Scenario-based Model Predictive Control: Recursive Feasibility and Stability

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1 Preprints o the 9th International Symposium on Advanced Control o Chemical Processes The International Federation o Automatic Control MoM1.6 Scenario-based Model Predictive Control: Recursive Feasibility and Stability Michael Maiworm Tobias Bäthge,,1 Rol Findeisen Institute or Automation Engineering, Otto-von-Guericke University, Magdeburg, Germany. {michael.maiworm, tobias.baethge, rol.indeisen}@ovgu.de Max Planck Institute or Dynamics o Complex Technical Systems, Magdeburg, Germany. Abstract: Many processes are inluenced by uncertain parameters or external disturbances, such as temperature changes. The control o such systems is in general challenging. In this work, we consider robust multi-scenario Model Predictive Control (MPC). Its central idea is to assume a inite number o possible values or the uncertainties and to model their combinations in a scenario tree. We adapt the classical dual mode approach o nominal MPC to establish recursive easibility and stability or the multi-scenario case, using a common terminal region and common terminal cost unction or all uncertainty realizations. For linear systems, the computation o these ingredients can be ormulated as a semideinite program. In a simulation, we apply the suggested approach to building climate control and show that it robustly stabilizes the system while a standard MPC controller violates state constraints and becomes ineasible. Keywords: Predictive control; robust control; stability; recursive easibility; uncertainty; scenario tree; building climate control. 1. INTRODUCTION Model Predictive Control (MPC) is an optimization-based control scheme naturally capable o dealing with multiinput multi-output systems. In addition, MPC allows to include input and state constraints in the controller design. For these reasons, MPC is increasingly used in industrial applications (Mayne (214); Di Cairano (212); Qin and Badgwell (23)). However, MPC needs an accurate model to predict the system behavior and to achieve good and reliable perormance. Unortunately, uncertainties (e.g. noise, unknown disturbances, model-plant-mismatch due to inaccurate modeling) are always present and deteriorate the perormance o the controller. They may also destroy important properties like recursive easibility and stability. For this reason, extensions to MPC have been developed over the last ew decades that allow to explicitly take uncertainties into account, guaranteeing constraint satisaction, recursive easibility, and stability. This includes e.g. min-max MPC (Rawlings and Mayne (29)), tubebased MPC (Mayne et al. (26, 29)), eedback MPC (Mayne et al. (2); Bertsekas (25)), and relaxationbased robust MPC (Strei et al. (214)). In the rame o this work, we consider scenario-based robust MPC, also denoted as Multi-Scenario Model Predictive Control (MS-MPC), see e.g. Lucia et al. (212). The central idea o this approach is to assume or approximate discrete values or an uncertainty and to capture their 1 Support by the International Max Plank Research School (IM- PRS) or Advanced Methods in Process and Systems Engineering, Magdeburg, Germany, is acknowledged. combinations over time by means o a scenario tree. Each scenario starts rom the root node (the initial value x(k)) and ends in one o the leaves (c. Fig. 1). Fig. 1. Scenario tree with a prediction horizon o N = 2 and the discrete uncertain parameter set P = {p 1, p 2 }. The time instant is set to k :=. While this is an approximation or continuous disturbances, the approach shows good perormance compared to standard MPC i the uncertainty values are selected suitably. In addition, it could be shown that the multiscenario optimization problem remains easible where standard MPC does not (Lucia et al. (212)). A major drawback o MS-MPC is that the problem size grows exponentially with the prediction horizon and the numbers o uncertain parameters and dierent values that these parameters can take. One way to circumvent this problem is to assume that the scenario tree stops branching urther ater a deined stage within the prediction horizon, called robust horizon N r (Lucia and Engell (212)). Copyright 215 IFAC 5

