Output-feedback finite-time stabilization of ... - Semantic Scholar
CONFIDENTIAL. Limited circulation. For review only
Output-feedback finite-time stabilization of disturbed feedback linearizable nonlinear systems Marco Tulio Anguloa,∗, Leonid Fridmanb , Jaime A. Morenoc a Facultad de Ingeniería, UAQ, México (e-mail: [email protected]) b Departamento de Control y Robótica, UNAM, México ([email protected]) c Eléctrica y Computación, Instituto de Ingeniería, UNAM, México (e-mail: [email protected])
Abstract A novel methodology for designing multivariable High-Order Sliding-Mode (HOSM) controllers for disturbed feedback linearizable nonlinear systems is introduced. It provides for the finite-time stabilization of the origin of the state-space by using output feedback. Only the additional assumptions of algebraic strong observability and smooth enough matched disturbances are required. The control problem is solved in two consecutive steps: firstly, designing an observer based on the measured output and, secondly, designing of a full-state controller computed from a new virtual output with vector relative degree. The introduced notion of algebraic strong observability allows recovering the state of the system using the measured output and its derivatives. By estimating the required derivatives through the HOSM differentiator, a finite-time convergent observer is constructed. Keywords: Sliding-mode control.
1. Introduction The design of robust controllers is an important topic in automatic control theory. Uncertainty is also manifested by only measuring a subset of the state variables of the system, namely, its “measured output”. In such scenario, Sliding-Mode (SM) output-feedback based controllers have shown to be very successful [1, 2, 3, 4]. Moreover, modern systems frequently mix continuos and discrete event dynamics for which the stabilization problems are much more intricate. However, hybrid systems with strictly positive dwell-time can be effectively controlled if the controller accomplishes the control objective before the next switching or impulse time. Therefore, robust output-feedback controllers providing finite-time state stabilization become relevant. High Order SMs (HOSMs) are useful in this context providing for the finite-time exact1 output stabilization using only output feedback [5]. In addition, in the presence of noise and sampling, HOSMs offer better accuracy than first-order SM. However, they were originally designed only for single-input single-output systems, see, e.g., [6, 7]. In [8], HOSM controllers are extended to Multi-Input Multi-Output (MIMO) nonlinear systems under the assumption of vector relative degree with respect to the
∗ Corresponding author: [email protected], May 3, 2013. 1 Exactness is more than robustness: the effect of (matched) distur-
bances is completely eliminated after a finite-time transient.
measured output. This last assumption also requires that the system has the same number of inputs as outputs. Recently, in [9], the authors introduced a methodology for the design of HOSM controllers for MIMO disturbed linear systems under necessary conditions: a known (affine) bound on the disturbance, controllability and strong observability [10]. The proposed methodology allows constructing an output-based HOSM controller guaranteeing the finite-time exact convergence of the whole state of the system. This brief extends the methodology introduced in [9] in the context of linear systems to a class of MIMO nonlinear systems. As in the linear case, the measured output does not necessarily has vector relative degree. In this form, HOSMs can be applied to a larger class of systems when compared to the strategy of [8]. Our approach requires introducing a suitable concept of observability despite disturbances. The notion of “algebraic strong observability” corresponds to the possibility of reconstructing the state as a function only of the measured output and a finite number of its derivatives. By using the HOSM differentiator [5] to estimate the required derivatives, a finite-time convergent observer is obtained. By combining the proposed observer with HOSMs controllers and dynamic feedback linearization the problem of exact finite-time state stabilization of nonlinear systems is solved. The proofs of all Theorems are collected in the Appendix. Main contributions. 1) An algorithm to construct an unknown input observer for nonlinear systems is pre-
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CONFIDENTIAL. Limited circulation. For review only sented based on the notion of algebraic strong observability and the use of HOSM differentiators; 2) by using the proposed observer, a novel HOSM-based output feedback control strategy for the finite-time state stabilization of (dynamic) feedback linearizable nonlinear systems with bounded matched disturbances is presented. 2. Problem statement Consider
x˙ = f ( x) + g( x)[ u + w],
y = h( x);
x(0) = x0 ,
(1)
where x( t) ∈ Rn , u( t) ∈ Rm , w( t) ∈ Rm and y( t) ∈ R p are the state, control input, disturbance and measured output, respectively. The vector fields f , g and the function h are assumed to be smooth. The control objective is to design the control input u( t), depending only on the output y( t), to provide finite-time state stability at zero despite the presence of the disturbance w( t). In order to present the assumptions we have made to solve the control problem, let us first introduce the following concept: Definition 1. System (1) is said to be algebraically strongly observable with respect to the output y( t) ∈ R p , if there exists a function F and integers k i , i = 1, . . . , p, such that (k1 )
x = F ( y1 , . . . , y1
(k p )
, · · · , yp , . . . , yp
).
(2)
Strong observability was introduced by Hautus [10] for linear systems to solve the Unknown Input Observer (UIO) design problem. It allows recovering the state of the system through the knowledge of the output and its derivatives only, irrespectively of the input. The notion of algebraic strong observability is a natural extension to the case of nonlinear systems. In Section 3, this property will be shown to be instrumental for the design of a UIO for nonlinear systems. The following assumptions about the system will be made throughout this paper: (A1) system (1) is algebraically strongly observable with respect to the measured output y;
A flat system is feedback linearizable and, moreover, its state and input can be written as a function of the flat output and its derivatives [12]. Without loss of generality, we assume that q(0) = 0. Note that, in general, the flat output z( t) may not coincide with the measured output y( t). The degree S of required smoothness for the disturbance depends on the “order” of the dynamic compensator used to linearize the system, as shown in Section 4. In the particular case when the system can be linearized by a static compensator, the disturbance needs to be only uniformly bounded. The approach we follow to solve the problem involves two steps. First, based on A1, we present an algorithm to compute the function F appearing in equation (2). This allows constructing a UIO for the system once the required derivatives are estimated. With assumptions A2 and A3, it becomes possible to estimate such derivatives using the HOSM differentiator [5]. In the second step, by using A4, it is shown that the finite-time state stabilization problem is equivalent to the finitetime stabilization of the flat output. This way, by using the estimated state of the previous step to evaluate the flat output, a multivariable HOSM controller that semiglobally stabilizes the flat output to zero in finite-time is presented. 3. Construction of the Unknown Input Observer Once the function F in (2) is known, the state can be determined when the derivatives of the output are estimated. By means of the HOSM differentiator, these derivatives can be estimated exactly and in finite time. In addition, the HOSM differentiator provides an estimation that is robust to measurement noise and has the best order of precision [5]. The following Subsection presents an algorithm to compute the function F for a class of nonlinear systems. Subsection 3.2, analyzes the use of the HOSM differentiator to estimate the required derivatives. 3.1. An Algorithm to construct an Unknown Input Observer. Let us consider the following additional assumption:
(A5) The Lie derivatives L g L kf h( x) are constant for k = (A2) for a given degree of smoothness S ∈ Z+ , there ex1, . . . , n − 1. ists a constant W + such that kw(k) ( t)k ≤ W + , ∀ t ≥ 0 This assumption means that in the time derivatives and all k = 0, . . . , S ; of the measured output, the matrix multiplying the in(A3) for all initial conditions x0 ∈ Rn and disturbances put is constant. This allows writing a constant orthogow(·) satisfying A2, the solution2 to system (1) with nal to such matrix. Assumption A5 is only introduced to u(·) ≡ 0 exists for all t ≥ 0 and remains upper- simplify the presentation of the algorithm: in the genbounded by a possibly unknown number; eral case when A5 is not satisfied, the orthogonal can be computed in the same way but now depending on (A4) there exists a function q : Rn → Rm such that z = the state. q( x) is a flat output for system (1). Let b⊥ denote a left annihilator for matrix b (i.e. ⊥ b b = 0). The presented algorithm computes the func2 Solutions to differential equations (and their associated inclution F introduced in equation (2), enabling the consions) are understood in Filippov’s sense [11]. struction of a UIO for system (1). Our algorithm is the
Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
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CONFIDENTIAL. Limited circulation. For review only “dual” of the standard Structure Algorithm (see [13, pp. 76]), since it looks for directions orthogonal to the input. Nevertheless, it is closely related to the algorithm presented in [14] for linear systems. Step 0. Set yˆ 0 := y and M0 ( x) := h( x). ˙ 0 ( x) = y˙ˆ 0 = L f M0 ( x) + Compute M Step 1. ˙ 0 ( x) = L g M0 ( x)w := a 1 ( x)+ b 1 ( u+w), introduce yˆ˙ 1 := b⊥ M 1
aˆ 1 ( x) and set ·
M1 ( x) :=
yˆ 0 y˙ˆ 1
¸
· =
h( x) aˆ 1 ( x)
¸
.
