Robust Controller Design Method for Systems with Parametric Uncertainties

Transcription

1 0 ICASE: he Institute of Control, Automation and Systems Engineers, KOREA Vol, No, June, 000 Robust Controller esign ethod for Systems with arametric Uncertainties Jietae Lee, oe-gyoon Koo, and homas F Edgar Abstract: his paper presents iterative schemes and continuation schemes for designing robust controllers which stabilize dynamic systems having bounded parametric uncertainties Utilizing results of the cheap control problem, some eistence conditions of the robust controller are obtained, which are different from the matching conditions Continuation schemes are used to overcome the divergence problem of iterative schemes he robust controller design method is etended to nonlinear system and easily implementable series solution is also obtained Results are illustrated with simple eamples Keywords: optimal control, robust control, linear system, cheap control, nonlinear system, series solution I Introduction here has been great interest recently in the influence of uncertainty in model parameters on the robustness of various process control strategies here are many cases where parameters may be changing during operations and their eact values are difficult to measure on-line In order to preserve closed-loop stability under parameter variations robust controllers can be designed if bounds on the parameter variations are nown any methods to obtain robust controllers have been reported hey are classified into three categories(silja, 989): frequency domain methods, algebraic methods by the analysis of characteristic polynomials, and methods by the Lyapunov stability theorem Frequency domain methods etend the classical methods for SISO systems based on gain and phase margins hey are now well-established and very powerful for linear multivariable uncertain systems (orato, 987) Algebraic methods etract the robustness from the characteristic polynomial; recent progress in mathematical programming methods and computer graphics mae the methods more attractive and promising (Silja, 989) Lyapunov methods have long been used in analyzing stability and robustness of control system since the wor of Kalman and Bertram (960) hey directly use the state equations and unlie the other two approaches mentioned above, they can easily be applied to nonlinear systems Using the Lyapunov method, Leitmann (979) obtained a robust controller for a class of uncertain systems which satisfy some conditions nown as the matching conditions his class of uncertain systems can be stabilized robustly for an arbitrarily large uncertainty bound Other researchers have characterized and enlarged this class of uncertain systems (see references in Corless and Leitmann, 988) Some results for nonlinear systems have been also reported, for eample, a cone type of nonlinear perturbation (Noldus, 98) and some nonlinear systems which are linearizable by the differential geometry method (Ha and Gilbert, 987; Kravaris and alani, 988) he Lyapunov method is very powerful but is highly anuscript received: April 3, 000, Accepted: June 3, 000 Jietae Lee: ept of Chem Eng, Kyungpoo National University oe-gyoon Koo: ept of Chem Eng, Kyungpoo National University homas F Edgar : ept of Chem Eng, University of eas, Austin dependent on the clever choice of a Lyapunov function which may sometimes be difficult to construct (Vannelli and Vidyasger, 98) Chang and eng (97) proposed a robust controller design method nown as guaranteed-cost control It is based on the optimal control method, and uses a suboptimal cost as the Lyapunov function In this method, an upper bound of the cost is obtained for bounded structured uncertainties (Rissanen, 966) For linear systems, they obtained a robust controller by solving Riccati equations iteratively Kosmidou and Bertrand (989) proposed a different iterative method whose convergence is guaranteed eterson and Hollot (986) proposed a robust controller design method which was also based on the Riccati equation but was non-iterative In this paper we further eploit the Kosmidou and Bertrand method and the Chang and eng method In section II, we have obtained a new convergent iterative scheme for the Kosmidou and Bertrand method and have