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 eciently 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 dicult 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 dicult 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 suciently 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 di€erent 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 di€erent 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 eciently 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 ˆ 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 ... A1n same time providing guaranteed robustness [143]. B6 . . 7C ...; det@4 . . 5A 4†

An1 ... Ann 3. The linear matrix inequality We apply this result to give that the LMI (2) is equiva- Here we de®ne the LMI and some of its basic prop- lent to: erties. We will use upper case Roman to refer to matrices, lower case Roman to refer to vectors or scalars, lower Xm F ‡ x F > 0 a linear inequality† case Greek to refer to scalars, and upper case calligraphic 0;11 i i;11 iˆ1 to refer to sets. The symbol 8 should be read ``for all'' and the symbol 2 should be read ``is an element of''. The ! ! notation Rm denotes the set of real vectors of length m, Xm Xm and RnÂn denotes the set of real n  n matrices. F0;11 ‡ xiFi;11 F0;22 ‡ xiFi;22 iˆ1 iˆ1 ! ! 3.1. De®nition Xm Xm F0;12 ‡ xiFi;12 F0;21 ‡ xiFi;21 > 0 A linear matrix inequality (LMI) has the form: iˆ1 iˆ1

Xm a quadratic inequality† Fx †ˆF0 ‡ xiFi > 0 2† iˆ1 . . where x 2Rm; F 2RnÂn. The inequality means that i 02 31 Fx †is a positive de®nite matrix, that is, Fx †11 ... Fx †1k B6 7C det . . > 0 zTFx †z > 0; 8z 6ˆ 0; z 2Rn: 3† @4 . . 5A Fx †k1 ... Fx †kk

The symmetric matrices Fi; i ˆ 0; 1; ...; m are ®xed kth order polynomial inequality† and x is the variable. Thus, Fx †is an ane function of the elements of x. . Eq. (2) is a strict LMI. Requiring only that Fx †be . positive semide®nite is referred to as a nonstrict LMI. The strict LMI is feasible if the set fgxjFx †> 0 is det †Fx †> 0 nth order polynomial inequality† nonempty (a similar de®nition applies to nonstrict LMIs). Any feasible nonstrict LMI can be reduced to an equiva- lent strict LMI that is feasible by eliminating implicit The n polynomial inequalities in x range from order 1 equality constraints and then reducing the resulting LMI to order n. by removing any constant nullspace ([22], page 19). We will therefore focus our attention on strict LMIs. 3.3. Convexity

3.2. LMI equivalence to polynomial inequalities A set C is said to be convex if lx ‡ †1 l y 2 C for all x; y 2 C and l 2 †0; 1 [107]. An important property It is informative to represent the LMI in terms of of LMIs is that the set fgxjFx †> 0 is convex, that is, scalar inequalities. More speci®cally, the LMI (2) is the LMI (2) forms a convex constraint on x. To see this, 366 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 let x and y be two vectors such that Fx †> 0 and where ÈÉ Fy †> 0, and let l 2 †0; 1 . Then 1 2 q Fi ˆ diag Fi ; Fi ; ...; Fi ; 8i ˆ 0; ...; m 12† Xm F †ˆlx ‡ †1 l y F ‡ †lx ‡ †1 l y F ÈÉ 0 i i i and diag X ; X ; ...; X is a block diagonal matrix iˆ1 1 2 q Xm with blocks X1; X2; ...; Xq. This result can be proved ˆ lF0 ‡ †1 l F0 ‡ l xiFi from the fact that the eigenvalues of a block diagonal iˆ1 matrix are equal to the union of the eigenvalues of the Xm blocks, or from the de®nition of positive de®niteness. ‡ †1 l yiFi iˆ1 ˆ lFx †‡ †1 l Fy † 4. The generality of LMIs and BMIs

> 0: 5† This section shows how many common inequalities can be written as LMIs. In addition, it shows how many control properties of interest can be written exactly in 3.4. LMIs are not unique terms of the feasibility of an LMI. Such a problem is referred to as an LMI feasibility problem. The same set of variables x can be represented as the feasible set of di€erent LMIs. For instance, if Ax †is 4.1. Linear constraints can be expressed as an LMI positive de®nite then Ax †subject to a congruence transformation (see section 14.7 of [154]) is also positive Linear constraints are ubiquitous in process control de®nite: applications. Model Predictive Control has become the most popular multivariable controller design method in A > 0 () xTAx > 0; 8x 6ˆ 0 6† many industries precisely because of its ability to address linear constraints on process variables [32,48,61,95,110, () zTMTAMz > 0; 8z 6ˆ 0; M nonsingular 7† 114]. The standard linear programming and quadratic programming model predictive control formulations () MTAM > 0 8† can be written in terms of LMIs. Here we show the ®rst step, which is to write the linear constraints on process variables as LMI constraints. This implies, for example, that some rearrangements Consider the general linear constraint Ax < b written of matrix elements do not change the feasible set of the as n scalar inequalities: LMI. Xm   AB 0 I AB 0 I bi Aijxj > 0; i ˆ 1; ...; n 13† > 0 () > 0 jˆ1 CD I 0 CD I 0  DC where b 2Rn, A 2RnÂm, and x 2Rm. Each of the n () > 0 9† scalar inequalities is an LMI. Since multiple LMIs can BA be written as a single LMI, the linear inequalities (13) can be expressed as a single LMI. 3.5. Multiple LMIs can be expressed as a single LMI 4.2. Stability of linear systems One of the advantages of representing process control problems with LMIs is the ability to consider multiple Stability is one of the most basic needs for any closed control requirements by appending additional LMIs. loop system. Some methods for analyzing the stability Consider a set de®ned by q LMIs: of linear systems are covered in undergraduate process control textbooks [102,133]. Moreover, some nonlinear F1 †x > 0; F2 †x > 0;...;Fq †x > 0 10† processes can be analyzed (at least to some degree) with linear techniques by performing a change of variables, Then an equivalent single LMI is given by such as in binary distillation [91] and pH neutralization [68,101]. Xm ÈÉThe Lyapunov method for analyzing stability is 1 2 q Fx †ˆF0 ‡ xiFi ˆ diag F †x ; F †x ; ...; F †x > 0; described in most texts on process dynamics [70,108]. iˆ1 The basic idea is to search for a positive de®nite func- 11† tion of the state (called the Lyapunov function) whose J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 367 time derivative is negative de®nite. A necessary and LTI techniques, including reactive ion etching [140], sucient condition for the linear system packed bed reactors [46], and most batch processes [9]. : In Section 4.2, we showed how testing the stability of x ˆ Ax 14† a linear system could be posed as an LMI feasibility problem. Now let us consider a generalization of that to be stable is the existence of a Lyapunov function problem to testing the stability of a set of linear time Vx †ˆxTPx where P is a symmetric positive de®nite varying systems that are described by a convex hull of matrix such that the time derivative of V is negative for matrices (a matrix polytope): all x 6ˆ 0 [108]: : x ˆ At †x; At †2CofgA1; ...; AL 19† dVx † : : ˆ xTPx ‡ xTPx dt An alternative way of writing this is [105]: Á ˆ xT ATP ‡ PA x < 0; 8x 6ˆ 0 15† : XL XL x ˆ At †x; At †ˆ liAi; 8li50; li ˆ 1: 20† () ATP ‡ PA < 0 16† iˆ1 iˆ1

