Robustification of time varying linear quadratic optimal control based on output integral sliding modes Rosalba Galván-Guerra, Leonid Fridman

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1 Techset Composition Ltd, Salisbury Doc: {IEE} CTASI tex 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: /iet-cta Robustification of time varying linear quadratic optimal control based on output integral sliding modes Rosalba Galván-Guerra, Leonid Fridman Facultad de Ingeniería, UNAM, CP 04510, Mexico DF, Mexico lfridman@unammx ISSN Abstract: A hierarchical observer for an uncertain linear time-varying system is designed generalising the output integral sliding-mode approach The observer is based on a step by step reconstruction of the output and its derivatives, to reconstruct the state theoretically exactly right after the initial time, eliminating completely the effects of the uncertainties/perturbations The observed state is used in an linear quadratic control with an output integral sliding-mode technique, allowing the uncertain system to behave as a nominal (uncertainties/perturbations free) one The proposed approach is tested by computer simulations Introduction The goals of this paper can be summarised as Linear time-varying systems are consequence of the presence of time-varying elements in the system or as a result of a linearisation procedure around a time-varying nominal trajectory [1] The time-varying characteristic makes them especially complex when it is compared with a linear time invariant process The linear quadratic (LQ) optimal control allows to control these systems by solving a Riccati differential equation [2, 3], under the hypothesis that the mathematical model is known and the states of the system can be measured from the very first moment When any control is implemented in a process, the presence of uncertainties/perturbations, missing in the mathematical model, makes it sometimes impossible to achieve the control objective The use of a sliding-mode technique [4, 5] makes the system theoretically insensitive with respect to any matched and bounded uncertainties/perturbations during the sliding phase [6] The integral sliding-mode (ISM) technique [7], offers the advantage of forcing the system to be on sliding phase from the first moment, making it insensitive to the uncertainties/perturbations right away But the main drawback is that it requires the complete knowledge of the state in the entire time interval Recently, in [7, 8] a robustified LQ output optimal control for uncertain linear time invariant systems is proposed, the state is reconstructed right after the initial time using a hierarchical observer, based on an output integral slidingmode (OISM) technique, with the restrictions: the uncertainties/perturbations are matched and bounded; the system has relative degree one with respect to the control input [9]; and it has more outputs than inputs This approach cannot be used if the system is varying in time, since the matrix gain is designed using the input and output matrices To propose a hierarchical observer for uncertain linear time-varying (ULTV) systems, based on a step by step reconstruction of the output and its derivatives, that allows to reconstruct the state theoretically right after the initial time To design an output-based robust methodology for a timevarying LQ optimal control problem To give sufficient conditions to assure that the system s observability properties are preserved With this aim the new methodology is developed ISM approach for the time-varying case An OISM for ULTV allowing to theoretically make the system insensitive to matched and bounded uncerainties/perturbations right after the initial time by using only output information This paper is organised as follows: Section 2 presents the problem formulation, together with some useful concepts and the assumptions needed in the paper In Section 3, the OISM methodology is proposed, with the sufficient conditions for the existence of the OISM observer In the same section, we revised a Kalman Bucy-based stabiliser to assure the boundedness of the error dynamics Section 4 is devoted to the constructive methodology of the hierarchical observer and gives the conditions for the realisability of the methodology Section 5 presents the simulation results of the robustification methodology applied to an aircraft and a spacecraft examples Section 6 concludes the paper IET Control Theory Appl, pp doi: /iet-cta The Institution of Engineering and Technology 2014