shown that eistence of the linear robust control law is closely related to the cheap control problem We also propose a continuation scheme to find the maimum allowable perturbation In section III, nonlinear uncertain systems have been considered and a nonlinear robust control law, which is in series form, has been obtained In section IV, for a low gain robust control law, the Chang and eng method has been treated II Linear uncertain system Consider the linear dynamic system: &( t) = ( A q ( t) B t) () n m where R, u R, q is an unnown constant and A, and B are nown matrices with appropriate dimensions he problem is to find a feedbac control law which stabilizes the system () for any q [ q, ] while not significantly degrading the cost functional: = J ( ( t0 ), )) ( Q u Ru) dt t0 where R > 0 and Q 0 Chang and eng (97) formulated this problem as the guaranteed cost control problem and obtained a robust control law: Here is the positive definite solution of = R B ()

2 ransaction on Control, Automation, and Systems Engineering Vol, No, June, 000 A A BR B Q Φ( = 0 (3) where for any q q, q ], [ q( ) Φ( hree candidates for Φ ( have been proposed: (a) (Chang and eng, 97) Φ( = S diag{ λ i( ) } S () where S is a diagonalization transformation matri such that = S (b) (eterson and Hollot, 986) diag{ λ ( )} S Φ ( = VV W W () where =VW and for some positive constant (c) (Kosmidou and Bertrand, 988) Φ ( = (6) Equation ()was originally proposed by Chang and eng (97) hey obtained a robust control law by solving the resulting Riccati equation iteratively he convergence of the iterative scheme was not proven due to the nonlinearity of equation () about Equation () results in A A ( BR B VV ) Q W W = 0 If BR B VV > 0, it has a positive definite solution and the control law () becomes a robust control law ore elaborate results were given by eterson and Hollot (986) Equation (6) was used by Kosmidou and Bertrand (988) hey proposed a convergent iterative scheme for the resulting Riccati equation heir iterative scheme is simple but it is somewhat hard to find an initial value for which convergence is guaranteed Here we propose a new convergent iterative scheme for the Riccati equation (3) with (6): A A BR B Q = 0 (7) and by applying the cheap control theorem to the iterative scheme, eistence conditions of a positive definite solution of (7) are eploited Iterative scheme o solve equation (7), we propose an iterative scheme: A A BR B Q = 0 0 = 0 (8) where A = ( A I) Lemma : Assume that ( A, B) is stabilizable he sequence is non-decreasing roof: Since for a b, the above a b i lemma follows from heorem of Wimmer (98) heorem : Assume that ( A, B) is stabilizable he iterative scheme (8) converges if and only if a positive definite solution of equation (7) eists roof: If a positive definite solution eist, the sequence is bounded above by the solution as 0 Λ Hence it converges to a positive definite solution Necessary part is self-evident he above theorem shows that we can obtain a positive definite solution of (7) if such a solution eists hat is, we can obtain a stabilizing controller (lubelle et al, 986) by the iterative scheme (8) Here some eistence results for equation (7) are sought hat is, using results on the cheap control problem (Sannuti, 983; Saberi and Sannuti, 989), a bound of the sequence is eploited Since is non-decreasing, an upper bound guarantees its convergence heorem : Let =VW he iterative scheme (8) is convergent for q 0, α ) where ( α = ( V ) (9) σ V and is the positive definite solution of A A BR B (0) Q W W = 0 where, the value of means some positive constant roof: Let = hen equation (0) becomes A A BR B Q W W = 0 he equality (9) means that W W If W W, and q W W Since 0 W W, for all hat is, the sequence is bounded by Convergence follows Corollary : Assume that the dynamic system = A Bu () y = W is stabilizable and detectable, and it has no transmission zeros with zero real parts hen α = σ ( V V o( )) where is the asymptotic solution of equation (0) as goes to zero roof: see Sannute(983) he equation for is available in Sannuti (983) It depends only on the system matrices ( A, B, W) of (), so computation of is straightforward However, its analytical description is rather complicated because it requires some magnitude scaling and transformation of states he above corollary shows that, if V V = 0, α grows to infinity as goes to zero he system with such uncertainties can be stabilized for arbitrarily large uncertainty bounds lie 0

