Game Theoretic Controllers that Confine a System to a Safe Region in the State Space

Transcription

1 Milano (Italy) August 8 - September, Game Theoretic Controllers that Confine a System to a Safe Region in the State Space R. B. Vinter EEE Department, Imperial College, London, SW7 AZ, UK ( r.vinter@imperial.ac.uk). Abstract: This paper addresses the problem of controlling a system, subject to disturbances, to prevent exit from a safe region of the state space. Problems of this nature are encountered in many different contexts, for example in the control of tanks to avoid overflow of operation of telecommunications systems to avoid the failing of a communications link. The proposed control methodology is based on solving a differential game, in which a feedback control is found that minimizes the disturbance energy to force the state out of the safe region. Such controls also have interpretations in terms of stochastic exit-time problems. We announce a new class ofexitproblems, including problems with high state dimension, forwhich therelated game can be simply computed. The proposed solution technique involves the on-line solution of N optimal control problems where, typically, N is small. Accordingly, the computational effort is several times larger, but still of the same order of magnitude, as that involved in implementing a stard non-linear model predictive controller. Keywords: Optimal control theory, Generalized solutions of Hamilton-Jacobi equations, Differential or dynamic games. INTRODUCTION We propose a methodology for controlling a system, subject to disturbances, to prevent the state exiting from a safe region of the state space. The system model is taken to be { ẋ(t) = f(x(t),u(t)) + σ(x(t))v(t) x() = x () u(t) Ω, the data for which comprise functions f : R n R m R n σ : R n R n m, a set Ω a point x R n. Here, x denotes the state variable. u is interpreted as a control variable, v as a disturbance variable. The safe region is an open subset A R n which has the functional inequality representation A = N j={x h j (x) < }, expressed in terms of a collection {h j (.) j =,...,N} of C functions, such that the gradients of active constraint functions at each boundary point of A are linearly independent. This paper deals with situations in which magnitude bounds on disturbances are not available the occurence of a state constraint violation, at some time, is inevitable. The object then is to postpone exit from the safe region for as long as possible. Notice that the control design problem is very different to that considered in the Robust Model Predictive control literature, which takes The numerical work reported in Section 6 was carried out by P. Falugi, supported the EPSRC grant Robust Model Predictive Control, ref. EP/PO6675. advantage of known bounds on disturbance signals to provide controllers which avoid exit from the safe region altogether steer the system to a neighborhood of the desired fixed point. See Mayne Rawlings (9). Our approach is to find closed loop control described by the mapping D U, compatible with a state feedback control u = χ(x), that solves the one-sided differential game: sup inf v dt + θ τ u U v V (P x ) subject to ẋ = f(x,u = D(v)) + σ(x)v a.e. t [,τ] x() = x. The data for this problem comprises the functions f, σ sets Ω A, as above, also a design parameter θ. V U denote open loop v-controls closed loop u-controls respectively; these concepts will be defined precisely below. τ is the first time exit time of the state trajectory from A. If θ =, the cost is v dt, that is, the minimum disturbance energy required to drive the state out of the safe region. When θ >, the disturbance energy cost is replaced by a weighted sum of energy time to exit. Thus the controller maximizes some measure of the effort that must be expended by the v-player to achieve violation of the state constraint in a short period of time. Copyright by the International Federation of Automatic Control (IFAC) 57