2 In this work, we investigate how to adapt MPC to guarantee recursive easibility and stability or the described multi-scenario case, using a terminal region, a terminal penalty, and a virtual local control law. The contributions o this paper can be summarized as ollows: Adaptation o the stability approaches o nominal MPC to guarantee recursive easibility and stability or the multi-scenario case. Computation o suitable terminal regions and terminal cost unctions or multiple scenarios. Application o MS-MPC to building climate control, namely the cooling o a cold storage house. The outline o the paper is as ollows. Section 2 presents the MS-MPC problem ormulation and the necessary adaptations to establish recursive easibility and stability. This leads to the deinition o a suitable common terminal region and common terminal cost unction, which are investigated in detail and computed in Section 3. Aterwards, the MS-MPC scheme is employed or an example o building climate control in Section 4. The work concludes with Section 5, which summarizes the results and highlights possible uture extensions. 2. MULTI-SCENARIO MODEL PREDICTIVE CONTROL We consider a general nonlinear discrete time system x + = (x, u, p) s.t. x X u U p P, where x R nx denotes the states, x + the states at the next time instant, u R nu the inputs, and p R np the uncertain parameters. The inputs and states are constrained by the closed and convex sets U and X. The undamental assumption o the MS-MPC approach is that, at each time step, the uncertainties p are drawn rom a discrete set o the orm P = {p i R np i = 1,..., s}, with s N denoting the number o possible values p can take. A discrete set P may be the true nature o a speciic uncertainty, e.g. the number o lost packages in a communication network or the possible aults in a system (ault-tolerant control). Likewise, a discrete set P may arise rom the discretization o a continuous set (Goodwin et al. (29)). We consider all unique combinatorial permutations o the elements o P over the prediction horizon. For a given prediction horizon N N and the total number s o possible values o the parameter p, the number o possible combinations is N s = s N. These unique combinations are represented by the parameter sequences p j (k) = { p j (k), p j (k + 1),..., p j (k + N 1) } with p j (k + i) P or all i I :N 1 = {,..., N 1} and all j I 1:Ns = {1,..., N s }. The superscript j is the scenario index and k denotes the current time step. We call each o these sequences p j (k) a parameter scenario as it resembles a unique realization o the uncertain parameter evolution over the prediction horizon. In total, we obtain N s dierent parameter scenarios and to each o them, we can associate a predicted input and state (1) sequence û j (k) and ˆx j (k). Predicted variables are denoted with a ˆ. To account or the dierent scenarios, we write the prediction model as ˆx j (k i) = (ˆx j (k + i), û j (k + i), p j (k + i) ) (2) with i I :N 1 and j I 1:Ns. Scenario tree complexity In the scenario tree, the controls leaving one node have to be the same or all scenarios that share that particular node because they are based on the same past inormation and no exact uncertainty realization can be anticipated. These are the so-called nonanticipatory constraints (Goodwin et al. (29)). They can be reormulated as linear equality constraints o the orm Γ x(k) = (3) Γũ(k) =, where Γ is a sparse matrix o appropriate dimensions and x(k) and ũ(k) are the extended state and input sequences containing the states and inputs o all scenarios. The nonanticipatory constraints matrices are the same or states and inputs because equal inputs are computed or equal states (or nodes) in the scenario tree. In the general case, ũ(k) consists o Ns N elements, the prediction horizon multiplied by the number o scenarios. Yet, the non-anticipatory constraints reduce the number o independent optimization variables in ũ(k). This number is equal to the sum o all nodes rom the root node to the nodes at the penultimate stage, N 1 i= si = sn 1 s 1. The inherent computational reduction can be expressed by N 1 i= si s N 1 s 1 Ns N = Ns N = sn 1 Ns N (4) (s 1) and can then be approximated or large s or N by s N Ns N (s 1) = 1 N(s 1). (5) Thus, or large s or N, (5) computes the percentage to which the number o optimization variables is reduced, compared to all elements in ũ(k). For instance, or s = 6 and N = 8, the number o optimization variables is reduced to 2.5 % o the total number o optimization variables contained in ũ(k) or that case. However, despite the reduction by the non-anticipatory constraints, the number o independent optimization variables still grows exponentially with the prediction horizon. Cost unction As it is unknown which o the scenarios will be the actual realization o the uncertainty evolution, all o them have to be taken into consideration, i.e., we have to optimize over the ull scenario tree. In this optimization, we use the cost unction with J j = N 1 i= N s V N (x(k), ũ(k)) = ω j J j (6) j=1 l (ˆx j (k + i), û j (k + i) ) + V (ˆx j (k + N) ), where J j is the cost associated with scenario j, i.e., the cost o one branch rom the root node x(k) to its inal lea node ˆx j (k + N). The stage cost is l( ) and V (x) denotes Copyright 215 IFAC 51

