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Chapter 4

PD Control with Neural Compensation

Abstract

In this chapter, we use a neural network to estimate the nonlinear terms of the exoskeleton robots, such as friction, gravity forces, and unmodeled dynamics, to guarantee stability of the closed-loop system with simple PD control law. This approach reduces the computation time with a simple neural network (NN). The learning rules obtained for the NN are very closed to the backpropagation rules [64]. This NN does not need offline previous learning, and the initial parameters are independent of the robot dynamics.

The other part of the chapter uses a modified algorithm to overcome the two drawbacks of PD control at the same time. First, the high-gain observer is joined with the normal PD control, which achieves stability with the knowledge of the friction and gravity. We use both the perturbation method and Lyapunov analysis to prove the stability, where we can see the relation between the observer error and the observer gain.

Second, we also use the RBF neural network to estimate the nonlinear terms of friction and gravity. The learning rules are obtained from the tracking error analysis. We show that the closed-loop system with high-gain observer and neural compensator is stable if the weights have certain learning rules and the observer is fast enough. Some experimental tests are carried out in order to validate the modified PD control.

Keywords

Neural networks; PD control; High-gain observer

4.1 PD control with high gain observer

The dynamics of a serial n-link rigid robot manipulator can be written as [131]

$M (q) \overset{\cdot \cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + F \overset{\cdot}{q} = τ$

(4.1)

where $q \in ℜ^{n}$ denotes the links positions, $\overset{\cdot}{q} \in ℜ^{n}$ denotes the links velocity, $M (q) \in ℜ^{n x n}$ is the inertia matrix, $C (q, \overset{\cdot}{q}) \in ℜ^{n x n}$ is the centripetal and Coriolis matrix, $G (q) \in ℜ^{n}$ is the gravity vector, $F \in R^{n x n}$ is a positive definite diagonal matrix of frictional terms (Coulomb friction), and $τ \in ℜ^{n}$ is the input control vector.

The following properties [131] of the robot's model (4.1) are used.

Property 4.1. The centripetal and Coriolis matrix can be written as

$C (q, \overset{\cdot}{q}) = \sum_{k = 1}^{n} C_{k} (q) \overset{\cdot}{q_{k}}$

where $C_{k, i j} (q) = (\frac{\partial B_{i j}}{\partial q_{k}} + \frac{\partial B_{i k}}{\partial q_{j}} - \frac{\partial B_{j k}}{\partial q_{i}})$ .

Property 4.2. The centripetal and Coriolis matrix satisfies the following relationship:

$‖ C (q, \overset{\cdot}{q}) ‖ ⩽ K_{c} ‖ \overset{\cdot}{q} ‖$

where $K_{c} = \frac{1}{2} \max_{q \in R^{n}} \sum_{k = 1}^{n} ‖ C_{k} (q) ‖$ .

Property 4.3. The centripetal and Coriolis matrix are bounded with respect to q.

4.1.1 Singular perturbation method

A nonlinear system is said to be singularly perturbed if it has the following form:

$\begin{matrix} \overset{\cdot}{x} = f (x, z) \\ ϵ \overset{\cdot}{z} = g (x, z, ϵ) \end{matrix}$

(4.2)

where $x \in R^{n}$ , $z \in R^{m}$ , $ϵ > 0$ is a small constant parameter.

It is assumed that (4.2) has a unique solution with a unique equilibrium point $(x, z) = (0, 0)$ , and has a unique root $z = ϕ (x)$ . (4.2) is in the standard form. Then it is possible to express the slow subsystem:

$\overset{\cdot}{x} = f (x, ϕ (x))$

(4.3)

Since the second equation of (4.2) can be written as $\overset{\cdot}{z} = \frac{g (x, z, ϵ)}{ϵ}$ , its velocity is very fast when $ϵ \to 0$ . Let $ϵ = 0$ in (4.2):

$\begin{matrix} \overset{\cdot}{x} = f (x, z) \\ 0 = g (x, z, 0) \end{matrix}$

(4.4)

The fast subsystem is

$\frac{d z}{d τ} = g (x, z (τ), 0)$

(4.5)

where $τ = \frac{t}{ϵ}$ and x is treated as a fixed unknown parameter. The slow subsystem is called the quasisteady-state system and the fast subsystem is called the boundary-layer system. For singular perturbation analysis the following two assumptions should be satisfied:

1. The equilibrium point $z = 0$ of (4.5) is asymptotically stable uniformly in x and $t_{0}$ (the initial time).
2. The eigenvalues of $\frac{δ g}{δ z}$ evaluated along $x (t)$ of (4.3) and $z = ϕ (x)$ have real parts smaller than a fixed negative number, i.e.,

$Re [λ (\frac{δ g}{δ z})] ⩽ - c < 0$

where c is a positive constant.

With these assumptions, the following theorem in the Vasileva's form is established [74].

Theorem 4.1

If the previous two assumptions are satisfied for (4.2), then the solutions of (4.3) and (4.5) approach the solutions of (4.2) in an interval $[t_{0}, T]$ and $[t_{1}, T]$ , respectively, where $0 ⩽ t_{0} ⩽ t_{1} < T$ .

The motion equations of the serial n-link rigid robot manipulator (4.1) can be rewritten in the state space form [100]:

$\begin{matrix} \overset{\cdot}{x_{1}} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f (x_{1}, x_{2}) + g (x_{1}) u \\ y = x_{1} \end{matrix}$

(4.6)

where $x_{1} = q \in ℜ^{n}$ is the vector of joint positions, $x_{2} = \overset{\cdot}{q} \in ℜ^{n}$ is the vector of joint velocities, $y \in ℜ^{n}$ is the measurable position vector.

$\begin{matrix} f (x_{1}, x_{2}) = M {(x_{1})}^{- 1} [- C (x_{1}, x_{2}) x_{2} - G (x_{1}) - F x_{2}] \\ g (x_{1}) = M {(x_{1})}^{- 1} u \end{matrix}$

(4.7)

The high-gain observer has the following form [100]:

$\begin{matrix} {\overset{\cdot}{\hat{x}}}_{1} = {\hat{x}}_{2} + \frac{1}{ε} H_{p} (y - {\hat{x}}_{1}) \\ {\overset{\cdot}{\hat{x}}}_{2} = \frac{1}{ε^{2}} H_{v} (y - {\hat{x}}_{1}) \end{matrix}$

(4.8)

where ${\hat{x}}_{1} \in ℜ^{n}$ , ${\hat{x}}_{2} \in ℜ^{n}$ denote the estimated values of $x_{1}$ , $x_{2}$ , respectively; ε is chosen as a small positive parameter; and $H_{p}$ , $H_{v}$ are positive definite matrices chosen such that $[\begin{matrix} - H_{p} & I \\ - H_{v} & 0 \end{matrix}]$ is a Hurwitz matrix. Let us define the observer error as

${\tilde{x}}_{1} = x_{1} - {\hat{x}}_{1}, {\tilde{x}}_{2} = x_{2} - {\hat{x}}_{2}$

From (4.6) and (4.8) the dynamic of observer error is

$\begin{matrix} {\overset{\cdot}{\tilde{x}}}_{1} = {\tilde{x}}_{2} - \frac{1}{ε} H_{p} {\tilde{x}}_{1} \\ {\overset{\cdot}{\tilde{x}}}_{2} = - \frac{1}{ε^{2}} H_{v} {\tilde{x}}_{1} + f (x_{1}, x_{2}) + g (x_{1}) u \end{matrix}$

(4.9)

If we define a new pair of variables,

${\tilde{z}}_{1} = {\tilde{x}}_{1}, {\tilde{z}}_{2} = ε {\tilde{x}}_{2}$

(4.9) can be rewritten as

$\begin{matrix} ε {\overset{\cdot}{\tilde{z}}}_{1} = {\tilde{z}}_{2} - H_{p} {\tilde{z}}_{1} \\ ε {\overset{\cdot}{\tilde{z}}}_{2} = - H_{v} {\tilde{z}}_{1} + ε^{2} [f (x_{1}, x_{2}) + g (x_{1}) u] \end{matrix}$

(4.10)

or in matrix form:

$ε \overset{\cdot}{\tilde{z}} = A \tilde{z} + ε^{2} B [f (x_{1}, x_{2}) + g (x_{1}) u]$

(4.11)

where $A = [\begin{matrix} - H_{p} & I \\ - H_{v} & 0 \end{matrix}]$ , $B = [\begin{matrix} 0 \\ I \end{matrix}]$ .

The PD control based on a high-gain observer is

$u = M (x_{1}) {\overset{\cdot}{x_{2}}}^{d} + C (x_{1}, {\hat{x}}_{2}) x_{2}^{d} + F {\hat{x}}_{2} + G (x_{1}) - K_{p} (x_{1} - x_{1}^{d}) - K_{d} ({\hat{x}}_{2} - x_{2}^{d})$

(4.12)

where $x_{1}^{d} \in ℜ^{n}$ is the desired position, $x_{2}^{d}$ is the desired velocity, and $K_{p}$ , $K_{d}$ ϵ $R^{n x n}$ are constant positive matrices. We assume that the desired trajectory and its first two derivatives are bounded. The compensation terms in (4.12) are $M (x_{1}) {\overset{\cdot}{x_{2}}}^{d} + C (x_{1}, {\hat{x}}_{2}) x_{2}^{d} + F {\hat{x}}_{2} + G (x_{1})$ .