2 Milano (Italy) August 8 - September, We mention that this differential game is linked to optimal exit time problems in stochastic control. Indeed it is known that the game-theoretic controller also minimizes a risk sensitized modification of the expected exit time τ from the safe region, namely, E x [e θ τ/ǫ], when the dynamics are described by the stochastic stochastic differential equation dx = f(x,u)dt + ǫ σ(x)dw, in which w(.) is Brownian motion, in the limit as the noise intensity ǫ vanishes (see Dupuis McEneaney (997), Bardi Capuzzo-Dolcetta (997), Fleming Soner (997)). There are clearly parallels between solving the above differential game the design of non-linear H controllers, controllers whose purpose is to minimize the norm of the mapping from disturbance signals to output deviations (see Van der Shaft (993)). We point out however that non-linear H controllers only indirectly address the containment problem, because theydonot take account of the specific safe region under consideration. The most general approach to solving thegame is to compute asolution to the associated Isaacs partial differential equation, thereby obtain the optimal strategy (see McEneaney (6), Falcone (987)). The computational dems of this approach increase rapidly with state space dimension. Ourapproach is, instead, torestrict thefamily ofdifferential games considered, insuchamannerthatthe solution can be obtained by analysis alone, by carrying out relatively simple online calculations, or by a combination of the two. According to our approach, the safe set A partitions into regions {D j } N i=, each associated with just one of the state constraints, for each j, it is possible to find a constant control value u j that pushes away from the j th state constraint boundary. The state feedback controller is a multi-valued bang-bang controller of the form: u = u j,where x D j. For the problems considered, the regions {D j } are characterized by examining the relative costs of N (one player) optimal control problems; the jth optimal control problem is constructed from the game, by fixing the u-control at u j, imposing only the j th boundary constraint h j (x(τ)) =. These optimal control problems are relatively simple. If the underlying dynamics are linear θ = then each problem is a linear quadratic problem with an endpointconstraint. Aspecialcase was examined earlier in Clark Vinter (3). We describe a very simple structure for solution to the games problem, namely that it reduces to the solution of a collection ofoptimal control problems indifferent regions of the state space. We give conditions for the solution to have such a structure: they are monotonicity nonintersecting extremals conditions. These conditions are satisfied in a number of engineering applications, some examples of which are analysed in the concluding sections of this paper. In summary then, we formulate the problem of controlling a system to prevent escape from a safe region as a differential game, identify a class of games arising in this way such that a state feedback control strategy can be obtained either by analysis, or by computing online the solution to a small number of optimal control problems. Finally, we supply some examples, pointing to potential areas of the application of the methodology. Within the space ofthis paper, the accompanying analysis is necessarily abbreviated; full details will appear in a forthcoming paper, along with elaborated treatment of the examples.. THE DIFFERENTIAL GAME A precise formulation of the differential game considered is as follows: Maximize inf (P x ) v V v dt + θ τ over D U. Here, τ is the first exit time from A of the state trajectory x(.) satisfying the equation ẋ(t) = f(x(t),u(t) = D(v(.))(t)) + σ(x(t))v(t), with initial state x. That is, we seek a closed loop control for the u-player which maximizes the specified cost, when the v-player, knowing the u-player s strategy, chooses an open loop control v V to minimize the cost. The value of (P x ), for arbitrary x A, is W(x ) = sup inf v dt + θ τ. v V D U This problem description requires definitions of the underlying control spaces. The spaces of open loop controls, U V, for the u v players are respectively U := {measurable functions u : [, ) R m s.t. u(t) Ω a.e. } V := L ([, );R m ). The class U of closed loop controls is taken to be the space of (Elliott-Kalton sense) non-anticipative controllers, namely U := {non-anticipative mappings D : V U} in which a non-anticipative mapping D : V U is a mapping such that, for each pair of open loop controls v, v V each T we have: v [,T] = v [,T] a.e. implies D(v) [,T] = D(v ) [,T] a.e. Denote by exit A the exit boundary from A. This is a subset of the boundary of A, comprising all exit points of state trajectories, originating in A corresponding to some u U v V. 3. DECOMPOSABLE PROBLEMS We introduce a class of games for which the optimal closedloop strategyhas aspecialstructure, which greatly simplifies its computation: the safe set can be partitioned into regions, on each of which the state feedback control takes a constant value. Furthermore, the regions can be identified by solving, for arbitrary initial states, a finite 57