3 the terminal cost unction which is assumed to be the same or all scenarios. For a speciic scenario, its probability to occur is given by ω j. The sum over all probabilities has to be one, i.e., N s j=1 ω j = Recursive Feasibility The scenario ormulation transorms the initially uncertain problem into a nominal MPC problem by spanning a scenario tree. Thereore, we can make use o nominal MPC theory to establish recursive easibility and stability or MS-MPC. In particular, we adapt the classical approach o nominal MPC, see e.g. Findeisen et al. (27) or Rawlings and Mayne (29), where the terminal state ends in a control invariant terminal region and is penalized by a suitable terminal cost unction. For the sake o clarity, we neglect the estimate nature, indicated by ˆ, o the state and input prediction. We assume that the ollowing two assumptions hold: Assumption 1. (Continuity). The unctions (x, u, p), l(x, u), and V (x) are continuous, with (,, p) = p P, l(, ) =, and V () =. Assumption 2. (Constraints). The sets X and X X are closed, and U is compact. Each set contains the origin. Establishing recursive easibility is equal to requiring that the terminal state o each scenario ends in a control invariant terminal region. We assume that this region is the same or all scenarios and denote it as common terminal region Ω (c. Fig. 2). Fig. 3. Scenario tree using system instances. Thus, a suitable terminal region must be invariant or the s dierent system instances. Assumption 3. (Common terminal region). The common terminal region Ω is control invariant or x + = j (x, u), j {1,..., s} with u U. Proposition 4. (Recursive easibility). Suppose that Assumptions 1, 2, and 3 hold, then the multi-scenario MPC is recursively easible. Proo. Following the proo or the nominal case and using the notation rom Rawlings and Mayne (29), let Ω N be the set o all initial conditions or which a solution to the multi-scenario optimal control problem exists. The underlying idea is to set Ω := Ω and to deine Ω 1 = { x X u U : j (x, u) Ω j {1,..., s} }. So, Ω 1 is the set o all states or which a control can be ound such that the state is driven to Ω, independent rom the exact system instance. This implies that Ω Ω 1. Since Ω is control invariant by Assumption 3, it ollows that Ω 1 is also control invariant. By backward recursion and induction, we arrive at the conclusion that Ω N is positive invariant and hence, multi-scenario MPC is recursively easible. 2.2 Stability Fig. 2. Scenario tree with s = 2 and N = 2 and appended terminal region. The question which arises is, how to deine and obtain a common control invariant terminal region or all scenarios. To this end, we look at the problem in a dierent way. Each realization o the parameter p P = {p 1,..., p s } can be interpreted as a realization or instance o the given system, i.e., x + = (x, u, p j ), j {1,..., s} x + = j (x, u), j {1,..., s}. This way, we can also reinterpret the scenario tree in such a way that the branching is caused by switching to another system realization (c. Fig. 3). At each node o the tree, the state evolves according to the current system instance. This is also true at the terminal state. Assuming that the state is in the terminal region Ω at i = N, recursive easibility translates to ensuring that the state stays in Ω or all possible system instances. To establish stability, we need more assumptions on the stage and terminal costs, l( ) and V (x), respectively. Since the proposed ormulation assigns an individual cost unction to each scenario, it also provides the ramework to include an individual terminal cost unction V j (x) to each scenario. Assumption 5. (Basic stability assumption). For all x Ω and or all j {1,..., N s }, min ũ(k) U { V j ((x, u, p)) + l(x, u) (x, u, p) Ω holds or all p P. } V j (x) This implies that, or each uncertainty realization, there exists an input such that the terminal region is control invariant, i.e., Assumption 5 implies Assumption 3. Note that Assumption 5 is satisied, i each V j (x) is a control Lyapunov unction or all p P. Furthermore, Assumption 5 only states that each terminal cost unction V j (x) has to satisy the descent property, but not whether all (x) have to be dierent or equal. Thus, i one Lyapunov unction V (x) that satisies this property can be ound, it can also be used or all remaining scenarios. Hence, as in V j Copyright 215 IFAC 52

