Journal of Process Control 10 (2000) 363±385 www.elsevier.com/locate/jprocont Review A tutorial on linear and bilinear matrix inequalities Jeremy G. VanAntwerp, Richard D. Braatz* Large Scale Systems Research Laboratory, Department of Chemical Engineering, University of Illinois at Urbana-Champaign, 600 South Mathews Avenue, Box C-3, Urbana, Illinois 61801-3792, USA Abstract This is a tutorial on the mathematical theory and process control applications of linear matrix inequalities (LMIs) and bilinear matrix inequalities (BMIs). Many convex inequalities common in process control applications are shown to be LMIs. Proofs are included to familiarize the reader with the mathematics of LMIs and BMIs. LMIs and BMIs are applied to several important process control applications including control structure selection, robust controller analysis and design, and optimal design of experiments. Software for solving LMI and BMI problems is reviewed. # 2000 Published by Elsevier Science Ltd. All rights reserved. 1. Introduction One of the main reasons for this is that process control engineers are generally unfamiliar with the mathematics A linear matrix inequality (LMI) is a convex con- of LMI/BMIs, and there is no introductory text avail- straint. Consequently, optimization problems with con- able to aid the control engineer in learning these vex objective functions and LMI constraints are mathematics. As of the writing of this paper, the only solvable relatively eciently with o-the-shelf software. text that covers LMIs in any depth is the research The form of an LMI is very general. Linear inequalities, monograph of Boyd and co-workers [22]. Although this convex quadratic inequalities, matrix norm inequalities, monograph is a useful roadmap for locating LMI and various constraints from control theory such as results scattered throughout the electrical engineering Lyapunov and Riccati inequalities can all be written as literature, it is not a textbook for teaching the concepts LMIs. Further, multiple LMIs can always be written as of LMIs to process control engineers. Furthermore, no a single LMI of larger dimension. Thus, LMIs are a existing text covers BMIs in any detail. useful tool for solving a wide variety of optimization This tutorial is an extension of a document used to and control problems. Most control problems of inter- train process control engineers at the University of Illi- est that cannot be written in terms of an LMI can be nois on the mathematical theory and applications of written in terms of a more general form known as a LMIs and BMIs. Besides training graduate students, the bilinear matrix inequality (BMI). Computations over tutorial is also intended for industrial process control BMI constraints are fundamentally more dicult than engineers who wish to understand the literature or use those over LMI constraints, and there does not exist o- LMI software, and experts from other ®elds (for exam- the-shelf algorithms for solving BMI problems. How- ple, process optimization) who wish to initiate investi- ever, algorithms are being developed for BMI problems, gations into LMI/BMIs. The only assumed background the best of which can be applied to process control is basic calculus, a course in state space control theory problems of modest complexity. [74,37], and a solid foundation in matrix theory [16,66]. The many ``nice'' theoretical properties of LMIs and The tutorial includes the proofs of several main BMIs have made them the emerging paradigm for for- results on LMIs. These are included for several reasons. mulating optimization and control problems. While First, many of the proofs are dicult to locate in the LMI/BMIs are gaining wide acceptance in academia, literature in the form that is most useful for applications they have had little impact in process control practice. to modern control problems. Second, the simplicity of the proofs provides some insights into the underlying * Corresponding author. Tel; +1-217-333-5073; fax: +1-217-33- geometry that manifests itself in terms of properties of 5052. the LMIs. Third, working through these proofs is the only E-mail address: [email protected] (R.D. Braatz). way to become suciently experienced in the algebraic 0959-1524/00/$ - see front matter # 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S0959-1524(99)00056-6 364 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 inputs. In many cases, it is impossible to even predict etch performance for the same system on two dierent runs. For this reason, it is impossible to maintain con- sistent etch quality without the use of feedback control. The feedback controller must be designed to be robust to the variability in process behavior as well as the Fig. 1. Reactive ion etcher in classical feedback form. nonlinear nature of the reactive ion etching process. Here, we