3 Milano (Italy) August 8 - September, number of optimal control problems, each involving just one of the functional inequality endpoint constraints a fixed control which pushes away from the boundary of this constraint. The definitions of this section concern the conditions under which this special structure arises. For j =,...,N write A j = {x h j (x) < } () (A j is a super-set of A, obtained by removing all but the j th state constraint). We have A = N i=a j. For j =,...,N, we select a u-control value u j which, in some sense, pushes away state trajectories from {x h j (x) = }, i.e. the boundary of A j ). For every j x A j consider the optimal control problem Minimize τ v dt + θ τ (Px) j over τ v L ([,τ];r m ) satisfying ẋ = f(x,u j ) + σ(x)v a.e. on [, ) x() = x, h j (x(τ)) =. It is important to observe that each (P j x) is a one player optimal control problem, because the u-player control value is frozen at u j. For each j let W j : A j R be the value function for (P j x) W j (x) = inf{(px)} j for x A. define D j = {x A W j (x) W j (x) for all j j}. (The D j s may overlap). It will be assumed that (H): For each j W j (.) is locally Lipschitz continuous on A j, continuously differentiable on {x A j W(x) > }. Also, for each x {x h j (x ) = } exit A seqence {x i } in A j such that x i x we have lim i W j (x i ) =.. Suppose that u j is the constant u-control associated with the D j region. Fix an initial state x in D j a v-player control v(.) such that the state trajectory x j (.), corresponding to v(.), u(.) u j the given initial state, exits from the domain A j of the (P j x ) problem at time τ. The monotonicity condition requires that the exit time τ for the trajectory x(.), corresponding to v(.) as above, any open loop u- control u the given initial state, will satisfy τ τ, i.e. the state trajectory x(.) will exit before the state trajectory x j (.). The property is illustrated in Fig.. This condition is satisfied, for example, in flow applications: if an inflow disturbance causes a tank to overflow when the outflow rate control is at a maximum, then this same inflow disturbance will certainly cause the tank to overflow, no later than before, if any other strategy is used to control the outflow. General conditions for monotonicity can be obtained, in the linear case for example, using methods of Farina Rinaldi ().. The non-intersecting extremals condition requires that, if the initial state is in region D j for some j, then the corresponding least escape energy trajectory x j (.) always remains in the same region D j. The property is illustrated in Fig.. u uj u(.) t x x(.) j h (x)= x (.) j τ τ t Definition 3.. We say that (P x ) is decomposable w.r.t. {(A j,u j )} if, in addition to (H), the following conditions are satisfied: For each j x A j, (P j x) has a minimizer ( v, τ), with corresponding state trajectory x, such that (a): (monotonicity) For any measurable u : [, τ] Ω the solution x(.) to ẋ(t) = f(x(t),u(t)) + σ(x(t)) v(t) x() = x satisfies h j (x(τ )) = for some τ [, τ], (b): (non-intersecting extremals) If for some j j we have W j (x) = W j (x) then (i): W j (x) > (ii): for all u Ω (W j W j )(x)(f(x,u) + σ(x) v()) <, (Under the hypotheses to be imposed, v(t) is continuous at t =.) (iii): for all u Ω v R m (W j W j )(x)(f(x,u) + σ(x)v) = implies f(x,u) + σ(x)v =. We comment on these conditions: Fig.. Monotonicity condition switching surface h (x)= h (x)= x (.) x (.) region D region D Fig.. Non-intersecting extremals condition 4. OPTIMAL STRATEGIES FOR DECOMPOSABLE PROBLEMS The theorem stated in this section supplies a complete description of the optimal closed loop control for decomposable games, where optimizing means optimizing 57

4 Milano (Italy) August 8 - September, over all non-anticipative u-player controls. Because the optimizing strategy will be described in terms of a state feedback, we must first provide a definition concerning the interpretation of state-feedbacks as non-anticipative controls. It will be helpful to consider the general, setvalued, state feedback relation: u χ(x) (3) where χ : R n Ω is a given set-valued function. (Allowing a set-valued feedback is convenient when dealing with discontinuous state feedback, when there is ambiguity over what control values to associate with the state feedback function at points on switching surfaces.) Definition 4.. A closed loop control D U, is said to be compatible with the feedback relation (3) if, for any v V, the solution x(.) to