4 the case o the terminal region, we can use one common terminal cost unction V (x). However, this is just a special case o the oregoing assumption. All V j (x) can be dierent, as long as Assumption 5 is satisied. Furthermore, we assume: Assumption 6. (Bounds on stage and terminal costs). The stage cost l(x, u) and the terminal costs V j (x) satisy l(x, u) α 1 ( x ) x Ω N, u U V j (x) αj 2 ( x ) x Ω and j {1,..., N s }, in which α 1 (x) and α j 2 (x) are K unctions. We can now ormulate the ollowing theorem: Theorem 7. (Multi-scenario MPC stability). Suppose that Assumption 1, 2, 3, 5, and 6 are satisied and that Ω contains the origin in its interior. Then, the origin is asymptotically stable with a region o attraction Ω N or the system x + = j (x, κ N (x)) or all j {1,..., s}. Proo. We again ollow the structure o the proo or the nominal case, see e.g. Rawlings and Mayne (29). The basic idea is that Assumptions 1, 2, and 3 need to be satisied or recursive easibility. Assumptions 5 and 6 ensure that the value unction is a Lyapunov unction or x + = j (x, κ N (x)) or all j {1,..., s} on the domain Ω N. Hence, the origin is asymptotically stable or all system instances. 2.3 Multi-scenario MPC ormulation The multi-scenario optimal control problem ormulation used in MPC becomes min ũ(k) V N (x(k), ũ(k)) s.t. i I :N 1 and j I 1:Ns : ˆx j (k + i + 1) = (ˆx j (k + i), û j (k + i), p j (k + i) ) ˆx j (k) = x(k) û j (k + i) U ˆx j (k + i) X ˆx j (k + N) Ω p j (k + i) P Γũ(k) =, where x(k) is the initial condition. The solution to (7) is denoted by ũ (k) and its irst element û (k) is applied to the plant. Hence, MPC establishes the control law u(k) = κ N (x(k)) = û (k). I the solution is inserted into the cost unction, we obtain the so-called value unction V N ( x(k), ũ (k) ). 3. TERMINAL REGION COMPUTATION In the previous section, we presented an approach that guarantees recursive easibility and stability or multiscenario MPC, using suitable terminal regions and terminal penalties. The determination o these ingredients is very challenging in the general nonlinear case. Thereore, (7) we ocus on the linear case in this section and show how to obtain a common terminal region and a common terminal cost unction, such that Assumptions 3, 5, and 6 are satisied. For that purpose, we use a quadratic unction as the common terminal cost and a linear terminal control law. A level set o the cost unction serves as the common terminal region. Constraint satisaction is achieved by using the support unction o the involved sets. The determination o the common terminal region is ormulated as a semideinite program. 3.1 Invariance o the Terminal Region We consider a linear discrete time system o the orm x + = A j x + B j u, j {1,..., s} s.t. x X (8) u U, where the matrices A j and B j are realizations o the uncertain parameter p P. For the terminal controller, we choose the linear control law u = κ (x) = Kx with K = RP and P = P T > o appropriate dimensions. For the common terminal cost, we use the quadratic unction V (x) = x T P x. The variables to be computed are the matrices R and P. The matrix P is determined such that V (x) is a Lyapunov unction or all closed-loop system instances. Thereore, P must be positive deinite and we need to ensure that V (x + ) V (x) is negative semideinite or all system instances. Inserting the system (8) and algebraic reormulation yields Q (A j Q + B j R) T Q 1 (A j Q + B j R) (9) or all j {1,..., s} with Q = P 1. The sought variables are Q and R and they appear in a nonlinear manner in (9), which makes their determination diicult. Thereore, the expression is reormulated into a set o linear matrix inequalities (LMIs) by means o the Schur complement (Boyd and Vandenberghe (24)). We obtain Q (A j Q + B j R) (A j Q + B j R) T, j {1,..., s}, Q (1) which is equivalent to (9) and hence, can be used instead o it. This is a set o LMIs, which are aine in the unknowns Q and R. Hence, (1) can be solved eiciently and yields the matrix P that on the one hand makes V (x) = x T P x a Lyapunov unction or the closed-loop system and, on the other hand, deines the terminal control law. We choose a level set o the Lyapunov unction as the common terminal region: Ω := { x R nx x T P x 1 }. (11) Hence, u = κ (x) = Kx with K = RP is a stabilizing terminal controller or all system instances on Ω. Note that (11) describes an ellipsoid in the state space. 3.2 Constraint Satisaction In the ollowing, we assume that the constraints can be represented by polyhedral sets o the orm X = { x R nx c T } i x r i, i = 1,..., N X (12a) U = { u R nu a T l u s l, l = 1,..., N U }, (12b) Copyright 215 IFAC 53