A necessary and sucient condition for the existence This is an LMI, where P is the variable. To see this, of a quadratic Lyapunov function Vx †ˆxTPx that select a basis for symmetric n  n matrices. proves the stability of (20) is the existence of P ˆ PT > As an example basis, for i5j de®ne Eij as the matrix 0 that satis®es: with its †i; j and †j; i elements equal to one, and all of dVx † : : its other elements equal to zero. There are m ˆ ˆ xTPx ‡ xTPx < 0; 8x 6ˆ 0; ij dt nn †‡ 1 =2 linearly independent matrices E and any 21† symmetric matrix P can be written uniquely as 8At †2CofgA1; ...; AL Xn Xn ij P ˆ PijE ; 17† Âà T T jˆ1 i5j () x At † P ‡ PA † t x < 0; 8x 6ˆ 0; 22† 8At †2CofgA1; ...; AL ij where Pij is the †i; j element of P. Thus the matrices E form a basis for symmetric n  n matrices (in fact, if the ij T columns of each E are stacked up as vectors, then the () At † P ‡ PA † t < 0; 8At †2CofgA1; ...; AL resulting vectors form an orthogonal basis, which could be made orthonormal by scaling). 23† Substituting for P in terms of its basis matrices gives ! ! the alternative form for the Lyapunov inequality XL T XL XL ! ! () liAi P ‡ P liAi < 0; 8li50; li ˆ 1 Xn Xn Xn Xn T T ij ij iˆ1 iˆ1 iˆ1 A P ‡ PA ˆ A PijE ‡ PijE A jˆ1 i5j jˆ1 i5j 24† Xn Xn Á ˆ P ATEij ‡ EijA < 0 XL Á XL ij T jˆ1 i5j () li Ai P ‡ PAi < 0; 8li50; li ˆ 1 25† 18† iˆ1 iˆ1

T which is in the form of an LMI (2), with F0 ˆ 0 and () Ai P ‡ PAi < 0; 8i ˆ 1; ...; L 26† T ij ij Fk ˆA E E A; for k ˆ 1; ...; m. The elements of the vector x in (2) are the Pij; i5j, stacked up on top of The search for P that satis®es these inequalities is an each other. LMI feasibility problem. This condition is also a su- cient condition for the stability of nonlinear time vary- 4.3. Stability of nonlinear and time varying systems ing systems where the Jacobian of the nonlinear system is contained within the convex hull in (20) [84]. There Many of the processes commonly encountered in are several diculties in applying the LMI condition for process control applications can be adequately modeled analyzing stability of nonlinear systems. First, it is very as being linear time invariant (LTI). However, many dicult to construct a convex hull for which the Jaco- chemical processes cannot be adequately analyzed using bian of a nonlinear system is provably contained within. 368 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385

Second, such a description will usually be highly con- Here Ax †corresponds to Sx †in the LMI (28), and servative, since the convex hull overbounds the Jacobian Qx †and Rx †correspond to I. of the real nonlinear system. Third, each new vertex adds another matrix inequality to the LMI feasibility 4.6. Ellipsoidal inequality problem (26). For a system with a large number of states (which is equal to the dimension of A) and ver- Ellipsoid constraints are important in process identi- tices (L), solving the LMI feasibility problem (26) can ®cation, parameter estimation, and statistics [15,27, become computationally prohibitive. The strength of 41,85]; as well as certain fast model predictive control the approach is that LMIs for controller synthesis for algorithms [138,139]. Applications recently described in systems of the form (20) are relatively easy to construct the literature include crystallization processes [88,93], [22,77,131]. polymer ®lm extruders [54], and paper machines [138, 139]. 4.4. The Schur complement lemma An ellipsoid described by

T 1 T The Schur complement lemma converts a class of †x xc P †x xc < 1; P ˆ P > 0 32† convex nonlinear inequalities that appears regularly in control problems to an LMI. The convex nonlinear can be expressed as an LMI using the Schur comple- inequalities are ment lemma with Qx †ˆ1, Rx †ˆP, and Sx †ˆ T †x xc : 1 T Rx †> 0; Qx †Sx †Rx † Sx †> 0; 27†  T 1 †x xc T T > 0: 33† where Qx †ˆQx †; Rx †ˆRx †,andSx †depend †x xc P anely on x. The Schur complement lemma converts this set of convex nonlinear inequalities into the equivalent LMI 4.7. Algebraic Riccati inequality  Qx † Sx † > 0: 28† Algebraic Riccati equations are used extensively in Sx †T Rx † optimal control, as described in textbooks on advanced process control [111,130], which describe applications to A proof of the Schur complement lemma using only chemical reactors, distillation columns, and other pro- elementary calculus is given in the Appendix. In what cesses. A result involving a Riccati equation can be follows, the Schur complement lemma is applied to replaced with an equivalent result where the equality is several inequalities that appear in process control. replaced by an inequality [151]. More speci®cally, these optimal controllers can be constructed by computing a 4.5. Maximum singular value positive de®nite symmetric matrix P that satis®es the algebraic Riccati inequality: The maximum singular value measures the maximum gain of a multivariable system, where the magnitude of ATP ‡ PA ‡ PBR1BTP ‡ Q < 0 34† the input and output vector is quanti®ed by the Eucli- dean norm [130]. It is also very useful for quantifying where A and B are ®xed, Q is a ®xed symmetric matrix, frequency-domain performance and robustness for and R is a ®xed symmetric positive de®nite matrix. multivariable systems [96,130]. Process applications are The Riccati inequality is quadratic in P but can be provided in many popular undergraduate process con- expressed as a linear matrix inequality by applying the trol textbooks [102,126]. Schur complement lemma: The maximum singular value of a matrix A which  ATP PA QPB anely depends on x is denoted by  †Ax †, which is > 0: 35† the square root of the largest eigenvalue of Ax †Ax †T. BTPR The inequality  †Ax †< 1 is a nonlinear convex con- straint on x that may be written as an LMI using the The next two sections provide examples of algebraic Schur complement lemma: Riccati inequalities for analyzing the properties of linear  †Ax †< 1 () Ax †Ax †T< I 29† or nonlinear systems. () I Ax †I1Ax †T> 0 30† 4.8. Bounded real lemma  IAx † The Bounded real lemma forms the basis for LMI () T > 0 31† Ax † I approaches to robust process control which have been J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 369 applied to reactive ion etching [140,143], polymer columns [129], packed bed reactors [46] and a reactive extruders [141], and paper machines [141], and gain ion etching [143]. A property that is regularly exploited scheduling which has been applied to chemical reactors in the development of robustness analysis tools [14,72] [12,13]. Although the Bounded real lemma has applica- for linear systems subject to linear or nonlinear pertur- tion to the control of both linear and nonlinear pro- bations is passivity. The linear system (36) is said to be cesses, the actual result is based on the state space passive if system representation of a linear system  : ut †Tyt †dt50 42† x ˆ Ax ‡ Bu; y ˆ Cx ‡ Du; x †ˆ0 0; 36† 0 where A 2RnÂn; B 2RnÂp; C 2RpÂn,andD 2RpÂp for all u and 50. This property is equivalent to the are given data. Assume that A is stable and that existence of P ˆ PT > 0 such that [22] †A; B; C is minimal [74]. The transfer function matrix is "# ATP ‡ PA PB CT Gs †ˆCsI † A 1B ‡ D: 37† 40: 43† BTP C DT D The worst-case performance of a system measured in terms of the integral squared errors of the input and It is instructive to show the connection between the output is quanti®ed by the H1 norm [157]: bounded real lemma and the positive real lemma [5], especially since it is often referred to in the robust con- k Gs †k1ˆ sup  †ˆGs † sup  †Gj †! : 38† Re †s >0 !2R trol literature. A standard result from network theory [10,45,125] is that passivity is equivalent to Gs †in (37) The H1 norm can be written in terms of an LMI. To being positive real, that is, see this, we will use a result from the literature [158] that à 2 Gs †‡Gs †50 8Refgs > 0 44† the H1 norm of Gs †is less than if and only if I DTD > 0 and there exists P ˆ PT > 0 such that ÁÁ where Gs †Ã is the complex conjugate transpose of Gs †. T T T A P ‡ PA ‡ C C ‡ PB ‡ C D The relationship between bounded real and positive 1 Á1Á real is that ‰ŠI Gs †‰ŠI ‡ Gs † is strictly positive real  2I DTD BTP ‡ DTC < 0 39† if and only if Gs †is strictly bounded real. This follows from [87] The Schur complement lemma implies that this Ric- à cati inequality is equivalent to the existence of P ˆ PT >  †A < 1 () A A < I 45† 0 such that the following LMI holds: à 1 à 1 "#ÂÃÂà () †I ‡ A †2I 2A A †I ‡ A > 0 46† ATP ‡ PA ‡ CTC PB ‡ CTD Âà > 0 40† 1 BTP ‡ DTC 2I DTD () †I ‡ Aà ‰Š †I Aà †‡I ‡ A †I ‡ Aà †I A which is equivalent to  †I ‡ A 1 > 0 47† "# ATP ‡ PA ‡ CTCPB‡ CTD < 0: 41† () †I ‡ Aà 1 †‡I Aà †I A †I ‡ A 1> 0 48† BTP ‡ DTCDTD 2I Âà () †I A †I ‡ A 1 Ç †I A †I ‡ A 1> 0 49† It is common to incorporate weights on the input u and output y so that the condition of interest is whether the H1 norm of W1 †s Gs †W2 †s is less than 1. A sys- 4.10. The S procedure tem with an H1 norm less than one is said to be strictly bounded real. This condition is checked by testing the The S procedure greatly extends the usefulness of feasibility of the LMI using the state-space matrices for LMIs by allowing non-LMI conditions that commonly the product W1 †s Gs †W2 †s . arise in nonlinear systems analysis to be represented as LMIs (although with some conservatism). This techni- 4.9. Positive real lemma que has been applied to the analysis of pH neutraliza- tion processes [119] and crystallization processes [116]. Robustness analysis has been widely applied in the First we will describe the S procedure as it applies to process control literature. Examples include distillation quadratic functions, and then discuss its application to 370 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 quadratic forms. Let 0; ...; p be quadratic scalar 4.11. Stability of linear systems with nonlinear functions of x 2Rn: perturbations

