Systems Theory

The main research contributions are primarily in the areas of robust control.

Multi-objective Control

Single-input Single-output systems

We have shown that that single-input single-output problems incorporating the l1 and the H2 norms can be solved by finite-dimensional convex optimization problems. This work established that in certain interesting cases, these problems can be solved by a single finite-dimensional convex problem. These, problems prior to the above mentioned work were considered infinite-dimensional and were therefore considered intractable. The mathematical strategy that proved most useful was the Kuhn-Tucker-Lagrange duality results and the subsequent analysis of the dual to unravel the structure of the optimal. Another crucial element was posing the optimization problem using the correct topology. It also became evident that the manner in which the different objectives appear in the problem influenced the solution structure in a non-trivial manner. The solution methods utilized in addressing the single-input single-output framework proved insightful in solving the multiple-input multiple-output problems. 
 

Relevant publications

• M. V. Salapaka, M. Dahleh, and P. Voulgaris., “Mixed objective control synthesis: Optimal l1/H2 control”, SIAM Journal on Control and Optimization, V35 no. 5:pp. 1672--1689, September, 1997. 

• M. V. Salapaka, P. Voulgaris, and M. Dahleh, “SISO controller design to minimize a positive combination of the l1 and the H2 norms”, Automatica, V33, no. 3:pp. 387--391, March 1997. 

• M. V. Salapaka, P. Voulgaris, and M. Dahleh, “ Controller design to optimize a composite performance measure”, Journal of Optimization Theory and Applications, V91, no. 1:pp. 91-113, October 1996.   
   

Multiple-input Multiple-output systems 

It was observed that the elegant structure in the single-input single-output case can be generalized to a class of mutiple-input multiple-output problem. Essentially we established that in the square case, the property of finite-dimensionality of the optimal can be concluded. Also, for a large class of square systems the control design task can be achieved by solving a single finite dimensional convex-optimization problem. This characterization of the optimal in the square case was utilized to obtain the solution in the non-square case by approximating the problem by a sequence of square problems using the delay-augmentation technique. 

Relevant publication

• M. V. Salapaka, M. Dahleh, and P. Voulgaris, “Mimo optimal control design: the interplay of the H2 and the l1 norms”, IEEE Transactions on Automatic Control, V43, no. 10:pp.1374-1388, October 1998.

However, with the setup of the previous work the achievability of closed-loop maps via stabilizing controllers was characterized by zero-interpolation conditions. This involved the computation of the multi-variable zeros and zero-directions. Also, the optimal solution of the problem yielded the closed-loop map as the solution and the task of retrieving the optimal controller still remained. This step often involved interpreting the closed-loop map appropriately and canceling the unstable zeros in obtaining the Youla parameter Q. 
Another issue left unsolved by the research community was a common paradigm where multi-objective concerns as needed by typical engineering applications could be accommodated naturally. Other groups had resolved the structures of H2/H∞ and l1/H∞ problems. But a common effective paradigm where H2/H∞/l1 together with performance measures with respect to specific trajectories (like the step-input) was still elusive. Recently, we were able to address most of the concerns indicated above. Unlike the previous approaches, our research has indicated that by making the Youla-parameter a variable in the optimization, by interpreting the optimization in the correct topology and utilizing the Banach-Alaoglu theorem, converging upper and lower bounds can be achieved to the optimal. Also, it was realized that the analysis no longer needed the Khun-Tucker-Lagrange duality results thereby streamlining the arguments and deducing the convergence results based on the primal alone. This approach was developed first solely for the H2/l1 case (see reference below).
 

Relevant publication

• M. V. Salapaka, M. Khammash, and M. Dahleh, “Solution of MIMO H2/l1 Problem without Zero Interpolation” SIAM Journal of Optimization and Control, vol. 37, No. 6, pp. 1865—1873, 1999. 

One of the important advantages of the mathematical paradigm developed in the above reference is that it can accommodate other performance measures in a natural manner. This is facilitated by restricting all the arguments of convergence (upper and lower bound) to the primal alone. 
In studies we have shown that H∞, H2, l1 and time domain constraints can be cast into a General Multiple Objective Framework for which upper and lower bound convergence can be established. Use of the above setup for controller design for practical applications has demonstrated the power of the tools developed. 

Relevant publications 

• Qi, X., M. H. Khammash, and M. V. Salapaka, “Optimal controller synthesis with multiple objectives”, To appear in ACC, 2001. · Qi, X., M. H. Khammash, and M. V. Salapaka, “A new approach to multiple-objective controller synthesis”, Proceedings of the 38th annual allerton conference on communication, control and computing, Allerton, Urbana, Illinois, 2000.   

  • The above work was instrumental in developing a paradigm where the general multi-objective problem can be effectively addressed. A software tool is being developed where control design with multiple objective concerns can be easily performed. A preliminary version of the software was used to design many practical control systems.

