2.1. Exact input-output feedback linearization

The input output linearization is based on two concepts: the concept of relative degree and the concept of state transformation.

The relative degree r of the system (1) is defined as the number of derivation of the output y needed to appear in the input u , such as ∀x ∈ ℜ ⁿ :

If r ≤ n , then system (1) can be feedback linearized into Byrnes-Isidori normal form (^{Isidori, 1995}) using the following steps:

Step 1: We apply the following control law

with v = y^(r)

This control law compensates the nonlinearities in the input-output behavior.

Step 2: First, system (1) is transformed into normal form (^{Isidori, 1995}) through a nonlinear change of coordinates:

with:

The resulting system with the transformed variables (4) can then be written as

where η is the state vector of the internal dynamics.

2.2. Singular perturbed system

A singularly perturbed system is a system that exhibits a two-time scale behavior, i.e. it has slow and fast dynamics, and it is modeled as follows (^{Glielmo & Corless, 2010}):

where ξ ∈ ℜ ^P and η ∈ ℜ ^m are respectively the slow and fast variables and ε > 0 is a small positive parameter. The functions F ₁ (·) and F ₂ (·) are assumed to be continuously differentiable.

ξ ₀ and η ₀ are respectively the initial conditions of the vectors ξ and η . If ε → 0, the dynamics of ξ acts quickly and leads to a time-scale separation. Such a separation can either represent the physics of the system or can be artificially created by the use of high-gain controllers.

As ε → 0, ξ can be approximated by its Quasi Steady State ξ-=ϑη,u obtained by solving

So, the reduced (slow) system is given by:

Note that the reduced system (8) is not necessarily affine in input.

In the next theorem, we establish the exponential stability of the singular perturbed system (7).

Theorem 1 (Khalil, 202). Assume that the following conditions are satisfied:

Then, there exists ε ^∗ > 0 such that, for all ε < ε ^∗, the origin of (7) is exponentially stable.

Theorem 2 (Khalil, 202). Given system (1), if there exists a Lyapunov function V (x ) and positive constants χ ₁ , χ ₂ and χ ₃ such that χ₁||x||² ≤ V(x) ≤ χ₂||x||² and V˙x≤-X3x2, then the origin is exponentially stable.

2.3. Basic results on the trajectory following method

The trajectory following method (^{Naiborhu & Shimizu, 2000}) is a numerical optimization method based on solving continuous differential equations.

The basic idea behind a "trajectory following" method is to form a set of differential equations from the gradient of the cost function.

Consider first the minimizing problem of the form:

subject to no constraint

where ϒ (u ) is a performance function of a control variable u .

Suppose that we use the local minimal point u * as an initial condition for integrating the differential equation,

where Λ (·) is a function at our disposal, to be determined shortly.

Calculate the time derivative of ϒ (u ) along the trajectory generated by the solution to (11). Then at u = u *:

Since we are interested in a trajectory that will search a minimum, the above observation suggests that we integrate (11) by choosing

and Eq. (12) becomes

Thus, in order to solve the problem of minimizing ϒ (·) in an unconstrained control space, one need only to choose an appropriate initial condition for u and integrate

until the equilibrium solution is achieved.

3.1. Boundary layer subsystem

Consider the nonlinear system described by (1), then we apply the control law (3) which is given by:

with k-i+1=ki+1εr-1

where y _ref is the reference trajectory for the output, ε → 0 a small positive parameter, and k _i > 0, ∀ i ∈ {1, 2, ..., n − 1} are the coefficients of a Hurwitz polynomial (^{Isidori, 1995}) and the internal dynamics are given by:

Under the assumption that the gains k-i are chosen large, such as for any choice of ε > 0, the closed loop is stable and ε can be used as a single tuning parameter, the system (16) and (17) can be written in the form of a singular perturbed system (7). So the fast state can be defined by:

If we replace (18) by (17), we obtain

and also by (16), such that

with ξref=yrefεy˙refε2y¨ref⋯εryref(r-1)T

thus, (17) can be written as follows:

3.2. Reduced subsystem

The internal dynamics depends on the output y and its derivatives ẏ,..., y^(r−1), such as:

In general, the input output linearization techniques decouple between the input-output behavior y and the internal dynamics η . On the other hand, the QSS assumption decouples between the internal dynamics η and the input output behavior y . Thus, y does not have any effect on η . Therefore, the output y ^{( r )} is used for the control of the internal dynamics. Thus, the boundary layer subsystem (21) and the reduced subsystem (22) can be manipulated separately.