2 141 2 Problem formulation 211 We consider an ULTV system defined on the time interval T 146 ẋ(t) = A(t)x(t) + B(t)(u(t) + φ(x, t)) y(t) = C(t)x(t), x(t 0 ) = x 0 (1) where x(t) R n is the state vector, u(t) R m is the control input vector and y(t) R p is the output vector The function φ : R n R R m represents the unknown uncertainties because of parameter variations, unmodelled dynamics and/or exogenous disturbances; and A : R R n n, B : R R n m and C : R R p n are continuously differentiable matrix functions As it is well known, in the presence of unknown signals, the LQ optimisation problem is not well-posed Hence, in order to design the optimal control, it is necessary to propose a nominal linear time-varying system assuming φ( ) = 0 ẋ n (t) = A(t)x n (t) + B(t)u n (t), x n (t 0 ) = x 0 (2) where u n is the nominal control Let the given continuously differentiable symmetric matrix functions Q : R R n n and R : R R m m be positive definite and positive semi-definite, respectively Moreover, assume the matrix Q f R p p is symmetric and positive definite Our aim is to minimise the cost function J (x n, u n, t) = x T n (t f )Q f x n (t f ) Fig 1 Robust Methodology for an LQ optimal control the error dynamics With this aim a Kalman Bucy-based stabiliser is designed The sliding-mode dynamics given by the OISM approach is used in the second part, for the design of a hierarchical observer It is mandatory to give the sufficient conditions ensuring the observability properties are not affected by the OISM methodology This hierarchical observer is based on a step by step reconstruction of the output and its derivatives making possible to reconstruct the state theoretically right after the initial time These two parts constitute a robust methodology for the LQ optimal control in the time-varying case In Fig 1 we present a block diagram with the structure of the proposed approach tf t 0 (x T n (t)q(t)x n(t) + u T n (t)r(t)u n(t)) (3) under all admissible trajectories of the nominal system under consideration Following the well known maximum principle [2, 3, 10], the optimal control takes the form [3] 21 Observability and controllability concepts In this subsection we recall some well-known observability and controllability concepts for LTV systems, that will become useful in the following sections u n (t) = R 1 (t)b T (t)p(t)x n (t) (4) where P(t) is solution of the differential Riccati equation Ṗ(t) = A(t)P(t) P(t)A T (t) + P(t)B(t)R 1 (t)b T (t)p(t) Q(t) (5) P(t f ) = Q f Note that when a complete state measure is not available the optimal control takes the form u n (t) = R 1 (t)b T (t)p(t)ˆx(t) (6) where ˆx(t) is an observed state This observed state converges to the original one right after the initial time Remark 1: It is important to emphasise that the proposed methodology can be applied to a nominal stabilising control This paper is divided in two parts The first part is devoted to the design methodology of the nominal control and the OISM control used to assure the elimination of the uncertainties/perturbations dynamics effects from the trajectory of the ULTV system using only output information The rest of the output information is used in an OISM-based observer, making necessary to assure the boundedness of Theorem 1 [1, 11] Observability of LTV systems: Let the given matrix functions A and C of the nominal system (2) be l o 2 and l o 1 times continuous differentiable, respectively, on the non-degenerate time interval T Let the observability index l o be the minimum integer such that rank(o lo (t)) = n, for t T, where the observability matrix is defined by O lo (t) = N 0 (t) N 1 (t) N lo 1(t) Rplc n (7) where N 0 (t) = C(t) and N i (t) = N i 1 (t)a(t) + dni 1(t) for i = 1,, l o Then, the pair (A(t), C(t)) is observable on this interval if and only if rank(o lo (t)) = n, for all t T Theorem 2 [1, 11] Controllability of LTV systems: Let the given matrix functions A and B of the nominal system (2) be l c 2 and l c 1 times continuous differentiable, respectively, on the non-degenerate time interval T Let the controllability index l c be the minimum integer such that rank(c lc (t)) = n, for t T, where the controllability matrix is defined by C lc (t) =[M 0 (t) M 1 (t) M lc 1(t)] R n mlc (8) IET Control Theory Appl, pp 1 10 The Institution of Engineering and Technology 2014 doi: /iet-cta

