Min-Max Output Integral Sliding Mode Control for Multiplant Linear Uncertain Systems

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1 Proceedings of the 27 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July -3, 27 FrC.4 Min-Max Output Integral Sliding Mode Control for Multiplant Linear Uncertain Systems F. J. Bejarano a, A. Poznyak a and L. Fridman b Abstract In this paper we consider the problem of using the min-max optimal control based on the LQ-index for a set of systems where only output information is available. We consider that each system is affected by matched uncertainties, and we propose to use an output integral sliding mode (OISM) to compensate the matched uncertainties right after the beginning of the process. For the case when the number of inputs is less than the number of outputs, a hierarchical sliding mode observer is proposed that converges to the original state with any small arbitrarily precision after any arbitrarily time. I. INTRODUCTION Here we presented the problem of controlling a set of systems using a unique control. The control used here is a robust optimal control based in a min-max LQ-index proposed in [ and [2. The optimal control based on the min-max LQ-index is a control designed under two basic assumptions, namely, the whole state is completely available and the system is free of any uncertainty. Therefore, if we have output information only, an observer for reconstructing the original states is necessary to take advantage of the state feedback robust optimal control. Furthermore, if each plant is influenced by a matched uncertainty, before applying any control action, we should ensure the compensation of such an uncertainty. In [3 and [4 were proposed two different forms for resolving the problem of matched uncertainty compensation for the case of a control based on the min-max LQ-index in the context of a multimodel system. In both cases were assumed the availability of the complete vector state; this assumption makes possible the application of the integral sliding mode technique (see [, [6, [7, [8). In the present paper we use the approach proposed in [9. There was suggested to use an output integral sliding mode (OISM) to compensate the matched uncertainties right after the beginning of the process. There was also suggested to design a hierarchical observer using sliding modes. In each step of the hierarchy one reconstruct a part of a vector composed of multiplying an observability matrix by the vector state. During the realization of such observer it is needed to use some filters. However, there was shown that the time of convergence and the observer error can be made arbitrarily small just by modifying the sample time and the filter constants used during the realization of the observer. That is why we take advantage of the OISM to overcome a CINVESTAV-IPN, Departamento de Control Automático A.P. 4-74, CP 7 México D.F. javbejarano@yahoo.com.mx, apoznyak@ctrl.cinvestav.mx. b Facultad de Ingenieria National Autonomous University of Mexico DEP-FI, UNAM, A. P. 7-26, CP 4, México, D.F. lfridman@verona.fi-p.unam.mx the restrictions imposed in the design of the robust optimal control. A. Basic assumptions and restrictions Since for each system the complete vector state is not unavailable, in this work: we consider a finite set of plants whose trajectories are suppose to be estimated; each plant of the system is described by a system of linear time-invariant ODE (ordinary differential equations) with matched uncertainties which may be of a nonlinear nature; the performance of each plant is characterized by a LQindex over a finite horizon; the optimal control action is assumed to be applied to all plants simultaneously. B. Main contribution It is shown that it is possible to apply the robust optimal control based on the min-max LQ-index even in the presence of matched uncertainties and even if only the output (not the complete state) of each system is measurable. To apply the robust optimal control to the system, we use an output integral sliding mode that allows to compensate the matched uncertainties, and to estimate the state with any small precision. II. PROBLEM STATEMENT Let us consider a set of linear time invariant uncertain systems ẋ α (t) = A α x α (t) + B α u (t) + B α γ (t) + d α (t) y α (t) = C α x α (t), x α () = x α where α =, N, (N is a positive integer), x α (t) R n is the state vector at time t [, T, u (t) R m is the control and y α (t) R p ( p < n) is the output. The vector d α (t) is assumed to be known for all t [, T. The current state x α (t) and the initial state x α are supposed to be non available. A α, B α, C α are known matrices of appropriate dimension with rankb α = m and rankc α = p. Here all the plants are running in parallel. Throughout the paper we will assume that: A. (A α, B α ) is controllable, (A α, C α ) is observable. A2. The vector γ (t) is upper bounded by a known scalar function q a (t), that is, () γ (t) q a (t) (2) A3. It is known a bound for every vector x α, that is, x α µ (3) /7/$2. 27 IEEE. 87 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 28, 28 at : from IEEE Xplore. Restrictions apply.