consider the laboratory reactive ion etcher studied by Vincent et al. [146]. The manipulated variables manipulations necessary to be able to formulate the LMI/ were the power of the applied rf voltage and the throttle BMIs to solve new process control problems. Finally, valve position which speci®es the input gas ¯owrate, and students learn far more by working through problems the controlled variables were the ¯uorine concentration or proofs than from reading theorem after theorem. and the bias voltage. Like many other chemical processes This paper is organized as follows. First an example is described in the literature, the plasma dynamics of a used to motivate studies in LMI/BMIs. The second sec- reactive ion etching process were reasonably well descri- tion de®nes the LMI and discusses some of its basic bed as a static input nonlinearity N followed by a linear properties. The third section shows how inequalities of time-invariant (LTI) plant PL (see Fig. 1), which is the many dierent types can be written as LMIs or BMIs. well known Hammerstein model structure [51,106,134]. The fourth section discusses optimization problems over This nonlinear model was identi®ed using an iterative LMI or BMI constraints, and why such optimization least squares algorithm with data obtained from an problems can be eciently solved numerically. The ®fth experimental system by exciting it with a pseudo-random section reviews algorithms and software packages used binary signal with varying amplitude [146]. The identi®ed to solve LMI/BMI optimization problems, and the sixth LTI plant for their experimental process was section lists LMI/BMI problems that are important in 2 3 process control applications. This is followed by con- 1:89e:5s s 38:2 35:9 s 37:8 6 7 cluding remarks. 6 s 5:37 s 0:160 s2 6:5s 20:2 7 6 7 PL6 7 1 4 0:0239e:5s s 9:6 0:143 s 38:9 5 2. Motivating example: a reactive ion etcher s 1:05 s 0:214 s2 3:28s 4:14 A large number of control problems can be written in The natural controller structure to use has the form ^ 1 terms of LMIs or BMIs that cannot be solved using K N KL where KL is designed to stabilize the linear ^ 1 Lyapunov equations, Riccati equations, spectral factor- portion of the plant PL and N is an approximate ization, or other classical techniques. The following is inverse of the static nonlinearity N. If the input non- an industrial process control problem in which the only linearity N were identi®ed perfectly then N^ 1 would be tractable solution is via an optimization over LMI and an exact inverse of N and there would be an identity BMI constaints. mapping from KL to PL. However, in practice the iden- Etching is known to be a highly nonlinear multi- ti®cation is not perfect, and there is a nonlinear map- variable process that is strongly dependent on reactor ping from KL to PL. Furthermore, it is highly unlikely geometry. Attempts to control etch characteristics that the system is nonlinear only at the process input. usually manipulate the reactor pressure, gas ¯ow rate, Output nonlinearity is also a probability. and the power applied to the electrodes. However, due Nonlinearities in both the input and the output can be to many disturbances, complicated reaction dynamics, rigorously accounted for by the uncertainty description and the general lack of detailed fundamental under- shown in Fig. 2. The operators ÁI and ÁO can vary standing of the plasma behavior, it is impossible to within set bounds as functions of time, and can achieve predict etch performance for a system given a set of an identical input-output mapping for any possible Fig. 2. Reactive ion etcher with input and output nonlinearities modeled as uncertainty. J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 365 nonlinearity within the magnitude of the bounds set by equivalent to n polynomial inequalities. To see this, the uncertainty weights WI and WO. consider the well-known result in matrix theory (e.g. The only known method for designing a globally page 951 of [154]) that an n by n real symmetric matrix optimal robust nonlinear controller for this process is A is positive de®nite if and only if all of its principal by formulating the controller design as an optimization minors are positive. Let Aij be the ijth element of A. over LMI and BMI contraints (see Section 7.5 for Recall that the principal minors of A are details). The formulation allows a direct optimization of 02 31 the worst-case closed loop performance over the set of A11 A12 A13 plants described by the nonlinear uncertainty descrip- A11 A12 B6 7C A11; det ; det@4 A21 A22 A23 5A; tion. The BMI-based controller responded more than A21 A22 twice as fast to set point changes than a carefully tuned A31 A32 A33 classical controller (linear quadratic control, whose 02 31 computation was via a Ricatti equation), while at the A11 ..