The tracking errors of PD control are

${\overline{x}}_{1} = x_{1} - x_{1}^{d}, {\overline{x}}_{2} = x_{2} - x_{2}^{d}$

From (4.6) and (4.12) the tracking errors are

$\begin{matrix} \overset{\cdot}{{\overline{x}}_{1}} = \overset{\cdot}{x_{1}} - {\overset{\cdot}{x}}_{1}^{d} = {\overline{x}}_{2} \\ \overset{\cdot}{{\overline{x}}_{2}} = \overset{\cdot}{x_{2}} - {\overset{\cdot}{x}}_{2}^{d} = f (x_{1}, x_{2}) + g (x_{1}) u (x_{1}, x_{1}^{d}, {\hat{x}}_{2}, x_{2}^{d}, {\overset{\cdot}{x}}_{2}^{d}) - {\overset{\cdot}{x}}_{2}^{d} \end{matrix}$

(4.13)

where

$\begin{matrix} H = M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} [- F {\tilde{x}}_{2} - K_{p} {\overline{x}}_{1} + K_{d} {\tilde{x}}_{2} - K_{d} {\overline{x}}_{2} \\ - C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d}) {\tilde{x}}_{2} - C ({\overline{x}}_{1} + x_{1}^{d}, {\overline{x}}_{2} + x_{2}^{d}) {\overline{x}}_{2}] \end{matrix}$

Including the control (4.12) into (4.10), then

$\begin{matrix} ε {\overset{\cdot}{\tilde{z}}}_{1} = {\tilde{z}}_{2} - H_{p} {\tilde{z}}_{1} \\ ε {\overset{\cdot}{\tilde{z}}}_{2} = - H_{v} {\tilde{z}}_{1} + ε^{2} H \end{matrix}$

The closed-loop is the combination of (4.6), (4.12), and (4.8),

$\begin{matrix} {\overset{\cdot}{\overline{x}}}_{1} = {\overline{x}}_{2} \\ {\overset{\cdot}{\overline{x}}}_{2} = H \\ ϵ {\overset{\cdot}{\tilde{z}}}_{1} = {\tilde{z}}_{2} - H_{p} {\tilde{z}}_{1} \\ ϵ {\overset{\cdot}{\tilde{z}}}_{2} = - H_{v} {\tilde{z}}_{1} + ϵ^{2} H \end{matrix}$

(4.14)

with the equilibrium point $({\overline{x}}_{1}, {\overline{x}}_{2}, {\tilde{z}}_{1}, {\tilde{z}}_{2}) = (0, 0, 0, 0)$ . Clearly, (4.14) has the singularly perturbed form (4.2).

Let $ϵ = 0$ :

$\begin{matrix} 0 = {\tilde{z}}_{2} - H_{p} {\tilde{z}}_{1} \\ 0 = - H_{v} {\tilde{z}}_{1} \end{matrix}$

which implies that

$\begin{matrix} {\tilde{z}}_{1} = 0 = x_{1} - {\hat{x}}_{1} \\ {\tilde{z}}_{2} = 0 = ϵ (x_{2} - {\hat{x}}_{2}) \end{matrix}$

which has an equilibrium point $(x_{1}, x_{2}) = ({\hat{x}}_{1}, {\hat{x}}_{2})$ . The system (4.14) therefore is in the standard form. Substituting the equilibrium point into the first two equations of (4.14), we obtain the quasisteady-state model:

$\begin{matrix} {\overset{\cdot}{\overline{x}}}_{1} = {\overline{x}}_{2} \\ {\overset{\cdot}{\overline{x}}}_{2} = H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, 0) \\ = M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} [- K_{p} {\overline{x}}_{1} - K_{d} {\overline{x}}_{2} - C ({\overline{x}}_{1} + x_{1}^{d}, {\overline{x}}_{2} + x_{2}^{d}) {\overline{x}}_{2}] \end{matrix}$

(4.15)

The boundary layer system of (4.14) is

$\begin{matrix} \frac{δ}{δ τ} {\tilde{z}}_{1} (τ) = {\tilde{z}}_{2} (τ) - H_{p} {\tilde{z}}_{1} (τ) \\ \frac{δ}{δ τ} {\tilde{z}}_{2} (τ) = - H_{v} {\tilde{z}}_{1} (τ) \end{matrix}$

(4.16)

where $τ = \frac{t}{ϵ}$ . (4.16) can be written as

$\frac{δ}{δ τ} \tilde{z} (τ) = A \tilde{z} (τ)$

(4.17)

where $\tilde{z} (τ) = {[{\tilde{z}}_{1}^{T} (τ), {\tilde{z}}_{2}^{T} (τ)]}^{T}$ and A is defined as in (4.11). The following theorem will show the stability properties of the equilibrium points $({\overline{x}}_{1}, {\overline{x}}_{2}) = (0, 0)$ and $({\tilde{z}}_{1}, {\tilde{z}}_{2}) = (0, 0)$ for (4.15) and (4.17), respectively.

Theorem 4.2

The equilibrium point $({\overline{x}}_{1}, {\overline{x}}_{2}) = (0, 0)$ of (4.15) and the equilibrium point $({\tilde{z}}_{1}, {\tilde{z}}_{2}) = (0, 0)$ of (4.17) are asymptotic stable.

Proof

For the first system, consider the following candidate Lyapunov function:

$V_{1} ({\overline{x}}_{1}, {\overline{x}}_{2}) = \frac{1}{2} {\overline{x}}_{2}^{T} M ({\overline{x}}_{1} + x_{1}^{d}) {\overline{x}}_{2} + \frac{1}{2} {\overline{x}}_{1}^{T} K_{P} {\overline{x}}_{1}$

(4.18)

with its derivative with respect to time and along (4.15),

$\begin{matrix} \overset{\cdot}{V_{1}} = {\overline{x}}_{2}^{T} [- K_{p} {\overline{x}}_{1} - K_{d} {\overline{x}}_{2} - C ({\overline{x}}_{1} + x_{1}^{d}, {\overline{x}}_{2} + x_{2}^{d}) {\overline{x}}_{2}] + \frac{1}{2} {\overline{x}}_{2}^{T} \overset{\cdot}{M} ({\overline{x}}_{1} + x_{1}^{d}) {\overline{x}}_{2} + {\overline{x}}_{1}^{T} K_{P} {\overline{x}}_{2} \\ = {\overline{x}}_{2}^{T} [\frac{1}{2} \overset{\cdot}{M} ({\overline{x}}_{1} + x_{1}^{d}) - C ({\overline{x}}_{1} + x_{1}^{d}, {\overline{x}}_{2} + x_{2}^{d})] {\overline{x}}_{2} - {\overline{x}}_{2}^{T} K_{d} {\overline{x}}_{2} \\ = - {\overline{x}}_{2}^{T} K_{d} {\overline{x}}_{2} ⩽ 0 \end{matrix}$

Applying LaSalle's theorem, the only solutions of (4.15) evolving in the set

$Ω_{V_{1}} = {({\overline{x}}_{1}, {\overline{x}}_{2}) | {\overline{x}}_{2} = 0}$

are

$\begin{matrix} {\overset{\cdot}{\overline{x}}}_{1} = 0 \\ 0 = - [M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} K_{p}] {\overline{x}}_{1} \end{matrix}$

which implies that ${\overline{x}}_{1} = 0$ (using Property 4.2 of the robot's dynamic).

For the second system, since A is a Hurwitz matrix, there exists a positive definite matrix P such that

$A^{T} P + P A = - Q$

where Q is a positive definite matrix.

Consider the candidate Lyapunov function:

$V_{2} ({\tilde{z}}_{1}, {\tilde{z}}_{2}) = {\tilde{z}}^{T} P \tilde{z}$

with its derivative with respect to time and along (4.17),

$\overset{\cdot}{V_{2}} = {\tilde{z}}^{T} (A^{T} P + P A) \tilde{z} = - {\tilde{z}}^{T} Q \tilde{z} < 0$

□

The advantage of this approach is that the singularly perturbed analysis may divide the original problem in two systems: the slow subsystem or quasisteady state system and the fast subsystem or boundary layer system. Then both systems can be studied independently with the boundary layer system faster enough compared with the slow subsystem. These two subsystems explain why the dynamic of the high-gain observer is faster than the dynamic of the robot and the PD control. If the slow subsystem is considered as static the high-gain observer in the fast time scale τ can make the estimated states $({\hat{x}}_{1}, {\hat{x}}_{2})$ converge to the real states $(x_{1}, x_{2})$ . In the slow time scale t the slow subsystem composed by the robot and the PD control can use then the estimated states $({\hat{x}}_{1}, {\hat{x}}_{2})$ .

Remark 4.1

From the point of the singular perturbation analysis, one can see that the high-gain observer (4.8) has a faster dynamic than the robot (4.6) and the PD control (4.12). Under the assumption of $ϵ = 0$ , the observer error and the tracking error of PD control are asymptotic stable if the joint velocities are measurable.

Since it is impossible for the high-gain observer (4.8) to have $ϵ = 0$ , it is necessary to find a positive value of ϵ for which the stability properties are valid. For this purpose, we proposed a modified version of [120].

Theorem 4.3

If there exists a continuous interval $Γ = (0, \overline{ϵ})$ such that for all $ϵ \in Γ$ satisfies

$\begin{matrix} (\frac{d^{2} K_{2}^{2}}{2}) ϵ^{4} + ((1 - d) d β_{1} K_{2}) ϵ^{2} + ((1 - d) α_{1} K_{1}) ϵ \\ + \frac{{(1 - d)}^{2} β_{1}^{2}}{2} - (1 - d) d α_{1} α_{2} < 0 \end{matrix}$

(4.19)

where $α_{1}$ , $α_{2}$ , $β_{1}$ , $K_{1}$ , $K_{2}$ are nonnegative constants, $0 < d < 1$ , $\overline{ϵ}$ is the positive root of (4.19), then the origin $(\overline{x}, \tilde{z}) = (0, 0)$ of (4.14) is asymptotically stable.