T T T i †ˆx x Tix ‡ 2ui x ‡ i; i ˆ 0; ...; p; Ti ˆ Ti The derivation of LMI feasibility problems to analyze the stability or performance of linear systems subject to 50† linear/nonlinear time invariant/varying perturbations is The existence of 150; ...;p50 such that rather straightforward conceptually [22], although the algebra can be messy for more complex systems Xp [115,118]. For continuous time systems, the basic †x  †x 50; 8x; 51† 0 i i approach is to postulate a positive de®nite Lyapunov iˆ1 function of the state and some undetermined matrices, implies that and then apply the S procedure (if necessary) to derive LMI conditions on the undetermined matrices which 0 †x 50; 8x such that i †x 50; i ˆ 1; ...; p: 52† imply that the time derivative of the Lyapunov function is negative de®nite. For discrete time systems, the divi- To see why this is true, assume there exists ded di€erence of the Lyapunov function is used instead 150; ...;p50 such that (51) holds for all of the time derivative. Here we show how this approach i †x 50; i ˆ 1; ...; p. Then is applied to a system of especial relevance to process control applications. Xp Consider a discrete time system subject to slope- †x 5  †x 50; 8x: 53† 0 i i restricted static nonlinearities: iˆ1 xk †ˆ‡ 1 Ax †‡ k B †qk † Note that (51) is equivalent to qk †ˆCx † k 61†

Xp  T0 u0 Ti ui with the nonlinearities described by T i T 50 54† u 0 u i 0 iˆ1 i i †qi †k ‰Ši †qi †k qi †k 40; for i ˆ 1; ...; m 62† since with the local slope restrictions xTTx ‡ 2uTx ‡ 50; 8x 55† i †qi †k ‡ 1 i †qi †k  0 < < Tii; for i ˆ 1; ...; m 63† x T Tu x qi †k ‡ 1 qi †k () 50; 8x 56† 1 uT 1 th where Tii is the maximum slope of the i nonlinearity.  This can be used to represent a linear process with x T Tu x () 50; 8x; 57† actuator limitation nonlinearities which is controlled by  uT  an antiwindup compensator [36,78,30], or a closed loop  system with each component being either a linear sys- Tu () 50: 58† tem or a dynamic arti®cial neural network [117,120]. uT Both of these types of closed loop systems have been extensively studied in the process control literature (see Hence the above S procedure can be equivalently the above references and citations therein). written in terms of quadratic forms. Instead of writing The Lur'e-Lyapunov function is de®ned by the above version which is completely in terms of non- m strict inequalities, we will provide here a version that X qi †k Vxk †ˆ † xT †k Px †‡ k 2  † Q d 64† applies to the case where the main inequality is strict i ii iˆ1 0 (the proof is similar). Let T0; ...; Tp be symmetric matrices. If there exists 150; ...;p50 such that where P is positive de®nite and the Qii are nonnegative Xp so that the Lyapunov function is positive de®nite. The T  T > 0; 59† 0 i i ®rst term is the standard quadratic Lyapunov function iˆ1 which is discussed in many state space systems text- then books [100] and in textbooks on process analysis [70,108], which describe applications to polymerization T T x T0x > 08x 6ˆ 0 such that x Tix50; i ˆ 1; ...; p: 60† and other chemical reactors. The second term was J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 371 introduced by Lur'e [86] to include the nonlinearities of ®nding some matrix  of compatible dimensions (62) explicitly in the Lyapunov function. such that The method of Lyapunov for discrete time systems is to write the divided di€erence for the Lyapunov function: A ‡ PTÂTQ ‡ QTÂP < 0 71†

