Robust and Optimal Control o Uncertain Dynamical Systems with State-Dependent Switchings Using Interval Arithmetic Andreas Rauh Chair o Mechatronics, University o Rostock, D-1859 Rostock, Germany Andreas.Rauh@uni-rostock.de Johanna Minisini and Eberhard P. Hoer Institute o Measurement, Control, and Microtechnology, University o Ulm, D-8969 Ulm, Germany {Johanna.Minisini,Eberhard.Hoer}@uni-ulm.de Harald Aschemann Chair o Mechatronics, University o Rostock, D-1859 Rostock, Germany Harald.Aschemann@uni-rostock.de Abstract In this paper, interval arithmetic techniques or controller design o nonlinear dynamical systems with uncertainties are summarized. The main reason or the application o interval techniques in this context is the quantiication o the inluence o uncertainties and modeling errors. They result rom neglecting nonlinear phenomena in the mathematical description o real-world systems. Furthermore, measured data usually do not provide exact inormation about the system parameters. Oten simpliications o nonlinear models or controller structures are necessary to enable the implementation o controllers. Thereore, these uncertainties have to be taken into account during the design o controllers or veriication o observability, controllability, stability, and robustness. Keywords: interval arithmetic, controller design, optimization, approximation techniques, saety-critical applications AMS subject classiications: 65G2, 65G4, 65K1, 34H5, 49J15, 49K15, 93C15, 93D9 Submitted: January 19, 29; Revised: February 9, 21; Accepted: March 1, 21. 333
334 A. Rauh et al., Robust and Optimal Control o Dynamical Systems 1 Introduction This paper gives an overview o general interval arithmetic procedures or the design o controllers or continuous-time dynamical systems [9]. To guarantee robustness with respect to uncertain initial conditions as well as uncertain system parameters, interval variables and veriied computational techniques are taken into account during all design stages. For that purpose, the notion o optimality o control strategies is extended to systems with uncertainties by minimization o guaranteed upper bounds o suitable perormance indices. In the irst part o this paper, techniques or the veriication o controllability, reachability, and observability o states in the presence o uncertainties as well as stabilizability o instable systems are demonstrated. These techniques are based on the veriied evaluation o the corresponding system theoretic criteria [6, 3]. Thereore, underlying computational approaches are required which yield guaranteed enclosures o all reachable states o dynamical systems. Especially or saety-critical applications, asymptotic stability and compliance with given restrictions or the state variables have to be assured or all possible operating conditions using analytical or numerical techniques. For both open-loop and closed-loop control systems, possibilities or the combination o veriied techniques with classical approaches or robust and stabilitybased controller design in the time-domain are highlighted (Sections 2 4). In the second part, procedures or structure and parameter optimization o continuous-time dynamical systems with bounded uncertainties are derived that rely on the above-mentioned basic concepts [8]. For that purpose, an interval-based algorithm is presented. This algorithm computes approximations o globally optimal openloop and closed-loop control laws or dynamical systems. Focusing on continuous-time applications, we present new strategies which allow to combine piecewise constant approximations o optimal control laws with continuous approximations (Sections 5,6). The ocus o the third part is the application o the previously introduced methods and procedures to the design o robust and optimal control strategies or dynamical systems with state-dependent switching characteristics [8]. In practically relevant applications, the act that control variables are usually bounded has to be taken into account directly during design. This aims at building a new uniied, general-purpose ramework or the veriied synthesis o dynamical systems integrating trajectory planning, unction-oriented and saety-related controller design, as well as robustness and optimality assessment and veriication. Computational routines or the previously mentioned tasks already exist. However, they are usually not integrated into a common tool veriying the design using interval arithmetic (Section 6). 2 Control-Oriented Modeling o Dynamical Systems In control engineering, commonly two tasks are distinguished: the design o open-loop control strategies and observer-based closed-loop control strategies. In the ollowing, we restrict the discussion to continuous-time dynamical system models which are described by sets o (nonlinear) ordinary dierential equations ẋ s (t) = (x s (t), p (t), u (t), t) (1)
