1 Introduction 1.1 Why Differential-Algebraic Equations? In system theory, a dynamical system is often considered as a set of ordi- nary differential or difference equations (ODE); these equations describe the relations between the system variables. As pointed out in [14], for the most general purpose of system analysis, one usually begins by defining the first order system F(x˙(t), x(t)) = 0, (1.1) where F and x are vector-valued functions. The equation (1.1) is termed as differential algebraic equations (DAE), since it contains differential equations as well as a set of algebraic constraints. For control and systems engineers, it is usually assumed that (1.1) can be rewritten in an explicit form x˙(t) =f (x(t)). (1.2) An ODE of the form (1.2) is called a state-variable (state-space) description inthesystemsandcontrolsociety.Sincethen,theoremsanddesigntechniques being developed are largely based on (1.2). In fact, the state-variable descrip- tions have been the predominant tool in systems and control theory. While therepresentation(1.2)willcontinuetobeveryimportant,therehasbeenan increasing interest in working directly with (1.1). If(1.1)can,inprinciple,berewrittenas(1.2)withthesamestatevariablesx, then it will be referred to as a system of implicit ODEs. In this monograph, we are especially interested in those problems for which this rewriting is im- possible or less desirable. We consider the general nonlinear DAEs which are linear in the derivative A(x(t))x˙(t) +f (x(t)) = 0. (1.3) H.S. Wang et al.: H Control for Nonlinear Descriptor Systems, LNCIS 326, pp. 1–11, 2006. ∞ © Springer-Verlag London Limited 2006 2 1 Introduction ∂A(x) Suppose that has a constant rank. Then, in principle, locally the ∂x system (1.3) can be put in the semi-explicit form x˙ (t) = f (x (t), x (t)) 1 1 1 2 (1.4) 0 = f (x (t), x (t)). 2 1 2 A large class of physical systems can be modeled by this kind of DAEs. The paper of Newcomb et al. [91] gives many practical examples–including circuit and system design, robotics, neural network, etc.–and presents an excellent review on nonlinear DAEs. Many other applications of DAEs as well as nu- merical treatments can be found in [14]. An existence and uniqueness the- ory for nonlinear DAEs has been well developed in [97] by exploiting their underlying differential geometric structure. Recently, Venkatasubramanian et al. [113] have extensively studied the bifurcation phenomena of DAEs. They have also thoroughly investigated feasibility regions in differential-algebraic systems. The notion of feasibility regions provides a natural gateway to the stability theory of DAEs. Depending on the area, system of the form (1.4). has different nomenclature in different fields. For example, control theorists and mathematicians have long been calling them singular systems[16][17][61][83], since the matrix on the derivative of the state-variables is generally singular, or sometimes they use the terminology generalized state-space systems[1][41][114], or at times extended state-space systems[55]. On the other hand, the name descriptor systems is most frequently used in the engineering economic systems community[78][79], since they give the natural description of the system, while numerical analysts call their descriptions differential-algebraic equations[14][81] [94], or differen- tial equations with algebraic constraints[73]. In the circuits area the original name was pseudostate[59] but more recently these systems have been called semistate systems[26] because they are almost state described. This usage is now somewhat obsolete. In this monograph, we will use the terms descriptor systems and DAEs interchangably. There are several reasons to consider systems of the form (1.4), rather than try to rewrite it as an ODE. Of great importance, we point out that, when physical problems are simulated, the model often takes the form of a DAE de- picting a collection of relationships between variables of interest and some of their derivatives, namely the algebraic constraints. These