5 where N X, N U N denote the respective numbers o inequalities by which the constraint sets are deined. Satisaction o state constraints translates to Ω X (13a) and since u = Kx U or all x Ω, satisaction o input constraints translates to KΩ U. (13b) These are general set conditions which, in general, are hard to veriy. In the case o closed convex sets, the support unction o the involved sets allows us to reormulate conditions (13) as LMIs (Blanchini and Miani (28); Kolmanovsky and Gilbert (1998)). The support unction o an ellipsoidal set as deined by (11) is given by h Ω (η) = ηt P 1 η with η R nx. Given the two closed convex sets Ω and X, checking the state constraint condition (13a), Ω X, is equivalent to checking i h Ω (η) h X (η) holds or all η. In case that X is a polyhedral set as deined in (12), checking Ω X simpliies even urther to veriying that h Ω (c i ) r i holds or all i {1,..., N X }. Putting all this together, we arrive at the equivalence Ω X c T i P 1 c i r i, i {1,..., N X }. The right hand expression can be reormulated by means o the Schur complement into the set o LMIs given by Q (Qci ) (Qc i ) T, i {1,..., N X }, (14) r 2 i with Q = P 1. Hence, the state constraints are satisied, i a Q can be ound that solves (14). Note that the Q in (14) is the same as in (1). The same procedure can be applied to transorm the input constraint condition (13b) into a set o LMIs. We get Q (R T a l ) (R T a l ) T s 2, l {1,..., N U }, (15) l with Q = P 1. Hence, the input constraints are satisied i a Q can be ound that satisies (15). 3.3 Optimization Problem In order to ind a common terminal region Ω with associated terminal controller κ (x), we use (1), (14), and (15) as constraints to an optimization problem whose objective it is to maximize the common terminal region. This can be achieved by maximizing log (det(q)) (see Boyd (1994); Boyd and Vandenberghe (24)): max log ( det(q) ) Q,R s.t. Q = Q T > Q (A j Q + B j R) T, j {1,..., s} (A j Q + B j R) Q Q (Qci ) (Qc i ) T, i {1,..., N X } r 2 i Q (R T a l ) (R T a l ) T s 2 l, l {1,..., N U }. (16) This optimization problem is a semideinite program. It can be eiciently solved, e.g. in MATLAB together with the YALMIP toolbox. We can now reormulate Theorem 7 or the linear case. Theorem 8. (Linear multi-scenario MPC stability). Suppose that Assumptions 1 and 2 are satisied and that a solution to (16) exists. Then, the origin is asymptotically stable with a region o attraction Ω N or the system x + = A j x + B j κ N (x(k)) or all j {1,..., s}. Proo. By construction, the solution to (16) yields the matrix P that makes x T P x a common control Lyapunov unction and Ω deined by (11) a common terminal region or the uncertain linear discrete time system (8). Hence, Assumptions 3, 5, and 6 are satisied and Theorem 7 guarantees asymptotic stability o the origin o the closed loop system x + = A j x+b j κ N (x(k)) or all j {1,..., s}. 4. BUILDING CLIMATE CONTROL We apply the MS-MPC approach (7) with the terminal ingredients computed by (16) to building climate control, which is a well-suited application or Model Predictive Control, see e.g. Oldewurtel et al. (212). Since MPC is a control method that uses predictions to determine a suitable input, weather predictions could be included easily. In our simulations, we use a model that is based on Bacher and Madsen (211). It can account or external inluences, such as ambient temperature and solar radiation. We ocus on the our-states version o the model, adapted to resemble a cold storage house. 4.1 Model The model equations, on the basis o Bacher and Madsen (211), are s = 1 (T i T s ) (17a) T s C s i = 1 (T s T i ) + 1 (T CCU T i ) T s C i R ih C i + 1 (T e T i ) + 1 (T a T i ) (17b) R ie C i R ia C i CCU = 1 (T i T CCU ) + 1 Φ CCU (17c) R ih C h C h e = 1 (T i T e ) + 1 (T a T e ), (17d) R ie C e R ea C e where T s, T i, T CCU, and T e are the temperatures o the sensor, the interior, the climate control unit (CCU), and the building envelope (walls). The input to the system is the CCU power Φ CCU. Moreover, the model underlies external inluences in the orm o the ambient temperature T a. Values or the model parameters can be ound in Bacher and Madsen (211). We use this model or a cold storage house in which e.g. ood is stored, and we consider the uncertain case o an open or closed door. This can be modeled as an uncertainty in the parameter R ia {1.77, 5.31}. The control task is to bring and keep the room temperature down to C as close as possible, without going below Copyright 215 IFAC 54