3 Ψ(, q ) = 0 ICASE: he Institute of Control, Automation and Systems Engineers, KOREA Vol, No, June, 000 uncertainties satisfying the matching conditions For some systems, V V is zero regardless of V Corollary : If the system () is stabilizable, detectable and of minimum phase, then α grows to infinity as goes to zero roof: Under the condition, = 0 (Sannuti, 983) Eample : Consider a system = q 0 0 u Q = and R = If one chooses between 0 and, the system satisfies the condition of Corollary Hence the iterative scheme (8) converges regardless of the magnitude of q he system matri used in Corollaries and is A Sometimes its structure is much different from A and is inadequate for the above analysis Results which can be applied directly to the system matri A are now sought heorem 3: he iterative scheme (8) is convergent for q ( 0, α ] and = µ σ ( µ ) where and µ is the solution of A A µ µ σ µ V α = µ ( σ ( ) ( V )) () µ µ BR B µ µ µ Q µ I W W = 0 roof: he equality () means that µ µ µ I W W Hence, as in heorem, µ µ bounds the sequence and convergence follows Eample : Consider a system 0 = 0 q 0 0 u R = and Q = dia,0) hen we have µ 0( µ ) = 3 0( µ ) 3 0( µ ) 0( µ ) Hence α grows to infinity as µ goes to zero hat is, a solution of (7) can be found by the iterative scheme (8) for an arbitrary large q, whenever is chosen as in heorem 3 Corollary 3: Assume that the system = A Bu y = W is stabilizable, detectable and of minimum phase If WB is non-singular, then α grows to infinity as µ goes to zero roof: Under the conditions, µ = O(µ ) (Sannuti, 983) As in eterson and Hollot (986), in consideration of the above analysis, the part of uncertainty matri which satisfies the matching conditions can be removed hat is, if can be decomposed as = BWa VW, then we can choose BR B Φ( = q WaRWa W V VW and the resulting Riccati equation has the same structure as that of equation (7) for V and W he above asymptotic analysis can be applied to the part VW of For some cone-bounded nonlinear perturbation such as = As qf ( Bu (3) the above results can be etended and a robust linear controller can be obtained heorem : If the nonlinear function f ( is bounded as, for any positive definite f( f( () and the solution of equation (7) eists, ;then the control law () stabilizes the system (3) robustly for any q [ q, ] roof: Under the condition (), becomes a Lyapunov function for the system (3) for any q [ q, ] Continuation scheme Sometimes there does not eist a solution of (7) for the required q In this case we should choose another method or reduce q for a robust controller Here, as a method illustrating the latter approach, a continuation method (Allgower and Georg, 980) is adopted to find a maimum possible q as well as a solution he continuation method has been used as a globally convergent method for nonlinear equations which are very hard to solve For eample, it has been applied to solve the Riccati equation (Jamshidi et al, 970) and a root-locus problem (an and Chao, 978) Assume that and q are functions of a dummy parameter τ and let Ψ( ( τ ), ( τ )) = A A () BR B Q We trac satisfying as q varies o do this, we differentiate equation () about the dummy parameter τ and equate as Ψ& ( ( τ ), ( τ )) = & ( A BR B ) ( A BR B ) & (6) & & = Ψ( ) τ Since Ψ( ( τ ), ( τ )) = e Ψ( (0), (0)), (τ ) becomes the solution of (7) for q (τ ) whenever (0) is the solution of (7) for q (0) By applying the Euler integration rule to the equation (6), we can obtain ( A, q q BR B ) ( A = ( BR BR B ), q& B Q (7a), =, q& (7b) We solve equation (7a) iteratively with a standard Lyapunov equation solver, that is,