Reliable Computing 15, 211 335 and nonlinear mathematical models speciying the sensor characteristics y (t) = h (x s (t), u (t), q (t), t). (2) In (1) and (2), the state vector is denoted by x s (t), the control vector by u (t), and the parameter vectors corresponding to uncertainties and disturbances by p (t) and q (t), resp. Finally, the reerence signal w (t) describes the trajectories o the desired system outputs, where y (t) = w (t) holds in the ideal case. For open-loop control strategies (Fig. 1) the control sequence u (w) only depends on the reerence signal w (t) and (oten) a inite number o its time derivatives. In contrast to the openloop case, inormation on the current system states x s (t) is ed back additionally in closed-loop control structures (Fig. 2). w control law u (w) u plant sensor characteristics x s ẋ s = (x s, p, u, t) y = h (x s, u, q, t) y Figure 1: Open-loop control o nonlinear dynamical systems. w control law u (ˆx, w) u plant ẋ s = (x s, p, u, t) x s sensor characteristics y = h (x s, u, q, t) y ˆx observer or state reconstruction Figure 2: Observer-based closed-loop control o nonlinear dynamical systems. Figure 2: Observer-based closed-loop control o nonlinear dynamical systems. I the complete state vector x s (t) and the parameters p (t) and q (t) are not directly accessible or measurements, estimated values ˆx (t) are used as a substitute or these quantities in the implementation o the eedback controller. These estimates are determined with the help o state observers (c. [6]). They are computed in terms o the known control u (t) and the measured output y (t). The goal o this paper is to highlight the components o a general-purpose sotware tool which uses interval arithmetic routines to quantiy the inluence o uncertainties resulting rom uncertain initial system states x s () [ x s, ; x s, ] as well as uncertain parameters p (t) [ p (t) ; p (t) ], q (t) [ q (t) ; q (t) ] at the earliest possible design stages. Thus, a system design becomes possible which does not rely on the implementation o controllers or a simpliied, nominal model with an a-posteriori assessment o the inluence o uncertainties. This oten leads to tedious iterations in the design until robustness o the complete control system can be shown or all possible values o the uncertain quantities. Instead, uncertainties are propagated through the com-
336 A. Rauh et al., Robust and Optimal Control o Dynamical Systems plete design process by guaranteed bounds on the system states computed by veriied techniques. The basic properties o control systems to be addressed are (global) asymptotic stability and stabilization o instable plants, improvement o the dynamics o the openloop system, robustness and optimality, as well as easibility o the necessary openloop and closed-loop control laws that are required to minimize the control errors w (t) y (t) and x s (t) ˆx (t), resp., or all t. 3 Classical Approaches or Robust Control In the ollowing, an overview o the most important design strategies or robust controllers o linear and nonlinear dynamical system models is given. As discussed in this section, the use o interval techniques and other veriied approaches during the complete control system design is not yet state-o-the-art in industrial control applications. Instead, approximations and simple design techniques based on linear system theoretic properties are oten employed. For linear inite-dimensional time-invariant system representations described by rational transer unctions, i.e., system models without time delay, pole placement is usually perormed to parameterize linear state eedback controllers or single-inputsingle-output systems using Ackermann s ormula (1972). This ormula provides a unique solution or the eedback gains K in u (t) = K x s (t). For multiple-inputmultiple-output systems, additional constraints (such as the decoupling o the inluence o input and output variables) are taken into account to ind a unique solution o the eedback gain matrix K. In both cases, controllability (as introduced by Kalman in 196) is a suicient condition or the assignment o arbitrary poles to the closed-loop control system. Furthermore, observability is necessary i not all state variables are directly accessible by measurements such that they have to be reconstructed using model-based estimators as, or example, shown in Fig. 2. Extensions that are commonly used or the design o controllers or systems with uncertainties and nonlinearities are: The Kharitonov criterion (1978) to prove asymptotic stability o characteristic polynomials with interval coeicients which is basically a generalization o the Hurwitz criterion or nominal system models. The parameter space approach by Ackermann and Kaesbauer [1], which is implemented in the Matlab toolbox PARADISE based on the boundary crossing theorem by Frazer and Duncan (1929). It shows that the poles o rational transer unctions depend continuously on continuous parameter variations (Γstability). Extensions to nonlinear systems with sector-bounded static nonlinearities on the basis o the Popov criterion (1962). Handling o constant time delays using the notion o B-stability. Design approaches based on control Lyapunov unctions as well as symbolic design approaches relying on state-eedback linearization, backstepping, and latness-based eedorward and eedback control [3, 6]. However, general approaches or controller design o nonlinear dynamical systems with bounded parameter uncertainties and bounded errors in the state reconstruction
Reliable Computing 15, 211 337 are not yet commonly used by engineers in industrial applications. Thereore, intervalbased implementations o both classical design approaches and novel numerical techniques are inevitable to obtain general sotware tools or veriication and validation o controller design. 4 An Interval-Based Framework or Robust and Optimal Controller Design 4.1 Overview In this section, selected components o an interval arithmetic ramework or veriied design o robust and optimal control strategies are presented [8, 9], see Fig. 3. They are extensions o classical techniques or the analysis o asymptotic stability o dynamical systems, parameter and structure optimization based on dynamic programming, and trajectory planning techniques using inormation about reachability and observability o states. Especially with respect to optimal control o dynamical systems with uncertainties, non-veriied approaches based on Bellman s dynamic programming and Pontryagin s maximum principle have also been used in other interval-based approaches [4, 5]. 4.2 Veriied Reachability Analysis Reachability analysis or nonlinear input-aine dynamical systems ẋ (t) = (x (t)) + g (x (t)) u (t), x R nx (3) is the prerequisite or the design o most closed-loop control strategies. To analyze the applicability o control sequences u (t) such that the state variables can be inluenced in a desired way, the Lie brackets o (x) and g(x) have to be evaluated, which are deined according to [ (x), g (x)] := g(x) x (x) (x) g (x). (4) x [ Using (4), the state-dependent reachability matrix P (x) deined by P (x) := P (x) P 1 (x)... P nx 1 (x) ] with P (x) := g (x), P 1 (x) := [ (x), g (x)],..., P k (x) := [ (x), P k 1 (x)], k = 2,..., n x 1 can be obtained. The rank o the matrix P (x) is usually state-dependent. To guarantee that all states can be inluenced in a desired way, P (x) must have ull rank n x along the complete trajectory or all possible values o the uncertain system parameters and all possible state variables. For linear systems, the rank criterion or the reachability matrix is identical to Kalman s criterion or state controllability. 4.3 Veriied Observability Analysis Similarly, the observability matrix Q(x) is deined by Q(x) := [ ( h(x) x ) T ( ) T L h(x)... x ( ) L n x 1 T ] T h(x) (5) x