relationships may even be generated automatically by a modeling or simulation program. In particular, the variables thus introduced usually have a physical significance. Changing the model to (1.2) may produce less meaningful state variables. If the original DAE can be solved directly, then it becomes easier for scientists or engineers to explore the effect of modeling changes and parameter variations. These advantages enable researchers to focus their attention on the physical problem of interest. On the other hand, although the state-space models are 1.1 Why Differential-Algebraic Equations? 3 very useful, but the state variables thus introduced often do not provide a physical meaning[29][111]. Besides, some physical phenomena, like impulse, hysterisis which are important in circuit theory, cannot be treated properly inthestate-spacemodels[67][114].Differential-algebraicequationsrepresenta- tionprovidesasuitablewaytohandlesuchproblems.Ithasbeenproveninthe literature that DAE systems have higher capability in describing a physical system[66][91][114]. In fact, DAE system models appear more convenient and natural than state-space models in large scale systems, economics, networks, power, neural systems and elsewhere [66][78][91]. Example 1. Constrained Variational Problems [14][33] ThefirstexamplethatcanbewellmodeledbyDAEsisavariationalproblem withconstraints.Consideraconstrainedmechanicalsystemwithpositionx(t), velocity v = x˙(t), kinetic energy T(x(t),v(t)), external force f(x(t),v(t),t) and constraint φ(x(t)) = 0. By the variational principle, the Euler-Lagrange formulation of the system can be put in the following form x˙(t)=v (t) d ∂ ∂T(x(t),v(t)) T(x(t),v(t))= +f(x(t),v(t),t)+GTλ dt ∂v ∂x 0=φ (x(t)), whereG= ∂φ,andλistheLagrangemultiplier.Thissystemcanberewritten ∂x as ∂2T v˙(t)=g (x(t),v(t),t)+GTλ (1.5a) ∂v2 x˙(t)=v (t) (1.5b) 0=φ (x), (1.5c) where ∂T ∂2T (cid:14) g(x,v,t)=f(x,v,t)+ − x˙. ∂x ∂x∂v ∂2T Inpracticalcase,thematrix isusuallypositivedefinite.Thenmultiplica- ∂v2 (cid:37) (cid:41) ∂2T −1 tion of (1.5a) by converts the previous system into a semi-explicit ∂v2 DAE. (cid:37) (cid:41) (cid:37) (cid:41) ∂2T −1 ∂2T −1 v˙(t)= g(x(t),v(t),t)+ GTλ ∂v2 ∂v2 4 1 Introduction x˙(t)=v (t) 0=φ (x). ♦♦♦ Example 2. Singular Dynamic Leontieff Systems [62][77] Consideraneconomicprocessthatinvolvesninterrelatedproductionsectors. The relationships between the levels of production of the sectors can be de- scribed by a so-called Leontieff Model: x(k) =Ax( k)+B(x(k+1)−x(k))+w(k). (1.6) Here the components of the n-dimensional vector x(k) are the levels of pro- duction of the sectors at time k. The vector Ax(k) should be interpreted as the capital that is required as direct input for production of x; a coefficient a of the flow coefficient matrix A indicates the amount of product i that is ij neededtoproduceoneunitofproductj.ThevectorBxstandsforthecapital that is required to be in stock to be able to produce x in the next time pe- riod.Acoefficientb ofthestock coefficient matrixB indicatestheamountof ij productithathastobeinstocktobeabletoproduceoneunitofproductj in thenexttimeperiod.Thevectorw(k)representsthelevelsofproductionthat are demanded. Econometric models of this type were considered by Leontieff in[64],inwhichbothdiscretetimeandcontinuoustimecaseswereconsidered. Usually, most of the elements in the stock coefficient matrix B are zero and B is often singular. This is because that productions in one sector does not requirecapitalinstockfromalltheothersectors.Furthermore,inmanycases, there are usually few sectors that offer capital in stock to other sectors. The representation (1.6) can be rewritten in the following form Bx(k+1) = (I−A+B)x(k)−w(k), which is a descriptor form. This serves as a example that a descriptor form can arise naturally in modelling a practical dynamical system. ♦♦♦ Example 3. Electrical Circuit with Operational Amplifier[14][80] Consider the electrical circuit of Fig. 1.1, which consists of a differential am- plifier,avoltagesource,andfourresistors.Toeachnode,thecircuitequations are derived from Kirchoff’s laws: 1. The algebraic sum of the currents into a node at any instant is zero. 2. The algebraic sum of the voltage drops around a loop at any instant is zero. By Kirchoff’s law, we can write down the circuit equations as 1.1 Why Differential-Algebraic Equations? 