Firstly, we define a novel state vector:

Such as the reduced subsystem (22) can be written by:

The system (24) will be expressed by the following state equation:

where

The stability of the internal state η is required to guarantee the output system y tracks the reference output trajectory y _ref . Based on the gradient descent control algorithm, we propose a method to make the internal state η tend to η _ref, η → η _ref if t → ∞ (internal state regulation).

Let η be a virtual output of the system and η _ref be the virtual desired output.

Then we find γ _η as relative degree of the system if η is the output of the system (1). We know that η ∈ R ^{( n − r )} then γη=γη1γη2⋯γηn-r. Based on Z and their derivatives, we construct the performance index as a descent function as follows

where the constants a₀, a₁..., a_r, b0i, b1i,..., byii, i=1,...,(n−r) will be chosen later.

Reasons why we define the descent function as in (26) are:

be a quadratic form and that ψ (0) be equal to zero. Accordingly, the minimum value of the descent function ψ ₀ (Z ) is zero.

If ψ (Z ) is zero, each term in the right side of Eq. (26) is positive ∀t , therefore we have:

By choosing the values of a ₀ , a ₁ ... , a _r such that the Eigen values of the polynomial

are real negative and the values of b0i, b1i,..., bγiii=1,...,(n−r) such that the Eigen values of the polynomial

are real negative, we obtain that:

if t → ∞.

In other words output tracking and regulation of internal state are achieved together (^{Artstein, 1983}). Our objective is to find a control law u _QSS such that the descent function ψ (Z ) becomes minimum. Then the output tracking problem can be written as:

In this paper, the output tracking problem (32) will be solved using the trajectory following method.

When u _QSS is a scalar,

Because the relative degree of the system is well defined,

The necessary condition for a local minimum

is satisfied if at the same time:

and it happens if the value of performance index ψ (Z ) = 0.

In the following we use a numerical technique to solve the minimization problem (32) and (33) at every point of a trajectory.

The idea of the trajectory following method is to solve the numerical optimization problem only once, at the initial point of the trajectory.

For all other points x along a trajectory, the control u _QSS is determined from the differential equation:

The control law in Eq. (37) is called the steepest descent control.

Calculate the time derivative of descent function (26)

we have

By substituting (21) into (38), we have

From Eq. (40) we see that the value of time derivative of descent function along the trajectory of (38) cannot be guaranteed to be less than zero for t ≥ 0. Consider the extended system (38) and time derivative of descent function (40). We do not have a variable which can be used to push the time derivative of descent function (40) less than zero.

Now we modify the steepest descent control (38) by adding an artificial input u _ar . Then the extended system (38) becomes:

from Sontag's formula (^{Sontag, 1989}), we get

The control law in Eq. (42) is called a modified steepest descent control. A similar method using artificial input can be seen in ^{Shimizu, Otsuka, and Naiborhu (1999)}.

4.1. Exponential stability of the boundary layer subsystem

Let us consider the error vector given by

Then, the boundary layer subsystem (21) becomes:

Letting τ = t /ε yields:

with A is defined by

Using the Theorem 2, the origin ξ-=0 is exponentially stable, and the Lyapunov function is

where _^ATP + PA = − Q and Q is a matrix defined positive.

4.2. Exponential stability of the reduced subsystem

To analyze the stability of the reduced subsystem, we use the following proposition:

Proposition 1: If the following statements are satisfied:

Assume that system (25) satisfies the following assumption.

Assumption 1. There exists a function V ₂ : ℜ ⁿ → ℜ, V ₂ (0) = 0, which is continuous, positive definite and radially unbounded such that the unforced dynamic system of (25), namely Z˙=Q-Z,0 is globally asymptotically stable, i.e., V˙2Z,uQSS<0, Z≠0.

First we define the performance index

where R is a matrix constant, R > 0. Then we determine the value of u _ar by Sontag's formula (^{Sontag, 1989}) such that the extended nonlinear system:

is asymptotically stable about (Z , u _QSS ) = (0, 0). The most important thing is to guarantee the existence of u _ar .