338 A. Rauh et al., Robust and Optimal Control o Dynamical Systems Reachability and observability analysis o open-loop control system with uncertainties Stability analysis o open-loop control system Coice between parameterization o predeined control structure or structure optimization Combination o structure optimization and eedback control N1 piecewise constant/ lin. controls Nm piecewise constant/ lin. controls task 1 task m Splitting o L1 control strategies Splitting o Lm control strategies Evaluation o state equations: piecewise constant/ linear control Evaluation o state equations: piecewise constant/ linear control Computation o guaranteed state enclosures Computation o guaranteed bounds or the cost unction Computation o guaranteed state enclosures Computation o guaranteed bounds or the cost unction Exclusion o control strategies (1) Exclusion o control strategies (1) Violation o state constaints Computation o upper bound J N1 or the perormance index Exclusion o non-optimal control sequences Violation o state constaints Computation o upper bound J Nm or the perormance index Exclusion o non-optimal control sequences no Has the desired number o iterations been reached? Has the desired number o iterations been reached? no yes yes Comparison o the upper bounds J or the perormance index obtained or control Nj strategies with dierent Nj: Calculation o the best upper bound J { } := min J Nj j=1,...,m Elimination o non-optimal control strategies i J <l> > J holds Nj Output o the best approximation or the optimal control strategy or continuation o the parallelized optimization routine Stability analysis o the closed-loop control system Veriied sensitivity analysis o the trajectories with respect to uncertain parameters Computation o regions o attraction and maximum admissible deviations rom desired trajectories to ensure asymptotic stability (1) Extension to robust trajectory planning by exclusion o inadmissible controls and inadmissible reerence signals Figure 3: Components o an interval-based ramework or the design o optimal Figure and robust 3: Components controllers. o an interval-based ramework or the design o optimal and robust controllers. 6
Reliable Computing 15, 211 339 with the Lie derivatives L i h(x) := L ( x L i 1 ) h(x) with L h(x) := h (x) and L h(x) := h(x) (x). These Lie derivatives provide inormation about the variation o the output y = h (x) o the dynamical system along the vector ield (x). Similar to the reachability analysis, the rank o the matrix Q(x) is usually state-dependent. For linear systems, the procedure leads to Kalman s observability matrix. Using techniques or automatic dierentiation, the matrices P (x) and Q(x) can also be constructed or systems with uncertainties, where enclosures o the sets o reachable states are determined by veriied ODE solvers such as VNODE, VSPODE, ValEncIA-IVP, or COSY VI. A suicient condition or the applicability o a speciic procedure or controller design is that both matrices have ull rank or the desired trajectories [9]. 4.4 Controller Design Using Input-to-State Linearization The use o veriied computational methods is shown exemplarily or the design o nonlinear controllers aiming at input-to-state linearization. The goal is to convert the input-aine dynamical system (3) into a set o linear ODEs ż (t) = A z (t) + B w (t) with y (t) = C z (t) (6) using dieomorphism T : D R n R n deining a dieomorphism z = τ (x) : D R nx R nx and a control law u = r (x) + W (x) w. Let δ i be the relative degree o the output y i, i = 1,..., m, i.e., the smallest order o the derivative d δ i y/dt δ i which explicitly depends on the control input u. Then, the state transormation z = τ(x) = [ τ 1 1 (x)... τ δ 1 1 (x) τ 1 2 (x)... ] T can be computed using the Lie derivatives τ r i i = L r i 1 h i(x), r i = 1,..., δ i. The eedback control law u = r(x) + W (x) w is given by In (8), the vector r(x) = D 1 (x)ϕ(x) and W (x) = D 1 (x). (8) ϕ(x) = [ ϕ 1(x) ϕ 2(x)... ϕ m(x) ] T x(t)=τ 1 (z) is deined by ϕ i(x) = L δ i hi(x) or all i = 1,..., m. matrix D(x) is computed by L g1 L δ 1 1 L g1 L δ 2 1 (7) (9) Furthermore, the decoupling h 1(x) L gm L δ 1 1 h 1(x) h 2(x) L gm L δ 2 1 h 2(x) D(x) =.. (1).. L g1 L δm 1 h m(x) L gm L δm 1 h m(x) Linear eedback controllers can be designed or (6) i rank{d (x)} = n x and δ = δ 1 +... + δ m = n x hold or all desired states x (t), all possible parameter values p, and uncertainties o x (t ). These prerequisites can be veriied by interval evaluation o D(x) or guaranteed enclosures [x (t)] o the sets o all reachable states. Note that the derivatives which are required to evaluate (9) and (1) have already been computed by automatic dierentiation or the reachability and observability matrices P (x) and Q (x).
