasized fact is this: It is extremely difficult to specify and impossible to implement a general multivariable function even if the function is known Generally speaking, a one variable function is a two-column table, a two-variable function is then a book of tables, a three-variable function is a library of books, a four-variable function is a universe of libraries, and so on. Thus, how does one store or specify a general arbitrary 100-variable function, nevermind implementing it even if the function is God given? No amount of hardware advances will overcome this fundamental impossibility even if mathematical advances provide the general solution. Exponential growth is one law that cannot be overcome in general. Our earlier successes with LQG theory and its extensions were enabled by the fact that the functions involved have a very special form, namely, they decompose into sums or products of functions of single variables or low dimensions. As we move from the control of continuous-variable dynamic systems into discrete-event systems or more complex human-made systems such as electric power grids, communication networks, huge manufacturing plants, and supply chains, there is no prior reason to expect that the optimal control law for such systems will have the convenient additive or multiplicative form. Even if in the unlikely scenario that we are lucky enough to have such a simple functional form for the control law of the systems under study,
Manuscript received August 26, 2004. Recommended by Associate Editor E. Bai. This work was supported in part by the U.S. Army Research Office under Contract DAAD19-01-1-0610, by the U.S. Air Force Office of Scientific Research under Contract F49620-01-1-0288, and in part by the National Science Foundation under Contract ECS-0323685. The author is with the Harvard University, Cambridge, MA 02138 USA, and also with the Center for Intelligent and Networked Systems (CFINS), Tsinghua University, Beijing 100084, China (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2005.844898 1Also known as decision rule, IF–THEN table, fuzzy logic, learning and adaptation algorithm, strategies, and a host of other names. However, nothing can be more general than the definition of a function that maps all available information into decision or action.
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then our efforts should be to concentrate on searches for and characterization of such systems as opposed to finding the more general form of control laws. By stating this, this author does not mean to imply that theorists are, in general, blind to this fact. Hilbert’s 13th problem, the Naïve approach to Bayesian Networks, the Product Distribution Theory, neural networks, and others are all examples of trying to approximate general multivariable functions by combinations of functions of single variables or low dimensions. However, this author submits that it is nevertheless our instinctive desire to look for this centralized solution theoretically or to attempt to incorporate all relevant information into decision making practically. The consequence of such instincts or unconscious desire can be misdirected effort and uneconomical utilization of resources.2 In this light, it is not surprising that “Divide and Conquer” or “Hierarchy” is a time-tested method that has successfully evolved over human history to tackle many complex problems. It is the only known antidote to exponential growth. Furthermore, by breaking down a large problem into ever smaller component problems, many useful tools that do not scale up well can be used on these smaller problems. Decomposing a multivariable function into weighted sum of one variable function is a simple example of this principle. In addition, nature has also appreciated this fundamental difficulty of multivariable dependence. Examples of adaptation using simple strategies based on local information and neighboring interactions to achieve successful global results abound (think globally, but act locally). Ants, bees, germs, and viruses are examples [1]. Recent research on the No-Free-Lunch Theorem [2] also points to the importance and feasibility of “simple” control laws for complex systems. As we venture into the sociological and psychological realm, there is even more evidence that striving for the “best” only leads to unhappiness and nonoptimality [5]. One can, of course, claim that all of this is obvious and needs no elaboration. However, the story of “The Emperor’s New Clothes” is still a fable worth telling. We hope pointing out that this “obvious” information can help to direct research efforts into complex systems.
Comments on “Variable Structure Control of Discretized Continuous-Time Systems” N. V. Paranjpe and S. B. Phadke Abstract—In this note, we point out that the control proposed inthe above paper can actually work under milder assumption with respect to the size of uncertainty in the control input vector. Index Terms—Discrete-time variable structure control, robust control, time-delay control, uncertain systems.
I. INTRODUCTION In [1], a novel discrete-time variable structure control technique is presented for uncertain continuous-time systems. In the design approach, the authors have used the variable structure control theory to bring the states into a certain sector and the concept of time-delay control (TDC) [2] to drive the states inside that sector, close to the switching surface. In [3], the authors have used the same methodology to control plants having no uncertainty in the control input vector. In this sense, [1] can be considered to be an improvement over [3]. However, a remark in [1] about the control inside the sector creates a doubt in the minds of the readers about the applicability of the control to systems in which the uncertainty in the control input vector is not too small. The purpose of this note is to show that, in fact, the control inside the sector works even when the uncertainty in the control input vector is not negligible. II. MAIN RESULT In this note, we comment on the control inside the sector. For ease of discussion, we repeat some equations from [1], but mostly omit any explanation of the notation used by the authors. The discretized system considered in [1] is given by x ^k+1
= A^x^k + B^ [(1 + b)uk + D(^xk ; k)]:
(1)
Using a control REFERENCES [1] F. Vertosick, The Genius Within: Discovering the Intelligence of Every Living Thing. New York: Harcourt, 2002. [2] Y. C. Ho and D. Pepyne, “Conceptual framework for optimization and distributed intelligence,” in Proc. IEEE Proc. Conf Decision and Control, Dec. 2004. [3] P. Kokotovic, “The joy of feedback,” IEEE Control Syst. Mag., pp. 7–17, Jun. 1992. [4] L. Chen and K. S. Narendra, “Nonlinear adaptive control using neural networks and multiple models,” Automatica, vol. 37, no. 8, pp. 1245–1255, 2001. [5] B. Schwartz, The Paradox of Choice –Why More is Less. New York: Harper-Collins, 2004.
=u +v
=
0R(A^ + I ) x^
eq k
(2)
k
with ukeq
^ RB
(3)
k
the expression for the sliding variable yields s^k+1
= RB^ [(1 + b)v + G(^x
k
k
; k)]
(4)
xk ; k) = bukeq + D(^ xk ; k) j 0(^ xk ; k). The where G(^ xk ; k) and j G(^ expressions for the control vk given by [1, (17a) and (17b)] are vk
2For example, approximating a control law does not necessarily guarantee similar approximation in performance unless some form of continuity property is also assumed.
uk
=
j0RB^ 0(^x ;k) ; j^s (1+ b )RB^ ^ ^s 0RBv ; ^ RB
0
^ 0(^xk ; k) if j s^k j> RB
(5a)
; k).
(5b)
^ 0(^x if j s^ j RB k
k
Manuscript received December 8, 2003; revised August 4, 2004 and September 1, 2004. Recommended by Associate Editor D. Nesic. The authors are with the Institute of Armament Technology, Pune 411 025, India (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2005.844904
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