4 ransaction on Control, Automation, and Systems Engineering Vol, No, June, 000 3, i ( A BR = ( BR B ) ( A BR, q& B Q B ),, i, 0 = (8), i heorem : Assume that is positive definite and ( A BR ) is stable he iterative scheme (8) is convergent if and only if equation (7a) has a positive definite solution roof of heorem is very similar to that of heorem and omitted here According to the heorem, we increase q to the point just before the iterative scheme (8) diverges III Etension to nonlinear systems In this section, the method of section II is etended to nonlinear systems Consider a nonlinear system described by with a cost functional: ( t) = f ( ( t)) qf ( ( t)) ( t)) t) (9) J ( ( t ), )) = ( m( u 0 Ru ) t0 dt Here f ( 0) = f (0) = 0 It is assumed that the above system is controllable and observable for any q [ q, ] (oylan and Anderson, 973) and all nonlinear functions are analytic A guaranteed cost control law is = R ) (0) where is the positive definite solution of following Hamilton-Jacobi equation: f ( m( Φ(, f R ( ) = 0 () heorem 6: Assume that the Hamilton-Jacobi equation () has a positive definite solution around the origin Let Ω be a level set of the function φ ( such that, for some positive constant v 0, 0 φ ( v0, and for any q q, q ], [ q f ( Φ(, f ( ) () hen the system (9) with the control law (0) is asymptotically stable in the region Ω such that every trajectory starting in Ω remains in Ω and converges to the origin roof: With in Ω, becomes a Lyapunov function for the system (9) with control law (0) his theorem is followed from the Lyapunov theorem (Chang and eng, 97; Hahn 967) It is remared that system (9) with control law (0) remains stable in the case of a 0% reduction in controller gain (Glad, 987) and hence the true attraction region may be larger than Ω We can obtain a robust control law by choosing a specific Φ (, f ( ) and sonving the Hamilton-Jacobi equation () If f ( is of the form f ( = r(, then we can choose and we have Φ(, f ( ) = R f ( m( r( Rr( = 0 r( Rr( ( ) R (3) he above equation satisfies condition () automatically and may have a positive definite solution around the origin Furthermore it can be solved via the standard series solution method for the nonlinear optimal regulation problem (Lues, 969) his uncertainty is included in the matching conditions For general uncertainties, we choose ( ( ), ( )) Φ φ f = [ f ( f ( ))] which is an etension of the linear quadratic problem of section II hen equation () becomes ( f ( m( f ( f R ( ) = 0 () We solve equation () by the power series method For this, we assume that all nonlinear functions are epanded as () f ( = A f ( Λ f () ( = f ( Λ () g ( = B g ( Λ (3) m( = Q m ( Λ and the linear quadratic problem of equation (7) has a solution Under the assumptions that the linear quadratic solution eists, the local power series solution of equation () may eist But its proof would require more elaborate study as in the nonlinear optimal regulation problem (Lues, 969) Assume that the solution of equation () is represented as (3) φ ( = φ ( Λ hen we have the equation (7) for and ( ) φ = h ( ) ( ) ( ( A BR (, = 3,,Λ B ) ( ) φ ( there h ( is a function of all nown and previously calculated variables [ j ] heorem 7:efine such that

5 ICASE: he Institute of Control, Automation and Systems Engineers, KOREA Vol, No, June, 000 = ( where is a leio-graphic listing vector for (j-)th order polynomials of (Brocett, 976) If ( A BR B ) is stable and λi ( ( A BR B ) ) λl ( ), for all i and l, then the solution of ()for j eists ( j) roof: Let φ ( = (Yoshida and Loparo, 989) Equation () becomes ( A BR = H B ) [ j ] ( j) [ J ] where H = h ( Hence we have ( A [ J ] [ J ] [ J ] [ J ] BR B ) = H [ j ] Under the above condition can be calculated (Bartell and stewart, 97) he above condition is easy to chec because it depends only on the linearized terms Higher order terms can be calculated with some symbolic operation of multivariable polynomials and solving systems of linear equations (Lee, 990) o use the series of φ (, it is truncated at some order and the region of attraction of the truncated control law should be determined IV Reducing control costand magnitude of feedbac gain High feedbac gain is not recommended when unmodeled high frequency parasitic dynamic eists So it would be important to reduce the controller gain while maintaining the robustness eterson and Barmish (987) studied properties of the lowest gain robust controller But such a controller is very hard to obtain he robust controller design method of Chang and eng (97) sometimes gives a much lower controller gain than that of section II Here we reformulate their scheme with the matri square root to