Step k + 1. Compute ˙ k ( x) = M
h
y˙ˆ 0
···
yˆ k(k+1)
iT
:= a k+1 ( x) + b k+1 ( u + w),
˙ ( x) = aˆ k+1 ( x), and set M introduce yˆ k(k++11) := b⊥ k+1 k yˆ 0 h( x) .. .. = M k+1 ( x) := . . . ( k+1) aˆ k+1 ( x) yˆ k+1 By applying the algorithm, one is able to compute a sequence of equations of the form [ yˆ 0 , y˙ˆ 1 , . . . , yˆ k(k) ]T = M k ( x) for k ≥ 1. The algorithm converges in the following sense. Theorem 1. Suppose that the vector fields f , g and the function h are meromorphic, rank h( x) = p and the system is algebraically strongly observable. Under these conditions rank d M n− p ( x) = n for almost all x ∈ Rn and the function F := M n−−1 p as in equation (2) exists locally. Recall that a meromorphic function is the quotient of two analytic functions. Computing the inverse function of M k−1 is, in general, a difficult problem. For this reason, it is useful to perform all possible simplifications at each step of the algorithm, replacing functions of the state by functions of the output. We have illustrated this idea in the example below by introducing the additional variables Nk , k ≥ 1. Example 3.1. Consider the following system not having vector relative degree with respect to the measured output:
x˙ 1 = x˙ 2 =
x1 x2 + w1 , x33 + w1 ,
x˙ 3 = x˙ 4 =
x4 (1 + x1 ) + w1 , sin( x1 x2 ) + w2 ,
y1 = y2 =
x1 , x2 .
Applying the algorithm yields: ·
¸ · ¸ x1 yˆ 0,1 Step 0: Set yˆ 0 = y and N0 ( x) = = yˆ 0 = . x2 yˆ 0,2 Trivially simplify it and set M0·( x) = N0¸( x).· ¸ ˙ 0 ( x) = x1 3x2 + 1 0 w = Step 1: Compute M 1 0 x £ 3 ¤ ⊥ − 1 1 a 1 ( x) + b w . Thus b = , introduce 1 £ 1 ¤ ˙ 0 ( x) = x3 − x1 x2 and set N1 ( x) := y˙ˆ 1 = −1 1 M 3 £ ¤T yˆ 0 y˙ˆ 1 . Proceed to simplify it as # " yˆ 0 £ ¤T ¢ 1 = x1 x2 x3 M 1 ( x ) := ¡ . ˙yˆ 1 + yˆ 0,1 yˆ 0,2 3
Step 2: Compute
x1 x2 1 ˙ 1 ( x) = + 1 x33 M 1 x4 (1 + x1 )
0 0 w := a 2 ( x) + b 2 w. 0
£ ¤ −1 0 1 Then b⊥ and introduce y¨ˆ 2 = 2 =¤ £ ˙ 1 ( x) = x4 (1 + x1 ) − x1 x2 , −1 0 1 M iT h ¡ ¢1 and simplify Set N2 ( x) = yˆ 0 , y˙ˆ 1 + yˆ 0,1 yˆ 0,2 3 , y¨ˆ 2 it as x1 yˆ 0 ¢ 1 x2 ¡ 3 ˙ M 2 ( x ) := = x3 . yˆ 1 ¨+ yˆ 0,1 yˆ 0,2 yˆ 2 + yˆ 0,1 yˆ 0,2 1+ yˆ 0,1 x4
Since rank d M2 ( x) = 4 the algorithm stops and the system is algebraically strongly observable. In addition, we have obtained an expression of all states as a function of the measured output and its derivatives.O This last example also shows that the condition of vector relative degree (with respect to the measured output) considered in [8] is not necessary for the system to be algebraic strongly observable. 3.2. Estimating the derivatives: a finite-time convergent UIO. Once equation (2) has been determined using the procedure described in the previous Subsection, the next step to construct the UIO is to evaluate the required derivatives of the measured output. This Subsection briefly describes the use of the HOSM differentiator to estimate them. Let yi ( t) be a component of the measured output to be differentiated k i -times. The k i -th order HOSM differentiator [5] takes the form 1
ki
k +1 ζ˙0 = ν0 = −λk i L i i |ζ0 − yi | k i +1 sign(ζ0 − yi ) + ζ1 , .. . 1 ¯ ¯1 ˙ζk −1 = νk −1 = −λ1 L 2 ¯ζk −1 − νk−2 ¯ 2 × i i i i × sign(ζk i −1 − νk i −2 ) + ζk i , ζ˙ k i = −λ0 L i sign(ζk i − νk i −1 ),
(3)
( j)
where ζ j is the estimation of the true derivative yi ( t). This differentiator provides for the finite-time exact differentiation under ideal conditions of exact measurement in continuous time. The only information needed ( k +1) is an a-priory known upper bound L i for | yi i |. Then a parametric sequence {λ j } > 0, j = 0, 1, . . . , k i , is recursively built, which provides for the convergence of the differentiators for each order k i . In particular, the parameters λ0 = 1.1, λ1 = 1.5, λ2 = 2, λ3 = 3, λ4 = 5, λ5 = 8 are enough till the 5-th differentiation order. With discrete sampling the differential equations are replaced by their Euler approximations. In addition, the differentiator (3) provides for the best possible order of accuracy in the presence of measurement noises [5].
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CONFIDENTIAL. Limited circulation. For review only Proposition 1. Suppose that the system (1) satisfies A1, A2 and A3. Then, for any large enough constant L i > 0, the HOSM differentiator (3) guarantees that ( j) ζ j ( t) = yi ( t) after a finite-time transient. Moreover, in the presence of measurement noise uniformly bounded by δ, the accuracies ( j)
j
|ζ j − yi | = O (L k i +1 δ
k i +1− j k i +1
),
j = 0, . . . , k i ,
are obtained after a finite-time transient. Using one HOSM differentiator per output, it is possible to obtain all the required derivatives of the measured output. Remark 1. The reliable derivatives’ estimations are available after a finite-time transient. Thus, a controller is to be applied only after the transient ends, avoiding unnecessarily increasing the disturbances in the system. This dwell-time strategy improves the stability and performance properties in the output-feedback control of strongly nonlinear systems [2, 3]. Detecting the convergence of the HOSM differentiators is done by checking how close to zero the differences |ζ0 ( t) − yi ( t)| are, for i = 1, . . . , p, as shown in [9]. This last criterion can be applied in the presence of measurement noise and discretization. It should also be noted that precluding finite-time escape of the trajectories (Assumption A3) is fundamental for the existence of the constants L i , i = 1, . . . , p, required for the HOSM differentiators. When this last assumption is not satisfied, it is still possible to use the HOSM differentiator but requires additional modifications [3]. 4. Control design Once the state of the system has been recovered as shown in the last section, the control problem can be solved by designing u( t) to keep the identity z( t) ≡ 0 after a finite-time transient. Proposition 2. For any input w(·) and initial state x0 , the condition z( t) ≡ 0, t ≥ 0 implies that x( t) ≡ 0, t ≥ 0. Since the flat output z( t) ∈ Rm has vector relative degree, each one of its components can be independently controlled by one distinct input. Let us recall that an undisturbed system (1) is said to be flat (or linearizable by dynamic state feedback) if there exists a feedback ξ˙ = M ( x, ξ) + N ( x, ξ)v, ξ( t) ∈ Rm¯ ;
u = α( x, ξ) + β( x, ξ)v, (4) where v( t) ∈ Rm is the “new input”, and a diffeomorphism η = Ξ( x, ξ) such that the closed loop system reads as η˙ = A η + Bv, where the pair ( A, B) is composed by Brunovsky blocks with only one input per block [12].