Vxk † †‡ 1 Vxk † † 65† This equation is solvable for some  if and only if the following two conditions hold: with the state vector substituted in using (61). The T overall system is globally asymptotically stable if the WPAWP < 0 72† undetermined matrices P and Qii can be computed so that Vxk † †‡ 1 Vxk † † is less than zero. The non- T WQAWQ < 0 73† linearities are bounded in the divided di€erence using (63) and the mean value theorem, and the S procedure is where WP and WQ are matrices whose columns are used to convert the divided di€erence subject to the bases for the null spaces of P and Q, respectively. A inequalities (62) to an LMI. With some algebra to col- proof of this result is given in [58]. lect the terms [115,118], it is found that a sucient con- dition for the global asymptotic stability of (61)±(63) is 4.13. Bilinear matrix inequality the existence of a positive-de®nite matrix P and diag- onal positive semide®nite matrices Q and R 2RhÂh such Bilinear inequalities arise in pooling and blending that problems [147], systems analysis [140], and nonlinear  programming. A bilinear matrix inequality (BMI) is of M M 1;1 1;2 > 0 66† the form: M2;1 M2;2 Xm Xn Xm Xn Fx †ˆ; y F0 ‡ xiFi ‡ yjGj ‡ xiyjHij > 0 where iˆ1 jˆ1 iˆ1 jˆ1 74† T T T M1;1 ˆA PA ‡ P †A I C TQC † A I 67† where Gj and Hij are symmetric matrices of the same T T T dimension as F , and y 2Rn. Bilinear matrix inequal- M1;2 ˆA PB †A I C TQCB i ities were popularized by Safonov and co-workers in a †A I TCTQ CTR 68† series of proceedings papers [63±65,125], and ®rst applied to a nontrivial process description (i.e., a che- mical reactive ion etcher) by VanAntwerp and Braatz T T T M2;1 ˆB PA B C TQC † A I QC † A I RC [140], and was later applied to paper machines [141]. A BMI is an LMI in x for ®xed y and an LMI in y for 69† ®xed x, and so is convex in x and convex in y. The bilinear terms make the set not jointly convex in x and y. T T T T T To see this, consider the simplest BMI which is the M2;2 ˆB PB B C TQCB QCB B C Q ‡ 2R bilinear inequality 70† 1 xy > 0; 75† and T ˆ diagfgTii . The new matrix R is introduced by the S procedure. This is an LMI feasibility problem that where x and y are scalar variables. One way to see that has been applied to the analysis of pH neutralization this set is nonconvex is to graph the set in the xy-plane processes [119] and crystallization processes [116] under and apply the de®nition of convexity. Another way to nonlinear feedback control. see this is to consider two elements of the set that con- tradict the de®nition of convexity. For example, con- 4.12. Variable reduction lemma sider †x; y equal to the values † 0:1; 7:9 and † 7:9; 0:1 . Both values satisfy the bilinear inequality since The variable reduction lemma allows the solution of 1 †0:1 †ˆ7:9 1 †7:9 †ˆ0:1 0:21 > 0. But the point algebraic Riccati inequalities that involve a matrix of on the line half way between the two values unknown dimension. This often arises when ®nding the †1=2 0:1; 7:9†‡1=27 †ˆ:9; 0:1 †4; 4 does not satisfy controller that minimizes the H1 norm (see Section 7.5 the bilinear inequality: 1 4Á4 ˆ15 < 0. for an example). Besides bilinear and general quadratic inequalities Given a symmetric matrix A 2RnÂn and two matrices P and Q of column dimension n, consider the problem xTQx ‡ cTx ‡ p > 0; 76† 372 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 general polynomial inequalities can also be written as important to understand that being NP-hard is a property BMIs. Consider, for example, the nonlinear inequality of the problem itself, not of any particular algorithm. It is also important to understand that having a problem x3 ‡ yz < 1 77† be NP-hard does not imply that practical algorithms are not possible. Practical algorithms for NP-hard problems By de®ning x2 ˆ w, and x ˆ v, this inequality is exist and typically involve approximation, heuristics, equivalent to: branch-and-bound, or local search [35,62,104]. Deter- mining whether a problem is polynomial time or NP- 1 xw yz > 0 78† hard informs the systems engineer what kind of accu- racy and speed can be expected by the best algorithms, x v50 and what kinds of algorithms to investigate for provid- v x50 ing practical solutions to the problem. Suppose that a real valued function fx †is de®ned on w vx50 a convex set C 2Rn. The function fx †is convex on C if vx w50 [107] f †lx ‡ †1 l y 4lfx †‡ †1 l fy † 79† Since a BMI describes sets that are not necessarily convex, they can describe much wider classes of con- for all x; y 2 C and l 2 ‰Š0; 1 .Aconvex optimization straint sets than LMIs, and can be used to represent problem has the form more types of optimization and control problems. The main drawback of BMIs is that they are much more inf fx †; 80† dicult to handle computationally than LMIs. x2C

where fx †is a convex function in x, C is a convex set, 5. Optimization problems and inf refers to the in®mum over C. If the in®mum is achieved by an element in C, then the minimization Many optimization and control problems can be problem will be written as min. written in terms of ®nding a feasible solution to a set of Well known problems that can be formulated as con- LMIs or BMIs. Most problems, however, are best writ- vex optimization problems include linear programming ten in terms of optimizing a simple objective function and convex quadratic programming. The advantage of over a set of LMIs or BMIs. There is a fundamental formulating control problems in terms of convex opti- di€erence between the computational requirements for mization problems (when possible) is that wide classes optimization problems over LMIs, and those over of convex optimization problems are in the class P [97]. BMIs. This section begins with an introduction to con- Being in P means that these problems can be provably vex optimization and computational complexity, which solved eciently on a computer. This makes convex provides a fundamental framework for understanding optimization problems desirable for solving large scale the relative complexities of optimization problems. This systems problems. Convex optimization problems often is followed by the de®nition of some optimization pro- occur in engineering practice and many can be written blems that appear when formulating and solving control as LMIs. This is the strength of using LMI formula- problems using LMIs/BMIs. tions. Convex optimizations over LMIs are solvable in polynomial time. 5.1. Computational complexity and convexity Other systems engineering problems cannot be written in terms of LMIs, but can be written in terms of BMIs. Optimization problems are generally characterized as Nearly every control problem of interest can be written being in one of two classes: P and NP-hard [62,104]. The in terms of an optimization problems over BMIs. These class P refers to problems in which the time needed to optimization problems, however, are NP-hard [135], exactly solve the problem can always be bounded by a which implies that it is highly unlikely that there exists a single function which is polynomial in the amount of polynomial-time algorithm for solving these problems. data needed to de®ne the problem. Such problems are This means that algorithms for solving optimization said to be solvable in polynomial time. Although the problems over BMIs are currently limited to problems exact consequences of a problem being NP-hard is still a of modest size. Algorithms and their expected perfor- fundamental open question in the theory of computa- mance will be discussed in more detail in Section 6. For tional complexity, it is generally accepted that a pro- the rest of this section we review the most common LMI blem being NP-hard means that its solution cannot be and BMI optimization problems that appear in control computed in polynomial time in the worst case. It is applications. J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 373

5.2. Semide®nite programming lB A > 0. As l is reduced from some suciently high value, at some point the matrix lB A will lose rank, at The following optimization problem is commonly which point there exists a nonzero vector y that solves referred to as a semide®nite program (SDP) [4]: (85), implying that this value of l is the largest general- ized eigenvalue. Hence inf cTx 81† x Fx †>0 lmax ˆ min l ˆ inf l 86† lBA50 lBA>0