6 States C 1 States C T s T i T CCU T e Time h (a) Evolution o the states. T s T i T CCU T e Time h (a) Evolution o the states. Input ΦCCU kw Input ΦCCU kw Time h (b) Input signal Φ CCU. Fig. 4. Simulation results or the nominal case. The room temperature T i violates the state constraints. The control input Φ h underlies high luctuations and switches between heating and cooling. Time h (b) Input signal Φ CCU. Fig. 5. Simulation results or the robust case. The room temperature T i stays above the reezing point o C. The control input Φ h varies only in a small range and does not switch between heating and cooling. that reezing point, and without letting the stored goods get warmer than 1 C, i.e., the room temperature T i is constrained to, 1 C. For the cooling or heating, the climate control unit can provide a power Φ CCU in the range o 25, 25 kw. Positive and negative values correspond to heating or cooling, respectively. Without loss o generality, we assume a constant ambient temperature o 3 C. The various temperatures in the room (initial state) shall also be 3 C, to show that the controllers are able to perorm a set point change. We compare a nominal MPC controller, which assumes the nominal value R ia = 1.77 or the whole simulation time, with a scenario-based MPC controller, which does not make assumptions about the actual uncertainties. The actual uncertainty realization is described by a randomly generated sequence o the two parameter values o R ia, i.e., the door is opened or closed randomly during the simulation. 4.2 Implementation The continuous model is discretized using Euler s method with a sampling time o t s = 18 s. The resulting discrete time system is used in a Quadratically-Constrained Quadratic Program (QCQP) which results rom the regular linear MPC to QP reormulation, and the quadratic terminal constraint. Both the nominal and the multi-scenario MPC controller use a prediction horizon o N = 2. The MS-MPC controller urthermore uses a robust horizon o N r = 3, ater which the scenario tree stops branching. In the considered example, this value is a good compromise between controller perormance and computational burden. In general, the determination o N r is still an open question. For details regarding the inluence o N r, see Lucia et al. (213). Using (16), the matrix P deining the terminal region or the considered case was computed to P = Simulation Results Fig. 4a shows that the nominal MPC controller is not robust, i.e., it violates the state constraints, whereas the scenario-based MPC controller is easible at all time steps and does not violate the state constraints (see Fig. 5a) and, hence, is robust. The price or the achieved robustness is a slight reduction o the perormance, as the average room temperature will be above C. Copyright 215 IFAC 55

7 The control inputs, shown in Fig. 4b and Fig. 5b, show equal behavior during the transition phase. Once the constraint is reached, the nominal MPC controller starts oscillating in magnitudes o up to 25 kw, whereas the MS-MPC controller only varies in a range o about 5 kw, resulting in less wear on the climate control unit. Using MATLAB s mincon on a standard desktop PC, the calculation times or solving the optimization problems in the MPC schemes lie in the range o seconds. Thus, realtime implementation o the MS-MPC controller is possible, given the used sampling time o 18 s. 5. SUMMARY AND CONCLUSION This work ocuses on recursive easibility and stability o Multi-Scenario Model Predictive Control. We consider the uncertain parameter case, assuming discrete uncertainty realizations modeled as a scenario tree. In order to establish recursive easibility and stability, we extend the classical dual mode approach o nominal MPC. This leads to the ormulation o a common control invariant terminal region and a common terminal cost unction or the optimal control problem. For the computation o these ingredients or the linear case, we set up a semideinite program accounting or the invariance property and the rigorous satisaction o input and state constraints, given by closed and convex sets. We apply this approach in a building climate control scenario. The simulations show that the MS-MPC controller with the computed terminal region ulills the task without violating the constraints, whereas a standard MPC controller violates the state constraints at several points. Future works could investigate the computation o a terminal region and terminal cost unction or certain classes o nonlinear systems, e.g. systems with sector-bounded nonlinearities. To extend the work on time-independent parameter uncertainties, urther research could also explore the case o uncertain additive disturbances. 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