obtain some analytical results and more attractive numerical methods Equation (3) combined with () can be written as A A BR B Q U = 0 (6a) U ( ) = 0 (6b) where U is symmetric and positive semi-definite We solve the equation (6) iteratively as q A A BR B Q [( ) ] = 0, 0 = 0 where [ ] means the principal matri square root (see Appendi for calculation by the transformation method) Equation (6b) is much easier to manipulate than the original one of Chang and eng (97) But it is still difficult to prove the convergence of its iterative scheme (7) ecept for a limited class of For eample, if is symmetric and orthogonal (that is, = I and = ), then the sequence is non-decreasing So it will converge if a positive definite solution of (6) eists Utilizing the solution of equation (7), following bounds for the sequence of iterative scheme (7) are obtained heorem 8: Assume that is symmetric If there eist such that σ ( ( σ ( ) σ m ( )) σ m ( ) (8) there is the solution of (7) and is the first iteration of (7), then the sequence of (7) is bounded between and roof: For any between, inequality (8) implies that ( Hence we have ( ) ( herefore, from the inequality about the matri square root (Bellman, 96), we have [( ) ] q ( heorem 8 follows heorem 8 shows that, for some uncertainties, iteration of (7) is bounded and the solution of (6) is not greater than some, a solution of (7) Since σ m( ) is independent on q and σ ( ) goes to some constant as q decreases, we may always find which satisfies the condition (8) for sufficiently small q Because we can differentate equation (6b) with respect to q and we can also apply the continuation method to the equation (6) as ( A BR B ) ( A BR BR B Q U = 0 U ( q q U U U )( q& (, q, =, q&, [ ) = 0 )] B ) )( ) Continuation will be used to prevent convergence problems of the iterative scheme (7) and to find the maimum possible q he method can be also etended to treat some nonlinear perturbations We will illustrate it with a simple eample in the following section V Eamples Here we compare schemes by applying them to the wellnown Van der ol oscillator problem Eample 3: Consider the dynamical system = with a cost functional = q u

6 ransaction on Control, Automation, and Systems Engineering Vol, No, June, 000 J ( ( t 0 ), u ( )) = ( u t 0 ) dt Let q = With restricting as, we can obtain three linear robust control laws as (from equation ()) (from equation ()) 3 97 = 97 0 u = (97 0 ) 80 = u = ( ), for (from equation (6)) = 9 07 = u = (07 076), for = 0 06 For this problem the cost and feedbac control gain obtained from the equation (6) is larger than those of the other two wo nonlinear robust control laws are computed as (from equation (3)) = = ( Λ ), Λ 3 for = the linear robust controller of equation (6), c) for the nonlinear robust controller of equation () (truncated at the -th order) (from equation ()) = = ( Λ ), Λ 3 for = Attraction regions for the linear controller of equation (6) and the nonlinear controller of equation () (truncated at the fifth order) are shown in Fig For this problem, the nonlinear controller has much larger attraction region and gives lower control cost than that of the linear controller Eample : Consider the dynamical system (Glad, 987) with a cost functional = q = u J ( ( t 0 ), u ( )) = ( u ) dt Let q = Restricting as, we can obtain a linear robust control law from equation (6) as 97 = 636 u = (636 t ), for = 0 he nonlinear robust control law is computed from equation () as = Λ Fig Attraction regions for the system of Eample 3 a) for the linear robust controller of equation (6) with 0% feedbac gain reduction tolerance, b) for Fig Attraction regions for the system of Eample