Equivalently, selecting each component of the flat output z i , i = 1, . . . , m, as the first variable of each Brunovsky block, the undisturbed system looks like a chain of integrators (n i )
zi
= vi ,
i = 1, . . . , m,
(5)
where n i , i = 1, . . . , m, is a list of integers. Note that unlike [8], the relative degree of the measured output plays no role in the control design. The only requirement is the invertibility of the augmented system with respect to the new input v. When the disturbance w is present, it perturbs system (5). The disturbance first appears in the derivative of the flat output corresponding to its relative degree r i , (r ) i.e., z i i = a i ( x) + b i ( x)( u + w), i = 1, . . . , m. From then on, it needs to be differentiated ( n i − r i )-times to linearize the system (n i )
zi
= vi +
Si X `=0
γ` ( x, ξ)w(`) ,
(6)
where S i = n i − r i and γ` ( x, ξ), ` = 0, . . . , S i , are known functions that can be computed from the parameters of the system. Therefore, the problem is now to design each input v i to provide exact finite-time stability of the output z i . For doing this, we shall assume that the required S = max{S 1 , . . . , S m } derivatives of the disturbance are all uniformly bounded (Assumption A2). Theorem 2. Consider system (1) under assumptions A2, A3 and A4. Then with any α i > 0 large enough and ² i > 0, the controller (4) with v = (v1 , · · · , vm ) selected as ( n i −1)
v i = −α i Φ i ( x, ξ) H n i ( z i , · · · , z i
),
i = 1, . . . , m,
(7)
where z i is the first variable of the i -th Brunovsky block and H n i is an n i -th order HOSM control algorithm with gain function Φ i selected as
Φ i ( x, ξ) = ² i +
S X `=0
γ` ( x, ξ)W + ,
i = 1, . . . , m,
provides for the finite-time stabilization of the origin. Moreover, if assumption A1 is also met and the function F in (2) has been determined then finite-time stability is preserved when all the variables in (7) are replaced by the ones estimated using Proposition 1. In expression (7), H n i denotes an n i -th order HOSM controller. It can be chosen from a predefined list of controllers using solely its order, see [15] and Section 5 for an example with n 1 = 1, n 2 = 2. When the function F appearing in (2) and the feedback linearization of (4) are globally valid, then our approach can provide for semi-global finite-time stability of the origin. This fact comes from noticing that given a ball of initial conditions for the system from which we want to stabilize it, it is always possible to design the gains L i , i = 1, . . . , p, large enough ensuring that the HOSM differentiators
Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
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CONFIDENTIAL. Limited circulation. For review only converge if the trajectories do not scape a larger ball, see Proposition 1. The proposed methodology can still be applied only when Assumption A3 is satisfied in the sense that the system’s trajectories exist for all t ≥ 0. In such a case, notice that for every R > 0 and T > 0, there exists r (R, T ) > 0 such that k x( t)k ≤ R for all t ∈ [0, T ] if k x0 k ≤ r for the uncontrolled system. According to Remark 1, the controller is turned on once the observer has converged, so it provides stability of the origin. Therefore, it always exists r 0 such that k x( t)k ≤ R for all t ≥ 0 if k x0 k ≤ r 0 when the state feedback control is applied. Hence if the observer is designed to be convergent when the trajectories are such that k x( t)k ≤ R and also such that it converges in a time T0 < T , when the trajectories are still in the set k x0 k ≤ r 0 , then the controller will assure that they satisfy k x( t)k ≤ R for all t ≥ 0. Hence, finite-time stability of the origin is obtained provided that the initial conditions of the system are small enough, i.e. k x0 k ≤ r ( r 0 , T0 ). These conditions for the observer can be always satisfied if the gain L i , i = 1, . . . , p, of each differentiator is sufficiently large [5]. This fact is illustrated in the example of Section 5. A constructive procedure to obtain the dynamic compensator of equation (4) is provided by the Inversion Algorithm [13], see also the example in Section 5. Remark 2. For a flat system the input can also be written as a function of the flat output and its derivatives. Together with the strong observability with respect to the measured output, this allows estimating the disturbance using the measured output. The only requirement is to take one more time-derivative. Due to the use of the HOSM differentiator, this adds restrictions in the disturbance and the trajectories of the system: it must be slow enough. Then it becomes possible to compensate the disturbance without applying a discontinuous HOSM controller. This last strategy has been recently explored in [16]. 5. Simulation Example Consider the following system x˙ 1 = x2 + u 1 + w1 , x˙ 2 = x33 + x3 + u 1 + w1 , x˙ 3 = sin( x1 x2 ) + x3 cos( x3 ) + u 2 + w2 ,
y1 = y2 =
x1 , x2 ,
with w1 = w2 = 0.5 sin( t) + 0.5 square(0.2 t) the sum of a sinusoid and a square signal of amplitude 0.5 and 0.2 seconds of period. The system above satisfies A3 only in the sense that their trajectories are bounded for each time instant. This happens since the x33 does not appear in the equation for x3 . The method of [8] is not applicable since y does not have vector relative degree. Using the algorithm of Section 3.1 yields £ ¤ £ ¤ x1 x2 x3 = y1 y2 φ1 ( y˙ 2 − y˙ 1 − y2 ) , where ´2/3 ³ p −23 + 2 9θ + 12 + 81θ 2 , φ1 (θ ) = ³ ´1/3 p 62/3 9θ + 12 + 81θ 2 1/3
1/3
and the required derivative was obtained using a firstorder HOSM differentiator with gain L = 250. We shall compare the control design for two distinct choices of flat output. For the first choice, we select the measured output as the flat output. In this case, it turns out that a dynamic compensator is required and that the first disturbance needs to be smooth. Since the selected disturbance does not satisfy this last requirement , it prevents applying the proposed methodology using the measured output as flat output. For the second selection we compute a new flat output that linearizes the system using a static compensator. This second choice outperforms the first one: both disturbances only need to be uniformly bounded, the controller is simpler and static, and it requires less derivatives. The simulation results for this second choice are also presented. For the first selection we use the measured output as flat output. Computing the second derivative of each output and using the dynamic compensator ξ˙ = −ξ − x33 + x3 + v1 with £ ¤ u 1 = ξ, u 2 = (3 x32 + 1)−1 v2 − γ( x) − v1 + ξ + x33 − x3 , and γ( x) = (3 x32 + 1)[sin( x1 x2 ) + x3 cos( x3 )] yields
y¨ 1 = v1 + w1 + w˙ 1 ,
y¨ 2 = v2 + (3 x32 + 1)w2 + w˙ 1 .