34 A. Rauh et al., Robust and Optimal Control o Dynamical Systems 4.5 Veriied Stability Analysis An approach or veriied stability analysis which is used in Matlab and C++ routines or controller design implemented by the authors is summarized in the ollowing. It is based on a procedure described in [2]. Ater computation o the interval enclosure [x ] o the equilibrium o the system (1), a double-valued approximate solution x [x ] is chosen. Then, an approximation A o the Jacobian is determined or x. Using this matrix, the Lyapunov equation A T P + P A = I with A := x (11) x= x is solved or the symmetric matrix P. I P is positive deinite, i.e., i the linearized system can be proven to be asymptotically stable, an estimate or the region o attraction o the asymptotically stable equilibrium o the original nonlinear system can be determined. For that purpose, an interval box [x ] or which [x ] [x ] holds is assumed. Additionally, this box must not contain urther equilibria. Typically, the initialization o the interval Newton iteration used to determine [x ] is chosen at this stage since it ulills the above-mentioned requirements. To analyze the stability o the dynamical system, the quadratic Lyapunov unction V (x, p) = (x x ) T P (x x ), (12) with P determined in (11), is used. For the time derivative o this Lyapunov unction, the properties V (x, p) = and V (x, p) x=x x = (13) x=x hold. Then, the Hessian H := 2 V (x, p) (14) x 2 has to be positive deinite or all x [x ]. This can be shown using a procedure described by Rohn in [11]. ( A symmetric ) interval ( matrix ) [H] = { H H c H H c + } with H c = 1 2 H + H and = 1 2 H H is positive deinite i the ollowing 2 n x 1 point matrices H z = H z are positive deinite. The matrices H z are deined according to H z := H c T z T z with T z := diag(z). The vector z has to be replaced by all possible combinations o the components z i = ±1, i = 1,..., n x. Thus, H z = H z holds. As shown in [2], the interval box [x] with center in [x ] and radius λ min n x d ([x ], [x ]) (15) λ max certainly belongs to the region o attraction o an asymptotically stable equilibrium x. In (15), λ min and λ max are the minimum and maximum eigenvalues o P, resp. Furthermore, d is a unction deined on IR nx IR nx with d : ([x], [y]) sup { r R B (r, [x]) [y]}, (16) an interval box [x] IR nx, and B (r, [x]) denoting the set { } x R nx min a x < r. (17) a [x]
o P, resp. Furthermore, d is a unction deined on IR nx IR nx with d : ([x], [y]) sup {r R B (r, [x]) [y]}, (16) an interval box [x] IR nx, and B (r, [x]) denoting the set { } Reliable Computing x 15, R nx 211 min a x < r. (17) 341 a [x] 5 Robustness 5 Robustness and Optimality and Optimality A general, Ainterval-arithmetic general, interval-arithmetic ramework ramework or structure or structure and parameter and parameter optimization o dynamical systems with both nominal and uncertain parameters has beenpresented presentedin in [8]. Accordingto to the deinitiono o optimalityor or uncertain systemsintroduced introduced therein, a control a control strategy strategy is optimal is optimal i it leads i it leads to thetosmallest smallest upper up- bound o the optimization o per boundperormance o the perormance index all index possible or all p possible [p]. Thispis depicted [p]. This exemplarily is depicted in Fig. 4 or exemplarily l max indierent Fig. 4 or control l laws. max dierent control laws. [ J <l> ] sup ([ J <l > ]) best approximation guaranteed non-optimal control sequences interval enclosure o the range o necessary costs 1 2 3 l l l max Figure 4: Figure Optimality 4: Optimality o control o strategies control strategies with interval with uncertainties. interval uncertainties. These controls can either result rom dierent parameterizations o a controller with These controls can either result rom dierent parameterizations o a controller with a a ixed ixed structure structure (parameter (parameter optimization) optimization) or or rom rom dierent dierent control control structures (determined (determined during structure during structure optimization). optimization). In the latterincase, thepiecewise