3 where M 0 (t) = B(t) and M i (t) = A(t)M i 1 (t) + dmi 1(t) for i = 1,, l c Then, the pair (A(t), B(t)) is controllable on this interval if and only if rank(c lc (t)) = n, for all t T Definition 1 [12 14]: A ULTV system (1) is uniformly completely observable if the following three conditions hold (any two implying the third), for some positive, α 1, α 2,α 3,α 4 and δ and for all s, t R + α 1 I W (s, s + δ) α 2 I α 3 I T (s, s + δ)w (s, s + δ) (s, s + δ) α 4 I (t, s) α 5 ( t s ) where ( ) is the state transition matrix for system (1), and W ( ) is the observability Gramian W (s, s + δ) = s+δ s T (t, s)c(t)c T (t) (t, s) and α 5 ( ) : R + R is bounded on bounded intervals We say that a model is uniformly completely controllable if its dual is uniformly completely observable 22 General assumptions Along this paper, we consider the following assumptions: A1 For simplicity in the notation let T =[t 0, t f ] A2 The uncertainties/perturbations φ(x, t) is bounded: φ(x, t) φ Max, for all t T, where φ Max R + is given Our aim is the construction of an optimal control law based and an OISM-based observer, hence as usual, we assume the following structural properties for system (1) A3 rank C(t)B(t) = m, t T A4 More outputs than inputs, that is, p > m A5 The matrices A(t), B(t) and C(t) are l 2, l 1, l 1 times continuously differentiable matrix functions in the time interval T These matrices and their derivatives are bounded and known A6 The pair (A(t), B(t)) is uniformly completely controllable in T, with controllability index l c A7 The pair (A(t), C(t)) is uniformly completely observable in T, with observability index l o A8 The pair (A(t), B(t), C(t)) is strongly observable in T A9 l = max{l c, l o } To assure the applicability of the proposed approach it is necessary to presume that A10 The initial condition is unknown but bounded, that is, x(t 0 ) χ Definition 2 [11] Strongly observable: The pair (A(t), B(t), C(t)) is called strongly observable if ẋ(t) = A(t)x + B(t)u, C(t)x(t) 0 on some non-degenerate time interval T for some control u, with Bu a continuous function, always implies that x(t) 0 Theorem 3 [11]: Let the matrix functions A, B and C be n 2, n 2, n 1 times continuously differentiable, respectively, and define the matrices B μ,ν = B μ,ν (t), t T, recursively by 3 Uncertainty compensation via time-varying output integral sliding-mode This section is devoted to the design of the output-based control methodology that will help us eliminate the effects of the uncertainties/perturbations right after the initial time moment Similar to the procedure in [7, 8], consider an ULTV system (1) and let u(t) = u n (t) + u 1 (t) (10) B μ,μ 1 := CB for 2 μ n B μ,1 := N μ 2 B + db μ 1,1 B μ,ν := B μ 1,ν 1 + db μ 1,ν for for 3 μ n 3 ν<μ n The the triple (A(t), B(t), C(t)) is strongly observable on T if and only if rank S(t) = rank S (t) for all t T, where the matrix function S : T R plo [n+(n 1)m] and S : T R (pn+n) [n+(n 1)m] are defined by where u n is any suitable nominal control, and u 1 (t) is the ISM control part part, which guarantees the compensation of the uncertainties/perturbations φ(x, t), on the time interval t T Now let us define a time-varying sliding surface of the form t s(y, t) = G(t)y(t) G(t 0 )y(t 0 ) (Ġ(τ)C(τ)ˆx(t) t 0 + G(τ)Ċ(τ)ˆx(t) + G(τ)C(τ)(A(τ)ˆx(τ) + B(τ)u n (τ))) dτ (11) with S(t) := [O n (t) J n (t)], S (t) := [ ] In 0 O n (t) J n (t) B 2,1 0 0 J n (t) := B 3,1 B 3,2 0 B n,1 B n,2 B n,n 1 (9) where ˆx(t) is the observed state and G(t) R m n is a continuously differentiable design matrix such that det(g(t)c(t)b(t)) = 0 Remark 2: Note that in contrast with the ISMs presented in [4, 5], we defined a time-varying sliding-mode surface, where matrix G is not assumed constant in t and Ġ is assumed to be known Also, note that the system is in the sliding-mode at the first moment, that is, s(y(t 0 ), t 0 ) = and I n the n n identity matrix, and where the matrices O n, N k are defined in Theorem 1 Taking the time derivative of the surface along the trajectories of (1), we obtain an equivalent control [6] of the 416 IET Control Theory Appl, pp doi: /iet-cta The Institution of Engineering and Technology 2014

