Min-Max Output Integral Sliding Mode Control for Multiplant Linear Uncertain Systems
|
|
- Tracey Cox
- 6 years ago
- Views:
Transcription
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.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 Robust Control With Exact Uncertainties Compensation: With or Without Chattering? Alejandra Ferreira, Member, IEEE, Francisco Javier Bejarano, and Leonid
More informationNonlinear Observers. Jaime A. Moreno. Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México
Nonlinear Observers Jaime A. Moreno JMorenoP@ii.unam.mx Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México XVI Congreso Latinoamericano de Control Automático October
More informationSliding Mode Regulator as Solution to Optimal Control Problem for Nonlinear Polynomial Systems
29 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June -2, 29 WeA3.5 Sliding Mode Regulator as Solution to Optimal Control Problem for Nonlinear Polynomial Systems Michael Basin
More informationEL 625 Lecture 10. Pole Placement and Observer Design. ẋ = Ax (1)
EL 625 Lecture 0 EL 625 Lecture 0 Pole Placement and Observer Design Pole Placement Consider the system ẋ Ax () The solution to this system is x(t) e At x(0) (2) If the eigenvalues of A all lie in the
More informationSYSTEMTEORI - KALMAN FILTER VS LQ CONTROL
SYSTEMTEORI - KALMAN FILTER VS LQ CONTROL 1. Optimal regulator with noisy measurement Consider the following system: ẋ = Ax + Bu + w, x(0) = x 0 where w(t) is white noise with Ew(t) = 0, and x 0 is a stochastic
More informationState estimation of uncertain multiple model with unknown inputs
State estimation of uncertain multiple model with unknown inputs Abdelkader Akhenak, Mohammed Chadli, Didier Maquin and José Ragot Centre de Recherche en Automatique de Nancy, CNRS UMR 79 Institut National
More informationACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016
ACM/CMS 17 Linear Analysis & Applications Fall 216 Assignment 4: Linear ODEs and Control Theory Due: 5th December 216 Introduction Systems of ordinary differential equations (ODEs) can be used to describe
More informationFINITE HORIZON ROBUST MODEL PREDICTIVE CONTROL USING LINEAR MATRIX INEQUALITIES. Danlei Chu, Tongwen Chen, Horacio J. Marquez
FINITE HORIZON ROBUST MODEL PREDICTIVE CONTROL USING LINEAR MATRIX INEQUALITIES Danlei Chu Tongwen Chen Horacio J Marquez Department of Electrical and Computer Engineering University of Alberta Edmonton
More informationPole-Placement in Higher-Order Sliding-Mode Control
Preprints of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. August 4-9, 14 Pole-Placement in Higher-Order Sliding-Mode Control Debbie Hernández Fernando
More informationA sub-optimal second order sliding mode controller for systems with saturating actuators
28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June -3, 28 FrB2.5 A sub-optimal second order sliding mode for systems with saturating actuators Antonella Ferrara and Matteo
More informationc 2009 Society for Industrial and Applied Mathematics
SIAM J. CONTROL OPTIM. Vol. 48, No. 2, pp. 1155 1178 c 29 Society for Industrial and Applied Mathematics UNKNOWN INPUT AND STATE ESTIMATION FOR UNOBSERVABLE SYSTEMS FRANCISCO J. BEJARANO, LEONID FRIDMAN,
More informationSteady State Kalman Filter
Steady State Kalman Filter Infinite Horizon LQ Control: ẋ = Ax + Bu R positive definite, Q = Q T 2Q 1 2. (A, B) stabilizable, (A, Q 1 2) detectable. Solve for the positive (semi-) definite P in the ARE:
More informationINVERSE MODEL APPROACH TO DISTURBANCE REJECTION AND DECOUPLING CONTROLLER DESIGN. Leonid Lyubchyk