Therefore, we require that w1 , w2 and w˙ 1 are all uniformly bounded. Since w1 ( t) contains a square signals, w˙ 1 is not uniformly bounded as a function of time and prevents applying the proposed methodology. For the second selection, we apply the procedure of [13, pp. 122] to find a flat output that linearizes the system using a static compensator. If possible, the only restriction would be that the disturbance is uniformly bounded. For finding the desired flat output, compute the subspaces H1 = span{d x}, H2 = span{d x1 − d x2 } and H3 = {0}. This fact confirms that the system can be linearized using a static compensator. Then set ω2 = d x1 − d x2 and compute ω˙ 2 = d(δ x1 ) − d(δ x2 ) = d x2 − (3 x32 + 1) d x3 . Then one may select ω1 = d x2 . This yields H1 = span{d x} = span{ω1 , ω2 , ω˙ 2 }, and therefore span{d z1 , d z2 } = span{ω1 , ω2 } is a set of flat outputs. That is z1 = x2 and z2 = x1 − x2 . Notice that z can be obtained using y through a nonsingular transformation, so the system is also strongly observable with respect to the output z. Computing the input-output dynamics for these two outputs one obtains
z˙ 1
=
x33 + x3 + u 1 + w1 ,
z¨ 2
=
x33 + x3 − γ( x) + u 1 + w1 − (3 x32 + 1)( u 2 + w2 ),
and setting
u 1 = − x33 − x3 +v1 , u 2 = −(3 x32 +1)−1 (− x32 − x3 −γ( x)− u 1 +v2 ), yields z˙ 1 = v1 + w1 , z¨ 2 = v2 − (3 x32 + 1)w2 . Now design the HOSM controller by selecting v1 = −α1 H1 ( z1 ) =
Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
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CONFIDENTIAL. Limited circulation. For review only −α1 sign( z1 ) and v2 = −α2 Φ2 ( x3 ) H2 ( z2 , z˙ 2 ) where
H2 ( z2 , z˙ 2 ) =
1 2
z˙ 2 + | z2 | sign( z2 ) 1
,
| z˙ 2 | + | z2 | 2
is a Quasi-Continous algorithm [15] with α1 = 1, α2 = 4.5, Φ2 ( x3 ) := 3 x32 + 0.5. The controller is switched on only when the convergence of the differentiator (i.e., the UIO) is detected according to [9]. All simulations were carried out using Euler’s method with 10−3 [s] of time step. Two simulations studies were carried out, see Figure 1. The first one presents the simulation results in the absence of noise. It shows the finite-time exact convergence of the state trajectories to zero. Secondly, measurement noises were added to both measured outputs. The noises were generated using Simulink’s “Uniform Random Number” generator with maximum amplitudes of 0.005 and 0.01 for y1 and y2 , respectively. The simulation results confirms that the robustness of HOSM controllers and differentiators to noises guarantees keeping a bounded state error despite the presence of noise. 6. Conclusions A methodology for the design of output feedback HOSM controllers for (dynamic) feedback linearizable disturbed nonlinear systems was presented. It provides finite-time exact state stabilization using only output feedback, provided that the system is algebraically strongly observable and the disturbances are smooth enough. The introduced notion of “strong” observability allows recovering the state by using only the output and its derivatives. When the derivatives are estimated using the HOSM differentiator, an Unknown Input Observer that converges in finite-time is obtained. Acknowledgements The authors would like to express their gratitude to Prof. Claude H. Moog (IRCCyN, France) for his assistance in the construction of the algorithm of Section 3-A. We also gratefully acknowledge the financial support from projects PAPIIT 17211 and IN111012, CONACyT 56819, 132125, 51244 and CVU 229959, Fondo de Colaboración del II-FI, UNAM, IISGBAS-165-2011.
Appendix: Proofs of Theorems and Propositions P ROOF (O F T HEOREM 1). Under the assumption that the vector fields are meromorphic, it is possible to use differential-algebraic tools to derive structural properties of the system [13]. This approach introduces finitedimensional spaces of 1-forms over the field of meromorphic functions K of x: Y k := spanK {d y, · · · , d y(k) }, X = spanK {d x}, W = spanK {d w( j) , j ≥ 0}. For our purposes, this means that the statement rank M k ( x) = n holds except (possibly) in a set of zero Lebesgue measure. Under this framework, Definition 1 of algebraic
strong observability is equivalent to the condition X ∩ Y k = X with k = max{k1 , · · · , k p }. By construction span{d M i ( x)} = X ∩ Y i . Notice that equality is guaranteed by construction. This way vectors M i ( x) define a chain of subspaces (X ∩ Y 0 ) ⊂ · · · ⊂ (X ∩ Y i ) ⊂ (X ∩ Y i+1 ) ⊂ · · · that is non-decreasing and bounded by X . Therefore, the sequence of subspaces has a unique limit. Since algebraic strong observability implies that X ∩ Y k = X for some finite k, then the upper bound of the sequence is attained and must coincide with the limit of such sequence. Now, since span{d M k ( x)} = X ∩ Y k = X this means that rank d M k ( x) = n. Moreover, the condition X ∩ Y i−1 = X ∩ Y i would imply that X ∩ Y i+1 = X ∩ Y i and that X ∩ Y i−1 is another limit, then the dimension of each subspace should increase by at least one at each step of the algorithm. Then, if we begin with a dimension p, at most n − p steps of the algorithm are required to reach the limit X ∩ Y n− p = X . Finally, since rank d M n− p ( x) = n then the Inverse Function Theorem guarantees that there exists, at least locally, an inverse for M n− p as claimed. P ROOF (O F P ROPOSITION 1). Due to Assumptions A2 and A3, x˙ ( t) is uniformly bounded as a function of time. By Assumption A1, the function F in formula (2) exists. From these two facts and taking one time deriva( k +1) tive of F and equating with x˙ ( t) obtain that yi i ( t), i = 1, . . . , p, are all uniformly bounded. Therefore, there exists large enough constants L i > 0, i = 1, . . . , p, such that each HOSM differentiator converges. This fact and the accuracy under measurement noise were proven in [5]. P ROOF (O F P ROPOSITION 2). By definition, the system is also algebraically strongly observable with respect to the flat output z( t). Now, on one hand, since q(0) = 0, the condition x( t) ≡ 0 implies that z( t) ≡ 0. Any other state trajectory satisfying z( t) ≡ 0 contradicts the strong observability of the system. P ROOF (O F T HEOREM 2). Using the diffeomorphism η = Ξ( x, ξ) and the dynamic compensator (4) yields a set of m disturbed chain of integrators (6). Moreover, the output z has vector relative degree with respect to v. This way the HOSM control design for v can be made exactly as shown in [9] to ensure that z( t) ≡ 0 after a finite-time T c . From Proposition 2, one concludes that x( t) ≡ 0, t ≥ T c . This completes the proof of the first part of the claim. For the second part, we assume that the system is algebraically strongly observable (A1). Then Proposition 1 ensures the finite-time exact estimation of the derivatives and hence, that controller (7) constructed using the estimated variables is applied only when it coincides with the controller constructed using the true variables. After the controller is applied, the correct estimation of the derivatives using the HOSM differentiator is preserved if theirs gains L i were large enough, as Proposition 1 ensures.
Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
7
CONFIDENTIAL. Limited circulation. For review only 1
Observation error
2
3.5
1 3
u1
0
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−1 −2 −3 −4
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time[s] 1
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State trajectories
2
0
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−2
0.8 0.6 0.4 0.2
−1 0
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Observation error
2 1
3
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time[s]
time[s]
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convergence
State trajectories
0
5
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0
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−2
0.8 0.6 0.4 0.2
−1 0
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a)
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0.5
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time[s]
c)
Figure 1: Simulation result with (bottom) and without (top) measurement noises. a) System trajectories x( t). b) Control input. c) Top: estimation error for state x3 ( t); bottom: 1 indicates that the differentiator (observer) has converged.
BIBLIOGRAPHY [1] T. Oliveira, A. Peixoto, and L. Hsu, “Sliding mode control of uncertain multivariable nonlinear systems with unknown control direction via switching and monitoring function,” Automatic Control, IEEE Transactions on, vol. 55, no. 4, pp. 1028 –1034, april 2010. [2] T. R. Oliveira, A. J. Peixoto, R. R. Costa, and L. Hsu, “Dwell-time and disturbance monitoring for peaking avoidance and performance improvement in high-gain observer based sliding mode control,” Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms, vol. 17, no. 6, pp. 839–874, 2010. [3] T. R. Oliveira, A. J. Peixoto, and L. Hsu, “Peaking free output-feedback exact tracking of uncertain nonlinear systems via dwell-time and norm observers,” International Journal of Robust and Nonlinear Control, vol. 23, no. 5, pp. 483–513, 2013. [4] A. J. Peixoto, T. R. Oliveira, L. Hsu, F. Lizarralde, and R. R. Costa, “Global tracking sliding mode control for a class of nonlinear systems via variable gain observer,” International Journal of Robust and Nonlinear Control, vol. 21, no. 2, pp. 177–196, 2011. [5] A. Levant, “High-order sliding modes: differentiation and output feedback control,” International Journal of Control, vol. 76, no. 9-10, pp. 1924–041, nov 2003. [6] F. Dinuzzo and A. Ferrara, “Higher order sliding mode controllers with optimal reaching,” IEEE Transactions on Automatic Control, vol. 54, no. 9, pp. 2126–2136, 2009.