ẋ(t) = f(x(t),u(t)) + σ(x(t))v(t), x() = x, in which u = D(v), satifies u(t) χ(x(t)) for a.e. t [, ). Conditions for compatibility are given, for example, in Clark et al. (4). Theorem 4.. Consider the differential game (P x ). Assume (H): f(.,.) is continuous, σ(.) is Lipschitz continuous f(.,u) Lipschitz continuous with respect to x, with Lipschitz constant independent of u Ω. Suppose there exist control values u,...,u N Ω such that, by defining A,...,A N according to (), (P x ) is decomposable w.r.t. {(A j,u j )}. Let W j (.), j =,...,N be the value functions for the optimal control problems introduced in Section 3.. Then the value function for (P x ) is the lower envelope of the family of optimal control value functions {W j (.)} N j= : W(x) = min j W j (x) for all x A. (4) Furthermore, anyclosed-loopu-control thatis compatible with the set-valued feedback χ(x) = co {u j j arg min j W j (x)} for x A (5) ( one such control exists) maximizes the cost for Problem (P x ). Sketch of Proof of Thm. 4.: Write J(D(v),v;x ) for the pay-off of the differential game (P x ), for given initial state x, D U v V, i.e. τ J(u = D(v),v;x ) = v dt + θ τ in which τ is the first exit time from A of the state trajectory corresponding to u = D(v) v, originating from x. Denote by D any closed loop u-control compatible with (5). Let W(.) be the function defined by (4). It will suffice to show (A): For any D U inf v V J(D,v;x ) W(x ). (6) (B): Take any non-anticipative control D compatible with the set-valued state feedback u = χ(x) any v V. Take any j arg min j W j (x ) let ( x, v, τ) be a minimizer for (P j x ). Then J( D(v),v;x ) W(x ) (7) J( D( v), v;x ) = W(x ). (8) Indeed, these relations will assure that, for arbitrary x A D U, W(x ) = inf J( D(v),v;x ) inf J(D(v),v;x ). v V v V Consider (A). Select any D U take any j arg min j {W j (x )}. By definition of decomposable problem the optimal control problem (Px j ) has a minimizer ( v(.), τ) with state trajectory x(.). Since τ is the first exit time from A j, h j ( x( τ)) =. Write u = D( v). Write also x(.) for the state trajectory obtained by solving ẋ(t) = f(x(t),u(t)) + σ(x) v(t) with initial value x. From the conditions for decomposability we have h j (x(τ)) = for some τ [, τ]. But this implies that inf J(D(v),v;x ) J(u, v;x ) = v V τ τ v dt + θτ v dt + θ τ = W j (x ) = W(x ). Relation (6) in (A) is confirmed. The function W(.) is not continuously differentiable, but it is locally Lipschitz continuous as a lower envelope of locally Lipschitz continuous functions. Relation (7) in (B), for arbitrary v, is proved by using W(.) as a nonsmooth verification function, making use of invariance techniques as in, for example, Chapter of Vinter (). Proof of the relation (8) in (B), for the open loop v-player control v, makes key use of the non-intersecting extremals condition. We give below an example, with state dimension n =, where the state feedback control function can be obtained by analysis. In other situations where pencil paper solution is not possible, typically when the underlying control system has a high dimensional state space, implementation ofthegame theoretic controlrequires numerical optimization, along the following lines: Let T be the sample period. At time t k = kt measure the state x(t k ) compute the N (optimal control problem) minimum costs: { V j (x(t k )) = Min v dt + θτ ẋ = f(x,uj ) + σ(x)v, } x() = x(t k ) h j (x(τ)) =. Implement the control u(t) = u j on [kt,(k + )T] where j = arg min j {W j (x(t k ))}. We can expect that this control will give an approximation to the game theoretic strategy, the accuracy of which will depend on the sampling period T, if the non-intersecting extremals condition is satisfied. Even if this condition 573

5 Milano (Italy) August 8 - September, is not satisfied, the procedure will yield a sub-optimal controller. Simple online tests are available for testing non-intersection of extremals. 