CINVESTAV Department of Automatic Control November 3, 20 INVERSE MODEL APPROACH TO DISTURBANCE REJECTION AND DECOUPLING CONTROLLER DESIGN Leonid Lyubchyk National Technical University of Ukraine Kharkov
More informationPOLE PLACEMENT. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 19
POLE PLACEMENT Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 19 Outline 1 State Feedback 2 Observer 3 Observer Feedback 4 Reduced Order
More informationObservability and state estimation
EE263 Autumn 2015 S Boyd and S Lall Observability and state estimation state estimation discrete-time observability observability controllability duality observers for noiseless case continuous-time observability
More informationLinear Quadratic Zero-Sum Two-Person Differential Games
Linear Quadratic Zero-Sum Two-Person Differential Games Pierre Bernhard To cite this version: Pierre Bernhard. Linear Quadratic Zero-Sum Two-Person Differential Games. Encyclopaedia of Systems and Control,
More informationLinear Quadratic Zero-Sum Two-Person Differential Games Pierre Bernhard June 15, 2013
Linear Quadratic Zero-Sum Two-Person Differential Games Pierre Bernhard June 15, 2013 Abstract As in optimal control theory, linear quadratic (LQ) differential games (DG) can be solved, even in high dimension,
More informationDESIGN OF OBSERVERS FOR SYSTEMS WITH SLOW AND FAST MODES
DESIGN OF OBSERVERS FOR SYSTEMS WITH SLOW AND FAST MODES by HEONJONG YOO A thesis submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey In partial fulfillment of the
More informationSTABILIZATION FOR A CLASS OF UNCERTAIN MULTI-TIME DELAYS SYSTEM USING SLIDING MODE CONTROLLER. Received April 2010; revised August 2010
International Journal of Innovative Computing, Information and Control ICIC International c 2011 ISSN 1349-4198 Volume 7, Number 7(B), July 2011 pp. 4195 4205 STABILIZATION FOR A CLASS OF UNCERTAIN MULTI-TIME
More informationIndex. A 1-form, 42, 45, 133 Amplitude-frequency tradeoff, 14. Drift field perturbation, 66 Drift vector field, 42
References 1. U. Beis An Introduction to Delta Sigma Converters Internet site http:// www.beis.de/ Elektronik/ DeltaSigma/ DeltaSigma.html, pp. 1 11, January 21, 2014. 2. B. Charlet, J. Lévine and R. Marino
More informationIntro. Computer Control Systems: F8
Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2
More informationLecture 19 Observability and state estimation
EE263 Autumn 2007-08 Stephen Boyd Lecture 19 Observability and state estimation state estimation discrete-time observability observability controllability duality observers for noiseless case continuous-time
More informationThe Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications
MAX PLANCK INSTITUTE Elgersburg Workshop Elgersburg February 11-14, 2013 The Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications Timo Reis 1 Matthias Voigt 2 1 Department
More informationSubject: Optimal Control Assignment-1 (Related to Lecture notes 1-10)
Subject: Optimal Control Assignment- (Related to Lecture notes -). Design a oil mug, shown in fig., to hold as much oil possible. The height and radius of the mug should not be more than 6cm. The mug must
More informationStability, Pole Placement, Observers and Stabilization
Stability, Pole Placement, Observers and Stabilization 1 1, The Netherlands DISC Course Mathematical Models of Systems Outline 1 Stability of autonomous systems 2 The pole placement problem 3 Stabilization
More informationLeader-Follower strategies for a Multi-Plant differential game
008 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June -3, 008 FrA066 Leader-Follower strategies for a Multi-Plant differential game Manuel Jiménez-Lizárraga, Alex Poznyak
More informationThe norms can also be characterized in terms of Riccati inequalities.