2 FrC.4 A. The control design challenge Before designing an optimal control, we must set free the system from the effects of the matched uncertainties. Therefore, the control design problem can be formulated as follows: design the control u in the form u = u + u (4) where the control u will be designed to compensate the uncertainty γ (t) from the initial time t =. And u ( ) u ( ), where u ( ) is a control function minimizing the LQindex: min max h α () u R m α,n + 2 T t= h α := 2 (xα (T),G α x α (T)) [(x α (t),q α x α (t)) + (u (t),ru (t)) dt Q α, G α, R > along the nominal system trajectories (6) ẋ α (t) = A α x α (t) + B α u + d α (7) The exact solution of () requires the availability of all the vector states x α (t) (see [2) at any t [, T, and the system must be free from any uncertainty. Therefore, to carried out this optimal control we firstly should ) ensure the compensation of the matched uncertainties γ (t), 2) design a state estimator that reconstruct every state vector x α (t). III. OUTPUT INTEGRAL SLIDING MODE (OISM) Substitution of the control law (4) into () yields ẋ α (t) = A α x α (t) + B α (u + u + γ (t)) + d α (t) (8) where α =, N. Let us define the following extended system where x :=. x N ẋ(t) = Ax(t) + B (u + u + γ) + d y (t) = Cx(t) x A B, A :=....., B :=. BN A N C. d C =....., d :=. d C N N () Now, we will assume that A4. rank(cb) = m Thus, define the auxiliary affine sliding function s : R pn R mn as follows (9) s (y (t)) := (CB) + y (t) + σ (t) () The term σ (t) includes an integral term which will be defined below. Thus, for the time derivative ṡ we have Define σ as ṡ = (CB) + CAx + u + u + γ + σ (2) σ = (CB) + C [Aˆx + d u σ () = (CB) + y () (3) The vector ˆx represents an observer whose form will be selected in the section IV. Substitution of σ into (2) gives ṡ = (CB) + CA (x ˆx) + u + γ, s () = The control u is designed in the following form u = β (t) s(t) s(t) with β (t) being a scalar gain that satisfies the condition β (t) q a (t) (CB) + CA x ˆx λ > (4) where λ is a constant. Notice that, by A3, an upper bound of x ˆx always can be estimated. Selecting the Lyapunov function as V = 2 s 2 and in view of (4) and (2) one gets ( V = (s, ṡ) = s, (CB) + CA (x ˆx) β s ) [ s + γ s β (CB) + CA x ˆx q a s λ ((s, ṡ) := s T ṡ). It means that V does not increase through the time. Hence, since s() =, this implies 2 s (t) 2 = V (s (t)) V (s ()) = 2 s () 2 = Thus, the identities s (t) = ṡ (t) = () hold for all t, i.e., there is no reaching phase. From (2) and in view of the equality () the equivalent control maintaining the trajectories on the surface is u eq = (CB) + CA (x ˆx) γ (6) Substitution of u eq into (9) yields where ẋ (t) = Ãx(t) + B (CB) + CAˆx(t) + Bu + d (t) y (t) = Cx(t) (7) Ã := [ I B (CB) + C A (8) Thus, our first objective was achieved, i.e., we have compensated the uncertainty γ. 876 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 28, 28 at : from IEEE Xplore. Restrictions apply.