latter case, constant and structures piecewise piecewise constantlinear and piecewise approximations linear o approximations the globally optimal o thecontrol globallysequence optimalare determined. are determined. control sequence In both cases, In bothveriication cases, veriication techniques techniques are used are used to evaluate to evaluate the the integral perormanceindex perormance t J = t (x (t ), p (t ), t ) + (x (t), p (t), u (t), t) dt (18) J = t (x (t ), p (t ), t ) + (x (t), p (t), u (t), t) dt (18) to be minimized. The term t (x (t ), p (t ), t ) corresponds to terminal costs at the prescribed inal point o time t i inal states x (t ) are not speciied exactly. The integrand (x (t), p (t), u (t), t) quantiies deviations between the current states and the desired trajectories o the state i variables inal states as wellx as (t the ) are required not speciied control eort u (t). to be minimized. The term t (x (t ), p (t ), t ) corresponds to terminal costs at the prescribed inal point o time t exactly. The integrand (x (t), p (t), u (t), t) quantiies deviations between the current states and the desired trajectories o the state variables as well as the required control eort u (t). 6 Optimal Control o a Mechanical Positioning System with Friction In the remainder o this paper, 9 optimal control o a mechanical positioning system with state-dependent switchings between dierent dynamical models is discussed [8]. For that purpose, a sliding mass m is considered. For a given initial position x 1 () and given initial velocity x 2 (), an approximation o an optimal control strategy or the accelerating orce u (t) = F ext (t) is determined under consideration o the riction orce F (x 2). The state equations or this system are given by ẋ (t) = t [ ] [ ] 1 x (t) + 1 m (Fext (t) F (x 2)) with x = [ x1 x 2 ]. (19)
342 A. Rauh et al., Robust and Optimal Control o Dynamical Systems The riction characteristic is modeled by l = 3 discrete states S = {S 1, S 2, S 3}, which are sliding riction or motion in negative (backward) direction (= S 1), static riction (= S 2), and sliding riction or motion in positive (orward) direction (= S 3). In this riction model, interval parameters are considered or both the static riction coeicient [F s] = [ ] [ ] F s ; F s and the sliding riction coeicient [µ] = µ ; µ. The resulting sliding riction orce is then given by F (x 2) = { [Fs] + [µ] x 2 or S 1 = true + [F s] + [µ] x 2 or S 3 = true (2) and the static riction by F (x 2) [F max s ] = [ F s ; F s ] or S 2 = true. (21) In Fig. 5, the results or an approximate solution o the optimal control problem are shown. I a piecewise constant approximation o the optimal control strategy is determined, the perormance index J = t ( (x1 (t) 1) 2 + x 2 (t) 2 + u (t) 2) dt + 1 T k max k=1 (u k u k 1 ) 2 (22) is minimized by a control strategy with a maximum number o 5 switchings in the considered time span. Analogously, the perormance index J = t ( (x1 (t) 1) 2 + x 2 (t) 2 + u (t) 2) dt + 1 T k max k=1 u 2 k (23) is minimized with u k := u (t), t k 1 < t < t k i piecewise linear control strategies are considered. Note that the absolute values o both perormance indices cannot be compared directly. However, these criteria have been chosen to guarantee a maximum similarity between both types o approximations. For the parameters F s =.15, µ =.1, u [ 1. ; 1.], and u [.5 ;.5] it can be shown that the inequalities J 2.56 and J 4.43 hold or the piecewise constant and piecewise linear approximation, resp. In this example, the range o admissible inal positions is described by the interval x 1 (t = 5) [.9 ; 1.1]. The admissible inal velocities are characterized by x 2 (t = 5) [.1 ;.1]. 7 Conclusions and Outlook on Future Research In this paper, interval-based techniques or the veriication o reachability, observability, and robust asymptotic stability o dynamical systems have been presented. They represent suicient conditions or the corresponding system theoretic properties. Overestimation in the computation o guaranteed lower bounds o the rank o the reachability and observability matrices might lead to conservative results i large domains are considered or state variables and uncertain parameters. Eicient routines employing subdivision o those regions are available which help to express the resulting dependencies during rank computation as precisely as necessary. Together with the routine or veriied stability analysis, they are included in a general-purpose ramework or optimal and robust controller design. In this ramework, also dynamical