One SDP which often arises in control applications is the LMI eigenvalue problem (EVP). It is the minimiza- Often it is desired to minimize the largest generalized tion of the maximum eigenvalue of a matrix that eigenvalue of two symmetric matrices which depend depends anely on the variable x, subject to an LMI anely on a variable x, subject to an LMI constraint on constraint on x. Many performance analysis tests, such x. as computing the H1 norm in (38), can be written in terms of an EVP [144]. Two common forms of the EVP inf lmax †Ax †; Bx †: 87† Bx †>0 are presented so that readers will recognize them: Cx †>0

inf l 82† Here lmax †Ax †; Bx † is the largest generalized eigen- x;l lIAx †>0 value of two matrices, A and B, each of which depend Bx †>0 anely on x. From (86) this optimization problem is equivalent to inf l 83† x;l inf l: 88† Ax †;l >0 lBx †Ax †>0 Bx †>0 Cx †>0 where Ax †; l is ane in x and l. The equivalence of (81), (82), and (83) will now be The problem of minimizing the maximum generalized demonstrated. The LMI eigenvalue problem (82) can be eigenvalue is a quasiconvex objective function subject to written in the form (83) by de®ning Ax †ˆ; l a convex constraint, where quasiconvexity means that diagfglI Ax †; Bx † (recall that multiple LMIs can be written as a single LMI of larger dimension). To show lmax †A †x ‡ †1  z ; B †x ‡ †1  z that a problem in the form (83) can be written in the ÈÉ 4 max lmax †Ax †; Bx †; lmax †Az †; Bz † 89† form (81), de®ne x^ ˆ ‰ŠxT l T, Fx †ˆ^ Ax †^ ,andcT ˆ ‰Š0T1 T where 0 is a vector of zeros. To see that (81) transforms to (82) consider for all  2 ‰Š0; 1 and all feasible x and z. To see that this is true, ®rst de®ne l^ equal to the right hand side of (89). inf cTx ˆ inf l ˆ inf l ˆ inf l: 84† Then Fx †>0 cTx0 lIAx †>0 Fx †>0 Fx †>0 Fx †>0 ^ ^ l5lmax †Ax †; Bx † and l5lmax †Az †; Bz †: 90† QED. From (86), this implies that 5.3. Generalized eigenvalue problems l^Bx †Ax †50 and l^Bz †Az †50: 91† A large number of the control properties can be computed as a generalized eigenvalue problem (GEVP), It follows that, for all  2 ‰Š0; 1 , including many robustness margins and the minimized hihi condition number discussed in Section 7. A GEVP is,  l^Bx †Ax † ‡ †1  l^Bz †Az †50 92† given square matrices A and B, B > 0, to ®nd scalars l and nonzero vectors y such that () l^B †x ‡ †1  z A †x ‡ †1  z 50: 93† Ay ˆ lBy 85† This and (86) imply that The computation of the largest generalized eigenvalue ^ can be written in terms of an optimization problem with l5lmax †A †x ‡ †1  z ; B †x ‡ †1  z : 94† LMI-like constraints. Consider that the positive de®- niteness of B implies that for suciently large l, QED. 374 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385

5.4. Convex determinant optimization problem 6.1. Solving LMI problems

We will refer to the following as a convex determinant The easiest algorithm to implement for solving LMI optimization problem (CDOP): problems is the ellipsoid algorithm (see Fig. 3) [18]. It Á solves a convex objective function with convex con- inf log det Ax †1 95† straints. In the ®rst step, an ellipsoid is computed that Ax †>0 Bx †>0 contains the optimum point. Often this means comput- ing an ellipsoid that covers the constraint set (see Fig. where A and B are symmetric matrices which are ane 3a). The next step is to compute a plane that passes functions of x. As we will see in Section 7, this problem through the center of the ellipsoid such that the solution appears in a variety of ellipsoidal approximation pro- is guaranteed to lie on one side of the plane (Fig. 3b). blems associated with state and parameter estimation Boyd et al. [22] gives analytical expressions for this cut- problems, and in optimal experimental design. The ting plane for each of the LMI problems. The main proof that log det Ax †1 ˆlog † det †A is con- point is that for each of the LMI problems there is a vex, which implies that a CDOP can be solved relatively half space which is de®nitely ``uphill,'' so that any points eciently on a computer, is given in the appendix. in that half space can be discarded. The remaining half ellipsoid is itself covered by an ellipsoid of minimal 5.5. BMI problem volume (Fig. 3c) and the process is repeated (Fig. 3d) until the algorithm converges to the optimal solution. An optimization over BMI constraints is called a BMI A more computationally ecient algorithm for sol- problem: ving LMI problems is the interior point method [97]. The interior point method uses the constraints to de®ne inf cTx ‡ dTy 96† a barrier function which is convex within the feasible x;y T region and in®nite outside it. This barrier function is Ax †‰ŠTyT >0 Fx †;y >0 incorporated into an objective function, which allows the constrained optimization problem to be replaced where Fx †; y is de®ned in (74). with an unconstrained optimization problem which can Many important problems in control that cannot be be solved using Newton's method. The analytic center is stated in terms of LMIs can be stated in terms of BMIs. de®ned to be the point which minimizes the uncon- Examples include robustness analysis [43,109], a large strained optimization problem. A scalar in the objective number of robust controller synthesis problems includ- to the unconstrained optimization problem is iterated ing low order and decentralized control [125,63], bilin- until the analytic center is optimal for the original problem. ear programming, and linear complementarity problems The interior point method is, in some ways, similar to [2,3,42]. Additionally, a large number of process design the penalty function method [107]. In both cases, the problems can be written exactly or approximately in this constraint set is incorporated into the objective function form [124,147,148]. of an unconstrained optimization problem which can be In the same way that the EVP (82) is an optimization solved using Newton's method. Also, in both cases a over LMI constraints, there is a corresponding optimi- scalar in the objective is iterated until the solution to the zation over BMI constraints called the BMI eigenvalue unconstrained optimization problem is equal to the problem (BEVP): solution to the original problem. However, both the objective functions and the scalar that is iterated are dif- inf 97† x;y; ferent in the two methods. The ellipsoid algorithm, on Ax †‰ŠTyT T >0 the other hand, works more like a standard branch and l max †Fx †;y < bound algorithm [90], in that it is continually discarding infeasible regions from the search. For an optimization where lmax is the maximum eigenvalue of Fx †; y . over a single scalar variable, the ellipsoid algorithm is Using algebra similar as for the LMI eigenvalue pro- equivalent to the bisection algorithm [44]. blem, it can be shown that this is a special case of the A modi®cation to the LMIs that can be critical for BMI optimization problem. obtaining convergence of these algorithms is to include a constraint that keeps the numerics well conditioned and the variables bounded. It is simplest to illustrate 6. Solving optimizations over LMI or BMI constraints this modi®cation with an example. Consider, for exam- ple, the search for a P ˆ PT > 0 that satis®es the LMI Here we outline the algorithms and review software feasibility problem used to solve optimization problems over LMIs and T BMIs. Ai P ‡ PAi < 0; 8i ˆ 1; ...; L 98† J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 375

Fig. 3. The ellipsoid algorithm algorithm (the vectors shown in (b) and (d) are perpendicular to the half spaces).