3 FrC.4 IV. DESIGN OF THE OBSERVER Now, with the system without uncertainties, we can recover the state vector. In order to design the observer, (Ã,C) must be observable. The following Lemma establishes the conditions in terms of (A,B,C) to determine when (Ã,C) is observable. ) Lemma : The pair (Ã,C is observable if and only if the triple (A,B,C) has no invariant zeros, i.e., {s C : rank(p (s)) < n + m} = (9) where P (s) is the Rosenbrock s matrix system defined as [ si A B P (s) = (2) C A proof of Lemma () has been given in [9. Thus, henceforth we assumed that A. the triple (A,B,C) has no zeros. It is known that when rank(cb) = m (assumption A4) and p = m the triple (A,B,C) has invariant zeros; therefore, A4 and A imply that p > m. The observer will be based on the recovering of the vectors Cx(t), CÃx(t) and so on until get Cà l x(t). Afterwards, the aim is to recover the vector Hx(t) where [ (CÃ) T H T = C T ( CÃl ) T (2) Here l is defined as the observability index that is the least positive integer such that rank(h) = n (see, e.g., [). Thus, after pre-multiplying Hx(t) by H +, the state vector x(t) can be recover by x(t) = H + Hx(t), where H + = }{{} gotten online ( H T H ) H T is the pseudo-inverse of H. Before designing the observer we need to ensure a bound required by the sliding mode algorithm. Design the following dynamic system x(t) = à x(t) + Bu (t) + B (CB) + CAˆx(t) +L(y (t) C x(t)) (22) where L must be designed such that the eigenvalues of  := (à LC) have negative real part. Let r(t) = x(t) x (t), from (7) and (22), the dynamic equations governing r(t) are ṙ(t) = [à LC r(t) = Âr (t) (23) Since the eigenvalues of  have negative real part, the equation (23) is exponentially stable, i.e., there exist some constants γ,η > such that ( ) r(t) γ exp ( ηt) Nµ + x () (24) Below it is shown that in the design of the observer we need to know a bound of r(t). Thus, (24) ensures that we can always satisfy such a requirement. A. Auxiliary Dynamic Systems and Output Injections The essential purpose in the design of the observer is to recover the vectors Cà k x, k =, l Firstly, to recover CÃx(t), let us introduce an auxiliary vector state x a (t) governed by the following dynamics equations ẋ a (t) = [u à x(t) + B (t) + (CB) + CAˆx(t) + L ( C L ) (2) v (t) + d (t) where x a() satisfies Cx a() = y () and L is any matrix such that det ( C L ). The vector ˆx(t) represents the observer we will design below. For the variable s R Np defined by we have s ( y (t),x a (t) ) = Cx(t) Cx a (t) (26) ṡ ( y (t),x a (t) ) = Cà (x(t) x(t)) v (t) (27) with v (t) defined as v s = M s. Here the scalar gain M must satisfy the condition M > Cà r to obtain the sliding mode regime. A bound of r can be estimated using (24). Then, repeating the procedure in III, we get s (t) =, ṡ (t) = t. Thus, from (26) we obtain that Cx(t) = C x (t), t (28) and from (27), the equivalent output injection is v eq (t) = CÃx (t) Cà x(t), t > (29) Thus, CÃx(t) is recovered from (29). Now, the next step is to recover the vector Cà 2 x(t). To do that, let us design the second auxiliary state vector x 2 a(t) generated by ẋ 2 a (t) = Ã2 x(t) + ÃBu (t) + L ( C L ) v 2 (t) +ÃB (CB) + CAˆx(t) + d (t) where x 2 a () satisfies CÃx a () + v eq () Cx2 a () =. Again, for s 2 R Np defined by s 2 ( v eq,x2 a) = Cà x(t) + v eq (t) Cx2 a and in view of (29), we have that s 2 takes the form s 2 ( v eq,x2 a) = CÃx(t) Cx 2 a (3) Hence, the time derivative of s 2 is ṡ 2 ( v eq,x 2 a) = Cà 2 x(t) Cà x(t) v 2 (t) (3) Now, take the output injection v 2 (t) as v 2 s 2 = M 2 s 2, M 2 > Cà 2 r (32) which implies that s 2 (t) = ṡ 2 (t) = (33) 877 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 28, 28 at : from IEEE Xplore. Restrictions apply.