9 Analysis of stability and H norms Consider the causal, linear, time-invariant system ẋ(t = Ax(t + Bu(t y(t = Cx(t Denote the transfer function G(s := C (si A 1 B. Theorem 85 The following statements
More informationEfficient robust optimization for robust control with constraints Paul Goulart, Eric Kerrigan and Danny Ralph
Efficient robust optimization for robust control with constraints p. 1 Efficient robust optimization for robust control with constraints Paul Goulart, Eric Kerrigan and Danny Ralph Efficient robust optimization
More informationGramians based model reduction for hybrid switched systems
Gramians based model reduction for hybrid switched systems Y. Chahlaoui Younes.Chahlaoui@manchester.ac.uk Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) School of Mathematics
More information1. Find the solution of the following uncontrolled linear system. 2 α 1 1
Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +
More informationHIGHER ORDER SLIDING MODES AND ARBITRARY-ORDER EXACT ROBUST DIFFERENTIATION
HIGHER ORDER SLIDING MODES AND ARBITRARY-ORDER EXACT ROBUST DIFFERENTIATION A. Levant Institute for Industrial Mathematics, 4/24 Yehuda Ha-Nachtom St., Beer-Sheva 843, Israel Fax: +972-7-232 and E-mail:
More informationROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF SINGULAR SYSTEMS
INTERNATIONAL JOURNAL OF INFORMATON AND SYSTEMS SCIENCES Volume 5 Number 3-4 Pages 480 487 c 2009 Institute for Scientific Computing and Information ROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF
More informationResearch Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components
Applied Mathematics Volume 202, Article ID 689820, 3 pages doi:0.55/202/689820 Research Article Convex Polyhedron Method to Stability of Continuous Systems with Two Additive Time-Varying Delay Components
More informationCONTROL DESIGN FOR SET POINT TRACKING
Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observer-based output feedback design to solve tracking problems. By tracking we mean that the output is commanded
More informationLinear System Theory
Linear System Theory Wonhee Kim Chapter 6: Controllability & Observability Chapter 7: Minimal Realizations May 2, 217 1 / 31 Recap State space equation Linear Algebra Solutions of LTI and LTV system Stability
More information6 OUTPUT FEEDBACK DESIGN
6 OUTPUT FEEDBACK DESIGN When the whole sate vector is not available for feedback, i.e, we can measure only y = Cx. 6.1 Review of observer design Recall from the first class in linear systems that a simple
More informationRobotics. Control Theory. Marc Toussaint U Stuttgart
Robotics Control Theory Topics in control theory, optimal control, HJB equation, infinite horizon case, Linear-Quadratic optimal control, Riccati equations (differential, algebraic, discrete-time), controllability,
More informationStability of linear time-varying systems through quadratically parameter-dependent Lyapunov functions
Stability of linear time-varying systems through quadratically parameter-dependent Lyapunov functions Vinícius F. Montagner Department of Telematics Pedro L. D. Peres School of Electrical and Computer
More informationSliding Modes in Control and Optimization
Vadim I. Utkin Sliding Modes in Control and Optimization With 24 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Contents Parti. Mathematical Tools 1
More informationObservability. It was the property in Lyapunov stability which allowed us to resolve that
Observability We have seen observability twice already It was the property which permitted us to retrieve the initial state from the initial data {u(0),y(0),u(1),y(1),...,u(n 1),y(n 1)} It was the property
More informationModeling and Analysis of Dynamic Systems
Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System - Stability
More informationThe ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 11, Issue 3 Ver. IV (May - Jun. 2015), PP 52-62 www.iosrjournals.org The ϵ-capacity of a gain matrix and tolerable disturbances:
More informationMULTI-AGENT TRACKING OF A HIGH-DIMENSIONAL ACTIVE LEADER WITH SWITCHING TOPOLOGY
Jrl Syst Sci & Complexity (2009) 22: 722 731 MULTI-AGENT TRACKING OF A HIGH-DIMENSIONAL ACTIVE LEADER WITH SWITCHING TOPOLOGY Yiguang HONG Xiaoli WANG Received: 11 May 2009 / Revised: 16 June 2009 c 2009
More informationRobustification of time varying linear quadratic optimal control based on output integral sliding modes Rosalba Galván-Guerra, Leonid Fridman
Techset Composition Ltd, Salisbury Doc: {IEE} CTASI20140095tex 1 6 Published in IET Control Theory and Applications Received on 24th January 2014 Revised on 22nd April 2014 Accepted on 22nd May 2014 doi:
More informationA Simple Derivation of Right Interactor for Tall Transfer Function Matrices and its Application to Inner-Outer Factorization Continuous-Time Case
A Simple Derivation of Right Interactor for Tall Transfer Function Matrices and its Application to Inner-Outer Factorization Continuous-Time Case ATARU KASE Osaka Institute of Technology Department of
More informationMulti-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures
Preprints of the 19th World Congress The International Federation of Automatic Control Multi-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures Eric Peterson Harry G.