This was a sucient condition for stability of a class Vandenberghe and Boyd produced the code SP [23] of systems described earlier (26). Now consider aug- which is an implementation of Nesterov and Todd's menting the above LMIs with primal-dual potential reduction method for semide®nite programming (this is an interior point algorithm). SP I < P < I 99† can be called from within Matlab [94]. Boyd and Wu extended the usefulness of the SP program by writing This limits the condition number of P to 1= while SDPSOL [25,26], which is a parser/solver that calls SP. bounding the set of feasible matrices P. The bounding The advantages of SDPSOL are that the problem can be of P does not a€ect the feasibility of the original problem, speci®ed in a high level language, and SDPSOL can run and the condition number limit does not appreciably without Matlab. SDPSOL can, in addition to linear restrict P provided that  is small. The advantage of the objective functions, handle trace and convex determi- condition number limit is that it will prevent the LMI nant objective functions. solution algorithm from converging to a P that could LMITOOL is another software package for solving lead to roundo€ problems [145]. LMI problems that uses the SP solver for its computa- tions [50]. LMITOOL interfaces with Matlab, and there 6.2. Numerical software for solving LMI problems is an associated graphical user interface known as TKLMITOOL [49]. The Induced-Norm Control Tool- Several research groups have produced publicly box [17] is a Matlab toolbox for robust and optimal available software packages for solving LMI problems. control based on LMITOOL. Gahinet and Nemirovskii wrote a software package The 1996 IEEE International Symposium on Com- called LMI-Lab [59] which evolved into the Matlab's puter Aided Control System Design in Dearborn, LMI Control Toolbox [60]. The LMI Control Toolbox Michigan [1] included a session on algorithms and soft- accepts problem statements in a high level mathematical ware for LMI problems. The presentations were focused form and solves the problem with a projective interior more on algorithms than providing comparisons point algorithm. Kojima, Shindoh, and Hara wrote SDPA between software packages or other computational (Semi-De®nite Programming Algorithm) [57], which is results. The numerical results that were presented based on a Mehrotra type predictor-corrector infeasible showed that the currently available software can handle primal-dual interior-point method. It does not allow the problems with Fx †in (2) up to size 100  100. The sol- user to state LMI problems in a high level language. vers that call SP are the easiest to use and can handle 376 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 bigger problems than the other software. As of the An LMI upper bound is derived by local optimization publication of this tutorial, none of the above LMI sol- or by ®xing some of the variables. For instance: vers exploit matrix sparsity to a high degree. inf 4 inf 105† Al †;r;x;y; >0 Al †;r;x;y; >0 6.3. Solving BMI problems  liˆl i l i4li4li  r 4r 4r rj4rj4rj j j j Consider the following BMI problem, where we have P P P P F ‡ l r F >0 F0‡ lirjFij>0 0 i j ij de®ned l and r as those variables which appear in the i j i j BMI constraint: inf 100† With these polynomial-time computable LMI upper AlP †;r;Px;y; >0 and lower bounds, the nonconvex optimization (100) is F0‡ li;rjFij>0 ideal for the application of the branch and bound algo- i j rithm. Interested readers are referred to [137] for more l 4l 4l i i i details. rj4rj4rj where A is jointly ane in l, r, x, y,and . A global optimization approach such as branch and 7. Applications bound is required for guaranteed convergence to the global optimum of a BMI problem because the BMI problem This section lists a variety of LMI and BMI problems is not convex. While several branch and bound algo- that have been or should be studied in process control. rithms have been developed for solving BMI problems [64,155,156], what appears to be the most ecient algo- 7.1. Control structure selection rithm was developed relatively recently [136,140,141, 142,143]. The art to developing an ecient branch and Assume that the matrix M 2RnÂm; n5m is full rank. bound algorithm is to derive tight upper and lower The condition number of M is the ratio of its largest bounds for the objective function over any given part of singular value to its smallest the domain. Reducing the ranges of all problem vari-  †M ables as much as possible is frequently the key to tight  †ˆM : 106† objective function bounding. The approach uses LMI  †M  relaxations as lower bounds for the BMI. inf ˆ inf 5 inf The condition number appears rather naturally in AlP †;r;Px;y; >0 Al †P;r;Px;y; >0 Al †P;r;Px;y; >0 many control problems, including control structure F0‡ li;rjFij>0 F0‡ wijFij>0 F0‡ wijFij>0 selection [121,96,102,130,149] and model identi®cation i j i j i j l 4l 4l l 4l 4l [121,54,83]. It is certainly the one of the most used (and i i i i i i l i4li4li      misused [82,28,29]) matrix functions in process control. rj4rj4rj rj4rj4rj  ri4ri4ri wijˆlirj hi Its application to chemical processes such as distillation w 2 w ;w columns is described in many undergraduate process ij  ij ij control textbooks [102,126]. 101† Another matrix function that is more relevant to many applications is the minimized condition number: where the overbar (underbar) indicates the upper (lower) bound for a variable and inf  †LMR 107† no R;L w ˆ min l r ; l r ; l r ; l r 102†  ij i j i j i j i j no where L 2RnÂn and R 2RmÂm are diagonal and non-   w ij ˆ max lir j; lirj; lirj; lirj : 103† singular. The minimized condition number (107) is used     for integral controllability tests based on steady-state Further, because wij is a bilinear term the following information [67,96] and for the selection of sensors and additional constraints may be included in the lower actuators using dynamic information [38,112,38,99,98]. bound (101) [90]: The sensitivity of stability to uncertainty in control sys- tems is given in terms of the minimized condition num-   wij4rjli ‡ lirj rjli ber in [127,128]. The minimized condition number is   applied regularly in the process industries, as part of the wij4lirj ‡ rjli lirj   : 104† Robust Multivariable Predictive Control Technology   wij5lirj ‡ rjli rjli sold by Honeywell [89]. The application to a fractio- w 5l r ‡ r l l r ij i j  j i i j nator and a paper machine is described in [89]. J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 377

It was shown in [30] how to pose the minimized con- Thus, the maximum volume ellipsoid E contained in dition number as a GEVP (88). Here we provide an the polytope P is given by alternative derivation that follows the later derivation in [22]. Note that the de®nition of the condition number max log det †B 118† BˆBT>0;d implies that it is greater than or equal to 1. For 51, we T kBAik‡Ai d4bi have that

 †LMR 4 () I4 †LMR T †LMR 4 2I 108† This optimization is convex in the variables B and d. For the case where the center of the ellipsoid is known  T (e.g. d ˆ 0 when Ax4b de®nes a symmetric polytope), () I4 L^ MR L^ MR 4 2I 109† (117) can be written as an LMI using the Schur com- plement lemma: Á  Á () RRT 14MT L^ TL^ M4 2 RRT 1 110† T T k BAi k‡Ai d4bi () bi Ai d50 and Á T 2 2 T 2 () Q4M PM4 Q 111† k BAi k 4 bi Ai d 119† Á for diagonal P > 0 2RnÂn and diagonal Q > 0 2RmÂm. T T 2 T 1 () bi Ai d50 and bi Ai d Ai BI BAi50 Therefore, solving the minimized condition number problem (107) is equivalent to solving the GEVP (88): 120†