4 FrC.4 In view of (33) and (3), v 2 eq (t) is v 2 eq (t) = CÃ2 x(t) CÃ x(t), t > (34) and the vector CÃ2 x(t) can be recovered from (34). Thus, iterating the same procedure, all the vectors CÃ i x can be recovered. In a summarizing form, the procedure above goes as follows: a) the dynamics of the auxiliary state x k a(t) at the k-th level is governed by ẋ k a(t) = Ã k x(t) + Ã k Bu (t) + L ( C L ) v k +Ãk B (CB) + CAˆx(t) + d (t) And the output injection v k at the k-th level is (3) v k s k = M k s k, M k > CÃ k r (36) where M k is a scalar gain, and a bound of r can be found using (24). b) Define s k at the k-level of the hierarchy as: { y Cx a, k = s k (t) = v k eq + CÃ k x Cx k a, k > (37) where veq k is the equivalent output injection whose general expression will be obtained in the following Lemma, but x k a () should be chosen such that sk () satisfies s k () =, k =,.., l (38) Lemma 2: If the auxiliary state vector x k a and the variable s k are designed as in (3) and (37), respectively, then v k eq (t) = CÃk [x(t) x(t) for all t (39) at each k =, l. Proof: It was shown that the following identity holds v eq (t) = CÃ [x(t) x(t) t > Now, suppose that the equivalent output injection veq k in (39). Then substitution of veq k in (37) gives is as s k ( v k eq (t),x k a (t) ) = CÃ k x(t) Cx k a (t) (4) The derivative of (4) yields ṡ k (t) = CÃ k [x(t) x(t) v k (t) (4) Thus, selecting v k (t) as in (36) one gets s k (t), ṡ k (t) for all t (42) Therefore, (42) and (4) implies (39). B. Observer in its Algebraic Form Now, we can design an observer with the properties required in the problem statement. From (28) and (39), we obtain the following algebraic equations arrangement Cx(t) = C x(t) + y (t) C x(t) CÃx(t) = CÃ x(t) + v eq (t). CÃl x(t) = CÃl x(t) + v l eq Thus, (43) yields the matrix equation (43) Hx(t) = H x(t) + v eq (t), t > (44) where H was defined in (2) and veq [(y T = (t) C x(t)) T ( ) v T ( ) eq v l T eq (4) ) Since the pair (Ã,C is observable, the matrix H has rank n. Thus, the left multiplication of (44) by H + yields x(t) x(t) + H + v eq (t), t > (46) That is why the observer can be designed as ˆx(t) := x(t) + H + v eq (t) (47) Therefore, we can formulate the following theorem. Theorem : Under the assumptions A-A ˆx(t) x(t) t > (48) Proof: It follows directly from (46) and (47). C. Observer Realization The achievement of the observer described by (47) requires the availability of the equivalent output injection v k eq. However, the non idealities in the implementation of v (k) cause the, so-called, chattering movements. Even though v k eq can not be directly measured, it may be indirectly measured. Namely, the first order-low pass-filter τ v k av (t) + v k av (t) = v k (t) ; v k av () = (49) gives an approach of veq k (see [). That is, lim vav k (t) = τ /τ v k (t),t >. eq Where is proportional to the sampling time (the time that v k lasts to pass from one state (M) to other ( M)). So, we can select τ = η ( < η < ). Hence, to realize the OISM observer we should: ) use a sampling interval very small; 2) substitute v k (t) into (37) and (4) by vk eq av (t); 3) chose x k a () in such a way that y () Cx a() =, for k = CÃ k x() Cx k a () =, for k > so we ensure the identity s k () =, k =,..., l. 878 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 28, 28 at : from IEEE Xplore. Restrictions apply.

5 FrC.4 V. OPTIMAL CONTROL DESIGN We return in this section to the problem of the optimal control u which resolves the problem (). Substitution of (48) into (7) yields the sliding motion equations for the state x that takes the form ẋ(t) = Ax(t) + B (t)u (x) + d Now the solution for the min-max optimal problem () can be given. Then, according [, [2, the control solving () for (7) is of the form: u (x) = R B (P λ x + p λ ) () where the matrix P λ R nn nn is the solution of the parameterized differential matrix Riccati equation: Ṗ λ +P λ A + A T P λ P λ BR B T P λ +ΛQ = P λ (T) = ΛG and the shifting vector p λ satisfies ṗ λ +A p λ P λ BR B p λ + P λ d = ; p λ (T) = where the weighting vector λ belongs to the simplex S N S N = λ R N : λ α, N λ α = α= () and the matrices Q, G, and Λ denote the extended matrices Q G Q :=....., G :=..... Q N GN λ I n n. Λ :=..... λ N I n n (2) The matrix Λ = Λ(λ ) is defined by (2) with the weight vector λ = λ solving the following finite dimensional optimization problem λ = arg min J (λ) λ S N J (λ) := max h α (3) α=,n From (48), the estimated state ˆx is used to realize the control u, i.e., the control u should be designed as u (t) = u (ˆx) = R B [P λ ˆx + p λ (4) with ˆx being designed according to (47). VI. EXAMPLE The following example shows the effectiveness of the suggested control method. Consider a case of N = 3 where the parameters are given by: A =..2 2, A 2 = A 3 =..4.3, B =., B 2 = B 3 =. [.2, C,2,3 = γ (t) = sin (t), d,2,3 =..2. For this example the weights are λ =.67, λ 2 =.3 and λ =.63, and the functional J (λ ) = 2.8. The trajectories for the three plants are shown in fig., fig. 2, and fig. 3; in these figures a comparison between the trajectories of the original vector state and the trajectories of the estimated state is made. The estimation error (e = x ˆx) is graphed in fig. 4; since we know the first and the second component of the state vector, then it is presented only the third component of the error vector. The fig. shows a comparison between the control law u (x, t) when all the state vector is available versus the control u (ˆx, t) when only output information is available. State x and observer xe Fig.. plant x xe Trajectories of the original state and the estimated one for the first CONCLUSIONS Here we have shown that for the case when the number of inputs is less than the number of outputs, the use of output integral sliding mode allows: first, to compensate the matched uncertainties right after the initial time (independently of the observation process), and second, as a consequence of 879 Authorized licensed use limited to: IEEE Xplore. Downloaded on October 28, 28 at : from IEEE Xplore. Restrictions apply.