5. EXAMPLE: SECOND ORDER SYSTEM We consider the problem of controlling a surge tank, a buffer device used in process systems engineering, to prevent excessive flow rate fluctuations, as fluids flow between reactors. Let y be the height of the fluid in a surge tank of uniform cross-section. y is related to the rate of change of inflow v the rate of change of outflow u according to: d y/dt = u + v. In this equation, v u are interpreted as a rom disturbance control signal, respectively. The constraints that the tank neither empties nor overflows are captured by the inequalities: y +. (9) On the other h, a condition that the maximum rate of change of outflow must be bounded in magnitude can be expressed as u +. () We now seek a state feedback control, with values in Ω, u = χ(y,ẏ), depending on the current height rate of change of height, to avoid violation of the state constraint (9). In this illustrative example, we set all constants to one. The differential game for this setup is sup D inf τ,v v dt + θ τ over D U, τ, v V subject to ẋ = Fx + b( D(v) + v) a.e. t [,τ] u(t) [,+] a.e. x() = x h j (x(τ)) = for some j, in which F = h j (x) = [ ] b = [ ] { c T x if j = c T x if j = in which c T = [ ]. The state constraint sets A A for the two optimal control problems are A = {x h (x) < } = (,+) R A = {x h (x) < } = (, ) R, with boundaries {x h (x) = } = {+} R {x h (x) = } = { } R. The controls pushing away from these boundaries are, respectively, u = + u =. It is a straightforward exercise to show that the game is monotone, in the sense of Def. 3.. We must also show that it has non-intersecting extremals. To this end, we need to examine the two optimal control problems (Px ) (Px ): inf v dt + θ τ(x) (Px v, τ ) over τ v L ([,τ];r) satisfying ẋ = Fx + b( + v) a.e. t [,τ] x() = x, x(τ) {+} R, inf v dt + θ τ(x) (Px v, τ ) over τ v L ([,τ];r) satisfying ẋ = Fx + b(+ + v) a.e. t [,τ] x() = x, x(τ) { } R. It is helpful to consider the two cases θ > θ = separately. Case (i): θ >. Define the mappings η : (, ) (, ) R η : (, ) (, ) R to be: η (q,τ) = ( τ + 6 qτ3 θτq,τ qτ + θq), η (r,σ) = ( + σ 6 rσ3 + θσr, σ + rσ θr). It is possible to show that there exists open sets O j, j =,, such that, for each j, η j is one-to-one on O j η j (O j ) = A j. Furthermore, for any intial state x A, the value function, first exit time optimal control for problem (Px) are W (x) = 6 q τ 3 + θτ first exit time =τ v(t) = q(τ t), for a.e. t [,τ], in which (q,τ) = (η ) (x). For any initial state x A on the other h, the value function, first exit time optimal control for problem (Px) are W (x) = 6 r σ 3 + θσ first exit time =σ v(t) = r(σ t), for a.e. t [,σ], in which (r,σ) = (η ) (x). We observe that optimal controls are continuous (one of the non-intersecting extremals conditions that needs to be checked), the set A decomposes into disjoint regions A = {x A W (x) < W (x)} Σ {x A W (x) < W (x)}. The switching set Σ, a smooth submanifold, is Σ = {x A 6 q τ 3 + θτ = 6 r σ 3 + θσ}, in which (q,τ) = (η ) (x) (r,σ) = (η ) (x)}. It can be shown that at points x A for which W (x) = W (x), we have W (x) >, (W W )(x)(f(x,u) + σ(x) v ()) <, (W W )(x)(f(x,u) + σ(x) v ()) <, 574

6 Milano (Italy) August 8 - September,, furthermore, f(x,u) + σ(x)v = if (W W )(x)(f(x,u) + σ(x)v) =. Here if v (.) v (.) are optimal controls for (P x) (P x) respectively, u v arbitrary points on Ω R m. These are the non-intersecting extremals conditions. Case(ii): θ=. The reason this case has to treated separately is that, now, W W are not continuously differentiable on the sets A = {x = (x,x ) x < +} A = {x = (x,x ) x > }, respectively. But checking theconditions for decomposable problems merely required that these functions are continuously differentiable on their support sets (sets on which they are strictly positive), as indeed they are. These two supports sets are: A support = {x A x < + x } A support = {x A x > + x } (The significance of these sets is that if, for either j = or, the initial condition x lies in A\A j support, then the related state trajectory with zero v control exits from A, whatever the u-player control happens to be, therefore incurs zero cost.) Take the functions η η as before (with θ = ). We can choose O O such that η j is one-to-one on O j for j =, η(o j ) = {(x,x ) x < } A support, η(o j ) = {(x,x ) x > } A support. It can be shown that, for any intial state x A support, the value function, first exit time optimal control for problem (Px ) are W (x) = 6 q τ 3 first exit time =τ v(t) = q(τ t), for a.e. t [,τ], in which (q,τ) = (η ) (x). Also, for any initial state x A support, the value function, first exit time optimal control for problem (Px ) are W (x) = 6 r σ 3 first exit time =σ v(t) = r(σ t), for a.e. t [,σ], in which (r,σ) = (η ) (x). These are the non-intersecting extremals conditions, also in this case. When θ =, the switching surface is generated by the points x (α) = ( + 3α α + α 3 3α 4 α 5 ) ( + 3α α α 3 + 3α 4 + α 5 ) α(α ) x (α) = ( + α) / ( + 3α α α 3 + 3α 4 + α 5 ), as α ranges of over the interval α (The denominators in these formulae are positive for all non-negative α s.) Comments: The Maximum Rate of Change of Outflow (MROC) is a significant