4 form u 1eq = φ(x, t) D 1 (t)(ġ(t)c(t) + G(t)Ċ(t) + G(t)C(t)A(t))(x(t) ˆx(t)) D(t) = G(t)C(t)B(t), t T (12) For simplicity, let us write the sliding-mode dynamics (13) as ẋ(t) = (A(t) + Ɣ(t))x(t) + B(t)ũ(t) y(t) = C(t)x(t); x(t 0 ) = x 0, t T (14) Substituting the equivalent control (12) in the ULTV system (1), we obtain the sliding-mode dynamics ẋ(t) = Ã(t)x(t) + B(t)u n (t) where ũ(t) = u n (t) D 1 (t)(ġ(t)c(t) + G(t)Ċ(t) + G(t)C(t)A(t))ˆx(t) B(t)D 1 (t)(ġ(t)c(t) + G(t)Ċ(t) + G(t)C(t)A(t))ˆx(t) y(t) = C(t)x(t); x(t 0 ) = x 0, t T where matrix Ã(t) is Ã(t) =[I B(t)D 1 (t)g(t)c(t)]a(t) B(t)D 1 (t)(ġ(t)c(t) + G(t)Ċ(t)) (13) Note that if the observed state is equal to the real one, these dynamics would be equivalent to the nominal ones (2) Then, if it was possible to reconstruct the state right after the initial time, we could make the ULTV system to behave as the nominal one As it should be inferred, the proposed approach is equivalent to the one proposed by Bejarano et al [8] and Fridman et al [7], for the time invariant case Then, the proposed methodology constitutes a generalisation to the time-varying case of the OISM methodology Note that the equivalent control has the effect of eliminate some dynamics of the system that could affect its observability properties The next theorem gives sufficient condition to assure that the pair (Ã(t), C(t)) preserves the observability properties of (1) Theorem 4: Under the Assumptions A4, A3, A8 and A5, the sliding-mode dynamics (13) is strongly observable if all the following conditions are fulfilled C1 dg(t)c(t) = 0 C2 rank(o l + (t)) = n for all t t 0 or (t) = 0, where = l 2, 0 = CƔ, i = i 1 A + d i 1 For simplicity, let us omit the parameter t from now until the end of the proof If we take the output and its l 1-derivatives, we see that where y dy d 2 y 2 d l 1y l 1 =[O l + J l + ϒ] }{{} S ϒ = ϒ 3,1 0 0 ϒ l,1 ϒ l,2 0 x u du d l 2u l 2 ϒ μ,μ 2 := 0 B for 3 μ n ϒ μ,1 := μ 3 B + dϒ μ 1,1 for 4 μ n ϒ μ,ν := ϒ μ 1,ν 1 + dϒ μ 1,ν for 4 ν<μ n Now, because of the form of the matrices J l and ϒ and the relative degree condition (Assumption A1), it is clear that (J l + ϒ) spano l and thus it is not in the space generated by O l + and since rank(o l + ) = n, then where rank S(t) = rank S (t) [ ] S I (t) := n 0 O l (t) + (t) J l (t) + ϒ(t) and Ɣ(t) = B(t)D 1 (t)g(t)c(t)a(t) B(t)D 1 (t)(ġ(t)c(t) + G(t)Ċ(t)) Proof: We know that if the ULTV system is strongly observable then rank S(t) = rank S (t) with S and S defined in Theorem 3 From this we know that ranko l = n and J l spano l Also recall Assumption A3, which implies that the system has relative degree one and the sliding-mode dynamics are strongly observable Now let us consider the case when Assumption A4 and Condition C1 are not fulfilled To achieve contradiction, consider the system (1) with p m and rank(c(t)b(t)) = p for all t T Assume the control law u(t) is designed as u(t) = u n (t) + u 1 (t), where u n is the nominal control used after the compensation of the matched uncertainties/perturbations and u 1 is designed to compensate the matched uncertainties/perturbations Let us analyse the case when p = m and later when p < m Consider the case when p = m Define an output integral sliding surface as in (11) The matrix G(t) must satisfy rank(g(t)c(t)b(t)) = m for all t T, but this is only satisfied when det(g(t)) = 0 As we saw in this section, the IET Control Theory Appl, pp 1 10 The Institution of Engineering and Technology 2014 doi: /iet-cta

5 equivalent control takes the form (12) Substituting this equivalent control in the system (1) yields (13) Premultiplying Ã(t) by G(t)C(t) we obtain G(t)C(t)Ã(t) = 0 This means Ã(t) belongs to the null space of G(t)C(t), whenever dg(t)c(t) = 0 and, since G(t) is a non-singular matrix for all t T, then Ã(t) belongs to the null space of C this would imply that (Ã(t), C(t)) is not observable Now suppose p < m, then Assumption A1 is not fulfilled and the proposed approach cannot be applied time stable and converge to the surface right after the first moment Remark 4: Assumption A2 is necessary to assure the existence of the scalar β, that is, the existence of the slidingmode 4 Time-varying output integral sliding mode observer design Remark 3: Observe that if the Assumptions A3 and A4 are not fulfilled the system dynamics under the sliding-mode will be unobservable Moreover, if the