As it has been mentioned in the Section 3.2, the blimp altitude control system is considered as aswitched system with a constant time-delay complemented with uncertain bounded disturbances, thus the controller has to compensate the distur-
bance estimated in real-time, deal with the time-delay and stabilize the switched system.
3.4. Controller design 81
3.4.1
Disturbance estimation
The disturbance term dz(t) in system description (3.2) represents the error be-
tween nominal model and blimp real situation.
With the intention of estimating the disturbance dz(t) in real-time, a filter is
designed: ˙
Xfil(t) = Az ˆσXfil(t) + Bz ˆσuz(t − τnom) + L(y(t) − yfil(t)) yfil(t) = CzXfil(t)
(3.25)
where L =h l1 l2 iT
is the gain of filter. It is obvious that the filter has a similar form as a Luenberger observer for nominal model, i.e. (3.2) without disturbance term.
Let e(t) = X(t) − Xfil(t) be the error between state vector of (3.2) and that of filter (3.25). Taking the time derivative of e(t), we get:
˙e(t) = (Az ˆσ −LC)e(t) + Bzσdz(t) + δ(t) (3.26)
where δ(t) = (Azσ−Az ˆσ)X(t). Thanks to the finite-time convergence of differen-
tiator, after a finite time T , we have ˆσ (t) = σ (t), there is
˙e1(t) = −l1e1(t) + e2(t) ˙e2(t) = −l2e1(t) + azσe2(t) + bzσdz(t)
Taking time-derivative on both sides for the first equation, we obtain: ¨e1(t) = −l1˙e1(t) + ˙e2(t)
Substituting ˙e2(t) by the second equation, and rearranging the terms, there is ˆ
dz(t) =
¨e1(t) + l1˙e1(t) + l2e1(t) − azσe2(t)
bzσ
82 CHAPTER 3. Altitude Control estimation: ˆ dz(t) = ¨e1(t) + (l1−azσ) ˙e1(t) + (l2−azσl1)e1(t) bzσ (3.27) Note that C =h 1 0 i, i.e. y(t) = x1(t), so e1 is the difference between altitude measurement and filter output, and it is available from measurements. The calculation of ˙e1(t) and ¨e1(t) is also realized by HOMD differentiator (3.23).
Aiming to determine the gain L of the filter, using the fact that the filter matrix Az ˆσ −LC needs to be Hurwitz, then we need to seek a symmetric and
positive definite matrix P and gain L satisfying the following LMIs (Linear Matrix Inequalities) [Geromel et al., 1998; Khalil, 1996]
P 0 (Az ˆσ−LC)TP + P (Az ˆσ−LC) ≺ 0, ∀σ ∈ P (3.28)
where the set P is defined in (3.2). Let W = P L, there is P 0 ATz ˆσP − CTWT+ P Az ˆσ −W C ≺ 0, ∀σ ∈ P (3.29)
A feasible solution of P and L can be solved by YALMIP toolbox of MATLAB [Löfberg, 2004]: P = 213.427 −30.826 −30.826 229.629 L =h 0.653 1.059 iT (3.30)
3.4.2
Predictor-based controller design
Many researchers have studied the problem of stability and stabilization of systems with control input delay, there are many possible approaches to deal with the problem. They can be classified into memoryless and memory controllers, the first type of controllers have feedback of the current state only, while the second one employs a feedback of the past control history as well as the current state [Moon et al., 2001].
3.4. Controller design 83 proaches to design the feedback, for instance, [Luo and Chung, 2002] proposed control based on the optimal control for delay-free linear system with quadratic performance index, [Kojima et al., 1994] used H∞control theory to investigate the robust stabilization problem for uncertain input-delay system, the authors of [Roh and Oh, 1999] proposed a sliding mode controller for the stabilization of uncertain input-delay systems with nonlinear parameter perturbations.
As for the memory controllers, [Cheres et al., 1990] designed a min-max control by using Razumikhin method which can handle a system with a fast time-varying delay, [Niculescu, 2001] used integral quadratic constraint method to design the memory controller but it is only available for a system with a constant delay, [Kwon and Pearson, 1980] designed a delayed feedback control by employing the reduction method which reduced the original system to a delay- free one, then [Moon et al., 2001] investigated the robustness of the reduction method controller by LMIs method, [Yue and Han, 2005] applied Lyapunov- Krasovskii functional approach to study the stability of uncertain system with time-varying delay and reduction method designed memory controller.
Since in our work the time-varying time-delay of the blimp NON-A prototype system is varying slowly, thus it is approximated by the constant nominal time- delay which is identified in tests as τnom = 0.6s. Then we chose to use the predictor-based controller which includes a Smith predictor to compensate the fixed nominal time-delay and transformed the system to a delay-free closed-loop system. The uncertainties caused by the time-delay approximation are included in the disturbance term and compensated in the controller. The predictor-based controller has two parts:
1) Predict state at time t + τnomwith Smith predictor [Smith, 1959]:
ˆ X(t + τnom) = eAz ˆστnomX(t) +ˆ 0 Z −τnom e−Az ˆσsB z ˆσuz(t + s)ds + 0 Z −τnom e−Az ˆσsB z ˆσdˆz(t + τnom+ s)ds (3.31)
84 CHAPTER 3. Altitude Control where ˆσ is estimated according to (3.24), and ˆdz is an estimate of the
disturbance obtained by (3.27).
2) Assign the controller output based on predictor result and the estimated disturbance term:
uz(t) = −Kz ˆσX(t + τˆ nom) − ˆdz(t + τnom) (3.32) From (3.31) and (3.32), it is clear that the disturbance term dz(t) has to be
estimated in real-time and predicted for the time interval [t, t + τnom]. As it has been observed in experiments, the estimated disturbance signal is rather noisy, in order to decrease the "chattering", we choose to use a time polynomial to fit
ˆ
dz(t) in a sliding window, then the polynomial is used to predict ˆdz(t + τnom). Remark 3.1. It is worth to mention that for the blimp NON-A V2 system, where the OptiTrack camera capturing system is implemented, the delay is greatly reduced and can be ignored with the assumption that blimp moves slowly. Thus the altitude controller for blimp V2 robot can be simply obtained by taking the
τnom= 0 in the equation (3.32), and replacing the nominal model parameters by the identification result shown in Table 2.2.
3.4.3
Determination of controller gain
As the system (3.2) is time-delayed with an uncertain bounded disturbances, a predictor-based controller (3.32) with disturbance compensation is designed. With the assumption that the switching signal is perfectly estimated ˆσ (t) = σ (t),
the problem remains to determine a gain of controller Kz ˆσ which can make the
closed-loop system ˙X(t) = (Azσ−BzσKz ˆσ)X(t) stable.
Assume the gains of controller of switched system are chosen to be the same for all σ , i.e. Kz ˆσ = Kz, to simplify calculation.
According to [Liberzon and Morse, 1999], if there exists a matrix P and gain
Kz, with P = PT, such that:
P 0 (Azσ−BzσKz)TP + P (Azσ−BzσKz) ≺ 0, ∀σ ∈ P (3.33)
3.5. Simulation 85