32 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 1, JANUARY 2002 A Fuzzy Logic Approach to LQG Design With Variance Constraints Emmanuel G. Collins, Jr., Senior Member, IEEE, and Majura F. Selekwa Abstract—One of the well-known deficiencies of most modern control methods [i.e., , , and (or ) design] is that they attempt to represent multiple criteria with scalar cost functions. Hence, in practice the (static or dynamic) weights in the scalar cost function must be determined by an iterative process in order to satisfy the multiple objectives. It is of great time and cost benefit to automate this iterative process, but these problems tend to be highly nonlinear and extremely difficult to model analytically. However, a good designer can often observe trends and develop effective weight selection methodologies. The designer’s logic is inherently “fuzzy” and it is hence natural to use fuzzy logic for algorithm implementation. This paper develops a fuzzy algorithm for selecting the weights in a linear quadratic Gaussian (LQG) cost functional such that constraints on the variances of the system are satisfied. This problem is denoted the variance con- strained LQG (VCLQG) problem. Variations of this problem are considered in the existing literature using crisp logic. Numerical experiments show that when both the input and output variances are constrained, the fuzzy algorithm converges faster and tends to be much more robust to new systems or constraints than the crisp algorithms. Index Terms—Fuzzy logic, linear quadratic Gaussian (LQG) control, stochastic optimal control. I. INTRODUCTION C ONTROL methods based on “modern” control theory are based upon finding a control law that minimizes or con- strains a scalar cost function. The use of optimization theories in the development of modern control laws leads to a degree of de- sign automation, since after the cost function is chosen, the con- trol law synthesis may be entirely performed by a computer-im- plemented numerical algorithm. However, as is well known, in most real-world problems the engineering objectives are mul- ticriteria and cannot be easily captured by scalar cost criteria. Hence, in practice the control engineer must iteratively choose a set of weights, which may be static or dynamic, so that the scalar cost function actually yields a control law that satisfies the mul- tiple objectives. So, despite greater utilization of the computer, modern control design has not yet reached a high degree of au- tomation. Because of the cost savings that automation affords, full au- tomation of the control design process, including weight selec- Manuscript received February 28, 2000. Manuscript received in final form March 20, 2001. Recommended by Associate Editor L. K. Mestha. This work was supported in part by the National Science Foundation under Grant CMS- 9802197. The authors are with the Department of Mechanical Engineering, Florida A&M University, Florida State University College of Engineering, Tallahassee, FL 32310 USA (e-mail: ecollins@eng.fsu.edu; majura@eng.fsu.edu). Publisher Item Identifier S 1063-6536(02)00328-7. tion is highly desirable. The subject of weight selection in con- trol design has attracted only very sporadic attention in the con- trol literature (e.g., [6], [8], [11], [16], [17], [20], [22], and [25]). This is, perhaps, largely due to the fact that these problems are highly nonlinear, and it is in general very difficult to analytically model the relationships between the cost function weights and the various design objectives. Fortunately, there often are observable relationships between the various cost function weights and the multiple criteria. In this case, with experience, a modern control designer can de- velop methodologies to pick the weights to satisfy the multiple criteria. Hence, it is possible to automate the design process by capturing the designer’s observations and experience and im- plementing the resulting design rules in iterative computer al- gorithms. This implementation can be attempted by using crisp logic. However, since the designer’s thinking and design princi- ples are in reality “fuzzy,” it is more natural to use fuzzy logic. This paper focuses on the use of fuzzy logic to choose weights in a weight selection problem that has received substantial at- tention in the literature. The problem considered is variance constrained linear quadratic Gaussian (VCLQG) design. The basic VCLQG problem is to pick the weights in an LQG (i.e., ) cost func- tion so that the variances of the system inputs and outputs are constrained by selected amounts. A variant of this problem is considered in [16], where the task of minimizing a preselected quadratic cost function subject to input and output variance constraints is considered. In other references [20], [25] the dual problems considered are: 1) minimize a weighted sum of the differences between each input variance and its constraint sub- ject to constraints on the output variances [the output variance assignment (OVA) problem] or 2) minimize a weighted sum of the difference between each output variance and its constraint subject to constraints on the input variances [the input variance assignment (IVA) problem]. The output covariance control (OCC) problem considered in [11] seeks to minimize an input cost subject to constraints on the output covariances. The OCC problem can be reduced to that of minimizing the input cost subject to constraints on output variances. Here, we consider the most general VCLQG problem and report a fuzzy weight selection methodology. It should be mentioned that variance constraints also have a deterministic interpretation since the variances bound the infinity norms of the inputs and outputs when the system is disturbed by a finite energy signal. This interpretation has practical value and the reader is referred to [11] and [25] for more details. All of the previous algorithms for variations of the VCLQG design problem are based on crisp logic. The algorithm in [16] is 1063–6536/02$17.00 © 2002 IEEE