Assumption A5 is not achieved in a non-degenerate interval, we cannot assure the robust observability of the pair (A(t), B(t), C(t)) (see Theorem 23) Theorem 5: Under the Assumptions A1 A6, the proposed sliding dynamics ṡ(y, t) = (Ġ(t)C(t) + G(t)Ċ(t) + G(t)C(t)A(t))(x(t) ˆx(t)) + D(t) (u 1 (t) + φ(x, t)) s(t 0 ) = 0 (15) This section is devoted to the design procedure of the timevarying hierarchical observer Without loss of generality let us fix the value of G(t) G(t) = (C(t)B(t)) + := [(C(t)B(t)) T (C(t)B(t))] 1 (C(t)B(t)) T Substituting the proposed G(t) into the matrix D( ) and Ã( ) on the LTV system (13) This leads to the following simplified version of the sliding-mode dynamics ẋ(t) = Ã(t)x(t) + B(t)u n (t) + B(t) ( Ġ(t)C(t) + G(t)Ċ(t) + G(t)C(t)A(t) ) ˆx(t) y(t) = C(t)x(t), x(t 0 ) = x 0 (18) are uniformly finite time stable [1, 15] on the time interval T if det(d(t)) = 0 Proof: Consider the time-varying Lyapunov candidate function V = 1 2 s( ) 2, t T The derivative of V along the trajectories of the sliding surface (11) is Recall Assumption A8, and that by Theorem 4 it was proven that the pair (Ã(t), C(t)) is strongly observable and the observability matrix (7) with A(t) = Ã(t) is well defined with observability index l = max{l c, l o } and has rank equal to n To reconstruct the state x(t), we need to recover step by step the vectors f (t) = O l (t)x(t) for all t T 41 Kalman Bucy-based stabiliser V (t) = s T (t)ṡ(t) = s T ((Ġ(t)C(t) + G(t)Ċ(t) + G(t)C(t)A(t)) (x(t) ˆx(t)) + D(t)(u 1 (t) + φ(x, t))) Assume a first order sliding-mode control in a unitary vector form [6], that is u 1 = β DT (t)s(t), t T (16) D T (t)s(t) where β is a scalar Then, the derivative of the Lyapunov candidate function along the trajectories of the sliding surface can be bounded as Before introducing the hierarchical observer it is necessary to assure that the error dynamics are bounded We use a methodology based on a Kalman Bucy filter allowing to stabilise the error, giving a very useful bound of the error dynamics Consider the following dynamical system x(t) = Ã(t) x(t) + B(t)u 0 + B(t)(Ġ(t)C(t) + G(t)Ċ(t) + G(t)C(t)A(t))ˆx(t) + K(t)(y(t) C(t) x(t)), x(t 0 ) = C + (t 0 )y(t 0 ) where K(t) must be designed in such a way that the dynamical observation error ṙ(t) = (Ã(t) K(t)C(t))r(t) = Â(t)r(t), r(t 0 ) = V (t) s T D(t) ( (t) x(t) ˆx(t) +β φ Max ) where (t) = D 1 (t)(ġ(t)c(t) + G(t)Ċ(t) + G(t)C(t) A(t)) Then, the proposed sliding-mode control assures uniform stability [1, 15] of the sliding surface (15) if the scalar β satisfies the inequality where r(t) = x(t) x(t), is exponentially stable Recall Assumptions A(6) A(7), then, from [13], if K(t) is such that K(t) = P KB (t)c T (t)r 1 (t) (19) where P KB (t) is the solution of the Riccati matrix (t) x(t) ˆx(t) +β φ Max λ>0 (17) where λ is a constant By the comparison principle [15] and knowing that for construction V (s(t 0 )) = 0, since s(y(t 0 ), t 0 ) = 0, the convergence time is given by t = t 0 and the proposed sliding-mode dynamics (15) is uniformly finite Ṗ KB (t) = A(t)P KB (t) + P KB (t)a T (t) P KB (t)c T (t)v 1 2 C(t)P KB (t) + V 1 (t) (20) with initial condition P KB (t 0 ) = P 0 and V 1, V 2 are positive semi-definite and positive definite matrix functions, respectively, the Kalman Bucy-based stabiliser is exponentially IET Control Theory Appl, pp doi: /iet-cta The Institution of Engineering and Technology 2014

6 701 stable Therefore there exist constants ψ, η such that r(t) ψe η(t t0) r(t 0 ) ψe η(t t0) (χ + x(t 0 ) ) satisfies C(t 0 )x a1 (t 0 ) = y(t 0 ) Using a sliding variable s 1 R p defined by (22), we have that the dynamics of the sliding surface are ruled by Remark 5: The Kalman Bucy filter is constructed for specific types of noises when their variances are known In the presence of time-varying uncertainties/perturbations with unknown variances, the use of a Kalman Bucy methodology acts as an error stabiliser allowing to assure the error dynamics will converge to a bounded region The slidingmode observer that we will implement needs a bounded observation error dynamics, to eliminate in finite time the uncertainties/perturbations dynamics, therefore the bound provided by the Kalman Bucy filter is very useful When the variances of the uncertainties/perturbation is known the stabiliser would work as a pre-filter allowing to diminish the effects