A compact observer-based control system is developed, utilizing functional observers and state feedback controllers to stabilize the inverted pendulum's motion.. This systemstands out fo
Dynamical Representation for Model of an Inverted Pendulum with a Cart
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Figure 2-01: The system includes an inverted pendulum and a cart.
Figure 2-01 shows a physical model of an engineering application consisting of an inverted pendulum and a cart In figure 2-01, ỉ(Ê) (deg) and x(t) are the pendulum's angle from the vertical axis and the cart's position ( or ground's position ), respectively M (kg), m (kg), L
(m) are weight of cart, weight and half length of pendulum, respectively I (kg.m”) denotes a mass moment of inertia of the pendulum In order to balance the system, a controlled force F is applied to the vehicle, and the frictional force is considered negligible while the vehicle is in motion (When calculating the parameters of the pendulum , the half-length of the pendulum is used because it represents the distance from the top of the bar to the center of the oscillation)
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In this part, we will develop a mathematical model of the study system The method we consider here is System analysis based on Newton's Law.
Figure 2-02 shows the force analysis for the physical model of an engineering application consisting of an inverted pendulum and a cart Here, 7g and 772g are weighted forces of the cart and pendulum, N are weighted force and force from the ground Ruy ,
Ry, Ro, Roằ are forces relating to the joint In Figure 2-02, F is the input force and F, is the friction force from the surface with the coefficient factor, related to velocity of the cart.
According to the analysis, dynamic equations of the system will be derived as follows
Let we consider the cart Forces: 47g (Weight), N( From the Ground), Ry, Ryằ, ( from Joint), F (Input Force) and F, (Friction Force).
= > where ay (t) is acceleration of the central point of the cart.
Taking equation (2-01) along x and y axises, we have:
The we consider the dynamic equations related to the pendulum.
Forces: mg (Weight), Ro, Roằ, (from Joint).
Applying Newton law: mg + N+ Rp, +Ro>=Mam (2-04) where a,,(t) is acceleration of the central point of the pendulum.
Taking equation (2-04) along x and y axises, we have mg(t) — Roi (t) =m ÿm() (2-05)
R22(t) = mềm(t), (2-06) where ym (t), Ÿm(Ð, Xm(t) and X(t) are
Ym(t) = Leosh(t), Ym(t)=—Lsin $(Đ$€), ÿŸm(Œ) = —Leos p(t)p(t)? —Lsin $()$(),
12 mg(t) — Rại(Q= m(—Leos$(t) g(t)? — Lsing(t) h(t), (2-07) Rạ;(t) = m(—Lsin p(t) b(t)? — Leosp(t) oi) + #@)), (2-08)
From (3), (8) and Riz = R21, we have
Now, we consider the rotation of the Pendulum: in this model, the pendulum is rotating around the pivot On the other hand, the pendulum is considered to be rotated around its center of mass.
The rotation of the pendulum can be expressed as follows:
Rạ+Lsin b(t) — Rạ;Lcos P(t) = Ie(), (2-10) where €(t) = #(t) is the angular acceleration.
From (2-07) : mgLsing(t) — R2,Lsin b(t) = mLsin $(t) (—Leos $(t)o(t)? — Lsind(t)d(t), (2-11)
From (2-08) and (2-11), mgLsing(t) = mLsing(t) (—Lcos p(t) p(t)? -— Lsin(t)$(t)) — m(—Lsin $ (t)¿()2 — Leos $(t)@(@) + X(t))Lcos b(t) + lậ(), (2-12)
(I+ ml?) f(t) — mgL sing(t) = mLx#(t)cos P(t) (2-13)
The mathematical representation of the studied system in Fig 2-01 can be obtained as follows:
(M + m) X(t) + bx(t) + mLQ()cos P(t) - mLg(t)?sin o(t) = F(t), (2 — 09)
(I + ml*) $(t) — mgL sing(t) = mLX(t)cos o(t) (2 — 13)
In the previous part, a mathematical non-linear model of the studied system has been obtained, nonetheless, the linear model is more popular for engineering analysis.
From small value of ở (t) ,cos cos f(t) = 1 , sin sin f(t) = $(t) and, @?() ~0, hence the system can be linearity about the equilibrium position as follows:
Denote q = ((M + m)(I + mL”) — m?1?) and from (2-14) and (2-15), a linear model of the studied system can be represented as follows:
#@)— PE r+ CMP OQ -TME oe) =0 (2-17)
In this part, we will develop a state space representation of the studied system In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output, and state variables related by first-order differential equations.
We denote X(t)= a i is the system state vector, andx(t p(t) u(t)=F(t) control input vector, Y(t)=CX(t) is the system output vector.
A state space presentation of the studied system can be obtained as follows:
Y(t) = Cx(o), (2-19) where A € R*** and B € R“*† are real matrices as
It is noted that if all state variables can be accessed, matrix C is an identity matrix.
The block diagram of dynamical representation of the studied system is shown Figure 2-03 as
Figure 2-03: The block diagram of dynamical representation of the studied system.
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The chapter commences with an exploration of the theoretical underpinnings governing state observers within the framework of dynamical systems, as described by state-space formulations In numerous control scenarios, direct access to the complete state vector is unattainable Consequently, the chapter elucidates methodologies for utilizing available system inputs and outputs to derive an approximation of the system's state vector Referred to as an observer, this mechanism facilitates the reconstruction of the state vector and operates as a time-invariant linear system, responsive to the inputs and outputs of the observed system.
Furthermore, the chapter delves into the nuanced process of selecting the observer's time constant, a parameter whose significance and implications are thoroughly examined.