More informationSet-based adaptive estimation for a class of nonlinear systems with time-varying parameters
Preprints of the 8th IFAC Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control Furama Riverfront, Singapore, July -3, Set-based adaptive estimation for
More informationBUMPLESS SWITCHING CONTROLLERS. William A. Wolovich and Alan B. Arehart 1. December 27, Abstract
BUMPLESS SWITCHING CONTROLLERS William A. Wolovich and Alan B. Arehart 1 December 7, 1995 Abstract This paper outlines the design of bumpless switching controllers that can be used to stabilize MIMO plants
More informationADAPTIVE OUTPUT FEEDBACK CONTROL OF NONLINEAR SYSTEMS YONGLIANG ZHU. Bachelor of Science Zhejiang University Hanzhou, Zhejiang, P.R.
ADAPTIVE OUTPUT FEEDBACK CONTROL OF NONLINEAR SYSTEMS By YONGLIANG ZHU Bachelor of Science Zhejiang University Hanzhou, Zhejiang, P.R. China 1988 Master of Science Oklahoma State University Stillwater,
More informationHigh-Gain Observers in Nonlinear Feedback Control. Lecture # 2 Separation Principle
High-Gain Observers in Nonlinear Feedback Control Lecture # 2 Separation Principle High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 1/4 The Class of Systems ẋ = Ax + Bφ(x,
More informationBalancing of Lossless and Passive Systems
Balancing of Lossless and Passive Systems Arjan van der Schaft Abstract Different balancing techniques are applied to lossless nonlinear systems, with open-loop balancing applied to their scattering representation.
More information5. Observer-based Controller Design
EE635 - Control System Theory 5. Observer-based Controller Design Jitkomut Songsiri state feedback pole-placement design regulation and tracking state observer feedback observer design LQR and LQG 5-1
More information16.31 Fall 2005 Lecture Presentation Mon 31-Oct-05 ver 1.1
16.31 Fall 2005 Lecture Presentation Mon 31-Oct-05 ver 1.1 Charles P. Coleman October 31, 2005 1 / 40 : Controllability Tests Observability Tests LEARNING OUTCOMES: Perform controllability tests Perform
More informationDelay-dependent Stability Analysis for Markovian Jump Systems with Interval Time-varying-delays
International Journal of Automation and Computing 7(2), May 2010, 224-229 DOI: 10.1007/s11633-010-0224-2 Delay-dependent Stability Analysis for Markovian Jump Systems with Interval Time-varying-delays
More informationI. D. Landau, A. Karimi: A Course on Adaptive Control Adaptive Control. Part 9: Adaptive Control with Multiple Models and Switching
I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 1 Adaptive Control Part 9: Adaptive Control with Multiple Models and Switching I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 2 Outline
More informationIMPULSIVE CONTROL OF DISCRETE-TIME NETWORKED SYSTEMS WITH COMMUNICATION DELAYS. Shumei Mu, Tianguang Chu, and Long Wang
IMPULSIVE CONTROL OF DISCRETE-TIME NETWORKED SYSTEMS WITH COMMUNICATION DELAYS Shumei Mu Tianguang Chu and Long Wang Intelligent Control Laboratory Center for Systems and Control Department of Mechanics
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. VII - System Characteristics: Stability, Controllability, Observability - Jerzy Klamka
SYSTEM CHARACTERISTICS: STABILITY, CONTROLLABILITY, OBSERVABILITY Jerzy Klamka Institute of Automatic Control, Technical University, Gliwice, Poland Keywords: stability, controllability, observability,
More informationFinite-Time Converging Jump Observer for Switched Linear Systems with Unknown Inputs
Finite-Time Converging Jump Observer for Switched Linear Systems with Unknown Inputs F.J. Bejarano a, A. Pisano b, E. Usai b a National Autonomous University of Mexico, Engineering Faculty, Division of
More informationLecture 2 and 3: Controllability of DT-LTI systems