Remark 1. Consider (47). If u _QSS = 0 then ψ (Z , u _QSS ) = V ₂ (Z ). In other words performance index becomes Lyapunov function and so we do not need to design control input u _QSS for only stabilizing system.

With u _QSS = 0 we can do nothing to increase the rate of convergence. However, by adding u _QSS to the system, we have freedom to accelerate the rate of convergence.

Remark 2. It would appear that when Z = 0 is globally asymptotically stable as assumed by Assumption 1, the global stabilization of the whole system should not be difficult.

4.3. Global stability

In order to illustrate the stability of the cascade control scheme, the following exponential stability theorem is introduced.

Theorem 3. For system (1), consider a controller where u _QSS is obtained by solving the optimization problem (41) and the input is computed using (3). If P and R are positive definite, then there exists ε > 0 that would exponentially stabilize (1).

Using Theorem 1, we can conclude that there exists ε* > 0 such that for all ε < ε*, the origin of (1) is exponentially stable. All the conditions of theorem 1 are satisfied such that:

5.1. Inverted cart-pendulum system

Consider the familiar inverted cart-pendulum system (^{Al-hiddabi, 2005}), depicted in Figure 2. The cart must be moved using the force u so that the pendulum remains in the upright position as the cart tracks varying positions at the desired time. The differential equations describing the motion are (^{Al-hiddabi, 2005}):

where θ is the angle of the pendulum, y _p is the displacement of the cart, and u is the control force, parallel to the rail, applied to the cart. The numerical parameters of the inverted pendulum system are given in Table 1.

Consider y _p as the output and let x=[θθ˙y_p ẏ_p ]^T. The inverted cart-pendulum can be written as the system (1). Where y _p represents the output, u is the input, x is the state-space vector. Hence, one has:

and h (x ) = y _p

The relative degree of the system is equal to r = 2 which is strictly lower than the system dimension n = 4.

Applying the procedure of input-output linearization to the system (49) of the inverted cart-pendulum, the boundary layer system is given by:

its control is given by:

and the internal dynamics is given by

Under the QSS assumption that θ˙=θ¨=υ=0 and θ → ≅0, sin (θ ) = θ and cos (θ ) = 1. Using Eq. (25), the reduced subsystem can be written as:

The control objective is to make the output θ track a desired reference trajectory θ _ref given that at the same time the displacement of the cart tracks the following trajectory:

The desired displacement (54) has smooth switching at t = 0, t = 2π and non-smooth switching at t = 4π .

The relative degree of η ₁ is equal to γn1=2 and the relative degree of η ₂ is equal to γn2=1.

By using Eq. (26), the descent function will be transformed under the following form:

with

the input u˙QSS=-∂ψZ∂ψQSS+1∂ψZduQSS-dψZdxx˙-dψZdxx˙2+dψZduQSS2 that stabilizes the internal dynamics is given by solving the problem (25). In simulation, the parameters used in the input-output linearization are: ε = 0.09, k ₁ = 3.6, k ₂ = 2.9. For the gradient descent control algorithm, the employed parameters are:

R=100000010000001000000100, p=4.1510.613.276.89, Q=1001 and a₀=5.4, a₁=2.1, a₂=3.4 b01=6, b11=9, b21=1.7, b02=2,b12=8.2. The initial conditions are y_p(0)= ẏ_p(0)= θ˙(0)=0 and θ (0) = − π , the downward position for the pendulum.

The simulation results are presented by Figures 3-5. Figure 3 shows the evolution of the displacement of the cart y _p compared to the desired one y _ref . The simulation results in Figure 3 show that the control scheme provides good tracking. In this figure, there is a perfect agreement between the two trajectories.

Figure 4 shows the evolution of the pendulum angle; indeed, it is a small variation around zero. The evolution of the stabilizing control law is shown in Figure 5. The dynamics of this control signal is quite satisfactory. In fact, there is no unacceptable physical overshoot. One can also see the reduced response time in which the control law stabilizes the controlled variable. This shows very interesting results given by the proposed cascade scheme control.