5 ✎☞ ✎☞ 1 2 ✍✌ ✍✌ ✉ ✉ ❆❆✁✁❆❆✁✁❆❆ ❆❆✁✁❆❆✁✁❆❆ R R 1 2 ❍ ✗✔ +❍❍ ✎☞ e ∞✟❍ ✉ 3 ✖✕ ✟✟ ✍✌ ✟ R R 3 4 ✉ ❆❆✁✁❆❆✁✁❆❆ ❆❆✁✁❆❆✁✁❆❆ ✎☞ 4 ✍✌ Fig. 1.1. Electrical circuit with Operational Amplifier(a)   1 1 − 0 0 10  R R       11 1 11 1  v 0 −R1 R1 +1R2 1 R2 1 01 00 vv12 00  0 − + − 01  3 =   (1.7)  R R R R  v  0  2 2 4 4   4    0 0 − 1 1 + 1 00 iE e  1 0 0R4 R3 0 R4 00 i0 0 0 1 0 −1 00 where i denotes the output current of the operational amplifier. Equation 0 (1.7) is a purely algebraic linear system. It is solvable if and only if the coef- ficient matrix is nonsingular, or equivalently, its determinant   1 1 − 0 0 10  R R   11 1 11 1    − + 0 00  R1 R1 1R2 1 R2 1 1  1 1 det 0 − + − 01 = −  R2 R2 1R4 1 R41  R2R3 R1R4  0 0 − + 00  R R R   4 3 4  1 0 0 0 00 0 1 0 −1 00 6 1 Introduction ✎☞ ✎☞ 1 2 ✍✌ ✍✌ ✉ ✉ ❆❆✁✁❆❆✁✁❆❆ ❆❆✁✁❆❆✁✁❆❆ R R 1 2 ❍ +❍❍ ✎☞ ∞❍ ✉ C ✟ 3 ✟ ✍✌ ✟ ✟ R R 3 4 ✉ ❆❆✁✁❆❆✁✁❆❆ ❆❆✁✁❆❆✁✁❆❆ ✎☞ 4 ✍✌ Fig. 1.2. Electrical circuit with Operational Amplifier (b) is nonzero. This shows that the solvability of the system depends on the spe- cificvaluesofthe resistances. In order to gain more insight of the differential- algebraic equations, let us replace the voltage source in the circuit given in Fig.1.1withacapacitor.Thecircuitequationsnowreadsasfollows(SeeFig. 1.2)   1 1 − 0 0 0    R R      Cv˙  11 1 11 1  v 0  001 + −R1 R1 +1R2 1 R2 1 01 0 vv12 = 00. (1.8)    0 − + − 1  3    0   R R R R  v  0  2 2 4 4  4 0  0 0 − 1 1 + 1 0 i0 0 R R R 4 3 4 0 1 0 −1 0 The determinant of the matrix pencil of system (1.8) is (cid:37) (cid:41) 1 1 1 1 1 1 1 (cid:14) sC − − =sG+H, R R R R R R R 1 4 2 3 1 2 3 whichisneveridenticallyzerofornonzeroresistances.Nowthenumberofstate variables is equal to the degree of the polynomial det[sG+H]. Equivalently, this is the number of independent initial conditions which must be specified. 1 1 1 1 1 1 Thus,if − =0,theonlysolutionisthezerosolution.If − R R R R R R 1 4 2 3 1 4 1.2 Control Problems Based on DAEs 7 1 1 (cid:46)= 0, there is one parameter family of solutions to (1.8). This shows R R 2 3 that the number of state variables are determined not only by the topology of the circuit, but also by the specific resistance values. ♦♦♦ 1.2 Control Problems Based on DAEs ThedesirabilityofworkingdirectlywithDAEshasbeenrecognizedfortwenty years by scientists and engineers in several areas. For the purpose of control, the to-be-controlled plant considered in this monograph is usually described in the following descriptor form: x˙ =F (x , x , w, u), (1.9) 1 1 1 2 0=F (x , x , u), (1.10) 2 1 2 z =Z(x, w, u), (1.11) y =Y(x, w, u), (1.12) Here u is the control input, w is the exogenous input (disturbances to-be- rejected or signals to-be-tracked), y is the measured output, and z denotes a setofthepenaltyvariables(trackingerrors,costvariables).Inthestate-space X, dynamic state variables x and instantaneous state variables x are dis- 1 2 tinguished. The control theory based on descriptor system models has been well esta- bilished for many years with the practical outcome that the shortcomings of state-variable theory are often overcome [91]. For the linear descriptor sys- tems, Cobb first gave a necessary and sufficient condition for the existence of an optimal solution to linear quadratic optimization problem[19] and also extensively studied the notions of controllability, observability and duality in descriptor systems[20]. Lewis[66], Bender et al.[10] and Takaba et al.