Nonlinear Observer-Based Control Allocation

Control allocation is the process of mapping virtual control inputs (such as torque and force) into actual actuator deflections in the design of control systems (Benosman et al., 2009; Bodson, 2002; Buffington et al., 1998; Liao et al., 2007; 2010). Essentially, it is considered as a constrained optimization problem as one usually wants to fully utilize all actuators in order to minimize power consumption, drag and other costs related to the use of control, subject to constraints such as actuator position and rate limits. In the design of control allocation, full state information is required. However, in practice, states may not be measurable. Hence, estimation of these unmeasurable states becomes inevitable.

and feedback-based methods (Bestle & Zeitz, 1983;Bornard & Hammouri, 1991;Gauthier & Kupka, 1994;Krener & Isidori, 1983;Krener & Respondek, 1985;Teel & Praly, 1994;Tsinias, 1989;1990). Optimization-based methods obtain an estimatê x(t) of the state x(t) by searching for the best estimatex(0) of x(0) (which can explain the evolution y(τ) over [0, t]) and integrating the deterministic nonlinear system fromx(0) and under u(τ). These methods take advantage of their systematic formulation, but suffer from usual drawbacks of nonlinear optimization (like computation burden, local minima, and so on). Feedback-based methods can correct on-line the estimationx(t) from the error between the measurement output and the estimated output. These methods include linearization methods (Bestle & Zeitz, 1983;Krener & Isidori, 1983;Krener & Respondek, 1985), Lyapunov-based approaches (Tsinias, 1989;1990), sliding mode observer approaches (Ahmed-Ali & Lamnabhi-Lagarrigue, 1999) and high gain observer approaches (Bornard & Hammouri, 1991;Gauthier & Kupka, 1994;Teel & Praly, 1994), and so on. Among them, linearization methods (Krener & Isidori, 1983) transform nonlinear systems into linear systems by change of state variables and output injection. It is applicable to a special class of nonlinear systems. Sliding mode observer approaches (Ahmed-Ali & Lamnabhi-Lagarrigue, 1999) is to force the estimation error to join a stabilizing variety. The difficulty is to find a variety attainable and having this property. High gain observer approaches (Besancon, 2007) use the uniform observability and weight a gain based on the linear part so as to make the linear dynamics of the observer error to dominate the nonlinear one. Due to the requirement of the uniform observability, these approaches can only be applied to a class of nonlinear systems with special structure. Interestingly, Lyapunov-based approaches (Tsinias, 1989;1990) provide a general sufficient Lyapunov condition for the observer design of a general class of nonlinear systems and the proposed observer is a direct extension of Luenberger observer in linear case.
In this chapter, we extend the control allocation approach developed in (Benosman et al., 2009;Liao et al., 2007;2010) from state feedback to output feedback and adopt the Lyapunov-type observer for a general class of nonlinear systems in (Tsinias, 1989;1990) to estimate the unmeasured states. Sufficient Lyapunov-like conditions in the form of the dynamic update law are proposed for the control allocation design via output feedback. The proposed approach ensures that the estimation error and its rate converge exponentially to zero as t → +∞ and the closed-loop system exponentially converges to the stable reference model as t → +∞. The advantage of the proposed approach is that it is applicable to a wide class of nonlinear systems with unmeasurable states, and it is computational efficiency as it is not necessary to optimize the control allocation problem exactly at each time instant.
This chapter is organized as follows. In Section 2, the observer-based control allocation problem is formulated where the control allocation design is based on the estimated states which exponentially converge to the true states as t → +∞. In Section 3, the main result of the observer-based control allocation design is presented in the form of dynamic update law. An illustrative example is given in Section 4, followed by some conclusions in Section 5.
Throughout this chapter, given a real map f (v, w), (v, w) ∈ R n × R m , D v f (v 0 , w 0 ) denotes its derivative with respect to v at the point (v 0 , w 0 ).F o rg i v e nr e a lm a ph(v) with v ∈ R n , Dh(v 0 ) denotes its derivative with respect to v at the point v 0 . In addition, · represent the induced 2-norm.

Problem formulation
Consider the following nonlinear system: where x ∈X ⊂R n is the state vector with X aopensubsetofR n , y ∈ R l is the measurement output vector, and u ∈ R m is the control input vector satisfying the constraints ···ū m ] T being vectors of lower and upper control limits, respectively.
We assume that the system (1) satisfies the following assumption:

Assumption 1. The function f (x, u) is smooth and the output function h(x) is continuously differentiable.
Since control allocation need full state information, the state estimation for the system (1) is required.
Consider a dynamic observer of the following forṁ Define the error e as To estimate the state x, we wish to design the mapping Φ(x, u) such that the trajectory of e with the dynamicsė exponentially converges to zero as t → +∞,u n i f o r m l yo nu ∈ Ω, for every x(0) subject to e(0)=x(0) −x(0) near zero.
The aim is to design a nonlinear control allocation law based on the state observer (3) such that a reference model that represents a predefined dynamics of the closed-loop system is tracked subject to the control constraint u ∈ Ω.
Given that the predefined dynamics of the closed-loop system is described by the following asymptotically stable reference modelẋ where A d ∈ R n×n , B d ∈ R n×n r and the reference r ∈ R n r satisfy the following assumption.
Assumption 2. A d is Hurwitz, and r ∈ Σ ⊂ R n r is continuously differentiable where Σ is an open subset defined by: for each r ∈ Σ,thereexistx ∈Xand u ∈ Ω such that the system (1) matches the reference system (6).