5.2. Ball and beam system

In this part, a ball and beam is considered. The ball and beam system is an unstable nonlinear system which is well known in automation; therefore, it is regarded as a perfect bench test for the design of control laws for non-minimum phase nonlinear systems.

The ball and beam system is composed of a rigid bar carrying a ball. The latter is characterized by its horizontal axis and its moment of inertia J . Its rotation angle θ compared to the horizontal one is controlled by an engine with direct current to which it applies a couple τ . A ball is placed on the beam where it is able to move with a certain freedom under the effect of gravity, as it is illustrated in Figure 6.

The dynamic model governing the behavior of the ball and beam system in open loop can be expressed by the following equations (^{Hauser et al., 1992}):

The numerical parameters of the ball and beam system are recapitulated in Table 2. To define a new input u the system can be written in state-space form as (^{Hauser et al., 1992}):

with

x=rbr˙bθθ˙T,y = h(x) and Bb=Mb/Jb/Rb2+Mb.

The relative degree of the system is equal to r = 3 which is strictly lower than the system dimension n = 4.

Applying the procedure of input-output linearization to the system (58) of the ball and beam, the boundary layer system is given by:

its control is given by:

with υ=y⃛ref+k3εy⃛ref-y¨+k2y˙ref-y˙+k1yref-y and the internal dynamics is given by

Using Eq. (25), the reduced subsystem can be written as:

The objective of the studied system control is to ensure that the ball always keeps contact with the beam and that its movement is carried out without slip what imposes a mechanical constraint on the acceleration of the beam. The desired trajectory is characterized by variable amplitude during time and it is given by:

The relative degree of η ₁ is equal to γη1=2. By using Eq. (26), the descent function will be transformed under the following form:

with

the input u˙QSS=-∂ψZ∂ψQSS+1∂ψZduQSS-dψZdxx˙-dψZdxx˙2+dψZduQSS2 that stabilizes the internal dynamics is given by solving the problem (25). In simulation, the parameters used for the input-output linearization are: ί = 0.098, k ₁ = 4.8, k ₂ = 1.7, k ₃ = 5.3. For the gradient descent control algorithm, the employed parameters are:

R=70000070000070000070, P=2.3311.438.434.218.665.633.221.344.88, Q=100010001, and b01=3, b11=2.1, b21=1.5, a₀=4, a₁=7.2, a₂=9.1, a₃=3.4.

The result of this study is shown by Figures 7-9. Figure 7 illustrates the evolution of the tracking trajectory compared to the desired one y _ref . In this figure, there is an agreement between the two trajectories. The amplitude variation of the desired trajectory did not hamper the precision of the real trajectory tracking.

Figure 8 represents the beam deviation angle compared to the horizontal one which is extremely reduced in spite of the consideration of rather severe constraints of displacement.

The control signal represented by Figure 9 is characterized by a very satisfactory dynamics. This explains the important results obtained by the proposed cascade control scheme, characterized by a satisfactory bounded control signal and a better trajectory tracking. This study proves the validity of the control scheme to ensure the high performance control for the ball and beam system.

In fact, the contribution of the cascade control scheme seems to be adapted to the amplitude variation of the desired trajectory.

5.3. Discussion

The main objective of this work was the control of single-input single-output nonlinear systems, using input-output feedback linearization technique.

The proposed methodology is based on input-output feedback linearization, gradient descent control method and singular perturbation theory.

The system is first input-output feedback linearized, separating the input-output system behavior from the internal dynamics. Gradient descent control is then used to stabilize the internal dynamics, using a reference trajectory of the system output. This results in a cascade-control scheme, where the outer loop consists of a gradient descent control of the internal dynamics, and the inner loop is the input-output feedback linearization.

The stability analysis of the cascade control scheme is provided using results of singular perturbation theory. The key idea is to introduce a time-scale separation, enforced by the introduction of a small parameter in the controller. Therefore, the quasi steady state assumption is made, allowing the input-output behavior of the system to be decoupled from its internal dynamics.

Thus, each subsystem is analyzed separately, providing a stability proof of the overall system.

The main contribution of the stability analysis is the global exponential stability of the gradient descent control scheme. The use of singular perturbation theory requires exponential stability results.

Indeed, the application of this methodology to the inverted cart-pendulum and ball and beam systems has shown excellent results.