3.1 Full-Order State Observers for Linear Systems
The full-order state observer aims to estimate all states within the system This section elucidates the definition of this observer and outlines the necessary conditions for its existence.
For comprehensive state estimation in a system with 'n' state variables, an observer of order 'n' is necessary Consider the linear system x(t) = Ax(t) + Bu(t), y(t) = Cx(t), where x(t) represents the state variable vector, u(t) the control vector, and y(t) the output.
In this context, it is assumed that direct measurement of all state variables encompassed within x7) is unfeasible, prompting the creation of an observer for state variable reconstruction.
To reconstruct all state variables, Ê(?) is introduced, representing an approximation of the state x(f) rather than its actual values An observer with the following dynamics is defined
X(t) = A#Œ) + Bu +L(y(t) — C#Œ)) (3-07) where L €R”*? is the observer gain matrix.
Figure 3.1 depicts the input-output behavior of a full-order state observer, emphasizing
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Full-Order State Observers for Linear Systems - ác + +ssssersrrske 16
The full-order state observer aims to estimate all states within the system This section elucidates the definition of this observer and outlines the necessary conditions for its existence.
It is deduced that a system featuring n state variables necessitates an observer of order n for comprehensive state estimation Let us consider a following linear system x(t) = Ax(t) + Bu(t), (3-05) y@) = Cx), (3-06) where x(t) € R” is state variable vector, u(t) € R” is control vector, y(t) € R” is output
In this context, it is assumed that direct measurement of all state variables encompassed within x7) is unfeasible, prompting the creation of an observer for state variable reconstruction.
To reconstruct all state variables, Ê(?) is introduced, representing an approximation of the state x(f) rather than its actual values An observer with the following dynamics is defined
X(t) = A#Œ) + Bu +L(y(t) — C#Œ)) (3-07) where L €R”*? is the observer gain matrix.
Figure 3.1 depicts the input-output behavior of a full-order state observer, emphasizing
To achieve feedback control, the extended Kalman filter (EKF) employs the estimated state 0,t > 00 We denote e(t) = &(t) — u(t) = Y§{(t) + LEN) — c0).
If condition (4-16) holds, further we obtain, e(t) = Y(§Œ) — FA(t)) > 0 (4 — 22)
Accordingly, the functional observer-based controller can be implemented in the following form u(t) = YếŒ) + Ly(t) (4— 23)
With the controller (4-23), the augmented closed-loop systems become
Theorem 4.3: The system (4-22) is asymptotically stable if exist matrices X, £, M, N, with appropritate dimension such the following conditions hold
To solve the functional state reconstruction problem, it is necessary to determine the appropriate observer parameters Y, £, M, F, and W These parameters should be chosen to minimize the order of the observer, denoted as q-th order, while still satisfying certain conditions The aim of this process is to obtain an observer representation #® from the relationship R = FB, where R and FB represent the residual and feedback signals, respectively.
29 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
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
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
30 inverse of K, and Ki € IR:~P)XŒ~?~?) denotes an orthogonal basis for the null-space of K., to give
From (4.37) there exists a solution for F, 1f and only if
Equation (4.37) has a solution for F,, where
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.
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.
31 e Step 6 Finally, obtain R = FB, where F = [F, Fy ]P7?.
Functional observer-based control for inverted pendulum eee 32
In this section, we undertake simulations to show the effectiveness of our proposed control scheme The simulation data be taken from isM = 500,mP,b = 0.1,1 = 3,] = 300 According to the data, matrices A,B can be obtained In this simulation, three scenarios of the initial values of pendulum's angle and vehicle's position are considered as e Scenario 1: - @ = 11° = 0.192 rad and xạ = —3 m. e Scenario 2: - dạ = 8° = 0.1396 rad and xạ = —2 m. e Scenario 3: - dạ = 5° = 0.083 rad and xạ = —1m.
To ensure system stability and maintain desired behavior, control action is crucial, especially when considering the pendulum's angle and vehicle's position Without such control, the system would remain unstable, making it essential to implement measures that restore and sustain stability.
At first, with the availability of the state space model, an optimal full state feedback controller can be derived in the form of
In this paper, the filter will estimate the functional state variable i = KQ(t) , hence the order of filter is only one By solving condition at Theorem 4.3, the functional observer's gain matrices can be obtained as
(m) ree, seme x(t)-Scenario | eo — RTr a()-Scenario | bị 5 ct % m= == x(t)-Scenario II | 7] rey —_ a():Scenario ll sẽ ` ~~ == x(t)-Scenario Ill 0.2 # 4% m— (\)-Scenario Ill | - ot tt
Fig 2: The responses of closed-loop system: (a)x(t); (b)a(t)
Fig 2 demonstrates the responses of pendulum’s angle, a(t) and vehicle’s position, x(t) of the closed-loop system embedded the proposed control method
Our proposed control strategy demonstrates its effectiveness by successfully guiding the pendulum's angle and vehicle's position back to their desired values within a brief settling period, regardless of initial conditions.
Low-order observers, such as functional observers, present a viable alternative to full-order observers by significantly reducing the computational burden while still maintaining the requisite accuracy for precise control Their implementation in managing systems like the inverted pendulum demonstrates their effectiveness in addressing the challenges of complex dynamic systems where complete state measurements are not feasible Functional observers achieve this by focusing on estimating only essential linear combinations of state variables relevant to the control objectives, rather than reconstructing the entire state vector This selective estimation allows for a more streamlined approach in system design, reducing the number of calculations required and thereby lowering the demands on computational resources.