Proof

Let us select Lyapunov function for (4.14) as

$V_{c l} (x, z) = (1 - d) V_{1} ({\overline{x}}_{1}, {\overline{x}}_{2}) + d V_{2} ({\tilde{z}}_{1}, {\tilde{z}}_{2})$

(4.20)

whose derivative with respect to time and along the trajectories of (4.14),

$\begin{matrix} \overset{\cdot}{V_{c l}} = (1 - d) [{\overline{x}}_{2}^{T} M ({\overline{x}}_{1} + x_{1}^{d}) \overset{\cdot}{{\overline{x}}_{2}} + \frac{1}{2} {\overline{x}}_{2}^{T} \overset{\cdot}{M} ({\overline{x}}_{1} + x_{1}^{d}) {\overline{x}}_{2} + {\overline{x}}_{1}^{T} K_{P} {\overline{x}}_{2}] + \frac{d}{ϵ} [{\tilde{z}}^{T} P \overset{\cdot}{\tilde{z}} + {\overset{\cdot}{\tilde{z}}}^{T} P \tilde{z}] \\ = (1 - d) [{\overline{x}}_{2}^{T} M ({\overline{x}}_{1} + x_{1}^{d}) H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, 0) + \frac{1}{2} {\overline{x}}_{2}^{T} \overset{\cdot}{M} ({\overline{x}}_{1} + x_{1}^{d}) {\overline{x}}_{2} + {\overline{x}}_{1}^{T} K_{P} {\overline{x}}_{2}] \\ + (1 - d) [{\overline{x}}_{2}^{T} M ({\overline{x}}_{1} + x_{1}^{d}) [H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, ϵ) - H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, 0)]] \\ + \frac{d}{ϵ} [{\tilde{z}}^{T} (P A + A^{T} P) \tilde{z}] + \frac{d}{ϵ} [2 {\tilde{z}}^{T} P ϵ^{2} B H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, ϵ)] \end{matrix}$

From Theorem 4.2, we have

$\begin{matrix} \overset{\cdot}{V_{c l}} = - (1 - d) {\overline{x}}^{T} [\begin{matrix} 0 & 0 \\ 0 & K_{d} \end{matrix}] \overline{x} \\ + (1 - d) [{\overline{x}}_{2}^{T} M ({\overline{x}}_{1} + x_{1}^{d}) [H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, ϵ) - H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, 0)]] \\ - \frac{d}{ϵ} {\tilde{z}}^{T} Q \tilde{z} + \frac{d}{ϵ} [2 {\tilde{z}}^{T} P ϵ^{2} B H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, ϵ)] \end{matrix}$

(4.21)

At this point, we need to solve

$\begin{matrix} [{\overline{x}}_{2}^{T} M ({\overline{x}}_{1} + x_{1}^{d}) [H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, ϵ) - H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, 0)]] \\ = {\overline{x}}_{2}^{T} [- F {\tilde{x}}_{2} + K_{d} {\tilde{x}}_{2} - C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d}) {\tilde{x}}_{2}] \\ = {\overline{x}}^{T} \frac{1}{ϵ} [\begin{matrix} 0 & 0 \\ 0 & - F + K_{d} - C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d}) \end{matrix}] \tilde{z} \end{matrix}$

(4.22)

Notice that this matrix is bounded because F and $K_{d}$ are constant matrices and property 6 of the robot's dynamic.

Using the robot's dynamic Property 4.1 and Property 4.2 and (4.22), we can conclude that

$\begin{matrix} [2 {\tilde{z}}^{T} P ϵ^{2} B H ({\overline{x}}_{1}, {\overline{x}}_{2}, x_{1}^{d}, x_{2}^{d}, ϵ)] \\ ⩽ {\tilde{z}}^{T} P ϵ [\begin{matrix} 0 & 0 \\ 0 & - 2 M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} (F - K_{d} + C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d})) \end{matrix}] \tilde{z} \\ + {\tilde{z}}^{T} P ϵ^{2} [\begin{matrix} 0 & 0 \\ - M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} K_{p} & - M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} (K_{d} + C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d})) \end{matrix}] \overline{x} \end{matrix}$

(4.23)

Applying (4.22) and (4.23) to (4.21),

$\begin{matrix} \overset{\cdot}{V_{c l}} = - (1 - d) {\overline{x}}^{T} [\begin{matrix} 0 & 0 \\ 0 & K_{d} \end{matrix}] \overline{x} + (1 - d) \frac{1}{ϵ} {\overline{x}}^{T} [\begin{matrix} 0 & 0 \\ 0 & - F + K_{d} - C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d}) \end{matrix}] \tilde{z} \\ - d {\tilde{z}}^{T} (\frac{1}{ϵ} Q - P [\begin{matrix} 0 & 0 \\ 0 & - 2 M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} (F - K_{d} + C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d})) \end{matrix}]) \tilde{z} \\ + d ϵ {\tilde{z}}^{T} P [\begin{matrix} 0 & 0 \\ - M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} K_{p} & - M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} (K_{d} + C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d})) \end{matrix}] \overline{x} \\ ⩽ - (1 - d) α_{1} ψ^{2} (\overline{x}) + (1 - d) \frac{1}{ϵ} β_{1} ψ (\overline{x}) ϕ (\tilde{z}) - d [\frac{α_{2}}{ϵ} - K_{1}] ϕ^{2} (\tilde{z}) + d ϵ K_{2} ψ (\overline{x}) ϕ (\tilde{z}) \end{matrix}$

(4.24)

where $α_{1} = ‖ [\begin{matrix} 0 & 0 \\ 0 & K_{d} \end{matrix}] ‖$ , $α_{2} = ‖ Q ‖$ , $β_{1} = ‖ [\begin{matrix} 0 & 0 \\ 0 & - F + K_{d} - C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d}) \end{matrix}] ‖$ , $ψ (\overline{x}) = ‖ \overline{x} ‖$ , $ϕ (\tilde{z}) = ‖ \tilde{z} ‖$

$\begin{matrix} K_{1} = ‖ P [\begin{matrix} 0 & 0 \\ 0 & - 2 M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} (F - K_{d} + C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d})) \end{matrix}] ‖ \\ K_{2} = ‖ P [\begin{matrix} 0 & 0 \\ - M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} K_{p} & - M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} (K_{d} + C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d})) \end{matrix}] ‖ \end{matrix}$

(4.24) can be written in a matrix form as

$\overset{\cdot}{V_{c l}} ⩽ - {[\begin{matrix} ψ (x) \\ ϕ (z) \end{matrix}]}^{T} T [\begin{matrix} ψ (x) \\ ϕ (z) \end{matrix}]$

where $T = [\begin{matrix} (1 - d) α_{1} & - \frac{(1 - d) β_{1}}{2 ϵ} - \frac{ϵ d K_{2}}{2} \\ - \frac{(1 - d) β_{1}}{2 ϵ} - \frac{ϵ d K_{2}}{2} & d [\frac{α_{2}}{ϵ} - K_{1}] \end{matrix}]$ . T has to be a positive definite matrix, and T is positive definite if there exists a continuous interval $Γ = (0, \overline{ϵ})$ such that for all ϵ∈ Γ satisfies

$\begin{matrix} (\frac{d^{2} K_{2}^{2}}{2}) ϵ^{4} + ((1 - d) d β_{1} K_{2}) ϵ^{2} \\ + ((1 - d) α_{1} K_{1}) ϵ + \frac{{(1 - d)}^{2} β_{1}^{2}}{2} - (1 - d) d α_{1} α_{2} < 0 \end{matrix}$

Then $\overline{ϵ}$ will be the upper bound of ϵ. □

Remark 4.2

Since (4.19) has 4 possible solutions the theorem will be valid if there exists a positive real root of (4.19) such that in the interval $[0, \overline{ϵ}]$ (4.19) is negative.

4.1.2 Lyapunov method

The motion equations of the serial n-link rigid robot manipulator (4.1) can be rewritten in the state space form (4.6). We assume u to be a bounded feedback control $u = η (x)$ , $\sup u = \overline{u}$ , so (4.6) can be rewritten as

$\begin{matrix} {\overset{\cdot}{x}}_{i} = x_{i - 1}, i = 1 \dots n - 1 \\ {\overset{\cdot}{x}}_{n} = Φ (x) \\ y = x_{1} \end{matrix}$

(4.25)

where $Φ (x) = F (x) + G (x) η (x)$ . If the solution of (4.25) exists, for any $t \in [0, \infty]$ , it is reasonable to make following assumption.

A4.1: The function $Φ (x)$ is bounded over $[0, \infty]$ .