of the uncertainties/perturbations ṡ 1 (y(t), x a1 (t)) = (C(t)Ã(t) + Ċ(t))(x(t) x(t)) v 1 (t) = N 1 (t)(x(t) x(t)) v 1 (t) with v 1 (t) defined as v 1 (t) = M 1 (t) s 1(t) s 1 (t), t T Here the scalar gain M 1 should satisfy the condition N 1 (t) x(t) x(t) < M 1 (t) Hierarchical observer design methodology Recall that the main idea in the design of our observer is to recover the vectors N i (t)x(t), i = 1,, l 1 Note that we are reconstructing the output and its derivatives The observer design is given by the following theorem To assure exponential stability, this gain should be chosen as M 1 (t) = N 1 (t) ψe η(t t0) (χ + x(t 0 ) ) Repeating the same stability proof in Section 4, we obtain s 1 ( ) =ṡ 1 ( ) = 0, t t 0 Thus, in view of (22), we have Theorem 6: If the auxiliary state vectors x ak, for all k = 1,, l 1 are designed as ẋ ak = Ã(t) x(t) + B(t)(u n (t) + (G(t)C(t)A(t) + Ġ(t)C(t) + G(t)Ċ(t))ˆx(t)) + L k (t)(n k 1 (t)l k (t)) 1 (v k (t) + Ṅ k 1 (t)( x(t) x ak )) (21) C(t)x(t) = C(t)x a1 (t) and the equivalent output injection is v 1eq (t) = N 1 (t) (x(t) x(t)), t T It follows that N 1 (t)x(t) is recovered by means of the following representation where L i (t) R n p is a is a matrix to be designed, such that det (N i 1 (t)l i (t)) = 0, and the initial conditions should satisfy C(t 0 )x a1 (t 0 ) = y(t 0 ) and N k 1 (t 0 ) x(t 0 ) + v k 1eq (t 0 ) = N k 1 (t 0 )x ak (t 0 ) Moreover, the variables s k are designed as N 1 (t)x(t) = N 1 (t) x(t) + v 1eq (t), t T Following the procedure presented in Theorem 6, we can reconstruct the vector O l (t)x(t) = O l (t) x(t) + v eq (t), t T and s 1 (y(t), x a1 (t)) = y(t) C(t)x a1 (t) (22) s k (y(t), x ak (t)) = N k 1 (t) x(t) + v k 1eq (t) N k 1 (t)x ak (t) (23) for 1 < k < l 1 Then, for all t T v keq (t) = N k (t)(x(t) x(t)) and k = 1 l 1 and it is possible to reconstruct completely all the vector functions N i (t)x(t), i = 1,, l 1 Proof: The main idea of this proof is based on the methodology proposed by Bejarano et al[8] and Fridman et al [7] A complete proof of this theorem can be found in [16] We only include here the first step of the proof and the full methodology is obtained by iteration To recover the first vector N 1 x(t), we need the auxiliary state vector x a1 governed by (21) with k = 1, where x a1 (t 0 ) where C(t)x a1 (t) C(t) x(t) v 1eq (t) v eq (t) = v 2eq (t) Rp v l 1eq (t) Note that we reconstruct the output and its l time-derivatives Since the pair (Ã, C) is strongly observable, the pseudo inverse of O l is well defined and the state can be recovered by means of the equation x(t) = x(t) + O + l (t)v eq (t), t T (24) The hierarchical ISM observer is designed as ˆx(t) = x(t) + O + l (t)v eq (t), t T (25) Remark 6: Note that the methodology proposed in this paper constitutes a generalisation of the algebraic hierarchical IET Control Theory Appl, pp 1 10 The Institution of Engineering and Technology 2014 doi: /iet-cta

7 841 observer proposed by Fridman et al [7] and Bejarano et al [8] 5 Illustrative examples 51 Aircraft model It is clear that under the assumptions of this paper and, assuming that the ideal OISM exists, it is satisfied that ˆx(t) x(t) for all t T The equivalent output injection v eq cannot be realised because of the intrinsic dependency on the unknown state x(t) We applied the proposed methodology to the aircraft model proposed by Fujimori and Ljung [17] ẋ(t) = A c (t)x(t) + B c (t)(u(t) + φ(t)) y(t) = C c x(t) (27) Remark 7: However, it is well known [6] that the output of the low-pass filter τv av (t) + v av (t) = v(t) tends to the equivalent control if the filter constant is small enough, that is where x 1, x 2, x 3 and x 4 are the x-axis velocity, the angle of attack, the pitch angle and the pitch rate of the aircraft, respectively, the input u(t) is the elevator angle We consider only matched perturbations with bound φ(t) 1, for all t 0 we assumed that this uncertainties/perturbations are caused by the altitude change of the flight and some errors in the model parameters The time-varying matrices in the model are given by lim v av (t) = v eq (t) τ 0 /τ 0 where τ = t η,0<η<1 and t is the sample time Let v kav (t) be the filtered signal for subsystem k and let τ = t be the time constant of all the filters Then, following an algorithm similar to the one proposed by Fridman et al [7] and Bejarano et al [8], the