Let us postulate that the feedback controller signal can be expressed as follows:
u(t) = KQ(t), (4 — 13)
Consequently, the closed system can be represented as:
A(t) = (A + BK)A(t) (4— 14)
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Numerous methodologies exist for determining the matrix gain, #, ensuring asymptotic stability of system (4-14). it is noteworthy, as discussed in the Introduction, that the proposed full-state feedback controller necessitates access to all system state variables. This requirement imposes practical challenges, particularly in scenarios where obtaining measurements for all state variables incurs significant costs and logistical complexities during implementation. Thus, there exists a pragmatic imperative to explore alternative control strategies that mitigate the reliance on full-state feedback. Motivated by such practical considerations, the principal aim of this research endeavor is to develop a novel functional-order observer (filter)-based feedback controller. This controller aims to regulate the vehicle to a desired horizontal position while concurrently maintaining the pendulum in a stable vertical orientation. By leveraging the principles of functional-order observers, this approach seeks to address the challenges associated with full-state feedback by estimating the system states from limited measurements, thereby
offering a promising avenue for practical implementation in real-world control systems.
In pursuit of this objective, we turn our attention to a functional observer, defined by the following set of equations
fi(t) = YZ(t) + Ly(t)
(4— 15)
¿Œ) = X£Œ) + My(t) + Ru(t)
where f(£) represents the estimated control input of u(t), 6(£) € R” denotes the observer state
vector, Y is the observer gain matrix, £ is a matrix, N governs the dynamics of the observer, M is another matrix related to measurements, R is a matrix related to control signal. The
schematic of a linear functional observer based control is given by Figure 4-02.
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Figure 4-02: The schematic of a linear functional observer based control.
Theorem 4.2 The controller signal estimation, 8(£) converge asymtotically to the desirable control signal, u(t) if the following conditions hold:
€(t) = Ne(t) is assymtotically stable, e(t) = f(t) — Z0Œ)
NF + MC - 7.41 = 0;
R-FB=0
% — Y7 — £C = 0.
Proof of Theorem 4.2:
eŒ) = ¿Œ)— FA(t)
NG(t) + MCQ(t) + Ru(t) — FAx(t) — FBu(t)
= Ne(t) + (NF + MC — FA)O(t) + (R — FB)u(t)
=0.
If condition (4-17) -(4-19) are satisfied, e(£) > 0,t > 00. We denote
e(t) = &(t) — u(t) = Y§{(t) + LEN) — c0).
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(4— 16)
(4-17)
(4-18) (4-19)
(4 — 20)
(4— 21)
If condition (4-16) holds, further we obtain,
e(t) = Y(§Œ) — FA(t)) > 0. (4 — 22)
Proof is completed.
Accordingly, the functional observer-based controller can be implemented in the following form
u(t) = YếŒ) + Ly(t) (4— 23)
C(t) = (W + ®Y)§Œ) + (M + ®£)y()
With the controller (4-23), the augmented closed-loop systems become
0Œ) = HY() (4— 24)
where
_[ A+BLC BY _ [2
#f =|(t + 2e Ore Ror YO = Lee)
Theorem 4.3: The system (4-22) is asymptotically stable if exist matrices X, £, M, N, with appropritate dimension such the following conditions hold
(A+ BK),N are Hurwitz,(4 — 25)
NF + MC — FA =0; (4 — 26) R—-FB=0; (4 — 27)
K — Y# — £€ =0 (4 — 28)
4.3 Algorithm Development
From (4-27), #® is obtained from R = FB. Consequently, the functional state reconstruction problem entails the determination of appropriate observer parametersY, £, M, F and W, with the aim of achieving the minimal order of observer, q-th order while satisfying the conditions
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outlined in Theorem 4.3, specifically equations (4.26), (4.27) & (3.28) of Theorem 4.3. Prior to delving into the determination of these observer parameters, it is prudent to initially simplify
both equations (4.26) & (4.28). This simplification can be accomplished as follows K € R'TM*", F € IR" and partition them according to the following
FAFP=[K, #;]
FP =([F, Fo] (4— 29)
where P is an invertible matrix, KH, € RTM?, K, € RTM®-?), F, € RI? and 7; € RMP),
Note that K and P are known matrices and therefore F and 7; are also known matrices.
Whereas F, and 7; are two unknown matrices, the determination of which will follow subsequently.
Now, by post-multiplying (4.26) & (4.28) by P where CP = [Jp 0],A = PTM1A4P =
lh Ay A A | then (4.26) & (4.28) are reduced to the following
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£=1 —Y#,(4— 31)
Ky = Y?,.(4— 33)
Equations (4.30) and (4.31) provide a clear pathway to directly derive matrices M and £ once F,, Fz, N (Hurwitz) and Y are resolved from the coupled matrix equations (4.32) and
(4.33). Thus, the development of a linear functional observer crucially hinges on the solvability of these equations. (4.32) and (4.33).
Without loss of generality, one can let Y = J,. and facilitates the resolution of equations (4.32) and (4.33), yielding:
To solve matrix equation (4.35) for K, and a stable N, let us post-multiply (4.35) by a full-
rank matrix [Kf Kt] € IRŒ-?)XŒ-P) where K} € IRŒ!-P)X” denotes the Moore-Penrose
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inverse of K, and Ki € IR:~P)XŒ~?~?) denotes an orthogonal basis for the null-space of K.,
to give
Fy Ay, Kz = —K2A22Kz. (4 — 37)
From (4.37) there exists a solution for F, 1f and only if
%;Aaz1z
rank Ai; | = rank[4,z2€3]. (4 — 38)
Equation (4.37) has a solution for F,, where
F, = Yt + E(1, — PY"), (4 — 39)
W = A, Ki, ® = —%;4;;?ÿ, ?¡s the Moore-Penrose inverse of and = € JR”XP is an
arbitrary matrix represents an arbitrary matrix, used to position eigenvalues of matrix NW at prescribed locations in the complex s-plane.
Finally, we obtain
N =N, — EN;,
Ni, = (®W*4¡; + K2Ag2) Kz (4 — 40)
Ny = (Wt — 1 )A;#ÿ.
The algorithm is summarised as follows
e Step 1: Obtain matrix A, B, C from the engineering system e Step 2: Obtain K such (A-B K) is Hurwitz
e Step 3: Generate transformation P and obtain 44, 4;, 4a, 4;;, K?,K>, W,t?,5
e Step 4: obtain Nj, N2 from 4.39, determine E such W in (4 — 39) be stable
e Step 5: Compute F, from (4.38). Obtain F, = Kz, Y = I,. Obtain M and £ from (3.23) and (3.24), respectively.
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e Step 6. Finally, obtain R = FB, where F = [F, Fy ]P7?.