2 Á inf 112† b ATd 2 ATB P>0 () i i i 50; 8i 2 ‰Š1; L 121† Q>0 BAi I Q4MTPM4 2Q where P and Q are diagonal. Hence in this case (118) can be written as the CDOP (95): 7.2. Parameter estimation and model predictive control max log det †B 122† BˆBT>0;d The approximation of polytopes with ellipsoids have T 2 T †biA d A B numerous applications, including parameter estimation i i 50 BA I [39,56,80] and model predictive control [34,138,139]. i The model predictive control application has been implemented on paper machine models constructed In the case where d is unknown, (118) is not an LMI from industrial data [139]. but is still a convex program that can be solved, for An ellipsoid has the form instance, by interior point methods [76,97]. ÈÉ A related problem of interest is to determine the " ˆ By ‡ djky k 41 113† smallest ellipsoid which encloses a given polytope. First de®ne the convex hull of a given set of points T in Rn as the set of all convex combinations of points in T .An equivalent de®nition is the smallest convex set contain- where B ˆ BT > 0. This ellipsoid is centered at d and ing T [105]. Let the polytope be described as the convex has volume proportional to det †B . Consider the poly- hull of its vertices tope PˆCofgv ; ...; v 123† ÈÉ 1 L T Pˆ xjAi x4bi; i ˆ 1; ...; L 114† and write the ellipsoid T where Ai is the ith row of the matrix A. An ellipsoid E is ÈÉ contained inside the polytope P if Eˆ xjkAx b k 41; A ˆ AT > 0 ; 124†

AT †By ‡ d 4b 8y; k y k 41 115† 1 i i whereÁ its center is A b and its volume is proportional to det A1 . Then the minimum volume ellipsoid which T T encloses the polytope is given by () max Ai By ‡ Ai d4bi 116† kyk41 Á inf log det A1 : 125† AˆAT>0 T kAv bk41 () k BAi k‡Ai d4bi 117† i 378 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385

This problem is convex in A and b, and can be written where M is a complex matrix and D is the set of com- as a CDOP (95) by applying the results of Section 4.6: plex nonsingular block diagonal matrices with some Á blocks possibly being repeated. This robustness margin inf log det A1 126† has been applied to numerous processes over the past 15 AˆAT>0 T years, including distillation columns [29,40,96,130], pH 1 †Avib >0 neutralization [119], packed bed reactors [47], paper Av bI i machines [81,122,33], polymer ®lm extrusion [52,55], and reactive ion etching [143]. 7.3. Optimal design of experiments This problem can be written in terms of a GEVP (88):

The goal of optimal experimental design is to max- 2 ˆ inf 2 131† imize the informativeness of data collected from the D2D à process [11]. Optimal experimental design algorithms †DMD1 †DMD1 4 2I have been applied to chemical kinetics [20,19,21,73,113], 2 synthetic ®ber manufacture [75], petroleum fractiona- ˆ inf 132† D2D tion [132], crystallization [92], distillation [79], and MÃDÃDM4 2DÃD polymer ®lm extrusion [53]. While most formulations require the solution of nonconvex optimization pro- ˆ inf 2 133† blems [69,150], here is presented a formulation for linear P2P MÃPM4 2P parameter estimation which requires only the solution of a CDOP (95). The goal is to estimate a vector of parameters x from where P is the set of complex symmetric positive de®nite some measurement y ˆ Ax ‡ w where A is a matrix of n  n block diagonal matrices with the corresponding inputs and w is zero-mean white measurement noise. blocks from D being repeated. The error covariance of the minimum variance esti- T 1 T mator is †A A . If the rows of A ˆ ‰Ša1; ...; aL are 7.5. Robust nonlinear controller synthesis chosen from a set of possible test vectors, BMI formulations arise naturally in the design of ai 2 fgv1; ...; vL ; i ˆ 1; ...; L; 127† robust optimal inversion-based controllers for nonlinear the goal of D-optimal experimental design is to select processes. Here we present the BMI formulation which was the vectors so that the determinant of the error covar- applied to the nonlinear simulation model of a reactive iance is minimized. ion etcher constructed from experimental data presented We can write in Section 2. The BMI-based controller demonstrated substantially improved performance and robustness XL T T over a traditional nonlinear controller [140,143]. A A ˆ livivi 128† iˆ1 After the nonlinear inversion technique removed the most signi®cant nonlinearities, the control synthesis where li50 is the fraction of rows equal to the vector vi, problem consisted of designing a linear controller for a PL which implies that iˆ1li ˆ 1. When L is a large num- linear plant subject to norm-bounded nonlinear time ber, the li can be treated as continuous variables instead varying perturbations. The state space realization for of integer multiples of 1/L. the plant transfer function Gs †ˆCsI † A 1B ‡ D Then the D-optimal design problem is the CDOP (95) was represented by [153]: 2 3 ! AB B XL 1 1 2 T G ˆ 4 5 134† inf log det livivi 129† C1 D11 D12 l 50 i iˆ1 PL C2 D21 D22 liˆ1 iˆ1 where D22 ˆ 0 without loss of generality [157]. The controller that optimizes the induced 2-norm perfor- 7.4. Robust control system analysis mance objective subject to the constraint of stability of the closed loop system with norm-bounded nonlinear The robustness margin for a variety of linear systems time varying perturbations was computed from the subject to linear or nonlinear perturbations solution of the BEVP (97) [140,143]: [31,71,96,123,130] can be computed by solving Á à ˆ inf 135†  ˆ inf  DMD1 130† †2BL;R;X;Y D2D lmax †L1R1 41 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 379 where B is the set such that L, R, X and Y are symmetric 7.6. Robust model predictive control matrices,  Here we describe an LMI-based robust model pre- L1 0 R1 0 dictive control algorithm which applied to a non- L ˆ ; R ˆ ; L1; R1 2D; 136† 0 I 0 I isothermal nonadiabatic continuous stirred tank reactor (CSTR) [77]. The LMI approach provided similar per- and formance as a non-LMI-based model predictive control algorithm, while having the capability of providing 2  3 ? 2 3 robustness to model uncertainty. B2 T T 0 AX ‡ XA XC1 B1 6 D 74 5 Consider the discrete time time varying linear system 4 12 5 C1X LD11 BT DT R 0 I 1 11 xk †ˆ‡ 1 Ak †xk †‡Bk †uk † 140† 2 3 ?T yk †ˆCk †xk † 141† B 6 2 0 7  6 D12 7< 0; 137† 4 5 where each state space matrix is arbitrarily time varying 0 I and lies within a polytope (see Section 4.3). De®ne xk †jk as the state of the uncertain system measured at 2 3 sampling time k, xk †‡ ijk as the state of the system at !? CT 2 3 time k ‡ i predicted at time k, uk †‡ ijk as the control 6 2 7 YA ‡ ATYYB CT 0 1 1 move at time k ‡ i computed at time k, and W and R 6 DT 74 T 5 6 21 7 B1 Y RD11 4 5 T are positive de®nite weighting matrices. For this control C1 D L 0 I 11 problem, the objective was to compute the state feed- 2 3 back matrix F: !?T CT 6 2 7 6 0 7 uk †ˆ‡ ijk Fx † k ‡ ijk 142†  6 DT 7< 0; 138† 4 21 5 0 I so as to minimize an upper bound on the in®nite hor- izon quadratic objective:  X1 XI T > 0: 139† xk †‡ ijk Wx †‡ k ‡ ijk uk †‡ ijk Ru † k ‡ ijk : 143† IY iˆ0