• Qi, X., M. H. Khammash, and M. V. Salapaka,” A Matlab Package for Multiobjective Control Synthesis”, IEEE conference on Decision and Control, 2001, Volume: 4 , 2001 Page(s): 3991 -3996   
   

Globally optimal robust controllers

The controller design task for nominal problems does not guarantee performance under structured uncertainty of the plant description. The task of synthesizing controllers that provide optimal robust performance is hard in any norm. In the H-infinity setting, one can perform the D-K iteration scheme to obtain reasonable solutions. However, this procedure does not guarantee any optimality. In the l1 setting, similar procedure has been developed; again with no guarantees on optimality. Indeed in the l1 setting due to certain characteristics of the l1 robust performance problem, the iteration typically converges to a solution far from the optimal. 
In an initial study, a mixed problem where a H2 norm is minimized while guaranteeing a pre-specified level of robust performance was analyzed. It was shown using sensitivity based arguments that when there is a single uncertainty a globally optimal solution can be achieved by solving a finite number of quadratic optimization problems. However, subsequent effort to generalize this method to accommodate higher number of uncertainty blocks were not fruitful. 
Recently we have developed a new approach to solve the l1 robust performance problem with an emphasis on obtaining global optimal solutions. This approach is based on using linear approximation for the bilinear non-convex constraints that result in the l1 robust performance problem. It was established that converging upper and lower bounds can be obtained by solving a sequence of linear programming problems. This method applies to problems with any number of uncertainty blocks. Further since the solution is obtained by using only linear programming (LP) problems, the existing LP tools can be effectively utilized. Software tool incorporating the method indicated above has been developed and all examples solved demonstrate the effectiveness of the solution methodology developed. This significant new development provides a solution to an important open problem 
 

Relevant publications

• M. V. Salapaka, M. Dahleh, A. Vicino, and A. Tesi, “Nominal H2 performance and l1 robust performance,” Proceedings of the IEEE Conference on Decision and Control. pp. 4034-4039, Kobe, Japan, December 1996.

• M. Khammash, M.V. Salapaka, T. VanVoorhis,”Robust Synthesis in l1: A globally optimal solution”, IEEE Trans. Automatic Control, Volume: 46 Issue: 11 , Nov. 2001, Page(s): 1744 -1754

Distributed Control Design

 In large scale dynamic systems, it is desirable to implement the controller in distributive manner, so that computational complexity can be shared among various parts of the system. This technology can be used in the implementation of large and complex sensor networks, employing massively parallel sets of sensors and actuators. For example, millions of micro size parallel cantilevers used for reading and writing on hard disk has potential to revolutionize the memory devices. To achieve this, we need to device intelligent methods to divide the controller into various stations such that the effect of communication noise on the system performance is appropriately characterized.                   Distributed implementation of a controller may also be required because of some structural constraints on the system, e.g. different actuators and sensors located at different geographical locations in networked control applications.
 

Relevant publications

• V. Yadav, P. Voulgaris, and M. V. Salapaka. “Architectures for Distributed Control for Performance Optimization in Presence of Sub-controller Communication Noise” Proceedings of the IEEE American Control Conference. Minneapolis, MN, USA June 2006.

  • In this paper, we use state space approach to give a sufficient condition for internal stability of the closed loop system when the centralized stabilizing controller is implemented in a distributive manner. Using this condition, we show that the centralized stabilizing controller for a 2-nest system can be split into two sub-controllers without affecting the internal stability. The effect of sub-controller to sub-controller communication noise on the performance is considered along with the constraint on strength of subcontroller to sub-controller communication signal. We take an input-output approach. In a 2-nest case, we obtain a sufficient condition for splitting the stabilizing controller such that the overall performance optimization can be cast as a convex problem in the Youla-Kucera parameter Q. We also present an architecture for distributive implementation of banded structure controllers such that all closed loop maps are affine in Q.

• V. Yadav, P. G. Voulgaris, and M. V. Salapaka, “Controller Architectures for Distributed Implementation and Performance Optimization,” Proceedings of the IEEE Conference on Decision and Control. Bahamas, December 2004.

  • In this paper we consider how to distribute and implement an unstructured or structured overall controller K to various stations (sub-controllers). In doing so In doing so we assume that noise is present in the sub-controller to subcontroller communication and thus, its effect on stability and performance has to be addressed. Using an observer based controller parameterization, we provide suitable stabilizing sub-controller architectures that directly take into account the effect of communication noise on performance. In particular, the overall performance optimization can be cast as a convex problem in the Youla-Kucera parameter Q. Similar results hold for banded controller structures, i.e., when there is also a delay in the subsystem to subsystem communication. 

• V. Yadav, P. G. Voulgaris, and M. V. Salapaka, “Stabilization of nested systems with uncertain subsystem communication channels,” Proceedings of the IEEE Conference on Decision and Control. pp. 2853-2858, Hawaii, December 2003.

  • This paper addresses the design of stabilizing controllers for a nested control system where the controller is realized in a distributed manner, considering uncertainty not only in the controller-to-plant and plant-to-controller channels but also in the nest-to-nest, subcontroller-to-subcontroller communication. An input-output approach is taken. Two appropriate controller architectures that stabilize the entire system with the various components subject to uncertainty and noise are addressed. We present controller synthesis procedures to address deterministic uncertainty and stochastic uncertainty, that model packet-loss in the Internet.