Let us construct the high-gain observer as

$\begin{matrix} {\overset{\cdot}{\hat{x}}}_{i} = {\hat{x}}_{i - 1} + \frac{α_{i}}{ε^{i}} (y - \hat{y}) \\ {\overset{\cdot}{\hat{x}}}_{n} = \frac{α_{n}}{ε^{n}} (y - \hat{y}) \\ \hat{y} = {\hat{x}}_{1} \end{matrix}$

(4.26)

where ε is a small positive parameter and positive constant $α_{i}$ are chosen such that the roots of

$s^{n} + α_{1} s^{n - 1} + \dots + a_{n - 1} s + a_{n} = 0$

(4.27)

have negative real parts. The observer error is defined as

$\tilde{x} = x - \hat{x}, x = {[x_{1} \dots x_{n}]}^{T}$

Theorem 4.4

Under the assumption A4.1, the observer error between the high gain observer (4.26) and nonlinear system (4.25) is asymptotically stable:

$\lim_{t \to \infty} \tilde{x} (t) = 0$

(4.28)

Proof

The error dynamics corresponding to (4.25) and (4.26) are

$\begin{matrix} {\overset{\cdot}{\tilde{x}}}_{i} = {\tilde{x}}_{i - 1} - \frac{α_{i}}{ε^{i}} {\tilde{x}}_{1} \\ {\overset{\cdot}{\tilde{x}}}_{n} = - \frac{α_{n}}{ε^{n}} {\tilde{x}}_{1} + Φ (x) \end{matrix}$

(4.29)

If we define $ξ = [{\tilde{x}}_{1}, ε {\tilde{x}}_{2}, \dots ε^{n - 1} {\tilde{x}}_{n}]$ , the error equation (4.29) is

$ε \overset{\cdot}{ξ} = A ξ + ε^{n} b Φ (x)$

(4.30)

where $A = [\begin{matrix} - a_{1} \\ ⋮ \\ - a_{n} \end{matrix} I]$ , $b = {[0 \dots 0, 1]}^{T}$ . Because $α_{i}$ satisfies (4.27), there exists a positive definite matrix P such that

$A^{T} P + P A = - I$

Now let us define a Lyapunov function as $V = ξ^{T} P ξ$ , considering (4.30) we have

$\begin{matrix} \overset{\cdot}{V} = \frac{1}{ε} ξ^{T} (A^{T} P + P A) ξ + 2 ε^{n - 1} Φ^{T} (x) b^{T} P ξ \\ = - \frac{1}{ε} ξ^{T} ξ + 2 ε^{n - 1} Φ^{T} (x) b^{T} P ξ \\ ⩽ - \frac{1}{ε} {‖ ξ ‖}^{2} + 2 ε^{n - 1} ‖ P b Φ (x) ‖ ‖ ξ ‖ \end{matrix}$

(4.31)

Under the assumption A4.1, we may conclude that $‖ P b Φ (x) ‖$ is bounded as

$c_{T} : = \sup_{t \in [0, T]} ‖ P b Φ (x) ‖$

So (4.31) is

$\overset{\cdot}{V} ⩽ - \frac{1}{ε} {‖ ξ ‖}^{2} + \frac{1}{ε} K_{s} ‖ ξ ‖, K_{s} = 2 ε^{n} c_{T}$

(4.32)

It is noted that if

$K_{s} < ‖ ξ ‖$

(4.33)

then $\overset{\cdot}{V} < 0$ , $\forall t \in [0, T]$ , ξ is bounded. It is easy to see that, for $ε < 1$ , $ξ_{i} ⩽ {\tilde{x}}_{i}$ , $i = 1 \dots n$ , the condition (4.33) means $K_{s} < ‖ \tilde{x} ‖$ . As $K_{s} = 2 ε^{n} c_{T}$ , for and finite T, $K_{s}$ can be made arbitrarily small for values of $ε < 1$ . Furthermore, if $K_{s}$ can be small as $K_{s} < α ‖ ξ ‖$ , $0 < α < 1$ , and $Φ (x)$ is bounded over $[0, \infty]$ (A4.1), then

$\overset{\cdot}{V} ⩽ - \frac{1}{ε} {‖ ξ ‖}^{2} + \frac{1}{ε} α {‖ ξ ‖}^{2} = - \frac{1}{ε} (1 - α) {‖ ξ ‖}^{2} < 0$

$\frac{1}{ε} (1 - α) \int_{0}^{\infty} {‖ ξ ‖}^{2} ⩽ V (0) - V (\infty) < \infty$

That means $ξ \in L_{2} \cap L_{\infty}$ , from the error equation (4.30) $\overset{\cdot}{ξ} \in L_{\infty}$ . Using Barlalat's lemma, we have (4.28). □

Remark 4.3

The high gain observer (4.26) is not dependent on the nonlinear plant (4.25) or (4.1); only output y is needed. The observer (4.26) can also be regarded as to estimate the derivatives of the output; see (4.1).

4.2 PD control with neural compensator

It is well known that the PD control with friction and gravity compensation may reach asymptotic stable [71]. Using neural networks to compensate the nonlinearity of the robot dynamic may be found in [85] and [78]. In [85] the authors used neural networks to approximate the whole nonlinearity of the robot dynamic. With this neural feedforward compensator and a PD control, they can guarantee a good track performance.

4.2.1 PD control with single layer neural compensation

If G and F are unknown, a neural network can be used to approximate them as

$G + F - f = {\hat{W}}_{t} σ (x) + η$

(4.34)

where η is bounded modeling error, $η^{T} Λ_{1} η ⩽ \overline{η}$ , $Λ_{1}$ is any matrix such that $Λ_{1} = Λ_{1}^{T} > 0$ , $σ (x)$ is the activation function, x is the input to the NN,

$x = {[q, \dot{q}, q^{d}, {\dot{q}}^{d}]}^{T}$

(4.35)

f is the other uncertainties.

The neural PD control is

$τ = - K_{v} r - {\hat{W}}_{t} σ (x)$

(4.36)

Let us define the tracking error as

${\overline{x}}_{1} = x_{1} - x_{1}^{d}, {\overline{x}}_{2} = x_{2} - x_{2}^{d}$

(4.37)

where $x_{1} = q$ , $x_{2} = \dot{q}$ , $x_{1}^{d} = q^{d}$ , $x_{2}^{d} = {\dot{q}}^{d}$ , and the auxiliary error is defined as

$r = {\overline{x}}_{2} + Λ {\overline{x}}_{1}, Λ = Λ^{T} > 0$

(4.38)

If the Lyapunov function candidate is

$V = \frac{1}{2} r^{T} M r + \frac{1}{2} t r ({\tilde{W}}_{t}^{T} K_{w}^{- 1} {\tilde{W}}_{t})$

and the updating law is $\overset{\cdot}{{\hat{W}}_{t} =} K_{w} σ (x) r^{T}$ , using

$r^{T} η ⩽ r^{T} Λ_{1}^{- 1} r + η^{T} Λ_{1} η ⩽ r^{T} Λ_{1}^{- 1} r + \overline{η}$

$\overset{\cdot}{V} ⩽ - r^{T} (K_{v} - Λ_{1}^{- 1}) r + \overline{η}$

(4.39)

where $\overline{η}$ is upper bound of η. To guarantee stability, we need $K_{v} > Λ_{1}$ . The tacking error is bounded, and ${‖ r ‖}_{K_{v} - Λ_{1}^{- 1}}^{2}$ converges to $\overline{η}$ .

Or as in [86], $r^{T} (K_{v} - Λ_{1}) r$ can be transformed into

$\overset{\cdot}{V} ⩽ - K_{v} {‖ r ‖}^{2} + α ‖ r ‖$

where α is a positive constant. In order to assure $\overset{\cdot}{V} ⩽ 0$ , we need $‖ r ‖ > \frac{α}{K_{v}}$ . This means

$r \to 0 when K_{v} \to \infty$

4.2.2 PD control with a multilayer feedforward neural compensator

The friction and gravity in (4.1) can be also approximated by the multilayer neural network:

$P (q, \overset{\cdot}{q}) = W^{⁎} Φ (V^{⁎} x) + \tilde{P}$

(4.40)

where $P (q, \overset{\cdot}{q}) = G (q) + F \overset{\cdot}{q}$ , where $W^{⁎}$ , $V^{⁎}$ are fixed bounded weights, $\tilde{P}$ is the approximated error, whose magnitude depends on the values of $W^{⁎}$ and $V^{⁎}$ .

The estimation of $P (q, \overset{\cdot}{q})$ may be defined as $\hat{P} (q, \overset{\cdot}{q})$ :

$\hat{P} (q, \overset{\cdot}{q}) = \hat{W} Φ (\hat{V} x)$

(4.41)

In order to implement the neural networks, the following assumption for $\tilde{P}$ in (4.40) is needed:

${\tilde{P}}^{T} Λ_{1} \tilde{P} ⩽ \overline{η}, \overline{η} > 0$

(4.42)

It is clear that Φ satisfies the Lipschitz condition:

$\begin{matrix} {\tilde{Φ}}_{t} : = Φ (V^{⁎ T} x) - Φ ({\hat{V}}_{t}^{T} x) = D_{σ} {\tilde{V}}_{t}^{T} x + ν_{σ}, \\ D_{σ} = \frac{\partial Φ^{T} (Z)}{\partial Z} |_{Z = {\hat{V}}_{t}^{T} x}, {‖ ν_{σ} ‖}_{Λ}^{2} ⩽ l {‖ {\tilde{V}}_{t}^{T} x ‖}_{Λ}^{2} \end{matrix}$

(4.43)

where Λ is a positive definite matrix, $l > 0$ ,

$\tilde{W} = W^{⁎} - \hat{W}, \tilde{V} = V^{⁎} - \hat{V}$

(4.44)

Remark 4.4

One can see that this condition is similar with [85] (Taylor series). The upper bound found for $ν_{σ}$ will be essential for proving stability of the PD control with a high gain observer and neural compensator.

We first assume the velocities are measurable, so the PD control is

$u = - K_{p} (x_{1} - x_{1}^{d}) - K_{d} (x_{2} - x_{2}^{d}) + \hat{W} Φ (\hat{V} x)$

(4.45)

where $x_{1}^{d} \in ℜ^{n}$ is the desired position and $x_{2}^{d}$ is the desired velocity. They are defined in (4.37). We assume the references are bounded as

$‖ x_{1}^{d} ‖ ⩽ d_{1}, ‖ x_{2}^{d} ‖ ⩽ d_{2}$

The input control vector is $τ = u$ . $K_{p}$ and $K_{d}$ are positive defined matrices corresponding to proportional and derivative coefficients.