realisation of the observer (25) takes the form ˆx(t) = x(t) + O l (t) + (t)v av (t), t T C(t)x a1 (t) C(t) x(t) v 1av (t) v av = v 2av (t) Rp (26) v l 1av (t) Algorithm 1: The proposed OISM robust methodology can be summarised as follows A c (t) = E 1 c (t)q c (t), B c (t) = E 1 c (t)r c (t) E c (t) = 0 0 V (t) 0, R c (t) = Z δe (t) 0 0 M α (t) 1 M δe (t) X u (t) g cos( 0 ) X α (t) Q c (t) = Z u (t) g sin( 0 ) Z α (t) V (t) + Z q (t) M u (t) 0 M α (t) M q (t) [ ] 0 C c = The time-varying parameters are defined as X u (t) = bv (t)s(c xu + 2C L tan( 0 )) 2m Z u (t) = bv (t)s(c zu 2C L ) 2m To assure a bounded observation error, design the matrix function K as in (19), assuring that the error dynamics of the Kalman Bucy-based stabiliser are exponentially stable 2 Compute the scalar gain β satisfying (17) and assuring the existence of the sliding-mode 3 Design the auxiliary systems x ak (21) with the sliding surfaces s k (23) and compute the constants M k, k = 1, 2,, l 1 This guarantees that the proposed the proposed observer would reconstruct the output and its derivatives right after the initial time 4 Before applying the observation methodology it is necessary to design first-order filters with a time constant τ = t These filters will reconstruct the control input vkeq from the high-frequency signal v k, that is, obtain the control input v kav Assure t is small enough 5 Design the nominal control and substitute x by ˆx, that is, use only output information 6 Run simultaneously the observer ˆx according to (26) and the controllers u n and u 1 as in (6) and (16), respectively We have now developed a robust methodology for the LQ nominal control u n Fig 2 Nominal Control (6) applied to the nominal aircraft system IET Control Theory Appl, pp doi: /iet-cta The Institution of Engineering and Technology 2014

8 Fig 3 Nominal control (6) applied to the perturbed (original) system (27) Fig 5 Observed state against real state t = 1e Fig 4 OISM (10) applied to the perturbed (original) system (27) Fig 6 Observed state against real state t = 1e M u (t) = bv (t)scc mu, X α (t) = bv 2 (t)sc xα 2I yy 2m Z α (t) = bv 2 (t)sc zα, M α (t) = bv 2 (t)scc mα 2m 2I yy M α (t) = bv (t)sc2 C m α 4I yy Z q (t) = bv (t)scc zq, M q (t) = bv (t)sc2 C mq 4m 4I yy Z δe (t) = bv 2 (t)sc zδe, M δe (t) = bv 2 (t)scc mδe 2m 2m where V (t) = 3(sin(t) + 49) is the flight velocity, m is the aircraft s mass, S is the main wing area, c is the main wing chord, b is the main wing span, C L is the lift coefficient The other parameters are the non-dimensional stability and control derivatives For more information about these parameters, consult [17] Applying the proposed approach, we obtain the following results: in Fig 3, we present the nominal behaviour of the aircraft under an LQ optimal Control Fig 3 presents the behaviour of the aircraft when only an LQ optimal control is applied In Fig 4, we present the behaviour of the system under the proposed OISM control Figs 5 7 presents the behaviour of the algebraic hierarchical observer under different values of sample times In Fig 8, we show the observation error norm for different values of sample times Note that the lower the sample time is, better is the response of the hierarchical observer we obtain Remark 8: Note that the proposed approach is able to reconstruct the state theoretically right after the initial time However, the first-order filter needed for the implementation affects the accuracy of the proposed observer 52 Spacecraft model Consider the spacecraft problem proposed in [18], with a time-varying inertia matrix I(t) = diag[i xx (t) I yy (t) I zz (t)] and equipped with a single-momentum wheel aligned to the IET Control Theory Appl, pp 1 10 The Institution of Engineering and Technology 2014 doi: /iet-cta

9 Fig 7 Observed state against real state t = 1e Fig 10 Observation error norm 1151 z-axis, with moment of inertia J and angular velocity relative to the body frame The linearised dynamics equation is given by 1221 ẋ(t) = A(t)x(t) + B(t)(u(t) + φ(t)) 1156 y(t) = Cx(t) Fig 8 Fig 9 Observation error norm Nominal system against robustified system where x =[δq T δω T ] T is formed with small displacements of the roll, yaw and pitch vector part q of the attitude quaternion from the nominal values q =[0 0 0] T and small deviations of the body rates from the nominal values ω x = ω y = 0, ω x = 0 (t), where 0 (t) is the orbital angular rate The control inputs u(t) are the magnetic torques and φ is the disturbance torque vector The matrices A,B and C are given by 0 0 (t) (t) A(t) = W x (t) 0 0 = 6k y (t) 2 0 (t) 0 W y(t) k z (t) 2 0 (t) B(t) = Ixx 1 (t) Iyy 1(t) Izz 1 (t) C = IET Control Theory Appl, pp doi: /iet-cta The Institution of Engineering and Technology 2014