6 FrC x 2 xe u using xe u using x State x 2 and observer xe Control law u Fig. 2. Trajectories of the original state and the estimated one for the second plant. State x 3 and observer xe 3 Fig. 3. plant. error e 3 error e 2 3 error e x 3 xe Trajectories of the original state and the estimated one for the third Fig. 4. Third component of the estimation error e = x ˆx Fig.. Comparison between u (x, t) and u (ˆx, t). the first, to design a hierarchical observer that reconstructs the system states. Using a low-pass filter for the observer realization, we have shown that the estimation error depends only on the sampling time and the filter time constant. It was proven that the time of convergence for the observation error can be made arbitrary small without any observer parameters adjustment only by decreasing the sampling step and the filter time constant. The use of an OISM might be a promising technique not only for the compensation of the matched uncertainties but also for making feasible the use of an optimal control as we saw in this manuscript with the design of an optimal control based on the min-max LQ-index using only output information. REFERENCES [ V. Boltyansky and A. Poznyak, Robust maximum principle in minimax control, Int. J. of Control, vol. 72, pp. 3 34, 999. [2 A. Poznyak, T. Duncan, B. Pasik-Duncan, and V. Boltyansky, Robust maximum principle for minimax linear quadratic problem, Int. J. of Control, vol. 7, no., pp. 7 77, 22. [3 A. Poznyak, L. Fridman, and F. Bejarano, Mini-max integral sliding mode control for multimodel linear uncertain systems, IEEE Transactions on Automatic Control, vol. 49, no., pp. 97 2, 24. [4 L. Fridman, A. Poznyak, and F. Bejarano, Decomposition of the min-max multimodel problem via integral sliding mode, International Journal of Robust and Nonlinear Control, vol., no. 3, pp. 9 74, 2. [ V. Utkin and J. Shi, Integral sliding mode in systems operating under uncertainty conditions, in Proceedings of the 3th IEEE Conference on Decision and Control, Kobe, Japan, 996, pp [6 V. Utkin, Guldner, and J. Shi, Sliding Mode Control in Electromechanical Systems. London: Taylor and Francis, 999. [7 M. Basin, J. Rodriguez, L. Fridman, and P. Acosta, Integral sliding mode design for robust filtering and control of linear stochastic time-delay systems, International Journal of Robust and Nonlinear Control, vol., no. 9, pp , 2. [8 F. Castaos and L. Fridman, Analysis and design of integral sliding manifolds for systems with unmatched perturbations, IEEE Transactions on Automatic Control, vol., no., pp , May 26. [9 F. Bejarano, L. Fridman, and A. Poznyak, Output integral sliding mode control based on algebraic hierarchical observer, Int. Journal of Control, vol. 8, no. 3, pp , March 27. [ C. Chen, Linear Systems: theory and design. New York: Oxford University Press, 999. [ V. Utkin, Sliding modes in control and optimization. Berlin, Germany: Springer Verlag, Authorized licensed use limited to: IEEE Xplore. Downloaded on October 28, 28 at : from IEEE Xplore. Restrictions apply.

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