[7] D. V. Efimov, A. Zolghadri, and T. Raïssi, “Actuator fault detection and compensation under feedback control,” Automatica, vol. 47, no. 8, pp. 1699–1705, 2011. [8] M. Defoort, T. Floquet, A. Kokosy, and W. Perruquetti, “A novel higher order sliding mode control scheme,” Syst. Control Lett., vol. 58, no. 2, pp. 102–108, 2009. [9] M. T. Angulo, L. Fridman, and A. Levant, “Outputfeedback finite-time stabilization of disturbed lti systems,” Automatica, vol. 48, no. 4, pp. 606 – 611, 2012. [10] M. Hautus, “Strong detectability and observers,” Linar Algebra and its Applications, vol. 50, pp. 353–468, apr 1983. [11] A. Filippov, Differential equations with discontinuous right-hand side. London: Kruwler, 1960. [12] H. Sira Ramírez and S. Agrawal, Differentially Flat Systems, ser. Control Engineering. Marcel Dekker, 2004. [13] G. Conte, C. Moog, and A. Perdon, Algebraic Methods for Nonlinear Control Systems, 2nd ed. Springer, may 2007. [14] B. P. Molinari, “A strong controllability and observability in linear multivariable control,” IEEE Transactions on Automatic Control, vol. 91, no. 5, pp. 761 – 764, 1976. [15] A. Levant and A. Michael, “Adjustment of high-order sliding-mode controllers,” International Journal of Robust and Nonlinear Control, vol. 19, no. 15, pp. 1657– 1672, Nov. 2009. [16] A. Ferreira, F. Bejarano, and L. Fridman, “Robust control with exact uncertainties compensation: With or without chattering?” Control Systems Technology, IEEE Transactions on, vol. 19, no. 5, pp. 969 – 975, 2011.
Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
Output-feedback finite-time stabilization of disturbed feedback linearizable nonlinear systems Marco Tulio Anguloa,∗, Leonid Fridmanb , Jaime A. Morenoc a Facultad de Ingeniería, UAQ, México (e-mail: [email protected]) b Departamento de Control y Robótica, UNAM, México ([email protected]) c Eléctrica y Computación, Instituto de Ingeniería, UNAM, México (e-mail: [email protected])
Abstract A novel methodology for designing multivariable High-Order Sliding-Mode (HOSM) controllers for disturbed feedback linearizable nonlinear systems is introduced. It provides for the finite-time stabilization of the origin of the state-space by using output feedback. Only the additional assumptions of algebraic strong observability and smooth enough matched disturbances are required. The control problem is solved in two consecutive steps: firstly, designing an observer based on the measured output and, secondly, designing of a full-state controller computed from a new virtual output with vector relative degree. The introduced notion of algebraic strong observability allows recovering the state of the system using the measured output and its derivatives. By estimating the required derivatives through the HOSM differentiator, a finite-time convergent observer is constructed. Keywords: Sliding-mode control.
1. Introduction The design of robust controllers is an important topic in automatic control theory. Uncertainty is also manifested by only measuring a subset of the state variables of the system, namely, its “measured output”. In such scenario, Sliding-Mode (SM) output-feedback based controllers have shown to be very successful [1, 2, 3, 4]. Moreover, modern systems frequently mix continuos and discrete event dynamics for which the stabilization problems are much more intricate. However, hybrid systems with strictly positive dwell-time can be effectively controlled if the controller accomplishes the control objective before the next switching or impulse time. Therefore, robust output-feedback controllers providing finite-time state stabilization become relevant. High Order SMs (HOSMs) are useful in this context providing for the finite-time exact1 output stabilization using only output feedback [5]. In addition, in the presence of noise and sampling, HOSMs offer better accuracy than first-order SM. However, they were originally designed only for single-input single-output systems, see, e.g., [6, 7]. In [8], HOSM controllers are extended to Multi-Input Multi-Output (MIMO) nonlinear systems under the assumption of vector relative degree with respect to the
∗ Corresponding author: [email protected], May 3, 2013. 1 Exactness is more than robustness: the effect of (matched) distur-
bances is completely eliminated after a finite-time transient.
measured output. This last assumption also requires that the system has the same number of inputs as outputs. Recently, in [9], the authors introduced a methodology for the design of HOSM controllers for MIMO disturbed linear systems under necessary conditions: a known (affine) bound on the disturbance, controllability and strong observability [10]. The proposed methodology allows constructing an output-based HOSM controller guaranteeing the finite-time exact convergence of the whole state of the system. This brief extends the methodology introduced in [9] in the context of linear systems to a class of MIMO nonlinear systems. As in the linear case, the measured output does not necessarily has vector relative degree. In this form, HOSMs can be applied to a larger class of systems when compared to the strategy of [8]. Our approach requires introducing a suitable concept of observability despite disturbances. The notion of “algebraic strong observability” corresponds to the possibility of reconstructing the state as a function only of the measured output and a finite number of its derivatives. By using the HOSM differentiator [5] to estimate the required derivatives, a finite-time convergent observer is obtained. By combining the proposed observer with HOSMs controllers and dynamic feedback linearization the problem of exact finite-time state stabilization of nonlinear systems is solved. The proofs of all Theorems are collected in the Appendix. Main contributions. 1) An algorithm to construct an unknown input observer for nonlinear systems is pre-
Preprint submitted to Automatica 1 Received May 3, 2013 07:55:45 PST
2
CONFIDENTIAL. Limited circulation. For review only sented based on the notion of algebraic strong observability and the use of HOSM differentiators; 2) by using the proposed observer, a novel HOSM-based output feedback control strategy for the finite-time state stabilization of (dynamic) feedback linearizable nonlinear systems with bounded matched disturbances is presented. 2. Problem statement Consider
x˙ = f ( x) + g( x)[ u + w],
y = h( x);
x(0) = x0 ,
(1)
where x( t) ∈ Rn , u( t) ∈ Rm , w( t) ∈ Rm and y( t) ∈ R p are the state, control input, disturbance and measured output, respectively. The vector fields f , g and the function h are assumed to be smooth. The control objective is to design the control input u( t), depending only on the output y( t), to provide finite-time state stability at zero despite the presence of the disturbance w( t). In order to present the assumptions we have made to solve the control problem, let us first introduce the following concept: Definition 1. System (1) is said to be algebraically strongly observable with respect to the output y( t) ∈ R p , if there exists a function F and integers k i , i = 1, . . . , p, such that (k1 )
x = F ( y1 , . . . , y1
(k p )
, · · · , yp , . . . , yp
).
(2)
Strong observability was introduced by Hautus [10] for linear systems to solve the Unknown Input Observer (UIO) design problem. It allows recovering the state of the system through the knowledge of the output and its derivatives only, irrespectively of the input. The notion of algebraic strong observability is a natural extension to the case of nonlinear systems. In Section 3, this property will be shown to be instrumental for the design of a UIO for nonlinear systems. The following assumptions about the system will be made throughout this paper: (A1) system (1) is algebraically strongly observable with respect to the measured output y;
A flat system is feedback linearizable and, moreover, its state and input can be written as a function of the flat output and its derivatives [12]. Without loss of generality, we assume that q(0) = 0. Note that, in general, the flat output z( t) may not coincide with the measured output y( t). The degree S of required smoothness for the disturbance depends on the “order” of the dynamic compensator used to linearize the system, as shown in Section 4. In the particular case when the system can be linearized by a static compensator, the disturbance needs to be only uniformly bounded. The approach we follow to solve the problem involves two steps. First, based on A1, we present an algorithm to compute the function F appearing in equation (2). This allows constructing a UIO for the system once the required derivatives are estimated. With assumptions A2 and A3, it becomes possible to estimate such derivatives using the HOSM differentiator [5]. In the second step, by using A4, it is shown that the finite-time state stabilization problem is equivalent to the finitetime stabilization of the flat output. This way, by using the estimated state of the previous step to evaluate the flat output, a multivariable HOSM controller that semiglobally stabilizes the flat output to zero in finite-time is presented. 3. Construction of the Unknown Input Observer Once the function F in (2) is known, the state can be determined when the derivatives of the output are estimated. By means of the HOSM differentiator, these derivatives can be estimated exactly and in finite time. In addition, the HOSM differentiator provides an estimation that is robust to measurement noise and has the best order of precision [5]. The following Subsection presents an algorithm to compute the function F for a class of nonlinear systems. Subsection 3.2, analyzes the use of the HOSM differentiator to estimate the required derivatives. 3.1. An Algorithm to construct an Unknown Input Observer. Let us consider the following additional assumption:
(A5) The Lie derivatives L g L kf h( x) are constant for k = (A2) for a given degree of smoothness S ∈ Z+ , there ex1, . . . , n − 1. ists a constant W + such that kw(k) ( t)k ≤ W + , ∀ t ≥ 0 This assumption means that in the time derivatives and all k = 0, . . . , S ; of the measured output, the matrix multiplying the in(A3) for all initial conditions x0 ∈ Rn and disturbances put is constant. This allows writing a constant orthogow(·) satisfying A2, the solution2 to system (1) with nal to such matrix. Assumption A5 is only introduced to u(·) ≡ 0 exists for all t ≥ 0 and remains upper- simplify the presentation of the algorithm: in the genbounded by a possibly unknown number; eral case when A5 is not satisfied, the orthogonal can be computed in the same way but now depending on (A4) there exists a function q : Rn → Rm such that z = the state. q( x) is a flat output for system (1). Let b⊥ denote a left annihilator for matrix b (i.e. ⊥ b b = 0). The presented algorithm computes the func2 Solutions to differential equations (and their associated inclution F introduced in equation (2), enabling the consions) are understood in Filippov’s sense [11]. struction of a UIO for system (1). Our algorithm is the
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3
CONFIDENTIAL. Limited circulation. For review only “dual” of the standard Structure Algorithm (see [13, pp. 76]), since it looks for directions orthogonal to the input. Nevertheless, it is closely related to the algorithm presented in [14] for linear systems. Step 0. Set yˆ 0 := y and M0 ( x) := h( x). ˙ 0 ( x) = y˙ˆ 0 = L f M0 ( x) + Compute M Step 1. ˙ 0 ( x) = L g M0 ( x)w := a 1 ( x)+ b 1 ( u+w), introduce yˆ˙ 1 := b⊥ M 1
aˆ 1 ( x) and set ·
M1 ( x) :=
yˆ 0 y˙ˆ 1
¸
· =
h( x) aˆ 1 ( x)
¸
.