Here A? is a matrix whose rows form a basis for the at sampling time k. This state feedback matrix is given null space of AT. The only nonconvexity in (135) is the by [77]: constraint lmax †L1R1 41 (which is a BMI). As the algebra of this derivation is lengthy and F ˆ YQ1 144† involved, only a summary is given here. The state space equations for the closed loop system are written as where Q > 0andY are solutions to the following EVP functions of the state space matrices of the plant and the (82): controller. A version of the Bounded real lemma is used inf 145† to write the induced 2-norm performance objective in ;Q;Y terms of matrix inequalities, and the variable reduction lemma of Section 4.12 is used to remove explicit depen- subject to dence of the matrix inequalities on the controller state  1 1 xk †jk T space matrices. Finally, D and D are replaced with L 50 146† and R and the additional constraint that lmax †R1L1 xk †jk Q < 1. Readers interested in a detailed derivation are referred to [137]. 2 3 T T T 1=2 T 1=2 A closely related formulation was used in the design QQAi ‡ Y Bi QW Y R 6 7 of linear controllers that optimize the robust perfor- 6 AiQ ‡ BiYQ 007 4 1=2 550; mance for large scale sheet and ®lm processes, such as W Q 0 I 0 1=2 polymer ®lm extruders and paper machines [141]. In R Y 00 I those particular applications, the only process non- 8i ˆ 1; 2; ...; L: linearities were perturbations about the nominal linear dynamics. 147† 380 J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385

The derivation uses a positive de®nite quadratic An interesting variation on the LPV approach is to function of the state to bound the performance objective treat the parameters as validity functions for linear (143), and then uses the Schur complement lemma to models used to represent the nonlinear process dynam- manipulate this inequality into the form of the LMI ics locally [9,12,13]. Each local model is assigned to an constraint (147). Input and output constraints can also element of the vector pk †which approaches 1 when the be handled by augmenting the EVP with additional plant moves into the local region of the model and LMI constraints (see [77] for further details). approaches 0 as the plant moves into other regions. The The state feedback matrix F is computed at each elements of pk †sum up to 1 at each time instance. sampling instance k, and used to compute the control While [13] applies an induced 2-norm approach simi- move uk †ˆuk †jk to be implemented. When a new lar to that described above, [9] proposes an LPV-based measurement is taken, F and the control move are re- model predictive control (MPC) design procedure which computed. The model predictive controller can be is an extension of the approach discussed in Section 7.6 shown to be stabilizing for all matrices within the matrix (the LMIs have a similar structure as those in Section polytope [77]. 7.6). The LPV-based MPC control algorithm is shown to asymptotically stabilize the closed loop LPV process. 7.7. Gain-scheduled/linear parameter varying systems The algorithm was applied to a continuous stirred tank reactor with output multiplicity, and to a semibatch Gain scheduling is discussed in undergraduate process reactor for free-radical polymerization of polymethyl control textbooks [102,126]. A relatively new approach methacrylate. Although the closed loop performance of to the design of gain-scheduled controllers is to repre- the LMI-based LPV-MPC algorithm was not quite as sent the process as being linear parameter varying good as an LPV-based quadratic programming algo- (LPV): rithm (similar to traditional MPC), it had the advantage of guaranteeing closed loop stability. xk †ˆ‡ 1 Apk † †xk †‡Bpk † †uk † 148† yk †ˆCpk † †xk †‡Dpk † †uk † 149† 8. Conclusions where the state space matrices are explicit functions of a A tutorial was provided on the mathematical theory time varying parameter vector pk †. An LPV process and process control applications of linear and bilinear reduces to a linear time varying process for a given tra- matrix inequalities. Many common convex inequalities jectory, and reduces to a linear time invariant system for occurring in nonlinear programming and several tests a constant parameter vector pk †. This model repre- for the stability of linear and nonlinear systems were sentation forms the basis for a solid theoretical frame- written in terms of LMI feasibility problems. Algo- work for the design of gain-scheduled controllers using rithms for solving optimization problems with LMI or LMIs [103,152,8,7]. BMI constraints and publicly available software were It is common to assume that the state space matrices reviewed. This was followed by a survey of applications are ane functions of pk †and that the time varying of LMIs and BMIs to control problems associated with parameter pk †varies within a polytope. Then the gain- chemical and mechanical processes. These included scheduled (or LPV) controller has a form control structure selection, parameter estimation, experimental design, and optimal linear and nonlinear ^ ^ x^ †ˆk ‡ 1 A †pk †x^ †‡k B †pk †yk † 150† feedback control. The authors believe that LMIs and BMIs form a set uk †ˆC^ †pk †x^ †‡k D^ †pk †yk † 151† of mathematical tools which are fundamental to the background of a process control engineer. It is hoped similar to that for the process. The controller is assumed that the many examples provided throughout the paper to be able to measure or estimate pk †on-line, so this provide a convincing justi®cation for this belief. information can be used by the controller to provide improved performance over controllers which do not exploit such information. The controller matrices that Appendix guarantee global asymptotic stability and minimize an induced 2-norm performance objective can be computed Proof of the Schur complement lemma as an EVP. The LMIs are derived using a quadratic Lyapunov function and a generalization of the Bounded ()) Assume real lemma. The EVP is somewhat similar to the BEVP  in Section 7.5, but with L ˆ R ˆ I, and so will not be Qx † Sx † T > 0 152† given here (see [6,7] for the exact form of the LMIs). Sx † Rx † J.G. VanAntwerp, R.D. Braatz / Journal of Process Control 10 (2000) 363±385 381 X  and de®ne 1=2 1=2 ˆlog † det †A0 log 1 ‡ tli A HA :  0 0 u T Qx † Sx † u i Fu †ˆ; v 153† 159† v Sx †T Rx † v then The last step follows because the determinant is the T product of the eigenvalues. The condition A ˆ A0 > 0 Fu †; v > 0 8‰Š6ˆuv 0 154† 0 implies that its matrix square root exists, and A0 ‡ tH > 1=2 1=2 0 implies that I ‡ tA0 HA0 > 0. Hence First, consider u ˆ 0. Then 1=2 1=2 1 ‡ tli A0 HA0 † > 0: F †ˆ0; v vTRx †v > 0; 8v 6ˆ 0 ) Rx †> 0: The ®rst and second derivatives of Next consider 1=2 1=2 log 1 ‡ tli A0 HA0 ††  ˆRx †1Sx †Tu; with u 6ˆ 0: are Then  d 1=2 1=2 Á log 1 ‡ tli A HA Fu †ˆ; v uT Qx †Sx †Rx †1Sx †T u > 0; 8u 6ˆ 0 dt 0 0  ˆl A1=2HA1=2 = 1 ‡ tl A1=2HA1=2 ; ) Qx †Sx †Rx †1Sx †T> 0: i 0 0 i 0 0 160† (() Now assume 2  d 1=2 1=2 log 1 ‡ tli A HA Qx †Sx †Rx †1Sx †T> 0; Rx †> 0: 155† dt2 0 0 2 ˆ l2 A1=2HA1=2 = 1 ‡ tl A1=2HA1=2 > 0: with Fu †; v de®ned as in (153). i 0 0 i 0 0 We will ®x u and optimize over v. 161†

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