performance indicator in surge tank control. Indeed, it is the rate of change of outflow, rather than the flow itself, that has adverse effects on downstream process units (disturbed sediments, turbulence, etc) (see McDonald McAvoy (986) Kantor (989)). Ourformulation of thecontrol problem (proposed in Clark Vinter (3)) is to maximize in some sense the elapsed time before the surge tank fills or empties, while observing a constraint on the maximum rate of change of outflow. Previous treatments have limited attention to deterministic disturbances with a particular profile (step changes in inflow rate). The formulation of this paper, by contrast, allows for rom inputs. Fig. 3 provides plots of the switching sets for a number of values ofconstantθ. Fig. 4illustrates the non-intersecting properties of extremals; we observe that optimal state trajectories forthetwo optimal controlproblems remainin regions where the u-control has constant value, they depart from the switching surface at a positive angle. x θ= θ= θ= Switching curve for θ=, θ = θ= x Fig. 3. Switching curves x u =- θ=, switching curve u = x Fig. 4. Non-intersecting extremals 575

7 Milano (Italy) August 8 - September, 6. A THIRD ORDER SYSTEM CONTROL REQUIRING NUMERICAL OPTIMIZATION Let us now look at a refinement of the surge tank control problem of the previous section, in which the disturbance acts on the system, not directly, but through a first order lag. (We set the time constant to one). The games theoretic controller is, in this case, the stochastic controller which minimizes a risk sensitive cost, in the limit as the noise intensity parameter vanishes, for the colored noise stochastic system The two problems were solved numerically for initial states at grid points in the box set { (x,x,x 3 ) x +, R x +R, R x 3 +R}, for some suitably large number R. The switching surface was then obtained bycomparing the minimum costs for the two problems for initial states at all points on the grid. The computed optimal trajectories confirm validity of the nonintersecting extremals condition. The switching surface is illustrated in Figure 5. dz = Fz + b( + e) dt de = e dt + ǫ / dw(t), where F b are as above. Changing the noise model in this manner gives rise to the following game, having state dimension n = 3: Minimize τ v dt + θτ subject to [ ] [ ] [ ] ẋ = x u + v u(t) [,+], x(t) (,+) for t [,τ) x() = x, [ ] x(τ) = { } {+}. The safe region in the state space is A = {x c T x < c T x < } = [,+] R R where, now, c T = []. The enlarged state constraint sets, resulting from retaining only one of the state constraints, are A = {x c T x < } = (,+) R R A = {x c T x < } = (, ) R R with boundaries {x c T x = } = {+} R R {x c T x = } = { } R R. The pushing away controls are u = + u =. It is straightforward to check the monotonicity condition is satisfied. The two optimal control problems to be considered are Minimize τ v dt + θτ subject to [ ] [ ] [ ] ẋ = x + v x(t) (,+) for t [,τ) x() = x, [ ] x(τ) = +. Minimize τ v dt + θτ subject to [ ] [ ] [ ] ẋ = x + + v x(t) (, ) for t [,τ) x() = x, [ ] x(τ) =. x x Fig. 5. Switching surface.5 x REFERENCES Bardi, M. Capuzzo-Dolcetta, I. (997). Optimal Control Viscosity Solutions of Hamilton-Jacobi Equations. Birkhäuser, Boston. Clark, J., James, M., Vinter, R. (4). The interpretation of discontinuous feedback strategies as nonanticipative feedback strategies in differential games. IEEE Transactions on Automatic Control, 49, Clark, J. Vinter, R. (3). A differential dynamic games approach to flow control. 43rd Conference on Decision Control, Hawaii. Dupuis, P. McEneaney, W. (997). Risk-sensitive robust escape criteria. SIAM Journal of Control Optimization, 35, 49. Falcone, M. (987). A numerical approach to the infinite horizon problem of deterministic control theory. Appl. Math. Optim., 5, 3. Farina, L. Rinaldi, S. (). Positive Linear Systems: Theory Applications. Wiley, New York. Fleming, W. Soner, H. (997). Controlled Markov Processes Viscosity Solutions. Birkhäuser, Boston,. Kantor, J. (989). Non-linear sliding mode controller objective function for surge tanks. Int. J. Control, 5, Mayne, D. Rawlings, J. (9). Model Predictive Control: Theory Design. Nob Hill Publishing, Madison. McDonald, K. McAvoy, T. (986). Optimal avering level control. AIChE Journal, 3, McEneaney, W. (6). Max-Plus Methods for Nonlinear Control Estimation. Birkhäuser, Boston. Van der Shaft, A. (993). Nonlinear state space H control theory. in Perspectives in Control, Birkhauser, Boston. Vinter, R. (). Optimal Control. Birkhäuser, Boston