7 6 ICASE: he Institute of Control, Automation and Systems Engineers, KOREA Vol, No, June, 000 a) for the linear robust controller of equation (6) with 0% feedbac gain reduction tolerance, b) for the linear robust controller of equation (6), c) for the nonlinear robust controller of equation () (truncated at the 3-rd order) = ( Λ ), for = 0 Attraction regions for the above controllers are shown in Fig For this problem, the nonlinear controller has a smaller attraction region but gives a much lower control cost than that of the linear controller VI Conclusions Guaranteed-cost control methods of Chang and eng (97) and Kosmidou and Bertrand (988) have been re-eamined for designing a robust controller We obtained a new convergent iterative scheme modifying the iterative scheme of Kosmidou and Bertrand By applying cheap control theorems to this scheme, we obtained some eplicit results about eistence of the robust controller In particular we obtained systems which can be stabilized for an arbitrarily large uncertainty bound, which are different from those of the matching conditions Since the cheap control theorems used are based on earlier papers (Sannuti, 983), our results can be improved by more recent results about cheap control (Saberi and Sannuti, 989) A continuation scheme was also proposed, which was especially useful for finding the maimum possible uncertainty bound as well as the robust controller he method was etended to nonlinear systems and an easily implementable series solution was obtained Convergence of the iterative scheme of Chang and eng (97) is hard to prove However, their method is still useful for some systems which contain unmodeled high frequency parasitic dynamics or suffer from stochastic noise, because it sometimes gives much lower feedbac gains he scheme was reformulated so that some analytical results and more reliable numerical methods including the continuation method could be applied he Lyapunov equations and the Riccati equations were solved by the Schur transformation method (Bartell and Stewart, 97; Laub, 979) hese may also be solved by the reliable matri sign function method (Roberts, 980; Bierman, 98; Shieh et al, 987) without rather lengthy programs to find Schur vectors A general program for the power series solution of the nonlinear problems was developed with subroutines using symbolic algebraic manipulation of multivariable polynomials (Lee, 990) All programs were written in FORRAN For low dimensional systems, we obtained robust controllers very efficiently However, a fast and reliable method for solving the equation A A Q = 0 would be required to speed up the calculations for large 3 dimensional systems Reference [] E Allogower and K Georg, Simplicial and continuation methods for approimating fied points and solutions of systems of equations, SIA Rev, vol, pp 8, 980 [] R H Bartell and G W Stewart, Solution of the matri equation AX XB=C, Commum AC, vol, pp 8, 97 [3] R Bellman, Introduction to matri analysis, cgraw- Hill, New Yor, 960 [] G J Bierman, Computational aspects of the matri sign function solution to the ARE, roc 3 rd Conf ecision Contr, pp, 98 [] R Brocett, Volterra series and geometric control theory, Automatica, vol, pp 67, 976 [6] S S L Chang and K C eng, Adaptive guaranteed cost control of systems with uncertain parameters, IEEE rans Automatic control, vol AC-7, pp 7, 97 [7] Corless and G Leirmann, Controller design for uncertain system via Lyapunov functions, roc ACC, Atlanta, GA, pp 09, 988 [8] orato, A historical review of robust control, IEEE Control Syst ag, vol 7, pp, 987 [9] S Glad, Robustness of nonlinear state feedbac a survey, Automatica, vol 3, pp, 987 [0] I J Ha and E G Gilbert, Robust tracing in nonlinear systems and its application to robotics, IEEE rans Automatic control, vol AC-3, pp 763, 987 [] W Hahn, Stability of motion, Springer-Verlag, New Yor, 967 [] Jamshidi, G C Ans and Kootovic, Application of a parameter-imbedded Riccati equation, IEEE rans Automatic Control, vol AC-, pp 68, 970 [3] Kailath, Linear systems, rentice-hall, New Jersey, 980 [] R E Kalman and J E Bertram, Control system analysis and design via second method of Lyapunov art I : continuous-time systems, ASE rans, 8, 37, 980 [] O I Kosmidou and Bertrand, Robust-controller design for systems with large parameter variations, Int J Control, vol, pp 97, 987 [6] C Karavaris and S alani, Robust nonlinear state feedbac under structured uncertainty, AIChE J vol 3, pp 9, 988 [7] A J Laub, A schur method for solving algebraic riccati equations, IEEE rans Automatic Control, vol AC-, pp 93, 979 [8] J Lee, A schur method to solve a generalized Lyapunov equation, IEEE rans Automatic Control, vol AC-3, pp, 990 [9] G Leitmann, Guaranteed asymptotic stability for some linear systems with bounded uncertainties, J ynam Syst eas Control, vol 0, pp, 979 [0] L Lues, Optimal regulation of nonlinear dynamical systems, SIA J Control, vol 7, pp 7, 969