Theorem 4.5

If the following learning laws for the weights of neural networks (4.41) are used

$\begin{matrix} {\overset{\cdot}{\hat{W}}}_{t} = - d_{t} K_{w} σ ({\hat{V}}_{t}^{T} x) {\overline{x}}_{2}^{T} - d_{t} K_{w} D_{σ} {\tilde{V}}_{t}^{T} x {\overline{x}}_{2}^{T} \\ {\overset{\cdot}{\hat{V}}}_{t} = - d_{t} K_{v} {\overline{x}}_{2}^{T} D_{σ} {\hat{W}}_{t}^{T} x + d_{t} l K_{v} x^{T} {\tilde{V}}_{t} x Λ_{3} \end{matrix}$

(4.46)

where $0 < Λ_{1} = Λ_{1}^{T} \in ℜ^{n \times n}$ ,

$\begin{matrix} d_{t} = {\begin{matrix} 0 & if & {‖ {\overline{x}}_{2} ‖}_{R}^{2} ⩽ \frac{χ^{2}}{4 λ_{\min} (Γ)} \\ 1 & if & {‖ {\overline{x}}_{2} ‖}_{R}^{2} > \frac{χ^{2}}{4 λ_{\min} (Γ)} \end{matrix} \\ Γ = K_{d} - \frac{1}{4} Λ_{3}^{- 1} - k_{c} ‖ d_{1} ‖ I - R, R = R^{T} > 0 \\ χ = \overline{η} + k_{c} {‖ d_{1} ‖}^{2} + λ_{\max} (M) ‖ d_{2} ‖ \end{matrix}$

then:

(I) The weights of neural networks ${\hat{W}}_{t}$ , ${\hat{V}}_{t}$ , and tracking error ${\overline{x}}_{2}$ are bounded.

(II) For any $T \in (0, \infty)$ the tracking error ${\overline{x}}_{2}$ converges to the residual set

$D_{{\overline{x}}_{2}} = {{\overline{x}}_{2} | {‖ {\overline{x}}_{2} ‖}_{R}^{2} ⩽ \frac{χ^{2}}{4 λ_{\min} (Γ)}}$

(4.47)

Proof

From (4.45) the closed-loop system is

$M (x_{1}) {\overset{\cdot}{x}}_{2} + C (x_{1}, \overset{\cdot}{x_{1}}) x_{2} + K_{p} {\overline{x}}_{1} + K_{d} {\overline{x}}_{2} - {\hat{W}}_{t} Φ ({\hat{V}}_{t} x) + W^{⁎} Φ (V^{⁎} x) + \tilde{P} = 0$

(4.48)

The proposed candidate Lyapunov function is

$V_{1} = \frac{1}{2} {\overline{x}}_{2}^{T} M {\overline{x}}_{2} + \frac{1}{2} {\overline{x}}_{1}^{T} K_{p} {\overline{x}}_{1} + \frac{1}{2} t r ({\tilde{W}}_{t}^{T} K_{w}^{- 1} {\tilde{W}}_{t}) + \frac{1}{2} t r ({\tilde{V}}_{t}^{T} K_{v}^{- 1} {\tilde{V}}_{t})$

(4.49)

where $K_{w}$ and $K_{v}$ are any positive definite constant matrices. The derivative of (4.49) is

$\begin{matrix} {\overset{\cdot}{V}}_{1} = {\overline{x}}_{2}^{T} M (x_{1}) {\overset{\cdot}{\overline{x}}}_{2} + \frac{1}{2} {\overline{x}}_{2}^{T} \overset{\cdot}{M} (x) {\overline{x}}_{2} + {\overline{x}}_{2}^{T} K_{p} {\overline{x}}_{1} \\ + t r ({\tilde{W}}_{t}^{T} K_{w}^{- 1} \overset{\cdot}{{\tilde{W}}_{t}}) + t r ({\tilde{V}}_{t}^{T} K_{v}^{- 1} {\overset{\cdot}{\tilde{V}}}_{t}) \end{matrix}$

(4.50)

Using (4.48),

$\begin{matrix} {\overline{x}}_{2}^{T} M {\overset{\cdot}{\overline{x}}}_{2} = - {\overline{x}}_{2}^{T} M {\overset{\cdot}{x}}_{2}^{d} - {\overline{x}}_{2}^{T} [C x_{2} + K_{p} {\overline{x}}_{1} + K_{d} {\overline{x}}_{2} - {\hat{W}}_{t} Φ ({\hat{V}}_{t} x) + W^{⁎} Φ (V^{⁎} x) + \tilde{P}] \\ = - {\overline{x}}_{2}^{T} M {\overset{\cdot}{x}}_{2}^{d} - {\overline{x}}_{2}^{T} C {\overline{x}}_{2} - {\overline{x}}_{2}^{T} C x_{2}^{d} - {\overline{x}}_{2}^{T} K_{p} {\overline{x}}_{1} - {\overline{x}}_{2}^{T} K_{d} {\overline{x}}_{2} \\ - {\overline{x}}_{2}^{T} [- {\hat{W}}_{t} Φ ({\hat{V}}_{t} x) + W^{⁎} Φ (V^{⁎} x) + \tilde{P}] \end{matrix}$

(4.51)

Using Property 4.2 and (4.51), (4.50) becomes

$\begin{matrix} {\overset{\cdot}{V}}_{1} = - {\overline{x}}_{2}^{T} M {\overset{\cdot}{x}}_{2}^{d} - {\overline{x}}_{2}^{T} C x_{2}^{d} - {\overline{x}}_{2}^{T} K_{d} {\overline{x}}_{2} \\ - {\overline{x}}_{2}^{T} [- {\hat{W}}_{t} Φ ({\hat{V}}_{t} \hat{x}) + W^{⁎} Φ (V^{⁎} \hat{x}) + \tilde{P}] + t r ({\tilde{W}}_{t}^{T} K_{w}^{- 1} \overset{\cdot}{{\tilde{W}}_{t}}) + t r ({\tilde{V}}_{t}^{T} K_{v}^{- 1} {\overset{\cdot}{\tilde{V}}}_{t}) \end{matrix}$

(4.52)

The term $- {\hat{W}}_{t} Φ ({\hat{V}}_{t} x) + W^{⁎} Φ (V^{⁎} x)$ can be expressed as

$\begin{matrix} W^{⁎} Φ (V^{⁎} x) - W^{⁎} Φ ({\hat{V}}_{t} x) + W^{⁎} Φ ({\hat{V}}_{t} x) - {\hat{W}}_{t} Φ ({\hat{V}}_{t} x) \\ = {\tilde{W}}_{t}^{T} Φ ({\hat{V}}_{t}^{T} x) + W^{⁎ T} D_{σ} {\tilde{V}}_{t}^{T} x + W^{⁎ T} ν_{σ} \\ = {\tilde{W}}_{t}^{T} Φ ({\hat{V}}_{t}^{T} x) + {\hat{W}}^{T} D_{σ} {\tilde{V}}_{t}^{T} x + {\tilde{W}}^{T} D_{σ} {\tilde{V}}_{t}^{T} x + W^{⁎ T} ν_{σ} \end{matrix}$

(4.53)

In view of the matrix inequality,

$X^{T} Y + {(X^{T} Y)}^{T} ⩽ X^{T} Λ^{- 1} X + Y^{T} Λ Y$

(4.54)

which is valid for any $X, Y \in ℜ^{n \times k}$ and for any positive defined matrix $0 < Λ = Λ^{T} \in ℜ^{n \times n}$ [153].

$- {\overline{x}}_{2}^{T} [\tilde{P} + W^{⁎ T} ν_{σ}]$ can be estimated as

$\begin{matrix} - {\overline{x}}_{2}^{T} [\tilde{P} + W^{⁎ T} ν_{σ}] ⩽ ‖ {\overline{x}}_{2} ‖ ‖ \tilde{P} ‖ + \frac{1}{4} {\overline{x}}_{2}^{T} Λ_{1}^{- 1} {\overline{x}}_{2} + {[W^{⁎ T} ν_{σ}]}^{T} Λ_{1} [W^{⁎ T} ν_{σ}] \\ ⩽ ‖ {\overline{x}}_{2} ‖ \overline{η} + \frac{1}{4} {\overline{x}}_{2}^{T} Λ_{1}^{- 1} {\overline{x}}_{2} + l {‖ {\tilde{V}}_{t}^{T} x ‖}_{Λ_{3}}^{2} \end{matrix}$

(4.55)

where $Λ_{3} : = W^{⁎} Λ_{1} W^{⁎ T}$ . Using

$\begin{matrix} - {\overline{x}}_{2}^{T} C (x_{1}, x_{2}) x_{2}^{d} ⩽ ‖ {\overline{x}}_{2} ‖ ‖ C (x_{1}, x_{2}) ‖ ‖ x_{2}^{d} ‖ ⩽ ‖ {\overline{x}}_{2}^{d} ‖ {\overline{x}}_{2}^{T} k_{c} I {\overline{x}}_{2} + k_{c} {‖ {\overline{x}}_{2}^{d} ‖}^{2} ‖ {\overline{x}}_{2} ‖ \\ - {\overline{x}}_{2}^{T} M (x_{1}) {\overset{\cdot}{x}}_{2}^{d} ⩽ λ_{\max} (M) ‖ {\overline{\overset{\cdot}{x}}}_{2}^{d} ‖ ‖ {\overline{x}}_{2} ‖ \end{matrix}$

(4.56)

$\begin{matrix} {\overset{\cdot}{V}}_{1} ⩽ - {\overline{x}}_{2}^{T} Γ {\overline{x}}_{2} + χ ‖ {\overline{x}}_{2} ‖ + L_{w} + L_{v} - {\overline{x}}_{2}^{T} R {\overline{x}}_{2} \\ ⩽ - λ_{\min} (Γ) {(‖ {\overline{x}}_{2} ‖ - \frac{χ}{2 λ_{\min} (Γ)})}^{2} + \frac{χ^{2}}{4 λ_{\min} (Γ)} - {\overline{x}}_{2}^{T} R {\overline{x}}_{2} + L_{w} + L_{v} \\ ⩽ \frac{χ^{2}}{4 λ_{\min} (Γ)} - {\overline{x}}_{2}^{T} R {\overline{x}}_{2} + L_{w} + L_{v} \end{matrix}$