Step k + 1. Compute ˙ k ( x) = M
h
y˙ˆ 0
···
yˆ k(k+1)
iT
:= a k+1 ( x) + b k+1 ( u + w),
˙ ( x) = aˆ k+1 ( x), and set M introduce yˆ k(k++11) := b⊥ k+1 k yˆ 0 h( x) .. .. = M k+1 ( x) := . . . ( k+1) aˆ k+1 ( x) yˆ k+1 By applying the algorithm, one is able to compute a sequence of equations of the form [ yˆ 0 , y˙ˆ 1 , . . . , yˆ k(k) ]T = M k ( x) for k ≥ 1. The algorithm converges in the following sense. Theorem 1. Suppose that the vector fields f , g and the function h are meromorphic, rank h( x) = p and the system is algebraically strongly observable. Under these conditions rank d M n− p ( x) = n for almost all x ∈ Rn and the function F := M n−−1 p as in equation (2) exists locally. Recall that a meromorphic function is the quotient of two analytic functions. Computing the inverse function of M k−1 is, in general, a difficult problem. For this reason, it is useful to perform all possible simplifications at each step of the algorithm, replacing functions of the state by functions of the output. We have illustrated this idea in the example below by introducing the additional variables Nk , k ≥ 1. Example 3.1. Consider the following system not having vector relative degree with respect to the measured output:
x˙ 1 = x˙ 2 =
x1 x2 + w1 , x33 + w1 ,
x˙ 3 = x˙ 4 =
x4 (1 + x1 ) + w1 , sin( x1 x2 ) + w2 ,
y1 = y2 =
x1 , x2 .
Applying the algorithm yields: ·
¸ · ¸ x1 yˆ 0,1 Step 0: Set yˆ 0 = y and N0 ( x) = = yˆ 0 = . x2 yˆ 0,2 Trivially simplify it and set M0·( x) = N0¸( x).· ¸ ˙ 0 ( x) = x1 3x2 + 1 0 w = Step 1: Compute M 1 0 x £ 3 ¤ ⊥ − 1 1 a 1 ( x) + b w . Thus b = , introduce 1 £ 1 ¤ ˙ 0 ( x) = x3 − x1 x2 and set N1 ( x) := y˙ˆ 1 = −1 1 M 3 £ ¤T yˆ 0 y˙ˆ 1 . Proceed to simplify it as # " yˆ 0 £ ¤T ¢ 1 = x1 x2 x3 M 1 ( x ) := ¡ . ˙yˆ 1 + yˆ 0,1 yˆ 0,2 3
Step 2: Compute
x1 x2 1 ˙ 1 ( x) = + 1 x33 M 1 x4 (1 + x1 )
0 0 w := a 2 ( x) + b 2 w. 0
£ ¤ −1 0 1 Then b⊥ and introduce y¨ˆ 2 = 2 =¤ £ ˙ 1 ( x) = x4 (1 + x1 ) − x1 x2 , −1 0 1 M iT h ¡ ¢1 and simplify Set N2 ( x) = yˆ 0 , y˙ˆ 1 + yˆ 0,1 yˆ 0,2 3 , y¨ˆ 2 it as x1 yˆ 0 ¢ 1 x2 ¡ 3 ˙ M 2 ( x ) := = x3 . yˆ 1 ¨+ yˆ 0,1 yˆ 0,2 yˆ 2 + yˆ 0,1 yˆ 0,2 1+ yˆ 0,1 x4
Since rank d M2 ( x) = 4 the algorithm stops and the system is algebraically strongly observable. In addition, we have obtained an expression of all states as a function of the measured output and its derivatives.O This last example also shows that the condition of vector relative degree (with respect to the measured output) considered in [8] is not necessary for the system to be algebraic strongly observable. 3.2. Estimating the derivatives: a finite-time convergent UIO. Once equation (2) has been determined using the procedure described in the previous Subsection, the next step to construct the UIO is to evaluate the required derivatives of the measured output. This Subsection briefly describes the use of the HOSM differentiator to estimate them. Let yi ( t) be a component of the measured output to be differentiated k i -times. The k i -th order HOSM differentiator [5] takes the form 1
ki
k +1 ζ˙0 = ν0 = −λk i L i i |ζ0 − yi | k i +1 sign(ζ0 − yi ) + ζ1 , .. . 1 ¯ ¯1 ˙ζk −1 = νk −1 = −λ1 L 2 ¯ζk −1 − νk−2 ¯ 2 × i i i i × sign(ζk i −1 − νk i −2 ) + ζk i , ζ˙ k i = −λ0 L i sign(ζk i − νk i −1 ),
(3)
( j)
where ζ j is the estimation of the true derivative yi ( t). This differentiator provides for the finite-time exact differentiation under ideal conditions of exact measurement in continuous time. The only information needed ( k +1) is an a-priory known upper bound L i for | yi i |. Then a parametric sequence {λ j } > 0, j = 0, 1, . . . , k i , is recursively built, which provides for the convergence of the differentiators for each order k i . In particular, the parameters λ0 = 1.1, λ1 = 1.5, λ2 = 2, λ3 = 3, λ4 = 5, λ5 = 8 are enough till the 5-th differentiation order. With discrete sampling the differential equations are replaced by their Euler approximations. In addition, the differentiator (3) provides for the best possible order of accuracy in the presence of measurement noises [5].
Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
4
CONFIDENTIAL. Limited circulation. For review only Proposition 1. Suppose that the system (1) satisfies A1, A2 and A3. Then, for any large enough constant L i > 0, the HOSM differentiator (3) guarantees that ( j) ζ j ( t) = yi ( t) after a finite-time transient. Moreover, in the presence of measurement noise uniformly bounded by δ, the accuracies ( j)
j
|ζ j − yi | = O (L k i +1 δ
k i +1− j k i +1
),
j = 0, . . . , k i ,
are obtained after a finite-time transient. Using one HOSM differentiator per output, it is possible to obtain all the required derivatives of the measured output. Remark 1. The reliable derivatives’ estimations are available after a finite-time transient. Thus, a controller is to be applied only after the transient ends, avoiding unnecessarily increasing the disturbances in the system. This dwell-time strategy improves the stability and performance properties in the output-feedback control of strongly nonlinear systems [2, 3]. Detecting the convergence of the HOSM differentiators is done by checking how close to zero the differences |ζ0 ( t) − yi ( t)| are, for i = 1, . . . , p, as shown in [9]. This last criterion can be applied in the presence of measurement noise and discretization. It should also be noted that precluding finite-time escape of the trajectories (Assumption A3) is fundamental for the existence of the constants L i , i = 1, . . . , p, required for the HOSM differentiators. When this last assumption is not satisfied, it is still possible to use the HOSM differentiator but requires additional modifications [3]. 4. Control design Once the state of the system has been recovered as shown in the last section, the control problem can be solved by designing u( t) to keep the identity z( t) ≡ 0 after a finite-time transient. Proposition 2. For any input w(·) and initial state x0 , the condition z( t) ≡ 0, t ≥ 0 implies that x( t) ≡ 0, t ≥ 0. Since the flat output z( t) ∈ Rm has vector relative degree, each one of its components can be independently controlled by one distinct input. Let us recall that an undisturbed system (1) is said to be flat (or linearizable by dynamic state feedback) if there exists a feedback ξ˙ = M ( x, ξ) + N ( x, ξ)v, ξ( t) ∈ Rm¯ ;
u = α( x, ξ) + β( x, ξ)v, (4) where v( t) ∈ Rm is the “new input”, and a diffeomorphism η = Ξ( x, ξ) such that the closed loop system reads as η˙ = A η + Bv, where the pair ( A, B) is composed by Brunovsky blocks with only one input per block [12].
Equivalently, selecting each component of the flat output z i , i = 1, . . . , m, as the first variable of each Brunovsky block, the undisturbed system looks like a chain of integrators (n i )
zi
= vi ,
i = 1, . . . , m,
(5)
where n i , i = 1, . . . , m, is a list of integers. Note that unlike [8], the relative degree of the measured output plays no role in the control design. The only requirement is the invertibility of the augmented system with respect to the new input v. When the disturbance w is present, it perturbs system (5). The disturbance first appears in the derivative of the flat output corresponding to its relative degree r i , (r ) i.e., z i i = a i ( x) + b i ( x)( u + w), i = 1, . . . , m. From then on, it needs to be differentiated ( n i − r i )-times to linearize the system (n i )
zi
= vi +
Si X `=0
γ` ( x, ξ)w(`) ,
(6)
where S i = n i − r i and γ` ( x, ξ), ` = 0, . . . , S i , are known functions that can be computed from the parameters of the system. Therefore, the problem is now to design each input v i to provide exact finite-time stability of the output z i . For doing this, we shall assume that the required S = max{S 1 , . . . , S m } derivatives of the disturbance are all uniformly bounded (Assumption A2). Theorem 2. Consider system (1) under assumptions A2, A3 and A4. Then with any α i > 0 large enough and ² i > 0, the controller (4) with v = (v1 , · · · , vm ) selected as ( n i −1)
v i = −α i Φ i ( x, ξ) H n i ( z i , · · · , z i
),
i = 1, . . . , m,
(7)
where z i is the first variable of the i -th Brunovsky block and H n i is an n i -th order HOSM control algorithm with gain function Φ i selected as
Φ i ( x, ξ) = ² i +
S X `=0
γ` ( x, ξ)W + ,
i = 1, . . . , m,
provides for the finite-time stabilization of the origin. Moreover, if assumption A1 is also met and the function F in (2) has been determined then finite-time stability is preserved when all the variables in (7) are replaced by the ones estimated using Proposition 1. In expression (7), H n i denotes an n i -th order HOSM controller. It can be chosen from a predefined list of controllers using solely its order, see [15] and Section 5 for an example with n 1 = 1, n 2 = 2. When the function F appearing in (2) and the feedback linearization of (4) are globally valid, then our approach can provide for semi-global finite-time stability of the origin. This fact comes from noticing that given a ball of initial conditions for the system from which we want to stabilize it, it is always possible to design the gains L i , i = 1, . . . , p, large enough ensuring that the HOSM differentiators
Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
5
CONFIDENTIAL. Limited circulation. For review only converge if the trajectories do not scape a larger ball, see Proposition 1. The proposed methodology can still be applied only when Assumption A3 is satisfied in the sense that the system’s trajectories exist for all t ≥ 0. In such a case, notice that for every R > 0 and T > 0, there exists r (R, T ) > 0 such that k x( t)k ≤ R for all t ∈ [0, T ] if k x0 k ≤ r for the uncontrolled system. According to Remark 1, the controller is turned on once the observer has converged, so it provides stability of the origin. Therefore, it always exists r 0 such that k x( t)k ≤ R for all t ≥ 0 if k x0 k ≤ r 0 when the state feedback control is applied. Hence if the observer is designed to be convergent when the trajectories are such that k x( t)k ≤ R and also such that it converges in a time T0 < T , when the trajectories are still in the set k x0 k ≤ r 0 , then the controller will assure that they satisfy k x( t)k ≤ R for all t ≥ 0. Hence, finite-time stability of the origin is obtained provided that the initial conditions of the system are small enough, i.e. k x0 k ≤ r ( r 0 , T0 ). These conditions for the observer can be always satisfied if the gain L i , i = 1, . . . , p, of each differentiator is sufficiently large [5]. This fact is illustrated in the example of Section 5. A constructive procedure to obtain the dynamic compensator of equation (4) is provided by the Inversion Algorithm [13], see also the example in Section 5. Remark 2. For a flat system the input can also be written as a function of the flat output and its derivatives. Together with the strong observability with respect to the measured output, this allows estimating the disturbance using the measured output. The only requirement is to take one more time-derivative. Due to the use of the HOSM differentiator, this adds restrictions in the disturbance and the trajectories of the system: it must be slow enough. Then it becomes possible to compensate the disturbance without applying a discontinuous HOSM controller. This last strategy has been recently explored in [16]. 5. Simulation Example Consider the following system x˙ 1 = x2 + u 1 + w1 , x˙ 2 = x33 + x3 + u 1 + w1 , x˙ 3 = sin( x1 x2 ) + x3 cos( x3 ) + u 2 + w2 ,
y1 = y2 =
x1 , x2 ,
with w1 = w2 = 0.5 sin( t) + 0.5 square(0.2 t) the sum of a sinusoid and a square signal of amplitude 0.5 and 0.2 seconds of period. The system above satisfies A3 only in the sense that their trajectories are bounded for each time instant. This happens since the x33 does not appear in the equation for x3 . The method of [8] is not applicable since y does not have vector relative degree. Using the algorithm of Section 3.1 yields £ ¤ £ ¤ x1 x2 x3 = y1 y2 φ1 ( y˙ 2 − y˙ 1 − y2 ) , where ´2/3 ³ p −23 + 2 9θ + 12 + 81θ 2 , φ1 (θ ) = ³ ´1/3 p 62/3 9θ + 12 + 81θ 2 1/3
1/3
and the required derivative was obtained using a firstorder HOSM differentiator with gain L = 250. We shall compare the control design for two distinct choices of flat output. For the first choice, we select the measured output as the flat output. In this case, it turns out that a dynamic compensator is required and that the first disturbance needs to be smooth. Since the selected disturbance does not satisfy this last requirement , it prevents applying the proposed methodology using the measured output as flat output. For the second selection we compute a new flat output that linearizes the system using a static compensator. This second choice outperforms the first one: both disturbances only need to be uniformly bounded, the controller is simpler and static, and it requires less derivatives. The simulation results for this second choice are also presented. For the first selection we use the measured output as flat output. Computing the second derivative of each output and using the dynamic compensator ξ˙ = −ξ − x33 + x3 + v1 with £ ¤ u 1 = ξ, u 2 = (3 x32 + 1)−1 v2 − γ( x) − v1 + ξ + x33 − x3 , and γ( x) = (3 x32 + 1)[sin( x1 x2 ) + x3 cos( x3 )] yields
y¨ 1 = v1 + w1 + w˙ 1 ,
y¨ 2 = v2 + (3 x32 + 1)w2 + w˙ 1 .