[111] constructed different kinds of Riccati equations for solving linear quadratic regulator problems based on certain assumptions. Some excellent results on pole placement[92] and robust control[29][119], to name only a few, were also obtained. Recently, Copeland and Safonov used the descriptor-system-like models to solvethesingularH andH controlproblemsinwhichtheplantshavepure 2 ∞ imaginary (including infinity) poles or zeros [21]. Solutions to H control ∞ problem for descriptor systems were given in Takaba et al. [111]. They dealt with the problem using aJ -spectral factorization, thus their proofs were in- volved. Moreover, only sufficient conditions for solutions to exist were given. Mostrecently,Masubuchietal.[82]haveconsideredasimilarproblembyusing amatrixinequalitiesapproach.Theytreatedamoregeneralproblemwithless 8 1 Introduction assumptions. Their solutions were obtained by use of a version of Bounded Real Lemma and given in terms of linear matrix inequalities(LMI) which may be solved by existing numerical tools. However, they gave a necessary and sufficientconditionintermsoftwogeneralizedalgebraicRiccatiinequal- ities(GARI) involving two unknown parameters plus two to-be-determined variables. On the other hand, compared to the existing results for linear case, less efforts have been made to investigate the problems of robust control for nonlineardescriptorsystems.Theproblemoffeedbackstabilizationofnonlin- ear descriptor systems have been investigated in [106], while the paper of Wu andMizukami[123]elaboratedthestabilityandrobuststabilizationofnonlin- eardescriptorsystemswithuncertainty.AHamilton–Jacobiliketheoremwas given in Xu and Mizukami[125]. Recently, Boutayeb and Darouach[13], and Shields[107] have developed some results concerning the observers for nonlin- ear descriptor systems. The monograph continues this line of research to study the H control prob- ∞ lem for descriptor systems. We give a comprehensive investigation on the contraction property of descriptor systems. The most general motivation of the H control problem stems from the small gain theorem, which is origi- ∞ nally addressed in the two celebrating papers by Professor Zames[137][138]. If the system uncertainties (linear and/or nonlinear) can be characterized by bounded real property, then the classical results in stability theory can be used to guarantee robust stability provided an appropriate closed-loop sys- tem has an L gain strictly less than (or equal to) a prescribed attenuation 2 level γ. Consider the interconnected system shown in Fig. 1.3. Suppose that M(s) ∈ Rp×q(s) is a stable transfer matrix and the interconnection is well defined. Then the classical Small Gain Theorem tells us that the loop M−Δ is well defined and internally stable, for all Δ(s)∈RH∞ with (cid:31)Δ(cid:31) ≤ 1, if ∞ γ and only if (cid:31)M(s)(cid:31) <γ. ∞ It is well known that, to solve the H control problem for conventional ∞ state-variable systems, we always need certain materials: Lyapunov stability theorem to verify closed-loop internal stability, Hamilton–Jacobi inequalities (equations)forobtainingH controllersfornonlinearsystems,algebraicRic- ∞ cati equation and bounded real lemma in deriving the central controller for linear systems; Youla parameterization in characterizing all stabilizing con- trollers [128]. (One often gets the impression that those materials in fact con- stitute the bottleneck of this problem.) This monograph can be regarded as an extension of those above mentioned results to the descriptor systems cir- cumstancesandweindeedusethisextensiontosolvetheH controlproblem ∞ for descriptor systems. Perhaps this is the first attempt in the literature to comprehensivelystudytheH controlproblemsforbothlinearandnonlinear ∞ descriptor systems. Most of the previous work for conventional state-variable systems can be generalized, mutatis mutandis, to descriptor systems. In par- ticular,wewillderivethenecessaryandsufficientconditionsfortheexistence 1.2 Control Problems Based on DAEs 9 v u 1 1 ✲ ❥ ✲ Δ ✻ y y 2 1 u v 2 ❄ 2 M ✛ ❥✛ Fig. 1.3. Small Gain Theorem of a controller solving the standard nonlinear H control problem. We first ∞ givevarioussufficientconditionsforthesolvabilityofH controlproblemfor ∞ nonlinear descriptor systems. Both state feedback and output feedback cases