1 Lecture 2 and 3: Controllability of DT-LTI systems Spring 2013 - EE 194, Advanced Control (Prof Khan) January 23 (Wed) and 28 (Mon), 2013 I LTI SYSTEMS Recall that continuous-time LTI systems can be
More informationAn Exact Stability Analysis Test for Single-Parameter. Polynomially-Dependent Linear Systems
An Exact Stability Analysis Test for Single-Parameter Polynomially-Dependent Linear Systems P. Tsiotras and P.-A. Bliman Abstract We provide a new condition for testing the stability of a single-parameter,
More informationTRACKING AND DISTURBANCE REJECTION
TRACKING AND DISTURBANCE REJECTION Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 13 General objective: The output to track a reference
More informationNonlinear and robust MPC with applications in robotics
Nonlinear and robust MPC with applications in robotics Boris Houska, Mario Villanueva, Benoît Chachuat ShanghaiTech, Texas A&M, Imperial College London 1 Overview Introduction to Robust MPC Min-Max Differential
More informationTopic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback
Topic #17 16.31 Feedback Control State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall
More informationResearch Article Stabilization Analysis and Synthesis of Discrete-Time Descriptor Markov Jump Systems with Partially Unknown Transition Probabilities
Research Journal of Applied Sciences, Engineering and Technology 7(4): 728-734, 214 DOI:1.1926/rjaset.7.39 ISSN: 24-7459; e-issn: 24-7467 214 Maxwell Scientific Publication Corp. Submitted: February 25,
More informationDynamic Integral Sliding Mode Control of Nonlinear SISO Systems with States Dependent Matched and Mismatched Uncertainties
Milano (Italy) August 28 - September 2, 2 Dynamic Integral Sliding Mode Control of Nonlinear SISO Systems with States Dependent Matched and Mismatched Uncertainties Qudrat Khan*, Aamer Iqbal Bhatti,* Qadeer
More informationChapter 3. LQ, LQG and Control System Design. Dutch Institute of Systems and Control
Chapter 3 LQ, LQG and Control System H 2 Design Overview LQ optimization state feedback LQG optimization output feedback H 2 optimization non-stochastic version of LQG Application to feedback system design
More informationChapter III. Stability of Linear Systems
1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,
More informationTrajectory Tracking Control of Bimodal Piecewise Affine Systems
25 American Control Conference June 8-1, 25. Portland, OR, USA ThB17.4 Trajectory Tracking Control of Bimodal Piecewise Affine Systems Kazunori Sakurama, Toshiharu Sugie and Kazushi Nakano Abstract This
More informationMultiple-mode switched observer-based unknown input estimation for a class of switched systems
Multiple-mode switched observer-based unknown input estimation for a class of switched systems Yantao Chen 1, Junqi Yang 1 *, Donglei Xie 1, Wei Zhang 2 1. College of Electrical Engineering and Automation,
More informationLyapunov Stability of Linear Predictor Feedback for Distributed Input Delays
IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL. 56 NO. 3 MARCH 2011 655 Lyapunov Stability of Linear Predictor Feedback for Distributed Input Delays Nikolaos Bekiaris-Liberis Miroslav Krstic In this case system
More informationME 233, UC Berkeley, Spring Background Parseval s Theorem Frequency-shaped LQ cost function Transformation to a standard LQ
ME 233, UC Berkeley, Spring 214 Xu Chen Lecture 1: LQ with Frequency Shaped Cost Function FSLQ Background Parseval s Theorem Frequency-shaped LQ cost function Transformation to a standard LQ Big picture
More informationARTICLE. Super-Twisting Algorithm in presence of time and state dependent perturbations
To appear in the International Journal of Control Vol., No., Month XX, 4 ARTICLE Super-Twisting Algorithm in presence of time and state dependent perturbations I. Castillo a and L. Fridman a and J. A.