Therefore, we require that w1 , w2 and w˙ 1 are all uniformly bounded. Since w1 ( t) contains a square signals, w˙ 1 is not uniformly bounded as a function of time and prevents applying the proposed methodology. For the second selection, we apply the procedure of [13, pp. 122] to find a flat output that linearizes the system using a static compensator. If possible, the only restriction would be that the disturbance is uniformly bounded. For finding the desired flat output, compute the subspaces H1 = span{d x}, H2 = span{d x1 − d x2 } and H3 = {0}. This fact confirms that the system can be linearized using a static compensator. Then set ω2 = d x1 − d x2 and compute ω˙ 2 = d(δ x1 ) − d(δ x2 ) = d x2 − (3 x32 + 1) d x3 . Then one may select ω1 = d x2 . This yields H1 = span{d x} = span{ω1 , ω2 , ω˙ 2 }, and therefore span{d z1 , d z2 } = span{ω1 , ω2 } is a set of flat outputs. That is z1 = x2 and z2 = x1 − x2 . Notice that z can be obtained using y through a nonsingular transformation, so the system is also strongly observable with respect to the output z. Computing the input-output dynamics for these two outputs one obtains
z˙ 1
=
x33 + x3 + u 1 + w1 ,
z¨ 2
=
x33 + x3 − γ( x) + u 1 + w1 − (3 x32 + 1)( u 2 + w2 ),
and setting
u 1 = − x33 − x3 +v1 , u 2 = −(3 x32 +1)−1 (− x32 − x3 −γ( x)− u 1 +v2 ), yields z˙ 1 = v1 + w1 , z¨ 2 = v2 − (3 x32 + 1)w2 . Now design the HOSM controller by selecting v1 = −α1 H1 ( z1 ) =
Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
6
CONFIDENTIAL. Limited circulation. For review only −α1 sign( z1 ) and v2 = −α2 Φ2 ( x3 ) H2 ( z2 , z˙ 2 ) where
H2 ( z2 , z˙ 2 ) =
1 2
z˙ 2 + | z2 | sign( z2 ) 1
,
| z˙ 2 | + | z2 | 2
is a Quasi-Continous algorithm [15] with α1 = 1, α2 = 4.5, Φ2 ( x3 ) := 3 x32 + 0.5. The controller is switched on only when the convergence of the differentiator (i.e., the UIO) is detected according to [9]. All simulations were carried out using Euler’s method with 10−3 [s] of time step. Two simulations studies were carried out, see Figure 1. The first one presents the simulation results in the absence of noise. It shows the finite-time exact convergence of the state trajectories to zero. Secondly, measurement noises were added to both measured outputs. The noises were generated using Simulink’s “Uniform Random Number” generator with maximum amplitudes of 0.005 and 0.01 for y1 and y2 , respectively. The simulation results confirms that the robustness of HOSM controllers and differentiators to noises guarantees keeping a bounded state error despite the presence of noise. 6. Conclusions A methodology for the design of output feedback HOSM controllers for (dynamic) feedback linearizable disturbed nonlinear systems was presented. It provides finite-time exact state stabilization using only output feedback, provided that the system is algebraically strongly observable and the disturbances are smooth enough. The introduced notion of “strong” observability allows recovering the state by using only the output and its derivatives. When the derivatives are estimated using the HOSM differentiator, an Unknown Input Observer that converges in finite-time is obtained. Acknowledgements The authors would like to express their gratitude to Prof. Claude H. Moog (IRCCyN, France) for his assistance in the construction of the algorithm of Section 3-A. We also gratefully acknowledge the financial support from projects PAPIIT 17211 and IN111012, CONACyT 56819, 132125, 51244 and CVU 229959, Fondo de Colaboración del II-FI, UNAM, IISGBAS-165-2011.
Appendix: Proofs of Theorems and Propositions P ROOF (O F T HEOREM 1). Under the assumption that the vector fields are meromorphic, it is possible to use differential-algebraic tools to derive structural properties of the system [13]. This approach introduces finitedimensional spaces of 1-forms over the field of meromorphic functions K of x: Y k := spanK {d y, · · · , d y(k) }, X = spanK {d x}, W = spanK {d w( j) , j ≥ 0}. For our purposes, this means that the statement rank M k ( x) = n holds except (possibly) in a set of zero Lebesgue measure. Under this framework, Definition 1 of algebraic
strong observability is equivalent to the condition X ∩ Y k = X with k = max{k1 , · · · , k p }. By construction span{d M i ( x)} = X ∩ Y i . Notice that equality is guaranteed by construction. This way vectors M i ( x) define a chain of subspaces (X ∩ Y 0 ) ⊂ · · · ⊂ (X ∩ Y i ) ⊂ (X ∩ Y i+1 ) ⊂ · · · that is non-decreasing and bounded by X . Therefore, the sequence of subspaces has a unique limit. Since algebraic strong observability implies that X ∩ Y k = X for some finite k, then the upper bound of the sequence is attained and must coincide with the limit of such sequence. Now, since span{d M k ( x)} = X ∩ Y k = X this means that rank d M k ( x) = n. Moreover, the condition X ∩ Y i−1 = X ∩ Y i would imply that X ∩ Y i+1 = X ∩ Y i and that X ∩ Y i−1 is another limit, then the dimension of each subspace should increase by at least one at each step of the algorithm. Then, if we begin with a dimension p, at most n − p steps of the algorithm are required to reach the limit X ∩ Y n− p = X . Finally, since rank d M n− p ( x) = n then the Inverse Function Theorem guarantees that there exists, at least locally, an inverse for M n− p as claimed. P ROOF (O F P ROPOSITION 1). Due to Assumptions A2 and A3, x˙ ( t) is uniformly bounded as a function of time. By Assumption A1, the function F in formula (2) exists. From these two facts and taking one time deriva( k +1) tive of F and equating with x˙ ( t) obtain that yi i ( t), i = 1, . . . , p, are all uniformly bounded. Therefore, there exists large enough constants L i > 0, i = 1, . . . , p, such that each HOSM differentiator converges. This fact and the accuracy under measurement noise were proven in [5]. P ROOF (O F P ROPOSITION 2). By definition, the system is also algebraically strongly observable with respect to the flat output z( t). Now, on one hand, since q(0) = 0, the condition x( t) ≡ 0 implies that z( t) ≡ 0. Any other state trajectory satisfying z( t) ≡ 0 contradicts the strong observability of the system. P ROOF (O F T HEOREM 2). Using the diffeomorphism η = Ξ( x, ξ) and the dynamic compensator (4) yields a set of m disturbed chain of integrators (6). Moreover, the output z has vector relative degree with respect to v. This way the HOSM control design for v can be made exactly as shown in [9] to ensure that z( t) ≡ 0 after a finite-time T c . From Proposition 2, one concludes that x( t) ≡ 0, t ≥ T c . This completes the proof of the first part of the claim. For the second part, we assume that the system is algebraically strongly observable (A1). Then Proposition 1 ensures the finite-time exact estimation of the derivatives and hence, that controller (7) constructed using the estimated variables is applied only when it coincides with the controller constructed using the true variables. After the controller is applied, the correct estimation of the derivatives using the HOSM differentiator is preserved if theirs gains L i were large enough, as Proposition 1 ensures.
Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
7
CONFIDENTIAL. Limited circulation. For review only 1
Observation error
2
3.5
1 3
u1
0
2.5
−1 −2 −3 −4
1.5
−2
−3 0
0.5
1
1.5
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3.5
4
4.5
0
5
0.5
1
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2
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3
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4
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4
4.5
5
time[s]
time[s] 1
1
4 0.5
convergence
State trajectories
2
0
−1
2
u2
0
0
−0.5
−2
0.8 0.6 0.4 0.2
−1 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
−4
5
0
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2
time[s]
2.5
3
3.5
4
4.5
0
0.5
1
1.5
2
2.5
1
Observation error
2 1
3
u1
0
2.5
−1 −2
2
−3 −4
1.5
3
time[s]
time[s]
3.5
0
−1
−2
−3 0
0.5
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5
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3
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5
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4
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time[s]
time[s] 1
1
4 0.5
convergence
State trajectories
0
5
2
u2
0
0
−0.5
−2
0.8 0.6 0.4 0.2
−1 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
−4
0
0.5
1
1.5
2
time[s]
a)
2.5
3
3.5
4
4.5
5
0
0
time[s]
b)
0.5
1
1.5
2
2.5
3
time[s]
c)
Figure 1: Simulation result with (bottom) and without (top) measurement noises. a) System trajectories x( t). b) Control input. c) Top: estimation error for state x3 ( t); bottom: 1 indicates that the differentiator (observer) has converged.
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Preprint submitted to Automatica Received May 3, 2013 07:55:45 PST