8 ransaction on Control, Automation, and Systems Engineering Vol, No, June, [] J oylan and B O Anderson, Nonlinear regulator theory and an inverse optimal control problem, IEEE rans Automatic Control, vol AC-8, pp 60, 973 [] E Noldus, esign of robust state feedbac laws, Int J Control, vol 3, pp 93, 98 [3] C an and K S Chao, A computer-aided root-locus method, IEEE rans Automatic Control, vol AC-3, pp 86, 978 [] I R eterson and B R Barmish, Control effort considerations in the stabilization of uncertain dynamical systems, Systems & Control Letters, vol 9, pp 7, 987 [] I R eterson and C V Hollot, A Riccati equation approach to the stabilization of uncertain linear systems, Automatica, vol, pp 397, 986 [6] A oubelle, I R eterson, R Gevers and R R Bitmead, A miscellany of results on an equation of count J F Riccati, IEEE rans Automatic Control, vol AC-3, pp 6, 986 [7] J J Rissanen, erformance deterioration of optimum systems, IEEE rans Automatic Control, vol AC-, pp 30, 966 [8] J Roberts, Linear model reduction and solution of the algebraic Riccai equation by use of the sign function, Int J Control, vol 3, pp 677, 980 [9] A Saberi and Sannuti, Cheap and singular controls for linear quadratic regulator, IEEE rans Automatic Control, vol AC-3, pp 08, 987 [30] Sannuti, irect singular perturbation analysis of high-gain and cheap control problems, Automatica, vol 9, pp, 983 [3] L S Shich, S R Lian and B C cinnis, Fast and stable algorithms for computing the principal square root of a comple matri, IEEE rans Automatic Control, vol AC-3, pp 80, 987 [3] Silja, arameter space methods for robust control design : a guided tour, IEEE rans Automatic Control, vol AC-3, pp 67, 989 [33] A Vannelli and Vidyasagar, aimal Lyapunov functions and domains of attraction for autonomous nonlinear systems, Automatica, vol, pp 69, 98 [3] H K Wimmer, onotonicity of maimal solutions of algebraic Riccati equations, Systems & Control Letters, vol, pp 37, 98 [3] Yoshida and K A Loparo, Quadratic regulatory theory for analytic non-linear systems with additive controls, Automatica, vol, pp 3, 989 Notations λ i ( ) : eigenvalue of a matri σ m ( ) : minimum singular value (minimum absolute eigenvalue) of a matri (Kailath, 980) σ ( ) : maimum singular value(maimum absolute eigenvalue) of a matri O ( ) : order of large O For eample, = O( ) means that goes to some matri as goes to zero o ( ) : order of small o For eample, = o( ) means that goes to zero as goes to zero diag ( ) : diagonal matri < Q( Q) : Q is positive definite (semi-definite) A : A I : gradient of φ ( about ( ) h j ( : j-th order terms in the power series epansion of an analytic function h( : leico-graphic listing vector for multivariable polynomials of (Brocett, 976; Yoshida and Loparo, 989) : a matri such that = ( ( ) : principal square root of a matri (Bellman, 960) Appendi (rincipal square root of a positive semi-definite matri We calculate the principal square root of a given positive semi-definite matri A by the Schur transformation method Since A and B = A have the same eigen structure (Bellman, 96), We have ~ ~ B = A ~ ~ where A = SAS and B = SBS If a transformation S is chosen as Schur vectors, A ~ and B ~ become upper triangular matrices and the upper triangular matri B ~ can be calculated as for = to n- ~ b ~ i ii j= i bii = ( a ~ ii ), i =,, Λ, n ~ ~ ~ ~ = ( a ~ ii bijb j i ) ( bii bi i ), i =,, Λ, n Because B satisfies that B = A and the eigenvalues of B are all non-negative real values, B becomes the principal square root of A All matrices used are real, because A is symmetric and positive semi-definite

9 8 ICASE: he Institute of Control, Automation and Systems Engineers, KOREA Vol, No, June, 000 Jietae Lee h; ept of Chem Eng, KAIS resent; rofessor in Kyungpoo National Univ Interest; rocess Identification and ulti-loop Control System, Robust Control oe-gyoon Koo BS; ept of Chem Eng, Yeungnam Univ S; ept of Chem Eng, Yeungnam Univ resent; ept of Chem Eng Kyungpoo National Univ Interest; rocess Identification, ultiloop Control System homas F Edgar h; rinceton University, (97) resent; rofessor in ept of Chem Eng, University of eas, Austin Interest; rocess Control