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Nonlinear Observer-Based Control Allocation

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Since the state x is unmeasurable, the control allocation design is then based on its estimatê x. In other words, we have to first choose the mapping Φ(x, u) in (3) such that the estimation error e exponentially converges to zero as t → +∞,uniformlyonu ∈ Ω, for every x(0) ∈X subject to e(0) near zero; then minimize the cost function where H 1 ∈ R m×m and H 2 ∈ R n×n are positive definite weighting matrices, and is the matching error between the actual dynamics and desired dynamics. Since power consumption minimization introduced by the term 1 2 u T H 1 u is a secondary objective, we Now the control allocation problem is formulated in terms of solving the following nonlinear static minimization problem: Then the constraint condition u ∈ Ω is equivalent to Introduce the Lagrangian where λ ∈ R m is a Lagrange multiplier. And assume that The following lemma is immediate ( (Wismer & Chattergy, 1978), p. 42). (13) (13) is convex in this case, Lemma 1 holds for a global minimum.

Lemma 1. If Assumptions 1 and 3 hold, the Lagrangian
To solve the control allocation problem (9) with the state estimatex from the observer (3), we consider the following control Lyapunov-like function where P > 0 is a known positive-definite matrix and Here the function V m is designed to attract (u, λ) so as to minimize the Lagrangian (13). The term 1 2 e T Pe forms a standard Lyapunov-like function for observer estimation error e which is required to exponentially converge to zero as t → +∞.
Following the observer design in (Tsinias, 1989), we define a neighborhood Q of zero with Q ⊂X, a neighborhood W of X with {x − e : x ∈X, e ∈ Q}⊂W,a n dac l o s e db a l lS of radius r > 0, centered at zero, such that S ⊂ Q. Then define the boundary of S as ∂S.F i g u r e1 illustrates the geometrical relationship of these defined sets.
Let H denote the set of the continuously differentiable output mappings h(x) : X→R l such that for every m 0 ∈ Q andx ∈ W, and where and assume that Assumption 5. There exist a positive definite matrix P ∈ R n x ×n x and a positive constant k 0 such that kerDh(x) ⊂ Nholdsforany(x, m 1 , u) ∈ W × Q × Ω. (5) is stable in the case of h(x)= h(x) and x =x. In particular, for linear systems, the condition in Assumption 5 is equivalent to detectability.

Denote
and define Let and the observer systemẋ are adopted. Here α, β ∈ R m are as in (20), and ξ 1 , ξ 2 ∈ R m satisfy with V m as in (15) and and the mapping with γ 1 (x, u) > 0 and γ 2 (x) > 0 defined as in (22) and (23).
For any nonzero e ∈ S,letν = r e −1 e. Obviously, ν ∈ ∂S.T h e nw eh a v ė In the following, we shall show that V converges exponentially to zero for all m 0 , m 1 ∈ S, x ∈ W, u ∈ Ω, e ∈ S, e = 0andν ∈ ∂S.
Consider now the issue of solving (26) with respect to ξ 1 and ξ 2 . One method to achieve a well-defined unique solution to the under-determined algebraic equation is to solve a least-square problem subject to (26). This leads to the Lagrangian where ρ ∈ R is a Lagrange multiplier. The first order optimality conditions leads to the following system of linear equations (47) always has a unique solution for ξ 1 and ξ 2 if any one of α and β is nonzero.
From Figure 2, it is observed that the estimated statesx 1 andx 2 converge to the actual states x 1 and x 2 and match the desired states x 1d and x 2d well, respectively, even when u 2 is stuck at −0.5. This observation is further verified by Figure 3 where both the state estimation errors e 1 (= x 1 −x 1 ) and e 2 (= x 2 −x 2 ) of the nonlinear observer as in (4) and the matching errors τ 1 (= 0) and τ 2 (= − sinx 1 + u 1 cosx 1 + u 2 sinx 1 + 25x 1 + 10x 2 − 25r) as in (8) exponentially converge to zero. Moreover, Figure 4 shows that the control u 1 roughly satisfies the control constraint u 1 ∈ [−1, 1] while the control u 2 strictly satisfies the control constraint u 2 ∈ [−0.5, 0.5]. This is because, in this example, the Lagrange multiplier λ 1 is first activated by the control u 1 < −1att = 0(seeFigure5whereλ 1 is no longer zero from t = 0), and then the proposed dynamic update law forces the control u 1 to satisfy the constraint u 1 ∈ [−1, 1]. It is also noted from Figure 5 that the Lagrange multiplier λ 2 is not activated in this example as the control u 2 is never beyond the range [−0.5, 0.5]. In addition, the output y and the Lyapunov-like function V m are shown in Figure 6. From Figure 6, it is observed that the Lyapunov-like function V m exponentially converges to zero.

Conclusions
Sufficient Lyapunov-like conditions have been proposed for the control allocation design via output feedback. The proposed approach is applicable to a wide class of nonlinear systems. As the initial estimation error e(0) need be near zero and the predefined dynamics of the 127 Nonlinear Observer-Based Control Allocation www.intechopen.com closed-loop is described by a linear stable reference model, the proposed approach will present a local nature.