More informationLINEAR QUADRATIC OPTIMAL CONTROL BASED ON DYNAMIC COMPENSATION. Received October 2010; revised March 2011
International Journal of Innovative Computing, Information and Control ICIC International c 22 ISSN 349-498 Volume 8, Number 5(B), May 22 pp. 3743 3754 LINEAR QUADRATIC OPTIMAL CONTROL BASED ON DYNAMIC
More informationRobust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers
28 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 28 WeC15.1 Robust Stabilization of Non-Minimum Phase Nonlinear Systems Using Extended High Gain Observers Shahid
More informationHigh order integral sliding mode control with gain adaptation
3 European Control Conference (ECC) July 7-9, 3, Zürich, Switzerland. High order integral sliding mode control with gain adaptation M. Taleb, F. Plestan, and B. Bououlid Abstract In this paper, an adaptive
More informationComparison of four state observer design algorithms for MIMO system
Archives of Control Sciences Volume 23(LIX), 2013 No. 2, pages 131 144 Comparison of four state observer design algorithms for MIMO system VINODH KUMAR. E, JOVITHA JEROME and S. AYYAPPAN A state observer
More informationME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ
ME 234, Lyapunov and Riccati Problems. This problem is to recall some facts and formulae you already know. (a) Let A and B be matrices of appropriate dimension. Show that (A, B) is controllable if and
More informationStochastic and Adaptive Optimal Control
Stochastic and Adaptive Optimal Control Robert Stengel Optimal Control and Estimation, MAE 546 Princeton University, 2018! Nonlinear systems with random inputs and perfect measurements! Stochastic neighboring-optimal
More informationAn Output Integral Sliding Mode FTC Scheme Using Control Allocation
5th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December -5, An Output Integral Sliding Mode FTC Scheme Using Control Allocation Mirza Tariq Hamayun,
More informationResearch Article A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
Mathematical Problems in Engineering Volume 1, Article ID 693453, 1 pages doi:11155/1/693453 Research Article A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
More informationON CHATTERING-FREE DISCRETE-TIME SLIDING MODE CONTROL DESIGN. Seung-Hi Lee
ON CHATTERING-FREE DISCRETE-TIME SLIDING MODE CONTROL DESIGN Seung-Hi Lee Samsung Advanced Institute of Technology, Suwon, KOREA shl@saitsamsungcokr Abstract: A sliding mode control method is presented
More informationEE C128 / ME C134 Final Exam Fall 2014
EE C128 / ME C134 Final Exam Fall 2014 December 19, 2014 Your PRINTED FULL NAME Your STUDENT ID NUMBER Number of additional sheets 1. No computers, no tablets, no connected device (phone etc.) 2. Pocket
More informationOPTIMAL CONTROL. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 28
OPTIMAL CONTROL Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 28 (Example from Optimal Control Theory, Kirk) Objective: To get from
More informationTopic # Feedback Control Systems
Topic #17 16.31 Feedback Control Systems Deterministic LQR Optimal control and the Riccati equation Weight Selection Fall 2007 16.31 17 1 Linear Quadratic Regulator (LQR) Have seen the solutions to the
More informationControl Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli
Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)
More informationThe Important State Coordinates of a Nonlinear System
The Important State Coordinates of a Nonlinear System Arthur J. Krener 1 University of California, Davis, CA and Naval Postgraduate School, Monterey, CA ajkrener@ucdavis.edu Summary. We offer an alternative
More informationTopic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis
Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of
More informationarzelier
COURSE ON LMI OPTIMIZATION WITH APPLICATIONS IN CONTROL PART II.1 LMIs IN SYSTEMS CONTROL STATE-SPACE METHODS STABILITY ANALYSIS Didier HENRION www.laas.fr/ henrion henrion@laas.fr Denis ARZELIER www.laas.fr/
More informationAdaptive Predictive Observer Design for Class of Uncertain Nonlinear Systems with Bounded Disturbance
International Journal of Control Science and Engineering 2018, 8(2): 31-35 DOI: 10.5923/j.control.20180802.01 Adaptive Predictive Observer Design for Class of Saeed Kashefi *, Majid Hajatipor Faculty of
More informationDelay-independent stability via a reset loop
Delay-independent stability via a reset loop S. Tarbouriech & L. Zaccarian (LAAS-CNRS) Joint work with F. Perez Rubio & A. Banos (Universidad de Murcia) L2S Paris, 20-22 November 2012 L2S Paris, 20-22
More informationand the nite horizon cost index with the nite terminal weighting matrix F > : N?1 X J(z r ; u; w) = [z(n)? z r (N)] T F [z(n)? z r (N)] + t= [kz? z r
Intervalwise Receding Horizon H 1 -Tracking Control for Discrete Linear Periodic Systems Ki Baek Kim, Jae-Won Lee, Young Il. Lee, and Wook Hyun Kwon School of Electrical Engineering Seoul National University,
More informationStability and composition of transfer functions
Problem 1.1 Stability and composition of transfer functions G. Fernández-Anaya Departamento de Ciencias Básicas Universidad Iberoaméricana Lomas de Santa Fe 01210 México D.F. México guillermo.fernandez@uia.mx
More informationOn Convergence of Nonlinear Active Disturbance Rejection for SISO Systems
On Convergence of Nonlinear Active Disturbance Rejection for SISO Systems Bao-Zhu Guo 1, Zhi-Liang Zhao 2, 1 Academy of Mathematics and Systems Science, Academia Sinica, Beijing, 100190, China E-mail:
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. XIII - Nonlinear Observers - A. J. Krener
NONLINEAR OBSERVERS A. J. Krener University of California, Davis, CA, USA Keywords: nonlinear observer, state estimation, nonlinear filtering, observability, high gain observers, minimum energy estimation,
More informationStabilization for Switched Linear Systems with Constant Input via Switched Observer
Stabilization for Switched Linear Systems with Constant Input via Switched Observer Takuya Soga and Naohisa Otsuka Graduate School of Advanced Science and Technology, Tokyo Denki University, Hatayama-Machi,
More informationMultivariable MRAC with State Feedback for Output Tracking
29 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 1-12, 29 WeA18.5 Multivariable MRAC with State Feedback for Output Tracking Jiaxing Guo, Yu Liu and Gang Tao Department
More informationOutput Regulation of the Tigan System
Output Regulation of the Tigan System Dr. V. Sundarapandian Professor (Systems & Control Eng.), Research and Development Centre Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-6 6, Tamil Nadu,
More informationA note on linear differential equations with periodic coefficients.
A note on linear differential equations with periodic coefficients. Maite Grau (1) and Daniel Peralta-Salas (2) (1) Departament de Matemàtica. Universitat de Lleida. Avda. Jaume II, 69. 251 Lleida, Spain.
More informationNoncausal Optimal Tracking of Linear Switched Systems
Noncausal Optimal Tracking of Linear Switched Systems Gou Nakura Osaka University, Department of Engineering 2-1, Yamadaoka, Suita, Osaka, 565-0871, Japan nakura@watt.mech.eng.osaka-u.ac.jp Abstract. In
More informationIntro. Computer Control Systems: F9
Intro. Computer Control Systems: F9 State-feedback control and observers Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 21 dave.zachariah@it.uu.se F8: Quiz! 2 / 21 dave.zachariah@it.uu.se
More information