(4.57)

where

$\begin{matrix} L_{w} = t r [(K_{w}^{- 1} {\overset{\cdot}{\tilde{W}}}_{t} - Φ ({\hat{V}}_{t}^{T} x) {\overline{x}}_{2}^{T} - D_{σ} {\tilde{V}}_{t}^{T} x {\overline{x}}_{2}^{T}) {\tilde{W}}^{T}] \\ L_{v} = t r [(K_{v}^{- 1} \overset{\cdot}{\tilde{V}} - {\hat{W}}^{T} D_{σ}^{T} x {\overline{x}}_{2}^{T} - l K_{v} x^{T} \tilde{V} x Λ_{3}) {\tilde{V}}^{T}] \end{matrix}$

If ${\overline{x}}_{2}^{T} R {\overline{x}}_{2} ⩾ \frac{χ^{2}}{4 λ_{\min} (Γ)}$ , using adaptive law (4.46),

${\overset{\cdot}{V}}_{1} ⩽ \frac{χ^{2}}{4 λ_{\min} (Γ)} - {\overline{x}}_{2}^{T} R {\overline{x}}_{2} ⩽ 0$

If ${\overline{x}}_{2}^{T} R {\overline{x}}_{2} < \frac{χ^{2}}{4 λ_{\min} (Γ)}$ , $V_{1}$ keeps constant, $V_{1}$ is bounded, that is, (I). So the total time during which ${‖ {\overline{x}}_{2} ‖}_{R}^{2} > \frac{χ^{2}}{4 λ_{\min} (Γ)}$ is finite. Let $T_{k}$ denote the time interval during which ${‖ {\overline{x}}_{2} ‖}_{R}^{2} > \frac{χ^{2}}{4 λ_{\min} (Γ)}$ :

• If only finite times that ${‖ {\overline{x}}_{2} ‖}_{R}^{2}$ stay outside the circle of radius $\frac{χ^{2}}{4 λ_{\min} (Γ)}$ (and then reenter), ${‖ {\overline{x}}_{2} ‖}_{R}^{2}$ will eventually stay inside of this circle.
• If ${‖ {\overline{x}}_{2} ‖}_{R}^{2}$ leave the circle infinite times, since the total time ${‖ {\overline{x}}_{2} ‖}_{R}^{2}$ leave the circle is finite,

$\sum_{k = 1}^{\infty} T_{k} < \infty, \lim_{k \to \infty} T_{k} = 0$

So ${‖ {\overline{x}}_{2} ‖}_{R}^{2}$ is bounded via an invariant set argument. From (4.48), $\overset{\cdot}{{\overline{x}}_{2}}$ is also bounded. Let ${‖ {\overline{x}}_{2} ‖}_{R}^{2}$ denote the largest tracking error during the $T_{k}$ interval. Bounded ${‖ {\overline{x}}_{2} ‖}_{R}^{2}$ imply that

$\lim_{k \to \infty} [{‖ {\overline{x}}_{2, k} ‖}_{R}^{2} - \frac{χ^{2}}{4 λ_{\min} (Γ)}] = 0$

So ${‖ {\overline{x}}_{2, k} ‖}_{R}^{2}$ will convergence to $\frac{χ^{2}}{4 λ_{\min} (Γ)}$ (II) is achieved. □

The neural networks compensation does not require structure information of the uncertainties. The RBF neural networks are more powerful [50]; we can also use RBF neural networks to approximate the friction and gravity in (4.1). The RBF neural networks is

$y_{j} = \sum_{i = 1}^{N} w_{i, j} ϕ_{i} (V x) + b$

(4.58)

where N is a hidden nodes number, $w_{i, j}$ is the weight connecting the hidden layer and the output layer. x is the input vector $x \in ℜ^{m}$ (m is input node number), $V \in ℜ^{N \times m}$ is the weight matrix in the hidden layer, $ϕ_{i} (V x)$ is the radial basis function, which we select it as the Gaussian function $ϕ_{i} (V x) = \exp {\frac{‖ V x - c_{j} ‖}{2 σ_{j}^{2}}}$ . Here, $c_{j}$ and $σ_{j}^{2}$ represent the center and spread of the basis function. b is the threshold. The significance of the threshold is that the output values have a nonzero mean. It can be combined with the first term as $w_{0, j} = b$ , $ϕ_{0} (V x) = 1$ , so $y_{j} = \sum_{i = 0}^{N} w_{i, j} ϕ_{i} (V x)$ . The friction and gravity of (4.1) can be approximated by a RBF neural network (4.58) as

$P (q, \overset{\cdot}{q}) = W^{⁎} Φ (V^{⁎} x) + \tilde{P}$

(4.59)

where $W^{⁎} \in ℜ^{M \times N}$ , $Φ (x) \in ℜ^{M}$ . $P (q, \overset{\cdot}{q}) = G (q) + F \overset{\cdot}{q}$ , $\tilde{P}$ is the approximated error. $W^{⁎}$ and $V^{⁎}$ are unknown optimal matrices to make $\tilde{P}$ minimum. The estimation of $P (q, \overset{\cdot}{q})$ may be defined as $\hat{P} (q, \overset{\cdot}{q})$ :

$\hat{P} (q, \overset{\cdot}{q}) = {\hat{W}}_{t} Φ ({\hat{V}}_{t} x)$

(4.60)

In order to implement neural networks, Assumption A4.1 for approximation error $\tilde{P}$ is necessary.

It is clear that the Gaussian function used in RBF neural networks satisfy the Lipschitz condition. So we may conclude that

${\tilde{Φ}}_{t} = Φ (V^{⁎ T} x) - Φ ({\hat{V}}_{t}^{T} x) = D_{σ} {\tilde{V}}_{t}^{T} x + ν_{σ}$

(4.61)

where ${‖ ν_{σ} ‖}_{Λ}^{2} ⩽ l {‖ {\tilde{V}}_{t}^{T} x ‖}_{Λ}^{2}$ , $D_{σ} = \frac{\partial Φ^{T} (Z)}{\partial Z} |_{Z = {\hat{V}}_{t}^{T} x}$ , Λ is a positive definite matrix, l is a positive constant, ${\tilde{W}}_{t} = {\hat{W}}_{t} - W^{⁎}$ , ${\tilde{V}}_{t} = {\hat{V}}_{t} - V^{⁎}$ .

The learning rules of ${\hat{W}}_{t}$ and ${\hat{V}}_{t}$ are derived from the stability analysis of the closed-loop system when the neural compensator (4.60) is combined with normal PD control as

$τ = - K_{p} (x_{1} - x_{1}^{d}) - K_{d} (x_{2} - x_{2}^{d}) + {\hat{W}}_{t} Φ ({\hat{V}}_{t} x)$

(4.62)

where $K_{p}$ and $K_{d}$ are positive defined matrices corresponding to proportional and derivative coefficients, $x_{1}^{d} \in ℜ^{n}$ is the desired position. $x_{2}^{d}$ is the desired velocity.

Remark 4.5

From the definition of the Lyapunov function (4.49), we may see that the learning rules (4.46) will minimize the tracking error ${\overline{x}}_{2}$ . This structure is different from normal neural networks which are used for approximation of nonlinear function. The first terms $- K_{w} {\overline{x}}_{2} Φ^{T} ({\hat{V}}_{t} x)$ and $- K_{v} D_{σ}^{T} {\hat{W}}_{t}^{T} {\overline{x}}_{2}$ are exactly corresponding to the backpropagation scheme. The second terms $- K_{w} x^{T} {\hat{V}}_{t}^{T} D_{σ}$ and $- K_{v} l Λ_{3} {\hat{V}}_{t} x x^{T}$ are additional ones, which can assure a stable learning. Deriving an update law with guaranteed stability is a very effective technique; some similar analysis may be found in [85].

4.3 PD control with velocity estimation and neural compensator

If neither the velocity $x_{2}$ nor the friction and gravity are known, the normal PD control should be combined with velocities estimation and neural compensations:

$τ = - K_{p} (x_{1} - x_{1}^{d}) - K_{d} ({\hat{x}}_{2} - x_{2}^{d}) + \hat{W} Φ (\hat{V} s)$

(4.63)

where $s = {(x_{1}^{T}, {\hat{x}}_{2}^{T})}^{T}$ . The structure of the new PD controller is shown in Fig. 4.1.

Figure 4.1 Neural PD control with high gain observer.