are considered. Then, a necessary condition for the output feedback control problemtobesolvableareobtainedintermsoftwoHamilton–Jacobiinequal- itiesplusaweakcouplingcondition.Moreover,aparameterizationofafamily of output feedback controllers solving the problem is also provided. All of the aforementioned results are then specialized to the linear case. For the linear case, the necessary and sufficient conditions for the corresponding problems tobesolvableareexpressedintermsoftwohierarchicallycoupledgeneralized algebraicRiccatiequationswhichmaybeconsideredtobethegeneralizations of the Riccati equations obtained by Doyle et al.[24]. When these conditions hold, state-space formulae for a controller solving the problem is also given. The approach used in this monograph is based on a generalized version of Bounded Real Lemma. Finally, the derivation of state-space formulae for all controllers solving the standard H control problem for descriptor systems ∞ are proposed. To establish the key formulae, a parameterization of all inter- nallystabilizingcontrollersfordescriptorsystemsisalsogiven(boththelinear and nonlinear cases are considered in this monograph). The results obtained may be considered as the generalization of the corresponding ones given in Doyle et al.[24] and Doyle[23]. 10 1 Introduction 1.3 Highlights of the Monograph The remainder of the monograph is organized in the following and the key results are highlighted accordingly. Chapter 2 reviews some elementary results concering descriptor systems, in- cluding controllability and obsrvability, existence and uniqueness of solution of DAEs, and the Lyapunov stability theorems. Furthermore, we explore cer- tain theorems and properties–namely the LaSalle’s Invariance Principle, dis- sipativeness, Bounded Real Lemma, and LQ optimization–in the descriptor systems case. In Chapter 3, we characterize all controllers that internally stabilize a given dynamicalsystemΣ,whichisdescribedbyasetofdifferential-algebraicequa- tions.Theconstructionofthecontrollerparameterizationisdonebyasimple change of variables and some direct algebraic calculations provided that a state feedback law and an observer gain are available. Moreover, to serve as an illustration, we give a complete parameterization of all solutions to the extended positive real control problems (ESPR) for linear descriptor systems by using a Youla parameterization approach. In this case, the state feedback law and the observer gain are constructed properly from the control gener- alized algebraic Riccati equation and the filter generalized algebraic Riccati equation, respectively. Chapter4isdevotedtotheH controlproblemsfordescriptorsystems.Both ∞ thenonlinearandlinearcasesarestudied.WefirstsolvetheH controlprob- ∞ lemsviastaticstatefeedback.Basedonthestatefeedbacklawthusobtained, the output feedback controllers are then obtained by a simple idea beginning with a change of variables and some algebraic calculations. It is shown that the solutions to the nonlinear H control problems are characterized by two ∞ Hamilton–Jacobi inequalities. In the linear case, the solutions are given in terms of two hierarchical-coupled generalized algebraic Riccati equations. Fi- nally, a family of output feedback controllers for nonlinear descriptor systems as well as a complete parameterization of output feedback controllers for lin- ear descriptor systems are given. Chapter 5 considers the problem of reducing the order of a descriptor system bythebalancedtruncationmethod.Thebalancedrealizationhastheproperty thatmodeiisequallycontrollableandobservable,withthesepropertiesbeing measured in terms of a number σ ≥ 0. As σ decreases, the corresponding i i level of controllability and observability becomes less important. The model reductionmethodthatappliesthetruncationoperationtoabalancedrealiza- tion is known as balanced truncation. For this algorithm, the absolute error of the reduced system with respect to the full order system is guaranteed to satisfy the twice-the-sum-of-the-tail bound