Reliable Computing 15, 211 343 1.5 1..4 position x1 1.1 1..9.5 1 2 3 4 5 t velocity x2.5.1..1.5 1 2 3 4 5 t orce Fext.2..2.4 1 2 3 4 5 t Figure Figure 5: 5: Approximate Approximate solution solution o o the the optimal optimal control control problem problem with with statedependent statedependent switchings. switchings. Solid Solid lines: lines: piecewise piecewise constant constant control, control, dashed dashed lines: lines: piecewise piecewise linear linear control. control. systems with state-dependent switchings between dierent dynamical models can be considered. In uture work, we will develop extensions towards the design o state and disturbance estimators using interval techniques. These estimators will be used or the computation o worst-case bounds or the inluence o approximation techniques in the design o both observers and controllers [7, 1]. A urther step will be real-time state estimation in model-predictive control approaches or dynamical systems with interval uncertainties. For that purpose, the use o veriied solvers or systems o dierential-algebraic equations (DAEs) will be included and relations between criteria or solvability o DAEs and the criteria or controllability and observability o states in control theory will be investigated. Reerences [1] J. Ackermann, P. Blue, T. Bünte, L. Güvenc, D. Kaesbauer, M. Kordt, M. Muhler, and D. Odenthal, Robust Control: The Parameter Space Approach. Springer- Verlag, London, 2nd edition, 22. [2] N. Delanoue, Algoritmes numriques pour l analyse topologique Analyse par intervalles et thorie des graphes, PhD thesis, cole Doctorale d Angers, 26, in French. [3] A. Isidori. Nonlinear Control Systems 1. An Introduction, Springer-Verlag, Berlin, 1995. [4] Y. Lin and M. A. Stadtherr, Deterministic Global Optimization or Dynamic Systems Using Interval Analysis, In CD-Proc. o the 12th GAMM-IMACS Intl. Symposium on Scientiic Computing, Computer Arithmetic, and Validated Numerics SCAN 26, Duisburg, Germany, IEEE Computer Society, 27. [5] Y. Lin and M. A. Stadtherr, Deterministic Global Optimization o Nonlinear Dynamic Systems AIChE Journal, vol. 53, no. 4, pp. 866 875, 27. [6] H. J. Marquez. Nonlinear Control Systems, John Wiley & Sons, Inc., New Jersey, 23. [7] J. Minisini, A. Rauh, and E. P. Hoer, Carleman Linearization or Approximate Solutions o Nonlinear Control Problems: Part 1 Theory, In F. L. Chernousko, 1
344 A. Rauh et al., Robust and Optimal Control o Dynamical Systems G. V. Kostin, and V. V. Saurin, editors, Advances in Mechanics: Dynamics and Control: Proc. 14th Intl. Workshop on Dynamics and Control, pp. 215 222, Moscow-Zvenigorod, Nauka Publ., Russia, 28. [8] A. Rauh and E. P. Hoer, Interval Methods or Optimal Control, In: A. Frediani and G. Buttazzo, editors, Proc. o the 47th Workshop on Variational Analysis and Aerospace Engineering 27, pp. 397 418, School o Mathematics, Erice, Italy, Springer Verlag, 29. [9] A. Rauh, J. Minisini, and E. P. Hoer, Veriication Techniques or Sensitivity Analysis and Design o Controllers or Nonlinear Dynamic Systems with Uncertainties, Intl. Journal o Applied Mathematics and Computer Science AMCS, vol. 19, no. 3, pp. 425 439, 29. [1] A. Rauh, J. Minisini, and E. P. Hoer, Carleman Linearization or Approximate Solutions o Nonlinear Control Problems: Part 2 Applications, In: F. L. Chernousko, G. V. Kostin, and V. V. Saurin, editors, Advances in Mechanics: Dynamics and Control: Proc. 14th Intl. Workshop on Dynamics and Control, pp. 266 273, Moscow-Zvenigorod, Russia, Nauka Publ., 28. [11] J. Rohn, Positive Deiniteness and Stability o Interval Matrices, SIAM Journal on Matrix Analysis and Applications, vol. 15, no. 1, pp. 175 184, 1994.