The dynamic of the tracking error can be formed from (4.6) and (4.63)

$\begin{matrix} {\overset{\cdot}{\overline{x}}}_{1} = {\overline{x}}_{2} \\ {\overset{\cdot}{\overline{x}}}_{2} = {\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{2}^{d} = H_{2} - {\overset{\cdot}{x}}_{2}^{d} \end{matrix}$

(4.64)

where

$\begin{matrix} H_{2} = M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} [- K_{p} {\overline{x}}_{1} + K_{d} {\tilde{x}}_{2} - K_{d} {\overline{x}}_{2} + \hat{W} Φ (\hat{V} s) \\ - W^{⁎} Φ (V^{⁎} s) - \tilde{P} - C ({\overline{x}}_{2} + x_{2}^{d})] \end{matrix}$

(4.65)

The closed-loop system is

$\begin{matrix} {\overset{\cdot}{\overline{x}}}_{1} = {\overline{x}}_{2}, {\overset{\cdot}{\overline{x}}}_{2} = H_{2} - {\overset{\cdot}{x}}_{2}^{d} \\ ε {\overset{\cdot}{\tilde{z}}}_{1} = {\tilde{z}}_{2} - K_{1} {\tilde{z}}_{1}, ε {\overset{\cdot}{\tilde{z}}}_{2} = - K_{2} {\tilde{z}}_{1} + ε^{2} H_{2} \end{matrix}$

(4.66)

The new updating law, which is obtained from next theorem is

$\begin{matrix} {\overset{\cdot}{\hat{W}}}_{t} = - d_{t} K_{w} Ψ [Φ^{T} ({\hat{V}}_{t} s) + s^{T} {\hat{V}}_{t}^{T} D_{σ}] \\ {\overset{\cdot}{\hat{V}}}_{t} = - d_{t} K_{v} [D_{σ}^{T} {\hat{W}}_{t}^{T} Ψ + l Λ_{3} {\hat{V}}_{t} s] s^{T} \end{matrix}$

(4.67)

where $Ψ = 2 d ε η_{M} {(x_{1} - {\hat{x}}_{1})}^{T} P_{12} M^{- 1} - 2 d ε^{2} η_{M} {\hat{x}}_{2}^{T} P_{22} M^{- 1} - (1 - d) x_{1}^{d}$ ,

$d_{t} = {\begin{matrix} 0 & if & {‖ y ‖}_{R_{1}}^{2} ⩽ a + \frac{λ_{\min}^{2} (K_{d})}{4 b^{2}} \\ 1 & if & {‖ y ‖}_{R_{1}}^{2} > a + \frac{λ_{\min}^{2} (K_{d})}{4 b^{2}} \end{matrix}$

(4.68)

$y = {[‖ \overline{x} ‖, ‖ \tilde{z} ‖]}^{T}$ , $a = \overline{Ψ} \overline{η} + \frac{λ_{\max} (Λ_{1}^{- 1})}{4} {\overline{Ψ}}^{2}$ , $b = k_{c}^{2} x_{1}^{d} + λ_{\max} (M) x_{2}^{d}$ . We can see that another condition is needed: M is known. This requirement is necessary if both velocity and friction are unknown (see [78]). When we realize the high-gain observer (4.8), we can see that the observer error is less than $2 ε^{2} C_{i, T}$ . Can we find the largest value of ε, which can assure that the closed-loop system is stable? The following theorem gives the answer to this question.

Theorem 4.6

If (a) the learning laws for the weights of neural networks (4.60) are used as in (4.67), (b) the gain of the observer satisfies

$ε < 1 / ‖ 2 η_{M} P_{1} [\begin{matrix} 0 & 0 \\ 0 & K_{d} \end{matrix}] ‖$

(4.69)

$P_{1}$ is the solution of the following Lyapunov equation:

$A^{T} P_{1} + P_{1} A = - δ I, δ > 0$

(4.70)

then we have:

(I) The weights of neural networks ${\hat{W}}_{t}$ , ${\hat{V}}_{t}$ , observer and tracking errors $‖ \tilde{z} ‖$ and $‖ \overline{x} ‖$ are bounded.

(II) For any $T \in (0, \infty)$ the tracking and observer error $y = {[‖ \overline{x} ‖, ‖ \tilde{z} ‖]}^{T}$ converges to the residual set

$D_{y} = {y | {‖ y ‖}_{R_{1}}^{2} ⩽ a + \frac{λ_{\min}^{2} (K_{d})}{4 b^{2}}}$

where a and b are defined in (4.68). $R_{1}$ is a positive defined matrix such that $R_{1} = [\begin{matrix} R_{11} & R_{21} \\ - R_{12} & R_{22} \end{matrix}]$ , $R_{i, j}$ are positive constants.

Proof

Let us select the following candidate Lyapunov function for (4.66):

$V_{2} = (1 - d) V_{1} + d {\tilde{z}}^{T} P_{1} \tilde{z}$

(4.71)

where $0 < d < 1$ , $V_{1}$ is the Lyapunov function in Theorem 4.4 as in (4.49). Since PD control (4.63) is different from (4.62), we cannot apply ${\overset{\cdot}{V}}_{1}$ in Theorem 4.5. From (4.66), we know $\overset{\cdot}{\tilde{z}} = \frac{1}{ε} A \tilde{z} + ε B H_{2}$ . Because

$\begin{matrix} H_{2} - H_{2, {\tilde{x}}_{2} = 0} = \frac{1}{ε} M^{- 1} K_{d} {\tilde{z}}_{2} \\ {\overset{\cdot}{\overline{x}}}_{2} = [H_{2, {\tilde{x}}_{2} = 0} - {\overset{\cdot}{x}}_{2}^{d}] + [H_{2} - H_{2, {\tilde{x}}_{2} = 0}] \end{matrix}$

the derivative of (4.71) is

$\begin{matrix} {\overset{\cdot}{V}}_{2} = (1 - d) [{\overline{x}}_{2}^{T} M H_{2, {\tilde{x}}_{2} = 0} + {\overline{x}}_{2}^{T} \overset{\cdot}{M} {\overline{x}}_{2} + {\overline{x}}_{2}^{T} K_{p} {\overline{x}}_{1}^{T} + t r ({\tilde{W}}_{t}^{T} K_{w}^{- 1} \overset{\cdot}{{\tilde{W}}_{t}}) + t r ({\tilde{V}}_{t}^{T} K_{v}^{- 1} {\overset{\cdot}{\tilde{V}}}_{t})] \\ + (1 - d) {\overline{x}}_{2}^{T} K_{d} {\tilde{x}}_{2} - \frac{d}{ε} {\tilde{z}}^{T} \tilde{z} + [2 d ε {\tilde{z}}^{T} P B H_{2}] \end{matrix}$

(4.72)

Compare (4.72) and (4.48) and the first term of right-hand side is the same as (4.50):

${\overset{\cdot}{V}}_{2} = (1 - d) ({\overset{\cdot}{V}}_{1} + {\overline{x}}_{2}^{T} K_{d} {\tilde{x}}_{2}) - \frac{d}{ε} {\tilde{z}}^{T} \tilde{z} + 2 d ε {\tilde{z}}^{T} P_{1} B H_{2}$

(4.73)

From (4.65),

$\begin{matrix} 2 d ε {\tilde{z}}^{T} P_{1} B H = - 2 d ε {\tilde{z}}^{T} P_{1} (C x_{2} + K_{p} {\overline{x}}_{1} + K_{d} {\overline{x}}_{2} - K_{d} {\tilde{x}}_{2}) \\ - 2 d ε {\tilde{z}}^{T} P_{1} B M^{- 1} (- \hat{W} Φ (\hat{V} s) + W^{⁎} Φ (V^{⁎} s) + \tilde{P}) \end{matrix}$

(4.74)

$[- {\overline{x}}_{2}^{T} M {\overset{\cdot}{x_{2}}}^{d} - {\overline{x}}_{2}^{T} C x_{2}^{d}]$ can be estimated as

$- {\overline{x}}_{2}^{T} M {\overset{\cdot}{x_{2}}}^{d} - {\overline{x}}_{2}^{T} C x_{2}^{d} ⩽ ‖ x_{2}^{d} ‖ {\overline{x}}_{2}^{T} k_{c} {\overline{x}}_{2} + k_{c} {‖ x_{2}^{d} ‖}^{2} ‖ {\overline{x}}_{2} ‖ + λ_{\max} (M) ‖ {\overset{\cdot}{x}}_{2}^{d} ‖ ‖ {\overline{x}}_{2}^{T} ‖$

$Ψ^{T} (W^{⁎} ν_{σ} - \tilde{P})$ may be estimated as

$Ψ^{T} (W^{⁎} ν_{σ} - \tilde{P}) ⩽ ‖ Ψ ‖ \overline{η} + \frac{1}{4} Ψ^{T} Λ_{1}^{- 1} Ψ + l {‖ {\tilde{V}}_{t}^{T} x ‖}_{Λ_{3}}^{2}$

(4.75)

$d ε {\tilde{z}}^{T} P_{1} (C x_{2})$ becomes

$d ε {\tilde{z}}^{T} P_{1} [C (x_{1}, x_{2}) x_{2}] ⩽ 2 d ε {\tilde{z}}^{T} P_{1} C_{0} (x_{1}) {\overline{x}}_{2}$

$\begin{matrix} 2 d ε {\tilde{z}}^{T} P_{1} [C (x_{1}, x_{2}) x_{2} + K_{p} {\overline{x}}_{1} + K_{d} {\overline{x}}_{2} - K_{d} {\tilde{x}}_{2}] \\ ⩽ 2 d ε {\tilde{z}}^{T} P_{1} [\begin{matrix} 0 & 0 \\ K_{p} & K_{d} + 2 C_{0} \end{matrix}] \overline{x} - 2 d {\tilde{z}}^{T} P_{1} [\begin{matrix} 0 & 0 \\ 0 & K_{d} \end{matrix}] \tilde{z} \end{matrix}$

(4.73) becomes

$\begin{matrix} {\overset{\cdot}{V}}_{2} ⩽ - y^{T} T y - (1 - d) ‖ {\overline{x}}_{2} ‖ (λ_{\min} (K_{d}) ‖ {\overline{x}}_{2} ‖ - b) \\ + a - y^{T} R_{1} y + L_{w 1} + L_{v 1} \end{matrix}$

(4.76)

We want to make T as a positive defined matrix, i.e.,

$\begin{matrix} 0 < R_{11} < (1 - d) α, R_{21} > \frac{(1 - d) β}{2 ε} + \frac{ε d k_{2}}{2} \\ 0 < R_{22} < d [\frac{1}{ε} - k_{1}], \frac{1}{ε} > k_{1} \end{matrix}$

(4.77)