10 where k x (t) = Iyy(t) Izz(t) I xx(t), k y (t) = Izz(t) Ixx(T) I yy(t), k z (t) = Ixx(t) Iyy(t) I zz(t), W x (t) = k x (t) 0 (t) + k wx (t), W y (t) = k y (t) 0 (t) + k wy (t), k wx = J I xx and k wy = J I yy Here is the nominal wheel speed Applying the proposed approach, a comparative behaviour between the nominal system and the robustified controlled system is presented in Fig 9 The observation error for different sample step values is shown in Fig 10 Note that as in the aircraft example, the lower the step value is, the lower will the observer error be 6 Concluding remarks A robust methodology for the solution of a time-varying LQ optimal control problem using only output information is proposed With this aim a hierarchical observer for ULTV system was proposed, allowing to reconstruct theoretically exactly the state of the system right after the initial time Sufficient conditions for the existence of the OISM were stated, assuring the system under the slidingmode dynamics does not lose its observability properties Moreover, the usage of an OISM technique ensures insensitivity of the ULTV system with respect to matched and bounded uncertainties/perturbations Obtained results are illustrated with the examples of the aircraft and the spacecraft 7 Acknowledgments The authors gratefully acknowledge the financial support from PAPIIT, UNAM, grant and from CONA- CyT Ciencias Básicas Rosalba Galván Guerra is thankful to UNAM for the fellowship received during her postdoctoral stage at Facultad de Ingeniería, UNAM 8 References 1 Rugh, WJ: Linear system theory (Prentice Hall, 1993) 2 Anderson, BD, Moore, JB: Optimal control : linear quadratic methods (Prentice Hall, 1990) 3 Bryson, AE, Ho, YC: Applied optimal control: optimization, estimation and control (Hemisphere Publishing Corp, New York, 1975) 4 Matthews, GP, DeCarlo, RA: Decentralized tracking for a class of interconnected nonlinear systems using variable structure control, Automatica, 1988, 24, (2), pp Utkin, V, Shi, J: Integral sliding mode in systems operating under uncertainty conditions Proc of the 35th IEEE Conf on Decision and Control, 1996, vol 4, pp Utkin, VI: Sliding modes in control and optimization (Springer- Verlag, 1992) 7 Fridman, L, Poznyak, A, Bejarano Rodríguez, FJ: Robust output LQ optimal control via integral sliding modes (Birkhäuser, 2014) 8 Bejarano, FJ, Fridman, L, Poznyak, AS: Output integral sliding mode control based on algebraic hierarchical observer, Int J Control, 2007, 80, pp Edwards, C, Spurgeon, S: Sliding Mode Control: Theory and Applications ser Series in Systems and Control Series Taylor & Francis, 1998 Available: commx/books?id=uh2rjhipsiyc 10 Pontryagin, LS, Boltyanskii, VG, Gamkrelidze, RV, Mishchenko, EF: The mathematical theory of optimal processes (Interscience Publishers, 1962) 11 Kratz, W, Liebscher, D: A local characterization of observability, Linear Algebr Appl, 1998, 269, (1 3), pp Anderson, BDO: Exponential stability of linear equations arising in adaptive identification, IEEE Trans Autom Control, 1977, 22, (1), pp Kalman, R, Bucy, RS: New results in linear filtering and prediction theory, J Fluids Eng, 1961, 83, (1), pp Kalman, RE: Contributions to the theory of optimal control, Bol Soc Mat Mexicana, 1960, 5, (2), pp Khalil, HK: Nonlinear systems 3rd edn, Upper Saddle River, New Jersey 07458, Prentice Hall, Galván-Guerra, R, Fridman, L: Output integral sliding mode observer for linear time variant systems, in Asociación mexicana de control automatico (AMCA), Fujimori, A, Ljung, L: Parameter estimation of polytopic models for a linear parameter varying aircraft system, Trans Japan Soc Aeronaut Space Sci, 2006, 49, (165), pp Corti, A, Lovera, M: Attitude regulation for spacecraft with magnetic actuators: an lpv approach, in Control of linear parameter varying systems with applications Springer, 2012, pp IET Control Theory Appl, pp 1 10 The Institution of Engineering and Technology 2014 doi: /iet-cta