Since $- \frac{(1 - d) β}{2 ε} - \frac{ε d k_{2}}{2} - R_{12} < 0$ , $R_{12}$ can be any positive constant. So if condition (4.69) is satisfied, $R_{1}$ is selected as (4.77), $y^{T} T y > 0$ . So

${\overset{\cdot}{V}}_{3} ⩽ (a + \frac{λ_{\min}^{2} (K_{d})}{4 b^{2}}) - y^{T} R_{1} y + L_{w 1} + L_{v 1}$

(4.78)

where (4.78) has a same structure as in Theorem 4.4, with the similar proof (I) and (II) may be established. □

Remark 4.6

Since $y = {[‖ \overline{x} ‖, ‖ \tilde{z} ‖]}^{T}$ is not measurable, the dead zone $d_{t}$ in (4.68) cannot be realized. We will use the available date $\overline{y} = \hat{x} - x^{d}$ to determine the dead zone. The dead zone can be represented as

${‖ \overline{y} ‖}_{R_{1}}^{2} > a + \frac{λ_{\min}^{2} (K_{d})}{4 b^{2}} + {‖ \overline{y} ‖}_{R_{1}}^{2} - {‖ y ‖}_{R_{1}}^{2}$

Because $\overline{y} = \overline{x} - \tilde{x}$ ,

$\begin{matrix} {‖ \overline{y} ‖}_{R_{1}}^{2} - {‖ y ‖}_{R_{1}}^{2} ⩽ {‖ \overline{x} ‖}_{R_{1}}^{2} + {‖ \tilde{x} ‖}_{R_{1}}^{2} - {‖ \overline{x} ‖}_{R_{1}}^{2} + {‖ \tilde{z} ‖}_{R_{1}}^{2} \\ = {‖ \tilde{x} ‖}_{R_{1}}^{2} + {‖ \tilde{z} ‖}_{R_{1}}^{2} ⩽ 4 ε^{2} λ_{\max} (R_{1}) C_{i, T} \end{matrix}$

So, the new dead zone is modified as

$d_{t} = {\begin{matrix} 0 & if & {‖ \overline{y} ‖}_{R_{1}}^{2} ⩽ a + \frac{λ_{\min}^{2} (K_{d}) + 4 ε^{2} λ_{\max} (R_{1}) C_{i, T}}{4 b^{2}} \\ 1 & if & {‖ \overline{y} ‖}_{R_{1}}^{2} > a + \frac{λ_{\min}^{2} (K_{d})}{4 b^{2}} + 4 ε^{2} λ_{\max} (R_{1}) C_{i, T} \end{matrix}$

4.4 Simulation

To develop the simulations, a two-link planar robot manipulator is considered. It is assumed that each link has its mass concentrated as a point at the end. The manipulator is in vertical position, with gravity and friction. The robot parameters are: $m_{1} = m_{2} = 1$ , $l_{1} = 1$ , $l_{2} = 2$ . The two friction coefficients are 0.3, and the gravity is 9.8. So the real matrices of (4.1) are

$\begin{matrix} M = [\begin{matrix} 4 \cos (q_{2}) + 6 & 2 \cos (q_{2}) + 4 \\ 2 \cos (q_{2}) + 4 & 4 \end{matrix}] \\ C = [\begin{matrix} - 4 \cos (\dot{q_{2}}) \sin (q_{2}) & 2 \overset{\cdot}{q_{1}} \sin (q_{2}) \\ - 2 \cos (\overset{\cdot}{q_{2}}) \sin (q_{2}) & 0 \end{matrix}] \\ G = [\begin{matrix} 19.6 \cos (q_{1} + q_{2}) + 19.6 \cos q_{1} \\ 19.6 \cos (q_{1} + q_{2}) \end{matrix}], F = [\begin{matrix} 0.3 & 0 \\ 0 & 0.3 \end{matrix}] \end{matrix}$

where $q = {[q_{1}, q_{2}]}^{T}$ . All data are given with the appropriate unities.

The following PD coefficients are chosen:

$K_{p} = [\begin{matrix} 31 & 0 \\ 0 & 45 \end{matrix}], K_{d} = [\begin{matrix} 60 & 0 \\ 0 & 80 \end{matrix}]$

The matrices P and Q are selected as

$P = [\begin{matrix} 5 & 0 & - \frac{1}{2} & 0 \\ 0 & 5 & 0 & - \frac{1}{2} \\ - 5 & 0 & 1 & 0 \\ 0 & - 5 & 0 & 1 \end{matrix}], Q = [\begin{matrix} 45 & 0 & 0 & 0 \\ 0 & 45 & 0 & 0 \\ - 45 & 0 & 5.5 & 0 \\ 0 & - 45 & 0 & 5.5 \end{matrix}]$

Let us calculate the constants in Theorem 4.5, $α_{1} = ‖ [\begin{matrix} 0 & 0 \\ 0 & K_{d} \end{matrix}] ‖ = 80$ , $α_{2} = ‖ Q ‖ = 45$ ,

$\begin{matrix} K_{1} = ‖ P [\begin{matrix} 0 & 0 \\ 0 & - 2 M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} (F - K_{d} + C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d})) \end{matrix}] ‖ = 42.71 \\ K_{2} = ‖ P [\begin{matrix} 0 & 0 \\ - M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} K_{p} & - M {({\overline{x}}_{1} + x_{1}^{d})}^{- 1} (K_{d} + C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d})) \end{matrix}] ‖ = 84.3242 \\ β_{1} = ‖ [\begin{matrix} 0 & 0 \\ 0 & - F + K_{d} - C ({\overline{x}}_{1} + x_{1}^{d}, x_{2}^{d}) \end{matrix}] ‖ = 80.0288 \end{matrix}$

where $‖ A ‖$ denotes the absolute value of the real part of the maximum eigenvalue of the matrix A. Then (4.19) becomes

$\begin{matrix} f (ϵ) = 3555.3 d^{2} ϵ^{4} + 6748.4 (1 - d) d ϵ^{2} \\ + 6745.9 (1 - d) ϵ + 3202.3 {(1 - d)}^{2} - 3600 (1 - d) d \end{matrix}$

(4.79)

With $d = 0.5$ the polynomial (4.79) is shown in Fig. 4.2.

One can see that for $0 < ϵ < 0.0558309561787$ (4.79) is negative. So $\overline{ϵ} = 0.056$ , i.e., $Γ = (0, 0.056)$ . The high-gain observer (4.8) is determined as $ϵ = 0.003$ .

Fig. 4.3 and Fig. 4.4 show rapid convergence of the observer to the link's velocities. The observer error is almost zero in this short interval. Fig. 4.3 shows that with a velocity observer (high-gain observer), PD control can make the links follow the desired trajectory.

The asymptotic stability assures that the desired positions may be reached with zero error if $t \to \infty$ . Rapid and accurate convergence of the observer is essential for the PD controller, because the observer is in the feedback loop. Since the high-gain observer (4.26) has these properties, we can use a simple controller (4.12) to compensate the uncertainties of the observer response.

Fig. 4.5 and Fig. 4.6 show the simulation results of the performance of the high-gain observer when a noise arises at $t = 3$ . The perturbation is a $20 %$ increment on the second link's mass. This is intended to see the robustness of the observer-based PD control. Fig. 4.7 shows the tracking done by both links of the robot with the same perturbation at $t = 3$ .

Figure 4.5 Links velocities with perturbation.

Figure 4.6 Links velocities with perturbation.

Since the observer and controller are independent of the robot's dynamics, the influence of perturbations are very small. All data are given with the appropriate unities. The following PD coefficients are chosen:

$K_{p} = [\begin{matrix} 31 & 0 \\ 0 & 45 \end{matrix}], K_{d} = [\begin{matrix} 60 & 0 \\ 0 & 80 \end{matrix}]$

Friction and gravity can be uniformly approximated by a radial basis function as in (4.60) with $N = 10$ . The Gaussian function is

$ϕ_{j} (V x) = \exp {- \frac{\sum_{i = 1}^{n} {(V x_{i} - c_{i})}^{2}}{100}},$

where the spread $σ_{l}$ was set to $\sqrt{50}$ and the center $c_{i}$ is a random number between 0 and 1. The control law applied is

$u = - K_{p} (x_{1} - x_{1}^{d}) - K_{d} (x_{2} - x_{2}^{d}) + \hat{W} Φ (\hat{V} x)$

starting with $W^{⁎} = 0.7$ and $V^{⁎} = 0.7$ as initial values. Even though some initial weights are needed for the controller to work, no special values are required nor is a previous investigation on the robot dynamics for the implementation of this control. Fig. 4.7 and Fig. 4.8 gave the comparison of the performance of PD controller neural compensation. The continuous line is the exact position, the dashed line is general PD control without friction and gravity, and the dashed-dotted line is the neural networks compensation.

We can see that PD control with the RBF networks compensator may improve the control performance.

4.5 Conclusions

The disadvantages of the popular PD control are overcome in the following two ways: (I) A high-gain observer is proposed for the estimation of the velocities of the joints. (II) The RBF neural networks are used to compensate the gravity and friction. By the singular perturbation and Lyapunov-like analysis, we give proofs of the high-gain observer and the stability of the closed-loop system with the velocity estimation and the RBF compensator.

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Table of Contents for Chapter 4: PD Control with Neural Compensation

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4.1 PD control with high gain observer

4.1.1 Singular perturbation method

4.1.2 Lyapunov method

4.2 PD control with neural compensator

4.2.1 PD control with single layer neural compensation

4.2.2 PD control with a multilayer feedforward neural compensator

4.3 PD control with velocity estimation and neural compensator

4.4 Simulation

4.5 Conclusions

Table of Contents for
Chapter 4: PD Control with Neural Compensation